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QCD and precision calculations Lecture 1: basics Gudrun Heinrich Max Planck Institute for Physics, Munich [email protected]

PREFIT School, March 2, 2020

1 Outline 1. Basics of QCD QCD Lagrangian Gauge invariance Colour algebra Feynman rules 2.Tree level amplitudes Application of Feynman rules Polarisation sums 3. NLO calculations (details see lectures 2-4) 4. Beyond NLO

2 Some literature

3 QCD QCD is a very rich !

confinement strong CP-problem QCD at finite temperature --plasma flavour puzzles spectroscopy

lattice disclaimer: I am not doing justice to it by just considering perturbative QCD, and even that only at fixed order However, as the focus will be on precision calculations, I will make a (highly biased) choice

4 Motivation • at colliders, QCD is everywhere • without QCD corrections, (most of) the data are not well described

, Grazzini, Kallweit, Rathlev ‘17

GH, Jahn, Jones, Kerner, Pires ‘17

5 QCD corrections perturbation theory in the strong coupling

next-to-next-to-leading order leading order next-to-leading order

NLO corrections typically NNLO corrections typically a few % but there are prominent exceptions (e.g. Higgs production, NLO corr. ~100%)

Why can we do such an expansion at all? important concepts: • asymptotic freedom • factorisation

6 Factorisation

parton distribution functions partonic cross power-suppressed non-perturbative (PDFs) section corrections

factorisation scale (separate short- and long-distance dynamics) renormalisation scale (UV divergences from loops, see later)

PDFs : from fits to data, only evolution with scale is calculable perturbatively

contains hard scattering matrix element, calculable in perturbation theory sum over parton types i, j

7 Asymptotic freedom

CMS

+0.0060 0.24 CMS Incl.Jet, s = 8TeV, α (M ) = 0.1164 (Q) S z -0.0043 S CMS Incl.Jet, s = 8TeV α 0.22 CMS R32 , s = 7TeV CMS Incl.Jet , s = 7TeV 0.2 CMS tt , s = 7TeV CMS 3-Jet Mass , s = 7TeV 0.18 D0 Incl.Jet D0 Angular Correlation 0.16 H1 ZEUS World Avg α (M ) = 0.1185 ± 0.0006 0.14 S z

0.12

0.1

0.08

5 6 7 8 10 20 30 40 100 200 300 1000 2000 Q (GeV) strong coupling becomes weaker as energy scale increases will be discussed in more detail later

8 LHC event

9 non-perturbative stuff

visualisation of pile-up in the ATLAS tracker (Run I)

10 QCD or New Physics?

a typical distribution, decreasing with energy

11 QCD or New Physics?

NLO LO

small discrepancy to data at max. available energy due to lack of NNLO corrections? or onset of New Physics? 12 New Physics! (Z- 1983)

13 matrix element

image: P.Uwer

Z-boson resonance note also: (see lecture)

14 Today

no sign of a resonance up to di-jet invariant masses of ~7 TeV if New Physics scale is very large, only indirect effects visible need precision

15 QCD Lagrangian strong interactions: gauge theory “colours” of non-Abelian structure of SU(3) related to gluon-self-interactions purely gluonic part: Yang-Mills Lagrangian

convention: doubly occurring indices are summed over

: structure constants of SU(3) (totally antisymmetric)

generators of SU(3)

8 , in the adjoint representation of SU(3)

16 QCD Lagrangian fermionic part: quark fields for flavour f: : colour index quarks are in the fundamental representation of SU(3)

, define

generators of SU(3) in fundamental representation

Gell-Mann matrices

17 Evidence for Colour hadronic R-ratio:

2 2 R(s)=Nc e ✓(s 4m ) N =3 f f c f=u,d,s,c,...X 2 1 1 threshold energy R =3 ( )2 +( )2 +( )2 + ... 3 3 3 to produce ✓ ◆ quark-antiquark-pair u d s 18 of mass Gauge invariance local gauge transformation with , ,

remember

covariant derivative

for to be invariant

19 gauge invariance

note: mass term ~

would break gauge invariance

20 Colour Algebra generators of SU(3): fundamental representation:

Gell-Mann matrices (traceless, hermitian)

21 Colour Algebra adjoint representation: dimension of the vector space on which it acts is equal to the dimension of the group itself ( dimension of the group = number of generators, ) matrices (8x8) - matrices group invariants:

: eigenvalues of Casimir operators in fundamental/adjoint representation

usually (convention)

22 Colour Algebra pictorially

23 Colour Algebra pictorially

more: see exercises

23 colour algebra exercise calculate the colour factors for the following diagrams

24 Colour Algebra explicit example for a quark-gluon vertex

figure: Peter Skands gluon represented as double line of two colours

(second term because therefore relation of traceless generators)

