Adjoint Quarks Research Articles

(1+1)D QCD with heavy adjoint quarks

In this paper, we determine at weak coupling the non-relativistic $n$-body Schr\"{o}dinger equation that describes the low-lying color singlet bound states of two dimensional adjoint $QCD$ with heavy quarks. In the case of three adjoint quarks, we show that the three-body equation reduces equivalently to the Schr\"{o}dinger equation that describes a point electric dipole in an electric field in a plane angular sector. We conjecture that the three-body problem is solvable. The eigenstates are given in terms of the triconfluent Heun functions. Our conjecture implies that a bound state of three adjoint quarks is described by a particle confined in a two dimensional Cornell potential. We expect the $n$-parton problem also to be solvable in a similar fashion.

Axial Anomaly in SU(N) Yang-Mills Matrix Models.

The SU(N) Yang-Mills matrix model admits self-dual and anti-self-dual instantons. When coupled to N_{f} flavors of massless quarks, the Euclidean Dirac equation in an instanton background has n_{+} positive and n_{-} negative chirality zero modes. The vacua of the gauge theory are N-dimensional representations of SU(2), and the (anti-) self-dual instantons tunnel between two commuting representations, the initial one composed of r_{0}^{(1)} irreps and the final one with r_{0}^{(2)} irreps. We show that the index (n_{+}-n_{-}) in such a background is equal to a new instanton charge T_{new}=±[r_{0}^{(2)}-r_{0}^{(1)}]. Thus T_{new}=(n_{+}-n_{-}) is the matrix model version of the Atiyah-Singer index theorem. Further, we show that the path integral measure is not invariant under a chiral rotation, and relate the noninvariance of the measure to the index of the Dirac operator. Axial symmetry is broken anomalously, with the residual symmetry being a finite group. For N_{f} fundamental fermions, this residual symmetry is Z_{2N_{f}}, whereas for adjoint quarks it is Z_{4N_{f}}.

Symmetries and strings of adjoint QCD2

We revisit the symmetries of massless two-dimensional adjoint QCD with gauge group SU(N). The dynamics is not sufficiently constrained by the ordinary symmetries and anomalies. Here we show that the theory in fact admits ∼ 22N non-invertible symmetries which severely constrain the possible infrared phases and massive excitations. We prove that for all N these new symmetries enforce deconfinement of the fundamental quark. When the adjoint quark has a small mass, m ≪ gYM, the theory confines and the non-invertible symmetries are softly broken. We use them to compute analytically the k-string tension for N ≤ 5. Our results suggest that the k-string tension, Tk, is Tk ∼ |m| sin(πk/N) for all N. We also consider the dynamics of adjoint QCD deformed by symmetric quartic fermion interactions. These operators are not generated by the RG flow due to the non-invertible symmetries, thus violating the ordinary notion of naturalness. We conjecture partial confinement for the deformed theory by these four-fermion interactions, and prove it for SU(N ≤ 5) gauge theory. Comparing the topological phases at zero and large mass, we find that a massless particle ought to appear on the string for some intermediate nonzero mass, consistent with an emergent supersymmetry at nonzero mass. We also study the possible infrared phases of adjoint QCD allowed by the non-invertible symmetries, which we are able to do exhaustively for small values of N. The paper contains detailed reviews of ideas from fusion category theory that are essential for the results we prove.

Continuum Goldstone spectrum of two-color QCD at finite density with staggered quarks

We carry out lattice simulations of two-color QCD and spectroscopy at finite density with two flavors of rooted-staggered quarks and a diquark source term. As in a previous four-flavor study, for small values of the inverse gauge coupling we observe a Goldstone spectrum which reflects the symmetry-breaking pattern of a Gaussian symplectic chiral random-matrix ensemble (GSE) with Dyson index $\beta_D=4$, which corresponds to any-color QCD with adjoint quarks in the continuum instead of QC$_2$D wih fundamental quarks. We show that this unphysical behavior occurs only inside of the bulk phase of $SU(2)$ gauge theory, where the density of $Z_2$ monopoles is high. Using an improved gauge action and a somewhat larger inverse coupling to suppress these monopoles, we demonstrate that the continuum Goldstone spectrum of two-color QCD, corresponding to a Gaussian orthogonal ensemble (GOE) with Dyson index $\beta_D=1$, is recovered also with rooted-staggered quarks once simulations are performed away from the bulk phase. We further demonstrate how this change of random-matrix ensemble is reflected in the distribution of eigenvalues of the Dirac operator. By computing the unfolded level spacings inside and outside of the bulk phase, we demonstrate that, starting with the low-lying eigenmodes which determine the infrared physics, the distribution of eigenmodes continuously changes from the GSE to the GOE one as monopoles are suppressed.

