Simulating the charged charmoniumlike structure $Z_c(4025)$

Inspired by recent observation of charged charmoniumlike structure $Z_c(4025)$, we explore the $Y(4260)\to (D^*\bar{D}^*)^- \pi^+$ decay through the initial-single-pion-emission mechanism, where the $D^*\bar{D}^*\to D^*\bar{D}^*$ interaction is studied by the ladder approximation including a non-interacting case. Our calculation of the differential decay width for $Y(4260)\to (D^*\bar{D}^*)^- \pi^+$ indicates that a charged enhancement structure around $D^*\bar{D}^*$ appears in the $D^*\bar{D}^*$ invariant mass spectrum for this process, which can correspond to newly observed $Z_c(4025)$ structure.

Before the observation of Z c (4025), the Belle Collaboration reported a charged bottomoniumlike structure Z b (10650) by studying Υ(10860) → (B * B * ) ± π ∓ [2], where Z b (10650) is near the (B * B * ) ± threshold. The similarity between Z c (4025) and Z b (10650) indicates that Z c (4025) can be as the counterpart of Z b (10650). In Ref. [3], we have proposed an explanation via the initial-single-pion-emission (ISPE) mechanism why there exists Z b (10650) in the decay Υ(10860) → (B * B * ) ± π ∓ . What is more important is that we have already predicted a charged structure near the D * D * threshold in the (D * D * ) ± invariant mass spectrum of ψ(4415) → (D * D * ) ± π ∓ in the same paper.
This recent experimental discovery of Z c (4025) provides us more chances to further reveal the underlying mechanism behind this novel phenomenon. In the past decades, experimental search for exotic states beyond the conventional hadron configurations is an important and intriguing research topic. The peculiarities of Z c (4025) immediately drive us to recognize that Z c (4025) can be the most reliable candidate as an exotic state. However, before giving a one-sided view, we need to exhaust all the possibilities under conventional frameworks.
Along this way, in this paper we analyze the decay process Y(4260) → (D * D * ) ± π ∓ via the ISPE mechanism or its extension to include higher orders. This mechanism [4] was first proposed to understand why two bottomoniumlike structures Z b (10610) and Z b (10650) can be found in the Υ(nS )π ± (n = 1, 2, 3) and h b (mP)π ± (m = 1, 2) invariant mass spectra of e + e − → Υ(nS )π + π − , h b (mP)π + π − at √ s = 10865 MeV [5]. Later, the ISPE mechanism has been extensively applied to study the hidden-charm dipion/dikaon decays of higher charmonia and charmoniumlike states [6][7][8], the hidden-bottom dipion decays of Υ(11020) [9], and the hidden-strange dipion decays of Y(2175) [10], where many novel phenomena of charged enhancement structures have been predicted. In this work, by studying the process Y(4260) → (D * D * ) − π + , we expect to answer whether a newly observed charged structure Z c (4025) can be explained by the ISPE mechanism, which is an intriguing research topic, too, to search for the underlying mechanism behind this kind of novel phenomena.
This work is organized as follows. After introduction, we present the calculation of Y(4260) → (D * D * ) ± π ∓ via extension of the ISPE mechanism, where description of the interaction D * 0 D * − → D * 0 D * − is given by the ladder diagrams applying an effective Lagrangian approach. In Sec. III, the numerical results are shown in comparison with the experimental data. The Last section is a short summary.
The ISPE mechanism for Y(4260) → (D * D * ) − π + is shown by the diagram in Fig. 1. Due to this mechanism, the emitted pion from the Y(4260) decay plays a very important role and has continuous energy distribution, which easily enables D * 0 and D * − with low momenta to interact with each other. Thus, in the following our main task is to describe the D * 0 D * − → D * 0 D * − interaction and combine this reaction with the corresponding Y(4260) decay. In this work, we include a tree diagram, i.e., the case that the kernel does not include an interaction.
To calculate the D * 0 D * − → D * 0 D * − interaction near the threshold as depicted by a grey kernel shown in Fig. 1, we adopt a ladder approximation presented in Fig. 3, where we borrow some ideas from the Bethe-Salpeter equation [11]. After expanding the amplitude by partial wave bases, the ladder diagrams with n loops can be expressed as a geometric series, which allows us to sum over all the ladder diagrams (see the first row in Fig. 3). This treatment is allowed when the higher loop contribution is as large as the lower-order one. In order to have a geometric series we make a further approximation for the ladder diagrams to insert cuts in between all the white kernels as shown in the first row of Fig. 1 this work we introduce a pion exchange and a contact term as the main contributions to the direct D * 0 D * − → D * 0 D * − interaction as listed in the second row of Fig. 3 depicted by the white kernel which is included in the grey kernel. The contact term can be regarded as an effective one due to the collection of heavier particle exchanges other than pion and hence there appears a relative phase to one pion exchange term. Here, we need to emphasize that we may use only a contact term to construct a much simpler model assuming one pion exchage can be approximated as a contact term. However, one pion exchange actually denotes the long-distant contribution while the contact term reflects the short-distant contrition from the heavier meson exchanges. Considering these facts, we would like to introduce both one pion exchange and contact term.
In the following, we first give the general formula describ- ing the two body → two body process. Before obtaining a total amplitude of Fig. 1 and all the ladder diagrams of the first row of Fig. 3, we need to consider the following items. Fig. 3 are directly described in terms of two-particle bases, |p 1 , p 2 , s 1 , s 2 , σ 1 , σ 2 , in the initial and final states. On the other hand, a simple relation between grey and white blobs is obtained if they are expressed in terms of a partial wave basis, |p, j, σ, ℓ, s . Detailed propertied of these bases which are used in deriving equations below are give in Appendix.

