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

\section{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.

\begin{center} \begin{tabular}{l*{6}{c}r} Quark & Spin & Parity & e_{q} & $I$ & $I_{3}$ & $S$ & $B$ \\

\hline

$u$ & 1/2 & +1 & +2/3 & 1/2 & +1/2 & 0 & +1/3\\ $d$ & 1/2 & +1 & -1/3 & 1/2 & -1/2 & 0 & +1/3\\ $s$ & 1/2 & +1 & -1/3 & 0 & 0 & -1 & +1/3\\

\bar{u}	     & 1/2 & -1  & -2/3 & 1/2 & -1/2 & 0 & -1/3\\

\bar{d} & 1/2 & -1 & +1/3 & 1/2 & +1/2 & 0 & -1/3\\ \bar{s} & 1/2 & -1 & +1/3 & 0 & 0 & +1 & -1/3\\

\hline

   \end{tabular}

\end{center} \begin{center} \caption{Table 1. Quarks in the Quark Model with their quantum numbers and electric charge n units of electron charge } \end{center}

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\cite{1}:

[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 $Q^2$ 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."\cite{2} 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 \cite{2} \cite{3} . It clearly shows the structure function dependence on [math]x_{B}[/math] variable and independence of the four-momentum transfer squared.

\begin{center} \includegraphics[scale=0.25]{Figure_1DependenceOnXbjorken} \end{center} \begin{center} \caption{Figure 1. Scaling behavior of $\nu W_2(1/x_{B})=F_2(1/x_{B})$ for various $Q^{2}$ ranges.} \end{center}

\begin{center} \includegraphics[scale=0.25]{Figure_2IndependenceOnQsqrd} \end{center} \begin{center} \caption{Figure 2.Value of $\nu W_2(Q^{2})=F_2(Q^{2})$ for $x_B=0.25$ .} \end{center}

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 \cite{4}:

[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].

\begin{center} \begin{tabular}{l*{6}{c}r} Particle & The Quark Model Prediction & Experimental Result \\

\hline

p & 3 & 2.79 \\ n & -2 & -1.91 \\ $\Lambda$ & -0.5 & -0.61\\ $\Sigma^+$ & 2.84 & 2.46 \\ $\Sigma^-$ & -1.16 & -1.16 \pm 0.03 \\ $\Xi^0$ & -1.33 & -1.25 \pm 0.01\\ $\Xi^-$ & -0.33 & -0.65 \pm 0.04 \\

\hline

   \end{tabular}

\end{center} \begin{center} \caption{Table 2. Magnetic moment of baryons in units of nuclear magnetons ($\frac{e}{2m_p}$). \cite{4} } \end{center}

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 \cite{4} :

[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]

\begin{center} \begin{tabular}{l*{6}{c}r} Particle & Prediction (MeV/c^{2}) & Experiment (MeV/c^{2}) \\

\hline

N & 930 & 940 \pm 2 \\ $\Delta$ & 1230 & 1232 \pm 2 \\ $\Sigma$ & 1178 & 1193 \pm 5 \\ $\Lambda$ & 1110 & 1116 \pm 1\\ $\Sigma^*$ & 1377 & 1385 \pm 4\\ $\Xi$ & 1329 & 1318 \pm 4\\ $\Xi^*$ & 1529 & 1533 \pm 4\\ $\Omega$ & 1675 & 1672 \pm 1\\

\hline

   \end{tabular}

\end{center} \begin{center} \caption{Table 3. Baryon mass predictions compared with experimental findings. \cite{4}\cite{5}} \end{center}

\bibliographystyle{abbrv} \bibliography{simple}

\begin{thebibliography}{[AHU]} \bibitem[1]{1}Close, F.E. (1979). \textit{An Introduction to Quarks and Patrons}. London, UK: Academic Press Inc. LTD. \bibitem[2]{2}Dissertori, G., Knowles, I.K., \& Schmelling, M. (2003). \textit{Quantum Chromodynamics: High Energy Experiments and Theory}. Oxford, UK: Oxford University Press. \bibitem[3]{3}Roberts, R. G. (1990). \textit{The structure of the proton. Cambridge Monographs on Mathematical Physics}. Cambridge, UK: Cambridge University Press. \bibitem[4]{4}Anisovich, V.V., Kobrinsky, M.N., Nyiri, J., \& Shabelski, Yu. M. (2004). \textit{Quark Model and High Energy Collisions}. World Scientific Publishing Co. Pte. Ltd. \bibitem[5]{5}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]

\end{thebibliography}

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

Semi-Inclusive Scattering

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:
350px

The fragmentation matrix, the probability of the struck quark fragmenting into a hadronic state [math]|P_h, S_h;X\gt [/math] can be written i the following way:

[math]\Xi_{i,j} = \Sigma_X \int \frac{d^3P_X}{(2\pi)^32E_X} \int d^4 \xi e^{ik \times \xi} \times \lt 0|\psi_i(\xi)|P_hS_h,X\gt \lt P_hS_h,X|\bar{\psi}_j(0)|0\gt [/math]

where

[math]P_X[/math] - the momentum of the final hadron

[math]E_X[/math] - the energy of the final hadron

The conservation of 4-momentum between the two states k=p+q and the momenta [math]P_h[/math] and [math]P_X[/math] 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):

[math]z=\frac{E_h}{\nu}[/math]

Semi-inclusive deep inelastic scattering experiments require a polarized lepton(electron) beam with a good duty factor??????, polarized targets and a spectrometer with wide range of acceptance for the created hadrons. Using CLAS (eg1)experiment, the produced hadrons are identified as well as pions from ammonia target(others).

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)

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)