Angular analysis of polarized top quark decay into $B$ mesons in two different helicity systems

We calculate the $\mathcal{O}({\alpha }_s)$ radiative corrections to the spin-dependent differential decay rates of the process $t\to b+W^{+}$. These are needed to study the angular distribution of the energy of hadrons produced in polarized top quark decays at next-to-leading order. In our previous work, we studied the angular distribution of the scaled energy of bottom-flavored hadrons ($B$) from polarized top quark decays, using a specific helicity coordinate system where the top quark spin was measured relative to the bottom momentum (system 1). Here, we study the angular distribution of the energy spectrum of $B$ hadrons in a different helicity system, where the top spin is specified relative to the $W$ momentum (system 2). These energy distributions are governed by the polarized and unpolarized rate functions which are related to the density matrix elements of the decay $t\to W^{+}+b$. Through this paper, we present our predictions of the $B$-hadron spectrum in the polarized and unpolarized top decay and shall compare the polarized results in two different helicity systems. These predictions can be used to determine the polarization states of top quarks and also provide direct access to the $B$-hadron fragmentation functions, and allow us to deepen our knowledge of the hadronization process.

Figure 1

Definition of the polar angle ${\theta }_P$ in two helicity systems, where (a) in the system 1, the three-momentum of the $b$-quark points into the direction of the positive $z$-axis, and (b) in the system 2, the momentum of the $W$-boson is defined along the positive $z$-axis. $\overrightarrow{P_t}$ is also the top polarization vector in both systems.

Figure 2

Definition of the azimuthal angle $\varphi$, the polarization vector of the top quark $\overrightarrow{P_t}$, and the polar angles $\theta$ and ${\theta }_P$ in the helicity coordinate system 1.

Figure 3

As in Fig. 2, but in the helicity coordinate system 2.

Figure 4

$d{\mathrm{\Gamma }}_B^{\mathbf{nlo}}/d{x}_B$ as a function of $x_B$ in the helicity system 1 (solid line) and the system 2 (dotted line). The polarized results are also compared to the unpolarized one $d{\mathrm{\Gamma }}_A^{\mathbf{nlo}}/d{x}_B$ (dashed line). The threshold at $x_B$ is shown.

Figure 5

$x_B$ spectrum at NLO in helicity system 2, normalized to the one in system 1.