ISU Coloq 11-3-2014

Elastic -vs- Inelastic Collisisons

Elastic Collisions: Conserve P and E

Inelastic : Only Conserve P

Definition of Momentum Transfer

4-Momentum vector definition using Ryder convention

[math]P_{\mu} \equiv \left ( \frac{E}{c} , - \vec p \right )[/math]

[math]P^{\mu} \equiv \left ( \frac{E}{c} , \vec p \right )[/math]

[math]P_{\mu} P^{\mu} = \left ( \frac{E}{c}\right )^2 - \vec p^2 = E^2-p^2 = m^2[/math] if you define the speed of light as unity

Note

Other conventions used by Perkins

[math]P_{\mu} \equiv \left ( \vec p, -E \right )[/math]

[math]P^{\mu} \equiv \left ( \vec p , E\right )[/math]

or Kollen

[math]P_{\mu} \equiv \left ( \vec p, iE \right )[/math]

[math]P^{\mu} \equiv \left ( \vec p , iE\right )[/math]

Momentum transfer is defined as

[math]q_{\mu} \equiv ( \omega, \vec q) = P^i_{\mu} - P^f_{\mu}[/math] : conservation of momentum

[math]q_{\mu}q^{\mu} = (E_i - E_f)^2 - (\vec {P}_i - \vec{P}_f) \cdot (\vec {P}_i - \vec{P}_f)[/math]

[math]= m_i^2 +m_f^2 - 2E_iE_f + 2 \left | \vec {P}_i \right | \left | \vec {P}_f \right |[/math]

References

Semi-inclusive deep inelastic scattering at small transverse momentum

TF_SIDIS_Physics