Spectroscopy of SU(4) composite Higgs theory with two distinct fermion representations

We have simulated the SU(4) lattice gauge theory coupled to dynamical fermions in the fundamental and two-index antisymmetric (sextet) representations simultaneously. Such theories arise naturally in the context of composite Higgs models that include a partially composite top quark. We describe the low-lying meson spectrum of the theory and fit the pseudoscalar masses and decay constants to chiral perturbation theory. We infer as well the mass and decay constant of the Goldstone boson corresponding to the nonanomalous U(1) symmetry of the model. Our results are broadly consistent with large-$N_c$ scaling and vector-meson dominance.

Figure 1

Map of our ensembles in the plane of pseudoscalar masses $M_{Pr}$. These are coarse lattices, with $0.94<\sqrt{t_0}/a<1.41$. We define arbitrarily $\sqrt{t_0}=(1.4\mathrm{GeV}{)}^{-1}$ for comparison with QCD. For most of these ensembles $1/a\simeq 1.45\mathrm{GeV}$ by this measure.

Figure 2

Same as Fig. 1, but here we plot ensembles on fine lattices, $\sqrt{t_0}/a>1.41$. If we fix $\sqrt{t_0}=(1.4\mathrm{GeV}{)}^{-1}$ then this means $1/a>2\mathrm{GeV}$. The blue squares are all at $1/a\simeq 2.1\mathrm{GeV}$.

Figure 3

Squared mass of the two pseudoscalar species, each plotted against the AWI mass of the corresponding fermion species, all in units of the flow scale $t_0$.

Figure 4

Decay constant of each pseudoscalar species plotted against the mass of the corresponding fermion species, in units of the flow scale $t_0$.

Figure 5

Breakdown of the contribution of lattice artifacts in the joint fit to $\chi \mathrm{PT}$ for the fundamental masses and decay constants.

Figure 6

Breakdown of the contribution of lattice artifacts in the joint fit to $\chi \mathrm{PT}$ for the sextet masses and decay constants.

Figure 7

Exploring the stability of leading-order LECs in chiral fits. We take the NLO result to define our central values, which appear at the bottom of each column. The variations are described in the text.

Figure 8

Mass squared ${\widehat{M}}_{\zeta }^2$ of the nonanomalous NG boson in the combined continuum ($\widehat{a}\to 0$) and chiral-sextet (${\widehat{m}}_6\to 0$) limits, as extracted using Eq. (3.18) and the central fit’s parameters, plotted against ${\widehat{m}}_4$. The pseudoscalar mass ${\widehat{M}}_4^2$ in the fundamental sector in the same limit is shown for comparison.

Figure 9

Vector masses vs fermion masses in units of the flow scale $t_0$.

Figure 10

Vector decay constants vs fermion masses in units of the flow scale $t_0$.

Figure 11

The mass ratio $M_{Pr}/M_{Vr}$ in a fixed representation.

Figure 12

Breakdown of the contribution of lattice artifacts in the empirical models for the vector masses and decay constants in the fundamental representation.

Figure 13

Breakdown of the contribution of lattice artifacts in the empirical models for the vector masses and decay constants in the sextet representation.

Figure 14

Ratio of the vector and pseudoscalar decay constants in each representation. The KSRF prediction is a constant value of $\sqrt{2}$.

Figure 15

Ratio of the vector mass and pseudoscalar decay constant in a fixed representation. KSRF identify this quantity with the coupling $g_{VPP}$. In QCD, this ratio is roughly 5.9.

Figure 16

Tree-level estimates for the width-to-mass ratio of the vector mesons according to KSRF. The KSRF estimate for this ratio is roughly 0.23 in QCD.

Figure 17

The ratio of the decay constants in the sextet and fundamental representations in both the pseudoscalar (dark green) and vector (light green) channels.

Figure 18

Representative plot showing the effective mass $ma$ extracted from a smeared-source, point-sink pseudoscalar correlator on a typical $16^3\times 18$ ensemble (top) and $16^3\times 32$ ensemble (bottom) used in this study. The black lines indicate the mass and error (including range-fit systematic uncertainty) extracted from a full nonlinear fit of the correlator. The horizontal width of the black lines indicates the starting and ending times $(t_i,t_f)$ used in the best range fit as determined by the MILC criterion. Note that in the lower panel, the small difference between the central value from the best fit and the effective mass at larger $t/a$ is covered by the $\mathrm{statistical}+\text{systematic error band}$, showing that our range-fit uncertainty is working as expected.

Figure 19

${\kappa }_c$ curves with uncertainties for both representations at two values of the bare coupling $\beta$, based on our fits to $\sqrt{t_0}{m}_r$ as described in the text. The red (horizontal) curve is where $m_6$ vanishes, defining the function ${\kappa }_6^c(\beta ,{\kappa }_4)$. The blue (vertical) curve is where $m_4$ vanishes, defining ${\kappa }_4^c(\beta ,{\kappa }_6)$.

Figure 20

Quantifying the size of leading-order finite volume corrections, Eq. (b7).

Figure 21

Explicit test of the dependence of the pseudoscalar masses and decay constants on spatial volume at bare parameters $(\beta ,{\kappa }_4,{\kappa }_6)=(7.75,0.128,0.131)$. The dashed lines indicate variations of $\pm 5\%$ with respect to the mean value of the rightmost $N_s=24$ point.

Figure 22

Explicit test of the dependence of the pseudoscalar masses and decay constants on spatial volume at bare parameters $(\beta ,{\kappa }_4,{\kappa }_6)=(7.75,0.129,0.1308)$. The dashed lines indicate variations of $\pm 5\%$ with respect to the mean value of the rightmost $N_s=24$ point.

Figure 23

Explicit test of the dependence of the Wilson flow scale $t_0/a^2$ on spatial volume at two sets of bare parameters, as described in the text. The dashed lines indicate variations of $\pm 5\%$ with respect to the mean value of the rightmost $N_s=24$ point.

Figure 24

Comparison of the potential for the NDS action, with $\gamma /\beta =1/125$ (diamonds), with that of the usual Wilson action (crosses).