Transparancies for Feynman Graphs
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Transcript Transparancies for Feynman Graphs
Particle Physics
2nd Handout
Feynman Graphs of QFT
•Relativistic Quantum Mechanics
•QED
•Standard model vertices
•Amplitudes and Probabilities
•QCD
•Running Coupling Constants
•Quark confinement
http://ppewww.ph.gla.ac.uk/~parkes/teaching/PP/PP.html
Chris Parkes
• See Advanced QM option
Adding Relativity to QM
p2
Apply QM prescription p i
E
2m
2 2
i
Get Schrödinger Equation
2m
dt
Missing phenomena:
Anti-particles, pair production, spin
Free particle
Or non relativistic
Whereas relativistically
1 2 p2
E mv
2
2m
E 2 p 2c 2 m2c 4
Applying QM prescription again gives:
Klein-Gordon Equation
1
mc
2
2
2
c dt
2
2
Quadratic equation 2 solutions
One for particle, one for anti-particle
Dirac Equation 4 solutions
particle, anti-particle each with spin up +1/2, spin down -1/2
2
Positron
KG as old as QM, originally dismissed. No spin 0 particles known.
Pion was only discovered in 1948.
Dirac equation of 1928 described known spin ½ electron.
Also described an anti-particle – Dirac boldly postulated existence of positron
Discovered by Anderson in 1933 using a cloud chamber (C.Wilson)
Track curves due to magnetic field F=qv×B
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Transition Probability
reactions will have transition probability
How likely that a particular initial state will transform to a specified final state
Interactions
e.g. decays
IV
Transition rateProby of decay/unit time
cross-section x incident flux
We want to calculate the transition rate between initial state i and final state f,
We Use Fermi’s golden rule
This tells us that fi (transition rate) is proportional to the
transition matrix element Tfi squared (Tfi 2)
T fi f H ' i
k i
f H' k k H'i
Ei Ek
....
This is what we calculate from our QFT, using Feynman graphs
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Quantum ElectroDynamics (QED)
• Developed ~1948 Feynman,Tomonaga,Schwinger
• Feynman illustrated with diagrams
Photon emission e-
Pair production
e-
annhilation
e-
e-
Time: Left to Right.
e+
e+
Anti-particles:backwards in time.
c.f. Dirac hole theory
M&S 1.3.1,1.3.2
Process broken down into basic components.
In this case all processes are same diagram rotated
We can draw lots of diagrams for electron scattering (see lecture)
Compare with
T fi f H ' i
k i
f H' k k H'i
Ei Ek
....
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Orders of
• The amplitude T is the sum of all
amplitudes from all possible diagrams
Feynman graphs are calculational tools, they have terms associated with them
Each vertex involves the emag coupling (=1/137) in its amplitude
So, we have a perturbation series – only lowest order terms needed
More precision more diagrams
There can be a lot of diagrams!
N photons, gives n in amplitude
c.f. anomalous magnetic moment:
After 1650 two-loop
Electroweak
diagrams Calculation accurate
at 10-10 level
and experimental
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precision also!
The main standard model vertices
s
W
s 0. 1
1
At low energy: 137
W
1
29
Strong:
All quarks (and
EM:
anti-quarks)
Weak neutral current: Weak charged current:
All charged particles
7
No change of
All particles
All particles
No change of flavour
flavour
No change of flavour Flavour changes
Amplitude Probability
|Tfi|2
The Feynman diagrams give us the amplitude,
c.f. in QM whereas probability is ||2
(1)
So, two emag vertices:
e.g.
e-e+ -+
amplitude gets factor from each vertex
And xsec gets amplitude squared
2
for e-e+ qq with quarks of charge q (1/3 or 2/3) ( q ) q
•Also remember : u,d,s,c,t,b quarks and they each come in 3 colours
•Scattering from a nucleus would have a Z term
2
(2)
If we have several diagrams contributing to same process,
we much consider interference between them e.g.
(b)
e(a)
eee+
e+
e+
2
2
ee+
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Same final state, get terms for (a+b)2=a2+b2+ab+ba
Massive particle exchange
Forces are due to exchange of virtual field quanta (,W,Z,g..)
E,p conserved overall in the process but not for exchanged bosons.
You can break Energy conservation as long as you do it for a short enough time that you don’t notice!
i.e. don’t break uncertainty principle.
