1.2 Beyond the Standard Model
The experimental success of the Standard Model and its main subcomponents QED, QCD, and EW unification and symmetry breaking is clearly incontestable, ranging from the confirmation of theoretical prognostication of the existence and some the properties of new particles (e.g. \(Z\), \(W^{\pm}\) and Higgs bosons or top quark) to the agreement of precise predictions with meticulous experimental observations. The fine structure constant \(\alpha\) at zero energy scale is an example of the latter, with its experimentally determined value consistent among independent physical measurements when the Standard Model based theoretical correction are accounted, down to 12 significant digits [28], [29]. In addition to describing natural phenomena with unprecedented accuracy, the SM is a self-consistent theory that provides non-divergent predictions at the highest energy scales probed to date.
1.2.1 Known Limitations
In spite of the successes mentioned above, several shortcomings of the Standard Model are known and hence the theory is not considered as a complete theory of natural phenomena at the most fundamental scales. Those concerns include unexplained empirically observed phenomena such as gravitational interactions, neutrino masses or dark matter particle candidates, theoretical considerations regarding the stability of vacuum or aesthetic principles such as naturalness. Hence, it is presumed that the Standard Model is an effective theory, able to successfully describe fundamental processes within a range of energies as an approximation of a more complete unified theory. For completeness, the main empirical and theoretical concerns are summarised:
Omission of gravitational interactions: the current formulation of the SM completely disregards the effect of gravity in fundamental interactions, because no consistent quantum descriptions for gravity matching the experimental predictions of the well-established theory of general relativity [30] have been developed to date. While several theoretical efforts are ongoing, such as loop quantum gravity [31] or string theory [32], the coupling for gravitational interactions at the current experimental high-energy reach is expected to be more than 30 times weaker than for weak interaction, and hence can be safely ignored when computing theoretical predictions.
Lack of a viable Dark Matter candidate: through a variety of astrophysical observations, including the observed galaxy rotation curves [33], gravitational lensing [34] and the Cosmic Microwave Background (CMB) [35], there is clear evidence indicating the presence of more gravitational interacting matter in the universe than what is expected by contrasting with the electromagnetic spectra. It has been thus estimated that about \(85\%\) of massive existing matter in the universe does not notably interact with ordinary matter and radiation, and therefore is referred as Dark Matter. While its particular nature is still unknown, scientific consensus seems to favour long-lived cold non-baryonic matter as an explanation, predominantly weakly-interacting massive particles (WIMPs). The three neutrino types are the only WIMP within the Standard Model, but considering the known upper limits on their masses, they can only account for a very small fraction of the total mass of dark matter in the universe.
Unexplained matter-antimatter asymmetry: as discussed in Section 1.1, each matter particle in the Standard Model has an identical anti-matter possessing opposite quantum numbers. Because pair creation and annihilation processes are symmetric, but our universe is manifestly dominated by what we refer as matter, some asymmetric interaction processes ought to exist. Within the SM, some electroweak processes are known to violate CP-symmetry and potentially explain a small part of the observed matter-antimatter asymmetry. New unknown CP-symmetry processes, potentially through interactions not included in the SM, are needed to resolve the mentioned disparity.
Origin of neutrino masses: the Standard Model was developed assuming that neutrinos were massless, yet is currently well established that neutrinos oscillate between different flavour eigenstates [36], [37], implying that flavour states mix and hence that neutrino masses are very small but different from zero. The SM Lagrangian can be extended to account for the masses of neutrinos in a similar fashion to what is done for leptons and quarks, but their Yukawa coupling has to be much smaller than of any of the other particles, and it requires the existence of very weakly interacting right-handed neutrinos. An alternative mechanism for including neutrino masses exists, and it is based on assuming that these particles are Majorana fermions and hence they are their own anti-particle. This hypothesis is currently being experimentally tested. It also worth noting that in order to explain the smallness of neutrino masses in a principled way, the Seesaw mechanism [38] has been proposed, which implicitly assumes that the SM is only a low-energy scale effective theory of a more complete unified theory.
Mismatch between vacuum energy and Dark Energy: in addition of providing evidence for dark matter, astrophysical observations such as studies of the properties of the Cosmic Microwave Background [35] or the redshift of type Ia supernovae [39], consistently point to the hypothesis of an accelerating expansion of the current universe. The simplest way to account for this in cosmological models is to include a cosmological constant, which should be understood as an intrinsic energy density of the vacuum, exerting a negative pressure and therefore driving the observed expansion of the universe. In fact, in order to reconcile the theoretical models with experimental observations, about \(68\%\) of the total energy in the present universe would correspond to this type of unknown energy density, generally referred to as Dark Energy. In most quantum field theories, such as the Standard Model, some non-zero zero-point energy originating from quantum fluctuations is expected. However, modern attempts to predict energy densities from QFT are at variance with the observed energy vacuum energy density, some of them differing by 120 orders of magnitude [40].
