[Paper Review] Single top quark differential decay rate formulae including detector effects

Joseph Boudreau, C. Escobar+3 more|arXiv (Cornell University)|Apr 20, 2013

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper develops analytic formulae for single top quark differential decay rates that incorporate detector effects via a spherical harmonic decomposition of the angular distributions. The method enables likelihood-based extraction of top quark decay amplitudes, phases, and polarization, including anomalous couplings and CP-violating phases, with a robust numerical minimization scheme using Hessian matrices and Lagrange multipliers for normalization constraints.

ABSTRACT

Since the discovery of parity violation in 1957, angular distributions of leptons coming from the weak decay of polarized fermions have been used to probe the structure of the Wqq' vertex. Vector and axial vector couplings reveal themselves in the angular distributions of both light and heavy polarized fermions, but tensor and pseudotensor couplings have a prominent influence on the angular distributions only for fermions heavier than the W boson; i.e. the top quark. The copious t-channel production of polarized single top quarks at the LHC provides an opportunity to study the angular distributions of leptons from polarized top quark decay. In this paper we develop formulae for differential rates intended to be used as a likelihood function in the simultaneous extraction of decay amplitudes, phases, and polarization. The incorporation of detector effects in these formulae is accomplished using a variant of the familiar convolution theorem applying to a decomposition of the differential rates in spherical harmonics.

Motivation & Objective

• To model the differential decay rates of polarized single top quarks at the LHC, including realistic detector effects.
• To enable simultaneous extraction of decay amplitudes, phases, and polarization from reconstructed collider data.
• To incorporate anomalous couplings and CP-violating phases in the likelihood framework for precision tests of the Standard Model.
• To develop a numerically stable minimization procedure using Hessian matrices and Lagrange multipliers for normalization constraints.
• To provide a complete analytic framework for likelihood-based analysis of top quark decay angular distributions in $t$-channel production.

Proposed method

• Uses the helicity formalism and Wigner D-functions to express the angular dependence of the $t \to W^+b$, $W^+ \to l^+\nu$ decay amplitude.
• Expands the differential decay rate in spherical harmonics to model detector resolution and acceptance effects via convolution theorems.
• Constructs a likelihood function in terms of complex coupling coefficients $g_{l,m}$, including real and imaginary parts as independent parameters.
• Applies a Newton-Raphson-like iterative minimization using first and second derivatives of the log-likelihood, derived analytically.
• Incorporates normalization constraints $|g_{l,m}|=1$ using Lagrange multipliers, modifying the Hessian and gradient for constrained optimization.
• Solves the minimization problem via matrix inversion $\vec{\eta}_0 = \mathbf{H}^{-1}(\mathbf{H}\vec{\eta} - \vec{\xi})$, with convergence checks and covariance matrix estimation.

Experimental results

Research questions

• RQ1 How can detector effects such as resolution and acceptance be systematically incorporated into the differential decay rate formulae for single top quarks?
• RQ2 What is the optimal likelihood-based method to extract complex decay amplitudes, phases, and polarization from angular distributions in top quark decays?
• RQ3 How can anomalous couplings and CP-violating phases in the $Wtb$ vertex be constrained using angular distributions in $t$-channel single top production?
• RQ4 What numerical strategy ensures stable and accurate minimization of the likelihood function under normalization and phase constraints?
• RQ5 How can the Hessian matrix be used to compute parameter uncertainties and correlations in the presence of complex coupling parameters?

Key findings

• The differential decay rate is expressed as a sum over spherical harmonics, enabling systematic inclusion of detector effects through convolution with detector response functions.
• The likelihood function is analytically differentiable with respect to all complex coupling coefficients $g_{l,m}$, allowing precise gradient-based minimization.
• The Hessian matrix of the log-likelihood is derived in closed form, enabling efficient computation of parameter uncertainties and correlations.
• The method successfully handles the $|g_{l,m}|=1$ normalization constraint via Lagrange multipliers, preserving physical consistency.
• The iterative minimization scheme converges reliably, with the final Hessian used to compute the covariance matrix as $\mathbf{C}^{-1} = \mathbf{H}/2$, ensuring statistical robustness.
• The framework is applicable to both signal and background fits, with phase freedom removed by fixing $g_{0,0}$ to zero.