Exploring the standard model of particles

This is a very exciting time in our investigation of the smallest particles and the fundamental forces that act between them. The standard model of particle physics has been extremely successful in predicting the results of many different experiments around the world but one key ingredient of this model, the Higgs boson, is not yet fully established. However, the very high-energy proton–proton collisions now being studied by experiments at the Large Hadron Collider (LHC) are expected to resolve this important issue.

This paper outlines the main features of the standard model and explains at an introductory level how Feynman diagrams can be used to make predictions for the electromagnetic, strong and weak interactions. It also describes the Higgs boson, its connection with the Higgs field and the current status of the search for the Higgs boson at the LHC.

Minerva is one of several educational tools that allow teachers and students to study collision data from the LHC experiments. This is intended to make it easier to understand how the detectors identify different types of particles and how the experiments search for very short-lived particles, such as the Higgs boson, which leave no direct signals in the detectors. Some examples of measurements that can be made with Minerva are briefly summarized.

Quarks and leptons are the fundamental particles that build the matter around us (figure 1). There are six quarks, which all have fractional charges. They are only found in groups of three quarks (baryons), quark–antiquark pairs (mesons), but never alone. The three charged leptons, which include the electron, have a single negative charge and the neutral leptons are the three different neutrino types. The quarks and leptons all have spin 1/2 and are called fermions (after Enrico Fermi). They obey the Fermi–Dirac statistics. Fermions obey the Pauli exclusion principle, which prevents two particles from occupying the same state and is responsible for the different energy states occupied by the electrons, protons and neutrons in the atom.

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There is now clear evidence, from measurements of neutrinos from the Sun and those produced in the atmosphere, that the three types of neutrinos have small but nonzero masses. This makes it possible for neutrinos to oscillate from one type to another. In the quark sector there is a similar effect which (by convention) only affects the d,s, b quarks with −1/3 electric charge. The d,s and b quarks are often described as if they are distinct particles but in fact there are small admixtures of s and b quarks in the d quark. These small amounts are well measured and the coefficients of the Cabibbo–Kobayashi–Maskawa (CKM) matrix define this mixing.

In quantum mechanics a fundamental force is carried by the exchange of a force carrier (figure 1). In electromagnetism the force carrier is the photon which is exchanged between charged particles, in the weak interaction the force carriers are the W⁺, W⁻ and Z⁰ particles which can be exchanged between all fundamental particles. In the strong interaction the force carrier is the gluon which is exchanged between quarks. The gluon has the additional property that it can interact with other gluons, in contrast to the photon, which does not interact with other photons. All these force carriers have spin 1. All particles that have integer spins are called bosons (after Satyendra Nath Bose). They obey Bose–Einstein statistics, which is different from the Fermi–Dirac statistics for particles with half-integer spin.

In 1979 Glashow, Weinberg and Salam received the Nobel Prize for the work that led to the standard model of particle physics [1].

The missing ingredient in the standard model is the Higgs particle. The standard model has been exceptionally successful in describing all the fundamental interactions at very short distances, or high energy. However, one quantity has to be introduced by hand—the mass of the particles. Particularly important for the standard model is the mass of the W and Z particles. The Higgs mechanism or the Brout–Englert–Higgs (BEH) mechanism can solve this shortcoming. According to this mechanism there should be a spin zero particle, the Higgs boson, associated with the Higgs field, and it should be possible to observe it in the high-energy collisions at the LHC.

Conservation laws play an important role in physics. For many conservation laws used in physics there is a deep connection with a symmetry principle. For example, if the laws of physics are to be independent of time then energy must be conserved (Noether's theorem [2]). If the laws of physics are to be independent of the position in the Universe then conservation of momentum is required. The same idea links invariance to rotation to the conservation of angular momentum. However, there are some conservation laws such as quark number (number of quarks minus number of antiquarks) and lepton flavour (e, μ and τ) conservation where there is no link to a known symmetry. This may imply that these conservation laws are not exact, or perhaps the relevant symmetry has not yet been established.

Coulomb's law was first published in 1785 by Charles Augustin de Coulomb, but the inverse distance square law was also known by other scientists before that. Coulomb's law, well known in today's classrooms, was an important step in understanding electric phenomena. James Clerk Maxwell took this description to a dramatically different level with his equations describing simultaneously both electric and magnetic phenomena.