25 QCD Lagrangian • the gluon is constructed from the inverse of the bilinear term in the gluon fields in the action

• in momentum space it should fulfil

(1)

however as Eq. (1) has zero modes, so [.] is not invertible

reason: contains physically equivalent configurations

need : add constraint on gluon fields with a Lagrange multiplier to the Lagrangian

26 QCD Lagrangian covariant gauges: add condition

: Feynman gauge

: Landau gauge

27 QCD Lagrangian

• however in covariant gauges unphysical (non-transverse) degrees of freedom also can propagate

• their effect is cancelled by ghost fields coloured complex scalars obeying Fermi statistics

Faddeev-Popov matrix in Feynman gauge:

28 QCD Lagrangian

• unphysical degrees of freedom and the ghost fields can be avoided by choosing axial (physical) gauges

in axial gauges: vector with

gluon propagator:

light-cone gauge:

29 Feynman rules : gluon quark ghost

covariant gauge

light-cone gauge

30 Feynman rules Vertices: quark-gluon

ghost-gluon

31 Feynman rules 3-gluon

4-gluon

32 Feynman rules spinors: incoming

incoming anti-fermion

outgoing fermion

outgoing anti-fermion polarisation vectors: incoming

outgoing vector boson

33 Feynman rules further rules: • momentum conservation at each vertex

• factor (-1) for each closed fermion loop

• factor (-1) for switching identical external

• integrate over loop momenta with

34 spinor relations useful identities:

()

(orthogonality)

(completeness)

35 colour decomposition using the “double line notation”, we can express gluon amplitudes entirely in termsColor of generators in pictures

Insertbased on and

where is color factor for qqg vertex

into typical string of fabc structure constants for a :

•Always single traces (at tree level) + all non-cyclic permutations •comes only from those planar(withdiagrams corresponding signs) with cyclic ordering of external legs fixed to 1,2,…,n

L. Dixon, similarly7/20/06 Higher Order QCD: Lect. 1 + permutations 20 A, µ B, ν A, µ p B, ν 1 p a b A B TR CA a b A B 2Nc T C 1 colour R A 2Nc = colour = 2 Nc − 1 2 CF = Nc − 1 2N CF = c 2Nc A T r(t ) = 0 T r(tA) = 0

A B AB A 1B AB 1 T r(t t ) = T δ , TT r(=t t ) = TRδ , TR = R R 2 2 A A A A tactcb = CF δab tactcb = CF δab A A ! ! CDA CDB AB CDA CDB ABf f = CA δ , CA = Nc f f = CA δC,D , CA = Nc C,D ! ! A A 1 1 (t )ab(t )cd = δadδbc − δabδcd A A 1 1 2 2Nc (t )ab(t )cd = δadδbc − δabδcd 2 2Nc f ABEf CDE + f BCEf ADE + f CAEf BDE = 0 f ABEf CDE + f BCEf ADE + f CAEf BDE = 0 A (t )ab ABC A i f (t )ab generators ofABSCU(Nc): 2 i f A Nc − 1 hermitean traceless matrices (t )ab generators of SU(Nc): ABC A B C N 2 − 1 hermitean traceless matrices (tA) f = −2 i Tr([t , t ] t ) c colourab decomposition A1 A2 A f ABC = −2TirT(r([t ttA, ·tB· ·]ttCn) + all non-cyclic permutations • therefore the colour part of the amplitude can be qq¯gggg . . . ⇒ T r(tA1 tA2 · · · tAn ) + permutations T r(tA1 tA2 · ·completely· tAn ) + all non-cyclicseparatedp ermfromutations the kinematicab part

tree n−2 A1 A2 A tree h1 h2 h M ({pA,1aA,2h }) =Ang T r(t t · · · t n ) M (1 , 2 . . . n n ) + all non-cyclic permutations qq¯gggg . . . ⇒nT r(t i ti ·i · · t )ab + permutations n momenta, colour, helicities colour ordered partial amplitude, colour factors stripped off tree n−2 A1 A2 An tree h1 h2 hn Mn ({pi, ai, hi}) = g usefulT r( tproperty:t · · · t as) Mn (1 , 2 . . . n ) comes+ all non-cyclic from diagramsperm withutations cyclic ordering of external legs, it only can have infrared singularities in adjacent invariants 2 si,i+1 = (pi + pi+1) • colour decomposition at loop level is straightforward, but then single and double trace structures are generated

