DOCSLIB.ORG

The Nucleon-Nucleon Interaction with the Leading Order Relativistic Effects

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Nucleon-Nucleon Interaction with the Leading Order Relativistic Effects The nucleon-nucleon interaction with the leading order relativistic effects Petr Vesel´y Institute of Particle and Nuclear Physics Faculty of Mathematics and Physics Charles University, Prague Preface This text was written as a part of the lecture course1 The Relativistic Description of Nuclear Systems. We briefly summarize the methods employed in a quantum-mechanical (QM) description of the nucleon-nucleon problem. The QM approach means the Hamil- tonian description in a Hilbert space with a fixed number of nucleons (from which all other degrees of freedom are “integrated out”). Here we consider only its version in which the relativistic effects are included in a Hamiltonian (both in its kinetic and interaction parts) perturbatively in their leading order. After a short historical introduction given in the first chapter we consider the one- scalar-meson-exchange (OSE) interaction between two nucleons. The relativistic OSE amplitude is derived from the corresponding Feynman diagram of the covariant field theory. We briefly discuss the relation of this amplitude to the quantum-mechanical potential, in particular the techniques of the off-energy-shell continuation. Separation of the center-of-mass motion in Hamiltonian with relativistic effects is also considered. In the third chapter we deal with the subsequent transformation of the OSE potential from the momentum to the coordinate representation. Dubna, March 2, 2006 Petr Vesel´y 1This course was read by Jiˇr´ıAdam in 2005/2006 at the Institute of Particle and Nuclear Physics i Contents 1 Brief review of the NN potentials 1 2 One scalar particle exchange - OSE 5 2.1 Amplitudeoftheexchangeprocess . 5 2.2 Off-energy-shellcontinuation. .... 9 2.3 Separation of the center-of-mass motion . ...... 10 2.3.1 ComparisontotheBonnpotential. 17 3 The OSE potential, x-representation 19 3.1 Central ~q-dependentpart ........................... 22 3.2 Spin-orbitpart ................................. 24 3.3 Centralanticommutatorpart. 25 3.4 The OSE potential in the Schr¨odinger equation . ....... 25 4 Summary 27 A Poincare and Galilean algebras 28 B Commutators needed for the c.m. separation 32 C Anticommutator term in the Schr¨odinger equation 34 ii Chapter 1 Brief review of the NN potentials Since the Rutherford’s discovery of nucleus and the subsequently confirmed nucleonic structure of nuclei the crucial task - the explanation of the origin of attractive nuclear forces - has been haunting nuclear physicists. In 1930’s the quantum field theory was emerging and in its spirit Yukawa suggested the first microscopic derivation. He suggested the massive elementary1 particle – meson – which would intermediate the finite range nucleon-nucleon (NN) interaction. In 1947 a pion - the particle identified later with the Yukawa meson - was found in cosmic rays. Subsequent effort of theoretical physicists led to introduction of the one-pion- exchange (OPE) and later the two-pion-exchange (TPE) models of the NN interaction. Comparison to experimental results (phase-shift analysis) established the OPE potential as the long range part (“tail”) of the nuclear force. The full OPE+TPE program failed, however: the historical circumstances are in more detail explained in the introduction of the survey [1]. From the theoretical point of view it is important that the way the TPE processes were included in 1950’s did not respect the chiral symmetry2. In 1960’s heavy mesons (in particular, the vector ρ- a ω-mesons) were discovered and the attention of physicists turned to newly established OBE (one-boson-exchange) models. Recall that although these days we consider (most of the) heavy mesons to be quark-antiquark bound states, they are experimentally seen as the two- or three-pion resonances either in a pion- pion scattering or in processes with several pions in a final state. One says that heavy meson exchanges between nucleons represent correlated multipion exchanges. The OBE potentials are based on the idea that the single one-meson exchanges (of various meson types) can realistically represent most features of the full NN interaction, to which also exchanges of multiple mesons contribute. The meson-nucleon interaction is strong, i.e., a perturbative expansion in powers of coupling constants cannot be expected to work and the potential has to be iterated to all orders with the help of some dynamical equation – either non-relativistic Lippmann-Schwinger equation or some of analogous relativistic ones. There is, nevertheless, an important organizing principle for various contributions to meson-mediated NN potential: their importance decreases with their decreasing range, which in turn is inversely proportional to the total exchanged mass 1Elementary until the discovery of the hadron substructure and of the quark model. 2Let us note that the OPE+TPE approach was not rejected forever. Recently, it was revived in the framework of the effective (chiral) field theory. 