Instituto Superior T´ecnico

Lectures in – Lecture 2

Jorge C. Rom˜ao Instituto Superior T´ecnico, Departamento de F´ısica & CFTP A. Rovisco Pais 1, 1049-001 Lisboa, Portugal

January 2014

Jorge C. Rom˜ao IDPASC School Braga – 1 IST Summary for Lecture 2: QED as an Example

Summary ❐ QED as a gauge theory QED as a gauge theory & GF ❐ Propagators and Green functions How to find QED F.R.? − Coulomb e ❐ Feynman rules for QED Coulomb scattering e+ γ in internal lines ◆ and in external lines Higher Orders ◆ in internal lines γ in external lines QED Feynman Rules ◆ Photons in external lines Simple Processes ◆ Higher orders Compton Scattering − − e e+ → µ µ+ ❐ Example 1: Compton scattering QFT Computations ❐ Example 2: e− + e+ µ− + µ+ in QED →

Jorge C. Rom˜ao IDPASC School Braga – 2 IST Local invariance of Dirac equation

Summary ❐ We start with Dirac Lagrangian QED as a gauge theory • Local invariance • QED Lagrangian = ψ(i∂/ m)ψ Propagators & GF L −

How to find QED F.R.? − ❐ Coulomb scattering e It is invariant under global phase transformations

Coulomb scattering e+ ′ iα in internal lines ψ = e ψ, α infinitesimal δψ = iαψ, δψ = iαψ γ → − Higher Orders γ in external lines ❐ What happens if the transformations are local, α = α(x)? QED Feynman Rules

Simple Processes δψ = iα(x)ψ ; δψ = iα(x)ψ Compton Scattering − − − e e+ → µ µ+ QFT Computations ❐ We have then

δ = ψγµψ ∂ α(x) L − µ and the Lagrangian is no longer invariant

Jorge C. Rom˜ao IDPASC School Braga – 3 IST Local invariance of Dirac equation ...

Summary ❐ We see that the problem is connected with the fact that ∂µψ do not QED as a gauge theory • Local invariance transform as ψ. We are then led to the concept of covariant derivative Dµ • QED Lagrangian that transforms as the fields, Propagators & GF

How to find QED F.R.? − δDµψ = iα(x)Dµψ Coulomb scattering e

Coulomb scattering e+

γ in internal lines ❐ For the Dirac field we define, in analogy with minimal prescription,

Higher Orders

γ in external lines Dµ = ∂µ + ieAµ QED Feynman Rules

Simple Processes ❐ Compton Scattering The vector field Aµ is a field that ensures that we can choose the phase − − e e+ → µ µ+ locally. Its transformation is chosen to compensate the term proportional to QFT Computations ∂µα 1 δA = ∂ α(x) µ − e µ

❐ We are then led to the introduction of the electromagnetic field Aµ satisfying the usual gauge invariance.

Jorge C. Rom˜ao IDPASC School Braga – 4 IST QED Lagrangian

Summary ❐ This new vector field Aµ needs a kinetic term. The only term quadratic that QED as a gauge theory • Local invariance is invariant under the local gauge transformations is • QED Lagrangian

Propagators & GF Fµν = ∂µAµ ∂ν Aµ, δFµν = 0 How to find QED F.R.? − − Coulomb scattering e ❐ µ Coulomb scattering e+ A mass term of the form A Aµ is not gauge invariant, so the field Aµ

γ in internal lines () is massless

Higher Orders ❐ γ in external lines The final Lagrangian is

QED Feynman Rules Simple Processes 1 µν QED = F F + ψ(iD/ m)ψ + Compton Scattering L −4 µν − ≡Lfree Linteraction − − e e+ → µ µ+ QFT Computations where 1 = F F µν + ψ(i∂/ m)ψ, = eψγ ψAµ Lfree −4 µν − Linteraction − µ ❐ This Lagrangian is invariant under local gauge transformations and describes the interactions of electrons (and positrons) with photons. The theory is called (QED)

Jorge C. Rom˜ao IDPASC School Braga – 5 IST The non-relativistic

Summary ❐ We will follow the method of Richard Feynman to arrive at the rules for QED as a gauge theory calculations in QED. Propagators & GF • Non-relativistic Prop. • GF as propagators ❐ As a warm up exercise we start with the non-relativistic Schr¨odinger equation • S Matrix • Relativistic Prop. • New processes ∂ • Green Function i H ψ(~x, t) = 0, H = H0 + V • S Matrix elements ∂t − • In & Out states   How to find QED F.R.? where H0 is the free particle Hamiltonian − Coulomb scattering e 2 Coulomb scattering e+ H0 = ∇ γ in internal lines −2m Higher Orders γ in external lines ❐ We can rewrite the equation in the form QED Feynman Rules Simple Processes ∂ Compton Scattering i H0 ψ = V ψ − − ∂t − e e+ → µ µ+   QFT Computations ❐ For arbitrary V this equation can normally only be solved in perturbation theory

Jorge C. Rom˜ao IDPASC School Braga – 6 IST The non-relativistic propagator ...

Summary ❐ For the scattering problems we are interested we will develop a perturbative QED as a gauge theory expansion using the technique of the Green’s Functions (GF). We introduce Propagators & GF • Non-relativistic Prop. the GF for the free Schr¨odinger equation with retarded boundary condition • GF as propagators • S Matrix • ∂ Relativistic Prop. i H (~x′) G (x′,x) = δ4(x′ x),G (x′,x) = 0 for t′

γ in external lines the most general solution of the original equation

QED Feynman Rules

Simple Processes ∂ i H0 ψ = V ψ Compton Scattering ∂t − − −   e e+ → µ µ+ QFT Computations is ′ ′ ′ ′ 4 ′ ψ(~x ,t ) = φi(~x ,t ) + d x G0(x ,x)V (x)ψ(x) Z

Jorge C. Rom˜ao IDPASC School Braga – 7 IST The non-relativistic propagator ...

Summary ❐ We can use this integral equation to establish a perturbative series. Consider QED as a gauge theory that the interaction is localized, that is V (~x, t) 0 as t . Then due Propagators & GF → → −∞ • Non-relativistic Prop. to the retarded GF properties we have • GF as propagators • S Matrix ′ ′ ′ ′ • Relativistic Prop. ′ lim ψ(~x ,t ) = φi(~x ,t ) • New processes t →−∞ • Green Function • S Matrix elements • In & Out states that is in the remote past we have a plane .

How to find QED F.R.? − ❐ Coulomb scattering e Now if V is small (in some sense) we can solve the integral equation

Coulomb scattering e+ perturbatively

γ in internal lines

Higher Orders ′ ′ ′ ′ 4 ′ ψ(~x ,t ) =φi(~x ,t ) + d x1 G0(x ,x1)V (x1)φi(x1) γ in external lines Z QED Feynman Rules 4 4 ′ Simple Processes + d x1d x2 G0(x ,x1)V (x1)G0(x1,x2)V (x2)φi(x2) Compton Scattering Z − + → − + 4 4 4 ′ e e µ µ + d x1d x2d x3 G0(x ,x1)V (x1)G0(x1,x2)V (x2)G0(x2,x3)V (x3)φi(x3) QFT Computations Z + ···

Jorge C. Rom˜ao IDPASC School Braga – 8 IST The non-relativistic propagator ...

