# Massless bosons but not massless fermions?

I noticed some article on massless Weyl fermions and it got me thinking. I'm wondering if there is any explanation for why bosons (specifically gauge bosons) can be massless (photon and gluon) but we don't see any fundamental massless fermions (working from the most likely confirmed hypothesis that neutrinos are massive).

I know that the $W^\pm$ and $Z$ get mass from spontaneous symmetry breaking, so obviously not all gauge bosons are massless, but why do we see no fundamental massless fermions?

The mechanism for "giving mass" to elementary bosons and fermions is different.

With bosons, it is related to the gauge symmetry ($SU(3)_c \times SU(2)_L \times U(1)_Y$) which is partially broken (and become $SU(3)_c \times U(1)_{em})$. The unbroken part imposes its associated bosons (gluons and photon) to be massless to respect this symmetry.

With fermions, there is no such constraint since their mass does not come from a gauge symmetry (with our current knowledge, fermions masses are put by hand via add hoc yukawa couplings). Therefore, the mass of the fermions is not predicted (contrary to the masses of bosons). So, asking "why do we see no fundamental massless fermions?", is equivalent as asking "why do we see fundamental fermions with their actual mass?". Answer: we don't know!

• Ahh I had forgotten that the masses we put in the lagrangian for fermions have to be experimentally measured. – CStarAlgebra Jul 18 '15 at 17:27

I'm wondering if there is any explanation for why bosons(specifically gauge bosons) can be massless (photon and gluon) but we don't see any fundamental massless fermions.

It's because a fermion is a "body", and because "the mass of a body is a measure of its energy content". See Einstein's E=mc² paper. He talks about a body and an electron here.

"The kinetic energy of the body with respect to ($\xi,\eta,\zeta$) diminishes as a result of the emission of light, and the amount of diminution is independent of the properties of the body. Moreover, the difference K0 − K1, like the kinetic energy of the electron (§ 10), depends on the velocity".

IMHO it's clear he thought of the electron as a body, and of radiation (eg photons) as energy. So a photon is "not a body". It travels at the speed of light, it's never at rest, so it doesn't have a rest mass.

I know that the W+/- and Z get mass from spontaneous symmetry breaking, so obviously not all gauge bosons are massless, but why do we see no fundamental massless fermions?

Because "elementary" fermions such as electrons and positrons aren't truly "fundamental". You can create them in gamma-gamma pair production. Each is akin to a 511keV photon in a gedanken mirror-box of its own making. In electron-positron annihilation you effectively open one box with another, whereupon each is a radiating body that loses mass. All of it. And then it's not there any more. Obviously there's a bit of a contradiction here between E=mc² and the Higgs mechanism, but that's one for another day.

• I don't understand this answer. This distinction between radiation and matter is outdated. In addition, you say that there is a contradiction between special relativity and higgs mechanism (?!?). What do you mean??? – Paganini Jul 18 '15 at 9:03
• You say a fermion is a "body". What does that even mean? A photon "is not a body" - Again, what does that mean? It travels at the speed of light so it doesn't have a rest mass - Au contraire. It has no mass, so it travels at the speed of light. And saying that "elementary" particles aren't "fundamental" because you can create them in gamma-gamma pair production? Really? You can also create the Higgs that way. You can also create the $W^\pm$ this way. – Omry Jul 18 '15 at 9:56
• @Paganini : with respect, Einstein's E=mc² paper is not outdated. And the mass of a body such as an electron is either a measure of its energy-content, or it's a measure of its interaction with a field. If it's the latter, it can't be the former. If you can take look a look at page 174 of A Zeptospace Odyssey where CERN physicist Gian Guidice says 98% of proton mass results from E=mc², whilst electromagnetic effects and the Higgs mechanism account for 1% each. – John Duffield Jul 18 '15 at 9:59
• @JohnDuffield: We're talking here about mass of elementary particles, not mass of composite ones. For composite ones, it is clear that interaction among its constituents contributes to its energy and thus its mass as for the proton. – Paganini Jul 18 '15 at 10:05
• @JohnDuffield: there is no well established theory (as the Standard model of elementary particles) on the market predicting the electron mass (or any fermion mass). The rest is simply speculation not supported by experimental evidences (or not having a predictive power as with string theory). – Paganini Jul 18 '15 at 11:23