The Standard Model of Particle Physics - II

Transcription

1 The Standard Model of Particle Physics II Lecture 4 Gauge Theory and Symmetries Quantum Chromodynamics Neutrinos Eram Rizvi Royal Institution London 6 th March 2012

2 Outline A Century of Particle Scattering scales and units overview of periodic table atomic theory Rutherford scattering birth of particle physics quantum mechanics a quick overview particle physics and the Big Bang A Particle Physicist's World The Exchange Model quantum particles particle detectors the exchange model Feynman diagrams The Standard Model of Particle Physics I quantum numbers spin statistics symmetries and conservation principles the weak interaction particle accelerators The Standard Model of Particle Physics II perturbation theory & gauge theory QCD and QED successes of the SM neutrino sector of the SM Beyond the Standard Model where the SM fails the Higgs boson the hierarchy problem supersymmetry The Energy Frontier large extra dimensions selected new results future experiments 2

3 Recap A quantum mechanical particle is associated with a wave function ψ The wave function encapsulates all information about the particle The wave function squared is proportional to probability of finding the particle at a particular place, time, energy, momentum etc.. kinetic energy + potential energy = total energy i(kx t) = Ae ~2 2m r2 + V (x, y, z) = Schrödinger equation describes the particle ψ behaves under influence of an energy field V(x,y,z) x,y,z,t are coordinates in space and time V(x,y,z) could be e.g. another particle s electric field The equation involves derivative operators: mathematical operators acting on wave function They calculate slopes or how the wave function changes per meter, or per second operators act on something, just like + or or In this case they act on the wave function ψ Symmetry: A transformation which leaves an experiment unchanged Each quantum symmetry is related to a conservation law Translation in time Translation in space Rotations Gauge Transformation Energy conservation Momentum conservation Angular Momentum conservation Charge conservation 3

4 Gauge Theory A gauge transformation is one in which a symmetry transformation leaves the physics unchanged +5V 0 V Both circuits behave identically Circuit is only sensitive to potential differences +513 V +508 V Change the ground potential of the earth and see no difference! Leads to concept of charge conservation In electromagnetism we are insensitive to phase δ of EM radiation All experiments can only measure phase differences Could globally change the phase at all points in universe Yields no observable change global gauge transformation (In electromagnetism this is the gauge symmetry expressed by the U(1) group) phase difference δ What happens if we demand local phase transformations? δ δ(x,t) i.e. δ is no longer a single number, it depends on position x and time t ~ 2 2m r2 + V (x, y, z) = i~ t Wave functions of all particles get an extra piece from the change in δ This spoils the Schrödinger equation (actually, relativistic versions are the KleinGordon and Dirac equations) δ(x,t) spoils the spatial & time derivatives 4

5 Gauge Theory (see lecture 2) = Ae i(kx t)! = Ae i (x,t) e i(kx t)... and since energy (E) and momentum (p) measurements are represented by operators in quantum mechanics i~ t = E ~ i x = p x The derivatives cause nuisance terms to appear in equations arising from δ(x,t) But we still want physics to work the way it did before the gauge transformation! We want the Schrödinger equation to still work! So add an additional term to the equation to cancel out those nuisance terms After adding these to the equation we ask ourselves: what do the new equation pieces look like? The alterations required to accommodate these changes introduce a new quantum field This field has a spin = 1 This field interacts with charged particles This field no charge itself The field particle has zero mass it is the photon! Our consideration of local symmetry leads us to predict the photon 5

