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Don't both masses require infinite corrections in their renormalization procedure?

It is my understanding that the electron self-energy in QED increases to infinity with increasing cutoff value on the loop momentum. The bare mass is then defined such that it cancels this divergence and creates a finite electron mass when added to the electron self energy.

As both quantities, the self-energy and the bare mass increase with the cutoff value, they can become X magnitutes greater than the physical electron mass, implying that they have to be chosen to cancel each other out to X digits.

Isn't this the same fine-tuning as with the higgs mass?

Where is the fundamental difference here that makes the higgs mass fine tuning a concern for physicists while the electron mass in QED is not see as problematic?

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    $\begingroup$ It has to do with "how bad" the divergences are. "OK" divergences are ones that depend logarithmically on the cutoff scale, proportional to $\log (\Lambda/E)$. "Bad" ones depend quadratically, proportional to $\Lambda^2 / E^2$. The QED electron mass divergence is of the former type and the Higgs mass of the second. $\endgroup$
    – jwimberley
    Commented Aug 31, 2014 at 11:59
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    $\begingroup$ Is this maybe the answer? $\endgroup$
    – Konstantin Schubert
    Commented Aug 31, 2014 at 17:43
  • $\begingroup$ Are you referring to the mass of the Higgs boson or the energy density of the spontaneously non-zero Higgs field? $\endgroup$
    – akrasia
    Commented Aug 31, 2014 at 21:37
  • $\begingroup$ A short and silly answer here would be that in QED with just one massive fermion species, the mass is the only dimensionful parameter, so there's nothing else to fine tune against... $\endgroup$
    – Holographer
    Commented Sep 1, 2014 at 10:00
  • $\begingroup$ @Konstantin I think it's mostly the answer, but I don't have the expertise to flesh this out into an answer. Maybe I'll try. $\endgroup$
    – jwimberley
    Commented Sep 3, 2014 at 12:47

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