How the Higgs Field Works: 4) Why the Higgs Field is Necessary
How the Higgs field works:
Until now, in a series of articles on the Higgs field, I explained to you the basic idea of how it works, and described how the Higgs field becomes nonzero and how the Higgs particle appears - at least for the simplest field type and the Higgs particle (from the Standard Model) . But I did not explain why there is no alternative for entering something resembling a Higgs field — why there are obstacles for entering masses of known particles in the absence of this field. This is what we will discuss in this article.
I explained that all elementary "particles" (that is, quanta) of nature are quanta of waves in fields. And, simply, all these fields satisfy an equation of class 1 of the form:
where Z (x, t) is the field, m is the mass of the particle, c is the speed of light, h is the Planck constant. If the particle is massless, then the corresponding field satisfies the same equation, where m = 0, which I called the equation of class 0.
Cases with m = 0 include photons, gluons, and gravitons — quanta of the electric, chromoelectric (or gluonic), and gravitational fields; all these are massless quanta (“particles”) moving at the universal speed limit c. For electrons, muons, tau, all quarks, all neutrinos, particles W, Z and the Higgs boson, each of which has its own mass, the corresponding field satisfies the equation of class 1 with the corresponding mass substituted into it.
Unfortunately, this is not the whole story. You see, for all the known elementary fields of nature that correspond to massive quanta, the equation written at the top is not satisfied — at least in the form in which I wrote it down. Why? The problem is that we have not introduced a weak interaction in our equations. And if we introduce it, then, as we shall see, these simple equations will be impossible to use. Instead, they will require more sophisticated equations capable of producing similar physical results.
Why?
The problem is this: the equations we write are necessary, but not enough. We need them to be executed, but this is not the only thing that should be done. We are missing something: weak interaction. And this interaction will not be able to make friends with the equation written above.
If I go into detail, the result will be too abstruse. I will explain this with the help of equations similar to those that are actually used, but not fully delving into the whole story.
To see the problem, consider it in the context of a specific field - for example, take the electronic field. The problem is that the electron field does not quite satisfy the equation above. An electron is a particle with spin -1/2, which means that it not only moves, but also rotates continuously, so that it is impossible to imagine - and it turns out that the equation above is only enough to describe the change in its position, but not to describe that what happens to his spin. As a result, it turns out that in fact an electron is formed from two fields, ψ (x, t) and χ (x, t), satisfying two equations:
Where I entered the constant μ = 2π mc² / h for short. Again, I’m keeping a few words back to you, because this equation of motion is only along one spatial dimension, the x axis; the full form of the equation is more complicated. But the essence is true; we will soon verify that these two equations imply the previous one mentioned at the beginning of the article.
Note: ψ and χ are often called "left-sided electronic" and "right-sided electronic" fields, but without introducing additional mathematics, such names are more confusing than clarified, so I will avoid them.
These two fields together form an electron field in the sense that the electron wave amplitudes χ and must be proportional to each other. This can be verified by making a wave of both of them:
where ψ 0 and χ 0 are the amplitudes of the waves, and ν and λ are their frequency and wavelength (which I assumed to be equal). Then we get:
Which means
These equations show the proportionality ψ 0 and χ 0 ; in general, if one is non-zero, then both are also, and if you increase one of them, the second will increase.
But keep in mind: these are two equations that describe two relationships that can easily contradict each other. Two equations can be consistent if there is an additional relationship between ν, -c / λ and μ. What is this attitude? Multiply the two equations and divide by ψ 0 χ 0 (which can be done until ψ 0 and χ 0 are not equal to zero - we assume that they are not equal), and we find:
What are the consequences of this equation? Suppose we have a single quantum wave in the fields ψ and χ - the wave of the minimum amplitude - in other words, the electron. Then the energy E = hν, and the momentum p = h / λ of this quantum can be obtained by multiplying this equation by h² and substituting μ = 2π mc² / h, obtaining
And this is Einstein's relation between the energy, momentum and mass of an object, which, naturally, an electron of mass m must satisfy.
