Lecture 4: Antiparticles

Transcription

1 Lecture 4: Antiparticles Relativistic wave equations have negative-energy solutions Antiparticles (Chap 3) Perturbation Theory Quantum Field Theories describe fundamental interactions. e.g., QED for electromagnetic interaction (Feynman rules from a Lagrangian) 1

2 Discovery of the Positron (1933) The positron candidate is found in 1933 by Anderson with cloud chamber (below). This positively charged particle has almost same mass with the electron (= anti-electron). This concludes that the particle is the positron. Anderson got a Nobel Prize with this discovery. Carl D. Anderson, "The Positive Electron". Physical Review 43 (6): (1933). DOI: /PhysRev mm lead Cloud chamber photograph of the first positron ever observed Original caption: A 63 million volt positron (Hρ = gauss-cm) passing through a 6 mm lead plate and emerging as a 23 million volt positron (Hρ = gauss-cm). The length of this latter path is at least ten times greater than the possible length of a proton path of this curvature. 1.4 T 23 MeV 63 MeV 2

3 Discovery of the Antiproton (1954) When the Bevatron switched on at Lawrence Berkeley National Laboratory in the fall of 1954, it was the largest particle accelerator ever built, capable of producing energies upwards of six billion electronvolts. The Bevatron s energy range wasn t chosen arbitrarily, but was specifically picked to provide the right conditions for creating antiprotons, then-theoretical particles as massive as protons but with negative electric charge. The discovery earned Nobel Prizes for Chamberlain and Segrè in Lawrence Berkeley National Laboratory 3

4 Continuity Equation to be Lorentz covariant 4

5 Schrondinger Equation [Classical Mechanics] E, p, and L of a particle are dynamical variables represented by time-dependent real numbers. [Schrodinger Picture of Quantum Mechanics] The wave function (Y) is postulated to contain all the information about particular state. Each time-dependent variable of classical dynamics is replaced by the corresponding operator acting on the time-dependent wave function, forming a differential equation. Assume a particle moving with momentum p in free space. It is described by a de Broglie wave function: i(p r E t)/ħ ψ r, t = Ne with frequency ν = E/h and wavelength λ = h/ p ; N is a normalization constant. 5

6 Continuity Eq. in Schronger Eq. 1) Mechanical equation of motion 2) Quantum Mechanical differential equation 3) Solve the equation: wavefunction f(x, t) 4) f(x, t) 2 as a probability of state 5) Quantum Mechanical continuity function: density and current vector (Not covariant) 6

7 r and j in Schrodinger Eq. 1) Begin with Mechanical equation of motion; Convert to Quantum Mechanical differential equation; Solve the equation: wavefunction Y(x, t) 2) Y(x, t) 2 = Y(x, t) * Y(x, t) as a probability of state ; Quantum Mechanical continuity equation: density and current vector 7

8 Klein-Gordon Equation 1) Mechanical equation of motion 2) Quantum Mechanical differential equation 3) Solve the equation: wavefunction f(x, t) 4) f(x, t) 2 as a probability of state 5) Quantum Mechanical continuity function: density and current vector 8

9 r and j in K-G Eq. 1) Begin with Mechanical equation of motion; Convert to Quantum Mechanical differential equation; Solve the equation: wavefunction Y(x, t) 2) Y(x, t) 2 = Y(x, t) * Y(x, t) as a probability of state ; Quantum Mechanical continuity equation: density and current vector 6) Lorentz-invariant normalization: 2E 9

10 Wave Equations Chap. 3, 4 Chap. 5 10

11 Wave Equations (cont d) 11

12 12 Current in K-G j fi μ = i e φ f μ φ i μ φ f φ i = e N f e N i e p i e + p f e e i p i e p f e x φ i e = N i e e ip i e x φ f e = N f e e ip f e x -T/2 x t φ i φ f -T/2 x t V φ i φ f? ) ( ) ( ) ( z p y p x p Et r p Et g x p x p px z y x t r r t g x x x x xx m p p p E p p E g p p p p pp z y x z y x

13 Spin-0 Electron vs. Spin-1/2 Electron A μ Completely different expression? Remember it often happened that you modify already-established solution when a new piece is introduced. 13

14 L q ഥψγ μ ψ A μ Invariance under local gauge transformation (LGT) 14

15 space Particles and Anti-particles By considering the propagation of the negative energy modes of the electron field backward in time, Ernst Stueckelberg reached a pictorial understanding of the fact that the particle and antiparticle have equal mass m and spin J but opposite charges q. This allowed him to rewrite perturbation theory precisely in the form of diagrams. Particle going back in time with negative momentum = time Feynman-Stuckelberg Interpretation Anti-particle going forward in time with positive momentum e - (-E) < 0 E > 0 Richard Feynman later gave an independent systematic derivation of these diagrams from a particle formalism, and they are now called Feynman diagrams. Each line of a diagram represents a particle propagating either backward or forward in time. This technique is the most widespread method of computing amplitudes in quantum field theory. = e + 15

16 Particles and Anti-particles 16

17 Protection from Divergence Remark on Hole theory in 30 s: the electron classical self-energy part to order e 2 was known to be linearly divergent. Dirac's hole theory softened the divergence of the self-energy to a logarithmical divergence, A consequence of the hole theory is that the positron, as first predicted by theory, has been observed. The position (particleantiparticle symmetry) would be a historic analogue to supersymmetry 17

18 Protection from Divergence (II) Anti-particleTransformation SUSY Transformation 18

19 Perturbation 19

20 Perturbation Pluto who? Caltech researchers discover possible ninth planet, January 26, :08 am 20

21 Nonrelativistic Perturbation Theory 4 21

22 H 0 ψ = a n t H 0 φ n (x) e ie nt = a n t E n φ n (x) e ie nt = a n t i t φ n(x)e ie nt = i t a n (t) φ n x e ie nt i da n t dt φ n (x)e ie nt LH of Eq. 2 = H 0 + V ψ = i t ψ i da n t dt φ n x e ie nt + Vψ 22

