Quantum ChromoDynamics

Quantum ChromoDynamics
Lectures by Gunnar Ingelman in April 2013 at course Advanced Particle Physics (1FA355) Starting point: QCD established as the theory for strong interactions, but non-perturbative region major unsolved problem Approach: Pragmatic – not formal/mathematical (detailed knowledge requires more time/studies) Goal: Understanding & intuition Basis for general physics knowledge & further studies G. Ingelman: QCD

QCD is vast field  overview / selections
Lecture 1: Basic theory - colour SU(3) - gauge principle - Lagrangian - Feynman rules - parton cross section - running coupling - asymptotic freedom - confinement - renormalisation Lecture 2: Theory  data - cross section - factorization - parton density fcn’s - matrix elements - parton branchings - jets - precision tests Lecture 3: Hadronisation - cluster model - Lund string model QCD at LHC - dominates cross-section - background for BSM - affecting BSM-signals - quark-gluon plasma G. Ingelman: QCD

Colour quantum number Quarks have colour quantum number in 3 values: Red, Green, Blue Original experimental evidences: Spin-3/2 baryon symmetric in flavour, space, spin But Fermi-Dirac statistics require totally anti-symmetric wave function, obtained through colour: Decay rate fits data =7.70.6 eV for charge factor including 3 colours with factor 3 from colour fits data + many checks in later detailed QCD tests … 0 G. Ingelman: QCD

Colour – SU(3) Colour ‘charge’ – source of strong force field
3 charges: ‘Red, Green, Blue’  basis states Arbitrary state Global gauge invariance: R G B everywhere  no change  Convention/choice of RGB-basis Transformations by 3x3 matrices 8 linearly indep. matrices (& unit matrix  no change) Conservation of probability/norm  unitary operator, i.e. UU†=1 or U†=U-1 Unitary operator where T is hermitean operator Infinitesimal transformation 1 gives U1+iT Here, T aTa= a a/2 where a=1,2…8 and a are Gell-Mann matrices i.e. the transformations form SU(3) - the special, unitary group in 3D with commutation relations [Ta,Tb]=ifabcTc , structure constants fabc define group G. Ingelman: QCD

Local gauge symmetry Invariance under conventions depending on space (US—EU) or time (am—pm), i.e. related by transformation U= exp(ia(x)Ta) Quark field: Last term spoils invariance of Lagrangian for free quark Restored by introducing 8 gluon fields with compensating transformation properties New Lagrangian density with gluon fields G and field strength tensor giving interactions with coupling gs Covariant derivative gives Gauge invariance  no gluon mass term  massless gluons Tully: eta corresponds to alpha through g_s = i.e. eta=alpha/g_s lambda_k /2 corresponds to T_a G. Ingelman: QCD

Gauge = standard of measure/calibration
Local symmetry  convention can be decided independently at every space-time point Recipe for local gauge symmetry: Global invariance (gauge symmetry) under a transformation Change to local (space-time dependent) transformation  destroys invariance Add new field(s) with transformation properties that compensates and restores the invariance  Lagrangian with local gauge invariance and interactions G. Ingelman: QCD

Comment: All forces are gauge interactions !
Strong interaction: global  local colour symmetry invariance under local colour transformations restored by gluon field Electro-magnetism: global  local charge symmetry E-field invariant to global change of zero point of scalar potential V  phase change of e- wave function Local phase change  physical effects, e.g. derivative in Schrödinger eq. acts on  (x,t) Restore invariance by introducing vector potential A  magn. field  Maxwell’s eqs. including magnetic interactions Weak interaction: global  local weak isospin symmetry invariance under local transformations restored by W,Z fields Gravity: global  local space-time coordinate system symmetry invariance under local coordinate transformations restored by gravitational field in general relativity G. Ingelman: QCD

QCD Lagrangian q kinetic energy and mass qg interaction, coupling gs
g kinetic energy and self interactions q propagator qg vertex R B  last term symbolically in gluon fields : g kinetic energy g self-interactions given by non-abelian SU(3) gauge group, only one coupling gs g propagator 3-gluon vertex 4-gluon vertex G. Ingelman: QCD