6 Gauge Theory This can be applied to other quantum interactions: local gauge invariance introduces new fields oscillations in the fields are the probability wave functions of particles Interaction Gauge particle Gauge group Symbol Felt by Electromagnetism photon U(1) q, W ±, e ±, μ ±, τ ± Strong force gluons SU(3) q, gluons Weak force W ± and Z 0 SU(2)L q, W ±, Z 0, e ±, μ ±, τ ±, all ν These are not simply abstract mathematical manipulations the particles exist! Weak bosons (spin 1 particles) discovered in 1983 at CERN s UA1 experiment March 4, : 55 DRAFT Physics 1984 Energy of electrons from the decay of the W particle: W eνe Entries / 2.5 GeV Data 2010 ( W eν QCD W τν Z ee Z τ tt WW 1983 L to dt = pb s = 7 TeV) e + 1 Entries / 2.5 GeV Data 2010 ( W eν QCD W τν Z ee Z τ tt WW 1 L dt = 36.2 pb s = 7 TeV) e Fig. 19a. The electron transverse energy distribution. The two curves show the results of a lit of the enhanced transverse mass distribution to the hypotheses W+ e+v and X+ C+V+V. The first hypothesis is clearly preferred..5 GeV Eram Rizvi Lecture W eν 4 Royal Institution London W eν 6 The three mass determinations gave very similar results. We preferred to Data 2010 ( s = 7 TeV) 7000 QCD e + E T [GeV] 1.5 GeV Data 2010 ( s = 7 TeV) QCD e E T [GeV] (In 2011 LHC has 40x more data than in 2010)

7 Perturbation Theory So far theory predicted new particle s existence How do we calculate particle reaction rates? e.g. reaction rate of electron positron scattering?? ~ 2 2m r2 + V (x, y, z) = i~ t In Schrödinger equation a particle interacts with a potential energy field V Potential energy is energy an object has by virtue of position Apple in a tree it has potential energy in Earth s gravitational field Apple falls it releases potential energy into kinetic energy Total energy is constant! potential energy kinetic energy as it falls In quantum mechanics the potential causes a transition from initial state to final state wave functions Potential = V + V' V gives rise to stable, time independent quantum states f and i V' is a weak additional perturbation leading to transitions between states i! f P = M fi 2 = R f V 0 i dv 2 P = probability of transition from initial to final state Mfi is known as the matrix element for the scattering process V' contains the standard model Lagrangian describes the dynamics of all interactions 7

8 Feynman Diagrams EPP Use of Feynman Diagr Although they are used pictorially to show what is going on, Feynm Diagrams are used more seriously to calculate cross sections or decay rates Transition due to the exchange of a gauge boson Exchanges momentum & quantum numbers Strength of the interaction is parameterised by couplings α One α for each fundamental force Draw all possible Feynman Diagrams for the process Draw all possible Feynman diagrams for your experiment: Simplest interaction is single boson exchange Propagator More complicated loop diagrams also contribute Sum over all allowed PHY306 particle EPPstates i.e. all quark flavours / colours / spins Eram Rizvi For each diagram calculate the transition amplitude Add all transition amplitudes Square the result to get the reaction rate Feynman Rules: start from left side } Assign values to each part of the diagram Calculate the amplitude by multiplying together Write a free particle wave function for each particle Lecture 4 Royal Institution London Feynman Diagrams Free particle i(kx t) Add the = Ae amplitudes for each diagram (including interference) Multiply by an exchanged boson write for particle of momentum p and mass m 1 p 2 +m 2 Diagrams are ordered in powers of α Form a power series Potentially infinite series of diagrams for 2 2 scattering process Square the amplitude to get the intensity/probability (cross section or decay rate) For each vertex multiply by coupling α order α order α 2 If perturbation is small i.e. α < 1 then contributions from extra loop diagrams is suppressed The sum to infinity of a power series can be finite! = X n=1 1 2 n =1 Ver cha 8

9 Assign values to each part of the diagram decay rates Free particle Draw all possible Feynman Diagrams for the process Propagator Vertex Although they are used pictorially to show what is going on, Feynman Diagrams are used more seriously to calculate cross sections or Feynman Diagrams M fi 2 = e4 1 q 4 4 in case you don t believe me... X {[ū(k 0 ) µ u(k)] [ū(k 0 ) u(k)] }{[ū(p 0 ) µ u(p)] {[ū(p 0 ) u(p)] } spin k, k = incoming, outgoing electron momentum p, p = incoming, outgoing muon momentum q = momentum transfer e = strength of electromagnetic interaction (electric charge) EPP Use of Feynman Diagrams e µ e k k q p p µ electron charge For electromagnetism αem = 1/137 ~ e 2 Small enough for perturbation theory to work For strong interaction αs ~ 0.1 Perturbation theory works but need to calc more diagrams for precision difficult! For QCD it took 10 years to calculate second order diagrams! 9