23 Scattering Amplitudes V is small. 4 a i = 1 (a ni = 0) Assume just one eigenstate i before interaction 23

24 Transition Probability Transition probability 24

25 Graphically

26 Perturbations 26

27 Time-Ordering Diagrams Any real process receives contributions from all possible virtual processes. e.g., e e e e scattering. virtual 27

28 Higher-order Processes Any real process receives contributions from all possible virtual processes. e.g., e e e e scattering. Multiphoton exchange processes are also contributed.... a 2 ~ 7x10-3 a 4 ~ 4x10-7 a 6 ~ 2x10-11 Amount of contribution of each diagram is determined by coupling constant (a = 1/137 in electromagnetic case) and number of vertices. If the coupling constant is small enough, higher-order contributions are negligible and total cross-section quickly converges to a finite value. Perturbation Theory 28

29 Perturbation for e - e - Elastic Collision = 29

30 Preview of Dirac Equation It says that pointlike particles with spin 1/2, mass m and electric charge q have a Dirac magnetic moment: Ԧμ D = q ԦS/m This relation is very well confirmed with leptons. 30

31 Anomalous Magnetic Moment Magnetic moment (MM) in Dirac theory: Ԧμ D = q ԦS/m Dirac MM modified to μ = g qsτm and deviation from Dirac-ness is a e = (g e 2)/2. Calculation of the anomalous MM of electron is one of the great triumphs of QED. In 1948 Schwinger calculated the first-order radiative correction to the naïve Dirac MM of the electron. The radiation and re-absorption of a single virtual photon, contributes a e = a/2p ~ Basic interaction with B field photon First order correction 3rd order corrections Today, a e is known to 4 parts per billion. Current theoretical limit is due to 4th order corrections (> dimensional integrals); a μ is known to 1.3 parts per million. 31

32 [1947] Small deviations from g = 2 Deviations from g = 2 for the point-like electron were observed at about the ~ 0.1% level. Schwinger calculates 1 st order radiative correction It agrees with experiment Higher-order terms are expansions in powers of a/p The set of radiative terms, represents the QED anomalous magnetic moment contribution for the leptons Schwinger Another story, but a e is calculated so precisely (and accurately) that we obtain the best a from it: 32

33 [2012] QED with 10 th -order terms * 12,672 diagrams * QED value here from 2010 Note: better than experiment. 33

34 a = (g 2)/2 depends on loops from ALL SM particles that couple to the muon QED Weak Had VP Had LbL Known well Theoretical work ongoing Critical The g-2 test : Compare experiment to theory. Is SM complete? 34

35 Hadronic VP arxiv:

36 Comparisons of Theory to Experiment arxiv: a exp a SM = ( ± 80) to 3.6 s Theory Expt Future Precision 36

37 Muo g-2 Experiment a exp a SM = ( ± 80) to 3.6 s Precession 37

38 Beyond the Standard Model 38

39 a exp a SM = ( ± 80) to 3.6 s [Q] What can you tell me? 39

40 g e + e - Pair Creation Need at least one other participant (for example, nucleus with charge +Ze) to make it as a real process. Remember, pure g e + e - process is a virtual process and cannot happen! For nucleus, the coupling is proportional to Z 2 α, hence the rate of this process is of order Z 2 α 3. Therefore, if there is a heavy material (larger Z), probability of g e + e - pair creation becomes higher. 40

41 Experimental View of Pair Creation Conversion point Data (2010) 41

42 CMS: Mammography of Tracker Material z <26 cm A complex activity is ongoing using many different, complementary methods: conversions, nuclear interactions, multiple scattering etc+ check of the energy loss and of the momentum scale using low mass resonances. Material uncertainty today (July 2010) better than 10% Systematics uncertainties on physics quantities related to material budget < 1%. 42

43 Beam pipe Pixel 2 Pixel 1 ATLAS: Mammography of Tracker Material Mapping the Inner Detector material with g e + e - conversions and hadron interactions and using data to find geometry imperfections in the simulation Goal is to know material to better than 5% Present understanding (2010) is at the level of ~ 10% Data s = 7 TeV Pixel support Structures to be fixed p 0 Dalitz decays Pixel 3 SCT 1 SCT 2 Reconstructed conversion point in the radial direction of g e + e - from minimum bias events (sensitive to X 0 ) 43

44 Reconstructed secondary vertices due to hadronic interactions in minimum-bias events in the first layer of the Pixel detector (sensitive to interaction length λ complementary to g conversion studies) φ Data Cables C-fiber shell φ Simulation Cooling pipe Bin size: 250 μm Pixel module Already very good, but can be improved! Vertex mass veto applied against g ee, K S0 and Λ Vertex (R, z) resolution ~ 250 μm (R <10 cm) to ~1 mm 44

45 Electron-Positron Scattering 45

46 Quiz It says that pointlike particles with spin 1/2, mass m and electric charge q have a Dirac magnetic moment: Ԧμ D = q ԦS/m This relation is very well confirmed with leptons. However for protons and neutrons, following momenta are observed by some experiments: magnetic What does this mean? 46

47 Quiz - Answer It says that pointlike particles with spin 1/2, mass m and electric charge q have a Dirac magnetic moment: Ԧμ D = q ԦS/m This relation is very well confirmed with leptons. However for protons and neutrons, following momenta are observed by some experiments: magnetic This was actually a first indication that the protons and neutrons are NOT the point-like particles (in 1933). 47