Gauge choice & ghost fields
Theoretical ‘complications’ due to additional freedom in Lagrangian Gluon field has more degrees of freedom than physical gluon: Spin 1  3 pol. states “0,1” for virtual gluon propagator mg=0  2 pol. states “1” for “real external” gluon Convention/choice  fix gauge  additional terms in Lagrangian covariant gauges e.g. =1  Feynman gauge axial/physical gauges e.g. n2 = 0  light-cone gauge Some gauges need ghost fields for unitarity (‘probability’) of cross-sections Calculation needs gauge fixing, but result independent of gauge due gauge invariance! Below: forget ghosts, use Feynman gauge G. Ingelman: QCD

Feynman rules of QCD: external, on-shell particles
object  diagram  in amplitude initial quark final quark initial anti-quark final anti-quark initial gluon final gluon u,v = spinor wave fcn,   = gluon polarisation vector p = 4-momentum (p with  =0,1,2,3  E,px,py,pz) s = spin, f = quark flavour,  = 1,2,3 for quark colour G. Ingelman: QCD

Feynman rules of QCD: basic interaction vertices
object  diagram  in amplitude a   quark-gluon vertex 3-gluon vertex 4-gluon vertex 1 3 2 1 2 3 4 SU(3): fabc str. const., , a=1,2…8 for g, , = 1,2,3 for q  = Dirac matrices, p = 4-momentum, g = metric tensor G. Ingelman: QCD

Feynman rules of QCD: internal, off-shell particles
object  diagram  in amplitude quark propagator gluon propagator quark loop gluon loop p p q p-q p,q = 4-momenta, ,  = Dirac matrices, g = metric tensor mf = mass of quark flavour f  = infinitesimal to handle poles for p on-shell SU(3)colour: a=1,2…8 for ‘gluon’, , = 1,2,3 for ‘quark’ = gauge fixing parameter, Feynman: =1  simple g-propagator G. Ingelman: QCD

Feynman rules  cross section: example
s-channel t-channel u-channel 1,a, 2,b, 3, 4, c,  Amplitudes: invariants  (cm energy) (mom. transfer)2 Cross-section = inc.flux | interfering ampl.|2 sum colour, average/sum initial/final spins  trace over  matrices s  gs2/4  final result: G. Ingelman: QCD

s=gs2/4 << 1  perturbation theory
Cross-section as power series in s : Each vertex has gs in amplitude  gs2 s in cross-section Higher orders  more vertices  more diagrams  n! fn hard to calculate (interference terms) But s<<1  higher order terms smaller  truncate series Smart developments: helicity amplitudes, twistors G. Ingelman: QCD

Comment: K-factor Cross-section as power series:
d K d (LO) is approximation of NLO effect as overall factor (still LO-shapes!) Example: + … + Smart developments: helicity amplitudes, twistors Leading order (LO) Next-to-leading order (NLO) normally K1+ s , but here K2 from s-correction on large (gg gg)  Important for heavy quark production! G. Ingelman: QCD

Comment: Heisenberg uncertainty relation at work
All external particles on-shell, i.e. Feynman diagrams: energy-momentum conservation in vertices  internal off-shell particle propagators ~ 1/p2 ‘Oldfashioned’ perturbation theory: internal particles on-shell  energy-conservation violated internally in time-ordered diagrams  energy denominators: probability ~ 1/(E)2 Both methods OK, intermediate, virtual state within p2=(p1+p2)2 > 0 time-like 1 2 4 3 p2=(p1-p4)2 < 0 space-like + G. Ingelman: QCD

Running couplings: QED vs QCD
QED: Quantum fluctuations polarise vacuum and screen electron charge at large dist. opposite field direction  weakens field QCD: Quark vacuum fluctuation  screening (as in QED) QCD: Gluon vacuum fluctuation  anti-screening (not in QED) Gluons dominate  net result is anti-screening same field direction  strengthens field G. Ingelman: QCD

