Hadronic three-body $D$ decays mediated by scalar resonances

In recent years many measurements of hadronic three-body and four-body decays of charmed mesons have been performed with Dalitz-plot amplitude analyses. Amplitudes describing $D$ meson decays into multibody final states are dominated by quasi-two-body processes, such as $D\to PP,VP,SP,AP$ and $TP$, where $P,V,S,A$ and $T$ denote pseudoscalar, vector, scalar, axial-vector and tensor mesons, respectively. Among various $S$-, $P$- and $D$-wave intermediate resonances, the identification of the scalar mesons is rather difficult due to their broad widths and flat angular distributions.

Scalar mesons with masses lower than 2 GeV can be classified into two nonets: one nonet with masses below or close to 1 GeV, including $\sigma/f_{0}(500)$, $f_{0}(980)$, $a_{0}(980)$ and $\kappa/K_{0}^{*}(700)$; and the other nonet with masses above 1 GeV, including $a_{0}(1450)$, $K^{*}_{0}(1430)$, $f_{0}(1370)$, $f_{0}(1500)$ and $f_{0}(1710)$. Since the last three are all isosinglet scalars and only two of them can be accommodated in the quark model, implying a dominant scalar glueball content in one of the three isosinglets.

In this work, we shall study the quasi-two-body $D\to SP$ decays and the three-body $D$ decays proceeding through intermediate scalar resonances. In Tables 1 and 2 we collect all the measured branching fractions of $D\to SP\to P_{1}P_{2}P$ decays available in the Particle Data Group (PDG) PDG . It is clear that $f_{0}(980)$ and the $f_{0}$ family such as $f_{0}(1370)$, $f_{0}(1500)$ and $f_{0}(1710)$ are observed in the three-body decays of $D^{+},D^{0}$ and $D_{s}^{+}$, while $a_{0}(980)$ is seen exclusively in three-body $D^{0}$ decays (except for $D_{s}^{+}\to a_{0}^{+,0}\pi^{0,+}$). Contrary to $f_{0}(980)$ and $a_{0}(980)$ which are relatively easy to identify experimentally, the establishment of $\sigma$ and $\kappa$ is very difficult and controversial because their widths are so broad that their shapes are not clearly resonant. Nevertheless, their signals in three-body $D$ decays have been identified in $D^{+,0}\to\sigma\pi^{+,0}\to\pi^{+}\pi^{-}\pi^{+,0}$, $D^{+}\to\bar{\kappa}^{0}\pi^{+}\to K_{S}\pi^{0}\pi^{+}$ and $D^{+}\to\bar{\kappa}^{0}K^{+}\to\pi^{+}K^{-}K^{+}$, respectively. Because of threshold and coupled-channel effects for $f_{0}(980)$ and $a_{0}(980)$ and the very broad widths for $\sigma$ and $\kappa$, it is no longer pertinent to use the conventional Breit-Wigner parametrization to describe their line shapes.

The $D\to SP$ decays and related three-body $D$ decays have been studied previously in Refs. Kamal ; Katoch ; Buccella96 ; Fajfer ; ChengSP ; ElBennich ; Boito ; Cheng:SAT ; Xie:2014tma ; Dedonder:2014xpa ; Loiseau:2016mdm ; Dedonder:2021dmb . In the $D\to SP$ decays, the flavor diagram of each topology has two possibilities: one with the spectator quark in the charmed meson going to the pseudoscalar meson in the final state, and the other with the spectator quark ending up in the scalar meson. We thus need two copies of each topological diagram to describe the decay processes. Many of these decays have been observed in recent years through dedicated experiments and powerful Dalitz plot analysis of multi-body decays. We will investigate whether an extraction of the sizes and relative strong phases of these amplitudes is possible.

One purpose of studying these decays is to check our understanding in the structures and properties of light even-parity scalar mesons. Another goal is to learn the final-state interaction pattern in view of the rich resonance spectrum around the $D$ meson mass range. Not only does this work update our previous study Cheng:SAT , we also study the finite-width effect in the three-body decays mediated by the scalar mesons. Such an effect is observed to be particularly important for decays involving $\sigma/f_{0}(500)$ and $\kappa/K_{0}^{*}(700)$ in the intermediate state because of their broad widths compared to their masses, respectively. Therefore, one should be careful in the use of the narrow width approximation (NWA) to extract the $D\to SP$ two-body decays from the three-body decay rates.

