The Standard Model Lagrangian is obtained by imposing three local gauge invariances on the quark and lepton field operators: symmetry: gauge boson
U(1)
SU(2)
SU(3)
“QED-like”
 neutral gauge boson weak
 3 heavy vector bosons color

8 gluons
This gives rise to 1 + 3 + 8 spin = 1 force carrying gauge particles.

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

= exp[i
 
/2
]
Expanding the group operation (rotation) …

“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 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.

flavor space electron field operator even terms

= (0,

’, 0) is along the “y” direction of flavor space.
3 odd terms

Real function of space and time covariant derivative
The final invariant L is given by:

generator of SU(2) rotations in flavor space!
covariant derivative
The final invariant L is given by: coupling constant generators of SU(2)


They commute with themselves, but not with each other:

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 a

1 + b


1
+ c


2
+ d


3
 operators:
These
 i are called the generators of the group.

The gauge bosons: W + W W 0
There is a surprise coming later: the W 0 is not the Z 0 .
Later we will see that the gauge particle from U(1) and the W 0 are linear combinations of the photon and the Z 0 .

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 red
= d red green
= d green blue = 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 local generators of SU(3
) red green blue 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
 a matrices (a = 1,2,3…8)

1
=

4
=


5
2
=
=


3
6
=
=

7
=

8
=
(n 2 – 1) = 3 2 - 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.

[

1
1
,

2
2
] the
 a don’t commute
= 2 i f 1
123
3

3
 f
123
= 1 = - f
213
= f
231
Likewise one can show: (for the graduate students) f abc
= -f bac
= f bca f
458
= f
678
= 
3 /2, f
147
= f
516
= f
246
= f
257
= f
345
= f
637
= 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
- 1 st term in cosine series.
red green blue looks like a rotation about z red and green “flip” red green blue
green red blue

SU(3) gauge invariance in the Standard Model generators of SU(3)
The invariant Lagrangian density is given by: generators of SU(3) interaction term

neutral vector boson heavy vectors bosons (W

, W 3 )
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
W

0

= cos
 w
= sin
 w
A
A


- sin
 w
+ cos
 w
Z 0
Z

0

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


Dirac Lagrangian density one obtains the SM interactions!
in the
… 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).