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Gauge theories with matter fields in various representations play an important role in different branches of physics. Recently, it was proposed that several aspects of the interesting pseudogap phase of cuprate superconductors near optimal doping may be explained by an emergent $SU(2)$ gauge symmetry. Around the transition with positive hole-doping, one can construct a $(2+1)-$dimensional $SU(2)$ gauge theory coupled to four adjoint scalar fields which gives rise to a rich phase diagram with a myriad of phases having different broken symmetries. We study the phase diagram of this model on the Euclidean lattice using the Hybrid Monte Carlo algorithm. We find the existence of multiple broken phases as predicted by previous mean field studies. Depending on the quartic couplings, the $SU(2)$ gauge symmetry is broken down either to $U(1)$ or $\mathbb{Z}_2$ in the perturbative description of the model. We further study the confinement-deconfinement transition in this theory, and find that both the broken phases are deconfining in the range of volumes that we studied. However, there exists a marked difference in the behavior of the Polyakov loop between the two phases., Comment: 11 pages, 11 figures, minor changes to results and conclusions

Using tensor network methods, we simulate the real-time evolution of the lattice Thirring model quenched out of equilibrium in both the critical and massive phases, and study the appearance of dynamical quantum phase transitions, as non-analyticities in the Loschmidt rate. Whereas the presence of a dynamical quantum phase transition in the model does not correspond to quenches across the critical line of the equilibrium phase diagram at zero temperature, we identify a threshold in the energy density of the initial state, necessary for a dynamical quantum phase transition to be present. Moreover, in the case of the gapped quench Hamiltonian, we unveil a connection of this threshold to a transition between different regions in the finite temperature phase diagram.

We provide the first extensive, numerical study of the non-trivial 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 $Sp(4)$ 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 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., Comment: 18 pages, 3 figures, 4 tables

Spectral densities encode non-perturbative 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 with lattice gauge theories, 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 Sp(4) 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 this theory, 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., Comment: 44 pages, 20 figures

In strongly coupled gauge theories that serve as completions of composite Higgs models, the fermionic bound states formed by fermions (hyperquarks) transforming in different representations, called chimera baryons, could serve as top partners, by embedding of the Standard Model appropriately. We report our results on the spectrum of chimera baryons in the Sp(4) gauge theory with hyperquarks transforming in fundamental and two-index antisymmetric representations. For this study, we adopt the quenched approximation. We investigate the mass hierarchy between the lightest chimera baryons with different quantum numbers, as a function of the lattice parameters. Inspired by baryon chiral effective field theory, and the Akaike Information Criterion, we perform a first extrapolation to the continuum and massless-hyperquark limit., Comment: 8 pages, 2 figures, contribution to the proceedings of the 40th International Symposium on Lattice Field Theory (LATTICE2023), July 31st - August 4th, 2023, Fermi National Accelerator Laboratory, Batavia, Illinois, USA

We report the findings of our extensive study of the spectra of flavoured mesons in lattice gauge theories with symplectic gauge group and fermion matter content treated in the quenched approximation. For the $Sp(4)$, $Sp(6)$, and $Sp(8)$ 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-$N$ limit, hence drawing a preliminary connection with gauge theories with unitary groups. We conclude with a brief discussion of the Weinberg sum rules., Comment: 31 pages, 28 figures, 7 tables. Version accepted for publication

We study a 3-dimensional SU(2) gauge theory with 4 Higgs fields which transform under the adjoint representation of the gauge group, that has been recently proposed by Sachdev et al. to explain the physics of cuprate superconductors near optimal doping. The symmetric confining phase of the theory corresponds to the usual Fermi-liquid phase while the broken (Higgs) phase is associated with the interesting pseudogap phase of cuprates. We employ the Hybrid Monte-Carlo algorithm to study the phase diagram of the theory. We find the existence of a variety of broken phases in qualitative accordance with earlier mean-field predictions and discuss their role in cuprates. In addition, we investigate the behavior of Polyakov loop to probe the confinement/deconfinement phase transition, and find that the Higgs phase hosts a stable deconfining phase consistent with previous studies., Comment: 7 pages, 4 figures, Proceedings of the 40th International Symposium on Lattice Field Theory (LATTICE2023), July 31st - August 4th, 2023, Fermi National Accelerator Laboratory

We study the most general Two Higgs Doublet Model with $SU(2)$ gauge fields on the lattice. The phase space is probed through the computation of gauge-invariant global observables serving as proxies for order parameters. In each phase, the spectrum of the theory is analysed for different combinations of bare couplings and different symmetry breaking patterns. The scale setting and determination of the running gauge coupling are performed through the Wilson flow computation of the action density.

