Predicted charged charmonium-like structures in the hidden-charm dipion decay of higher charmonia

# Predicted charged charmonium-like structures in the hidden-charm dipion decay of higher charmonia

Dian-Yong Chen    Xiang Liu111Corresponding author Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China
School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
Nuclear Theory Group, Institute of Modern Physics of CAS, Lanzhou 730000, China
July 18, 2019
###### Abstract

In this work, we predict two charged charmonium-like enhancement structures close to the and thresholds, where the Initial Single Pion Emission mechanism is introduced in the hidden-charm dipion decays of higher charmonia , , and charmonium-like state . We suggest BESIII to search for these structures in the , and invariant mass spectra of the decays into , and . In addition, the experimental search for these enhancement structures in the , and invariant mass spectra of the hidden-charm dipion decays will be accessible at Belle and BaBar.

###### pacs:
13.25.Gv, 14.40.Pq, 13.75.Lb

## I introduction

In the past years, experimentalist has made big progress on the search for the charmonium-like states, the so-called XYZ states, in the meson decay, the collision, the fusion process, which have aroused extensive interest in revealing the underlying properties of the observed charmonium-like states (see Refs. Swanson:2006st (); Zhu:2007wz (); Godfrey:2008nc (); Nielsen:2009uh (); Brambilla:2010cs () for a review). The study of charmonium-like states is a research field full of challenges and opportunities in hadron physics.

Very recently the Belle Collaboration Collaboration:2011gj () reported two charged structures around 10610 MeV and 10650 MeV by studying the () and () invariant mass spectra of hidden-bottom decay channels (see Refs. Liu:2008fh (); Liu:2008tn (); Bondar:2011ev (); Chen:2011zv (); Zhang:2011jj (); Yang:2011rp (); Bugg:2011jr (); russian (); Guo:2011gu (); Sun:2011uh (); Chen:2011pv () for theoretical progress). In Ref. Chen:2011pv (), we proposed the Initial Single Pion Emission (ISPE) mechanism to explain the observed structures. By emitting a pion, decays into and mesons with low momentum. Then, and mesons interact with each other by exchanging meson and transit into or . Here, two structures near the and thresholds appear in the and invariant mass spectra, which could correspond to and Collaboration:2011gj ().

Just indicated in Ref. Chen:2011pv (), if the ISPE mechanism is an universal mechanism existing in heavy quarkonium decay, we can naturally extend such physical picture to study hidden-charm decays of higher vector charmonia due to the similarity between charmonium and bottomonium families, and predict some novel phenomena similar to the structures.

In Particle Data Book Nakamura:2010zzi (), six vector charmonia are established well, which are , , , , and with . Among these charmonia, only , and are higher than the thresholds of , and . Thus, we study the hidden-charm decays of , and via the ISPE mechanism. From the analysis of such modes, one indicates that enhancement structures similar to the charged also exist in the charm case.

In addition, in this work we will study the hidden-charm decays of , which is an important charmonium-like state observed by the BaBar Collaboration in the process Aubert:2005rm (). Its mass, width and are MeV, MeV and Nakamura:2010zzi (). The study presented in Ref. Chen:2010nv () indicates that can be related to charmonia and , where the structure can be reproduced by the interference of production amplitudes of the processes via direct annihilation and through intermediate charmonia and Chen:2010nv (). We naturally apply the ISPE mechanism existing in and decays to discuss the hidden-charm dipion decays of . Thus, studying hidden-charm decays through the ISPE mechanism is an intriguing issue, where we will also predict some enhancement structure similar to the charged . As announced by BaBar Aubert:2005rm (), was first observed in its decay channel. Searching enhancement structure in the invariant mass spectrum of will be accessible in future experiments, which could provide a direct test to the non-resonant explanation for proposed in Chen:2010nv ().

This work is organized as follows. After the Introduction, we illustrate the hidden-charm dipion decays of higher charmonia under the ISPE mechanisms. In Sec. III, the numerical results are presented. The last section is the discussion and conclusion.

