We study in detail the effect of the finite current quark mass on chiral symmetry breaking, in the framework of truncated Coulomb gauge QCD with a linear confining quark-antiquark potential. In the chiral limit of massless current quarks, the breaking of chiral symmetry is spontaneous. But for a finite current quark mass, some dynamical symmetry breaking continues to add to the explicit breaking caused by the quark mass. Moreover, using as order parameter the mass gap, i. e. the quark mass at vanishing moment or the quark condensate, a finite quark mass transforms the chiral symmetry breaking from a phase transition into a crossover. For the study of the QCD phase diagram it thus is relevant to determine how the current quark mass affects chiral symmetry breaking. Since the current quark masses of the six standard flavours u, d, s, c, b, t span over five orders of magnitude from 1.5 MeV to 171 GeV, we develop an accurate numerical method to study the running quark mass gap and the quark vacuum energy density from very small to very large current quark masses.
In this paper we study the power-like confining potentials r^alpha. The region of allowed alphas is identified, the mass-gap equation is constructed for an arbitrary alpha and solved for several values of the latter, and the vacuum energy and the chiral condensate are calculated. The question of replica solutions to the mass-gap equation for such potentials is addressed and it is demonstrated that the number of replicas is infinite for any alpha, as a consequence of the peculiar behaviour of the quark self-energy in the infrared domain.
The QCD beta-function and the anomalous dimensions for the Coulomb potential and the static potential first differ at three loop order. We evaluate the three loop ultrasoft anomalous dimension for the Coulomb potential and give the complete three loop running. Using this result, we calculate the leading logarithmic Lamb shift for a heavy quark-antiquark bound state, which includes all contributions to the binding energies of the form mα_s^4(α_s lnα_s)^k, k\ge 0.
In this paper we study the breaking of chiral symmetry with non-confining power-like potentials. The region of allowed exponents is identified and, after the previous study of confining (positive exponent) potentials, we now specialize in shorter range non-confining potentials, with a negative exponent. These non-confining potentials are close to the Coulomb potential, and they are also relevant as corrections to the linear confinement, and as models for the quark potential at the deconfinement transition. The mass-gap equation is constructed and solved, and the quarks mass, the chiral angle a…
We analyze the fully relativistic, field-theoretical treatment of the scalar Coulomb problem. We work in a truncated Hilbert-Fock space containing the two-constituent states and the two-constituent-and-one-massless-exchange-particle states. Self-energy contributions are dominant for large values of the coupling constant. Using the covariant formulation of the self-energy does not alter these results significantly. Both the weak-coupling limit and the heavy-mass limit lead to the non-relativistic results.
We calculate the lattice quark propagator in Coulomb gauge both from dynamical and quenched configurations. We show that in the continuum limit both the static and full quark propagator are multiplicatively renormalizable. From the propagator we extract the quark renormalization function Z(|p|) and the running mass M(|p|) and extrapolate the latter to the chiral limit. We find that M(|p|) practically coincides with the corresponding Landau gauge function for small momenta. The computation of M(|p|) can be however made more efficient in Coulomb gauge; this can lead to a better determination of …
We use the functional form of the QCD running coupling to modify the conformal metric in AdS/CFT mapping the fifth-dimensional z-coordinate to the energy scale in the four-dimensional QCD. The resulting type-0 string theory in five dimensions is solved with the Nambu-Goto action giving good agreement with the Coulombic and confinement $Q \bar Q$ potential.
We consider the effects of the Faddeev-Popov determinant in the Coulomb gauge on the confinement properties of the QCD vacuum. We show that the the determinant is needed to regularize the otherwise divergent functional integrals near the Gribov horizon but still enables large field configurations to generate IR enhanced running coupling. The physical gluon propagator is found to be strongly suppressed in the IR consistent with expectations from lattice gauge calculations.
