# Can the \(\psi (4040)\) explain the peak associated with *Y*(4008)?

- 55 Downloads

## Abstract

We study the well-known resonance \(\psi (4040)\), corresponding to a \(3^{3}S_{1}\) charm–anticharm vector state \(\psi (3S)\), within a QFT approach, in which the decay channels into *DD*, \(D^{*}D\), \(D^{*}D^{*}\), \(D_{s}D_{s}\) and \(D_{s}^{*}D_{s}\) are considered. The spectral function shows sizable deviations from a Breit–Wigner shape (an enhancement, mostly generated by \(DD^{*}\) loops, occurs); moreover, besides the \(c{\bar{c}}\) pole of \(\psi (4040)\), a second dynamically generated broad pole at 4 GeV emerges. Naively, it is tempting to identify this new pole with the unconfirmed state *Y*(4008). Yet, this state was not seen in the reaction \(e^{+}e^{-}\rightarrow \psi (4040)\rightarrow DD^{*}\), but in processes with \(\pi ^{+}\pi ^{-}J/\psi \) in the final state. A detailed study shows a related but different mechanism: a broad peak at 4 GeV in the process \(e^{+}e^{-}\rightarrow \psi (4040)\rightarrow DD^{*}\rightarrow \pi ^{+}\pi ^{-}J/\psi \) appears when \(DD^{*}\) loops are considered. Its existence in this reaction is not necessarily connected to the existence of a dynamically generated pole, but the underlying mechanism – the strong coupling of \(c{\bar{c}}\) to \(DD^{*}\) loops – can generate both of them. Thus, the controversial state *Y*(4008) may not be a genuine resonance, but a peak generated by the \(\psi (4040)\) and \(D^{*}D\) loops with \(\pi ^{+}\pi ^{-}J/\psi \) in the final state.

## 1 Introduction

The understanding of the nature and properties of hadronic states is a substantial challenge for both experimentalists and theorists. Remarkable progress in the field of charmonium spectroscopy was provided in the past decades: while various resonances emerge as conventional charmonium states (ordinary \({\bar{c}}c\) objects), the so-called *X*, *Y*, and *Z* states are candidates for exotic hadrons (such as molecules, hybrids, multi-quarks objects or glueballs; see Refs. [1, 2, 3, 4] and refs. therein).

In this work, we shall concentrate on the vector sector in the energy region close to 4 GeV. Here, the well established charmonium vector state \(\psi (4040)\) is listed in the Particle Data Group (PDG) [5] (it has \(J^{PC}=1^{--}\) where, as usual, *P* refers to parity and *C* to charge conjugation). This resonance can be successfully interpreted as a charmonium state with \((n,L,S,J)=(3,0,1,1)\), where *n* is the principal number, *L* the angular momentum, *S* the spin and *J* the total spin); hence, the nonrelativistic spectroscopic notation reads *n* \(^{2S+1}L_{J}=\) 3 \(^{3}S_{1}\) (see e.g. Refs. [6, 7, 8, 9, 10, 11] and refs. therein).

Very close to 4 GeV, the enigmatic (and not yet confirmed) resonance *Y*(4008) was also observed as a significant enhancement by the Belle Collaboration when measuring the cross section of \(e^{+}e^{-}\rightarrow \pi ^{+}\pi ^{-}J/\psi \) via initial state radiation (ISR) technique [12] and later on confirmed by the same group [13]: its mass was determined as \(4008\pm 40_{-28}^{+114}\) MeV and the decay width as \(\Gamma =226\pm 44\pm 87\) MeV. Moreover, a broad *Y*(4008) was also found in the recent analysis of Ref. [14]. However, the statistic at Belle was pretty limited and *Y*(4008) could *not* be confirmed by subsequent experiments studying the same production process at BaBar [15] and BESIII [16], making its existence rather controversial. Nevertheless, several possible theoretical assignments on its nature have been suggested, including non-conventional scenarios as \(D^{*}{\bar{D}}^{*}\) molecular state [17, 18] (see however also Ref. [19]), tetraquark [20, 21] or even an interference effect with background [22]. Moreover, in Refs. [23, 24] it was proposed to identify *Y*(4008) as \(\psi (3S)\) charmonium state, but this assignment is not favoured, since, as mentioned above, \(\psi (4040)\) is well described by a standard \(\psi (3S)\) state. The unexplained status of the observed structure corresponding to *Y*(4008) makes it an interesting subject that deserves clarification, hence we aim to perform a detailed study of the nearby energy region.

To this end, we develop a quantum field theoretical effective model in which a *single* \(c{\bar{c}}\) seed state, to be identified with \(\psi (4040),\) couples to *DD*, \(DD^{*},\) and \(D^{*}D^{*}.\) The immediate question is if we can describe both resonances \(\psi (4040)\) and *Y*(4008) at the same time and within a unique setup. The idea that we test is somewhat reminiscent of studies in the light scalar sector, in which the state \(a_{0}(980)\) can be seen as a companion pole of the predominantly \(q{\bar{q}}\) state \(a_{0}(1450)\) [25, 26, 27, 28] as well as the light \(\kappa \) state, now named \(K_{0}^{*}(700),\) as a companion pole of the \(K_{0}^{*}(1430)\) [29]. Quite interestingly, two poles emerge also in the study of the charmonium resonance \(\psi (3770)\) [30, 31]. As we shall see, some similarities, but also some important differences, will emerge between those studies and the one that we are going to present.

