# Higgs boson and mass

• I
The Higgs boson with mass couples to quarks and leptons to give them mass. What was the nature of these particles before they acquired mass? Were they virtual particles?

PeterDonis

Demystifier
Gold Member
The Higgs boson with mass couples to quarks and leptons to give them mass. What was the nature of these particles before they acquired mass? Were they virtual particles?
No, before that they were just fields, not particles at all.

No, before that they were just fields, not particles at all.
Can't fields be quantized as particles, just as the Higgs field can be quantized as Higgs particles?

Demystifier
Gold Member
Can't fields be quantized as particles, just as the Higgs field can be quantized as Higgs particles?
In many situations, especially when interactions are strong, fields cannot be easily quantized as particles. More precisely, the Hamiltonian eigenstates are not states with a definite number of particles.

In many situations, especially when interactions are strong, fields cannot be easily quantized as particles. More precisely, the Hamiltonian eigenstates are not states with a definite number of particles.
So then what is it that the Higgs particle is coupling with to give mass to quarks and leptons - is it the field-state of the future quarks and leptons?

Demystifier
Gold Member
So then what is it that the Higgs particle is coupling with to give mass to quarks and leptons - is it the field-state of the future quarks and leptons?
You misunderstood. We always have fields. Sometimes those fields look like particles, but they are still fields.

PeterDonis
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2020 Award
The Higgs boson with mass couples to quarks and leptons to give them mass.
More precisely, a part of the pre-electroweak symmetry breaking Higgs field couples to the pre-electroweak symmetry breaking massless quark and lepton fields (and a part of the pre-electroweak symmetry breaking massless electroweak gauge boson fields) to form new massive quark and lepton fields (and new massive weak gauge boson fields). Schematically, it looks something like this:

Fields before electroweak symmetry breaking: ##H^+, H^-, H^0, \bar{H}^0, W^1, W^2, W^3, B, q^0, e^0##. Here the ##H## fields are the Higgs fields, the ##W## and ##B## fields are the electroweak gauge boson fields, and the ##q## and ##e## fields are the quark and lepton fields.

Fields after electroweak symmetry breaking: ##W^+ = H^+ \left( W^1 + i W^2 \right)##, ##W^- = H^- \left( W^1 - i W^2 \right)##, ##Z = \bar{H}^0 \left( \cos \theta W^3 + \sin \theta B \right)##, ##\gamma = \sin \theta W^3 - \cos \theta B##, ##q##, ##e##, ##H##. Here we have three new massive weak gauge boson fields, which have "eaten" three of the four original Higgs fields to become massive; the one massless gauge boson field left over is the photon ##\gamma##. We also have that the quark and lepton fields now are coupled to the Higgs fields in a more complicated way to give them mass. The Higgs boson ##H## is what we observe in experiments at the LHC; it is actually the "leftover" part of the Higgs field that did not get eaten.

What was the nature of these particles before they acquired mass?
See above. Basically, what you have is two different ways of grouping together the same underlying fields. One works well at high energies (before electroweak symmetry breaking), and describes a bunch of massless fields, so before the fields "acquired mass" they were simply massless. The other grouping works well at low energies (after electroweak symmetry breaking), and describes a bunch of massive fields that are combinations of the original massless ones, with one massless one (the photon) left over.

andrew s 1905, Ranku, nrqed and 2 others
DarMM
Gold Member
A crude way of looking at it is that essentially the LSZ formalism only allows us to associate particles with fields with vanishing vacuum expectation value:
$$\langle\phi\rangle = 0$$
What happens in symmetry breaking is that the vacuum picks up a charge and some set of fields no longer have a vanishing expectation value and so cannot be associated with particle states anymore. They don't map directly to the clicks you see in detectors. So you "shift" to a new set of fields that do. These shifted fields often have a different set of masses and interactions from the unshifted fields.

So essentially before the vacuum gained a Higgs charge particle states were associated with one set of fields and after they were associated with another set of fields of different masses and interactions. They're not really the same particles as they're quanta of different sets of fields. The Higgs shifted which fields have quanta.

Ranku and nrqed
Peter, does it matter (apart from convention) whether ##Z## "eats" ##\bar H^0## or ##H^0## or an arbitrary linear combo of the two?

DarMM
Gold Member
Peter, does it matter (apart from convention) whether ##Z## "eats" ##\bar H^0## or ##H^0## or an arbitrary linear combo of the two?
You have to be careful with this picture, as although it is a good demonstration of the effect it is essentially a classical argument. Even at the classical level it tends not to work if you push it too much.

Strocchi's "An introduction to Non-perturbative Foundations of Quantum Field Theory" section 7.6 contains a good discussion, but I won't go into it too much since this is an I level thread.

weirdoguy
PeterDonis
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does it matter (apart from convention) whether ##Z## "eats" ##bar{H}^0## or ##H^0## or an arbitrary linear combo of the two?
The picture I gave was highly schematic. To answer this question you would need to dig in much more detail into the math. As @DarMM said, that's really beyond the scope of an "I" level thread--it's an "A" level topic.

The Higgs boson ##H## is what we observe in experiments at the LHC; it is actually the "leftover" part of the Higgs field that did not get eaten.
So how does the 'leftover' Higgs boson acquire mass by itself? Or, is it that it remains a massless field and the high energies at LHC transfer sufficient energy to it to become massive and be detectable as such?

PeterDonis
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2020 Award
how does the 'leftover' Higgs boson acquire mass by itself?
The process of electroweak symmetry breaking makes the "leftover" Higgs boson massive.

is it that it remains a massless field and the high energies at LHC transfer sufficient energy to it to become massive
This can't happen; exciting a massless field in a high-energy experiment doesn't make it massive. "Mass" here means invariant mass, which is not changed by exciting the field.

Ranku
We also have that the quark and lepton fields now are coupled to the Higgs fields in a more complicated way to give them mass. The Higgs boson ##H## is what we observe in experiments at the LHC; it is actually the "leftover" part of the Higgs field that did not get eaten
Wanted to clarify one more thing: When the Higgs fields couple to the quark and lepton fields, does that happen after the Higgs boson that represents the Higgs fields has already acquired mass?

PeterDonis
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2020 Award
When the Higgs fields couple to the quark and lepton fields, does that happen after the Higgs boson that represents the Higgs fields has already acquired mass?
Electroweak symmetry breaking is one thing, and all of the things you mention are part of electroweak symmetry breaking. So they all happen "at the same time".

vanhees71 and Ranku