The SU(3) gauge theory with fermions in the sextet representation is one of several theories of interest for technicolor models. We have carried out a Schrodinger functional (SF) calculation for the lattice theory with two flavors of Wilson fermions. We find that the discrete beta function changes sign when the SF renormalized coupling is in the neighborhood of g^2 = 2.0, showing a breakdown of the perturbative picture even though the coupling is weak. The most straightforward interpretation is an infrared-stable fixed point.
I report the results of Schroedinger functional calculations in the SU(3) gauge theory with two flavors of color sextet fermions, defined with the Wilson-clover action using nHYP fat links. While we cannot confirm the infrared fixed point seen with thin links, we find very slow evolution of the coupling constant, so slow that extraction of the mass anomalous dimension is straightforward.
As a follow up to the previous talk about the beta function of SU(3) gauge theory with Nf=2 symmetric representation (clover) fermions [arXiv:0809.2885 [hep-lat]], we describe our explorations of the beta-kappa plane, away from the massless limit. Our simulations are mostly done on lattices of length L=8 and 12. We observe a phase transition from a strong coupling confined phase to a deconfined, chirally restored phase. The line of transitions avoids (so far) the location of the infrared fixed point discussed in the last talk.
We study QCD with two flavours of colour-sextet quarks as a candidate walking-Technicolor theory. We simulate lattice QCD with two flavours of colour-sextet staggered quarks at finite temperatures to observe the scales of confinement and chiral-symmetry breaking. These should give us some indication as to whether the massless theory has an infrared fixed point making it a conformal field theory, or whether it exhibits confinement and chiral symmetry breaking with a slowly varying coupling constant, i.e. `walks'. We find that unlike the case with fundamental quarks, the deconfinement and chiral…
We present new results from our ongoing study of the SU(3) sextet model with two flavors in the two-index symmetric representation of the gauge group. In the simulations use unimproved Wilson fermions to investigate the infrared properties of the model. We have previously presented results for the spectrum of the model in the weak coupling regime. Here, to better understand the overall behavior of the lattice model, we map its non-trivial phase structure in the space of bare parameters. At strong coupling, we observe a first order phase transition when decreasing the bare quark mass. This firs…
We extend our previous study of the SU(3) gauge theory with N_f=2 flavors of fermions in the sextet representation of color. Our tool is the Schroedinger functional method. By changing the lattice action, we push the bulk transition of the lattice theory to stronger couplings and thus reveal the beta function and the mass anomalous dimension gamma_m over a wider range of coupling, out to g^2 ~ 11. Our results are consistent with an infrared fixed point, but walking is not ruled out. Our main result is that gamma_m never exceeds 0.45, making the model unsuitable for walking technicolor. We use …
We have carried out a Schrodinger functional (SF) calculation for the SU(3) lattice gauge theory with two flavors of Wilson fermions in the sextet representation of the gauge group. We find that the discrete beta function, which governs the change in the running coupling under a discrete change of spatial scale, changes sign when the SF renormalized coupling is in the neighborhood of g^2=2.0. The simplest explanation is that the theory has an infrared-attractive fixed point, but more complicated possibilities are allowed by the data. While we compare rescalings by factors of 2 and 4/3, we work…
Recent progress in understanding the emergence of confinement and other nonperturbative effects in the strong interaction vacuum is reviewed. Special emphasis is placed on the role of different types of collective infrared gluonic degrees of freedom in this respect. After a survey of complementary approaches, models of the QCD vacuum based on center vortices, Abelian magnetic monopoles and topological charge lumps such as instantons, merons and calorons are examined. Both the physical mechanisms governing these models as well as recent lattice studies of the respective degrees of freedom are r…
We have been studying QCD with 2 flavours of colour-sextet quarks as a candidate walking-Technicolor theory using lattice-QCD simulations. The evolution of the coupling constant with lattice spacing is measured at the finite-temperature chiral transition to determine if this theory is asymptotically free and hence QCD-like. The lattice spacing is varied by changing the number of lattice sites, $N_t$, in the Euclidean time direction. QCD with 3 flavours is studied for comparison. Since this theory is expected to be conformal, with an infrared fixed point, the coupling constant at the chiral tra…
