(Max Planck Institute Für Physik) 1St Lecture

Home , Eightfold way (physics), Isospin multiplet

QCD

Giulia Zanderighi (Max Planck Institute für Physik)

1st Lecture

ESHEP School September 2019 Today’s high energy colliders

Today’s high energy physics program relies mainly on results from Today’s high energy colliders Collider Process Today’s high statusenergy colliders + T-oday’s high energy colliders Collider LEP/LEP2Process setaetus Today’s high1989-2000energy colliders Today’s chighurrenenergyt and ucpolliderscoming ex- Hera e±p Today’s high1992-2007energy colliders HERCAoll(idAe&r B) Preo±cpess rsutanntuinsg periments collide protons Collider TevatronProcess statupps curre1983-2011nt and upcoming ex- Collider Process status current and upcoming ex- THeERvCaotrlAloidn(eAr(I&&B)II) Proceep±sp¯sp startuunsning cpuerrrimenetntsancdolluidpecopmroitongnsex- HERA (A & B)LHC-e ±Runp I runninppg cpuerrrimenetntsa2010-2012ncadolllluidipnecvopomrolvitonegnQseCDx- Collider Process status THeERvatrALoHn(AC(I&&B)II) ep±p¯p starurtsnn2in0g07 pe⇒riments collide protons THeERvatrAon(A(I&&B)II) ep±p¯p running perimcuernretsnctolalinded puroptoconms ing ex- LHC- Run II pp astartedll inavlloilnv 2015evoQlvCDe QCD THeERvatrALoHn(AC(I&&B)II) ep±p¯p starurtsnn2in0g07 pe⇒riments collide protons TevatrLoHnC(I & II) pp¯ starurtsnn2in0g07 ⇒ all involve QCD LEPHER highA: m precisionainly mea measurementssurements of pa ofrto masses,n denaslilti couplings,iensvaonlvdedQiffr CDEWacti oparametersn ... TevaLtrHLoHCnC(I & II) pppp¯ stasrtstarur2tsn0n02i7n0g07 ⇒ ⇒ Hera: mainly measurements of proton structureal l/ inpartonvolve densitiesQCD HTHEReERLvA:HatrA:Cmonma:inamliynalmyinemlyaesdpuaipsrsecumoreveemnrtseysntaotsffrptsthoafer2top0toan0rp7todaennsddietirene⇒sslaiatitenedds mdainffredaasdcutiiffrroenamcetionnts THTevatron:eLTHERveHERavCtrA:aotrA:dnmoe:nmasm :imainlyngaamlniyinnaelmlyidynemldtoya isdiscoveryesdcuaiosrsevcuemorreveyemnortseyf nthooftsfef pthtoptooaferptopto aaandnnrpdtodaern nemanyslddaietiterenedssla imQCDatiteneeddsasmdauinffr ermeasurementsdeaamsdcuetiiffrronentsamcetionnts LTLHCeHLTveHCERavCtradd odesignedtrA:eidsnsoec:ingmsom:nivgaaemeniidnrnaelthltoyidyn toemldtoyiHsedcaiigossvgcueosrrevyaemnoredyf nthomtsefethtooafespputoaarenrpdtoit’arnsnelddpaerteronedpslaeimtiteretiedseassmaunerdeamsdueiffrrnetsamcetionnts LHLTediscoverHCvdCaiddustreincdssorecaivngsove:nivegrem elnththedrpaeethtooidn sHelHiggstosyigiHbdgiligsesgcap os[donenhvdayesnmridycesoma insfbeuthe arRunyeesouintot’rsde pI]pthit’arosenpdpSMerroetipelaestertidesmeasurements discover the Higgs and measure it’s properties unravelundurinasrvcaeovl vepossiblepelorpsthossiebsleiHb pilgeBSMhgypsshiacy sphysicssnidbcesmyboeenayd s[elusiveouthnrde thSMit’sep SMuprop toer tinow]es LHOuC rdeasbigilintyedtotodiscover new particles and to measure their unravel possible physics beyond th2e SM duinsrcaovveelrpthosesiHbilgegpshaysnidcsmbeeaysounrde thit’sepSMroperties OupOurrorapbeairlbitityielitystolitomdistedcidosvcbeoyrvthenreewnqeupwaalriptitycaloretifscolaeunsrduantonddemrtosetaasmnuderieansgthuoreefirQCtheDir pOurorpuenarrbtiaielvitysellitopmoitesdsidsibcbloyevtheprheynqseuiwcasliptybaeorytifocolneudsr uthanneddeSMrtostamndeiansguoref QCtheDir pOurorpearbtiielitys litomitedidscboyvtherenqeuwaliptyaortifcoleusr uannddertostamndeiansguoref QCtheDir properties limited by the quality of our understanThde oinne-lgoop oamfplitQCude forDsix gluon scattering - April 2006 – p.2/20 pOurorpearbtiielitys litomitedidscboyvtherenqeuwaliptyaortifcoleusr uaTnhenodndee-loorptosamtaplitumnde dfoer siainx gsgluoun osrcaeftterQCingth- AperDil 2i0r06 – p.2/20 The one-loop amplitude for six gluon scattering - April 2006 – p.2/20 properties limited by the quality of our uTnhe odnee-loorpsamtaplitunde dfor siinx ggluon oscaftterQCing - AprDil 2006 – p.2/20

The one-loop amplitude for six gluon scattering - April 2006 – p.2/20

The one-loop amplitude for six gluon scattering - April 2006 – p.2/20 Today’s status of particle physics

• The Higgs boson: the last missing ingredient of the Standard Model of particle physics • The SM is a consistent theory up to very high energy, but it can not be a complete theory because of many theory and experimental puzzles (Dark matter and dark energy, neutrino masses, ﬂavour physics, hierarchy problem, no theory of quantum gravity…)

The days of guaranteed discoveries or no-lose theorems are over. Progress will be driven by LHC data. The LHC program in the coming decades will focus on • Direct searches: production of new (heavy status) … • Indirect searches: Higgs couplings and branchings, rare decay modes … • Consistency tests: precision electroweak data

3 Today’s status of particle physics

Improving our theoretical description of data crucial to enhance sensitivity when looking for new physics. Example: extraction of Higgs couplings limited by theory uncertainties

Sharpen the tools ⇒ get more accurate predictions ⇒ get improved indirect sensitivity to New Physics (which would modify SM couplings) 4 Propose of these lectures

In these lectures: perturbative QCD as a tool for precision QCD at colliders • Introduction to QCD (or refresh your knowledge of QCD) • Basic concepts which appear over and over in different contexts • Understand the terminology • Recent developments in the ﬁeld

5 Some bibliography • A QCD primer G. Altarelli (TASI lectures 2002) hep-ph/0204179 • QCD and Collider Physics (a.k.a. The Pink Book) R.K. Ellis, W. J. Stirling, B. R. Webber, Cambridge University Press (1999) • Foundations of Quantum Chromodynamics T. Muta, World Scientiﬁc (1998) • Quantum Chromodynamics: High Energy Experiments and Theory G. Dissertori, I. Knowles, M. Schmelling, International Series of Monograph on Physics (2009) • The theory of quark and gluon interactions F. J. Yndurain, Springer-Verlag (1999) • Gauge Theory and Elementary Particle Physics T. Cheng and L. Li, Oxford Science Publications (1984) • The Black Book of Quantum Chromodynamics: A Primer for the LHC Era J. Campbell, J, Huston, F. Krauss (2018) • many write-ups of QCD lectures given at previous CERN schools ... 6 The beginning: the eightfold way

Organise hadron spectrum to manifest some symmetry-pattern

One hadron missing, but predicted following the pattern

But what is the reason for this pattern?

7 Quarks (1964)

Gell-Mann and Zweig propose the existence of elementary spin 1/2 particles (the quarks). Three types of quarks, up, down, strange (plus their antiparticles) can explain the composition of all observed hadrons

8 First experimental evidence for colour

I. Existence of Δ++ particle: particle with three up quarks of the same spin and with symmetric spacial wave function. Without an additional quantum number Pauli’s principle would be violated ⇒ color quantum number

u1 u2 u3

++ = ijkuiujuk

New quantum number solves spin-statistics problem (wave function becomes asymmetric)

9 U †U = UU † = 1 det(U) = 1

ψ∗ψ U ∗ ψ U ψ = ψ∗ψ i i → ij j ik k k k !i !ijk !k

ψiψjψk U ∗ Ujj Ukk ψi ψj ψk = ψi ψj ψk First→ experimentalii′ ′ ′ ′ evidence′ ′ for′ ′ colour′ !ijk ijk!i′j′k′ i!′j′k′ n n = n 3 with n integer II.R-ratio: ratioq of− (eq¯+e- → ·hadrons)/(e+e- → µ+µ-) e+ p1 + e e− hadrons 2 R + → + Nc Qf ≡ e e− µ µ− ∝ → !f e p2

Experimental data conﬁrms Nc = 3. (Will come back to R later.)

