Quantum Chromodynamics, Colliders & Jetsstaff.washington.edu/sdellis/QCD08Lect_1.pdfQuantum Chromodynamics, Colliders & Jets Stephen D. Ellis University of Washington Maria Laach September

Quantum

Chromodynamics,

Colliders & Jets Stephen D. Ellis

University of Washington

Maria Laach September 2008

Lecture 1: The Big Picture – How do we Think About and Calculate processes at Colliders?The Parton Model + QCD

ASIDE: I am here to provide historical context, i.e.,

I can remember particle physics before QCD, jets

or colliders!

S. D. Ellis Maria Laach 2008 Lecture 1 2

Historical art

work by Siggi

© S. Bethke June 1993


Outline

1. Introduction – The Big Picture

pQCD - e+e- Physics and Perturbation Theory

(the Improved Parton Model);

pQCD - Hadrons in the Initial State and PDFs

2. pQCD - Hadrons and Jets in the Final State

3. Colliders & Jets at Work


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 – Z0, W+, W- (massive)

* Of course, in 3 hours we won’t really cover everything – actually nearly nothing in detail!


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()pe+hadrons, pphadrons (jets), pp+X, pp+-+X, ppB(eyond the)SM

SMSM


(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 PiTP 2007 Lectureshttp://www.phys.washington.edu/users/ellis/PiTP%20July%2007.htm

• My TSI 2006 lectures -http://www.phys.washington.edu/users/ellis/TSI%20July%2006.htm

• “Jets in Hadron-Hadron Collisions” by S.D. Ellis, J. Huston, K. Hatakeyama, P. Loch, M. Toennesmann, arXiv:hep-ph/0712.2447v1

• The “Primer for LHC Physics” by J.M. Campbell, J.W. Huston, W.J. Stirling, arXiv:hep-ph/0611148v1

• Lectures by George Sterman, et al. – (for more references in formal details) arXiv:hep-ph/0807.5118v1, arXiv:hep-ph/0412013v1, arXiv:hep-ph/0409313v1

http://www.phys.washington.edu/users/ellis/TSI July 06.htm



http://arxiv.org/abs/hep-ph/0611148v1







http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.5118v1.pdf










(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

http://pdg.lbl.gov/2008/reviews/qcdrpp.pdf

http://pdg.lbl.gov/2008/reviews/qcdrpp.pdf

http://www.phys.psu.edu/~cteq/

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 (based on experiment – no YM theory)

• Interact via scale invariant, perturbative (weak) interactions dominated by

exchanges with momenta k ≤ m ~ 1 GeV (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 zS. 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’


Recall the EXclusive case: ep → ep

= p’s anomalous magnetic moment,

2 2

1 2

22 2 2 2

22

2

2

1 1

4 21

4

p

E M M E

p p

p

J ieu p F q i q F q u pm

qieu p G q G q i q G q G q u p

qm m

m

General EM vertex (see HW) symmetries 2 functions

22 20

22 2 2 2 2

0 0~ 0.7 GeV

0 1, 0 1, 1

,

i E M

i j q q

F G G

F q G q q q q

Form Factors – The harder you hit a proton, the more likely it is to fall apart !

Electric Form Factor Magnetic Form Factor

S. D. Ellis Maria Laach 2008

Lecture 1

11

Now the INclusive case: ep→eX

In the proton rest frame (Lab) the

kinematics look like:

22 2

2 2

2 2

2

2 2

2

2

2 1

1

X

p p

p p

p

M W p q

q p q p

m m Q

m x m

Q m

2 2

,0 , , , ,

1

2 2

1

p

p

p p

p m k E k k E k

p qE E

m

q Qx

m m E E

q p Ey

k p E

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)

2 2

22 2 2 2

1 22 4

2sin , cos ,2 2

4 sin2

L L

Q qLL ep ep

dW Q W Q

dE d E


Lecture 1

12

2 new dimensionless functions (see the

HW) in Relativistic Notation

2

2 2 2 2

1 2 12 4

4 11 1 , , 2 ,

d yy F x Q F x Q xF x Q

dxdQ Q x

F1 absorption of transversely polarized photons

F2 – 2x F1 longitudinally polarized photons (in the high energy limit)

Recall: for an elementary fermion (onshell ), electric charge ef

2 2p q p

22

2

2 4

4| 1 1 1

2

f

f

edy x

dxdQ Q

2 2 2 2

1 1 2 2, , , , ,F x Q mW Q F x Q W Q


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.

2, ;Q x

2 2 2,i i iF x Q F x m Q F x

• Interpretation – the proton is composed of essentially free, point-like charged partons = quarks (?) with x as the fraction of the proton’s moment carried by the scattered parton - what could be simpler!

