1. Spin precession note T-BMT equation, Spin motion study (Analytic and GEANT4) – g-2 motion in unique M-field in “J-PARC g-2 case”, – EDM motion for J-PARC g-2 case, Spin-dependent muon decay revisit Expected positron time spectra for EDM case 2010/04/14 Hiromi Iinuma 1

2. Spin equation (T-BMT equation + EDM) Our case: 2

3. Spin equation (T-BMT equation + EDM) Our case: 3

4. Spin motion study 4

5. G-2 motion in M-field (1) nsec 5

6. G-2 motion in M-field (2) Precession : nsec  sec Sx component in the  rest frame 6

7. S=(Sx, Sy, Sz)  =(momvx, momvy, momvz) S  =cos(  a t) nsec  sec GEANT4 check: g-2 motion 7

8. JPARC EDM (Analytically-approach) s x and s y components are the sum of T-BMT and EDM effects Initially s x =s y =0  later s x  0, s y  0  J-PARC Initially s z =0  later s z  0  E821 8

9. GEANT4 Anal. Check! 9

10. We confirmed s z is correct. Then I have  s z /  t =s z (t 1 )-s z (t 2 )/(t 1 -t 2 ) by GEANT4. Extract from GEANT4 calculation Left-hand side Right-hand side Left-hand side  (1-s z 2 ) Check! Although I can not figure out sx and sy by analytically, but I check their scalar product is correct! 10

11. S  Precession : Amplitude does not growth as a function of time. s x, s y J-PARC EDM (GEANT4)  (1-Sz 2 ) Sx component in the  rest frame 11

12. S  /|s||  | S  /|s||  | comparison between G-2 and EDM X=0.00175 X=1 12

13. S  /|s||  | comparison S  /|s||  | comparison PSI vs. JPARC 13

14. Spin-dependent muon decay revisit & Expected positron time spectra for EDM case 14

15. ++ e + momentum spin cos  S * g-2 条件の時、 EDM 条 件の時、共に同じ。 15

16. Kinematic in the  rest frame  th =0.75 16

17. Expected time spectrum (1) 0.65  0.75  0.00175=8.5E-4 17

18. Expected time spectrum (2) 0.0017 0.6M,  + 18

19. Expected time spectrum (3) 0.65  /2  2  3E-6=6E-6 19

20. PSI EDM (try spin frozen by GEANT4) 20

21. Cyclotron motion Try Spin-frozen (PSI) 21 I use “G4EqEMFieldWithSpin” of Geant4.9.2.p02 (bug fixed) Apply B=(0, 0, Bz), Bz =1 Tesla, R=0.42 m  =1.55 (125MeV/c) and radial electric field |E R |=0.642663E+6 volt/m  11.416… nsec G-2 precession should be frozen! Parameters from hep-ph/0606034v1 Spin motion expectation: I will explain how to get spin and momentum vector information in the next page.

22. Yes, frozen (PSI) 22 S=(Sx, Sy, Sz) p/|p|=(momvx, momvy, momvz) S  =cos(  a t) G-2 precession is completely canceled:Spin-Frozen!! I set initial (t=0) values: s/|s|=(0,1,0)  /|  |=(0,1,0) Precession :

23. PSI EDM 23

24. GEANT4 check: PSI EDM S  /|s||  | szsz E R is exact value. But s  0, because  0 !! Sz  0 and s  0 How to distinguish between “  0” and “E R error”? 24

25. If  + stays in storage ring forever? PSI-EDM 25

26. In case of E R =E R  0.995,  =0 26 s z =0 No Frozen!! Sz=0, but s  0 S  0.5% level control Precession :

27. Backup 27

28. But, I got wrong statement… Sx, Sy, Sz (Sx0, Sy0, Sz0)=(0,0,1) I tried very large  value to check behavior and have wrong expectation. wrong? Envelope is sin function 28