Fluxes of cosmic-ray muons and atmospheric neutrinos at high energies

The problem of charm generation in the interactions of nucleons with nuclei of air atoms at energies inaccessible at present-day accelerators is discussed. Both experimental data on cosmic-ray muons and the predictions of QCD-based theoretical models are used in analyzing the behavior of the differential cross sections for charmed-particle production at high energies. The calculated fluxes of muons and neutrinos arriving at sea level both along the vertical and along the horizontal direction are presented, together with their approximations for the interval 2×10²–10¹⁰ GeV.     

Energy spectrum of cosmic-ray muons

Summary The energy spectrum of cosmic-ray muons at sea-level is calculated by the numerical integration of a diffusion equation by assuming a scaling model. The calculated spectrum is in good agreement with the observed data of muons with the zenith angles of 0° and nearly 90°. Furthermore, by this comparison, it is concluded that the scaling model is valid up to the muon energy of 6.9 TeV. It is also found that the survival probability of muons at large zenith angles is in favour of that calculated by Ramana Murthy rather than that obtained by Dau, Carstensen and Jokisch. Two production spectra of muons obtained by Thompson and Whalley and Dauet al. are analysed in reference to the recent experimental data. As a result, it is concluded that the power law exponent of the differential-energy spectrum of muon parent particles is 2.75±0.15 in the muon energy range from 100 GeV to 6.9 TeV.

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Riassunto Si calcola lo spettro di energia dei muoni dei raggi cosmici per mezzo dell'integrazione numerica di un'equazione di diffusione assumendo un modello a scala variabile. Lo spettro calcolato è in buon accordo con i dati osservati dei muoni con angoli di zenith di 0° e quasi 90°. Inoltre da questo confronto si conclude che il modello a scala variabile è valido fino ad energie muoniche di 6.9 TeV. Si trova anche che la probabilità di sopravvivenza dei muoni ad angoli di zenith grandi è in favore di quella calcolata da Ramana Murthy piuttosto che di quella ottenuta da Dau, Cartensen e Jokisch. Si analizzano due spettri di produzione dei muoni ottenuti da Thomson e Whalley e Dauet al. rispetto a recenti dati sperimentali. Come risultato si conclude che l'esponente della legge di potenza dello spettro di energia differenziale delle particelle genitrici dei muoni è 2.75±0.15 nell'intervallo di energia muonica compreso tra 110 GeV e 6.9 TeV.

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Резюме Вычисляется энергетический спектр мюонов космических лучей на уровне моря, использия численное интегрирование уравнения диффузии и предполагая модель подобия. Вычисленный спектр хорошо согласуется с наблюденными данными для мюонов с зенитными углами вблизи 0° и 90°. Из этого сравнения делается вывод, что модель подобия справедлива для энергии мюонов вплоть до 6.9 ТэВ. Отмечается, что вероятность выживания мюонов при больших зенитных углах отдает предпочтение вычислениям Рамана Мурти, а не Дау, Карстенсена и Джокиша. Два спектра образования мюонов, полученные Томпсоном и Валлеем и Дау и др., сравниваются с недавними экспериментальными данными. Из сравнения получено, что показатель степенного закона в дифференциальном энергетическом спектре равен 2.75±0.15 в области энергий мюонов от 110 ГэВ до 6.9 ТэВ.

     

High-energy hadrons and cosmic-ray muons in the atmosphere

Summary The experimental data on fluxes of high-energy hadrons and muons in the atmosphere are compared with the predictions based on the latest information about the properties of high-energy interactions and the primary cosmic-ray composition. It is demonstrated that the expected fluxes are in good agreement with the observations if protons dominate at energies up to 10¹⁵ eV.

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Riassunto Si confrontano i dati sperimentali sui flussi degli adroni e muoni ad alta energia nell'atmosfera con de previsioni basate sulle informazioni piú recenti sulle proprietà delle interazioni ad alta energia e la composizione primaria dei raggi cosmici. Si dimostra che i flussi attesi sono in buon accordo con le osservazioni se i protoni dominano ad energie fino a 10¹⁵ eV.

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Резюме Экспериментальные данные для потоков высокоэнергетических адронов и мюонов сравниваются с предсказаниями, основанными на последней информации относительно свойств взаимодействий при высоких энергиях и составе первичных космических лучей. Показывается, что ожидаемые потоки хорошо согласуются с наблюдениями, если протоны доминируют при энергиях вплоть до 10¹⁵ эВ.