10 Observation of Quarks Quarks discovered in scattering experiments at Stanford Linear Accelerator Laboratory California e ± e ± photon / Z 0 Q 2 proton quark ep! e + X High energy electron beam strikes stationary proton High energy = small wavelength Quarks kicked out of proton Seen as jet / spray of new particles why? What keeps quarks bound inside proton? Taylor Friedman Kendall Nobel Laureates

11 Observation of Quarks Reaction rate ep! e + X SLAC ep scattering in 1968 The data demonstrate proton has: pointlike constituents they have fractional electric charge 1.20 they are spin 1 /2 fermions behave as free unbound particles 1.00 P Calcula Additional measurements 0.60 showed only 50% of proton s momentum is carried by 0.40 quarks! M 0.20 (Momentum Transfer) 2 W 2 (or F 2 ) as a function of q 2 at x = For this choice of x, there is practically no dependence on q 2, that is, there 15 is exact BscalingC. (After Friedman and Kendall 1972.) However, this is not the whole story! 11 It is found that quarks carry only half the momentum of

12 Observation of Quarks ep! H1 e and + ZEUS X in 2011 r,nc (x,q 2 ) x 2 i x = , i=21 x = , i=20 x = , i=19 x = , i=18 x = , i=17 x = , i=16 x = , i=15 x = , i=14 x = , i=13 x = , i=12 x = 0.005, i=11 x = 0.008, i=10 x = 0.013, i=9 x = 0.02, i=8 HERA I NC e + p Fixed Target HERAPDF1.0 x = 0.032, i=7 x = 0.05, i=6 x = 0.08, i=5 x = 0.13, i=4 x = 0.18, i=3 x = 0.25, i=2 x = 0.40, i= x = 0.65, i= Q 2 / GeV 2 12

13 There are no free ends Quantum Chromodynamics If you pull hard enough you get two bits of elastic but still no free ends If you pull hard enough you get two bits of elastic but still no free ends LEP at CERN: Large Electron Positron Collider q! e q q! q! e + q Jets q! q! q! q! Hadrons PHY306 EPP Colour and QCD Slide 306 EPP Colour and QCD Several particles make up two distinct jets as quarks fragment New particles always consist of: 3 quarks (baryons e.g. proton & neutron) or quark and antiquark (mesons e.g. pion, kaon) 12 Two quark jets from e + e annihilation at LEP Seen in the OPAL experiment 13

14 Quantum Chromodynamics One baryon was seen called Δ ++ Known to consist of 3 u quarks each has charge + 2 /3 Spin = + 3 /2 i.e. all quark spins aligned in same direction All three quarks have same quantum state! The quarks violate the Pauli Exclusion Principle This led to development of theory of strong interactions A new quantum number was invented to circumvent the problem: colour The new quantum number also explained why quarks are bound in composite particles The colour quantum number is the charge of the strong interaction takes 3 values: red green blue u u u Δ ++ antiquarks take anticolours: antired antigreen antiblue 14

15 Quantum Chromodynamics Quark colour is an analogy not actual colour Composite particles must remain colourless overall EPP Quark Confinement The reason there are red no + free green quarks, + blue and = white the only stable configurations are qqq and q) (and colour possible + anticolour gg 'glueball' = black states) is due to the fact that * decreases with increasing energy and increases with increasing distance Only white / black particles are observable in nature Mediated by gluons: Particles with colour feel the strong force At short distance, inside a Try and pull one out and * proton, Electron * has is no small colour and quantum the number ignores increases strong and force the force quarks are 'free' tries to keep it in As quarks separate, field strength increases PHY306 EPP Colour and QCD Slide Known as quantum chromodynamics: QCD 11 15