1 CHAPTER 1. BRIEF REVIEW OF THE NN POTENTIALS 2 (i.e., sum of the masses of all exchanged mesons). In the framework of OBE potentials, the scalar, pseudoscalar and vector mesons with masses 1 GeV are usually taken into account. All mesons, or rather all their parameters, need≤ not necessarily be identified with physical mesons seen in experiments and listed in the Particle Data tables: some of them can actually effectively represent a sum of all (also multiple) exchanges in a particular spin/isospin channel. Thus, the scalar (isoscalar) field plays rather important role in the OPE models: it delivers a necessary intermediate range attraction. Still, the existence of the corresponding physical particle was not confirmed, at best it is a very broad and unstable resonance. The Bonn group (mentioned below) demonstrated (among other similar facts) that one should consider the scalar field as an effective field describing the correlated two pion exchange (with spin and isospin equal to zero). Although it was possible to fit the NN scattering data well by using OBE potentials, it was not actually clear that the uncorrelated multiparticle exchanges can be completely neglected. Also the role of nucleon excitation was not very clear. In the mid 1970’s the Bonn group [2, 1] started to evaluate the multipion exchange diagrams and in the following decade they computed all 2π diagrams (including those with virtual ∆ - resonances), relevant 3π and 4π diagrams and also combined crossed meson exchanges - ωπ, ρπ. This calculations demonstrated – though admittedly in a model-dependent way – that with rather good precision the NN potential can be indeed represented by OBE potential, with some of the exchanges being effective in a sense described above. This developments and other successful descriptions of the NN interaction in terms of meson exchanges (e.g., [3]) encouraged a birth of a new generation of realistic semi- phenomenological NN models called high-precision potentials. In chronological order they are Nijmegen Nijm-I, Nijm-II and Reid93 potentials [4]; • Argonne V potential [5]; • 18 CD-Bonn potential [6]. • All these potentials are inspired by the meson-exchange picture, in particular, all of them include the OPE part. They are also all charge-dependent, as required by deviations of the pp and np data. The main differences among them are in meson exchanges explicitly included and in parts of the force which are described completely phenomenologically. Also in their OBE components various approximations are made, in particular, to en- able the Fourier transform of the corresponding Feynman amplitudes the non-relativistic reduction (resulting in the local approximation) is often employed. The models Nijm-I and Nijm-II are based on the Nijmegen78 OBE potential [3]. Nijm- II uses the local approximation for all OBE amplitudes, while Nijm-I keeps some terms non-local. The Reid93 and V18 do not use meson exchanges for the intermediate and short-range parts of the potentials and describe them purely phenomenologically. The V18 potential employs the functions of the Wood-Saxon type, while Reid93 uses local Yukawa functions of multiples of the pion mass. Unlike other models, the CD-Bonn employs the full nonlocal Feynman amplitudes for the OBE potentials. Apart from the pion, the physical vector mesons ρ(769) and ω(783) and less important δ and η exchanges, two fictitious scalar-isoscalar σ-mesons are used. CHAPTER 1. BRIEF REVIEW OF THE NN POTENTIALS 3 To sum it up, the current model descriptions of the NN interaction, based on the meson-exchange picture with semi-phenomenological adjustments at the short-range can fit very successfully the NN data below pion production threshold. Extension of these ideas to the 3N force and e.m. and weak nuclear currents is, in principle, also phenomenologically successful, but not yet as conclusive. More important, it is rather difficult to deduce from these models some unambiguous information on related aspects of hadronic dynamics, e.g., on the importance of various physical (heavy) mesons and/or nucleon resonances, on their coupling constants, on whether it is adequate to include them as stable particles etc. It is also difficult to relate assumptions and the phenomenological input of these models to the underlying fundamental theory, the QCD. Even the important symmetries of the QCD – the approximate isospin symmetry and the approximate spontaneously-broken chiral symmetry – are reflected only purely phenomenologically by adopting the experimental values of the meson masses and coupling constants, considering the phenomenological mixing of heavier mesons and recognizing the leading role of the pion as the lightest meson. Recently, the Chiral Perturbation Theory (ChPT) has been established as an attempt to describe the hadron dynamics in the low-energy regime in more systematic and fun- damental way. The ChPT in relation to the fundamental theory of strong interaction - QCD - is an effective field theory satisfying all symmetries included in QCD with nucleons and pions (and in some formulations also ∆-isobars) used as the degrees of freedom. The applicability of this effective theory is then limited by the momentum scale of Λ 1 GeV.