Running coupling: QED theory
e- charge defined by e- coupling: but also … ‘Bare’ charge e0 at ‘bare’ vertex (without loops) in physical process, e.g. e- e- e- e- scattering amplitude: = q … sum of geometric series 1-x+x2…= 1/(1+x)  Integral over loop momentum • M cut-off for  loop momentum • bare charge/coupling  not measurable, unphysical … G. Ingelman: QCD

Renormalised QED coupling
Re-parameterise (‘renormalise’) to physical reference momentum scale  : Set Q2= 2 giving , solve for insert back in (keep leading terms)  Running coupling  unphysical bare charge & scale M replaced by charge/coupling at measurable reference scale 2 Landau pole (Q2) increases with scale Q2 (me2)=1/137…, (mZ2)=1/128… G. Ingelman: QCD

Running coupling: QCD Procedure as for QED, but 3-gluon vertex  gluon loops in gluon propagator  loop integral nf <16  opposite sign to QED Running coupling  (orR) is ‘renormalisation scale’ s(Q2) decreases with increasing Q2 asymptotic freedom, perturbation theory OK at large Q2 Nobel prize 2004  Gross,Wilczek ; Politzer G. Ingelman: QCD

Q2-dependence of s s(Q2)  for instead of  as free parameter 
Q2 2 : s(Q2)   infrared slavery, confinement fits to data  0.2 GeV cf. ‘Heisenberg’ i.e.  scale ~ hadron size Q2 : s(Q2)  0 asymptotic freedom Q2 > ~ 1 GeV2  s small perturbation theory works series converges and can be truncated G. Ingelman: QCD

‘Higher’ loops  sub-leading corrections
q p-q 1-loop corrections: 1 2 1 N ~1 Sum of 1,2…N  leading log approx.  ‘resummed’ 2-loop correction: example q p1 p2 Order s2, but shared propagator  not 2 indep. loops  only one log with above  resummation  next-to-leading log approx. Renormalisation scheme defines how much of finite parts absorbed into coupling, e.g. MS G. Ingelman: QCD

Digression on renormalisation
Renormalisation allows computation of physical effects due to loops in the perturbative expansion of amplitudes. Infinities from loops consequence of naïve definition of charge. Re-parameterisation from bare charge to physical charge give finite results in agreement with experiment. Physical charge (coupling) measured at one momentum scale, allows calculation of charge (coupling) at any other scale. Other quantities (mass, wave function) also affected by loops, need renormalisation leading to momentum scale (Q2) dependence. G. Ingelman: QCD

Renormalisation group invariance
Perturbative expansion of physical observable R must be renormalised to remove infinities and get meaningful results Renormalisation scheme defines how much of the finite parts are kept at each order (MS, MS, …) Renormalisation scale defines at which momentum scale the subtraction of infinite parts are made Physical result independent of scheme and scale Renormalisation group invariance Independence of choice of renormalisation scale  Renormalisation group equation -function G. Ingelman: QCD

-function  running of s
The QCD -function pert. expansion with calculated ?? 0, 1 fixed, but 2, 3, … depend on renormalisation scheme n=0 term  leading log approx. as before  n=0 & 1 terms  NLO approx. as before Physical observable depends on  via s Renormalisation Group eq.  Q2-dependence of R in terms of s(Q2) running coupling  large log’s of Q2/2 resummed order-by-order Sub-leading next-to-leading log’s, with fewer log’s per power in s , can also be included G. Ingelman: QCD

Renormalisation scale dependence
Full pert. series independent of renormalisation scheme & scale BUT truncation  scheme & scale dependence  To calculate, fix scale using FAC – Fastest Apparent Convergence choose R so RNLO= RLO , i.e. NLO-correction = 0 PMS – Principle of Minimal Sensitivity impose scale independence for truncated series, i.e. BLM – Brodsky-Lepage-Mackenzie follow QED and absorb all quark and gluon vacuum polarisations into running of s Scheme & scale not defined in LO, need NLO. Scheme change at NLO  scale change Ex: DIS 2+1-jets: *q qg, *g qq  LO +NLO s corrections (real+virtual) G. Ingelman: QCD

