We review arguments that chiral symmetry breaking is triggered when the quark bilinear condensate's dimension passes through one ($γ=1$). This is supported by gap equations and more recently holographic models. Confinement may then be a separate property of the pure Yang-Mills theory below the scale of the dynamically generated quark mass, occurring at the scale of the pole in the deep IR running. Here, we use perturbative results for the running of the gauge coupling and $γ$ in asymptotically free SU($N_c$) gauge theories with matter in higher dimension representations to seek the best candidate theories where confinement and chiral symmetry breaking can be maximally separated. For example, SU(2) gauge theory with a single Weyl quark in the $S_3$ (dimension 4) representation may have a factor of 20 separation in scale. Such a theory could be simulated on the lattice to test the separation. We also propose studying multi-representation theories where the higher dimension representation forms a condensate at one scale that can be quite separate from the condensation scale of the second representation matter. The confinement scale would presumably be below the second scale. For example, SU(3) gauge theory with a Weyl adjoint fermion and ten fundamental quarks may have a separation of a factor of 20 also.
We analytically study the relation between quark confinement and spontaneous chiral-symmetry breaking in QCD. In terms of the Dirac eigenmodes, we derive some formulae for the Polyakov loop, its fluctuations, and the string tension from the Wilson loop. We also investigate the Polyakov loop in terms of the eigenmodes of the Wilson, the clover and the domain wall fermion kernels, respectively. For the confinement quantities, the low-lying Dirac/fermion eigenmodes are found to give negligible contribution, while they are essential for chiral symmetry breaking. These relations indicate no direct …
Gauge-independent transition separating confinement and Higgs phases in lattice SU(2) gauge theory with a scalar field in the fundamental representation
Using the eigen-mode of the QCD Dirac operator $\Slash D=γ^μD^μ$, we develop a manifestly gauge-covariant expansion and projection of the QCD operators such as the Wilson loop and the Polyakov loop. With this method, we perform a direct analysis of the correlation between confinement and chiral symmetry breaking in lattice QCD Monte Carlo calculations. Even after removing the low-lying Dirac modes, which are responsible to chiral symmetry breaking, we find that the Wilson loop obeys the area law, and the string tension or the confinement force is almost unchanged. We find also that the Polyako…
We develop a manifestly gauge-covariant expansion and projection using the eigen-mode of the QCD Dirac operator. Applying this method to the Wilson loop and the Polyakov loop, we perform a direct analysis of the correlation between confinement and chiral symmetry breaking in SU(3) lattice QCD calculation on 6^4 at beta=5.6 at the quenched level. Notably, the Wilson loop is found to obey the area law, and the slope parameter corresponding to the string tension or the confinement force is almost unchanged, even after removing the low-lying Dirac modes, which are responsible to chiral symmetry br…
Using the eigen-mode of the QCD Dirac operator $\Slash D=γ^μD^μ$, we develop a manifestly gauge-covariant expansion and projection of the QCD operators such as the Wilson loop and the Polyakov loop. With this method, we perform a direct analysis of the correlation between confinement and chiral symmetry breaking in lattice QCD Monte Carlo calculation on $6^4$ at $β$=5.6. Even after removing the low-lying Dirac modes, which are responsible to chiral symmetry breaking, we find that the Wilson loop obeys the area law, and the slope parameter corresponding to the string tension or the confinement …
We construct an effective Lagrangian which illustrates why color deconfines when chiral symmetry is restored in hot gauge theories with quarks in the fundamental representation. For quarks in the adjoint representation we show that while deconfinement and the chiral transition do not need to coincide, entanglement between them is still present. Extension to the chemical potential driven transition is discussed.
