Proton−Proton Total Cross−Section Based On New Data of Colliders and Cosmic Rays
Jorge Pérez-Peraza^{*}, Alejandro Sánchez Hertz
Instituto de Geofísica, Universidad Nacional Autónoma de México, Coyoacán, México
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To cite this article:
Jorge Pérez-Peraza, Alejandro Sánchez Hertz. Proton−Proton Total Cross−Section Based On New Data of Colliders and Cosmic Rays. International Journal of High Energy Physics. Vol. 2, No. 2, 2015, pp. 27-33. doi: 10.11648/j.ijhep.20150202.12
Abstract: High energy colliders (accelerators) are fundamental tools in many branches of science. Similarly, cosmic rays observatories are one of the windows to study the universe and high energy particle processes. The last advances in these fields are respectively the LHC (Large Hadron Collider) and the Pierre Auger Observatory. Among the main subjects studied in hadronic physics is the proton-proton (pp) elastic scattering. The Total Cross-Section (σ_{pp}), has been recently measured at 7 and 8 TeV in the LHC, and at 57 TeV in the Pierre Auger Observatory. Importance of the σ_{pp} lies in studies of elastic and diffractive scattering of protons, and to model the development of showers induced by the interaction of ultra high energy cosmic rays in the atmosphere. The gap in data between accelerators and cosmic ray experiment energies does not allow for the exact knowledge of σ_{pp} with energy. Furthermore, since cosmic rays results are of indirect nature, there is consequently a high dispersion in predictions of different authors at this regard. Using the new data, we show here that within the frame of the first-order Glauber multiple diffraction theory the overall data fits very successfully. Our results shows that σ_{pp} grows more slowly (compared with previous predictions), within narrow error bands that avoid any fast slope change. We predict that the future experimental value at 13 TeV from the LHC will fall nicely within our fitting curve. Our phenomenological approach allows for the calculation of σ_{pp} for any other energy value either at the colliders or cosmic ray energies. A deep knowledge, control and handle of hadron-hadron interactions at very high energies will have useful implications in many branches of physics.
Keywords: Cosmic Rays, Elastic Scattering, Hadronic Interactions
1. Introduction
An important process in the hadron physics is the pp elastic scattering. In spite of the amount of currently available data and descriptive models of these data, actually there is not a satisfactory description based on pure Quantum Chromodynamics (QCD), that would be widely accepted in considering this dynamic process. The QCD perturbative theory cannot be extended to the weak interactions region and the QCD non-perturbative theory is not able to predict dispersion states. There are approaches based on QCD that try to explain the phenomenon which have been successful in describing processes where there is much transfer momentum, where quarks, which are the particles that compose hadrons behave as if they were free particles. In this case, the perturbative approach can be applied. On the other hand, in the region with low transfer momentum (the namely region of soft collisions), the effective coupling constant of strong interactions is large and therefore, perturbative approach cannot be applied. Historically, the study of the total cross section, which measure the total interaction probability has played a fundamental role in nuclear and particle physics. For energies of only a few GeV, the total cross section in hadrons scattering usually has a complicated structure composed with peaks or resonances, which reveals the formation of excited hadronic states. On the other side, for higher energies, the total cross section have a softer behaviour.
Also it has been extensively investigated the pp elastic dispersion, mainly in the region where there is a small transfer momentum, which is where there is a great number of experimental data available, although in some specific energies data have been obtained in the region where there is large transfer momentum. An important feature that have resulted of the analysis of experimental data, is the discovery that the effective range of interaction in hadron collision increases in accordance with the energy growth. In the same way, it has been discovered that the probability of absorption also increases; namely, that particles appear to expand and become blacker for high energies.
In this work we develop a prediction based on a purely diffractive model that approximates reasonably the existing experimental data of particle accelerators and cosmic ray observatories for and total cross section in the Center of Mass Energy range: 10−10^{5} GeV, including the values of 7 and 8 TeV obtained in the LHC at the CERN in 2011 and 2013 [1–4]. We also consider here the new value at 57 TeV from the Pierre Auger Observatory obtained in July 2012 [5].
