Constrains of hadronic interaction models from the cosmic muon observations
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Constrains of hadronic interaction models from the cosmic muon observa-tions.
Dedenko L.G. , , a , Lukyashin A.V. , b , Fedorova G.F. , c , and Roganova T.M. , d Faculty of Physics M.V. Lomonosov Moscow State University, Leninskie Gory, 119991 Moscow, Russia. Skobeltsin Institute of Nuclear Physics, Lomonosov Moscow State University, 119234 Moscow, Russia.
Abstract.
A simple method of the vertical muon energy spectrum simulations have been suggested. Thesecalculations have been carried out in terms of various models of hadronic interactions. The most energetic π ± -mesons and K ± -mesons produced in hadron interactions contribute mainly in to this energy spectrum of muonsdue to the very steep energy spectrum of the primary particles. So, some constraints on the hadronic interactionmodels may be set from a comparison of calculated results with the cosmic data on the vertical muon energyspectrum. This comparison showed that the most energetic secondary particles production is too high in caseof the QGSJET II-04 model and rather low in case of the QGSJET II-03 model. These conclusion have beensupported by the LHC data. The longitudinal development of extensive air showers(EAS), in particular, the depth X max of its maximum de-termined by the rate of fragmentation of energy E of theprimary particle. This rate depends on the interaction crosssections of shower particles, and on the energy spectra ofsecondary particles are generated in interactions. Obvi-ously, if the probability of particle production with ener-gies close to the energy of the incident particle is highthen the development of the cascade slows down. Con-versely, in the case of rapid fragmentation of the incidentenergy cascade develop rather rapidly. The depth X max of shower maximum in many studies is the main param-eter for determining the composition of primary cosmicradiation (PCR) at ultra-high energies. It should also benoted that in the case of the slow rate of development cas-cade and hence large values of depth X max lateral distri-bution of the shower particles at the observation level be-comes narrower. Therefore, the values of signals in thesurface and underground detectors located at large dis-tances from the shower core are decreased. It must betaken into account when determining the density of muonsat large distances from the shower core and the composi-tion of the PCR found from the muon lateral distribution.Studies of the composition and characteristics of the en-ergy spectrum of the PCR are important components oftheories of the origin of cosmic rays at ultra-high ener-gies. Interpretation of experimental data on the depth ofthe shower maximum X max and the observed fraction of a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] d e-mail: [email protected] muons at a fixed distance from the shower core are carriedout in terms of di ff erent hadronic interaction models. Inthe case of "soft" interaction generation of secondary par-ticles emitted at mostly small angles with respect to theprojectile particle (the highest values of pseudorapidity η )is of importance. The models based on the Gribov-Reggetheory [1, 2] are commonly used. The dominant contribu-tion of the pomeron at ultra-high energies and other e ff ectsare taken into account di ff erently in various models [3-5]. Therefore, testing of models at the highest energies ofsecondary particles is very important for understanding ofphysics of hadron interactions and for the interpretation ofthe EAS data. At the LHC this testing is carried out in theexperiments LHCf [6] and TOTEM [7]. In cosmic rays,we suggest testing of models of hadron interactions withthe help of atmospheric muon spectrum. In this case, dueto the higher slope of the energy spectrum of the primaryparticles (a coe ffi cient of the slope of the di ff erence spec-trum is γ = .
75) generation of secondary particles ( π ± -mesons and K ± -mesons) with the highest energies is ofimportance. It was shown [8] that the model QGSJET II-03 [4] leads to a spectrum of vertical muons, the inten-sity of which is about a factor f = . + Cosmic [9], MACRO [10]and LVD [11]. The result [8] was obtained as a solutionof transport equations. The π ± and K ± mesons decay into µ ± -mesons. These π ± and K ± mesons are generated bythe parent particles of several generations. Therefore, ifsome excess of the secondary particle production in sin-gular interaction of hadrons is determined by the coe ffi -cient k , then, in the case of i generations, the factor f will be f ≈ k i . Thus, the spectra of atmospheric muonsare very useful tool to test the models of hadronic inter- a r X i v : . [ a s t r o - ph . H E ] A p r SVHECRI 2014
Figure 1.
