<table><caption>TABLE XLIV. Numerical results for the chimera baryons overlap factors in ensemble M5. G stands for Gaussian fit, C stands for Cauchy function fit, C indicates the mesonic channel considered, $K_{B,0}$ is the result of the correlator fitting analysis, $\sigma_G$ is the energy smearing radius used for the Gaussian fits, $\sigma_C$ for the Cauchy fit.</caption><thead><tr><th>C</th><th>a<sup>3</sup>K<sub>B,0</sub>-G</th><th>a<sup>3</sup>K<sub>B,0</sub>-C</th><th>a<sup>3</sup>K<sub>B,0</sub>(corr)</th><th>σ<sub>G</sub>/m<sub>C</sub></th><th>σ<sub>C</sub>/m<sub>C</sub></th></tr></thead><tbody><tr><td>Λ<sup>+</sup><sub>CB</sub></td><td>0.000503(7)</td><td>0.000503(8)</td><td>0.000503(11)</td><td>0.25</td><td>0.25</td></tr><tr><td>Σ<sup>+</sup><sub>CB</sub></td><td>0.002312(10)</td><td>0.002312(11)</td><td>0.002312(13)</td><td>0.3</td><td>0.3</td></tr><tr><td>Σ<sup>+</sup><sub>CB</sub></td><td>0.002498(11)</td><td>0.002498(12)</td><td>0.002498(18)</td><td>0.2</td><td>0.2</td></tr><tr><td>Λ<sup>+</sup><sub>CB</sub></td><td>0.000392(6)</td><td>0.000392(6)</td><td>0.000392(16)</td><td>0.3</td><td>0.3</td></tr><tr><td>Σ<sup>+</sup><sub>CB</sub></td><td>0.001525(7)</td><td>0.001525(7)</td><td>0.001525(22)</td><td>0.2</td><td>0.2</td></tr><tr><td>Σ<sup>+</sup><sub>CB</sub></td><td>0.001551(7)</td><td>0.001553(7)</td><td>0.001553(31)</td><td>0.25</td><td>0.25</td></tr></tbody></table>

in Appendix A. In this Appendix we compare these results to a determination of the hadron-vacuum matrix elements (i.e., the decay constants) based on operators that have not been smeared, but rather built by inverting the Dirac operator with the use of stochastic wall sources with $Z_2 \otimes Z_2$ noise [191], and without APE smearing.

To measure the pseudoscalar decay constants, we perform a simultaneous fit to the pseudoscalar correlation function,

$C_{\text{PS}}(t) = \langle \mathcal{O}_{\text{PS}}(t) \bar{\mathcal{O}}_{\text{PS}}(0) \rangle$, and the correlation function between pseudoscalar and axial-vector operators, restricted to the $\mu = 0$ component, $C_{\text{AV,PS}}(t) = \langle \mathcal{O}_{\text{AV}}^{\mu=0}(t) \bar{\mathcal{O}}_{\text{PS}}(0) \rangle$, that, at large Euclidean times, takes the form

$$C_{\text{AV,PS}}(t) \rightarrow \frac{f_{\text{PS}} \langle 0 | \mathcal{O}_{\text{PS}} | \text{PS} \rangle^*}{\sqrt{2}} (e^{-m_{\text{PS}}t} + e^{-m_{\text{PS}}(T-t)}). \quad (\text{D1})$$

The decay constant, $f_{\text{PS}}$, is then extracted from this behavior, combined with the matrix element of the pseudoscalar correlation function, $C_{\text{PS}}(t)$, and its mass. We proceed in the same manner for (f)-type as well as (as)-type fermions.

The fitting results for the $Z_2 \otimes Z_2$ stochastic wall case for ensembles M1–M5 are renormalized according to Eq. (30), and then reported in Table XLV. A comparison with simultaneous fits of smeared-smeared and smeared-unsmeared two-point correlation functions is shown in Fig. 9. From this study, it appears that the two methods present compatible results, with no particular gain with either choice.

<table><caption>TABLE XLV. Numerical results for decay constants obtained by using stochastic wall sources (denoted by the label “loc”) from ensembles M1–M5 and pseudoscalar/vector meson channels, for both types of fermions, and comparison with the results obtained by using different Wuppertal smearing levels (labeled by “smear”).</caption><thead><tr><th>Ensemble</th><th>af<sup>loc</sup><sub>PS</sub></th><th>af<sup>smear</sup><sub>PS</sub></th><th>af<sup>loc</sup><sub>PS</sub></th><th>af<sup>smear</sup><sub>PS</sub></th><th>af<sup>loc</sup><sub>V</sub></th><th>af<sup>smear</sup><sub>V</sub></th><th>af<sup>loc</sup><sub>V</sub></th><th>af<sup>smear</sup><sub>V</sub></th></tr></thead><tbody><tr><td>M1</td><td>0.0500(2)</td><td>0.0503(3)</td><td>0.0832(2)</td><td>0.0813(7)</td><td>0.0770(3)</td><td>0.0771(3)</td><td>0.1028(4)</td><td>0.1015(8)</td></tr><tr><td>M2</td><td>0.0494(3)</td><td>0.0487(3)</td><td>0.0822(3)</td><td>0.0805(8)</td><td>0.0729(4)</td><td>0.0729(4)</td><td>0.099(1)</td><td>0.0984(6)</td></tr><tr><td>M3</td><td>0.0497(2)</td><td>0.0499(4)</td><td>0.0828(2)</td><td>0.0820(9)</td><td>0.0717(4)</td><td>0.073(2)</td><td>0.1002(8)</td><td>0.0988(6)</td></tr><tr><td>M4</td><td>0.0540(2)</td><td>0.0538(2)</td><td>0.0868(2)</td><td>0.0872(2)</td><td>0.0745(5)</td><td>0.076(1)</td><td>0.1068(5)</td><td>0.107(1)</td></tr><tr><td>M5</td><td>0.0445(5)</td><td>0.0437(4)</td><td>0.0787(9)</td><td>0.0770(6)</td><td>0.0639(8)</td><td>0.0634(4)</td><td>0.0952(4)</td><td>0.0942(7)</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>FIG. 9. Comparison between matrix elements found through simultaneous fits of smeared-smeared and smeared-unsmeared correlation functions against the adoption of stochastic wall sources, for all available ensembles, M1–M5, and for both type of fermions. We restrict our attention to the pseudoscalar and vector channel, for which we obtained the most precise measurements.</figcaption></figure>
[1] M. Hansen, A. Lupo, and N. Tantalo, Extraction of spectral densities from lattice correlators, Phys. Rev. D **99**, 094508 (2019).

