Mesonic decay constant and mass ratios and the conformal window

Abstract

Two particular ratios related to mesons are proposed for the study of the conformal window in 𝑆𝑈 (3) gauge theory and fundamental fermions. Lattice and other studies indicate that the lower end, 𝑁 * 𝑓 , is at around 7 -13 flavors which is a wide range without a clear consensus. Here we propose the decay constant to mass ratios of mesons, 𝑓 𝑃𝑆,𝑉 /𝑚 𝑉 , as a proxy since below the conformal window lattice studies have shown that they are largely 𝑁 𝑓 -independent while at the upper end of the conformal window they are vanishing. The drop from the non-zero constant value to zero at 𝑁 𝑓 = 16.5 might be indicative of 𝑁 * 𝑓 . We compute 𝑓 𝑉 /𝑚 𝑉 to N 3 LO and 𝑓 𝑃𝑆 /𝑚 𝑉 to NNLO order in (p)NRQCD. The results are unambiguously reliable just below 𝑁 𝑓 = 16.5, hence the results are expanded á la Banks-Zaks in 𝜀 = 16.5 − 𝑁 𝑓 . The convergence properties of the series and matching with the non-perturbative infinite volume, continuum and chiral extrapolated lattice results at 𝑁 𝑓 = 10 suggest that the perturbative results might be reliable down to 𝑁 𝑓 = 12. A sudden drop is observed at 𝑁 𝑓 = 12 and 𝑁 𝑓 = 13 in 𝑓 𝑉 /𝑚 𝑉 and 𝑓 𝑃𝑆 /𝑚 𝑉 , respectively.