Load more

Recommended publications
  • Higgs Mechanism and Debye Screening in the Generalized Electrodynamics
    Higgs Mechanism and Debye Screening in the Generalized Electrodynamics C. A. Bonin1,∗ G. B. de Gracia2,y A. A. Nogueira3,z and B. M. Pimentel2x 1Department of Mathematics & Statistics, State University of Ponta Grossa (DEMAT/UEPG), Avenida Carlos Cavalcanti 4748,84030-900 Ponta Grossa, PR, Brazil 2 Instituto de F´ısica Teorica´ (IFT), Universidade Estadual Paulista Rua Dr. Bento Teobaldo Ferraz 271, Bloco II Barra Funda, CEP 01140-070 Sao˜ Paulo, SP, Brazil 3Universidade Federal de Goias,´ Instituto de F´ısica, Av. Esperanc¸a, 74690-900, Goiania,ˆ Goias,´ Brasil September 12, 2019 Abstract In this work we study the Higgs mechanism and the Debye shielding for the Bopp-Podolsky theory of electrodynamics. We find that not only the massless sector of the Podolsky theory acquires a mass in both these phenomena, but also that its massive sector has its mass changed. Furthermore, we find a mathematical analogy in the way these masses change between these two mechanisms. Besides exploring the behaviour of the screened potentials, we find a temperature for which the presence of the generalized gauge field may be experimentally detected. Keywords: Quantum Field Theory; Finite Temperature; Spontaneous Symmetry Breaking, Debye Screening arXiv:1909.04731v1 [hep-th] 10 Sep 2019 ∗[email protected] [email protected] [email protected] (corresponding author) [email protected] 1 1 Introduction The mechanisms behind mass generation for particles play a crucial role in the advancement of our understanding of the fundamental forces of Nature. For instance, our current description of three out of five fundamental interactions is through the action of gauge fields [1].
    [Show full text]
  • Fermion Potential, Yukawa and Photon Exchange Copyright C 2007 by Joel A
    Last Latexed: October 24, 2013 at 10:23 1 Lecture 15 Oct. 24, 2013 Fermion Potential, Yukawa and Photon Exchange Copyright c 2007 by Joel A. Shapiro Last week we learned how to calculate cross sections in terms of the S- matrix or the invariant matrix element . Last time we learned, pretty M much, how to evaluate in terms of matrix element of the time ordered product of free fields (i.e.M interaction representation fields) between states given na¨ıvely by creation and annihilation operators for the incoming and outgoing particles. Today we are going to address the first “real” applications. First we will consider the low energy scattering of two different fermions • each coupled to a single scalar field, as proposed by Yukawa to explain the nuclear potential1. Then we will give Feynman rules for the electromagnetic field, that is, • for photons. This will not be rigorously derived until next semester, but we can motivate a set of correct rules, which will get elaborated as we go on. Finally, we will couple the fermions with photons instead of Yukawa • scalar particles, and see that this differs from the Yukawa case not only because the photon is massless, and therefore the range of the Coulomb potential is infinite, but also in being repulsive between two identical particles, and attractive only between particle - antiparticle pairs. We also see that gravity should, like the Yukawa potential, be always attractive. Read Peskin and Schroeder pp. 118–126. Let us consider the scattering of two distinguishable fermions 1This is often mentioned as a great success, with the known range of the nuclear potential predicting the existance of the pion.
