## Forewords

The hunt officially started on Oct 19th, 1964 when Peter Higgs published a paper called “*Broken symmetries and the masses of gauge bosons*” on Physical Review Letters (Vol 13, No. 16). Nowadays, his brilliant explanation about the origin of the mass of elementary particles is being tested in two laboratories: at Fermilab, 80 miles south of Chicago (USA), and at CERN, in the outskirts of Geneva (Switzerland).

Friday Jul, 22th marks an important date on the long quest for the Higgs boson. At the EPS conference in Grenoble (France) the four main experiments in particle physics gathered for a showdown. Has the god~~damn~~ boson finally cornered?

## The Battlefield

The most successful theoretical explanation of known phenomena in particle physics is called the **Standard Model**(SM).

It is a *Model*, because it tries to make testable predictions but keeps away from making firm statements about the reality itself. And it is *Standard* because it is widely accepted from the scientific community. In fact, in the last decades, its predictions have always been verified within experimental uncertainties. Anyway, physicists find this model unsatisfactory. Too many free parameters, too many *ad hoc* assumptions (“*it just works fine*“) and, last but not least, it makes the worst prediction ever in physics! (see: cosmological constant and vacuum catastrophe)

At the core of the Standard Model lies the concept of **field**. A field is a an “entity” which is defined everywhere in the spacetime. For instance, weather forecasts show temperature and wind fields across the Country. In our case, a field is associated to every kind of elementary particle: electrons, photons, quarks and the like. Fields are classified in two groups according to an intrinsic property called **spin**: when measured in units of “reduced” Planck constant , integer-spin particles are called *h***bosons**, and half-integer spin particles are called **fermions**. It turns out that matter is made of fermions (electrons, quarks) while forces (electromagnetic, strong, weak) are mediated by the exchange of bosons among fermions. Not all the particles are elementary: for instance, the proton is made of three quarks soaked in a sea of gluons and other, fainter quarks. At the LHC, these constituents of the proton interact among each other. We aim to understand the underlying physics looking at what comes out of these high-energy accidents.

Since a long time, experiments show that there are some quantities that are **conserved** through interactions, such as the sum of all electric charges of a system. To account for this fact, the mathematical description of the fields is scrutinized in the search for invariant (=preserved) properties. This idea is encoded in a theorem due to Amalie Noether (she was among the first women who taught in a German University in 1915!). According to this theorem, *to every symmetry there is an associated conserved current*. In turn, thanks to Gauss’ theorem, for every conserved current there is a conserved quantity that we call **charge**.

In the simplest case, a symmetry called U(1) explains the origin of the electric charge, a property owned by several fields (*e.g.* electron, quark) *but not all*. Add Einstein’s Special Relativity and *voil*á you get a description of the **electromagnetic force**. A more complicated symmetry called SU(2) gives rise to the so called *weak isospin*, and its related charge *T3* explain why in nuclear beta decays only certain types of particle can be created, and not others at will. Those decays are due to the **weak force**. Finally, the SU(3) symmetry introduces the *color interaction*, whose by-product is the **strong nuclear force** that binds protons and neutrons together inside atomic nuclei. It is so strong (hence its name) that it even overcomes electrostatic repulsion. As the photon mediates the electromagnetic force, so the gluon mediates the strong force. The main difference between the two is that the photon carries no electric charge, and so there is no photon-photon interaction. On the other hand, gluons alwasy carry a color charge: gluon-gluon interactions *can* happen. This fact brings funny things inside the proton, and it is the main ingredient for the physics at the LHC!

Finally, some theoreticians believe that some other symmetry could explain the **gravitational force**. Its mediator would be a spin-2 boson called graviton. So far, nor this symmetry nor any graviton have ever been seen (or is it?).

## Breaking Rules

Why so many interactions? In 1865 Maxwell published **“***A Dynamical Theory of the Electromagnetic Field*” in which he first proposed an explanation of light as the result of the unification of electricity and magnetism. You can still buy yourself a copy of this book on Amazon! Try to imagine what a breakthrough it was: for centuries, electricity, magnets and light was seen as three completely different phenomena.

Following this school of thought, Salam, Glashow and Weinberg proposed the unification of electromagnetic and weak interactions in what we call now **electroweak force**. If blending electricity and magnetism explains the origin of light, what could show up merging electromagnetism with the weak force? A long-time unresolved puzzle concerns the typical length of the interactions. While light has an infinte range of actions (photons arrive to Earth from stars millions of light-years away from us!), weak interaction seems to be very **short-ranged**. A possible explanation would be that the force is mediated by some kind of heavy photon, that fades away in a very short time (a billionth of billionth of a second). As a joke, this is sometimes referred to as “heavy light”!

A good hint came from the theory that explains why we can make permanent magnets with some metals, but not with others. Every atom can be seen as tiny magnet. When all the magnet axes point toward the same direction, the overall effect is enhanced and becomes macroscopic. Otherwise, if chaos rule, no net effect can be seen in the material. Scientists discovered that there is a temperature above which this property can be imposed, using an external magnetic field. An interesting effect is the discovery that Earth’s magnetic field reversed its polarity several times in the past!

