LFTC-24-01/84 $B_{c}^{\pm}$-¹²C states and detailed study of momentum space method for $\Upsilon$- and $\eta_{b}$-nucleus bound states

Studies of hadron properties under extreme conditions indicate that the Lorentz scalar effective masses of mesons are expected to decrease in a nuclear medium, as a consequence of partial restoration of chiral symmetry Krein:2010vp ; Tsushima:2011kh ; Ko:1992tp ; Asakawa:1992ht . This negative effective mass shift (Lorentz scalar) of the meson can be regarded as an attractive Lorentz scalar potential, and when the sum of the Lorentz scalar and the Lorentz vector potentials (total potential in a nonrelativistic sense) is sufficiently attractive, the mesons can be bound to atomic nuclei. In the case of heavy quarkonia (no vector potentials arise in the lowest order), charmonium-nucleus systems were proposed in 1989 Brodsky:1989jd , which led to various subsequent studies Krein:2010vp ; Tsushima:2011kh ; Hosaka:2016ypm ; Metag:2017yuh ; Krein:2017usp ; Lee:2000csl ; Krein:2013rha ; Klingl:1998sr ; Hayashigaki:1998ey ; Kumar:2010hs ; Belyaev:2006vn ; Yokota:2013sfa ; Peskin:1979va ; Kharzeev:1995ij ; Kaidalov:1992hd ; Luke:1992tm ; deTeramond:1997ny ; Brodsky:1997gh ; Sibirtsev:2005ex ; Voloshin:2007dx ; TarrusCastella:2018php ; Cobos-Martinez:2020ynh and lattice QCD calculations Yokokawa:2006td ; Kawanai:2010ev ; Skerbis:2018lew on these states. Recent lattice results show also a possibility of $\phi$-nucleon (N) bound states Chizzali:2022pjd , although $\phi$ is not the heavy quarkonium. Furthermore, by the developed Faddeev three-body approach using the lattice extracted $\phi$-N potential, a possible $\phi$-NN bound state is predicted Etminan:2024vkv . We have studied further the bottomonium-nucleus systems Krein:2010vp ; Cobos-Martinez:2020ynh , and predicted that the $\Upsilon$ and $\eta_{b}$ mesons can form strong nuclear bound states with various nuclei Zeminiani:2020aho ; Zeminiani:2021vaq ; Zeminiani:2021xvw ; Cobos-Martinez:2022fmt , provided that they are produced inside a nucleus with very low relative momenta to the nucleus.

The Okubo-Zweig-Iizuka (OZI) rule dictates the suppression of the light hadron exchange in the heavy quarkonium-nucleus interaction, then the interaction must occur primarily by multigluon exchange, namely, a QCD van der Waals type of interaction Brodsky:1997gh . Looking at different possibilities, we consider an alternative (effective) mechanism for the quarkonium-nucleus interaction. We estimated the charmonia ($J/\Psi$ and $\eta_{c}$) and bottomonia ($\Upsilon$ and $\eta_{b}$) mass shifts by the enhanced in-medium self-energies via the excitations of intermediate-state hadrons with light quarks Cobos-Martinez:2020ynh .

The focus of the present study is firstly the bottomonia, for which the self-energies were calculated including only the minimal loop contributions for the $\Upsilon$ and $\eta_{b}$ mesons, namely, the $BB$ loop for the $\Upsilon$, and the $BB^{*}$ loop for the $\eta_{b}$ Zeminiani:2020aho . The calculations were performed neglecting any possible imaginary part of the self-energies. Secondly, we study the $B_{c}$-nucleus bound states for the first time, with and without the Coulomb force.

To obtain the bound-state energies and the corresponding bound meson state wave functions of the meson-nucleus systems in the coordinate space, we solve the Klein-Gordon equation in momentum space, where for the $\Upsilon$ case, the Proca equation is approximated and reduced to the Klein-Gordon equation assuming the longitudinal and transverse modes are nearly equal with low momentum. It is very convenient to deal with the kinetic term in the Klein-Gordon equation in momentum space, and the search for the bound-state energies is also much more simple than performing in coordinate space. This can be done by solving the differential wave equation as an eigenvalue equation, and using a method for finding selected eigenvalues and eigenfunctions (eigen vectors). This makes it easy to extend the calculation for complex Hamiltonians, e.g., when including the meson widths and/or imaginary potentials. In coordinate space, dealing with complex eigenvalues is tricky, as it involves searching for a convergence point in the real and imaginary parts in energy plane, as well as the real and imaginary parts of the coordinate space wave functions.

