Computational Methods for High Energy Physics

Advances in High Energy Physics

Guest Editors: Carlo Cattani, Mehmet Bektasoglu, and Gongnan Xie Computational Methods for High Energy Physics Advances in High Energy Physics

Computational Methods for High Energy Physics

This is a special issue published in “Advances in High Energy Physics.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Botio Betev, Switzerland Ian Jack, UK Neil Spooner, UK Duncan L. Carlsmith, USA Filipe R. Joaquim, Portugal Luca Stanco, Italy Kingman Cheung, Taiwan Piero Nicolini, Germany EliasC.Vagenas,Kuwait Shi-Hai Dong, Mexico Seog H. Oh, USA Nikos Varelas, USA Edmond C. Dukes, USA Sandip Pakvasa, USA Kadayam S. Viswanathan, Canada Amir H. Fatollahi, Iran Anastasios Petkou, Greece Yau W. Wah, USA Frank Filthaut, The Netherlands Alexey A. Petrov, USA Moran Wang, China Joseph Formaggio, USA Frederik Scholtz, South Africa Gongnan Xie, China Chao-Qiang Geng, Taiwan George Siopsis, USA Hong-Jian He, China Terry Sloan, UK Contents

Computational Methods for High Energy Physics, Carlo Cattani, Mehmet Bektasoglu, and Gongnan Xie Volume 2014, Article ID 458025, 2 pages High Performance Numerical Computing for High Energy Physics: A New Challenge for Big Data Science,FlorinPop Volume 2014, Article ID 507690, 13 pages

Angular Dependence of 𝜙 Meson Production for Different Photon Beam Energies, Jun-Hui Kang, Ya-Zhou Wang, and Bao-Chun Li Volume 2014, Article ID 837121, 6 pages

Radio-Wave Propagation in Salt Domes: Implications for a UHE Cosmic Neutrino Detector, Alina-Mihaela Badescu and Alexandra Saftoiu Volume 2014, Article ID 901434, 9 pages

A Nonparametric Peak Finder Algorithm and Its Application in Searches for New Physics, S. V. Chekanov and M. Erickson Volume 2013, Article ID 162986, 4 pages

Neutron Noise Analysis with Flash-Fourier Algorithm at the IBR-2M Reactor,MihaiO.Dima, Yuri N. Pepelyshev, and Lachin Tayibov Volume 2013, Article ID 821709, 6 pages

FLUKA Monte Carlo Simulations about Cosmic Rays Interactions with Kaidun Meteorite, Turgay Korkut Volume 2013, Article ID 826730, 7 pages

Computational Determination of the Dirac-Theory Adjunctator,M.Dima Volume 2013, Article ID 391741, 4 pages

𝑋(1870) and 𝜂2(1870): Which Can Be Assigned as a Hybrid State?,BingChen,Ke-WeiWei, and Ailin Zhang Volume 2013, Article ID 217858, 14 pages

Maxwell’s Equations on Cantor Sets: A Local Fractional Approach, Yang Zhao, Dumitru Baleanu, CarloCattani,De-FuCheng,andXiao-JunYang Volume 2013, Article ID 686371, 6 pages Transverse Momentum Distributions of Final-State Particles Produced in Soft Excitation Process in High Energy Collisions, Fu-Hu Liu, Ya-Hui Chen, Hua-Rong Wei, and Bao-Chun Li Volume 2013, Article ID 965735, 15 pages

Three-Dimensional Dirac Oscillator with Minimal Length: Novel Phenomena for Quantized Energy, Malika Betrouche, Mustapha Maamache, and Jeong Ryeol Choi Volume 2013, Article ID 383957, 10 pages

Numerical Simulations on Nonlinear Dynamics in Lasers as Related High Energy Physics Phenomena, Andreea Rodica Sterian Volume 2013, Article ID 516396, 8 pages

Regular Solutions in Vacuum Brans-Dicke Theory Compared to Vacuum Einstein Theory, Alina Khaybullina, Ramil Izmailov, Kamal K. Nandi, and Carlo Cattani Volume 2013, Article ID 367029, 7 pages The Duals of Fusion Frames for Experimental Data Transmission Coding of High Energy Physics, Jinsong Leng, Qixun Guo, and Tingzhu Huang Volume 2013, Article ID 837129, 9 pages

Characteristic Roots of a Class of Fractional Oscillators, Ming Li, S. C. Lim, Carlo Cattani, and Massimo Scalia Volume 2013, Article ID 853925, 7 pages Geant4 Simulation Study of Deep Underground Muons: Vertical Intensity and Angular Distribution, Halil Arslan and Mehmet Bektasoglu Volume 2013, Article ID 391573, 4 pages

Design Study of an Underground Detector for Measurements of the Differential Muon Flux, Bogdan Mitrica Volume 2013, Article ID 641584, 9 pages

EssayonKolmogorovLawofMinus5over3ViewedwithGoldenRatio,MingLiandWeiZhao Volume 2013, Article ID 680678, 3 pages Wavelets-Computational Aspects of Sterian Realistic Approach to Uncertainty Principle in High Energy Physics: A Transient Approach,CristianToma Volume 2013, Article ID 735452, 6 pages VOF Modeling and Analysis of the Segmented Flow in Y-Shaped Microchannels for Microreactor Systems, Xian Wang, Hiroyuki Hirano, Gongnan Xie, and Ding Xu Volume 2013, Article ID 732682, 6 pages Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 458025, 2 pages http://dx.doi.org/10.1155/2014/458025

Editorial Computational Methods for High Energy Physics

Carlo Cattani,1 Mehmet Bektasoglu,2 and Gongnan Xie3

1 Department of Mathematics, University of Salerno, 84084 Fisciano (SA), Italy 2 Sakarya University, 54187 Sakarya, Turkey 3 Northwestern Polytechnical University, Xi’an 710072, China

Correspondence should be addressed to Carlo Cattani; [email protected]

Received 31 March 2014; Accepted 31 March 2014; Published 24 April 2014

Copyright © 2014 Carlo Cattani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

New discoveries in high energy physics are leading to some fully explored, from the point of view of their physical- intriguing and complex methods of analysis which demand mathematical consequences. Some of them are focusing for extremely advanced computational approaches. In order on some general aspects of fundamental research (such as to face the complexity of the contemporary frontiers of GEANT4,FLUKA,etc.)whilesomeothersaremainly Physics, computational methods and algorithms are becom- devoted to different areas of interest like CORSIKA, AIRES ing more and more important for a truthful interpretation in cosmic rays theory and GENIE in neutrino physics. Each of physical results. So that the computational tools are the simulation code is based on several interaction models at high key point for the proper understanding of physical results. or low energy (e.g., QGSJET, EPOS, QGSP, etc.). Each model Modern algorithms to be used in robust computer simu- is based on the results obtained at different laboratories over lations and complex computations are then the pillars of the world, and the parameters are extrapolated to energies the physical theory. Since the birth of the quantum theory, higher than an experiment could reach. general relativity or astrophysical problems, researchers were The main goal of this special issue is to bring together dealing with problems where the physical evidence has been important information about modern simulation codes and shown only as a consequence of advanced computational to collect the main streams in the computational methods methods. Thus the computational approach became not only devoted to high energy problems, such as Monte-Carlo tools the most representative of any modern physical theory but andinteractionmodels.Inthisissuewehavecollectedsome also the epistemological setting of any modern physical original research papers on the frontier of computational theory. physics covering a wide range of topics. Recent discoveries in high energy physics have opened We have invited scientists active in the field to contribute new frontiers in Science and technologies, thus increasing with original research papers or review papers (both exper- the interests for new challenging computational methods. imental and theoretical), which have included: Methods, It should be noted that this interest is not restricted to algorithms, and codes, which have been discussed in the high energy physics although many areas of modern physics papers of F.Pop on “High performance numerical computing is covered by this field. Research fields like cosmic rays, for high energy physics: a new challenge for big data science”; dark matter, neutrino physics, and so forth are subjects of S. V. Chekanov and M. Erickson, on “A nonparametric peak interest for many scientists over the world. Together with finder algorithm and its application in searches for new experimental physics, computational methods like Monte- physics”; M. Dima, “Computational determination of the Carlo simulations represent very important and useful tools dirac-theory adjunctator”; Y. Zhao, D. Baleanu, C. Cattani, for a better understanding of many phenomena. During De-Fu Cheng, and Xiao-Jun Yang, “Maxwell’s equations on the last decades, an increasing huge number of algorithms cantor sets: a local fractional approach”; Ming Li, S. C. Lim, and simulation codes have been developed but not yet C. Cattani, and M. Scalia, “Characteristic roots of a class of 2 Advances in High Energy Physics fractional oscillators”; Jinsong Leng, QixunGuo, and Tingzhu Huang, “The duals of fusion frames for experimental data transmission coding of high energy physics”. ThepaperofTurgayKorkut“FLUKAMonteCarlosimu- lations about cosmic rays interactions with kaidun meteorite”, deals with Astrophysics and cosmic rays, while Cosmology isthemainconcernsofthepapersofAlinaKhaybullina, Ramil Izmailov, Kamal K. Nandi, and Carlo Cattani “Regular solutions in vacuum brans-dicke theory compared to vacuum Einstein theory”, and Ming Li and Wei Zhao “Essay on Kolmogorovlawofminus5over3viewedwithgoldenratio”. Related nonlinear mathematical models and methods and high energy questions, such as neutrino, photon, quan- tizedenergy,arediscussedinthepapersofJun-HuiKang, Ya-Zhou Wang, and Bao-Chun Li “Angular dependence of Meson production for different photon beam energies”; MihaiO.Dima,YuriN.Pepelyshev,andLachinTayibov “Neutron noise analysis with flash-fourier algorithm at the IBR-2M reactor”; Bing Chen, Ke-Wei Wei, and Ailin Zhang “X(1870) and 𝜂2(1870): which can be assigned as a hybrid state?”; Fu-Hu Liu, Ya-Hui Chen, Hua-Rong Wei, and Bao- Chun Li “Transverse momentum distributions of final-state particles produced in soft excitation process in high energy collisions”; Malika Betrouche, Mustapha Maamache, and Jeong Ryeol Choi, “Three-dimensional dirac oscillator with minimal length: novel phenomena for quantized energy”; Andreea Rodica Sterian, “Numerical simulations on nonlinear dynamics in lasers as related high energy physics phenomena”; Halil Arslan and Mehmet Bektasoglu, “Geant4 simulation study of deep underground muons: vertical intensity and angular distribution”; Bogdan Mitrica “Design study of an underground detector for measurements of the differential muon flux”; Cristian Toma, “Wavelets-computational aspects of sterian realistic approach to uncertainty principle in high energy physics: a transient approach”; Xian Wang, Hiroyuki Hirano, Gongnan Xie, and Ding Xu “VOF modeling andanalysisofthesegmentedflowinY-shapedmicrochan- nels for microreactor systems”; Alina-Mihaela Badescu and Alexandra Saftoiu, “Radio-wave propagation in salt domes: implications for a UHE cosmic neutrino detector”. Althoughawiderangeoftopicsaretouchedupon,itisnot possible to cover the entire field of computational methods in high energy physics into a single journal issue, also because of the continuous further progress both in the computational methods and in the high energy physics fields. We hope that this topics could be continued to track the updated trends years by years. Finally, we would like to express our thanks to all the contributors of this special issue for their support and cooperation and to the qualified reviewers who helped the authors to improve the quality of their papers. Carlo Cattani Mehmet Bektasoglu Gongnan Xie Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 507690, 13 pages http://dx.doi.org/10.1155/2014/507690

Research Article High Performance Numerical Computing for High Energy Physics: A New Challenge for Big Data Science

Florin Pop

Computer Science Department, Faculty of Automatic Control and Computers, University Politehnica of Bucharest, Splaiul Independentei 313, Bucharest 060042, Romania

Correspondence should be addressed to Florin Pop; [email protected]

Received 20 September 2013; Revised 23 December 2013; Accepted 26 December 2013

Academic Editor: Carlo Cattani

Copyright © 2014 Florin Pop. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Modern physics is based on both theoretical analysis and experimental validation. Complex scenarios like subatomic dimensions, high energy, and lower absolute temperature are frontiers for many theoretical models. Simulation with stable numerical methods represents an excellent instrument for high accuracy analysis, experimental validation, and visualization. High performance computing support offers possibility to make simulations at large scale, in parallel, but the volume of data generated by these experiments creates a new challenge for Big Data Science. This paper presents existing computational methods for high energy physics (HEP) analyzed from two perspectives: numerical methods and high performance computing. The computational methods presented are Monte Carlo methods and simulations of HEP processes, Markovian Monte Carlo, unfolding methods in particle physics, kernel estimation in HEP, and Random Matrix Theory used in analysis of particles spectrum. All of these methods produce data-intensive applications, which introduce new challenges and requirements for ICT systems architecture, programming paradigms, and storage capabilities.

1. Introduction wavelets-computational aspects [3]), new processes (cancer therapy [4], food preservation, or nuclear waste treatment), High Energy Physics (HEP) experiments are probably the or even the birth of a new industry (Internet) [5]. main consumers of High Performance Computing (HPC) in This paper analyzes two aspects: the computational meth- the area of e-Science, considering numerical methods in real odsusedinHEP(MonteCarlomethodsandsimulations, experiments and assisted analysis using complex simulation. Markovian Monte Carlo, unfolding methods in particle Starting with quarks discovery in the last century to Higgs physics, kernel estimation, and Random Matrix Theory) and Bosonin2012[1], all HEP experiments were modeled using the challenges and requirements for ICT systems to deal with numerical algorithms: numerical integration, interpolation, processing of Big Data generated by HEP experiments and random number generation, eigenvalues computation, and simulations. so forth. Data collection from HEP experiments generates a The motivation of using numerical methods in HEP hugevolume,withahighvelocity,variety,andvariabilityand simulations is based on special problems which can be for- passes the common upper bounds to be considered Big Data. mulated using integral or differential-integral equations (or The numerical experiments using HPC for HEP represent a systems of such equations), like quantum chromodynamics new challenge for Big Data Science. evolution of parton distributions inside a proton which can Theoretical research in HEP is related to matter (funda- be described by the Gribov-Lipatov-Altarelli-Parisi (GLAP) mental particles and Standard Model) and Universe forma- equations [6], estimation of cross section for a typical tion basic knowledge. Beyond this, the practical research in HEP interaction (numerical integration problem), and data HEP has led to the development of new analysis tools (syn- representation using histograms (numerical interpolation chrotron radiation, medical imaging or hybrid models [2], problem). Numerical methods used for solving differential 2 Advances in High Energy Physics

Comparison and correction Physics model → simulation process → Simulated Events event generator →vectors of events detectors

Vector of Reconstruction events

Nature Real (physical experiments) detectors Events

Figure 1: General approach of event generation, detection, and reconstruction. equations or integrals are based on classical quadratures and are modeled using vector of events. This section presents Monte Carlo (MC) techniques. These allow generating events general approach of event generation, simulation methods intermsofparticleflavorsandfour-momenta,whichispar- based on Monte Carlo algorithms, Markovian Monte Carlo ticularly useful for experimental applications. For example, chains, methods that describe unfolding processes in particle MC techniques for solving the GLAP equations are based physics, Random Matrix Theory as support for particle on simulated Markov chains (random walks), which have spectrum, and kernel estimation that produce continuous the advantage of filtering and smoothing the state vector for estimates of the parent distribution from the empirical prob- estimating parameters. ability density function. The section ends with performance In practice, several MC event generators and simulation analysis of parallel numerical algorithms used in HEP. tools are used. For example, HERWIG (http://projects .hepforge.org/herwig/) project considers angular-ordered 2.1. General Approach of Event Generation. The most impor- parton shower, cluster hadronization (the tool is imple- tant aspect in simulation for HEP experiments is event gener- mented using Fortran), PYTHIA (http://www.thep.lu.se/ ation. This process can be split into multiple steps, according torbjorn/Pythia.html) project is oriented on dipole- to physical models. For example, structure of LHC (Large type parton shower and string hadronization (the tool Hadron Collider) events: (1) hard process; (2) parton shower; is implemented in Fortran and C++), and SHERPA (3) hadronization; (4) underlying event. According to official (http://projects.hepforge.org/sherpa/) considers dipole-type LHC website (http://home.web.cern.ch/about/computing): parton shower and cluster hadronization (the tool is “approximately 600 million times per second, particles collide implemented in C++). An important tool for MC simulations within the LHC...Experiments at CERN generate colossal is GATE (GEANT4 Application for Tomographic Emission), amounts of data. The Data Centre stores it, and sends it a generic simulation platform based on GEANT4. GATE around the world for analysis.” Th e an a l y s i s mu s t pro du c e provides new features for nuclear imaging applications valuable data and the simulation results must be correlated and includes specific modules that have been developed to with physical experiments. meet specific requirements encountered in SPECT (Single Figure 1 presents the general approach of event gener- Photon Emission Tomography) and PET (Positron Emission ation, detection, and reconstruction. The physical model is Tomography). used to create simulation process that produces different type The main contributions of this paper are as follows: of events, clustered in vector of events (e.g., the fourth type of (i) introduction and analysis of most important model- events in LHC experiments). ing methods used in High Energy Physics; In parallel, the real experiments are performed. The (ii) identifying and describing of the computational detectors identify the most relevant events and, based on numerical methods for High Energy Physics; reconstruction techniques, vector of events is created. The (iii) presentation of the main challenges for Big Data detectors can be real or simulated (software tools) and processing. the reconstruction phase combines real events with events detectedinsimulation.Attheend,thefinalresultiscompared Thepaperisstructuredasfollows.Section2 introduces with the simulation model (especially with generated vectors the computational methods used in HEP and describes the of events). The model can be corrected for further experi- performance evaluation of parallel numerical algorithms. ments. The goal is to obtain high accuracy and precision of Section 3 discusses the new challenge for Big Data Science measured and processed data. generatedbyHEPandHPC.Section4 presents the conclu- Software tools for event generation are based on ran- sions and general open issues. dom number generators. There are three types of random numbers: truly random numbers (from physical generators), 2. Computational Methods Used in pseudorandom numbers (from mathematical generators), High Energy Physics and quasirandom numbers (special correlated sequences of numbers, used only for integration). For example, numerical Computational methods are used in HEP in parallel with integration using quasirandom numbers usually gives faster physical experiments to generate particle interactions that convergencethanthestandardintegrationmethodsbasedon Advances in High Energy Physics 3

(1) procedure Random Generator Poisson(𝜇) (2) 𝑛𝑢𝑚𝑏𝑒𝑟 ←−1; (3) 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟; ←1.0 (4) 𝑞←exp {−𝜇}; (5) while 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 >𝑞 do (6) 𝑟𝑛𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 ←𝑅𝑁𝐷() ; (7) 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 ← 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 ∗𝑟𝑛𝑑 𝑛𝑢𝑚𝑏𝑒𝑟; (8) 𝑛𝑢𝑚𝑏𝑒𝑟 ← 𝑛𝑢𝑚𝑏𝑒𝑟 +1; (9) end while (10) return number; (11) end procedure

Algorithm 1: Random number generation for Poisson distribution using many random generated numbers with normal distribution (RND). quadratures. In event generation pseudorandom numbers are Random generator Poisson (Alg. 1) used most often. 350000 300000 The most popular HEP application uses Poisson distribu- 250000 tion combined with a basic normal distribution. The Poisson 200000 distributioncanbeformulatedas 150000 100000 Frequency 50000 𝑘 0 𝜇 0 5 10 15 20 25 30 𝑃 [𝑋=𝑘] = exp {−𝜇}, 𝑘=0,1,..., (1) 𝑘! Generted numbers (a) with 𝐸(𝑘) = 𝑉(𝑘) =𝜇 (𝑉 is variance and 𝐸 is expectation Random generator Poisson RND (Alg. 2) 400000 value). Having a uniform random number generator called 350000 RND() (Random based on Normal Distribution) we can use 300000 250000 the following two algorithms for event generation techniques. 200000 The result of running Algorithms 1 and 2 to generate 150000

106 Frequency 100000 around random numbers is presented in Figure 2.In 50000 general, the second algorithm has better result for Poisson 0 distribution. General recommendation for HEP experiments 0 5 10 15 20 25 30 indicates the use of popular random number generators like Generted numbers TRNG (True Random Number Generators), RANMAR (Fast (b) Uniform Random Number Generator used in CERN experiments), RANLUX (algorithm developed by Luscher used Figure 2: General approach of event generation, detection, and by Unix random number generators), and Mersenne Twister reconstruction. (the “industry standard”). Random number generators provided with compilers, operating system, and programming advance. If 𝜌(𝑟) is the probability density function, 𝜌(𝑟)𝑑𝑟 = language libraries can have serious problem because they are 𝑃[𝑟󸀠 <𝑟 <𝑟+𝑑𝑟] based on system clock and suffer from lack of uniformity , the cumulative distributed function is of distribution for large amounts of generated numbers and 𝑟 𝑑𝐶 (𝑟) correlation of successive values. 𝐶 (𝑟) = ∫ 𝜌 (𝑥) 𝑑𝑥 󳨐⇒ 𝜌 (𝑟) = . (2) −∞ 𝑑𝑟 Theartofeventgenerationistouseappropriatecombina- tions of various random number generation methods in order 𝐶(𝑟) is a monotonically nondecreasing function with all to construct an efficient event generation algorithm being values in [0, 1]. The expectation value is solution to a given problem in HEP. 𝐸(𝑓)=∫ 𝑓 (𝑟) 𝑑𝐶 (𝑟) = ∫ 𝑓 (𝑟) 𝜌 (𝑟) 𝑑𝑟. (3) 2.2. Monte Carlo Simulation and Markovian Monte Carlo Chains in HEP. In general, a Monte Carlo (MC) method is And the variance is any simulation technique that uses random numbers to solve 2 2 2 a well-defined problem, 𝑃.If𝐹 is a solution of the problem 𝑉(𝑓)=𝐸[𝑓−𝐸(𝑓)] =𝐸(𝑓)−𝐸 (𝑓) . (4) 𝑛 𝑃 (e.g., 𝐹∈𝑅 or 𝐹 has a Boolean value), we define 𝐹̂,an ̂ estimation of 𝐹,as𝐹=𝑓({𝑟1,𝑟2,...,𝑟𝑛},...),where{𝑟𝑖}1≤𝑖≤𝑛 2.2.1. Monte Carlo Event Generation and Simulation. To is a random variable that can take more than one value and define a MC estimator the “Law of Large Numbers (LLN)” for which any value that will be taken cannot be predicted in is used. LLN can be described as follows: let one choose 𝑛 4 Advances in High Energy Physics

(1) procedure Random Generator Poisson RND(𝜇, 𝑟) (2) 𝑛𝑢𝑚𝑏𝑒𝑟 ←0; (3) 𝑞←exp {−𝜇}; (4) 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟; ←𝑞 (5) 𝑝←𝑞; (6) while 𝑟 > 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 do (7) 𝑛𝑢𝑚𝑏𝑒𝑟 ← 𝑛𝑢𝑚𝑏𝑒𝑟 +1; (8) 𝑝 ← 𝑝 ∗ 𝜇/𝑛𝑢𝑚𝑏𝑒𝑟; (9) 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 ← 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟; +𝑝 (10) end while (11) return number; (12) end procedure

Algorithm 2: Random number generation for Poisson distribution using one random generated number with normal distribution.

𝑟 + numbers 𝑖 randomly, with the probability density function 𝜇 (q1) uniform on a specific interval (𝑎, 𝑏), each 𝑟𝑖 being used to evaluate 𝑓(𝑟𝑖).Forlarge𝑛 (consistent estimator),

1 𝑛 1 𝑏 𝜃 e+(p ) e−(p ) ∑𝑓(𝑟𝑖)󳨀→𝐸(𝑓)= ∫ 𝑓 (𝑟) 𝑑𝑟. (5) 1 2 𝑛 𝑖=1 𝑏−𝑎 𝑎

z The properties of a MC estimator are being normally Φ distributed (with Gaussian density); the standard deviation is 𝜇−(q ) 𝜎=√𝑉(𝑓)/𝑛;MCisunbiasedforall𝑛 (the expectation value 2 is the real value of the integral); the estimator is consistent if − Figure 3: Example of particle interaction: 𝑒(𝑝1)+𝑒(𝑝2)→ 𝑉(𝑓) <∞ + − (the estimator converges to the true value of the 𝜇 (𝑞1)𝜇 (𝑞2). integral for every large 𝑛);asamplingphasecanbeapplied to compute the estimator if we do not know anything about the function 𝑓; it is just suitable for integration. The sampling where 𝑃 is total four-momentum of the 𝑛-particle system; 𝑝𝑘 phase can be expressed, in a stratified way, as and 𝑚𝑘 are four-momenta and mass of the final state particles; 𝛿(4)(𝑃−∑𝑛 𝑝 ) 𝑏 𝑟1 𝑟2 𝑏 𝑘=1 𝑘 is the total energy momentum conservation; ∫ 𝑓 𝑟 𝑑𝑟 = ∫ 𝑓 𝑟 𝑑𝑟 + ∫ 𝑓 𝑟 𝑑𝑟+⋅⋅⋅+∫ 𝑓 𝑟 𝑑𝑟. 2 2 ( ) ( ) ( ) ( ) 𝛿(𝑝𝑘 −𝑚𝑘) is the on-mass-shell condition for the final state 𝑎 𝑎 𝑟 𝑟 1 𝑛 system. Based on the integration formula (6) 𝑑3𝑝 ∫ 𝛿(𝑝2 −𝑚2)Θ(𝑝0)𝑑4𝑝 = 𝑘 , MC estimations and MC event generators are necessary 𝑘 𝑘 𝑘 𝑘 2𝑝0 (9) tools in most of HEP experiments being used at all their 𝑘 steps: experiments preparation, simulation running, and data obtain the iterative form for cross section: analysis. 𝑅 (𝑃, 𝑝 ,𝑝 ,...,𝑝 ) An example of MC estimation is the Lorentz invariant 𝑛 1 2 𝑛 phase space (LIPS) that describes the cross section for a 𝑑3𝑝 (10) typical HEP process with 𝑛 particle in the final state. = ∫ 𝑅 (𝑃 − 𝑝 ,𝑝 ,𝑝 ,...,𝑝 ) 𝑛 , 𝑛−1 𝑛 1 2 𝑛−1 2𝑝0 Consider 𝑛 which can be numerical integrated by using the recurrence 𝜎 ∼ ∫ |𝑀|2𝑑𝑅 , 𝑛 𝑛 (7) relation. As result, we can construct a general MC algorithm for particle collision processes. where 𝑀 is the matrix describing the interaction between 𝑒+𝑒− →𝜇+𝜇− particles and 𝑑𝑅𝑛 istheelementofLIPS.Wehavethe Example 1. Let us consider the interaction: following estimation: where Higgs boson contribution is numerically negligible. Figure 3 describes this interaction (Φ is the azimuthal angle, 𝑅𝑛 (𝑃,1 𝑝 ,𝑝2,...,𝑝𝑛) 𝜃 the polar angle, and 𝑝1,𝑝2,𝑞1,𝑞2 are the four-momenta for particles). 𝑛 𝑛 (4) 2 2 0 4 The cross section is = ∫ 𝛿 (𝑃 − ∑𝑝𝑘) ∏ (𝛿 (𝑝 −𝑚 )Θ(𝑝 )𝑑 𝑝𝑘), 𝑘 𝑘 𝑘 2 𝑘=1 𝑘=1 𝛼 𝑑𝜎 = [𝑊 (𝑠) (1 + 2𝜃) + 𝑊 (𝑠) 𝜃] 𝑑Ω, (11) (8) 4𝑠 1 cos 2 cos Advances in High Energy Physics 5

2 where 𝑑Ω = 𝑑 cos 𝜃𝑑Φ, 𝛼=𝑒/4𝜋 (fine structure constant), be provided in terms of Monte Carlo event generators, which 0 0 2 𝑠=(𝑝1 +𝑝2) is the center of mass energy squared, and directly simulate these processes and can provide unweighted 𝑊1(𝑠) and 𝑊2(𝑠) are constant functions. For pure processes (weight = 1) events. A good Monte Carlo algorithm should we have 𝑊1(𝑠) = 1 and 𝑊2(𝑠) =,andthetotalcrosssection 0 be used not only for numerical integration [7](i.e.,pro- becomes vide weighted events) but also for efficient generation of unweighted events, which is very important issue for HEP. 2𝜋 1 𝑑2𝜎 𝜎=∫ 𝑑Φ ∫ 𝑑 cos 𝜃 . (12) 0 −1 𝑑Φ𝑑 cos 𝜃 2.2.2. Markovian Monte-Carlo Chains. AclassicalMonte Carlo method estimates a function 𝐹 with 𝐹̂ by using a We introduce the following notation: random variable. The main problem with this approach is that 𝑑2𝜎 we cannot predict any value in advance for a random variable. 𝜌 ( 𝜃, Φ) = , (13) cos 𝑑Φ𝑑 𝜃 In HEP simulation experiments the systems are described cos in states [8]. Let us consider a system with a finite set of 𝑆 ,𝑆 ,... 𝑆 𝑡 and let us consider 𝜌(̃ cos 𝜃, Φ) an approximation of possible states 1 2 ,and 𝑡 the state at the moment .The 𝜌(cos 𝜃, .ThenΦ) 𝜎=∬𝑑Φ𝑑̃ cos 𝜃𝜌̃.Now,wecancompute conditional probability is defined as

2𝜋 1 𝑃(𝑆 =𝑆 |𝑆 =𝑆 ,𝑆 =𝑆 ,...,𝑆 =𝑆 ), 𝑡 𝑗 𝑡1 𝑖1 𝑡2 𝑖2 𝑡𝑛 𝑖𝑛 (18) 𝜎=∫ 𝑑Φ ∫ 𝑑 cos 𝜃𝜌 (cos 𝜃, Φ) 0 −1 where the mappings (𝑡1,𝑖1), . . . ,(𝑡𝑛,𝑖𝑛) canbeinterpretedas 2𝜋 1 the description of system evolution in time by specifying a = ∫ 𝑑Φ ∫ 𝑑 cos 𝜃𝑤 (cos 𝜃, Φ) 𝜌̃(cos 𝜃, Φ) (14) 0 −1 specific state for each moment of time. The system is a Markov chain if the distribution of 2𝜋 1 𝑆𝑡 depends only on immediate predecessor 𝑆𝑡−1 and it is ≈ ⟨𝑤⟩𝜌̃ ∫ 𝑑Φ ∫ 𝑑 cos 𝜃𝜌̃(cos 𝜃, Φ) = 𝜎̃⟨𝑤⟩𝜌̃, 0 −1 independent of all previous states as follows: 𝑤( 𝜃, Φ) = 𝜌( 𝜃, Φ)/𝜌(̃ 𝜃, Φ) ⟨𝑤⟩ where cos cos cos and 𝜌̃ is the 𝑃(𝑆𝑡 =𝑆𝑗 |𝑆𝑡−1 =𝑆𝑖 ,...,𝑆𝑡 =𝑆𝑖 ,𝑆𝑡 =𝑆𝑖 ) ̃ 𝑡−1 2 2 1 1 estimation of 𝑤 based on 𝜌.Here,theMCestimatoris (19) =𝑃(𝑆𝑡 =𝑆𝑗 |𝑆𝑡−1 =𝑆𝑖 ). 1 𝑛 𝑡−1 ⟨𝑤⟩ = ∑𝑤 , MC 𝑖 (15) 𝑛 𝑖=1 To generate the time steps (𝑡1,𝑡2,...,𝑡𝑛) we use the probability of a single forward Markovian step given by 𝑝(𝑡 | ∞ and the standard deviation is 𝑡𝑛) with the property ∫ 𝑝(𝑡𝑛 |𝑡 )𝑑𝑡 =1 and we define 𝑡𝑛 𝑛 1/2 𝑝(𝑡) = 𝑝(𝑡 |0).The1-dimensionalMonteCarloMarkovian 1 2 𝑠 =( ∑(𝑤 − ⟨𝑤⟩ ) ) . (16) Algorithm used to generate the time steps is presented in MC 𝑛(𝑛 − 1) 𝑖 MC 𝑖=1 Algorithm 3. The main result of Algorithm 3 is that 𝑃(𝑡max) follows a The final numerical result based on MC estimator is Poisson distribution:

𝜎 = 𝜎̃⟨𝑤⟩ ± 𝜎𝑠̃ . 𝑡max 𝑡max MC MC MC (17) 𝑃𝑁 = ∫ 𝑝(𝑡1 |𝑡0)𝑑𝑡1 × ∫ 𝑝(𝑡2 |𝑡1)𝑑𝑡2 0 𝑡 As we can show, the principle of a Monte Carlo estimator 1 𝑡 in physics is to simulate the cross section in interaction and max ×⋅⋅⋅×∫ 𝑝(𝑡𝑁 |𝑡𝑁−1)𝑑𝑡𝑁 (20) radiation transport knowing the probability distributions (or 𝑡 an approximation) governing each interaction of elementary 𝑁−1 ∞ 1 𝑁 −𝑡 particles. × ∫ 𝑝(𝑡 |𝑡 )𝑑𝑡 = (𝑡 ) 𝑒 max . 𝑁+1 𝑁 𝑁+1 𝑁! max Basedonthisresult,theMonteCarloalgorithmusedto 𝑡max generate events is as follows. It takes as input 𝜌(̃ cos 𝜃, Φ) and in a main loop considers the following steps: (1) gen- We can consider the 1-dimensional Monte Carlo Marko- erate (cos 𝜃, Φ) peer from 𝜌̃; (2) compute four-momenta vian Algorithm as a method used to iteratively generate the 𝑝1,𝑝2,𝑞1,𝑞2; (3) compute 𝑤=𝜌/𝜌̃.Theloopcanbestopped systems’ states (codified as a Markov chain) in simulation inthecaseofunweightedevents,andwewillstayintheloop experiments. According to the Ergodic Theorem for Markov for weighted events. As output, the algorithm returns four- chains, the chain defined has a unique stationary probability momenta for particle for weighted events and four-momenta distribution [9, 10]. and an array of weights for unweighted events. The main issue Figures 4 and 5 present the running of Algorithm 3. is how to initialize the input of the algorithm. Based on 𝑑𝜎 According to different values of parameter 𝑠 used to generate formula (for 𝑊1(𝑠) = 1 and 𝑊2(𝑠) =), 0 we can consider as the next step, the results are very different, for 1000 iterations. 2 2 2 input 𝜌(̃ cos 𝜃, Φ) = (𝛼 /4𝑠)(1 + cos 𝜃).Then𝜎=4𝜋𝛼̃ /3𝑠. Figure 4 for 𝑠=1shows a profile of the type of noise. For In HEP theoretical predictions used for particle collision 𝑠 = 10, 100, 1000 profile looks like some of the information processes modeling (as shown in presented example) should is filtered and lost. The best results are obtained for 𝑠 = 0.01 6 Advances in High Energy Physics

(1) Generate 𝑡1 according with 𝑝(𝑡1)=𝑝(𝑡1 |𝑡0 =0) (2) if 𝑡1 <𝑡max then ⊳ Generate the initial state. 𝑃 =∫𝑡max 𝑝(𝑡 |𝑡)𝑑𝑡 ⊳ (3) 𝑁≥1 0 1 0 1; Compute the initial probability. (4) Retain 𝑡1; (5) end if (6) if 𝑡1 >𝑡max then ⊳ Discard all generated and computed data. ∞ −𝑡max (7) 𝑁=0; 𝑃0 =∫ 𝑝(𝑡1 |𝑡0)𝑑𝑡1 =𝑒 ; 𝑡max (8) Delete 𝑡1; (9) EXIT. ⊳ The algorithm ends here. (10) end if (11) 𝑖=2; (12) while (1) do ⊳ Infinite loop until a successful EXIT. (13) Generate 𝑡𝑖 according with 𝑝(𝑡𝑖 |𝑡𝑖−1) (14) if 𝑡𝑖 <𝑡max then ⊳ Generate a new state and new probability. 𝑡max (15) 𝑃𝑁≥𝑖 =∫ 𝑝(𝑡𝑖 |𝑡𝑖−1)𝑑𝑡𝑖; 𝑡𝑖 (16) Retain 𝑡𝑖; (17) end if (18) if 𝑡𝑖 >𝑡max then ⊳ Discard all generated and computed data. ∞ (19) 𝑁=𝑖−1; 𝑃𝑖 =∫ 𝑝(𝑡𝑖 |𝑡𝑖−1)𝑑𝑡𝑖; 𝑡max (20) Retain (𝑡1,𝑡2,...,𝑡𝑖−1); Delete 𝑡𝑖; (21) EXIT. ⊳ The algorithm ends here. (22) end if (23) 𝑖=𝑖+1; (24) end while

Algorithm 3: 1-Dimensional Monte Carlo Markovian Algorithm.

and 𝑠 = 0.1 and the generated values can be easily accepted Table 1: Efficiency of integration methods for 1 dimension and for for MC simulation in HEP experiments. 𝑑 dimensions. Figure 5 shows the acceptance rate of values generated Method 1 dimension 𝑑 dimensions with parameter 𝑠 used in the algorithm. And parameter values −1/2 −1/2 Monte Carlo 𝑛 𝑛 are correlated with Figure 4. Results in Figure 5 show that 𝑛−2 𝑛−2/𝑑 the acceptance rate decreases rapidly with increasing value of Trapezoidal rule −4 −4/𝑑 parameter 𝑠. The conclusion is that values must be kept small Simpson’s rule 𝑛 𝑛 −2𝑚 −2𝑚/𝑑 to obtain meaningful data. A correlation with the normal 𝑚-points Gauss rule (𝑚<𝑛) 𝑛 𝑛 distribution is evident, showing that a small value for the mean square deviation provides useful results.

2.2.3. Performance of Numerical Algorithms Used in MC Simulations. Numerical methods used to compute MC esti- Aspracticalexample,inatypicalhigh-energyparticlecol- lision there can be many final-state particles (even hundreds). mator use numerical quadratures to approximate the value 𝑛 𝑑=3𝑛−4 of the integral for function 𝑓 on a specific domain by a If we have final state particle, we face with {𝑤 } dimensional phase space. As numerical example, for 𝑛=4 linear compilation of function values and weights 𝑖 1≤𝑖≤𝑚 𝑑=8 as follows: we have dimensions, which is very difficult approach for classical numerical quadratures. 𝑏 𝑚 Full decomposition integration volume for one double 𝑑 ∫ 𝑓 (𝑟) 𝑑𝑟 = ∑𝑤𝑖𝑓(𝑟𝑖). (21) number (10 Bytes) per volume unit is 𝑛 ×10Bytes. For the 𝑎 𝑖=1 example considered with 𝑑=8and 𝑛=10divisions for interval [0, 1] we have, for one numerical integration, We can consider a consistent MC estimator 𝑎 aclas- 𝑤 =1 sical numerical quadrature with all 𝑖 .Efficiencyof 108 ×10 integration methods for 1 dimension and for 𝑑 dimensions 𝑛𝑑 ×10 = 𝐺 ≈ 0.93 𝐺 . Bytes 3 Bytes Bytes (22) is presented in Table 1. We can conclude that quadrature 1024 methods are difficult to apply in many dimensions for variate 6 integration domains (regions) and the integral is not easy to Considering 10 events per second, one integration per event, be estimated. thedataproducedinonehourwillbe≈3197.4 𝑃 Bytes. Advances in High Energy Physics 7

0.7 0.5 100 0.3 0.1

s = 0.01 0 0 200 400 600 800 1000 80 State (iteration) (a) 60 4 2 0

s = 0.1 −2 0 200 400 600 800 1000 40

State (iteration) (%) rate Acceptance (b) 20 3 1 −1 s=1 −3 0 −3 −2 −1 0 200 400 600 800 1000 0 1 2 3 4 (s) State (iteration) log10 (c) Figure 5: Analysis of acceptance rate for 1-dimensional Monte Carlo 3 Markovian algorithm for different 𝑠 values. 1 −1

s=10 −3 0 200 400 600 800 1000 a well-confirmed theory. An inverse process starts with a State (iteration) measured distribution and tries to identify the true distri- (d) bution. These unfolding processes are used to identify new 2 theories based on experiments [11]. 1 0 −1 s = 100 0 200 400 600 800 1000 2.3.1. Unfolding Processes in Particle Physics. The theory of unfolding processes in particle physics is as follows [12]. For a State (iteration) physics variable 𝑡 we have a true distribution 𝑓(𝑡) mapped in (e) 𝑥 and an 𝑛-vector of unknowns and a measured distribution 1 𝑔(𝑠) (for a measured variable 𝑠)mappedinan𝑚-vector of 𝑚×𝑛 0 measured data. A response matrix 𝐴∈𝑅 encodes a Kernel −1 𝐾(𝑠, 𝑡)

s = 1000 function describing the physical measurement process 0 200 400 600 800 1000 [12–15]. The direct and inverse processes are described by the State (iteration) Fredholm integral equation [16] of the first kind, for a specific (f) domain Ω, Figure 4: Example of 1-dimensional Monte Carlo Markovian ∫ 𝐾 (𝑠,) 𝑡 𝑓 (𝑡) 𝑑𝑡 =𝑔 (𝑠) . (24) algorithm. Ω In particle physics the Kernel function 𝐾(𝑠, 𝑡) is usually The previous assumption is only for multidimensional knownfromaMonteCarlosampleobtainedfromsimulation. arrays. But due to the factorization assumption, A numerical solution is obtained using the following linear equation: 𝐴𝑥=𝑏. Vectors 𝑥 and 𝑦 are assumed to 𝑝(𝑟1,𝑟2,...,𝑟𝑛)=𝑝(𝑟1)𝑝(𝑟2)⋅⋅⋅𝑝(𝑟𝑛), we obtain for one integration be 1-dimensional in theory, but they can be multidimensional in practice (considering multiple independent linear 𝑛×𝑑×10Bytes = 800 Bytes, (23) equations). In practice, also the statistical properties of the measurements are well known and often they follow the which means ≈2.62 𝑇 Bytesofdataproduceforonehourof Poisson statistics [17]. To solve the linear systems we have simulations. different numerical methods. # First method is based on linear transformation 𝑥=𝐴𝑦. # −1 2.3. Unfolding Processes in Particle Physics and Kernel Estima- If 𝑚=𝑛then 𝐴 =𝐴 and we can use direct Gaussian tion in HEP. In particle physics analysis we have two types methods, iterative methods (Gauss-Siedel, Jacobi or SOR), or of distributions: true distribution (considered in theoretical orthogonal methods (based on Householder transformation, models) and measured distribution (considered in experi- Givens methods, or Gram-Schmidt algorithm). If 𝑚>𝑛 mental models, which are affected by finite resolution and (the most frequent scenario) we will construct the matrix # 𝑇 −1 𝑇 limited acceptance of existing detectors). A HEP interaction 𝐴 =(𝐴 𝐴) 𝐴 (called pseudoinverse Penrose-Moore). In process starts with a true knows distribution and generate these cases the orthogonal methods offer very good and stable a measured distribution, corresponding to an experiment of numerical solutions. 8 Advances in High Energy Physics

Second method considers the singular value decomposi- where 𝐾 is an estimator. For example, a Gauss estimator with tion: mean 𝜇 and standard deviation 𝜎 is 𝑛 𝑇 𝑇 2 𝐴=𝑈Σ𝑉 = ∑𝜎𝑖𝑢𝑖V𝑖 , (25) 1 (𝑥−𝜇) 𝑖=1 𝐾 (𝑥) = exp {− }, (29) 𝜎√2𝜋 2𝜎2 𝑚×𝑛 𝑛×𝑛 where 𝑈∈𝑅 and 𝑉∈𝑅 are matrices with orthonormal columns and the diagonal matrix Σ=diag{𝜎1,...,𝜎𝑛}= and has the following properties: positive definite and 𝑇 𝑈 𝐴𝑉.Thesolutionis infinitely differentiable (due to the exp function), and it can 𝑛 1 𝑛 1 be defined for an infinite supports (𝑛→∞). The kernel is a 𝑥=𝐴#𝑦=𝑉Σ−1 (𝑈𝑇𝑦) = ∑ (𝑢𝑇𝑦) V = ∑ 𝑐 V , ℎ 𝜎 𝑖 𝑖 𝜎 𝑖 𝑖 (26) nonparametric method, which means that is independent of 𝑖=1 𝑖 𝑖=1 𝑖 dataset and for large amount of normally distributed data we 𝑐 =𝑢𝑇𝑦 𝑖=1,...,𝑛 can find a value for ℎ that minimizes the integrated squared where 𝑖 𝑖 , , are called Fourier coefficients. ̂ error of 𝑓(𝑥). This value for bandwidth is computed as 2.3.2. Random Matrix Theory. Analysis of particle spectrum 4 1/5 (e.g., neutrino spectrum) faces with Random Matrix Theory ℎ∗ =( ) 𝜎. (30) 3𝑛 (RMT), especially if we consider anarchic neutrino masses. The RMT means the study of the statistical properties of The main problem in Kernel Estimation is that the eigenvalues of very large matrices [18]. For an interaction set of data {𝑥𝑖}1≤𝑖≤𝑛 is not normally distributed and in matrix 𝐴 (with size 𝑁), where 𝐴𝑖𝑗 is an independent dis- 𝐻 real experiments the optimal bandwidth it is not known. tributed random variable and 𝐴 is the complex conjugate An improvement of presented method considers adaptive 𝐻 andtransposematrix,wedefine𝑀=𝐴+𝐴 , which describes Kernel Estimation proposed by Abramson [20], where ℎ𝑖 = a Gaussian Unitary Ensemble (GUE). The GUE properties ℎ/√𝑓(𝑥𝑖) and 𝜎 are considered global qualities for dataset. are described by the probability distribution 𝑃(𝑀)𝑑𝑀: (1) The new form is 𝑃(𝑀)𝑑𝑚 = it is invariant under unitary transformation, 𝑛 󸀠 󸀠 󸀠 𝐻 1 1 𝑥−𝑥 𝑃(𝑀 )𝑑𝑀 ,where𝑀 =𝑈 𝑀𝑈, 𝑈 is a Hermitian matrix 𝑓̂ (𝑥) = ∑ 𝐾 ( 𝑖 ) , 𝐻 𝑎 𝑛 ℎ ℎ (31) (𝑈 𝑈=𝐼); (2) the elements of 𝑀 matrix are statistically 𝑖=1 𝑖 𝑖 independent, 𝑃(𝑀) =∏𝑖≤𝑗𝑃𝑖𝑗 (𝑀𝑖𝑗 );and(3) the matrix 𝑀 can 𝐻 and the local bandwidth value that minimizes the integrated be diagonalized as 𝑀=𝑈𝐷𝑈 ,where𝑈=diag{𝜆1,...,𝜆𝑁}, 𝑓̂ (𝑥) 𝜆𝑖 is the eigenvalue of 𝑀 and 𝜆𝑖 ≤𝜆𝑗 if 𝑖<𝑗 squared error of 𝑎 is 𝑁 1/5 𝐻 ∗ 4 𝜎 Propability (2): 𝑃 (𝑀) 𝑑𝑀 ∼ 𝑑𝑀 exp {− 𝑇𝑟 (𝑀 𝑀)} ; ℎ =( ) √ , 2 𝑖 3𝑛 ̂ (32) 𝑓(𝑥𝑖) 2 (3) 𝑃 (𝑀) 𝑑𝑀 ∼ 𝑑𝑈∏𝑑𝜆 ∏(𝜆 −𝜆 ) Propability : 𝑖 𝑖 𝑗 ̂ 𝑖 𝑖<𝑗 where 𝑓 is the normal estimator. Kernel estimation is used for event selection to confidence 𝑁 level evaluation, for example, in Markovian Monte Carlo × {− ∑ (𝜆2)} . exp 𝑖 chains or in selection of neural network output used in 2 𝑖 experiments for reconstructed Higgs mass. In general, the (27) main usage of Kernel estimation in HEP is searching for new The numerical methods used for eigenvalues computa- particle, by finding relevant data in a large dataset. tion are the QR method and Power methods (direct and A method based on Kernel estimation is the graphical indirect). The QR method is a numerical stable algorithm representation of datasets using advanced shifted histogram andPowermethodisaniterativeone.TheRMTcanbeused algorithm (ASH). This is a numerical interpolation for large for many body systems, quantum chaos, disordered systems, datasets with the main aim of creating a set of 𝑛𝑏𝑖𝑛 histograms quantum chromodynamics, and so forth. 𝐻={𝐻𝑖},withthesamebin-widthℎ.Algorithm4 presents the steps of histograms generation starting with a 2.3.3. Kernel Estimation in HEP. Kernel estimation is a very specific interval [𝑎, 𝑏], a number of points 𝑛 in this interval, powerful solution and relevant method for HEP when it and a number of bins and a number of values used for kernel estimation, 𝑚.Figure6 showstheresultsofkernel is necessary to combine data from heterogeneous sources 2 like MC datasets obtained by simulation and from Standard estimation if function 𝑓 = −(1/2)𝑥 on [0, 1] and graphical Model expectation, obtained from real experiments [19]. For representation with a different number of bins. The values asetofdata{𝑥𝑖}1≤𝑖≤𝑛 with a constant bandwidth ℎ (the onverticalaxisareaggregatedinstep17ofAlgorithm4 and difference between two consecutive data values), called the increase with the number of bins. smoothing parameter, we have the estimation 1 𝑛 𝑥−𝑥 2.3.4. Performance of Numerical Algorithms Used in Particle 𝑓̂(𝑥) = ∑𝐾( 𝑖 ), 𝑛ℎ ℎ (28) Physics. All operations used in presented methods for par- 𝑖=1 ticle physics (Unfolding Processes, Random Matrix Theory, Advances in High Energy Physics 9

(1) procedure ASH (𝑎,𝑏,𝑛,𝑥,𝑛bin,𝑚,𝑓𝑚) (2) 𝛿=(𝑏−𝑎) /𝑛bin; ℎ=𝑚𝛿; (3) for 𝑘=1...𝑛bin do (4) V𝑘 =0; 𝑦𝑘 =0; (5) end for (6) for 𝑖=1...𝑛do (7) 𝑘=(𝑥𝑖 −𝑎)/𝛿+1; (8) if 𝑘∈[1, 𝑛bin] then (9) V𝑘 = V𝑘 +1; (10) end if (11) end for (12) for 𝑘=1...𝑛bin do (13) if V𝑘 =0then (14) 𝑘=𝑘+1; (15) end if (16) for 𝑖=max {1, 𝑘 − 𝑚} +1 ...min {𝑛bin,𝑘+𝑚−1} do (17) 𝑦𝑖 =𝑦𝑖 + V𝑘𝑓𝑚 (𝑖−𝑘); (18) end for (19) end for (20) for 𝑘=1...𝑛bin do (21) 𝑦𝑘 =𝑦𝑘/(𝑛ℎ) ; (22) 𝑡𝑘 =𝑎+(𝑘 − 0.5) 𝛿; (23) end for return {𝑡 } {𝑦 } (24) 𝑘 1≤𝑘≤𝑛bin, 𝑘 1≤𝑘≤𝑛bin. (25) end procedure

Algorithm 4: Advanced shifted histogram (1D algorithm). and Kernel Estimation) can be reduced to scalar products, 𝑇(𝑛) = Θ(𝑓(𝑛)),where𝑓(𝑛) is a fundamental function 𝛼 𝑛 matrix-vector products, and matrix-matrix products. In [21] (𝑓(𝑛) ∈ {1, 𝑛 ,𝑎 , log 𝑛, √𝑛,...}). In parallel systems (mul- thedesignofnewstandardfortheBLAS(BasicLinear tiprocessor systems, with 𝑝 processors) we have the serial ∗ Algebra Subroutines) in C language by extension of precision processing time 𝑇 (𝑛) =1 𝑇 (𝑛) and parallel processing is described. This permits higher internal precision and time 𝑇𝑝(𝑛). The performance of parallel algorithms can be mixed input/output types. The precision allows implemen- analyzed using speed-up, efficiency, and isoefficiency metrics. tation of some algorithms that are simpler, more accurate, 𝑆(𝑝) and sometimes faster than possible without these features. (i) The speed-up, ,representshowaparallelalgo- Regarding the precision of numerical computing, Dongarra rithm is faster than a corresponding sequential algo- 𝑆(𝑝) =𝑇 (𝑛)/𝑇 (𝑛) and Langou established in [22]anupperboundforthe rithm. The speed-up is defined as 1 𝑝 . 𝑛×𝑛 𝑆(𝑝) ≤ residual check for 𝐴𝑥=𝑦system, with 𝐴∈𝑅 a dense There are special bounds for speed-up23 [ ]: 𝑝𝑝/(𝑝̃ + 𝑝−1)̃ 𝑝=𝑇̃ /𝑇 matrix. The residual check is defined as ,where 1 ∞ is the average 󵄩 󵄩 󵄩𝐴𝑥 −𝑦󵄩 parallelism (the average number of busy processors 𝑟 = 󵄩 󵄩∞ < 16, given unbounded number of processors). Usually ∞ 󵄩 󵄩 (33) 𝑛𝜖 (‖𝐴‖∞‖𝑥‖∞ + 󵄩𝑦󵄩∞) 𝑆(𝑝) ≤𝑝, but under special circumstances the speed-up can be 𝑆(𝑝) >𝑝 [24]. Another upper where 𝜖 is the relative machine precision for the IEEE bound is established by the Amdahls law: 𝑆(𝑝) = representation standard; ‖𝑦‖∞ is the infinite norm of a vector: 1/2 (𝑠 + ((1 − 𝑠)/𝑝)) ≤1/𝑠where 𝑠 is the fraction of ‖𝑦‖∞ = max1≤𝑖≤𝑛{|𝑦𝑖|};and‖𝐴‖∞ is the infinite norm of a 𝑛 matrix ‖𝐴‖∞ = max1≤𝑖≤𝑛{∑ |𝐴𝑖𝑗 |}. a program that is sequential. The upper bound is 𝑗=1 0 Figure 7 presents the graphical representation of Dongar- considered for a time of parallel fraction. ras result (using logarithmic scales) for simple and double (ii) The efficiency is the average utilization of 𝑝 proces- −24 𝐸(𝑝) = 𝑆(𝑝)/𝑝 precision. For simple precision, 𝜖𝑠 =2 ,forall𝑛 ≥ 1.05 × sors: . 6 10 the residual check is always lower than imposed upper (iii) The isoefficiency is the growth rate of workload 𝜖 =2−53 bound, similarly for double precision with 𝑑 ,forall 𝑊𝑝(𝑛) = 𝑝𝑝𝑇 (𝑛) intermsofnumberofprocessorsto 14 𝑛 ≥ 5.63 × 10 . If matrix size is greater than these values, it keep efficiency fixed. If we consider 𝑊1(𝑛)−𝐸𝑊𝑝(𝑛) = will not be possible to detect if the solution is correct or not. 0 for any fixed efficiency 𝐸 we obtain 𝑝=𝑝(𝑛).This These results establish upper bounds for data volume in this means that we can establish a relation between needed model. number of processors and problem size. For example In a single-processor system, the complexity of algo- for the parallel sum of 𝑛 numbers using 𝑝 processors rithms depends only on the problem size, 𝑛.Wecanassume we have 𝑛≈𝐸(𝑛+𝑝log 𝑝),so𝑛 = Θ(𝑝 log 𝑝). 10 Advances in High Energy Physics

0 Table 2: Isoefficiency for a hypercube architecture: 𝑛=Θ(𝑝log 𝑝) 2 −5 and 𝑛=Θ(𝑝(log 𝑝) ).Wemarkedwith(∗) the limitations imposed −10 by Formula (33). −15

nbin = 10 2 −20 Scenario Architecture size (𝑝) 𝑛=Θ(𝑝log 𝑝) 𝑛 = logΘ(𝑝( 𝑝) ) 0 0.2 0.4 0.6 0.8 1 1 1 1 1 10 1.0 × 10 1.00 × 10 a, b 2 2 2 [ ] 2 10 2.0 × 10 8.00 × 10 3 3 4 (a) 3 10 3.0 × 10 2.70 × 10 4 4 5∗ 0 4 10 4.0 × 10 6.40 × 10 −10 105 5.0 × 105∗ 1.25 × 107 −20 5 6 6 8 −30 6 10 6.0 × 10 2.16 × 10 −40 7 7 9

nbin = 100 10 7.0 × 10 3.43 × 10 −50 7 8 8 10 0 0.2 0.4 0.6 0.8 1 8 10 8.0 × 10 5.12 × 10 9 9 11 [a, b] 9 10 9.0 × 10 7.29 × 10 (b) 0 1/𝑡 −100 is the time needed to send a word ( 𝑤 is called bandwidth), −200 and 𝐿 is the message length (expressed in number of words). −300 The word size depends on processing architecture (usually it −400 𝑡 nbin = 1000 −500 is two bytes). We define 𝑐 as the processing time per word for 0 0.2 0.4 0.6 0.8 1 a processor. We have the following results. a, b 𝑇 [ ] (i) External product 𝑥𝑦 . The isoefficiency is written as (c) 𝑡𝑐𝑛≈𝐸(𝑡𝑐𝑛+(𝑡𝑠 +𝑡𝑤)𝑝log 𝑝) 󳨐⇒ 𝑛 = Θ(𝑝 log 𝑝) . (34) Figure 6: Example of advanced shifted histogram algorithm run- 𝑇 =𝑡𝑛/𝑝+(𝑡 +𝑡 ) 𝑝 ning for different bins: 10, 100, and 1000. Parallel processing time is 𝑝 𝑐 𝑠 𝑤 log . The optimality is computed using 𝑑𝑇 𝑛 𝑡 +𝑡 𝑡 𝑛 𝑝 =0󳨐⇒−𝑡 + 𝑠 𝑤 =0󳨐⇒𝑝≈ 𝑐 . 15 𝑐 2 (35) 𝑑𝑝 𝑝 𝑝 𝑡𝑠 +𝑡𝑤

10 𝑇 𝑛 (ii) Scalar product (internal product) 𝑥 𝑦=∑𝑖=1 𝑥𝑖𝑦𝑖. The isoefficiency is written as 5 𝑡 𝑡 𝑡 𝑛2 ≈𝐸(𝑡𝑛2 + 𝑠 𝑝 𝑝+ 𝑤 𝑛√𝑝 𝑝) 𝑐 𝑐 2 log 2 log 0 (36) (1/e∗n)) 2 10 󳨐⇒ 𝑛 = Θ ( 𝑝( log 𝑝) ). log −5 𝑛 (iii) Matrix-vector product 𝑦=𝐴𝑥,𝑦𝑖 =∑𝑗=1 𝐴𝑖𝑗 𝑥𝑗.The −10 isoefficiency is written as 2 2 𝑡𝑐𝑛 ≈𝐸(𝑡𝑐𝑛 +𝑡𝑠𝑝 log 𝑝+𝑡𝑤𝑛√𝑝 log 𝑝) 󳨐⇒ 𝑛 −15 (37) 2 0 5 10 15 20 25 =Θ(𝑝(log 𝑝) ) . (n) log10 Simple precision Table 2 presentedtheamountofdatathatcanbepro- Double precision cessed for a specific size. The cases that meet the upper bound 6 HPL2.0 bound 𝑛 ≥ 1.05 × 10 are marked with (∗). To keep the efficiency Figure 7: Residual check analysis for solving 𝐴𝑥 =𝑦 system in high for a specific parallel architecture, HPC algorithms for HPL2.0 using simple and double precision representation. particle physics introduce upper limits for the amount of data, which means that we have also an upper bound for Big Data volume in this case. The factors that determine the efficiency of parallel algo- Numerical algorithms use for implementation a hyper- rithms are task balancing (work-load distribution between cube architecture. We analyze the performance of different all used processors in a system → to be maximized); numerical operations using the isoefficiency metric. For the concurrency (the number/percentage of processors working hypercube architecture a simple model for intertask commu- simultaneously → to be maximized); and overhead (extra 𝑇 =𝑡 +𝐿𝑡 𝑡 nication considers com 𝑠 𝑤 where 𝑠 is the latency (the work for introduce by parallel processing that does not appear time needed by a message to cross through the network), 𝑡𝑤 in serial processing → to be minimized). Advances in High Energy Physics 11

Experimental data Modeling and Analysis reconstruction (reduce result to (app. TB of data → big data (find properties of observed simple statistical problem) particles) distribution)

Compare

Simulation Modeling and Analysis (generate MC data based on reconstruction (reduce result to theoretical models-Big (find properties of observed simple statistical Data problem) particles) distribution)

Figure 8: Processing flows for HEP experiments.

3. New Challenges for Big Data Science [28]. There are other several processing platforms for Big Data: Mesos [29], YARN (Hortonworks, Hadoop YARN: A TherearealotofapplicationsthatgenerateBigData,like next-generation framework for Hadoop data processing, 2013 social networking profiles, social influence, SaaS & Cloud (http://hortonworks.com/hadoop/yarn/)), Corona (Corona, Apps, public web information, MapReduce scientific exper- Under the Hood: Scheduling MapReduce jobs more effi- iments and simulations (especially HEP simulations), data ciently with Corona, 2012 (Facebook)), and so forth. A review warehouse, monitoring technologies, and e-government ser- of various parallel and distributed programming paradigms, vices. Data grow rapidly, since applications produce contin- analyzing how they fit into the Big Data era is presented in uously increasing volumes of both unstructured and struc- [30]. The challenges that are described for Big Data Science on tured data. The impact on the approach to data processing, the modern and future Scientific Data Infrastructure are pre- transfer, and storage is the need to reevaluate the way and sented in [31]. The paper introduces the Scientific Data Life- solutionstobetteranswertheusers’needs[25]. In this con- cycle Management (SDLM) model that includes all the major text, scheduling models and algorithms for data processing stages and reflects specifics in data management in modern have an important role becoming a new challenge for Big e-Science. The paper proposes the SDI generic architecture Data Science. model that provides a basis for building interoperable data or HEP applications consider both experimental data (that project centric SDI using modern technologies and best prac- are application with TB of valuable data) and simulation tices. This analysis highlights in the same time performance data (with data generated using MC based on theoretical and limitations of existing solutions in the context of Big models). The processing phase is represented by modeling Data. Hadoop can handle many types of data from disparate andreconstructioninordertofindpropertiesofobserved systems: structured, unstructured, logs, pictures, audio files, particles (see Figure 8). Then, the data are analyzed a reduced communications records, emails, and so forth. Hadoop relies to a simple statistical distribution. The comparison of results on an internal redundant data structure with cost advantages obtained will validate how realistic is a simulation experiment and is deployed on industry standard servers rather than on andvalidateitforuseinothernewmodels. expensive specialized data storage systems [32]. The main Since we face a large variety of solutions for specific challenges for scheduling in Hadoop are to improve existing applications and platforms, a thorough and systematic anal- algorithms for Big Data processing: capacity scheduling, ysis of existing solutions for scheduling models, methods, fair scheduling, delay scheduling, longest approximate time andalgorithmsusedinBigDataprocessingandstorage to end (LATE) speculative execution, deadline constraint environments is needed. The challenges for scheduling scheduler, and resource aware scheduling. impose specific requirements in distributed systems: the Data transfer scheduling in Grids, Cloud, P2P, and so claimsoftheresourceconsumers,therestrictionsimposedby forth represents a new challenge that is the subject to Big resource owners, the need to continuously adapt to changes Data. In many cases, depending on applications architecture, of resources’ availability, and so forth. We will pay special data must be transported to the place where tasks will attention to Cloud Systems and HPC clusters (datacenters) be executed [33]. Consequently, scheduling schemes should as reliable solutions for Big Data [26]. Based on these consider not only the task execution time, but also the data requirements, a number of challenging issues are maximiza- transfer time for finding a more convenient mapping of tasks tion of system throughput, sites’ autonomy, scalability, fault- [34]. Only a handful of current research efforts consider tolerance, and quality of services. the simultaneous optimization of computation and data When discussing Big Data we have in mind the 5 Vs: Vol- transfer scheduling. The big-data I/O scheduler [35]offersa ume, Velocity, Variety, Variability, and Value. There is a clear solution for applications that compete for I/O resources in need of many organizations, companies, and researchers to a shared MapReduce-type Big Data system [36]. The paper deal with Big Data volumes efficiently. Examples include web [37] reviews Big Data challenges from a data management analytics applications, scientific applications, and social net- respective and addresses Big Data diversity, Big Data reduc- works. For these examples, a popular data processing engine tion, Big Data integration and cleaning, Big Data indexing for Big Data is Hadoop MapReduce [27]. The main problem and query, and finally Big Data analysis and mining. On the is that data arrives too fast for optimal storage and indexing opposite side, business analytics, occupying the intersection 12 Advances in High Energy Physics of the worlds of management science, computer science, and Conflict of Interests statistical science, is a potent force for innovation in both the private and public sectors. The conclusion is that the data is The author declares that there is no conflict of interests too heterogeneous to fit into a rigid schema38 [ ]. regarding the publication of this paper. Another challenge is the scheduling policies used to determine the relative ordering of requests. Large distributed Acknowledgments systems with different administrative domains will most likely have different resource utilization policies. For example, The research presented in this paper is supported by a policy can take into consideration the deadlines and thefollowingprojects:“SideSTEP—Scheduling Methods for budgets, and also the dynamic behavior [39]. HEP experi- Dynamic Distributed Systems: a self-∗ approach”, ( P N - ments are usually performed in private Clouds, considering II-CT-RO-FR-2012-1-0084); “ERRIC—Empowering Roma- dynamicschedulingwithsoftdeadlines,whichisanopen nian Research on Intelligent Information Technologies,” F P 7- issue. REGPOT-2010-1, ID: 264207; CyberWater Grant of the The optimization techniques for the scheduling process Romanian National Authority for Scientific Research, CNDI- represent an important aspect because the scheduling is a UEFISCDI, Project no. 47/2012. The author would like to main building block for making datacenters more available to thank the reviewers for their time and expertise, constructive user communities, being energy-aware [40]andsupporting comments, and valuable insights. multicriteria optimization [41]. An example of optimization is multiobjective and multiconstrained scheduling of many References tasks in Hadoop [42] or optimizing short jobs [43]. The cost effectiveness, scalability, and streamlined architectures [1] H. 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Research Article Angular Dependence of 𝜙 Meson Production for Different Photon Beam Energies

Jun-Hui Kang,1 Ya-Zhou Wang,2 and Bao-Chun Li2

1 Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China 2 Department of Physics, Shanxi University, Taiyuan, Shanxi 030006, China

Correspondence should be addressed to Bao-Chun Li; [email protected]

Received 19 September 2013; Revised 25 November 2013; Accepted 27 November 2013; Published 21 January 2014

Academic Editor: Gongnan Xie

Copyright © 2014 Jun-Hui Kang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

The dependence of 𝜙-meson photoproduction on the polar angle is investigated in the framework of a multisource thermal model. We present a detailed comparison between our results and experimental data of the neutral decay mode in the reaction 𝛾𝑝 → 𝑝𝜙(𝐾𝑆𝐾𝐿). The results are in good agreement with the experimental data. It is found that the movement factor 𝑏𝑧 increases linearly with the photon beam energies.

1. Introduction has been studied by the azimuthal anisotropy of final-state particles produced in high energy collisions. In our previous More than 20 years ago, strangeness enhancement was work [13], the multisource thermal model has been used to predicted to be a unique signature of the quark-gluon plasma describe the elliptic flows of final-state hadrons produced (QGP) formation in high energy collisions [1]. So far, many in nucleus-nucleus collisions at RHIC energies. It involves experimental results from the Super Proton Synchrotron the anisotropic expansion of the participant area in the (SPS) and Relativistic Heavy Ion Collider (RHIC) have shown transverse momentum space. The model is successful in the the enhancement of strange particles [2]. At low photon description of (pseudo)rapidity and multiplicity distributions energies, strangeness production starting from threshold of produced particles [14].Inthepresentwork,wefocusour is also helpful to understand underlying photoproduction attention on the dependence of 𝜙 meson photoproduction mechanisms because it is far below the region of perturbative on the polar angle. A purpose of this paper is to check Quantum Chromodynamics (QCD) and is an energy domain whether the model can fit the angular dependence of 𝜙 meson well above the region controlled by low energy theorems [3– photoproduction. 𝜙 5]. As a particle with hidden strangeness, meson can be The paper is organized as follows: in Section 2,the produced without an associated hyperon. Recently, in the multisource thermal model is introduced; in Section 3,we reaction 𝛾𝑝 → 𝑆𝑝𝜙(𝐾 𝐾𝐿), a photoproduction cross-section 𝜙 present our results, which are compared with the experimen- of the meson in its neutral decay mode was measured by tal data; at the end, we provide discussions and conclusions the CLAS Collaboration. The experiment is conducted by a in Section 4. tagged photon beam of energy 1.65 ≤ 𝐸𝛾 ≤ 3.6 GeV on a liquid hydrogen target at the Thomas Jefferson National Accelerator Facility (TJNAF) [6]. 2. Angular Distribution in the Multisource In order to explain the abundant experimental data, some Thermal Model phenomenological models of initial-coherent multiple interactions and particle transport were proposed and developed According to the multisource model [14], some emission in recent years [7–12]. The dynamics of the system evolution sources of 𝜙 mesons are assumed to be formed in the reaction 2 Advances in High Energy Physics

1 101

10 ) ) b

b E = 1.65 GeV E = 1.75GeV 𝛾 𝜇 𝛾 𝜇 ( ( 0 100

10 cm cm 𝜃 𝜃 −1

−1 cos cos 10 10 d𝜎/d

d𝜎/d −2 10−2 10

−1 −0.5 0.0 0.5 1.0 −1 −0.5 0.0 0.5 1.0

cos 𝜃cm cos 𝜃cm (a) (b)

1 E = 1.85GeV 1 10 𝛾 10 E𝛾 = 1.95GeV ) ) b b 𝜇 𝜇 ( 100 ( 100 cm cm 𝜃 𝜃 −1 −1 cos 10 cos 10 d𝜎/d 10−2 d𝜎/d 10−2

−1 −0.5 0.0 0.5 1.0 −1 −0.5 0.0 0.5 1.0

cos 𝜃cm cos 𝜃cm (c) (d) 𝜃 𝜙 1.65 ≤ 𝐸 ≤ 1.95 Figure 1: The cos cm dependence of meson differential cross-section for different photon beam energy 𝛾 GeV.The scattered symbols represent the experimental data. The solid lines are the calculated results.

process. Each source is considered to emit particles isotrop- Generally speaking, 𝑎𝑧 >1denotes the expansion of the ically in a source rest frame. Let the incident beam direction emission source along the 𝑜𝑧-axis and 𝑏𝑧 >0or 𝑏𝑧 <0 be 𝑜𝑧-axis and let the reaction plane be the 𝑥𝑜𝑧 plane, denotes the movement of the source along the positive or which is scanned by the vector of the beam direction and negative axes. impact parameter. In the source rest frame, the momentum 󸀠 󸀠 󸀠 components 𝑃𝑥, 𝑃𝑦,and𝑃𝑧 of a considered particle have the same Gaussian distributions with a width 𝜎.Then,a 3. Comparison with the CLAS Results 𝑃󸀠 = 𝑃󸀠2 +𝑃󸀠2 transverse momentum 𝑇 √ 𝑥 𝑦 obeys a Rayleigh The cos 𝜃 dependences of 𝜙 meson differential cross-section distribution. 𝑑𝜎/𝑑 cos 𝜃 over a photon beam energy range from 1.6 GeV to Due to the interactions among the emission sources, the 2.8 GeV is presented in Figure 1,whichisplottedfor0.1GeV sources will depart from isotropic emissions at the direction photon energy bins. The symbols denote the experimental of 𝑧. The corresponding momentum component 𝑃𝑧 of the data of the photoproduction cross-section of the 𝜙 meson particle is produced in its neutral decay mode in the reaction 𝛾𝑝 → 𝑝𝜙(𝐾𝑆𝐾𝐿).The𝑑𝜎/𝑑 cos 𝜃 increases with cos 𝜃 for the 󸀠 cm 𝑃𝑧 =𝑎𝑧𝑃𝑧 +𝑏𝑧, (1) photon energy range 1.65 ≤ 𝐸𝛾 ≤1.95GeV. The solid linesdenotetheresultscalculatedbythemultisourcemodel. where 𝑎𝑧 and 𝑏𝑧 indicate a deformation and movement of the One can see that the results are in agreement with the emitted source, respectively. As a standard treatment in the experimental data in the whole observed cos 𝜃 region for the Monte Carlo calculation, the random variable 𝑃𝑇 and 𝑃𝑧 are four energy bins. The values of 𝑎𝑧 and 𝑏𝑧 are obtained by fitting the data. The 𝑏𝑧 increases linearly with the incident

𝑃𝑇 =𝜎√−2 ln 𝑅1, photon energies. In order to see the structure of the sources (2) corresponding to different parameter values, the expansions 𝑃 =𝑎𝜎√−2 𝑅 (2𝜋𝑅 )+𝑏, and deformations of the modeling source are shown in 𝑧 𝑧 ln 2 cos 3 𝑧 Figure 6(a). In Figure 2, we show the 𝜙 meson differential cross- 𝑅 𝑅 𝑅 where 1, 2,and 3 are random variables distributed in section 𝑑𝜎/𝑑 cos 𝜃 as a function of cos 𝜃 .Thesymbols [0, 1] cm . So the polar angle is given by indicate the experimental data for the photon beam energy bins 2.05 ≤ 𝐸𝛾 ≤ 2.35 GeV. The solid lines indicate 𝑃 √−2 𝑅 𝜃= 𝑇 = ln 1 . the results calculated by the multisource thermal model. arctan 𝑃 arctan (3) 𝑧 𝑎𝑧√−2 ln 𝑅2 cos (2𝜋𝑅3)+𝑏𝑧 In the calculation, we take the different expansion and Advances in High Energy Physics 3

1 1 10 10 E = 2.15GeV E𝛾 = 2.05GeV 𝛾 ) ) b b 𝜇 𝜇

( 0 ( 100 10 cm cm 𝜃 𝜃 −1 −1

cos 10

cos 10 d𝜎/d d𝜎/d 10−2 10−2

−1 −0.5 0.0 0.5 1.0 −1 −0.5 0.0 0.5 1.0

cos 𝜃cm cos 𝜃cm (a) (b)

1 101 10 E = 2.35GeV E𝛾 = 2.25GeV 𝛾 ) ) b b 𝜇 𝜇 0 0 ( ( 10 10 cm cm 𝜃 𝜃 −1 −1

cos 10

cos 10 d𝜎/d d𝜎/d 10−2 10−2

−1 −0.5 0.0 0.5 1.0 −1 −0.5 0.0 0.5 1.0

cos 𝜃cm cos 𝜃cm (c) (d)

Figure 2: Same as Figure 1, but for the photon energy range 2.05 ≤ 𝐸𝛾 ≤ 2.35 GeV.

101 101 ) )

E = 2.45GeV b b 𝛾 E = 2.55GeV

𝜇 𝛾 𝜇 ( ( 100 100 cm cm 𝜃 𝜃 −1 10−1 10 cos cos d𝜎/d d𝜎/d 10−2 10−2

−1 −0.5 0.0 0.5 1.0 −1 −0.5 0.0 0.5 1.0

cos 𝜃cm cos 𝜃cm (a) (b)

101 101 ) ) b b E = 2.65GeV 𝜇 𝜇 𝛾 E = 2.75GeV ( ( 𝛾 100 100 cm cm 𝜃 𝜃 10−1 10−1 cos cos d𝜎/d d𝜎/d 10−2 10−2

−1 −0.5 0.0 0.5 1.0 −1 −0.5 0.0 0.5 1.0

cos 𝜃cm cos 𝜃cm (c) (d)

Figure 3: Same as Figure 1, but for the photon energy range 2.45 ≤ 𝐸𝛾 ≤ 2.75 GeV. 4 Advances in High Energy Physics

101 101 ) ) b 0 E = 2.85GeV b 𝜇 10 𝛾 𝜇 0 E = 2.95GeV

( 𝛾 ( 10 cm cm 𝜃 −1 𝜃 10 10−1 cos cos −2 10 10−2 d𝜎/d d𝜎/d 10−3 10−3 −1 −0.5 0.0 0.5 1.0 −1 −0.5 0.0 0.5 1.0

cos 𝜃cm cos 𝜃cm (a) (b)

101 101 ) ) b E = 3.05GeV b E = 3.15GeV 𝜇 0 𝛾 𝜇 0 𝛾 ( 10 ( 10 cm cm 𝜃 10−1 𝜃 10−1 cos cos 10−2 10−2 d𝜎/d d𝜎/d 10−3 10−3 −1 −0.5 0.0 0.5 1.0 −1 −0.5 0.0 0.5 1.0

cos 𝜃cm cos 𝜃cm (c) (d)

Figure 4: Same as Figure 1, but for the photon energy range 2.85 ≤ 𝐸𝛾 ≤ 3.15 GeV.

101 101 ) ) b b

𝜇 E = 3.35GeV 0 E = 3.25GeV 𝜇 0 𝛾 ( 10 𝛾 ( 10 cm cm 𝜃 10−1 𝜃 10−1 cos cos 10−2 10−2 d𝜎/d d𝜎/d 10−3 10−3 −1 −0.5 0.0 0.5 1.0 −1 −0.5 0.0 0.5 1.0

cos 𝜃cm cos 𝜃cm (a) (b)

101 101 ) ) b b E𝛾 = 3.45GeV E𝛾 = 3.55GeV 𝜇 𝜇 0 0 ( ( 10 10 cm cm 𝜃 𝜃 10−1 10−1 cos cos 10−2 10−2 d𝜎/d d𝜎/d 10−3 10−3 −1 −0.5 0.0 0.5 1.0 −1 −0.5 0.0 0.5 1.0

cos 𝜃cm cos 𝜃cm (c) (d)

Figure 5: Same as Figure 1, but for the photon energy range 3.25 ≤ 𝐸𝛾 ≤ 3.55 GeV. Advances in High Energy Physics 5

4.0 7 5.6 3.2 4.8 6 2.4 4.0 5 4 1.6 3.2 3

(a.u.) 2.4 (a.u.) (a.u.) z 0.8

z 1.6 2 p z p 0.0 0.8 p 1 0.0 −0.8 0 −0.8 −1.6 −1 −1.6 −2 −3.2 −1.6 0.0 1.6 3.2 −5−3 −1 135 −6 −4 −2 0246 p (a.u.) py (a.u.) y py (a.u.)

