# Information for 3+1 transverse lattice: glueballs

## Assignment of multiplets associated with the 2 dimensional lattice

Multiplets are as follows:

• In the following P_1 and P_2 are reflections, R is a 90° rotation, and E is the identity.
• Note that: P_2 = P_1 R^2 and R^2 = P_1 P_2.
• Charge conjugation: C = R^2 O, where O is orientation reversal
• A blank means that the state remains indefinite under the associated symmetry.
``` group    |                 Multiplet number
element  | 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
---------|---------------------------------------------------------
E       | 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
P_1     |          1 -1  1 -1  1 -1  1 -1
P_2     |          1 -1  1 -1 -1  1  1 -1              1 -1
R^2     |    1 -1  1  1  1  1 -1 -1  1  1  1 -1  1 -1
|
P_1 R   |          1 -1 -1  1              1  1 -1 -1        1 -1
R       |          1  1 -1 -1
R^3     |          1  1 -1 -1
P_1 R^3 |          1 -1 -1  1              1 -1 -1  1
-------------------------------------------------------------------
J_z^P_1 |         0+ 0- 2+ 2- 1+ 1-

The multiplets have the following use:

9,  8    P_perp = (c,0),    P_2 = +1
7, 10    P_perp = (c,0),    P_2 = -1
11,12    P_perp = (c,c),  P_1 R = +1
14,13    P_perp = (c,c),  P_1 R = -1
15       n = (c,0)          P_2 = +1
16       n = (c,0)          P_2 = -1
17       n = (c,c)        P_1 R = +1
18       n = (c,c)        P_1 R = -1
```

To produce these results, we wrote lots of C-code. This code is linked to standard packages BLAS, LAPACK, and (optionally) ARPACK along with some other standard routines, and a parallel lanczos solver.

## Results from 1999

These are our most important results for glueballs.

## Scaling trajectory for various methods.

For the mesons, one is forced to use periodic boundary conditions for the link fields.  This introduces some serious problems with convergence in K.

The due to finite-K errors, the couplings that produce the best Lorentz covariance are slightly shifted for different methods:

• Improved matrix elements, anti-periodic boundary conditions (this is what we used to obtain our glueball results in the past).
• Anti-periodic boundary conditions, plain DLCQ.
• Periodic boundary conditions, plain DLCQ.  Since convergence in K is so poor, we resort to a 1/K extrapolation.
• BIG basis, periodic boundary conditions, plain DLCQ.  Since convergence in K is so poor, we resort to a 1/K extrapolation.  Some spectra are printed incorrectly although chi2 and couplings are OK.
• 2003 BIG basis, with K up to 20 and periodic boundary conditions, plain DLCQ.  Since convergence in K is so poor, we resort to a 1/K extrapolation.

## Compare methods

Plots of the results, comparing the various methods:

## 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 the best fit glueball data ttraj_1_composite.out.  In order to make the calculations manageable, the spectrum for each sector was calculated separately:
Now calculate this for all the data in ttraj_1_composite.out

 E-mail: bvds@pitt.edu 