Particle Physics WS 2012/13 ( )
|
|
- Basil Richardson
- 5 years ago
- Views:
Transcription
1 Particle Physics WS 2012/13 ( ) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 226, 3.101
2 2 2
3 3 3
4 4 4
5 5
6 Where are we? W fi = 2π 4 LI matrix element M i (2Ei) fi 2 ρ f (E i ) LI phase space i> f> M fi = M a c+x M(b+x d) q 2 mx 2 e.g. e + μ e + μ Matrix element describing two interaction vertices and one intermediate particle can be computed as the product of the matrix elements of the both vertices and 1 a factor for the intermediate particle (propagator) q 2 mx 2 Next step: compute matrix element of three prongue vertex (e.g. e e + γ) Up to now, everything was independent of type of interaction, to compute vertex, we need to study a specific interaction 6
7 Where do we want to go? Program of next two lectures Derive potential of electromagnetic IA Derive general rules for computation of Matrix element (here will come back to the free spin=1/2 particle wave functions) e μ + Calculate cross-section for e + e + μ + μ + Helicity vs. Chirality Calculate cross-section for related process exploiting Mandelstamm variables e + μ e + μ e + e μ e and e + e + e + e + e e e e + + μ μ e + e e + e + 7
8 8 8
9 x -i 9 9
10 Matrix Element for e - τ - Scattering e e M fi = M a c+x M(b+x d) q 2 mx 2 V = qγ 0 γ μ A μ p 1 p 2 p 4 p 3 M = u e (p 2 ) q e γ 0 γ μ u e (p 1 ) λ ε μ λ (ε v λ ) q 2 u τ (p 4 )q τ γ 0 γ ν u τ (p 3 ) τ τ <Ψ V Ψ > e <Ψ V Ψ > τ Ψ = u(p)ei(px Et) Ψ = u(p) e i(px Et) = -q e q τ u e (p 2 )γ μ u e (p 1 ) g μν q 2 u τ (p 4 )γ ν u τ (p 3 ) u e (p 2 )γ 0 u τ (p 2 )γ 0 10
11 Reminder Probability Density and Currents x ψ (iγ μ μ m )ψ = 0 i μ ψγ μ + m ψ= 0 x ψ iψγ μ μ m ψψ + i( μ ψ)γ μ ψ+ m ψψ= 0 i μ (ψγ μ ψ)= 0 Continuity equation: μ j μ = 0 with j μ = (ρ, j) j μ = ψγ μ ψ ρ = ψγ 0 ψ = ψ ψ = particle current 4 4 i=1 ψ i ψi = i=1 ψ i 2 > 0 Historically the positive density ρ was the original motivation for Dirac s work Charge current for an electron: j μ = -eψγ μ ψ 11
12 Feynman Rules e e M = -q e q τ u e (p 2 )γ μ u e (p 1 ) g μν q 2 u τ (p 4 )γ ν u τ (p 3 ) = - q e q τ q 2 j eμ g μν j τ v p 1 p 2 p 3 p 4 M = - q e q τ q 2 j e j τ τ τ Matrix element is four-vector scalar product thus it is Lorentz invariant! This expression hides a lot of complexity. Needed to sum over all possible time-orderings and summed over all polarization states. Fortunately don t need to redo this for each new Feynman diagram! We can write down matrix elements using a set of simple rules Feynman rules! Propagator factor for each internal line (e.g. each internal virtual particles) Dirac Spinor for each external line (e.g. each real real incoming/outgoing particle) Vertex factor for each vertex 12
13 Feynman Rules 13
14 Feynman Rules e p 1 e p 3 -im = i q e v e p 2 γ μ u e (p 1 ) igμν q 2 iqeu e p 3 γ ν v e (p 4 ) p 2 p 4 e + e + u p 1 u p 3 -im = i q u u u (p 2 ) γ μ u u (p 3 ) igμν q 2 iqdv d p 4 γ ν v d (p 2 ) p 2 p 4 d d e - p 1 p 3 -im = i q e ε(p 3 ) μ* γ μ u e (p 1 ) i(γ ν q ν +m) q 2 m 2 iqeε p 4 ξ γ ξ v e (p 3 ) e + p 2 p 4 14
15 Calculation of cross section e + e - μ + μ - Lecture 3, diff. cross-section in CMS: e μ p 1 p 3 dσ = 1 p f M dω 64 π 2 s p fi 2 i p 1 = p 2 = pi p 3 = p 4 = pf s = (E 1 + E 2 ) 2 p 2 e + μ + p 4 lowest order diagram M ~ e 2 ~α em (NU: e= 4πα; α ~ 1 ) 137 & many second order diagrams e μ + p 1 p 3 e μ + p 1 p 3 e + p 2 M = Γ 1 + Γ 2 + μ p 4 Summing amplitudes not amplitudes square, therefore interference effects possible. Higher order in elm. IA however suppressed (α~ ), thus consider only 1st order here e + p 2 μ p 4 15
