---
address: |
    Theory Division, CERN, CH-1211 Geneva 23, Switzerland\
    E-mail: [martin.beneke@cern.ch]{}
author:
- 'M Beneke $^\dag$'
title: ' QCD factorization for exclusive, non-leptonic $B$ decays '
---

Introduction
============

Measuring asymmetries in exclusive decays of $B$ vs. $\bar{B}$ mesons is the primary task of four experimental facilities soon in operation. While a non-zero asymmetry establishes CP violation unambiguously, the interpretation of such a measurement in terms of fundamental theory parameters is mostly rather difficult. Hence many strategies that use different measurements in an attempt to eliminate or minimize theoretical input on strong interaction dynamics, essentially trading theoretical uncertainties for experimental ones. It would be clearly useful to have some theoretical guidance as well.

In this talk I describe the approach put forward in [@BBNS99]. The result is that most non-leptonic, two-body $B$ decays can be treated as hard processes in QCD and become simple in the heavy quark limit. The naive factorization approximation follows as the leading term in a systematic expansion. A particularly interesting consequence is that soft final state interactions vanish in the heavy quark limit. Strong phases are therefore calculable, if the form factors and light-cone distribution amplitudes of mesons are known. A simple factorized (in a sense to be explained below) expression holds only for the leading term in an expansion in $\Lambda_{\rm QCD}/m_b$. Whether the heavy quark limit is an adequate approximation for $B$ mesons is quite a different matter, which I do not address in this talk. For this reason, all numbers that may follow should be considered as illustrative of what can be done, once we are convinced that the heavy quark limit is good.

The factorization formula
=========================

The basic objects are the matrix elements $\langle M_1 M_2| {\cal O}_i| 
B\rangle$, where $M_1$ and $M_2$ denote the two final state mesons and ${\cal O}_i$ an operator in the weak effective Hamiltonian. Such matrix elements contain both short-distance (‘hard’) effects, related to the large scale $m_b$, and long-distance (‘soft’, ‘collinear’) effects. The idea is to factorize (and compute) hard contributions and to parameterize soft and collinear ones, hoping that this leads to some simplification. I take the heavy quark limit, i.e. relative corrections of order $\Lambda_{\rm QCD}/m_b$ are neglected. Hard and infrared contributions will be separated by looking at Feynman diagrams, a method familiar from other hard QCD processes (hadron-hadron collisions, fragmentation, jets, hard diffraction ...). It is then assumed that factorization holds non-perturbatively. This assumption that soft effects that may not be visible from Feynman diagrams do not destroy the power counting is always implicit in factorization ‘theorems’ and I do not know of any example, where it would go wrong – but it is an assumption.

Before applying these considerations to the case at hand, note that two light final state mesons carry energy and momentum $M_B/2$. If all the mesons’s constituents have large momentum, the meson is appropriately described by light-cone distribution amplitudes. There is a finite probability for asymmetric partonic fluctuations in which a subset of partons carries almost all momentum. For example, for a pion with momentum $E$ there is a probability of order $1/E^2$ for a $q\bar{q}$ component, in which one of the quarks carries momentum of order $\Lambda_{\rm QCD}$. This probability can be estimated from the endpoint behaviour of the asymptotic distribution amplitude, since the distribution amplitude approaches the asymptotic one in the high energy limit. Such endpoint contributions are not appropriately described by overlap integrals of light-cone distribution amplitudes. If unsuppressed, they require introducing more general non-perturbative parameters.

The main results are summarized as follows. Let $M_1$ be the meson that absorbs the light spectator quark from the $B$ meson. $M_1$ can be a light meson ($M_{M_1}/m_b\to 0$ in the heavy quark limit) or a heavy meson ($M_{M_1}/m_b \to\,$finite). The second meson $M_2$ is required to be light. Then:

\(1) ‘Non-factorizable’ contributions, i.e. contributions that do not belong to the $B\to M_1$ form factor or the decay constant $f_{M_2}$ of $M_2$, are dominated by hard gluon exchange. This effect can be calculated. Since hard gluons transfer large momentum to $M_2$, one obtains a convolution with the light-cone distribution amplitude of $M_2$ rather than $f_{M_2}$ as in naive factorization. ‘Non-factorizable’ soft exchange is suppressed, because the $q\bar{q}$ pair that forms $M_2$ is produced as a small colour dipole [@BJ]. The cancellation of soft gluons is not enough to arrive at a factorization formula. It is important that ‘non-factorizable’, collinear gluons cancel as well.