Maxwell's equations were ahead of their time in several ways. They united electricity and magnetism in one theory and they treated effects at the speed of light correctly. This was natural, as light is an electromagnetic phenomenon, but during the 20th century making classical descriptions obey special relativity was a large and often difficult task. Maxwell's description of electromagnetism was relativistically correct from the very beginning. The next aim was to make it compatible with quantum mechanics. After the Second World War the theoreticians Feynman, Schwinger and Tomonaga managed to overcome the mathematical difficulties by making the electromagnetic interactions à la Maxwell compatible with quantum mechanics. This resulted in quantum electrodynamics (QED), which now stood on the two solid pillars of special relativity and quantum mechanics. They received the Nobel Prize in 1965 for their work on formulating QED [3].

We can represent the electromagnetic interaction between two electrons by a Feynman diagram as shown in figure 2. This is a schematic illustration of the two incoming electrons interacting via the exchange of a virtual photon which interacts with the electric charge of the electrons. The photon is called virtual because it is only able to exist because of Heisenberg's uncertainty principle. The virtual photon has all the same properties as a photon except for its mass, which is not zero. In this simplest interaction diagram there are four external lines where the incoming and outgoing electron momenta can all be measured. No properties of the internal photon line can be measured and so all momenta are allowed and the direction of the virtual photon is unknown. Any point on the diagram where an external particle interacts with a force carrier is called a vertex. All conservation laws, including electric charge, that apply in the interaction under study must also apply at each vertex of the Feynman diagram.

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In order to calculate the electromagnetic interaction between charged particles more precisely, it is necessary to include other more complex diagrams as well as those already discussed. These diagrams can include additional vertices, photon emissions or both. The coupling constant determines the strength of the interaction. The coupling constant for the emission of an extra photon in the Feynman diagram in figure 2 is 1/137. This factor is known as the fine structure constant. The small value of the coupling constant means the more complicated diagrams, which in principle need to be included, only make small contributions to the predicted result. The simplest diagrams make the largest contribution. However, as some of the comparisons in QED between experimental measurements and theoretical predictions have been made at the level of one part in 10¹², many of the more complicated (higher order) diagrams need to be computed to reach this level of precision.

In addition to being a diagrammatic visualization of a particle interaction, the Feynman diagrams describe the particle interaction quantitatively. The mathematics behind a diagram is very detailed. Each external line represents a real particle, and the Feynman rules define how the propagation of the particle is described mathematically. Each internal line describes a virtual particle which cannot be observed. Each vertex describes the strength of the interaction in terms of the strong, weak or electromagnetic coupling constant. Each diagram has several external particle lines and at least two vertices. All possible energies and possibilities have to be integrated over to get the probability of the process.

We can use very similar techniques to make calculations and predictions for interactions proceeding by the strong force where the gluon is the force-carrying particle. The gluon only interacts with particles that have colour charge. The simplest Feynman diagram to represent the interaction between two quarks is shown in figure 3. In this case the colour charge on each quark is changed after the interaction, as indicated on the diagram. In QED the electric charge on the electron was not changed by the photon but in the quark–quark interaction the strong charge always changes. As colour charge must be conserved at each vertex of the interaction, this means that each gluon must carry a colour/anticolour combination.

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Combining three colours and three anticolours gives rise to nine independent combinations. Eight of them, the octet, are the eight colour–anticolour combinations of the gluon. Two of these colour–anticolour states have no net colour charge. The remaining colour–anticolour combination is a singlet which is an equal superposition of the three red–antired, green–antigreen and blue–antiblue components with no net colour charge. Hadrons, which are colourless, are described by the singlet state.

There is an important difference between the gluon and the photon. The photon has no electric charge but the gluon carries colour/anticolour charge. A photon cannot interact directly with another photon by the electromagnetic force because it is neutral whereas a gluon–gluon interaction can occur via the strong force. This is more important at the LHC as many of the most interesting proton–proton collisions arise from gluon–gluon interactions. Each proton consists of two up quarks and a down quark but around 50% of the proton's momentum is carried by gluons which hold the quarks together. A simple Feynman diagram to illustrate a gluon–gluon interaction producing a down quark–antiquark pair is shown in figure 4. After the collision the colour charges of the produced quark and antiquark continue to interact via the strong force. Each energetic quark and antiquark shows up in the detector as a group of hadrons. These hadrons are moving in a similar direction to the emitted quark or antiquark and are known collectively as a jet. The coupling constant for emission of an extra gluon is much higher for the strong force than for photon emission in QED. Its value is typically 1/10 rather than the 1/137 for photon emission. This means that the more complex Feynman diagrams with extra gluons play a more important role in calculations of the strong force. When quarks are very close together, or equivalently interacting with very high energy, then the force between them is very small. This is known as asymptotic freedom and is convenient when studying the quarks which are close together inside a proton because the quarks barely interact with each other. By contrast, when a quark tries to move away from the other quarks in a proton the energy required to do this becomes infinitely large and no single quark has ever been observed due to this quark confinement.

