This completes a brief review of the mathematical formulation of the Standard Model, sum- marising only its features salient to our work in this thesis. It is a theory which contains 30 elementary particles (counting antiparticles as well as particles) and 26 parameters - 9 fermion masses, 3 quark mixing angles, 1 (Charge Parity (CP)-violating) phase, 3 gauge couplings, 1 further angle (QCD vacuum angle), 1 Higgs mass and one Higgs vacuum expectation value, 3 neutrino masses7 and 4 neutrino mixing matrix (PMNS matrix) parameters. Its development and experimental verification have been the subjects of extensive efforts throughout the past century and it has proved remarkably successful, both in extending our theoretical knowledge of fundamental particle physics and in describing experimental results at colliders and elsewhere, often up to unprecedented accuracy via loop calculations. Nonetheless, despite its obvious suc- cess, the Standard Model is known to be incomplete, having several theoretical and experimental problems and absences. A brief summary of several of the key issues is presented here; again targeted on those most relevant to our work, nonetheless more comprehensive reviews can be found in [20–22].
1. Technical Hierarchy Problem - As demonstrated in the discussion of renormalisation and running of parameters within the Standard Model, loop corrections can offer interest- ing problems for the Standard Model as a quantum field theory. In an exactly analogous manner to the computations of loop corrections to fermion masses and gauge couplings in the context of running, loop corrections to the Higgs mass must also be considered in order to determine their effect upon its mass. The Higgs boson couples to all particles with mass, therefore there are corrections from scalar loops, fermion loops and vector boson loops as demonstrated in Figure 1.3.
Figure 1.3: 1-loop corrections to the Higgs mass arising in the Standard Model; starting from the top left and proceeding anticlockwise there are corrections from scalars (i.e. the Higgs itself) due to the Higgs self-interaction, fermions due to the Higgs Yukawa couplings, and Vector bosons due to the 3- and 4-point interactions resulting from Electroweak Symmetry Breaking (EWSB).
7In fact, as only the mass squared differences of the 3 neutrinos are known, the lightest may be massless,
The Higgs however couples to particles in proportion to their masses, therefore the domi- nant correction to the Higgs mass comes from a top-antitop fermion loop. Using the Stan- dard Model Feynman rules this 1-loop Higgs mass correction can be written down and evaluated, here the correction is ultimately determined at zero Higgs momentum (q = 0) for simplicity, we introduce a UV cut-off to demonstrate that the divergence naturally pushes the Higgs mass to the largest scale in the theory:
(δm(t)h)2 =−|yt| 2 2 Z Λ d4k (2π)4T r h k + /q + m/ t (k + q)2− m2 t / k + mt k2− m2 t i ∼ −2|yt|2 Z Λ d4k (2π)4 h 1 (k2− m2 t) + 2m 2 t (k2− m2 t)2 i ∼ −2|yt|2 Z Λ 2π2k3dk (2π)4 1 (k2− m2 t) ∼ −2|yt|2 Z Λ kdk 8π2 ∼ − |yt|2 8π2Λ 2. (1.35)
Therefore the Higgs boson mass-squared receives corrections at 1-loop which are quadratic in the UV cut-off of the Standard Model and this indicates that the Higgs mass has a very sensitive dependence upon higher scale physics. This implies that either new physics should be seen very soon at energy scales being probed at colliders and elsewhere, or there must be some delicate cancellation present at the higher-than-expected new physics scale whereby new physics particle loop corrections to the Higgs mass are fine-tuned to be very close to one another and hence delicately cancel to provide a Higgs boson at the lower scale of electroweak physics. There are in fact two related but subtly different questions here - first, is the Higgs boson mass stable with respect to loop corrections? As we have just seen it is not in the Standard Model, this is called the “Technical” Hierarchy Problem. Second of all, why do these different scales arise in the first place, i.e. why is the scale of the Higgs boson (and hence electroweak physics) significantly lower than the scale of new physics even if the Higgs boson mass satisfies this Technical Hierarchy Problem (is “technically natural”)? This is the Hierarchy or Naturalness Problem. There are many potential new physics solutions to these hierarchy and naturalness issues, however often to avoid constraints (such as smallness of observed flavour-changing-neutral currents, small CP violation (CPV), precision electroweak tests or collider search bounds) the new physics in these models is pushed to higher energies, thereby reintroducing a “little” hierarchy problem between this scale and the electroweak (EW) scale.