Feynman diagrams in
2. Insert cuts in all the places in a grey blob where two propagators connecting two white blobs like the first row in Fig. 3 appear.
3. Expand the grey blob in terms of the white blobs.

5.
One cut is now inserted between the vertex and the grey blob as in Fig. 2 so that a total amplitude in Fig. 1 can be a multiplication of the tree vertex, Y(4260)π + D * 0 D * − , and the approximate ladder diagrams.
Using Eq. (A7), the grey and white blobs are expressed as q,j,σ,l,s |T 0 | p, j, σ, ℓ, s = T ( j) 0 (p)l ,s ℓ,s δ 4 (p − q) , (2) which are expressed in partial wave bases and T and T 0 are the corresponding T matrices, respectively. Following the items 2 and 3 above, the diagrams listed in the first row of Fig. 3 can be finally described by the series where β = (2π) 4 /2, all the indices are suppressed and the amplitude T ( j) 0 (p)l ,s ℓ,s which consist of the one-pion exchange contribution and the contact term. All the quantities included in Eq. (3) should be tacitly understood to be matrices and be accordingly multiplied with each other.
Having the above preparation and using the effective Lagrangian approach, we illustrate how to obtain a matrix element p 1 , s 1 , σ 1 ; p 2 , s 2 , σ 2 |T 0 |p ′ 1 , s ′ 1 , σ ′ 1 ; p ′ 2 , s ′ 2 , σ ′ 2 for the discussed D * 0 D * − → D * 0 D * − interaction. The involved effective Lagrangians are given by where these Lorentz structures are given in Refs. [12][13][14]. These effective Lagrangians are obtained by assuming the S U(2) invariance among couplings of S U(2) pseudoscalar and vector multiplets as usual. The coupling constant g D * D * π can be related to the D * → Dπ decay [15] and hence, g D * D * π = 8.94 GeV −1 is obtained [16]. However, the coupling constants g YD * D * π and h YD * D * π cannot be constrained since they are related to the inner structure of Y(4260). In this work, we will discuss the line shapes of the D * 0 D ( * −) invariant mass spectra when taking different values of ξ = h YD * D * π /g YD * D * π . The amplitude for the interaction D * 0 (p 2 , ǫ 2 )D * − (p 1 , ǫ 1 ) → D * 0 (p 4 , ǫ 4 )D * − (p 3 , ǫ 3 ) by exchanging one pion is while the amplitude for the contact term reads as In addition, the tree amplitude of the direct Y(4260)(p, ǫ Y ) → D * 0 (p 2 , ǫ 2 )D * − (p 1 , ǫ 1 )π + (k) decay is As for the process D * 0 D * − → D * 0 D * − , a matrix element can be further expressed as where the phase φ is introduced. Following the item 3, we need to insert the cut in between the tree vertex and the grey blob as in Fig. 2. There have been a couple of examples to calculate the decay amplitudes by inserting a cut between a tree vertex and other diagrams. See, e.g., Refs. [17,18] and [19]. Finally, by using Eq. (3), the total partial wave amplitude for the process Y(4260) → π + D * 0 D * − discussed in this work becomes where T ( j) YD * D * π (p) ℓ ′ ,s ′ ℓ,s is the tree amplitude given by Eq. (10) expressed in partial wave bases like in Eq. (4).
Summing over all T ( j) total (p)l ,s ℓ,s partial amplitudes with different quantum numbers, we get the total amplitude M. The differential decay width reads as where p π is a three-momentum of the emitted pion in the rest frame of Y(4260), while (p * D * , Ω * D * ) is a momentum and angle of D * in the cms rest frame of the D * 0 and D * − mesons. Ω π denotes the angle of π in the rest frame of Y(4260) and m D * D * is the D * 0 D * − invariant mass.

