Kaotik davranış mekanizmaları ve aşırı inelastik saçılmada ölçeklenme
Mechanisms of chaotic behaviour and scaling in deep inelastic scattering
- Tez No: 46482
- Danışmanlar: PROF.DR. AVADİS HACINLIYAN
- Tez Türü: Yüksek Lisans
- Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 80
Özet
ÖZET Hadronlarm iç yapısı hakkındaki kuark parton modeli çerçevesinde aşırı inelastik lepton-hadron saçılması olayında boyutsuzlaşma ve boyutsuzlaşmadan sapmalar değişik bir bakış açısıyla incelenmiştir. Bu ise son yıllarda oldukça önem kazanıp popüler olan kaotik davranıştır. Bu çalışmada, kaotik davranışla kendine benzerlik arasında ilişkiden yararlanılmıştır.“”Aşırı inelastik lepton-hadron saçılması, Bjorken ölçeklenmesi gösterdi ğinden ve saçılan leptonlarm dağılımı parton modeli ile verildiğinden, bu konu lara değinilmiş ve faz dağılım bağıntılarından bahsedilmiştir. Kendine benzer yapıları inceleyebilmek için dallanma modeli ile ke- siklilik tarif edilmiştir. Parton modeli üzerinde, kesiklilik ve fraktal boyut hesaplan yapılmış ve bunlar için gerekli olan spektrum fonksiyonu bulunmuştur. Çağlayan oluşturulurken matematiksel yönden ilginç olan Verhulst tasvirinin genelleştirilmiş şekli, ayrılma fonksiyonu olarak alınmıştır. Olçeklenmeden sapmalar incelenirken, ölçeklenme değişkeni a; 'e ek ola rak ölçeklenmeden sapmaların parametrize edilmesi için probleme foton virtual dörtlü-momentum karesi q2 de ikinci değişken olarak dahil edilmiştir. Dallanma modelinde, ayrılma fonksiyonu olarak genelleştirilmiş Verhulst 'un yanısıra lite ratürdeki üç farklı fonksiyon da kullanılmıştır. Çağlayan eteğindeki dağılımın, parton yapı fonksiyonlarının karakteris tik davranışlarını gösterdiği belirlendiğinden, deneysel olarak CERN'de ölçülen yapı fonksiyonu değerleri ile karşılaştırma yapılmıştır. En iyi uyumu, genelleşti rilmiş Verhulst 'un ayrılma fonksiyonu olarak kullanıldığı çağlayanın eteğindeki dağılım vermiştir. iv
Özet (Çeviri)
SUMMARY MECHANISMS OF CHAOTIC BEHAVIOUR AND SCALING IN DEEP INELASTIC SCATTERING Convincing evidence of an internal structure in the proton was provided by experiments on the scattering of fast electrons by protons conducted at SL AC (Stanford) and DESY (Hamburg) since the late 1960s. In these experiments a stationary target was bombarded with electrons with energies of 20GeV or more. When such a high-energy electron hit a proton head on, it was able to penetrate the proton and probe its inner structure. It was found that most of the electrons that passed through the target underwent only small deviations from their original path. However, the number of large-angle deviations was significantly larger than what would have been expected if the proton mass and charge were uniformly“distributed”throughout its volume. These results are reminiscent of Rutherford's experiment which probed the atom and revealed its nucleus. Indeed, analysis of the SLAC and DESY results showed that they were consistent with most of the proton being electrically neutral with just several charged“nuclei”, with diameters very small in comparison with the proton's diameter. Richard Feynman of Caltech initially named these objects“partons”. Bjorken and Paschos conjectured that they could be bound quarks. It turned out that they could be taken as the constituent quarks of spin 1/2 to a first approximation. Quantum Chromodynamics (QCD) confirmed this quark- parton model as its zero order approximation. It (QCD) also contained spin 1 gluons which constituted the integer spin background. Furthermore, since the gluons could also interact among themselves, they also provided the mechanism of confinement. Most of the partons turned out to have spin 1/2, and could be the confined quarks [25]. Inelastic lepton-nucleon scattering can be interpreted in terms of the in coherent sum of elastic scattering of the lepton by pointlike parton constituents and these can be identified with the quarks. At high momentum transfers, the elastic form factor is very small, and inelastic scattering of the incident electron is much more probable than elastic scattering. In a general inelastic scatter ing process, there is an extra variable because the space and time components (q, iv)1 of the momentum transfer q are no longer related by q2 = 2Mi/, where M is the proton mass. If we denote the three-momentum, energy and invariant mass of the final hadron state by p', E' and W, we obtain xThe Pauli metric for four vectors xM = (x, ict) with an imaginary timelike component is used. Natural units such that h = 1, c = 1 are used.q2 = (p' - O)2 - (£' - Mf and v = E' - M, W2 = E'2 -p'2 so that ç2 = 2Mu - W2 + M2. (1) W = M corresponds to elastic scattering, with q2 = 2Mv. The cross-section can be written as oo, v - > oo, the function F(q2, v) remains finite, it can depend only on the dimensionless and finite ratio of these two quantities, that is, oni = q2 j2Mv. Since x is dimensionless, there is no scale of mass or length; hence the term scale invariance. Even at larger values of x (corresponding to small v) or smaller values of x, the ç2-dependence is weak, we have essentially very weak ç2-dependence. The scale invariance proposed by Bjorken in the limit q2 - >? oo seems then to hold approximately in the region of q2 of a few times M2. Imagine a reference frame in which the target proton has very large three-momentum-the so-called infinite momentum frame, Figure (1). The pro ton mass can be neglected, so it has four-momentum P = (p, 0, 0, ip) and is vi sualized as consisting of a parallel stream of partons, each with four-momentum xP, (0 < x < 1). If P is large, masses and transverse momentum components of VIinvolve the renormalization group, it is natural to observe various types of chaotic behaviour in processes described by QCD. Indeed, intermittency has been observed in strong interactions. Evidence has been found for power-law behaviour of normalized factorial moments Fi, when the bin width of rapidity interval 6 is decreased. The phenomenon therefore suggests self-similarity in multiplicity fluctuations in a range of resolution scales. It also suggests the existence of fractal properties in the problem [48-51]. Unlike the usual fractal types of behaviour in geometrical and statistical systems [30], the multiparticle- production processes have dynamical and kinematic features that pose special problems. The formulation of this problem begins with the branching model. Con sider an initial state of a highly virtual parton produced in a high energy colli sion. It has a large momentum squared. Through a cascade process, final state partons with squared momentum below a certain cutoff value are produced. For the cascade process, we work in the infinite momentum frame. De note the longitudinal momentum fraction by a;. At each vertex, a parent parton goes into two daughter partons with the longitudinal momentum splitting into fractions: x and (1 - x). The selection of the x value is done in a stochastically consistent manner with the probability distribution P(x). Three types of distri butions used in the literature [34] are Twin, !