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The comparison between the muon and the neutrino as probes of the nucleon structure is presented. The prediction of the structure functions, quark distributions, leptonic currents, and cross section led us to obtain some of the features of the electro-weak interactions in the deep inelastic scattering. A perturbation technique is used to evaluate the leptonic current that is assumed to be a complex quantity. The imaginary part of which represents the rate of absorption. On the other hand, the quarks wave functions forming the nucleon are extracted from experimental data for neutrino-nucleon and muon-nucleon collisions. A numerical technique is applied to analyze the data of the experiments CERN-NA-2 and CERN-WA25, to evaluate the quark functions and hence to calculate the hadronic current. It is found that the quark distribution functions predicted by the muon as a probe is slightly shifted up compared with that of the neutrino. Finally, the differential cross section is calculated in terms of leptonic and hadronic currents.

Electroweak measurements are a very important part of the physics program targeting the measurement of electroweak precision observables [

The study of structure functions of the nucleon offers a unique window on the internal quark structure of stable baryons. This provides insight into the two defining features of QCD. From measurements of structure functions, we can deduce the fraction of the nucleon momentum and spin carried by quarks.

During more than three decades of measurements at many accelerator facilities worldwide, appreciable amount of data has been collected, covering several orders of magnitude in both kinematic variables (x, the fraction of the nucleon momentum carried by the struck quark, and the momentum transfer squared, Q^{2}).

Although a large body of structure function data exists over a wide range of x and Q^{2}, the region x > 0.6 is not well explored. For x ≥ 0.4 the contributions from the

If the interaction between quarks that are spectators to the deep inelastic collision is dominated by one-gluon exchange [

Determining d/u experimentally would lead to important insights into the mechanisms responsible for spin- flavor symmetry breaking. In addition, quark distributions at large x are a crucial input for estimating backgrounds in searches for new physics beyond the Standard Model at high energy colliders.

From this point of view, we are looking forward to use the data of muon (as electroweak particle) and neutrino (as weak particle) to probe the structure of the nucleon.

The electroweak theory is the unified description of both electromagnetic force and weak force. Although the weak force is 10^{6} weaker than the electromagnetic force at low energies, they would merge into a single electroweak force at energies of the order of 100 GeV.

The W and Z bosons are the mediators of the weak interaction, as in the case of the photon, which is mediator of the electromagnetic interaction. There are several papers [

The paper is organized as follows: After this introduction, Section 2 discusses the general features of neutrino-nucleon and muon-nucleon interactions. In Section 3, a prediction of the quark functions is presented. Section 4 looks to the Simple model for calculation of the DIS. Finally concluding Remarks are given in Section 5.

We use the data of the experiment CERN-WA25 [_{2} and F_{3} in the variables, Bjorken scale x, and the square of momentum transfer Q^{2}. We carry out the Exponential fitting by MATHEMATICA9.0 to parameterize the function F_{2} and F_{3} for neutrino-proton and neutrino-neutron interactions in the form

The data in ^{2}. The figures show also that ^{2} and increases rapidly toward the DIS (x®0).

We used the data of the experiment CERN-NA-2 [_{2}(x, Q^{2}) for muon-proton and muon-neutron interactions using the previous technique.

From

functions of the nucleon F_{2} in both cases of muon and neutrino interactions. In each value of x, F_{2} fluctuates about a certain value showing scaling behavior with Q^{2}. F_{2} increases with x that fits the deep inelastic behavior of the reaction.

_{2} in both cases of neutrino and muon with the proton as a target, at two scales Q^{2} = 2 (Gev/c)^{2} and Q^{2} = 40 (Gev/c)^{2} as a function of x.

Apart from the small changes, F_{2} is approximately independent of Q^{2} and independent of any length scale. Such scaling proves that the nucleon is composed of point-like constituents.

The difference between the distributions of neutrino and muon at very DIS (small values of x) is large. As x increases, both come close. i.e. at large x the features of both weak and electromagnetic interaction disappear. In addition, as Q^{2} increases, the structure function of neutrino increases quickly at very low x, while the structure function of muon increases very slowly with energy.

In this section, we will apply an empirical method to determine the quark functions for (u & d) and their anti-

quarks in terms of the structure functions F_{2}(x) and xF_{3}(x) that were extracted from the neutrino deep inelastic scattering experiments. Assumption (SU (2) Isospin); neutron is just proton with u⇔d: Proton = uud; Neutron = ddu. Making the approximation that sets the Cabibbo angle to zero, we obtain the following correspondence relations;

where

Neutrino-nucleon collision (F_{2}) | ||
---|---|---|

Parameter | Proton | Neutron |

a_{0}_{ } a_{1 } a_{2 } b_{0 } b_{1 } b_{2}_{ } | 0.975 −1.364 −0.273 −0.035 −0.266 −0.358 | 1.722 −1.850 −0.595 0.031 −0.174 0.268 |

interactions respectively. By solving last equations algebraically, we get for the quark and anti-quark functions as;