The infrared behaviour of the gluon and ghost propagators in Landau gauge QCD is reviewed. The Kugo-Ojima confinement criterion and the Gribov-Zwanziger horizon condition result from quite general properties of the ghost Dyson-Schwinger equation. The numerical solutions for the gluon and ghost propagators obtained from a truncated set of Dyson-Schwinger equations provide an explicit example for the anticipated infrared behaviour. The results are in good agreement with corresponding lattice data obtained recently. The resulting running coupling approaches a fix point in the infrared, $α(0) = 8.…
We consider the class of gauges that interpolates between Landau- and Coulomb-gauge QCD, and show the non-renormalisation of the two independent ghost-gluon vertices. This implies the existence of two RG-invariant running couplings, one of which is interpreted as an RG-invariant gauge parameter. We also present the asymptotic infrared limit of solutions of the Dyson-Schwinger equations in interpolating gauges. The infrared critical exponents of these solutions as well as the resulting infrared fixed point of one of the couplings are independent of the gauge parameter. This coupling also has a …
It is shown that a Coulomb potential using a running coupling slightly modified from the perturbative form can produce an interquark potential that appears nearly linear over a large distance range. Recent high-statistics SU(2) lattice gauge theory data fit well to this potential without the need for a linear string-tension term. This calls into question the accuracy of string tension measurements which are based on the assumption of a constant coefficient for the Coulomb term. It also opens up the possibility of obtaining an effectively confining potential from gluon exchange alone.
Truncated Dyson-Schwinger equations represent finite subsets of the equations of motion for Green's functions. Solutions to these non-linear integral equations can account for non-perturbative correlations. A closed set of coupled Dyson-Schwinger equations for the propagators of gluons and ghosts in Landau gauge QCD is obtained by neglecting all contributions from irreducible 4-point correlations and by implementing the Slavnov-Taylor identities for the 3-point vertex functions. We solve this coupled set in an one-dimensional approximation which allows for an analytic infrared expansion necess…
Recently evidence has been found that the perturbative QCD potential agrees well with phenomenological potentials and lattice computations of the QCD potential. We review the present status of the perturbative QCD potential and theoretical backgrounds. We also report our recent analysis which shows analytically, on the basis of renormalon dominance picture, that the perturbative QCD potential quickly ``converges'' to a Coulomb-plus-linear form. The Coulomb-plus-linear potential can be computed systematically as we include more terms of the perturbative series; up to three-loop running (our cur…
The ground and P-wave excited states of nnn, nns and ssn baryons are studied in the framework of the Field Correlator Method using the running strong coupling constant in the Coulomb-like part of the three-quark potential. The running coupling is calculated up to two loops in the background perturbation theory. The three-quark problem has been solved using the hyperspherical functions method. The masses of the S- and P-wave baryons are presented. Our approach reproduces and improves the previous results for the baryon masses obtained for the freezing value of the coupling constant. The string …
We present approximate non-perturbative solutions for the propagators as well as the running coupling of QCD from a coupled system of renormalised, truncated Dyson--Schwinger equations. We employ ansaetze for the dressed vertices such that the running coupling and the quark mass function are independent of the renormalisation point. Our solutions are in good agreement with the results of recent lattice calculations. We find a weakly vanishing gluon propagator at small momenta and an infrared singular ghost propagator in agreement with Zwanziger's horizon condition and the Kugo-Ojima confinemen…
We analyze the approach to chiral symmetry breaking in QCD at finite temperature, using the functional renormalization group. We compute the running gauge coupling in QCD for all temperatures and scales within a simple truncated renormalization flow. At finite temperature, the coupling is governed by a fixed point of the 3-dimensional theory for scales smaller than the corresponding temperature. Chiral symmetry breaking is approached if the running coupling drives the quark sector to criticality. We quantitatively determine the phase boundary in the plane of temperature and number of flavors a…
We present the first direct determination of the running top-quark mass based on the total cross section of top-quark pair-production as measured at the Tevatron. Our theory prediction for the cross section includes various next-to-next-to-leading order QCD contributions, in particular all logarithmically enhanced terms near threshold, the Coulomb corrections at two loops and all explicitly scale dependent terms at NNLO accuracy. The result allows for an exact and independent variation of the renormalization and factorization scales. For Tevatron and LHC we study its dependence on all scales, …
We show analytically that the QCD potential can be expressed, up to an O(Lambda_QCD^3 r^2) uncertainty, as the sum of a ``Coulomb'' potential (with log corrections at short distances) and a linear potential, within an approximation based on perturbative expansion in alpha_S and the renormalon dominance picture. The expansion of V_QCD(r) is truncated at O(alpha_S^N) [N = 6 pi/(beta_0 alpha_S)], where the term becomes minimal according to the estimate by NLO renormalon, and is studied for N >> 1. Analytic expressions for the linear potential are obtained in some cases.
