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: 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 245 247 Recall, for a scalar field theory we defined an operator: that creates a particle localized in the momentum space near wave packet with width! 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 Similarly, let s define an operator: that creates a particle localized in the momentum space near for separated in the past. In the interacting theory is a state describing two particles widely is not time independent wave packet with width! 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. 246 248
A guess for a suitable initial state: Thus we have: Similarly, let s consider a final state: we can normalize the wave packets so that or for its hermitian conjugate: where again and Similarly for d-type states: The scattering amplitude is then: and similarly for d-type particles The scattering amplitude: we put in time-ordering symbol (without changing anything) extra minus sign for each exchange of operators! 249 251 A useful formula: The scattering amplitude can be written as: Integration by parts, surface term = 0, particle is localized, (wave packet needed). is 0 in free theory, but not in an interacting one! Lehmann-Symanzik-Zimmermann formula (LSZ) 250 252
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 we require: there is no conserved charge Renormalization of fields requires including the Z-factors in the lagrangian, e.g.: For a Majorana field,, everything we derived holds. we can use expressions for b or d, whichever is more convenient We obtained a formula for scattering amplitudes given in terms of correlation functions: now we need to calculate them... 253 255 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 free Dirac field we have: The free fermion propagator based on S-42 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 all other vanish The simplest correlation function is the Feynman propagator: 254 256
following the same calculation as for the harmonic oscillator: we find: we find: a polynomial function and we can write the Feynman propagator in a compact form: 257 259 Note: similarly: is a Green s function of the Dirac wave operator and similarly: Finally, all other two point functions vanish: 258 260
For a free Majorana field we have: The path integral for fermion fields based on S-43 Recall that for a real scalar field we have: the same as for the Dirac field but because of the Majorana condition, the other correlation functions do not vanish: and the correlation functions are given by where similarly: and 261 263 Correlation functions of more than two fields: Dirac field: For a complex scalar field we have: all fields must pair up to form propagators extra minus sign from ordering and the correlation functions are given by Majorana field: where note: 262 264
Define functional derivatives for anticommuting source variables: we include interactions in the same way as for a complex scalar field: then for a free Dirac field we get the formula for the path integral: the same procedure as for a complex scalar field and the correlation functions are given by will lead to a Feynman diagram expansion 265 267 Thus we have: Similarly, for a Majorana field we have where and the formula for the path integral: where in analogy with the real scalar field is the Feynman propagator, a Green s function for the Dirac equation: the inverse of the Majorana wave operator the minus sign, since we take 1/i for all fields 266 268
The Feynman rules for Dirac fields based on S-45 Let s look at the term in the expansion that has one vertex, two fermion propagators and one scalar propagators: 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 spin indices contracted in the obvious way 269 271 The path integral for this theory is: where Imposing we can write it as: the arrow rule: sum of connected Feynman diagrams with sources! (no tadpoles) 270 272
First few tree diagrams that contribute to iw: Similarly: S = 2 contributes e.g. to: corresponding diagrams with sources removed: we need to calculate connected correlation functions: odd permutation of final states 273 275 Let s start with: Scattering amplitude for : S = 1 for each external particle there is a Dirac or K-G operator acting on the corresponding propagator that generates delta functions; corresponding diagrams with sources removed: collecting exponentials and integrating over internal coordinates generates delta functions that conserve four-momentum in each vertex: 274 276
we defined: and we get: Feynman rules to calculate : external lines: incoming electron outgoing electron vertex and the rest of the diagram incoming positron outgoing positron incoming scalar we don t have to write since the denominators never vanish outgoing scalar 277 279 Similarly scattering amplitude for : vertex one arrow in and one out the arrow for scalars can point both ways (represent momenta) 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 is given as: for each internal scalar for each internal fermion 278 280
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. follow arrows backwards! Let s apply the rules to calculate the scattering amplitude for assign proper relative signs to different diagrams 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 we easily find: additional rules for counterterms and loops 281 283 Let s apply the rules to calculate the scattering amplitude for Let s apply the rules to calculate the scattering amplitude for to determine the relative minus sign we easily find: we easily find: 282 284
Gamma matrix technology based on S-47 Important formulas for gamma matrices: Proof: Trace formulas: in the same way we can show: equal by the cyclic property of the trace 285 287 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 a simple consequence: homework follows from: 286 288
Formulas for contracted gamma matrices: In order to calculate the cross section we need : we easily find: d = # of space-time dimensions cyclic property of the trace homework and we can plug in the formulas for spinor products: homework Part of the homework set - 5 will be 47.1, 47.2, 47.3 289 291 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: 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 can simplify it using ; and write it as: we have the result in terms of traces of products of up to 4 gamma matrices. 290 292
we can write it as: where: Something harder: For the scattering amplitude of then and we found: 293 295 averaging over the initial spins and summing over the final spins: we find: we get: we can plug the result to the formulae for differential cross section... 294 296
we get: 297 the remaining two: putting it all together we get: we can plug the result to the formulae for differential cross section... 298