[2] M. T. Hansen, H. B. Meyer, and D. Robaina, From deep inelastic scattering to heavy-flavor semileptonic decays: Total rates into multihadron final states from lattice QCD, Phys. Rev. D **96**, 094513 (2017).

[3] J. Bulava and M. T. Hansen, Scattering amplitudes from finite-volume spectral functions, Phys. Rev. D **100**, 034521 (2019).

[4] L. Kades, J. M. Pawlowski, A. Rothkopf, M. Scherzer, J. M. Urban, S. J. Wetzel, N. Wink, and F. P. G. Ziegler, Spectral reconstruction with deep neural networks, Phys. Rev. D **102**, 096001 (2020).

[5] G. Bailas, S. Hashimoto, and T. Ishikawa, Reconstruction of smeared spectral function from Euclidean correlation functions, Prog. Theor. Exp. Phys. **2020**, 043B07 (2020).

[6] P. Gambino and S. Hashimoto, Inclusive semileptonic decays from lattice QCD, Phys. Rev. Lett. **125**, 032001 (2020).

[7] M. Bruno and M. T. Hansen, Variations on the Maiani-Testa approach and the inverse problem, J. High Energy Phys. 06 (2021) 043.

[8] A. Lupo, M. Panero, N. Tantalo, and L. Del Debbio, Spectral reconstruction in SU(4) gauge theory with fermions in multiple representations, Proc. Sci., LATICE2021 (2022) 092 [arXiv:2112.01158].

[9] L. Del Debbio, T. Giani, and M. Wilson, Bayesian approach to inverse problems: An application to NNPDF closure testing, Eur. Phys. J. C **82**, 330 (2022).

[10] L. Del Debbio, A. Lupo, M. Panero, and N. Tantalo, Multi-representation dynamics of SU(4) composite Higgs models: Chiral limit and spectral reconstructions, Eur. Phys. J. C **83**, 220 (2023).

[11] J. Bulava, M. T. Hansen, M. W. Hansen, A. Patella, and N. Tantalo, Inclusive rates from smeared spectral densities in the two-dimensional O(3) non-linear $\sigma$-model, J. High Energy Phys. 07 (2022) 034.

[12] J. M. Pawlowski, C. S. Schneider, J. Turnwald, J. M. Urban, and N. Wink, Yang-Mills glueball masses from spectral reconstruction, Phys. Rev. D **108**, 076018 (2023).

[13] P. Gambino, S. Hashimoto, S. Mächler, M. Panero, F. Sanfilippo, S. Simula, A. Smecca, and N. Tantalo, Lattice QCD study of inclusive semileptonic decays of heavy mesons, J. High Energy Phys. 07 (2022) 083.

[14] A. Lupo, L. Del Debbio, M. Panero, and N. Tantalo, Fits of finite-volume smeared spectral densities, Proc. Sci., LATICE2022 (2023) 215 [arXiv:2212.08019].

[15] J. Bulava, Spectral reconstruction of Euclidean correlator moments in lattice QCD, Nuovo Cimento Soc. Ital. Fis. **47C**, 199 (2024).

[16] A. De Santis, M. Buzzicotti, and N. Tantalo, Extraction of lattice QCD spectral densities from an ensemble of trained machines, Proc. Sci., LATTICE2023 (2024) 003 [arXiv:2401.05344].

[17] A. Lupo, L. Del Debbio, M. Panero, and N. Tantalo, Bayesian interpretation of Backus-Gilbert methods, Proc. Sci., LATTICE2023 (2024) 004 [arXiv:2311.18125].

[18] T. Bergamaschi, W. I. Jay, and P. R. Oare, Hadronic structure, conformal maps, and analytic continuation, Phys. Rev. D **108**, 074516 (2023).

[19] C. Bonanno, F. D'Angelo, M. D'Elia, L. Maio, and M. Naviglio, Sphaleron rate of Nf = 2 + 1 QCD, Phys. Rev. Lett. **132**, 051903 (2024).

[20] R. Frezzotti, N. Tantalo, G. Gagliardi, F. Sanfilippo, S. Simula, and V. Lubicz, Spectral-function determination of complex electroweak amplitudes with lattice QCD, Phys. Rev. D **108**, 074510 (2023).

[21] R. Frezzotti, N. Tantalo, G. Gagliardi, F. Sanfilippo, S. Simula, V. Lubicz, G. Martinelli, and C. T. Sachrajda, $B_s \to \mu^+\mu^-\gamma$ decay rate at large $q^2$ from lattice QCD, Phys. Rev. D **109**, 114506 (2024).

[22] M. Bruno, L. Giusti, and M. Saccardi, Spectral densities from Euclidean lattice correlators via the Mellin transform, Phys. Rev. D **111**, 094515 (2025).

[23] L. Del Debbio, A. Lupo, M. Panero, and N. Tantalo, Bayesian solution to the inverse problem and its relation to Backus-Gilbert methods, Eur. Phys. J. C **85**, 185 (2025).

[24] A. Patella and N. Tantalo, Scattering amplitudes from Euclidean correlators: Haag-Ruelle theory and approximation formulae, J. High Energy Phys. 01 (2025) 091.

[25] Extended Twisted Mass Collaboration (ETMC), Probing the energy-smeared R ratio using lattice QCD, Phys. Rev. Lett. **130**, 241901 (2023).

[26] T. Blum, W. I. Jay, L. Jin, A. S. Kronfeld, and D. B. A. Stewart, Toward inclusive observables with staggered quarks: The smeared R ratio, Proc. Sci., LATTICE2024 (2025) 126 [arXiv:2411.14300].