This paper is organized as follows. In Section II, we review the current experimental status about how various $D\to SP$ decay branching fractions are extracted using the NWA from three-body decay rates. In Section III, we discuss the two-quark $q\bar{q}$ and tetraquark pictures of the scalar nonet near or below 1 GeV along with the associated conundrums. The decay constants and form factors required for subsequent numerical calculations are given in this section, too. Section IV sets up the notation and formalism of flavor amplitude analysis, for both quark-antiquark and tetraquark pictures. In Section V, we take the factorization approach as an alternative toward analyzing these decays. We also introduce line shapes for the scalar resonances when describing various three-body decays. Section VI gives the results obtained based upon the approaches in the previous two sections for a comparison. Section VI.2 is devoted to the study of finite-width effect and how the NWA should be modified. We summarize our findings in Section VII.

It is known that three- and four-body decays of heavy mesons provide a rich laboratory for studying the intermediate-state resonances. The Dalitz plot analysis of three-body or four-body decays of charmed mesons is a very useful technique for this purpose. We are interested in $D\to SP$ decays followed by $S\to P_{1}P_{2}$. The results of various experiments are summarized in Tables 1 and 2. To extract the branching fraction for a $D\to SP$ decay, it is the usual practice to use the NWA:

$\displaystyle\Gamma(D\to SP\to P_{1}P_{2}P)=\Gamma(D\to SP)_{\rm NWA}{\cal B}(S\to P_{1}P_{2})~{}.$

(1)

Since this relation holds only in the $\Gamma_{S}\to 0$ limit, we put the subscript NWA to emphasize that ${\cal B}(D\to SP)$ thus obtained is under this limit. Finite width effects will be discussed in Section VI.2. For the branching fractions of two-body decays of scalar mesons, we shall use PDG

	$\displaystyle{\cal B}(a_{0}(980)\to\pi\eta)=0.850\pm 0.017~{},$		$\displaystyle{\cal B}(\sigma(500)\to\pi^{+}\pi^{-})={2\over 3}~{},$
	$\displaystyle{\cal B}(f_{0}(1500)\to\pi\pi)=0.345\pm 0.022~{},$		$\displaystyle{\cal B}(f_{0}(1710)\to K^{+}K^{-})=0.292\pm 0.027~{},$		(2)
	$\displaystyle{\cal B}(K_{0}^{*0}(1430)\to K^{+}\pi^{-})={2\over 3}(0.93\pm 0.10)~{},$		$\displaystyle{\cal B}(\kappa(700)\to K^{+}\pi^{-})={2\over 3}~{},$

where we have applied the average of $\Gamma(a_{0}(980)\to K\overline{K})/\Gamma(a_{0}(980)\to\pi\eta)=0.177\pm 0.024$ from PDG PDG to extract the branching fraction of $a_{0}(980)\to\pi\eta$, assuming that its width is saturated by the $K\overline{K}$ and $\pi\eta$ modes. For $f_{0}(1710)$ we have used the values of $\Gamma(f_{0}(1710)\to\pi\pi)/\Gamma(f_{0}(1710)\to K\overline{K})=0.23\pm 0.05$ and $\Gamma(f_{0}(1710)\to\eta\eta)/\Gamma(f_{0}(1710)\to K\overline{K})=0.48\pm 0.15$ from PDG together with the assumption of its width being saturated by $\pi\pi$, $K\overline{K}$ and $\eta\eta$ modes. For $S=f_{0}(980)$ or $a_{0}(980)$, we are not able to extract the branching fractions of $D\to SP$ due to the lack of information of ${\cal B}(S\to P_{1}P_{2})$ (except for $a_{0}(980)\to\pi\eta$), especially for ${\cal B}(S\to K\overline{K})$ where the threshold effect must be taken into account. For example, the NWA relation

$\displaystyle\Gamma(D^{+}\to f_{0}(980)K^{+}\to K^{+}K^{-}K^{+})=\Gamma(D^{+}\to f_{0}(980)K^{+}){\cal B}(f_{0}(980)\to K^{+}K^{-})$

(3)

cannot be applied to extract the branching fraction of $D^{+}\to f_{0}(980)K^{+}$ due to the unknown ${\cal B}(f_{0}(980)\to K^{+}K^{-})$. Therefore, we will calculate the branching fractions of ${\cal B}(D\to SP\to P_{1}P_{2}P)$ directly and compare them with experiment (see Table 8 below).

