Evolution equations beyond one loop from conformal symmetry

  • V. M. BraunEmail author
  • A. N. Manashov
Regular Article - Theoretical Physics


We study implications of exact conformal invariance of scalar quantum field theories at the critical point in non-integer dimensions for the evolution kernels of the light-ray operators in physical (integer) dimensions. We demonstrate that all constraints due the conformal symmetry are encoded in the form of the generators of the collinear sl(2) subgroup. Two of them, S and S 0, can be fixed at all loops in terms of the evolution kernel, while the generator of special conformal transformations, S +, receives nontrivial corrections which can be calculated order by order in perturbation theory. Provided that the generator S + is known at the −1 loop order, one can fix the evolution kernel in physical dimension to the -loop accuracy up to terms that are invariant with respect to the tree-level generators. The invariant parts can easily be restored from the anomalous dimensions. The method is illustrated on two examples: The O(n)-symmetric φ 4 theory in d=4 to the three-loop accuracy, and the su(n) matrix φ 3 theory in d=6 to the two-loop accuracy. We expect that the same technique can be used in gauge theories e.g. in QCD.


Anomalous Dimension Conformal Symmetry Casimir Operator Critical Coupling Invariant Part 
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A.M. is grateful to Dieter Müller and Sergey Derkachov for helpful discussions. This work was supported by the DFG, grant BR2021/5-2.


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© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversity of RegensburgRegensburgGermany
  2. 2.Department of Theoretical PhysicsSt.-Petersburg State UniversitySt.-PetersburgRussia

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