WHY THE HIGGS PARTICLE IS SO IMPORTANT!

• Why the Higgs particle?

DETAILS:

• Overview: building blocks and forces of matter
• Why the Higgs particle: elaborate
• References

1. WHY THE HIGGS PARTICLE?

We have said that it is relatively easy to picture the Higgs field and it is like we seem heavier as we move through water. In general this is good enough but it is also sometimes too simple. For example, if we row through the water and then stop rowing, the boot will come to a rest after a while due the friction with the water. So why do not all particles (or stars or planets) come to a rest due to the “friction” with the Higgs-field? This can be understood a little bit better by considering the motion of a particle through a field generated by a lattice of point sources. See figure 4.

Figure 4. The motion of the blue spheres through the “Higgs field” of the red spheres. The red spheres repel the blue sphere in the plain.

In figure 4 the red spheres are fixed lattice points that repel the blue spheres in the plain. We can now see that a blue sphere that in point 1 moves upwards won't come to a rest. Because the blue sphere, moving from point 1 to point 2, is closer to the red spheres A and B than to the other red spheres, there will be an upward directed force exerted on the blue sphere (the horizontal forces cancel). The blue sphere will increase its speed a little and will have reached its maximum speed in point 3. In point 3 the force on the blue sphere will be zero again, as is clear from the symmetry of the picture. Now the blue sphere moves towards point 4 and is closer to the red spheres C and D than to the other red spheres and now there will be a downward directed force on the blue sphere and so the blue sphere will decrease its speed. In point 5 the blue sphere will have exactly the same speed as it had in point 1 and will continue in the same way, as depicted in point 6. The initial speed of the blue sphere will not increase or decrease on a macroscopic scale and the blue sphere will not come to a stop.
If we want to deviate the blue sphere from its straight path and want to let it move in a particular way, as shown by the points 7,8 and 9, then we do feel the repulsion of the red spheres. The blue sphere will seem heavier because we will have to overcome the force that is exerted on it because we come closer to one of the red spheres. The above sketch looks more like how mass reveals itself. As long as we allow a body to move in a straight line with constant speed we don't notice whether it is heavy or not. The body will not come to a rest and continues to move in a straight line with, on a macroscopic scale, the same velocity. It is only when we try to deviate it from its straight path that mass feels “sticky” and it takes a certain effort.

3.2 RELATIVITY THEORY

We are now ready for the real story: why the introduction of the Higgs field was inevitable. Let us start with Einstein's famous equation E=mc². It states that energy equals mass times the speed of light times once again the speed of light. It is perhaps the most beautiful formula of physics! Yet there is something strange about it. If we put m=0 in this formula, it tells us that a particle without mass has no energy. We know that this cannot be true: light has no mass but it has energy (directly proportional to its frequency). We can say that Einstein's famous equation is valid only for objects or particles with mass. That is correct but that doesn't save us completely. One would expect that if the mass becomes smaller, one would end up with the energy formula for light but not that the answer goes to zero (for the experts: the √(1-v²/c²) doesn't help us either because there is no law that tells us how v-> c as m-> 0). So we come to the conclusion that particles with a tiny amount of mass are still fundamentally different from particles without mass.

And that's right: particles with mass are completely different compared to massless particles. There is another way to see this. For this we will use a so-called thought experiment. That is an experiment that is possible in principle but in practice difficult to carry out and that is why we do it in our mind. If a train runs past us, then of course the train has some speed. We can stop the train relative to us by running alongside the train. If we run fast enough, the train will be at rest relative to us. Not relative to the earth, of course, but relative to someone running alongside the train. Thus it is possible to find an observer relative to which the train is at rest. The same goes for an elementary particle with mass. It can go very fast, perhaps nearly at the speed of light, but we can always make it come to rest by flying alongside with the same speed. But now it comes: a particle with no mass always moves at the speed of light and since ordinary observers with mass (read: people) can never move at the speed of light, we can never make a particle without mass come to rest! Indeed, however fast we run after a massless particle, a massless particle will always move relative to us at the speed of light (which is the basis of relativity theory). The strange thing is that this is not a gradual process: each particle with mass, however small, can in principle always be made to come to rest while a particle without mass will always move relative to us with the speed of light. All fundamental equations in physics will in one way or the other have to respect this principle. (To be absolutely clear: E=mc² is, of course, correct for massive particles at rest. We just want to stress that massless particles are really different from massive particles and that it is not allowed to substitute m=0 in this equation because a massless particle has to move with the velocity of light.)

3.3 SPIN AND QUANTUM MECHANICS

Elementary particles have the characteristic that beside mass or charge they rotate about their own axis. This is called spin. Electrons and quarks spin with a speed of a 1/2 around their own axis (it's not important what this 1/2 actually means). The force carrying particles, photon, gluon, W and Z, spin with a speed of 1 around their axis. The graviton (if it exists) with a velocity of 2. The Higgs particle doesn't rotate around its axis. If found, the Higgs particle will be the only elementary particle that doesn't spin!

