1 W 1 2 W 2 3 W 3 Substitute B cos W A sin W Z 0 Sum over first generation particles Flavor changing interactions ID: 261950
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Slide1
Derivation of Electro-Weak Unification and Final Form of Standard Model with QCD and Gluons
1
W
1
+
2
W
2
+
3
W
3Slide2
Substitute B
=
cos
W
A
+ sin
W Z0
Sum over first generation particles.
Flavor changing interactions.
Left handed only
Flavor up
Flavor down
up
down
text error
text errorSlide3
Weak interaction terms
flavor changing: leptons
flavor changing: quarksSlide4
We want the coefficient for the electron-photon term to be
-e
-e
f
= 0 for neutrino and = 1 for others
A
A
Z
0
Z
0
Slide5
Consider only the A
term:
e
a
1
e
a2
gives agreement with experiment.
C
f
= 2T
3
= -1
RSlide6
The following values for the constants
give the correct charge for all the particles. Slide7
Coefficients for the
Z0
term
Z
0
Z0
g
1
=
g
2
sin
W
/
cos W
f
Y
fR = 2
Q f
C
f
cos
W
= 2T
3
[
1-sin
2
W
]/
cos
W
= -1
Y
f
L
=
2 [
Q
f
– T
3
]
Slide8
A
Z
0
Final Form for Electro-Weak Interaction Terms(the QCD terms have yet to be written in terms of the color/anti-color gluons)
In the text by Gordon Kane, p.92, Eq. 7.32, there appears to be a typo in the sign of the third term. Slide9
(E & M) QED interactions
weak neutral current interactions
weak flavor changing interactions
QCD color interactions
-
+
The Standard Model Interaction
Lagrangian
for the 1
st
generationSlide10
Weak neutral current interactions
Z
0
Z
0
Z
0
Z
0
note: no flavor changesSlide11
quarks
leptons
Weak charged flavor changing interactions
-g
2
-g
2Slide12
Quantum
Chromodynamics
(QCD): color forces
Only non-zero
components of
contribute. Slide13
To find the final form of the QCD terms, we rewrite the above sum,
collecting similar quark “color” combinations.Slide14
The QCD interaction
Lagrangian
density Slide15
The
red
, anti-green gluon
The green
, anti-blue gluon
Note that there are only 8 possibilities:
r
g
g
r
g
-
g
g
b
-Slide16
At any time the proton is color neutral. That is,
it contains one
red
, one blue
and one green quark.
The gluon forces hold the
proton together
protonSlide17
neutron
proton
beta decay
u
d
u
d
d
W doesn’t see color
u
W
-Slide18
decay of
-
-
u
d
-Slide19
p
p
d
u
u
u
u
d
-
-
-
W
production from
p
-
p
p
p
-
-
W
+Slide20
The nuclear force
n
p
u
d
d
u
d
u
u
u
d
u
p
n
d
d
u
W
-
Note that W
-
d
+
u
=
-
In older theories, one would consider rather the exchange of a
-
between the n and p.
-Slide21Slide22
Cross sections and Feynman diagrams
everything happens here
transition probability amplitude
must sum over all possible Feynman diagram amplitudes with the same initial and final states .Slide23Slide24
Feynman rules applied to a 2-vertex electron positron scattering diagram
left vertex function
right vertex function
M
fi
=
spin
spin
time
propagator
metric tensor
The next steps are to do the sum over
and and carry out the matrix multiplications.
Note that
is a 4x4 matrix and the
spinors
are 4-component vectors. The result is a
a function of the
momenta
only, and the four spin (
helicity
) states.
coupling constant –
one for each vertex
Note that each vertex is
generated by the interaction
Lagrangian
density. Slide25
Confinement of quarks
free quark terms
free gluon terms
quark- gluon interactions
The free gluon terms have products of 2, 3 and 4 gluon field operators. These
terms lead to the interaction of gluons with other gluons.
Slide26
G
G
quark
loop
gluon
loop
N
f
N
c
N
f
= # flavors
N
c
= # colors
normal free gluon term
3-gluon vertex
Note signSlide27
momentum squared of exchanged gluon
N
f
N
c
N
c
N
f
In QED one has no terms corresponding to the number of colors (the 3-gluon) vertex.
This term has a negative sign.
-7
M
2
quark
The terms given explicitly in M are only those loops shown in the previous diagram. Higher order terms are indicated by “ + … “Slide28
Quark confinement arises from the increasing strength of the interaction at
long range. At short range the gluon force is weak; at long range it is strong.
This confinement arises from the SU(3) symmetry – with it’s non-commuting
(non-
abelian) group elements. This non-commuting property generatesterms in the Lagrangian density which produce 3-gluon vertices – and gluonloops in the exchanged gluon “propagator”. Slide29
Conditions on the SU(2) Gauge Particle Fields which
complete the Invariance of the Lagrangian DensitySlide30
D’
’ =
D
’
[
ei/2 ] =
ei/2
D
=
cancel
We want to find
W
such that the following is satisfied: Slide31
0 and the
k
are linearly independent s
o [ …] = 0 and the following is the expression for
W
Slide32
The Higgs
Lagrangian
Contribution