## Abstract

The Standard Model is the theory used to describe the interactions between fundamental particles and fundamental forces. It is remarkably successful at predicting the outcome of particle physics experiments. However, the theory has not yet been completely verified. In particular, one of the most vital constituents, the Higgs boson, has not yet been observed. This paper describes the Standard Model, the experimental tests of the theory that have led to its acceptance and its shortcomings.

## 1. Introduction

The Standard Model of particle physics is the theory used to describe the interactions of fundamental particles (or fermions) and fundamental forces (which are conveyed by particles called bosons). In total, it describes the behaviour of 12 fermions (six quarks and six leptons, which are often arranged into the three ‘generations’ shown below owing to the similarities observed in behaviour between generations), under the electromagnetic, weak and strong forces. Each fermion has an antimatter counterpart, which the Standard Model treats in the same way as its matter equivalent. The theory is remarkably successful at predicting the outcome of experiment. However, it is also an incomplete theory. It has not yet been completely verified, and it lacks explanation for many features of the universe. The three generations of quarks are: up and down; charm and strange; top and bottom. The three generations of leptons are: electron and electron neutrino; muon and muon neutrino; tau and tau neutrino.

This paper provides a brief survey of the Standard Model (a fuller description of the theory may be found in earlier studies [1–3], and various textbooks on the subject). The structure of the Standard Model is described in §2. Section 3 outlines the experimental tests of the theory that have led to its widespread acceptance. Section 4 provides an introduction to the known shortcomings of the Standard Model, which will be explored in greater depth experimentally and theoretically in other papers in this issue. Finally, §5 concludes this survey.

## 2. The structure of the Standard Model

The Standard Model is a quantum field theory that describes interactions between fundamental fermions and the electromagnetic, weak and strong forces. These interactions are encapsulated in Lagrangian equations,^{1} and the action of each force is described by a Lagrangian of similar form.

The precise form of the equations arises from imposing a condition known as ‘local gauge invariance’. Mathematically, this corresponds to requiring that the Lagrangian remain invariant when the fermion wave function is multiplied by an arbitrary phase, which can vary with space and time. Physically, this corresponds to requiring that predictions made with the Standard Model have the same value anywhere in the universe. This very general requirement places strict constraints on the form of the equations that can satisfy it. The Lagrangian can only remain invariant if the forces interact in a particular way, and if the bosons conveying the force are massless. This reasoning was a great triumph when applied to the quantum theory of electromagnetism, as it predicted the existence of the massless photon. It is compelling that the same argument can also apply to the weak and strong forces, whose Lagrangian equations are derived in a similar way. The Standard Model Lagrangian is given by the sum of the Lagrangians describing the action of each force, which differ only in the details of the individual force fields and their interactions. For example, electromagnetism has one source, electric charge, and one boson, the photon; the weak force has two weak charges and three bosons, W^{+}, W^{−} and Z; the strong force has three strong charges and eight bosons (gluons).

### (a) Drawbacks

Although the idea behind the Standard Model is simple and elegant, real life behaves somewhat differently. Any observed differences to the boson and fermion behaviour assumed by the Standard Model must be added to the theory to ensure that it describes reality. In fact, the Standard Model predicts very little about the forces and particles whose behaviour it describes, and information on force strengths, particle mass and so on must be input from experimental measurements.

Some of the biggest differences are found in the Standard Model description of the weak force. Here, the observed bosons are massive, rather than massless. Only left-handed fermions (those with their spin aligned against their direction of motion) are observed to participate in weak interactions. Quarks participating in weak interactions are further observed to be quantum mechanical admixtures of the quarks that participate in strong interactions. Neutrinos exhibit a similar mixing behaviour. The fermion behaviour is parametrized and imposed on the theory. Including massive W and Z bosons requires adding a new field—the Higgs field—to the Standard Model Lagrangian.