The treatment in momentum space requires the nuclear potentials to be transformed from coordinate space to momentum space, and then decomposed into partial waves. For this purpose we compare three different methods, namely, (i) the spherical Bessel transform of the numerically obtained original potential in coordinate space, (ii) the partial wave decomposition of the Fourier transform of the Woods-Saxon approximated form for the original potential, and (iii) the spherical Bessel transform of the Woods-Saxon approximation of the numerically obtained original potential.

The bound-state energies calculated for $\Upsilon$-¹²C, $\Upsilon$-⁴He, $\eta_{b}$-¹²C and $\eta_{b}$-⁴He systems reported in Cobos-Martinez:2022fmt are reproduced here and compared with new values obtained through the different methods mentioned above, for which we now present for the first time the wave functions in coordinate space associated with each energy level of the bound state. In this article we also include the first calculation of the $B_{c}$-nucleus bound states, for $B_{c}$-⁴He and $B_{c}$-¹²C with no Coulomb interaction, followed by a realistic study of $B_{c}^{\pm}$-¹²C including the Coulomb potentials, which are obtained self-consistently in the ¹²C nucleus when the $B_{c}^{\pm}$ mesons are absent, within the quark-meson coupling (QMC) model treatment for finite nuclei Guichon:1987jp ; Guichon:1995ue ; Tsushima:2002cc ; Tsushima:1997df ; Saito:1996sf ; Saito:2005rv .

This article is organized as follows. In Sec. II we discuss different methods used to solve the Klein-Gordon equation in momentum space. The numerical procedure of the calculation is explained in Sec. III, and the results are presented in Sec. IV. An initial study of the $B_{c}$-⁴He (no Coulomb potential) and $B_{c}^{\pm}$-¹²C bound states (with and without the Coulomb potentials) is made in Sec. V. The summary and conclusions are given in Sec. VI.

The meson-nucleus bound states are studied in momentum space by solving the Klein-Gordon (K.G.) equation. While the K.G. equation is solved for the pseudoscalar $\eta_{b}$ meson, we also solve the K.G. equation for the spin-1 $\Upsilon$ meson. This is based on the approximation that the $\Upsilon$-nucleus relative momentum is nearly zero. In this approximation, the transverse and longitudinal components in the Proca equation are expected to be very similar, which allows us to reduce it to solving a single-component K.G. equation, as practiced in Ref. Zeminiani:2021vaq ; Zeminiani:2021xvw ; Cobos-Martinez:2022fmt .

The $\eta_{b}$- and $\Upsilon$-nuclear potentials were previously calculated in coordinate space in Ref. Cobos-Martinez:2022fmt , in which they were transformed into the momentum-space by one particular method for that study (“Woods-Saxon Fourier transform” method to be explained later). For the purpose of the present study, we will transform the coordinate space potentials using a few methods, including the same method used in the past.

The K.G. equation will be solved for each value of the angular momentum $\ell$ of the meson-nucleus system to calculate the bound-state energies and to get the momentum space eigen wave function corresponding to each bound state energy level of the system. This will require a partial wave decomposition for either the momentum-space or the coordinate-space potentials. We will compare three different methods to obtain the partial-wave decomposed momentum-space potentials.

The partial wave expansion of Eq. (2) is given by

The discretized Eq. (8) can be written in matrix form as

where $\hat{\psi}_{i}=\psi_{n\ell}(P_{i})$. The matrix elements of the Hamiltonian are

with $NP$ being the number of points for the momentum grid, and $P_{i}$ and $w_{j}$ are respectively the $i$-th momentum point and $j$-th weight.