The primary spectrum of protons and he-lium nuclei (p + He). Dashed line - modified Gaisser-Honda [13] approximation. Data: solid line - AMS02 [14]; ◦ ATIC 2 [15]; • CREAM [16]; (cid:4)
WCFTA [17]; (cid:77)
ARGO [18]; (cid:3)
RUN JOB [19]; (cid:5)
TUNKA [21] (all par-ticles); (cid:78)
SPHERE 2 [22] (all particles); × KASKADE [20](all particles, QGSJET II-03); + KASKADE [20] (all particles,SIBYLL 2.1).
Figure 2.
The energy spectrum of the primary protons. Dashedline - modified Gaisser-Honda [13] approximation. Various data- see the text. (cid:78)
AMS02 [14]; ◦ ATIC 2 [15]; • CREAM [16]; (cid:3)
RUN JOB [19]. actions. However, other factors (cross sections of interac-tions, etc.) are a ff ects result of comparison. In this articlewe propose to test the hadron interaction models by thevery simple original method [12] with the help of observedatmospheric muon spectrum. The very simple original method of simulations of the en-ergy spectrum of vertical muons can be described as fol-low [12]. Let the ( dI p / dE ) and ( dI He / dE ) be the di ff er-ential energy spectra of the primary protons and heliumnuclei. As for the spectrum of muons the energy per nu-cleon is of importance. So, the heavier nuclei give to this Figure 3.
The energy spectrum of the primary helium nu-clei. Dashed line - modified Gaisser-Honda [13] approxima-tion. Various data - see the text. (cid:78)
AMS02 [14]; ◦ ATIC 2 [15]; • CREAM [16]; (cid:3)
RUN JOB [19]. spectra negligible contribution. In the energy range of 10 - 3 · GeV, we used the approximations ( dI p / dE ) GH and ( dI He / dE ) GH by Gaisser-Honda [13]. At energiesabove E = · GeV, these spectra were multipliedby a factor ( E / E ) . and are refered as modified GH ap-proximation. Figure 1 shows a comparison of the sumof modified GH approximations [13] for the primary pro-tons and helium nuclei (dotted line) with the experimentaldata AMS02 [14] - solid line, ATIC2 [15] - hollow cir-cles ( ◦ ), CREAM [16] - dark circles ( • ), WCFTA [17]- dark squares ( (cid:4) ), ARGO [18] - hollow triangles ( (cid:77) ),RUN JOB [19] - hollow squares ( (cid:3) ). The experimentaldata KASKADE [20], interpreted in terms of the model ofQGSJET II-03 shows the oblique crosses ( × ), and in termsof the model SIBYLL 2.1 - straight crosses ( + ). The ex-perimental data TUNKA [21] for all primary particles areshown by hollow diamonds ( (cid:5) ), and data SPHERE 2 [22] -dark triangles ( (cid:78) ). The spectra of the primary protons andprimary helium nuclei are presented separately in Figure2 and Figure 3 with various data mentioned above for acomparison. The notation: ( (cid:78) ) is applied in Figure 2 andFigure 3 only for the data of AMS02 [14]. From com-parison with data it can be concluded that the acceptedapproximation [13] does not overestimate the flux of theprimary protons and helium nuclei. This is important forthe conclusions on possible uncertainties of tested mod-els. The energy spectra of vertical muons D p ( E µ ) dE µ and D He ( E µ ) dE µ induced by the primary protons and heliumnuclei are expressed by simple integrals over the energy E of the primary particles as follows: D p (cid:16) E µ (cid:17) · dE µ = (cid:90) dE · (cid:32) dI p dE (cid:33) · S p (cid:16) E µ , E (cid:17) · dE µ D He (cid:16) E µ (cid:17) · dE µ = (cid:90) dE · (cid:32) dI He dE (cid:33) · S He (cid:16) E µ , E (cid:17) · dE µ The sum of these spectra: D (cid:16) E µ (cid:17) = (cid:16) D p (cid:16) E µ (cid:17) + D He (cid:16) E µ (cid:17)(cid:17) onstrains of hadronic interaction models from the cosmic muon observations. Figure 4.
