Infrared Behavior of Three-Point Functions in Landau Gauge Yang-Mills Theory
Reinhard Alkofer, Markus Q. Huber, Kai Schwenzer
备注：
21 pages, 29 figures; numerical data of the infrared dressing functions can be obtained from the authors v2: a few minor changes, corresponds to version appearing in EPJC
Analytic solutions for the three-gluon and ghost-gluon vertices in Landau gauge Yang-Mills theory at low momenta are presented in terms of hypergeometric series. They do not only show the expected scaling behavior but also additional kinematic divergences when only one momentum goes to zero. These singularities, which have also been proposed previously, induce a strong dependence on the kinematics in many dressing functions. The results are generalized to two and three dimensions and a range of values for the ghost propagator's infrared exponent kappa.
Correlation functions of Landau gauge Yang-Mills theory are calculated from their equations of motion. Solutions for the three- and four-gluon vertices and the role of two-loop diagrams in the gluon propagator equation are discussed.
Examples of normalizable zero modes of the Faddeev-Popov operator in SU(2) Euclidean Yang-Mills theories in the Landau gauge are constructed in d=2,3,4 dimensions.
G_2 Yang--Mills theory is an interesting laboratory to investigate non-perturbative effects. On one hand, no conventional quark confinement via a linearly rising potential is present. On the other hand, its thermodynamic properties are similar to ordinary SU(N) Yang--Mills theory. Finally, it has been conjectured that gluons are removed from the physical spectrum in the same way as in SU(N) Yang--Mills theory. The last claim will be explored by determining the Landau-gauge ghost and gluon propagators, as well as the Faddeev--Popov operator eigenspectrum, in G_2 lattice gauge theory in two and …
We prove that the Faddeev-Popov ghost dressing function in the Yang-Mills theory is non-zero and finite in the limit of vanishing momenta and hence the ghost propagator behaves like free in the deep infrared regime, within the Gribov-Zwanziger framework of the $D$-dimensional SU(N) Yang-Mills theory in the Landau gauge for any $D>2$. This result implies that the Kugo-Ojima color confinement criterion is not satisfied in its original form. We point out that the result crucially depends on the explicit form of the non-local horizon term adopted. The original Gribov prediction in the Landau gauge…
We discuss the properties of ghost and gluon propagators in the deep infrared momentum region of Landau gauge Yang-Mills theory. Within the framework of Dyson-Schwinger equations and the functional renormalization group we demonstrate that it is only a matter of infrared boundary conditions whether infrared scaling or decoupling occurs. We argue that the second possibility is at odds with global BRST symmetry in the confining phase. For this purpose we improve upon existing truncation schemes in particular with respect to transversality and renormalization.
We investigate Landau gauge $SU(3)$ Yang-Mills theory in a systematic vertex expansion scheme for the effective action with the functional renormalisation group. Particular focus is put on the dynamical creation of the gluon mass gap at non-perturbative momenta and the consistent treatment of quadratic divergences. The non-perturbative ghost and transverse gluon propagators as well as the momentum-dependent ghost-gluon, three-gluon and four-gluon vertices are calculated self-consistently with the classical action as only input. The apparent convergence of the expansion scheme is discussed and …
We develop a general power counting scheme for the infrared limit of Landau gauge SU(N) Yang-Mills theory in arbitrary dimensions. Employing a skeleton expansion, we find that the infrared behavior is qualitatively independent of the spacetime dimension d. In the cases d=2, 3 and 4 even the quantitative results for the infrared exponents of the vertices differ only slightly. Therefore, corresponding lattice simulations provide interesting qualitative information for the physical case. We furthermore find that the loop integrals depend only weakly on the numerical values of the IR exponents.
