## Open Questions: Higgs Physics

[Home] [Up] [Glossary] [Topic Index] [Site Map]

Prerequisites: The standard model

### Introduction

In physics, symmetry is a good thing. Even a partial, or "broken", symmetry is better than none at all.

As we saw in our discussion of the standard model, solid theories of three out of the four known fundamental forces -- electromagnetism, the weak force, and the strong force -- have been constructed as "Yang-Mills gauge theories". Further, the first two of these forces have been unified into a single theory -- the theory of the "electroweak" force, and it is a theory of the same type. Symmetry plays an essential role in each of these theories.

The theory of the electroweak force, however, has some peculiarities.

For one thing, the gauge bosons of the theory -- the W and Z particles which mediate the weak force -- have large masses, unlike the gauge bosons of electromagnetism (photons) or of the strong force (gluons). This is not only strange. It, in fact, threatens to make the theory non-renormalizable. That is, without some additional provisions, calculations using the theory can produce infinite values, so the theory is not mathematically consistent.

The electroweak theory does provide a unified theory of electromagnetism and the weak force. That is, a single set of equations describes both forces. (Just as the much older Maxwell's equations describe both electricity and magnetism, which were unified long ago.) Nevertheless, the observed properties of the two forces are far from being symmetrical.

In particular, gauge bosons of the two theories are very different. As noted, photons are massless while W and Z particles are very massive. And as a direct consequence of this, the electromagnetic force has "infinite" extent (proportional to 1/r2), while the weak force has a very short range. Another asymmetry is that electrons are much more massive than their electroweak symmetry partners the neutrinos. The pairs of quarks in each generation (up and down, charm and strange, top and bottom quarks), which are also electroweak symmetry partners, likewise have different masses. Finally, the electric charges of particles in each of these pairs are also different.

All these varied asymmetries are examples of what is called "spontaneous symmetry breaking". That is, equations which have a high degree of symmetry can have solutions which possess lower symmetry. This isn't in itself an inconsistency, but it does compel us to look for an explanation of how it comes about.

Physicists have made various attempts to come up with a mechanism that can address these problems and explain the asymmetries. The best known of these is the Higgs mechanism, named after Peter Higgs who first proposed something like it in 1964. It's not the only possible answer, but it has a number of agreeable properties. Fortunately, from the standpoint of the electroweak theory, what counts is having some way to explain the mass of the gauge bosons and the breaking of the symmetry. The exact details of the mechanism are of lesser importance.

We say "fortunately", because Higgs theory makes certain predictions which are still not verified experimentally -- the primary example of which is the existence of (at least) one massive spin 0 boson (i. e. a "scalar" boson) that has not yet been observed, despite intensive experimental searches -- the Higgs particle.

### The mass problem

It's no accident that the gauge bosons of electromagnetism and the strong force have zero rest mass. There are serious difficulties with renormalizability and mathematical consistency when a gauge field theory has carrier particles with non-zero mass. The problem arises because a particle with mass can't have arbitrarily small energy (as, say a massless photon can). The Heisenberg relations require the product of the uncertainty in energy of a particle and the uncertainty of time when the particle is observed to be larger than Planck's constant. So there is an upper limit on how long a virtual force carrier particle can be "real" in the process of an interaction. And consequently (since the particle can't travel faster than light), there is a limit to the range of the force.

Unfortunately, with the weak force we know experimentally that the carrier particles have mass and that there is limit to the range of the force. We have to live with that.

Intuitively, the theoretical problem here is that if the gauge force has finite range, it is hard to see how it can act to account for the ability to apply a local gauge symmetry throughout space, as a Yang-Mills type theory requires. The mathematics of Yang-Mills theories and local gauge symmetries works only when the gauge boson has zero mass. If the boson were to have mass, then the equations of the theory would not be invariant under local gauge transformations.

It is possible to be a little more specific about the nature of the theoretical problem here. A key requirement of any reasonable physical theory is that it obey the rules of special relativity. (It was this need for consistency with special relativity which motivated Dirac to do his original work with quantum electrodynamics -- the relativistic theory of electrons -- around 1930.) In practical terms, what this requires is that all equations of a theory behave properly under what is called a Lorentz transformation. This is to ensure that the theory does not depend on any special choice of a uniformly moving coordinate system.

Particle spin is a property which categorizes how the motion of a particle behaves under a Lorentz transformation. Particles with a spin of 1 transform like a three dimensional vector (which is why spin 1 bosons are sometimes called "vector bosons"). In other words, there are three independent degrees of freedom that define the orientation of the spin. (This is sometimes known as the "polarization".) However, it can be shown that if a spin 1 particle is massless, then the particle has only two actual degrees of freedom as far as its polarization is concerned. Intuitively, the spin has components only in the plane transverse (i. e., at right angles) to its direction of motion. This is why light -- which is made up of massless photons -- has the property of being polarized in a plane transverse to its motion. In short, then, a massive spin 1 particle has one more degree of freedom than does a massless spin 1 particle.

Physicists do field theory calculations related to particle interactions using a technique called perturbation theory. This entails computing an infinite sum. Each term in the sum represents some particular distinct way in which the interaction could take place. (These terms can be represented pictorially by "Feynman diagrams".)

It turns out that when the gauge bosons of a theory are massless, the theory can be renormalized. This means that everything can be arranged so that the infinite sums converge, i. e. the computations do not produce infinite results. However, when the gauge bosons have mass, and therefore an extra degree of freedom, the sums may diverge.

This is the "mass problem".

### The weak force

Important concepts about the weak force have been introduced in our discussion of the standard model. You should review the relevant parts of that discussion if some of the ideas here are not familiar.

