Invariant Mass

Temporary version:

If a heavy particle A decays to two very lightweight particles B and C, what you need to do to find the mass of A is measure the energy E_B and E_C, and the angle theta_BC between the directions of motion of the two particles; then compute

2 E_B E_C (1 – cos theta_BC),

take the square root, and divide by the speed of light squared.  The answer is called the invariant mass of particles B and C, and it equals the mass of particle A.  Thus you can determine the mass of particle A simply by knowing E_B, E_C and theta_BC.

If particles B and C aren’t that lightweight, a more precise answer is to replace 2 E_B E_C (1 – cos theta_BC) with

(m_B)^2 c^4 + (m_C)^2 c^4 + 2 E_B E_C (1 – v_B v_C/c^2 cos theta_BC),

where m_B and v_B are the mass and velocity of particle B (and similarly for C), and c is the speed of light.  Again the square root of this quantity, divided by c^2, is the invariant mass, and this will be the mass of the parent particle A.

4 responses to “Invariant Mass

  1. FYI, you might find it useful to know that WordPress supports inline LaTeX. For example, the first expression above could be rendered by coding it as

    2 E_B E_C (1 – cos \Theta_BC)

    (Frankly, I won’t know until I click the post button whether the above LaTeX code will get formatted with the LaTeX engine or if the source will be shown. Let’s see….)

  2. And it looks like it tried to parse, but I had a typo. Oops….

  3. Ah, here it is:

    2 E_B E_C (1-\cos{\theta}_{BC})

    This was rendered using the following HTML source:

    $$latex 2 E_B E_C (1-\cos{\theta}_{BC})$

    For more info, see http://en.support.wordpress.com/latex/ .

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