Friday, July 16, 2021

Serendipitous sum rules

Today I shall report that I am rather more positive than I was, about the second "wrong" idea in the previous post. The reason is that the sum of all the masses in a multiplet, is a quite natural item to appear in a sum rule! So the relation would be, that the sum of the masses of the charged leptons, considered as a multiplet of a flavor or family symmetry, equals the sum of the neutron and proton masses, with the neutron and proton to be considered as an isospin doublet. 

What I still lack is a mechanism. I believe that (for example, in the linear sigma model), nucleon mass can be regarded as originating in the spontaneous breaking of chiral symmetry. Meanwhile, the Higgs-yukawa interactions in the standard model give the fermions their masses, once electroweak symmetry breaking occurs; so one might consider models in which chiral symmetry breaking triggers electroweak symmetry breaking (something which might also explain the order-of-magnitude similarity between the QCD scale and the Fermi scale). 

On the other hand, since the variations among the fermion masses derive from the yukawas, it might seem that the relevant symmetry-to-be-broken is the family symmetry, not the electroweak symmetry... Then there are other hints, like obtaining electroweak symmetry by gauging part of chiral symmetry (something which is formally common in chiral perturbation theory, I think), and the Rivero idea that the leptons are Goldstone fermions, superpartners of mesons. 

In other news, I will mention that tweeted out a fictitious paper whose content is serendipitously close to other things I have been thinking about. The very first posts on this blog were discussions of snarxiv papers, and if there was nothing else to blog about, I might have talked more about "Special Lagrangian Branes Wrapped on the Moduli Space of Squashed Lens Spaces". But I'll save that for another time.