LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free theory: one particle states: with Lorentz invariant normalization:
Recall, for a scalar field theory we defined an operator: that creates a particle localized in the momentum space near wave packet with width σ and localized in the position space near the origin. (go back to position space by Fourier transformation) is a state that evolves with time (in the Schrödinger picture), wave packet propagates and spreads out and so the particle is localized far from the origin in at. for in the past. In the interacting theory REVIEW is a state describing two particles widely separated is not time independent
A guess for a suitable initial state: Similarly, let s consider a final state: The scattering amplitude is then: we can normalize the wave packets so that where again and REVIEW
Similarly, let s define an operator: that creates a particle localized in the momentum space near wave packet with width σ and localized in the position space near the origin. (go back to position space by Fourier transformation) is a state that evolves with time (in the Schrödinger picture), wave packet propagates and spreads out and so the particle is localized far from the origin in at. for separated in the past. In the interacting theory is a state describing two particles widely is not time independent
A guess for a suitable initial state: Similarly, let s consider a final state: we can normalize the wave packets so that where again and The scattering amplitude is then: and similarly for d-type particles
A useful formula: Integration by parts, surface term = 0, particle is localized, (wave packet needed). is 0 in free theory, but not in an interacting one!
Thus we have: or for its hermitian conjugate: Similarly for d-type states: The scattering amplitude: we put in time-ordering symbol (without changing anything) extra minus sign for each exchange of operators!
The scattering amplitude can be written as: Lehmann-Symanzik-Zimmermann formula (LSZ)
The scattering amplitude for any process can be obtained from the time ordered product of creation and annihilation operators representing the initial and final states with the following replacements: For a Majorana field,, everything we derived holds. we can use expressions for b or d, whichever is more convenient
In the derivation of the LSZ formula we assumed that creation operators of free field theory work the same way in the interacting theory. The LSZ formula holds provided the field is properly normalized. For a Dirac field we require: required by Lorenz invariance required by charge conservation properly normalized one particle state, it can be achieved by rescaling the field
For a Majorana field we require: there is no conserved charge Renormalization of fields requires including the Z-factors in the lagrangian, e.g.: We obtained a formula for scattering amplitudes given in terms of correlation functions: now we need to calculate them...
For a free Dirac field we have: The free fermion propagator based on S-42 The simplest correlation function is the Feynman propagator: all other vanish
we find:
similarly:
following the same calculation as for the harmonic oscillator: we find: a polynomial function and we can write the Feynman propagator in a compact form:
Note: is a Green s function of the Dirac wave operator and similarly: Finally, all other two point functions vanish:
For a free Majorana field we have: the same as for the Dirac field but because of the Majorana condition, the other correlation functions do not vanish: similarly:
Correlation functions of more than two fields: Dirac field: all fields must pair up to form propagators extra minus sign from ordering Majorana field: note:
The path integral for fermion fields based on S-43 Recall that for a real scalar field we have: and the correlation functions are given by where and
For a complex scalar field we have: and the correlation functions are given by where
Define functional derivatives for anticommuting source variables: then for a free Dirac field we get the formula for the path integral: the same procedure as for a complex scalar field
Thus we have: where is the Feynman propagator, a Green s function for the Dirac equation:
we include interactions in the same way as for a complex scalar field: and the correlation functions are given by will lead to a Feynman diagram expansion
Similarly, for a Majorana field we have and the formula for the path integral: in analogy with the real scalar field where the inverse of the Majorana wave operator the minus sign, since we take 1/i for all fields
The Feynman rules for Dirac fields based on S-45 Consider a theory with a real scalar field interacting with a Dirac field: Yukawa theory the lagrangian is still invariant under the U(1) symmetry and so there is the corresponding Noether current: The path integral for this theory is: we can identify the b-type particle as and the d-type particle as.
The path integral for this theory is: where Imposing we can write it as: sum of connected Feynman diagrams with sources! (no tadpoles)
Let s look at the term in the expansion that has one vertex, two fermion propagators and one scalar propagators: the spin indices contracted in the obvious way
the arrow rule:
First few tree diagrams that contribute to iw: contributes e.g. to: we need to calculate connected correlation functions:
Let s start with: S = 1 corresponding diagrams with sources removed:
Similarly: S = 2 corresponding diagrams with sources removed: odd permutation of final states
Scattering amplitude for : for each external particle there is a Dirac or K-G operator acting on the corresponding propagator that generates delta functions; collecting exponentials and integrating over internal coordinates generates delta functions that conserve four-momentum in each vertex:
we defined: and we get: we don t have to write since the denominators never vanish
Similarly scattering amplitude for : is given as:
Feynman rules to calculate : external lines: vertex and the rest of the diagram incoming electron outgoing electron incoming positron outgoing positron incoming scalar outgoing scalar
vertex one arrow in and one out draw all topologically inequivalent diagrams for internal lines assign momenta so that momentum is conserved in each vertex (the four-momentum is flowing along the arrows) propagators the arrow for scalars can point both ways (represent momenta) for each internal scalar for each internal fermion
spinor indices are contracted by starting at the end of the fermion line that has the arrow pointing away from the vertex, write or ; follow the fermion line, write factors associated with vertices and propagators and end up with spinors or. assign proper relative signs to different diagrams follow arrows backwards! draw all fermion lines horizontally with arrows from left to right; with left end points labeled in the same way for all diagrams; if the ordering of the labels on the right endpoints is an even (odd) permutation of an arbitrarily chosen ordering then the sign of that diagram is positive (negative). sum over all the diagrams and get additional rules for counterterms and loops
Let s apply the rules to calculate the scattering amplitude for we easily find:
Let s apply the rules to calculate the scattering amplitude for we easily find:
Let s apply the rules to calculate the scattering amplitude for to determine the relative minus sign we easily find:
Gamma matrix technology based on S-47 Important formulas for gamma matrices: Trace formulas: in the same way we can show:
a simple consequence:
Proof: equal by the cyclic property of the trace
Traces involving gamma-5 matrices: follows from: traces with more gamma-5 matrices can be easily written as traces with zero or one gamma-5 in addition: so we only have to consider terms with even number of gamma matrices homework follows from:
Formulas for contracted gamma matrices: d = # of space-time dimensions homework homework Part of the homework set - 5 will be 47.1, 47.2, 47.3
Spin sums and spin averaged cross sections based on S-46,48 For the scattering amplitude of with momenta p k p k we found: we can simplify it using ; and write it as:
In order to calculate the cross section we need : we easily find: cyclic property of the trace and we can plug in the formulas for spinor products:
if we are not interested in specific spins of initial or final states, or we simply cannot measure them, we should sum over spins of final states and average over possible spins of initial states: 2 possible spins of the incoming electron we have the result in terms of traces of products of up to 4 gamma matrices.
we can write it as: where:
we find: we can plug the result to the formulae for differential cross section...
Something harder: For the scattering amplitude of we found: then and
averaging over the initial spins and summing over the final spins: we get:
we get:
the remaining two: putting it all together we get: we can plug the result to the formulae for differential cross section...