QCD with 2 flavours of massless colour-sextet quarks is studied as a possible walking-Technicolor candidate. We simulate the lattice version of this model at finite temperatures near to the chiral-symmetry restoration transition, to determine whether it is indeed a walking theory (QCD-like with a running coupling which evolves slowly over an appreciable range of length scales) or if it has an infrared fixed point, making it a conformal field theory. The lattice spacing at this transition is decreased towards zero by increasing the number $N_t$ of lattice sites in the temporal direction. Our si…
We compute the renormalized running coupling of SU(3) gauge theory coupled to N_f = 2 flavors of massless Dirac fermions in the 2-index-symmetric (sextet) representation. This model is of particular interest as a minimal realization of the strongly interacting composite Higgs scenario. A recently proposed finite volume gradient flow scheme is used. The calculations are performed at several lattice spacings with two different implementations of the gradient flow allowing for a controlled continuum extrapolation and particular attention is paid to estimating the systematic uncertainties. For sma…
The properties of a generalized version of the Borel Transform in infrared unstable theories with dynamical mass generation are studied. The reconstruction of the nonperturbative structure is unambiguous in this version. Various methods for extracting the singularity structure of the Borel Transform for lattice formulations of such theories are explored, and illustrated explicitly with the O(N) sigma model. The status of the first infrared renormalon in QCD is discussed. The feasibility of a proposed technique for analytically continuing from the left hand Borel plane (where nonperturbative in…
QCD with two flavours of massless colour-sextet quarks is considered as a model for conformal/walking Technicolor. If this theory possess an infrared fixed point, as indicated by 2-loop perturbation theory, it is a conformal(unparticle) field theory. If, on the other hand, a chiral condensate forms on the weak-coupling side of this would-be fixed point, the theory remains confining. The only difference between such a theory and regular QCD is that there is a range of momentum scales over which the coupling constant runs very slowly (walks). In this first analysis, we simulate the lattice versi…
Hierarchical renormalization group (RG) transformations are related to nonassociative algebras. These algebras serve as a new basic tool for a rigorous treatment of global RG flows and the search of nontrivial infrared fixed points. Convergent expansion methods are presented and analyzed in terms of algebra norms. It is shown that the infrared fixed points are solutions of a quadratic equation with an infinite number of unknowns. A continuous manifold of two dimensional periodic nontrivial fixed points is presented in terms of theta functions. Local Borel summability of the $ε$- expansion for …
We continue our lattice simulations of QCD with 2 flavours of colour-sextet quarks as a model for conformal or walking technicolor. A 2-loop perturbative calculation of the $β$-function which describes the evolution of this theory's running coupling constant predicts that it has a second zero at a finite coupling. This non-trivial zero would be an infrared stable fixed point, in which case the theory with massless quarks would be a conformal field theory. However, if the interaction between quarks and antiquarks becomes strong enough that a chiral condensate forms before this IR fixed point is…
We test here our recently introduced new lattice method for the $β$-function defined over infinite Euclidean space-time in the continuum from scale changes generated by infinitesimal or finite steps of the renormalized gauge coupling on the gradient flow. Harlander and Neumann calculated in this scheme the three-loop approximation to the continuum $β$-function. Our goal is the nonperturbative lattice implementation of the scheme which we tested originally in the chiral limit of the sextet model and in multi-flavor QCD with ten and twelve flavors of massless fermions. Results are reported here …
We study the Landau gauge correlators of Yang-Mills fields for infrared Euclidean momenta in the context of a massive extension of the Faddeev-Popov Lagrangian which, we argue, underlies a variety of continuum approaches. Standard (perturbative) renormalization group techniques with a specific, infrared-safe renormalization scheme produce so-called decoupling and scaling solutions for the ghost and gluon propagators, which correspond to nontrivial infrared fixed points. The decoupling fixed point is infrared stable and weakly coupled, while the scaling fixed point is unstable and generically s…