The Dyson-Schwinger equations arising from minimizing the vacuum energy density in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge are solved numerically. A new solution is presented which gives rise to a strictly linearly rising static quark potential and whose existence was previously observed in the infrared analysis of the Dyson-Schwinger equations. For the new solution we also present the static quark potential and calculate the running coupling constant from the ghost-gluon vertex.
We argue that although at asymptotically high temperatures the QGP in bulk behaves as a gas of weakly interacting quasiparticles (modulo long-range magnetism), at temperatures up to few times the critical temperature $T_c$ it displays different properties. If the running of the QCD coupling constant continues in the Coulomb phase till the screening length scale, it reaches the strong coupling treshold $α_s(m_D)\sim 1$. As a result, the Coulomb phase supports weakly bound Coulombic s-wave $\bar c c$, light quark and even $gg$ states. The existence of shallow bound states dramatically increases …
We present yet another composite model explaining the relatively broad peak in the diphoton invariant mass distribution around 750 GeV recently observed at the LHC experiments. We consider the excess originates in bound states of vector-like fermions Q transforming under the electroweak group (but not QCD) of the standard model and which are also charged under a new SU(N_{TC}) QCD-like confining force. Since the new uncolored fields transform as SU(2) doublets they can acquire a mass through Yukawa interactions with the electroweak Higgs doublet, as quarks and leptons. We parametrize the Q-oni…
We study the conjectured "Insensitivity to Chiral Symmetry Breaking" in the highly excited light baryon spectrum. While the experimental spectrum is being measured at JLab and CBELSA/TAPS, this insensitivity remains to be computed theoretically in detail. As the only existing option to have both confinement, highly excited states and chiral symmetry, we adopt the truncated Coulomb gauge formulation of QCD, considering a linearly confining Coulomb term. Adopting a systematic and numerically intensive variational treatment up to 12 harmonic oscillator shells we are able to access several angular…
We start with the explicit solution, in terms of the Lambert W function, of the renormalization group equation (RGE) for the gauge coupling in the supersymmetric Yang-Mills theory described by the well-known beta function of Novikov et al.(NSVZ). We then construct a class of beta functions for which the RGE can be solved in terms of the Lambert W function. These beta functions are expressed in terms of a function which is a truncated Laurent series in the inverse of the gauge coupling. The parameters in the Laurent series can be adjusted so that the first coefficients of the Taylor expansion o…
Perturbative QCD (pQCD) running coupling a(Q^2) (=alpha_s(Q^2)/pi) is expected to get modified at low spacelike momenta 0 < Q^2 < 1 GeV^2 so that, instead of having unphysical (Landau) singularities it remains smooth and finite there, due to infared (IR) fixed point. This behavior is suggested by: Gribov-Zwanziger approach, Dyson-Schwinger equations (DSE) and other functional methods, lattice calculations, light-front holographic mapping AdS/CFT modified by a dilaton background, and by most of the analytic (holomorphic) QCD models. All such couplings, A(Q^2), differ from the pQCD couplings a(Q…
We analyze the free energy of a static quark anti-quark pair in quenched QCD at short and large distances. From this we deduce running couplings, g^2(r,T), and determine the length scale that separates at high temperature the short distance perturbative regime from the large distance non-perturbative regime in the QCD plasma phase. Ambiguities in the definition of a coupling beyond the perturbative regime are discussed in their relation to phenomenological considerations on heavy quark bound states in the quark gluon plasma. Our analysis suggests that it is more appropriate to characterize the…
Some features of the solutions to the truncated Dyson-Schwinger equations(DSEs) for the propagators of QCD in Landau gauge are summarized. In particular, the Kugo-Ojima confinement criterion is realized, and positivity of transverse gluons is manifestly violated in these solutions. In Landau gauge, the gluon-ghost vertex function offers a convenient possibility to define a nonperturbative running coupling. The infrared fixed point obtained from this coupling which determines the 2-point interactions of color-octet quark currents implies the existence of unphysical massless states which are nec…
A large approximate degeneracy of the excited hadron spectra, appears in the light meson spectra measured at LEAR, suggesting a novel principal quantum number n+j in QCD spectra. We recently showed that the large degeneracy could not be understood with state of the art confining and chiral invariant quark models, derived in a truncated Coulomb gauge. To search for a solution to this problem, here we add the gluon or string degrees of freedom. Although independently the quarks or the gluons would lead to a 2 n +j or 2 n +l spectrum, adding them may lead to the desired n+j pattern.
