Weak Interactions in the
Nucleus I
Summer School, Tennessee
June 2003
Using the nucleus to search for new physics
Summary
Historical Introduction
Dirac Equation. E&M and Weak Int.
Non-VA forces in weak decays
 Measure e-n correlation
Non-Unitarty of the CKM matrix
Isospin Breaking
Measure b asymmetry with UCN
Time-Reversal Invariance Violation
Measure TRIV correlation
The Weak Interaction: A Drama in
Many Acts

1890’s: Roentgen discovers b rays
Thought Uranium salts were affected by the sun but rainy Paris soon
helped showing otherwise.

1920’s: Pauli proposes n
To explain continuous b spectrum: only way to save conservation of
energy.

1950’s Parity Violation
To explain identical properties of q and t particles. Then clearly proven
in Madame Wu’s experiment.

1960’s CP-Violation
Parity Violation




x  -x
p  -p
r×p  r×p
JJ
Parity Mirror
Madame Wu’s experiment:
Polarize 60Co and look at the direction of the emitted b’s.
In a Parity-symmetric world we
would see as many electrons
emitted in the direction of J as
opposite J.
pe
pe
Parity Violation




x  -x
p  -p
r×p  r×p
JJ
Parity Mirror
Madame Wu’s experiment:
Polarize 60Co and look at the direction of the emitted b’s.
But in the real world we see
only electrons emitted in the
direction opposite J.
pe
Parity Violation




x  -x
p  -p
r×p  r×p
JJ
Parity Mirror
Other experiments:
Look at the helicity of neutrinos.
In a Parity-symmetric world
we would see as many n’s
with left-handed helicity (p
opposite S) as right-handed
helicity (p parallel to S).
Sn
pn
Sn
pn
Parity Violation




x  -x
p  -p
r×p  r×p
JJ
Parity Mirror
Other experiments:
Look at the helicity of neutrinos.
But in the real world we see
as only n’s with left-handed
helicity (p oppsite S).
Sn
pn
Schroedinger Equation
How do we get a wave equation that yields conservation of energy and
the correct deBroglie relations between particles and associated waves?
2
p
V  E
2m
p  k
Schroedinger Equation
How do we get a wave equation that yields conservation of energy and
the correct deBroglie relations between particles and associated waves?
2
p
V  E
2m

H  ih
t
h
p 
i
p  k
 h2 2


  V    ih 

t
 2m

Schroedinger Equation: perturbation theory

( H0  V )  i 
t
H0  n  En n
How do we get
 ( x, t ) 
?
 a (t ) ( x) e
 iEn t / 
n
n
n
Replacing in Schrodinger’s Equation and integrating:
T
1
iEn t / 
 iEi t / 
3
*
an (t ) 
dt  d x  n ( x) e
V i ( x) e

 
i  T
Lorentz-invariant form:
1
4
*
Tfi   d x  n ( x ) V i ( x )
i
Schroedinger Equation: decaying rate
Tfi  V fi 2 ( E f  Ei )
If V(x) is time-independent:
The transition probability per unit time:
2

|V fi |2  ( E f  Ei )
T

2
Wfi 
| Tfi |
In a decay, like np e n we have to sum over final states
FERMI’s golden rule:
2
2 dN
Wfi 
|V fi |

dE
Examples of phase-space calculations
Using the neutron mean-life (t ≈ 900 s) estimate the anti-neutrino
absorption cross section on protons n+p  n+e+:
For neutron decay:
2
2 dN
Wfi 
|V fi |

dE
d pn d pe
2
2

|V fi | 
3
3  ( Emax  Ee  En )
t 
(2) (2)
1
3
3
For anti-neutrino absorption (Kamland):
2
d pe
2

|V fi | 
3  ( Ee  En  M p  M n )

(2)
3
Dirac Equation
How do we get a wave equation that is relativistically correct?
p m  E
2
2
2
Dirac showed that one can start with a linear equation
h


 

 1
   bm  ih 
 2
 3
i   x1
 x2
 x3 
t
2

2 2
2
2
from where one regains   h   m    h
2 
t
 i k   k  i  2ik
for which the coefficients
 i b  b i  0
and the wave function can
not be simply scalars
b2  1
Dirac Equation
The matrices alphas have to be at minimum of dimension 4:
 0 i 
i  


0
 i

1 0 
b

0

1


where
 0 1
x  

 1 0
 i are the Pauli matrices
 0  i
y  

 i 0
1 0
z  

 0  1
The wave function now has 4 components. For a free particle with p=0:
 1
 0
 
 
0
1
 imt / h  
 imt / h  
1  e
 0  2  e
 0
 
 
 0
 0
 0
 0
 
 
0
0
 imt / h  
 imt / h  
4  e
3  e
 0
 1
 
 
 1
 0
We write it in terms of 2-comp spinors. For a free particle with p=0:
 1
 0
1
2
   ;    
 0
 1
E0
E0
 s
u  
 0
 0
v   s
 
For p ≠ 0:
 m
Hu  
  p
E0
  p  u A 
 uA 
    E 
 m   uB 
 uB 


s


u      p  s


  E  m 
  . p uB  ( E  m)u A
 
  . p u A  ( E  m)uB
E0
     p s

 
u    | E | m  


s



Dirac Eq. and E&M
Dirac Equation without E&M
  b i
i
  p   b m  0
where
 b
0
   n   n    2g n
with E&M (for electron)
e
 ( p  A )   b m  0
c

