New Schwinger-Dyson equations for non-Abelian gauge theories

Physical Review D, 1972

Feynman rules for the massless Yang-Mills field interacting with nucleons ar_e derived using th? well-known method of canonical quantization. In the radiation gauge for the b, field, i.e., iib,=O, the interaction Hamiltonian is an infinite series of noncovariant terms containing the b, field and the I) field thus giving an infinity of noncovariant vertices. It is shown that in spite of a seemingly infinite sequence of primary diagrams, it is possible to describe all the diagrams to any order by a set of noncovariant Feynman rules involving three propagators SF, D,, , and F ,, and three vertices I ' ; , T ;yX, and r ; Tf , . The same set of noncovariant rules holds for diagrams with any number of loops provided that a loop with all F,,-type propagators is multiplied by a weight factor (1 -1/2"-'); all other loops have coefficients given by the symmetry number. It is proved that tree diagrams can be covariantized to any order thus giving a simple covariant set of Feynman rules to describe them to a y order. Similarly the one loop to order gZ can also be made covariant. Only the closed b, field loops result in an extra scalar loop with weight ( -2) relative to the vector loop on covariantization.

Nuclear Physics B, 2003

We demonstrate the effectivity of the covariant background field method by means of an explicit calculation of the 3-loop β-function for a pure Yang-Mills theory. To maintain manifest background invariance throughout our calculation, we stay in coordinate space and treat the background field non-perturbatively. In this way the presence of a background field does not increase the number of vertices and leads to a relatively small number of vacuum graphs in the effective action. Restricting to a covariantly constant background field in Fock-Schwinger gauge permits explicit expansion of all quantum field propagators in powers of the field strength only. Hence, Feynman graphs are at most logarithmically divergent. At 2-loop order only a single Feynman graph without subdivergences needs to be calculated. At 3-loop order 24 graphs remain. Insisting on manifest background gauge invariance at all stages of a calculation is thus shown to be a major labor saving device. All calculations were performed with Mathematica in view of its superior pattern matching capabilities. Finally, we describe briefly the extension of such covariant methods to the case of supergravity theories.

Phys Lett B, 1997

Yang-Mills theory in the first order formalism appears as the deformation of a topological field theory, the pure BF theory. We discuss this formulation at the quantum level, giving the Feynman rules of the BF-YM theory, the structure of the renormalization and checking its uv-behaviour in the computation of the β-function which agrees with the expected result.

1993

We use the S-matrix pinch technique to derive to one loop-order gauge independent γ W^+ W^- and Z W^+ W^- vertices in the context of the Standard Model,with all three incoming momenta off-shell. We show that the γ W^+ W^- vertex so constructed is related to the gauge-independent W self-energy,derived by Degrassi and Sirlin,by a very simple QED-like Ward identity.The same results are obtained by the pinch technique applied directly to the process e^+e^-→ W^+W^-. Explicit calculations give rise to expressions for the static properties of the W gauge bosons like magnetic dipole and electric quadruple moments which being gauge independent satisfy such crucial properties as infrared finiteness and perturbative unitarity.

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 relatively mild impact, whereas in d = 3 they are linear, giving rise to enhanced effects. In the case of the gluon propagator, these effects do not interfere with its finiteness, but make its first derivative diverge at the origin, and introduce a maximum in the region of infrared momenta. The three-gluon vertex is also affected, and the induced divergent behavior is clearly exposed in certain special kinematic configurations, usually considered in lattice simulations; the sign of the corresponding divergence is unambiguously determined. The main underlying concepts are developed in the context of a simple toy model, which demonstrates clearly the interconnected nature of the various effects. The picture that emerges is subsequently corroborated by a detailed nonperturbative analysis, combining lattice results with the dynamical integral equations governing the relevant ingredients, such as the nonperturbative ghost loop and the momentum-dependent gluon mass.

Journal of Physics G: Nuclear and Particle Physics, 2004

The all-order construction of the pinch technique gluon self-energy and quark-gluon vertex is presented in detail within the class of linear covariant gauges. The main ingredients in our analysis are the identification of a special Green's function, which serves as a common kernel to all self-energy and vertex diagrams, and the judicious use of the Slavnov-Taylor identity it satisfies. In particular, it is shown that the ghost-Green's functions appearing in this identity capture precisely the result of the pinching action at arbitrary order. By virtue of this observation the construction of the quarkgluon vertex becomes particularly compact. It turns out that the aforementioned ghost-Green's functions play a crucial role, their net effect being the non-trivial modification of the ghost diagrams of the quark-gluon vertex in such a way as to reproduce dynamically the characteristic ghost sector of the background field method. The gluon self-energy is also constructed following two different procedures. First, an indirect derivation is given, by resorting to the strong induction method and the assumption of the uniqueness of the S-matrix. Second, an explicit construction based on the intrinsic pinch technique is provided, using the Slavnov-Taylor identity satisfied by the all-order three-gluon vertex nested inside the self-energy diagrams. The process-independence of the gluon self-energy is also demonstrated, by using gluons instead of quark as external test particles, and identifying the corresponding kernel function, together with its Slavnov-Taylor identity. Finally, the general methodology for carrying out the renormalization of the resulting Green's functions is outlined, and various open questions are briefly discussed.

