# How particles interact with the Higgs Field?

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1. May 25, 2014

### webb202

Reading around gives me some idea of WHY particles interact with the Higgs field but not How!
I am looking for a model that I can use to explain the process to my High school students. Can anyone help please.
John

2. May 25, 2014

### vanhees71

I don't know, how to explain the inner workings of the Standard Model to high school students, because for this you need pretty advanced math on groups to describe symmetries and quantum field theory including renormalization of non-Abelian gauge symmetries.

The point is that quantum flavor dynamics (aka. Glashow-Salam-Weinberg model) is based on a local chiral gauge symmetry with gauge group $\mathrm{SU}(2)_{\text{wiso}} \times \mathrm{U}(1)_{\text{Y}}$. Where "wiso" stands for "weak isospin" and "Y" for hypercharge.

The "matter particles" involved are the leptons and quarks, that are grouped into three families. The model works with any number of families. For the first family the left handed parts of the lepton and quark fields build isospin doublets
$$\psi_{Le}=\begin{pmatrix} e \\ \nu_e \end{pmatrix}, \quad \psi_{Lq}=\begin{pmatrix} u \\d \end{pmatrix}.$$
The right-handed parts are iso singulets. The very same structure is used for the other two families grouping together $\mu,\nu_{\mu}$; $c,s$ (2nd generation) and $\tau,\nu_{\tau}$; $t,b$.

There are three gauge bosons for the isospin symmetry and one for hypercharge symmetry. For all these particles it is forbidden to write down naive mass terms, because this would break the local gauge invariance. That's immediately clear for the gauge fields, because this would involve mass terms $\propto A_{\mu} A^{\mu}$, which is not a gauge-invariant expression. For the leptons and quarks it's due to the chiral nature of the interaction. Since the gauge fields couple differently to the left and righthanded parts, there must not be any terms that mix these terms in a naive way.

To provide mass to the particles as observed, one must necessarily use the Higgs mechanism. In the minimal version (which seems to be the one realized in nature as is indicated by the newest results on the Higgs boson at the LHC) you introduce a isospin doublet of scalar-boson fields, but with a mass term of "the wrong sign". Together with a isospin symmetric four-boson coupling you apparently get spontaneous symmetry breaking, i.e., the scalar boson fields must have a non-vanishing vacuum expectation value, which can be chosen arbitrarily in any direction of the Higgs-field doublet (which are given by 2 complex, i.e., 4 real field-degrees of freedom). The gauge group is thus spontaneously broken to $\mathrm{U}(1)_{\text{em}}$.

Via the minimal couplings of the gauge bosons to the Higgs doublet the constant vaccum expectation value (vev) leads to a mass term for three of the four gauge fields, one of which turns out to be electrically neutral as it turns out when expressing the gauge fields in terms of their mass eigenstates, the Z boson, and the other two form the two charged W bosons. The remaining gauge field stays massless and is identified with the photon field.

Now, the leptons and quarks remain still massless, but that can be "cured" by writing down couplings between the leptons and quarks with the Higgs doublet, i.e., one forms an iso doublet with the left- and righthanded pieces and combines it in a gauge symmetric with the Higgs doublet. After the spontaneous symmetry breaking the leptons and quarks acquire a mass, but for the quarks the isospin eigenstates do not coincide with the mass eigenstates (as we know now, that also holds true for the leptons, because one has observed neutrino mixing which is only possible for massive neutrinos; in the here discussed original GSW model the neutrinos are kept massless and thus there's no mixing in this approximation).

The key observation of Anderson, Brout, Englert, Higgs, Kibble, Guralnik, and Hagen was that the spontaneous breakdown of a local gauge symmetry does not imply the existence of massless Nambu-Goldstone boson as is the case for the spontaneous breaking of a global symmetry.

The latter mechanism is at work in approximate form in the chiral symmetry in context with the strong interaction, where the quark-condensate vev is responsible for the spontaneous breaking of the approximate chiral symmetry in the light-quark sector. There the pions turn out to be the (pseudo-)Goldstone bosons of the breaking of this global symmetry.

For the case of a local gauge symmetry the "would-be Goldstone bosons" can be "gauged away", i.e., by the choice of a particular gauge, one can lump three of the four real Higgs-field degrees of freedom into the gauge boson fields. This is necessary, because a massless vector field has only two physical degrees of freedom (there are only two polarization states of the electromagnetic field, i.e., the massless photon) but a massive vector field has three physical spin-degrees of freedom. Thus in "absorbing" the would-be Goldstone modes into the gauge bosons they provide the necessary third physical spin component for these gauge bosons.