    [Show full text]
  • Spontaneous Symmetry Breaking in the Higgs Mechanism
    Spontaneous symmetry breaking in the Higgs mechanism August 2012 Abstract The Higgs mechanism is very powerful: it furnishes a description of the elec- troweak theory in the Standard Model which has a convincing experimental ver- ification. But although the Higgs mechanism had been applied successfully, the conceptual background is not clear. The Higgs mechanism is often presented as spontaneous breaking of a local gauge symmetry. But a local gauge symmetry is rooted in redundancy of description: gauge transformations connect states that cannot be physically distinguished. A gauge symmetry is therefore not a sym- metry of nature, but of our description of nature. The spontaneous breaking of such a symmetry cannot be expected to have physical e↵ects since asymmetries are not reflected in the physics. If spontaneous gauge symmetry breaking cannot have physical e↵ects, this causes conceptual problems for the Higgs mechanism, if taken to be described as spontaneous gauge symmetry breaking. In a gauge invariant theory, gauge fixing is necessary to retrieve the physics from the theory. This means that also in a theory with spontaneous gauge sym- metry breaking, a gauge should be fixed. But gauge fixing itself breaks the gauge symmetry, and thereby obscures the spontaneous breaking of the symmetry. It suggests that spontaneous gauge symmetry breaking is not part of the physics, but an unphysical artifact of the redundancy in description. However, the Higgs mechanism can be formulated in a gauge independent way, without spontaneous symmetry breaking. The same outcome as in the account with spontaneous symmetry breaking is obtained. It is concluded that even though spontaneous gauge symmetry breaking cannot have physical consequences, the Higgs mechanism is not in conceptual danger.
    [Show full text]
  • Applications of Quantum Mechanics University of Cambridge Part II Mathematical Tripos
    Preprint typeset in JHEP style - HYPER VERSION Lent Term, 2017 Applications of Quantum Mechanics University of Cambridge Part II Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/aqm.html [email protected] { 1 { Recommended Books and Resources There are many good books on quantum mechanics. Here's a selection that I like: • Griffiths, Introduction to Quantum Mechanics An excellent way to ease yourself into quantum mechanics, with uniformly clear expla- nations. For this course, it covers both approximation methods and scattering. • Shankar, Principles of Quantum Mechanics • James Binney and David Skinner, The Physics of Quantum Mechanics • Weinberg, Lectures on Quantum Mechanics These are all good books, giving plenty of detail and covering more advanced topics. Shankar is expansive, Binney and Skinner clear and concise. Weinberg likes his own notation more than you will like his notation, but it's worth persevering. This course also contains topics that cannot be found in traditional quantum text- books. This is especially true for the condensed matter aspects of the course, covered in Sections 3, 4 and 5. Some good books include • Ashcroft and Mermin, Solid State Physics • Kittel, Introduction to Solid State Physics • Steve Simon, Solid State Physics Basics Ashcroft & Mermin and Kittel are the two standard introductions to condensed matter physics, both of which go substantially beyond the material covered in this course. I have a slight preference for the verbosity of Ashcroft and Mermin. The book by Steve Simon covers only the basics, but does so very well.
    [Show full text]
  • 3. Interacting Fields
    3. Interacting Fields The free field theories that we’ve discussed so far are very special: we can determine their spectrum, but nothing interesting then happens. They have particle excitations, but these particles don’t interact with each other. Here we’ll start to examine more complicated theories that include interaction terms. These will take the form of higher order terms in the Lagrangian. We’ll start by asking what kind of small perturbations we can add to the theory. For example, consider the Lagrangian for a real scalar field, 1 1 λ = @ @µφ m2φ2 n φn (3.1) L 2 µ − 2 − n! n 3 X≥ The coefficients λn are called coupling constants. What restrictions do we have on λn to ensure that the additional terms are small perturbations? You might think that we need simply make “λ 1”. But this isn’t quite right. To see why this is the case, let’s n ⌧ do some dimensional analysis. Firstly, note that the action has dimensions of angular momentum or, equivalently, the same dimensions as ~.Sincewe’veset~ =1,using the convention described in the introduction, we have [S] = 0. With S = d4x ,and L [d4x]= 4, the Lagrangian density must therefore have − R [ ]=4 (3.2) L What does this mean for the Lagrangian (3.1)? Since [@µ]=1,wecanreado↵the mass dimensions of all the factors to find, [φ]=1 , [m]=1 , [λ ]=4 n (3.3) n − So now we see why we can’t simply say we need λ 1, because this statement only n ⌧ makes sense for dimensionless quantities.
