Electromagnetic Interactions
Dimensionless coupling constant specifying strenght of
interaction between charged particles
and photons
2
e
1
Fine structure costant:   4c  137
(it determines spin-orbit splitting in atomic spectra)

t
Photoelectric effect: absorption (or emission)
of a photon by an electron (for an electron
bound in an atom to ensure momentum
Conservation.
Em fields have vector transformation properties.
Photon is a vector particle  spin parity JP = 1In the example seen, the photoelectric cross section (or matrix
elements squared) is proportional to a first order process
The Rutherford scattering is a second order process
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e e  
 μμ
e+
-
e-
+

~
Other examples at higher orders:
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 
2
2
 em  4em

em
s



3s 
Renormalization and gauge invariance
QED: quantum field theory to compute
cross sections for em processes
e

renormalisability
Gauge invariance

Electron line: represents a”bare” electron
Real observable particles: “bare” particles “dressed” by these
virtual processes (“self-energy” terms) which contribute to the
mass and charge.
No limitation on the momentum
k of these virtual particles 
dk
logarithmically divergent  k term
As a consequence the theoretically calculated “bare” mass or
charge (m0, e0) becomes infinite
Divergent terms of this type are present in all QCD calculations.
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RENORMALIZATION:
“Bare” mass or charge are replaced by physical values e,m as
determined from the experiment.
A consequence of renormalization procedure:
Coupling constants (such ) are not constants: depends on log of
measurements energy scale
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EM(0) =1/137.0359895(61)
1/EM
GAUGE INVARIANCE:
To have renormalisability:theory must be gauge invariant.
In electrostatics, the interaction energy which can be measured,
depends only on changes in the static potential and not on its
absolute magnitude
invariant under arbitrary changes in the potential scale or gauge
In quantum mechanics, the phase of an electron wavefunction is
arbitrary: can be changed at any local point in space-time and
physics does not change
This local gauge invariance leads to the conservation of currents and
of the electric charge.
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• Conserved quantum numbers associated with cc are colour charges
in strong interactions they play similar role to the electric charge
in em interactions.
• A quark can carry one of the three colours (red, blue, green). An
anti-quark one of the three anti-colours
• All the observable particles are “white” (they do not carry colour)
Hadrons: neutral mix of r,g,b colours
Anti-hadrons: neutral mix of r,g,b anti-colours
Mesons:
neutral mix of colours and anti-colours
• Quarks have to be confined within the hadrons since non-zero
colour states are forbidden.
1
 0
 0
• 3 independent colour wavefunctions
are represented by colour spinor
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 
r   0 ,
 0
 
 
g   1 ,
 0
 
 
b   0
1
 
• These spinors are acted on by 8 independent “colour operators ”
which are represented by a set of 3-dimensional matrices
(analogues of Pauli matrices)
• Colour charges Ic3 and Yc are eigenvalues of corresponding
operators
• Colour hypercharge Yc and colour isospin Ic3 charge are additive
quantum numbers, having opposite sign for quark and antiquark.
Confinement condition for the total colour charges of a hadron:
Ic3 = Yc = 0
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Colour
• Experimental data confirm predictions based on the assumption
of symmetric wave functions
Problem: ++ is made out of 3 quarks, and has spin J=3/2 (= 3
quarks of s= ½ in same state?) This is forbidden by Fermi
statistics (Pauli principle)!
Solution: there is a new internal degree of freedom (colour) which
differentiate the quarks: ++=urugub
• This means that apart of space and spin degrees of freedom,
quarks have yet another attribute
• In 1964-65, Greenberg and Nambu proposed the new property –
the colour – with 3 possible states, and associated with the

corresponding wavefunction c    ( x ) cc C
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Strong Interactions
• Take place between quarks which make up the hadrons
• Magnitude of coupling can be estimated from decay probability (or
width G) of unstable baryons.

