Non-perturbative determination of improvement coefficients $$b_\mathrm{m}$$ and $$b_\mathrm{A}-b_\mathrm{P}$$ and normalisation factor $$Z_\mathrm{m}Z_\mathrm{P}/Z_\mathrm{A}$$ with $$N_\mathrm{f}= 3$$ Wilson fermions

We determine non-perturbatively the normalisation parameter $$Z_\mathrm{m}Z_\mathrm{P}/Z_\mathrm{A}$$ZmZP/ZA as well as the Symanzik coefficients $$b_\mathrm{m}$$bm and $$b_\mathrm{A}-b_\mathrm{P}$$bA-bP, required in $$\mathrm{O}(a)$$O(a) improved quark mass renormalisation with Wilson fermions. The strategy underlying their computation involves simulations in $$N_\mathrm{f}=3$$Nf=3 QCD with $$\mathrm{O}(a)$$O(a) improved massless sea and non-degenerate valence quarks in the finite-volume Schrödinger functional scheme. Our results, which cover the typical gauge coupling range of large-volume $$N_\mathrm{f}=2+1$$Nf=2+1 QCD simulations with Wilson fermions at lattice spacings below $$0.1\,\mathrm{fm}$$0.1fm, are of particular use for the non-perturbative calculation of $$\mathrm{O}(a)$$O(a) improved renormalised quark masses.