A review of quarkonia under strong magnetic fields

Abstract

We review the properties of quarkonia under strong magnetic fields. The main phenomena are (i) mixing between different spin eigenstates, (ii) quark Landau levels and deformation of wave function, (iii) modification of $\bar{Q}Q$ potential, and (iv) the motional Stark effect. For theoretical approaches, we review (i) constituent quark models, (ii) effective Lagrangians, (iii) QCD sum rules, and (iv) holographic approaches.

Appendices

Appendix A: S-wave charmonia

In this Appendix, we briefly review the mass spectra of S-wave charmonia and wave function deformation as a characteristic phenomenon of quarkonia in a magnetic field. In Refs. [19, 20], the authors evaluated the mass spectra and wave functions of quarkonia from the constituent quark model under a constant (or static) magnetic field [17] and the cylindrical Gaussian expansion method (CGEM) [19, 20] which is a numerical approach to solve the anisotropic few-body systems.

Fig. 6

Mass spectra and probability densities of the S-wave charmonia in a magnetic field [20]. Upper: $S_z=\pm \,1$ states from $J/\psi$ and $\psi (2S)$. Lower: $S_z=0$ states from $\eta _c(1S)$, $J/\psi$, $\eta _c(2S)$, and $\psi (2S)$

In Fig. 6, we show the mass spectra and probability densities (namely, the squares of wave functions) on the $\rho$-z plane. The S-wave charmonia has the four states below the two-meson ($D{\bar{D}}$) threshold: the spin-singlet $\eta _c(1S)$-$\eta _c(2S)$ and the spin-triplet $J/\psi$-$\psi (2S)$. The upper panel of Fig. 6 shows the magnetic-field dependence of the masses of the two lowest vector charmonia, $J/\psi$ (red points) and $\psi (2S)$ (blue points), with $S_z=\pm \,1$. These states do not mix with different spin eigenstates. With increasing magnetic field, the masses increase by quark Landau levels (or harmonic-oscillator-type potential in the $\rho$ direction). At the same time, their wave functions are squeezed on the $\rho$ plane, and then, they are stretched along the z axis, as shown in small windows. Here, the row of three small windows shows the probability densities at $eB = 0$, 1.0, and $10.0 \ \mathrm {GeV}^2$. For example, at $eB = 0$ (the left), we can obtain the spherical (or isotropic) wave functions. At $eB = 1.0 \ \mathrm {GeV}^2$ (the middle), the 1S state is not significantly modified, while the 2S state is drastically deformed and its form is the first excitation not in the radial direction but in the z direction. At $eB = 10.0 \ \mathrm {GeV}^2$ (the right), the 1S state becomes a “cigar-shaped” wave function, and 2S state becomes “rod-shaped”. Note that these deformation is equivalent to mixing between the S-wave and higher partial waves with an even number of orbital angular momentum ($L=2,4,\ldots$). In this sense, radial eigenstates such as 1S and 2S are no longer “true” eigenstates, and the true eigenstates in finite magnetic fields are represented as mixing states between different partial waves, which reflects the fact that a magnetic field violates the spherical symmetry, and the orbital angular momentum is no longer a good quantum number.

As another interesting behavior in the upper panel of Fig. 6, one can see a linear increase of the charmonium masse in the strong-magnetic-field region. This behavior can be easily understood by the picture of quark Landau levels. The energies of nonrelativistic Landau levels for a one-body quark with a mass m are represented as $(|qB|/m)(n+1/2)$ with an integer n. In the strong-field limit, this energy is dominated by the lowest Landau level ($n=0$). By substituting $|qB|=(2/3)|eB|$ and $m_c=1.8$ GeV into this expression, the sum of the mass shifts of the two quarks is estimated to be 0.37|eB| GeV, which is consistent with the slope obtained from the numerical results in Fig. 6. Thus, the magnetic behavior of $S_z=\pm \,1$ states in the strong-field limit can be well approximated by a picture with single quarks. In addition, such a rough estimate seems to be slightly larger than the slope of the 1S state. This tendency may be caused by an attractive mass shift from the Coulomb potential.

The lower panel of Fig. 6 shows the mass spectra of $S_z=0$ states, where the spin-singlet eigenstates are mixed with the $S_z=0$ components of the spin-triplet states. Due to the level repulsion between the two states, the mass of the first state (red points) starting from $\eta _c(1S)$ decreases in nonzero magnetic fields, while that of the second state (blue points) starting from $J/\psi$ increases. At $eB = 1.0{-}1.1 \ \mathrm {GeV}^2$, the second state approaches the third state (magenta points) from $\eta _c(2S)$. After the crossing, the second state becomes a 2S-like wave function, and the third state does 1S-like. Similarly, we find the crossing between the third and fourth states at $eB = 1.8$-$1.9 \ \mathrm {GeV}^2$ and the crossing between the fourth and fifth states at $eB = 0.6$-$0.7 \ \mathrm {GeV}^2$. Thus, the wave functions of excited states are more sensitive to a magnetic field than the ground states. We emphasize that the position of a crossing point between two states is useful as a visible guideline to determine the mass spectrum under a magnetic field.

