Bjorken

We would like to point out some steps of derivation of the nonrelativistic limit Hamiltonians by means of the Foldy-Wouthuyisen transformation (Bjorken and Drell, 1964). The method is based on the transformation of a relativistic equation of motion to the Schrodinger equation form. 

Bjorken J. and Drell S. Relativistic quantum mechanics, McGraw-Hill, New York,... 

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill Book Co., NY, 1964. 

J.B. Bjorken, S.D. Drell, Relativistic Quantum Fields (Me Graw-Hill, New York, 1964)... 

Beutler Hans, 592 Bienko Dariusz C., 650 Bielejewska Anna, 978 Bird R. Byron, 881 Bishop David M., 592 Bixon Mordekhai, 340 Bjorken James D., 124... 

The textbook by Bjorken and DreU is very popular for good reason and can be highly recommended for further reference. The book is accompanied by a more introductory volume on relativistic quantum mechanics by the same authors [98]. 

The techniques of field theory are highly sophisticated involving subtle questions of renormalization etc. (see e.g. Bjorken and Drell, 1964, 1965 Itzykson and Zuber, 1980). At this point we require only a simple heuristic appreciation of certain features. 

Further details, in particular the rules for calculating Feynman diagrams in QED, QCD and in weak interactions, are given in Appendix 2. For a complete treatment the reader is referred to Bjorken and Drell (1964, 1965) and Itzykson and Zuber (1980). 

When a Dirac particle is moving fast enough so that its energy E m it can be shown that its spinor becomes an eigenstate of 75 (see, for example, Section 7.10 of Bjorken and DreU, 1964). If and v are spinors for particles and antiparticles respectively, and if we denote by R (right-handed) states with hehcity +5 and by L (left-handed) states with helicity — then one has... 

The production rate for the Bjorken process (Bjorken, 1977) Z Hjji. pr compared with Z —> is shown in Fig. 6 3. This is expected... 

The surprising lack of dependence of vW2 and Wi on was in a sense predicted by Bjorken (1969). He studied Wap y, in the following mathematical limit (now referred to as the Bjorken limit )... 

In analogy to the spin independent case [see eqn (15.5.4)) the functions Gi,2(t > Q ) multiplied by kinematic factors are expected to obey Bjorken scaling. One thus writes... 

In the following chapter we shall see how the quark-parton model explains the occurrence of Bjorken scaling and we shall obtain expressions for all the scaling functions in terms of number densities for the quark-partons in a hadron. 

Q /2mN ) plays a role. We have thus obtained the remarkable features of the deep inelastic structure functions mentioned earlier, and satisfied the Bjorken result (15.5.4). Indeed, in the above naive parton model we have... 

We shall see later that diagrams of this type play a crucial role in the breaking of Bjorken scaling and in providing a slow dependence for the scaling functions. This behaviour will turn out to be one of the pieces of evidence in favour of QCD. 

The original Bjorken analysis when applied to Gi,2 would suggest... 

To end this brief survey we show how to derive the scaling version of the Bjorken sum rule (15.6.35) from the parton model (Hey, 1974). 

The quark-paxton model was developed in great detail in the previous chapter. Here we discuss the experimental situation and confront the theory with the vast amoimt of data on deep inelastic scattering and related reactions. It must be borne in mind that we have not yet discussed the QCD corrections to the model. These are not small, but their dominant effect can be taken into account by allowing the parton number densities to depend upon in a way calculable in QCD, so that, fortuitously, the entire formalism is basically unchanged, except that each qj x) —> qj x,Q ). As mentioned earlier this implies a dynamical breaking of perfect Bjorken scaling. It also implies that if one is seeking accurate information about the qj x) from experiment then care must be taken to specify the involved. 

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