# Higgs 101

The Higgs boson..the “God Particle” (ick…sorry Leon). The Holy Grail of particle physics. (Um, also there’s that dark matter stuff but who knows what it might be…)

Where to begin? The Higgs boson is one thing, or many, there, or not, waiting to be discovered, if we can. But what is it? Why is it? Does it exist at all?

We know a lot, now, after decades of experiments at the great accelerators. Working backwards: the Tevatron at Fermilab, LEP 1 and 2 at CERN, KEK-B at KEK in Japan, PEP2 at SLAC, CESR at Cornell, the SLC at SLAC, the SppS at CERN, HERA at DESY, Tristan at KEK, PEP at SLAC, PETRA at DESY. But it’s all coming down, sooner or later, to the Big One: the LHC at CERN. This machine will, in all likelihood, answer the question: what is the origin of electroweak symmetry breaking? That is, why do the carriers of the weak nuclear force, the W and Z bosons, have mass (and large mass at that) and the photon does not have mass at all? If we are really lucky we may start to get an answer to the question: why do fundamental particles have masses at all, and the rather peculiar masses they do have? And even if we know all this, so what? What then?

That was a lot of jargon and acronyms for one paragraph…so we’ll start there. Firstly, the word “electroweak”. We believe that all of the particles we know about interact “electroweakly” in the sense that they all partake of the weak nuclear force and, if they have electric charge, interact electromagnetically as well. We believe that the electromagnetic and weak nuclear forces are manifestations of a single underlying force of nature. The word “electromagnetic” should start to give you the flavor of the enterprise. Electricity=magnetism has been with us for nearly 150 years, first unified in the form of the “classical” electromagnetic relations of James Clerk Maxwell in 1861. This feat gave humanity its first taste of victory over matter, energy, space, and time, and propelled us headlong into the modern age. Surely one could unify electromagnetism with the other obvious force of nature, gravity, the universal nature of which was established in the late 1600′s by Isaac Newton.

But no. Gravity has stubbornly resisted unification to this day. Perhaps more on that later. Meanwhile, the late 1800′s and early 1900′s saw an incredible unfolding of unexpected events: the establishment of an apparently absolutely empty vacuum (no “ether” to serve as host to electromangetic waves), the undeniable ultimate universal speed limit namely that of light. Then in rather quick succession came the quantum, special relativity, the atom, the nucleus, and finally the dawning of the quantum mechanical description of our world.

By 1930 the quantum revolution was complete, and we could set about the usual human pattern of (intellectual) colonization, militarization, exploitation, and capitalization. All the modern technology we now enjoy derives principally on our ability to first understand and then technologize the quantum world.

We’d done it: unlocked the secrets of the atom! And we got the nucleus as a bonus! So much energy in so small a place…unbelievable. Could there be more? We humans never stop – we probe ever deeper, and in this case deeper means smaller, and smaller means more energy, and more energy means bigger, and more expensive. What government, what power, would want to miss out on this one, though? If you could build nuclear weapons maybe something even smaller was inside protons and neutrons and you could build an even bigger or better weapon? No question – the military drove this one. And in the US, with a nuclear-weapon economy much larger than the auto industry, the scientists had free rein in the 1950s and 1960s to pursue these questions: were there smaller things still inside the nucleons (the protons and neutrons?)

Yes. Lots. As the energies of accelerators grew ever larger, we produced more new and strange particles, never before seen. We already knew since the 1930′s, or at least believed we knew, that there exists antimatter in the world. Antimatter looks just like the regular stuff, but has opposite charge. The jargon is “quantum numbers”: for every particle with its known quantum numbers, there are antiparticles with all the same properties but the opposite quantum numbers. The electron has electric charge -1 and its antiparticle has the same mass and interactions but has electric charge +1. The antielectron does not have “anti-mass” in any sense, but looking at the equations you can sort of tilt your head, and if you hold you mouth just right, the antielectron looks like an electron with negative energy moving backward in time. But never mind that. Anti-electrons, or positrons, do indeed exist, they move forward in time, and they have positive energy. (Lost yet? Don’t worry…)

All the fundamental matter particles we know about have antiparticle cognates. But what are the *fundamental* particle/antiparticles? Which are fundamental in the sense of being truly indivisible, unexcitable, basic, utterly elementary?