2. Elastic Hadronic Scattering Amplitude
To know how much the hadrons disperse during an elastic collision, an hadronic scattering amplitude must be constructed. One way to construct the hadronic amplitude from the most elemental frame is by means of Glauber’s multiple diffraction theory. The approach is based on the impact parameter and eikonal formalisms as follows: assuming azimuthal symmetry in the collision of two hadrons (neglecting the spin), in our case two protons, we have the following expression for the elastic hadronic scattering amplitude:
(1)
here q^{2} = -t is the four−momentum transfer squared, is the center of mass energy, b the impact parameter, J_{0} the zero−order Bessel and Ω is a function of real values which is used to describe the opacity of hadrons. The equation describing Ω is the following:
(2)
where G is the hadronic form factor:
(3)
and Imf(q,s) is the imaginary part of the elementary parton-parton amplitude:
(4)
where α^{2}, β^{2}, a^{2} and C are energy dependency parameters of real values. The proportionality factor C, is known as the "absorption factor".
3. The ρ Parameter
The ρ parameter is an experimental value that is obtained in particle accelerators, and is equal to the ratio of the real part to the imaginary part of the hadronic scattering amplitude F at q = 0:
(5)
On high energies this amplitude is mainly imaginary, but a knowledge of the real part allows us to obtain predictions of the total cross section (even for high energies) by means of scattering relations.
For obtain the fit of ρ we have used the experimental data that have been obtained in particle accelerators in the Center of Mass (CM) energy range: 13.8−1800 GeV, that is, from the Alternating Gradient Synchrotron (AGS) to the Tevatron. With these experimental data we have calculated the fit for the whole range of energy: 10−10^{5} GeV, and thus, we have predicted reasonably the value obtained at 7 TeV in the LHC of CERN. The form of the fit is the following:
(6)
where s_{0} = 400 GeV^{2} controls the point where ρ reaches the zero and where the coefficients A_{1}, A_{2} and A_{3} control the maximum and the asymptotic behaviour, which values are the next:
A_{1} = 0.0702, A_{2} = 0.3691, A_{3} = 1.502x10^{−3}. (7)
The results are shown in table 1 and figure 1. As we can see, the ρ parameter has negative values at CM energies less than approximately 21 GeV, and moreover, this present an asymptotic growth for energies greater than 4 TeV. In the same way we can see that the fit presents a reasonably approximation to the value obtained by the LHC at 7 TeV.
(GeV) | ρ (pred.) | ρ (exp.) | References |
13.8 | -0.072 | -0.074 ± 0.018 | AGS [6] |
19.4 | -0.004 | 0.019 ± 0.016 | Fermilab [6] |
23.5 | 0.020 | 0.020 ± 0.050 | ISR [7] |
30.7 | 0.046 | 0.042 ± 0.011 | ISR [7] |
44.7 | 0.071 | 0.062 ± 0.011 | ISR [7] |
52.8 | 0.080 | 0.078 ± 0.010 | ISR [7] |
62.5 | 0.086 | 0.095 ± 0.011 | ISR [7] |
541 | 0.132 | 0.135 ± 0.015 | SPS UA4/2 [8] |
1800 | 0.142 | 0.140 ± 0.069 | Tevatron E-710 [9] |
7000 | 0.149 | 0.145 ± 0.091 | LHC TOTEM [3] |
4. Obtaining the Fits for the Energy Dependent Parameters
With the values that were obtained for the energy dependent parameters (table 2), we have calculated the fits (parametrizations) for each one of those, which allow us to extrapolate our calculations to energies greater than 1800 GeV.
Table 2. Calculated values for the energy dependent parameters: C, α^{−2} and λ. On the second column, s_{0} = 1 GeV^{2}. For the pp dispersion we have used the energies: 13.8 to 62.5 GeV, and for the dispersion: 546, 630 y 1800 GeV.
(GeV) | ln(s/s0) | C (s) (GeV−2) | α−2(s) (GeV−2) | λ(s) | Cα2 | Dispersion Type | References |
13.8 | 5.25 | 9.86 | 2.092 | -0.094 | 4.713 |
| Fermilab [10–13] |
19.4 | 5.93 | 9.96 | 2.128 | 0.024 | 4.680 |
| Fermilab [10–13] |
23.5 | 6.31 | 10.16 | 2.174 | 0.025 | 4.673 |
| ISR [14] |
30.7 | 6.85 | 10.37 | 2.222 | 0.053 | 4.667 |
| ISR [14] |
44.7 | 7.60 | 10.82 | 2.299 | 0.079 | 4.706 |
| ISR [14] |
52.8 | 7.93 | 11.12 | 2.350 | 0.099 | 4.732 |
| ISR [14] |
62.5 | 8.27 | 11.42 | 2.400 | 0.121 | 4.758 |
| ISR [14] |
546 | 12.60 | 17.44 | 2.915 | 0.180 | 5.983 |
| SPS UA4 [15, 16] |
630 | 12.89 | 17.80 | 2.948 | 0.184 | 6.038 |
| SPS UA4 [17] |
1800 | 14.99 | 22.41 | 3.310 | 0.199 | 6.770 |
| Tevatron E-710 [18] |
Based on the behavior that is shown by both C and α^{−2} parameters, we can see that their data sets are statistically consistent with quadratic polynomials for [ln(s)]^{2,}so that using linear regression we have obtained the following fits for each one of them:
(8)
(9)
where we have taken s_{0} = 1 GeV^{2}. In figures 2 and 3 we show the previous fits together with the corresponding values of table 2. As we can see, both present positive curves with the energy increase. The dimensionless product Cα^{2} provides us with information of the blackening and expansion in elastic hadron scattering, the plot of this product can be seen in figure 4.
