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在相对论重离子碰撞早期, 会产生一个极强的磁场. 初始碰撞产生的粲偶素会受到磁场的影响, 进而携带磁场的信息. 本文利用磁场下的两体薛定谔方程研究磁场对粲偶素的影响. 利用角动量展开的方法, 数值计算了不同磁场强度下粲夸克偶素的能谱. 采取的方案是把三维波函数展开成不同轨道角动量以及自旋态的叠加, 实际计算过程中发现, 当
$n\leqslant 2$ ,$l\leqslant 7$ 时能很好地满足精确度. 进一步, 哈密顿量可以写成$H=H_0+(qB)^2 H_1+qBP_{{\rm ps},\perp} H_2$ 形式, 其中$H_{0}$ ,$H_{1}$ ,$H_{2}$ 不依赖于B和$P_{{\rm ps},\perp}$ , 因此只要计算出$H_{0}$ ,$H_{1}$ ,$H_{2}$ 就能求出任意B和$P_{{\rm ps},\perp}$ 下的哈密顿量. 这样的数值方法在保证计算精度的同时显著减少了计算量. 计算结果表明随着磁场和总动量的增加, 粲偶素的质量增大, 在磁场强度为$20m_{\pi}^{2}$ , 总动量为$1.8\;{\rm {GeV}}$ 时, 质量的增加量为20%.Heavy ion collisions are an important method to study the quantum chromodynamics. In the early stage of relativistic heavy ion collisions, an extremely strong magnetic field is generated. The magnetic field will induce novel phenomena such as the chiral magnetic effect. However, the magnetic field will decrease rapidly, so it is difficult to measure its effect on the system. Charmonium states which are created by the initial scattering will be affected by the magnetic field and carry the information about it. We use the two-body Schrodinger equation with magnetic field to study the influence of the magnetic field on the charmonium state. The magnetic field is introduced via minimal coupling and its effect breaks the conservation of momentum and the conservation of angular momentum as well. The energy of the charmonium state depends not only on the magnetic field, but also on the momentum of the charmonium, thereby leading the final charmonium yield to be anisotropic. For a constant and homogeneous magnetic field, using the method of angular momentum expansion, we numerically calculate the energy spectra of the charm quark bound states with different magnetic field strengths and total momentum. The method is used to expand the three-dimensional wave function on the basis of different orbital angular momentum and spin states whose wave functions are numerically calculated first. In the actual calculation process, it is found that a good accuracy is achieved when taking$n\leqslant 2$ ,$l\leqslant 7$ . Furthermore, the dependence of the Hamiltonian on the magnetic field and total momentum is analytically determined to be$H=H_0+(qB)^2 H_1+qBP_{{\rm{ps}},\perp} H_2$ . Therefore, only the coefficient matrices$H_{1}$ and$H_{2}$ need to be numerically calculated once and the Hamiltonian with arbitrary magnetic field and momentum can be determined. The inverse power method is then used to find the lowest eigenvalue in the angular momentum space. Such a numerical method significantly reduces the amount of calculation and still ensures the accuracy of the calculation as well. The calculation results show that as the magnetic field and the total momentum increase, the mass of the charm element increases. The increase of the mass can be as large as$20\%$ , when we take$eB = 20 m_{\rm{\pi}}^2$ and$P_{{\rm{ps}}}=1.8 \;{\rm{GeV }}$ , which can be easily achieved in RHIC collisions. Therefore there should exist significant magnetic effect on the$J/\psi$ production in heavy ion collisions.-
Keywords:
- bound state /
- spin /
- heavy particle collision /
- charm quark
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图 1
$qB=10 m_{\pi}^{2}$ ,$P_{\rm kin}=1\; {\rm GeV}$ 时$l=7$ 和$l=6$ 的本征态Fig. 1. The eigenstates of
$l=7$ and$l=6$ at$qB=10 m_{\pi}^{2}$ and$P_{\rm kin}=1 \; {\rm GeV}$ 
图 2
$qB=0 m_{\pi}^{2}$ (红色),$qB=5 m_{\pi}^{2}$ (蓝色),$qB=10 m_{\pi}^{2}$ (黑色),$qB=15 m_{\pi}^{2}$ (橙色),$qB=20 m_{\pi}^{2}$ (绿色)下,$J/\psi^{}$ 粒子的质量随$\langle P_{{\rm kin}, \perp}\rangle$ 的变化图像Fig. 2. The momentum
$\langle P_{{\rm kin}, \perp}\rangle$ dependence of mass m and electric dipole moment$q \langle y \rangle$ for$J/\psi^{\pm}$ in magnet field with$qB = 0$ (dashed black), 5(red), 10 (blue), 15 (violet) and 20 (orange)$m_{\pi}^{2}$ -
[1] Collins J C, Perry M 1975 Phys. Rev. Lett 34 1353
Google Scholar
[2] Shuryak E V 1980 Phys. Rep. 61 71
Google Scholar
[3] Matsui T, Satz H 1986 Phys. Lett. B 178 416
Google Scholar
[4] Guo X Y, Shi S Z, Xu N, Xu Z, Zhuang P F 2015 Phys. Lett. B 751 215
Google Scholar
[5] Crater H W 1994 J. Comput. Phys. 115 470
Google Scholar
[6] Li G S, Zhou K, Chen B Y 2012 Phys. Rev. C 85 044907
Google Scholar
[7] Guo X Y, Shi S Z, Zhuang P F 2012 Phys. Lett. B 718 143
Google Scholar
[8] Asakawa M, Majumder A, Muller B 2010 Phys. Rev. C 81 064912
Google Scholar
[9] Fingberg J 1998 Phys. Lett. B 424 343
Google Scholar
[10] Peskina U, Steinberg M 1998 J. Chem. Phys. 109 704
Google Scholar
[11] 施舒哲 2015 硕士学位论文 (北京: 清华大学物理系)
Shi S Z 2015 M. S. Thesis (Beijing: Tsinghua University) (in Chinese)
[12] Alford J, Strickland M 2013 Phys. Rev. D 88 105017
Google Scholar
[13] Rafelski J, Muller B 1976 Phys. Rev. Lett. 36 517
Google Scholar
[14] 何航 2016 博士学位论文(北京: 清华大学物理系)
He H 2016 Ph.D. Dissertation (Beijing: Tsinghua University) ( in Chinese)
[15] Kharzeev D E, McLerran L D, Warringa H J 2008 Nucl. Phys. A 803 227
Google Scholar
[16] Kharzeev D E 2010 Annals Phys. 325 205
Google Scholar
[17] Gusynin V P, Miransky V A, Shovkovy I A 1994 Phys. Rev. Lett. 73 3499
Google Scholar
[18] Teller E 1937 J. Phys. Chem. 41 109
Google Scholar
[19] Vogt R 2002 Nucl. Phys. A 700 539
Google Scholar
[20] Olver F W, Daniel L W, Ronald F B, Charles W C 2010 NIST Handbook of Mathematical Functions (Cambridge: Cambridge University Press) pp351−382
[21] Kawanai T, Sasaki S 2012 Phys. Rev. D 85 091503
Google Scholar
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