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In quantum field theory, we are looking for a Lagrangian that is, amongst other, renormalizable. But how do we determine whether or not a theory is renormalizable? Is this purely done by power counting due to Weinberg? This question is already answered in a previous question.

My question result from the fact that the Yang-Mills Lagrangian was considered to be non-renormalizable, and thus non-physical, for a decade until Veltman and 't Hooft found a method to regularize the theory. Keeping this in mind, is it possible that there are theories that we today consider to be non-renormalizable, and thus non-physical, which actually are renormalizable but we haven't (yet) discovered a way to do this?

I apologize in advance if my question is vague and if I'm using the wrong terminology, but I'm very new to the idea of renormalizability.

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    $\begingroup$ related post physics.stackexchange.com/q/88884 ; P.S. non-renormalizable Lagrangians are acceptable as effective theories $\endgroup$ – user26143 Feb 24 '14 at 10:13
  • $\begingroup$ @user26143 thanks for that! I will now update my question (I don't think it answers my second question). $\endgroup$ – Hunter Feb 24 '14 at 11:41
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    $\begingroup$ If a theory is power counting non-renormalizable, then for sure it is not renormalizable. $\endgroup$ – Jia Yiyang Feb 26 '14 at 10:50
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    $\begingroup$ I answered this at physicsoverflow.org/13666 $\endgroup$ – Arnold Neumaier Nov 8 '15 at 19:05
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    $\begingroup$ @JiaYiyang: this is only true at a perturbative level. $\endgroup$ – Adam Nov 8 '15 at 21:05

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