And this is not by chance, since the Einstein relation holds for a quantum wave satisfying a class 1 equation, and the two equations for ψ and χ imply that ψ and χ satisfy an equation of class 1! To see this, multiply the first equation by –μ and substitute it into the second:
What gives (if we consider that d / dx (dχ / dt) = d / dt (dχ / dx)) an equation of class 1 for χ (a similar trick gives an equation of class 1 for):
Two equations instead of one — a cunning way (invented by Dirac) to make particles with spin -1/2 satisfy the Einstein relation for energy, momentum, and mass. An electron is a quantum of a wave in the fields ψ and χ that together make up the electron field, and this quantum acts as a particle with mass m and spin 1/2. The same is true for muon, tau, and six quarks.
Unfortunately, this beautiful set of equations written in 1930 turned out to be incompatible with experiments. In the 1950s and 1960s, we found that weak interaction only affects χ, but not ψ! This means that the equation
It does not make sense; the time variation of the field χ under the influence of a weak interaction cannot be proportional to the field ψ, which does not depend on the weak interaction. In other words, the field W can turn the field χ (x, t) into the neutrino field ν (x, t), but cannot turn ψ (x, t) into nothing, so the version of this equation that appears after combining the field with it W is not defined and does not make sense:
W ↓ field
This failure of the equations in combination with weak interaction tells us (as the 1960s physicists said) that it is necessary to find a new set of equations. Solving this problem will require a new idea. And the new idea is the Higgs field.
At this stage, the equations will become more complex (so I did not give a detailed explanation from the very beginning). The article, without technical details, which describes what the world would be like with a zero Higgs field , indicates the structure that will appear in the equations below.
We will need equations for electrons and neutrinos, which allow the possibility of the transformation by means of a particle W of an electron into a neutrino and vice versa - but only when interacting with χ (the so-called "left-side electron field"), and not with.
For this it is necessary to recall one subtlety: before the Higgs field becomes nonzero, there are four Higgs fields, and not one. Three of them disappear as a result. It can be confusing that there are several ways to call them - and each of the methods is useful in its context. In my article on the world with the zero field of Higgs, I called these four fields, each of which is a real number in space and time, the names H 0 , A 0 , H + and H - . The Higgs field H (x, t), which I refer to in this series of articles, is H 0 (x, t). Here I will call them two complex fields — that is, functions that have a real and imaginary value at every point of space and time. I will call these two complex fields H + and H 0 ; and the Higgs field H (x, t), which I refer to in this series of articles, will be the real part of H 0 (x, t). After the Higgs field becomes nonzero, H + is absorbed by what we call the field W + , and the imaginary part of H 0 is absorbed by what we call the field Z. [The complex part of H + is called H - ; and since W + absorbs H + , its imaginary part W - absorbs H - ].
The weak fact is connected with the weak interaction: the particles of nature and the equations with which they satisfy must be symmetrical when exchanging some fields between themselves. Full symmetry is quite complicated, but the part we need looks like this:
ψ does not change
χ ⇆ ν
H + ⇆ H 0
H - ⇆ H 0 * (complex part)
W + ⇆ W -
χ ⇆ ν reflects the fact that a weak interaction affects these fields. The fact that ψ does not change reflects the fact that this interaction does not affect it. Without this symmetry, and without its more general form, quantum versions of equations for weak interactions have no meaning: they lead to predictions, from which it follows that the probability of certain events is greater than one or less than zero.
It turns out that the equations we need look like this (here y is the Yukawa parameter, g is a constant that determines the strength of the weak interaction):
Notice that these equations satisfy the symmetry mentioned above . Experts will notice that I have simplified these equations, but I hope that they will agree that they still describe the essence of the problem. Notice that t and x are time and space (although I simplify, tracking only one of the three spatial dimensions); c, h, y, and g are constants independent of space and time; ψ, χ, W, H, etc. - These are fields, functions of space and time.
