# Mesons on the transverse lattice

Here are the lowest meson isovector states for each
`J`_{z}^{P1 C} sector.
The operator `P`_{1} generates reflections
in the `x`^{1} direction.
`J`_{z}^{P1 C} |
state | mass (MeV) |
state | mass (MeV) |

0^{++} | a_{0} | 983 |
a_{2} | 1320 |

0^{+-} | rho | 768 |
rho(1450) | 1450 |

0^{-+} | pion | 135 |
a_{1} | 1260 |

0^{--} | b_{1} | 1235 |
| |

2^{++} | a_{2} | 1320 |
| |

2^{+-} | rho_{3} | 1690 |
| |

2^{-+} | a_{2} | 1320 |
| |

2^{--} | rho_{3} | 1690 |
| |

1^{?-} | rho | 768 |
b_{1} | 1235 |

1^{?+} | a_{1} | 1260 |
a_{2} | 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`) =
L_{i}(`K`/10)*`kappa_5[``i`]

;

where L_{i}(`x`) is the Laguerre
polynomial and we sum over `i`.
### three-link truncation

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
`J`_{z}=±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.