The puzzle of excessive non-$D\bar D$ component of the inclusive $\psi(3770)$ decay and the long-distant contribution

In this letter we suggest that the obvious discrepancy between theoretical prediction on the $\mathrm{non}-D\bar D$ decays of $\psi(3770)$ and data is to be alleviated by taking final state interaction (FSI) into account. By assuming that $\psi(3770)$ overwhelmingly dissociates into $D\bar D$, then the final state interaction induces a secondary process, we calculate the branching ratios of $\psi(3770)\to D\bar D\to J/\psi\eta, \rho\pi, \omega\eta, K^*K$. Our results show that the branching ratio of $\psi(3770)\to \mathrm{non}-D\bar{D}$ can reach up to $\mathcal{B}_{\mathrm{non}-D\bar{D}}^{FSI}=(0.2\sim1.1)%$ while typical parameters $I=0.4$ GeV$^{-2}$ and $\alpha=0.8\sim 1.3$ are adopted. This indicates that the FSI is obviously non-negligible.

Generally, one can categorize the strong decay modes of ψ(3770) into three types: open charm decay (DD), hidden charm decay J/ψX (X = light mesons) and the decay into light hadrons (L-H decay). One can be more confident that the rates of hidden charm decays are properly evaluated in terms of the QCD multi-expansion, and the L-H decay occurs via three-gluon emission mechanism cc → 3g.
There is an alternative explanation to the puzzle. Twenty years ago, Lipkin proposed that the non-DD strong decays of ψ(3770) realize via DD intermediate states, and further suggested that ψ(3770) does not 100% decay into DD [22]. Later Achasov and Kozhevnikov calculated the non-DD channels of ψ(3770) only considering the contribution from the imaginary part of the decay amplitude [23]. Namely such final state interactions which are involved in the hadronic loop effects, do contribute to both the hidden charm and L-H decays. The essential point of the loop effect is attributed to the coupled channel effects. A quark-level process is explicitly illustrated in the left diagram of Fig. 1. Such a mechanism should exist in all hidden charm and L-H decays of charmonia [24,25]. As shown in Fig. 1, ψ(3770) → J/ψX and ψ(3770) → Light Hadrons processes do not suffer from the Okubo-Zweig-Iizuka (OZI) suppression. Since ψ(3770) → DD takes place near the energy threshold, one can expect that the FSI may be significant. In this letter, we focus on two-body hidden charm decay modes (J/ψη) and two-body L-H decay modes (ρπ, ωη and K * K) which obviously are the main ones. Here K * K denotes K * K +K * K.
In order to calculate the hadronic loop effect in strong decays of ψ(3770), we consider the diagrams depicted by Fig. 2, which are an alternative description in the hadron-level language. ψ(3770) first dissolves into two charmed mesons, then by exchanging D * in t-channel, they turn into two on-shell real hadrons A and B. Since the dissociation does not suffer from the OZI suppression, one can expect it to be dominant. One can obtain the absorptive part of the decay amplitude of ψ(3770) → D +D → A + B (AB = J/ψη, ρπ, ωη and K * K) which compensates the off-shell effect of exchanged meson and describes the structure effect of the interaction vertex. As a free parameter, Λ can be parameterized as Λ(m i ) = m i + αΛ QCD [26]. m i denotes the mass of exchanged meson, Λ QCD = 220 MeV. The range of dimensionless phenomenological parameter α is around 0.8 < α < 2.2 [26]. As a matter of fact, there are other possible forms for F m 2 D ( * ) , q 2 , such as the exponential one etc. in literature. Generally they are equivalent somehow, as long as their asymptotic behaviors are the same.
Since the mass of ψ(3770) is close to the threshold of DD production, the dispersive part of the amplitude of ψ(3770) → D +D → A + B makes a large contribution to the decay width. By unitarity, one can obtain the dispersive part in terms of the dispersion relation. The total decay amplitude of ψ(3770) → D +D → A + B which includes both absorptive and dispersive parts is expressed by [27,28,29] where r 2 DD = 4m 2 D . After replacing M 2 ψ in the amplitude in eq. (1) with r, we get the amplitude A AB (r). The energy dependent factor R(r) is defined as R(r) = exp (−I |q(r)| 2 ), which not only reflects the |q(r)|−dependence of the interaction between ψ(3770) and DD mesons, but also plays the role of ultraviolet cutoff. Meanwhile, R(r) can be understood as the coupled channel effect summing up all the bubbles from the charmed meson loops [30]. Here |q(r)| denotes the three momentum of D meson in the rest frame of ψ(3770) with the mass M ψ ≈ √ r. The interaction length factor I is related to the radius of the interaction by I = R/6 [31]. Pennington and Wilson indicated that I = 0.4 GeV −2 corresponding to R = 0.3 fm is favorable when studying the charmonium mass shift [31].