As a first step of our analysis, we calculate the spectral function of \(\psi (4040)\). As expected, it is peaked at about 4.04 GeV, but it cannot be approximated by a standard Breit–Wigner shape; most remarkably, it may develop an additional enhancement below 4 GeV (this is due to the strong coupling of the bare \({\bar{c}}c\) state to the \(DD^{*}\) channel). Moreover, two poles appear in the complex plane: one corresponding to the peak in the spectral function (hence to \(\psi (4040)\)) and an additional companion pole, dynamically generated by meson–meson (mostly \(DD^{*}\)) quantum fluctuations. A large-\(N_{c}\) study shows that \(\psi (4040)\) behaves as a conventional quark-antiquark state while the enhancement does not fit into this standard picture.

At a first sight, it appears quite natural to assign the state *Y*(4008) to this additional dynamically generated pole. Yet, a closer inspection is necessary: the study of the decay chain in which *Y*(4008) was seen, \(e^+e^- \rightarrow \psi (4040)\rightarrow DD^{*}\rightarrow \pi ^{+}\pi ^{-}J/\psi \) (the latter can occur via a light scalar state, most notably \(f_{0}(500)\), but not only), which shows that a broad peak at about 3.9 GeV emerges for the cross-section \(e^{+}e^{-}\rightarrow \pi ^{+}\pi ^{-}J/\psi \) (also when \(e^{+}e^{-}\) comes from a previous ISR process, as observed in experiment). This is due to the fact that the loop contribution of \(DD^{*}\) is peaked at about \(m_{D} +m_{D^{*}}\simeq 3.9\) GeV. As we shall explain in detail later on, this contribution is multiplied by the modulus squared of the propagator of \(\psi (4040)\), centered at 4.04 GeV and about 80 MeV large, hence a sizable overlap is present. As we shall show, the emergent peak at about 3.9 GeV is very far from a Breit–Wigner state, but is rather distorted. Strictly speaking, the very existence of an additional companion pole is not necessary for the emergence of this signal, but both phenomena arise from a strong coupling of the seed state to \(DD^{*}\), hence it is rather natural that they both take place at the same time.

The paper is organized as follows: in Sect. 2 we introduce theoretical model, in particular the Lagrangians, the possible decays channels of \(\psi (4040)\) resonance with corresponding theoretical expression for decay widths, loop function (hence, the propagator) and spectral function. Moreover, we show in detail the determination of the parameters of our model. In Sect. 3 we present our results. Summary and outlooks are presented in Sect. 4. Additional results for different parameter values are reported in the Appendices.

## 2 The model

In this section we present the theoretical model used to analyze the energy region close to 4 GeV. In our approach, only a *single* standard \(c{\bar{c}}\) seed state corresponding to \(\psi (4040)\) is included.

### 2.1 Theoretical framework

*DD*and \(D_sD_s\)), one vector and one pseudoscalar meson (\(DD^{*}\) and \(D^*_sD_s\)), and two vector mesons (\(D^{*}D^{*}\))]:

*m*)

*A*or

*B*, with masses \(m_{A}\) and \(m_{B}\) respectively, in the rest frame of the decaying particle with mass

*m*. The tree-level on-shell decay width for the state \(\psi (4040)\) are obtained by setting \(m=m_{\psi (4040)}=4.04\) GeV (here and in the following, we use the average mass \(4039.6\pm 4.3\) MeV [5] rounded to \(4.040\pm 0.004\) GeV).

*smooth*behavior (a step function is not an admissible choice) and a

*sufficiently fast*decrease on the real positive axis. For completeness, another smooth form factor

What is rather important is the numerical value of \(\Lambda \). We expect a value between \(\sim 0.4\) and \(\sim 0.8\) GeV. Namely, for the light \(\kappa \) meson, \(\Lambda \simeq 0.5\) GeV was obtained by a fit to data [29]. In the recent work of Refs. [30, 31], a even smaller value \(\Lambda \approx 0.3\) GeV is found (but a value of about 0.4 GeV also delivers results compatible with data). A comparison with the \(^{3}P_{0}\) model induces a cutoff of \(\Lambda \approx 0.8\) GeV [38, 39] (but that approach was typically not employed to calculate meson–meson loops). In this work, we test how the results vary upon changing \(\Lambda \) in the range from 0.4 to 0.8 GeV (and for different form factors), but only up to 0.6 GeV physically acceptable results are obtained.

It should be stressed that our approach is an effective model of QCD, therefore the value of \(\Lambda \) does *not* represent the maximal value for the possible values of the momentum *k*. When *k* is larger than \(\Lambda \), then that particular decay is suppressed, but this is a physical consequence of the nonlocal interaction between the decaying meson and its decay products (all of them being extended objects). The momentum *k* can take any value from 0 to \(\infty \), even arbitrarily larger than \(\Lambda .\) In particular, the normalization of the spectral function (a crucial feature of our approach, see below) involves an integration up to \(k\rightarrow \infty \). Of course, even if it is allowed to take *k* arbitrarily large, this does not imply that the model is physically complete: since we take into account a single resonance, the \(\psi (4040)\), our approach can describe some of the features around 4 GeV (and up to about 4.15 GeV). Above that, one should include the resonance \(\psi (4160)\) and, even further, the resonance \(\psi (4415).\) (For completeness, we have tested the case in which \(\psi (4040)\) and \(\psi (4160)\) are present at the same time. As we shall comment later on, including \(\psi (4160)\) does not substantially change the results for \(\psi (4040)\)).

*m*and \(m+\mathrm {dm}\). It must fulfill the normalization condition

*m*and \(m+\mathrm {dm}\) and decays in the channel \(DD^{*}\) [42]. Similar interpretations hold for all other channels. These quantities are physically interesting since they emerge when different channels are studied; if, for instance, the process \(e^{+}e^{-}\rightarrow DD^{*}\) is considered, the corresponding cross section is proportional to \(d_{\psi \rightarrow DD^{*}}(m)\).