[27] Extended Twisted Mass Collaboration, Inclusive hadronic decay rate of the $\tau$ lepton from lattice QCD, Phys. Rev. D **108**, 074513 (2023).

[28] Extended Twisted Mass Collaboration, Inclusive hadronic decay rate of the $\tau$ lepton from lattice QCD: The $\bar{u}s$ flavor channel and the Cabibbo angle, Phys. Rev. Lett. **132**, 261901 (2024).

[29] M. Panero, A. Smecca, N. Tantalo, and D. Vadacchino, The quenched glueball spectrum from smeared spectral densities, Proc. Sci., LATTICE2023 (2024) 050 [arXiv:2311.14806].

[30] M. Asakawa, T. Hatsuda, and Y. Nakahara, Maximum entropy analysis of the spectral functions in lattice QCD, Prog. Part. Nucl. Phys. **46**, 459 (2001).

[31] H. B. Meyer, Transport properties of the quark-gluon plasma: A lattice QCD perspective, Eur. Phys. J. A **47**, 86 (2011).

[32] C. Ratti, Bulk properties of QCD matter from lattice simulations, J. Phys. Conf. Ser. **779**, 012016 (2017).

[33] G. Aarts et al., Phase transitions in particle physics: Results and perspectives from lattice quantum chromodynamics, Prog. Part. Nucl. Phys. **133**, 104070 (2023).

[34] G. Aarts, C. Allton, J. Foley, S. Hands, and S. Kim, Spectral functions at small energies and the electrical conductivity in hot, quenched lattice QCD, Phys. Rev. Lett. **99**, 022002 (2007).

[35] G. Aarts, C. Allton, A. Amato, P. Giudice, S. Hands, and J.-I. Skullerud, Electrical conductivity and charge diffusion in thermal QCD from the lattice, J. High Energy Phys. 02 (2015) 186.
[36] G. Almirante, N. Astrakhantsev, V. V. Braguta, M. D'Elia, L. Maio, M. Naviglio, F. Sanfilippo, and A. Trunin, Electrical conductivity of the quark-gluon plasma in the presence of strong magnetic fields, Phys. Rev. D **111**, 034505 (2025).

[37] R. Bignell et al., Anisotropic excited bottomonia from a basis of smeared operators, Proc. Sci., LATTICE2024 (2025) 202 [arXiv:2502.03185].

[38] A. Smecca et al., The NRQCD Υ spectrum at non-zero temperatures using Backus-Gilbert regularisations, Proc. Sci., LATTICE2024 (2025) 197 [arXiv:2502.03060].

[39] G. Backus and F. Gilbert, The resolving power of gross earth data, Geophys. J. Int. **16**, 169 (1968).

[40] M. Aizenman and H. Duminil-Copin, Marginal triviality of the scaling limits of critical 4D Ising and $\phi^4$ models, Ann. Math. **194**, 163 (2021).

[41] M. Luscher and P. Weisz, Scaling laws and triviality bounds in the lattice $\phi^4$ theory. 1. One component model in the symmetric phase, Nucl. Phys. **B290**, 25 (1987).

[42] M. Luscher and P. Weisz, Scaling laws and triviality bounds in the lattice $\phi^4$ theory. 2. One component model in the phase with spontaneous symmetry breaking, Nucl. Phys. **B295**, 65 (1988).

[43] M. Luscher and P. Weisz, Scaling laws and triviality bounds in the lattice $\phi^4$ theory. 3. N component model, Nucl. Phys. **B318**, 705 (1989).

[44] J. Bulava, P. Gerhold, K. Jansen, J. Kallarackal, B. Knippschild, C. J. D. Lin et al., Higgs-Yukawa model in chirally-invariant lattice field theory, Adv. High Energy Phys. **2013**, 875612 (2013).

[45] E. Molgaard and R. Shrock, Renormalization-group flows and fixed points in Yukawa theories, Phys. Rev. D **89**, 105007 (2014).

[46] D. Y. J. Chu, K. Jansen, B. Knippschild, and C. J. D. Lin, Finite-size scaling for four-dimensional Higgs-Yukawa model near the Gaussian fixed point, J. High Energy Phys. 01 (2019) 110.

[47] K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov, Is there a hot electroweak phase transition at $m_H \gtrsim m_W$?, Phys. Rev. Lett. **77**, 2887 (1996).

[48] M. Laine, G. Nardini, and K. Rummukainen, Lattice study of an electroweak phase transition at $m_h \simeq 126$ GeV, J. Cosmol. Astropart. Phys. 01 (2013) 011.

[49] M. Cirelli, A. Strumia, and J. Zupan, Dark matter, arXiv: 2406.01705.

[50] Planck Collaboration, Planck 2013 results. I. Overview of products and scientific results, Astron. Astrophys. **571**, A1 (2014).

[51] D. B. Kaplan and H. Georgi, SU(2) × U(1) breaking by vacuum misalignment, Phys. Lett. **136B**, 183 (1984).

[52] H. Georgi and D. B. Kaplan, Composite Higgs and custodial SU(2), Phys. Lett. **145B**, 216 (1984).

[53] M. J. Dugan, H. Georgi, and D. B. Kaplan, Anatomy of a composite Higgs model, Nucl. Phys. **B254**, 299 (1985).

[54] G. Panico and A. Wulzer, The composite Nambu-Goldstone Higgs, Lect. Notes Phys. **913**, 1 (2016).

[55] O. Witzel, Review on composite Higgs models, Proc. Sci., LATTICE2018 (2019) 006 [arXiv:1901.08216].

[56] G. Cacciapaglia, C. Pica, and F. Sannino, Fundamental composite dynamics: A review, Phys. Rep. **877**, 1 (2020).

[57] E. Bennett, J. Holligan, D. K. Hong, H. Hsiao, J.-W. Lee, C. J. D. Lin et al., Sp(2N) lattice gauge theories and extensions of the standard model of particle physics, Universe **9**, 236 (2023).

[58] G. Ferretti and D. Karateev, Fermionic UV completions of composite Higgs models, J. High Energy Phys. 03 (2014) 077.

[59] G. Ferretti, Gauge theories of partial compositeness: Scenarios for Run-II of the LHC, J. High Energy Phys. 06 (2016) 107.

