Derivation of Electro-Weak Unification and Final
Form of Standard Model with QCD and Gluons
 1W1+  2W2 +  3W3
Substitute B = cosW A  + sin W Z0

Sum over first generation particles.
up down
Left handed only
Flavor up
Flavor down
Flavor changing interactions.
Weak interaction terms
flavor changing: leptons
flavor changing: quarks
We want the coefficient for the electron-photon term to be -e
f=0 for neutrino and = 1 for others
A
-e
A
Z0
Z0
Consider only the A term:
ea1
ea2
gives agreement with experiment.
Cf = 2T3
 = -1
The following values for the constants
gives the correct charge for all the particles.
A
Z0
The Standard Model Interaction Lagrangian for the 1st generation
(E & M) QED interactions
weak neutral current interactions
weak flavor changing interactions
+
QCD color interactions
Weak neutral current interactions
Z0
Z0

Z0
Z0
Weak charged flavor changing interactions
quarks
leptons
g2
g2
Quantum Chromodynamics (QCD): color forces
Only non-zero
components of 
contribute.
To find the final form of the QCD terms, we rewrite the above sum,
collecting similar quark “color” combinations.
The QCD interaction Lagrangian density
Note that there are only 8 possibilities:
grrg-g
ggb-
The red, anti-green gluon
The green, anti-blue gluon
The gluon forces hold the
proton together
proton
At any time the proton
is color neutral. That is,
it contains one red, one
blue and one green
quark.
beta decay
u
d
u
d
d
u
W-
neutron
W doesn’t see color
proton
decay of -
-u
d
-
W production from
d
p
u
u
p-
-d
-u
-u
W+
ppp--
The nuclear force
u
n
u
d
d
d
u
u
p
W-
p
u
d
d
d
u
u
Note that W-  d + u- = - In older theories, one would
consider rather the exchange of a - between the n and p.
n
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 .
Feynman rules applied to a 2-vertex electron positron scattering diagram
Note that each vertex is
generated by the interaction
Lagrangian density.
time
spin
spin
metric tensor
Mfi =
left vertex function
coupling constant –
one for each vertex
right vertex function
propagator
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.
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.
G
normal free gluon term
Nf= # flavors
G
Note sign
3-gluon vertex
Nc= # colors
Nf
quark
loop
Nc
gluon
loop
momentum squared of exchanged gluon
Nf
Nc
 M2quark
Nf
Nc
-7
In QED one has no terms corresponding to the number of colors (the 3-gluon) vertex.
This term aslo has a negative sign.
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 generates
terms in the Lagrangian density which produce 3-gluon vertices – and gluon
loops in the exchanged gluon “propagator”.
The Higgs Lagrangian Contribution