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:

$M_{T}^2 = (E_{T, 1} + E_{T, 2})^2 - (\overrightarrow{p}_{T, 1} + \overrightarrow{p}_{T, 2})^2$

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.