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

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 antileptons and antiquarks, all signs in the above tables would be reversed for them. Also, quarks and antiquarks 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 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 [math]q^2=(k-k^')[/math] decreases and it is function of [math]x=\frac{-(k-k^')^2} {2p(k-k^')}[/math]

Assuming that [math]x_i[/math] is the fraction of the proton carried by the quark [math]x_i = \frac{k^+}{p^+}[/math]. Because [math]k^2 = 2k^+ k^- - k^2 _T = 0[/math], [math]k^- = \frac{k^2 _T} {2x_i p^+}[/math] and scalar product of k and p is [math]k^+ q^- + k^- q^+ = x_i p^+ q^- = x_i p q = x_i M \nu[/math]

[math]\delta [(k + q)^2] = \frac{1} {2M\nu}\delta(x_i - x)[/math] (3)

Using (1) and (2)(????????????) the hardonic tensor can be expressed as

[math]W_{\mu \nu}(q, p) = \sum_i e^2 _i \int \frac{d^4k}{2M\nu}[f^i _+ (p.k) + f^i _- (p.k)] \delta (x_i - x) [2k_\mu k_\nu + k_\mu q_\nu + q_\mu k_\nu - g_{\mu \nu}k.q][/math] (4)

For example, take [math]\mu = \nu = 2[/math] and go to the proton rest frame(??????????/) in this case [math]W_{22} = W_1[/math] and using [math]d^4k = \frac{\pi}{2} \frac{dx}{x} dk^2 dk_T ^2[/math],

[math]W_{22} = \sum_i e^2 _i \int \frac{dxdk^2 _T}{2xM\nu}[f^i _+ (p.k) + f^i _- (p.k)] \delta (x_i - x) xM\nu[/math] (5)

or

[math]MW_1 (p, q) \rightarrow F_1 (x) = \sum_i e_i ^2 q_i (x)[/math] (6)

where

[math]q_i(x) = \frac{\pi}{4} \int dk^2 _T [f^i _+ (p.k) + f^i _- (p.k)][/math] (7)

Inclusive Scattering

Kinematic variables in 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

Quark distribution Functions

define and describe [math] q_v (x)[/math] and [math]\Delta q_v (x)[/math] here

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
(1)
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,
[math]\int u(x)dx = 2[/math] (2)
[math]\int d(x)dx = 1[/math] (3)

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,

[math]u(x) = u_v (x) + u_s (x)[/math]
[math]d(x) = d_v (x) + d_s (x)[/math]
q(x) is the unpolarized distribution function and [math]\Delta q(x)[/math] - the polarized.

The structure functions in the quark parton model can be written in terms of quark distribution functions,

[math]F_1 (x) = \frac {1} {2}[/math] [math]\sum[/math][math]e_q^{2} (q^+ (x) + q^- (x)) = \frac{1} {2} \sum e_p^{2} q(x)[/math] (4)

[math]g_1 (x) = \frac {1} {2}[/math] [math]\sum[/math][math]e_q^{2} (q^+ (x) - q^- (x)) = \frac{1} {2} \sum e_p^{2} \Delta q(x)[/math] (5)

The unpolarized structure function [math]F_1 (x)[/math] - measures the total quark number density in the nucleon, [math]g_1 (x)[/math] - the polarized structure function is helicity difference quark number density.

The unpolarized structure functions [math]F_i (x)[/math]( i= L, R, O ) should satisfy the following inequalities,
[math]xF_3 (x)\leq 2xF_1 (x)\leq F_2 (x)[/math](1)

If [math]Q^2[/math] is increased so that the weak part of the natural current will be included, that means we have [math]\gamma[/math]-exchange, Z-exchange and [math]\gamma[/math]-Z interference. The cross-section can be expressed as

[math]\frac {d\sigma (e_L^{-} p)} {dxdQ^2} = \frac {4\pi \alpha^2} {xQ^4} [xy^2F_1 (x) + (1-y)F_2(x) + xy(1-\frac{1} {2} y)F_3(x)][/math] (2)

The structure functions in terms of the parton distributions can be written as,

[math]F_2 (x) = 2xF_1 (x) = \sum_i A_i (Q^2) [xq_i (x) + xq_i^{'}(x)][/math](3)
[math]F_3 (x) = \sum_i B_i (Q^2) [q_i (x) - q_i^{'}(x)][/math](4)

where

[math]A_i (Q^2) = e_i^{2} - e_i g_{Le} (g_{Li} + g_{Ri})P_z + \frac {1}{2}g_{Le}^2(g_{Li}^2 + g_{Ri}^2)P_z^2[/math] (5)

[math]B_i (Q^2) = e_i g_{Le} ( - g_{Li} + g_{Ri})P_z - \frac {1}{2}g_{Le}^2( - g_{Li}^2 + g_{Ri}^2)P_z^2[/math] (6)
where [math]e_i[/math], [math]q_{Li}[/math], [math]q_{Ri}[/math] are the charge, left- and right-handed weak couplings of a ith type quark and [math]g_{Le}[/math], [math]g_{Re}[/math] corresponding couplings for the electron.

Polarized

Both models, pQCD and a hyperfine perturbed constituent quark model(CQD), show that as the scaling variable [math]x_{Bj}[/math] goes to one the double spin asymmetry [math]A_{1,N}[/math] is unity. On the other hand, CQM with SU(6) symmetry predicts that at [math]x_{Bj}[/math] = 1, [math]A_{1,n}[/math] = 5/9 for the proton, [math]A_{1,n}[/math] = 0 for the neutron and [math]A_{1,d}[/math] = 1/3 for the deuteron. The double spin asymmetry and the ratio of the polarized valence down quark distribution function to the unpolarized [math]({\Delta d_v} / {d_v}) [/math] can give knowledge of these two different results.

