derivation of electro-weak unification and final form of standard model with qcd and gluons 1 w 1 +...
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Derivation of Electro-Weak Unification and Final Form of Standard Model with QCD and Gluons
1W1+ 2W2 + 3W3
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Substitute B = cosW A + sin W Z0
Sum over first generation particles.
Flavor changing interactions.
Left handed only
Flavor up Flavor down
up down
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Weak interaction terms
flavor changing: leptons flavor changing: quarks
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We want the coefficient for the electron-photon term to be -e
-e
f=0 for neutrino and = 1 for others
A
A
Z0
Z0
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Consider only the A term:
ea1 ea2
gives agreement with experiment.
Cf = 2T3
= -1
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The following values for the constants gives the correct charge for all the particles.
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A
Z0
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(E & M) QED interactions
weak neutral current interactions
weak flavor changing interactions
QCD color interactions
-
+
The Standard Model Interaction Lagrangian for the 1st generation
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Weak neutral current interactions
Z0
Z0
Z0
Z0
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quarks
leptons
Weak charged flavor changing interactions
g2
g2
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Quantum Chromodynamics (QCD): color forces
Only non-zerocomponents of contribute.
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To find the final form of the QCD terms, we rewrite the above sum,collecting similar quark “color” combinations.
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The QCD interaction Lagrangian density
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The red, anti-green gluon
The green, anti-blue gluon
Note that there are only 8 possibilities:
r ggrg-ggb
-
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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
proton
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neutron
proton
beta decay
ud
u
d
d
W doesn’t see color
u
W-
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decay of -
-
u
d
-
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p
p
duu
uud---
W production from
p-
p
p p
-
-
W+
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The nuclear force
np
u
d
d
u
d
u
u
u
d
u
pn
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.
-
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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 .
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Feynman rules applied to a 2-vertex electron positron scattering diagram
left vertex function right vertex function
Mfi =
spinspin
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 aa function of the momenta only, and the four spin (helicity) states.
coupling constant –one for each vertex
Note that each vertex isgenerated by the interactionLagrangian density.
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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. Theseterms lead to the interaction of gluons with other gluons.
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G G
quarkloop
gluonloop
NfNc
Nf= # flavors Nc= # colors
normal free gluon term3-gluon vertex
Note sign
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momentum squared of exchanged gluon
Nf Nc
Nc
Nf
In QED one has no terms corresponding to the number of colors (the 3-gluon) vertex. This term aslo has a negative sign.
-7
M2quark
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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”.
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The Higgs Lagrangian Contribution