Particles and interactions

1. Particles and interactions Bytheend of thistopicyoushould be able to: statethemeaning of thetermelementaryparticle; identifythethreeclasses of elementary, the quarks, theleptons and theexchangeparticles; understandthemeaning of quantum numbers; statethemeaning of thetermantiparticle; classifyparticlesaccordingtotheir spin; understandthePauliexclusionprinciple and howitisapplied; understand and applytheHeisenberguncertaintyprincipleforenergy and time; appreciatethemeaning of theterm virtual particle, describe the fundamental interactions; statethemeaning of theterminteractionvertex; understandwhatismeantbyFeynmandiagrams; drawFeynmandiagrams in ordertorepresentvariousphysicalprocesses; applytheHeisenberguncertaintyprinciple in orderto derive therange of aninteraction.

2. In 1896, the British physicist J. J. Thompson discovered the electron, performing experiments with cathode rays. In 1911, Ernest Rutherford discovered the atomic nucleus. In 1917, (in experiments reported in 1919) Rutherford proved that the hydrogen nucleus is present in other nuclei, a result usually described as the discovery of the proton. The neutron was not discovered until 1932 when James Chadwick used scattering data to calculate the mass of this neutral particle. Since the time of Rutherford it had been known that the atomic mass number A of nuclei is a bit more than twice the atomic number Z for most atoms and that essentially all the mass of the atom is concentrated in the relatively tiny nucleus. As of about 1930 it was presumed that the fundamental particles were protons and electrons, but that required that somehow a number of electrons were bound in the nucleus to partially cancel the charge of A protons.

3. Elementaryparticles Anelementaryparticleiscalledelementaryifitisnotmadeout of anysmallercomponentparticles.

5. Particle zoo

6. Quarks There are sixtypes(or “flavours”) of quarks. They are denotedby u, d, s, c, b and t, and are called up, down, strange, charmed, bottom and top, respectively. All of thesehaveelectriccharge. The up quark isthelightest and the top quark istheheaviest. Thereissolid experimental evidencefortheexistence of allsixflavours of quarks. A quark can combine withanantiquarktoform a meson. Three quarks can combine toform a baryon.

7. Leptons There are six of these: theelectron and its neutrino, the muon and its neutrino, and the tau and its neutrino. They are denotedby e-, νe, μ-, νμ, τ-, ντ. The muon isheavierthantheelectron, and the tau isheavierthanthe muon. Thethree neutrinos were once thoughttobemassless, butthereisnowconclusiveevidencethat in facttheyhave a verysmallmass.

8. Exchange particles Thisclass of elementaryparticlescontainsthephoton (denotedbyγ). Thephotonisintimatelyrelatedtotheelectomagneticinteraction. There are alsotheparticles W+, W- and Z0, calledthe W and Z bosons. Theseparticles are intimatelyrelatedtotheweakinteraction. Thenthere are eightparticlescalledgluonsthat are relatedtothestronginteraction. Finally, thereisthegraviton, whichisrelatedtothegravitationalforceorinteraction.

9. Higgsboson

10. Standard Model

11. Antiparticles In additiontotheelementaryparticles, wehavetheantiparticles of all of theabove. Toeveryparticletherecorrespondsanantiparticle of thesamemass as theparticlebut of oppositeelectriccharge (and oppositeallother quantum numbers). Theexistence of antiparticleswaspredictedtheoreticallyby Paul Dirac in 1928. Thefirstantiparticletobediscoveredexperimentallywasthepositronin 1932 by Carl Anderson.

12. Quantum numbers Quantum ‘numbers’ are numbers (orproperties) usedtocharacterizeparticles. Thereisone quantum numberyouknowalready—theelectriccharge. Some (butnotall) quantum numbers are conserved in interactions. The quantum numberforelectricchargeisalwaysconserved. Other quantum numbers: flavor, colour, strangeness, baryonnumber and generationleptonnumber.

13. Spin In classicalmechanics, a body of massmmovingalong a circle of radiurrwithspeedv has a propertycalled angular momentum. Thisisdefinedtobe L = mvr. Thisquantity has units of Js.Ifthebodyspinsarounditsown axis, it has additional angular momentum. Particlesappeartohave a similar property, measuredalso in units of Js, and thispropertywascalled spin byanalogywith a spinning bodymechanics. But a particle’s spin isnotthesamething as the angular momentum of a spinning body. Forelementaryparticles, spin is a consquence of Einstein’stheory of relativity. The spinning bodyisjust a usefulanalogy. Allknownparticleshave a spin thatis a multiple of a basicunit. Unit of spin = h/2π h = 6.62 x 10-34Js

14. Spin Alltheknownparticleshave a spin thatiseitheran integral multiple of thebasicunitor a half-integral multiple. Particles are calledbosonsiftheyhavean integral spin, and they are calledfermionsiftheyhave a half-integral spin.