16 Calculation of cross section e + e - μ + μ - e μ + e + p 1 p 2 μ p 4 p 3 -im = i q e v e p 2 γ μ u e (p 1 ) igμν iq q 2 μ u μ p 3 γ ν v μ (p 4 ) M = - e 2 [v s e p 2 γ ν u e (p 1 )] [ u μ p 3 γ ν v μ (p 4 )] In general electron and positron will not be polarized, i.e. there will be equal numbers of positive and negative helicity states there are four possible combinations of helicities in the initial state! (right handed helicity helicity=+1; left handed helicity helicity=-1) Similarly there are four possible helicity combinations in the final state In total there are 16 combinations e.g. RL RR, RL RL, Need to sum over all 16 possible helicity combinations and then average over number of Initial helicity states: < M 2 > = 1 4 ( M RL RR 2 + M RL LR 2 + ) elicity 16 spin averaged matrix element particle helicity antiparticle helicity
17 Calculation of cross section e + e - μ + μ - Helicity eigenstates: u h=1 = N c e iφ s p E+m c p E+m eiφ s u h=-1 = N s e iφ c p E+m s p E+m eiφ c v h=1 = N p E+m s p E+m eiφ c s e iφ c v h=-1 = N p E+m c p E+m eiφ s c e iφ s N = E + m, c=cosθ/2, s=sinθ/2 z p = p sin θ cos φ sin θ sin φ cos θ φ θ y x In limit of E>>m: u h=1 = E c e iφ s c e iφ s u h=-1 = E s e iφ c s ei φ c v h=1 = E s ei φ c s e iφ c v h=-1 = E c e iφ s c e17 iφ s
18 x p 1 z p 4 Calculation of cross section e + e - μ + μ - p 3 e e + θ μ + u h=1 = E π-θ c e iφ s c e iφ s μ p 2 φ=0 u h=-1 = E p 1 = (E, 0, 0, E) p 2 = (E, 0, 0, -E) p 3 = (E, Esin θ, 0, E cos θ) p 4 = (E, -Esin θ, 0, E cos θ) s e iφ c s ei φ c v h=1 = E s ei φ c s e iφ c [θ in CMS, E>>m μ ] v h=-1 = E c e iφ s c e iφ s Initial state electron (φ=0, θ=0): u h=1 = E u h=-1 = E Final state muon (φ=0,θ): c u h=1 = E s c s u h=-1 = E s c s c Initial state positron (φ=0, θ=π): v h=1 = E v h=-1 = E Final state anti-muon (φ=0,θ-π): v h=1 = E c s c s v h=-1 = E s c s c 18
19 muon current: j μ = u p 3 γ ν v(p 4 ) Muon Current e μ + p 1 p 4 For arbitrary spinors it is straight forward to show: e + p 2 μ p 3 Ψ γ 0 γ 0 Φ Ψ 1 Φ1 + Ψ 2 Φ2 + Ψ 3 Φ3 + Ψ 4 Φ4 Ψ γ ν Φ= Ψ γ 0 γ 1 Φ Ψ γ 0 γ 2 Φ = Ψ 1 Φ4 + Ψ 2 Φ3 + Ψ 3 Φ2 + Ψ 4 Φ1 i(ψ 1 Φ4 Ψ 2 Φ3 + Ψ 3 Φ2 Ψ 4 Φ1 ) [eq.1] Ψ γ 0 γ 3 Φ Consider μ - R, μ + L using Ψ=u R, φ=v L Ψ 1 Φ3 Ψ 2 Φ4 + Ψ 3 Φ1 Ψ 4 Φ2 u h=1 = E j μ RL= u p 3 R γ ν v p 4 L = 2E(0,sinθ/2 2 -cosθ/2 2,i,2Esinθ/2 cosθ/2) = 2E(0,-cosθ, i,sinθ) c s c s v h=-1 = E s c s c 19
20 Muon Current analogously compute: j μ RL= u p 3 R γ ν v p 4 L = 2E(0,-cosθ,i,sinθ) j μ RR= u p 3 R γ ν v p 4 R = (0,0,0,0) j μ LL= u p 3 L γ ν v p 4 L = (0, 0, 0, 0) j μ LR= u p 3 L γ ν v p 4 R = 2E(0, -cosθ,-i, sinθ) In the limit of E>>m, only two non vanishing helicity combinations for final state! Due to angular momentum conservation, need to look at 4 combinations of initial and final state only: Next step: needs to compute, j e RL j e LR 20
21 Electron Current The electron current can either be directly derived from eq[1] or from the expression of the muon current: j μ RL = [u p 3 R γ ν v p 4 L ] = [u(p 3 ) R γ 0 γ ν v p 4 L ] = v(p 4 ) L γ ν γ 0 u p 3 R = v(p 4 ) L γ 0 γ ν u p 3 R = v p 4 L γ ν u p 3 R p 1 e μ + p 4 γ μ = -γ μ p 1 p 3 μ e e + θ p 2 e + p 2 μ p 3 π-θ p 4 φ=0 j e RL = v p 2 L γ ν u p 1 R = Θ =0 2E(0,-cosθ,-i,sinθ) = 2E(0,-1,-i,0) j e LR = v p 2 R γ ν u p 1 L = 2E(0,-cosθ,i,sinθ) = 2E (0, -1,i,0) 21
22 Matrix Element Calculation Can now calculate M =- e2 s j ej μ for the four possible helicity combinations. e.g. for M(RL RL) particle helicity antiparticle helicity j e RL = v p2 L γ ν u p 1 R = 2E (0, -1,-i,0) j μ RL= u p 3 R γ ν v p 4 L = 2E(0, -cosθ,i, sinθ) M(RL RL) = e2 s 4E2 (0, 1, i, 0)(0, cos θ, i, sin θ) = -e 2 (cosθ+1) s = 2E = -4πα(1+cosθ) M(RL RL) 2 = e cosθ 2 22
23 Matrix Element Calculation M(RL RL) 2 M(RL LR) 2 M(LR RL) 2 M(LR LR) 2 Assuming that the incoming electrons are unpolarized, all 4 possibilities are equally likely. 23
