Goal:

  1. Learn the basic elements in the standard model:
    1. fundamental particles
    2. Conservation laws
    3. Interactions
    4. Quark states

                                                    i.     mesons

                                                  ii.     hadrons

                                                iii.     exotics

  1. Understand reactions
    1. apply Feynman diagram methods as a starting point to understand reactions
    2. Identify symmetry constraints
    3. recognize important processes and their implications

 

 

To start lets take a simple world that has a charged particle and a E&M field. As in all particle physics our formulation needs to incorporate realativity.

 

If we imagine that the particle is an electron we will need to specify some parameters that identify the particle. Above we find charge, mass and spin as particle labels. In addition we will also need to specify other aspects of the state that characterize the particles motion. This could be characterized by providing for example the particle momentum.

 

In relativity the representation of position and momentum is as 4-vectors.

[Wikipedia: In terms of covariance and contravariance of vectors, lower indices represent (components of!) covariant vectors (covectors), while upper indices represent (components of!) contravariant vectors (vectors): they transform covariantly (resp., contravariantly) with respect to change of coordinates. Subtle pointThe autohor wants to highlight the fact that you can define the two vectors as components time basis vectors. He prefers to keep the idea of a basis vector separated. Thus you define one basis and two types of vectors. Only the components of these vectors are presented as values.]

Wolfram

Contravariant

There are Lorentz invariants

 

 

E&M

Here the electric and magnetic fields are grouped to form a second rank tensor. However the field can also be represented in terms of fields

 

These two equations for electricity reduce to

\partial_{\beta} F^{\alpha\beta} = \mu_0 J^{\alpha} \,

where

J^{\alpha} = ( c \, \rho , \vec{J} ) \,is the 4-current.

The same holds for magnetism. If we take the magnetostatic equation

\vec{\nabla} \cdot \vec{B} = 0

which tells us that there are no "true" magnetic charges, and the magnetodynamics equation

\frac{ \partial \vec{B}}{ \partial t } + \vec{\nabla} \times \vec{E} = 0

which tells us the change of the magnetic field with respect to time plus the curl of the Electric field is equal to zero (or, alternatively, the curl of the electric field is equal to the negative change of the magnetic field with respect to time). With the electromagnetic tensor, the equations for magnetism reduce to

 

Decide that the ability to provide the first order Feynman will be important. Therefore the students should be able to draw the diagrams for any interaction. The focus will be primarily on the quark and lepton level but strong interactions will be mediated on the long-range scales by quark exchange. Simplest model for proton neutron interactions is pion exchange.

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Following excerpted from Stanford website

http://www2.slac.stanford.edu/vvc/theory/feynman.html

   Left-to-right in the diagram represents time; a process begins on the left and ends on the right.

   Every line in the diagram represents a particle; the three types of particles in the simplest theory (QED) are:

NOTE: (There are different conventions for the direction of time. Some choose to have time develop vertically and so the diagrams are rotated.

 

 

Image

Description

Particle Represented

f-electron.gif (164 bytes)

straight line, arrow to the right

electronGlossary Term

f-positron.gif (159 bytes)

straight line, arrow to the left

positron

f-photon.gif (211 bytes)

wavy line

photon

f-a.gif (344 bytes)

An electron emits a photon

 

 

f-b.gif (335 bytes)

An electron absorbs a photon

 

 

f-c.gif (333 bytes)

A positron emits a photon

 

 

f-d.gif (344 bytes)

A positron absorbs a photon

 

 

f-e.gif (322 bytes)

A photon produces an electron and a positron (an electron-positron pair)

 

 

f-f.gif (331 bytes)

An electron and a positron meet and annihilate (disappear), producing a photon

 

 

 

 

 

 

 

 

 

 

Need to identify internal lines and external lines when diagrams are assembled to describe processes. Lines that ultimately enter or leave are real particles and must have a physical mass corresponding to the particle type. Internal lines are mediating the interaction. That is they are exchanged in order to have a momentum and energy transfer or interaction between two physical particles. These field particles do not have to have a correct mass based on the particle type. At all vertices the momentum and energy are conserved. Sum of all the incoming is equal to the outgoing for the three (four) particles that combine to from a vertex.

