A first-order, confinement/deconfinement phase transition appears in the finite temperature behavior of many non-Abelian gauge theories. These theories play an important role in proposals for completion of the Standard Model of particle physics, hence the phase transition might have occurred in the early stages of evolution of our Universe, leaving behind a detectable relic stochastic background of gravitational waves. Lattice field theory studies implementing the density of states method have the potential to provide detailed information about the phase transition, and measure the parameters determining the gravitational-wave power spectrum, by overcoming some of the challenges faced by importance-sampling methods. We assess this potential for a representative choice of Yang-Mills theory with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>p</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> gauge group. We characterize its finite-temperature, first-order phase transition, in the thermodynamic (infinite volume) limit, for two different choices of number of sites in the compact time direction, hence taking the first steps toward the continuum limit extrapolation. We demonstrate the persistence of nonperturbative phenomena associated to the first-order phase transition: coexistence of states, metastability, latent heat, surface tension. We find consistency between several different strategies for the extraction of the volume-dependent critical coupling, hence assessing the size of systematic effects. We also determine the minimum choice of ratio between spatial and time extent of the lattice that allows one to identify the contribution of the surface tension to the free energy. We observe that this ratio scales nontrivially with the time extent of the lattice, and comment on the implications for future high-precision numerical studies.

United Kingdom

Ed Bennett

Biagio Lucini

Germany

David Mason

Maurizio Piai

Japan

Enrico Rinaldi

Davide Vadacchino

Fabian Zierler

Finite-temperature Yang-Mills theories with the density of states method: Toward the continuum limit

We provide an extended lattice study of the SU(2) gauge theory coupled to one Dirac fermion flavor (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi mathvariant="normal">f</mi></msub><mo>=</mo><mn>1</mn></math>) transforming in the adjoint representation as the continuum limit is approached. This investigation is supplemented by results obtained for the SU(2) gauge theory with two Dirac fermion flavors (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi mathvariant="normal">f</mi></msub><mo>=</mo><mn>2</mn></math>) transforming in the adjoint representation, for which we perform numerical investigations at three values of the lattice spacing. The purpose of our study is to advance the characterization of the infrared properties of both theories, which previous investigations have concluded to be in the conformal window. For both, we determine the mass spectrum and the anomalous dimension of the fermion condensate using finite-size hyperscaling of the spectrum, mode number analysis of the Dirac operator (for which we improve on our previous proposal) and the ratio of masses of the lightest spin-2 particle over the lightest scalar. All methods provide a consistent picture, with the anomalous dimension of the condensate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>γ</mi><mo>*</mo></msub></math> decreasing significantly as one approaches the continuum limit for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi mathvariant="normal">f</mi></msub><mo>=</mo><mn>1</mn></math> theory towards a value consistent with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>γ</mi><mo>*</mo></msub><mo>=</mo><mn>0.170</mn><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></math>, while for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi mathvariant="normal">f</mi></msub><mo>=</mo><mn>2</mn></math> the anomalous dimension converges more rapidly with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>β</mi></math> to a value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>γ</mi><mo>*</mo></msub><mo>=</mo><mn>0.291</mn><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo></math>. A chiral perturbation theory analysis shows that the infrared behavior of both theories is incompatible with the breaking of chiral symmetry.

Cyprus

Andreas Athenodorou

Georg Bergner

Spain

Denmark

Pietro Butti

Julian Lenz

SU(2) gauge theory with one and two adjoint fermions toward the continuum limit

SU(3) gauge theories with an increasing number of light fermions are the templates of strongly interacting sectors and studying their low-energy dynamics and spectrum is important, both for understanding the strong dynamics of QCD itself, and for discovering viable UV completions of beyond the Standard Model physics. In order to contrast many-flavor strongly interacting theories with QCD on a quantitative footing, we use lattice field theory simulations. We focus on the study of the flavor-singlet spectrum in the scalar and pseudoscalar channels: This is an interesting probe of the dynamics of the strongly interacting sector, as reminded by the QCD case with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>500</mn><mo stretchy="false">)</mo></mrow></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>η</mi><mo>′</mo></msup></math> mesons. The hierarchy of the spectrum of a strongly coupled new gauge sector of the Standard Model defines the potential reach of future colliders for new physics discoveries. In addition to a novel hierarchy with light scalars, introducing many light flavors at fixed number of colors can influence the dynamics of the lightest flavor-singlet pseudoscalar. We present a complete lattice study of both of these flavor-singlet channels on high-statistics gauge ensembles generated by the LatKMI Collaboration with 4, 8, and 12 copies of light mass-degenerate fermions. We also present other hadron masses on the lightest ensemble for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>8</mn></math> generated by the LatKMI Collaboration and discuss the chiral extrapolation of the spectrum in this particular theory. We contrast the results to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>4</mn></math> simulations and previous results of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>12</mn></math> simulations.

