$Z_c(4430)$ and $Z_c(4200)$ as triangle singularities

$Z_c(4430)$ discovered by the Belle and confirmed by the LHCb in $\bar{B}^0\to\psi(2S)K^-\pi^+$ is generally considered to be a charged charmonium-like state that includes minimally two quarks and two antiquarks. $Z_c(4200)$ found in $\bar{B}^0\to J/\psi K^-\pi^+$ by the Belle is also a good candidate of a charged charmonium-like state. We demonstrate that kinematical singularities in triangle loop diagrams induce a resonance-like behavior that can consistently explain the properties (mass, width, and Argand plot) of $Z_c(4430)$ and $Z_c(4200)$ from the experimental analyses. The triangle diagrams include only experimentally well-established hadrons. Applying this idea to $\Lambda_b^0\to J/\psi p\pi^-$, we also identify triangle singularities that behave like $Z_c(4200)$, but no triangle diagram is available for $Z_c(4430)$. This is consistent with the LHCb's finding that their description of the $\Lambda_b^0\to J/\psi p\pi^-$ data is significantly improved by including a $Z_c(4200)$ contribution while $Z_c(4430)$ seems to hardly contribute. Even though the proposed mechanisms have uncertainty in the absolute strengths which are currently difficult to estimate, they are certainly a compelling alternative to tetraquark-based interpretations of $Z_c(4430)$ and $Z_c(4200)$.

Z c (4430) was discovered by the Belle Collaboration as a bump in the ψ(2S)π + invariant mass distribution ofB 0 → ψ(2S)K − π + [3]; charge conjugate modes are implicitly included throughout. Many theoretical interpretations of Z c (4430) have been proposed: diquarkantidiquark, hadronic molecule, hadro-charmonium, hybrid, and kinematical cusp, as summarized in reviews [4][5][6][7]. The experimental determination of the spin-parity (J P = 1 + ) ruled out many of the scenarios [8,9]; in particular, the threshold cup has been eliminated. After the LHCb Collaboration found a resonance behavior in the Z c (4430) Argand plot [9], a consensus is that Z c (4430) is a genuine tetraquark state [10]. Z c (4200) is also a good tetraquark candidate. It was observed by the Belle inB 0 → J/ψK − π + [11]. The LHCb also found Z c (4200)-like contributions inB 0 → J/ψK − π + [12] and Λ 0 b → J/ψ p π − [13]. Meanwhile, triangle singularities (TS) [14][15][16] have been considered to interpret several resonance(like) states such as the hidden charm pentaquark P c (4450) + [17][18][19] and a 1 (1420) [20,21]. The TS is a kinematical effect that arises in a triangle diagram like Fig. 1 when a special kinematical condition is reached: three intermediate particles are allowed to be on-shell at the same time. A mathematical detail how the singularity shows up is well illustrated in Ref. [19]. Although it was claimed in Ref. [22,23] that a kinematical effect from a triangle diagram can induce a spectrum bump of Z c (4430), this effect has nothing to do with the above TS and relies on the existence of an experimentally unobserved hadron.
In this work, we give a new insight into Z c (4430) and Z c (4200) by showing that these exotic candidates can be consistently interpreted as TS if the TS have absolute strengths detectable in the experiments. First we point out that triangle diagrams in Fig. 2, formed by experimentally well-established hadrons, meet the kinematical condition to cause the TS (in the zero width limit of unstable particles). Then we demonstrate that the diagram of Fig. 2(a) [ Fig. 2 The Breit-Wigner masses and widths extracted from the spectra turn out to be in very good agreement with those of Z c (4430) and Z c (4200). The Z c (4430) Argand plot from the LHCb [9] is also well reproduced by the triangle diagram. Finally, we give a natural explanation for the absence of Z c (4430) in Λ 0 b → J/ψpπ − and e + e − annihilations in terms of the TS.
We use a simple and reasonable model to calculate the triangle diagrams of Fig. 2. Let us use labeling of particles and their momenta in Fig. 1 to generally express the triangle amplitudes: where E denotes the total energy in the center-of-mass (CM) frame. The quantity E x (p x ) = p 2 x + m 2 x is the energy of a particle x with the mass m x and momentum p x . An exception is applied to unstable intermediate where Γ j is the width. It is particularly important to consider the vector charmonium width in Fig. 2(a) where ψ(4260) and K * (892) have comparable widths. We take the mass and width values from Ref. [2].
