Download Lecture 12 – Asymptotic freedom and the electrodynamics of quarks

Lecture 12 – Asymptotic freedom and the
electrodynamics of quarks
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●
Asymptotic freedom and the running of coupling
”constants”.
Testing the theory of the quarks and strong
force in e- e+ reactions.
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The theory of the strong force QCD
Its a bit embarrassing.
We talk about quarks and gluons but we haven't
even attempted a calculation/estimate for a process
using Feynman diagrams.
q
q
αs
q
q
αs
Problem is that α S >> 1 for most of the processes we've been interested
in. However, α S can be low depending on how close the quarks
are to each other.
Easiest to see how this all works by first considering electromagnetic interactions.
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Capacitors
Field in capacitor with vacuum between
Q
plates : E0 =
Aε 0
Dielectric between plates is polarised
(+- dipoles produced)
Field in capacitor between plates:
Q
E=
εr
< E0
Aε 0
εr > 1
The presence of +- dipoles lowers the field.
Plate area =A
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What is the charge of a particle ?
+ve charged particle q in a dielectric.
The material surrounding the particle is composed of
molecules which become polarised by the electric
field of q. Produce a dipole field which reduces the
electric field from q.
q
q
εr
E=
→
(12.01)
4πε 0 r 2
4πε 0 r 2
Anyone making a measurement of q in the dielectric
would see a screened charge q . Screening reduced as
εr
measurement is made closer to the particle than molecular
separation.
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Screening in a dielectric
qeff
q/εr
Intermolecular separation
r
The effective charge increases at small distances
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Vacuum polarisation
The "vacuum" consists of virtual particles fluctuating into and out of existence.
An electron is surrounded by virtual particles which act to shield the charge as
in a polarised dielectric.
eg e − , e + pairs (lightest and easiest to make).
Feynman diagram formalism shown as photon coupling to e − , e + pairs.
Screening reduced for distances shorter than λc =
λc =
h
(12.02)
me
h
= Compton wavelength=2.43 ×10−12 m (12.03)
me
e-
e-
e-
e-
e+,e+
e+,e-
e-
e-
e+,ee-
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Screened charges
Electromagnetic force (QED)
Electric charge screened: distance > Compton wavelength λc
α
Strong force (QCD)
e2
=
Implications: coupling constant
at vertex:
(1.24)
Effective
colour α
charge
grows
at larger distances.
4πε 0
Small charge over distances < fm lead to asymptotically free
If the charges are screened then coupling, α changes according to interaction distance.
quarks in hadrons.
Alternatively interaction distance d ∼
1
1
∼
(12.04)
momentum exchange Q
⇒ α depends on interaction distance or momentum exchange.
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The electromagnetic coupling
4πε 0
= 2
α
e
1
Barely changes
~(Momentum transfer=Q)2 /GeV2
Interaction distance
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What about the strong force ?
q
q
q
q
q
q
q
q
q,q
+ other higher
order diagrams
q,q
q,q
q
q
q
q
q
q
q
q
Similar story as for electromagnetism except that gluons
can self-interact (they carry colour - the photon carries
no charge!)
This turns out to be critical.....
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Asymptotic freedom
Strong force (Quantum chromodynamics: QCD)
Effective colour charge grows at larger distances.
Small charge over distances < fm lead to quasi-free
quarks in hadrons.
Asymptotic freedom!
Nobel prize (2004) for Gross, Politzer and Wilczek.
 ( 33 − 2 N f )
 Q
α s ( Q ) = α s ( M Z ) 1 +
α s ( M Z ) ln 
6π

