1. Hadronic Paschen-Back Effect in Charmonium
S. Iwasaki 𝐴 , M. Oka 𝐴,𝐵 , K. Suzuki 𝐶 , T. Yoshida 𝐴
Tokyo Inst. of Tech. 𝐴 , JAE A 𝐵 , KEK 𝐶

2. Index Calculation method Results Motivation, Purpose
What’s Paschen-Bach effect?
System we consider
Relative Hamiltonian in MF / Separating the channels
Calculation method
CGEM / Approximation to solve Schrodinger equation / Generalized eigenvalue problem
Results
Overall results we have / 𝐽 𝑧 =±2 / 𝐽 𝑧 =±1 PB effect / 𝐽 𝑧 =±1 spectra & deformation of wave function
Summary, Prospect

3. Motivation
New phenomenon emerges such as chiral magnetic effect; chiral current appears in a magnetic field (MF).
A strong magnetic field is predicted in heavy ion collision(HIC) but strength yet to be measured
Charmonium is quickly produced in QGP in HIC appropriate to measure quickly disappearing MF
P-wave is more sensitive to MF than S-wave
Purpose
cf: WF’s in vauum↓
S-wave→
To calculate spectra and deformed wave functions (WF)
of P-wave charmonia in a strong MF and propose probe of MF
P-wave→ 1

4. What‘s Paschen-Bach effect?
anomalous Zeeman eff. : spilit by 𝑆 𝑧
normal Zeeman eff. : split by 𝐿 𝑧
↑strong MF range : Paschen-Back effect
Wave functions is separated by 𝐿 𝑧 , 𝑆 𝑧 as PB eff. when MF gets stronger than the scale of LS coupling.
𝐿 𝑧 , 𝑆 𝑧 : 0, ,0
and are mixed in vacuum by 1:1 but
they are separated in strong MF
easy to observe deformation of overall WF
We maybe can measure MF by the deformation
Zeeman effect

5. Relative Hamiltonian in MF
𝜌 2 = 𝑟 2 − 𝑧 2 , 𝜇= 𝑚 𝑐 2 :reduced mass, 𝑩= 0,0,𝐵 , 𝑺= 𝑺 𝟏 + 𝑺 2
q : electric charge of charm quark
parameters: 𝜎 = Ge V 2 , 𝛼 𝑠 =0.5461, Λ= GeV, 𝑚 𝑐 = GeV
Phys. Rev. D 72, (2005) Barnes, Godfrey, Swanson 2

6. Separating the channels
𝑉 𝐿𝑆 ∝𝑳⋅𝑺= 𝐿 𝑧 𝑆 𝑧 𝐿 + 𝑆 − + 𝐿 − 𝑆 +
LS term mixes the states with different 𝐿 𝑧 , 𝑆 𝑧
𝑉 m.m. =− 𝑖=1 2 𝝁 𝑖 ⋅𝑩 ∝ 𝑺 𝟏𝒛 − 𝑺 𝟐𝒛
Magnetic moment mixes total 𝑆
But still … 𝐽 𝑧 is a good quantum number. States with different 𝐽 𝑧 do not mix we can separately calculate by 𝐽 𝑧 3

7. Cylindrical Gaussian Expansion Method (CGEM) for P-wave
GEM: to use spherical Gaussian basis: Ψ spherical (𝑟)=𝑁 𝑒 −𝛼 𝑟 2 Spherical symmetry Violates by MF Previous studies: cylindrically symmetric one for S-wave: Ψ cylindrical S (𝜌,𝑧,𝜙)=𝑁 𝑒 −𝛽 𝜌 2 −𝛾 𝑧 2 Furthermore…… This study: cylindrically symmetric one for P-wave: Ψ cyl P 𝜌,𝑧,𝜙; 𝐿 𝑧 = 𝑁 𝑧 𝑒 −𝛽 𝜌 2 −𝛾 𝑧 2 for 𝐿 𝑧 =0 𝑁 ∓𝜌 𝑒 ±𝑖𝜙 𝑒 −𝛽 𝜌 2 −𝛾 𝑧 2 for L z =±1
※ 𝛽,𝛾 : range parameters, spin functions omitted, 𝑁: normalization factor,
cf.) E. Hiyama, Y.Kino and M. Kamimura, Prog.Part.Nucl.Phys (2003).
K. Suzuki and T.Yoshida, Phys.Rev.D93, (2016). 7

