Quantum Chromodynamics Colliders Jets Stephen D Ellis University

Quantum Chromodynamics, Colliders & Jets Stephen D. Ellis University of Washington Lecture 1: The Big Picture – How do we Think About and Calculate processes at Colliders? The Parton Model + QCD Maria Laach September 2008

ASIDE: I am here to provide historical context, i. e. , I can remember particle physics before QCD, jets or colliders! Historical art work by Siggi © S. Bethke June 1993 S. D. Ellis Maria Laach 2008 Lecture 1 2

Outline 1. Introduction – The Big Picture p. QCD - e+e- Physics and Perturbation Theory (the Improved Parton Model); p. QCD - Hadrons in the Initial State and PDFs 2. p. QCD - Hadrons and Jets in the Final State 3. Colliders & Jets at Work S. D. Ellis Maria Laach 2008 Lecture 1 3

Concepts/Vocabulary* • Matter – quarks & leptons, quark model of states and resonances • Parton Model – parton distribution functions (pdf’s), fragmentation functions • Symmetries – global and local – SU(3) of QCD (local, unbroken), U(1) of E&M (local unbroken), SU(2)L of Weak (local, broken), SU(2) (to SU(6)) of Flavor (global, approximate) • Interactions – mediated by gauge bosons (local symmetry) Strong – gluons (massless) Electromagnetic – photons (massless) Weak – Z 0, W+, W- (massive) * Of course, in 3 hours we won’t really cover everything – actually nearly nothing in detail! S. D. Ellis Maria Laach 2008 Lecture 1 4

Concepts/Vocabulary II • Quantum Field Theories – local non-Abelian gauge symmetries, UV singularities, running couplings UV freedom & IR slavery, perturbative expansions, IR & collinear singularities, (leading to) renormalization (scale and scheme, e. g. , MSbar) of PDFs factorization (scale), power corrections, log resummation • Experimental quantities – exclusive cross sections, inclusive cross sections, IR safe quantities, jets • Experimental processes – e+e- hadrons, e( )p e+hadrons, pp hadrons (jets), pp +X, pp + -+X, pp B(eyond the)SM S. D. Ellis Maria Laach 2008 Lecture 1 5

(Incomplete) References: (I’ll focus on concepts/images) • “The Pink Book” – QCD and Collider Physics, R. K. Ellis, W. J. Stirling and B. R. Webber (Cambridge University Press, 1996) • My Pi. TP 2007 Lectures http: //www. phys. washington. edu/users/ellis/Pi. TP%20 July%2007. htm • My TSI 2006 lectures http: //www. phys. washington. edu/users/ellis/TSI%20 July%2006. htm • “Jets in Hadron-Hadron Collisions” by S. D. Ellis, J. Huston, K. Hatakeyama, P. Loch, M. Toennesmann, ar. Xiv: hep-ph/0712. 2447 • The “Primer for LHC Physics” by J. M. Campbell, J. W. Huston, W. J. Stirling, ar. Xiv: hep-ph/0611148 v 1 • Lectures by George Sterman, et al. – (for more references in formal details) ar. Xiv: hep-ph/0412013 v 1, ar. Xiv: hep-ph/0409313 v 1 S. D. Ellis Maria Laach 2008 Lecture 1 6

(more) References: • QCD Summary on the Web at the Particle Data Group site: http: //pdg. lbl. gov/2008/reviews/qcdrpp. pdf • The CTEQ Handbook in Rev. Mod. Phys. Volume 67, Number 1, January 1995, (pp. 157 -248) and on the Web: http: //www. phys. psu. edu/~cteq/#Handbook S. D. Ellis Maria Laach 2008 Lecture 1 7

