The study of the B_c system is important since it is the only one that contains two different heavy flavors in the Standard Model. The $c\bar{b}$ states, unlike their counterparts in the J/ψ and ϒ families, cannot annihilate into gluons, making them more stable. The ground B_c(1¹S₀) state was discovered by the CDF Collaboration at the Tevatron collider in 1998 [1, 2]. In 2014, the ATLAS Collaboration observed a peak at 6842 ± 4 ± 5 MeV [3], which could be interpreted either as the ${B}_{c}^{* }({2}^{3}{S}_{1})$ excited state or a pair of peaks resulting from the decays of B_c(2¹S₀) → B_c(1¹S₀)π⁺π^- and ${B}_{c}^{* }({2}^{3}{S}_{1})\to {B}_{c}^{* }({1}^{3}{S}_{1}){\pi }^{+}{\pi }^{-}$ followed by ${B}_{c}^{* }({1}^{3}{S}_{1})\to {B}_{c}({1}^{1}{S}_{0})\gamma $. In 2019, the CMS [4] and LHCb [5] Collaborations identified signals consistent with the B_c(2S) and ${B}_{c}^{* }(2S)$ states in the B_c(1S)π⁺π^- invariant mass spectrum. Shortly thereafter, the normalized invariant mass distributions of the pions pair emitted in the ${B}_{c}^{(* )}(2S)\to {B}_{c}^{(* )}{\pi }^{+}{\pi }^{-}$ decays were presented [6]. The dipion transition to the ground B_c state is the dominant decay mode of the B_c(2S) state and will be key to characterizing those states, which we focus on in this study. The dipion transitions between heavy quarkonia are typically described as a two-step process in which the soft gluons are first emitted from the heavy quarks and then recombine into light quarks. The heavy quarkonia are expected to be compact, and the method of QCD multipole expansion [7-10], combined with soft-pion theorems, has been successfully used to study dipion transitions between the equal-mass $Q\bar{Q}$ systems, i.e., ψ(2S) → ψ(1S)ππ and ϒ(2S) → ϒ(1S)ππ. Since the electricmultipole transitions are only sensitive to the relative position of the quark and anti-quark, the same method has been used to study the hadronic transition for the unequal-mass $Q{\bar{Q}}^{{\prime} }$ systems B_c(2S) → B_c(1S)ππ [11-16]. However, the ππ final-state interaction (FSI) was not considered in [11-16]. In the B_c(2S) → B_c(1S)ππ process, the dipion invariant mass reaches nearly 600 MeV, making the ππ FSI significant and requiring careful consideration. We will employ dispersion theory to account for the ππ FSI and study the decay B_c(2S) → B_c(1S)ππ. The same method has been successfully used to describe transitions of ψ(2S) → ψ(1S)ππ and ϒ(2S) → ϒ(1S)ππ [17-20].

The chromopolarizability α of a heavy quarkonium quantifies its effective interaction with soft gluons and is an important parameter for describing quarkonium interactions. Recently, the chromopolarizability of heavy quarkonium has garnered increased interest due to its relevance in interpreting the structures of multiquark hadrons containing heavy quark and anti-quark. For reasonable values of the chromopolarizabilities of the charmonia, bound states such as the double-J/ψ state, several hadro-charmonium bound states, and baryo-charmonium bound states have been identified with certain XYZ states and the ${P}_{c}^{+}$ pentaquark states [21-27]. Similarly, for appropriate values of the chromopolarizabilities of bottomonia, several hadro-bottomonium and baryo-bottomonium bound states have been predicted [28, 29]. The $c\bar{b}$ states are intermediate between the $c\bar{c}$ and $b\bar{b}$ states, both in mass and in size. Therefore, it is expected that the values of the chromopolarizabilities of the $c\bar{b}$ states will fall between the values of the chromopolarizabilities of the $c\bar{c}$ and the $b\bar{b}$ states. For certain values of the chromopolarizabilities of the $c\bar{b}$ states, multiquark hadrons containing a $c\bar{b}$ component may also exist. While the diagonal chromopolarizabilities α_AA, with A representing a heavy quarkonium, cannot be determined directly from current experimental data. One possible method to estimate α_AA assumes that the heavy quarkonia are purely color-Coulombic systems. This assumption works well for ground-state bottomonia but introduces significant uncertainties for charmonia and excited bottomonia [29, 30]. On the other hand, the determination of the transitional chromopolarizability ${\alpha }_{{A}^{{\prime} }A}\equiv {\alpha }_{{A}^{{\prime} }\to A}$ is of importance as it serves as a reference benchmark for each of the diagonal terms due to the Schwartz inequality: ${\alpha }_{AA}{\alpha }_{{A}^{{\prime} }{A}^{{\prime} }}\geqslant {\alpha }_{{A}^{{\prime} }A}^{2}$ [21, 31]. In this work, we will extract the value of ${\alpha }_{{B}_{c}(2S){B}_{c}}$ from the B_c(2S) → B_c(1S)ππ transition.

