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Elementarteilchenphysik

Elementarteilchenphysik. Antonio Ereditato LHEP University of Bern. Lesson on: Interactions and fields (2) Exercise: Rutherford scattering. Classical and quantum interactions.

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Elementarteilchenphysik

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  1. Elementarteilchenphysik Antonio Ereditato LHEP University of Bern Lesson on: Interactions and fields (2) Exercise: Rutherford scattering A.Ereditato SS 2008

  2. Classical and quantum interactions Classical electromagnetism: distance interaction among two particles. A field is established due to the charge of one particle, the other interacts with the field. The field covers the whole space around the field source. A.Ereditato SS 2008

  3. The electron and the proton feel each other by exchanging a photon, carrying an energy E. This is allowed by conservation principles only if the interaction lasts a time t for which: The quantum is then said VIRTUAL. Note that both interpretations are acceptable: we do not directly detect neither the field nor the quantum but only the effects of the FORCE. When dealing with elementary particles we follow the “particle” exchange interpretation Alternative figure: the two particles feel the interaction through exchange of a “mediator” carrying momentum from one particle to the other. by the way, how to explain attractive forces ? A.Ereditato SS 2008

  4. Relativistic Klein-Gordon equation: For m = 0 reduces to the wave propagation equation, with either the wave function of the photon or the potential U(r). For a static potential we obtain: The solution is: with The analogous equation for EM (Laplace): gives Yukawa Theory (1935) This implies that g can be called the “nuclear charge” For a typical R=10-13 cm one obtains mc2 ~ 100 MeV The  (140 MeV) was then assumed to be the mediator of the strong interaction at short distances. Today we know that the actual mediator is the massles gluon (boson). A.Ereditato SS 2008

  5. The propagator A particle is deflected by a potential being scattered with a momentum transfer q (e.g. Rutherford scattering) A scattering amplitude f(q) corresponds to the potential U(r) (Fourier transform) For a central potential: U(r) = U(r) one can integrate by setting: Replacing the Yukawa potential U(r) we obtain a relation describing the potential in momentum coordinates: i.e. scattering of a particle of coupling g0by the static potential generated by a massive source of strength g. In the general case also energy E is transferred with the momentum p = q (4-momentum transfer): q2 = p2 - E2 (relativistically invariant) A.Ereditato SS 2008

  6. Therefore, the amplitude (matrix element) for a scattering process is the product of two coupling constants (of the boson with scattered and scattering particles) multiplied by a propagator term: The rate W of a given reaction (collision, decay) is given by |f(q2)|2 multiplied by the phase space (Fermi Golden rule): (example given in the exercise: Rutherford scattering Formula) A.Ereditato SS 2008

  7. e+ e- elastic scattering propagator space time Interactions in Quantum Field Theory: Feynman diagrams R. Feynman Nobel Prize 1965 Q, E and p are conserved at the vertices Lines joining vertices are virtual particles A.Ereditato SS 2008

  8. e- e-  e- e-  e- e-  e-   Electromagnetic interactions: QED The EM coupling constant is: Coulomb scattering between two electrons: amplitude proportional to x  The virtual photon has a “mass”: m2 = -q2 (propagator 1/q2) Matrix element proportional to /q2 Cross section proportional to 2/q4 Leading order processes (in ) Photon Bremsstrahlung: electron emitting a real photon when accelerated in the Coulomb field of a nucleus. The electron in the second line is virtual (off mass shell) A virtual photon is exchanged with the nucleus to conserve the momentum Cross section proportional to 3 A.Ereditato SS 2008

  9.   B B e- e-    B  e-     e- B  e-  Self-energy and electron magnetic moment At the leading order term: eis equal to the Bohr magneton (Dirac) There are, however, higher order corrections For next-to-leading graphs, the (bare) electron charge still stays with the electron and the field will interact with it, but part the electron mass-energy goes into photons or virtual electron loops. The e/mratio slightly increases. If we define eequal to gBs (with s the electron spin ±1/2) the difference between the actual magnetic moment, including higher order contributions, and the Bohr magneton is expressed by the deviation of g from 2: Astonishing agreement with the experimental results! Indication of disagreement at the level of <1/108 A.Ereditato SS 2008

  10. Effective electron charge, renormalization in QED “Bare” electron (low energy) different from “dressed” electron (high energy). Infinite terms contribute to the bare electron charge and mass (logarithmically divergent). Renormalization: replace the un-measurable bare quantities m0 and e0 (assuming no self energy contributions) with the experimentally measured values for m and e. Consequence from the renormalization:  is not a constant but “runs” with the energy: = 1/137 (low energy) 1/128 at ~100 GeV Screening effect Renormalization can only be applied to “gauge invariant theories” (such as QED). Examples are: in electrostatics the potential can be re-defined without modifying the physics; in Q.M. the phase of a wave-function can be changed without affecting the observables. We will come back on this when talking about electroweak and QCD theories. A.Ereditato SS 2008