Particles with a half valued spin (1/2, 3/2, 5/2,...) are called fermions. Particles with an integer spin value (0, 1, 2) are called bosons. That is why the Higgs particle is sometimes also called the Higgs boson. If one wants to emphasis that the Higgs particle has spin 0, or in other words doesn't spin and therefore doesn't generate a preferred direction, it is called a Higgs scalar. I consider the term “Higgs boson” an unnecessary complicated technical expression and will not use it here.

The axis about which particles spin can be oriented in different directions but here also massless particles, like photons and gluons, have something in particular. The axis around which they rotate must always be directed in the direction of motion. Spin is the result of combining quantum mechanics and relativity theory and hard to describe in simple terms but we will nevertheless try to explain why the photon can only have two polarization directions. Intuitively, this can be understood by picturing light not as a point particle but as infinitely thin and infinitely small balloon or as an empty little sphere, see figure 5.

Figure 5.
We imagine light as an infinitely thin and infinitely small balloon and assume that the light in the picture is coming straight towards us with a velocity c.
A: Light cannot rotate around an axis perpendicular to the direction of motion
B: Light can still rotate around an axis parallel to its direction of motion.

The surface of the light sphere can't move into the direction of motion (so rotate around an axis perpendicular to its direction of motion) because then a point on its surface would sometimes go faster than the speed of light and this is not possible. A point on the surface of the light sphere only has the freedom to move perpendicularly to the direction of motion of light because it can never move in the same direction as the light. So light can only rotate around it's direction of morion.

In other words, if a photon is coming towards us we see it spin clockwise or counterclockwise.

Now suppose we want to make a particle that spins like light (it spins with a velocity 1) but has mass. Because such a particle has mass we can make it come to rest by running alongside. The above argument that the particle should rotate around the direction of motion is not longer valid. It is at rest and can rotate in all directions. Well, not in all directions, due to quantum mechanics. Quantum mechanics tells us that all things in nature come in pieces or quanta. It says that a particle with a speed of rotation of 1 can rotate in any direction, opposite to that direction or perpendicular to that direction. Thus such a light-particle with mass is allowed to rotate in 3 directions according to quantum mechanics and we have just seen that real light (without mass) according to the theory of relativity can only rotate in 2 directions. We will call the direction in which a massless particle isn't allowed to rotate 'the third direction of spin”. It will play a major role in our story.

Not all particles with mass cause problems in the formulas if we let the mass approach zero. The electrons and quarks are spin 1/2 particles. This means (according to quantum mechanics) that they can only rotate in two directions (at a rate of a 1/2) whether they have mass or not. However, the above explanation that light can only spin about its direction of motion applies to all massless particles, also to massless spin 1/2 particles. This is not a problem because if for a massive fast-moving spin 1/2 particle we let the mass in the formulas approach zero, the axis about which they spin moves automatically in the direction of motion of the particles. Since there are always only two directions of rotation for a spin ½ particle, there is no problem if we make it come to rest by moving alongside. There exist an elementary particle, the neutrino, an (almost) massless spin 1/2 particle and it is not a problem to describe this particle. The problems with the different number of spin directions occur only for particles of spin 1 (and spin 2 particles like the graviton).

3.4 THE UNCERTAINTY RELATION

A photon (particle of light) is a force carrying particle that transmits the electromagnetic force (see Figure 2). It is a very important particles in our daily lives. It keeps the atoms together by binding the electron to the nucleus and makes the light shine in our living room. The photon is a particle with no mass, no electric charge (no direct interaction with itself) and a spin 1 particle. We have discussed that, because it is massless, the photon can only spin around its direction of motion.

A photon with mass would spin in three directions, but why are we interested in a photon with mass? This has to do with radioactivity. Not all matter turned out to be stable. Some substances fall apart and cause the dangerous radioactivity. It turned out that a certain kind of radioactivity could not be explained by the electromagnetic force (and not by gravity or the strong force). It was therefore assumed that there exists another force in nature. This force is called the weak force or the weak interaction. Gravity and the electromagnetic force are always around us, we deal with them daily. The weak force, responsible for a certain kind of radioactivity, fortunately, we normally don't notice.

The reason is that gravity and the electromagnetic force work over long distances. The earth is attracted by the gravity of the Sun over a distance of 150 million kilometers and the light of the stars (photon) comes from even further away. For one reason or another, the weak force is only effective over short distances.

The reason is that the force carrying particles that transmit the weak force, unlike the photon, do have mass. But why does the fact that if something works over short distances mean that the particle that transmits the force must be heavy. This has to do with quantum mechanics. A particle that transmits a force, uses a kind of borrowed energy. According to quantum mechanics, nature cannot exactly determine whether energy is conserved. This follows from the Heisenberg uncertainty relation which states that nothing in nature is exactly determined but only approximately. This principle allows a force carrying particle to borrow from the vacuum a small amount of energy for a long time or a large amount of energy for a short time. Because the photon is massless, it can borrow from the vacuum a very small amount of energy for a long time or in other words over a large distance (distance is time times the speed of light). A particle with mass doesn't have this option. Because it has to borrow at least the amount of energy corresponding to its mass (E=mc^2), it can only travel a short distance. If a massive particle would borrow from the vacuum that amount of energy for a long time, nature would find out and that is not allowed. Consequently a heavy particle cannot exist for long and therefore has not enough time to travel large distances.

