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.

Germany

Georg Bergner

Stefano Piemonte

United States

Mithat Ünsal

Investigating two-dimensional adjoint QCD on the lattice

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

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

We present the first lattice investigation of coupled-channel   <math> <mi>D</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ D\overline{D} $$  and   <math> <msub> <mi>D</mi> <mi>s</mi> </msub> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {D}_s{\overline{D}}_s $$  scattering in the J  PC  = 0++ and 2++ channels. The scattering matrix for partial waves l = 0, 2 and isospin zero is determined using multiple volumes and inertial frames via Lüscher’s formalism. Lattice QCD ensembles from the CLS consortium with m  π  ≃ 280 MeV, a ≃ 0.09 fm and L/a = 24, 32 are utilized. The resulting scattering matrix suggests the existence of three charmonium-like states with J  PC  = 0++ in the energy region ranging from slightly below 2 mD  up to 4.13 GeV. We find a so far unobserved   <math> <mi>D</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ D\overline{D} $$  bound state just below threshold and a   <math> <mi>D</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ D\overline{D} $$  resonance likely related to χ  c0(3860), which is believed to be χ  c0(2P). In addition, there is an indication for a narrow 0++ resonance just below the   <math> <msub> <mi>D</mi> <mi>s</mi> </msub> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {D}_s{\overline{D}}_s $$  threshold with a large coupling to   <math> <msub> <mi>D</mi> <mi>s</mi> </msub> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {D}_s{\overline{D}}_s $$  and a very small coupling to   <math> <mi>D</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ D\overline{D} $$ . This resonance is possibly related to the narrow X(3915)/χ  c0(3930) observed in experiment also just below   <math> <msub> <mi>D</mi> <mi>s</mi> </msub> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {D}_s{\overline{D}}_s $$ . The partial wave l = 2 features a resonance likely related to χ  c2(3930). We work with several assumptions, such as the omission of J/ψω, η  c  η and three-particle channels. Only statistical uncertainties are quantified, while the extrapolations to the physical quark-masses and the continuum limit are challenges for the future.

Slovenia

Sasa Prelovsek

Sara Collins

Daniel Mohler

M. Padmanath

Charmonium-like resonances with JPC = 0++, 2++ in coupled   <math> <mi>D</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{D}\overline{\mathrm{D}} $$ ,   <math> <msub> <mi>D</mi> <mi>s</mi> </msub> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {\mathrm{D}}_{\mathrm{s}}{\overline{\mathrm{D}}}_{\mathrm{s}} $$  scattering on the lattice

In this paper, we summarize our efforts in simulating Yang-Mills theories coupled to matter fields transforming under the fundamental and adjoint representations of the gauge group. In the context of composite Higgs scenarios, gauge theories with mixed representation fields have been suggested to describe the fundamental interactions well beyond the electroweak unification scale, and they are also closely related to supersymmetric QCD. In addition, they are studied as deformations of theories with pure adjoint matter in the context of adiabatic continuity. We provide some first results for bare parameter tuning and interdependence of the two representations. We also investigate how the chiral symmetry breaking or a conformal scenario can be realized and checked in such theories.

Lattice simulations of a gauge theory with mixed adjoint-fundamental matter

We determine the magnetic susceptibility of thermal QCD matter by means of first principles lattice simulations using staggered quarks with physical masses. A novel method is employed that only requires simulations at zero background field, thereby circumventing problems related to magnetic flux quantization. After a careful continuum limit extrapolation, diamagnetic behavior (negative susceptibility) is found at low temperatures and strong paramagnetism (positive susceptibility) at high temperatures. We revisit the decomposition of the magnetic susceptibility into spin- and orbital angular momentum- related contributions. The spin term — related to the normalization of the photon lightcone distribution amplitude at zero temperature — is calculated non-perturbatively and extrapolated to the continuum limit. Having access to both the full magnetic susceptibility and the spin term, we calculate the orbital angular momentum contribution for the first time. The results reveal the opposite of what might be expected based on a free fermion picture. We provide a simple parametrization of the temperature- and magnetic field-dependence of the QCD equation of state that can be used in phenomenological studies.