Chiral dynamics is investigated using the chiral confining Lagrangian (CCL), previously derived from QCD with confinement interaction. Based on the calculations of the quark condensate, which is defined entirely by confinement in the zero quark mass limit, one can assert that chiral symmetry breaking occurs not spontaneously but is predetermined by confinement. The expansion of the CCL provides the GMOR relations and the effective chiral Lagrangian, which is used to provide the masses and decay constants of all chiral mesons, including $η,η'$. For the latter one needs to define a non-chiral co…
In this paper we study the properties of QCD at nonzero chiral density $ρ_5$, which is introduced through chiral chemical potential $μ_5$. The study is performed within lattice simulation of QCD with dynamical rooted staggered fermions. We first check that $ρ_5$ is generated at nonzero $μ_5$ and in the chiral limit observe $ρ_5 \sim Λ_{QCD}^2 μ_5$. We also test the possible connection between confinement and topological fluctuations. To this end, we measured the topological susceptibility $χ_{\mbox{\footnotesize top}}$ and string tension $σ$ for various values of $μ_5$. We observed that both s…
Confinement without symmetry breaking in chiral gauge theories
School of Physics, Sun Yat-Sen University（中山大学物理学院）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学中心（iTHEMS））；Centre for Cosmology, Particle Physics and Phenomenology (CP3), Université catholique de Louvain（天主教鲁汶大学宇宙学、粒子物理与现象学中心（CP3））
We analyze the vacuum structure of SU(2) QCD with multiple massless adjoint representation fermions formulated on a small spatial $S^1 \times \R^3$. The absence of thermal fluctuations, and the fact that quantum fluctuations favoring the vacuum with unbroken center symmetry in a weakly coupled regime renders the interesting dynamics of these theories analytically calculable. Confinement, the area law behavior for large Wilson loops, and the generation of the mass gap in the gluonic sector are shown analytically. By abelian duality transformation, the long distance effective theory of QCD is ma…
Two-dimensional QED with $N$ flavor fermions is solved at zero and finite temperature with arbitrary fermion masses to explore QCD physics such as chiral condensate and string tension. The problem is reduced to solving a Schrödinger equation for $N$ degrees of freedom with a specific potential determined by the ground state of the Schrödinger problem itself.
We study the relation between quark confinement and chiral symmetry breaking in QCD. Using lattice QCD formalism, we analytically express the various "confinement indicators", such as the Polyakov loop, its fluctuations, the Wilson loop, the inter-quark potential and the string tension, in terms of the Dirac eigenmodes. In the Dirac spectral representation, there appears a power of the Dirac eigenvalue $λ_n$ such as $λ_n^{N_t-1}$, which behaves as a reduction factor for small $λ_n$. Consequently, since this reduction factor cannot be cancelled, the low-lying Dirac eigenmodes give negligibly sm…
Inspired by the construction of the Gribov-Zwanziger action in the Landau gauge, we introduce a quark model exhibiting both confinement and chiral symmetry aspects. An important feature is the incorporation of spontaneous chiral symmetry breaking in a renormalizable fashion. The quark propagator in the condensed vacuum turns out to be of a confining type. Besides a real pole, it exhibits complex conjugate poles. The resulting spectral form is explicitly shown to violate positivity, indicative of its unphysical character. Moreover, the ensuing quark mass function fits well to existing lattice d…
Employing functional approaches the infrared behaviour of Landau gauge QCD vertex functions is investigated. Results for the ghost-gluon, three-gluon and quark-gluon vertex functions are presented. As can be analytically shown a linear rising potential between heavy quarks is generated by infrared singularities in the dressed quark-gluon vertex. The selfconsistent mechanism that generates these singularities implies the existence of scalar Dirac amplitudes of the full vertex and the quark propagator. These amplitudes can only be present when chiral symmetry is broken, either explicitly or dyna…
Two crucial properties of QCD, confinement and chiral symmetry breaking, cannot be understood within the context of conventional Feynman perturbation theory. Non-perturbative phenomena enter the theory in a fundamental way at both the classical and quantum level. Over the years a coherent qualitative picture of the interplay between chiral symmetry, quantum mechanical anomalies, and the lattice has emerged and is reviewed here.
A self-consistent holographic QCD model is proposed which can realize both chiral symmetry breaking and confinement, two most important phenomena of QCD. It is pointed out that the model can accommodate both Regge spectra of hadrons and the linear potential between quarks. The model is formulated in the framework of graviton-dilaton-scalar system, where the dilaton field is dual to the dimension-2 gluon condensate and can lead to the linear confinement, while the scalar field corresponds to the quark anti-quark condensate and can explain the property of chiral dynamics.