We set up an infrared-based moment problem to obtain estimates of the masses of the scalar, pseudoscalar and tensor glueballs in Euclidean Yang-Mills theories using the Refined Gribov-Zwanziger (RGZ) version of the Landau gauge, which takes into account nonperturbative physics related to gauge copies. Employing lattice input for the mass scales of the RGZ gluon propagator, the lowest order moment problem approximation gives the values m_{0++}\approx 1.96 GeV, m_{2++} \approx 2.04 GeV and m_{0-+}\approx 2.19 GeV in the SU(3) case, all within a 20% range of the corresponding lattice values. We a…
We present a study of the IR behaviour of a three-dimensional super-renormalisable quantum field theory (QFT) consisting of a scalar field in the adjoint of $SU(N)$ with a $φ^4$ interaction. A bare mass is required for the theory to be massless at the quantum level. In perturbation theory the critical mass is ambiguous due to infrared (IR) divergences and we indeed find that at two-loops in lattice perturbation theory the critical mass diverges logarithmically. It was conjectured long ago in [Jackiw 1980, Appelquist 1981] that super-renormalisable theories are nonperturbatively IR finite, with…
We present updated results for the non-perturbative $β$-function of SU(3) gauge theories with $N_f = 12$ or 10 massless flavors in the fundamental rep or $N_f = 2$ in the sextet rep, measured with staggered fermions. New data at finer lattice spacing and our previously introduced method, the infinitesimal $β$-function, strengthen the case that the $N_f = 12$ model has no infrared fixed point up to $g^2 = 7.2$. We show how underestimated cutoff dependence in one domain wall study for $N_f = 10$ has been corrected, which is now consistent with staggered results showing a monotonically increasing…
Perturbative QCD uses the Faddeev-Popov gauge-fixing procedure, which leads to ghosts and the local BRST invariance of the gauge-fixed perturbative QCD action. In the asymptotic regime, where perturbative QCD is relevant, Gribov copies can be neglected. In the nonperturbative regime, one must adopt either a nonlocal Gribov-copy free gauge (e.g., Laplacian gauge) or attempt to maintain local BRST invariance at the expense of admitting Gribov copies. These issues are explored. In addition, we discuss the relationship between recent Dyson-Schwinger based model calculations of the infrared behavio…
QCD with 2 flavours of massless colour-sextet quarks is studied as a theory which might exhibit a range of scales over which the running coupling constant evolves very slowly (walks). We simulate lattice QCD with 2 flavours of sextet staggered quarks to determine whether walks, or if it has an infrared fixed point, making it a conformal field theory. Our initial simulations are performed at finite temperatures $T=1/N_ta$ ($N_t=4$ and $N_t=6$), which allows us to identify the scales of confinement and chiral-symmetry breaking from the deconfinement and chiral-symmetry restoring transitions. Unl…
The coupled Dyson-Schwinger equations for the gluon and ghost propagators are investigated in the Landau gauge using a two-loop improved truncation that preserves the multiplicative renormalizability of the propagators. In this truncation all diagrams contribute to the leading order infrared analysis. The infrared contributions of the nonperturbative two-loop diagrams to the gluon vacuum polarization are computed analytically, and this reveals that infrared power behaved propagator solutions only exist when the squint diagram contribution is taken into account. For small momenta the gluon and …
In this thesis, several aspects of Yang-Mills theory are studied. It begins with the constrained quantization in the Coulomb gauge, using the Dirac bracket formalism. A nonperturbative analysis of the infrared asymptotics of propagators in any spatial dimension follows, and a connection to the Landau gauge is given. In the Coulomb gauge Hamiltonian approach, a linearly rising static color Coulomb potential is found, along with an infrared diverging gluon energy, both signaling confinement. The propagators and vertices in the entire momentum regime are calculated with the variational principle.…
We study a manifestly gauge invariant set of Schwinger-Dyson equations to determine the nonperturbative dynamics of the gluon and ghost propagators in $d=3$ Yang-Mills. The use of the well-known Schwinger mechanism, in the Landau gauge, leads to the dynamical generation of a mass for the gauge boson (gluon in $d=3$), which, in turn, gives rise to an infrared finite gluon propagator and ghost dressing function. The propagators obtained from the numerical solution of these nonperturbative equations are in very good agreement with the results of $SU(2)$ lattice simulations.