2. Dark Matter - Evidence from a variety of astrophysical distance scales clearly indicates the presence of some non-Standard Model mass component in the universe which has so far only been detected interacting gravitationally. This evidence comes from a variety of sources; from rotational velocity curves of stars around galaxies through cluster dynamics (the Bullet Cluster being a classic example) to large-scale structure formation. There are many reviews on this subject [23, 24] so the details are skipped here. Nonetheless the conclusion is that there is an additional fundamental component of the universe not accounted for by the Standard Model. There are many potential suggestions for what this
component could be; from new Weakly Interacting Massive Particles (WIMPS) at around the electroweak and collider scale, to axions which are a very light pseudoscalar particle behaving as a collectively oscillating field (as a result of the low mass), to primordial black holes. One of many reviews on the subject is given in [25]. Many new physics models include various dark matter candidate particles, for example the Lightest (stable) Susy Particles (LSPs) of supersymmetry, see Chapter 2. The common features of these dark matter candidates are that they are either too light/heavy, too weakly interacting, or both, to have so far been detected at experiments; nevertheless this is an active area of research with many current and proposed experiments aiming to target different candidates and regions of parameter space in the search for the nature of dark matter.
3. Matter-Antimatter Asymmetry - It is observed astrophysically that the universe has a discrepancy between the number of baryons (nB) and the number of antibaryons (nB¯):
ξB = nBn−nγ B¯ = 10−9, with nγ the number of photons. However, given it is assumed
that the Big Bang produced equal numbers of baryons and antibaryons and that these were in equilibrium with photons, the question of how such an asymmetry could have emerged arises. As the universe expanded in its early history we expect γ + γ ⇔ B +
¯
B backward and forward processes to be in equilibrium as the photon temperature is initially high. As the temperature drops the forward process becomes disfavoured and so only the reverse annihilation reaction of baryon-antibaryon annihilation to photons remains, depleting the number of baryons and antibaryons in favour of photons. This continues until the baryon and antibaryon density becomes such that the reverse reaction freezes out as it eventually becomes slower than the expansion rate of the universe, as set by the Hubble scale. Therefore it is expected that the number of photons be much greater than the number of baryons and antibaryons, but also naively that the baryon and antibaryon densities in the universe are equal. In order to create a matter-antimatter asymmetry 3 “Sakharov” conditions [26] must be satisfied. In the Standard Model there is allowance for the number of baryons to exceed the number of antibaryons and so create a small asymmetry as a result of CP violation, arising via the complex phase of the Cabbibo-Kobayashi-Maskawa (CKM) matrix which relates the mass and gauge eigenstates of quarks. In addition to this measured CP violation in the quark sector, there may also be CPV in the lepton sector, nonetheless the scale of the CPV in the Standard Model is ξSM
B = 10−18, much lower than observed in the universe. As a result, new sources of CPV
are required beyond the Standard Model in order to explain the observed asymmetry. 4. Neutrino Masses - The observation of neutrino oscillations at a variety of experiments
around the world [27–32] (a review is presented in [33]) means that the 3 neutrino mass eigenstates must have different masses, i.e. ∆m2
126= 0 and ∆m2136= 0. Masses for at least
2 of the 3 neutrinos must therefore be incorporated into the Standard Model. As seen previously in this chapter, Dirac particle mass terms can be generated in EWSB of the form m(D)f ( ¯fLφfR+ ¯fRφfL), where m(D)f =
y√fv
neutrinos to obtain a neutrino mass. However this would not explain the smallness of the neutrino masses without accepting a correspondingly small Yukawa coupling for the neu- trinos. Given the right-handed neutrinos are gauge singlets however, arbitrary additional terms involving them may be added to LSM whilst respecting the overall gauge symme-
tries. Consequently, “Majorana” mass terms of the form−12M ¯νc
RνRmay be added8. Such
Majorana terms can be understood to be allowed as a result of the fact the right-handed neutrinos are singlets and so may act as their own antiparticles, such terms therefore vi- olate lepton number. Consequently the overall Lagrangian for the neutrino masses would contain Dirac and Majorana mass contributions, and the diagonalisation of the mass ma- trix can then generate a “see-saw” mechanism [36–40] (a review is available in [41]) pushing the left-handed neutrinos to small masses and the right-handed neutrinos to large masses, explaining the suppressed masses of the former and the lack of experimental observation of the latter.
