Ratio of Real to Imaginary for pp and pp Elastic Scatterings in QCD Inspired Model
|
|
- Sandra Dixon
- 6 years ago
- Views:
Transcription
1 Coun. Theor. Phys. Beijing, China pp c International Acadeic Publishers Vol. 47, No. 3, March 15, 2007 Ratio of Real to Iaginary for pp and pp Elastic Scatterings in QCD Inspired Model LU Juan, 1,2 MA Wei-Xing, 1,3 and HE Xiao-Rong 2 1 Collaboration Group of Hadron Physics and Non-perturbative QCD Study, Guangxi University of Technology, Liuzhou , China 2 Institute of Physics Science and Engineering Technology, Guangxi University, Nanning , China 3 Institute of High Energy Physics, the Chinese Acadey of Sciences, Beijing , China Received April 28, 2006 Abstract We use the QCD inspired odel to analyze the ratio of the real to the iaginary for pp and pp elastic scatterings. A calculation for the ratio of the real to the iaginary is perfored in which the contributions fro gluongluon interaction, quark-quark interaction, quark-gluon interaction, and eikonal profile function are included. Our results show that the QCD inspired odel gives a good fit to the LHC experiental data. PACS nubers: p, Gg, Ve Key words: QCD inspired odel, eikonal profile function, ratio of real to iaginary 1 Introduction Measureents of ρ-values ratio of real to iaginary for pp and pp elastic scatterings have had a rich history. AGS and FNAL had been easured the value of, at 0 < < 40 GeV energy region. When the ISR was turned on in 1971, soe of the first experients done were an elastic scattering easureent of the ratio of the real to the iaginary ρ for pp and pp elastic scatterings at 30 GeV < s < 70 GeV energy region by CERN Roe group. Eagerly awaited high energy collision at the CERN large hadron collider LHC 1] will give access not only yet unexplored sall distances but also siultaneously to large distances that were neither explored. Now, the easureents of the ratio of the real to the iaginary ρ for pp and pp elastic scatterings have reached = 2 TeV at LHC. Recently soe odels with ulti-poeron structures were proposed, 2 4] soe of these 2,3] used Born aplitudes with two Poeron structures as single 2] or double poles. 3] The foral violation of the Froissart Martin bound in soe of these odels is considered as practically negligible, though in ters of particle-wave aplitudes unitarity violation is flagrant at present energies. Soe of these 5] used eikonal odel with three Poerons. The odel shows quite a good agreeent with LHC data. But the odel is polypoeron hypothesis what happens if one adits a fourth ect. Moreover, the origin and nature of the Poeron has not been known so far and the Poeron coupling to the proton has a vector for, γ µ, siilar to that of C = +1 isoscalar photon, which is in contradiction to the vacuu property of the Poeron. The principles of analyticity, unitarity and crossing syetry are truly fundaental to our understanding of particle physics. A requireent of analyticity is that the forward scattering aplitudes for elastic nuclear scattering coe fro the sae analytic function. Further, unitarity provides a relationship between the al cross section and the iaginary part of the forward scattering aplitude the optical theore. Cheng and Wu 9] proposed initially that eikonalization should properly unitarize odels. Now we used a QCD-inspired, and eikonalized odel to predict the ratio of the real to the iaginary at the energies of LHC. Using the ipact paraeter representation, the QCD-inspired odel predicts the experiental data by understanding of eleental scattering theory fro non relativistic quantu echanics. The eikonal odels that are capable of describing the data for non-zero transferred oenta are developed in Refs. 6] and 7], using an eikonal structure for the pp and pp scattering aplitude, Block and Kaidalov 8] have derived factorization on theore for pp, pp at high energies. In this paper, we will base on QCD inspired eikonal odel. Fro the conventional eikonal is suppleented with a QCD otivated part consisting of three ters. 7] A calculation for the ratio of the real to the iaginary ρ for pp and pp elastic scatterings is perfored in which the contributions fro gluon-gluon, quark-quark and gluonquark interactions are included. 2 Eikonal Model We introduce the eikonal forulis in the twodiensional transverse ipact paraeter space b, we use the coplex analytic eikonals, χ pp = χ even χ and χ pp = χ even +χ. even or under the transforation E E, E = k where E is the proton laboratory energy and is the proton ass. The data both for pp and pp are fitted using the al coplex analytic eikonal profile functions, i.e. phase transition, in ters of The project supported in part by National Natural Science Foundation of China under Grant Nos and and the Science Foundation of Guangxi Province of China under Grant Nos , , and
2 No. 3 Ratio of Real to Iaginary for pp and pp Elastic Scatterings in QCD Inspired Model 551 the even and forward scattering aplitudes f + and f even and under crossing], which is related to the even eikonal profile function χ even and eikonal profile functions χ respectively. We write the center-of-ass pp and pp forward scattering aplitude, 10] f c. s, t = k e i q b ab, sd 2 b, 1 π where k is the oentu in center-of-ass syste, t = 2k 2 1 cos θ is the invariant four-oentu transfer. θ is the center-of- ass syste scattering angle. q is a twodiensional vector in the ipact paraeter space b such that q 2 = t, where ab, s is the scattering aplitude in ipact paraeter space, d 2 b = 2πbdb. Our coplex eikonal, χ = χ R + iχ I, is defined. So that the coplex forward scattering aplitude in ipact paraeter space b is given by ab, s = i 2 1 e iχ = i 2 1 e χ I b,s+iχ R b,s. 2 Substituting Eq. 2 into Eq. 1, we arrive at f c. s, t = k e i q b i π 2 1 e χ I b,s+iχ R b,s d 2 b, 3 f c. s, t = 0 = k i π 2 1 e χ I b,s+iχ R b,s d 2 b. 4 To calculate ρ, the ratio of the real to the iaginary part of the forward pp and pp scattering aplitude, we write ρs = Re {f c.s, 0} I {f c. s, 0} = Re {i 1 e χ I b,s+iχ R b,s d 2 b} I {i 1 e χ I b,s+iχ R b,s d 2 b}. 