The infrared behavior of the quark-gluon vertex of quenched Landau gauge QCD is studied by analyzing its Dyson-Schwinger equation. Building on previously obtained results for Green functions in the Yang-Mills sector we analytically derive the existence of power-law infrared singularities for this vertex. We establish that dynamical chiral symmetry breaking leads to the self-consistent generation of components of the quark-gluon vertex forbidden when chiral symmetry is forced to stay in the Wigner-Weyl mode. In the latter case the running strong coupling assumes an infrared fixed point. If chir…
In the lattice QCD formalism, we derive a gauge-invariant analytical relation connecting the Polyakov loop and the Dirac modes on a temporally odd-number lattice, where the temporal lattice size is odd, with the normal (nontwisted) periodic boundary condition. This analytical relation indicates that low-lying Dirac modes have little contribution to the Polyakov loop. Using lattice QCD simulations, we numerically confirm the analytical relation and the negligible contribution of low-lying Dirac modes to the Polyakov loop at the quenched level, i.e., the Polyakov loop is almost unchanged by remo…
We report our recent progress on the search of Chiral Magnetic Effect (CME) by developing new measurements as well as by hydrodynamic simulations of CME and background effects, with both approaches addressing the pressing issue of separating flow-driven background contributions and possible CME signal in current heavy ion collision measurements.
We investigate the contribution from each Dirac modes to the Polyakov loop based on a gauge-invariant analytical relation connecting the Polyakov loop and the Dirac modes on a temporally odd-number lattice, where the temporal lattice size is odd, with the normal (nontwisted) periodic boundary condition. The dumping factor in the relation plays crucial role for the negligible contribution of low-lying Dirac modes to the Polyakov loop. The zero-value of the Polyakov loop in the confinement phase is due to the ``positive/negative symmetry'' of the Dirac-mode contribution to the Polyakov loop. In …
We demonstrate that $SO(N_{c})$ gauge theories with matter fields in the vector representation confine due to monopole condensation and break the $SU(N_{F})$ chiral symmetry to $SO(N_{F})$ via the quark bilinear. Our results are obtained by perturbing the ${\cal N}=1$ supersymmetric theory with anomaly-mediated supersymmetry breaking.
We study the QCD phase diagram, in particular we study the critical points of the two main QCD phase transitions, confinement and chiral symmetry breaking. Confinement drives chiral symmetry breaking, and, due to the finite quark mass, at small density both transitions are a crossover, while they are a first or second order phase transition in large density. We study the QCD phase diagram with a quark potential model including both confinement and chiral symmetry. This formalism, in the Coulomb gauge hamiltonian formalism of QCD, is presently the only one able to microscopically include both a…
We study the relation between quark confinement and spontaneous chiral-symmetry breaking directly in QCD. In lattice QCD formalism, we derive an analytical gauge-invariant relation between the Polyakov loop $\langle L_P \rangle$ and the Dirac eigenvalues $λ_n$, i.e., $\langle L_P \rangle \propto \sum_n λ_n^{N_t -1} \langle n|\hat U_4|n \rangle$, on a temporally odd-number lattice, where the temporal lattice size $N_t$ is odd. Here, $|n \rangle$ denotes the Dirac eigenmode, i.e., $\not D|n \rangle=iλ_n|n \rangle$, and $\hat U_4$ the temporal link-variable operator. We here use an ordinary squar…
We study confinement and chiral symmetry breaking phase transitions for QCD and supersymmetric QCD. The method is based on some justified assumption regarding the behavior of the Green function relevant for the potential between two charges or a gap equation. We determine the number of flavors at which the transitions to confinement and chiral symmetry breaking take place. We find that whereas for chiral symmetry breaking the results are in perfect agreement with those in the literature for confinement the situation is more complex.