The methods of lattice gauge theory may be applied to gauge theories besides QCD, in fact to any gauge group and any representation of matter fields (as long as the coupling is not chiral). Such theories are useful for model building beyond the Standard Model, for instance in technicolor models. We have carried out Monte Carlo simulations of the SU(3) gauge theory with color sextet fermions. Our result for its discrete beta function indicate an infrared fixed point that makes the theory conformal rather than confining. The lattice theory's phase diagram shows no separation of chiral and confin…
In gauge theories, charged particles obey modified dispersion due to medium interactions (forward scattering), leading at high energies $E \geq T$ to an asymptotic mass-squared $m^2_\infty$. We calculate the infrared part of this mass nonperturbatively for the theory of the strong interactions, QCD, through a lattice treatment of its low-energy effective description, Electrostatic QCD (EQCD). Incorporation of these results into a nonperturbative determination of the effective thermal mass will require a still-incomplete next-to-leading order perturbative matching of this quantity to full QCD.
Nonperturbative lattice field theory simulations provide a systematic framework to investigate properties of conformal systems at strong couplings. These simulations can be performed using different lattice discretizations. Here we present numerical results for the step scaling beta function in SU(3) gauge theories with ten and twelve fundamental flavors. We calculate the renormalized $β$ function in the finite volume gradient flow renormalization scheme. Using Möbius domain wall fermions with Symanzik gauge action, Zeuthen gradient flow, and perturbative tree-level improvement, we implement a…
We present a nonperturbative lattice formulation of noncommutative Yang-Mills theories in arbitrary even dimension. We show that lattice regularization of a noncommutative field theory requires finite lattice volume which automatically provides both an ultraviolet and an infrared cutoff. We demonstrate explicitly Morita equivalence of commutative U(p) gauge theory with (p*n_f) flavours of fundamental matter fields on a lattice of size L with twisted boundary conditions and noncommutative U(1) gauge theory with n_f species of matter on a lattice of size (p*L) with single-valued fields. We discu…
We have calculated the running coupling in SU(2), SU(3), and SU(4) gauge theories to see whether they have infrared fixed points. An infrared fixed point means no confinement: It means that the long-distance physics is conformal, without a mass scale and indeed without a particle spectrum. All these theories run slowly in perturbation theory, so a nonperturbative determination is interesting.
The complete tensor structure of the quark--gluon vertex in Landau gauge is determined at two kinematical points (`asymmetric' and `symmetric') from lattice QCD in the quenched approximation. The simulations are carried out at beta=6.0, using a mean-field improved Sheikholeslami-Wohlert fermion action, with two quark masses ~ 60 and 115 MeV. We find substantial deviations from the abelian form at the asymmetric point. The mass dependence is found to be negligible. At the symmetric point, the form factor related to the chromomagnetic moment is determined and found to contribute significantly to…
We show that the O(Lambda) ambiguity in the pole mass can be fixed in a natural way by introducing a modified nonperturbative V-scheme momentum space coupling tilde-alphaV(q) where the confining contributions have been subtracted out. The method used is in the spirit of the infrared finite coupling approach to power corrections, and gives a non perturbative definition of the `potential subtracted' mass. The short distance expansion of the static potential is derived, taking into account an hypothetical short distance linear term. The magnitude of the standard OPE contributions are estimated in…
We investigate different aspects of lattice QCD in Landau gauge using Monte Carlo simulations. In particular, we focus on the low momentum behavior of gluon and ghost propagators. The gauge group is SU(3). Different systematic effects on the gluon and ghost propagators are studied, e.g. the dependence on the choice of Gribov copies or the influence of dynamical Wilson fermions. We compare our data with results from studies of Dyson-Schwinger equations for the gluon and ghost propagators. We demonstrate that the infrared behavior of both propagators, as found in this thesis, is consistent with …
SU(2) gauge theory with two fermions transforming under the adjoint representation may appear conformal or almost conformal in the infrared, and is one of the candidate theories for building models for technicolor. Early lattice Monte Carlo studies of this model have used unimproved Wilson fermion formulation, which can be expected to have large lattice cutoff effects. In this paper we present the calculation of the O(a) improved lattice Wilson-clover action of the theory. The Sheikholeslami-Wohlert coefficient has been determined non-perturbatively, and various boundary improvement terms, nee…
A model for the infrared sector of SU(2) Yang-Mills theory, based on magnetic vortex degrees of freedom represented by (closed) random world-surfaces, is presented. The model quantitatively describes both the confinement properties (including the finite-temperature transition to a deconfined phase) and the topological susceptibility of the Yang-Mills ensemble. A (quenched) study of the spectrum of the Dirac operator furthermore yields a behavior for the chiral condensate which is compatible with results obtained in lattice gauge theory.