Since we know that the parameter λ has a very similar behaviour of the ρ parameter, we have used a similar fit:
(10)
where s_{0} = 400 GeV^{2} controls the point where λ reaches zero, and the coefficients A_{1}, A_{2} and A_{3} control the maximum and the asymptotic behaviour of the fit, whose values are:
A_{1} = 0.088, A_{2} = 0.334, A_{3} = 2.737x10^{−7}. (11)
In figure 5 we can see the obtained fit for λ.
Figure 5. Predictions for the l parameter (10). The dots are the data of table 2.
5. Derivation of the pp Total Cross Section
In the absence of a QCD description for this phenomenon, a number of models and phenomenological approximations have been developed to describe the available data. Though these formalisms do not give a final answer to the basic involved processes, they are however useful tools that allow for geometric and dynamic assumptions, that lead to reproduce the experimental data. Geometrical models based on the Multiple Diffraction theory of Glauber [19, 20] have proved to be good phenomenological approaches. An essential feature in the multiple diffraction formalism is the connection of the elastic dispersion cross-sections for composite particles (originally for nuclei and after for nucleons) with the dispersion amplitudes of their individual components.
Following this theory, we present here a prediction of the pp dispersion based on an eikonal (a symmetrical two-dimen- sional Fourier transform) that depends on parameters describing the hadronic form factor and the elementary parton-parton amplitude. By means of this eikonal, we then calculate the real and imaginary parts of the hadronic scattering amplitude. With this amplitude and with the fits for the parameters associated with the eikonal, we have obtained a prediction curve for the pp total cross section.
For the calculation of the total cross-section σ_{pp} we have first obtained the differential cross section (dσ/dt) for each one of the experimental data of the accelerators whose operation were previous to the LHC: which corresponds to the energies range 13.8−1800 GeV. The fit for dσ/dt have been obtained by means of the real and imaginary part of the hadronic scattering amplitude F (equation (1)), which is dependent of the energy and transfer momentum, such as is described by the following equation:
(12)
Figure 6 shows an example for the energy of 52.8 GeV, where we can see a reasonably concordance with the experimental data. Figures for the remaining energies can be consulted in [21].
Figure 6. Fit for the pp differential cross section (dσ/dt) at = 52.8 GeV compared with the experimental data.
The knowledge of dσ/dt is also very important for the study of pp elastic dispersion.
With the fits for the energy dependent parameters: C, α^{−2} and λ, we have thus completely determined all the parameters associated with the eikonal, and then, we can now to calculate the pp total cross section by means of the following expression:
(13)
where the integrand represents the imaginary part of the elastic hadronic scattering amplitude and λ is, as we mentioned before, an energy dependent parameter describing the proportionality between the real and imaginary parts. The procedure to derive the previous equation is somewhat complex and was extensively described in [21-24].
In figure 7 we show the result obtained for the pp total cross section, calculated by means of (13) and the fits for the energy dependent parameters (8-11), in the energies range: 10 to 10^{5} GeV. We have calculated Error Bands with 95% of prediction for each of the involved parameters. Further, we know that the significance δ is related with the prediction percentage by means of: 100(1−δ)% = 95%. Therefore, we have that δ = 0.05 and the Student’s t that we require is for n−2 = 8 degrees of freedom, which is equal to 2.306.