### 2.2 Determination of the parameters

Position of the poles in the complex plane for different parameters used in the model. The coupling constants \(g_{\psi DD}\) and \(g_{\psi D^{*}D^{*}}\) are dimensionless, \(g_{\psi D^{*}D}\) has dimensions GeV\(^{-1}\), \(\Lambda \) and \(M_{\psi }\) are in GeV

\(\Lambda \) | Gaussian form factor | Dipolar form factor | ||
---|---|---|---|---|

Parameters | Pole(s) [GeV] | Parameters | Pole(s) [GeV] | |

0.4 | \(g_{\psi DD}=48.8 \pm 4.6\) | \(\bullet ~ (4.052 \pm 0.003)\) | \(g_{\psi DD}=25.4 \pm 5.0\) | \(\bullet ~ (4.058 \pm 0.019)\) |

\(g_{\psi D^{*}D}=3.60 \pm 0.95\) | \(~ -i (0.035 \pm 0.005)\) | \(g_{\psi D^{*}D}=3.50 \pm 0.58\) | \(~ -i(0.050 \pm 0.014)\) | |

\(g_{\psi D^{*}D^{*}}=1.65 \pm 0.86\) | \(\blacklozenge ~ (3.936 \pm 0.005)\) | \(g_{\psi D^{*}D^{*}}=1.93 \pm 0.89\) | \(\blacklozenge ~ (3.941 \pm 0.003)\) | |

\(M_{\psi }=4.00\) | \(~ -i(0.022 \pm 0.001)\) | \(M_{\psi }=4.02\) | \(~ -i(0.045 \pm 0.010)\) | |

0.42 | \(g_{\psi DD}=39.8 \pm 5.0\) | \(\bullet ~ (4.053 \pm 0.004)\) | \(g_{\psi DD}=21.7 \pm 4.4\) | \(\bullet ~ (4.062 \pm 0.023)\) |

\(g_{\psi D^{*}D}=3.44 \pm 0.80\) | \(~ -i(0.039 \pm 0.009)\) | \(g_{\psi D^{*}D}=3.06 \pm 0.49\) | \(~ -i(0.056 \pm 0.011)\) | |

\(g_{\psi D^{*}D^{*}}=1.85 \pm 0.93\) | \(\blacklozenge ~(3.934 \pm 0.006)\) | \(g_{\psi D^{*}D^{*}}=1.94 \pm 0.89\) | \(\blacklozenge ~ (3.942 \pm 0.004)\) | |

\(M_{\psi }=4.01\) | \(~ -i(0.030 \pm 0.001)\) | \(M_{\psi }=4.03\) | \(~ -i(0.052 \pm 0.010)\) | |

0.45 | \(g_{\psi DD}=29.9 \pm 5.0\) | \(\bullet ~ (4.055 \pm 0.005)\) | \(g_{\psi DD}=17.4 \pm 3.8\) | \(\bullet ~(4.070 \pm 0.027) \) |

\(g_{\psi D^{*}D}=3.14 \pm 0.61\) | \(~ -i(0.047 \pm 0.018)\) | \(g_{\psi D^{*}D}=2.57 \pm 0.38\) | \(~ -i(0.066 \pm 0.008)\) | |

\(g_{\psi D^{*}D^{*}}=2.07 \pm 0.99\) | \(\blacklozenge ~ (3.928 \pm 0.008)\) | \(g_{\psi D^{*}D^{*}}=1.97 \pm 0.89\) | \(\blacklozenge ~ (3.943 \pm 0.006)\) | |

\(M_{\psi }=4.02\) | \(~ -i(0.042 \pm 0.002)\) | \(M_{\psi }=4.04\) | \(~ -i(0.064 \pm 0.011)\) | |

0.5 | \(g_{\psi DD}=19.5 \pm 4.2\) | \(\bullet ~ (4.055 \pm 0.009)\) | \(g_{\psi DD}=12.6 \pm 3.0\) | \(\bullet ~ (4.087 \pm 0.033)\) |

\(g_{\psi D^{*}D}=2.64 \pm 0.39\) | \(~ -i(0.066 \pm 0.054)\) | \(g_{\psi D^{*}D}=2.02\pm 0.27 \) | \(~ -i(0.083 \pm 0.006)\) | |

\(g_{\psi D^{*}D^{*}}=2.3 \pm 1.0 \) | \(\blacklozenge ~ (3.918 \pm 0.007)\) | \(g_{\psi D^{*}D^{*}}=2.02 \pm 0.89\) | \(\blacklozenge ~ (3.943 \pm 0.011)\) | |

\(M_{\psi }=4.04\) | \(~-i(0.063 \pm 0.004)\) | \(M_{\psi }=4.05\) | \(~ -i(0.085 \pm 0.014)\) | |

0.6 | \(g_{\psi DD}=10.4 \pm 2.7\) | \(\bullet ~ (4.025 \pm 0.015)\) | \(g_{\psi DD}=7.4 \pm 2.0\) | \(\bullet ~ (4.032 \pm 0.019)\) |

\(g_{\psi D^{*}D}=1.95 \pm 0.22\) | \(~ -i(0.041 \pm 0.031)\) | \(g_{\psi D^{*}D}=1.44 \pm 0.16\) | \(~ -i(0.035 \pm 0.020)\) | |

\(g_{\psi D^{*}D^{*}}=2.3 \pm 1.0\) | \(\blacklozenge ~ (4.056 \pm 0.017)\) | \(g_{\psi D^{*}D^{*}}=2.09 \pm 0.90\) | \(\blacklozenge ~ (4.056 \pm 0.023)\) | |

\(M_{\psi }=4.07\) | \(~ -i(0.032 \pm 0.007)\) | \(M_{\psi }=4.08\) | \(~ -i(0.029 \pm 0.006)\) |