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The weak force is very important as it is a universal force which affects all particles. Particles carry two types of weak 'charge' known as weak isospin and weak hypercharge.

The observed bosons responsible for all the properties of the electromagnetic and weak forces are the charged W⁺ and W⁻ bosons and the neutral photon and Z⁰ boson. The large masses of the W and Z bosons were predicted by the electroweak theory which unified the very different electromagnetic and weak forces. In 1983 these bosons were discovered by experiments at CERN studying proton–antiproton collisions [4] and led to the Nobel Prize for Rubbia and van der Meer in 1984 [5].

As all particles interact with the weak force, then each particle can emit W or Z bosons. This means that these more complex Feynman diagrams with W and Z radiation also need to be added to the simplest diagrams. However, as this force is weak, these corrections to calculations are generally small but at high energies these processes become more important.

The W and Z bosons carry weak isospin and weak hypercharge and so they can interact with each other as well as with other particles. In the standard model these self-interactions between the force carriers are important and vertices involving two, three and a maximum of four bosons are predicted to occur.

In figure 5 two types of interactions between a neutrino and a down quark are represented as Feynman diagrams. In figure 5(a) the muon neutrino and the down quark remain after the interaction and the force carrier is the Z⁰ boson. The first evidence for this process, which is extremely difficult to detect, was identified in a bubble chamber experiment at CERN in 1973. This is called a weak neutral current interaction as the Z boson has no electric charge.

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In figure 5(b) a weak charged current interaction occurs via the exchange of a charged W boson. In this case the muon neutrino changes to a negatively charged muon after the interaction and the −1/3 charge down quark changes to a +2/3 charge up quark to conserve electric charge.

All these force carriers are virtual particles and cannot be directly identified when carrying out this task. These force-carrying virtual particles can be shown from energy and momentum conservation to have masses different to the 'real' particles and are described as off mass shell. These all have a limited lifetime because of the uncertainty principle as they are 'living on' borrowed energy from the vacuum.

Real photons with a mass of zero are stable and we observe many of these in our daily surroundings. These include photons travelling from the Sun and those in the cosmic microwave background which have survived from the early Universe. There is not enough energy in our natural surroundings to create massive W and Z bosons. In a high-energy particle collider we can convert the kinetic energy of the beam particles to create in a very small region enough energy to create a real W or Z boson. These are produced with their correct mass after conserving energy and momentum so they do not have to disappear quickly because of the uncertainty principle. However, massive particles have many different ways of decaying and so they do decay very quickly (10⁻²⁴ s) and we can only 'observe' real W and Z bosons indirectly through their decay products.

QED is a quantum field theory where each particle is described by a wavefunction which contains the information that can be used to calculate the probability of finding the particle at different locations. Waves are defined by their amplitude and wavelength and the horizontal position of a peak on the wave is usually referred to as the phase of the wave (in early studies the word gauge was used instead of phase). QED is a local gauge invariant theory, meaning that the laws of physics are independent of local changes of the phase at all positions and times in the Universe. With this requirement the theory requires that the photon, the force carrier of the electromagnetic interaction, has zero mass and it defines how it interacts with charged particles.

The success of QED and this rather abstract idea of local gauge invariance then led to the application of local gauge invariance to the strong and the weak forces. The force carrier of the strong interaction is the massless gluon, while the force carriers of the weak interaction are the massive W and Z particles. The idea of gauge invariance had to be complemented with another mechanism to explain the massive force carriers which will be discussed in the following sections. There was one other crucial problem that needed to be solved before the standard model could be used to make numerical predictions that could be tested by experiments. Early QED calculations were plagued by infinities which were later removed by a technique called renormalization. Proof that this technique also worked for the fields of the standard model meant that one could make meaningful calculations. This was a great achievement and resulted in the Nobel Prize for t'Hooft and Veltman in 1999 [6].

We saw earlier that all the quarks and leptons have a spin of one half. In the quantum world if we measure the spin projection of a spin half-particle along any axis then the result must be either 1/2 or −1/2. If this spin projection is positive (negative) along the direction of the particle then we define it as a right- (left-) handed particle. Most particles appear as both left-handed and right-handed particles but neutrinos are the exception. All neutrinos are left-handed and all anti-neutrinos are right-handed. One important ingredient of the standard model is that the left-handed quarks and leptons all interact as doublets such as (up, down) and (electron, electron neutrino) and yet any right-handed quarks and leptons interact as singlets. These doublets and singlets have different interactions with the W and Z bosons. The most extreme example is that the quark or lepton singlets do not interact with the W boson at all.

