Predicting the singlet vector channel in a partially broken gauge-Higgs theory

We study a toy version of a grand-unified theory on the lattice: An $SU(3)$ gauge theory, which experiences a Brout-Englert-Higgs effect due to a single Higgs field in the fundamental representation. This yields a perturbative breaking pattern $SU(3)\to SU(2)$. We investigate the singlet vector channel, finding a nondegenerate and massive ground state. This is in contradistinction to the perturbative prediction of three massless and five massive vector states, even though the correlation functions of the gauge bosons exhibit a weak-coupling behavior, being almost tree-level-like. However, a combination of perturbation theory with the Fröhlich-Morchio-Strocchi mechanism, and thus passing to gauge-invariant perturbation theory, allows one to predict the physical spectrum in this channel.

Figure 1

The top panel shows the “phase” diagram of the theory. The blue triangles show no BEH effect in ’t Hooft-Landau gauge, while the red points do (The red point in the lower left corner in the “blue region” shows indeed a BEH effect. This effect has also been observed in [6] for large values of $\beta$ and low values of $\lambda$.). The big black point denotes our primary simulation point at $\beta =9.59055$, $\kappa =0.444462$, $\lambda =0.4118$. The lower panel shows examples for (6) as a function of volume, the Higgs case being the simulation point. This demonstrates how the different phases are identified.

Figure 2

Top: propagators in the unbroken $SU(2)$ subgroup and the coset $SU(3)/SU(2)$ for a $V=20^4$ lattice. The momenta are along the edges, and along all possible diagonals of the lattice, i.e. $(p,0,0,0),\dots ,(p,p,p,p)$. Results are for the parameters $\beta =9.59055$, $\kappa =0.444462$, $\lambda =0.4118$. The dashed lines denote the fits discussed in the text. Bottom: differences of the lattice propagators and the fits.

Figure 3

Results from the variational analysis for the energy levels of the $J^P=1^{-}$ singlet channel are shown for the largest lattices ($V=16^4$, $20^4$). The first three energy levels are shown and only the ground state gives a reliable result. Points with too large errors haven been suppressed.

Figure 4

Volume dependence of the effective masses for the massive $W$-propagator and the $1^{-}$ channel ground state. The dashed lines are fits to $a{m}_{\mathrm{eff}}=A+B{e}^{-cV}$.