Chromopolarizability of charmonium and $\pi\pi$ final state interaction revisited

The chromopolarizability of a quarknonium describes the quarknonium's interaction with soft gluonic fields and can be measured in the heavy quarkonium decays. Within the framework of dispersion theory which consider the $\pi\pi$ final state interaction (FSI) model-independently, we analyze the transition $\psi^\prime\to J/\psi\pi^+\pi^-$ and obtain the chromopolarizability $\alpha_{\psi^\prime \psi}$ and the parameter $\kappa$. It is found that the $\pi\pi$ FSI plays an important role in extracting the chromopolarizability from the experimental data. The obtained chromopolarizability with the FSI is reduced to about 1/2 of that without the FSI. With the FSI, we determine the chromopolarizability $|\alpha_{\psi^\prime\psi}|=(1.44\pm 0.02)$ GeV$^{-3}$ and the parameter $\kappa=0.139\pm 0.005.$ Our results could be useful in studying the interactions of charmonium with light hadrons.


I. INTRODUCTION
The chromopolarizability α of a quarkonium parametrizes the quarkonium's effective interaction with soft gluons, and it is an important quantity in the heavy quark effective theory. Within the multipole expansion in QCD in terms of the chromopolarizability, many processes can be described, including the hadronic transtions between quarkonium resonances [1,2] and the interaction of slow quarkonium with a nuclear medium [3]. A recent interest of the chromopolarizabilities of J/ψ and ψ ′ comes from the hadrocharmonium [3][4][5][6][7] interpretation of the P + c (4380) and P + c (4450) observed by the LHCb Collaboration, and it is found that the P + c (4450) can be interpreted as a ψ ′ -nucleon bound state if α ψ ′ /α J/ψ ≃ 15 [8].
There are a few studies of the chromopolarizabilities of J/ψ and ψ ′ , some of which are not in line with each others. Calculated in the large-N c limit in the heavy quark approximation, the values of the chromopolarizabilities of the J/ψ and ψ ′ are obtained: α J/ψ ≈ 0.2 GeV −3 and α ψ ′ ≈ 12 GeV −3 [6,9]. Within a quarkonium-nucleon effective field theory, the chromopolarizability of the J/ψ is determined through fitting the lattice QCD data [10] of the J/ψ-nucleon potential, and the result is α J/ψ = 0.24 GeV −3 [11,12]. While based on an effective potential formalism given in Ref. [13] and a recent lattice QCD calculation [14], the chromopolarizabilities of J/ψ is extracted to be α J/ψ = (1.6 ± 0.8) GeV −3 [8]. On the other hand, the determination of the transitional chromopolarizability α ψ ′ ψ ≡ α ψ ′ →J/ψ is of importance since it acts a reference benchmark for either of the diagonal terms due to the Schwartz inequality: α J/ψ α ψ ′ ≥ α 2 ψ ′ ψ [4]. The perturbative prediction in the large N c limit is α ψ ′ ψ ≈ −0.6 GeV −3 [6,9]. While extracted from the process of ψ ′ → J/ψππ, the result is |α ψ ′ ψ | ≈ 2 GeV −3 [4,15]. Taking account of the ππ FSI in a chiral unitary approach, it is found that the value of |α ψ ′ ψ | may be reduced to about 1/3 of that without the ππ FSI [16].
Since the FSI plays an important role in the heavy quarkonium transitions and modifies the value of α ψ ′ ψ significantly, it is thus necessary to account for the FSI properly. In this work we will use the dispersion theory to take into account of the ππ FSI and extract the value of α ψ ′ ψ . Instead of the chiral unitary approach [16,17], in which the scalar mesons (σ, f 0 (980), and a 0 (980)) are dynamically generated, in the dispersion theory the ππ FSI is treated in a model-independent way consistent with ππ scattering data. Another update of our calculation is that we consider the FSIs of separate partial waves, namely the S-and D-waves, instead of only accounting for the S-wave as in the parametrization in Refs [15,16].
The theoretical framework is described in detail in Sec. II. In Sec. III, we fit the decay amplitudes to the data for the ψ ′ → J/ψπ + π − transition, and determine the the chromopolarizability α ψ ′ ψ and the parameter κ. A brief summary will be presented in Sec. IV.