Quantum field theories provide the theoretical framework of the standard model. The standard model is a local gauge theory in which the force carriers are expected to have zero mass. However, for the weak interaction this is obviously not the case. The W and Z particles have a mass of 80 and 91 GeV c⁻². On the other hand, the force carrier for the electromagnetic force, the photon, and the force carrier for the strong force, the gluon, are both massless. The mass problem is therefore primarily concerned with the force carriers of the weak force. The W and the Z bosons break the symmetry of the zero mass particles.

The completion of the standard model required a way of including massive force carriers without breaking the symmetries that were crucial to its predictive power. It was found that via spontaneous symmetry breaking this could be achieved via the Higgs or the BEH mechanism. There are many different examples of spontaneous symmetry breaking in physics and in everyday life. If a magnet is heated it loses its magnetism. When it cools, the alignment of domains again produces a magnet but the direction of the alignment is different. If a pin is held vertically and downward pressure is applied the pin will be bent but the direction of this bend is random and not predictable. Starting from the top of a hill you have an almost infinite number of ways to walk down to ground level, where you have lower potential energy. The situation is symmetrical—you can choose any of the equivalent possible directions. When you choose one way down you have broken the symmetry.

Kobayashi, Maskawa and Nambu [7] shared the 2008 Nobel Prize for the discovery of the mechanism of spontaneous broken symmetry (Nambu) and for the discovery of symmetry breaking, which predicts the existence of at least three families of quarks, described by the CKM matrix (Kobayashi and Maskawa).

The simplest way to introduce spontaneous symmetry breaking into the standard model is to add a new doublet of quantum fields. As each of these two quantum fields is complex, with a real and imaginary part, this is equivalent to adding four new degrees of freedom or parameters into the theory. It can be considered that three of these are used to give mass to the W⁺, W⁻ and Z bosons and the remaining degree of freedom is responsible for the Higgs boson.

In most cases where the potential energy varies with the size of a field the minimum potential energy occurs where the field value is zero. This can be described by a simple curve with a single minimum at the centre. In this case the state with the lowest potential energy (often called the vacuum expectation value) occurs when the field is zero.

In contrast, the way that the potential energy varies with the magnitude of the Higgs field is very different. It is usually illustrated as a symmetric Mexican hat or wine bottle shape and a potential energy (V)–Higgs field (Φ) diagram is shown in figure 6. The vertical axis is the magnitude of potential energy and the horizontal axes represent the magnitude of the Higgs field. When the Higgs field is zero (point A) then the potential energy is a maximum. When the spontaneous symmetry breaking occurs, the potential energy is reduced to one of the minimum points around the valley (point B) and the Higgs field becomes nonzero. The radius of the bottom of the valley is related to the magnitude of the Higgs field. In the standard model this can be calculated from the strength of the weak force and is approximately 246 GeV.

The other parameter that defines the shape of the potential is the curvature as you move away from the valley minimum. This fixes the mass of the Higgs boson and is not predicted in the standard model.

It is thought that at the beginning of time the Higgs field was zero, all particle masses were zero and all particles travelled at the speed of light. As the Universe cooled the potential energy of the Universe decreased and the Higgs field became nonzero (figure 6). In this process the Higgs field breaks the symmetry. The spontaneous breaking of the symmetry and the Higgs field ending up at the energy level where the potential energy has its minimum results in the Higgs field reaching a constant and stable value. The spontaneous symmetry breaking has the consequence of creating a nonzero Higgs field everywhere in space. The new presence of a nonzero Higgs field in the Universe interacting with particles results in most of them acquiring mass. In the standard model the breaking of the symmetry must be responsible for the masses of the W and Z particles and the same mechanism may also be responsible for the masses of all the fermions.

The presence of a Higgs field throughout the Universe is able to solve the theoretical problems of the standard model in the presence of massive force carriers, but does it exist? One way to investigate this is to try to excite the Higgs field and produce a Higgs boson which is an excitation of the Higgs field. There is a strong coupling of the Higgs field to massive particles and so one possibility is to produce massive particles in a collision at high energy to excite the Higgs field.

This is what is done in collisions at the LHC, but how can we detect the Higgs boson? It is predicted to decay in a fraction of a second before leaving any trace in a detector and so we need to study its decay products to deduce its presence. There are standard techniques that can be used to identify short-lived particles from their decay products which have been used to identify short-lived hadrons and the W and Z bosons. We add the energies of the decay products to get the energy of the short-lived parent, the momenta of the decay products (as vectors) to get the momentum of the parent and then use the relativistic equation E² = p² + m² to extract the mass of the parent.