[60] G. Cacciapaglia, G. Ferretti, T. Flacke, and H. Serôdio, Light scalars in composite Higgs models, Front. Phys. **7**, 22 (2019).

[61] D. B. Kaplan, Flavor at SSC energies: A new mechanism for dynamically generated fermion masses, Nucl. Phys. **B365**, 259 (1991).

[62] Y. Grossman and M. Neubert, Neutrino masses and mixings in nonfactorizable geometry, Phys. Lett. **B 474**, 361 (2000).

[63] T. Gherghetta and A. Pomarol, Bulk fields and supersymmetry in a slice of AdS, Nucl. Phys. **B586**, 141 (2000).

[64] Z. Chacko and R. K. Mishra, Effective theory of a light dilaton, Phys. Rev. D **87**, 115006 (2013).

[65] E. Katz, A. E. Nelson, and D. G. E. Walker, The intermediate Higgs, J. High Energy Phys. 08 (2005) 074.

[66] R. Barbieri, B. Bellazzini, V. S. Rychkov, and A. Varagnolo, The Higgs boson from an extended symmetry, Phys. Rev. D **76**, 115008 (2007).

[67] P. Lodone, Vector-like quarks in a 'composite' Higgs model, J. High Energy Phys. 12 (2008) 029.

[68] B. Gripaios, A. Pomarol, F. Riva, and J. Serra, Beyond the minimal composite Higgs model, J. High Energy Phys. 04 (2009) 070.

[69] J. Mrazek, A. Pomarol, R. Rattazzi, M. Redi, J. Serra, and A. Wulzer, The other natural two Higgs doublet model, Nucl. Phys. **B853**, 1 (2011).

[70] D. Marzocca, M. Serone, and J. Shu, General composite Higgs models, J. High Energy Phys. 08 (2012) 013.

[71] C. Grojean, O. Matsedonskyi, and G. Panico, Light top partners and precision physics, J. High Energy Phys. 10 (2013) 160.

[72] G. Cacciapaglia and F. Sannino, Fundamental composite (Goldstone) Higgs dynamics, J. High Energy Phys. 04 (2014) 111.

[73] G. Ferretti, UV completions of partial compositeness: The case for a SU(4) gauge group, J. High Energy Phys. 06 (2014) 142.

[74] A. Arbey, G. Cacciapaglia, H. Cai, A. Deandrea, S. Le Corre, and F. Sannino, Fundamental composite electroweak dynamics: Status at the LHC, Phys. Rev. D **95**, 015028 (2017).

[75] G. Cacciapaglia, H. Cai, A. Deandrea, T. Flacke, S. J. Lee, and A. Parolini, Composite scalars at the LHC: the Higgs, the Sextet and the Octet, J. High Energy Phys. 11 (2015) 201.

[76] L. Vecchi, A dangerous irrelevant UV-completion of the composite Higgs, J. High Energy Phys. 02 (2017) 094.

[77] T. Ma and G. Cacciapaglia, Fundamental composite 2HDM: SU(N) with 4 flavours, J. High Energy Phys. 03 (2016) 211.
[78] F. Feruglio, B. Gavela, K. Kanshin, P. A. N. Machado, S. Rigolin, and S. Saa, The minimal linear sigma model for the Goldstone Higgs, J. High Energy Phys. 06 (2016) 038.

[79] T. DeGrand, M. Golterman, E. T. Neil, and Y. Shamir, One-loop chiral perturbation theory with two fermion representations, Phys. Rev. D **94**, 025020 (2016).

[80] S. Fichet, G. von Gersdorff, E. Pontón, and R. Rosenfeld, The excitation of the global symmetry-breaking vacuum in composite Higgs models, J. High Energy Phys. 09 (2016) 158.

[81] J. Galloway, A. L. Kagan, and A. Martin, A UV complete partially composite-pNGB Higgs, Phys. Rev. D **95**, 035038 (2017).

[82] A. Agugliaro, O. Antipin, D. Becciolini, S. De Curtis, and M. Redi, UV complete composite Higgs models, Phys. Rev. D **95**, 035019 (2017).

[83] A. Belyaev, G. Cacciapaglia, H. Cai, G. Ferretti, T. Flacke, A. Parolini, and H. Serodio, Di-boson signatures as standard candles for partial compositeness, J. High Energy Phys. 01 (2017) 094.

[84] C. Csáki, T. Ma, and J. Shu, Maximally symmetric composite Higgs models, Phys. Rev. Lett. **119**, 131803 (2017).

[85] M. Chala, G. Durieux, C. Grojean, L. de Lima, and O. Matsedonskyi, Minimally extended SILH, J. High Energy Phys. 06 (2017) 088.

[86] M. Golterman and Y. Shamir, Effective potential in ultra-violet completions for composite Higgs models, Phys. Rev. D **97**, 095005 (2018).

[87] C. Csáki, T. Ma, and J. Shu, Trigonometric parity for composite Higgs models, Phys. Rev. Lett. **121**, 231801 (2018).

[88] T. Alanne, D. Buarque Franzosi, and M. T. Frandsen, A partially composite Goldstone Higgs, Phys. Rev. D **96**, 095012 (2017).

[89] T. Alanne, D. Buarque Franzosi, M. T. Frandsen, M. L. A. Kristensen, A. Meroni, and M. Rosenlyst, Partially composite Higgs models: Phenomenology and RG analysis, J. High Energy Phys. 01 (2018) 051.

[90] F. Sannino, P. Stangl, D. M. Straub, and A. E. Thomsen, Flavor physics and flavor anomalies in minimal fundamental partial compositeness, Phys. Rev. D **97**, 115046 (2018).

[91] T. Alanne, N. Bizot, G. Cacciapaglia, and F. Sannino, Classification of NLO operators for composite Higgs models, Phys. Rev. D **97**, 075028 (2018).

[92] N. Bizot, G. Cacciapaglia, and T. Flacke, Common exotic decays of top partners, J. High Energy Phys. 06 (2018) 065.

[93] C. Cai, G. Cacciapaglia, and H.-H. Zhang, Vacuum alignment in a composite 2HDM, J. High Energy Phys. 01 (2019) 130.

