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Consider the following two diagrams involving a loop correction to some process $q \bar q \rightarrow q \bar q \gamma$. In a special case where I assume that the momenta $p_1$ and $p_2$ are identical (that is $p_1=p_2$), would these two diagrams come with a relative minus sign in determining their contribution to the amplitude for such a process?

The two fermion lines in the upper part of each diagram are representing a quark and antiquark. Because these are not identical species then I think the statistics is not in play here but I am not certain.

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Whether $p_1$ and $p_2$ are equal or not is irrelevant: if the two particles were the same, the idea would be that we would not know whether to assign $p_1$ to one and $p_2$ to the other, or the other way around. But they are not identical particle in your case, so no.

Beware though that if you measure e.g. $\gamma + \text{jet}$, and a jet coming from the hadronisation of a quark is not distinguished from a jet coming from the hadronisation of an antiquark, you would need to add both contribution, but at the cross-section level, not at the amplitude level.

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