What happens if the Higgs field becomes nonzero? The H field - and the imaginary part of H 0 will disappear (why I won't write here), being absorbed by other fields. The real part of H 0 becomes nonzero, with the average value of v; as described in the article on how the Higgs field works, we write:
where h (x, t) is a field whose quantum, the Higgs physical particle, we observe in nature. After this equation take the form:
These equations, after the Higgs field takes a non-zero value v, describe the interactions between:
• An electron field whose quanta are electrons of mass m e = yv;
• One of the three neutron fields whose quanta are neutrinos (they are massless in these equations. To add mass, we have to change the equations a little, which I will not write here).
• The field W, whose quanta are W particles, and whose presence implies the participation of weak interaction.
• The Higgs field h (x, t), whose quanta are Higgs particles.
Notice that the equations already seem not to satisfy the mentioned symmetry. This symmetry is “hidden” or “broken.” Her presence is no longer apparent when the Higgs field becomes nonzero. And yet, everything works as it should to fit the experiments:
• If the fields h, W and ν are zero in some region of space and time, the equations turn into the original equations of the electronic field, but as a combination of ψ and χ.
• If the field W in some region is zero, the terms where h enters indicate that the interaction between electrons and Higgs particles is proportional to y, and therefore proportional to the mass of the electron.
• If the field h is zero at some site, then the terms where W - and W + come in show that a weak interaction can turn electrons into neutrinos and vice versa, specifically turning χ into ν without affecting.
Let's summarize. For particles with spin -1/2, simple equations of class 1
which we have studied so far, we have to complicate, as Dirac understood at the time. The description of an electron and its mass requires several equations implying an equation of class 1, but with additional properties. Unfortunately, the simple Dirac equations are not enough, since their structure does not coincide with the behavior of the weak interaction. The solution is to complicate the equations by entering the Higgs field, which, taking an average nonzero value, can give the electron a mass, without interfering with the work of the weak interaction.
We saw how this works with the mass of an electron, up to the equations for the electron field. Similar equations work for the half brothers of the electron, muon, and tau, and for all quark fields; a small change allows them to work for neutrino fields. The masses of particles W and Z appear in different equations, but some of the similar problems — the need to maintain a certain symmetry so that weak interaction makes sense — play their part here too.
In any case, the behavior of the weak interaction, judging by the experiments, and the masses of the known elementary (seemingly) particles observed in the experiments, would not coincide with each other if it were not for something like the Higgs field. Recent experiments at the Large Hadron Collider provided the necessary confirmation that the equations and concepts I described are the ones that they are based on are more or less correct. We are waiting for a new experimental study of the Higgs particle to find out if there are other Higgs fields, and whether the Higgs field will be more complicated than I describe it.
- main idea
- Why the Higgs field is nonzero on average
- How does the Higgs particle appear
- Why the Higgs Field is Necessary
Until now, in a series of articles on the Higgs field, I explained to you the basic idea of how it works, and described how the Higgs field becomes nonzero and how the Higgs particle appears - at least for the simplest field type and the Higgs particle (from the Standard Model) . But I did not explain why there is no alternative for entering something resembling a Higgs field — why there are obstacles for entering masses of known particles in the absence of this field. This is what we will discuss in this article.
I explained that all elementary "particles" (that is, quanta) of nature are quanta of waves in fields. And, simply, all these fields satisfy an equation of class 1 of the form:
d/dt(dZ(x,t)/dt)−c2d/dx(dZ(x,t)/dx)=−(2 pic2/h)2m2Z(x,t)
where Z (x, t) is the field, m is the mass of the particle, c is the speed of light, h is the Planck constant. If the particle is massless, then the corresponding field satisfies the same equation, where m = 0, which I called the equation of class 0.