The energy spectra of muons in showers induced bythe primary protons with various fixed energies E (simulationsin terms of QGSJET II-04 model): 1 − , · ; 2 − ; 3 − ; 4 − ; 5 − ; 6 − GeV. will be used for comparison with data [9-11]. Functions S p ( E µ , E ) · dE µ and S He ( E µ , E ) · dE µ are the di ff erentialenergy spectrum of muons in showers induced by the pri-mary protons and helium nuclei with the fixed energy E .These spectra were calculated for 24 and 19 values of theenergy E of the primary protons and helium nuclei, re-spectively, in the range of 10 ÷ GeV. Calculationshave been carried out in terms of two models of hadroninteractions (QGSJET II-03 [4] and QGSJET II-04 [5])using the package CORSIKA 7.4 [23]. The calculationswere performed with the statistics of 10 events in the lowenergy region and up to 10 events at the highest ener-gies of the primary particles. Figure 4 and Figure 5show examples of the muon energy spectra S p (cid:16) E µ , E (cid:17) and S He (cid:16) E µ , E (cid:17) in the energy range of 10 ÷ GeV calcu-lated in the terms of model QGSJET II-04 [5] for the 6values of proton energies and the 5 values of helium nu-clei energies, respectively. It is seen that in the energyrange of 10 ÷ GeV statistics is small. Therefore, wewill use the energy range of 10 ÷ GeV for a compari-son. Dependence of the spectra S p (cid:16) E µ , E (cid:17) and S He (cid:16) E µ , E (cid:17) on the models of hadron interactions (QGSJET II-03 ( • )and QGSJET II-04 ( ◦ )) are shown in Figure 6 and Figure7 for the primary protons and helium nuclei, respectivelyfor the energy E = GeV. It is evident that the modelQGSJET II-04 [5] predicts the greatest density of muons,while the model QGSJET II-03 [4] - the lowest one.
The spectra of vertical muons D ( E µ ) in the energy rangeof 10 - 10 GeV for the hadron interaction modelsQGSJET II-03 [4] and QGSJET II-04 [5] are presented inFigure 8. It can be seen that the model QGSJET II-04 [5]predicts the intensity of muon flux by a factor 2 ÷ Figure 5.
The energy spectra of muons in showers induced by theprimary helium nuclei with various fixed energies E (simulationsin terms of QGSJET II-04 model): 1 − ; 2 − ; 3 − ; 4 − ; 5 − GeV. than the intensity of the muon flux calculated in terms ofQGSJET II-03 [4] model. This finding is consistent withthe results shown in Figure 6 and Figure 7. Figure 8 is alsovery clearly demonstrates an increase in steepness of themuon spectrum at energies E above 100 GeV. The constant B π (cid:39)
100 GeV is the decay constant for π -mesons in the at-mosphere. Therefore, the muon spectrum become steeperbecause π -mesons at energies E > B π rather interact withnuclei in the atmosphere than decay into muons. Compar-ison of the calculated spectra with experimental data al-lows us to test models. The ratios MC / DATA of the resultsof calculations for models [4] and [5] to the smooth ap-proximation of the data of collaborations L3 + Cosmic [9],MACRO [10] and LVD [11] are shown in Figure 9. It canbe seen that these ratios are increasing from ∼ ∼ ∼ ∼ ÷ GeV. The mostimportant fact is that these increase becomes higher at en-ergies E µ above 10 GeV for the QGSJET II-04 model.For the QGSJET II-03 model the ratios MC / DATA becomeconstant at the level of ∼ . E µ above 10 GeV. No slowing of this increase is seen at higher ener-gies of muons. Thus, Figure 9 demonstrates a very seriousdi ff erence between the calculated spectra and the data re-ported in [9], [10] and [11]. This di ff erence is associatedwith a di ff erent rate of energy fragmentation of projectileparticles in events of its interactions with nuclei in the at-mosphere. Thus, the model QGSJET II-03 underestimatethe probability of secondary particles production at high-est energies. The model QGSJET II-04 overestimate thisprobability of secondary particles production at the highestenergies. According to calculations, the main contributionto integrals D p (cid:16) E µ (cid:17) · dE µ and D He (cid:16) E µ (cid:17) · dE µ comes fromsecondary particles with energies in the ranges of (0.01 SVHECRI 2014
Figure 6.
The energy spectra of muons in showers induced by theprimary protons with the fixed energy E = GeV calculatedin terms of two models: • QGSJET II-03, ◦ QGSJET II-04.
