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Factorization in non-leptonic decays of heavy mesons

We propose a QCD-based model for calculation of the non-perturbative corrections to the factorization approximation in the decays of heavy mesons. In the framework of the model, factorization in pseudoscalar transitions holds exactly at the leading order leaving the opportunity to calculate non-leading corrections consistently.


I. INTRODUCTION
The factorization approximation is widely used in the heavy quark physics. As a calculational scheme, it is very convenient and there are numerous models based on it. However, the theoretical basis for the approximation itself is not completely understood. The usual motivation for it is a color transparency argument due to Bjorken [1] which suggests that the quark pair produced in the decay does not have enough time to evolve to the real-size hadronic entity, being a small size bound state with a small chromomagnetic moment. That is taken to imply that the QCD interaction is suppressed. For the heavy-to-heavy-light transition (we shall hereafter refer to the color allowed decay B → D + π − for the sake of clarity) the matrix elements in question are Here the usual notations are used where we have explicitly written out color indices. In the usual factorization approximation one simply separates out currents by inserting the vacuum state and disregards any QCD interactions between them on the basis of Bjorken's conjecture. That implies that the M 8 contribution in (1) is dropped, as it has no factorizable color singlet form. New nonfactorizable corrections appearing in the M 8 matrix element can arise if a hard or soft gluon is emitted or absorbed by the escaping pair. Obviously, a more quantitative estimate of these contributions is needed.
The first attempt to put a factorization approximation on the solid theoretical base was made in [2]. The authors considered a limit of large energy transferred to light quarks Here v is the velocity of the decaying heavy quark p b = m b v + k and q is the momentum transferred to the light quark pair. In this limit the energy scales as a heavy quark mass. Proposing a Large Energy Effective Theory (LEET) as an extension of the Heavy Quark Effective Theory and making use of a convenient gauge condition n · A = 0 (with n being a null vector, n 2 =0) they proved factorization for the physical amplitude dominated by collinear quarks. It is not clear, however, how the pair of collinear quarks hadronizes into the pion. Also, the validity of the LEET itself is not yet established.
The important step towards the sought theoretical description was made in [3]. The authors considered a specific small velocity limit where m b −m c ∼ const which implies different scaling of (3): E ∼ const as m b → ∞. It turns out that this limit is theoretically "clean" for the application of the QCD sum rule method and provides a theoretical justification In what follows we develop a QCD-based model which works in the limit E ∼ m b as The idea is to study the propagation of the light quarks emerging to form the final hadron while passing through the (perturbative or non-perturbative) gluonic fields present in the decay. Interestingly, results can be obtained which are independent of the nature of the gluonic fields. The light quark pair, produced at x = 0, interacts strongly with the background gluonic field while escaping from the point they were produced and hadronizes into the pion. Since the energy transferred to the quark pair is large it is not necessary that the hadronization occurs at x ∼ 0. As a result, it is not sufficient to take the operators of the lowest dimension in the Operator Product Expansion (OPE) but rather one must sum up a whole series of operators. The novel technical point of our method is to use introduce "generating (distribution) functions" which incorporate this infinite series of matrix elements, and then to use Heavy Quark Symmetry to restrict the form of the matrix elements enough that useful statements can be made. In principle, these functions could be modeled or perhaps fixed by other measurements. However, at the leading order in q 2 = 0, x 2 ∼ 0 the color octet matrix element vanishes due to cancellations and factorization holds. The method should be able to be extended beyond leading order, allowing one to also see corrections to factorization.