The Higgs field, and its associated Higgs boson, confers mass to W and Z bosons and fermions, in a manner that (by construction) conserves local gauge invariance. The importance of the Higgs to the Standard Model cannot be overstated. Besides conferring mass, the Higgs mechanism introduces a degree of unification between the electromagnetic and weak forces. The observed, massive W and Z bosons turn out to be mixtures of the original, massless pre-Higgs weak and electromagnetic bosons. The Standard Model does not predict the mass of the Higgs boson or the individual masses of any particles, but it does predict the ratio of the Z and W boson masses. Intriguingly, this is linked to the ratio of the weak and electromagnetic force strengths. The Higgs mechanism also ensures that Standard Model predictions are finite. Without it, predictions make no sense.

Although the Standard Model is based on a simple idea, it can be complicated to use and calculate. Nobel prizes have been awarded for demonstrating that the theory is renormalizable (see http://nobelprize.org/nobel_prizes/physics/laureates/1999/) (i.e. that realistic predictions can be made), and that the theory describing the strong force works (see http://nobelprize.org/nobel_prizes/physics/laureates/2004/). Historically, most precision tests of the Standard Model have been carried out in the electroweak sector, as these interactions are usually easier both to predict and to measure experimentally.

## 3. Tests of the Standard Model

The tests of the Standard Model that have been carried out to date fall into three categories: testing the validity of assumptions regarding fermion and force behaviour; using the Standard Model to extract values for any unknown parameters from a variety of measurements, and checking that they are consistent; and checking the internal consistency of the overall framework by comparing predictions with measurements.

### (a) Validating assumptions

Some assumptions in the Standard Model are implicit. For example, the theory has been constructed, assuming that there are three generations of fermions, and that the different leptons have the same electroweak couplings (a feature known as ‘lepton universality’). The weak and strong forces are also assumed to be non-Abelian in nature (in other words, that the carriers of the force can interact with themselves, as well as with fermions), a feature that arises because of the underlying mathematical framework assumed for the Standard Model. These assumptions can be tested experimentally.

Experiments at the Large Electron–Positron Collider (LEP) have provided proof that these assumptions are valid. LEP was an electron–positron collider that operated at energies varying between the mass of the Z boson up to 209 GeV, in the 1990s. Lepton universality was tested by measuring how often Z bosons decayed to pairs of electrons, muons and tau leptons. In each case, the strength of the boson–lepton couplings could be extracted, and compared with each other and the Standard Model prediction. These were found to be consistent with each other [4], proving that universality is a valid assumption, at least at the energies tested. The Standard Model prediction for the couplings can also be compared with the experimental determination, and was consistent for each generation.

The assumption that there are three generations of fermions rests on the fact that no others have yet been observed. Although the Standard Model does not predict or constrain the number of generations, it can provide a check of the number of generations that are consistent with the experimental data. The width of the Z boson production cross section, when plotted as a function of the energy of the experiment, is related to the number of different possible final Z decay states. The width of this ‘lineshape’ can be predicted theoretically. It is the sum of the visible Z decay contributions, which can be measured directly, and the invisible contributions where the Z decays to undetected pairs of neutrinos. The latter are unmeasured, but can be predicted, such that the only unknown is the number of neutrino generations. Each generation contains one neutrino, which has the lowest mass of all fermion species. It seems reasonable that if any more generations exist, then the corresponding neutrino may also be light and a potential decay product of the Z boson. The experimental data were fitted to a prediction of the lineshape, where the number of neutrino generations was allowed to vary. The fitted number of neutrino generations is consistent with the three observed [4] (figure 1).

The non-Abelian nature of the weak and strong force can be checked by comparing measured production rates with predictions, in processes that involve contributions where bosons self-interact. For example, the DELPHI experiment measured the rate at which Z bosons produced four jets of particles (which arise from quarks or gluons) [5]. This can occur via several processes, one of which involves self-interaction of gluons. Coupling strengths between quarks and gluons, and gluons and gluons were derived from the experimental measurement. The data showed that gluon self-interactions contributed, and that the force was therefore non-Abelian in nature.

