The first-order formalism for Yang–Mills theory

1994 ◽  
Vol 72 (9-10) ◽  
pp. 601-607 ◽  
Author(s):  
D. G. C. McKeon
Keyword(s):  
Background Field ◽  
Loop Order ◽  
Point Function ◽  
Yang Mills ◽  
First Order ◽  
Order Form ◽  
Effective Lagrangian ◽  
Gauge Fixing ◽  
Vector Formula ◽  
Field Quantization

It is possible to replace the second-order Yang–Mills Lagrangian [Formula: see text] with the first-order Lagrangian [Formula: see text]. In this form, the interaction term in the Lagrangian, [Formula: see text] is very simple; the only disadvantage is that now not only the vector [Formula: see text] but also the auxiliary field [Formula: see text] propagate. The gauge-fixing and ghost contributions to the effective Lagrangian can similarly be reduced to first-order form by the introduction of auxiliary fields. We demonstrate the procedure by computing the two-point function to one-loop order using background field quantization.

Convex decomposition of the gauge field and background-field quantization

1992 ◽  
Vol 70 (6) ◽  
pp. 470-474 ◽  
Author(s):  
N. C. A. Hill
Keyword(s):  
Gauge Field ◽  
Background Field ◽  
Field Method ◽  
Point Function ◽  
Yang Mills ◽  
Convex Decomposition ◽  
Perturbative Method ◽  
Gauge Fixing ◽  
The Matrix ◽  
Background Field Method

The 1PI (one-particle-irreducible) two-point function of a pure Yang–Mills gauge theory is computed. The background-field method is employed in a slightly altered form that makes use of the convexity of the space of gauge fields. It is shown how this avoids the singularity of the matrix ∂2S/∂V∂V thereby allowing the calculation of the Gaussian integral in the generating functional without having to fix the gauge. It is also shown how, when it comes to actually calculating the 1PI two-point function by a perturbative method, a singularity in a particular term, [Formula: see text] of the total matrix [Formula: see text] necessitates the introduction of a gauge-fixing term. The 1PI two-point function is shown to be identical to that of the conventional background-field method except for the presence of a new parameter, t, introduced by the convex decomposition of the gauge field.

scholarly journals Chiral perturbation theory for GR

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Kirill Krasnov ◽  
Yuri Shtanov
Keyword(s):  
Perturbation Theory ◽  
Chiral Perturbation ◽  
Perturbative Calculation ◽  
New Formalism ◽  
Yang Mills ◽  
First Order ◽  
Starting Point ◽  
Gauge Fixing ◽  
New Perturbation ◽  
Effective Description

Abstract We describe a new perturbation theory for General Relativity, with the chiral first-order Einstein-Cartan action as the starting point. Our main result is a new gauge-fixing procedure that eliminates the connection-to-connection propagator. All other known first-order formalisms have this propagator non-zero, which significantly increases the combinatorial complexity of any perturbative calculation. In contrast, in the absence of the connection-to-connection propagator, our formalism leads to an effective description in which only the metric (or tetrad) propagates, there are only cubic and quartic vertices, but some vertex legs are special in that they cannot be connected by the propagator. The new formalism is the gravity analog of the well-known and powerful chiral description of Yang-Mills theory.

scholarly journals A FORMULATION OF THE YANG–MILLS THEORY AS A DEFORMATION OF A TOPOLOGICAL FIELD THEORY BASED ON THE BACKGROUND FIELD METHOD AND QUARK CONFINEMENT PROBLEM

2001 ◽  
Vol 16 (07) ◽  
pp. 1303-1346 ◽  
Author(s):  
KEI-ICHI KONDO
Keyword(s):  
Field Theory ◽  
Magnetic Monopole ◽  
Background Field ◽  
Field Method ◽  
Quark Confinement ◽  
Mass Generation ◽  
Yang Mills ◽  
Topological Quantum ◽  
Gauge Fixing ◽  
Background Field Method

