The chromopolarizability of a quarkonium describes the quarkonium’s interaction with soft gluonic fields and can be measured in the heavy quarkonium decay. Within the framework of dispersion theory which considers the ππ final state interaction (FSI) model-independently, we analyze the transition ψ′→J/ψπ+π- and obtain the chromopolarizability αψ′ψ and the parameter κ. It is found that the ππ 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 αψ′ψ=(1.44±0.02)GeV-3 and the parameter κ=0.139±0.005. Our results could be useful in studying the interactions of charmonium with light hadrons.

Fundamental Research Funds for the Central Universities065000771. 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 transitions 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 [3–8] interpretation of the Pc+(4380) and Pc+(4450) observed by the LHCb Collaboration, and it is found that the Pc+(4450) can be interpreted as a ψ′-nucleon bound state if αψ′/αJ/ψ≃15 [9].

There are a few studies of the chromopolarizabilities of J/ψ and ψ′, some of which are not in line with each other. Calculated in the large-Nc limit in the heavy quark approximation, the values of the chromopolarizabilities of the J/ψ and ψ′ are obtained: αJ/ψ≈0.2GeV-3 and αψ′≈12GeV-3 [6, 7, 10, 11]. Within a quarkonium-nucleon effective field theory, the chromopolarizability of the J/ψ is determined through fitting the lattice QCD data [12] of the J/ψ-nucleon potential, and the result is αJ/ψ=0.24GeV-3 [13, 14]. Based on an effective potential formalism given in [15] and a recent lattice QCD calculation [16], the chromopolarizabilities of J/ψ are extracted to be αJ/ψ=(1.6±0.8)GeV-3 [9]. On the other hand, the determination of the transitional chromopolarizability αψ′ψ≡αψ′→J/ψ is of importance since it acts a reference benchmark for either of the diagonal terms due to the Schwartz inequality: αJ/ψαψ′≥αψ′ψ2 [4]. The perturbative prediction in the large Nc limit is αψ′ψ≈-0.6GeV-3 [6, 7, 10, 11]. While being extracted from the process of ψ′→J/ψππ, the result is αψ′ψ≈2GeV-3 [4, 17]. Taking account of the ππ FSI in a chiral unitary approach, it is found that the value of αψ′ψ may be reduced to about 1/3 of that without the ππ FSI [18].

Since the FSI plays an important role in the heavy quarkonium transitions and modifies the value of αψ′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 αψ′ψ. Instead of the chiral unitary approach [18, 19], in which the scalar mesons (σ, f0(980), and a0(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 [17, 18].

The theoretical framework is described in detail in Section 2. In Section 3, we fit the decay amplitudes to the data for the ψ′→J/ψπ+π- transition and determine the chromopolarizability αψ′ψ and the parameter κ. A brief summary will be presented in Section 4.

2. Theoretical Framework

First we define the Mandelstam variables of the decay process ψ′(pa)→J/ψ(pb)π(pc)π(pd)(1)s=pc+pd2,t=pa-pc2,u=pa-pd2.

The amplitude for the π+π- transition between S-wave states A and B of heavy quarkonium can be written as [4, 13](2)MAB=2mAmBαABπ+pcπ-pd12Ea·Ea0=8π2bmAmBαABκ1pc0pd0-κ2pcipdi,where the factor 2mAmB appears due to the relativistic normalization of the decay amplitude, αAB is the chromopolarizability, and Ea denotes the chromoelectric field. b is the first coefficient of the QCD beta function, b=11/3Nc-2/3Nf, where Nc=3 and Nf=3 are the number of colors and of light flavors, respectively. κ1=2-9κ/2, and κ2=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 terms of spin multiplets, and one has J≡ψ·σ+ηc, where σ contains the Pauli matrices and ψ and ηc annihilates the ψ and ηc states, respectively [20]. The effective Lagrangian, at the leading order in the chiral as well as the heavy quark nonrelativistic expansion, reads [21–23](3)Lψψ′ππ=c12J†J′uμuμ+c22J†J′uμuνvμvν+h.c.,where vμ=(1,0) is the velocity of the heavy quark. The Goldstone bosons of the spontaneous breaking of chiral symmetry can be parametrized according to (4)uμ=iu†∂μu-u∂μu†,u2=eiΦ/Fπ,Φ=π02π+2π--π0,where Fπ=92.2MeV denotes the pion decay constant.

The amplitude obtained by using the effective Lagrangians in (3) is(5)Ms,t,u=-4Fπ2c1pc·pd+c2pc0pd0.Matching the amplitude in (2) to that in (5), we can express the low-energy couplings in the chiral effective Lagrangian in terms of the chromopolarizability αAB and the parameter κ(6)c1=-π2mψ′mψFπ2bαψ′ψ4+3κ,c2=12π2mψ′mψFπ2bαψ′ψκ.

The partial-wave decomposition of M(s,t,u) can be easily performed by using the relation(7)pc0pd0=14s+q2-14q2σπ2cos2θ,where q is the 3-momentum of the final vector meson in the rest frame of the initial state with q={[(mψ′+mψ)2-s]/[(mψ′-mψ)2-s]}1/2/(2mψ′), σπ≡1-4mπ2/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 (8)M0χs=-2Fπ2c1s-2mπ2+c22s+q21-σπ23,M2χs=23Fπ2c2q2σπ2.