Yasumichi Aoki

Tatsumi Aoyama

Toshihide Maskawa

Kohtaroh Miura

Hiroshi Ohki

Akihiro Shibata

Koichi Yamawaki

Takeshi Yamazaki

Novel view of the flavor-singlet spectrum from multi-flavor QCD on the lattice

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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>p</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> gauge theory with fermion matter fields transforming in distinct representations, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi mathvariant="normal">f</mi></msub><mo>=</mo><mn>2</mn></math> Dirac fermions in the fundamental and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mtext>as</mtext></msub><mo>=</mo><mn>3</mn></math> 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.

Luigi Del Debbio

Niccolò Forzano

Ryan C. Hill

South Korea

Deog Ki Hong

Taiwan

Ho Hsiao

Jong-Wan Lee

C.J. David Lin

France

Alessandro Lupo

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

We study the finite-temperature behavior of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>p</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> Yang-Mills lattice theory in four dimensions, by applying the logarithmic linear relaxation algorithm. We demonstrate the presence of coexisting (metastable) phases, when the system is in the proximity of the transition. We measure observables such as the free energy, the expectation value of the plaquette operator and of the Polyakov loop, as well as the specific heat, and the Binder cumulant. We use these results to obtain a high-precision measurement of the critical coupling at the confinement-deconfinement transition, and assess its systematic uncertainty, for one value of the lattice extent in the time direction. Furthermore, we perform an extensive study of the finite-volume behavior of the lattice system, by repeating the measurements for fixed lattice time extent, while increasing the spatial size of the lattice. We hence characterize the first-order transition on the lattice and present the first results in the literature on this theory for the infinite volume extrapolation of lattice quantities related to latent heat and interface tension. Gauge theories with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>p</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> group have been proposed as new dark sectors to provide a fundamental origin for the current phenomenological evidence of dark matter. A phase transition at high temperature, in such a new dark sector, occurring in the early Universe, might have left a relic stochastic background of gravitational waves. Our results represent a milestone toward establishing whether such a new physics signal is detectable in future experiments, as they enter the calculation of the parameters, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>β</mi></math>, controlling the power spectrum of gravitational waves. We also outline the process needed in the continuum extrapolation of our measurements and test its feasibility on one additional choice of temporal extent of the lattice.

Density of states method for symplectic gauge theories at finite temperature

We report the results of an extensive numerical study of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>p</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> lattice gauge theory coupled to fermion matter content consisting of three (Dirac) flavors, transforming in the two-index antisymmetric representation of the gauge group. In the presence of (degenerate) fermion masses, the theory has an enhanced global <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></math> symmetry, broken explicitly and spontaneously to its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></math> subgroup. This symmetry breaking pattern makes the theory interesting for applications in the context of composite Higgs models, as well as for the implementation of top partial compositeness. Alternatively, it can also provide a dynamical realization of the strongly interacting massive particle paradigm for the origin of dark matter. We adopt the standard plaquette gauge action, along with the Wilson-Dirac formulation for the fermions, and apply the (rational) hybrid Monte Carlo algorithm in our ensemble generation process. We monitor the autocorrelation and topology of the ensembles. We explore the bare parameter space, and identify the weak and strong coupling regimes, which are separated by a line of first-order bulk phase transitions. We measure two-point correlation functions between meson operators that transform as nontrivial representations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></math>, and extract the ground-state masses, in all accessible spin and parity channels. We assess the size of finite volume effects, and restrict attention to measurements in which these systematic effects are negligibly small compared to the statistical uncertainties. The accuracy of our data enables us to extract the decay constants of the composite particles in the pseudoscalar, vector and axial-vector channels. In addition, we measure the mass of the first excited state for one of the channels, the vector meson, by performing a generalized eigenvalue problem analysis involving two different meson operators. Spectral quantities show a mass dependence that is compatible with the expectation that, at long distances, the theory undergoes confinement, accompanied by the spontaneous breaking of the approximate global symmetries acting on the matter fields. Finally, we discuss the continuum and massless extrapolations within the framework of Wilson chiral perturbation theory, after setting the physical scale using the gradient flow method, and compare the results to those of existing studies in the quenched approximation, as well as to the literature on closely related theories.

Meson spectroscopy in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>p</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> gauge theory with three antisymmetric fermions

Spectral densities encode nonperturbative information that enters the calculation of a plethora of physical observables in strongly coupled field theories. Phenomenological applications encompass aspects of standard-model hadronic physics, observable at current colliders, as well as correlation functions characterizing new physics proposals, testable in future experiments. By making use of numerical data produced in a <math><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> lattice gauge theory with matter transforming in an admixture of fundamental and 2-index antisymmetric representations of the gauge group, we perform a systematic study to demonstrate the effectiveness of recent technological progress in the reconstruction of spectral densities.To this purpose, we write and test new software packages that use energy-smeared spectral densities to analyze the mass spectrum of mesons. We assess the effectiveness of different smearing kernels and optimize the smearing parameters to the characteristics of available lattice ensembles. For concreteness, we analyze the <math><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> lattice gauge theory with matter transforming in an admixture of fundamental and 2-index antisymmetric representations of the gauge group. We generate new ensembles for the theory in consideration, with lattices that have a longer extent in the time direction with respect to the spatial ones. We run our tests on these ensembles, obtaining new results about the spectrum of light mesons and their excitations. We make available our algorithm and software for the extraction of spectral densities, that can be applied to theories with other gauge groups, including the theory of strong interactions (QCD) governing hadronic physics in the standard model.

