# Transverse mass

The transverse mass is a useful quantity to define for use in particle physics as it is invariant under Lorentz boost along the z direction. In natural units it is:

$m_{T}^2 = m^2 + p_{x}^2 + p_{y}^2 \,$
where the z-direction is along the beam pipe and so
$p_x$ and $p_y$ are the momentum perpendicular to the beam pipe and
$m$ is the mass.

Hadron collider physicists use another definition of transverse mass, in the case of a decay into two particles. This is often used when one particle cannot be detected directly but is only indicated by missing transverse energy. In that case, the total energy is unknown and the above definition cannot be used.

$M_{T}^2 = (E_{T, 1} + E_{T, 2})^2 - (\overrightarrow{p}_{T, 1} + \overrightarrow{p}_{T, 2})^2$
where $E_{T}$ is the transverse energy of each daughter, a positive quantity defined using its true invariant mass $m$ as:
$E_{T}^2 = m^2 + (\overrightarrow{p}_{T})^2$

So 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

$M_{T}^2 \rightarrow 2 E_{T, 1} E_{T, 2} \left( 1 - \cos \phi \right)$
where $\phi$ is the angle between the daughters in the transverse plane:

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

## References

• J.D. Jackson (2008). "Kinematics". Particle Data Group. - See sections 38.5.2 ($m_{T}$) and 38.6.1 ($M_{T}$) for definitions of transverse mass.
• J. Beringer et al. (2012). "Review of Particle Physics". Particle Data Group. - See sections 43.5.2 ($m_{T}$) and 43.6.1 ($M_{T}$) for definitions of transverse mass.