[Paper Review] Inconsistences of the k_T-Factorization in Exclusive B-Meson Decays

Feng Feng, J. P.+1 more|ArXiv.org|Jan 20, 2009

Particle physics theoretical and experimental studies1 references3 citations

TL;DR

This paper identifies fundamental theoretical inconsistencies in the $k_T$-factorization approach for exclusive B-meson decays, demonstrating that perturbative coefficients derived from off-shell parton amplitudes are gauge-dependent due to light-cone singularities, violating gauge invariance in general covariant gauges. It further shows that two distinct B-meson wave functions exist with different evolution equations, rendering the standard $k_T$-factorization's use of a single wave function inconsistent for resumming large logarithms at leading log accuracy.

ABSTRACT

The k_T-factorization has been widely used for exclusive decays of B-mesons. In this factorization the pertubative coefficients are extracted from scattering of off-shell partons. Because the off-shellness of partons the extracted pertubative coefficients in general are not gauge-invariant. We show that these perturbative coefficients contain gauge-dependent singularities beyond tree-level. This leads to that the k_T-factorization is gauge-dependent and violated in the general covariant gauge. This is not consistent with expectations of physics. In the factorization there exist two different wave functions for a B-meson in general. We show that the two wave functions satisfy different evolution equations whose solutions are used to re-sum large log's. Based on this fact, the rseummation of large log's at leading log approximation in the widely used k_T-factorization with only one wave function is inconsistent.

Motivation & Objective

• To investigate the gauge invariance of the $k_T$-factorization framework in exclusive B-meson decays beyond tree-level.
• To examine whether the widely used single B-meson wave function in $k_T$-factorization is theoretically consistent with the existence of two distinct wave functions in the heavy quark limit.
• To assess the consistency of large logarithm resummation in $k_T$-factorization when only one wave function is employed.
• To clarify the physical implications of gauge-dependent singularities arising from off-shell parton amplitudes in the factorization scheme.

Proposed method

• Analyzing the semi-leptonic decay $B \to \pi\ell\nu_\ell$ using Heavy Quark Effective Theory (HQET) for the b-quark.
• Deriving the leading-order amplitude in the $k_T$-factorization framework, expressing it via matrix elements involving two distinct B-meson wave functions $\phi_+$ and $\phi_-$.
• Computing one-loop corrections to the perturbative coefficients, identifying gauge-dependent light-cone singularities originating solely from the pion and B-meson wave functions.
• Deriving evolution equations for $\phi_+$ and $\phi_-$ in the $\zeta$-variable, showing they differ in structure and logarithmic dependence.
• Comparing the resummation of large logarithms via the solutions of these distinct evolution equations, revealing inconsistency in the single-wave-function approximation.
• Using the Feynman gauge as a reference to show that singularities vanish, confirming gauge dependence in general covariant gauges.

Experimental results

Research questions

• RQ1 Are the perturbative coefficients in $k_T$-factorization gauge-invariant when computed beyond tree-level for exclusive B-decays?
• RQ2 Do the two distinct B-meson wave functions $\phi_+$ and $\phi_-$ in the heavy quark limit satisfy the same evolution equations in the $k_T$-factorization framework?
• RQ3 Can the resummation of large logarithms in $k_T$-factorization be consistently performed using only one wave function, given that the two wave functions evolve differently?
• RQ4 What is the origin of gauge-dependent singularities in the $k_T$-factorization, and do they originate from the wave functions or the hard scattering amplitudes?
• RQ5 Is the widely used $k_T$-factorization framework theoretically consistent, given its reliance on a single wave function and gauge-dependent coefficients?

Key findings

• The $k_T$-factorization for exclusive B-decays is gauge-dependent due to light-cone singularities in the perturbative coefficients, which appear in general covariant gauges but vanish in Feynman gauge.
• These singularities arise exclusively from the B-meson and pion wave functions, not from the hard scattering amplitudes, indicating a non-gauge-invariant structure.
• Two distinct B-meson wave functions $\phi_+$ and $\phi_-$ exist in the heavy quark limit, and they satisfy different evolution equations with different logarithmic dependencies on the $\zeta$-variable.
• The evolution equation for $\phi_+$ includes a double-logarithmic term $\ln^2\zeta^2$ from specific diagrams, while $\phi_-$ lacks this term, leading to different resummation properties.
• The resummation of large logarithms in the standard $k_T$-factorization, which uses only one wave function, is inconsistent because the two wave functions and their evolution equations are fundamentally different.
• The $k_T$-factorization framework is therefore theoretically inconsistent beyond tree-level, as it violates gauge invariance and fails to properly account for the dual wave function structure and their distinct evolution.