LieAlgebras in Elementary Particle Physics

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Lie-Algebras in Elementary Particle Physics

• Alexander Merle
• Max-Planck-Institut für Kernphysik
• University of Heidelberg
• E-Mail Alexander.Merle_at_mpi-hd.mpg.de

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• Users Manual how to attend this talk
• as a High School Student dont be disappointed
  if you dont understand all the formulae try to
  fol-low the basic principles and to get a clue of
  the idea behind
• as a University Student try also to understand
  the mathematics a bit deeper and maybe to
  cal-culate some things yourself
• as a lecturer always keep in mind that Im a
  physicist so please forgive me the following
  equations!
• Lie Algebra Lie Group Symmetry ep3

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• Table of Contents
• The particles in our nature and where we know
  them from
• The description of particles by fields
• The gauge principle
• QED (U(1)) vs. QCD (SU(3))
• Effects of the group structure
• Conclusions

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I. The particles in our nature and where we know
them from
exchange bosons
matter
Spin ½
Spin 1
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Question How are forces mediated??
from http//www.physics.ohio-state.edu/klaus/ph
ys780.20/lecture_notes/forces.gif
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Feynman diagrams
Feynman diagrams serve as a nice short
description for complicated mathematics.
e.g. electron-positron scattering via
electromagnetic interaction exchange boson
photon
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Particle Accelerators
LHC_at_CERN (Geneva/Switzerland) p p
HERA_at_DESY (Hamburg/Germany) p e-
and many more
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• Some properties of the elementary particles
• u-quarks (up, charm, top) electrical charge
  2/3 (fractional elementary charge!!!), color
  charge (red, green, blue)
• d-quarks (down, strange, bottom) electrical
  charge -1/3, color charge
• charged leptons (electron, muon, tauon)
  electrical charge -1, no color charge
• neutrinos (electron-neutrino, muon-neutrino,
  tau-neutrino) no electrical no color charge
• ? Question Where do we know that from???
• ?Answer R-ratio, particle production,

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The R-ratio
? one of many proofs for number of colors
frac-tional electrical charges
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II. The description of particles by fields
The Pauli principle
All particles have either integer spin (s0,1,
bosons) or half-integer spin (s1/2, 3/2,
fermions). ? Fermions obey the Pauli exclusion
principle two fermions can NEVER have exactly
the same state (space-time point, spin
(projections), orbital angular momentum,) ? this
is one of the reasons that our world looks like
it does electrons are fermions ? existence of
atomic shells due to the Pauli principle ?
chemistry as we know it
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Mathematical notation

• space-time point
• metric
• scalar product of 4-vectors
• momentum operator in QM
• dAlembert operator
• Feynman slash
• ?-matrices
• Pauli matrices
• 2x2 unity matrix

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Lagrangians

• Lagrangian density energy density
• it contains fields F(x) they are functions of
  the space-time and describe the different
  particles ? one needs dif-ferent fields depending
  on the spin of the particle
• there exist 3 types of terms in the Lagrangian ?
  struc-tures
• -kinetic terms e.g. (field) x (field) or
  (field) x (field)
• -mass terms (field)2
• -interaction terms contain 3 or more fields
• ? these terms are enough to describe all
  particles and in-teractions

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Scalar fields spin s0

• equation of motion Klein-Gordon equation
• free Lagrangian
• -real scalar field neutral particles
• -complex scalar field charged particles
• physical particles Higgs, pion, kaon,

kinetic term
mass term
kinetic term
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Spinor fields spin s1/2

• equation of motion Dirac equation
• has 4 degrees of freedom particle
  anti-particle, each with spin up or spin down
• 4-component object needed spinor (?vector!!!)
• free Lagrangian
• physical particles electron, neutrino, quarks,

kinetic term
mass term
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Massless vector fields spin s1

• equation of motion Maxwell equations
• field strength tensor
• free Lagrangian
• ? has NO mass term ? reason will become clear
  later
• physical particles photon, gluon,

(only kinetic term)
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III. The gauge principle
SU(N)-groups

• S special ? unit determinant
• U unitary ?
• N dimension ? NxN-matrix
• every group element can be written as
• Ts generators? fulfill the Lie-Algebra
• ? x,yxy-yx commutator
• (N2-1) Hermitean (TT) traceless (Tr(T)0)
  generators
• fs structure constants ? characterize the
  group

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example SU(2)

• unitarity
• ? this is fulfilled for Hermitean generators
• determinant relation
• ? this is fulfilled for traceless generators
• for N2, these generators are simply the 22-13
  Pauli mat-rices (times ½) ? every group element
  can be written as
• commutation relation of the Pauli-matrices
• ? e is the totally antisymmetric tensor of 3rd
  level

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• hence, on gets for the commutation relation of
  the genera-tors
• and for the structure constants
• hence, we now know how every group element U
  from SU(2) has to look like!
• the basic principles remain the same for higher
  values of N, but the number of generators grows
  like (N2-1)
• - generator complex NxN matrix ? 2N2 degrees of
  freedom
• - Hermitean generators ? 1 equation for each
  element of the matrix ? only 2N2-N2N2 degrees of
  freedom left
• - Traceless generators ? 1 more equation for
  all entries together ? finally N2-1 degrees of
  freedom left ? there can be
  only N2-1 independent generators!