The theoretical framework is outlined in section 2. In section 3, we fit the decay amplitudes to the data of the ππ invariant mass distribution of the B_c(2S) → B_c(1S)ππ process, and determine the chromopolarizability ${\alpha }_{{B}_{c}(2S){B}_{c}}$ and the parameter κ. A summary is provided in section 4.

The low-energy interaction of heavy quarkonia with light hadrons is mediated by soft gluons. The heavy quarkonia are expected to be compact, so that the method of QCD multipole expansion is often employed to study the interaction of heavy quarkonia with soft gluons. The leading E1 chromo-electric dipole term in the expansion is simple ${H}_{E1}=-\frac{1}{2}{\xi }^{a}{\boldsymbol{r}}\cdot {{\boldsymbol{E}}}^{a}$, where ${\xi }^{a}={t}_{1}^{a}-{t}_{2}^{a}$ is the difference between the SU(3) color generators acting on the quark and anti-quark, r is the relative position vector between the quark and anti-quark, and E^a is the chromo-electric field. For the transition between two S-wave heavy quarkonium states A and B with an emitted gluon system which possesses the quantum numbers of two pions, the leading order amplitude is in the second order of H_E1 [22] where α_AB denotes the chromopolarizability, quantifying the strength of the E1-E1 chromo-electric interaction of the quarkonium with the soft gluon system.

The amplitude for the dipion transition between two S-wave heavy quarkonium states, A and B, is expressed as [22, 32] where the factor $2\sqrt{{m}_{A}{m}_{B}}$ arises from the relativistic normalization of the amplitude. b denotes the first coefficient of the QCD beta function, and its expression is $b=\frac{11}{3}{N}_{c}-\frac{2}{3}{N}_{f},$ where N_c = 3 and N_f = 3 are the number of colors and of light flavors, respectively. The parameters κ₁ and κ₂ are not independent, since κ₁ = 2 - 9κ/2 and κ₂ = 2 + 3κ/2, where the value of κ can be determined from experimental data.

The above expression of the QCD multipole expansion together with the soft-pion theorem, can be derived by constructing a chiral effective Lagrangian for the B_c(2S) → B_c(1S)ππ transition [18, 33] where v^μ = (1, 0) represents the velocity of the heavy quark. The Goldstone bosons of the spontaneous breaking of chiral symmetry can be parametrized as where F represents the pseudo-Goldstone boson decay constant, and we will use F_π = 92.2 MeV for the pions and F_K = 113.0 MeV for the kaons.

The contact term amplitude for the B_c(2S)(p_a) → B_c(p_b)π(p_c)π(p_d) process, derived using the chiral effective Lagrangians in equation (3) is where $s={({p}_{c}+{p}_{d})}^{2}$, and θ is the angle between the 3-momentum of the π⁺ in the rest frame of the dipion system and that of the dipion system in the rest frame of the initial B_c(2S). By matching the amplitude in equation (2) to that in equation (5), the chiral effective Lagrangian coupling constants c₁ and c₂ can be expressed in terms of the chromopolarizability α_AB and the parameter κ,

The strong FSIs between two pseudoscalar mesons can be incorporated model-independently using dispersion theory. Since the invariant mass of the pion pair approaches to the $K\bar{K}$ threshold, we will consider the coupled-channel (ππ and $K\bar{K}$) FSI for the dominant S-wave component, while for the D-wave we will only take account of the single-channel FSI.