5 Fro Eq. 5, 1 e iχ iχ, so that equation 5 becoes = Re { χ R b, s + iχ I b, sd 2 b} I { χ R b, s + iχ I b, sd 2 b}. 6 Because of χb, s = χ R b, s + iχ I b, s, so that ρ = Re { χb, sd 2 b} I { χb, sd 2 b}. 7 The al eikonal profile functions for pp and pp elastic scattering are defined as χ pp = χ even χ and χ pp = χ even + χ. The ratio of the real to the iaginary for pp and pp elastic scattering can be written, = Re { χ even b, s χ b, sd 2 b} I { χ even b, s χ b, sd 2 b}, 8 = Re { χ even b, s + χ b, sd 2 b} I { χ even b, s + χ b, sd 2 b}. 9 Using the optical theore, we can write the al cross section σ in the center-of-ass syste as σ = 4π k I {f c.s, t = 0}. 10 Substituting Eq. 4 into Eq. 10, we arrive at { } σ = 2 I χ even b, s + χ b, sd 2 b. 11 Substituting Eq. 11 into Eqs. 8 and 9, the ratios of the real to the iaginary for pp and pp elastic scattering becoe = 2 Re { χ even b, s χ b, sd 2 b} = 2 Re { χ even b, s + χ b, sd 2 b}, Equations 12 and 13 are starting point of our study. Now, let us consider the quarks and gluons degrees of freedo to contribute to the ratios of the real to the iaginary for proton and antiproton elastic scattering processes in QCD. 3 QCD Inspire Model 3.1 Even Eikonal Profile Function The even QCD inspired eikonal profile function χ even for nucleon-nucleon scattering is given by the su of three contributions fro gluon-gluon, quark-gluon, quark-quark sectors. They are individually factorizable into a product of a cross section σs ties an ipact paraeter space distribution function W b; µ, i.e., χ even = χ gg b, s + χ qg b, s + χ qq b, s = iσ gg sw b; µ gg + σ qg sw b; µ qq µ gg + σ qq sw b; µ qq ]. 14 The factor i is inserted in Eq. 14 since the high energy eikonal profile function is largely iaginary. σ ij are the cross section of the colliding partons. The ipact paraeter space distribution functions used in Eq. 14 were taken to be convolutions of two dipole for factors, i.e. we paraeterize W b; µ as the Fourier transfor of two dipole for factors of the nucleon, 11] W b; µ = µ2 96π bµ3 K 3 bµ, 15 where K 3 χ is the odified Bessel function of the second kind. It is noralized so that W b; µd 2 b = 1. Studying wb; µ indicates that the al cross sections are essentially independent of the choice of for factor shape. As a consequence of both factorization and the noralization chosen for the W b; µ, the following identity should be held χ even s, bd 2 b = iσgg s + σ qg s + σ qq s]. 16 In the QCD inspired odel, it allows one to reforulate the Froissart bounding in axioatic field theory. We found that the al cross section contributed fro the gluongluon interaction ter can be given asypically 7] by ε σ gg = 2π µ gg 2 log 2 s s 0. 17
3 552 LU Juan, MA Wei-Xing, and HE Xiao-Rong Vol. 47 The quark-quark interaction is siulated by a constant cross section plus a -even falling down cross section. It can be approxiated by σ qq = Σ gg C + C even s is the threshold ass, which is deterined by experient and takes the value of = 0.6 GeV. The contribution fro quark-gluon interaction is also siulated by σ qg s = Σ gg C log qg log s s For large s, the even aplitude in Eq. 14 is ade analytic by the substitution s s e iπ/2. So that gluongluon, quark-gluon, quark-quark, eikonal profile function contributions for the al cross section and the ratio of the real to iaginary can be rewritten as ε 2 σ gg = 2π log 2 s 4 iπ 2 ε 2 s log, 20 σ qq = Σ gg C + C even ] + iσ gg C even sin, 21 s 4 σ qg s = Σ gg Cqg log log s iσ gg C log π qg s The values of all paraeters are given in Table 1 and are used coonly. Table 1 Fitted Values of the paraeters used in the fit. Fixed = 0.6 GeV C = 5.65 ± 0.14 ɛ = 0.3 Cqg log = ± µ qq = 0.89 GeV Σ gg = 9πα 2 s /2 0 µ gg = 0.73 GeV C even = 2.53 ± 0.2 µ = 0.53 GeV C = 7.62 ± 0.28 α s = 0.5 s 0 = 1.0 GeV Odd Eikonal Profile Function The eikonal profile function, χ b, s = σ W b; µ, accounts for difference between pp and pp elastic scatterings, and ust vanish at high energies. A behaved analytic eikonal profile function can be paraetrized as follows: χ b, s = σ W b; µ = C Σ gg W b; µ, 23 the aplitude in Eq. 23 is ade analytic by the substitution s s e iπ/2. So that χ b, s = C Σ gg + ic Σ gg in. 24 s 4 The paraeters and functions in the above equations, equations 20 24, are given in Table 1, and explained in the sae way as before. However, the eikonal profile function is copletely different fro the even one. 4 gg, qq, qg and Odd Profile Function Contributions for Ratio of Real to Iaginary The eikonal profile function is given by four ters: gluon-gluon ter, quark-quark ters, quark-gluon ters and eikonal profile function ter. So that we consider the ratio of the real to the iaginary for pp and pp elastic scatterings is divided into four ters, = 2σ gg + σ qg + σ qq σ ε 2 = 4π log 2 s 4 + 2Σ gg C + C even ] + 2Σ gg Cqg log log s 2C Σ gg in, 25 s 0 s 4 = 2σ gg + σ qg + σ qq + σ ε 2 = 4π log 2 s 4 + 2Σ gg C + C even ] + 2Σ gg Cqg log log s + 2C Σ gg in. 26 s 0 s 4 The ratio of the real to the iaginary fro gluon-gluon contribution can be rewritten, gg = 2π2 ε/µ gg 2 logs/s 0, 27 gg = 2π2 ε/µ gg 2 logs/s The ratio of the real to the iaginary fro quark-quark qq = 2Σ ggc even / sinπ/4, 29 qq = 2Σ ggc even / sinπ/4. 30 The ratio of the real to the iaginary fro quark-gluon qg π/2 qg = 2Σ ggc log, 31 qg = 2Σ ggcqg log π/2. 32 The ratio of the real to the iaginary fro eikonal = 2C Σ gg / cosπ/4, 33