Summary ❐ We can look at the perturbative series in another way, in terms of the full QED as a gauge theory GF of the theory with interactions, G(x′,x) Propagators & GF • Non-relativistic Prop. • GF as propagators ∂ ′ ′ ′ 4 ′ • S Matrix i H0(x ) V (x ) G(x ,x) δ (x x) • Relativistic Prop. ∂t − − ≡ − • New processes   • Green Function • S Matrix elements ❐ It satisfies • In & Out states

How to find QED F.R.? − ′ ′ 4 ′ Coulomb scattering e G(x ,x) = G0(x ,x) + d yG0(x , y)V (y)G(y,x) Coulomb scattering e+ Z γ in internal lines

Higher Orders ❐ This leads to the perturbative series (small V )

γ in external lines QED Feynman Rules ′ ′ 4 ′ G(x ,x) =G0(x ,x) + d x1 G0(x ,x1)V (x1)G0(x1,x) Simple Processes

Compton Scattering Z 4 4 ′ − − e e+ → µ µ+ + d x1d x2 G0(x ,x2)V (x2)G0(x2,x1)V (x1)G0(x1,x) QFT Computations Z + ···

Jorge C. Rom˜ao IDPASC School Braga – 9 IST Green Functions as Propagators

Summary ❐ The last equation allows for suggestive graphical interpretation. We notice QED as a gauge theory ′0 0 0 0 0 that the retarded character of G0 implies x > x3 >x2 >x1 >x . Propagators & GF ··· • Non-relativistic Prop. • GF as propagators ❐ So we have the situation of the following diagrams for the first 3 terms • S Matrix • Relativistic Prop. • New processes • Green Function G(x′,x) =G (x′,x) + d4x G (x′,x )V (x )G (x ,x) • S Matrix elements 0 1 0 1 1 0 1 • In & Out states Z How to find QED F.R.? 4 4 ′ − + d x1d x2 G0(x ,x2)V (x2)G0(x2,x1)V (x1)G0(x1,x) Coulomb scattering e Z Coulomb scattering e+ + γ in internal lines ··· Higher Orders ′ ′ ′ ′ γ in external lines (~x ,t ) (~x ,t ) t ′ ′ t t QED Feynman Rules (~x ,t ) V (~x2,t2) Simple Processes ′ V (~x1,t1) Compton Scattering G0(x ,x) − − V (~x1,t1) e e+ → µ µ+

QFT Computations (~x, t) (~x, t) (~x, t) x x x

Jorge C. Rom˜ao IDPASC School Braga – 10 IST Scattering processes and the S Matrix

Summary ❐ We are interested in scattering processes. This means that in the past we QED as a gauge theory have a solution of the free equation, a plane wave with ~ki Propagators & GF • Non-relativistic Prop. • GF as propagators 1 i~ki·~x−iωit • S Matrix φi(~x, t) = 3/2 e • Relativistic Prop. (2π) • New processes • Green Function • S Matrix elements ❐ ~ • In & Out states In the future (detector) we have another plane wave with momentum kf

How to find QED F.R.? ′ ′ − 1 ~ Coulomb scattering e ′ ′ ikf ·~x −iωf t φf (~x ,t ) = 3/2 e Coulomb scattering e+ (2π)

γ in internal lines Higher Orders ❐ The relevant quantity is S matrix element (transition amplitude) γ in external lines

QED Feynman Rules 3 ′ ∗ ′ ′ ′ ′ Sfi =′ lim d x φf (~x ,t )ψ(~x ,t ) Simple Processes t →∞ Compton Scattering Z − − 3 ′ ∗ ′ ′ ′ ′ 4 ′ e e+ → µ µ+ = lim d x φf (~x ,t ) φi(~x ,t ) + d x1 G0(x ,x1)V (x1)φi(x1) + t′→∞ ··· QFT Computations Z  Z  3 ~ ~ 3 ′ 4 ∗ ′ ′ ′ =δ (kf ki)+ lim d x d x1 φf (~x ,t )G0(x ,x1)V (x1)φi(x1) + − t′→∞ ··· Z Jorge C. Rom˜ao IDPASC School Braga – 11 IST The propagator for the Relativistic Theory

′ Summary ❐ The starting point is the interpretation of G(x ,x) as the probability ′ QED as a gauge theory amplitude to propagate the particle from x to x Propagators & GF • Non-relativistic Prop. • GF as propagators ′ ′ 4 ′ G(x ,x) =G0(x ,x) + d x1 G0(x ,x1)V (x1)G0(x1,x) • S Matrix • Relativistic Prop. Z • New processes 4 4 ′ • Green Function + d x1d x2 G0(x ,x2)V (x2)G0(x2,x1)V (x1)G0(x1,x) • S Matrix elements • In & Out states Z How to find QED F.R.? ··· − Coulomb scattering e ❐ The contribution of order n corresponds to the diagram Coulomb scattering e+

γ in internal lines ◆ t x′ A particle is created at x, propagates to Higher Orders x1, interacts with the potential V (x1), γ in external lines xn propagates to x2 and so on. QED Feynman Rules Simple Processes x3 ◆ This interpretation is suited to the rela- Compton Scattering x2 − − tivistic theory because of the space-time e e+ → µ µ+ x1 emphasis instead of the Hamiltonian evo- QFT Computations x lution. x

Jorge C. Rom˜ao IDPASC School Braga – 12 IST The propagator for the Relativistic Theory: New Processes

t t t ′ Summary x x′ QED as a gauge theory x x′ Propagators & GF • Non-relativistic Prop. 3 e− − + • GF as propagators − + e e • S Matrix e e e+ • Relativistic Prop. − 2 e − • New processes e • Green Function x x • S Matrix elements 1 x 1 • In & Out states x x x How to find QED F.R.? − Coulomb scattering e ❐ The existence of a is associated with the absence of an of Coulomb scattering e+ negative γ in internal lines Higher Orders ❐ Therefore we can interpret the destruction of an positron at 3 as being the γ in external lines creation of an electron of negative energy at that point QED Feynman Rules Simple Processes ❐ This suggests (Feynman) the possibility that the amplitude to create a Compton Scattering positron at 1 and destroy it at 3 be related to the amplitude to create an − − e e+ → µ µ+ electron of negative energy at 3 and destroy it at 1 QFT Computations ❐ Then electrons of positive energy propagate to the future and electrons of negative energy (positrons) propagate back in time

Jorge C. Rom˜ao IDPASC School Braga – 13 IST The propagator for the Relativistic Theory: Green Function

Summary ❐ Let us then look for the GF of the Dirac equation in interaction with the QED as a gauge theory electromagnetic field Propagators & GF • Non-relativistic Prop. • GF as propagators • S Matrix (i∂/ eA/ m)ψ(x) = 0 • Relativistic Prop. − − • New processes • Green Function ❐ It is the solution of the equation • S Matrix elements • In & Out states ′ ′ ′ 4 ′ How to find QED F.R.? (i∂/ eA/ m)SF (x ,x) = iδ (x x) − − − − Coulomb scattering e Coulomb scattering e+ ❐ The full GF can only be obtained in perturbation theory. For the free theory γ in internal lines we have Higher Orders

γ in external lines (i∂/′ m)S (x′,x) = iδ4(x′ x) QED Feynman Rules − F − Simple Processes ′ ′ Compton Scattering ❐ Noticing that SF (x ,x) = SF (x x) and applying the Fourier transform − − e e+ → µ µ+ − 4 QFT Computations d p ′ S (x′ x) = e−ip·(x −x)S (p) F − (2π)4 F Z

Jorge C. Rom˜ao IDPASC School Braga – 14 IST The propagator for the Relativistic Theory: Green Function ...

❐ Substituting in the equation we get for SF (p) Summary

QED as a gauge theory i(p/ + m) 2 2 Propagators & GF (p/ m)SF (p) = i SF (p) = 2 2 , p = m • Non-relativistic Prop. − → p m 6 • GF as propagators − ❐ • S Matrix To complete the definition we need a prescription on how to deal with the • Relativistic Prop. singularity. This is related with the boundary conditions we want to impose • New processes • Green Function on the GF, positive propagate into the future and negative energies • S Matrix elements • In & Out states back in time.

How to find QED F.R.? − ❐ Coulomb scattering e The inverse Fourier transform is calculated using the residue theorem + Coulomb scattering e 0 3 dp d p 0 ′ 0 ′ i γ in internal lines ′ −ip (x −x) i~p·(~x −~x) SF (x x) = 3 e e 0 2 2 2 Higher Orders − 2π (2π) (p ) ( ~p + m ) Z Z − | | γ in external lines 0 QED Feynman Rules m(p ) Simple Processes I

Compton Scattering ′ − + → − + t >t lower semi-plane e e µ µ ~p 2 + m2 ′ QFT Computations | | t

Summary ❐ The localization of the poles is obtained giving a negative infinitesimal part QED as a gauge theory to m2 Propagators & GF • Non-relativistic Prop. • GF as propagators • S Matrix m2 m2 iε • Relativistic Prop. → − • New processes • Green Function • S Matrix elements ❐ With this prescription (due to Feynman) the propagator is • In & Out states

How to find QED F.R.? − (p/ + m) Coulomb scattering e 2 2 SF (p) = i 2 2 , p0 = ~p + m iε Coulomb scattering e+ p m + iε → ± | | − − γ in internal lines p  Higher Orders ❐ We can do now the integration in p0 to obtain γ in external lines

QED Feynman Rules 3 ′ Simple Processes ′ d p 1 −ip·(x −x) ′ SF (x x) = (p/ + m) e θ(t t) Compton Scattering − (2π)3 2E − − − e e+ → µ µ+ Z h ′ +( p/ + m) eip·(x −x) θ(t t′) QFT Computations − − i

Jorge C. Rom˜ao IDPASC School Braga – 16 IST The propagator for the Relativistic Theory: Green Function ...