3.5 GAUGE INVARIANCE

We have discussed that the electromagnetic force is transmitted by massless photons and the weak force by massive W particles. We also discussed that particles with mass have to be described differently than particles without mass. The electromagnetic force is well described by the massless spin 1 photon, but in the formulas an essential ingredient is needed to make them work: gauge invariance.

When the theory of relativity and quantum mechanics are combined, it turns out that it is impossible to describe a free massless spin 1 particle (like the photon) in the same manner as a free spin 1/2 particle (like a free electron or a free quark) is described. The formulas of relativistic quantum field theory simply don't allow this. When passing from one observer to another (say from an observer at rest to a moving observer) massless spin 1 particles tend to develop again the third direction of spin. The same is true for a spin 2 particle, like the graviton.

Because a free massless spin 1 particles is not allowed to have a third direction of spin, the formulas must be setup in such a way that they consist of two terms that are subtracted from each other. When the two terms are subtracted, the third direction of spin cancels. This is called gauge invariance or gauge freedom, because canceling the two terms can also be seen as an invariance or freedom. By subtracting two terms we have introduced the freedom to add the same number to the two terms because when we subtract the two terms the numbers cancel. It is like the distance between the floors of a building. The distance between the sixth and eighth floor, for example, can be four stairs. Whether we add 10 new floors on top of the building or (in one way or another) put 10 more floors beneath the building, the distance between the original floors remains four stairs. So we have the freedom to add floors beneath or on top of the building but that freedom is not unlimited because we are not allowed to put floors between the sixth and eighth floor because then we have to walk more stairs.

If we want to describe spin 1 particles not only as free particles but also as force carrying particles, like the photon in Figure 2, then we must also avoid that the third direction of spin is going to give us problems. This is also obtained by allowing a certain freedom into the formulas. In this case it is the particle that emits the force carrying particle that sets the third direction of spin to zero. One could say that it is the other particle (the electron in figure 2) that takes upon itself the rotation that the massless photon is not allowed to have.

This freedom in the formulas, for a free spin 1 particle to be insensitive to the third direction of spin and for a force carrying spin 1 particle the fact that the other particle can put the third direction of spin to zero, are both part of gauge invariance. Gauge invariance is essential in theories of massless spin 1 or spin 2 particles. It is one of the most abstract concepts in physics.

We have described the theory behind the (massless) photon. We have seen that quite abstract concepts were needed. The justification for all this comes from the fact that the electromagnetic force, the theory of the photon, called quantum electrodynamics, agrees to many decimal places with experiment. For example, the motion of the electron around the nucleus of the hydrogen atom can be described very accurately, including the effects of the rotation of the electron around its own axis (with rate 1/2). There is no doubt that quantum electrodynamics describes the electromagnetic force correctly. A deviation has never been found! The principles behind the development of the theory of the electromagnetic force have become a guideline for the development of the theories of the other forces. However, to describe the electromagnetic force the Higgs field was initially not required because it was not needed to give mass to the photon and the mass terms for the quarks and electrons could easily be put in by hand.

3.6 RENORMALIZATION

We have so far discussed a number of pillars of physics, relativity, quantum mechanics, gauge invariance. To this we will now add a final pillar.

There is another reason why physicist were so pleased with the description of the electromagnetic force. The mathematical calculations that to many decimal places correspond to experiment, are just that good thanks to a trick. Initially, the calculations turned out to result in infinity or infinitely large numbers. The physicists were of course very worried about these infinities, until they came up with a trick to get rid of them.

Despite how beautifully the electromagnetic force is described by theory, it still has a number of parameters that first have to be measured experimentally. The mass and charge of the quarks and the electron, for example, must still be determined experimentally, and also the strength of a force must be measured. Once these parameters are determined, it is possible to describe the interaction between quarks, electrons and light, with great precision. It turned out that for the electromagnetic force it was possible to put all the infinities into these parameters that have to be measured anyway. The infinities are then eliminated by substituting at that point the experimentally measured values. This trick is called renormalization and whenever this is possible the theory is called a renormalizable theory. Everything that we don't know about the particles and the forces of nature, all our ignorance, is put into the parameters and all the knowledge we don't possess yet is obtained experimentally by measuring the value of the parameters.

The origin of the infinities is probably due to the fact that we describe the force carrying particles and quarks and electrons as point particles without any underlying structure but the real reasons why renormalization is necessary is not really understood. We will therefore mainly discuss how renormalization works and much less why it is required in physics.

The beauty of it all is that it is possible to put all the infinities in the parameters that have to be measured, thanks to the freedom that was put into the electromagnetic theory in order to be insensitive to the third direction of spin of the massless photon. So thanks to gauge invariance. The gauge invariance, that maybe in the beginning we didn't really wanted, turns out to be an essential ingredient to get rid of the infinities in the calculations.