E𝛾 = 1.65GeV E𝛾 = 1.85GeV E𝛾 = 2.05GeV E𝛾 = 2.25GeV E𝛾 = 2.45GeV E𝛾 = 2.65GeV az = 1.34 az = 1.56 az = 1.58 az = 1.54 az = 1.38 az = 1.62 bz = 0.59 bz = 1.06 bz = 1.92 bz = 2.42 bz = 2.65 bz = 3.45 E = 1.75GeV E = 2.15GeV E = 2.55GeV 𝛾 E𝛾 = 1.95GeV 𝛾 E𝛾 = 2.35GeV 𝛾 E𝛾 = 2.75GeV az = 1.08 az = 1.96 az = 1.48 az = 1.62 az = 1.08 az = 1.63 bz = 0.63 bz = 1.40 bz = 2.32 bz = 2.68 bz = 2.25 bz = 3.65

(a) (b) (c) 7 7 6 6 5 5 4 4 3 3 (a.u.) (a.u.) 2 z z 2 p p 1 1 0 0 −1 −1 −2 −2 −6 −4 −2 0 24 6 −6 −4 −2 0 2 4 6

py (a.u.) py (a.u.)

E𝛾 = 2.85GeV E𝛾 = 3.05GeV E𝛾 = 3.25GeV E𝛾 = 3.45GeV az = 1.93 az = 1.86 az = 1.53 az = 1.54 bz = 3.81 bz = 4.40 bz = 4.06 bz = 4.28 E = 2.95GeV E = 3.35GeV 𝛾 E𝛾 = 3.15GeV 𝛾 E𝛾 = 3.55GeV az = 1.82 az = 1.88 az = 1.39 az = 1.50 bz = 3.87 bz = 4.50 bz = 3.86 bz = 4.36

(d) (e)

Figure 6: The source deformations and movements in the reaction plane corresponding to the different photon beam energies 𝐸𝛾.

3.0 5 2.5 4 2.0 3 z z

1.5 b 2 a 1.0 1 0.5 0 0.0 −1 1.5 2.0 2.5 3.0 3.5 4.0 1.5 2.0 2.5 3.0 3.5 4.0

E𝛾 E𝛾

(a) (b)

Figure 7: The dependence of 𝑎𝑧 and 𝑏𝑧 on the photon beam energies 𝐸𝛾. The symbols represent the parameter values used in Figures 1–5.The solid lines are the fitted results. 6 Advances in High Energy Physics deformation sources as marked in Figure 6(b).Figures3, Acknowledgments 4,and5 are detailed comparisons between the results and the experimental data for 2.45 ≤ 𝐸𝛾 ≤ 2.75 GeV, 2.85 ≤ This work is supported by the National Natural Science 𝐸𝛾 ≤ 3.15 GeV, and 3.25 ≤ 𝐸𝛾 ≤ 3.55 GeV, respectively. Foundation of China under Grants nos. 11247250, 11005071, The results are also in agreement with the data from the and 10975095, the National Fundamental Fund of Personnel CLAS Collaboration. The corresponding anisotropic sources Training under Grant no. J1103210, the Shanxi Provincial are given in Figures 6(c)–6(e). Natural Science Foundation under Grants nos. 2013021006 The parameters 𝑎𝑧 and 𝑏𝑧 obtained in the above com- and no. 2011011001, the Open Research Subject of the Chinese parisons are given in Figure 7.Thevalueof𝑎𝑧 is around 1.5 Academy of Sciences Large-Scale Scientific Facility under and has no obvious regularity. Except for a saturation region, Grant no. 2060205, and the Shanxi Scholarship Council of the values of 𝑏𝑧 exhibit a linear dependence on the photon China. beam energies, which are best fitted by the function 𝑏𝑧 = (3.02 ± 0.04)𝐸 − (4.5 ± 0.08) 𝛾 . The movement of the sources References along the positive 𝑧 direction increases approximately with increasing photon beam energy. [1] J. Rafelski and B. Muller,¨ “Strangeness production in the quark- gluon plasma,” Physical Review Letters, vol. 48, no. 16, pp. 1066– 1069, 1982. 4. Discussions and Conclusions [2] J. Adams, M. M. Aggarwal, Z. Ahammed et al., “Multistrange baryon elliptic flow in Au+Au collisions at √𝑆𝑁𝑁 = 200 GeV,” In the framework of the multisource thermal model, we Physical Review Letters,vol.95,no.12,ArticleID122301,6pages, have investigated the differential cross-section versus the 2005. polar angle of 𝜙 meson photoproduction on hydrogen in [3]J.Barth,W.Braun,J.Ernstetal.,“Low-energyphotoproduction the neutral decay mode. The incident photon beam energy of 𝜙-mesons,” The European Physical Journal A,vol.17,pp.269– range is 1.65–3.55 GeV. The results calculated in the model 274, 2003. approximately agree with the data of the CLAS Collaboration. [4] T. Mibe, W. C. Chang, T. Nakano et al., “Near-threshold In the above discussions, the deformation and movement of diffractive 𝜙-meson photoproduction from the proton,” Phys- the sources are obtained by fitting the experimental data. ical Review Letters,vol.95,no.18,ArticleID182001,5pages, The local sources of the final-state particles are formed in 2005. the reaction plane. Their interactions, which are related to [5]E.Anciant,T.Auger,G.Auditetal.,“Photoproductionof the matter in the sources, result in the anisotropic expansion 𝜙(1020) mesons on the proton at large momentum transfer,” and the movement along the direction of the beam [12]. For Physical Review Letters,vol.85,pp.4682–4686,2000. 𝜙 the longitudinal structure of the sources, 𝑎𝑧 >1presents [6]H.Seraydaryan,M.J.Amaryan,G.Gavalianetal.,“ -Meson a longitudinal expansion of the source along 𝑜𝑧-axis. When photoproductiononHydrogenintheneutraldecaymode,” http://arxiv.org/abs/1308.1363. 𝑏𝑧 =0̸, the source has a movement along 𝑜𝑧-axis. The model involves the anisotropic expansion of the participant area in [7] M.L.Miller,K.Reygers,S.J.Sanders,andP.Steinberg,“Glauber the transverse momentum space. Therefore, the multisource modeling in high-energy nuclear collisions,” Annual Review of thermalmodelcanbeusedtostudytheellipticflowof Nuclear and Particle Science,vol.57,pp.205–243,2007. charged hadrons produced in high energy collisions at RHIC [8]H.DeVries,C.W.DeJager,andC.DeVries,“Nuclear energies. The analysis of the elliptic flow shows that the charge-density-distribution parameters from elastic electron expansion factor is characterized by the impact parameter, scattering,” Atomic Data and Nuclear Data Tables,vol.36,pp. 495–536, 1987. which is related to participating nucleons. Moreover, the system expansion can be quantified in the momentum space. [9] P. Braun-Munzinger, D. Magestro, K. Redlich, and J. Stachel, “HadronproductioninAu–AucollisionsatRHIC,”Physics In the description of (pseudo)rapidity and multiplicity dis- Letters B,vol.518,no.1-2,pp.41–46,2001. tributions for the final-state particles [14], this model is also [10] J. Rafelski and J. Letessier, “Testing limits of statistical successful. hadronization,” Nuclear Physics A, vol. 715, pp. 98c–107c, 2003. All final-state particles are emitted from the sources formed in the reaction. The polar angle distributions can [11] V. Roy and A. K. Chaudhuri, “Transverse momentum spectra and elliptic flow in ideal hydrodynamics and geometric scaling,” be obtained by the statistical method. The movement factor 𝑏 Physical Review C,vol.81,no.6,ArticleID067901,4pages,2010. 𝑧 used in the calculation exhibits a linear dependence on 𝜓 𝐸 < 3.0 [12] T. Song, C. M. Ko, S. H. Lee, and J. Xu, “J/ production and thephotonbeamenergy 𝛾 GeV. Our previous work elliptic flow in relativistic heavy-ion collisions,” Physical Review mostly described particle production in intermediate energy C, vol. 83, Article ID 014914, 13 pages, 2011. and high energy collisions. In the description of 𝜙 meson 𝛾𝑝 → 𝑝𝜙(𝐾 𝐾 ) [13]B.C.Li,Y.Y.Fu,L.L.Wang,andF.-H.Liu,“Dependence photoproduction in the reaction 𝑆 𝐿 ,thiswork of elliptic flows on transverse momentum and number of is a good attempt. participants in Au+Au collisions at √𝑆𝑁𝑁 = 200 GeV,” Journal of Physics G,vol.40,no.2,ArticleID025104,2013. [14] F. H. Liu, X. Y. Yin, J. L. Tian, and N. N. Abd Allah, “Charged- + − Conflict of Interests particle (pseudo)rapidity distributions in 𝑒 𝑒 , 𝑝𝑝,andAA The authors declare that there is no conflict of interests collisions at high energies,” Physical Review C,vol.69,Article ID 034905, 5 pages, 2004. regarding the publication of this paper. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 901434, 9 pages http://dx.doi.org/10.1155/2014/901434

Research Article Radio-Wave Propagation in Salt Domes: Implications for a UHE Cosmic Neutrino Detector

Alina-Mihaela Badescu1 and Alexandra Saftoiu2 1 University Politehnica of Bucharest, Boulevard Iuliu-Maniu, No. 1-3, Bucharest, Romania 2 IFIN-Horia Hulubei, Strada Reactorului No. 30, P.O. Box MG-6, Bucharest-Magurele, Romania

Correspondence should be addressed to Alina-Mihaela Badescu; [email protected]

Received 3 September 2013; Revised 21 December 2013; Accepted 24 December 2013; Published 20 January 2014

Academic Editor: Mehmet Bektasoglu

Copyright © 2014 A.-M. Badescu and A. Saftoiu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Salt deposits can be used as a natural dielectric medium for a UHE cosmic neutrino radio detector. Such a detector relies on the capability of reconstructing the initial characteristics of the cosmic neutrino from the measured radio electrical field produced at neutrino’s interaction in salt by the subsequent particle shower. A rigorous characterization of the propagation medium becomes compulsory. It is shown here that the amplitude of the electric field vector is attenuated by almost 90% after 100 m of propagation in a typical salt rock volume. The heterogeneities in salt also determine the minimal uncertainty (estimated at 19%) and the resolution of the detector.

1. Introduction Thus, ice can serve as the detecting medium for optical and radio waves (an example is the IceCube detector [3]thatuses Ultra high energy neutrinos can be a proof of the theoretical photomultipliers to measure the EM field) and salt—for radio upper limit on the energy of cosmic rays from distant sources, waves. The latter was tested at Stanford Linear Accelerator play an important role in the Big Bang scenario, and also where radio waves from high energy particles interacting in unveil the mystery of the cosmic accelerator (pulsars, active synthetic rock salt were detected [4]. galactic nuclei, etc.). In the same time they travel undeflected One of the key problems associated with a neutrino by intervening magnetic fields and interact very weakly. This radio detector in salt is the capability of reconstructing the makes their observation a scientifical and technical challenge. characteristicsofthecosmicneutrinosthatinteractinthe A cosmic neutrino detector images the sky using interac- salt volume from measurements of the radio radiation which tions of a nearly massless subatomic particle called neutrino. resulted in their interaction. Detection can be performed Neutrinos are weakly interacting particles that cannot be using arrays of (standard) radio antennas placed in boreholes: detected directly. Their properties are deduced by analyzing the more transparent the medium, the larger the usable the showers resulted from their interaction with nuclei in the distance between antennas. The radio waves produced by the medium. neutrino-induced shower travel through salt so the prop- One detection method was proposed by Askaryan [1]. agation medium has a huge impact on measurements and He suggested that if a particle, including neutrinos, interacts results. Most of the medium’s properties can be described within a volume of dielectric, a broadband electromagnetic using the relative permittivity. In order to achieve the highest (EM) field (including radio frequencies), that can be mea- detection performances, antennas should be placed away sured, will be generated. In order to compensate for the small from the periphery of the dome (including its cap rock). interaction probability of the neutrino [2], a huge volume of The existing optical neutrino detector at the South Pole detecting material is required that can be found in natural IceCube pointed out serious problems in detection and data dielectric volumes, such as the ice sheets at the poles or analysis due to the effects that the nonideal medium has naturalsaltdomes.Themediumshouldbetransparentfor on wave propagation. A detailed study of the properties of the produced waves to ensure large propagation distances. theglacialiceattheSouthPolehasbeenperformedbythe 2 Advances in High Energy Physics

AMANDA collaboration [5]. They found that ice is very clear in the optical and near UV regions but both scattering and absorption are strongly depth dependent. At each 10 m depth interval, the effective scattering and absorption lengths as a function of wavelength were determined. In this work, we investigate the possibility of detecting cosmic neutrinos in salt by measuring and analyzing their interaction products: radio waves [6, 7]. Radio waves are the result of the Askaryan effect [1]. The broadband frequency spectrumpeaksatafewGHz,butduetoseveralreasons (attenuation, temperature dependence, etc.) we decided to select an operating frequency of about 200 MHz [6]. As waves propagate in a nonideal medium before being measured, it is mandatory to have first a good geophysical material description for radio waves propagation. Hapke already remarked that this regime is not well understood [8]. It was suggested that an effective-medium theory should be applied to calcu- Figure 1: Genesis Room (54 m height, 208 m depth) of the “Unirea” late the permittivity. salt mine, Slanic Prahova, Romania. Photo courtesy of © Andrei Niculescu. The influence of the medium can be quantified by transmission—a parameter that incorporates all propagation effects (more details are given in Section 3). Transmission should be estimated because the data to be analyzed—for heterogeneities and other factors that can affect propagation. example, measurements recorded by each antenna—is pro- In the next section, natural occurring impurities and main portional to the product of transmission and the radio field heterogeneities in a salt dome and their effects on radio generated at interaction. For the latter, the model in [9, 10] wave propagation are analyzed. The last part summarizes our will be used. results. One method to achieve medium characterization is by downhole geophysical logging. This technique involves 2. Geological Properties of Salt Mines installing sensing devices into a borehole to record physical parameters that may be interpreted as specific rock character- Rock salt deposits are widely distributed throughout the istics. The geological mapping of underground mine working world. The salt accumulations in Romania are among the and geological logging of core samples are insufficient to largest in Europe and thus construction of a neutrino detector make an identification of the internal structure of evaporites wouldbewelljustifiedhere.Thesaltthatformsdiapirsstud- [11]. ied here is Early Miocene in age [22].Inthefollowing,wewill Another method to obtain information about the spa- only refer to Unirea mine, in Slanic Prahova (Romania), char- tial variation in dielectric properties is ground penetrating acterized by a domal folding structure (Figure 1). radar. A pulse radar emits an electromagnetic pulse from a This salt dome represents a gigantic plug that has risen dipole antenna into a rock and the returned signal contains upward diapirically, because of its low density, into the overly- reflections caused by subsurface contrasts in electromagnetic ing strata [22]. Originally the sedimentary salt deposits were impedance. The travel time reflects the depth to the impe- at depths of several kilometers. Under hydrostatic pressure, dance contrasts [12]. the very plastic salt of lower density starts to flow upward The same issue—good geophysical material description through sediments of greater density. Salt flow is controlled for radio waves propagation—has been addressed by other through the fault intersections resulting in a domal structure. types of applications: locating buried utilities [13–15], detect- The shape of the salt deposit depends on the specific condi- ing buried land mines [16–18], profiling the subsurface tions of its emplacement in geological environment (“Unirea” of highway pavement, and so forth. Although theoretical dome has a lenticular shape) [22]. approaches have been reported by [19, 20]andothers,models Following [11], we considered four basic heterogeneous for propagation are rare because the heterogeneity in salt internal structures of the salt deposits. clearly affects radio waves. First tests on determining the attenuation length of a radio signal at 400 and 800 MHz on 2.1. Domal Heterogeneity. The main characteristic of domal samples from the “Unirea” salt dome in Slanic Prahova, heterogeneity is the mixture of halite and anhydrite. Darker Romania,showedalargedeviationfromtheidealmedium salts owe their color to dissemination of anhydrite. A mixture case [21]. The “Unirea” salt mine will be considered a conser- of salt and anhydrite can be found in various proportions vative case. It was chosen in our analysis because we already at short distance, which will result in extreme heterogeneity. have a working laboratory installed so we had access to Generally, the content of anhydrite in salt dome varies from different results concerning the purity of salt. 1% to 80%. Inthefollowingweinvestigateandquantifytheeffect The effect of the main domal heterogeneities (listed in ofthemediumonwavepropagation.InSection 2 we briefly Table 1) was simulated and their influence on the properties review the geological properties of salt domes, including main of the medium was presented in [23]. Advances in High Energy Physics 3

Table 1: Concentration of main impurities in Slanic Prahova mine, where 𝑟12𝑠,𝑝 istheratiooftheamplitudesofthereflectedelec- according to [38]. tric field to the incident one when the field vector is perpen- dicular (index 𝑠)/parallel (index 𝑝)tothepropagationplane. Al Ca Ti Fe V Mn Cu Br The reflection coefficients are given by[8] 𝐶 [𝜇g/kg] 14.7 2190 2260 33 14.9 61.7 7.9 126 2 cos 𝛼 𝑟12𝑠 = , cos 𝛼+𝑚cos 𝛼𝑡 (5) 2 𝛼 n cos Layer 𝑟12𝑝 = , 𝑚 cos 𝛼+cos 𝛼𝑡

where 𝑚 is the ratio of the refractive indexes (the refractive index of a medium is defined as the square root of the medium’spermittivity) and 𝛼𝑡 is the angle that the transmitted wave makes with the normal at the interface with the second 2 medium. It can be calculated using Snell’s law of refraction: 𝛼t Layer dL2 sin 𝛼=𝑚sin 𝛼𝑡. (6) 𝛼 Layer 1 If the permittivity is a complex number, 𝛼𝑡 will also be dL1 complex. The physical meaning of a complex number isa Figure 2: An unaltered sedimentary structure of a salt dome [23]. phase shift of the transmitted wave. The solid thin lines represent waves that propagate in each layer. The The sedimentary heterogeneity effect (the transmitted dashed lines show the reflected ones. The thick solid line shows the fraction from layer 1 to layer 𝑛)canbeevaluatedusing wave that enters in a layer after multiple reflections. The thickness of the model described by (1)–(6). As shown in Figure 2,the layer 𝑖 is 𝑑𝐿𝑖. electromagnetic radiation crosses the first layer and only a fraction of it enters the second layer. The reminder is reflected and can be regarded as a loss. In order to show that the loss 2.2. Sedimentary Heterogeneity. On ancient oceans bottoms, assumption is well justified, the power of the transmitted layers of sediment were deposited as a sequence of horizontal waves in layer two (thick arrow in Figure 2) that resides after beds. An unaltered sedimentary structure of a dome is multiple reflections in layer 1 (dashed arrow in Figure 2)has represented by a column of various beds which could be been calculated. As it is more than three orders of smaller grouped in function of their chemical composition or facial magnitude it can be ignored [7]. development (Figure 2)[11]. Thick sedimentary beds are classified as relatively homogeneous, nondispersive, isotropic, 2.3. Mineral Heterogeneity. With respect to mineral het- and linear medium. In this model, we ignored interbedded erogeneity, the spatial relationship between halite salts and layers as they are not a common feature in salt diapirs. The potassium salts is primarily important. The multimineral 𝜏 transmission coefficient duetowavepropagationfromlayer development is of a complex nature and is usually located 1 to layer 3 (that is the fraction of wave energy that reaches at the boundary of a sedimentary basin. The development layer 3) is given by [24] of potassium salts in a majority of cases occurs like a frame 2 around the rock salt body. 𝜏=|𝑤| , (1) The mineral heterogeneity will not be further investigated where 𝑤 is in this paper, due to lack of experimental probes. For further processing, multiple samples should be collected from differ- 1−𝑟2 𝑤= 12 , ent locations and chemically analyzed. We estimate that this 2 (2) exp (−𝑖Ψ) −𝑟12 exp (𝑖Ψ) heterogeneity (associated with peripheral areas of the dome) can be disregarded in the case of a neutrino detector since with dataacquisitionwillbemadeonlyinthecentralpartsofthe 𝜔 Ψ= ℎ √𝜀 − 2𝛼 salt block. 𝑐 2 𝑟2 sin (3) being a measure of the absorption in medium 2; ℎ2 is the 2.4. Structural Heterogeneities. Structural heterogeneities of height of the second medium (layer’s thickness), 𝜀𝑟2 is the evaporated strata are the product of tectonic movements of permittivity of medium 2, and 𝛼 istheincidenceanglewith the earth’s crust in the region of their deposition. respect to the normal (Figure 2). The structural heterogeneities in evaporated strata could The coefficient 𝑟12 describes the reflections at the first vary significantly due to differences in elasticity of individual interface. If one considers nonpolarized waves, then layers. For example, gypsum, anhydrite, marl, and others might have developed an apparent internal blocky structure. 𝑟12𝑠 +𝑟12𝑝 𝑟= , (4) It is specific to cap rocks or other types of folding structures 2 and thus it will not represent a case study here. 4 Advances in High Energy Physics

Another geological property of a salt mine is the faulting Salt is impermeable and when it reaches a layer of per- structures. A problem associated with the step faults is that meable rock, in which hydrocarbons are migrating, it blocks they can be a water feeder to the salt deposits. the pathway in much the same manner as a fault trap. Due Investigations of radio wave propagation in salt are diffi- to that, the region around the perimeter of the salt dome is an cult because a priori knowledge of the location of water- or ideal geologic environment for hydrocarbon traps [32]. brine-bearing zones is seldom available. The fracturing of surrounding rocks due to the intruding Rock salt is a conductor for electric current mostly due to saltandtheliftingoftherocksabovethesaltdomealso internally contained water in the pores’ interstices. The water provide an environment for the existence of fault traps and induced conductivity in rock salt can be determined using anticlinal traps in addition to the salt dome traps around the the model in [25].Thevaluefortheconductivityinducedby perimeter of the dome. A salt dome region, therefore, is an −6 −1 −1 a 0.02% water content is 2 × 10 Ω m [26]. Such small excellent geologic environment for all types of traps [33]. water content is not expected to decrease significantly the Moreover, association with evaporite minerals can provide transmission in salt beds. excellent sealing capabilities. The monitoring of reservoir The effect of brines in forms of fluid inclusions was also productioncanbeperformedusingEMmethods[34]. estimated in [23]. The brine induced radio wave attenuation Since in the case of a cosmic neutrino detector the detect- was calculated following [27, 28]. If the waves travel a distance ing elements are placed away from periphery of the dome, we of 1.1 cm in brine, about 70% of their energy will be lost. can conclude that neither EMI nor IP will affect radio wave If the distance increases to 1.4 cm, only 20% of energy will propagation. Moreover, these effects are important at kHz remain. For propagation distances larger than 2.1 cm, the frequencies, so observations at ∼200 MHz will not be affected. transmission coefficient will be smaller than 10%. It is concluded that connate water and secondary trapped 3.2. Radio Transmission Estimates. We consider long-range water [11] will absorb all radio electromagnetic radiation. propagation of radio waves, which implies a large number of Location of such caverns must be determined a priori. They crossed sedimentary layers. For this purpose, we will intro- can be traced only experimentally (i.e., using GPR measure- duce the transmission—that is, the ratio of the transmitted ments, as suggested in [29]). Another groundwater source electric field strength after propagation through the entire salt tracercanbeachievedbychemicalcompositionanalysis(e.g., medium to the initial electrical field strength. Absorption and a high MgCl2 content indicates connate water, the Cl/Br scatteringeffectsarealsoincludedinthisquantity. coefficient-primary trapped water etc.11 [ ]). Forthesimulationspresentedinthispaper,weconsidered the classical propagation model. We assumed that there are 3. Propagation Studies only a forward and a backward traveling wave in each layer. A limitation of the model that would require further investi- 3.1. Effect of Heterogeneous Zones on Electromagnetic Waves. gation is connected to the evanescent fields at the separation Pure sodium chloride is an isotropic crystal. Entering radi- borders between layers which could react with the evanescent ation is refracted at a constant angle and passes through fields of adjacent layers. the crystal at a single velocity without being polarized by In real situations, one can neither predict nor measure the interaction with the electronic components of the crystalline permittivities of all the salt layers (especially when the volume lattice.Thus,foraslongasimpuritiesareinsmallnumber,one of interest is of the order of cubic kilometers) and extended can expect no birefringence phenomenon in all frequency invasive procedures are not an option before excavating bands. theboreholes.Thus,oneshouldinvestigatethepossibility The electromagnetic (EM) data observed in geophysical of approximating the real environment by considering an experiments in heterogeneous media generally reflect two “equivalent permittivity” of the medium (given by the mean phenomena: electromagnetic induction (EMI) in the earth and of the permittivities of the constituents layers, here consid- the induced polarization (IP) effect related to the relaxation of ered equal in size). The assumption of a media build of homo- polarized charges in rock formations. The IP effect is caused geneous layers with constant thickness does not fit the in situ by the complex electrochemical reactions phenomena that characteristics of a salt dome in most cases, but it is treated accompany current flow in the earth and it is manifested here as a conservative case. by accumulating electric charges on the surface of different To study the effect of the impurities on long-range prop- grains forming the rock. agation, we simulated a situation where only one impurity EMI and IP phenomena occur in rock formations in is changing its concentration from layer to layer. If the areas of mineralization (areas with mineralized particles) and layer to layer variations in concentration are small (about 𝐶 𝐶 hydrocarbon reservoirs [30]. 0.1 element—where element is the concentration of the element Areas of mineralization areobservedinsaltdome’scap given in Table 1), the results are independent of the impurity rock. The cap rock is composed mainly of anhydrite, gypsum, type that varies its concentration. This is due to the fact that and calcite arranged in heterogeneous layers. Cap rock most of the power is lost by reflections between layers while layering is irregular and varies greatly from dome to dome. absorption and scattering due to impurities are very small. Structural deformation and fracturing are common, as are Absorption remains small compared to reflections even cavernous voids. Analysis of other features as the locations of if layer to layer difference in concentration is large (about 0.5𝐶 kimberlites, faulting, including zones of mineralization, can element). The transmission is poorer for impurities of be determined from, for example, aeromagnetic data [31]. higher physical dimension. Most losses are associated with Advances in High Energy Physics 5

0.385

0.94 0.38

0.93 0.375 ), Br (layered) ), Br 0.92 i 0.37 ), Br (equivalent) ), Br /E i t E /E ( t E ( 0.91 0.365

0.9 0.36 10 20 30 40 10 20 30 40 Number of layers Number of layers (a) (b)

Figure 3: Effect of wave reflection when the concentration of one impurity varies randomly from layer to layer. All layers have the same thickness (20 cm—thinner lines—or 2 m—thicker lines marked by “+”). The continuous lines (b) mark the actual transmitted field and the dotted ones (a) mark the transmitted field in the “equivalent” situation.

reflections at interfaces between layers. The higher the num- to the product of transmission and the neutrino generated ber of layers is, the higher the losses (reflections) are. radio field at interaction in salt. Moreover, the resolution and In order to evaluate the effects of the number of layers sensitivity of the detector depend on this factor. and of a single type of impurity on long-range propagation, In previous studies, the propagation medium was consid- we have performed a simulation in which all layers have a ered homogeneous and the attenuation length was defined constant thickness 𝑑𝐿. From layer to layer, only one impurity straightforward [35]: varies its concentration, uniformly distributed in the interval 𝑐 [0, 𝐶 ]. The rest of impurities have the same concentra- 𝐿= 0 , element (7) tion in all layers (again given in Table 1). For each number 𝜋𝑓√R {𝜀𝑟} tan 𝛿 of layers 𝑁, an “equivalent” situation has been calculated (in 𝑁×𝑑 which the permittivity of a single layer of length 𝐿 is where the loss tangent is determined using the mean volume averaged concentration of impurities), represented in Figure 3 by dotted lines. In I {𝜀 } 𝛿= 𝑟 . Figure 3, both the actual transmission and the “equivalent” tan R {𝜀 } (8) transmission are shown. One can see that the “equivalent” 𝑟 transmission is overestimating the actual situation. Moreover, In both equations, R{𝜀𝑟} denotes the real part of the permit- if only one impurity varies its concentration layer to layer, tivity and I{𝜀𝑟} is the imaginary part. reflections at interfaces are extremely small. Results are Forpuresalt,theattenuationlengthreachesmorethan independent of the impurity type but dependent on their 1 km at a few hundreds MHz. This is the case reported in volume concentration. We have shown results for just one [36] where attenuation lengths of 900 m were measured at type of impurity (Br) because results are much alike for the 200 MHz in Hockley Mine, USA. One of the most precise rest. measurements of radio attenuation in a natural salt formation To make an estimate of the magnitude of absorption only, performed in the Cote Blanche salt mine found attenuation we have calculated the difference between the transmission lengthsfrom93m(at150MHz)to63m(at300MHz)[37]. with and without absorption, when the total propagation However, salt in other deposits in North America showed distance is up to 100 m and the layers are of thickness 2, −9 dielectric constants ranging 5–7 and loss tangents 0.015– 4, or 10 cm. Absorption accounts for less than 10 of the 0.030 or more at 300 MHz, implying attenuation lengths total loss of energy. Thus, the main mechanism that causes below 10 m [35]. thedecreaseofawave’samplitudeisreflectionatinterfaces Intheprevioussection,itwasshownthatthepermit- between sedimented layers. tivity cannot be approximated by any kind of “mean” value (Figure 3) of permittivities from different (possibly collected) 3.3. Propagation Induced Limitations in a Cosmic Neutrino samples. This has direct consequences on the construction Detector. Transmission through salt layers should be care- and predictions for the detector. A realistic model for prop- fully estimated because the data to be analyzed—for example, agation should consider a layered medium and in each layer measurements recorded by each antenna—is proportional the impurities should have a random concentration. To this 6 Advances in High Energy Physics

0.3 140

0.06 0.02

0.25 120 0.02 0.04 0.06

0.04 0.1 0.08 0.04 0.06 0.02 100 0.08 0.2 0.06

i 0.1

(m) 80 0.02 0.12 /E 0.15 t

d 0.04

E 0.12

0.08 0.14 0.1 0.08

60 0.06 0.16 0.04

0.02 0.06

0.1 0.18 0.16 0.1 0.12 0.12 0.1 0.14 0.14 0.060.08 0.04 40 0.02

0.05 0.02 0.08 0.12 0.140.16 0.04 0.12 0.1 0.08

0.06 0.14 0.04 20 0.1 0.16 0.060.02 0 246810 40 50 60 70 80 90 100 30 dL (cm) Propagation distance (m) Figure 6: The isosurfaces for transmission when different layer dL =2cm thickness’ and propagation distances were considered. dL =4cm dL =10cm

Figure 4: Mean and standard deviation for transmission when different propagation distances and layers thickness are considered (details are given in text). (e.g., distance between the neutrino interaction and radio detector element) but also the thickness of the sedimentary 0.3 layers can be determined. Results in Figure 4 are obtained in the assumption of equal sedimentary layers. 0.25 As a next step, we considered simultaneous random vari- 0.4 ations in propagation distance and layer thickness. Figure 5 0.2 shows the mean values of the transmission factors for each 0.3 combination of layer thickness-propagation distance. The widest layers and smallest propagation length produce the i 0.15 highest transmission. /E 0.2 t

E Depending on the layer’s thickness, the 1𝜎 standard 0.1 deviation of the transmission varies. For 1 cm thick layers, it is 0.1 10 equalto18.75%anditdecreaseswiththeincreaseinthelayer’s 5 0.05 thickness (for layers of thickness 2 cm it becomes 17.9%). 0 The minimal resolution of the detector is given by the 150 (cm) 100 0 d L standard deviation. The larger the standard deviation is, the 50 0 d (m) larger the ambiguity in determining the thickness of the layers is. However, for particular propagation distances, it is still Figure 5: Mean transmission values versus different propagation possible to uniquely estimate the thickness of the layers. distances and thickness of constituents (equal) layers. In Figure 6, the isosurfaces of transmission are shown. From the cost point of view, we are interested in a detector that allows maximum propagation distance (i.e., distance between antennas) but is capable of good reconstruction of end,weallowedeachimpuritytypeineachlayertotakearan- signals (thus neutrino properties) in a given medium. If we 𝐶 𝐶 dom concentration in the interval [0.5 element,1.5 element]— impose a maximum propagation distance of 100 m and a 𝐶 where element is the nominal concentration given in Table 1— threshold of 10% (the medium itself produces 90% losses), and repeated each simulation multiple times. themediumshouldbeformedbylayersofminimum9cm We considered sets of 30 simulations, in each of which we thickness. A smaller distance between receivers of 80 m used random values for concentrations of all impurity types, allows a threshold of 12% when the minimum thickness of different from layer to layer. The transmission was calculated the layers is 8 cm. in each simulation and a mean value and standard deviation An important observation is the phase change. When were determined for the set of 30 simulations. Figure 4 shows crossing interfaces between different layers, not only the results when equal layers of thickness 2 cm, 4 cm, and 6 cm amplitude of the wave is reduced but also a phase shift occurs. were considered. Figure 7 showsthephaseshiftversusthenumberoflayersof Figure 4 shows that in the case of a known value of thicknessequalto20cm.Themeasuredphasecannotbeused transmission (ratio 𝐸𝑡/𝐸𝑖) not only the propagation distance to determine the initial particle direction. Advances in High Energy Physics 7

×10−6 10−2 2

1 10−4 0 ) i

−1 /E t 10−6

−2 Δ(E

−3 Phase shift (deg) 10−8 −4

−5 10−10 −6 30 40 50 60 70 80 10 20 30 40 Propagation distance (m) Number of layers dL =2cm Ca dL =4cm Ti dL =10cm

Figure 7: The phase shift as function of the number of layers of Figure 8: Difference between transmission at 187.5 MHz and 1 GHz. thickness 20 cm. With dashed line we presented results correspond- Layers have a thickness of 2 cm (continuous line), 4 cm (dash-dotted ing to variations in the concentration of calcium and with dotted line), and 10 cm (dotted line, marked by “+”). line—concentration of titanium.

To obtain the optimal detection equipment characteristics (antenna and filters), we simulated propagation through of radio wave propagation in salt is necessary for frequencies layers of different thickness, at two frequencies: 187.5MHz around 200 MHz. (the corresponding value for a half wave dipole of 80 cm) The subject requires a deep investigation as the propa- and 1 GHz, and plotted the difference. We considered that gating environment is the main factor that determines the the total propagation distance was formed by equal layers of energy threshold of the detector, its resolution, and the uncer- thickness 𝑑𝐿 (in the first simulation 2 cm, followed by 4cm tainties in determining the primary particle characteristics. and 10 cm). In each layer, the concentration of each impurity Up to date, there are no extensive studies on the effects that type varied randomly, having values of 0.5 to 1.5 times the the layered medium has on the performances of the detector. nominal concentration given in Table 1.AsshowninFigure 8, We analyzed all possible types of salt heterogeneities the difference in transmission for the two considered fre- and concluded that in the specific case of a radio neutrino quencies is extremely small, especially for layers of greater detector in “Unirea” salt mine only sedimentary and domal thickness. When 𝑑𝐿 is smaller, one should choose the smaller heterogeneities will affect radio wave propagation. Peripheral frequency as central frequency for the electronic equipment. associated effects can be excluded because the detector would bebuiltinthecentralpartofthedome.Moreover,theatten- 4. Conclusions uation length for propagation in salt is small compared to ice and thus for this type of neutrino detector, we expect an 3 3 Inthispaper,wesimulatedsomeeffectsthatinfluenceradio extension of only about 0.2 km for a 1 km instrumented wave propagation in layered media. Our research was driven volume. by the desire to build a neutrino radio detector in a Romanian For homogeneous medium, the equivalent dielectric con- salt mine. “Unirea” mine was chosen because we already have stantofsaltwascalculatedat187.5MHz:𝜀𝑟 = 7.4747+𝑖0.0003. a working laboratory installed. As stated in the first section, If the concentration of all elements increases 5 times their neutrino-salt interactions generate an electromagnetic field value in [38], the imaginary part of permittivity increases that can be measured by radio antennas. By studying this almost 10 times (up to 0.002) [23]. This would correspond interaction, the main characteristics of the cosmic particle to an attenuation length greater than 1 km. The distance can be found: the energy can be determined from the ampli- between detecting elements—antennas here—could be at tude of the EM field, the direction from the arrival time of that order. Moreover the threshold energy for the cosmic the radio pulse at different antennas. The neutrino type may neutrino would be less than 1 PeV [6]. be inferred from the signature of the shower (the three flavors Compared to absorption caused by impurities, the scat- have distinct interactions characteristics). The issue of deter- tering effect is 40 orders of magnitude smaller, so it can be mining the flavour of the primary neutrino in a radio experi- easily ignored. Waves at higher frequencies are scattered more ment is a complex problem and is not the scope of this study. efficiently as the corresponding wavelengths are smaller and A good theoretical description and mathematical model approach the size of impurities. However, at ∼200 MHz, the 8 Advances in High Energy Physics dimensions of any heterogeneity that can be found in natural than 100 m. A distance between receivers of 80 m allows a rock salt (including anhydrite, hydrocarbons, etc.) are small threshold of 12% of the original EM field but only when the compared to the wavelength and thus scattering in the Mie minimum thickness of the layers is 8 cm. regime is insignificant. The consequences on construction of the detector are When considering a sedimentary layered medium, sim- straightforward: the detection elements should be as close as ulations have shown that absorption accounts for less than possible to each other. On the other hand, this will increase −9 10 of the total loss of energy and thus the main mechanism thecostofelectronicsasthevolumewillhavetobeinstru- that causes the decrease of wave amplitude is reflections at mentedwithalargernumberofstations.However,since interfaces between layers of different permittivity. Multiple the EM field is proportional to the energy of the primary reflections due to back-reflected waves within one layer can neutrino, the spacing between antennas will be determined be directly considered to be losses. by the threshold energy we select to measure. A gradual change in concentration of impurities within We estimate that for such a neutrino detector the main layers diminishes reflections. The transmission becomes factor of uncertainty is represented by the properties of the almost independent of number of crossed layers if only one propagation medium (e.g., by the thickness of sedimented element varies in concentration from layer to layer. layers and their constituents). In our analysis, we considered As mentioned, simulations have shown that the main data from “Unirea” salt mine (e.g., impurities concentration factor that contributes to the decrease of signal was reflections andthicknessoflayers).However,thereisnoobviousreason at interfaces between layers. The worst case scenario (the not to extend the results to other salt mines that were formed smallest layers, here considered to be 1 cm, together with by the same geological mechanisms, especially since we concentration of elements unknown by a factor of 2) leads to a allowed a loose variation for the concentration of impurities. transmission factor unknown by ∼20%. Our main conclusion is that for a cosmic neutrino detector in Another factor to consider for radio wave propagation is salt the uncertainty due solely to the medium is 19%. thepresenceofwaterinthesaltstructure:ifthewavestravel adistancehigherthan2cminabrinebubble,about90%of their energy will be lost [23]. It is clear that secondary trapped Conflict of Interests water in caverns will absorb all electromagnetic radiation. It The authors declare that there is no conflict of interests becomes mandatory to locate such caverns by complemen- regarding the publication of this paper. tary techniques (such as geoelectrics, GPR, etc.). It has been shown that wave polarization information is completely lost in propagation. This is due to the fact that Acknowledgments at each layer crossing a phase shift occurs. Given that the number of layers between production and reception of the Part of this work was supported by the Projects CORONA electromagnetic wave cannot be accurately known, the initial (no. 194/2012) and AugerNext (Contract no. 1 ASPERA 2 phasecannotbeinferredcorrectlybasedonthefinalmea- ERA-NET). The authors would like to thank the referees for sured phase. their useful comments. The medium itself determines the optimal observational frequency. If layers of sediments are thin (a case closer to References reality), one should choose a smaller frequency as central frequency for the electronic equipment (propagation losses [1] G. Askaryan, “Excess negative charge of electron-photon are smaller). shower and the coherent radiation originating from it. 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Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 162986, 4 pages http://dx.doi.org/10.1155/2013/162986

Research Article A Nonparametric Peak Finder Algorithm and Its Application in Searches for New Physics

S. V. Chekanov1 and M. Erickson1,2 1 HEP Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA 2 Physics Department, The College of New Jersey, 2000 Pennington Road, Ewing, NJ 08628-0718, USA

Correspondence should be addressed to S. V. Chekanov; [email protected]

Received 13 September 2013; Accepted 6 December 2013

Academic Editor: Mehmet Bektasoglu

Copyright © 2013 S. V. Chekanov and M. Erickson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We have developed an algorithm for nonparametric fitting and extraction of statistically significant peaks in the presence of statistical and systematic uncertainties. Applications of this algorithm for analysis of high-energy collision data are discussed. In particular, we illustrate how to use this algorithm in general searches for new physics in invariant-mass spectra using 𝑝𝑝 Monte Carlo simulations.

1. Introduction uncertain in the regions of interest, difficult to use for background simulation, or entirely nonexistent. Even for a simple Searching for peaks in particle spectra is a task which is jet-jet invariant mass, finding an analytical background func- becoming increasingly popular at the Large-Hadron col- tion that fits the QCD-driven background spanning many lider that focuses on new physics beyond TeV-scale. Bump orders in magnitude and which can be used to extract pos- searches can be performed either in single-particle (such sible excess of events due to new physics requires a careful as 𝑝𝑇 distributions) or invariant-mass spectra. For instance, examination. Attempts to fit two-jet and three-jet invariant searches for new particles decaying into a two-body final masses have been discussed in CMS [3, 4]andATLAS[5] state (jet-jet, gamma-gamma, etc.) and multibody decays are papers; while both experiments have reached the necessary typically done by examining invariant masses of final-state precision for such fits using initial low statistical data, objects (jets, leptons, missing transverse momenta, etc.). For theusedanalyticalfunctionsareratherdifferentandhave example, assuming seven identified particles (jets, photons, many free parameters. This task becomes even more difficult electrons, muons, taus, 𝑍-bosons, and missing 𝑝𝑇), a search considering multiple channels (invariant-mass distributions) canbemadeforparentparticlesdecayinginto2,3,or4 with various cuts or detector-selection criteria (like 𝑏-tag- daughter particles. This leads to 322 unique daughter groups. ging). Each such channel requires a careful selection of ana- Thus, the task of analyzing such invariant-mass combinations lytical functions for background fit and adjustments of their becomes rather tedious and difficult to handle. Considering initial values for convergence of a nonlinear regression while a “blind” analysis technique for scanning many channels determining an expectantly smooth background shape. A [1], any cut variation increases the number of channels that fully automated approach to searches for new physics has need to be investigated. Finally, similar challenges exist for been discussed elsewhere [6]. automatic searches for new hadronic resonances combining One technically attractive approach is to find a nonpara- tracks [2]. metric way to extract statistically significant peaks without The task of finding bumps is ultimately related to the task aprioriassumptions on background shapes. Such approach of determining a correct background shape using theoretical is popular in many areas, from image processing to studies orknowncrosssections.However,atheorycanberather of financial market, where a typical peak-identification task 2 Advances in High Energy Physics isreducedtodatasmoothinginordertocreateafunction 2. Nonparametric Peak Finder Algorithm that attempts to approximate the background. The smoothing can be achieved using the moving average [7], Lowess [8], Due to the reasons discussed above, the program called Non- and Splines [9] algorithms. Statistically significant deviations parametric Peak Finder (NPFinder) was developed using a from smoothed distributions can be considered as peaks. numerical, iterative approach to detect statistically significant Such technique is certainly adequate for the peak extraction, peaks in event-counting distributions. In short, NPFinder butitdoesnotpursuethegoalofpeakidentificationwitha iterates through bins of input histograms and, using only one correct treatment of statistical (or systematic) uncertainties. sensitivity parameter, determines the location and statistical The later can be asymmetric. significance of possible peaks. Unlike the known smoothing The closest peak-search approach for high-energy- algorithms, the main focus of this method is not how to physics applications has been developed for studies of 𝛾-ray smooth data and then extract peaks, but rather how to extract spectra where the usual features of interest are the energies peaks by comparing neighboring points and then calling what and intensities of photo-peaks. Several techniques have been is left over the “background.” Below we discuss the major developed, such as those based on least squares [10], second elements of this algorithm and then we illustrate and discuss differences with least-squares fitting11 [ ], the Fourier trans- its limitations and possible improvements. For each point 𝑖 in a histogram, the first-order derivative formation [12], Markov chain [13], and convolution [14], (just 𝛼 naming a few). While such approaches are well suited for 𝑖 is found taking into account possible (statistical or/and counting-type observables, they typically focus on narrow systematic) uncertainties. This is done by calculating the peaks on top of small and often flat-shaped background. slope between two points including their experimental uncertainties: if point 𝑖+1is lower than point 𝑖,theuppererror For example, the ROOT analysis framework [15]usedin 𝑖+1 𝑖 high-energy physics contains the TSpectrum package based is used, while if point is higher than point ,the on a smoothing method developed for 𝛾-ray spectra [14]. The lowererroruncertaintyisused.Thisisdoneinordertobe latter typically have narrow peaks on a smooth background. always on a conservative side while reducing statistical noise. This algorithm is efficient in finding sharp peaks, while Mathematically, this can be written as detection of wide peaks requires a visual examination of 𝑦 +𝛿𝑦 −𝑦 𝛼 = 𝑖+1 𝑖+1 𝑖 , datatoadjustseveralfreeparametersofthistool.Thusthis 𝑖 𝑥 −𝑥 (1) approach is not well suited for a completely automatic peak 𝑖+1 𝑖 search. In addition, systematic uncertainties on data points where the uncertainty 𝛿𝑦𝑖+1 istakenwithnegativesignfor are not easy to incorporate in this approach. 𝑦𝑖+1 >𝑦𝑖 and with positive sign otherwise. The uncertainty In high-energy collisions, a typical standard-model back- may not need to be symmetric, but for simplicity we assume ground distribution has a falling shape spanning many orders that they are symmetric as this is usually the case for statistical of magnitude in event counts. A typical example is jet-jet nature of uncertainties. The derivatives are averaged calculat- invariantmassesusedfornewparticlesearches[3, 5]. For ingarunningaverageforanygivenposition𝑁: such spectra, the most interesting regions are the tails of the 𝑁 exponentially suppressed distributions where a new high-𝑝𝑇 1 𝛼𝑁 = ∑𝛼𝑖. (2) physics may show up. This means that there should be rather 𝑁 𝑖=0 different thresholds to statistical noise, depending on the phase-space region, and, as result, a correct treatment of The algorithm triggers the beginning of a peak if the local statistical and systematic uncertainties is obligatory. Unlike derivatives satisfy the 𝛾-ray spectra where peaks are rather common and subject 𝛿𝛼 =𝛼 − 𝛼 >Δ, of various classification techniques, peaks in high-energy 𝑁+1 𝑁+1 𝑁 collisions are rather rare. As a consequence, relatively little (3) 𝛿𝛼𝑁+2 =𝛼𝑁+2 − 𝛼𝑁 >Δ, progress has been made to develop a nonparametric fitting technique for high-energy physics applications where an where Δ is a free positive parameter that reflects (unknown) observation of peaks is typically a subject for searches for new slopeofthepeak.Thisparametershouldbefoundempirically physics rather than for peak-classification purposes. and we will discuss below a possible range for its value. When The above discussion leads to the need for a nonpara- the above conditions are true, NPFinder registers a possible metric way of background estimation together with the peak and begins classifying next points as a part of the peak. peak extraction mechanism which can be suited for high- The running average equation (2) is not accumulated for the energy collision distributions, such as invariant masses. The points which belong to a possible peak. Δ is the only free algorithmshouldbeabletotakeintoaccountthediscrete parameter which specifies the sensitivity to the peak finding. nature of input distributions with their uncertainties. The This parameter should decrease with increase of sensitivity proposed algorithm is less ambiguous compared to the to the peaks (and likely will increase sensitivity to statistical smoothing methods (such as that used in ROOT [14, 15]), fluctuations). since it uses only one free parameter. In addition, it can take NPFinder continues to walk over data points until 𝛿𝛼𝑁+1 into account systematic uncertainties on data points (that can and 𝛿𝛼𝑁+2 are both negative, which signifies that the max- be asymmetric) and thus can estimate statistical significance imum of the peak has been reached. The double condition of possible peaks in the presence of systematic uncertainties. in (3) is used to reinforce the peak-search robustness. When Advances in High Energy Physics 3

6 Upper error 10 105 𝜎 = 401.3

x(i + 2), y(i + 2) 104 𝜎 = 104.7 Lower error x(i), y(i) 3

Entries x(i + 1), y(i + 1) 10 Events 𝛼(i + 1) 𝜎 = 8.0 102

1 0 500 1000 1500 2000 2500 3000 3500 4000 M (jet-jet) (GeV) X Peak Figure 1: A graphical illustration of the NPFinder algorithm. Each Background data point is characterized by a coordinate (𝑥𝑖,𝑦𝑖), with (optional) upper and lower uncertainties on the 𝑦𝑖 values. See (1)forthe Figure 2: Invariant mass of two jets generated with the PYTHIA definition of the slopes 𝛼𝑖. Monte Carlo model. Several peaks seen in this figure were added using the Gaussian distributions with different widths and peak values (see the text). The peaks are found using the NPFinder this condition is met, NPFinder exits the peak and adds an algorithm which also estimates their statistical significance values equalnumberofpointstotherightsidefromthepeakcenter. as discussed in the text. The requirement of having the same number of points implies that the peaks are expected to be symmetric, which is the most common case. For steeply falling distributions, such as Finally, NPFinder uses the background points to calculate transverse-momenta spectra or dijet mass distributions, this thestatisticalsignificanceofeachpeakinagivenhistogram. assumption usually means that we somewhat underestimate This is done by summing up the differences 𝑟𝑖 of the original the peak significance. Figure 1 illustrates the NPFinder algo- points in a peak with respect to the calculated background rithm for a falling invariant-mass distribution. Each point of points and then dividing this value by its own square root. the distribution can have an upper and lower statistical (or Foragivenpeak,itcanbeapproximatedby systematic) uncertainty. ∑𝑟 After detecting all peak candidates, NPFinder iterates 𝜎= 𝑖 , √ (5) through the list of possible peaks in order to form a back- ∑ 𝑟𝑖 ground for each peak. This is achieved by performing a linear where the sum runs over all points in the peaks. The regression of points between the first and last points in the algorithmrunsoveraninputhistogramorgraph,buildsalist peak, that is, applying the function 𝑦𝑖 =𝑚𝑥𝑖 +𝑏,where𝑚 and of peaks, and estimates their statistical significance. A typical 𝑏 are the slope and intercept of the linear regression, which statistically significant peak in this approach has 𝜎>5–7.A in this case is rather trivial as it is performed via the two first peak is usually ignored as it corresponds to the kinematic points only. It should be noted that the linear regression is peak of background distributions. also performed taking into account uncertainties: Belowweillustratetheaboveapproachbygeneratingfully (𝑦 +𝛿𝑦)−(𝑦 +𝛿𝑦) inclusive 𝑝𝑝 collision events using the PYTHIA generator 𝑚= 2 2 1 1 , −1 (4) [16]. The required integrated luminosity was 200 pb .Jetsare 𝑥2 −𝑥1 reconstructed with the anti-𝑘𝑇 algorithm [17]withadistance where 𝑦1 is the first point of the peak, 𝑦2 is the final point of parameterof0.6usingthecut𝑝𝑇 > 100 GeV. Then, the dijet the peak, and 𝛿𝑦1 and 𝛿𝑦2 are their statistical uncertainties, invariant-mass distribution is calculated and the NPFinder respectively. Here the statistical uncertainties are added in finder is applied using the parameter Δ=1. As expected, no order to always be on the conservative side in estimation of peaks with 𝜎>5were found. the background level under the peak. The intercept parameter Next,afewfakepeaksweregeneratedusingtheGaussian then is 𝑏=𝑦1 +𝛿𝑦1 −𝑚𝑥1. distributions with different peak positions and widths. The It should be mentioned that the technique of the peak peaks were added to the original background histogram. finding considered above is somewhat similar to that dis- Figure 2 shows an example with 3 peaks generated at cussed for 𝛾-ray applications [11]. But there are several impor- 1000 GeV (20 GeV widths, 200000 events), 1500 GeV (50 GeV tant differences of NPFinder compared to this algorithm: widths, 30000 events), and at 2800 GeV (40 GeV widths, NPFinder can detect peaks of arbitrary shapes (not only 1200 events). The algorithm found all three peaks and gave Gaussian-shaped peaks as in [11]), no fitting or smoothing correct estimates of their positions, widths, and approximate procedure is used, and statistical and systematic uncertainties statistical significance using the input parameter Δ=1. for data points are included during the peak-finding proce- For a comparison, the same distribution was used to test dure. The algorithm [11]wasnottestedsinceitssourcecode the TSpectrum package of the ROOT program discussed is not publicly available. in the introduction. It was found that TSpectrum can 4 Advances in High Energy Physics also detect such peaks, but several iterations with a visual [2] S. Chekanov, “A C++ framework for automatic search and examination of the data have been required to adjust the identification of resonances,”in Proceedings of the HERA and the free parameters of this algorithm, 𝜎 (an effective sigma of LHC: A Workshop on the Implications of HERA for LHC Physics, searched peaks), and the amplitude of the expected peaks. Part B, CERN-2005-014, DESYPROC-2005-01, p. 624, 2005. After the first TSpectrum pass, an additional analytic fit was [3] V.Khachatryan, A. M. Sirunyan, A. Tumasyan et al., “Search for required to determine the statistical significance of each peak. dijet resonances in 7 TeV pp collisions at CMS,” Physical Review Thisapproachwasfoundtobedifficulttoimplementinafully Letters,vol.105,no.21,ArticleID211801,14pages,2010. [4] S. Chatrchyan, V. Khachatryan, A. M. Sirunyan et al., “Search automatic peak search. √𝑠=7 It should be noted that the peak statistical significance for three-jet resonances in pp collisions at TeV,” Physical Review Letters, vol. 107, no. 10, Article ID 101801, 15 pages, 2011. of the proposed nonparametric method might be smaller [5]G.Aad,B.Abbott,J.Abdallahetal.,“Asearchfornewphysics than that calculated using more conventional approaches, √𝑠= 𝜒2 in dijet mass and angular distributions in pp collisions at such as those based on a minimisation with appropriate 7 TeV measured with the ATLAS detector,” New Journal of background and signal functions. This can be due to the Physics,vol.13,no.5,ArticleID053044,2011. assumption on the symmetric form of the extracted peaks, [6] T. Aaltonen, J. Adelman, T. Akimoto et al., “Global search for −1 the linear approximation for the background under the peaks, new physics with 2.0 fb at CDF,” Physical Review D,vol.79, andthewayinwhichtheuncertaintyisincorporatedinthe no. 1, Article ID 011101(R), 9 pages, 2009. peak-significance calculation. An influence of experimen- [7] E. S. Kenney, Mathematics of Statistics, Part 1, 3rd edition, 1962. tal resolution can also be an issue [18], which can only be [8]W.S.Cleveland,“Robustlocallyweightedregressionand addressed via correctly identified signal and background smoothing scatterplots,” Journal of the American Statistical functions. Such drawbacks are especially well seen for the Association, vol. 74, no. 368, pp. 829–836, 1979. highest mass peak shown in Figure 2 where a statistical fluc- [9] J. Schoenberg, “Contributions to the problem of approximation tuation to the right of the peak pulls the background level of equidistant data by analytic functions, parts A and B,” up compared to the expected falling shape. Given the approx- Quarterly of Applied Mathematics,vol.4,pp.45–99,112–141, imate nature of the statistical significance calculations which 1946. only serve to trigger attention of analyzers who need to [10] F. C. P. Huang, C. H. Osman, and T. R. Ophel, “Line shape study the found peaks in more detail, the performance of the analysis of spectra obtained with Ge(Li) detectors,” Nuclear algorithmwasfoundtobereasonable. Instruments and Methods,vol.68,no.1,pp.141–148,1969. It should be noted that there is a correlation between the [11] M. Mariscotti, “A method for automatic identification of peaks in the presence of background and its application to spectrum peak width and the input parameter Δ: a detection of broader Δ analysis,” Nuclear Instruments and Methods,vol.50,no.2,pp. peaks typically requires a smaller value of (which can be as 309–320, 1967. low as 0.2). [12] K. J. Blinowska and E. F. Wessner, “A method of on-line spectra In conclusion, a peak-detection algorithm has been devel- evaluation by means of a small computer employing Fourier oped which can be used for extraction of statistically sig- transforms,” Nuclear Instruments and Methods,vol.118,no.2, nificant peaks in event-counting distributions taking into pp. 597–604, 1974. account statistical (and potentially systematic) uncertainties. [13] Z. K. Silagadze, “A new algorithm for automatic photopeak The method can be used for new physics searches in high- searches ,” Nuclear Instruments and Methods in Physics Research energy particle experiments where a correct treatment of A,vol.376,no.3,pp.451–454,1996. such uncertainties is one of the most important issues. The [14] M. Morha´c,ˇ J. Kliman, V. Matouˇsek, M. Veselsky,´ and I. Turzo, nonparametric peak finder has only one free parameter which “Identification of peaks in multidimensional coincidence 𝛾-ray is fairly independent of input background distributions. The spectra,” Nuclear Instruments and Methods in Physics Research algorithm was tested and found to perform well. The code A,vol.443,no.1,pp.108–125,2000. is implemented in the Python programming language with [15] I. Antcheva, M. Ballintijn, B. Bellenot et al., “ROOT—a C++ the graphical output using either ROOT (C++) [15]orSCaVis framework for petabyte data storage, statistical analysis and (Java) [19]. The code example is available for download [20]. visualization,” Computer Physics Communications,vol.180,no. 12, pp. 2499–2512, 2009. [16] T. Sjostrand, S. Mrenna, and P.Z. Skands, “PYTHIA 6.4 physics Acknowledgments and manual,” JournalofHighEnergyPhysics,vol.2006,no.5, article 26, 2006. The authors would like to thank J. Proudfoot for discussion [17] M. Cacciari, G. P. Salam, and G. Soyez, “The anti-kt jet and comments. The submitted paper has been created by clustering algorithm,” Journal of High Energy Physics,vol.2008, UChicago Argonne, LLC, Operator of Argonne National Lab- no.4,article63,2008. oratory (Argonne). Argonne, a U.S. Department of Energy [18] S. V.Chekanov and B. B. Levchenko, “Influence of experimental Office of Science Laboratory, is operated under Contract no. resolution on the statistical significance of a signal: implication DE-AC02-06CH11357. for pentaquark searches,” Physical Review D,vol.76,no.7, Article ID 074025, 7 pages, 2007. [19] S. Chekanov, Scientific Data Analysis Using Jython Scripting and References Java, Springer, London, UK, 2010, http://jwork.org/scavis/. [20] M. Erickson and S. Chekanov, “Non-parametric peak [1] G. Choudalakis, Model independent search for new physics at the finder algorithm,” http://atlaswww.hep.anl.gov/asc/packages/ tevatron [Ph.D. thesis],2008,http://arxiv.org/abs/0805.3954. NPFinder/. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 821709, 6 pages http://dx.doi.org/10.1155/2013/821709

Research Article Neutron Noise Analysis with Flash-Fourier Algorithm at the IBR-2M Reactor

Mihai O. Dima,1 Yuri N. Pepelyshev,2 and Lachin Tayibov2

1 Institute for Physics and Nuclear Engineering, DFCTI, Magurele, 76900 Bucharest, Romania 2 Joint Institute for Nuclear Research, FLNP, Dubna, Moscow 141980, Russia

Correspondence should be addressed to Mihai O. Dima; [email protected]

Received 19 September 2013; Accepted 17 November 2013

Academic Editor: Carlo Cattani

Copyright © 2013 Mihai O. Dima et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Neutronnoisespectrainnuclearreactorsareaconvolutionofmultipleeffects.FortheIBR-2Mpulsedreactor(JINR,Dubna),one part of these is represented by the reactivities induced by the two moving auxiliary reflectors and another part of these by other sources that are moderately stable. The study of neutron noise involves, foremostly, knowing its frequency spectral distribution, hence Fourier transforms of the noise. Traditional methods compute the Fourier transform of the autocorrelation function. We show in the present study that this is neither natural nor real-time adapted, for both the autocorrelation function and the Fourier transform are highly CPU intensive. We present flash algorithms for processing the Fourier-like transforms of the noise spectra.

1. Introduction which shows the fluctuations (1) being over Poissonian— due to correlations of neutrons in the same chain—and (2) Neutronic processes in nuclear reactors have a probabilistic theneutronsproducedinthesamegroupstatisticallydis- character due to the quantum mechanics of scattering and the appearing (exponentially) all at the same time. The modern stochastic of propagation in materials. Design is mostly per- theoretical approach is given by the Pal-Bell equation [8– formed on the equations of neutron flux transport (in energy 14], as an applied case of the Chapman-Kolmogorov master and space) and associated effects (fission, thermal fluxes, etc.). equation to Markov-chain neutron processes. Statistical deviations from average quantities give however On top of said, neutron stochastic behavior is the modu- the complete image of neutron physics in the reactor—the lation of the neutron flux by various reactivities: some due so-termed neutron noise, described by large using Markov- to 2-phase liquid flow (bubbling), or fuel embrittlement, chain theories. The theories associated with the underlying mechanically induced reactivities, and so forth. Any addition stochasticity that produces the said fluctuations are actually to the spectrum may be detected and classified, issuing a century old [1, 2], stemming from population studies. It specificwarning.Inthisrespect,neutronnoisespectrum was thus shown that (Alphonse) de Candolle’s conjecture on analysis is a very far reaching tool in nuclear safety. the extinction of family names [3–5] leads to nonergodic To analyze the noise, usually a Fourier transform of the behavior in the case of neutron chains. Such theories were autocorrelation function is performed. This is basically the picked up in nuclear physics by Feynman, de Hoffmann and same as the square of the Fourier transformed signal. In others to describe neutron processes in fission6 [ , 7], leading practice, one also needs a time window, given by ad hoc 𝛼 to the Feynman -formula: apodisation functions. This is rather arbitrary and nonnat- ural. We show a more natural approach passing the noise through a resonant analyzing instrument together with flash 𝜎2 (𝑡) 1−𝑒−𝛼𝑡 algorithms for processing the Fourier-like transforms of the 𝑍 =1+𝜖(1− ) (1) ⟨𝑍 (𝑡)⟩ 𝛼𝑡 noise spectra. 2 Advances in High Energy Physics

100 Core Stationary reﬂector Water 10−1 moderators

Auxiliary 10−2 moving reﬂector Main moving 10−3

reﬂector (a.u.) density ﬂux Neutron

−4 Figure 1: Details of the IBR-2M reactor showing the active core and 10 the two movable reflectors. 0 50000 100000 150000 200000 t (𝜇s) 2. The IBR-2M Reactor Figure 2: Power versus time between two subsequent pulses. Data The IBR-2M [15–17] is a modification of the IBR-2 (2 MW is normalized to the maximum pulse. nominal power) pulsed research fast reactor with PuO2 fuel elements. The reactor coolant is liquid sodium. The pulsed mode operation is enabled by a reactivity modulator consisting of two rotating parts—a main movable reflector of the number of prompt neutrons, and the average number (OPO, at 1500 rpm) and an auxiliary movable reflector (DPO, of neutrons in a single fission event, respectively; 𝜏 is the 𝑆=𝑆 +𝑆 at 300 rpm)—shown in Figure 1,thatcreatereactivitypulses. lifetime of prompt neutrons in the core; sp del is the For nominal DPO rotation speed, the reactivity of every fifth neutron source intensity (that operates continuously during 𝑆 pulse is positive; that is, the reactor becomes prompt neutron the pulse); sp is the intensity of spontaneous neutrons and 18 supercritical for ca. 0.400 ms (0.215 ms half-width), with a those from the (𝛼, 𝑛) reaction on oxygen O(partofthe repetition frequency of 5 Hz (Figure 2). As a result, powerful 𝑆 =𝛽]𝐹 𝛽 oxide fuel); del eff and eff are the intensity, and power pulses of 5 Hz repetition frequency take place in the effective fraction of delayed neutrons respectively; 𝐹=𝐿⋅𝑊 reactor. The energy range of the fast neutrons is up to 10 MeV, is the fission rate; 𝐿 isthenumberoffissioneventsina with an almost linear logarithmic distribution versus energy. second per watt; and 𝑊 istheabsolutereactorpower.At The pulsed operation mode of the reactor is established high power (for IBR-2M above 100 W) the fluctuations of the (𝛿𝑘−𝛽) 2 when the prompt neutron supercriticality reaches external reactivity (component 𝛿 ) dominates. Basically, the 𝜀 =𝜀 ∼1⋅10−3 0 the “equilibrium” value 𝑚 𝑚0 (at5Hz)atwhich mechanical vibrations of the two moving auxiliary reflectors the reactor can be periodically pulsed. For supercriticality causethis.Allnoisediagnosticsofthereactorisbasedon smaller than the “equilibrium” value, the amplitude (and con- research of this noise component, which naturally degrades sequently the energy) of each subsequent pulse is smaller in time. The power is dependent on the neutron flux; hence, than that of the previous one, which means that the reactor monitoring reactor power indicates the size of the neutron is attenuating. There are two main causes for pulse energy flux, together with its fluctuations. fluctuation in the reactor: the stochastic character of fission and neutron multiplication processes, and the fluctuation of external reactivity. Stochastic noise dominates power 3. Power Noise Baseline and fluctuations at low neutron intensity for powers below 1 W. Fourier Transform Pulse energy fluctuations at high power have adverse effects on the operation of the reactor: in the dynamics, startup For a power noise analysis [18–21], we must first extract any and adjustment process, performance of the experimental reactor dynamics influences from the recorded waveforms equipment, and so forth. However, power fluctuations also (overcompensation by the AR power control rod, etc.). In have a positive aspect, being as a tool for reactor diagnosis. this respect, we want to determine a baseline with respect The most important characteristic of a pulsed reactor is the to which reference the instantaneous power by subtraction relative dispersion of pulse energy fluctuations: obtains the power noise. Such method, generally for any power-generating systems, has been presented elsewhere 𝜎 2 𝑄 2 2 2 [22]. The method applied here is a spline of third order over ( ) =Δ (1 + 𝛿0)+𝛿0, (2) 𝑄 st ±30 bins around each current point. Basically, the order of the spline and the number of bins were so determined such that Δ2 = ]Γ/2𝑆𝜏 where st is the relative dispersion of stochastic theeffectivecut-offfrequencyisbelowthenaturalcut-offof 2 fluctuations; 𝛿0 istherelativedispersionofnoisecausedby the mechanical vibrations of the moving reflectors in our case external reactivity fluctuations; Γ and ] are the dispersion 0.2 Hz. The result is presented in Figure 3. Advances in High Energy Physics 3

2500 coefficients for said wave-vectors. What such an analysis instrument does is to use a set of resonance phenomena, 2250 tuned on various frequencies. The instantaneous mechanical 2000 quantity producing measurable effect is (in the case of sound) theairvelocity,orinelectricalcircuits,thecurrent: 1750 𝑡 2𝑤 𝑤 ((𝜏−𝑡)/2𝑄) 𝑖= 0 ∫ 𝜖 (𝜏) 𝑒 0 1500 𝑅√4𝑄2 −1 0

1250 1 × (𝑤 (𝜏−𝑡) √1− −𝜙)𝑑𝜏, 650 sin 0 2 Power (a.u.) Power 4𝑄 1000 600 550 750 500 1 sin 𝜙=√1− , 500 450 4𝑄2 400 250 −1 350 𝜙= , 0.74 0.75 0.76 0.77 cos 2𝑄 0 0.4 0.6 0.8 1 1.2 1.4 (5) Time (h) where 𝑄 is the quality factor, 𝜔0 the resonant pulsation, 𝑅 the 𝜀(𝑡) Figure 3: Reactor power during power up (red line) with fitted bas- electrical circuit’s resistance, and is the driving voltage. eline (blue curve of spline order 𝑘=3over a time interval [−30, +30] Itcanbeseenthatthe“analysisinstrument”performs“some- bins). thing like” a Fourier Transform with an asymmetrical Poisson apodising function, looking into the “past.” As shown above, this is a natural choice—with a definite The concept of Fourier transform in engineering can be model in sight—for an apodising function, containing what quite different from its mathematical analogue. In mathemat- is natural as analysis instrument. ics, a given signal (function of time) is transformed to another The remaining problem is that such an instrument is function of frequency, basically the coefficients of the signal highly computationally the intensive. in the canonically conjugate image. It can be recognized that the sine function can be written In engineering, a signal is long, and we need an “instanta- as sum of exponentials and that the mathematical construct neous” Fourier Transform at each given time, thus a function to be computed is principally 𝑇 that provides a time—window-termed apodisation function. 𝑖𝑤𝑡 𝑖𝑤(𝑛−𝑛 )Δ The question is, for discrete Fourier transforms (DFFT-[23]), ∫ 𝑓 (𝑡) 𝑒 𝑑𝑡 = ∑ 𝑓 (𝑛Δ) 𝑒 0 ⋅Δ, (6) 0 how to perform said clipping, as the apodisation function introduces sidelobes to the central lobe describins the signal- where Δ is the bin size and 𝑛 the bin number over which sum- best seen for the Parzen apodisation function, in Figure 4. mation occurs. Traditionally, the Fourier transform of the power noise is There are two cases for how the summation padding is computed from the autocorrelation function: advantageous, depending on how {2𝜋/𝜔Δ} compares to 1/2, {⋅⋅⋅} [⋅⋅⋅] +∞ where and are the fractional, respectively integer 2 1 𝑖𝑤𝜏 part: 𝑃𝑤 ∼𝑉𝑤 = ∫ 𝑒 𝑟𝑉 (𝜏) 𝑑𝜏, (3) 2𝜋 −∞ (i) {2𝜋/𝜔Δ} ≥:thesumatagivenbin 1/2 𝑛0 is 𝑟 (𝜏) where 𝑉 is the autocorrelation function: 𝑛0+[2𝜋/𝑤Δ] 𝑖𝑤Δ(𝑛−𝑛 +(1/2){2𝜋/𝑤Δ}) 𝑆 = ∑ 𝑓 (𝑛Δ) 𝑒 0 . +∞ 𝑛0 (7) ∗ 𝑛=𝑛 𝑟𝑉 (𝜏) = ∫ 𝑉 (𝑡) 𝑉 (𝑡+𝜏) 𝑑𝑡. (4) 0 −∞ The recurrence formula for obtaining the sum at the It is evident that although the Fourier transform is a single in- next point is obtained by shifting the exponentials tegral, the problem of calculating a double integral (high CPU with 1 bin, subtracting the lower end and adding the cost), has not been eliminated, neither that of calculating the upper end: autocorrelation function. −𝑖𝑤Δ 2𝜋 𝑖𝜙 −𝑖𝜙 Beyond the above considerations, special tools are needed 𝑆𝑛 +1 =𝑒 𝑆𝑛 +𝑓(𝑛0 +1+[ ]) 𝑒 −𝑓(𝑛0)𝑒 , 0 0 𝑤Δ to compute—what is mathematically improper termed— (8) Fourier Transform of noise signals. The basic image is that of sound noise, perceived as higher or lower pitch. In cate- where 𝜙 = 𝜔Δ/2(2 − {2𝜋/𝜔Δ}).Indoingso,mostof gorizingnoiseassuch,theearforinstance(asoneexample thetermsareknownandcanbecomputedinadvance; of analysis instrument) does not work with orthogonal wave- thepreviousbin’ssumisalsoknown,soverylittle vectors, or mathematically complete spaces, nor does it obtain needstobecomputedforthecurrentbin. 4 Advances in High Energy Physics

Parzen window Fourier transform 1 0 −10 0.9 −20 0.8 −30 0.7 −40

0.6 −50 −60 0.5 −70 Decibels

Amplitude 0.4 −80 0.3 −90 −100 0.2 −110 0.1 −120 0 −130 0 N−1 −40 −30 −20 −10 0 10203040 Samples Bins

(a) (b) SRS ﬂat top window 5 Fourier transform 0 −10 4 −20 −30 3 −40 −50 2 −60 −70 Decibels Amplitude −80 1 −90 −100 0 −110 −120 −1 −130 0 N−1 −40 −30 −20 −10 0 10203040 Samples Bins

Figure 4: Fourier spectrum for various apodising functions. The Parzen function shows best the sidelobes that occur, while the SRS function provides a relatively flattop spectrum with good side-rejection of the lobes.

(ii) {2𝜋/𝜔Δ} <:thesumatagivenbin 1/2 𝑛0 is where 𝜙 = 𝜔Δ/2(1 − {2𝜋/𝜔Δ}).Indoingso,mostof thetermsareknownandcanbecomputedinadvance; 𝑛0−1+[2𝜋/𝑤Δ] 𝑖𝑤Δ(𝑛−𝑛0+1/2+(1/2){2𝜋/𝑤Δ}) thepreviousbin’ssumisalsoknown,soverylittle 𝑆𝑛 = ∑ 𝑓 (𝑛Δ) 𝑒 . (9) 0 needstobecomputedforthecurrentbin. 𝑛=𝑛0

The recurrence formula for obtaining the sum at the As mentioned in the introduction, there are beneficial next point is obtained by shifting the exponentials aspects in detecting and subtracting the baseline. Apart from with 1 bin, subtracting the lower end and adding the extracting from the low frequency end reactor dynamics upper end: features, also technical aspects are achieved, such as rejecting −𝑖𝑤Δ 2𝜋 𝑖𝜙 −𝑖𝜙 aliasing. In general, such unwanted phenomenon is due to 𝑆𝑛 +1 =𝑒 𝑆𝑛 +𝑓(𝑛0 +[ ]) 𝑒 −𝑓(𝑛0)𝑒 , (10) 0 0 𝑤Δ the fact that some frequency is much higher than the Nyquist Advances in High Energy Physics 5

2000 104 1800 1600 103 1400 1200 102 1000 800 10 600 FFT amplitude 1 400 FFT amplitude 200 10−1 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Frequency (Hz) Frequency (Hz)

Baseline Power noise Power (a) (b)

Figure 5: Fourier transform of reactor power, baseline, and noise. It can be seen that the raw power and baseline both are affected by aliasing in the same amount. The strong peaks at 1 Hz and 2 Hz disappear tracelessly in the difference (power noise spectrum), proving the efficiency of the method.

