# Mesons on the transverse lattice

Here are the lowest meson isovector states for each JzP1 C sector.  The operator P1 generates reflections in the x1 direction.
 state mass (MeV) state mass (MeV) JzP1 C 0++ a0 983 a2 1320 0+- rho 768 rho(1450) 1450 0-+ pion 135 a1 1260 0-- b1 1235 2++ a2 1320 2+- rho3 1690 2-+ a2 1320 2-- rho3 1690 1?- rho 768 b1 1235 1?+ a1 1260 a2 1320

## Current results

In the following calculations, the pseudoscalar 4-fermion interaction is included in the Hamiltonian.  This was done with the help of Geneva Undergraduate Justin Lambright, project summary (PDF).  The renormalization of the associated coupling kappa_5 is complicated.  Our strategy is to make it an arbitrary function of K and determine its value by fitting some observable, either the pion mass or decay constant at each value of K

The coupling kappa_5 at a given K is specified by an array of numbers, with the value given by the formula:

kappa_5(K) = Li(K/10)*`kappa_5[i]`;
where Li(x) is the Laguerre polynomial and we sum over i.

In the following, the glueball parameters are obtained from the best fit glueball data ytraj_1_composite.out with periodic boundary conditions.
• kappa_5 determined by fitting the pion mass to the experimental value at each K. The results don't look too good: ggg_1.out and ggg_2.out
• kappa_5 determined by demanding, at each K, that the pion mass equal the average value and fitting the extrapolated value to the experimental value. This works somewhat better: ggg_3.out, ggg_4.out, and the best results ggg_5.out.  The coefficient of the log divergence of the rho is large.
• kappa_5 determined by demanding, at each K, that f_pi equal the average value.  Also, minimize pion and rho log divergence. ggg_6.out
In all cases, the largest source of error is the log(K) coefficient of the rho.  This may be simply an artifact of the low values of K used in these calculations.

In the following, the glueball parameters are obtained from the best fit glueball data ttraj_1_composite.out with periodic boundary conditions.

• kappa_5 determined by demanding, at each K, that f_pi equal the average value.  Also, minimize pion and rho log divergence. ggg_10.out.  (neither converged)
• kappa_5 determined by demanding, at each K, that the pion mass equal the average value and fitting the extrapolated value to the experimental value. ggg_11.out.

We switch computational technique to use explicit constrained fields.  Thus, the kappa_5 coupling is renamed kappa_P; it is still found by the equation above.  We also introduce the transverse-vector-four-fermion interaction, with coupling kappa_T which acts on the Jz=±1 components of the rho.  In both cases, we determine the couplings by demanding the associated decay constant be independent of K

## Entire spectrum

To find the entire spectrum, we find all the states for a given K with no additional truncation in particle number.  The coupling constants are from ggg_10.out above.  Although the kappa_5 coupling is included, it should have a negligible effect on the excited states.  In order to make the calculations manageable, the spectrum for each sector is calculated separately.
The mass shifts are based on the number of particles.

## Code

To produce these results, we wrote lots of C-code davecode.tgz.  This code is linked to standard packages BLAS, LAPACK, and (optionally) ARPACK along with other routines long.tgz, and a parallel Lanczos package plan3.tgz.
 E-mail: bvds@pitt.edu 