Simple Initial Picture – The Naive Parton Model (~1970, BSM = Before the SM) Imagine a “theory” of hadrons composed of (nearly massless) quarks and (massless) gluons which • Interact via scale invariant, perturbative (weak) interactions dominated by exchanges with momenta k ≤ m ~ 1 Ge. V (typical hadron mass scale) • Are never seen as isolated states • Interact with Electro-weak currents in expected way (quark model charges) • Inside (relativistic) hadrons are described by (scale invariant) parton distribution functions: q(x) = Fq/h(x) = probability to find quark (of flavor q) with (collinear) momentum fraction x in hadron (and little transverse momentum ~1/hadron size) → PDF • When isolated in phase space, fragment into hadrons as described by (scale invariant) fragmentation functions: Dh/q(z) = probability to find hadron in (collinear) debris of quark with momentum fraction z S. D. Ellis Maria Laach 2008 Lecture 1 8

Parton Model – partons are the building blocks of hadrons and play a role in the dynamics (even if we didn’t understand it!). Consider the inclusive deeply inelastic scattering of electrons from protons – DIS, (e. g. , SLAC). e’ e S. D. Ellis Maria Laach 2008 Lecture 1 9

Recall the EXclusive case: ep → ep General EM vertex (see HW) symmetries 2 functions = p’s anomalous magnetic moment, Electric Form Factor Magnetic Form Factors – The harder you hit a proton, the more likely it is to fall apart ! S. D. Ellis Maria Laach 2008 Lecture 1 10

Now the INclusive case: ep→e. X In the proton rest frame (Lab) the kinematics look like: A new degree of freedom, x (MX or ), but still two (dimensionful) functions (allowed by symmetries) describing the scattering (x→ 1 = elastic) (See the HW) S. D. Ellis Maria Laach 2008 Lecture 1 11

2 new dimensionless functions (see the HW) in Relativistic Notation F 1 absorption of transversely polarized photons F 2 – 2 x F 1 longitudinally polarized photons (in the high energy limit) Recall: for an elementary fermion (onshell electric charge ef S. D. Ellis Maria Laach 2008 Lecture 1 ), 12

Scaling, the bj limit • Limit fixed (the “bj” or scaling limit), if there is no large hadronic scale (the hadronic physics is soft or “slow”), we naively expect , , i. e. , scaling. Plot versus x for different Q 2, Not falling off rapidly with Q 2 like form factors The proton is not filled with mush! • Interpretation – the proton is composed of essentially free, pointlike charged partons = quarks (? ) with x as the fraction of the proton’s moment carried by the scattered parton - what could be simpler! S. D. Ellis Maria Laach 2008 Lecture 1 13

Sum of individual (incoherent) quark (parton) contributions (the parton model) x. F = the momentum fraction carried by the quark, For the proton we have distribution of quarks within the proton. Callan-Gross relation (spin ½). Experimentally (approximately) true - evidence that partons are quarks (or at least fermions). S. D. Ellis Maria Laach 2008 Lecture 1 14

Flavors: SU(3) hadrons in 8’s, 10’s and 1’s Define distributions for each flavor, with valence quarks and a flavor neutral sea: Valence quarks “Ocean” quarks So that with (experimentally correct) S. D. Ellis Maria Laach 2008 Lecture 1 15

Momentum: But total momentum (in DIS) - • Only 50% of the momentum is carried by quarks, the rest is glue! • Typical parton distribution functions look like PDFs Note factors of x Note that the sea is NOT SU(3) or even SU(2) symmetric. S. D. Ellis Maria Laach 2008 Lecture 1 16

Collider Lessons • pp collisions really parton-parton • Small x is dominated by glue • SM (< 1 Te. V) Physics at the LHC is dominantly from gluon-gluon collisions -not like the Tevatron! S. D. Ellis Maria Laach 2008 Lecture 1 17

Other Processes: e+e- hadrons (final state) Think of this inclusive process in terms of section is thus . The total cross < data ! The picture looks like – again factor short and long distances At “long distances” the scattered quarks pull further quarks and anti-quarks out of the vacuum that somehow reassemble into hadrons. S. D. Ellis Maria Laach 2008 Lecture 1 18