 

 

So the QED vertex

One can rotate any arm as follows:

 

 

We will need to examine which rotations are significant. In order to do this we need to identify internal and external lines. Therefore we draw the lowest order diagram that represents a process:

 

A+BC+D

 

Rotating external lines is only significant if the line move across the 90o line. This causes the particle to turn into its antiparticle and the reaction changes.

A antiB + C + D as shown above.

Moving particle from one side to the other or rotating external arms is based on crossing symmetry. If one is actually performing a calculation the question remains do elements of the underlying calculation change when one explores the crossing symmetry related diagrams.

 

  1. change the sign of the four momentum so that the particle is now a negative energy state: p is the four momnetum

A(p1)+B(p2)C(p3)+D(p4) same amplitude A(p1) C(p3)+D(p4) +antiB(-p2)

  1. fermions have an additional factor of -1 for the amplitude

 

Rotating internal lines simply changes the time ordering of the vertices. All such time orderings are assumed to be included so only one of the diagrams is drawn.

 

 

Therefore there is only the need to draw on of these drawings to describe the process A+BC+D

 

However there may be other drawings. These can be found by deciding which of the external lines are meeting at a vertex.

 

If we look at the possibilities and calculate the 4-momentum given to the internal line then we have:

 

A B vertex

PAμ + PBμ

s channel

A C vertex

PAμ - PCμ

t channel

A D vertex

PAμ - PDμ

u channel

The allowable vertices depend on the types of particles.

 

e+ + e- e+ + e- (E&M)

 

Allows two of the diagrams above, the incoming particles (A,B) can form a vertex or outgoing (A,C) can form a vertex but not (A,D).

 

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Weak vertices

 

ud, cs, νe- , e+ νbar

These emission graphs are W+

du, sc, e- ν , νbar e+

These emission graphs are W-

uu, dd, ν ν , e+ e+

These emission graphs are Zo

ud, cs, νe- , e+ νbar

These absorption graphs are W-

du, sc, e- ν , νbar e+

These absorption graphs are W+

uu, dd, ν ν , e+ e+

These absorption graphs are Zo

ubard, cbars, e- νbar

These production graphs are W-

dbaru, sbarc, e+ ν

These production graphs are W+

ubard, cbars, e- νbar

These annihilation graphs are W-

dbaru, sbarc, e+ ν

These annihilation graphs are W+

ubaru, cbarc, ν νbar, e- e+

These annihilation graphs are Zo

ubaru, cbarc, ν νbar, e- e+

These production graphs are Zo

Due to quark mixing the any d,s,b can be replaced by a d,s,b.

 

 

Here are some weak processes from the lepton sector. The external lines show the fundamental fields to be the (e,νe) and (μ,ν μ). For the weak interaction there is no difference between electron and neutrino. They are manifestations of the same particle. The interactions will need W+, W- ,Zo to mediate the interaction so they will be included.

(ASIDE: Also to carefully define the states the electron and muon need to have right and left handed versions. eR, eL, μR, μL. These are spin projection that are chosen by projecting onto the momentum direction. Only left handed leptons are present in the above diagrams. This detail can be neglected without difficulty when establishing and overview of the interactions.)

Most of the hadrons will interact over long distances by pion exchange because color singlets are not allowed. Typically we can follow these by showing the quark lines. The gluons have not been shown in the following diagrams.

Very close range interaction Standard nuclear force

P+NP+N P+NP+N

 

Diagrams on the left are not color singlets so they must be very short range. While the diagrams on the right exchange a particle that is a color singlet.

Very close range interaction Standard nuclear force

P+NP+N P+NP+N

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Weak interaction

 

 

fundamental vertex spectator quarks added

 

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Weak

 

fundamental vertex spectator quarks added

 

The QCD interactions are complicated because the quarks come in three colors but the color structure of objects that make up composite systems is not typically shown. In principle one can simply add gluons in the same manner that one adds photons. The only change will be in the color which is not shown. The usefulness of the Feynman diagrams requires a bit more care. Since there can be appreciable gluon exchange an expansion in terms of graphs is only appropriate in some situations.