For the B_c(2S)(p_a) → B_c(p_b)π⁺(p_c)π^-(p_d) processes, the partial-wave expansion of the amplitude including FSI is expressed as

For the dominant S-wave, we consider the two-channel rescattering effects. The two-channel unitarity conditions reads where the two-dimensional vector M₀(s) contains both the ππ and the $K\bar{K}$ final states, The two-dimensional matrices ${T}_{0}^{0}(s)$ and Σ(s) are defined as and ${\rm{\Sigma }}(s)\equiv \,\rm{diag}\,\left(\right.{\sigma }_{\pi }(s)\theta (s-4{m}_{\pi }^{2}),{\sigma }_{K}(s)\theta (s-4{m}_{K}^{2})\left)\right.$. The ${T}_{0}^{0}(s)$ matrix incorporates three input functions: the ππ S-wave isoscalar phase shift ${\delta }_{0}^{0}(s)$ and the $\pi \pi \to K\bar{K}$ S-wave amplitude ${g}_{0}^{0}(s)=| {g}_{0}^{0}(s)| {{\rm{e}}}^{{\rm{i}}{\psi }_{0}^{0}(s)}$ with its modulus and phase. We will use the parameterization of the ${T}_{0}^{0}(s)$ matrices given in [34, 35]. These inputs are used up to $\sqrt{{s}_{0}}=1.3\,\,\rm{GeV}\,$ (the f₀(1370) resonance is known to have a significant coupling to 4π [36]). Beyond s₀, the phases ${\delta }_{0}^{0}(s)$ and ${\psi }_{0}^{0}$ are smoothly guided to 2π via [37] The inelasticity ${\eta }_{0}^{0}(s)$ in equation (10) is related to the modulus $| {g}_{0}^{0}(s)| $ by

The solution to the two-channel unitarity condition in equation (8) is given by where Ω(s) satisfies the homogeneous two-channel unitarity relation Numerical results for Ω(s) are available in [37-40].

For the D-wave, we consider the single-channel FSI. In the elastic ππ rescattering region, the partial-wave unitarity conditions is expressed as In the elastic region, the phase of the D-wave isoscalar amplitude ${\delta }_{2}^{0}$ equals the ππ elastic phase shifts modulo nπ, as required by Watson's theorem [41, 42]. The Omnès function is defined as [43] Using the Omnès function, the solution of equation (15) is given by where the polynomial ${P}_{2}^{n-1}(s)$ is a subtraction function. We use the result of the D-wave phase shift ${\delta }_{2}^{0}(s)$ given in [44], and extend it smoothly to π as s → ∞.

On the other hand, at low energies, the amplitudes M₂(s) and M₀(s) should match the results from chiral perturbation theory. Namely, if one turn off the final-state interactions, ${{\rm{\Omega }}}_{2}^{0}(s)=1$ and Ω(0) = 1, the subtraction functions should agree with the partial-wave projections of the chiral result obtained using the Lagrangian in equation (3). For the S-wave, this leads to the integral equation: where

For the D-wave, the integral equation is given by where

The ππ invariant mass distribution for the B_c(2S) → B_cπ⁺π^- is expressed as where the Legendre polynomial ${P}_{2}(\cos \theta )=(3{\cos }^{2}\theta -1)/2.$

Since for the B_c(2S) → B_cπ⁺π^- process the CMS Collaboration [6] only provides the normalized dipion invariant mass distributions, we use the theoretical prediction of ${{\rm{\Gamma }}}_{{B}_{c}(2S)\to {B}_{c}\pi \pi }$ = ${{\rm{\Gamma }}}_{{B}_{c}(2S)}\times $Br(B_c(2S) → B_cππ) = 73.1 × 0.811keV = 59.3keV in [13] to transform them into the differential widths over the dipion invariant mass, which will be fitted in our study. Note that the theoretical prediction of the decay rate ${{\rm{\Gamma }}}_{{B}_{c}(2S)\to {B}_{c}\pi \pi }$ in [13] is the sum of the decay modes with charged and neutral pions, and we take the decay rate of the neutral pions mode to be half of the decay rate of the charged pions mode.

