This page will serve as an outline of the material covered.  Consider this a

rough draft.  Some details may need to be corrected but I wanted to be able to provide a rough outline since we are not following closely with the book at this point.

There are many sources that layout the basic aspects for the standard model. For example,

This week we reviewed elements of the Standard Model:

• Charge conservation is an example of a U(1) transformation that doesn’t change the physics. This is nothing more than the statement that one has freedom to change the overall phase of a wf

without changing the physics.  This is of course conservation of probability and with such a conservation law you have a rule for how the stuff moves around. We should recognize

as a simple rule that if stuff flows in or out of a volume then the density increases or decreases. The overall amount of stuff remains the same. Currents are an important concept for particle physics. They are normally treated in 4-vector notation.  The commonly know electric charge current becomes

a combination of the three vector current and the charge density. The U(1) transformation is important. It will be used to generate the E&M force.

• Lepton sector
• leptons (e,μ,τ, charge, mass) (ν, neutrinos, neutral, small mass)
• Three generations and Conservation of Lepton Number (Le Lμ, Lτ  )
• The states of  (e, ν) as an SU(2) representation and the presence of this structure in the SM. This structure is important because making this transformation local (Gauge transformations), the weak interaction of the standard model will mix partners in these 2-particle multiplets. The Weak interaction in some sense sees this pairing as the natural structure for the leptons. Examine

The weak interaction will use the pairs as currents (more obvious when we add Feynam diagrams and more interaction structure). Clearly for weak processes a u quark is not a stable entity. The u-d particle however is.

• Six quarks u,d,s,c,t,b as discussed above grouped together as SU(2) partners.
• The quark generations are not labeled by a conservation law but there are three generations just as for the leptons.
• We begin to recognize that grouping things together and describing the transformations that mix members of these groups is important.  The example of spin 1/2 as an SU(2) grouping is very helpful. The rotations of SU(2) will in general mix a spin ½ state such that all possible orientations of the spin vector are included and therefore all possible linear combinations of projections are included. So rotation mixes the + and – spin states. The mixing of these states is due to the non-commutivity of the generators or in other words we cannot choose Jx, Jy and Jz as simultaneously observable. J and Jz label the states usually chosen. These are not mixed by rotation about the z-axis. With momentum and energy Px, Py, Pz, H all commute.  A state can be labeled by all these observables. The irreducible representations are trivial 1-plets. Each label,fore example the energy, is independent and has only a single value as the state label. SU(2) is more complex and can requires j-plets for a particle description.

• More discussion: (Further discussion also at http://www.upscale.utoronto.ca/GeneralInterest/DBailey/SubAtomic/Lectures/LectF13/Lect13.htm )
• Transformation mixes or makes changes.
• Rotations, Translations, SU(2) Isospin
• Physics is the same è Conservation law and a current rule as to how the quantity moves around in space so that the overall amount stays the same.
• The stuff that doesn’t change in the physics reactions is sometimes the stuff that is mixed in a way by the transformations but our goal is to choose labels that do not change for the global transformations.
• To be specific, for all quantum states the transformations need to be unitary (anti unitary perhaps).

Therefore the transformation U has to be expressible in the above form where the Z’s represent a set of operators that are connected with observables (charge, momentum, spin, upness, downess). There can be several of these operators connected with one class of transformations.

·        Spatial translations è momentum px, py, pz therefore 3

·        Time translation è energy H, 1

·        Rotationsè angular momentum Jx,Jy,Jz

·        Isospin rotation è momentum τx, τy, τz (note z component is upness or downess for the u-d pair)

·        Charge conservation, phase change è Q

The observables are conserved if the physics doesn’t change under the U transformation of the wave function. Also remember that the generators may not all be simultaneously observed since the generators can be a commuting or a non-commuting set. The relationships between these generators determine the structure of the mixing of the states. (A rotation about x mixes the z-components but a rotation about z doesn’t).

o       We review the original motivation for SU(2) and isospin to clarify the similarity between the proton and the neutron for the nuclear force.  As quarks are added to the hadron sector the group will grow. SU(2) (u,d) becomes SU(3) (u,d,s) when the strange flavored quark is added.  Is this a true symmetry? No it is due to the mass degeneracy when compared to the scale of the strong forces. For example, the proton and neutron are approximately the same mass, which is perhaps evidence of a very small difference between up and down quarks.  The strange quark is not as small an influence but it is small enough to be included into a useful approximate symmetry.  Interestingly the SU(2) introduced for the nuclear force is the same pairing that we have used for the Weak pairing. However it is a different symmetry.

o       SU(2) weak isospin (e,ν) (u,d) appears to be the same at some level as original isospin proposed to understand neutron proton similarity. It combines (u,d). However it is based on proprieties of the weak interaction.

o       SU(2) increases to SU(3) flavor for (u, d, s). Weak Isospin groups the same particles together but as we will see reflects a more fundamental relationship that underlies the interaction structure of the elementary particles.

o       Additional quarks are too massive to be included in this scheme.

o       SU(3) flavor is a reflection of the fact that under the strong interaction bound quark systems are quite similar even though the quark content varies. If quark flavor is irrelevant than SU(3) becomes an exact symmetry.

o       RHH  and LHH (right handed and left handed helicity).  We look at the polarization of the classical transverse E&M field.  The electric field vector is perpendicular to the direction of propagation (assume z). So that the linear polarization is a two dimension vector in x-y basis. Normally traveling waves are treated in complex notation.

This allows one to use a different basis

The circularly polarized light has an E field that rotates around the z-direction Cw and CCW.  This corresponds to the LHH and RHH states. The spin of the particle is projected onto the momentum direction. These are called helicity states. Notice that the transverse photon doesn’t have a projection of zero spin. For massless particles the helicity states are good state labels. For particles with mass, the helicity changes with boosts. (A particle could be boosted to a frame which reversed the direction of momentum but not spin.) Helicity is therefore not a Lorentz invariant label for particle with mass.

There is another right handed and left handed identifier. This RHC and LHC.  This is a different spin projection. The fermions and quarks that make up the standard model are all spin ½ particles so under the Dirac treatment using 4-vector spinors this projection operators become [1± γ5]. There is a projection operator that takes any state |Φ > and projects out the left or right handed pieces such that the state can be written.

|Φ > =  L > +  R >.

This projections is referred to as chirality.

Left handed Chirality: [1- γ5]

Right handed chirality: [1+ γ5]

RHC and LHC states are Lorentz invariant as long as I don’t allow for reflections (Parity).

NOTE:

so the [1± γ5] projection combines a scalar and psuedoscalar piece.

For massless particles the chiral states and the helicity states are the same. As mentioned particles that have mass mix the helicity states when reference frames are changed but the chiral states are Lorentz invariants (Parity not allowed).

It should be noted that when there is no mass the states of different helicity do not mix and under these conditions SU(2) of spin will become SU(2)LxSU(2)R

(see Dan Kabat, http://www.phys.columbia.edu/~kabat/particles/index.html , lecture 4.) LH and RH sometimes refers to spin along the momentum or helicity.

LH spinor or RH spinor is chirality the behaviour under the Lorentz transformation ([1± γ5] behaviour). These are different notions that converge for massless particles.

The reason we are discussing helicity and the notion of Right Handed and Left Handed particles is that the SU(2) of weak isospin is only applied to a Left Handed multiplet.  The SU(2) of weak isospin is denoted SU(2)L. The actual states then become

(eLL), eR (uL,dL) uR,dR

where the left handed particles are paired and the right handed partners are not.  The resulting interaction of the left handed piece is maximally parity violating because it contains these chiral states.