Exercise to treat spin-dependent decays
I.
Goal:
–
–
Study the relationship between momentum pe
accuracy/precision and a, Analyzing power <A>.
Estimate the required performance of the detector.
II. Exercise to check basic kinetics:
Today’s contents
1. Energy and momentum conservation,
2. 2D event yield distribution as functions of y and cmS
y = pcme/pmax
cmS is an angle between spin-axis and momentum direction of
decay-e+ at the center-of-mass system. ( see next page)
III. Check wiggle plots:
“usual” wiggle plot,
“Beam-loss free” wiggle plot.
1
Center-of-Mass system
Magnetic field
Z
Y


Scm
a t
m
cm
Spin-direction

B  0,0, BZ 

S  cos a t, sin a t,0
Direction of decay-positron
pe
m
m
m
  cos cm , sin cm cos , sin cm sin 
pe

X,
Momentum 
Angle between spin-axis and momentum direction of decay-e+ at the center-of-mass system:
cos Scm
 
pe
m
m
   S  sin cm
cos  sin a t  cos cm
cos a t
pe
m

We measure Lab , .
Lorentz boost  cos m
Lab 
m
  cos cm
m
1   cos cm
2

1
dP
n y 1  Ay  cos sin ,

dyd 2
2y 1
2
.
n y   y 3  2 y , Ay  
3  2y
y  p e p e max
p e max

m
2
3
4
Condition:
3T
B
P=300MeV/c ,
=3,
Tc =7.4nsec,
R=333mm,
Ta=2/a=2.2sec.
Positron energies
28 ~191 MeV
5
8.6MeV positron
B＝3T
50.4MeV positron
102MeV positron
6
II. Check basic kinetic values
from GEANT4
7
Probing Spin-dependent Decay Info.


I. To be more simple, I set 100%   e  e  !
II. Probe “decay process” information in the lab frame
directly. (I use “UserSteppingAction”.)


Spin vector, momentum of  at previous step of decay
process.
Momentum and energies of daughters.
III. Check momentum/energy conservation.

Within few eV at =1, within few keV at =3.  why?
IV. Apply Lorentz transformation to get values in the
center-of-mass system.
V. Cook values as I want!!
8
p T  p 2Y  p 2Z
pL  pX
y  p e p e max
p e max

m
2
p e  p 2X  p 2Y  p 2Z
X axis is always momentum direction.
9
y = pcme/pmax
 is an angle between spin-axis and
momentum direction of
decay-e+ at the center-of-mass
system.
10
11
III. Wiggle plots made by GEANT4
“Usual” wiggle plot and
“Beam-loss free” wiggle plot
12
 t
Ft   N exp 
 
  
9.5
105 ,
E> 200 MeV
1.3105e+

1  A cosa t   


4 free parameters
13
Covariant matrix is OK.
mom
“Beam loss free” wiggle plot by knowing cm

An angle between + and e+ momentum
direction in the center-of-mass system.

dP
1

n y  1  a y  cos Scm ,
dyd cm 4
mom
mom
here , cos Scm  sin cm
cos  sin a t     cos cm
cosa t   .

dP
dP
n y 
mom


1  a y  cos cm
cosa t   
S
S
dyd cm sin cm dydcm
2





1
S
S 
mom
Nt     n y dy    n y a y sin cm dydcm t   cos cm cosa t   
2 y

 y Scm



N
m
Measure!

1  A cos cm
cosa t   
2

Lt  
N
2

1 
A  cosa t  , R t  
N
2
Lt   R t 
 Asym t  
  A  cosa t   
Lt   R t 
1 
A  cosa t  ,
No exponential
14
term!
LEFT
RIGHT
Asym t    A  cosa t  
 No worry about -beam loss!
 But, need to handle left-right
detector asymmetry.
9.5 105 , 1.9 105 e+
y> 0.6 , LEFT: 1  cos   0.7
RIGHT:1 cos  1  0.7
15
A big advantage to measure 
Lab-frame
A LAB 
S
S




n
y
a
y
cos

dyd
cos

cm
cm


y cos Scm
 n y dy  d cos 
cos 
th
E Lab

.
 y
If we can measure cm S event-byevent, ”Effective Analyzing Power”
is NOT smeared by cos cm S!
 n y a y dy
y
 n y dy
y
,
”Effective Analyzing Power” is smeared by cos cm S
Center-of-mass frame
A CM 
S
cm
cos Scm
y
S
cm
Spin
cm
.
We have bigger effective
16
Analyzing Power
Next things….
Now, I am ready to think about detector performance.
I.
II.
Study the relationship between measured
momentum accuracy/precision and a, Analyzing
power <A>.
Estimate the required performance of the detector.
I, also, will play with G4-beamline to think about beam line. (Need a time to learn it, though.)
17
How many positrons we need for EDM ?
statistics
Value [e cm]
comment
Exp.
( 3.7  3.4 ) 1019
11.4106 e+, e CERN (1974~76)
results (0.04  1.6 0.17) 1019 9.4106 e+
E821 (1999, 2000, Trace back
detector, Fig .7)*
( 0.1  0.2 1.07 )1019 975 106 e
E821（2001, PSD1-5, Tbl. IV)*
Predic (1.4  1.5 ) 1025
-tion
> 10 -23
Our
goal
Mass scale of lepton EDMs
Extended SM model
~1013 @ magic=29.3
~1017@ magic=29.3
1022 level
1024 level
~ 31014 @ =3
~ 31018@ =3
 e
14
EDM  
   4.7 10 e  cm
2 2m c
EDM sensitivity:
2t m

Bt e
 
1
A1A   2 N 0e

1
A1A 2    N sum
e
0
0


N sum
~
N
exp

t





N
e
 
  e
 e
t
“Improved Limit on the Muon Electric
Dipole Moment “ 2EAPS/123-QCD
,
eB
A1 
2m
18
y vs. cos 
cos
dP
1

n y 1  Ay  cos 
dyd cos  2
2y  1
2






n y  y 3  2y , A y 
.
3  2y
y
19
 t 
N e t   N 0 exp    cosa t ,
  
Relationship
between a and 

N total   N e t dt  N 0 
0
a 
1
2
 N 0 A 2
ミュービーム強度はによらず、一定だとし、(Ntotal=const.)
N total  N 0   N*0  *,
 N*0 

N0 ,
*

1
2
1
2
*a  *

  N*0 A 2  * N 0 A 2
*a


a

*
*
,

I checked with Toy Monte Carlo