𝑓 =1/2Δ𝑡=2.5 sampling frequency; in our case, Ny Hz. Imag- that is being presented. Analog active filters have a high ine the effect of 50 Hz ripple for instance, or that of certain Q; thus, broadband signals have components that bypass reactor chirps. Active analog filters would come to mind as them. Passive very low-pass filters are extremely bulky, as the first resource since they have a somewhat higher Q-factor inductors need to reach into thousands of mH range. than passive filters. (Passive filters in the range [0, 2.5 Hz] The method presented is feasible in real time, via a flash are too bulky in our view of a DAQ setup.) Still, even active updating procedure of the data on the fly, allowing a physi- filters do not have that high Q-factors, and if they do, 50 Hz cally credible estimate of noise spectral density. sources do have a certain bandwidth, rendering very high- It is interesting to note that FFT of noise is a very useful Q filters useless. As for reactor chirps, they are of broad tool also in monitoring the electrical grid, for instance in bandwidth and impossible to filter. All these phenomena observing hysteresis noise in transformer cores (leading create unwanted aliasing that is impossible to remove in any to magnetic embrittlement and anomalous heating of the traditional way. Fortunately, the effect is present for both raw core, resp., transformer power losses). Other types of losses data and for the baseline fit. Subtracting one from the other (dielectric polarization losses in areas with high humidity) yields a clean noise signal reliable for physics studies of the can also be well monitored with this method. reactor. Figure 5 shows the Fourier transform calculated with Other nuclear reactor types could also benefit from the the above method for the reactor power and baseline and method, in principle power reactors which have significant noise.Itcanbeseenthattherawpowerandbaselineboth water cooling systems, as these reactors produce vacuum are affected by aliasing in the same amount. The strong peaks reactivity noise if boiling occurs in the pipes. at1Hzand2Hzdisappeartracelesslyinthedifference(the power noise spectrum), proving the efficiency of the method. Acknowledgments A toolkit with spectrum processing various utility codes has been created to help do all the steps required in a modular One of the authors (M. Dima) acknowledges the support by fashion. a grant from the Romanian National Authority for Scientific Research ANCS Grant PN09370104 and two of the authors 4. Conclusions (Y. N. Pepelyshev and L. Tayibov) acknowledge the support by a grant from JINR, Dubna, Order No. 71, Item 21/2012. For low frequency power noise monitoring, it is in certain contexts impossible to remove aliasing frequency compo- References nents, presampling. Although aliasing collision is not present 𝑓 in the high end of the spectrum, close to Ny—this is being a [1] D. G. Kendall, “Branching processes since 1873,” Journalofthe naturally free region—in the low region, the post-DAQ pro- London Mathematical Society,vol.41,pp.385–406,1966. cessing method presented here of raw-baseline subtraction [2] T. E. Harris, The Theory of Branching Processes, Springer, Berlin, is the only effective means of aliasing collision suppression. Germany, 1963. Such context appears for instance when the sensors delivering [3] A. de Candolle, Histoire des Sciences et des Savants,Geneve:` H. the data have saturation and blind-times after high pulses (for Georg, Paris, France, 1873. instance of radiation), acting as an effective sampling that is [4] F. Galton, Problem 4001, The Educational Times, 1873. not avoidable through filters. Also, very low-pass filters are [5] H. W. Watson and F. Galton, “On the probability of the extinc- difficult to design for 2.5 Hz in the case presented. Digital tion of families,” The Journal of the Anthropological Institute of filters need high sampling rates to avoid the exact condition Great Britain and Ireland,vol.4,p.138,1875. 6 Advances in High Energy Physics

[6] R. P. Feynman, F. de Hoffmann, and R. Serber, “Dispersion [22] M. O. Dima, Y. N. Pepelyshev, and T. Lachin, “Flash algorithms of the neutron emission in U-235 fission,” Journal of Nuclear for power noise analysis at the IBR-2M Reactor,” AIE,vol.2,no. Energy,vol.3,no.1-2,pp.64–69,1956. 1,pp.21–25,2014. [7] L. Pal´ and I. Pazsit,´ “Theory of neutron noise in a temporaly [23] J. O. Smith, Mathematics of the Discrete Fourier Transform fluctuating multiplying medium,” Nuclear Science and Engineer- (DFT) with Audio Applications, Center for Computer Re- ing,vol.155,pp.425–440,2007. search in Music and Acoustics (CCRMA), 2nd edition, 2007, ∼ [8] I. Pazsit´ and L. Pal,´ Neutron Fluctuations: A Treatise on the http://ccrma.stanford.edu/ jos/mdft/. Physics of Branching Processes, Elsevier, Amsterdam, The Neth- erlands, 2007. [9] I. Pazsit and A. Enqvist, Neutron Noise in Zero Power Systems: A Primer in the Physics of Branching Processes, Chalmers Univer- sity, Gothenburg, Sweden, 2008. [10] F. Zinzani, C. Demaziere,` and C. Sunde, “Calculation of the eigenfunctions of the two-group neutron diffusion equation and application to modal decomposition of BWR instabilities,” Annals of Nuclear Energy,vol.35,no.11,pp.2109–2125,2008. [11] L. Pal´ and I. Pazsit,´ “Neutron fluctuations in a multiplying medium randomly varying in time,” Physica Scripta,vol.74,no. 1,pp.62–70,2006. [12] I. Pazsit,´ Transport Theory and Stochastic Processes,Lecture Notes, Department of Nuclear Engineering, Chalmers Univer- sity of Technology, Gothenburg, Sweden, 2007. [13] I. Pazsit,´ C. Demaziere, C. Sunde, A. Hernandez-Solis, and P. Bernitt, “Final report on the research project ringhals diagnostics and monitoring, stage-12,” Tech. Rep. CTH-RF-194/RR- 14, Chalmers University of Technology, Gothenburg, Sweden, 2008. [14] C. Sunde, Noise diagnostics of stationary and non-stationary reactor processes [Ph.D. thesis], Department of Nuclear Engi- neering, Chalmers University of Technology, Gothenburg, Swe- den, 2007. [15] Y. N. Pepelyshev and A. K. Popov, “Investigation of dynamical reactivity effects of IBR-2 moving reflectors,” Atomic Energy,vol. 101, no. 2, pp. 549–554, 2006. [16] I. M. Frank and P.Pacher, “First experience on the high intensity pulsed reactor IBR-2,” Physica B,vol.120,no.1–3,pp.37–44, 1983. [17] M. Makai, Z. Kalya, I. Nemes, I. Pos, and G. Por, “Evaluat- ing new methods for direct measurement of the moderator temperature coefficient in nuclear power plants during normal operation,” in Proceedings of the 17th Symposium of AER on VVER Reactor Physics and Reactor Safety,Yalta,Ukraine, September 2007. [18] Y. N. Pepyolyshev, “Method of experimental estimation of the effective delayed neutron fraction and of the neutron generation lifetime in the IBR-2 pulsed reactor,” Annals of Nuclear Energy, vol.35,no.7,pp.1301–1305,2008. [19] W. Dzwinel, Y. N. Pepyolyshev, and K. Janiczak, “Predicting of slow noise and vibration spectra degradation in the IBR- 2 pulsed neutron source using a neural network simulator,” Progress in Nuclear Energy,vol.43,no.1–4,pp.145–150,2003. [20] W. Dzwinel, Pepyolyshev, and N. Yu, “Pattern recognition, neutral networks, genetic algorithms and high performance computing in nuclear reactor diagnostics—results and perspectives,” in Proceedings of the 7th Symposium on Nuclear Reactor Surveillance and Diagnostics (SMORN ’95),Avignon,France, June 1995. [21] Y. N. Pepyolyshev and W. Dzwinel, “Pattern recognition application for surveillance of abnormal conditions in a nuclear reactor,” Annals of Nuclear Energy,vol.18,no.3,pp.117–123,1991. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 826730, 7 pages http://dx.doi.org/10.1155/2013/826730

Research Article FLUKA Monte Carlo Simulations about Cosmic Rays Interactions with Kaidun Meteorite

Turgay Korkut

Faculty of Science and Art, Department of Physics, Agrı˘ Ibrahim˙ C¸ec¸en University, 04100 Agrı,˘ Turkey

Correspondence should be addressed to Turgay Korkut; [email protected]

Received 19 September 2013; Accepted 20 November 2013

Academic Editor: Mehmet Bektasoglu

Copyright © 2013 Turgay Korkut. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

∘ 󸀠 ∘ 󸀠 An asteroid called Kaidun fell on December 3, 1980, in Yemen (15 0 N, 48 18 E). Investigations on this large-sized meteorite are ongoing today. In this paper, interactions between cosmic rays-earth atmosphere and cosmic rays-Kaidun meteorite were modeled using a cosmic ray generator FLUKA Monte Carlo code. Isotope distributions and produced particles were given after these interactions. Also, simulation results were compared for these two types of interactions.

1. Introduction called Andreyivanovite and Florenskyite were discovered after research on the meteorite. There are several studies Cosmic rays (CRs) are high-energy charged particles, coming about Kaidun meteorite and its minerals (Andreyivanovite from space which voyage at close to the speed of light and and Florenskyite). These studies are generally focused on hit the Earth’s atmosphere from anywhere. Essentially, all of the chronology of meteorite [2–6], the geochemistry [7–16], the elements in the periodic table; about 89% of the nuclei mineralogy [17–19], and the space research [20–25]. are hydrogen (protons), 10% helium, and about 1% heavier FLUKA makes Monte Carlo estimations about particle- elements, generate CRs. These heavier elements (such as matter interactions. FLUKA Monte Carlo package has already carbon, oxygen, magnesium, silicon, and iron) are present been used to simulate CRs generation and interactions in in about the same relative abundance as in the solar system, several research papers [26–29]. Interactions between CRs- but there are important differences in elemental and isotopic Kaidun meteorite and CRs-earth atmosphere were simulated composition that provides information on the origin and in the present paper. history of galactic cosmic rays. For example, there is a significant excess of the rare elements (Li, Be, and B) produced when heavier CRs such as C, N, and O fragment into lighter 2. Methodology nuclei during collisions with the interstellar gas. The isotope 22 Ne is also excessive, showing that the nucleosynthesis of FLUKA is a FORTRAN-based Monte Carlo tool used for cosmic rays and solar system material has differed. Electrons calculations of particle transportations and interactions with constituteabout1%ofgalacticcosmicrays.Itisnotknown different materials used in many different application areas why electrons are apparently less efficiently accelerated than such as calorimetry, activation, dosimetry, detector design, nuclei. accelerators, cosmic rays, neutrino physics, and radiotherapy. ∘ 󸀠 ∘ 󸀠 Kaidun meteorite fell in Yemen (15 0 N, 48 18 E) In this code, 60 different subatomic particles are described on 3 December, 1980. It is in the carbonaceous chondrite and almost all of the materials used in many applications meteorite class and its mass is about 2 kg. It holds a special can be defined using suitable cards30 [ , 31]. In 2010, FLUKA place in the world meteorite collection. It is characterized Monte Carlo code was presented as a new high-energy cosmic by an unprecedentedly wide variety of meteorite material ray generator. With this new release, FLUKA is able to in its makeup. It is believed that Kaidun originated from simulate interactions between high-energy cosmic rays and Phobos, one of two Martian moons [1]. Two new minerals Earth’s atmosphere. In this code using the combinational 2 Advances in High Energy Physics

Exosphere Thermosphere: 320 km 100 layers of Mesosphere: 80 km atmosphere Stratosphere: 51 km (FLUKA model)

Troposphere: 9 km Earth surface

REarth = 12756 km

Earth surface Troposphere: 9 km Stratosphere: 51 km 100 layers of Mesosphere: 80 km atmosphere Thermosphere: 320 km (FLUKA model)

Exosphere

Figure 1: 100 layered FLUKA atmosphere model. geometry package included into the general code, Earth 0.01 atmosphere is modeled with a set of 100 concentric spherical shells (Figure 1). For the material components of Earth 0.001 atmosphere, the same material at different pressure and temperature conditions for each layer was used. Composition 0.0001

of Kaidun meteorite was taken in the literature. Hadronic per primary) interactions were activated using DPMJET package [32]. As −1 1e − 05 Sr

for primary cosmic ray beam primary mass composition −1 1e − 06 model is used such as 90% of protons and 9% of alphas Cm Primary proton flux [33]. As for scores of simulations, we wanted primary proton −1 fluency, produced radioisotopes, and secondary particles 1e − 07 (GeV existing after interactions. The general FLUKA code exists on the website at http://www.fluka.org/. 1e − 08 0.1 1 10 100 1000 10000 100000 Energy (GeV)

3. Results Figure 2: The differential primary proton fluence spectra. In terms of performance of FLUKA code about capability of cosmic ray-atmosphere interactions, we plotted primary proton fluency as a function of primary proton energies (Figure 2).Ascanbeseeninthisfigure,protonenergiescan distribution (atomic number from 1 to 79) is seen unlike 56 reach up to 10 TeV after interactions between CRs and Earth’s Table 1.Intheseproducts,themostabundanceis Fe because atmosphere. of Kaidun’s high Fe content. Other important isotopes are 55 55 54 49 82 108 In Table 1, radioisotopes produced after interactions Fe, Mn, and Mn. Small amounts of Ca, Br, and Ag between CRs and Earth’s atmosphere are seen. 66 different arealsoproduced. cosmogenic isotopes were produced as atomic number range Table 3 showsnumberofparticlesproducedafterCRs- 4 14 14 from 1 to 20. Abundance of He, C, and Nisotopesis atmosphere and CRs-Kaidun. In this table, neutrons and 37 18 34 remarkable. Trace isotopes are S, O, and P. protons are outstanding. Antisigma (+), antisigma (−), xi (0), In Table 2, isotope products derived from CRs-Kaidun anti-xi (0), and anti-xi (+) are trace particles. Looking at the meteorite interactions were given. A wide range of isotope relative differences, it is seen that the biggest changes occur Advances in High Energy Physics 3

Table 1: Isotope distribution after interactions between CR and Table 1: Continued. Earth’s atmosphere. Isotope Number of nuclei Isotope Number of nuclei Cl-35 1,42E − 03 H-1 4,63E +02 S-36 6,77E − 02 H-2 6,84E +01 Cl-36 1,13E − 02 H-3 2,24E +01 Ar-36 9,21E − 05 He-3 2,12E +01 S-37 5,74E − 05 He-4 4,62E +02 Cl-37 1,64E − 02 He-6 4,53E − 02 Ar-37 4,41E − 02 Li-6 2,27E +01 Cl-38 7,48E − 02 Li-7 1,60E +01 Ar-38 1,72E +00 Be-7 6,78E +00 Cl-39 5,60E − 03 He-8 3,25E − 04 Ar-39 8,78E − 01 Li-8 3,51E +00 Cl-40 5,99E − 04 B-8 1,10E − 01 Ar-40 1,32E +00 Li-9 6,16E − 03 K-40 1,10E − 01 Be-9 2,10E +01 Ar-41 1,36E − 01 C-9 4,01E − 02 Ca-42 2,44E − 04 Be-10 1,49E +01 B-10 1,99E +01 3 C-10 1,03E +00 for xi (0), sigma (−), and antilambda. The amount of He has Be-11 2,83E − 01 not changed. B-11 6,93E +01 In Table 4, numbers of secondary particles generated in C-11 1,22E +01 inelastic interactions were given. 47 different particles for B-12 3,64E +00 atmosphere and 50 for Kaidun meteorite were detected by C-12 7,27E +01 simulations. After each two interactions with CRs, produced N-12 3,11E − 04 maximum particle is neutrons as up to three millions. Also, B-13 5,01E − 01 a small amount of D mesons (D+) was produced. Judging by the relative differences, maximum changes occur for D- C-13 9,50E +01 meson (D+), delta baryon (D0BAR), and omega baryon. N-13 1,73E +01 Charmed omega baryons (OMEGAC), tau neutrinos (NEU- C-14 1,10E +02 TRIT), and their antiparticles (ANEUTRIT) were formed N-14 2,19E +02 only after CRs-Kaidun interactions. O-14 3,97E − 01 Finally in Table 5,numbersofdecayproductshavebeen C-15 7,48E − 04 shown. 26 different decay products for atmosphere and 31 N-15 2,40E +01 of Kaidun interactions were detected. In these products, O-15 8,70E +00 there are many photons, electrons, positrons, muons, and N-16 7,91E − 01 neutrinos (electron and tau). Relative difference of anti-xi (+), O-16 5,42E +01 xi (0), and negative kaon particles is higher than that of other products. O-17 1,99E − 03 O-18 6,53E − 05 Ne-20 1,25E − 04 4. Conclusions Ne-21 9,71E − 01 Na-22 1,53E − 04 Recently, the interest in dark matter investigations, Mars mission studies, and actual cosmological studies is increas- Al-28 2,52E − 01 − ing rapidly. On these issues, meteorites, cosmic dust, and Si-28 1,81E 01 cosmogenic subatomic particles from outer space give some − Si-30 3,50E 04 hints for researchers. In this paper, we simulated CRs inter- − Si-31 4,01E 01 actions with Earth’s atmosphere and Kaidun meteorite by S-32 3,22E − 04 FLUKA code which is a popular CR generator. For this P-33 8,92E − 04 aim, Earth’s atmosphere, Kaidun sample, and primary cosmic S-33 1,86E − 02 rays consisting of protons, alphas, and a small amount of P-34 7,41E − 05 other particles were modeled. As for outputs of simulations, S-34 5,15E − 03 produced isotopes and subatomic particles by atmosphere and Kaidun were obtained. Looking at the isotope distribu- S-35 5,42E − 02 tions, 185 types of isotopes were produced after CRs-Kaidun 4 Advances in High Energy Physics

Table 2: Isotope distribution after interactions between CR and Table 2: Continued. Kaidun meteorite. Isotope Number of nuclei Isotope Number of nuclei Si-31 2,43E − 06 H-1 2,89E +01 P-31 2,05E − 02 H-2 1,41E+00 P-32 4,95E − 02 H-3 3,34E − 01 S-32 1,06E − 01 He-3 5,62E − 02 P-33 1,66E − 04 He-4 5,41E+00 S-33 6,57E − 02 He-6 4,09E − 02 S-34 1,07E − 01 Li-6 2,37E − 04 Cl-34 1,18E − 03 Li-7 3,73E − 04 S-35 3,28E − 05 Be-7 1,85E − 05 Cl-35 5,72E − 02 Li-9 2,43E − 06 S-36 3,15E − 04 Be-9 4,22E − 06 Cl-36 8,91E − 02 B-10 8,91E − 06 Ar-36 7,90E − 02 B-11 2,66E − 04 Cl-37 6,31E − 02 C-11 4,70E − 04 Ar-37 3,54E − 01 B-12 2,70E − 05 K-37 4,75E − 06 C-12 5,41E − 04 Cl-38 1,30E − 04 C-13 2,60E − 04 Ar-38 1,20E − 01 N-13 3,94E − 06 K-38 5,57E − 03 C-14 1,09E − 02 Ca-38 4,01E − 06 N-14 2,16E − 03 Cl-39 1,22E − 03 C-15 1,09E − 05 Ar-39 2,31E − 02 N-15 8,53E − 02 K-39 3,78E − 01 O-15 1,20E − 05 Ca-39 3,47E − 02 N-16 3,86E − 05 Cl-40 4,40E − 06 O-16 5,72E − 02 Ar-40 5,99E − 05 O-17 6,20E − 02 K-40 4,53E − 01 F-17 3,64E − 05 Ca-40 3,61E − 01 O-18 2,04E − 02 K-41 1,13E − 01 F-18 8,53E − 04 Ca-41 1,57E − 02 F-19 1,58E − 02 Sc-41 1,51E − 05 O-20 3,72E − 06 K-42 1,16E − 03 F-20 1,30E − 03 Ca-42 8,91E − 02 Ne-20 2,39E − 02 Sc-42 9,01E − 05 F-21 2,99E − 04 K-43 1,08E − 04 Ne-21 3,12E − 01 Ca-43 2,50E − 02 Ne-22 4,22E − 01 Sc-43 4,28E − 04 Ne-23 2,40E − 02 Ca-44 5,46E − 02 Na-21 5,72E − 05 Sc-44 8,85E − 02 Na-22 8,06E − 02 Ca-45 1,22E − 03 Na-23 2,09E+00 Sc-45 5,30E − 02 Mg-23 6,04E − 06 Ca-46 8,53E − 06 Na-24 3,09E − 02 Sc-46 2,51E − 02 Mg-24 1,99E − 04 Ca-47 4,37E − 04 Al-26 4,17E − 05 Sc-47 1,28E − 03 Mg-27 4,05E − 08 Sc-48 6,73E − 05 Al-27 2,21E − 02 Ca-49 5,88E − 09 Al-28 1,24E − 03 Sc-49 1,17E − 05 Si-28 5,30E − 03 Ti-44 3,27E − 03 Al-29 1,30E − 05 Ti-45 2,85E − 02 Si-29 5,07E − 03 Ti-46 3,60E − 01 Al-30 3,38E − 03 V-46 7,42E − 06 Si-30 6,41E − 02 Ti-47 3,41E − 01 P-30 5,35E − 04 V-47 2,82E − 03 Advances in High Energy Physics 5

Table 2: Continued. Table 2: Continued. Isotope Number of nuclei Isotope Number of nuclei Ti-48 2,99E − 01 Ni-64 2,49E − 02 V-48 2,24E − 01 Cu-64 6,52E − 06 Cr-48 5,46E − 02 Zn-64 2,98E − 04 Ti-49 6,89E − 02 Ni-65 1,08E − 04 V-49 5,99E − 01 Cu-65 2,99E − 04 Cr-49 1,25E − 01 Zn-65 5,99E − 04 Ti-50 7,63E − 02 Zn-66 5,94E − 04 V-50 6,63E − 01 Zn-67 8,53E − 04 Cr-50 7,95E − 01 Zn-68 1,81E − 03 Mn-50 1,23E − 01 Zn-69 2,60E − 04 V-51 2,11E − 01 Ga-69 5,62E − 06 Cr-51 8,80E − 01 Ga-72 2,75E − 05 Mn-51 3,54E − 01 As-74 2,82E − 04 V-52 8,06E − 02 As-75 4,13E − 06 Cr-52 1,72E+00 As-76 4,62E − 05 Mn-52 6,52E − 01 Br-82 7,74E − 07 Fe-52 8,80E − 02 Kr-83 6,89E − 05 V-53 1,26E − 05 Kr-84 1,85E − 04 Cr-53 7,00E − 01 Rb-84 8,69E − 02 Mn-53 1,58E+00 Ag-108 8,11E − 07 Fe-53 4,64E − 01 Ag-110 2,16E − 05 Cr-54 2,06E − 01 Cd-114 3,57E − 05 Mn-54 2,83E+00 Ir-192 5,09E − 06 Fe-54 2,48E+00 Au-197 2,33E − 06 Co-54 1,11E − 05 Cr-55 1,13E − 02 Table 3: Number of particles generated by interactions. Mn-55 3,07E+00 Fe-55 3,75E+00 Number of particle − Particle name Relative Co-55 1,23E 01 Atmosphere Kaidun Mn-56 1,02E+00 difference (%) Fe-56 2,46E+01 4-HELIUM 8,84E +03 9,03E+03 2,13 Co-56 2,03E − 01 3-HELIUM 1,60E+03 1,60E+03 — Ni-56 3,50E − 05 TRITON 2,13E+03 2,14E+03 0,47 DEUTERON 3,20E+03 3,26E+03 1,86 Mn-57 3,11E − 03 HEAVYION 1,03E+03 1,09E+03 5,66 Fe-57 2,60E+00 PROTON 4,57E+05 4,61E+05 0,87 Co-57 2,88E − 01 2,53 − APROTON 1,56E+03 1,60E+03 Ni-57 7,05E 02 NEUTRON 1,67E+06 1,69E+06 1,19 − Mn-58 5,72E 06 ANEUTRON 1,55E+03 1,57E+03 1,28 − Fe-58 7,74E 02 MUON+ 4,48E+01 4,52E+01 0,89 Co-58 3,33E − 01 MUON− 4,18E+03 4,43E+03 5,81 Ni-58 7,32E − 01 PION+ 3,32E+03 3,27E+03 1,52 Fe-59 2,25E − 03 PION− 3,22E+03 3,20E+03 0,62 Co-59 9,75E − 02 KAON+ 2,47E+01 2,18E+01 12,5 Ni-59 2,91E − 01 KAON− 1,24E+01 1,39E+01 11,4 Co-60 2,43E − 01 LAMBDA 6,10E − 01 3,83E − 01 45,72 − − Ni-60 5,41E − 01 ALAMBDA 3,95E 03 7,26E 03 59,05 − − − Cu-60 5,08E − 06 SIGMA 1,12E 03 2,99E 03 91,00 − − Co-61 2,13E − 03 SIGMA+ 2,51E 03 3,24E 03 25,39 KAONZERO 2,97E+01 3,43E+01 14,38 Ni-61 7,32E − 02 − AKAONZER 3,82E+01 4,33E+01 12,52 Cu-61 2,34E 02 ASIGMA− 1,51E − 04 2,09E − 04 32,22 − Co-62 2,34E 02 ASIGMA+ 2,75E − 04 1,71E − 04 46,64 Ni-62 2,90E − 02 XSIZERO 1,18E − 04 4,48E − 04 116,61 Ni-63 2,93E − 02 AXSIZERO 1,09E − 04 1,88E − 04 53,20 Cu-63 2,85E − 04 AXSI+ 1,25E − 04 6 Advances in High Energy Physics

Table 4: Number of secondaries generated in inelastic interactions. Table 5: Decay products generated in inelastic interactions.

Number of particle Number of particle Particle name Relative Particle name Atmosphere Kaidun Atmosphere Kaidun Relative difference (%) difference (%) 4-HELIUM 1,80E+06 1,82E+06 1,10 PROTON 7,96E+03 8,05E+03 1,12 3-HELIUM 1,31E+05 1,33E+05 1,52 APROTON 1,55E+02 1,52E+02 1,95 TRITON 1,30E+05 1,31E+05 0,77 ELECTRON 2,92E+05 2,94E+05 0,68 DEUTERON 3,00E+05 3,02E+05 0,66 PROTON 2,41E+06 2,44E+06 1,24 POSITRON 3,10E+05 3,12E+05 0,64 APROTON 1,44E+03 1,47E+03 2,06 NEUTRIE 3,05E+05 3,07E+05 0,65 ELECTRON 1,38E+02 1,35E+02 2,20 ANEUTRIE 2,87E+05 2,89E+05 0,69 POSITRON 1,38E+02 1,34E+02 2,94 PHOTON 7,70E+05 7,75E+05 0,65 − − NEUTRIE 2,89E 02 7,12E 02 84,52 NEUTRON 6,90E+03 6,89E+03 0,15 − − ANEUTRIE 4,89E 01 9,96E 01 68,28 ANEUTRON 1,34E+02 1,41E+02 5,09 PHOTON 7,90E+05 8,03E+05 1,63 MUON+ 3,13E+05 3,15E+05 0,64 NEUTRON 3,09E+06 3,13E+06 1,29 MUON− 2,97E+05 3,00E+05 ANEUTRON 1,44E+03 1,46E+03 1,38 1,01 MUON+ 5,68E+01 5,16E+01 9,59 PION+ 1,48E+04 1,47E+04 0,68 MUON− 4,26E+01 4,12E+01 3,34 PION− 2,01E+04 2,02E+04 0,50 KAONLONG 2,04E+02 1,74E+02 15,87 KAON+ 5,43E − 01 7,37E − 01 30,31 PION+ 2,92E+05 2,94E+05 0,68 KAON− 2,30E − 01 4,76E − 02 131,41 PION− 2,77E+05 2,79E+05 0,72 LAMBDA 3,79E+03 3,76E+03 0,79 KAON+ 1,19E+04 1,20E+04 0,84 ALAMBDA 6,04E+01 6,32E+01 4,53 − KAON 4,32E+03 4,37E+03 1,15 SIGMA+ 6,47E − 05 — LAMBDA 6,70E+03 6,86E+03 2,36 PIZERO 2,16E+04 2,14E+04 0,93 ALAMBDA 1,41E+02 1,39E+02 1,43 KAONZERO 2,33E − 04 5,30E − 04 77,85 KAONSHRT 2,03E+02 1,79E+02 12,57 AKAONZER 4,09E − 04 7,94E − 04 SIGMA− 2,01E+03 1,98E+03 1,50 64,01 SIGMA+ 2,35E+03 2,34E+03 0,43 NEUTRIM 5,98E+05 6,02E+05 0,66 SIGMAZER 3,55E+03 3,53E+03 0,56 ANEUTRIM 6,01E+05 6,05E+05 0,66 PIZERO 3,64E+05 3,67E+05 0,82 XSIZERO 1,64E − 01 8,05E − 04 198,05 KAONZERO 1,14E+04 1,14E+04 — AXSIZERO 3,08E − 01 1,96E − 01 44,44 AKAONZER 4,11E+03 4,16E+03 1,21 AXSI+ 1,54E − 04 1,67E − 01 199,63 NEUTRIM 4,15E+03 4,39E+03 5,62 KAONLONG — 5,74E − 05 − − ANEUTRIM 2,97E 01 3,08E 01 3,64 KAONSHRT — 5,74E − 05 ASIGMA− 4,62E+01 4,46E+01 3,52 XSI− —2,35E − 02 ASIGMAZE 4,26E+01 4,25E+01 0,24 − − ASIGMA+ 4,15E+01 4,68E+01 12,00 TAU — 8,29E 05 − XSIZERO 1,28E+02 1,26E+02 1,57 NEUTRIT — 8,29E 05 AXSIZERO 8,77E+00 1,08E+01 20,75 ANEUTRIT — 8,29E − 05 XSI− 1,09E+02 1,09E+02 — AXSI+ 8,21E+00 8,82E+00 7,16 OMEGA− 3,93E − 01 6,92E − 02 140,11 AOMEGA+ 8,51E − 01 1,10E+00 25,53 interactions. Also, differences in the number of particles produced from atmosphere and Kaidun interactions are D+ 1,34E − 04 2,40E − 03 178,85 outstanding. For example, tau neutrinos which do not occur D− 2,29E − 04 2,35E − 04 2,59 − − after CRs-atmosphere interactions have been observed after D0 6,17E 04 3,45E 04 56,55 CRs-Kaidun interactions. Given results as a result of this − − D0BAR 3,51E 04 2,98E 03 157,85 paper may be useful for space and particle researchers. DS+ 6,74E − 05 1,45E − 04 73,07 − − − DS 2,54E 04 8,29E 05 101,57 References LAMBDAC+ 1,34E − 04 3,17E − 04 81,15 NEUTRIT 2,30E − 03 [1] A. V. Ivanov, “Is the Kaidun meteorite a sample from Phobos?” ANEUTRIT 2,30E − 03 Solar System Research, vol. 38, no. 2, pp. 97–107, 2004. [2] A. V. Ivanov, N. N. Kononkova, S. V. Yang, and M. E. Zolensky, OMEGAC0 9,17E − 05 “The Kaidun meteorite: clasts of alkaline-rich fractionated Advances in High Energy Physics 7

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Research Article Computational Determination of the Dirac-Theory Adjunctator

M. Dima

Institute for Physics and Nuclear Engineering, Atomistilor Street 407, P.O. Box MG-6, 76900 Bucharest, Romania

Correspondence should be addressed to M. Dima; [email protected]

Received 9 June 2013; Accepted 15 August 2013

Academic Editor: Carlo Cattani

Copyright © 2013 M. Dima. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 A number of particle properties stem from the use of 𝛾 as adjunctator (Bargmann-Pauli) in the Dirac theory (spin alignment, 0 Dirac current, etc.). The early motivations for accepting 𝛾 as adjunctator were representation-dependent, mildly bearing relation 0 to the actual conditions forcing 𝛾 as adjunctator. Representation-independent approaches to the physical predictions of the Dirac 0 equation are somewhat new, here presented as being the reasons for 𝛾 as adjunctator of the Dirac theory, together with the essential roleofthelatterinthephysicalaspectsofthetheory.

1. Introduction Leaving such “physics” arguments behind, the real aspects can be now explored. Little attention was given to this 𝛽, 𝛼⃗ 𝛾0 1, 𝛾⃗ The definition of the Dirac operators ( )as ( ), topic; however Pauli [3] did have interest in it at the time. [𝛾𝜇,𝛾]] =2𝑔𝜇] ⋅ 1 satisfying + ,thatis,aCliffordalgebra His arguments though representation dependent (with one C𝑙 (𝐶) 1,3 , does not include any information on how adjoint minor inaccuracy) offer the only explanation to date on this operators are constructed. In representation-dependent 𝜇† topic. The only representation-independent study4 [ , 5]is form, as matrices, the adjoints are constructed as (𝛾 )𝛼,𝛽 about the unicity of the adjunctator together with a pleiad of 𝜇 ∗ =(𝛾)𝛼,𝛽; however, such representation-dependent solu- related studies [6, 7]. 𝜇 tions also include representation-specific artifacts, often non- In the Dirac algebra (of the 𝛾 matrices), the adjoint physical. The importance of presenting properties in repre- operators form another representation of the same algebra, sentation-independent form, stemming from minimal infor- as the arguments of Pauli [3] show the following: mation in the abstract definition of the foundations, thus [𝛾𝜇†,𝛾]†] =−[𝛾𝜇,𝛾]] † =2𝑔𝜇] ⋅ 1 󳨐⇒ 𝛾 𝜇† = B𝛾𝜇B−1. cannot be emphasized enough. − − The adjunctator (algebra element that constructs the (2) adjoint) is present in a number of particle properties (current definition,spinalignment,etc.).InDiractheory,thisisthe The two representations are thus related through a similarity 0 Bargmann-Pauli [1–3]adjunctator𝛾 and its unicity is crucial transformation B termed adjunctator or hermitiser [1, 2], to knowing that the physical quantities used are well defined. with the following properties: The initial demand in determining how the adjoint is 𝜇† † 𝜇 −1† 𝜇 † −1† 𝜇 −1† −1 𝛾0 𝛾⃗ (i) (𝛾 ) =𝛾 = B 𝛾 B =(B B)𝛾 (B B) ⇒ constructed for ( , ) was from the Dirac-1928 equation itself −1† 𝜇 † as follows: [B B,𝛾 ]− =0for all 𝜇=0, 3,respectively,B = 𝜆B. E = H = 𝛼⋅⃗p⃗+𝛽𝑚𝑐| , † † † 2 𝜓 (1) (ii) (B ) =(𝜆B) =|𝜆|B = B ⇒|𝜆|=1. 0 where E is the energy, H the Hamiltonian, (𝛽, 𝛼⃗) Dirac’s initial The reasons for the consecrated B =𝛾 mustthuslieinthe † matrices, and 𝜓 the system’s wave function. Since E = E it mathematical foundations of the bispinor space and cannot † seemed natural to ask that H = H . This is rather ad-hoc, all be collected from the Lagrangian conditions already using † 0 that the above relation has to offer being Im{⟨H − H⟩𝜓}=0. 𝛾 as adjunctator, other ad hoc conditions (shown in the 2 Advances in High Energy Physics

† beginning), or representation-specific arguments leading to The only effect of 𝑧 is that of choice: with 𝑧=1, B = B, † 𝑖𝜃/2 † 𝑖𝜃 generalisations not pertinent to the problem. with 𝑧=𝑖, B =−B,with𝑧=𝑒 , B =𝑒 B—therefore † The representation-independent conditions applicable 𝑧 canbechosen1,henceB standard and B = B.The are property of significance is that the adjunctator is, up to a unit 2 complex multiplier, standard. The arguments in3 [ ]ofcount- (i) self-adjointness: any X = 1 has real eigenvalues; † 0 5 ing symmetric and antisymmetric matrices, though reaching hence X = X . For example, X =𝜎𝑖,𝛾 ,𝛾 ,andso † the same conclusion, forbid B =−B which, as shown above, forth; couldhaveservedequallywell. (ii) positive definiteness: for any X,itsnormsquared 2 ‖X𝜓‖ =⟨X𝜓|X𝜓⟩ ≥;hence 0 B must be such that † 3. Self-Adjoint Operators X X ≥0, ∀X. Respectively they are the mathematical translation of under- The simplest proof from this point on is by looking at the (𝜎,⃗𝑔)⃗ 𝜎=𝛾⃗ 0𝛾𝛾⃗5 standing the scalar-product. The early arguments of Pauli generators of the Dirac algebra- ,where and 𝑔=(𝛾⃗ 5,−𝑖𝛾0𝛾5,𝛾0) [3] relied on counting representation-dependent anti-/sym- ; the two sets defining two isomorphic metric matrices. This eludes the above two conditions and halves of the space [8] are shown as follows: B leads to one minor misconclusion on . 2 𝜎×⃗ 𝜎=2𝑖⃗ 𝜎,⃗ with 𝜎𝑖 = 1, (7) 2. Structure of the Scalar Product 2 𝑔×⃗ 𝑔=2𝑖⃗ 𝑔,⃗ with 𝑔𝑖 = 1. 𝜇 ] 𝜌 † 𝜌 ] 𝜇 −1 −1 Let Γ=𝛾 𝛾 ⋅⋅⋅𝛾 ;thenΓ = B𝛾 ⋅⋅⋅𝛾 𝛾 B = sign ⋅BΓB . The two halves of the space admit various other represen- It is useful thus to define 5 5 tations, such as (1 −𝛾)𝜎/2⃗ and (1 +𝛾)𝜎/2⃗ ;however,the def Γ = B−1Γ†B, (3) one presented is the most convenient for the task at hand. The endomorphism between the two halves is S = 1 + 𝜎⃗𝑔=⃗ 0 1 2 0 1 2 with the trivial action of complex conjugation on scalars and 1 +𝛾 𝛾 −𝑖𝛾 +𝑖𝛾 𝛾 𝛾 . a generalisation of this on operators as follows: Thetwohalvesseemidentical;howeverfromtheadjunc- tation point of view they differ as follows: (i) XY = YX, −1 −1 (ii) X = X , (𝜎1,𝜎2,𝜎3)=(𝜎1,𝜎2,𝜎3), (iii) R +𝑖C = R−𝑖C,forR and C “standard”, that is, R = R (8) (𝑔 ,𝑔 ,𝑔 )=(−𝑔,−𝑔 ,𝑔 ). and C = C; 1 2 3 1 2 3 (iv) ΓΓ = ΓΓ;thatis,ΓΓ is standard; 2 2 Encouraged by the fact that 𝜎𝑖 = 1 and 𝑔𝑖 = 1,weexplore 𝜆X +𝜇Y = 𝜆X + 𝜇Y † † (v) ; under what conditions 𝜎⃗= 𝜎⃗and 𝑔⃗= 𝑔.⃗ (vi) if R +𝑖C =0;thenbothR and C are zero since 0= For the first half of the space, we seek an adjunctator in 𝐵=𝜆+𝑎𝜎⃗⃗ [𝜆, 𝜎]⃗ =0 [𝑎,⃗𝜎]⃗ =0 R −𝑖C =0. the form of ,where − and − . With some loss of generality, we let [𝜆, 𝑎]⃗− =0and 𝑎×⃗ 𝑎=0⃗ , With this notation, it is easier to understand the action of B for the sake of exemplifying the mechanisms in place. Under as follows: this adjunctator; † −1 Γ = B Γ B (4) 2 𝜎⃗† = 𝜎+⃗ [𝑎2⃗𝜎⃗ −𝜆𝑎×⃗ 𝜎]⃗ . 2 2 ⊥ (9) † −1 𝜆 − 𝑎⃗ and to further discover the adjunctator: B =𝜆B = BBB ⇒ B =𝜆B.TheoperatorB is comprised of a standard and 𝑋=𝑏⃗𝜎⃗ 𝑋†𝑋≥0 𝑎=0⃗ B = R +𝑖C R C For , in order that , it is evident that , antistandard part: ,whereboth and are leaving 𝜆=scalar + LIN{𝑔}⃗. B =𝜆B R −𝑖C =𝜆R +𝑖𝜆C standard. Therefore translates to , For the second half of the space, similar arguments follow, which takes into account that 𝜆=𝜆𝑅 +𝑖𝜆𝐶 andalsoinview only that in this case the 𝑔𝑖’s are not similar to the 𝜎𝑖’s. Further of property (vi) becomes. restricting generality, assume 𝑎=𝑎⃗ 𝑒3⃗;then 𝜆 −1 −𝜆 R ( 𝑅 𝐶 )( )=0. 2 𝜆 𝜆 +1 C (5) 𝑔†⃗= 𝑔−⃗ [𝜆2𝑔⃗ −𝜆𝑎×⃗ 𝑔]⃗ 𝐶 𝑅 𝜆2 − 𝑎2⃗ ⊥ (10) Respectively, the determinant must vanish (which for |𝜆| =, 1 The role of 𝑎=0⃗ in the first half transfers here to 𝜆=0. the case at hand, actually does). The condition from (5)isthat 𝐵=𝑔 𝜎⃗† = 𝜎⃗ 𝑔†⃗ = 𝑔⃗ R ∼ C B =𝑧R 𝑧 Itcanbeseenthat 3 performs and , ,or where is a unit complex number relevant † which for any operator in the space satisfies 𝑋 𝑋≥0. only for aesthetical purposes as follows: † † Conversely, without proof, if 𝜎⃗= 𝜎⃗and 𝑔⃗= 𝑔,thenany⃗ † −1 −1 † Γ = (𝑧B) Γ(𝑧B) = BΓB . (6) operator in the space satisfies 𝑋 𝑋≥0. Advances in High Energy Physics 3

0 0 Let 𝑋=𝛾̃ 𝑋𝛾 be an operation with the same properties Expanding the above, X as , mainly just complex conjugating the coefficients in a −1 0 † −1 ̃ −1 2 2 sumofoperators.IfA = B𝛾 ,then𝑋 = A𝑋̃A and A𝑋A 𝑋=(𝛼 −4𝛽 ) † −1 −1 ̃ 2 2 𝜎𝑖 = A𝜎𝑖A = A𝜎𝑖A =𝜎𝑖, ×[(𝜆𝜇+𝜆𝜇) 𝛼 S + (𝜆𝜇−𝜆𝜇) 𝛽 S (11) 𝑔† = A𝑔̃ A−1 = A𝑔 A−1 =𝑔, 𝑖 𝑖 𝑖 𝑖 2 2 2 + |𝜆| (𝛼 −𝛽 )+4(𝜆𝜇+𝜆𝜇) 𝛼𝛽 (17) as assumed, hinted by the fact that 𝜎𝑖 and 𝑔𝑖 squared are unity 󵄨 󵄨2 󵄨 󵄨2 2 2 and have real eigenvalues. This implies that +8𝛼𝛽󵄨𝜇󵄨 S +4󵄨𝜇󵄨 (𝛼 +𝛽 ) [A,𝜎] =0, ∀𝑖=1, 3, 𝑖 − + 4(𝜆𝜇−𝜆𝜇) 𝛼𝛽 ]≥0, (12) [A,𝑔𝑖]− =0, ∀𝑖=1, 3, where the imaginary boxed terms disappear for 𝛽=0;hence 0 andinturn A ∼ 1 and B =𝛾 (qed). Note that the expectation values of A𝛾0𝛾𝑖𝛾5 =𝛾0𝛾𝑖𝛾5𝐴, S = 1+𝜎⋅⃗𝑔⃗can be 1/4 and 7/4, stemming from the product of two independent 1/2 angular momenta. 0 0 A𝛾 =𝛾A, (13) In general form (without the anisotropy argument), the proof ends annihilating all 𝑎⃗and nondiagonal 𝑀𝑖𝑗 coefficients A𝛾5 =𝛾5A. for the same reason above: the existence of imaginary terms in a relation that needs to be real and positive. 𝑖 0 Respectively, both [A,𝛾]− =0and [A,𝛾 ]− =0;thatis, The unicity issue has been addressed by direct construc- 𝜇 [A,𝛾 ]− =0, ∀𝜇 = 0, 1,orA = 1. tion, although proof in this respect can be found in [4, 5]. All these exemplifications point to the asymmetry—with Anotheraspectofinterestisthepossibilityofusing † respecttoadjunctation—ofthetwohalvesofthespaceandto the bar in place of (adjunctation) on the helicity-positive 0 the fact that 𝑔3 =𝛾 may be the Bargmann-Pauli adjunctator. subspace as follows:

def † 4. Positive Definiteness of 𝑋 𝑋 (𝜓 | 𝜙) =⟨C𝜓|C𝜙⟩ . (18) † AmoreinvolvedargumentisthatB should be such that 𝑋 𝑋 In this case, B = CC>0,theC operator taking the in-between −1 0 is positive definite for any 𝑋;thatis:A𝑋̃A 𝑋≥0,which bearing role of 𝛾 . Evidently, any operator satisfying SS = 1 is the matrix version of understanding the scalar product. invaries the new scalar product of (𝜓 | 𝜙).Theparticularityof Since (𝜎,⃗𝑔)⃗have the same properties, and the arbitrariness this context is that any C satisfies the scalar product and norm of 𝑋 imposes equal constraints on both, that is, any property conditions (i.e., absolutely arbitrary scalar product), leading holding for 𝑋(𝜎,⃗𝑔)⃗must also hold for 𝑋(𝑔,⃗𝜎)⃗,itfollowsthat from the physics point of view to arbitrary many Lorentz A is symmetric in (𝜎,⃗𝑔)⃗as follows: invariant conserved currents. A =𝑏⋅1 + 𝑎(⃗𝜎+⃗ 𝑔)⃗ + 𝑀 (𝜎 𝑔 +𝜎𝑔 ). 𝑖𝑗 𝑖 𝑗 𝑗 𝑖 (14) 5. Dirac Currents 𝑎⃗ 𝑀 In the above, and 𝑖𝑗 are unjustifiable anisotropies; hence Another interesting aspect is that two Lorentz invariant 𝑎=0⃗ 𝑀 =𝛽𝛿 A A=𝛼⋅1+𝛽S 2 and 𝑖𝑗 𝑖,𝑗; therefore reduces to ,with quantities which include charge density ⟨𝜌⟩ = |𝜓(𝑥)| can be −1 2 2 2 ̃ −1 𝜇 𝜇 5 A =(𝛼 −S 𝛽 )/(𝛼 −4𝛽 ), where S =4 ⋅ 1.TheA𝑋A 𝑋 ≥ 0 defined: 𝜌B𝛾 and 𝜌B𝛾 𝛾 . Of these two, only the former is 𝜇 inequality becomes conserved: 𝜕𝜇⟨𝜌B𝛾 ⟩𝜓 =0. It is clear that if the adjunctator B is not unique, more ̃ −1 2 2 −1 A𝑋A 𝑋=(𝛼 −4𝛽 ) than one such conserved Lorentz invariant quantity can be defined. The transformation properties under space-, time-, ×[𝛼2𝑋̃ −𝛽2𝑋̃ +𝛼𝛽S (𝑋̃ − 𝑋̃ )] 𝑋 ≥0, 𝜇 0 𝜇 𝜎 𝑔 𝜎 𝑔 𝜎 andspace-timeinversionfor𝑗 =⟨𝜌𝛾𝛾 ⟩𝜓 are (15) P 𝑗𝜇 (𝜓) 󳨀→ − 𝑗 𝜇̂ (𝜓) , where 𝑋𝜎 =𝑋(𝜎,⃗𝑔)⃗and 𝑋𝑔 =𝑋(𝑔,⃗𝜎)⃗, S playing the go- between role: S𝑋𝜎 =𝑋𝑔S and 𝑋𝜎S = S𝑋𝑔.Let𝑋𝜎 =𝑋𝑔 = T 𝑗𝜇 (𝜓) 󳨀→ − 𝑗 𝜇̂ (𝜓) , (19) 𝜆+𝜇S;then(15)(usingS̃ = S)becomes −1 𝜇 PT 𝜇 A𝑋̃A−1𝑋=(𝛼2 −4𝛽2) 𝑗 (𝜓) 󳨀→ 𝑗 (𝜓) ,

† † † 2 2 2 2 U =−U U =−U U =U ×[𝜆(𝛼 −𝛽 )+𝜇(𝛼 +𝛽 )+8𝜇𝛼𝛽] (16) where 𝑃 𝑃, 𝑇 𝑇 and 𝑃𝑇 𝑃𝑇 were evidently 𝜇̂ 𝜇] 𝜇 included. The notation 𝑎 refers to 𝑔 𝑎].Above,𝑗 is not 𝜇 𝜇 ×(𝜆+𝜇S)≥0. an operator, but rather the product: 𝑗 =⟨𝜌⟩𝜓 ⋅⟨k ⟩𝜓.Its 4 Advances in High Energy Physics transformation properties seem to differ from textbook elec- [8]P.Caban,J.Rembielinski,andM.Wlodarczyk,“Spinoperator trodynamics by the minus sign. This comes from the fact that in the Dirac theory,” Physical Review A,vol.88,no.2,ArticleID charge density is better defined as a pseudoscalar under both 022119, 2013. space- and time-inversion [9–12]. Incidentally, this also solves [9] R. M. Kiehn, “Charge is a pseudo scalar,” http://www22.pair the problem of not finding a representation-independent .com/csdc/pdf/ptpost.pdf. charge-conjugation operator C, since time reversal does this [10] J. Schwinger, “Quantum electrodynamics. I: a covariant formu- automatically for all currents, without the need of a specific lation,” Physical Review, vol. 74, no. 10, pp. 1439–1461, 1948. charge-conjugation operator for each current in its own [11] R. P. Feynman, “The theory of positrons,” Physical Review,vol. space. 76,no.6,pp.749–759,1949. [12] M. Dima, “Representation independent CPT transformations,” Internal Note IFIN-PUB-0487,Institute for Nuclear Physics and 6. Conclusion Engineering, 2012. It has been shown in representation-independent form that 0 the adjunctator for the Dirac space is the 𝛾 matrix. This is an important result, as the adjunctator enters the definition of the Dirac current and multiple adjunctators would imply the possibility of defining multiple currents, which, incidentally, is possible for the positive helicity subspace. As the adjunctator plays also the role of parity transformation, its unicity is essential to the theory. Furthermore, together with time inversion, it defines complete coordinate inversion, which reverses all currents—without the need for a charge-conjugation operator. The latter is impossible to define in representation-independent form, which in itself is a statement over why it does not exist and why it is important to have representation-independent results.

Acknowledgments This work was supported by a Grant of the Roma- nian National Authority for Scientific Research, CNCS- UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0323.

References

[1] W. Pauli, “Uber¨ die Formulierung der Naturgesetze mit funf¨ homogenen Koordinaten. Teil II: Die Diracschen Gleichungen fur¨ die Materiewellen,” Annalen Der Physik,vol.410,no.4,pp. 337–372, 1933. [2] W. Pauli, “On the formulation of the laws of Nature with five homogeneous components. Part II: the Dirac equation for matter waves,” Annals of Physics,vol.18,pp.337–372,1933. [3]W.Pauli,“Contributionsmathematiques´ alath` eorie´ des matrices de Dirac,” Annales de L’Institut Henri Poincare´,vol.6,pp. 109–136, 1936. [4] M. Arminjon and F.Reifler, “Four-vector versus four-scalar representation of the Dirac wave function,” International Journal of Geometric Methods in Modern Physics,vol.9,no.4,ArticleID 1250026, 2012. [5] M. Arminjon and F. Reifler, “Dirac equation: representation independence and tensor transformation,” Brazilian Journal of Physics, vol. 38, no. 2, pp. 248–258, 2008. [6] M. Arminjon, “A simpler solution of the non-uniqueness problem of the covariant Dirac theory,” International Journal of Geo- metric Methods in Modern Physics,vol.10,no.7,ArticleID 1350027, 2013. [7] M. Arminjon and F. Reifler, “A non-uniqueness problem of the Dirac theory in a curved spacetime,” Annalen Der Physik,vol. 523, no. 7, pp. 531–551, 2011. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 217858, 14 pages http://dx.doi.org/10.1155/2013/217858

Research Article 𝑋(1870) and 𝜂2(1870): Which Can Be Assigned as a Hybrid State?

Bing Chen,1 Ke-Wei Wei,1 and Ailin Zhang2

1 School of Physics and Electrical Engineering, Anyang Normal University, Anyang 455000, China 2 Department of Physics, Shanghai University, Shanghai 200444, China

Correspondence should be addressed to Bing Chen; [email protected]

Received 11 July 2013; Revised 15 September 2013; Accepted 23 September 2013

Academic Editor: Gongnan Xie

Copyright © 2013 Bing Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The mass spectrum and strong decays of the 𝑋(1870) and 𝜂2(1870) are analyzed. Our results indicate that 𝑋(1870) and 𝜂2(1870) are 1 the two different resonances. The narrower 𝑋(1870) seems likely a good hybrid candidate. We support the 𝜂2(1870) as the 𝜂2(2 𝐷2) quarkonium. We suggest to search the isospin partner of 𝑋(1870) in the channels of 𝐽/𝜓 󳨀→0 𝜌𝑓 (980)𝜋 and 𝐽/𝜓 󳨀→1 𝜌𝑏 (1235)𝜋 in the future. The latter channel is very important for testing the hybrid scenario.

1. Introduction reaction. By contrast, the hadronic 𝐽/𝜓 decay are considered “hybrid rich” (Figure 1(b)). 𝑋(1870) An isoscalar resonant structure of was observed by Furthermore, the branching ratio R1 =Γ(𝜂2(1870) → 𝜎 the BESIII Collaboration with a statistical significance of 7.2 𝑎2(1320)𝜋)/Γ(𝜂2(1870) → 𝑎0(980)𝜋) = 32.6 ± 12.6 reported 𝐽/𝜓 → 𝜔𝑋(1870)+ →𝜔𝜂𝜋 𝜋− in the processes recently [1]. by the WA102 Collaboration indicates that the decay channel Its mass and width were given as of 𝑎0(980)𝜋 is tiny for 𝜂2(1870) [7]. This has been confirmed 𝑀 = 1877.3 ±+3.4 6.3 ,Γ=57±12+19 . by an extensive reanalysis of the Crystal Barrel data [10]. −7.4 MeV −4 MeV (1) Differently, the analysis of BESIII Collaboration indicates 𝑋(1870) 𝑎 (980)𝜋 Here the first errors are statistical and the second ones are that the primarily decays via the 0 channel systematic. The product branching fraction of B(𝐽/𝜓 → [1]. Then the present measurements of the decay widths, ± ∓ ± productions, and decay properties suggest that 𝜂2(1870) and 𝜔𝑋(1870)) ⋅ B(𝑋(1870) →0 𝑎 (980)𝜋 )⋅B(𝑎0 (980) → ± +0.72 −4 𝑋(1870) are two different isoscalar mesons. 𝜂𝜋 ) = [1.5 ± 0.26(stat) (syst)] × 10 was also presented −0.36 If the production process 𝐽/𝜓 → 𝜔𝑋(1870) is mainly [1]. But the quantum numbers of 𝑋(1870) are still unknown, + −+ hadronic, the quantum numbers of 𝑋(1870) should be 0 0 , then the partial wave analysis is required in future. + ++ + −+ 0 1 ,or0 2 . One notices that the predicted masses for The mass of 𝑋(1870) is consistent with the 𝜂2(1870),but + −+ + ++ + −+ the light 0 0 , 0 1 and 0 2 hybrids overlap 1.8 GeV in the width is much narrower than the 𝜂2(1870).Inthetables the Bag model [11, 12], the flux tube model [13, 14], and the of the Particle Data Group (PDG) [2], the available mass and constituent gluon model [15]. In addition, the decay width of width of 𝜂2(1870) are −+ isoscalar 2 hybridisexpectedtobenarrow[16]. Therefore, −+ 𝑀 = 1842 ±8 , Γ = 225 ± 14 . 𝑋(1870) becomes a possible 2 hybrid candidate. MeV MeV (2) −+ −+ In addition, the predicted masses of 0 and 2 glueball The 𝜂2(1870) hasbeenobservedin𝛾𝛾 reactions [3, 4], aremuchhigherthan1.8GeVbylatticegaugetheory[17–19]. 𝑝𝑝 annihilation [5–8], and radiative 𝐽/𝜓 decays [9]. It should Therefore, 𝑋(1870) is not likely to be a glueball state. More- be stressed that radiative 𝐽/𝜓 decay channels (Figure 1(a)) over, the molecule and four-quark states are not expected in and 𝑝𝑝 annihilation processes are the ideal glueball hunting this region [16]. Then the unclear structure 𝑋(1870) looks grounds. But the glueball production is suppressed in 𝛾𝛾 more like a good hybrid candidate. But the actual situation 2 Advances in High Energy Physics

𝛾 𝜔, 𝜌

J/𝜓 J/𝜓 Glueball Hybrid

(a) (b)

Figure 1: (a) A prior production of glueball in the 𝐽/𝜓 →𝐺 𝛾𝑋 ; (b) a prior production of hybrid in the 𝐽/𝜓 →𝐻 𝜔𝑋 .

is much complicated because the nature of 𝜂2(1870) is still proposed by the Cornell group (Cornell potential) with the ambiguous. form [29] 4 𝛼 (i) Since no lines of evidence have been found in 𝑉 (𝑟) =− 𝑠 +𝜎𝑟+𝐶. 𝑞𝑞 3 𝑟 (4) the decay mode of 𝐾𝐾𝜋,the𝜂2(1870) disfavors 1 the 1 𝐷2𝑠𝑠 quarkonium assignment. The mass of The strong coupling constant 𝛼𝑠, the string tension 𝜎, 1 𝐶 𝜂2(1870) seems much smaller for the 2 𝐷2𝑛𝑛 (𝑛𝑛≡ and the constant are the model parameters which can (𝑢𝑢+𝑑𝑑)/√2) state in the Godfrey-Isgur (GI) quark be fixed by the well-established experimental states. The model [20]. Therefore the 𝜂2(1870) has been assigned remaining spin-dependent terms for mass shifts are usually −+ as the 2 hybrid state [10, 21–23]. treated as the leading-order perturbations which include the spin-spin contact hyperfine interaction, spin-orbit and tensor (ii) However, Li and Wang pointed out that the mass, interactions, and a longer-ranged inverted spin-orbit term. production, total decay width, and decay pattern of 𝜂 (1870) They arise from one gluon exchange (OGE) forces and the the 2 do not appear to contradict with the assumed Lorentz scalar confinement. The expressions for 21𝐷 𝑛𝑛 picture of it as being the conventional 2 state these terms may be found in [20]. [24]. It should be pointed out that the nonperturbative con- Therefore, systematical study of the mass spectrum and tribution may dominate for the hyperfine splitting of light strong decay properties is urgently required for 𝑋(1870) and mesons, which is not like the heavy quarkonium [30]. For example, the hyperfine shift of the ℎ𝑐(1𝑃) meson with respect 𝜂2(1870). Some valuable suggestions for the experiments in to the center gravity of the 𝜒𝑐(1𝑃) mesons is much small: future are also needed. 𝑀 (𝜒 )−𝑀(ℎ) = −0.02 ± 0.19 ± 0.13 The paper is organized as follows. In Section 2,themasses cog 𝑐 𝑐 MeV [31]. However, for the light isovector mesons 𝑎0(1450), 𝑎1(1260), 𝑎2(1320), of 𝑋(1870) and 𝜂2(1870) will be explored in the GI relativized and 𝑏1(1235),thehyperfineshiftis76.7 ± 44.4 MeV. Here quark model and the Regge trajectories (RTs) framework. In 𝑎 𝑎 𝑎 𝑏 Section 3, the decay processes that an isoscalar meson decays the masses of 0, 1, 2,and 1 aretakenfromPDG[2]. For into light scalar (below 1 GeV) and pseudoscalar mesons will the complexities of nonperturbative interactions, we are not be discussed. The two-body strong decays of 𝑋(1870) and going to calculate the hyperfine splitting. 3 Now, the spin-averaged mass, 𝑀𝑛𝑙,of𝑛𝐿 multiplet can be 𝜂2(1870) will be calculated within the 𝑃0 model and the flux- tube model. Finally, our discussions and conclusions will be obtained by solving the spinless Salpeter equation: presented in Section 4. [√𝑝⃗2 +𝑚2 + 𝑝⃗2 +𝑚2 +𝑉 𝑟 ]𝜓 𝑟 =𝐸𝜓 𝑟 . 𝑞 𝑞 √ 𝑞 𝑞 𝑞𝑞 ( ) ( ) ( ) (5) 2. Mass Spectrum Here we employ a variational approach described in [32] to solve (5). This variational approach has been applied well in IntheGodfrey-Isgurrelativizedpotentialmodel[20], the solving the Salpeter equation for 𝑐𝑠 [33], 𝑐𝑐 and 𝑏𝑏 [34]mass Hamiltonian 𝐻 consists of a central potential and the kinetic spectrum. term in a “relativized” form In the calculations, the basic simple harmonic oscillator (SHO) functions are taken as the trial wave functions. It is 𝐻=√𝑝⃗2 +𝑚2 + √𝑝⃗2 +𝑚2 +𝑉 (𝑟) . 𝑞 𝑞 𝑞 𝑞 𝑞𝑞 (3) given by 𝜓 (𝑟, 𝛽) The funnel-shaped potentials which include a color 𝑛𝑙 coulomb term at short distances and a linear scalar confining 2 (2𝑛 − 1)! 2 2 (6) term at large distances are usually incorporated as the zeroth- 3/2√ 𝑙 −𝛽 𝑟 /2 𝑙+1/2 2 2 =𝛽 (𝛽𝑟) 𝑒 𝐿𝑛−1 (𝛽 𝑟 ) order potential. The typical funnel-shaped potential was Γ (𝑛+𝑙+(1/2)) Advances in High Energy Physics 3 in the position space. Here the SHO function scale 𝛽 is the Table 1: The spin-averaged mass (unit: GeV) and the harmonic − variational parameter. oscillator parameter 𝛽 (unit: GeV 1)ofthestates2𝑆, 3𝑆, 4𝑆, 1𝑃, 2𝑃, By the Fourier transform, the SHO radial wave function 3𝑃, 1𝐷, 2𝐷, 3𝐷, 1𝐹, 2𝐹, 1𝐺,and1𝐻. in the momentum is States 𝑀𝑛𝑙(𝑛𝑛) 𝛽 Expt. [2] 𝑀𝑛𝑙(𝑠𝑠) 𝛽 Expt. [2] 𝜓𝑛𝑙 (𝑝, 𝛽) 1𝑆 — 0.44/0.34 — — 0.42/0.39 — 2𝑆 0.310 1.389 1.631 0.330 1.629 𝑛 𝑙 2 1.399 (−1) 2 (2𝑛 − 1)! 𝑝 2 2 𝑝 = √ ( ) 𝑒−𝑝 /2𝛽 𝐿𝑙+1/2 ( ). 3𝑆 1.859 0.295 2.069 0.310 𝛽3/2 Γ (𝑛+𝑙+(1/2)) 𝛽 𝑛−1 𝛽2 4𝑆 2.240 0.290 2.436 0.300 (7) 1𝑃 1.252 0.310 1.257 1.460 0.340 1.478 2𝑃 1.711 0.294 1.926 0.315 The wave functions of 𝜓𝑛𝑙(𝑟, 𝛽) and 𝜓𝑛𝑙(𝑝, 𝛽) meet the 3𝑃 normalization conditions: 2.110 0.290 2.308 0.300 1𝐷 1.883 ∞ ∞ 1.661 0.280 1.672 0.300 2 2 2 2 2𝐷 2.067 ∫ 𝜓𝑛𝑙 (𝑟, 𝛽) 𝑟 𝑑𝑟 = 1, ∫ 𝜓𝑛𝑙 (𝑝, 𝛽) 𝑝 𝑑𝑝 = 1. (8) 0.276 2.272 0.292 0 0 3𝐷 2.417 0.275 2.609 0.288 1𝐹 1.924 0.277 2.128 0.295 In the variational approach, the corresponding 𝑀𝑛𝑙 are given by minimizing the expectation value of 𝐻 2𝐹 2.287 0.275 2.478 0.290 1𝐺 2.161 0.275 2.350 0.292 𝑑 𝐸 (𝛽) = 0, 1𝐻 2.377 0.273 2.554 0.287 𝑑𝛽 𝑛𝑙 (9) where

𝐸𝑛𝑙 (𝛽) ≡⟨𝐻⟩𝑛𝑙 = ⟨𝜓𝑛𝑙 |𝐻| 𝜓𝑛𝑙⟩ . (10) here are also reasonable, for example, 𝑎4(2040) and 𝑓4(2050) are very possible the 𝐹-wave 𝑛𝑛 isovector and isoscalar When all the parameters of the potential model are +10 mesons with the masses of 1996 MeV and 2018 ± 11 MeV, known, the values of the harmonic oscillator parameter 𝛽 can −9 respectively [2]. The predicted spin-averaged mass of 1𝐹 𝛽 be fixed directly. With the values of ,allthespin-averaged is not incompatible with experiments. Our results are also 𝑀 𝑀 masses 𝑛𝑙 will be obtained easily. 𝑛𝑙 obtained in this way overall in good agreement with the expectations from [37]. trend to be better for the higher-excited states [35]. The trend that a higher excited state corresponds to a smaller It is unreasonable to treat the spin-spin contact hyperfine 𝛽 coincides with [38–40]. For considering the spin-spin interaction as a perturbation for the ground states, because contact hyperfine interaction, there are two 𝛽sforthe1𝑆 the mass splitting between pseudoscalar mesons and vector 1 mesons. The larger one corresponds to the 1 𝑆0 state and the mesons are much large. Then we consider the contributions 3 1 𝑆1 of 𝑉𝑠⋅⃗𝑠(𝑟)⃗ for the 1𝑆 mesons.ThefollowingGaussian-smeared smaller one the state. 𝑉 (𝑟) contact hyperfine interaction36 [ ] is taken for convenience: Asshownin[37, 41], the confinement potential conf is determinant for the properties of higher excited states. In 3 𝜎 = 0.143 2 𝑠⋅⃗𝑠⃗ 32𝜋𝛼 𝜅 −𝜅2𝑟2 [37], the masses for higher excited states with GeV 𝑉 (𝑟) = 𝑠 ( ) 𝑒 𝑆⃗⋅ 𝑆⃗. 𝛼 =0 𝑞𝑞 9𝑚2 √𝜋 𝑞 𝑞 (11) and 𝑠 are closer to experimental data than the results 𝑞 given in [20]. Then we ignored the Coulomb interaction for 1𝐷 2𝐷 1𝐹 1𝐺 1𝐻 𝑀 In this work, we choose the model parameters as follows: , , , ,and states. In this way, 𝑛𝑙 for these states increase about 100 MeV. 𝑚𝑢 =𝑚𝑑 = 0.220 GeV, 𝑚𝑠 = 0.428 GeV, 𝛼𝑠 = 0.6, 𝜎= 󸀠 1 󸀠 3 󸀠 1 1 0.143 2 𝜅 = 0.37 𝐶 = −0.37 The masses of 𝜂 (3 𝑆0), 𝑓1(2 𝑃1), 𝜂2(1 𝐷2),and𝜂2(2 𝐷2) GeV , GeV, and GeV. We take ∼ the smaller value of 𝜎 here rather than the value in [20]. are usually within 1.8 2.1 GeV in various quark potential The smaller 𝜎 was obtained by the relation between the slope models [20, 25–27](seeinTable 2). The predicted spin- 󸀠 3𝑆(𝑠𝑠) 2𝑃(𝑠𝑠) 1𝐷(𝑠𝑠) 2𝐷(𝑛𝑛) of the Regge trajectory for the Salpeter equation 𝛼 and the averaged masses of , , ,and are 𝛼󸀠 also within this mass regions (bold ones in Table 1). Due to slope st in the string picture [30]. The Gaussian smearing 𝜅 the uncertainty of the potential models, absolute deviation parameter seems a little smaller than that in [20]. However, ∼ the 𝜅 is usually fitted by the hyperfine splitting of low-excited from experimental data are usually about 100 150 MeV for 𝑛𝑆 states in the works of literature with a certain arbitrariness. the higher excited states. Compared with these predicted 𝑋(1870) 𝜂󸀠(31𝑆 ) The values of 𝑀𝑛𝐿 and 𝛽 for the states 2𝑆, 3𝑆, 4𝑆, 1𝑃, 2𝑃, masses, disfavors the 0 assignment for its low 𝑓󸀠(23𝑃 ) 𝜂󸀠 (11𝐷 ) 𝜂 (21𝐷 ) 3𝑃, 1𝐷, 2𝐷, 3𝐷, 1𝐹, 2𝐹, 1𝐺,and1𝐻 are listed in Table 1.The mass. But the possibilities of 1 1 , 2 2 ,and 2 2 still exist. Here we do not consider the possibility of 𝑋(1870) experimental masses for the relative mesons are taken from 1 PDG [2]. as the 𝜂(3 𝑆0) state because 𝜂(1760) looks more like a good 1 Obviously, the spin-averaged masses of the 2𝑆, 1𝑃, 1𝐷𝑛𝑛 𝜂(3 𝑆0) candidate [42–44]. and 1𝑃,and2𝑆𝑠𝑠 mesons are consistent with the experimental Regge trajectories (RTs) are another useful tool for study- data. Indeed, the predicted masses of higher excited states ing the mass spectrum of the light flavor mesons. In [45], the 4 Advances in High Energy Physics

1 󸀠 3 󸀠 1 󸀠 + −+ + ++ Table 2: The masses predicted for 3 𝑆0(𝜂 ), 2 𝑃1(𝜂 ), 1 𝐷2(𝜂 ), and Table 4: The masses predicted for 𝜂𝐻(0 0 ), 𝑓𝐻(0 1 ) and 1 † + −+ 2 𝐷2(𝜂)in[20, 25–27]. The value denoted by“ ” was predicted for 𝜂𝐻(0 2 ) hybrid states in [11–15]. 21𝐷 the mass of isovector 2 state in [20]. + −+ + ++ + −+ States 𝜂𝐻(0 0 )𝑓𝐻(0 1 )𝜂𝐻(0 2 ) 𝜂󸀠(31𝑆 )𝑓󸀠(23𝑃 )𝜂󸀠 (11𝐷 )𝜂(21𝐷 ) States 0 1 1 2 2 2 2 Bag [11, 12] 1.3 Heavier 1.9 † [20] — 2030 1890 2130 Flux tube [13, 14]1.7∼1.9 1.7∼1.9 1.7∼1.9 [25]2085201619091960Constituent gluon [15]1.8∼2.2 1.3∼1.8 1.8∼2.2 [26] 2099 1988 1851 — [27]— — 18531863 Y

Table 3: 𝑋(1870) calculated in RTs for different states. The masses 𝜂󸀠(1475) 𝑓 (1420) ℎ (1380) 𝜂 (1645) of , 1 , 1 ,and 2 are taken from PDG 𝜅0 +1 𝜅+ [2].

