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.