QCD β -function on the string worldsheet

We consider confining strings in pure gluodynamics and its extensions with adjoint (s)quarks. We argue that there is a direct map between the set of bulk fields and the worldsheet degrees of freedom. This suggests a close link between the worldsheet $S$-matrix and parton scattering amplitudes. We report an amusing relation between the Polchinski--Strominger amplitude responsible for the breakdown of integrability on the string worldsheet and the Yang--Mills $\ensuremath{\beta}$-function ${b}_{0}=\frac{{D}_{cr}\ensuremath{-}{D}_{ph}}{6}.$ Here ${b}_{0}=11/3$ is the one-loop $\ensuremath{\beta}$-function coefficient in the pure Yang--Mills theory, ${D}_{cr}=26$ is the critical dimension of bosonic strings and ${D}_{ph}=4$ is the dimensionality of the physical space-time we live in. A natural extension of this relation continues to hold in the presence of adjoint (s)quarks, connecting two of the most celebrated anomalies---the scale anomaly in quantum chromodynamics (QCD) and the Weyl anomaly in string theory.

Light adjoint quarks in the instanton-dyon liquid model. IV.

We discuss the instanton-dyon liquid model with $N_f$ Majorana quark flavors in the adjoint representation of color $SU_c(2)$ at finite temperature. We briefly recall the index theorem on $S^1\times R^3$ for twisted adjoint fermions in a BPS dyon background of arbitrary holonomy, and use the ADHM construction to explicit the adjoint anti-periodic zero modes. We use these results to derive the partition function of an interacting instanton-dyon ensemble with $N_f$ light and anti-periodic adjoint quarks. We develop the model in details by mapping the theory on a 3-dimensional quantum effective theory with adjoint quarks with manifest $SU(N_f)\times Z_{4N_f}$ symmetry. Using a mean-field analysis at weak coupling and strong screening, we show that center symmetry requires the spontaneous breaking of chiral symmetry, which is shown to only take place for $N_f=1$. For a sufficiently dense liquid, we find that the ground state is center symmetric and breaks spontaneously flavor symmetry through $SU(N_f)\times Z_{4N_f}\rightarrow O(N_f)$. As the liquid dilutes with increasing temperature, center symmetry and chiral symmetry are restored. We present numerical and analytical estimates for the transition temperatures.

On the definition and interpretation of a static quark-antiquark potential in the color-adjoint channel

We study possibilities to define a static quark-antiquark pair in a color-adjoint orientation based on Wilson loops with generator insertions, using both lattice QCD and leading-order perturbation theory in various gauges. Nonperturbatively, the only way to obtain nonzero results while maintaining positivity of the Hamiltonian is by using either temporal gauge or Coulomb gauge with some additional constraint on the temporal links removing the residual gauge symmetry. In this case the correlator is equivalent to a gauge invariant correlation function of a static quark-antiquark pair and a static adjoint quark; the resulting three-point potential is attractive. Saturating open color indices with color magnetic fields instead also leads to a gauge invariant correlator. However, this object is found to couple to the singlet sector only and results in a hybrid potential. None of the considered lattice observables reproduces the repulsive adjoint potential predicted by perturbation theory in Lorenz or Coulomb gauges.

Infrared conformality and bulk critical points: SU(2) with heavy adjoint quarks

Abstract The lattice phase structure of a gauge theory can be a serious obstruction to Monte Carlo studies of its continuum behaviour. This issue is particularly delicate when numerical studies are performed to determine whether a theory is in a (near-)conformal phase. In this work we investigate the heavy mass limit of the SU(2) gauge theory with N f = 2 adjoint fermions and its lattice phase diagram, showing the presence of a critical point ending a line of first order bulk phase transition. The relevant gauge observables and the low-lying spectrum are monitored in the vicinity of the critical point with very good control over different systematic effects. The scaling properties of masses and susceptibilities open the possibility that the effective theory at criticality is a scalar theory in the universality class of the four-dimensional Gaussian model. This behaviour is clearly different from what is observed for SU(2) gauge theory with two dynamical adjoint fermions, whose (near-)conformal numerical signature is hence free from strong-coupling bulk effects.

