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:

${\displaystyle m_{T}^{2}=m^{2}+p_{x}^{2}+p_{y}^{2}=E^{2}-p_{z}^{2}\,}$

where the z-direction is along the beam pipe and so

${\displaystyle p_{x}}$ and ${\displaystyle p_{y}}$ are the momentum perpendicular to the beam pipe and

${\displaystyle m}$ is the (invariant) mass.

This definition of the transverse mass is used in conjunction with the definition of the (directed) transverse energy

${\displaystyle {\vec {E}}_{T}=E{\frac {{\vec {p}}_{T}}{|{\vec {p}}|}}={\frac {E}{\sqrt {E^{2}-m^{2}}}}{\vec {p}}_{T}}$

with the transverse momentum vector ${\displaystyle {\vec {p}}_{T}=(p_{x},p_{y})}$. It is easy to see that for vanishing mass (${\displaystyle m=0}$) the three quantities are the same: ${\displaystyle E_{T}=p_{T}=m_{T}}$. The transverse mass is used together with the rapidity, transverse momentum and polar angle in the parameterization of the four-momentum of a single particle:

${\displaystyle (E,p_{x},p_{y},p_{z})=(m_{T}\cosh y,\ p_{T}\cos \phi ,\ p_{T}\sin \phi ,\ m_{T}\sinh y)}$

Using these definitions (in particular for ${\displaystyle E_{T}}$) gives for the mass of a two particle system:

${\displaystyle M_{ab}^{2}=(p_{a}+p_{b})^{2}=p_{a}^{2}+p_{b}^{2}+2p_{a}p_{b}=m_{a}^{2}+m_{b}^{2}+2(E_{a}E_{b}-{\vec {p}}_{a}\cdot {\vec {p}}_{b})}$

${\displaystyle M_{ab}^{2}=m_{a}^{2}+m_{b}^{2}+2\left(E_{T,a}{\frac {\sqrt {p_{a,x}^{2}+p_{a,y}^{2}+p_{a,z}^{2}}}{p_{T,a}}}E_{T,b}{\frac {\sqrt {p_{b,x}^{2}+p_{b,y}^{2}+p_{b,z}^{2}}}{p_{T,b}}}-{\vec {p}}_{T,a}\cdot {\vec {p}}_{T,b}-p_{z,a}p_{z,b}\right)}$

${\displaystyle M_{ab}^{2}=m_{a}^{2}+m_{b}^{2}+2\left(E_{T,a}E_{T,b}{\sqrt {1+p_{a,z}^{2}/p_{T,a}^{2}}}{\sqrt {1+p_{b,z}^{2}/p_{T,b}^{2}}}-{\vec {p}}_{T,a}\cdot {\vec {p}}_{T,b}-p_{z,a}p_{z,b}\right)}$

Looking at the transverse projection of this system (by setting ${\displaystyle p_{a,z}=p_{b,z}=0}$) gives:

${\displaystyle (M_{ab}^{2})_{T}=m_{a}^{2}+m_{b}^{2}+2\left(E_{T,a}E_{T,b}-{\vec {p}}_{T,a}\cdot {\vec {p}}_{T,b}\right)}$

These are also the definitions that are used by the software package ROOT, which is commonly used in high energy physics.

Transverse mass in two-particle systems

Hadron collider physicists use another definition of transverse mass (and transverse energy), 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.

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

where ${\displaystyle E_{T}}$ is the transverse energy of each daughter, a positive quantity defined using its true invariant mass ${\displaystyle m}$ as:

${\displaystyle E_{T}^{2}=m^{2}+({\vec {p}}_{T})^{2}}$,

which is coincidentally the definition of the transverse mass for a single particle given above. Using these two definitions, one also gets the form:

${\displaystyle M_{T}^{2}=m_{1}^{2}+m_{2}^{2}+2\left(E_{T,1}E_{T,2}-{\vec {p}}_{T,1}\cdot {\vec {p}}_{T,2}\right)}$

(but with slightly different definitions for ${\displaystyle E_{T}}$!)

For massless daughters, where ${\displaystyle m_{1}=m_{2}=0}$, we again have ${\displaystyle E_{T}=p_{T}}$, and the transverse mass of the two particle system becomes:

${\displaystyle M_{T}^{2}\rightarrow 2E_{T,1}E_{T,2}\left(1-\cos \phi \right)}$

where ${\displaystyle \phi }$ is the angle between the daughters in the transverse plane. The distribution of ${\displaystyle M_{T}}$ has an end-point at the invariant mass ${\displaystyle M}$ of the system with ${\displaystyle M_{T}\leq M}$. This has been used to determine the ${\displaystyle W}$ mass at the Tevatron.

References

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