Collaborators:
Asit K. De
Dipankar Chakrabarti
James P. Vary
There are various and well-known motivations to study QCD in the light-front Hamiltonian formalism. In fact, there have been many attempts recently to study relativistic bound state problem in the Hamiltonian formalism in a light-front Fock space basis. It has been realized that a major impediment to a straightforward diagonalization of the Hamiltonian is the rapid growth of the dimension of the Hamiltonian matrix with particle number. An alternative approach will be to use an effective Hamiltonian that operates in a few particle basis. A challenging problem here is that, for a successful description of low energy observables, the effective Hamiltonian must incorporate main features of strong interaction dynamics.
When it comes to the study of bound states, QCD poses challenging problems. To overcome many pitfalls of standard effective Hamiltonians, similarity renormalization was proposed in the literature. It avoids vanishing energy denominators and thus provides an improvement over Bloch effective Hamiltonian. Before embarking on a detailed study of effective Hamiltonian in the similarity renormalization approach which is a modification of the Bloch effective Hamiltonian, it is quite instructive to study the Bloch effective Hamiltonian itself. This will provide us quantitative measures on the strengths and weaknesses of numerical procedures in handling singular interactions on the computer. It is crucial to have such quantitative measures in order to study the effects of similarity cutoff factors on the nature of the spectrum.
We first studied [1] the meson sector of 2+1 dimensional light-front QCD using a Bloch effective Hamiltonian in the first non-trivial order. The resulting two dimensional integral equation was converted into a matrix equation and solved numerically. We investigated the efficiency of Gaussian quadrature in achieving the cancellation of linear and logarithmic light-front infrared divergences. The vanishing energy denominator problem which leads to severe infrared divergences in 2+1 dimensions was investigated in detail. Our study indicated that in the context of Fock space based effective Hamiltonian methods to tackle gauge theories in 2+1 dimensions, approaches like similarity renormalization method may be mandatory due to uncanceled infrared divergences caused by the vanishing energy denominator problem. We defined and studied numerically a reduced model which is relativistic, free from infrared divergences, and exhibits logarithmic confinement. The manifestation and violation of rotational symmetry as a function of the coupling were studied quantitatively.
Next, we studied [2] similarity renormalization approach to the same problem. By performing analytical calculations with a step function form for the similarity factor, we showed that in addition to curing the vanishing energy denominator problem, similarity approach generates linear confining interaction for large transverse separations. However, for large longitudinal separations, the generated interaction grows only as the square root of the longitudinal separation and hence produces violations of rotational symmetry in the spectrum. We carried out numerical studies in the G{\l}azek-Wilson and Wegner formalisms and presented low lying eigenvalues and wavefunctions. We investigated the sensitivity of the spectra to various parameterizations of the similarity factor and other parameters of the effective Hamiltonian, especially the scale sigma. Our results illustrated the need for higher order calculations of the effective Hamiltonian in the similarity renormalization scheme.
A complementary approach to hadron spectra in light front formulation is Transverse Lattice QCD which has many interesting features. With the gauge choice A^{+a}=0 and the elimination of the constrained variable A^-, it uses minimal gauge degrees of freedom in a manifestly gauge invariant formulation exploiting the residual gauge symmetry in this gauge. So far encouraging results have been obtained in the pure gauge sector and in the meson sector with particle number truncation.
In Ref. [3] we addressed the problems of fermions in light front QCD on a transverse lattice. We proposed and numerically investigated different approaches of formulating fermions on the light front transverse lattice. In one approach we used forward and backward derivatives. There is no fermion doubling and the helicity flip term proportional to the fermion mass in the full light front QCD becomes an irrelevant term in the free field limit. In the second approach with symmetric derivative (which has been employed previously in the literature), doublers appear and their occurrence is due to the decoupling of even and odd lattice sites. We studied their removal from the spectrum in two ways namely, light front staggered formulation and the Wilson fermion formulation. The numerical calculations in free field limit were carried out with both fixed and periodic boundary conditions on the transverse lattice and finite volume effects are studied. We found that an even-odd helicity flip symmetry on the light front transverse lattice is relevant for fermion doubling.
Subsequently, we investigated [4] q{\bar q} spectra and wavefunctions of light front transverse lattice Hamiltonians that result from different methods of formulating fermions on the transverse lattice. We adopted the one link approximation for the transverse lattice and Discrete Light Cone Quantization (DLCQ) to handle longitudinal dynamics. We performed a detailed study of the continuum limit of DLCQ and associated techniques to manage severe light front infrared divergences. We explored the effects of various parameters of the theory, especially, the strength of the helicity-flip interaction and the link mass on spectra and wavefunctions.
[1] Mesons in (2+1) Dimensional Light front QCD: Investigation of a Bloch Effective
Hamiltonian, Dipankar Chakrabarti and A. Harindranath, Phys. Rev. D
64, 105002 (2001).
[2] Mesons in (2+1) dimensional Light front QCD.II: Similarity Renormalization
Approach, Dipankar Chakrabarti and A. Harindranath, Phys. Rev. D 65,
045001 (2002).
[3] Fermions on the Light Front Transverse Lattice, Dipankar Chakrabarti, Asit
K. De, A. Harindranath, Phys. Rev. D 67, 076004 (2003).
[4] A Study of Quark - Antiquark States in Transverse Lattice QCD Using
Alternative Fermion Formulations , Dipankar Chakrabarti, A. Harindranath,
and James P. Vary, Phys. Rev. D 69, 034502 (2004).