on the sum of 4 momentum and excited mass

when we have a decay process, there are many fragments, we can measure their momentum and energy and construct the 4-momentum

P_i = ( E_i , p_i )

we use the c = 1 unit as usual.

to find out the mass before the decay, we can use

Sqrt[ Sum[E_i]^2 - Sum[p_i]^2 ] = excited mass.

the reason for the term "excited mass", we can see by the following illustration.

consider a head on collision of 2 particles in C.M. frame, with momentum p and energy E1 and E2.

the mass for each one is given by

m1=Sqrt[ E1^2 - p^2 ]
m2=Sqrt[ E2^2 - p^2 ]

but if we use the sum of the 4 momentum and calculate the mass,

Sqrt[ (E1+E2) ^2 - (p - p)^2 ] = E1 + E2

which is not equal to Sqrt[ E1^2 - p^2 ] + Sqrt[ E2^2 - p^2 ]

in fact, it is larger.

the reason for its larger is, when using the sum of 4 momentum, we actually assumed the produce of collision is just 1 particles, and the collision is inelastic. Thus, if we think about the time-reverse process, which is a decay, thus, some of the mass will convert to K.E. for the decay product.