# A What is mass?

1. Nov 8, 2018

### Nusc

In Abraham Pais book (which was first published in 1982), he states the following:

"Quantum field theory has taught us that particles nevertheless have structure, arising from quantum fluctuations. Recently, unified field theories have taught us that the mass of the electron is certainly not purely electromagnetic in nature. But we still do not know what causes the electron to weigh."

Despite the developments in particle physics, what did he mean by this?

By the equivalence principle, heavy and inertial mass must be identical. What does the Higg's Boson tell us about mass?

If you can provide references for further reading that would be useful.

Thanks

2. Nov 10, 2018

### vanhees71

A somewhat more modern book is Ledermann, "The God Particle" or Veltman, "Mysteries and Facts in Elementary Particle Physics".

Today the socalled bare masses of the elementary particles (quarks and leptons) is due to their coupling to the Higgs field and the fact that the electroweak symmetry is "Higgsed", i.e., the local gauge symmetry is hidden by a non-vanishing vacuum expectation value of the Higgs field. The Higgs field has been introduced in the mid 1960ies by very many people to provide masses not only to the quarks and leptons but also to the gauge bosons of the electroweak interaction (describing in a kind of unification the weak and the electromagnetic interaction in one model) without violating the very foundation of the symmetry, namely the chiral local gauge theory of the electroweak interaction. Violating this fundamental symmetry is a absolute no-go for the theory, because then it becomes intrinsically inconsistent and there'd be no way to make any physics sense out of this inconsistent construct.

The Higgs boson is named a bit unjustly only after Higgs, but on the other hand Higgs was the first who realized that with the introduction of the Higgs field with non-vanishing vacuum expectation value resulting in the realization of the electroweak gauge symmetry as a (partially) hidden local symmetry unavoidably implies that there must be a corresponding particle, the so-called Higgs boson.

The equivalence of inertial and heavy mass is the foundation of Einstein's theory of general relativity. According to this symmetry there's not even a conceptual difference in the inertial and heavy mass at all. The problem (in my opinion the most important open problem of fundamental theoretical physics today) is that GR and thus the gravitational interaction is not yet described on the quantum level, i.e., there's no consistent unified theory for all interactions.

3. Nov 10, 2018

### ZapperZ

Staff Emeritus
Mass is not all "higgs". That is a misconception.

The Higgs may explain the origin of mass in leptons, or those involved in the electroweak interaction, but for a proton, for example, neither the sum of all the masses of quarks, nor the sum of the masses of the Higgs, all of them combined will not produce the mass of a proton.

BTW, Don Lincoln did a video on inertial and gravitational mass earlier this year:

Zz.

4. Nov 10, 2018

### vanhees71

That's an important point. That's why I stressed that the socalled "bare masses" of the elementary particles in the standard model come from the vacuum expectation value of the Higgs field (NOT from the Higgs boson, which is a certain state of the Higgs field).

The bulk (about 98-99%) of the mass of the matter around us is dynamically generated by the strong interaction and part of the somewhat enigmatic mechanism called "confinement". One way to at least check this theoretically is lattice QCD, i.e., numerically evaluating the hadron masses from QCD by calculating corresponding expectation values of quantum fields defined on a discretized space-time lattice and then taking the "continuum limit". This quite tricky business leads to a very good description of the known hadron spectrum, which makes us pretty confident that QCD works not only in the "asymptotic free limit" (i.e., for deep inelastic scattering), where perturbation theory is applicable, but also as a theory at low energies, describing not quarks and gluons but the hadrons, within which they are "confined".

From a symmetry point of view, there are in principle two mechanisms of mass generation. In the light-quark sector it's a good approximation to neglect the quark masses at all. In this massless limit of QCD, there's no length scale at all. The only dimensionful quantities are the fields themselves. If QCD were a classical field theory there'd be no way to get "mass without mass" (a famous saying by Wilczek), but first the theory is quantized, and the conformal symmetry of the classical massless-QCD Lagrangian is anomalously broken. In perturbation theory you formally see this that you have to renormalize the theory by subtracting divergences of Feynman diagrams forming closed loops. This subtraction leads to a redefinition of the coupling and the wave-function normalizations. However, you cannot subtract the corresponding one-particle irreducible diagrams (proper vertex functions) with all the external momenta vanishing since this introduces new divergences because all particles in the theory (quarks and gluons) are massless. You thus necessarily have to choose a scale $\Lambda_{\text{QCD}}$ and corresponding (space-like) momenta at this scale to subtract the divergencies. This introduces necessarily an energy scale and thus breaks the conformal symmetry. That leads to a non-vanishing trace of the energy-momentum tensor and that's why this anomaly (i.e., a symmetry of a classical field thwory which gets broken in the corresponding quantized field theory) is called trace anomaly.

Another way to generate "mass without mass" is hinted at by chiral symmetry. This refers to the fact that with vanishing quark masses the quark sector of QCD becomes chirally symmetric. Now the hadron spectrum (i.e., "QCD at low energies") doesn't show this chiral symmetry, because it implies that for each hadron there should be another hadron with the same mass but with opposite parity ("chiral partners"), but looking at the particles in the particle data book one always finds possible chiral partner states but with different masses. The explanation for this phenomenon is that chiral symmetry is spontaneously broken. Since this is a global symmetry (not a local gauge symmetry!) this implies the existence of massless states (Nambu-Goldstone bosons of the spontaneously broken symmetry). Now, there are no massless hadrons either, but there are the pions with a mass of about 140 MeV, which is pretty small compared to the usual hadron masses of around 1GeV=1000 MeV. This suggests that chiral symmetry is also a bit explicitly broken and thus the pions become a bit massive, but can be taken as approximate Nambu-Goldstone modes of broken chiral symmetry. Indeed it turns out that chiral perturbation theory is a successful effective theory of the hadrons as far as the light quarks (and also the strange quark) are concerned.

From a fundamental point of view, i.e., spoken in terms of quarks, lattice QCD shows that the spontaneous chiral symmetry breaking is due to the formation of a scalar "quark condensate", i.e., because of a non-vanishing vacuum expectation value $\langle \bar{q} q \rangle$. Both the trace anomaly and the quark condensate are thus the prime culprits for the generation of "mass without mass". For a nice review, see

Roberts, C.D. Few-Body Syst (2017) 58: 5. https://doi.org/10.1007/s00601-016-1168-z
https://arxiv.org/abs/1606.03909