Nonperturbative running of the tensor operator for $N_\rm{f}=3$ QCD from the chirally rotated Schrödinger Functional

Abstract

We study the Renormalisation Group (RG) running of the non-singlet tensor operator, for $N_{\mathrm{f}}=3$ QCD with Wilson fermions in a mixed action setup, with standard Schrödinger Functional (SF) boundary conditions for sea quarks and chirally rotated Schrödinger Functional ($\ddot{I}‡$SF) boundary conditions for valence quarks. Based on a recursive finite-size scaling technique we compute non-perturbatively the tensor step-scaling function for an energy range between a hadronic scale and an electroweak scale, above which perturbation theory may be safely applied. Our result is expressed as the RG-running factor $T^{\mathrm{RGI}}/[T(\hat{I}{¼}_{\mathrm{had}}){]}_{\mathrm{R}}$, where the numerator is the scale independent (Renormalisation Group Invariant - RGI) tensor operator and the denominator is its renormalised counterpart at a hadronic scale $\hat{I}{¼}_{\mathrm{had}}=233(8)$~MeV in a given scheme. We determine the step-scaling function in four distinct renormalisation schemes. We also compute the renormalisation parameters of these schemes at $\hat{I}{¼}_{\mathrm{had}}$ which, combined with the RG-running factor, gives the scheme-independent quantity $Z_{\mathrm{T}}^{\mathrm{RGI}}(g_0^2)$ in four schemes and for a range of bare gauge couplings in which large volume hadronic matrix element simulations are performed by the CLS consortium in $N_{\mathrm{f}}=2+1$ QCD. All four results are compatible and also agree with a recent determination based on a unitary setup for Wilson quarks with Schrödinger Functional boundary conditions~arXiv:2309.04314 . This provides a strong universality test.

35 pages, 28 figures