Chimera baryons and mesons on the lattice: a spectral density analysis

TL;DR

This work develops and tests a spectral-density analysis based on smeared energy kernels to extract masses and matrix elements from lattice two-point functions, applied to an $Sp(4)$ gauge theory with mixed fermion representations relevant for composite Higgs models with partial top compositeness. By combining Hansen-Lupo-Tantalo spectral-density reconstruction with variational GEVP spectroscopy, the authors obtain ground and excited-state spectra for flavored mesons and, for the first time with dynamical fermions, chimera baryons across spin, parity, and representation channels, along with renormalized decay constants and overlap factors. The study uses five ensembles with enhanced statistics, provides systematic controls for the spectral-density method, and releases the LSDensities software for public use, demonstrating the method’s consistency with traditional analyses while offering off-shell information via spectral densities. The results yield phenomenologically relevant benchmarks, including the overlap factor ratios $K_B/f_{PS}^3$ that inform partial top compositeness, and indicate the viability of the $Sp(4)$ theory as a CHM/TPC candidate, while outlining clear paths toward continuum extrapolations and broader applications.

Abstract

We develop and test a spectral-density analysis method, based on the introduction of smeared energy kernels, to extract physical information from two-point correlation functions computed numerically in lattice field theory. We apply it to a $Sp(4)$ gauge theory and fermion matter fields transforming in distinct representations, with $N_{\rm f}=2$ Dirac fermions in the fundamental and $N_{\rm as}=3$ in the 2-index antisymmetric representation. The corresponding continuum theory provides the minimal candidate model for a composite Higgs boson with partial top compositeness. We consider a broad class of composite operators, that source flavored mesons and (chimera) baryons, for several finite choices of lattice bare parameters. For the chimera baryons, which include candidate top-quark partners, we provide the first measurements, obtained with dynamical fermions, of the ground state and the lowest excited state masses, in all channels of spin, isospin, and parity. We also measure matrix elements and overlap factors, that are important to realize viable models of partial top compositeness, by implementing an innovative way of extracting this information from the spectral densities. For the mesons, among which the pseudoscalars can be reinterpreted to provide an extension of the Higgs sector of the Standard Model of particle physics, our measurements of the renormalized matrix elements and decay constants are new results. We complement them with an update of existing measurements of the meson masses, obtained with higher statistics and improved analysis. The analysis software is made publicly available, and can be used in other lattice studies, including application to quantum chromodynamics (QCD).

Figures (9)