Cases with m = 0 include photons, gluons, and gravitons — quanta of the electric, chromoelectric (or gluonic), and gravitational fields; all these are massless quanta (“particles”) moving at the universal speed limit c. For electrons, muons, tau, all quarks, all neutrinos, particles W, Z and the Higgs boson, each of which has its own mass, the corresponding field satisfies the equation of class 1 with the corresponding mass substituted into it.
Unfortunately, this is not the whole story. You see, for all the known elementary fields of nature that correspond to massive quanta, the equation written at the top is not satisfied — at least in the form in which I wrote it down. Why? The problem is that we have not introduced a weak interaction in our equations. And if we introduce it, then, as we shall see, these simple equations will be impossible to use. Instead, they will require more sophisticated equations capable of producing similar physical results.
Why?
The problem is this: the equations we write are necessary, but not enough. We need them to be executed, but this is not the only thing that should be done. We are missing something: weak interaction. And this interaction will not be able to make friends with the equation written above.
If I go into detail, the result will be too abstruse. I will explain this with the help of equations similar to those that are actually used, but not fully delving into the whole story.
More complex equations for an electron
To see the problem, consider it in the context of a specific field - for example, take the electronic field. The problem is that the electron field does not quite satisfy the equation above. An electron is a particle with spin -1/2, which means that it not only moves, but also rotates continuously, so that it is impossible to imagine - and it turns out that the equation above is only enough to describe the change in its position, but not to describe that what happens to his spin. As a result, it turns out that in fact an electron is formed from two fields, ψ (x, t) and χ (x, t), satisfying two equations:
d psi/dt−cd psi/dx= mu chid chi/dt+cd chi/dx=− mu psi
Where I entered the constant μ = 2π mc² / h for short. Again, I’m keeping a few words back to you, because this equation of motion is only along one spatial dimension, the x axis; the full form of the equation is more complicated. But the essence is true; we will soon verify that these two equations imply the previous one mentioned at the beginning of the article.
Note: ψ and χ are often called "left-sided electronic" and "right-sided electronic" fields, but without introducing additional mathematics, such names are more confusing than clarified, so I will avoid them.
These two fields together form an electron field in the sense that the electron wave amplitudes χ and must be proportional to each other. This can be verified by making a wave of both of them:
psi= psi0cos(2 pi[ nut+x/ lambda]) chi= chi0sin(2 pi[ nut+x/ lambda])
where ψ 0 and χ 0 are the amplitudes of the waves, and ν and λ are their frequency and wavelength (which I assumed to be equal). Then we get:
(2 pi)( nu−c/ lambda) psi0sin(2 pi[ nut+x/ lambda])= mu chi0sin(2 pi[ nut+x/ lambda])0−(2 pi)( nu+c/ lambda) chi0cos(2 pi[ nut+x/ lambda])=− mu psi0cos(2 pi[ nut+x/ lambda])
Which means
( nu+c/ lambda) psi0=( mu/2 pi) chi0( nu−c/ lambda) chi0=( mu/2 pi) psi0
These equations show the proportionality ψ 0 and χ 0 ; in general, if one is non-zero, then both are also, and if you increase one of them, the second will increase.
But keep in mind: these are two equations that describe two relationships that can easily contradict each other. Two equations can be consistent if there is an additional relationship between ν, -c / λ and μ. What is this attitude? Multiply the two equations and divide by ψ 0 χ 0 (which can be done until ψ 0 and χ 0 are not equal to zero - we assume that they are not equal), and we find:
nu2−(c/ lambda)2=( mu/2 pi)2
What are the consequences of this equation? Suppose we have a single quantum wave in the fields ψ and χ - the wave of the minimum amplitude - in other words, the electron. Then the energy E = hν, and the momentum p = h / λ of this quantum can be obtained by multiplying this equation by h² and substituting μ = 2π mc² / h, obtaining
E2−(pc)2=(mc2)2
And this is Einstein's relation between the energy, momentum and mass of an object, which, naturally, an electron of mass m must satisfy.