[94] A. Agugliaro, G. Cacciapaglia, A. Deandrea, and S. De Curtis, Vacuum misalignment and pattern of scalar masses in the SU(5)/SO(5) composite Higgs model, J. High Energy Phys. 02 (2019) 089.

[95] G. Cacciapaglia, T. Ma, S. Vatani, and Y. Wu, Towards a fundamental safe theory of composite Higgs and dark matter, Eur. Phys. J. C **80**, 1088 (2020).

[96] D. Buarque Franzosi, G. Cacciapaglia, and A. Deandrea, Sigma-assisted low scale composite Goldstone-Higgs, Eur. Phys. J. C **80**, 28 (2020).

[97] H. Gertov, A. E. Nelson, A. Perko, and D. G. E. Walker, Lattice-friendly gauge completion of a composite Higgs with top partners, J. High Energy Phys. 02 (2019) 181.

[98] V. Ayyar, M. F. Golterman, D. C. Hackett, W. Jay, E. T. Neil, Y. Shamir et al., Radiative contribution to the composite-Higgs potential in a two-representation lattice model, Phys. Rev. D **99**, 094504 (2019).

[99] G. Cacciapaglia, H. Cai, A. Deandrea, and A. Kushwaha, Composite Higgs and dark matter model in SU(6)/SO(6), J. High Energy Phys. 10 (2019) 035.

[100] D. Buarque Franzosi and G. Ferretti, Anomalous dimensions of potential top-partners, SciPost Phys. **7**, 027 (2019).

[101] G. Cacciapaglia, S. Vatani, and C. Zhang, Composite Higgs meets Planck scale: Partial compositeness from partial unification, Phys. Lett. B **815**, 136177 (2021).

[102] G. Cacciapaglia, A. Deandrea, T. Flacke, and A. M. Iyer, Gluon-photon signatures for color octet at the LHC (and beyond), J. High Energy Phys. 05 (2020) 027.

[103] T. Appelquist, J. Ingoldby, and M. Piai, Nearly conformal composite Higgs model, Phys. Rev. Lett. **126**, 191804 (2021).

[104] Z.-Y. Dong, C.-S. Guan, T. Ma, J. Shu, and X. Xue, UV completed composite Higgs model with heavy composite partners, Phys. Rev. D **104**, 035013 (2021).

[105] G. Cacciapaglia, T. Flacke, M. Kunkel, and W. Porod, Phenomenology of unusual top partners in composite Higgs models, J. High Energy Phys. 02 (2022) 208.

[106] A. Banerjee, D. B. Franzosi, and G. Ferretti, Modelling vector-like quarks in partial compositeness framework, J. High Energy Phys. 03 (2022) 200.

[107] G. Ferretti, Compositeness above the electroweak scale and a proposed test at LHCb, EPJ Web Conf. **258**, 08002 (2022).

[108] H. Cai and G. Cacciapaglia, Partial compositeness under precision scrutiny, J. High Energy Phys. 12 (2022) 104.

[109] T. Appelquist, J. Ingoldby, and M. Piai, Composite two-Higgs doublet model from dilaton effective field theory, Nucl. Phys. B**983**, 115930 (2022).

[110] G. Cacciapaglia, A. Deandrea, M. Kunkel, and W. Porod, Coloured spin-1 states in composite Higgs models, J. High Energy Phys. 06 (2024) 092.

[111] A. Banerjee, E. Bergeaas Kuutmann, V. Ellajosyula, R. Enberg, G. Ferretti, and L. Panizzi, Vector-like quarks: Status and new directions at the LHC, SciPost Phys. Core **7**, 079 (2024).

[112] R. Contino, Y. Nomura, and A. Pomarol, Higgs as a holographic pseudoGoldstone boson, Nucl. Phys. B**671**, 148 (2003).

[113] K. Agashe, R. Contino, and A. Pomarol, The minimal composite Higgs model, Nucl. Phys. B**719**, 165 (2005).

[114] K. Agashe and R. Contino, The minimal composite Higgs model and electroweak precision tests, Nucl. Phys. B**742**, 59 (2006).

[115] K. Agashe, R. Contino, L. Da Rold, and A. Pomarol, A custodial symmetry for $Zb\bar{b}$, Phys. Lett. B **641**, 62 (2006).
[116] R. Contino, L. Da Rold, and A. Pomarol, Light custodians in natural composite Higgs models, Phys. Rev. D **75**, 055014 (2007).

[117] A. Falkowski and M. Perez-Victoria, Electroweak breaking on a soft wall, J. High Energy Phys. 12 (2008) 107.

[118] R. Contino, The Higgs as a composite Nambu-Goldstone boson, arXiv:1005.4269.

[119] R. Contino, D. Marzocca, D. Pappadopulo, and R. Rattazzi, On the effect of resonances in composite Higgs phenomenology, J. High Energy Phys. 10 (2011) 081.

[120] J. Erdmenger, N. Evans, W. Porod, and K. S. Rigatos, Gauge/gravity dynamics for composite Higgs models and the top mass, Phys. Rev. Lett. **126**, 071602 (2021).

[121] J. Erdmenger, N. Evans, W. Porod, and K. S. Rigatos, Gauge/gravity dual dynamics for the strongly coupled sector of composite Higgs models, J. High Energy Phys. 02 (2021) 058.

[122] D. Elander, M. Frigerio, M. Knecht, and J.-L. Kneur, Holographic models of composite Higgs in the Veneziano limit. Part I. Bosonic sector, J. High Energy Phys. 03 (2021) 182.

[123] D. Elander, M. Frigerio, M. Knecht, and J.-L. Kneur, Holographic models of composite Higgs in the Veneziano limit. Part II. Fermionic sector, J. High Energy Phys. 05 (2022) 066.

[124] D. Elander, A. Fatemiabhari, and M. Piai, Toward minimal composite Higgs models from regular geometries in bottom-up holography, Phys. Rev. D **107**, 115021 (2023).

[125] J. Erdmenger, N. Evans, Y. Liu, and W. Porod, Holographic non-Abelian flavour symmetry breaking, Universe **9**, 289 (2023).

