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We discuss the renormalisation group (RG) evolution for the $\Delta S = 1$operators in unquenched QCD with $N_f = 3$ ($m_u=m_d=m_s$) or, more generally,$N_f = 2+1$ ($m_u=m_d \ne m_s$) flavors. In particular, we focus on thespecific problem of how to treat the singularities which show up only for$N_f=3$ or $N_f = 2+1$ in the original solution of Buras {\it et al.} for theRG evolution matrix at next-to-leading order. On top of Buras {\it et al.}'soriginal treatment, we use a new method of analytic continuation to obtain thecorrect solution in this case. It is free of singularities and can therefore beused in numerical analysis of data sets calculated in lattice QCD.Comment: 7 pages, minor revisions, to appear in PR

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Renormalization group evolution for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Δ</mml:mi><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math>effective Hamiltonian with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>N</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>

Renormalization group evolution for theΔS=1effective Hamiltonian withNf=2+1