And this is not by chance, since the Einstein relation holds for a quantum wave satisfying a class 1 equation, and the two equations for ψ and χ imply that ψ and χ satisfy an equation of class 1! To see this, multiply the first equation by –μ and substitute it into the second:
− mu(d psi/dt−cd psi/dx)=(d/dt−cd/dx)(d chi/dt+cd chi/dx)=− mu2 chi
What gives (if we consider that d / dx (dχ / dt) = d / dt (dχ / dx)) an equation of class 1 for χ (a similar trick gives an equation of class 1 for):
d/dt(d chi/dt)−c2d/dx(d chi/dx)=− mu2 chi
Two equations instead of one — a cunning way (invented by Dirac) to make particles with spin -1/2 satisfy the Einstein relation for energy, momentum, and mass. An electron is a quantum of a wave in the fields ψ and χ that together make up the electron field, and this quantum acts as a particle with mass m and spin 1/2. The same is true for muon, tau, and six quarks.
The mass of the electron, calculated "in the forehead", and the weak interaction contradict each other
Unfortunately, this beautiful set of equations written in 1930 turned out to be incompatible with experiments. In the 1950s and 1960s, we found that weak interaction only affects χ, but not ψ! This means that the equation
d chi/dt−d chi/dx=− mu psi
It does not make sense; the time variation of the field χ under the influence of a weak interaction cannot be proportional to the field ψ, which does not depend on the weak interaction. In other words, the field W can turn the field χ (x, t) into the neutrino field ν (x, t), but cannot turn ψ (x, t) into nothing, so the version of this equation that appears after combining the field with it W is not defined and does not make sense:
d chi/dt−d chi/dx=− mu psi
W ↓ field
d nu/dt−d nu/dx=???
This failure of the equations in combination with weak interaction tells us (as the 1960s physicists said) that it is necessary to find a new set of equations. Solving this problem will require a new idea. And the new idea is the Higgs field.
Higgs field enters: correct equations for electron mass
At this stage, the equations will become more complex (so I did not give a detailed explanation from the very beginning). The article, without technical details, which describes what the world would be like with a zero Higgs field , indicates the structure that will appear in the equations below.
We will need equations for electrons and neutrinos, which allow the possibility of the transformation by means of a particle W of an electron into a neutrino and vice versa - but only when interacting with χ (the so-called "left-side electron field"), and not with.
For this it is necessary to recall one subtlety: before the Higgs field becomes nonzero, there are four Higgs fields, and not one. Three of them disappear as a result. It can be confusing that there are several ways to call them - and each of the methods is useful in its context. In my article on the world with the zero field of Higgs, I called these four fields, each of which is a real number in space and time, the names H 0 , A 0 , H + and H - . The Higgs field H (x, t), which I refer to in this series of articles, is H 0 (x, t). Here I will call them two complex fields — that is, functions that have a real and imaginary value at every point of space and time. I will call these two complex fields H + and H 0 ; and the Higgs field H (x, t), which I refer to in this series of articles, will be the real part of H 0 (x, t). After the Higgs field becomes nonzero, H + is absorbed by what we call the field W + , and the imaginary part of H 0 is absorbed by what we call the field Z. [The complex part of H + is called H - ; and since W + absorbs H + , its imaginary part W - absorbs H - ].
The weak fact is connected with the weak interaction: the particles of nature and the equations with which they satisfy must be symmetrical when exchanging some fields between themselves. Full symmetry is quite complicated, but the part we need looks like this:
ψ does not change
χ ⇆ ν
H + ⇆ H 0
H - ⇆ H 0 * (complex part)
W + ⇆ W -
χ ⇆ ν reflects the fact that a weak interaction affects these fields. The fact that ψ does not change reflects the fact that this interaction does not affect it. Without this symmetry, and without its more general form, quantum versions of equations for weak interactions have no meaning: they lead to predictions, from which it follows that the probability of certain events is greater than one or less than zero.
