PHYS402: Particle Physics II

1
PHYS402 Particle Physics II

• Lecture 11
• Gauge Theories and the Search for the Higgs Boson

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Gauge Theories and the Higgs

• Gauge Invariance
• Quantum Electrodynamics (QED)
• Quantum Chromodynamics (QCD)
• The Electroweak Gauge Theory
• Gauge Invariance and the Higgs Boson
• Higgs Boson Phenomenology
• Search for the Higgs Boson at LEP
• Higgs Prospects at Hadron Colliders
• A detailed account of Gauges Theories may be
  found in Appendix C of Martin and Shaw

3
Gauge Invariance

• Gauge Invariance is a fundamental symmetry
  associated with theories in which force carriers
  are spin-1 bosons
• Plays an important role in unified EW theory
  where it is needed to ensure the cancellation of
  divergences which occur in individual Feynman
  diagrams
• Because the W and Z are massive it leads to the
  prediction of a new spin-0 boson - the Higgs
  boson
• The simplest type of Gauge Invariance arises in
  QED
• can use QED to introduce the Gauge Principle
  which makes Gauge Invariance the fundamental
  property from which the detailed properties of
  the interaction are deduced

4
Electromagnetic Interactions

• The wave equation for a particle of mass m and
  charge q moving non-relativistically in an EM
  field is
• H?(x,t) -(? - iqA(x,t))²/2m q?(x,t) ?
  i??(x,t)/?t
• where ?(x,t) and A(x,t) are the scalar and vector
  EM potentials
• E - ?? - ? A /?t and B ?xA
• or, in more convenient form
• i ?/?t iq? ? - (1/2m) (? - iqA)²
  ?

5
Gauge Transformations

• EM potentials are not unique - the E and B fields
  remain unchanged by the transformation
• ? ? ? ? ?f/?t A ? A A -
  ?f (check this!)
• where f(x,t) is an arbitrary scalar function
• this is a gauge transformation and a theory whose
  physical predictions remain unaltered by this is
  gauge invariant
• However, the wave equation does not appear to be
  gauge invariant because ? ? ? and A ? A results
  in
• i ?/?t iq? -iq?f/?t ? - (1/2m) (? -
  iqA -iq?f )² ?
• This is because it was assumed that ? remains the
  same. The remedy is that ? simultaneously
  transforms
• ?(x,t) ? ?(x,t) exp-iqf(x,t)
  ?(x,t)
• consequently ?/?t iq? ?
  exp-iqf(x,t) ?/?t iq? ?
• (? - iqA)² ? (? - iqA) exp-iqf(x,t) (? -
  iqA) ?
• exp-iqf(x,t) (? -
  iqA)² ?

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Gauge Invariance and the Photon Mass

• The E and B Fields satisfy Maxwells equations,
  from which can be derived wave equations for ?
  and A
• (?²/?t² - ?²) ? - (?/?t)(??/?t ?.A) 0
  (check this!)
• (?²/?t² - ?²) A ?(??/?t ?.A) 0
  (and this!)
• These equations are gauge invariant and their
  interpretation is facilitated by choosing a gauge
• for any set of fields obeying these equations can
  always find a function f(x,t) such that the gauge
  transformed fields satisfy the Lorentz condition
• ??/?t ?.A 0
• Choosing the Lorentz Gauge does not alter the
  physical content of the theory - the wave
  equations are now
• (?²/?t² - ?²) ? 0 and (?²/?t² - ?²) A
  0
• Their interpretation follows directly from
  comparison with the Klein-Gordan equation for a
  particle of mass m (see lecture 5)

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Gauge Invariance and the Photon Mass

• (?²/?t² - ?²) ? m² ? 0
  (Klein-Gordan Equation)
• Equations for ?(x,t) and A(x,t) are of exactly
  the same form except that the associated
  particles - photons - are massless
• However, photons are Spin-1 particles which
  actually satisfy the Proca Equations for
  particles of mass m
• (?²/?t² - ?²) ? - (?/?t)(??/?t ?.A) m² ? 0
• (?²/?t² - ?²) A ?(??/?t ?.A) m² A 0
• Adding ?/?t of 1st eqn and div of 2nd eqn gives
• m² ??/?t ?.A 0
• and if m ? 0 then Lorentz condition must be
  satisfied and Proca equations reduce to KG eqns
  for ? and A
• Proca equations are only gauge invariant if m0
  since the m²? and m²A terms arent invariant
  under ? ? ? and A ? A so, gauge invariance
  requires the photon to be massless !

