NSVZ Relation and NSVZ Scheme in $$\mathcal{N} = 1$$ Non-Abelian Supersymmetric Gauge Theories
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SVZ Relation and NSVZ Scheme in 1 =1 Non-Abelian Supersymmetric Gauge Theories K. V. Stepanyantz* Moscow State University, Faculty of Physics, Department of Theoretical Physics, Moscow, 119991 Russia *e-mail: [email protected] Received December 20, 2019; revised January 16, 2020; accepted January 29, 2020
Abstract—We briefly describe the perturbative derivation of the exact NSVZ β -function for 1 = 1 supersymmetric non-Abelian gauge theories in the case of using the higher covariant derivative regularization. It is demonstrated that with this regularization the NSVZ equation written in the form of a relation between the β -function and the anomalous dimensions of the quantum superfields is satisfied by the renormalization group functions defined in terms of the bare couplings independently of a renormalization prescription. For the renormalization group functions defined in terms of the renormalized couplings this implies that one of the NSVZ schemes in all loops is given by the HD+MSL prescription, when the theory is regularized by higher derivatives and the renormalization constants contain only powers of ln Λ /μ . All these statements are verified by comparing certain terms in the three-loop β -function with the corresponding contributions to anomalous dimensions of the quantum superfields. DOI: 10.1134/S1063779620040693
1. INTRODUCTION In 1 = 1 supersymmetric gauge theories the β-function is related to the anomalous dimension of the matter superfields in all loops by the exact Novikov, Shifman, Vainshtein, and Zakharov (NSVZ) β-function [1–4],
α2(3C2 − T (R) + C (R)ij γ ij (α, λ)/r ) β(α, λ) = − . 2π(1 − C2α/2π)
(1)
Here α and λ are the gauge and Yukawa couplings, respectively, tr(T AT B ) ≡ T (R)δ AB , (T A )ik (T A )kj ≡ C (R)ij , C2 = T ( Adj ) , and r denotes the dimension of a simple gauge group G . This implies that for the 1 = 1 supersymmetric Yang–Mills theory (without matter superfields) the β-function is given by the geometric series. Three- and four-loop calculations made for 1 = 1 supersymmetric theories in the DR -scheme (i.e. with the help of dimensional reduction supplemented by the modified minimal subtractions) revealed [5–7, 9] that the NSVZ relation is valid only for the one- and two-loop β-function, where the scheme dependence is not essential. However, by a proper finite renormalization of the gauge coupling constant the NSVZ equation can also be restored in the three- and fourloop approximations. This implies that the disagreement can be explained by the scheme-dependence of
the NSVZ relation, which is therefore valid only in certain (NSVZ) subtraction schemes, which do not include the DR -scheme. However, the NSVZ scheme can be naturally constructed with the higher covariant derivative regularization [10, 11] in the supersymmetric version [12, 13]. The main advantage of this regularizationis that the renormalization group functions (RGFs) defined in terms of the bare couplings satisfy the NSVZ equation in all loops independently of a renormalization prescription. (Note that with the
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