Theory

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

Generation	q	Q	D	U	S	C	B	T
Generation 1	d(down)	-1/3	-1	0	0	0	0	0
Generation 1	u(up)	2/3	0	1	0	0	0	0
Generation 2	s(strange)	-1/3	0	0	-1	0	0	0
Generation 2	c(charm)	2/3	0	0	0	1	0	0
Generation 3	b(bottom)	-1/3	0	0	0	0	-1	0
Generation 3	t(top)	2/3	0	0	0	0	0	1

There are six leptons electron, muon and tau, with their partner neutrino. Leptons are classified by their charge(Q), electron number([math]L_e[/math]), muon number([math]L_\mu[/math]) and tau number([math]L_\tau[/math]).

THE LEPTON CLASSIFICATION

Generation	Lepton	Q	[math]L_e[/math]	[math]L_\mu[/math]	[math]L_\tau[/math]	Mass([math]\frac{MeV}{c^2}[/math])
Generation 1	e	-1	1	0	0	0.511003
Generation 1	[math]\nu_e[/math]	0	1	0	0	0
Generation 2	[math]\mu[/math]	-1	0	1	0	105.659
Generation 2	[math]\nu_\mu[/math]	0	0	1	0	0
Generation 3	[math]\tau[/math]	-1	0	0	1	1784
Generation 3	[math]\nu_\tau[/math]	0	0	0	1	0

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 [math]W^+[/math], [math]W^-[/math] and [math]Z^0[/math] gauge bosons mediate the weak nuclear interactions between particles of different flavors.
- The gluons mediate the strong nuclear interaction between quarks.

THE MEDIATOR CLASSIFICATION

Mediator	Q(charge)	Mass([math]\frac{MeV}{c^2}[/math])	Force
gluon	0	0	strong
photon [math]\gamma[/math]	0	0	electromagnetic
[math]W^{+,-}[/math]	+1, -1	81,800	weak(charged)
[math]Z^0[/math]	0	92,600	weak(neutral)

The Quark Parton Model

In 1964, Gell-Mann and Zweig, against all experimental result, suggested that the fundamental triplet does exist and it contains three so called quarks. The quarks are the building blocks of the baryons and mesons and they cant be found in isolation. The quarks come with three different flavours: up([math]u[/math]), down([math]d[/math]) and strange([math]s[/math]) and their antiparticles. This set of three quarks corresponds to the fundamental SU(3) representation. The quantum numbers of quarks with their antiparticles are given in Table 1. The quarks carry electric charges [math]\pm \frac{2}{3}[/math] and [math]\pm \frac{1}{3}[/math] of the electron charge, which has never been observed before.

Quark	Spin	Parity	[math]e_{q}[/math]	[math]I[/math]	[math]I_{3}[/math]	S	B
[math]u[/math]	1/2	+1	+2/3	1/2	+1/2	0	+1/3
[math]d[/math]	1/2	+1	-1/3	1/2	-1/2	0	+1/3
[math]s[/math]	1/2	+1	-1/3	0	0	-1	+1/3
[math]\bar{u}[/math]	1/2	-1	-2/3	1/2	-1/2	0	-1/3
[math]\bar{d}[/math]	1/2	-1	+1/3	1/2	+1/2	0	-1/3
[math]\bar{s}[/math]	1/2	-1	+1/3	0	0	+1	-1/3

Table 1. Quarks in the Quark Model with their quantum numbers and electric charge n units of electron

Baryons are obtained by as a combination of three quarks([math]qqq[/math]) and mesons by combining a quark and an antiquark ([math]q\bar{q}[/math]). From the rules for combining representation of SU(3) one can show the patterns of baryons and mesons <ref name="Close">Close, F.E. (1979). An Introduction to Quarks and Patrons. London, UK: Academic Press Inc. LTD.</ref>:

[math]q\bar{q} = \bold{3} \otimes \bold{3} = 1 \oplus 8[/math]

[math]qqq = 3 \otimes 3 \otimes 3 = 1 \oplus 8 \oplus 8 \oplus 10[/math]

The constituent quark model describes a nucleon as a combination of three quarks. According to the quark model, two of the three quarks in a proton are labeled as having a flavor ``up" and the remaining quark a flavor ``down". The two up quarks have fractional charge [math]+\frac{2}{3}e[/math] while the down quark has a charge [math]-\frac{1}{3}e[/math]. All quarks are spin [math]\frac{1}{2}[/math] particles. In the quark model each quark carries one third of the nucleon mass.

