QCD sum rule study on excited light meson operators

TL;DR

This work targets the spectroscopy of excited light mesons by constructing twelve covariant-derivative operators $J = \bar{q}_a {\overset{\leftrightarrow}{D}}_\alpha \Gamma q_a$ and applying QCD sum rules to ten of them across $\bar{q}q$, $\bar{q}s$, and $\bar{s}s$ content. The authors compute masses and decay constants via operator product expansion and Borel-transformed sum rules, enforcing quark-hadron duality with thresholds and OPE convergence criteria. They identify established SU(3) flavor nonets, notably the $2^{++}$ nonet $a_2(1320)$, $f_2(1270)$, $f_2'(1525)$, and $K_2^*(1430)$, and provide robust predictions for several excited states, with results in good agreement with lattice QCD (Dudek 2013). The analysis also highlights potential hybrid assignments for exotic quantum numbers in the $1^{-+}$ sector (e.g., $\pi_1(1600)$, $\eta_1(1855)$) and delivers masses and decay constants that can guide future experiments and lattice calculations. Overall, the study advances excited-meson spectroscopy by combining covariant-derivative operator construction with nonperturbative QCD sum rules and cross-validating with lattice results.

Abstract

We apply the QCD sum rule method to systematically study excited light meson operators and calculate their decay constants. These operators are constructed by explicitly adding one covariant derivative to the quark-antiquark pair. In total, twelve such operators are constructed, among which ten are subjected to detailed numerical analyses. The considered quark contents include $\bar{q}q$, $\bar{q}s$, and $\bar{s}s$ ($q = u/d$), allowing the formation of various $SU(3)$ flavor nonets. For instance, our results support the interpretation that the $a_2(1320)$, $f_2(1270)$, $f_2^\prime(1525)$, and $K_2^*(1430)$ constitute a flavor nonet with quantum numbers $J^{P(C)} = 2^{+(+)}$. In addition, we predict several excited meson states, whose masses and decay constants are determined using the QCD sum rule method.

Figures (4)

• Figure 1: Categorization of excited light meson operators: Each operator is constructed from a quark field, an antiquark field, and a covariant derivative, combined with an appropriate Lorentz structure $\Gamma$. The symbol $\mathcal{A}[\cdots]$ denotes anti-symmetrization over the index set $\{\mu\nu\}$, $\mathcal{S}[\cdots]$ represents symmetrization and trace subtraction over $\{\mu\nu\}$, and $\mathcal{S^\prime}[\cdots]$ indicates symmetrization and trace subtraction over the set $\{\mu\alpha\}$.
• Figure 2: Feynman diagrams for excited light meson operators. Diagrams (a) and (b--i) are proportional to $g_s^{N=0}$; diagrams (c--i) and (d--i) are proportional to $g_s^{N=1}$; and diagrams (e--i) are proportional to $g_s^{N=2}$. The gluons appearing in diagrams (c--2), (d--2), and (e--2) originate (partly) from the covariant derivative operator.
• Figure 3: CVG$_{8}$ (short dashed curve defined in Eq. (\ref{['eq:convergence8']})), CVG$_{6}$ (middle dashed curve defined in Eq. (\ref{['eq:convergence6']})), and the pole contribution (solid curve defined in Eq. (\ref{['eq:pole']})) as functions of the Borel mass $M_B$. These curves are obtained using the operator $J_{\mu \nu}^{2{++}}$ with the quark content $\bar{q}s$ ($q = u/d$), under the threshold setting $s_0 = 2.9$ GeV$^2$.
• Figure 4: Mass of the state $| \bar{q}s; 2^{+(+)} \rangle$, extracted from the operator $J_{\mu \nu}^{2{++}}$ with the quark content $\bar{q}s$ ($q = u/d$), as a function of the threshold value $s_0$ (left) and the Borel mass $M_B$ (right). In the left panel the short-dashed/solid/long-dashed curves are plotted by setting $M_B^2 = 0.90/1.02/1.14$ GeV$^2$, respectively. In the right panel the short-dashed/solid/long-dashed curves are plotted by setting $s_0 = 2.4/2.9/3.4$ GeV$^2$, respectively.