Meson spectroscopy from spectral densities in lattice gauge theories

We provide the first extensive, numerical study of the nontrivial problem of mixing between flavor-singlet composite states emerging in strongly coupled lattice field theories with matter field content consisting of fermions transforming in different representations of the gauge group. The theory of interest is the minimal candidate for a composite Higgs model that also accommodates a mechanism for top partial compositeness: the <math><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> gauge theory coupled to two (Dirac) fermions transforming as the fundamental and three as the two-index antisymmetric representation of the gauge group, respectively. We apply an admixture of APE smearing and Wuppertal smearings, as well as the generalized eigenvalue problem approach, to two-point functions involving flavor-singlet mesons, for ensembles having time extent longer than the space extent. We demonstrate that, in the region of lattice parameter space accessible to this study, both masses and mixing angles can be measured effectively, despite the presence of (numerically noisy) contributions from disconnected diagrams.

Mixing between flavor singlets in lattice gauge theories coupled to matter fields in multiple representations

We report the findings of our extensive study of the spectra of flavored mesons in lattice gauge theories with symplectic gauge group and fermion matter content treated in the quenched approximation. For the <math><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>)</mo></math>, <math><mi>S</mi><mi>p</mi><mo>(</mo><mn>6</mn><mo>)</mo></math>, and <math><mi>S</mi><mi>p</mi><mo>(</mo><mn>8</mn><mo>)</mo></math> gauge groups, the (Dirac) fermions transform in either the fundamental, or the 2-index, antisymmetric or symmetric, representations. This study sets the stage for future precision calculations with dynamical fermions in the low-mass region of lattice parameter space. Our results have potential phenomenological applications ranging from composite Higgs models, to top (partial) compositeness, to dark matter models with composite, strong-coupling dynamical origin. Having adopted the Wilson flow as a scale-setting procedure, we apply Wilson chiral perturbation theory to extract the continuum and massless limits for the observables of interest. The resulting measurements are used to perform a simplified extrapolation to the large-<math><mi>N</mi></math> limit, hence drawing a preliminary connection with gauge theories with unitary groups. We conclude with a brief discussion of the Weinberg sum rules.

United States

Jack Holligan

Spectrum of mesons in quenched <math><mrow><mi>S</mi><mi>p</mi><mo>(</mo><mn>2</mn><mi>N</mi><mo>)</mo></mrow></math> gauge theories

We report the results of lattice numerical studies of the <math><mrow><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>)</mo></mrow></math> gauge theory coupled to fermions (hyperquarks) transforming in the fundamental and two-index antisymmetric representations of the gauge group. This strongly coupled theory is the minimal candidate for the ultraviolet completion of composite Higgs models that facilitate the mechanism of partial compositeness for generating the top-quark mass. We measure the spectrum of the low-lying, half-integer spin, bound states composed of two fundamental and one antisymmetric hyperquarks, dubbed chimera baryons, in the quenched approximation. In this first systematic, nonperturbative study, we focus on the three lightest parity-even chimera-baryon states, in analogy with QCD, denoted as <math><msub><mi>Λ</mi><mrow><mi>CB</mi></mrow></msub></math>, <math><msub><mi>Σ</mi><mrow><mi>CB</mi></mrow></msub></math> (both with spin <math><mn>1</mn><mo>/</mo><mn>2</mn></math>), and <math><mrow><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>CB</mi></mrow><mrow><mo>*</mo></mrow></msubsup></mrow></math> (with spin <math><mn>3</mn><mo>/</mo><mn>2</mn></math>). The spin-<math><mn>1</mn><mo>/</mo><mn>2</mn></math> such states are candidates of the top partners. The extrapolation of our results to the continuum and massless-hyperquark limit is performed using formulas inspired by QCD heavy-baryon Wilson chiral perturbation theory. Within the range of hyperquark masses in our simulations, we find that <math><msub><mi>Σ</mi><mi>CB</mi></msub></math> is not heavier than <math><msub><mi>Λ</mi><mi>CB</mi></msub></math>.

Lattice investigations of the chimera baryon spectrum in the <math><mrow><mi>S</mi><mi>p</mi><mo>(</mo><mn>4</mn><mo>)</mo></mrow></math> gauge theory