This is equivalent to the previous plus an interaction:
e

A 
c

Quantizing the fields
 h2 2


  V    ih 

t
 2m

Schroedinger Equation
Take:
 ( x, t ) 
 b (t )
n
n
H n ( x)  En n ( x)
( x);
n
dbn  i
 bn En
dt
h
Then:
The Hamiltonian that yields the previous is:
2


h
3
2
H   d x  *( x, t ) 
  V   ( x, t )
 2m


Interpreting

bn
as an operator:
bn ,bn'  nn'

^
H
Eb
n n

bn
Quantizing the fields
 h2 2


  V    ih 

t
 2m

Schroedinger Equation
Take:
 ( x, t ) 
 b (t )
n
n
( x);
n
Then:
H n ( x)  En n ( x)
dbn  i
 bn En
dt
h
The Hamiltonian that yields the previous is:
2


h
3
2
H   d x  *( x, t ) 
  V   ( x, t )
 2m

Interpreting
bn
b ,b   

n
n'
as an operator:
nn '
H 

Enbn  bn
Quantizing the fields
For bosons:
For fermions:
 
b ,b   
bn ,bn'   nn'

n
n'
nn '
The various wave functions are generated by applying
the creator operator on the vacuum state:
| C1 , t    d x C1 ( x) ( x, t )|0 
3

Current-current interaction
E&M interaction in Dirac’s Equation
e  
3

Hint    d x  ( x , t )  A   ( x , t )
c

The vector potential should satisfy Maxwell’s equations:
A  j  A (q )  
2
j
q2
e 
j   ( x , t )   ( x , t )
c
Example of Feynman diagram: e- scattering.
2
e
M         ( p) 2  e    e
q


e
e
E&M vs. WEAK
Same order of
magnitude:
g 0.22 ≈ e
E&M
e2
       ( p) 2  e    e
q
Weak
g2

     (1   5 )   ( p) 2


(1   5 )  e
2
e
q  MW
W’s are left
handed
Example of Feynman diagram: e- scattering.


e
e
E&M vs. Weak
Order of magnitude of the Weak
coupling at very low energies:
Order of magnitude of the E&M
coupling :
Ratio Weak/E&M:
q
5 
2
g

2  g 
 80,000
MW
2
2
e
2
 107
2
E&M vs. Weak: helicity
 
 .p
H =
|p|
Helicity is defined as:
The helicity of
leptons produced
in Weak decays:
p
H ( 1- 5 )   ( 1- 5 )
E
Konopinski’s argument:
( t  0   E   )
instant velocity against
momentum can only be c
| a|2 (  c)  (1 | a|2 )( c)  v
Then helicity:
| a| (  1)  (1 | a| )(  1)   v/c
2
2
Allowed approximation
(0+,1-)

p
    (1, )  H   t i  T
m
i
 
p 

  5   (
,  )  H   t i i
m
i
(0-,1+)
As a result, two types of allowed transitions:
I f  Ii ;  f   i ;
Fermi:
J f  Ji ;
Gamow-Teller:
J f  Ji  1;
I f  Ii  1;  f   i ;
Looking for Physics Beyond the
Standard Model
Standard Model
Looking for Physics Beyond the
Standard Model
Non-VA currents in Weak decays
e+
d
ne
Are weak decays carried only by W’s?
W
u
d
u
e+
e+
ne
Higgs
e+
Or is there something new?
u
d
ne
Lepto-Quark
Non-VA currents in Weak decays
e+
d
ne
Are weak decays carried only by W’s?
W
u
e+
Vector
d
u
e+
ne
Higgs
e+
Or is there something new?
Scalar
u
d
ne
Lepto-Quark
Detecting Scalar currents in weak decays
The e-n correlation depends strongly on the nature of the carrier
(we take a 0+  0+ transition).
spins have to
couple to zero
Standard Model
Vector Currents
e+
ne
dW/dW = 1+ pe.pn/Ee En
e+
n
spins
momenta
New Physics?
Scalar Currents
e+
ne
dW/dW = 1- pe.pn/Ee En
A trick to avoid detecting the neutrino
32Ar
Instead of detecting
the neutrino
31S+p
32Cl
A trick to avoid detecting the neutrino
32Ar
Instead of detecting
the neutrino
31S+p
32Cl
We detect the proton
that contains the info
about the 32Cl recoil
(Doppler)
A trick to avoid detecting the neutrino
32Ar
Instead of detecting
the neutrino
31S+p
32Cl
We detect the proton
that contains the info
about the 32Cl recoil
(Doppler)
Monte-Carlo calculation
of proton energy
scalar
vector
Experimental set-up
Super-conducting solenoid
B=3.5 Tesla
Data
Problem: Isol-trap fellows measured a mass of
33Ar and found in disagreement with parabola
for A=33 system.
Problem: Isol-trap fellows measured a mass of
33Ar and found in disagreement with parabola
for A=33 system.
Solution: we found out the mass of 33Cl(T=3/2)
they were using was incorrect (Pyle et al.
PRL 88, 122501 (2002).)
Using the correct mass for 33Cl(T=3/2) one
obtains an excellent agreement with the
Isospin parabola.
Assuming the parabola works for A=32 one
obtains M(32Ar)= -2197.0+-4.2 keV
The Isol-trap new determination of the mass of
32Ar is: -2200.1 +- 1.8 keV.
New Isol-trap data shows excellent agreement
with the Isospin parabola but several
quantities that affect our determination of the
(e,n) correlation have changed.
QEC, Energy calibration
We are presently re-doing all the data analysis
to extract the correlation coefficient and
systematic uncertainties.
Widths and spins of 33Cl from decay of 33Ar
Limits for scalar couplings