One-loop β-functions in 4-derivative gauge theory in 6 dimensions

Abstract

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 .

Higher Derivative Gauge theory in d = 6 and the $$ \mathbb{C}{\mathbb{P}}^{\left({N}_f-1\right)} $$ NLSM

“…, Nc is the field strength of the SU(Nc) gauge field), which unlike to the QED case doesn't vanish. See also [24]. We thank to John Gracey for drawing our attention to these points.…”

“…We also mention that in [9][10][11][14][15][16][17][18][19][20] the vector model, tensor models, fermionic QED and fermionic QCD have been studied in the dimension 4 < d < 6. Also see [21][22][23][24], where the supersymmetric higher derivative gauge theories have been considered.…”

Low-energy 6D, N=(1,1) SYM effective action beyond the leading approximation

“…As the final remark, we emphasize that there are two interesting further directions of applying the background field method used here. One such direction amounts to studying the structure of the effective action in six-dimensional N = (1, 0) SYM theory with higher derivatives [42], [43], [44], another one concerns deriving the Born-Infeld-type effective action associated with D5-brane. The latter problem will require carrying out the superspace multi-loop calculations (see the relevant discussion in [45] on the Born-Infeld-type action related to D3-brane in the framework of 4D, N = 4 SYM theory).…”

Supergraph calculation of one-loop divergences in higher-derivative 6D SYM theory

“…[34]. 1 In the case under consideration, the issue of renormalizabity was studied in [34] (see also [35]) in a component formulation and in [36] in a superfield formalism. The actual calculations definitely show that the theory is renormalizable at one loop.…”

“…2 The one-loop divergences, renormalization of the coupling constant and the corresponding beta-function in this theory were calculated in refs. [34,35] by two different methods based on the component formulations.…”