[126] D. Elander, A. Fatemiabhari, and M. Piai, Holographic vacuum misalignment, Phys. Rev. D **111**, 015040 (2025).

[127] J. Erdmenger, N. Evans, Y. Liu, and W. Porod, Holography for $Sp(2N_c)$ gauge dynamics: From composite Higgs to technicolour, J. High Energy Phys. 07 (2024) 169.

[128] T. Imoto, T. Sakai, and S. Sugimoto, $O(N(c))$ and $USp(N(c))$ QCD from string theory, Prog. Theor. Phys. **122**, 1433 (2010).

[129] D. Elander and M. Piai, Towards top-down holographic composite Higgs: Minimal coset from maximal supergravity, J. High Energy Phys. 03 (2022) 049.

[130] T. Das, G. S. Guralnik, V. S. Mathur, F. E. Low, and J. E. Young, Electromagnetic mass difference of pions, Phys. Rev. Lett. **18**, 759 (1967).

[131] M. E. Peskin, The alignment of the vacuum in theories of technicolor, Nucl. Phys. **B175**, 197 (1980).

[132] J. Preskill, Subgroup alignment in hypercolor theories, Nucl. Phys. **B177**, 21 (1981).

[133] A. Hietanen, R. Lewis, C. Pica, and F. Sannino, Fundamental composite Higgs dynamics on the lattice: $SU(2)$ with two flavors, J. High Energy Phys. 07 (2014) 116.

[134] W. Detmold, M. McCullough, and A. Pochinsky, Dark nuclei. II. Nuclear spectroscopy in two-color QCD, Phys. Rev. D **90**, 114506 (2014).

[135] R. Arthur, V. Drach, M. Hansen, A. Hietanen, C. Pica, and F. Sannino, $SU(2)$ gauge theory with two fundamental flavors: A minimal template for model building, Phys. Rev. D **94**, 094507 (2016).

[136] R. Arthur, V. Drach, A. Hietanen, C. Pica, and F. Sannino, $SU(2)$ gauge theory with two fundamental flavours: Scalar and pseudoscalar spectrum, arXiv:1607.06654.

[137] C. Pica, V. Drach, M. Hansen, and F. Sannino, Composite Higgs dynamics on the lattice, EPJ Web Conf. **137**, 10005 (2017).

[138] J.-W. Lee, B. Lucini, and M. Piai, Symmetry restoration at high-temperature in two-color and two-flavor lattice gauge theories, J. High Energy Phys. 04 (2017) 036.

[139] V. Drach, T. Janowski, and C. Pica, Update on $SU(2)$ gauge theory with $\mathrm{NF} = 2$ fundamental flavours, EPJ Web Conf. **175**, 08020 (2018).

[140] V. Drach, T. Janowski, C. Pica, and S. Prelovsek, Scattering of Goldstone bosons and resonance production in a composite Higgs model on the lattice, J. High Energy Phys. 04 (2021) 117.

[141] V. Drach, P. Fritzsch, A. Rago, and F. Romero-López, Singlet channel scattering in a composite Higgs model on the lattice, Eur. Phys. J. C **82**, 47 (2022).

[142] L. S. Bowes, V. Drach, P. Fritzsch, A. Rago, and F. Romero-Lopez, 2-flavour $SU(2)$ gauge theory with exponential clover Wilson fermions, Proc. Sci., LATICE2023 (2024) 094 [arXiv:2401.00589].

[143] T. DeGrand, Y. Liu, E. T. Neil, Y. Shamir, and B. Svetitsky, Spectroscopy of $SU(4)$ gauge theory with two flavors of sextet fermions, Phys. Rev. D **91**, 114502 (2015).

[144] T. A. DeGrand, M. Golterman, W. I. Jay, E. T. Neil, Y. Shamir, and B. Svetitsky, Radiative contribution to the effective potential in composite Higgs models from lattice gauge theory, Phys. Rev. D **94**, 054501 (2016).

[145] V. Ayyar, T. DeGrand, M. Golterman, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir, and B. Svetitsky, Spectroscopy of $SU(4)$ composite Higgs theory with two distinct fermion representations, Phys. Rev. D **97**, 074505 (2018).

[146] V. Ayyar, T. DeGrand, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir, and B. Svetitsky, Baryon spectrum of $SU(4)$ composite Higgs theory with two distinct fermion representations, Phys. Rev. D **97**, 114505 (2018).

[147] V. Ayyar, T. DeGrand, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir, and B. Svetitsky, Finite-temperature phase structure of $SU(4)$ gauge theory with multiple fermion representations, Phys. Rev. D **97**, 114502 (2018).

[148] V. Ayyar, T. DeGrand, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir et al., Partial compositeness and baryon matrix elements on the lattice, Phys. Rev. D **99**, 094502 (2019).

[149] G. Cossu, L. Del Debbio, M. Panero, and D. Preti, Strong dynamics with matter in multiple representations: $SU(4)$ gauge theory with fundamental and sextet fermions, Eur. Phys. J. C **79**, 638 (2019).

[150] A. Hasenfratz, E. T. Neil, Y. Shamir, B. Svetitsky, and O. Witzel, Infrared fixed point and anomalous dimensions in a composite Higgs model, Phys. Rev. D **107**, 114504 (2023).

[151] G. Bergner and S. Piemonte, Lattice simulations of a gauge theory with mixed adjoint-fundamental matter, Phys. Rev. D **103**, 014503 (2021).

[152] G. Bergner and S. Piemonte, Mixed adjoint-fundamental matter and applications towards SQCD and beyond, Proc. Sci., LATTICE2021 (2022) 242 [arXiv:2111.15335].

[153] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, M. Piai, and D. Vadacchino, $Sp(4)$ gauge theory
on the lattice: towards SU(4)/Sp(4) composite Higgs (and beyond), J. High Energy Phys. 03 (2018) 185.

[154] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, and D. Vadacchino, Sp(4) gauge theories on the lattice: $N_f = 2$ dynamical fundamental fermions, J. High Energy Phys. 12 (2019) 053.