Since only three of the four Higgs-field degrees of freedom are absorbed in this socalled unitary gauge, first of all the fourth gauge boson remains massless (as it should be, because the photon is with an amazing precision found to be indeed massless). Second, and this was the unique contribution by Higgs, there must remain one scalar boson in the physical particle spectrum, which justifies the name "Higgs boson" for this physical particle and after all the Nobel prize to be given only to Englert and Higgs and not to the other people, who also published the idea of spontaneous breaking of a local non-Abelian gauge symmetry to provide mass to gauge bosons without explicit breaking of this gauge symmetry, which would spoil the entire consistency of the model making it useless as a physical model of elementary particles.

The real breakthrough of this model came when 't Hooft and Veltman could prove that non-abelian gauge theories (no matter if the gauge symmetry is spontaneously broken or not) is renormalizable as long as one only admits "superficially renormalizable" terms in the original Lagrangian. The key issue in the broken case is, not to choose the unitary gauge here, but another gauge, invented by 't Hooft and Veltman (the socalled $R_{\xi}$ gauges). This leads to the usual power counting of diagrams and thus the renormalizability of this kind of gauge theories. Since the unitary gauge is a smooth limit of the $R_{\xi}$, the resulting S-matrix, which is gauge invariant, is renormalizable and unitary.

Last but not least one must mention that the Standard Model's gauge group is also free of anomalies, thanks to the specific charge pattern of the leptons and quarks. Here it is very important that this constraint matches with the -1/3 and 2/3 charges of the quarks and their additional color charge (each quarks comes in three colors). All together the Standard Model based on the here summarized quantum flavor dynamics and Quantum Chromodynamics for the strong interactions, builds a pretty beautiful model that in addition withstood all attempts to find deviations of the behavior of elementary particles, and this is true although the particle physicists would really like to find deviations from the Standard Model to solve some issues this model still has on the theoretical level, but that's another story.

For a somewhat more explicit explanation about this fascinating model, see my transparencies of lectures (see Lecture 1 for a brief review of the Standard Model), I've given recently to graduate students:

http://fias.uni-frankfurt.de/~hees/hqm-lectweek14/index.html

For a very clear exposition of the whole Standard Model, see

O. Nachtmann. Elementary Particle Physics - Concepts and Phenomenology. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1990.

As I already said above: I do not know, how you can translate this to high-school level. Perhaps the following popular physics books help:

L. Lederman, D. Teresi, The God Particle

and

F. Close, The Infinity Puzzle: Quantum Field Theory and the Hunt for an Orderly Universe

3. May 26, 2014

### webb202

I agree that the math is daunting but dont underestimate the ability of able students to get a flavour of the theory. 'To provide mass to the particles as observed' -your response dodges the 'How' of the mechanism.

4. May 26, 2014

### nikkkom

He did explain the "how". Fermions acquire mass by having non-zero Yukawa coupling to the Higgs field.

5. May 26, 2014

Staff Emeritus
Don't confuse "I didn't understand it" with "you didn't explain it".

6. May 26, 2014

### ChrisVer

Really, what do you mean by "how"?
They interact with the Higgs field, in the same way they interact with other fields.... eg how electrons interact with the electromagnetic field?
Higgs field is everywhere, but gets a vacuum value (that's what I guess you mean by "why"), and so particles moving in that field/interacting with it, get their masses.

7. May 26, 2014

### Bill_K

In addition to being uniform and everywhere, the Higgs field is Lorentz invariant. After all, it's an attribute of the vacuum state, which itself is Lorentz invariant. The Higgs field looks the same from every reference frame.

The popsci picture is that the Higgs field gives particles mass by acting like molasses and slowing them down.
But "moving in the Higgs field" cannot happen in the first place. A particle, no matter what its state of motion, is always at rest WRT the Higgs field.

Last edited: May 26, 2014
8. May 26, 2014

### ChrisVer

and a highschool student would ask, what is Lorentz Invariant?
That's why I don't think he understood what the people above mentioned about Yukawa terms or Lagrangian.... you couldn't explain the Higgs to a highscool student by telling him 1000 unknown words and terminology, which needs ~3years of undergrad just to get a glimpse of...
Better try to give a rough and bad optimization of what happens. So yes, molasses is a nice way

9. May 26, 2014

### Bill_K

OMG, well in that case we totally disagree. Since the truth is rather complicated, you want to tell them something which is flatly false. Might as well tell them the Earth is 10,000 years old.

