Creation of Two Charginos and Neutralino Via Different Propagators

We investigated the creation of two Charginos ( (cid:1)(cid:2) ± ) and neutralino ( (cid:1)(cid:2) (cid:4) ) owing to electron-positron annihilation via the process (cid:5) (cid:6) (cid:7)(cid:8) (cid:9) (cid:10) + (cid:5) (cid:12) (cid:7)(cid:8) (cid:13) (cid:10) → (cid:1)(cid:2) (cid:15)(cid:6) (cid:7)(cid:8) (cid:16) (cid:10) + (cid:1)(cid:2) (cid:17)(cid:12) (cid:7)(cid:8) (cid:18) (cid:10) + (cid:1)(cid:2) ℓ° (cid:7)(cid:8) (cid:21) (cid:10) and estimated the cross section for this interaction in the Minimal Supersymmetric Standard Model (MSSM). There are three gatherings of Feynman graphs which taken by the sorts of the propagators. Group (I) when (cid:1)(cid:2) and (cid:4) boson are propagators, Group (II) when (cid:1)(cid:2) (cid:4) and h (cid:4) (lighter Higgs boson) propagators and Group (III) (cid:1)(cid:2) (cid:4) and H (cid:4) Higgs boson) propagators, (cid:10) in group III At S interval (1000- 2100) Gev, the best value of σ is (cid:7) 0.072 (cid:10) Pb in-group (I). When masses of Charginos are m /0 12 = 700 GeV , m /0 67 = 700GeV and mass of neutralino is m 8(cid:2) ℓ9 = 800 GeV

Which leads to about 40 GeV increase [12], where m h mass of the top quark From these constraints, it also follows that m H < m J < m I , m I,H = [M J + M W ± k M J + M W − 4M W M J cos 2β] Where the masses m I , m H , m J , m I ± of the Higgs particles H°, h°, A°, H ± respectively. The two angles β and α fixed in terms of the Higgs boson masses [13]. The angel α lie in the interval −π 2 s ≤ α ≤ 0. And the angel β can be taken to lie in the interval 0 ≤ β ≤ π 2 s . [14] In particle physics, slepton is a Superpartner of a lepton that described by Supersymmetry. It has the same flavor and electric charge alike leptons and their spin is zero. For example, selectron ̃u is superpartner of electron [4].

Cross-sections meaning
The term "cross section" its technical meaning is very different from the common usage. In everyday speech, "cross section" refers to a slice of an object. A particle physicist might use the word this way, but more often it is used to mean the probability that two particles will collide and react in a certain way. For instance, when CMS physicists measure the "proton-proton to top-antitop" cross section, they are counting how many top-antitop pairs were created when a given number of protons were fired at each other. But why use "cross section" when alternatives like "probability" and "reaction rate" exist? Cross section is independent of the intensity and focus of the particle beams, so cross section numbers measured at one accelerator can be directly compared with numbers measured at another, regardless of how powerful the accelerators are.

Scattering cross-sections
Almost everything we know about nuclear and atomic physics has been discovered by scattering experiments, e.g., Rutherford's discovery of the nucleus, the discovery of sub-atomic particles (such as quarks), etc. In low energy physics, scattering phenomena provide the standard tool to explore solid state systems, e.g., neutron, electron, xray scattering, etc. As a general topic, it therefore remains central to any advanced course on quantum mechanic A scattering cross-section, σ, is a quantity proportional to the rate at which a particular radiation-target interaction occurs. More specifically, if the incoming radiation is considered as being composed of quanta or 'particles' (for example, photons or neutrons), a cross-section is a scattering rate (number of scattering events per unit time) per unit incident radiation flux, where the latter is the number of incident particles striking the target surface per unit time per unit area. In cases where the radiation is being treated as a continuous classical wave, as in the case of long-wavelength electromagnetic radiation, scattering cross-sections are determined by dividing the power of the scattered wave by the intensity of the incident wave. Dimensionally, a cross-section represents an area, with the basic unit being the barn, which represents an area of 10−28 m 2 . A scattering cross-section should not be interpreted as a true geometric cross-sectional area, but as an effective area that is proportional to the probability of interaction between the radiation and target.
In a real scattering experiment, information about the scatterer can be figured out from the different rates of scattering to different angles. Detectors are placed at various angles (y, z Of course, a physical detector collects scattered particles over some nonzero solid angle. The usual notation for infinitesimal solid angle is "{ = | !y"y"z. The full solid angle (all possible scatterings) is ∫ "{ = 4~ the area of a sphere of unit radius.
The differential cross section, written "•/"{ is the fraction of the total number of scattered particles that come out in the solid angle "{, so the rate of particle scattering to this detector is !"•/"{, with n the beam intensity as defined above. From the differential, we can obtain the total cross section by integrating over all solid angles The cross section depends sensitively on energy of incoming particles

