Feynman Diagrams of the Standard Model Sedigheh Jowzaee

Transcription

1 tglied der Helmholtz-Gemeinschaft Feynman Diagrams of the Standard Model Sedigheh Jowzaee PhD Seminar, 5 July 01

2 Outlook Introduction to the standard model Basic information Feynman diagram Feynman rules Feynman element factors Feynman amlitude Examles

3 The Standard Model The Standard Model of articles is a theory concerning the electromagnetic, weak and strong nuclear interactions Collaborative effort of scientists around the world Glashow's electroweak theory in 1960, Weinberg and Salam effort for Higgs mechanism in 1967 Formulated in the 1970s Incomlete theory Does not incororate the full theory of gravitation or redict the accelerating exansion of the universe Does not contain any viable dark matter article Does not account neutrino oscillations and their non-zero masses

4 The Standard Model Generations of matter Gauge bosons Higgs bosons The standard model has 61 elementary articles The common material of the resent universe is the stable articles, e, u, d

5 Gauge Bosons Force carriers that mediate the strong, weak and electromagnetic fundamental interactions Photons: mediate the electromagnetic force between charged articles W, Z: mediate the weak interactions between articles of different flavors quarks & letons Gluons: mediate the strong interactions between color charged quarks Forces are resulting from matter articles exchanging force mediating articles Feynman diagram calculations are a grahical reresentation of the erturbation theory aroximation, invoke force mediating articles 5

6 Feynman diagram Schematic reresentation of the behavior of subatomic articles interactions Nobel rize-winning American hysicist Richard Feynman, 198 A Feynman diagram is a reresentation of quantum field theory rocesses in terms of article aths Feynman gave a rescrition for calculation the transition amlitude or matrix elements from a field theory Lagrangian M is the Feynman invariant amlitude Transition amlitudes matrix elements must be summed over indistinguishable initial and final states and different order of erturbation theory 6

7 What do we study? Reactions A+BC+D+ Exerimental observables: Cross sections, Decay width, scattering angles etc Calculation of or based on Fermi s Golden rule: _ decay rates 1++ +n _ cross sections n Calculation of observable quantity consists of two stes: 1. Determination of M we use the method of Feynman diagrams. Integration over the Lorentz invariant hase sace δ E d E d E d M = dγ n n n... π π... π π E δ E d E d E d mm = M dσ n n n... π π... π π

8 Feynman rules different tyes of lines: Incoming lines: extend from the ast to a vertex and reresents an initial state Outgoing lines: extend from a vertex to the future and reresent the final state Incoming and outgoing lines carry an energy, momentum and sin Internal lines connect vertices a oint where lines connect to another lines is an interaction vertex Quantum numbers are conserved in each vertex e.g. electric charge, leton number, energy, momentum Particle going forwards in time, antiarticle backward in time Intermediate articles are virtual and are called roagators Virtual Particles do not conserve E, for s: E - 0 At each vertex there is a couling constant In all cases only standard model vertices allowed sace time sace time They are urely symbolic! Horizontal dimension is time but the other dimension DOES NOT reresent article trajectories! 8

9 Feynman interactions from the standard model Because gluons carry color charge, there are three-gluon and four-gluon vertices as well as quark-quarkgluon vertices. 9

10 We construct all ossible diagrams with fixed outer articles Examle: for scattering of scalar articles: 1/ Tree diagram Since each vertex corresonds to one interaction Lagrangian term in the S matrix, diagrams with loos corresond to higher orders of erturbation theory We classify diagrams by the order of the couling constant this is just erturbation Theory!! 1/ 1/ 1/ 1/ 1/ 1 st order erturbation nd order erturbation 1/ th order erturbation For a given order of the couling constant there can be many diagrams Must add/subtract diagram together to get the total amlitude, total amlitude must reflect the symmetry of the rocess e + e - identical bosons in final state, amlitude symmetric under exchange of, Moller scattering: e i1- e i- e f1- e f- identical fermions in initial and final state, amlitude anti-symmetric under exchange of i1,i and f1,f : M=M 1 -M 10

11 Feynman diagram element factors Associate factors with elements of the Feynman diagram to write down the amlitude The vertex factor Couling constant is just the i times the interaction term in the momentum sace Lagrangian with all fields removed The internal line factor roagator is i times the inverse of kinetic oerator by free equation of motion in the momentum sace Sin 0 : scalar field Higgs, ions, Sin ½: Dirac field electrons, quarks, letons scalar roagator multilies by the olarization sum Sin 1: Vector field Massive W, Z weak bosons Massless hotons External lines are reresented by the aroriate olarization vector or sinor e.g. Fermions ingoing,outgoing u, ū ; antifermion v, v ; hoton ; scalar 1, 1 11

12 Feynman rules to extract M 1- Label all incoming/outgoing -momenta 1,,, n ; Label internal -momenta q 1,q,q n. - Write Couling constant for each vertex - Write Proagator factor for each internal line - write E/ conservation for each vertex k 1 +k +k ; k s are the - momenta at the vertex +/ if incoming/outgoing 5- Integration over internal momenta: add 1/ d q for each internal line and integrate over all internal momenta 6- Cancel the overall Delta function that is left: 1 + n What remains is: -im 1

13 First order rocess Simle examle: -theory We have just one scalar field and one vertex We will work only to the lowest order momentum sace sace The tree-level contribution to the scalar-scalar scattering amlitude in this -theory 1

14 Second order rocesses in QED There is only one tree-level diagram 1

15 tglied der Helmholtz-Gemeinschaft Thank you