The TMD soft function can be obtained by formulating the Wilson line in terms of auxiliary 1-dimensional fermion fields on the lattice. In this formulation, the directional vector of the auxiliary field in Euclidean space has the form $\tilde n = (in^0, \vec 0_\perp, n^3)$, where the time component is purely imaginary. The components of these complex directional vectors in the Euclidean space can be mapped directly to the rapidities of the Minkowski space soft function. We present the results of the one-loop calculation of the Euclidean space analog to the soft function using these complex directional vectors. As a result, we show that the calculation is valid only when the directional vectors obey the relation: $|r| = |n^3/n^0| > 1$, and that this result corresponds to a computation in Minkowski space with space-like directed Wilson lines. Finally, we show that a lattice calculable object can be constructed that has the desired properties of the soft function., Comment: The 40th International Symposium on Lattice Field Theory (Lattice 2023)

We report the results of lattice numerical studies of the Sp(4) 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, non-perturbative study, we focus on the three lightest parity-even chimera-baryon states, in analogy with QCD, denoted as $\Lambda_{\rm CB}$, $\Sigma_{\rm CB}$ (both with spin 1/2), and $\Sigma_{\rm CB}^\ast$(with spin 3/2). The spin-1/2 such states are candidates of the top partners. The extrapolation of our results to the continuum and massless-hyperquark limit is performed using formulae inspired by QCD heavy-baryon Wilson chiral perturbation theory. Within the range of hyperquark masses in our simulations, we find that $\Sigma_{\mathrm{CB}}$ is not heavier than $\Lambda_{\mathrm{CB}}$., Comment: 36 pages, 12 figures, 19 tables; minor improvements and corrections; aligning numbers in Data release; version accepted for publication in PRD

The light-cone distribution amplitude (LCDA) of the pion contains information about the parton momentum carried by the quarks and is an important theoretical input for various predictions of exclusive processes at high energy, including the pion electromagnetic form factor. Progress towards constraining the fourth Mellin moment of the LCDA using the heavy-quark operator product expansion (HOPE) method is presented., Comment: 10 pages, 4 figures, proceedings to talk presented at the 40th International Symposium on Lattice Field Theory, July 31st - August 4th 2023, Fermi National Accelerator Laboratory, Batavia, Illinois, USA. arXiv admin note: text overlap with arXiv:2211.17009

Four-dimensional gauge theories based on symplectic Lie groups provide elegant realisations of the microscopic origin of several new physics models. Numerical studies pursued on the lattice provide quantitative information necessary for phenomenological applications. To this purpose, we implemented Sp(2N) gauge theories using Monte Carlo techniques within Grid, a performant framework designed for the numerical study of quantum field theories on the lattice. We show the first results obtained using this library, focusing on the case-study provided by the Sp(4) theory coupled to Nas = 4 Wilson-Dirac fermions transforming in the 2-index antisymmetric representation. In particular, we discuss preliminary tests of the algorithm and we test some of its main functionalities., Comment: 9 pages, 7 figures, contribution for the 40th International Symposium on Lattice Field Theory (Lattice 2023), July 31st - August 4th, 2023, Fermi National Accelerator Laboratory, presenting the results of the paper: arXiv:2306.11649

Symplectic gauge theories coupled to matter fields lead to symmetry enhancement phenomena that have potential applications in such diverse contexts as composite Higgs, top partial compositeness, strongly interacting dark matter, and dilaton-Higgs models. These theories are also interesting on theoretical grounds, for example in reference to the approach to the large-N limit. A particularly compelling research aim is the determination of the extent of the conformal window in gauge theories with symplectic groups coupled to matter, for different groups and for field content consisting of fermions transforming in different representations. Such determination would have far-reaching implications, but requires overcoming huge technical challenges. Numerical studies based on lattice field theory can provide the quantitative information necessary to this endeavour. We developed new software to implement symplectic groups in the Monte Carlo algorithms within the Grid framework. In this paper, we focus most of our attention on the Sp(4) lattice gauge theory coupled to four (Wilson-Dirac) fermions transforming in the 2-index antisymmetric representation, as a case study. We discuss an extensive catalogue of technical tests of the algorithms and present preliminary measurements to set the stage for future large-scale numerical investigations. We also include the scan of parameter space of all asymptotically free Sp(4) lattice gauge theories coupled to varying number of fermions transforming in the antisymmetric representation., Comment: 42 pages, 16 figures. Version accepted for publication

We review the current status of the long-term programme of numerical investigation of $Sp(2N)$ gauge theories with and without fermionic matter content. We start by introducing the phenomenological as well as theoretical motivations for this research programme, which are related to composite Higgs models, models of partial top compositeness, dark matter models, and in general to the physics of strongly coupled theories and their approach to the large-N limit. We summarise the results of lattice studies conducted so far in the $Sp(2N)$ Yang-Mills theories, measuring the string tension, the mass spectrum of glueballs and the topological susceptibility, and discuss their large-N extrapolation. We then focus our discussion on $Sp(4)$, and summarise numerical measurements of mass and decay constant of mesons in the theories with fermion matter in either the fundamental or the antisymmetric representation, first in the quenched approximation, and then with dynamical fermions. We finally discuss the case of dynamical fermions in mixed representations, and exotic composite fermion states such as the chimera baryons. We conclude by sketching the future stages of the programme. And we describe our approach to open access., Comment: 75 pages, 22 figures, 3 tables; contribution to the special issue of Universe "Numerical Studies of Strongly Coupled Gauge Theories (SCGTs) in the Search of New Physics"; version accepted for publication

