Introduction to QCD and perturbative QCD ELFT Summer School, May 24, 2005

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Introduction to QCD and perturbative QCD ELFT
Summer School, May 24, 2005

• Three lectures
• Symmetries
• exact
• approximate
• Asymptotic freedom
• renormalisation group
• ß function
• Basics of pQCD
• fixed order
• Resummation
• Should be understood by everybody, so will be
  trivial

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• When you measure what you are speaking about and
  express it in numbers, you know something about
  it, but when you cannot express it in numbers
  your knowledge about is of a meagre and
  unsatisfactory kind.
• Lord Kelvin

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Field content

• Quark fields qf f1,,6 B1/3

f 1 2 3 4 5 6
qf u d s c b t
mf 5MeV 7MeV 100MeV 1.2GeV 4.2GeV 174GeV
mf running masses (see later) at ?2GeV,
approximate values Nf is the number of light
quarks (e.g. 3)

• Gluon fields A?? ?1,,8

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The QCD Lagrangian

• QCD is a QFT, part of the SM
• The SM is a gauge theory with underlying
• SUc(3)? SUL(2)? UY(1)

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The Classical Lagrangian
are the SU(Nc) generators with algebra
fundamental representation
adjoint representation
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The Classical Lagrangian
Colour factors (like eigenvalues of J2)

• in the fundamental representation

• in the adjoint representation

source of non-Abelian nature gluon self coupling
Note in lattice QCD
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Exact symmetries of the classical Lagrangian

• Quantum effects may violate
• (e.g. scale invariance, axial anomaly)
• Continuous local gauge invariance
• (suppress flavour and Dirac indices)

• The covariant derivative transforms as the field
  itself

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Exact symmetries of the classical Lagrangian

• Almost supersymmetric
• massless QCD for one flavour

• SUSY Yang-Mills

• q transforms under the fundamental, ? under the
  adjoint representation of SU(Nc)
• Advantageous in deriving matrix elements

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Exact symmetries of the classical Lagrangian

• The quark mass term is, the gluon mass term is
  not gauge invariant
• Discrete C, P and T in agreement with observed
  properties of strong interactions (C, P and T
  violating strong decays are not observed)
• Note ? additional gauge invariant dimension-four
  operator, the ?-term

Conventional normalisation

• violates P and T (corresponds to E?B in
  electrodyn.)
• ? ? is small (lt10-9 experimentally), set ?0 in
  pQCD

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Approximate symmetries of the classical Lagrangian

• Related to the quark mass matrix

Introduce
and from Dirac algebra
Let
Eigenvector of ?5
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Approximate symmetries of the classical Lagrangian
The quark sector of the Lagrangian can be written

• would not work if gluons were not vectors (in D)
• the left- and right-handed fields are not coupled
  ? LChir is invariant under UL(Nf)? UR(Nf)
• the group elements can be parametrised in terms
  of 2 Nf2 real numbers

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Approximate symmetries of the classical Lagrangian

• This symmetry acts separately on left- and
  right-handed fields chiral symmetry
• Has vector subgroups SUV(Nf)? UV(1)

• The axial transformations do not form a subgroup

• Chiral symmetry is not observed in the QCD
  spectrum, it is spontaneously broken to
• SUV(Nf)? UV(1)

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Chiral perturbation theory

• In QCD it is believed that the vacuum has a
  non-zero VEV of the light-quark operator

• This quark condensate breaks chiral symmetry
  because it connects left- and right-handed fields

• The SSB of chiral symmetry implies the existence
  of Nf2-1massless Goldstone bosons
• The light quarks are not exactly massless ? the
  chiral symmetry is not exact, the Goldstone
  bosons are not massless pseudoscalar meson octet
• The mf are treated as perturbation ? ?PT

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Topics of QCD (T0)

• Low-energy properties (ltGeV)
• High energy collisions (gtGeV)
• Perturbative
• Non-perturbative

• ?PT (?light quark masses)
• Jet physics
• Sum rules, lattice QCD

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Approximate symmetries of the classical Lagrangian
Choose Weyl representation
two-component Weyl spinors
helicity eigenstates if m0, g0
Define
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Asymptotic freedom

• At the heart of QCD ? Nobel prize 2004
• Consider a dimensionless physical observable
  RR(Q), with Q being a large energy scale,
• Q ? any other dimensionful parameter (e.g. mf)
• ? set mf 0 (check later if R (mf 0) is OK)
• Classically dimR 1 ?

• In a renormalized QFT we need an additional
  scale ? renormalization scale ? R R (Q2/?2) is
  not a constant scaling violation
• ? the small parameter in the perturbative
  expansion of R, ?s(?) also depends on the scale
  choice

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Asymptotic freedom

• But ? is an arbitrary, non-physical parameter
  (LCl does not depend on it) ? physical quantites
  cannot depend on ?