One important physical fact about the weak force is that a quantity called "lepton number" is conserved in weak interactions. What this means is simply that the total number of leptons involved in a weak interaction does not change as a result of the interaction, as long as leptons are each counted with a positive number, and their antiparticles are counted with a negative number. For instance, in one of the simplest interactions, a neutron decays into a proton plus an electron, plus an anti-neutrino. There are no leptons to start with. At the end, there is one lepton (the electron) and an anti-lepton (the anti-neutrino), but their lepton numbers cancel out, leaving a net of 0.

This conservation law corresponds to a symmetry between electrons and neutrinos. The particles differ in a quantity called "weak isospin", because it is quite analogous to the quantity called "isospin" that Heisenberg introduced in 1932 to differentiate the two nucleons (protons and neutrons). The SU(2) symmetry group has a representation that involves permutations of a doublet consisting of an electron and a neutrino. This symmetry arises because the particles can be regarded as equivalent under a rotation in "weak isospin space".

If a Yang-Mills gauge theory could be developed which used this SU(2) symmerty as a local gauge symmetry we would have a straightforward explanation for the observed conservation of lepton number, since gauge symmetries lead naturally to conservation laws.

Further, if we had such a theory, it should be renormalizable. Unfortunately, we have a physical situation where the boson which mediates the weak force (the W particle) is known to be massive rather than massless.

What we need is some mechanism which can give the W boson mass while still deriving its existence from a locally gauge invariant theory.

There is a broader issue lurking here. Although the electromagnetic and weak forces have been unified into a single theory, the symmetry of this theory is not exact. As noted above, the symmetry is broken in various ways. In practice what this means is that we have a single set of equations which describe both forces. The symmetry means that we can interchange particles as they appear in the equations without invalidating the equations. Since the particles in fact have different masses (e. g. electrons and neutrinos or up and down quarks), the masses can't appear directly in the equations. In other words, the particle masses come from outside the theory itself.

Determining how this breaking of electroweak symmmetry comes about is one of the top priority problems in particle physics. The mechanism which causes it, whatever that may be, must come from outside the standard model. The best candidate that provides the required explanation is called the Higgs mechanism.

As we will see a little later, there are a number of other related issues in particle physics and its implications for cosmology which are dependent on the details of this mechanism.

### The Higgs mechanism

Let's review where we stand so far.
• We have a nice, well-behaved (i. e., mathematically consistent, renormalizable) Yang-Mills gauge theory of the electromagnetic force, based on U(1) gauge symmetry.
• We would like to have an equally nice Yang-Mills gauge theory of the weak force, and it should be based on a SU(2) symmetry.
• Experimentally, it is known that the particles which mediate the weak force are massive, instead of massless as required in a Yang-Mills theory.
• The electromagnetic and weak forces are intertwined, because the weak SU(2) symmetry exchanges particles that have different amounts of electric charge.
• Yet any potential symmetry between electromagnetic and weak forces can't be exact, since the forces have different strengths.

A series of profound insights by Sheldon Glashow, Steven Weinberg, and Abdus Salam, mostly as independent contributions, led to the unified theory of the electroweak force. This was accomplished by taking the above givens, making a few inspired assumptions, and synthesizing everything in a new -- and quite effective -- way.

The insights were as follows:

1. Most of the theoretical difficulties result from the existence of nonzero rest masses of the various particles. The masses break the symmetry between electrons and neutrinos (and other particle pairs), they are incompatible with a straightforward Yang-Mills gauge theory, and they are the root of the problems with renormalizability.
2. At very high energies, the energy contributed by a particle's rest mass becomes insignificant compared to the total energy. So at sufficiently high energy, assuming a particle rest mass of zero is a very good approximation.
3. A consistent, unified Yang-Mills theory of electromagnetism and the weak force can be formulated for the very high energy situation where particle rest masses are effectively zero.
4. At "low" energies (including almost all levels of energy which are actually accessible to experiment), the symmetries of the high energy theory are broken, and at the same time most particles acquire a nonzero rest mass. These two "problems" appear simultaneously when symmetry is lost at low energy, much as symmetry is lost when matter changes state from a gas to a liquid to a solid at low temperature.

The "Higgs mechanism" is basically nothing more than a means of making all of this mathematically precise.

The key ingredient not yet specified is to assume there is a new quantum field -- the Higgs field -- and a corresponding quantum of the field -- the Higgs particle. (Actually, there could be more than one field/particle combination, but for the purposes of exposition, one will suffice.) The Higgs particle must have spin 0, so that its interaction with other particles does not depend on direction. (If the Higgs particle had a non-zero spin, its field would be a vector field which has a particular direction at each point. Since the Higgs particle generates the mass of all other particles that couple to it, their mass would depend on their orientation with respect to the Higgs field.) Hence the Higgs particle is a boson, a "scalar" boson, since having spin 0 means that it behaves like a scalar under Lorentz transformations.

The Higgs field must have a rather unusual (but not impossible) property. Namely, the lowest energy state of the field does not occur when the field itself has a value of zero, but when the field has some nonzero value. Think of the graph of energy vs. field strength has having the shape of a "W". There is an energy peak when the strength is 0, while the actual minimum energy (the y-coordinate) occurs at some nonzero point on the x-axis. The value of the field at which the minimum occurs is said to be its "vacuum" value, because the physical vacuum is defined as the state of lowest energy.

This trick wasn't created out of thin air just for particle theory. It was actually suggested by similar circumstances in the theory of superconductivity. In that case, spinless particles that form a "Bose condensate" also figure prominently.

The next step is to add the Higgs field to the equations describing the electromagnetic and weak fields. At this point, all particles involved are assumed to have zero rest mass, so a proper Yang-Mills theory can be developed for the symmetry group U(1)xSU(2) that incorporates both the electromagnetic and weak symmetries. The equations are invariant under the symmetry group, so all is well.