The pion light-cone distribution amplitude (LCDA) is a central non-perturbative object of interest for the calculation of high-energy exclusive processes in quantum chromodynamics. This article describes the progress in the lattice QCD calculation of the fourth Mellin moment of the pion LCDA using a heavy-quark operator product expansion (HOPE)., Comment: 10 pages, 3 figures, contribution to the proceedings of the 39th International Symposium on Lattice Field Theory (LATTICE2022), 8th-13th August 2022, Bonn, Germany

Sp($2N$) gauge theories with fermonic matter provide an ideal laboratory to build extensions of the standard model based on novel composite dynamics. Examples include composite Higgs along with top partial compositeness and composite dark matter. Without fermions, their study also complements those based on SU($N_c$) gauge theories with which they share a common sector in the large $N_c=2N$ limit. We report on our recent progress in the numerical studies of Sp($2N$) gauge theories discretised on a four-dimensional Euclidean lattice. In particular, we present preliminary results for the low-lying mass spectra of mesons and chimera baryons in the theories with $N=2$. We also compute the topological susceptibility for various values of $N$, extrapolate the results to the large $N$ limit, and discuss certain universal properties in Yang-Mills theories., Comment: 8 pages, 6 figures, Contribution to the proceedings of the 15th Quark Confinement and the Hadron Spectrum conference (ConfXV), 1st-6th August 2022, Stavanger, Norway

Chimera baryons are an important element of strongly coupled theories that provide a microscopic origin for UV complete composite Higgs models (CHMs), since they play the role of top partners in top partial compositeness. In a particular interesting realisation of CHMs based upon an underlying $Sp(4)$ gauge theory, such exotic objects are composed of two fermion constituents transforming on the fundamental, and one on the 2-index antisymmetric representations. We perform lattice computations of the chimera baryon spectrum in the quenched approximation. We present preliminary results for the masses of various chimera baryons with different quantum numbers, including the one interpreted as the top partner. We test the technology needed for future calculations with dynamical fermions., Comment: 9 pages, 5 figures, contribution to the proceedings of the 39th International Symposium on Lattice Field Theory (LATTICE2022), 8th-13th August 2022, Bonn, Germany. Minor changes: two references included, clarification of the choice of the top partner

In this contribution, we report on our study of the properties of the Wilson flow and on the calculation of the topological susceptibility of $Sp(N_c)$ gauge theories for $N_c=2,\,4,\,6,\,8$. The Wilson flow is shown to scale according to the quadratic Casimir operator of the gauge group, as was already observed for $SU(N_c)$, and the commonly used scales $t_0$ and $w_0$ are obtained for a large interval of the inverse coupling for each probed value of $N_c$. The continuum limit of the topological susceptibility is computed and we conjecture that it scales with the dimension of the group. The lattice measurements performed in the $SU(N_c)$ Yang-Mills theories by several independent collaborations allow us to test this conjecture and to obtain a universal large-$N_c$ limit of the rescaled topological susceptibility., Comment: 9 pages, 3 figures. Proceeding for the 39th International Symposium on Lattice Field Theory, 8th-13th August 2022, Bonn, Germany

We study two different SU(2) gauge-scalar theories in 3 and 4 spacetime dimensions. Firstly, we focus on the 3 dimensional SU(2) theory with multiple Higgs fields in the adjoint representation, that can be mapped to cuprate systems in condensed matter physics which host a rich phase diagram including high-Tc superconductivity. It has been proposed that the theory with 4 adjoint Higgs fields can be used to explain the physics of hole-doped cuprates for a wide range of parameters. We show exploratory results on the phase diagram of the theory. On the other hand, we are interested in the 4 dimensional theory with 2 sets of fundamental scalar (Higgs) fields, which is relevant to the 2 Higgs Doublet Model (2HDM), a proposed extension to the Standard Model of particle physics. The goal is to understand the particle spectrum of the theory at zero temperature and the electroweak phase transition at finite temperature. We present exploratory results on scale setting and the multi-parameter phase diagram of this theory., Comment: 10 pages, 5 figures, Proceedings of the 39th International Symposium on Lattice Field Theory (LATTICE2022), 8th-13th August 2022, Bonn, Germany

We perform numerical calculations of masses and decay constants of the lightest (flavoured) pseudoscalar, vector and axial vector mesons in the $Sp(4)$ lattice gauge theory with three Dirac fermions in the antisymmetric representation. The corresponding continuum theory plays an important role in certain ultra-violet complete realisations of composite Higgs, partial top compositeness, and composite dark matter models. In addition, we measure the masses of other flavoured mesons in spin-$0$ and $1$ channels, as well as the first excited state of the vector mesons. Using the gradient flow to set the scale, we carry out the continuum extrapolation and show preliminary results for the meson spectrum of the theory., Comment: 8 pages, 4 figures, 1 Table, Proceedings of the 39th International Symposium on Lattice Field Theory (LATTICE2022), 8th-13th August 2022, Bonn, Germany