We investigate the infrared dynamics of a nonsupersymmetric SU(X) gauge theory featuring an adjoint fermion, Nf Dirac flavors and an Higgs-like complex Nf x Nf scalar which is a gauge singlet. We first establish the existence of an infrared stable perturbative fixed point and then investigate the spectrum near this point. We demonstrate that this theory naturally features a light scalar degree of freedom to be identified with the dilaton and elucidate its physical properties. We compute the spectrum and demonstrate that at low energy the nonperturbative part of the spectrum of the theory is th…
Using a parametrization of the Wilson loop with the minimal-area law, we calculate the polarization operator of a valence gluon, which propagates in the confining background. This enables us to obtain the infrared freezing (i.e. finiteness) of the running strong coupling in the confinement phase, as well as in the deconfinement phase up to the temperature of dimensional reduction. The momentum scale defining the onset of freezing is found both analytically and numerically. The nonperturbative contribution to the thrust variable, originating from the freezing, makes the value of this variable c…
We report the nonperturbative behavior of the twisted Polyakov loop (TPL) coupling constant for the SU(3) gauge theories defined by the ratio of Polyakov loop correlators in finite volume with twisted boundary condition. We reveal the vacuum structures and the phase structure for the lattice gauge theory with the twisted boundary condition. Carrying out the numerical simulations, we determine the nonperturbative running coupling constant in this renormalization scheme for the quenched QCD and Nf=12 SU(3) gauge theories. At first, we study the quenched QCD theory using the plaquette gauge actio…
Dynamical chiral symmetry breaking is a nonperturbative phenomenon that may be studied using QCD's gap equation. Model-independent results can be obtained with a nonperturbative and symmetry preserving truncation. The gap equation yields the massive dressed-quark propagator, which has a spectral representation when considered as a function of the current-quark mass. This enables an explication of the connection between the infrared limit of the QCD Dirac operator's spectrum and the quark condensate appearing in the operator product expansion.
In a recent paper, we developed and applied a dilaton-based effective field theory (EFT) to the analysis of lattice-simulation data for a class of confining gauge theories with near-conformal infrared behavior. It was employed there at the classical level to the SU(3) gauge theory with eight Dirac fermions in the fundamental representation. Here, we explore the structure of the EFT further. We examine its application to lattice data (newly updated) for the SU(3) theory with eight Dirac fermions in the fundamental representation, and the SU(3) theory with two Dirac fermions in the sextet repres…
We report the nonperturbative behavior of the twisted Polyakov loop (TPL) coupling constant for the SU(3) gauge theories defined by the ratio of Polyakov loop correlators in finite volume with twisted boundary condition. Carrying out the numerical simulations, we determine the nonperturbative running coupling constant in this renormalization scheme for Nf=12 SU(3) gauge theory. According to the perturbative two loop analysis, the Nf=12 SU(3) gauge theory might have a conformal fixed point in the infrared region. However, the recent lattice studies show controversial results for the existence o…
In the present work we discuss certain characteristic features encoded in some of the fundamental QCD Green's functions, whose origin can be traced back to the nonperturbative masslessness of the ghost field, in the Landau gauge. Specifically, the ghost loops that contribute to these Green's functions display infrared divergences, akin to those encountered in the perturbative treatment, in contradistinction to the gluonic loops, whose perturbative divergences are tamed by the dynamical generation of an effective gluon mass. In d=4, the aforementioned divergences are logarithmic, thus causing a…
The spontaneous breaking of chiral symmetry (SBCS) is one of the most important phenomena of hadron physics. It defines the properties of all the light mesons and baryons. The Chiral Perturbation Theory (ChPT) encodes QCD hadronic correlators at low-energy region in the terms the low-energy constants (LEC) -- the expansion parameters on light quark current masses $m$ and external momenta $p$. The LEC's can be extracted from the phenomenology or from QCD lattice calculations. On the other hand, QCD instanton vacuum/instanton liquid model provides a very natural nonperturbative explanation of th…