The inclusive double polarization asymmetries [math]A_N[/math] in the valence region, where the scaling variable [math]x_{Bj} \gt 0.3[/math]can be written in terms of polarized [math]\Delta q_v (x)[/math] and unpolarized [math] q_v (x)[/math] valence quark distributions,

[math]A_{1, p}^{I} = \frac {4\Delta u_v (x) + \Delta d_v (x)} {4 u_v (x) + d_v (x)} [/math] (1)
[math]A_{1, n}^{I} = \frac {\Delta u_v (x) + 4\Delta d_v (x)} {u_v (x) + 4d_v (x)} [/math] (2)

The semi-inclusive pion electro-production asymmetries can be written in terms of the valence quark distributions
[math]A_{1, p}[/math]^{[math]\pi^+ - \pi^-[/math]} = [math]\frac {4\Delta u_v (x) - \Delta d_v (x)} {4 u_v (x) - d_v (x)} [/math] (3)

[math]A_{1,2H}[/math]^{[math]\pi^+ - \pi^-[/math]} = [math]\frac {\Delta u_v (x) + \Delta d_v (x)} { u_v (x) + d_v (x)} [/math] (4)

where

[math]A[/math]^{[math]\pi^+ - \pi^-[/math]} =[math]\frac {\sigma^{\pi^+ - \pi^-}_{\uparrow \downarrow} - \sigma^{\pi^+ - \pi^-}_{\uparrow \uparrow}} {\sigma^{\pi^+ - \pi^-}_{\uparrow \downarrow} + \sigma^{\pi^+ - \pi^-}_{\uparrow \uparrow}} [/math] (5)

where [math]\sigma^{\pi^+ - \pi^-}[/math] 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 [math]\pi^+[/math] and [math]\pi^-[/math] asymmetries:

[math]A_{1,2H}^{\pi^+ - \pi^-} = \frac {A^{\pi^+}} {1 + \frac {1} {R_p^{{\pi^+}/{\pi^-}}} } [/math] - [math]\frac {A^{\pi^-}} {1 + R_p^{{\pi^+}/{\pi^-}} } [/math] (6)

where [math]R_{2H}^{\pi^+/\pi^-} = \frac{\sigma^{\pi^+}} {\sigma^{\pi^-}}[/math] and

[math]A^{\pi^+ (\pi^-)} = \frac {\sigma^{\pi^+ (\pi^-)}_{\uparrow \downarrow} - \sigma^{\pi^+(\pi^-)}_{\uparrow \uparrow}} {\sigma^{\pi^+ (\pi^-)}_{\uparrow \downarrow} + \sigma^{\pi^+(\pi^-)}_{\uparrow \uparrow}} [/math] (7)

An asymmetry [math]\Delta R_{np} ^{\pi^+ + \pi^-} = \frac {\Delta\sigma_p^{\pi^+ + \pi^-} - \Delta\sigma_n^{\pi^+ + \pi^-}} {\sigma_p^{\pi^+ + \pi^-} - \sigma_n^{\pi^+ + \pi^-}} = \frac {g_1^p - g_1^n} {F_1^p - F_1^n} (x, Q^2)[/math] (8)
where [math]F_1[/math] is the unpolarized structure function and [math]g_1[/math] the scaling polarized structure function.

The last equation can be expressed as
[math]\triangle R_{np} ^{\pi^+ + \pi^-} = R_{n/p}[\frac {A_p^{\pi^+}} {1 + \frac {1} {R_p^{{\pi^+}/{\pi^-}}} } + \frac {A_p^{\pi^-}} {1 + R_p^{{\pi^+}/{\pi^-}} } ] + R_{p/n}[\frac {A_n^{\pi^+}} {1 + \frac {1} {R_n^{{\pi^+}/{\pi^-}}} } + \frac {A_n^{\pi^-}} {1 + R_n^{{\pi^+}/{\pi^-}} } ][/math] (9)

using the nomenclature of (6) equation, we have

[math]R_{i/j} = \frac {\frac {1 + (1-y)^2} {2y(2 - y)} } {1 - \frac {R_{i/j}^{\pi^+}} {1 + \frac{1}{R_j^{\pi^+/\pi^-} }} - \frac {R_{i/j}^{\pi^+}} {1 + R_j^{\pi^+/\pi^-} }} [/math]

[math]R_{i/j}^{\pi^c} = \frac {\sigma_i ^{\pi^c}} {\sigma_j ^{\pi^c}} [/math]

Appendix

The hardonic tensor [math]W_{\mu \nu}[/math] can be written for unpolarized scattering in the following form

[math]W_{\mu \nu} = \sum_i \sum_s \int d^4k f_s ^i (p,k) w_{\mu \nu} ^i (q,k) \delta [(k + q)^2][/math] (1)

i - is the summation over the quark flavors and s - the quarks helicities. [math]w_{\mu \nu}[/math] term is the interaction of the virtual photon with the quark of momentum k and in case of massless on-shell quarks is

[math]w_{\mu \nu}^i = \frac{1}{4} e_i ^2Tr[k\gamma_\mu (k+q) \gamma_\nu][/math](2)