CDF @ UCSD Frank Würthwein Computing (finished since 8/2006
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Transcript CDF @ UCSD Frank Würthwein Computing (finished since 8/2006
Physics 222 UCSD/225b UCSB
Lecture 9
Weak Neutral Currents
Chapter 13 in H&M.
Weak Neutral Currents
• “Observation of neutrino-like interactions
without muon or electron in the gargamelle
neutrino experiment” Phys.Lett.B46:138-140,1973.
• This established weak neutral currents.
4G
M
2 J NC J NC
2
1
NC
J (q) u cV c A 5 u
2
allows for different coupling
from charged current.
cv = cA = 1 for neutrinos, but
not for quarks.
Experimentally: NC has small right handed component.
EWK Currents thus far
• Charged current is strictly left handed.
• EM current has left and right handed component.
• NC has left and right handed component.
=> Try to symmetrize the currents such that we get one
SU(2)L triplet that is strictly left-handed, and a singlet.
Reminder on Pauli Matrices
0
1
1
0
2
i
1
3
0
1
1
0
i
2
0
0
3
1
0 1
1
(1 i 2 )
2
0 0
0 0
1
(1 i 2 )
2
1 0
We will do the same
constructions we did last quarter
for isospin, using the same formalism. Though, this time,
the symmetry operations are identified with a “multiplet of
weak currents” . The states are leptons and quarks.
Starting with Charged Current
• Follow what we know from isospin, to form
doublets:
0 1
0 0
L ;
;
e L
0 0
1 0
J (x) L L
1
1
1
J (x) L 3 L L L eL eL
2
2
2
3
We thus have a triplet of left handed currents W+,W-,W3 .
Hypercharge, T3, and Q
• We next take the EM current, and decompose it such
as to satisfy:
Q = T3 + Y/2
em
j
1 Y
J j
2
3
• The symmetry group is thus: SU(2)L x U(1)Y
• And the generator of Y must commute with the
generators Ti, i=1,2,3
of SU(2)L .
• All members of a weak isospin multiplet thus have
the same eigenvalues for Y.
Resulting Quantum Numbers
Lepton T T3 Q Y
1/2 1/2 0 -1
e -L
1/2 -1/2 -1 -1
e -R
0
0
-1 -2
Quark
uL
dL
uR
dR
T
1/2
1/2
0
0
T3
1/2
-1/2
0
0
Q
2/3
-1/3
2/3
-1/3
Note the difference in Y quantum numbers for
left and right handed fermions of the same flavor.
You get to verify the quark quantum numbers in HW3.
Y
1/3
1/3
4/3
-2/3
Now back to the currents
• Based on the group theory generators, we
have a triplet of W currents for SU(2)L and a
singlet “B” neutral current for U(1)Y .
Basic EWK interaction:
g Y
igJ W i J B
2
i
i
• The two neutral currents B and W3 can, and
do mix to give
us the mass eigenstates of
photon and Z boson.
W3 and B mixing
• The physical photon and Z are obtained as:
3
A W sin W B cos W
Z W 3 cosW B sin W
• And the neutral interaction as a whole
becomes:
g Y
3
3
igJ
W
i
J B
2
Y
J
igsin W J 3 gcosW A
2
Y
J
igcos W J 3 gsin W Z
2
Constraints from EM
ej em
Y
Y
J
J
eJ 3 igsin W J 3 gcos W
2
2
gsin W gcos W e
sin W
g
g
cosW
We now eliminate g’ and write the weak NC interaction as:
g
g
3
2
em
NC
i
J
sin
j
Z
i
J
W
Z
cosW
cosW
Summary on Neutral Currents
1 Y
j J j
2
NC
3
2
em
J J sin W j
em
3
This thus re-expresses the “physical” currents for
photon and Z in form of the “fundamental” symmetries.
Vertex Factors:
ieQ f
Charge of fermion
g
1
f
f 5
i
(cV c A )
cosW 2
cV and cA differ according to
Quantum numbers of fermions.
Q, cV,cA
fermion Q
cA
cV
neutrino 0
1/2
1/2
e,mu
-1
-1/2
-1/2 + 2 sin2W ~ -0.03
u,c,t
+2/3
1/2
1/2 - 4/3 sin2W ~ 0.19
d,s,b
-1/3
-1/2
-1/2 + 2/3 sin2W ~ -0.34
Accordingly, the coupling of the Z is sensitive to sin2W .
You will verify this as part of HW3.
Origin of these values
The neutral current weak interaction is given by:
1
g
5
3
2
i
f (1 )T sin W Q f Z
2
cosW
Comparing this with:
Leeds to:
g
1
f
f 5
i
(cV c A )
cosW 2
c T 2Q f sin W
f
V
c T
f
A
3
f
3
f
2
Effective Currents
• In Chapter 12 of H&M, we discussed effective
currents leading to matrix elements of the
form:
4G *T
J J
2
G
g2
2
8M
2
W
M CC
4G
2 J NC J NC *T
2
G
g2
2
2
8M
cos
W
2
Z
M NC
From this we get the relative strength of NC vs CC:
2
M
2 W2
M Z cos W
EWK Feynman Rules
Photon vertex:
ieQ f
Z vertex:
g
1
i
(cVf c Af 5 )
cosW 2
W vertex:
g 1
i
(1 5 )
2 2
Chapter 14 Outline
• Reminder of Lagrangian formalism
– Lagrange density in field theory
• Aside on how Feynman rules are derived from
Lagrange density.
• Reminder of Noether’s theorem
• Local Phase Symmetry of Lagrange Density leads to
the interaction terms, and thus a massless boson
propagator.
– Philosophically pleasing …
– … and require to keep theory renormalizable.
• Higgs mechanism to give mass to boson propagator.
Reminder of Lagrange Formalism
• In classical mechanics the particle equations
of motion can be obtained from the Lagrange
equation:
d L L
0
dt qÝ q
• The Lagrangian in classical mechanics is
given by:
L = T - V = Ekinetic - Epotential
Lagrangian in Field Theory
• We go from the generalized discrete
coordinates qi(t) to continuous fields (x,t), and
thus a Lagrange density, and covariant
derivatives:
L(q, qÝ,t) L(, , x )
d L L
L
L
0
0
dt qÝ q
x ( )
Let’s look at examples (1)
• Klein-Gordon Equation:
L
L
0
x ( )
1
1 2 2
L m
2
2
2
m 0
Note: This works just as well for the Dirac equation. See H&M.
Let’s look at examples (2)
Maxwell Equation:
L L
( A )
A 0
1
=>
L F F j A
4
L
j
A
L
1
A A A A
( A ) ( A ) 4
F
j
2
1
1
g g
A
A
A
2
A
A
2
( A )
2
F
Aside on current conservation
• From this result we can conclude that the EM
current is conserved:
F
F
j
A A
A A ( A A ) 0
• Where I used:
A A
Aside on mass term
• If we added a mass term to allow for a
massive photon field, we’d get:
( m )A j
2
This is easily shown from what we have done.
Leave it to you as an exercise.