Obviously, at first the physicists weren't very happy with the need for renormalization but later they even turned the argument around. A theory can only be taken seriously if all infinities can be put into the parameters that have to be measured anyway. The incredible good agreement between these theories and experiment justifies this principle. We therefore assume that only renormalizable theories describe nature sufficiently well.

3.7 RENORMALIZABILITY

How can we be certain that a theory is renormalizable, or in other words that it is possible to put all the infinities into parameters like mass or charge? A lot of math is needed to really prove that the electromagnetic force is renormalizable but after much effort the physicists eventually succeed in proving this. It is much easier to show that a particular theory is not renormalizable. One looks at a certain term in the formulas, and lets the energy in that term go to infinity. If the term goes to infinity too quickly, the theory is not renormalizable. This means that it can be known in advance that it is impossible to put the infinities into the parameters.. A non renormalizable theory still yields reasonable answers at low energies or not too precise calculations, but at higher energies or accurate calculations it will eventually go completely wrong. In short: only renormalizable terms are allowed otherwise at some point the calculations will certainly go wrong.

The reverse is not true. If a theory consists only of renormalizable terms this does not mean that it will give sensible answers. It only tells us that if we let the energy go to infinity, the terms aren't approaching infinity too fast. The renormalizable terms also go to infinity (not just too fast). It is also necessary that all infinities can be put into the parameters that are measured experimentally. All we can say is that if a term is not renormalizable it will surely go wrong, with renormalizable terms there is still hope.

Maybe it sounds strange that when a term doesn't approaches infinity too fast if we let the energy go to infinity, the infinities can be put into the parameters of the theory while if a term goes to infinity too quickly this is no longer possible. It works approximately like this: for more accurate calculations, the terms are damped, that is, they go increasingly slower to infinity when the energy goes to infinity. For non renormalizable terms this damping doesn't go fast enough and they continue to go to infinity. The renormalizable terms will dampen fast enough and after a few steps in the calculation, they no longer go to infinity. So for renormalizable terms only a few steps in the calculation will be plagued by infinities. If in those steps, where we suffer from infinities, it is possible to put the infinities into the parameters, we get a sensible theory with sensible answers.

The current view on renormalizability is that it will work up to any energy scale. There is however a subtlety. It is not true that we do not allow non renormalizable terms in the theory, it's more so that it appears that we can neglect their contributions. The non renormalizable terms are present but are not as strong as renormalizable terms. How do we know that renormalizable terms exist if they can't be observed? Without gravity that would indeed be an empty statement. Since gravity is carried by a spin 2 particle it can be demonstrated that no renormalizable terms can exist. For a massless spin 2 particle, so many spin directions have to be eliminated (as the graviton is massless, it can also only have two spin directions) that so much freedom is needed in the formulas that no renormalizable terms can fulfill this requirement.

Since gravity contains only non renormalizable terms it is not well understood at very high energy, such as during the creation of the universe or in very accurate calculations. If gravity only consists of non renormalizable terms and non renormalizable terms are almost negligible compared to renormalizable terms, then the gravitational force should be very weak compared to the other forces that do contain renormalizable terms (and negligible non renormalizable terms). Gravity is indeed very weak and compared to the other three forces it is a completely negligible force between elementary particles. This is a strong indication that non renormalizable terms exist but are negligible compared to renormalizable terms. So renormalization will work as long as we can neglect the non renormalizable terms. However, neglecting the non renormalizable terms (except for gravity) seems to be, even up to the CERN energy scale, a very good approximation because for all four forces there has never been observed a deviation from experiment. There are indications that the current theory may only go wrong at energies billions of times higher than the energies used at the CERN particle accelerator but recently there are also indications that it may even continue to work at those energy scales.

3.8 SUPERCONDUCTIVITY

The gauge freedom needed in the electromagnetic interaction because the photon is massless, should not be needed for the weak interaction. The weak interaction is mediated by massive particles because it is effective over short distances only and may very well be described without gauge invariance. All three polarization directions are allowed and the freedom provided by gauge invariance to get rid of the unwanted polarization direction is not needed. UNFORTUNATELY: the term needed to describe the mass of the W particle is not renormalizable. If we would add such a term to the equations, we would introduce a term that approaches infinity too fast when we let the energy go to infinity and we would thus introduce infinities that can no longer be eliminated in the calculations. Such a term in the theory would produce infinity as the outcome of the calculations and thus give insensible results.

So we have a nice theory for massless force carrying particles in which the gauge freedom allows us to eliminate the infinities and the unwanted spin direction but we cannot make a theory of weak interactions with a massive force carrying particle, while the mass is needed to explain why the weak interaction only works at short range. So the question was: how can we introduce a mass term without introducing infinities that cannot be eliminated. At first sight this seems like an almost impossible task: introduce mass without introducing a mass term!