New “long Distance” Concepts: • Jets - A jet is a “spray” of essentially collinear hadrons whose total momentum and even flavor quantum numbers track (but don’t equal) those of the fragmenting quark – the “footprint” of the quark. Based on observation, e. g. , in high energy cosmic ray collisions, that hadron collisions produce mostly hadrons in longitudinal direction, low relative k. T (reason for parton model). ■ Expect 2 jets in electron-positron annihilation. ■ One “current” jet (from scattered quark) and remnant of target in DIS. • Fragmentation - the fragmentation function Dh/q(z) describes the probability to find a hadron h in the collinear debris of the fragmenting quark q with momentum fraction z of the original quark, assuming cutoff in transverse momentum, k. T < 500 Me. V/c. Hard hadrons unlikely Soft hadrons likely Momentum conservation S. D. Ellis Maria Laach 2008 Lecture 1 19

Jets in e+e- Physics • Study the kinematics of the produced quarks by studying the kinematics of the leading hadrons – forming 2 jets. • The angular cross section for electrons to quarks, i. e. , spin ½ fermions, should track the angular distribution of the jets (or at least the leading hadrons) – and it (approximately) agrees with the data! This is another indication that the charged partons are really quarks! h = hadron, not Higgs S. D. Ellis Maria Laach 2008 Lecture 1 20

Hadron-Hadron collisions Large Transverse Momentum (Large PT) Inclusive Cross section e. g. , pp→ 0 +X z • Treat as FACTORING into 4 independent components!! [Factor short distance/large momentum from long distance/low momentum] x 1 x 2 {Dimensional analysis} [As observed (incorrectly) in 1972] S. D. Ellis Maria Laach 2008 Lecture 1 21

For Example: (More Later!) S. D. Ellis Maria Laach 2008 Lecture 1 22

Unfortunately no (known) quantum field theory has all of these properties exactly! • Fortunately QCD, an SU(3) non-Abelian gauge theory, has approximately these properties (and thus explains the observations)! • Unfortunately proving this resemblance requires the calculation of (too) many Feynman diagrams, a careful choices of gauges, a thorough understanding of the Renormalization Group, etc. , the proof has taken 30 years and still needs work. • Fortunately there are many smart people doing the hard work (including string theorists)! Big issue is that in a gauge theory there are (sometimes) relevant interactions at all momentum scales! • Fortunately the dominant dynamics is (approximately) local in momentum space and FACTORIZATION still works (for the right questions); we can approximate full dynamics as a convolution of several factors involving different momentum scales. S. D. Ellis Maria Laach 2008 Lecture 1 23

Same form – more Details – be explicit about scale dependence Short distance, UV physics, k > running coupling s( ) in perturbative calculation of Long distance, IR physics, k. T < F (collinear) scale dependent, universal PDFs and Fragmentation functions NOTE: Full Physics is independent of scale choices, scale dependences must match (order-by-order in Pert. Thy) S. D. Ellis Maria Laach 2008 Lecture 1 24

The (Classical) QCD Lagrangian (+ gauge fix + counter terms) Acting on the triplet and octet, respectively, the covariant derivative is The matrices for the fundamental (tab. B) and adjoint (TCDB) representations carry the information about the Lie algebra (f. BCD is the structure constant of the group) S. D. Ellis Maria Laach 2008 Lecture 1 25

Yang-Mills – first easy (algebraic) improvements to parton model Quarks have a previously hidden quantum number, COLOR, that comes in 3 values (quarks are a 3 under the corresponding SU(3)) so that • Color singlet ground state - meson - baryon with 3 quarks is anti-symmetric ( ) • Must sum over colors in e+e- final state factor of 3, Re+e- 2! • Extra partons holding proton together gluons, carrying the rest of the momentum (a LOCAL SU(3) symmetry), but only “small” corrections to parton model • QCD Dynamics – look at where perturbation theory is large (divergences) – UV, soft and/or collinear configurations (propagators ~ on-shell) – what is in the Monte Carlos S. D. Ellis Maria Laach 2008 Lecture 1 26

Feynman Rules: Propagators – (in a general gauge represented by the parameter �, Feynman gauge is � = 1; this form does not include axial gauges) Vertices – Quark – gluon 3 gluons S. D. Ellis Maria Laach 2008 Lecture 1 27