1996

Yang-Mills theory in the first order formalism appears as the deformation of a topological field theory, the pure BF theory. In this approach new non local observables are inherited from the topological theory and the operators entering the t'Hooft algebra find an explicit realization. A calculation of the {\it vev}'s of these operators is performed in the Abelian Projection gauge.

Journal of High Energy Physics, 2006

In this article we study the general structure and special properties of the Schwinger-Dyson equation for the gluon propagator constructed with the pinch technique, together with the question of how to obtain infrared finite solutions, associated with the generation of an effective gluon mass. Exploiting the known all-order correspondence between the pinch technique and the background field method, we demonstrate that, contrary to the standard formulation, the non-perturbative gluon self-energy is transverse order-by-order in the dressed loop expansion, and separately for gluonic and ghost contributions. We next present a comprehensive review of several subtle issues relevant to the search of infrared finite solutions, paying particular attention to the role of the seagull graph in enforcing transversality, the necessity of introducing massless poles in the three-gluon vertex, and the incorporation of the correct renormalization group properties. In addition, we present a method for regulating the seagull-type contributions based on dimensional regularization; its applicability depends crucially on the asymptotic behavior of the solutions in the deep ultraviolet, and in particular on the anomalous dimension of the dynamically generated gluon mass. A linearized version of the truncated Schwinger-Dyson equation is derived, using a vertex that satisfies the required Ward identity and contains massless poles belonging to different Lorentz structures. The resulting integral equation is then solved numerically, the infrared and ultraviolet properties of the obtained solutions are examined in detail, and the allowed range for the effective gluon mass is determined. Various open questions and possible connections with different approaches in the literature are discussed.

We calculate the massive fermion propagator at one-loop order in QED3. The Ward-Takahashi identity (WTI) relates the propagator to the vertex. This allows us to split the vertex into its longitudinal and transverse parts. The former is fixed by the WTI. Following the scheme of Ball and Chiu later modified by Kizilersii et. al., we calculate the full vertex at one-loop order. A mere subtraction of the longitudinal part of the vertex gives us the transverse part. The a dependence in the transverse vertex can be eliminated by making use of the perturbative expressions for the wavefunction renormalization function and the mass function of complicated arguments of the incoming and outgoing fermion momenta. This leads us to a vertex which is nonperturbative in nature. We also calculate an effective vertex for which the arguments of the unknown functions have no angular dependence, making it particularly suitable for numerical studies of dynamical symmetry breaking.

Journal of the Franklin Institute, 1985

Journal of High Energy Physics, 2011

We construct a general Ansatz for the three-particle vertex describing the interaction of one background and two quantum gluons, by simultaneously solving the Ward and Slavnov-Taylor identities it satisfies. This vertex is known to be essential for the gauge-invariant truncation of the Schwinger-Dyson equations of QCD, based on the pinch technique and the background field method. A key step in this construction is the formal derivation of a set of crucial constraints (shown to be valid to all orders), relating the various form factors of the ghost Green's functions appearing in the aforementioned Slavnov-Taylor identity. When inserted into the Schwinger-Dyson equation for the gluon propagator, this vertex gives rise to a number of highly non-trivial cancellations, which are absolutely indispensable for the self-consistency of the entire approach.

2000

2005

An approximate procedure for performing nonperturbative calculations in quantum field theories is presented. The focus will be quantum non-Abelian gauge theories with the goal of understanding some of the open questions of these theories such as the confinement phenomenon and glueballs. One aspect of this nonperturbative method is the breaking down of the non-Abelian gauge group into smaller pieces. For example SU(2)→ U(1) + coset or SU(3)→ SU(2) + coset. The procedure also uses some aspects of an old method by Heisenberg to calculate the n-point Green’s function of a strongly interacting, non-linear theory. Using these ideas we will give approximate calculations of the 2 and 4-points Green’s function of the theories considered.

2012

We calculate gauge theory one-loop amplitudes with the aid of the complex shift used in the Britto-Cachazo-Feng-Witten (BCFW) recursion relations of tree amplitudes. We apply the shift to the integrand and show that the contribution from the limit of infinite shift vanishes after integrating over the loop momentum, by a judicious choice of basis for polarization vectors. This enables us to write the one-loop amplitude in terms of on-shell tree and lower point one-loop amplitudes. Some of the tree amplitudes are forward amplitudes. We show that their potential singularities do not contribute and the BCFW recursion relations can be applied in such a way as to avoid these singularities altogether. We calculate in detail n-point one-loop amplitudes for n=2,3,4, and outline the generalization of our method to n>4.

Acta Physica Polonica B Proceedings Supplement, 2015

We review the status of calculations of Yang-Mills Green functions from Dyson-Schwinger equations. The role of truncations is discussed and results for the four-gluon vertex are presented.