The ghost and gluon propagator and the ghost-gluon and three-gluon vertex of two-dimensional SU(2) Yang-Mills theory in (minimal) Landau gauge are studied using lattice gauge theory. It is found that the results are qualitatively similar to the ones in three and four dimensions. The propagators and the Faddeev-Popov operator behave as expected from the Gribov-Zwanziger scenario. In addition, finite volume effects affecting these Green's functions are investigated systematically. The critical infrared exponents of the propagators, as proposed in calculations using stochastic quantization and Dy…
Expanding the Landau gauge gluon and ghost two-point functions in a power series we investigate their infrared behavior. The corresponding powers are constrained through the ghost Dyson-Schwinger equation by exploiting multiplicative renormalizability. Without recourse to any specific truncation we demonstrate that the infrared powers of the gluon and ghost propagators are uniquely related to each other. Constraints for these powers are derived, and the resulting infrared enhancement of the ghost propagator signals that the Kugo-Ojima confinement criterion is fulfilled in Landau gauge Yang-Mil…
Fixing a gauge in the non-perturbative domain of Yang-Mills theory is a non-trivial problem due to the presence of Gribov copies. In particular, there are different gauges in the non-perturbative regime which all correspond to the same definition of a gauge in the perturbative domain. Gauge-dependent correlation functions may differ in these gauges. Two such gauges are the minimal and absolute Landau gauge, both corresponding to the perturbative Landau gauge. These, and their numerical implementation, are described and presented in detail. Other choices will also be discussed. This investigati…
Recently it has been shown that infrared singularities of Landau gauge QCD can confine static quarks via a linearly rising potential. We show that the same mechanism can also provide a confining interaction between charged scalar fields in the fundamental representation. This confirms that within this scenario static confinement is a universal property of the gauge sector even though it is formally represented in the functional equations of the matter sector. The simplifications compared to the fermionic case make the scalar system an ideal laboratory for a detailed analysis of the confinement…
We solve the Dyson-Schwinger equations of the ghost and gluon propagators of Landau gauge Yang-Mills theory together with that of the ghost-gluon vertex. The latter plays a central role in many truncation schemes for functional equations. By including it dynamically we can determine its influence on the propagators. We also suggest a new model for the three-gluon vertex motivated by lattice data which plays a crucial role to obtain stable solutions when the ghost-gluon vertex is included. We find that both vertices have a sizable quantitative impact on the mid-momentum regime and contribute to…
Yang-Mills theory can be formulated for any semi-simple Lie algebra, and thus any semi-simple Lie group. In principle, the dynamics could be different for each one. However, functional studies predict that the propagators in Landau gauge depend only quantitatively on the gauge algebra. In particular, genuine non-perturbative effects should be present even in the large N-limit for su(N) gauge algebras. Lattice gauge theory is used to investigate this in detail. The propagators are determined for the gauge groups SU(2), SU(3), SU(4), SU(5), SU(6) and G2, in two and three dimensions. In accordanc…
We present a more detailed picture of the infrared regime of Landau gauge Yang-Mills theory. This is done within a novel framework that allows one to take into account the influence of finite scales within an infrared power counting analysis. We find that there are two qualitatively different infrared fixed points of the full system of Dyson-Schwinger equations. The first extends the known scaling solution, where the ghost dynamics is dominant and gluon propagation is strongly suppressed. It features in addition to the strong divergences of gluonic vertex functions in the previously considered…
We study domain walls in two different extensions of super Yang--Mills characterized by the absence of a logarithmic term in their effective superpotential. The models, defined by the usual gaugino condensate and an extra field Y, give different patterns of domain walls despite both leading to the same effective limit for heavy Y, i.e. the Veneziano--Yankielowicz effective Lagrangian of super Yang--Mills. We explain the origin of those differences and also give a physical motivation for introducing the field Y.
Using elementary considerations of Lorentz invariance, Bose symmetry and BRST invariance, we argue why the decay of a massive color-octet vector state into a pair of on-shell massless gluons is possible in a non-Abelian SU(N) Yang-Mills theory, we constrain the form of the amplitude of the process and offer a simple understanding of these results in terms of effective-action operators.
The topological susceptibility is calculated within the Hamiltonian approach to Yang-Mills theory in Coulomb gauge, using the vacuum wave functional previously determined by a variational solution of the Yang-Mills Schroedinger equation. The numerical result agrees qualitatively with the predictions of lattice simulations.
Vertices describe the interactions between the fundamental degrees of freedom, and are therefore of vital importance in many ab-initio descriptions of field theory, especially using functional methods. To this end, we present the first lattice study of the thermal behavior of (minimal) Landau-gauge SU(2) Yang--Mills three-point functions, i.e. three-gluon and ghost-gluon vertices. Focusing on the chromomagnetic sector, we find that the phase transition mainly affects the three-gluon vertex, while the ghost-gluon vertex is relatively inert.
Using a spin-charge separation of the gluon field in the Landau gauge we show that the SU(2) Yang-Mills theory in the low-temperature phase can be considered as a nematic liquid crystal. The ground state of the nematic crystal is characterized by the A^2 condensate of the gluon field. The liquid crystal possesses various topological defects (instantons, monopoles and vortices) which are suggested to play a role in non-perturbative features of the theory.
The Landau-Yang theorem, forbidding transition amplitudes between a massive spin-1 particle and two photons, is widely assumed to apply to other massless spin-1 particles. We show that this is not true in Standard-Model QCD, so that for instance antisymmetric colour-octet spin-1 quarkonia can be formed by two on-shell gluons in second-order perturbative QCD.