15. ThePauliexclusionprinciple Itisimpossiblefortwoidenticalfermions (particleswithhalf-integral spin) tooccupythesame quantum stateiftheyhavethesame quantum numbers. Thisiswhytheinnershell of anyatom can contain at mosttwoelectrons. Electrons are fermions and so thePauliexclusionprincipleappliestothem. In theinnershelltheone quantum numberthat can distinguishtwoelectronsisthe spin. Sincethe spin of theelectronis ½, there are justtwo quantum statesavailable: one in whichthe spin is “up” and another in whichitis “down”.

16. Heisenberguncertaintyprinciple In 1928 theGermanphysicist Werner Heisenbergdiscoveredone of the fundamental principles of quantum mechanics. Theuncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as position x and momentum p, can be known simultaneously. The more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.

17. Thissaysthat, theshorterthe time intervalwithinwhichthemeasurementismade, thegreatertheuncertainty in themeasuredvalue of theenergy. Tohave a verysmalluncertainty in energywouldrequire a verylong time forthemeasurement of energy. Theversion of theprinciplethatwillconcernusisthatwhichappliestosimultaneousmeasurements of energy and time. Measurements of theenergy of a particleor of anenergylevel are subjecttoanuncertainty. Thisuncertaintyisnottheresult of randomorsystematicerrors. Themeasurement of theenergy of a particlemustbecompletedwithin a certaininterval of time thatwemaycallΔt. Heisenbergprovedthattheuncertainty in themeasurement of theenergyΔE isrelatedtoΔt through ΔE Δt ≥ h/4π

18. Thereis, however, a subtler and more usefulinterpretation of theenergy-time Heisenberguncertaintyprinciple. Weknowthat total energyisalwaysconserved. Butsuppose, for a moment, that in a certainprocessenergyconservationisviolated. Forexample, assumethat in a certaincollisionthe total energyafterthecollisionislargerthantheenergybeforebyanamountΔE. TheHeisenberguncertaintyprincipleclaimsthatthis in factpossibleprovidedtheprocessdoesnotlastlongerthan a time intervalΔt givenbyΔt ≈ h/4πΔE. In otherwords, energyconservation can beviolatedprovidedthe time ittakesforthattohappenisnottoolong.

19. Virtual particles Thisprocessactuallyviolatesthelaw of conservation of energy. Itcannottake place unlessthephotonthatisemittedisveryquickly absorbed bysomethingelse so thattheenergyviolation (and thephotonitself) becomesundetectable. Preciselybecausethisphotonviolatesenergyconservation, itiscalled a virtualphoton.

20. Interactions and exchangeparticles Becausethefirstelectronemitted a photon, itchangeddirection a bit in orderto conserve momentum. Similarly, thesecondphotonalsochangeddirection, sinceit absorbed a photon. Looked at from a largedistanceaway, thechange in direction of thetwoelectrons can beinterpreted as theresult of a forceorinteractionbetweenthetwoelectrons. Theelectromagneticinteractionistheexchange of a virtual photonbetweenchargedparticles. Theexchangedphotonisnot observable.

21. Basic interactionvertices At a fundamental level, particlephysicsviewsaninteractionbetweentwoelementaryparticles in terms of interactionvertices. The fundamental interactionvertex of theelectromagneticinteractionis:

23. Notallowedvertices

24. Feynmandiagrams Notjust a picture. Itrepresents a verydefinitmathematicalexpressioncalledtheamplitude of theprocess. Thesquare of theamplitudegivestheprobability of theprocessactuallytaking place. Fortheelectromagneticinteraction, thebasicvertexisassignedthevalue √αEM, whereαEM≈ 1/37 and iscloselyrelatedtothecharge of theelectron. Theamplitude of thediagramisthentheproduct of the √αEMforeachvertexthatappears.

25. SinceαEMis a smallnumberlessthan 1, theprocesseswithfourinteractionvertices are lesslikelytooccur. To a firstapproximation, itissufficientto examine thediagramwithtwoverticesonly.

26. BuildingFeynmandiagrams • Allyouneed: • thebasicinteractionvertex; • lineswitharrowstorepresentelectrons and positrons; • wavylinestorepresentphotons.

27. Feynmandiagramsforotherinteractions Basic interactionverticesfortheweakforce. Beta decay

29. Therange of aninteraction Considerthediagram in whichtwoparticleinteractthroughtheexchange of theparticleshownbythewavy line. Letthemass of thisparticlebem. Thefastestthe virtual particle can travelisthespeed of light c. IfRistherange of theinteraction, thenthe virtual particlewillreachthesecondparticle in a time no smallerthanR/c. Theenergythatwillbeexchangedwillbe of theorder of mc2. Forthepurpose of theestimate, takinguncertainties of orderR/c in the time and mc2 in theenergy, wethenhavethatbytheHeisenberguncertaintyprinciple: mc2 x R/c ≈ h/4π and hencetherange of theinteractionisapproximatelygivenby R ≈ h/4πmc