Perturbative calculation of the clover term for Wilson fermions in any representation of the gauge group SU(N)

We calculate the Sheikholeslami-Wohlert coefficient of the O(a) improvement term for Wilson fermions in any representation of the gauge group SU(N) perturbatively at the one-loop level. The result applies to QCD with adjoint quarks and to N=1 supersymmetric Yang-Mills theory on the lattice.

QCD-like theories at nonzero temperature and density

We investigate the properties of hot and/or dense matter in QCD-like theories with quarks in a (pseudo)real representation of the gauge group using the Nambu-Jona-Lasinio model. The gauge dynamics is modeled using a simple lattice spin model with nearest-neighbor interactions. We first keep our discussion as general as possible, and only later focus on theories with adjoint quarks of two or three colors. Calculating the phase diagram in the plane of temperature and quark chemical potential, it is qualitatively confirmed that the critical temperature of the chiral phase transition is much higher than the deconfinement transition temperature. At a chemical potential equal to half of the diquark mass in the vacuum, a diquark Bose-Einstein condensation (BEC) phase transition occurs. In the two-color case, a Ginzburg-Landau expansion is used to study the tetracritical behavior around the intersection point of the deconfinement and BEC transition lines, which are both of second order. We obtain a compact expression for the expectation value of the Polyakov loop in an arbitrary representation of the gauge group (for any number of colors), which allows us to study Casimir scaling at both nonzero temperature and chemical potential.

Low-energy limit of Yang–Mills with massless adjoint quarks: chiral Lagrangian and Skyrmions

If the fundamental quarks of QCD are replaced by massless adjoint quarks, the pattern of the chiral symmetry breaking drastically changes compared to the standard one. It becomes SU(Nf) → SO(Nf). While for Nf = 2, the chiral Lagrangian describing the ‘pion’ dynamics is well known, this is not the case at Nf > 2. We outline a general strategy for deriving chiral Lagrangians for the coset spaces and study in detail the case of Nf = k = 3. We obtain two- and four-derivatives terms in the chiral Lagrangian on the coset space = SU(3)/SO(3), as well as the Wess–Zumino–Novikov–Witten term, in terms of an explicit parameterization of the quotient manifold. Then we discuss stable topological solitons supported by this Lagrangian. Aspects of relevant topological considerations scattered in the literature are reviewed. The same analysis applies to SO(N) gauge theories with Nf Weyl flavours in the vector representation.

Scaling and Goldstone effects in a QCD with two flavours of adjoint quarks

We study QCD with two Dirac fermions in the adjoint representation at finite temperature by Monte Carlo simulations. In such a theory the deconfinement and chiral phase transitions occur at different temperatures. We locate the second-order chiral transition point at β c = 5.624 ( 2 ) and show that the scaling behaviour of the chiral condensate in the vicinity of β c is in full agreement with that of the 3d O ( 2 ) universality class, and to a smaller extent comparable to the 3d O ( 6 ) class. From the previously determined first-order deconfinement transition point β d = 5.236 ( 3 ) and the two-loop beta function we find the ratio T c / T d ≈ 7.8 ( 2 ) . In the region between the two phase transitions we explicitly confirm the quark mass dependence of the chiral condensate which is expected due to the existence of Goldstone modes like in 3d O ( N ) spin models. At the deconfinement transition 〈 ψ ¯ ψ 〉 shows a gap, and below β d , it is nearly mass-independent for fixed β.

Relation between the Polyakov loop and the chiral order parameter at strong coupling

We discuss the relation between the Polyakov loop and the chiral order parameter at finite temperature by using the Gocksch-Ogilvie model with fundamental or adjoint quarks. The model is based on the double expansion of strong coupling and large dimensionality on the lattice. In an analytic way, using the mean field approximation, we show that the confined phase must be accompanied by the spontaneous breaking of chiral symmetry for both fundamental and adjoint quarks. Then we proceed to numerical analyses to look into the coupled dynamics of the Polyakov loop and the chiral order parameter. In the case of fundamental quarks, the pseudocritical temperature inferred from the Polyakov loop behavior turns out to coincide with the pseudocritical temperature of the chiral phase transition. We discuss the physical implication of the coincidence of the pseudocritical temperatures in two extreme cases: one is deconfinement dominance and the other is chiral dominance. As for adjoint quarks, the deconfinement transition of first order persists and the chiral phase transition occurs at distinctly higher temperature than the deconfinement transition. The present model study gives us a plausible picture to understand the results from lattice QCD and aQCD simulations.