We study Yang-Mills lattice theories with $Sp(N_c)$ gauge group, with $N_c=2N$, for $N=1,\,\cdots,\,4$. We show that if we divide the renormalised couplings appearing in the Wilson flow by the quadratic Casimir $C_2(F)$ of the $Sp(N_c)$ group, then the resulting quantities display a good agreement among all values of $N_c$ considered, over a finite interval in flow time. We use this scaled version of the Wilson flow as a scale-setting procedure, compute the topological susceptibility of the $Sp(N_c)$ theories, and extrapolate the results to the continuum limit for each $N_c$., Comment: 19 pages, 17 figures. v4: Typos corrected in eq. 35, Figures 3 and 4. Results unchanged

For Yang-Mills theories in four dimensions, we propose to rescale the ratio between topological susceptibility and string tension squared in a universal way, dependent only on group factors. We apply this suggestion to $SU(N_c)$ and $Sp(N_c)$ groups, and compare lattice measurements performed by several independent collaborations. We show that the two sequences of (rescaled) numerical results in these two families of groups are compatible with each other. We hence perform a combined fit, and extrapolate to the common large-$N_c$ limit., Comment: 8 pages, 2 figures; v2 minor change, a reference added; v3. minor changes to match the version accepted in PLB

We consider the $Sp(4)$ gauge theory coupled to $N_f=2$ fundamental and $n_f=3$ antisymmetric flavours of Dirac fermions in four dimensions. This theory serves as the microscopic origin for composite Higgs models with $SU(4)/Sp(4)$ coset, supplemented by partial top compositeness. We study numerically its lattice realisation, and couple the fundamental plaquette action to Wilson-Dirac fermions in mixed representations, by adopting a (rational) hybrid Monte Carlo method, to perform non-trivial tests of the properties of the resulting lattice theory. We find evidence of a surface (with boundaries) of first-order bulk phase transitions in the three-dimensional space of bare parameters (one coupling and two masses). Explicit evaluation of the Dirac eigenvalues confirms the expected patterns of global symmetry breaking. After investigating finite volume effects in the weak-coupling phase of the theory, for the largest available lattice we study the mass spectra of the lightest spin-0 and spin-1 flavoured mesons composed of fermions in each representation, and of the lightest half-integer spin composite particle made of fermions in different representations -- the chimera baryon. This work sets the stage for future systematical studies of the non-perturbative dynamics in phenomenologically relevant regions of parameter space., Comment: 44 pages, 20 figures, 6 tables; minor improvements and references added; version accepted for publication in PRD

The pion light-cone distribution amplitude (LCDA) is a central non-perturbative object of interest for the calculation of high-energy exclusive processes in quantum chromodynamics. In this article, we discuss the calculation of the second and fourth Mellin moment of the pion LCDA using a heavy-quark operator product expansion. The resulting value for the second Mellin moment is $ \langle{ \xi^2 }\rangle(\mu = 2~\text{GeV})= 0.210 \pm 0.013\text{ (stat.)} \pm 0.034\text{ (sys.)}$. This result is compatible with those from previous determinations of this quantity., Comment: 17 pages, 8 Figures, 3 Tables; Talk presented at The 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

Lattice studies of gauge theories with symplectic gauge groups provide valuable information about gauge dynamics, and complement the results of lattice investigations focused on unitary gauge groups. These theories play a central role in phenomenological contexts such as composite Higgs and strongly interacting dark matter models. We report on recent progress of our lattice research programme, starting from the glueball mass spectrum and the topology of the pure gauge theory. We present our results on the mass spectrum of mesons in the quenched approximation, by varying the number of colours in the symplectic group. For the $Sp(4)$ theory, we focus on results obtained with dynamical fermion matter content comprising both fundamental and 2-index antisymmetric representations of the gauge group, as dictated by a well known model of composite Higgs with partial top compositeness., Comment: 39 pages, 29 figures, 5 tables, combined proceedings of the 38th International Symposium on Lattice Field Theory (LATTICE2021), 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We review numerical results for models with gauge group Sp(2N), discussing the glueball spectrum in the large-N limit, the quenched meson spectrum of Sp(4) with Dirac fermions in the fundamental and in the antisymmetric representation and the Sp(4) gauge model with two dynamical Dirac flavours. We also present preliminary results for the meson spectrum in the Sp(4) gauge theory with two fundamental and three antisymmetric Dirac flavours. The main motivation of our programme is to test whether this latter model is viable as a realisation of Higgs compositeness via the pseudo Nambu Goldstone mechanism and at the same time can provide partial top compositeness. In this respect, we report and briefly discuss preliminary results for the mass of the composite baryon made with two fundamental and one antisymmetric fermion (chimera baryon), whose physical properties are highly constrained if partial top compositeness is at work. Our investigation shows that a fully non-perturbative study of Higgs compositeness and partial top compositeness in Sp(4) is within reach with our current lattice methodology., Comment: Corrected notations in Eq. 1 and redrawn Fig 2, results unchanged; 8 pages, 4 figures, contribution to the proceedings of A Virtual Tribute to Quark Confinement and the Hadron Spectrum (vConf21), August 2nd-6th 2021

