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.

Older data from 1998

Results from 1999

Scaling trajectory for various methods.

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). 
  • ytraj_imp_11.out
  • ytraj_imp_12.out
  • ytraj_imp_13.out
  • ytraj_imp_14.out
  • ytraj_imp_15.out
• Anti-periodic boundary conditions, plain DLCQ.
  • ytraj_0_11.out
  • ytraj_0_12.out
  • ytraj_0_13.out
• Periodic boundary conditions, plain DLCQ.  Since convergence in K is so poor, we resort to a 1/K extrapolation.
  • Entire trajectory: ytraj_1_10.out
  • Re-run entire trajectory: ytraj_1_21.out
  • Re-run entire trajectory: ytraj_1_31.out
  • Re-run entire trajectory: ytraj_1_41.out
  • Re-run entire trajectory: ytraj_1_51.out
  • Using above runs, lowest chi^2 for each mass: ytraj_1_composite.out
  • Re-run entire trajectory: ytraj_1_61.out
• 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.
  • Entire trajectory: ttraj_1_10.out
  • Rerun with slightly larger K and different starting couplings: ttraj_1_20.out
  • Rerun using fit to above as starting points: ttraj_1_30.out (generally poor)
  • Rerun using ytraj_1_* composite data as starting points: ttraj_1_40.out
  • Rerun using ttraj_1_* composite data as starting points: ttraj_1_50.out
  • Composite best chi² of the above: ttraj_1_composite.out
• 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.
  • First try: ttraj_2_10.out and ttraj_2_11.out.  Some spectra are printed incorrectly although chi² and couplings are OK. 
  • Second try: ttraj_2_20.out.  Some spectra are printed incorrectly, although the couplings and chi² are ok.
  • Third try: ttraj_2_30.out.
  • Fourth try: ttraj_2_40.out.
  • Fifth try, fitting to teper's 0⁺⁺ mass: ttraj_2_45.out.
  • Best chi² for each mass: ttraj_2_composite.out.

Compare methods

• K_convergence_dlcq.ps: K convergence of a two link state, winding=(2,0), for various DLCQ methods. The coupling constants are from the best fit data ytraj_1_composite.out.
• spectrum_even.ps (LaTeX source) spectrum from the best fit datum in ytraj_1_composite.out.
• ttraj_spectrum.ps (LaTeX source) spectrum from the best fit datum in ttraj_1_composite.out.

Entire spectrum

• dense_spectrum_6.out
• dense_spectrum_8.out
• dense_spectrum_10.out
• full_spectrum_12.out; less than the full number of states in a sector.

• dense_6_0.out dense_8_0.out dense_10_0.out full_12_0.out
• dense_6_1.out dense_8_1.out dense_10_1.out full_12_1.out
• dense_6_2.out dense_8_2.out dense_10_2.out full_12_2.out
• dense_6_3.out dense_8_3.out dense_10_3.out full_12_3.out
• dense_6_4.out dense_8_4.out dense_10_4.out full_12_4.out
• dense_6_5.out dense_8_5.out dense_10_5.out full_12_5.out
• dense_6_6.out dense_8_6.out dense_10_6.out full_12_6.out
• dense_6_7.out dense_8_7.out dense_10_7.out full_12_7.out
• dense_6_8.out dense_8_8.out dense_10_8.out full_12_8.out (some sectors missing)
• dense_6_9.out dense_8_9.out dense_10_9.out full_12_9.out (some sectors missing)
• dense_6_10.out dense_8_10.out dense_10_10.out full_12_10.out (some sectors missing)

E-mail: bvds@pitt.edu