[155] E. Bennett, D. K. Hong, J.-W. Lee, C.-J. D. Lin, B. Lucini, M. Mesiti, M. Piai, J. Rantaharju, and D. Vadacchino, Sp(4) gauge theories on the lattice: Quenched fundamental and antisymmetric fermions, Phys. Rev. D **101**, 074516 (2020).

[156] E. Bennett, J. Holligan, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini et al., Color dependence of tensor and scalar glueball masses in Yang-Mills theories, Phys. Rev. D **102**, 011501 (2020).

[157] E. Bennett, J. Holligan, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini et al., Glueballs and strings in Sp(2N) Yang-Mills theories, Phys. Rev. D **103**, 054509 (2021).

[158] E. Bennett, D. K. Hong, H. Hsiao, J.-W. Lee, C. J. D. Lin, B. Lucini et al., Lattice studies of the Sp(4) gauge theory with two fundamental and three antisymmetric Dirac fermions, Phys. Rev. D **106**, 014501 (2022).

[159] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, and D. Vadacchino, Color dependence of the topological susceptibility in Yang-Mills theories, Phys. Lett. B **835**, 137504 (2022).

[160] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai et al., Sp(2N) Yang-Mills theories on the lattice: Scale setting and topology, Phys. Rev. D **106**, 094503 (2022).

[161] E. Bennett et al., Symplectic lattice gauge theories in the grid framework: Approaching the conformal window, Phys. Rev. D **108**, 094508 (2023).

[162] E. Bennett, D. K. Hong, H. Hsiao, J.-W. Lee, C. J. D. Lin, B. Lucini et al., Lattice investigations of the chimera baryon spectrum in the Sp(4) gauge theory, Phys. Rev. D **109**, 094512 (2024).

[163] E. Bennett, J. Holligan, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini et al., Spectrum of mesons in quenched Sp(2N) gauge theories, Phys. Rev. D **109**, 094517 (2024).

[164] E. Bennett et al., Meson spectroscopy from spectral densities in lattice gauge theories, Phys. Rev. D **110**, 074509 (2024).

[165] E. Bennett, N. Forzano, D. K. Hong, H. Hsiao, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, D. Vadacchino, and F. Zierler, Mixing between flavor singlets in lattice gauge theories coupled to matter fields in multiple representations, Phys. Rev. D **110**, 074504 (2024).

[166] E. Bennett, D. K. Hong, H. Hsiao, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, and D. Vadacchino, Meson spectroscopy in the Sp(4) gauge theory with three antisymmetric fermions, Phys. Rev. D **111**, 074511 (2025).

[167] D. K. Hong, J.-W. Lee, B. Lucini, M. Piai, and D. Vadacchino, Casimir scaling and Yang-Mills glueballs, Phys. Lett. B **775**, 89 (2017).

[168] S. Kulkarni, A. Maas, S. Mee, M. Nikolic, J. Pradler, and F. Zierler, Low-energy effective description of dark Sp(4) theories, SciPost Phys. **14**, 044 (2023).

[169] E. Bennett, H. Hsiao, J.-W. Lee, B. Lucini, A. Maas, M. Piai, and F. Zierler, Singlets in gauge theories with fundamental matter, Phys. Rev. D **109**, 034504 (2024).

[170] Y. Dengler, A. Maas, and F. Zierler, Scattering of dark pions in Sp(4) gauge theory, Phys. Rev. D **110**, 054513 (2024).

[171] E. Bennett, B. Lucini, D. Mason, M. Piai, E. Rinaldi, and D. Vadacchino, The density of states method for symplectic gauge theories at finite temperature, Phys. Rev. D **111**, 114511 (2025).

[172] J. Barnard, T. Gherghetta, and T. S. Ray, UV descriptions of composite Higgs models without elementary scalars, J. High Energy Phys. 02 (2014) 002.

[173] H. Cai, T. Flacke, and M. Lespinasse, A composite scalar hint from di-boson resonances?, arXiv:1512.04508.

[174] A. Belyaev, G. Cacciapaglia, H. Cai, T. Flacke, A. Parolini, and H. Serôdio, Singlets in composite Higgs models in light of the LHC 750 GeV diphoton excess, Phys. Rev. D **94**, 015004 (2016).

[175] G. Cacciapaglia, G. Ferretti, T. Flacke, and H. Serodio, Revealing timid pseudo-scalars with taus at the LHC, Eur. Phys. J. C **78**, 724 (2018).

[176] E. Bennett et al., Mixing between meson flavor singlets and glueballs in lattice gauge theories coupled to matter fields in multiple representations (to be published).

[177] S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B **195**, 216 (1987).

[178] M. A. Clark and A. D. Kennedy, Accelerating dynamical fermion computations using the rational hybrid Monte Carlo (RHMC) algorithm with multiple pseudo-fermion fields, Phys. Rev. Lett. **98**, 051601 (2007).

[179] P. Boyle, A. Yamaguchi, G. Cossu, and A. Portelli, Grid: A next generation data parallel C++ QCD library, Proc. Sci., LATTICE2015 (2016) 023 [arXiv:1512.03487].

[180] P. A. Boyle, G. Cossu, A. Yamaguchi, and A. Portelli, Grid: A next generation data parallel C++ QCD library, Proc. Sci., LATTICE2015 (2016) 023.

[181] A. Yamaguchi, P. Boyle, G. Cossu, G. Filaci, C. Lehner, and A. Portelli, Grid: OneCode and FourAPIs, Proc. Sci., LATTICE2021 (2022) 035 [arXiv:2203.06777].

[182] L. Del Debbio, A. Patella, and C. Pica, Higher representations on the lattice: Numerical simulations. SU(2) with adjoint fermions, Phys. Rev. D **81**, 094503 (2010).

[183] "GitHub—claudiopica/HiRep: HiRep repository—github.com." https://github.com/claudiopica/HiRep.

[184] "GitHub-sa2c/HiRep: HiRep repository—github.com. https://github.com/sa2c/HiRep.

[185] S. Gusken, A study of smearing techniques for hadron correlation functions, Nucl. Phys. B, Proc. Suppl. **17**, 361 (1990).

[186] D. S. Roberts, W. Kamleh, D. B. Leinweber, M. S. Mahbub, and B. J. Menadue, Accessing high momentum states in lattice QCD, Phys. Rev. D **86**, 074504 (2012).