Summary of Lecture 1 Colour ‘charge’ – source of strong field
Local colour gauge invariance  gluon field in Lagrangian Lagrangian  Feynman rules  cross-section = flux |amplitude|2 ‘Bare’ charge unphysical (not measurable) Renormalisation: re-parametrization to physical coupling  running coupling due to q,g-loops in gluon propagator (mediating interaction) gluon loops   sign in -fcn  asymptotic freedom  perturbation theory G. Ingelman: QCD

QCD-improved parton model
pf p´e e p pe P  pi q Point-like probe in Deep Inelastic Scattering (DIS) mom.transf , Bjorken- hadronic mass , Cross-section (-exchange, LO) neglected: masses and bound state ‘higher twist’ effects For i.e. energy-momentum fraction e.g. light cone fraction invariant under z-boosts Proton structure function probability to find quark (flavour f, fraction x) when probed by Q2 Parton densities q are not observables – proton structure function F2 is … G. Ingelman: QCD

DIS data Rise of F2 at low x  High quark (gluon) density,
No Q2 dependence = scaling  point-like quarks log Q2 dependence = scaling violations  perturbative QCD High quark (gluon) density, increasing with increasing Q2  QCD dynamics G. Ingelman: QCD

QCD in proton structure function
Field theory  continuous energy levels More quantum fluctuations larger virtuality requires larger Q2 to be realized More partons with smaller x seen at larger Q2 A A* QCD evolution  parton ladder q(x,Q2) from very complicated (interferences) Approximations: separate branchings |Ai|2 summed to all orders (with conditions)  leading contribution G. Ingelman: QCD

Standard Q2 evolution: DGLAP equations
Dokshitzer, Gribov, Lipatov, Altarelli, Parisi Apply for strongly ordered virtualities (or ) Splitting functions: z z regularisation from virtual corrections How derive/understand DGLAP? … G. Ingelman: QCD

O(s) QCD matrix elements
gluon radiation boson-gluon fusion 3 new degrees of freedom where  splitting functions Leading part of ME’s longitudinal z-distr.  splitting functions G. Ingelman: QCD

0th + 1st +… order QCD matrix elements
* x * x  to be added … unmeasurable ‘bare’ distr. pT0 soft, non-pert. QCD collinear singularity absorb G. Ingelman: QCD

Factorisation Collinear singularities for pT 0 ‘factorised out’ and absorbed into ‘bare’ parton distribution at a ‘factorisation scale’ F giving ‘renormalised’ distribution containing non-pert. QCD dynamics, cannot be calculated but obtained from fit to data, then giving Factorisation  well defined prescription to handle logarithmic singularities Factorisation theorem proven to all orders in perturbation theory. G. Ingelman: QCD

Factorisation scheme In …
… arbitrariness how finite contributions treated  factorisation schemes DIS scheme: whole C(z) absorbed into q(x,F)  no s -corrections to any order MS-scheme: only special part of C(z) absorbed Now added , should also be done above for G. Ingelman: QCD

Digression on factorisation
Parton densities (& structure functions) depend on factorisation scheme and scale  Choose scheme & scale Well-defined parton density functions which are universal Applicable in other processes if same scheme used in pert. QCD calculations Higher order corrections depend on scheme used MS most commonly used G. Ingelman: QCD

DGLAP in (Next-to)n-Leading-Log Approximation
j kT i e.g. qqg N 1 Q2 Sum of leading powers [s log Q2]N dominant contribution for strongly ordered kT (virtualities) Leading log DGLAP eq’s  Next-to-leading log approximation: i j i j kT k’T e.g. kT  k’T  s2 logQ2 contrib. sN logN-1Q2 Sum all orders  splitting fnc’s and DGLAP eq’s Complicated but now standard to order NLL in DGLAP  data … G. Ingelman: QCD

DIS F2(x,Q2) data  precision test of QCD
in NLO MS scheme Q2-dep. dF2/logQ2  from DGLAP eq. Fit to data: s(Q2) or MS-bar quark densities enter directly gluon density only indirectly, but dominates Q2-dep. at small x MC study x G. Ingelman: QCD