Color becomes the charge of strong interaction. Interaction is so strong, that the only observed hadrons in nature are those where quarks are combined in colour singlet states

Colour SU(3) is an exact symmetry of nature

1 QCD

Model for strong interactions: non-abelian gauge theory SU(3)

U †U = UU † = 1 det(U) = 1

ψ∗ψ U ∗ ψ U ψ = ψ∗ψ Hadron spectrum fully classiﬁedi i → withij thej ik followingk assumptionsk k !i !ijk !k - hadrons (baryons,mesons): made of spin 1/2 quarks ψ ψ ψ U ∗ U U ψ ψ ψ = ψ ψ ψ i j k ii′ jj′ kk′ i′ j′ k′ i′ j′ k′ ijk → ijki j k i j k - each! quark of a given! ′ﬂavour′ ′ comes in Nc=3 colors!′ ′ ′ + e e− hadrons color SU(3) is an exactR symmetry→ Q2 - ≡ e+e µ+mu ∝ − → − - hadrons are colour neutral, i.e. colour singlet under SU(3) - observed hadrons are colour neutral ⇒ hadrons have integer charge

1 Color singlet hadrons

Quarks can be combined in 2 elementary ways into color singlets of the SUc(3) group

Mesons (bosons, e.g. pion ...)

U U = i i ij ik j k k k i ijk k

Baryons (fermions, e.g. proton, neutrons ...)

⇥ ⇥ ⇥ U U U ⇥ ⇥ ⇥ = det(U)⇥ ⇥ ⇥ ijk i j k ijk ii jj kk i j k ijk i j k ijk iijjkk ijk

12 Quark mass spectrum

100 up-type quarks t 1 down-type quarks u c t 10 b 2/3 up charm top 2 c

+ 10 top /m

q 3 s 10 d s b m electric charge down strange bottom − 1/3 4 10 d u 5 1st 2nd 3rd 10 quark generation 6 10

charge 2/3 up charm top mass= few MeV ~1.6 GeV ~172 GeV charge -1/3 down strange bottom mass = few MeV ~100 MeV ~5 GeV 13 The R-ratio: comparison to data Renormalisation group QCD beta function Short-distance observables 2 Comparison of RˆR tochangesexperi mcrossingental d aquarkta ﬂavour thresholds R = Nc eq q 10 2 φ u, d, s ω 3 loop pQCD 10 Naive quark model ρ ρ′ nf =3 R =2 1

Sum of exclusive Inclusive measurements measurements -1 10 0.5 1 1.5 2 2.5 3 7 ψ(2S) J/ψ ψ4160 c 6 Mark-I Mark-I + LGW ψ4415 Mark-II ψ4040 5 PLUTO ψ3770 R DASP 10 4 Crystal Ball BES nf =3 4 R = 3 3

3 3.5 4 4.5 5

8 Υ(1S) Υ(3S) b 7 Υ(2S) Υ(4S) 6 5 11 4 nf =4 5 R = 3 ARGUS CLEO CUSB DHHM 3 MD-1 2 Crystal Ball CLEO II DASP LENA

9.5 10 10.5 11 √s [GeV] 14

Andrea Banﬁ Lecture 2 QCD matter sector

e e u 2/3 up + q d s q electric charge

down strange − 1/3 p X 1st 2nd Feynman diagram describing DIS of an quark generation electron on a proton

• The light quark's existence was validated by the SLAC's deep inelastic scattering (DIS) experiments in 1968: strange was a necessary component of Gell-Mann and Zweig's three-quark model, it also provided an explanation for the kaon and pion mesons discovered 1947 in cosmic rays in 1947 15 QCD matter sector

Feynman diagram describing the mixing of a kaon into its anti-particle. The black boxes d s indicate weak eﬀective four- u c fermion interactions 2/3 up charm + K c c K¯

d s s¯ d¯ electric charge

down strange − 1/3 2 2 2 mc 1st 2nd GF sin c 2 MW quark generation

• In 1970 Glashow, Iliopoulos, and Maiani (GIM mechanism) presented strong theoretical arguments for the existence of the as-yet undiscovered charm quark, based on the absence of ﬂavour-changing neutral currents

[S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2 (1970) 2] 16 QCD matter sector

u c 2/3 up charm + d s electric charge

down strange − 1/3

Computer reconstruction of st 1 2nd a ψ′ decay in the Mark I detector at SLAC, making a quark generation near-perfect image of the Greek letter ψ

• Charm quarks were observed almost simultaneously in November 1974 at SLAC and at BNL as charm anti-charm bound states (charmonium). The two groups had assigned the discovered meson two different symbols, J and ψ. Thus, it became known as the J/ψ (Nobel Prize 1976)

17 1.5 excluded at CL > 0.95 excluded area has CL > 0.95 ! 1.0 QCD matter sector #md & #ms sin 2' u c 2/3 up0.5 charm + #md d s $K b " electric charge

down strange bottom − 1/3 ! '

) 0.0 st nd rd 1 2 " 3 Unitarity triangle V measuring the amount quark generation ub of CP violation in the SL Standard Model V • The− bottom0.5 quark wasub postulated& % in 1973 by Kobayashi and Maskawa" to describe the phenomenon of CP violation, which requires the existence of at least three generations of quarks in Nature (Nobel Prize 2008)

[M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652] 18 −1.0 $K CKM f i t t e r ! sol. w/ cos 2' < 0 Beauty 09 (excl. at CL > 0.95)

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ( QCD matter sector

The “bump” at 9.5 GeV that lead to the discovery of the bottom quark at u c FNAL in 1977 2/3 up charm + d s b electric charge

down strange bottom − 1/3

1st 2nd 3rd quark generation

• In 1977, physicists working at the ﬁxed target experiment E288 at FNAL discovered Υ (Upsilon) meson. This discovery was eventually understood as the bound state of the bottom/anti-bottom (bottomonium)

19 QCD matter sector

Diagram involving the virtual exchange of top quarks that induces a mass diﬀerence in the B meson system u c t d W b 2/3 up charm top + B t t B¯ d s b electric charge

down strange bottom − 1/3 ¯b W + d¯

1st 2nd 3rd M G2 m f 2 V 2 m2 quark generation B F B B | td| t

• The measurement of the oscillations of B mesons into its own antiparticles in 1987 by ARGUS led to the conclusion that the top-quark mass has to be larger than 50 GeV. This was a big surprise at that time, because in 1987 the top quark was generally believed to be much lighter

20 QCD matter sector

t, b u c t

2/3 Z, W Z, W up charm top + t d s b t b electric charge

down strange bottom − 1/3 W Z t ¯ 1st 2nd 3rd b quark generation Diagrams that feature a quadratic dependence on the top-quark mass

• It was also realized that certain precision measurements of the EW vector-boson masses and couplings are very sensitive to the mass of the top quark. By 1994 the precision of these indirect measurements led to a prediction of the top-quark mass between 145 GeV and 185 GeV

21 QCD matter sector

jet µ+

Top anti-top production in proton anti-proton collision at the Tevatron µ u c t b 2/3 up charm top +

+ W t p p¯ d s b ¯ electric charge t down strange bottom − 1/3 ¯ t¯ b jet W 1st 2nd 3rd b jet quark generation

jet

• The top quark was ﬁnally discovered in 1995 by CDF and D0 at FNAL. It’s mass is today measured to be mt = (173.1 ± 0.6) GeV

22 QCD matter sector

100 u c t up-type quarks t 1 down-type quarks 2/3 up charm top 10

+ b 2 10 c d s b 3 s

electric charge 10

down strange bottom − 1/3

Yukawa coupling Yukawa 10 d 1st 2nd 3rd u 5 quark generation 10

6 10

• The masses of the six different quarks range from 2 MeV for the up quark to 172 GeV for the top quark. Why these masses are split by almost six orders of magnitude is one of the big mysteries of particle physics

23 QCD matter sector

100 u c t up-type quarks t 1 down-type quarks 2/3 up charm top 10

+ b 2 10 c d s b 3 s

electric charge 10

down strange bottom − 1/3

Yukawa coupling Yukawa 10 d 1st 2nd 3rd u 5 10 quark generation proton 6 10

• The up, down, and strange quark are much lighter than the proton. If one takes them to have an identical mass, the quarks become indistinguishable under QCD, and one obtains an effective SU(3)f symmetry

24 QED and QCD

QED and QCD are very similar, yet very different theories quarks are a bit like leptons, but there are three of each gluons are a bit like photons, but there are eight of them gluons interact with themselves the QCD coupling is also small at collider energies, but larger then the QED one the similarities and differences are evident from the two Lagrangians

