Abstract
Decoupling of heavy quarks at low energies can be described by means of an effective theory as shown by S. Weinberg in Ref. [1]. We study the decoupling of the charm quark by lattice simulations. We simulate a model, QCD with two degenerate charm quarks. In this case the leading order term in the effective theory is a pure gauge theory. The higher order terms are proportional to inverse powers of the charm quark mass M starting at M−2. Ratios of hadronic scales are equal to their value in the pure gauge theory up to power corrections. We show, by precise measurements of ratios of scales defined from the Wilson flow, that these corrections are very small and that they can be described by a term proportional to M−2 down to masses in the region of the charm quark mass.
Highlights
In a field theory which contains light fields and fields of a heavy mass M, the functional integral over the latter can be performed resulting in an effective theory for the light fields which was formulated by Weinberg [1]
By precise measurements of ratios of scales defined from the Wilson flow, that these corrections are very small and that they can be described by a term proportional to M−2 down to masses in the region of the charm quark mass
On the ensembles generated with two flavors of O(a) improved Wilson fermions reported in [6,21] and those generated in the twisted mass simulations at maximal twist and the pure gauge simulation√s wh√ich are listed in Table 1, we measure the hadronic scales t0, tc and w0
Summary
In a field theory which contains light (mass-less) fields and fields of a heavy mass M, the functional integral over the latter can be performed resulting in an effective theory for the light fields which was formulated by Weinberg [1]. The action of the effective theory contains the action of the light fields (without the heavy fields) and an infinite number of non-renormalizable terms. The latter are suppressed by powers of E/M at low energies E M. The non-renormalizable couplings do not contribute to the renormalization group equations of the renormalizable couplings of the light fields. This property holds for mass-independent renormalization schemes like the MS scheme as shown in [1]. The heavy fields still affect the value of the renormalized couplings of the light fields through the decoupling relations, which result from the matching of the effective and the fundamental theory at low energies
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