It is known that the underlying structure of scalar mesons is not well established theoretically (see, e.g., Refs. Amsler ; Close for a review). Scalar mesons with masses lower than 2 GeV can be classified into two nonets: one nonet with masses below or close to 1 GeV, including the isoscalars $f_{0}(500)$ (or $\sigma$), $f_{0}(980)$, the isodoublet $K_{0}^{*}(700)$ (or $\kappa$) and the isovector $a_{0}(980)$; and the other nonet with masses above 1 GeV, including $f_{0}(1370)$, $a_{0}(1450)$, $K^{*}_{0}(1430)$ and $f_{0}(1500)/f_{0}(1710)$. If the scalar meson states below or near 1 GeV are identified as the conventional low-lying $0^{+}$ $q\bar{q}$ nonet, then the nonet states above 1 GeV could be excited $q\bar{q}$ states.

In the naïve quark model, the flavor wave functions of the light scalars read

	$\displaystyle\sigma={1\over\sqrt{2}}(u\bar{u}+d\bar{d})~{},\qquad\qquad~{}f_{0}=s\bar{s}~{},$
	$\displaystyle a_{0}^{0}={1\over\sqrt{2}}(u\bar{u}-d\bar{d})~{},\qquad\qquad a_{0}^{+}=u\bar{d}~{},\qquad a_{0}^{-}=d\bar{u}~{},$		(4)
	$\displaystyle\kappa^{+}=u\bar{s}~{},\qquad\kappa^{0}=d\bar{s}~{},\qquad~{}\bar{\kappa}^{0}=s\bar{d}~{},\qquad~{}\kappa^{-}=s\bar{u}~{},$

where an ideal mixing for $f_{0}$ and $\sigma$ is assumed as $f_{0}(980)$ is the heaviest one and $\sigma$ the lightest one in the light scalar nonet. However, as summarized in Ref. Cheng:SAT , this simple picture encounters several serious problems:

1. 1.

   It is impossible to understand the mass degeneracy between $f_{0}(980)$ and $a_{0}(980)$, which is the so-called “inverted spectrum problem.”

2. 2.

   The $P$-wave $0^{+}$ meson has one unit of orbital angular momentum which costs an energy around 500 MeV. Hence, it should have a mass lying above rather than below 1 GeV.

3. 3.

   It is difficult to explain why $\sigma$ and $\kappa$ are much broader than $f_{0}(980)$ and $a_{0}(980)$ in width.

4. 4.

   The $\gamma\gamma$ widths of $a_{0}(980)$ and $f_{0}(980)$ are much smaller than naïvely expected for a $q\bar{q}$ state bar85 .

5. 5.

   The radiative decay $\phi\to a_{0}(980)\gamma$, which cannot proceed if $a_{0}(980)$ is a pure $q\bar{q}$ state, can be nicely described by the four-quark nature of $a_{0}(980)$ Achasov:1987ts ; Achasov:2003cn or the kaon loop mechanism Schechter06 . Likewise, the observation of the radiative decay $\phi\to f_{0}(980)\gamma\to\pi\pi\gamma$ is also accounted for by the four-quark state of $f_{0}(980)$ Achasov:2003cn .

It turns out that these difficulties can be readily resolved in the tetraquark scenario where the four-quark flavor wave functions of light scalar mesons are symbolically given by Jaffe

	$\displaystyle\sigma=u\bar{u}d\bar{d}~{},\qquad\qquad\qquad~{}~{}f_{0}=\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})s\bar{s}~{},$
	$\displaystyle a_{0}^{0}=\frac{1}{\sqrt{2}}(u\bar{u}-d\bar{d})s\bar{s}~{},\qquad a_{0}^{+}=u\bar{d}s\bar{s}~{},\qquad a_{0}^{-}=d\bar{u}s\bar{s}~{},$
	$\displaystyle\kappa^{+}=u\bar{s}d\bar{d}~{},\qquad\kappa^{0}=d\bar{s}u\bar{u}~{},\qquad\bar{\kappa}^{0}=s\bar{d}u\bar{u}~{},\qquad\kappa^{-}=s\bar{u}d\bar{d}~{}.$		(5)