10. May 26, 2014

### ChrisVer

Not false.. Always optimizing quantum stuff is falsely done... You cannot think of an electron as a small ball going around the proton, however you can't tell a highschool student "a that's the Dirac equation which has this solution and from the solution you see...."
The technical stuff should be used according to your audience. You can't go to 1st year undegrads and start building the SM lagrangian, that's what a fool would do IMO. I mean what's the point of doing that? Even if the poster here is able to understand the technical stuff behind Higgs Mech (he'd be a genius child and should be in Standord or I don't know in which institute with scholarship), the class won't get it. The school is not to produce physicists, but to arise the interest for studying fields (1 of which is physics)
Although I also found "Richard Feynman refuses to try to explain the Higgs particle in any terms the viewer would understand. For Feynman, a workaday metaphor would do more harm with its inaccuracy than good by attempting to explain." So for the poster, I give bullet (1) below...
Nevertheless, it's not the right topic, we could do that in the lounge or I don't know....

1. there are material coming from CERN about Higgs, which refer to highschool students (because it hosts schools). If you search on google, you'll probably find some matterial. It's better looking through them and once you have questions concerning them, come and ask them specifically.
2. there is some group who made some nice videos on youtube about CERN-LHC, in a rap way...

(of course I'm still expecting what thread poster meant by "how")

Last edited by a moderator: Sep 25, 2014
11. May 26, 2014

### dauto

I think it is OK to tell your students that the math is beyond their abilities (for now). It is the truth and the truth is always better than some made up explanation such as the molasses "analogy".

12. May 26, 2014

### The_Duck

For most particles it costs energy to be in a region of space where the value of the Higgs field is nonzero*. The energy of a particle when it at rest is related to its mass by $E=mc^2$. Therefore most particles acquire mass when they are in a region where the Higgs field is not zero. And in fact the Higgs field is nonzero everywhere in the universe.

The energy of some particles is unaffected by the Higgs field. Photons are an example; they have zero mass.

(*) There's not really a mechanism for why it costs energy to be in a region of nonzero Higgs field; it is just postulated in the fundamental equations we use to describe the Higgs field. However that does not mean that we just pulled this idea out of thin air. The mathematical structure of our theory of particle physics essentially requires it.

Last edited: May 26, 2014
13. May 27, 2014

### webb202

Thanks

Thank you all for such terrific responses. I am working at the edges of my own understanding and you have given me much material to probe.
As an educator I disagree about explaining complex concepts. Back in about the 16th century it was believed that the blood supply moved like the tides, it wasnt untill piped water became generally available that the understanding of the process moved on to the heart pumping a continuous supply. It is by the mental models that we have that enable us to advance. You will be amazes by the growth in comlexity of mental models young people are being exposed to. I can recall the problems explaining the concept of push-on - push -off buttons to my father -he expected to do a different action to turn off i.e. a toggle switch.
I next want to continu asking about the 'how' of the interaction with the Higgs Field. As I understand the proceses of interaction with such as the e/m field it can be explained as vibration in the field - I presume that QM rules apply to its position, and intensity. Is the Higgs field interaction similar?

14. May 27, 2014

### ChrisVer

About the heart, it's not only the heart pumbing but also the elasticity of the veins that explains how the blood can reach through all your body. The heart pumping is not enough to explain of why blood can run through the whole body without damping... The motion of blood is subject to non-linear dynamics, and its motion throughout the body is self-supported by the characteristics of the veins.

Suppose that you have the Higgs potential, and the Higgs then acquires a vev... That means that the field of the higgs, being all over the space, at its ground state will not be zero. As such, any field interacting with the Higgs field, even when the last is at the ground state (no excitations) will get some extra contribution. And because matter fields are coupled to the Higgs in quadratic forms, this will lead to masses...
$g H \bar{f}f$
is a Yukawa term... if $H$ has a vev, then it means that the field $f$ will be able even when Higgs is at the ground state as a constant, to "interact with itself" leading to mass term, then the above becomes:
$g h_{vev} \bar{f}f$
the mass of the $f$ field is then $g h_{vev}$
that is "how" they get mass...
Is there an answer to why they do? the answer to why they do, is because they are allowed to...