4 No. 3 Ratio of Real to Iaginary for pp and pp Elastic Scatterings in QCD Inspired Model 553 = 2C Σ gg / cosπ/4. 34 The under crossing forward scattering aplitude accounts for the difference between pp and pp elastic scatterings, respectively. The ratio of the real to iaginary for pp and pp elastic scatterings fro eikonal contribution also accounts for difference of pp and pp. It should be noticed that all the paraeters in the above equations given in Table 1 are fixed by fitting experients. Now, we can plot gg, qq, qg, against the energy, the theoretical curves are given in Fig. 1, respectively, for proton-proton elastic scattering. Fro Fig. 1, we see that the gluon-gluon interaction contribution asypically grows as logs/s 0, when we go up in energy, we can see that the gluon-gluon interaction ter doinates the other three ter contributions. Both quark-quark interaction contribution and eikonal profile function contribution contained factor of 1/, because of the different fitting paraeters, we can see that the eikonal profile function contribution goes up ore quickly than the quark-quark interaction contribution. Finally they becoe zero at the high energies region. The quark-gluon interaction contribution actually is a constant ters C, and alost equal to zero at the whole energy region. quark-gluon interaction contribution. On the contrary, eikonal profile function contribution to pp is positive. The forward scattering aplitude accounts for the difference between the pp and pp elastic scattering. Fig. 2 The ratio of the real to iaginary ρ for pp elastic scattering. The solid curve is gluon-gluon contribution, the dotted-dashed curve is quark-quark contribution, the dashed curve is quark-gluon contribution and eikonal profile function contribution is the dotted curve and it is positive. All the behaviors of individual ters tell us that the gluon-gluon interaction contribution doinates quarkquark interaction ter, quark-gluon interference ter and eikonal profile function contributions. Fig. 1 The ratio of the real to iaginary ρ for pp elastic scattering. The solid curve is gluon-gluon contribution, the dotted-dished curve is quark-quark contribution, the dashed curve is quark-gluon contribution, and eikonal profile function contribution is the dotted curve and it is negative. We can plot gg, qq, against the energy, the theoretical curves are given in Fig. 2, respectively, for antiproton-proton elastic scattering. We see that the ratio of the real to the iaginary for pp elastic scattering fro the gluon-gluon interaction contribution, quark-quark interaction contribution, quark-gluon interaction contribution is asypically equal to the ratio of the real to the iaginary fro the gluon-gluon interaction contribution, quark-quark interaction contribution, qg, 5 Predictions for Total Ratio of Real to Iaginary Our theoretical predictions for the al ratio of the real to iaginary of pp and pp elastic scattering at high energies are given by the following forulae: = gg + = 1 qq + qg 2π 2 ε 2 s log 2Σ gg C even + πσ gg C log qg 2C Σ gg cos = gg + = 1 qq + qg 2π 2 ε 2 s log 2Σ gg C even + πσ gg C log qg + 2C Σ gg cos in s 4 π ], 35 4 π in s 4 π ] With Eqs. 35 and 36, the nuerical calculations of al ratio of the real to the iaginary are perfored. The predictions are shown in Fig. 3. By the lower curve of Fig. 3, we plot the al ratio of the real to the iaginary of pp elastic scattering vs. energy s. The up curve in Fig. 3 is the result for pp elastic scattering. Clearly, we
5 554 LU Juan, MA Wei-Xing, and HE Xiao-Rong Vol. 47 have gotten excellent fits to experiental data both for pp and pp elastic scatterings at high energies. Fig. 3 The al ratio of the real to iaginary for pp and pp scattering against the center-of-ass energy in GeV. The solid line and circles points are for pp scattering and the dotted-dashed line and square black points are for pp scattering. The points on the figure are experiental data of al ratio of the real to iaginary at FNAL and LHC. 6 Conclusion In this paper, we used QCD-inspired eikonal odel to analyze the ratio of the real to the iaginary for pp and pp elastic scatterings, which is based on fundaental theory of strong interaction QCD. We consider contribution fro quark and gluon degrees of freedo of QCD to the ratio of the real to the iaginary for pp and pp elastic scatterings. Because the eikonal is consisting of four ters, the ratio of the real to the iaginary ρ includes four ters. The gluon-gluon ter ρ gg, which rises as logs/s 0 s 0 is an energy scale paraeters doinates the other ters at the whole energies. It akes doinant contribution to the al ratio of the real to the iaginary ρ. The quark-quark ter contains a factor 1/, it goes up slowly with energy increasing and eventually becoes zero. The quark-gluon ter is equal to zero. In other words, it is no contribution to the al ratio of the real to the iaginary ρ. The eikonal profile function ter also contains a factor 1/, but it is negative for the ratio of the real to the iaginary, it rises slowly with energy increasing. The eikonal profile function ter for the ratio of the real to the iaginary is positive and eventually becoes zero. In conclusion, we clais that gluon-gluon interaction akes doinant contributions to the al ratio of the real to the iaginary ρ for pp and pp elastic scattering, as we have done to the al cross section. References 1] A. Faus-Golfe, J. Velasco, and M. Haguenauer, hepex/ ] P. Gayron and B. Nicolescu, Phys. Lett. B , hep-ph/ ] K. Kontros, A. Lengyel, and Z. Tarics, hep-ph/ ] A. Donnachie and P.V. Landshoff, Phys. Lett. B ] V.A. Petrov and A.V. Prokudin, Eur. Phys. J. C ] P. Degrolard, M. Giffon, E. Martynov, and E. Predazzi, Eur. Phys. J. C ] M.M. Block, E.M. Gregores, F. Halzen, and G. Pancheri, Phys. Rev. Lett. D ] M.M. Block, F. Halzen, and G. Pancheri, hepph/ ] H. Cheng and T.T. Wu, Phys. Rev. Lett ] M.M. Block and R.N. Cahn, Rev. Mod. Phys ] D. Cline, F. Halzen, and J. Luthe, Phys. Rev. Lett ; P. L Heureux, B. Margolis, and P. Valin, Phys. Rev. D ; L. Durand and H. Pi, Phys. Rev. Lett ; Phys. Rev. D ; V. Innocente, A. Capella, and J.T.T. Van, Phys. Rev. B ; B. Margolis, et al., ibid ; B.Z. Kopeliovich, N.N. Nikolaev, and I.K. Potashnikova, Phys. Rev. D ; J.C. Collins and G.A. Ladinsky, ibid