Fig. 7 also shows the experimental data that have been obtained in particle accelerators and cosmic ray observatories (see table 3). With the aim of better resolution in figures 8 and 9 we can see the same results, but for two different energy intervals. As we can see from the previous figures and table 3, the fit that we have calculated for the pp total cross section reasonably agrees with the particle accelerators data: from that obtained by the Fermilab to the LHC. With respect to the results of the cosmic rays observatories, our prediction is quite consistent with the Akeno’s data, the value of Fly’s Eye, and the result of Pierre Augier Observatory.
In table 4 are shown the calculated values for the upper and lower prediction bands of energies in which there exist experimental data. In the fifth and sixth columns we can see the absolute differences that there is between each one of the prediction bands with respect to the central prediction, that is, ∆σ_{1} = and ∆σ_{2} = . We can observe that for energies less or equal than 1800 GeV the absolute differences are practically the same, that is , we can say that we have an ±; while, for energies greater than 1800 GeV, the absolute differences between the upper and lower prediction bands begins to slightly increment. This is mainly due to the fact that the dispersion in the measurements is not uniform for each one of the energy dependent parameters. On the other hand, we can also see that the error bands begin to get wider for energies greater than 1800 GeV. This occurs because from this energy is where the extrapolation of the energy dependent fits begins, and therefore, the uncertainty in the prediction tends to increase for higher energies.
It should be mentioned that for the energy values (546, 630 and 1800 GeV) we used data ofdispersion, because there are not pp experimental data for higher energies than the ISR energies and less than the LHC energies, namely, for 62.5 GeV < < 7 TeV. However, from the analysis of the existing experimental data at present, for both reactions, it is known that for higher energies than approximately 35 GeV, the p and pp total cross sections tend to be equal [25], which justifies our calculations.
6. Comparison with a Previous Fit
In a previous work [24]. a fit was obtained using the same methodology that we have employed in the present work, except that the fits for the energy dependent parameters were different, since these were made using a distinct approach for the experimental data of the differential cross sections. In figure 10 are shown both fits, together with the experimental data. The black line corresponds to the prediction of this work and the red line to that published in [24]. In table 5 we can see the obtained values for both fits at energies in which there is experimental data. In the fourth column is shown the absolute difference between both predictions. As we can see, this difference begins to increase from energies greater than 18 TeV. For the energy of 57 TeV, where the Pierre Augier Observatory has presented a value of 133 mb, we can see that in the present work we have obtained a value of 135.93 mb, while in [24] it was obtained a value of 139.31 mb, so, that with the fit of the present work we are more close to the result of the Pierre Augier experiment. For the energies of 7 and 8 TeV, where the LHC present its results, both fits agree reasonably. In the same way, a good match are obtained for the energy range: 10-200 GeV, that is the energy region that corresponds to the Fermilab and ISR experimental data.
7. Conclusions
Our results adequately describe the experimental data obtained in particle accelerators and cosmic rays observatories for the energy range: 10-10^{5} GeV, including the most recent values of 7 and 8 TeV published by the LHC (CERN) and the value presented by the Observatory Pierre Auger at 57 TeV. Our results have also improved the previous approaches, published a few years before the current data of the LHC and the Pierre Auger Observatory were found: especially those in [24], whose fits showed a good prediction for the data in the range of energies: 1-100 TeV, few years before the current data of the LHC and the Pierre Auger Observatory were found. Due to the fact that at present our results are the most consistent with experimental data, both from accelerators and cosmic ray observatories, we expect that the same fitting trend will continue when new high energy data values will be available. At the moment we predict the next experimental value at 13 TeV (at CERN) to be mb, which falls nicely within our fitting curve. Our model allows now to predict any value of at any other energy.