    [Show full text]
  • Yukawa Potential & Invariance Principles
    PHY304 Particle Physics – 4 Dr. C N Booth Yukawa Potential and the Propagator Term Consider the electrostatic potential about a charged point particle. This is given by ∇2φ = 0, e which has the solution φ = . This describes the potential for a force mediated by massless πε 4 0r particles, the photons. For a particle with mass, the relativistic equation E2 = p2c2 + m2c4 can be converted into a wave equation by the substitutions ∂ ∂ E → iℏ ; p → −iℏ etc. ∂t x ∂x ∂2 φ Hence, −ℏ2 =()m 24 c −∇φ ℏ 222 c . ∂t 2 Or, in the static, time independent case, this leads to m2c2 ∇2 − φ = 0 , ℏ2 (which gives ∇2φ = 0 for the massless case, as required). For a point source with spherical symmetry, the differential operator can be written as 1d dφ 1d 2 d2 m2c2 ∇φ→2r 2 ≡() r φ , so ()rφ = rφ rr2d d r rr d 2 d r 2 ℏ2 with solution −r e R φ = g2 r = ℏ where g is a constant (the coupling strength) and R mc is the range of the force. This is known as the Yukawa form of the potential, and was originally introduced to describe the nuclear interaction between protons and neutrons due to pion exchange. Using this form of potential and the Born approximation leads, after some manipulation (see the homework!), to a matrix element given by π 2ℏ2 = 4 g Mfi 2 2 2 . q + m c 1 Returning to our normal convention of setting c = 1, the terms in the denominator give and q2 + m2 this is called the propagator term .
    [Show full text]
  • The Yukawa Potential: from Quantum Mechanics to Quantum Field Theory
    The Yukawa potential: from Quantum Mechanics to Quantum Field Theory Author: Christian Y. Arce Borro Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Catalonia, Spain. Advisor: Josep Taron Roca Abstract: Quantum Mechanics and Quantum Field Theory treatment for the scattering of two fermions is compared to find out an instantaneous effective Yukawa potential describing the inter- action in the non relativistic limit. The comparison is made between the S matrix element for the scattering of 2 particles in time independent QM and the non relativistic limit of the scattering amplitude of 2 fermions due to a Yukawa interaction in QFT. I. INTRODUCTION This connection will be made introducing an S matrix element as follows [1]: The understanding of interactions between particles is essential for any theory that attempts to explain the fun- damental structure and behaviour of matter, because it's Sfi = h f j ii (3) this understanding which allows to make predictions that can be then observed (or not) in real life, and thus test Manipulating eqs. (1) and (2) in order to work out eq. a theory. (3), we can obtain the following relations between j ai Quantum Mechanics explains that particles are sources of and jφai: potential, and this potential covers some region of space that if another particle were to enter it, an atractive or ± 1 ± j i = jφai + V^ j i (4) repulsive interaction depending on the potential would a (E − H^ ± i) a then occur. a 0 1 Quantum Field Theory describes interactions from the j ±i = jφ i + V^ jφ i (5) a a ^ a perspective of virtual particle exchange between the par- (Ea − H ± i) ticles that are interacting, so instead of an instantaneous potential, there's a deeper understanding on how the Where the ±i have been introduced to avoid the sin- propagation of information is carried.