     

Hadrons and other secondaries generated by cosmic-ray muons underground

Summary The production of nuclear-active and electromagnetic particles, as well as long-lived isotopes in nuclear and electromagnetic showers generated by cosmic-ray muons at different depths underground is discussed. Characteristics of the secondary components and their dependence on the depth and the muon energy, obtained in experiments and calculations, are presented.     

The multiple scattering of cosmic-ray muons in magnetic spectrometers

Summary The multiple scattering of high-energy cosmic-ray muons in solid-iron magnetic-momentum spectrometers is investigated theoretically and the results are compared with experimental measurements. In particular, spectrometers which record orthogonal views of the muon trajectory are considered. In spectrometers of this type typically two projected angles are measured: 1) an angle ϕ_T in a plane perpendicular to the magnetic field and 2) another angle, ϕ _y , in a plane containing the magnetic-field vector. In the present work a new parameter α=ϕ _y /σϕT is defined and the theory of Molière is used to derive a multiple-scattering probability distribution,f(α), which can be directly compared to spectrometer data. Here σ is the constant noise-to-signal ratio of the spectrometer as calculated by the Molière theory. The new multiple-scattering distributionf(α) is independent of the cosmic-ray spectrum, thus allowing for the first time cosmic-ray spectrometer data to be used directly to investigate the predictions of multiple-scattering theory. Good agreement between theory and experiment is demonstrated for a sample of 8000 muon events measured with the AMH magnetic spectrometer.

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Riassunto Si studia dal punto di vista teorico lo scattering multiplo di muoni dei raggi cosmici ad alta energia negli spettrometri ad impulso magnetico di ferro solido e i risultati sono confrontati con le misurazioni sperimentali. In particolare, si considerano spettrometri che registrano vitse ortogonali della traiettoria del muone. In spettrometri di questo tipo si misurano tipicamente due angoli proiettati: 1) un angolo ϕ_T in un piano perpendicolare al campo magnetico e 2) un altro angolo, ϕ _y , in un piano contenente il vettore del campo magnetico. In questo lavoro, si definisce un nuovo parametro α=ϕ _y /σϕ_T e la teoria di Molière è usata per derivare una distribuzione di probabilità per lo scattering multiplo,f(α), che può essere direttamente confrontata con i dati dello spettrometro. Qui σ è il rapporto costante rumore-segnale dello spettrometro come calcolato con la teoria di Molière. La nuova distribuzione dello scattering multiplof(α) è indipendente dallo spettro dei raggi cosmici, permettendo così di usare per la prima volta direttamente i dati dello spettrometro dei raggi cosmici per studiare le predizioni della teoria dello scattering multiplo. Si mette in evidenza un buon accordo tra dati teorici e sperimentali per un campione di 8000 eventi muonici misurati con uno spettrometro magnetico AMH.

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Резюме Теоретически исследуется многократное рассеяние космических мюонов высоких энергий в магнитных импульсных спектрометрах. Полученные результаты сравниваются с экспериментом. В частности, рассматриваются спектрометры, которые регистрируют мюонную траекторию с взаимноперпендикулярных точек зрения. В спектрометрах этого типа измеряются два угла: 1) угол ϕ_T в плоскости, перпендикулярной магнитному полю и 2) другой угол ϕ _y в плоскости, содержащей вектор магнитного поля. В настоящей работе определяется новый параметр α=ϕ _y /σϕ_T и используется теория Мольера для вывода распределения вероятности многократного рассеяния,f(α), которое можно непосредственно сравнить со спектрометрическими данными. В этой работе с представляет постоянное отношение шума к сигналу для спектрометра, вычисленное с помощью теории Мольера. Новое распределение многократного рассеянияf(α) не зависит от спектра космических лучей. Таким образом, впервые данные спектрометра для космических лучей непосредственно исполязуются для исследования предсказаний теории многократного рассеяния. Обнаружено хорошее согласие между теорией и экспериментом на примере 8000 мюонных событий, измеренных с помощью АМН магнитного спектрометра.

     

Response functions for cosmic-ray muons at various depths underground

Summary Response functions for underground muons are obtained by solving numerically the equations of hadronic cascades in the atmosphere, by assuming Feynman scaling for hadronic interactions over the whole energy region. For practical use of the response functions in cosmic-ray modulation study, calculations are made for various combinations of parameters such as zenith angle, atmospheric depth and muon threshold energy. The present response functions are consistent with cosmic-ray measurements and can be well utilized for the study on the cosmic-ray modulation.