Four possible states for X(1870) 󸀠 1 󸀠 3 󸀠 1 1 𝜂 (3 𝑆0)𝑓1 (2 𝑃1)𝜂2(1 𝐷2)𝜂2(2 𝐷2) 󸀠 𝜂 (1475) 𝑓1(1420) ℎ1(1380) 𝜂2(1645) 𝜇2 = +0.04 𝜇2 = +0.04 𝛽2 = +0.06 𝜇2 = +0.04 a− 0 a+ 1.34−0.03 1.38−0.03 1.60−0.06 0.91−0.03 0 a0 0 I3 −1 𝜎 f0 +1 authors fitted the RTs for all light-quark meson states listed in the PDG tables. A global description was constructed as

2 𝑀 = 1.38 (4) 𝑛 + 1.12 (4) 𝐽 − 1.25 (4) . (12) 𝜅− −1 𝜅0 Here, 𝑛 and 𝐽 mean the the radial and angular- momentum quantum number. Recently, the authors of [45] repeated their fits with the subset mesons of the paper 2 Figure 2: The “S”nonetbelow1GeVshownin𝑌-𝐼3 plane. [46]. They found a little smaller averaged slopes of 𝜇 = 2 2 2 1.28(5) GeV and 𝛽 = 1.09(6) GeV to be compared with 2 2 2 2 2 𝜇 = 1.38(4) GeV and 𝛽 = 1.12(4) GeV in (12). Here the 𝜇 𝛽2 nearly equal, the possible assignments of 𝑋(1870) also suit and are the weighed averaged slopes for radial and angular- 𝜂 (1870) momentum RTs [45, 47]. 2 . The investigations of the strong decay properties 󸀠 will be more helpful to distinguish the 𝜂2(1870) and 𝑋(1870). Now ℎ1(1380), 𝑓1(1420),and𝜂 (1475) have been estab- 1 3 1 lished as the 1 𝑃1, 1 𝑃1,and2 𝑆0𝑠𝑠 states in PDG [2]. With the differences between the mass squared of 𝑋(1870) and 3. The Strong Decay these states (Table 3), 𝑋(1870) couldbeassignedforthe 󸀠 1 󸀠 1 3.1. The Final Mesons Include the Scalar Mesons below 1 GeV. 𝜂 (3 𝑆0) and 𝑓1(2 𝑃1).Themassof𝑋(1870) is too large for 󸀠 1 Despite many theoretical efforts, the scalar nonet of 𝑞𝑞 the 𝜂2(1 𝐷2) stateintheRTs.𝜂2(1645) has been assigned as 1 2 2 mesons has never established well. The lowest-lying scalar the 1 𝐷2𝑛𝑛 meson [2]. Since 𝑀 (𝑋(1870))−𝑀 (𝜂2(1640)) = +0.04 2 2 mesons including 𝜎(500) (or 𝑓0(600)), 𝜅(800), 𝑎0(980),and 0.91−0.03 GeV which is much smaller than 1.38(4) GeV , 𝑓0(980) are difficult to be described as 𝑞𝑞 states; for example, 𝑋(1870) 21𝐷 𝑛𝑛 looks unlike the 2 state for its low mass. 𝑎0(980) is associated with nonstrange quarks in the 𝑞𝑞 𝑀2(𝑋(1870)) −2 𝑀 (ℎ (1170)) = However, the difference of 1 scheme.Ifthisistrue,itshighmassanddecaypropertiesare 2.16+0.06 2 2.37(11) 2 −0.05 GeV matches the slopes GeV well. Then difficult to be understood simultaneously. So interpretations the RTs can not exclude the possibility of 𝑋(1870) as the 1 as exotic states were triggered, For examples, two clusters 2 𝐷2𝑛𝑛 state. of two quarks and two antiquarks [48], particular quasi- As mentioned in Section 1, 𝑋(1870) is also a good hybrid molecular states [49–51], and uncorrelated four-quark states candidate since its mass overlaps the predictions given by 𝑞𝑞𝑞 𝑞 + −+ + ++ [52–54]havebeenproposed. different models. The predicted masses for 0 0 , 0 1 ,and + −+ Though the structures of these scalar mesons below 1 GeV 0 2 𝑛𝑛𝑔 states by these models are collected in Table 4. are still in dispute, the viewpoint that these scalar mesons In this section, the mass of 𝑋(1870) has been studied can constitute a complete nonet states has been reached in intheGIquarkpotentialmodelandtheRTsframework.In most works of the literature (as illustrated in Figure 2). In the GI quark potential model, 𝑋(1870) can be interpreted as thefollowing,wewilldenotethisnonetas“S”multipletfor 𝑓󸀠(23𝑃 ) 𝜂󸀠 (11𝐷 ) 𝜂 (21𝐷 ) the 1 1 , 2 2 ,or 2 2 state with a reasonable convenience. 󸀠 1 uncertainty. In the RTs, 𝑋(1870) favors the 𝜂 (3 𝑆0) and Due to the unclear nature of the S mesons, it seems much 󸀠 3 1 𝑓1(2 𝑃1) assignments. But the 𝜂2(2 𝐷2) assignment can not difficulttostudythedecayprocesseswhenthefinalmesons be excluded thoroughly. 𝑋(1870) is also a good hybrid state include a S member. As an approximation, 𝑎0(980), 𝜎(500), 3 candidate. Since the masses of 𝑋(1870) and 𝜂2(1870) are and 𝑓0(980) were treated as 1 𝑃0 𝑞𝑞 mesons in [44, 55]. In Advances in High Energy Physics 5

[24, 43], this kind of decay channel was ignored. However, In order to determine the relations between these cou- this kind of decay mode may be predominant for some pling constants, we will assume the process that the 𝑞𝑞 or mesons. For example, the observations indicate that 𝑓1(1285), 𝑞𝑞𝑔 meson decays into a S and another 𝑞𝑞 mesons obey the 𝜂(1405),and𝑋(1870) primarily decay via the 𝑎0(980)𝜋 OZI (Okubo-Zweig-Iizuka) rule; that is, the two quarks in channel [1]. the mother meson go into two daughter mesons, respectively. In what follows, we will extract some useful information Therefore, there are four forbidden processes: 𝑋((1/√2)(𝑢𝑢− (3) √ aboutthiskindofdecaymodebytheSU flavor symmetry. 𝑑𝑑))󴀂󴀠𝑎 󴀀 0 +𝑠𝑠, 𝑋((1/ 2)(𝑢𝑢+𝑑𝑑)) 󴀀󴀂󴀠𝜎0 +𝑠𝑠, 𝑋(𝑠𝑠) 󴀀󴀂󴀠 We will show that 𝑎0(980)𝜋, 𝜎0𝜂,and𝑓0𝜂 are the main decay √ 𝑓0 +(1/ 2)(𝑢𝑢+𝑑𝑑) and 𝑋(𝑠𝑠) 󴀀󴀂󴀠𝜎0 +𝑠𝑠. With the help of the channels for the isoscalar 𝑛𝑛 and the 𝑛𝑛𝑔 mesons when they (3) S +𝑃 𝑃 SU Clebsch-Gordan coefficients59 [ ], the ratios between decay primarily through “ ” mesons, where the sign “ ” the five coupling constants are extracted as denotes a light pseudoscalar meson. This will explain why 𝑋(1870) 𝜂𝜋+𝜋− has been first observed in the channel. 𝑔𝐴81 :𝑔𝐴18 :𝑔𝐵88 :𝑔𝐵11 :𝑔𝐴88 We noticed that the S nonet could be interpreted like the 𝑞𝑞 nonet in the diquark-antidiquark scenario. In Wilczek and 2 2 = √2:−√ (√5+1):− (√5+1): Jaffe’s terminology [56, 57], the S mesons consist of a “good” 5 √5 diquark and a “good” antidiquark. When 𝑢, 𝑑 quarks form (17) a “good” diquark, it means that the two light quarks, 𝑢 and 2 − √ (√5+1):1 𝑑, could be treated as a quasiparticle in color 3,flavor3,and 5 thespinsinglet.The“good”𝑢, 𝑑 diquark is usually denoted as ≈ 1.41 : −2.05 : −2.89 : −2.05 : 1.00. [𝑢𝑑]. In the diquark-antidiquark limit, the parity of a 𝐿 It is well known that the physical states, 𝜂(548) and tetraquark is determined by 𝑃=(−1)12−34 [58]wherethe 󸀠 𝜂 (958), are the mixture of the SU(3) flavor octet and singlet. 𝐿12−34 refers to the relative angular momentum between two They can be written in terms of a mixing angle, 𝜃𝑝, as follows: clusters. Thus the S mesons are the lightest tetraquark states in the diquark-antidiquark model with 𝐿12−34 =0.TheS 𝜂 (548) cos 𝜃𝑝 − sin 𝜃𝑝 |8⟩𝐼=0 nonet in the full set of flavor representations is ( 󸀠 )=( )( ). (18) 𝜂 (958) sin 𝜃𝑝 cos 𝜃𝑝 |1⟩𝐼=0 (3⊗3) ⊗ (3 ⊗ 3) =8⊕1. 3 3 (13) The mixing angle 𝜃𝑝 has been measured by various means. (3) Because the SU flavor symmetry is not exact, the two However, there is still uncertainty for 𝜃𝑝.Anexcellentfit physical isoscalar mesons, 𝜎0 and 𝑓0,areusuallythemixing to the tensor meson decay widths was performed under the |8⟩ |1⟩ ∘ states of the 𝐼=0 and 𝐼=0 states [48], SU(3) symmetry, and 𝜃𝑝 ≃−17 was obtained [23]. In our 𝜃 −17∘ 𝑛𝑛 𝑓0 cos 𝜗 sin 𝜗 |8⟩𝐼=0 calculation, 𝑝 is taken as . The excited mixtures of ( )=( )( ). (14) 𝑠𝑠 𝜎0 − sin 𝜗 cos 𝜗 |1⟩𝐼=0 and are denoted as 𝜉 When the mixing angle 𝜗 equals the so-called ideal mixing cos 𝜃 sin 𝜃 |1⟩𝐼=0 ∘ ( 󸀠)=( )( ). (19) angle, that is, 𝜗 = 54.74 , the composition of the 𝜎(500) and 𝜉 − sin 𝜃 cos 𝜃 |8⟩𝐼=0 𝑓0(980) is 󵄨 In this scheme, the ideal mixing occurs with the choice 󵄨 1 ∘ 󸀠 𝑓 󵄨 ([𝑠𝑢][𝑠 𝑢] + [𝑠𝑑] [𝑠𝑑])⟩ of 𝜃 = 35.3 .When𝜉 and 𝜉 decay into a S and pseudoscalar ( 0)=(󵄨√ ). 𝜎 󵄨 2 󵄨 (15) mesons, the relations of decay amplitudes are governed by the 0 󵄨 2 󵄨[𝑢𝑑] [𝑢𝑑]⟩ coefficients 𝜁 which are model-independent in the limitation of SU(3)𝑓 symmetry. With the coupling constants in hand, It seems that the deviation from the ideal mixing angle 2 󸀠 𝜎(500) 𝑓 (980) the coefficients 𝜁 of 𝜉 and 𝜉 versus the mixing angle 𝜃 are of the and 0 is small [48]. In the following 󸀠 shownintheFigures3 and 4.When𝜉 and 𝜉 occur in the calculations, we will treat them in the ideal mixing scheme. 2 Under the SU(3) flavor assumption, all the members of ideal mixing, the values of 𝜁 are presented in Table 5.Inthe theoctethavethesamebasiccouplingconstantinonetype factorization framework, the decay difference of a hybrid and excited 𝑞𝑞 mesons comes from the spatial contraction [60]. of reaction, while the singlet member has a different coupling 2 constant. Particularly, when a quarkonium decays into S and Then the coefficients 𝜁 for hybrid states are the same as these 𝑞𝑞 of 𝑞𝑞 quarkonia. mesons, there are five independent coupling constants, 󸀠 that is, 𝑔𝐴88, 𝑔𝐴81, 𝑔𝐴18, 𝑔𝐵88,and𝑔𝐵11, corresponding to five Here the mixing of 𝜂(548) and 𝜂 (958) has been consid- 2 󸀠 different channels ered. It is sure that the 𝜁 are zero for the processes 𝜉 →𝑎0𝜋, 󸀠 󸀠 󸀠 2 𝜉 →𝜎𝜂,and𝜉 →𝜎𝜂, since they are OZI-forbidden. 𝜁 of |8⟩𝑞𝑞 󳨀→ |8⟩S ⊗ |8⟩𝑞𝑞 :𝑔𝐴88, 󸀠 󸀠 𝜉 →𝑓0𝜂 has not been considered in Table 5 since 𝑋(1870) |8⟩ 󳨀→ |8⟩ ⊗ |1⟩ :𝑔, 󸀠 𝑞𝑞 S 𝑞𝑞 𝐴81 lies below the threshold of 𝑓0𝜂 . |8⟩𝑞𝑞 󳨀→ |1⟩S ⊗ |8⟩𝑞𝑞 :𝑔𝐴18, (16) As illustrated in Figures 3 and 4,theprimarydecay 𝑠𝑠 𝑠𝑠𝑔 𝑓 𝜂 |1⟩ 󳨀→ |8⟩ ⊗ |8⟩ :𝑔, channels of a or predominant excitation are 0 and 𝑞𝑞 S 𝑞𝑞 𝐵88 𝜅𝐾.Ifthedeviationof𝜃 from the ideal mixing angle is not |1⟩𝑞𝑞 󳨀→ |1⟩S ⊗ |1⟩𝑞𝑞 :𝑔𝐵11. large, 𝑋(1870) should be a 𝑛𝑛 or 𝑛𝑛𝑔 predominant state 6 Advances in High Energy Physics

3.5 3.2. The Strong Decays of 𝜂2(1870) and 𝑋(1870). In [24], 𝜉→𝜎𝜂 3 1 the 𝑃0 model [61–63]andtheflux-tubemodel[64]were 3.0 (uu + dd) √2 employed to study the two-body strong decays of 𝜂2(1870). 2.5 35.3∘ There, the pair production (creation) strength 𝛾 and the sim- 𝜉→f0𝜂 𝜉→a0𝜋 2.0 ple harmonic oscillator (SHO) wave function scale parameter,

2 𝛽 𝜁 s, were taken as constants. 1.5 𝜉→𝜅K However, a series of studies indicate that the strength 𝛾 1.0 may depend on both the flavor and the relative momentum 𝜉→𝜎𝜂󳰀 𝛾 0.5 of the produced quarks [28, 65]. may also depend on the reduced mass of quark-antiquark pair of the decaying meson 0.0 3 [66]. Firstly, the relations of the 𝑃0 model to “microscopic” 0 50 100 150 QCD decay mechanisms have been studied in [65]. There, the 𝜃 (deg) authors found that the constant 𝛾 corresponds approximately 𝜎/𝑚 𝛽 𝑚 𝜁2 𝜉 to the dimensionless combination, 𝑞 ,where 𝑞 is the Figure 3: The coefficients of the isoscalar meson versus the 𝛽 mixing angle 𝜃. mass of produced quark, means the meson wave function scale, and 𝜎 is the string tension. Secondly, the momentum dependent manner of 𝛾 has been studied in [28]. It was found that 𝛾 is dependent on the relative momentum of the 3.5 𝑞𝑞 𝛾(𝑘) = 𝐴 +𝐵 (−𝐶𝑘2) ss 𝜉󳰀 → 𝜎𝜂 created pair,andtheformof exp with ⃗ ⃗ 3.0 𝑘=|𝑘3 − 𝑘4| was suggested. Thirdly, Segovia et al. proposed 35.3∘ 󳰀 𝜉 → a0𝜋 that 𝛾 isafunctionofthereducedmassofquark-antiquark 2.5 󳰀 𝜉 → f0𝜂 pair of the decaying meson [66].Basedonthefirstand 2.0 third points above, 𝛾 will depend on the flavors of both the 2 𝜁 1.5 𝜉󳰀 → 𝜅K decaying meson and produced pairs. In our calculations, we will treat the 𝛾 as a free parameter and fix it by the 1.0 𝜉󳰀 → 𝜎𝜂󳰀 well-measured partial decay widths. 3 0.5 In addition, the amplitudes given by the 𝑃0 model and 0.0 the flux-tube model often contain the nodal-type Gaussian 0 50 100 150 form factors which can lead to a dynamic suppression for 𝛽 𝜃 (deg) some channels. Then the values of are important to extract the decay width for the higher excited mesons in these two 2 󸀠 Figure 4: The coefficients 𝜁 of the isoscalar meson 𝜉 versus the strong decay models. mixing angle 𝜃. In the following, the two-body strong decay of 𝑋(1870) 3 will be investigated in the 𝑃0 model where the strength 𝛾 will be extracted by fitting the experimental data. The SHO wave 2 󸀠 𝛽 Table 5: The coefficients 𝜁 of 𝜉 and 𝜉 in the ideal mixing. function scale parameter, s, will be borrowed from Table 1 which are extracted by the GI relativized potential model. We 󸀠 Decay channels 𝑎0𝜋𝜎𝜂𝜅𝐾𝑓0𝜂𝜎𝜂will also check the possibility of 𝑋(1870) as a possible hybrid 2 𝜁 [𝑛𝑛(𝑔)] 2.17 3.56 0.47 1.07 0.44 state by the flux-tube model. 2 ̂ 𝜁 [𝑠𝑠(𝑔)] 0.00 0.00 0.72 1.08 0.00 In the nonrelativistic limit, the transition operator T of 3 the 𝑃0 model is depicted as

T̂ =−3𝛾∑ ⟨1,𝑚;1,−𝑚 |0,0⟩ since 𝑋(1870) primarily decays via the 𝑎0(980)𝜋 channel. At −+ ++ 𝑚 present, only the ground 0 and the 0 isoscalar mesons 3 ⃗ 3 ⃗ 3 ⃗ ⃗ deviate from the ideal mixing distinctly. In addition, if the × ∬ 𝑑 𝑘3𝑑 𝑘4𝛿 (𝑘3 + 𝑘4) 𝑋(1870) is produced via a diagram of Figure 1(b),itshould 𝑘⃗ − 𝑘⃗ also be 𝑛𝑛 or 𝑛𝑛𝑔 predominant state. 𝑚 3 4 (3,4) (3,4) (3,4) † ⃗ † ⃗ × Y1 ( )𝜔0 𝜑0 𝜒1,−𝑚𝑑3𝑖(𝑘3)𝑑4𝑗(𝑘4), Of course, the SU(3)𝑓 symmetry breaking will affect the 2 ratios of these channels listed in Table 5, because the three- (20) momentum of the these products are different. However, the 2 coefficients 𝜁 have presented the valuable information for (3,4) (3,4) where the 𝜔0 and 𝜑0 arethecolorandflavorwave these specific decay channels. When 𝜂2(1870) occupies the (3,4) 1 functions of the 𝑞3𝑞4 pair created from vacuum. Thus, 𝜔0 = 2 𝐷2𝑛𝑛 state, 𝑋(1870) becomes a good 𝑛𝑛𝑔 candidate. In the (𝑅𝑅+𝐺𝐺+𝐵𝐵)/√3 𝜑(3,4) =(𝑢𝑢+𝑑𝑑+𝑠𝑠)/√3 following subsection, we will explore the two-body strong , 0 are color and 3 decays of 𝑋(1870) within the 𝑃 model and the flux-tube flavor singlets. The pair is also assumed to carry the quantum 0 0++ 3𝑃 model. Of course, the analysis of 𝑋(1870) also suits 𝜂2(1870) numbers of , suggesting that they are in a 0 state. Then (3,4) for their nearly equal masses. 𝜒1,−𝑚 represents the pair production in a spin triplet state. Advances in High Energy Physics 7

𝑚 ⃗ ⃗ 𝑚 𝛾 The solid harmonic polynomial Y1 (𝑘) ≡𝑘| | Y1 (𝜃𝑘,𝜙𝑘) Table 6: Values of in different channels and comparison with the 𝜌(770) 𝑎 (1320) 𝑓󸀠(1420) 𝑓 (1270) reflects the momentum-space distribution of the 𝑞3𝑞4. results given in [28]. Here, , 2 , 1 , 2 , 𝑀 ,𝑀 ,𝑀 𝑓󸀠(1525) 𝜌 (1690) The helicity amplitude M 𝐽𝐴 𝐽𝐵 𝐽𝐶 (𝑝) of 𝐴→𝐵+𝐶 2 and 3 have been studied. is given by Decay channels p (GeV) 𝛾 (103) 𝛾 [28] 󵄨 󵄨 3 𝑀 ,𝑀 ,𝑀 𝜌→𝜋𝜋 󵄨̂󵄨 ⃗ ⃗ ⃗ 𝐽𝐴 𝐽𝐵 𝐽𝐶 0.362 17.8 9.18 ⟨𝐵𝐶 󵄨T󵄨 𝐴⟩ = 𝛿 (𝑃𝐴 − 𝑃𝐵 − 𝑃𝐶) M (𝑝) , (21) 𝑎2 →𝜂𝜋 0.535 11.5 — 𝑓 →𝜋𝜋 where 𝑝 represents the momentum of the outgoing meson in 2 0.622 7.8 7.13 the rest frame of the meson 𝐴. When the mock state [67]is 𝜌3 →𝜋𝜋 0.833 4.2 — 󸀠 ∗ adoptedtodescribethespatialwavefunctionofameson,the 𝑓1 →𝐾𝐾 0.158 4.9 — 𝑀 ,𝑀 ,𝑀 M 𝐽𝐴 𝐽𝐵 𝐽𝐶 (𝑝) helicity amplitude can be constructed in the 𝑓2 →𝐾𝐾 0.401 2.9 6.11 𝐿−𝑆 𝐴 basis easily [62, 63]. The mock state for meson is 𝑎2 →𝐾𝐾 0.434 2.3 3.91 󵄨 𝑓󸀠 →𝐾𝐾 󵄨 2𝑆 +1 𝐽𝐴𝑀𝐽 2 0.579 2.0 5.66 󵄨𝐴(𝑛 𝐴 𝐿 𝐴 (𝑃⃗))⟩ 󵄨 𝐴 𝐴 𝐴 25 ≡ √2𝐸 ∑ ⟨𝐿 𝑀 𝑆 |𝐽 𝑀 ⟩𝜔12𝜙12𝜒12 𝐴 𝐴 𝐿𝐴 𝐴 𝐴 𝐽𝐴 𝐴 𝐴 𝑆𝐴𝑀𝑆 n: u, d quark 𝑀 𝑀 𝐴 𝐿𝐴 𝑆𝐴 20 nn →nn+nn 𝐿 𝑀 󵄨 ⃗ 𝐴 𝐿𝐴 ⃗ ⃗ 󵄨 ⃗ ⃗ × ∫ 𝑑𝑃 𝜓𝑛 (𝑘 , 𝑘 ) 󵄨𝑞 (𝑘 )𝑞 (𝑘 )⟩ . ) 15 𝐴 𝐴 1 2 󵄨 1 1 2 2 3 (22) 10 𝛾(p)(10 nn →ns+sn To obtain the analytical amplitudes, the SHO wave func- ss →ns+sn 𝐿 𝑀 𝜓 𝐴 𝐿𝐴 (𝑘⃗, 𝑘⃗) 5 tions are usually employed for 𝑛𝐴 1 2 .Forcompar- ison with experiments, one obtains the partial decay width 𝐽𝐿 M (𝑝) via the Jacob-Wick formula [68]: 0 0.2 0.4 0.6 0.8 √2𝐿 + 1 p (GeV) M𝐿𝑆 (𝑝) = ∑ ⟨𝐿0𝐽𝑀𝐽 |𝐽𝐴𝑀𝐽 ⟩ 2𝐽 +1 𝐴 𝐴 2 𝐴 𝑀 ,𝑀 Figure 5: The functions of 𝛾(𝑝) = 𝐴+𝐵 exp(−𝐶𝑝 ) in different decay 𝐽𝐵 𝐽𝐶 processes. The symbols of red “e”andblack“◼”denote𝛾 values 𝑀 ,𝑀 ,𝑀 ×⟨𝐽 ,𝑀 𝐽 ,𝑀 |𝐽𝑀 ⟩ M 𝐽𝐴 𝐽𝐵 𝐽𝐶 (𝑝) . determined by the experimental data. 𝐵 𝐽𝐵 𝐶 𝐽𝐶 𝐽𝐴 (23) 󸀠 Finally, the decay width Γ(𝐴 → 𝐵𝐶) is derived For example, values of 𝛾 fixed by 𝑎2 →𝐾𝐾 and 𝑓2 →𝐾𝐾 analytically in terms of the partial wave amplitudes are roughly equal. In the following calculations, we assume that the values 𝐸 𝐸 𝐵 𝐶 󵄨 󵄨2 of 𝛾 corresponding to the processes of 𝑠𝑠→𝑛𝑠+𝑠𝑛 and Γ (𝐴󳨀→𝐵𝐶) =2𝜋 𝑝∑󵄨M𝐿𝑆 (𝑝)󵄨 . (24) 𝑀𝐴 𝐿𝑆 𝑛𝑛→𝑛𝑠+𝑠𝑛 are determined by one function. Similarly, we 2 take the function, 𝛾(𝑝) = 𝐴 +𝐵 exp(−𝐶𝑝 ),forthecreation 3 More technical details of the 𝑃0 model can be found in vertex. The function of the creation vertex here is different from 3 [63]. The inherent uncertainties of the 𝑃0 decay model itself the one used in [28]. With the four decay channels listed in the fifth column of Table 6,wefixthefunctionas𝛾(𝑝) = have been discussed in [28, 69, 70]. 2 The dimensionless parameter 𝛾 will be fixed by the 8 well- 1.8 + 4 exp(−10𝑝 ). For the processes of 𝑛𝑛→𝑛𝑛+𝑛𝑛 (the measured partial decay widths which are listed in Table 6. first column of Table 6), we fix the creation vertex function 𝛾(𝑝) = 3.0 +25 (−4𝑝2) 𝛾 The M𝐿𝑆 amplitudes of these decay channels are presented as exp . The dependence of on the explicitly in Appendix A. momentum 𝑝 is plotted in Figure 5. Obviously the functions As mentioned before, 𝛾 may depend on the flavors of both can describe the dependence of 𝛾 and 𝑝 well. The functions the decaying meson and produced pairs. Then we divide the of creation vertex given here need further test. 8decaychannelsintotwogroups:oneis𝑛𝑛→𝑛𝑛+𝑛𝑛, Since we neglected the mass splitting within the isospin and the other includes 𝑠𝑠→𝑛𝑠+𝑠𝑛 and 𝑛𝑛→𝑛𝑠+𝑠𝑛. multiplet, the partial width into the specific charge channel The values of 𝛾 here are a little different from these given in should be multiplied by the flavor multiplicity factor F [28]wherean𝐴𝐿1 potential (for details of 𝐴𝐿1 potential, see (Table 7). This F factor also incorporates the statistical factor [35]) was selected to determine the meson wave functions. Of 1/2 if the final state mesons 𝐵 and 𝐶 are identical (as illustrated course, the meson wave function given by different potentials in Figure 6). More details of F can be found in the Appendix will influence the values of 𝛾. Aof[71]. It is clear in Table 6 that 𝛾 decrease with 𝑝 increase. In The partial decay widths of 𝑋(1870) are shown in Table 8 addition, our calculation indicates that 𝛾 depends on flavors except the channels of S+𝑃 mesons. 𝑎2(1320)𝜋 and 𝑓2(1275)𝜂 1 of both the decaying meson and the produced quark pairs. are large channels for the 𝜂2(2 𝐷2)𝑛𝑛 state in our work and 8 Advances in High Energy Physics

3 4 C C

1 2

2 2 2 3 A B A B

1 4 11 d1 d2

(a) (b)

3 Figure 6: Two topological diagrams for a 𝑞𝑞 meson decay in the 𝑃0 decay model. We refer to the left one as 𝑑1 where the produced quark goes into meson 𝐶 and 𝑑2 where it goes into 𝐵.

Table 7: The second and third columns for the flavor weight factors a (980) corresponding to two topological diagrams shown in Figure 6.The 0 last column for the flavor multiplicity factor F.Here,|𝜂⟩ = (|𝑛𝑛⟩ − |𝑠𝑠⟩)/√2 |𝜂󸀠⟩=(|𝑛𝑛⟩ + |𝑠𝑠⟩)/√2 and have been taken for simplicity. a0(qq) I (𝑑1) I (𝑑1) F X(1870) Decay channels flavor flavor 𝜌 → 𝜋𝜋√ +1/ 2−1/√2 1 𝜂(1295) √ √ 𝜋 𝑓2 →𝜋𝜋 −1/ 2−1/2 3/2 √ 𝑓2 →𝐾𝐾 0 −1/ 2 2 󸀠 ∗ 𝑓1 →𝐾𝐾 +1 0 4 󸀠 S +𝑃 𝑓2 →𝐾𝐾 +1 0 2 Figure 7: The diagram for the “ ” channels through a virtual 13𝑃 𝑞𝑞 𝑎2 →𝐾𝐾 0 −11intermediate 0 meson.

𝑎2 → 𝜂𝜋 +1/2 +1/2 1 √ √ 𝜂2 →𝜔𝜔 −1/ 2−1/2 1/2 √ √ 𝜂2 →𝑎𝑖𝜋−1/2−1/2 3 3 be canceled in the ratio of (Γ(𝑋(1870) →0 𝑎 (1 𝑃0)𝜋) ⋅ 𝜂2 →𝑓𝑖𝜂 +1/2 +1/2 1 3 3 𝜀(1 𝑃0(𝑞𝑞)→𝑎0(980)))/(Γ(𝜂(1295) →0 𝑎 (1 𝑃0)𝜋) ⋅ 3 𝜀(1 𝑃0(𝑞𝑞) →0 𝑎 (980))),thevalueof(Γ(𝑋(1870) → 𝑎0(980)𝜋))/(Γ(𝜂(1295)0 →𝑎 (980)𝜋)) canbeextracted [24], which are consistent with the experimental observations 𝜂 (1870) 𝐾∗𝐾 𝜌𝜌 𝜔𝜔 roughly. Although the assumption above seems a little rough, of the 2 .Thepartialwidthsof , ,and we just need to evaluate magnitudes of these decay channels. are narrower in our work than the expectations from [24]. 𝜂(1295) is proposed to be the first radial excited state of 𝜂2(1870) hasbeenobservedbytheBESCollaborationinthe 𝜂(550) Γ(𝑋(1870) → S +𝑃) 𝐽/𝜓 → 𝛾𝜂𝜋𝜋 . Then the total decay widths of radiative decay channel of [24]. However, no are evaluated no more than 12.6 MeV and Γ(𝑋(1870) → apparent 𝜂2(1870) signals were detected in the channels of 𝑎0(980)𝜋) ≤ 3.8 MeV. The BESIII Collaboration claimed that 𝐽/𝜓 → 𝛾𝜌𝜌 [72]and𝐽/𝜓 → 𝛾𝜔𝜔 [73, 74]. Therefore, 𝑋(1870) primarily decay via 𝑎0(980)𝜋 [1]. The small partial improved experimental measurements of the radiative 𝐽/𝜓 width of Γ(𝑋(1870) →0 𝑎 (980)𝜋) also indicates that the decay channels are needed for the 𝜂2(1870) in future. 𝑋(1870) 21𝐷 𝑞𝑞 S + can not be interpreted as the 2 state. Next, we will evaluate the partial widths of “ In addition, our results do not support 𝑋(1870) as the 𝑃” channels which have not been listed in Table 8. The 1 𝜂2(2 𝐷2)𝑛𝑛 statesinceitsobserveddecaywidthismuch scheme is proposed as follows. As illustrated in Figure 7, smaller than the theoretical estimate. The 𝑎2(1320)𝜋 is the we assume that 𝑋(1870) decays into 𝑎0(980)𝜋 via a virtual 3 largest decay channel in our numerical results and in [24] intermediate 1 𝑃0𝑞𝑞 meson. We notice that the 𝜂(1295) also 1 for the 𝜂2(2 𝐷2)𝑛𝑛 state (Table 8). If the partial width of dominantly decays into the 𝜂𝜋𝜋 [1]. Its three-body decay 𝑎0(980)𝜋 channelisaslargeas𝑎2(1320)𝜋, the predicted width can occur via three intermediate processes: 𝜂(1295) → 𝑋(1870) 𝜂𝜎/𝑎 (980)𝜋/𝜂(𝜋𝜋) →𝜂𝜋𝜋 of will be much larger than the observed value. 0 𝑆-wave [2]. With the ratio Γ(𝑎 (980)𝜋)/Γ(𝜂𝜋𝜋) = 0.65 ±0.10 Γ(𝜂(1295)) = 55 ± Weadoptthefluxtubemodeltocheckthepossibilityof 0 and 𝑋(1870) as a hybrid meson. The partial widths are also listed 5 MeV, the partial width of 𝜂(1295) decaying into 𝑎0(980)𝜋 is 3 in Table 8 for the comparison. Details of the flux model are estimatednomorethan45MeV.Bythe 𝑃0 model, the ratio 3 3 collected in Appendix B. of (Γ(𝑋(1870) →0 𝑎 (1 𝑃0)𝜋))/(Γ(𝜂(1295)0 →𝑎 (1 𝑃0)𝜋)) Two groups of the partial widths predicted in [16]are can be reached easily. If the uncertainty of the coupling vertex quoted in the Table 8. The left column was given by the flux 3 of 𝜀(1 𝑃0(𝑞𝑞) →0 𝑎 (980)) (see in Figure 7) is assumed to tube decay model of Isgur, Kokoski, and Paton (IKP) with Advances in High Energy Physics 9

Table 8: The partial widths of 𝑋(1870) and compared with results from [16, 24].

1 + −+ + ++ + −+ Decay 𝜂2(2 𝐷2)𝜂𝐻(0 0 )𝑓𝐻(0 1 )𝜂𝐻(0 2 ) channels Our [24] Our [16] Our [16] Our [16] 𝐾∗𝐾 0.5 17.7 19.3 12.6 10 5 4.9 24.1 18.0 3.2 2.0 1.0 𝜌𝜌 12.9 52.2 56.8 ××× × × × × × × 𝜔𝜔 4.2 16.9 18.4 ××× × × × × × × 𝐾∗𝐾∗ 0.2 2.1 2.3 ××× × × × × × ×

𝑎0(1450)𝜋 16.0 2.4 2.6 56.3 70 175 0.5 × 6 0.5 0.0 0.6

𝑎1(1260)𝜋 0.0 15.2 16.6 ×××57.3 14 232 × 0.3 ×

𝑓1(1280)𝜂 0.0 0.0 0.0 ×××2.5 — — × 0.0 ×

𝑎2(1320)𝜋 54.2 102.5 111.6 8.8 1 16 35.1 5.0 179.4 26.7 25.1 67

𝑓2(1275)𝜂 15.1 17.5 19.0 0.0 ×× 1.0 — — 4.6 0.0 0.0

∑ Γ𝑖 103.3 26.5 246.7 77.7 81 196 101.3 43.1 435.4 35.0 27.4 68.6 +19 Expt (MeV) 225 ± 14 [2] 57 ± 12−4 [1] The symbol “×” indicates that the decay modes are forbidden and “—” denotes that the decay channels can be ignored. Here, we collected the results given by 3 −+ ++ −+ the 𝑃0 model from [24] in the left column and the right column by the flux-tube model. In [16], the masses are taken as 1.8 GeV for the 0 , 1 ,and2 hybrid states. the “standard parameters” [75]. The right column was by the of 𝑎0(980)𝜋. However, this channel seems much small if 1 developed flux tube decay model of Swanson-Szczepaniak 𝜂2(1870) is a pure 2 𝐷2𝑛𝑛 meson. The mixing effect will also −+ ++ (SS). In [16],themassesaretakenas1.8GeVforthe0 , 1 , enlarge this partial width. Here, we do not plan to discuss −+ and 2 hybrid states. the mixing of 𝑋(1870) and 𝜂2(1870) further for the complex For a hybrid meson, 𝑋(1870) seems most possible to be mechanism. + −+ the 𝜂𝐻(0 2 ) state because the total widths which exclude the channels of S +𝑃are much narrow in our work and in [16]. It is consistent with the narrow width of 𝑋(1870). 4. Discussions and Conclusions As shown in Table 8, 𝑋(1870) is impossible to be the + −+ A isoscalar resonant structure of 𝑋(1870) was observed by 𝜂𝐻(0 0 ) hybrid state. The predicted width in both our work 𝐽/𝜓 → 𝜔𝑋(1870) →𝜔𝜂𝜋+𝜋− and in [16] is broader. In addition, 𝜂𝜋 is a visible channel BESIII in the channels recently. Although the mass of 𝑋(1870) is consistent with the for both 𝑎0(1450). A week signal was found in the region ± 𝜂 (1870) of 1200∼1400MeV in the analysis of 𝜂𝜋 (Figure 2(b) of 2 , the production, decay width, and decay properties are much different. In this paper, the mass spectrum and [1]), which contradicts the large 𝑎0(1450)𝜋 channel of the + −+ strong decays of the 𝑋(1870) and 𝜂2(1870) are analyzed. 𝜂𝐻(0 0 ) state. We can exclude the possibility of 𝑋(1870) + −+ Firstly, the mass spectrum is studied in the GI potential as the 𝜂𝐻(0 0 ) hybrid state preliminarily. + ++ model and the RTs framework. In the GI potential model, The assignment for 𝑋(1870) as the 𝑓𝐻(0 1 ) hybrid 󸀠 1 󸀠 3 both 𝑋(1870) and 𝜂2(1870) could be the 𝜂 (1 𝐷2), 𝑓 (2 𝑃1), seems impossible since the theoretical width of 𝑎1(1260)𝜋 is 2 1 𝜂 (21𝐷 ) rather broad in our results and in the IKP model. If the partial and 2 2 states. In RTs, the possible assignments are the 𝜂(31𝑆 ) 𝑓󸀠(23𝑃 ) 𝜂 (21𝐷 ) width of 𝑎0(980)𝜋 channel is as large as 𝑎1(1260)𝜋,thetotal 0 , 1 1 and 2 2 states. For the mass spectrum, widths of 𝑋(1870) will be much broader than the experimen- they are also good hybrid candidates since the masses overlap tal value. But the width given by the 𝑆𝑆 flux tube decay model the predictions given by different models (see Table 4). + ++ 𝑛𝑛 𝑛𝑛𝑔 for the 𝑓𝐻(0 1 ) hybrid is much small. So the possibility Secondly, the processes of a quarkonium or a + ++ S +𝑃 of 𝑋(1870) as a 𝑓𝐻(0 1 ) hybrid cannot be excluded. We hybrid meson decaying into the “ ”mesonsarestudied (3) suggest to detect the decay channel of 𝑎1(1260)𝜋 because this under the SU 𝑓 symmetry and the diquark-antidiquark + −+ S channel is forbidden for the 𝜂𝐻(0 2 ) state in the IKP flux description of the mesons. We assumed that the processes tube decay model and very small in the 𝑆𝑆 flux tube decay obey the OZI rule. We find that the channels of 𝑎0𝜋, 𝜎𝜂,and 𝑓 𝜂 𝑛𝑛 𝑛𝑛𝑔 model (see Table 8). Then the channel of 𝑎1(1260)𝜋 can dis- 0 are the dominant when a quarkonium or a hybrid + ++ + −+ criminate the states 𝑓𝐻(0 1 ) and 𝜂𝐻(0 2 ) for 𝑋(1870). meson decays primarily through this kind of processes. This 1 𝑋(1870) Finally, if 𝜂2(1870) is the 𝜂2(2 𝐷2) state, its decay width result can explain why hasbeenfirstobservedinthe is predicted about 100 MeV which is much smaller than the 𝜂𝜋𝜋 channel. Thirdly, the two-body strong decay of 𝑋(1870) is com- experiments. However, the difference can be explained by 3 1 the remedy of mixing effect. If 𝑋(1870) and 𝜂2(1870) have puted in the 𝑃0 model. As the 𝜂2(2 𝐷2) quarkonium, the + −+ the same quantum numbers, 0 2 , they should mix with predicted width of 𝑋(1870) looks much larger than the each other with a visible mixing angle. Then the interference observations. The broad resonance, 𝜂2(1870),canbeanatural 1 enhancement will enlarge the width of 𝜂2(1870).Thebroad candidate for the 2 𝐷2𝑛𝑛 meson. There, we fix the creation decay width of 𝜂2(1870) could be explained naturally. On strength, 𝛾,intwokindsofprocesses:(1) 𝑛𝑛→𝑛𝑛+𝑛𝑛; the other hand, 𝜂2(1870) has been observed in the channel (2) 𝑛𝑛→𝑛𝑠+𝑠𝑛 and 𝑠𝑠→𝑛𝑠+𝑠𝑛. The functions of creation 10 Advances in High Energy Physics

2 3 1 1 vertex are determined as 𝛾(𝑝) = 3.0 +25 exp(−4𝑝 ) and For 1 𝑆1 →1𝑆0 +1 𝑆0, 2 𝛾(𝑝) = 1.8 +4 exp(−10𝑝 ), respectively. Meanwhile, the SHO wave function scale, 𝛽s, is obtained by the GI potential model. 2𝜇 − ] M10 = 𝑝. We have evaluated the magnitude of the partial widths √ 5/4 5/2 3/2 (A.3) 8 3𝜋 𝜇 (𝛽𝐴𝛽𝐵𝛽𝐶) of “S +𝑃”channelsbytheratio,(Γ(𝑋(1870) → 𝑎0(980)𝜋))/(Γ(𝜂(1295)0 →𝑎 (980)𝜋)), under a rather crude 13𝑃 →11𝑆 +11𝑆 assumption that 𝜂(1295)/𝑋(1870)0 →𝑎 (980)𝜋 through For 2 0 0, 3 a virtual intermediate 1 𝑃0𝑞𝑞 meson (see Figure 7). Then 3 3/2 2 2 2 3/2 1 𝑃 (𝑞𝑞) → 2𝜇𝛽𝐵 −(𝑝 ] +2𝜇(1−𝑝 ])) 𝛽𝐶 the uncertainties of the coupling vertex for 0 M = . 20 5/2 3/2 (A.4) 𝑎0(980) areassumedtobecanceledintheratio.Thetotal √ 5/4 7/2 3 8 15𝜋 𝜇 𝛽𝐴 𝛽𝐵 𝛽𝐶 widths of “S +𝑃”areevaluatednomorethan12.6MeV and Γ(𝑋(1870) →0 𝑎 (980)𝜋) ≤ 3.8 MeV. Since 𝑋(1870) 3 3 1 For 1 𝑃2 →1𝑆1 +1 𝑆0, M21 =−√3/2M20. primarily decays via 𝑎0(980)𝜋, it also indicated that the 3 3 1 1 For 1 𝑃1 →1𝑆1 +1 𝑆0, 𝑋(1870) cannot be interpreted as the 2 𝐷2𝑛𝑛 state. We also study the 𝑋(1870) as a hybrid state in the flux 3/2 2 2 2 3/2 tube model. Our results agree well with most of predictions 4𝜇𝛽𝐵 +(𝑝 ] +2𝜇(1−𝑝 ])) 𝛽𝐶 𝑋(1870) 𝜂 (0+2−+) M01 = , given by [16]. looks most like the 𝐻 state 24𝜋5/4𝜇7/2𝛽5/2𝛽3/2𝛽3 for the narrow predicted width, which is consistent with the 𝐴 𝐵 𝐶 + ++ (A.5) experiments. But we cannot exclude the possibility of 0 1 . (2𝜇 − ]) ] 𝑎 (1260)𝜋 M = 𝑝2. A precise measurement of 1 is suggested to pin down 21 24√2𝜋5/4𝜇7/2𝛽5/2𝛽3/2𝛽3 this uncertainty. 𝐴 𝐵 𝐶 Finally, some important arguments and useful sugges- 1 13𝐷 →11𝑆 +11𝑆 tions are given as follows. (1) If 𝜂2(1870) is the 𝜂2(2 𝐷2) For 3 0 0, state, the broad 𝜋2(1880) should be isovector partner of 𝜂 (1870) 𝜋 (1880) 2𝜇 − ] 2 . 2 has been interpreted as the conventional M =− ]2𝑝3. 1 30 9/2 3/2 3/2 (A.6) 2 𝐷2 𝑞𝑞 𝜔𝜌 √ 5/4 9/2 mesonin[76]. Indeed, the decay channel of is 16 35𝜋 𝜇 𝛽𝐴 𝛽𝐵 𝛽𝐶 large enough for 𝜋2(1880) [77]. This observation disfavors 𝜋 (1880) 2−+ the 2 as a hybrid candidate for the selection rule 21𝑆 →13𝑃 +11𝑆 that a hybrid meson decaying into two 𝑆-wave mesons is For 0 0 0, (2) 𝑋(1870) strongly suppressed [78]. If is a hybrid meson, 1 wesuggesttosearchitsisospinpartnerinthedecaychannels M = 11 5/4 11/2 7/2 5/2 3/2 of 𝐽/𝜓→𝜌𝑓0(980)𝜋 and 𝐽/𝜓→𝜌𝑏1(1235)𝜋,whichare 96𝜋 𝜇 𝛽𝐴 𝛽𝐵 𝛽𝐶 accessible at BESIII, Belle, and BABAR Collaborations. The 1 ×(−24𝑝2𝜂𝜇3 −𝑝4]4 +2𝑝2𝜇]2 (−10 + 𝑝2 (1 + 𝜂) ]) decay channel of 𝑏1(1235)𝜋 is forbidden for the 𝜋2(2 𝐷2) quarkoniumduetothe“spinselectionrule”[60, 79]. We −4𝜇2 (15 − 5𝑝2 (1 + 𝜂) ] +𝑝4𝜂]2)+6𝜇2 also suggest to search the 𝜂2(1870) in the decay channels of 𝐽/𝜓 → 𝛾𝜌𝜌 𝐽/𝜓 → 𝛾𝜔𝜔 and since these channels are 2 2 2 2 2 2 forbidden for the hybrid production. × (4𝑝 𝜂𝜇 +𝑝 ] −2𝜇(−3 + 𝑝 (1+ 𝜂) ])) 𝛽𝐴). (A.7) Appendices 1 3 1 For 2 𝐷2 →1𝑆1 +1 𝑆0, A. The Expressions of Amplitudes M = ((−𝑝4]4 +2𝑝2]2𝜇(]𝑝2 − 14) + 28𝜇2 (]𝑝2 −5) We have omitted an exponential factor in the following decay 11 amplitudes M𝐿𝑆 for compactness: + 14𝜇2 (𝑝2]2 −2(]𝑝2 −5)𝜇𝛽2 )) 2 𝐴 2𝜆𝜇 − ] 2 (− 𝑝 ), −1 exp 4𝜇 (A.1) √ 5/4 13/2 11/2 3/2 3/2 ×(160 21𝜋 𝜇 𝛽𝐴 𝛽𝐵 𝛽𝐶 ) ) ]𝑝, where we defined M = ( (−28𝜇2 + ]3𝑝2 −2𝜇] (]𝑝2 −9) 1 1 1 1 31 𝜇= ( + + ), 2 2 2 2 2 2 𝛽𝐴 𝛽𝐵 𝛽𝐶 +14(2𝜇−])𝜇 𝛽 ) 𝑚 𝑚 𝐴 ] = 1 + 2 , (𝑚+ 𝑚 )𝛽2 (𝑚+ 𝑚 )𝛽2 √ 5/4 13/2 11/2 3/2 3/2 −1 2 3 1 𝐵 2 𝐶 × (160 14𝜋 𝜇 𝛽𝐴 𝛽𝐵 𝛽𝐶 ) )×] 𝑝 . 𝑚2 𝑚2 𝑚 𝜆= 1 + 2 ,𝜂=1 . (A.8) 2 2 2 2 𝑚+𝑚 (𝑚+ 𝑚 1) 𝛽𝐵 (𝑚+ 𝑚 2) 𝛽𝐶 1 1 3 3 󸀠 √ 󸀠 √ (A.2) For 2 𝐷2 →1𝑆1 +1 𝑆1, M11 = 2M11 and M31 = 2M31. Advances in High Energy Physics 11

1 3 1 1 3 1 For 2 𝐷2 →1𝑃0 +1 𝑆0 For 2 𝐷2 →2𝑃1 +1 𝑆0 1 M20 =− 192√35𝜋5/4𝜇15/2𝛽11/2𝛽5/2𝛽3/2 1 𝐴 𝐵 𝐶 M = 21 160√14𝜋5/4𝜇15/2𝛽11/2𝛽9/2𝛽3/2 ×(𝑝4]5 −2𝑝2𝜇]3 (−18 + 𝑝2 (1 + 𝜂) ]) 𝐴 𝐵 𝐶

3 2 2 2 3 +4𝜇2] (63 − 11𝑝2 (1 + 𝜂) ] +𝑝4𝜂]2) × (−112𝜂𝜇 + 252𝜇 ] + 56𝑝 𝜂 𝜇 ]

3 2 + 56𝜇 (−2 + 𝜂 (−2 + 𝑝 ])) − 80𝑝2𝜂𝜇2]2 + 36𝑝2𝜇]3

− 14𝜇2 (𝑝2]3 −2𝜇] (−7 + 𝑝2 (1 + 𝜂) ]) +4𝑝4𝜂2𝜇2]3 −4𝑝4𝜂𝜇]4 +𝑝4]5

2 2 2 2 +4𝜇 (−2 + 𝜂 (−2 + 𝑝 ]))) 𝛽𝐴) ]𝑝 . 2 2 2 2 − 10𝜇 ] (14𝜇 + 𝑝 ] )𝛽𝐵 (A.9) 1 3 1 2 2 2 2 2 2 3 For 2 𝐷2 →1𝑃1 +1 𝑆0, − 14𝜇 𝛽𝐴 (14𝜇] +4𝑝 𝜂 𝜇 ] +𝑝 ] 𝑝2]2 − 14𝜇2𝛽2 + 14𝜇 𝐴 2 2 2 2 M21 =− − 4𝜂𝜇 (2𝜇 +𝑝 ] ) − 10𝜇 ]𝛽 )) 16√35𝜋5/4𝜇11/2𝛽11/2𝛽5/2𝛽3/2 𝐵 𝐴 𝐵 𝐶 (A.10) ×(𝜂−1)]𝑝2. ×(𝜂−1)𝑝2.

1 3 1 (A.12) For 2 𝐷2 →1𝑃2 +1 𝑆0, 1 M02 = √ 5/4 15/2 11/2 5/2 3/2 𝑚1 and 𝑚2 are the masses of quarks in the decaying meson 480 14𝜋 𝜇 𝛽 𝛽𝐵 𝛽 𝐴 𝐶 𝐴. 𝑚 isthemassofthecreatedquarkfromthevacuum.For ×(𝑝6𝜇6 −2𝑝4𝜇]4 (𝑝2 (1 + 𝜂) ] − 21) calculating the decay widths, the masses of quarks are taken as 𝑚𝑢 =𝑚𝑑 = 0.220 GeV, 𝑚𝑠 = 0.428 GeV, which are the 2 2 2 2 4 2 +4𝑝 𝜇 ] (105 − 14𝑝 (1 + 𝜂) ] +𝑝 𝜂] ) same as those in Section 2. The above amplitudes, M𝐿𝑆,can be reduced further in the approximation of 𝑚1 =𝑚2 =𝑚 3 2 4 2 + 56𝜇 (15 − 5𝑝 (1 + 𝜂) ] +𝑝 𝜂] ) and 𝛽𝐴 =𝛽𝐵 =𝛽𝐶 =𝛽. The reduced M𝐿𝑆 are consistent with 9/2 those given by [71] except for an unimportant factor, −2 , − 14𝜇2 (𝑝4]4 −2𝑝2𝜇]2 (−10 + 𝑝2 (1 + 𝜂) ]) since this factor can be absorbed into the coefficient 𝛾.

+4𝜇2 (15 − 5𝑝2 (1 + 𝜂) ] +𝑝4𝜂]2)) 𝛽2 ), 𝐴 B. Hybrid Decay in the Flux Tube Model 1 M =− 22 672√5𝜋5/4𝜇15/2𝛽15/2𝛽5/2𝛽3/2 The flux tube model was motivated by the strong coupling 𝐴 𝐵 𝐶 expansion of the lattice QCD. In this model, decay occurs 4 5 2 3 2 when the flux-tube breaks at any point along its length, with ×(𝑝 ] −2𝑝 𝜇] (−18 + 𝑝 (1 + 𝜂) ]) 𝑃𝐶 ++ a 𝑞𝑞 pair production in a relative 𝐽 =0 state. It is similar 2 2 4 2 3 +2𝜇 ] (126 − 25𝑝 (1 + 𝜂) ] +2𝑝 𝜂] ) to the 𝑃0 decay model but with an essential difference. The flux tube model extends the nonrelativistic constituent quark + 28𝜇3 (−7 + 𝜂 (−7 + 2𝑝2])) model to include gluonic degrees of freedom in a very simple and intuitive way, where the gluonic field is regarded as tubes −14(𝜇2)(𝑝2]3 −2𝜇] (−7 + 𝑝2 (1 + 𝜂) ]) of color flux. Then it can be extended to the hybrid research. When the hybrid mesons are assumed to be narrow, and the 2 2 2 2 threshold effects are not taken into account, the partial decay +2𝜇 (−7 + 𝜂 (−7 + 2𝑝 ]))) 𝛽𝐴) ]𝑝 , width Γ𝐿𝑆(𝐻 → 𝐵𝐶) isgivenbythefluxmodelas[79] 1 M =− 42 1120𝜋5/4𝜇15/2𝛽11/2𝛽5/2𝛽3/2 𝐴 𝐵 𝐶 ̃ ̃ 𝑝 𝑀 𝑀 󵄨 󵄨2 Γ (𝐻󳨀→𝐵𝐶) = 𝐵 𝐶 󵄨M (𝐻󳨀→𝐵𝐶)󵄨 , 2 3 2 2 𝐿𝑆 ̃ 󵄨 𝐿𝑆 󵄨 ×(] (𝑝 ] − 36𝜇 +2𝜇] (11 − 𝑝 ])) (2𝐽𝐴 +1)𝜋 𝑀𝐴 (B.1) + 2𝜂𝜇 (28𝜇2 −𝑝2]3 +2𝜇] (𝑝2] −9))

−14𝜇2 (2𝜇 − ])(2𝜂𝜇−])𝛽2 ) ]2𝑝4. ̃ ̃ ̃ 𝐴 where 𝑀𝐴, 𝑀𝐵, 𝑀𝐶 are the “mock-meson” masses of 𝐴, (A.11) 𝐵, 𝐶 [64]. When a hybrid meson decays into 𝑃-wave and 12 Advances in High Energy Physics

̃ 𝑀̃𝐼=0 = 𝑀̃𝐼=1 = 1250 𝑀̃𝐼=0 = 720 𝑀̃𝐼=1 = Table 9: Partial wave amplitudes M𝐿(𝐻 → 𝐵𝐶) for an initial 𝐵 𝐵 MeV, 𝐶 MeV, 𝐶 2, hybrid 𝐻 decaying into a 𝑃-wave and pseudoscalar mesons. 850 MeV, 𝛽𝐴 = 0.27 GeV, 𝛿 = 0.62, 𝑏 = 0.18 GeV and 𝜅= ̃ + −+ + ++ + −+ 0.9.Finalstatescontaining𝜋 have 𝛽 = 0.36 GeV; otherwise BH(0 0 ) H(0 1 ) 0 2 H( ) 𝛽̃ = 0.40 0++ +√2M /3 −√2M /√3 +M /3 GeV. 𝑆 𝑃2 𝐷 1++ −M /√2 — 𝑃1 — 2++ +M /3 −M /√30 −√5M /√18 Acknowledgments 𝐷 𝑃4 𝑆 +M /√5 √ 𝐹 − 7M𝐷/3 Bing Chen thanks Jun-Long Tian and D. V. Bugg for very helpful discussions. This work is supported by the Key Program of the He’nan Educational Committee of China M (𝐻 → pseudoscalar mesons, the partial wave amplitude 𝐿 (nos. 13A140014 and 12B140001), the National Natural Science 𝐵𝐶) 𝑆=𝑆 (with 𝐵)isgivenasthefollowingform: Foundation of China under Grant no. 11305003, 11075102, M (𝐻󳨀→𝐵𝐶) 11005003, and U1204115, the Innovation Program of Shanghai 𝐿 Municipal Education Commission under Grant no. 13ZZ066, andtheProgramofHe’nanTechnologyDepartment(no. 𝑎𝑐̃ 1 𝑓𝑏 𝜅√𝑏 =⟨𝜙 𝜙 |𝜙 𝜙 ⟩( 𝐴0 √ ) 11147201). 𝐵 𝐶 𝐴 0 00 2 9√3 2 𝜋 (1 + 𝑓𝑏/(2𝛽̃2)) References 2𝜋 𝛽3/2+𝛿 𝐴 ̃ + − × M𝐿 (𝐻󳨀→𝐵𝐶) . [1] M. Ablikim, M. N. Achasov, D. Alberto et al., “𝜂𝜋 𝜋 Resonant √ 3 ̃ 2 + − 3Γ ( +𝛿) 𝛽 Structure around 1. 8 GeV/c and 𝜂(1405) in 𝐽/𝜓 → 𝜔𝜂𝜋 𝜋 ,” 2 Physical Review Letters,vol.107,no.18,ArticleID182001,2011. (B.2) [2] J. Beringer, J. F. Arguin, R. M. Barnett et al., “Review of particle physics,” Physical Review D,vol.86,no.1,ArticleID010001, The flavor matrix element ⟨𝜙𝐵𝜙𝐶 |𝜙𝐴𝜙0⟩ has been 2012. ̃ discussed before. M𝐿(𝐻 → 𝐵𝐶) are listed in Table 9 for the [3]K.Karch,D.Antreasyan,H.W.Bartelsetal.,“Analysisofthe + −+ + ++ + −+ 0 0 states of 𝜂𝐻(0 0 ), 𝑓𝐻(0 1 ),and𝜂𝐻(0 2 ). 𝜂𝜋 𝜋 final state in photon-photon collisions,” Zeitschriftur f¨ M M M M M = Here the 𝑆, 𝐷, 𝑃𝑖 ,and 𝐹 are defined as 𝑆 Physik C,vol.54,no.1,pp.33–44,1992. −(3̃ℎ −𝑔̃ +4̃ℎ ) M =(𝑔̃ +5̃ℎ ) M =−𝑖(2𝑔̃ +3̃ℎ −𝑔̃ ) [4] M. Feindt, DESY-90-128 (1990), to appear in Proceedings of 0 1 2 , 𝐷 1 2 , 𝑃1 0 1 2 , M =−𝑖(𝑔̃ + 𝑔̃ ) M =−𝑖(10𝑔̃ +9̃ℎ + 𝑔̃ ) M = the 25th International Conference on High Energy Physics, 𝑃2 0 2 , 𝑃4 0 1 2 ,and 𝐹 Singapore 1990. ̃ ̃ −3𝑖(𝑔̃2 + ℎ3). The analytical expressions of 𝑔̃𝑖 and ℎ𝑖 are given [5]J.Adomeit,C.Amsler,D.S.Armstrongetal.,“Evidencefortwo 𝑃𝐶 −+ as isospin zero 𝐽 =2 mesons at 1645 and 1875 MeV,” Zeitschrift 𝑛 fur¨ Physik C,vol.71,no.1,pp.227–238,1996. 𝑀 𝑚 −(𝑛+𝛿+3)/2 𝑛+𝛿+3 𝑃𝐶 𝑔̃ =23+𝛿 (2𝛽2 + 𝛽̃2) Γ( ) [6]A.V.Anisovich,C.A.Baker,C.J.Battyetal.,“Three𝐼=0𝐽 = 𝑛 𝑛+1 𝐴 −+ (𝑀+𝑚) 2 1 2 mesons,” Physics Letters B,vol.477,no.1–3,pp.19–27,2000. 𝜂𝜋+𝜋− 2 [7] D.Barberis,F.G.Binon,F.E.Closeetal.,“Astudyofthe 𝑛+𝛿+3 𝑀 2 channel produced in central pp interactions at 450 GeV/c ,” ×𝐹1 [ ,𝑛+1,−( ) 2 𝑀+𝑚 Physics Letters B,vol.471,no.4,pp.435–439,2000.

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Research Article Maxwell’s Equations on Cantor Sets: A Local Fractional Approach

Yang Zhao,1,2 Dumitru Baleanu,3,4,5 Carlo Cattani,6 De-Fu Cheng,1 and Xiao-Jun Yang7

1 College of Instrumentation & Electrical Engineering, Jilin University, Changchun 130061, China 2 Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China 3 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia 4 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey 5 Institute of Space Sciences, Magurele, 077125 Bucharest, Romania 6 Department of Mathematics, University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy 7 Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China

Correspondence should be addressed to De-Fu Cheng; [email protected]

Received 3 October 2013; Accepted 7 November 2013

Academic Editor: Gongnan Xie

Copyright © 2013 Yang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Maxwell’s equations on Cantor sets are derived from the local fractional vector calculus. It is shown that Maxwell’s equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Local fractional differential forms of Maxwell’s equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are obtained. Maxwell’s equations on Cantor set with local fractional operators are the first step towards a unified theory of Maxwell’s equations for the dynamics of cold dark matter.

1. Introduction electric potential was developed. Recently, based on the Hausdorff derivative, fractal continuum electrodynamics in 4𝛼 4 Nondifferentiability, complexity, and similarity represent the time-space Ω ⊂Ωwas proposed [18]. The fractional basic properties of the nature. Fractals [1]arethebasic differential form of Maxwell’s equations on fractal sets was characteristics of nature, which are that fractal geometry suggested in [19]. In [20], the Maxwell equations of fractional of substances generalizes to noninteger dimensions. Micro- Ω3+𝛼 ⊂Ω4 physics reveals the fractal behaviors of matter distribution in electrodynamics in time-space were considered. theuniverse[2]andsoftmaterials[3]. The Maxwell equations on anisotropic fractal media in time- Ω3𝛼+1 ⊂Ω4 Fractal time was used to describe the transport of charges space were developed in [21]. and defects in the condensed matter [4]. In fractal space-time, The local fractional calculus theory [22, 23]wasapplied the geometric analogue of relativistic quantum mechanics to model some dynamics systems with nondifferentiable was presented in [5–8]. In fractal-Cantorian space-time characteristics. In [22–26], the heat-conduction equation 4𝛼 4 Ω ⊂Ω [9, 10], the unified field theory, quantum physics, on Cantor sets was considered. In [27], the Navier-Stokes cosmology, and chaotic systems were discussed in [11–13]. equations on Cantor sets based on local fractional vector Basedonthefractaldistributionofchargedparticles,the calculus were proposed. Helmholtz and diffusion equations 3𝛼+1 4 electric and magnetic fields in time-space Ω ⊂Ω were via local fractional vector calculus were reported in [28]. The developedin[14]andfractionalMaxwell’sequationswere Fokker-Planck equation with local fractional space derivative proposed in [15]. In [16, 17], the concept of static fractional was suggested in [29]. In [30], Lagrangian and Hamiltonian 2 Advances in High Energy Physics mechanics with local fractional space derivative was pre- 1 sented. The measuring structures of time in fractal, fractional, 0.9 classical, and discrete electrodynamics are shown in Figure 1. The aim of this paper is to structure Maxwell’s equations 0.8 on Cantor sets from the local fractional calculus theory [23, 0.7 27, 28] point of view. This paper is structured as follows. In 0.6 Section 2, we introduce the basic definitions and theorems for local fractional vector calculus. In Section 3,Maxwell’s ta 0.5 equations on Cantor sets in the local fractional vector integral 0.4 form are presented. Maxwell’s equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are given 0.3 in Section 4.Finally,Section 5 is devoted to conclusions. 0.2 0.1 2. Fundaments 0 0 0.2 0.4 0.6 0.8 1 In this section, we recall the basic definitions and theorems t for local fractional vector calculus, which are used throughout the paper. Fractal space Local fractional gradient of the scale function 𝜙 is defined Fractional space as [23, 27] Euclidean space 1 ∇𝛼𝜙= ( ∯ 𝜙𝑑S(𝛽)) , Figure 1: Graph for comparison of the measuring structures of time lim (𝛾) (1) 𝑑𝑉(𝛾) →0 𝑑𝑉 𝑆(𝛽) in fractal, fractional, and classical electrodynamics. (𝛽) where S is its bounding fractal surface, 𝑉 is a small fractal 𝑃 (𝛼) (𝛼) volume enclosing , and the local fractional surface integral with the elements of line Δl𝑃 requiring that all |Δ𝑙𝑃 |→0 is given by [23, 27, 28] as 𝑁→∞and 𝛽=2𝛼. 𝑁 The local fractional Gauss theorem of the fractal vector (𝛽) (𝛽) field states that [23, 27] ∬ 𝑢(𝑟𝑃)𝑑S = lim ∑𝑢(𝑟𝑃) n𝑃Δ𝑆 , (2) 𝑁→∞ 𝑃 𝑃=1 𝛼 (𝛾) (𝛽) ∭ ∇ ⋅ u𝑑𝑉 = ∯ u ⋅𝑑S , (7) with 𝑁 elements of area with a unit normal local fractional 𝑉(𝛾) 𝑆(𝛽) (𝛽) vector n𝑃, Δ𝑆𝑃 →0as 𝑁→∞for 𝛾 = (3/2)𝛽 =3𝛼,and u 𝛼 where the local fractional volume integral of the function is ∇ is denoted as the local fractional Laplace operator [22, 23]. written as [23] u The local fractional divergence of the vector function is 𝑁 defined through [23] (𝛾) (𝛾) ∭ u (𝑟𝑃)𝑑𝑉 = lim ∑u (𝑟𝑃)Δ𝑉 , (8) 𝑁→∞ 𝑃 1 𝑃=1 ∇𝛼 ⋅ u = ( ∯ u ⋅𝑑S(𝛽)) , lim (𝛾) (3) 𝑑𝑉(𝛾) →0 𝑑𝑉 𝑆(2𝛼) (𝛾) with the elements of volume Δ𝑉𝑃 →0as 𝑁→∞and where the local fractional surface integral is suggested by [23, 𝛾 = (3/2)𝛽 =3𝛼. 27, 28] The local fractional Stokes theorem of the fractal field states that [23, 27] 𝑁 (𝛽) (𝛽) ∬ u (𝑟𝑃)⋅𝑑S = lim ∑u (𝑟𝑃)⋅n𝑃Δ𝑆 , (4) (𝛼) 𝛼 (𝛽) 𝑁→∞ 𝑃 ∮ u ⋅𝑑l = ∬ (∇ × u)⋅𝑑S . (9) 𝑃=1 𝑙(𝛼) 𝑆(𝛽) with 𝑁 elements of area with a unit normal local fractional The Reynolds transport theorem of the local fractional vector (𝛽) vector n𝑃, Δ𝑆𝑃 →0as 𝑁→∞for 𝛾 = (3/2)𝛽 =3𝛼. field states that [27] The local fractional curl of the vector function u [23]is 𝐷𝛼 ∭ 𝐹 (𝑥,) 𝑡 𝑑𝑉(𝛾) defined as follows: 𝛼 𝐷𝑡 𝑉(𝛾) 𝛼 1 (𝛼) ∇ × u = ( ∮ u ⋅𝑑l ) n , 𝛼 (10) lim (𝛽) 𝑃 (5) 𝜕 𝑑𝑆(𝛽) →0 𝑑𝑆 𝑙(𝛼) = ∭ 𝐹 (𝑥,) 𝑡 𝑑𝑉(𝛾) + ∯ 𝐹 (𝑥,) 𝑡 𝜐 ⋅𝑑S(𝛽), 𝛼 𝑉(𝛾) 𝜕𝑡 𝑆(𝛽) where the local fractional line integral of the function u along 𝛼 𝜐 afractalline𝑙 is given by [23] where is the fractal fluid velocity.

(𝛼) ∫ u (𝑥𝑃,𝑦𝑃,𝑧𝑃)⋅𝑑l 3. Local Fractional Integral Forms of 𝑙(𝛼) Maxwell’s Equations on Cantor Sets 𝑁 (6) (𝛼) = lim ∑u (𝑥𝑃,𝑦𝑃,𝑧𝑃)⋅Δl , 𝑁→∞ 𝑃 According to fractional complex transform method [31], the 𝑃=1 fact that the classical differential equations always transform Advances in High Energy Physics 3 into the local fractional differential equations leads to the idea 3.2.1. Gauss’s Law for the Fractal Electric Field. From (3), the of yielding Maxwell’s equations on Cantor sets using the local electric charge density can be written as fractional vector calculus. 1 ∇𝛼 ⋅𝐷= ( ∯ 𝐷⋅𝑑S(𝛽)) =𝜌, lim (𝛾) (18) 𝑑𝑉(𝛾) →0 𝑑𝑉 𝑆(2𝛼) 3.1. Charge Conservations in Local Fractional Field. Let us consider the total charge, which is described as follows: where 𝐷 is electric displacement in the fractal electric field. From (7), (18)becomes 𝑄=∭ 𝜌 (𝑟,) 𝑡 𝑑𝑉(𝛾), (11) 𝛼 (𝛾) (𝛽) 𝑉(𝛾) ∭ ∇ ⋅𝐷𝑑𝑉 = ∯ 𝐷⋅𝑑S , 𝑉(𝛾) 𝑆(𝛽) and the total electrical current is as follows (19) ∭ ∇𝛼 ⋅𝐷𝑑𝑉(𝛾) = ∭ 𝜌𝑑𝑉(𝛾). (𝛽) 𝐼=∬ 𝐽 (𝑟,) 𝑡 ⋅𝑑S , (12) 𝑉(𝛾) 𝑉(𝛾) 𝑆(𝛽) Hence, we obtain Gauss’s law for the fractal electric field where 𝜌(𝑟, 𝑡) is the fractal electric charge density and 𝐽(𝑟, 𝑡) is in the form the fractal electric current density. (𝛽) (𝛾) The Reynolds transport theorem in the fractal field gives ∯ 𝐷⋅𝑑S = ∭ 𝜌𝑑𝑉 . (20) 𝑆(𝛽) 𝑉(𝛾) 𝐷𝛼𝑄 𝐷𝛼 = ∭ 𝜌 (𝑟,) 𝑡 𝑑𝑉(𝛾) 𝛼 𝛼 3.2.2. Ampere’s Law in the Fractal Magnetic Field. Mathe- 𝐷𝑡 𝐷𝑡 𝑉(𝛾) matically, Ampere’s law in the fractal magnetic field can be 𝜕𝛼𝜌 (𝑟,) 𝑡 = ∭ 𝑑𝑉(𝛾) + ∯ 𝜌 (𝑟,) 𝑡 𝜐 ⋅𝑑S(𝛽) (13) suggested as [23] 𝛼 𝑉(𝛾) 𝜕𝑡 𝑆(𝛽) (𝛼) ∮ 𝐻⋅𝑑l =𝐼, (21) =0, 𝑙(𝛼) where 𝐻 is the magnetic field strength in the fractal field. which leads to The current density in the fractal field can be written as 𝛼 𝜕 𝜌 (𝑟,) 𝑡 (𝛾) (𝛽) ∭ 𝑑𝑉 + ∯ 𝜌 (𝑟,) 𝑡 𝜐 ⋅𝑑S =0. 𝛼 1 (𝛼) 𝛼 (14) ∇ ×𝐻= ( ∮ 𝐻⋅𝑑l ) n =𝐽, 𝑉(𝛾) 𝜕𝑡 𝑆(𝛽) lim (𝛽) 𝑃 (22) 𝑑𝑆(𝛽) →0 𝑑𝑆 𝑙(𝛼) From (14), we have which leads to 𝜕𝛼𝜌 (𝑟,) 𝑡 ∭ 𝑑𝑉(𝛾) + ∯ 𝐽 (𝑟,) 𝑡 ⋅𝑑S(𝛽) ∮ 𝐻⋅𝑑l(𝛼) = ∬ (∇𝛼 ×𝐻)⋅𝑑S(𝛽), 𝛼 𝑉(𝛾) 𝜕𝑡 𝑆(𝛽) 𝑙(𝛼) 𝑆(𝛽) (23) 𝛼 𝜕 𝜌 (𝑟,) 𝑡 (𝛽) 𝛼 (𝛽) = ∭ ( +∇𝛼 ⋅𝐽(𝑟,) 𝑡 )𝑑𝑉(𝛾) (15) ∬ 𝐽⋅𝑑S = ∬ (∇ ×𝐻)⋅𝑑S . 𝛼 (𝛽) (𝛽) 𝑉(𝛾) 𝜕𝑡 𝑆 𝑆 =0, The total current in the fractal fried reads (𝛽) 𝐽=𝜌𝜐 𝐼=∬ (𝐽𝑎 +𝐽𝑏)⋅𝑑S , (24) where represents the current density in the fractal 𝑆(𝛽) field. 𝐽 Hence, from (15), we get where 𝑎 is the conductive current and 𝛼 𝛼 𝜕 𝐷 𝜕 𝜌 (𝑟,) 𝑡 𝛼 𝐽 = , (25) +∇ ⋅𝐽(𝑟,) 𝑡 =0. (16) 𝑏 𝜕𝑡𝛼 𝜕𝑡𝛼 which satisfies the following condition: By analogy with electric charge density in the fractal field, we obtain the conservation of fractal magnetic charge, (𝛽) 𝐼𝑏 = ∯ 𝐽𝑏 ⋅𝑑S namely, 𝑆(𝛽) 𝛼 𝛼 𝜕 𝜌 𝑟, 𝑡 𝜕 (𝛽) 𝑚 ( ) 𝛼 = ∯ 𝐷⋅𝑑S +∇ ⋅𝐽 (𝑟,) 𝑡 =0, (17) 𝛼 (26) 𝜕𝑡𝛼 𝑚 𝑆(𝛽) 𝜕𝑡 𝜕𝛼 where 𝜌𝑚(𝑟, 𝑡) is the fractal magnetic charge density and = ∯ 𝐷⋅𝑑S(𝛽). 𝛼 𝐽𝑚(𝑟, 𝑡) is the fractal magnetic current density in the fractal 𝜕𝑡 𝑆(𝛽) field. Hence, Ampere’s law in the fractal field is expressed as follows: 3.2. Formulation of Maxwell’s Equations on Cantor Sets. We 𝜕𝛼𝐷 now derive Maxwell’s equations on Cantor set based on the ∮ 𝐻⋅𝑑l(𝛼) = ∬ (𝐽 + )⋅𝑑S(𝛽). 𝑎 𝛼 (27) local fractional vector calculus. 𝑙(𝛼) 𝑆(𝛽) 𝜕𝑡 4 Advances in High Energy Physics

3.2.3. Faraday’s Law in the Fractal Electric Field. Mathemat- The Cantor-type cylindrical coordinates can be written as ically, Faraday’s law in the local fractional field is expressed follows [26]: as 𝛼 𝛼 𝛼 𝛼 𝜕 𝜑 𝑥 =𝑅 cos𝛼𝜃 , 𝐸=− , (28) 𝜕𝑡𝛼 𝛼 𝛼 𝛼 𝑦 =𝑅 sin𝛼𝜃 , (35) where 𝜑 is the magnetic potential in the fractal field and 𝐸 is 𝑧𝛼 =𝑧𝛼, theelectricalfieldstrengthinthefractalfield. From (28), we have 2𝛼 2𝛼 1 with 𝑅∈(0,+∞), 𝑧∈(−∞,+∞), 𝜃 ∈ (0, 𝜋],and𝑥 +𝑦 = ∇𝛼 ×𝐸= ( ∮ 𝐸⋅𝑑l(𝛼)) n 2𝛼 lim (𝛽) 𝑃 𝑅 𝑑𝑆(𝛽) →0 𝑑𝑆 𝑙(𝛼) . (29) Making use of (35), we have 𝜕𝛼 𝜕𝛼𝐵 =− (∇𝛼 ×𝜑)=− , 𝜕𝑡𝛼 𝜕𝑡𝛼 𝜕𝛼𝑟 1 𝜕𝛼𝑟 𝑟 𝜕𝛼𝑟 ∇𝛼 ⋅ r = 𝑅 + 𝜃 + 𝑅 + 𝑧 , 𝛼 𝛼 𝛼 𝛼 𝛼 where 𝐵 is the magnetic induction in the fractal field. 𝜕𝑅 𝑅 𝜕𝜃 𝑅 𝜕𝑧 In view of (9), we rewrite (29)as 1 𝜕𝛼𝑟 𝜕𝛼𝑟 ∇𝛼 × r =( 𝜃 − 𝜃 ) e𝛼 𝑅𝛼 𝜕𝜃𝛼 𝜕𝑧𝛼 𝑅 ∮ 𝐸⋅𝑑l(𝛼) = ∬ (∇𝛼 ×𝐸)⋅𝑑S(𝛽), 𝑙(𝛼) 𝑆(𝛽) 𝛼 𝛼 (36) 𝜕 𝑟𝑅 𝜕 𝑟𝑧 𝛼 𝛼 +( − ) e 𝜕 𝐵 𝛼 𝛼 𝜃 ∬ (∇𝛼 ×𝐸)⋅𝑑S(𝛽) = ∬ (− )⋅𝑑S(𝛽) 𝜕𝑧 𝜕𝑅 𝛼 (30) 𝑆(𝛽) 𝑆(𝛽) 𝜕𝑡 𝛼 𝛼 𝜕 𝑟𝜃 𝑟𝑅 1 𝜕 𝑟𝑅 𝛼 𝛼 +( + − ) e , 𝜕 𝛼 𝛼 𝛼 𝛼 𝑧 =− ∬ 𝐵⋅𝑑S(𝛽). 𝜕𝑅 𝑅 𝑅 𝜕𝜃 𝛼 𝜕𝑡 𝑆(𝛽) So, from (30), Faraday’s law in the fractal field reads as where 𝛼 𝛼 𝛼 𝛼 𝛼 𝛼 𝛼 𝛼 𝛼 (𝛼) 𝜕 (𝛽) r =𝑅 𝜃 e +𝑅 𝜃 e +𝑧 e ∮ 𝐸⋅𝑑l + ∬ 𝐵⋅𝑑S =0. (31) cos𝛼 1 sin𝛼 2 3 (𝛼) 𝛼 (𝛽) 𝑙 𝜕𝑡 𝑆 𝛼 𝛼 𝛼 (37) =𝑟𝑅e𝑅 +𝑟𝜃e𝜃 +𝑟𝑧e𝑧 . 3.2.4. Magnetic Gauss’s Law for the Fractal Magnetic Field. From (3), we derive the local fractional divergence of the From (7)and(20), the local fractional differential form of magnetic induction in the fractal field, namely, Gauss’s law for the fractal electric field is expressed by 1 ∇𝛼 ⋅𝐵= ( ∯ 𝐵⋅𝑑S(𝛽))=0. 𝛼 lim (𝛾) (32) ∇ ⋅𝐷=𝜌, (38) 𝑑𝑉(𝛾) →0 𝑑𝑉 𝑆(2𝛼)

Furthermore, the magnetic Gauss’ law for the fractal which leads to magnetic field reads as 𝜕𝛼𝐷 1 𝜕𝛼𝐷 𝐷 𝜕𝛼𝐷 (𝛽) 𝑅 + 𝜃 + 𝑅 + 𝑧 =𝜌(𝑅, 𝜃,) 𝑧,𝑡 , ∯ 𝐵⋅𝑑S =0. (33) 𝛼 𝛼 𝛼 𝛼 𝛼 (39) 𝑆(𝛽) 𝜕𝑅 𝑅 𝜕𝜃 𝑅 𝜕𝑧

𝐷=𝐷 e𝛼 +𝐷 e𝛼 +𝐷 e𝛼 3.2.5. The Constitutive Equations in the Fractal Field. Similar where 𝑅 𝑅 𝜃 𝜃 𝑧 𝑧 . to the constitutive relations in fractal continuous medium In view of (9)and(27), we can present the local fractional mechanics [23], the constitutive relationships in fractal elec- differential forms of Ampere’s law in the fractal magnetic field tromagnetic can be written as 𝜕𝛼𝐷 𝐷=𝜀 𝐸, ∇𝛼 ×𝐻=𝐽 + , 𝑓 𝑎 𝜕𝑡𝛼 (34) 𝛼 𝛼 𝛼 𝛼 𝐻=𝜇𝑓𝐵, 1 𝜕 𝐻 𝜕 𝐻 𝜕 𝐻 𝜕 𝐻 ( 𝜃 − 𝜃 ) e𝛼 +( 𝑅 − 𝑧 ) e𝛼 𝛼 𝛼 𝛼 𝑅 𝛼 𝛼 𝜃 𝜀 𝜇 𝑅 𝜕𝜃 𝜕𝑧 𝜕𝑧 𝜕𝑅 where 𝑓 is the fractal dielectric permittivity and 𝑓 is the (40) fractal magnetic permeability. 𝜕𝛼𝐻 𝐻 1 𝜕𝛼𝐻 +( 𝜃 + 𝑅 − 𝑅 ) e𝛼 𝜕𝑅𝛼 𝑅𝛼 𝑅𝛼 𝜕𝜃𝛼 𝑧 4. Local Fractional Differential Forms of 𝜕𝛼𝐷 (𝑅, 𝜃,) 𝑧,𝑡 Maxwell’s Equations on Cantor Sets =𝐽 (𝑅, 𝜃,) 𝑧,𝑡 + , 𝑎 𝜕𝑡𝛼 In this section, we investigate the local fractional differential 𝛼 𝛼 𝛼 forms of Maxwell’s equations on Cantor sets. where 𝐻=𝐻𝑅e𝑅 +𝐻𝜃e𝜃 +𝐻𝑧e𝑧 . Advances in High Energy Physics 5

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Research Article Transverse Momentum Distributions of Final-State Particles Produced in Soft Excitation Process in High Energy Collisions

Fu-Hu Liu, Ya-Hui Chen, Hua-Rong Wei, and Bao-Chun Li

Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China

Correspondence should be addressed to Fu-Hu Liu; [email protected]

Received 17 July 2013; Revised 5 September 2013; Accepted 20 September 2013

Academic Editor: Gonang Xie

Copyright © 2013 Fu-Hu Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Transverse momentum distributions of final-state particles produced in soft process in proton-proton (pp) and nucleus-nucleus (AA) collisions at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies are studied by using a multisource thermal model. Each source in the model is treated as a relativistic and quantum ideal gas. Because the quantum effect can be neglected in investigation on the transverse momentum distribution in high energy collisions, we consider onlythe relativistic effect. The concerned distribution is finally described by the Boltzmann or two-component Boltzmann distribution. Our modeling results are in agreement with available experimental data.

1. Introduction system and give a judgment on formation and property of the new matters. High energy collisions are an important research topic in The final-state particle spectra include rapidity 𝑦 (or particle and nuclear physics. The Relativistic Heavy Ion pseudorapidity 𝜂)[8, 9], transverse momentum 𝑝𝑇 (or trans- Collider (RHIC) at Brookhaven National Laboratory (BNL) verse mass) [10, 11], transverse energy [12, 13], and other did firstly collider experiments on heavy ions [1], and the distributions [14]. It is known that 𝑦 and 𝑝𝑇 distributions ( 𝑠 ) center-of-mass energy per nucleon pair √ 𝑁𝑁 at the RHIC reflect, respectively, the degrees of longitudinal extension reached highly 200 GeV [2]. The Large Hadron Collider and transverse excitation of interacting system. Especially, (LHC) at European Laboratory for Particle Physics (CERN) for transverse excitation, soft excitation and hard scattering 𝑠 renovated value of √ 𝑁𝑁 to TeV region [3]. It seems that a processes can affect, respectively, distributions in low- and new state of matter, namely, Quark-Gluon Plasma (QGP), is high-𝑝𝑇 ranges.Thesoftandhardprocessescorrespond possibly formed in heavy ion collisions at RHIC and LHC to different physics mechanisms and distribution laws [15]. energies due to high temperature and density [4, 5]. At initial In low energy collisions, the soft process is main process, stage of high energy collisions, another possible new state of and the hard process can be neglected due to almost zero matter, namely, color glass condensate (CGC), is caused by contribution. In high energy collisions, although the hard strong color fields in the low-x gluon realm [6, 7], where x process cannot be neglected, the soft process is still main denotes the ratio of quark or gluon momentum to hadron process. one. A CGC is in fact a region of the nuclear wave function at 2 To understand the transverse excitation, we need firstly to low-x and Q and exists already before the collisions, where study the soft excitation process. A lot of models have been 2 Q denotes the square momentum of virtual photon. On introduced to describe the soft process, although some of the other hand, the CGC may not be a new state, but more them can be used to describe the hard process too [16, 17]. like a model or calculation for initial state hadron behavior. Among the models, the multisource thermal model proposed One cannot measure the QGP and CGC directly. However, by us is a very simple one and can be used to describe one can measure final-state particle spectra at freeze-out 𝑝𝑇 spectra in both the soft and hard processes if source’s to extract thermal and other characteristics of interacting contribution is given by an Erlang distribution [18]. Finally, 2 Advances in High Energy Physics

10 10 pp, 200 GeV |y| < 0.1 |y| < 0.1 ) 1 ) 1 −2 −2 ((GeV/c) ((GeV/c) dy dy T 10−1 T 10−1 N/dp N/dp 2 2 )d )d T T

10−2 10−2 (1/2𝜋p (1/2𝜋p

10−3 10−3 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2

pT (GeV/c) pT (GeV/c)

𝜋− 𝜋+

K− K+ p p

(a) (b) 1 1 1 pp, 200 GeV, 𝜋+ K+ p (e) 3.00 ± 0.10, ×0.01 10−1 y = (a) 0.00 ± 0.10 10−1 (c) 1.20 ± 0.10, ×0.1 10−1 (f) (b) 0.90 ± 0.10, ×0.3 (d) 2.95 ± 0.05, ×0.03 3.30 ± 0.10, ×0.003 ) ) ) 10−2 10−2 10−2 −2 −2 −2 (g) 3.50 ± 0.10, ×0.001

10−3 10−3 10−3 ((GeV/c) ((GeV/c) ((GeV/c) −4 −4 −4 dy dy dy 10 10 10 T T T

10−5 10−5 10−5 N/dp N/dp N/dp (a) 2 2 2 (a) )d )d )d T T T (b) 10−6 (b) 10−6 10−6 (a) (c) (c) (b) (c) (1/2𝜋p (1/2𝜋p (1/2𝜋p 10−7 10−7 10−7 (d) (d) 10−8 10−8 10−8 (d) (e) (e) (e) (g) (f) (g)(f) (g) (f) 10−9 10−9 10−9 0123 0123 0123

pT (GeV/c) pT (GeV/c) pT (GeV/c)

− − + + + + Figure 1: The transverse momentum distributions of (a) 𝜋 , 𝐾 ,and𝑝,(b)𝜋 , 𝐾 ,and𝑝,(c)𝜋 ,(d)𝐾 ,and(e)𝑝 produced in 𝑝𝑝 collision at √𝑠 = 200 GeV with different y ranges and magnifications are shown in the figure. The symbols represent the experimental dataofthe STAR [26] and BRAHMS Collaborations [27] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution. the considered distribution is described by a multicomponent aMonteCarlomethodtogiveanumericalresult.Ourmodel Erlang distribution [19, 20]. Different from some simulation is easy to be used by experimental experts. codes, our model gives directly a few statistical laws by Due to significances of the considered model and topic, analytical expressions in describing some quantities. In the in this paper, based on Boltzmann distribution for a sin- case of being incapable of analytical expressions, we could use gle source, we describe 𝑝𝑇 spectra of final-state particles Advances in High Energy Physics 3

10 10−7 pp, 200 GeV, J/𝜓 pp, 200 GeV, J/𝜓

1 −8 )

10 −2

10−1 10−9 −2.2 < y < −1.2 (nb(GeV/c) dy ±4% GSU T 10−2 10−10 𝜎/dp 2

invariant yield (a.u.) invariant 1.2 < |y| < 2.2

)Bd −3 T J/𝜓 −11 10 ±10.1% GSU 10 |y| < 0.35 ±3% GSU, ×10−1 |y| < 0.35 (1/2𝜋p −12 −4 ±10.1% GSU, ×10−1 10 1.2 < y < 2.2 10 ±4% GSU, ×10−2 10−13 10−5 0 1234567 012345697 8

pT (GeV/c) pT (GeV/c)

(a) (b)

10−6 10−1 d-Au, 200 GeV, J/𝜓 400 GeV p-Pb, J/𝜓

10−7 −0.425 < y < 0.575 10−2 2 2.9 < M𝜇+ 𝜇− <3.3GeV/c 10−8 ) −1 −2.2 < y < −1.2 ±4% GSU −9 10 ((GeV/c) 10−3 T invariant yield (a.u.) invariant

10−10 |y| < 0.35 dN/dp J/𝜓 −1 ±3% GSU, ×10 10−4

10−11 1.2 < y < 2.2 ±4% GSU, ×10−2

−5 10−12 10 01234567 01234567

pT (GeV/c) pT (GeV/c)

Figure 2: The transverse momentum distributions of 𝐽/𝜓 produced in (a) and (b) 𝑝𝑝 collision at √𝑠 = 200 GeV, (c) d-Au collisions at √𝑠𝑁𝑁 = 200 GeV,and (d) p-Pb collisions at beam energy being 400 GeV with different y ranges, GSU (or 𝑀𝜇+𝜇− ) ranges, and magnifications areshowninthefigure.ThesymbolsrepresenttheexperimentaldataofthePHENIX[28–30] and NA50 Collaborations [31] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution. produced in soft process in proton-proton𝑝𝑝 ( )andnucleus- can be different regions in the overlap region or different nucleus (𝐴𝐴) collisions at RHIC and LHC energies. Some mechanisms, and these particles can be created/emitted at interesting results are obtained. different times in the collisions. In fact, these sub-sources can be divided into different groups (sources) due to different 2. The Model interacting mechanisms or event samples. Obviously, soft process corresponds to sources with low degree of excita- According to the multisource thermal model, many emission tion or to particles with low transverse momentum, and sub-sources of final-state particles are assumed to form in hard process corresponds to sources with high degree of high energy collisions [19, 20]. These multiple sub-sources excitation or to particles with high transverse momentum, 4 Advances in High Energy Physics

d-Au, 200 GeV |y| < 0.1 107 |y| < 0.1 107

6 6 0–100% 10 0–100% 10 6 ×106 ×10 ) ) −2 105 −2 105

104 104 ((GeV/c) 40–100% ((GeV/c) 40–100% 4 4 dy 3 ×10 dy 3 ×10 T 10 T 10 N/dp 2 N/dp 2 2 10 2 10 )d )d 20–40% T 20–40% T 10 ×102 10 ×102 (1/2𝜋p 1 (1/2𝜋p 1 0–20% 0–20% ×100 ×100 10−1 10−1

10−2 10−2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2

pT (GeV/c) pT (GeV/c) 𝜋− 𝜋+ K− K+ p p

(a) (b)

− − + + Figure 3: The transverse momentum distributions of (a) 𝜋 , 𝐾 ,and𝑝 as well as (b) 𝜋 , 𝐾 ,and𝑝 produced in d-Au collisions at √𝑠𝑁𝑁 = 200 GeV with |𝑦| < 0.1 and different centrality classes and magnifications are shown in the figure. The symbols represent the experimental data of the STAR Collaboration [26] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution.

