The $\beta$-function of $\mathcal{N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives as an integral of double total derivatives

Abstract

For a general $\mathcal{N}=1$ supersymmetric gauge theory regularized by higher covariant derivatives we prove in all orders that the $\beta$-function defined in terms of the bare couplings is given by integrals of double total derivatives with respect to loop momenta. With the help of the technique used for this proof it is possible to construct a method for obtaining these loop integrals, which essentially simplifies the calculations. As an illustration of this method, we find the expression for the three-loop contribution to the $\beta$-function containing the Yukawa couplings and compare it with the result of the standard calculations made earlier. Also we briefly discuss, how the structure of the loop integrals for the $\beta$-function considered in this paper can be used for the all-loop perturbative derivation of the NSVZ relation in the non-Abelian case.

43 pages, 4 figures

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Cited by in corpus (14)

• The all-loop perturbative derivation of the NSVZ $\beta$-function and the NSVZ scheme in the non-Abelian case by summing singular contributions
• Two-loop renormalization of the matter superfields and finiteness of $\mathcal{N}=1$ supersymmetric gauge theories regularized by higher derivatives
• The NSVZ relations for $\mathcal{N}=1$ supersymmetric theories with multiple gauge couplings
• The NSVZ $\beta$-function for theories regularized by higher covariant derivatives: the all-loop sum of matter and ghost singularities
• The higher covariant derivative regularization as a tool for revealing the structure of quantum corrections in supersymmetric gauge theories
• The three-loop anomalous dimension and the four-loop $\beta$-function for $\mathcal{N}=1$ SQED regularized by higher derivatives
• Three-loop $\beta$-functions and two-loop anomalous dimensions for MSSM regularized by higher covariant derivatives in an arbitrary supersymmetric subtraction scheme
• On supersymmetric gauge theories with higher derivatives and nonlocal terms in the matter sector
• Finiteness of the two-loop matter contribution to the triple gauge-ghost vertices in $\mathcal{N}=1$ supersymmetric gauge theories regularized by higher derivatives
• Exact $\beta$-functions for $\mathcal{N}=1$ supersymmetric theories finite in the lowest loops
• Finiteness of the triple gauge-ghost vertices in $\mathcal{N}=1$ supersymmetric gauge theories: the two-loop verification
• The four-loop $\beta$-function from vacuum supergraphs and the NSVZ relation for $\mathcal{N}=1$ SQED regularized by higher derivatives
• The $\beta$-function of supersymmetric theories from vacuum supergraphs: a three-loop example
• Quantum properties of gauge theories with extended supersymmetry formulated in $\mathcal{N}=1$ superspace