Invariant mass ( $m$ )

Rest mass is the invariant mass

m

in the relativistic energy-momentum relation. It equals the energy measured in the particle's rest frame divided by

c^{2}

Table of contents
- Diphoton invariant mass Invariant mass
  - Correlated diphoton mass resolution Diphoton invariant mass
- Cascade-decay invariant-mass endpoint Invariant mass
  - Complementary invariant masses in a sequential decay Cascade-decay invariant-mass endpoint

Diphoton invariant mass ( $M^{2} c^{4} = 2 E_{1} E_{2} (1 - cos θ)$ )

 0  0

Adding two null photon four-momenta gives the displayed invariant mass, with opening angle

θ

measured in the same frame as the energies. A nonzero mass of the pair does not give either photon a rest mass. At fixed energies it vanishes for parallel photons and is greatest for opposite directions. A reconstructed mass peak must be assessed with background, resolution and identification controls before assigning a parent particle.

Correlated diphoton mass resolution ( $σ_{M}^{2} / M^{2} = (s_{1}^{2} + s_{2}^{2} + 2 ρ s_{1} s_{2}) /4$ at fixed opening angle)

 0  0

Logarithmic differentiation of the diphoton mass gives

d M / M = (d E_{1} / E_{1} + d E_{2} / E_{2}) /2

at fixed angle. If

s_{i}

are fractional energy standard deviations and

ρ

their correlation coefficient, covariance propagation gives the displayed formula. A common multiplicative calibration error transfers directly to the reconstructed mass; independent errors combine differently. Angular errors, if present, add the corresponding derivative and covariance terms.

Cascade-decay invariant-mass endpoint ( $(m_{s t}^{2})_{m a x} = (m_{C}^{2} - m_{B}^{2}) (m_{B}^{2} - m_{A}^{2}) / m_{B}^{2}$ )

 0  0

For sequential on-shell two-body decays

C \to B + t

and

B \to A + s

, with massless

s, t

, evaluate their energies in the B rest frame. Their pair mass is

2 E_{s} E_{t} (1 - cos θ)

, so its squared invariant mass ranges from zero to the displayed endpoint. The mass ordering is

m_{C} > m_{B} > m_{A} > 0

. spin correlations can alter the distribution inside these limits, but cannot move the kinematic endpoints while the same on-shell decay chain applies.

Complementary invariant masses in a sequential decay ( $m_{A t}^{2} + m_{s t}^{2} = m_{C}^{2} + m_{A}^{2} - m_{B}^{2}$ )

 0  0

For the sequential decay defining the cascade-decay invariant-mass endpoint, momentum conservation gives the displayed eventwise identity. Thus

m_{A t}^{2}

has minimum

m_{A}^{2} m_{C}^{2} / m_{B}^{2}

and maximum

m_{A}^{2} + m_{C}^{2} - m_{B}^{2}

, attained at the opposite relative angles from the

s, t

pair endpoints. The final A particle is massive, so its lower pair-mass bound need not vanish.

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Invariant mass is a concept from physics, particularly in the context of special relativity and particle physics. It refers to the mass of a system of particles as measured in a specific reference frame, and it remains constant regardless of the motion of the observer. Invariant mass is particularly useful for understanding systems involving multiple particles or decaying particles. In technical terms, the invariant mass $ M $ of a system of particles can be calculated using the energy and momentum of those particles.

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