  11. Strong interactions: color We can compare the (uds) EM life time (~10-19 s) with that of its strong decay mode (~10-23 s). The decay rate is then proportional to the square of the coupling constants: The carrier of the strong force is a massless boson: the gluon. The gluon has 6 possible “strong charge” (color) values. Unlike the photon the gluon carries color charge (self interaction, triple and quadruple gluon vertices). This is a key feature of QCD: it is a non-Abelian Theory Color symmetry is exact: the force is independent on the color charge. However, since only a unique proton, neutron, or pion state exist, hadrons must be “colorless”. In accordance with the theory of visible light, this can be done by combining quarks of the three fundamental colors (b, g, r) or by adding up quarks with a given color and its “anticolor” (r, anti-r, b, anti-b or g,anti-g). MESONS BARYONS A.Ereditato SS 2008

  12. Q Q s s g g g Q Q s u u { { u u p  u d { d  u The diagrams on the right correspond to a single-gluon exchange, but since s ~1 one can have with the same probability multi-gluon graphs (terms in s2 , s3,…) For close collisions (large q2) s <<1 and single-gluon dominates, while for distant collisions s > 1 and the the theory cannot be computed. This is related to the quark confinement. Strong interactions can then be described by a potential with two terms, the first active for close collisions, the second for distant. k is of the order of 1 GeV/fm: equivalent to a force of ~15 ton between the two quarks pulling apart. A.Ereditato SS 2008

  13. The string energykrincreases with increasing the distance between quarks. At a certain point the transformation of this energy in a QQ pair becomes more “economical”. If we create quarks, as in e+e-  QQ processes, the produced high-energy quarks, fragment into collimated jets of hadrons. A.Ereditato SS 2008

  14. Jet production in electron-positron collisions at high energy at the CERN LEP accelerator A.Ereditato SS 2008

  15. A.Ereditato SS 2008

  16. In QED the bare electron is surrounded by a cloud of virtual e+e- with the e+ closer to the electron, determining a screening effect. Therefore, if we want to measure the charge of the electron with a test particle, the measured charge will be larger by going closer to the electron, for high-energy interactions. Contrary to QED, the possibility of self-coupling of the gluon (non-Abelian Theory) allows to spread the color of a given gluon to other gluons nearby: a red charge is generally surrounded by other red charges. Then, a “close” inspection around a red quark will measure less red charge (anti-screening effect). One obtains the so-called asymptotic freedom: two red quarks feel smaller and smaller force by getting closer, and viceversa, they experience confinement. Anti-screening effect A.Ereditato SS 2008

  17. Charged current - g e-  g  W - g Z0 g g e- e- g Neutral current Neutron decay Weak interactions All quarks and leptons experience weak interactions By comparing the weak and EM decay modes of the and of the The weak “charge” is g. Weak interactions can be practically detected only if the more probable strong and EM reactions are forbidden by some conservation rules: This implies the presence of neutrinos (not feeling strong and EM interactions) and the decay of quarks with flavor changing. Remember that strong and EM reactions conserve the quark flavor number: S = 0, C = 0, B = 0, T = 0. “Text book” weak reactions: direct and inverse -decay A.Ereditato SS 2008

  18.  electron    - W- Z0 e X ’ e X Neutral Current Gargamelle bubble chamber at CERN (1973) (discovery of neutral currents) Charged Current A.Ereditato SS 2008

  19. The “weakness” of weak interaction: large mass of the mediating bosons and respectively 80.403  0.029 and 91.1876  0.0021GeV Recalling the amplitude transition form, we have in this case: For small q2 the amplitude is independent of q2 (pointlike) and coincide with the original interpretation of the  decay given by Fermi in 1934, with a coupling constant u d g n p e-   e- g The unifying EW theory by Glashow, Weinberg and Salam (1968) assumes g = e and predicts the value of the weak boson masses to good accuracy. The bosons were actually discovered in 1983 by Rubbia and coworkers at CERN. A.Ereditato SS 2008

  20. Cross-section where a beam of particles awith density a hits a target of thickness dx containing nb particles b per unit volume. c and d are the reaction products Assume a reaction Incident flux: particles/unit area x unit time: Each target particle has a geometrical cross-section:  Probability for any beam particle to hit a target particle:  nbdx Number of interactions per unit time:  nbdx Reaction rate per target particle: W =  The cross-section is measured in barn: 1 b = 10-24 cm2 From the Fermi Golden’s rule one can calculate the cross-section: A.Ereditato SS 2008

  21. 1 max 0.5  The mean life time of a state is: Recalling: M0 E the width of a state, i.e. the energy spread of the decaying states is given by: If a set of particles A decays at a rate proportional to the number of particles at time t, one has: with solution given by (exponential decay) A particle that can decay in several modes has a total width: Resonance: states with defined width and life time that can be formed by interacting particles. The shape of the resonance is determined by the exponential decay. Relativistic (Lorentz invariant) Breit-Wigner formula A.Ereditato SS 2008

  22. e Z0 e Measurement of the Z0 width performed at the LEP electron storage ring at CERN (1991): there are only three ‘light ‘ neutrinos A.Ereditato SS 2008

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