Introduction to Particle Physics I

Risto Orava Spring 2016

electromagnetic interactions

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the standard model - electromagnetic interactions

•  the matter families •  properties of electromagnetic interactions •  the basic process •  Feynman diagrams for the EM interaction •  R = σ(e+e- → hadrons)/σ(e+e- → µ+µ-)

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electromagnetic interaction

•  properties of the EM interaction –  exchange of photons –  conserves all quantities

–  flavour – parity –  charge parity

•  only involves charged particles (no neutrinos)

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the matter families SM: 3 families of fundamental fermions & 3 forces and their force carriers (NOTE: (1) gravity left out here, (2) antimatter families are there, as well)

I II III Q colour weak charge

νe νµ νt 0 no yes e- µ- τ- -1 no yes u c t 2/3 yes yes d s b -1/3 yes yes

families em strong weak

force carriers: γ 8×g W±/Z

+ the antimatter families

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the electromagnetic interaction - Feynman

•  base calculations on Feynman diagrams

•  the Feynman diagrams has the key ingredients of a cross section

•  if particle = antiparticle –> no arrow.

•  vertex represents the coupling of various

particles √α:

fermion

vertex

photon 036.1371

4 0

2

EM ===c

e!πε

αα

6

f

f

γ

•  the only basic process in QED:

the electromagnetic interaction

•  a fermion comes in, emits a photon and exits •  the time flows to the right (the vertical axis has here no meaning) •  a Feynman diagram represents a quantum mechanical amplitude •  the amplitude for a process is the sum of all possible amplitudes •  the cross section or decay width for a process is proportional to the square of the

amplitude •  since only what goes into an interaction and what comes out is measured, must

sum over all possible internal interactions

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the electromagnetic interaction...

•  the theory specifies the fundamental interactions – vertices •  a Feynman diagram made up of combinations of fundamental vertices •  an example:e-e- scattering (Møller scattering) •  note that it is not necessary to specify which electron emits and which

electron absorbs the photon

e- e-

e- e-

γ

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•  the next most complicated diagrams are: •  a redifinition of the electron… •  a set of Feynman diagrams represents a quantum mechanical perturbation calculation -it only makes sense if the perturbation series converges •  a set of Feynman rules can be deduced which state a factor for each line and vertex - in QED, the factor contains where for an electron .

the electromagnetic interaction...

παπ 4)/4( =ce !q) chargefor 4( παq137/1=α

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•  energy and momentum are conserved at each vertex •  E2 –p2 ≠ m2 for internal lines – for virtual particles – a price to

pay for violating the energy-mass relation, just as one pays for 1/ΔE –terms in non-relativistic perturbation theory! •  a process can be rotated or twisted to give related processes - a particle going backward in time is an antiparticle going forward in time •  an example: Møller scattering rotated by 90o is Bhabha scattering (e+e-)

the electromagnetic interaction...

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Note: the basic interaction does not change flavour, nor does it change colour - the photon does not recognize colour •  EM interactions do not change flavour or colour •  µ- → e-γ does not proceed through the EM interaction, since there is

no way of destroying the µ- or creating the e-. -  conservation of electron/muon number for the EM interactions, isthus a consequence of the

fundamental interactions. -  neutrinos do not interact electromagnetically since they do possess electric charge!

•  calculate one of the fundamental numbers of particle physics, the ratio

R:

the electromagnetic interaction...

)(/)( −+−+−+ →→= µµσσ eehadronseeR

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R = σ(e+e- → hadrons)/σ(e+e- → µ+µ-)

•  the ratio:

•  R for c.m. energy of Ecm = 36 GeV? (i.e. quark flavours: u, d, s, c and b are accesible, not top)

•  this in case the colour did not exist – since quarks appear in 3

colours, we have to multiply the above by 3* (Note: the above is a not entirely correct - if we cannot distinguish between amplitudes, we have to sum the amplitudes and then square. But the colour wave function for a quark-antiquark state is (1/√3)δij. Summing, we get 3/√3 = √3, and squaring, 3.)

22 / µqqRi

i∑=

911

31

32

31

31

32 22222

=⎟⎠

⎞⎜⎝

⎛−+⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛−+⎟⎠

⎞⎜⎝

⎛−+⎟⎠

⎞⎜⎝

⎛=R

311

=⇒ R

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e-e+ → hadrons

R=11/3

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µ+e- scattering •  one possible Feynman Diagram (topological restraints) •  the process has two vertices linking the photon with the muon

and the electron •  the photon introduces a propagator term that depends on the four

momentum transfer: 1/q2

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µ+e- scattering •  the total cross section is effected by the spin of in and outgoing

particles.

•  it is also effected by the various possible diagrams (only one in this case, which is why it has been chosen)

( )φθ

αµµ

σ

dddtus

see

dd

cos2 2

222

=Ω

+=→

Ω−+−+

:where

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γe scattering

•  another similar situation. –  only look at the couplings.

–  ignore spin states and absolute value of cross section

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e-e+ → µ-µ+•  fundamental annihilation process.

–  only one diagram

–  sum over final state spins

–  average over initial state spin states

–  related to µ+e- Scattering

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e-e+ → µ-µ+

•  the differential cross section is given by:

( )

( )

( )s

ee

s

sut

see

dd

2

22

2

222

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cos14

2

απµµσ

θα

αµµ

σ

=→

+=

+=→

Ω

−+−+

−+−+

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e-e+ → µ-µ+

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•  identical to the process e-e+ → µ-µ+ except for the quark charges.

–  in case of a up-type quark q=2e/3

–  in case of a down-type quark q=-e/3

qqee →−+