The answer was found by looking at a different area of physics. It is possible to run an electric current through a ring of a certain kind of material. If the material is cooled to about a few hundred degrees Celsius below zero, then at a given moment the electric current no longer feels any resistance. It can circulate inside the ring forever. This is called superconductivity. It turns out to be impossible for radiation to penetrate such a superconducting ring . When the radiation tries to enter, the current that runs inside the ring will go faster and a field opposing the penetrating radiation is generated. This corresponds to the disappearance of a radio signal inside a tunnel. The electrons in the metal construction of the tunnel regroup in such a way that an opposing field is generated that cancels the radio signal. To be more precise, an electromagnetic field can penetrate a superconducting ring but only over a very short distance. But wait a minute, that was what we were looking for: a photon (light-particle or radiation) that is only effective over a short range.

It was indeed possible to describe the failure of the radiation and thus of the photon to penetrate the superconductor as if the photon had become massive due to its interaction with the field inside the superconductor. What kind of field is there inside a superconductor? The bottom line is that the material has become so cold that the electrons in the material form pairs (called Cooper pairs). The electrons in the pair can spin in the same direction or in opposite directions. If the electrons spin in opposite directions the rotations cancel each other and form a spin 0 particle. As a non-rotating (or spin 0) particle has no preferred direction, the physicists were especially interested in this combination. Because it's not rotating, a spin zero particle is fairly easy to describe. The field associated with the particle is in accordance a spin 0 field.

3.9 THE HIGGS FIELD

Physicists who heard about this mechanism wanted to apply it to the theory of the weak interactions (containing at this point massless force carrying spin 1 particles). It was therefore necessary to add a spin 0 field to the formulas, but in such a way that the original freedom, needed for the massless spin 1 particles to keep the third direction of spin at zero and also needed to eliminate the infinities, would be preserved.

The question now arises what kind of terms for the Higgs field are allowed in the formulas. Indeed, only terms that preserve gauge invariance and are renormalizable are allowed. For the energy of the Higgs field only two terms are permitted. The energy that these two terms form together has the shape of a sombrero hat, or in other words a Mexican hat (see Figure 6).

Figure 6. The shape of the energy of the Higgs field: a sombrero or Mexican hat. The lower circle of the sombrero we will be referred to as the channel. (From Wikipedia)

The energy that the Higgs field can have is obviously not a sombrero, it just has the shape of a sombrero. Since everything in nature tends to go to the least possible energy, the energy of the Higgs field will evolve to the channel of the sombrero. For a sombrero, the channel is the lowest point of the hat. The part for the head of a sombrero is higher than the channel and so is the brim of the sombrero. With a little rain a small puddle of water would remain in the channel of the sombrero and this is also where the Higgs field remained. Droplets on the sombrero would slide towards the channel and so did the Higgs field slide towards its lowest value as the universe cooled.

The energy on the channel of the sombrero is the lowest energy that the Higgs field can have but it is not zero. It's an energy that fills the entire vacuum and thus the entire universe!

Each point in the channel of the hat is at the same height and is therefore just as good as any other point in the channel. This represents the original freedom needed to be insensitive to the third direction of spin and to get rid of the infinities.

3.10 THE BIRTH OF THE HIGGS PARTICLE

When Peter Higgs wanted to publish his idea of the Higgs field, it was rejected by the editors of the scientific journal. There were at that time many theories in circulation and the magazine only wanted to publish new theories that implied new particles in order that the theory was new and could be tested experimentally. Higgs added that if there is a Higgs field, there must also exist a Higgs particle. After that remark the article was published. In fact one of the two terms of the energy of the Higgs field represents the mass of a particle. This is why Higgs could write that if the Higgs field existed, there should also exist a Higgs particle.

3.11 SPONTANEOUS SYMMETRY BREAKING

We have discussed that the energy of the Higgs field looks like a sombrero and we have added this field to the theory of the weak interactions containing at this point only massless force carrying particles. To do calculations, one point in the channel of the Mexican hat has to be chosen. By choosing one point in the channel of the hat, the symmetry that allows us to choose any point in the channel of the hat is broken. This is called spontaneous symmetry breaking. It is like a pen that is put vertically on its tip. Even if we succeed in doing this, then by a random fluctuation (that are common in quantum mechanics) the pen will fall in some direction. All directions are equally probable, but by coincidence the pen will fall in one particular direction. By falling down the original symmetry, that the pen could fall to all directions, obviously no longer exist: the symmetry is spontaneously broken. As in the case of the Higgs field, the pen reaches its lowest energy but that energy is not zero. The pen could still fall from the table. In the same way that a pen falls to a particular direction we have to choose a particular point from the channel to which the Higgs field has evolved, yet all points are equivalent.

The symmetry needed in order to be insensitive to the third direction of spin of the massless W particles breaks when we choose one particular point from all possible points in the channel of the Mexican hat. Each point in the channel of the Mexican hat is the lowest value the Higgs field can have but it is not zero and it is an energy that fills the vacuum. Thus it seems like the vacuum energy is not zero, as would be a normal value for a real vacuum. By subtracting the energy of the Higgs field from the non-zero vacuum value we set the energy of the vacuum to zero, and abracadabra: the additional terms obtained by breaking the symmetry and setting the vacuum value to zero, have exactly the same form as the third direction of spin of the massive W particles. The massless W particles thereby obtain exactly the third direction of spin that they would have if they were to have mass. They now look the same and behave the same as W particles with mass. They have now become massive W particles but with terms added by the Higgs field that eliminate the negative effects of a mass-term (the infinities that it introduces)!