Nuclear Slope Parameter of pp and pp Elastic Scattering in QCD Inspired Model
Commun. Theor. Phys. (Beijing, China) 49 (28) pp. 456 46 c Chinese Physical Society Vol. 49, No. 2, Feruary 15, 28 Nuclear Slope Parameter of pp and pp Elastic Scattering in QCD Inspired Model LU Juan,
More information2 Q 10. Likewise, in case of multiple particles, the corresponding density in 2 must be averaged over all
Lecture 6 Introduction to kinetic theory of plasa waves Introduction to kinetic theory So far we have been odeling plasa dynaics using fluid equations. The assuption has been that the pressure can be either
More informationarxiv:hep-ph/ v2 5 Dec 2001
Northwestern University: N.U.H.E.P. Report No. 901 University of Wisconsin: MADPH-01-1251 revised: November 26, 2001 On factorization, quark counting and vector dominance M. M. Block Department of Physics
More informationScattering and bound states
Chapter Scattering and bound states In this chapter we give a review of quantu-echanical scattering theory. We focus on the relation between the scattering aplitude of a potential and its bound states
More informationProton Structure and Prediction of Elastic Scattering at LHC at Center-of-Mass Energy 7 TeV
Proton Structure and Prediction of Elastic Scattering at LHC at Center-of-Mass Energy 7 TeV M. M. Islam 1, J. Kašpar 2,3, R. J. Luddy 1 1 Department of Physics, University of Connecticut, Storrs, CT 06269
More information7. Renormalization and universality in pionless EFT
Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 7. Renoralization and universality in pionless EFT Recall the scales of nuclear forces fro Section 5: Pionless EFT is applicable
More informationStern-Gerlach Experiment
Stern-Gerlach Experient HOE: The Physics of Bruce Harvey This is the experient that is said to prove that the electron has an intrinsic agnetic oent. Hydrogen like atos are projected in a bea through a
More informationChapter 6 1-D Continuous Groups
Chapter 6 1-D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well- defined range. This chapter explores:
More informationarxiv: v2 [hep-ph] 9 Jan 2014
Fluctuation induced equality of ulti-particle eccentricities for four or ore particles Ada Bzdak a, Piotr Bozek b,c, Larry McLerran d,a,e arxiv:1311.7325v2 [hep-ph] 9 Jan 2014 a RIKEN BNL Research Center,
More informationPHYSICS 110A : CLASSICAL MECHANICS MIDTERM EXAM #2
PHYSICS 110A : CLASSICAL MECHANICS MIDTERM EXAM #2 [1] Two blocks connected by a spring of spring constant k are free to slide frictionlessly along a horizontal surface, as shown in Fig. 1. The unstretched
More informationP (t) = P (t = 0) + F t Conclusion: If we wait long enough, the velocity of an electron will diverge, which is obviously impossible and wrong.
4 Phys520.nb 2 Drude theory ~ Chapter in textbook 2.. The relaxation tie approxiation Here we treat electrons as a free ideal gas (classical) 2... Totally ignore interactions/scatterings Under a static
More informationElastic scattering of protons and their structure
Journal of Physics: Conference Series PAPER OPEN ACCESS Elastic scattering of protons and their structure To cite this article: I M Dremin 215 J. Phys.: Conf. Ser. 67 125 View the article online for updates
More informationDoes the quark cluster model predict any isospin two dibaryon. resonance? (1) Grupo defsica Nuclear
FUSAL - 4/95 Does the quark cluster odel predict any isospin two dibaryon resonance? A. Valcarce (1), H. Garcilazo (),F.Fernandez (1) and E. Moro (1) (1) Grupo defsica uclear Universidad de Salaanca, E-37008
More informationδ 12. We find a highly accurate analytic description of the functions δ 11 ( δ 0, n)
Coplete-return spectru for a generalied Rosen-Zener two-state ter-crossing odel T.A. Shahverdyan, D.S. Mogilevtsev, V.M. Red kov, and A.M Ishkhanyan 3 Moscow Institute of Physics and Technology, 47 Dolgoprudni,
More informationMass Spectrum and Decay Constants of Conventional Mesons within an Infrared Confinement Model
Mass Spectru and Decay Constants of Conventional Mesons within an Infrared Confineent Model Gurjav Ganbold (BLTP, JINR; IPT MAS (Mongolia)) in collaboration with: T. Gutsche (Tuebingen) M. A. Ivanov (Dubna)
More informationIII. Quantization of electromagnetic field
III. Quantization of electroagnetic field Using the fraework presented in the previous chapter, this chapter describes lightwave in ters of quantu echanics. First, how to write a physical quantity operator
More informationIn the session you will be divided into groups and perform four separate experiments:
Mechanics Lab (Civil Engineers) Nae (please print): Tutor (please print): Lab group: Date of lab: Experients In the session you will be divided into groups and perfor four separate experients: (1) air-track
More informationDETECTION OF NONLINEARITY IN VIBRATIONAL SYSTEMS USING THE SECOND TIME DERIVATIVE OF ABSOLUTE ACCELERATION
DETECTION OF NONLINEARITY IN VIBRATIONAL SYSTEMS USING THE SECOND TIME DERIVATIVE OF ABSOLUTE ACCELERATION Masaki WAKUI 1 and Jun IYAMA and Tsuyoshi KOYAMA 3 ABSTRACT This paper shows a criteria to detect
More informationSOLUTIONS. PROBLEM 1. The Hamiltonian of the particle in the gravitational field can be written as, x 0, + U(x), U(x) =
SOLUTIONS PROBLEM 1. The Hailtonian of the particle in the gravitational field can be written as { Ĥ = ˆp2, x 0, + U(x), U(x) = (1) 2 gx, x > 0. The siplest estiate coes fro the uncertainty relation. If
More informationGLOBAL ANALYSIS OF NEUTRINO DATA
GLOBAL ANALYSIS OF NEUTRINO DATA Stony Brook & IFIC-Valencia) Nobel Syposiu, August 4 Introduction: The Paraeters of the New Minial Standard Model Orthodox Global Fits Solar and Reactor Neutrinos Atospheric