*DD*, \(D_{s}D_{s}\), \(D^{*}D\), \(D_{s}^{*}D_{s}\) and \(D^{*}D^{*}\). The corresponding theoretical expression for the total decay width of \(\psi (4040)\) state is given by

The numerical values of \(g_{\psi DD}\), \(g_{\psi D^{*}D}\), \(g_{\psi D^{*}D^{*}}\) and of bare mass \(M_{\psi }\) are reported in Table 1 in Sect. 3.1 for given values of the cutoff \(\Lambda \). As expected, \(g_{\psi D^{*}D}\), \(g_{\psi D^{*}D^{*}}\) depend rather mildly on \(\Lambda \), but \(g_{\psi DD}\) quite strongly. This is so because the *DD* threshold is the most distant from the on-shell mass of the state \(\psi (4040)\) and the corresponding momentum \(k_{\psi DD}=\sqrt{\frac{m_{\psi (4040)}^{2}}{4}-m_{D^{+}}^{2}} \simeq 0.76\) GeV is comparable to \(\Lambda .\) As a consequence, a variation of \( \Lambda \) implies a sizable variation of the corresponding coupling constant. However, the decay width into *DD* is quite small and affects only slightly the overall picture. For completeness, we report in Appendix A also the partial decay widths for various choices of the cutoff and for different form factors. While the results are basically compatible with each other, future experimental determination of the channel \(\psi (4040)\rightarrow D_{s}^{+}D_{s}^{-}\) would be very helpful to constrain our model.

## 3 Results

In this section we show the results and comment on them. First, in Sect. 3.1 we concentrate on the form of the full spectral function (as well as the partial ones into *DD*, \(D^{*}D\), \(D^{*}D^{*}\) channels) of the resonance \(\psi (4040)\). Moreover we determine the position of the pole(s) in the complex plane. Then, in Sect. 3.2 we present the discussion of the important process \(e^{+}e^{-}\rightarrow J/\psi \pi ^{+}\pi ^{-}\) and the possible generation of a peak for *Y*(4008).

### 3.1 Spectral function and pole positions

Since scattering data have still quite large errors, it is not yet possible to determine the value of \(\Lambda \) through a fit. Moreover, such a fit would also need to include an unknown background contribution. This is why \(\Lambda \) has been varied in a quite large range in Table 1, in which the positions of the poles have also been reported. As already mentioned in the introduction, we always find two poles in the complex plane, one corresponding to the maximum of the spectral function and an additional dynamically generated one. For \(\Lambda \) up to \(\sim 0.5\) GeV similar results are obtained, but for larger values the second pole (even if always present) appears at higher values. We have also tested values larger than 0.6 GeV, but they do not generate satisfactory results. (This outcome is in agreement with the results of Sect. 3.2 and Appendix B, see later on).

In the following, we choose for the numerical value \(\Lambda =0.42,\) since it generates a pole in terms of the variable \(\sqrt{s}\) whose imaginary part is 40 MeV, then \(\Gamma _{\psi (4040)}^{\text {pole}}=80\) MeV. (Note, through all this work we use \(\sqrt{s}\) , hence the definition \(\sqrt{s_{\psi (4040)}^{\text {pole}}}=m_{\psi (4040)}^{\text {pole}}-i\Gamma _{\psi (4040)}^{\text {pole}}/2\) holds.) We then use this value for illustration and for the presentation of the plots. (Yet, it should not be considered as a sharp value for the cutoff). The spectral function \(d_{\psi }(m)\) defined in Eq. (18) is shown in Fig. 2a together with a standard Breit–Wigner function peaked at 4.04 GeV and with a width of 80 MeV, which serves for comparison.

In Fig. 2b we present the contributions of individual channels (*DD*, \(D^{*}D\) and \(D^{*}D^{*}\)) to the total spectral function. The \(D^{*}D\) channel turns out to be the most important for the deformation on the l.h.s. of the spectral function. In the complex plane we found two poles: one for 4.053–0.039 *i* GeV, corresponding to \(\psi (4040)\) resonance, and one for 3.934–0.030 *i* GeV. Thus, even if only one single seed state identified with \(\psi (4040)\) was included into the calculations, two poles naturally emerge.

At a first sight, it is tempting to identify the additional pole with the controversial state *Y*(4008). Moreover, a look at Table 1 shows that a second additional pole always exists, and up to values of about 0.5 GeV the dynamically generated pole is not far from 4 GeV. However, care is needed for the following reasons: the pole width of the additional state is too small when compared to the experimental value (about 200 MeV), but it should be stressed that a direct comparison of this pole width and the experiment is misleading, since very different reactions were measured, see later on. If the enhancement in Fig. 1 is real, it should be visible in the cross-section of the channel \(e^{+}e^{-}\rightarrow \psi (4040)\rightarrow D^{*}D,\) which is proportional to \(d_{\psi \rightarrow DD^{*}}(m)\) (see Fig. 1b); presently, the data have too large errors to resolve such a complicated structure. Quite importantly, the state *Y*(4008) has been observed in the ISR reaction \(e^{+}e^{-}\rightarrow \gamma e^{+}e^{-}\rightarrow \gamma \pi ^{+}\pi ^{-}J/\psi \) and not in the \(DD^{*}\) channel, see next section for the discussion of this important point.

The large-\(N_{c}\) study shows that for smaller \(\lambda \) (hence, larger \(N_{c}\)), the left enhancement in the spectral function becomes smaller and finally disappears, while the spectral function becomes narrower. For what concerns the pole trajectory, the seed pole corresponding to \(\psi (4040)\) resonance moves towards to real energy axis, while the additional companion pole moves away from it. This behavior confirms that the resonance \(\psi (4040)\) is a conventional \(q{\bar{q}}\) meson while the second pole is dynamically generated.