Plot versus x for different Q2,

Not falling off rapidly with Q2 like form factors

The proton is not filled with mush!


Sum of individual (incoherent) quark (parton)

contributions (the parton model)

xF = the momentum fraction carried by the quark,

For the proton we have

2

2 1ˆ ˆ2bj bj q bj bj FF x xF x e x x x

2

2 1 /

,

2 q q p

q q

F x xF x e xF x

/q pF x q x distribution of quarks within the proton (factored from rest of the event).

1 22xF x F x Callan-Gross relation (spin ½).

Experimentally (approximately) true - evidence that partons are quarks (or at least fermions).

,q Fp x p 2 2bjx Q m


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:

/ /;u p V d p VF x u x u x S x F x d x d x S x

S x u x d x s x s x

So that

2

4 1

9 9F x x u x u x d x d x s x s x

with (experimentally correct)

1 1

0 0

2; 1V Vdxu x dxd x

Valence quarks

“Ocean” quarks


Momentum:

But total momentum (in DIS) -

1 1

0 0

6 0.5V V dataq

dxx q x q x dxx u x d x S x

• 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.

Collider Lessons

• pp collisions really parton-parton

• Small x is dominated by glue

• SM (< 1 TeV) Physics at the LHC is dominantly

from gluon-gluon collisions

-not like the Tevatron!



Other Processes: e+e- hadrons (final state)

Think of this inclusive process in terms of . The total cross

section is thus

e e qq

22 2

0 2

4 2;

3 3q qe e

q q

e e qqe R e

Q e e

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.

< data !


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-hadron collisions produce mostly hadrons in longitudinal direction, low relative kT (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, kT < 500 MeV/c.

1

0

1h q

h

dz zD z 1

~

h qn

h q h q

zD z N

z

Hard hadrons unlikely

Soft hadrons likelyMomentum conservation


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) –

2

2 2

21 cos

cos 2q

q

de

d Q

and it (approximately) agrees with the data! This is another indication that the charged partons are really quarks!

,

ˆ ˆ2 2

h q h q

qh

h h h h h

q jet jet

de e hX e e qq D z D z

dz

E E p j E p jz

E Q Q E p

h = hadron, not Higgs

1 2 1 22 2 2

2

2

4 0

2

ˆ~ , , , ,

1

T

T T T

T

T

T

d d sF x F x p x x z D z

dp dp p

m

p

dp pp X

dp


Hadron-Hadron collisions Large Transverse

Momentum (Large PT) Inclusive Cross section

• Treat as FACTORING into 4

independent components!!

[Factor short distance/large

momentum from long

distance/low momentum]

e.g., pp→0 +X

[As observed (incorrectly) in 1972]

x1

x2

z

{Dimensional analysis}


For Example: (More Later!)

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.


Same form – more Details – be explicit about scale

dependence


2

1 2 1 22 2 2 2

2

1 22 2

2

/

ˆ~ , , , , 1

,

, ,

ˆ 1,, , , , ,

pp

a p a F b p b F

T

ab

T

T T T T

T TT

T

c c F

F T

dF x F x

d d s mF x F x p x x z D z

dp dp p p

p ps mp x x z

p

dp

D zp

Short distance, UV physics, k > running coupling s() in perturbative calculation of ̂

Long distance, IR physics, kT < 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 PertThy)

20

T

d d

d dp


The (Classical) QCD Lagrangian (+ gauge fix + counter

terms)

Acting on the triplet and octet, respectively, the covariant derivative is

,

, ,

1

4

B

QCD B f a f f babf

B B B BCD C D

L F F q iD m q

F A A gf A A

;C C B B

ab CDab CDab CDD ig t A D ig T A

The matrices for the fundamental (tabB) and adjoint (TCD

B) representations carry the information about the Lie algebra

, ; , ; ;

4; ;

2 3

3

B C BCD D B C BCD D B BCD

CD

BCB C BC B B

R ab bc ac F ac

B C BC BC

A

t t if t T T if T T if

Tr t t T t t C

Tr T T C

(fBCD is the structure constant of the group)


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

a aq q

abc a b cq q q


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


Feynman Rules II:

4 gluons


Lecture 1

29

pQCD 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” dimensionfulscales (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 the theory

does not specify the strength of the coupling in terms of the “bare” coupling in the Lagrangian

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-model-like, so-called physical gauges.