Appendix B: S-wave bottomonia

Next, we compare the magnetic responses of charmonia and bottomonia. Significant differences between the charm and bottom quarks are (i) the electric charges and (ii) the quark masses.¹ The absolute value of the electric charge of bottom quarks is $|q_b| = (1/3)|e|$ which is two times smaller than that of charm quarks $|q_c| = (2/3)|e|$. The masses of bottom quarks are approximately three times heavier than those of charm quarks. Hence, we can expect that bottomonia are less sensitive to a magnetic field than charmonia.

Fig. 7

Mass spectra and probability densities of the S-wave bottomonia in a magnetic field [20]. Upper: $S_z=\pm \,1$ states from $\varUpsilon (1S)$, $\varUpsilon (2S)$, and $\varUpsilon (3S)$. Lower: $S_z=0$ states from $\eta _b(1S)$, $\varUpsilon (1S)$, $\eta _b(2S)$, $\varUpsilon (2S)$, $\eta _b(3S)$, and $\varUpsilon (3S)$

In Fig. 7, we show the mass spectra of bottmonia, obtained in Ref. [20]. The S-wave bottomonia have six states below the two-meson ($B{\bar{B}}$) threshold: the spin-singlet $\eta _b(1S)$-$\eta _b(2S)$-$\eta _b(3S)$ and spin-triplet $\varUpsilon (1S)$-$\varUpsilon (2S)$-$\varUpsilon (3S)$. The upper panel of Fig. 7 shows the spectra of $S_z=\pm \,1$ states. At $eB = 1.0 \ \mathrm {GeV}^2$, the wave functions of 1S, 2S, and 3S states are almost spherical, which is a situation different from the charmonum spectrum, where $\psi (2S)$ is significantly deformed. Even at $eB = 10.0 \ \mathrm {GeV}^2$, the wave function of the ground state is still spherical. The insensitivity of bottomonia to magnetic fields can be seen also in the magnitude of the mass shift.

The lower panel of Fig. 7 shows the mass spectra of $S_z=0$ states. With the same mechanism as the charmonia, one can see the level crossing between different levels. The crossing point between the second and third states is at $eB = 5.5 \ \mathrm {GeV}^2$, while the corresponding point in the charmonium spectrum is $eB = 1.0{-}1.1 \ \mathrm {GeV}^2$. Similarly, one can see that $eB = 8.5 \ \mathrm {GeV}^2$ for the third and fourth states, $eB = 3.0 \ \mathrm {GeV}^2$ for the fourth and fifth states, and $eB = 5.0 \ \mathrm {GeV}^2$ for the fifth and sixth states.

Appendix C: P-wave charmonia and hadronic Paschen–Back effect

In this Appendix, we briefly review the hadronic Paschen–Back effect (HPBE) [21] which is analogous to a phenomenon observed in atomic physics, the Paschen–Back effect [61]. The HPBE is realized by interplay between an orbital angular momentum of hadrons and a magnetic field. The HPBE occurs in various hadrons with a finite orbital angular momentum, but here we focus on the P-wave charmonia: the spin-singlet $h_c$ ($^1 \! P_1$) and spin-triplet $\chi _{c0}$ ($^3 \! P_0$), $\chi _{c1}$ ($^3 \! P_1$), and $\chi _{c2}$ ($^3 \! P_2$), where we used a notation $^{2S+1} \! P_J$ with the total angular momentum $J=L+S$, orbital angular momentum L, and spin angular momentum S. In zero magnetic field, $L_z$ and $S_z$ are not conserved by the existence of the LS and tensor coupling, and the good quantum numbers are J, L, and S.

When a magnetic field is switched on, only the z component of the total angular momentum, $J_z=L_z+ S_z$, is a conserved quantity. In addition, if the magnetic field becomes stronger than the scale of the spin-orbit splitting, $L_z$ and $S_z$ are approximately good quantum numbers. Then, instead of the $|{J;LS}\rangle$ bases used in weak fields, it may be convenient to use a new basis $|{J_z;L_zS_z}\rangle$ in a strong field. We call such a strong magnetic-field region the PB region (or the PB limit for the strong-field limit).² In particular, the wave functions of P-wave charmonia in the PB region are approximately expressed as

$$\begin{aligned} \varPsi _{L_z;S_{1z}S_{2z}}(\rho ,z,\phi ) = \varPhi _{L_z}(\rho , z) Y_{1 L_z}(\theta ,\phi )\chi (S_{1z}, S_{2z}), \end{aligned}$$

(C.1)

where $(\rho ,z,\phi )$ is the cylindrical coordinate, and $\tan \theta =\rho /z$. This configuration is composed of two parts: (i) the spatial part and (ii) spin part. The spatial wave function is defined by $\varPhi (\rho ,z)$ and the spherical harmonics $Y_{1L_z}(\theta ,\phi )$.³ The spin wave function is defined as $\chi (S_{1z}, S_{2z})$, where $S_{1z}$ and $S_{2z}$ are the third components of the spin of the charm quarks.