II. THEORETICAL FRAMEWORK
First we define the the Mandelstam variables of the decay process ψ ′ (p a ) → J/ψ(p b )π(p c )π(p d ) The amplitude for the π + π − transition between S-wave states A and B of heavy quarkonium can be written as [4,11] where the factor 2 √ m A m B appears due to the relativistic normalization of the decay amplitude, α AB is the chromopolarizability, and E a denotes the chromoelectric field. b is the first coefficient are the number of colors and of light flavors, respectively. κ 1 = 2 − 9κ/2, and κ 2 = 2 + 3κ/2, where κ is a parameter that can be determined from the data.
The above result of the QCD multipole expansion together with the soft-pion theorem can be reproduced by constructing a chiral effective Lagrangian for the ψ ′ → J/ψππ transition. Since the spin-dependent interactions are suppressed by the charm mass, the heavy quarkonia can be expressed in term of spin multiplets, and one has J ≡ ψ · σ + η c , where σ contains the Pauli matrices and ψ and η c annihilate the ψ and η c states, respectively [18]. The effective Lagrangian, at the leading order in the chiral as well as the heavy-quark nonrelativistic expansion, reads [19][20][21] where v µ = (1, 0) is the velocity of the heavy quark. The Goldstone bosons of the spontaneous breaking of chiral symmetry can be parametrized according to where F π = 92.2 MeV denotes the pion decay constant.
The amplitude obtained by using the effective Lagrangians in Eq. (3) is Matching the amplitude in Eq. (2) to that in Eq. (5), we can express the low-energy couplings in the chiral effective Lagrangian in terms of the chromopolarizability α AB and the parameter κ The partial-wave decomposition of M (s, t, u) can be easily performed by using the relation where q is the 3-momentum of the final vector meson in the rest frame of the initial state with π /s, and θ is the angle between the 3-momentum of the π + in the rest frame of the ππ system and that of the ππ system in the rest frame of the initial ψ ′ .
Parity and C-parity conservations require the pion pair to have even relative angular momentum l. We only consider the S-and D-wave components in this study, neglecting the effects of higher partial waves. Explicitly, the S-and D-wave components of the amplitude read There are strong FSI in the ππ system especially in the isospin-0 S-wave, which can be taken into account model-independently using dispersion theory [20][21][22][23][24][25][26]. We will use the Omnès solution to obtain the amplitude including FSI. In the region of elastic ππ rescattering, the partial-wave unitarity conditions read Below the inelastic threshold, the phases δ I l of the partial-wave amplitudes of isospin I and angular momentum l coincide with the ππ elastic phase shifts modulo nπ, as required by Watson's theorem [27,28]. It is known that the standard Omnès solution of Eq. (9), where the P n l (s) is a polynomial, and the Omnès function is defined as [29] Ω I l (s) = exp At low energies, M 0 (s) and M 2 (s) can be matched to the chiral representation. Namely in the limit of switching off the ππ FSI , i.e. Ω 0 l (s) ≡ 1, the polynomials P n l (s) can be identified exactly with the expressions given in Eq. (8). Therefore, the amplitudes including the FSI take the form Now we discuss the ππ phase shifts used in the calculation of the Omnès functions. For the S-wave, we use the phase of the nonstrange pion scalar form factor as determined in Ref. [30], which yields a good description below the onset of the KK threshold. For the D-wave, we employ the parametrization for δ 0 2 given by the Madrid-Kraków collaboration [31]. Both phases are guided smoothly to π for s → ∞.

III. PHENOMENOLOGICAL DISCUSSION
The unknown parameters are the low-energy constants c 1 and c 2 in the chiral Lagrangian (3), which can be expressed in terms of the chromopolarizability α ψ ′ ψ and the parameter κ as in Eq. (6).
In order to determine α ψ ′ ψ and κ, we fit the theoretical results to the experimental π + π − invariant mass spectra and the helicity angular distribution from the BES ψ ′ → J/ψπ + π − decay data [32], and the corresponding decay width Γ(ψ ′ → J/ψπ + π − ) [33]. The fit results are plotted in Fig. 1, where the red solid and blue dashed curves represent the results with or without the ππ FSI, respectively. The fit parameters as well as the χ 2 /d.o.f. are shown in Table I.
We observe that the ππ FSI modifies the value of the chromopolarizability α ψ ′ ψ significantly, and resultant value with the FSI is almost 1/2 of that without the FSI. The obtained value with the FSI, |α ψ ′ ψ | = (1.44 ± 0.02) GeV −3 , coincides with the suspicion α J/ψ ≥ |α ψ ′ ψ | [3] with the value α J/ψ = (1.6 ± 0.8) GeV −3 from the calculation [8] based on the recent lattice QCD data of J/ψ-nucleon potential [14]. It should be mentioned that the value of α ψ ′ ψ with the FSI obtained here is different from the one in Ref. [16], |α ψ ′ ψ | = (0.83 ± 0.01) GeV −3 , and also our result without the FSI slightly differs from those in Refs. [15,16]. The reasons are that the chiral unitary dΓ/d s  approach instead of dispersion theory is used to account for the FSI in [16], and we use the updated experimental data [32,33] and a general theoretical amplitude rather than the one only containing the S-wave as employed in Refs [15,16].
For the parameter κ, as shown in Table I its value is affected little by the ππ FSI. One notes that a detailed study of the ψ ′ → J/ψπ + π − process using the Novikov-Shifman model [2] have been performed by BES [34], and based on the joint m π + π − -cos θ π + distribution this parameter was determined as κ = 0.183 ± 0.002 ± 0.003. We have tried fitting the same old BES data [34], and our κ changes slightly and is still much smaller than the BES one. In the Novikov-Shifman model, the ψ ′ → J/ψπ + π − amplitude reads [2] If we make the same approximation, namely neglect the O(m 2 π ) terms except the m 2 π /s ones, as in Ref. [2] and set (m ψ ′ + m ψ ) 2 − s ≈ (m ψ ′ + m ψ ) 2 in the expression of 3-momentum q, our amplitude without the ππ FSI agrees with Eq. (15). While numerically we find that some neglected O(m 2 π ) terms are at the same order as the κ m ψ ′ − m ψ 2 term in Eq. (15), and this may account for the difference of κ between ours and that in Ref. [34]. On the other hand, one can readily see from Eqs. (6) and (8) or Eq. (15), while the chromopolarizability α ψ ′ ψ determines the overall decay rate, the parameter κ characterizes the D-wave contribution. We have checked that the contribution of the D-wave to the total rate is less than two percent, and the same observation has been made in [34].

IV. CONCLUSIONS
We have used dispersion theory to study the ππ FSI in the decay ψ ′ → J/ψπ + π − . Through fitting the data of the ππ mass spectra and the angular cos θ distributions, the values of the chromopolarizability α ψ ′ ψ and the parameter κ are determined. It is found that the effect of the ππ FSI is quite sizeable in the chromopolarizability α ψ ′ ψ , and the one with FSI is almost 1/2 of that without FSI. While the parameter κ, which accounts for the D-wave contribution, is affected little by the ππ FSI. The results obtained in this work would be valuable to understand the chromopolarizability of charmonia, and will have applications for the studies of the nucleoncharmonia interaction.