0
o


K

p


1385




• Consider:
G=36 MeV, t = 10-23 s
If we compare this with the em decay:  0 1192     , t = 10-19 s
g s2
1
We get for the coupling of the strong charge  s 
4
1
 10 19  2
s 
  100


23
  10 
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QCD, Jets and gluons
• Quantum Chromodynamics (QCD): theory of strong interactions
 Interactions are carried out by a massless spin-1 particle- gauge
boson
 In quantum electrodynamics (QED) gauge bosons are photons, in
QCD, gluons
 Gauge bosons couple to conserved charges: photons in QED- to
conserved charges, and gluons in QCD – to colour charges.
Gluons do not have electric charge and
couple to colour charges  strong
nteractions are flavour-independent
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• Colour simmetry is supposed to be exact  quark-quark force is
independent of the colours involved
• Colour quantum numbers of the gluon are: r,g,b
qb
grb
qr
Neutral combinations:
rgb, rgb, rr, …
• Gluons carry 2 colours, and there are 8 possible combinations
colour-anticolour
• Gluons can have self-interactions
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- Gluons can couple to other gluons
- Bound colourless states of gluons are called glueballs (not detected
experimentally yet).
-Gluons are massless  long-range interaction
Principle of asymptotic freedom
-At short distances, strong interactions are sufficiently weak
(lowest order diagrams) quarks and gluons are essentially free
particles
-At large distances, higher-order diagramsdominate 
interaction is very strong
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• For violent collisions (high q2), as < 1 and single gluon exchange is a
good approximation.
• At low q2 (= larger distances) the coupling becomes large and the
theory is not calculable. This large-distance behavior is linked with
confinement of quarks and gluons inside hadrons.
• Potential between two quarks often taken as:
4 s
Vs  
 kr
3 r
Single gluon exchange
Confinment
• Attempts to free a quark from a hadron results in production of
new mesons. In the limit of high quark energies the confining
potential is responsible for the production of the so-called “jets
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Running of s
The s constant is the QCD analogue of em and is a measure of the
interaction strenght.
However s is a “running constant”, increases with increase of r,
becoming divergent at very big distances.
- At large distances, quarks are subject to the “confining potential”
which grows with r:
V(r) ~ l r (r > 1 fm)
-Short distance interactions are associated with the large

q
 O( r 1 )
momentum transfer
2
2
Lorentz-invariant momentum transfer Q is defined as: Q  q  Eq
2
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-In the leading order of QCD, s is given by:
s 
12
(33  2 N f ) ln(Q 2 / 2 )
Nf = number of allowed quark flavours
 ~ 0.2 GeV is the QCD scale parameter which
has to be defined experimentally
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QCD jets in
+
ee
collisions
- A clean laboratory to study QCD:
e   e    *  hadrons
-At energies between 15 GeV and 40 GeV, e+e- annihilation produces
a photon which converts into a quark-antiquark pair
- Quark and antiquark fragment into observable hadrons
- Since quark and antiquark momenta are equal and counterparallel,
hadrons are produced in two opposite jets of equal energies
- Direction of a jet reflects direction of a corresponding quarks.
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e+
q
 EM
e-
S
q
2 collimated jets of hadrons travelling in
Opposite direction and following the
momentum vectors of the original quarks
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Colliding e+ and e- can give 2
quarks in final state. Then, they
fragment in hadrons
Comparison of the process with the reaction
e  e   *      
must show the same angular distribution both
for muons and jets
2
d

2
(e  e       ) 
(
1

cos
q)
2
d cosq
2Q
where q is the production angle with respect to the initial electron
direction in CM frame
For a quark-antiquark pair:
d
d
 
2
(e e  qq )  3eq
(e  e       )
d cosq
d cosq
Where the fractional charge of a quark eq is taken into account and
factor 3 arises from number of colours. If quarks have spin ½,
angular distribution goes like (1+cos2q); if they have spin 0, like (1cos2q)
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Angular distribution of the quark jet in e+e- annihilation, compared
with models
-Experimentally measured angular dependence is clearly proportional
to (1+cos2q) jets are aligned with spin-1/2 quarks
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If a high momentum (hard) gluon is emitted by the quark or the anti
-quark, it fragments to a jet, leading to a 3-jet events
A 3-jet event seen in a e+e- annihilation at the DELPHI experiment
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-In 3-jet events it is difficult to understand which jet come from
the quarks and which from the gluon
-Observed rate of 3-jet and 2-jet events can be used to determine
value of s (probability for a quark to emit a gluon determined by s)
s= 0.15  0.03 for ECM = 30-40 GeV
Principal scheme of hadroproduction in e+e- hadronization begins at
distances of 1 fm between partons.
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The total cross section of e+e-  hadrons is often expressed as:
 (e  e   hadrons)
R
 (e  e       )
Where:

 

 e e  


4 2
3Q 2
Assuming that the main contribution comes from quark-antiquark
two jet events:
 (e  e   hadrons) 

