2-body scattering and Mandelstam Variables

[Pizza Pasta physics]

Homework Statement

In a 2-body scattering event, A + B → C + D, it is convenient to introduce the Mandelstam variables,
s ≡ −(PA + PB)² ,
t ≡ −(PA − PC) ² ,
u ≡ −(PA − PD) ² ,
where PA,...,D are the 4-momenta of the particles A, . . . , D respectively, (· · ·) ² = (· · ·) · (· · ·) denotes a scalar product, and we are using natural units in this problem. The Mandelstam variables are useful in theoretical calculations because they are invariant under Lorentz transformation.

Demonstrate that in the centre of mass frame of A and B, the total CM energy, i.e., E_total ≡ E_A +E_B = E_C + E_D , is equal to √ s.

Homework Equations

s + t + u = m_A² + m_B² + m_C² + m_D² (I had to show this before which I did, not sure if its relevant or not).
P_A = -P_B (due to being in a CM frame)

The Attempt at a Solution

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Using the scalar product notation for s, I managed to reduce s to -(E_A + E_B) however I still can't take the square root to show √ s = E_A + E_B due to the pesky negative sign.

Apart from me doing something wrong with my algebra, I was wondering if the given Mandelstam variables are correct. From all the secondary sources I've looked at none give them with the negative signs.

 

[Orodruin]

This depends on whether or not you use a +--- or -+++ signature on your metric. Most particle physicists use +--- and it seems like your source does not. Check what convention is used.