Mesons on the transverse lattice

Results from 2002 and 2003

Here is the mode expansion of the pseudoscalar four fermion interaction (postscript). If we keep track of flavor indices we get a real mess (postscript).

two-link truncation

In the following, the glueball parameters are obtained form the best fit glueball data ytraj_1_composite.out with periodic boundary conditions.

ll_1.out, no log(K) fit criterion.
ll_2.out, this isn't working too well probably because we are fitting the masses to the wrong function of K.
ll_5.out. Concentrate on the rho and pion states. The no log(K) fit criterion forces kappa_A to zero.
Remove Yukawa four-point interactions: ll_6.out, no log(K) fit criterion. Concentrate on the rho and pion states. Try again: ll_7.out.
ll_8.out, no log(K) fit criterion, concentrate more on the pi.
ll_9.out, no log(K) fit criterion, concentrate more on the pi, c in both directions.
Scan holding mu_F fixed: scan_mf_1.out, no log(K) fit criterion. Second try: scan_mf_2.out.
Scan holding mu_F fixed: scan_mf_3.out, no log(K) fit criterion. Concentrate on the rho and pion states.
Scan holding mu₁² fixed, Concentrate on the rho and pion states.
- scan_m1_1.out, no log(K) fit criterion.
- Second try: scan_m1_2.out, no log(K) fit criterion.
- Third try: scan_m1_3.out.
- Again: scan_m1_4.out, no log(K) fit criterion.
- Again: scan_m1_5.out, no log(K) fit criterion.
- finer mesh: scan_m1_6.out, no log(K) fit criterion.

Instead, fit both of the rho components to experiment and drop the degeneracy criterion. The chi^2 tries to gives 20% errors for the pion and rho c² and mass² (the pion mass² error is larger).

ll_10.out, different weights.
ll_11.out, Now adding c^2 criterion for all components of the rho, different weights.
ll_12.out, with log(K) criterion, different weights.
ll_13.out. also ll_14.out and ll_15.out. Wavefunctions for a point in ll_15.out.
Scan holding mu₁² fixed: scan_m1_7.out. A much larger range of mass: scan_m1_8.out.
Scans holding mu₀² and mu₁² fixed:
- scan_m0_m1_1.out. Wavefunctions from several points in this scan.
- scan_m0_m1_2.out, emphasizing mu₀²<0 region.
- scan_m0_m1_3.out, emphasizing mu₀²>0, mu₁²>0 region.
Here are some plots from the best fit data from these scans:
ll_16.out: some best fit points based on these scans. ll_17.out: re-do points that haven't converged. The starting couplings from:
- The first four points: mu₀²=-0.5 and mu₁²=0.2.
- The second four points: mu₀²=0.1 and mu₁²=-0.4.
- The third set of four points: mu₀²=0.1 and mu₁²=0.2.
- Start with best fit of ll_15.out.
- Start with best fit of ll_14.out.

three-link truncation

In the following, the glueball parameters are obtained form the best fit glueball data ytraj_1_composite.out with periodic boundary conditions.

lll_1.out, no log(K) fit criterion.
lll_2.out, no log(K) fit criterion, concentrate on rho and pi states.
lll_3.out, no log(K) fit criterion, concentrate more on the pi.
lll_4.out matching ll_11.out above.
lll_5.out, starting couplings from ll_16.out above.
lll_6.out and lll_7.out: include log(K) fit criterion. Starting couplings from ll_16.out above. Wavefunctions for lowest chi² point in lll_6.out and lll_7.out.
lll_8.out : include log(K) fit criterion, random starting couplings. Wavefunctions for points in lll_8.out.
Fix the (mass)² shift in the q-qbar sector to various values: lll_9.out. Second try: lll_10.out

Using the m₀²=0.5 datum from lll_10.out above, wavefunctions for the pion:

`K`	Probabilities	data file
10	2: 0.0446357, 3: 0.769965, 4: 0.147486, 5: 0.0379137	meson_5_20+4_4.m
12	2: 0.0514797, 3: 0.756411, 4: 0.148293, 5: 0.0438168	meson_5_24+4_4.m
15	2: 0.0618235, 3: 0.739964, 4: 0.147654, 5: 0.050559	meson_5_30+4_4.m
20	2: 0.0786231, 3: 0.719079, 4: 0.144348, 5: 0.0579499	meson_5_40+4_4.m

Using the m₀²=1.0 datum from lll_10.out above, wavefunctions for the pion:

