Lecture 10: Standard Model Lagrangian The Standard Model Lagrangian is obtained by imposing three local gauge invariances on the quark and lepton field operators: symmetry: gauge boson U(1) “QED-like”  neutral gauge boson SU(2) weak  3 heavy vector bosons SU(3) color  8 gluons This gives rise to spin = 1 force carrying gauge particles.

n = 2  3 components  3 gauge particles SU(2) and SU(n) dot product The  are called the generators of the group. Pauli spin matrix functions of x,y,z,t

Group of operators, U = exp[i  /2 ] Expanding the group operation (rotation) …

“rotated” flavor stateoriginal flavor state SU(2): rotations in Flavor Space These are the Pauli spin matrices,  1  2  3 local  depends on x, y, z, and t.

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 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.

Example of a rotation in flavor space: even termsodd terms electron field operator flavor space  = (0,  ’, 0) is along the “y” direction of flavor space. 3

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 The final invariant L is given by: generators of SU(2) interaction term coupling constant generator of SU(2) covariant derivative rotations in flavor space! 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 - 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”: red green blue = d red green 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. quark field operators

A general “rotation” in color space can be written as a local, (non-abelian) SU(3) gauge transformation local generators of SU(3 ) Since the a don’t commute, the SU(3) gauge transformations are non-abelian. a = 1,2,3,…8 red green blue

The generators of SU(3): eight 3x3 a matrices (a = 1,2,3…8) All 3x3 matrix elements of SU(3) can be written as a linear combination of these 8 a plus the identity matrix. (n 2 – 1) = = 8 generators 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 =

1  f 123 = 1 = - f 213 = f = 2 i f [ 1, 2 ] the a don’t commute 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

odd power of 2 looks like a rotation about z - 1 st term in cosine series. red green blue red green blue - green red red and green “flip” even power of 2

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

The Lagrangian density with the U(1), SU(2) and SU(3) gauge particle interactions neutral vector boson heavy vectors bosons (W , W 3 ) 8 gluons Y

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 electro- magnetic force field using Weinberg’s angle,  w B  = cos  w A  - sin  w Z 0  W 0  = sin  w A  + cos  w 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.

Standard Model covariant derivative gauge particles Standard Model: Summary of the 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). *