Equation (9) can be solved as an eigenvalue equation using iterative methods, such as the inverse iteration Heddle:1985dh . It is a method for finding selected eigenvalues and eigenfunctions, that has the advantage of only requiring the inversion of an $NP\times NP$ matrix. The equation to be solved is of the type

We begin with some guess value $\tau_{n}$, and constructing the operator

that contains the Hamiltonian of Eq. (11). Next, we construct a state $\ket{\chi^{0}_{n}}$ that can be expressed using the complete set of eigenfunctions $\ket{\psi_{n}}$

which is multiplied $N$ times by the operator $B_{n}$, resulting in

For nondegenerate eigenstates and for a proper guess $\tau_{n}$, $n^{\prime}=n$ will rapidly take over the sum. Then, for a large enough N

which can be written in a momentum space calculation as

To facilitate the iterative numerical calculation, $\chi^{N}_{n}(P)$ is divided by its element with maximum modulus, $\tilde{\chi}^{N}_{n}(P)=\chi^{N}_{n}(P)/|\chi^{N}_{n}(P)|_{\text{max}}$, and $\chi^{N+1}_{n}(P)$ is redefined as

and for sufficiently large $N$,

The eigenvalue is then given by

In the same way, the eigenvector $\psi_{n}(P)$ is determined by Eq. (18). The obtained eigenvalue and eigenvector are the system’s eigenenergy and momentum space wave function, respectively, for each $n$ and $\ell$.

After establishing and confirming the momentum-space methods by three different ways, as well as to obtain the wave functions in the coordinate space, we now proceed to an initial study of the $B_{c}^{\pm}$-⁴He and $B_{c}^{\pm}$-¹²C bound states. The mass shift of $B_{c}$ meson in symmetric nuclear matter was calculated in Ref. Zeminiani:2023gqc , and we obtain the $B_{c}$-nucleus strong potential using a local density approximation. For a realistic calculation, the Coulomb potentials should enter for $B_{c}^{\pm}$-nucleus potentials, but they are absent in the $\Upsilon$ and $\eta_{b}$ cases. First, we include only the strong interaction part of the $B_{c}^{\pm}$-⁴He and $B_{c}^{\pm}$-¹²C potentials, without the Coulomb potentials. Second, including the Coulomb potentials, we study the $B_{c}^{\pm}$-¹²C bound states, and focus on the role of the Coulomb potentials. This is because the ¹²C nucleus is constructed self-consistently by the QMC model (relativistic, quark-based nuclear shell model), but this is not done for the ⁴He nucleus.

The strong interaction potentials for $B_{c}$ are presented in Fig. 15. These are calculated in the same manner as those for the $\Upsilon$ and $\eta_{b}$ cases. The parametrization of the fitted Woods-Saxon form of these potentials is given in Table. 4. We solve the K.G. equation with the three methods mentioned previously and obtain the bound-state energies, which we present in Table 5. The corresponding wave functions are shown in Figs. 16 to 21.

We now study the more realistic case for the $B_{c}^{\pm}$-¹²C, where we include the Coulomb interaction. Including the Coulomb potential $V_{c}$ in Eq. (1), it becomes

and the Hamiltonian for the numerical calculation in Eq. (10) now becomes

The Coulomb potentials in the ¹²C nucleus (nucleon density distribution and Coulomb mean field) is obtained within the QMC model Saito:1996sf self-consistently. We show the Coulomb potentials side-by-side with the nuclear potentials for this nucleus in Fig. 22. Since the mean-field approximation of the QMC model is not good for a light nucleus such as the ⁴He, we do not investigate the $B_{c}^{\pm}$-⁴He system with the Coulomb potentials in this initial study. In the near future, we plan to include the Coulomb potentials for the $B_{c}^{\pm}$-⁴He system with a different approach, although the effects are expected to be small.

The treatment of the Coulomb potential in momentum space is known to be difficult due to the logarithmic singularity that appears in its partial wave decomposition, requiring the use of some regularization technique, such as the Lande subtraction method Heddle:1985dh ; Kwan:1978zh . In this calculation, however, the treatment is much more simple. The reason is that we can calculate the $B_{c}^{\pm}$-¹²C Coulomb potentials in coordinate space using the self-consistent mean Coulomb field in the ¹²C nucleus by neglecting the feedback of the Coulomb force from $B_{c}^{\pm}$ (which should give very small effects). Thus, we simply get the direct partial wave decomposition of the Coulomb potentials in momentum space according to the methods already explained in Sec. II.3.

