As many know, in 2012 the Higgs Boson was "detected" at the LHC. I have read that the Higgs boson was not actually directly observed, but the existence of the Higgs boson in the standard model helped to predict what was observed, and thus their existence was taken to be true. However, I have read in this stack exchange answer by Conifold (https://hsm.stackexchange.com/a/2128) that there are so-called "Higgless Theories" that include an effective version of the Higgs Boson, and are therefore "not ruled out by the LHC detection". My question is then, how do these Higgsless Theories (that are consistent with what was actually measured at the LHC) work?
Quantum Field Theory – Higgsless Theories Explaining Higgs Boson Detection at the LHC
effective-field-theoryhiggslarge-hadron-colliderquantum-field-theory
Related Solutions
I will try to address your question, though, as David says in the comments, it is evident that you have very little background in elementary particle physics. I will bring over an event much simpler than a display of an event that could show a Higgs particle decay.
Here is a simple antiproton annihilation event whose end particles are recorded by their passage through a bubble chamber which also has a magnetic field perpendicular to the picture. The antiproton enters from below and hits a proton which is at rest, so not visible, in the bubble chamber liquid. It annihilates and eight pions come out, their momentum measured by the curvature, their mass by the ionisation track.
Where is the Higgs field in this picture? It permeates everything and at the point of interaction when the pions materialize it has supplied the masses to the quarks and antiquarks that they are made up of.
The simulated Higgs event display you have attached shows the decay products of the Higgs Boson. This particle is predicted by the Standard Model and it is necessary to find and confirm it in order to validate the SM. It appears because a Higgs field exists, but it is a particle in the data set of particles predicted and mostly found by the SM. In the real experiment, a number of events
with two photons, for example, have been accumulated so that the claim of seeing a Higgs like particle has been established statistically.
A lot of work remains to make sure that the bump seen has really the decay branching ratios and spin and statistics expected from the SM before the discovery of the Higgs boson is established unequivocally. Then we could state with some certainty that the Standard Model which depends on the existence of a Higgs field is validated.
So it should be clear that each individual event is not like a spider that can be dissected. It is an instant of the materialization of the fields and the experiment has to accumulate enough events to statistically establish an observation that validates a hypothesis.
Best Answer
Have you looked at the list your link gave for theories without a Higgs field?
If you do you will see a long list of disparate theories that try to model the same data that the standard model,SM, of particle data fits, with various successes as you will see reading the list and its references.
The SM can be thought as a data bank of 99% of measurements and observations at present, and its success is due to also being predictive for new data, as was the discovery of the Higgs boson.
The SM is a quantum field theory, and the introduction of the Higgs field in order to break the symmetry the model has at very high energies , to the mass spectrum we observe in the laboratory, carries with it the necessity of the existence of a Higgs boson, which was a prediction until the Higgs boson was discovered at the LHC.
Alternative theories attempting to model the same data have to embed the SM or show that the SM is derivable from the new theory, in order to fit the existing data.
Each alternate theory works with its own mathematics which cannot be described in a page on a question and answers site.