Right at this point, you redefine the Higgs field so that it does attain its vacuum value (i. e., its minimum energy) when the (redefined) field is 0. This redefinition, at one fell swoop, has the following results: the gauge symmetry is broken, the Higgs particle acquires a nonzero mass, and most of the other particles covered by the theory do too. And all this is precisely what is required for consistency with what is actually observed.

In fact, the tricky part is to ensure that the photon, the quantum of the electromagnetic force, remains massless, since that is what is in fact observed. It turns out that this can be arranged. In fact, the photon turns out to be a mixture of a weak force boson and a massive electromagnetic boson that falls out of the theory. The exact proportion of these two bosons that have to be mixed to yield a photon is given by a mysterious parameter called the "electroweak mixing angle". It's mysterious, since the theory doesn't specify what it needs to be, but it can be measured experimentally.

So, the Higgs mechanism is a clever mathematical trick applied to a theory which starts by assuming all particles have zero rest mass. This is especially an issue for the bosons which mediate the electroweak force, since a Yang-Mills theory wants such bosons to be massless. While the photon is massless, the W and Z particles definitely aren't. Where, then does their mass come from? Recall that we observed that spin-1 bosons have 3 "degrees of freedom" if they are massive, while only 2 otherwise. It turns out that this extra degree of freedom comes from combining the massless boson with a massive spin-0 Higgs boson. That Higgs boson provides both the mass for the W and Z, as well as the extra degree of freedom.

In fact, the mechanism furnishes mass to all particles which have a nonzero rest mass. This occurs because all the fermions -- quarks as well as leptons -- feel the weak force and are permuted by the SU(2) symmetry. And since quarks acquire mass this way, so too do hadrons composed of quarks, such as protons and neutrons, which compose ordinary matter as we know it.

But this mechanism is more than just a trick. If the whole theory is valid, then the Higgs boson (or possibly more than one), must be a real, observable particle with a nonzero mass of its own. This is why the search for the Higgs boson has become the top priority in experimental particle physics.

What about renormalizability? Has this been achieved in spite of all the machinations? It seemed plausible that the answer was "yes", which was of course the intention, since the high-energy form of the theory has the proper gauge symmetry. But it took several years until a proper proof could be supplied, in 1971, by Gerard 't Hooft.

### Supersymmetry

It should be pretty clear by now that Higgs physics is very much tied in to the standard model. Indeed, it's necessary in some form to make sense of many features of the standard model -- such as electroweak symmetry breaking and particle masses. In fact, it -- or something very like it -- seems to be necessary just to make the theory consistent.

And yet it's not quite a part of the standard model either. It has a bit of an ad hoc feel to it. If, in fact, the Higgs mechanism exists in more or less the form outlined here, then the standard model certainly has no explanation for why it's there, for what makes it happen. We shall want more than that. We want to know the source of the Higgs physics itself.

There may be a number of ways to do that (which might be related among themselves). But there is one body of theory which can provide exactly the explanation of Higgs physics we're looking for, and which has been in gestation since the early 1970s (i. e., since the time the standard model assumed its present form). It's called supersymmetry.

We'll discuss it in much more detail elsewhere. All we need to say about it here can be put very simply. The essential idea is to postulate one more symmetry, but of a radical sort. This new symmetry relates bosons (particles with integral spin) to fermions (particles with half-integral spin). The symmetry associates to each fermion and boson a particle of the opposite type, known as its "superpartner". The equations of the theory are set up so that they remain true when a symmetry operation exchanges any fermion or boson with its superpartner. This is a radical step, because none of the postulated superpartners can be identfied with any known particle, so the theory immediately doubles the number of particles which must exist. Even the Higgs boson has a supersymmetric parther, the higgsino fermion.

One justification for taking such a radical step is this: When the mathematics of supersymmetry is worked through, it turns out that the whole Higgs physics -- the Higgs field, the Higgs boson(s), and the Higgs mechanism -- falls out as a necessary consequence. This is great for Higgs physics, if in fact supersymmetry is a correct theory. But the other side of the coin is that if the Higgs physics can't be verified experimentally, then supersymmetry can't be correct. This is yet another reason why Higgs physics is of such urgent concern to particle physicists.

The fact alone that the Higgs physics is a mathematical consequence of supersymmetry is quite striking. It doesn't seem likely to be just a concidence. Further, the discovery of any supersymmetric particles would validate the theory of supersymmetry, and thereby validate the Higgs physics also. On the other hand, the Higgs mechanism could still exist even if supersymmetry doesn't exist in nature. But it would have serious problems, such as the "hierarchy problem", and the lack of any obvious source or cause of the Higgs field.

If supersymmetry is correct, then, so is the Higgs mechanism. And in fact, there are more detailed predictions. Most notably, there will be not one Higgs boson, but several, each with a different mass. All of the "extra" Higgs bosons could be quite a bit heavier than the lightest one which is needed by the standard model. In particular, they might be so heavy that they would not be detected soon, if at all. There are additional details predictable by supersymmetry which further constrain the mass of the lightest Higgs boson beyond what we might guess from the standard model alone.

If supersymmetric particles are detected before the Higgs boson, than will be confirmation of supersymmetry, so the Higgs particle must show up eventually as well. But what about the converse? Suppose the Higgs boson is detected first. Will that be evidence for supersymmetry? Yes, probably.

The reason lies in what we have alluded to, namely that the Higgs physics by itself leaves something to be desired, as long as it is an ad hoc addition to the standard model. We really want to have a good explanation for the physics itself. Supersymmetry provides this. It automatically contains fields which behave as a Higgs field should, and hence entails the existence of Higgs bosons. It also says something about how standard model particles interact with these fields, which elucidates the mechanism.