The branching ratio of ψ(3770) → non − DD including all J/ψη, ρπ, ωη, K * K modes with a fixed value I = 0.4 GeV −2 is shown in Fig. 4 within the range of α = 0.8 ∼ 2.2. Furthermore, let us compare our result with the BES data [5] and the result of ψ(3770) → Light Hadrons including the color-octet mechanism calculated up to next to leading order in the approach of NRQCD [20]. Fig. 4 shows that when FSI effects are taken into account, the NRQCD results plus FSI contribution can be very close to the BES data as long as α takes a value of 2.0∼2.2. Since our results heavily depend on the parameter α, which is fully determined by the non-perturbative QCD effects and therefore cannot be determined based on a first principle, one can only phenomenologically fix it by fitting data. We also notice that the amplitudes in eqs. (4)-(5) are dependent on the values of coupling constant in every vertex, which results in that the decay width is proportional to the square product of all of the coupling constants. If the uncertainty is 20% for each coupling constant, the maximum of uncertainty the decay width is 4%.
The BES data B[ψ(3770) → ρπ] < 2.4 × 10 −3 with corresponding width Γ[ψ(3770) → ρπ] < 65 keV and the CLEO data B[ψ(3770) → ρπ] < 4.0 × 10 −3 with corresponding width Γ[ψ(3770) → ρπ] < 109 keV [12] help to further constrain the range of α to 0.8 < α < 1.1 and 0.8 < α < 1.3, respectively. The relevant values of the decay widths and the branching fraction are listed in Table I. It is noted that when the FSI is taken into account and α is much restricted, the prediction of the branching ratio  [5] and the result of ψ(3770) → Light Hadrons (dash-dotted line with shadowed band) by the color-octet mechanism calculated up to next to leading order within the framework of NRQCD [20]. Here the red line with green shadowed band is the total result including the NLO NRQCD effects and FSI contribution. The green shadowed band corresponds to the error tolerance, coming from the NRQCD estimate in Ref. [20]. The orange and light blue shadowed bands are the suitable window for α, which is respectively determined by the BES data [35] and CELO data [12] of ψ(3770) → ρπ.   of ψ(3770) → non − DD caused by the FSI can reach up to B FS I non−DD = (0.2 ∼ 1.1)% (taking CLEO data of ψ(3770) → ρπ to constrain α). It indicates that even though FSI is significant, it cannot make a drastic change as long as α is restricted to be less than 2.1. Furthermore, the upper limit of the total contribution of the NRQCD and FSI is up to 4.6%. The branching ratios of E1 transition ψ(3770) → γχ cJ (J=0,1,2) and ψ(3770) → J/ψππ, J/ψη are about (1.5 ∼ 1.8)% [10,14,15]. If summing up all the above non-DD contributions, the branching ratio of the channels with non-DD final states can be as large as 6.4%, which is still smaller than the experimental value B[ψ(3770) → non − DD] = (13.4 ± 5.0 ± 3.6)% but near its lower bound [5].
As a short summary, let us emphasize a few points. First, even including contributions of color-octet, the NRQCD prediction on the branching ratio of ψ(3770) → non − DD which is calculated up to NLO, cannot coincide with the data of BES [20]. At the energy range, the FSI obviously is significant and this allegation has been confirmed by many earlier phenomenological studies on other processes. When the FSI effects are taken into account, the discrepancy between theoretical prediction and data is significantly alleviated, even though not sufficient. Considering the rather large error range in measurements of both inclusive non − DD decay of ψ(3770) and the exclusive mode ψ(3770) → ρπ, one would still be able to obtain a value for the parameters α which does not conflict with the data, by which the theorical prediction and data might be consistent. The more accurate mesurements which will be conducted in the future will provide more information which can help to make a definite conclusion if the FSI indeed solves the "puzzle" or not. Secondly, our result shows that the FSI can make significant contribution to all the channels of ψ(3770) → ρπ, K * K, and each of them should be searched in future experiments. Thirdly, no doubt, more accurate measurements on ψ(3770) → non − DD, especially ψ(3770) → Light Hadrons, are necessary. Thanks to the great improvement of facility and technology of detection at the charm-tau energy region, the BESIII [36] will provide much more precise data, by which we may gain more information. Furthermore, along the other lines more theoretical studies which may involve other mechanics, even new physics beyond standard model are badly needed.