The infrared behaviour of gluon and ghost propagators, ghost-gluon vertex and three-gluon vertex is investigated for both the covariant Landau and the non-covariant Coulomb gauge. Assuming infrared ghost dominance, we find a unique infrared exponent in the d=4 Landau gauge, while in the d=3+1 Coulomb gauge we find two different infrared exponents. We also show that a finite dressing of the ghost-gluon vertex has no influence on the infrared exponents. Finally, we determine the infrared behaviour of the three-gluon vertex analytically and calculate it numerically at the symmetric point in the C…
The issue of the Gribov copies is discussed in Euclidean N=4 Super Yang-Mills theory quantized in the Landau gauge. As a consequence of the absence of a renormalization group invariant scale, it turns out that it is not possible to attach a dynamical meaning to the Gribov parameter. This implies that, in N=4 Super Yang-Mills, there is no need for the restriction of the domain of integration in the functional integral to the Gribov region, and no non-perturbative dynamical scale is generated. This result is in agreement with the absence of a confining phase observed from the Coulomb behaviour o…
Vertices are of central importance for constructing QCD bound states out of the individual constituents of the theory, i.e. quarks and gluons. In particular, the determination of three-point vertices is crucial in non-perturbative investigations of QCD. We use numerical simulations of lattice gauge theory to obtain results for the 3-point vertices in Landau-gauge SU(2) Yang-Mills theory in three and four space-time dimensions for various kinematic configurations. In all cases considered, the ghost-gluon vertex is found to be essentially tree-level-like, while the three-gluon vertex is suppress…
The infrared properties of the high-temperature limit of Landau-gauge Yang-Mills theory are investigated. In a first step the high-temperature limit of the Dyson-Schwinger equations is taken. The resulting equations are identical to the Dyson-Schwinger equations of the dimensionally reduced theory, a three-dimensional Yang-Mills theory coupled to an effective adjoint Higgs field. These equations are solved analytically in the infrared and ultraviolet, and numerically for all Euclidean momenta. We find infrared enhancement for the Faddeev-Popov ghosts, infrared suppression for transverse gluons…
The Dyson-Schwinger equations of Landau-gauge Yang-Mills theory for the gluon and ghost propagators are investigated. Numerical results are obtained within a truncation scheme which has proven to be successful at vanishing temperature. For temperatures up to 250 MeV we find only minor quantitative changes in the infrared behaviour of the gluon and ghost propagators. The effective action calculated from these propagators is temperature-independent within the numerical uncertainty.
The infrared behavior of Yang-Mills theory at finite temperature provides access to the role of confinement. In this review recent results on this topic from lattice calculations and especially Dyson-Schwinger studies are discussed. These indicate persistence of a residual confinement even in the high-temperature phase. The confinement mechanism is very similar to the one in the vacuum for the chromomagnetic sector. In the chromoelectric sector screening occurs at the soft scale g^2T, although not leading to a perturbative behavior.
The infrared behaviour of vertex functions in an SU(N) Yang-Mills theory in Landau gauge is investigated employing a skeleton expansion of the Dyson-Schwinger equations. The three- and four-gluon vertices become singular if and only if all external momenta vanish while the dressing of the ghost-gluon vertex remains finite in this limit. The running coupling as extracted from either of these vertex functions possesses an infrared fixed point. In general, diagrams including ghost-loops dominate in the infrared over purely gluonic ones.
Green's functions are a central element in the attempt to understand non-perturbative phenomena in Yang-Mills theory. Besides the propagators, 3-point Green's functions play a significant role, since they permit access to the running coupling constant and are an important input in functional methods. Here we present numerical results for the two non-vanishing 3-point Green's functions in 3d pure SU(2) Yang-Mills theory in (minimal) Landau gauge, i.e. the three-gluon vertex and the ghost-gluon vertex, considering various kinematical regimes. In this exploratory investigation the lattice volumes…
We calculate the dressed gluon and ghost propagators of Landau gauge Yang-Mills theory in the complex momentum plane from their Dyson-Schwinger equations. To this end, we develop techniques for a direct calculation such that no mathematically ill-posed inverse problem needs to be solved. We provide a detailed account of the employed ray technique and discuss a range of tools to monitor the stability of the numerical calculation. Within a truncation employing model ansaetze for the three-point vertices and neglecting effects due to four-point functions, we find a singularity in the gluon propag…
Possible mechanisms for gluon confinement in Yang-Mills theory in covariant gauges are discussed briefly. Implications for the infrared behavior of the Green's functions are outlined. Results for the gluon and Faddeev-Popov ghost propagators from Dyson-Schwinger studies and lattice calculations are presented. The importance of the ghost-gluon vertex in these studies is discussed. Numerical results for a non-perturbatively constructed ghost-gluon vertex in four and three space-time dimensions are presented.