… and parton density functions (PDF’s)
DGLAP: Q2-evolution up from Q20 ~ 1GeV2 where perturbation theory applicable. Need input PDF’s i.e. assumed x-shapes at Q02 accounting for bound state proton wave function, typically with ~5 parameters/distribution Half proton energy carried by gluons: x-weighted  huge # density of small-x gluons ‘overlapping’ interacting partons Rp~1 fm R~1/Q Q2 G. Ingelman: QCD

Fit of DGLAP to many data sets  PDF-parametrizations
CTEQ6M: MS, Q0=1.3 GeV, input forms: g u d s c Q2= 4 GeV2 x xf(x) 20 free shape parameters + 5 for normalisation, but constraints from sum rules p=uud  momentum sum rule Use DGLAP evolution in Next-to-Leading Log approx. and fit to ‘lots’ of data  PDF’s in NLL G. Ingelman: QCD

 (hadron) = PDF   (parton)
Factorisation  universal PDF’s  applicable for different processes Consistency: same order in approximations, e.g. NLL-PDF and NLO-ME Example: jets in pp Hadronisation Jet finding LO: 22 s2 processes qq qq, qg qg, gg qq, gg gg, ... pT gluon jets dominate at small pT quark jets at large pT NLO: s3 processes real 23 processes virtual corrections ANLO A*LO G. Ingelman: QCD

Observation of jets: ATLAS@LHC
hadronisation ? Hadronic showers EM showers ET Mjj= 4 TeV The highest mass event passing our event selection. The dijet mass is 4040 GeV while the leading jets have (pT, eta, phi, color) of (1850 GeV, 0.32, 2.2, red) and (1840 GeV, -0.53, -0.92, green). pT=1.8 TeV pseudo- rapidity = ln tan /2 azimuthal angle G. Ingelman: QCD

Jet reconstruction Cluster ‘particles’ to jets  hard partons
test perturbative QCD ‘particles’=calorimeter cells or tracks Cone algorithm: combine particles in a cone around jet axis, kT algorithm: combine particles according to their relative transverse momentum (kT) Advantages: Infrared/collinear safe at all orders in pQCD  good for precision NLO QCD analysis No bias from seed towers. Every particle assigned to a jet. G. Ingelman: QCD

 (high-pT jets)  QCD test
Jet transverse momentum Di-jet invariant mass Data/theory Figure 10: Inclusive jet double-differential cross section as a function of jet pT in different regions of $|y| for jets identified using the anti-kt algorithm with R=0.6. For convenience, the cross sections are multiplied by the factors indicated in the legend. The data are compared to NLO pQCD calculations using NLOJET++ to which non-perturbative corrections have been applied. The theoretical and experimental uncertainties indicated are calculated as described in Fig. 9. Figure 11: Ratio of inclusive jet cross section to the theoretical prediction obtained using NLOJET++ with the CT10 PDF set. The ratio is shown as a function of jet pT in the rapidity region |y|<0.3, for jets identified using the anti-kt algorithm with R=0.6. The current result is compared to that published in Ref. [21]. Figure 9: Inclusive jet double-differential cross section as a function of jet pT in different regions of |y| for jets identified using the anti-kt algorithm with R=0.4. For convenience, the cross sections are multiplied by the factors indicated in the legend. The data are compared to NLO pQCD calculations using NLOJET++ to which non-perturbative corrections have been applied. The error bars, which are usually smaller than the symbols, indicate the statistical uncertainty on the measurement. The dark-shaded band indicates the quadratic sum of the experimental systematic uncertainties, dominated by the jet energy scale uncertainty. There is an additional overall uncertainty of 3.4\% due to the luminosity measurement that is not shown. The theory uncertainty, shown as the light, hatched band, is the quadratic sum of uncertainties from the choice of the renormalisation and factorisation scales, parton distribution functions, alphaS(MZ), and the modeling of non-perturbative effects, as described in the text. Figure 15: Dijet double-differential cross section as a function of dijet mass, binned in half the rapidity separation between the two leading jets, y*=|y1-y2|/2. The results are shown for jets identified using the anti-kt algorithm with R=0.6. For convenience, the cross sections are multiplied by the factors indicated in the legend. The data are compared to NLO pQCD calculations using NLOJET++ to which non-perturbative corrections have been applied. The theoretical and experimental uncertainties indicated are calculated as described in Fig. 14. Data well described by QCD over 9 orders of magnitude !! NLO matrix elements  PDF’s in NLL Uncertainties: PDF’s, factorisation & renormalisation scales G. Ingelman: QCD