Table 1: Parameter values corresponding to the curves in Figure 1.

2 Figure Collision Particle Rapidity 𝑇1 (GeV) 𝑘1 𝑇2 (GeV) 𝜒 /dof − + 𝜋 /𝜋 |𝑦| < 0.1 0.165 1.000 — 1.128/1.212 − + Figure 1(a)/Figure 1(b) pp 200 GeV 𝐾 /𝐾 |𝑦| < 0.1 0.187 1.000 — 0.211/0.243 𝑝/𝑝 |𝑦| < 0.1 0.197/0.212 1.000 — 0.340/0.274 0.00 ± 0.10 0.175 0.896 0.360 0.849 0.90 ± 0.10 0.175 0.930 0.360 0.958 1.20 ± 0.10 0.175 0.912 0.360 0.968 + Figure 1(c) pp 200 GeV 𝜋 2.95 ± 0.05 0.175 0.823 0.270 1.415 3.00 ± 0.10 0.175 0.856 0.270 1.170 3.30 ± 0.10 0.166 0.888 0.270 1.669 3.50 ± 0.10 0.205 1.000 — 1.273 0.00 ± 0.10 0.160 0.700 0.360 0.499 0.90 ± 0.10 0.160 0.600 0.360 0.786 1.20 ± 0.10 0.160 0.600 0.360 0.437 + Figure 1(d) pp 200 GeV 𝐾 2.95 ± 0.05 0.198 0.800 0.360 1.048 3.00 ± 0.10 0.176 0.788 0.300 0.758 3.30 ± 0.10 0.160 0.780 0.273 1.689 3.50 ± 0.10 0.160 0.667 0.253 0.714 0.00 ± 0.10 0.160 0.400 0.265 1.028 0.90 ± 0.10 0.160 0.540 0.290 1.240 1.20 ± 0.10 0.160 0.540 0.290 0.821 Figure 1(e) pp 200 GeV 𝑝 2.95 ± 0.05 0.156 0.875 0.280 1.457 3.00 ± 0.10 0.156 0.872 0.280 0.750 3.30 ± 0.10 0.190 0.859 0.280 1.561 3.50 ± 0.10 0.176 1.000 — 0.907 Advances in High Energy Physics 5 ) ) 104 104 −2 −2 Cu-Cu, 22.5 GeV, 𝜋− 𝜋+

102 102 ((GeV/c) ((GeV/c) dy dy T T 1 1 N/dp N/dp 2 2 )d )d T T 10−2 10−2 (1/2𝜋p (1/2𝜋p 10−4 10−4 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3

pT (GeV/c) pT (GeV/c)

(a) (b) ) 3 − ) 3 + −2 10 K −2 10 K 102 102 ((GeV/c) ((GeV/c) dy dy T 1 T 1 N/dp N/dp 2 2 )d −2 )d −2 T 10 T 10 (1/2𝜋p (1/2𝜋p 10−4 10−4 0 0.5 1 1.5 2 0 0.5 1 1.5 2

pT (GeV/c) pT (GeV/c)

dy −1 dy −1 T 10 T 10 N/dp N/dp 2 2

)d −3 )d −3 T 10 T 10 (1/2𝜋p (1/2𝜋p 10−5 10−5 01234 01234

pT (GeV/c) pT (GeV/c)

0–100%, ×300 30–60%, ×5 0–100%, ×300 30–60%, ×5 0–10%, ×20 60–100%, ×10 0–10%, ×20 60–100%, ×10 10–30%, ×10 10–30%, ×10

(e) (f) − + − + Figure 4: The transverse momentum distributions of (a) 𝜋 ,(b)𝜋 ,(c)𝐾 ,(d)𝐾 ,(e)𝑝,and(f)𝑝 produced in Cu-Cu collisions at √𝑠𝑁𝑁 = 22.5 GeV with different centrality classes and magnifications are shown in the figure. The symbols represent the experimental dataofthe PHENIX Collaboration [32] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution.

−1 where the excitation means to create particles through √𝑝2 +𝑚2 −𝜇 1 𝑑𝑁 2[ 0 ] string breaking, direct scattering, recombination, and their 𝑓𝑝 (𝑝) = =𝐶0𝑝 [exp ( ) ±1] , hybrid. 𝑁 𝑑𝑝 𝑇 In the rest frame of a source, we consider the source as a [ ] thermodynamic system of relativistic and quantum ideal gas. (1) The momentum (𝑝) distribution of final-state particles in the where 𝑁 is the number of particles, 𝐶0 is the normalization natural unit system is given by [21] constant, 𝑚0 is the rest mass of a considered particle, 𝜇 6 Advances in High Energy Physics

Table 2: Parameter values corresponding to the curves in Figure 2.

2 Figure Collision Particle Rapidity 𝑇1 (GeV) 𝑘1 𝑇2 (GeV) 𝜒 /dof −2.2<𝑦<−1.2 0.462 1.000 — 0.663 Figure 2(a) pp 200 GeV 𝐽/𝜓 in pp |𝑦| < 0.35 0.462 1.000 — 1.157 1.2 < 𝑦 < 2.2 0.462 1.000 — 1.383 1.2 < |𝑦| < 2.2 0.160 0.175 0.541 1.371 Figure 2(b) pp 200 GeV 𝐽/𝜓 in pp |𝑦| < 0.35 0.160 0.175 0.654 1.638 −2.2<𝑦<−1.2 0.525 1.000 — 1.332 Figure 2(c) d-Au 200 GeV 𝐽/𝜓 in d-Au |𝑦| < 0.35 0.525 1.000 — 1.125 1.2 < 𝑦 < 2.2 0.525 1.000 — 1.845 Figure 2(d) 400 GeV p-Pb 𝐽/𝜓 in p-Pb −0.425 < 𝑦 < 0.575 0.205 0.650 0.368 1.041

Table 3: Parameter values corresponding to the curves in movement of the source; that is, we use 𝑎𝑝𝑇 +𝑏instead of 𝑝𝑇 Figure 3(a)/Figure 3(b).Thecollisionsared +Auat200GeV. in (3). The revised 𝑝𝑇 distribution can be given by 2 Particle Centrality 𝑇1 (GeV) 𝜒 /dof 1 𝑑𝑁 𝑓 (𝑝 )= 𝑝𝑇 𝑇 0–20% 0.178 0.564/0.704 𝑁 𝑑𝑝𝑇 𝜋−/𝜋+ 20–40% 0.170/0.175 1.053/0.779 40–100% 0.170 0.879/0.905 √ 2 −2 2 [ (𝑝𝑇 −𝑏) 𝑎 +𝑚0 ] =𝐶(𝑝 −𝑏)𝑎−2 [− ] . 0–100% 0.173 0.796/0.771 𝑇 exp 𝑇 0–20% 0.206/0.228 0.119/0.052 [ ] 𝐾−/𝐾+ 20–40% 0.206/0.218 0.112/0.114 (4) 40–100% 0.206/0.213 0.216/0.074 0–100% 0.206/0.217 0.137/0.131 In the case of considering multiple sources, we have 0–20% 0.243/0.239 0.036/0.032 𝑛 √ 2 2 1 𝑑𝑁 𝑝𝑇 +𝑚0 𝑝/𝑝 20–40% 0.232/0.229 0.047/0.036 𝑓 (𝑝 )= = ∑𝑘 𝐶 𝑝 (− ) 𝑝𝑇 𝑇 𝑖 𝑖 𝑇 exp (5) 40–100% 0.215/0.212 0.091/0.082 𝑁 𝑑𝑝𝑇 𝑖=1 𝑇𝑖 0–100% 0.229 0.032/0.013 or

𝑓𝑝 (𝑝𝑇) is the chemical potential, 𝑇 is the temperature parameter, 𝑇 +1 denotes fermions, and −1 denotes bosons, respectively. 1 𝑑𝑁 Our calculations show that at RHIC and LHC energies the = 𝑁 𝑑𝑝𝑇 quantum effect and chemical potential can be neglected (6) compared to the relativistic effect [22]. Then, we have a simple 𝑛 √(𝑝 −𝑏)2𝑎−2 +𝑚2 expression for momentum distribution to be [23, 24] −2 [ 𝑇 𝑖 𝑖 0 ] = ∑𝑘𝑖𝐶𝑖 (𝑝𝑇 −𝑏𝑖)𝑎𝑖 exp [− ] , 𝑖=1 𝑇𝑖 [ ] √ 2 2 1 𝑑𝑁 𝑝 +𝑚0 𝑓 (𝑝) = =𝐶𝑝2 (− ), 𝑘 𝐶 𝑇 𝑝 𝑁 𝑑𝑝 0 exp 𝑇 (2) where 𝑖, 𝑖,and 𝑖 denote the contribution ratio, normalization constant, and temperature of the ith source, respectively. Because the effects of deformation and movement of the 𝐶 =(1/𝑚2𝑇)(1/𝐾 (𝑚 /𝑇)) 𝐾 (𝑚 /𝑇) source can be neglected in the calculation of transverse where 0 0 2 0 and 2 0 is the 𝑎 =1 𝑏 =0 modified Bessel function of order 2. momentum, we take the default values of 𝑖 and 𝑖 in the revised 𝑝𝑇 distribution, which results from the Boltzmann The 𝑝𝑇 distribution can be written as a Boltzmann distribution [25]: distribution or a multicomponent Boltzmann distribution. We should have a few sources to describe the soft and hard processes. For the soft process, the number of sources is √ 2 2 1 𝑑𝑁 𝑝𝑇 +𝑚0 𝑓 (𝑝 )= =𝐶𝑝 (− ), generally 1 or 2. For the hard process, the number of sources 𝑝𝑇 𝑇 𝑇 exp (3) 𝑁 𝑑𝑝𝑇 𝑇 is also 1 or 2. The total number of sources will be from 2 to4 for a wide 𝑝𝑇 distribution. In this paper, we pay our attention on the soft process which has a narrow 𝑝𝑇 distribution. It where 𝐶 is the normalization constant. Because of interac- is hard to say that what the 𝑝𝑇 distribution range is for the tions among different sources, the considered source has a soft process. What we can say is that for low energy collisions deformation and/or movement in the transverse plane. Let the distribution range is narrower. In the present work, we a denote a relative deformation and let b denote an absolute regard the distribution range as 0–10 GeV/c. The difference Advances in High Energy Physics 7

Table 4: Parameter values corresponding to the curves in Figure 4.

2 Figure Collision Particle Centrality 𝑇1 (GeV) 𝑘1 𝑇2 (GeV) 𝜒 /dof 0–10% 0.183 0.934/0.912 0.310 1.079/1.008 10–30% 0.183 0.934/0.917 0.310 1.195/0.848 Cu-Cu − + Figure 4(a)/Figure 4(b) 𝜋 /𝜋 22.5 GeV 30–60% 0.173/0.183 0.934/0.931 0.300/0.310 0.992/1.810 60–100% 0.162/0.160 0.934/0.900 0.293/0.270 1.394/0.959 0–100% 0.183 0.934/0.920 0.310 1.229/1.084 0–10% 0.183 0.860/0.747 0.310 0.636/0.593 10–30% 0.183 0.840/0.732 0.310 0.632/0.818 Cu-Cu − + Figure 4(c)/Figure 4(d) 𝐾 /𝐾 22.5 GeV 30–60% 0.183 0.900/0.755 0.310 1.087/1.440 60–100% 0.183 0.975/0.747 0.310 1.677/1.311 0–100% 0.183 0.860/0.747 0.310 0.632/0.515 0–10% 0.183/0.200 0.760/0.510 0.310 1.898/0.904 10–30% 0.183/0.200 0.860/0.567 0.310 0.825/1.159 Cu-Cu Figure 4(e)/Figure 4(f) 𝑝/𝑝 22.5 GeV 30–60% 0.183/0.200 0.883/0.580 0.310 1.849/1.747 60–100% 0.160/0.200 1.000/0.888 —/0.310 1.953/1.855 0–100% 0.183/0.200 0.862/0.566 0.310 1.090/1.466

Table 5: Parameter values corresponding to the curves in Table 6: Parameter values corresponding to the curves in Figure 5(a)/Figure 5(b).ThecollisionsareAu+Auat62.4GeV. Figure 6(a)/Figure 6(b).ThecollisionsareAu+Auat130GeV.

2 2 Particle Centrality 𝑇1 (GeV) 𝜒 /dof Particle Centrality 𝑇1 (GeV) 𝜒 /dof 0–5% 0.185 0.184/0.184 0–6% 0.185 0.210/0.268 5–10% 0.185 0.192/0.156 6–11% 0.182 0.290/0.232 10–20% 0.182 0.224/0.246 11–18% 0.180 0.430/0.328 20–30% 0.180 0.230/0.228 − + 18–26% 0.180 0.394/0.310 − + 𝜋 /𝜋 𝜋 /𝜋 30–40% 0.180 0.276/0.302 26–34% 0.175 0.222/0.172 40–50% 0.175 0.112/0.128 34–45% 0.170 0.126/0.144 50–60% 0.170 0.118/0.146 45–58% 0.170 0.192/0.234 60–70% 0.170 0.138/0.234 58–85% 0.165 0.334/0.354 70–80% 0.165 0.254/0.292 0–6% 0.275 0.088/0.146 0–5% 0.275 0.056/0.010 6–11% 0.275 0.268/0.087 5–10% 0.275 0.024/0.018 11–18% 0.273/0.263 0.151/0.584 10–20% 0.275 0.040/0.018 − + 18–26% 0.283 0.189/0.584 𝐾 /𝐾 20–30% 0.253 0.064/0.046 26–34% 0.255/0.235 0.101/0.598 − + 𝐾 /𝐾 30–40% 0.243 0.102/0.154 34–45% 0.260/0.245 0.846/0.293 40–50% 0.235 0.082/0.132 45–58% 0.245/0.205 0.639/0.448 50–60% 0.215 0.166/0.084 58–85% 0.205 0.311/0.543 60–70% 0.205 0.100/0.080 0–6% 0.580 0.177/0.177 70–80% 0.205 0.362/0.130 6–11% 0.530 0.157/0.128 0–5% 0.512 0.114/0.166 11–18% 0.450 0.123/0.104 5–10% 0.475 0.068/0.120 18–26% 0.410 0.105/0.118 𝑝/𝑝 10–20% 0.442 0.048/0.096 26–34% 0.410 0.103/0.068 20–30% 0.393 0.040/0.136 34–45% 0.363 0.071/0.121 𝑝/𝑝 30–40% 0.355 0.076/0.082 45–58% 0.303 0.076/0.065 40–50% 0.305 0.104/0.080 58–85% 0.265 0.029/0.055 50–60% 0.270 0.070/0.086 60–70% 0.250 0.038/0.058 70–80% 0.220 0.020/0.084 correspondstoanequilibriumstatewithalowerdegreeof excitation. The latter one describes a wider distribution which between the single and multisource models is obvious. corresponds to a few local equilibrium sates with different The former one describes a narrower distribution which excitations. 8 Advances in High Energy Physics

1010 Table 7: Parameter values corresponding to the curves in Au-Au, 62.4 GeV Figure 7(a)/Figure 7(b).ThecollisionsareAu+Auat200GeV. 9 10 2 Particle Centrality 𝑇1 (GeV) 𝜒 /dof 8 10 0–5% 0.193 0.940/0.746 ) −2 107 5–10% 0.193 0.865/0.907 (a) 10–20% 0.193/0.190 0.780/0.444 106 ((GeV/c) (b) 20–30% 0.190/0.188 0.628/0.661 𝜋−/𝜋+ dy 5 30–40% 0.188 1.432/1.449 T 10 (c) 40–50% 0.188 0.570/0.577 4 (d) N/dp 10

2 50–60% 0.180 0.974/0.914 (e) )d

T 60–70% 0.175 1.269/1.295 103 (f) 70–80% 0.170 1.269/1.311 2 (1/2𝜋p 10 (g) 0–5% 0.327 0.019/0.020 10 (h) 5–10% 0.327 0.019/0.032 10–20% 0.285 0.044/0.034 1 (i) 20–30% 0.285 0.029/0.022 − + 0 0.2 0.4 0.6 0.8 1 1.2 𝐾 /𝐾 30–40% 0.243 0.112/0.107

pT (GeV/c) 40–50% 0.243 0.082/0.079 8 50–60% 0.243 0.007/0.021 (a) 70–80%, ×10 𝜋− 7 (b) 60–70%, ×10 K− 60–70% 0.215 0.081/0.071 6 (c) 50–60%, ×10 p 70–80% 0.200 0.235/0.143 5 (d) 40–50%, ×10 0–5% 0.570 0.046/0.031 (a) 5–10% 0.516 0.071/0.013 1010 10–20% 0.516 0.037/0.045 20–30% 0.435 0.047/0.035 9 10 𝑝/𝑝 30–40% 0.396 0.039/0.036 108 40–50% 0.338/0.345 0.035/0.032 ) 50–60% 0.312/0.320 0.025/0.031 −2 7 10 60–70% 0.270 0.086/0.057 (a) 106 70-80% 0.246 0.045/0.022

((GeV/c) (b)

dy 5 (c) T 10 (d) 4 3. Comparisons with Experimental Data N/dp 10 2 (e) )d

T Figure 1 presents the transverse momentum distributions of 103 − − + + + + (f) (a) 𝜋 , 𝐾 and 𝑝,(b)𝜋 , 𝐾 and 𝑝,(c)𝜋 ,(d)𝐾 , and (e) 2 (g) 𝑝 𝑝𝑝 √𝑠= (1/2𝜋p 10 produced in collision at center-of-mass energy 200 𝑦 (h) GeV with different ranges and magnifications shown 10 in the figure. The symbols represent the experimental data of (i) 1 theSTAR[26] (Figures 1(a) and 1(b)) and BRAHMS Collabo- rations [27] (Figures 1(c)–1(e)), and the curves are our results 0 0.2 0.4 0.6 0.8 1 1.2 calculated by the Boltzmann or two-component Boltzmann pT (GeV/c) distribution. In the calculation, we have used a fitting method (e) 30–40%, ×104 𝜋+ to obtain parameter values which are shown in Table 1 with 𝜒2 𝜒2/ (f) 20–30%, ×103 K+ values of per degree of freedom ( dof). To give a short (g) 10–20%, ×102 p presentation, the values corresponding to “negative/positive” (h) 5–10%, ×101 charged particles are given in terms of “the first value/the (i) 0–50%, ×100 second value” or “value” in the case of the first value and (b) thesecondvaluebeingthesame.Wewouldliketopoint out that the presenting styles of rapidity ranges for Figures − − 1(a) and 1(b) as well as for Figures 1(c)–1(e) are different due Figure 5: The transverse momentum distributions of (a) 𝜋 , 𝐾 ,and + + to different presentations in [26, 27]. One can see that the 𝑝,and(b)𝜋 , 𝐾 ,and𝑝 produced in Au-Au collisions at √𝑠𝑁𝑁 = 62.4 GeV with different centrality classes and magnifications are modeling results with 1 or 2 sources are in agreement with the 𝜋−/𝜋+ 𝐾−/𝐾+ 𝑝/𝑝 shown in the figure. The symbols represent the experimental data experimental data. For emissions of , and , of the STAR Collaboration [26]andthecurvesareourresultscalcu- the temperature parameter increases with increase of particle lated by the Boltzmann or two-component Boltzmann distribution. mass (Figures 1(a) and 1(b)), which indicates the impact of Advances in High Energy Physics 9

1010 1010 Au-Au, 130 GeV 109 109

108 108 ) ) −2 −2 107 107 (a) (a) 106 106

((GeV/c) (b) ((GeV/c) (b) dy dy 5 5 (c) T T 10 (c) 10

4 (d) 4 (d) N/dp N/dp 10 10 2 2 )d )d (e) T T (e) 3 103 10 (f) 2 (f) 2 (1/2𝜋p (1/2𝜋p 10 10 (g) (g) 10 10 (h) (h) 1 1 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2

pT (GeV/c) pT (GeV/c) + (a) 58–85%, ×107 𝜋− (e) 18–26%, ×103 𝜋 6 2 (b) 45–58%, ×10 K− (f) 11–18%, ×10 K+ 5 1 (c) 34–45%, ×10 p (g) 6–11%, ×10 p (d) 26–34%, ×104 (h) 0–6%, ×100

(a) (b)

Figure 6: The same as Figure 5, but showing the results at √𝑠𝑁𝑁 = 130 GeV.

1010 1010 Au-Au, 200 GeV 109 109

108 108 ) ) −2 107 −2 107 (a) (a) 106 (b) 106 (b) ((GeV/c) ((GeV/c) (c) dy 5 (c) dy 5 T 10 T 10 (d) (d) 4 4 N/dp 10 N/dp 10 2 (e) 2 (e) )d )d T 103 (f) T 103 (f)

2 (g) 2 (g) (1/2𝜋p 10 (1/2𝜋p 10 (h) (h) 10 10 (i) (i) 1 1 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2

pT (GeV/c) pT (GeV/c) (a) 70–80%, ×108 𝜋− (e) 30–40%, ×104 𝜋+ 7 (b) 60–70%, ×10 K− (f) 20–30%, ×103 K+ (c) 50–60%, ×106 p (g) 10–20%,×102 p (d) 40–50%, ×105 (h) 5–10%,×101 (i) 0–50%, ×100

(a) (b)

Figure 7: The same as for Figure 5, but showing the results at √𝑠𝑁𝑁 = 200 GeV. 10 Advances in High Energy Physics

10 10

pp, 900 GeV

1 1 ) ) −1 −1

−1 ((GeV/c) −1 10 ((GeV/c) 10 ) ) T T N/dydp N/dydp

2 −2 10 2 10−2 )(d )(d EV EV (1/N 10−3 (1/N 10−3

10−4 10−4 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3

pT (GeV/c) pT (GeV/c) 𝜋− 𝜋+ K− K+ p p

(a) (b) 10 10 )

1 −2 1 ) −1 10−1 ((GeV/c) )

−1 T ((GeV/c) 10 −2 ) 10 T /d𝜂dp

ch −3

N 10 2 N/dydp 2 10−2 )(d

T −4 )(d

p 10 EV EV −5

(1/N 10 10−3 (1/2𝜋N 10−6

10−4 10−7 0 0.5 1 1.5 22.5 3 3.5 4 024681012

pT (GeV/c) pT (GeV/c) 0 KS (×10) 𝜙(×1) NSD, charged particles, |𝜂| < 0.8 Λ(×5) Ξ+Ξ+ − (×0.5) Λ(×2)

− − + + 0 + − Figure 8: The transverse momentum distributions of (a) 𝜋 , 𝐾 ,and𝑝,(b)𝜋 , 𝐾 ,and𝑝,(c)𝐾𝑆 , Λ, Λ, 𝜙,andΞ +Ξ ,and(d)charged particles in range of |𝜂| < 0.8 in NSD produced in pp collisions at √𝑠 = 900 GeV with different magnifications are shown in the figure. The symbols represent the experimental data of the ALICE Collaboration [33, 34] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution. Advances in High Energy Physics 11

5 Pb-Pb, 2.76 TeV, 0–5% 10 Pb-Pb, 2.76 TeV, 𝜋0 3 10 104 ) 3 −2 10 102 ) 2 −1

10 ((GeV/c)

) 10 T pp 2.76 TeV, 𝜋0 1 ((GeV/c)

T −1 N/dydp 10 10 2 −2 )(d 10 T p N/dydp

2 −3 d EV 10

−4 1 10 (1/2𝜋N 10−5 10−6

−7 10−1 10 0 0.5 1 1.5 22.5 3 3.5 4 0246810

pT (GeV/c) pT (GeV/c) 𝜋− K− 0–20% (×500) 60–80% (×1) 0 20–40% (×50) Inelastic (×1) KS p 40–60% (×4)

(a) (b) 105

104 Pb-Pb, 2.76 TeV, p, |y| < 0.5 pp 7 TeV, |y| < 0.5 10−3 ) 3 10 −1 )

−1 2 −4 10 ((GeV/c) 10 ) T

((GeV/c) 10 T N/dydp 2 10−5 1 )(d N/dydp 2 EV d 10−1 −6 (1/2𝜋N 10 10−2

10−3 10−7 0 0.5 1 1.5 22.5 3 3.5 4 0123 45 689 7

pT (GeV/c) pT (GeV/c) 0–5% (×1) 30–40% (×2) Ξ 5–10% (×8) 40–50% (×1.5) Ω 10–20% (×5) 50–60% (×1) 20–30% (×3) 60–70% (×1)

− 0 − 0 Figure 9: The transverse momentum distributions of (a) 𝜋 , 𝐾𝑆 +𝐾 ,and𝑝,(b)𝜋 ,(c)𝑝,and(d)Ξ and Ω produced in Pb-Pb collisions at √𝑠𝑁𝑁 = 2.76 TeV ((a), (b), and (c)) and pp collision at √𝑠 = 2.76 and 7 TeV ((b) and (d)) with different y ranges, centrality classes, and magnifications are shown in the figure. The symbols represent the experimental data of the ALICE Collaboration [35–38]andthecurvesare our results calculated by the Boltzmann or two-component Boltzmann distribution. 12 Advances in High Energy Physics

Table 8: Parameter values corresponding to the curves in Figure 8.

2 Figure Collision Particle 𝑇1 (GeV) 𝑘1 𝑇2 (GeV) 𝜒 /dof − + 𝜋 /𝜋 0.148/0.142 0.803/0.765 0.349/0.341 0.153/0.369 − + Figure 8(a)/Figure 8(b) pp 900 GeV 𝐾 /𝐾 0.185 0.670 0.427 0.158/0.375 𝑝/𝑝 0.185 0.700/0.670 0.390/0.427 0.103/0.156 0 𝐾𝑆 0.132 0.505 0.369 0.375 Λ 0.323 1.000 — 1.026 Figure 8(c) pp 900 GeV Λ 0.311 1.000 — 1.375 𝜙 0.362 1.000 — 0.890 + − Ξ +Ξ 0.325 1.000 — 0.664 Figure 8(d) pp 900 GeV charged 0.259 0.985 0.754 1.903 radial flow and/or the early emission of heavy hadrons. The or two-component Boltzmann distribution. The values of 2 temperature does depend nonobviously on rapidity range parameters and 𝜒 /dof are given in Table 8. We see that the (Figures 1(c)–1(e)). model with 1 or 2 sources describes the experimental data. Figure 2 shows the 𝑝𝑇 distributions of 𝐽/𝜓 produced For emissions of charged hadrons, the temperature parameter in (a) and (b) 𝑝𝑝 collision at √𝑠 = 200 GeV, (c) d-Au increases with increase of particle mass. collisions at √𝑠𝑁𝑁 = 200 GeV,and (d) p-Pb collisions at beam The 𝑝𝑇 distributions of identified particles produced in energy being 400 GeV with different y ranges, global scale (a) central Pb-Pb collisions at √𝑠𝑁𝑁 =2.76TeV, (b) Pb- uncertainty (GSU) (or invariant mass (𝑀𝜇+𝜇− )) ranges, and Pb collisions with different centralities at √𝑠𝑁𝑁 =2.76TeV magnifications shown in the figure, where 𝜎 and B denote and inelastic 𝑝𝑝 collision at √𝑠 = 2.76 TeV, (c) central the cross section and dilepton branching ratio, respectively. rapidity region in Pb-Pb collisions with different centralities The symbols represent the experimental data of the PHENIX at √𝑠𝑁𝑁 = 2.76 TeV, and (d) central rapidity region in 𝑝𝑝 [28] (Figures 2(a) and 2(c))[29, 30](Figure 2(b))andNA50 collision at √𝑠=7TeV are presented in Figure 9.Thesymbols Collaborations [31](Figure 2(d)), and the curves are our represent the experimental data of the ALICE Collaboration results calculated by the Boltzmann or two-component Boltz- [35–38] and the curves are our results calculated by the 𝜒2/ Boltzmann or two-component Boltzmann distribution. The mann distributions. The values of parameters and dof 2 are given in Table 2.Weseeagainthatthemodelwith1or values of parameters and 𝜒 /dof are given in Table 9.We 2 sources describes the experimental data. For emission of seethatinmostcasesthemodelwith1or2sources 𝐽/𝜓 describes the experimental data. Especially, for emissions the temperature parameter does depend nonobviously on − − rapidity range (Figures 2(a) and 2(c)). of 𝜋 , 𝐾 ,and𝑝 (Figure 9(a)), the temperature parameter 0 The 𝑝𝑇 distributions of identified charged particles pro- increaseswithincreaseofparticlemass;foremissionof𝜋 duced in d-Au collisions at √𝑠𝑁𝑁 = 200 GeV, Cu-Cu (Figure 9(b)), the temperature parameter does not depend collisions at √𝑠𝑁𝑁 = 22.5 GeV, Au-Au collisions at √𝑠𝑁𝑁 = on impact centrality; and for emission of 𝑝 (Figure 9(c)), 62.4,130,and200GeVwithdifferentcentralityclassesare the temperature parameter increases with increase of impact displayed in Figures 3–7, respectively. The symbols represent centrality (or with decrease of impact parameter). 2 the experimental data of the STAR [26] (Figures 3, 5, 6, From Tables 1–9 we see that some values of 𝜒 /dof are and 7) and PHENIX Collaboration [32](Figure 4), and the too low pointing to overestimated errors of the experimental curves are our results calculated by the Boltzmann or two- points. In fact, in the case of errors being not available in component Boltzmann distributions. Correspondingly, the related references, we have used a half size of the experimental 2 values of parameters and 𝜒 /dofaregiveninTables3– points to give the errors. This treatment may cause larger 7, respectively. Once more, the model with 1 or 2 sources errors in some cases. describes the experimental data. From Tables 5, 6,and7 − − we see clearly that for emissions of 𝜋 , 𝐾 ,and𝑝 (Figures + + 4. Discussions and Conclusions 5(a), 6(a),and7(a)), as well as 𝜋 , 𝐾 ,and𝑝 (Figures 5(b), 6(b),and7(b)), the temperature parameter increases with From the above discussions we see that the model used increases of particle mass, impact centrality, and √𝑠𝑁𝑁,where in the present work is just a simple phenomenology which we would like to point out that a large centrality (a small does not contain other processes such as parton-hadron percentage) corresponds to a small impact parameter. The string dynamics, hydrodynamic flows, and resonances. The similarconclusionscanbeobtainedfromTables3 and 4. successful description renders that the mentioned processes Figure 8 gives the 𝑝𝑇 distributions of (a)–(c) identified should contribute a higher transverse momentum at multi- particles and (d) charged particles in range of |𝜂| < 0.8 GeV energy or a refined structure in distribution curve. in nonsingle diffraction (NSD) produced in 𝑝𝑝 collision at In the concerned transverse momentum region and for the √𝑠 = 900 𝑁 𝑁 GeV,where EV and ch denote numbers of events concerned distribution curves, we just need to consider the and charged particles, respectively. The symbols represent Maxwell-Boltzmann thermal law. the experimental data of the ALICE Collaboration [33, 34] Thepresentworkisjustifiedtocomparefitsinalowtrans- and the curves are our results calculated by the Boltzmann verse momentum region (<10 GeV/c) for different particles Advances in High Energy Physics 13

Table 9: Parameter values corresponding to the curves in Figure 9.

2 Figure Collision Particle Type 𝑇1 (GeV) 𝑘1 𝑇2 (GeV) 𝜒 /dof − 𝜋 0–5% 0.169 0.595 0.364 0.858 0 − Figure 9(a) Pb-Pb 2.76 TeV 𝐾𝑆 +𝐾 0–5% 0.355 0.900 0.460 0.141 𝑝 0–5% 0.697 1.000 — 0.832 0–20% 0.332 0.985 0.900 1.321 0 20–40% 0.332 0.985 0.900 1.760 Pb-Pb 2.76 TeV 𝜋 Figure 9(b) 40–60% 0.332 0.985 0.900 1.324 60–80% 0.332 0.975 0.900 0.583 pp 2.76 TeV Inelastic 0.332 0.965 0.900 1.816 0–5%, |𝑦| < 0.5 0.632 1.000 — 0.517 5–10%, |𝑦| < 0.5 0.600 1.000 — 0.324 10–20%, |𝑦| < 0.5 0.580 1.000 — 0.275 20–30%, |𝑦| < 0.5 0.540 1.000 — 0.200 Figure 9(c) Pb-Pb 2.76 TeV 𝑝 30–40%, |𝑦| < 0.5 0.532 1.000 — 0.275 40–50%, |𝑦| < 0.5 0.461 1.000 — 0.432 50–60%, |𝑦| < 0.5 0.450 1.000 — 0.296 60–70%, |𝑦| < 0.5 0.385 1.000 — 0.371 Ξ |𝑦| < 0.5 0.350 0.700 0.730 0.510 Figure 9(d) pp 7TeV Ω |𝑦| < 0.5 0.565 1.000 — 0.666 by the same thermal law. Although the difference for charged Especially, there are Coulomb corrections for emissions of andneutralparticlesisunlikelyduetoCoulombeffectswhich charged particles, which affects the extraction of temperature are important for very soft charged particles only, both the [39]. charged and neutral particles obey the same thermal law. The The values of temperature parameter for emissions of − + transverse momentum can extend to more than 100 GeV/c 𝜋 /𝜋 areabout160–190MeVwhichreachesthetemperature at multi-GeV energy. The distribution in the low transverse (166–172MeV)ofcreatingQGPatzerobaryon-chemical momentum region is mainly contributed by the soft pro- potential, where 172 MeV is the equilibrium phase transition cesses. The hard processes which contribute high transverse temperature and 166 MeV is due to finite hadron size [40]. In momentums can be partly described by the thermal law. most cases the temperature for emission of heavy hadrons is Toconclude,wehaveusedthemultisourcethermal greater than that for pions, which renders the impact of radial model to describe the transverse momentum distributions of flow and/or the early emission of heavy hadrons in collisions. particlesproducedinthesoftprocessin𝑝𝑝 and 𝐴𝐴 collisions at RHIC and LHC energies. For single source, the relativistic Conflict of Interests ideal gas model is applied in description of particle behavior. The concerned distribution is finally described by single The authors declare that there is no conflict of interests source or two sources which result from a Boltzmann or two- regarding the publication of this paper. component Boltzmann distributions. The modeling results are in agreement with available experimental data, which renders that an equilibrium or two local equilibriums are Acknowledgments reached in high energy collisions. Because of the evolvement This work was supported by the National Natural Science time of interesting system in collisions being very short, the Foundation of China under Grant no. 10975095 and no. particles should reach rapidly to the state of equilibrium. 11247250, the China National Fundamental Fund of Person- The present work can be used to extract nuclear tempera- nel Training under Grant no. J1103210, the Open Research tureforsoftprocess.Foremissionsofchargedhadrons,the Subject of the Chinese Academy of Sciences Large-Scale temperature parameter increases with increases of particle Scientific Facility under Grant no. 2060205, and the Shanxi mass, impact centrality, and center-of-mass energy and does Scholarship Council of China. depend nonobviously on rapidity range. That the temperature increases with particle mass indicates the impact of radial flow and/or the early emission of heavy hadrons. The References 𝐽/𝜓 temperature parameter for emission of does depend [1]B.B.Back,M.D.Baker,D.S.Bartonetal.,“Charged-particle nonobviously on rapidity range too, which is consistent with pseudorapidity density distributions from Au + Au collisions at 𝜋0 charged hadrons. However, for emission of the tempera- √𝑠𝑁𝑁 = 130 GeV,” Physical Review Letters,vol.87,no.10,Article ture parameter does not depend on impact centrality, which ID 102303, 2001. is inconsistent with charged hadrons. Different behaviors 0 − + [2]I.G.Bearden,D.Beavis,C.Besliuetal.,“Pseudorapidity for 𝜋 and 𝜋 /𝜋 render different production mechanisms. distributions of charged particles from Au+Au collisions at the 14 Advances in High Energy Physics

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Research Article Three-Dimensional Dirac Oscillator with Minimal Length: Novel Phenomena for Quantized Energy

Malika Betrouche,1 Mustapha Maamache,2 and Jeong Ryeol Choi3

1 LaboratoiredePhysiqueMathematique´ et Subatomique (lpmps), Department´ de Physique, Faculte´ des Sciences Exactes, Universite´ Constantine1, 25000 Constantine, Algeria 2 LaboratoiredePhysiqueQuantiqueetSystemes´ Dynamiques, Departement´ de Physique, Faculte´ des Sciences, UniversiteFerhatAbbasS´ etif´ 1, 19000 Setif, Algeria 3 Department of Radiologic Technology, Daegu Health College, Yeongsong-ro 15, Daegu 702-722, Republic of Korea

Correspondence should be addressed to Jeong Ryeol Choi; [email protected]

Received 3 July 2013; Accepted 17 August 2013

Academic Editor: Gongnan Xie

Copyright © 2013 Malika Betrouche et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study quantum features of the Dirac oscillator under the condition that the position and the momentum operators obey generalized commutationrelations that lead to the appearance of minimal length with the order of the Planck length, 󳵻𝑥min = 󸀠 ℎ√3𝛽 + 𝛽󸀠,where𝛽 and 𝛽 are two positive small parameters. Wave functions of the system and the corresponding energy spectrum are derived rigorously. The presence of the minimal length accompanies a quadratic dependence of the energy spectrum on quantum number n, implying the property of hard confinement of the system. It is shown that the infinite degeneracy of energy levels appearing in the usual Dirac oscillator is vanished by the presence of the minimal length so long as 𝛽 =0̸ . Not only in the 󸀠 nonrelativistic limit but also in the limit of the standard case (𝛽=𝛽 =0), our results reduce to well known usual ones.

1. Introduction alsobeeninitiatedbysomeauthors[15–18]. The introduction of minimal length scale leads to a generalization of the It is widely accepted that the consideration of a minimal Heisenberg uncertainty principle in a way that it incorporates lengthscaleinnature,whichisusuallyexpectedtobeof gravitationally induced uncertainty [19]. Kempf et al. intro- √ 3 −35 the order of the Planck length, 𝑙𝑝 = 𝐺ℎ/𝑐 ∼10 m, is duced, through a series of their papers [20–23], a deformed necessary for a consistent formulation of quantum theory of quantum mechanics on the basis of modified commutation gravity. According to this, the development of mathematical relations between position and momentum operators. These manipulation to handle the minimal length became a central commutation relations are characterized with non-zero min- issue in modern quantum gravity. The several research fields imal length and lead to a generalized uncertainty principle in which the concept of an observable minimal length plays (GUP) which includes some corrections from the ordinary an important role in their complete description are string Heisenberg uncertainty relation. There are plentiful reports theory [1–5], loop quantum gravity [6], noncommutative for the consequences of the GUP. Among them, we quote geometry [7], noncommutative field theories [8–10], and the works of Kempf et al. [21, 22]andChangetal.[24] black-hole physics [11, 12]. Standard formulation of quantum that are relevant to the study for solving the Schrodinger¨ mechanics with minimal length for these systems has been equation in momentum space for the harmonic oscillator in carried out starting from the modified Heisenberg algebra D-dimensions. Besides, the effects of the minimal length on with a deformed commutation relation between position and the energy spectrum and momentum wave functions of the momentum operators, which arises from intrinsic noncom- Coulomb potential in one dimension and three dimensions mutativity in geometries [13, 14]. Other similar constructions have been studied, respectively, in [25]and[26–28]. The that accompanies the concept of minimal length scale have quantization scheme in the presence of a minimal length 2 Advances in High Energy Physics is also applied to the problem of the Casimir force for the (1)and(2)thattheHeisenberguncertaintyprincipletakesa electromagnetic field29 [ , 30], the magnetization of electron modified form (GUP) that is given by [31], the Pauli equation for a charged particle in a magnetic ℎ field [32], and the cosmological constant problem [33, 34]. 2 󸀠 2 Δx𝑖Δp𝑖 ≥ (1 + 3𝛽(Δp𝑖) +𝛽(Δp𝑖) ). (6) Further noteworthy studies in this direction include the high 2 temperature properties of the one-dimensional Dirac oscil- Hence, the specific correction of the quantum commutation lator (DO) [35] and supersymmetric quantum mechanics relations between x𝑖 and p𝑖 leads to the extension of the of the three-dimensional Dirac oscillator [14]. The Dirac usual uncertainty relations clarifying a lower bound. The oscillator plays an important role in the description of rela- additional terms in this principle are the consequence of the tivistic many-body problems and supersymmetric relativistic modification of ordinary space of position and momentum, quantum mechanics [36–41]. Dirac oscillator representation which is nonnegligible especially in the high energy quantum is also proposed in quantum chromodynamics, particularly, regime with the energy comparable to the Planck mass in connection with quark confinement models in mesons and √ℎ𝑐/𝐺 ∼19 10 baryons [42]. GeV. Such GUP is also obtained by analyzing the Gedanken experiment [43, 44]. Thepurposeofthisworkistoinvestigatethemath- Δp ematical formulation of the Dirac oscillator problem and By minimizing position uncertainty with respect to 𝑖 its consequences by solving fundamental equations in the in the limit that GUP is saturated, we obtain an isotropic framework of relativistic quantum mechanics with minimal minimal length such that length. In Section 2,wegiveabriefsummaryofthemain √ 󸀠 features of quantum mechanics with generalized commuta- Δxmin =ℎ 3𝛽 + 𝛽 . (7) tion relations. We solve, in Section 3, the three-dimensional Diracoscillatorequationinmomentumspacecompletelyand This relation implies that there is a loss of the notion of derive the corresponding wave functions and the spectrum localization in the position space. Since we are going to work of quantized energy for 𝑠=1/2(spin up) and 𝑠=−1/2 in momentum space, it is convenient to use the following (spindown).Themainconsequencesinthenonrelativistic representation of the position and momentum operators: limit are discussed and it is shown that our results reduce to 2 𝜕 󸀠 𝜕 the standard ones when we remove the scale of the minimal x𝑖 =𝑖ℎ[(1+𝛽𝑝) +𝛽𝑝𝑖𝑝𝑗 +𝛾𝑝𝑖], length. The concluding remarks are given in Section 4 which 𝜕𝑝𝑖 𝜕𝑝𝑗 is the last section. p𝑖 =𝑝𝑖, (8) 𝜕 2. Quantum Mechanics with Minimal Length L = −𝑖ℎ𝜖 𝑝 . 𝑖 𝑖𝑗𝑘 𝑗 𝜕𝑝 The presence of a minimal length as a reflection of dynamical 𝑘 phenomenon stems from the fundamental fluctuations of the The parameter 𝛾 can be taken arbitrarily, and it just modi- background metric at the Planck scales. The Planck length fies the squeezing factor of the momentum space measure shouldbetakenintoaccountwhenwewanttodescribe without affecting the commutation relations. In fact, the inner gravitational fields with its quantum fluctuations. Let us product is now defined as start this section with a brief review of deformed quantum mechanics in 3D. Indeed, following [14, 21], we have 𝑑3𝑝 󵄨 󵄨 ∫ 󵄨𝑝⟩⟨𝑝󵄨 =1. 󸀠 󸀠 󵄨 󵄨 (9) 2 󸀠 [1+(𝛽+𝛽󸀠) 𝑝2]1−((𝛾−𝛽 )/(𝛽+𝛽 )) [x𝑖, p𝑗]=𝑖ℎ[𝛿𝑖𝑗 (1 + 𝛽p )+𝛽p𝑖p𝑗], (1)

󸀠 󸀠 2 [x𝑖, x𝑗]=−𝑖ℎ(2𝛽−𝛽 + (2𝛽 + 𝛽 )𝛽p )𝜖𝑖𝑗𝑘 L𝑘, (2) 3. The Dirac Oscillator with Minimal Length

[p𝑖, p𝑗]=0, (3) The modification of the wave function and the quantum energy that has taken place due to the existence of minimal 󸀠 where 𝛽 and 𝛽 are two very small nonnegative parameters. length will be investigated here by a rigorous procedure for The components of the angular momentum are given by their evaluation. To do this, we consider momentum space problem which is more easier than that in position space due −1 L =(1+𝛽p2) 𝜖 x p ,𝑖=1,2,3. to the fact that position operators do not commute each other 𝑖 𝑖𝑗𝑘 𝑗 𝑘 (4) ̃ [see, (2)]. The replacement p → p −𝑖𝛽𝑚𝜔r in the Dirac These satisfy the usual commutation relations of the form equation for a free particle gives the Dirac oscillator equation [42]suchthat [L , x ] = 𝑖ℎ𝜖 x ,[L , p ] = 𝑖ℎ𝜖 p . 𝑖 𝑗 𝑖𝑗𝑘 𝑘 𝑖 𝑗 𝑖𝑗𝑘 𝑘 (5) ̃ ̃ 2 [𝑐𝛼(̃ p −𝑖𝛽𝑚𝜔r)+𝛽𝑚𝑐 ]𝜓=𝑊𝜓, (10) The GUP can be established in this context if we take into account the presence of the minimal length scale. where 𝑚 is the rest mass, 𝜔 is the frequency of the Dirac 𝜓 ⟨p⟩=0 𝛼̃ 𝛽̃ 𝜓=( 𝑎 ) Considering physical states with and the fact that oscillator, and are the Dirac matrices, and 𝜓𝑏 is the momentum uncertainties Δp𝑖 are isotropic, we see from a two-component spinor. Advances in High Energy Physics 3

̃ Using the following representation of 𝛼̃ and 𝛽: where p̂ = p/|p| isaunitvector.Forfurtherevaluation,we needthesquareofthepositionoperator: 0𝜎 10 𝛼=(̃ ), 𝛽=(̃ ), 𝜕 2 𝜎0 0−1 (11) r2 =−ℎ2 {[(1 + (𝛽 +󸀠 𝛽 )𝑝2) ] 𝜕𝑝 we get the following two simultaneous equations: 2 𝜕 +[ +2(𝛽+𝛾)𝑝]×[(1+(𝛽+𝛽󸀠)𝑝2) ] 𝑝 𝜕p 𝑊𝜓 (p) =𝑐𝜎(p +𝑖𝑚𝜔r) 𝜓 (p) +𝑚𝑐2𝜓 (p) , 𝑎 𝑏 𝑎 2 (12) 𝐿 2 2 − −(2𝛽𝐿 −3𝛾) 2 𝑊𝜓𝑏 (p) =𝑐𝜎(p −𝑖𝑚𝜔r) 𝜓𝑎 (p) −𝑚𝑐 𝜓𝑏 (p) . 𝑝

This system gives the following factorized equation for the + [𝛾 (3𝛽 +𝛽󸀠 +𝛾)−𝛽2𝐿2]𝑝2}. large component 𝜓𝑎(p): (17) 2 2 4 (𝑊 −𝑚 𝑐 )𝜓𝑎 (p) 𝑚𝑗 The action of (𝜎L) on the spin-angular function 𝜘𝜅 (p) results 2 2 2 2 2 2 2 in [45]: =𝑐 {p +𝑚 𝜔 r +𝑖𝑚𝜔[𝜎r,𝜎p] +𝑖𝑚 𝜔 𝜎 (r ∧ r)}𝜓𝑎 (p) . 𝑚𝑗 𝑚𝑗 (13) 𝜎L𝜘𝜅 (p̂) = ℎ𝜅𝜘𝜅 (p̂) , (18) where the quantum number 𝜅 is equal to [𝑠(2𝑗 + 1) −1]. Using (1)and(2), we easily show that By substituting (17)and(18)into(15), we have 2𝜎L 1 [𝜎r,𝜎p] =𝑖ℎ(1+(𝛽+𝛽󸀠)𝑝2)( +3), (𝑊2 −𝑚2𝑐4 + 2𝑚𝜔ℎ𝜅𝑐2 ℎ 𝑐2 (14) r ∧ r = −𝑖ℎ (2𝛽 −𝛽󸀠 +(2𝛽+𝛽󸀠)𝛽𝑝2) L. +3𝑚𝜔ℎ𝑐2 −𝑚2𝜔2ℎ2𝜅𝑐2 (2𝛽 − 𝛽󸀠)) 𝐹 (𝑝)

=−𝑚2𝜔2ℎ2 The substitution of the above equations into13 ( )gives 2 󸀠 2 𝜕 2 (𝑊2 −𝑚2𝑐4) ×{[(1+(𝛽+𝛽)𝑝 ) ] +[ +2(𝛽+𝛾)𝑝] 𝜓 (p) 𝜕𝑝 𝑝 𝑐2 𝑎 𝜕 𝐿2 (19) 2 2 2 2 ×[(1+(𝛽+𝛽󸀠)𝑝2) ]− − (2𝛽𝐿2 −3𝛾) ={p +𝑚 𝜔 r 𝜕𝑝 𝑝2 + [−2𝑚𝜔 (1 + (𝛽 +𝛽󸀠)𝑝2) (15) −1 +[ +𝛾(3𝛽+𝛽󸀠 +𝛾)−𝛽2𝐿2 𝑚2𝜔2ℎ2 +𝑚2𝜔2ℎ(2𝛽−𝛽󸀠 + (2𝛽 + 𝛽󸀠)𝛽𝑝2)] 𝜎L 1 + (2𝜅 + 3) (𝛽 + 𝛽󸀠) 󸀠 2 𝑚𝜔ℎ −3𝑚𝜔ℎ (1 + (𝛽 +𝛽 )𝑝 )} 𝜓𝑎 (p) . −(2𝛽+𝛽󸀠)𝛽𝜅]𝑝2}𝐹(𝑝). The first two terms in the right-hand side constitute the Hamiltonian of the 3-dimensional harmonic oscillator. The third term is the spin-orbit contribution with a momen- Although this is a very complicated equation at a glance, we can simplify it via the following standard procedure. Let tum dependent coupling strength. The fourth term is a 𝜉 momentum-dependent shift function which affects both the us first introduce a useful parameter in the form energy levels and wave functions. It is obvious that the usual 𝜌 󸀠 𝜉= , Dirac oscillator equation is reproduced in the limit 𝛽=𝛽 = √𝑚𝜔ℎ (20) 0. where 1 3.1. Wave Functions. Let us now derive wave functions of the 𝜌= arctan 𝑝√𝛽+𝛽󸀠. Dirac oscillator in momentum space by solving (15). We make √𝛽+𝛽󸀠 (21) the decomposition of the wave function into radial part and a spin-angular part as Then, it is able to change the variable from 𝑝 ∈ (0, ∞) to 𝜉 ∈ (0, 𝜋/(2√𝑘)) where 𝑚𝑗 𝜓𝑎 (p) 𝐹(𝑝)𝜘𝜅 (p̂) 𝜓 (p) =( )=( 𝑚 ), 𝜓 (p) 𝑗 (16) 𝑘=√𝑚𝜔ℎ (𝛽 +𝛽󸀠). 𝑏 𝐺(𝑝)𝜘−𝜅 (p̂) (22) 4 Advances in High Energy Physics

󸀠 Notice that the relation between 𝜉 and 𝑘 is as follows: where 𝜂 = 𝛽/(𝛽 +𝛽 ).Atthisstage,weeliminatethe centrifugal barrier by imposing a condition that 𝜆 satisfies 1 𝜉= (𝑝√𝛽+𝛽󸀠). equation 𝑘 arctan (23) (2𝜅 + 3) 1 𝜆2 −𝜆(1+2𝜂)−𝜂2𝐿2 + −𝜂(𝜂+1)𝜅− =0. Now, (19)canberewrittenas 𝑘2 𝑘4 𝜁 (29) − 𝐹(𝑝) 𝑘2 The corresponding solutions are 󸀠 1 𝜕2 [ 2𝑚𝜔ℎ√𝛽+𝛽 2𝑚𝜔ℎ 1+2𝜂 ={ + [ + 𝜆 = 2 2 ± 2 𝑘 𝜕𝜉 𝑘2 (𝜌√𝛽+𝛽󸀠) 𝑘2√𝛽+𝛽󸀠 [ tan (1 + 2𝜂)2 (2𝜅 + 3) 1 ± √ +𝜂2𝐿2 − +𝜂(𝜂+1)𝜅+ . ] 𝜕 𝑚𝜔ℎ 4 𝑘2 𝑘4 × (𝜌√𝛽+𝛽󸀠) ] − (𝛽 + 𝛽󸀠)𝐿2 tan 𝜕𝜌 𝑘2 (30) ] To proceed further, it is convenient to introduce a parameter 2 2 𝑙 󸀠 as 𝑧=2𝑆 −1and to choose 𝑓(𝑆) =𝑆 𝑔(𝑆).Then(28)becomes ×[cot (𝜌√𝛽+𝛽)] (1 − 𝑧2)𝑔󸀠󸀠 (𝑧) + [(𝑏−𝑎) − (𝑎+𝑏+2) 𝑧] 𝑔󸀠 (𝑧) 𝑚𝜔ℎ 𝑚𝜔ℎ − (2𝛽𝐿2 −3𝛾)+ 𝑘2 𝑘2 (𝛽 + 𝛽󸀠) 1 𝜁 + [ −2𝜂𝐿2 − (2𝑙 + 3) 𝜆+𝑙(2𝜂−1)]𝑔(𝑧) =0, −1 4 𝑘2 ×[ +𝛾(3𝛽+𝛽󸀠 +𝛾)−𝛽2𝐿2 𝑚2𝜔2ℎ2 (31) 1 𝑎 𝑏 + (2𝜅 + 3) (𝛽 + 𝛽󸀠)−(2𝛽+𝛽󸀠)𝛽𝜅] where the parameters and are defined by 𝑚𝜔ℎ 1 1 𝑎=𝜆 −𝜂− ,𝑏=+𝑙. 2 + 2 2 (32) ×[tan (𝜌√𝛽+𝛽󸀠)] }𝐹(𝑝), Notice that 𝑎 iswrittenintermsof𝜆+ givenin(30), and it will (24) be justified later. To solve this differential equation, we set where 1 𝜁 [ −2𝜂𝐿2 − (2𝑙 + 3) 𝜆+𝑙(2𝜂−1)] 2 𝑊2 −𝑚2𝑐4 4 𝑘 𝜁= + 2𝜅 + 3 − 𝑚𝜔ℎ𝜅 (2𝛽 − 𝛽󸀠) . (25) (33) 𝑚𝜔ℎ𝑐2 =𝑛󸀠 (𝑛󸀠 +𝑎+𝑏+1),

For further simplifications, we put 󸀠 where 𝑛 =(𝑛−𝑙)/2. This is the spectral condition from which 𝜆+𝛿 𝐹=𝐶 𝑓 (𝑆) , (26) we can extract the energy spectrum. Thus, we obtain 2 󸀠󸀠 󸀠 where 𝑆 and 𝐶 are given by (1 − 𝑧 )𝑔 (𝑧) + [(𝑏−𝑎) − (𝑎+𝑏+2) 𝑧] 𝑔 (𝑧) (34) 󸀠 󸀠 𝑆=sin (𝑘𝜉) ,𝐶=cos (𝑘𝜉) , (27) +𝑛 (𝑛 +𝑎+𝑏+1)𝑔(𝑧) =0, and 𝜆 is a constant that will be determined later, while 𝛿= 󸀠 whose corresponding solution is given in terms of the Jacobi 𝛾/(𝛽 +𝛽 ).Fromsubstitutionof(26)into(24), we get polynomials:

2 󸀠󸀠 2 󸀠 (𝑎,𝑏) (1 − 𝑆 )𝑓 −[(2𝜆+1+2(1−𝜂))𝑆− ]𝑓 𝑔 (𝑧) =𝑁𝑃󸀠 (𝑧) , (35) 𝑆 𝑛 𝜁 with a normalization constant 𝑁. +{( −(2𝜂−1)×𝐿2 −3𝜆) 𝑘2 Then, the large radial component momentum wave func- (28) tion is given by 𝐿2 (2𝜅 + 3) − +(𝜆2 −𝜆(1+2𝜂)−𝜂2𝐿2 + 𝑆2 𝑘2 𝐹 (𝑧) 1 𝑆2 =𝑁2−(𝑎+𝑏+𝜂+𝛿)/2(1−𝑧)(𝑎+𝜂+𝛿+1/2)/2(1+𝑧)(𝑏−1/2)/2𝑃(𝑎,𝑏) (𝑧) . −𝜂 (𝜂 + 1) 𝜅− ) }𝑓=0, 𝑛󸀠 𝑘4 𝐶2 (36) Advances in High Energy Physics 5

Upon returning to the old variable 𝑝,wehave At this stage we use the following decomposition, which will be useful in the calculation of both 𝐺(𝑝) and the −(𝑎+𝑏+𝜂+𝛿)/2 𝑏−(1/2) 𝐹(𝑝)= 𝑁[1+(𝛽+𝛽󸀠)𝑝2] (√𝛽+𝛽󸀠𝑝) normalization constant: 1 (𝛽 + 𝛽󸀠)𝑝2 −1 𝐹(𝑝)= 𝑓−(𝛿+𝜂−1)/2𝑅 (𝑝) , ×𝑃(𝑎,𝑏) ( ), 𝑝 1 (44) 𝑛󸀠 (𝛽 + 𝛽󸀠)𝑝2 +1 1 (37) 𝐺(𝑝)= 𝑓−(𝛿+𝜂−1)/2𝑅 (𝑝) , 𝑝 2 (45) which leads to the large component of the DO wave function: 󸀠 2 −(𝑎+𝑏+𝜂+𝛿)/2 𝑏−(1/2) where 𝑓(𝑝) = 1 + (𝛽 +𝛽 )𝑝 .Bycomparing(44)with(37), 𝜓 (p) = 𝑁[1 + (𝛽 +𝛽󸀠)𝑝2] (√𝛽+𝛽󸀠𝑝) 𝑎 one immediately gets (𝛽 +󸀠 𝛽 )𝑝2 −1 (𝑎,𝑏) 𝑚 𝑏−1/2 𝑗 󸀠 𝑏+1/2 −(𝑎+𝑏+1)/2 (𝑎,𝑏) ×𝑃󸀠 ( )𝜘𝜅 (p̂) . 𝑅 (𝑝) = 𝑁√𝛽+𝛽 𝑝 𝑓 𝑃 (𝑧) . (46) 𝑛 (𝛽 +󸀠 𝛽 )𝑝2 +1 1 𝑛󸀠 (38) In the meantime, 𝑅2(𝑝) is different depending on spin. Using Let us now turn to the calculation of the small component of the properties of the Jacobi polynomials given in [46], we 𝜓 (p) the DO wave function 𝑏 given by have the following results for each spin: 𝑐𝜎 (p −𝑖𝑚𝜔r) 𝜓 (p) = 𝜓 (p) . (39) 𝑏 𝑊+𝑚𝑐2 𝑎 (i) 𝑠=1/2

With the aid of the following relations [14]: 𝑏−1/2 −𝑚𝜔ℎ𝑐𝑁√𝛽+𝛽󸀠 𝜕 𝜕 2 󸀠 𝑅2 (𝑝) = 𝜎r = 𝑖ℎ𝜎𝑖 ((1 + 𝛽𝑝 ) +𝛽𝑝𝑖𝑝𝑗 +𝛾𝑝𝑖), 𝑊+𝑚𝑐2 𝜕𝑝𝑖 𝜕𝑝𝑗 𝜕 1 ℎ𝜅 𝜕 𝜕 𝜎L +2 𝜕 𝜎L ×[𝑓 +( − 𝛽ℎ𝜅) 𝑝− ] 𝜎𝑖 =( + )𝜎𝑝 =𝜎𝑝 ( − ), 𝜕𝑝 𝑚𝜔ℎ 𝑝 𝜕𝑝𝑖 𝜕𝑝 𝑝 𝜕𝑝 𝑝 (40) ×𝑝𝑏+1/2𝑓−(𝑎+𝑏+1)/2𝑃(𝑎,𝑏) (𝑧) 𝜕 𝜕 𝜕 𝜕 𝑛󸀠 𝜎 𝑝 𝑝 = (𝜎⋅p) (𝑝 )=𝜎 𝑝2 =𝑝2 𝜎 , 𝑖 𝑖 𝑗 𝜕𝑝 𝜕𝑝 𝑝 𝜕𝑝 𝜕𝑝 𝑝 𝑏−1/2 𝑗 −2𝑚𝜔ℎ𝑐𝑁√𝛽+𝛽󸀠 𝜕 𝜕 = 𝜎 =𝜎 , 𝑊+𝑚𝑐2 𝜕𝑝 𝑝 𝑝 𝜕𝑝 𝑏−1/2 −(𝑎+𝑏+1)/2 𝑑 (𝑎,𝑏) ×𝑝 𝑓 (1+𝑧) 𝑃 󸀠 (𝑧) where 𝜎𝑝 =(𝜎⋅p)/𝑝,andusing(18), one finds 𝑑𝑧 𝑛

𝑐 𝑏−1/2 𝜓 (p) = 𝜎 󸀠 󸀠 󸀠 𝑏 𝑊+𝑚𝑐2 𝑝 −2𝑚𝜔ℎ𝑐 (𝛽 +𝛽 )(𝑎+𝑏+𝑛 +1)𝑁√𝛽+𝛽 = 2 𝜕 𝑊+𝑚𝑐 × [𝑝 + 𝑚𝜔ℎ ((1 + (𝛽+𝛽󸀠)𝑝2) 𝜕𝑝 ×𝑝𝑏+3/2𝑓−(𝑎+𝑏+3)/2𝑃(𝑎+1,𝑏+1) (𝑧) , 𝑛󸀠−1 ℎ𝜅 (47) − + (𝛾 − 𝛽ℎ𝜅) 𝑝)]𝜓 (p) . 𝑝 𝑎 (41) (ii) 𝑠=−1/2 Through the introduction of the following property: 𝑏−1/2 𝑚 𝑚 𝑗 𝑗 √ 󸀠 𝜎𝑝𝜘𝜅 (p̂) =−𝜘−𝜅 (p̂) , (42) −𝑚𝜔ℎ𝑐𝑁 𝛽+𝛽 𝑅 (𝑝) = 2 𝑊+𝑚𝑐2 the small radial wave function 𝐺(𝑝) is represented as follows: −𝑚𝜔ℎ𝑐 𝜕 1 ℎ (𝜅+1) 𝐺(𝑝)= ×[𝑓 +( −𝛽ℎ(𝜅+1))𝑝− ] 𝑊+𝑚𝑐2 𝜕𝑝 𝑚𝜔ℎ 𝑝 1 ×𝑝𝑏+1/2𝑓−(𝑎+𝑏+1)/2𝑃(𝑎,𝑏) (𝑧) ×[( +𝛾−𝛽ℎ𝜅)𝑝 𝑛󸀠 𝑚𝜔ℎ (43) 𝑏−1/2 󸀠 󸀠 2 𝜕 ℎ𝜅 −2𝑚𝜔ℎ𝑐𝑁√𝛽+𝛽 +(1+(𝛽+𝛽 )𝑝 ) − ]𝐹(𝑝). = 𝑝𝑏−1/2𝑓−(𝑎+𝑏+1)/2 𝜕𝑝 𝑝 𝑊+𝑚𝑐2 6 Advances in High Energy Physics

𝑑 ×[(1+𝑧) +𝑏]𝑃(𝑎,𝑏) (𝑧) It is easy to see that the integrand with the small radial wave 𝑛󸀠 −(2𝜆−2𝜂−1) 𝑑𝑧 function behaves like 𝑝 for 𝑝→∞,whichrequires 𝑏−1/2 𝜆 > 𝜂+(1/2). This condition can only be fulfilled by choosing 󸀠 󸀠 −2𝑚𝜔ℎ𝑐 (𝑏 +𝑛 )𝑁√𝛽+𝛽 the upper sign in (30). For the case 𝜆=𝜆+,theintegrandwith = −(2𝜆 −2𝜂+1) 𝑊+𝑚𝑐2 the large radial wave function behaves like 𝑝 + ,and then the convergence criterion requires 𝜆+ > 𝜂 − (1/2) which ×𝑝𝑏−1/2𝑓−(𝑎+𝑏+1)/2𝑃(𝑎+1,𝑏−1) (𝑧) . 𝑛󸀠 is automatically satisfied in our treatment. Let us derive the nonrelativistic limit of the Dirac oscil- (48) 2 lator equation with minimal length by setting 𝑊=𝑚𝑐 +𝐸 2 Now, the normalization constant 𝑁 canbecalculatedusing with 𝐸≪𝑚𝑐 and dividing by 2𝑚 from (13). Then, one finds themodifiedclosurerelation ∞ 𝐸𝜓𝑎 (p) 𝑑𝑝 󵄨 󵄨2 󵄨 󵄨2 ∫ (󵄨𝑅 (𝑝)󵄨 + 󵄨𝑅 (𝑝)󵄨 )=1. 󵄨 1 󵄨 󵄨 2 󵄨 (49) 2 0 𝑓(𝑝) p 1 2 2 3 󸀠 2 =[ + 𝑚𝜔 r −𝜇𝑆𝐿 (𝑝)L 𝜎 − ℎ𝜔 (1+(𝛽+𝛽 )𝑝 )] 𝜓𝑎 (p) , If we substitute (46)–(48)into(44)and(45), the final 2𝑚 2 2 expressions of the radial components of normalized wave (54) functions are obtained as follows: 2 1/2 where 𝑊+𝑚𝑐 𝜔 𝐹(𝑝)= ( ) 𝜇 (𝑝) = (1 + (𝛽 + 𝛽󸀠)𝑝2) 2𝑊 𝑆𝐿 ℎ

(𝑛󸀠) 𝑏−1/2 −(1/2)(𝑎+𝑏+𝜂+𝛿) (𝑎,𝑏) 2 (55) ×𝐴 (𝑎,) 𝑏 𝑝 𝑓 𝑃 (𝑧) , 𝜔 󸀠 󸀠 2 𝑛󸀠 − (2𝛽 − 𝛽 + (2𝛽 + 𝛽 )𝛽𝑝 ). (50) 4 𝑊−𝑚𝑐2 1/2 𝐺(𝑝)= −𝜀( ) We observe that, besides the usual shifted 3D harmonic 2𝑊 oscillator, the Hamiltonian contains an additional interaction 󸀠 ̃ ̃ ̃ term that is originated from the presence of the minimal ×𝐴(𝑛̃ ) (𝑎,̃ ̃𝑏)𝑏−1/2 𝑝 𝑓−(1/2)(𝑎+̃ 𝑏+𝜂+𝛿)𝑃(𝑎,̃ 𝑏) (𝑧) , 𝑛̃󸀠 length. Unlike the usual Dirac oscillator, the second term in (54) exhibits a momentum-dependent spin-orbit coupling where strength given by 𝜇𝑆𝐿(𝑝). As it is well known, the spin-orbit (𝑛󸀠) 𝐴 (𝑎,) 𝑏 interaction is generated by the scalar potential in the usual case. In our setup such a scalar potential can be attributed to 𝑏+1 1/2 2(𝛽+𝛽󸀠) (𝑎+𝑏+2𝑛󸀠 +1)𝑛󸀠!Γ(𝑎+𝑏+𝑛󸀠 +1) a gravitation-like field generated by the perturbation of the =( ) , space in the presence of the minimal length. Γ(𝑎+𝑛󸀠 +1)Γ(𝑏+𝑛󸀠 +1)

1 𝑊 3.2. Energy Spectrum. The existence of a nonvanishing min- 𝑎=𝑎+1,̃ ̃𝑏=𝑏+2𝑠, 𝑛̃󸀠 =𝑛󸀠 −𝑠− ,𝜖= , imal length implies the modification of energy level for the 2 |𝑊| usual treatment of the Dirac oscillator. In order to derive the (51) energy spectrum of the Dirac oscillator with minimal length, 󸀠 𝑛󸀠 𝑎 𝑏 𝜆 𝜁 and 𝑛 =0,1,2,...,exceptwhen𝑠=1/2and 𝜖=−1,inwhich we substitute the expressions of , , , +,and into (33)and 󸀠 𝑊 case 𝑛 =1,2,3,.... solve for .Indeed,afterastraightforwardcalculations,we At this stage, we turn back to the problem of justifying obtain that 𝑎 is expressed in terms of 𝜆+ asshownin(32). It 𝑊2 −𝑚2𝑐4 was pointed in [21] that the normalization condition alone does not always guarantee physically relevant wave functions 3 = 𝑚𝜔ℎ𝑐2 {2 (𝑛 + ) for the case considered the existence of minimal length. 2 However, the physical wave functions of our system must be recognizable in the domain of momentum since p𝑖 and p𝑗 1/2 󸀠 2 commutes mutually as shown in (3). This physically means (𝛽 +3𝛽) × [1+(𝑚𝜔ℎ)2 ( +𝛽2𝐿2)−𝜖 ] that there is a finite uncertainty in momentum, leading to the 4 𝜅1 following condition: [ ] ∞ 𝑝2𝑑𝑝 2 2 󵄨 󵄨2 󵄨 󵄨2 󸀠 3 󸀠 ⟨𝑝 ⟩=∫ (󵄨𝑅1 (𝑝)󵄨 + 󵄨𝑅2 (𝑝)󵄨 )<∞, (52) +𝑚𝜔ℎ[(𝛽+𝛽)(𝑛+ ) +(𝛽−𝛽) 0 𝑓(𝑝) 2 or ∞ 2 ∞ 2 9 3 𝑝 𝑑𝑝 󵄨 󵄨2 𝑝 𝑑𝑝 󵄨 󵄨2 ×(𝐿2 + )+ 𝛽󸀠]−𝜖 }, ∫ 󵄨𝑅1 (𝑝)󵄨 <∞, ∫ 󵄨𝑅2 (𝑝)󵄨 <∞. 4 2 𝜅2 0 𝑓(𝑝) 0 𝑓(𝑝) (53) (56) Advances in High Energy Physics 7 where problem has been mapped into the motion of a point particle 2 near the surface of a sphere which is in essence a motion in 𝜖 =3(1+ 𝜅) 𝑚𝜔ℎ󸀠 (𝛽+𝛽 )−𝑚2𝜔2ℎ2𝛽 (2𝛽 +𝛽󸀠)𝜅, potential wells. In our setup the boundaries of the well are 𝜅1 3 localized at 0 and 𝜋/(2√𝑚𝜔ℎ(𝛽 +𝛽󸀠)). 2 𝜖 =3(1+ 𝜅) − 𝑚𝜔ℎ𝜅 (2𝛽󸀠 −𝛽 ), The nonrelativistic limit of the energy spectrum in the 𝜅2 3 presence of minimal length is obtained from (56)bysetting (57) 2 2 𝑊=𝑚𝑐 +𝑊𝑛𝑟 with the assumption that 𝑊𝑛𝑟 ≪𝑚𝑐.From which are the terms expressing the spin-orbit coupling. alittlecalculation,weget Notice that (56) does not depend on the parameter 𝛾,itbut 󸀠 3 depends on 𝛽 and 𝛽 . By expanding (56) up to first order in 𝛽 𝑊𝑛𝑟 =ℎ𝜔{(𝑛+ ) 󸀠 2 and 𝛽 ,weget

2 2 4 2 2 1/2 𝑊 =𝑚𝑐 + 𝑚𝜔ℎ𝑐 (𝛽󸀠 +3𝛽) × [1 +(𝑚𝜔ℎ)2 ( +𝛽2𝐿2)− 𝜖 ] 4 𝜅1 1 𝑚𝜔ℎ𝛽 1 ×{2(𝑛−𝑗+ )[1+ (𝑛 − 𝑗 + )] [ ] 2 2 2 𝑚𝜔ℎ 3 2 + [(𝛽 +󸀠 𝛽 )(𝑛+ ) +(𝛽−𝛽󸀠) 𝑚𝜔ℎ𝛽󸀠 9 (58) 2 2 + (𝑛 + 𝑗 + ) 2 2 9 3 𝜖 ×(𝐿2 + )+ 𝛽󸀠]− 𝜅2 }. 4 2 2 +𝑚𝜔ℎ𝛽󸀠 (−4𝑛−2𝑛𝑗−3)}, (61) If we ignore the contribution of the spin-orbit coupling for 𝑗 = 𝑙 + (1/2),and 𝜖𝜅1 and 𝜖𝜅2, this exactly coincides with the 3D harmonic 𝑊2 =𝑚2𝑐4 + 𝑚𝜔ℎ𝑐2 oscillator energy in deformed space with minimal length [24]. In order to compare our results with the ones considered 𝛽 3 𝑚𝜔ℎ𝛽 3 up to the first order in ,thatisobtainedin[14], using super- ×{2(𝑛+𝑗+ )[1+ (𝑛 + 𝑗 + )] symmetric quantum mechanical methods, let us expand (61) 󸀠 2 2 2 to the first order in 𝛽 and 𝛽 such that 𝑚𝜔ℎ𝛽󸀠 7 (59) 1 + (𝑛 − 𝑗 + ) 𝑊 =ℎ𝜔{(𝑛−𝑗+ ) 2 2 𝑛𝑟 2

𝑚𝜔ℎ𝛽 1 +𝑚𝜔ℎ𝛽󸀠 (−2𝑛+2𝑛𝑗−3)}, ×[1+ (𝑛 − 𝑗 + )] 2 2 𝑚𝜔ℎ𝛽󸀠 for 𝑗 = 𝑙 − (1/2). 9 󸀠 󸀠 + (𝑛 + 𝑗 + ) − 𝑚𝜔ℎ𝛽 In the standard setup where 𝛽, 𝛽 →0,wecaneasily 4 2 show that the following equations recover the usual energy 3 1 spectrum of the Dirac oscillator [45]: ×(2𝑛+𝑛𝑗+ )}, 𝑗=𝑙+ , 2 2 1 1 𝑊2 −𝑚2𝑐4 = 2𝑚𝜔ℎ𝑐2 [𝑛 − 𝑗 + ], 𝑗=𝑙+ , (62) 2 2 3 (60) 𝑊 =ℎ𝜔{(𝑛+𝑗+ ) 3 1 𝑛𝑟 2 𝑊2 −𝑚2𝑐4 = 2𝑚𝜔ℎ𝑐2 [𝑛 + 𝑗 + ], 𝑗=𝑙− , 2 2 𝑚𝜔ℎ𝛽 3 ×[1+ (𝑛 + 𝑗 + )] which shows a degeneracy of all the states with 𝑛±𝑞and 𝑗±𝑞 2 2 for 𝑗 = 𝑙 + (1/2),where𝑞 is an integer, and of all the states 𝑛±𝑞 𝑗∓𝑞 𝑗 = 𝑙 − (1/2) 𝑚𝜔ℎ𝛽󸀠 7 with and for .Aswecanseefrom + (𝑛 − 𝑗 + ) − 𝑚𝜔ℎ𝛽󸀠 (58)and(59), a remarkable feature of the incorporation of the 4 2 minimal length in the DO equation is that these degeneracy 𝛽󸀠 =0̸ 3 1 is completely removed when .FromFigure1,wesee ×(𝑛−𝑛𝑗+ )}, 𝑗=𝑙− , that the intervals between adjacent energy levels that are 2 2 󸀠 󸀠 degenerated under 𝛽 =0become large as 𝛽 grows. 󸀠 It is important to note that the energy spectrum contains respectively. Clearly, in the limit 𝛽 =0,theseequations 2 additional terms proportional to 𝑛 ,whichindicateshard become identical with the ones obtained in [14]. The discrep- 󸀠 confinement. This behavior is expected since the original ancy when 𝛽 =0̸ canbeattributedtothemethodusedbythe 8 Advances in High Energy Physics authors of [14] where there is a manifest distinction between 2.2 large and small values of 𝑗. 𝛽󸀠 =0 𝛽=0 Now, by setting and from (62), one 2.1 obtains the energy levels of the standard Dirac oscillator in 4 the nonrelativistic limit as follows: c 2 2

1 1 −m 𝑊𝑛 =ℎ𝜔(𝑛−𝑗+ ), 𝑗=𝑙+ , 2

2 2 W (63) 1.9 3 1 𝑊 =ℎ𝜔(𝑛+𝑗+ ), 𝑗=𝑙− . 𝑛 2 2 1.8 󸀠 Using 𝑛=2𝑛+𝑙, we confirm that the average energy between 0 0.0002 0.0004 0.0006 0.0008 0.001 ̃ 󸀠 theup-spinandthedown-spinstatesis𝑊𝑛 = ℎ𝜔(2𝑛 +𝑙+ 𝛽󳰀 (1/2)) which differs from the usual nonrelativistic harmonic 󸀠 oscillator eigenvalue by ℎ𝜔,whichisattributedtothespin- Figure 1: Splitting of degenerated energies in the presence of 𝛽 .The (𝑛, 𝑙) orbit coupling. In the presence of the minimal length, the value of , in turn from upper line, are (2,1), (3,2), (4,3), (5,4), 𝑗=𝑙+1/2𝛽 = 0.001 𝑐=1𝑚=1 following can be shown: and (6,5). We have taken , , , , 𝜔=1,andℎ=1and all these values are chosen dimensionlessly for 1 convenience. 𝑊̃ =ℎ𝜔{(2𝑛󸀠 +𝑙+ ) 𝑛 2

𝑚𝜔ℎ𝛽 1 nonrelativistic expression of the energy levels exactly reduces ×[1+ (2𝑛󸀠 +𝑙+ )] 2 2 (64) to that for the energy of 3D harmonic oscillator with the minimal length scale [24]. 𝑚𝜔ℎ𝛽󸀠 1 Finally, it is important to note that our results were − (2𝑛󸀠 +𝑙− )}, 2 2 obtained using a noncovariant deformed algebra given by (1)– (3), and it is legitimate to ask whether the different properties of the energy spectrum obtained in this work remain valid if where additional spin-orbit-like contribution proportional to 𝛽 𝛽󸀠 oneusesacovariantdeformedalgebraliketheoneintroduced and appears. in [47, 48]. This issue will be addressed in our forthcoming contribution on the subject of this context. Indeed, the 4. Conclusion concept of minimal length became one of common factors for many different formulations of quantum mechanics with In this paper, we have exactly solved the Dirac oscillator equa- general relativity, due to the fact that minimal length is an tion in 3 dimensions in the framework of relativistic quantum intrinsic scale characterizing the physically meaningful finite mechanics with minimal length. Quantum features of the minimal size in string theory. system such as wave functions and the energy spectrum are investigated using common techniques for noncommutative geometry algebra. Acknowledgments 󸀠 Although our wave functions for 𝛽 =0̸ and 𝛽 =0̸ exactly coincide with the ones obtained in [14]usingsupersymmetric Malika Betrouche is indebted to Professor Khireddine quantum mechanical (SUSYQM) formalism, it is shown that Nouicer (Laboratory of Theoretical Physics and the Depart- the energy spectrum we have obtained here is different from ment of Physics, University of Jijel, Algeria) for his interest theirs. An important point of our result, that distinguishes his generous assistance and fruitful discussions during this it from previous works, is that the presence of the minimal work.TheworkofM.Maamachewassupportedbythe length reveals a quadratic dependence of the energy spectrum Agence Thematique´ de Recherche en Sciences et Technologie on quantum number 𝑛,aswellastheappearanceoftheusual (ATRST) contract of May 2, 2011 (code pnr: 8/u19/882). term that is linearly dependent on 𝑛, implying the property The work of J. R. Choi was supported by the Basic Science 󸀠 of hard confinement of the system. However, for 𝛽 =0,the Research Program through the National Research Founda- energy spectrum to the first order in 𝛽 becomes the same as tion of Korea (NRF) funded by the Ministry of Education the one obtained in [14]. (Grant no. 2013R1A1A2062907). An interesting feature that we have shown in this work is that the well-known usual infinite degeneracy of the References usualDiracoscillatorenergylevelsiscompletelyremoved 󸀠 bythepresenceoftheminimallengthinthecase𝛽 =0̸ . [1] G. Veneziano, “A stringy nature needs just two constants,” We see from Figure 1 that all of degenerated energies under Europhysics Letters,vol.2,no.3,pp.199–204,1986. 󸀠 󸀠 𝛽 =0are splitting in the presence of 𝛽 ,andtheinterval 󸀠 [2] D. Amati, M. Ciafaloni, and G. Veneziano, “Can spacetime be between split energies becomes large as 𝛽 increases. It is also probed below the string size?” Physics Letters B,vol.216,no.1-2, confirmed that, when we ignore the spin-orbit coupling, the pp.41–47,1989. Advances in High Energy Physics 9

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Hossenfelder, “The minimal length and large extra dimen- oscillator in the presence of minimal lengths,” JournalofPhysics sions,” Modern Physics Letters A,vol.19,no.37,pp.2727–2744, A,vol.39,no.18,pp.5125–5134,2006. 2004. [36] M. Moshinsky and Y. F. Smirnov, The Harmonic Oscillator in [16] S. Hossenfelder, “Running coupling with minimal length,” Modern Physics, Harwood, Amsterdam, 1996. Physical Review D, vol. 70, no. 10, Article ID 105003, 11 pages, [37] R. P. Mart´ınez-y-Romero, H. N. Nuńez-Y˜ epez,´ and A. L. Salas- 2004. Brito, “Relativistic quantum mechanics of a Dirac oscillator,” [17] Hossenfelder, “Suppressed black hole production from minimal European Journal of Physics,vol.16,no.3,pp.135–141,1995. length,” Physics Letters B, vol. 598, no. 1-2, pp. 92–98, 2004. [38] J. Ben´ıtez, R. P. Mart´ınez y Romero, H. N. Nuńez˜ Yepez,´ and A. [18] R. R. Sastry, “Quantum mechanics of smeared particles,” Journal L. 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[43] M. Maggiore, “A generalized uncertainty principle in quantum gravity,” Physics Letters B, vol. 304, no. 1-2, pp. 65–69, 1993. [44] M. Maggiore, “Quantum groups, gravity, and the generalized uncertainty principle,” Physical Review D,vol.49,no.10,pp. 5182–5187, 1994. [45] P.Strange, Relativistic Quantum Mechanics, Cambridge Univer- sity Press, Cambridge, UK, 1998. [46] A. Erdelyi,W.Magnus,F.Oberhettinger,andF.G.Triconi,´ Higher Tran-Scendental Functions,vol.2,McGraw-Hill,NY, USA, 1953. [47] C. Quesne and V. M. Tkachuk, “Lorentz-covariant deformed algebra with minimal length and application to the (1 + 1)- dimensional Dirac oscillator,” Journal of Physics A,vol.39,no. 34, pp. 10909–10922, 2006. [48] C. Quesne and V. M. Tkachuk, “Lorentz-covariant deformed algebra with minimal length,” Czechoslovak Journal of Physics, vol. 56, no. 10-11, pp. 1269–1274, 2006. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 516396, 8 pages http://dx.doi.org/10.1155/2013/516396

Research Article Numerical Simulations on Nonlinear Dynamics in Lasers as Related High Energy Physics Phenomena

Andreea Rodica Sterian

Academic Center of Optical Engineering and Photonics, Polytechnic University of Bucharest, 313 Splaiul Independentei, 060042 Bucharest, Romania Correspondence should be addressed to Andreea Rodica Sterian; [email protected]

Received 4 August 2013; Accepted 16 September 2013

Academic Editor: Mehmet Bektasoglu

Copyright © 2013 Andreea Rodica Sterian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper aims to present some results on nonlinear dynamics in active nanostructures as lasers with quantum wells and erbium doped laser systems using mathematical models, methods, and numerical simulations for some related high energy physics phenomena. We discuss nonlinear dynamics of laser with quantum wells and of fiber optics laser and soliton interactions. The results presented have important implications in particle detection and postdetection processing of information as well as in soliton generation and amplification or in the case that these simulations are thought to be useful in the experiments concerning the high energy particles. The soliton behaviour as particle offers the possibility to use solitons for better understanding of real particles in this field. The developed numerical models concerning nonlinear dynamics in nanostructured lasers, erbium doped laser systems, the soliton interactions, and the obtained results are consistent with the existing data in the literature.