A semi-perturbative calculation of the ghost-gluon vertex in Landau gauge Yang-Mills theory in four and three Euclidean space-time dimensions is presented. Non-perturbative gluon and ghost propagators are employed, which have previously been calculated from a truncated set of Dyson--Schwinger equations and which are in qualitative and quantitative agreement with corresponding lattice results. Our results for the ghost-gluon vertex show only relatively small deviations from the tree-level one in agreement with recent lattice data. In particular, we do not see any sign for a singular behaviour o…
We summarise results for the running coupling from the four-gluon vertex in Landau gauge, SU($N_c$) Yang-Mills theory as given by a combination of dressing functions of the vertex and the gluon propagator. These functions have been determined numerically from the corresponding set of Dyson-Schwinger equations. In the infrared we obtain a nontrivial infrared fixed point which is three orders of magnitude smaller than the corresponding one in the coupling of the ghost-gluon vertex.
It is pointed out that the Parke-Taylor or maximally helicity violating amplitudes in the pure Yang-Mills can, after some specifications, be interpreted as amplitudes of scattering of massive vector bosons in the Higgs-Yang-Mills system.
Non-Abelian gauge theories with fermions transforming in the adjoint representation of the gauge group (AdjQCD) are a fundamental ingredient of many models that describe the physics beyond the Standard Model. Two relevant examples are N=1 Supersymmetric Yang-Mills (SYM) theory and Minimal Walking Technicolor, which are gauge theories coupled to one adjoint Majorana and two adjoint Dirac fermions, respectively. While confinement is a property of N=1 SYM, Minimal Walking Technicolor is expected to be infrared conformal. We study the propagators of ghost and gluon fields in the Landau gauge to co…
We review earlier proposals for $E_8$ family unification, and discuss why recent work of Kovner and Shifman on condensates in supersymmetric Yang-Mills theories suggests the reconsideration of $E_8$ supersymmetric Yang-Mills as a family unification theory.
The three-gluon and ghost-gluon vertices of Landau gauge Yang-Mills theory are investigated in the low momentum regime. Due to ghost dominance in the infrared we can use the known power law behavior for the propagators to determine analytically the complete momentum dependence of the dressing functions. Besides a uniform, i. e. all momenta going to zero, divergence, we find additional singularities, if one momentum alone goes to zero, while the other two remain constant. At these asymmetric points we can extract additional infrared exponents, which corroborate previous results and expand the k…
I discuss various situations in which perturbative expansions are used in Yang-Mills theories with asymptotic freedom and establish the limits of its applicability.
The gluon and ghost propagators in Landau Gauge Yang-Mills Theory are investigated. Self-consistent solutions are obtained from their equations of motion above and below the presumed phase transition. Gluon confinement is manifest in these solutions and can be read off the infrared behavior of the gluon and ghost propagators. Confinement prevails below the presumed phase transition. Above and in the infinite temperature limit, a qualitative change is observed: The chromoelectric sector exhibits screening, while long-range chromomagnetic interactions, mediated by soft modes, are still observed.…
The properties of the high-temperature phase of Yang-Mills theory in Landau gauge are investigated by extending an earlier study on the infinite-temperature limit to finite temperatures. To this end the Dyson-Schwinger equations for the propagators of the gluon and the Faddeev-Popov ghost are solved analytically in the infrared and numerically at non-vanishing momenta. Gluons, polarized transversely with respect to the heat bath are found to comply with the Gribov-Zwanziger and the Kugo-Ojima scenario, while longitudinally polarized gluons are screened. Therefore the high-temperature phase is …
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…
The high-temperature phase of Yang-Mills theory in Landau gauge is studied using Dyson-Schwinger equations. The propagators of the gluon and the Faddeev-Popov ghosts are obtained at finite and infinite temperature, partially analytically. The results are in good agreement with lattice results. It is found that the infrared properties are only quantitatively affected by temperature. The confinement of at least soft gluons transverse to the heat bath is established, in accordance with the Kugo-Ojima and Gribov-Zwanziger confinement scenarios. This is confirmed by investigating the corresponding …
${\cal N}=4$ Super Yang-Mills theory is a highly constrained theory, and therefore a valuable tool to test the understanding of less constrained Yang-Mills theories. Our aim is to use it to test our understanding of both the Landau gauge beyond perturbation theory as well as truncations of Dyson-Schwinger equations in ordinary Yang-Mills theories. We derive the corresponding equations within the usual one-loop truncation for the propagators after imposing the Landau gauge. We find a conformal solution in this approximation, which surprisingly resembles many aspects of ordinary Yang-Mills theor…
