Goal:
i. mesons
ii. hadrons
iii. exotics
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.
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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 4vectors.
[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 pointèThe 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
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There are Lorentz invariants
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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
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These two equations for
electricity reduce to
where
is the 4current.
The same holds for magnetism. If
we take the magnetostatic equation
which tells us that there are no
"true" magnetic charges, and the magnetodynamics equation
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 longrange 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
·
Lefttoright 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 


straight line,
arrow to the right 


straight line,
arrow to the left 
positron 


wavy line 
photon 


An electron emits a photon 




An electron absorbs a photon 




A positron emits a photon 




A positron absorbs a photon 




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




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+BèC+D
Rotating external lines is only significant if the line move across the 90^{o} line. This causes the particle to turn into its antiparticle and the reaction changes.
Aè 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.
A(p1)+B(p2)èC(p3)+D(p4) same amplitude A(p1) èC(p3)+D(p4) +antiB(p2)
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+BèC+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 4momentum given to the internal line then we have:
A B vertex 
P_{A}^{μ} + P_{B}^{μ} 
s channel 
A C vertex 
P_{A}^{μ}  P_{C}^{μ} 
t channel 
A D vertex 
P_{A}^{μ}  P_{D}^{μ} 
u channel 
The allowable vertices depend on the types of particles.
e^{+} + e^{} è 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
uèd, cès, νèe^{ }, e^{+}è ν_{bar} 
These emission graphs are W_{+} 
dèu, sèc, e^{}è ν^{ }, ν_{bar} è e^{+} 
These emission graphs are W_{} 
uèu, dèd, νè ν^{ }, e^{+}è e^{+} 
These emission graphs are Z_{o} 
uèd, cès, νèe^{ }, e^{+}è ν_{bar} 
These absorption graphs are W_{} 
dèu, sèc, e^{}è ν^{ }, ν_{bar} è e^{+} 
These absorption graphs are W_{+} 
uèu, dèd, νè ν^{ }, e^{+}è e^{+} 
These absorption graphs are Z_{o} 
èu_{bar}d, èc_{bar}s, è e^{} ν_{bar} 
These production graphs are W_{} 
èd_{bar}u, è s_{bar}c, èe^{+} ν 
These production graphs are W_{+} 
u_{bar}dè, c_{bar}sè, e^{} ν_{bar}è 
These annihilation graphs are W_{} 
d_{bar}uè, s_{bar}cè, e^{+} νè^{ } 
These annihilation graphs are W_{+} 
èu_{bar}u, èc_{bar}c, è ν ν_{bar},_{ }è e^{} e^{+} 
These annihilation graphs are Z_{o} 
u_{bar}uè, c_{bar}cè, ν ν_{bar}è,_{ } e^{} e^{+}è 
These production graphs are Z_{o} 
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,ν_{e}) 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^{} ,Z^{o} 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. e_{R}, e_{L}, μ_{R},_{ }μ_{L}. 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+NèP+N P+NèP+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+NèP+N P+NèP+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.