A whole zoo of particles emerged in the 50′s, 60′s and 70s, but let’s cut to the chase: here is a picture of all the *fundamental* matter particles, and every bit of matter we know about (except dark matter; again more later on that) is made of these things:

There are six quarks (and their antiquark mirror images), and six leptons. They are apparently arranged in three “families” or “generations” in order of increasing mass. Within each generation we find the same four apparently fundamental types: one neutrino, one charged lepton, and two quarks. The quarks interact strongly, that is, by the force that holds the nucleus itself together. All the particles interact weakly, that is, via the weak nuclear force. And all of them except the neutrinos interact electromagnetically – they carry electric charge.

I believe that the greatest (and I mean THE greatest) discovery of the 20th century was to recognize that every symmetry in nature coresponds to some conserved physical quantity. It is a great sorrow that Emmy Noether did not win the Nobel Prize for this profound work. Symmetries are all around us – some are very simple, and some not so simple. For example, consider symmetry in time. The laws of physics are (we presume) the same now as they are at the time you finish reading this sentence, and will be the same 100 years from now. If you move (translate) in time, the rules stay the same. This symmetry in fact leads to conservation of energy. Likewise, if you move in space, the laws of physics are the same. This leads to conservation of momentum. If you rewrite the laws of physics in a frame of reference rotated 42.6 degrees from the one where you are writing them now, they are the same…conservation of angular momentum! Wow!

But it gets much better. We know, since the beginning of the last century, that you need to obey the rules of relativity. Fine. Also quantum mechanics. Okay…how? We write down wavefunctions that obey all the rules. Then we make them also obey all the symmetries we know about.

It turns out that wavefunctions are represented by complex numbers, which have real and imaginary parts. (By imaginary, we mean multiples of the square root of negative 1, not “eleventy-forty seven” which is perhaps another imaginary number.) Complex numbers are rich and fascinating, and the calculus of complex functions of complex numbers is great fun and beautiful. Most remarkably, this calculus describes the world perfectly when combined with quantum mechanics and relativity!

“Describes the world perfectly.”? Okay, well, it describes a broad swath of phenomena at the subatomic particle level really well…freakily well. We call it the Standard Model. I give it capital letters here because it has become so deeply established, and has such predictive power over such a broad range of phenomena that you have to give it a proper name. And I and most of my friends live for the day when we can destroy it and return it to lower case letters, the formerly standard model.

It’s only a model, after all. To build the Standard Model we assume that our wavefunctions obey certain peculiar symmetries, called gauge symmetries. The simplest to understand is called U(1). With complex wavefunctions, the probability for finding a particle at a certain place/time is calculated by finding modulus of the wavefunction, sort of its absolute value or magnitude. You square it, basically, in the complex space in which it lives. It turns out, though, that you get *exactly the same answer* for this probability if you square the wavefunction, or the wavefunction multiplied by a complex number of modulus 1 (e to the something imaginary). This something imaginary can in fact be a function of position in spacetime. It is, in fact, the same as a rotation of the wavefunciton in the complex plane…and that’s U(1) symmetry.

Here comes the miracle: if you impose upon our relativistic, complex, quantum-mechanical wavefunctions the requirement that they be invariant under these U(1) transformations, then you get electromagnetism. Conservation of electric charge. A massless photon. QED – quantum electrodynamics, in all its 12-digit precision glory. *Electromagnetism is a simple consequence of the U(1) symmetry of any wavefunction. *

When you have a hammer, especially one like this, everything starts to look like a nail. So in the 60′s and 70′s, the theorists starting banging away, and came up with the next level of symmetry, which (surprise) was called SU(2). If we impose SU(2) on our wave functions we get…well, a puzzle. Because whereas U(1) appears to be exactly obeyed, and though particles appear to obey SU(2), it’s only approximate. (Particle friends, please close your virtual ears/eyes here because I am guilty of oversimplification here.) It’s a “broken” symmetry. Maybe at very, very high energies it appears exact, but down here on cold hard Earth it’s broken, sort of badly.

Then, BREAKING NEWS, circa 1970s: It *does* appear that the strong nuclear force can be described by an exact SU(3) symmetry! But, alas, our ability to calculate anything in the quantum mechanics of SU(3) is stymied by the facts that a) the carrier of the strong force, the “gluon” (analogous to the photon) can interact directly with itself, and b) the strong force is damned strong and we cannot use our favorite quantum mechanical approximation trick, perturbation theory, to calculate much of anything that we can measure. A subculture of particle theorists and experimentalists is born, and continue to sport 70′s sideburns (not the women) well into the 90′s, when they mysteriously return to fashion. Quantum chromodynamics is what our theory of the strong force is called, “color charge” is what’s conserved, and it explains the strong interactions rather perfectly.

