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I have heard a bit about the Higgs field and I know that it is a quantum field that assigns mass to everything but recently I heard about the Higgs boson and I was wondering how that particle differed from the quantum field.

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It's all in the math of the σ-model, but I gather you are asking for a non-technical explanation. The short answer is that the Higgs boson is the only particle-type degree of freedom of the Higgs field not that closely involved in mass generation.

The Higgs scalar field consists of 4 boson degrees of freedom, arranged in a complex 2-vector signaling how the various pieces rotate into each other by internal space symmetry transformations (SU(2), or rather SO(4), really... don't worry about it), $$ \begin{pmatrix} \pi^+(x) \\ v+ h(x)+i\pi^0(x) \end{pmatrix}.$$ $\pi^+$ is really two fields, so it "knows" about its complex conjugate $\pi^-$ and represents it, implicitly.

So, there are 4 particles (bosons) in all, $\pi^+,\pi^-,\pi^0, h $, and v is just a number, a quarter of a TeV, in our world.

In a world without gauge bosons, this field would give masses to all fermions through its Yukawa couplings to them, and the Higgs boson h would have a mass, while the three πs would be massless.

But in our world, the three πs are degrees of freedom of the W and Z vector bosons, and, in the same breath give them a mass, an apparently impossible feat. But where there is a will there is a way, and Englert, Brout, and Higgs got there 57 years ago. So, now, not that much really depends on what h does and how it couples, but, still, its properties and couplings shed quite some light on the finer aspects of the theory, a tightly fitting clockwork structure.

A parting reminded: while the masses of the leptons by coupling to the Higgs field are the whole story, the quarks, with a mass achieved this way, get much more mass through the couplings to gluons, completely different gauge bosons, and in fact, the hadrons they underlie (the nucleons, the mesons, etc...) get most of their mass through such interactions, not the Higgs field couplings mentioned above.

This is a decent summary of what I sketched the trail map for, above.

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Someone else has already given a more technical answer to your question but to put it into more simple terms. The Higgs Boson is an excitation of the Higgs field. It is actually the interaction with the Higgs Field that "gives" particles mass. The Higgs particle are the excitations in the field that we can measurably detect in particle accelerators.

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The two are essentially synonymous. After spontaneous symmetry breaking, the Standard Model Lagrangian contains terms in which fermions and massive bosons couple to the Higgs field in the following ways: $$\overline{\psi}\psi h, ~~W^\pm W^\pm h, ~~W^\pm W^\pm h^2, ~~Z Z h, ~~ZZ h^2$$ which are interpreted as "mass terms". The coefficients of these terms tell you about the masses of the particle.

The Higgs field can also couple to itself (the SM Lagrangian also contains terms such as $h^3, h^4$), which make the Higgs boson itself massive.

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A laypersons explanation is easier if we go back a step or two, and avoid the complexities of mathematics. So this isn't technically precise, but will give a feeling for the answer.

Originally, early physicists recognised "forces". There were forces and there were objects, and forces acted on objects. Gravity acted on matter, magnets acted on iron objects, and so on.

Gradually we started to consider fields rather than forces. That revolution in thinking started with the electromagnetic field, where the force and its interactions, arose from a pervasive field that existed and interacted.

By the 20th century all known forces were conceived in terms of fields that could interact, father than just forces. Three of them, the electromagnetic, strong and weak interactions, had also been merged with/formed into a form that fitted with quantum theory - known as quantum fields and the theory known as quantum field theory, and covered by the Standard Model. (The fourth, gravity, hasn't yet been merged, its seen as related to the changing geometry of spacetime according to general relativity, even though we often discuss a gravitational field).

In quantum field theory, the particles we see, are manifestations of those fields. (Not exactly, but it'll do). They aren't separate independent objects. If a field changes, the particles that manifest as a result of the field will fundamentally change too, so will the forces and interactions we seem to observe in the world.

The problem with the first three fields, is that they could explain most things we saw particles and forces do, except they couldn't explain mass. Features of the underlying quantum field theory required a property called "gauge invariance", but a gauge invariant theory would have had massless particles, or particles we know don't exist (called Goldstone bosons). Clearly wrong. Big headache in the 1950s - 1960s.

The solution was spotted in the early 1960s. An odd kind of field might be able to interact in a way that (the problematic) particles would end up with mass even if they didn't have mass originally. That's the Higgs Field and it is responsible for some particles being massive when we look around us. The Higgs Field modifies the quantum phenomena that give rise to those particles, at all temperatures other than "trillions of degrees and higher".

That would mean yet another quantum field existed, although unsuspected so far by physicists. Big news if so.

But if the Higgs Field exists, how can we detect it, and prove it?

Well, all quantum fields have particles they can cause to be manifested. You might see those described as their quanta, or gauge bosons, or excitations. But every quantum field we know of, has at least one particle it can manifest as. The Higgs field is a quantum field, so it too has a particle it can manifest. That's the Higgs Boson.

(Technically the Higgs Boson is "left over" when the Higgs field "breaks" other fields and changes them, causing some of their particles to become massive - we call that kind of interaction "symmetry breaking". The Higgs Field has 4 parts to it, but only 3 parts take part in that interaction. Those 3 parts get "swallowed up"/merged when particles acquire mass. The 4th part of the Higgs field is left out alone, and that part manifests as a particle, the Higgs boson. When we detect that particle, and confirm its properties, we know a Higgs field exists)

The Highs Boson's main interest is that we can routinely produce and study Higgs bosons in a particle collider like the Large Hadron Collider, and learn their properties. That teaches us what a Higgs Field is like, the details of its properties and how it can behave. Which teaches us how our universe works.

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