So, let’s start by looking at the QED Lagrangian

25 The QED Lagrangian

electromagnetic vector potential

ﬁeld strength tensor covariant derivative

26 QED Feynman rules

27 QED gauge invariance

A crucial property of the QED Lagrangian is that it is invariant under

which acts on the Dirac ﬁeld as a local phase transformation Exercise: Check that the QED Lagrangian is invariant under the above transformations

Yang and Mills (1954) proposed that the local phase rotation in QED could be generalised to invariance under a continuous symmetry [C. N. Yang and R. L. Mills, Phys. Rep. 96 (1954) 191]

28 U †U = UU † = 1 det(U) = 1

ψU∗†ψU = UU † U=∗1ψ Udetψ(U)== 1 ψ∗ψ i i → ij j ik k k k !i !ijk !k

ψi∗ψi Uij∗ ψjUikψk = ψk∗ψk ψiψjψk → Uii∗ Ujj′ Ukk′ ψi′ ψj′ ψk′ = ψi′ ψj′ ψk′ !i→ !ijk ′ !k !ijk ijk!i′j′k′ i!′j′k′ ψ ψ ψ U ∗ U U ψ ψ ψ = ψ ψ ψ i j k ii′ jj′ kk′ i′ j′ k′ i′ j′ k′ →nq nq¯ = n 3 with n integer !ijk ijk!−i′j′k′ · i!′j′k′

nq n+q¯ = n 3 with n integer −e e− · hadrons 2 R + → + Nc Qf ≡ +e e− µ µ− ∝ f e e− →hadrons !2 RThe QCD→ LagrangianN Q ≡ e+e µ+µ ∝ c f 1 µν − a −¯(f) f (f) QCD = Fa Fµ→ν + ψi (iD/!ij mf δij) ψj L −14 f − = F µνF a + !ψ¯(f) (iD/ m δ ) ψ(f) LQCD −4 a µν i ij − f ij j !f Dµ ∂µδ + ig ta Aµ , F a ∂ Aa ∂ Aa g f Ab Ac ij ≡ ij s ij a µν ≡ µ ν − ν µ − s abc µ ν ⇒ covariant derivative ⇒ ﬁeld strength

1 µ A 2 gauge ﬁxing = ∂ Aµ L − −2λ only one QCD parameter gs regulating" the strength# of the interaction (quark masses have EW origin) ψ(f) U ff ′ ψ(f ′) setting gs = 0 one obtains the→ free Lagrangian (free propagation of f ′ quarks and gluons without interaction)!

terms proportional(f) (f) to g ( fin) the (ﬁeldf) strength cause(f) self-interaction(f) (f) (f) = ψ¯ D/ψ + ψ¯s D/ψ m ψ¯ ψ + ψ¯ ψ LFbetween gluonsL (makesL theR differenceR − w.r.t. fQED)R L L R !f " # !f " # a color matrices t ij are the generators of SU(3) SU (N ) SU (N ) U (1) U (1) QCD ﬂavour blindL (differencesf × R onlyf × dueL to EW)× R 29 1 ψ = P ψ , ψ = P ψ , P = (1 γ ) L L R R L/R 2 ∓ 5

σ0(γ∗ hadrons) 2 R0 = → + = Nc qf σ0(γ∗ µ µ−) → !f αs R1 = R0 1 + $ π % 1 1 U †U = UU † = 1 det(U) = 1

ψ∗ψ U ∗ ψ U ψ = ψ∗ψ i i → ij j ik k k k !i !ijk !k

ψiψjψk UThe∗ Ujj U generatorskk ψi ψj ψk = ofψ SU(N)i ψj ψk → ii′ ′ ′ ′ ′ ′ ′ ′ ′ !ijk ijk!i′j′k′ i!′j′k′ The gaugen groupn = ofn QCD3 wisi tSU(N)h n in withtege rN =3 q − q¯ ·

+ 2 2 N×N complexe e −generichadrons matrix ⇒ N complex2 values, i.e. 2 N real ones R + → + Nc Qf ≡ e e− µ µ− ∝ f UU † = U †→U = 1N N ! det(U) = 1 1 µν a ¯(f) (f) QCD = F F⇒ +2 ψi (iD/ij mf δij) ψj ⇒ L ☛unitarity−4 a µν N conditions −☛unit determinant 1 condition !f

µ µ a µ a a a 2 b c a Dij ∂ δSo,ij + theig sfundamentaltijAa , representationFµν ∂µAν of SU(N)∂ν Aµ hasgs Nfab-1cA generatorsµAν t : ≡ N×N traceless hermitian matrices≡ ⇒ − N2-1 gluons−

a 2 U = eiθa(x)t a = 1, N 1 · · ·

c [ta, tb] = fabc t 30

f = f = f abc − acb − bac

1 µ A 2 gauge ﬁxing = ∂ Aµ L − −2λ " # ψ(f) U ff ′ ψ(f ′) → !f ′

= ψ¯(f)D/ψ(f) + ψ¯(f)D/ψ(f) m ψ¯(f)ψ(f) + ψ¯(f)ψ(f) LF L L R R − f R L L R !f " # !f " #

SU (N ) SU (N ) U (1) U (1) L f × R f × L × R

1 Phenomenology: lecture 3 (p. 56) QCD basics Lagrangian + colour Lagrangian

ψ1 Quarks — 3 colours: ψa = ψ ⎛ 2 ⎞ ψ3 Quark part of LagrTheangian : Gell-Mann⎝ ⎠ matrices ¯ µ µ C C q = ψa(iγ ∂µδab gs γ1 tab µ m)ψb One explicit representation:L tA−= A A − SU(3) local gauge symmetry 8 (= 322 1) generators t1 . . . t8 ↔ − ab ab corresponding to 8 gluons 1 . . . 8 . A Aµ Aµ λ are the Gell-MannA matrices1 A A representation is: t = 2 λ , 0 1 0 0 −i 0 1 0 0 0 0 1 λ1 = 0 1 0 0 1 , λ2 = 0 i 0 0 1 , λ3 = 0 0 −1 0 1 , λ4 = 0 0 0 0 1 , @ 0 0 0 A @ 0 0 0 A @ 0 0 0 A @ 1 0 0 A

0 0 −i 0 0 0 0 0 0 1 0 0 0√3 1 λ5 = 0 0 0 0 1 , λ6 = 0 0 0 1 1 , λ7 = 0 0 0 −i 1 , λ8 = 0 1 0 , √3 2 @ i 0 0 A @ 0 1 0 A @ 0 i 0 A B 0 0 − C @ √3 A 1 Standard normalization: Tr(tatb) = T ab T = R R 2

Notice that the ﬁrst three Gell-Mann matrices contain the three Pauli matrices in the upper-left corner 31 The generators of SU(N)

Inﬁnitesimal transformations (close to the identity) give complete information about the group structure. The most important characteristic of a group is the commutator of two transformations:

[U( ), U( )] U( )U( ) U( )U( ) 1 2 ⇥ 1 2 2 1 = (ia) (ib) [ta, tb] + (3) 1 2 O

The two matrices to not commute, therefore the transformations don’t. Such a group is called non-abelian.

• Familiar abelian groups: translations, phase transformations U(1) ...

• Familiar non-abelian groups: 3D-rotations

32 The generators of SU(N)

Consider the commutator

c Tr([t ,t ]) = 0 [ta,tb]=ifabct a b The Lie algebra of the group is deﬁned by the commutation relation of the generators if the group. fabc are the structure constants of the SU(Nc) algebra, they generate the adjoint representation of the algebra

Clearly, fabc is anti-symmetric in (ab). It is easy to show that it is fully antisymmetric

ifabc = 2 Tr ([ta, tb]tc) and that hence it is fully antisymmetric

f = f = f abc bac acb 33 g2 α = s s 4π

2 α α 2 M 2 gs s s UV αs = R2 = R0 1 + + c + πb0 ln 2 4π ! π " π # ! Q $$ 2 2 M 2 2 αs αs MUV bare UV bare R = R 1 + + c + πb ln αs(µ) = αs + b0 ln αs 2 0 0 2 µ2 ! π " π # ! Q $$ % & M 2 2 11Nc 4nf TR α (µ) = αbare + b ln UV αbare b0 = − s s 0 µ2 s 12π % & 11N 4n T c f R 2 b0 = − 2 12π αs(µ) αs(µ) µ 3 R = R0 1 + + c + πb0 ln + (αs(µ)) ⎛ π ! π $ ! Q2 $ O ⎞

