Introduction to quantum electrodynamics
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Introduction to Quantum Electrodynamics. Properties of Dirac Spinors Description of photons Feynman rules for QED Simple examples Spins and traces. Summary – Spin-1/2. u (1) , u (2) , v (1) , v (2) need not be pure spin states, but their sum is still “complete”.

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Introduction to Quantum Electrodynamics

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Introduction to quantum electrodynamics

Introduction to Quantum Electrodynamics

Properties of Dirac Spinors

Description of photons

Feynman rules for QED

Simple examples

Spins and traces

Much of the content of these slides is acknowledged to come from

“Introduction to Elementary Particles”

By David Griffiths, Wiley, 1987


Summary spin 1 2

Summary – Spin-1/2

u(1), u(2), v(1), v(2) need not be pure spin

states, but their sum is still “complete”.

Brian Meadows, U. Cincinnati


Summary photons

Summary - Photons

  • Photons have two spin projections (s):

Brian Meadows, U. Cincinnati


Feynman rules for qed

Feynman Rules for QED

Label:

  • Label each external line with 4-momenta p1, … pn.

    Label theirspins s1, … sn.

    Label internal lines with 4-momenta q1, … qn

    Directions:

    Arrows on external indicate Fermion or anti-Fermion

    Arrows on internal lines preserve flow

    Arrows on internal lines preserve flow

    External photon arrows point in direction of motion

    Internal photon arrows do not matter

Brian Meadows, U. Cincinnati


Feynman rules for qed1

Feynman Rules for QED

  • For external lines, write factor

  • For each vertex write a factor

Always follow a Fermion line

To obtain a product:

(adjoint-spinor)()(spinor)

E.g.:

pj , sj

pk , sk

e -

e -

q

u (sk)(k) ige u(s))(j)

Brian Meadows, U. Cincinnati


Feynman rules for qed2

Feynman Rules for QED

  • Write a propagator factor for each internal line

    NOTE:qj2 = mj2c2 for internal lines.

    NOTE also:use of the ”slash – q” a 4 x 4 matrix rather than a 4-vector!

Brian Meadows, U. Cincinnati


Feynman rules for qed3

Feynman Rules for QED

Now conserve momentum (at each vertex)

5.Include a d function to conserve momentum at each vertex.

where the k's are the 4-momenta entering the vertex

6.Integrate over all internal 4-momenta qj. I.e. write a factor

For each internal line.

  • Cancel the d function. Result will include factor

  • Erase the d function and you are left with

  • Anti-symmetrize (“-” sign between diagrams with swapped Fermions)

Brian Meadows, U. Cincinnati


Example e e scattering

p1 , s1

p3 , s3

e -

e -

u (s3 )(3) ige u(s1 )(1)

q

u (s4 )(4) ige u(s2 )(2)

-

-

p2 , s2

p4 , s4

NOTE u(k) is short

for u(sk )(pk)

Example – e-- e--Scattering

  • We already wrote down one vertex:

    Use index “”

  • The other is similar:

    BUT use index “”

  • Leads to

Brian Meadows, U. Cincinnati


Evaluate e e scattering griffiths problem 7 24

Evaluate – e-- e--Scattering(Griffiths, problem 7:24)

  • Assume e - and  - move along the z-axis, each with helicity +1. After collision, they return likewise. Therefore

    where

Brian Meadows, U. Cincinnati


The electron and muon vertex functions

The electron and muon vertex functions:

  • So the inner product is

Brian Meadows, U. Cincinnati


Example e e e e moller scattering

p1 , s1

p1 , s1

p4 , s4

p3 , s3

e -

e -

e -

e -

q

q

e-

p3 , s3

e-

e-

p2 , s2

e-

p2 , s2

p4 , s4

Example – e-e- e-e- (Moller Scattering)

  • One other diagram required in which 3  4 are interchanged (not possible in e-- scattering)

  • Anti-symmetrization leads to:

Brian Meadows, U. Cincinnati


Spin summation and averaging

Spin Summation and Averaging

  • We have, so far:

  • Consider e- scattering. To obtain a cross-section or decay rate, we need to evaluate

    and (usually) sum over final spins and average over initial ones.

  • Each term in […] is a number, as seen above, so we can re-order them and evaluate in pairs like

    whereis a 4 x 4 matrix (in this case,).

Brian Meadows, U. Cincinnati


Spin summation and averaging1

Spin Summation and Averaging

  • It is easy to show that

  • Therefore we can re-write V with the u(1)’s together:

4 x 4 matrix

Brian Meadows, U. Cincinnati


Spin summation and averaging2

Spin Summation and Averaging

  • To sum over spins for particle 1, use the completeness property:

  • Now sum over spins for particle 3:

4 x 4 matrix

WHY ?

scalar

Scalar

NO u’s !!

Brian Meadows, U. Cincinnati


Spin summation and averaging3

Spin Summation and Averaging

  • Why is:

  • Pre-multiply by u(3) then post-multiply by u(3)

  • So

4 x 4 matrix

scalar

?

4 x 4 matrix

Brian Meadows, U. Cincinnati


Spin summation and averaging4

Spin Summation and Averaging

  • Repeat for other vertex to get:

  • For the specific case of e- scattering:

    where m is mass of electron and M is mass of .

Average over

4 initial spins

Brian Meadows, U. Cincinnati


Dirac g matrices reminder

Dirac g Matrices - Reminder

  • Almost done, but we need to use gm,g5, p, etc..

Brian Meadows, U. Cincinnati


About the traces

… About the Traces

  • Almost done, but the traces need some ready results:

Brian Meadows, U. Cincinnati


Evaluate traces for e scattering

Evaluate Traces for e- Scattering

  • We obtained:

  • Expand the first factor:

Spin average over initial states

4 g

0

0

Brian Meadows, U. Cincinnati


Evaluate traces for e scattering1

Evaluate Traces for e- Scattering

  • Therefore

  • So the first factor is:

  • The second factor is therefore:

Brian Meadows, U. Cincinnati


Evaluate traces for e scattering2

Evaluate Traces for e- Scattering

  • The product is:

  • Contracting terms:

  • Done with traces!

Brian Meadows, U. Cincinnati


Evaluate cross section for e scattering

e -

e -

-

Evaluate Cross-Section for e- Scattering

  • Work in the frame where the  is at rest.

    Assume so that we can ignore the recoil of the, and therefore |p1| = |p3| = |p| and |p2| = |p4| = 0

  • Computing terms in matrix element:

Brian Meadows, U. Cincinnati


Evaluate cross section for e scattering1

Evaluate Cross-Section for e- Scattering

  • Insert in matrix element:

  • To get the cross-section:

Brian Meadows, U. Cincinnati


Limiting cases

Limiting Cases:

  • Relativistically, we have Mott scattering (originally for e- p):

  • Low energy we get Rutherford scattering:

  • High energy limit

Brian Meadows, U. Cincinnati


Compton scattering

Compton Scattering

  • Two diagrams in lowest order:

  • Apply Feynman rules (first diagram – second is similar):

p2 , s2

p4 , s4

p2 , s2

p4 , s4

+

Time

p1 , s1

p3 , s3

p1 , s1

p3 , s3

Fermion line (backwards)

g in

g out

Brian Meadows, U. Cincinnati


Compton scattering1

Compton Scattering

  • Conserve energy-momentum, etc.:

    where q = p1-p3, etc..

  • Add term for other diagram.

  • Write as trace

  • Evaluate trace

  • Evaluate cross-section

Brian Meadows, U. Cincinnati


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