Physics 56400 Introduction to Elementary I

Lecture 16 Fall 2019 Semester Prof. Matthew Jones Review of Lecture 15

• When we introduced a (classical) electromagnetic field, the Dirac equation became inhomogeneous: 휇 휇 0 훾 푝휇 − 푚 휓 푝 = −푒훾 퐴휇휓 푝 = 훾 푉휓 푝 • We wanted to solve this using time dependent perturbation theory. • We expressed matrix elements in the basis of asymptotically .free states, |푖ۧ 3 2휋 푑 푝푖 2 푑Γ = ෑ 3 푓 푉 푖 푀 2휋 2퐸푖 푖 3 2휋 푑 푝푖 2 푑휎 = ෑ 3 푓 푉 푖 |푣퐴퐵| ∙ 2퐸퐴 ∙ 2퐸퐵 2휋 2퐸푖 푖 Free Photon States

• Next, we need a way to describe free photons. • Photons are quanta of the electromagnetic field and are manifest as oscillating 퐸 and 퐵 vector field components. • These components must transform as expected under rotations and Lorentz boosts. • Photons are massless so there are only two polarization states, right-circular (+) and left-circular (-). • If we describe a photon state by a 4-vector 휀휇(푘) then we also need to specify constraints to reduce the number of degrees of freedom from 4 to 2. Free Photon States

• The 퐸 and 퐵 fields can be expressed in terms of the vector potential:

0 −퐸푥 −퐸푦 −퐸푧 퐸푥 0 −퐵푧 퐵푦 퐹휇휈 = 휕휇퐴휈 − 휕휈퐴휇 = 퐸푦 퐵푧 0 −퐵푥 퐸푧 −퐵푦 퐵푥 0 • Maxwell’s equations: 휇휈 휈 휕휇퐹 = 푗 • Gauge condition: 휇 휕휇퐴 = 0 • In free space, 푗휈 = 0 휇휈 휇 휈 휕휇퐹 = 휕휇휕 퐴 Free Photon States

• We can look for solutions that are of the form 퐴휇 = 휀휇푒−푖푘∙푥 • Gauge condition: 휇 휇 −푖푘∙푥 휕휇퐴 = 푘휇휀 푒 = 0 • Polarization vectors: 휀(+) = −(0,1, 푖, 0)/ 2 휀(−) = (0,1, −푖, 0)/ 2 • Inhomogeneous equation: 휇휈 휇 휈 휈 휈 휕휇퐹 = 휕휇휕 퐴 = 푗 = −푒(휓ത푓훾 휓푖) • Fourier transform: 2 휈 휈 푞 퐴 = −푒 푢푓훾 푢푖 Feynman Rules for QED

• An current creates an electromagnetic field, which interacts with another electron current: 푢푖 푢ത 푖푒훾휇 푓

푖푔휇휈 − 푞2

휈 푖푒훾 푢ഥ′ 푢′푖 푓 Feynman Rules for QED

• The structure of the transition amplitude can be expressed graphically using the following rules: – External fermions: • Incoming fermion 푢 • Outgoing fermion 푢 – External anti-fermions: • Incoming anti-fermion 푣 • Outgoing anti-fermion 푣 – External photons • Incoming photon 휀휇 • ∗ Outgoing photon 휀휇 Feynman Rules for QED

• The structure of the transition amplitude can be expressed graphically using the following rules: – There is only one kind of vertex in QED:

푖푒훾휇 Feynman Rules for QED

• The structure of the transition amplitude can be expressed graphically using the following rules: – Internal photon line (photon propagator):

푖푔휇휈 − 푞2

– Internal fermion line (fermion propagator):

푖(푝 + 푚) 푝2 − 푚2 Examples of Feynman Amplitudes

• Electron- scattering

′ 푢(푘) 푖푒훾휇 푢(푘 )

푖푔휇휈 − 푞2

휈 ′ 푢(푝) 푖푒훾 푢(푝 ) 푖푔휇휈 −푖ℳ = 푢 푘′ 푖푒훾휇 푢 푘 − 푢 푝′ 푖푒훾휈 푢 푝 푞2 Examples of Feynman Amplitudes

• Electron-electron scattering

′ 푢(푘) 푖푒훾휇 푢(푘 )

푖푔휇휈 − 푞2

휈 ′ 푢(푝) 푖푒훾 푢(푝 ) 푖푔휇휈 −푖ℳ = 푢 푘′ 푖푒훾휇 푢 푘 − 푢 푝′ 푖푒훾휈 푢 푝 1 푞2 Examples of Feynman Amplitudes