Fig. 8

Mass spectra of the P-wave charmonia in a magnetic field [21]. Upper: $J_z=\pm \,2$ states from $\chi _{c2}(1P)$ and $\chi _{c2}(2P)$ and $J_z=\pm \,1$ states from $\chi _{c1}(1P)$, $h_c(1P)$, $\chi _{c2}(1P)$, and $\chi _{c1}(2P)$. Lower: $J_z=0$ states from $\chi _{c0}(1P)$, $\chi _{c1}(1P)$, $h_c(1P)$, $\chi _{c2}(1P)$, and $\chi _{c0}(2P)$

The mass spectra of P-wave charmonia in a magnetic field is shown in Fig. 8. The upper panel shows both the $J_z = \pm \,2$ and $J_z = \pm \,1$ states, and the lower panel shows $J_z = 0$. The $J_z = \pm \,2$ states starting from $\chi _{c2}$ are not mixed with other states, so that their masses continue to increase, which is similar to the $S_z=\pm \, 1$ states of the S-waves. The $J_z = \pm \,1,0$ states exhibit the mixing between spin eigenstates, which is similar to the $S_z=0$ states of the S-waves. As a result of the mixing and level repulsion, the masses of the lowest states decrease as the magnetic field increases, while higher states becomes heavier. We find the level crossings between the 1P-like state and 2P-like state at $eB = 0.6 \ \mathrm {GeV}^2$ for $J_z=\pm \,1$, and at $eB = 0.5$ and $0.8\ \mathrm {GeV}^2$ for $J_z=0$. Thus, the magnetic field for the crossing between 1P and 2P states is smaller than that for the corresponding 1S and 2S states.

Fig. 9

Probability densities of wave functions of P-wave charmonia for $J_z=\pm \,1$ in magnetic fields [21]. The vertical axis is $|\varPsi (\rho , z, \phi )|^2$, and the horizontal plane is represented by the $\rho$ and z axes, where $\rho$ (z) is the spatial direction perpendicular (parallel) to the magnetic field

In Fig. 9, we show the probability densities on the $\rho$-z plane, defined as $|\varPsi (\rho , z, \phi )|^2$ for the $J_z=\pm \,1$ states. At $eB=0$ [(a), (d), and (g)], the “1st”, “2nd”, and “3rd” wave functions correspond to $\chi _{c1}, h_c$, and $\chi _{c2}$ states, respectively. With increasing magnetic field, the wave functions are gradually deformed. At $eB=0.1 \ \mathrm {GeV}^2$ [(c), (f), and (i)], one can see probability densities characterized by $|L_z|$. Note that the Hamiltonian has inversion symmetry along the z-axis, so that we see the same physics for both the $L_z=\pm \, 1$ states. For example, the basis function with $L_z=\pm \,1$ is characterized by the factor of $r Y_{1\,\pm \,1}(\theta , \phi )\propto r\cos \theta e^{\pm i\phi }=\rho e^{\pm i\phi }$. This factor means that the magnitude of the wave function on the z axis is zero [see (c) and (i)]. On the other hand, the basis function with $L_z=0$ is based on the factor of $r Y_{10}(\theta , \phi ) \propto r\sin \theta =z$. This means that the wave function amplitude on the $\rho$ axis is zero [see (f)]. Thus, in the PB limit, the spatial wave function of an eigenstate is purified to either wave function with $L_z=\pm \,1$ or $L_z=0$. The magnetic field of $eB=0.1 \ \mathrm {GeV}^2$ can be well approximated as the PB limit, and thus one can observe drastic deformation by the HPBE even in small magnetic fields.

We emphasize that the deformation mechanism by the HPBE is quite different from that in S-wave quarkonia. The deformation in S-waves is induced by quark Landau levels, or equivalently the mixing between an S-wave and higher partial waves with an even orbital angular momentum, $L=2,4,\ldots$. On the other hand, the origin of the HPBE is “resolving” the mixed states originating from the LS coupling between two quarks. Therefore, the scale of magnetic fields that induces the HPBE corresponds to the LS splitting of hadron masses, which leads to drastic deformation in relatively smaller magnetic field than that for S-waves.