The two free parameters are the low-energy coupling constants c₁ and c₂ in the chiral Lagrangian (3), which can be expressed in terms of the chromopolarizability ${\alpha }_{{B}_{c}(2S){B}_{c}}$ and the parameter κ as in equation (6). By performing the χ² fit to the dipion invariant mass distributions of B_c(2S) → B_cπ⁺π^-, we can determine the values of the parameters: with χ²/d.o.f = 6.89/(8 - 2) = 1.15. The fit results are plotted in figure 2, and it can be seen that the dipion invariant mass distributions of B_c(2S) → B_cπ⁺π^- can be well described. In the following, we discuss the fit results in detail.

For the transition chromopolarizability ${\alpha }_{{B}_{c}(2S){B}_{c}}$, first, as shown in equation (6), it depends on the overall decay rate. In this study, we use the theoretical prediction of the decay rate of ${{\rm{\Gamma }}}_{{B}_{c}(2S)\to {B}_{c}\pi \pi }$ provided in [13] as input, and note that the variations in decay rate predictions across [12-14, 16] are minor. A more precise value of ${\alpha }_{{B}_{c}(2S){B}_{c}}$ can be determined when the experimental data of the decay rate of ${{\rm{\Gamma }}}_{{B}_{c}(2S)\to {B}_{c}\pi \pi }$ becomes available in the future. Second, [6] also provides the normalized dipion invariant mass distributions of the ${B}_{c}^{* }(2S)\to {B}_{c}^{* }{\pi }^{+}{\pi }^{-}$ process, which are compatible with the normalized dipion invariant mass distributions of the B_c(2S) → B_cπ⁺π^- process within the uncertainties. However, since the ${B}_{c}^{* }$ state has not been observed yet and the masses of ${B}_{c}^{* }$ and ${B}_{c}^{* }(2S)$ have not been measured experimentally [36], we choose not to fit the data of the normalized dipion mass spectra of the ${B}_{c}^{* }(2S)\to {B}_{c}^{* }{\pi }^{+}{\pi }^{-}$ process. Nevertheless, since the vector charm-beauty mesons are the spin partners of the pseudoscalar ones, their polarizabilities should be the same as those of the pseudoscalar mesons at leading order, as a consequence of heavy quark spin symmetry. Also note that, theoretically, the decay rates of ${{\rm{\Gamma }}}_{{B}_{c}(2S)\to {B}_{c}{\pi }^{+}{\pi }^{-}}$ and ${{\rm{\Gamma }}}_{{B}_{c}^{* }(2S)\to {B}_{c}^{* }{\pi }^{+}{\pi }^{-}}$ are expected to be similar, and both the mass differences between B_c and ${B}_{c}^{* }$ and the mass difference between B_c(2S) and ${B}_{c}^{* }(2S)$ are expected to be below a few percent [12-14, 16]. Therefore, it is expected that the values of chromopolarizabilities ${\alpha }_{{B}_{c}(2S){B}_{c}}$ and ${\alpha }_{{B}_{c}^{* }(2S){B}_{c}^{* }}$ are of the same order of magnitude. Third, since both in mass and in size, the $c\bar{b}$ states are intermediate between the $c\bar{c}$ and $b\bar{b}$ states, it is natural to expect that the diagonal and nondiagonal (transitional) chromopolarizabilities of the $c\bar{b}$ states should lies between corresponding diagonal and nondiagonal chromopolarizabilities of the $c\bar{c}$ and $b\bar{b}$ states, respectively. In table 1, the transitional chromopolarizabilities α_AB and the parameters κ extracted from ψ(2S) → J/ψππ [27], B_c(2S) → B_cππ, and ϒ(2S) → ϒππ [19], respectively, are given. Assuming ${\alpha }_{{B}_{c}(2S){B}_{c}}\simeq {\alpha }_{{B}_{c}^{* }(2S){B}_{c}^{* }}$, we observe the following hierarchy: $| {\alpha }_{\Upsilon (2S)\Upsilon (1S)}| \lt | {\alpha }_{{B}_{c}^{(* )}(2S){B}_{c}^{(* )}}| \lt | {\alpha }_{\psi (2S)\psi }| $, which agrees with above expectation for the transition 2³S₁ → 1³S₁ππ. Note that in [29], the bottomonium diagonal chromopolarizabilities are calculated by considering them as pure Coulombic systems. Based on the result that the value of |α_ϒ(2S)ϒ(2S)| lies in the range of (23 ~ 33) GeV^-3, several possible ϒ(2S)-N bound states are predicted [29]. It is natural to expect that $| {\alpha }_{{B}_{c}^{* }(2S){B}_{c}^{* }(2S)}| \gt | {\alpha }_{\Upsilon (2S)\Upsilon (2S)}| $, and therefore the ${B}_{c}^{* }(2S)$-N bound states may also exit. Also note that in [27], assuming that the external current couples to a single charm (anti-) quark only, the authors estimate the ratio $\xi \equiv {\alpha }_{J/\psi J/\psi }/{\alpha }_{\psi (2S)J/\psi }\unicode{x0007E}{ \mathcal O }(10)$ through study of the overlap integral of the spatial wave functions of the initial- and final-state charmonia. Furthermore, they find that even with a conservative estimate of ξ = 2, it is possible for two J/ψ mesons to form a bound state. If a significant contribution to the chromopolarizabilities of the B_c states arises from the coupling of the charm (anti-) quark, one might also estimate that the ratio ${\alpha }_{{B}_{c}^{* }{B}_{c}^{* }}/{\alpha }_{{B}_{c}^{* }(2S){B}_{c}^{* }}\unicode{x0007E}{ \mathcal O }(10)$. As shown in table 1, the value of $| {\alpha }_{{B}_{c}^{(* )}(2S){B}_{c}^{(* )}}| $ is about one third of the value of α_ψ(2S)J/ψ, suggesting that the value of ${\alpha }_{{B}_{c}^{* }{B}_{c}^{* }}$ may also be large enough to form a two ${B}_{c}^{* }$ bound state.