 MZ
N f = number of quark flavours
αs


 
−1
(12.05)
α s ( M Z ) = 0.118 ± 0.002 (12.06)
Varies strongly with momentum!!
αs
Q
αs
Momentum transfer, Q (GeV)
Interaction distance
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Running of the coupling ”constants”
Similar story for weak force.
If interactions occur over
distance scales which
suppress screening
The three forces would be
of (roughly) the same size
Measurements
Predicted behaviour (tested
with measurements at lower
energies)
The coupling strengths of the strong,electromagnetic, and
Weak forces converge at higher energies. Is this evidence that
they are part of a single force ? Later lecture.
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Consequences
2
From lecture 11 : F2 constant with Q at fixed x.
Quasi-free quark is struck by photon.
⇒ Bjorken scaling.
1
Short distance/time interaction ∆t ∼
Q
QCD corrections imply scaling violations.
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Scaling violations
Described by QCD
over several orders
of magnitude in Q2.
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OZI suppression
In OZI-suppressed processes, the gluons carry all of the momenta
of some or all of the final particles and there is consequently a
lower probability of emitting such ”hard” gluons in comparison with
”soft” gluons in non-OZI suppressed reactions.
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Testing what we know about the quarks and their interactions
Use what we've learned from hadron properties and
DIS on a totally different reaction:
e − + e + → hadrons.
Study electomagnetic quark interactions
(quark electrodynamics) and strong interactions..
Show
(1) 3 colours
2
1
(2) Quarks (+ antiquarks) with charges ± e and ± e.
3
3
(3) QCD makes precise predictions for short distance reactions.
Also discuss how long-distance effects produce the observed hadrons
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Ratio of hadronic to muonic production in e+e- collisions
R=
σ ( e− e+ → hadrons )
σ (e e → µ µ
− +
−
+
)
∼
.
Centre-of-mass energy ECM (GeV)
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Understand the features of the graph.
(1) e − e + → hadrons ≡ e− e+ → qq (bound state -resonance)+ e − e + → qq (non-bound state)
≡
+
Resonance : Ecm = resonance rest mass . Particle must be produced at rest in c.m frame.
eg e− e + → ρ ; ρ mass ∼ 770 MeV , width ∼ 200 MeV
⇒ Mass-localised peak around Ecm ∼ 770 MeV
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Bound + non-bound states
Non-bound states
Centre-of-mass energy ECM (GeV)
Five quarks and three colours
Away from the resonance region 15 < Ecm < 40 GeV
σ ( e− e + → hadrons ) = σ ( e− e+ → qq ) = N C ea2 ( e − e + → µ − µ + ) (12.08)
NC = number of colours. Five quarks u, d , s, c, b can be produced (top is too heavy):
11
NC
9
11
No. colours N C = 3 ⇒ R =
3
Five quarks and three colours!
R = N C ( eu2 + ed2 + es2 + ec2 + eb2 ) =
R = 5 ⇔ 6 quarks,3 colours
Bound + non-bound states
Non-bound states
R=
11
⇔ 5 quarks,3 colours
3
R=
11
⇔ 5 quarks,1 colour
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Centre-of-mass energy ECM (GeV)
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Taking a closer look
QCD demands a correction to (12.08) by taking into
account an additional diagram (gluon emission)
Gluon momentum >> 1 GeV
⇒ α S << 1 GeV
⇒ use QCD.
with correction
without correction
Bound + non-bound states
Non-bound states
Centre-of-mass energy ECM (GeV)
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”Observing” gluon emission
Gluons and quarks carry colour so are never seen.
jet
The process by which they convert into hadrons
is known as hadronisation/fragmentation. High
energy quarks and gluons convert into jets of hadrons.
Models of hadronisation exist but are simply very good
jet
"best guesses". We don't yet understand the process
jet
by which quarks are confined and therefore the process
through which jets are formed. An example of how we
think hadronisation is given in the next lecture when the
jet
top quark discovery is discussed.
Computer visulation of 3 jets
jet
reconstructed in an e − e + reaction
jet
at the PETRA collider (1979).
First measurement of gluon-jets.
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Summary
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Vacuum polarisation makes the couplings of the
fundamental forces ”run” with energy
−
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Asymptotically free quarks!
At high energies QCD makes precise
calculations
−
Scaling violations
−
Hadronic to muonic ratio
Tested our picture of quarks, gluons and the
strong force on e- e+ interactions.
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