24 Differential Cross Section The cross section is obtained by averaging over the initial states and summing over all final states. (E>>m μ p f = p i ) = dσ dω = 1 4 = 1 64 π 2 s p f p i M fi 2 1 ( M(RL RL) 2 + M(LR LR) 2 + M(RL LR) 2 + M(LR RL) 2 ) 256 π 2 s 4πα π 2 s (2(1+cosθ)2 + 2(1-cosθ) 2 ) dσ = α2 dω 4s (1+cos2 θ) Example: e + e - μ + μ, s = 29 GeV pure QED (α 3 ) calculation QED + Z contribution 24
25 Total Cross Section Total cross section obtained by intregrating over θ and φ using 1 + cos 2 θ dω = 2π (1 + cos 2 θ) dθ = 16π 3 total cross section for (e + e - μ + μ ): σ = 4πα2 3s Lowest order computation provides good description of data Very impressive result: from first principle computation we have reached a 1% precise result! 25
26 Helicity and Chirality = ς 0 0 ς = ς 1 ς 1 ς 2 ς 2 ς 3 ς 3 h = Σp [H,h] = 0 p s p s h Ψ= + Ψ h Ψ= - Ψ right handed (RH) helicity right handed (RH) helicity Helicity is a conserved quantity! Gedankenexperiment: A massive particle (e.g. electron) with helicity +1 is moving with v c p,v s h=1 An observer overtakes the electron by moving with v obs > v In reference system of observer velocity and momentum will have opposite sign, while spin will be the same. Electron helicity in the frame of the observer will be LH (h=+1) Helicity is not a LI quantity! 26
27 Helicity Projector Projector on helicity eigenstates (see homework) P ± = 1 2 (1 ± Σp) unpolarized electron Ψ(e - ) = a Ψ h=+1 + b Ψ h=-1 P + (Ψ(e - )) = a Ψ h=+1 P - (Ψ(e - )) = b Ψ h=-1 P +2 (Ψ(e - )) = a Ψ h=+1 P -2 (Ψ(e - )) = b Ψ h=-1 Definition of any projector P 2 Ψ = P Ψ 27
28 Chirality chirality operator: γ 5 = i γ 0 γ 1 γ 2 γ 3 = 0 I I 0 in Dirac representation chirality projector: P ± = 1 2 (1 ± γ5 ) P ± 2 = ± γ5 2 = ± γ5 1 ± γ 5 = ± 2γ5 + γ 5 2 = ± 2γ5 + 1 = 1 2 (1 ± γ5 ) define LH chirality state: Ψ L = 1 2 (1 γ5 )Ψ define RH chirality state: Ψ R = 1 2 (1 + γ5 )Ψ [H,γ5] 0 (see homeworks) chirality is not a conserved quantity chirality is however LI! 28
29 Chirality-Eigenstates In QED, basic IA between fermion and photon is: qψ γ μ Φ φ = φ R + φ L Ψ = Ψ R + Ψ L qψ γμ Φ = q Ψ R + Ψ L γ μ Φ R + Φ L exploiting following relations: (γ 5 ) 2 =1 γ 5 =γ 5 γ 5 γ μ = -γ μ γ 5 Ψ R γμ Φ L = γ5 Ψ γ 0 γ μ γ5 Φ = 1 4 Ψ + Ψ γ 5 γ 0 γ μ [1 γ 5 ]Φ = 1 4 Ψ γ 0 γ μ Φ + Ψ γ 5 γ 0 γ μ Φ Ψ γ 5 γ 0 γ μ Φ Ψ γ 0 γ μ Φ analogous: Ψ L γμ Φ R =0 = 0 only certain chirality eigenstates contribute to IA (always true statement)! only for E>>m : γ 5 = Σp (see homework) LH chirality eigenstates LH helicity eigenstates RH chirality eigenstates RH helicity eigenstates For massive particles (not ultra relativisitc): helicity states are combination of chirality states! 29
30 Intermediate Summary 30
31 Mandelstamm Variables 31
32 Lorentz Invariant Representation of X-Section 32
Particle Physics WS 2012/13 ( )
Particle Physics WS 2012/13 (9.11.2012) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 226, 3.101 QED Feyman Rules Starting from elm potential exploiting Fermi s gold rule derived QED Feyman
More informationParticle Physics WS 2012/13 ( )
Particle Physics WS /3 (3..) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 6, 3. How to describe a free particle? i> initial state x (t,x) V(x) f> final state. Non-relativistic particles Schrödinger
More informationOutline. Charged Leptonic Weak Interaction. Charged Weak Interactions of Quarks. Neutral Weak Interaction. Electroweak Unification
Weak Interactions Outline Charged Leptonic Weak Interaction Decay of the Muon Decay of the Neutron Decay of the Pion Charged Weak Interactions of Quarks Cabibbo-GIM Mechanism Cabibbo-Kobayashi-Maskawa
More informationParticle Physics WS 2012/13 ( )
Particle Physics WS 01/13 (3.11.01) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 6, 3.101 Content of Today Structure of the proton: Inelastic proton scattering can be described by elastic scattering
More information3.3 Lagrangian and symmetries for a spin- 1 2 field