As a last point, we comment on mixing with other vector state, in particular with the closest quarkonium state \(\psi (4160).\) By studying the mix propagator (see Refs. [47, 48] for some formal equation), we tested how the spectral function of \(\psi (4040)\) changes when taking into account the processes \(\psi (4040)\rightarrow DD^{*}\) \(\rightarrow \psi (4160)\rightarrow DD^{*}\rightarrow \psi (4040)\) (this is so because \(\psi (4160)\) also couples to \(DD^*\); analogous processes with *DD* and \(D^{*}D^{*}\) are possible). The spectral function of \(\psi (4040)\) turns out to be only slightly affected in the region of interest, thus the results here presented would hold also in the enlarged scenario in which more \({\bar{c}}c\) states are considered.

### 3.2 Decay into \(J/\psi \pi ^{+}\pi ^{-}\)

An important decay channel, in which various *Y* states have been observed, among which the *Y*(4008) state is one, is the \(e^{+}e^{-}\rightarrow \gamma J/\psi \pi ^{+}\pi ^{-}\) decay, where the photon \(\gamma \) comes from ISR. Hence, the reaction may be recasted into two steps: \(e^{+}e^{-}\rightarrow \gamma \left( e^{+} e^{-}\right) _{\text {off-shell}}\) and \(\left( e^{+}e^{-}\right) _{\text {off-shell}}\rightarrow J/\psi \pi ^{+}\pi .\ \)

*c*-quark and their subsequent conversion into light quarks implies \(g_{QCD}^{4}\) with \( g_{QCD}\propto N_{c}^{-1/2}.\) Finally, the factor \(N_{c}^{2}\) comes from the fact that there are two closed quark lines in this process. For the regular \( {\bar{q}}q\) amount into \(f_{0}(500),\) one may use \(g_{f_{0}(500)\text {-}{\bar{q}} q\text { }}\propto N_{c}^{-1/2},\) hence the amplitude \(1/N_{c}^{3/2}\) follows. Within this context, for the final state with two pions, one obtains \(A_{\psi (4040)\rightarrow j/\psi \text {-}gg\rightarrow j/\psi \pi ^{+}\pi ^{-}}\propto N_{c}^{-2}.\) For other non-conventional components in \( f_{0}(500),\) this decay amplitude should be even more further suppressed. Note, the leading term of the decay width into \(j/\psi \pi ^{+}\pi ^{-}\) scales as \(N_{c}^{-4},\) thus explaining why such decay is very small.

Summarizing, in Eq. (36) the product of two functions is present: \(\left| \Delta _{\psi }(m^{2})\right| ^{2},\) peaked at 4.04 GeV, and \(\Gamma _{\psi (4040)\rightarrow J/\psi f_{0}(500)}(m),\) peaked at 3.9 GeV. Of course, other channels, such as *DD*, would also couple \(J/\psi f_{0}(500)\), but (i) the coupling of \(\psi (4040)\) to *DD* is sizably smaller and (ii) the function \(\left| \Sigma _{DD}(m^{2})\right| ^{2}\) is peaked at the *DD* threshold, hence the overlap with \(\left| \Delta _{\psi }(m^{2})\right| ^{2}\) is negligible.

*Y*(4008) is a manifestation of the standard state \(\psi (4040),\) which arises due to the decay into \(J/\psi \pi ^{+}\pi ^{-}\) through the nearby \(DD^{*}\) loop. It is important to stress that this conclusion is

*independent*of the presence of a dynamically generated pole and its precise position, but it is a consequence of the strong coupling to \(DD^{*}\) and the fact that the \(DD^{*}\) threshold is not far from the peak of \(\psi (4040)\). We call the phenomenon leading to Fig. 5b as ‘real-part-loop generated peak’.

This mechanism for the generation of the state *Y*(4008) does not necessarily correspond to a resonance. Moreover, the width of the dynamical pole is not related to the width of the signal of Fig. 5b. In Appendix B we also present the results for variations of the parameters, and for a cutoff up to 0.5 GeV a very similar outcome is obtained.

## 4 Summary and discussion

We studied the energy region close to the resonance \(\psi (4040)\) in the framework of a QFT effective model. We evaluated its spectral function and found that, besides the expected resonance pole of \(\psi (4040)\) (corresponding to peak in the spectral function and to the underlying seed \(c{\bar{c}}\) state), an additional companion pole (no peak, but an enhancement in the spectral function) emerges naturally within our approach (see Table 1). Illustrative result in agreement with phenomenology are: \(4.053-i0.039\) GeV for the seed state \(\psi (4040)\) and \(3.934-i0.030\) GeV for the enhancement. A large-\(N_{c}\) study confirms that \(\psi (4040)\) resonance is predominantly a charm–anticharm state, while the second pole is dynamically generated by meson–meson quantum fluctuations.

The pole itself cannot be directly associated to the \(Y(4008).\ \)Namely, this pole would be mostly visible in a two-peak structure of the cross-section \(e^{+}e^{-}\rightarrow \psi (4040)\rightarrow DD^{*}\) (whose data precision is not good enough). Yet, the chain \(e^{+}e^{-}\rightarrow \psi (4040)\rightarrow DD^{*}\rightarrow J/\psi +f_{0}(500)\rightarrow J/\psi \pi ^{+}\pi ^{-}\)is quite promising: the \(DD^{*}\) loop generates a peak at about 3.9 GeV in the cross-section. The strong coupling of \(\psi (4040)\) to \(DD^{*}\) and the overlap of the \(DD^{*}\)-loop function with the modulus square of the propagator are responsible for a quite broad peak in the corresponding spectral function, which can be identified with *Y*(4008). The important point is that the existence of an additional pole corresponding to *Y*(4008) is possible (and indeed it does exist for the parameters of our model) but is actually not necessary for the process that we describe. The very same mechanism can be also investigated in the future in other channels, as for instance in connection with the states *Y*(4260) and \(\psi (4160)\).