[187] C. Alexandrou, F. Jegerlehner, S. Gusken, K. Schilling, and R. Sommer, B meson properties from lattice QCD, Phys. Lett. B **256**, 60 (1991).

[188] APE Collaboration, Glueball masses and string tension in lattice QCD, Phys. Lett. B **192**, 163 (1987).

[189] M. Falcioni, M. L. Paciello, G. Parisi, and B. Taglienti, Again on SU(3) glueball mass, Nucl. Phys. B**251**, 624 (1985).

[190] B. Blossier, M. Della Morte, G. von Hippel, T. Mendes, and R. Sommer, On the generalized eigenvalue method for
energies and matrix elements in lattice field theory, J. High Energy Phys. 04 (2009) 094.

[191] P. A. Boyle, A. Juttner, C. Kelly, and R. D. Kenway, Use of stochastic sources for the lattice determination of light quark physics, J. High Energy Phys. 08 (2008) 086.

[192] G. Martinelli and Y.-C. Zhang, The connection between local operators on the lattice and in the continuum and its relation to meson decay constants, Phys. Lett. **123B**, 433 (1983).

[193] G. P. Lepage and P. B. Mackenzie, On the viability of lattice perturbation theory, Phys. Rev. D **48**, 2250 (1993).

[194] S. Weinberg, Precise relations between the spectra of vector and axial vector mesons, Phys. Rev. Lett. **18**, 507 (1967).

[195] T. Dash, V. S. Mathur, and S. Okubo, Symmetry, super-convergence, and sum rules for spectral functions, Phys. Rev. Lett. **18**, 761 (1967).

[196] C. W. Bernard, A. Duncan, J. LoSecco, and S. Weinberg, Exact spectral function sum rules, Phys. Rev. D **12**, 792 (1975).

[197] D. A. Kosower, Symmetry breaking patterns in pseudoreal and real gauge theories, Phys. Lett. **144B**, 215 (1984).

[198] K. G. Wilson, Confinement of quarks, Phys. Rev. D **10**, 2445 (1974).

[199] S. Hands, I. Montvay, S. Morrison, M. Oevers, L. Scorzato, and J. Skullerud, Numerical study of dense adjoint matter in two color QCD, Eur. Phys. J. C **17**, 285 (2000).

[200] M. Lüscher and P. Weisz, Perturbative analysis of the gradient flow in non-abelian gauge theories, J. High Energy Phys. 02 (2011) 051.

[201] BMW Collaboration, High-precision scale setting in lattice QCD, J. High Energy Phys. 09 (2012) 010.

[202] M. Lüscher, Future applications of the Yang-Mills gradient flow in lattice QCD, Proc. Sci., LATTICE2013 (**2014**) 016 [arXiv:1308.5598].

[203] M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, J. High Energy Phys. 08 (2010) 071.

[204] Z. Fodor, K. Holland, J. Kuti, D. Nogradi, and C. H. Wong, The Yang-Mills gradient flow in finite volume, J. High Energy Phys. 11 (2012) 007.

[205] N. Madras and A. D. Sokal, The Pivot algorithm: A highly efficient Monte Carlo method for selfavoiding walk, J. Stat. Phys. **50**, 109 (1988).

[206] ALPHA Collaboration, Monte Carlo errors with less errors, Comput. Phys. Commun. **156**, 143 (2004).

[207] M. Lüscher, Schwarz-preconditioned HMC algorithm for two-flavour lattice QCD, Comput. Phys. Commun. **165**, 199 (2005).

[208] “GitHub—RJHudspith/GLU: GaugeLinkUtility repository —github.com.” https://github.com/RJHudspith/GLU.

[209] F. Erben, V. Gülpers, M. T. Hansen, R. Hodgson, and A. Portelli, Prospects for a lattice calculation of the rare decay $\Sigma^+ \to p\ell^+\ell^-$, J. High Energy Phys. 04 (2023) 108.

[210] N. Forzano and A. Lupo, “Is densities: Lattice spectral densities.” https://github.com/LupoA/Isdensities, 04, 2024.

[211] Y. Aoki, T. Izubuchi, E. Shintani, and A. Soni, Improved lattice computation of proton decay matrix elements, Phys. Rev. D **96**, 014506 (2017).

[212] M. E. Peskin and T. Takeuchi, Estimation of oblique electroweak corrections, Phys. Rev. D **46**, 381 (1992).

[213] E. Bennett et al., Symplectic lattice gauge theories in the Grid framework: Domain wall fermions and continuum extrapolations (to be published).

[214] www.dirac.ac.uk.

[215] E. Bennett, L. D. Debbio, N. Forzano, R. Hill, D. K. Hong, H. Hsiao et al., Chimera baryons and mesons on the lattice: A spectral density analysis— workflow release, 10.5281/ zenodo.15804889 (2025).

[216] E. Bennett, L. D. Debbio, N. Forzano, R. Hill, D. K. Hong, H. Hsiao et al., Chimera baryons and mesons on the lattice: A spectral density analysis—data release, 10.5281/zen-odo.15804848 (2025).

[217] E. Bennett, The TELOS Collaboration approach to reproducibility and open science, arXiv:2504.01876.

[218] J. Kuipers, T. Ueda, J. Vermaseren, and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. **184**, 1453 (2013).

[219] P. Breitenlohner and D. Maison, Dimensionally renormalized Green's functions for theories with massless particles. 1., Commun. Math. Phys. **52**, 39 (1977).

[220] P. Breitenlohner and D. Maison, Dimensional renormalization and the action principle, Commun. Math. Phys. **52**, 11 (1977).

[221] G. 't Hooft and M. J. G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. **B44**, 189 (1972).

[222] P. Lepage, gplepage/vegas: vegas version 6.2.1, 10.5281/zenodo.14834979 (2025).

[223] G. Martinelli and Y.-C. Zhang, One loop corrections to extended operators on the lattice, Phys. Lett. **125B**, 77 (1983).

[224] S. Capitani, Lattice perturbation theory, Phys. Rep. **382**, 113 (2003). www.dirac.ac.uk.