The pion light-cone distribution amplitude (LCDA) is a central non-perturbative object of interest for high-energy exclusive processes in quantum chromodynamics. In this article, the second Mellin moment of the pion LCDA is determined as a proof-of-concept calculation for the first numerical implementation of the heavy-quark operator product expansion (HOPE) method. The resulting value for the second Mellin moment, determined in quenched QCD at a pion mass of $m_\pi=550$ MeV at a factorization scale of 2 GeV is $ \langle \xi^2 \rangle = 0.210 \pm 0.013\text{ (stat.)} \pm 0.034\text{ (sys.)}$. This result is compatible with those from previous determinations of this quantity., Comment: 30 pages, 16 figures

Parton distribution functions (PDFs) and light-cone distribution amplitudes (LCDAs) are central non-perturbative objects of interest in high-energy inelastic and elastic scattering, respectively. As a result, an ab-initio determination of these objects is highly desirable. In this paper we present theoretical details for the calculation of the PDFs and LCDAs using a heavy-quark operator product expansion method. This strategy was proposed in a previous paper [Phys. Rev. D 73, 014501 (2006)] for computing higher moments of the PDFs using lattice QCD. Its central feature is the introduction of a fictitious, valence heavy quark. In the current article, we show that the operator product expansion (OPE) of the hadronic matrix element we study can also be expressed as the convolution of a perturbative matching kernel and the corresponding light-cone distribution, which in principle can be inverted to determine the parton momentum fraction dependence. Regarding the extraction of higher moments, this work also provides the one-loop Wilson coefficients in the OPE formulas for the unpolarized PDF, helicity PDF and pseudo-scalar meson LCDAs. Although these Wilson coefficients for the PDFs can be inferred from existing results in the literature, those for the LCDAs are new., Comment: 22 pages, 3 figures; Text updated to be consistent with published version

Motivated in part by the pseudo-Nambu Goldstone Boson mechanism of electroweak symmetry breaking in Composite Higgs Models, in part by dark matter scenarios with strongly coupled origin, as well as by general theoretical considerations related to the large-N extrapolation, we perform lattice studies of the Yang-Mills theories with $Sp(2N)$ gauge groups. We measure the string tension and the mass spectrum of glueballs, extracted from appropriate 2-point correlation functions of operators organised as irreducible representations of the octahedral symmetry group. We perform the continuum extrapolation and study the magnitude of finite-size effects, showing that they are negligible in our calculation. We present new numerical results for $N=1$, $2$, $3$, $4$, combine them with data previously obtained for $N=2$, and extrapolate towards $N\rightarrow \infty$. We confirm explicitly the expectation that, as already known for $N=1,2$ also for $N=3,4$ a confining potential rising linearly with the distance binds a static quark to its antiquark. We compare our results to the existing literature on other gauge groups, with particular attention devoted to the large-$N$ limit. We find agreement with the known values of the mass of the $0^{++}$, $0^{++*}$ and $2^{++}$ glueballs obtained taking the large-$N$ limit in the $SU(N)$ groups. In addition, we determine for the first time the mass of some heavier glueball states at finite $N$ in $Sp(2N)$ and extrapolate the results towards $N \rightarrow +\infty$ taking the limit in the latter groups. Since the large-$N$ limit of $Sp(2N)$ is the same as in $SU(N)$, our results are relevant also for the study of QCD-like theories., Comment: 42 pages, 21 figures, 23 tables. References added. Published version

We explore the feasibility of determining Mellin moments of the pion's light cone distribution amplitude using the heavy quark operator product expansion (HOPE) method. As the first step of a proof of principle study we pursue a determination of the second Mellin moment. We discuss our choice of kinematics which allows us to successfully extract the moment at low pion momentum. We describe the numerical simulation, and describe the data analysis, which leads us to a preliminary determination of the second Mellin moment in the continuum limit in the quenched approximation as $\langle\xi^2\rangle=0.19(7)$ in the $\bar{\text{MS}}$ scheme at 2 GeV., Comment: 9 pages, 8 figures. Contribution to the proceedings of the 2020 Asia-Pacific Symposium on Lattice Field Theory (APLAT 2020). Several citations and additional affiliation added

We report the masses of the lightest spin-0 and spin-2 glueballs obtained in an extensive lattice study of the continuum and infinite volume limits of $Sp(N_c)$ gauge theories for $N_c=2,4,6,8$. We also extrapolate the combined results towards the large-$N_c$ limit. We compute the ratio of scalar and tensor masses, and observe evidence that this ratio is independent of $N_{c}$. Other lattice studies of Yang-Mills theories at the same space-time dimension provide a compatible ratio. We further compare these results to various analytical ones and discuss them in view of symmetry-based arguments related to the breaking of scale invariance in the underlying dynamics, showing that a constant ratio might emerge in a scenario in which the $0^{++}$ glueball is interpreted as a dilaton state., Comment: 6 pages, 1 figure: v2. minor changes in Sp(4) data and the title, to appear in PRD

