Limits based on Mandelstam Variables

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Limits based on Mandelstam Variables

Since the Mandelstam variables are the scalar product of 4-momenta, which are invariants, they are invariants as well. The sum of these invariant variables must also be invariant as well. Find the sum of the 3 Mandelstam variables when the two particles have equal mass in the center of mass frame gives:

Since

This implies

In turn, this implies

At the condition both t and u are equal to zero, we find

Holding u constant at zero we can find the maximum of t

The domain of the arccos function is from −1 to +1 inclusive and the range is from 0 to π radians inclusive (or from 0° to 180°). This implies for arccos -1, the range will include imaginary numbers. Knowing that the range of the cosine function is -1 to +1 inclusive and the domain to be any angle

From Euler's formula

Multiply with

Letting

We get an quadratic equation:

Apply the natural log on both sides gives the solution for arccos 3:

Converting to polar coordinates:

Since

Since there is only an imaginary component.

Figure 2.1: Angle theta measured with respect to imaginary plane.