Mass and Decay-Constant Evolution of Heavy Quarkonia and $B_c$ States from Thermal QCD Sum Rules

TL;DR

The paper advances finite-temperature QCD sum-rule analyses of heavy-quarkonia ($J/\psi$, $\Upsilon$, and $B_c$) by updating inputs to PDG $2024$ values and incorporating lattice-informed gluon-condensate evolution. It constrains the temperature dependence of the continuum threshold $s_0(T)$ via vacuum stability and uses a Borel-transformed, LO$+$D$=4 framework to predict in-medium masses $m(T)$ and decay constants $f(T)$ up to $T/T_c \lesssim 0.9$, validated against vacuum benchmarks. The results reveal a sequential suppression pattern with $\Upsilon$ most robust, then $J/\psi$, and $B_c$ most affected, and predict a $1P$–$1S$ splitting for the $B_c$ system of $\Delta m \approx 0.477$ GeV in agreement with LHCb. The work provides a coherent finite-temperature baseline for future extensions, including radiative corrections, higher-dimensional operators, and finite-width effects, and aligns with lattice thermodynamics and experimental spectroscopy.

Abstract

We analyze the thermal behavior of heavy vector and axial-vector mesons ($J/ψ$, $Υ$, and $B_c$) within the finite-temperature QCD sum-rule framework. Using updated PDG-2024 quark masses, modern lattice-informed gluon condensates, and a temperature-dependent continuum threshold constrained by vacuum stability, we compute the evolution of the masses $m(T)$ and decay constants $f(T)$ up to $T/T_c \lesssim 0.9$. At $T=0$ the sum rules are calibrated to reproduce the experimental and LHCb masses and reference decay constants within the expected $\mathcal{O}(10\%)$ accuracy of a leading-order $+$ $D{=}4$ analysis, and we interpret the subsequent finite-temperature evolution as a genuine prediction of the framework. Near the critical temperature, the relative suppression follows a clear hierarchy $Υ< J/ψ< B_c$, consistent with their binding energies and lattice spectral trends. The predicted $1P$--$1S$ splitting for the $B_c$ system, $0.477~\mathrm{GeV}$, matches the LHCb observation of orbitally excited $B_c^{+}$ states. The results provide a coherent finite-temperature baseline for future extensions including radiative, higher-dimensional, and width effects.

Figures (3)

• Figure 1: Temperature dependence of the masses $m(T)$ for all channels up to $T_c$. Error bands are omitted for clarity at LO + D = 4. The effects of the Borel-window and continuum-threshold variations are discussed in Systematic uncertainties
• Figure 2: Relative decay constants $f(T)/f(0)$ up to $T_c$ for all channels. The dashed (dotted) line indicates 1.0 (0.5).
• Figure 3: Extracted mass $m(M^2)$ as a function of the Borel parameter at $T{=}0$ for all channels. The horizontal band indicates the PDG value $\pm3\%$. The working windows (shaded) exhibit plateaus consistent with the criteria of Table 1.