Summary ❐ We define the normalized plane QED as a gauge theory

Propagators & GF • Non-relativistic Prop. r 1 −3/2 r −iεr p·x • GF as propagators ψp(x) = (2π) w (~p) e • S Matrix √2E • Relativistic Prop. • New processes • Green Function ❐ Then we obtain • S Matrix elements • In & Out states How to find QED F.R.? 2 − r Coulomb scattering e ′ ′ 3 r ′ SF (x x) =θ(t t) d p ψp(x )ψp(x) + − − Coulomb scattering e r=1 Z X γ in internal lines 4 Higher Orders ′ 3 r ′ r θ(t t ) d p ψp(x )ψp(x) γ in external lines − − Z r=3 QED Feynman Rules X Simple Processes ❐ ′ Compton Scattering This expresses SF (x x) as a sum of eigenfunctions of the free Dirac − − − e e+ → µ µ+ operator. From this expression is clear that the negative energy solutions ′ QFT Computations (r = 3, 4) are propagated back in time (t t)

Jorge C. Rom˜ao IDPASC School Braga – 17 IST S Matrix elements

Summary ❐ As we will be interested in scattering problems, we will be focusing in the QED as a gauge theory elements of the S matrix. To find these we start by noticing the solution of Propagators & GF • Non-relativistic Prop. the Dirac equation with interactions, • GF as propagators • S Matrix • Relativistic Prop. (i∂/ m)Ψ = eA/Ψ • New processes − • Green Function • S Matrix elements can be written, in analogy with the non-relativistic case, • In & Out states

How to find QED F.R.? − 4 Coulomb scattering e Ψ(x) = ψ(x) ie d y S (x y)A/(y)Ψ(y) − F − Coulomb scattering e+ Z γ in internal lines ❐ Using the expression for S (x y) we get Higher Orders F − γ in external lines 2 QED Feynman Rules 3 r 4 r Simple Processes lim Ψ(x) ψ(x) = d p ψp(x) ie d y ψp(y)A/(y)Ψ(y) t→+∞ − − r=1 Compton Scattering Z X  Z  − − e e+ → µ µ+ 4 QFT Computations 3 r 4 r lim Ψ(x) ψ(x) = d p ψp(x) +ie d y ψp(y)A/(y)Ψ(y) t→−∞ − r=3 Z X  Z 

Jorge C. Rom˜ao IDPASC School Braga – 18 IST S Matrix elements ...

Summary ❐ This again shows that positive energies are scattered into the future and QED as a gauge theory negative energy solutions in the past. Propagators & GF • Non-relativistic Prop. • GF as propagators ❐ Using now the S matrix definition • S Matrix • Relativistic Prop. • New processes • Green Function 3 † • Sfi = lim d x ψf (x)Ψi(x) S Matrix elements t→εf ∞ • In & Out states Z How to find QED F.R.? we get − Coulomb scattering e

Coulomb scattering e+ γ in internal lines 4 Sfi = δfi ieεf d y ψ (y)A/(y)Ψi(y) Higher Orders − f γ in external lines Z QED Feynman Rules where εf = +1 for positive energies in the future (final state) and εf = 1 Simple Processes − Compton Scattering for negative energies into the past (initial state). ψf is a plane wave with the − − e e+ → µ µ+ appropriate quantum numbers for the final state.

QFT Computations ❐ This is the main result the we use in the following

Jorge C. Rom˜ao IDPASC School Braga – 19 IST Initial and Final states

Summary ❐ The description of initial and final states is as follows QED as a gauge theory Propagators & GF ◆ • Non-relativistic Prop. Initial state • GF as propagators • S Matrix • Relativistic Prop. • New processes 1 1 −ipi·x • Green Function electron ψi = u(pi,si) e • S Matrix elements → √2E √V • In & Out states

How to find QED F.R.? − 1 1 ipf ·x Coulomb scattering e positron ψ = v(p ,s ) e → i √ √ f f Coulomb scattering e+ 2E V

γ in internal lines ◆ Higher Orders Final state

γ in external lines

QED Feynman Rules

Simple Processes 1 1 −ipf ·x electron ψf = u(pf ,sf ) e Compton Scattering → √2E √V − − e e+ → µ µ+

QFT Computations 1 1 ipi·x positron ψf = v(pi,si) e → √2E √V

Jorge C. Rom˜ao IDPASC School Braga – 20 IST Initial and Final states

Summary ❐ The conventions are spelled out in the following figure QED as a gauge theory

Propagators & GF • Non-relativistic Prop. + • GF as propagators − (pf ,sf ) e (pf ,sf ) • S Matrix e • Relativistic Prop. • New processes • Green Function • S Matrix elements • In & Out states

How to find QED F.R.? − Coulomb scattering e

Coulomb scattering e+

γ in internal lines − + Higher Orders e e (pi,si) (pi,si) γ in external lines

QED Feynman Rules Simple Processes ❐ We have chosen the normalization in a box of volume V Compton Scattering − − e e+ → µ µ+ 1 1 1 QFT Computations d3x ψ†ψ = u†(p ,s )u(p ,s ) d3x = d3x = 1 i i V 2E i i i i V ZV i ZV ZV

Jorge C. Rom˜ao IDPASC School Braga – 21 IST How to find Feynman Rules for QED

Summary ❐ We are going here to start from the central result QED as a gauge theory

Propagators & GF

How to find QED F.R.? 4 Sfi = ieεf d yψ (y)A/ (y)Ψi(y)(i = f) − f Coulomb scattering e − 6 Z Coulomb scattering e+ γ in internal lines and derive a set of rules (Feynman Rules) that will show us how to calculate Higher Orders in QED γ in external lines QED Feynman Rules ❐ For that we will consider: Simple Processes Compton Scattering ◆ Electrons in external legs: Coulomb scattering for e−: − − e e+ → µ µ+ e−+ Nuclei(Z) e−+ Nuclei(Z) QFT Computations → ◆ Positrons in external legs: Coulomb scattering for e+: e++ Nuclei(Z) e++ Nuclei(Z) → ◆ Photons in internal lines: e−µ− e−µ− → ◆ Higher order processes: e−µ− e−µ− → ◆ Photons in external legs: Compton scattering: γ + e− γ + e− →

Jorge C. Rom˜ao IDPASC School Braga – 22 IST Coulomb Scattering for Electrons

Summary ❐ We consider Coulomb scattering by a fixed Nuclei(Z), that is, by a classical QED as a gauge theory electromagnetic Coulomb potential (not quantized) Propagators & GF

How to find QED F.R.? − 0 Ze Coulomb scattering e A (x) = − , A~(x) = 0, e < 0 • Amplitude 4π ~x • Probability | | • Rate & Flux • Cross Section ❐ In lowest order we approximate Ψ (x) by a plane wave • Traces i • Theorems on traces • Mott formula 1 1 −ipi·x Coulomb scattering e+ Ψi(x) = u(pi,si)e √2Ei √V γ in internal lines

Higher Orders ❐ γ in external lines For the final state we take

QED Feynman Rules

Simple Processes Compton Scattering 1 1 ψ (x) = u(p ,s )eipf ·x − + → − + f f f e e µ µ 2Ef √V QFT Computations p

Jorge C. Rom˜ao IDPASC School Braga – 23 IST Coulomb Scattering for Electrons: The Amplitude & ...

Summary ❐ The S matrix amplitude between the initial and final state is QED as a gauge theory

Propagators & GF 2 i(pf −pi)·x ie Z 1 1 0 4 e How to find QED F.R.? Sfi = u(pf ,sf )γ u(pi,si) d x − Coulomb scattering e 4π V 2Ei2Ef ~x • Amplitude Z | | • Probability p • Rate & Flux ❐ The integration can done (~q = ~pf ~pi is the transferred momentum) and we • Cross Section − • Traces get the final result • Theorems on traces • Mott formula 0 1 1 u(pf ,sf )γ u(pi,si) Coulomb scattering e+ 2 Sfi = iZe 2 2πδ(Ef Ei) γ in internal lines V 4E E ~q − i f | | Higher Orders p γ in external lines ❐ We notice that we are assuming the nuclei fixed, so we have only energy QED Feynman Rules conservation Simple Processes 3 Compton Scattering ❐ 3 d pf The number of final states in the interval d pf is V 3 , and therefore the − + → − + (2π) e e µ µ probability for the particle to go into one of these states is QFT Computations d3p P = S 2 V f fi | fi | (2π)3

Jorge C. Rom˜ao IDPASC School Braga – 24 IST Coulomb Scattering for Electrons: The Amplitude & ...

