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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 1 + 3 + 8 spin = 1 force carrying gauge particles.

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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

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

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“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.

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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.

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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

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Flavor flipping “rotation”:

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Summary: QED local gauge symmetry Real function of space and time covariant derivative The final invariant L is given by:

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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

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The matrices don’t commute! They commute with themselves, but not with each other:

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Non-Abelian Gauge Field Theory Non-Abelian means the SU(n) group has non-commuting elements.

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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.

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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.

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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

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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

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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) = 3 2 - 1 = 8 generators 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 =

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1 f 123 = 1 = - f 213 = f 231 12 3 = 2 i f 123 3 [ 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.

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Example of a “color rotation” on the down quark color triplet Components of determine the “rotation” angles

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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

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

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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

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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.

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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.

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*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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