• Let t ln (Q 2/?2), ?(?s) ?

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Asymptotic freedom

• To solve this renormalization-group equtaion, we
  introduce the running coupling ?s(Q 2)

?

• If ?2 Q 2 ? et 1, the scale-dependence in R
  enters through ?s(Q 2)
• All this was non-perturbative yet

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The ? function in perturbation theory

• We solve

in PT (we analyse the validity of PT a little
later)

• known coefficients

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The ? function in perturbation theory

• if ?s(Q 2) can be treated as small parameter, we
  can truncate the series, keep the first two terms

with

• LO

if t??

• Relation between ?s(Q 2) and ?s(? 2) if both
  small
• Q 2 ? 0 ? ?s(Q 2) ? 0 asymptotic freedom (sign!)

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The ? function in perturbation theory

• The running coupling resums logs
• if R R1 ?sO(?s2) ?

• R2 ?s2 gives one less log in each term
• NLO (b1?0)

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?QCD

• A more traditional approach to solving the
  renormalization-group equation introduce ?

• ? indicates the scale at which ?s(Q 2) gets
  strong
• LO (b0?0, bi0)

• NLO (b1?0)

• The two solutions differ by subleading terms that
  are important in present day precision
  measurements

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The running coupling
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The quark masses

• Assume one flavour with renormalized mass m yet
  another mass scale

• ?m is the mass anomalous dimension, in PT

• R is dimensionless ?

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The running quark mass

• To solve this renormalization-group equtaion, we
  introduce the running quark mass m(Q 2)

• the derivative terms (if finite) are suppressed
  by at least an inverse power of Q at high Q 2
• ? dropping the quark masses is justified
• ? only IR-safe observables can be computed

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The running quark mass

• All non-trivial scale dependence of R can be
  included in the running of mass and coupling

Solution

• c/b gt 0 ? the running mass vanishes with the
  running coupling at high Q 2

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2 hard photons in CMS
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4 muons in CMS
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4 muons in CMS
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Basics of Perturbative QCD

• Vast subject only give the flavour
• Will use a specific example

• 2?2 scattering has one free kinematical
  parameter, the ? scattering angle
• The differential cross section for

• The total cross section
• below the Z pole
• on the Z pole

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The total hadronic cross section

• LO the hadronic cross section is obtained by
  counting the possible final states

• With q u,d,s,c,b R 11/33.67 and RZ 20.09
• The measured value at LEP is RZ 20.790.04
• The 3.5 difference is mainly due to QCD effects
• Real
• virtual
• gluon emission

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NLO real gluon emission

• Three-body phase space has 5 independent
  variables 2 energies and 3 angles
• Integrate over the angles and use yij 2pipj/s
  scaled two-particle subenergies, y12y13y231
• The real contribution to the total cross section
  

• Divergent along the boundaries at yi3 0
• Unphyisical singularities - quarks and gluons are
  never on (zero) mass shell Breakdown of PT

• Divergent when E3?0 (soft gluon), or ?i3 ?0
  (collinear gluon)

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NLO the real and virtual contribution in d ? 4

• To make sense of the real contribution, we use
  dimensional regularization

• Has to be combined with the virtual contribution

• The sum of the real and virtual contributions is
  finite in d 4
• (same for RZ)

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The total hadronic cross section at O(as3)

• The total cross section can be computed more
  easily using the optical theorem

• Satisfies the renormalization-group equation to
  order as4

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The total hadronic cross section at O(as3)
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The total hadronic cross section at
O(as4)(non-singlet contribution)
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Jet cross sections
Typical final states in high energy
electron-positron collisions 2 jets 3 jets
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Modelling of events with jets
Production probability pattern 2jets 3jets
4jets O(as0) O(a s1) O(a s2) ? jets reflect
the partonic structure
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Jet cross sections

• We average over event orientation ? M22 has no
  dependence on the parton momenta

• NLO corrections

• Cannot combine the integrands (like for stot)

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The subtraction scheme

• Process and observable independent solution

• Made possible by the process-independent
  factorisation properties of QCD matrix elements

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The subtraction scheme

• The approximate cross section

I(e) M22

• with universal factorisation

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The subtraction scheme

• The integrated approximate cross section

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Parton showers and resummation

• The universal factorisation
• can be used to describe parton showers
  (neglecting colour correlation of soft emissions
  and azimuthal correlations of gluon splitting -
  not transparent in the simple example considered
  here)

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General picture of high-energy collisions
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R at low energies