From equations (6), (20), and (22), it is clear that the parameter κ characterizes the ππD-wave contribution. As shown in table 1, the central values of the results indicate the hierarchy $| {\kappa }_{{B}_{c}(2S){B}_{c}}| \lt | {\kappa }_{\psi (2S)\psi }| \lt | {\kappa }_{\Upsilon (2S)\Upsilon (1S)}| $. In figure 1, we plot the normalized dipion mass spectra for the transitions of ψ(2S) → J/ψππ (green dashed), B_c(2S) → B_cππ (red solid), and ϒ(2S) → ϒππ (blue dotted), respectively, based on the unified theoretical framework employing the chiral effective theory and the dispersive theory. It is observed that, compared to the ψ(2S) → J/ψππ and the ϒ(2S) → ϒππ cases, the line shape of the normalized dipion mass spectra of B_c(2S) → B_cππ is more low-fat. We have checked that the ππD-wave contribution to the total rate is negligible for the B_c(2S) → B_cππ process and accounts for less than two percent of the total rate in the ψ(2S) → J/ψππ process.

We employed dispersion theory to investigate the ππ FSI in the B_c(2S) → B_cπ⁺π^- transition. By fitting the ππ mass spectra data, the values of the transitional chromopolarizability ${\alpha }_{{B}_{c}(2S){B}_{c}}$ and the parameter κ are determined. Our results indicate that the transitional chromopolarizability of the $c\bar{b}$ state lies between the transitional chromopolarizabilities of the $c\bar{c}$ and $b\bar{b}$ states. Additionally, the parameter κ, which accounts for the ππD-wave contribution, is tiny. The findings of this study provide valuable insights into the chromopolarizability of B_c states and have potential applications in the investigation of multiquark hadrons containing $c\bar{b}$.