• Electron-electron scattering

′ 푢(푘) 푖푒훾휇 푢(푘 )

푖푔휇휈 − 푞2

휈 ′ 푢(푝) 푖푒훾 푢(푝 ) 푖푔휇휈 −푖ℳ = 푢 푝′ 푖푒훾휇 푢 푘 − 푢 푝′ 푖푒훾휈 푢 푝 2 푞2 Examples of Feynman Amplitudes

• 푒+푒− → 푓푓 (but not 푒+푒− → 푒+푒−) ′ 푣(푘′) 푢(푝 )

푖푔휇휈 − 푞2 푖푒훾휇 푖푒훾휈

푢(푘) 푣(푝)

푖푔휇휈 −푖ℳ = 푣 푘 푖푒훾휇 푢 푘 − 푢 푝′ 푖푒훾휈 푣 푝 푞2 Examples of Feynman Amplitudes

• : 푒+푒− → 푒+푒− ′ 푣(푘) 푢(푝 )

푖푔휇휈 − 푞2 푖푒훾휇 푖푒훾휈

′ 푢(푝) 푣(푘 )

푖푔휇휈 −푖ℳ = 푣 푘 푖푒훾휇 푢 푝 − 푢 푝′ 푖푒훾휈 푣 푘′ 1 푞2 Examples of Feynman Amplitudes

• Bhabha scattering: 푒+푒− → 푒+푒− ′ 푢(푘) 푖푒훾휇 푢(푘 )

푖푔휇휈 − 푞2

휈 ′ 푣(푝) 푖푒훾 푣(푝 )

푖푔휇휈 −푖ℳ = 푢 푘′ 푖푒훾휇 푢 푘 − 푣 푝 푖푒훾휈 푣 푝′ 2 푞2 Examples of Feynman Amplitudes

• : 휀∗(푘′) 휀휇(푘) 휈 푖(푞 + 푚) 푞2 − 푚2 푖푒훾휇 푖푒훾휈

푢(푝) 푢(푝′)

푖 푞+푚 −푖ℳ = 푢 푝′ 푖푒훾휈 푖푒훾휇 푢 푝 휀∗(푘′)휀 (푘) 1 푞2−푚2 휈 휇 푞 = 푝 + 푘 = 푝′ + 푘′ Examples of Feynman Amplitudes

• Compton scattering: 휀∗(푘′) 휀휇(푘) 휈

휇 푖푒훾휈 푖푒훾 푖(푞 + 푚) 푞2 − 푚2

푢(푝) 푢(푝′) 푖 푞+푚 −푖ℳ = 푢 푝′ 푖푒훾휈 푖푒훾휇 푢 푝 휀∗(푘′)휀 (푘) 2 푞2−푚2 휈 휇 푞 = 푝 − 푘′ = 푝′ + 푘 Probability Amplitudes

• Expressions for cross sections and decay rates include the term ℳ 2 • When more than one Feynman diagram contributes the amplitudes are summed: 2 2 2 2 ∗ ℳ = ℳ1 + ℳ2 = ℳ1 + ℳ2 + 2Re ℳ1 ℳ2 • Sum over initial and final state spins/polarizations:

푠 푠 휇 ෍ 푢 푝 푢ത 푝 = 푝 훾휇 + 푚 푠

푠 푠 휇 ෍ 푣 푝 푣ҧ 푝 = 푝 훾휇 − 푚 푠

휆 ∗ 휆 ෍ 휀휇 휀휈 = −푔휇휈 휆 Probability Amplitudes

• The spin-averaged matrix elements can be simplified using trace algebra. • For electron-muon scattering 푖푔휇휈 −푖ℳ = 푢 푘′ 푖푒훾휇 푢 푘 − 푢 푝′ 푖푒훾휈 푢 푝 푞2 4 2 푒 ℳ = ෍ 푢 푘′ 훾휇푢 푘 푢 푘 훾휈푢 푘′ ∙ 푢 푝′ 훾 푢 푝 푢 푝 훾 푢 푝′ 푞4 휇 휈 푠푝푖푛푠 푒4 = Tr 푘′ + 푚 훾휇 푘 + 푚 훾휈 ∙ Tr 푝′ + 푀 훾 푝 + 푀 훾 푞4 휇 휈 • This needs to be divided by 4 to average over initial state spins. Probability Amplitudes