The chirally symmetric overlap quark propagator is explored in Coulomb gauge for the first time. This gauge is especially well suited for studying the interrelation between confinement and chiral symmetry breaking, since confinement can be attributed to the infrared divergence of the vector dressing function of the inverse quark propagator. Using quenched gauge field configurations on a $20^4$ lattice, the dressing functions of the quark propagator as well as the dynamical quark mass function are evaluated, also in the chiral limit. Chiral symmetry is artificially restored by removing the low …
We study the QCD phase diagram, in particular we study the critical points of the two main QCD phase transitions, confinement and chiral symmetry breaking. Confinement drives chiral symmetry breaking, and, due to the finite quark mass, at small density both transitions are a crossover, while they are a first or second order phase transition in large density. We study the QCD phase diagram with a quark potential model including both confinement and chiral symmetry. This formalism, in the Coulomb gauge hamiltonian formalism of QCD, is presently the only one able to microscopically include both a…
We study interplay between confinement/deconfinement and chiral properties. We derive some analytical relations of the Dirac modes with the confinement quantities, such as the Polyakov loop, its susceptibility and the string tension. For the confinement quantities, the low-lying Dirac eigenmodes are found to give negligible contribution, while they are essential for chiral symmetry breaking. This indicates no direct, one-to-one correspondence between confinement/deconfinement and chiral properties in QCD. We also investigate the Polyakov loop in terms of the eigenmodes of the Wilson, the clove…
A key question to QCD is what mechanism generates the hadron mass in the light quark sector, where both confinement and chiral symmetry breaking are in the game. Are confinement and chiral symmetry breaking in the vacuum uniquely interconnected? Can hadrons survive chiral symmetry restoration? If yes, what happens with their mass and what symmetries beyond the chiral symmetry are there? We review our recent insights. In particular, in a dynamical lattice simulation we artificially restore chiral symmetry by removing the low-lying Dirac modes of the valence quark propagators, which is a well de…
Gribov's scenario of confinement caused by supercritical charges is described and the present status of Gribov's equation for the Green function of light quarks is discussed.
Spontaneous chiral symmetry breaking is accepted to occur in low energy hadronic physics, resulting in the several successful theorems of PCAC. On the other hand scalar confinement is suggested both by the spectroscopy of hadrons and by the string picture of confinement. However these two evidences are apparently conflicting, because chiral symmetry breaking requires a chiral invariant coupling to the quarks, say a vector coupling like in QCD. Here we reformulate the coupling of the quarks to the string, and we are able to comply with chiral symmetry breaking, using scalar confinement. The res…
We study QCD-like four dimensional theories in the theoretically controlled framework of deformation theory and/or twisted partition function on S*1 x R*3. By using duality, we show that a class of one-flavor theories exhibit new physical phenomena: discrete chiral symmetry breaking induced by the condensation of topological disorder operators, and confinement and the generation of mass gap due to new non-selfdual topological excitations. In the R*4 limit, we argue that the mass gap disappears, the chiral symmetry breaking vacua are of runaway type, and the theory flows to a CFT. We also study…
To clarify the relation between confinement and chiral symmetry breaking in QCD, we consider a temporally odd-number lattice, with the temporal lattice size $N_t$ being odd. We here use an ordinary square lattice with the normal (nontwisted) periodic boundary condition for link-variables in the temporal direction. By considering ${\rm Tr} (\hat{U}_4\hat{\not D}^{N_t-1})$, we analytically derive a gauge-invariant relation between the Polyakov loop $\langle L_P \rangle$ and the Dirac eigenvalues $λ_n$ in QCD, i.e., $\langle L_P \rangle \propto \sum_n λ_n^{N_t -1} \langle n|\hat U_4|n \rangle$, w…
Spectral sums of the Dirac-Wilson operator and their relation to the Polyakov loop are thoroughly investigated. The approach by Gattringer is generalized to mode sums which reconstruct the Polyakov loop locally. This opens the possibility to study the mode sum approximation to the Polyakov loop correlator. The approach is re-derived for the ab initio continuum formulation of Yang-Mills theories, and the convergence of the mode sum is studied in detail. The mode sums are then explicitly calculated for the Schwinger model and SU(2) gauge theory in a homogeneous background field. Using SU(2) latt…
We study the gravitational-wave signal stemming from strongly coupled models featuring both, dark chiral and confinement phase transitions. We therefore identify strongly coupled theories that can feature a first-order phase transition. Employing the Polyakov-Nambu-Jona-Lasinio model, we focus our attention on SU(3) Yang-Mills theories featuring fermions in fundamental, adjoint, and two-index symmetric representations. We discover that for the gravitational-wave signals analysis, there are significant differences between the various representations. Interestingly we also observe that the two-i…
Monopole operators play a central role in 3 dimensional supersymmetric dualities: a careful understanding of their spectrum is necessary to match chiral operators on either sides of a conjectured duality. In Chern-Simons theories ($k\neq0$), monopole operators acquire an electric charge, thus they need to be "dressed" by chiral matter superfields to be made gauge-invariant. Here we present strong evidence that "dressed" monopoles appear in $SU(N)$ chiral theories even for $k=0$ because of mixed CS terms generated along certain Coulomb branch directions. Our analysis is based on the dimensional…
We study the phase diagram of QCD with the help of order parameters for chiral symmetry breaking and quark confinement. We also introduce a new order parameter for the confinement phase transition, which is related to the quark density. It is easily accessible by different theoretical approaches, such as functional approaches or lattice simulations. Its relation to the Polyakov loop expectation value is discussed and the QCD phase diagram is analysed. Our results suggest a close relation between the chiral and the confinement phase transition.