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 [22–31]. 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(9)ImMls=Mlssinδl0se-iδl0s.Below the inelastic threshold, the phases δlI 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 [32, 33]. It is known that the standard Omnès solution of (9) is as follows: (10)Mls=PlnsΩl0s,where the Pln(s) is a polynomial, and the Omnès function is defined as [34](11)ΩlIs=expsπ∫4mπ2∞dxxδlIxx-s.

At low energies, M0(s) and M2(s) can be matched to the chiral representation. Namely, in the limit of switching off the ππ FSI, i.e., Ωl0(s)≡1, the polynomials Pln(s) can be identified exactly with the expressions given in (8). Therefore, the amplitudes including the FSI take the form (12)M0s=-2Fπ2c1s-2mπ2+c22s+q21-σπ23Ω00s,(13)M2s=23Fπ2c2q2σπ2Ω20s.

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 [35], which yields a good description below the onset of the KK¯ threshold. For the D-wave, we employ the parametrization for δ20 given by the Madrid–Kraków collaboration [36]. Both phases are guided smoothly to π for s→∞.

It is then straightforward to calculate the ππ invariant mass spectrum and helicity angular distribution for ψ′→J/ψπ+π- using(14)dΓdsdcosθ=sσπq128π3mψ′2M0s+M2sP2cosθ2,where the Legendre polynomial P2(cosθ)=(3cos2θ-1)/2.

3. Phenomenological Discussion

The unknown parameters are the low-energy constants c1 and c2 in the chiral Lagrangian (3), which can be expressed in terms of the chromopolarizability αψ′ψ and the parameter κ as in (6). In order to determine αψ′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 [37] and the corresponding decay width Γ(ψ′→J/ψπ+π-) [38]. The fit results are plotted in Figure 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 χ2/d.o.f. are shown in Table 1. One observes that the experimental data can be well described regardless of whether the FSI is included. This is due to the simple shapes of the ππ invariant mass distribution and the helicity angular distribution in this process and does not mean the FSI is not important. Since the dipion mass invariant mass reaches about 600 MeV in such a decay, the ππ FSI is known to be strong in this energy range and needs to be considered. On the other hand, one can readily see from (6) and (8) that while the chromopolarizability αψ′ψ determines the overall decay rate, the parameter κ characterizes the D-wave contribution, and we do not find significant correlation between αψ′ψ and κ.

The parameter results from the fits of the ψ′→ψππ processes with and without the ππ FSI.

Without ππ FSI

With ππ FSI

αψ′ψ (GeV^{−3})

2.37±0.02

1.44±0.02

κ

0.135±0.005

0.139±0.005

χ2d.o.f

115.3120-2=0.98

117.6120-2=1.00

Simultaneous fit to the ππ invariant mass distributions and the helicity angle distributions in ψ′→J/ψπ+π-. The red solid and blue dashed curves represent the theoretical fit results with ππ FSI and without ππ FSI cases, respectively. The data are taken from [37].

We observe that the ππ FSI modifies the value of the chromopolarizability αψ′ψ significantly, and resultant value with the FSI is almost 1/2 of that without the FSI. The obtained value with the FSI, αψ′ψ=(1.44±0.02)GeV-3, coincides with the suspicion αJ/ψ≥αψ′ψ [3] with the value αJ/ψ=(1.6±0.8)GeV-3 from the calculation [9] based on the recent lattice QCD data of J/ψ-nucleon potential [16]. It should be mentioned that the value of αψ′ψ with the FSI obtained here is different from the one in [18], αψ′ψ=(0.83±0.01)GeV-3, and also our result without the FSI slightly differs from those in [17, 18]. The reasons are that the chiral unitary approach instead of dispersion theory is used to account for the FSI in [18], and we use the updated experimental data [37, 38] and a general theoretical amplitude rather than the one only containing the S-wave as employed in [17, 18].

For the parameter κ, as shown in Table 1 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] has been performed by BES [39], and based on the joint mπ+π--cosθπ+ distribution this parameter was determined as κ=0.183±0.002±0.003. We have tried fitting the same old BES data [39] 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](15)M∝s-κmψ′-mψ21+2mπ2s+32κmψ′-mψ2-sσπ2cos2θ-13.If we make the same approximation, namely, neglect the O(mπ2) terms except the mπ2/s ones, as in [2] and set (mψ′+mψ)2-s≈(mψ′+mψ)2 in the expression of 3-momentum q, our amplitude without the ππ FSI agrees with (15). While numerically we find that some neglected O(mπ2) terms are at the same order as the κmψ′-mψ2 term in (15), this may account for the difference of κ between ours and that in [39]. On the other hand, we have checked that the contribution of the D-wave, which is characterized by the parameter κ, to the total rate is less than two percent, and the same observation has been made in [39].

4. 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 cosθ distributions, the values of the chromopolarizability αψ′ψ and the parameter κ are determined. It is found that the effect of the ππ FSI is quite sizeable in the chromopolarizability αψ′ψ, and the one with FSI is almost 1/2 of that without FSI. 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 nucleon-charmonia interaction.

Data Availability

All the data used in this work are from [37, 38].

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

We acknowledge Feng-Kun Guo for the proposal for this work and for the useful comments on the manuscript. This research is supported in part by the Fundamental Research Funds for the Central Universities under Grant no. 06500077.

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