Power-suppressed corrections to factorization in class-I two-body decays $\bar B\to D^{(*)}\pi$: completing the Politzer-Wise program

In 1991 Politzer and Wise computed the one-loop perturbative corrections to naive factorization for the class-I (color-allowed) non-leptonic decays $\bar B\to D\pi^-$ and $\bar B\to D^*\pi^-$, expressing the non-factorizable vertex corrections as a convolution of hard kernels (depending on $r=m_c/m_b$) with the pion light-cone distribution amplitude, and listed what remained unresolved: the subtraction-scheme dependence of the one-loop result pending two-loop anomalous dimensions of the four-quark operators, the unknown sign and magnitude of the correction given the then-unknown pion distribution amplitude, the uncomputed higher-order perturbative and power-suppressed terms, and the absence of an all-orders factorization proof. Subsequent work has resolved the scheme issue, computed the leading-power kernels through two loops in $\alpha_s$, constrained the pion distribution amplitude from lattice QCD, and proven factorization at leading power. What survives as a genuine open problem is the quantitative, first-principles control of the power-suppressed $O(\Lambda_{\mathrm{QCD}}/m_b)$ corrections - spectator scattering, weak annihilation, higher-twist and soft contributions - needed to bring predictions of the decay amplitudes and of ratios such as $\Gamma(\bar B\to D\pi^-)/\Gamma(\bar B\to D^*\pi^-)$ to the few-percent level; current data versus theory indicate, but do not establish, such corrections at the 10-15 percent level on the amplitude.

Background

Non-leptonic two-body decays of B mesons (transitions of a b quark into a c quark plus a light quark-antiquark pair, e.g. $b\to c\bar u d$) are the benchmark channels for testing how the weak interaction is dressed by the strong interaction in purely hadronic processes. In the naive factorization approximation the decay amplitude is written as a product of a semileptonic form factor (the function parameterizing the $B\to D^{(*)}$ transition, obtained from the semileptonic decay $B\to D^{(*)}e\nu$) and a decay constant (the parameter measuring how a light meson couples to its valence-quark current); combined with heavy-quark symmetry (the approximate spin-flavor symmetry of QCD in the limit where the heavy-quark mass is much larger than the intrinsic hadronic scale $\Lambda_{\mathrm{QCD}}$, which relates the $B\to D$ and $B\to D^*$ form factors), this yields absolute predictions for ratios of decay rates, for instance $\Gamma(B\to D\pi^-)/\Gamma(B\to D^*\pi^-)=1$. Politzer and Wise (Phys. Lett. B 257 (1991) 399) computed the leading perturbative strong corrections to this picture for the class-I decays, in which the light meson is emitted by the weak current while the spectator quark of the B meson joins the charm quark in the heavy final-state meson. They expressed the one-loop, non-factorizable vertex corrections (gluons exchanged between the quarks forming the emitted pion and the heavy-quark line) as a convolution of hard coefficient functions with the pion light-cone distribution amplitude $\phi_\pi(x,m_b)$ - the probability amplitude for the valence quarks of a fast-moving pion to share its momentum in the fraction $x$ - with the kernels depending on the mass ratio $r=m_c/m_b$; in modern notation the operator basis is the color-singlet current-current operator $O_1$ (the four-quark $(V-A)\otimes(V-A)$ weak operator with color-singlet currents) and its color-octet counterpart $O_8$. Their paper closed with four unresolved issues: (i) the $x$-independent one-loop combination $T_8^{(\mathrm{S})}+T_8^{(\mathrm{P})}$ of the color-octet hard-scattering kernel is subtraction-scheme dependent, and a scheme-independent next-to-leading-order statement requires the two-loop anomalous dimensions (the matrices governing the logarithmic scale dependence of the operators) of $O_1$ and $O_8$; (ii) the sign and magnitude of the convolution integral with $\phi_\pi$ were indeterminate because the pion light-cone distribution amplitude was not known; (iii) only the leading perturbative order and the leading power in $\Lambda_{\mathrm{QCD}}/m_b$ were computed, leaving higher-order corrections in $\alpha_s$ and power-suppressed non-perturbative terms unquantified; and (iv) no all-orders control of infrared-sensitive contributions, i.e. no factorization proof, was given. These questions seeded the modern theory of exclusive non-leptonic B decays. The QCD factorization (QCDF) framework of Beneke, Buchalla, Neubert and Sachrajda established that, at leading power in $\Lambda_{\mathrm{QCD}}/m_b$ and to all orders in $\alpha_s$, the matrix element factorizes into a $B\to D^{(*)}$ form factor and the convolution of a perturbatively computable hard kernel with the light-meson distribution amplitude; Bauer, Pirjol and Stewart gave an all-orders proof in soft-collinear effective theory (SCET, the effective field theory describing highly energetic light particles moving close to the light cone, together with their soft-gluon interactions, in heavy-meson decays) for $B\to D\pi$. The hard kernels are now known through two loops (order $\alpha_s^2$), the pion distribution amplitude is constrained by lattice QCD, and the two-loop (and now four-loop) anomalous dimensions of the four-quark operators are available, resolving items (i), the perturbative part of (iii), and the leading-power part of (iv). However, beyond leading power the factorization arguments break down: the corrections of order $\Lambda_{\mathrm{QCD}}/m_b$ receive contributions from soft and collinear physics that are not dominated by hard gluon exchange and cannot be computed within the leading-power factorization formula. Only heuristic estimates exist, while the comparison of the (now perturbatively controlled) leading-power predictions with measured branching fractions and decay-rate ratios shows persistent few-sigma differences that point to negative power corrections of order 10-15 percent at the amplitude level. The problem thus reduces to the last open item of the 1991 program: the quantitative first-principles account of the power-suppressed corrections.