2 ⎝2 ⎠ αs(µ) αs(µ) µ ren 3 ren 2 αs R = R0 1 + + c + πb0 ln + (αs(µ))β(α ) µ ⎛ π ! π $ ! Q2 $ O ⎞ s ≡ dµ2 ⎝ ⎠ αren 2 β(αren) µ2 s β = b0 αs(µ) + . . . Colors ≡ algebra:dµ2 fundamental identities− 2 2 1 µ 1 β = b0 αs(µ) + . . . = b0 ln 2 + Fundamental representation− 3: αs(µ) µ0 αs(µ0) 2 1 1 µ 1 a = b ln + αs(µ) = 2 i 0 j 2 = ij i jµ = tij αs(µ) µ0 αs(µ0) b0 ln Λ2 1 2 i β = αs(µ) biαs(µ) αs(µ) = 2 Adjoint representation 8:µ − c i b0 ln Λ2 + 11N 4n T 2 i b = c − f R βa= αs(µ) biαs(µ) 0 − b = ab a 12πb = ifabc +i 2 11Nc 4nf TR 17Nc 5Ncnf 3CF nf b = − b1 = − − Trace identities:0 12π 24π2

17N 2 5N n 3C n a b =a c − c f=−0 F f a Tr(tb)==T0R 1 24π2

a b ab Tr(ta) = 0 Tr(t t ) = TRδ

34 a b ab Tr(t t ) = TRδ 2

2 What do color identities mean physically

¯ A i tij j 010 0 (1, 0, 0) 100 1 000 0 ¯ 1 What does this really mean? i tij j

Gluons carry color and anti-color. They repaint quarks and other gluons.

35 QCD Lagrangian Feynman rules Pictorial representation of SU(Nc) identities Casimir factors

Fundamental representation 3:

2 − a a Nc 1 t t = CF δij CF = ik kj 2N Xa c = CF

Adjoint representation 8: N 2 1 a a acd bcd ab c (tij)(tkfj) =f CF δ=ijCAδ CF C=A = N−c a Color algebra:2 NCasimirsc & Fierz =identityC ! Xcd A Fierz identity:acd bdc ab f f Fie=rzCidAeδntity: CA = Nc cd ! QCD Lagrangian 1 1 Feynman rules aa i i a al l 1 i il l −1 i 1l l 1 (t )k (t )j = Pδicjtorδiakl representationδkof SδjU(Nc) identities 1 (t )k(t )j = δ2jδk 2Nδckδj = − 2 − 2Nc QCD Lagrangian 2 2Nc Casimir factors Feynman rules Pictorial representation of SU(Nc) identities Gluons as carriers of colour in the large-Nc limit FundamentalCasimirF frepresentationuancdtaomresntal represe n3:tation 3: 1 N 2 − 1 = + O(1/Nc ) a Faundamental representacti2on 3: 2 tiktkj = CF δij CF = N 1 a a 2Nc c (tXa)(t ) = CF δij CF = − = CF ij kj 2 − a a a Nc 2N1c ! tiktkj = CF δij CF = Adjoint representation28N: c C Xa acd bdc ab Andrea Banﬁ Lecture 1 = F f f = CAδ CA = Nc cd acd bcd ab N 2 1 Adjoint!a representationfa Adfjoint=repCrAesδe n8:ta t CioAn 8=c : Nc (tij)(tkj) = CF δij CF = − = C Xcd A a a i a l 1 i l 1 i 2lNc ! (t a)ckd(t b)cjd δjδk ab δkδj f f 2= CA−δ 2NCA = Nc acd bdFcierz idenatbity: c = C Xcdf f = CAδ CA = Nc A !cd a i a l 1 i l − 1 1 l 1 (t a)ki (Ftiae)rljz1=idienlδtijtyδ:k 1 i l δk δj 1 Exercises:(t ) (t ) δ δ2 δ2Nδc = − k j 2 j k − 2N k j 2 2Nc 1 c 1 1) derive(ta)i (ta )Fierzl = δidentityi δl − δ1 δl 1 1 k Gljuons2asj ckarrie2rsNof ckolojur in the large-Nc lim=it − 2) use the Fierz identityc to derive the value of 2CF and CA 2Nc 36 1 Gluons as carriers of colour in th=e large-Nc limit + O(1/N ) 2 c

= 1 + O(1/N ) 2 c

Andrea Banﬁ Lecture 1

3 3 U †U = UU † = 1 det(U) = 1

ψi∗ψi Uij∗ ψjUikψk = ψk∗ψk i → ijk k ! U †U =!UU † = 1 det(U) =!1

ψ ψ ψ U ∗ U U ψ ψ ψ = ψ ψ ψ i j k ii′ jj′ kk′ i′ j′ k′ i′ j′ k′ ijk → ijψk∗iψji k U ∗ ψjUikψk = ψ∗ψikj k ! i!′ ′ →′ ij k !′ ′ ′ !i !ijk !k nq nq¯ = n 3 with n integer ψiψjψk − Uii∗ U·jj′ Ukk′ ψi′ ψj′ ψk′ = ψi′ ψj′ ψk′ → ′ !ijk ijk!i′j′k′ i!′j′k′ + e e− hadrons 2 R Un†qU =+nq¯U=→Un† =3+1 wditeht(nUi)Nn=tceg1er Qf ≡ −e e− ·µ µ− ∝ → !f + 1 e e− hadrons 2 Rψ∗ψ µν a U ∗ ψ U (ψf) =N ψQ∗ψ (f) = i Fi +F →+ij +j ψ¯ik k(iD/c kmfk δ ) ψ QCD Gauge≡ a→e eµ−ν invarianceµ µ−i ∝ ij f ij j L !i −4 !ijk → !k!f− !f 1 (f) (f) ψ ψ ψ µνU ∗a U U ¯ψ ψ ψ = ψ ψ ψ The QCD Lagrangiani Qj CkD is= invariantFa Fiiµ′ν +underjj′ kkψ′ i ilocal′ (ijD′/ijk ′gaugemf δij transformations,) iψ′ j j′ k′ i.e. ijk L → −ijk4i j k − i j k one can Dredefineµ !∂µδ the+ iquarkg ta A!′µ and′ ,′ gluon!Ff a fields∂ Aindependentlya ∂!′ ′ A′ a g f atA everyb Ac point ij ≡ ij s ij a µν ≡ µ ν − ν µ − s abc µ ν nq nq¯ = n 3 with n integer in space and µtime µwithouta− changingµ · thea physicala contenta of theb c theory Dij ∂ δij + igstijAa , Fµν ∂µAν ∂ν Aµ gsfabcAµAν ≡ ≡ a − − + U = eiθa(x)t Gauge transformation fore ethe− quarkhadrons field 2 • R → Nc Q + + a f ≡ e e− µ µiθ−a(x)t∝ →U = e !f ψ ψ′ = U(x)ψ 1 µν a ¯(f) (f) QCD = Fa Fµν→+ ψi (iD/ij mf δij) ψj L −4 ψ ψ′ = U(x)ψ − → !f a a a 1 i a µ 1 • The covariantt A aderivativet Aa′ = ( UD ( µx )) ijt A = a U µ− (ij x )+ + ig s t (ij ∂ AU a ( xmust)) U − transform(x) as µ aµ → a a µ a a 1 aigs a 1 b c Dij ∂t Aδija + itgsAtija′ A=a U, (x)t AFaUµν− (x∂)µ+Aν (∂νUA(xµ)) Ugs−fa(bxc)AµAν (covariant =≡ transforms→ “with” the field)≡ gs− − Dµψ Dµ′ ψ′ = U(x)Dµψ D ψ→ D′ ψ′ = Ua (x)D ψ µ →U =µeiθa(x)t µ c [ta, tb] = fabc tc [ta, tb] = fabc t ψ ψ′ = U (x)ψ • From which one derives thei → transformationi jk k property of the gluon field

ffaabcbc== ffacbb == ffbabcac a a a −− 1 −− i 1 t Aa t Aa′ = U(x)t AaU − (x) + (∂U(x)) U − (x) → gs 11 µ A 2 2 gauge ﬁxing = ∂ µA A gauge ﬁxing =37 c ∂ Aµ µ LL −[−ta, tb] =−fa22bλcλt "" # #

fabc = facb = fbac − 1 − 1 1 µ A 2 gauge ﬁxing = ∂ Aµ L − −2λ " # ψ(f) U ff ′ ψ(f ′) → !f ′

