Transverse mass

 ( Particle Physics ) 

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: $$m\_T^2\; =\; m^2\; +\; p\_x^2\; +\; p\_y^2\; =\; E^2\; -\; p\_z^2$$

• 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 (invariant) mass.

This definition of the transverse mass is used in conjunction with the definition of the (directed) transverse energy $$\backslash vec\{E\}\_T\; =\; E\; \backslash frac\{\backslash vec\{p\}\_T\}$$ = \frac{E}{\sqrt{E^2-m^2}}\vec{p}_T with the transverse momentum vector $\backslash vec\{p\}\_T\; =\; (p\_x,\; p\_y)$. It is easy to see that for vanishing mass ($m\; =\; 0$) the three quantities are the same: $E\_T\; =\; p\_T\; =\; m\_T$. 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: $$(E,\; p\_x,\; p\_y,\; p\_z)\; =\; (m\_T\; \backslash cosh\; y,\backslash \; p\_T\; \backslash cos\backslash phi,\backslash \; p\_T\; \backslash sin\backslash phi,\backslash \; m\_T\; \backslash sinh\; y)$$

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

$M\_\{ab\}^2\; =\; (p\_a\; +\; p\_b)^2\; =\; p\_a^2\; +\; p\_b^2\; +\; 2\; p\_a\; p\_b\; =\; m\_a^2\; +\; m\_b^2\; +\; 2\; (E\_a\; E\_b\; -\; \backslash vec\{p\}\_a\backslash cdot\; \backslash vec\{p\}\_b)$

$M\_\{ab\}^2\; =\; m\_a^2\; +\; m\_b^2\; +\; 2\; \backslash left(E\_\{T,a\}\backslash frac\{\backslash sqrt\{p\_\{a,x\}^2+p\_\{a,y\}^2+p\_\{a,z\}^2\}\}\{p\_\{T,a\}\}\; E\_\{T,b\}\backslash frac\{\backslash sqrt\{p\_\{b,x\}^2+p\_\{b,y\}^2+p\_\{b,z\}^2\}\}\{p\_\{T,b\}\}\; -\; \backslash vec\{p\}\_\{T,a\}\backslash cdot\; \backslash vec\{p\}\_\{T,b\}\; -\; p\_\{z,a\}p\_\{z,b\}\backslash right)$

$M\_\{ab\}^2\; =\; m\_a^2\; +\; m\_b^2\; +\; 2\; \backslash left(E\_\{T,a\}E\_\{T,b\}\backslash sqrt\{1+p\_\{a,z\}^2/p\_\{T,a\}^2\}\backslash sqrt\{1+p\_\{b,z\}^2/p\_\{T,b\}^2\}\; -\; \backslash vec\{p\}\_\{T,a\}\backslash cdot\; \backslash vec\{p\}\_\{T,b\}\; -\; p\_\{z,a\}p\_\{z,b\}\backslash right)$

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

$(M\_\{ab\}^2)\_T\; =\; m\_a^2\; +\; m\_b^2\; +\; 2\; \backslash left(E\_\{T,a\}E\_\{T,b\}\; -\; \backslash vec\{p\}\_\{T,a\}\backslash cdot\; \backslash vec\{p\}\_\{T,b\}\backslash right)$

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

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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.

$M\_\{T\}^2\; =\; (E\_\{T,\; 1\}\; +\; E\_\{T,\; 2\})^2\; -\; (\backslash vec\{p\}\_\{T,\; 1\}\; +\; \backslash vec\{p\}\_\{T,\; 2\})^2$

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

$E\_\{T\}^2\; =\; m^2\; +\; (\backslash vec\{p\}\_\{T\})^2$,

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

$M\_\{T\}^2\; =\; m\_1^2\; +\; m\_2^2\; +\; 2\; \backslash left(E\_\{T,\; 1\}\; E\_\{T,\; 2\}\; -\; \backslash vec\{p\}\_\{T,\; 1\}\; \backslash cdot\; \backslash vec\{p\}\_\{T,\; 2\}\; \backslash right)$

(but with slightly different definitions for $E\_T$!)

For massless daughters, where $m\_1\; =\; m\_2\; =\; 0$, we again have $E\_\{T\}\; =\; p\_T$, and the transverse mass of the two particle system becomes:

$M\_\{T\}^2\; \backslash rightarrow\; 2\; E\_\{T,\; 1\}\; E\_\{T,\; 2\}\; \backslash left(\; 1\; -\; \backslash cos\; \backslash phi\; \backslash right)$

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

• - See sections 38.5.2 ($m\_\{T\}$) and 38.6.1 ($M\_\{T\}$) for definitions of transverse mass.
• - See sections 43.5.2 ($m\_\{T\}$) and 43.6.1 ($M\_\{T\}$) for definitions of transverse mass.

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