<p>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.</p>

Cyprus

Andreas Athenodorou

United Kingdom

Ed Bennett

Germany

Georg Bergner

Spain

Denmark

Pietro Butti

Julian Lenz

Biagio Lucini

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

We study the behaviour of SU(2) Yang-Mills fields on a $\textit{T}$ × $\textit{R}$ geometry where the two-torus is equipped with twisted boundary conditions. We monitor the evolution of the dynamics of the system as a function of the torus size $\textit{l}$ . For small sizes the behaviour of the system is well understood in terms of semiclassical predictions. In our case, the long distance structure is that of a two-dimensional gas of vortex-like fractional instantons with size and density growing with $\textit{l}$ . Our lattice Monte Carlo simulations confirm the semiclassical predictions and allow the determination of the relevant scale signalling the transition to the non-dilute situation. At low densities the string tension takes the standard value of a 2D center-vortex gas, growing with the density and approaching the value measured at infinite volume. Our work includes preliminary studies of the extension to SU(N) and to the region of large sizes in which boundary conditions are irrelevant, and all physical scales are determined by the Λ parameter.

Antonio González-Arroyo

Italy

Ivan Soler

A $\textit{T}$

Linear confinement with Casimir scaling of the string tension in confining gauge theories is a consequence of a certain property of the Polyakov loop related to random matrices. This mechanism does not depend on the details of the theories (neither the gauge group nor dimensions) and explains approximate Casimir scaling below string-breaking length. In this paper, we study 3d SU(2) pure Yang-Mills theory numerically and find the same random-matrix behavior for rectangular Wilson loops. We conjecture that this is a universal feature of strongly coupled confining gauge theories.

Vaibhav Gautam

Masanori Hanada

China

Jack Holden

Wilson loops and random matrices

We present our investigations of SU(N) adjoint QCD in two dimensions with one Majorana fermion on the lattice. We determine the relevant parameter range for the simulations with Wilson fermions and present results for Polyakov loop, chiral condensate, and string tension. In the theory with massive fermions, all observables we checked show qualitative agreement between numerical lattice data and theory, while the massless limit is more subtle since chiral and non-invertible symmetry of the continuum theory are explicitly broken by lattice regularization. In thermal compactification, we observe N perturbative vacua for the holonomy potential at high-T with instanton events connecting them, and a unique vacuum at low-T. At finite-N, this is a cross-over and it turns to a phase transition at large-N thermodynamic limit. In circle compactification with periodic boundary conditions, we observe a unique center-symmetric minimum at any radius. In continuum, the instantons in the thermal case carry zero modes (for even N) and indeed, in the lattice simulations, we observe that chiral condensate is dominated by instanton centers, where zero modes are localized. We present lattice results on the issue of confinement vs. screening in the theory and comment on the roles of chiral symmetry and non-invertible symmetry.

Stefano Piemonte

United States

Mithat Ünsal

Investigating two-dimensional adjoint QCD on the lattice

We propose an orbifold lattice formulation of QCD suitable for quantum simulations. We show explicitly how to encode gauge degrees of freedom into qubits using noncompact variables, and how to write down a simple truncated Hamiltonian in the coordinate basis. We show that SU(3) gauge group variables and quarks in the fundamental representation can be implemented straightforwardly on qubits, for arbitrary truncation of the gauge manifold.

Japan

Enrico Rinaldi

Andreas Schäfer

Toward QCD on quantum computer: orbifold lattice approach

We present new investigations on the Adjoint Filtering Method (AFM), a proposal for filtering gauge configurations by using adjoint zero modes. This method relies on the existence of the Supersymmetric Zero Mode (SZM), whose density follows the gauge action density for classical configurations. We review how a similar construction on the lattice is implemented based on the overlap operator and test the method against smooth configurations showing a remarkable agreement with the expected densities even when pairs of fractionals instanton/anti-instantons are present and noise is added. Then we explore the application of the method to Monte Carlo generated configurations based on SU(2) gauge group. The tuning of the parameters and quantitative results are explicitly shown for a T 3 × R lattice. We show explicit examples comparing the AFM to the density obtained from the Gradient Flow. The agreement is remarkable for some specific configurations containing fractional instantons with the advantage that the AFM does not modify the underlying structures.

