Theoretical Descriptions of the Nucleon
The Standard Model
The Standard Model of particle physics, a Quantum Field Theory, was developed between 1970 and 1973. The Standard Model describes all of the known elementary particles interactions except gravity. It is the collection of the following related theories quantum electrodynamics, the Glashow-Weinberg-Salam theory of electroweak processes and quantum chromodynamics.
As it is known the matter is made out of three types of elementary particles: quarks, leptons and mediators.In the Standard Model there are six quarks,including the up(u) and down(d) quarks, which make up the neutron and proton. They are classified according to charge(Q), strangeness(S), charm(C), beauty(B) and truth(T). All quarks are spin- 1/2 fermions(1)
THE QUARK CLASSIFICATION
There are six leptons electron, muon and tau, with their partner neutrino. Leptons are classified by their charge(Q), electron number(
THE LEPTON CLASSIFICATION
There are six anti leptons and anti quarks, all signs in the above tables would be reversed for them. Also, quarks and anti quarks can carry three color charges(red, green and ) that enables them to participate in strong interactions.
In the Standard Model the mediator elementary particles with spin-1 are bosons.
- The electromagnetic interaction is mediated by the photons, which are massless.
- The , and gauge bosons mediate the weak nuclear interactions between particles of different flavors.
- The gluons mediate the strong nuclear interaction between quarks.
THE MEDIATOR CLASSIFICATION
The Quark Parton Model
In the Quark Parton Model(QPM), partons are hypothetical fundamental particles which are the constituents of hadrons. It was shown that hadrons contain quarks and gluons. (Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, ISBN-10: 0-471-60386-4 ) Insert Reference here
However, the Parton Model is used to describe short-distance interactions.
Bjiorken predicted that at very high energies the inelastic structure function dependence on decreases and it is function of
Assuming that is the fraction of the proton carried by the quark . Because , and scalar product of k and p is
For example, take and go to the proton rest frame(??????????/) in this case and using ,
Kinematic variables in deep inelastic scattering
Kinematic variables in deep inelastic scattering
|,||4 - momenta of the initial and final state leptons|
|,||Polar and azimuthal angle of the scattered lepton|
|4 - momentum of the initial target nucleon|
|4 - momentum of the virtual photon|
|Negative squared 4 - momentum transfer|
|Energy of the virtual photon|
|Bjorken scaling variable|
|Fractional energy of the virtual photon|
|Squared invariant mass of the photon-nucleon system|
|4 - momentum of a hadron in the final state|
|Fractional energy of the observed final state hadron|
The semi-inclusive scattering, where the scattered electron and one or more hadrons in coincidence are detected, provides reliable? information on the individual contributions of u, d and s quarks in the nucleon as well as the separate contributions of valence and sea quarks(wigni_1). "Within the Quark-Parton Model(QPM) the charge of the hadron and its valence quark composition provide sensitivity to the flavor of the struck quark"(wigni_1). The diagram of the semi-inclusive deep inelastic scattering is shown below:
The fragmentation matrix, the probability of the struck quark fragmenting into a hadronic statecan be written i the following way:
- the momentum of the final hadron
- the energy of the final hadron
The conservation of 4-momentum between the two states k=p+q and the momentaand can be used to describe a new variable z, the energy fraction transferred by the virtual photon which is contained in the detected hadron(main.pdf):
After not taking in consideration transverse hadron momentum, can be related to k(main.pdf):
Quark distribution Functions
define and describe
Quark distribution function q(x) is the probability(density) of finding a quark with fraction x of the proton momentum. It can be expressed as
It is known that the proton contains up(u) and down(d) quarks. Accordingly, We have up u(x) and down d(x) quark distribution functions in the proton. u(x) is the probability that momentum fraction x is carried by a u type quark and d(x) - for a d type quark. Moreover,
u(x)dx ( d(x)dx ) is the average number of up (down) quarks which have a momentum fraction between x and x+dx.
Actually, the proton can contain an extra pair of quark - anti quarks. The original(u, d) quarks are called valence quarks and the extra ones sea quarks.we are allowed to separate the quark distribution function into a valence and a sea part,
q(x) is the unpolarized distribution function and - the polarized.
The structure functions in the quark parton model can be written in terms of quark distribution functions,
The unpolarized structure function- measures the total quark number density in the nucleon, - the polarized structure function is helicity difference quark number density.
The unpolarized structure functions ( i= L, R, O ) should satisfy the following inequalities,
The structure functions in terms of the parton distributions can be written as,
where , , are the charge, left- and right-handed weak couplings of a ith type quark and , corresponding couplings for the electron.
Both models, pQCD and a hyperfine perturbed constituent quark model(CQD), show that as the scaling variable
The inclusive double polarization asymmetriesin the valence region, where the scaling variable can be written in terms of polarized and unpolarized valence quark distributions,
The semi-inclusive pion electro-production asymmetries can be written in terms of the valence quark distributions
where is the measured difference of the yield from oppositely charged pions. Using the first four equation (1), (2), (3) and (4) one can construct the valence quark distribution functions.
The semi - inclusive asymmetry can be rewritten in terms of the measured semi-inclusive and asymmetries:
where is the unpolarized structure function and the scaling polarized structure function.
The last equation can be expressed as
using the nomenclature of (6) equation, we have
The hardonic tensor
i - is the summation over the quark flavors and s - the quarks helicities. term is the interaction of the virtual photon with the quark of momentum k and in case of massless on-shell quarks is
Using (1) and (2) the hardonic tensor can be expressed as