16 onsider a proton made up of one red, one green and one blue quark Consider a proton made up of one red, one green and one blue quark Quantum Chromodynamics Do a global colour transformation red Proton is colourless Contains 3 quarks Do a green global everywhere colour transformation red green everywhere Proton is colourless Perform global colour A green No observable difference EPP A transform r g red blue C blue C Now do a local colour transformation red green at one q B red B green Local Colour Trans Still Perform 'colourless' local colour nothing has changed transform Still 'colourless' r g at A emits nothing has changed Can rg be ur Transformations gluon made colourless if A one quark only C absorbs emits an gluon r& to gluon become which r is 306 EPP Colour and QCD Slide 6 absorbed at C. When the PHY306 EPP Colour and QCD Slide 6 reen at one quark only gluon is absorbed Local transformations at C the yield No long gluon exchange green and antigreen annihilate A green A green to make C red Gluons carry the strong force A blue EPP Self Interaction blue C C Since gluons are coloured they B B green Gluons green selfinteract that carry colour g + charges g + rcan r& interact # r with each other (NonAbelian) Colourless again Proton No has longer gained colourless Modify equations to force no colour Introduce a spin Local 1 field colour with colour transformation & # exchange of gluons # f late Need to make it colourless again anticolour PHY306 EPP 16 Colour gluons and QCD Eram Rizvi Lecture 4 Royal Institution London 8 gluons 16

17 Quantum Chromodynamics Analogy: Gluons are like bits of elas There are no free ends TASSO experiment 1979 at DESY e + e! q qg If you pull hard enough you get two bits o still no free ends e q e + q This 3jet event demonstrates existence of gluons As quark is accelerated it radiates a gluon First evidence for the gluon PHY306 EPP Measurement of angular distribution of jets confirms gluons are spin 1 particles Gluons make up the missing 50% of proton s momentum Colour and Q 17

18 Quantum Chromodynamics Experiment performed at energy below charm quark mass PHY306 EPP Experiments observed fragmenting quarks but... how do we know colour exists? EPP Use of F Measure reaction produced, rate ratio: the value o Although they are used pictorially to show w Diagrams are used R = e+ e more! qq seriously to Rcalcul 3 decay rates e + e! compatible µ + µ with the v Draw all possible Feynman Diagrams for the e +, ū, d, s e,u,d,s collision energy The ratio R of the crosssection section for e + e G hadrons, Calculate Feynman divided diagrams. by that In the for ratio e + e G everything + G. cancels except coupling (i.e. charge) and number of quarks Assign values to each part of the diagram Calculate the amplitude by multiplying toget R = ( 2 3 )2 +( 1 3 )2 +( 1 3 )2 1 2 = 2 3 since u,d,s Add quarks the have amplitudes charges of 2 /3, 1 /3 and 1 /3 for each diagram (includ But: quarks come in 3 colours, so we need to multiply by 3 for each red, green and blue quark Square the amplitude to get the intensity/pro section or Colour decay predicts rate) R = 2 Well verified by experiment! Feynman Diagrams Note that since each q sum must be over bot At low energies, wher

19 The Couplings α ons that carry colour charges can interact with h other (NonAbelian) quark quark 16 gluons Probing smaller distances (i.e. higher energies) 12 gluons we see less gluons! Strong force get weaker 4 gluons 16 gluons 8 gluons 4 gluons 'Strength' of the colour charge appears to decrease as distance decreases (energy increases) 6 EPP Colour and QCD Slide coupling given by the electronic charge e Electromagnetic Interactions electron Running of! Q + 4q + Q + 8q + creened by +ve charges. Charge appears to nce increases (lower energy) Q At low energy the strong force binds quarks into colourless objects confinement At high energy strong force is feeble and quarks are free asymptotic freedom Slide 12 8 At low energy an electron s charge Q is screened by quantum fluctuations of the vacuum many e + e pairs formed At high energy less screening occurs EM force is stronger charge = Q Probing smaller distances charge = Q + 4q + (i.e. higher energies) we see more electron charge EM force get weaker charge = Q + 8q + 19