Non-perturbative aspects of the physics of $Sp(2N)$ gauge theories are interesting for phenomenological and theoretical reasons, and little studied so far, particularly in the approach to the large-$N$ limit. We examine the spectrum of glueballs and the string tension of Yang-Mills theories based upon these groups. Glueball masses are calculated numerically with a variational method from Monte-Carlo generated lattice gauge configurations. After taking continuum limits for $N$ = 1, 2, 3 and 4, we extrapolate the results towards large $N$. We compare the resulting spectrum with that of $SU(N)$ gauge theories, both at finite $N$ and as $N$ approaches infinity., Comment: 1+6 pages. 2 figures. Proceedings of the 37th International Symposium on Lattice Field Theory - Lattice 2019, Wuhan (China) V2: typo in list of authors has been corrected

We present concluding results from our study for zero-temperature phase structure of the massive Thirring model in 1+1 dimensions with staggered regularisation. Employing the method of matrix product states, several quantities, including two types of correlators, are investigated, leading to numerical evidence of a Berezinskii-Kosterlitz-Thouless phase transition. Exploratory results for real-time dynamics pertaining to this transition, obtained using the approaches of variational uniform matrix product state and time-dependent variational principle, are also discussed., Comment: 7 pages, 4 figures, contribution to the 37th International Symposium on Lattice Field Theory (LATTICE2019), 16-22 June 2019, Wuhan, China

We perform lattice studies of meson mass spectra and decay constants of the $Sp(4)$ gauge theory in the quenched approximation. We consider two species of (Dirac) fermions as matter field content, transforming in the 2-index antisymmetric and the fundamental representation of the gauge group, respectively. All matter fields are formulated as Wilson fermions. We extrapolate to the continuum and massless limits, and compare to each other the results obtained for the two species of mesons. In the case of two fundamental and three antisymmetric fermions, the long-distance dynamics is relevant for composite Higgs models. This is the first lattice study of this class of theories. The global $SU(4) \times SU(6)$ symmetry is broken to the $Sp(4) \times SO(6)$ subgroup, and the condensates align with the explicit mass terms present in the lattice formulation of the theory. The main results of our quenched calculations are that, with fermions in the 2-index antisymmetric representation of the group, the masses squared and decay constant squared of all the mesons we considered are larger than the corresponding quantities for the fundamental representation, by factors that vary between $\sim$1.2 and $\sim$2.7. We also present technical results that will be useful for future lattice investigations of dynamical simulations, of composite chimera baryons, and of the approach to large $N$ in the $Sp(2N)$ theories considered. We briefly discuss their high-temperature behaviour, where symmetry restoration and enhancement are expected., Comment: 49 pages, 11 figures, 20 tables; Version accepted for publication in PRD

The SU(2) gauge theory with 8 fundamental fermions is studied using unimproved staggered regularization. A phase transition or a crossover at strong coupling, which can be a bulk transition. By using chiral random matrix model we analyze the chiral condensate of this system. We also report the chiral susceptibility and its volume dependence near the transition point., Comment: 7 pages, 3 figures, talk presented at 37th International Symposium on Lattice Field Theory - Lattice2019, 16 - 22 June 2019, Wuhan, China

We calculate the meson spectrum of the Sp(4) lattice gauge theory coupled to two fundamental flavours of dynamical Dirac fermions. We focus on some of the lightest (flavoured) spin-0 and spin-1 states. This theory provides an ultraviolet completion for composite Higgs models based upon the SU(4)/Sp(4) coset. We analyse the strongly coupled dynamics in isolation, without explicit coupling to the standard model. We carry out continuum extrapolations using dynamical ensembles generated at five different values of bare lattice coupling, and for several values of the bare fermion mass. We fit the resulting meson masses and decay constants to a low-energy effective field theory built along the ideas of hidden local symmetry. We also compare our results to those of other closely related lattice gauge theories, which have matter content consisting of two fundamental Dirac flavours., Comment: 7 pages, 1 table, 4 figures, Contribution to the 37th International Symposium on Lattice Field Theory - LATTICE2019, 16-22 June 2019, Wuhan, China

We perform lattice studies of the gauge theory with Sp(4) gauge group and two flavours of (Dirac) fundamental matter. The global SU(4) symmetry is spontaneously broken by the fermion condensate. The dynamical Wilson fermions in the lattice action introduce a mass that breaks the global symmetry also explicitly. The resulting pseudo-Nambu-Goldstone bosons describe the SU(4)/Sp(4) coset, and are relevant, in the context of physics beyond the Standard Model, for composite Higgs models. We discuss scale setting, continuum extrapolation and finite volume effects in the lattice theory. We study mesonic composite states, which span representations of the unbroken Sp(4) global symmetry, and we measure masses and decay constants of the (flavoured) spin-0 and spin-1 states accessible to the numerical treatment, as a function of the fermion mass. With help from the effective field theory treatment of such mesons, we perform a first extrapolation towards the massless limit. We assess our results by critically comparing to the literature on other models and to the quenched results, and we conclude by outlining future avenues for further exploration. The results of our spectroscopic analysis provide new input data for future phenomenological studies in the contexts of composite Higgs models, and of dark matter models with a strongly coupled dynamical origin., Comment: 61 pages, 23 figures, 13 tables; Revised version; Accepted for publication in JHEP