III. NUMERICAL RESULTS
With the above analytical calculations, in the following we present the differential decay width of Y(4260) → π + D * 0 D * − dependent on the m D * D * invariant mass (see Fig. 4), where three free parameters g c , ξ and φ are involved in our calculation. By this study, we want to answer whether the newly observed Z c (4025) can be reproduced by our model. In our numerical calculation, we only take the ℓ = 0 partial wave since its contribution is dominant. The comparison between our theoretical result and the experimental data indicates that we can simulate the enhancement structure near the D * D * threshold, which is similar to the Z c (4025) structure observed by BESIII just shown in Fig. 4. Here, we notice that there appear experimental data below the threshold, which is due to the adopted experimental method, i.e., BESIII has studied the π ∓ recoil mass spectrum [1]. The above comparison between theoretical and experimental results provides a direct evidence that newly observed Z c (4025) structure can be well understood via the ISPE mechanism.

IV. SUMMARY
In summary, a new charged enhancement Z c (4025) near the D * D * threshold has been reported by BESIII. This intriguing experimental observation not only makes the family of the charged charmoniumlike structure become abundant, but also stimulates our interest in revealing the underlying mechanism behind this novel phenomenon. In this work, we study the Y(4260) → π + D * 0 D * − decay via the ISPE mechanism, where the involved D * 0 D * − → D * 0 D * − interaction is considered by introducing the ladder diagrams. Our result shows that there exists an enhancement structure near D * D * threshold appearing in the D * D * invariant mass spectrum of Y(4260) → π + D * 0 D * − , which can correspond to the newly observed Z c (4025). This fact indicates that the ISPE mechanism existing in the Y(4260) decays can be as one of the possible mechanisms to explain this new BESIII's observation.
At present, experiment has made big progress on searching for charged bottomoniumlike and charoniumlike structures. Studying these phenomena is an interesting research topic with full of challenges and opportunities. Further theoretical and experimental efforts will be helpful to finally understand what is the reason resulting in these observations. Before closing this section, we need to discuss further developments of our model: 1. In this work, we have neglected the coupled-channel effect arising from one-loop box diagrams like D * 0 D * − → D 0 D − /D * 0 D − /D 0 D * − → D * 0 D * − via two π 0 exchanges. Hence, in the next step, we need to include the coupledchannel effect in our model.
2. In this work, we introduce only the imaginary part of the loop when calculating the diagrams listed in Fig. 3. To some extent, this treatment is an approximation. Hence, we need to develop our model to include the real part of loop diagrams.