The mass of the lightest gluelump

TL;DR

This work advances the precision understanding of renormalon normalizations and the leading nonperturbative input in heavy-quark hybrids by combining hyperasymptotic expansions, principal-value resummation, and lattice data. It provides two independent, renormalization-group-invariant determinations of the gluelump scale ΛB^PV, from lattice gluelump energies and from short-distance hybrid energies, converging to a robust combined value. The final result, ΛB^PV ≈ 2.44 r0^-1, enables absolute, scheme- and scale-independent gluelump masses and strengthens first-principles predictions for heavy quarkonium hybrid spectra. The analysis also clarifies subleading renormalons and demonstrates consistency with Casimir scaling, offering a solid foundation for future spectroscopic studies of gluonic excitations in QCD.

Abstract

We give the most up-to-date determinations of the normalization of the leading renormalons of the pole mass, the singlet static potential, the octet static potential, and the gluelump energy. They read $Z^{\rm MS}_m=-Z^{\rm MS}_{V_s}/2=\{0.604(17),0.551(20)\}$, $Z^{\rm MS}_{V_o}=\{0.136(8),0.121(13)\}$, and $Z^{\rm MS}_A=\{-1.343(36),-1.224(43)\}$, for $n_f=0$ and $n_f=3$ respectively. We obtain two independent renormalization group invariant and renormalization scale independent determinations of the energy of the ground state gluelump in the principal value summation scheme: $Λ_{B}^{\rm PV}=2.47(9)r_0^{-1}$ and $Λ_{B}^{\rm PV}=2.38(11)r_0^{-1}$. They combine in $Λ_{B}^{\rm PV}=2.44(7)r_0^{-1}$.

Figures (13)

• Figure 1: Plot of $\sqrt{n_0}\frac{1}{a}[c_{A,n}(1/a)-c_{A,n}^{\rm (as)}(1/a)]\alpha_L^{n+1}(a)$ in $r_0$ units for different values of $n_0$. See the main text for details.
• Figure 2: Determination of $-Z_{V_s}/2$ using $v_n/v_n^{(as)}Z_{V_s}$ as a function of $x=\nu_s r$ and for different values of $n$ in the $\overline{\rm MS}$ scheme. The gray continuous line is $v_3/v_3^{(as)}Z_{V_s}$ withouth the ultrasoft logarithmically related term (see Ref. Ayala:2014yxa for details). The black horizontal line is our final prediction and the blue band our final error estimate. (a) are determinations with $n_f=0$ and (b) with $n_f=3$.
• Figure 3: Determination of $Z_{m}$ using $r_n/r_n^{(as)}Z_{m}$ as a function of $x=\nu/{\overline m}$ and for different values of $n$ in the $\overline{\rm MS}$ scheme. The black horizontal line is our final prediction and the blue band our final error estimate. (a) are determinations with $n_f=0$ and (b) with $n_f=3$. For extra details see the main text and Ref. Ayala:2014yxa.
• Figure 4: Determination of $Z_{V_o}$ using $v^{(o)}_n/v_n^{(o,as)}Z_{V_o}$ as a function of $x=\nu_s r$ and for different values of $n$ in the $\overline{\rm MS}$ scheme. The gray continuous line is $v^{(o)}_3/v_3^{(o,as)}Z_{V_o}$ without the ultrasoft logarithmically related term. The black horizontal line is our final prediction and the blue band our final error estimate. (a) are determinations with $n_f=0$ and (b) with $n_f=3$.
• Figure 5: Determination of $Z_{V_A}$ using $v^{(A)}_n/v_n^{(A,as)}Z_{V_A}$ as a function of $x=\nu_s r$ and for different values of $n$ in the $\overline{\rm MS}$ scheme. The gray continuous line is $v^{(A)}_3/v_3^{(A,as)}Z_{V_A}$ without the ultrasoft logarithmically related term. The black horizontal line is our final prediction and the blue band our final error estimate. (a) are determinations with $n_f=0$ and (b) with $n_f=3$.