Mesons on the transverse lattice

Student involvement

During the summer of 2001, two Geneva students, Beth Watson and Jonathan Bratt, worked on this project. Beth Watson wrote a Summary of Project Activities (PDF version).

The Hamiltonian operator

The Hamitonian operator is rather lengthy for this system. It is evaluated in a Mathematica notebook (Postscript version) using notation explained in some notes on the inchworm model and in the paper Mesons on a transverse lattice.

Limit of small Bjorken-`x`

We investigated the behavior of the wavefunctions in the limit of small Bjorken-x. The results of this investigation are summarized in some notes on the inchworm model.

Zero modes

If we include fermions, the gluon fields must have periodic boundary conditions. This causes some difficulties for the numerical solution of the bound state equations. I discuss a way of solving this problem using zero modes.

Test Cases

Here are some test cases for the DLCQ meson code. This uses Simon's form for the kappa self-intertia term.

K=2, one gluon: Hamiltonian matrix (2 P⁺P^-) and eigenvalues for various couplings, Mathematica notebook and postscript file.
Various values of K, one gluon.
K=3, 2 gluons:
K=5, 2 gluons:
K=2, 1 gluon, Nonzero transverse momentum and with various couplings turned off.
K=4, 1 gluon, Nonzero transverse momentum
K=3, 2 gluons, Nonzero transverse momentum and with various couplings turned off and more combinations.
K=4, 2 gluons, Nonzero transverse momentum.
K=5, 2 gluons, Nonzero transverse momentum
K=4, 3 gluons, Nonzero transverse momentum
K=5, 3 gluons, Nonzero transverse momentum

Results

Various values of K, one gluon using our form for the kappa self-inertia.

Here are some of the structure functions for the pion (J_z = 0^-), using our form for the kappa self-inertia. We used:
{MuB, MuF, G², KappaS, KappaA, Beta, Lambda1, Lambda2, Lambda3, Lambda4, Lambda5} =
{0.1803, 0.362, 1, -0.323, 0.162, 0.688, -0.038, -0.09, 374, 0.158, 143}

Structure function plots for zero gluons with K=10 and K=50 (solid line): plot data
Structure function plots for up to one gluon with K=10 and K=50 (solid line): plot data
Structure function plot for up to two gluons with K=10: plot data

And here are various plots of the lowest eigenvalue of the Hamiltonian (using same couplings as above):

Smallest eigenvalue as a function of K, holding number of links constant:
- Plot for 1 gluon, data points
- Plot for 2 gluons, data points
- Plot for 3 gluons, data points
- Combined plots.
Smallest eigenvalue as a function of number of gluons, holding K constant:
- Plot for K=4, data points
- Plot for K=5, data points
- Plot for K=6, data points
- Plot for K=7, data points
- Combined plots.

Probability distribution for gluons, with gluon truncation at 3 and K = 4, 5, 6: Plot.

Here is some timing information for the construction of hamiltonian matrices.

E-mail: bvds@pitt.edu