Probing the substructure of $f_0(1370)$

Within an effective nonlinear chiral Lagrangian framework, the substructure of $f_0(1370)$ is studied. The investigation is conducted in the context of a global picture of scalar mesons in which the importance of the underlying connections among scalar mesons below and above 1 GeV is recognized and implemented. These connections are due to the mixings among various quark-antiquarks, four-quarks, and glue components and play a central role in understanding the properties of scalar mesons. Iterative Monte Carlo simulations are first performed on the 14-dimensional parameter space of the model and sets of points in this parameter space (the global sets) that give an overall agreement with all experimental data on mass spectrum, various decay widths, and decay ratios of all isosinglet scalar states below 2 GeV are determined. Then within each global set, subsets that give closest agreement for the properties of $f_0(1370)$ are studied. Unlike the properties of other isosinglet states that show a range of variation within each global set, it is found that there is a clear signal for $f_0(1370)$ to be predominantly a quark-antiquark state with a substantial $s\overline{s}$ component, together with small remnants of four-quark and glue components.

Figure 1

Mass and decay properties (GeV) of $f_0(1370)$ obtained from Monte Carlo simulation for global fits I (left), II (middle), and III (right) defined in Sec. 4a. The results of simulations (squares) and their averages and 1 standard deviation around the averages (triangles and error bars) are compared with experiment (filled circles and error bars).

Figure 2

Percentages of quark and glue components of $f_0(1370)$ obtained from Monte Carlo simulation for global fits I (left), II (middle), and III (right) defined in Sec. 4a. In the first row, the components 1 to 5 on the horizontal axes respectively represent $\overline{u}\overline{d}ud$, $(\overline{d}\overline{s}ds+\overline{s}\overline{u}su)/\sqrt{2}$, $s\overline{s}$, $(u\overline{u}+d\overline{d})/\sqrt{2}$, and glueball $G$. In the second row, the percentages of total four-quark ($1+2$), total quark-antiquark ($3+4$), and glue (5) are given.

Figure 3

Histograms of Monte Carlo simulations versus each component of $f_0(1370)$ obtained in global fits I (left set of figures), II (middle set of figures), and III (right set of figures). Each set consists of five figures, the vertical axes represent the percentages of Monte Carlo simulations, the horizontal axes (in each set) from left to right respectively represent the percentages of components $\overline{u}\overline{d}ud$, $(\overline{d}\overline{s}ds+\overline{s}\overline{u}su)/\sqrt{2}$, $s\overline{s}$, $(u\overline{u}+d\overline{d})/\sqrt{2}$, and glueball $G$. Each component is divided into five percentage intervals (0–20%, 20%–40%, etc). For example, the first figure on the left shows that more than 90% of Monte Carlo simulations for global fit I, resulted in estimating the $\overline{u}\overline{d}ud$ component of $f_0(1370)$ to be below 20%, and less than 10% of the simulations resulted in estimating the $(\overline{d}\overline{s}ds+\overline{s}\overline{u}su)/\sqrt{2}$ component between 20%–40%, etc.

Figure 4

Percentages of the quark and glue components of $f_0(1370)$ obtained obtained over subsets $S_{\mathrm{IA}}$ [relation (33)], $S_{\mathrm{IIA}}$ [relation (36)], and $S_{\mathrm{III}}$ [relation (39)]. Components 1 to 5 respectively represent $\overline{u}\overline{d}ud$, $(\overline{d}\overline{s}ds+\overline{s}\overline{u}su)/\sqrt{2}$, $s\overline{s}$, $(u\overline{u}+d\overline{d})/\sqrt{2}$, and glue. In the second row, the percentages of total four-quark ($1+2$), total quark-antiquark ($3+4$), and glue (5) are given. The averages (triangles) and standard deviations (error bars) are shown.

Figure 5

Histograms for Monte Carlo simulations versus each component of $f_0(1370)$ obtained over subsets $S_{\mathrm{IA}}$ [relation (33), the left set of figures], $S_{\mathrm{IIA}}$ [relation (36), the middle set of figures], and $S_{\text{IIIA}}$ [relation (39), the right set of figures]. Each set consists of five figures; the vertical axes represent the percentages of Monte Carlo simulations, the horizontal axes (in each set) from left to right respectively represent the percentages of components $\overline{u}\overline{d}ud$, $(\overline{d}\overline{s}ds+\overline{s}\overline{u}su)/\sqrt{2}$, $s\overline{s}$, $(u\overline{u}+d\overline{d})/\sqrt{2}$, and glue. Each component is divided into five percentage intervals (0–20%, 20%–40%, etc). For example, the first figure on the left shows that more than 90% of Monte Carlo simulations in zoom IA resulted in estimating the $\overline{u}\overline{d}ud$ component of $f_0(1370)$ to be below 20%, and less than 10% of the simulations resulted in estimating the $(\overline{d}\overline{s}ds+\overline{s}\overline{u}su)/\sqrt{2}$ component between 20%–40%, etc.

Figure 6

Percentages of the quark and glue components of $f_0(1370)$ obtained over subsets $S_{\mathrm{IB}}$ [relation (40)]. Components 1 to 5 respectively represent $\overline{u}\overline{d}ud$, $(\overline{d}\overline{s}ds+\overline{s}\overline{u}su)/\sqrt{2}$, $s\overline{s}$, $(u\overline{u}+d\overline{d})/\sqrt{2}$, and glue. The percentages of total four-quark ($1+2$), total quark-antiquark ($3+4$), and glue (5) are also given (right figure).

Figure 7

Comparing the $s\overline{s}$ and $n\overline{n}$ components of $f_0(1370)$. Vertical axis shows the percentage of simulations in which the $s\overline{s}$ component is greater than the $n\overline{n}$ component. The horizontal axis gives from left to right the three global fits followed by their zooms A and zoom IB.

Figure 8

Percentages of the quark and glue components of $f_0(1370)$ obtained over subset $S_{\text{IIIB}}$ [relation (41)]. Components 1 to 5 respectively represent $\overline{u}\overline{d}ud$, $(\overline{d}\overline{s}ds+\overline{s}\overline{u}su)/\sqrt{2}$, $s\overline{s}$, $(u\overline{u}+d\overline{d})/\sqrt{2}$, and glue.

Figure 9

Production of $f_0(1370)$ in decay $B_s\to J/\psi {\pi }^{+}{\pi }^{-}$ reported by LHCb [127]. The decay proceeds via production of an $s\overline{s}$ pair and is consistent with the prediction of a large $s\overline{s}$ component in the substructure of $f_0(1370)$ presented in this work.

Figure 10

Mixing mechanism for isodoublet and isotriplet scalar mesons below and above 1 GeV. Existence of a pure four-quark scalar meson nonet beneath a quark-antiquark nonet (a) results in a mass spectrum for $I=1/2,1$ states where the two isotriplets are closer to each other than the two isodoublets (i.e., ${\delta }_1<\delta 1/2$). This is due to the inverted mass spectrum in a four-quark nonet compared to that of a quark-antiquark nonet. Allowing the states with the same quantum numbers to mix leads to a level repulsion in which the isotriplets split more than the isodoublets and consequently a level crossing occurs that results in $a_0(1450)$ becoming heavier than the $K_0^{*}(1430)$. This mechanism also works when the two isotriplets are degenerate in mass (b). The mechanism developed in Ref. [99], in the leading order, favored situation (b). The level repulsion also shows that the light scalar mesons below 1 GeV get pushed down and become lighter than expected.

Figure 11

Schematic diagram for a possible rearrangement of the quark lines in the mixing process.