❐ Putting everything together we have Summary QED as a gauge theory 2 2 0 2 3 Z (4πα) u(pf ,sf )γ u(pi,si) d pf 2 Propagators & GF P = | | [2πδ(E E )] fi 2E V ~q 4 (2π)32E f − i How to find QED F.R.? i | | f − Coulomb scattering e • Amplitude ❐ • Probability The square of the delta function needs some clarification. We define a • Rate & Flux transition time T and then • Cross Section • Traces • Theorems on traces T/2 i(Ef −Ei)t • Mott formula (2π)δ(Ef Ei)= lim dte + − T →∞ Coulomb scattering e Z−T/2 γ in internal lines Higher Orders ❐ Then γ in external lines

QED Feynman Rules T/2 Simple Processes 2πδ(0)= lim dt = lim T T →∞ T →∞ Compton Scattering ZT/2 − − e e+ → µ µ+ QFT Computations ❐ Therefore

[2πδ(E E )]2 = 2πδ(0)2πδ(E E ) = 2πTδ(E E ) f − i f − i f − i

Jorge C. Rom˜ao IDPASC School Braga – 25 IST Coulomb Scattering for Electrons: The Transition Rate

Summary ❐ Dividing by T we obtain the transition rate QED as a gauge theory Propagators & GF 2 2 0 2 3 4Z α u(pf sf )γ u(pisi) d pf QED How to find F.R.? Rfi = | 4 | δ(Ef Ei) − 2EiV ~q 2Ef − Coulomb scattering e | | • Amplitude • Probability • Rate & Flux ❐ To get the cross section we have to divide by the incident flux. Using • Cross Section • Traces • Theorems on traces • Mott formula χ(s) Coulomb scattering e+ 1 √E + m ~ i −ipi·x γ in internal lines Jinc = ψi(x)~γψi(x), with ψi = e √V √2Ei  ~σ·~p χ(s)  Higher Orders Ei+m γ in external lines   QED Feynman Rules we get

Simple Processes

Compton Scattering 1 1 1 ~pi J~inc = 2 ~pi = | | − − e e+ → µ µ+ | | V 2Ei | | V Ei QFT Computations with the usual interpretation: density, 1/V , times velocity, ~pi/Ei

Jorge C. Rom˜ao IDPASC School Braga – 26 IST Coulomb Scattering for Electrons: The Cross Section

Summary ❐ The differential cross section is then QED as a gauge theory Propagators & GF 2 2 0 2 2 dσ Z α u(pf )γ u(pi) pf dpf How to find QED F.R.? = | | δ(Ef Ei) − 4 Coulomb scattering e dΩ ~pi ~q Ef − • Amplitude Z | | | | • Probability • Rate & Flux ❐ Finally using pf dpf = Ef dEf we get • Cross Section • Traces 2 2 • Theorems on traces dσ Z α 0 2 • Mott formula = u(p ,s )γ u(p ,s ) dΩ ~q 4 | f f i i | Coulomb scattering e+ | | γ in internal lines

Higher Orders ❐ In practice we normally do not have polarized beams and do not measure the γ in external lines polarization of the final state. So we want the unpolarized cross section QED Feynman Rules given by Simple Processes Compton Scattering dσ 1 dσ − − = e e+ → µ µ+ dΩ 2 dΩ s ,s QFT Computations Xi f

Jorge C. Rom˜ao IDPASC School Braga – 27 IST Coulomb Scattering for Electrons: Sums into Traces

Summary ❐ The spin sums can be transformed into traces, the Casimir’s trick. We have QED as a gauge theory for any matrix Γ Propagators & GF

QED How to find F.R.? 2 − Coulomb scattering e u(pf ,sf )Γu(pi,si) = • Amplitude | | si,sf • Probability X • Rate & Flux • Cross Section = uσ(pf ,sf )uα(pf ,sf )Γαβ uβ(pi,si)uδ(pi,si)Γδσ • Traces sf si • Theorems on traces X X • Mott formula 0 † 0 + = Tr (p/f + m)Γ(p/i + m)Γ , with Γ γ Γ γ Coulomb scattering e ≡ γ in internal lines h i Higher Orders where we used

γ in external lines QED Feynman Rules uα(p,s)uβ(p,s)=(p/ + m)αβ Simple Processes ±s Compton Scattering X − − e e+ → µ µ+ ❐ So the final result is QFT Computations dσ Z2α2 = Tr (p/ + m)γ0(p/ + m)γ0 dΩ 2 ~q 4 f i | | h i Jorge C. Rom˜ao IDPASC School Braga – 28 IST Theorems on traces of Dirac γ matrices

Summary ❐ Due to equivalence relation γ′µ = U −1γµU and the cyclic property, traces QED as a gauge theory are independent of the representation of the γ matrices Propagators & GF How to find QED F.R.? ❐ − The trace of an odd number of γ matrices vanishes Coulomb scattering e • Amplitude • Probability ❐ For 0 and 2 matrices we have • Rate & Flux • Cross Section • Traces Tr1 = 4 • Theorems on traces Tr[a/b/] = Tr[(b/a/)] = 1 Tr[(a/b/ + b/a/)] = a b Tr1 = 4a b • Mott formula 2 · · Coulomb scattering e+ γ in internal lines ❐ We have the recurrence form (n even) Higher Orders

γ in external lines Tr [a/1 a/n] =a1 a2 Tr [a/3 a/n] a1 a3 Tr [a/2a/4 a/n] QED Feynman Rules ··· · ··· − · ··· Simple Processes + a1 an Tr a/2 a/n−1 Compton Scattering · ··· − + − +   e e → µ µ ❐ An important corollary is QFT Computations Tr [a/ a/ a/ a/ ] =a a Tr [a/ a/ ] a a Tr [a/ a/ ] + a a Tr [a/ a/ ] 1 2 3 4 1 · 2 3 4 − 1 · 3 2 4 1 · 4 2 3 =4 [a a a a a a a a + a a a a ] 1 · 2 3 · 4 − 1 · 3 2 · 4 1 · 4 2 · 3 Jorge C. Rom˜ao IDPASC School Braga – 29 IST Theorems on traces of Dirac γ matrices ...

Summary ❐ For traces with γ5 (needed for the SM) QED as a gauge theory

Propagators & GF Tr [γ ] = 0, Tr [γ a/b/] = 0, Tr [γ a/b/c/d/] = 4iε aµbν cρdσ How to find QED F.R.? 5 5 5 − µνρσ − Coulomb scattering e • Amplitude ❐ • Probability Sometimes it is useful to reduce he number of γ matrices before taking the • Rate & Flux trace. Useful results are • Cross Section • Traces µ • Theorems on traces γµγ = 4 • Mott formula µ + γ a/γ = 2a/ Coulomb scattering e µ − γ in internal lines µ γµa/b/γ = 4a.b Higher Orders µ γ in external lines γµa/b/c/γ = 2c/b/a/ µ − QED Feynman Rules γµa/b/c/d/γ =2[d/a/b/c/ + c/b/a/d/] Simple Processes Compton Scattering ❐ − − In practice when the number of γ matrices is bigger than 4 we use specific e e+ → µ µ+ software to evaluate the traces. QFT Computations

Jorge C. Rom˜ao IDPASC School Braga – 30 IST Coulomb Scattering for Electrons: The Mott Cross Section

❐ Finally we calculate the differential cross section for Coulomb scattering Summary QED as a gauge theory dσ Z2α2 Propagators & GF 0 0 = 4 Tr (p/f + m)γ (p/i + m)γ How to find QED F.R.? dΩ 2 ~q − | | h i Coulomb scattering e ❐ • Amplitude The trace gives • Probability • Rate & Flux 0 0 0 0 2 0 0 • Cross Section Tr (p/f + m)γ (p/i + m)γ =Tr p/f γ p/iγ + m Tr γ γ • Traces • Theorems on traces h i =8EhE 4p i p + 4m2  • Mott formula i f − i · f Coulomb scattering e+

