Derivation of Electro-Weak Unification in the Standard Model
Derivation of Electro-Weak Unification and Final Form of Standard Model with QCD and Gluons 1W 1+ 2W 2 + 3W 3 Substitute B = cosW A + sin W Z0 Sum over first generation particles. up down
text error Left handed only text error 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: R
ea1 ea2 gives agreement with experiment. Cf = 2T3 = -1 The following values for the constants give the correct charge for all the particles.
Coefficients for the Z0 term g1 = g2 sinW/cos W YfL = 2 [ Q f T3] f YfR = 2 Q f Z0 Z0 Cf cos W = 2T3 [1-sin2 W]/ cos W
= -1 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. 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 Z 0 note: no flavor changes 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 doesnt see color proton decay of -
-u d - W production from p d
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 quark loop Nc gluon loop 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. 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 its 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. Conditions on the SU(2) Gauge Particle Fields which complete the Invariance of the Lagrangian Density We want to find W such that the following is satisfied: D = D [ei/2 ] = ei/2 D cancel
= 0 and the are linearly independent so [ ] = 0 k and the following is the expression for W The Higgs Lagrangian Contribution
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