It turns out that the equations we need look like this (here y is the Yukawa parameter, g is a constant that determines the strength of the weak interaction):
d psi/dt−d psi/dx=(2 pic2/h)y(H0∗ chi+H− nu)d chi/dt+d chi/dx+gW− nu=−(2 pic2/h)yH0 psid nu/dt+d nu/dx+gW+ chi=−(2 pic2/h)yH+ psi
Notice that these equations satisfy the symmetry mentioned above . Experts will notice that I have simplified these equations, but I hope that they will agree that they still describe the essence of the problem. Notice that t and x are time and space (although I simplify, tracking only one of the three spatial dimensions); c, h, y, and g are constants independent of space and time; ψ, χ, W, H, etc. - These are fields, functions of space and time.
What happens if the Higgs field becomes nonzero? The H field - and the imaginary part of H 0 will disappear (why I won't write here), being absorbed by other fields. The real part of H 0 becomes nonzero, with the average value of v; as described in the article on how the Higgs field works, we write:
Real[H0(x,t)]=H(x,t)=v+h(x,t)
where h (x, t) is a field whose quantum, the Higgs physical particle, we observe in nature. After this equation take the form:
d psi/dt−d psi/dx=(2 pic2/h)y(v+h) chid chi/dt+d chi/dx+gW− nu=−(2 pic2/h)y(v+h) psid nu/dt+d nu/dx+gW+ chi=0
These equations, after the Higgs field takes a non-zero value v, describe the interactions between:
• An electron field whose quanta are electrons of mass m e = yv;
• One of the three neutron fields whose quanta are neutrinos (they are massless in these equations. To add mass, we have to change the equations a little, which I will not write here).
• The field W, whose quanta are W particles, and whose presence implies the participation of weak interaction.
• The Higgs field h (x, t), whose quanta are Higgs particles.
Notice that the equations already seem not to satisfy the mentioned symmetry. This symmetry is “hidden” or “broken.” Her presence is no longer apparent when the Higgs field becomes nonzero. And yet, everything works as it should to fit the experiments:
• If the fields h, W and ν are zero in some region of space and time, the equations turn into the original equations of the electronic field, but as a combination of ψ and χ.
• If the field W in some region is zero, the terms where h enters indicate that the interaction between electrons and Higgs particles is proportional to y, and therefore proportional to the mass of the electron.
• If the field h is zero at some site, then the terms where W - and W + come in show that a weak interaction can turn electrons into neutrinos and vice versa, specifically turning χ into ν without affecting.
Results
Let's summarize. For particles with spin -1/2, simple equations of class 1
d/dt(dZ(x,t)/dt)−c2d/dx(dZ(x,t)/dx)=−(2 pic2/h)2m2Z(x,t)
which we have studied so far, we have to complicate, as Dirac understood at the time. The description of an electron and its mass requires several equations implying an equation of class 1, but with additional properties. Unfortunately, the simple Dirac equations are not enough, since their structure does not coincide with the behavior of the weak interaction. The solution is to complicate the equations by entering the Higgs field, which, taking an average nonzero value, can give the electron a mass, without interfering with the work of the weak interaction.
We saw how this works with the mass of an electron, up to the equations for the electron field. Similar equations work for the half brothers of the electron, muon, and tau, and for all quark fields; a small change allows them to work for neutrino fields. The masses of particles W and Z appear in different equations, but some of the similar problems — the need to maintain a certain symmetry so that weak interaction makes sense — play their part here too.
In any case, the behavior of the weak interaction, judging by the experiments, and the masses of the known elementary (seemingly) particles observed in the experiments, would not coincide with each other if it were not for something like the Higgs field. Recent experiments at the Large Hadron Collider provided the necessary confirmation that the equations and concepts I described are the ones that they are based on are more or less correct. We are waiting for a new experimental study of the Higgs particle to find out if there are other Higgs fields, and whether the Higgs field will be more complicated than I describe it.
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