γ in internal lines ❐ Using (recall that E = Ei = Ej, and θ is the scattering angle) Higher Orders γ in external lines 2 2 2 2 2 2 2 2 2 pi pf = E ~p cos θ = m +2β E sin (θ/2) , ~q = 4 ~p sin (θ/2) QED Feynman Rules · −| | | | | | Simple Processes Compton Scattering ❐ We get the final result, the Mott cross section − − e e+ → µ µ+ 2 2 QFT Computations dσ Z α = 1 β2 sin2 (θ/2) dΩ 4 ~p 2 β2 sin4 (θ/2) − | |   in the limit β 0 it reduces to Rutherford non-relativistic formula → Jorge C. Rom˜ao IDPASC School Braga – 31 IST Coulomb Scattering for Positrons

Summary ❐ We start now from QED as a gauge theory

Propagators & GF 4 How to find QED F.R.? Sfi = ie d xψf (x)A/(x)ψi(x) − Coulomb scattering e Z Coulomb scattering e+ where + γ in internal lines e (pf ,sf ) Higher Orders 1 1 ipf ·x γ in external lines ψi(x) = v(pf ,sf )e 2Ef √V QED Feynman Rules

Simple Processes 1 1 ipi·x ψf (x) =p v(pisi)e Compton Scattering √2Ei √V − − e e+ → µ µ+ + e (p ,s ) QFT Computations i i

❐ Then the S matrix element is

2 4 Ze 1 1 0 d x i(pf −pi)·x Sfi = i v(pi,si)γ v(pf ,sf ) e − 4π V 2E 2E ~x i f Z | | p

Jorge C. Rom˜ao IDPASC School Braga – 32 IST Coulomb Scattering for Positrons ...

Summary ❐ We get now QED as a gauge theory Propagators & GF 2 2 dσ Z α 0 2 QED How to find F.R.? = 4 v(pi,si)γ v(pf ,sf ) − dΩ + 2 ~q | | Coulomb scattering e e s ,s   | | Xf i Coulomb scattering e+ γ in internal lines ❐ Using the relation for v spinors Higher Orders

γ in external lines v(p,s)v(p,s)=(p/ m) QED Feynman Rules − s Simple Processes X Compton Scattering − − we finally get e e+ → µ µ+ QFT Computations 2 2 dσ Z α 0 0 = 4 Tr (p/f m)γ (p/i m)γ dΩ e+ 2 ~q − −   | | h i ❐ This is the same result as for electrons with m m. As, in lowest order in α, the Mott cross section only depends in m2,→ the − cross section is the same for electrons and positrons

Jorge C. Rom˜ao IDPASC School Braga – 33 IST Scattering e− + µ− e− + µ− →

Summary ❐ We want now to consider the situation when the electromagnetic field is not − − − − QED as a gauge theory static but is also quantized. As the process e + e e + e would bring → Propagators & GF an unnecessary complication due to identical particles we choose the process How to find QED F.R.? with the µ−, a kind of heavy electron interacting in the same way as the e− − Coulomb scattering e Coulomb scattering e+ ❐ We start from the fundamental relation γ in internal lines • Photon propagator • S Matrix 4 µ Sfi = ie d xψ (x)γ ψi(x)Aµ(x) • − f • Road to xs Z • The Cross Section

Higher Orders where ψi and ψf refer to the electron γ in external lines ❐ QED Feynman Rules We have to calculate Aµ(x). This is the field created by the muon. It is

Simple Processes given by the solution of the equation (in the Lorentz gauge)

Compton Scattering − − µ µ e e+ → µ µ+ A (x) = J (x) QFT Computations ⊔⊓ ❐ J µ(x) is the current due to the muon, given by

− − µ µ µ µ J (x) = eψf (x)γ ψi (x), e < 0

Jorge C. Rom˜ao IDPASC School Braga – 34 IST Scattering e− + µ− e− + µ−: Photon Propagator →

Summary ❐ The solution of the equation for Aµ(x) is obtained with the GF technique QED as a gauge theory leading to the photon propagator. We have Propagators & GF

How to find QED F.R.? µν µν 4 − D (x y) = ig δ (x y) Coulomb scattering e ⊔⊓ F − − Coulomb scattering e+ γ in internal lines ❐ We get for the Fourier transform • Photon propagator • S Matrix µν • Feynman Diagram µν g • Road to xs DF (k) = i− 2 • The Cross Section k

Higher Orders 2 γ in external lines ❐ We have to decide what to do at the pole k = 0. A similar study, as done QED Feynman Rules for the electrons, shows that the correct choice is Simple Processes g Compton Scattering D (k) = i µν − + − + Fµν 2 e e → µ µ − k + iε QFT Computations

❐ The solution for Aµ(x) is then (we neglect the solution of the free equation)

Aµ(x) = i d4yDµν (x y)J (y) − F − ν Z Jorge C. Rom˜ao IDPASC School Braga – 35 IST Scattering e− + µ− e− + µ−: S Matrix → ❐ Substituting we get the amplitude for the S matrix Summary

QED as a gauge theory − − 2 4 4 µν µ µ Propagators & GF S =( ie) d xd y ψ (x)γ ψ (x)D (x y)ψ (y)γ ψ (y) fi − f µ i F − f ν i How to find QED F.R.? Z − Coulomb scattering e ❐ Coulomb scattering e+ After introducing the plane waves for initial and final states we get

γ in internal lines • Photon propagator 2 • S Matrix ie 4 4 1 1 • Feynman Diagram S =− (2π) δ (p + p p p ) fi 2 1 2 3 4 − − − − • V − − e e µ µ Road to xs 2E 2E 2E 2E • The Cross Section i f f f

Higher Orders ′ q1 q ′ µ − − γ in external lines u(p4,se)γµu(p2,se) 2 u(p3,sµ )γ u(p1,sµ ) (p3 p1) + iε QED Feynman Rules − 1 h 1 1 i h i Simple Processes = (2π)4δ4(p + p p p ) M 2 − − − − 1 2 3 4 fi Compton Scattering V 2Ee 2Ee 2Eµ 2Eµ − − − − i f i f e e+ → µ µ+

QFT Computations q q ❐ Where Mfi is given by

′ µ igµν ′ ν − − Mfi = u(p4,se)( ieγ )u(p2,se) − 2 u(p3,sµ )( ieγ )u(p1,sµ ) − (p3 p1) + iε − h i − h i Jorge C. Rom˜ao IDPASC School Braga – 36 IST Feynman Diagram

❐ At this point Feynman had a genius idea that completely changed the way of Summary making calculations in QFT. He made a one-to-one correspondence between QED as a gauge theory

Propagators & GF the matrix element

How to find QED F.R.? − ′ µ igµν ′ ν Coulomb scattering e − − Mfi = u(p4,se)( ieγ )u(p2,se) − 2 u(p3,sµ )( ieγ )u(p1,sµ ) + − (p3 p1) + iε − Coulomb scattering e h i − h i γ in internal lines • Photon propagator and a diagram describing the process. • S Matrix • Feynman Diagram − − • e µ Road to xs − • The Cross Section p4 p3 e γ γ Higher Orders ( ie γµ) γ in external lines time p3 p1 − QED Feynman Rules − − − p2 p1 − e Simple Processes e µ

Compton Scattering − − ❐ e e+ → µ µ+ To each fermion line entering the diagram we have a spinor u QFT Computations ❐ To each fermion line leaving the diagram we have a spinor u ❐ The internal line corresponds to the virtual (k2 = 0) photon propagator 6 ❐ Each vertex corresponds to the quantity ( ieγ ), as indicated on the right − µ Jorge C. Rom˜ao IDPASC School Braga – 37 IST On the Road to the Cross Section: δ2 and Phase Space

Summary ❐ Like in the case of Coulomb scattering we have to deal with the square of QED as a gauge theory the delta function. A generalization of Propagators & GF

How to find QED F.R.? 2 − [2πδ(E E )] 2πTδ(E E ) Coulomb scattering e f − i ⇒ f − i Coulomb scattering e+

γ in internal lines gives • Photon propagator • S Matrix 2 • 4 4 4 4 Feynman Diagram (2π) δ pf pi VT (2π) δ pf pi • Road to xs − ⇒ − • The Cross Section h X X i X X  Higher Orders where, as before, T is the interaction time and V is the volume of the box γ in external lines where we normalize the wave functions. QED Feynman Rules Simple Processes ❐ To evaluate the cross section we have to sum over all the momenta states Compton Scattering available. The number of states between ~p3 and ~p3 + d~p3 and between ~p4 − − e e+ → µ µ+ and ~p4 + d~p4 is QFT Computations d3p d3p V 3 V 4 (2π)3 (2π)3