8
The Gauge Principle

• In Electromagnetism invariance of the wave
  equation under a gauge transformation of the EM
  potentials gt the wavefunction also undergoes
  such a transformation
• With the development of the EW Theory and QCD it
  has become fashionable to reverse the argument
• invariance of the wave equation under the gauge
  transformation ?(x,t) ? ?(x,t) is taken as the
  fundamental requirement and is used to infer the
  form of the interactions
• known as the the principle of minimal gauge
  invariance (or simply the gauge principle)
• To understand it, consider EM interactions of
  Spin-1/2 relativistic particles

9
The Gauge Principle

• In the absence of interactions relativistic
  Spin-1/2 particles satisfy the Dirac Equation
  (see lecture 5)
• i??/?t -i?.?? ?m?
• where ?.? ?i1,3 ?i ?/?xi
• This equation is not invariant under the
  fundamental gauge transformation since ?
  satisfies
• i ?/?t iq?f/?t ? -i?.(? iq?f) ?
  ?m?
• This can be fixed by adding just those terms
  involving the EM potentials which are needed to
  make the Dirac equation gauge invariant the
  minimal substitutions are
• ?/?t ? ?/?t iq? and ? ? ? -
  iqA
• Resulting in the equation of motion of the
  electron in QED if q -e
• i ?/?t - ie? ? -i?.(? ieA) ?
  ?m?

10
Quantum Chromodynamics

• In QCD the gauge principle was used to infer
  detailed form of interactions which were
  previously unknown !
• Require a new type of gauge transformation which
  not only changes the phase of the quark
  wavefunctions (as in QED) but also changes the
  colour state
• ?(x,t) ? ?(x,t) exp -igs ?i Fi?i(x,t)
  ?(x,t)
• where gs is the strong coupling constant
• the ?i(x,t) are 8 arbitrary gauge functions
  corresponding to the 8 colour charge operators Fi
  (i1,8)
• The scalar and vector components of the colour
  potential satisfy
• ?i ? ?i ?i ??i/?t gs?jk fijk ?j ?k
• Ai ? Ai Ai - ??i gs?jk fijk ?j Ak
• where fijk are QCD structure constants (see C.5
  MartinShaw)

11
Quantum Chromodynamics

• Form of the quark-gluon and gluon-gluon
  interaction can be derived from the equation of
  motion of the quarks (deduced in an analogous way
  to the QED equation of motion)
• i ?/?t igs ?i Fi ?i ? -i?.(? - igs ?i Fi
  Ai ) ? ?m?
• e.g. consider a red quark ? ?(x,t)?c ?(x,t)r
  ...
• The equation of motion contains terms like
• - gs F1 ?1 ?(x,t)r - gs (1/2) ?1 ?(x,t)g
• because the colour charge operator F1 changes the
  quark colour!
• By colour conservation this is only possible if
  the gluon fields (?i,Ai) can carry away colour ?
  gluons are coloured
• If the colour charges Fi are the sources of the
  gluon fields (?i,Ai) and gluons are coloured then
  gluons can interact with other gluons !

12
Electroweak Interactions Weak Isospin

• Require a set of gauge transformations which
  transform e- and ?e into themselves or into
  eachother
• i.e. neutral current and charged current
  interactions respectively
• QCD contains gauge transformations which
  transform different colour states into eachother
  so begin by setting up a joint description of e-
  and ?e
• Fermions (e- and ?e) carry a weak isospin charge
  IW 1/2 with 3rd component IW3 1/2 (?e) and
  -1/2 (e-)
• 2x2 matrices represent the weak isospin operators
  I1 , I2 and I3
• and can show that I1 ? (1/2) e , I2 e
  (-i/2) ? , etc, as required
• Wavefunctions become ? ?(x,t) ?f where ?f is
  weak isospin (flavour dependent) part
• In the QCD the colour charge operators Fi are
  represented by 3x3 matrices because there are 3
  colour charges (r,g,b)

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Gauge Invariance and Charged Currents

• Description of e- and ?e states differ from
  description of coloured states by the
  replacements
• ? ?(x,t) ?c ? ? ?(x,t) ?f
• Fi (i1,2,8) ? Ii (i1,2,3)
• fijk ? ?ijk (weak structure constants
  defined in C6.1 MartinShaw)
• The gauge transformations required to deduce a
  gauge invariant equation of motion for weakly
  interacting fermions follow from these relations
  see C6.2 of MartinShaw but are of similar
  form to QCD
• the scalar and vector weak potential each have 3
  components
• ? (?1 i?2)/?(2) and W (W1 iW2)/?(2)
  represent the W and W- bosons which mediate
  charged current interactions
• but, NC interactions involving (?3,W3) have
  similar form and strength to the CC interactions
  - disagreeing with experiment !