\(2) The $B\to M_1$ form factor is dominated by soft gluon exchange for both heavy and light $M_1$ by power counting. The reason for this is that the $B$ meson contains a soft spectator quark. If $M_1$ is light, the endpoint suppression of the light-cone distribution amplitude of $M_1$ is not sufficient to render the soft contribution power suppressed. In fact, the hard gluon correction is suppressed by one power of $\alpha_s$ relative to the soft one. If $M_2$ is heavy, the light-cone distribution amplitude favours the absorption of a soft quark and the hard contribution is suppressed even further. This discussion refers to counting powers. It ignores the possibility that resummation of Sudakov logarithms suppresses the soft contribution beyond naive power counting. This possibility deserves further investigation, even though it appears unrealistic for realistic $B$ mesons.

\(3) ‘Non-factorizable’, hard gluon exchange between $M_2$ and the $B$ meson spectator quark is a leading power effect. Because the gluon is hard, the interaction is local in transverse distance and can be described by the convolution of three light-cone distribution amplitudes. If the ‘non-factorizable’, hard gluon exchange occurs between $M_2$ and the other quarks of $B$ and $M_1$, the spectator quark can be a distance $1/\Lambda_{\rm QCD}$ away. This implies that in this case we must keep the $B\to M_1$ form factor.

\(4) Annihilation topologies and higher Fock states of $M_2$ give contributions suppressed by powers of $\Lambda_{\rm QCD}/m_b$.

The observations collected above are expressed by the factorization formula $$\begin{aligned}
\label{fff}
&&\hspace*{-0.6cm}
\langle M_1(p') M_2(q)|{\cal O}_i|B(p)\rangle =
\nonumber\\
&&\hspace*{-0.2cm}
F_{B\to M_1}(0)\,\int\limits_0^1 \!dx\,T_{i}^I(x)\,\Phi_{M_2}\!(x) 
\\[-0.3cm]
&&\hspace*{-0.2cm}
+\,\int\limits_0^1 \!d\xi dx dy \,T_i^{II}(\xi,x,y)\,
\Phi_B(\xi)\Phi_{M_1}\!(y)\Phi_{M_2}\!(x),
\nonumber\end{aligned}$$ where the last line accounts for the hard interaction with the spectator quark in the $B$ meson and the equality sign is valid up to corrections of order $\Lambda_{\rm QCD}/m_b$. $T_i$ denote hard scattering functions that depend on longitudinal momentum fractions and $\Phi$ label the light-cone distribution amplitudes. See Fig. \[fig1\] for a graphical illustration of (\[fff\]). If $M_1$ is heavy, the last line is suppressed and should be dropped. In this case an equation like (\[fff\]) has already been used in [@PW]. If $M_2$ is a heavy, onium-like meson, factorization still holds. If, on the other hand, $M_2$ is a heavy-light meson, factorization does not occur.

=13.5cm

The validity of (\[fff\]) has been demonstrated by an explicit 1-loop calculation [@BBNS99]. General arguments support factorization to all orders in perturbation theory. For a heavy-light final state (such as $D\pi$), an explicit demonstration of the cancellation of soft and collinear divergences at two loops exists [@BBNS].

One of the consequences of (\[fff\]) is that in those cases where it applies, it also justifies naive factorization as the limit in which corrections of order $\alpha_s$ and $\Lambda_{\rm QCD}/m_b$ are neglected. The underlying physical picture is that the meson $M_1$ is produced in an asymmetric configuration, in which one quark carries almost all of the meson’s momentum, while the other meson $M_2$ is produced in a symmetric $q\bar{q}$ configuration which leaves the decay region with little probability for interactions.

The possibility to compute systematically logarithmic corrections to naive factorization solves the problem of scheme-dependence often discussed in the literature in the context of naive factorization. The hard scattering kernels require ultraviolet subtractions related to the renormalization of the operator ${\cal O}_i$. If performed consistently, these subtractions compensate the scheme- and scale-dependence of the Wilson coefficients in the weak effective Hamiltonian.

