Boosting QED and QCD bound states in the path integral formalism

Wave functions and energy eigenvalues of the path integral Hamiltonian are studied in the Lorentz frame moving with velocity $v$. The instantaneous interaction produced by the Wilson loop is shown to be reduced by an overall factor $\sqrt{1-(\frac{v}{c}{)}^2}$. As a result, one obtains the boosted energy eigenvalues in the Lorentz covariant form $E=\sqrt{{\mathbf{P}}^2+M_0^2}$, where $M_0$ is the c.m. energy, and this form is tested for two free particles and for the Coulomb and linear interaction. Using Lorentz-contracted wave functions of the bound states, one obtains the scaled-parton wave functions and valence quark distributions for large $P$. Matrix elements containing wave functions moving with different velocities strongly decrease with growing relative momentum; e.g., for the timelike form factors, one obtains $F_h(Q_0)\sim {(\frac{M_h}{Q_0})}^{2n_h}$ with $n_h=1$ and 2 for mesons and baryons, as in the “quark counting rule.”

Figure 1

The amplitude of the process $e^{+}{e}^{-}\to B\overline{B}$ with an intermediate state $Q\overline{Q}(Q=u,d,s,\dots )$. The step $X$ in the process is a nonperturbative double pair creation during a small time interval $\mathrm{\Delta }t\sim \lambda \sim 0.1\mathrm{fm}$.