A similar test was applied to the weak force, by measuring the rate at which pairs of W bosons were produced at LEP. The measured production rate was compared with predictions that contained contributions from Abelian and non-Abelian processes. Only the prediction containing all non-Abelian contributions described the measurements made in data, proving that the weak force too is non-Abelian in nature [6].

### (b) Consistency of input parameters

The Standard Model parameters that quantify force strengths, quark mixing and so on can be extracted from the experimental measurements that are sensitive to them. The consistency of the Standard Model description can be tested by comparing the values of a given parameter extracted from different observables, with different theoretical dependencies.

The Standard Model does not predict the strengths of any force, and requires this information to be input from experimental measurements. The strong force strength, for example, can be determined from a measurement of any process that produces quarks and gluons. All measurements so far give consistent values (see Nakamura *et al*. [7] for an experimental review). The dependence of the strong force strength with energy can also be tested, and is correctly described. This app- lies not just to the strong force either, but to the electromagnetic and weak forces too. The Standard Model appears to describe correctly the behaviour of them all.

The relative strengths of the weak and electromagnetic force, and the relative masses of the W and Z bosons, are described in the Standard Model by one parameter, *θ*_{W}, whose value is unknown. This parameter can also be extracted from measurements of the Z boson couplings to different fermion species. Figure 2 shows a selection of determinations from different experimental measurements. The data points, which are averages of the four LEP experiment measurements, are consistent with the shaded band (average value), and so give consistent determinations. The Standard Model predicted value is shown in the lower part of figure 2, which is given as a function of the unknown Higgs mass. This is consistent with the experimental determinations, which also place a constraint on the values of Higgs mass consistent with this value of *θ*_{W}.

The quark mixing parameters too can be determined from experiments. These are usually extracted from measurements of the decay rates of particles containing b or s quarks, measurements of the rate at which these particles turn into their antimatter partners and back again, measurements of the differences in behaviour between matter and antimatter (a phenomenon known as ‘CP violation’, where C and P refer to charge and parity, the variables that distinguish matter from antimatter) and so forth. Despite these measurements having been made in different experimental environments, and for a variety of processes, values extracted for the Standard Model parameters describing quark mixing and CP violation have consistent values. This is remarkable, for it means that the parametrization employed in the Standard Model does indeed reflect experimental data. Measurements of the Standard Model parameters that describe neutrino mixing are also underway, although these are experimentally harder to measure and are not known to high precision yet.

#### (i) Predictions

Once values for (some of) the unknown parameters have been added to the theory, the Standard Model can be used to make predictions. Figure 3*a* shows measurements and predictions of the W boson mass. This has been measured directly by experiments at LEP and the Tevatron (a proton–antiproton collider that operates at energies close to 2000 GeV), and the world average mass is shown by the shaded band. Also shown is a value that has been inferred from a measurement of neutrino scattering by a NuTeV experiment, and Standard Model predictions based on information from measurements of (i) Z boson properties and (ii) Z and top quark properties. The Standard Model predictions match the measurements well. The inferred value from NuTeV is low, and we will return to this observation later in the review.

Figure 3*b* shows the equivalent comparison for the top quark mass, which has been measured by the two experiments, CDF and D0, at the Tevatron collider. Standard Model predictions based on measurements of the Z boson, or Z and W bosons, are also shown. These predictions were in place before the top quark was discovered in 1994, and are consistent with the measured values.

With more experimental measurements than free parameters, the internal consistency of the theory can be tested. Figure 4 lists a selection of experimental observables, their measured values and the values predicted by the Standard Model. The difference between measurement and prediction is shown in the rightmost column, and it can be seen that the overall agreement is very good. The Standard Model describes experimental data very well indeed.