A Higgs mechanism without supersymmetry would also introduce what is known as the hierarchy problem. This problem arises if (as seems likely) the strong and electroweak forces are unified just as the electromagnetic and weak forces are -- but at a much higher energy scale -- around 1016 GeV. The problem is to explain how this can be so much higher than the electroweak unification scale of 100 GeV, or, alternatively, how the latter scale, and the masses of the W and Z bosons, can be so small.

In short, if Higgs bosons are observed, we will have evidence for supersymmetry, as that is the only theory we know of that makes good sense of Higgs physics.

### Where does the Higgs field come from?

OK. It's all well and good to say that mass comes from the Higgs field. But where does that come from? What is it exactly? Why is it there?

The Higgs field, in some sense, answers the question of where mass comes from. But that merely shifts the question of explaining mass to that of explaining the Higgs field.

This is still an open question, but there are some plausible answers, of different sorts.

There is, first of all, purely a mathematical and theoretical answer. It so happens that there is a theorem, called Goldstone's theorem, after Jeffrey Goldstone, who came up with it around 1960. The theorem says that when a continuous global symmetry is spontaneously broken, there must exist a massless spin-0 boson. The particle is called (generically) a Goldstone boson. Unfortunately, such a particle has never been detected. Something's fishy.

Oddly enough, there is also this puzzle regarding a massless spin-1 boson which Yang-Mills theory requires in order to carry a gauge force. Physicists were going crazy because that could not be found either, for the weak force. They spent a lot of time trying to get around the apparent requirement for both of these non-existent particles in the theory of the weak force.

Eventually it was realized that there was a way to combine the two inadequate answers mathematically in order to concoct an answer that worked. This is basically what Weinberg and Salam did in coming up with the theory of the electroweak force. They found that by adding yet another particle -- the Higgs -- they could make the Goldstone boson disappear and make the electroweak bosons massive. The electroweak bosons are said to "eat" the Goldstone boson and thereby put on weight. In the presence of the Higgs field, the Goldstone boson, in effect, becomes the third polarization state of a gauge boson. (Recall that massless spin-1 bosons have only two polarization states or degrees of freedom.)

There is a second type of theoretical way to explain the Higgs mechanism. Recall that a basic postulate about the Higgs field was that when the energy of the field is plotted against the strength of the field, the resulting graph has a W shape. The simplest mathematical curve with such a shape is a fourth degree polynomial of the form E = x4 + Bx2, where E is energy and x is field strength. (E is plotted on the y-axis.) If B is negative, then for values of x close to 0 (but not exactly 0), E will be negative. Hence for such values, you actually get a lower energy with a non-zero field.

Now, in the standard model, all this just needs to be taken as a given. But it turns out that in theories with supersymmetry, it is actually possible to compute how the coefficient B in this equation behaves as a function of temperature. It is found that at high temperatures (say, corresponding to an energy of 1000 GeV), B is positive. The polynomial expression for E in that case has just a single minimum value (of 0) when the field strength is 0. On the other hand, at lower temperatures (such as what we have in the universe at present), B is negative. In that case, there are two minima of the polynomial for E, at nonzero value of the field strength, which is just what we need.

This mathematical behavior reflects exactly what is required to have a nonzero Higgs field appear "from nowhere" at relatively low temperatures. That is, the field doesn't exist at high temperatures, because minimizing energy requires it to not exist. Yet at lower temperatures it does exist, because in the changed circumstances, that is what yields a minimum energy.

This puzzling behavior becomes much more plausible by analogy with a number of other physical phenomena. All of these involve a change of state, a "phase transition", in matter when the temperature of the system changes. Among the many examples are:

• A magnetized piece of iron retains its magnetism up to a temperature of about 768° C but loses it above that point. Upon cooling below that point, the magnetic field reappears.
• A number of materials have the property of superconductivity at very low temperatures, but lose this property at a few tens of degrees above absolute zero.
• A crystal has a small number of distinct symmetry axes at low temperature, but loses these axes, and becomes more symmetrical, when the temperature is high enough to melt the crystal. Water, in the form of an ice crystal or snowflake is a perfect example.
What all these examples have in common is that a piece of matter exhibits a higher amount of symmetry at higher temperatures. In addition, this phase transition occurs at a definite point. Finally, the higher symmetry is lost if the matter is cooled below the critical point. This phenomenon is so familar we have various names for it (in different contexts), such as "precipitation" (e. g. rain), condensation, crystallization, etc.

This is precisely what happens with the Higgs field. It is "really" there all along. However, at high temperatures the equations governing the field are such that it does not affect matter. As the temperature decreases, at some critical point the equations change and the field condenses into a new state where it does affect matter. It suddenly causes matter to have mass, because under the new equations the overall system has lower energy when matter has mass than when it does not.

This new state at lower temperature also corresponds to the breaking of previous symmetry -- which is exactly what the Higgs mechanism is supposed to do. In fact, the mechanism was, originally, consciously invented to account for the breaking of symmetry which explains the phenomenon of superconductivity, as we mentioned earlier.

### Searching for Higgs bosons

Why has it been so difficult to find the Higgs particle experimentally? The answer is that it must be fairly massive, so that very high energy particle accelerators are required for the search.

Well, then, how massive is it? The answer is: the expected mass isn't very well constrained by the theory, which makes the search even harder. It becomes necessary to search systematically at every possible energy level, which becomes all the more tedious since the searches must be done at the limits of current accelerator capability.