Jorge C. Rom˜ao IDPASC School Braga – 38 IST On the Road to the Cross Section: The Incident Flux

Summary ❐ The incident flux is QED as a gauge theory

Propagators & GF ~ 1 ~p1 ~p2 1 How to find QED F.R.? Jinc = 0 0 = ~vrelative − | | V p − p V | | Coulomb scattering e 1 2

Coulomb scattering e+

γ in internal lines ❐ For future use we note that the combination V J~inc multiplied by the • Photon propagator | | • S Matrix energy of the incoming particles is • Feynman Diagram • − Road to xs − µ • ~ e 0 0 The Cross Section V Jinc 2Ei 2Ei =4 p1~p2 p2~p1 Higher Orders | | | − |

γ in external lines 2 2 2 =4 (p1 p2) memµ QED Feynman Rules · − q Simple Processes where the last expression shows that it is a Lorentz invariant. To derive this Compton Scattering − − expression we have to assume that ~p1 and ~p2 are collinear, as is the situation e e+ → µ µ+

QFT Computations in normal scattering experiments

Jorge C. Rom˜ao IDPASC School Braga – 39 IST The Cross Section

Summary ❐ We have now all the ingredients to evaluate the cross section. First we QED as a gauge theory determine the transition rate by unit time and unit volume Propagators & GF How to find QED F.R.? 1 − 2 Coulomb scattering e lim Sfi = wfi V,T →∞ VT | | Coulomb scattering e+

γ in internal lines • Photon propagator ❐ Using the previous results we get • S Matrix • Feynman Diagram • Road to xs 4 4 1 1 2 • w = (2π) δ (p + p p p ) M The Cross Section fi 1 2 − 3 − 4 V 4 2p0 2p0 2p0 2p0 | fi | Higher Orders 1 2 3 4

γ in external lines ❐ QED Feynman Rules Finally we divide by the incident flux and by the number density of particles Simple Processes in the target (just 1/V with our normalization) and sum over the final states Compton Scattering to get − − e e+ → µ µ+ 3 3 QFT Computations d p d p V σ = 3 4 V 2 w (2π)3 (2π)3 ~ fi Z Jinc 3 3 | | d p3 d p4 1 1 4 4 2 = 3 3 0 0 0 0 (2π) δ (p1 + p2 p3 p4) Mfi (2π) (2π) 2p1 2p2 2p3 2p4 V J~ − − | | Z | inc | Jorge C. Rom˜ao IDPASC School Braga – 40 IST The Cross Section ...

1 d3p d3p Summary 2 4 4 3 4 σ = Mfi (2π) δ (p1 +q2 p3 p4) 2 2 2 3 0 3 0 QED as a gauge theory 4 (p p ) m m | | − − (2π) 2p (2π) 2p Z 1 2 1 2 3 4 Propagators & GF · − How to find QED F.R.? ❐ Initialp State: The factor − Coulomb scattering e 1 Coulomb scattering e+ 2 2 2 γ in internal lines 4 (p1 p2) m1m2 • Photon propagator · − • S Matrix p • Feynman Diagram ❐ Final State: The factor • Road to xs 3 3 • The Cross Section d p d p (2π)4δ4(p + p p p ) 3 4 Higher Orders 1 2 3 4 3 0 3 0 − − (2π) 2p3 (2π) 2p4 γ in external lines QED Feynman Rules This factor is also Lorentz invariant because Simple Processes

Compton Scattering 3 d p 4 2 2 0 − + → − + = d p δ(p m )θ(p ) e e µ µ 2E − QFT Computations Z Z ❐ Matrix Element: M 2 | fi | The Physics is in Mfi and this is evaluated through the Feynman diagrams and Feynman Rules.

Jorge C. Rom˜ao IDPASC School Braga – 41 IST Higher Order Corrections to e−µ− e−µ− → ❐ To have the full Feynman rules we need to know how to evaluate higher Summary orders in perturbation theory. We go back to the master equation QED as a gauge theory

Propagators & GF 4 How to find QED F.R.? Sfi = ie d yψf (y)A/(y)Ψi(y) − − Coulomb scattering e Z Coulomb scattering e+ ❐ γ in internal lines Instead of the plane wave we use now the next order to Ψi, that is Higher Orders

γ in external lines 4 Ψi(y) = ie d xSF (y x)A/(x)ψi(x) QED Feynman Rules − − Simple Processes Z Compton Scattering and − − e e+ → µ µ+ (2) 4 4 µ ν S = d yd xψf (y)( ieγ )SF (y x)( ieγ )ψi(x)Aµ(y)Aν(x) QFT Computations fi − − − Z

( ie γµ) Aµ(y) −

( ie γν ) Aν (x) −

Jorge C. Rom˜ao IDPASC School Braga – 42 IST Higher Order Corrections to e−µ− e−µ− ... →

Summary ❐ The origin of the terms Aµ and Aν is the current of the muon. So we should QED as a gauge theory have Propagators & GF

How to find QED F.R.? − 4 4 Coulomb scattering e A (y)A (x)= d zd w D ′ (y z)D ′ (x w)+D ′ (y w)D ′ (x z) µ ν Fµµ − Fνν − Fµν − Fνµ − Coulomb scattering e+ Z h − ′ ′ − i γ in internal lines µ µ ν µ ψf (z)( ieγ )SF (z w)( ieγ )ψi (w) Higher Orders − − − γ in external lines ❐ QED Feynman Rules This corresponds to the diagrams

Simple Processes

Compton Scattering − − e e+ → µ µ+ ′ ′ QFT Computations y ( ie γµ ) y ( ie γµ ) − −

′ ′ x ( ie γν ) x ( ie γν ) − −

Jorge C. Rom˜ao IDPASC School Braga – 43 IST Higher Order Corrections to e−µ− e−µ− ... →

Summary ❐ Putting all together QED as a gauge theory

Propagators & GF (2) 4 4 4 4 µ ν How to find QED F.R.? Sfi = d yd xd zd w ψf (y)( ieγ )SF (y x)( ieγ )ψi(x) − − − − Coulomb scattering e Z + Coulomb scattering e DFµµ′ (y z)DFνν′ (x w) + DFµν′ (y w)DFνµ′ (x z) γ in internal lines − − − − − h µ ′ ′ − i Higher Orders µ ν µ ψf (z)( ieγ )SF (z w)( ieγ )ψi (w) γ in external lines − − − QED Feynman Rules ❐ Introducing ψ , ψ and the Fourier transforms of the propagators we are Simple Processes i f ··· Compton Scattering lead to the final expression − − e e+ → µ µ+ 1 1 1 QFT Computations S(2) = (2π)4δ4(p + p p p ) M fi − − 2 1 2 3 4 fi e− e− µ µ V − − 2Ei 2Ef 2Ei 2Ef q q ❐ With

a b Mfi = Mfi + Mfi

Jorge C. Rom˜ao IDPASC School Braga – 44 IST Higher Order Corrections to e−µ− e−µ− ... →

Summary 4 a d k µ i(p/4 k/ + me) ν QED as a gauge theory M = u(p )( ieγ ) − ( ieγ )u(p ) ~p4 ~ ~p3 fi (2π)4 4 − (p k)2 m2 + iε − 2 k Propagators & GF Z  4 − − e How to find QED F.R.? − ′ ′ Coulomb scattering e u i(p/3 + k/ + mµ) ν u(p3)( ieγ ) ( ieγ )u(p1) + 2 2 Coulomb scattering e − (p3 + k) mµ + iε − γ in internal lines − Higher Orders ′ 1 1 µµ ′ γ in external lines ( ig ) 2 ( igνν ) 2 − k + iε − (p2 p4 + k) + iε ~p2 ~p1 QED Feynman Rules −  Simple Processes

Compton Scattering 4 b d k µ i(p/4 k/ + me) ν − + − + e e → µ µ Mfi = 4 u(p4)( ieγ ) −2 2 ( ieγ )u(p2) (2π) − (p4 k) me + iε − ~p4 ~p3 QFT Computations Z  − − ~k ′ i(p/ k/ + m ) ′ u(p )( ieγµ ) 1 − µ ( ieγν )u(p ) 3 − (p k)2 m2 + iε − 1 1 − − µ 1 1 ′ ′ ( igµν ) 2 ( igνµ ) 2 − k + iε − (p2 p4 + k) + iε −  ~p2 ~p1

Jorge C. Rom˜ao IDPASC School Braga – 45 IST Photons in external lines: Compton Scattering

❐ To complete our Feynman rules we have to consider photons in external Summary lines. The idea is to represent the photon in external lines by a plane wave. QED as a gauge theory

Propagators & GF We have

How to find QED F.R.? − µ 1 1 µ −ik·x ∗µ ik·x Coulomb scattering e A (x) = ε (k) e + ε (k) e 0 Coulomb scattering e+ √V √2k

γ in internal lines  

Higher Orders where the first term corresponds to the initial state and the second to the

γ in external lines final state

QED Feynman Rules ❐ Simple Processes The polarization vectors satisfy

Compton Scattering − − µ µ ∗ µ e e+ → µ µ+ k k = 0, ε k = 0, ε ε = 1 µ µ µ − QFT Computations ❐ ′ ′ Compton scattering ǫ′∗, k′ p ǫ′∗, k′ p e− + γ e− + γ → We should have the diagrams: ǫ, k p ǫ, k p

Jorge C. Rom˜ao IDPASC School Braga – 46 IST Photons in external lines: Compton Scattering ...