1. Introduction high power of petawatt and ultrashort pulses together with a gamma beam system, highly collimated and very brilliant and Numerical simulations on the nonlinear dynamics in lasers tunable [10, 11], so that numerical simulations on nonlinear arelargelypresentedintheliterature,buttheresearch dynamics are essential and inevitable. work for their development has determined unprecedented The mathematical models, the methods, and the numer- progress in particle physics [1–8]. ical simulations from nonlinear dynamics are important The research put into evidence the existence of relevant in processing of data concerning particle physics. Some conditions and limitations which are important in the particle numerical results concerning the dynamics of a two-level detectors and data processing after detection and in the medium under the action of short optical pulses are presented experimental studies in high energy physics also. These in the papers [12–18]. We analyse in the paper the nonlinear unprecedented advances offer opportunities in theoretical dynamics of the lasers with quantum wells and optic fibers as and numerical predictions in high energy physics and also well as the soliton collisions and the behavior of particles in in the related fields. The particles having soliton behavior, as high energy physics as solitons. The used methods are typical being localized and preserved under collisions, lead to a large for the study of quantum fields and particles, particularly number of applications but offer the possibility to use solitons cosmic rays [19]. for better understanding of the real particles, in high energy physics. On the other hand, the research developing in Roma- 2. Modulation Capabilities and nia concerning the Project “Extreme Light Infrastructure- Bifurcation Diagrams Nuclear Physics” (ELI-NP) demonstrate the complexity of problems in the physics and engineering of high energy The numerical simulations of the lasers with quantum particles [9]. The ELI-NP facilities, imply as main tools for the wells (QWL) are based on solving the nonlinear differential research directions, and the realization of the laser systems of equationswhichdescribethebehaviourofthiskindof 2 Advances in High Energy Physics

25 For a suitable modulation index, there are two coexisting stable attractors as it is showed in Figures 2(a) and 2(b),but

ph in the presence of the noise, the induced transitions from one n 20 attractor to another can occur (Figure 2(c)). The contribution oftheincludednoisehadprovedtobeanimportantfactor 15 that limits the design of the format of modulation.

10 3. Case Study on Nonlinear Dynamics A case study in nonlinear dynamics of lasers with quantum 5 wells is analysed systematically based on the model with two

Normalized peak photon density peak photon Normalized equations proposed by Anghel et al. [29–31]. These equations were solved numerically using typical parameters of a laser 0 with multiple quantum wells. The parameter values used 0.3 0.4 0.5 0.6 0.7 0.8 0.9 in numerical simulations are presented in the papers [20, Modulation index m 21]. Simulations were performed in MATLAB programming language using a scheme Runge-Kutta of order four. Detailed Figure 1: Bifurcation diagram: photon density dependence of numerical investigations showed that for the lower frequency modulation index. compared to the frequency of relaxation, dynamics of the system is periodic with the period of the radio-frequency modulating signal. devices [20–31]. Study of functional characteristics of QWL For a sufficiently low modulation index is obtained a is possibly based on the models present in [20], by imple- sinusoidal periodic regime, so that laser output power is menting programs to integrate these equations in MathCad sinusoidal. For a growth of the modulation index and the usingappropriatealgorithms.TheMathCadsoftwarehasthe frequencyofmodulationnearthefrequencyoftherelaxation advantage of solving complicated systems starting from the oscillations, a pulsing mode of operation is obtained with a usual mathematical form by comparison with other methods duration of tens of picoseconds (ultrashort pulses), so that of integration of the nonlinear equations. The modulation the laser system can be used as a source of pulses for opti- studies are important, and for solitons generation in optical cal communications. Higher frequency modulation causes fibers also, the phase self-modulation phenomena are the multiple bifurcation points including a chaotic dynamics for essential mechanism of this process [26]. In general, all the certain parameter values (Figure 3). Also, in Figure 3 critical analysed models had given satisfactory results for the study dependence of the modulation index (which leads to the of transient and dynamics operational regimes at the small bifurcation) of frequency is observed. signal of injection. At high level of injection, the obtained The behavior is similar for other values of the injection numerical results had showed the limits of specific model current above the threshold. However, an increasing depen- used but are consistent with the theoretical analysis. dence of the modulation index for the doubling of period with the current by the polarization can be observed. We have considered the amplitude modulation of the current of injection [27].Theoutputpowerofthelaserversus the modulation index was investigated (Figure 1). 4. Nonlinear Dynamics in Optical Graphical representation of the modulated signal simul- Fibers Systems taneously with the evolution of the system in the phase space highlights also the route to chaos with the increase of 𝑚 by In recent years much attention has been paid to the study of doubling the period. nonlinear dynamics in optical fiber lasers [32–39]. We have examined the diagram of bifurcation of the Numerical simulations discussed here refer to nonlin- photon density versus the modulation index directly (with ear effects in such kind of systems. The chaotic dynamics the increase of 𝑚) and vice versa (with decreasing of 𝑚)for was studied numerically using the rate equations, based a cycle of modulation as an indicator of the system progress. on phenomenon of formation of the ion pairs [40–42]. It The chaotic region has periodic windows and a dynamic type was demonstrated the stable, nonchaotic operation of the of hysteresis. analyzed laser system and discussed the performance of their We have considered also the additional noise sources modulation using the methods of communications theory. in the rate equations [28]. In the Markov approximation, Numerical modeling was realized using the MATLAB the noise sources are random Gaussian variables with zero programmingmedium.Thebaseelementoftheprogram mean and delta-correlated quantities [24]. For numerical was the function ode 45, which realize the integration of representation of the dynamics including the noise sources, the right side expressions of the nonlinear coupled equation we had used the random Gaussian numbers applied by using the method Runge-Kutta. The self-pulsing and chaotic MATLAB functions. Numerical calculations showed that a functioning of the EDFLs have been reported in various small noise disturbs and can also occur transitions between experimental conditions including in the case the pumping is coexisting attractors (Figure 2). near the laser threshold [32–35]. A model for the single-mode Advances in High Energy Physics 3

5 p n 4

1 A 0

3.4 3.5 3.6 3.4 3.5 3.6 3.4 3.5 3.6 ne ne ne

(a) (b) (c)

Figure 2: Gaussian noise influence on the system evolution in the phase plane.

12 8

10 f=7GHz 6 8

f=8GHz 6 4 mW) mW) P ( P ( 4 2 2

0 0 0 2 4 6 8 10 12 024681012 m m

(a) (b)

Figure 3: The laser bifurcation diagrams for different modulation frequencies. laser was analysed firstly taking into account the presence of the ion pairs, the laser system is in a steady state or self- of the erbium ion pairs that act as a saturable absorber. For pulsing operation regime, depending on the value of the thistypeoflasersystemdon’tyetcompletelyareknown pumping parameter. For larger values of the ion pairs, there is the interaction mechanisms, in spite of many published a range of self-pulsation dynamics. This is showed by means works. The nonlinear dynamics of an erbium doped fiber ofabifurcationdiagraminwhichthemodeintensityofthe laser is explained based on a simple model of the ion pairs laser is plotted against the pumping parameter. This dynamics present in heavily doped fibers. The dynamics is reducible has also been encountered in the case of the one-mode model, to four nonlinear differential equations. Next, the nonlinear the change from a stationary stable state to a self-pulsing one dynamics of a two-mode laser with optical fiber is explained being a Hopf bifurcation. in terms of a classical two-modes laser model and two Figure 4(a) shows the maxima of intensity at a larger additional equations of the quantum states of ion pairs [40], concentration of the ion pairs. There exists a region of two so that the laser dynamics is described through six coupled maximum intensities, that is, a period doubling. A small differential equations. At a suitable level of the concentration increase of the ion pairs concentration (Figure 4(b))leads 4 Advances in High Energy Physics

3 )

) 2

units 2 units −3 −3 (10 1 (10 I 1 I 1 1

0 0 1.0 1.4 1.8 2.2 1.0 1.4 1.8 2.2 r r

(a) (b)

Figure 4: Bifurcation diagram of the mode intensity for two values of ion pairs versus the pumping strength 𝑟.

to the appearance of a window of a tripled period. Further a model of the interaction between two solitons, taking increase of the ion-pairs concentrations leads to a more into account both the fiber dispersion and nonlinearity. The complicated bifurcation diagram, including of the regions numerical simulations show how the solitons are propagating with multiple stable states (generalized bistability) and routes in optical fibers and how two solitons interact, passing one tochaoticdynamicssuchastheperiod-doublingrouteora through another, with only a phase change. These simulations quasiperiodic one. are thought to be useful for the designers working in the The used method of the fourth-order Runge-Kutta for digital data transmission and in particles physics as well. the numerical simulation demonstrate the importance of The program Maple10 helps us plot solutions (we can use “computer experiments” in the designing, improving, and animation or individual frames) for different values of the optimization of these coherent optical systems. wave number 𝑘, for different moments 𝑡, and for an arbitrary Some new features of the computer analysed systems interval for 𝑥. For the study of the interaction between two and the existence of some new situations have been put solitons, we need a solution of the KdV equation present in into evidence for their use by the designers in different [3, 50]: applications [42–48]. The obtained results are important not 2 only for the optimization of the functioning conditions of this 𝑢2 := 2 (−𝑘2𝑘2 (𝑘 (𝑥−4𝑘2𝑡)) kind of devices but also for designing of the systems for the 2 1 cosh 1 1 processing of data in the particle physics and in the systems 2 2 2 2 engineering of high energy. +𝑘2𝑘1 cosh (𝑘2 (𝑥−4𝑘2𝑡))

5. Numerical Simulation of 2 2 4 2 2 −𝑘2𝑘1 −𝑘1 cosh (𝑘2 (𝑥−4𝑘2𝑡)) Soliton Propagation 4 2 2 4 In the paper [49] the specific features of the soliton pulses +𝑘1 + cosh (𝑘1 (𝑥−4𝑘1𝑡)) 𝑘2) were studied, so: the type of the nonlinearity, dispersive properties of the propagation medium and the propagation ×(( (𝑘 (𝑥 − 4𝑘2𝑡)) (𝑘 (𝑥−4𝑘2𝑡)) 𝑘 equations, for the optical and acoustic solitons. Were analysed cosh 1 1 cosh 2 2 2 the specific equations as quadratic nonlinear Schrodinger (QNLS), Kerr cubic (CNLS), Saturable (SNLS), discretized 2 − sinh (𝑘2 (𝑥 − 4𝑘 𝑡)) (DNLS), Korteveg de Vries (KdV), Boussinesq, Burgers, and 2 sine-Gordon. Also a study of the methods of discretization −1 2 2 used by the different numerical simulations was realised. × sinh (𝑘1 (𝑥 − 4𝑘1𝑡))1 𝑘 ) ) . A numerical simulation of soliton propagation, based on the Korteweg-de Vries equation, using a powerful PC (1) program (Maple10) which permits to perform numerical calculations, plot or animate functions, and manage ana- With that solution, Maple10 helps us visualize the interaction lytical expressions is presented in paper [50]. We discuss between the two solitons as in Figure 5. Advances in High Energy Physics 5

1.6 1.6 1.2 1.4 1.4 1 1.2 1.2

1 1 0.8

0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2

0 0 0 −40 −20 20 40 −40 −20 20 40 −30 −20 −10 10 20 30 x x x

1.6 1.6

1.4 1.4

1.2 1.2

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 −40 −20 20 40 −40 −20 20 40 x x

Figure 5: The interaction of two solitons for 𝑘1 = 0.4; 𝑘2 = 0.9; 𝑥 = −50 ⋅ ⋅ ⋅ 50; 𝑡 = −10; −2;. 0;2;10

We can see at both solitons the regaining of their shapes 6. Solitons Like Behavior of Particles in High after collision (after they pass one through another). Obvi- Energy Physics ously, it is better to use the animation feature of the Maple10 program by modification of the parameters and observing the The behavior as particle of the solitons (they are localised changes. and preserved in the collision) leads to a large number of For designers working in digital data transmission, the applications and is important in the particle physics also simulation of the solitons propagation along optical fibers [50]. It is well known that we hope the solitons to be is useful and may be performed successfully also with used for a better understanding of real particles. There are Maple10. The animated images are some useful features of this already soliton models of nuclei. On the other hand, there are theories which permit us to see particles such as electrons computer program. Especially, the collision of two solitons and positrons as being solitons in appropriate cases. In the and of 𝑘 dependence of the soliton propagation can be studied paper [52] they had proposed a general semiclassical method by this means, respectively. for computing of the probability of the production of soliton- As it was stressed above, after the collision, the solitons antisoliton pairs in the particle collisions. They calculated also regain their original shape and velocity [8, 50, 51]. The only the probability of this process at high energies. In the paper remaining effect of the scattering is a phase shift (i.e., a change [53] they had studied the creation of the solitons from the in the position they would have reached without interac- particles, using an suitable model used as a prototype. It was tion). For the trisolitonic model, using Maple11 similarly, considered the scattering of the small identical wave pulses, the solution is plotted in Figure 6, showing the collision of which are equivalent to a sequence of particles, and they three solitons [49, 51], by choosing various values for the found that kink-antikink pairs are created in a large region parameters. oftheparametersspace.Thedissertation[54]isdevoted 6 Advances in High Energy Physics

t=0. t = 1.6667

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

−10 −5 0 5 10 −10 −5 0 5 10 x x

t = 2.0833 t = 3.3333 0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

−10 −5 0 5 10 −10 −5 0 5 10 xx

Figure 6: The interaction of three solitons.

to the study of monopoles and solitons in particle physics. (ii) A computational software was elaborated to study the The monopoles and solitons are of relevance as intermediate nonlinear phenomena in quantum well lasers (QWL). and final states of the particles in the collision experiments. Mathcadprovedtobeapowerfultoolinanalyzing The study of monopoles and solitons in the theory of strong and designing of these models. For the QWL, we interactions had shown the possible interpretation of the put into evidence their complex nonlinear dynamics, solitons as baryons. However, so far there is no general characterized by bifurcation points and chaos, the theory in which the particles are described as solitons. critical values of the parameters being determined. There are macroscopic phenomena such as the spontaneous transparency and the behavior of light in fiber optics that are (iii) The classical Runge-Kutta method was used success- understood only in terms of solitons. The methods specified fully for numerical simulation of the complex nonlin- above are typical for the quantum fields and the particles ear dynamics in lasers with optical fibers. The chaotic study, particularly of the cosmic rays [55–61]. dynamics was studied numerically based on the rate equations and on the phenomenon of formation of the ion pairs. 7. Conclusions (i) The nonlinear dynamics in lasers was treated taking (iv) Our results put into evidence the existence of the new into account the complexity of the phenomena which situations which are important for the optimization govern the field-substance interaction with applica- of the functioning conditions for this kind of devices. tions in particle detectors and data processing after The nonlinear dynamics can be controlled, and unde- detection, but these methods are typical for the study sired situations can be avoided if it is necessary, by and for quantum fields and particles also. suitable parameters choosing. Advances in High Energy Physics 7

(v) The studies on laser modulation are important for the [11] O. Tesileanu, D. M. Filipescu, Gh. Cata-Danil, and N. V.Zamfir, generation of solitons in optical fibers, the phase self- “Nuclear photonics at ELI-NP,” Romanian Reports in Physics, modulation phenomenon being the essential mech- vol. 64, supplement, pp. 1373–1379, 2012. anismofthisprocess.Thestudyoftheinteraction [12] M. E. Crenshaw, M. Scalora, and C. M. Bowden, “Ultrafast intr- between two solitons using a powerful PC program insic optical switching in a dense medium of two-level atoms,” (Maple10) is of great interest for a better understand- Physical Review Letters,vol.68,no.7,pp.911–914,1992. ing of real particles in spite of the fact that now there is [13]V.Ninulescu,I.Sterian,andA.-R.Sterian,“Switchingeffectin no general theory in which the particles are described dense media,” in Advanced Laser Technologies (ALT ’01),vol. as solitons. 4762 of Proceedings of SPIE, pp. 227–234, September 2001. [14] V. 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Research Article Regular Solutions in Vacuum Brans-Dicke Theory Compared to Vacuum Einstein Theory

Alina Khaybullina,1 Ramil Izmailov,1 Kamal K. Nandi,1,2 and Carlo Cattani1,3

1 Ya.B. Zel’dovich International Center for Astrophysics, BSPU, Ufa 450000, Russia 2 Department of Mathematics, University of North Bengal, Siliguri 734013, India 3 Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

Correspondence should be addressed to Carlo Cattani; [email protected]

Received 28 June 2013; Accepted 24 August 2013

Academic Editor: Gongnan Xie

Copyright © 2013 Alina Khaybullina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We will confront some static spherically symmetric vacuum Brans-Dicke solutions in the Jordan and Einstein Frames with the Robertson parameters. While the regular solution in the vacuum Einstein theory is just the Schwarzschild black hole, the same in the Jordan frame Brans-Dicke theory is shown to represent not a black hole but a traversable wormhole. But, in this case, the valid range of 𝜔 becomes too narrow to yield the observed weak field Robertson parameters at the positive mass mouth. The corresponding solution in the Einstein frame also provides a regular wormhole, and it yields the correct parametric values but only up to “one and half order.” We argue that a second-order contribution can in principle distinguish between the signatures of the regular wormhole and the singular Buchdahl solution in the Einstein frame. Thus, at the level of regular solutions, Brans-Dicke theory in each frame predicts effects very different from those of Einstein’s theory. To our knowledge, these theoretical distinctions seem not to have received adequate attention so far.

1. Introduction the singular Brans Class I solution, and nothing is asked if regular solutions exist at all in the theory. However, the It is known from history that Einstein was much influenced singularity in Brans Class I solution for unrestricted 𝜔 should by Mach’s principle1 and that he tried to incorporate the be mandatorily admitted if one wants the solution to test principle in his theory of general relativity. The principle has againsttheweakfieldsolarsystemtests. many variants, but within the field theory framework, the TheBDfieldequationsaswellastheClassIsolutionyield simplest one is that a static distribution of matter should post-Newtonian (pN) expansion that coincides at |𝜔| → ∞ cause only a static geometry. However, Godel¨ [1]in1949 with the same expansion of Einstein’s field equations and so showed that this is not the case: Einstein’s general relativity the latter has gained popularity as the local solution. The fact equations produce a rotating universe even though the matter that it has a naked singularity is noted, but the solution is not distribution is static! The incorporation of Mach’s principle abandoned as unphysical. Moreover, the current limit on the in a field theory framework was accomplished by Brans and coupling constant |𝜔| ≥ 50000 creates the impression that BD Dickein1961[2]. There has been a lot of important works theory no longer stands as a distinct alternative to Einstein’s in wormholes and cold black hole spacetimes initiated by theory. However, in this context, we mention that the belief Bronnikov et al. [3–5] earlier, Brans et al. listed four classes that Einstein’s theory is recovered from BD theory at |𝜔| → of static spherically symmetric solutions in their theory ∞ isnotalwaystrue,butwewillnotdiscussithere(see,for [6]. On the other hand, enormous work in the literature details, [7]). exists exploring various features of the Brans-Dicke (BD) If we want the naked singularity in Class I solution to theory.However,mostoftheworksdealingwithspherically disappear, additional inequalities connecting the integration symmetric solutions in the BD theory concentrate only on constants 𝐶 and 𝜆 must be introduced reducing the solution 2 Advances in High Energy Physics

𝑀 𝑀2 3 𝑀3 to what were termed as cold black hole spacetimes. Cold =−(1−2𝛼 +2𝛽 − 𝜖 +⋅⋅⋅)𝑑𝑡2 because the Hawking temperature is zero. This leads to several 1 𝑟 1 𝑟2 2 1 𝑟3 difficulties on physical grounds. First, the inequality severely 𝜔 𝑀 3𝛿 𝑀2 restricts the range of to an absurdly limited range so that +(1+2𝛾 + 1 +⋅⋅⋅) the weak field tests are out of question. Second, although 1 𝑟 2𝑟2 coordinate patch can be extended beyond the cbh horizon, it 2 2 2 2 2 leads to disruption of freely falling particles caused by large ×[𝑑𝑟 +𝑟 (𝑑𝜃 + sin 𝜃𝑑𝜓 )] , tidal forces just above the horizon. The cbh then becomes (1) naked black hole alaHorowitz and Ross. For Campanellu Lousto BH condition, namely, 𝐶+1 ≥ 2𝜆,allcurvature 𝛼 𝛽 𝜖 𝛾 𝛿 𝑟=𝐵 𝐶=−1/(𝜔+2) where 1, 1, 1, 1,and 1 are constant parameters to be invariants vanish at .With , this implies found by fitting with observations. As well known, Brans a very restrictive range −(2 + 1/√3) < 𝜔 < −2, while 𝑟=𝐵 𝐶+2>𝜆⇒𝜔<−2 Class I solution [6]intheJFisgiveninisotropiccoordinates being an outgoing null surface requires . (𝑡, 𝑟,)by 𝜃,𝜓 Excess tidal forces become finite if 𝜉=𝐶/2𝜆>1,but 𝑟=𝐵 𝐶/2𝜆 <1 Hawking temperature finite at requires . 𝑑𝜏2 =−𝑒2𝛼(𝑟)𝑑𝑡2 +𝑒2𝛽(𝑟) 𝑎 = (1/𝜆)((𝐶 + 2)/2).Fortherange−(2 + 1/√3)<𝜔<−2, 𝜉=𝐶/2𝜆<1 2 2 2 2 2 we see that ;hence,tidalforcestearapartatest ×[𝑑𝑟 +𝑟 (𝑑𝜃 + sin 𝜃𝑑𝜓 )] , body. But solar tests are out of question. 1/𝜆 For regular Class II solution, which is a wormhole, weak 𝛼(𝑟) 𝛼 1−𝐵/𝑟 −2 < 𝜔 < −3/2 𝑒 =𝑒 0 [ ] , field tests are not possible because ,buttidal 1+𝐵/𝑟 forcesarefiniteatthethroat. (2) 2 (𝜆−𝐶−1)/𝜆 The question is if all observed tests boil down to only 𝛽(𝑟) 𝛽 𝐵 1−𝐵/𝑟 𝑒 =𝑒 0 [1 + ] [ ] , tallying with the pN or for that matter model independent 𝑟 1+𝐵/𝑟 Robertson parameters, what is the need for dealing with exact solutions at all in both the theories? The answer is that the 1−𝐵/𝑟 𝐶/𝜆 𝜑 (𝑟) =𝜑[ ] . exact solutions offer richer, and testable, geometric structures 0 1+𝐵/𝑟 of black holes, naked singularities, wormholes, and so forth. It is along this thought that we want to investigate one BD When this solution is put in the BD field equations, it yields regular solution and its conformal version and ask what kind the constraint equation as of geometry it represents, and where exactly it differs from 𝜔𝐶 naked singularities or black holes. Previously, it was reported 𝜆2 ≡ (𝐶+1)2 −𝐶(1− )>0, in the literature (see, for details, [8, 9]) that the desired regular 2 (3) solution of the vacuum BD theory in the Jordan Frame (JF) as well as in the conformally rescaled Einstein Frame (EF) where 𝜆, 𝛼0, 𝛽0, 𝐵, 𝐶,and𝜑0 are real constants. The constants represents traversable wormhole geometries.2 𝛼0 and 𝛽0 are determined by asymptotic flatness at 𝑟=+∞ Our purpose here is to confront the singular and regular as 𝛼0 =𝛽0 =0. solutions with the generic Robertson parameters. We argue Under the usual Newtonian limit giving 𝑀=2𝐵/𝜆and how a second-order contribution to deflection can in princi- 𝐶(𝜔) = −(1/(𝜔 +2)),theabovesolutionexpandssuchthat ple differentiate between a wormhole and a singular solution 𝛼1 =1, 𝛽1 =1,and𝛾1 = (𝜔+1)/(𝜔+2).Forfinite𝜔, in the EF. The practical problems about measuring the the solution has a naked singularity at 𝑟=𝐵.Further,the second-order contribution in model independent situation constraint (3)yields are also addressed briefly. The general conclusion is that those 2𝜔 + 3 regular natural wormhole spacetimes would never reduce at 𝜆=±√ . (4) any limit to Schwarzschild black hole of the vacuum Einstein 2𝜔 + 4 theory. (We take 𝐺=1, 𝑐=1.) 𝛼(𝑟) The negative 𝜆 is ruled out as it makes 𝑒 singular at 𝑟=𝐵 and the mass 𝑀 negative. The solutions (2)and(3)withthe positive value of 𝜆, even though singular, immediately yield 2. Brans Class I Singular Solution in JF the above weak field Robertson expansion1 ( ) that depends on arbitrary values of 𝜔. Indeed even the limit 𝜔→±∞ We begin by starting with the Robertson expansion for a is possible, making the solution eventually indistinguishable centrally symmetric configuration in isotropic coordinates fromtheEinsteinvalues𝛼1 =1, 𝛽1 =1,and 𝛾1 =1,and given by [10, 11] of course from (4)theultimateSchwarzschildlimit𝜆=1, 𝐶=0isachieved.WeknowthattheSchwarzschildsolution is a regular black hole solution in Einstein’s theory. A natural question is to ask whether or not we can find a regular 2 2 𝑑𝜏 =−𝐵(𝑟) 𝑑𝑡 +𝐴(𝑟) solution in the JF Brans-Dicke theory. The answer is we can, and it is exactly this regular solution that makes the passage 󸀠2 2 2 2 2 ×[𝑑𝑟 +𝑟 (𝑑𝜃 + sin 𝜃𝑑𝜓 )] to Schwarzschild limit impossible. Advances in High Energy Physics 3

󸀠 3. Regular Wormhole in JF (wormhole throat). The solution in the positive mass2𝐵 ( /Λ) side formally expands exactly as desired in the Robertson The regular solution can be obtained from the Class I solution expansion (1) leading to parameter values 𝛼1 =𝛽1 =1, by the following Wick-like rotations on it: 4 2 𝛾1 = (𝜔+1)/(𝜔+2),and𝛿1 =(𝜔 +23𝜔+14)/6(𝜔+2) ,similar 1 𝑖 to those of singular solution of Section 2.Butthestoryisnot 𝑟󳨀→ ,𝐵󳨀→,𝜆󳨀→−𝑖Λ, 𝑟󸀠 𝐵󸀠 complete yet. Note that we have not yet specified the range of (5) values of 𝜔.With𝐶 = −(1/(𝜔+2)),thisrangewillcomefrom 󸀠 𝛼0 󳨀→ 𝜖 0,𝛽0 󳨀→ 𝛿 0 +2ln 𝐵 , the constraint (3), namely, 𝐵󸀠 Λ 2𝜔 + 3 where and are real. Using the identity Λ=√−( ), (14) 2𝜔 + 4 −1 𝑖 1−𝑖𝑥 tan (𝑥) = ln ( ), (6) 2 1+𝑖𝑥 so that 󸀠 we arrive at the metric functions and the scalar field as + 2𝐵 𝑀 = . (15) follows: √− ((2𝜔 + 3) / (2𝜔 + 4)) 2 2𝛼(𝑟󸀠) 2 2𝛽(𝑟󸀠) 𝑑𝜏 =−𝑒 𝑑𝑡 +𝑒 + Now Λ has to be real in order for the mass 𝑀 to be real, (7) 𝜔 ×[𝑑𝑟󸀠2 +𝑟󸀠2 (𝑑𝜃2 + 2𝜃𝑑𝜓2)] , and it is exactly from here that a narrow range for appears, sin namely, −2<𝜔<−3/2. Consequently, it does not yield the desired value of positive 𝛾1 ∼1, hence inconsistent with solar where system tests. Even the possibility 𝜔→±∞is ruled out in 󸀠 + 󸀠 2 −1 𝑟 this case since Λ and 𝑀 become imaginary. Alternatively, 𝛼(𝑟 )=𝜖 + ( ), 󸀠 + 0 Λtan 𝐵󸀠 (8) we may allow 𝐵 and Λ to be imaginary to keep 𝑀 real, but then we have to sacrifice regularity because we would get back 2 (𝐶+1) 𝑟󸀠 𝑟󸀠2 the same singular Class I solution as soon as these imaginary 𝛽 (𝑟󸀠) =𝛿 − −1 ( ) − ( ) , 0 Λ tan 𝐵󸀠 ln 𝑟󸀠2 +𝐵󸀠2 (9) quantities are put into (5)and(11). Hence, in the JF,the regular Brans-Dicke solution does not explain solar system tests nor 2𝐶 𝑟󸀠 does it go over to Schwarzschild black hole (no horizon). It is 𝜑(𝑟󸀠)=𝜑 [ −1 ( )], 0 exp Λ tan 𝐵󸀠 (10) just a regular wormhole born as such and existing ever since. This is the price of having a regular BD solution! We now 𝜔𝐶 Λ2 ≡𝐶(1− )−(𝐶+1)2 >0. concentrateonthepictureintheEF. 2 (11)

󸀠 Asymptotic flatness at 𝑟 =∞requires that 4. Buchdahl Singular Solution in EF 𝜋 𝜋 (𝐶+1) The picture in the EF is completely different due to the fact 𝜖 =− ,𝛿= . (12) 𝜔 0 Λ 0 Λ that does not appear in this frame. Under the substitution 󸀠 𝑔̃ =𝜑𝑔 , The above solution set is regular everywhere including at 𝑟 = 𝜇] 𝜇] 󸀠 𝐵 and is an exact nonsingular traversable wormhole solution 2𝜔 + 3 𝑑𝜑 (16) (see, for details, [9]). Incidentally, Brans [6]mentionedthe 𝑑𝜙 = √ ,𝜇=0,̸ above solution as Class II among his list of four classes but is 2𝜇 𝜑 not recognized as such.3 However, though not independent, we see that Class I and II solutions are fundamentally thevacuumBDactiongoesintotheactionintheEFinthe different as the former is singular, while the latter is a regular variables (𝑔̃𝜇],𝜙) as follows: wormhole. Does there anyway exist a transition from this regular Brans-Dicke to Schwarzschild solution under 𝜔→ 𝑆 = ∫ 𝑑4𝑥√−𝑔[̃ R̃ −𝜇𝑔̃ 𝜙 𝜙 ], EF 𝜇] ,𝜇 ,] (17) ±∞?No,asweshowbelow. 󸀠 𝑒2𝛼(𝑟 ) It can be verified that the weak field expansions of , where we have introduced a constant arbitrary parameter using relevant identities, yield, respectively, the Schwarzs- 𝜇(=±1)thatcanhaveanysign.Thefieldequationsfrom 𝑀± child masses action (17), which are just Einstein’s equations with source, 󸀠 + 2𝐵 󸀠 are given by 𝑀 = as 𝑟 󳨀→ + ∞, Λ 𝑅̃ =𝜇𝜙 𝜙 , (13) 𝜇] ,𝜇 ,] 2𝐵󸀠 2𝜋 (18) 𝑀− =− [ ] 𝑟󳨀→−∞,̂ 2 Λ exp Λ as ◻ 𝜙=0. 𝜇] of two asymptotically flat spacetimes on two different coordi- We see that if the kinetic term 𝜇𝑔 𝜙,𝜇𝜙,] in the action has an nate charts lying on either side of the minimal circumference overall positive sign (𝜇=+1), the stresses satisfy all energy 4 Advances in High Energy Physics conditions. The counterpart of the singular Brans Class I them. Note that unlike 𝜔 in Section 3, 𝛾 here is unconstrained. solution (2)and(3)intheconformallyrescaledEFthusis TheabovesolutionisjustavariantoftheregularEllis- Bronnikov wormhole.4 2𝛾 −1 󸀠 󸀠 −1 󸀠 󸀠 2 1 − 𝑚/2𝑟 2 Using the identity tan (𝑟 /𝐵 )=𝜋/2−tan (𝐵 /𝑟 ) for 𝑑𝜏 =−( ) 𝑑𝑡 󸀠 Buch 1+𝐵/2𝑟 𝑟 >0, which corresponds to the positive mass mouth 𝑀(= 󸀠 2𝐵 𝛾), the metric functions (22)and(23) expand as 𝑚 2(1+𝛾) 𝑚 2(1−𝛾) +(1− ) (1 + ) (19) 2𝑟 2𝑟 2 3 󸀠 2𝛼1𝑀 2𝛽1𝑀 3 𝑀 2 2 2 2 2 2 𝑃(𝑟 )=1− + − 𝜖 +⋅⋅⋅, ×[𝑑𝑟 +𝑟 𝑑𝜃 +𝑟 sin 𝜃𝑑𝜓 ], 𝑟󸀠 𝑟󸀠2 2 1 𝑟󸀠3 (25) 2𝛾 𝑀 𝑀2 1 2 𝑄(𝑟󸀠)=1+ 1 + (2 + )+⋅⋅⋅ , 2(1−𝛾 ) 1 − 𝑚/2𝑟 󸀠 󸀠2 2 𝜙 (𝑟) = √ [ ], (20) 𝑟 𝑟 2𝛾 Buch 𝜇 ln 1 + 𝑚/2𝑟

giving values 𝛼1 =𝛽1 =1, 𝛾1 =1, indicating that the solution where 𝜇, 𝑚,and𝛾 are arbitrary positive constants. This is yields exactly the same parametric values as in the vacuum known as Buchdahl solution [12]. The metric shares all the Einsteintheorybutonlyuptothe“oneandhalforder.”Bythat characters of the Class I solution. Though singular at 𝑟= 󸀠 term, we mean first order in 𝑀 in 𝑄(𝑟 ) and second order in 𝑚/2, the Buchdahl solution also yields the weak field solar 󸀠 𝑀 in 𝑃(𝑟 ).Thereasonisthattheparametervalues𝛼1 =𝛾1 = system Robertson parameter values up to “one and half order” 1 are enough to explain the weak field light deflection, but (see below) with the definition of 𝑀=𝑚𝛾.At𝛾=1, the perihelion advance of planets in the solar system, though Schwarzschild black hole is recovered. The result of Section 2 2 still a weak field test, requires information up to 𝑀 order in and the present one indicate that the local tests and the 𝑃(𝑟󸀠) 𝛽 =1 possibility of ultimate Schwarzschild limit go hand in hand ;thatis, 1 . In that sense, this particular test might with allowing a singularity in respective spacetimes. be termed as a “one and half order” test. One might wonder 𝜇] why the previous JF solutions (7)–(11)involvingnarrow𝜔 If however the kinetic term 𝜇𝑔 𝜙,𝜇𝜙,] has an overall negative sign (𝜇=−1), one has massless ghost matter do not support solar system tests at any order, while the present EF solutions (21)–(24)atleastsupportthemupto source for which all the energy conditions including the 1.5 dominant energy condition are violated giving rise to the order. In our opinion, this fact clearly underscores the possibility of wormholes. This is discovered by Ellis [13]and fundamental distinction between Machian BD and Einstein at the same time independently by Bronnikov [3]. For details theory at the level of regular solutions. No such distinction about the relation between ghost and wormholes, see [4, 5]. exists between singular solutions in either frame—both BD Class I and Buchdahl solutions support those tests to any However, such wormholes are unstable with respect to linear 𝜔 𝛾→1 perturbations [14, 15]. order for arbitrarily large in the former and at in the latter. These successes of the singular solutions led many tobelievethatitisonlyamatteroflarge𝜔,thelargerthevalue 5. Regular Wormhole in EF the closer it is to Einstein’s theory; the impossibility of having 𝜔 𝜇=−1 large in the regular solution (Section 3) has gone unnoticed. Choosing and using (16), the metric in the EF 𝑀2 𝑄(𝑟󸀠) becomes Atthetruesecondorder,comparingthe term in with the Robertson expansion (1), we see that 𝑑𝜏2 = 𝑔̃ 𝑑𝑥𝜇𝑑𝑥] EF 𝜇] 3𝛿 𝑀2 𝑀2 1 4 1 =−𝑃(𝑟󸀠)𝑑𝑡2 +𝑄(𝑟󸀠) 1 = (2+ ) 󳨐⇒ 𝛿 = + . (21) 2𝑟󸀠2 𝑟󸀠2 2𝛾2 1 3 3𝛾2 (26) 󸀠2 󸀠2 2 2 2 ×[𝑑𝑟 +𝑟 (𝑑𝜃 + sin 𝜃𝑑𝜓 )] , In contrast, it can be verified that the Robertson expansion of where singular Buchdahl (𝑀=𝑚𝛾)solution(19)yields𝛼1 =𝛽1 =1, 𝛾 =1 𝑟󸀠 1 , but at the second order gives 𝑃(𝑟󸀠)= [2𝜖 + 4𝛾 −1 ( )], exp tan 𝐵󸀠 (22) 4 1 𝛿1 = − , (27) 𝐵󸀠2 2 𝑟󸀠 3 3𝛾2 𝑄(𝑟󸀠)=(1+ ) [2𝜁 − 4𝛾 −1 ( )], (23) 𝑟󸀠2 exp tan 𝐵󸀠 which means that one recovers the other Schwarzschild value 󸀠 󸀠 −1 𝑟 2 2 𝛿1 =1at 𝛾=1. This Schwarzschild value is simply impossible 𝜙(𝑟 )=4𝜆tan ( ), where 2𝜆 =1+𝛾. (24) 𝐵󸀠 to obtain in the regular Ellis-Bronnikov wormhole (21)–(24) in the EF that instead gives 𝛿1 =5/3at 𝛾=1.Hence, Asymptotic flatness requires that 𝜖=−𝜋𝛾and 𝜁=𝜋𝛾.The significant differences in prediction would appear in the 2 2 constraint equation 2𝜆 =1+𝛾 among free constants comes second-order contribution distinguishing between singular from the EF field equations when the solution is put into and regular solutions in the EF. Advances in High Energy Physics 5

6. Second Order Deflection trajectory giving the same deflection angle; there is no extra control parameter (since 𝑉=1)toregulatethatangle.Thus We will start our discussion from the model independent massless unbound trajectories are of no use to measure the 𝐴 𝐵 Robertson expansion (1)with and , generic of any spher- second-order deflection. ically symmetric configuration. Now, for particles orbiting On the other hand, the deflection of a massive unbound 𝑉 𝜃=𝜋/2 with velocity on the equatorial plane ( ), the particle trajectory can always be controlled by choosing deflection angle measured by an asymptotic observer is given different velocities 𝑉<1,andfromcurvefittingall by [10] the parameters 𝑟0, 𝛾1, 𝛽1,and𝛿1 can be unambiguously 𝛼(𝑟 )=𝐼(𝑟)−𝜋, determined. The resulting parameter values can then be 0 0 (28) compared with the theoretical predictions of deflection from any spherically symmetric configuration, including those where 𝑟0 is the distance of closest approach, and of Einstein or Brans-Dicke theory. It is exactly here that ∞ 𝑑𝑟 1 1 1 −(1/2) the upper limit on 𝛿1 obtained from fitting can distinguish 𝐼(𝑟 )=2∫ 𝐴−(1/2) (𝑟) [ { −𝐸}− ] , 𝛾=1 0 𝑟2 𝐽2 𝐵 (𝑟) 𝐴𝑟2 between the values in (26)and(27). Choose in the 𝑟0 solution.Ifitturnsoutfromthefitthat𝛿1 ≃1,thenweare (29) possibly looking at a Schwarzschild black hole, and if 𝛿1 ≃ 5/3 where ,thenwearelookingattheEllis-Bronnikovwormhole. A very speculative natural scenario for the observation of 1 1/2 material orbit deflection could be provided by the trajectory 𝐽=𝑟[𝐴 (𝑟 ){ −𝐸}] ,𝐸=1−𝑉2. (30) 0 0 𝐵(𝑟 ) of unstable H1 orbits with nonzero angular momentum 0 moving in and out to asymptotes enclosing the galactic center. 5 Using the expressions for 𝐴(𝑟) and 𝐵(𝑟) from (1), the deflec- Of course, it is technically a very challenging task. tion for unbound particles up to the second-pN order, when 2 2𝑀 ≪ 𝑉 ,becomes[16] 2 𝑀 𝑀 7. Conclusion 𝛼 (𝑟0) =𝑎( ) +𝑏( ) , (31) 𝑟0 𝑟0 The contents of Section 2 show that, in singular BD I, 𝜔 can where assume any value leading to the Schwarzschild limit at 𝜔→ ±∞ 1 . On the other hand, we have just seen in Section 3 that 𝑎=2(𝛾 + ), the regular solution (7)–(11)allowsonlytonarrowarange 1 𝑉2 −2<𝜔<−3/2, which prevents any passage to solar system (32) 3𝜋𝛿 𝜋 1 2 values as is evident from the Robertson parameters. It is also 𝑏= 1 +(2+2𝛾 −𝛽 ) −2(𝛾 + ) . 4 1 1 𝑉2 1 𝑉2 evident that the solution has no horizon, so it cannot tend to a black hole. However, the spacetime can still represent a For bound orbits, one would need to take into account also very different object, a traversable wormhole. (For a classic the effect of 𝜖1,butwearefocusinghereonlyontheunbound treatise on wormholes, see [17].) This wormhole structure orbits. For photon, 𝑉=1,andoneobtainstheknown occurs naturally,inbothJFandEF,asopposedtomany Schwarzschild expression at the values 𝛿1 =𝛾1 =𝛽1 =1. artificially engineered wormholes presented in the literature. However,thereisgreatambiguityintheliteratureabouthow (Bynatural,wemeanawell-definedLagrangiantheory.)In the deflection variables are to be identified. For example, the EF (Section 5), which is just Einstein’s general relativity, consider a light ray just grazing the limb of the sun. If the the wormhole spacetime yields even the known weak field isotropic 𝑟0 is identified with the independently measured solar system parameters up to 1.5 order, but significant Euclidean solar radius, the second-order contribution gives differences appear in the Robertson parameter 𝛿1.Whilein ∼3.5 𝜇sec. If similar calculations are done in the standard the singular Buchdahl solution (Section 4), one can reduce Schwarzschild radial variable, say 𝜌, and identifying the 𝛿1 totheSchwarzschildvalueofunity;thereisnochance distance of closest approach 𝜌0 with the Euclidean solar to do so in the case of wormhole solution. This could in radius, the second-order deflection will become ∼7 𝜇sec. principle mean a decisive test about the observable signature Andfinally,iftheimpactparameter𝑏 is identified with the of the nonsingular Ellis-Bronnikov wormhole vis-a-vis the Euclidean solar radius, the contribution becomes ∼11 𝜇sec singular Buchdahl solution (Section 6). We are aware that [16]. The reason for such ambiguity lies in the fact that instability of wormhole solutions in the massless minimally Euclidean solar radius works well only up to first order, but coupled theory [14, 15] spoils any chance of their very physical not up to the second order as the error involved is also of existence, let alone the prospect of such measurements. the same order. To resolve the ambiguity, one has to regard However, our aim was to point out the theoretical distinctions 𝑟0 or 𝜌0 as another variable together with others like 𝛾1, between different kinds of geometric structures as exposed 𝛽1,and𝛿1 and compute them from the curve fitting to by the Robertson expansion. Note also that we had not observed photon deflection data. Unfortunately, even this enteredtheissueastowhichframeJForEFisphysical.For schemewouldfailduetothefactthatthelightraypassing informative discussions on this issue, see, for example, [18– through 𝑟=𝑟0 (or 𝜌=𝜌0) would always follow the same 22]. 6 Advances in High Energy Physics

Finally, we return to the topic in the title. The answer 5. We are thankful to Bhadra et al. for pointing out that seems to be largely a matter of choice. Depending on the the deflection of ionized hydrogen by the intergalactic attitude, we might accept the singular Class I solution as the magnetic field would mask the small second-order con- solution prizing its weak field successes. Or we may accept tribution. However, the realization that the deflection the nonsingular Class II solution as the solution, which would data, not of light but of only unbound material orbits, then amount to simply denying vacuum Brans-Dicke theory can allow us to test the second-order contribution is very of its observational successes because of narrow range of fundamental and was first advanced in16 [ ]. All planned 𝜔. But it would nonetheless offer us a natural traversable experiments to measure that contribution should take wormhole. The third option is to accept both the solutions into account this proposal. (BD I and II) as describing two different scenarios, but Schwarschild black holes are certainly ruled out in either Acknowledgment framesincethecurvaturesdivergeinBDIandnohorizon in BD II. Since neither black holes nor wormholes have been The authors thank Sonali Sarkar for technical assistance with conclusively observed, we think that the best option is to some of the past literature. keepthechoiceopen.DismissingthevacuumBDtheory as effectively nondistinct from Einstein theory because of large 𝜔 would blind us to the distinct geometric structures References discussed here. [1] K. Godel,¨ “An example of a new type of cosmological solutions of einstein’s field equations of gravitation,” Reviews of Modern Endnotes Physics,vol.21,pp.447–450,1949. [2]C.H.BransandR.H.Dicke,“Mach’sprincipleandarelativistic 1. Ernst Mach, a nineteenth century philosopher, argued theory of gravitation,” Physical Review,vol.124,no.3,pp.925– that local inertial properties of matter are determined 935, 1961. bythematterdistributioninthewholeuniverse.Thisis [3]K.A.Bronnikov,Acta Physica Polonica B,vol.4,p.251,1973. known as Mach’s principle. [4] K. A. Bronnikov, M. V. Skvortsova, and A. A. Starobinsky, 2. In the EF, vacuum BD equations reduce to minimally “Notes on wormhole existence in scalar-tensor and F(R) grav- coupledscalarfieldEinsteintheory.Sinceintroducing ity,” Gravitation and Cosmology,vol.16,no.3,pp.216–222,2010. scalar fields into many theories is the order of the day, [5] K. A. Bronnikov and A. A. Starobinsky, “No realistic wormholes itwillbeusefultoseehowtheregularsolutionsbehave from ghost-free scalar-tensor phantom dark energy,” JETP in the simplest scalar field gravity theory. Letters,vol.85,no.1,pp.1–5,2007. → [6]C.H.Brans,“Mach’sprincipleandarelativistictheoryof 3. The reverse route (Class II Class I) was first shown gravitation. II,” Physical Review,vol.125,no.6,pp.2194–2201, by Bhadra and Sarkar [23] with the conclusion that 1962. the two solutions are not independent. However, the [7] A. Bhadra and K. K. Nandi, “𝜔 dependence of the scalar field detailed wormhole features of their seed Class II were in Brans-Dicke theory,” Physical Review D,vol.64,ArticleID not investigated. 087501, 3 pages, 2001. 4. To put the solution into a more familiar form, transform [8] K. K. Nandi, I. Nigmatzyanov, R. Izmailov, and N. G. Migranov, the radial variable as “New features of extended wormhole solutions in the scalar field gravity theories,” Classical and Quantum Gravity,vol.25,no.16, 𝐵󸀠2 Article ID 165020, 2008. ℓ=𝑟󸀠 − . (33) 𝑟󸀠 [9] A. Bhattacharya, R. Izmailov, E. Laserra, and K. K. Nandi, “A nonsingular Brans wormhole: an analogue to naked black 󸀠 With 𝐵 =𝑚/2, and using the identity holes,” Classical and Quantum Gravity,vol.28,ArticleID 155009, 2011. ℓ+√ℓ2 +𝑚2 𝜋 ℓ [10] S. Weinberg, Gravitation and Cosmology,JohnWiley&Sons, 2 −1 ( )≡ + −1 ( ), tan 𝑚 2 tan 𝑚 (34) New York, NY, USA, 1972. [11] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation,W. we see that the solution set (21)–(24)becomesjustthe H. Freeman, San Francisco, Calif, USA, 1973. Ellis-Bronnikov (EB) wormhole as follows: [12] H. A. Buchdahl, “Reciprocal static metrics and scalar fields in the general theory of relativity,” Physical Review, vol. 115, no. 5, 𝑑𝜏2 =−𝐹 𝑑𝑡2 +𝐹−1 pp. 1325–1328, 1959. EB EB EB [13]H.G.Ellis,“Etherflowthroughadrainhole:aparticlemodel 2 2 2 2 2 2 ×[𝑑ℓ +(ℓ +𝑚 ) (𝑑𝜃 + sin 𝜃𝑑𝜓 )] in general relativity,” Journal of Mathematical Physics,vol.14,p. 104, 1973, Erratum in Journal of Mathematical Physics,vol.15,p. −1 ℓ (35) 520, 1974. 𝐹 = exp [−𝜋𝛾 + 2𝛾 tan ( )] EB 𝑚 [14] J. A. Gonzalez, F. S. Guzman,´ and O. Sarbach, “Instability of wormholes supported by a ghost scalar field: I. Linear stability −1 ℓ 2 2 𝜙 =𝜆[𝜋+2tan ( )] , 2𝜆 =1+𝛾. analysis,” Classical and Quantum Gravity,vol.26,ArticleID EB 𝑚 015010, 2009. Advances in High Energy Physics 7

[15] J. A. Gonzalez, F. S. Guzman,´ and O. Sarbach, “Instability ofwormholes supported by a ghost scalar field: II. Nonlinear evolution,” Classical and Quantum Gravity,vol.26,ArticleID 015011, 2009. [16] A. Bhadra, K. Sarkar, and K. K. Nandi, “Testing gravity at the second post-Newtonian level through gravitational deflection of massive particles,” Physical Review D,vol.75,ArticleID 123004, 2007. [17] M. Visser, Lorentzian Wormholes—From Einstein to Hawking, AIP, New York, NY, USA, 1995. [18] E.´ E.´ Flanagan, “The conformal frame freedom in theories of gravitation,” Classical and Quantum Gravity,vol.21,pp.3817– 3829, 2004. [19] J. E. Lidsey, D. Wands, and E. J. Copeland, “Superstring cosmology,” Physics Report,vol.337,no.4-5,pp.343–492,2000. [20] M. Gasperini and G. Veneziano, “The pre-big bang scenario in string cosmology,” Physics Report,vol.373,no.1-2,pp.1–212, 2003. [21] V. Faraoni, “The 𝜔→∞limit of Brans-Dicke theory,” Physics Letters A, vol. 245, pp. 26–30, 1998. [22] A. Bhadra, K. Sarkar, D. P.Datta, and K. K. Nandi, “Brans-Dicke theory: jordan versus einstein frame,” Modern Physics Letters A, vol. 22, no. 5, pp. 367–375, 2007. [23] A. Bhadra and K. Sarkar, “On static spherically symmetric solutions of the vacuum Brans-Dicke theory,” General Relativity and Gravitation,vol.37,pp.2189–2199,2005. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 837129, 9 pages http://dx.doi.org/10.1155/2013/837129

Research Article The Duals of Fusion Frames for Experimental Data Transmission Coding of High Energy Physics

Jinsong Leng, Qixun Guo, and Tingzhu Huang

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China

Correspondence should be addressed to Jinsong Leng; [email protected]

Received 9 August 2013; Accepted 27 August 2013

Academic Editor: Carlo Cattani

Copyright © 2013 Jinsong Leng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The experimental data transmission is an important part of high energy physics experiment. In this paper, we connect fusion frames with the experimental data transmission implement of high energy physics. And we research the utilization of fusion frames for data transmission coding which can enhance the transmission efficiency, robust against erasures, and so forth. For this application, we first characterize a class of alternate fusion frames which are duals of a given fusion frame in a Hilbert space. Then, we obtain the matrix representation of the fusion frame operator of a given fusion frame system in a finite-dimensional Hilbert space. By using the matrix representation, we provide an algorithm for constructing the dual fusion frame system with its local dual frames which can be used as data transmission coder in the high energy physics experiments. Finally, we present a simulation example of data coding to show the practicability and validity of our results.

1. Introduction when erasures occur in signal transmissions. So, the theory of frames has been developed rapidly in mathematics and Because of the cross-regional feature of high energy physics achieved successful applications in various areas of pure and experiment, there exists a huge amount of data produced applied sciences and engineering in the past twenty years. in the experiment procedure which needs to be transmitted We only mention some applications of frames here such as from each experimental field to the remote center for process- signal and image processing [2], quantization [3], capacity of ing synthetically every day. The relevant technologies in many transmission channel [4–6], coding theory [7–12], and data current data transmission systems are the data transmission transmission technology [13]. protocol GridFTP, the object-related database management With the development of signal processing systems, system PostgreSQL, the application sever JBoss, and so forth frames are restricted and fusion frames appear. The utility in order to ensure the real-time, reliable, and efficient data of fusion frames in handling missing data packet erasures transmission [1]. But in many transmission systems, all the problem is shown in [14]. The theory of fusion frames signalshavetoberetransmittedwhenoneormorevectors was systematically introduced in [15, 16]. Since then, many of data are lost in the transmission process, which leads to excellent results about the theory and application of fusion wasting a lot of time and resources. Why not use fusion frames have been obtained in an amazing speed [15–22]. In frames? We find that they are natural suitable tools for fact, fusion frames are generalization of conventional frames the experimental data transmission coding of high energy andgobeyondthem,andtheyhavebeenfoundtobegood physics. In fact, this application of fusion frames can save tools in large signal processing systems in which distributed more time and resources caused by the retransmission. or parallel processing is required. For instance, in a coding Redundancy is an interesting and attractive feature of transmission process, the encoded and quantized data must frames, because it has at least two advantages. First, it be put in numbers of packets. When one or more packets makes the construction of various classes of frames flexible; arescrumped,lost,ordelayed,fusionframescanenhance secondly, it can enhance the robustness of encoding data the robustness to the packet erasures. Furthermore, we can 2 Advances in High Energy Physics see the successful applications of fusion frames in sensors the Fetcher finds that there are new data directories in the network [23], filter bank24 [ ], transmission coding [14, 25, Dropbox, the original data will be encoded by using a local 26],andsoforth.Wereferto[27] and the reference therein frame, quantized, and stored into some packets in it. Once for more details about the applications of fusion frames. the Sending Directories receives these encoded and quantized We first describe how to use fusion frames for transmis- packets which consist of some vectors from the Dropbox, sion coding in experiment of high energy physics to enhance it will send all these vectors to the processing center and the transmission efficiency and robust against erasures. wait the feedback from the receiver. The feedback is sent by According to the characteristics of the experimental data the Data Checking Module of the processing center when transmission system of high energy physics, the distributed, it confirms that all data from the sender are received. Then parallel, and fused processing are required in the transmis- it submits all these vectors to the Receiving Directories in sion process. Hence, fusion frames can be applied to coding which these vectors will be decoded and fusing processed by a in the transmission scheme to improve the transmission fusion frame system and its dual. Finally, all decoded signals efficiency, stability, and robust of the whole system. are submitted to the Warehouse, and the procedure is over. On the other hand, however, some related problems In the old transmission model, the Data Checking Module of about fusion frames, especially in applications, are still open. the receiver will check the integrity of these received packets. Many excellent results about conventional frames have been When it finds that some vectors or coefficients are lost in achieved and applied successfully, but how to generalize them the transmission process, it will ask the sender retransmit to fusion frames? It is a tempting subject because of the all signals. The re-transmission procedure is unnecessary complexity of the structure of fusion frames compared with if a fusion frame and its dual are used for data coding, conventional frames. For the application of data transmis- and a lot of time and resources are saved. Thus, applying sion, we study mainly the dual fusion frames of a given fusion fusion frames for data coding in the transmission process can frameandthematrixrepresentationsoffusionframesystems enhance the reliability, efficiency, and robust for erasures. of in finite-dimensional Hilbert spaces for constructing dual the transmission system. fusion frame systems in this paper. Then, let us recall and introduce some basic notations, Weoutline this paper as follows. In Section 2,werecallthe concepts, and results about frames and fusion frames that are experimental data transmission course of high energy physics needed for this paper. Let us begin with the concept of frames. and propose a new transmission model in which fusion frame Let H be a separable (real or complex) Hilbert space. A and its dual are used for data coding. Then, we introduce and collection of vectors 𝐹={𝑓𝑖}𝑖∈𝐼 ⊂ H is called a frame for H recall some notations, conceptions, and some basic theory if there exist constants 0<𝐴≤𝐵<∞such that about frames and fusion frame systems. In Section 3,we 󵄩 󵄩2 󵄨 󵄨2 󵄩 󵄩2 𝐴󵄩𝑓󵄩 ≤ ∑󵄨⟨𝑓,𝑓⟩󵄨 ≤𝐵󵄩𝑓󵄩 ∀𝑓 ∈ H firstintroduceakindofalternatefusiondualsbasedon 󵄩 󵄩 󵄨 𝑖 󵄨 󵄩 󵄩 (1) the definition given in20 [ ]. We investigate and characterize 𝑖∈𝐼 these alternate fusion duals. Then, we consider how to get holds for every 𝑓∈H,where𝐼 denote an index set. The the matrix representation of the fusion frame operator of a optimal constants (maximal for 𝐴 and minimal for 𝐵)are given fusion frame system in a finite-dimensional Hilbert called lower and upper frame bounds.If𝐴=𝐵,then𝐹 is called space. So that, based on this matrix representation, a method a tight frame,anditiscalledaParseval frame (Sometimes, a for construction of the dual fusion frame system with its Parseval frame is also called a normalized tight frame)when local dual frames is prescribed. A simulation example is 𝐴=𝐵=1.Auniform frame is a frame when all the elements given to show the practicality and validity of these results in in the frame sequence have the same norm. 2 experimental data transmission. Given a frame 𝐹={𝑓𝑖}𝑖∈𝐼,theoperatorΘ𝐹 : H →ℓ(𝐼) defined by 2. Fusion Frames for Experimental Data Θ (𝑓) = ∑ ⟨𝑓,𝑓⟩𝑒 𝐹 𝑖 𝑖 (2) Transmission Coding of High Energy 𝑖∈𝐼 Physics and Preliminaries is called the analysis operator of 𝐹,where{𝑒𝑖}𝑖∈𝐼 is the standard ℓ2(𝐼) Θ∗ Θ The main function of fusion frame in data transmission orthonormal basis for . The adjoint operator 𝐹 of 𝐹 procedure is data coding to implement distributed, parallel, given by and fused processing of the whole transmission system. A large amount of data produced by experiment sites of ∗ Θ𝐹 (∑𝑐𝑖𝑒𝑖)=∑𝑐𝑖𝑓𝑖 (3) high energy physics is encoded by local frames and stored 𝑖∈𝐼 𝑖∈𝐼 in some packets in the sender sides; the packets from all 𝐹 𝑆 =Θ∗Θ experiment sites are decoded/processed by dual fusion frame is called the synthesis operator of .Ifwelet 𝐹 𝐹 𝐹,then in the center. Based on the conventional transmission system, we have we establish the structure scheme of the data transmission 𝑆 (𝑓) = ∑ ⟨𝑓,𝑓⟩𝑓,𝑓∈𝐻. 𝐹 𝑖 𝑖 (4) system by using fusion frames (see Figure 1)andprecise 𝑖∈𝐼 the transmission procedure briefly as follows. The original data from the Data Acquisition System is transmitted to the Thus, 𝑆𝐹 is a positive invertible bounded linear operator on Dropbox for Temporary Directories by the Data Buffer. When H,whichiscalledtheframe operator of 𝐹. Advances in High Energy Physics 3

Copying Dropbox for Encoding by a local frame Data acquisition system Data buffer temporary Fetcher Sending directories directories (gridFTP) Transmission Sending feedback Sending Decoding/processing by a fusion frame system and its dual Warehouse Receiving directories Data checking module

Figure 1: The experimental data transmission course of high energy physics.

̃ ̃ A collection of vectors 𝐹={𝑓𝑖}𝑖∈𝐼 in H is called a dual of weights {𝑤𝑖}𝑖∈𝐼 where 𝑤𝑖 >0for all 𝑖∈𝐼.Then,{(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 ̃ ̃ H 0<𝐶≤ frame for 𝐹={𝑓𝑖}𝑖∈𝐼 if 𝐹={𝑓𝑖}𝑖∈𝐼 satisfies the reconstruction is called a fusion frame for if there exist constants formula 𝐷<∞such that 󵄩 󵄩2 ̃ ̃ 󵄩 󵄩2 2󵄩 󵄩 󵄩 󵄩2 𝑓=∑ ⟨𝑓,𝑓⟩ 𝑓 = ∑ ⟨𝑓, 𝑓⟩𝑓 ∀𝑓 ∈ H. 𝐶󵄩𝑓󵄩 ≤ ∑𝑤𝑖 󵄩𝑃𝑊 (𝑓)󵄩 ≤𝐷󵄩𝑓󵄩 ∀𝑓 ∈ H, 𝑖 𝑖 𝑖 𝑖 (5) 󵄩 𝑖 󵄩 (7) 𝑖∈𝐼 𝑖∈𝐼 𝑖∈𝐼

where 𝑃𝑊 denotes the orthogonal projection onto the 𝑊𝑖.The Adirectcalculationyields 𝑖 constants 𝐶 and 𝐷 are called the lower and upper fusion frame 𝑓=∑ ⟨𝑓,𝑓⟩𝑆−1 (𝑓)=∑ ⟨𝑓,𝑆−1 (𝑓)⟩ 𝑓 ∀𝑓 ∈ H. bounds.Thefamily{(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 is called a C-tight fusion frame 𝑖 𝐹 𝑖 𝐹 𝑖 𝑖 (6) 𝑖∈𝐼 𝑖∈𝐼 if 𝐶=𝐷,anditiscalledaParseval fusion frame if 𝐶=𝐷=1. The family {(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 is called an orthonormal fusion basis −1 H =⊕ 𝑊 This implies that {𝑆𝐹 (𝑓𝑖)}𝑖∈𝐼 is a dual frame of {𝑓𝑖}𝑖∈𝐼.The if 𝑖∈𝐼 𝑖.ABessel fusion sequence refers to the case −1 when {(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 has an upper fusion frame bound, but not frame {𝑆𝐹 (𝑓𝑖)}𝑖∈𝐼 is called the canonical dual frame of {𝑓𝑖}𝑖∈𝐼. ̃ necessarily a lower bound. If a dual frame {𝑓𝑖}𝑖∈𝐼 is not the canonical dual frame, it is also called an alternate dual frame. {(𝑊,𝑤)} H 𝐹 𝑆 =Θ∗ ⋅Θ = Definition 2. Let 𝑖 𝑖 𝑖∈𝐼 be a fusion frame for ,and Aframe is a tight frame if and only if 𝐹 𝐹 𝐹 {𝑓 } 𝑊 𝐽 𝑖∈𝐼 𝑖𝑗 𝑗∈𝐽𝑖 be a frame of 𝑖 where 𝑖 are index sets for .Then, 𝐴𝐼H for some positive constant 𝐴,where𝐼H is the identity {𝑓 } 𝑖∈𝐼 {(𝑊,𝑤,{𝑓 } )} 𝑖𝑗 𝑗∈𝐽𝑖 , are called local frames, and 𝑖 𝑖 𝑖𝑗 𝑗∈𝐽𝑖 𝑖∈𝐼 operator. A frame 𝐹 is a Parseval frame if and only if 𝑆𝐹 = H 𝐶 𝐷 Θ∗ ⋅Θ =𝐼 𝐹 is called a fusion frame system for .Theconstants and 𝐹 𝐹 H; that is, the canonical dual of is itself. So, are the associated lower and upper fusion frame bounds if they the analysis operator Θ𝐹 of a Parseval frame is an isometry are the fusion frame bounds for {(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼,and𝐴 and 𝐵 are operator. A linear operator 𝑃 from a Hilbert space H to H is 2 the local frame bounds if there are the common frame bounds called an orthogonal projection if 𝑃 is self-adjoint and 𝑃 =𝑃. {𝑓 } 𝑖∈𝐼 𝑘 for the local frames 𝑖𝑗 𝑗∈𝐽𝑖 for each .Thedualframes Given a finite frame 𝐹={𝑓𝑖}𝑖=1 in an 𝑛-dimensional ̃ {𝑓𝑖𝑗 }𝑗∈𝐽 , 𝑖∈𝐼of the local frames in 𝑊𝑖 are called local dual Hilbert space H,thenwenecessarilyhave𝑘≥𝑛.When𝑘=𝑛, 𝑖 frames. 𝐹 is automatically a basis of H. F We will use the notation when the result being stated The following result shows the relationship between a holds for both the real number field R and the complex C H = F 𝑛 𝑓 (𝑖= 1,2,...,𝑘) fusionframesystemanditslocalframes,aswellastheirframe number field .When ,then 𝑖 are bounds. column vectors, and the analysis operator Θ𝐹 for the frame 𝐹={𝑓}𝑘 𝑓∗ 𝑖 𝑖 𝑖=1 is a matrix with the row vector 𝑖 as the th Theorem 3 (c.f. [16], Theorem 2.3). For each 𝑖∈𝐼,let𝑊𝑖 row of the matrix for 𝑖 = 1,2,...,𝑘,wherethesuperscript H {𝑓 } ∗ be a closed subspace for ,andlet 𝑖𝑗 𝑗∈𝐽𝑖 be a frame for “ ” denotes the conjugate-transpose of a vector or a matrix. 𝑊𝑖 with frame bounds 𝐴𝑖 and 𝐵𝑖.Supposethat0<𝐴= Θ∗ 𝐹 Relatively, the synthesis operator 𝐹 for the frame is the inf𝑖∈𝐼𝐴𝑖 ≤ sup𝑖∈𝐼𝐵𝑖 =𝐵<∞.Then,thefollowingconditions conjugate-transpose matrix of Θ𝐹, and the frame operator ∗ are equivalent. 𝑆𝐹 =Θ ⋅Θ𝐹 is an 𝑛×𝑛positive invertible matrix. With respect 𝐹 {(𝑊,𝑤)} H to a fixed orthonormal basis of H, any element of H and any (i) 𝑖 𝑖 𝑖∈𝐼 is a fusion frame for . {𝑤 𝑓 } H linear operator can be expressed by the coordinate vector and (ii) 𝑖 𝑖𝑗 𝑗∈𝐽𝑖,𝑖∈𝐼 is a frame for . the matrix representation. So in most cases we will identify an 𝑛 {(𝑊,𝑤,{𝑓 } )} 𝑛 F In particular, if 𝑖 𝑖 𝑖𝑗 𝑗∈𝐽𝑖 𝑖∈𝐼 is a fusion frame -dimensional Hilbert space with . H 𝐶 𝐷 Let us now recall the definitions and basic results about system for with fusion frame bounds and ,then {𝑤𝑖𝑓𝑖𝑗 }𝑗∈𝐽 ,𝑖∈𝐼 H 𝐴𝐶 𝐵𝐷 fusion frames which are mostly adopted from [15, 16]. 𝑖 is a frame for with frame bounds and . {𝑤 𝑓 } H 𝐶 𝐷 If 𝑖 𝑖𝑗 𝑗∈𝐽𝑖,𝑖∈𝐼 is a frame for with frame bounds and , 𝐼 {𝑊} {(𝑊,𝑤,{𝑓 } )} H Definition 1. Let denote an index set, and let 𝑖 𝑖∈𝐼 be a then 𝑖 𝑖 𝑖𝑗 𝑗∈𝐽𝑖 𝑖∈𝐼 is a fusion frame system for with family of closed subspaces of a Hilbert space H with a family fusion frame bounds 𝐶/𝐵 and 𝐷/𝐴. 4 Advances in High Energy Physics

Let W ={(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 be a fusion frame for H.The 3. A Class of Alternate Fusion Duals and Θ analysis operator W is defined by Construction of Dual Fusion Frame Systems We first introduce a class of alternate fusion duals V = Θ : H 󳨀→ ( ∑ ⊕𝑊) Θ (𝑓) = {𝑤 𝑃 (𝑓)} , {(𝑉,𝑤)} W ={(𝑊,𝑤)} W 𝑖 with W 𝑖 𝑊𝑖 𝑖∈𝐼 𝑖 𝑖 𝑖∈𝐼 of a given fusion frame 𝑖 𝑖 𝑖∈𝐼 which 𝑖∈𝐼 ℓ2 satisfy (8) −1 −1 𝑃 𝑆 𝑃 (𝑓) = 𝑃 −1 𝑆 𝑃 (𝑓) , ∀𝑓 ∈ H,𝑖∈𝐼. 𝑉𝑖 W 𝑊𝑖 𝑆W 𝑊𝑖 W 𝑊𝑖 (14) where A Bessel fusion sequence which satisfies14 ( )naturallysatis- fies (13). So, we can obtain the following obvious result. 󵄩 󵄩 (∑ ⊕𝑊) ={{𝑓} |𝑓∈𝑊,{󵄩𝑓󵄩} ∈ℓ2 𝐼 } 𝑖 𝑖 𝑖∈𝐼 𝑖 𝑖 󵄩 𝑖󵄩 𝑖∈𝐼 ( ) (9) 𝑖∈𝐼 Proposition 6. Let W ={(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 be a fusion frame for ℓ2 space H with fusion frame operator 𝑆W. If a Bessel fusion ∗ V ={(𝑉,𝑤)} is called the representation space. The synthesis operator Θ sequence 𝑖 𝑖 𝑖∈𝐼 satisfies (14),thenitisanalternate W fusion dual of W. (the adjoint operator of ΘW) can be defined by All Bessel fusion sequences satisfying (14)formaspecial ∗ ∗ kind of alternate fusion duals of W.Thefollowingproposition ΘW :(∑ ⊕𝑊𝑖) 󳨀→ H with ΘW (𝑓) = ∑𝑤𝑖𝑓𝑖, 𝑖∈𝐼 𝑖∈𝐼 will characterize these duals. ℓ2 (10) Proposition 7. Let W ={(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 be a fusion frame H 𝑆 𝑓={𝑓𝑖}𝑖∈𝐼 ∈(∑ ⊕𝑊𝑖) . for space with fusion frame operator W. A Bessel fusion 𝑖∈𝐼 V ={(𝑉,𝑤)} ℓ2 sequence 𝑖 𝑖 𝑖∈𝐼 satisfies (14) if and only if it has the form The fusion frame operator 𝑆W for W is defined by −1 𝑉𝑖 =𝑆W𝑊𝑖 ⊕𝑈𝑖, 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑠𝑢𝑏𝑠𝑝𝑎𝑐𝑒𝑖 𝑈 𝑜𝑓 H ∗ 2 (15) 𝑆W (𝑓) = Θ ΘW (𝑓) = ∑𝑤 𝑃𝑊 (𝑓) . 𝑎𝑛𝑑 𝑒𝑎𝑐ℎ 𝑖∈𝐼. W 𝑖 𝑖 (11) 𝑖∈𝐼 Proof. If (14)holdsforany𝑖∈𝐼,thenwehave About dual fusion frames, the following definition was −1 −1 −1 𝑃𝑉 𝑆 𝑃𝑊 (𝑓) =𝑃𝑆−1𝑊 𝑆 𝑃𝑊 (𝑓) =𝑆 𝑃𝑊 (𝑓) ,∀𝑓∈H, given in [15]. 𝑖 W 𝑖 W 𝑖 W 𝑖 W 𝑖 (16) Definition 4. Let {(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 be a fusion frame for space H −1 which follows with fusion frame operator 𝑆W.Then,{(𝑆W𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 is called the dual fusion frame of {(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼. −1 −1 (𝐼H −𝑃𝑉 )𝑆W𝑃𝑊 (𝑓) =𝑉 𝑃 ⊥ 𝑆W𝑃𝑊 (𝑓) = 0, ∀𝑓 ∈ H. The dual fusion frame defined previously satisfies the 𝑖 𝑖 𝑖 𝑖 following reconstruction formula (17) −1 −1 For any 𝑓∈𝑆W𝑊𝑖, there exists 𝑔∈𝑊𝑖 such that 𝑓=𝑆W𝑔. 𝑓=∑𝑤2𝑆−1𝑃 (𝑓) 𝑖 W 𝑊𝑖 Hence, 𝑖∈𝐼 −1 −1 (12) 𝑃 ⊥ 𝑓=𝑃 ⊥ 𝑆 𝑔=𝑃 ⊥ 𝑆 𝑃 𝑔=0, 2 −1 𝑉𝑖 𝑉𝑖 W 𝑉𝑖 W 𝑊𝑖 (18) = ∑𝑤 𝑃 −1 𝑆 𝑃 (𝑓) , 𝑓 ∈ H. 𝑖 𝑆W 𝑊𝑖 W 𝑊𝑖 𝑖∈𝐼 ⊥ −1 ⊥ which implies that 𝑓⊥𝑉𝑖 .Hence,𝑆W𝑊𝑖 ⊥𝑉𝑖 ,which −1 −1 follows 𝑆W𝑊𝑖 ⊂𝑉𝑖.Let𝑈𝑖 =𝑉𝑖 −𝑆W𝑊𝑖;thenforany Basedon(12), the following definition about alternate duals −1 −1 𝑓=𝑆 𝑔∈𝑆 𝑊𝑖,wehave was introduced in [20]. W W −1 𝑃𝑈 𝑓=(𝑃𝑉 −𝑃𝑆−1𝑊 )𝑆W𝑃𝑊 (𝑔) = 0, (19) Definition 5. Let W ={(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 be a fusion frame for 𝑖 𝑖 W 𝑖 𝑖 H 𝑆 V ={(𝑉, V )} space with fusion frame operator W,and, 𝑖 𝑖 𝑖∈𝐼 which implies be a Bessel fusion sequence. Then, V is called an alternate W −1 dual of if 𝑆W𝑊𝑖 ⊥𝑈𝑖. (20) −1 𝑉 =𝑆−1𝑊 ⊕𝑈 𝑓=∑𝑤𝑖V𝑖𝑃𝑉 𝑆 𝑃𝑊 (𝑓) , Hence, 𝑖 W 𝑖 𝑖. 𝑖 W 𝑖 (13) 𝑖∈𝐼 Conversely, assume that V satisfies (15). Then for any 𝑓∈ H 𝑖∈𝐼 𝑆−1𝑃 (𝑓) ∈−1 𝑆 𝑊 and ,since W 𝑊𝑖 W 𝑖,wehave holds for all 𝑓∈H. −1 −1 𝑃 𝑆 𝑃 (𝑓) = 𝑃 −1 𝑆 𝑃 (𝑓) , 𝑉𝑖 W 𝑊𝑖 𝑆 𝑊𝑖 W 𝑊𝑖 (21) Then, it was proved that V is also a fusion frame [20]. We W will call it an alternate fusion dual of W in this paper. which implies that (14)holds. Advances in High Energy Physics 5

