Lecture 10: Standard Model Lagrangian The Standard Model Lagrangian is obtained by imposing three...
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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
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.
Slide 2
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
Slide 3
Group of operators, U = exp[i /2 ] Expanding the group
operation (rotation)
Slide 4
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.
Slide 5
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.
Slide 6
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
Slide 7
Flavor flipping rotation:
Slide 8
Summary: QED local gauge symmetry Real function of space and
time covariant derivative The final invariant L is given by:
Slide 9
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
Slide 10
The matrices dont commute! They commute with themselves, but
not with each other:
Slide 11
Non-Abelian Gauge Field Theory Non-Abelian means the SU(n)
group has non-commuting elements.
Slide 12
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.
Slide 13
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.
Slide 14
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 wont need the t and b. quark
field operators
Slide 15
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 dont commute, the SU(3) gauge transformations are
non-abelian. a = 1,2,3,8 red green blue
Slide 16
The generators of SU(3): eight 3x3 a matrices (a = 1,2,38) 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 =
Slide 17
1 f 123 = 1 = - f 213 = f 231 12 3 = 2 i f 123 3 [ 1, 2 ] the a
dont 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 = all the rest = 0.
Slide 18
Example of a color rotation on the down quark color triplet
Components of determine the rotation angles
Slide 19
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
Slide 20
SU(3) gauge invariance in the Standard Model The invariant
Lagrangian density is given by: interaction term generators of
SU(3)
Slide 21
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
Slide 22
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 Weinbergs 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.
Slide 23
Slide 24
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.
Slide 25
*SO(3,1) has 6 generators: 3 for rotations, 3 for boosts. It is
isomorphic to SU(2) x SU(2). *