Fortunately, there are upper limits on the possible mass, given reasonable assumptions. The standard model itself and existing experimental results imply that the upper limit on a Higgs particle mass is about 8 times that of the Z boson. Since that is about 91.2 GeV, the upper limit on the Higgs is around 700 GeV. Under some plausible further assumptions, the limit can be lowered to around 3 times the mass of a Z, or about 270 GeV.

Experimental results already obtained place further limits on the expected mass of a Higgs boson. The way this works is to assume some particular value for this mass and derive various experimental consequences from that. Then consider experimental results actually obtained. If you look at what the mass needs to be in order to agree with all the results simultaneously, you find that the mass of the Higgs can't be more than about 2 times the mass of a Z, or about 180 GeV.

In the best case, if the simplest form of supersymmetry is correct, the limit must be even lower, perhaps about 1.5 times the mass of the Z, or 135 GeV. Although there may be more than one Higgs boson in a supersymmetric theory, this limit can be derived for the lightest Higgs boson. (There aren't similar constraints on the heavier Higgs bosons.)

Even if a more complicated supersymmetric model is required to describe the real world (because there are additional interactions and particles and forces), it appears the mass limit on the lightest Higgs is still no more than 2 times the Z mass.

The very latest experimental results rule out any Higgs particles up to a mass of about 115 GeV, so there is actually rather little range left to search. Perhaps only to 135 GeV, or 180 GeV at most.

We should expect some answers pretty soon.

What sort of evidence is being sought in order to detect Higgs bosons? Explaining this gives a good illustration of how experimental particle physics works. To begin with, theory says the Higgs particles must decay into particle-antiparticle fermion pairs. Any supersymmetric particles, as well as the top quark (at about 155 GeV) would be too heavy.

Further, since the Higgs generates the mass of other particles by its interaction with them, theory says its probability of interaction is proportional to the mass. Thus the probability of decaying into any particular (allowable) particle-antiparticle pair is in proportion to the particle mass. The next three heaviest standard model fermions are the bottom (or b) quark, the tau lepton, and the charm quark. All other fermions are much lighter. The bottom quark is the heaviest, so most of the time a Higgs will decay into b and anti-b pairs. Therefore, experiments seeking to detect the Higgs will look for events that generate mostly b, tau, and charm pairs in the appropriate ratios.

There are only three accelerators in the world which could in principle detect a Higgs boson. Two are at CERN in Geneva. The first of these is the Large Electron Positron Collider (LEP), which has already been decommissioned to make room for the second, the Large Hadron Collider (LHC), which won't be ready to work before 2005 (or later). Just before the LEP was shut down late in 2000 there were hints that Higgs particles might have been detected. Subsequent analysis of the data indicated that this was a false alarm.

That leaves only the Tevatron at Fermilab in Illinois. A good deal of time at that facility is now devoted to searching for the Higgs boson. If it is a real particle, it ought to be detected very soon -- given that experiments are quickly reaching the upper limit of the plausible mass range. By 2006 a large number of Higgs events should have been observed (again supposing the particle exists). This will permit even low probability decay modes to be studied and should produce enough information to discriminate among possible theoretical alternatives.

### Related issues

Higgs physics may seem like an esoteric issue. Except for fairly superficial references to the search for Higgs particles and occasionally an allusion to the role that the Higgs field plays in explaining the source of particle mass, the subject is rarely discussed in publications intended for a general audience. While it's hard to disagree that the origin of mass is an important issue, the concerns about the mechanism of symmetry breaking and renormalizability might seem to be merely technical details only physicists worry about.

And yet it turns out that Higgs physics is involved in an astonishing -- almost an alarming -- number of aspects of frontier questions of physics and (especially) cosmology. In addition to the various topics touched on already, here are a goodly number of others.

#### Grand unified theories and the hierarchy problem

Following the succesful unification of the electomagnetic and weak forces in the electoweak theory around 1970, there was much enthusiasm to seek a similar unification of the electroweak and strong forces in a similar sort of Yang-Mills theory, called a "grand unified theory" (GUT), We discuss this in more detail elsewhere, but a central part of any such effort is the introduction of additional Higgs fields to account for the spontaneous breaking of the symmetry of this (hypothetical) unified theory.

Suffice it to say that, for a variety of reasons, the search for a GUT has not yet proven successful. One of the problems is related to the vast difference in the energy levels that would be involved. If there were such a unification of the electroweak and strong forces, it would be manifest only at extremely high energies -- at least 1015 GeV. In contrast, the breaking of the electoweak symmetry occurs around 100 GeV.

This is a difference of a factor of at least 1013. There would have to exist many new bosons analogous to the photon, W, Z, and gluons. These bosons are collectively called X bosons, and they would have masses at least 1015 GeV. The Higgs particles to account for such massive bosons would need to be of a similar mass.

It is theoretically difficult to understand how there could be such a huge mass difference between the lightest Higgs particle(s) which occur in the electroweak theory and these other hypothetical particles. This is an aspect of what is known as the "hierarchy problem". It is especially acute for Higgs particles, because they are scalar bosons, which reflect relationships between different energy scales. In particular, the masses of such bosons are related by equations whose parameters would require extreme "fine tuning" to account for particles of such vastly different masses. This problem can be handled if the theory of supersymmetry is correct.

#### Inflationary cosmology

As we noted above, systems of matter and energy tend to undergo what are called phase transitions as the temperature of the system varies. At a very early time in the existence of the universe (when it was about 10-36 seconds old, to be more precise), it is suspected that an extremely important phase transition took place. The temperature at that time corresponded to an energy of about 1015 GeV.

