Provide NNLO inputs needed for small-flow-time expansions with flavor separation

Develop next-to-next-to-leading order (NNLO, O(α_s^2)) results that separate heavy- and light-quark flavor contributions in the small-flow-time expansions of S(t) = ⟨\bar{χ}(t,x) χ(t,x)⟩ and R(t) = ⟨\bar{χ}(t,x) \overleftrightarrow{\slashed{\mathcal{D}}} χ(t,x)⟩, and compute the full NNLO expression for the vacuum expectation value ⟨\bar{ψ}(x) ψ(x)⟩ in multi-flavor QCD, so these inputs can be used for complete NNLO matching.

Background

To cross-check and analytically extract terms in the small-flow-time expansions of the flowed bilinears, the authors use matching coefficients and unflowed operator vacuum expectation values. At NLO, flavor mixing between heavy and light quarks does not arise and existing results suffice. At NNLO, however, flavor-specific contributions become relevant.

The authors point out that published NNLO results do not provide enough information to disentangle heavy- and light-flavor pieces and that the full NNLO vacuum expectation value of ⟨\bar{ψ}ψ⟩ in the presence of additional massless quarks is not yet available. Supplying these NNLO inputs would enable full analytic NNLO extractions and improve the precision of the proposed mass determination method.