NNLO Vertex Corrections in charmless hadronic B decays: Imaginary part

TL;DR

This work delivers the first complete NNLO evaluation of the imaginary part of vertex corrections in the QCD Factorization framework for charmless two-body B decays, demonstrating factorization through explicit cancellation of IR divergences and providing analytic results with charm-mass dependence. By performing the 2-loop calculation in the CMM basis and translating to the traditional basis, the authors obtain the NNLO imaginary parts of the topological tree amplitudes, including detailed Gegenbauer-convoluted kernels and their scale dependences. The numerical analysis, including spectator scattering in LL resummation, shows that NNLO corrections substantially modify strong phases, particularly enhancing $ ext{Im}oldsymbol{α}_2$, with total NNLO contributions being significant though subject to power- and scale-related uncertainties. This work advances the precision of direct CP-violation predictions in $B ooldsymbol{ππ}$ and lays groundwork for a full NNLO treatment of penguin amplitudes and real parts.

Abstract

We compute the imaginary part of the 2-loop vertex corrections in the QCD Factorization framework for hadronic two-body decays as B -> pi pi. This completes the NNLO calculation of the imaginary part of the topological tree amplitudes and represents an important step towards a NNLO prediction of direct CP asymmetries in QCD Factorization. Concerning the technical aspects, we find that soft and collinear infrared divergences cancel in the hard-scattering kernels which demonstrates factorization at the 2-loop order. All results are obtained analytically including the dependence on the charm quark mass. The numerical impact of the NNLO corrections is found to be significant, in particular they lead to an enhancement of the strong phase of the colour-suppressed tree amplitude.

Figures (6)

• Figure 1: Generic 1-loop diagram with different insertions of a four-quark operator $Q_i$. The upper lines go into the emitted meson $M_2$, the quark to the right of the vertex and the spectator antiquark in the $\bar{B}$ meson (not drawn) form the recoil meson $M_1$.
• Figure 2: Full set of (naively) non-factorizable 2-loop diagrams. The bubble in the last four diagrams represents the 1-loop gluon self-energy. Only diagrams with final state inter-actions, i.e. with at least one gluon connecting the line to the right of the vertex with one of the upper lines, give rise to an imaginary part.
• Figure 3: Scalar Master Integrals that appear in our calculation. We use dashed lines for massless propagators and double (wavy) lines for the ones with mass $m_b$ ($m_c$). Dashed/solid/double external lines correspond to virtualities $0\,/\,u m_b^2\,/\,m_b^2$, respectively. Dotted propagators are taken to be squared.
• Figure 4: Tree level diagram (a), naively factorizable (b) and non-factorizable (c) NLO diagrams.
• Figure 5: 1-loop contribution to the form factor correction $F^{(1)}_\text{amp}$.