Renormalization of Yang-Mills Theories

Abstract

It has already been said in Chapter I that the Yang-Mills field Aμ(x) is a vector function of x = {x₀, x₁, x₂, x₃} which takes values in the space of the adjoint representation of the Lie algebra of a compact semisimple group G:

$$ {\rm{A\mu }}\left( {\rm{x}} \right)\;\;{\rm{ = }}\;\;{\rm{A}}_{\rm{\mu }}^{\rm{d}}{{\rm{T}}^{\rm{d}}}\;{\rm{,}} $$

(1)

where T^d are the Hermitean generators of the group. The Yang-Mills field (or the gauge field) is simultaneously a Poincaré vector with μ being the Lorentz index. The matrix function U(x) (where for every x one has U(x) ∈ G) is an element of the gauge group (multiplication law being (U₁U₂) (x) = U₁ (x)U₂ (x)). Under the transformation U(x) of the gauge group the Yang-Mills field transforms according to the rule

$$ {\rm{A\mu }}\left( {\rm{x}} \right)\; \to \;{\rm{A}}_{\rm{\mu }}^{\rm{U}}\;\left( {\rm{x}} \right)\;\;{\rm{ = }}\;\;{\rm{U}}\left( {\rm{x}} \right){{\rm{A}}_{\rm{\mu }}}\;\left( {\rm{x}} \right){{\rm{U}}^{\rm{ + }}}\;\left( {\rm{x}} \right)\;\;{\rm{ - }}\;\;{\rm{i}}{{\rm{g}}^{{\rm{ - 1}}}}{\partial _{\rm{u}}}{\rm{U}}\;\left( {\rm{x}} \right)\;{{\rm{U}}^{\rm{ + }}}\;\left( {\rm{x}} \right) $$

(2)

with g interpreted as a coupling constant of Yang-Mills theory. If U(x) is an infinitesimal transformation of the gauge group — that is, if

$$ {\rm{U}}\left( {\rm{x}} \right)\;\;{\rm{ = }}\;\;{\rm{1}}\;\;{\rm{ + }}\;\;{\rm{ig\varepsilon }}\;\left( {\rm{x}} \right)\;\;\; \equiv \;\;{\rm{1}}\;\;{\rm{ + }}\;\;{\rm{ig}}{{\rm{\varepsilon }}^{\rm{d}}}\;\left( {\rm{x}} \right)\;{{\rm{T}}^{\rm{d}}}\;{\rm{,}} $$

(3)

where ɛ^d(x) are infinitely small real functions — then the transformation (2) takes the form

$$ \begin{array}{l} {{\rm{A}}_{\rm{\mu }}}\; \to \;{{\rm{A}}_{\rm{\mu }}}\;\;{\rm{ + }}\;\;{\rm{\delta }}{{\rm{A}}_{\rm{\mu }}}\;{\rm{,}}\\ {\rm{\delta }}{{\rm{A}}_{\rm{\mu }}}\;\left( {\rm{x}} \right)\;\;{\rm{ = }}\;\;{\partial _{\rm{\mu }}}{\rm{\varepsilon }}\left( {\rm{x}} \right)\;\;{\rm{ + }}\;\;{\rm{ig}}\left[ {{\rm{\varepsilon }}\left( {\rm{x}} \right)\;{\rm{,}}\;\;{{\rm{A}}_{\rm{\mu }}}\;\left( {\rm{x}} \right)} \right]\;{\rm{,}} \end{array} $$

(4)