Abstract
The contributions of the matter superfields and of the Faddeev-Popov ghosts to the β-function of mathcal{N} = 1 supersymmetric gauge theories defined in terms of the bare couplings are calculated in all orders in the case of using the higher covariant derivative regularization. For this purpose we use the recently proved statement that the β-function in these theories is given by integrals of double total derivatives with respect to the loop momenta. These integrals do not vanish due to singularities of the integrands. This implies that the β-function beyond the one-loop approximation is given by the sum of the singular contributions, which is calculated in all orders for singularities produced by the matter superfields and by the Faddeev-Popov ghosts. The result is expressed in terms of the anomalous dimensions of these superfields. It coincides with the corresponding part of the new form of the NSVZ equation, which can be reduced to the original one with the help of the non-renormalization theorem for the triple gauge-ghost vertices.
Highlights
The constant C2 is defined by the equation C2δAB = f ACDf BCD
The main observation leading to this statement is that the renormalization group functions (RGFs) defined in terms of the bare couplings presumably satisfy the NSVZ equation in all orders in the case of using the higher covariant derivative regularization
For N = 1 supersymmetric gauge theories regularized by dimensional reduction the NSVZ equation for RGFs defined in terms of the bare couplings is not valid starting from the order O(α02, α0λ20, λ40), where the dependence on a regularization becomes essential [30]
Summary
The expression (2.8) allows regularizing all divergences beyond the one-loop approximation, for the all-loop derivation of the NSVZ β-function we need to introduce some auxiliary parameters, namely, the complex coordinate independent parameter g and the chiral superfield g(xμ, θ).. The expression (2.8) allows regularizing all divergences beyond the one-loop approximation, for the all-loop derivation of the NSVZ β-function we need to introduce some auxiliary parameters, namely, the complex coordinate independent parameter g and the chiral superfield g(xμ, θ).4 The former one relates the original theory corresponding to g = 1 to the theory in which quantum superfields interact only with the background gauge superfield corresponding to g → 0. This action contains higher covariant derivatives inside the regulator function K(x) which satisfies the standard conditions K(0) = 1, K(x → ∞) → ∞. Parameterized by a chiral superfield A which takes values in the Lie algebra of the gauge group
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