    [Show full text]
  • Yukawa Potential from QFT
    PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 12 Wednesday, January 22, 2020 Topics: Yukawa Potential from QFT. Yukawa Potential from QFT Let us compute the scattering amplitudes in Yukawa theory. A simple application of the rules we derived would be scattering of distinguishable fermions, in the non-relativistic limit. By comparing the amplitude for this process to the Born approximation formula from non-relativistic quantum mechanics, we can determine the potential V (r) created by the Yukawa interaction. If the two interacting particles are distinguishable, only the first diagram in Fig. ?? contributes. Figure 1: Scattering diagram giving rise to the Yukawa potential in non-relativistic limit. In the non-relativistic limit, we keep terms only in lowest order in the 3-momenta. That is, up to O(p2; p02; ··· ). Noting that M p we have p2 p0 = m + ≈ m (1) 2m PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 For the momenta involved in the scattering process p = (m; p); k = (m; k); p0 = (m; p0); k0 = (m; k0) (2) Using these expressions we have (p0 − p)2 = −|p0 − pj2; (3) and we have relations such as ! p ξs us(p) = m ; (4) ξs where ξs is a two-component constant spinor normalized to 0 0 (ξs )yξs = δss (5) The spinor product in the expression for the scattering amplitude 2 0 1 0 iM = (−ig) u¯(p )u(p) 0 2 2 u¯(k )u(k) (6) (p − p) − mφ becomes 0 0 0 u¯s (p0)us(p) = 2m(ξs )yξs = 2mδss (7) 0 0 0 u¯r (k0)ur(k) = 2m(ξr )yξr = 2mδrr : (8) We see that the spin of each particle is separately conserved in this non-relativistic scattering interaction.
    [Show full text]
  • Non-Relativistic Scattering
    Non-relativistic scattering Contents 1 Scattering theory 2 1.1 Scattering amplitudes . 3 1.2 The Born approximation . 5 2 Virtual Particles 5 3 The Yukawa Potential 6 Appendices 10 .A Beyond Bornx 10 1 Non-relativistic scattering 1 Scattering theory We are interested in a theory that can describe the scattering of a particle from a potential V (x). Our Hamiltonian is H = H0 + V: where H0 is the free-particle kinetic energy operator p2 H0 = : 2m In the absence of the potential V the solutions of the Hamiltonian could be written as the free-particle states satisfying H0jφi = Ejφi: These free-particle eigenstates could be written as momentum eigenstates jpi, but since that isn't the only possibility we hold off writing an explicit form for jφi for now. The full Schr¨odingerequation is (H0 + V )j i = Ej i: We define the eigenstates of H such that in the limit where the potential disappears (V !0), we have j i!jφi, where jφi and j i are states with the same energy eigenvalue. (We are able to do this since the spectra of both H and H + V are continuous.) A possible solution is1 1 j i = V j i + jφi: (1) E − H0 By multiplying by (E − H0) we can show that this agrees with the definitions above. There is, however the problem of the operator 1=(E − H0) being singular. The singular behaviour in (1) can be fixed by making E slightly complex and defining ( ) 1 ( ) j ± i = jφi + V j ± i : (2) E − H0 ± i This is the Lippmann-Schwinger equation.
