1. Derivation of Electro-Weak Unification and Final Form of Standard Model with QCD and Gluons
 1W1+  2W2 +  3W3

2. Substitute B = cosW A  + sin W Z0 
Sum over first generation particles.
text error up
down
Left handed only
text error
Flavor up
Flavor down
Flavor changing interactions.

3. Weak interaction terms
flavor changing: leptons
flavor changing: quarks

4. 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

5. Consider only the A term:
gives agreement with experiment.
Cf = 2T3
 = -1

6. The following values for the constants
give the correct charge for all the particles.

7. Coefficients for the Z0 term
g1 = g2 sinW/cos W
YfL = 2 [ Q f – T3]
f YfR = 2 Q f
Z0
Z0
Cf cos W = 2T3 [1-sin2 W]/ cos W
 = -1

8. Final Form for Electro-Weak Interaction Terms
(the QCD terms have yet to be written in terms of the color/anti-color gluons) A
Z0
In the text by Gordon Kane, p.92, Eq. 7.32, there
appears to be a typo in the sign of the third term.

9. - The Standard Model Interaction Lagrangian for the 1st generation +
(E & M) QED interactions
weak neutral current interactions -
weak flavor changing interactions +
QCD color interactions

10. Weak neutral current interactions
Z0 Z0 Z0
Z0
note: no flavor changes

11. Weak charged flavor changing interactions
quarks
-g2
leptons

12. Quantum Chromodynamics (QCD): color forces
Only non-zero
components of 
contribute.

13. To find the final form of the QCD terms, we rewrite the above sum,
collecting similar quark “color” combinations.

14. The QCD interaction Lagrangian density

15. grg - - ggb Note that there are only 8 possibilities: r g
The red, anti-green gluon -
ggb
The green, anti-blue gluon

16. At any time the proton is color neutral. That is,
The gluon forces hold the
proton together
At any time the proton is color neutral. That is,
it contains one red, one blue and one green quark.
proton

17. beta decay u u d d d u W-
proton
neutron
W doesn’t see color

18. decay of - - u d -

19. -
W production from -
p p p d u p u W+ - d - u - p p - u

20. n p p n The nuclear force u u d d d u u W- 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. -

22. 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 .

24. 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
right vertex function
coupling constant –
one for each vertex
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.

25. 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.

26. G G Nf= # flavors Nc= # colors Nc Nf Note sign
normal free gluon term
3-gluon vertex
Nf= # flavors
Nc= # colors Nc Nf
quark
loop
gluon
loop

27. The terms given explicitly in M are only those loops shown in the previous diagram. Higher order terms are indicated by “ + … “
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 has a negative sign.

28. 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”.

29. Conditions on the SU(2) Gauge Particle Fields which
complete the Invariance of the Lagrangian Density

30. = D’ ’ = D’ [ei/2 ] = ei/2 D 
We want to find  W such that the following is satisfied:
D’ ’ = D’ [ei/2 ] = ei/2 D 
cancel =

31.  0 and the k are linearly independent so [ …] = 0 and the following is the expression for  W

32. The Higgs Lagrangian Contribution