# Difference between revisions of "Limits based on Mandelstam Variables"

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<center><math> \theta_{max} \equiv \arccos 3</math></center> | <center><math> \theta_{max} \equiv \arccos 3</math></center> | ||

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+ | 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 3, the range will include imaginary numbers. We can approximate this value using a Taylor series expansion | ||

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+ | <center><math>\arccos{x} \approx \frac{π}{2} - x - \frac{x^3}{6 - \frac{3 x^5}{40} </math></center> |

## Revision as of 00:34, 10 June 2017

# 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 3, the range will include imaginary numbers. We can approximate this value using a Taylor series expansion