We give a general definition of the conformal anomaly for theories that are not classically Weyl invariant and show that this definition yields a quantity that is both finite and local. As an example we study the conformal anomaly for a non-minimally coupled massless scalar and show that our definition coincides with results obtained using the heat kernel method.

A classically scale-invariant 6d analog of the 4d Yang-Mills theory is the 4-derivative (∇F ) 2 + F 3 gauge theory with two independent couplings. Motivated by a search for a perturbatively conformal but possibly non-unitary 6d models we compute the one-loop β-functions in this theory. A systematic way of doing this using the background field method requires the (previously unknown) expression for the b 6 Seeley-DeWitt coefficient for a generic 4-derivative operator; we derive it here. As an application, we also compute the one-loop β-function in the (1,0) supersymmetric (∇F ) 2 6d gauge theory constructed in hep-th/0505082. 1 lorenzo.casarin@aei.mpg.de 2 Also at the Lebedev Institute, Moscow. tseytlin@imperial.ac.uk 1 In 4 dimensions the F 2 + (∇F ) 2 + F 3 theory was studied in [2] and later in [3]. The result of [2] for the one-loop divergences in this 4d theory was corrected in [4] making it in agreement with that of [3].2 We use m, n, k, ... = 1, ..., 6 for coordinate indices and flat Euclidean 6d metric so that the position of contracted indices is irrelevant. The gauge group generators are normalized as tr(t a t b ) = −TRδ ab , [t a , t b ] = f abc t c , where TR = 1 2 in the fundamental representation of SU (N ) (we denote the trace in this case as Tr) and TR = C2 = N in the adjoint representation.3 Two other possible 4-derivative ∇F ∇F invariants are related to the above two by the Bianchi identity, e.g.,Fmn∇ 2 Fmn = −2 (∇mFmn) 2 + 4FmnF nk F km + total derivative. 4 Here tr is the trace over the matrix indices of a particular representation to which the quantum field belongs;for example, in the gauge theory case it is in the adjoint representation A ab m = f acb A c m , f acd f bcd = C2δ ab .

Motivated by the goal of applying the average null energy condition (ANEC) to renormalisation group flows, we calculate in λϕ4 theory the expectation value of the ANEC operator in a particular scalar state perturbatively up to third order in the quartic coupling and verify the expected CFT answer. The work provides the technical tools for studying the expectation value of the ANEC operator in more interesting states, for example tensorial states relevant to the Hofman-Maldacena collider bounds, away from critical points.

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