• Anti-commutation relations: 훾휇, 훾휈 = 2푔휇휈 • Trace theorems: – Trace of an odd number of gamma matrices vanishes – 푇푟 퐼 = 4 – 푇푟 훾휇훾휈 = 4푔휇휈  푇푟 푎푏 = 4푎 ∙ 푏 – 푇푟 푎푏푐푑 = 4 푎 ∙ 푏 푐 ∙ 푑 − 푎 ∙ 푐 푏 ∙ 푑 + 푎 ∙ 푑 푏 ∙ 푐 – 푇푟 푎훾휇푏훾휈 = 4[푎휇푏휈 + 푎휈푏휇 − 푎 ∙ 푏푔휇휈] Probability Amplitudes

• At high energies we might ignore the masses. • Electron-muon scattering: 4 2 8푒 ℳ = 푘′ ∙ 푝′ 푘 ∙ 푝 + 푘′ ∙ 푝 푘 ∙ 푝′ 푞4 • Mandlestam variables: 푠 ≡ 푘 + 푝 2 ≈ 2푘 ∙ 푝 ≈ 2푘′ ∙ 푝′ 푡 ≡ 푘 − 푘′ 2 ≈ −2푘 ∙ 푘′ ≈ −2푝 ∙ 푝′ 푢 ≡ 푘 − 푝′ 2 ≈ −2푘 ∙ 푝′ ≈ −2푘′ ∙ 푝 • Recall that in this case, 푞2 = 푘 − 푘′ 2 = 푡 Probability Amplitudes

• Electron-muon scattering at high energies: 2 2 2 푠 + 푢 ℳ = 2푒4 푡2 • Other probability amplitudes can be deduced by relabeling the 4-vectors: 푢(푘) 푖푒훾휇 푢(푘′) 푣(푝) 푢(푘′) 푖푔휇휈 − 2 2 ′ 2 푖푔휇휈 푞 푞 = 푘 − 푘 − 휇 휈 푞2 푖푒훾 푖푒훾

휈 ′ 푞2 = 푘 + 푝 2 푢(푝) 푖푒훾 푢(푝 ) 푢(푘) 푣(푝′) Probability Amplitudes

• Fermion pair-production by 푒+푒− scattering: 2 2 2 푡 + 푢 ℳ = 2푒4 푠2 • Center-of-mass energy is 퐸푐푚 = 푠 • Assume the initial particles collide along the z-axis 푡 = −2푘 ∙ 푘′ = −2푠 1 − cos 휃 푢 = −2푘 ∙ 푝′ = −2s 1 + cos 휃 2 ℳ = 16푒4(1 + cos2 휃) Probability Amplitudes

• Fermion pair-production by 푒+푒− scattering: 2 2 2 푡 + 푢 ℳ = 2푒4 푠2 • Phase-space for 22 scattering in the center-of-mass frame: 1 푝푓 푑푄 = 푑Ω 4휋2 4 푠 • Initial flux in the center-of-mass frame: 퐹 = 4푝푖 푠 • Differential : 2 4 푝 ℳ 16푒 푓 2 푑휎 = 푑푄 = 2 ∙ 1 + cos 휃 푑Ω 퐹 64휋 푠 푝푖 Differential Cross Sections

• Fermion anti-fermion pair production: 푑휎 훼2 = (1 + cos2 휃) 푑훺 4푠 • Total cross section: 4휋훼2 휎 푒+푒− → 푓푓 = 3푠 • This applies when 푓 = 휇, 휏 but when the final state consists of , we need to fix this up. Hadronic Cross Section

• Quarks can have three colors, so we have to sum over these final states. • Quarks have fractional charges which modifies one of the vertex factors. 4휋훼2 휎 푒+푒− → 푞푞 = 3푄2 × 푞 3푠 • Quarks will eventually turn into , so we just need to count events containing hadrons. • Unfortunately, we can’t tell what kind of was produced (unless we are clever) so we need to sum over all possible quark flavors. Hadronic Cross Section

• Consider the ratio of cross sections: 휎(푒+푒− → ℎ푎푑푟표푛푠) 푅 = = 3 ෍ 푄2 휎(푒+푒− → 휇+휇−) 푞 푞 • If the center-of-mass energy is insufficient to produce charm quarks, then 2 2 1 2 1 2 푅 = 3 + + = 2 3 3 3 • After crossing the charm production threshold, 2 2 1 2 1 2 2 2 10 푅 = 3 + + + = 3 3 3 3 3

• After crossing the 푏푏 production threshold, 2 2 1 2 1 2 2 2 1 2 11 푅 = 3 + + + + = 3 3 3 3 3 3 풆+풆− → 퐡퐚퐝퐫퐨퐧퐬

• The simple picture is modified slightly due to o Threshold effects o Resonances o Radiative corrections (which can be calculated) • Deviations from this picture must be due to new physics not included in the model.