Meanwhile, a number of models with SU(2)-type symmetries are proposed, but one in particular wins out in the intense intellectual battle that ensues. This one, the creation of Sheldon Glashow, Abdus Salam, and Stephen Weinberg, combines SU(2)xU(1) with the then-little-known Higgs mechanism to produce a theory that conserves “weak hypercharge”, and accounts for both the weak interactions and the photon (electromagnetism). Awesome in its sweeping scope and power, this theory predicts the existence of two heavy “weak vector bosons”, the particles that transmit the weak nuclear force: the W and Z bosons. The weak nuclear force is weak, and short range, because the W and Z are very heavy: 80 and 91 GeV, respectively, nearly two orders of magnitude times the mass of the proton.

The W and Z are then discovered at CERN in 1983 in the debris of collisions of protons and antiprotons. Weinberg/Salam/Glashow had already won the Nobel prize in 1979 for their work, and Carlo Rubbia and Simon van der Meer won it for their discovery (along with hundreds of colleagues on the UA1 and UA2 experiments at CERN) the next year, 1984. A new field, high energy hadron collider physics, is firmly established. The hunt is on for the Higgs boson!

And so at last we come to the Higgs boson. The electroweak theory posits a “Higgs field” permeating all spacetime. You just paste it into the “Lagrangian”, the equation that tells you how all possible wavefunctions behave in the presence of the U(1), SU(2), and SU(3) symmetries that we believe must be obeyed. The Higgs field has a certain potential energy form that leads to the SU(2) symmetry being broken, and only imperfectly obeyed, at low energies (that is, our world). I you wrote out the full electroweak Lagrangian, this is what you get (oops, there is a missing parenthesis…can you spot it?):

The Standard Model is not pretty, but it does work. As we said before, it’s only a model. Does the Higgs field exist? Well, if it does, and if it’s broken the way we think it is, then a massless photon exists and so do massive W and Z bosons. Since these do exist, and behave in precisely the predicted manner, then can we but conclude that the Higgs must exist as well?

If so, we should try to find it, and as I have said in Cosmic Variance before, I have been searching for it for about twenty years, with no success yet. Should I or my hundreds of colleagues in this quest be discouraged? No, because if it does exist we don’t really think we should have been quite able to see it yet.

Why not? The basic reason is that all our best data indicate it should have a mass of about 120 GeV, give or take, and the Higgs boson is very feebly interacting. Therefore it’s hard to make lots of Higgs bosons.

Nevertheless, at the LEP 1 and LEP 2 accelerators we had a real chance to produce and detect Higgs bosons, because in addition to interacting with our matter particles (the quarks and leptons) the Higgs interacts with the W and Z bosons. At the LEP accelerator, electrons and positrons collided at very high energies, equivalent to the mass of the Z boson at LEP 1 and up to more than twice that energy at LEP 2. When you produce a Z boson, it can occasionally give you a Higgs boson and a “virtual” Z that almost immediately decay. The LEP experiments all looked feverishly for the Higgs in the 1990s, but in the end there was no evidence for a Higgs boson with mass less than 114.4 GeV. This was a major accomplishment, and if the Higgs boson did exist with a mass less than this value, I am certain LEP would have found it. There were some tantalizing collision events at the very end, but not enough to say “this is it”.

In the mean time we refined what we know about the Standard Model, and discovered the top quark at the Tevatron in 1995. This was an important key to unlocking the Higgs boson, because the Higgs boson mass can be calculated, somewhat roughly, if we know the mass of the top quark, the W boson, and how the Z behaves in its production and decays. The answer that we get is that the Higgs boson should have a mass which is probably less than the limit we’ve set! I say probably because the uncertainties in this calculation allow some possibility that the Higgs boson mass could be larger than the present limit, say, 120 GeV, without causing too much difficulty with the other existing measurements.

In the end, then, the Higgs boson of the Standard Model could be just out of range of the LEP accelerator, which was decommissioned at the end of 2000 to make way for the Large Hadron Collider, the LHC at CERN, which will begin operating next year if all goes well. Meanwhile, the Tevatron proton-antiproton collider at Fermilab in Illinois has had an opportunity to observe the Higgs boson before the LHC turns on.

