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We determine the missing finite-size corrections to the asymptotic one-loop dilatation operator of the real $\beta$-deformed $\mathcal{N}=4$ SYM theory for the gauge groups $U(N)$ and $SU(N)$ in the 't Hooft limit.
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S. Frolov, R. Roiban, and A. A. Tseytlin, “Gauge-string duality for (non)supersymmetric deformations of 𝒩 = 4 \mathcal{N}=4 super Yang-Mills theory,” Nucl.Phys
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