1 U †U = UU † = 1 det(U) = 1

ψ∗ψ U ∗ ψ U ψ = ψ∗ψ i i → ij j ik k k k !i !ijk !k U †U = UU † =U1†Ud=etU(UU)† = 11 det(U) = 1 ψiψjψk U ∗ Ujj Ukk ψi ψj ψk = ψi ψj ψk → U †U = iiU′ U † ′= 1′ d′ et(′ U)′ = 1 ′ ′ ′ !ijk ijk!i′j′k′ i!′j′k′ ψ∗ψ U ∗ ψ∗Uψi ψ =U ∗ ψψjU∗ψikψk = ψ∗ψk i i ij ij ik→k ij k k k i → ijk i ijk k k ! ! !nq nq¯ = !n 3!with n int!eger ψ−i∗ψi ·Uij∗ ψjUikψk = ψk∗ψk i → ijk k ψiψjψk U ∗ Ujj Ukk ψi ψj ψk = ψi ψj ψk ψiψjψk Uii∗ !Ujj Ukk ψi ψ!jii′ψk ′ = ′ ′ ψ′i ψ!j′ ψk ′ ′ ′ ijk ′ →′ ijk+i′ j k′ ′ ′ ′ ′ i′ j k ijk → ijk!i j k e!e′ −′ ′ hadronsi j k 2!′ ′ ′ ! !′ ψ′ ′ψ ψ R →U ∗ U U !′ ψ′ ′ψNcψ =Q ψ ψ ψ i j k + ii′ j+j′ kk′ i′ j′ k′ f i′ j′ k′ →≡ e e− µ µ− ∝ f !ijk nijqk!i′jn′kq¯′ =→n 3 with n in!tegeri!′j′k′ nq nq¯ = n 3 w−ith n in·teger − · 1 µν a ¯(f) (f) QCD = nqFa n+Fq¯µ=ν +n 3 ψiwit(hiDn/ijintemgefrδij) ψj −e e− ·hadrons 2 L+ −R4 f N − Q e e− hadrons+ → !+ 2 c f R → ≡ +e e−N µ µQ− ∝ f + + e e →chadronsf ! ≡ e e− µ µ− −∝ 2 R → f Nc Q µ µ → a 1µ + !a+ a af b c D ∂ δ + ig t ≡A ,eµνe−a Fµ µ¯−(f∂) ∝A ∂ A g (ff) A A ij ij QCD =s ij aFa Fµν→+ µν ψi µ(iDν/ij !fνmfµδij) ψsj abc µ ν ≡1 µνL a −4¯(f) ≡ (−f−) − QCD = F F + ψ (iD/ij f mf δij) ψ a µν 1 iµν a ! (f) j (f) L −4 = f F F + − ψ¯ (iD/ m δ ) ψ QCD a µν i a ij f ij j L !−4 iθa(x)t − µ µ a µ U = e!f a a a b c Dij ∂ δij + igstijAa , Fµν ∂µAν ∂ν Aµ gsfabcAµAν µ µ a ≡µ a a ≡a − b −c Dij ∂ δij + igsµtijAa µ, Faµν µ ∂µAν ∂aν Aµ gasfabcAµaAν b c ≡ Dij ∂ δij + igstijA≡a ψ, ψ′−=FµUν (x)a∂ψ−µAν ∂ν Aµ gsfabcAµAν ≡ Gauge→U =invarianceeiθa(≡x)t − − iθ (x)ta a a a Ua = e a i1θa(x)t i 1 t Aa t A′ = U(x)t UAa=Ue− (x) + (∂U(x)) U − (x) a ψ ψ′ = U(x)ψ The QCD Lagrangian→ is invariant→ under localgs gauge transformations, i.e. one can redeﬁne theψ quarkψ′ = Uand(x) ψgluon ﬁelds iindependently at every point a → a Dµψ ψ Daµ′ ψψ′′ ==1UU((xx))ψDµψ 1 t Aa t Aa′ = U→(x)→t AaU − (x) + (∂U(x)) U − (x) in space and time without→ changing the physicalgs content of the theory i a a a a a 1 ¯ a¯ ¯1 i1 1 t Aa t Aa′ =t UA(x)t tAAaU′ −= (Uxψ()x+)t ψA′ =(U∂−ψUU((xx†())x)+)U − (x∂U) (x)) U − (x) a a Dµψ D′aψ′ = U(x)Dµψ • It follows→ that → →→ gsµ gs

a a Dµψ Dµ′ ψ′ D=iµUgψsψ(¯txF)DDµνψµ¯µ′ ′=ψψ=′ [=Dψ¯UµU,†D((xxν))]Dµψ → →→

a a a¯ aa′ ¯a ¯ a a 1 ψ¯ tψ¯F′ µ=ν ψ¯Uitψ†gF(xµt)νF=ψ′ U==(xψ[D)Ut †,F(Dxµν)U] − (x) → → s→ µν µ ν

a a a a a′ a ca a 1 a at F i g[ttatF, tFb] ===Ufa([bDxc)tt, FD ]U − (x) e.g. because i gs t Fµν =µν[D→µ,sDνµ]νµν µ µνν

a a a a′ 1 ac a 1 a a a a′ t Fµν ta[Ftaµa,νtb=] =U1 f(axb)ctt Fµν U − (x) Thereforet F theµν QCDt Fµν Lagrangian= U(x)t Fµν Uis− indeed(x) gauge invariant • → → 1 1 1 F ′cµν F ′a = F µνF a [ta, tb] =−fa4bc ta µν −4 a µν

′(f) ′(f) 1 (f) (f) ψ¯ iD/′ 1 m δ ψ = ψ¯ (iD/ m δ ) ψ i ij − f ij j i ij − f ij j !f " # !f 38 c [ta, tb] = fabc t

f = f = f abc − acb − bac

1 µ A 2 gauge ﬁxing = ∂ Aµ L − −2λ " # ψ(f) U ff ′ ψ(f ′) → !f ′

= ψ¯(f)D/ψ(f) + ψ¯(f)D/ψ(f) m ψ¯(f)ψ(f) + ψ¯(f)ψ(f) LF L L R R − f R L L R !f " # !f " #

SU (N ) SU (N ) U (1) U (1) L f × R f × L × R 1 ψ = P ψ , ψ = P ψ , P = (1 γ ) L L R R L/R 2 ∓ 5

σ0(γ∗ hadrons) 2 R0 = → + = Nc qf σ0(γ∗ µ µ−) → !f αs R1 = R0 1 + $ π % g2 α = s s 4π

2 2 αs αs MUV R2 = R0 1 + + c + πb0 ln 2 & π $ π % & Q ''

M 2 2 α (µ) = αbare + b ln UV αbare s s 0 µ2 s " # 11N 4n T b = c − f R 0 12π

2 Gauge invariance

The QCD Lagrangian is invariant under local gauge transformations, i.e. one can redeﬁne the quark and gluon ﬁelds independently at every point in space and time without changing the physical content of the theory

Remarks:

• the ﬁeld strength alone is not gauge invariant in QCD (unlike in QED) because of self interacting gluons (carries of the force carry colour, unlike the photon)

• a gluon mass term violate gauge invariance and is therefore forbidden (as for the photon). On the other hand quark mass terms are gauge invariant.

2 µ m AµA

39 Isospin symmetry

Isospin SU(2) symmetry: relates the content in terms of up and down quarks. Isospin invariance means invariance under u ↔ d 1 I = [(n n ) (n n ¯)] 3 2 u u¯ d d

• Proton has isospin I3=1/2, while the neutron I3=-1/2. Protons and neutrons form an iso-spin multiplet

+ - • Pions have I3=1 (� = ud), I3=0 (�0 =1/√2(uu+dd), I3=0 (� = ud). They are also in an iso-spin multiplet

Particles in the same isospin multiplet have very similar masses (proton and neutron, neutral and charged pions)