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The Unification Condition

• Problem of erroneous neutral currents is resolved
  by unification with the electromagnetic
  interaction
• Introduce the weak hypercharge YW via the
  relation
• Q IW3 YW
  (Q is electric charge)
• and then treat YW as a source (with coupling g
  different from IW coupling g) by requiring gauge
  invariance under suitable gauge transformations
  (see MartinShaw C6.3)
• The electromagnetic fields (?,A) and the Z
  fields (?z,Z) appear as arbitrary linear
  combinations of (?3,W3) and the (?B,B) fields
  associated with the YW (?W is the weak mixing
  angle)
• A(x,t) B(x,t)cos?W W3(x,t)sin?W
• Z(x,t) -B(x,t)sin?W W3(x,t)cos?W
• To avoid EM coupling to ? and to ensure correct
  QED coupling
• e g sin?W g cos?W (the unification
  condition )

15
Gauge Invariance and the Higgs Boson

• Gauge invariance has been shown to imply that
  Spin-1 gauge bosons are massless if they are the
  only bosons in the theory - this is OK for QED
  and QCD which both involve massless gauge bosons
  (photons and gluons) -however, the W and Z bosons
  are very heavy !
• This apparent contradiction is overcome by
  assuming that the various particles interact with
  a new kind of scalar field called the Higgs
  Field
• interactions of the Higgs field with the gauge
  bosons are gauge invariant - but it differs from
  the other fields in its behaviour in the vacuum
  state where it has a non-zero vacuum expectation
  value, ?0, which is not invariant under a gauge
  transformation - for this reason the gauge
  invariance is referred to as hidden or
  spontaneously broken (although the gauge
  invariance of the Higgs interactions, as opposed
  to the vacuum state, remains exact)

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The Higgs Potential

• Exactly how the Higgs Mechanism gives mass to the
  gauge bosons is beyond the scope of this course
  (and MartinShaw!)
• but, anyway, heres a taste of the Higgs
  potential ...

17
Higgs Phenomenology

• Existence of the Higgs field has 3 main
  consequences
• W and Z bosons acquire mass in the ratio MW/MZ
  cos?W
• neutral Higgs bosons H must exist !
• interactions with the Higgs field can generate
  fermion masses
• This is the most important prediction of the
  Standard Model which has not been verified by
  experiment
• Designing experiments to search for H is not
  easy ...
• the theory does not predict the mass of the Higgs
  boson
• however, the theory does predict its couplings to
  other particles
• e.g. coupling to fermions gffH mf
• ? H?bb is likely to be the decay mode for Higgs
  discovery
• Since H couples strongly to W and Z the best
  places to search for it are at LEP and at Hadron
  Colliders
• Precise EW data places important limits on MH ...

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Indirect Higgs Search
Strongly favours a light Higgs Boson with MH lt
200 GeV
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Higgs Branching Fractions
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Higgs Search at LEP
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Hint of a Higgs at LEP ?
The ALEPH Collaboration claimed in November 2000
to have evidence for the existence of a Higgs
boson with mass 115 GeV the evidence is not
overwhelming (!) and the other 3 collaborations
do not see an excess of events around 115 GeV
Large background sample
Lower background sample
Lowest background sample
ee- ? ZH ?4 jets
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Higgs Search at the Tevatron
Gluon fusion dominates but WH/ZH more accessible
(gg?bb channel has huge QCD background !)
Gluon Fusion
Associated Production WH or ZH
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Low Mass Higgs at the Tevatron
For MH ? 135 GeV use the same basic strategy as
LEP study associated production of ZH and WH
To the standard leptonic HZ channels add W ? l ?
with H ? bb ...
the qqbb channel is very difficult as the QCD
backgrounds are severe

• Low mass Higgs sensitivity depends on
• the integrated luminosity collected
• b-quark jet tagging performance
• mass resolution of reconstructed bb jets

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Example of Light Higgs Signal at the Tevatron
ZH ? ??bb / eebb/ ??bb
Simulation!
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Tevatron Higgs Discovery Potential
95 CL exclusion to 185 GeV for 10 fb-1 ? 3?
observation to 185 GeV for 20 fb-1 5?
discovery up to 125 GeV for 30 fb-1
Tevatron expected to accumulate 10 fb-1 by 2005
and 15 fb-1 by 2007
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LHC Higgs Discovery Potential
The LHC will certainly find the Higgs boson(s) if
it(they) exists()!