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 suggest that Hamiltonian matrix elements between physical states in QCD might be approximated, in Coulomb gauge, by "lattice-improved" tree diagrams; i.e. tree diagram contributions with dressed ghost, transverse gluon, and Coulomb propagators obtained from lattice simulations. Such matrix elements can be applied to a truncated-basis treatment of hadronic states which include constituent gluons. As an illustration, we apply this hybrid approach to the heavy quark potential, for quark-antiquark separations up to 2.4 fm. The Coulomb string tension in SU(3) gauge theory is about a factor of fo…
We summarize recent results on the nonperturbative quark-gluon interaction in Landau gauge QCD. Our analytical analysis of the infrared behaviour of the quark-gluon vertex reveals infrared singularities, which lead to an infrared divergent running coupling and a linear rising quark-antiquark potential when chiral symmetry is broken. In the chirally symmetric case we find an infrared fixed point of the coupling and, correspondingly, a Coulomb potential. These findings provide a new link betwen dynamical chiral symmetry breaking and confinement.
The thermal history of a large class of running vacuum models in which the effective cosmological term is described by a truncated power series of the Hubble rate, whose dominant term is $Λ(H) \propto H^{n+2}$, is discussed in detail. Specifically, by assuming that the ultra-relativistic particles produced by the vacuum decay emerge into space-time in such a way that its energy density $ρ_r \propto T^{4}$, the temperature evolution law and the increasing entropy function are analytically calculated. For the whole class of vacuum models explored here we findthat the primeval value of the comovi…
We study the chiral crossover, the spectra of light-light and of static-light mesons and the deconfinement crossover at finite temperature T. Our framework is the confining and chiral invariant quark model, related to truncated Coulomb gauge QCD. Since we are dealing with light quarks, where the linear potential dominates the quark condensate and the spectrum, we only specialize in the linear confining potential for the quark-antiquark interaction. We utilize T dependent string tensions previously fitted from lattice QCD data, and a fit of previously computed dynamically generated constituent …
We construct a manifestly gauge invariant Lagrangian in 3+1 dimensions for N Kaluza-Klein modes of an SU(m) gauge theory in the bulk. For example, if the bulk is 4+1, the effective theory is Π_{i=1}^{N+1} SU(m)_i with N chiral (\bar{m},m) fields connecting the groups sequentially. This can be viewed as a Wilson action for a transverse lattice in x^5, and is shown explicitly to match the continuum 4+1 compactifed Lagrangian truncated in momentum space. Scale dependence of the gauge couplings is described by the standard renormalization group technique with threshold matching, leading to effecti…
We analyze the phases of supersymmetric chiral gauge theories with an antisymmetric tensor and (anti)fundamental flavors, in the presence of a classically marginal superpotential deformation. Varying the number of flavors that appear in the superpotential reveals rich infrared chiral dynamics and novel dualities. The dualities are characterized by an infinite family of magnetic duals with arbitrarily large gauge groups describing the same fixed point, correlated with arbitrarily large classical global symmetries that are truncated nonperturbatively. At the origin of moduli space, these theorie…