More informationMechanics Physics 151
Mechanics Physics 5 Lecture Oscillations (Chapter 6) What We Did Last Tie Analyzed the otion of a heavy top Reduced into -diensional proble of θ Qualitative behavior Precession + nutation Initial condition
More informationarxiv: v2 [hep-th] 16 Mar 2017
SLAC-PUB-6904 Angular Moentu Conservation Law in Light-Front Quantu Field Theory arxiv:70.07v [hep-th] 6 Mar 07 Kelly Yu-Ju Chiu and Stanley J. Brodsky SLAC National Accelerator Laboratory, Stanford University,
More informationPhysics 139B Solutions to Homework Set 3 Fall 2009
Physics 139B Solutions to Hoework Set 3 Fall 009 1. Consider a particle of ass attached to a rigid assless rod of fixed length R whose other end is fixed at the origin. The rod is free to rotate about
More informationMESONIC SUPER-ČERENKOV-LIKE EFFECTS IN HADRONIC AND NUCLEAR MEDIA
MESONIC SUPER-ČERENKOV-LIKE EFFECTS IN HADRONIC AND NUCLEAR MEDIA D.B. ION 1,2), M.L. ION 3) 1) National Institute for Physics and Nuclear Engineering Horia Hulubei, IFIN-HH, Bucharest, P.O. Box MG-6,
More informationarxiv: v2 [hep-ph] 30 Jan 2018
IPPP/17/89 January 31, 2018 Elastic proton-proton scattering at 13 TeV arxiv:1712.00325v2 [hep-ph] 30 Jan 2018 V.A. Khoze a,b, A.D. Martin a and M.G. Ryskin a,b a Institute for Particle Physics Phenomenology,
More informationProc. of the IEEE/OES Seventh Working Conference on Current Measurement Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES
Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES Belinda Lipa Codar Ocean Sensors 15 La Sandra Way, Portola Valley, CA 98 blipa@pogo.co
More informationM. Kirchbach. Germany. L. Tiator. Abstract. The pseudoscalar and pseudovector N coupling constants are calculated
ON THE COUPLING OF THE MESON TO THE NUCLEON M. Kirchbach Institut fur Kernphysik, TH Darstadt, D{64289 Darstadt, Gerany L. Tiator Institut fur Kernphysik, Universitat Mainz, D{5599 Mainz, Gerany Abstract
More informationCausality and the Kramers Kronig relations
Causality and the Kraers Kronig relations Causality describes the teporal relationship between cause and effect. A bell rings after you strike it, not before you strike it. This eans that the function
More informationFIRST MEASUREMENTS OF PROTON-PROTON ELASTIC SCATTERING AND TOTAL CROSS-SECTION AT THE LHC BY TOTEM
FIRST MEASUREMENTS OF PROTON-PROTON ELASTIC SCATTERING AND TOTAL CROSS-SECTION AT THE LHC BY TOTEM M. DEILE on behalf of the TOTEM Collaboration CERN, 111 Genève 3, Switzerland The TOTEM experiment at
More informationOptical Properties of Plasmas of High-Z Elements
Forschungszentru Karlsruhe Techni und Uwelt Wissenschaftlishe Berichte FZK Optical Properties of Plasas of High-Z Eleents V.Tolach 1, G.Miloshevsy 1, H.Würz Project Kernfusion 1 Heat and Mass Transfer
More informationPhys463.nb. Many electrons in 1D at T = 0. For a large system (L ), ΕF =? (6.7) The solutions of this equation are plane waves (6.
â â x Ψn Hx Ε Ψn Hx 35 (6.7) he solutions of this equation are plane waves Ψn Hx A exphä n x (6.8) he eigen-energy Εn is n (6.9) Εn For a D syste with length and periodic boundary conditions, Ψn Hx Ψn
More informationScalar meson photoproduction
Scalar eson photoproduction Šukasz Bibrzycki H. Niewodnicza«ski Institute of Nuclear Physics PAN, Cracow, Poland Nov. 11, 29 Š. Bibrzycki (IFJ PAN) Scalar eson photoproduction Nov. 11, 29 1 / 2 Outline
More informationUNIT HOMEWORK MOMENTUM ANSWER KEY
UNIT HOMEWORK MOMENTUM ANSWER KEY MOMENTUM FORMULA & STUFF FROM THE PAST: p = v, TKE = ½v 2, d = v t 1. An ostrich with a ass of 146 kg is running to the right with a velocity of 17 /s. a. Calculate the
More informationOutline THE EXOTIC Y STATES. UPDATE OF THE π + π - ψ(2s) ISR ANALYSIS AT BABAR
Outline THE EXOTIC Y STATES UPDATE OF THE π + π - J/ψ ISR ANALYSIS AT BABAR Context New π + π - J/ψ results Analysis of π + π - in π + π - J/ψ Suary UPDATE OF THE π + π - ψ(s) ISR ANALYSIS AT BABAR Context
More informationHadron interactions and hadron structure
+ q 10.1098/rsta.2000.0725 Hadron interactions and hadron structure By W. Ja es Stirling Departents of Matheatical Sciences and Physics, University of Durha, Durha DH1 3LE, UK While deep-inelastic scattering
More informationEntangling characterization of (SWAP) 1/m and Controlled unitary gates
Entangling characterization of (SWAP) / and Controlled unitary gates S.Balakrishnan and R.Sankaranarayanan Departent of Physics, National Institute of Technology, Tiruchirappalli 65, India. We study the
More informationAn Approximate Model for the Theoretical Prediction of the Velocity Increase in the Intermediate Ballistics Period
An Approxiate Model for the Theoretical Prediction of the Velocity... 77 Central European Journal of Energetic Materials, 205, 2(), 77-88 ISSN 2353-843 An Approxiate Model for the Theoretical Prediction
More informationMeasuring orbital angular momentum superpositions of light by mode transformation
CHAPTER 7 Measuring orbital angular oentu superpositions of light by ode transforation In chapter 6 we reported on a ethod for easuring orbital angular oentu (OAM) states of light based on the transforation
More informationor hadrons. In the latter case the state was observed in production in a Drell-Yan-like channel pn J X X X
Physics 557 Lecture 12 Even More particles heavy flavors: The exciteent of the 1970 s continued with the observation in 1974 (continuing through 1976) of a new quark flavor (recall that the sae period
More informationNumerical Solution of the MRLW Equation Using Finite Difference Method. 1 Introduction
ISSN 1749-3889 print, 1749-3897 online International Journal of Nonlinear Science Vol.1401 No.3,pp.355-361 Nuerical Solution of the MRLW Equation Using Finite Difference Method Pınar Keskin, Dursun Irk
More informationReading from Young & Freedman: For this topic, read the introduction to chapter 25 and sections 25.1 to 25.3 & 25.6.