Factorisation & renormalisation scale  uncertainty
Factorisation / renormalisation removes infinities into PDF’s / s Scheme define how much of finite parts are kept at each order Scale defines at which momentum scale subtraction of infinities are made Full perturbative series independent of scheme & scale but truncation  scheme and scale dependence Proper uncertainty can only be found through higher order calculation, e.g. NLO  NNLO Uncertainty usually estimated simply by varying the scales e.g. pT/2 <  R,  F < 2pT  remaining uncertainty G. Ingelman: QCD

Summary of Lecture 2 Factorisation separates hard from soft QCD dynamics scheme and scale dependence in truncated pert. series Parton density functions fq,g/p(x,Q2) parametrized Q2-dependence from DGLAP (next-to)-leading logQ2 evolution x-shape parametrized non-pert. bound state p dynamics Large density of gluons at small-x Jets for QCD tests, works over range 109 in cross-section G. Ingelman: QCD

From partons to hadrons – hadronisation
Hadronisation is non-perturbative QCD dynamics  models: Independent hadronisation of each parton: First, simplest model, but unphysical with separate colour charges and no proper energy-momentum conservation String model colour field  forming hadrons Cluster model forced  clusters decaying into hadrons G. Ingelman: QCD

Cluster hadronisation model
1. Force g  qq at end of pert.QCD phase 2. Form colour singlet clusters 3. Decay cluster  2 hadrons isotropically (with phase space weights) Nice model: simple & clean but with some problems e.g. large mass clusters decaying isotropic  bad fit to data Solved by introducing longitudinal (string-like) decay for clusters with mass above ~2-3 GeV Successfully used in HERWIG Monte Carlo Mcluster G. Ingelman: QCD

Lund string model Gluon self-interaction  colour flux tube
described by mass-less relativistic string Colour triplet field between 3 and 3 (q and q)  potential confinement !   1 GeV/fm from cc & bb spectroscopy and lattice calculation New screens field  lower potential energy virtual in a point, ‘eats’ field energy  on-shell treated as quantum mechanical tunneling G. Ingelman: QCD

Lund model: energy-momentum distribution
Use z-boost invariant light-cone variables Iterate string-breaking ( MC): on-shell hadron takes fraction z of available ‘string’ E+pz leaving for the remaining string Left-right symmetry gives probability distribution inside-out cascade larger z for heavy hadron Iterate until Termination: small-mass system  1-2 hadrons with E,p conservation String-breaking is local phenomenon  cms energy not essential model works well down to Mstring = few GeV G. Ingelman: QCD

Lund model: gluon = kink on string
carrying energy-momentum Force ratio gluon/quark = 2 QCD: NC/CF=9/4  2 for NC   No new parameters for gluon hadronisation! String effect: lower particle and energy flow in region without string First observed by JADE 1980 many later tests favour string picture  coherence in non-pert. context  = indep.fragm.  G. Ingelman: QCD

Observed fragmentation function D(z)
Probability to find hadron h in a quark-jet q, with fraction given by for h from first or later string break-up Fragmentation function D(z,Q2) similar to parton density q(x,Q2). DGLAP Q2-dependence added as analytic formulas or explicit parton shower evolution in Monte Carlo event generator q G. Ingelman: QCD

Many Higgs but large background !
 Total number b-quarks Jets W or Z top-quarks Higgs bosons Cross section   probability  events/year QCD at LHC Strong interactions dominate: Total # events b-quarks  B hadrons Jets; multijet events common  (QCD jets) >> (new physics) Important background for signals for new physics; new heavy particles decaying into quarks  jets top quarks  jets 1015 1013 1011 109 107 105 103 101 mH =125 GeV Higgs 105 Higgs/year of totally 1015 collisions, i.e. 1 in 1010 Many Higgs but large background ! collision energy= LHC G. Ingelman: QCD