### (c) How well will the standard model work at the Large Hadron Collider?

How accurately will the Standard Model predict experimental measurements at the Large Hadron Collider (LHC)? Predicting processes here is more complicated than at an electron–positron collider such as LEP, as the colliding protons at the LHC are not fundamental objects. They consist of quarks and gluons (collectively known as ‘partons’), so that in a collision a parton from one proton will interact with a parton from the other. A description of the energy carried by these partons must be included in Standard Model predictions of LHC processes. Partonic behaviour is not known analytically. Instead, it is parametrized from fits to experimental data. Owing to the higher energy of the collider, interactions at the LHC occur in a slightly different kinematic region to that explored before; so the parton description must be evolved to this region to be used. However, for predictions of W and Z production, the uncertainty involved is only 1 per cent or 2 per cent over much of the region detectable experimentally. So precise predictions can be made at the LHC. If this precision can be matched experimentally, then precise probes of the Standard Model are possible. Experimental data from the LHC will determine whether the Standard Model describes data at these higher energies.

## 4. Shortcomings of the Standard Model

Despite the successful prediction and description of existing data, it is still not clear if the Standard Model is correct or not. Experimentally, there is no confirmation that the Higgs exists, and there are a couple of potential discrepancies between prediction and measurement that must be investigated. Worse still, many phenomena exist that are not described or predicted in the Standard Model. This leads many particle physicists to suspect that the theory is effective at best, and that a deeper, more fundamental explanation of the universe may be necessary.

### (a) Experimental issues

The most obvious experimental shortcoming in the Standard Model is the absence of the Higgs boson. The Higgs is an integral part of the theory, and if the theory is correct then the Higgs boson must exist and should be found.

Although a value for the Higgs boson mass is not predicted by the Standard Model, it can be constrained through its dependence on the W boson and top quark masses. The wide band in figure 5*a* shows the dependence of the Higgs mass on these two variables. The experimentally observed values of W boson and top quark masses are represented by the solid ellipse. The ellipse overlaps with the band and coincides with the lighter values of predicted Higgs mass, implying that these are favoured. Also shown, by the dotted ellipse, are inferred values for the W boson and top quark masses based on measurements made with Z bosons, and that do not have as much sensitivity to Higgs mass as the observed values.

This information can be converted into the probability of observing the Standard Model Higgs boson, as a function of its mass. This is shown graphically in figure 5*b*, where the parabola illustrates a measure of the probability of the Higgs existing as a function of its mass. As the Standard Model prediction is not absolute but relies on other variables, the additional parabolae show the effect on the probability when these values are changed within their experimental uncertainties. The most probable value of the Higgs mass that is consistent with the experimental measurement corresponds to the minimum of the parabola.

Experiments have also performed direct searches for the Higgs boson. Although no evidence of Higgs production has yet been found, these searches can rule out the masses such a boson could have. Experiments at LEP have ruled out the existence of a Higgs boson with a mass less than 114 GeV c^{−2} [8], and at the Tevatron masses between 158 and 173 GeV c^{−2} [9]. These exclusions are shown by the shaded regions. In other words, if the Higgs exists, and it is as described in the Standard Model, then we know that its mass must lie between 114 and 158 GeV c^{−2} at a 95% confidence level.

The Standard Model predicts how the Higgs will decay (and so be visible in experiments), as a function of its mass. In the lower allowed mass regions, we expect the Higgs to decay predominately to a ‘b’ and ‘anti-b’ quark pair. At the higher end, we expect it is most likely to decay to pairs of W bosons. Both decays involve very different experimental signatures and are subject to different background processes mimicking the signal. Nonetheless, experiments at the Tevatron and the LHC have used this information to devise strategies to best distinguish Higgs bosons in data, and have estimated how much data would be needed to either discover or rule out the existence of the particle. Figure 6*a* shows projections for the Tevatron, which will soon cease operations, and which has collected approximately 9 fb^{−1} of data per experiment to date. With this much data, it can be seen that the Tevatron is capable of excluding, but not discovering, a Higgs boson over the mass region preferred by the Standard Model fits [10]. Results from a similar study by the CMS experiment [11] are shown in figure 6*b* (the ATLAS experiment has similar findings [12]). The figure shows that at current LHC energies (7 TeV), all Standard Model preferred Higgs masses can be excluded with about 2 fb^{−1} of data. A discovery would require rather more data, of order 10 fb^{−1} or even more, depending on the boson mass. Nonetheless, it is hoped that the current data taking run, which continues until the end of 2012, will provide sufficient data to either rule out or observe the first indications of the elusive boson.