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Global transformations
e.g. global SU(N)-transformation of a spinor
field Important CONSTANT coefficients ?a!!!
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Local (gauge) transformations
e.g. local SU(N)-transformation of a spinor
field Important the ?a have to be a
CONTINOUS function of x!!!
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The invariance of the action

• the action S is defined as the space-time
  integral over the Lagrangian density
• hence, it has the dimension of (energy) x
  (time), like ?
• basic principle The action has to be invariant
  under all gauge transformations!!!
• reason a gauge is nothing physical, just
  something like giv-ing names to certain things
  (e.g. strong interaction should not depend on
  what we call red, green, and blue, even if
  this naming is different for different points in
  space-time)
• this means for the Lagrangian, that it (in
  principle) also has to be invariant (up to total
  derivatives) ? no artificial pro-duction of
  energy

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IV. QED (U(1)) vs. QCD (SU(3))

• QED
• interaction electromagn.
• fermions electron positron
• exchange boson photon (electrically neutral)
• spinor
• U(1)-symmetry (scalar pre-factor)

• QCD
• interaction strong
• fermions quarks and anti-quarks, each with 3
  possible (anti-)colors
• exchange bosons 8 gluons (carry color
  anti-color!!!)
• spinor
• SU(3)-symmetry (3 components ? 3x3 pre-factor)
  needed

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Lagrangians without interactions

• QED
• particle content 1 fer-mion ( anti-particle)
  and 1 boson (photon)
• QED-Lagrangian
• no terms that couple the photon with the
  electron (all contain either one or the other
  one) ? No interactions!!! (yet)

• QCD
• particle content 6 fermions (
  anti-particles) and 8 bosons (gluons)
• QCD-Lagrangian
• no terms that couple gluons with the quarks (all
  contain either one or the other one)
  weiß
  ? No interactions!!! (yet)

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The rise of interactions

• QED
• interaction terms that couples electron with
  photons ? by covariant
  derivative
• new term

• QCD
• interaction terms that couples quarks with
  gluons
  weiß
  ? by covariant derivative
• new term

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The gauge transformation

• QED
• U(1)
• question What happens to the Lagrangian??
• electron mass term
• ? invariant! v

• QCD
• SU(3)
• question What happens to the Lagrangian??
• quark mass term
• ? invariant! v

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• QED
• electron kinetic and in-teraction term
• only invariant, if the photon also transforms

• QCD
• quark kinetic and interaction term
• only invariant if the gluons also transform

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V. Effects of the group structure
Physical consequence massless gauge bosons

• QED
• a photon mass term would look like
• How does such a term behave?
  weiß

• QCD
• mass terms for gluons would look like
• How does such a term behave under a gauge
  transformation? Weiß

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• QED
• a photon mass term would NOT be gauge-invariant!
• NOT ALLOWED!!!
• consequence
• Photons have to be mass-less to ensure
  gauge-in-variance of the QED-La-grangian under
  U(1)!
• ? This is exactly what is seen in all
  experiments!
• ? (Free) photons always propagate with the speed
  of light!

• QCD
• mass terms for the gluons would NOT be
  gauge-invariant! weiß
• NOT ALLOWED!!!
• consequence
• Gluons have to be massless to ensure
  gauge-invariance of the QCD-Lagrangian under
  SU(3)! weiß
• ? This is exactly what is seen in all
  experiments!
• ? Free gluons (if they existed) would always
  propagate with the speed of light!

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Further physical consequence self-interaction
gauge bosons

• QED
• alternative definition of the field strength
  with the covariant derivative

• QCD
• here, this definition of the field strength is
  in fact the more natural one

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• QED
• Photon-Lagrangian
• only kinetic terms (quadratic in A) weiß
• no-interaction between photons (at lowest order)
• Photons are electrical-ly neutral!
• weiß
• weiß weiß

• QCD
• Gluon-Lagrangian
• additionally to the kinetic terms cubic (G3)
  and quartic (G4) terms weiß
• weiß
  weiß
• weiß
  weiß
• weiß
• Self-interaction Gluons carry color!

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VI. Conclusions

• we have a good knowledge about the particles in
  our world coming from excellent experiments
• gauge transformations seem to be the right way
  to describe interactions physical properties of
  the ex-change particles as well as the
  possibility and im-possibility, respectively, of
  certain processes are predicted correctly
• direct consequences of the group structure can
  be seen directly in nature
• the understanding of Lie algebras is necessary
  for the understanding of symmetries in nature

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References

• B. Schrempp Theory of Elementary particles
• L. Ryder Quantum Field Theory
• T.-P. Cheng L.-F. Li Gauge Theory of
  Elementary Par-ticles
• B. Povh et al. Teilchen und Kerne

THANKS FOR YOUR ATTENTION!!!
Further Questions??? ? Contact me Alexander.Merle
_at_mpi-hd.mpg.de