How create & find Higgs boson at LHC ?
Proton-proton collisions gives quark-gluon processes where energy create mass, E=mc2  probability to create H0 x2 x1 top anti- H0 particle Higgs boson decays to particle-antiparticle pair and is reconstructed from decay products G. Ingelman: QCD

Need precision QCD smaller x ! PDF’s at larger Q2  DGLAP in NNLA
PDF’s at lower x  BFKL, GLR… Gluon density may be high  ‘saturation scale’ affecting mini-jets & underlying event Matrix elements in higher orders to reduce theoretical uncertainty and better matching partons  hadrons NNLO is frontier DGLAP evolution smaller x ! G. Ingelman: QCD

Event structure  Monte Carlo simulation – step 1
Incoming beams  cms energy G. Ingelman: QCD

parton densities  parton cross-section factorisation of soft and hard QCD processes Differential cross-section  probability distribution (normalised to )  Monte Carlo simulation G. Ingelman: QCD

W-decay given by weak interaction theory  momenta of decay products G. Ingelman: QCD

Initial state gluon radiation  parton shower (space-like) less hard processes which are factorised from the initial, hardest subprocess Unfold DGLAP equations  one branching at a time  iterate G. Ingelman: QCD

Final state gluon radiation  parton shower (time-like) less hard processes which are factorised from the initial, hardest subprocess One branching at a time  iterate Offshell (time-like virtual) parton from hard subprocess generate radiation Similar to initial state radiation but no parton densities G. Ingelman: QCD

Multiple parton-parton hard interactions … d  parton densities  parton cross-section but 1st interaction has taken away energy & partons G. Ingelman: QCD

… with its initial- and final-state parton radiation G. Ingelman: QCD

Outgoing final state partons & beam remnants … … are colour-charged and cannot appear as free objects … G. Ingelman: QCD

Colour-charged objects form interconnecting colour-triplet string-fields 1 fm Not to scale! Strings have hadronic widths  1 fm G. Ingelman: QCD

Strings fragment (hadronise) to primary hadrons G. Ingelman: QCD

Unstable hadrons decay producing final observable state G. Ingelman: QCD

Simulated Higgs event G. Ingelman: QCD

Simulated complete events  powerful physics analysis tool
Any experimental observable can be extracted from simulated events detailed comparison of models/theory in MC with data high-precision physics analyses Simulated or real event ? G. Ingelman: QCD

QCD  Quark-Gluon-Plasma
Phase diagram Big Bang AuAu collision in RHIC: 8668 particle tracks Heavy ion collision: G. Ingelman: QCD

Jet quenching as signal for QGP
Hadron flow in azimuth  vs trigger particle/jet Uppsala-Heidelberg: “Soft Colour Interaction” model elastic scatterings   half observed effect But should add medium-induced radiation RHIC peripheral central Quark/gluon from hard scattering multiple scattered in plasma energy loss hadrons/jet gets lower energi back-to-back jet disappears! G. Ingelman: QCD

Conclusions Perturbative QCD well treated from 1st principles:
infinities well handled through renormalization & factorization high precision predictions: NLO matrix elements, resummation of log corrections agreement with data: many observables & large kinematic ranges Non-pert. QCD is the major unsolved problem within the Standard Model: parton density functions within hadrons  PDF:s parametrization colour fields and hadronisation  Lund string model mass spectrum of hadrons  progress from lattice theory QCD at LHC: new effects: high gluon density at small-x, quark-gluon plasma dominates cross-section  large backgrounds for new physics searches affects signals for new physics (beyond SM) G. Ingelman: QCD

Ab initio lattice QCD  observed hadron mass spectrum
after 20+ years, Science vol 322 3 parameters: mud , ms , absolute mass scale G. Ingelman: QCD

