User:BHBrunt/drafts/Transverse mass

The transverse mass is a useful quantity in particle physics. For a particle of invariant mass $$m$$, it is defined (in natural units) as
 * $$m_{T}^2 = m^2 + p_{x}^2 + p_{y}^2 \,$$
 * where the z-direction is taken along the beam-line, giving $$p_x$$ and $$p_y$$ as the momentum-components perpendicular to the beam-line.

This definition has the property that it is invariant under Lorentz boosts along the beam direction, and that it is independent of $$p_{z}$$, which is poorly known in hadron collider experiments.

For the case of a decay of a mother particle into two daughters in hadron collider experiments, the transverse mass of the mother particle is often defined as
 * $$M_{T}^2 = (E_{T, 1} + E_{T, 2})^2 - (\overrightarrow{p}_{T, 1} + \overrightarrow{p}_{T, 2})^2$$
 * where the transverse energy $$E_{T}$$ is a positive quantity defined for each daughter as $$E_{T}^2 = m^2 + {{p}_{T}}^2$$

Equivalently,
 * $$M_{T}^2 = m_1^2 + m_2^2 + 2 \left(E_{T, 1} E_{T, 2}  - \overrightarrow{p}_{T, 1} \cdot \overrightarrow{p}_{T, 2} \right) $$

For massless daughters, where $$m_1 = m_2 = 0$$, the transverse energy simplifies to $$E_{T} = | \overrightarrow{p}_T |$$, and the transverse mass becomes
 * $$M_{T}^2 = 2 E_{T, 1} E_{T, 2} \left( 1 - \cos \phi \right)$$
 * where $$\phi$$ is the angle between the daughters in the transverse plane:

A distribution of $$M_T$$ has an upper-bound at the true mother mass: $$M_T \leq M$$. This has been used at the Tevatron to determine the mass of the W boson.