∗ 𝑙 ∗ The following proposition shows that the dual fusion Proof. For any 𝑓=(𝑎1,𝑎2,...,𝑎𝑙) ∈ F ,wehave𝐿 (𝑓) = {(𝑆−1𝑊,𝑤)} 𝑙 ∗ 2 𝑙 2 2 ∗ frame W 𝑖 𝑖 𝑖∈𝐼 can minimize the projection norm of ∑𝑖=1 𝑎𝑖𝑒𝑖 ∈𝑊,and‖𝐿 (𝑓)‖ =∑𝑖=1 |𝑎𝑖| =‖𝑓‖,where𝐿 is any 𝑓∈H in the class of alternate fusion duals introduced the conjugate-transpose of 𝐿. Therefore, previously. The property is analogous to Theorem 6.8 of[28] 󵄩 󵄩2 󵄩 ∗ 󵄩2 in the case of traditional frames, and its proof is trivial. 𝐴󵄩𝑓󵄩 =𝐴󵄩𝐿 (𝑓)󵄩

𝑘 𝑘 Proposition 8. Let W ={(𝑊𝑖,𝑤𝑖)}𝑖∈𝐼 be a fusion frame for 󵄨 󵄨2 󵄨 󵄨2 ≤ ∑󵄨⟨𝐿∗ (𝑓) , 𝑓⟩󵄨 = ∑󵄨⟨𝑓,𝐿 (𝑓)⟩󵄨 space H with fusion frame operator 𝑆W.Thenforanyalternate 󵄨 𝑖 󵄨 󵄨 𝑖 󵄨 𝑖=1 𝑖=1 (23) fusion dual V ={(𝑉𝑖,𝑤𝑖)}𝑖∈𝐼 of W which satisfies (14),one have 𝑘 󵄨 󵄨2 󵄩 ∗ 󵄩2 󵄩 󵄩2 = ∑󵄨⟨𝑓,𝑔𝑖⟩󵄨 ≤𝐵󵄩𝐿 (𝑓)󵄩 =𝐵󵄩𝑓󵄩 , 2󵄩 󵄩2 2󵄩 󵄩2 ∑𝑤 󵄩𝑃𝑆−1𝑊 (𝑓)󵄩 ≤ ∑𝑤 󵄩𝑃𝑉 (𝑓)󵄩 ,∀𝑓∈H. 𝑖=1 𝑖 󵄩 W 𝑖 󵄩 𝑖 󵄩 𝑖 󵄩 (22) 𝑖∈𝐼 𝑖∈𝐼 as required. Then, we consider the construction of the dual fusion Theorem 12. 𝑊 𝑙 F 𝑛 frame in a finite-dimensional Hilbert space H.InSection 2, Let be an -dimensional subspace of with {𝑒 }𝑙 𝐹={𝑓}𝑘 𝐿 we will see that any 𝑛-dimensional Hilbert space can be an orthonormal basis 𝑖 𝑖=1 and a frame 𝑖 𝑖=1. is 𝑛 𝑆 identified with F , and the analysis, synthesis, and frame defined as the previously mentiond lemma. 𝐹 is the frame 𝐹 operator of any conventional frame can be expressed by operator of .Then, their matrix representations, respectively. It is essential for −1 ∗ ∗ −1 𝑆 =𝐿 (𝐿𝑆 𝐿 ) 𝐿, (24) this construction to obtain the matrix representation of 𝐹 𝐹 𝑆 the fusion frame operator W and its inverse which need is the inverse of 𝑆𝐹 in 𝑊.Moreover,theorthogonalprojection the local frames. Hence, we will study the construction 𝑃 𝐹𝑛 𝑊 𝑃 =𝑆−1𝑆 =𝑆 𝑆−1 =𝐿∗𝐿 −1 −1 𝑊 from onto is 𝑊 𝐹 𝐹 𝐹 𝐹 . of the dual fusion frame system {(𝑆W𝑊𝑖,𝑤𝑖,{𝑆W𝑓𝑖𝑗 }𝑗∈𝐽 )}𝑖∈𝐼 𝑖 ∗ with its local dual frames of a given fusion frame system Proof. Let Θ𝐹 and Θ𝐹 be the analysis operator and synthesis {(𝑊,𝑤,{𝑓 } )} 𝐹 𝐿Θ∗ =(𝐿(𝑓), 𝐿(𝑓 ),..., 𝑖 𝑖 𝑖𝑗 𝑗∈𝐽𝑖 𝑖∈𝐼. operator of , respectively; then 𝐹 1 2 𝑛 {(𝑊,𝑤,{𝑓 } 𝑖 )}𝑚 𝑘 Let 𝑖 𝑖 𝑖𝑗 𝑗=1 𝑖=1 be a fusion frame system for 𝐿(𝑓𝑘)) is the synthesis operator of 𝐺={𝑔𝑖 =𝐿(𝑓𝑖)}𝑖=1 which 𝑛 ∗ space F . Then the analysis operator of the local frame of 𝑊𝑖 is is denoted by Θ𝐺.Bythepreviouslemmalemma,𝐺 is the 𝑛 ×𝑛 Θ 𝑓∗ 𝑗 𝑛×𝑛 F 𝑙 𝑆 =Θ∗ Θ =𝐿Θ∗Θ 𝐿∗ = a 𝑖 matrix 𝐹𝑖 with 𝑖𝑗 as its th row, and the 𝑖 matrix frame of ;hence,thematrix 𝐺 𝐺 𝐺 𝐹 𝐹 ∗ 𝐿𝑆 𝐿∗ 𝐺 Θ𝐹 is its synthesis operator. Furthermore, the 𝑖th local frame 𝐹 which is the frame operator of is invertible. Denote 𝑖 ∗ ∗ ∗ −1 −1 𝑛×𝑛 𝑆 =Θ Θ 𝐿 (𝐿𝑆𝐹𝐿 ) 𝐿 by 𝑆𝐹 . operator is an matrix 𝐹𝑖 𝐹𝑖 𝐹𝑖 . For any 𝑓∈𝑊,wehave 𝑛 Remark 9. Forthepurposeofcodingofany𝑓∈F ,any 𝑛 𝑙 vector of the subspaces of F we consider has 𝑛 elements, and 𝑓=∑ ⟨𝑓, 𝑒 ⟩ 𝑒 =𝐿∗𝐿 (𝑓) . 𝑘 𝑖 𝑖 (25) Θ {𝑓 } 𝑖 𝑖=1 𝐹𝑖 always denotes the analysis operator of the system 𝑖𝑗 𝑗=1 F 𝑛 𝑘 ×𝑛 in throughout this paper. Hence, it is a 𝑖 matrix, not a Therefore, we can get 𝑘𝑖 ×(dim 𝑊𝑖) matrix. −1 𝑛 𝑆𝐹 𝑆𝐹 (𝑓) Definition 10. Let 𝑊 be an 𝑙-dimensional subspace of F with 𝐹={𝑓}𝑘 𝑆 𝐹 ∗ ∗ −1 alocalframe 𝑖 𝑖=1. 𝐹 isthelocalframeoperatorof . =𝐿 (𝐿𝑆𝐹𝐿 ) 𝐿𝑆𝐹 (𝑓) If there exists an operator 𝐴 such that 𝑓=𝑆𝐹𝐴(𝑓) =𝐹 𝐴𝑆 (𝑓) 𝑓∈𝑊 𝐴 𝑆 𝑊 ∗ ∗ −1 ∗ ∗ holds for all ,wecall the inverse of 𝐹 in and =𝐿 (𝐿𝑆𝐹𝐿 ) 𝐿𝑆𝐹𝐿 𝐿(𝑓)=𝐿 𝐿(𝑓)=𝑓, 𝑆−1 (26) denote it by 𝐹 . −1 𝑆𝐹𝑆𝐹 (𝑓) ̃ −1 For obtaining the local dual frames 𝑓𝑖𝑗 =𝑆𝐹 (𝑓𝑖𝑗 ) (𝑗= 𝑖 =𝑆 𝐿∗(𝐿𝑆 𝐿∗)−1𝐿(𝑓) 1,2,...,𝑛𝑖, 𝑖 = 1,2,...,𝑚) of a given fusion frame system 𝐹 𝐹 −1 {(𝑊𝑖,𝑤𝑖,{𝑓𝑖𝑗 }𝑗∈𝐽 )}𝑖∈𝐼,where𝑆𝐹 is the inverse of 𝑆𝐹 in 𝑊𝑖, ∗ ∗ ∗ −1 ∗ 𝑖 𝑖 𝑖 =𝐿 𝐿𝑆 𝐿 (𝐿𝑆 𝐿 ) 𝐿(𝑓)=𝐿 𝐿(𝑓)=𝑓; 𝑆−1 𝐹 𝐹 we must calculate 𝐹𝑖 at first. For this purpose, the following −1 lemma holds. hence, 𝑆𝐹 is the inverse of 𝑆𝐹 in 𝑊. 𝑛 Moreover, for any 𝑓∈F , its orthogonal projection onto Lemma 11. 𝑊 𝑙 F 𝑛 Let be an -dimensional subspace of with an 𝑊 is {𝑒 }𝑙 𝐹={𝑓}𝑘 orthonormal basis 𝑖 𝑖=1 and a frame 𝑖 𝑖=1 with frame 𝑘 bounds 𝐴, 𝐵.Define𝐿 to be an 𝑙×𝑛matrix with the vector −1 ∗ ∗ 𝑃𝑊 (𝑓) = ∑ ⟨𝑃𝑊 (𝑓) ,𝑖 𝑓⟩𝑆𝐹 (𝑓𝑖) 𝑒𝑖 as its 𝑖th row for 𝑖 = 1,2,...,𝑙,where𝑒𝑖 is the conjugate- 𝑖=1 𝑘 transpose of 𝑒𝑖. The sequence 𝐺={𝑔𝑖}𝑖=1 is given by 𝑔𝑖 =𝐿(𝑓𝑖) 𝑙 𝑘 for 𝑖=1,2,...,𝑘.Then,𝐺 is a frame of F with the same frame −1 −1 ∗ −1 = ∑ ⟨𝑓,𝑓𝑖⟩𝑆𝐹 (𝑓𝑖)=𝑆𝐹 Θ𝐹Θ𝐹 (𝑓) =𝐹 𝑆 𝑆𝐹 (𝑓) bounds as 𝐹. 𝑖=1 6 Advances in High Energy Physics

𝑘 −1 −1 −1 ∗ −1 Step 6. Calculatethelocaldualframes𝑆𝑆 (𝑆W𝐹𝑖)= W−1𝐹 = ∑ ⟨𝑓,𝑆𝐹 (𝑓𝑖)⟩ 𝑓𝑖 =Θ𝐹Θ𝐹𝑆𝐹 (𝑓) 𝑖 −1 −1 𝑛𝑖 −1 𝑚 𝑖=1 {𝑆 (𝑆 (𝑓 ))} {(𝑆 𝑊,𝑤)} 𝑖 = 1,2,...,𝑚 𝑆 −1 W 𝑖𝑗 𝑗=1,of W 𝑖 𝑖 𝑖=1 for ,as W 𝐹𝑖 𝑙 required. −1 ∗ =𝑆𝐹𝑆𝐹 (𝑓) = ∑ ⟨𝑓,𝑒𝑖⟩𝑒𝑖 =𝐿 𝐿(𝑓), The fusion frame system of the following example is given 𝑖=1 in [21]. (27) Example 14. Assume that 𝑛=4, 𝑚=2,and 𝑛1 =𝑛2 =3. {(𝑊,𝑤,𝐹 ={𝑓}3 )}2 as claimed. The fusion frame system 𝑖 𝑖 𝑖 𝑖𝑗 𝑗=1 𝑖=1 is given by 𝑤1 =𝑤2 =1and

The proof of the following proposition is straightforward, √6 √6 √6 √3 𝑇 by using Proposition 2.6 of [16]andtheprevioustheorem,we 𝑓 =( ,− , ,− ) , 11 6 6 6 3 omit it. 𝑛 𝑇 𝑖 𝑚 √6 √6 √6 √3 Proposition 13. Let {(𝑊𝑖,𝑤𝑖,𝐹𝑖 ={𝑓𝑖𝑗 }𝑗=1)}𝑖=1 be a fusion 𝑓12 =( ,− ,− , ) , F 𝑛 𝐹̃ ={𝑓̃ } 𝑖∈𝐼 6 6 6 3 frame system for ,andlet 𝑖 𝑖𝑗 𝑗∈𝐽𝑖 , be the local ̃ −1 𝑓 =𝑆 (𝑓 ) 𝑗 = 1,2,...,𝑛 𝑇 𝑇 dual frames given by 𝑖𝑗 𝐹𝑖 𝑖𝑗 for all 𝑖, √6 √6 6 6 𝑖 = 1,2,...,𝑚 𝑓 =( ,− ,0,0) ,𝑓=( , ,0,0) , . Then, the matrix representation of the fusion 13 6 6 21 6 6 frame operator is given by √6 √3 𝑇 √3 √3 𝑇 2 ∗ 2 ∗ 𝑓 = (0, 0, − ,− ) ,𝑓=( , ,0,0) . 𝑆W = ∑𝑤𝑖 Θ ̃ Θ𝐹 = ∑𝑤𝑖 Θ𝐹 Θ𝐹̃ 22 23 𝐹𝑖 𝑖 𝑖 𝑖 3 3 3 3 𝑖∈𝐼 𝑖∈𝐼 (28) (30) = ∑𝑤2𝑆−1𝑆 = ∑𝑤2𝑆 𝑆−1, 𝑖 𝐹𝑖 𝐹𝑖 𝑖 𝐹𝑖 𝐹𝑖 𝑖∈𝐼 𝑖∈𝐼 The maximally linear independent subsets of 𝐹1 and 𝐹2 are {𝑓11,𝑓12} and {𝑓21,𝑓22}, respectively. Applying the Gram- ̃ Θ Θ ̃ 𝐹 𝐹 Schmidt process on them, we get the orthonormal bases of where 𝐹𝑖 and 𝐹𝑖 are the analysis operators of 𝑖 and 𝑖, 𝑆 𝐹 𝑖∈𝐼 𝑊1 and 𝑊2 as follows: respectively, and 𝐹𝑖 is the frame operator of 𝑖 for each .

𝑛𝑖 𝑚 𝑇 Given a fusion frame system {(𝑊𝑖,𝑤𝑖,𝐹𝑖 ={𝑓𝑖𝑗 }𝑗=1)}𝑖=1 1 1 1 2 𝑛 𝑒 =(√ ,−√ , √ ,−√ ) , of a finite-dimensional Hilbert space F ,wesummarize 11 5 5 5 5 the previous results to provide the concrete algorithm to 𝑇 {(𝑆−1𝑊,𝑤,𝑆−1𝐹 = 3 3 2 4 construct its dual fusion frame system W 𝑖 𝑖 W 𝑖 √ √ √ √ −1 𝑛𝑖 𝑚 𝑒12 =( ,− ,− , ) , {𝑆W𝑓𝑖𝑗 }𝑗=1)}𝑖=1 with its local dual frames as follows. 10 10 15 15

𝑇 Step 1. For each 𝑖 = 1,2,...,𝑚, search the maximally linear 1 1 𝑇 √6 √3 𝑙𝑖 𝑒21 =( , ,0,0) ,𝑒22 = (0, 0, − ,− ) . independent subset of 𝐹𝑖; we denote it by 𝐺𝑖 ={𝑔𝑖𝑗 }𝑗=1. √2 √2 3 3 (31) Step 2. Use the Gram-Schmidt process on 𝐺𝑖 to compute 𝑙𝑖 an orthonormal basis for 𝑊𝑖;wedenoteitby𝑅𝑖 ={𝑒𝑖𝑗 }𝑗=1. So that Construct the matrix 𝐿𝑖 constituted by this basis as follows: 1 1 1 2 [ √ −√ √ −√ ] ←󳨀 𝑒 ∗ 󳨀→ [ 5 5 5 5] 𝑖1 𝐿 = [ ] , [←󳨀 𝑒 ∗ 󳨀→ ] 1 [ ] [ 𝑖2 ] 3 3 2 4 𝐿 = . √ −√ −√ √ 𝑖 [ . ] (29) 10 10 15 15 [ . ] [ ] ∗ (32) [←󳨀 𝑒 𝑖𝑙 󳨀→ ] 1 1 𝑖 00 [√ √ ] 𝐿 = [ 2 2 ] . 𝑃 =𝐿∗𝐿 𝑆 = 2 [ √6 √3] Step 3. Since 𝑊𝑖 𝑖 𝑖 by Theorem 12,wehave W 𝑚 2 ∗ 00− − ∑𝑖=1 𝑤𝑖 𝐿𝑖 𝐿𝑖. [ 3 3 ]

−1 −1 𝑛𝑖 2 2 ∗ Step 4. Compute the local frames 𝑆W𝐹𝑖 ={𝑆W(𝑓𝑖𝑗 )}𝑗=1 of Then, we have 𝑆W =∑𝑖=1 𝑤𝑖 𝐿𝑖 𝐿𝑖 =𝐼,where𝐼 is the {(𝑆−1𝑊,𝑤)}𝑚 𝑖=1,2,...,𝑚 identity matrix, which implies that the fusion frame W 𝑖 𝑖 𝑖=1 for . 3 2 system {(𝑊𝑖,𝑤𝑖,{𝑓𝑖𝑗 }𝑗=1)}𝑖=1 is a Parseval fusion frame, Step 5. Use formula (24) to calculate the inverse of all local −1 and the local frames of the dual fusion frame system 𝑆 𝑆 −1 𝑊 𝑖=1,2,...,𝑚 −1 −1 3 2 −1 frame operators 𝑆 −1 in W 𝑖 for . {(𝑆 𝑊,𝑤,{𝑆 𝑓 } )} 𝑆 𝐹 =𝐹 𝑖=1,2 W 𝐹𝑖 W 𝑖 𝑖 W 𝑖𝑗 𝑗=1 𝑖=1 are W 𝑖 𝑖 for . Advances in High Energy Physics 7

4 4 Since [ 00] [3 3 ] [4 4 ] [ 00] [3 3 ] [ ] √ √ √ √ ∗ [ ] 6 6 6 3 𝑆𝐹 =Θ𝐹 Θ𝐹 = [ 2 √2] . [ − − ] 2 2 2 [00 ] [ 6 6 6 3 ] [ 3 3 ] [ ] [ ] [ ] [ ] [√6 √6 √6 √3 ] [ √2 1 ] Θ𝐹 = [ − − ] , [00 ] 1 [ 6 6 6 3 ] 3 3 [√6 √6 ] [ ] [ − 00] 6 6 (34) [ ]

6 6 By using (24), we obtain 00 [ ] [ 6 6 ] 1 1 [ √6 √3] [ − 00] [ 00− − ] [ 2 2 ] Θ = [ 3 3 ] , [ ] 𝐹2 [ ] [ 1 1 ] [ ] [− 00] [√3 √3 ] −1 ∗ ∗ −1 [ 2 2 ] [ ] 𝑆𝐹 =𝐿1(𝐿1𝑆𝐹 𝐿1) 𝐿1 = [ ] , 00 1 1 [ 1 √2] 3 3 [ 00 − ] [ ] [ 3 3 ] [ √2 2 ] 00− √6 √6 √6 [ 3 3 ] [ ] [ 6 6 6 ] (35) [ ] (33) 3 3 [ √6 √6 √6] 00 [− − − ] [16 16 ] [ 6 6 6 ] [ ] [ ] [ 3 3 ] ∗ [ ] [ 00] Θ𝐹 = [ √6 √6 ] , [ ] 1 [ − 0 ] −1 ∗ ∗ −1 [16 16 ] [ ] 𝑆𝐹 =𝐿2(𝐿2𝑆𝐹 𝐿2) 𝐿2 = [ 2 √2] . [ 6 6 ] 2 2 [ 00 ] [ ] [ ] [ √ √ ] [ 3 3 ] [ 3 3 ] [ ] − 0 [ √2 1 ] 3 3 00 [ ] [ 3 3 ] 6 √3 [ 0 ] Hence, the local dual frame of 𝐹1 is [6 3 ] [ ] [6 √3] ̃ −1 ̃ −1 [ 0 ] 𝑓11 =𝑆𝐹 𝑓11 =𝑓11, 𝑓12 =𝑆𝐹 𝑓12 =𝑓12, [6 3 ] 1 1 [ ] (36) ∗ [ ] ̃ −1 Θ𝐹 = [ √6 ] , 𝑓13 =𝑆𝐹 𝑓13 =𝑓13, 2 [0− 0 ] 1 [ ] [ 3 ] [ ] the local dual frame of 𝐹2 is [ √ ] [ 3 ] 0− 0 𝑇 3 ̃ −1 3 3 ̃ −1 [ ] 𝑓21 =𝑆𝐹 𝑓21 =( , ,0,0) , 𝑓22 =𝑆𝐹 𝑓22 =𝑓22, 2 8 8 2 (37) √ √ 𝑇 ̃ −1 3 3 𝑓23 =𝑆𝐹 𝑓23 =( , ,0,0) . 2 8 8 we can get They are also the local dual frames of the dual fusion frame −1 −1 3 2 system {(𝑆W𝑊𝑖,𝑤𝑖,{𝑆W𝑓𝑖𝑗 }𝑗=1)}𝑖=1. 1 1 The quark-gluon plasma is a state of the extremely dense − 00 [ 2 2 ] matter which contains the quarks and gluons in high energy [ ] [ ] physics. The gray image of quark-gluon plasma is shown in [ 1 1 ] [− 00] Figure 2. We encode the data of the image by using the local [ 2 2 ] frames of the fusion frame given by this example. Suppose ∗ [ √ ] 𝑆𝐹 =Θ𝐹 Θ𝐹 = [ 1 2] , 1 1 1 [ 00 − ] that the fist element of every local vector is lost in the [ 3 3 ] [ ] transmission process. Then, we decode the received data by [ ] using the dual fusion frame computed by this example. The [ √2 2 ] 00− reconstructed image is shown in Figure 3.Onecanobserve 3 3 [ ] the reconstruction effect by comparing the two figures. 8 Advances in High Energy Physics

Acknowledgments The authors are thankful to the anonymous reviewers for their valuable comments and suggestions that have improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (11271001, 61170311), 973 Program (2013CB329404), and Sichuan Province Science and Technology Research Project (12ZC1802). be the analysis operator and synthesis operator of 𝐹, respectively; then

References

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Research Article Characteristic Roots of a Class of Fractional Oscillators

Ming Li,1,2 S. C. Lim,3 Carlo Cattani,4 and Massimo Scalia5

1 School of Information Science & Technology, East China Normal University No. 500, Dong-Chuan Road, Shanghai 200241, China 2 Department of Computer and Information Science, University of Macau, Avenue Padre Tomas Pereira, Taipa 1356, Macau SAR, China 3 Faculty of Engineering, Multimedia University, Selangor Darul Ehsan, 63100 Cyberjaya, Malaysia 4 Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy 5 Department of Mathematics, University of Rome, la Sapienza Piazzale Aldo Moro, 00185 Rome, Italy

Correspondence should be addressed to Ming Li; ming [email protected]

Received 9 August 2013; Accepted 13 September 2013

Academic Editor: Gongnan Xie

Copyright © 2013 Ming Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The fundamental theorem of algebra determines the number of characteristic roots of an ordinary differential equation of integer order. This may cease to be true for a differential equation of fractional order. The results given in this paper suggest that the number of the characteristic roots of a class of oscillators of fractional order may in general be infinitely great. Further, we infer that it may also be the case for the characteristic roots of a differential equation of fractional order greater than 1. The relationship between the range of the fractional order and the locations of characteristic roots of oscillators in the complex plane is considered.

1. Introduction [38], Papoulis [39], Bendat and Piersol [40], Devasahayam [41], Karrenberg [42], Edson [43], and Balaban et al. [44]). Oscillators are an essential component in devices in electron This research is in the domain of fractional oscillators positron collider systems (see, e.g., Zhao et al. [1], Ma et al. that attract increasing interests of physicists and engineers. [2], Zang et al. [3], Ding et al. [4], Marder et al. [5], Barroso More specifically, we aim at revealing specific properties of [6], Miller et al. [7], and Lemke [8], just citing a few). As a characteristic roots of a class of fractional oscillators. In doing matter of fact, oscillations are phenomena widely observed so, we first consider an ordinary differential equation of order in sciences and engineering relating to high energy physics 𝑛 given by (see, e.g., Akhmediev et al. [9], Bachas [10], Winter et al. [11], 𝑛 𝑛−1 Dodonov [12], Tan [13], Diamandis et al. [14], Greenwald et 𝑑 𝑦 (𝑡) 𝑑 𝑦 (𝑡) 𝑏 +𝑏 +⋅⋅⋅+𝑏 =𝑥(𝑡) , (1) al. [15], Mathews et al. [16], Faiman [17], Cocho et al. [18], 𝑛 𝑑𝑡𝑛 𝑛−1 𝑑𝑡𝑛−1 0 Baldiotti et al. [19], Kyu Shin [20], Kirson [21], Clement [22], 𝑛 𝑏 Sikstrom¨ et al. [23], Asghari et al. [24], Um et al. [25], Bahar where is a natural number and 𝑛 is any complex number. We always assume that at least one of the higher coefficients and Yasuk [26], Hassanabadi et al. [27], Bhattacharya and Roy 𝑏 =0̸ 𝑛>1 [28], and Saad et al. [29], simply mentioning a few). 𝑛 for . The characteristic equation of (1)isgivenby There are various structures of oscillators, such as Math- 𝑛 𝑛−1 𝐵 (𝛼) =𝑏𝑛𝛼 +𝑏𝑛−1𝛼 +⋅⋅⋅+𝑏0 =0. (2) ieu oscillator (Floris [30]), Lienardtypeoscillator(Yas´ ¸ar [31]), relativistic oscillator (Osborne [32]), Schrodinger¨ equation The fundamental theorem of algebra says that the number type oscillator (Cornwall and Tiktopoulos [33]), and Duff- of roots of (2)is𝑛 (G. A. Korn and T. M. Korn [45]). This ing oscillator (Baltanas´ et al. [34] and Erturk and Inman theorem is stated in the domain of complex variables (Krantz [35]). In fact, oscillators play a role in various fields, rang- [46]). ing from experimental physics to electronics engineering Suppose that the 𝑛 roots of 𝐵(𝛼) are 𝛼1,𝛼2,...,𝛼𝑛.For (see, e.g., Riley et al. [36], Soong and Grigoriu [37], Harris each root 𝑟 of multiplicity of 𝑚,eitherrealorcomplex,we 2 Advances in High Energy Physics always consider 𝑟 the 𝑚 roots in what follows unless otherwise However, we shall show that the number of the roots in stated. Using the partial fraction expansion, 𝐵(𝛼) can be the above expression dramatically differs from what in the fol- expressed by lowing expression:

𝑛 2 (𝑚𝑗𝛼𝑗,12 +𝑐𝑗𝛼𝑗,12 +𝑘𝑗)=0. (11) 𝐵 (𝛼) =𝑏𝑛 (𝛼 −1 𝛼 )(𝛼−𝛼2)⋅⋅⋅(𝛼−𝛼𝑛)=𝑏𝑛∏ (𝛼 −𝑖 𝛼 ). 𝑖=1 (3) The contributions of this paper are twofold. One is to exhibit that the number of the characteristic roots of (8)isin Now, we rewrite (3)bythefollowingexpression: general infinitely great. The other is to reveal the relationship between the range of 𝛽 and the locations of the characteristic { 𝑛/2 𝛼 𝑗= {𝐾∏ (𝑚 𝛼2 +𝑐𝛼 +𝑘), 𝑛 , roots of (8) in a complex plane. In addition, if all 𝑗 ( { 𝑗 𝑗 𝑗 𝑗 𝑗 is even 1,...,𝑛 { 𝑗=1 ) are simple complex pair of roots, the ordinary 𝐵 (𝛼) = { differential equation of order 𝑛 (1) and its generalization given { (𝑛−1)/2 { 2 by {𝐾(𝛼−𝛼𝑛) ∏ (𝑚𝑗𝛼𝑗 +𝑐𝑗𝛼𝑗 +𝑘𝑗), 𝑛 is odd, { 𝑗=1 𝑛 𝑛−1 𝛽 (4) (𝑏𝑛𝐷 𝑦(𝑡)𝑛−1 +𝑏 𝐷 𝑦(𝑡)+⋅⋅⋅+𝑏0) (12) where 𝑚𝑗, 𝑐𝑗, 𝑘𝑗,and𝐾 are constants. Without loss of gener- =𝑥(𝑡) ,𝛽>0 ality, we can suppose that the only simple zero of 𝐵(𝛼) is 𝛼𝑛 if 𝑛 is odd. may be taken as the product of oscillators of integer order and 2 The factor (𝑚𝑗𝛼𝑗 +𝑐𝑗𝛼𝑗 +𝑘𝑗) in (4) corresponds to the fractional order in series in the wide sense for 𝑛 being even, oscillator equation in the form respectively. The rest of the paper is organized as follows. We shall 𝑑2𝑦 (𝑡) 𝑑𝑦 (𝑡) 𝑚 +𝑐 +𝑘𝑦 (𝑡) =𝑥(𝑡) . (5) give the results in Section 2, including the proof that there are 𝑗 𝑑𝑡2 𝑗 𝑑𝑡 𝑗 infinite characteristic roots regarding (8), and the explanation that (1)and(12) may be taken as oscillators in series in the The characteristic equation of (5)isintheform wide sense. Discussions are given in Section 3,whichis 2 followed by Conclusions. (𝑚𝑗𝛼𝑗 +𝑐𝑗𝛼𝑗 +𝑘𝑗)=0. (6)

There are two classes of fractional oscillators. One is in the 2. Results form (Ryabov and Puzenko [47, Eq. (1)], Ahmad et al. [48], 2.1. Result 1. The number of the characteristic roots of (8) Radwan et al. [49], Drozdov [50, Eq. (9)], Tofighi and Pour may be infinitely great. [51], Tofighi [52,Eq.(2)],Blaszczyketal.[53, Eq. (10)], and Denote by C the set of complex numbers. Let 𝑧∈C. Narahari Achar et al. [54, 55]) Suppose that a power function is given by 𝑑2±𝜀𝑦 (𝑡) 𝑑±𝜀𝑦 (𝑡) 𝑚 +𝑐 +𝑘𝑦 (𝑡) 𝑤=𝑧𝑏 =𝑒𝑏ln(𝑧). 𝑗 2±𝜀 𝑗 ±𝜀 𝑗 (13) 𝑑𝑡 𝑑𝑡 (7) 𝑤 =𝑥(𝑡) for 0≤𝜀<1. Then,thenumberofdifferentvaluesof relies on the value of 𝑏 for a given 𝑧. More precisely, we express that by the The other is expressed with the form (Lim et al.56 [ –58], following lemmas, which can be found in the literature, such Muniandy and Lim [59], Eab and Lim [60], and Li et al. [61]) as [45]orYu[62].

𝛽 (𝑚 𝐷2𝑦(𝑡) +𝑐 𝐷𝑦(𝑡) +𝑘 𝑦(𝑡)) =𝑥(𝑡) 𝛽>0. (8) Lemma 1. If 𝑏 is a rational number expressed by the irreducible 𝑗 𝑗 𝑗 for 𝑏 fraction 𝑙/𝑚,where𝑚≥1, the number of values of 𝑧 is 𝑚. This research utilizes the form of (8). According to the fundamental theorem of algebra, there Lemma 2. If 𝑏 is an irrational number or imaginary number, 𝑏 are only two characteristic roots with respect to the oscillator the number of values of 𝑧 is infinitely great. equation (5). They are The general expression of 𝑤 is in the form 2 −𝑐𝑗 ± √𝑐𝑗 −4𝑚𝑗𝑘𝑗 𝛼 = . (9) 𝑤=𝑧𝑏 =𝑒𝑏ln(𝑧) 𝑗,12 2𝑚 𝑗 (14) =𝑒𝑏[ln |𝑧|+𝑖(arg 𝑧+2𝑚𝜋)],𝑖=√−1. One might be carelessly misled to consider that there exist only two characteristic roots regarding the fractional Therefore, from Lemma 2,wehavethetheorembelow. oscillator equation (8)because

𝛽 Theorem 3. The number of the characteristic roots of the (𝑚 𝛼2 +𝑐𝛼 +𝑘) =0. 𝑗 𝑗,12 𝑗 𝑗,12 𝑗 (10) fractional oscillator (8) is infinitely great if 𝛽 =1̸. Advances in High Energy Physics 3

x(t) y(t) Proof. Let 𝑐𝑗 = 0. Then, the characteristic roots 𝛼𝑗,12 in (9) B (𝛼) B (𝛼) B (𝛼) become the imaginary numbers expressed by 1 2 n/2

±𝑖√𝑚𝑗𝑘𝑗 Figure 1: Oscillators of 2-order in concatenation. 𝛼𝑗,12 = . (15) 𝑚𝑗 3. Discussions According to Lemma 2, the number of the roots of either 𝛽 𝛽 The previous section says that there are infinite roots in (𝛼−𝛼 ) (𝛼 − 𝛼 ) 𝛽 𝛽 𝑗,1 or 𝑗,2 is infinitely great. Thus, Theorem 3 𝐵 (𝛼) if 𝛽 =1̸.Inthecaseof𝛽=1, 𝐵 (𝛼) reduces to the results. 𝑗 𝑗 characteristic equation of the conventional oscillator (5)with two roots only. Thus, the fraction 𝛽 =1̸ dramatically alters the 2.2. Result 2. Equations (1)and(12)maybetakenasoscilla- behavior of characteristic roots of oscillators. For facilitating tors in series. 𝑗 (𝑚 𝛼2 +𝑐𝛼 +𝑘) 𝐵 (𝛼) our discussions, we omit the subscript in what follows if Denote 𝑗 𝑗 𝑗 𝑗 𝑗 in (4)by 𝑗 : not confused. More precisely, we specifically consider the 2 fractional oscillator in the form 𝐵𝑗 (𝛼) = (𝑚𝑗𝛼𝑗 +𝑐𝑗𝛼𝑗 +𝑘𝑗) . (16) 𝛽 (𝑚𝐷2𝑦(𝑡) + 𝑐𝐷𝑦(𝑡) + 𝑘𝑦(𝑡)) Without loss of generality, 𝑛 isassumedtobeeven.Inaddi- 𝛼 𝑗=1,...,𝑛 (20) tion, we suppose that all 𝑗 ( ) are simple complex =𝑥(𝑡) 𝛽>0. pair of roots. Then, we have the theorem below. for Theorem 4. Figure 2 shows an RLC resonance circuit in series, where The ordinary differential equation (1) may be 𝑅 𝐿 𝐶 takenasanoscillator(i.e.,productofoscillators)inthewide , ,and represent resistor, inductor, and capacity, respectively. In Figure 2, 𝐼(𝑡) is the electronic current and V(𝑡) the sense if 𝑛 is even and all 𝛼𝑗 (𝑗 = 1,...,𝑛) are simple complex pair of roots. By wide sense, one means that it is a system power source. According to the Kirchhoff voltage law, one has consisting of the product of a series of conventional 2-order 2 𝑅 1 1 𝑑V (𝑡) 𝐷 𝐼 (𝑡) + 𝐷𝐼 (𝑡) + 𝐼 (𝑡) = . (21) oscillators. 𝐿 𝐿𝐶 𝐿 𝑑𝑡

Proof. On the one hand, 𝐵𝑗(𝛼) stands for the characteristic 𝜔=√1/𝐿𝐶 𝑅/𝐿 = 2𝑏 (1/𝐿)(𝑑V(𝑡)/𝑑𝑡) 𝑗 𝑛 Let and .Denote by equation of the th oscillator of order 2 since is even and 𝑒(𝑡).Then,(21) becomes the form all 𝛼𝑗 (𝑗 = 1,...,𝑛) are simple complex pair of roots. On the 2 2 other hand, the characteristic equation of (1) can be expressed 𝐷 𝐼 (𝑡) +2𝑏𝐷𝐼(𝑡) +𝜔 𝐼 (𝑡) =𝑒(𝑡) . (22) by 𝛽 𝑛/2 Generalizing (22) to the fractional order yields {𝐾∏𝐵 (𝛼) ,𝑛, { 𝑗 is even 2 2 𝛽 { (𝐷 𝐼(𝑡) + 2𝑏𝐷𝐼(𝑡) +𝜔 𝐼(𝑡)) =𝑒(𝑡) . (23) 𝐵 (𝛼) = 𝑗=1 { (𝑛−1)/2 (17) { {𝐾(𝛼−𝛼𝑛) ∏ 𝐵𝑗 (𝛼) ,𝑛is odd. Below, we specifically study the circuit in Figure 2 with { 𝑗=1 𝑅=0, as indicated in Figure 3. In the case of Figure 3,(23) becomes the form

𝛽 Based on the theory of filter design (Mitra and Kaiser (𝐷2𝐼(𝑡) +𝜔2𝐼(𝑡)) =𝑒(𝑡) . (24) [63]), the system (1)inthecaseof𝑛 being even may be expressed by Figure 1. Denote by ℎ(𝑡) the impulse response function of (24). Therefore, the system (1) may be expressed by the product Then, using the techniques in fractional calculus and differ- of a series of 2-order oscillators. ential equations [64–81], we have (see [61] for details) 𝛽 Denote by 𝐵 (𝛼) the characteristic equation of (12). Then, √𝜋 𝛽−1/2 𝑛/2 ℎ (𝑡) = 𝑡 𝐽 (𝜔𝑡) , 𝛽−1/2 𝛽−1/2 {𝐾∏𝐵𝛽 (𝛼) ,𝑛, Γ(𝛽)(2𝜔) { 𝑗 is even (25) 𝛽 { 𝑗=1 𝐵 (𝛼) = { (18) 𝛽 > 0, 𝑡 ≥0, { (𝑛−1)/2 {𝐾(𝛼−𝛼 ) ∏ 𝐵𝛽 (𝛼) ,𝑛 , 𝑛 𝑗 is odd 𝐽 (𝜔𝑡) { 𝑗=1 where 𝛽−1/2 is the Bessel function of the first kind of order 𝛽−1/2. where The following theorems reflect the particularity of roots 𝛽 𝛽 2 𝛽 of 𝐵 (𝛼). 𝐵𝑗 (𝛼) =(𝑚𝑗𝛼𝑗 +𝑐𝑗𝛼𝑗 +𝑘𝑗) . (19) Theorem 5. 0<𝛽<1 𝐵𝛽(𝛼) Thus, the system of fractional order expressed by (12)maybe If ,allrootsof are located in the the product of a series of fractional oscillators of (8). left side of the complex plane. 4 Advances in High Energy Physics

L From the above, we have the following: I(t) + V 1 R −(𝑡/2) 2 V−1/2 (t) C ∫ (1 − 𝑢 ) 𝑑𝑢 Γ (V +1/2) Γ (1/2) −1 − V 1 (𝑡/2) 2 V−1/2 Figure 2: Illustration of RLC resonance circuit in series. ≤𝐽V (𝑡) ≤ ∫ (1 − 𝑢 ) 𝑑𝑢. Γ (V +1/2) Γ (1/2) −1 (31) L I(t) + 𝛽>1 V >1/2 (t) C Since implies ,weimmediatelyseethatboth the right side and the left one on the above expression are − respectively unbounded when 𝑡→∞.Thus,for𝛽>1,the fractional oscillator (24) is nonstable according to the theory Figure 3: Illustration of LC resonance circuit in series with damping 𝑅=0 of systems (Gabel and Roberts [77], Dorf and Bishop [78]). . Consequently, at least, some of poles of 𝐻(𝑠) are in the right 𝛽 of the 𝑠 plane. Therefore, at least, parts of roots of 𝐵 (𝛼) are locatedintherightsideofthecomplexplane. Proof. Note that Most of previous discussions take oscillators of fractional order (24) as a specific object. Note that the number of the ∞ (−1)𝑚 𝑡 2𝑚+V characteristic roots of differential equation in general in the 𝐽V (𝑡) = ∑ ( ) . (26) 𝑚=0 𝑚!Γ (𝑚+V +1) 2 form of (12) may also be infinitely great. Hence, comes the following theorem in passing.

Theorem 7. Fractional-order differential equation expressed Fromtheabove,wehavetheasymptoticexpressioninthe 𝑛>1 form by (12) has infinite characteristic roots if and if there is at least a pair of roots that are simple complex. 1 𝐽V (𝑡) ∼ for 𝑡󳨀→∞. √𝑡 (27) Proof. The characteristic equation of (12)maybedecom- posedintheformof(18)dueto𝑛>1.Becausethereisat Applying (27)to(25)produces least a pair of roots that are simple complex, the number of the characteristic roots of (19) is infinitely great. Thus, the √𝜋 𝛽−1/2 number of characteristic roots of (12) is infinitely great. This lim ℎ (𝑡) = lim 𝑡 𝐽𝛽−1/2 (𝜔𝑡) 𝑡→∞ 𝑡→∞Γ(𝛽)(2𝜔)𝛽−1/2 completes the proof. (28) =0, 0<𝛽<1. The previous discussions exhibit interesting phenomena of the characteristic roots of the oscillators of the fractional Denote by 𝐻(𝑠) the Laplace transform of ℎ(𝑡).Then, type of (24). In the future, we will work on exploring the according to the final-value theorem, we have answers of the questions described below. lim 𝑠𝐻 (𝑠) =0, 0<𝛽<1. 𝑠→0 (29) (i) Are all poles of 𝐻(𝑠) with respect to (24)intheright of the 𝑠 plane when 𝛽>1? The above implies that all poles of 𝐻(𝑠) except the origin are strictly in the left side of 𝑠 plane. In the right of the 𝑠 plane, (ii) Might there be interesting oscillation behavior of (12) 𝑐 =0 𝑛 𝐻(𝑠) is analytic. This completes the proof. if all 𝑗 in (18)andif is even?

Theorem 6. 𝛽>1 𝐵𝛽(𝛼) If ,atleast,partsofrootsof are located 4. Conclusions in the right side of the complex plane. We have explained that the number of the characteristic Proof. Note that roots of fractional-order oscillators of (24)isusuallyinfinitely great. This conclusion has been further inferred to the case (𝑡/2)V 𝐽V (𝑡) = of fractional-order differential equation of12 ( ). We have Γ (V +1/2) Γ (1/2) exhibited that all characteristic roots of (24)arestrictly 1 locatedintheleftsideofthecomplexplaneif0<𝛽<1and at 2 V−1/2 1 × ∫ (1 − 𝑢 ) cos (𝑡𝑢) 𝑑𝑢, Re V >− . least some of characteristic roots of (24) are in the right side −1 2 of the complex plane if 𝛽>1.Inthecaseof𝛽=1,(24)reduce (30) to an ordinary damping-free oscillator. Advances in High Energy Physics 5

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Research Article Geant4 Simulation Study of Deep Underground Muons: Vertical Intensity and Angular Distribution

Halil Arslan and Mehmet Bektasoglu

Department of Physics, Sakarya University, 54187 Sakarya, Turkey

Correspondence should be addressed to Mehmet Bektasoglu; [email protected]

Received 28 July 2013; Accepted 19 September 2013

Academic Editor: Carlo Cattani

Copyright © 2013 H. Arslan and M. Bektasoglu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Underground muon intensities up to 10000 m.w.e. and angular distribution up to 6500 m.w.e. in standard rock have been investigated using Geant4 simulation package. Muons with energies above 100 GeV were distributed from the ground level taking into account the muon charge ratio of ∼1.3 at sea level. The simulated differential muon intensities are in good agreement with the intensities given in the literature. Furthermore, the simulation results for the integrated intensities are consistent with the experimental data, particularly at depths above 4000 m.w.e., where the simulation gives slightly smaller intensities than the experimental ones. In addition, the simulated exponent n at different underground depths agrees well with the experimental points, especially above ∼2000 m.w.e.

14 1. Introduction ∼10 eV can be obtained from deep underground muon data. Therefore, underground muon studies have been Interaction of the primary cosmic rays (mostly protons attracting a continuous attention for the past several decades and alpha particles) with the atmospheric nuclei produces (see, for instance, [3–10]). Recently, Monte Carlo simulations secondary cosmic rays that are able to reach the Earth’s have also been utilized in order to investigate intensity surface. Among the mentioned secondary cosmic rays are [11] and angular distributions [12] of muons at relatively muons,whicharethemostabundantchargedparticlesatsea shallower depths. level. They interact weakly with the media they propagate in As the high energy muons are not affected by the mag- and can even penetrate large thicknesses of water, ice, or rock. netic field of the Earth, they show no azimuthal dependence The energy loss mechanisms, such as ionization and atomic at sea level. Since only the energetic muons reach significant excitation, bremsstrahlung, electron pair production, and depths, underground muons have no azimuthal dependence photonuclear interactions, play a role while muons propagate as well. On the other hand, underground muons do show in the matter [1]. a zenithal dependence. This is because muons with larger Significant pieces of information can be achieved from zenith angles, compared to the ones with smaller angles, underground or underwater muon investigations. For lose more energy in the medium they propagate through. instance, a direct correlation exists between underground Therefore, the intensity of muons incident with a large zenith muon intensities and meson production in the stratosphere angle reaching a particular depth is expected to be smaller [2] due to different sensitivities of pions and kaons against than that of vertical muons. The zenith angle distribution of temperature changes. Furthermore, deep underground underground muons could be expressed by muon data are useful for background estimations in the underground areas housing neutrino experiments, and muon 𝐼 (𝜃,) ℎ =𝐼(0∘,ℎ) 𝑛 (𝜃) , intensity measurements at various depths yield information cos (1) ontheelectromagneticprocessesthatreducetheflux[1]. ∘ Additionally, information on the spectrum, as well as the where 𝜃 is the zenith angle, 𝐼(0 ,ℎ) is the vertical intensity, composition, of the primary cosmic rays with energies above and 𝑛 is an exponent at the depth of ℎ. The exponent 𝑛,which 2 Advances in High Energy Physics

∘ works for zenith angles 𝜃≤40 and depths up to ∼5000 m.w.e. 10−5

[4], is given by ) 10−6 −1

−4 −7 𝑛 = 1.53 + 8.0 ⋅10 ℎ+𝜖, (2) 10

(GeV/c) −8

−1 10 2 s where the depth ℎ is in hg/cm (m.w.e.) and 𝜖 is a small −1 −9

sr 10

correction coming from muon decay and ionization losses at −2 shallower depths [1]. Since at greater slant depths the neutrino 10−10 induced muons dominate, the usage of the cosine power low 10−11 of angular distribution becomes inadequate. In the present study, deep underground muon intensity 10−12 up to 10000 m.w.e. and angular distribution up to 6500 m.w.e. 10−13 in standard rock have been investigated using Monte Carlo simulations, and the simulation results have been compared 10−14 Differential intensity (cm intensity Differential with measurements performed by different groups. The limit 10−15 of 10000 m.w.e. was selected due to the reason that for deeper 102 103 104 depths neutrino-induced muons start to contribute to the Muon momentum (GeV/c) muon intensity dominantly. Geant4 Rastin 2. Simulation Figure 1: Sea level muon distribution: measurements by Rastin [13] Geant4 (for GEometry ANd Tracking) is an object-oriented and primary muons used in Geant4 simulation (color online). toolkit, simulating the passage of particles through matter [14] using Monte Carlo methods. This simulation package is being intensively used in several areas including nuclear and 3. Results and Discussion particle physics, medical and space physics. Its usage for the studies of various cosmic muon properties, both at sea level The simulation results for the normalized local spectrum, [15–18] and underground [11, 12], has recently grown. which was defined as the differential energy spectrum nor- In this work, deep underground vertical intensity and malized to the corresponding integrated intensity, of under- angulardistributionofcosmicmuonsareinvestigatedusing ground muons are given in Figure 2 for various depths in Geant4 simulation package (release 4.9.3.p01). The Earth’s standardrock.Thefigure,inwhichtheerrorbarsaresmaller crust was modeled as a rectangular box made of standard than the marker size for most of the bins, illustrates that for ∼ rock, which has been defined as a mixture of CaCO3 and relativelylowmuonenergies(upto 100 GeV) the spectrum 3 MgCO3 with an average density of 𝜌 = 2.65 g/cm , 𝑍/𝐴 = has a tendency to become flat with increasing depth. For 2 ℎ > 2500 0.50,and𝑍 /𝐴 = 5.50 (𝑍 and 𝐴 being the atomic and mass the depth m.w.e., the underground differential number, resp.) [19]. spectrum of muons is again constant for low muon energies Positively and negatively charged muons with energies but, independent of the depth, steepens to reflect the surface ∘ ∼ above 100 GeV were distributed isotropically [18](𝜃≤60) muon spectrum at muon energies above 500 GeV. One can from the ground level taking into account the muon charge also notice that the shape of the spectrum also seems to ratio of ∼1.3 at sea level (see, e.g., [20]). The threshold energy become independent of depth as the depth gets bigger than ℎ > 2500 of 100 GeV is selected in order to eliminate the low energy m.w.e. These results are in good agreement with the muons that do not have enough energy to reach the under- information in the literature [22, 23]. ground depth of interest. Energy distribution of the primary The integral intensity of muons underground obtained muons used as input in the simulations was resembled to from the Geant4 simulation is illustrated in Figure 3 as themeasurementsbyRastin[13] below 3 TeV. For muons a function of depth in the range 500–10000 m.w.e. with with energies high enough to reach deep underground, the 500 m.w.e. increment. Also in the figure are shown several distribution was extrapolated up to 20 TeV (see Figure 1). data points from various experiments [3, 5, 8]. One may The Low and High Energy Parameterization (LHEP) conclude that the simulation reproduces the experimental and the standard electromagnetic (emstandard) models have results well especially at depths above 4000 m.w.e, below been used for hadronic and electromagnetic interactions, which the simulation yields slightly smaller intensities than respectively. Detailed information on these models can be those the experiments give. The red solid line indicates the fit found in the Geant4 web page [21]. made to the simulation results using the Frejus´ function [24]: Energies and zenith angles of the muons reaching the ℎ 2 depth up to 10000 m.w.e. in the simulated rock sample 0 −ℎ/ℎ0 𝐼𝜇 (ℎ) =𝐴( ) 𝑒 . (3) were recorded for 20 levels with 500 m.w.e. increment. The ℎ integrated intensity for each level up to 10000 m.w.e. and the −6 −2 −1 −1 angular distribution up to 6500 m.w.e. were determined. The The fit yields 𝐴 = (0.89 ± 0.07) ×10 cm s sr and following section discusses the results. ℎ0 = 1307 ± 3 m.w.e. The fit parameters obtained from Advances in High Energy Physics 3

10−2

) 8 −1 10−3 n 6 10−4

Exponent 4 10−5

2 10−6

Normalized local spectrum (GeV Normalized 103 10−7 Depth (m.w.e.) Geant4 simulation 1 10 102 103 104 Stockel (1969) Muon energy (GeV) Miyake et al. (1964) 500 m.w.e. 3500 m.w.e. 𝑛 1500 m.w.e. 5000 m.w.e. Figure 4: The simulated and experimental exponent as a function 2500 m.w.e. of depth in standard rock (color online).

Figure 2: Normalized differential muon intensities in selected depths of standard rock (color online). Lastly, the exponent 𝑛 has been extracted for every 500 m.w.e., starting from 500 m.w.e. up to 6500 m.w.e. depth, by fitting the simulated zenith angular distribution of under- −5 ∘ 𝑛 10 ground muons with the expression 𝐼(𝜃) = 𝐼(0 )cos (𝜃) for ∘ ) −6 𝜃≤40.Theobtainedvaluesof𝑛 have been drawn as a func-

−1 10 s tion of depth in standard rock (see Figure 4). The figure also −1 −7 sr 10 includes the exponent values from two experiments [3, 5]. −2 Comparison between the exponent values obtained from the −8 10 simulation with the ones obtained from the measurements 10−9 shows that the simulation gives very consistent results with the experimental ones, especially above ∼2000 m.w.e. Below 10−10 that depth, the simulation yields slightly larger exponent values.Theredsolidlinerepresentsthefunctiongivenin(2). 10−11 Integrated intensity (cm intensity Integrated 10−12 4. Conclusions 0 2000 4000 6000 8000 10000 Depth (m.w.e.) In the present simulation study, underground muon intensity up to 10000 m.w.e. and angular distribution up to Geant4 simulation Stockel (1969) LVD experiment Miyake et al. (1964) 6500 m.w.e. in standard rock have been investigated using Geant4 simulation package. The differential muon intensities Figure 3: Integrated intensity of muons as a function of depth (color obtained from the simulation are in good agreement with online). the ones in the literature. Also, the simulation results for the integrated intensities are consistent with the experimental data especially at depths above 4000 m.w.e., below which Table 1: Fit parameters from intensity-depth relation in (3). the simulation yields slightly smaller intensities than the −6 −2 −1 −1 𝑛 Experiment 𝐴 (×10 cm s sr ) ℎ0 (m.w.e.) experiments. Furthermore, the exponent for each depth underground agrees, in general, well with the experimen- Frejus´ 1989 1.96 ± 0.09 1184 ± 8 tal points. Slight disagreement between the simulated and LVD 1995 1.77 ± 0.02 1211 ± 3 𝑛 ∼ 2.18 ± 0.05 1127 ± 4 experimental values for depths below 2000 m.w.e. could Frejus´ 1996 be attributed to the interaction models and is the subject of 1.81 ± 0.06 1231 ± 1 MACRO 1995 further investigations. Geant4 0.89 ± 0.07 1307 ± 3 References various experiments (Frejus´ 1989 [24], LVD 1995 [25], Frejus´ [1] P. K. F. Grieder, Cosmic Rays at Earth: Researcher’s Reference 1996 [26], and MACRO 1995 [7]) together with the ones from Manual and Data Book, Elsevier, Amsterdam, The Nether- thepresentworkaregiveninTable 1. lands, 2001. 4 Advances in High Energy Physics

[2]E.W.Grashorn,J.K.deJong,M.C.Goodmanetal.,“The [20] S. Haino, T. Sanuki, K. Abe et al., “Measurements of primary atmospheric charged kaon/pion ratio using seasonal variation and atmospheric cosmic-ray spectra with the BESS-TeV spec- methods,” Astroparticle Physics,vol.33,no.3,pp.140–145,2010. trometer,” Physics Letters B,vol.594,no.1-2,pp.35–46,2004. [3] S. Miyake, V. S. Narasimham, and P. V. Ramana Murthy, [21] Geant4, 2013, http://geant4.cern.ch/support/. “Cosmic-ray intensity measurements deep undergound at [22] T. Gaisser, Cosmic Rays and Particle Physics, Cambridge Univer- depths of (800–8400) m w.e.,” Il Nuovo Cimento,vol.32,no.6, sity Press, Cambridge, UK, 1990. pp.1505–1523,1964. [23] G.L.Cassiday,J.W.Keuffel,andJ.A.Thompson,“Calculationof [4]M.G.K.Menon,S.Naranan,V.S.Narasimhametal.,“Muon the stopping-muon rate underground,” Physical Review D,vol. intensities and angular distributions deep underground,” Pro- 7,no.7,pp.2022–2031,1973. ceedings of the Physical Society, vol. 90, no. 3, pp. 649–656, 1967. [24] Ch. Berger, M. Frohlich,¨ H. Monch¨ et al., “Experimental study [5]C.T.Stockel,“Astudyofmuonsdeepunderground.I.Angular of muon bundles observed in the Frejus´ detector,” Physical distribution and vertical intensity,” JournalofPhysicsA,vol.2, Review D,vol.40,no.7,pp.2163–2171,1989. no. 6, pp. 639–649, 1969. [25] M. Aglietta, B. Alpat, E. D. Alyea et al., “Neutrino-induced and [6] J. N. Crookes and B. C. Rastin, “The absolute intensity of muons −2 atmospheric single-muon fluxes measured over five decades at 31.6 hg cm below sea-level,” Nuclear Physics B,vol.58,no.1, of intensity by LVD at Gran Sasso Laboratory,” Astroparticle pp. 93–109, 1973. Physics,vol.3,no.4,pp.311–320,1995. [7] M. Ambrosio, R. Antolini, G. Auriemma et al., “Vertical muon [26] W.Rhode, K. Daum, P.Bareyre et al., “Limits on the flux of very intensitymeasuredwithMACROattheGranSassolaboratory,” high energy neutrinos with the Frejus´ detector,” Astroparticle Physical Review D,vol.52,no.7,pp.3793–3802,1995. Physics,vol.4,no.3,pp.217–225,1996. [8] M. Aglietta, B. Alpat, E. D. Alyea et al., “Muon “depth-intensity” relation measured by the LVD underground experiment and cosmic-ray muon spectrum at sea level,” Physical Review D,vol. 58,no.9,ArticleID092005,6pages,1998. [9]P.Adamson,C.Andreopoulos,D.J.Autyetal.,“Measurement of the underground atmospheric muon charge ratio using the MINOS near detector,” Physical Review D,vol.83,no.3,Article ID 032011, 10 pages, 2011. [10] B. Mitrica, “Design study of an underground detector for measurements of the differential muon flux,” Advances in High Energy Physics.Inpress. [11] M. Bektasoglu, H. Arslan, and D. Stanca, “Simulations of muon flux in slanic salt mine,” Advances in High Energy Physics,vol. 2012,ArticleID751762,8pages,2012. [12] M. Bektasoglu and H. Arslan, “Simulation of the zenith angle dependence of cosmic muon intensity in Slanic salt mine with Geant4,” Romanian Journal of Physics,vol.59,no.1-2,2014. [13] B. C. Rastin, “An accurate measurement of the sea-level muon spectrum within the range 4 to 3000 GeV/c,” Journal of Physics G,vol.10,no.11,pp.1609–1628,1984. [14] S. Agostinelli, J. Allison, K. Amako et al., “Geant4—a simulation toolkit,” Nuclear Instruments and Methods in Physics Research A, vol. 506, no. 3, pp. 250–303, 2003. [15] M. Bektasoglu and H. Arslan, “Estimation of the effects of the Earth’s electric and magnetic fields on cosmic muons at sea level by Geant4,” Journal of Atmospheric and Solar-Terrestrial Physics, vol. 74, pp. 212–216, 2012. [16] H. Arslan and M. Bektasoglu, “Azimuthal angular dependence study of the atmospheric muon charge ratio at sea level using Geant4,” Journal of Physics G,vol.39,no.5,ArticleID055201, 2012. [17] M. Bektasoglu and H. Arslan, “Investigation of the zenith angle dependence of cosmic-ray muons at sea level,” Pramana,vol.80, no. 5, pp. 837–846, 2013. [18] H. Arslan and M. Bektasoglu, “Geant4 simulation for the zenith angle dependence of cosmic muon intensities at two different geomagnetic locations,” International Journal of Modern Physics A,vol.28,no.16,ArticleID1350071,11pages,2013. [19] J. Reichenbacher and J. de Jong, “Calculation of the underground muon inensity crouch curve from a parameterization of the flux at surface,” in Proceeding of the 30th International Cosmic Ray Conference, Mexico City, Mexico, 2008. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 641584, 9 pages http://dx.doi.org/10.1155/2013/641584

Research Article Design Study of an Underground Detector for Measurements of the Differential Muon Flux

Bogdan Mitrica

Horia Hulubei National Institute of Physics and Nuclear Engineering-IFIN HH, P.O. Box MG-6, 077125 Magurele, Romania

Correspondence should be addressed to Bogdan Mitrica; [email protected]

Received 3 July 2013; Accepted 9 July 2013

Academic Editor: Carlo Cattani

Copyright © 2013 Bogdan Mitrica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Since 2006 an underground laboratory is in operation in Unirea salt mine from Slanic Prahova Romania. A new rotatable detector for measurements of the directional variation of the muon flux has been designed and will be put in operation at the end of 2013. The detector will be used to investigate the possible presence of unknown cavities in the salt ore. Preliminary muon flux measurements performed in the underground of Slanic Prahova salt mine show an important variation of the flux with the thickness of the rock but indicate also that more precise data are necessary. Based on that, a modern detector using 4 layers of plastic scintillators bars has been designed. The detector is installed on a rotatable and mobile frame which allows precise directional measurements of the muon flux on different locations in the mine. In order to investigate the performances of the detector, detailed Monte Carlo simulations have been performed using several codes available on the market. The simulations show that the detector can be used for measurements of the differential flux of cosmic ray muons and for the detection of hidden cavities in the ore.

1. Introduction Difficulties regarding the environmental problems are also influencing the prices and the feasibility of the methods. During the last decade, a number of events implying col- Weproposeaninnovativeand100%safemethodtodetect lapsed cavities have been observed all over the world [1, 2]. the hidden cavities based on the measurements of the muons These events illustrate the major risk represented by the from the secondary cosmic radiation, which is the most presence of unknown cavities in different ores, especially important source of natural radiation. on areas with historical mining activity. Starting from this problem waiting to be solved, an underground muon detector 1.1. The Cosmic Rays. From 1912, when Victor Hess has was built and will be put in operation in Slanic Prahova salt discovered the cosmic rays [3], until the present days, mine. an important advance was made in this field of research. Since the detection of hidden cavities is very important Shaded for a period of time by the parallel development andactualespeciallyforthelocationswithminingactivity,a of particle accelerators, an increasing interest is granted number of methods based on different techniques are avail- lately to cosmic rays, largely due to the higher energetic able in the market. Those methods are based on ultrasounds possibilities and a much lower cost of the infrastructure. The (US), X-ray, or even gamma ray detection. Several companies particle accelerators make it possible to obtain and investigate (e.g., http://www.socon.info/) over the world are presently particles of very high energy [4], but that is still very far performing this kind of measurements on request, but the from the energies observed in the cosmic rays spectrum 20 results are affected by secondary effects and experimental (up to 10 eV) [5]. The primary cosmic rays consist of uncertainties. The ultrasounds could affect the stability of the protons (90%), He nuclei (7%), and other nuclei (3%). The rocks, and the gamma and X-rays could produce important energy spectrum covers more than 21 orders of magnitude. changes in the radioactive background of the investigated The measurements of the spectrum of the primary cosmic area. Another problem of these methods is represented by rays could be performed by direct measurements only for 14 20 the relative high cost of the detectors and of the services. energies <10 eV. For higher energy, up to 10 eV, where 2 Advances in High Energy Physics

2 thefluxislowerthan1particle/km century, the spectrum In this paper we report about the design and construction could be measured only by indirect experiments that cover of an underground muon detector that could be used to detect large areas at the surface of the Earth [6–9]. There are a unknown cavities in different areas with mining activity. few open problems addressed by the researchers regarding Detailed Monte Carlo simulation using MUSIC [19]and the cosmic rays, such as the investigation of the hadronic CORSIKA [11] codes has been used in order to establish interaction models at high energies [10], where no data the performances of this new device. The simulations are from the accelerators are available, or the origin of cosmic controlled by muon flux measurements performed with the ray particles that are still far from a clear solution. Many mobile muon detector of IFIN-HH [20, 21]. simulation codes and computational models are available on the market [11–13], making possible detailed calculations for different processes at high and low energies. 2. Measurements of the Muon Flux in the Underground 1.2. The Muons. Interacting with the atmosphere of the Measurements of the cosmic ray muon flux have been Earth, cosmic rays break into numerous components, also performed in the underground of Slanic Prahova salt mine named the secondary cosmic radiation, generating Extended using the mobile detector of IFIN-HH. The detector has been air showers that propagate until they reach the ground. designed for underground and ground level measurements of Excepting the photons and neutrinos, muons are the most the total muon intensity. abundant component of the secondary cosmic radiation at Preliminary muon flux measurements have been per- the sea level. As leptons, muons are less affected by hadronic formed using the mobile detector of IFIN-HH [20, 21]. The interactions, also they show reduced electromagnetic interac- detector consists of 2 modules each one having 4 scintillator 2 tions and hence interact weakly with matter. They penetrate plates of 25 × 100 × 10 cm covering a surface of 1 m .The large thicknesses of matter before they are stopped and decay, systemismountedonavanwhichallowsalargemobilityon and they are historically known as penetrating̈ componentsof¨ ground level or in the underground. The signal from the each secondary cosmic rays, even detectable in deep underground − PMT is read out independently. Further, the four signals from sites. At sea level the inclusive flux is about 120 muonss 1 ⋅ −2 −1 each plate are ored, and then the results from both modules m ⋅ sr and comprises about 80% of the total atmospheric are put in coincidence. Directional measurements combining flux of secondary charged cosmic ray particles. Consequently 2 plates from both modules could be performed in order to much more information could be given by muons than on see the angular variation of the flux. Due to the large angular any other component, such as information on high energy acceptance, no precise directional variation of the muon flux processes in the atmosphere or on the primary radiation, is possible. in particular on its spectrum and composition. It is evident The angular acceptance of the detector was determined by that the flux of atmospheric muons is a quantity of high Monte Carlo simulation using GEANT4 [22] and CORSIKA interest and information which characterizes the site of a [11] codes. A correction factor of 1.56 should be applied to the nuclear physics laboratory which is planned for far-reaching raw data in the case of the detector mounted on the van and investigations. Another reason for measuring the flux of 1.11 in the case of the plates placed one on top of each other. the atmospheric muons in the underground arises from the practical necessity of information on the cosmic radiation background for different sites, for experiments like [14, 15], 2.1. The Slanic Prahova Salt Ore. The salt ore from Slanic and so forth. Prahova consists of a lens of 500 m thickness, few kilometers In the near past, the idea to use to use muons for the long and wide (see Figure 1). The salt is extracted from the detection of hidden cavities has been used in the search for Slanic mine continuously since ancient times, and due to hidden chambers in ancient pyramids from Giza [16]orthe this fact, many galleries (i.e., shaped caverns) are already AztecPyramidoftheSunatTeotihuacan[17]. Since these excavated. experiments reported interesting results, one can conclude Presently few mines are well known in Slanic Prahova salt that the cosmic ray muons could be used for the purpose of ore. Some of them are used for touristic activities (Unirea this project with high expectation. In the present days, some and Mihai), one of them (Cantacuzino) is still used to extract attempts to use muons for geophysical application have been salt, and others (Victoria and Carol) are closed for security done [18], but no device for the detection of hidden cavities is reasons. Some other ancient cavities close to the surface are availableinthemarket.Weproposetofillthisemptyposition used also for touristic activities. The existing cavities will be and will bring a new and modern solution for the security of used to calibrate the detector in the underground. people over the world. The method will be further extended Since 2006 a modern underground laboratory (𝜇Bqlab) for new application as geophysical data or speleology. [23, 24] for low background measurements was developed The idea of using cosmic ray muons to improve the quality by IFIN-HH in the Unirea mine from Slanic Prahova. The of life has great potential but is underestimated and largely location of the laboratory is indicated in Figure 1.During2012 unexplored. During the last years some preliminary results the laboratory has been upgraded and new scientific infras- have been obtained in IFIN-HH for measurements of the tructureshavebeenachieved.Oneofthemisrepresented muon flux in underground of Unirea mine, at sea level, and by a new rotatable mobile detector for measurements of the at ground level at different elevations. directional variation of the muon flux. Advances in High Energy Physics 3

The “Unirea” mine is characterized by temperature: 12.0- ∘ 3 13.0 C, humidity: 65–70%, excavated volume: 2.9 million m ,

430 2 241 257 + 177 273 304 + 319 225 193 + + 289 335 209 < 𝜇 + + + + + 351 + + +

+ floor area: 70000 m , average high: 52–57 m, aerosols 10 m: 8 3 ETAJ I ETAJ ETAJ VIII ETAJ IX ETAJ ETAJ VII ETAJ X ETAJ XI ETAJ ETAJ VI ETAJ XII ETAJ ETAJ V ETAJ ETAJ II ETAJ III ETAJ IV ETAJ 2⋅10 part/m , and distance between walls: 32–36 m, existing infrastructure:electricity,roads,railway,elevator,phone,

N Internet, and GSM networks (also inside the galleries). Slanic Prahova is also the Romanian proposal for LAGUNA-LBNO [14] experiment which is intent to build a huge detector with Mina Victoria 3 an active volume around 100.000 m . From the Slanic site a +188 +246 huge volume of material has been already excavated, but the XIII Camera (tavan) shallowdepthcouldinduceaproblem.Following[14], for Bloc Moldova XI 353 +374 + the GLACIER [25] experiment, the Slanic mine could be a Camera GENEZEI

II feasible location as for this technique a depth of only 600 mwe Camera moldova Camera Camera

I is necessary, comparable with the one of Unirea mine. Camera +398 +191 Jomp put Put extractie Put unite

VIII 2.2. Preliminary Muon Flux Measurements. The muon flux Camera Mina Unirea Mina Mina principatele Mina +410 hasbeenmeasuredinUnirea,Mihai,andCantacuzinomines oltenia Camera I

Camera at different levels, and some of the results was previously

VII reported in [20, 21]. New measurements in the main and Camera in a small cavity of Mihai mine have been performed. The results are presented in Table 1. The theoretical depth of each mine has been determined taking in to account the XIV I +188,5 Camera Camera presence of a standard rock hill above the entrance in the mine. The real depth has been obtained considering the geometrical characteristics of each cavity. Unfortunately, due Salina Slanic Prahova Salina Slanic Put De ExtractiePut to the nonuniform shape of the cavities, the results need to be

Profil Sud-Nord Prin Exploatari Sud-Nord Profil corrected and will be estimated based on directional variation ofthemuonflux.TheresultsarecomparedwithMonteCarlo simulations based on MUSIC code [19]. From Table 1 one can observe that the muon flux is quite +145 Mina Cantacuzino Mina +193 sensitive to the variation of the rock thickness. Unfortunately, the direction of the muons and the location of different caverns are not easy to be determined. For more precise Et. V Et. VI Et. VII Et. VIII Et. IX Et. X Et. XI Et. ocnele vechi measurements, it is mandatory to have a detector which Pilier protectie Pilier allows to establish the direction of the incoming muon and which permits to establish the presence and the location of possible unknown cavities. ˘ anic ˘ a 3. The New Rotatable Detector ¸¸ ˘ anic si orasul sl si orasul anic In order to perform precise measurements of the directional Galerie De Coast ̂ aul sl variation of the muon flux, a new detector was designed ̂ ar and constructed; the measurements are planned to start at the Autumn of 2013. The detector consists in 160 plastic scintillator bars type UPS 923A of 26 × 1000 × 100 mm (see Figure 2)producedbyAMCRYSLtd.Ukraine[26]. Each one has a stripe right on the middle where an optical fiber will Piler protectie pentru p protectie Piler transfer the optical pulse to the PMT [27]. The scintillator bars are mounted in 4 boxes (40 pcs in each box) and the signal from each one is transferred to one of the two micro-PMTs of type R11265-112-M64 [28](see S Figure 3)mountedoneachbox(seeFigure 4). The 4 boxes are mounted 2 by 2 at top of each other and orientated in cross. Then each group of 2 is separated by the 0 60 40 20 20 40 60 80 80 100 480 460 440 420 400 380 360 340 320 300 280 260 240 220 200 180 160 140 120 100 120 140 other one by a distance of 0.8 m, and the entire system is placed on a rotatable and mobile frame (see Figure 5). The direction of each incident muon is determined using the 𝑥 Figure 1: Artistic view of the salt ore of Slanic. and 𝑦 positions given by the signals from each scintillator bar from both layers of 2. 4 Advances in High Energy Physics

Table 1: The muon flux data obtained in underground measurements.

Muon flux for the Muon flux for Theoretical depth Real depth Muon flux Location theoretical depth the real depth −2 −1 (m) − − (m) − − (m s ) (m 2s 1) (m 2s 1) Unirea 297 0.051 170 0.18 0.18 Mihai—main cavity 204 0.12 90 0.51 0.52 Mihai—small cavity 204 0.12 125 0.32 0.33 Cantacuzino mine—level 5 173 0.17 165 0.19 Cantacuzino mine—level 6 189 0.15 173 0.17 Cantacuzino mine—level 7 205 0.12 181 0.16 Cantacuzino mine—level 8 221 0.11 189 0.15 0.19 Cantacuzino mine—level 9 237 0.10 197 0.13 Cantacuzino mine—level 10 253 0.08 205 0.12 Cantacuzino mine—level 11 269 0.06 213 0.11 0.09

UPS-923A Reﬂective coating h

s†0,1 ± 1,6

B ± 0,1 2,3 ± 0,1 h

l±1 A±0,1

Figure 2: The scintillator bar [26].

The information from PMT is read out by modular at high depths (more than 500 m.w.e.). Since the necessity of electronic devices produced by CAEN [29]andNational the detection of hidden cavities requires an estimation of the Instruments [30]. The detector permits directional measure- muon flux at low depths, in this paper, the code has been ∘ ments of the muon flux up to 45 zenith angle. We estimate modified in order to overlap its limits. For this task, it is that for the Slanic Prahova salt ore a realistic time of data importanttoknowthemuonfluxatgroundlevel.Toestimate acquisition will be from 2 to 3 days for a zenith angle from the muon flux at ground level, one can choose from several ∘ ∘ 0to10 up to 3 months for 45 . options represented by semianalytical formulas or detailed Monte Carlo simulation. The muon flux is defined as the number of muons 4. Simulation of the Detector’s Performances transversing a horizontal element of area per unit of time [5]:

For the studies regarding the detector performances, detailed 𝑑𝑁 −2 −1 −1 −1 𝐽𝜇 = (cm ⋅ s ⋅ sr ⋅ (GeV/c) ). Monte Carlo simulation is necessary. The directional varia- 𝑑𝑡 ⋅ 𝑑𝐴 ⋅ 𝑑Ω ⋅𝑑𝑃 tion of the muon flux in the underground should be accu- (1) rately simulated in order to investigate the detector response. There are several empirical approximations describing For this task, several options represented by computational the fluxes by analytical expressions like power-law distribu- codes are available. Some of them (e.g., GEANT4 [31]or tions (see Grieder [5]). Recent approaches by Gaisser [34] FLUKA [32, 33]) can give more precise results but request display explicitly the dependence on primary energy but a longer running time. Instead, the MUSIC code could give with complicated mathematical procedures and valid only realistic results in a shorter time. formuonenergiesabove10GeV.Thisholdsalsoforthe MUon SImulation Code (MUSIC) [19]isacomputa- simplification given in Gaisser’s Book [35] tional model for 3-dimensional simulations of the muon propagation through rock. It takes into account energy 0.14 −2.7 𝜙𝜇 = ⋅ (𝐸/GeV) losses of muons by pair production, inelastic scattering, cm2 ⋅ s ⋅ sr ⋅GeV bremsstrahlung, ionization and the angular deflection by 1 0.37 multiple scattering. The program uses the standard CERN ×[ + ] 1+(𝐸⋅ 𝜃/110 ) 1+(𝐸⋅ 𝜃/760 ) library routines and random number generators. The code cos GeV cos GeV has been originally designed for simulation of the muon flux (2) Advances in High Energy Physics 5

26,2 6,6 18,4 1

1 8

Useful area 26,2

57 64

Top view Side view

Figure 3: The microphotomultiplier tube [28].

1600 266 530 1070 57

66 526 1000

Figure 4: The detection module.

