Bjorken scaling

Bjorken scaling (after J. Bjorken , who introduced it in 1969) describes in physics a dependence of the structure functions in the case of deep inelastic scattering (e.g. of electron and proton ) on only one kinematic variable.

This behavior corresponds to an elastic scattering at point objects, which led to the development of the Parton model .

In the case of inelastic scattering, a dependency on two independent kinematic variables is actually expected; However, this does not occur due to the internal structure of the proton, since it is effectively scattered on individual quarks .

Mathematical formulation

For inelastic electron-proton scattering, the cross-section can generally be written with the structure functions as: ${\displaystyle W_{1},W_{2}}$

${\displaystyle {\frac {d^{2}\sigma }{d\Omega \,dE^{\prime }}}=\left({\frac {d\sigma }{d\Omega }}\right)_{\text{Mott}}\left[2W_{1}(Q^{2},\nu )\,\tan ^{2}(\theta /2)+W_{2}(Q^{2},\nu )\right]}$.

It is

• ${\displaystyle \left({\tfrac {d\sigma }{d\Omega }}\right)_{\text{Mott}}}$the Mott cross section
• ${\displaystyle Q^{2}=-q^{2}=(p_{e}-p_{e}^{\prime })^{2}}$the ( four ) momentum transfer
  • ${\displaystyle p_{e}}$the electron impulse
• ${\displaystyle \nu ={\tfrac {Pq}{M}}=E_{\text{Labor}}-E_{\text{Labor}}^{\prime }}$ the energy transfer
  • ${\displaystyle P}$the quadruple momentum of the target (e.g. a proton)
  • ${\displaystyle M}$ the mass of the target
• ${\displaystyle \theta }$the scattering angle .

In the elastic case

${\displaystyle {\frac {d\sigma }{d\Omega }}=\left({\frac {d\sigma }{d\Omega }}\right)_{\text{Mott}}{\frac {E^{\prime }}{E}}\left[2K_{1}\sin ^{2}(\theta /2)+K_{2}\cos ^{2}(\theta /2)\right]}$

the structure functions only depend on one variable. ${\displaystyle K_{1},K_{2}}$

The variable can be used instead of or as an independent variable. In the quark model, it indicates the momentum fraction of a quark in the proton. ${\displaystyle x_{\text{Bjorken}}\equiv x=-{\tfrac {q^{2}}{2qP}}=-{\tfrac {q^{2}}{2\nu M}}}$ ${\displaystyle \nu }$ ${\displaystyle Q^{2}}$ ${\displaystyle xP}$

James Bjorken predicted that at high energies the structure functions behave like

${\displaystyle MW_{1}(q^{2},x)\rightarrow F_{1}(x)}$

${\displaystyle {\frac {-q^{2}}{2Mc^{2}x}}W_{2}(q^{2},x)\rightarrow F_{2}(x)}$,

so only depend on one variable . This behavior, with the dependency on only one variable, is called Bjorken scaling. ${\displaystyle x=x_{\text{Bjorken}}}$

Scale violation

For extreme values by a function of the structure function occurs by scale injury to: ${\displaystyle x}$ ${\displaystyle F_{2}}$ ${\displaystyle Q^{2}}$

• with small increases with (increasing) ${\displaystyle x}$ ${\displaystyle F_{2}}$ ${\displaystyle Q^{2}}$
• with large falls with (increasing) . ${\displaystyle x}$ ${\displaystyle F_{2}}$ ${\displaystyle Q^{2}}$

This is due to how the structural functions of the proton depend on the energy scale :

• with small ones the relative proportion of sea ​​quarks and gluons increases with large ones ${\displaystyle x}$ ${\displaystyle Q^{2}}$
• for large , the relative proportion of valence quarks decreases for large . ${\displaystyle x}$ ${\displaystyle Q^{2}}$

literature

• David Griffiths: "Introduction to elementary particles". Wiley-VCH Verlag, Weinheim (2004).
• Bogdan Povh, et al .: "Particles and Cores", Springer Verlag Berlin (2014).

Web links

• Wu-Ki Tung, Bjorken scaling, Scholarpedia

Individual evidence

1. ^ J. Bjorken, Asymptotic Sum Rules at Infinite Momentum. Phys. Rev., Volume 179, 1969, pp. 1547-1553.