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

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

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

  • Photons have two spin projections (s):

Brian Meadows, U. Cincinnati


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 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 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 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


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

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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

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The electron and muon vertex functions:

  • So the inner product is

Brian Meadows, U. Cincinnati


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

  • 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 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 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 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 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

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

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… About the Traces

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

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Evaluate Traces for e- Scattering

  • We obtained:

  • Expand the first factor:

Spin average over initial states

4 g

0

0

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Evaluate Traces for e- Scattering

  • Therefore

  • So the first factor is:

  • The second factor is therefore:

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Evaluate Traces for e- Scattering

  • The product is:

  • Contracting terms:

  • Done with traces!

Brian Meadows, U. Cincinnati


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- Scattering

  • Insert in matrix element:

  • To get the cross-section:

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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

  • 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 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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