ˆ 0n A


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 the theory may change)

Short distance Long distance

(non-perturbative)(perturbative)(new)


Diagrammatically

• Corrections to the parton model come from

adding gluon interactions, including

LO

1 Loop2 Loop

Loops are UV divergent like dk4/k4 - keep (logarithmic) contributions from the range to M as a “formal” series for the effective coupling in terms of the initial coupling.

22 2

2 2 30 0

2 2

0

~ ln ln4 4

11 2 211

3 3

s s s s

A f

f

M M MM M

C nn


Interpret as screening/anti-screening of color

charge in volume (1/)3

2/ + +

+


This result is more compactly specified by the renormalization

group equation, which can be evaluated order-by-order in

perturbation theory (PDG notation)

Sum (reorganize) the (cutoff) calculation results as an effective (renormalized) coupling**

02

02

11 2 2; 11

3 31 ln

4

A fs

s f

s

C nMn

MM

**Masses and wave functions also exhibit renormalization.

2 3 4

0 1 22 3

1

2

2

22 4 64

1951

3

5033 3252857

9 27

s s s ss

f

f f

d

d

n

n n


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

0 0

2

0

0

2

ln

s

s QCD

QCD

e

Dimensional transmutation !!

* In any case is an asymptotic expansion, not convergent series


Lecture 1

35

The first form above is the “one-loop” solution for s (keeping only the 0 term). pQCD allows one to systematically include the higher loop corrections, as expansion in inverse powers of ln[].

2

2

1

22 20

2 20

22 2

1 2 02 22

4 2 120

ln ln4 2

1

ln ln

4 1 5ln ln

2 8 4ln

QCD

s

QCD QCD

QCD

QCD

Beyond 1-Loop


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 ~ 21625 MeV (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(mZ) = 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(Q)

• EM(Q) – only fermion

loops contribute, runs the

other way (0EM < 0!)

NOTE -1


Lecture 1

38

But Note!!

• Physical quantities, (Q), cannot depend on

• This is essential to QCD engineering!

, , 0s

d Q

d m


Lecture 1

39

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), the 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 < 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

2 21 2

22 1 or 2

3 1 2 0

1 2 1 2

0, soft0

, 0, collineark k

kk k k

k k k k


pQCD II - Perturbative Corrections to Parton

Model – e+e- Annihilation (illustrative example)

• Revisit e+e- scattering (massless partons!) – real emission

• Define handy variables (q2=Q2=s)

3

221

2

2 20 2

2 11 cos

2

i ii i i

i

i j k k

ij

i j i j i j

E p qx x x

qq

p p q p x

E E E E x x


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)

+ + +

2 2

1 20

1 2 2 3 1 3

2 2 221 2

0 0

1 2

2

43

2 1 1 3

sF

sF f

f

E EdC s

dx dx p p p p

x xC e

x x s


e+e- Annihilation cont’d

• The cross section looks like (see

HW)

Hence the singular regions are:

• Collinear gluon -- 130, x21

230, x11

• Soft gluon – x30 (x11 and

x21) with (1- x1)/(1- x2) fixed

2 2

1 20

1 2 1 22 1 1

sF

d x xC

dx dx x x

Phase Space

The red singularities arise from a propagator above going on-shell – either 1+3 or 2+3 !


Long Distance Collinear/Soft Singularities

• On-shell propagators – long distance propagation – perturbative

expansion fails – 1+ s x big + s2 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

+


ASIDE: Dim(emsional) Reg(ulation)

• Say we want –

• Consider – [44-2, Wick rotate to Euclidean space]

• Calculate –

4

4 22 2

1

2

d kI

k m

4 4 22 2 3 24 2

4 4 2 4 22 2 2

E E

d k d k ddk k

4 2

4 2 2

2 13 22 11

2 22 2

0 0

2

2 1

22 4

12

21

2 2

EE

E

d

kdk dzz z

mk m

m


Simplify

• Using –

• Find –

singular bits, plus finite bits 0, plus log singularity as m0

• Define Scheme – subtract (absorb) 1/ , E and ln(4) bits****

1 ; 1 0.5772Ez z z

2

2 20

1 1ln 4 2ln

4 4EI O

mm

MS

**** You can hid anything in infinity!


pQCD Calculation III: Apply Dim-Reg to total

e+e- cross section ( )

• Real emission

2 2

1 2 1 2

0 1 2 1 1

1 2 1 2

2

0 2

2

1 2 12

2 1 1 1

2 3 19

2 2

3 1 41

3 2 2 2

qqg F s

F s

x x x xCH dx dx

x x x x

CH O

H O

Dependence of Born

Numerator - Dependence of matrix element

Denominator - Dependence of phase space

4 2D


Lecture 1

47

Virtual -

• Virtual emission (interference, < 0!)

