Supersymmetric QCD on the lattice: Fine-tuning of the Yukawa couplings

We determine the fine-tuning of the Yukawa couplings of supersymmetric QCD, discretized on a lattice. We use perturbation theory at one-loop level. The modified minimal subtraction scheme (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mover accent="true"><a:mi>MS</a:mi><a:mo stretchy="true">¯</a:mo></a:mover></a:math>) is employed; by its definition, this scheme requires perturbative calculations, in the continuum and/or on the lattice. On the lattice, we utilize the Wilson formulation for gluon, quark, and gluino fields; for squark fields we use naive discretization. The sheer difficulties of this study lie in the fact that different components of squark fields mix among themselves at the quantum level and the action’s symmetries, such as parity and charge conjugation, allow an additional Yukawa coupling. Consequently, for an appropriate fine-tuning of the Yukawa terms, these mixings must be taken into account in the renormalization conditions. All Green’s functions and renormalization factors are analytic expressions depending on the number of colors, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>N</e:mi><e:mi>c</e:mi></e:msub></e:math>, the number of flavors, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>N</g:mi><g:mi>f</g:mi></g:msub></g:math>, and the gauge parameter, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>α</i:mi></i:math>, which are left unspecified. Knowledge of these renormalization factors is necessary in order to relate numerical results, coming from nonperturbative studies, to the renormalized, “physical” Green’s functions of the theory. Published by the American Physical Society 2024