# How the Higgs field works: basic idea

Understanding particle physics:
1. A ball on a spring, the Newtonian version
2. Quantum ball on spring
3. Waves, classic look
4. Waves, the classical equation of motion
5. Quantum waves
6. Fields
7. Particles are quanta
8. How particles interact with fields

How the Higgs field works:

If you read my series of articles about particle and field physics , you know that everything is so-called. “Elementary particles” are in fact quanta (waves whose amplitude and energy are minimally admissible by quantum mechanics) of relativistic quantum fields. Such fields usually satisfy the equations of motion of class 1 (or their generalization) of the form



Where Z (x, t) is the field, Z 0 is the equilibrium state, x is the space, t is the time, d 2 Z / dt 2 represents the time variation of the time variation of Z (d 2 Z / dx 2 is the same for the space ), c is the universal speed limit (often called the “speed of light”), and ν min is the minimum permissible frequency for a wave in a field. Some fields satisfy a class 0 equation, which is simply a class 1 equation, in which the value of ν min is zero. A quantum of such a field has mass



Where h is Planck's constant. In other words,



All this is true only up to a certain limit. If all the fields satisfy the equations of class 0 or class 1, nothing would happen in the Universe. Quanta would just fly past each other and do nothing. Neither scattering, nor collisions, nor the formation of such interesting things as protons or atoms. So let's introduce a common, interesting and required addition according to the experiments.

Imagine two fields, S (x, t) and Z (x, t). Imagine that the equations of motion for S (x, t) and Z (x, t) will be modified versions of the equations of class 1 and 0, respectively, that is, the particles S will be massive, and the particles Z will be massless. So far, we assume that the equilibrium values ​​of S 0 and Z 0 are zero.



Complicate the equation in a way that is everywhere in the real world. Specifically, they contain additional terms in which S (x, t) is multiplied with Z (x, t).



Let me remind you that S and Z are abbreviations for S (x, t) and Z (x, t), varying in space and time. All the rest (c, h, y, m S ) are constants independent of space and time. The parameter y is a number, usually between 0 and 1, called the “ Yukawa parameter” for historical reasons.

In almost all cases in particle physics, the deviations of the fields S (x, t) and Z (x, t) from their equilibrium states S 0 and Z 0 are extremely small. Since we assume that S 0 = 0 and Z 0 = 0, this means that S and Z are small in themselves: any waves in S and Z will have small amplitudes (usually they will consist of one quantum) and although spontaneous quantum Perturbations occur constantly (they are often called virtual particles and are described in articles on particles and fields as a quantum tremor), these disturbances are also small in amplitude (although sometimes very important). If S is small, Z is small, then SZ is really small. Since y is small, the terms y 2 SZ 2 and y 2 S 2 Z are small enough to be ignored in many cases.

Specifically, they can be ignored when calculating the mass of "particles" (that is, quanta) S and Z. To understand what a particle S is, we need to consider the wave S (x, t), considering Z (x, t) to be very small. To understand what the particle Z is, we need to consider the wave Z (x, t), considering that S (x, t) is very small. As soon as we ignore the additional terms y 2 SZ 2 and y 2 S 2 Z, both fields S and Z will satisfy simple equations of motion of class 0 or 1, from which we started, from which we deduce that the particle S has mass m S , and the particle Z has a mass of zero.

Now imagine a world in which Z 0 is zero and S 0 is not. We slightly change the equations:



Again, S and Z are functions of space and time, but everything else, including S 0 , are constants. In this case, Z (x, t) is very small, but S (x, t) is not! In such cases, it is helpful to write



$S\left(x,t\right)={S}_{0}+s\left(x,t\right)$

Where s is the variation of S from the equilibrium state S 0 . We can say that s (x, t) is a shifted version of the field S (x, t). The statement that the fields in particle physics most of the time remain near their equilibrium states is equivalent to the fact that s (x, t) is very small, and not to the fact that S (x, t) is very small. Substituting the last equation into a set of equations for S and Z, and remembering that S 0 is a constant, therefore d S 0 / dt = 0 and dS 0 / dx = 0, we get:





As before, if we need to know the masses of the quanta of the fields S and Z, we can discard any term in the equations, which contains multiplication of two or more small fields - members like Z 2 or s Z 2 or sZ or s 2 Z. Let's see what remains if we leave only members with only one field:





(“+ ...” reminds us that we have excluded something). The equation for the field s has not changed much, since all the new terms, y 2 [S 0 + s] Z 2 contain at least two degrees Z. But in the equation for the field Z we cannot ignore the term y 2 [S 0 + s] 2 Z, because it contains a member of the form y 2 S 0 2 Z, containing only one field. Consequently, although the quantum of the field S still satisfies the equation of class 1 and has a mass m S , the quantum of the field Z no longer satisfies the equation of class 0! He now satisfies the equation of class 1:



Therefore, the quantum of the field Z now has a mass!



${m}_{Z}=y{S}_{0}$

Due to simple interactions of the fields S and Z with the force y, a nonzero equilibrium value S 0 for the field S gives the quantum Z a mass proportional to y and S 0 .

Non-zero value of the field S gave mass to the particle of the field Z!

Small print: even if for some reason the mass m Z of the particle Z was initially non-zero, then the mass of the particle Z will move.



${m}_{{Z}_{new}}=\left[{m}_{Z}^{2}+{y}^{2}{S}_{0}^{2}{\right]}^{1/2}$

(recall that x 1/2 means the same as √x).

So, in fact, the Higgs field H (x, t) and adds mass to the particles. It turns out that for all known particles σ (except for the Higgs particle itself), the equation of motion for the field Σ (x, t) corresponding to it is an equation of class 0, which, at first glance, means that particle σ is massless. However, in the equations of motion many of these fields have additional terms, including the member



Where y σ is the Yukawa parameter, its own for each field, denoting the interaction force between the fields H and Σ. In such cases, a non-zero average value of the Higgs field H (x, t) = H 0 shifts the minimum frequency of the waves Σ, and, consequently, the mass of particles σ, from zero to a non-zero value: $m_ \ sigma = y_ \ sigma H_0$. A variety of Yukawa parameters for different fields of nature leads to a variety of masses among the "particles" (more precisely, quanta) of nature.

Note that the Higgs particle has nothing to do with this. The Higgs particle - the quantum of the Higgs field - the ripples of the minimum energy in H (x, t), a small wave, depending on space and time. A non-zero Higgs field equilibrium constant, H (x, t) = H 0 , extending over the entire Universe gives a mass to other known particles of nature. This timeless and omnipresent constant is very different from the Higgs particles, which are ripples varying in space and time, localized and ephemeral.

That is the basic idea. In this article, I did not reveal many obvious questions - why are there necessarily members in the equations that include the products of two or more fields (the importance of these members can be read here )? Why would known particles be massless if there were no Higgs field? Why does the Higgs field have a nonzero equilibrium value, although this is not the case for most other fields? How does the Higgs particle relate to all this? In the following articles I will try to cover these and other topics.

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