Conformal Contact Terms and Semi-local Terms
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Annales Henri Poincar´ e
Conformal Contact Terms and Semi-local Terms Yu Nakayama Abstract. We study conformal properties of local terms such as contact terms and semi-local terms in correlation functions of a conformal field theory. Not all of them are universal observables, but they do appear in physically important correlation functions such as (anomalous) Ward– Takahashi identities or Schwinger–Dyson equations. We develop some tools such as embedding space delta functions and effective action to examine conformal invariance of these local terms.
1. Introduction Correlation functions of local operators are fundamental observables in conformal field theories. They are non-local functions of space, and they possess severe constraints from the conformal symmetry and the operator product expansions. These constraints are the basis of the conformal bootstrap approach to conformal field theories, which played a pivotal role in our current understanding of critical phenomena in higher dimensions than two. See, e.g. [1] for a recent review. They are relatively less studied, but local terms in the correlation functions are sometimes physically important. By “local” we mean they have at least one delta function (appropriately regularized) in the coordinate space correlation functions. When the support of the correlation functions is just one point, we may call them “contact terms”. Otherwise we may call them “semi-local”. To list a few physically relevant local terms, we have canonical commutation relations, Schwinger–Dyson equations, and (anomalous) Ward–Takahashi identities. It is noted that they are more or less related to the “Lagrangian picture” of the quantum field theory. It is therefore a good meeting point to understand the role of local terms in axiomatized conformal field theories, which tries to dismiss the smell of “Lagrangian”.
Y. Nakayama
Ann. Henri Poincar´e
More recently, we have learned that some local terms play important roles to understand the renormalization group flow and the moduli space of coupling constants. The effects of local terms have played crucial roles in the momentum space correlation functions in particular to understand their ultraviolet behavior. For example, the dilaton scattering amplitudes [2] that are used to understand the renormalization group flow may be affected by the semi-local terms [3–5]. The semi-local terms are also important in our understanding of shortening anomalies or anomalies in uplifting coupling constants to background fields [6–9]. Local terms in correlation functions are delicate objects because they are often affected by local counter-terms as well as a definition of the operator itself. Some local terms, however, have no such ambiguities from the symmetry consideration, and they are universal (see some examples in [10–12]). The lack of universality does not necessarily mean that they are physically irrelevant. They are important, similarly to the cosmological constant, as a part of the physical model, but they are simply independent of the framework of the conformal
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