 (e  e   qq )
q

3
eq2 (e  e       )
q
And hence: R = 3q e2q
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R allows to check the number of colours in QCD and number of
quark flavours allowed to be produced at a given Q
R(u,d,s)=2 R(u,d,s,c)=10/3 R(u,d,s,c,b)=11/3
If the radiation of hard gluons is taken into account,
the extra
2
factor proportional to s arises: R  3 eq2 (1   s (Q ) )

q
Measured R with
theoretical predictions
for five available
flavours (u,d,s,c,b),
using two different
s calculations
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Elastic electron scattering
- Beams of structureless leptons are a good tool for investigating
properties of hadrons
- Elastic lepton hadron scattering can be used to measure sizes of
hadrons
m
 d 

 
 d  R 4 p 2 sin 4 (q )
2
2
2
-Angular distribution of an electron momentum p scattered by a
static electric charge e is given by the Rutherford formula,
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-If the electric charge is not point-like, but is spread with a
spherically symmetric density distribution e er(r), where r(r) is
normalized  r (r )d 3 x  1
Then the differential cross-section is replaced:
Where the electric form factor GE (q )   r (r )e
2
d  d  2 2

 GE (q )
d  d  R

iq x
3
d x
- For q=0, GE(0) = 1 (low momentum transfer)
- For q2, GE(q2) 0 (large mom. transfer)
Measurements of d/d determine the form-factor and hence the
charge distribution of the proton. At high momentum transfers,

q
the recoil energyof the proton is not negligible, and is replaced by
  2
the Lorentz-invariant Q
2
Q  ( p  p' )  (E  E' ) 2
At high Q, static interpretation of charge distribution breaks down.
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Inelastic lepton scattering
-Historically, was first to give evidence of quark constituents of the
proton. In what follows only one-photon exchange is considered
-The exchanged photon acts as a probe of the proton structure
-The momentum transfer corresponds to the photon wavelenght
which must be small enough to probe a proton  big momentum
transfer is needed.
-When a photon resolves a quark within a proton,the total leptonproton scattering is a two-step process
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Deep inelastic
lepton-proton scattering
1)First step: elastic scattering of the lepton from one of the quarks:
l-+q l-+q
2) Fragmentationof the recoil quark and the proton remnant into
observable hadrons
Angular distributions of recoil leptons reflect properties of quarks
from which they are scattered
For further studies, some new variables have to be defined:
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-Lorentz-invariant generalization for the transferred energy n,
defined by:
2 M pn  W 2  Q 2  M 2p
where W is the invariant mass of the final hadron state; in the rest
frame of the proton n = E-E’
2
Q
-Dimensionless scaling variable x: x 
2 M pn
- For Q >> Mp and a very large proton momentum P >>Mp, x is fraction
of the proton momentum carried by the struck quark; 0x 1
-Energy E’ and angle q of scattered lepton are independent variables,
describing inelastic process.
2
d
2
1
2q 
2
2q  Q
2

[cos   F2 ( x, Q )  sin  
F
(
x
,
Q
)
1
2
q
dE' d' 4 E 2 sin 4 ( ) n
2
 2  xM p
2
This is a generalization of the elastic scattering formula
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-Structure functions F1 and F2 parameterize the interaction at
the quark-photon vertex (just like G1 and G2 parameterize the
elastic scattering)
-Bjorken scaling: F1,2(x,Q2) ~F1,2(x)
- At Q >> Mp, structure functions are approx. independent on Q2
-If all particle masses, energies and momenta are multiplied by a
scale factor, structure functions at any given x remain unchanged
F2
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-SLAC data from ’69 were first evidence of quarks
-Scaling is observed at very small and very big x
Approximate scaling behaviour can be explained if protons are
considered as composite objects
- The trivial parton model: proton consists of some partons.
Interaction between partons are not taken into account
- The parton model can be valid if the target p has a sufficiently
big momentum, so that: z = x
- Measured cross section at any given x is prop. to the probability
of finding a parton with a fraction z = x of the proton momentum,
If there are several partons; F2(x,Q2) = ae2axfa(x)
With fa(x) dx = probability of finding a parton a with fractional
momentum between x and x+dx
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 Parton distributions fa(x) are not known theoretically F2(x)
has to be measured experimentally
-Predictions for F1 depend on the spin of a parton:
F1(x,Q2) = 0 (spin-0)
2x F2(x,Q2) = F2(x,Q2) (spin-1/2)
 The expressionfor spin-1/2 is called Callan-Gross relation and is
very well confirmed by experiments  partons are quarks
 Comparing proton and neutron structure functions and those
from n scattering, e2a can be evaluated: it appears to be
consistent with square charges of quarks.
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