Extra material G. Ingelman: QCD

Feynman rules of QCD: be aware of different conventions
Book of Ellis, Stirling, Webber: A,B… = 1,2…8 colour index  tA instead of Ta a,b… = 1,2,3 colour index  (tA)ab instead of Ta ,…= 0,1,2,3 Lorentz vector index  g instead of g Choose one convention and stick to it! G. Ingelman: QCD

Factorisation & renormalisation scale  uncertainty
• Factorisation / renormalisation removes infinities into PDF’s / s • Scheme define how much of finite parts are kept at each order • Scale defines at which momentum scale subtraction of infinities is made • Full perturbative series independent of scheme & scale but truncation  scheme and scale dependence Proper uncertainty can only be found through higher order calculation, NLO  NNLO Usually estimated simply by varying the scales e.g. pT/2 <  R,  F < 2pT  Remaining uncertainty G. Ingelman: QCD

Uncertainties in PDF’s
CTEQ5M1 CTEQ5HJ MRS2001 g(x,Q2=10)/[CTEQ6-g(x,Q2=10)] x u(x,Q2=10)/[CTEQ6-u(x,Q2=10)] Different parameterisations differ significantly !  Use latest state-of-art parameterisations Still, uncertainties in x-section predictions +20 % -20 mH s PDF total % uncertainty due to Ex: Higg production at LHC gg H H G. Ingelman: QCD

Small-x  BFKL evolution
Radiation of gluon has soft divergence ~1/Eg ~1/z small x  large log’s in x from gluon ladder diagram with x0<< … << xi << … << x e p ~1 need to resum for all N no ordering in kT,i or ki2 unfold k2 integration and use un-integrated gluon distribution  gluon with x and virtuality …  Balitsky-Fadin-Kuraev-Lipatov eq. Lipatov kernel Solve from starting distr. Evolution paths into non-pQCD G. Ingelman: QCD

Summing log’s  ladder diagrams
Double Leading Log Approx. DGLAP BFKL DLLA Recent developments: CCFM G. Ingelman: QCD

? Screening at small-x ?   ~ F(x)~xq(x) or xg(x)
violates unitarity ! Increase must stop ? ? ‘overlapping’ interacting partons parton- > Rp Confinement non-perturbative QCD dynamics  Lattice QCD ? Regge ‘theory’ ? Continuous field 0x = Rp~1 fm R~1/Q Q2 parton fusion ‘recombination’ parton splitting DGLAP or BFKL ‘free’ partons G. Ingelman: QCD

Gluon recombination Sum of fan diagrams Gribov-Levin-Ryskin eq.
Gluon recombination ladder  fan diagrams A A* Sum of fan diagrams Gribov-Levin-Ryskin eq. in simplified form (Mueller-Qiu) Negative correction term Non-linear in gluon field Normal DGLAP size of gluon-region Reduces growth of gluon density at small x Damping stronger if hot spots in the proton, i.e. smaller regions with higher gluon density  Speculative! G. Ingelman: QCD

Generalised parton distributions
Conventional PDF fi(x,Q2) probability of parton type i with momentum fraction x when measured at scale Q2 Unintegrated distr. fg(x,k2) for gluon with fraction x and virtuality k2 Generalised parton distr. (GPD) defined by with operator O of quark/gluon fields,  = |A|2 different longitudinal momentum fractions e.g. ‘deep virtual Compton scattering (DVCS) pT  impact parameter distribution  info on transverse distr. of partons large-x partons more central in p than small-x partons  sea cloud around valence core ? fm  3D-image of proton structure! x G. Ingelman: QCD

Model for non-perturbative x-shape of PDF’s
DGLAP need input xf(x,Q02) for non-pert. bound state proton dynamics Simple model (Alwall+GI) works and gives insights: p rest frame  parton momenta spherically symmetric Heisenberg  Assume Gaussian Use z-boost invariant momentum fraction  Apply kinematic constraints Sea distr’s from hadronic quantum fluctuations Few parameters  many distributions G. Ingelman: QCD

… comparison to data Add standard DGLAP for Q2-evolution, fit to data  few parameters many in CTEQ,MSRx F2 in DIS at HERA & fixed target from Tevatron strange sea from Tevatron  Not bad for simple model! G. Ingelman: QCD

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