Employing matrix product states as an ansatz, we study the non-thermal phase structure of the (1+1)-dimensional massive Thirring model in the sector of vanishing total fermion number with staggered regularization. In this paper, details of the implementation for this project are described. To depict the phase diagram of the model, we examine the entanglement entropy, the fermion bilinear condensate and two types of correlation functions. Our investigation shows the existence of two phases, with one of them being critical and the other gapped. An interesting feature of the phase structure is that the theory with non-zero fermion mass can be conformal. We also find clear numerical evidence that these phases are separated by a transition of the Berezinskii-Kosterlitz-Thouless type. Results presented in this paper establish the possibility of using the matrix product states for probing this type of phase transition in quantum field theories. They can provide information for further exploration of scaling behaviour, and serve as an important ingredient for controlling the continuum extrapolation of the model., Comment: 31 pages, 18 figures; minor changes to the text, typos corrected, references added; version published in Physical Review D

We review lattice results related to pion, kaon, $D$-meson, $B$-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_+(0)$ arising in the semileptonic $K \to \pi$ transition at zero momentum transfer, as well as the decay constant ratio $f_K/f_\pi$ and its consequences for the CKM matrix elements $V_{us}$ and $V_{ud}$. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $SU(2)_L\times SU(2)_R$ and $SU(3)_L\times SU(3)_R$ Chiral Perturbation Theory. We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $m_c$ and $m_b$ as well as those for $D$- and $B$-meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $\alpha_s$. Finally, in this review we have added a new section reviewing results for nucleon matrix elements of the axial, scalar and tensor bilinears, both isovector and flavor diagonal., Comment: 458 pages, 46 figures, 209 tables, 1146 references. Minor changes, version as published in EPJC. arXiv admin note: substantial text overlap with arXiv:1607.00299, arXiv:1310.8555

We analyse finite-size scaling behaviour of a four-dimensional Higgs-Yukawa model near the Gaussian infrared fixed point. Through improving the mean-field scaling laws by solving one-loop renormalisation group equations, the triviality property of this model can be manifested in the volume-dependence of moments of the scalar-field zero mode. The scaling formulae for the moments are derived in this work with the inclusion of the leading-logarithmic corrections. To test these formulae, we confront them with data from lattice simulations in a simpler model, namely the O(4) pure scalar theory, and find numerical evidence of good agreement. Our results of the finite-size scaling can in principle be employed to establish triviality of Higgs-Yukawa models, or to search for alternative scenarios in studying their fixed-point structure, if sufficiently large lattices can be reached., Comment: 20 pages, 5 figures; minor revision with typos corrected and clarification added; version published in JHEP

We report on the status of our programme to simulate Sp($2N$) gauge theories on the lattice. Motivated by the potential realization of an SU($4$)/Sp($4$)$\sim$SO($6$)/SO($5$) composite Higgs model and the applications to self interacting dark matter, we first perform dynamical simulations of Sp($4$) theories with two Dirac flavors in the fundamental representation. Preliminary results of the meson spectrum are presented, along with discussion of the lattice systematics. We also introduce two-index anti-symmetric Dirac fermions. Such fermions are relevant in the context of partial top compositeness, provided they carry SU($3$) color quantum numbers, and hence we introduce three (Dirac) copies of them. We present the quenched meson spectrum and explore the phase space of bare lattice parameters. For all the numerical simulations we use the standard Wilson lattice gauge and fermion actions., Comment: 7 pages, 6 figures, Contribution to The 36th Annual International Symposium on Lattice Field Theory - LATTICE2018, 22-28 July 2018, East Lansing, Michigan, USA

We report an exploratory study of the current-current matrix elements that are relevant to the extraction of moments of the pion light-cone distribution amplitude, employing the method of introducing a valence relativistic heavy quark. The numerical investigation is carried out in the quenched approximation with the physical volume $L\approx 2.4$ fm at two values of lattice spacing (0.05 and 0.075 fm). We obtain clean signals for the relevant Euclidean hadronic tensor with reasonable statistics, but observe that the lattice artefacts are non-negligible in our results. The key conclusion from the analysis hitherto is that although our approach has the potential for making significant contributions to parton physics, data at finer lattice spacings that are currently being produced are needed in order to control the continuum extrapolation., Comment: 7 pages, 5 figures; contribution to the proceedings of Lattice 2018 conference

We present preliminary results of a study on the non-thermal phase structure of the (1+1) dimensional massive Thirring model, employing the method of matrix product states. Through investigating the entanglement entropy, the fermion correlators and the chiral condensate, it is found that this approach enables us to observe numerical evidence of a Kosterlitz-Thouless phase transition in the model., Comment: 7 pages, 4 figures; contribution to the proceedings of Lattice 2018 conference

