# 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).