Effects of Chiral Restoration on the Behavior of the Polyakov Loop

We discuss the relation between the Polyakov loop and the chiral order parameter at finite temperature by using the Gocksch-Ogilvie model with either fundamental or adjoint quarks. For fundamental quarks, the pseudo-critical temperature read from the Polyakov loop behavior turns out to coincide with the pseudo-critical temperature of the chiral phase transition. As for adjoint quarks, it is found that the chiral phase transition occurs distinctly at higher temperature than the deconfinement transition does.

Higgs description of two-flavor QCD vacuum

A Higgs description for the QCD vacuum in the limit of two quark flavors is proposed. Complete “spontaneous breaking” of color symmetry is realized by condensation of an adjoint quark–antiquark and a quark–quark pair. The vacuum is invariant under isospin symmetry applying color and flavor transformations simultaneously. All elementary excitations acquire integral charges. The dressed microscopic quark and gluon degrees of freedom can be identified with the macroscopic baryon and vector meson degrees of freedom. In addition to the nucleons we find soliton-type fermions with integer isospin. Their presence provides an important test accessible to lattice QCD simulations.

Massive random matrix ensembles atβ=1and 4: QCD in three dimensions

The zero momentum sectors in effective theories of three dimensional QCD coupled to pseudoreal (two colors) and real (adjoint) quarks in a classically parity-invariant manner have alternative descriptions in terms of orthogonal and symplectic ensembles of random matrices. Using this correspondence, we compute finite-volume QCD partition functions and correlation functions of Dirac operator eigenvalues in a presence of finite quark masses of the order of the smallest Dirac eigenvalue. These novel correlation functions, expressed in terms of quaternion determinants, are reduced to conventional results for the Gaussian ensembles in the quenched limit.

“Gluelump” spectrum and adjoint source potential in lattice QCD 3

We calculate the potential between “quarks” which are in the adjoint representation of SU(2) color in the three-dimensional lattice theory. We work in the scaling region of the theory and at large quark separations R. We also calculate the masses M Qg of color-singlet bound states formed by coupling an adjoint quark to adjoint glue (“gluelumps”). Good scaling behavior is found for the masses of both magnetic (angular momentum J = 0) and electric ( J = 1) gluelumps, and the magnetic gluelump is found to be the lowest-lying state. It is naively expected that the potential for adjoint quarks should saturate above a separation R scr where it becomes energetically favorable to produce a pair of gluelumps. We obtain a good estimate of the naive screening distance R scr. However we find little evidence of saturation in the potential out to separations R of about 1.5 R scr.

Abelian dominance and adjoint sources in lattice QCD.

Certain properties of maximal abelian projection are derived which suggest that the fundamental and adjoint SU(2) string tensions are reproduced by singly and doubly charged abelian Wilson loops, respectively. Thus, abelian dominance, which has been observed for color sources (quarks) in the fundamental representation, can be extended to higher representations. Numerical evidence in support of this conjecture also for adjoint quarks is presented. The difference between maximal abelian and local projections is elucidated and the role of non-Wilson-like terms in the effective abelian action is discussed.

Adjoint “quarks” and the physics of confinement

The quark-antiquark potential and the chromoelectric fields generated by ``quarks'' in the adjoint representation of SU(2) color are analyzed in the scaling region of the theory. New results with interesting implications for our understanding of the confinement mechanism are presented. In particular, the formation of color-electric flux-tubes between adjoint quarks is demonstrated in four-dimensional SU(2) color. The flux-tubes for fundamental and adjoint representation quarks are shown to have very similar cross-sections. This result could imply that the QCD vacuum is dual to a type I superconductor.

Adjoint “quark” color fields in four-dimensional lattice gauge theory: Vacuum screening and penetration

The fields generated by “quarks” in the adjoint representation of SU(2) color are analyzed in the scaling region of the four-dimensional lattice theory. Evidence of vacuum screening of adjoint quarks is obtained from a comparison of quark-antiquark ( Q Q ) flux-tubes for quarks in the adjoint (“isospin” j = 1) and fundamental ( j = 1 2 ) representations. The component ε j of the color-electric field strength in the direction parallel to the Q Q axis is calculated. Near the quarks the ratio of fields ε j=1 ε j= 1 2 approaches the value 8 3 , which is equal to the ratio of SU(2) Casimirs. In between the quarks, the ratio falls well below 8 3 at large R. ε j also falls off rapidly as a function of distance x ⊥ perpendicular to the Q Q axis. However, the ratio ε j=1 ε j= 1 2 depends very weakly on x ⊥. The flux-tubes in the two representations thus appear to have very similar cross-sections. This result could imply that the QCD vacuum is dual to a type I superconductor.