PHY10 Electricity Topic 6 (Lectures 9 & 10) Electric Current and Resistance n this topic, we will cover: 1) Current in a conductor ) Resistivity 3) Resistance 4) Oh s Law 5) The Drude Model of conduction
More informationHee = ~ dxdy\jj+ (x) 'IJ+ (y) u (x- y) \jj (y) \jj (x), V, = ~ dx 'IJ+ (x) \jj (x) V (x), Hii = Z 2 ~ dx dy cp+ (x) cp+ (y) u (x- y) cp (y) cp (x),
SOVIET PHYSICS JETP VOLUME 14, NUMBER 4 APRIL, 1962 SHIFT OF ATOMIC ENERGY LEVELS IN A PLASMA L. E. PARGAMANIK Khar'kov State University Subitted to JETP editor February 16, 1961; resubitted June 19, 1961
More informationPattern Recognition and Machine Learning. Artificial Neural networks
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3 - MMIS Fall Seester 2017 Lessons 7 20 Dec 2017 Outline Artificial Neural networks Notation...2 Introduction...3 Key Equations... 3 Artificial
More informationarxiv:hep-ph/ v1 6 Nov 2006
arxiv:hep-ph/0611063v1 6 Nov 2006 in diffractive reactions Institute of Nuclear Physics PAN, ul. Radzikowskiego 152 PL-31-342 Cracow, Poland and University of Rzeszów, ul. Rejtana 16 PL-35-959 Rzeszów,
More informationarxiv: v1 [hep-ph] 26 Apr 2018
Evidence for maximality of strong interactions from LHC forward data E. Martynov a, B. Nicolescu b a Bogolyubov Institute for Theoretical Physics, Metrologichna 14b, Kiev, 368 Ukraine b Faculty of European
More informationTruncated BFKL Series and Hadronic Collisions
Preprint typeset in JHEP style. - PAPER VERSION arxiv:hep-ph/0009011v2 15 Jan 2001 Truncated BFKL Series and Hadronic Collisions M.B. Gay Ducati and M.V.T. Machado Instituto de Física, Univ. Federal do
More informationPHYS 102 Previous Exam Problems
PHYS 102 Previous Exa Probles CHAPTER 16 Waves Transverse waves on a string Power Interference of waves Standing waves Resonance on a string 1. The displaceent of a string carrying a traveling sinusoidal
More informationChapter 1: Basics of Vibrations for Simple Mechanical Systems
Chapter 1: Basics of Vibrations for Siple Mechanical Systes Introduction: The fundaentals of Sound and Vibrations are part of the broader field of echanics, with strong connections to classical echanics,
More informationarxiv:hep-ph/ v1 4 Feb 1997
DOUBLE SPIN TRANSVERSE ASYMMETRIES IN DRELL YAN PROCESSES V. Barone a,b, T. Calarco c and A. Drago c a Dipartimento di Fisica Teorica, Università di Torino and INFN, Sezione di Torino, 10125 Torino, Italy
More informationQuantization of magnetoelectric fields
Quantization of agnetoelectric fields E. O. Kaenetskii Microwave Magnetic Laboratory, Departent of Electrical and Coputer Engineering, Ben Gurion University of the Negev, Beer Sheva, Israel January 22,
More informationFlavor Asymmetry of the Nucleon Sea and W-Boson Production*
Flavor Asymmetry of the Nucleon Sea and W-Boson Production* Department of Physics University of Illinois 7 December 2012 *R. Yang, J.C. Peng, M. Grosse-Perdekamp, Phys. Lett. B 680 (2009) 231-234 What
More informationDispersion. February 12, 2014
Dispersion February 1, 014 In aterials, the dielectric constant and pereability are actually frequency dependent. This does not affect our results for single frequency odes, but when we have a superposition
More informationACTIVE VIBRATION CONTROL FOR STRUCTURE HAVING NON- LINEAR BEHAVIOR UNDER EARTHQUAKE EXCITATION
International onference on Earthquae Engineering and Disaster itigation, Jaarta, April 14-15, 8 ATIVE VIBRATION ONTROL FOR TRUTURE HAVING NON- LINEAR BEHAVIOR UNDER EARTHQUAE EXITATION Herlien D. etio
More informationCalculation of the Gluon Distribution Function Using Alternative Method for the Proton Structure Function
Commun. Theor. Phys. (Beijing, China 40 (2003 pp. 551 557 c International Academic Publishers Vol. 40, No. 5, November 15, 2003 Calculation of the Gluon Distribution Function Using Alternative Method for
More informationProton-Proton Total Cross Sections from the Window of Cosmic Ray Experiments
Proton-Proton Total Cross Sections from the Window of Cosmic Ray Experiments A.A. Arkhipov a a Theoretical Physics Division, Institute for High Energy Physics, 142284 Protvino, Moscow Region, Russia The
More informationMatching collinear and small x factorization calculations for inclusive hadron production in pa collisions
Matching collinear and small x factorization calculations for inclusive hadron production in pa collisions The Pennsylvania State University, Physics Department, University Park, PA 16802 H. Niewodniczański
More informationChapter 6 Aberrations
EE90F Chapter 6 Aberrations As we have seen, spherical lenses only obey Gaussian lens law in the paraxial approxiation. Deviations fro this ideal are called aberrations. F Rays toward the edge of the pupil
More informationHigh Energy Physics. Lecture 9. Deep Inelastic Scattering Scaling Violation. HEP Lecture 9 1
High Energy Physics Lecture 9 Deep Inelastic Scattering Scaling Violation HEP Lecture 9 1 Deep Inelastic Scattering: The reaction equation of DIS is written e+ p e+ X where X is a system of outgoing hadrons
More information13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 0-1 Matrices
CS71 Randoness & Coputation Spring 018 Instructor: Alistair Sinclair Lecture 13: February 7 Disclaier: These notes have not been subjected to the usual scrutiny accorded to foral publications. They ay
More informationwhich is the moment of inertia mm -- the center of mass is given by: m11 r m2r 2
Chapter 6: The Rigid Rotator * Energy Levels of the Rigid Rotator - this is the odel for icrowave/rotational spectroscopy - a rotating diatoic is odeled as a rigid rotator -- we have two atos with asses
More information821. Study on analysis method for deepwater TTR coupled vibration of parameter vibration and vortex-induced vibration
81. Study on analysis ethod for deepwater TTR coupled vibration of paraeter vibration and vortex-induced vibration Wu Xue-Min 1, Huang Wei-Ping Shandong Key aboratory of Ocean Engineering, Ocean University
More informationLecture 16: Scattering States and the Step Potential. 1 The Step Potential 1. 4 Wavepackets in the step potential 6
Lecture 16: Scattering States and the Step Potential B. Zwiebach April 19, 2016 Contents 1 The Step Potential 1 2 Step Potential with E>V 0 2 3 Step Potential with E
More informationSupporting Information for Supression of Auger Processes in Confined Structures
Supporting Inforation for Supression of Auger Processes in Confined Structures George E. Cragg and Alexander. Efros Naval Research aboratory, Washington, DC 20375, USA 1 Solution of the Coupled, Two-band
More informationNuclear Physics (10 th lecture)
~Theta Nuclear Physics ( th lecture) Content Nuclear Collective Model: Rainwater approx. (reinder) Consequences of nuclear deforation o Rotational states High spin states and back bending o Vibrational
More informationAngular Momentum Properties
Cheistry 460 Fall 017 Dr. Jean M. Standard October 30, 017 Angular Moentu Properties Classical Definition of Angular Moentu In classical echanics, the angular oentu vector L is defined as L = r p, (1)
More informationAccuracy of the Scaling Law for Experimental Natural Frequencies of Rectangular Thin Plates
The 9th Conference of Mechanical Engineering Network of Thailand 9- October 005, Phuket, Thailand Accuracy of the caling Law for Experiental Natural Frequencies of Rectangular Thin Plates Anawat Na songkhla
More information12 Towards hydrodynamic equations J Nonlinear Dynamics II: Continuum Systems Lecture 12 Spring 2015
18.354J Nonlinear Dynaics II: Continuu Systes Lecture 12 Spring 2015 12 Towards hydrodynaic equations The previous classes focussed on the continuu description of static (tie-independent) elastic systes.