We said that any point in the channel of the sombrero may be chosen and the vacuum value may be put to zero. Now, the additional terms that occur by zeroing the vacuum value, are not as easily recognizable as mass terms for all points in the channel of the sombrero. For an arbitrary point in the channel of the sombrero, the extra terms look like a kind of massless spin 0 particles, called Goldstone bosons. Only for one special point in the channel arethe terms easily recognized as mass terms. These massless Goldstone bosons have caused the physicist a lot of problems. They could not get rid of them and it seemed that spontaneous symmetry breaking would only give rise to massless particles while they were trying to produce mass terms. Thus, by using the original symmetry that allows us to choose any point we like (they're all equivalent) and then choosing precisely that particular point for which the mass terms are easily recognizable, it is as if the massless Goldstone bosons are eaten by the W and Z particles which thereby become massive. Another way to interpret this is that the massless Goldstone bosons produce the third direction of spin that the massless W and Z particles need to become massive.

It possible to discuss in some more detail that we have to introduce 4 degrees of freedom to obtain two massive W particles, one massive Z particle and one massive Higgs particle after the symmetry breaking. The channel in Figure 6 is a circle. Each point on the circle consists of an x and y coordinate. Furthermore, each x and y coordinate is a complex number (as is almost always the case in quantum mechanics) and complex numbers are made up of two coordinates them selfs. The complex numbers represents the symmetry of the photon that will not be broken and therefore the photon will remain massless. So to maintain the original gauge symmetry, we must introduce the Higgs field with 4 coordinates. After the partial breaking of the symmetry, three coordinates become the massless Goldstone bosons and the fourth coordinate becomes the massive spin 0 Higgs particle. By choosing a special point in the circle, we recognize the Goldstone bosons as the third direction of spin of the two W particles and Z particle which then become massive and the fourth coordinate will remain the massive spin 0 Higgs particle.

At that time there were many potential theories going around that in some way would give mass to the W particles and the above manner was just one of many theories. It was necessary to introduce a spin 0 field and furthermore it was not possible to give mass to the chatrged W particles without introducing another massive neutral W particle. Besides the massless uncharged photon there had to exist a massive neutral W particle about as heavy as the charged W particle. So three W particles, a positively charged W, a negatively charged W and a neutral W. The neutral W was called the W0 in the beginning but later it was named the Z particle. The gauge symmetry had to be broken only partially because the massless photon had to remain massless after the symmetry breaking. The symmetry of the Z particle was broken, but the symmetry of the photon had to remain intact. Furthermore, the beautiful symmetry, needed to remove the infinities, was not longer completely present after the spontaneous symmetry breaking with all possible consequences. Moreover, because the theory mixes the Z particle of the weak force with the photon of the electromagnetic force, the two forces had to be united into a single force called: the electroweak force.

It is only fair to say that one of the other forces, the strong interaction, which binds the quarks, also works over a short distance but is transferred by massless gluons. But we have just said that if force carrying particles are only effective over short distances they must be massive. It is now clear that the strong interaction can be described in the same manner as the electromagnetic force (and thus also in the same way as the weak force) but the force is much stronger. The force is so strong that the gluons and the quarks cannot escape. Unlike photons the gluons feel their own field directly and is so strong that they cannot escape. For this reason the strong interaction is only effective over short distances. It was not at all clear whether the weak interaction should not also be described in a similar fashion as the strong interactions.

3.12 THE BIRTH OF THE STANDARD MODEL

The model of the weak interactions including the Higgs mechanism was developed by physicist Steven Weinberg, but because the renormalizability of the theory was not clear in the beginning it didn't get the attention it deserved (it was not cited in the first year and only twice in the following two years). The Dutch physicist Martinus Veltman kept stubbornly believing in this version of the theory of the electroweak interactions, even after almost everyone had lost interest. At that time, Gerard 't Hooft was a PhD student of Veltman and obviously had to be assigned a task. Veltman asked 't Hooft to take a look at the infinities that occurred in that complicated model with its W and Z particles, broken by the Higgs field. To everyone's surprise, Gerard 't Hooft was able to prove that before and after spontaneous symmetry breaking all infinities could be eliminated or in other words all infinities could be put into the parameters that are obtained by experiment. He accomplished this by consistently applying a complicated mathematical technique developed earlier by Feynman and by clever use of the original gauge freedom of the model. The bottom line is that although we break the symmetry by selecting a point from the channel of the sombrero, the symmetry is still hidden somewhere in the formulas. This hidden legacy of the old symmetry is then just sufficient to eliminate all infinities.