figure: Lance Dixon

37 A A tactcb = CF δab !A CDA CDB AB f f = CA δ , CA = Nc C,D ! A A 1 1 (t )ab(t )cd = δadδbc δabδcd 2 − 2Nc

f ABEf CDE + f BCEf ADE + f CAEf BDE = 0

A (At A)ab tactcb = CF δab ABC !A i f CDA CDB AB f f = CA δ , CA = Nc generators of SU(Nc): 2 C,D A Nc 1 hermitean traceless! matrices (t )ab − A A 1 1 (t )ab(t )cd = δadδbc δabδcd f ABC = 2 i2Tr([tA−, t2BN] ctC ) − f ABEf CDE + f BCEf ADE + f CAEf BDE = 0 T r(tA1 tA2 tAn ) + all non-cyclic permutations Color· · · sumsA Color sums(t )ab qq¯gggg . . . T r(tA1 tA2i f ABtCAn ) + permutations ⇒ · · · ab In theIn the end, end, we we want generatorswant to tosum/average ofsum/averageSU(Nc): over over final/initial final/initial colors colors N 2 1 hermitean traceless matrices (tA) (as(as well well as ashelicities): helicities):c − ab tree( p , a , h ) = gn−2T r(tA1 tA2 tAn ) M tree(1h1 , 2h2 . . . nhn ) + all non-cyclic permutations Mn { i i i} colourf ABC = · ·2·i T expansionr([tA, ntB] tC ) Inserting:Inserting: −

A1 A2 An 2 dσtreeT r((tp ,ta , h t ) ) + all non-cyclicM tree(permp ,utationsa , h ) { i i · · ·i} ∼ n { i i i} ai hi A1!A2 !An" " insert colourqq¯ggg gordered. . . T amplituder(t t andt "perform)ab + ptheerm colourutations sum:" ⇒ · · · andand doing doing the the color color sums sums diagrammatically: diagrammatically: 1 2 2 dσtree( p , a , h ) N n M tree(σ(1h ), σ(2h ) . . . σ(nhn )) + (1/N 2) { i i i} ∼ c n O c tree n−2σ∈SnA/Z1 nA2 hi An tree h1 h2 hn n ( pi, ai, hi ) = g! T r!(t t ! t" ) Mn (1 , 2 . . . n ) + all non-cyclic" permutations M { } · · · " " 2 si,i+1 = (pi + pi+1) 2 wewe get: get: dσtree( p , a , h ) M tree( p , a , h ) { i i i} ∼ n { i i i} ai h ! !i2 " " g " " αs = 4π 2 dσtree( p , a , h ) N n M tree(σ(1h1 ), σ(2h2 ) . . . σ(nhn )) + (N n−2) { i i i} ∼ c n O c σ∈S /Z h Up to 1/N2 2 suppressed effects,!n n !i " squared subamplitudes" have Up to 1/Nc suppressedc effects, squared" subamplitudes" have 2 definitedefinite color color flow flow– important–Non-planarimportant topologiesforsi,ifor +1handoff =handoff(p arei + psubleadingi+1 to) to parton parton in colour shower shower programsprograms L. Dixon, 7/20/06 Higher Order QCD: Lect. 1 24 L. Dixon, 7/20/06 Higher Order QCD:g2 Lect. 1 24 α = Note: parton showers usually dos not4π take subleading colour into account 38 Tree level amplitudes the non-Abelian structure of QCD leads to important differences compared to QED (physical polarisations, beta-function, …)

consider first a simple QED process:

,

39 Tree level amplitudes

consider the current charge conservation implies

QCD analogue:

40 Tree level amplitudes

use

term in square brackets is the same as in QED, so

for the remaining terms we find

cancels with contribution from vanishes only when contracted with the polarisation vector of a physical gluon, i.e. if 41 Polarisation sums QCD: vanishes only for physical gluons

QED: regardless whether or not

for cross sections we need

let us consider just (the second boson is treated analogously)

42 Polarisation sums In QED, we can make the replacement

In QCD, this will in general lead to the wrong result. Why? sum over physical polarisations:

define , in QED second part of pol. sum will not contribute 43 Polarisation sums in QCD: therefore, if we can not just use for the polarisation sum

it can be shown that

calculating this ghost contribution results in

ghost degrees of freedom cancel the unphysical gluon polarisations!

true in polarisation sums as well as virtual loop diagrams

when calculation in a non-physical gauge (e.g. Feynman gauge), need also ghost loops in virtual corrections 44 Summary

• Hadron Collider experiments need QCD corrections

• Description as SU(3) local gauge theory has important consequences

• We got familiar with the basics of the colour algebra and the QCD Feynman rules

• Application to tree level example

We saw how to sum over gluon polarisations when squaring an amplitude

next: NLO!

45 Exercise calculate for - scattering

e(p )+µ(p ) e(p )+µ(p ) 1 2 ! 3 4

p3 p4

q

p1 e µ p2

46