Since late 1960, inelastic scattering experiments were used to probe a nucleon's excited states. Performed experiments suggested that the charge of the nucleon is distributed on a pointlike constituents of the nucleon. The experiments at the SLAC used high energy electrons scattered by nucleons, where virtual photon is the mediator between the target nucleon and coulomb scattering of an electron. The four-momentum, Q, of the virtual photon serves as a measure of the resolution of the scattering and may be formulated as:

[math]d = \frac{\hbar c}{Q} = \frac{0.2 \;\mbox {GeV} \cdot \mbox{ fm}}{Q}[/math] .

The electron scattering data taken during the SLAC experiments revealed a scaling behavior, which was later defined as Bjorken scaling. The inelastic cross section was anticipated to fall sharply with $Q^2$ like the elastic cross section. However, the observed limited dependence on [math]Q^2[/math] suggested that the nucleons constituents are pointlike dimensionless scattering centers. Independently, Richard Feynman introduced the quark parton model where the nucleons are constructed by three point like constituents, called partons.

Shortly afterwards, it was discovered that partons and quarks are the same particles. In the QPM the mass of the quark is much smaller than in the naive quark model. In the parton model the inelastic electron nucleon interaction via the virtual photon is understood as an incoherent elastic scattering processes between the electron and the constituents of the target nucleon. ``In other words, one assumes that a single interaction does not happen with the nucleon as a whole, but with exactly one of its constituents."<ref name="QCDHighEnergyExperimentsandTheory">Dissertori, G., Knowles, I.K., & Schmelling, M. (2003). Quantum Chromodynamics: High Energy Experiments and Theory. Oxford, UK: Oxford University Press.</ref> In addition, two categories of quarks were introduced, ``sea" and ``valence" quarks. The macroscopic properties of the particle are determined by its valence quarks. On the other hand, the so called sea quarks, virtual quarks and antiquarks, are constantly emitted and absorbed by the vacuum.

The inelastic scattering between the electron and the nucleon can be described by the two structure function, which only depend on [math]x_{B}[/math] Bjorken scaling variable - the fraction of nucleon four-momentum carried by the partons.

It was experimentally shown, that the measured croos section of inelastic lepton-nucleon scattering depends only on $x_{B}$, as it was mention above it is reffered as scaling. If there where additional objects inside the nucleon beside the main building partons, it would introduce new energy scale. The experimental observation of scaling phenomenon was the first evidence of the statement that the quarks are the constituents of the hadron. The results which were obtained from MIT-SLAC Collaboration(1970)are presented below on Figure 1 and 2 <ref name="QCDHighEnergyExperimentsandTheory">Dissertori, G., Knowles, I.K., & Schmelling, M. (2003). Quantum Chromodynamics: High Energy Experiments and Theory. Oxford, UK: Oxford University Press.</ref> <ref name="RobertsSpinStructure">Roberts, R. G. (1990). The structure of the proton. Cambridge Monographs on Mathematical Physics. Cambridge, UK: Cambridge University Press.</ref> . It clearly shows the structure function dependence on [math]x_{B}[/math] variable and independence of the four-momentum transfer squared.

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Figure 1. Scaling behavior of [math]\nu W_2(1/x_{B})=F_2(1/x_{B})[/math] for various [math]Q^{2}[/math] ranges.

Figure 2.Value of [math]\nu W_2(Q^{2})=F_2(Q^{2})[/math] for [math]x_B=0.25[/math] .

The quark parton model predictions are in agreement with the experimental results. One of those predictions is the magnet moments of baryons. For example, the magnet moment of the proton should be the sum of the magnetic moments of the constituent quarks according to the naive quark model <ref name="Anisovich">Anisovich, V.V., Kobrinsky, M.N., Nyiri, J., & Shabelski, Yu. M. (2004). Quark Model and High Energy Collisions. World Scientific Publishing Co. Pte. Ltd.</ref> :

[math]\frac{e}{2m_p} \mu_p= \Sigma_{i=1,2,3} \lt P_{\frac{1}{2}}|\frac{e_q(i)\sigma_z(i)}{2m_p(i)}|P_{\frac{1}{2}}\gt [/math]

Assuming that the masses of light non-strange quarks are just one third of the total nucleon mass [math]m_d=m_u=\frac{m_p}{3}=\frac{m_n}{3}[/math] and expressing the magnetic moment in units of [math]\frac{e}{2m_p}[/math] we get the following result [math]\mu_p=3[/math], which agrees with the findings of experiment. In addition, the quark parton model predictions of magnetic moments of the other baryons are compared with the experimental results below in Table 2. As it can be observed, it is in agreement with the experiment within the accuracy of 20 - 25 [math]%[/math].