The four quarks $q^{2}\bar{q}^{2}$ can form an $S$-wave (rather than $P$-wave) $0^{+}$ meson without introducing one unit of orbital angular momentum. This four-quark description explains naturally the inverted mass spectrum of the light nonet, ¹¹1However, it has been claimed recently in Ref. Kuroda:2019jzm that the inverse mass hierarchy can be realized in the $q\bar{q}$ picture through a $U(1)$ axial anomaly including explicit $SU(3)_{F}$ breaking. The anomaly term contributes to $a_{0}(980)$ with the strange quark mass and to $\kappa/K_{0}^{*}(700)$ with the up or down quark mass due to its flavor singlet nature. The current mass of the strange quark makes the $a_{0}$ meson heavier than the $\kappa$ meson. especially the mass degeneracy between $f_{0}(980)$ and $a_{0}(980)$, and accounts for the broad widths of $\sigma$ and $\kappa$ while $f_{0}(980)$ and $a_{0}(980)$ are narrow because of the suppressed phase space for their decays to the kaon pairs. Lattice calculations have confirmed that $a_{0}(1450)$ and $K_{0}^{*}(1430)$ are $q\bar{q}$ mesons, and suggested that $\sigma$, $\kappa$ and $a_{0}(980)$ are tetraquark mesonia Prelovsek ; Mathur ; Wakayama:scalar ; Alexandrou:a0kappa ; Alexandrou:a0 .

The inverted spectrum problem can also be alleviated in the scenario where the light scalars are dynamically generated from the meson-meson interaction, with the $f_{0}(980)$ and the $a_{0}(980)$ coupling strongly to the $K\overline{K}$ channel with isospin 0 and 1, respectively. Indeed, the whole light scalar nonet appears naturally from properly unitarized chiral amplitudes for pseudoscalar-pseudoscalar scatterings Oller:1997ng ; Oller:1998hw . Consequently, both $f_{0}(980)$ and $a_{0}(980)$ are good candidates of $K\overline{K}$ molecular states Weinstein:1990gu , while $\sigma$ and $\kappa$ can be considered as the bound states of $\pi\pi$ and $K\pi$, respectively.

In the naïve two-quark model with ideal mixing for $f_{0}(980)$ and $\sigma(500)$, $f_{0}(980)$ is purely an $s\bar{s}$ state, while $\sigma(500)$ is an $n\bar{n}$ state with $n\bar{n}\equiv(\bar{u}u+\bar{d}d)/\sqrt{2}$. However, there also exists some experimental evidence indicating that $f_{0}(980)$ is not a purely $s\bar{s}$ state. For example, the observation of $\Gamma(J/\psi\to f_{0}\omega)\approx{1\over 2}\Gamma(J/\psi\to f_{0}\phi)$ PDG clearly shows the existence of the non-strange and strange quark contents in $f_{0}(980)$. Therefore, isoscalars $\sigma(500)$ and $f_{0}(980)$ must have a mixing

$\displaystyle|f_{0}(980)\rangle=|s\bar{s}\rangle\cos\theta+|n\bar{n}\rangle\sin\theta~{},\qquad|\sigma(500)\rangle=-|s\bar{s}\rangle\sin\theta+|n\bar{n}\rangle\cos\theta~{}.$

(6)

Various mixing angle measurements have been discussed in the literature and summarized in Refs. CCY ; Fleischer:2011au . A recent measurement of the upper limit on the branching fraction product ${\cal B}(\overline{B}^{0}\to J/\psi f_{0}(980))\times{\cal B}(f_{0}(980)\to\pi^{+}\pi^{-})$ by LHCb leads to $|\theta|<30^{\circ}$ LHCb:theta . Likewise, in the four-quark scenario for light scalar mesons, one can also define a similar $f_{0}$-$\sigma$ mixing angle

$\displaystyle|f_{0}(980)\rangle=|n\bar{n}s\bar{s}\rangle\cos\phi+|u\bar{u}d\bar{d}\rangle\sin\phi~{},\qquad|\sigma(500)\rangle=-|n\bar{n}s\bar{s}\rangle\sin\phi+|u\bar{u}d\bar{d}\rangle\cos\phi~{}.$

(7)

It has been shown that $\phi=174.6^{\circ}$ Maiani .