Yang-Mills theories undergo a deconfining phase transition at a critical temperature. In lattice calculations the temporal Wilson loop and Z_3 order parameter show above this temperature a behavior typical of deconfinement. A quantity of interest in the study of this transition is the gluon propagator and its evolution with temperature. This contribution describes the current status of an investigation of the finite temperature gluon propagator in Landau gauge. It analyzes the high temperature case. The resulting equations are compared to the corresponding ones of three-dimensional Yang-Mills …
We study the decay of a Z' boson into two Z bosons by extending the Landau-Yang theorem to a parent particle decaying into two Z bosons. For a spin-1 parent the theorem predicts: 1) there are only two possible couplings and 2) the normalized differential cross-section depends on kinematics only through a phase shift in the azimuthal angle between the two decay planes of the Z boson. When the parent is a Z' the two possible couplings are anomaly-induced and CP-violating, respectively. At the Large Hadron Collider (LHC) effects of the two couplings could be disentangled when both Z bosons decay …
Landau-Yang theorem is sometimes formulated as a selection rule forbidding two real (that is, non-virtual) photons with zero total momentum to be in the state of the total angular momentum J=1. In this paper we discuss whether the theorem itself and this particular formulation can be extended to a pair of two {\em twisted} photons, which carry orbital angular momentum with respect to their propagation direction. We point out possible sources of confusion, which may arise both from the unusual features of twisted photons and from the fact that usual proofs of the Landau-Yang theorem operate in …
We discuss exclusive elastic double diffractive axial-vector χ_c(1^{+}) meson production in proton-antiproton collisions at the Tevatron. The amplitude for the process is derived within the k_t-factorisation approach with unintegrated gluon distribution functions (UGDFs). We show that the famous Landau-Yang theorem is not applicable in the case of off-shell gluons. Differential cross sections for different UGDFs are calculated. We compare exclusive production of χ_c(1^+) and χ_c(0^+). The contribution of χ_c(1^+) to the J/Ψ+ γchannel is smaller than that of the χ_c(0^+) decay, but not negligib…
We discuss a non-minimal Einstein-Yang-Mills-Higgs model with uniaxial anisotropy in the group space associated with the Higgs field. We apply this theory to the problem of propagation of color and color-acoustic waves in the gravitational background related to the non-minimal regular Wu-Yang monopole.
Recently it has been argued that infrared singularities of the quark-gluon vertex of Landau gauge QCD can confine static quarks via a linear potential. It is demonstrated that the same mechanism also may confine fundamental scalar fields. This opens the possibility that within functional approaches static confinement is an universal property of the gauge sector even though it is formally represented in the functional equations of the matter sector. The colour structure of Dyson-Schwinger equations for fundamental and adjoint scalar fields is determined for the gauge groups SU(N) and G(2) exhib…
Correlation functions of Yang-Mills theory in the Landau gauge are calculated from their equations of motion. The employed setup is completely parameter free and leads, within errors, to good quantitative agreement with corresponding lattice results for the ghost and gluon propagators as well as the ghost-gluon and three-gluon vertices. Also the four-gluon vertex is calculated. The present setup allows for the first time for a unique subtraction of quadratic divergences in the gluon propagator Dyson-Schwinger equation. Thus, there are no ambiguities which can arise due to the use of models or …
We consider the running coupling from the four-gluon vertex in Landau gauge, SU($N_c$) Yang-Mills theory as given by a combination of dressing functions of the vertex and the gluon propagator. We determine these functions numerically from a coupled set of Dyson-Schwinger equations. We reproduce asymptotic freedom in the ultraviolet momentum region and find a coupling of order one at mid-momenta. In the infrared we find a nontrivial (i.e. nonzero) fixed point which is three orders of magnitude smaller than the corresponding fixed point in the coupling of the ghost-gluon vertex. This result expl…
An analytical and inductive approach to hot SU(N) Yang-Mills dynamics is developed. For N=2,3 pressure and energy density are pointwise compared with lattice data.
The gauge invariant definition of the spin dependent gluon distribution function from first principle is necessary to study the proton spin crisis at high energy colliders. In this paper we derive the gauge invariant Noether's theorem in Yang-Mills theory by using combined Lorentz transformation plus local non-abelian gauge transformation. We find that the definition of the gauge invariant spin (or orbital) angular momentum of the Yang-Mills field does not exist in Yang-Mills theory although the definition of the gauge invariant spin (or orbital) angular momentum of the quark exists. We show t…
