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The Manifestly Gauge Invariant

Exact Renormalisation Group

Abstract

We construct a manifestly gauge invariant Exact Renormalisation Group (ERG)
whose form is suitable for
computation in SU(N) Yang-Mills theory, beyond one-loop.
An effective cutoff is implemented by embedding the physical
SU(N) theory in a
spontaneously broken SU(N|N) Yang-Mills theory.

To facilitate computations within this scheme, which proceed at every
step without fixing the gauge, we develop a set of diagrammatic
techniques. As an initial test of the formalism, the one-loop SU(N) Yang-Mills
beta-function, beta_1, is computed, and the standard, universal answer
is reproduced.

It is recognised that the computational technique can be greatly
simplified. Using these simplifications, a partial proof
is given that, to all orders in perturbation theory,
the explicit dependence of perturbative beta-function coefficients, beta_n,
on certain non-universal elements of the manifestly gauge invariant
ERG cancels out. This partial proof yields an extremely
compact, diagrammatic form for the surviving contributions to
arbitrary beta_n, up to a set of terms which are
yet to be dealt with. The validity of the
compact expression is reliant on an unproven assertion
at the third loop order and above.

Starting from the compact expression for beta_n, we
specialise to beta_2 and explicitly construct the
set of terms yet to be dealt with. From the resulting diagrammatic
expression for beta_2, we extract a numerical coefficient which,
in the limit that the coupling of one of the unphysical
regulator fields is tuned to zero, yields the standard,
universal answer. Thus, we have performed the very first
two-loop, continuum calculation in Yang-Mills
theory, without fixing the gauge.

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Thesis pdf (3178kB)

Last modified: Thu Aug 11 11:35:48 BST 2005