Feynman Rules II: 4 gluons S. D. Ellis Maria Laach 2008 Lecture 1 28

p. QCD I - Use QCD Lagrangian to Correct the Parton Model • Naïve QCD Feynman diagrams exhibit infinities at nearly every turn, as they must in a conformal theory with no “bare” dimensionful scales (ignore quark masses for now). *** First consider life in the Ultra-Violet – short distance/times or large momenta (the Renormalization Group at work): • The singular UV behavior means that theory Ø does not specify the strength of the coupling in terms of the “bare” coupling in the Lagrangian Ø does specify how the coupling varies with scale [ s( ) measures the “charge inside” a sphere of radius 1/ ] *** Typical of any renormalizable gauge field theory. We will not discuss the issue of choice of gauge. Typically axial gauges ( ) yield diagrams that are more parton-modellike, so-called physical gauges. S. D. Ellis Maria Laach 2008 Lecture 1 29

Consider a range of distance/time scales – 1/ • use the renormalization group below some (distance) scale 1/m (perhaps down to a GUT scale 1/M where theory changes? ) to sum large logarithms ln[M/ ] and ln[ /m] • use fixed order perturbation theory around the physical scale 1/ ~ 1/Q (at hadronic scale 1/m things become non-perturbative, above the scale M theory may change) renormalization group fixed order Short distance (new) renormalization group Long distance (perturbative) S. D. Ellis Maria Laach 2008 (non-perturbative) Lecture 1 30

Diagrammatically • Corrections to the parton model come from adding gluon interactions, including LO 1 Loop 2 Loops are UV divergent like dk 4/k 4 - keep (logarithmic) contributions from the range to M as a “formal” series for the effective coupling in terms of the initial coupling. S. D. Ellis Maria Laach 2008 Lecture 1 31

Interpret as screening/anti-screening of color charge in volume (1/ )3 2/ + + + S. D. Ellis Maria Laach 2008 Lecture 1 32

Sum (reorganize) the (cutoff) calculation results as an effective (renormalized) coupling** This result is more compactly specified by the renormalization group equation, which can be evaluated order-by-order in perturbation theory (PDG notation) **Masses and wave functions also exhibit renormalization. S. D. Ellis Maria Laach 2008 Lecture 1 33

Lesson: Can Sum Large Logarithms The “running” coupling illustrates typical features of QCD – • expanding to a fixed power of s is often not enough* • large logarithms (the remnants of the infinities) must be resumed to all orders by some technique • By measuring s at some scale 0 can define a dimensionful parameter QCD Dimensional transmutation !! * In any case is an asymptotic expansion, not convergent series S. D. Ellis Maria Laach 2008 Lecture 1 34

Beyond 1 -Loop The first form above is the “one-loop” solution for s (keeping only the 0 term). p. QCD allows one to systematically include the higher loop corrections, as expansion in inverse powers of ln[ ]. S. D. Ellis Maria Laach 2008 Lecture 1 35

Asymptotic Freedom/Infrared Slavery Our knowledge of the behavior in the UV is now encoded in QCD. Note that the precise value of QCD will to depend on the order of the function used (1 -loop, 2 -loop, etc. ) and the scheme. The data does not change, only the internal theoretical parameters. • Experimentally QCD ~ 216 25 Me. V (using 5 “active” flavors at the Z pole) • The running of the coupling is clear in the data, as is the precision of our knowledge of s, e. g. , s(m. Z) = 0. 1176 0. 002. Look at the (amazing!) behavior of running s – As increases, s decreases – asymptotic freedom! As decreases, s increases – infrared slavery! Just what we wanted in the parton model!!!!! S. D. Ellis Maria Laach 2008 Lecture 1 36

• s(Q) NOTE -1 • EM(Q) – only fermion loops contribute, runs the other way ( 0 EM < 0!) S. D. Ellis Maria Laach 2008 Lecture 1 37

But Note!! • Physical quantities, (Q), cannot depend on • This is essential to QCD engineering! S. D. Ellis Maria Laach 2008 Lecture 1 38