3.3 Lagrangian and symmetries for a spin- 1 2 field The Lagrangian for the free spin- 1 2 field is The corresponding Hamiltonian density is L = ψ(i/ µ m)ψ. (3.31) H = ψ( γ p + m)ψ. (3.32) The Lagrangian
More informationElectroweak Physics. Krishna S. Kumar. University of Massachusetts, Amherst
Electroweak Physics Krishna S. Kumar University of Massachusetts, Amherst Acknowledgements: M. Grunewald, C. Horowitz, W. Marciano, C. Quigg, M. Ramsey-Musolf, www.particleadventure.org Electroweak Physics
More informationQuantum Field Theory Spring 2019 Problem sheet 3 (Part I)
Quantum Field Theory Spring 2019 Problem sheet 3 (Part I) Part I is based on material that has come up in class, you can do it at home. Go straight to Part II. 1. This question will be part of a take-home
More informationIntroduction to Elementary Particle Physics I
Physics 56400 Introduction to Elementary Particle Physics I Lecture 16 Fall 018 Semester Prof. Matthew Jones Review of Lecture 15 When we introduced a (classical) electromagnetic field, the Dirac equation
More informationOUTLINE. CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion
Weak Interactions OUTLINE CHARGED LEPTONIC WEAK INTERACTION - Decay of the Muon - Decay of the Neutron - Decay of the Pion CHARGED WEAK INTERACTIONS OF QUARKS - Cabibbo-GIM Mechanism - Cabibbo-Kobayashi-Maskawa
More informationLecture 3. Experimental Methods & Feynman Diagrams
Lecture 3 Experimental Methods & Feynman Diagrams Natural Units & the Planck Scale Review of Relativistic Kinematics Cross-Sections, Matrix Elements & Phase Space Decay Rates, Lifetimes & Branching Fractions
More information. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω. Friday April 1 ± ǁ
. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Friday April 1 ± ǁ 1 Chapter 5. Photons: Covariant Theory 5.1. The classical fields 5.2. Covariant
More informationParticle Physics Dr M.A. Thomson Part II, Lent Term 2004 HANDOUT V
Particle Physics Dr M.A. Thomson (ifl μ @ μ m)ψ = Part II, Lent Term 24 HANDOUT V Dr M.A. Thomson Lent 24 2 Spin, Helicity and the Dirac Equation Upto this point we have taken a hands-off approach to spin.
More information. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω. Wednesday March 30 ± ǁ
. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Wednesday March 30 ± ǁ 1 Chapter 5. Photons: Covariant Theory 5.1. The classical fields 5.2. Covariant
More informationOutline. Charged Leptonic Weak Interaction. Charged Weak Interactions of Quarks. Neutral Weak Interaction. Electroweak Unification
Weak Interactions Outline Charged Leptonic Weak Interaction Decay of the Muon Decay of the Neutron Decay of the Pion Charged Weak Interactions of Quarks Cabibbo-GIM Mechanism Cabibbo-Kobayashi-Maskawa
More informationQuantum ElectroDynamics III
Quantum ElectroDynamics III Feynman diagram Dr.Farida Tahir Physics department CIIT, Islamabad Human Instinct What? Why? Feynman diagrams Feynman diagrams Feynman diagrams How? What? Graphic way to represent
More information11 Spinor solutions and CPT
11 Spinor solutions and CPT 184 In the previous chapter, we cataloged the irreducible representations of the Lorentz group O(1, 3. We found that in addition to the obvious tensor representations, φ, A
More information129 Lecture Notes More on Dirac Equation
19 Lecture Notes More on Dirac Equation 1 Ultra-relativistic Limit We have solved the Diraction in the Lecture Notes on Relativistic Quantum Mechanics, and saw that the upper lower two components are large
More informationLecture 6:Feynman diagrams and QED
Lecture 6:Feynman diagrams and QED 0 Introduction to current particle physics 1 The Yukawa potential and transition amplitudes 2 Scattering processes and phase space 3 Feynman diagrams and QED 4 The weak
More informationPhysics 444: Quantum Field Theory 2. Homework 2.