When the physicist heard about this, they were very eager to look at Weinberg's model again (it is now the most cited paper in particle physics). The infinities could be eliminated due to the original gauge freedom in the theory. By breaking the symmetry the W (and Z) particles obtained a mass term but in such a way that the original symmetry still ensured that the infinities could be put into the parameters. Now that the infinities could be eliminated, it was possible to make precise calculations and rigorously test the theory. The model also gave various relationships between all kinds of parameters that could also be tested. A key test would be whether the Z particle, which was needed in that theory, really existed and could be found and would have approximately the mass that should be expected given the range at which the weak force is effective.

Because the electromagnetic, weak and strong interactions are all described in the same manner, they are collectively called the Standard Model. Now only the gravitational force is left over. We have discussed that gravity contains only non renormalizable terms. So for gravity it is not possible to eliminate the infinities. Now, fortunately, under normal conditions this is not necessary. The infinities occur only in precision calculations and because the gravitational force between elementary particles is negligible compared to the other forces, we normally don't encounter any problems with infinities. Gravity is only noticeable when large amounts of particles build up a gravitational field, like for the Sun or the earth. Gravity only gets stronger because there exists, loosely speaking, only positive gravity and no negative gravity that could cancel the positive gravity. Gravity is a weak force but can become very strong under extreme circumstances, like on the edge of a black hole or during the creation of the universe. It's only there that we may suffer from the infinities in the calculations.

It is possible to describe gravity as transmitted by a massless graviton in flat space, which leads to the same formulas as Einstein's theory of gravity which describes gravity as curved space-time. In this sense there is no real difference between gravity and the other forces, and they will (possibly) all go wrong at extremely high energy. It is the personal opinion of the writer that gravity should be considered as part of the Standard Model but normally the Standard Model refers only to the other three forces.

Of course, it is not elegant that gravity is not well understood at the quantum level and describing all four forces from one principle or by one model is the holy grail of physics. There are indications that at very high energy the weaker forces become stronger and the stronger forces become weaker, so at some energy scale they become equally strong and can be seen as a single force.

3.13 THE MEANING OF LEFT-HANDEDNESS

We have given mass to the light-particles via the Higgs mechanism. In the electromagnetic theory it was no problem to add mass-terms for the electrons and quarks to the formulas. We thus expect that it will also be possible to add mass-terms to the electroweak theory. This, however, would be very strange because we needed the Higgs field to give mass to the light-particles but then electrons and quarks get their mass through a different (unknown) mechanism. Or maybe the electrons and quarks get part of their mass via the Higgs field and another part via some unknown mechanism. This would lead to a very ambiguous meaning of the Higgs field. Luckily, now that the light-particles get their mass through the Higgs field, it is no longer possible to add mass-terms to the formulas and also the electrons and quarks have to get their mass through the Higgs field. The reason is that the weak interaction only has an effect on left-handed particles (particles that rotate to the left). The right-handed particles don't feel the weak force. It is of course quite strange that nature treats left-handed particles differently from right-handed particles and it shocked the physics community but it has by now been confirmed experimentally.

If we want to make symmetry breaking in the electroweak theory possible, we have to treat left-handed particles differently from right-handed particles. The mass-terms in the electromagnetic theory made no distinction between left-handed and right-handed particles and for that reason could no longer be added to the new electroweak theory with symmetry breaking because in this theory it is necessary to treat left-handed and right-handed particles differently. The only symmetric and renormalizable term that can be added to the formulas consists of a left-handed pair of particles (an electron and a neutrino pair or an up quark and a down quark pair), a Higgs particle and a right-handed electron or quark. The fact that left-handed electrons or quarks have to exist in pairs while right handed electrons or quarks do not (the weak force has no effect and cannot mix them), shows that left-handed particles are treated differently from right-handed particles. After the symmetry breaking and setting the vacuum energy to zero, this so called Yukawa term gives mass to the electrons and quarks. So thanks to the left-handedness of the weak force, electrons and quarks also have to get their mass through the Higgs field! It is likely that behind all of this, lays another layer of complexity that can explain how it all comes about but for more than 40 years nobody has been able to demonstrate if this is true and how it should work.

3.14 THE ANOMALY THREAT

At a certain point, the physicists discovered that the infinities, that in the electroweak theory could all be eliminated so beautifully, still could arise somewhere in the formulas. This is called an anomaly and indicates that something could be seriously wrong. The anomaly is absent if the charges of all species of spin 1/2 particles add up to zero. In that case, the expressions that go to infinity are multiplied by zero and rendered harmless.

Figure 7. Example of a diagram where an anomaly might occur. In the triangle, all species of spin 1/2 particles can go around. The wavy lines depict spin 1 gauge bosons.

The reason that all the spin 1/2 particles are of interest is shown by a Feynman diagram in Figure 7. In this diagram three spin 1 particles are combined with three spin 1/2 particles (this shows that also spin 1/2 particles, like electrons and quarks, can act as force carrying particles, a fact that is not often mentioned). Due to the specific form of the Feynman diagram all species of spin 1/2 particles can rotate the triangular part of the diagram. This diagram will produce infinities unless the electric charge of all the species of spin 1/2 particles cancel each other out.