More informationLecture #8-3 Oscillations, Simple Harmonic Motion
Lecture #8-3 Oscillations Siple Haronic Motion So far we have considered two basic types of otion: translation and rotation. But these are not the only two types of otion we can observe in every day life.
More informationPhysically Based Modeling CS Notes Spring 1997 Particle Collision and Contact
Physically Based Modeling CS 15-863 Notes Spring 1997 Particle Collision and Contact 1 Collisions with Springs Suppose we wanted to ipleent a particle siulator with a floor : a solid horizontal plane which
More informationASYMPTOTIC BEHAVIOR OF NUCLEON ELECTROMAGNETIC FORM FACTORS IN SPACE- AND TIME-LIKE REGIONS
ASYMPTOTIC BEHAVIOR OF NUCLEON ELECTROMAGNETIC FORM FACTORS IN SPACE- AND TIME-LIKE REGIONS Egle Tomasi-Gustafsson (1) and Michail P. Rekalo (2) (1) DAPNIA/SPhN, CEA/Saclay, 91191 Gif-sur-Yvette Cedex,
More informationCorrelated Signals at the Energy and Intensity Frontiers from Nonabelian Kinetic Mixing
Correlated Signals at the Energy and Intensity Frontiers fro Nonabelian Kinetic Mixing G. Barello, Spencer Chang and Christopher A. Newby Departent of Physics and Institute of Theoretical Science, University
More informationRe-study of Nucleon Pole Contribution in J/ψ N Nπ Decay
Commun. Theor. Phys. Beijing, China 46 26 pp. 57 53 c International Academic Publishers Vol. 46, No. 3, September 5, 26 Re-study of Nucleon Pole Contribution in J/ψ N Nπ Decay ZONG Yuan-Yuan,,2 SHEN Peng-Nian,,3,4
More informationProblem T1. Main sequence stars (11 points)
Proble T1. Main sequence stars 11 points Part. Lifetie of Sun points i..7 pts Since the Sun behaves as a perfectly black body it s total radiation power can be expressed fro the Stefan- Boltzann law as
More informationUniversity of Bath DOI: /OE Publication date: Link to publication
Citation for published version: Chen, L & Bird, DM 2011, 'Guidance in Kagoe-like photonic crystal fibres II: perturbation theory for a realistic fibre structure' Optics Express, vol. 19, no. 7, pp. 6957-6968.
More informationImaging the Proton via Hard Exclusive Production in Diffractive pp Scattering
Exclusive Reactions at High Momentum Transfer Jefferson Lab, Newport News, VA May 21-24, 2007 Imaging the Proton via Hard Exclusive Production in Diffractive pp Scattering Charles Earl Hyde Old Dominion
More information(a) As a reminder, the classical definition of angular momentum is: l = r p
PHYSICS T8: Standard Model Midter Exa Solution Key (216) 1. [2 points] Short Answer ( points each) (a) As a reinder, the classical definition of angular oentu is: l r p Based on this, what are the units
More informationBALLISTIC PENDULUM. EXPERIMENT: Measuring the Projectile Speed Consider a steel ball of mass
BALLISTIC PENDULUM INTRODUCTION: In this experient you will use the principles of conservation of oentu and energy to deterine the speed of a horizontally projected ball and use this speed to predict the
More informationPhysics with Hadron Beams at COMPASS
Physics with Hadron Beams at COMPASS Bernhard Ketzer Technische Universität München MAMI and Beyond 2009 International Workshop on Hadron Structure and Spectroscopy 2009 30 March 2009 The Goal Understand
More informationGeneral Properties of Radiation Detectors Supplements
Phys. 649: Nuclear Techniques Physics Departent Yarouk University Chapter 4: General Properties of Radiation Detectors Suppleents Dr. Nidal M. Ershaidat Overview Phys. 649: Nuclear Techniques Physics Departent
More informationNonmonotonic Networks. a. IRST, I Povo (Trento) Italy, b. Univ. of Trento, Physics Dept., I Povo (Trento) Italy
Storage Capacity and Dynaics of Nononotonic Networks Bruno Crespi a and Ignazio Lazzizzera b a. IRST, I-38050 Povo (Trento) Italy, b. Univ. of Trento, Physics Dept., I-38050 Povo (Trento) Italy INFN Gruppo
More informationAll Excuses must be taken to 233 Loomis before 4:15, Monday, April 30.
