# Basic Collision Theory

Please take care (cross-check) while following any of this hints.

-- JuanCastillo - 25 Mar 2008

Asymptotic Freedom: The interactions between 2 quarks become weaker when closer. Asymptotic freedom can be derived by calculating (see beta function) describing the variation of the coupling constant under the renormalization scale . Calculating the beta-function is a matter of evaluating Feynman diagrams contributing to the interaction of a quark emitting or absorbing a gluon.
For >>> , integrating the beta function we reach:

That is, grows with growing . For ,

Baryon number: Approximate conserved quantum number. Each quark contributes with B = 1/3, each antiquark with B = -1/3. If the particle has no quarks (leptons, photon, W and Z bosons) B =0. See also Wikipedia

Bosons: S integer particles. Like neutrons or protons. Hadrons.

Central Rapidity region: Region of rapidity about midway between the projectile rapidity and the target rapidity. See also rapidity.

Charge Independence: Since the masses of a proton and a neutron are nearly the same, they can be considered as degenerated states. Any arbitrary combination of proton and a neutron wave functions is equivalent to a single 'proton' or 'neutron'. See also Isospin.

Colour Degree of Freedom: (Greenberg 1964) From baryon spectroscopy. For a baryon made of 3 q of spin 1/2:

It was realized (Dalitz 1965) than the product of these wavefunctions for the ground-state baryons was symmetric under exchange of any two quarks. But the quantum field theory requires fermions to follow the exclusion principle (that is, that the total wavefunction should be antisymmetric under quark interchange). The easy solution is supposing that quarks are having an additional DOF, called colour. Ex: , made of 3 up quarks (flavour symmetric), in state

Confinement: Colour charged particles cannot be isolated. They must be confined.

Current: Called j(x),if u(x), v(x) are fermionic field operators, and O a combination of matrixes including Dirac matrixes (), j(x) = u(x)Ov(x)

Dimensional Transmutation: The appearance of a scale in the quantum theory, where there was NONE in the classical counterpart.

Energy distribution: Probability that a particle is in energy state E. Mathematically: f(E). It is used the Maxwell-Boltzmann energy distribution, the Fermi-Dirac and the Bose-Einstein ones. You can compare them in Hyperphisics. When we speak about Boltzmann distribution, we refer of a fractional number of particles (Ni/N) occupying a set of states i which each respectively possess energy Ei. See article also in wikipedia.

Electromagnetic probes: In HIC, photons, electron pairs and muon pairs.

Explicit Symmetry Breaking: In chiral symmetry, the Goldstone boson associated with the spontaneous symmetry breaking is the pion. Since the pion is NOT massless, we speak about explicit symmetry breaking. See also spontaneous symmetry breaking.

Fermion: S half integer particles. Like the electron, muons and neutrinos (that are also fundamental) and the quarks.

Forward Direction: In a hadron collider experiment, it refers to regions of the detector that are close to the beam axis, at high ||.

Hadron: Group of quarks. Like the proton and the neutron (baryons) or the pion and the kaon (mesons). H not 0, therefore, they feel strong interaction.

Hard Probes: (HP) Jets, heavy quarks, direct photons. Created in a very early phase after collisions. See also electromagnetic (EM) probes and direct photons.

Heisenberg picture: Operators (observables and others) are time-dependent and eigenvectors are time-independent. Mathematical details you will find here. See also Schrödinger picture.

Helicity: Projection of the spin of the particle onto the direction of momentum. More info here

Hypercharge(Y): Y = B + S, being B= baryon number, S= strangeness.

Isospin: Introuced by Heisemberg (1932) and discovered by Chadwick (1932).
Historycally known as "isotopic spin".
Since the neutron an the proton are having equal masses, you can think of them as the same particle (nucleon) with different isospin. So, the proton has isospin "up" and the neutron "down".
Thus, for A nucleons in a nucleous, we define a total isospin T as:

For a givent we have 2T+1 degenerate states (isospin multiplets).
Succesive values of correspond on changin one proton onto neutron or viceversa.
States in the same multiplet must have the same quantum numbers.

Isotopic Invariance: Independece from charge of the strong interactions.

Isotopic multiplet: Particles with the same spin, same parity, same mass (more or less), same behaviour under strong interactions, but different charge. EX: triplet , ,

J/Psi: charm-anticharm (charmoniun-type) flavor neutral meson. The J/; has a rest mass of 3096.9 MeV/c2 and a half life of 7.2×10-21 s.

Kaon: (AKA K-meson). Mesons with strangeness (a single strange or antistrange quark).
Quantum numbers: .
Structure: , ,,, analogous.
Rest mass around 494 MeV.
( for is the charge conjugate) main decay channels:
(63%)
(21%)
(5.6%)

Kink: (Math) A twist in the variable φ which takes the system from one solution φ = 0 to an adjacent with φ = 2π. You can find more info readign about the Sine-Gordon Equation

Kink vertex: Historically, from bubble chambers, shape of the vertex created after a K+(K-) decay into a muon(+/-) and its neutrino.

Lepton: Fundamental fermions. No color (hadronic charge). Leptons are the electron, muon, tau and its neutrinos.

Meson: 2-quark particles: quark and an antiquark.

Parton Distribution Function: (PDF) Distribution function of the partons in a nucleon. Generically , where is the "Bjorken x" and is the hardness (energy transfer).
dependence of PDF is given in DIS by DGLAP. The standart approach is parametrize it experimentally :

Where the coefficients are depending of the quarks and defined from DIS experiments.
We consider 2N+1 different PDF, being N the number of flavors.

Pion: Particle exchanged between protons (p) and neutrons (n) inside the nucleus. Yukawa 1947. The neutral pions (high energy) decay in photons and protons. Quantum numbers: .
Structure: , ,,.
Rest mass around 139 MeV ( for is the charge conjugate) main decay channels:
(99.9877%)
(0.0123%)
main decay channels:
(98.798%)
(Dalitz decay)(1.198%)

Spontaneous Symmetry Breaking: When a system that is symmetric in some symmetry group goes onto a non-symmetric state. This implies the appearance of a massless (Goldstone) boson. Example: the mexican hat potential (spontaneously broken) vs the parabolic potential. See wikipedia for more details.

Schrödinger picture: The eigenstates are constant while the observables evolve in time. Mathematical details you will find here. See also Heisenberg picture.

Schwinger Mechanism: The Schwinger mechanism of particle production was first used for examine the production of e+e- pairs in a strong uniform electromagnetic field.
Applied to quarks, the field between a quark and an antiquark is represented by an Abelian gauge field equivalent to a constant electric field between 2 plates of a condenser in QED. A particle-antiparticle pair is produced when a particle tunnels from the negative energy continuum to the positive energy one.
The field in this case is a color field defined inside a color flux tube. The interaction energy between a quark and an antiquark is then proportional to their separation, being the constant of proportionality , where = 1 Gev/fm is the string tension, = magnitude of the "electric color field" and A = cross section of the tube.

Width: In particle physics, . In a first approximation, is coming from the uncertain principle , in a more detailed description, is coming from the Breit-Wigner energy distribution. More info and an width calculator in Hyperphisics
Topic revision: r2 - 2008-04-08, JuanCastillo

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