Physics 444: Quantum Field Theory Homework. 1. Compute the differential cross section, dσ/d cos θ, for unpolarized Bhabha scattering e + e e + e. Express your results in s, t and u variables. Compute the
More information4. The Standard Model
4. The Standard Model Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 4. The Standard Model 1 In this section... Standard Model particle content Klein-Gordon equation Antimatter Interaction
More informationOrganisatorial Issues: Exam
Organisatorial Issues: Exam Date: - current date: Tuesday 24.07.2012-14:00 16:00h, gr. HS - alternative option to be discussed: - Tuesday 24.07.2012-13:00 15:00h, gr. HS - Friday 27.07.2012-14:00 16:00h,
More informationParticle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 5 : Electron-Proton Elastic Scattering. Electron-Proton Scattering
Particle Physics Michaelmas Term 2011 Prof Mark Thomson Handout 5 : Electron-Proton Elastic Scattering Prof. M.A. Thomson Michaelmas 2011 149 i.e. the QED part of ( q q) Electron-Proton Scattering In this
More informationWeak interactions, parity, helicity
Lecture 10 Weak interactions, parity, helicity SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 Weak decay of particles The weak interaction is also responsible for the β + -decay of atomic
More information6. QED. Particle and Nuclear Physics. Dr. Tina Potter. Dr. Tina Potter 6. QED 1
6. QED Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 6. QED 1 In this section... Gauge invariance Allowed vertices + examples Scattering Experimental tests Running of alpha Dr. Tina Potter
More informationStandard Model of Particle Physics SS 2013
Lecture: Standard Model of Particle Physics Heidelberg SS 23 Fermi Theory Standard Model of Particle Physics SS 23 2 Standard Model of Particle Physics SS 23 Weak Force Decay of strange particles Nuclear
More informationPARTICLE PHYSICS Major Option
PATICE PHYSICS Major Option Michaelmas Term 00 ichard Batley Handout No 8 QED Maxwell s equations are invariant under the gauge transformation A A A χ where A ( φ, A) and χ χ ( t, x) is the 4-vector potential
More information1 Spinor-Scalar Scattering in Yukawa Theory
Physics 610 Homework 9 Solutions 1 Spinor-Scalar Scattering in Yukawa Theory Consider Yukawa theory, with one Dirac fermion ψ and one real scalar field φ, with Lagrangian L = ψ(i/ m)ψ 1 ( µφ)( µ φ) M φ
More informationStandard Model of Particle Physics SS 2012
Lecture: Standard Model of Particle Physics Heidelberg SS 22 Fermi Theory Standard Model of Particle Physics SS 22 2 Standard Model of Particle Physics SS 22 Fermi Theory Unified description of all kind
More informationLecture 3 (Part 1) Physics 4213/5213
September 8, 2000 1 FUNDAMENTAL QED FEYNMAN DIAGRAM Lecture 3 (Part 1) Physics 4213/5213 1 Fundamental QED Feynman Diagram The most fundamental process in QED, is give by the definition of how the field
More informationIntroduction to Neutrino Physics. TRAN Minh Tâm
Introduction to Neutrino Physics TRAN Minh Tâm LPHE/IPEP/SB/EPFL This first lecture is a phenomenological introduction to the following lessons which will go into details of the most recent experimental
More informationLecture 7 From Dirac equation to Feynman diagramms. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 7 From Dirac equation to Feynman diagramms SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 Dirac equation* The Dirac equation - the wave-equation for free relativistic fermions
More informationParticle Physics WS 2012/13 ( )
Particle Physics WS 2012/13 (13.11.2012) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 226, 3.101 Content of Today Up to now: first order non-resonant e+e- cross-section dσ e.g. (e + e μ + μ
More informationSECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS
A047W SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS TRINITY TERM 05 Thursday, 8 June,.30 pm 5.45 pm 5 minutes
More informationPhysics 214 UCSD/225a UCSB Lecture 11 Finish Halzen & Martin Chapter 4
Physics 24 UCSD/225a UCSB Lecture Finish Halzen Martin Chapter 4 origin of the propagator Halzen Martin Chapter 5 Continue Review of Dirac Equation Halzen Martin Chapter 6 start with it if time permits
More information1 Introduction. 2 Relativistic Kinematics. 2.1 Particle Decay
1 Introduction Relativistic Kinematics.1 Particle Decay Due to time dilation, the decay-time (i.e. lifetime) of the particle in its restframe is related to the decay-time in the lab frame via the following
More informationStandard Model of Particle Physics SS 2013
Lecture: Standard Model of Particle Physics Heidelberg SS 23 Weak Interactions I Standard Model of Particle Physics SS 23 ors and Helicity States momentum vector in z direction u R = p, = / 2 u L = p,
More informationDr Victoria Martin, Spring Semester 2013
Particle Physics Dr Victoria Martin, Spring Semester 2013 Lecture 3: Feynman Diagrams, Decays and Scattering Feynman Diagrams continued Decays, Scattering and Fermi s Golden Rule Anti-matter? 1 Notation
More informationiδ jk q 2 m 2 +i0. φ j φ j) 2 φ φ = j
PHY 396 K. Solutions for problem set #8. Problem : The Feynman propagators of a theory follow from the free part of its Lagrangian. For the problem at hand, we have N scalar fields φ i (x of similar mass
More informationInelastic scattering
Inelastic scattering When the scattering is not elastic (new particles are produced) the energy and direction of the scattered electron are independent variables, unlike the elastic scattering situation.
More informationParticle Physics. Michaelmas Term 2011 Prof. Mark Thomson. Handout 2 : The Dirac Equation. Non-Relativistic QM (Revision)
Particle Physics Michaelmas Term 2011 Prof. Mark Thomson + e - e + - + e - e + - + e - e + - + e - e + - Handout 2 : The Dirac Equation Prof. M.A. Thomson Michaelmas 2011 45 Non-Relativistic QM (Revision)
More informationPhysics 217 Solution Set #5 Fall 2016
Physics 217 Solution Set #5 Fall 2016 1. Repeat the computation of problem 3 of Problem Set 4, but this time use the full relativistic expression for the matrix element. Show that the resulting spin-averaged
More information2 Feynman rules, decay widths and cross sections
2 Feynman rules, decay widths and cross sections 2.1 Feynman rules Normalization In non-relativistic quantum mechanics, wave functions of free particles are normalized so that there is one particle in
More informationSECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS
754 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS TRINITY TERM 04 Thursday, 9 June,.30 pm 5.45 pm 5 minutes
More informationParticle Physics Dr. Alexander Mitov Handout 2 : The Dirac Equation
Dr. A. Mitov Particle Physics 45 Particle Physics Dr. Alexander Mitov µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - Handout 2 : The Dirac Equation Dr. A. Mitov Particle Physics 46 Non-Relativistic
More informationH&M Chapter 5 Review of Dirac Equation
HM Chapter 5 Review of Dirac Equation Dirac s Quandary Notation Reminder Dirac Equation for free particle Mostly an exercise in notation Define currents Make a complete list of all possible currents Aside
More informationPHY 396 K. Solutions for homework set #9.
