## LATEX

### Basics of phenomenology

The physical world is constituted of elementary particles.
There are four types of forces governing all known interactions between these elementary particles:
1-The gravitational force which is not relevant to the energy scales of particle physics so we will not discuss it any further.
2-The strong nuclear force which is crucial in the construction of the standard model of elementary particles. It is a short-range interaction with an interaction radius equal $10^{-13}$ cm.
It is also a force of great relevance to star formation, astrophysics and cosmology of the early Universe.
For example, the heat produced in the Sun is due to the strong nuclear force when deuterium nuclei and protons are combined into helium nuclei.
Particles which feel the strong nuclear force are called hadrons.
These are bound states of quarks subjected to color confinement which is one of the most fundamental characteristic signatures of the strong nuclear force.
Those hadrons with integer spins are called mesons (bound states of a quark and anti-quark) whereas those hadrons with half-integer spins are called baryons (bound states of three quarks or three anti-quarks).
Particles which decay by means of the strong force are called resonances.
The strong nuclear force is described by the theory of quantum chromodynamics or QCD for short which is a local gauge theory based on the gauge group $SU(3)_c$. This is to be contrasted with the electromagnetic force described by the theory of quantum electrodynamics or QED which is also a local gauge theory based on the gauge group $U(1)_{em}$.
A local gauge symmetry means a group of local transformations, i.e. they depend on the space-time locations, which leave the theory (action, local Lagrangian, the quantum vacuum, the spectrum, etc) invariant.
The lower index c in $SU(3)_c$ denotes the strong nuclear charges carried by the quarks which are called colors. There are precisely 3 different charges or colors denoted for example by R (red), G (green) and B (blue) and their conjugates, i.e. anti-charges or anti-colors.
The hadrons are however colorless and we say that they are singlet states under the gauge group. This condition is precisely why hadrons can only come as either baryons (three quarks) or mesons (quark and anti-quark).
We stress the fact that these colors have nothing to do with visual colors but they simply denote extra degrees of freedom which characterize the quarks besides their usual quantum numbers.
In a strict sense these extra degrees of freedom are the charges acted upon by the strong nuclear force in the same way that the electric charge $+$ (and its conjugate $-$) is acted upon by the electromagnetic force.
This is why the electromagnetic gauge group^is $U(1)_{em}$ because there is a single type of electromagnetic charge as opposed to the three types of charge of the the strong nuclear force and the gauge group $SU(3)_c$.
Indeed, the color charges are the sources of the strong nuclear gauge field (also called gluon fields) in the same way that the electric charge is the source of the electromagnetic field (also called photon field).
However, the photon field is chargeless whereas the gluon field is NOT colorless since it carries a color and anti-color charges and as a consequence its absorption and emission by a quark will change the color charge on the quark.
There are therefore $3x3=9$ gluon fields a priori but the color trace combination can not change the color charge on the quark and therefore we are left with only $3x3-1=8$ gluon fields.
This is one of the reason why color gauge symmetry is a strongly interacting force and hence highly non-perturbative whereas electric gauge symmetry is largely weak and perturbative.
The $8$ gluons like the photon are neutral and massless vector particles.
Gauge symmetry (both strong nuclear and electromagnetic) are exact symmetries of Nature (which translates to the masslessness of the gluons and photon) which is also, as it turns out, mathematically equivalent to the conditions of unitarity and renormalizability.
Hadrons are however composite objects and the fundamental degrees of freedom which appear in the QCD Lagrangian are the quarks and the gluons and not hadrons. As a consequence the force between color-neutral (singlet) hadrons is a residual nuclear force which arises from the fundamental color force in the same way that Van der Waals forces between neutral atoms arise from electromagnetism between the constituents electrons.
The photons, as we have said, do not carry electric charge and therefore they create no new electromagnetic field around them (the photon gauge field is not self-interacting).
The electromagnetic field is therefore the strongest around the charge which created it and then it becomes weaker as we go further away from the charge.
In contrast, the gluon gauge field is self-interacting and gluons and quarks exhibit another remarkable phenomena (beside color confinement) called asymptotic freedom.
Gluons carry color charges and therefore they create around them a new color field corresponding to new gluons and these will create new gluons and so on and so forth.
Therefore the color field created by a color charge will be enhanced by the new color fields of the generated virtual gluons and the field therefore tend to increase as we move away from the color charge and not decrease as in the case of the electromagnetic field.
The color charge inside is thus masked by vacuum polarization and becomes effectively larger as seen from larger distances. This is the so-called vacuum polarization effect.
This can also be characterized by the running of the coupling constant $\alpha_s$ of QCD with energy $q^2$ (that is why it is running) which is given by the famous perturbative formula in the image.
In this equation $n_f$ is the number of quark flavors and $\Lambda_{QCD}$ is the QCD scale at which the almost free quarks at high energies (where we can use effectively perturbation theory) become confined in bound states of the known hadrons.
Indeed, for $q^2>>\Lambda_{QCD}^2$ the effective coupling constant $\alpha_s(q)$ becomes small indicating that quarks and gluons are almost free and perturbation theory is a valid prescription.
Whereas when $q^2~\Lambda_{QCD}^2$ we see that the effective coupling constant becomes infinite indicating that bound states of hadrons have formed, quarks and any color state are confined, and perturbation theory is no more a valid approximation.
The QCD scale is not a free parameter of the theory (like the gauge coupling constant $\alpha_s=g_s/4\pi$) but it is determined from experiment to be in the range between 100 and 200 Mev.
The renormalization group evolution of the running gauge coupling constant $\alpha_s$ given by the formula in the image is therefore stating that the strength of the strong nuclear force (the color interaction) as measured by $\alpha_s$ becomes negligible at high-energies $q^2>>\Lambda_{QCD}^2$ and therefore quarks and gluons become approximately free in that regime.
This is the so-called asymptotic freedom.