❐ (2) Summary The rules for the diagrams follow from the second order Sfi

QED as a gauge theory

Propagators & GF (2) 4 4 µ ν Sfi = d yd xψf (y)( ieQeγ )SF (y x)( ieQeγ )ψi(x)Aµ(y)Aν(x) How to find QED F.R.? − − − − Z Coulomb scattering e

Coulomb scattering e+ substituting Aµ(x) and Aν (y) by plane waves. For instance for diagram a)

γ in internal lines ′ Higher Orders 1 1 ′∗ ik ·y 1 1 −ik·x γ in external lines Aµ(y) = εµ e ,Aν(x) = εν e √V √2k′0 √V √2k0 QED Feynman Rules

Simple Processes ❐ ′∗ ′ ′ Compton Scattering The amplitudes are then ǫ , k p − − e e+ → µ µ+ i(p/ + k/ + me) QFT Computations M a = u(p′)(ieγµ) (ieγν )u(p) ε′∗(k′)ε (k) fi (p + k)2 m2 µ ν − e ǫ, k p ′ ǫ′∗, k′ p i(p/ k/ + m ) M b = u(p′)(ieγν) − e (ieγµ)u(p) ε′∗(k′)ε (k) fi (p′ k)2 m2 µ ν − − e ǫ, k p Jorge C. Rom˜ao IDPASC School Braga – 47 IST Summary of Feynman Rules for QED

Summary 1. For a given process draw all topologically distinct diagrams

QED as a gauge theory

Propagators & GF 2. For each electron entering the diagram a factor u(p,s). If it leaves the How to find QED F.R.? diagram a factor u(p,s) − Coulomb scattering e

Coulomb scattering e+ 3. For each positron leaving the diagram a factor v(p,s). If it enters the γ in internal lines diagram a factor v(p,s) Higher Orders

γ in external lines 4. For each photon in the initial state a polarization vector εµ(k). In the final ∗ QED Feynman Rules state εµ(k) Simple Processes

Compton Scattering 5. For each electron internal line the propagator − − e e+ → µ µ+ (p/ + m) QFT Computations β α S (p) = i αβ F αβ p2 m2 + iε p − 6. For each internal photon line the propagator (in the Feynman gauge)

gµν µ ν DFµν (k) = i k − k2 + iε

Jorge C. Rom˜ao IDPASC School Braga – 48 IST Summary of Feynman Rules for QED ...

Summary

QED as a gauge theory 7. For each vertex the factor

Propagators & GF − How to find QED F.R.? e − γ Coulomb scattering e µ ( ieQeγ )αβ Coulomb scattering e+ − γ in internal lines e− Higher Orders γ in external lines 8. For each internal momentum not fixed by energy-momentum conservation QED Feynman Rules (in loops) a factor Simple Processes

Compton Scattering

− + → − + e e µ µ 4 QFT Computations d q (2π)4 Z 9. For each loop of fermions a minus sign 10. A factor of -1 between diagrams that differ but odd permutations of fermions lines

Jorge C. Rom˜ao IDPASC School Braga – 49 IST Simple Processes in QED

Summary ❐ If we restrict the processes to two particles in final state the number of QED as a gauge theory processes is very small. Propagators & GF

How to find QED F.R.? − Coulomb scattering e Process Comment + Coulomb scattering e γ + e− γ + e− Compton Scattering γ in internal lines − +→ − + Higher Orders e + e µ + µ in QED → γ in external lines µ− + e− µ− + e− in QED QED Feynman Rules → e− + e+ e− + e+ Simple Processes → Compton Scattering e−+ Nuclei(Z) e−+ Nuclei(Z) +γ − + − + → e e → µ µ e− + e+ γ + γ Pair Annihilation QFT Computations → e− + e− e− + e− M¨oller Scattering → γ + γ e− + e+ Pair Creation → γ+ Nuclei(Z) Nuclei(Z) +e− + e+ Pair Creation →

❐ We will discuss γ + e− γ + e− and e− + e+ µ− + µ+ in QED → →

Jorge C. Rom˜ao IDPASC School Braga – 50 IST Compton Scattering

❐ Diagrams and kinematics Summary ε, kε′, k′ ε, k ε′, k′ QED as a gauge theory Propagators & GF p =(m,~0) k =(k, 0, 0, k) How to find QED F.R.? − Coulomb scattering e ′ ′ ′ ′ ′ ′ ′ + p =(E , ~p ) k =(k , k sin θ, 0, k cos θ) Coulomb scattering e p p′ p p′ γ in internal lines time Higher Orders ❐ γ in external lines The amplitude is M = M1 + M2

QED Feynman Rules

Simple Processes 2 i ′ µ ′ν∗ ′ M1 =(ie) 2 2 u(p )γν (p/ + k/ + m)γµu(p)ε (k)ε (k ) Compton Scattering (p + k) m • Amplitudes − • Spin Sums 2 i ′ ′ µ ′ν∗ ′ • Cross Section M2 =(ie) ′ 2 2 u(p )γµ(p/ k/ + m)γνu(p)ε (k)ε (k ) − − (p k ) m − e e+ → µ µ+ − − QFT Computations ❐ We write M iu(p′,s′)Γ u(p,s) i ≡ − i e2 Γ = γ (p/ + k/ + m)γ εµ(k,λ)ε′ν∗(k′,λ′) 1 2p k ν µ · e2 Γ = − γ (p/ k/′ + m)γ εµ(k,λ)ε′ν∗(k′,λ′) 2 2p k′ µ − ν · Jorge C. Rom˜ao IDPASC School Braga – 51 IST Spin Sums

❐ We want to calculate Summary

QED as a gauge theory 1 2 1 2 2 † † Propagators & GF M = M1 + M2 + M1 M2 + M1M2 4 ′ ′ | | 4 ′ ′ | | | | How to find QED F.R.? s,s λ,λ s,s λ,λ − X X X X h i Coulomb scattering e Coulomb scattering e+ ❐ We have (i = 1, 2) γ in internal lines

Higher Orders 2 ′ ′ † † 0 ′ ′ γ in external lines Mi = u(p ,s )Γiu(p,s)u (p,s)Γi γ u(p ,s ) ′ | | ′ QED Feynman Rules Xs,s Xs,s Simple Processes ′ ′ ′ ′ 0 † 0 = u(p ,s )Γiu(p,s)u(p,s)Γiu(p ,s ) Γi γ Γi γ Compton Scattering ′ ≡ • Amplitudes Xs,s • Spin Sums ′ • Cross Section =Tr (p/ + m)Γi(p/ + m)Γi − − e e+ → µ µ+   QFT Computations ❐ Where we have used

uα(p,s)uβ(p,s)=(p/ + m)αβ s X

Jorge C. Rom˜ao IDPASC School Braga – 52 IST Spin Sums ...