## Notes

### Acknowledgements

The authors thank S. Coito for useful discussions. M.P. and F.G. acknowledge financial support from the Polish National Science Centre (NCN) through the OPUS project no. 2015/17/B/ST2/01625. P. K. were supported by the Hungarian OTKA fund K109462 and by the ExtreMe Matter Institute EMMI at the GSI Helmholtzzentrum fur Schwerionenforschung, Darmstadt, Germany.

## Supplementary material

## References

- 1.N. Brambilla, Heavy quarkonium: progress, puzzles, and opportunities. Eur. Phys. J. C
**71**, 1534 (2011). https://doi.org/10.1140/epjc/s10052-010-1534-9. arXiv:1010.5827 [hep-ph]ADSCrossRefGoogle Scholar - 2.H.X. Chen, W. Chen, X. Liu, S.L. Zhu, The hidden-charm pentaquark and tetraquark states. Phys. Rep.
**639**, 1 (2016). https://doi.org/10.1016/j.physrep.2016.05.004. arXiv:1601.02092 [hep-ph]ADSMathSciNetCrossRefGoogle Scholar - 3.A. Esposito, A.L. Guerrieri, F. Piccinini, A. Pilloni, A.D. Polosa, Four-quark hadrons: an updated review. Int. J. Mod. Phys. A
**30**, 1530002 (2015). https://doi.org/10.1142/S0217751X15300021. arXiv:1411.5997 [hep-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar - 4.M. Nielsen, F.S. Navarra, S.H. Lee, New charmonium states in QCD sum rules: a concise review. Phys. Rep.
**497**, 41 (2010). https://doi.org/10.1016/j.physrep.2010.07.005. arXiv:0911.1958 [hep-ph]ADSCrossRefGoogle Scholar - 5.M. Tanabashi et al., (Particle Data Group). Phys. Rev. D
**98**, 030001 (2018). https://doi.org/10.1103/PhysRevD.98.030001 - 6.S. Godfrey, N. Isgur, Mesons in a relativized quark model with chromodynamics. Phys. Rev. D
**32**, 189 (1985). https://doi.org/10.1103/PhysRevD.32.189 ADSCrossRefGoogle Scholar - 7.E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, T.M. Yan, Charmonium: the model. Phys. Rev. D
**17**(1978) 3090 Erratum: [Phys. Rev. D**21**(1980) 313]. https://doi.org/10.1103/PhysRevD.17.3090. https://doi.org/10.1103/physrevd.21.313.2 - 8.E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, T.M. Yan, Charmonium: comparison with experiment. Phys. Rev. D
**21**, 203 (1980). https://doi.org/10.1103/PhysRevD.21.203 ADSCrossRefGoogle Scholar - 9.E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, T.M. Yan, Phys. Rev. Lett.
**36**, 500 (1976). https://doi.org/10.1103/PhysRevLett.36.500 ADSCrossRefGoogle Scholar - 10.J. Segovia, D.R. Entem, F. Fernandez, E. Hernandez, Constituent quark model description of charmonium phenomenology. Int. J. Mod. Phys. E
**22**, 1330026 (2013). https://doi.org/10.1142/S0218301313300269. arXiv:1309.6926 [hep-ph]ADSCrossRefGoogle Scholar - 11.P.G. Ortega, J. Segovia, D.R. Entem, F. Fernández, Charmonium resonances in the 3.9 GeV/\(c^{2}\) energy region and the \(X(3915)/X(3930)\) puzzle. Phys. Lett. B
**778**, 1 (2018). https://doi.org/10.1016/j.physletb.2018.01.005. arXiv:1706.02639 ADSCrossRefGoogle Scholar - 12.C.Z. Yuan et al., [Belle Collaboration], Measurement of e+ e-: \(>\) pi+ pi- J/psi cross-section via initial state radiation at Belle. Phys. Rev. Lett.
**99**, 182004 (2007). https://doi.org/10.1103/PhysRevLett.99.182004. [arXiv:0707.2541 [hep-ex]] - 13.Z.Q. Liu et al., [Belle Collaboration], Study of \(e^{+}e\) and Observation of a Charged Charmoniumlike State at Belle. Phys. Rev. Lett.
**110**, 252002 (2013) https://doi.org/10.1103/PhysRevLett.110.252002. arXiv:1304.0121 [hep-ex] - 14.X.Y. Gao, C.P. Shen, C.Z. Yuan, Resonant parameters of the \(Y(4220)\). Phys. Rev. D
**95**(9), 092007 (2017). https://doi.org/10.1103/PhysRevD.95.092007. arXiv:1703.10351 [hep-ex]ADSCrossRefGoogle Scholar - 15.B. Aubert et al., [BaBar Collaboration], Exclusive initial-state-radiation production of the D anti-D, D* anti-D*, and D* anti-D* systems. Phys. Rev. D
**79**, 092001 (2009). https://doi.org/10.1103/PhysRevD.79.092001. arXiv:0903.1597 [hep-ex] - 16.M. Ablikim et al., [BESIII Collaboration], Precise measurement of the \(e^{+}e^{-}\rightarrow \pi ^{+} \pi ^{-}J/\psi \) cross section at center-of-mass energies from 3.77 to 4.60 GeV. Phys. Rev. Lett.