PHY 396 K. Solutions for homework set #9. Problem 2(a): The γ 0 matrix commutes with itself but anticommutes with the space-indexed γ 1,2,3. At the same time, the parity reflects the space coordinates
More information1 The muon decay in the Fermi theory
Quantum Field Theory-I Problem Set n. 9 UZH and ETH, HS-015 Prof. G. Isidori Assistants: K. Ferreira, A. Greljo, D. Marzocca, A. Pattori, M. Soni Due: 03-1-015 http://www.physik.uzh.ch/lectures/qft/index1.html
More informationQFT Perturbation Theory
QFT Perturbation Theory Ling-Fong Li (Institute) Slide_04 1 / 43 Interaction Theory As an illustration, take electromagnetic interaction. Lagrangian density is The combination L = ψ (x ) γ µ ( i µ ea µ
More informationChapter 17 The bilinear covariant fields of the Dirac electron. from my book: Understanding Relativistic Quantum Field Theory.
Chapter 17 The bilinear covariant fields of the Dirac electron from my book: Understanding Relativistic Quantum Field Theory Hans de Vries November 10, 008 Chapter Contents 17 The bilinear covariant fields
More information7 Quantized Free Dirac Fields
7 Quantized Free Dirac Fields 7.1 The Dirac Equation and Quantum Field Theory The Dirac equation is a relativistic wave equation which describes the quantum dynamics of spinors. We will see in this section
More informationIntroduction to particle physics Lecture 6
Introduction to particle physics Lecture 6 Frank Krauss IPPP Durham U Durham, Epiphany term 2009 Outline 1 Fermi s theory, once more 2 From effective to full theory: Weak gauge bosons 3 Massive gauge bosons:
More informationPhysics 217 FINAL EXAM SOLUTIONS Fall u(p,λ) by any method of your choosing.
Physics 27 FINAL EXAM SOLUTIONS Fall 206. The helicity spinor u(p, λ satisfies u(p,λu(p,λ = 2m. ( In parts (a and (b, you may assume that m 0. (a Evaluate u(p,λ by any method of your choosing. Using the
More informationIntercollegiate post-graduate course in High Energy Physics. Paper 1: The Standard Model
Brunel University Queen Mary, University of London Royal Holloway, University of London University College London Intercollegiate post-graduate course in High Energy Physics Paper 1: The Standard Model
More informationParticle Physics I Lecture Exam Question Sheet
Particle Physics I Lecture Exam Question Sheet Five out of these 16 questions will be given to you at the beginning of the exam. (1) (a) Which are the different fundamental interactions that exist in Nature?
More informationStandard Model of Particle Physics SS 2013
Lecture: Standard Model of Particle Physics Heidelberg SS 013 Weak Interactions II 1 Important Experiments Wu-Experiment (1957): radioactive decay of Co60 Goldhaber-Experiment (1958): radioactive decay
More informationCurrents and scattering
Chapter 4 Currents and scattering The goal of this remaining chapter is to investigate hadronic scattering processes, either with leptons or with other hadrons. These are important for illuminating the
More information5 Infrared Divergences
5 Infrared Divergences We have already seen that some QED graphs have a divergence associated with the masslessness of the photon. The divergence occurs at small values of the photon momentum k. In a general
More informationJackson, Classical Electrodynamics, Section 14.8 Thomson Scattering of Radiation
High Energy Cross Sections by Monte Carlo Quadrature Thomson Scattering in Electrodynamics Jackson, Classical Electrodynamics, Section 14.8 Thomson Scattering of Radiation Jackson Figures 14.17 and 14.18:
More informationAs usual, these notes are intended for use by class participants only, and are not for circulation. Week 6: Lectures 11, 12
As usual, these notes are intended for use by class participants only, and are not for circulation Week 6: Lectures, The Dirac equation and algebra March 5, 0 The Lagrange density for the Dirac equation
More informationQuantum Physics 2006/07
Quantum Physics 6/7 Lecture 7: More on the Dirac Equation In the last lecture we showed that the Dirac equation for a free particle i h t ψr, t = i hc α + β mc ψr, t has plane wave solutions ψr, t = exp
More informationLecture 4 Quantum Electrodynamics (QED)
Lecture 4 Quantum Electrodynamics (QED) An introduc9on to the quantum field theory of the electromagne9c interac9on 22/1/10 Par9cle Physics Lecture 4 Steve Playfer 1 The Feynman Rules for QED Incoming
More informationQuantum Electrodynamics 1 D. E. Soper 2 University of Oregon Physics 666, Quantum Field Theory April 2001
Quantum Electrodynamics D. E. Soper University of Oregon Physics 666, Quantum Field Theory April The action We begin with an argument that quantum electrodynamics is a natural extension of the theory of
More informationLecture 4 - Relativistic wave equations. Relativistic wave equations must satisfy several general postulates. These are;
Lecture 4 - Relativistic wave equations Postulates Relativistic wave equations must satisfy several general postulates. These are;. The equation is developed for a field amplitude function, ψ 2. The normal
More informationInteractions... + similar terms for µ and τ Feynman rule: gauge-boson propagator: ig 2 2 γ λ(1 γ 5 ) = i(g µν k µ k ν /M 2 W ) k 2 M 2 W
Interactions... L W-l = g [ νγµ (1 γ 5 )ew µ + +ēγ µ (1 γ 5 )νwµ ] + similar terms for µ and τ Feynman rule: e λ ig γ λ(1 γ 5 ) ν gauge-boson propagator: W = i(g µν k µ k ν /M W ) k M W. Chris Quigg Electroweak
More informationLecture 8 Feynman diagramms. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 8 Feynman diagramms SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 Photon propagator Electron-proton scattering by an exchange of virtual photons ( Dirac-photons ) (1) e - virtual
More informationFundamental Interactions (Forces) of Nature
Chapter 14 Fundamental Interactions (Forces) of Nature Interaction Gauge Boson Gauge Boson Mass Interaction Range (Force carrier) Strong Gluon 0 short-range (a few fm) Weak W ±, Z M W = 80.4 GeV/c 2 short-range
More informationLecture 4 - Dirac Spinors
Lecture 4 - Dirac Spinors Schrödinger & Klein-Gordon Equations Dirac Equation Gamma & Pauli spin matrices Solutions of Dirac Equation Fermion & Antifermion states Left and Right-handedness Non-Relativistic
More informationDerivation of Electro Weak Unification and Final Form of Standard Model with QCD and Gluons 1 W W W 3
Derivation of Electro Weak Unification and Final Form of Standard Model with QCD and Gluons 1 W 1 + 2 W 2 + 3 W 3 Substitute B = cos W A + sin W Z 0 Sum over first generation particles. up down Left handed
More information752 Final. April 16, Fadeev Popov Ghosts and Non-Abelian Gauge Fields. Tim Wendler BYU Physics and Astronomy. The standard model Lagrangian
752 Final April 16, 2010 Tim Wendler BYU Physics and Astronomy Fadeev Popov Ghosts and Non-Abelian Gauge Fields The standard model Lagrangian L SM = L Y M + L W D + L Y u + L H The rst term, the Yang Mills
More informationRotation Eigenvectors and Spin 1/2
Rotation Eigenvectors and Spin 1/2 Richard Shurtleff March 28, 1999 Abstract It is an easily deduced fact that any four-component spin 1/2 state for a massive particle is a linear combination of pairs
More informationParticle Physics Lecture 1 : Introduction Fall 2015 Seon-Hee Seo
Particle Physics Lecture 1 : Introduction Fall 2015 Seon-Hee Seo Particle Physics Fall 2015 1 Course Overview Lecture 1: Introduction, Decay Rates and Cross Sections Lecture 2: The Dirac Equation and Spin
More informationFeynman Diagrams. e + e µ + µ scattering
Feynman Diagrams Pictorial representations of amplitudes of particle reactions, i.e scatterings or decays. Greatly reduce the computation involved in calculating rate or cross section of a physical process,
More informationBack to Gauge Symmetry. The Standard Model of Par0cle Physics
Back to Gauge Symmetry The Standard Model of Par0cle Physics Laws of physics are phase invariant. Probability: P = ψ ( r,t) 2 = ψ * ( r,t)ψ ( r,t) Unitary scalar transformation: U( r,t) = e iaf ( r,t)
More informationQFT Perturbation Theory
QFT Perturbation Theory Ling-Fong Li Institute) Slide_04 1 / 44 Interaction Theory As an illustration, take electromagnetic interaction. Lagrangian density is The combination is the covariant derivative.
More informationParticle Notes. Ryan D. Reece
Particle Notes Ryan D. Reece July 9, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation that
More informationMaxwell s equations. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationLecture notes for FYS610 Many particle Quantum Mechanics
UNIVERSITETET I STAVANGER Institutt for matematikk og naturvitenskap Lecture notes for FYS610 Many particle Quantum Mechanics Note 20, 19.4 2017 Additions and comments to Quantum Field Theory and the Standard
More informationPHY 396 K. Solutions for problem set #11. Problem 1: At the tree level, the σ ππ decay proceeds via the Feynman diagram
PHY 396 K. Solutions for problem set #. Problem : At the tree level, the σ ππ decay proceeds via the Feynman diagram π i σ / \ πj which gives im(σ π i + π j iλvδ ij. The two pions must have same flavor
More informationThe path integral for photons
The path integral for photons based on S-57 We will discuss the path integral for photons and the photon propagator more carefully using the Lorentz gauge: as in the case of scalar field we Fourier-transform
More informationLoop corrections in Yukawa theory based on S-51
Loop corrections in Yukawa theory based on S-51 Similarly, the exact Dirac propagator can be written as: Let s consider the theory of a pseudoscalar field and a Dirac field: the only couplings allowed
More informationParticle Physics. experimental insight. Paula Eerola Division of High Energy Physics 2005 Spring Semester Based on lectures by O. Smirnova spring 2002
experimental insight e + e - W + W - µνqq Paula Eerola Division of High Energy Physics 2005 Spring Semester Based on lectures by O. Smirnova spring 2002 Lund University I. Basic concepts Particle physics
More informationarxiv: v1 [hep-ph] 31 Jan 2018
Noname manuscript No. (will be inserted by the editor) Polarization and dilepton angular distribution in pion-nucleon collisions Miklós Zétényi Enrico Speranza Bengt Friman Received: date / Accepted: date
More informationQuantum Electrodynamics Test
MSc in Quantum Fields and Fundamental Forces Quantum Electrodynamics Test Monday, 11th January 2010 Please answer all three questions. All questions are worth 20 marks. Use a separate booklet for each
More informationFinal Exam Solutions
Final Exam Solutions. Pair production of electrons from two photons. a) I refer to the initial four-momentum of the cosmic ray photon by q µ and the photon in the background q µ. The requirement for the
More informationRelativistic quantum mechanics
Relativistic quantum mechanics Lectures at the HADES summerschool Oberreifenberg 2002 Bengt Friman GSI 1 Introduction When we are dealing with microscopic problems at relativistic energies, we have to
More informationQED and the Standard Model Autumn 2014
QED and the Standard Model Autumn 2014 Joel Goldstein University of Bristol Joel.Goldstein@bristol.ac.uk These lectures are designed to give an introduction to the gauge theories of the standard model
More informationDiscrete Transformations: Parity
Phy489 Lecture 8 0 Discrete Transformations: Parity Parity operation inverts the sign of all spatial coordinates: Position vector (x, y, z) goes to (-x, -y, -z) (eg P(r) = -r ) Clearly P 2 = I (so eigenvalues
More informatione e Collisions at ELIC
Physics With Collisions at ELIC Collisions at ELIC E. Chudakov (JLab), June 26, 26 Opportunity to build a collider using the ELIC ring Physics motivation for a high luminosity, polarized collider Discussion
More informationL = 1 2 µφ µ φ m2 2 φ2 λ 0
Physics 6 Homework solutions Renormalization Consider scalar φ 4 theory, with one real scalar field and Lagrangian L = µφ µ φ m φ λ 4 φ4. () We have seen many times that the lowest-order matrix element
More information3P1a Quantum Field Theory: Example Sheet 1 Michaelmas 2016
3P1a Quantum Field Theory: Example Sheet 1 Michaelmas 016 Corrections and suggestions should be emailed to B.C.Allanach@damtp.cam.ac.uk. Starred questions may be handed in to your supervisor for feedback
More information3 Quantization of the Dirac equation
3 Quantization of the Dirac equation 3.1 Identical particles As is well known, quantum mechanics implies that no measurement can be performed to distinguish particles in the same quantum state. Elementary
More informationParticle Physics: Introduction to the Standard Model
Particle Physics: Introduction to the Standard Model Overview of the Standard Model Frédéric Machefert frederic@cern.ch Laboratoire de l accélérateur linéaire (CNRS) Cours de l École Normale Supérieure
More informationLecture 7. both processes have characteristic associated time Consequence strong interactions conserve more quantum numbers then weak interactions
Lecture 7 Conserved quantities: energy, momentum, angular momentum Conserved quantum numbers: baryon number, strangeness, Particles can be produced by strong interactions eg. pair of K mesons with opposite
More informationRelativistic Waves and Quantum Fields
Relativistic Waves and Quantum Fields (SPA7018U & SPA7018P) Gabriele Travaglini December 10, 2014 1 Lorentz group Lectures 1 3. Galileo s principle of Relativity. Einstein s principle. Events. Invariant
More informationFeynman Amplitude for Dirac and Majorana Neutrinos
EJTP 13, No. 35 (2016) 73 78 Electronic Journal of Theoretical Physics Feynman Amplitude for Dirac and Majorana Neutrinos Asan Damanik Department of Physics Education Faculty of Teacher Training and Education,
More informationPHYSICS OF NEUTRINOS. Concha Gonzalez-Garcia. Nufact07 Summer Institute, July (ICREA-University of Barcelona & YITP-Stony Brook)
PHYSICS OF MASSIVE NEUTRINOS (ICREA-University of Barcelona & YITP-Stony Brook) Nufact07 Summer Institute, July 2007 Plan of Lectures I. Standard Neutrino Properties and Mass Terms (Beyond Standard) II.
More information1.7 Plane-wave Solutions of the Dirac Equation
0 Version of February 7, 005 CHAPTER. DIRAC EQUATION It is evident that W µ is translationally invariant, [P µ, W ν ] 0. W is a Lorentz scalar, [J µν, W ], as you will explicitly show in homework. Here
More informationCornell University, Department of Physics
Cornell University, Department of Physics May 18th, 2017 PHYS 4444, Particle physics, Final exam You have two and a half hours for the exam. The questions are short and do not require long calculations.
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Monday 7 June, 004 1.30 to 4.30 PAPER 48 THE STANDARD MODEL Attempt THREE questions. There are four questions in total. The questions carry equal weight. You may not start
More informationPHY 396 K. Solutions for problem set #6. Problem 1(a): Starting with eq. (3) proved in class and applying the Leibniz rule, we obtain
PHY 396 K. Solutions for problem set #6. Problem 1(a): Starting with eq. (3) proved in class and applying the Leibniz rule, we obtain γ κ γ λ, S µν] = γ κ γ λ, S µν] + γ κ, S µν] γ λ = γ κ( ig λµ γ ν ig
More information