Fermion Mass Structure: Explaining the Reversal of Up/Down Quarks

[michael879]

I was taking a look at the SM particle table and I noticed that the up/down quarks seem to be "reversed" from the usual structure. Neglecting those two quarks, the "up" family is the most massive, then the "down" family, then the "electron" family, then the "neutrino" family. However for some reason the down quark is more massive than the up quark, breaking this pattern.

So I was wondering if there are any theories that attempt to explain this mass structure of the fermions (or coupling constants if the Higgs generates their mass)?

 

[mitchell porter]

The discussion of the fermion masses is usually dominated by other issues, like "why is the top quark so much heavier than everything else?" The different tilt of the first-generation quarks (u<d) compared to the higher generations (t>b, c>s) does occasionally get a mention, but often it has to be separately accounted for, i.e. a model builder will add a whole extra epicycle in order to produce this difference.

In this regard, it's interesting that http://arxiv.org/abs/hep-ph/9211209 states (page 25) "An interesting feature of many models with masses generated through radiative corrections is that they couple the up quark to the strange and bottom quarks and couple the down quark to the charm and top quarks through the higher-order corrections, thus providing a natural explanation for the observation m_u < m_d."

So in effect this is saying that the heaviness of the top is making a bigger virtual contribution to the down mass, than the bottom makes to the up mass. (Let me emphasize for general readers that this is not a statement about how mass works in the standard model, where these masses come solely from a coupling to the Higgs; this is a beyond-standard-model hypothesis, that will presumably become testable as we measure the magnitude of the direct couplings between fermions and Higgs.)

 

[ohwilleke]

So I was wondering if there are any theories that attempt to explain this mass structure of the fermions (or coupling constants if the Higgs generates their mass)?

Short answer: Yes there are theories that attempt to explain it, and no there aren't any that have widespread acceptance (and as your question implies, the SM explains masses as the product of Higgs Yukawas and other stuff, but this is essentially question begging because it doesn't explain the Yukawas themselves).

Long answer: There are two basic approaches. One is phenomonology: look at the relationships empirically and find an interesting looking formula that describes those relationships in fact, then look for a theory that could explain it. The other is theoretical: hypothesize a theoretical source of a constant and see if you can make it fit the numbers possibly with a smaller set of constants than we started with.

In phenomenology (where this enterprise is often called the study of mass matrix "textures"), some of the most notable research efforts include the following observations:

* There are strong hints that there is a functional relationship between the quark masses (or their square roots) and the elements of the CKM mixing matrix.
* There are also theories (quark-lepton complentarity) that argue that the lepton mixing matrix elements when properly parameterized have a simple relationship to the CKM mixing matrix elements, possibly also implying that knowing the elements of one mixing matrix is enough to fix all of the other mixing matrix elements and all of the fermion masses relative to each other.
* The charged leptons masses are related to each other via "Koide's formula." Koide's formula cannot be fit to the neutrino masses (even given current uncertainty about their values) without modification in some form.
* Proposed extensions of Koide's formula seem to hold for all of the quark masses except the up quark and for the neutrino masses. This doesn't necessarily mean a lot, however, because there is immense experimental uncertainty in the know values of the quark masses since they are always confined and QCD calculations that relate hadron masses to fundamental quark masses are very hard to make accurately.
* Some theories see potential in multiplying several, mostly empty matrixes by each other and then normalizing them so the matrixes are unitary are suggestive results.
* There are some potential relationships between quark masses and lepton masses (the sum of the strange-charm-bottom quark masses is about three times the sum of the three charged lepton masses).
* There are some approximate empirical relationships between the top quark mass, the weak force boson masses, and the Higgs boson mass.
* The Higgs boson mass is almost exactly equal to the sum of the three weak force boson masses.
* Nobody has any really solid theory to explain why neutrino masses are so much smaller than the charged particle masses, although a "see-saw" model and Majorana mass terms can help to explain this with the cost of additional BSM physics.
* constrained SUSY theories often try to set masses of sparticles based in part on two constants (m zero and m 1/2).
* Some theories derive Higgs boson or other masses from a value (like zero) at a boundary condition (e.g. the running of the mass brings it to zero at the GUT scale), and work back using the running of masses with energy scale - and this can provide a baseline mass scale for the fermions.
* Some people have observed a similarity between the ratios of masses in the mass matrix and Standard Model coupling constants, and between the top-Higgs boson yukawa and the strong force coupling constant.
* Theoretically, there is lack of clarity over the mass at which formulas should be applied (rest mass, a consistent energy scale, W or Z or Higgs boson scale, etc.)
* One theory attempted to build up masses from a preon model and a special formula for adding up preon masses with heavier fermions having complex preonic structures.
* The crumdgen theory is the "anarchic" model that holds that the masses have values such that they fit no simple formula and are effectively random subject to some bare hierarchy relationships.