40 U †U = UU † = 1 det(U) = 1

ψ∗ψ U ∗ ψ U ψ = ψ∗ψ i i → ij j ik k k k !i !ijk !k U †U = UU † = 1 det(U) = 1 ψ ψ ψ U ∗ U U ψ ψ ψ = ψ ψ ψ i j k ii′ jj′ kk′ i′ j′ k′ i′ j′ k′ ijk → ijki j k i j k ! U †U!′ =′ ′ UU † = 1 det(U) = 1 !′ ′ ′ ψ∗ψi U ∗ ψjUikψk = ψ∗ψk ni q →nq¯ = n ij3 with n integer k !i − !ijk · !k ψi∗ψi Uij∗ ψjUikψk = ψk∗ψk i e+→e ijk hadrons k ψ ψ ψ ! U−!∗ U U ψ ψ ψ!= 2 ψ ψ ψ i j k R ii→′ jj′ kk′ i′ Nj′ k′ Q i′ j′ k′ → + + c f ijk ij≡ki′j′ek′ e− µ µ− ∝ f i′j′k′ ! ψiψjψk ! U ∗ →Ujj Ukk ψi ψj ψk!= !ψi ψj ψk → ii′ ′ ′ ′ ′ ′ ′ ′ ′ ijk ij1ki′j′k′ (f) i′j′k′ (f) ! nq !Fnµq¯ν=F an 3 ψw¯ithinD/integmer!δ ψ QCD = − a µν +· i ( ij f ij) j L −4 f − nq nq¯ = n 3! with n integer −+ · e e− hadrons 2 µ µ R a ++µ → +a aNc Qa f b c D i∂ δij ≡gst eeAee, hadronsµF µ ∂µ∝A ∂ν A gsfabcA A ij ≡ − ij a−− µν ≡− ν − f µ2− µ ν R + →→ + Nc !Qf ≡ e e− µ µ− ∝ f 1 Isospinµν a → symmetry(f) ! (f) = F F + ψ¯1 (iD/ 2 m δ ) ψ QCD 1 a µν i µ ijA f ij j L −4 gaµuνge afixing =f ¯(f) ∂ Aµ− (f) QCD = LFa F−µν + !−ψ2i λ (iD/ij mf δij) ψj Isospin arisesL because− QCD4 interactions are" flavour−# blind and the !f (f) ff ′ (f ′) accidentalµ factµ that up anda downµ ψ havea veryU closeψ a massesa b c Dij i∂ δij gstijAa , →Fµfν ∂µAν ∂ν Aµ gsfabcAµAν µ≡ µ − a µ !a′ ≡ a − a − b c CheckD: theij QCDi∂ δij LagrangiangstijAa has, isospinFµν symmetry∂µAν ∂ifν mAµ = mgsf orabc Amµ,A mν → 0 ≡ − ≡ − u − d u d = ψ¯(f)D/ψ(f) + ψ¯(f)D/ψ(f) 1 m ψ¯2(f)ψ(f) + ψ¯(f)ψ(f) In this limitF of vanishingL Lmu, m d Rmasses,=R one can∂µ separateAf A R Lthe fermionL R L f gauge fixing −1 f µ 2 fields into left! and" rightL chiralities− −#2λ! µ A" # gauge fixing = "∂ Aµ # L − −2λ " #1 (f) ff ′ (f ′) ψL = PLψ , ψ ψR = PRψU, PψL/R = (1 γ5) (f) ff ′ (f ′) 2 ∓ ψ → f U ψ → !′ The QCD Lagrangian in terms of leftf ′and right states becomes σ0(γ∗ h!adrons) 2 R0 = → = Nc q (f) (f) σ ((fγ) (µf)+µ ) (ff) (f) (f) (f) = ψ¯ D/ψ + ψ¯0 ∗D/ψ − m fψ¯ ψ + ψ¯ ψ F L¯(f) L(f) ¯R(f) →R(f) f !¯(Rf) (fL) ¯(fL) (fR) L F =f ψ D/ψ + ψ D/ψ − f mf ψ ψ + ψ ψ ! " L L R R # !αs " R L L R # L f R = R 1−+ f So neglecting! "fermion masses the1 Lagrangian0 # !π has the" larger symmetry # $ % 1 ψL = PLψ , ψR = PRψ , 2 PL/R = (1 γ5) SUL(Nf ) SUR(Nf )gs UL(1) U2R(1∓) × αs = × × 41 4π σ (γ∗ hadrons) R = 0 → = N 1 q2 ψL = P0 Lψ , ψR = PR1+ψ , PL/R =c (1 f γ5) σ0(γ∗ µ µ−) 2f → ! ∓ α σ0(γ∗ hadrons) s R = R1 =→R0 1 + = N q2 0 + π c f σ0(γ∗ µ$ µ−) % → !f 2 gs αs R1 =αRs 0= 1 + $ 4π π % 11 Isospin symmetry

The fact that left-handed and right-handed terms in the Lagrangian are separately invariant is a direct consequence of the fact that the chirality of massless fermions is conserved This symmetry of the QCD is known as chiral symmetry and it is spontaneous broken by the QCD vacuum (much as the Higgs mechanism). The (approximately massless) pions are the Goldstone boson of the broken symmetry This happens when the vacuum state of the theory is not invariant under the same symmetries as the Lagrangian. In the case of QCD it is known that 0 qq¯ 0 = 0 uu¯ + dd¯0 (250MeV)3 h | | i h | | i⇡ Similar mechanisms to the one that break the chiral symmetry in QCD have been proposed to explain why the Higgs boson so is light in composite Higgs scenarios … but this is a topic for a different lecture!

42 Feynman rules: propagators

Obtain quark/gluon propagators from free piece of the Lagrangian

′(f) ′(f) (f) (f) ψ¯ iDQuark/′ m propagator:δ ψ = replaceψ¯ i∂( i→D/ k andm takeδ ) ψthe i × inverse i ij − f ij j i ij − f ij j !f " # !f ¯(f) (f) p i q,free = ψi (i∂/ mf ) δijψj i j L − /p m ij !f c [ta, tb] = fabc t Gluon propagator: replace i∂ → k and take the i × inverse ?

fabc = facb = fbac 1 µ ⇥ − g,−free = A ( gµ⇥ µ⇥ ) A L 2

1 µ A 2 ➥ inversegauge doesﬁxing not= exist,∂ sinceAµ ( g ) = =0 L − −2λ µ µ µ " # How can one to deﬁne the propagator ? ψ(f) U ff ′ ψ(f ′) → 43 !f ′

= ψ¯(f)D/ψ(f) + ψ¯(f)D/ψ(f) m ψ¯(f)ψ(f) + ψ¯(f)ψ(f) LF L L R R − f R L L R !f " # !f " #

SU (N ) SU (N ) U (1) U (1) L f × R f × L × R 1 ψ = P ψ , ψ = P ψ , P = (1 γ ) L L R R L/R 2 ∓ 5

σ0(γ∗ hadrons) 2 R0 = → + = Nc qf σ0(γ∗ µ µ−) → !f αs R1 = R0 1 + $ π % g2 α = s s 4π

2 2 αs αs MUV R2 = R0 1 + + c + πb0 ln 2 & π $ π % & Q ''

M 2 2 α (µ) = αbare + b ln UV αbare s s 0 µ2 s " #

2 Gauge ﬁxing

Solution: add to the Lagrangian a gauge ﬁxing term which depends on an arbitrary parameter ξ In covariant gauges:

1 2 ξ=1 Feynman gauge = µAA Lgauge ﬁxing µ ξ=0 Landau gauge Gluon propagator:

Gauge ﬁxing explicitly breaks gauge invariance. However, in the end physical results are independent of the gauge choice. Powerful check of higher order calculations: verify that the ξ dependence fully cancels in the ﬁnal result 44 QCD Lagrangian Feynman rules Pictorial representation of SU(Nc) identities Covariant gauge

Gauge ﬁxing condition: ∂ Aµ = 0 µ a Ghosts 1 µ 2 ab i LGF = − (∂µA ) ⇒ ∆ (k) = dµν 2α a µν k2 ∗ kµkν In covariantdµν =gaugesϵ gauge(k, λ)ϵ ﬁxingν (k, λ )term= − gmustµν + be(1 −supplementedα) with ghost µ k2 term to cancelX λunphysical longitudinal degrees of freedom which should not propagate Ghost Lagrangian: a µ b k i ghost = µ †D a b ab L µ ab abµ µ a b c k2 LF P = ∂ c¯aDµ cb = ∂ c¯a∂µca − gfabc∂ c¯ Aµc

η: complexQuantum scalarcorrec tﬁeldions iwhichntrodu cobeyse non -Fermiphysic astatisticsl polarisa tions whose contribution is cancelled by ghost-gluon interactions

2 2 2 − = λ=+1,−1,0 λ=+1,−1

Andrea Banﬁ 45 Lecture 1 Axial gauges

Alternative: choose an axial gauge (introduce an arbitrary direction n)

1 2 = nµAA Laxial gauge µ The gluon propagator becomes i n k + n k (n2 + k2)k k d = g + µ µ + µ µ k2 µ n k (n k)2 ab · · Light cone gauge: n2 = 0 and ξ = 0

Axial gauges for k2 0 µ µ dµ k = dµ n =0 i.e. only two physical polarizations propagate, that’s why often the term physical gauge is used

46 QCD Feynman rules: the vertices

47 U †U = UU † = 1 det(U) = 1

ψi∗ψi Uij∗ ψjUikψk = ψk∗ψk i → ijk k Perturbative! expansion! of! the R-ratio ψiψjψk U ∗ Ujj Ukk ψi ψj ψk = ψi ψj ψk → ii′ ′ ′ ′ ′ ′ ′ ′ ′ !ijk ijk!i′j′k′ i!′j′k′ The R-ratio is deﬁned as nq nq¯ = n 3 with n integere+ − · * + γ /Z (e e hadrons)+ e e− hadrons R + R + N Q2 (e e µ µ +) → + c f ≡ e e− µ µ− ∝ → !f - 1 e = F µνF a + ψ¯(f) (iD/ m δ ) ψ(f) LQCD −4 a µν i ij − f ij j At lowest order in perturbation theory!f (PT)