## Ii The hidden-charm decays of higher charmonia

### ii.1 The ISPE mechanism

With as an example, we first illustrate the possible decay mechanisms in the dipion hidden-charm decay of higher charmonium. can directly decay into . In Refs. Kuang:1981se (); Yan:1980uh (); Novikov:1980fa (), the QCD Multipole Expansion method was proposed and can be applied to calculate such direct decay process. The second mechanism is that the dipion is from the intermediate scalar () or tensor () meson, where the hadronic loops constructed by the mesons could be as a bridge to connect and (see Ref. Chen:2011qx () for more details).

The remaining decay mechanism existing in the hidden-charm dipion decays of higher charmonia is the ISPE mechanism, which was first proposed in Ref. Chen:2011pv (). By the quark-level diagram we give an explicit description (left-side diagram in Fig. 1) of the ISPE mechanism in decay. The physical picture is that with a pion emission first dissolves into and mesons with low momentum, which further turn into . Here, transition occurs via exchanging meson Chen:2011pv ().

An equivalent hadron-level description is also presented in the right-side diagram of Fig. 1, which can be as an effective approach for dealing with the practical calculations.

### ii.2 Effective Lagrangian and coupling constant

We adopt effective Lagrangian approach to calculate these hadron-level diagrams listed in Fig. 1. Here, the effective Lagrangians involved in the interaction vertexes in Fig. 1 include Oh:2000qr (); Casalbuoni:1996pg (); Colangelo:2002mj ()

 Lψ′D(∗)D(∗)π =−igψ′DDπεμναβψ′μ∂νD∂απ∂β¯D+gψ′D∗Dπψ′μ(Dπ¯D∗μ+D∗μπ¯D) −igψ′D∗D∗πεμναβψ′μD∗ν∂απ¯D∗β−ihψ′D∗D∗πεμναβ∂μψ′νD∗απ¯D∗β,

where denotes the initial state charmonium (one of the , , or ). This Lagrangian reflects the initial state charmonium decays into .

 LD∗D(∗)π =igD∗Dπ(D∗μ∂μπ¯D−D∂μπ¯D∗μ)−gD∗D∗πεμναβ∂μD∗νπ∂α¯D∗β,
 LψD(∗)D(∗) =igψDDψμ(∂μD¯D−D∂μ¯D)−gψD∗Dεμναβ∂μψν(∂αD∗β¯D +D∂α¯D∗β)−igψD∗D∗{ψμ(∂μD∗ν¯D∗ν−D∗ν∂μ¯D∗ν) +(∂μψνD∗ν−ψν∂μD∗ν)¯D∗μ+D∗μ(ψν∂μ¯D∗ν−∂μψν¯D∗ν)},
 LhcD(∗)D(∗) =ghcD∗Dhμc(¯D∗μD+D∗μ¯D)+ighcD∗D∗εμναβ∂μhcνD∗α¯D∗β,

which will be applied to describe a rescattering mechanism involving in the two charmed mesons into or by exchanging a meson. In the above Lagrangians, and are grouped together on the basis of heavy quark symmetry while that pions appear as the result of a representation of the chiral symmetry. In addition, we define charm meson iso-doublets as , and Oh:2000qr ().

The values of the coupling constants can be determined by the relations

 gψDD = gψD∗D∗mDm∗D=gψD∗Dmψ√mDm∗D=mψfψ, ghcDD∗ = −2g1√mhcmDmD∗,  ghcD∗D∗=2g1mD∗√mhc, gD∗D∗π = gD∗Dπ√mDmD∗=2gfπ,  g1=−√mχc031fχc0,

where GeV and GeV are the decay constants of and , respectively. In addition, GeV can be approximately determined by the QCD sum rule approach Colangelo:2002mj (). With the measured branching ratio of by CLEO-c Anastassov:2001cw () and MeV, one gets Isola:2003fh ().

