Confinement in Light-front Qcd

Numerical results for the calculation of the (rest-frame) QQ̄ potential, for light-front quantized QCD2+1 on a ⊥ lattice are presented. Both in the longitudinal as well as the ⊥ spatial directions one obtains linear confinement. The resulting potential is almost rotational symmetric. Light-front (LF) quantization is the most physical approach to calculating parton distributions on the basis of QCD . In the transverse lattice formulation of QCD , one keeps the time direction and one spatial direction continuous and discretizes the transverse spatial directions. The space time geometry is thus an array of 1+1 dimensional sheets. One the one hand, this provide both UV and IV cutoffs in the transverse directions and on the other hand one can still perform LF quantization since the longitudinal directions are continuous. Furthermore, in the compact formulation, it is straightforward to implement Gauss’ law as a constraint on the states, which helps avoid troublesome divergences for k → 0, which plague many other formulations of LF QCD. Another advantage of the transverse lattice is that confinement is manifest in the limit of large ⊥ lattice spacing a⊥. The mechanism differs for the longitudinal and the ⊥ directions: if one separates a QQ̄ pair longitudinally then, since a⊥ is large, the fields in different sheets couple only weakly and the quarks interact only with fields in the same sheet, i.e. effectively the theory reduces to 1 + 1 dimensional QCD, where confinement is known to be linear. In contrast, when one separates the QQ̄ pair transversely, gauge invariance demands that they are connected by a chain of (gluon) link fields. For large a⊥, where there are only little fluctuations, this implies that the energy of such a configuration is given by the energy for creating one link quantum times the number of link quanta, i.e. linear confinement also in the ⊥ direction. Since the confinement mechanisms are very different for these two cases, one might ask whether a rotationally invariant QQ̄ potential results in the continuum limit. In fact, in the limit of large a⊥ one finds in 2 + 1 dimensions : V (xL, x⊥) = σ (|xL|+ |x⊥|) , where σ is the string tension, which is clearly not rotationally invariant. In order to investigate this issue, I used DLCQ and a Lanczos algorithm to calculate the rest frame QQ̄ potential from the LF Hamiltonian for QCD2+1 on a ⊥ lattice. The procedure for computing the rest frame potential in this formalism follows Ref.. An approximation, where one allows at most one link field quantum per link, was used. Within this approximation, one obtains only a first order phase transition at the critical aContributed to PANIC96