`K`	Probabilities	data file
10	2: 0.0255917, 3: 0.770274, 4: 0.164213, 5: 0.0399208	meson_5_20+4_5.m
12	2: 0.0299683, 3: 0.757692, 4: 0.166077, 5: 0.0462625	meson_5_24+4_5.m
15	2: 0.0365303, 3: 0.743192, 4: 0.166705, 5: 0.0535731	meson_5_30+4_5.m
20	2: 0.0472461, 3: 0.725968, 4: 0.16498, 5: 0.0618059	meson_5_40+4_5.m

Likewise, for the J_z=0 component of the rho:

`K`	Probabilities	data file
10	2: 0.0149028, 3: 0.790438, 4: 0.151007, 5: 0.0436527	meson_5_20-3_5.m
12	2: 0.0182405, 3: 0.774614, 4: 0.15595, 5: 0.0511955	meson_5_24-3_5.m
15	2: 0.0242197, 3: 0.753845, 4: 0.161249, 5: 0.0606859	meson_5_30-3_5.m
20	2: 0.0365393, 3: 0.72514, 4: 0.16562, 5: 0.0727004	meson_5_40-3_5.m

Likewise, for the J_z=1 component of the rho:

`K`	Probabilities	data file
10	2: 0.0223294, 3: 0.778436, 4: 0.160621, 5: 0.038613	meson_5_20-7_5.m
12	2: 0.026616, 3: 0.765037, 4: 0.163238, 5: 0.0451094	meson_5_24-7_5.m
15	2: 0.0332713, 3: 0.749218, 4: 0.164757, 5: 0.0527536	meson_5_30-7_5.m
20	2: 0.0445898, 3: 0.730043, 4: 0.163851, 5: 0.0615159	meson_5_40-7_5.m

Using the m₀²=1.5 datum from lll_10.out above, wavefunctions for the pion:

`K`	Probabilities	data file
10	2: 0.0174693, 3: 0.776035, 4: 0.166369, 5: 0.0401269	meson_5_20+4_6.m
12	2: 0.0204997, 3: 0.763594, 4: 0.169292, 5: 0.046615	meson_5_24+4_6.m
15	2: 0.0250157, 3: 0.749565, 4: 0.17127, 5: 0.0541496	meson_5_30+4_6.m
20	2: 0.0323865, 3: 0.733499, 4: 0.171331, 5: 0.0627838	meson_5_40+4_6.m

Using the m₀²=2.0 datum from lll_10.out above, wavefunctions for the pion:

`K`	Probabilities	data file
10	2: 0.012776, 3: 0.779957, 4: 0.167068, 5: 0.0401985	meson_5_20+4_7.m
12	2: 0.014994, 3: 0.767431, 4: 0.170783, 5: 0.046792	meson_5_24+4_7.m
15	2: 0.0182851, 3: 0.753433, 4: 0.173781, 5: 0.0545005	meson_5_30+4_7.m
20	2: 0.0236483, 3: 0.737681, 4: 0.175213, 5: 0.0634575	meson_5_40+4_7.m

The pion

Here, we take a closer look at the m₀²=1.0 datum from lll_10.out above.

Quark distribution functions and spin-dependant distrubution functions.

Results for much larger values of K along with some fits to the quark distribution and a plot of the distribution function and a plot of mass². Here are some results where each of the sector dependant masses are shifted, in turn:

m₀² smaller; fits to the quark distribution jan_2a_fits.m and plot jan_2a_quark.ps.
m₀² larger; fits to the quark distribution jan_2b_fits.m and plot jan_2b_quark.ps.
m₁² smaller; fits to the quark distribution jan_3a_fits.m and plot jan_3a_quark.ps.
m₁² larger; fits to the quark distribution jan_3b_fits.m and plot jan_3b_quark.ps.
m₂² smaller; fits to the quark distribution jan_4a_fits.m and plot jan_4a_quark.ps.
m₂² larger; fits to the quark distribution jan_4b_fits.m and plot jan_4b_quark.ps.

Pion wavefunctions at nonzero transverse momenta for the m₀²=1.0 datum:

The associated output file perp_10_0.tmp contains information about the run.

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

Fix the (mass)² shift in the q-qbar sector to various values: lll_11.out; try again lll_12.out (at work).

four-link truncation

In the following, the glueball parameters are obtained form the best fit glueball data ytraj_1_composite.out with periodic boundary conditions. No log(K) fit, because matrices are too large.

E-mail: bvds@pitt.edu