Lecture 10:
Standard Model Lagrangian
The Standard Model Lagrangian is obtained by imposing
three local gauge invariances on the quark and
lepton field operators:
U(1) “QED-like”
SU(2) weak
SU(3) color
gauge boson
 neutral gauge boson
 3 heavy vector bosons
 8 gluons
This gives rise to 1 + 3 + 8 spin = 1 force carrying gauge particles.
SU(2) and SU(n)
dot product
Pauli spin matrix
functions of x,y,z,t
The  are called the
generators of the group.
n = 2  3 components
 3 gauge particles
Group of operators, U = exp[i /2]
Expanding the group operation (rotation) …
SU(2): rotations in Flavor Space
“rotated” flavor state
original flavor state
These are the Pauli spin
matrices, 1 2 3
local  depends on x, y, z, and t.
Flavor Space
Flavor space is used to describe an intrinsic property of a particle. While this is not
(x,y,z,t) space we can use the same mathematical tools to describe it.
Flavor space can be thought of as a three dimensional space.
The particle eigenstates we know about (quarks and leptons)
are “doublets” with flavor up or down – along the “3” axis.
Example of a rotation in flavor space:
flavor space
electron field operator
 = (0, ’, 0) is along the “y”
direction of flavor space.
even terms
odd terms
Flavor flipping “rotation”:
Summary: QED local gauge symmetry
Real function of space and time
covariant derivative
The final invariant L is given by:
SU(2) local gauge symmetry
generator of SU(2)
rotations in
flavor space!
covariant derivative
The final invariant L is given by:
coupling constant
generators of SU(2)
interaction term
The  matrices don’t commute!
They commute with themselves,
but not with each other:
Non-Abelian Gauge Field Theory
Non-Abelian means the SU(n) group has non-commuting elements.
Rotations in flavor space (SU(2) operations) are local and non-abelian.
The group SU(2) has an infinite number of elements, but all operations can
“generated” from a linear combination of the three  operators:
a1 + b  1 + c  2 + d  3
These i are called the generators of the group.
The gauge bosons: W+ W- W0
There is a surprise coming later: the W0 is not the Z0.
Later we will see that the gauge particle from U(1)
and the W0 are linear combinations of the photon
and the Z0 .
Rotations (on quark states) in color space: SU(3)
The quarks are assumed to carry an additional property called color. So,
for the down quark, d, we have the “down quark color triplet”:
quark field operators
= d red
= d green
= d blue
There is a color triplet for each quark: u, d, c, s, t, and b,
but, for now we won’t need the t and b.
A general “rotation” in color space can be written as a local,
(non-abelian) SU(3) gauge transformation
generators of SU(3)
a = 1,2,3,…8
Since the a don’t commute, the SU(3) gauge transformations
are non-abelian.
The generators of SU(3): eight 3x3
matrices (a = 1,2,3…8)
1 =
2 =
3 =
4 =
5 =
6 =
7 =
8 =
(n2 – 1) = 32 - 1 = 8 generators
All 3x3 matrix elements of SU(3) can be written as a linear combination
of these 8 a plus the identity matrix.
[ 11, 22]
the a don’t commute
= 2i f1123 33
 f123 = 1 = - f213 = f231
Likewise one can show: (for the graduate students)
fabc = -fbac = fbca
f458 = f678 =
3 /2,
f147 = f516 = f246 = f257 = f345 = f637 = ½
… all the rest = 0.
Example of a “color rotation” on the down quark color triplet
Components of  determine the “rotation” angles
even power of 2
odd power of 2
- 1st term in cosine series.
looks like a rotation about z
red and green “flip”
SU(3) gauge invariance in the Standard Model
generators of SU(3)
generators of SU(3)
The invariant Lagrangian density is given by:
interaction term
The Lagrangian density with
the U(1), SU(2) and SU(3)
gauge particle interactions
neutral vector boson
heavy vectors bosons (W, W3)
8 gluons
What we have left to sort out:
1. The Standard Model assumes that the neutrinos have
no mass and appear only in a left-handed state. This
breaks the left/right symmetry – and one must divide
all the quarks and leptons into their left handed and
right handed parts. The W interacts only with the
left handed parts of the quarks and leptons.
2. Incorporate “unification” of the weak and electromagnetic force field using Weinberg’s angle, w
B = cos w A  - sin w Z0 
W0 = sin w A  + cos w Z0 
3. Sort out the coupling constants so that in all interactions
involving the photon and charged particles the coupling
will be proportional to e, the electronic charge.
Summary of the Standard Model covariant derivative:
Standard Model:
gauge particles
Standard Model
covariant derivative
When this Standard Model (SM) covariant derivative is substituted for  in the
Dirac Lagrangian density one obtains the SM interactions!
… more about the color rotations to follow.
*SO(3,1) has 6 generators: 3 for rotations, 3 for boosts.
It is isomorphic to SU(2) x SU(2).

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