Introduction to QCD and perturbative QCD ELFT Summer

Introduction to QCD and perturbative QCD ELFT Summer School, May 24, 2005 Three lectures: 1. Symmetries – exact – approximate 2. Asymptotic freedom – renormalisation group – β function 3. Basics of p. QCD – fixed order – Resummation Should be understood by everybody, so will be trivial

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

Field content • Quark fields: qf f=1, …, 6 B=1/3 f 1 2 3 4 5 6 qf u d s c b t mf ~5 Me. V ~7 Me. V ~100 Me. V 1. 2 Ge. V 4. 2 Ge. V 174 Ge. V mf running masses (see later) at =2 Ge. V, approximate values; Nf is the number of light quarks (e. g. 3) • Gluon fields: A ɑ ɑ=1, …, 8

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)

The Classical Lagrangian are the SU(Nc) generators with algebra: fundamental representation: adjoint representation:

The Classical Lagrangian Colour factors (like eigenvalues of J 2): • in the fundamental representation: • in the adjoint representation: source of non-Abelian nature: gluon self coupling Note: in lattice QCD

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

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

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 dimensionfour operator, the -term: Conventional normalisation • violates P and T (corresponds to E B in electrodyn. ) is small (<10 -9 experimentally), set =0 in p. QCD

Approximate symmetries of the classical Lagrangian Related to the quark mass matrix Introduce: …and from Dirac algebra: Let: Eigenvector of 5:

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 Nf 2 real numbers:

Approximate symmetries of the classical Lagrangian • This symmetry acts separately on left- and righthanded 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)

Chiral perturbation theory • In QCD it is believed that the vacuum has a nonzero 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 Nf 2 -1 massless 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

Topics of QCD (T=0) • Low-energy properties (<Ge. V) • PT ( light quark masses) • High energy collisions (>Ge. V) • Perturbative • Non-perturbative • Jet physics • Sum rules, lattice QCD

Approximate symmetries of the classical Lagrangian Choose Weyl representation: two-component Weyl spinors Define helicity eigenstates if m=0, g=0

Asymptotic freedom • At the heart of QCD Nobel prize 2004 • Consider a dimensionless physical observable R=R(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 dim. R = 1 • In a renormalized QFT we need an additional scale: renormalization scale R = R (Q 2/ 2) is not a constant: scaling violation • the „small” parameter in the perturbative expansion of R, s( ) also depends on the scale choice

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) =

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

The function in perturbation theory • We solve in PT (we analyse the validity of PT a little later) • known coefficients:

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

The function in perturbation theory • The running coupling resums logs: if R = R 1 s+O( s 2) • R 2 s 2 gives one less log in each term • NLO (b 1 0):

QCD • A more traditional approach to solving the renormalization-group equation: introduce • indicates the scale at which s(Q 2) gets strong • LO (b 0 0, bi=0): • NLO (b 1 0): • The two solutions differ by subleading terms that are important in present day precision measurements

The running coupling

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

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

The running quark mass • All non-trivial scale dependence of R can be included in the running of mass and coupling: Solution: • c/b > 0 the running mass vanishes with the running coupling at high Q 2

2 hard photons in CMS

4 muons in CMS

4 muons in CMS

Basics of Perturbative QCD • Vast subject – only give the flavour • Will use a specific example: • 2→ 2 scattering has one free kinematical parameter, the θ scattering angle • The differential cross section for The total cross section • below the Z pole • on the Z pole

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 =11/3=3. 67 and RZ =20. 09 • The measured value at LEP is RZ =20. 79± 0. 04 • The 3. 5% difference is mainly due to QCD effects: – Real – virtual gluon emission

NLO: real gluon emission • Three-body phase space has 5 independent variables: 2 energies and 3 angles • Integrate over the angles and use yij = 2 pi·pj/s scaled two-particle subenergies, y 12+y 13+y 23=1 • The real contribution to the total cross section: • Divergent along the boundaries at yi 3 = 0: • Unphyisical singularities - quarks and gluons are never on (zero) mass shell: Breakdown of PT • Divergent when E 3→ 0 (soft gluon), or θi 3 → 0 (collinear gluon)

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)

The total hadronic cross section at O(αs 3) • The total cross section can be computed more easily using the optical theorem • Satisfies the renormalization-group equation to order αs 4

The total hadronic cross section at O(αs 3)

The total hadronic cross section at O(αs 4) (non-singlet contribution)

Jet cross sections Typical final states in high energy electron-positron collisions 2 jets 3 jets

Modelling of events with jets Production probability pattern: 2 jets : 3 jets : 4 jets ~ O(αs 0) : O(α s 1) : O(α ⇒ jets reflect the partonic structure s 2)

Jet cross sections • We average over event orientation ⇒ |M 2|2 has no dependence on the parton momenta • NLO corrections: • Cannot combine the integrands (like for σtot)

The subtraction scheme • Process and observable independent solution • Made possible by the process-independent factorisation properties of QCD matrix elements

The subtraction scheme • The approximate cross section: = I(ε) |M 2|2 • with universal factorisation:

The subtraction scheme • The integrated approximate cross section:

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)

General picture of high-energy collisions

R at low energies