The so-called decoupling and scaling solutions of functional equations of Landau gauge Yang-Mills theory are briefly reviewed. In both types of solutions the positivity violation seen in the gluon propagator is taken as an indication of gluon confinement. In the scaling solution the resulting infrared singularities of the quark-gluon vertex are responsible for the linear potential between static quarks and are therefore signaling quark confinement. A corresponding description of the UA(1) anomaly in functional approaches is only known for the scaling solution. Nevertheless, it seems puzzling a…
In the physical vacuum of QCD, the energy density of light-quark fields strongly coupled to slowly varying gluon fields can be negative, and so a condensate of pairs of quarks and antiquarks of nearly opposite momenta forms which breaks chiral symmetry, confines quarks, and makes gluons massive.
We illustrate why color deconfines when chiral symmetry is restored in gauge theories with quarks in the fundamental representation, and while these transitions do not need to coincide when quarks are in the adjoint representation, entanglement between them is still present.
The mass function of the nonperturbative quark propagator in SU(3) gauge theory shows only a weak dependence on the vortex content of the gauge configurations. Of particular note is the survival of dynamical mass generation on vortex-free configurations having a vanishing string tension. This admits the possibility that mass generation associated with dynamical chiral symmetry breaking persists without confinement. In this presentation, we examine the low-lying ground-state hadron spectrum of the pi, rho, N and Delta and discover that while dynamical mass generation persists in the vortex-free…
The gap equation is a cornerstone in understanding dynamical chiral symmetry breaking and may also provide clues to confinement. A symmetry-preserving truncation of its kernel enables proofs of important results and the development of an efficacious phenomenology. We describe a model of the kernel that yields: a momentum-dependent dressed-quark propagator in fair agreement with quenched lattice-QCD results; and chiral limit values: f_pi= 68 MeV and <q-bar q> = -(190 MeV)^3. It is compared with models inferred from studies of the gauge sector.
We investigate the relevant gluon-momentum region for confinement in lattice QCD on $16^4$ at $β$=5.7, 5.8 and 6.0, based on the Fourier expansion. We find that the string tension $σ$, i.e., the confining force, is almost unchanged even after removing the high-momentum gluon component above 1.5GeV in the Landau gauge. In fact, the confinement property originates from the low-momentum gluon component below 1.5GeV, which is the upper limit to contribute to $σ$. In the relevant region, smaller gluon momentum component is more important for confinement. Next, we develop a manifestly gauge-covarian…
Some of the difficulties arising when one tries to understand confinement as well as dynamical and anomalous chiral symmetry breaking are reviewed. Criteria to be fulfilled by a successful and complete picture of these phenomena are presented, and a few of the suggested explanations are listed.
