2 Gauge Theories and the Standard Model
13
2.3 Overview of the Standard Model
The SM is a gauge field theory based on the symmetry group SU (3)⊗SU (2)⊗U(1).
The transformations of the group act on the basic fields. This group has 8+3+1=
12 generators with a non trivial commutator algebra (if all generators commute
the gauge theory is said to be “abelian”, while the SM is a “non abelian” gauge
theory). SU (3) is the “colour” group of the theory of strong interactions (QCD:
Quantum Chromo-Dynamics [1–3]). SU (2)⊗U(1) describes the electroweak (EW)
interactions [4–6] and the electric charge Q, the generator of the QED gauge group
U(1) Q , is the sum of T 3 , one of the SU (2) generators and of Y/2, where Y is the
U(1) generator: Q = T 3 + Y/2.
In a gauge theory to each generator T is associated a vector boson (also said
gauge boson) with the same quantum numbers as T , and, if the gauge symmetry is
unbroken, this boson is of vanishing mass. These vector (i.e. of spin 1) bosons act as
mediators of the corresponding interactions. For example, in QED the vector boson
associated to the generator Q is the photon γ . The interaction between two charged
particles in QED, for example two electrons, is mediated by the exchange of one
(or seldom more than one) photon emitted by one electron and reabsorbed by the
other one. Similarly in the SM there are 8 massless gluons associated to the SU (3)
colour generators, while for SU (2) ⊗ U(1) there are 4 gauge bosons W + , W − , Z 0
and γ . Of these, only the photon γ is massless because the symmetry induced by
the other 3 generators is actually spontaneously broken. The masses of W + , W −
and Z 0 are quite large indeed on the scale of elementary particles: m W ∼ 80.4 GeV,
m Z ∼ 91.2 GeV are as heavy as atoms of intermediate size like rubidium and
molibdenum, respectively. In the electroweak theory the breaking of the symmetry is
of a particular type, denoted as spontaneous symmetry breaking. In this case charges
and currents are as dictated by the symmetry but the fundamental state of minimum
energy, the vacuum, is not unique and there is a continuum of degenerate states
that all together respect the symmetry (in the sense that the whole vacuum orbit is
spanned by applying the symmetry transformations). The symmetry breaking is due
to the fact that the system (with infinite volume and infinite number of degrees of
freedom) is found in one particular vacuum state, and this choice, which for the SM
occurred in the first instants of the Universe life, makes the symmetry violated in
the spectrum of states. In a gauge theory like the SM the spontaneous symmetry
breaking is realized by the Higgs mechanism (described in detail in Sect. (2.7)):
there are a number of scalar (i.e. of zero spin) Higgs bosons with a potential that
produces an orbit of degenerate vacuum states. One or more of these scalar Higgs
particles must necessarily be present in the spectrum of physical states with masses
very close to the range so far explored. It is expected that the Higgs particle(s) will
be found at the LHC thus completing the experimental verification of the SM.
The fermionic (all of spin 1/2) matter fields of the SM are quarks and leptons.
Each type of quark is a colour triplet (i.e. each quark flavour comes in three colours)
and also carries electroweak charges, in particular electric charges +2/3 for up-type
quarks and −1/3 for down-type quarks. So quarks are subject to all SM interactions.
13
2.3 Overview of the Standard Model
The SM is a gauge field theory based on the symmetry group SU (3)⊗SU (2)⊗U(1).
The transformations of the group act on the basic fields. This group has 8+3+1=
12 generators with a non trivial commutator algebra (if all generators commute
the gauge theory is said to be “abelian”, while the SM is a “non abelian” gauge
theory). SU (3) is the “colour” group of the theory of strong interactions (QCD:
Quantum Chromo-Dynamics [1–3]). SU (2)⊗U(1) describes the electroweak (EW)
interactions [4–6] and the electric charge Q, the generator of the QED gauge group
U(1) Q , is the sum of T 3 , one of the SU (2) generators and of Y/2, where Y is the
U(1) generator: Q = T 3 + Y/2.
In a gauge theory to each generator T is associated a vector boson (also said
gauge boson) with the same quantum numbers as T , and, if the gauge symmetry is
unbroken, this boson is of vanishing mass. These vector (i.e. of spin 1) bosons act as
mediators of the corresponding interactions. For example, in QED the vector boson
associated to the generator Q is the photon γ . The interaction between two charged
particles in QED, for example two electrons, is mediated by the exchange of one
(or seldom more than one) photon emitted by one electron and reabsorbed by the
other one. Similarly in the SM there are 8 massless gluons associated to the SU (3)
colour generators, while for SU (2) ⊗ U(1) there are 4 gauge bosons W + , W − , Z 0
and γ . Of these, only the photon γ is massless because the symmetry induced by
the other 3 generators is actually spontaneously broken. The masses of W + , W −
and Z 0 are quite large indeed on the scale of elementary particles: m W ∼ 80.4 GeV,
m Z ∼ 91.2 GeV are as heavy as atoms of intermediate size like rubidium and
molibdenum, respectively. In the electroweak theory the breaking of the symmetry is
of a particular type, denoted as spontaneous symmetry breaking. In this case charges
and currents are as dictated by the symmetry but the fundamental state of minimum
energy, the vacuum, is not unique and there is a continuum of degenerate states
that all together respect the symmetry (in the sense that the whole vacuum orbit is
spanned by applying the symmetry transformations). The symmetry breaking is due
to the fact that the system (with infinite volume and infinite number of degrees of
freedom) is found in one particular vacuum state, and this choice, which for the SM
occurred in the first instants of the Universe life, makes the symmetry violated in
the spectrum of states. In a gauge theory like the SM the spontaneous symmetry
breaking is realized by the Higgs mechanism (described in detail in Sect. (2.7)):
there are a number of scalar (i.e. of zero spin) Higgs bosons with a potential that
produces an orbit of degenerate vacuum states. One or more of these scalar Higgs
particles must necessarily be present in the spectrum of physical states with masses
very close to the range so far explored. It is expected that the Higgs particle(s) will
be found at the LHC thus completing the experimental verification of the SM.
The fermionic (all of spin 1/2) matter fields of the SM are quarks and leptons.
Each type of quark is a colour triplet (i.e. each quark flavour comes in three colours)
and also carries electroweak charges, in particular electric charges +2/3 for up-type
quarks and −1/3 for down-type quarks. So quarks are subject to all SM interactions.