### ii.3 Decay Amplitudes

With these Lagrangians just listed above, we write out the decay amplitude for the dipion transition between and , i.e., there are three interaction vertexes and three propagators, which are obtained by the effective Lagrangian presented in Sec. II.2. Additionally, we also introduce the monopole form factor in decay amplitudes, which is taken as . Here, is the mass of the exchanged meson while the phenomenological parameter can be parameterized as with MeV. Such monopole form factor is introduced to describe the structure effects of the interaction vertexes as well as the off-shell effects of the exchanged charmed mesons for transitions in decays.

When only considering the intermediate contributions to , there are twelve diagrams just shown in Fig. 2. Among these diagrams, there are only six independent diagrams if considering symmetry, i.e., Fig. 2 (i) can be transferred into Fig. 2 (i+6) () by transformations and . Thus, the total decay amplitude for with the intermediate contributions are expressed as

 M[ψ(4040)→J/ψπ+π−]D∗¯D+h.c.=2∑i=1,⋯,6M(i)D∗¯D+h.c., (1)

where we mark the four momenta of the corresponding mesons. Factor 2 reflects symmetry mentioned above. The subscript denotes that occurs via the intermediate . The expressions of decay amplitudes () read as

 M(1)D∗¯D+h.c.=(i)3∫d4q(2π)4[gψ′D∗Dπϵμψ][igD∗D∗π(iPρ4)] ×[−igJ/ψD∗D∗ϵνJ/ψ((−iqν+ip2ν)gθϕ+(iP5ϕ+iqϕ)gνθ −(ip2θip5θ)gνϕ)]1p21−m2D−gϕμ+p1μpϕ1/m2D∗p22−m2D∗ ×−gθρ+qρqθ/m2D∗q2−m2D∗F2(q2), (2)
 M(2)D∗¯D+h.c.=(i)3∫d4q(2π)4[gψ′D∗Dπϵμψ][igD∗Dπ(−ipρ4)] ×[igJ/ψDDϵνJ/ψ(ip2ν−iqν)]−gμρ+p1μp1ρ/m2D∗p21−m2D∗ ×1p22−m2D1q2−m2DF2(q2), (3)
 M(3)D∗¯D+h.c.=(i)3∫d4q(2π)4[gψ′D∗Dπϵμψ][−gD∗D∗πεθϕδτ(iqθ) ×(−ipδ1)][−gJ/ψD∗Dερναβ(ip5ρ)ϵJ/ψν(−iqα)] ×−gμτ+p1μp1τ/m2D∗p21−m2D∗1p22−m2D ×−gβϕ+qβqϕ/m2D∗q2−m2D∗F2(q2), (4)

which correspond to the dipion transitions between and with a initial single pion () emission. , and can be obtained by , and respectively if making the replacement in Eqs. (2)-(4). Here, () are decay amplitudes of the dipion transitions between and with a initial single pion () emission.

We also present the decay amplitude of via the intermediate .

 M[ψ(4040)→J/ψπ+π−]D∗¯D∗=2∑α=1,⋯,4M(α)D∗¯D∗. (5)

We list all diagrams contributing to in Fig. 3. Among these eight eight diagrams, Fig. 2 () can be obtained by Fig. 2 () () if making the transformations and , which results in factor 2 in Eq. (5) due to symmetry.

The decay amplitudes and are expressed as

 M(1)D∗¯D∗=(i)3∫d4q(2π)4[−igψ′D∗D∗πεμραβϵψμ(ip3α) −ihψ′D∗D∗πεαμρβϵψμ(−ip0α)][igD∗Dπ(−ip4λ)] ×[−gJ/ψD∗Dεδνθϕ(ipδ5)ϵνJ/ψ(−ipθ2)]−gλρ+p1ρpλ1/m2D∗p21−m2D∗ ×−gϕβ+p2βpϕ2/m2D∗p22−m2D∗1q2−m2DF2(q2), (6)
 M(2)D∗¯D∗=(i)3∫d4q(2π)4[−igψ′D∗D∗πεμραβϵψμ(ip3α) −ihψ′D∗D∗πεαμρβϵψμ(−ip0α)][−gD∗D∗πεδτθϕ(−ip1δ)(iqθ)] ×[−igJ/ψD∗D∗ϵνJ/ψ((−iqν+ip2ν))gωλ+(ip5ω+iqω)gνλ +(−ip2λ−ip5λ)gνω]−gρτ+p1ρp1τ/m2D∗p21−m2D∗ ×−gωβ+p2βpω2/m2D∗p22−m2D∗−gλϕ+qϕqλ/m2D∗q2−m2D∗F2(q2). (7)

Thus, by Eqs. (6) and (7) we can easily obtain decay amplitudes and corresponding to Fig. 3 (3) and (4), where the transformation is performed.