The Tevatron began operating in 1985 and has been the highest energy proton-antiproton collider in the world ever since. In the early 1990s plans were underway to construct the gargantuan SSC, the Superconducting Supercollider, in Texas, to begin operating in about 1999. Miles of tunnel underground were excavated, and the SSC laboratory was taking shape when, in late 1993, the US Congress de-funded the project in the wake of a ballooning cost estimate. The cancellation of the SSC meant that the next big accelerator would be the LHC at CERN, but it was going to take over a decade to complete it. This meant that by the mid-1990s the only place to look for the Higgs was the Tevatron. The CDF and D0 experiments were ready, flush with the victory of the top quark discovery and eager to join the hunt, underway at LEP, for the Higgs.

As we mentioned above, though, the Higgs interacts rather feebly with quarks, leptons, and the W and Z bosons. The “coupling” to fermions like quarks and leptons is in fact proportional to their mass, and so the Higgs boson favors the heaviest ones: the top quark, the bottom quark, the tau lepton, and the charm quark. If you produce a Higgs boson, it will decay to the heaviest quark or lepton paris (for example a bottom quark and antiquark) that it can. If it can decay to pairs of W or Z bosons, though, it prefers to do that. The Higgs boson is produced at the Tevatron in an analogous fashion to how it produced at LEP. At the Tevatron, quarks in the protons collide with antiquarks in the antiproton, and make virtual W or Z bosons, which in turn can give a real W or Z plus a Higgs boson. We call this the VH process, because the W and Z are vector bosons (meaning they have spin 1).

At the Tevatron, though, the challenge was to get a high enough proton-antiproton collision rate to make the Higgs, and so the whole Fermilab complex underwent a major upgrade in the late 1990s, adding a new Main Injector to get more beam into the Tevatron, and numerous improvements to the rest of the machines. By 2001 new data started to roll in and the hunt resumed, having ended the year before at LEP.

The Tevatron luminosity (the number of collisions per second), steadily grew until the machine hit its stride in the past few years. If the Higgs boson is there, then we should have been able to produce thousands of Higgs boson events. But at the Tevatron, one is also producing billions of non-Higgs events! This gives you a very difficult, nearly overwhelming “background” of events at the Tevatron that look just like Higgs events but aren’t. The detectors, furthermore, have imperfect energy resolution for the Higgs decay products, and so it’s hard to get a nice sharp mass peak right at the Higgs mass.

Nevertheless, lots of extremely talented people are working very hard to analyze the data we’ve collected and perform the search for the Standard Model Higgs boson. I’ve been working on the CDF experiment since 1993, and have done quite a bit o this work myself…it is extremely challenging, and takes the collaboration of dozens of people to succeed. The Tevatron experiments are inching closer to being able to extend the limit set by LEP (if the Higgs is not there) and maybe, just maybe, get a glimpse of the Higgs before the LHC turns on and acquires enough data to do the trick.

Can the Tevatron do it? I say “maybe” above and that’s always the right thing to say because it’s true. My own bet is that with about four times the present data sample, and with new ideas and methods, we can surely extend the LEP limit and get a glimpse if it is there at a mass less than about 125 GeV, or (oddly) has a mass close to 160 GeV, where it decays predominantly to WW. Can the Tevatron get four times more data? Yes, if all goes well, and the Tevatron runs through 2010. Is that too late? Maybe… The LHC could turn on in 2008, but will probably not produce enough collisions to see the SM Higgs. What about 2009? It is much more likely that this would be the first year that the LHC could conceivably discover the Higgs, if the detectors are well-understood, the LHC machine works very well, and the Higgs has a favorable mass, around 120 GeV. It actually could happen earlier if the Higgs mass is larger and its ZZ decays become significant, but as we noted above, the theoretically favored mass for an SM Higgs is not much larger than the present limit, 114.4 GeV.

Should the Tevatron run in 2010 or not? This question is beginning to be addressed by the community and the government. It would cost about $30 million to run the machine in 2010, according to what I have heard, and would require a staff of accelerator people and people from the experiments to operate the detectors and analyze the data. Considering the huge investment the US and countries around the world have made in this beautifully running machine and well-understood and efficient experiments, it’s a bargain.

There are good physics reasons to do this. If the LHC does see an SM Higgs signal it will be in a very rare decay mode at first: its decays to pairs of gamma rays. This distinct signal occurs in one of every 500 Higgs boson decays, and will give a sharp peak at the mass of the Higgs in the ATLAS And CMS detectors, once about a year’s worth of collisions is analyzed. The Tevatron, by contrast, would see a Higgs boson via its decays to b quark pairs; the LHC will not be able to do this for several years due to the overwhelming backgrounds. Seeing *both* decay modes is an important test of whether in fact it is a Higgs boson we are seeing at all, and a good cross check of its mass and other parameters.