Other Potential Singularities – Infrared (after renormalizing, i. e. , removing, the UV singularities, formally with counter terms) • Soft & Collinear! (Massless) Propagators can go on shell due to emission of soft & collinear gluons – See SCET (Effective Theories) • Infrared (soft) familiar from QED – e. g. , since the photon is zero mass (in the gauge symmetric theory), theory wants to emit an infinite number of zero energy photons and the exclusive (electron) cross section diverges. Fix with inclusive cross section that sums over soft photons over 0 < E leading to Ln[ E/Q] dependence • Collinear, mq 0, still gives ln[Q/ mq] which is large for mq Q, and here we will think about mq → 0 S. D. Ellis Maria Laach 2008 Lecture 1 39

p. QCD II - Perturbative Corrections to Parton Model – e+e- Annihilation (illustrative example) • Revisit e+e‑ scattering (massless partons!) – real emission • Define handy variables (q 2=Q 2=s) S. D. Ellis Maria Laach 2008 Lecture 1 40

Sum & Square • Sum amplitudes and square, visualized (ignoring the lepton part) as the (3 -body) imaginary (absorptive) parts of the following loop diagrams (the vertical dashed line identifies the particles that are put on the mass shell, i. e. , that are the “real” particles in the final state) + + S. D. Ellis Maria Laach 2008 + Lecture 1 41

e+e- Annihilation cont’d Phase Space • The cross section looks like (see HW) Hence the singular regions are: • Collinear gluon -- 13 0, x 2 1 23 0, x 1 1 • Soft gluon – x 3 0 (x 1 1 and x 2 1) with (1 - x 1)/(1 - x 2) fixed The red singularities arise from a propagator above going onshell – either 1+3 or 2+3 ! S. D. Ellis Maria Laach 2008 Lecture 1 42

Long Distance Collinear/Soft Singularities • On-shell propagators – long distance propagation – perturbative expansion fails – 1+ s x big + s 2 x bigger … (No Surprise) • Still parton model-like picture – short distance/time simple, long distance complex. How do we proceed? • Ask questions that are insensitive to long distance structure (IR Safe) e. g. , TOT which receives contributions from all states – details cannot matter – in detail the singularities in the virtual graph (interfering with LO) cancel with those above S. D. Ellis Maria Laach 2008 + Lecture 1 43

ASIDE: Dim(emsional) Reg(ulation) • Say we want – • Consider – [4 4 -2 , Wick rotate to Euclidean space] • Calculate – S. D. Ellis Maria Laach 2008 Lecture 1 44

Simplify • Using – • Find – singular bits, plus finite bits 0, plus log singularity as m 0 • Define Scheme – subtract (absorb) 1/ , E and ln(4 ) bits**** You can hid anything in infinity! S. D. Ellis Maria Laach 2008 Lecture 1 45

p. QCD Calculation III: Apply Dim-Reg to total e+e- cross section ( ) • Real emission Numerator - Dependence of matrix element Dependence of Born S. D. Ellis Maria Laach 2008 Denominator - Dependence of phase space Lecture 1 46

Virtual • Virtual emission (interference, < 0!) • Sum and set 0, R = (e+e- hadrons)/ (e+e- + -) Parton Model (with color) NLO QCD Correction S. D. Ellis Maria Laach 2008 Lecture 1 47

Well behaved as promised ! • Finite and well behaved – more work, higher order corrections S. D. Ellis Maria Laach 2008 Lecture 1 48

Higher Orders Typical Behavior When -n Cancel • Physical quantity is INdependent !! Fixed Order p. QCD is NOT!! • p. QCD higher orders exhibit explicit ln( /Q) factors • of higher orders exhibits reduced dependence on unphysical parameter • at order sn the residual explicit ln( ) dependence is order sn+1 • dependence is an artifact of the truncation of the perturbative expansion S. D. Ellis Maria Laach 2008 Lecture 1 49

Standardize the Real – Virtual Cancellation with Concept - Infra. Red Safety!! • Define Infra. Red Safe (IRS) quantities – insensitive to collinear and soft emissions, i. e. , real and virtual emissions contribute to same value of quantity and the infinites cancel! (can really set quark masses to zero here) • Powerful tools exist to study the appearance of infrared poles (in dim reg) in complicated momentum integrals viewed as contour integrals in the complex (momentum) plane. For a true singularity the contour must be “pinched” between (at least) 2 such poles (else Cauchy will allow us to avoid the issue). We will not review these tools in detail here. • See Lecture 2 S. D. Ellis Maria Laach 2008 Lecture 1 50