µ µ + a µ a a + a b c D i∂ δ(ije egst A hadrons), F =∂µA(e e∂ν A qgq¯s)fabcA A ij ≡ − ija µν ≡ 0 ν − µ − µ ν

1 µ A 2 Since common factors cancelgauge inﬁ xnumerator/denominator,ing = ∂ Aµ to lowest order L − −2λ one ﬁnds " # σ0(γ∗ hadrons) 2 R0 = → + = Nc qf σ0(γ∗ µ µ−) → !f αs R1 = R0 1 + 48$ π % 2 2 αs αs MUV R2 = R0 1 + c + πb0 ln 2 & π $ π % & Q ''

1 The R-ratio: perturbative expansion

First order correction

virtual real

Real and virtual do not interfere since they have a different # of particles. The amplitude squared becomes 2 2 2 2 2 gs A = A + A + 2Re A A + ( ) s = | 1| | 0| s | 1,r| { 0 1,v} O s 4⇥ ⇥ Integrating over phase space, the ﬁrst order result reads R = R 1 + s 1 0 ⇥ ⇥

49 U †U = UU † = 1 det(U) = 1 U †U = UU † = 1 det(U) = 1

ψi∗ψi Uij∗ ψjUikψk = ψk∗ψk i → ijk k ψi∗ψi Uij∗ ψjUikψk = ψk∗ψk ! ! ! i → ijk k ! ! ! ψiψjψk U ∗ Ujj Ukk ψi ψj ψk = ψi ψj ψk → ii′ ′ ′ ′ ′ ′ ′ ′ ′ !ijk ijk!i′j′k′ i!′j′k′ ψ ψ ψ U ∗ U U ψ ψ ψ = ψ ψ ψ i j k ii′ jj′ kk′ i′ j′ k′ i′ j′ k′ ijk → ijki j k i j k n n = n 3 with n integer ! !′ ′ ′ !′ ′ ′ q − q¯ ·

nq nq¯ = n 3 with n integer e+e hadrons − · − 2 R + → + Nc Qf ≡ e e− µ µ− ∝ f + → ! e e− hadrons 2 1 R → Nc Q µν a ¯(f) (f) + + f QCD = Fa Fµν + ψi (iD/ij mf δij) ψj ≡ e e− µ µ− ∝ f L −4 − → ! !f 1 µν a ¯(f) (f) QCD = Fa Fµν + ψi (iD/ij µ mf δijµ) ψj a µ a a a b c L −4 D−ij i∂ δij gstijAa , Fµν ∂µAν ∂ν Aµ gsfabcAµAν !f ≡ − ≡ − −

R-ratio and UV divergences 1 µ A 2 µ gauge ﬁxing = ∂ Aµ D i∂µδ g ta Aµ , F a ∂ Aa ∂ Aa g f ALb Ac − −2λ ij ≡ ij − s ij a µν ≡ µ ν − ν µ − s abc µ ν " # σ0(γ∗ hadrons) 2 To compute the second order correction one hasR0 to= compute→ diagrams+ = Nc qf 1 2 σ0(γ∗ µ µ−) f µ A → ! like these and gmanyauge ﬁ xmoreing = ∂ Aµ L − −2λ αs " # R1 = R0 1 + $ π % σ0(γ∗ hadrons) 2 R0 = → = Nc q 2 2 + f αs αs MUV σ0(γ∗ µ µ−) R = R 1 + + c + πb ln → !f 2 0 ... 0 2 & π $ π % & Q '' αs 2 R1 = R0 1 + bare MUV bare 2 One gets π αs(µ) = α + b0 ln α $ % s µ2 s " # 2 2 αs αs MUV 11Nc 4nf TR R2 = R0 1 + + c + πb0 ln 2 b0 = − & π $ π % & Q '' 12π

2 2 αs(µ) αs(µ) µ 3 R = R0 1 + + c + πb0 ln + (αs(µ)) ⎛ π & π ' & Q2 ' O ⎞ Ultra-violet divergences do not cancel. Result⎝ depends on UV cut-off. ⎠

1 50

1 Renormalization

Loop corrections in QCD are (often) divergent. Divergences originate from regions of very large momenta

QCD is a renormalizable theory. This means that that one can 1. regularize the divergence (e.g. using dimensional regularization) 4 2 4 2 d k µ d k 2. absorbe all UV divergences into a universal redeﬁnition of a ﬁnite number of the bare parameters of QCD

51 U †U = UU † = 1 det(U) = 1

ψ∗Uψ†U = UUU†∗=ψ1U dψet(=U) = 1ψ∗ψ i i → ij j ik k k k !i !ijk !k

ψi∗ψi Uij∗ ψjUikψk = ψk∗ψk ψiψjψk Uii∗ Ujj′ Ukk′ ψi′ ψj′ ψk′ = ψi′ ψj′ ψk′ →i → ij′k k !ijk !ijk!i′j′k′ ! ! i!′j′k′

ψiψjψk U ∗ Ujj Ukk ψi ψj ψk = ψi ψj ψk n→q nq¯ = nii′ 3 ′ wi′th′ n ′int′eger ′ ′ ′ !ijk −ijk!i′j′k′ · i!′j′k′ + nqe enq¯ = nhadrons3 with n integer − − · 2 R + → + Nc Qf ≡ e e− µ µ− ∝ f + → ! e e− hadrons R → N Q2 1 µν+ a + (f) c f (f) = ≡F eFe− + µ ψµ¯− (∝iD/ f m δ ) ψ LQCD −4 a µν → i ij !− f ij j !f 1 µν a ¯(f) (f) QCD = Fa Fµν + ψi (iD/ij mf δij) ψj L −4 f − Dµ i∂µδ g ta Aµ , F!a ∂ Aa ∂ Aa g f Ab Ac ij ≡ ij − s ij a µν ≡ µ ν − ν µ − s abc µ ν Dµ i∂µδ g ta Aµ , F a ∂ Aa ∂ Aa g f Ab Ac ij ≡ ij − s ij a µν ≡ µ ν − ν µ − s abc µ ν 1 µ A 2 gauge fixing = ∂ Aµ L − −2λ 1 " µ A #2 gauge fixing = ∂ Aµ Lσ (γ − had−ro2nλs) 0 ∗ " # 2 R0 = → + = Nc qf σ0(γ∗ µ µ−) σ0(γ∗ →hadrons) !f 2 R0 = → + = Nc qf σ0(γ∗ µ µ−)α R = →R 1 + s !f 1 0 π Renormalization and$ runningαs % coupling R1 = R0 1 + α α $ 2 π % M 2 For the R-ratio,R the= divergenceR 1 + s +is dealts withc + byπb renormalizationln UV of the 2 0 π π 2 0 Q22 coupling constant & αs $ αs% & MUV '' R2 = R0 1 + + c + πb0 ln 2 & π $ π % & 2 Q '' bare MUV bare 2 αs(µ) = αs + b0 ln 2 αs bare MµUV bare 2 αs(µ) = α + b0 ln "α # s µ2 s R expressed in terms of the renormalized coupling" # is finite

2 2 αs(µ) αs(µ) µ 3 R = R0 1 + + c + πb0 ln + (αs(µ)) ⎛ π & π ' & Q2 ' O ⎞ ⎝ ⎠ Renormalizability of the theory guarantees that the same redeﬁnition of the 1 coupling removes all UV divergences from1 all physical quantities (massless case) Renormalization achieved by replacing bare masses and the bare coupling with renormalized ones. Masses and coupling become dependent on the renormalization scale. The dependence is fully predicted in pQCD • the coupling ⇒ β function • the masses ⇒ anomalous dimensions γm 52 U †U = UU † = 1 det(U) = 1

ψ∗ψ U ∗ ψ U ψ = ψ∗ψ i i → ij j ik k k k !i !ijk !k

ψiψjψk U ∗ Ujj Ukk ψi ψj ψk = ψi ψj ψk → ii′ ′ ′ ′ ′ ′ ′ ′ ′ !ijk ijk!i′j′k′ i!′j′k′ n n = n 3 with n integer q − q¯ ·

+ e e− hadrons 2 R + → + Nc Qf ≡ e e− µ µ− ∝ → !f 1 = F µνF a + ψ¯(f) (iD/ m δ ) ψ(f) LQCD −4 a µν i ij − f ij j !f

Dµ i∂µδ g ta Aµ , F a ∂ Aa ∂ Aa g f Ab Ac ij ≡ ij − s ij a µν ≡ µ ν − ν µ − s abc µ ν