II. THE OCTET CONTRIBUTION
The amplitude for the singlet or octet part is where we set A µ ≡ A µ = the external field. As one can see, in this approximation we neglect higher Fock state contributions. Our normalization is such that T r t a t b = δ ab /2. Thus we can rewrite the equation (5) as with non-local correlators Shifting the x-dependence of the second correlator We shall work in the Fock-Shwinger gauge [5] which gives us the advantage of expressing the A µ in terms of gauge-invariant quantities (such as the gluon stress tensor G a ρµ ): We treat the light quarks as massless, so the bare propagator of a light quark is given by This allows us to rewrite (6) as The correlator G(0, ±x) entering the expression above is non-perturbative and nonlocal but can be related to the wave function of the forming pion. Following [6] we expand it near the light cone (x 2 ∼ 0). The general form of expansion is with Γ α being the full set of Dirac matrices. We would like to note, however, that since G(0, x) (or G(x, 0)) contributes to the trace, only γ α and γ 5 γ α terms enter the final expression for G(0, x) (G(x, 0)). In Eq. (12), the non-gauge invariant terms are exactly zero by the FS gauge condition (9) which allows us to use analysis similar to [6]. Moreover, in this gauge Parameterizing unknown matrix elements in the conventional form where q is a momentum of the pion we arrive at the simple form for the G-correlators with the function φ(z) introduced through its moments The zeroth moment gives the overall normalization of φ(z): dz φ(z) = f π /(2 √ N c ) with f π being the decay constant of π-meson. Inserting (15) into the (6) we obtain This gives for the octet amplitude of B → Dπ transition: 1 Note that the transferred momentum q can be written as q = En where n is a null vector (as in the approximation of collinear u and d -quarks) and E is energy transferred to the light quark system (it is large in the kinematical limit under consideration). Observe that contrary to [2] there are terms in (18) which are not proportional to n · A.
This amplitude is also nonlocal, so we use the same idea as in Eq. (12) expanding it around x 2 ∼ 0 and introducing "generating functions". It is clear that the only matrix element to be parameterized is In FS gauge one can expand A µ (x) about x = 0 using "gauge invariant" decomposition [5] A This implies the matrix elements In order to parameterize the matrix elements appearing in the expression above we note that there are a total of n + 3 indices and that the whole expression is anti-symmetric with respect to ρ ↔ ν interchange. Also, there are only two vectors (or their combinations) available, so we choose them to be p B and q. This leads to with a i and b i being unknown coefficients. Performing the contraction and dropping terms proportional to q 2 and x 2 we have for the first matrix element in (18) Note that in this limit the dependence of the matrix element on b i cancels out. These sums can be parameterized in terms of some distribution function f (n, ν). Using the binomial formula, we obtain where C i n is a binomial coefficient. This gives for the matrix element Performing resummation with respect to n we arrive at This results in Similarly, for the second matrix element one obtains At this stage we have classified the infinite number of the unknown matrix elements in terms of a few continuous functions. The strategy at this point in general would be to consider the structure of these functions and make reasonable smooth guesses as to their form. For example φ(z) is related to wavefunction used in the studies of the perturbative limit of the pion form factor and has the asymptotic form φ(z) = 2f π z(1 − z)/ √ 6 [6]. However, we find that an important cancellation occurs which is independent of the form of these structure functions. Taking a β = (α + z)q β + ανp B β , b β = (1 + z + α)q β + ανp B β and integrating over x we obtain for the M The key integral is at the leading order in q 2 for any φ and ψ. This implies that the semi-soft-one-gluon matrix elements do not bring new contributions to the factorization approximation result in the kinematical limit under consideration.
We would like to point out that the two-pseudoscalar final state is somewhat special: for instance, B → Dρ does not fall into the same category -there is a good chance that under the similar conditions there is no complete cancellation of the diagrams. This is due to the fact that there can be extra terms in (22) that include polarization vectors of ρ meson.
In conclusion, we have constructed a QCD-based model for the factorization approximation in the limit E ∼ m bm b →∞ . In the framework of the model the complete cancellation of the non-factorizable terms is observed at the leading order in q 2 , x 2 . The OPE is resummed by introducing two generating (distribution) functions that can be independently parameterized. The model gives a consistent way to estimate non-leading (non-factorizable) contributions.