Although it seems irrelevant to the problem of the electroweak unification, it turned out that the same mathematical equations can be used to explain a number of phenomena in particle physics. First, remember that in order to rise the temperature of an object, you have to give it energy, as a kettle on a gas stove. In our case, temperature is proportional to the energy of the interacting particles, the constituents of the proton: quarks and gluons. So, at extremely high temperatures, only the electroweak force exists. Mathematically, it can be described by a symmetry called U(1)×SU(2). When the temperature gets lower (*e.g.* at room temperature), *if* this symmetry gets “broken” for some unknown reason, we can see two completely different forces: electromagnetism and weak force. So the question is: what breaks the symmetry?

## Enter: Peter Higgs

Well, the answer is not so simple but one man suggested a minimal solution.

Some years before, Landau and Ginzburg proposed a model to exaplain superconductivity, the lack of electrical resistance shown by some materials when the temperature drops under a certain threshold. In their model, the energy of the system is described by a quartic function whose *unique* minimum is trivially zero at room temperature. Wherever the energy reaches a minimum, the system is said to be in equilibrium.

Superconductivity can appear if the temperature gets low enough: in this case other, more stable minima appear (the function, with properly adjusted coefficients, is called *mexican hat*). It looks like a ball sitting at the top of a hill: even a slight kick (in this case given by quantum fluctuations) makes the system to go downhill. Physically, this triggrers a phase transition. Such a process is sometimes dubbed **spontaneous symmetry breaking**.

Peter Higgs worked out how to make use of this process in order to explain why the electroweak force is broken at room temperature.

First, let’s suppose that everything in immersed with a scalar field whose energy looks like the mexican hat. Some of the particles are drenched with this field, other don’t. The ones who do, have weight. The ones who don’t, are weightless. When the mediators of the electroweak force “eat” this scalar field they acquire weight and the symmetry is broken. The leaftover crumbles are the **Higgs bosons**. Two birds, two stones: this explains why the weak interaction is short-ranged, and why we do not see the unified electroweak symmetry in everyday life! Anyway, it is important to stress that this is just one explanation of the phenomenon. Nature could have chosen a more complicated one, involvin for instance more than one scalar field. Only experiments can wipe out wrong theories.

Merging all togheter, theoreticians predicted the existence of 5 particles: the already-known massless photon (which is in fact…light) plus four massive bosons, never seend befor. They were the three mediators of the weak interactions (W+, W- and Z), and one Higgs boson (H).

The Z boson has been discovered at CERN in 1973 by the Gargamelle experiment (now on display in CERN’s garden). In 1983, again at CERN, the UA1 and UA2 experiments finally catched the charged W bosons. Salam, Weinberg and Glashow were awarded with the Noble prize in 1979. Rubbia and Van der Meer shared another Nobel prize in 1984. The Standard Model was firmly established in 1995 when the top quark was eventually discoverd at Fermilab.

So only one tile in the puzzle is missing.

## The Quest for Go(l)d Particle

As the water flows downhill from its spring to the sea following different paths, so a particle can decay through different “**channels**” until stable grounds are reached. To our present knowledge, only photons, protons, electrons and to some extent neutrinos are stable. All the others crumble down unitl stable particles are eventually produced. The decay process is not completely arbitrary: it must fulfill a set of rules described by quantum mechanics (QM). We can use these rules backward and make an educated guess of what is more likely to have produced the particle we observed. The sum of all the probabilities *must* give 100%: this fact is so important that it even has got a name: *unitarity*.

Anyway, with QM we can calculate the probability that a path is followed, but which one is actually taken is a matter of chance and is *intrinsecally* unpredictable. If you toss a quantum coin you have a 50% probability of getting head or tail. Apparently, Nature prevents ourselves from predicting *exactly* which side of the coin will show up.

So it is for the Higgs boson. The first thing we want to do sounds quite simple: just count how many of them we see! The quest is twofold: theory predicts a number of different way to produce it, and several decay paths it can follow. What is actually predicted is a quantity called **cross-section**. It can be loosely interpreted as the probability to generate a desired particle given certain conditions. This probability depends strongly on the type of particles you collide, on the energy of the collision and on the mass of the particle you aim to produce. While you can control the first two using different kinds of accelerators (*e.g.* Tevatron or the LHC), the third is set by Nature. In our case, we don’t know which is the mass of the Higgs boson: at the end of the game, we will discover its value by inference.

Once you managed to produce some Higgs bosons (congratulations!), you still have to calculate the chance to see them decaying into each possible channel. Most importantly, it turns out that *the probability of each channel depends on the mass* of the Higgs boson. But this is not the end of the story: the Higgs is likely to produce intermediate particles, which in turn decay until stable ones are generated. A definite set of stable particles describing the end of a decay chain is called **final state**. Some final states are considered extremely appealing from an experimental point of view: they are easy to study and so they are dubbed **golden channels**.

