What precisely and mathematically does it mean to say gauge bosons as elementary particles?

Question

In standard particle physics textbook, we say that photons, gluons and $W$ and $Z$ bosons are gauge bosons as elementary particles.

However the gauge bosons are vector bosons and they carry the form of one form gauge field, and have gauge invariant form such as a closed line as $$ tr_R [\exp(i \oint A)], $$ for some representation $R$.

So gauge field is better defined precisely and mathematically as 1 dimensional loop, instead of 0 dimensional particle.

My Question

So why can we regard photons, gluons and $W$ and $Z$ bosons are gauge bosons as 0 dimensional elementary particles? In fact, it makes better reasons to consider $$ tr_R [\exp(i \oint W^{\pm})], $$ $$ tr_R [\exp(i \oint Z^{0})], $$ And how do we measure them as particles in experiments? (the lifetime width of the excitations) instead of regarding gauge bosons as 1 dimensional loop?

This post imported from StackExchange Physics at 2020-12-13 12:44 (UTC), posted by SE-user annie marie heart

Comments

Why do you want to "regard gauge bosons as 1 dimensional loop"? (These Wilson lines do have physical meaning, but usually not that of being particles) What about the standard quantization procedures of e.g. QED (Gupta-Bleuler or BRST) that produce the (physical, gauge-invariant!) photon states is unclear to you?

This post imported from StackExchange Physics at 2020-12-13 12:44 (UTC), posted by SE-user ACuriousMind

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This reminds me of Kaluza-Klein theory.

This post imported from StackExchange Physics at 2020-12-13 12:44 (UTC), posted by SE-user PM 2Ring

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Isn't this just the same as the eternal issue of potentials vs fields?

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user Javier

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If you find my question meaningless: The dimensionally of scalar boson like Higgs is 0 dim, but the dimensionally of gauge boson like W or Z is 1 dim vector, better to integrate out some loop. So it is very different to say W or Z as particles (weird due to 1 dim vector) than the Higgs or leptons or quarks as particles which are indeed 0 dim.

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user annie marie heart

Answer 1

This is my understanding of the Quantum Field Theory (QFT) as used in particle physics models, after years working in experiments in particle physics to verify or find discrepancies in the Standard Model.

To the point particles in the table of the standard model, the QFT mathematical model assigns a field, an electron field, an electron neutrino field....., which mathematically is the plane wave solution of the corresponding quantum mechanical equation. The fields cover all of space time. On these fields creation and annihilation operators work in creating and annihilating the particles in the table, so an interaction of real particles can be modeled finally with Feynman diagrams.

For the theory to be correct, there should exist the corresponding particle as a real particle too, in order to be created and annihilated, on par with the existence of electrons, muons, etc.in the table. They are called gauge bosons because of the way they are used in the theory of the SM, as the carriers of the weak interaction.

In my time the W and the Z bosons were found with the masses in the table, confirming the predictions of the standard model .

That is why we were desperately looking for a particle Higgs, because the Higgs field of the SM had to have the corresponding particle, which finally the LHC managed to find in the data of the CMS and Atlas experiments.

Here is the measurement of the Higgs boson by CMS, verifying the standard model hypotheses , in the invariant mass of the channel of two gammas.

In comments you ask:

compare the W/Z vs Higgs.

The gauge bosons have vector spin 1, the Higgs spin 0. (intrinsic spin is introduced for elementary particles in the table in order for angular momentum to be conserved in interactions and decays). They have different functions within the model. The gauge bosons are the virtual particles exchanged in the lowest order Feynman diagram for the given force (electromagnetic, strong, weak), and allowing for change in quantum numbers at the vertices in all orders. The Higgs mechanism is introduced in the model in order to describe mathematically electroweak symmetry breaking . ( this has the history, of how experimentally observed partial symmetries led to the model)The structure of a QFT model is such that fields cannot exist without the corresponding real (with an on mass four vector) particle. That is why it was necessary to find a Higgs particle, to confirm the standard model. There are other models than the SM, where the Higgs is a composite particle, technicolor for example.

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user anna v

Comments

Really nice answer.

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user Árpád Szendrei

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Thanks!!! --- If you find my question meaningless: The dimensionally of scalar boson like Higgs is 0 dim, but the dimensionally of gauge boson like W or Z is 1 dim vector, better to integrate out some loop. So it is very different to say W or Z as particles (weird due to 1 dim vector) than the Higgs or leptons or quarks as particles which are indeed 0 dim. So I will be more willing to accept your answer if you compare the W/Z vs Higgs.

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user annie marie heart

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Of course I voted up - I like experiments

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user annie marie heart

Answer 2

Fundamental particles in quantum mechanics are defined to be the quanta of fundamental fields. For example, when we apply the laws of quantum mechanics to the electromagentic field, we find out the the energy and momentum of the field can only come in discrete bundles (quanta) which we call photons. For electrons, we say that there is an electron field. When we apply the laws of quantum mechanics to this field, we again get discrete bundles that correspond to electrons and positrons. For quarks, we say that there is a gauge field, and when we quantize the gauge field, we again get discrete bundles.

In every case, we start by simply declaring that a field exists. In this sense, the field is fundamental. It's existence is inferred from experiment and not derived from some other theory. Then, the quanta of these fields are referred to as fundamental particles. They are fundamental because they are the quanta of fundamental fields.

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user JoshuaTS

Comments

Thanks --- If you find my question meaningless: The dimensionally of scalar boson like Higgs is 0 dim, but the dimensionally of gauge boson like W or Z is 1 dim vector, better to integrate out some loop. So it is very different to say W or Z as particles (weird due to 1 dim vector) than the Higgs or leptons or quarks as particles which are indeed 0 dim.

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user annie marie heart

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I'm now sure what you mean when you say that a scalar particle is 0-dimensional. Are you talking about spin? The Higgs boson is spin-0, and the W and Z boson are spin-1.

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user JoshuaTS

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more than that, the spin-1 particle has gauge transformation and it is not a gauge invariant object by itself. But it can be gauge invariant only integrating over a loop

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user annie marie heart

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of course, one can ask the spin-2 graviton case.

This post imported from StackExchange Physics at 2020-12-13 12:45 (UTC), posted by SE-user annie marie heart