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Fundamentals of Neutrino Physics and Astrophysics$

Carlo Giunti and Chung W. Kim

Print publication date: 2007

Print ISBN-13: 9780198508717

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198508717.001.0001

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(p.664) Appendix E Feynman Rules Of The Standard Electroweak Model

(p.664) Appendix E Feynman Rules Of The Standard Electroweak Model

Source:
Fundamentals of Neutrino Physics and Astrophysics
Publisher:
Oxford University Press

The Feynman rules allow one to calculate the transition amplitude A of a given process. In the following, we summarize the Feynman rules of the Standard Electroweak Model in the unitary gauge, for the calculation of tree diagrams. This is sufficient for the topics discussed in the book. The calculation of higher order diagrams with loops is much more complicated (see, for example, Refs. [25, 634, 314, 917, 720, 721]). In section E.4 we give general formulas suitable for the calculation of the cross-section or decay rate of a process from its amplitude.

The Feynman rules for the calculation of the tree-level amplitude A of a given process are as follows:

  1. 1. Draw all connected tree diagrams which contribute to the process under consideration by using the external lines in section E.1, the internal lines in section E.2 and the vertices in section E.2.

  2. 2. For each external line, write down the corresponding quantity in section E.1.

  3. 3. For each internal line, write down the corresponding propagator in section E.2.

  4. 4. For each vertex, write down the corresponding quantity in section E.3.

  5. 5. Enforce energy–momentum conservation in each vertex.

  6. 6. Assign a relative factor –1 to diagrams which differ only by an interchange of two external lines (the overall sign is irrelevant).

E.1 External lines

p is the particle momentum, r is the spin index for fermions and α is the spin index for spin-one bosons.

(E.1) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.2) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.3) Appendix E Feynman Rules Of The Standard Electroweak Model
(p.665)
(E.4) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.5) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.6) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.7) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.8) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.9) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.10) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.11) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.12) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.13) Appendix E Feynman Rules Of The Standard Electroweak Model

E.2 Internal lines

(E.14) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.15) Appendix E Feynman Rules Of The Standard Electroweak Model
(p.666)
(E.16) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.17) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.18) Appendix E Feynman Rules Of The Standard Electroweak Model

E.3 Vertices

The coefficient q f is the charge of the fermion f in units of the elementary charge e: q ν = 0, q e = q µ = q τ = –1, q u = q c = q t = 2/3, q d = q s = q b = –1/3. The coefficients g V f and g A f are given in Table 3.6 (page 78). From eqn (3.42) we have gsinϑw = e.

(E.19) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.20) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.21) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.22) Appendix E Feynman Rules Of The Standard Electroweak Model
(p.667)
(E.23) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.24) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.25) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.26) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.27) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.28) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.29) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.30) Appendix E Feynman Rules Of The Standard Electroweak Model
(p.668)
(E.31) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.32) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.33) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.34) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.35) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.36) Appendix E Feynman Rules Of The Standard Electroweak Model

E.4 Cross-sections and decay rates

The differential cross-section of a process with two particles a, b in the initial state and N f particles in the final state is given by

(E.37) Appendix E Feynman Rules Of The Standard Electroweak Model
where p a, p b are the four-momenta of the two initial particles, m a, m b are their masses, p f = ( E f , p f ) is the four-momentum of the fth final particle, P i =p a+p b and (p.669) P f = f=1 N f are, respectively, the total four-moment a of the initial and final states, A is the total amplitude of the process (the sum of the amplitudes of the diagrams contributing to the process), and the symbol spin ¯ indicates an average over the unobserved spin states of the initial particles and a sum over the unobserved spin states of the final particles. S is a statistical factor given by the product of a factor 1/n! for each set of n identical particles.

The cross-section of a process of the type

(E.38) Appendix E Feynman Rules Of The Standard Electroweak Model
with unpolarized particles, depends on the four-momenta of the particles only through the three Lorentz-invariant Mandelstam variables
(E.39) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.40) Appendix E Feynman Rules Of The Standard Electroweak Model
(E.41) Appendix E Feynman Rules Of The Standard Electroweak Model
where we used the energy–momentum conservation
(E.42) Appendix E Feynman Rules Of The Standard Electroweak Model
Only two Mandelstam variables are independent, since117
(E.43) Appendix E Feynman Rules Of The Standard Electroweak Model

The decay width of a particle with mass m in a final state with N f particles is given by

(E.44) Appendix E Feynman Rules Of The Standard Electroweak Model
where p is the four-momentum of the initial fermion. (p.670)

Notes:

(117) This relation can be straightforwardly obtained by summing all the expressions for the three Mandelstam variables in eqns (E.39)(E.41)).