(GeV) | (pred.) (mb) | (exp.) (mb) | References |
13.8 | 38.02 | 38.46 ± 0.04 | Fermilab [25,26] |
19.4 | 38.68 | 38.98 ± 0.04 | Fermilab [25,26] |
23.5 | 39.22 | 38.94 ± 0.17 | ISR [27] |
30.7 | 40.18 | 40.14 ± 0.17 | ISR [27] |
44.7 | 41.87 | 41.79 ± 0.16 | ISR [27] |
52.8 | 42.74 | 42.67 ± 0.19 | ISR [27] |
62.5 | 43.71 | 43.32 ± 0.23 | ISR [27] |
546 | 62.64 | 61.26 ± 0.93 | Tevatron CDF [28] |
61.9 ± 1.5 | SPS UA4 [29] | ||
900 | 68.56 | 65.3 ± 1.5 | SPS UA5 [30] |
1800 | 77.62 | 71.71 ± 2.02 | Tevatron E-811 [31] |
72.8 ± 3.1 | Tevatron E-710 [32] | ||
78.3 ± 5.9 | Tevatron E-710 [33] | ||
80.03 ± 2.24 | Tevatron CDF [29] | ||
6000 | 95.55 | 91.13 ± 14 | Akeno [34] |
7000 | 98.04 | 98.3 ± 2.8 | LHC TOTEM [1] |
98.6 ± 2.2 | LHC TOTEM [2] | ||
98.0 ± 2.5 | LHC TOTEM [3] | ||
99.1 ± 4.3 | LHC TOTEM [3] | ||
8000 | 100.23 | 98.12 ± 15 | Akeno [35] |
101.7 ± 2.9 | LHC TOTEM [4] | ||
10000 | 103.95 | 116.4 ± 17 | Akeno [35] |
14000 | 109.74 | 103.49 ± 26 | Akeno [35] |
18000 | 114.18 | 100.27 ± 28 | Akeno [35] |
24000 | 119.40 | 122.85 ± 35 | Akeno [35] |
30000 | 123.54 | 120 ± 15 | Fly’s Eye [36] |
57000 | 135.93 | 133 ± 13 | Pierre Auger [5] |
(GeV) | (mb) | ^{ }(mb) | ^{ }(mb) | ∆σ1 (mb) | ∆σ2(mb) |
13.8 | 38.02 | 38.58 | 37.47 | 0.56 | 0.55 |
19.4 | 38.68 | 39.26 | 38.11 | 0.58 | 0.57 |
23.5 | 39.22 | 39.81 | 38.65 | 0.59 | 0.57 |
30.7 | 40.18 | 40.77 | 39.59 | 0.59 | 0.59 |
44.7 | 41.87 | 42.47 | 41.27 | 0.60 | 0.60 |
52.8 | 42.74 | 43.35 | 42.15 | 0.61 | 0.59 |
62.5 | 43.71 | 44.33 | 43.11 | 0.62 | 0.60 |
546 | 62.64 | 63.39 | 61.93 | 0.75 | 0.71 |
900 | 68.56 | 69.36 | 67.80 | 0.80 | 0.76 |
1800 | 77.62 | 78.51 | 76.77 | 0.89 | 0.85 |
6000 | 95.55 | 96.64 | 94.52 | 1.09 | 1.03 |
7000 | 98.04 | 99.16 | 96.98 | 1.12 | 1.06 |
8000 | 100.23 | 101.38 | 99.14 | 1.15 | 1.09 |
10000 | 103.95 | 105.15 | 102.83 | 1.20 | 1.12 |
14000 | 109.74 | 111.01 | 108.54 | 1.27 | 1.20 |
18000 | 114.18 | 115.51 | 112.93 | 1.33 | 1.25 |
24000 | 119.40 | 120.80 | 118.09 | 1.40 | 1.31 |
30000 | 123.54 | 125.01 | 122.18 | 1.47 | 1.36 |
40000 | 129.01 | 130.55 | 127.58 | 1.54 | 1.43 |
57000 | 135.93 | 137.57 | 134.41 | 1.64 | 1.52 |
(GeV) | (present work) (mb) | (NJP) (mb) | Absolute difference (mb) |
13.8 | 38.02 | 38.18 | 0.16 |
19.4 | 38.68 | 38.90 | 0.22 |
23.5 | 39.22 | 39.46 | 0.24 |
30.7 | 40.18 | 40.41 | 0.23 |
44.7 | 41.87 | 42.05 | 0.18 |
52.8 | 42.74 | 42.89 | 0.15 |
62.5 | 43.71 | 43.81 | 0.10 |
546 | 62.64 | 61.86 | 0.78 |
900 | 68.56 | 67.66 | 0.90 |
1800 | 77.62 | 76.60 | 0.92 |
6000 | 95.55 | 95.21 | 0.34 |
7000 | 98.04 | 97.83 | 0.21 |
8000 | 100.23 | 100.15 | 0.08 |
10000 | 103.95 | 104.13 | 0.18 |
14000 | 109.74 | 110.35 | 0.61 |
18000 | 114.18 | 115.18 | 1.00 |
24000 | 119.40 | 120.90 | 1.50 |
30000 | 123.54 | 125.47 | 1.93 |
57000 | 135.93 | 139.31 | 3.38 |
Acknowledgements
We thank the UNAM for supporting this research by means of the grant IN106214 DGAPA-PAPIIT.
References