Miscellaneous Notes he end is near don t get behind. All Excuses ust be taken to 233 Loois before 4:15, Monday, April 30. he PHYS 213 final exa ties are * 8-10 AM, Monday, May 7 * 8-10 AM, uesday, May
More informationStructured Illumination Super-Resolution Imaging Achieved by Two Steps based on the Modulation of Background Light Field
017 nd International Seinar on Applied Physics, Optoelectronics and Photonics (APOP 017) ISBN: 978-1-60595-5-3 Structured Illuination Super-Resolution Iaging Achieved by Two Steps based on the Modulation
More informationSupplemental Material for Correlation between Length and Tilt of Lipid Tails
Suppleental Material for Correlation between Length and Tilt of Lipid Tails Ditry I. Kopelevich and John F. Nagle I. RESULTS FOR ALTERNATIVE DIRECTOR DEFINITIONS A. Alternative Director Definitions The
More informationMeasuringtheFifthStructureFunctionin
CLAS Collaboration Meeting, Oct, 1-1, 1 p. 1/ MeasuringtheFifthStructureFunctionin H( e, e p)n G.P. Gilfoyle, R. Burrell, C. Copos, K. Gill, K. Greenholt, M. Jordan, (Richond) W.K. Brooks, (USM) J.D. Lachniet,
More informationImplications of G p E(Q 2 )/G p M(Q 2 ).
Implications of G p E(Q 2 )/G p M(Q 2 ). S. Dubnička 1, A. Z. Dubničková 2, OUTLINE: 1. JLab proton polarization data puzzle 2. Existence of two different G p E(t) behaviors in spacelike region 3. Consequences
More informationThe Transactional Nature of Quantum Information
The Transactional Nature of Quantu Inforation Subhash Kak Departent of Coputer Science Oklahoa State University Stillwater, OK 7478 ABSTRACT Inforation, in its counications sense, is a transactional property.
More informationNow multiply the left-hand-side by ω and the right-hand side by dδ/dt (recall ω= dδ/dt) to get:
Equal Area Criterion.0 Developent of equal area criterion As in previous notes, all powers are in per-unit. I want to show you the equal area criterion a little differently than the book does it. Let s
More informationMA304 Differential Geometry
MA304 Differential Geoetry Hoework 4 solutions Spring 018 6% of the final ark 1. The paraeterised curve αt = t cosh t for t R is called the catenary. Find the curvature of αt. Solution. Fro hoework question
More informationarxiv: v2 [hep-ph] 19 Feb 2016
TWIST EXPANSION OF FORWARD DRE YAN PROCESS Tomasz Stebel, eszek Motyka, Mariusz Sadzikowski arxiv:1602.01762v2 [hep-ph] 19 Feb 2016 The Marian Smoluchowski Institute of Physics, Jagiellonian University
More informationTHE POMERON IN EXCLUSIVE VECTOR MESON PRODUCTION. Academy of Science of Ukraine UA Kiev, Ukraine c Dipartimento di Fisica, Università di Padova
DFCAL-TH 03/3 February 2003 THE POMERON IN EXCLUSIVE VECTOR MESON PRODUCTION R. Fiore a, L.L. Jenkovszky b, F. Paccanoni c, A. Prokudin d a Dipartimento di Fisica, Università della Calabria Instituto Nazionale
More informationEnergy and Momentum: The Ballistic Pendulum
Physics Departent Handout -10 Energy and Moentu: The Ballistic Pendulu The ballistic pendulu, first described in the id-eighteenth century, applies principles of echanics to the proble of easuring the
More informationIntelligent Systems: Reasoning and Recognition. Artificial Neural Networks
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley MOSIG M1 Winter Seester 2018 Lesson 7 1 March 2018 Outline Artificial Neural Networks Notation...2 Introduction...3 Key Equations... 3 Artificial
More informationJohn K. Elwood and Mark B. Wise. California Institute of Technology, Pasadena, CA and. Martin J. Savage
π e e * John K. Elwood and Mark B. Wise California Institute of Technology, Pasadena, CA 9115 arxiv:hep-ph/95488v1 11 Apr 1995 and Martin J. Savage Departent of Physics, Carnegie Mellon University, Pittsburgh
More informationResearch Article Approximate Multidegree Reduction of λ-bézier Curves
Matheatical Probles in Engineering Volue 6 Article ID 87 pages http://dxdoiorg//6/87 Research Article Approxiate Multidegree Reduction of λ-bézier Curves Gang Hu Huanxin Cao and Suxia Zhang Departent of
More informationA Simplified Analytical Approach for Efficiency Evaluation of the Weaving Machines with Automatic Filling Repair
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 0 A Siplified Analytical Approach for Efficiency Evaluation of the eaving
More informationReggeization of the Phillips-Barger model of high-energy hadron scattering
IL NUOVO CIMENTO Vol. C, N. Marzo-Aprile 0 DOI.9/ncc/i0-- Colloquia: LC Reggeization of the Phillips-Barger model of high-energy hadron scattering L. Jenkovszky BITP, National Academy of Sciences of Ukraine
More informationModern Physics Letters A Vol. 24, Nos (2009) c World Scientific Publishing Company
Modern Physics Letters A Vol. 24, Nos. 11 13 (2009) 816 822 c World cientific Publishing Copany TOWARD A THREE-DIMENIONAL OLUTION FOR 3N BOUND TATE WITH 3NFs M. R. HADIZADEH and. BAYEGAN Departent of Physics,
More informationQCD and Rescattering in Nuclear Targets Lecture 2
QCD and Rescattering in Nuclear Targets Lecture Jianwei Qiu Iowa State University The 1 st Annual Hampton University Graduate Studies Program (HUGS 006) June 5-3, 006 Jefferson Lab, Newport News, Virginia
More informationPhase transition theory of pulse formation in passively mode-locked lasers with dispersion and Kerr nonlinearity
Optics Counications 223 (2003) 151 156 www.elsevier.co/locate/optco Phase transition theory of pulse foration in passively ode-locked lasers with dispersion and Kerr nonlinearity Ariel Gordon, Baruch Fischer
More informationSeeking the Shadowing in ea Processes. M. B. Gay Ducati. V. P. Gonçalves
Seeking the Shadowing in ea Processes M. B. Gay Ducati and V. P. Gonçalves InstitutodeFísica, Univ. Federal do Rio Grande do Sul Caixa Postal 15051, 91501-970 Porto Alegre, RS, BRAZIL Abstract: We consider
More information