Download A first view on the mathematical structure of the standard

A first view on the mathematical structure of the standard model
F. M. Rohrhofer∗
Institution for physics, Karl-Franzens-Universität
(Dated: 8.6.2014)
For a considerable time physicist wonder at the composition of materia, they are anxious for
finding every single particle that for all intents and purposes plays a role in our life. We have developed a new theory to describes all known elementary particles and the interaction between them,
including the strong, the weak and the electromagnetic interaction. One fundamental component is
the classification of the elementary particles in fermions, gauge bosons and the Higgs boson. With
these particles every nucleus, every atom, in other words the whole materia can be build. In this
publication we want to present some mathematical details of the theories which are essential for
the standard model: gauge theories like the electroweak theory, or Quantum Chromodynamics. We
discuss different symmetries, their properties in group theory, and how we identify their generators
as gauge bosons. Up to date, there is no challenging theory which describes nature on a fundamental level better than the standard model. Nevertheless, gravity is not included. Even worse,
gravity cannot be described as gauge theory and therefore the mathematical framework of gauge
theories cannot be the language of a true fundamental theory, or Grand Unified Theory (GUT).
Diverse extensions are currently being discussed by the international community. As an alternative
String theories have been a subject of research for years now. Although there is no way to check
the predictions produced by String theories, they supply the same results for known phenomena.
Keywords: standard model, mathematical structure, particle physics, gauge theories, QCD
INTRODUCTION
From today’s prospect particles physics is welldescribed in the standard model of elementary particles
and the fundamental interactions. With the word elementary particle we understand the point-shaped component of the materia without any substructure. The radius
of these particles add up to 10−18 to 10−19 m. There are
three types, the particles which form the materia, the
particles that are in charge of the interaction and the
Higgs-boson, which is essential for the mass of the particles. The former are fermions and they have half-integer
spin. Fermions are divided into leptons and quarks.
Within the leptons there are the neutrinos with charge
zero and the electron, the myon and the tauon with
charge -1. The neutrinos exclusively underlie the weak
interaction, whereas the quarks also interact strongly.
Equal the leptons there are six different quarks or flavors. All quarks have a charge that is a multiple of 1/3.
The second type of elementary particles are the so
called gauge bosons. If we disregard the gravitation,
all relevant interactions are described with the exchange
of the bosons that have Spin 1. We have the photon,
which carries the electromagnetic interaction, the 8 gluons, which are in charge of the strong interaction and
there are the W+,W- and Z Boson which carry the weak
interaction.
The electromagnetic interaction is well-described
within QED [1] . As oldest quantum field theory the
QED is extremely successful till now. Two essential properties of the QED are the invariance of the Lagrangian
and the renormalization. The former calls for the possibility to choose the phase of a fermionic field, whereas
the renormalization causes the abolishment of divergent
terms based on self-energy-content, which consequently
allows accurate calculations. The big success of the QED
hypothesize that all fundamental field theories have to be
invariant of the Lagrangian and renormalizable.
If one compares the ranges and forces of the three interactions, there are big differences relevant to the properties of the gauge bosons. The massless photon causes
an endless range of the electromagnetic interaction, the
very short distance of the weak interaction (1018 m) corresponds with massive bosons. The strong interaction
has no endless range like the exchange of massless bosons
implicits. The additional property of the so called ”confinement” [2] leads to a finite distance of about 10−15
m. The force of the electromagnetic interaction is described through the coupling constant e or equivalent alpha, whereat alpha for low energies is given through the
fine-structure constant:
α=
1
e2
=
4πε0 ~c
137
(1)
The weak interaction, also for low energies, has a force
given with the fermi constant
GF = 1.167 · 10−5 GeV −2 .
(2)
The name of the strong interaction is because of the comparatively stronger coupling, given through the coupling
constant αs , which has the value 1 for low energies and
minimize by high energies.
SYMMETRY
From the theoretical point of view the standard model
is a quantum field theory, which is based on the SU(3)C
× SU(2)L × U(1)Y gauge symmetry. This gauge group
includes the symmetry group of the strong interaction
2
SU(3)C and the symmetry group of the electroweak interaction, SU(2)L × U(1)Y . The symmetry group of the
electromagnetic interaction, U(1)em appears in the standard model as a sub-group of SU(2)L × U(1)Y . Because
of this the weak and the electromagnetic interaction are
combined to the electroweak interaction. The existence
of symmetries plays an essential role in particle physics.
A symmetry U exists if the considered physical system
stays invariant under the transformation U, this means
if the Hamiltonian [3] is invariant: UHU+ = H. The independent generatrix of a symmetry form the algebraic
structure of a group. These are called symmetry groups.
The symmetry groups appeared in the standard model
are continuous symmetries, this means the parameters
take continuous values compared to discrete symmetries.
Additionally these symmetries refer to the inner quantum numbers and not to the for example the space-timecoordinates like it is with rotations. Inner symmetries so
transform a particles in another particles with different
inner quantum numbers but the same mass.
GAUGE THEORY
Gauge theories are types of field theories in which the
Lagrangian is invariant under a continuous group of local transformations. They are called gauge because the
degrees of freedom in the Lagrangian are redundant. So
called gauge transformations which are transformations
between possible gauges, form a Lie group. Implicated
with any Lie group is the Lie algebra of generators. Every
group generator causes the existence of a corresponding
vector field. These vector fields is called gauge fields and
to assure its invariance under the local group transformation, also called gauge invariance, they are included
in the Lagrangian. If a gauge theory is quantized, the
quanta of gauge fields are called gauge bosons. When
the symmetry group is non-commutative, the gauge theory is called non-abelian, like in our case the standard
model is. Another example is the Yang-Mills-Theory. [4]
Gauge theories are said to have a global symmetry if they
are invariant under a transformation identically applied
at every point in the space in which the physical processes
occur. If a symmetry is called local, then one can apply a
local transformation which means that the representation
of the symmetry group is a function of the manifold and
can thus be taken to act differently on different points of
spacetime.
QCD
The QCD is based on the gauge symmetry of the strong
interaction, more accurate of the local transformation in
color space (3 dimensions), which let Lagrangian invariant. The gauge group, which is caused by the color transformation, is the non-abelian Lie group SU(3)C . The letter ”C”stands for color and the number 3 for the three
possible color states of the quarks. The gluons are the
gauge bosons of this symmetry and there are eight different ones, correspondent to the number of the SU(3)
generators.
CONCLUSION
The standard model of particle physics is one of the
most successful theories in physics. Based on a gauge
theory many new and important predictions have been
made. Disregarding from the trueness of the Higgsmechanism one can note from the theoretical aspect, that
indeed a quite sophisticated mathematical theory is the
bases of the standard model, but especially the quite big
number of parameters and open questions like for example the number of fermion families and interaction overall
provide an ungratified image. There are a lot of alternative theories, for example the both continuative theories
GUT and super symmetry [5]. They partially can answer some open questions in the standard model, but no
one so far provide a consistent image and above all for
each of this alternative theories any experimental proof
is missing.
∗
Electronic address: [email protected]
[1] C. Schubert, AIPConf.Proc.917:178-194 (2007).
[2] M.Engelhardt, B.Sperisen, AIPConf.Proc.892:176-179
(2007).
[3] V. M. Villanueva et al., J.Phys. A38 (2005) 7183-7196.
[4] S.A. Larin, Physics of Particles and Nuclei 44 (2013) 386390.
[5] Jerome Margueron, Philippe Chomaz, Phys.Rev. C71
(2005) 024318.