According to GUT models, somewhere around there is the critical point where the electromagnetic, weak, and nuclear forces have the same strength. Above that energy (and earlier in time), there was just one unified force. Below that energy, the electroweak force and the strong force become distinct. It is hypothesized that several Higgs fields exist which account for this symmetry breaking. (They are different from the Higgs field that breaks the electroweak symmetry at a much lower energy.)

As the universe cooled through the critical temperature (about 1028° K) at first nothing happened. But the universe was not energetically stable. It was in a state resembling a supersaturated solution or water cooled below the freezing point. This state has been called the "false vacuum". Then a phase transition took place and -- in technical terms -- all hell broke loose. So much energy was released by the phase transition (just as occurs when water freezes, but a lot more dramatically) that the universe quickly inflated in size by a factor of 1050. This is the event known as "cosmic inflation".

Of course, it's still just a hypothesis. Yet it accounts for a number of features which can be observed in the universe today, which we discuss elsewhere. Indeed, the evidence for the correctness of this inflationary cosmology is good, and getting better all the time. The evidence for inflation, in fact, is much better than that for the Higgs mechanism. It seems pretty clear that inflation really did occur. It's less clear what the exact mechanism was. But the best guess is that various Higgs fields which account for the breaking of GUT symmetry were involved. If so, this is indirect evidence for the Higgs mechanism.

#### Magnetic monopoles

There is another complication related to the use of a Higgs field in grand unified theories. In some of those theories, such as the one based on SU(5) symmetry, if the Higgs field does exist, magnetic monopoles should have been created during the first 10-35 second after the big bang -- during the phase transition responsible for cosmic inflation.

Magnetic monopoles would basically be constructed out of Higgs fields. Suppose there are three such fields. At each point in space, each field is described by a single number, since it's a scalar field. But with three fields, you need three numbers, so we have, essentially, a three-component vector at each point. During the chaos of the phase transition these vectors will tend to line up with each other at nearby points. But at a few points, conditions may be so chaotic that no consistent direction can be established. A magnetic monopole would develop at that point, with the magnetic field arising from the interaction of the various Higgs fields.

A magnetic monopole is a type of 0-dimensional singularity. 1-dimensional and 2-dimensional singularities could also develop under these conditions. Such singularities are called "cosmic strings" and "domain walls", respectively. Objects of this sort are also called, collectively, "topological defects". Just as when a liquid cools very rapidly to a crystalline solid, different regions may crystallize in different orientations, resulting in a discontinuous boundary between the regions. This boundary would become a domain wall. The intersection of two walls would be a cosmic string. Such objects, if they exist, would be exceedingly massive, and could have acted to "seed" the clumping of matter when inflation ended.

Despite numerous experimental searches, magnetic monopoles have never been conclusively observed. Cosmic strings and domain walls haven't either. However, this is not necessarily a fatal problem, since inflation itself handily disposes of it. If inflation occurred, all the monopoles that were created in the first instant would have been dispersed so thoroughly in the subsequent inflation that they would be very sparsely distributed in the present universe, and hence observation of them would be most unlikely.

#### Gravity

If "empty" space is actually filled with Higgs fields, and hence with rather massive Higgs particles, how is it that they apparently have no gravitational effect at all? Yes, there is some sort of "dark matter" out there, apparently quite a bit of it. But physicists have ruled out any contribution in the form of Higgs particles to this dark matter.

What's really going on here is concealed from us because we lack a viable quantum theory of gravity. Indeed, it certainly makes sense that if Higgs particles really do explain why particles of matter have mass, they there should be a very close connection with gravity -- which is a theory all about the reciprocal effects of mass and space on each other.

#### The cosmological constant, vacuum energy density

Although we do not yet possess a consistent quantum theory of gravity, some essential properties of such a theory are known. If there is a quantum theory of gravity at all, it must be mediated by a spin 2 boson, the graviton. The graviton must couple to anything which has mass or (by the equivalence of mass and energy) anything which carries energy, including the Higgs field.

Computations of this hypothetical coupling indicate that the cosmological constant -- which occurs in Einstein's fundamental equation of general relativity -- should have a huge value far in excess of what is observed. In fact, the constant should be so large that the entire universe would curl up to have a diameter less than a meter.

It's hard to see how this could be. Theoretical explanations are forced to assume that if there were no Higgs field in the vacuum, then spacetime would have a huge negative curvature precisely sized to cancel out almost exactly the positive curvature caused by the Higgs field.

This does not feel like an aesthetically satisfying solution to the problem of the cosmological constant. We must, presumably, wait for a satisfactory quantum theory of gravity to really understand what goes on here.

#### Axions

A Higgs mechanism has been used to address a symmetry breaking problem quite different from that of the electroweak theory. The symmetry involved is called CP, which is a combination of two discrete symmetries: charge conjugation (C) and parity (P). There are various interesting issues associated with these symmetries and a third -- time reversal (T).

We discuss these issues elsewhere, but the basic situation is that there's a basic theorem which states the combination of all three symmetries (CPT) is always preserved in nature. That is, if you take any particle interaction and simultaneously apply all three symmetry operations, the result will be another interaction that is exactly as likely to occur as the original one. This is not necessarily the case if you take only two symmetries at a time, however. CP symmetry, for instance, is often violated in weak interactions.

But with interactions involving the strong force, the probability of CP violation is extremely small, possibly zero. There are two ways the strong force could violate CP symmetry. (One is inherent in the equations of the theory, and the other follows from the fact quarks have mass, which is a consequence of the electroweak force.) If the actual violation is very small or zero, the two effects would cancel each other almost exactly, which is curious. This situation is known as the "strong CP problem".

