## Feynman Rules

In this Appendix we collect together all of the rules for the calculation of relativistic cross sections and decay rates. The rules fall into two parts. There are rules for the calculation of the cross section and decay rates from the relativistic scattering matrix, M, and then there are rules for calculating M in a given theory. The former are quite general, but the latter, referred to as the Feynman rules, depend on the specific theory.

B.l DECAY RATES AND CROSS SECTIONS

The rules for calculation of relativistic decay rates and cross sections were derived in Sees. 9.2 and 9.3.

The differential n-body decay rate, dWn, for a particle with energy E is rbtained from the following factors:

• a factor of (27r)4<54(p/ -p,), where p/ is the total four-momentum of the n decay products and pi is the four-momentum of the decaying particle,

• a factor of d3kj (2n)32Eki for each particle in the final state, where kx and E^ are the momentum and energy of the ith particle,

• a factor of 1 /2E for the initial particle which is decaying, and

• the absolute square of the .M-matrix.

The differential decay rate is then where s and s' are the spins of the initial particles.

Finally, in calculating both decay rates and differential cross sections, for each set of m identical particles in the final state, the integrals over momenta must either be divided by m! or limited to the restricted cone 0\ < 02 < ■ ■ ■ < 0m.

### B.2 GENERAL RULES

The Feynman rules for calculation of the .M-matrix depend of the theory used to do the calculation. The basic rules are given first, and then the forms required for specific theories. Any diagram will either be a tree diagram (with no loops) or will have one or more closed loops.

• The diagrams consist of lines and vertices.

• Each internal line represents the propagation of a particular particle from one space-time point to another, and the vertices are the points in spacetime where particles are created or destroyed, as described by the interaction Lagrangian of the theory.

• Label the momenta of each external particle, and use energy-momentum conservation to determine the four-momentum of each internal line. Tree diagrams have no closed loops, and each internal momentum can be fixed in terms of the external momenta. Loop diagrams have momenta which cannot be uniquely specified, and these must be integrated over. There will be one undetermined four-momentum for each loop.

The Feynman rules tell how to associate a number with each Feynman diagram. There are several basic rules from which the number is constructed:

Rule 1: an operator for each vertex, the precise form of which depends on the theory and the particular particles involved. Rule 2: a propagator for each internal line with four-momentum k, the precise form of which depends on the particle propagating. For spin zero bosons with isospin indices i,j, for fermions with Dirac indices a, /3, and for photons or massive vector bosons with polarization indices /i, v, the forms are:

Fig. B.l Symmetry factors for bubbles with identical neutral bosons.

In all cases, k is constrained by momentum conservation, and for the photon or gluon, a is the gauge parameter (which could be unity).

Rule 3: for fermions, assemble the incoming fermion spinors, vertex operators, propagators, and outgoing fermion spinors in order along each fermion line to make a well-formed matrix element. In particular:

• multiply from the left by u(p_,s_) for each outgoing fermion with momentum p_ and spin s_.

• multiply from the right by u(k-, s_) for each incoming fermion with momentum Jfc_ and spin s_.

• multiply from the right by v(p+is+) for each outgoing an-tifermion with momentum p+ and spin s+.

• multiply from the left by v(k+,s+) for each incoming antifermion with momentum k+ and spin s+.

for photons and vector bosons, construct well-formed vector products by saturating any free vector polarization indices p on current operators by:

• multiplying by e* for each outgoing particle with polarization index p.

• multiplying by for each incoming particle with polarization index p.

Rule 4: • symmetrize between identical bosons in the initial or final state.

• antisymmetrize between identical fermions in the initial or final state.

Rule 5: integrate over each internal four-momentum k not fixed by energy-momentum conservation with a weight f d4k J (2tt)« '

Rule 6: for each closed fermion loop, a minus sign. Rule 7: multilpy by the proper symmetry factor, which is \ for bubbles with two identical neutral bosons of the type shown in Fig. B.l A and 1/3! for bubbles with three identical neutral bosons of the type shown in Fig. B.1B.  QCD:

 ^ a, ß g fabc [gnu a,H,k +g»
 Y 7M[l-4Sin20w]-75) e J e (W) V 1 V lw~ -igw cosOw-y* (l - j5) « I V

B.3 SPECIAL RULES 605 Standard Model (boson self-couplings and Higgs couplings in unitary gauge):  606

where T(a) is the generalization of (a - 1)! to non-integer numbers. It has the properties r(a+ 1) = ar(a) r(2) = r(i) = i

where 7 = 0.5772 • • • is Euler's constant and the last relation holds when e is infinitesimal. We will also have use for the 5-function, defined as follows: