1.1 The Standard Model of Particle Physics
1.1.8 Unsolved Problems in the Standard Model
The Standard Model is an extremely powerful predictive framework which is com- patible with experimental data in all areas except in the case of neutrino masses in recent years. It is highly accurate in estimating the cross sections of its inter- actions, and it has successfully predicted the existence of the b-quark [32, 33], the W and Z bosons [34–37], the top quark [38, 39], the τ neutrino [40] and the Higgs boson [20, 21]. Many of its free parameters have been experimentally determined both directly and indirectly to the extent of being over-constrained with a high level of consistency between the estimates. As an example, figure 1.5 shows several direct measurements of the top mass from current and previous experiments as well as SM fits including and excluding these measurements. All measurements and fit results are consistent within the errors. The precision of theoretical predictions has also improved through new techniques that have made higher order cross section calcula- tions feasible. However, there are theoretical considerations that make it improbable that the SM is the final answer in the search for the laws governing particle physics. This problem goes much deeper than just the glaring absence of gravity in the the- ory. Many aspects of the framework such as the number of particles and generations as well as the relative scale of the particle masses were not implemented based on rigorous theoretical insights, but in order to describe the observed data phenomeno- logically, which they do with impressive accuracy. Many fundamental problems are still left unsolved, some of which will be discussed briefly in the following.
Baryon Asymmetry in the Universe
The observable universe is heavily dominated by matter. For this matter-antimatter asymmetry to occur, the three Sakharov conditions [43] need to be fulfilled. Among these is the requirement that baryon number violation must occur, which does not happen in any SM processes. Additionally, processes that violate C and CP sym- metries must exist, but the amount of CP violation present in the weak interaction is too small to explain the degree of baryon asymmetry that is observed. These con- siderations hint at physics beyond the SM that incorporate stronger CP violation and baryon number violation, which might become relevant at higher energies than those currently experimentally accessible.
Masses and Fine-tuning of Parameters
The masses of the fermions are parameters of the theory which are not known a priori and need to be added by hand from experimental data. Furthermore, the observed hierarchy of masses between the individual particles and between the three generations has not been explained. Such a model based on fundamental principles has long been sought within the physics community. As mentioned in section 1.1.1 even the neutrinos seem to have mass. They were previously thought to exist only in their left-handed state making it impossible to add a renormalisable mass term [23, p. 713-715].
Several parameters in the SM seem to be fine-tuned, including the bare Higgs mass (the mass at infinitesimal distances). It receives large quantum loop corrections fromW,Z,H and top quark loop contributions that diverge quadratically with the renormalisation scale used in the procedure, which can be interpreted as the energy at which the SM breaks down. Although counterterms are guaranteed to cancel these divergencies, choosing values for the renormalisation scale near the Planck scale (1019 GeV) requires the tree diagram to have almost exactly the same value (difference needs to be order 1017smaller) to counteract the loop contributions and arrive at a Higgs boson mass at the electroweak scale. This is an example of a ‘fine-tuning’ problem.
Several experiments have observed neutrino disappearance effects indicating that neutrinos can oscillate between the physically observable flavour states in di- rect contradiction with the SM prediction of massless neutrinos. The flux of electron neutrinos νe from the sun has been studied by the Homestake [44] and GNO [45]
experiments, anti-neutrinos ¯νe generated by reactors are the focus of other experi-
Super-Kamiokande [47]. Disappearance effects have been observed in all of these cases where the observed neutrino flux is lower than the SM prediction. Appear- ance of surplus amounts of νe and ντ in beams of muon neutrinos has also been
detected [48, 49].
Neutrino oscillations can be described theoretically by defining mass eigen- states|νii that are linear combinations of the flavour eigenstates|ναi:
|νii= X
j
Uαi|ναi. (1.42)
The probability of a particular flavour oscillation|ναi → |να0idepends on the size of
the mass difference ∆m2α,α0, the distance travelled since the flavour eigenstate was
created L, the energy of the system E and the mixing matrix U. Assuming there are three mass eigenstates, fits to experimental data are able to constrain the mass differences between each state resulting in [5]
|∆m212| ≈7.5·10−5 eV,
|∆m231| ≈2.3·10−3 eV,
|∆m221|/|∆m231| ≈0.032.
(1.43)
Due to their lack of an electric charge it is still an open question whether neutrinos are their own antiparticle (Majorana fermion) or if they exhibit the same behaviour as the charged leptons having distinct antiparticles (Dirac fermion). In the former case lepton number is not a conserved quantity, which opens the possibility of ex- perimentally determining their Majorana or Dirac nature through searches for e.g. neutrinoless double beta decay.
Unification of the Forces
The coupling constants of the three fundamental forces vary with the cut-off energy scale chosen when performing renormalisation. Plotting the coupling strengths as a function of the energy scale as seen in figure 1.6, the coupling strengths seem to coalesce into one unified force at very high energies of ∼ 1016 GeV. This has led to several Grand Unified Theories (GUTs) postulating that at energies above this GUT scale the three SM symmetries are replaced by a new single symmetry. Within the SM the strengths of the forces do not meet at exactly the same point. One way to make this happen is by introducing another symmetry which sets fermions and bosons on equal footing. This is called supersymmetry [50, 51] and is one out of
10
log Q
1/
α
iMSSM
10log Q
1/
α
i 0 10 20 30 40 50 60 0 5 10 15 0 10 20 30 40 50 60 0 5 10 15Figure 1.6: The evolution of the inverse SM coupling constants (left) and in the minimal supersymmetric extension of the SM (right). Unification of all three forces
only happens with the inclusion of supersymmetry. Theαirepresent, in numerically
ascending order, the EM, weak and strong coupling constants [52].
many contenders for a theory beyond the SM.
Dark Matter
Evidence for the existence of dark matter has been found primarily in measurements of the properties of galaxies. Their rotational velocity as a function of the distance from their centre, the strength of gravitational lensing and the way galaxies cluster together cannot be explained by the presence of only visible matter. Dark matter is hypothesised to be a new form of matter that acts weakly or not at all through the electromagnetic force and which is stable over the age of the universe. Fluctuations in the cosmic microwave background (CMB) also provide evidence for dark matter. According to fits from the Planck Collaboration [53] the universe contains only 4.9% ordinary matter, while 26.8% is composed of dark matter. The remaining energy density is called dark energy and can be described as a cosmological constant that acts to accelerate the expansion of the universe. Models exist to describe the nature of dark matter including weakly interacting massive particles (WIMPs), axions and sterile neutrinos. Dark matter candidates can also be constructed through super- symmetric models that can be searched for at the LHC. Experimental evidence has not yet resolved which of these models, if any, are an accurate description of dark matter.