**118**(9), 092001 (2017). https://doi.org/10.1103/PhysRevLett.118.092001. arXiv:1611.01317 [hep-ex] - 17.X. Liu, Understanding the newly observed Y(4008) by Belle. Eur. Phys. J. C
**54**, 471 (2008). https://doi.org/10.1140/epjc/s10052-008-0551-4. arXiv:0708.4167 [hep-ph]ADSCrossRefGoogle Scholar - 18.W. Xie, L.Q. Mo, P. Wang, S.R. Cotanch, Coulomb gauge model for hidden charm tetraquarks. Phys. Lett. B
**725**, 148 (2013). https://doi.org/10.1016/j.physletb.2013.07.003. arXiv:1302.5737 [hep-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar - 19.G.J. Ding, Bound states of the heavy flavor vector mesons and Y(4008) and Z+(1)(4050). Phys. Rev. D
**80**, 034005 (2009). https://doi.org/10.1103/PhysRevD.80.034005. arXiv:0905.1188 [hep-ph]ADSCrossRefGoogle Scholar - 20.L. Maiani, F. Piccinini, A.D. Polosa, V. Riquer, The Z(4430) and a new paradigm for spin interactions in tetraquarks. Phys. Rev. D
**89**, 114010 (2014). https://doi.org/10.1103/PhysRevD.89.114010. arXiv:1405.1551 [hep-ph]ADSCrossRefGoogle Scholar - 21.P. Zhou, C.R. Deng, J.L. Ping, Identification of Y (4008), Y (4140), Y (4260), and Y (4360) as tetraquark states. Chin. Phys. Lett.
**32**(10), 101201 (2015). https://doi.org/10.1088/0256-307X/32/10/101201 ADSCrossRefGoogle Scholar - 22.D.Y. Chen, X. Liu, X.Q. Li, H.W. Ke, Unified fano-like interference picture for charmoniumlike states Y(4008), Y(4260) and Y(4360). Phys. Rev. D
**93**, 014011 (2016). https://doi.org/10.1103/PhysRevD.93.014011. arXiv:1512.04157 [hep-ph]ADSCrossRefGoogle Scholar - 23.B.Q. Li, K.T. Chao, Higher charmonia and X, Y, Z states with screened potential. Phys. Rev. D
**79**, 094004 (2009). https://doi.org/10.1103/PhysRevD.79.094004. arXiv:0903.5506 [hep-ph]ADSCrossRefGoogle Scholar - 24.L.J. Chen, D.D. Ye, A. Zhang, Is \(Y(4008)\) possibly a \(1^{--}\psi (3^{3}S_{1})\) state? Eur. Phys. J. C
**74**(8), 3031 (2014). https://doi.org/10.1140/epjc/s10052-014-3031-z. arXiv:1402.5470 [hep-ph]ADSCrossRefGoogle Scholar - 25.E. van Beveren, T.A. Rijken, K. Metzger, C. Dullemond, G. Rupp, J.E. Ribeiro, A low lying scalar meson nonet in a unitarized meson model. Z. Phys. C
**30**, 615 (1986). https://doi.org/10.1007/BF01571811. arXiv:0710.4067 [hep-ph]ADSCrossRefGoogle Scholar - 26.N.A. Tornqvist, Understanding the scalar meson q anti-q nonet. Z. Phys. C
**68**, 647 (1995). https://doi.org/10.1007/BF01565264. arXiv:hep-ph/9504372 ADSCrossRefGoogle Scholar - 27.M. Boglione, M.R. Pennington, Dynamical generation of scalar mesons. Phys. Rev. D
**65**, 114010 (2002). https://doi.org/10.1103/PhysRevD.65.114010. arXiv:hep-ph/0203149 ADSCrossRefGoogle Scholar - 28.T. Wolkanowski, F. Giacosa, D.H. Rischke, \(a_{0}(980)\) revisited. Phys. Rev. D
**93**(1), 014002 (2016). https://doi.org/10.1103/PhysRevD.93.014002. arXiv:1508.00372 [hep-ph]ADSCrossRefGoogle Scholar - 29.T. Wolkanowski, M. Soltysiak, F. Giacosa, \(K_{0}^{\ast }(800)\) as a companion pole of \(K_{0}^{\ast }(1430)\). Nucl. Phys. B
**909**, 418 (2016). https://doi.org/10.1016/j.nuclphysb.2016.05.025. arXiv:1512.01071 [hep-ph]ADSCrossRefzbMATHGoogle Scholar - 30.S. Coito, F. Giacosa, Line-shape and poles of the \(\psi (3770)\), arXiv:1712.00969 [hep-ph]
- 31.S. Coito, F. Giacosa, Formation and deformation of the \(\psi (3770)\). Acta Phys. Pol. Suppl.
**10**, 1049 (2017). https://doi.org/10.5506/APhysPolBSupp.10.1049. arXiv:1708.02041 [hep-ph]CrossRefGoogle Scholar - 32.J. Terning, Phys. Rev. D
**44**, 887 (1991)ADSCrossRefGoogle Scholar - 33.G.V. Efimov, M.A. Ivanov,
*The Quark confinement model of hadrons*(Bristol, UK: IOP 1993)Google Scholar - 34.Y.V. Burdanov, G.V. Efimov, S.N. Nedelko, S.A. Solunin, Phys. Rev. D
