Lecture 21: Apr 10th 2012
Reading Griffiths chapter 7
HW due Thursday: 7.37, 7.39
Calculate the e+e- +- CM frame matrix element. In the ultra-relativistic limit show that it obeys the s,t,u crossing relationship with e-- e--
0) Feynman rules for iM
a) Draw a diagram labeling all the incoming and outgoing momentum and spins. Use arrows to indicate particles (with direction of time) and antiparticles (opposite direction of time) and internal lines (preserving continuity of the direction of flow). Note down which spins are known and which will be summed over (final or internal state spins) or averaged over (initial state spins if unknown).
b) Write down spinors for particles and antiparticles. u for incoming electrons and for outgoing electrons. for incoming positrons and for outgoing positrons. for incoming photons and for outgoing photons. For instance for an incoming electron, 1, we write and for an outgoing electron, 3,
c) Vertex factors of ige, use a different index for each vertex term.
Each vertex represents a separate interaction and has its own index
As an example with particles a vertex factor is written between the outgoing (first) and incoming (last) particle to form a current term.
b and c form the current terms, example , which will reduce to factors proportional to momentum just as in our scalar particle transition rate. This is the basic unit for a particle in transition and is called a particle current, i.e. the J we derived.
Where there is an initial or final state photon at the vertex the vertex factor will be summed over with the photon if there is one. = .
d) Propagators for the photons or electrons/positrons: -ig/q2,i(q + m)/(q2 - m2)
Note in the photon propagator case the g matrix converts to so it is summed with of the first current. The propegator is what relates the two interactions in this case.
In the case of the electron propagator each current has a photon which is summed with thevertex and the propagator has it’s own sum over q, so no further sums have to be done to get a scalar result for the matrix element probability.q is often written as
e) Add an integral over the internal four momentum d4q/(2)4 and a four dimensional delta function for each vertex and use one to integrate over the internal four momentum and cancel the other since it corresponds to overall four momentum conservation.
f) antisymmetrization. Crossed diagrams will exist for identical sets of particles in the final state. The second diagram will get a negative sign (Fermi statistics).
Diagrams can have their electrons moved from the initial to the final state where they occur as a positron producing an equivalent in probability diagram ( a type of crossing symmetry).To have a consistent convention you have to be careful about the signs in those cases.
1) Non relativistic scattering of unpolarized identical particles
Before tackling the more “complex” relativistic case where we must evaluate all the matrix and vector multiplications it is instructive to consider a non relativistic case where the matrix math simplifies.
,,, N=(|E|+m)
,,
If p<m then all the p terms in the spinors go to 0 because they are divided by m. Then using iuj = 2mij,= 1,2 spins and the fact that only the 0 matrix will contribute since the other are anti-block diagonal.
Then using the result
Now consider all the various spin combinations so we can later average over the initial states and sum over the final states.
identical particles in the end state so both the t and u channel diagrams contribute
and
Note that the summation condition over and u does not allow this interaction to flip the spins. This is expected at first order for slow moving electrons, since they will only interact via the electric field, which we know from the atom does not continually flip spins. However, at higher energy spin flips are possible because the moving electrons are starting to represent a strong current and can interact via the magnetic part of the field. Also for the first set of spins we have true identical particles we must include crossed diagrams in M. In the second two matrix elements we also do not flip the spins since that would create a sum over and u that is 0, but the crossed u channel diagram leaves particle 4 with the spin of particle 1 so that matrix multiplication is not 0.
To evaluate the full matrix element for unpolarized we have to average over the number of initial spin states and sum over the final spin states.
The probability will be proportional to the matrix element times it’s complex conjugate. Each of the three processes above is independent so we will square each and then add them. Averaging over initial spin states introduces two factors or ½. Summing over the final states each matrix element there are three distinct possibilities that occur twice each.
substituting in our expressions for t and u and simplifying
and putting this in:
d/d = (1/8)2 |M|2 pi/pf 1/E2
where in the non relativistic limit E = 2me
This type of result will be typical even in the full treatment in that we will end up with a number of s, t and u type scattering or annihilation terms.
3) Unpolarized relativistic electron-electron or electron-muon scattering
For results where we will sum over spins the terms of the matrix element squared can also be substantially simplified.
The matrix element is:
consider just the first term. The first term also is relevant for electron-muon scattering
Each term is a scalar and the gamma matrix indices indicate any necessary sums
This allows us to group terms with the same particles together.
Consider terms such as
where the adjoint gamma matrix
Evaluate the sum over the spins of particle 1 and then write indices of the matrix multiplication explicitly to understand how to contract the sum over the spins of particle 3
Contracting the row and column matrices of particle three will force i=j. Move the ajoint operator for particle 3 to the opposite end and then sum over the spins of particle 3.
Thus in general
where all the traces of gamma matrices and momentum vectors have been worked out.
Thus in our example
4 is from the averaging over initial spins
Using trace identities
10) Trace of an odd number of gamma matrices is 0
12)
17)
Often we evaluate these matrix elements in relativistic limits.
In the ultra relativistic limit the Madelstam variables become:
s = ,
t = ,
u = ,
and
note t =
which is valid if the lepton masses can be taken as the same.
This cross section includes and infinity at theta = 0 which corresponds to infinite range scattering.
4) Crossing symmetry and e-e+ -+
Starting from e-- e-- you can get to e-e+ -+ by moving the final state electron electron (p3) to the initial state as a positron (-p2) and by moving the initial state muon (p2) to become a final antimuon (-p3).
Since you are just switching p3 with –p2 or p2 with –p3 then this is equivalent to changing s and t
Which is quite interesting considering one of the interactions is a scattering interaction and the other annihilation! This only works at high energy. At low energy there is not enough energy to form the muon pair and there is no s channel process. Also notice that the annihilation interaction is not divergent. This makes sense since you can’t have annihilation at infinite range. In fact we can calculate the total cross section.
In CM d/d = (1/8)2 |M|2 pi/pf 1/E2
using the exact expression for s, t and u in CM
s = ,
t = , and
u = ,
which can be integrated!
The cross section is finite and goes down at higher energy, which iswhat is observed.
The t channel scattering cross section is still divergent at =0. The point =0 corresponds to the point where b=. That the total cross section is infinite just represents the fact that a 1/r2 potential still has a small effect no matter how far away you get. However, we are interested in computing scattering cross section or probabilities that for a particle between b and b+db to scatter to angle theta. As you get to very large b you are probing very small scattering angles between ~0 and +d. The cross section and the probability to go to any of the angles will still be finite for any finite b.