Feynman diagram
Richard Feynman developed a technique referred to as Feynman diagrams. The essence of these diagrams is that they portrayed quantum events as trajectories. For example, along the time axis an electron and positron (antielectron) particle annihilate each other producing a virtual photon that becomes a quark-antiquark pair. Feynman diagrams are in common use in particle physics. The value of these diagrams is to facilitate the calculation of interactions between particles. The introduction of these diagrams contributed to the theory of QED, first introduced by Dirac. Along with two others, Richard Feynman was awarded the Nobel Prize in 1965 for work in electrodynamics and consequences for the physics of elementary particles.
These diagrams are one of the fundamental tools used to make precise calculations for the probability of occurrence of any process by physicists. Different diagrams can represent a single interaction process, and the contribution from each diagram is taken into consideration while calculating this probability. Although the mathematical expressions involved in calculating these probabilities are quite complex, a lot simpler as compared to other techniques.
Although the American theoretical physicist Richard Feynman first introduced these diagrams only as a bookkeeping device for simplifying lengthy calculations in the area of quantum electrodynamics, these diagrams have come a long way now. Even David Kaiser once quoted, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations. Feynman diagrams have revolutionized nearly every aspect of theoretical physics." Undoubtedly, these diagrams are one of Feynman's finest contributions ever made to the Physics fraternity.
• In nature neutralino dark matter observed experimentally either indirectly by using … ray and neutrino telescopes or directly by using an array of semiconductor detectors and through experiments such as Cryogenic dark matter search (CDMS) it is a series of experiments designed to detect particle dark matter directly in the form of WIMPs [18,19]. • The heavier neutralinos typically decay through a neutral Z boson to a lighter neutralino or through a charged W boson to a light chargino [20] • Produced in pairs via s-channel …/& exchange [21,22]. The expression of the mean lifetime ( † / 0 g ±) of ± in terms of ∆R / ĝ and expected to be typically a fraction of a nanosecond. Lifetime of charginos between 0.1 and 10 ns [23]. • The charginos decay into the lightest neutralino °, which is taken to be stable, and a pair of fermions (‰) which are quarks and antiquarks or leptons and neutrinos: [24] • ± → °+ ‰‰‹ • The lightest chargino mass greater than 103.5 GeV [25] • The second chargino ± is generally expected to be significantly heavier than the first state. [24] • There are three variables or soft terms ( > , OE and tan •) in chargino mass matrix and four variables or soft terms ( > , > , OE and tan •) in neutralino mass matrix [26] Where: > is the soft-breaking bino mass > is the soft-breaking wino mass OE is the Higgsino superpotential mass parameter tan • is the ratio of the two Higgs vacuum expectation values The relation in grand unification theory GUT between > and > : Where y • is the Weinberg angle and p + p = p + p + p (1) s = σ + p (2) The cross section (σ) for the process e p + e p → χ w p + χ " p + χ ℓ° p can be written in the form σ = • π |M| dx dy dσ Λ S, m , m Λ S, σ, m 3

Calculation Cross sections in (Pb
Where M is the matrix element, by applying Feynman rules we can write the M-matrix for the Feynman diagram and the trace thermos used to calculate the square matrix (|>| ), the integration performed using a simple approximation obtained by an improved Weizsacker-Williamson procedure [27,28]. Where: Λ x, y, z = [x + y + z − 2x y − 2x z − 2y z ] ⁄ (4) Then, by using Mathematica program the integration simplifying and the limit of integration are

Calculation Cross Sections in (Pb) for Group (I):
By applying Feynman rules and using equation (3) and Mathematica program, the cross sections calculated as a function of center of mass energy for the Feynman diagram of fig. (2). the results given in figs.3 (a-d) by interchanging the mass of charginos (m 8 ® 2 , m 8 7 ) at different mass of Neutralino m / 0 ℓ°f or the process e p + e p → χ w p + χ " p + χ ℓ° p m°8

Results for Group (II):
After studding the Feynman rules and calculate the cross sections as a function of center of mass energy (S) for the process + → + + ℓ° via χ and h propagators in fig.5 (a-d) Fig. (3.a)  Fig. (3.b)  Fig. (3.c)  Fig. (3.d)

Results for Group (III):
After studding the Feynman rules and calculate the cross sections as a function of center of mass energy (S) for the process + → + + ℓ° via χ and H propagators in fig.7 (a-d) we found that: At S increase from 1450 to 2200 we have different maximum values from the cross-sections at different values of Chargino mass (m / 1 2 , m / 6 7 ) and different value of neutralino mass m 8 ℓ 9 . From table (3) the best value of σ is 1 × 10 = Pb when masses of Charginos are m / 1 2 = 700 GeV, m / 6 7 = 600GeV and m 8 ℓ 9 = 800 GeV 6. Discussion Figs. (3, 5, 7). Shows the cross-sections calculation for the process e P + e P → P + P + χ ℓ = P as a function of center of mass energy S, via ° and Z boson propagators group (I), via ° and h boson propagators group (II) and via ° and H boson propagators (III) respectively. If center of mass energy S increases the cross-sections increase, but after certain value of S the value of cross sections decrease and the range of center of mass energy from (1000-2100) The following table shows that the peak values of the cross section σ (Pb) for each mode and the corresponding center of mass energy S (GeV) at different masses of neutralino χ ℓ = and different masses of charginos , to determine the best value of cross-section for the reaction e P + e P → P + P + χ ℓ = P

Conclusion
From table (4), we have success to identify the scenario for highest cross section for the reactione P + e P → P + P + χ ℓ = P .