The center-stabilized multiflavor QCD* theories formulated on R_3 x S_1 exhibit both Abelian and non-Abelian confinement as a function of the S_1 radius, similar to the Seiberg-Witten theory as a function of the mass deformation parameter. For sufficiently small number of flavors and small r(S_1), we show occurence of a mass gap in gauge fluctuations, and linear confinement. This is a regime of confinement without continuous chiral symmetry breaking (χSB). Unlike one-flavor theories where there is no phase transition in r(S_1), the multiflavor theories possess a single phase transition associa…
The Lorentz nature of confinement in a heavy-light quarkonium is investigated. It is demonstrated that an effective scalar interaction is generated selfconsistently as a result of chiral symmetry breaking, and this effective scalar interaction is responsible for the QCD string formation between the quark and the antiquark.
Continuum strong QCD is the application of models and continuum quantum field theory to the study of phenomena in hadronic physics, which includes; e.g., the spectrum of QCD bound states and their interactions. Herein I provide a Dyson-Schwinger equation perspective, focusing on qualitative aspects of confinement and dynamical chiral symmetry breaking in cold, sparse QCD, and also elucidating consequences of the axial-vector Ward-Takahashi identity and features of the heavy-quark limit.
We consider a holographic quark model where the confinement is a consequence of the quark condensate. Surprisingly, the equation of motion of our holographic model can be mapped to the old spin-less bag model. Both models correctly reproduce the linear Regge trajectory of hadrons for zero quark mass. For the case of non-zero quark mass, the model lead us to Heun's equation. The mass term is precisely the origin of the higher singularity, which changes the system behavior drastically. Our result can shed some light on why the chiral transition is so close to the confinement transition. In the m…
We consider the holographic abelian Higgs model and show that, in the absence of the scale symmetry breaking effect, chiral symmetry breaking gives linear confinement where the slope is given by the value of the chiral condensation. The model can be considered either as the theory of superconductivity or as the axial sector of QCD depending on the interpretation of the charge. We also provided a few models with linear confinement.
The breaking of chiral symmetry in holographic light-front QCD is encoded in its longitudinal dynamics with its chiral limit protected by the superconformal algebraic structure which governs its transverse dynamics. The scale in the longitudinal light-front Hamiltonian determines the confinement strength in this direction: It is also responsible for most of the light meson ground state mass consistent with the Gell-Mann-Oakes-Renner constraint. Longitudinal confinement and the breaking of chiral symmetry are found to be different manifestations of the same underlying dynamics as found in 't Ho…
The effective quark Lagrangian is derived from the QCD action by means of averaging over the vacuum fields. In the limit of large N_c and keeping for simplicity the lowest order gluonic correlator one obtains nonlinear equations for the quark propagator, exhibiting scalar confinement. Connection to the quark zero mode density is discussed.
I discuss the chiral dynamics of Gribov's theory of confinement. At a critical coupling $α_s^c$ the light quark vacuum undergoes a series of phase transitions which should be taken into account in the application of chiral perturbation theory. The Gribov theory offers a simple explanation of the value of the pion nucleon sigma term without need to invoke a large strange quark component in the nucleon.
In a solvable model of two dimensional SU(N) (N \to \infty) gauge fields interacting with matter in both adjoint and fundamental representations we investigate the nature of the phase transition separating the strong and weak coupling regions of the phase diagram. By interpreting the large N solution of the model in terms of SU(N) representations it is shown that the strong coupling phase corresponds to a region where a gap occurs in the spectrum of irreducible representations. We identify a gauge invariant order parameter for the generalized confinement-deconfinement transition and give a phy…
We construct an effective model for the chiral field and the Polyakov loop in which we can investigate the interplay between the approximate chiral symmetry restoration and the deconfinement of color in a thermal SU(3) gauge theory with three flavors of massive quarks. The phenomenological couplings between these two sectors can then be related to the recent lattice data on the renormalized Polyakov loop and the chiral condensate close to the critical region.
We study the QCD phase diagram, in particular we study the critical points of the two main QCD phase transitions, confinement and chiral symmetry breaking. Confinement drives chiral symmetry breaking, and, due to the finite quark mass, at small density both transitions are a crossover, while they are a first or second order phase transition in large density. We study the QCD phase diagram with a quark potential model including both confinement and chiral symmetry. This formalism, in the Coulomb gauge hamiltonian formalism of QCD, is presently the only one able to microscopically include both a…
