Flux Determination through Ultra High Energy Muons by Using Pair-Meter Technique

Cosmic rays are free source of elementary particles and have enormous range of energies, still they produce small amount of striking rate at the detector level. Uncontrollable fluxes have been produced, which is the major disadvantage of cosmic rays. The study of cosmic ray spectrum enhances our knowledge about both astrophysics and particle physics [1-5], and also gives a signature of the existence of the new particles and some physics behind it, which was confirmed by some accelerator experiments. These experiments provide us to understand the structure of matter and interaction between its building blocks. The interaction of high energy cosmic rays with the earth’s atmospheric nuclei produces extensive air showers, which are continuously rains on the earth through all the directions. These extensive air showers have been generated by the following interactions-


Introduction
Cosmic rays are free source of elementary particles and have enormous range of energies, still they produce small amount of striking rate at the detector level. Uncontrollable fluxes have been produced, which is the major disadvantage of cosmic rays. The study of cosmic ray spectrum enhances our knowledge about both astrophysics and particle physics [1][2][3][4][5], and also gives a signature of the existence of the new particles and some physics behind it, which was confirmed by some accelerator experiments. These experiments provide us to understand the structure of matter and interaction between its building blocks. The interaction of high energy cosmic rays with the earth's atmospheric nuclei produces extensive air showers, which are continuously rains on the earth through all the directions. These extensive air showers have been generated by the following interactions-• Electromagnetic interactions of charged particles, which gives electrons and photons.
• Inelastic hadronic interactions, which gives the secondary fluxes of particles.
• Nuclear interactions, which gives the compositional changes between chemical and isotropic composition of cosmic ray nuclei.
Here we consider hadronic interactions only and focus on to determine the muon fluxes, generated from the decay of produced secondary hadrons. The range of cosmic ray energy lies some eV to 1020 eV and its spectrum obeys a power law behaviour of differential flux where the value of spectral index γ = 2.7 at knee and γ = 3.1 beyond knee. This behaviour changes at two points in the spectrum [6][7][8][9][10][11].
1. The steepening of the spectrum known as knee region occurs at energy ≈ 106 GeV.
2. The flattening of the spectrum known as ankle region occurs at energy ≈ 109 GeV.
This type of changes in the transition at knee and ankle is highly intriguing and not clear to understand because the ground array experiments do not give a reliable reconstruction of the energy. This problem can be solved by taking two ways. The first way to study about the generation of some new heavy particles at knee energy and the second way to measure the compositional change of primary cosmic rays at knee energy. Here we have adopted the first way. Pair-meter technique for the measurement of muon energy is useful for the large iron detectors. This technique provides a reliable reconstruction of muon energy and energy resolution is not affected with the increase in muon energy.

Pair-meter technique
This technique is useful for large size iron detectors and by means of which one can measure the individual muon energy. This technique is not useful for the small size detectors. Since muons have the high penetrating power ability due to their massive nature, therefore for the energy measurement, it requires some different technique from other particles which provides a better resolution of energy measurement of muons this technique also provides the measurement of frequency as well as energy of electron and positron pair production produced by high energy muons traversed in dense matter. The energy measurement process includes-1. The differential cross section of pair production and bremsstrahlung processes are generated by muon of energy E µ above a threshold E 0 . *Corresponding author: Sharda Pandey, Department of Physics, University of Lucknow, Lucknow-226007, Uttar Pradesh, India, Tel: 088086 15285; E-mail: sharda2012536@gmail.com 2. A relative energy transfer above threshold, v =E 0 /E µ , where E 0 is threshold energy and E µ is muon energy above threshold.
3. Calculation of energy loss for each cascade resulting from Bremsstrahlung and Pair production. 4. Inferring the muon energy with the help of total number of cascades at detector having energies above a threshold E 0 .

Pair production cross section
The differential pair production cross section can be calculated by taking some approximation of relative energy transfer and threshold energy is given by [4,5,12,13].
where m e is the mass of electron, e the electron charge and Z is the atomic number. Here we are taking Z = 26 for the iron nuclei. Therefore, the differential cross-section of pair-production process through muon is [12].
Where α fine structure constant and k = 1.8 and t 0 is the radiation length (rl).
The radiation length is the average amount of matter for both pair production and bremsstrahlung energy loss. The general form of expression of radiation length is: Approximate values of t 0 can be calculated by the Hayakawa formula [14].

3
-2 0 t 10 gcm 6Z Z +1 where N av is Avogadro number, r e classical electron radius and A is the Atomic weight for iron nuclei (A=56). Therefore, the radiation length is [6] t 0 = 13.75 g cm -2 The Figure 1 shows the behaviour of differential cross-section of pair production and brems strahlung above the threshold energy E 0 . The integral cross-section of pair production is: Where correction term C 1.4 [14].

Bremsstrahlung cross section
The brems strahlung is the process in which the interaction of charged particles with the electromagnetic field of atomic nuclei produces photons. The expression of differential cross-section for bremsstrahlung is given by: f(z) is coulomb correction function. The integral cross section of bremsstrahlung:

Number of interactions of muons
The number of interaction of muons can be calculated by the total cross-section of pair production as well as bremsstrahlung, which gives the total number of cascades of muons (M).
For pair production process the total number of cascades above the threshold energy E 0 can be calculated by: All calculations have been done for the 100 kton iron detector. We have preferred the dimensions of 100 kton iron detector are 15.6 m × 15.6 m × 78 m. We have taken an average value of path-length T = 1000 rl for a muon, traversing 15 m in an iron detector. The average number of muon cascades at different threshold energies has been shown by Figure 2, which indicates that the reduction in muon cascades with the increment in threshold energy. At E 0 = 1 GeV and E µ = 100 TeV there is approximately more than 70 cascades of muons in comparison to E µ = 100 TeV at E 0 = 100 GeV.

