In the Standard Model, I understand that the mass of the electron is assume to arise from two effects:

  1. A bare mass given by Yukawa interaction with the Higgs field, and

  2. A mass correction from mass renormalization effects

In this framework, why do we need to assume 1? Could mass renormalization explain the mass of fermions in general?


In the actual local quantum field theories, theories of point-like particles, the mass correction due to the renormalization effects from (2) is divergent. It has a short-distance divergence so it is infinite. One needs to cancel the "infinite part" so that there's a finite leftover. What is the separation of the physical observed mass to (1) and (2) depends on various choices, "renormalization scheme", etc.

However, there's one more fundamental misconception implicitly included in your setup. In (2), you seem to assume that the loop effects only correct the mass but they don't affect the Yukawa coupling. But this assumption is invalid. The loop corrections affect the Yukawa coupling as well, so that the proportionality essentially holds before as well as after renormalization. So because we know that the physical mass of the electron is nonzero, we also know that the Yukawa coupling is nonzero. Both of them "run" i.e. depend on the renormalization scale but this running is logarithmically slow.

Note that the non-renormalized, bare parameters – whether we talk about masses or Yukawa or other couplings – are always infinite and only when the infinite counterterms are added, these infinite counterterms cancel the infinite part of the loop corrections coming from the bare parameters and we obtain a finite result. Again, this is true for electron's mass, Yukawa coupling, gauge couplings, and other couplings.

  • 1
    $\begingroup$ Aah, that is a nice and clear explanation :-)! Prof. Strassler probably talked about the same issue, but because he always wants to talk to laypeople and avoid "technical jargon" etc, he in this case produced so much fog and fuzz that I really could not figure out what exactly he was talking about then when reading it ... :-/ $\endgroup$ – Dilaton Nov 9 '12 at 14:48
  • $\begingroup$ The loop correction being infinite is not related to the "infiniteness of bare parameters", but caused with a "too singular interaction". The counter-terms serve to modify this interaction in each perturbative order. $\endgroup$ – Vladimir Kalitvianski Nov 9 '12 at 16:43
  • $\begingroup$ I heard in one of Susskind's lectures about renormalisation that fermion mass is a special case in that they only get a mass renormalisation if the bare mass is non-zero. So, with a zero mass to start with, it would stay zero. The argument was that the self-energy can't act as a mass-term since the vertices comes in pairs and won't therefore mix left/right states like a mass-interaction would. Wouldn't this rule out the OP's suggestion? (I felt a bit unsure on the lecture's result though, but it seemed convincing :) $\endgroup$ – BjornW Jun 26 '17 at 20:24
  • $\begingroup$ Yes, @BjornW, a vanishing mixing of the two 2-component spinors will ban a mass even after renormalization, by the chiral symmetry. Maybe you should write an answer and I will upvote it... $\endgroup$ – Luboš Motl Jun 27 '17 at 10:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.