Particle	The Quark Model Prediction	Experimental Result
p	3	2.79
n	-2	-1.91
[math]\Lambda[/math]	-0.5	-0.61
[math]\Sigma^+[/math]	2.84	2.46
[math]\Sigma^-[/math]	-1.16	-1.16 [math]\pm[/math] 0.03
[math]\Xi^0[/math]	-1.33	-1.25[math] \pm[/math] 0.01
[math]\Xi^-[/math]	-0.33	-0.65 [math]\pm[/math] 0.04

Table 2. Magnetic moment of baryons in units of nuclear magnetons ([math]\frac{e}{2m_p}[/math]). <ref name="Anisovich">Anisovich, V.V., Kobrinsky, M.N., Nyiri, J., & Shabelski, Yu. M. (2004). Quark Model and High Energy Collisions. World Scientific Publishing Co. Pte. Ltd.</ref>

The Quark Parton Model was succesful explaining the mass of baryons. The baryon masses can be expressed in the quark model using the de Rujula-Georgi-Glashow approach:

[math]m_B = \Sigma_i m_q(i)+ b \Sigma_{i\neq j}\frac{\sigma(i)\sigma(j)}{m_q(i)m_q(j)}[/math]

The difference between the actual experimental results and the predictions is in order of 5 -6 MeV. On the other hand, the similar formula for meson masses fails. The difference in meson mass case, between the experiment and calculation is approximately 100 MeV. This can be explained, by calculating the average mass of the quark in a baryon and meson <ref name="Anisovich">Anisovich, V.V., Kobrinsky, M.N., Nyiri, J., & Shabelski, Yu. M. (2004). Quark Model and High Energy Collisions. World Scientific Publishing Co. Pte. Ltd.</ref> :

[math]\lt m_q\gt _M = \frac{1}{2}(\frac{1}{4}m_\pi + \frac{3}{4}m_p)=303 MeV[/math]

[math]\lt m_q\gt _B = \frac{1}{3}(\frac{1}{2}m_N + \frac{1}{2}m_{\Delta})=363 MeV[/math]

Particle	Prediction ([math]MeV/c^{2}[/math])	Experiment ([math]MeV/c^{2}[/math])
N	930	940 [math]\pm[/math] 2
[math]\Delta[/math]	1230	1232 [math]\pm[/math] 2
[math]\Sigma[/math]	1178	1193 [math]\pm[/math] 5
[math]\Lambda[/math]	1110	1116 [math]\pm[/math] 1
[math]\Sigma^*[/math]	1377	1385 [math]\pm[/math] 4
[math]\Xi[/math]	1329	1318 [math]\pm[/math] 4
[math]\Xi^*[/math]	1529	1533 [math]\pm[/math] 4
[math]\Omega[/math]	1675	1672 [math]\pm[/math] 1

Table 3. Baryon mass predictions compared with experimental findings.<ref name="Anisovich">Anisovich, V.V., Kobrinsky, M.N., Nyiri, J., & Shabelski, Yu. M. (2004). Quark Model and High Energy Collisions. World Scientific Publishing Co. Pte. Ltd.</ref> <ref name="Camalich"> Camalich, J.M., Geng, L.S., & Vicente Vacas, M.J. (2010). The lowest-lying baryon masses in covariant SU(3)-flavor chiral perturbation theory. arXiv:1003.1929v1 [hep-lat]</ref>

Notes

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Semi Inclusive Deep Inelastic Scattering

Kinematic variables in deep inelastic scattering

Kinematic variable	Description
[math]k = (E, k)[/math], [math]k = (E^', k^')[/math]	4 - momenta of the initial and final state leptons
[math]\theta[/math], [math]\phi[/math]	Polar and azimuthal angle of the scattered lepton
[math]P^{lab} = (M, 0)[/math]	4 - momentum of the initial target nucleon
[math]q = k - k^'[/math]	4 - momentum of the virtual photon
[math]Q^2 = - {q^2} ^{lab} = 4EE^'\sin^2 \frac{\theta}{2}[/math]	Negative squared 4 - momentum transfer
[math]\nu = \frac {P q}{M} =^{lab} E - E^'[/math]	Energy of the virtual photon
[math]x = \frac {Q^2}{2 P q} = \frac {Q^2}{2M \nu}[/math]	Bjorken scaling variable
[math]y = \frac{Pq}{Pk} =^{lab} \frac{\nu}{E}[/math]	Fractional energy of the virtual photon
[math]W^2 = (P + q)^2 = M^2 + 2M\nu - Q^2[/math]	Squared invariant mass of the photon-nucleon system
[math]p = (E_h, p)[/math]	4 - momentum of a hadron in the final state
[math]z = \frac{Pp}{Pq} =^{lab} \frac{E_h}{\nu}[/math]	Fractional energy of the observed final state hadron