Estimation of muon energy loss and surface muon energy
As muon traverses through the rock between the earth's surface and detector, it produces energy losses. These losses have been originated by means of ionization, bremsstrahlung, pair production and photo nuclear processes. They can be effectively parametrized by [6,[12][13][14] for the muons at higher energy range from [1T eV −100000T eV]. For high energy muons at depth, the discrete energy losses become more important which are mainly bremsstrahlung, pair production and photo nuclear processes. Due to these processes a calculation of muon energy loss is required to correlate the muon energies at detector level with their surface energies.
Since the average energy loss with respect to depth is directly proportional to the muon energy. Therefore the total muon energy loss is: where α is the contribution in energy loss due to the ionization and β is the contribution in energy loss comes from bremsstrahlung, pairproduction, photo nuclear processes i.e., br pair ph β β β β → + + The average muon energy at detector level on traversing a distance x: where s E µ is the surface muon energy.
From equation (17) we get the relation between surface muon energy and muon energy which traverse depth X through the rock is: From equation (18) the minimum surface energy can be calculated by assuming muon energy Eµ ≈ 0 The differential flux relation of muon traversing through the rock at a depth X is obtained by using the (18

Conventional and prompt muon fluxes
As we have mentioned earlier in introduction part of this paper that we are looking for the generation of new particle having mass (nearly GeV) at knee by the means of which we will be able to explain the changes in the spectrum. For this purpose, we have taken two flux models TIG and PRS for the conventional and prompt muon flux calculations (Figures 4 and 5) [7-11].

TIG model
The differential equation of conventional and prompt muon fluxes have been given by the expression as For the conventional and prompt muon fluxes, we have used the some fixed values which have been given by table [9].   log(E flux) = (-a + bx + cx -dx ) µ × Where The values of other PDF's and re-normalization scales have been given by the following tables (Tables 1 and 2). By using these flux models we have calculated the surface muon fluxes as well as muon fluxes at underground level. The plot 4 and plot 5 shows the renormalized surface muon fluxes and muon fluxes respectively by considering the depth: X = 1.89 × 10 5 cm −2 g.
On seeing these plots we may conclude that the fluxes of prompt muons cross the conventional muon fluxes beyond 10 4 T eV muon energy.       events rates which generates due to uncertainties in the muon fluxes at higher energy ranges. We may see from the Table 3 at energy 1 TeV we obtain an observable number of muons and as we are moving towards the higher energies we will get 2-3 muons only.

Results and Discussion
We have calculated the event rates by using two different models for five years of exposer time of iron detector. The results of this calculation have been shown by the Figure 8 and tabulated our data in Table 3. We see from Table 3 that at lowest energy range i.e., at 1 TeV, the event rate is very high for five years of exposer time of detector and as we move towards the higher energy ranges at 100000 TeV; we will unable to obtain an observable number of event rate. We have also calculated cascade numbers of muon by using equation 11 at different-2 threshold energies i.e., E 0 = 5, 10, 50, 100, 300, 500, 1000, 5000, 10000, 50000 GeV respectively which has tabulated in Table 4.
Here we have used two flux models TIG and PRS and both flux models have based on the perturbation QCD values but they differ in their event rates. A great variation occurs in similar perterbative QCD based models PRS1, PRS2, PRS3 and our model too. Hence muon event rate produces a large uncertainties in the prediction of QCD models (i.e., charm production), which helps us to understand about the produced spectral changes in the cosmic ray at knee.

Conclusion
In this work we have attempted to study the vertical flux of ultrahigh energy cosmic ray muons at surface as well as underground at detector level. We have presented our main results in Figures 2 and 8 and Tables 3 and 4. From these results, we have obtained significantly larger muon fluxes than TIG [9] and PRS [10]. The dominance behaviour of prompt muon flux can be seen above 10 2 TeV by Figures 4-7. Hence we may conclude that the underground muon energy measurements for an energy range Eµ from 1−10000 TeV are possible with 100 kton iron detector running over 5 years. This will give a better deal on the ultra-high energy muon fluxes in between the range of several TeV to 10 PeV. This will reduce the uncertainties present in charm production models and it will be also helps us to improve our knowledge about the ultra-high energy neutrino astronomy. In our results we have discussed the observable muon energy range 1 − 10000 TeV, which is crucial of the origin of knee. In this process our calculations of muon fluxes with and without slant depth show feasible demonstration to get deeper knowledge about the knee origin. In this paper in the first place we want to demonstrate the observational feasibility in comparison with the precise prediction through our calculation of muon measurement. Thus our results help to know more about cosmic ray physics, ultrahigh energy neutrino astronomy and charm production models.

Conventional and prompt muon fluxes at slant depth
The topography of the rock for an iron detector gives the slant depth. Basically it is calculated by: Where w 0 is the slope of the rock.
Here we have used the rock density

Calculation of event rates
We are using iron calorimeter detector. The number of event rate above a threshold energy entering in the iron detector over five years is given by: where A is the exposed area of the detector having the value 2.4 × 10 7 cm 2 . This is given by the Table 3 which shows a large variation in the