Figure 7.
The energy spectra of muons in showers induced by theprimary helium nuclei with the fixed energy E = GeV calcu-lated in terms of two models: • QGSJET II-03, ◦ QGSJET II-04. – 0.6) E and (0.001 – 0.1) E , where E is the energy of aprojectile particle for the primary protons and helium nu-clei, respectively. The results of this comparison are alsoconfirmed by the data of the LHCf [6] and TOTEM [7].For example, the QGSJET II-04 model [5] overestimatesthe density of charged particles dN ch / d η per unit of pseu-dorapidity at the pseudorapidity η = .
345 by a factor of k ≈ . ff er-ence increases at large values η because of the di ff erencebetween the slopes of the calculated curve and the datafrom [7]. The QGSJET II-04 [5] model predicts the den-sity dN ch / d η which is (18–30)% higher than the data [7]in the range 5 . ≤ η ≤ .
4. It is also important to notethat for di ff erence of rapidity ∆ y (cid:39)
0, the value of the av-
Figure 8.
The calculated energy spectra of near vertical muonsfor various models: • QGSJET II-03, ◦ QGSJET II-04.
Figure 9.
Comparison of calculated muon energy spectrum withdata [9-11]. The ratios MC / DATA are shown: • QGSJET II-03, ◦ QGSJET II-04. erage transverse momentum (cid:104) p T (cid:105) for π -mesons is about50 MeV / c less than data [24]. Under the assumption thata similar decreasing of the (cid:104) p T (cid:105) dependence is also validfor charged π mesons. The calculated density of muonsat large distances from the axis shower will be underesti-mated. This underestimation was observed by the PierreAuger Collaboration [25] and in Yakutsk [26]. The en-ergy spectra of photons in p – p collisions at the energy of √ s = . ≤ η ≤ . ÷ The atmospheric muon energy spectra calculated in termsof the QGSJET II-04 [5] model is by a factor f = , E µ = GeV. The atmospheric muon energy spectrum calculatedin terms of the QGSJET II-03 [4] is by a factor f = , onstrains of hadronic interaction models from the cosmic muon observations. lower then data [9], [10], [11] at energy E µ = GeV.So, we can conclude, these models of hadronic interac-tions should be updated at very high energies of secondaryparticles.
Acknowledgements
Authors thank LSS grant (grant 3110.2014.2) for support!Speaker thank organizing committee of the ISVHECRI 2014 forwell organised conference and administrative support!
References [1] V.Gribov. Sov. Phys. JETP 26. 414 (1968).[2] T.Regge. Nuovo Cimento 14, 951 (1959).[3] E.-J. Ahn, R. Engel, T. K. Gaisser, P. Lipari, andT. Stanev, Phys. Rev.
D 80 , 094003 (2009).[4] S. S. Ostapchenko, Phys. Rev.
D 74 , 014 026 (2006).[5] S. S. Ostapchenko, Phys. Rev.
D 83 , 014 018 (2011).[6] H. Menjo, O. Adriani, and M. Bongi (for LHCf Col-laboration), Nucl. Instrum. Methods Phys. Res.
A 692 ,224 (2012).[7] G. Latino (on behalf of TOTEM Collab.), arXiv: hep-ex / , 219 (2008).[9] The L3 Collab., arXiv: hep-ex / D 52 , 3793 (1995).[11] M. Aglietta, B. Alpat, E. D. Alieva et al. (The LVDCollaboration), arXiv: hep-ex / // / indico / contributionDisplay.py?contribId = = = , 153 (2002). [14] V. Choutko (on behalf AMS Collaboration),in Proceedings of the 33-th International Cos-mic Ray Conference (Rio-de-Janeiro, 2013);https: // / indico / contributionDisplay.py?contribId = = = , 494(2007); A. D. Panov, J. H. Adams Jr., H. S. Ahn, et al.,Bull. Russ. Acad. Sci.: Phys. , 564 (2009).[16] H. S. Ahn (for the CREAM Collaboration), Astro-phys. J. Lett. , L89 (2010).[17] S. S. Zhang (for the WFCTA Collaboration), Nucl.Instrum. Methods Phys. Res. A 629 , 57 (2011).[18] B. Bartoli (for the ARGO_YBJ Collaboration), Phys.Rev.