A quite strong assumption is that the production and decay probabilities are independent of each other. This means that *daughter particles have only a loose memory of their parents*. If you think about it, a lot of animals behave this way! So far, it always proved to be true for all know particles: there is no reason to believe that the Higgs might have a different opinion. It cannot say “I accept to decay to a couple of Z bosons, but only if they promise me they will decay into four charged leptons!”. But you never know…

Unfortunately, the Higgs boson shares its possible final states with other less interesting processes. In jargon it is said that what you’re looking for is the **signal**, while all the other stuff is the **background**. Most of the intersting signals are very tiny, and most of the events are rubbish. The proportion is usually less than 1 in one billion.

Being an experimentalists consist mostly in trying to extract signals out of the background. There are many techniques to do it. The simplest one is to apply “**cuts**“: final states are sifted and only the very interesting ones end up in your histograms. If your analysis is set up correctly, some peak should rise above the background. For the die-hard nerds, more advanced techniques involve neural networks and other pattern recognition techniques.

## Pushing to the Limits

Now that you have all the tools of the trade, you can join the Search for Gold. Pick your preferred final state, calculate its probability as a function of Higgs boson’s mass and start looking for it digging into data. Some suggestions are:

- A very light Higgs is more likely to produce two photons, or two b-quarks, or two tau leptons;
- A moderately heavy Higgs could produce a couple of W bosons. In turn, the can decay into two pairs of charged leptons and neutrinos;
- A heavy Higgs could decay into two Z bosons. The final state with four charged leptons is the golden channel;
- An extremely heavy Higgs breaks unitarity so it’s not believed to exist!

Do not jump into conclusions too fast! The golden channel has a very clean signature (few processes end up with four charged leptons!) but it is very very rare and it works fine for a heavy Higgs. Other channels have a higher probability in some mass ranges, but they suffer from a larger background – so it’s harder for you to teach your computers how to recognize this particle!

If the Higgs boson really exists, with an infinite amount of data we would be able to see it in every channel. However, with a limited number of events we must play smart. The key to success in the search for the Higgs boson is in the combination of all channels. This is done using statistical procedures that take into account what is shared among the channels, and what is specific of each one. At the end of the day, one hope to see a combined signal rising above the background. But what if we see no signal at all? Again, do not jump into conclusions too early. The calculation of the cross-section, along with machine parameters such as luminosity, center-of-mass energy and detector efficiency tells you how many events you must collect before you can say Eureka. If you have acquired enough data and you still have seen nothing, perhaps the particle you are looking for does not exist at all! Othere possibilities are that its production rate is lower than you thought, or it’s just that you have been unlucky: a downward statistical fluctuation produced less Higgs bosons than expected and you just happened to see none. In these unfortunate cases, what’s done is to set *limits*, which is to say: “*I collected a certain amount of data, but the signal did not show up. I conclude that its strength s* must be f times smaller than that as predicted by the theory s_th*“. In order to take into account statistical fluctuations,

*is presented along with an uncertainty. This range of possible values gives you a*

**f***confidence level*or

*credible interval*. Their meaining is roughly this: “

*The strength of the signal has an unknown but fixed value*“. Most common intervals are 68% (stringent), 90% and 95% (loose) confidence levels. If you test your hypothesis about

**s***. If I could run an infinite number of experiments, 68% of these would say that**s*** lies within the quoted uncertainty**as a function of the mass of the Higgs boson, you obtain a graph that looks like this:**

*f*The observed data are tested against two hypotheses for each mass point: background-only and signal+background. Before taking data, one runs the simulation for the background-only scenario, obtaining the dashed curve, surrounded by it uncertainty (the green-yellow “Brazil band”).When the dashed line drops under the horizontal level (fraction = 1 that means * s* *=

*) theory predicts that you are supposed either to find or not the signal. Once these data are taken, limits on the signal strength are calculated (the solid black line). If there is no signal at all, the black line is supposed to be very close to the dashed line. If this is not the case, something is going on! The more the two lines are separated, the higher the probability that what you are seeing is not due to a statistical fluke!*

**s_SM**So, what do you see in this graph? The black line drops under * f*=1 between 155 and 190 GeV. We saw no Higgs there. However, there is a broad excess between 120 and 150 GeV (the black line is still over

*=1, but it is clearly outside the yellow band). How to interpret this observation? At the moment, the only thing that we can say is that experiments seem to contradict the background-only hypothesis. Whether it is consistent or not with the presence of the Higgs boson is a matter of debate. The statistical significance is low (it’s still too close to the yellow band to tell), but at the end of the 2011 there should be 5 times more data to be analyzed. The answer is near!*

**f**## Conclusions

After almost 40 years we will see if Peter Higgs’ explanation of the electroweak symmetry breaking is right or wrong. Either way, this answer will be a turning point in the history of particle physics. For some theoretical reasons and astrophysical observation, the Standard Model is already in dire straits. Understanding what lies beyond is LHC’s business.