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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Renormalization Of Composite Operators. Short Distance Expansion

Renormalization Of Composite Operators. Short Distance Expansion

Quantum Field Theory and Critical Phenomena


Oxford University Press

This chapter discusses two related problems: the renormalization of composite operators, that is, local polynomials of the field, and the short distance expansion (SDE) of the product of operators. It first discusses the renormalization of composite operators in general from the point of view of power counting. It uses the relations between bare and renormalized operators to establish a set of CS equations for the insertion of operators of dimension 4 in the φ4 4 field theory. It then shows that in a given field theory there exist linear relations between operators due to the equations of motion and relations derived in Section 7.3. The second part of the chapter is devoted to the SDE. It first establishes the existence of a SDE for the product of two basic fields, and discusses the SDE at leading order in the φ4 4 field theory. It then passes from short distance behaviour to large momentum behaviour and derives a CS equation for the coefficient of the expansion at leading order. Finally, the chapter briefly discusses the generalization of this analysis to the SDE beyond leading order — to the SDE of arbitrary operators, and to the light cone expansion which appears when one studies the large momentum behaviour of real time correlation functions (in contrast to euclidean imaginary time).

Keywords:   renormalization, composite operators, short distance expansion, large momentum behaviour, light cone expansion, CS equations

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