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Quantum ChromodynamicsHigh Energy Experiments and Theory$

Günther Dissertori, Ian G. Knowles, and Michael Schmelling

Print publication date: 2009

Print ISBN-13: 9780199566419

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199566419.001.0001

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(p.425) Appendix B Building Blocks Of Theoretical Predictions

(p.425) Appendix B Building Blocks Of Theoretical Predictions

Quantum Chromodynamics
Oxford University Press

(p.425) Appendix B

Building Blocks Of Theoretical Predictions

B.1 The Feynman rules of Qcd

Feynman diagrams provide a very useful pictorial device in which each component of a diagram represents a part of the algebraic expressions for the corresponding S-matrix amplitude. When using the Feynman rules, to go from a diagram to an algebraic expression, you are advised to pay careful attention to the directions of the momenta and the order of any indices. These details are important because they impact on the relative signs of the various terms which in turn help to ensure gauge invariance.

External quarks and gluons correspond to basis spinors and polarization vectors as shown below. Ghosts are scalars and therefore have trivial unit basis states.

Appendix B Building Blocks Of Theoretical Predictions

Internal particles correspond to propagators, which are colour diagonal, as shown. The sign of the infinitesimal, imaginary part iє is chosen so as to ensure causality. In the case of the gluon the Lorentz tensor d µν(p) depends on the choice of the gauge fixing term and the gauge parameter ξ. Two common choices are:

(B.1) Appendix B Building Blocks Of Theoretical Predictions
In the physical gauges ghosts do not appear in the Feynman diagrams, but in the covariant gauges they are required in order to preserve unitarity; see Section 3.3.3

(p.426) Particle interactions are represented by vertices. We have the gluon-quark, gluon-ghost, triple-gluon and quartic-gluon vertices corresponding to the following algebraic factors:

Appendix B Building Blocks Of Theoretical Predictions
In all these graphs the convention is that all momenta are outgoing and so sum to zero. In the gluon-ghost vertex the momentum is that of the outgoing ghost. In the gluon self-couplings, observe that both vertices are symmetric upon interchange of all the labels on any pair of legs. Implicit in each of these vertices is a four-momentum conserving δ-function. Each internal line is accompanied by an integral over its four-momentum. This results in an overall four-momentum conserving δ-function which is absorbed into the phase space definition.

Since quarks and ghosts are fermionic, for every closed loop involving them in a diagram, an additional factor −1 must be included. Furthermore, when a pair of identical fermions is present in the external state of two diagrams, then the ‘crossed’ diagram acquires a minus sign relative to the ‘uncrossed’ diagram, again to account for the anticommutativity of fermions. Finally, when n identical particles are present in the final state of a set of diagrams, then a symmetry factor 1/n! must be included in the amplitude.

Note that we have suppressed the spinor indices on the quark propagators and vertices. The correct ordering of these terms is given by working backwards along the individual fermion lines. This prescription will also give the correct (p.427) ordering of any colour matrices.

For completeness we also include some relevant Feynman rules from the standard electroweak theory. Here the propagators for the vector bosons, V = γ, W± or Z, are given by

(B.2) Appendix B Building Blocks Of Theoretical Predictions
whilst their couplings to fermions take the form
(B.3) Appendix B Building Blocks Of Theoretical Predictions

The individual coefficients appearing in this expression are collected in the following table:






e f




1/(2 sinθ w cos θ w)

I 3 f - 2ef sin2 θ w

I 3 f

V f   f / ( 2 2 sin θ w )



Here for the fermion f, ef is its electric charge measured in units of the positron charge e > 0; I 3 f is its third component of weak isospin, h = + 1 2 for up-type quarks or neutrinos, and h = 1 2 for down-type quarks or charged leptons. For the corresponding antiparticles the signs are reversed. For charged current interactions involving quarks, the coefficients V ff are the respective elements of the Cabbibo-Kobayashi-Maskawa matrix, the dominant elements of which are Vud ≈ 0.975 ≈ V cs, V us ≈ 0.222 ≈ -V cd and V tb ≈ 1. For leptons one effectively has V νit. ≈ δee’ The parameter 6 w is the weak mixing angle with sin2θw ≈ 0.223.

B.2 Phase space and cross section formulae

Once the amplitude squared for a process has been evaluated, it is necessary to include the flux factor and the (differential) phase space in order to obtain the (differential) cross section. We consider the general process p a+p bp 1+…+p n for which the cross section is given schematically by

(B.4) Appendix B Building Blocks Of Theoretical Predictions
Here it should be understood that the cross section and phase space are typically multi-differential quantities. For head-on collisions the flux factor is given by
(B.5) Appendix B Building Blocks Of Theoretical Predictions
In the second line the flux is given in terms of the C.o.M. momenta, p a * = p b * and the laboratory variables p a lab and p b lab = ( m b , 0 ) . The third line is appropriate (p.428) in the limit of negligible particle masses. In the case of a particle decay the flux factor is given by twice the decaying particle’s mass, flux = 2M.

A differential element of the Lorentz invariant n-body phase space for the outgoing particles is given by

(B.6) Appendix B Building Blocks Of Theoretical Predictions
In the second version the on mass-shell δ-function has been explicitly integrated out and the positive energy solution E i = + p i 2 + m i 2 selected. Using eqn (B.6) and eqn (B.4) it is easy to verify that the dimensionality of the phase space is given by 3n, whilst the mass dimension of |M|2 must be 4–2n.

In many practical situations the incoming particles are unpolarized and the spins of the final state particles are not measured. The same applies for their colours. To take this into account one has to sum the amplitude squared over the spins and colours of the outgoing particles and average over the spins and colours of the incoming particles. Thus, in eqn (B.4) we use

(B.7) Appendix B Building Blocks Of Theoretical Predictions
where the colour degeneracy is N R = N c for a quark or an antiquark and N c 2 -1 for a gluon and where we allow two spin polarizations for the external fermions and massless external gluons or photons.