transverse mass
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The transverse mass is a useful quantity to define for use in
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
as it is invariant under
Lorentz boost In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
along the z direction. In
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a Coherence (units of measurement), coherent unit of a quantity. For e ...
, 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 \vec_T = E \frac = \frac\vec_T with the transverse momentum vector \vec_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 \cosh y,\ p_T \cos\phi,\ p_T \sin\phi,\ m_T \sinh y) Using these definitions (in particular for E_) gives for the mass of a two particle system: :M_^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 - \vec_a\cdot \vec_b) :M_^2 = m_a^2 + m_b^2 + 2 \left(E_\frac E_\frac - \vec_\cdot \vec_ - p_p_\right) :M_^2 = m_a^2 + m_b^2 + 2 \left(E_E_\sqrt\sqrt - \vec_\cdot \vec_ - p_p_\right) Looking at the transverse projection of this system (by setting p_ = p_ = 0) gives: :(M_^2)_T = m_a^2 + m_b^2 + 2 \left(E_E_ - \vec_\cdot \vec_\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. :M_^2 = (E_ + E_)^2 - (\vec_ + \vec_)^2 where E_ is the transverse energy of each daughter, a positive quantity defined using its true
invariant mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
m as: :E_^2 = m^2 + (\vec_)^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_^2 = m_1^2 + m_2^2 + 2 \left(E_ E_ - \vec_ \cdot \vec_ \right) (but with slightly different definitions for E_T!) For massless daughters, where m_1 = m_2 = 0, we again have E_ = p_T, and the transverse mass of the two particle system becomes: :M_^2 \rightarrow 2 E_ E_ \left( 1 - \cos \phi \right) where \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 \leq M. This has been used to determine the W mass at the Tevatron.


References

* - See sections 38.5.2 (m_) and 38.6.1 (M_) for definitions of transverse mass. * - See sections 43.5.2 (m_) and 43.6.1 (M_) for definitions of transverse mass. Particle physics Kinematics Special relativity {{particle-stub