Weak Interactions in the Nucleus I Summer School, Tennessee June 2003 Using the nucleus to search for new physics Summary Historical Introduction Dirac Equation.
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Transcript Weak Interactions in the Nucleus I Summer School, Tennessee June 2003 Using the nucleus to search for new physics Summary Historical Introduction Dirac Equation.
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
JJ
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
JJ
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
JJ
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
JJ
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 Enn
How do we get
( x, t )
?
a (t ) ( x) e
iEnt /
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 Vfi 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 np 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 Mn )
(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 2ik
for which the coefficients
i b bi 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
i are the Pauli matrices
0 1
0 i
1 0
x
y
z
1 0
i 0
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:
p uA
m
uA
Hu
E
p m uB
uB
E 0
s
u p s
E m
. p uB ( E m)u A
. p uA ( 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 bm 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
E
b
n n 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
107
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:
Fermi:
J f Ji ; I f I i ; f i ;
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