Identifying topological structures with adjoint mode filtering

We explain the microscopic origin of linear confinement potential with the Casimir scaling in generic confining gauge theories. In the low-temperature regime of confining gauge theories such as QCD, Polyakov lines are slowly varying Haar random modulo exponentially small corrections with respect to the inverse temperature, as shown by one of the authors (M. H.) and Watanabe. With exact Haar randomness, computation of the two-point correlator of Polyakov loops reduces to the problem of random walk on group manifold. Linear confinement potential with approximate Casimir scaling except at short distances follows naturally from slowly varying Haar randomness. With exponentially small corrections to Haar randomness, string breaking and loss of Casimir scaling at long distance follow. Hence we obtain the Casimir scaling which is only approximate and holds only at intermediate distance, which is precisely needed to explain the results of lattice simulations. For (1 + 1)-dimensional theories, there is a simplification that admits the Casimir scaling at short distances as well.

Color confinement and random matrices. A random walk down group manifold toward Casimir scaling

We extend our analysis of bound states in  $${\mathcal {N}}=1$$  <math> <mrow> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>   supersymmetric Yang–Mills theory by the consideration of baryonic operators, which are composed of three gluino fields. The corresponding states are similar to the baryons in QCD, but due to the difference between gluino and quark fields, their properties and the fermion line contractions involved in their correlation functions are different from QCD. In this work, we first explain the derivation of these operators and the contractions needed in numerical calculations of their correlators. In contrast to QCD the correlators contain a spectacle piece, which requires methods for all-to-all propagators. We provide a first estimate of the two-point function and the mass of the lightest baryonic state in  $${\mathcal {N}}=1$$  <math> <mrow> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>   supersymmetric Yang–Mills theory.

Pakistan

Sajid Ali

Camilo López

Istvan Montvay

Gernot Münster

Baryonic states in  $$\mathbf {{\mathcal {N}}=1}$$  <math> <mrow> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>   supersymmetric SU(2) Yang–Mills theory on the lattice

We test the gauge/gravity duality between the matrix model and type IIA string theory at low temperatures with unprecedented accuracy. To this end, we perform lattice Monte Carlo simulations of the Berenstein-Maldacena-Nastase (BMN) matrix model, which is the one-parameter deformation of the Banks-Fischler-Shenker-Susskind (BFSS) matrix model, taking both the large N and continuum limits. We leverage the fact that sufficiently small flux parameters in the BMN matrix model have a negligible impact on the energy of the system while stabilizing the flat directions so that simulations at smaller N than in the BFSS matrix model are possible. Hence, we can perform a precision measurement of the large N continuum energy at the lowest temperatures to date. The energy is in perfect agreement with supergravity predictions including estimations of α′-corrections from previous simulations. At the lowest temperature where we can simulate efficiently (T = 0.25λ 1/3, where λ is the ’t Hooft coupling), the difference in energy to the pure supergravity prediction is less than 10%. Furthermore, we can extract the coefficient of the 1/N 4 corrections at a fixed temperature with good accuracy, which was previously unknown.

Stratos Pateloudis

Pavlos Vranas

Hiromasa Watanabe

Norbert Bodendorfer

Precision test of gauge/gravity duality in D0-brane matrix model at low temperature

In this work we investigate the infrared behavior of a Yang-Mills theory coupled to a massless fermion in the adjoint representation of the gauge group SU(2). This model has many interesting properties, corresponding to the <math><mi>N</mi><mo>=</mo><mn>2</mn></math> super-Yang-Mills theory without scalars and in the recent years there has been an increasing interest toward understanding whether confinement and fermion condensation occur at low energy. We simulate the theory on the lattice close to the massless limit using the overlap discretization of the fermion action, allowing a precise and clean study of the chiral symmetry-breaking pattern and of the fermion condensate. We present results for the scale setting, the condensate, and the running of the coupling constant through the gradient flow—all of them pointing to a theory without an infrared fixed point and remaining confined deep in the infrared regime.

Juan Camilo Lopez

Ivan Soler Calero

Lattice simulations of adjoint QCD with one Dirac overlap fermion