Symmetries in -decay
•
•
•
•
•
•
•
Specifications of the -decay
Parity in -decay
Handedness of leptons
Current-Current interactions and intermediate Bosons
Universality of the Weak Interaction and CKM matrix
Symmetries and observables in -decay
Muon decay and TWIST experiment
-decay properties
-decay is a three-body decay.
Beta spectrum:
Life time of the -decay is orders of magnitude
longer than the EM decays
Strong decays life time  10-21-10-16 s
EM decays life time
 10-9-10-6 s
WI decays life time
 10-3-106 s
Choosing definite handedness of leptons
Third particle
()
Fermi coupling
Constant (G)
P
Fermi coupling constant for WI
Fermi’s golden rule:
2
2

H fi  ( E0 )

H fi  M fi
where
Matrix element of EM interaction:
Considering interaction between electron current and proton current
M fi  j (e)
1 
j ( p)
2
q
e
e
j(p)
p

p
When Maxwell equation holds:
j(e)
□2Aµ=j µ (p)
Fermi proposed to use the same expression
with different coupling constant for WI.
M fi  G j ( L) j  ( H )
n
p
j(H)
e
j(L)
e
Discrete symmetries
Parity:


ˆ
P(r , t )   p (r , t )
•Introduced by Wigner in 1927/1928
•No classical analogue
•Conserved quantum number
Time reversal:

* 
T  (r , t )  T  (r ,t )
•Introduced by Wigner in 1932
•Relates initial and final states
•No conserved quantum number!
Charge conjugation: C
•Introduced by Kramers in 1937
•Particle-antiparticle interchange
•Requires quantum field theory
Parity symmetry
What is parity


ˆ
P r  r i.e. (x,y,z)  ( x, y, z )


ˆ
P (r )   (r )
Eigenvalues
Pˆ 2  1̂
Parity conservation
QM tells us:
If

H,Pˆ   0 ,
Eigenvalue s p  1
d Pˆ
then
dt
0
Examples of operators:
Pˆ s   s
Pˆ p   p
Pˆ V  V
(scalar : a number)
Pˆ A   A
(axial vector : angular momentum  postion  velocity)
(psuedoscalar : spin  momentum)
(vector : momentum)
Parity in WI
Everybody expected parity to be conserved in weak interaction.
First hint that something was wrong from the θ- puzzle!
  2
L  0 P  P2 (1) L  1
  3
L  0 P  P (1)  1
3
L
• But both particle have same mass and lifetime, i.e. must be the
same particle.
• Parity is violated ???
(Questioned by Lee and Yang)
Parity in WI; Wu experiment
Measure decay spectrum of polarized Cobalt -decay
60
Co 60 Ni *  e  e
 
 P   exp(.B / kT )
•60Co
at T=0.01 K
•all spins are parallel in external field
Measure electron angular distribution
v
I ( )  1  cos( )
c
e-
θ
J
Now calculate:
 
 . p  cos  0
But, this is a pseudo-scalar and has to be 0, if parity is
conserved!
Parity in WI; Wu experiment
C.S. Wu, Phys. Rev. 105 (1957)1413
Handedness (chirality) of Leptons

Dirac equation for free particle: (α.p   m) u ( p)  E u ( p)
When
 0 i 
I 0 
i  
,  



0
0

I


 i

If particle is at rest (p=0):
mI
0

For E>0 there are two answers:
Represents Particles.
u (1)
For E<0 also there are two answers:
Represents Anti-Particles.
0 
u  Eu

 mI 
1
 
0
 
0
 
0
 
u ( 2)
0
 
1
 
0
 
0
 
u ( 3)
0
 
0
 
1
 
0
 
u ( 4)
 0
 
 0
 
0
 
1
 
Handedness (chirality) of Leptons
 
u A 
 m  . p  u A 
. p  m  u   E u 

 B
 B
For non zero p :
 
Gives two coupled equation:
At limit of p0 for E>0:
Then:
 1 


 0 
(1)
u ( p)  N   . p 
 E m
 0 


And similarly for E<0:
 . p uB  ( E  m) u A
 
 . p u A  ( E  m) uB
 
1
 0


.p 
u A   , uB 
, 1     2   
Em
 0
1
 0 


 1 
( 2)
u ( p)  N  0 
 . p 

    . p 
 0  
 E m 

 .p 
 | E | m 


(3)
( 4)
0
|
E
|

m


u ( p)  N
u ( p)  N 

 1 
 0 


 1 
 0 




Handedness (chirality) of Leptons
1

 3 0   0
  
  
 0 3  0
0

Helicity operator is:
0 0 0

1 0 0 
0 1 0

0 0  1
u (1)  1u (1) and u ( 2)  1u ( 2) for E  0
Applying to the spinors:
u (3)  1u (3) and u ( 4)  1u ( 4) for E  0

(α.p ) u( p)  E u ( p)
For mass-less particles:
  i 0 
i  

0

i

 0 k 
0 I 
 I
k
5
 and   
 ,
  
  
We also define:
 I 0
 0
  k 0 
 
 


.
p
0


 
    . p   E
It gives two set of equations: 

E
   
 
 
0

.
p

 . p   E

 
 
0
0

I
Handedness (chirality) of Leptons
For the positive energy solution of the first equation:
 
.p
When
 is the Helicity operator.
| p|
 
.p 
  
| p|
Therefore this solution can be interpreted as Left handed particle.
The negative energy solution for the first equation :
 
.p 
 
| p|
That shows Right handed anti-particle.
Similarly one can show that  presents the Right handed particle (E>0)
and Left handed anti-particle (E<0).


