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Bernd Surrow Bernd Surrow Introduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 1 Introduc tion to Nuclear and Particle PhysicsIntroduc tion to Nuclear and Particle Physics8.7018.701Lecture 2Lecture 2SM interactions / Relativistic KinematicsSM interactions / Relativistic Kinematics 2 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Standard Model (SM) Standard Model (SM) InteractionsInteractionsSM ForcesSM ForcesQuantum Electrodynamics (QED) Quantum Electrodynamics (QED) inclincl. vacuum polarization and. vacuum polarization andbehaviour of coupling constantbehaviour of coupling constantQuantum Quantum Chromodynamics Chromodynamics (QCD) (QCD) inclincl. vacuum polarization and. vacuum polarization andbehaviour of QCD coupling constantbehaviour of QCD coupling constantWeak InteractionsWeak InteractionsLeptonsLeptonsQuarksQuarksRelativistic KinematicsRelativistic KinematicsReview of basic conceptsReview of basic conceptsCentre-of-mass frame Centre-of-mass frame vsvs. laboratory frame. laboratory frameMandelstam variablesMandelstam variablesDecay lengthDecay lengthCollider Collider kinematicskinematicsSummarySummary OutlineOutline 3 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 SM forces•As far as we know, there are just four fundamental forces in nature:•The Standard Model (SM) refers to three of these fundamental forces: 10-610-21Typical couplingTypical coupling10-11 mb10-3 mb10 mbTypical cross-Typical cross-sectionsection10-12 s10-20 - 10-16 s10-23 sTypical life timeTypical life time10-3 ≈ 1/mw∞1F ≈ 1/mπColor confinementrangeRangeRangeleptons, quarks, W±, Zcharged particlesquarks, gluonsParticlesParticlesexperiencing itexperiencing itflavorElectric chargeColor chargeActs onActs onW±(m≈80GeV/c2), Z(m≈90GeV/c2)photon (m=0)gluon g (m=0)MediatorMediatorWeakWeakElectromagneticElectromagneticStrongStrong Standard Model InteractionsStandard Model Interactions 4 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 •All electromagnetic phenomena are ultimately reducible to the following QED elementarycoupling: Two elements: Charged particle, photonMeaning: Charged particle e enters, emits (or absorbs) a photon γ and exits.•Example of a complete process: Moeller scattering Meaning: Interaction of two electrons which is mediated by a photonClassical case: Coulomb repulsion Standard Model InteractionsStandard Model InteractionsQED: Fundamental Feynman graph TimeU(1) gauge theory 1 charge with 1 boson (photon) Coupling: Fine structure constant α Time Note:Amplitude ofprocess i 5 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Rule: Particles which are running “backward in time” are to beinterpreted as the corresponding anti-particle!Bhabha scattering is related to Moeller scattering by a general principle which is known as crossingsymmetry:A + B  C + DThese processes are dynamically allowed,but not necessarily kinematically!Bhabha scattering and Moeller scatteringare related by crossing symmetry!Example: If A weighs less than B, C and D,then this process is kinematically notallowed! Standard Model InteractionsStandard Model Interactions•Example of a complete process: Bhabha scattering Time Total amplitude: Rule: Any particle can be “crossed” over the other side at the equation, provided it turns into its anti-particle:A  B + C + DA + C  B + DC + D  A + B 6 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 •Example of a complete process: Pair annihilation •Example of a complete process: Pair production •Example of a complete process: Compton scattering Standard Model InteractionsStandard Model Interactions Time Time Time 7 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 •Remarks on Feynman diagramsFeynman diagrams are purely symbolic in place of of mathematical expression of a scattering amplitudeFeynman diagrams do not represent particle trajectories!Example: “Components” of a Feynman diagram using M1 of Moeller scattering Each Feynman diagram represents a matrix, Mi, to account for a particular process.Feynman rules enforce conservation of energy and momentum at each vertex, and hence for the diagram as a whole.Procedure:Draw all diagrams, i.e. amplitudes, that have the appropriate external lines.Sum of all diagrams, i.e. amplitudes, with the given external lines represents the actual physical process.Problem: There are infinite many Feynman diagrams! Standard Model InteractionsStandard Model Interactions Time External lines:External lines:Real (on-shell)particlesSpinor ofelectron Internal lines: Vertex factor:Strength of coupling: Here: Twovertices 8 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Calculations get very complicated!BUT: Each vertex contributes a factorα: QED fine structure constant (general: QED coupling constant)Higher order terms are suppressed since α << 1 (perturbation theory)!What matters “mainly” is the leading order contribution!However: There are many examples where measurements are so precise that higher order termshave to be taken into account!  TEST OF QEDExample: Measurement of anomalous magnetic moment of muon! Standard Model InteractionsStandard Model Interactions•Example: Moeller scattering at higher ordersBesides above two leading order Feynman diagrams for Moeller scattering there are other higher-order diagrams: higher orderdiagrams…! 