A strategy for B-physics observables in the continuum limit

Abstract

In a somewhat forgotten paper [1] it was shown how to perform interpolations between relativistic and static computations in order to obtain results for heavy-light observables for masses from, say, m charm to m bottom . All quantities are first continuum extrapolated and then interpolated in 1/m h = 1/m heavy . Large volume computations are combined with finite volume ones where a relativistic bottom quark is accessible with small am bottom . We discuss how this strategy is extended to semi-leptonic form factors and other quantities of phenomenological interest. The essential point is to form quantities where the limit m h → ∞ is approached with power corrections O(1/m h ) only. Perturbative corrections ∼ α s (m h ) γ+n are cancelled in the construction of the observables. We also point out how such an approach can help to control systematics in semileptonic decays with just large volume data. First numerical results with N f = 2 + 1 and lattice spacings down to 0.039 fm are presented in [2].

B-physics observables in the continuum from a combination of static and relativistic results

“…The design principle of our approach, detailed in [1,7] consists in reaching the b-quark scale by interpolation through a combination of static and relativistic results in the continuum. The difficulty in the straight forward application is that the matching of the static theory to QCD introduces massdependent matching functions 𝐶 𝑥 (𝑚 RGI ℎ /Λ) that diverge logarithmically in the static limit.…”

“…B-physics observables in the continuum from a combination of static and relativistic results determined from the large volume input 𝑦 𝐵 = 𝐿 ref 𝑚 𝐵 through the chain [7]…”

“…Here S 𝑚 and S 0 denote the massive and massless Lines of Constant Physics (LCP) as defined in [7]. The b-quark mass proxies 𝑦 𝑖 are obtained recursively from the large volume physical input…”

“…with 𝐴 0 , ì 𝑉 denoting the axial and vector heavy-light currents respectively, we obtain a step scaling chain identical in form to Eq. (1), namely [7]…”

“…A step forward was made in [2] by combining Step Scaling Functions (SSFs) in the static approximation [3,4] with the relativistic ones at finite heavy quark mass 𝑚 ℎ [5,6], where the available lattice spacings require 𝑚 ℎ ≤ 𝑚 𝑏 /2 to avoid the 𝑎𝑚 ℎ > 1 regime. We have further developed this original proposal in [7] by showing how to generalise the strategy to the extraction of semi-leptonic decays, and we presented in [1] preliminary results for the b-quark mass and the 𝐵 (★) meson decay constants. Our prescription relies on building suitable quantities with a continuum limit and simple behaviour in 1/𝑚 ℎ , such that the renormalisation and matching factor in the static theory cancel completely.…”