India

Gunnar Bali

Gergely Endrődi

Magnetic susceptibility of QCD matter and its decomposition from the lattice

Supersymmetric Yang-Mills (SYM) theories in four dimensions exhibit many interesting nonperturbative phenomena that can be studied by means of Monte Carlo lattice simulations. However, the lattice regularization breaks supersymmetry explicitly, and in general a fine-tuning of a large number of parameters is required to correctly extrapolate the theory to the continuum limit. From this perspective, it is important to preserve on the lattice as many symmetries of the original continuum action as possible. Chiral symmetry for instance prevents an additive renormalization of the fermion mass. A (modified) version of chiral symmetry can be preserved exactly if the Dirac operator fulfills the Ginsparg-Wilson relation. In this contribution, we present an exploratory nonperturbative study of <math><mi>N</mi><mo>=</mo><mn>1</mn></math> supersymmetric Yang-Mills theory using the overlap formalism to preserve chiral symmetry at nonzero lattice spacings. <math><mi>N</mi><mo>=</mo><mn>1</mn></math> SYM is an ideal benchmark toward the extension of our studies to more complex supersymmetric theories, as the only parameter to be tuned is the gluino mass. Overlap fermions allow us therefore to simulate the theory without fine-tuning. We compare our approach to previous investigations of the same theory, and we present clear evidences for gluino condensation.

Monte Carlo simulations of overlap Majorana fermions

In $${\mathcal {N}}=1$$ <math><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow></math>  supersymmetric Yang–Mills theory, regularised on a space-time lattice, in addition to the breaking by the gluino mass term, supersymmetry is broken explicitly by the lattice regulator. In addition to the parameter tuning in the theory, the supersymmetric Ward identities can be used as a tool to investigate lattice artefacts as well as to check whether supersymmetry can be recovered in the chiral and continuum limits. In this paper we present the numerical results of an analysis of the supersymmetric Ward identities for our available gauge ensembles at different values of the inverse gauge coupling $$\beta $$ <math><mi>β</mi></math>  and of the hopping parameter $$\kappa $$ <math><mi>κ</mi></math> . The results clearly indicate that the lattice artefacts vanish in the continuum limit, confirming the restoration of supersymmetry.

Henning Gerber

Philipp Scior

Continuum extrapolation of Ward identities in $${{\mathcal {N}}=1}$$ <math><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow></math>  supersymmetric SU(3) Yang–Mills theory