Naturalness, hierarchy and fine-tuning concerns: as discussed at the beginning of Section 1.1, the SM can be thought of the most general theory based on a set symmetries, and its 19 parameters (or 26 accounting for neutrino masses and mixing angles) are not obtained from first principles but measured experimentally. Having such a large number of free parameters and observing large differences among their relative magnitude has been viewed as a theoretical concern from an aesthetic perspective. A related issue is why the electroweak energy scale (epitomised by the Higgs mass) is much smaller than the assumed cut-off scale of the SM, where gravitational interactions become relevant at \(M_{\textrm{Planck}} \approx 10^{19} \textrm{GeV}\), which is generally referred as the hierarchy problem. In the absence of New Physics or additional interaction mechanisms, the only way to obtain the observed Higgs mass from the bare Higgs mass (at zero energies) is through a very precise cancellation of divergences, which is regarded as an unnatural or fine-tuned property of the SM theory.
Other possible issues, in some cases related with those discussed, have also been raised. One of them is the apparent vacuum meta-stability [41] and other the so-called strong CP problem [42]. Many of these questions can be clarified once the higher precision measurements of the SM become available, which are mainly obtained in particle collider experiments.
1.2.2 Possible Extensions
The known limitations stated in the previous section have motivated the development of alternative theories for describing fundamental interactions. Given the quantitative success of the Standard Model, most of the known proposed theoretical models are either extensions of the SM or its associated predictions can be effectively reduced to those of the SM at the energy range current being explored in particle physics experiments. The set of alternatives that have been proposed is too substantial to be exhaustively listed here, especially given that many of the alternatives include additional free parameters that greatly modify the expected theoretical observables.
1.2.2.1 Precision Measurements of the SM
Due the existing large space of alternatives to the SM from a theoretical standpoint, the exploration of all possibilities through dedicated searches becomes unattainable. An alternative way to possibly obtain quantitative information pointing to extension of the SM is to measure its most relevant observables with high precision. If significant discrepancies are found between the experimental measurement and the theoretical prediction of those observables, it could be evidence pointing to New Physics outside the SM.
1.2.2.2 Effective Field Theories
In addition to carrying out precision measurements and model-specific searches, there exists a practical way to consider possible extensions due to New Physics phenomena occurring at a higher energy scale \(\Lambda\) than the one being probed, which will be denoted by \(E\). The model-independent approach often referred to as effective field theory (EFT) [43], [44] allows to compute observables by extending the SM Lagrangian terms from Section 1.1 with additional operators: \[ \mathcal{L}_\textrm{EFT} = \mathcal{L}_\textrm{SM} + \sum_i \frac{c_i} { \Lambda^{d_i - 4}} \mathcal{O}_i \qquad(1.31)\] where \(\mathcal{O}_i\) are referred to as effective operators, describing the characteristics of the new interactions that are considered in the extended theory and \(c_i\) are the the EFT or Wilson coefficients that parametrise the strength of those new interactions. The integer \(d_i\) defines the dimension of the operator \(\textrm{dim} \left( \mathcal{O}_i \right) = \left[ E \right]^{d_i}\), and while in principle an infinite set of operators with any dimension \(d_i > 4\) can be considered, their effects is expected to be suppressed by \((E/\Lambda)^{d_i - 4}\) thus high-dimensional operators may be neglected when studying the dominant effects of an EFT extension of the SM.
If all the EFT coefficients \(c_i\) are zero or the new energy scale \(\Lambda\) is infinite, the EFT theory reduces to the SM Lagrangian. Instead, if \(\Lambda \approx E\), the effective approximation in Equation 1.31 does not hold, and the interactions have to be realistically modelled using a complete theoretical description of the New Physics scenario under study. While in general effective field theories are not renormalisable, observables and higher-order corrections can be computed, because of the well-defined cutoff energy scale \(\Lambda\). The best-known example of an EFT that has been used in practice is Fermi theory, which is a useful simplification to compute EW observables at low-energies \(E \approx 10\ \textrm{MeV}\) rather than an extension of the SM, given that the detailed structure of electroweak interactions due to \(\textrm{W}^{\pm}\) boson mediating \(\beta\) decays was unknown at the time.
At the LHC and other collider experiments, the main use case of EFT is to describe generic extensions of the SM that could arise due to New Physics at energy scales that are not directly accessible. From an experimental standpoint, the goal is thus to constraint the values of the EFT operator coefficients using experimental data. Because the for \(d_i=5\) the only possible operator is relevant for neutrino phenomenology [45], the set of Lagrangian operators of interest at collider experiments often corresponds to \(d_i=6\) dimension operators. The large set of possible dimension six operators can be greatly reduced by requiring that the main experimentally verified properties of the SM are respected, such as the gauge and Poincaré symmetries, or baryon number conservation. In Chapter 5, a subset of dimension six EFT operators are used to study non-resonant extensions of Higgs pair production in a model-independent manner.