Consider exchange of particle X, mass mx, in CM of A:
B
X
A
Uncertainty principle
Particle range R c c / E
A(mA ,0) A( EA , p) X ( EX ,p)
E E X E A m A
E 2 p : p
E m X : p 0
E m X For all p, energy not conserved
/mx c
So for real photon, mass 0, range is infinite
For W (80.4 GeV/c2) or Z (91.2 GeV/c2), range is 2x10-3 fm
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Virtual particles
This particle exchanged is virtual (off mass shell)
e.g.
e-
(E,p)
symmetric
Electron-positron
(E,-p)
collider
E 2 E
+
e+
(E , p)
p 0
m
*2
E p 0
2
2
Yukawa Potential
Strong Force was explained in previous course as neutral pion exchange
Consider again:
•Spin-0 boson exchanged, so obeys Klein-Gordon equation
See M&S 1.4.2, can show solution is
g 2 er / R
V (r )
4 r
Can rewrite in terms of dimensionless
strength parameter
g2
X
2
4c
e 1
For mx0, get coulomb potential V (r )
10
40 r
R is range
7.1 M&S
Quantum Chromodynamics (QCD)
QED – mediated by spin 1 bosons (photons) coupling to conserved electric charge
QCD – mediated by spin 1 bosons (gluons) coupling to conserved colour charge
u,d,c,s,t,b have same 3 colours (red,green,blue), so identical strong interactions
[c.f. isospin symmetry for u,d], leptons are colourless so don’t feel strong force
•Significant difference from QED:
• photons have no electric charge
• But gluons do have colour charge – eight different colour mixtures.
Hence, gluons interact with each other. Additional Feynman graph vertices:
Self-interaction
s
3-gluon
4-gluon
These diagrams and the difference in size of the coupling constants are responsible
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for the difference between EM and QCD
Running Coupling Constants - QED
+
-
+Q
-
Charge +Q in dielectric medium
Molecules nearby screened,
At large distances don’t see full charge
Only at small distances see +Q
+
Also happens in vacuum – due to spontaneous production of virtual e+e- pairs
e+
e+ e-
And diagrams with
two loops ,three loops….
each with smaller effect: ,2….
QED – small variation
eAs a result coupling strength grows with |q2| of photon,
1/128
1/137
higher energy smaller wavelength gets closer to bare charge
0
|q2|
(90GeV)2
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Coupling constant in QCD
•Exactly same replacing photons with gluons and electrons with quarks
•But also have gluon splitting diagrams
g
g
g
This gives anti-screening effect.
Coupling strength falls as |q2| increases
Grand Unification ?
g
LEP data
Strong variation in strong coupling
From s 1 at |q2| of 1 GeV2
To s at |q2| of 104 GeV2
Hence:
•Quarks scatter freely at
high energy
•Perturbation theory converges very
Slowly as s 0.1 at current expts
And lots of gluon self interaction diagrams
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Range of Strong Force
Gluons are massless, hence expect a QED like long range force
But potential is changed by gluon self coupling
Qualitatively:
QED
Form of QCD potential:
QCD
VQCD
+
-
Standard EM field
q
q
Field lines pulled into strings
By gluon self interaction
4
3
s
r
kr
Coulomb like to start with,
but on ~1 fermi scale energy
sufficient for fragmentation
QCD – energy/unit length stored in field ~ constant.
Need infinite energy to separate qqbar pair.
Instead energy in colour field exceeds 2mq and new
q qbar pair created in vacuum
This explains absence of free quarks in nature.
Instead jets (fragmentation) of mesons/baryons
NB Hadrons are colourless, Force between
hadrons due to pion exchange. 140MeV1.4fm
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Formation of jets
1. Quantum Field Theory – calculation
2. Parton shower development
3. Hadronisation
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Summary
Add Relativity to QM anti-particles,spin
Quantum Field Theory of Emag – QED
1.
2.
•
•
3.
Feynman graphs represent terms in perturbation series in powers
of α
Couples to electric charge
Standard Model vertices for Emag, Weak,Strong
•
Diagrams only exist if coupling exists
•
e.g. neutrino no electric charge, so no emag diagram
QCD – like QED but..
4.
•
•
5.
Gluon self-coupling diagrams
α strong larger than α emag
Running Coupling Constants
•
α strong varies, perturbation series approach breaks down
QCD potential – differ from QED due to gluon interactions
6.
•
Absence of free quarks, fragmentation into colourless hadrons
Now, consider evidence for quarks, gluons….
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