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: 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. 3 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 interaction term 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: a1 + 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 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 = 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 33 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” red green blue red green blue green - red blue 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 Y 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).