By making use of the background field method, we derive a novel reformulation of the Yang–Mills theory which was proposed recently by the author to derive quark confinement in QCD. This reformulation identifies the Yang–Mills theory with a deformation of a topological quantum field theory. The relevant background is given by the topologically nontrivial field configuration, especially, the topological soliton which can be identified with the magnetic monopole current in four dimensions. We argue that the gauge fixing term becomes dynamical and that the gluon mass generation takes place by a spontaneous breakdown of the hidden supersymmetry caused by the dimensional reduction. We also propose a numerical simulation to confirm the validity of the scheme we have proposed. Finally we point out that the gauge fixing part may have a geometric meaning from the viewpoint of global topology where the magnetic monopole solution represents the critical point of a Morse function in the space of field configurations.

scholarly journals Two-loop renormalisation of gauge theories in 4D implicit regularisation and connections to dimensional methods

2021 ◽  
Vol 81 (5) ◽  
Author(s):  
A. Cherchiglia ◽  
D. C. Arias-Perdomo ◽  
A. R. Vieira ◽  
M. Sampaio ◽  
B. Hiller
Keyword(s):  
Quantum Electrodynamics ◽  
Dimensional Regularization ◽  
Background Field ◽  
Field Method ◽  
Loop Order ◽  
Abelian Gauge ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Lorentz Algebra ◽  
Background Field Method

AbstractWe compute the two-loop $$\beta $$ β -function of scalar and spinorial quantum electrodynamics as well as pure Yang–Mills and quantum chromodynamics using the background field method in a fully quadridimensional setup using implicit regularization (IREG). Moreover, a thorough comparison with dimensional approaches such as conventional dimensional regularization (CDR) and dimensional reduction (DRED) is presented. Subtleties related to Lorentz algebra contractions/symmetric integrations inside divergent integrals as well as renormalisation schemes are carefully discussed within IREG where the renormalisation constants are fully defined as basic divergent integrals to arbitrary loop order. Moreover, we confirm the hypothesis that momentum routing invariance in the loops of Feynman diagrams implemented via setting well-defined surface terms to zero deliver non-abelian gauge invariant amplitudes within IREG just as it has been proven for abelian theories.

One-loop effects in N = 4 supersymmetry

1996 ◽  
Vol 74 (3-4) ◽  
pp. 176-181
Author(s):  
D. G. C. McKeon
Keyword(s):  
Radiative Corrections ◽  
Loop Order ◽  
Four Dimensions ◽  
Yang Mills ◽  
Yukawa Couplings ◽  
Gauge Fixing ◽  
Component Field ◽  
Β Function ◽  
Supersymmetric Theories ◽  
Invariant Gauge

It has been demonstrated that in massless supersymmetric theories, finite radiative corrections to the superpotential can occur (viz. the nonrenormalization theorems can be circumvented). In this paper, we examine the consequences of this in N = 4 supersymmetric Yang–Mills theory, a model in which the β function is known to be zero. It is shown that radiative corrections to the superpotential arise at one loop order in this theory contrary to the expectations of the nonrenormalization theorem, but that their form depends on which formulation of the model is used. When one uses a superfield formulation involving an N = 1 vector superfield and three N = 1 chiral superfields in conjunction with a supersymmetric (but not SU(4)) invariant gauge fixing, then at one-loop order, the radiative generation of terms in the superpotential means that the equality of the gauge and Yukawa couplings and indeed of different Yukawa couplings is lost. If one uses the component field formulation of the N = 4 model in the Wess–Zumino gauge with a covariant, SU(4) invariant (but not supersymmetric invariant) gauge fixing, then the SU(4) invariance is maintained, but the gauge and Yukawa couplings are no longer equal. We also consider computations in the component field formulation in the Wess–Zumino gauge using an N = 1 super Yang–Mills theory in ten dimensions, dimensionally reduced to four dimensions, with a ten-dimensional covariant gauge fixing condition. This formulation ensures that there is no distinction between gauge and Yukawa couplings and that SU(4) invariance is automatically preserved; however, supersymmetry is broken by the gauge fixing procedure.

