Exploratory study of the off-shell properties of the weak vector bosons

Gauge invariance requires even in the weak interactions that physical, observable particles are described by gauge-invariant composite operators. Such operators have the same structure as those describing bound states, and consequently the physical versions of the $W^{\pm }$, the $Z$, and the Higgs should have some kind of substructure. To test this consequence, we use lattice gauge theory to study the physical weak vector bosons off shell, especially their form factor and weak radius, and compare the results to the ones for the elementary particles. We find that the physical particles show substantial deviations from the structure of a pointlike particle. At the same time the gauge-dependent elementary particles exhibit unphysical behavior.

Figure 1

On the left-hand side we show the transverse part of the propagator and on the right-hand side the longitudinal part of the propagator as a function of momentum with two nonvanishing components. Results are plotted with (black circles) and without (red squares) lattice corrections for the largest lattice volume $20^4$ and for set A1.

Figure 2

The transverse propagators as function of momentum for the largest lattice volume $20^4$ for all parameter sets available. The lattice-corrected transverse gauge-invariant propagators are shown as open symbols, the propagators of elementary fields are displayed with the corresponding closed symbols, and the optimal ground-state propagator (19) is also shown. On the right-hand side the ratio of the gauge-invariant as well as gauge-variant propagators to the optimal propagator are shown. All error bars are much smaller than the symbol sizes.

Figure 3

The form factor in the symmetric momentum configuration on the $20^4$ lattices for all parameter sets without (top-left panel) and with (top-right panel) the phenomenological lattice improvement. The bottom-left panel shows the correction factor alone, and the bottom-right panel a magnification of the corrected and uncorrected vertex for the A1 set.

Figure 4

The $3\text{−}W$ vertex form factor (left-hand side) and the physical 3-vector vertex form factor (15) (right-hand side) for set A1 for all volumes and momentum configurations. Here and hereafter, for the ratio the results from the largest volume have been linearly interpolated and the error determined by error propagation.

Figure 5

The $3\text{−}W$ vertex form factor (left-hand side) and the physical 3-vector vertex form factor (15) (right-hand side) for set A2 for all volumes and momentum configurations.

Figure 6

The $3\text{−}W$ vertex form factor (left-hand side) and the physical 3-vector vertex form factor (15) (right-hand side) for set A3 for all volumes and momentum configurations.

Figure 7

The $3\text{−}W$ vertex form factor (left-hand side) and the physical 3-vector vertex form factor (15) (right-hand side) for set B1 for all volumes and momentum configurations.

Figure 8

The $3\text{−}W$ vertex form factor (left-hand side) and the physical 3-vector vertex form factor (15) (right-hand side) for set B2 for all volumes and momentum configurations.

Figure 9

The $3\text{−}W$ vertex form factor (top panels) and the physical 3-vector vertex form factor (15) (middle panels) in the symmetric (right panels) and back-to-back momentum configuration (left panels) for all sets on the largest volume. The bottom panels show the ratio of the physical form factor to the $3\text{−}W$ form factor. The form factors have been renormalized in the back-to-back configurations to one at large momenta utilizing the fit results in Sec. 5c.

Figure 10

The fit parameters $a$ (top panels) and $b$ (bottom panels) for the $3\text{−}W$ form factor (left panels) and the physical form factor (right panels) for all lattice setups. Open symbols are from the back-to-back configuration and closed symbols from the symmetric configuration.

Figure 11

Comparison of the fit (18) using the fit parameters shown in Fig. 10 compared to the data in the back-to-back configuration for the $3\text{−}W$ form factor (left panel) and physical form factor (right panel). Note that the plot is against $p^2$ down to the pole located at $p^2=-m_W^2$ to illustrate its closeness.

Figure 12

The derived reduced effective coupling strength (top panels) and radius (bottom panels). The left panels show the $3\text{−}W$ case and the right panels the physical case. For negative radius squared $\mathrm{sign}(r^2)\sqrt{|r^2|}$ is plotted. For the $3\text{−}W$ case the radius in fm and for the physical case the inverse radius in GeV is plotted, see text. Note the different scales. One point in the physical case and a few points in the $3\text{−}W$ case at small volumes are outside the plot range.