Bjorken scaling

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Author: Prof. Wu-Ki Tung, Department of Physics and Astronomy Michigan State University

Bjorken Scaling refers to an important simplifying feature (scaling) of a large class of dimensionless physical quantities in elementary particles, notably the structure functions in deep inelastic scattering, that implies observed strongly interacting particles (hadrons) are made of point-like constituents. It was first proposed by James Bjorken in 1967. This idea, along with the contemporaneous concept of partons by Feynman, and the subsequent experimental discovery of such behavior, inspired the formulation of Quantum Chromodynamics (QCD), the modern fundamental theory of strong interactions, in 1974. Bjorken scaling is, however, not exact; it is mildly broken. The QCD theory can predict the logarithmic scale-breaking behavior of the relevant physical quantities; and these predictions have been fully confirmed by modern high energy experiments.

1 Introduction
2 Origins
- 2.1 The Bjorken Limit and Bjorken Scaling
- 2.2 Breaking of Bjorken Limit and Scaling
3 Experimental Observation of Bjorken Scaling
4 Scale Breaking According to QCD
5 Experimental Evidence of QCD Predictions on Scaling

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Introduction

In 1968, Bjorken proposed that the structure functions measured in lepton-nucleon deep inelastic scattering (DIS, Cf. Fig. 1), $W_{i}(Q^{2},\nu )$ , may exhibit scaling behavior in the asymptotic limit,

(1)

$\lim_{Q^{2}\rightarrow \infty ,\,\nu /Q^{2}\,\mathrm{fixed}}~\nu W_{2}(Q^{2},\nu ) =MF_{2}(x) \quad ; \quad \lim_{Q^{2}\rightarrow \infty ,\,\nu /Q^{2}\,\mathrm{fixed}}~MW_{1}(Q^{2},\nu ) =F_{1}(x)$

Figure 1: Inclusive lepton-hadron deep inelastic scattering.

where $Q^{2}$ represents the squared 4-momentum-transfer vector $q$ of the exchanged virtual vector boson ( $\gamma ,W,Z$ ), $\nu =q\cdot p/M$ the energy loss of the scattering leptons ( $l_{1,2}$ ), $M$ the target nucleon ( $p$ ) mass; and the dimensionless variable $x=Q^{2}/2M\nu$ is the Bjorken $x$ scaling variable.

The cross section for inclusive DIS of an lepton (electron, muon, or neutrino) on a nucleon, depicted in Fig.\,1, is given in terms of the structure functions as

(2)

$\sigma _{\mathrm{DIS}}\sim \sigma _{0}\left[ W_{2}+2W_{1}\tan ^{2}(\frac{ \theta }{2})\right]$

where $\sigma _{0}$ is the well known Mott cross sections for scattering of an electron on a point-like charged particle, and $\theta$ is the scattering angle of the lepton $l_{2}$ in the laboratory frame. This formula resembles that of elastic scattering of an electron on a nucleon, with $W_{1,2}$ taking the place of the electromagnetic form factors of the nucleon, $F_{i}(Q^{2})$ , $i=1,2$ . $F_{i}(Q^{2})$ had been known to fall rapidly as a function of $Q^{2}$ , reflecting the finite size of the nucleon charge distribution. Therefore, the general expectation for $\sigma _{\mathrm{DIS}}$ before its measurement was that it should also be a fast falling function of $Q^{2}$ . Bjorken's scaling proposition, expressed by the $Q$ -independence of the right-hand side of Eq. 1, would contradict this expectation. It would imply that the nucleon target appears as a collection of point-like constituents when probed at very high energies in DIS (implied by the $Q^{2}\rightarrow \infty$ limit on the left-hand side of Eq. 1).

The famous SLAC-MIT experiment on DIS, carried out at the Stanford Linear Accelerator Center at about the same time as the theoretical proposal of scaling, discovered that the measured $\sigma _{\mathrm{DIS}}$ indeed exhibit approximately the scaling behavior of Eqs. 1 & 2. Fig. shows some early results of this experiment. The DIS data points at three different center-of-energy energies are plotted against the variable $Q^{2}$ . The approximately $Q$ -independent behavior is in sharp contrast to the fast falloff of the elastic form factor shown in the same plot for comparison.

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