scholarly journals First order formulation of the Yang–Mills theory in a background field

2019 ◽  
Vol 409 ◽  
pp. 167932 ◽  
Author(s):  
F.T. Brandt ◽  
J. Frenkel ◽  
D.G.C. McKeon
Keyword(s):  
Background Field ◽  
Yang Mills ◽  
First Order

scholarly journals PERTURBATIVE ANALYSIS OF CHERN–SIMONS FIELD THEORY IN THE COULOMB GAUGE

1998 ◽  
Vol 13 (11) ◽  
pp. 1773-1783 ◽  
Author(s):  
FRANCO FERRARI ◽  
IGNAZIO LAZZIZZERA
Keyword(s):  
Perturbative Expansion ◽  
Coulomb Gauge ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Green Functions ◽  
Loop Order ◽  
Yang Mills ◽  
Gauge Fixing ◽  
Infrared Divergences ◽  
Chern Simons

In this paper, we analyse the perturbative aspects of Chern–Simons field theories in the Coulomb gauge. We show that in the perturbative expansion of the Green functions there are neither ultraviolet nor infrared divergences. Moreover, all the radiative corrections are zero at any loop order. Some problems connected with the Coulomb gauge fixing, like the appearance of spurious singularities in the computation of the Feynman diagrams, are discussed and solved. The regularization used here for the spurious singularities can be easily applied also to the Yang–Mills case, which is affected by similar divergences.

scholarly journals ONE-LOOP EFFECTIVE MULTIGLUON LAGRANGIAN IN ARBITRARY DIMENSIONS

1999 ◽  
Vol 14 (28) ◽  
pp. 4457-4471 ◽  
Author(s):  
E. RODULFO ◽  
R. DELBOURGO
Keyword(s):  
Field Theory ◽  
Background Field ◽  
Space Time ◽  
Yang Mills ◽  
Effective Lagrangian ◽  
Arbitrary Dimensions ◽  
Field Formalism ◽  
The One

We exhibit the one-loop multigluon effective Lagrangian in any dimension for a field theory with a quasilocal background, using the background-field formalism. Specific results, including counterterms (up to 12 space–time dimensions), have been derived, applied to the Yang–Mills theory and found to be in agreement with other string-inspired approaches.

A functional approach to loop diagrams

1996 ◽  
Vol 74 (9-10) ◽  
pp. 614-617 ◽  
Author(s):  
D. G. C. McKeon
Keyword(s):  
Heat Kernel ◽  
Background Field ◽  
Functional Approach ◽  
Closed Form Expression ◽  
Quantum Mechanical ◽  
Loop Order ◽  
Point Function ◽  
Form Expression ◽  
Generating Functional ◽  
Loop Contribution

A closed-form expression for the N-loop contribution to the generating functional can be written down using the heat kernel [Formula: see text]. If [Formula: see text] where Aμ and V are functions of the background field, then by using quantum-mechanical techniques, this heat kernel can be expanded in powers of the background field, allowing one to compute Green's functions. We demonstrate that one can also employ to this end a distinct functional approach developed by Onofri, which circumvents both loop-momentum integrals and the quantum-mechanical path integral. We illustrate the technique by computing the two-point function in scalar electrodynamics to one-loop order.

scholarly journals On restricting first order form of gauge theories to one-loop order

2021 ◽  
pp. 168426
Author(s):  
D.G.C. McKeon ◽  
F.T. Brandt ◽  
J. Frenkel ◽  
S. Martins-Filho
Keyword(s):  
Gauge Theories ◽  
Loop Order ◽  
First Order ◽  
Order Form
Sign in / Sign up

Export Citation Format

Share Document