Neutrino-nucleon collision (F_{3}) | ||
---|---|---|

Parameter | Proton | Neutron |

a_{0}_{ } a_{1 } a_{2 } b_{0 } b_{1 } b_{2}_{ } | 0.490 0.684 −2.471 −0.029 0.274 −0.399 | 0.539 2.977 −5.361 0.031 −0.205 0.345 |

Muon-nucleon collision (F_{2}) | ||
---|---|---|

Parameter | Proton | Neutron |

a_{0}_{ } a_{1 } a_{2 } b_{0 } b_{1 } b_{2}_{ } | 0.353 −0.012 −0.918 −0.004 0.034 −0.045 | 0.301 −0.379 −0.221 −0.012 0.082 −0.117 |

We shall use the features of the structure functions F_{2}(x) that extracted from the deep inelastic scattering experiments of muon-nucleon collision. Assumption (SU (2) Isospin): neutron is just proton with u⇔d: Proton = uud; Neutron = ddu. Making the approximation that sets the Cabibbo angle to zero, we obtain the correspondent euations;

where

and for anti-quarks we can use the correspondence

which lead to the anti-quark functions as

First judge in

The quark and anti-quark wave functions deduced by neutrino and muon are displayed in

In this section, the goal is to use the quark functions, deduced by neutrino and muon, in Section 3, into a simple

model for calculating the hadronic current and consequently the deep inelastic differential cross section in both cases. For this aim, we study the lepton current and quark current. In the leptonic current, we view a neutrino as weak particle and a muon as electromagnetic particle.

In this model we assume that the lepton interacts with nucleons via the intermediate vector boson (IVB) as shown in

It is assumed that the interactions go through electromagnetic field mediated by a virtual photon or weak field mediated by Z^{0} or W^{±} bosons. Our strategy for studying these interactions is to use Feynman diagram of two vertices. The first is related to the field of the lepton and can be either electromagnetic interaction in the case of muon as probe, or weak interaction for the case of neutrino. The second vertex is a strong interaction which excites the nucleon that decays producing many hadrons in the final state.

In the presence of currents, the total interaction matrix element is

The leptonic current describes the transition of leptons from initial to final state during the scattering by the field and is defined as;

where

The 4-component matrix u describes neutrino or muon with spin 1/2 is;

Since the neutrino is massless and the muon has small mass relative to the incident energy, so we can neglect it. Considering the projectile is initially moving in the z-direction, then,

On the other hand, we used the perturbation technique to get the scattered wave function of the lepton as;

The sum runs over the possible orders of perturbation. And r is the distance from the scattering center,

Feynman representation of the neutrino-nucleon scattering is shown in ^{±} or Z^{0} with effective mass about 80 GeV.

In this case the scattering is due to weak field, then it is sufficient to consider only one term in the perturbation series (where g_{w} is the weak coupling constant)

Then the first component of the leptonic current J_{x} corresponding to µ = 1 in Equation (8), is given by

The integrals in Equation (13) are regarded as the average of the current allowed in the available space inside the nucleon of radius R. This leads to an analytical form of the current J_{x} as

Similarly, for the y and z components, corresponding to µ = 2, 3 respectively, we get;

Feynman representation of the muon-nucleon scattering is shown in

(17)

where g_{EM} is the electromagnetic coupling constant.

Number of exchange photons | Probability (f^{2}) |
---|---|

One photon | |

Two photons | |

Three photons |

Then the first component of the leptonic current J_{x} corresponding to µ = 1 in Equation (8), is given by

The integrals in Equation (18) are regarded as the average of the current allowed in the available space inside the nucleon of radius R. this leads to an analytical form of the current J_{x} as

Similarly, J_{y} and J_{z} corresponding to µ = 2, 3 respectively, are found to be

The leptonic current density for the both cases is a complex function of the momentum transfer q. The imaginary part measures the absorption rate. The absolute values of the current components J_{x} and J_{y} are equal due to the azimuthal symmetry of the problem (

trino is proportional inversely with the square of the mass of the IVB as well as with q^{4}, while in the case of muon the leptonic current density is proportional directly with the factor

The results of the quark and anti-quark functions are used in calculating the quark currents as,

As mentioned above at the beginning of Section 4, the matrix element associated with the propagator of the field as well as with the leptonic current and hadronic (quark) current, then the matrix element

no interaction that is associated with the propagator

And, the matrix element (M_{µ}) for the muon interaction that is associated with the propagator

The differential cross section decreases with x. It has the same behavior for neutrino but with lower magnitude.

In this analysis we studied the deep inelastic scattering using neutrino and muon as probes.

_{2} of the neutrino and muon diverges at very DIS (small-x). As x increases, the two come close i.e. at large x the features of both weak and electromagnetic interaction disappear.

^{2}, although the structure function of neutrino increases quickly at very low x, however, the structure function of muon increases very slowly with energy.

^{4}, while in the case of muon the leptonic current density is proportional directly with the factor

^{17}. This is due to the relative weight factor of coupling constant of both fields.

Z with a propagator of the form

^{2}.