−𝛾 𝐴 ⋅𝑊 ⋅𝐸 𝜋 ⋅𝐻 used, for example, by Unger et al. [36]. In Figure 6 this 𝜋 𝜇 𝜋 𝜋 𝐷𝜋 (𝐸𝜇,𝜃)= , formula is compared with the results of the Monte Carlo 𝐸𝜋 ⋅ cos 𝜃+𝐻𝜋 simulations, displaying the disagreement in particular at −𝛾 (3) 𝐴 ⋅𝑊 ⋅𝐸 𝑘 ⋅𝐻 lower energies. 𝑘 𝜇 𝑘 𝑘 𝐷𝑘 (𝐸𝜇,𝜃)= . The approach by Judge and Nash [37]usesasinputthe 𝐸𝑘 ⋅ cos 𝜃+𝐻𝑘 production spectra of parent pions and kaons and calculates There 𝐻𝜋,𝑘 and 𝐻𝜇 are parameters accounting for the prop- the flux resulting from pion and kaon decay by agation of the particles in the atmosphere, and 𝜃 is the 6 Advances in High Energy Physics

1600 2492 513 280

2300

(a) (b)

Figure 5: The rotatable detector. ) 2 1 of the Monte Carlo simulations (see [39]) and to experimental data from the BESS experiment [40]. −1 (Gev/c) 10 Thespectraofpionsandkaonsaregivenbytherelation −1

sr −2 −2.97

−1 10 𝑁(𝑝𝜋,𝑘)=𝐴𝜋,𝑘 ⋅𝑝𝜋,𝑘 , (4) s −2 10−3 𝐴 𝐴 𝐴 = 0.373 (cm where the factors 𝜋 and 𝑘 have the values 𝜋 and 3

P 𝐴 =1 𝑘 . ∗

−4 10 Semianalytical approaches are able to reproduce globally

−5 the results of Monte Carlo simulations and experimental 10 data, and in particular the approach of Judge and Nash does account for muon energies <10 GeV. However, these 10−6

Differential ﬂux Differential approaches can be hardly modified in order to take into 10−1 1 10 102 103 account also finer effect like the influence of the geomagnetic Muon momentum (GeV/c) field. For that detailed Monte Carlo simulations have to be

Δ CORSIKA simulations for Hiroshima invoked. CORSIKA simulations for Bucharest The simulation tool COsmic Rays SImulation for KAS- Semianalytical formula of Gaisser CADE (CORSIKA) has been originally designed for the Semianalytical formula of Nash four-dimensional simulations of extensive air showers with 15 Experimental data BESS primary energies around 10 eV.Theparticletransport WILLI (preliminary) includes the particle ranges defined by the life time of the particle and its cross-section with air. The density profile Figure 6: Comparison between the muon flux obtained by COR- oftheatmosphereishandledascontinuousfunctionthus SIKA with experimental results obtained by BESS [40] and semianalytical formulae of Nash [37]andGaisser[35, 39, 41, 42]. not sampled in layers of constant density. Ionization losses, multiple scattering, and the deflection in the local magnetic field are considered. The decay of particles is simulated in exact kinematics, and the muon polarization is taken into ∘ azimuthal angle of the muons (0 for vertical). The parameters account. In contrast to other air shower simulations tools 𝐴𝜋,𝑘 are the normalizations of the pion and kaon production CORSIKA offers alternatively six different models for the spectra. There are several other parameters entering the description of the high energy hadronic interaction and approximation: the absorption lengths of the primary particle three different models for the description of the low energy 𝜆𝑝,ofthepions𝜆𝜋, and kaons 𝜆𝑘.Thereisclearlysome hadronic interaction. The threshold between the high and low 𝐸 =80 influence on pion and kaon’s production rate [38], but in energy models is set by default to Lab GeV/n. the present investigation, the values have been fixed along Figure 6 shows the comparison between the muon flux the original proposal [37]. Only the 𝐴𝜋,𝑘 values have been simulated with CORSIKA compared with the one using semi- changed in order to adjust the calculated fluxes to the results analytical approaches of Nash and Gaisser and experimental Advances in High Energy Physics 7

10−5 10−5 ) −1 ) sr −1 −1 sr s −1 −2 s

−2 −6 10 10−6 Muon flux (cm flux Muon Muon flux (cm flux Muon 10−7 0 10 20 30 40 50 0 10 20 30 40 50 Θ (deg) Θ (deg)

Muon ﬂux in Unirea mine for the real depth Muon flux in Cantacuzino-11 mine for the real depth Δ Muon ﬂux in Unirea mine for the physical depth Δ Muon flux in Cantacuzino-11 mine for the physical depth

(a) (b)

Figure 7: The directional variation of muon flux for the real and theoretical depths of Unirea mine (a) the directional variation of muon flux for the real and theoretical depths of Cantacuzino mine, level 11 (b).

10−4

) Necessary time for measurements

−1 2

sr 10 −1 s −5 −2 10

−6 (hours) Time

Muon ﬂux (cm ﬂux Muon 10

1 0 10 20 30 40 50 200 300 400 500 600 700 800 900 1000 1100 Θ (deg) Depth (m)

Δ Muon flux in Mihai mine for the real depth Muon flux in Mihai mine-gallery for the real depth Figure 9: Simulation of the muon flux in the underground of Slanic Muon flux in Unirea mine for the real depth Prahova salt mine. Muon flux in Cantacuzino-11 mine for the real depth

Figure 8: Simulation of the directional variation of the differential muon flux in the underground of Slanic Prahova salt mines. Mihai mines is presented. The differences are significant and are caused by different depths. The possibility do detect hidden cavities in the salt dome has been investigated using the MUSIC code. The requested data given by BESS experiment. From this picture one can measurement time for the detection of a 10 m gap starting see that the best solution for the muon flux at sea level is from 50 m depth has been calculated taking into account the represented by CORSIKA simulation code. necessary statistics to obtain valid results. In Figure 9 the In order to perform accurate simulation of the muon flux variation of the requested measurements time is represented. intheunderground,CORSIKAresultshavebeenusedas From this picture one can observe that for depths up to input data for MUSIC program and for its extension MUSUN 1000 m the necessary time is not bigger than 250 hours [19]. (aprox. 10 days) which is more than reasonable for the The directional variation of the muon flux in the under- purposes of this project. ground has been simulated taking into account the angular acceptance of the new detector. The results obtained for the real and for the theoretical depths of Cantacuzino and 5. Conclusions Unirea mines are displayed in Figure 7.Onecanobserve that important variation of the differential flux for the real A new detector has been developed which permit directional and theoretical depths is present. In Figure 8 the directional measurements of the muon flux. The muon flux measure- variation of the muon flux for Unirea, Cantacuzino, and mentswillbeusedtostudythepossibilitytodetectunknown 8 Advances in High Energy Physics cavities in the old mining sites (e.g., Slanic Prahova). Mea- [10] J. Wentz, A. F. Badea, A. Bercuci et al., “The relevance of surements could be performed on different sites from Europe the muon charge ratio for the atmospheric neutrino anomaly,” using the mobile detector, but one should take into account Journal of Physics G,vol.27,no.7,p.1699,2001. that the for the biggest depths a much longer time is necessary. [11] D. Heck, J. Knapp, J. N. Capdevielle, G. Schatz, and T. Thouw, A new version of MUSIC computational code has been ReportFZKA,6019ForschungszentrumKarlsruhe,1998. developed and allows simulation of the muon flux for shallow [12] A. Geranios et al., “AIRES EAS simulations for the energy depths. The code could be adapted for any site and rock estimation of UHECR,” in Proceedings of the 29th International composition from the world. Cosmic Ray Conference, Pune, INDIA, August 2005. Such muon flux measurements could be also used for [13] X. Duan et al., “A shannon-runge-kutta-gill method for geological studies, for example, to explore variations in the convection-diffusion equations,” Mathematical Problems in rock density above the observation level. The mobility of the Engineering,vol.2013,ArticleID163734,5pages,2013. detector implies a considerable practical flexibility of using [14] A. Rubbia, “Underground neutrino detectors for particle and the procedure of measuring muon flux differences for various astroparticle Science: The Giant Liquid Argon Charge Imaging aspects. ExpeRiment (GLACIER),” Journal of Physics,vol.171,no.1, Article ID 012020. [15] P. Schreiner and M. Goodman, “Measurement of the muon Acknowledgments charge ratio in MINOS,”in Proceedings of the 29th International Cosmic Ray Conference, vol. 9, pp. 97–98, Pune, India, 2005. The present work has been possible due to the support of the [16] L. W. Alvarez, J. A. Anderson, F. El Bedwei et al., “Search for Romanian Authority for Scientific Research by the projects hidden chambers in the pyramids,” Science,vol.167,no.3919, PN 09 37 01 05 and CAPACITATI MARI, 7PM/I/2008, IFIN- pp. 832–839, 1970. Dezvoltarea Infrastructurii de Cercetare.Theauthorthanks [17] R. Alfaro, E. Belmont-Moreno, A. Cervantes et al., “A muon Dr. Denis Stanca, Eng. Mihai Niculescu-Oglinzanu, and Eng. detector to be installed at the Pyramid of the Sun,” Revista Bogdan Cautisanu for their useful advices and help. 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[29] http://www.caen.it/. [30] http://www.ni.com/. [31] M. Bektasoglu, H. Arslan, and D. Stanca, “Simulations of muon flux in slanic salt mine,” Advances in High Energy Physics,vol. 2012,ArticleID751762,8pages,2012. [32] A. Fasso,` A. Ferrari, and P. R. Sala, “Electronphoton transport in FLUKA: status,” in Proceedings of the MonteCarlo 2000 Conference,A.Kling,F.Barao,M.Nakagawa,L.Tavora,and P. Vaz, Eds., pp. 159–164, Springer, Berlin, Germany, October 2000. [33] http://www.fluka.org/. [34] T. K. Gaisser, “Semi-analytic approximations for production of atmospheric muons and neutrinos,” Astroparticle Physics,vol. 16, no. 3, pp. 285–294, 2002. [35] T. K. Gaisser, Cosmic Rays and Particle Physics, Cambridge University Press, Cambridge, UK, 1992. [36] M. Unger et al., “Results from L3+C,” International Journal of Modern Physics A ,vol.20,p.6928,2005. [37] R. J. R. Judge and W. F. Nash, “Measurements on the muon flux at various zenith angles,” Il Nuovo Cimento Series 10,vol.35,no. 4, pp. 999–1024, 1965. [38] I. M. Brancus, H. Rebele, A. Haungs et al., “WILLI—a scintillator detector setup for studies of the zenith and azimuth variation of charge ratio and flux of atmospheric muons,” Nuclear Physics B—Proceedings Supplements,vol.175-176,pp.370–373,2008. [39] B. Mitrica, “Asymmetry of charge ratio for low energetic muons,” AIP Conference Proceedings, vol. 972, pp. 500–504, 2008. [40] M. Motoki et al., “Precise measurements of atmospheric muon fluxes with the BESS spectrometer,” Astroparticle Physics,vol.19, no.1,pp.113–126,2003. [41] B. Mitrica et al., “KASCADE-Grande, Forschungszentrum Karlsruhe,” Internal Report 2004-01, 2004. [42] H. Rebel, J. Wentz, I. M. Brancus, O. Sima, B. Mitrica, and G. Toma, “Scientific report on the NATO project,” Tech. Rep. Nr.PDD(CP) (PST.CLG, 979737), 2004. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 680678, 3 pages http://dx.doi.org/10.1155/2013/680678

Research Article Essay on Kolmogorov Law of Minus 5 over 3 Viewed with Golden Ratio

Ming Li1 and Wei Zhao2

1 School of Information Science & Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China 2 Department of Computer and Information Science, University of Macau, Avenue Padre Tomas Pereira, Taipa 1356, Macau

Correspondence should be addressed to Ming Li; ming [email protected]

Received 14 July 2013; Accepted 7 August 2013

Academic Editor: Carlo Cattani

Copyright © 2013 M. Li and W. Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The golden ratio is an astonishing number in high-energy physics, neutrino physics, and cosmology. The Kolmogorov −5/3 law plays a role in describing energy transfer of random data or random functions. The contributions of this essay are in twofold. One is to express the Kolmogorov −5/3 law by using the golden ratio. The other is to represent the fractal dimension of random data following the Kolmogorov −5/3 law with the golden ratio. It is our hope that this essay may be helpful to provide a new outlook of the Kolmogorov −5/3 law from the point of view of the golden ratio.

1. Instruction The rest of paper is organized as follows. The preliminar- 𝜑 (1 + √5)/2 ies are briefed in Section 2. The results that the Kolmogorov Let be the golden ratio. It equals to . Its inverse −5/3 1/𝜑 = (√5−1)/2 law and the self-similarity are explained from a view of iscalledthegoldenmean.Bothareirrational thegoldenratioaregiveninSection 3,whichisfollowedby numbers. Approximately, they are conclusions. 1 𝜑 ≈ 1.618, ≈ 0.618. 𝜑 (1) 2. Preliminaries The number 𝜑 has wide applications to various fields, ranging from physics (Wurm and Martini [1], King [2], Ding 2.1. Golden Ratio. Consider a straight line in Figure 1.Arra- et al. [3], Feruglio and Paris [4]), to cosmology (Livio [5], nge three points A, B, and C, there such that the ratio below Boeyens [6]). AC AB In addition to the golden ratio, fractal is a mathematical = (2) model attracting interests of scientists and physicists in the CB AC field of high-energy physics (Ghosh et al. [7]). While studying energy transfer, the Kolmogorov −5/3 law introduced by equals to 𝜑.Then,wesaythatthestraightlineiscutinextreme Kolmogorov [8]playsaroleinthefield(Qian[9], Brun et andmeanratio([5], Ackermann [20], Kaygn et al. [21]). al. [10, Chapter 7], Hillebrandt and Kupka [11,Chapter4], There are various ways to synthesize the number 𝜑. Acco- Gomes-Fernandes et al. [12,page309],LumleyandYaglom rding to the definition of extreme and mean ratio, one has [13], Warhaft [14], Geipel et al. [15]). Motivated by those, −5/3 𝑥+1 this essay aims at exhibiting the Kolmogorov law and 𝑥= . (3) the self-similarity, which is an important fractal property 𝑥 (Mandelbrot [16, 17], Cattani et al. [18, 19]), from the point of the golden ratio. It is our expectation that this essay may Multiplying 𝑥 on the both sides of the above yields be helpful to describe the nature of random data that follows 2 the Kolmogorov −5/3 law. 𝑥 −𝑥−1=0. (4) 2 Advances in High Energy Physics

ABC In order to connect the Kolmogorov −5/3 law with the golden ratio, we write Figure 1: Illustration of a straight line for the golden ratio. 5 +𝑒=𝜑. 3 (12) Solving (4)produces From the previous discussions, we have 1+√5 1−√5 𝑥 = ,𝑥= . 1 2 (5) 5 1+√5 5 √5−1 2 1 2 2 2 𝑒=𝜑− = − = − = − . 3 2 3 2 3 𝜑 3 (13) Conventionally, 𝑥1 is denoted by 𝜑,and𝑥2 is denoted by −1/𝜑 . Thus, the Kolmogorov −5/3 law expressed by (11)maybe rewritten by 2.2. Fractal Dimension. Let the autocorrelation function (ACF) of a random function 𝑥(𝑡) be 𝑟𝑥𝑥(𝜏),where𝑟𝑥𝑥(𝜏) = −(𝜑−𝑒) 𝑆𝑥𝑥 (𝜔) ∼𝑐|𝜔| for 𝜔󳨀→∞. (14) 𝐸[𝑥(𝑡)𝑥(𝑡 +𝜏)].Then,𝑟𝑥𝑥(𝜏) for 𝜏→0represents the small- scaling phenomenon of 𝑥(𝑡).FollowingDaviesandHall[22], From (10), we see that the fractal index may be expressed if 𝑟𝑥𝑥(𝜏) is sufficiently smooth on (0, ∞)andif by using the golden ratio as 𝛼 𝑟𝑥𝑥 (0) −𝑟𝑥𝑥 (𝜏) ∼𝑐1|𝜏| for |𝜏| 󳨀→ 0, (6) 𝛼=𝜑−𝑒−1. (15) where 𝑐1 is a constant and 0<𝛼≤2is the fractal index of 𝑥(𝑡), the fractal dimension, which is denoted by D,ofx(t)is Thus, we attain the fractal dimension from the point of view in the form ofthegoldenratiointheform 𝛼 𝐷=2− . 𝜑−𝑒−1 2 (7) 𝐷=2− . 2 (16) Note that fractal dimension 1≤𝐷<2is a measure to characterize the local self-similarity or local roughness of 𝑥(𝑡) Fromthepreviousdiscussions,wehavethefollowingtheo- (Mandelbrot [17], Gneiting and Schlather [23]). rems. Denote the power spectrum density (PSD) function of 𝑥(𝑡) 𝑆 (𝜔) Theorem 1. Let 𝑥(𝑡) betherandomfunctionthatobeysthe by 𝑥𝑥 .DenotebyFtheoperatorofFouriertransform. −5/3 Then, Kolmogorov law expressed by (11). Then, its PSD may be expressed using the golden ratio in (14). ∞ −𝑗𝜔𝑡 √ 𝑆𝑥𝑥 (𝜔) = F [𝑟𝑥𝑥 (𝑡)]=∫ 𝑟𝑥𝑥 (𝑡) 𝑒 𝑑𝑡, 𝑗= −1. (8) Theorem 2. 𝑥(𝑡) −∞ Let be the random function that obeys the Kolmogorov −5/3 law expressed by (11). Then, its fractal Inthedomainofgeneralizedfunctions,wehave(Kanwal dimensionmayberepresentedbyusingthegoldenratioasthat [24], Gelfand and Vilenkin [25], Li and Lim [26]) in (16).Theconcretevalueof𝐷 in (16) is 5/3.

𝛼 𝜋𝛼 −(𝛼+1) F (|𝜏| )=−2sin ( )Γ(1+𝛼) |𝜔| , (9) Finally, we note that the significance of the present results 2 lies in expressing the Kolmogorov −5/3 law, which plays a role Γ in energy transfer, by using the golden ratio, which attracts where is the Gamma function. Therefore, we have an ana- interests of researchers and scientists in high-energy physics, logy of (6)inthefrequencydomainintheform neutrino physics, cosmology, and many others. 𝛼 −(𝛼+1) F (|𝜏| ) ∼𝑐2|𝜔| for 𝜔󳨀→∞, (10) 4. Conclusions where 𝑐2 is a constant (Chan et al. [27]). We have explained our results in expressing the Kolmogorov −5/3 law with the golden ratio. In addition, we have expressed 3. Results the fractal dimension of random data obeying the Kol- −5/3 The Kolmogorov −5/3 law implies that the PSD of a random mogorov law based on the golden ratio. function has the asymptotic behavior expressed by

−5/3 Acknowledgments 𝑆𝑥𝑥 (𝜔) ∼𝑐|𝜔| for 𝜔󳨀→∞, (11) This work was supported in part by the National Natural where 𝑐 is a constant (Monin and Yaglom [28, 29]). The above Science Foundation of China under the Project Grant nos. well characterizes the local irregularity of turbulence function 61272402,61070214,and60873264,andbythe973planunder 𝑥(𝑡) (Tropea et al. [30]). the Project Grant no. 2011CB302800. Advances in High Energy Physics 3

References [23] T. Gneiting and M. Schlather, “Stochastic models that separate fractal dimension and the Hurst effect,” SIAM Review,vol.46, [1] A. Wurm and K. M. Martini, “Breakup of inverse golden mean no. 2, pp. 269–282, 2004. shearless tori in the two-frequency standard nontwist map,” [24] R. P. Kanwal, Generalized Functions: Theory and Applications, Physics Letters A,vol.377,no.8,pp.622–627,2013. Birkhauser,¨ 3rd edition, 2004. [2] S. F. King, “Tri-bimaximal-Cabibbo mixing,” Physics Letters B, [25] I. M. Gelfand and K. Vilenkin, Generalized Functions,vol.1, vol. 718, no. 1, pp. 136–142, 2012. Academic Press, New York, NY, USA, 1964. [3] G. J. Ding, L. L. Everett, and A. J. Stuart, “Golden ratio neutrino [26] M. Li and S. C. Lim, “A rigorous derivation of power spectrum mixing and A5 flavor symmetry,” Nuclear Physics B,vol.857,no. of fractional Gaussian noise,” Fluctuation and Noise Letters,vol. 3, pp. 219–253, 2012. 6, no. 4, pp. C33–C36, 2006. [4] F. Feruglio and A. Paris, “The golden ratio prediction for the [27]G.Chan,P.Hall,andD.S.Poskitt,“Periodogram-basedestima- solar angle from a natural model with A5 flavour symmetry,” tors of fractal properties,” The Annals of Statistics,vol.23,no.5, Journal of High Energy Physics, vol. 2011, no. 3, article 101, 2011. pp. 1684–1711, 1995. [5] M. Livio, The Golden Ration,RandomHouse,2003. [28] A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: [6]J.C.A.Boeyens,Chemical Cosmology, Springer, 2010. Mechanics of Turbulence, vol. 1, The MIT Press, Cambridge , [7]D.Ghosh,A.Deb,S.Paletal.,“Evidenceoffractalbehaviorof Mass, USA, 1971. pions and protons in high energy interactions—an experimen- [29] A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: tal investigation,” Fractals,vol.13,no.4,pp.325–339,2005. Mechanics of Turbulence, vol. 2, The MIT Press, Cambridge , [8] A. N. Kolmogorov, “Local structure of turbulence in an incom- Mass, USA, 1971. pressible viscous fluid at very high Reynolds numbers,” Soviet [30]C.Tropea,A.L.Yarin,andF.JohnFoss,Springer Handbook of Physics Uspekhi, vol. 10, no. 6, pp. 734–736, 1968. Experimental Fluid Mechanics, Springer, 2007. [9] J. Qian, “Generalization of the Kolmogorov −5/3 law of turbulence,” Physical Review E,vol.50,no.1,pp.611–613,1994. [10] C. Brun, D. Juve,´ M. Manhart, and C. D. Munz, Numerical Simu- lation of Turbulent Flows and Noise Generation, Springer, 2009. [11] W.Hillebrandt and F. Kupka, Interdisciplinary Aspects of Turbu- lence, Springer, Berlin, Germany, 2009. [12] R. Gomes-Fernandes, B. Ganapathisubramani, and J. C. Vas- silicos, “Particle image velocimetry study of fractal-generated turbulence,” Journal of Fluid Mechanics,vol.711,pp.306–3336, 2012. [13] J. L. Lumley and A. M. Yaglom, “ACentury of Turbulence,” Flow, Turbulence and Combustion,vol.66,no.3,pp.241–286,2001. [14] Z. Warhaft, “Turbulence in nature and in the laboratory,” Proceedings of the National Academy of Sciences of the United States of America,vol.99,no.1,pp.2481–2486,2002. [15]P.Geipel,K.H.H.Goh,andR.P.Lindstedt,“Fractal-generated turbulence in opposed jet flows,” Flow, Turbulence and Combus- tion,vol.85,no.3-4,pp.397–419,2010. [16] B. B. Mandelbrot, Gaussian Self-Affinity and Fractals,Springer, New York, NY, USA, 2002. [17] B. B. Mandelbrot, The Fractal Geometry of Nature,W.H. Freeman, New York, NY, USA, 1982. [18] C. Cattani, G. Pierro, and G. Altieri, “Entropy and multifrac- tality for the myeloma multiple TET 2 gene,” Mathematical Problems in Engineering,vol.2012,ArticleID193761,14pages, 2012. [19] C. Cattani, “Fractional calculus and Shannon wavelet,” Math- ematical Problems in Engineering,ArticleID502812,26pages, 2012. [20] E. C. Ackermann, “The Golden Section,” The American Mathe- matical Monthly,vol.2,no.9-10,pp.260–264,1895. [21] B. Kaygn, B. Balc¸in, C. Yildiz, and S. Arslan, “The effect of teach- ing the subject of Fibonacci numbers and golden ratio through the history of mathematics,” Procedia—Social and Behav- ioral Sciences,vol.15,pp.961–965,2011. [22] S. Davies and P. Hall, “Fractal analysis of surface roughness by using spatial data,” JournaloftheRoyalStatisticalSocietyB,vol. 61,no.1,pp.3–37,1999. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 735452, 6 pages http://dx.doi.org/10.1155/2013/735452

Research Article Wavelets-Computational Aspects of Sterian Realistic Approach to Uncertainty Principle in High Energy Physics: A Transient Approach

Cristian Toma

Faculty of Applied Sciences, Politehnica University, 313 Splaiul Independentei, 060042 Bucharest, Romania

Correspondence should be addressed to Cristian Toma; [email protected]

Received 13 August 2013; Accepted 21 August 2013

Academic Editor: Carlo Cattani

Copyright © 2013 Cristian Toma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This study presents wavelets-computational aspects of Sterian-realistic approach to uncertainty principle in high energy physics. According to this approach, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in reciprocal Fourier spaces. However, such aspects regarding the use of conjugate Fourier spaces can be also noticed in quantum field theory, where the position representation of a quantum wave is replaced by momentum representation before computing the interaction in a certain point of space, at a certain moment of time. For this reason, certain properties regarding the switch from one representation to another in these conjugate Fourier spaces should be established. It is shown that the best results can be obtained using wavelets aspects and support macroscopic functions for computing (i) wave-train nonlinear relativistic transformation, (ii) reflection/refraction with a constant shift, (iii) diffraction considered as interaction with a null phase shift without annihilation of associated wave, (iv) deflection by external electromagnetic fields without phase loss, and (v) annihilation of associated wave-train through fast and spatially extended phenomena according to uncertainty principle.

1. Introduction leading to the following statement. The product of the inac- curacies arising from simultaneous determination of two AccordingtotheSterianrealisticapproach[1], one cannot canonical conjugate variables is of the order of magnitude of make a device for the simultaneous measuring of the canon- Planck’s constant (see [2] for more details). ical conjugate variables in the conjugate Fourier spaces. Theserelationscanbewrittenalsointheformofcom- Generally, Heisenberg’s uncertainty principle states that mutation relationships between corresponding incompatible incompatible dynamic variables in relation to the measuring observables according to the theorem: being given two her- process satisfy the following relations: mitic operators 𝐴̂ and 𝐵̂ and their commutator 𝐶̂,onecan ℎ demonstrate the relationship Δ𝑥Δ𝑝𝑥 ≥ , 2 ⟨𝐶⟩̂ ℎ Δ𝐴Δ𝐵 ≥ , (2) Δ𝑦Δ𝑝 ≥ , 2 𝑦 2 (1) ℎ where the brackets for 𝐶̂indicate an expectation value (aspect Δ𝑧Δ𝑝 ≥ , 𝑧 2 emphasized in [1]). By this moment, no actual physical mea- ℎ surements can avoid the limitations of these relationships. Δ𝑡Δ𝐸 ≥ , TheSterianrealisticapproachisbasedontheanalysisofa 2 wave packet (the usual mathematical model for propagating where Δ𝑥, Δ𝑦,andΔ𝑧 are the uncertainties in position for associated wave function). From the classical theory of wave each spatial axis, Δ𝑝𝑥, Δ𝑝𝑦,andΔ𝑝𝑧 are the uncertainties propagation, it is known that the width of a wave packet in momentum, and ℎ is the reduced Planck constant, thus Δ𝑥 involves a certain width Δ𝑘 in the reciprocal Fourier 2 Advances in High Energy Physics transform space, which corresponds to the wave vector 𝑘. (i) The space-time coordinates 𝑥, 𝑦, 𝑧,and𝑡 correspond- 󸀠 Standard Fourier analysis involves ing to the received wave are transformed into the 𝑥 , 𝑦󸀠 𝑧󸀠 𝑡󸀠 Δ𝑥Δ𝑘 ≥ 2𝜋. (3) , ,and coordinates of the transformed wave according to the action of the Lorentz transformation Δ𝑥 Δ𝑘 𝑇 Thus uncertainties for and are interconnected. Since matrix 𝐿 upon the cuadrivector [𝑥,𝑦,𝑧,𝑖𝑡] of this 𝑝 momentum for a quantum particle is proportional to the supposed coordinates 𝑥, 𝑦, 𝑧,and𝑡 (the coordinates the 𝑝=ℎ𝑘 wave vector of the associated wave ( ), it results in that wave would have had in the absence of interaction), Δ𝑥 Δ𝑝 alowervalue involves a greater value for uncertainty with the space-time origin considered in the point of upon momentum. Similarly, for finite duration disturbances, space and at the moment of time where the received the standard Fourier analysis involves wave first time interacts with the observer’s material Δ𝑡Δ𝜔 ≥ 2𝜋 (4) medium (in fact 𝐿 matrix multiplies the column [𝑥,𝑦,𝑧,𝑖𝑡]𝑇 against reciprocal Fourier transform spaces of a pair of cuadrivector so as to result in the column [𝑥󸀠,𝑦󸀠,𝑧󸀠,𝑡󸀠]𝑇 signals, where Δ𝜔 corresponds to the spectral width. cuadrivector ). 󸀠 Further it is considered that in quantum physics simul- (ii) The transformed wave function 𝜙 is represented by taneous measurements of the canonical conjugate variables a vector or a higher-order tensor which describes (as space momentum, time energy) cannot be performed, the quantum field/particle. For an electromagnetic because these variables correspond to the reciprocal Fourier wave, 𝜙 corresponds to the cuadridimensional vector 𝑇 spaces. The time interval required by any device for per- [𝐴, 𝑖𝑉] . In the most general case, 𝜙 corresponds to forming any measurement upon a certain quantum particle a state vector describing the quantum particle. The or system cannot be avoided. The standard experiments are Lorentz matrix 𝐿 multiplies the vector of higher-order not meant for simultaneous measurements of position and tensor 𝜙 ofthereceivedwavesoastoresultinthe 󸀠 momentum of a quantum system, as it is usually admit- vector or higher-order tensor 𝜙 of the transformed ted, being a consequence of the used measurement device wave. (Fourier transformer), through which the signals pass having a finite speed. Any quantum system is subject to uncertainty As a consequence, each Lorentz transformation is specific relations, proving its dual nature. Due to the requirements of to a certain wave train, with the zero moment of time con- the principle of causality in the theory of relativity, one cannot sidered when the received wave first time interacts with the make a device for simultaneous measuring of the canonical observer’s material medium. Thus this relativistic transfor- conjugate variables in the conjugate Fourier spaces. Due to mation is connected to transient phenomena (as the propa- finite speed of propagation of interactions, signal switch- gation of associated waves), and no memory of previous mea- ing within physical system to perform Fourier transform surements is involved regarding space-time measurements has a finite duration. Moreover, uncertainty relations are for events in different reference (material) systems. Logical considered to confirm the principle of causality due to the contradictions as clock paradox do not appear any more. finite speed of propagation of interactions required by any This aspect implies a very important property of mea- measurement process. suring device to be added to Sterian realistic approach, with However, this approach is far from being rigorous. Spe- significant consequences upon computational methods: the cific aspects regarding relativistic transformation of waves observer’s material medium acts in a nonlinear manner upon implied by measuring procedures should be added. Next a a superposition of received wave trains 𝜙𝑖 in a certain area. rigorous analysis for coherence aspects implied by different Each wave train has its own amplitudes, frequency, and wave interaction phenomena should be performed (the Fourier vector,butithasalsoitsownzeromomentoftimetobe transformation being just a mathematical tool for analyzing considered within Lorentz transformation. Thus at a certain this coherence for wave packets corresponding to a certain moment of time there will be different time intervals Δ𝑡𝑖 for quantum particle). Finally, aspects regarding creation annihi- each wave train considered from each specific time origin 𝑇0𝑖 lation of quantum particles during the measurement process as should be analyzed by taking into consideration correlation aspectsinquantumfieldtheoryandthechangefromcoor- Δ𝑡𝑖 =𝑇−𝑇0𝑖. (6) dinate space to momentum space (reciprocal Fourier spaces) According to Lorentz transformation of space-time coordi- required by the mathematical model. 󸀠 nates, the time interval Δ𝑡𝑖 for the wave train transformed by 2. Supplementary Aspects regarding the observer’smaterial medium (considered from this specific 𝑇 Relativistic Wavelets Transformation time origin 0𝑖)willbe 1 As has been shown in [3], a certain wave function received Δ𝑡󸀠 = Δ𝑡 . 𝑖 𝑖 (7) by a reference system 𝑆 (represented by a material medium) √1−(V/𝑐)2 is transformed according to Lorentz transformation as 󸀠 󸀠 󸀠 󸀠 󸀠 𝜙 𝜙 (𝑥 ,𝑦 ,𝑧 ,𝑡 )=𝐿𝜙(𝑥,𝑦,𝑧,𝑡), (5) Thus all parts of the wave trains 𝑖 received by the observer’s material medium at a certain moment of time 𝑇 will be where the following hold. translated in time with different values, depending on these Advances in High Energy Physics 3

󸀠 time differences Δ𝑡𝑖 considered from different time origins The effect of interface nonuniformities could be consid- 𝑇0𝑖 as eredasvanishingbydrawingatangentlineasaglobalapprox-

󸀠 imation through the front surface for each point-source wave. 𝜙𝑖 (𝑇) 󳨀→ 𝜙 𝑖 (𝑇0𝑖 +Δ𝑡𝑖 ) (8) Yet there is no valid argument regarding a minimum value for the radii of curvature of this tangent line. Theoretically, it (a certain delay time; there is no possible anticipation of a couldbeverysmall,andthustheglobaltangentlinecould future event). This translation is a nonlinear transformation, consistofalotoflocalcurveswithsignificantcurvatures since parts of the reviewed wave train are translated differ- which are joined together. In this way a lot of divergent light ently. After this translation is performed, it can be considered beams could be created along the reflected/refracted trajec- that the Lorentz matrix [𝐿] acts in a linear manner upon the tory, and the directionality would be lost very quickly. 𝜙 vector or higher-order tensor specific to the wave train 𝑖 so A more rigorous standard quantum model considers that 𝜙󸀠 as to result in the final transformed wave train 𝑖 as photons interact with collectivised electrons of the solid crystalline lattice before being reemitted. Since the associated 𝜙󸀠 (𝑇 +Δ𝑡󸀠)=[𝐿] 𝜙 (𝑇 +Δ𝑡󸀠). 𝑖 0𝑖 𝑖 𝑖 0𝑖 𝑖 (9) wave function for the collectivised electrons is represented in position for large space intervals, the influence of local Thus it results in that high energy (relativistic) corrections nonuniformities is decreased. Thus a tangent line local radius regarding finite speed for propagation of interaction for mea- of curvature greater than a certain value can be drawn, and surements based on quantum aspects (presented in Sterian a better directionality for reflected/refracted wave can be realistic approach) should be completed with considerations obtained. about nonlinear transformations for superposition of wave However, this standard quantum model does not take into trains (as shown above) so as to result in a rigorous and account the phase shift between the incident and the reemit- complete transient approach. With each wave train being ted wave for different points of the interface. A complete described by specific frequency and wave vector, it results in analysisbasedonquantumtheoryshouldconsiderthatwaves that temporal and spatial correlations are involved in these reemitted from different points of the interface are part of the relativistic transformations for part of wave trains. Supple- wavet rain corresponding to a certain photon, with the prob- mentary aspects regarding phase will be presented in the next ability of detecting a reflected/refracted photon being deter- paragraph. mined by the coherent plane-wave compounding method (it is well known that a particle interferes just with itself). There 3. Aspects regarding Phase Changes for isnovalidargumentbasedexclusivelyonstandardquantum Sterian Realistic Approach theory regarding the constant phase shift between the local incident wave and the corresponding local reemitted wave The most usual transformations performed by a certain mate- in each interface point. The wave function for collectivised rial medium upon a wave function are reflection and refrac- electrons of the crystal lattice is far from being constant in tion. A preliminary analysis of reflection/refraction phe- space time along this interface. nomena is based on classical electromagnetic field, which For this reason, space correlations for the incident wave corresponds in fact to the wave function associated with a trainandthereemittedwavetrainshouldbetakeninto photon (the electric field E,magneticfieldB,vectorpotential account. This could suggest that lattice quantum functions A,andscalarpotentialV are the main quantities used). interact in a global manner with parts of the incident wave on This wave function can explain basic aspects in wave theory a large spatial area of the interface, with a certain correlation of light as reflection/refraction angles and the influence between phases of reflected/refracted wave being noticed of polarization upon electric and magnetic fields for both even for surface points separated by crystal defects. Some reflected and refracted beams. support functions generated on this interface able to correlate At first view this classical model is just an approximation, the phase of reemitted waves by remote interface points could since according to the rigorous quantum theory the energy be taken into consideration for a complete computational can be transmitted just in a discontinuous manner, with model. amounts of ℎ𝜔. Parts of the incident (received) wave undergo In the same study [4] showed that space correlations an interaction with a certain interface, so quantum aspects achieved within a very short time interval for nonadjacent should be involved. However, as was shown in [4], an anal- spatial intervals which interact with wavefronts (part of the ysis based exclusively on standard quantum aspects cannot reflected/refracted wave) become a key issue within an exclu- explain important phase aspects. Within such a model, light sively quantum model if we consider that reflected/refracted consists of photons which are packets of energy that primarily wave can undergo diffraction at a later time. The requirement interact with interface atoms. Through this interaction, the of constant phase shift for parts of the associated wave energy of the photon is absorbed by collectivised electrons generated by points or edges of a diffraction grating is still of the solid crystalline lattice, and the photon ceases to exist. valid—yet the points or edges of such a diffraction grating are Then the electron will return to a lower energy state by emit- situatedonnonadjacentareas,sotheycannotbecorrelatedby ting a photon. Each photon behaves more like a point source, any surface quantum wave functions. as if the light was originating right there. These emitted These aspects imply either the use of space-propagation spherical waves generate the total wavefront as the envelope properties for the support functions previously mentioned that encloses all these point-source waves. or the use of the assumption that the reemitted wavefronts 4 Advances in High Energy Physics areinphasewiththereceivedwavefronts(partsofthesame bytheelectricormagneticfieldofthemeasurementdevice. incident wave), with the phase shift being null anywhere, at As was shown in [7], the standard second quantification any time. cannot describe these interactions without the use of certain The second choice is far more attractive (being more support functions. The use of virtual photons for analyzing simple and connected to quantum field theory). However, it the trajectory of an electron generates the phenomenon of implies the extension of phase conservation from reception/ phase loss due to multiple interactions to be computed along emission of associated waves to reception/emission of parts of the trajectory. For example, an electrostatic field should be thesameassociatedwave.Moreover,itrequiresadistinction decomposed (using the Fourier transformation) in a set of between these reflection/refraction or diffraction phenomena waves with a certain angular frequency 𝜔 and a certain wave (which does not alter coherence, but modifies momentum) vector 𝑘: and the annihilation/creation phenomena involved by any 1 𝐴𝑒 (𝑥 , 𝑥⃗) = ∫ 𝑑𝑞𝐴𝑒 (𝑞) 𝑒−𝑖𝑞𝑥, interaction in quantum field theory; otherwise any reflec- 𝜇 0 2 𝜇 (10) tion/refraction or diffraction of a part of the incident wave (2𝜋) (considered as a sequence annihilation-creation) would gen- 𝑞𝑥 erate the annihilation of the entire associated wave function in where the product stands for any other point of space (the wave function vanishes instan- 𝑞𝑥0 =𝑞 𝑥0 − 𝑞𝑥⃗⃗ (11) taneously). However, the use of some support functions for com- with 𝑥0 representing the time coordinate, 𝑥=(𝑥⃗ 1,𝑥2,𝑥3) rep- putationalaspects(similartowaveletspresentedin[5, 6]) resenting the vector of position, 𝑞0 representing the angular is still required in certain circumstances, usually when the frequency, and 𝑞=(𝑞⃗ 1,𝑞2,𝑞3) representing the wave vector. properties of the material medium imply a nonzero phase The measurement system is chosen soas 𝑐=1, ℎ/(2𝜋) = shift (the same in any point where the wave function interacts 1 𝜋 for performing a better correspondence from the angular with it). An example is represented by the phase shift frequency to the energy and from the wave vector to the (corresponding to 1/2 of the wavelength) for reflection on momentum of the quantum particle (see [8] for more details). metallic surfaces, when the electric field E should vanish very 𝑒 𝑒 The quantities 𝐴𝜇(𝑥) and 𝐴𝜇(𝑝) do not correspond to quickly, in any point of the metallic surface, such that all parts standard photons, and for this reason they cannot be sub- of the wave function are to be shifted by 𝜋 before spatial stituted by operators as required by second quantification reflection. theory. However, experimental facts have shown that the For this reason a complete computational model cannot electromagnetic field effect can be studied using the pertur- be based exclusively on standard quantum aspects. Certain bations method from quantum theory. support functions corresponding to spatial coherence on Using this method [8] the matrix element (for the first extended space intervals should be added for a correct com- order of perturbation) corresponding to the electromagnetic putation of wave trajectories in case of reflection/refraction or interaction between the electron and the electromagnetic diffraction phenomena. Moreover, the assumption regarding field is represented by nonannihilation of the entire wave function when certain parts of it undergo interaction with phase conservation (or 𝑆(1) even constant phase shift) should be also added for a complete 𝑓𝑖 model. These phase aspects should be added to Sterian realis- (+) 𝑖𝑝 𝑥 𝑒 −𝑖𝑞𝑥 (−) −𝑖𝑝 𝑥 =−𝑖𝑒∫ 𝑑𝑥V (𝑝⃗)𝑒 𝑓 ∫ 𝑑𝑞𝐴 (𝑞) 𝛾 𝑒 V (𝑝⃗)𝑒 𝑖 , tic approach for correct determination of wave space proper- 𝑟󸀠 𝑓 𝜇 𝜇 𝑟 𝑖 ties (as the wave vector or the momentum of the associated (12) wave) in a more general and rigorous transient approach. At this moment, Sterian realistic approach uses just spatial (±) where V stands for the basic vectors of an electron with selection for computing the wave vector (a certain direction is 𝑟 positive or negative frequencies (energies)—usually column selected) without taking into consideration phase properties. (±) vectors, V𝑟 stands for their Dirac conjugate vectors—usually 𝑟 4. Implications of Using Quantum Field line vectors, index denotes the projection of the spin along a certain axis, and 𝑝𝑓⃗and 𝑝𝑖⃗correspond to the final and initial Theory upon the Realistic Approach three-dimensional momenta for the electron, respectively. Any measurement for the wave vector (proportional to the These can be also written as momentum of the quantum particle) cannot be based exclu- 𝑆(1) =−𝑖𝑒∫ 𝑑𝑞V(+) (𝑝⃗)𝛾 𝐴𝑒 (𝑞) V(−) (𝑝⃗)𝛿(𝑞−𝑝 +𝑝). sively on the spatial selection of certain directions (as is 𝑓𝑖 𝑟󸀠 𝑓 𝜇 𝜇 𝑟 𝑖 𝑓 𝑖 considered in the Sterian realistic approach in an implicit (13) manner, due to the use of Fourier space). Any direction of 𝑒 propagation generated by a diffraction grating, for example, In both previous equations the expression 𝐴𝜇𝛾𝜇 corre- implies a validation by means of a final interaction with the sponds to the sum material medium (the particle is annihilated while it interacts, generating a specific signal for the measurement device). 3 𝑒 𝑒 𝑒 Firstwemusttakeintoaccountthepossibilityforthepar- 𝐴𝜇𝛾𝜇 =𝐴0𝛾0 + ∑𝐴𝑛𝛾𝑛, (14) ticletobedeflectedorcaptured(foraveryshorttimeinterval) 𝑛=1 Advances in High Energy Physics 5

𝑒 where 𝐴𝜇 represents the cuadrivector of the electromagnetic of the electron with the electromagnetic field, according to field and 𝛾𝜇 represents the Dirac matrices (its argument cor- the dynamical equations: responds to the energy and momentum conservation laws). 2 If the electromagnetic field does not depend on time, the 𝜕 2 𝑒 (− +∇ )𝐴 (𝑥) =−𝑒𝜓𝛾 𝜓, 2 𝜇 𝜇 cuadripotential 𝐴𝜇(𝑥) can be presented as 𝜕𝑡

1 𝜕 𝐴𝑒 (𝑥) =𝐴𝑒 (𝑥⃗) = ∫ 𝑑𝑞𝑒𝑖𝑞𝑥⃗⃗𝐴𝑒 (𝑞)⃗ (𝑖𝛾 −𝑒𝛾 𝐴 )𝜓(𝑥) −𝑚𝜓(𝑥) =0, 𝜇 𝜇 3/2 𝜇 (15) 𝜇 𝜇 𝜇 (18) (2𝜋) 𝜕𝑥𝜇 𝜕 which shows that the virtual photons composing the electro- 𝜓 (𝑥) (𝑖𝛾𝜇 +𝑒𝛾𝜇𝐴𝜇)+𝑚𝜓 (𝑥) =0 magnetic field have a nonzero value just for the momentum 𝜕𝑥𝜇 (the energy corresponding to quantity 𝑞0 is equal to zero). As a consequence, the matrix element for the first order of (see [8]).Thusthewavefunctioncorrespondingtotheelec- perturbationcanbewrittenas tron is modified in a continuous manner along its trajectory (according to the first quantification theory). The exterior 𝑆(1) =−𝑖𝑒∫ 𝑑𝑞V(+) (𝑝⃗)𝛾 𝐴𝑒 (𝑞)⃗V(−) (𝑝⃗) field acts as a support function (as mentioned in previous 𝑓𝑖 𝑟󸀠 𝑓 𝜇 𝜇 𝑟 𝑖 (16) paragraph), determining the evolution of the electron wave train. Moreover it shows the need for a certain reference sys- ×𝛿(𝑞−⃗ 𝑝⃗+ 𝑝⃗)𝛿(𝑝 −𝑝 ). 𝑓 𝑖 𝑓0 𝑖0 tem which acts upon the wave trains corresponding to quantum particles and their associated waves in any rigorous By performing the integration on 𝑑𝑞, it results in computational model for transient phenomena (the exterior electromagnetic field is part of it). This aspect is supported 𝑆(1) =−𝑖𝑒V(+) (𝑝⃗)𝛾 𝐴𝑒 (𝑞)⃗V(−) (𝑝⃗)𝛿(𝑝 −𝑝 ). 𝑓𝑖 𝑟󸀠 𝑓 𝜇 𝜇 𝑟 𝑖 𝑓0 𝑖0 (17) bythehighermagnitudeandaslowtimevariationofthese exterior electromagnetic fields and by the lack of reversibility According to standard interpretation of quantum field the- for diffraction phenomena in quantum physics (we can notice ory, this first order element from perturbation method is an electron diffraction phenomenon when an electron beam connected to the probability of an interaction between an interacts with a motionless crystal lattice, but we cannot imagine a diffraction of an entire crystal lattice when it inter- electron with initial momentum 𝑝𝑖⃗and energy 𝑝𝑖0 and a virtual photon with momentum 𝑞⃗so as to result in an electron acts with a motionless spatial distribution of electrons). Similar to reflection/refraction and diffraction phenom- with momentum 𝑝𝑓⃗and energy 𝑝𝑓0. However, for computing the whole interaction for an ena previously presented, these deflections do not alter coher- electron in an electrostatic field (so as to determine the ence. They just alter the spatial directions (the wave vector) of trajectory of the associated wave train) we have to consider certain wave trains previously selected (e.g., by a diffraction that the electron has to undergo multiple interactions with grating). After corrections due to possible deflections are such virtual photons until the action of the exterior field computed,weshouldanalyzetheannihilationphenomenon vanishes (such multiple interactions are allowed by second which validates the detection of a particle (wave train) with quantification theory). According to the quantum laws, each certain characteristics. The rigorous model is based on quan- interaction transforms the initial electron (its wave train tumfieldtheory(novirtualparticlesaretakenintoconsid- used in computation) into a new wave train (the final elec- eration). According to this model, the wave train is analyzed tron) with different characteristics (another momentum and using the momentum space. The switch from position space energy) and so on. to momentum space is performed using the Fourier transfor- Considering that the wave train corresponding to the mation.Thespaceandtimeoriginsareselectedinthispoint electron undergoes a set of interactions with virtual photons, of interaction, when the associated wave train is received it results in that a certain transient time is required by by the material medium. The annihilation of the particle each interaction, so as physical quantities as wavelength and implies the instant annihilation of all parts of this wave train, angular frequency to be transferred in this local phenomenon irrespective of the distance to this interaction point. It is true (these quantities are used for further computing the matrix that this seems to contradict the relativity postulated (no elements for creation/annihilation phenomena). This tran- speed can surpass the light speed), but in fact any speed can be sient time causes a phase loss for the initial wave train, so noticed just by an emission and a reception of a certain signal itresultsinthatthegreatnumberofinteractionsofthe or particle. The annihilation of a certain quantum particle initial electron in an electrostatic (Coulombian) field would cannot be noticed in two different space areas. Thus there finally cause significant phase loss (the time length of the is no contradiction. Moreover, the existence of some high wave train associated with the final electron tends to zero). speed support functions which cancel the wave in an instant This corresponds to a vanishing phenomenon for the initial manner can be supported by uncertainty principle also. Since electron, in contradiction with experimental facts. the particle ceases to exist, its final momentum is zero. Being This phenomenon can be avoided into a complete and rig- no uncertainty, it results in that orous transient approach if the whole trajectory is computed Δ𝑝 =Δ𝑝 =Δ𝑝 =0, using just one Lagrangian function for the whole interaction 𝑥 𝑦 𝑧 (19) 6 Advances in High Energy Physics and, according to uncertainty principle, quantum particle are represented by instant phenomena ℎ supported by uncertainty principle (unlike considerations Δ𝑥 ≥ 󳨀→ ∞, Δ𝑝 in Sterian realistic approach); since there is no uncertainty 𝑥 for final momentum when the particle is annihilated, an ℎ infinitely extended phenomenon is generated (an instant Δ𝑦 ≥ 󳨀→ ∞, Δ𝑝 (20) propagating spatial noise is generated, similar to temporal 𝑦 Gaussian noise—see [11], and frequency dependent noise— ℎ see [12]). Δ𝑧 ≥ 󳨀→ ∞. Δ𝑝𝑧 References Thus this annihilation phenomenon can be considered as being infinitely extended. Unlike standard phenomena con- [1] P. E. Sterian, “Realistic approach of the relations of uncertainty sidered by Sterian realistic approach, the uncertainty for of Heisenberg,” Advances in High Energy Physics,vol.2013, momentum determines the uncertainty for space intervals. ArticleID872507,7pages,2013. However, for computational aspects it can be simply noticed [2] L.D.LandauandE.M.Lifshitz,QuantumMechanics:Non-Rela- that discontinuities cannot be analyzed using the differential tivistic Theory,vol.3,PergamonPress,Oxford.UK,3rdedition, form of wave equations. A discontinuity looks like a limita- 1974. tion, but physically they are represented by fracture phenom- [3] E. G. Bakhoum and C. Toma, “Relativistic short range phenomena for certain quantities. An avalanche process can be easily ena and space-time aspects of pulse measurements,” Mathemat- imagined and computed (similar to mechanical engineering). ical Problems in Engineering, vol. 2008, Article ID 410156, 20 pages, 2008. 5. Conclusions [4] E. Bakhoum and C. Toma, “Transient aspects of wave propagation connected with spatial coherence,” Mathematical Problems This study has presented supplementary aspects for the Ste- in Engineering,vol.2013,ArticleID691257,5pages,2013. rian realistic approach to uncertainty principle. It was shown [5] C. Cattani, “Fractional calculus and Shannon wavelet,” Mathe- that high energy (relativistic) corrections regarding finite matical Problems in Engineering, vol. 2012, Article ID 502812, 26 speed for propagation of interaction for measurements based pages, 2012. on quantum aspects should be completed with considerations [6] J. Leng, T. Huang, and C. Cattani, “Construction of bivariate about nonlinear transformations for superposition of wave nonseparable compactly supported orthogonal wavelets,” Math- ematical Problems in Engineering,vol.2013,ArticleID624957, trains so as to result in a rigorous and complete transient 11 pages, 2013. approach. With each wave train being described by a specific frequency, wave vector, and its own Lorentz transformation [7] E. G. Bakhoum and C. Toma, “Mathematical transform of traveling-wave equations and phase aspects of quantum interac- (with specific time origin), it results in that temporal and tion,” Mathematical Problems in Engineering,vol.2010,Article spatial correlations are involved in these relativistic transfor- ID695208,15pages,2010. mations for part of wave trains. [8] N. Nelipa, Physique des Particules Elementaires, Editions MIR, It was also shown that a complete computational model Moscou, Russia, 1981. for a transient approach cannot be based exclusively on [9] J.J.Rushchitsky,C.Cattani,andE.V.Terletskaya,“Waveletanal- standard quantum aspects. Certain support functions corre- ysis of the evolution of a solitary wave in a composite material,” sponding to spatial coherence on extended space intervals International Applied Mechanics,vol.40,no.3,pp.311–318,2004. (similar to wavelets propagation in composite materials [9]or [10] C. Cattani, “Multiscale analysis of wave propagation in compos- to multiscale phenomena [10]) should be added for a correct ite materials,” Mathematical Modelling and Analysis,vol.8,no. computation of wave trajectories in case of reflection/refrac- 4, pp. 267–282, 2003. tion or diffraction phenomena. Moreover, an assumption [11] M. Li and W.Zhao, “On bandlimitedness and lag-limitedness of regarding nonannihilation of the entire wave function when fractional Gaussian noise,” Physica A,vol.392,no.9,pp.1955– certain parts of it undergo interaction with phase conserva- 1961, 2013. tion (or even constant phase shift) should be also added for [12] M. Li and W. Zhao, “On 1/𝑓 noise,” Mathematical Problems in a complete model. The need for a certain exterior reference Engineering,vol.2012,ArticleID673648,23pages,2012. system acting upon the quantum wave train is also presented (based on some diffraction aspects). Computational aspects for deflection of associated waves in electromagnetic fields were also studied. It emphasized the need for a certain reference system which acts upon the wave trains corresponding to quantum particles and their associated waves in any rigorous computational model for transient phenomena (the exterior electromagnetic field is part of it as support functions), so that phase-loss phenomenon generated by the use of second quantification is to be avoided. Finally it was shown that both the switch from position space to momentum space and the annihilation of a certain Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 732682, 6 pages http://dx.doi.org/10.1155/2013/732682

Research Article VOF Modeling and Analysis of the Segmented Flow in Y-Shaped Microchannels for Microreactor Systems

Xian Wang,1,2 Hiroyuki Hirano,2 Gongnan Xie,3 and Ding Xu1

1 State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China 2 Department of Applied Chemistry, Okayama University of Science, Okayama 700-0005, Japan 3 Engineering Simulation and Aerospace Computing (ESAC), School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China

Correspondence should be addressed to Xian Wang; [email protected]

Received 28 May 2013; Accepted 7 July 2013

Academic Editor: Carlo Cattani

Copyright © 2013 Xian Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Microscaled devices receive great attention in microreactor systems for producing high renewable energy due to higher surface-to- volume, higher transport rates (heat or/and mass transfer rates), and other advantages over conventional-size reactors. In this paper, the two-phase liquid-liquid flow in a microchannel with various Y-shaped junctions has been studied numerically. Two kinds of immiscible liquids were injected into a microchannel from the Y-shaped junctions to generate the segment flow mode. The segment length was studied. The volume of fluid (VOF) method was used to track the liquid-liquid interface and the piecewise-liner interface construction (PLIC) technique was adopted to get a sharp interface. The interfacial tension was simulated with continuum surface force (CSF) model and the wall adhesion boundary condition was taken into consideration. The simulated flow pattern presents consistence with our experimental one. The numerical results show that a segmented flow mode appears in the main channel. Under the same inlet velocities of two liquids, the segment lengths of the two liquids are the same and depend on the inclined angles of two lateral channels. The effect of inlet velocity is studied in a typical T-shaped microchannel. It is found that the ratio between the lengths of two liquids is almost equal to the ratio between their inlet velocities.

1. Introduction thereby perfect mixing within seconds are achieved. Exten- sive reviews of advances in microreaction technology can be To save energy or decrease energy consumption, the develop- found in [2–6]. ment of new advanced technologies is desired. Among them, The spectrum of applications includes gas and liquid the microreaction technology is considered an attractive flowaswellasgas-liquidandliquid-liquidmultiphaseflows. solution way to replace the traditional energy resources like Recently, immiscible liquid-liquid slug flows in Y-shaped fossil energy resources. Nowadays, micro-reaction technol- microchannels received extensive attention due to the seg- ogyiswidelyusedduetotheexcellentmassandheattransfer mentationofthetwofluids,inwhichthechemicalreaction, properties as well as uniform flow patterns and residence time mixing, and diffusion of the solutes can be enhanced greatly distributions of the small microfabricated reactors [1]. The [7–9]. On the other hand, microdroplets can be produced in microreaction technology has other merits of perfect mixing, such a liquid-liquid system in Y-shaped microchannels under effective heat exchange, and small reaction volume. The somecertainconditions.Suchcanbewidelyusedintheareas representations in better performance of heat exchange are of pharmacy, cosmetics, food, and various powder technolo- higher surface-to-volume ratio and better heat transfer mag- gies. The efficiency of Y-shaped microchannels for various nitudes compared to normal equipment. The better mixing is purposes depends on the segment lengths of both phases. attributed to the laminar flow where fast diffusion and For example, when the segment length is small, the enhanced 2 Advances in High Energy Physics

Fluid 2 reactions and small-sized particles can be obtained. Besides, →− u2 xl there are various parameters affecting the segment length, Lateral channel yl such as the channel geometry, viscosity/density ratio between 8l 𝜑2 two fluids, interfacial tension, and the flux of two liquids. 𝜑 h The various affecting parameters mean that optimizing the 1 Main channel segment flow system in microchannels needs extensive exper- →− l u1 imental works, which would cause a large amount of human Fluid 1 Lateral channel and financial resources. In this sense, numerical investigation (including simulation and prediction) on such flow system Figure 1: Computational model system. is indispensable to provide a reasonable and economical designing process for microreactor systems. get a sharp interface. The interfacial tension was simulated The variety and complexity of multiphase flow phenom- with continuum surface force (CSF) model. ena clearly pose major challenges to the modeling computational approaches. Tang et al. [10] presented the detailed 2. Physical Model review of the computational methods for multiphase flow simulation. Among the various approaches of multiphase Figure 1 shows the present model system. Two immiscible flow simulation, volume of fluid (VOF) [11, 12], level set fluids were simultaneously injected into a Y-shaped micro- method [13], front-tracking method [14], and lattice boltz- channel from two lateral channels with the inlet velocities 𝑢⃗01 mann method [15] are known as useful tools for modeling and 𝑢⃗02, respectively. Both phases were assumed to be incom- ∘ multiphase flow phenomena. Further new developments on pressible and Newtonian at 1 atm and 25 C. The flow was capturing interface include particle-mesh method [16]and treated as laminar one, which proves correct for the micro- constrained interpolation profile (CIP) method [17]. flow under current situations1 [ ]. The gravitational force was The VOF method is widely adopted by in-house codes not taken into consideration due to its unimportance in the and built-in in commercial codes. It is a popular interface microflow. tracking algorithm and has proved to be a useful and robust The present computations are two-dimensional and tool. The idea in VOF method is that a color function 𝐶𝑘 steady, and the geometrical parameters of the computational is set as 1 for the 𝑘th fluid and 0 for the void, so that the domain (𝑥𝑙 × 𝑦𝑙) are also shown in Figure 1.Thewidthsofthe interfacelocatesinthecomputationalcellsinwhich0< lateral and main channels ℎ are the same and equal to 200 𝜇m. 𝐶𝑘 <1. However, a sharp interface is hard to obtain when The lengths of the lateral channels and the main channel are solving the color function equation due to the numerical 𝑙=5hand8𝑙=40h, respectively. The inclined angles of diffusion. This is the key point for this method. Therefore, the two lateral channels (Y-shaped junction) are indicated as 𝜑1 reconstruction of the interface is required. Fortunately, sev- and 𝜑2, respectively. An independent grid system of 800 × eral interface reconstruction techniques were developed suc- 220 cells is adopted for all simulations. The motivation of this cessfully to fix the requirement, for example, Donor Acceptor, study is to study the segmented flow at various inclined angles FLAIR (flux line-segment model for advection and interface or inlet velocities of a typical Y-shaped microchannel with reconstruction), SLIC (simple line interface calculation) and fixed inclined angles. PLIC (piecewise linear interface calculation). The historical perspective and comparison of these interface reconstruction 3. Governing Equations techniques were introduced in [18]. In the multiphase flow, interfacial tension force plays a The governing equations for the fluids are as follows. crucial role. A popular model for the interfacial tension force Color function 𝜕𝐶 𝜕 (𝑢𝐶) 𝜕 (V𝐶) is the continuum surface force (CSF) model, which was first + + =0. developed by Brackbill et al. in 1992 [19]. In the CSF model, 𝜕𝑡 𝜕𝑥 𝜕𝑦 (1) the interfacial tension was interpreted as a continuous, three- dimensional effect across an interface, rather than as a pres- Continuity equation sure boundary condition on the interface. It is widely used in 𝜕𝑢 𝜕V + =0. the multiphase simulation due to its easy implementation and 𝜕𝑥 𝜕𝑦 (2) high accuracy. On the other hand, using the CSF model, the wall adhesion/wetting effect can be conveniently simulated by Momentum equations a simple boundary condition on the interfacial tension force, 𝜕𝑢 𝜕𝑢 𝜕𝑢 1 𝜕𝑝 𝜇 𝜕2𝑢 𝜕2𝑢 1 +𝑢 + V =− + ( + ) + 𝑓 , which is an important phenomenon in a microsystem [20]. 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑥 𝜌 𝜕2𝑥 𝜕2𝑦 𝜌 sv𝑥 In this study, the influence of two factors on the segment 𝜕V 𝜕V 𝜕V 1 𝜕𝑝 length of a liquid-liquid flow system in Y-shaped microchan- +𝑢 + V =− nels is numerically studied. These are the inclined angles of 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜌 𝜕𝑦 Y-shaped junctions and the inlet flow rate of fluids. In the 𝜇 𝜕2V 𝜕2V 1 + ( + )+ 𝑓 , presentsimulation,thevolumeoffluid(VOF)methodwas 𝜌 𝜕2𝑥 𝜕2𝑦 𝜌 sv𝑦 used to track the liquid-liquid interface and the piecewise- liner interface construction (PLIC) technique was adopted to (3) Advances in High Energy Physics 3

where 𝜌=𝜌2 +𝐶(𝜌1 −𝜌2) and 𝜇=𝜇2 +𝐶(𝜇1 −𝜇2).Subscripts1 and2denotefluid1and2,respectively.𝐶 is the color function, whichissetto1forfluid1and0forfluid2,respectively. Ontheinterfaceoftwofluids,thevalueof𝐶 is taken at (0, 1). In this study, the physical properties of the two fluids are 𝜌 = 1000 3 𝜇 = 0.001 ⋅ 𝜌 = assumed to be 1 kg/m , 1 Pa sand 2 Figure 2: Experimental photograph for the segment flow in a Y- 800 3 𝜇 = 0.02 ⋅ kg/m , 2 Pa s. The interfacial tension is set to be shaped microchannel at ℎ = 200 𝜇m, |𝑢⃗01|=|𝑢⃗02| = 0.025 m/s and 𝜎 = 0.05 |𝑢⃗ |= ∘ N/m. The magnitudes of inlet velocities are 01 𝜑1 =𝜑2 =30. Red: cyclohexane; green: water. |𝑢⃗02| = 0.005 m/s, and their directions are parallel to the lateral channels (Figure 1). Accordingly, the corresponding typical dimensionless groups for microchannel are Re = 2 −4 𝜌1𝑢01ℎ/𝜇1 = 1.0 and Ca =𝜇1 /(𝜎𝜌1ℎ) = 10 . The flow field was solved with finite difference methodon a staggered mesh system, and the HSMAC (highly simplified marker and cell) algorithm [21]isemployedtosolvethepres- sure field and improve the velocity field. For HSMAC, only Figure 3: Numerical simulation result for the segment flow in a Y- 𝑛+1 󸀠 ℎ = 200 𝜇 |𝑢⃗ |=|𝑢⃗ | = 0.025 the local pressure value 𝑝 (or pressure correction 𝑝 )is shaped microchannel at m, 01 02 m/s, and 𝑖,𝑗 𝑖,𝑗 𝜑 =𝜑 =30∘ regarded as unknown in the Poisson equation of pressure and 1 2 . Blue: cyclohexane; green: water. is solved by Newton-Raphson method. The velocities in the 󸀠 neighbor grids are corrected by 𝑝𝑖,𝑗 until the continuity equa- where n̂ istheunitnormaltotheinterfaceandn̂ = n/|n|.The tion is satisfied. n =∇C Since 𝐶 is not a continuous variable, (1)cannotbesolved normal vector . The detailed computational equations as usual. To get a sharp interface, the PLIC technique [22]is of every term in (7)areshownin[20]. adopted, in which a cell-centered line segment is used to sim- The effect of wall adhesion at fluid interfaces in contact ulatetheinterfaceandthefluxoffluidiscomputedgeomet- with rigid boundaries, which is very important in a micro- rically according to the interface location and slope. The line flow, can be estimated with CSF model by applying a bound- segment can be described as ary condition in (7) as follows: 𝑛 𝑥+𝑛 𝑦+𝜂=0. n̂ = ( 𝜃 ) n̂ + ( 𝜃 ) n̂ , 𝑥 𝑦 (4) cos ca 𝑤 sin ca 𝑡 (8) 𝑛 𝑛 𝜂 𝐶 𝑥, 𝑦,and can be computed according to .Basedonthe where 𝜃 is the contact angle between fluid and solid wall. It 𝑛 𝑛 𝜂 ca ∘ values of 𝑥, 𝑦,and ,thefluxoffluid1,whosecolorfunction is set to be 90 in the present computation. n̂𝑤 is the unit wall is equal to 1, can be calculated analytically. In the present com- normal directed into the solid wall and n̂𝑡 is the unit tangent putation, the outflow flux in a grid is computed, and the in- normal pointing into fluid 1. flow flux is corrected by the out-flow fluxes of the four neigh- boringgrids.Thisisbecausethein-flowfluxofagridshould be computed according to the values of 𝐶 in its neighboring 4. Results and Discussion grids. Our previous work [23, 24] shows the detailed imple- 4.1. Comparison with Experiment. The validity of our compu- mentation and performance of our PLIC-reconstruction tational code was proved by comparing with our experimen- method. 𝑓⃗ tal result. Figure 2 is the experimental photograph. The red sv in (3) is the surface tension force per volume. Its direc- liquid is cyclohexane (with red color dye) and the green one is tion is pointing into the concave side of the interface. Assum- water (with blue color dye). The channel width is 200 𝜇m, the ingthecenteroftheinterfacecurveisinfluid1,fortheCSF ∘ inlet velocities are 𝑢⃗01 = 𝑢⃗02 = 0.025 m/s, and 𝜑1 =𝜑2 =30. method, the surface tension is modeled according to [15]: The interfacial tension is 0.025 N/m. The segment lengths of ∇𝜌 𝑓⃗ =𝜎𝜅 =𝜎𝜅∇𝐶, cyclohexane and water are about 0.67 and 0.7 mm, respec- sv (5) 𝜌1 −𝜌2 tively. It should be noted that the exact segment length is very hard to be measured because the real interfaces shown in Fig- 𝜎 𝜅 where is the interfacial tension coefficient and is the local ure 2 are not as clear as the computational one, which is due curvature of the interface curve. To avoid the acceleration due to the irregular contact angle caused by the complicated solid to surface tension depending on the density, (5)isimproved wall condition. Figure 3 shows the corresponding numerical as follow: result of our code under the same conditions with exper- 𝜌 𝑓⃗ = 𝜎𝜅∇𝐶 iment. The blue and green liquids in Figure 3 are organic sv (6) (𝜌1 +𝜌2)/2 and aquatic phases, respectively. The computational segment Thus, when (6)isappliedon(3), the last term (i.e., accelera- lengths of cyclohexane and water are 0.68 and 0.71 mm. The tion due to surface tension) is independent of the density. The images of slug generation near the Y-shaped junction, which curvature 𝜅 is calculated from is indicated in the circle in Figure 2,arequitesimilarfor 1 n experimental and numerical results. Therefore, from both 𝜅=−∇⋅n̂ = [( ⋅∇) |n| − (∇⋅n)] , (7) qualitative and quantitative points of view, the numerical and |n| |n| experimental results agree well. 4 Advances in High Energy Physics

Table 1: Relative segment length 𝑤/ℎ at various inclined angles 3 of Y-shaped junction at 𝜌1 = 1000 kg/m , 𝜇1 = 0.001 Pa⋅s, 𝜌2 = h 3 800 kg/m , 𝜇2 = 0.02 Pa⋅s, 𝜎 = 0.05 N/m, and |𝑢⃗01|=|𝑢⃗02|= w1 w2 0.005 m/s.

𝜑 =𝜑 =45∘ 𝜑 w/h (a) 1 2 2 ∘ 𝜑1 =0 𝜑1 =𝜑2 ∘ 15 6.70 4.32 ∘ 30 4.13 3.46 ∘ 45 3.64 3.44 ∘ 𝜑 =𝜑 = 150∘ 60 3.30 3.90 (b) 1 2 ∘ 75 3.50 3.45 ∘ 90 3.55 3.45 ∘ 150 3.60 3.62

∘ ∘ (c) 𝜑1 =0,𝜑2 =45

(a) |𝑢⃗01| = 0.001 m/s ∘ ∘ (d) 𝜑1 =0,𝜑2 =90

Figure 4: Simulated segment flow mode in a Y-shaped microchannel at various inclined angles 𝜑1 and 𝜑2. (b) |𝑢⃗01| = 0.02 m/s Figure 5: Simulated segment flow mode in a T-shaped microchannel at |𝑢⃗02| = 0.005 m/s, and (a) |𝑢⃗01| = 0.001 m/s; (b) |𝑢⃗01|= 4.2. The Effect of Inclined Angles of Y-Shaped Junction. Fig- 0.02 m/s. ure 4 shows three typical images of simulated segment flow modes in the Y-shaped microchannels. Table 1 shows the ∘ ∘ relative segment length at various inclined angles of Y-shaped of 𝜑1 =0 and 𝜑2 =90 (hereafter called the T-shaped micro- ℎ 𝑤= junction. Here, is the channel width (Figure 4)and channel). Figure 5 shows the simulated segment flow modes (𝑤 +𝑤 )/2 𝑤 𝑤 1 2 . 1 and 2 are the segment lengths of liquid 1 and at |𝑢⃗01| = 0.001 and 0.02 m/s. |𝑢⃗02| is not changed and kept to 2, respectively (Figure 4). The two values should be same the- 0.005 m/s. It can be seen that the segment length of phase 1, oreticallysincetheinletfluxesarethesame.Weusedaverage 𝑤1, increases with its inlet velocity |𝑢⃗01|, while 𝑤2 decreases. 𝑤 value as the result in Table 1 forthesakeofaslightdifference The segment length can be controlled easily by the inlet flux 𝑤 𝑤 between 1 and 2 caused by some unavoidable computa- of the liquids. Figure 6 shows the plot of 𝑤 against 𝑢 for the tional errors. T-shaped microchannel. Here, 𝑤 is the ratio of the segment ItcanbeseenfromTable1, under current conditions, lengths of two phases, 𝑤1/𝑤2,and𝑢 is the ratio of their inlet 𝜑 ∘ when 2 is equal to 15 , the segment length is largest for both velocities |𝑢⃗01|/|𝑢⃗02|, which is equal to their inlet volume flow 𝜑 =0∘ 𝜑 =𝜑 𝜑 ∘ cases ( 1 and 1 2). When 2 is equal to 90 ,although rates. Here, different values of 𝑢 are obtained by different the segment length is not the smallest, it keeps to a small value |𝑢⃗01|.FromFigure6,itcanbeseenthat𝑤 is almost linearly 𝜑 =0∘ 𝜑 =𝜑 𝜑 =0∘ for both cases ( 1 and 1 2). For the cases of 1 proportional to 𝑢. This means the ratio between the segment 𝜑 =0∘ 𝜑 (Table 1 column 1 ), the value of 2 affects the segment volumes of two phases is almost equal to the ratio of their 𝜑 <60∘ length greatly. When 2 , the segment length decreases inlet volume flow rates. It should be noted that under current 𝜑 𝜑 =60∘ with the increase in 2.When 2 , the segment length conditions, when |𝑢⃗01| ≥ 0.04 m/s, the segment flow mode 𝜑 >60∘ reaches its smallest value. When 2 , the segment length cannot be obtained. Instead, a parallel flow mode appears in 𝜑 𝜑 =𝜑 ∘ ∘ increases with 2.However,when 1 2 (Table 1 column theT-shapedmicrochannel(𝜑1 =0 and 𝜑2 =90). 𝜑1 =𝜑2),thevalueoftheinclinedangledoesnotaffectthe ∘ slug length so much. Except the cases of 𝜑1 =𝜑2 =15 and ∘ 60 , the segment lengths for other cases are almost the same. 5. Conclusion Thus, the segment length can be controlled by adjusting Microchannels are gradually used in microreactor systems the inclined angles of Y-shaped junction. T-shaped junction with microreaction technology, where a typical application 𝜃 =90∘ tends to generate more interfaces of two liquids ( ca ) includes liquid-liquid multiphase flows. In the present work, 𝜃 =90̸ ∘ or get the small slugs and droplets ( ca ), which is very a two-phase liquid-liquid segmented flow in a microchannel important for the mixing, chemical reaction, and making with various Y-shaped junctions has been studied numeri- micro/nanosized particles using microreactor. callybymeansofvolumeoffluid(VOF)approach.Usingliner interface construction (PLIC) technique, a sharp interface 4.3. The Effect of Inlet Flow Rate. In this section, the effects between two phases was simulated. The influence of the of the inlet velocities of two phases (𝑢⃗01 and 𝑢⃗02)onsegment inclined angle of Y-shaped junction and the inlet velocity on lengths (𝑤1 and 𝑤2)arestudiedwithatypicalY-shapedcase the segmented flow length has been observed. Advances in High Energy Physics 5

7 𝜑: The angle between lateral channel and the extension 6 central line of main channel in its opposite direction ∘ ( ). 5

4 Subscripts w 3 𝑖,: 𝑗 Indicators of cell-centered value 𝑜: Orangic phase 2 𝑡:Walltangentcomponent 1 𝑤: Water phase, or wall nomal component 𝑥,:Componentsin 𝑦 𝑥 and 𝑦 directions 0 1, 2: Indicators for fluids 1 (water) and 2 (organic fluid). 0123456 u Acknowledgments 𝑤 𝑢 |𝑢⃗ | = 0.005 Figure 6: The plot of against at 02 m/s for the T- Thewouldliketoacknowledgefinancialsupportforthiswork 𝜑 =0∘ 𝜑 =90∘ shaped microchannel of 1 and 2 . provided by the Advanced Research in Evolution Area by Okayama Prefecture Industrial Promotion Foundation and QOL Innovative Research (2012–2016) from the Ministry of AsegmentflowmodeisfoundintheY-shapedmicro- Education, Culture, Sports, Science and Technology of Japan ∘ channel when the contact angle is set to 90 . Under current and the National Natural Science Foundation of China (no. ∘ situations, the segment length is largest at 𝜑2 =15,andis 11242010). ∘ ∘ kept to a small value at 𝜑2 =90 for both cases (𝜑1 =0 and 𝜑 =𝜑 1 2). This means using T-shaped junction, small slug, References and more interfaces of two fluids can be obtained. The ratio between the segment lengths of two liquids is almost equal to [1] W.Ehrfeld,V.Hessel,andH.Lowe, Microreactors, New Technol- the ratio of their inlet velocities for the case of Y-shaped ogy for Modern Chemistry,vol.D-69469,Wiley-VCHGmbH, ∘ ∘ microchannel (𝜑1 =0 and 𝜑2 =90). Weinheim, Germany, 2000. [2] P. Watts and C. Wiles, “Recent advances in synthetic micro reaction technology,” Chemical Communications,no.5,pp.443– Nomenclature 467, 2007. 𝐶: Color function (−) [3] C. Wiles and P.Watts, “Recent advances in micro reaction tech- 𝑓 3 nology,” Chemical Communications,vol.47,no.23,pp.6512– sv: Total external force per volume on the fluid (N/m ) ℎ:Thewidthofmicrochannel(𝜇m) 6535, 2011. 𝑙: The length of the lateral channel (𝜇m) [4] P.WattsandC.Wiles,“Microreactors,flowreactorsandcontin- 𝐿: The length of the slug (𝜇m) uous flow synthesis,” JournalofChemicalResearch,vol.36,no.4, 𝐿: The ratio between organic slug length and water slug pp.181–193,2012. length (−) [5] J. M. Kohler,T.Henkel,A.Grodrianetal.,“Digitalreaction¨ n: Normal vector to the interface (−) technology by micro segmented flow—components, concepts and applications,” Chemical Engineering Journal,vol.101,no.1– n̂: Unit normal vector to the interface (−) 3, pp. 201–216, 2004. 𝑡:Time(s) [6] T. Nisisako, T. Torii, and T. Higuchi, “Novel microreactors for Δ𝑡:Timeinterval(s) 𝑢⃗ functional polymer beads,” Chemical Engineering Journal,vol. :Velocityvector(m/s) 101, no. 1–3, pp. 23–29, 2004. 𝑢:Velocitycomponentin𝑥 direction (m/s) 𝑢 𝑢=|𝑢⃗ |/|𝑢⃗ |(−) [7] L. Wu, M. Tsutahara, L. S. Kim, and M. Ha, “Three-dimensional : 02 01 lattice Boltzmann simulations of droplet formation in a cross- V:Velocitycomponentin𝑦 direction (m/s) 𝑥𝑙 junction microchannel,” International Journal of Multiphase : Thelengthofsolutiondomain(m) Flow,vol.34,no.9,pp.852–864,2008. 𝑦𝑙 : The width of solution domain (m). [8] A.-L. Dessimoz, L. Cavin, A. Renken, and L. Kiwi-Minsker, “Liquid-liquid two-phase flow patterns and mass transfer Greeks characteristics in rectangular glass microreactors,” Chemical Engineering Science,vol.63,no.16,pp.4035–4044,2008. 𝜂: A constant term in the line segment equation (−) 𝜃 ∘ [9]Y.Okubo,T.Maki,N.Aoki,T.HongKhoo,Y.Ohmukai,andK. ca:Contactangle() −1 Mae, “Liquid-liquid extraction for efficient synthesis and sepa- 𝜅: Thecurvatureoftheinterface(m ) ration by utilizing micro spaces,” Chemical Engineering Science, 𝜇: Viscosity of fluid (Pa⋅s) vol.63,no.16,pp.4070–4077,2008. 3 𝜌: Density of fluid (kg/m ) [10] H. Tang, L. C. Wrobel, and Z. Fan, “Tracking of immiscible 𝜎: Interfacial tension coefficient between water and interfaces in multiple-material mixing processes,” Computa- organic fluid (N/m) tional Materials Science,vol.29,no.1,pp.103–118,2004. 6 Advances in High Energy Physics

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