5. Many other issues - In addition to these issues, there are a number of other problems and absences of the Standard Model which are listed here for brevity and in no particular order. There is no Standard Model explanation for the manner in which fermions are repli- cated into 3 near-identical copies differing only by mass, the complicated flavour structure of the Standard Model and the highly hierarchical nature of the CKM matrix are unex- plained as they are input parameters in the Standard Model, and no reason behind the apparent quantisation of the electromagnetic charges is offered. Why there are 3 gauge groups and the combined SU (2)L× U(1)Y is chiral are also not answered, furthermore
there is no inclusion of gravity or dark energy (on top of the exclusion of a viable dark matter candidate). Similarly, the strong CP problem of why the θQCD parameter in the
Lagrangian term θQCDα8πsFµνF˜µν is observed to be smaller than 10−11 (this results in no
measurable electric dipole moment for the neutron) is unexplained. Subsets of such issues may be explained by a variety of Beyond Standard Model theories, many of which are not relevant for the discussion of the work undertaken in this thesis and are therefore not detailed. Grand Unified Theories may offer solutions for charge quantisation and for the existence of 3 generations and 3 gauge groups, the strong CP problem can be accounted for via the introduction of the axion through the Peccei-Quinn mechanism [42] perhaps also offering a dark matter candidate, flavour structure may be explained by a variety of new physics theories, and the list goes on. Several reviews of some of the issues of the Standard Model and their possible solution in Beyond Standard Model theories are available, for example in [20–22].
The Standard Model therefore suffers from many issues. Nonetheless it also has great scope for improvement and adaptation, hopefully explaining many of these matters whilst retaining the successes of our predecessors in developing such an accurate description of physics up to collider scales. This has therefore led to a wide and blossoming field of Beyond Standard Model
8
physics, with many theories, adaptations and hypotheses built to resolve various subsets of these issues.
In general, the aims of these Beyond Standard Model theories are to offer some minimal extension or additional framework within which to set the Standard Model in order to provide all its innumerable successes on top of further resolutions of some of these issues and intricacies left unresolved. There are correspondingly two approaches, the first are UV complete models offering a “top-down” approach with the well-tested Standard Model physics at collider scales and lower arising naturally out of these models as a lower energy scale manifestation of some more fundamental picture. The second are those offering minimal theoretical or phenomenological extensions (“bottom-up” models) to explicitly maintain the Standard Model as a fundamental basis for particle physics but with slight modifications to rectify some of its issues and absences. The wide range of Beyond Standard Model theories will not be reviewed here, the only one of specific relevance to the work discussed will be supersymmetry (one of the most popular of these theories), a UV complete model that serves as an extension of the Standard Model at low scales and which we shall therefore describe in Chapter 2. Further information and more detailed discussions of the Standard Model, its issues and Beyond Standard Model theories may be found in the books [14, 16, 34].