4 vertices: 9 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 In QED, the vacuum behaves like a dielectric, creating e-/e+ pairs out of the vacuum: Standard Model InteractionsStandard Model Interactions•QED vacuum polarization Bare electroncharge isscreenedEffective charge<< bare charge Or view it like this: e-low-energyprobe Distance from e-chargeCoupling electron chargehigh-energyprobelow-energyprobe 10 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 In Quantum Chromodynamics: SU(3) gauge theory  3 charges with 8 bosons (gluons)Color plays the role of charge and the fundamental process is: quark  quark + gluonFundamental vertex:Note: Color is always conserved! gluon is carrying away thedifference: Here b and anti-red! Types of gluons: Standard Model InteractionsStandard Model Interactions•QCD: Quantum Chromodynamics (1) No net color…like a photon!color singlet color octet SU(3)Vertex factor:Strong couplingconstant αs! 11 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Confinement requires that all naturally occurring particles be color singlets!|9> is a color singlet state. If it exists as mediator, it should also occur as a free particleAs such a mediator: Exchange between color singlet particles, e.g. a proton and neutron. Therefore: Long-range force with strong coupling.HoweverHowever: Strong force is of very short range. Therefore: Experiments tell us that there are only 8 gluons ⇒ color octets: SU(3)Because gluons themselves carry color (in contrast to the photon which is electrically neutral): gluons couple directlyto one another: QCD calculationsQCD calculations: Apply QCD Feynman rules to calculate various processes!How does the QCD coupling constant behave compared to the QED coupling constant?3 gluon vertices 4 gluon vertices Standard Model InteractionsStandard Model Interactions•QCD: Quantum Chromodynamics (2) 12 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Clear difference between QED and QCD Self-coupling of gluons which is absent in case of QED.QCD vacuum polarization diagrams:But also:NoteNote: Behavior is opposite for αsQuark polarization: αs is large at short distances!Gluon polarization: αs is small at short distances!Quark polarizationQuark polarization Standard Model InteractionsStandard Model Interactions•QCD vacuum polarization (1) Gluon polarizationGluon polarizationA priori notclear who wins! 13 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 The winner depends on the relative number at flavors (f) and colors (n):Critical parameter:Standard Model:Consequence: QCD coupling constant decreases at short distances!Another “last picture”: Asymptotic freedomAsymptotic freedomQuarks are quasi-free Small coupling at short distances(high-energy)! Standard Model InteractionsStandard Model Interactions•QCD vacuum polarization (2) Discovery of asymptotic freedom in thetheory of strong interaction (QuantumChromo Dynamics): Nobel prize in physics2004But also: Distance from barequark color chargeCoupling color chargeHigh-energyprobe!Confinementbarrier! 14 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 SU(2) gauge theory: 3 mediators: Z0 W±Electroweak unification: SU(2) X U(1): 4 mediators: Z0, W± and γQuarks and leptons take part in weak interactions2 types of interactions:a) charged current: Involving W±b) neutral current: Involving Z0 and γLeptons:charged vertex: neutral vertex:Example: Standard Model InteractionsStandard Model Interactions•Weak interactions (1) First “picture” of neutral weakprocess discovered at CERN in1973 Vertex factor:Weak coupling constants αw and αZ! Vertex factordepends on l and ν! 15 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Note: The Glashow-Weinberg-Salam (GWS) model (SU(2) X U(1)) includesneutral weak processes as an essential ingredient. Their existence wasconfirmed experimentally at CERN in 1973! … established 3 families in nature…:Production at Z0 at LEP: Standard Model InteractionsStandard Model Interactions •Weak interactions (2) Time 16 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 eeProduction of Z0 bosons at LEP: Standard Model InteractionsStandard Model Interactions•Weak interactions (3) µµ Jet-Jet 17 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QuarksFundamental vertex: W± Note: Flavor is not conserved in weak interactions! Since the quark flavor changes at aweak vertex (Recall: Color change of quarks at QCD vertex!), weak interactions aresometimes called flavordynamics!Note: A quark of charge -1/3(d,s,b) converts into the corresponding quark withcharge +2/3 (u,c,t) with the emission at W- (vice versa for W+)Example: Standard Model InteractionsStandard Model Interactions•Weak interactions (3) Time 18 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Example: Beta-decay of the neutron: Example:Fundamental vertex: Z0 Standard Model InteractionsStandard Model Interactions Note: Quark flavor is notchanged! Vertex factor for (u,c,t) and(d,s,b) different (GSWprediction!) 19 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007  Flavor changing reactions•In the spirit of the charged weak coupling with respect to leptons which yieldonly changes within each family, i.e. :••However:However: The following observed decays involve the conversion of a strange quark s into anup-quark:•One might assume that this also holds for quarks: Standard Model InteractionsStandard Model Interactions •Flavor changes do occuracross quark families! 