1 µ A 2 gauge ﬁxing = ∂ Aµ L − −2λ " #

σ0(γ∗ hadrons) 2 R0 = → + = Nc qf σ0(γ∗ µ µ−) The beta-function→ !f αs R1 = R0 1 + 2 ren 2 ds(µ ) ( ) $ µ π % s dµ2 α α 2 M 2 R = R 1 + s + s c + πb ln UV 2 0 π π 0 Q2 The renormalized coupling& is $ % & '' 2 bare MUV bare 2 αs(µ) = αs + b0 ln 2ren αs ren 2µαs ren " #ren 2 αs β(αs ) µ 2 β(α ) µ So, one immediately gets ≡ dµ s ≡ dµ2 ren ren 2 αs 2 2 β(α ) µ β = b0αs(µ) + . . β. = b0αs(µ) + . . . s ≡ dµ2 − − 2 2 2 1 µ 11 µ 1 Integrating theβ = differentialb0αs(µ) equation+ .=. . b lonen ﬁnds+ at lowest= b0 orderln + − 0 2 2 αs(µ) µ0 ααss((µµ)0) µ0 αs(µ0) 1 1 µ2 1 1 1 = b0 ln + ⇒ αs(µ) = 2 2 αs(µ) = µ2 µ αs(µ) µ0 αs(µ0) b0 ln 2 b0 ln Λ2 Λ 1 53 αs(µ) = µ2 b0 ln Λ2

2 2 More on the beta-function Roughly speaking: (a) quark loop vacuum polarization diagram gives a negative contribution to b0 ∼ - 2nf /12�

(a) (b) gluon loop gives a positive contribution to b0 ∼ 11Nc /12�

(b)

Since (b) > (a) ⇒ b0,QCD > 0 ⇒ overall negative beta-function in QCD While in QED (b) = 0 ⇒ b < 0 1 0,QED = 2 + ... QED 3 54 More on the beta-function

• QCD: perturbative picture valid for scales � >> �QCD (about 300 MeV) • QED: perturbative picture valid for scales � << �QED

Question: why does nobody talk about �QED?

Answer: 1 ⇤ = m exp 1090GeV >> M QED e 2b ↵(m ) ⇠ Planck ⇢ 0 e

(Note that the fact that QED is not a consistent theory up to very high scales implies that it must be an effective theory)

55 ren ren 2 αs β(αs ) µ 2 ≡ drµen ren 2 αrsen β(αren) µ2 αs β(αs ) µ 2 s ≡2 dµ2 β = b0 αs(µ) + . . . − 2 β = b0 α2s(µ) + . . . β = −b0 αs(µ) + . . . 1 − µ2 1 = b ln 2 + 1 Back0 µ 22to the1 QCD beta-function αs(µ1 ) µ αs1(µ0) = b0 ln 0 + = b0 ln 2 + αs(µ) µ20 αs(µ0) s 0 1 s 0 α (µ) = 1 Perturbatives expansion1 ofµ 2the beta-function: αs(µ) = b0 ln µ22 αs(µ) = µΛ2 b0 ln Λ2 b0 ln Λ2 2 i β = αs2(µ) biαis(µ) β = α2s(µ) biαis(µ) β =−−αs(µ) i biαs(µ) − !i !i 1111NNc 44nnfTTR 11Nc 4nf TR b0b0== − 0 122ππ 2 17N22 5Ncnf 3CF nf b =1177NNcc − 55Nccnnff − 33CCFFnnf f b b1== −− 2 −− 1 244ππ22

• nf is the number of active flavours (depends on the scale) • today, the beta-function known up to five loops, but only first two coefficients are independent of the renormalization scheme

Exercise: proof the above statement [hint: use the fact that at O(αs) the coupling in two different schemes is related by a ﬁnite change] 56

2 Active ﬂavours & running coupling

The active ﬁeld content of a theory modiﬁes the running of the couplings

Constrain New Physics by measuring the running at high scales?

57 Renormalization Group Equation

Consider a dimensionless quantity A, function of a single scale Q. The dimensionless quantity should be independent of Q. However in quantum ﬁeld theory this is not true, as renormalization introduces a second scale µ But the renormalization scale is arbitrary. The dependence on it must cancel in physical observables up to the order to which one does the calculation.

So, for any observable A one can write a renormalization group equation

⌅ ⌅ ⌅ Q2 µ2 + µ2 s A , (µ2) = 0 ⌅µ2 ⌅µ2 ⌅ µ2 s ⇤ s ⌅ ⇥

⌅s = (µ2) ⇥( ) = µ2 s s s ⌅µ2 Scale dependence of A enters through the running of the coupling: 2 knowledge of A (1 , s ( Q )) allows one to compute the variation of A with Q given the beta-function 58 Current experimental results on αS Measurements of the runningBeth couplingke,hep-ph/0407021 α (M ) = 0.1182 0.0027, MS, NNLO S Z ±

DIS [pol. strct. fctn.] DIS [Bj-SR] DIS [GLS-SR] Summarizing: o-decays [LEP]

xF3 [i -DIS] F [e-, µ-DIS] • overall consistent picture: αs from very 2 DIS [ep –> jets] QQ + lattice QCD different observables compatible [ decays + _ a αS is large at current scales. e e_ F 2 • αs is not so small at current scales e+ e [m ] _ had e+e [jets & shapes 14 GeV] _ Measurement is stable, e+e [jets & shapes 22 GeV] αS _ • αs decreases slowly at higher energies e+ e [jets & shapes 35 GeV] + _ (α (M ) = 0.1183 0.0027 in 2002). e e_ [mhad] S Z e+e [jets & shapes 44 GeV] _ ± (logarithmic only) e+ e [jets & shapes 58 GeV] pp --> bb X The decrease of αS is quite slow – as the higher order corrections are and will pp, pp --> a X • m(pp --> jets) inverse power of a logarithm. K(Z0--> had.) [LEP] _ remain important e+ e [scaling. viol.] _ e+ e [4-jet rate] Higher order corrections are and will con- jets & shapes 91.2 GeV jets & shapes 133 GeV tinue to be important. jets & shapes 161 GeV jets & shapes 172 GeV jets & shapes 183 GeV World average jets & shapes 189 GeV jets & shapes 195 GeV jets & shapes 201 GeV jets & shapes 206 GeV s(MZ0 )=0.1184 0.007 0.08 0.10 0.12 0.14 ± _s (S Z)

Quantum Chromodynamics at the LHCLecture I: Proton structure and Parton Showers – p.10/58 Current experimental results on αS Measurements of the runningBeth couplingke,hep-ph/0407021 α (M ) = 0.1182 0.0027, MS, NNLO S Z ±

DIS [pol. strct. fctn.] DIS [Bj-SR] DIS [GLS-SR] o-decays [LEP]

xF3 [i -DIS] F2 [e-, µ-DIS] DIS [ep –> jets] Questions: QQ + lattice QCD [ decays _ a α is large at current scales. e+ e F S Why is the determination of αs from _ 2 • e+ e [m ] _ had e+e [jets & shapes 14 GeV] _ Measurement is stable, e+e [jets & shapes 22 GeV] αS t-decays so accurate? _ e+ e [jets & shapes 35 GeV] + _ (α (M ) = 0.1183 0.0027 in 2002). e e_ [mhad] S Z e+e [jets & shapes 44 GeV] Why is the determination of αs from _ ± • e+ e [jets & shapes 58 GeV] pp --> bb X The decrease of αS is quite slow – as the the four-jet rate so accurate? pp, pp --> a X m(pp --> jets) inverse power of a logarithm. K(Z0--> had.) [LEP] _ e+ e [scaling. viol.] _ e+ e [4-jet rate] Higher order corrections are and will con- jets & shapes 91.2 GeV jets & shapes 133 GeV tinue to be important. jets & shapes 161 GeV jets & shapes 172 GeV jets & shapes 183 GeV World average jets & shapes 189 GeV jets & shapes 195 GeV jets & shapes 201 GeV jets & shapes 206 GeV s(MZ0 )=0.1184 0.007 0.08 0.10 0.12 0.14 ± _s (S Z)

Quantum Chromodynamics at the LHCLecture I: Proton structure and Parton Showers – p.10/58 The two faces of QCD

Conﬁnement asymptotic freedom (large distance) (short distance) NB: no proof of conﬁnement. We simply never observed quarks as free particles

61 Recap

The formulation of QCD as a non-abelian Quantum Field Theory allows to • Describe the hadron spectrum • Explain experimentally the observed symmetries in the strong iteration • Avoid mixing between strong and weak interactions • Obtain a ﬁeld-theoretical description of the strong force, opening the path to a uniﬁed formalism of all fundamental interactions

We have then discussed the UV behaviour of QCD • discussed renormalisation of UV divergences • introduced the running of the coupling constant and the beta-function • discussed measurements of the coupling constant As we will see, the perturbative description of QCD is very predictive but we understand much less the regime governed by strong dynamics.

62 Next

Next we’ll discuss generic properties of QCD amplitudes • Soft-collinear divergences (and how they are dealt with) • Kinoshita-Lee-Nauenberg theorem • The concept of infrared ﬁniteness • Sterman Weinberg jets