In the following, we extend the same framework to study the dipion transition between and . By replacing with in Fig. 2 (1), (3), (4), (6), (7), (9), (10) and (12) and Fig. 3, we obtain all diagrams relevant to decay. The total decay amplitudes of via and are

 M[ψ(4040)→hc(1P)π+π−]D∗¯D+h.c. = 2∑β=1,⋯,4A(β)D∗¯D+h.c., (8) M[ψ(4040)→hc(1P)π+π−]D∗¯D∗ = 2∑κ=1,⋯,4A(κ)D∗¯D∗, (9)

respectively, where the concrete amplitude expressions are

 A(1)D∗¯D+h.c.=(i)3∫d4q(2π)4[gψ′D∗D∗πϵμψ] ×[igD∗Dπ(−ipρ4)][ighcD∗D∗εδνθϕ(ipδ5)ϵνhc]1p21−m2D ×−gϕμ+p2μpϕ2/m2D∗p22−m2D∗−gθρ+qρqθ/m2D∗q2−m2D∗F2(q2), (10)
 A(2)D∗¯D+h.c.=(i)3∫d4q(2π)4[gψ′D∗D∗πϵμψ][−gD∗D∗πεθϕδτ(iqθ) ×(−ipδ1)][ghcD∗Dϵhcν]−gμϕ+p1μpϕ1/m2D∗p21−m2D∗ ×1p22−m2D−gντ+qνqτ/m2D∗q2−m2D∗F2(q2), (11)

and

 A(1)D∗¯D∗=(i)3∫d4q(2π)4[−igψ′D∗D∗πεμραβϵμψ(ipα3−ipα0)] ×[igD∗Dπ(−ip4λ)][ghcD∗Dϵhcν]−gρλ+pρ1pλ1/m2D∗p21−m2D∗ ×−gβν+pβ2pν2/m2D∗p22−m2D∗1q2−m2DF2(q2), (12)
 A(2)D∗¯D∗=(i)3∫d4q(2π)4[−igψ′D∗D∗πεμραβϵμψ(ipα3−ipα0)] ×[−gD∗D∗πϵδτθϕ(−ipδ1)(iqθ)][ighcD∗D∗εκνλω(ipκ5)ϵνhc] ×−gρτ+pρ1pτ1/m2D∗p21−m2D∗−gβω+pβ2pω2/m2D∗p22−m2D∗ ×−gϕλ+qϕqλ/m2D∗q2−m2D∗F2(q2). (13)

After performing the transformation , , , and can be transferred into , , and respectively.

The differential decay width for reads as

 dΓ=131(2π)3132m3ψ(4040)¯¯¯¯¯¯¯¯¯¯¯¯|M2|dm2J/ψπ+dm2π+π− (14)

with and , where the overline indicates the average over the polarizations of the in the initial state and the sum over the polarization of in the final state. Replacing with or , we obtain the differential decay width for or .

When studying the hidden-charm dipion decay of other higher charmonia , and charmonium-like state , we only need to replace the relevant coupling constants and the masses in the formulism of the decays.

## Iii numerical result

In this work, we are mainly concerned with the line shapes of the differential decay widths of , , and charmonium-like state decays into , and , which are dependent on the invariant mass spectra of , and . Thus, we set the coupling constants of as 1 in our calculation. Besides these coupling constants listed in Sec. II.2, other input parameters are the masses involved in our calculation, which are taken from Particle Data Book Nakamura:2010zzi ().