8. Approximation to solve Schodinger equation
To represent Hamiltonian as matrix form with basis Range parameters: 𝛽 1 , ⋯, 𝛽 𝑁 ; 𝛾 1 , ⋯, 𝛾 𝑁 connect them with geometric series 𝛽 𝑛 = 𝛽 1 𝛽 𝑁 𝛽 1 𝑛−1 𝑁−1 , 𝛾 𝑛 = 𝛾 1 𝛾 𝑁 𝛾 1 𝑛−1 𝑁−1 𝑛=1,⋯,𝑁 We have 𝑁 bases: Ψ 𝑛 𝜌, 𝜙,𝑧 = 𝑁 𝑛 𝑟 𝑌 𝐿 𝑧 1 𝜃,𝜙 𝑒 − 𝛽 𝑛 𝜌 2 − 𝛾 𝑛 𝑧 2 , 𝑛=1, ⋯,𝑁 8

9. Generalized eigenvalue problem
To rewrite wave function, Hamiltonian and norm with the bases Ψ 𝑛 𝜌, 𝑧, 𝜙 : Ψ 𝜌, 𝑧, 𝜙 = 𝑛=1 𝑁 𝑐 𝑛 Ψ 𝑛 𝜌, 𝑧, 𝜙 𝐻 𝑖𝑗 = Ψ 𝑖 ∗ 𝐻 Ψ j d𝑉 , 𝑁 𝑖𝑗 = Ψ 𝑖 ∗ Ψ j d𝑉 𝐻 𝑖𝑗 𝑐 𝑗 =𝐸 𝑁 𝑖𝑗 𝑐 𝑗 𝐻𝒄=𝐸𝑁𝒄 generalized eigenvalue problem 9

10. Overall results we have ( 𝐽 𝑧 =±2,±1)
𝒉 𝒄
𝝌 𝒄𝟎
𝝌 𝒄𝟏
𝝌 𝒄𝟐 ±2
fin. ±1
not
yet
↓finished til here
We’ve calculated
matrix elements
In vacuum, each particle above is degenerated with different 𝐽 𝑧 channels in each column. But we can calculate on fixed 𝐽 𝑧 channels in each row.
Summation of all wave functions should be spherically symmetric
顔の意味 10

11. Results ( J z =±2) 1/2 mass mass 𝑒𝐵=0~5.0 GeV 2 ↑ 𝑒𝐵=0~1.0 GeV 2 ↑
SI et al. in preparation
𝐻 rel ⊃ 𝑒 𝑐 2 𝐵 2 8𝜇 𝜌 2
mass
mass
𝑒𝐵=0~5.0 GeV 2 ↑
𝑒𝐵=0~1.0 GeV 2 ↑
Impose MF mass increased 11

12. Results ( J z =±2) 2/2 3D GIF of WF↑ Impose MF
SI et al. in preparation
Ψ 2
3D GIF of WF↑
Impose MF
wave function shrinks along 𝜌
CGEM can produce reasonable results also for P-wave
cf: WF
in vacuum 12

13. Results ( J z =±1, PB effect)
・WF’s start to deform or get anisotropic from 𝑒𝐵~0.05Ge V 2
・ MF in LHC is predicted that 𝑒𝐵~0.3Ge V Deformations of wave functions are detectable
It is possible to estimate MF
from deformation of wave functions

14. Results ( J z =±1, spectra & deformation of WF)
3rd:
2nd:
PB effect
MF in LHC
1st :

15. Summary Prospect Motivation: to measure MF in HIC
Object: P-wave charmonium in strong MF
We prepared bases for P-wave: Ψ cyl 𝑃 𝜌,𝑧,𝜙 =𝑁𝑟 𝑌 𝐿 𝑧 𝐿=1 𝜃,𝜙 𝑒 −𝛽 𝜌 2 −𝛾 𝑧 2
We’ve gotten results on 𝐽 𝑧 =±2,±1
Prospect
To finish calculation on 𝐽 𝑧 =0,
To discuss anisotropic decay from anisotropic wave function 14

16. Slide in case：PB effect on 𝐽 𝑧 =±1, 3rd

17. Slide in case ：results on 𝐽 𝑧 =±1 0.0, 0.1, …, 1.0 GeV のGIF(1st~3rd)

18. Slide in case ：results on 𝐽 𝑧 =±1 GIF of 0.0, 0.1, …, 1.0 GeV (4th)