• Figure 1: Illustrative examples of the plateaux in the spectral density reconstructed with the HLT method, for a fixed value of the energy, $\omega$. The underlying data correspond to the correlation function of pseudoscalar mesons made of $({\rm f})$-type fermions, in ensemble M1 in Tab. \ref{['tab:ensembles']}, using $t_{\rm max} =a N_t / 2$, $\sigma = 0.33 a m_{\mathrm{PS}}$. In the top panel, the reconstructed spectral density, $\hat{\rho}_{\sigma}(\omega)$, is shown as a function of the trade-off parameter, $\lambda$, for three choices of $\alpha = 0, \, 1.00, \, 1.99$. The horizontal band is our best estimate, its width representing the statistical error. All estimates obtained for asymptotically small choices of $\lambda$ are compatible within statistical errors, but affected by larger uncertainties, which decrease with larger $\lambda$, until the discrepancy exceeds the statistical uncertainty, for large enough $\lambda$. The bottom panel shows the same results, but restricted to one value of $\alpha$, and plotted as a function of $A[\vec{g}] /(A_0 = A[\vec{g} = \vec{0}])$, obtained by minimizing the functional $W[\vec{g}]$, for the same selection of values of $\lambda$ as in the top panel. Again, the plot shows that for small values of $\lambda$ the reconstructed density is independent of the correspondingly small value of $A[\vec{g}] /(A_0)$, but discrepancies larger than the statistical uncertainty are visible for large $\lambda$, in which case also $A[\vec{g}] /A_0$ is large.
• Figure 2: Examples of the result of a simultaneous fit of the spectral densities, involving different smearing levels (Tab. \ref{['table:E1_matrix_mesons']}), defined in Eqs. (\ref{['eq:sp_dens_N80_N80']}) and (\ref{['eq:sp_dens_N80_N0']}). The two spectral densities and the best fit functions are shown as a function of the energy, $E$, normalized to the mass of the ground state, $m_0$, in the relevant channel. The height of the fitting bands is our estimate of the uncertainty, inclusive of both statistical and systematic components, summed in quadrature. In these examples, we are measuring the correlation functions involving vector meson, ${\rm V}$, composed of $({\rm f})$-type fermions. The underlying numerical data is taken from ensemble M1 in Tab. \ref{['tab:ensembles']}. These measurements use a Gaussian kernel with $\sigma / m_0 = 0.33$. The two analysis differ by the smearing level of the sink, $N_{\rm sink}=80$ (left panel) and $N_{\rm sink}=0$ (right panel).
• Figure 3: Representative examples of mass measurements in the $Sp(4)$ theory with $N_{\rm f}=2$ and $N_{\rm as}=3$ hyperquarks, for the lightest flavored mesons composed of $({\rm f})$-type (${\rm PS}$ mesons, top panel) and $({\rm as})$-type (${\rm ps}$ mesons, middle panel) fermions, as well as the lightest chimera baryons ($\Lambda^{+}_{\rm CB}$ and $\Sigma^{+}_{\rm CB}$, bottom panel), in all available ensembles, as summarised in Tab. \ref{['tab:ensembles']}. The masses are expressed in lattice units, and the uncertainties displayed in these plots include only the statistical component. The five measurements of each bound-state mass (horizontally offset for presentation purposes) are obtained with five different methodologies: the result of the conventional GEVP analysis of (APE and Wuppertal smeared) correlation functions based on the variational method is compared to those obtained with four different choices of smearing kernel, in the HLT reconstruction of the spectral densities---for more details, including all the other mass measurements performed, see Tabs. \ref{['table:E1_results_ground_mesons']}--\ref{['table:E5_results_second_mesons']} for mesons, and Tabs. \ref{['table:E1_results_ground_CB']}--\ref{['table:E5_results_second_CB']} for chimera baryons.
• Figure 4: Representative examples of meson ($\text{T}$) and chimera baryon ($\Sigma^{*+}_{\rm CB}$) mass spectra in the ensembles M1, M2, M3, M4, and M5, characterized in Tab. \ref{['tab:ensembles']}. All measurements have been obtained by fitting the spectral densities. For each channel, a tower of mass eigenvalues, expressed in units of the Wilson flow, $\hat{m} \equiv w_0 \cdot m$, is shown, the elements of which correspond to ground, first and (where available) second excited state. The vertical midpoint of each color block is the numerical result, whereas the height is the uncertainty, comprehensive of statistical and systematic errors, summed in quadrature. Horizontal offsets in the towers have no physical meaning, but are used to distinguish graphically the different ensembles. The choice of filling color is used to identify ensembles M1-M3, while different patterns are used to indicate ensembles M4 and M5, as shown in the legend.
• Figure 5: Flavored meson and chimera baryon mass spectra in all available ensembles, as summarized in Tab. \ref{['tab:ensembles']}. The meson spectra include both composite states made of $({\rm f})$-type (Fundamental) and $({\rm as})$-type (Antisymmetric) fermions. The spectrum is found through the fitting analysis of the spectral densities described in the text. For each channel, we show a tower of masses, $\hat{m} \equiv w_0 \cdot m$, expressed in units of the Wilson flow scale, $w_0$. The particles correspond to ground, first and (where available) second excited states. The vertical midpoint of each color block is the numerical result, while the height is the uncertainty, inclusive of statistical and systematic errors, summed in quadrature. Horizontal offsets are used to distinguish different ensembles. Different shadings of the same colors differentiate ensembles that differ in time extents ($N_t = 48, \, 64$ and $96$ for ensembles M1, M2, and M3, respectively), while different patterns are used to indicate ensembles that differ also in bare parameters (ensembles M4 and M5). Six colors distinguish different meson channels. The colors match meson operators built with the same gamma-matrix structure, but different fermion constituents. We include also the three chimera baryons channels, split according to their two parity projections.