Finite field equation of Yang–Mills theory

• Published in: Journal of mathematical physics 1980-03, Vol.21 (3), p.547-560
• Main Authors: Brandt, Richard A., Wing‐Chiu, Ng, Yeung, Wai‐Bong
• Format: Article
• Language: English
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.524453

We consider the finite local field equation −{[1+1/α (1+f 4)]g μν⧠−∂μ∂ν}A νa =−(1+f 3) g 2 N[A cν A aμ A ν c ] +⋅⋅⋅+(1−s)2 M 2 A aμ, introduced by Lowenstein to rigorously describe SU(2) Yang–Mills theory, which is written in terms of normal products. We also consider the operator product expansion A cν(x+ξ) A aμ(x) A bλ(x−ξ) ∼ΣM c a bνμλ c′a′b′ν′μ′λ′ (ξ) N[A ν′c′ A μ′a′ A λ′b′](x ), and using asymptotic freedom, we compute the leading behavior of the Wilson coefficients M ... ...(ξ) with the help of a computer, and express the normal products in the field equation in terms of products of the c‐number Wilson coefficients and of operator products like A cν(x+ξ) A aμ(x) A bλ(x−ξ) at separated points. Our result is −{[1+(1/α)(1+f 4)]g μν⧠−∂μ∂ν }A νa =−(1+f 3) g 2limξ→0{ (lnξ)−0.28/2b [A cν (x+ξ) A aμ(x) A ν c (x−ξ) +ε a b c A μc (x+ξ) ∂ν A b ν(x)+⋅⋅⋅] +⋅⋅⋅}+(1−s)2 M 2 A aμ, where β (g) =−b g 3, and so (lnξ)−0.28/2b is the leading behavior of the c‐number coefficient multiplying the operator products in the field equation.