- (8) IXdefined for each constituent, if more than one species of constituents are present. The distribution at the tail of the cascade increases for small x values and decreases for large x values as a function of Q2. In addition, it indicates a regular variation with Q2 for a fixed x value. At the same time, it should be resonable to compare them to the experimentally observed values of ^(a;, Q2). Experimental data is from the measurements of the European Muon Collaboration (EMC) at CERN [59]. As it shown in graphics of section (6) (Figure 6.2 -a,b,c), we observed very good agreement between the simulation and the experimental data on the proton structure functions for three kinds of parametrizations of the scaling variable (x', x“ and x'”) Since such simple cascade models show intermittency, it is clear that in- termittency is an important aspect of scaling in the structure functions. There is also a logical connection between a restriction in the phase space and in termittency. Scaling imposes a restriction on the dependence of the structure functions on the allowed kinematic variables and therefore, is related to inter mit t ene v. xninvolve the renormalization group, it is natural to observe various types of chaotic behaviour in processes described by QCD. Indeed, intermittency has been observed in strong interactions. Evidence has been found for power-law behaviour of normalized factorial moments Fi, when the bin width of rapidity interval 6 is decreased. The phenomenon therefore suggests self-similarity in multiplicity fluctuations in a range of resolution scales. It also suggests the existence of fractal properties in the problem [48-51]. Unlike the usual fractal types of behaviour in geometrical and statistical systems [30], the multiparticle- production processes have dynamical and kinematic features that pose special problems. The formulation of this problem begins with the branching model. Con sider an initial state of a highly virtual parton produced in a high energy colli sion. It has a large momentum squared. Through a cascade process, final state partons with squared momentum below a certain cutoff value are produced. For the cascade process, we work in the infinite momentum frame. De note the longitudinal momentum fraction by a;. At each vertex, a parent parton goes into two daughter partons with the longitudinal momentum splitting into fractions: x and (1 - x). The selection of the x value is done in a stochastically consistent manner with the probability distribution P(x). Three types of distri butions used in the literature [34] are Twin, !- (8) IXdefined for each constituent, if more than one species of constituents are present. The distribution at the tail of the cascade increases for small x values and decreases for large x values as a function of Q2. In addition, it indicates a regular variation with Q2 for a fixed x value. At the same time, it should be resonable to compare them to the experimentally observed values of ^(a;, Q2). Experimental data is from the measurements of the European Muon Collaboration (EMC) at CERN [59]. As it shown in graphics of section (6) (Figure 6.2 -a,b,c), we observed very good agreement between the simulation and the experimental data on the proton structure functions for three kinds of parametrizations of the scaling variable (x', x“ and x'”) Since such simple cascade models show intermittency, it is clear that in- termittency is an important aspect of scaling in the structure functions. There is also a logical connection between a restriction in the phase space and in termittency. Scaling imposes a restriction on the dependence of the structure functions on the allowed kinematic variables and therefore, is related to inter mit t ene v. xnthe partons can be neglected. Suppose now that one parton of mass m is scat tered elastically by absorbing the current four-momentum q from the scattered lepton. Then (xP + q)2 = m2 ~ 0, xlP2 + ql + 2xP ? q ~ 0. (3) If | x2P2 |= x2M2 < q2, we obtain x - 2P ? q 2Mv (4) where the invariant scalar product P. q has been evaluated in the laboratory system in which the energy transfer is v and the nucleon is at rest. (1-x)P Figure 1. The parton model of a deep-inelastic collision. Thus, x in equation (4) represents the fractional, three-momentum of the parton in the infinite-momentum frame. It is as if we had a hypothetical parton of mass m, stationary in the laboratary system, with the elastic q2 = 2mi/, so that, provided always that q2 >? M2, x = 2Mv m M (5) vııthe partons can be neglected. Suppose now that one parton of mass m is scat tered elastically by absorbing the current four-momentum q from the scattered lepton. Then (xP + q)2 = m2 ~ 0, xlP2 + ql + 2xP ? q ~ 0. (3) If | x2P2 |= x2M2 < q2, we obtain x - 2P ? q 2Mv (4) where the invariant scalar product P. q has been evaluated in the laboratory system in which the energy transfer is v and the nucleon is at rest. (1-x)P Figure 1. The parton model of a deep-inelastic collision. Thus, x in equation (4) represents the fractional, three-momentum of the parton in the infinite-momentum frame. It is as if we had a hypothetical parton of mass m, stationary in the laboratary system, with the elastic q2 = 2mi/, so that, provided always that q2 >? M2, x = 2Mv m M (5) vıı
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