It turns out that the probability of CP violation in a strong force interaction can be interpreted as the average value of a spinless quantum field, and the quantum of this field is a particle called the "axion". The mathematics behind this result is basically the same as that of the Higgs mechanism employed in the electroweak theory. It involves the spontaneous breaking of a global symmetry called the Peccei-Quinn symmetry. The Higgs field which causes this symmetry breaking may have been one that contributed to the formation of domain walls.

Like Higgs particles, axions have not yet been observed. Unlike the Higgs particles, however, they are expected to be extremely light -- less than 1/100 the mass of an electron. In spite of their light weight, some theorists think axions could be so numerous in the universe that they might be a prime candidate to constitute "dark matter".

### Alternatives to the Higgs mechanism

In light of all that's been said about the importance of the Higgs field and the Higgs boson to particle physics, would it be a disaster if (as appears possible) no Higgs particle is actually found?

No. There are alternatives to the Higgs mechanism for explaining electroweak symmetry breaking and particle mass, even though each has problems of its own. What we do know is that if no Higgs boson exists, then there must be some other particles or forces -- of an unknown type -- which play the same role. The symmetry breaking isn't simply an "accident".

The typical form of such alternatives involves new particles and forces that bind together in such a way as to produce a composite particle which behaves in essential ways like the Higgs boson. Thus, although such a particle is not elementary, it still interacts with known particles to slow them down and give them mass.

In any case, there would be no reason, based on current experimental evidence, to give up the present standard model. It is not in conflict with experiment. There are certainly many things which still require explanation. If something like the Higgs mechanism isn't true of the real world, then there will be other causes. It just may take a little longer to find them.

#### Technicolor

One of the more noteworthy alternatives developed in the late 1970s was an entirely new type of force called a "technicolor" force. The basic idea was to construct Higgs bosons as composite particles -- like mesons and hadrons -- rather than assume they are elementary particles like leptons and quarks. Essentially, this idea would hypothesize a new force rather like the color force, but at a scale about a thosand times smaller. The force was called technicolor because of the analogy with the color force.

In this scheme there would be a new set of spin 1/2 particles called (of course) technifermions. A bound state of one of these with its antiparticle would be a spin 0 particle (a boson) analogous to a pion (which consists of a quark and an anti-quark, bound by the color force). Naturally, this would be called a technipion. One such particle would play the role of the Higgs boson in lending mass to the gauge bosons of the weak force.

There are a variety of problems with technicolor theory in its various forms. Just to begin with, while it explains the mass of the weak gauge bosons, it does not explain how fermions acquire mass. Although the theory predicts a large number of additional particles should exist, no evidence has been found for any of them, or any other effects of the hypothetical technicolor force. There are many other problems of a techni-cal nature, such as problems reproducing known phenomena of weak interactions. Efforts to extend the theory to deal with such problems have only made it even more baroque and artificial than it was to begin with.

In short, theories of this kind are still pursued by some who dislike the Higgs mechanism for one reason or another. But deficiences and inelegance of such theories makes them unpopular with most physicists.

### Open questions

To sum it all up, physicists have pursued an understanding of the Higgs mechanism for three related purposes:
• To make the Yang-Mills theory of the electroweak force renormalizable and mathematically consistent
• To provide an explanation for the fact most known particles (except for photons and gluons) have mass
• To explain why spontaneous symmetry breaking occurs in the theory of the electroweak force (and the asymmetry of the electroweak and the strong force in a grand unified theory)
Theoretically, this effort has been successful on all counts. Experimentally, however, until Higgs particles are actually observed, there remains substantial room for doubt. Some of the causes for concern, aside from the lack of direct evidence for Higgs particles, are as follows:
• Introduction of new fields and particles to solve theoretical problems, without independent evidence, seems a little ad hoc and contrived.
• There is little explanation of what causes or generates the Higgs field itself. (Perhaps another way of saying it is ad hoc.) This can be remedied with the help of more ambitions theories, such as supersymmetry, but such theories are themselves unverified.
• Computations of the cosmological constant, assuming the existence of Higgs fields, produce a result that is absurdly large.

#### Where are the Higgs particles?

This is the biggest concern at the moment. There should exist at least one Higgs boson with a mass less than about 135 GeV under reasonable assumptions. Actual experiments have already ruled out any Higgs bosons with masses close to this limit.

#### What are the theoretical implications if Higgs bosons can't be found?

The standard model would survive. The Higgs mechanism solves various problems for the standard model, but it is not actually predicted by the model. That is, the mechanism provides a sufficient, but not necessary, means of resolving the problems. The nonexistence of Higgs bosons would not lead to any conflict between theory and experimental results.

The standard model is essentially a theory of massless particles. The Higgs mechanism provides a means of explaining the masses of particles, through their coupling with the Higgs field, without sacrificing mathematical consistency of the standard model. If Higgs particles do not actually exist, it may still be possible that there is a Higgs field which provides for mass. If there is no Higgs field at all (which would greatly mitigate the cosmological constant puzzle), then the explanation for particle mass would be a major mystery, yet the standard model itself wouldn't fall.

#### What is the origin of mass?

Assuming that the theory of the Higgs mechanism is essentially correct, and that Higgs particles are eventually observed, then all particles that "couple" with the Higgs field will acquire a certain amount of mass. Here then is an explanation of where mass comes from. In fact, none of the particles which occur in the standard model could have any mass that does not come from coupling with the Higgs field if the theory is to be mathematically consistent.

But even if all this is correct, there are still puzzles. Where do the masses of the Higgs particles themselves come from? For any other particle, their observed mass is proportional to the strength with which they couple to the Higgs particle. But what is it that determines the strength of this coupling, and hence the specific mass of each particle?

Most mysteriously of all, since gravity is preeminently the theory of the interaction of mass with spacetime, how is gravity related to the Higgs mechanism?