20 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 •Therefore:The flavor eigenstate of u is not the partner to the flavor eigenstate of d, but to alinear combination of d, s and b here written as d’: Note:The matrix is called after their “inventors”: 3 X 3 Kobayashi – Maskawa matrix!Earlier scheme (2 families) by Cabibbo (1963): Linear-combination: Standard Model InteractionsStandard Model Interactions Cabibbo angle: 21 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Self coupling of weak-bosoms and photon: Standard Model InteractionsStandard Model Interactions Experimental data: K. Hagiwara et al., Phys. Rev. D66 (2002) 010001. 22 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 ••Last remark:Last remark:Grand Unified Theories(GUT): One ForceE Standard Model InteractionsStandard Model Interactions••Summary: SM interactionsSummary: SM interactions Flavor change! No Flavor change!αSαW/Zα1015GeV 23 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Standard Model (SM) InteractionsStandard Model (SM) InteractionsSM ForcesSM ForcesQuantum Electrodynamics (QED) incl. vacuum polarization andQuantum Electrodynamics (QED) incl. vacuum polarization andbehaviour of coupling constantbehaviour of coupling constantQuantum Chromodynamics (QCD) incl. vacuum polarization andQuantum Chromodynamics (QCD) incl. vacuum polarization andbehaviour of QCD coupling constantbehaviour of QCD coupling constantWeak InteractionsWeak InteractionsLeptonsLeptonsQuarksQuarksRelativistic KinematicsRelativistic KinematicsReview of basic conceptsReview of basic conceptsCentre-of-mass frame vs. laboratory frameCentre-of-mass frame vs. laboratory frameMandelstam variablesMandelstam variablesDecay lengthDecay lengthCollider kinematicsCollider kinematicsSummarySummary OutlineOutline 24 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 (1) Lorentz TransformationsGiven two inertial frames S and S’, with S’ moving at uniform speed v with respect to S: Events are related in space-time as follows (Lorentz TransformationsLorentz Transformations):xyzS Relativistic KinematicsRelativistic KinematicsRelativistic Kinematics•Review of basic concepts With:x’z’S’ y’ 25 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 a. Relativity at simultaneity:If two events A and B occur at the same time in S, but at different locations, thenthey do not occur at the same time in S’:b. Lorentz contraction:A moving object is shortened by a factor:c. Time dilation:Moving clocks run slow:(2) Consequences Relativistic KinematicsRelativistic Kinematics Decay length of decaying particle: Example: Muon decay in earth athmosphere 26 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 If u’=c then u=c! Relativistic KinematicsRelativistic Kinematicsd. Velocity addition:Particle moves with speed u’ with respect to S’. What is the speed, u, with respectto S? Note: S’ moves with speed v with respect to S: (3) Four vectors Einstein conventionEinstein convention:Sum over double occurring indices! 27 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 This expression, I, is a relativistic invariant quantity! Covariant four vector:Contra variant four vector:With Metric Tensor: Relativistic KinematicsRelativistic Kinematics(4) Invariant quantitiesEasy to show that the following expression is invariant in any inertial system: Short: 28 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Product of any two four vectors:timelikespacelike(5) Energy and Momentumlightlike Relativistic KinematicsRelativistic Kinematics With: Invariant mass squared!Invariant mass squared! 29 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Relativistic KinematicsRelativistic Kinematics•Note on conserved and invariant quantitiesA conserved quantity remains the same, in a particularframe, before and after an eventAn invariant quantity is the same in all inertial referenceframesEnergy/momentum is conserved, but not invariantMass is invariant, but not conserved 30 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Relativistic KinematicsRelativistic Kinematics•Centre-of-mass frame / lab frameLorentz transformation: Energy and momentum viewed from a frameEnergy and momentum viewed from a framevoving voving with velocity:with velocity: Consider the following process: Mandelstam variables: 31 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Relativistic kinematicsRelativistic kinematics The Mandelstam variables s, t and u are related to three distincttopological channels with which Feynman diagrams might bedrawn to represent the interaction: Let’s work out an example using the Mandelstamvariable s:Lab frameCM frame stu CM frameLab frame 32 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Relativistic kinematicsRelativistic kinematicsRelate CM system and Lab system: Calculate CM energy •Collider configuration:•Fixed-target configuration: For HERA:Same CM energy as HERA for: The CM system is yourfriend! 33 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 1. Energy and momentum2. Angular momentum3. Electric charge4. Color charge5. Baryon number: A = 1 for Baryons; A = -1 for Anti-Baryons; A = 0 for non Baryons.6. Lepton number:Particles at each generation (Leptons) are conserved:7. Flavor is conserved in strong and electromag. interactions, but not in weakinteractions!Conservation laws: Analysis of particle reactions: SummarySummary ParticleAnti-Particle Example: