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Feynman Diagrams and How to Use Them

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Feynman Diagrams and How to Use Them Introduc)on to the famous Feynman diagrams and how to use them Rosi Reed First Diagram Diagrams introduced at the the Pocono Conference in early April, 1948 • An hours drive from here! • Discussion over the problems in QED First published in May, 1949 Rosi Reed – Feynman 2 Feynman Van Rosi Reed – Feynman 3 Imagery and Understanding Feynman always thought about the representaon of the problems he studied • Invented his own trigonometry symbols in high-school • He had great creave power as a problem solver Who else would have thought of drawing pictures to calculate quanIIes? Rosi Reed – Feynman 4 Quantum Electrodynamics RelavisIc quantum field theory of electrodynamics IniIal descripIon by Dirac à ensemble of harmonic oscillators w/creaon + annihilaon operators • Calculaons γ+charged parIcles were reliable only to 1st order in perturbaon theory – At higher orders infini)es emerged – Fundamental incompability between special relavity and quantum mechanics Rosi Reed – Feynman 5 Quantum Electrodynamics Shin'ichirō Tomonaga, Julian Schwinger, Richard Feynman à Fully covariant formulaons finite at any order • Nobel Prize in 1965 • Schwinger and Tomonaga’s approach was field- theoreIc and operator-based • Feynman's approach was based on his diagrams • Freeman Dyson à 2 approaches are equivalent Rosi Reed – Feynman 6 How are the Diagrams used? Rosi Reed – Feynman 7 Fermi’s Golden Rule Describes the transiIon rate from one energy eigenstate of a quantum system into other energy eigenstates in a conInuum 2 Γ fi = 2π | Tfi | ρ(Ei ) Rosi Reed – Feynman 8 Fermi’s Golden Rule Describes the transion rate from one energy eigenstate of a quantum system into other energy eigenstates in a conInuum 2 Γ fi = 2π | Tfi | ρ(Ei ) Density of States Natural Units (Kinemacs) c = ! =1 TransiIon Matrix (Diagrams) Rosi Reed – Feynman 9 Fermi’s Golden Rule Describes the transion rate from one energy eigenstate of a quantum system into other energy eigenstates in a conInuum 2 Γ fi = 2π | Tfi | ρ(Ei ) TransiIon Matrix Derived using Time Dependent (Diagrams) Perturbaon Theory ˆ ˆ ˆ Hamiltonian is split into a me H = H0 + H ' independent (H0) and an interac)on (H’) part Rosi Reed – Feynman 10 Fermi’s Golden Rule Describes the transion rate from one energy eigenstate of a quantum system into other energy eigenstates in a conInuum 2 Γ fi = 2π | Tfi | ρ(Ei ) ˆ ˆ ˆ T | Hˆ ' | H = H0 + H ' fi =< ψ f ψi > 1st order calculaon of the transiIon matrix from the iniIal state (ψi) to the final state (ψf) Rosi Reed – Feynman 11 Fermi’s Golden Rule Describes the transion rate from one energy eigenstate of a quantum system into other energy eigenstates in a conInuum 2 Γ fi = 2π | Tfi | ρ(Ei ) T =< ψ | Hˆ ' |ψ > fi f i ρ(Ei ) = ∫ δ(Ei − E)dn dn = # of states with momentum p Combine together + some normalizaon from quanIzaon (and insure Lorentz Invariance!) Rosi Reed – Feynman 12 Lecture 22 Last time we discussed the scenario a 1 + 2, but since much particle physics is done in accelerators, we want to be! able to calculate things for the scenario a + b 1 + 2. First! we should define a few things: # of particles Φ =flux= Unit time Unit area Cross-Secons ⇥# of particles n =numberdensity= Unit Volume σ = cross-section = e↵ective area The transiIon probability is related to Γ fi →σ the cross-secon (σ) σ depends on the momentum of the parIcles and the Figure 1:parIcular interacIon force On the left are the two tubes (”beams”) of particles a and b, colliding together. On the right is the cross-section area, showing the overlap between the beam and particles b. Rosi Reed – Feynman 13 From Figure 1 we can see that the volume of the overlap region can be determined as: V = A(va + vb)δt The total number of b particles in the overlap volume is: δNb = nbV = nbA(va + vb)δt The probability that a particle a interacts is the area covered by the b particles divided by the total area of the beam, also shown in 1: δNbσ nbA(va + vb)δtσ δP = = = n A(v + v )δtσ A A b a b The interaction rate is the probability of interacting per unit time: δP r = = n (v + v )σ a δt b a b The reaction rate for the entire beam of a particles is: Ra = δNara = naVra =(na(va + vb))(nbV )σ = ΦNbσ From here we can see that Rate = Flux # target particles cross-section. ⇥ ⇥ 1 Cross-Sec)on (Center of Mass) 1 p* f | M |2 d * σ = 2 * ∫ fi Ω 64π s pi p* Tfi → M fi f * p i s à Total energy in the center of mass frame Rosi Reed – Feynman 14 Cross-Sec)on (Center of Mass) 1 p* f | M |2 d * σ = 2 * ∫ fi Ω 64π s pi Calculate using diagrams! 1 α ≈ g2 ≈ EM EM 137 QED Vertex Element Time Rosi Reed – Feynman 15 Figure 1: Standard model vertices. Time runs from left to right. For the left- 2 1 most vertex (EM) the coupling constant is ↵EM gEM 137 . All charged parti- cles can interact through the EM force, and particle⇡ flavor⇡ must be conserved in all interactions. The second vertex represents the strong (QCD) force. the cou- 2 pling constant is ↵s gs 1. Only quarks can interact through QCD, but like the EM force particle⇡ flavor⇡ must be conserved in all interactions. The next two vertices are for the weak force, with the W ± being the EM charged boson and Z 2 1 being the neutral boson. The weak coupling constant is ↵W—Z gW—Z 30 . All fermions can interact through the weak force, with the W the⇡ particle flavor⇡ is always changed and with the Z the flavor is never changed. Figure 2: This is a LO diagram for the e− + e− e− + e− process in the t-channel configuration. There is a second diagram! that also contributes, the u-channel, since the particles in the final state are identical. 2 2 -hne,sneteprilsi h nlsaeaeidentical. are state final the in particles the since u-channel, -hne ofiuain hr sascn iga htas otiue,the contributes, also that diagram second a is There configuration. t-channel ! − − − − rcs nthe in process e + e e + e the for diagram LO a is This 2: Figure h ao snvrchanged. never is flavor the Z the with and changed always is l emoscnitrc hog h ekfre ihteWtepril flavor particle the W the with force, weak the through interact can fermions All ⇡ ⇡ 30 W—Z W—Z . g ↵ is constant coupling weak The boson. neutral the being 2 1 ± Z and boson charged EM the being W the with force, weak the for are vertices h Mfrepril ao utb osre nalitrcin.Tenx two next The interactions. all in conserved be must flavor particle force EM the ⇡ ⇡ s s .Ol urscnitrc hog C,btlike but QCD, through interact can quarks Only 1. g ↵ is constant pling 2 l neatos h eodvre ersnstesrn QD oc.tecou- the force. (QCD) strong the represents vertex second The QED Vertex interactions. all lscnitrc hog h Mfre n atceflvrms ecnevdin conserved be must flavor particle and force, EM the through interact can cles ⇡ ⇡ 137 EM EM l hre parti- charged . All g ↵ is constant coupling the (EM) vertex most 2 1 Fermions are parIcles iue1 tnadmdlvrie.Tm usfo ett ih.Frteleft- the For right. to left from runs Time vertices. model Standard 1: traveling forward in Ime Figure AnI-fermions are parIcles traveling backwards in Ime • Stueckelberg interpretaon Figure 1: Standard model vertices. Time runs from left to right. For the left- 2 1 most vertex (EM) the coupling constant is ↵EMTime g . All charged parti- + e+ EM 137 cles can interact throughe the EM force, and particle⇡ flavor⇡ must be conserved in all interactions. The second vertex represents the strong (QCD) force. the cou- Rosi Reed – Feynman 2 16 pling constant is ↵s gs 1. Only quarks can interact through QCD, but like the EM force particle⇡ flavor⇡ must be conserved in all interactions. The next two vertices are for the weak force, with the W ± being the EM charged boson and Z 2 1 being the neutral boson. The weak coupling constant is ↵W—Z gW—Z 30 . All fermions can interact through the weak force, with the W the⇡ particle flavor⇡ is always changed and with the Z the flavor is never changed. Figure 2: This is a LO diagram for the e− + e− e− + e− process in the t-channel configuration. There is a second diagram! that also contributes, the u-channel, since the particles in the final state are identical. 2 2 -hne,sneteprilsi h nlsaeaeidentical. are state final the in particles the since u-channel, -hne ofiuain hr sascn iga htas otiue,the contributes, also that diagram second a is There configuration. t-channel ! − − − − rcs nthe in process e + e e + e the for diagram LO a is This 2: Figure . Z 1 30 ⇡ 2 W—Z g process in the ⇡ − e . All charged parti- W—Z + 1 ↵ 137 − e h ao snvrchanged. never is flavor the Z sawy hne n ihthe with and changed always is ⇡ l emoscnitrc hog h ekfre ihteWtepril flavor particle the W the with force, weak the through interact can fermions All ! ⇡ ⇡ 2 EM 30 W—Z W—Z g . g ↵ en h eta oo.Tewa opigcntn is constant coupling weak The boson. neutral the being − 2 1 e ⇡ ± Z and boson charged EM the being W etcsaefrtewa oc,wt the with force, weak the for are vertices + being the EM charged boson and h Mfrepril ao utb osre nalitrcin.Tenx two next The interactions.
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