Another consequence of (\[fff\]) is that strong final state interactions are computable. The imaginary part of the amplitude is generated by the imaginary parts of the hard scattering amplitudes only. As all other soft effects, soft final state interactions are also suppressed by a power of $\Lambda_{\rm QCD}/m_b$ and so soft strong phases disappear in the heavy quark limit. Since this statement may be a point of controversy [@DGPS96], it requires further discussion.

Final state interactions
========================

When discussing final state interactions, we may choose to use a partonic or hadronic language. The partonic language is justified by the dominance of hard rescattering in the heavy quark limit. In this limit the number of physical intermediate states is arbitrarily large. We may then argue on the grounds of parton-hadron duality that their average is described well enough (say, up to $\Lambda_{\rm QCD}/
m_b$ corrections) by a partonic calculation. This is the picture implied by (\[fff\]). The hadronic language is in principle exact. However, the large number of intermediate states makes it almost impossible to observe systematic cancellations, which usually occur in an inclusive sum of intermediate states.

To be specific, consider the decay of a $B$ meson into two pions. Unitarity implies that $$\label{unitarity}
\mbox{Im}\,{\cal A}_{B\to \pi\pi} \sim \sum_n 
{\cal A}_{B\to n}{\cal A}_{n\to \pi\pi}^*.$$ The elastic rescattering contribution is related to the $\pi\pi$ scattering amplitude, which exhibits Regge behaviour in the high-energy ($m_b\to\infty$) limit. Hence the soft, elastic rescattering phase increases slowly in the heavy quark limit [@DGPS96]. On general grounds, it is rather improbable that elastic rescattering gives an appropriate description in the heavy quark limit. This expectation is also borne out in the framework of Regge behaviour, see [@DGPS96], where the importance of inelastic rescattering is emphasized. However, the approach pursued in [@DGPS96] leaves open the possibility of soft rescattering phases that do not vanish in the heavy quark limit, as well as the possibility of systematic cancellations, for which the Regge language does not provide an appropriate theoretical framework.

Eq. (\[fff\]) implies that such systematic cancellations do occur in the sum over all intermediate states $n$. It is worth recalling that such cancellations are not uncommon for hard processes. Consider the example of $e^+ e^-\to\,$hadrons at large energy $q$. While the production of any hadronic final state occurs on a time scale of order $1/\Lambda_{\rm QCD}$ (and would lead to infrared divergences if we attempted to describe it in perturbation theory), the inclusive cross section given by the sum over all hadronic final states is described very well by a $q\bar{q}$ pair that lives over a short time scale of order $1/q$. In close analogy, while each particular hadronic intermediate state $n$ in (\[unitarity\]) cannot be described partonically, the sum over all intermediate states is accurately represented by a $q\bar{q}$ fluctuation of small transverse size of order $1/m_b$, which therefore interacts little with its environment. Note that precisely because the $q\bar{q}$ pair is small, the physical picture of rescattering is very different from elastic $\pi\pi$ scattering – hence the Regge picture is difficult to justify in the heavy quark limit. Technically, 2-gluon exchange (plus ladder graphs) between a compact $q\bar{q}$ pair with energy of order $m_b$ and transverse size of order $1/m_b$ and the other pion does not lead to large logarithms and hence no possibility to construct the pomeron. (Notice the difference with elastic vector meson production through a virtual photon, which also involves a compact $q\bar{q}$ pair. However, in this case one considers $s\gg Q^2$ and this implies that the $q\bar{q}$ fluctuation is born long before it hits the proton. Hence the possibility of pomeron exchange.)

It follows from (\[fff\]) that the leading strong interaction phase is of order $\alpha_s$ in the heavy quark limit. The same statement holds for rescattering in general. For instance, according to the duality argument, a penguin contraction with a charm loop represents the sum over all intermediate states of the form $D\bar{D}$, $J/\Psi\rho$, etc. that rescatter into two pions.