#### (i) What if no Higgs is found with these masses?

It is important to remember that we should only expect to find a Higgs boson at these masses, if the assumptions regarding the Higgs field in the Standard Model are correct. The Higgs field, if it exists, could be more complex than that assumed in the Standard Model, and if so, the assumptions made in the determination of the allowed masses will be incorrect.

However, there is an independent and strong upper limit on the mass a possible Higgs boson can have, if the Standard Model is to remain consistent with experimental observation. It comes from considering a process whereby W bosons scatter off each other at high energies. In the absence of a Higgs boson, the predicted cross section quickly grows and becomes unphysically large. It is unlikely that this cross section will grow so quickly with energy in practice, but the LHC has been designed to provide data at these energies so that the predictions can be tested there. In order for these predictions to make sense, the Higgs mass must be lower than about 1 TeV c^{−2} [13,14], which is a boson mass that the LHC experiments should still be able to discover, given sufficient data. So in other words, if no Higgs boson is found with a mass below 160 GeV c^{−2}, the Standard Model can survive with a modified description of the Higgs field, provided that a Higgs exists with a mass below 1 TeV c^{−2}. If the LHC rules out the Higgs at this mass and elsewhere, new physics mechanisms must come into play.

#### (ii) Other potential discrepancies

There are no significant discrepancies between Standard Model predictions and experimental measurements in the electroweak sector. However, a couple of measurements show differences when compared with prediction at a level just below three standard deviations. These are the NuTeV inferred W mass (shown in figure 3*a*), and the forward–backward asymmetry of Z bosons *A*_{FB} (see http://lepewwg.web.cern.ch/LEPEWWG/plots/summer2010/). No experimental explanations for these discrepancies have been found, and neither are yet at the level that would signify that the Standard Model is breaking down. It is hoped that measurements made at the LHC will help understand these differences—either by remeasuring the observables themselves and resolving differences, or by making measurements that better constrain the theory used to infer predictions.

### (b) Philosophical issues

Despite this experimental success, the Standard Model is an incomplete theory. It describes only experimentally observed particles, and so offers no explanation of or prediction for dark matter, or dark energy. It does not explain why there are so many generations of matter, or fundamental forces (or indeed how to describe gravity). It is tantalizing that when the force strengths are measured at higher energies they start to converge, but do not coincide, at a common value. Coincidence would signal unification of the forces, a feature that can be achieved with modified theories (for example, supersymmetry), but not with the Standard Model.

The Standard Model also does not offer explanations for many of the observed features of the universe that are described consistently. For example, the amount of CP violation in the Standard Model is described very consistently when confronted with the vast amount of experimental data described in §3*b*. However, the Standard Model does not explain the origin of CP violation, and despite this consistency, predicts a value for it that is far smaller than the amount needed to explain the observed matter–antimatter asymmetry of the universe. None of these features represent an error in the theory. Rather, they signal that a deeper understanding has yet to emerge. A further discussion of these issues can be found in Ellis [15].

## 5. Conclusions

The Standard Model is the theory that describes interactions between fundamental particles and forces in particle physics. It has proved to be very successful at consistently describing and predicting processes in an experiment. However, one prediction, the existence of the Higgs boson, has not yet been proved and, without this, the Standard Model cannot be completely verified. In addition, the theory is known to be incomplete. The LHC has been constructed to search for the Higgs boson and phenomena not yet described in the Standard Model, in order to provide the data needed to establish a more complete theory of the subatomic universe.

## Footnotes

One contribution of 15 to a Discussion Meeting Issue ‘Physics at the high-energy frontier: the Large Hadron Collider project’.

↵1 Classically, these equations describe the way in which energy is distributed in a system, and tabulate the difference between its kinetic and potential energy.

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