We solve Eq. (22) using the direct double spherical Bessel transform for the following cases: (i) Nuclear and attractive Coulomb potentials ($B_{c}^{-}$-¹²C), (ii) nuclear and repulsive Coulomb potentials ($B_{c}^{+}$-¹²C)), (iii) only nuclear potentials, (iv) only Coulomb potentials, and (v) the cases of the term $2EV_{c}\to 0$ in the right hand side of Eq. (22), in the right hand of the expansion, $(E-V_{c})^{2}=E^{2}+V_{c}^{2}-2EV_{c}$ (see Eq. (23) with $E\leftrightarrow\varepsilon_{n\ell}$). The results are presented in Table 6. We make this comparison to see the interference effect between the dominant nuclear and Coulomb potentials, which arises naturally for the relativistic K.G. equation case but not for the lowest order Schrödinger equation. Solving Eq. (22) for $n=1$, $\ell=0$ and the Coulomb potential alone, we get energies, $\varepsilon_{n\ell}=\varepsilon_{10}\approx\pm 1.4$ MeV for $\pm|V_{c}|$, with a small asymmetry between the cases of the attractive and repulsive Coulomb potentials, that arises from the $V^{2}_{c}$ term and diminishes the energy $E$ in both cases. Naively one could expect that the bound-state energies obtained including the nuclear and attractive (or repulsive) Coulomb potentials to be around 1.4 MeV deeper (or shallower) than the case of the nuclear potential alone, however, in fact we obtain more than 3 to 4 MeV difference for the 1s state. This happens because the nuclear potential dominates $E$ in the term $2EV_{c}$ that appears when we include the Coulomb potential. The ”interference” between the nuclear and Coulomb potentials is responsible for this larger difference. By making the term $2EV_{c}\to 0$, we nearly recover the energies found for the case of only nuclear interaction, aside from a small difference due to the $V^{2}_{c}$ term. This shows that when including the Coulomb interaction, the nuclear potential dominates $E$, as one can expect. Although the Coulomb effect is small, such behavior described above cannot be seen in a naive treatment using the Schrödinger equation without including up to the $V_{c}^{2}$ term.

The coordinate space wave functions for the 1s and 1p states for the $B^{\pm}_{c}$-¹²C system are presented in Fig. 23.

By solving the Klein-Gordon equation in momentum space, we have calculated the single-particle energies and the corresponding coordinate-space wave functions for the quarkonium-nucleus systems, for the $\Upsilon$ and $\eta_{b}$ mesons and the ⁴He and ¹²C nuclei. The meson-nucleus potentials have been calculated from the mass shift of the mesons in nuclear matter using a local density approximation without the effects of the meson widths. The results depend on the cutoff parameter $\Lambda$ values, introduced in the regularization of the meson self-energies.

We have studied and compared in detail the momentum space methods for solving the Klein-Gordon equation, as well as the resultant coordinate space wave functions by three methods to construct the momentum-space potentials: (i) double spherical Bessel transform of the numerically obtained coordinate-space original potential, (ii) Fourier transform of the fitted Woods-Saxon form potential in the coordinate space, and (iii) double spherical Bessel transform of the fitted Woods-Saxon form potential in the coordinate space.

When using the double spherical Bessel transform, the choice of a suitable momentum grid is crucial to guarantee the corresponding proper coordinate space regions where the potential is relevant.

It is important to use the same momentum grid when transforming the momentum-space wave function into the coordinate space in order to obtain a stable, smooth and proper wave function. Furthermore, there is no need to interpolate.

The double spherical Bessel transform of the fitted Woods-Saxon form potential in the coordinate-space yields nearly the same results for the bound-state energies and coordinate space wave functions. Thus, we can provide the corresponding parameter sets of the fitted Woods-Saxon form potentials, instead of the original meson-nucleus potentials obtained numerically.

We also have calculated for the first time the $B_{c}$-⁴He and $B_{c}$-¹²C bound-state energies and coordinate space wave functions, with only the strong interaction potentials first, and then, for $B_{c}^{\pm}$-¹²C system, we have studied including the realistic Coulomb potentials, and analyzed the ”interference” effect between the nuclear and Coulomb potentials.

In the future, more realistic studies can be made for the $\Upsilon$-, $\eta_{b}$- and $B_{c}^{\pm}$-nucleus systems for various nuclei, including the possible imaginary part of the potentials for the $\Upsilon$-, $\eta_{b}$- and $B_{c}^{\pm}$-nucleus bound states. In particular, the realistic Coulomb potentials will be included for the $B_{c}^{\pm}$- and $B_{c}^{*\pm}$-nucleus systems.