Problem Statement

Establish the quantitative, first-principles account of the power-suppressed corrections to the leading-power factorization formula for the class-I two-body non-leptonic decays $\bar B\to D\pi^-$ and $\bar B\to D^*\pi^-$ computed by Politzer and Wise, together with their vector-meson analogues $\bar B\to D\rho^-$ and $\bar B\to D^*\rho^-$, of which $\bar B^0\to D^+\pi^-$ and $\bar B^0\to D^{*+}\pi^-$ are the paradigm (class-I meaning that the light meson is emitted by the weak current while the spectator quark of the B meson joins the charm quark in the heavy final-state meson). At leading power in $\Lambda_{\mathrm{QCD}}/m_b$ the amplitude is known to factorize,

$$\langle D^{(*)}L|Q_i|\bar B\rangle = \sum_j F_j^{B\to D^{(*)}}(m_L^2)\int_0^1 du\, T^{I}_{ij}(u,z)\,\Phi_L(u) + O\!\left(\frac{\Lambda_{\mathrm{QCD}}}{m_b}\right),$$

with $z=m_c/m_b$, the hard kernels $T^I$ now computed through $O(\alpha_s^2)$ and factorization proven at leading power. Determine the corrections of order $\Lambda_{\mathrm{QCD}}/m_b$ to this formula, including power-suppressed spectator scattering (gluon exchange involving the spectator quark of the B meson), weak annihilation (annihilation into gluons of the light quark-antiquark pair produced by the weak current), higher-twist light-cone contributions (corrections to the leading light-cone distribution amplitude involving additional partons or transverse momentum), and soft-gluon exchange between the $B\to D^{(*)}$ transition and the emitted light meson. The four contribution classes just listed jointly define the scope of the correction rather than four independent sub-deliverables, and the required non-perturbative inputs (such as the $B$- and $D$-meson light-cone distribution amplitudes) may be taken from existing determinations or obtained within the chosen framework. The answer must fix the sign and magnitude of the total correction to the color-allowed tree amplitudes $a_1(D^{(*)}L)$ with $L=\pi,\rho$ (the effective coefficient by which the naive-factorization amplitude is multiplied in these decays) and to the decay-rate ratios, in particular $\Gamma(\bar B^0\to D^+\pi^-)/\Gamma(\bar B^0\to D^{*+}\pi^-)$, or provide a rigorous model-independent bound on it.

References

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