Nonperturbative calculational method in quantum field theory

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.

2005

An approximate technique for performing nonperturbative calculations in quantum SU(3) gauge theory is presented. One aspect of this nonperturbative method is the breaking down $SU(3) \to SU(2) + coset$. The procedure also uses some aspects of an old method by Heisenberg to calculate the n-point Green's function of strongly interacting, non-linear fields. Using these ideas we give approximate calculations of the 2 and 4-points Green's functions.

EPJ Web of Conferences

The nonperturbative quantization technique à la Heisenberg is applied for the SU(3) gauge theory. The operator Yang-Mills equation and corresponding infinite set of equations for all Green's functions are considered. Gauge degrees of freedom are splitted into two groups: (1) A a μ ∈ S U(2) × U(1) ⊂ S U(3); (2) coset degrees of freedom S U(3)/S U(2) × U(1). Using some assumptions about 2-and 4-point Green's functions, the infinite set of equations is truncated to two equations. The first equation is the S U(2)× U(1) Yang-Mills equation, and the second equation describes a gluon condensate formed by coset fields. A flux tube solution describing longitudinal color electric fields stretched between quark and antiquark located at the ± infinities is obtained. It is shown that the dual Meissner effect appears in this solution: the electric field is pushed out from the gluon condensate. 2 Nonperturbative quantization à la Heisenberg for non-Abelian gauge theories According to Heisenberg, the SU(3) Yang-Mills operator equations can be written as follows D ν F aμν = 0, (1)

2017

The nonperturbative quantization technique à la Heisenberg is applied for non-Abelian gauge theories. The operator Yang-Mills equation is written, which on the corresponding averaging gives an infinite set of equations for all Green functions. We split all degrees of freedom into two groups: in the former, we have A^a_μ∈ G ⊂ SU(N), and in the second group we have coset degrees of freedom SU(N) / G. Using such splitting and some assumptions about 2- and 4-point Green functions, we truncate the infinite set of equations to two equations. The first equation is for the gauge fields from the subgroup G, and the second equation is for a gluon condensate which is the dispersion of quantum fluctuations of the coset fields. Two examples are considered: The first one is a flux tube solution describing longitudinal color electric fields stretched between quark and antiquark located at the ± infinities. The second one is a flux tube stretched between two quarks (antiquarks) located at ±∞. A spe...

OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), 2018

Cornell University - arXiv, 1995

Previous results on fermion chirality-ipping four-point functions are extended to non-abelian gauge theories. The problem is purely non-perturbative, and the analytical formalism used is based on the Schwinger-Dyson hierarchy. This is truncated, and the resulting equations are solved numerically by relaxation techniques. Taking the large-N limit in SU(N) theories simpli es the problem substantially, allowing the formulation of conjectures on the behavior of n-point and chirality-ipping fermion Green functions for general n.

2018

Journal of High Energy Physics, 2008

We show that the application of the pinch technique to the conventional Schwinger-Dyson equations for the gluon propagator, gluon-quark vertex, and three-gluon vertex, gives rise to new equations endowed with special properties. The new series coincides with the one obtained in the Feynman gauge of the background field method, thus capturing the extensive gauge cancellations implemented by the pinch technique at the level of individual Green's functions. Its building blocks are the fully dressed pinch technique Green's functions obeying Abelian all-order Ward identities instead of the Slavnov-Taylor identites satisfied by their conventional counterparts. As a result, and contrary to the standard case, the new equation for the gluon self-energy can be truncated gauge invariantly at any order in the dressed loop expansion. The construction is streamlined by resorting to the Batalin-Vilkovisky formalism which allows for a concise treatment of all the quantities appearing in the intermediate steps. The theoretical and phenomenological implications of this novel non-perturbative framework are discussed in detail.

2021

Yang-Mills theories based on the symplectic groups – denoted by Sp(2N) – are interesting for both theoretical and phenomenological reasons. Sp(2N) theories with two fundamental Dirac fermions give rise to pseudo-Nambu-Goldstone bosons which can be interpreted as a composite Higgs particle. This framework can describe the existing Higgs boson without the need for unnatural fine-tuning. This justifies a programme of wider investigations of Sp(2N) gauge theories aimed at understanding their general behaviour. In this work, we study the glueball mass spectrum for Sp(2N) Yang-Mills theories using the variational method applied to Monte-Carlo generated gauge configurations. This is carried out both for finite N and in the limit N → ∞. The results are compared to existing results for SU(N) Yang-Mills theories, again, for finite- and large-N. Our glueball analysis is then used to investigate some conjectures related to the behaviour of the spectrum in Yang-Mills theories based on a generic non-Abelian gauge group G. As well as studying the glueball spectrum, we examine the quenched-meson spectrum for fermions in the fundamental, antisymmetric and symmetric representations for N = 2 and N = 3. This study enables us to provide a first account of how the related observables vary with N. The investigations presented in this work contribute to our understanding of the non-perturbative dynamics of Sp(2N) gauge theories in connection with Higgs compositeness and, more in general, with fundamental open problems in non-Abelian gauge theories such as confinement and global symmetry breaking.

Cornell University - arXiv, 2020

A formal expansion for the Green's functions of an interacting quantum field theory in a parameter that somehow encodes its "distance" from the corresponding non-interacting one was introduced more than thirty years ago, and has been recently reconsidered in connection with its possible application to the renormalization of non-hermitian theories. Besides this new and interesting application, this expansion has special properties already when applied to ordinary (i.e. hermitian) theories, and in order to disentangle the peculiarities of the expansion itself from those of non-hermitian theories, it is worth to push further the investigation limiting first the analysis to ordinary theories. In the present work we study some aspects related to the renormalization of a scalar theory within the framework of such an expansion. Due to its peculiar properties, it turns out that at any finite order in the expansion parameter the theory looks as non-interacting. We show that when diagrams of appropriate classes are resummed, this apparent drawback disappears and the theory recovers its interacting character. In particular we have seen that with a certain class of diagrams, the weak-coupling expansion results are recovered, thus establishing a bridge between the two expansions.