Communications in Mathematical Physics

, Volume 313, Issue 1, pp 257–290 | Cite as

The Operator Product Expansion Converges in Perturbative Field Theory

  • Stefan Hollands
  • Christoph KopperEmail author


We show, within the framework of the massive Euclidean \({\varphi^4}\) -quantum field theory in four dimensions, that the Wilson operator product expansion (OPE) is not only an asymptotic expansion at short distances as previously believed, but even converges at arbitrary finite distances. Our proof rests on a detailed estimation of the remainder term in the OPE, of an arbitrary product of composite fields, inserted as usual into a correlation function with further “spectator fields”. The estimates are obtained using a suitably adapted version of the method of renormalization group flow equations. Convergence follows because the remainder is seen to become arbitrarily small as the OPE is carried out to sufficiently high order, i.e. to operators of sufficiently high dimension. Our results hold for arbitrary, but finite, loop orders. As an interesting side-result of our estimates, we can also prove that the “gradient expansion” of the effective action is convergent.


Renormalization Group Normal Product Formal Power Series Operator Product Expansion Loop Order 
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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.School of MathematicsCardiff UniversityCardiffUK
  2. 2.Centre de Physique Théorique, CNRS, UMR 7644, Ecole PolytechniquePalaiseauFrance

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