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Riassunto Si ottengono funzioni di risposta per muoni sotterranei risolvendo numericamente le equazioni per le cascate adroniche nell'atmosfera, assumendo lo scaling di Feynman per interazioni adroniche nell'intera regione di energia. Per l'uso pratico di funzioni di risposta nello studio di modulazione dei raggi cosmici, si fanno calcoli per varie combicazioni di parametri come l'angolo di zenit, la profondità atmosferica e l'energia muonica di soglia. Le attuali funzioni di risposta sono coerenti con le misurazioni dei raggi cosmici e possono essere bene utilizzate per lo studio della modulazione dei raggi cosmici.

     

Integral and differential absolute intensity measurements of cosmic-ray muons below 1 GeV

Summary Absolute differential and integral muon intensities below 1 Ge V/c have been measured with a flash-tube range spectrograph in which stopping muons are identified by the decay sequence. The differential intensities at 0.314 GeV/c and 0.805 GeV/c are (3.25±0.17)·10⁻³ cm⁻² sr⁻¹·s⁻¹ (GeV/c)⁻¹ and (3.60±0.18)·10⁻³ cm⁻² sr⁻¹ s⁻¹ (GeV/c)⁻¹, respectively,i.e. 16% and 33% higher than that of Rossi. These measurements confirm the rencent results of other authors according to which the Rossi reference point at 1 GeV/c is too low of about 25%. Our integral measurements at 0.457 GeV/c and 0.918 GeV/c support this evidence. Резюме Измерены абсолютные дифферецниальные и интегральные интенсивности мюонов ниже 1 ГэB/c, используя спектрограф, в котором остановившиеся мюоны идентифицируются по цепочке распадов. Диффенциальные интенсивностн при 0.314 ГэB/c и 0.805 ГэB/c соответственно составляют (3.25±0.17)·10⁻³ см⁻² ср⁻¹ с⁻¹ (ГэВ/с)⁻¹ и (3.60±0.18)·10⁻³ см⁻² ср⁻¹ с⁻¹ (ГэВ/с)⁻¹, т.е. нв 16% и 33% выше результатов Росси. Проведенные измерения подтвершдают недавние результаты других авторов, согласно которым точка отсчета Росси при 1 ГэВ/с расподожена ниже на 25%. Наши интегральные измерения при 0.457 ГэВ/с и 0.918 ГэВ/с подтверждают этот результат.

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Riassunto Con uno spettrografo a range di tubi a flash sono state misurate le intensità differenziali e integrali sotto 1 GeV/c rivelando la catena del decadimento del muone. Le intensità differenziali a 0.314 GeV/c e 0.805 GeV/c sono rispettivamente (3.25±0.17)·10⁻³ cm² sr⁻¹ s⁻¹ (GeV/c)⁻¹ e (3.60±0.18)·10⁻³ cm⁻² sr⁻¹ s⁻¹ (GeV/c)⁻¹, cioè 16% e 33% più alte di quella di Rossi. Queste misure confermano i recenti risultati di altri autori secondo i quali il punto di riferimento di Rossi a 1 GeV/c è più basso di∼25%. Le nostre misure d'intensità integrale a 0.457 GeV/c e 0.918 GeV/c sostengono queste evidenze.

     

Investigation of the zenith angle dependence of cosmic-ray muons at sea level

Angular distribution of cosmic-ray muons at sea level has been investigated using the Geant4 simulation package. The model used in the simulations was tested by comparing the simulation results with the measurements made using the Berkeley Lab cosmic ray detector. Primary particles’ energy and fluxes were obtained from the experimental measurements. Simulations were run at each zenith angle starting from θ?=?0° up to θ?=?70° with 5° increment. The angular distribution of muons at sea level has been estimated to be in the form I(θ)?=?I(0°) cos ⁿ (θ), where I(0°) is the muon intensity at 0° and n is a function of the muon momentum. The exponent n?=?1.95±0.08 for muons with energies above 1 GeV is in good agreement, within error, with the values reported in the literature.     

Temperature effect of the integral flux of cosmic-ray muons at high energies

The temperature coefficients of the integral fluxes of cosmic-ray muons arriving at sea level vertically and horizontally with energies of 10², 10⁴, and 3 × 10⁶ GeV are calculated. Decays of pions, kaons, and charmed particles are considered as sources of muon generation in the atmosphere (according to current data on the generation cross sections of pions, kaons, and charmed particles in interactions between nucleons and the nuclei of atmospheric atoms, obtained in experiments on accelerators and in quantum chromodynamics models). The uncertainties in the generation cross sections of charmed particles are quite high.     