The Tevatron can also continue to refine the knowledge of the top quark mass and the mass of the W boson, so that when a Higgs is observed we can test whether its mass agrees with the theory – another important test.

Perhaps more importantly is the fact that it might not be a Standard Model Higgs boson at all that we seek. There are very good reasons to believe that, theoretically, the Standard Model Higgs cannot be the only fundamental particle remaining to be discovered. Theoretically, the calculations of its mass suffer from corrections that grow dramatically with the mass of any new heavy fermion that might exist, and in fact the Higgs receives corrections to its mass from *itself* that are unstable and need to somehow be finely tuned to cancel to many orders of magnitude. It is, as the thoerists say, “unnatural” to expect nature to have done this.

Solutions to this problem have been sought, and found, for decades now. If the universe obeys a symmetry called “supersymmetry” which relates bosons (integer spin particles) to fermions (half-integer spin particles) then there must exist a supersymmetric partner for every particle we know about. Supersymmetry is clearly not perfect; it must be broken if it is there, otherwise we would know about all the supersymmetric particles, which would have the same masses as their ordinary particle partners. The presence of these particles cancels off the instability in the Higgs mass in a neat way, greatly reducing the unnaturalness of the situation. The cost, though, is to add many many new particles and parameters to the theory…Occam would perhaps not be impressed.

And, though we’ve looked for supersymmetry for decades, we have not found any evidence of any supersymmetric particles…nevertheless some theorists are convinced it must be there, and we continue that hunt as well. If supersymmetry does exist, the Higgs boson gets more interesting. There are not one but at least two Higgs fields permeating spacetime, one giving mass to the up type quarks, and the other giving mass to the down-type quarks and the leptons. There would be not one but three neutral Higgs boson states to look for, plus a charged Higgs boson. The lightest neutral Higgs boson could have properties very similar to those of the Standard Model Higgs boson, and therefore if we do see a Standard-Model-like Higgs boson we need to measure it carefully to see if it is the one from supersymmetry or not.

Even better, if supersymmetry exists then the Higgs bosons it predicts could be produced at a rate greatly enhanced compared with the rate for the Standard Model one. This means that the Tevatron, and the LHC, can produce more massive Higgs bosons more rapidly. For the Tevatron this could be the way that it could discover the Higgs boson and get evidence for supersymmetry before the LHC turns on.

This was the subject of m previous “Bump Hunting” series of posts at CV, and the latest results are that there is no evidence for production of supersymmetric Higgs bosons at an enhanced rate. This null result allows us to rule out regions of possible supersymmetry parameters. There are numerous variants of the supersymmetry, however, with even more complicated Higgs boson families. It’s fair to say we ave not looked for all of them, and even fairer to say that we have not explored all possible supersymmetric Higgs phenomena theoretically.

It is also possible that there isn’t a Standard Model Higgs boson or a supersymmetric one. A number of theories have been proposed recently (an not so recently) that would explain electroweak symmetry breaking as a byproduct of the existence of some new force, even stronger than the strong nuclear force, which comes into play at a high energy scale, possibly even with new particles.

Other theories propose to solve the naturalness problem by the addition of a minimal set of new particles and/or extra spatial dimensions. Typically one still gets a Higgs or Higgs-like boson, but at a possibly higher mass than we’ve been sensitive to so far.

In short, if you’ve slogged your way through to this point, you might get the idea we don’t really know what we are looking for at all! There is some truth to that. It is important to keep our minds open to new ideas and to be ready to look for as-yet-un-thought-of predictions that come from them. And we do try to do that…yet…

It is hard to look at the beautiful agreement between indirect determinations of the Higgs mass and the prediction from the measurement of the top quark mass and W mass, made in the context of the idea of the Higgs field, and think that this is all just an accident, an random approximation of something else entirely. My own gut feeling is that there really is an excellent chance that in a few years here, at the LHC and/or the LHC, we will see something that looks like a Higgs boson, quite likely at a mass of 120 GeV or thereabouts.

If we don’t see a Higgs boson (SM-like or supersymmetric) with mass less than about 130 GeV, things will get very interesting – it’s almost a certainty then that some other model is the correct explanation. That would be amazing, too, and may be very difficult to unravel at the LHC, with such large background rates.

This whole post barely scratches the surface of the present state of affairs in the search for the Higgs boson and the explanation of electroweak symmetry breaking. I shall leave to a future post the larger question: If we do discover a Higgs boson, of what use might it be, as J.J. Thomson wondered about his great discovery, that of the electron, 110 years ago?

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