#### What is the origin of the Higgs field itself?

We have noted above various ways in which this question can partially be answered. But even if these answers are correct as far as they go, they don't seem like a "final" answer. The situation is somewhat similar to that of questions like "where does space come from?" or "where does time come from?" Physics may at some point be able to provide answers to questions like this. Or at least, to questions of where the hypothetical single unified force and the Higgs fields come from. (Ironically, though, the number of necessary Higgs fields seems to increase even as the number of independent forces decreases.)

#### If there is a Higgs mechanism, what solves the hierarchy problem?

Although the Higgs mechanism handles a number of puzzles fairly well it creates a rather nasty problem of its own in grand unified theories, which unify three of the four fundamental forces (excepting only gravity). This hierarchy problem, though rather technical, doesn't seem capable of being dismissed as a mere aesthetic blemish. That would entail a fantastically improbable circumstance. Supersymmetry offers a solution, but supersymmetry itself currently lacks critical experimental evidence. If supersymmetry is real, many puzzles are solved. In particular, we have a way to explain the origins of the Higgs mechanism and to handle the hierarchy problem. But without supersymmetry, we must find alternative solutions to both problems.

#### If there is a Higgs mechanism, what keeps the cosmological constant small?

The problem is, in short, that the Higgs mechanism is a bit too efficient. If the vacuum is actually as full of nonzero Higgs fields as it seemingly must be to account for particle mass and spontaneous symmetry breaking, then the cosmological constant (i. e., vacuum energy density) must be enormous -- 120 orders of magnitude larger than what observation seems to allow. Somehow, the effects of all the Higgs fields need to cancel each other out almost (but not quite) entirely. It's a "fine tuning" situation that could hardly happen by chance. Even supersymmetry does not appear to help out.

### Recommended references: Web sites

#### Sites with general resources

LEP Higgs Working Group
Home page of 4 research groups searching for Higgs bosons.

#### Surveys, overviews, tutorials

Higgs boson
Article from Wikipedia. See also Technicolor (physics).
Physics with ATLAS: The Higgs Particle
Overview of the role of the Higgs field in accounting for the mass of Standard Model particles.
The Higgs Mechanism
An elementary explanation in cartoon form, based on ideas by David J. Miller. The original brief article is here.
The Waldegrave Higgs Challenge
The best 5 one-page particle essays on Higgs physics, written in response to a challenge by UK Science Minister, William Waldegrave.
My Life as a Boson: The Story of 'the Higgs'
A slide presentation by Peter Higgs, given at the 2001: A Spacetime Odyssey conference.
Introducing the little Higgs
November 2002 article from Physics World, by Martin Schmaltz. Describes "little Higgs" theories, which are alternatives to supersymmetry. "For many years a theory called supersymmetry offered the most compelling solution to the hierarchy problem in particle physics, but now it has a rival."
CERN chases the Higgs
September 2000 news article from Physics World, about the decision to allow the Large Electron Positron collider at CERN to operate for another month in hopes of gathering more data that might confirm observation of a Higgs boson.
Higgs boson on the horizon
September 2000 news article from Physics World. Describes possible evidence of the Higgs boson.
Net tightens round the Higgs boson
September 1999 article from Physics World, by Philip Burrows. Describes state of the search for the Higgs boson.
The search for a standard model Higgs at the LHC
PhD thesis by Ulrik Egede. Detailed technical treatment of theoretical and experimental Higgs physics. Look in particular at Higgs physics at the LHC.
The Linear Collider Opportunity
An essay by Gordon Kane on the need for construction of a new linear collider. The essence of the matter is that an understanding of electroweak symmetry breaking and the Higgs mechanism is a top priority in theoretical particle physics and that a NLC will provide experimental data not obtainable any other way.
What exactly is the Higgs boson?
Question and answers from Scientific American's Ask the Experts section.
How does the Higgs boson affect string theory?
Question and answer (by Gordon Kane) from Scientific American's Ask the Experts section.
What is a Goldstone Boson?
Goldstone bosons play a technical role in symmetry breaking via the Higgs mechanism. The question is answered by Jeffrey Goldstone.
The Higgs Boson
Brief introductory information.
Higgs Boson: One Page Explanation
Five articles that explain the Higgs boson in a page or less.
Signs of mass-giving particle get stronger
November 2000 news article from Science News, about the observation of possible signs of the Higgs boson at the LEP collider.
LEP shutdown postponed by one month
September 2000 news release about postponement of the LEP collider shutdown to allow for additional investigation of possible signals of the Higgs boson.
Most-Wanted Particle Appears, Perhaps
September 2000 news article from Science News about the observation of possible signals of the Higgs boson at the LEP collider.

### Recommended references: Magazine/journal articles

The Mysteries of Mass
Gordon Kane
Scientific Amerian, July 2005
Jiggling the Cosmic Ooze
Peter Weiss
Science News, March 10, 2001, pp. 152-154
The Higgs particle is thought to be responsible for the existence of mass in the standard model. Detection of the Higgs particle is the highest priority objective in current high-energy physics.
Higgs Won't Fly
Graham P. Collins
Scientific American, February 2001
CERN declines a massive opportunity to find the Higgs particle.
The Higgs Boson
Martinus J. G. Veltman
Scientific American, November 1986, pp. 76-84
Historically, physicists have developed the theory of Higgs fields for two different reasons: to account for masses of elementary particles, and to give consistency to the mathematics of elementary particle theory. Actual existence of Higgs fields and bosons would solve some problems, but pose others.

### Recommended references: Books

Abdus Salam -- Unification of Fundamental Forces
Cambridge University Press, 1990
An introductory lecture by one of the co-recipients of a Nobel prize for work on the unification of the weak and electromagnetic forces.