Regarding the 23 → ab interaction v ab;23 in Eq. (1), where the particles 2 and a are vector charmoniums while 3 and b are pions, we use an s-wave interaction: where ǫ a and ǫ 2 are polarization vectors for the particles a and 2, respectively. The form factors f 01 ab (p ab ) and f 01 23 (p 23 ) will be defined in Eq. (4); the momentum of the particle i in the ij-CM frame is denoted by p ij and p ij = |p ij |. An s-wave pair of ψ f π coming out from this interaction has J P = 1 + , which is consistent with the experimentally determined spin-parity of Z c (4430) and Z c (4200), and also with the insignificant d-wave contribution in the Z c (4430)-region [9].
The R → ij decay vertex Γ ij,R in Eq. (1) is explicitly given as where Y LM is spherical harmonics. Clebsch-Gordan coefficients are written as (abcd|ef ), and the spin and its z-component of a particle x are denoted by s x and s z x , respectively. The form factor f LS ij (p ij ) is parameterized as where we use the cutoff Λ = 1 GeV throughout; main conclusions in this work are essentially determined by the kinematical singularities and are robust in a reasonable cutoff range: Λ = 0.7 − 1.3 GeV. For the 1 → 3c and 23 → ab interactions, there is only one available set of {L, S} for which we set g LS ij = 1. Regarding the H → 12 decay, meanwhile, several sets of {L, S} are available. The H → 12 decay vertices are currently unknown but details would not change the main conclusions. Thus we assume simple structures and detectable strengths. For theB 0 decays, we set g LS ij = 1 only for S = |s 1 − s 2 | and the lowest allowed L; g LS ij = 0 for the other {L, S}. Because of using the above v ab;23 , theB 0 decays are necessarily parity-violating. For the Λ 0 b decays, on the other hand, both parity-conserving and -violating interactions are possible. We choose the parity-conserving one and set g LS ij = 1 only for S = |s 1 − s 2 | and the lowest allowed L; g LS ij = 0 otherwise. We first present the ψ f π invariant mass distributions forB 0 → ψ(2S)K − π + andB 0 → J/ψK − π + . The red solid curves in Figs. 3(a) and (b) are solely from the triangle diagrams of Figs. 2(a) and (b), respectively. For comparison, we also plot the phase-space distributions by the black dotted curves. A clear resonance-like peak appears at m ψ(2S)π ∼ 4.45 GeV in panel (a) (m J/ψπ ∼ 4.2 GeV in panel (b)) due to the TS. We also calculated the m J/ψπ spectrum forB 0 → J/ψK − π + from the triangle diagram of Fig. 2(a), and obtained a result very similar to Fig. 3(a) after the normalization explained in the caption.
We interpret the peaks associated with the TS in terms of Z c -excitation mechanisms. We fit the Dalitz plot distributions from the triangle diagrams of Figs. 2(a) and (b) using the mechanism ofB 0 → Z c K − followed by Z c → ψ f π + ; the Z c propagation is expressed by the Breit-Wigner form used in Ref. [8]. In the fit, we include the kinematical region where the magnitude of the Dalitz plot distribution is larger than 20% of the peak height. The obtained fits of reasonable quality are shown by the blue dash-dotted curves in Figs. 3(a) and (b). Because the spectrum shape from the triangle diagrams is somewhat different from the Breit-Wigner, their peak positions are slightly different. The Breit-Wigner parameters resulting from the fits are given in Table I along with those from experimental data. Their agreement is remarkable.
Next we confront the triangle amplitude with the Z c (4430) Argand plot from the LHCb [9]. Because Z c and K − are relatively in p-wave, the angle-independent part of the amplitude (A) to be compared with the Argand plot is   Fig. 2(a) [(b)]. The parameters from the experimental analyses are also shown; the first (second) errors are statistical (systematic).

