Top Momentum Distribution

Next we consider the top-quark momentum ( $\vert{\bf {p}}_t\vert$) distribution near $t\bar{t}$ threshold [8,9]. It has been shown that experimentally it will be possible to reconstruct the top-quark momentum ${\bf {p}}_t$ from its decay products with reasonable resolution and detection efficiency. Fig. 4.2(a) shows a comparison of reconstructed top momenta (solid circles) with that of generated ones (histogram), where the events are generated by a Monte Carlo generator and are reconstructed after going through detector simulators and selection cuts; see Subsection 4.2.4 for details. The figure demonstrates that the agreement is fairly good.

Theoretically, the top-quark momentum distribution is given by

 

$$\frac{d\sigma}{d\vert{\bf {p}}_t\vert} \propto \biggl\vert \sum_n \frac{\phi_n ({\bf {p}}_t)\psi_n^* ({\bf {0}})}{E-E_n+i\Gamma_t} \biggl\vert^2 + \mbox{(sub-leading)} .$$ (4.3)

The $\vert{\bf {p}}_t\vert$-distribution is thus governed by the momentum-space wave functions of the resonances. By measuring the momentum distribution, essentially we measure (a superposition of) the wave functions of the toponium resonances. Shown in Fig. 4.2(b) are the top momentum distributions for various energies.

  

Figure: (a) Reconstructed momentum distribution (solid circles) for the lepton-plus-4-jet mode, compared with the generated distribution (histogram). The Monte Carlo events were generated with $\alpha _s(M_Z)=0.12$ and m_t = 150 GeV [5]. (b) Top-quark momentum distributions $d\sigma/d\vert{\bf {p}}_t\vert$for various c.m. energies measured from the lowest lying resonance, $\Delta E = \sqrt{s} - M_{1S}$, taking $\alpha _s(M_Z) = 0.118$ and m_t = 175 GeV.

Eps files of left figure and left figure

One may also vary the magnitude of $\alpha _s$ and confirm that the distribution is indeed sensitive to the resonance wave functions [8,9]. Hence, the momentum distribution provides information independent of that from the total cross section.