20 various inputs to this combination, evolved to ight) The provides Couplings strongest evidence α for the correct ce of the strong coupling. ) 0.5 α s (Q) Strong Coupling Deep Inelastic Scattering e + e? Annihilation Heavy Quarkonia QCD α s(μ Z) = ± Q [GeV] Momentum Transfer July 2009 surements of α s (MZ 2 ), used as input for the of measurements The strong force of α s coupling as a function decreases of the with aretakenfromref.172. increasing energy Conclusio Electromagnetic (inverse) Coupling Six y LEP st result Runn electr estab Runn electr precis Can into t!" hadr The electromagnetic force coupling increases with (Compton increasing energy scattering studied at the This is why quarks are free at high energy and bound into composite particles at low energy Napoli 9/6/2006 Salvatore Mele 20 0, :57

21 The Weak Force electron neutron proton Mediated by charged W exchange antineutrino The weak force is mediated by W ± and Z 0 bosons Beta decay is due to weak interactions All particles experience weak force except gluons n d d u W e u d u e p 3 flavours of charged lepton 3 flavours of neutrino e µ νe νµ Neutrinos only interact via weak force very inert PHY306 EPP Feynman Diagrams Slide involved in weak beta decays powering solar fusion Solar neutrino flux is large ~ 10 6 through your thumbnail every second! 10 τ ντ calculated rate Sun produces calculable rate of ν e measure ~30% less than expected measured rates Experiments sensitive to different neutrino flavours sensitive to all 22 flavours! 21

22 Neutrino Oscillations We assumed neutrinos of fixed flavour have fixed mass This does not need to be the case Can describe neutrino wave functions by an alternative but equivalent set of wave functions Purely quantum mechanical effect One set has definite flavour but indefinite mass (νe νμ ντ) One set had definite mass but indefinite flavour (ν1 ν2 ν3) defined: total flow rate temperature defined: hot flow rate cold flow rate Solar neutrinos are produced as electron neutrinos: νe These oscillate into different flavours as they propagate to Earth Only experiments sensitive to all flavours of neutrino see full solar neutrino flux But both are equivalent 22

23 The Standard Model Quarks Leptons +2/3 1/3 1 0 u u u d d d e νe I Fermions c t c t c t s b s b s b µ τ νµ ντ II III g g g g g g g g 8 gluons Bosons W Z γ 3 electroweak bosons H 1 higgs boson 23

24 The Standard Model Quantum mechanics predicts the gyromagnetic ratio of the electron g=2 (ratio of magnetic dipole moment to it's spin) Experiment measures g exp = ± Discrepancy of g2 due to radiative corrections Electron emits and reabsorbs additional photons Corresponds to higher terms in perturbative series expansion g theory 2 2 g exp 2 2 = ± = ± Phenomenal agreement between theory and experiment! 4 parts in 10 8 QED (quantum electrodynamics) is humanity's most successful theory Demonstrates understanding of our universe to unprecedented precision Equivalent to measuring distance from me to centre of moon and asking if we should measure from top of head or my waist! 24

25 The Standard Model Perl Gross Rubbia van der Meer Reines Lederman Gellman Cronin Steinberger Feynman Glashow Taylor Friedman Hofstadter Schwinger Higgs Veltman Kendall Politzer Ting Alvarez Fitch Schwarz Richter Weinberg Yang 29 Nobel prizes awarded for the Standard Model Wilczek Salam Lee t Hooft 1 more yet to come? 25

26 The Standard Model This equation is the sum of all small perturbations V describing all interactions of quarks and leptons and the electroweak and strong gauge fields Welcome to the Standard Model of particle physics! 26