LHCC

1
Electromagnetic Interactions
Dimensionless coupling constant specifying
strenght of interaction between charged
particles and photons Fine structure
costant (it determines spin-orbit splitting in
atomic spectra) Em fields have vector
transformation properties. Photon is a vector
particle ? spin parity JP 1- In 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
Photoelectric effect absorption (or emission) of
a photon by an electron (for an electron bound
in an atom to ensure momentum Conservation.

2
e
?-
e-
?
Other examples at higher orders
3
Renormalization and gauge invariance
QED quantum field theory to compute
cross sections for em processes
a
a
Electron line represents abare 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 ?? logarithmically divergent
termAs a consequence the theoretically
calculated bare mass or charge (m0, e0)
becomes infinite
Divergent terms of this type are present in all
QCD calculations.
4
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 a) are not constants depends on log of
measurements energy scale
5
GAUGE INVARIANCE
To have renormalisabilitytheory 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.
6

• 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)
• Quarks have to be confined within the hadrons
  since non-zero colour states are forbidden.
• 3 independent colour wavefunctions
• are represented by colour spinor

Hadrons neutral mix of r,g,b colours Anti-hadron
s neutral mix of r,g,b anti-colours Mesons
neutral mix of colours and anti-colours
7

• 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

8
Colour

• Experimental data confirm predictions based on
  the assumption of symmetric wave functions
• Problem ? is made out of 3 quarks, and has
  spin J3/2 ( 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

9
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.
• Consider
  
• G36 MeV, t 10-23 s
• If we compare this with the em decay
  , t 10-19 s
• We get for the coupling of the strong charge

10
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

11

• 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
• Gluons carry 2 colours, and there are 8 possible
  combinations
• colour-anticolour
• Gluons can have self-interactions

qb
grb
Neutral combinations rgb, rgb, rr,
qr
12

• 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

13

• For violent collisions (high q2), as lt 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
• 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

Single gluon exchange
Confinment
14
Running of as

• The as constant is the QCD analogue of aem and is
  a measure of the
• interaction strenght.
• However as 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 gt 1 fm)
• Short distance interactions are associated with
  the large
• momentum transfer
• Lorentz-invariant momentum transfer Q is defined
  as

15

• In the leading order of QCD, as is given by
• Nf number of allowed quark flavours
• L 0.2 GeV is the QCD scale parameter which
• has to be defined experimentally

16
QCD jets in ee- collisions

• - A clean laboratory to study QCD
• At energies between 15 GeV and 40 GeV, ee-
  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.

17
Colliding e and e- can give 2 quarks in final
state. Then, they fragment in hadrons
e
q
e-
2 collimated jets of hadrons travelling in
Opposite direction and following the momentum
vectors of the original quarks
18
Comparison of the process with the
reaction must show the same angular
distribution both for muons and jets where q
is the production angle with respect to the
initial electron direction in CM frame For a
quark-antiquark pair 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
(1cos2q) if they have spin 0, like (1-cos2q)
19

• Angular distribution of the quark jet in ee-
  annihilation, compared
• with models
• Experimentally measured angular dependence is
  clearly proportional
• to (1cos2q) ?jets are aligned with spin-1/2
  quarks

20
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 ee- annihilation at the DELPHI
experiment
21

• 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 as (probability for a quark to emit a
  gluon determined by as)
• as 0.15 ? 0.03 for ECM 30-40 GeV

Principal scheme of hadroproduction in ee-
hadronization begins at distances of 1 fm
between partons.
22
The total cross section of ee- ? hadrons is
often expressed as Where Assuming that
the main contribution comes from quark-antiquark
two jet events And hence R
3Sq e2q
23
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 factor proportional to as
arises Measured R with theoretical
predictions for five available flavours
(u,d,s,c,b), using two different as
calculations
24
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
• Angular distribution of an electron momentum p
  scattered by a
• static electric charge e is given by the
  Rutherford formula,

25

• 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
• Then the differential cross-section is replaced
• Where the electric form factor
• For q0, GE(0) 1 (low momentum transfer)
• For q2??, GE(q2) ?0 (large mom. transfer)
• Measurements of ds/dW determine the form-factor
  and hence the
• charge distribution of the proton. At high
  momentum transfers,
• the recoil energyof the proton is not negligible,
  and is replaced by
• the Lorentz-invariant Q
• At high Q, static interpretation of charge
  distribution breaks down.

26
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 lepton-
• proton scattering is a two-step process

27

• Deep inelastic
• lepton-proton scattering
• 1)First step elastic scattering of the lepton
  from one of the quarks
• l-q ?l-q
• 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

28

• Lorentz-invariant generalization for the
  transferred energy n,
• defined by
• where W is the invariant mass of the final
  hadron state in the rest
• frame of the proton n E-E
• Dimensionless scaling variable x
• For Q gtgt Mp and a very large proton momentum P
  gtgtMp, 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.
• This is a generalization of the elastic
  scattering formula

29

• 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 gtgt 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
30

• 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)
  Sae2axfa(x)
• With fa(x) dx probability of finding a parton
  a with fractional
• momentum between x and xdx

31

• 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.