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Can the result on cross section of some process be dependent on the renormalization scheme used?

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If one calculates it to all orders, it shouldn't. In particular, if the coupling constant is small, the $O(g^k)$ accuracy calculation of the cross section obtained in different schemes can differ at most by $o(g^k)$, negligible amounts.

However, when one truncates the cross section at some order, the results may really depend on the renormalization scheme by subleading, higher-order terms.

Physically, cross sections are measurable so a valid theory must predict unequivocal values for it. And a renormalization scheme doesn't really "change the theory".

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