As is clear from the discussion, parton-hadron duality is crucial for the validity of (\[fff\]) beyond perturbative factorization. Proving quantitatively to what accuracy we can expect duality to hold is a yet unsolved problem in QCD. Short of a solution, it is worth noting that the same (often implicit) assumption is fundamental to many successful QCD predictions in jet physics, hadron-hadron physics and heavy quark decays.

CP asymmetry in $B_d\to \pi^+\pi^-$ decay
=========================================

I now use the factorization formula to compute the time-dependent, mixing-induced asymmetry in $B_d\to \pi^+\pi^-$ decay. For reasons discussed below I do not consider this result final. However, it illustrates the predictivity of a systematic approach to calculate non-leptonic decay amplitudes.

The complete expressions for the decay amplitudes into two pions of any charge are given in [@BBNS99] including $\alpha_s$ corrections to naive factorization. The relevant Feynman diagrams are shown in Fig. \[fig2\]. In the following I use the same input parameters as in [@BBNS99]. The time-dependent asymmetry can be expressed as $$A(t) = -S\cdot \sin(\Delta M_{B_d}t)+C\cdot\cos(\Delta M_{B_d} t).$$ In the absence of a penguin contribution (defined as the contribution to the amplitude which does not carry the weak phase $\gamma$ in standard phase conventions) $S=\sin 2\alpha$ (where $\alpha$ refers to one of the angles of the CKM unitarity triangle) and $C=0$. Fig. \[alpha\] shows $S$ as a function of $\sin 2\alpha$ with the amplitudes computed according to (\[fff\]). The central of the solid lines refers to the heavy quark limit including $\alpha_s$ corrections to naive factorization and including the power-suppressed term $a_6 r_\chi$ (for notation see [@BBNS99]) that is usually also kept in naive factorization. The other two solid lines correspond to dropping this term or multiplying it by a factor of 2. This exercise shows that formally power-suppressed terms can be non-negligible (see below), but it also shows that a measurement of $S$ can be converted into a range for $\sin 2\alpha$ which may already provide a very useful constraint on CP violation.

Outlook
=======

Much work remains to be done on the theoretical and phenomenological side. On the theoretical side, the proof of factorization for a final state of two light mesons has to be completed. Then, to define the heavy quark limit completely, one must control all logarithms of $m_b$. While this is straightforward for the first term on the right hand side of (\[fff\]), this is less trivial for the hard scattering term, since $\xi\sim \Lambda_{\rm QCD}/m_b$. This means that one must control logarithms of $\xi$ that may appear in higher orders. To do this, one has to define the appropriate $B$ meson wave function, since $\Phi_B(\xi)$ as it stands still depends on $m_b$. Work on these issues is in progress.

Power corrections are an important issue, as $m_b$ is not particularly large. There exist ‘chirally enhanced’ corrections that involve the formally power suppressed, but numerically large parameter $r_\chi=2 m_\pi^2/(m_b (m_u+m_d))$. All terms involving such chiral enhancements can be identified, but they involve non-factorizable soft gluons. The size of these terms has to be estimated to arrive at a realistic phenomenology.

If this can be done, we expect promising constraints and predictions for a large number of non-leptonic, two-body final states.

[**Acknowledgements.**]{} It is a pleasure to thank G. Buchalla, M. Neubert and C.T. Sachrajda for an on-going collaboration on the subject of this talk. I also wish to thank J. Donoghue for interesting discussions. This work was supported in part by the EU Fourth Framework Programme ‘Training and Mobility of Researchers’, Network ‘Quantum Chromodynamics and the Deep Structure of Elementary Particles’, contract FMRX-CT98-0194 (DG 12 - MIHT).

[9]{} M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, Phys. Rev. Lett. [**83**]{} (1999) 1914. J.D. Bjorken, Nucl. Phys. (Proc. Suppl.) [**B11**]{}, (1989) 325; M.J. Dugan and B. Grinstein, Phys. Lett. [**B255**]{} (1991) 583. H.D. Politzer and M.B. Wise, Phys. Lett. [**B257**]{} (1991) 399. M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, in preparation. J.F. Donoghue, E. Golowich, A.A. Petrov and J.M. Soares, Phys. Rev. Lett. [**77**]{} (1996) 2178.