The Sp(4) gauge theory with two Dirac fundamental flavours provides a candidate for the microscopic origin of composite-Higgs models based on the SU(4)/Sp(4) coset. We employ a combination of two different, complementary strategies for the numerical lattice calculations, based on the Hybrid Monte Carlo and on the Heat Bath algorithms. We perform pure Yang-Mills, quenched computations and exploratory studies with dynamical Wilson fermions. We present the first results in the literature for the spectrum of glueballs of the pure Sp(4) Yang-Mills theory, an EFT framework for the interpretation of the masses and decay constants of the lightest pion, vector and axial-vector mesons, and a preliminary calculation of the latter in the quenched approximation. We show the first numerical evidence of a bulk phase transition in the lattice theory with dynamical Wilson fermions, and perform the technical steps necessary to set up future investigations of the mesonic spectrum of the full theory., Comment: References added, discussions expanded, figures improved, typos fixed. Version accepted for publication on JHEP

Tensor Network methods have been established as a powerful technique for simulating low dimensional strongly-correlated systems for over two decades. Employing the formalism of Matrix Product States, we investigate the phase diagram of the massive Thirring model. We also show the possibility of studying soliton dynamics and topological phase transition via the Thirring model., Comment: 8 pages, 3 figures; contribution to the proceedings of the 35th International Symposium on Lattice Field Theory, 18 - 24 June 2017, Granada, Spain

We present results from two projects on lattice calculations for the Higgs-Yukawa model. First we report progress on the search of first-order thermal phase transitions in the presence of a dimension-six operator, with the choices of bare couplings that lead to viable phenomenological predictions. In this project the simulations are performed using overlap fermions to implement the required chiral symmetry. Secondly, our study for applying finite-size scaling techniques near the Gaussian fixed point of the Higgs-Yukawa model is presented. We discuss the analytical formulae for the Higgs Yukawa model and show results for a first numerical study in the pure $O(4)$ scalar sector of the theory., Comment: 8 pages, 4 figures; Contribution to the proceedings of the 35th International Symposium on Lattice Field Theory, 18 - 24 June 2017, Granada, Spain

As a first step in the study of $\mathrm{Sp}(2N)$ composite Higgs models, we obtained a set of novel numerical results for the pure gauge $\mathrm{Sp}(4)$ lattice theory in 3+1 space-time dimensions. Results for the continuum extrapolations of the string tension and the glueball mass spectrum are presented and their values are compared with the same quantities in neighbouring $\mathrm{SU}(N)$ models., Comment: 8 pages, 3 tables, 3 figures. Talk presented at the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain

As a first step towards a quantitative understanding of the SU(4)/Sp(4) composite Higgs model through lattice calculations, we discuss the low energy effective field theory resulting from the SU(4) $\to$ Sp(4) global symmetry breaking pattern. We then consider an Sp(4) gauge theory with two Dirac fermion flavours in the fundamental representation on a lattice, which provides a concrete example of the microscopic realisation of the SU(4)/Sp(4) composite Higgs model. For this system, we outline a programme of numerical simulations aiming at the determination of the low-energy constants of the effective field theory and we test the method on the quenched theory. We also report early results from dynamical simulations, focussing on the phase structure of the lattice theory and a calculation of the lowest-lying meson spectrum at coarse lattice spacing., Comment: 16 pages, 7 figures, 4 tables. Combined contributions of B. Lucini and J.-W. Lee to the Lattice 2017 Symposium. To appear in EPJ Web Conf

As part of an ongoing programme to study $\mathrm{Sp}(2N)$ gauge theories as potential realisations of composite Higgs models, we consider the case of $\mathrm{Sp}(4)$ on the lattice, both as a pure gauge theory, and with two Dirac fermion flavors in the fundamental representation. In order to compare results between these two cases and maintain control of lattice artefacts, we make use of the gradient flow to set the scale of the simulations. We present some technical aspects of the simulations, including preliminary results for the scale setting in the two cases and results for the topological charge history., Comment: 8 pages, 6 figures; talk presented at the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain

Motivated by the absence of signals of new physics at the LHC, which seems to imply the presence of large mass hierarchies, we investigate the theoretical possibility that these could arise dynamically in new strongly-coupled gauge theories extending the standard model of particle physics. To this purpose, we study lattice data on non-Abelian gauge theories in the (near-)conformal regime---specifically, $\mathrm{SU}(2)$ with $N_{\mathrm{f}}=1$ and $2$ dynamical fermion flavours in the adjoint representation. We focus our attention on the ratio $R$ between the masses of the lightest spin-2 and spin-0 resonances, and draw comparisons with a simple toy model in the context of gauge/gravity dualities. For models in which large anomalous dimensions arise dynamically, we show indications that this mass ratio can be large, with $R > 5$. Moreover, our results suggest that $R$ might be related to universal properties of the IR fixed point. Our findings provide an interesting step towards understanding large mass ratios in the non-perturbative regime of quantum field theories with (near) IR conformal behaviour., Comment: 6 pages, 4 figures; presented at the 34th annual International Symposium on Lattice Field Theory (LATTICE2016)