Standard Model and the Higgs

Standard Model and the Higgs

The Standard Model (SM) is a description of nature in terms of fundamental particles and their interactions. It has been developed over a number of decades, and its development has been guided both by theoretical predictions and experimental discoveries. The SM encompasses three of the four fundamental forces of nature: electromagnetism, the strong interaction and the weak interaction. Apart from gravity, the interactions described by the SM are responsible for all aspects of daily life. Electromagnetism describes the interaction of electrons with nuclei and is thus responsible for all of chemistry and biology. The strong force describes the interactions within the nucleus. The weak force provides a description of radioactivity and nuclear fusion, which powers the stars. The SM describes nature using a mathematical formalism known as quantum field theory.
The fundamental particles are represented by the states of quantized fields. Quarks and leptons constitute matter and are associated with fields of half integer spin, referred to as “fermion” fields. The dynamics of this system, i.e., the motion and interactions of excitations in the fields, is governed by a mathematical quantity referred to as the Lagrangian. The SM is a particular type of quantum field theory known as a gauge theory.The Lagrangian of the SM is invariant under space-time dependent continuous internal transformations of the group SU(3)×SU(2)×U(1).This invariance is referred to as gauge invariance and is critical for ensuring that the theory is renormalizable. Renormalizability is a necessary form of consistency; theories which are not renormalizable lack predictive power. Additional quantum fields are required to ensure gauge invariance. These fields are have spin one and are referred to as “gauge fields”. The excitations of the gauge fields correspond toparticles referred to as “gauge bosons”. In the standard model twelve gauge fields are included in the Lagrangian, eight for the generators of SU(3), three for the generators of SU(2), and one for theU(1)generator. In principle, what has been described above is enough to define a theory of particles and the irinteractions. Infact, the SU(3) gauge symmetry coupled to the quarks correctly describes the strong interaction, with the eight SU(3) gauge fields associated to the different colored states of the gluon. Gluons have been observed experimentally and interact with quarks as predicted in the SM. A problem arises when considering the part of the SM that describes the electromagnetic and weak interactions, governed by the SU(2)×U(1) symmetry. To preserve gauge invariance, the gauge fields must be added without mass terms. This implies that the gauge bosons should appear as mass-less particles, as is the case for gluons. However, to properly describe the weak force, the gauge bosons associated to it are required to have a large mass, seemingly in contradiction with the prediction. The masses of the quarks and leptons pose another problem.The weak interaction violates parity, coupling differently to left and right-handed quark and lepton helicity states. To account for this in the SM, the left and right-handed fermions are treated as different fields with different couplings. A fermion mass term in the Lagrangian would couple these different field sand thus break gauge invariance. A gauge invariant left-handed weak interaction implies that the fermion fields should not have mass terms and that the quarks and leptons which appear in nature should be mass-less particles. This, again, is in direct conflict with observation. From a theoretical point of view, both of the these problems can be overcome by what is referred to as “spontaneous symmetry breaking”. The idea is that additional quantum fields are added to the theory that couple to the electroweak SU(2)×U(1) gauge fields. These fields have zero spin and are referred to as “scalar”fields.The scalar fields are included in a way that respects the SU(2)×U(1) symmetry and preserves the gauge invariance of the Lagrangian.The trick is that the scalar fields are added with a special form of interaction such that zero values of the fields do not correspond to the lowes tenergy state.While the actual interaction in the Lagrangian preserves the SU(2)×U(1) symmetry,the ground state of the field will necessarily break it. As a result, the Lagrangian preserves gauge invariance, despite the fact that the particular state that describes nature does not exhibit SU(2)×U(1) symmetry. In this sense the symmetry is said to be “spontaneously broken”. The upshot of the spontaneous symmetry breaking is that in nature the scalar fields will take on a non-zero value, referred to as the “vacuum expectation value”, or vev.The vev will couple to the fermion and gauge fields in away that is equivalent to having mass terms, but nevertheless preserves gauge invariance. As a result, the fermions and weak gauge bosons can appear in nature as massive particles,consistent with observation. The masses of the gauge bosons are set by the vev and by the couplings associated to the gauge symmetry and are thus constrained by the theory. The fermion masses, on the other-hand, depend on arbitrary coupling parameters that must be input to the theory. Through spontaneous symmetry breaking, massive fermions and weak bosons can be accommodated in a gauge invariant way. The SM as sketched above provides a theory for describing massive fermions interacting via the electromagnetic, the strong, and the parity-violating weak force. The predictions of the SM have been tested over many years, by many different experiments, and have been shown to accurately describe all of the observed data. Focusing on the electro-weak sector, examples of the impressive agreement of SM predictions with observed data are shown. The hadronic cross-section in e+e− collisions as a function of the center-of-mass energy. The black curve shows the cross section of electron-positron collisions to fermions prediction by the SM; the points give the measurements from various different experiments. The falling cross-section at low center-of-mass energy and the peak due to Z boson production are accurately described by the SM. The figure also shows the agreement of the observed LEP-II data with the SM prediction for e+e− → WW. This process is sensitive to the ZWW coupling, which is a direct consequence of the gauge structure of the theory. Figure1.2 shows a summary of various SM cross section predictions and their measurements in √s = 7 TeVpp collisions at the LHC. An impressive agreement is found over many orders of magnitude. Another consequence of the spontaneous symmetry breaking is the prediction of a massive scalar particle.The interactions that generate the vev give mass to one of the additional scalar fields. This field should appear in nature as a neutral massive spinzero boson, referred to as the “Higgs” boson. The mass of the Higgs boson depends on an arbitrary parameter associated to the symmetry breaking and is thus an input to the theory. The interactions of the Higgs boson with the fermions and gauge bosons are, however, fixed by the theory. The couplings to gauge bosons are fixed by the gauge couplings, and the couplings to fermions are fixed by the fermion masses; the Higgs boson couples to fermions proportionally to their mass. As of the beginning of the LHC running, the Higgs boson had not been observed experimentally. As mentioned above, the mass of the Higgs boson is not predicted by the SM. There are no rigorous bounds on the Higgs mass from theory alone. The Higgs must be massive to generate the spontaneous symmetry breaking, and Iitis assumed that perturbation theory is valid, the mass of the Higgs should be below about a TeV. The next section will describe constraints on the Higgs mass from measurements of the other electro-weak parameters. The Higgs boson is a necessary ingredient in the SM for ensuring gauge invariance. Masses for the fermions and gauge bosons are allowed at the price of an additional scalar particle, the Higgs boson. A search for the Higgs bosons at the LHC is the subject of this thesis. The following section describes constraints and experimental limits on the Higgs boson mass prior to 2011. The SM presented above is the minimal version that spontaneously breaks the electro-weak symmetry. More complex arrangements of scalar fields can be added to the theory. In general, these lead to additional physical particles, but serve the purpose of gauge invariant mass generation. These more complicated extensions are not considered in this thesis.



ΑΝΑΔΗΜΟΣΙΕΥΣΗ ΑΠΟ ΤΟ ΒΙΒΛΊΟ «Τhe Road to Discovery Detector Alignment, Electron Identification, Particle Misidentification, WW Physics, and the Discovery of the Higgs Boson»

Doctoral Thesis accepted by the University of Pennsylvania, USA

Author Dr. John Alison Enrico Fermi Institute University of Chicago Chicago, IL USA

25/3/2016