We present a lattice QCD study of charmonium resonances and bound states with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> and <math><msup><mn>3</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> near the open-charm threshold, taking into account their strong transitions to <math><mover><mi>D</mi><mo>¯</mo></mover><mi>D</mi></math>. Vector charmonia are the most abundant in the experimentally established charmonium spectrum, while recently LHCb reported also the first discovery of a charmonium with likely spin three. The <math><mover><mi>D</mi><mo>¯</mo></mover><mi>D</mi></math> scattering amplitudes for partial waves <math><mi>l</mi><mo>=</mo><mn>1</mn></math> and <math><mi>l</mi><mo>=</mo><mn>3</mn></math> are extracted on the lattice by means of the Lüscher formalism, using multiple volumes and inertial frames. Parametrizations of the scattering amplitudes provide masses and widths of the resonances, as well as the masses of bound states. CLS ensembles with <math><mn>2</mn><mo>+</mo><mn>1</mn></math> dynamical flavors of nonperturbatively <math><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></math> improved Wilson quarks are employed with <math><msub><mi>m</mi><mi>π</mi></msub><mo>≃</mo><mn>280</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, a single lattice spacing of <math><mi>a</mi><mo>=</mo><mn>0.09</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and two lattice spatial extents of <math><mi>L</mi><mo>=</mo><mn>24</mn></math> and 32. Two values of the charm quark mass are considered to examine the influence of the position of the <math><mover><mi>D</mi><mo>¯</mo></mover><mi>D</mi></math> threshold on the hadron masses. For the lighter charm quark mass we find the vector resonance <math><mi>ψ</mi><mo>(</mo><mn>3770</mn><mo>)</mo></math> with mass <math><mi>m</mi><mo>=</mo><mn>3780</mn><mo>(</mo><mn>7</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and coupling <math><mi>g</mi><mo>=</mo><mn>16.0</mn><msubsup><mo>(</mo><mrow><mo>−</mo><mn>0.2</mn></mrow><mrow><mo>+</mo><mn>2.1</mn></mrow></msubsup><mo>)</mo></math> (related to the width by <math><mi>Γ</mi><mo>=</mo><msup><mi>g</mi><mn>2</mn></msup><msup><mi>p</mi><mn>3</mn></msup><mo>/</mo><mn>6</mn><mi>π</mi><msup><mi>m</mi><mn>2</mn></msup></math>). Both quantities are consistent with their experimental values, <math><msup><mi>m</mi><mi>exp</mi></msup><mo>=</mo><mn>3773.13</mn><mo>(</mo><mn>35</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math><msup><mi>g</mi><mi>exp</mi></msup><mo>=</mo><mn>18.7</mn><mo>(</mo><mn>9</mn><mo>)</mo></math>. The vector <math><mi>ψ</mi><mo>(</mo><mn>2</mn><mi>S</mi><mo>)</mo></math> appears as a bound state with <math><mi>m</mi><mo>=</mo><mn>3666</mn><mo>(</mo><mn>10</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. The charmonium resonance with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>3</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> is found at <math><mrow><mi>m</mi><mo>=</mo><mn>3831</mn><msubsup><mrow><mo>(</mo></mrow><mrow><mo>−</mo><mn>16</mn></mrow><mrow><mo>+</mo><mn>10</mn></mrow></msubsup><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, consistent with the <math><mi>X</mi><mo>(</mo><mn>3842</mn><mo>)</mo></math> recently discovered by LHCb. At our heavier charm-quark mass the <math><mi>ψ</mi><mo>(</mo><mn>2</mn><mi>S</mi><mo>)</mo></math> as well as the <math><mi>ψ</mi><mo>(</mo><mn>3770</mn><mo>)</mo></math> are bound states and the <math><mi>X</mi><mo>(</mo><mn>3842</mn><mo>)</mo></math> remains a resonance. We stress that all quoted uncertainties are only statistical, while lattice spacing effects and the approach to the physical point (for the light and strange quarks) still need to be explored. This study of conventional charmonia sets the stage for more challenging future studies of unconventional charmoniumlike states.

Charmonium resonances with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> and <math><msup><mn>3</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> from <math><mover><mi>D</mi><mo>¯</mo></mover><mi>D</mi></math> scattering on the lattice

The realization of center and chiral symmetries in <math><mi>N</mi><mo>=</mo><mn>1</mn></math> super Yang-Mills theory (SYM) is investigated on a four-dimensional Euclidean lattice by means of Monte Carlo methods. At zero temperature this theory is expected to confine external fundamental charges and to have a nonvanishing gaugino condensate, which breaks the nonanomalous <math><msub><mi>Z</mi><mrow><mn>2</mn><msub><mi>N</mi><mi>c</mi></msub></mrow></msub></math> chiral symmetry. In this work we find, for the first time, a nonvanishing condensate at zero temperature and zero gaugino mass with Wilson fermions. This is achieved by means of the renormalization properties of the Yang-Mills and fermion gradient flows, which are independent of the lattice regularization. At finite temperatures, we find that the phase transitions corresponding to deconfinement and chiral restoration occur at roughly the same critical temperature for SU(2) gauge group, implying the saturation of the bound <math><msubsup><mi>T</mi><mtext>chiral</mtext><mi>c</mi></msubsup><mo>≤</mo><msubsup><mi>T</mi><mi>dec</mi><mi>c</mi></msubsup></math>, recently obtained through anomaly matching. We furthermore discuss the agreement of our findings with conjectures from superstring theory.