At first glance the sum of the electric charges of all species of spin 1/2 particles is not zero. The electrons have a charge of -1, a down quark a charge -1/3 and an up quark a charge 2/3 (and the neutrino a charge 0) and -1-1/3+2/3 equals -2/3 and so is not zero at all. However, in the strong interaction, the force that holds the quarks together, the quarks have a charge similar to electric charge. This is called color charge. Each quark can posses three distinct color charges denoted by red, green and blue (that's why they are depicted with these colors in figure 1). We must therefore, according to the strong force, base the electroweak force on six types of quarks. Just as we have counted an up quark of charge 2/3 differently from a down quark of charge -1/3, we must also treat quarks with different color charges as different particles. If we now sum all species of spin 1/2 particles we get: -1+2/3+2/3+2/3-1/3-1/3-1/3=0. This is a very strong indication that the Standard Model of electromagnetic, weak and strong interactions is rock solid!

3.15 EXPERIMENTAL TESTS OF THE STANDARD MODEL

The W and Z particles have by now been discovered experimentally and their masses can be determined very accurately. Especially the discovery of the Z particle can be seen as a very strong confirmation of the correctness of the Standard Model and thus of the existence of the Higgs field and the Higgs mechanism. It has also been established that the way in which the W, Z and photons interact, is correctly described by the Standard Model. The decay of particles into other particles, such as the decay of a down quark into an up quark plus an electron and an anti-neutrino (see Figure 3), can be precisely calculated.

The description of the electroweak interaction including the Higgs field is probably correct. It is important to study the Higgs field in detail. The study of the Higgs particle gives immediate insight into the Higgs field, for example by examining how fast and to what kind of particles the Higgs particle decays and whether this corresponds to the predictions of the Standard Model. The discovery of the Higgs particle would be a very strong confirmation of the Standard Model!

Figure 8. First indications of the existence of a Higgs particle with a mass of approximately 126 GeV. Results are shown for three decay modes of the Higgs particle. Note: the blue and green point almost lay on top of each other. The bands indicate the various uncertainties in the measurements.

Many years after the introduction of the Higgs particle in physics there are now the first signs of the existence of a Higgs particle with a mass around 126 GeV (about 126 times the mass of a hydrogen atom). See figure 8. The figure more or less speaks for itself. This is the result of the ATLAS detector at CERN, the CMS detector at CERN shows similar results.

Also the strong interaction, the force that holds the quarks together, has been carefully tested. There exist other combinations of quarks than the proton and the neutron (of figure 1). The masses (including the binding energy) of such new composite particles can be calculated using the theory of the strong interactions and agree to the values found experimentally. These calculations require an incredible computing power because the approximate description of a force as being transmitted by the exchange of force carrying particles (like in figure 2), no longer holds for the strong interaction and it is necessary to work with the full mathematical formulas.

3.16 THE FUTURE OF THE STANDARD MODEL

To see what we can expect for the future of high energy physics, it is instructive to look at the past. The decay of the neutron into a proton via the weak interaction was described for the first time by Fermi. The description of neutron decay by Fermi was a breakthrough and can be seen as one of the highlights of elementary particle physics. However, the term by which Fermi described neutron decay was not renormalizable. As we know this means that such a term will go wrong at high energy. At lower energies the Fermi theory could explain many of the characteristics of the weak interaction but it was only after the introduction of the Higgs mechanism that the weak interaction was described by renormalizable terms in the equations and gave correct answers up to the energies reachable today.

It is interesting that the non renormalizable terms could be substituted by renormalizable terms after introduction of more substructure for the theory. It was vital that the neutron and proton were made out of quarks and also the W and Z particles had to be added to the theory. In the modern theory of the electroweak interaction including the Higgs mechanism, we subtract the energy of the Higgs field from the vacuum value. It is then clear that there exists a strong connection between the good old Fermi theory of the weak interactions and the modern version with the Higgs mechanism, as the energy of the Higgs field is directly related to the most important constant of the Fermi theory which was already known at the time and didn't have to be remeasured.

We have said that in the formulation of the forces the non renormalizable terms could be neglected compared to renormalizable terms (except gravity that contains only non renormalizable terms). From the above, we can expect that at very high energy the non renormalizable terms can no longer be neglected relative to the renormalizable terms. For not to high very high energy the non renormalizable terms can still give reasonable answers but at even higher energies it will eventually go wrong. The energy scale at which this happens will be so high that all forces (also gravity) will probably be approximately equal in strength. It is likely that at that energy scale more substructure of the particles will become visible or a completely different kind of theory will emerge. For now it looks like this will happen at energies unreachable to humanity.

We haven't treated it here but a more extensive substructure of the particles is already known described by the Standard Model. We know that there are 3 species of electrons and 3 species of up and down quarks. A model with more substructure should explain in a natural way why there are exactly 3 species of quarks and electrons and should also predict their masses. No theory is known that is even approximately able to show this.

It looks like the Standard Model will keep its current form for many years to come. The last 40 years nothing essential to the Standard Model has changed and the current experimental results at CERN have only validated the Standard Model even further. We will have to get used to be pleased with a theory that, given the ingredients (particles and forces), gives such excellent results at the currently reachable energies!

4. REFERENCES