E>0
-
+
E<0
+
-
Handedness (chirality) of Leptons
Let’s apply this operator:
 I 0      
    
1 / 2(1   ) u  
 0 0     0 
and
5
 I 0     0 
    
1 / 2(1   ) u  
 0 0      
5
If we apply
•
•
•
•
5
5
5





,

 
on individual states:
5 is defining handedness for mass less particles.
It is a Chiral operator and neutrinos are it’s eigen states.
Massive particles are not eigen state of Chiral operator.
-decay produces just Left handed neutrinos and Right
handed anti-neutrinos  1-  5
Handedness (chirality) of Leptons
Applying 1±  5 on pure states:
1 / 2(1   5 )   1 / 2(   )  0
1 / 2(1   5 )   1 / 2(    )  
1 / 2(1   5 )   1 / 2(   )  
1 / 2(1   5 )   1 / 2(    )  0
Experimental observation shows that -decay produces
just Left handed neutrinos and Right handed anti-neutrinos.
M. Goldhaber, Phys. Rev. 109(1958)1015
Therefore WI is coupled to the  states not to the  states  1-  5
If 1-  5 is contributing in WI, parity is violated
because  5 is a pseudoscalar operator.
Current-current interaction in EM and WI
Fermi’s golden rule:
2
2

H fi  ( E0 )

H fi  M fi
when
Matrix element of EM interaction:
Considering interaction between electron current and proton current
1 
M fi  j (e) 2 j ( p)
q
e
□2Aµ=j µ (p)
e
j(p)
p

p
When Maxwell equation holds:
j(e)
Current-current interaction in EM and WI
If we write the currents in terms of the Dirac spinors: e
1
M fi  e (u p u p ) ( 2 ) (u e  ue )
q
2


p
One can write the similar relation for the WI in -decay :
p
≈
EM current is a Vector current expressed by µ.
e
n
p
M fi  G (u p  un ) (u e  u )
When G is weak coupling constant.
e e
• Fermi proposed to use Vector type current for WI also.
• This gives successful description of some decay properties in special
type of the -decay (Fermi decays).
• It does not explain why WI chooses one type of handedness.
• It does not explain source of the parity violation in -decay.
Which currents are available?
One can build different type of the currents using  matrixes:
Parity
1
Scalar
S
1
5
Pseudoscalar
P
1

Vector
V
1
 5 
Axial vector
A
1
i   
Tensor
T
Most general form of any matrix element and also WI is:
M fi   Ci (u p Oi un ) (u eOi u )
i
When Ci are current coupling constants and Oi are current operators
Parity violation in WI  pseudoscalar term
To include the fact that parity is violated in the WI, we can add
a Pseudoscalar term to the interaction:
M fi   Ci (u pOiun ) (u eOiu )   Ci(u pOiun ) (u eOi 5u )
i
if
i

Ci 5 
M fi   Ci (u pOi un ) u eOi (1   )u 
Ci
i


Ci  Ci

G
5
M fi 
C
(
u
O
u
)
u
O
(
1


)u
p
e

i
i n
i
2 i

We know that 1- 5 gives observed Left handed neutrinos
and right handed anti-neutrinos in the -decay . Therefore:
G
5
M fi 
C
(
u
O
u
)
u
O
(
1


)u
p
e

i
i n
i
2 i
Is most general form of the WI matrix element


Currents and handedness of electrons
Let’s have a look at the correlation between the particles in the -decay
Mediated by different currents
• V and A currents are coupled to the right handed positrons
while S and T currents are coupled the Left handed positrons.
• Experiments show that just right handed positrons are produced
in the -decay. L. Goldzins, Prog. Nucl. Phys. 7 (1959)163
Therefore WI is mediated by the V and A currents.
V-A  Left handed electrons
Including V and A currents in terms of the Dirac Spinors:
G
M fi 
{CV (u p  un ) [u e  (1   5 )u ]
2
 C A (u p i   5un ) [u ei   5 (1   5 )u ] }
G

[u p  (CV  C A 5 )un ] [u e  (1   5 )u ]
2
Experimental evidence shows:
Therefore:
C A  CV
G
M fi 
(u p  (1   5 )un ) [u e  (1   5 )u ]
2
WI matrix element in the Standard Model (SM)
Intermediate Bosons
For a pure leptonic WI, like neutrino scattering, one can calculate
the cross section including the fact that in high energies mass of
the electron is negligible.
 e
 ( e e   e e) 
e
2
G s

e
e
Where the s is the center of mass energy of the scattering.
This shows that by going to high energies of the scattering,
cross section diverges. To fix this problem we introduce
the intermediate bosons to mediate the WI.
e
M fi  [
g
1
g