Summary: • Fix issues of Parton Model with QCD! • “Physical” picture remains the same – partons at short distances, hadrons at long distances • Some changes – both coupling and distributions now vary with scale in predictable way - must be measured experimentally at some scale • Physics still factorizes into convolution of factors depending on different scales • Corrections to Parton Model are “small” for the “right” (IRS) quantities! • Next Lecture – More about calculating in QCD, IRS & Jets S. D. Ellis Maria Laach 2008 Lecture 1 51

Extra Detail Slides S. D. Ellis Maria Laach 2008 Lecture 1 52

The Big Picture of the Big Four (except Dark Energy): Interaction Observed Strength Range Carrier Theory Strong ~1 ~10 -15 m pion SM <1 1/r 2 (< 10 -15 m) Gluon Nuclear Forces EM Atomic Systems ~10 -2 1/r 2 Photon SM Weak Decays ~10 -5 ~10 -18 m W , Z SM Gravity Astronomy ~10 -39 1/r 2 Graviton SUSY Strings S. D. Ellis Maria Laach 2008 Lecture 1 53

First a Bit of History • Pre. QCD I (< ~1968) Quarks “algebraic” with no dynamics (of course, we had dynamics in the form of Regge theory, the bootstrap picture and dual models but …). Hadron quantum numbers correct if Property/Quark u d s Electric Charge +2/3 -1/3 Isospin, I 3 +1/2 -1/2 0 0 0 -1 Strangeness S. D. Ellis Maria Laach 2008 Lecture 1 54

Hadrons appear in multiplets of SU(2), e. g. , mesons S. D. Ellis Maria Laach 2008 Lecture 1 55

1970 s Include c quark - SU(4) Representations Mesons S = 0 - (↑↓) Baryons S = 1/2+(↑↑↓) S = 1 - (↑↑) S = 3/2+(↑↑↑) S. D. Ellis Maria Laach 2008 Lecture 1 56

ASIDE: Early Duality If no fundamental particles, • then particles are made out of themselves = “bootstrap” • resonances “made” in the s-channel are equivalent (dual) to resonances exchanged in the t-channel s-channel t-channel = Topologically equivalent, like stretching an elastic sheet String Theory (eventually) S. D. Ellis Maria Laach 2008 Lecture 1 57

Review Elastic Scattering of “elementary” fermions: e → e (see HW) • Kinematics S. D. Ellis Maria Laach 2008 Lecture 1 58

ASIDE: Viewed in Rest Frame Rutherford recoil spin nonflip spin flip Mott (electron on scalar) S. D. Ellis Maria Laach 2008 Lecture 1 59

ep → ep Scattering Kinematics (in p rest frame) S. D. Ellis Maria Laach 2008 Lecture 1 60

Impact Scenario in Space-Time • electron and quark interaction on short time scale (~1/Q); quarks interact on long time scale (~1/mass). Essentially free during scattering • Aside Also useful to consider the “infinite momentum frame” where P infinity, the proton is “mostly” contracted (except wee partons ~dx/x), internal interactions are “frozen” (dilated) S. D. Ellis Maria Laach 2008 Lecture 1 61

QCD: • QCD Field Theory – unbroken SU(3) • quarks in the fundamental representation – the triplet, • anti-quarks in the complex conjugate representation, • Recall that we now have 6 quarks (with masses spread over an enormous range – NOT explained by QCD – why not? ) quark u d s c b t charge 2/3 -1/3 2/3 ~ 4 Me. V ~ 7 Me. V ~ 135 Me. V mass ~1. 5 Ge. V ~ 178 Ge. V S. D. Ellis Maria Laach 2008 Lecture 1 62

Local Symmetry vector gluons in the Adjoint Representation, the 8 • Gauge Transformation coupling • Quarks have both a flavor and a color index (and spin) • Color singlet states (1) with no indices - just what we wanted!! - mesons - baryons S. D. Ellis Maria Laach 2008 Lecture 1 63