Summary ❐ For the interference terms QED as a gauge theory Propagators & GF † † ′ ′ (M1M +M M2) = Tr (p/ + m)Γ1(p/ + m)Γ2 +Tr (p/ + m)Γ2(p/ + m)Γ1 How to find QED F.R.? 2 1 ′ − s,s Coulomb scattering e X     Coulomb scattering e+ ❐ γ in internal lines The sum over photon polarizations is

Higher Orders µ ∗ν µν γ in external lines ε (k,λ)ε (k,λ) = g + terms proportional to k − QED Feynman Rules λ Simple Processes X Compton Scattering ❐ • Amplitudes Terms proportional to k do not contribute to the amplitude due to gauge • Spin Sums invariance and therefore we will use the simplified form • Cross Section − − e e+ → µ µ+ εµ(k,λ)ε∗ν(k,λ) = gµν QFT Computations − Xλ

Jorge C. Rom˜ao IDPASC School Braga – 53 IST Compton Cross Section

❐ In the rest frame of the electron the cross section is Summary QED as a gauge theory 3 ′ 3 ′ 1 4 4 ′ ′ d p d k Propagators & GF 2 dσ = (2π) δ (p + k p k ) M 3 ′0 3 ′0 How to find QED F.R.? 4mk − − | | (2π) 2p (2π) 2k − Coulomb scattering e 3 ′ Coulomb scattering e+ ❐ Using the delta function we integrate over d p . We get γ in internal lines ′2 Higher Orders dσ 1 1 k = dk′ δ(m + k E′ k′) M 2 γ in external lines 2 ′ ′ dΩk′ 4mk (2π) 2k 2E − − | | QED Feynman Rules Z Simple Processes ❐ ′ ′ Compton Scattering To use the last delta function we note that E is related to k . In fact from ′ ′ ′ ′ • Amplitudes δ3(~p + ~k ~p ~k ) we have ~p = ~k ~k , and therefore • Spin Sums − − − • Cross Section − + − + ′ ′2 2 2 ′2 ′ 2 e e → µ µ E = ~p + m = k + k 2kk cos θ + m QFT Computations − p p ❐ This implies ′ k δ k ′ ′ 1+ k (1−cos θ) ′ ′ − m dE k k cos θ δ(m + k E k ) = dE′ with ′ = − ′  ′  dk E − − 1 + dk

Jorge C. Rom˜ao IDPASC School Braga – 54 IST Compton Cross Section ...

❐ And therefore Summary ′ ′ ′ QED as a gauge theory dE E + k k cos θ m + k(1 cos θ) m k 1 + ′ = | −′ | = −′ = ′ ′ Propagators & GF dk E E E k

How to find QED F.R.? − Coulomb scattering e ❐ Putting all together Coulomb scattering e+ ′ 2 γ in internal lines dσ 1 1 k 1 = M 2 where M 2 = M 2 Higher Orders ′ 2 2 dΩk 64π m k | | | | 4 ′ | |   s,s λ,λ′ γ in external lines X X QED Feynman Rules Simple Processes ❐ Calculating the traces Compton Scattering 4 • Amplitudes 2 4 2 ′ ′ ′ e • Spin Sums M1 = 8 2 m + m ( p p p k + 2p k)+(p k)(p k) 2 • Cross Section | | − · − · · · · (2p k) − − · e e+ → µ µ+   4 4 2 ′ ′ ′ ′ ′ ′ ′ e QFT Computations M 2 = 8 2m + m ( p p + p k 2p k )+(p k )(p k ) | 2| − · · − · · · (2p k′)2 ·   8e4 [M M † + M †M ] = [2(k p)(p p′) 2(k k′)(p p′) 2(p p′)(p k′) 1 2 1 2 4(k p)(k′ p) · · − · · − · · · · +m2( 2k p k p′ + k k′ p p′ + 2p k′ + p′ k′) m4 − · − · · − · · · −  Jorge C. Rom˜ao IDPASC School Braga – 55 IST Compton Cross Section ...

Summary ❐ Now we use the kinematics of the rest frame of the electron QED as a gauge theory Propagators & GF p′ = p + k k′ p k = mk How to find QED F.R.? − · − ′ ′ ′ ′ ′ Coulomb scattering e p k = mk k k = kk (1 cos θ) = m(k k ) · · − − Coulomb scattering e+

γ in internal lines to obtain

Higher Orders ′ γ in external lines 1 2 2 † † 4 k k 2 M1 + M2 +M1M2 +M1 M2 = 2e + sin θ QED Feynman Rules 4 {| | | | } k′ k − s,s′ λ,λ′      Simple Processes X X Compton Scattering • Amplitudes ❐ Finally we put everything together to get the Klein-Nishina formula for the • Spin Sums • Cross Section differential cross section of the Compton scattering. − − e e+ → µ µ+ QFT Computations dσ α2 k′ 2 k′ k = + sin2 θ dΩ 2 m2 k k k′ −       

Jorge C. Rom˜ao IDPASC School Braga – 56 IST Scattering e−e+ µ−µ+ in QED → ❐ Diagram and kinematics Summary 2 QED as a gauge theory p2 q1 p1 =√s/2 (1, 0, 0, 1) 4mµ + − β = 1 s Propagators & GF e µ − p2 =√s/2 (1, 0, 0, 1) q How to find QED F.R.? − − Coulomb scattering e − q =√s/2 (1,β sin θ, 0,β cos θ) e µ+ 1 Coulomb scattering e+ p1 q2 q2 =√s/2 (1, β sin θ, 0, β cos θ) γ in internal lines − − ❐ Higher Orders Amplitude

γ in external lines

QED Feynman Rules µ i gµν ν M =v(p2)( ieγ )u(p1) − 2 u(q1)( ieγ )v(q2) Simple Processes − (p1 + p2) + iε − Compton Scattering 2 1 µ − − e e+ → µ µ+ =ie 2 v(p2)γ u(p1) u(q1)γµv(q2) • Amplitude (p1 + p2) + iε • Cross Section QFT Computations ❐ Spin averaged amplitude squared

4 1 2 e µ ν M = 4 Tr [(p/2 me)γ (p/1 + me)γ ] Tr [(q/1 + mµ)γµ(q/2 mµ)γν ] 4 | | 4(p1 + p2) − − spinsX 4 8e 2 2 2 2 = 4 p1 p2mµ + p1 q1p2 q2 + p1 q2p2 q1 + q1 q2me + 2memµ (p1 + p2) · · · · · ·   Jorge C. Rom˜ao IDPASC School Braga – 57 IST Cross Section

❐ The general formula for the cross section is Summary QED as a gauge theory 2 1 d3q Propagators & GF 2 4 4 i σ = M (2π) δ (p1 + p2 q1 q2) 2 4 3 0 How to find QED F.R.? 4 (p1 p2) me | | − − (2π) 2qi − Z i=1 Coulomb scattering e · − Y + p Coulomb scattering e ❐ We get the differential cross section γ in internal lines

Higher Orders dσ 1 ~q1 γ in external lines 2 = 2 | | M QED Feynman Rules dΩ 32π s √s | | Simple Processes 2 2 α 2 2 4mµ 3 Compton Scattering = β β cos θ + 1 + 10 − − 4s s e e+ → µ µ+ ! • Amplitude 2 • Cross Section α 2 2 2 2 = β 1 + cos θ + (1 β ) sin θ 10 QFT Computations 4s − (pb)

  σ ❐ And finally the total cross section 101

2πα2 σ = β(3 β2) 3s − 100 0 50 100 150 200

ECM (GeV)

Jorge C. Rom˜ao IDPASC School Braga – 58 IST Computational Techniques in Quantum Field Theory

❐ Mathematica Summary QED as a gauge theory ◆ FeynArts Propagators & GF Program to draw Feynman diagrams. Can be obtained from How to find QED F.R.? http://www.feynarts.de − Coulomb scattering e ◆ FeynCalc Coulomb scattering e+ Lorentz and Dirac algebra and calculations at one–loop. Can have as γ in internal lines

Higher Orders input FeynArts. Can be obtained from http://www.feyncalc.org

γ in external lines ❐ QED Feynman Rules QGRAF Simple Processes Very efficient program to generate Feynman diagrams for any theory to any Compton Scattering loop order done by Paulo Nogueira. Can be downloaded from − − e e+ → µ µ+ http://cfif.ist.utl.pt/~paulo/qgraf.html QFT Computations ❐ Numerics: C/C++ or Fortran To do efficient numerics one has to use the power of C/C++ or Fortran.A special useful package is CUBA with routines for numerical integration can be obtained from http://www.feynarts.de/cuba/ ❐ My CTQFT Home Page: http://porthos.ist.utl.pt/CTQFT/ Here you can find all the links and many programs for standard processes in QED and in the SM.

Jorge C. Rom˜ao IDPASC School Braga – 59