5
u e  (1   5 )u ]
[
u

(
1


)u ]
e
2
2
2
MW  q
e
W
e
e
Intermediate Bosons
n
p
A typical -decay mediated by charged boson
W
e
WI happens at the level of quarks
n
d
u
d
W
d
WI violates the strange quantum number.  u
s
Also, strength of the WI is smaller in the
s quark decay in compare with the d quark decay.
W
e


d
u
u
p
d
u
u
p
ee
ee
Universality of WI; CKM matrix
To hold the universality of the WI, Cabbibo proposed to include
the mixing between the quark flavours (s and d).
N. Cabibbo, Phys. Rev. Lett. 10 (1963)531
u

u 

   
 d    d cos(C )  s sin( C ) 
By finding more quark flavours, all the different possible mixings are
collected in the CKM matrix.
“weak eigenstates”
 d   Vud
  
 s   Vcd
 b  V
   td
“flavor eigenstates”
Vus
Vcs
Vts
Vub 

Vcb 
Vtb 
Flavour eigenstates are not the weak eigenstates.
d 
 
s 
b 
 
General derivation of the WI matrix element (-decay )
To have a general expression for the WI in the -decay
one can consider all possible currents contributing in the WI

G
5
M fi 
C
(
u
O
u
)
u
O
(
1


)u
p i n
e i

i
2 i

Calculating the differential decay rate and parameterizing the expression
with observables gives the matrix element part equal to:
All the coefficients can be written in terms of the
current coupling constants.
Symmetry Violation; sources and observables
Non SM correlations:
a, b, Q
A, B, G
D
R
Permanent Electric Dipole Moment (EDM):
P
E
O
E
O
T
E
E
O
O
C
E
O
O
E
General decay rate for Muon
Most general interaction does not presuppose the W

e


rate ~


 S,V ,T
i, jR,L



2
gij ei   e      j

• S,V,T = scalar, vector or tensor interactions
• R, L = right and left handed leptons (e, , or  )
Couplings in the present Standard Model
2
rate ~

 S,V ,T

gij ei   e
     j
i, jR,L
S
gRR
0
gV
RR  0
gTRR  0
S
gLR
0
gV
LR  0
gTLR  0
S
gRL
0
gV
RL  0
gTRL  0
S
gLL
0
gV
LL  1
gTLL  0
Current measured couplings
2
rate ~

 S,V ,T

gij ei   e
     j
i, jR,L
S
gRR
 0.066
gV
RR  0.033
gTRR  0
S
gLR
 0.125
gV
LR  0.060
gTLR  0.036
S
gRL
 0.424
gV
RL  0.110
gTRL  0.122
S
gLL
 0.55
gV
LL  0.96
gTLL  0
Michel parameters
2
1 x 


2
2
rate ~ x 3  3x   4 x  3  3 xo 
  P cose 1 x   4 x  3
 x 


3
3

x
Spectral shape in x, cose is characterized
in terms of four parameters -- ,  , , 
P is the muon polarization
  0.0
(L. Michel, A. Sirlin)
and


pe
s


P
 1.0

E emax
e

SM prediction for Michel parameters:
  3 / 4,   3 / 4,   1.0
Ee
xo 
E emax 
me
E emax
m2  me2
2m
The TWIST experiment
 Collect high precision data to obtain the e+ spectrum
from -decay as a function of x and cose
 Extract the best values of the spectral parameters
,  , ,  simultaneously (the first time this has
been done)
 Obtain a precision in , ,  (a) of 10-3 and
(b) a few parts in 104 (~10-3 precision for )
 Compare ,  , ,  from our fit with Standard
Model values
Obtain high precision data on the e+ spectrum
Highly polarized +
+ stop in Al target
(several kHz)
Unbiased +
(scintillator)
trigger
Chambers & half detector
Planar drift chambers sample
positron track

Use 44 drift planes,
and 12 PC planes
Typical decay event

e+
e+ spectrum in x, cose
1 x 


2
2
rate ~ x 3  3x   4 x  3  3 xo 
  P cose 1 x   4 x  3
 x 


3
3

rate
2
Determination of , , , 
Accelerator data
are collected
Monte Carlo data
are generated
Event Analysis
• Event classification (31 types)
• Helix fit to events within fiducial volume
• Extract e+ momentum and angle spectrum
in bins of x and cose
Accelerator data
spectrum
Fit
, , , 
Monte Carlo
data spectrum
TWIST experiment: first results

• Number of events: 6109
• Four data set:
B=1.96,2.0 and 2.04 T
• =0.749640.00066(stat.)
0.00112(syst.)
• SM Prediction: =0.75

• Number of events: 6109
• Four data set:
B=1.96,2.0 and 2.04 T
• =0.750800.00032(stat.)
0.00097(syst.)
• SM Prediction: =0.75
These two value are consistent with SM