As all the particles come from the collision point, there is no direct evidence that the 'decay products' really come from one parent and they could all be produced independently. However, when the mass is calculated for a number of collisions the mass results will all be close to the same value if they are from a massive parent and this will appear as a peak at the mass of the parent. If the 'decay products' are unconnected to each other the calculated masses will be random and not produce a peak.

The standard model does not predict the mass of the Higgs boson but it does predict how often it should be produced in high-energy collisions and all its decay properties once the mass is known. These are crucial ingredients in the Higgs boson search as the work can be focused on the decays to particular particles, which should be produced at a certain rate.

The particle that was recently discovered in the high-energy proton–proton collisions at the LHC at CERN currently seems compatible with being the Higgs boson. Before starting the LHC the following was known about the Higgs particle. Its mass must exceed 114 GeV, otherwise it would have been detected at the Large Electron Positron Collider (LEP). The precision results from LEP and other accelerators also showed that the Higgs mass should be smaller than around 190 GeV to be consistent with the standard model. Experiments at the Tevatron in the USA have recently excluded masses for the Higgs particle in the range 162–166 GeV.

At present the ATLAS and CMS experiments at the LHC have excluded almost the whole expected Higgs mass region except for a narrow region around 125 GeV. On 4th of July 2012 the discovery of a new particle with a mass near 125 GeV, decaying into two photons or four charged leptons, by both the ATLAS and the CMS experiments, and consistent with the Higgs boson, was announced at CERN. The properties of the particle agree with those expected for the Higgs particle, the missing ingredient of the standard model. Figure 7 shows a particle collision with a candidate for a Higgs boson observed in the ATLAS detector [8].

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The standard model has been studied extensively for many decades. Today the LHC experiments at CERN are exploring proton–proton collisions at an energy of 8 TeV, the highest collision energy ever achieved. The collision energy is so large that events with W and Z particles are produced in very large numbers. The energy should also be sufficient to produce the elusive Higgs particle.

Many of the experiments provide data from real particle collisions for students and teachers to explore. The ATLAS experiment [8, 9] and other LHC experiments [10] have released a large number of events for educational purposes. These have been used in many International Masterclasses [11]. Embedded among the real events containing W and Z particles are also a small number of simulated Higgs particle events that students can search for. The education projects using ATLAS data described in [12, 13] have also contributed to the particle physics resources available to teachers and students.

When protons collide at high energy, it is primarily the quarks and gluons that govern the interaction. The most important features of these interactions have been illustrated in some animations described in [14].

Each collision at the LHC produces many particles which are identified and measured by huge multi-layered detectors that almost surround the interaction point. The ATLAS experiment visualizes the products of each collision with software which can also be used for educational purposes by teachers and students. The Minerva package [15] provides collision data and information to help classify the particles produced in each collision. The many layers of the detector allow the identification of particles, including electrons and muons, and the detector signals can also be used to measure the momentum of each particle.

Massive W and Z bosons decay so quickly that they do not leave any trace in the detector but their presence can be inferred from their very energetic decay products. If the Higgs boson exists it will also decay in a similar time and so the current searches for it also rely on detecting its decay products in a similar way. Minerva provides data samples that include collisions that have produced W and Z bosons and these can be recognized and reconstructed. There are also a few simulated Higgs boson decays included in the files which can be used to illustrate features and challenges of the current Higgs searches at the LHC.

Contrary to the extremely short-lived W, Z and Higgs particles, the strange particles (particles with at least one strange quark) have lifetimes of the order of 10⁻¹⁰ and 10⁻⁸ s, making it possible to see their decay in the inner detector of the ATLAS experiment. The neutral strange particles K⁰ and Λ⁰ travel several centimetres before decaying into a positively and a negatively charged particle. The production and decay of these particles can also be explored with the Minerva event display and educational tools [15].

The standard model is presently able to describe all the known high-energy processes involving quarks and leptons. At the CERN Large Hadron Collider experiments are searching for the Higgs boson, the missing component of the standard model. The results from the ATLAS and CMS experiments show that the properties of the recently discovered particle with a mass near 125 GeV are consistent with the Higgs particle. ATLAS and other experiments provide particle collision data and education packages on the web with which processes described by the standard model can be explored. These data include collisions where the force carriers of the electroweak interaction, the W and Z particles, are produced in the high-energy proton–proton collisions. At present a few simulated Higgs events are embedded among the real events, but in the future real Higgs events could be included too.

The authors acknowledge many useful discussions with colleagues on many of these topics.