A proof of factorization for B -> D pi

TL;DR

It is proved that the matrix elements of four fermion operators mediating the decays B macro (0)-->D(+)pi(-) and B--->D(0)pi(+) factor into the product of a form factor describing the B-->D transition and a convolution of a short distance coefficient with the nonperturbative pion light-cone wave function.

Abstract

We prove that the matrix elements of four fermion operators mediating the decay B^0 -> D^+ π^- and B^- -> D^0 π^- factor into the product of a form factor describing the B -> D transition and a convolution of a short distance coefficient with the nonperturbative pion light-cone wave function. This is shown to all orders in alpha_s, up to corrections suppressed by factors of 1/mb, 1/mc, and 1/E_pi. It is not necessary to assume that the pion state is dominated by the q-qbar Fock state.

Figures (3)

• Figure 1: How the factorization of modes takes place.
• Figure 2: Matching for the order $\lambda^0$ Feynman rule with a heavy quark and $m$ collinear gluons.
• Figure 3: Order $\lambda^0$ Feynman rules for coupling ultrasoft or soft gluons (spring lines) to "collinear" fermions (thick dashed lines) and "collinear" gluons (thick spring+solid lines) (c.f. section 5). As an example, coupling an ultrasoft gluon to two collinear gluons in background field Feynman gauge one finds $F^{\nu\lambda}(q_1,q_2) = 2 \bar{n} \hbox{$\cdot$} q_1\, g^{\nu\lambda}$, which is $V_4^c=1$.