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An object of the crucial importance in quantum field theory is the effective action. By definition, it is a generating functional for one-particle irreducible Green's functions, which contains complete information about the quantum properties of this model. To find an effective action, it is convenient to use the technique of harmonic superspace.

Superspace[1] is a generalization of Minkowski space and allows explicitly and in a simple form to implement supersymmetry transformations. In theories with extended supersymmetry, the concept of harmonic superspace[2] plays an important role, which allows one to realize the symmetries of the theory off shell. Its important property is the presence of an analytic subspace invariant with respect to supersymmetry.

When studying $4D, \mathcal N = 4$, and $5D, \mathcal N = 2$ of supersymmetric Yang-Mills theories in harmonic superspace, some of the transformations of supersymmetry and, accordingly, $R$-symmetry are realized hiddenly[3,4]. The standard approach for obtaining low-energy effective action is as follows[4,5]. First, the leading one-loop quantum corrections for to the effective actions of $4D, \mathcal N = 2$ and $5D, \mathcal N = 1$ SYM theories are calculated. Then, by calculating the expansion of this contribution with respect to hidden supersymmetry, expression is obtained for the effective action for $ 4D, \mathcal N = 4$, and $5D, \mathcal N = 2$ theories, respectively.

In this paper, a different approach is used. The hidden $R$-symmetry transformations was found. Then, by calculating the expansion of the leading term with respect to $R$-symmetry, the expression for the effective action was obtained. This procedure was applied for $4D, \mathcal N = 4$ and $5D, \mathcal N = 2$ SYM theories. The nontrivial result is that the effective action is not only $R$-symmetric invariant, but can also be obtained from the requirement of the presence of $R$-symmetry.

It is expected that this approach may be useful in the study of other supersymmetric theories, for example, $6D, \mathcal N=(2,0)$ SYM theory.

This reseerch based on the parer[6].

References

1. I. L. Buchbinder, S. M. Kuzenko. Ideas and Methods of Supersymmetry and Supergravity or a Walk

through Superspace// (IOP, Bristol, UK, 1995).

2. A.S. Galperin, E.A. Ivanov, V.I. Ogievetsky, E.S. Sokatchev. Harmonic superspace//Cambridge, UK: Univ. Pr. (2001)

3. I.L. Buchbinder, E.A. Ivanov , I.B. Samsonov. Low-energy effective action in 5D, N=2

supersymmetric gauge theory//Nucl. Phys. B 940 (2019) 54–62

4. I. L. Buchbinder, E. A. Ivanov, I. B. Samsonov. The Low-Energy N=4 SYM Effective Action in Diverse Harmonic Superspaces// Phys. Part. Nucl. 48 (3) (2017) 333

5. I. L. Buchbinder and E. A. Ivanov. Complete N = 4 structure of low-energy effective action in N = 4 super Yang-Mills theories// Phys. Lett. B 524, 208 (2002)

6. I.L. Buchbinder, E.A. Ivanov, V.A. Ivanovskiy. Superfield realization of hidden R-symmetry in extended supersymmetric gauge theories and its applications. arXiv:2001.01649 [hep-th]