The European Physical Journal C | 2021
Exact $$\\beta $$-functions for $$\\mathcal{N}=1$$ supersymmetric theories finite in the lowest loops
Abstract
<jats:p>We consider a one-loop finite <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal{N}=1$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>N</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> supersymmetric theory in such a renormalization scheme that the first <jats:italic>L</jats:italic> contributions to the gauge <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>β</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-function and the first <jats:inline-formula><jats:alternatives><jats:tex-math>$$(L-1)$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>L</mml:mi>\n <mml:mo>-</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> contributions to the anomalous dimension of the matter superfields and to the Yukawa <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>β</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-function vanish. It is demonstrated that in this case the NSVZ equation and the exact equation for the Yukawa <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>β</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-function in the first nontrivial order are valid for an arbitrary renormalization prescription respecting the above assumption. This implies that under this assumption the <jats:inline-formula><jats:alternatives><jats:tex-math>$$(L+1)$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>L</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>-loop contribution to the gauge <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>β</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-function and the <jats:italic>L</jats:italic>-loop contribution to the Yukawa <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>β</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-function are always expressed in terms of the <jats:italic>L</jats:italic>-loop contribution to the anomalous dimension of the matter superfields. This statement generalizes the result of Grisaru, Milewski, and Zanon that for a theory finite in <jats:italic>L</jats:italic> loops the <jats:inline-formula><jats:alternatives><jats:tex-math>$$(L+1)$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>L</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>-loop contribution to the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>β</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-function also vanishes. In particular, it gives a simple explanation why their result is valid although the NSVZ equation does not hold in an arbitrary subtraction scheme.</jats:p>