    [Show full text]
  • The Strong Interaction
    The Strong Interaction What is the quantum of the strong interaction? The range is finite, ~ 1 fm. Therefore, it must be a massive boson. Yukawa theory of the strong interaction Relativistic equation for a massive particle field. Scaler (Klein-Gordon) equation: E2 − p2c2 − m2c4 = 0 ∂2Φ − 2 + 2c2∇2Φ − m2c4Φ = 0 ∂t2 m2c2 1 ∂2Φ ∇2Φ − Φ = + 0 2 c2 ∂t2 Compare with Schroedinger equation 2 ∂Ψ - ∇2Ψ = i 2m ∂t Relativistic equation for a massive particle field. 1 ∂2Φ m2c2 ∂2Φ − + ∇2Φ − Φ = 0 Steady state = 0 Φ(r,t) → φ(r) c2 ∂t2 2 ∂t2 m2c2 Add source term gδ (r − 0) ∇2φ − φ = gδ (r − 0) 2 mc 2 2 − r 2 1 ∂ ⎛ 2 ∂φ ⎞ m c g g −r/R Away from r = 0 ⎜ r ⎟ − φ = 0 ⇒ φ ∝ − e = e r2 ∂r ⎝ ∂r ⎠ 2 r r g2 Yukawa potential: φ(r) = e−r/R r Exercise: verify φ is a solution. mc g − r g2 Spherically symmetric solution φ ∝ − e = e−r/R r r 2 2 g −0r g 2 e Photons field: m = 0 φ ∝ − e = . g = r r 4πε0 Strong field: R 1.5 fm=1×10−15 m. Exercise: Predict the mass of the Yukawa particle. hc hc 1240 eV-nm R mc2 123 MeV = = 2 = = −6 = mc 2πmc 2π R 2π × 1.5 × 10 nm 1937 • µ lepton (muon) discovered in cosmic rays. • Mass of µ is about 105 MeV. • Ini@ally assumed to be Yukawa's meson but it was too penetra@ng. • Meanlife: ~ 2.2 µs this is too long for a strongly interac@ng obJect – or is it? Lattes, C.M.G.; Muirhead, H.; Occhialini, G.P.S.; Powell, C.F.; Processes Involving Charged Mesons Nature 159 (1947) 694; Motivation In recent investigations with the photographic method, it has been shown that slow charged particles of small mass, present as a component of the cosmic radiation at high altitudes, can enter nuclei and produce disintegrations with the emission of heavy particles.
    [Show full text]
  • Lecture 3: Propagators
    Lecture 3: Propagators 0 Introduction to current particle physics 1 The Yukawa potential and transition amplitudes 2 Scattering processes and phase space 3 Feynman diagrams and QED 4 The weak interaction and the CKM matrix 5 CP violation and the B-factories 6 Neutrino masses and oscillations 7 Quantum chromodynamics (QCD) 8 Deep inelastic scattering, structure functions and scaling violations 9 Electroweak unification: the Standard Model and the W and Z boson 10 Electroweak symmetry breaking in the SM, Higgs boson 11 LHC experiments 1 Recap of Lecture 2 : A massive spinless particle being exchanged as part of a force field can be described by a quantum mechanical wavefunction that dies faster than 1/r Yukawa potential g0 −mr (spherical symmetric ψ(r) = e static case) 4πr m=1/R = mass of particle carrying the force R = range g0 = fundamental strength (coupling) of the force € Mass and range are related: the larger the mass exchanged, the shorter the range … and if m = 0, R = ∞ (though with 1/r2 decay of the force strength) Today: calculate scattering amplitude for particles incident on a potential, taking Yukawa potential as an example 2 (From the end of Lecture 2 …) Yukawa Potential In case of photon, electric potential, m=0 and R=∞ For m>0, force has range R: beyond R, V tends to zero quickly because of exponential factor g ψ(r) = e−mr 4πr The larger the mass exchanged, the shorter the range … … and now we have a model for force strength versus distance € 3 Scattering of Plane Waves Much of particle physics is about calculating how likely
    [Show full text]
  • The Physical Origin of the Nucleon-Nucleon Attraction by Pion Exchange Nicolae Mandache, Dragos Palade
    The physical origin of the nucleon-nucleon attraction by pion exchange Nicolae Mandache, Dragos Palade To cite this version: Nicolae Mandache, Dragos Palade. The physical origin of the nucleon-nucleon attraction by pion exchange. 2017. hal-01655838 HAL Id: hal-01655838 https://hal.archives-ouvertes.fr/hal-01655838 Preprint submitted on 5 Dec 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The physical origin of the nucleon - nucleon attraction by pion exchange Nicolae Bogdan Mandache and Dragos Iustin Palade National Institute for Laser, Plasma and Radiation Physics, Plasma Physics and Nuclear Fusion L aboratory, Bucharest - Magurele, Romania E - mail: [email protected] Abstrac t A formula is derived for the central nucleon - nucleon potential, based on an analysis of t he physical origin of the nucleon - nuc l eon attraction by pion exchange . T he decrease of the dynamical mass of the interaction fiel d, exchange d pion in this case, is the principal mechanism respons ible for the nuclear attraction in a similar way the decrease of the kinetic energy of the exchange electron in the diatomic molecule is directly responsible for the covalent molecular attra ction.
    [Show full text]