**54**, 4483 (1996). arXiv:hep-ph/9601344 ADSCrossRefGoogle Scholar - 35.A. Faessler, T. Gutsche, M.A. Ivanov, V.E. Lyubovitskij, P. Wang, Phys. Rev. D
**68**, 014011 (2003). arXiv:hep-ph/0304031 ADSCrossRefGoogle Scholar - 36.F. Giacosa, T. Gutsche, A. Faessler, Phys. Rev. C
**71**, 025202 (2005). arXiv:hep-ph/0408085 ADSCrossRefGoogle Scholar - 37.M. Soltysiak, F. Giacosa, A covariant nonlocal Lagrangian for the description of the scalar kaonic sector. Acta Phys. Pol. Suppl.
**9**, 467 (2016). https://doi.org/10.5506/APhysPolBSupp.9.467. arXiv:1607.01593 [hep-ph]CrossRefGoogle Scholar - 38.E.S. Ackleh, T. Barnes, E.S. Swanson, On the mechanism of open flavor strong decays. Phys. Rev. D
**54**, 6811 (1996). https://doi.org/10.1103/PhysRevD.54.6811. arXiv:hep-ph/9604355 ADSCrossRefGoogle Scholar - 39.Z.G. Luo, X.L. Chen, X. Liu, B(s1)(5830) and B*(s2)(5840). Phys. Rev. D
**79**, 074020 (2009). https://doi.org/10.1103/PhysRevD.79.074020. arXiv:0901.0505 [hep-ph]ADSCrossRefGoogle Scholar - 40.F. Giacosa, G. Pagliara, On the spectral functions of scalar mesons. Phys. Rev. C
**76**, 065204 (2007). https://doi.org/10.1103/PhysRevC.76.065204. arXiv:0707.3594 [hep-ph]ADSCrossRefGoogle Scholar - 41.F. Giacosa, G. Pagliara, Spectral function of a scalar boson coupled to fermions. Phys. Rev. D
**88**(2), 025010 (2013). https://doi.org/10.1103/PhysRevD.88.025010. arXiv:1210.4192 [hep-ph]ADSCrossRefGoogle Scholar - 42.F. Giacosa, Non-exponential decay in quantum field theory and in quantum mechanics: the case of two (or more) decay channels. Found. Phys.
**42**, 1262 (2012). https://doi.org/10.1007/s10701-012-9667-3. arXiv:1110.5923 [nucl-th]ADSCrossRefzbMATHGoogle Scholar - 43.X.L. Wang et al., [Belle Collaboration], Observation of \(\psi (4040)\) and \(\psi (4160)\) decay into Î . J/\(\psi \). Phys. Rev. D
**87**(5), 051101 (2013). https://doi.org/10.1103/PhysRevD.87.051101. arXiv:1210.7550 [hep-ex] - 44.M.N. Anwar, Y. Lu, B.S. Zou, Modeling charmonium-\(\eta \) Decays of \(J^{PC}=1^{--}\) higher charmonia. Phys. Rev. D
**95**(11), 114031 (2017). https://doi.org/10.1103/PhysRevD.95.114031. arXiv:1612.05396 [hep-ph]ADSCrossRefGoogle Scholar - 45.T.E. Coan et al., [CLEO Collaboration], Charmonium decays of Y(4260), psi(4160) and psi(4040). Phys. Rev. Lett.
**96**, 162003 (2006) https://doi.org/10.1103/PhysRevLett.96.162003. arXiv:hep-ex/0602034 - 46.E. Witten, Baryons in the 1/n expansion. Nucl. Phys. B
**160**, 57 (1979). https://doi.org/10.1016/0550-3213(79)90232-3 ADSCrossRefGoogle Scholar - 47.G.V. Baryshevskii, V.I. Lyuboshitz, M.I. Podgorerskii, Nonorthogonal quasisationary states. Sov. Phys. JETP
**30**(1), 91–94 (1970)Google Scholar - 48.N.N. Achasov, G.N. Shestakov, Line shape of \(\psi (3770)\) in \(e^{+}e^{-}\rightarrow D{\bar{D}}\). Phys. Rev. D
**86**, 114013 (2012). https://doi.org/10.1103/PhysRevD.86.114013. arXiv:1208.4240 [hep-ph]ADSCrossRefGoogle Scholar - 49.J.R. Pelaez, From controversy to precision on the sigma meson: a review on the status of the non-ordinary \(f_{0}(500)\) resonance. Phys. Rep.
**658**, 1 (2016). https://doi.org/10.1016/j.physrep.2016.09.001. arXiv:1510.00653 [hep-ph]ADSCrossRefGoogle Scholar - 50.R.F. Lebed, Phenomenology of large N(c) QCD. Czechoslov. J. Phys.
**49**, 1273 (1999). https://doi.org/10.1023/A:1022820227262. arXiv:nucl-th/9810080 ADSCrossRefGoogle Scholar - 51.J.R. Pelaez, On the Nature of light scalar mesons from their large N(c) behavior. Phys. Rev. Lett.
**92**, 102001 (2004). https://doi.org/10.1103/PhysRevLett.92.102001. arXiv:hep-ph/0309292 ADSCrossRefGoogle Scholar - 52.F. Giacosa, Dynamical generation and dynamical reconstruction. Phys. Rev. D
**80**, 074028 (2009). https://doi.org/10.1103/PhysRevD.80.074028. arXiv:0903.4481 [hep-ph]ADSCrossRefGoogle Scholar - 53.S. Weinberg, Tetraquark mesons in large \(N\) quantum chromodynamics. Phys. Rev. Lett.
**110**, 261601 (2013). https://doi.org/10.1103/PhysRevLett.110.261601. arXiv:1303.0342 [hep-ph]ADSCrossRefGoogle Scholar - 54.J.R. Peláez, J. Nebreda, G. Ríos, J. Ruiz de Elvira, Properties and composition of the \(f_0(500)\) resonance. Acta Phys. Pol. Suppl.
**6**(3), 735 (2013). https://doi.org/10.5506/APhysPolBSupp.6.735. arXiv:1304.5121 [hep-ph]CrossRefGoogle Scholar - 55.T. Cohen, F.J. Llanes-Estrada, J.R. Pelaez, J. Ruiz de Elvira, Nonordinary light meson couplings and the \(1/N_c\) expansion. Phys. Rev. D
**90**(3), 036003 (2014). https://doi.org/10.1103/PhysRevD.90.036003. arXiv:1405.4831 [hep-ph]ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}