Internal particle width effects on the triangle singularity mechanism in the study of the $\eta (1405)$ and $\eta (1475)$ puzzle

In this article, the analyticity of triangle loop integral with complex masses of internal particles is discussed in a new perspective, based on which we obtain the explicit width dependence of the absorptive part of the triangle amplitude. We reanalyze the decay pattern of $\eta (1405/1475)$ with the width effects included in the triangle singularity (TS) mechanism. Based on the present experimental information, we provide a self-consistent description of the $K\overline{K}\pi$, $\eta \pi \pi$, and $3\pi$ decay channels for $\eta (1405/1475)$. Our results confirm the claim that the TS mechanism plays a decisive role in the understanding of the $\eta (1405)$ and $\eta (1475)$ puzzle. Namely, the observed differences of $\eta$ resonances within the mass region of 1.40–1.48 GeV are originated from the same state. For the isospin violated process $J/\psi \to \gamma \eta (1405/1475)\to f_0(980)\pi \to 3\pi$, we identify an additional contribution to the $a_0(980)-f_0(980)$ mixing via the TS mechanism.

Figure 1

Typical triangle diagram with kinematic variables.

Figure 2

The motions of functions $y_0^{(i)}$ and $y_j^{(i)}$ ($j=1$, 2, and $i=1$, 2, 3) on the complex plane indicating the TS conditions via the Spence function variables $z_{1j}^{(i)}$ and $z_{2j}^{(i)}$. The detailed descriptions of these three pairing functions are given in the context. For a given value of $s_1$, the motions of the $y_0^{(i)}$ and $y_j^{(i)}$ with the variation of $s_3$ from the normal threshold $s_3=(m_2+m_3{)}^2$ to $s_3^{-}$ [Note $s_3^{-}=s_{3c}$ if $s_1=(m_1+m_3{)}^2$] are plotted. The dashed vertical lines indicate the TS conditions that the pairing functions overlap horizontally with $s_3=s_3^{-}$ for any $s_1\in [(m_1+m_3{)}^2,s_{1c}]$.

Figure 3

Calculations of the imaginary part of scalar loop corresponding to ${\eta }^{\prime \prime }\to K^{*}\overline{K}+\mathrm{c}.\mathrm{c}.\to K\overline{K}\pi \to f_0(980)\pi$. The plots on the left panel from upper to lower correspond to the $\Im M$ dependence of $\sqrt{s_1}$ at $\sqrt{s_3}=1\mathrm{GeV}$ with $\mathrm{\Gamma }=200$, 50, and 0 MeV, respectively. The plots on the right panel from upper to lower correspond to the $\Im M$ dependence of $\sqrt{s_3}$ at $\sqrt{s_1}=1.4\mathrm{GeV}$ with $\mathrm{\Gamma }=200$, 50, and 0 MeV, respectively. The thick red solid, thin blue long-dashed, thin green short-dashed and yellow dotted lines are calculated using Eq. (71), LoopTools, Eqs. (69) and (14), respectively.

Figure 4

Variation of $\max \Im M$ when $\mathrm{\Gamma }=10\mathrm{MeV}$ (left) and 50 MeV (right), with $\delta \mathrm{\Gamma }=\mathrm{\Gamma }/10\mathrm{MeV}$.

Figure 5

The value of $\beta$ of the scalar loop containing $K^{*}K\overline{K}$ as internal particles.

Figure 6

The maximum value of $\Im M$ as a function of $s_1$. The blue solid line is the value calculated using Eq. (71) and the orange dashed line is the approximated value using Eq. (73). The left panel corresponds to $\mathrm{\Gamma }=10\mathrm{MeV}$, while the right one corresponds to $\mathrm{\Gamma }=50\mathrm{MeV}$.

Figure 7

Isospin violating mechanisms for ${\eta }^{\prime \prime }\to {\pi }^{+}{\pi }^{-}{\pi }^0$. (a) describes the direct isospin breaking via the TS mechanism, (b) describes the TS contribution to the production of the $a_0(980)$ which then mixes with the $f_0(980)$ via the $K\overline{K}$ loops, and (c) represents the tree-level contributions to the intermediate $a_0(980)$ production, which will then mixing with the $f_0(980)$ via the $K\overline{K}$ loops.

Figure 8

Transition mechanisms for ${\eta }^{\prime \prime }\to K\overline{K}\pi$. (a) describes the tree-level transition of ${\eta }^{\prime \prime }\to K^{*}\overline{K}+\mathrm{c}.\mathrm{c}.\to K\overline{K}\pi$, (b) describes ${\eta }^{\prime \prime }\to a_0(980)\pi \to K\overline{K}\pi$, and (c) describes the TS mechanism which enhances the production of the intermediate $a_0(980)\pi$.

Figure 9

Transition mechanisms for ${\eta }^{\prime \prime }\to \eta \pi \pi$. (a) describes the tree-level transition of ${\eta }^{\prime \prime }\to a_0(980)\pi \to \eta \pi \pi$, and (b) describes the TS mechanism which enhances the production of the intermediate $a_0(980)\pi$.

Figure 10

Invariant mass spectra of $K\overline{K}$ (left panel) and $K\pi$ (right panel) at $\sqrt{s}=1.42\mathrm{GeV}$ in ${\eta }^{\prime \prime }\to K\overline{K}\pi$. The full calculations are denoted by the red solid lines. The blue dashed lines are the contributions from the tree-level $a_0(980)\pi$ amplitude. The green dotted lines denote the contribution of triangle diagrams, and the orange dot-dashed lines are from tree diagrams of $K^{*}\overline{K}$.

Figure 11

The invariant mass spectrum of $\eta {\pi }^0$ at $\sqrt{s}=1.42\mathrm{GeV}$ in ${\eta }^{\prime \prime }\to \eta \pi \pi$. The full calculation is denoted by the red solid line. The green dotted and blue dashed lines represent contributions from the triangle and tree-level diagrams, respectively.

Figure 12

The ${\pi }^{+}{\pi }^{-}$ invariant mass spectrum at $\sqrt{s}=1.42\mathrm{GeV}$. The full calculation is denoted by the red solid line. The blue dot-dot-dashed line is the spectrum with only triangle diagram of the $f_0\pi$ channel [Fig. 7]. The green dot-dashed line is the contribution from the $a_0-f_0$ mixing with the bare $g_{{\eta }^{\prime \prime }{a}_0\pi }$ coupling [Fig. 7], and the orange dashed line is the contribution from the $a_0-f_0$ mixing via the TS production of the $a_0(980)$ [Fig. 7]. The black dotted line is the total contribution from the $a_0-f_0$ mixing.

Figure 13

Invariant mass spectra of the $K\overline{K}\pi$ (red solid), $\eta \pi \pi$ (blue dashed and multiplied by 10) and ${\pi }^{+}{\pi }^{-}{\pi }^0$ (orange dot-dashed and multiplied by 20). The physical mass of the ${\eta }^{\prime \prime }$ applied in the propagator is 1.475 GeV.