Asymptotic freedom in the front-form Hamiltonian for quantum chromodynamics of gluons

Asymptotic freedom of gluons in pure-gauge QCD is obtained in the leading terms of their renormalized Hamiltonian in the Fock space, instead of considering virtual Green’s functions or scattering amplitudes. Namely, we calculate the three-gluon interaction term in the effective front-form Hamiltonian for gluons in the Minkowski space-time using the renormalization group procedure for effective particles (RGPEP), with a new generator. The resulting three-gluon vertex is a function of the scale parameter, $s$, that has an interpretation of the size of effective gluons. The corresponding Hamiltonian running coupling constant, $g_{\lambda }$, depending on the associated momentum scale $\lambda =1/s$, is calculated in the series expansion in powers of $g_0=g_{{\lambda }_0}$ up to the terms of third order, assuming some small value for $g_0$ at some large ${\lambda }_0$. The result exhibits the same finite sensitivity to small-$x$ regularization as the one obtained in an earlier RGPEP calculation, but the new calculation is simpler than the earlier one because of a simpler generator. This result establishes a degree of universality for pure-gauge QCD in the RGPEP.

Figure 1

The FF Hamiltonian third-order RGPEP running coupling constant for effective gluons, $g_{\lambda }$ of Eq. (68), is drawn using different dashed lines for different values of $x_0$, as a function of $\lambda$ in GeV, starting from an arbitrarily chosen value $g_0=1.1$ at ${\lambda }_0=100\mathrm{GeV}$. Plots (a) and (b) correspond to small-$x$ regularizations in Eqs. (23) and (24). The thick continuous lines in both plots show one and the same result for the regularization in Eq. (25), which exhibits no dependence on $x_0$.

Figure 2

Bare three-gluon vertex.

Figure 3

Graphical representation of terms (b5) and (b8).

Figure 4

Graphical representation of terms (b6) and (b9).

Figure 5

First-order term of the three-gluon vertex.

Figure 6

Graphical representation of the second-order RGPEP contribution to the Hamiltonian effective gluon mass term. Thin internal lines correspond to intermediate bare gluons and thick external to the effective gluons.

Figure 7

Third-order contributions to the three-gluon vertex, including counterterm, ($j$). The labels (a) to (j) correspond to contributions ${\gamma }_{(a)}$ to ${\gamma }_{(j)}$ in Eq. (48).

Figure 8

Effective three-gluon vertex (expansion up to third order).