The finite-temperature behavior of gluon and of Faddeev-Popov-ghost propagators is investigated for pure SU(2) Yang-Mills theory in Landau gauge. We present nonperturbative results, obtained using lattice simulations and Dyson-Schwinger equations. Possible limitations of these two approaches, such as finite-volume effects and truncation artifacts, are extensively discussed. Both methods suggest a very different temperature dependence for the magnetic sector when compared to the electric one. In particular, a clear thermodynamic transition seems to affect only the electric sector. These results…
In this paper we study the nonperturbative structure of the SU(3) four-gluon vertex in the Landau gauge, concentrating on contributions quadratic in the metric. We employ an approximation scheme where "one-loop" diagrams are computed using fully dressed gluon and ghost propagators, and tree-level vertices. When a suitable kinematical configuration depending on a single momentum scale $p$ is chosen, only two structures emerge: the tree-level four-gluon vertex, and a tensor orthogonal to it. A detailed numerical analysis reveals that the form factor associated with this latter tensor displays a …
The strong CP problem is inseparably connected with the topology of gauge fields and the mechanism of color confinement, which requires nonperturbative tools to solve it. In this talk I present results of a recent lattice investigation of QCD with the $θ$ term in collaboration with Yoshifumi Nakamura. The tool we are using to address the nonperturbative properties of the theory is the gradient flow, which is a particular realization of momentum space RG transformations. The novel result is that within QCD the vacuum angle $θ$ is renormalized, together with the strong coupling constant, and flo…
Perturbation theory of thermodynamic potentials in QCD at $T>T_c$ is considered with the nonperturbative background vacuum taken into account. It is shown that the magnetic confinement in the QCD vacuum prevents the infrared catastrophe of the perturbation theory present in the case of the free vacuum (the "Linde problem"). A short discussion is given of the applicability of the nonperturbative formalism at large $T$ and of the relation with the HTL theory.The observation of A.D.Linde, that the terms $O(g^n), n>6$ contribute to the order $O(g^6)$ is confirmed also with the account of the color…
We investigate the gauge dynamics of nonsupersymmetric SU(N) gauge theories featuring the simultaneous presence of fermionic matter transforming according to two distinct representations of the underlying gauge group. We bound the regions of flavors and colors which can yield a physical infrared fixed point. As a consistency check we recover the previously investigated conformal windows bounds when restricting to a single matter representation. The earlier conformal windows can be imagined to be part now of the new conformal house. We predict the nonperturbative anomalous dimensions at the inf…
Resembling the renormalization group from ultraviolet to infrared, we construct a dynamical holographic model in the graviton-dilaton-scalar framework, where the dilaton background field $Φ$ and scalar field $X$ are responsible for the gluodynamics and chiral dynamics, respectively. At the UV boundary, the dilaton field is dual to the dimension-4 gluon operator, and the scalar field is dual to the dimension-3 quark-antiquark operator. The metric structure at IR is automatically deformed by the nonperturbative gluon condensation and chiral condensation in the vacuum. The produced scalar gluebal…
We propose an ansatz for the nonperturbative beta function of a generic non-supersymmetric Yang-Mills theory with or without fermions in an arbitrary representation of the gauge group. While our construction is similar to the recently proposed Ryttov-Sannino all order beta function, the essential difference is that it allows for the existence of an unstable ultraviolet fixed point in addition to the predicted Bank-Zaks -like infrared stable fixed point. Our beta function preserves all of the tested features with respect to the non-supersymmetric Yang-Mills theories. We predict the conformal wi…
We uncover a novel solution of the 't Hooft anomaly matching conditions for QCD. Interestingly in the perturbative regime the new gauge theory, if interpreted as a possible QCD dual, predicts the critical number of flavors above which QCD in the nonperturbative regime, develops an infrared stable fixed point. Remarkably this value is identical to the maximum bound predicted in the nonpertubative regime via the all-orders conjectured beta function for nonsupersymmetric gauge theories.