Quark-gluon state of matter and positive excess of cosmic-ray muons at high energies

The problem of the relationship between the numbers of positively and negatively charged particles in the flux of cosmic-ray muons arriving at sea level with energies in excess of 0.1 TeV (up to 100 TeV) is discussed. It is shown that the formation of quark—gluon matter as the result of high-energy nuclear interactions leads to a reduction of the positive excess in cosmic-ray muons at the above energies. At the present time, the quark-gluon state of matter is studied in accelerator experiments at colliding-particle energies of up to √s = 200 GeV per nucleon. Estimates presented in this article for the positive excess of muons having energies of up to 3 or 4 TeV are based on available data from accelerator experiments; at higher muon energies, the respective estimates are based on extrapolating these data.     

The photonuclear energy loss of cosmic-ray muons and its relation with photon-nucleon cross-section

Summary The coefficientb _N of the photonuclear energy loss for high-energy muons is calculated by the generalized vector-dominance model with the different energy dependences in real photon-nucleon cross-sections and by the parton model with a modified scaling variable. The intensities of cosmic-ray muons at great depths are converted into the energy spectrum by taking range fluctuation into account. By comparing the exponents of the spectrum with those from other experiments, it is concluded that the real photon-nucleon cross-section has a ln² ν dependence at higher energies than TeV regions.

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Riassunto Il coefficienteb _N della perdita di energia fotonucleare per muoni ad alta energia è calcolata mediante il modello generalizzato di dominanza vettoriale con diverse dipendenze dall'energia nelle sezioni d'urto fotone-nucleone e mediante il modello a partoni con una variabile di scala modificata. Le intensità dei muoni dei raggi cosmici a grandi profondità sono convertite in spettro di energia prendendo in considerazione la fluttuazione dell'intervallo. confrontando gli esponenti dello spettro con quelli ottenuti da altri esperimenti, si conclude che la sezione d'urto reale fotone-nucleone ha dipendenza ln² v ad energie più alte delle regioni del TeV.

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Резюме Вычисляется коэффициентb _N фотоядерных потерь энергии для мюонов высоких энергий, используя модель обобщенной векторной доминантности с различными энергетическими зависимостями в реальных фотон-протонных поперечных сечениях и партонную модель с модифицированной переменной масштаба. Интенсивности мюонов космических личей на больших глубинах, с учетом флуктуаций пробегов, преобразовывают энергетический спектр. Сравнивая экспоненты спектра с экспонентами из друтих экспериментов, делается утверждение, что реальное фотоннуклонное поперечное сечение имеет зависимость ln² ν при высоких энергиях, в ТэВ-ной области.

     

Effect of primary nuclei composition on the response functions for cosmic-ray muons

Summary The integral response functions for cosmic-ray muons, at different muon threshold energies, are calculated by taking into account the nuclear composition of the primary cosmic radiation, and showing their contribution also at large zenith angles.

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Riassunto Sono state calcolate le funzioni di risposta dei muoni della radiazione cosmica per differenti energie di soglia, introducendo la composizione nucleare della radiazione cosmica primaria e valutando il suo contributo anche a grandi angoli zenitali.

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Резуме Вычисллутся функции интегрального отклика для муонов космического излучения при различных пороговых энергиях муонов, учитывая ядерний состав первичного космического излучения и оценивая их вклад при вольших зенитных углах.

     

On knock-on electrons of cosmic-ray muons in condensed matter (at 12° N)

The dependence of knock-on electron probability on the momenta of muons producing the knock-on electrons in condensed matter has been investigated by using a counter controlled cloud chamber. A search has also been made on the nature of the energy spectrum of the knock-on electrons from muons in the momentum region (0.80±0.05) to (1.70±0.05) Bev/c. It appears from the results of the measurements that there is a very weak dependence of knock-on electron probability on the primary muon momentum in the investigated muon spectral region (i.e. (0.80±0.05) to (1.70±0.05) Bev/c). The angular distributions and hence the energy spectra of the emergent electrons are found to be independent of muon momentum.     

Spectrum of ultrahigh-energy cosmic rays

Detailed calculation of the energy spectra of ultrahigh-energy cosmic rays has been performed. The spectral features related to the interaction of protons with cosmic microwave background have been analyzed. The calculated spectra are compared with the experimental data obtained at the giant detectors for ultrahigh-energy cosmic rays.