Zc(4430)
Zc (4200) (a) Belle [8] LHCb [9] (b) Belle [11] MBW In the LHCb analysis, a complex value representing the Z c (4430) amplitude is fitted to dataset in a m 2 ψ(2S)π bin with a bin size ∆. To take account of the bin size, we simply average our amplitude without pursuing a theoretical rigor:Ā where m 2 ab (i) is the central value of an i-th bin. As shown in Fig. 4, the empirical Z c (4430) Argand plot is fitted well withĀ(m 2 ab (i)) from the triangle diagram of Fig. 2(a); c norm = −0.16−0.79i, c bg = 0.16+0.02i in Eq. (5). This demonstrates that the counterclockwise behavior found in Ref. [9] does not necessarily indicate the existence of a resonance state. A similar statement has also been made in Ref. [17]. We also confirmed a counterclockwise behavior of the Argand plot from the triangle diagram of Fig. 2(b), as the Belle [11] found the Z c (4200) amplitude to behave so.

FIG. 4.
Zc(4430) Argand plot. Six curved segments are from triangle diagram Fig. 2(a). Six data points from Ref. [9] are from fitting data in six bins equally-separating the range of 18 GeV 2 ≤ m 2 ψ(2S)π ≤ 21.5 GeV 2 ; m 2 ψ(2S)π increases counterclockwise. A curved segment and a data point of the same color belong to the same bin. A solid circle is an average of the curved segment of the same color. See Eq. (6) for averaging.
by ∼ 40%, and thus R model ψ(4260) ∼ 0.1 × |c R ψπ | 2 . Therefore, the model reproduces R exp ψ(4260) ∼ 0.3 with |c R ψπ | ∼ 1.7, and the puzzling R exp Zc(4430) ∼ 11 is also reproduced with the same |c R ψπ |. Now we discuss the J/ψπ invariant mass distributions for Λ 0 b → J/ψpπ − induced by the triangle diagram of Fig. 2(c). In the Z c (4200)-region, the TS is expected to create a spectrum bump. Interestingly, several nucleon resonances (N * ) of 1400−1800 MeV can contribute to the singularities and, depending on the mass and width of N * , the position and width of the bump can vary. In Fig. 3(c), we show results obtained with some representative four-star resonances: N * = N (1440) 1/2 + , N (1520) 3/2 − , and N (1680) 5/2 + . As expected, the triangle diagrams including different N * generate different spectrum bumps in the Z c (4200)-region. In reality, these bumps may coherently interfere with each other to create a single broad bump. The LHCb analysis [13] found that the Λ 0 b → J/ψpπ − decay data is significantly better described by including the Z c (4200) amplitude. Because of limited statistics of the Λ 0 b → pπ − J/ψ data, the mass and width of Z c (4200) were assumed to be the same as those inB 0 → J/ψK − π + [11]. Therefore, the spectrum bumps shown in Fig. 3(c), some of which extend to the lower end of the Z c (4200)-region, are still consistent with the LHCb's finding.
Another important finding in the LHCb analysis [13] is that Z c (4430) seems to hardly contribute to Λ 0 b → J/ψpπ − . If Z c (4430) found inB 0 → ψ(2S)K − π + is associated with the TS, a natural explanation follows: within experimentally observed hadrons, no combination of a charmonium and a nucleon resonance is available to form a triangle diagram like Fig. 2(c) that causes TS at the Z c (4430) position. This idea can be further generalized. At present, a puzzling situation about Z c is that those observed in e + e − annihilations and in B decays are mutually exclusive. If the Z c states are due to TS, the answer is simple: a TS in a B decay does not exist or is highly suppressed in e + e − annihilations, and vice versa. Therefore, a key to establishing a genuine tetraquark state is to identify it in different processes including different initial states. However, there are still cases where, as we have seen in Figs. 3(b) and (c), different TS could induce similar resonance-like behaviors.
In summary, we demonstrated that Z c (4430) and Z c (4200), which are often regarded as genuine tetraquark states, can be consistently interpreted as singularities from the triangle diagrams we identified. The Breit-Wigner parameters extracted from the TS-induced spectrum bumps ofB 0 → ψ f K − π + are in very good agreement with those of Z c (4430) and Z c (4200) from the Belle and LHCb analyses. The Z c (4430) Argand plot from the LHCb is also well reproduced. We also explained in terms of TS why Z c (4200)-like contribution was observed in Λ 0 b → J/ψpπ − but Z c (4430) was not. In future, we look for more TS that would be responsible for the other Z c and other seemingly exotic hadrons.