• Sum and set 0,

R = (e+e-hadrons)/(e+e-+-)

2

0 2

2 38

2

qq g F sCH O

0

2

2 20

2

33

3

4

1

1

f

f

qq gqqg

F s

f s

f

s

s

CR e

e

O

O

Parton Model (with color)

NLO QCD Correction


Well behaved as promised !

• Finite and well behaved – more work, higher order corrections

22

2 2

32 2

2

2 2

4

1 1.4092 1.9167ln3

12.805 7.8179ln 3.674ln

s s

f

f

s

s

RK

e Q

Q Q

O


Higher Orders Typical Behavior When -n Cancel -

• Physical quantity is INdependent !!Fixed Order pQCD is NOT!!

• pQCD 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 s

n+1

• dependence is an artifact of the truncation of the perturbative expansion


Standardize the Real – Virtual Cancellation with

Concept - InfraRed Safety!!

• Define InfraRed Safe (IRS) quantities – insensitive to collinear and

soft emissions, i.e., real and virtual emissions contribute to same

value of quantity and the infinites can 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


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


Extra Detail Slides


Lecture 1

53

The Big Picture of the Big Four (except Dark

Energy):

Interaction Observed Strength Range Carrier Theory

Strong Nuclear

Forces

~1

<1

~10-15 m

1/r2 (< 10-15 m)

pion

Gluon

SM

EM Atomic

Systems

~10-2 1/r2 Photon SM

Weak Decays ~10-5 ~10-18 m W, Z SM

Gravity Astronomy ~10-39 1/r2 Graviton SUSY

Strings


Lecture 1

54

First a Bit of History

• PreQCD 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 baryon, meson,

approximate FLAVOR SU(3)

qqq qq

Property/Quark u d s

Electric Charge +2/3 -1/3 -1/3

Isospin, I3 +1/2 -1/2 0

Strangeness 0 0 -1


Hadrons appear in multiplets of SU(2),

e.g.,mesons

1 1 1 12 3 2 3

23

.

, ,

0,

z

z

Y S B I Y

d u

I

s

3

23

1 1 1 12 3 2 3

,

0,

, ,

z

z

Y I Y

s

I

u d

3

0 0 1 12 2

0 01 12 2

1

,1 ,1

2 0,0

11,0 1,02 6

00,02

, 1 , 1

z

Y

K K ds K K us

uu dd

Idu ud Iuu dd ss

I

uu dd

K K su K K sd

-8 0

3uu dd ss

ss

1


1970s Include c quark - SU(4) Representations

Mesons Baryons

S = 0- (↑↓)

S = 1- (↑↑)

S = 1/2+(↑↑↓)

S = 3/2+(↑↑↑)


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

t-channels-channel

=

Topologically equivalent, like stretching an elastic sheet String Theory (eventually)


Lecture 1

58

Review Elastic Scattering of “elementary”

fermions: e → e (see HW)

• Kinematics

2 2 2 2 2 2

2 2

2 22

2 2

2 2

2 2 2 2

2 2

,

2 2

1

e

e

e e

e

p p m k k m

s p k p k

t q k k p p

s m m

u k p k p

u

y

ym m s m m

s t u m m

2 2

2 2 2 22

2 22 2, 2 2

2 41 1

es m m

e e

d s uy

dq sq q


Lecture 1

59

ASIDE: Viewed in Rest Frame

2 2

rest frame: ,0 , , , , ,

, 4 sin2

L

p m k E k k E k

E Ey q EE

E

2 2

22 4

2,2 4

rest frame

2 2 22

22 4

2cos sin

2 24 sin

2

cos2 1 tan

224 sin2

e

L Ls m m

LLab

L

L

L

d E EE

d E mE

E q

E mE

spin flipRutherford spin nonfliprecoil

Mott (electron on scalar)


ep → ep Scattering

2 2

2 2 222 2 2

1 2 1 22 22 4

22 2 2 2

22 22 2 2 2

2 22 4

2

cos2 tan

24 24 sin2

4cos sin

2 224 sin 124

L

L

LL p pep ep

E M

p L LM

L p

pQ q

d E t tF t F t F t F t

d E m mE

QG Q G Q

mE QG Q

QE mEm

Kinematics (in p rest frame)

22 2

22 2 2

,0 , , , ,

4 sin ,2

p

L

p

p m k E k k E k

E Eq k k EE y

E

m p p p q


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 Pinfinity, the proton is “mostly” contracted (except wee partons ~dx/x), internal interactions are “frozen” (dilated)


Lecture 1

62

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 -1/3 2/3 -1/3 2/3

mass ~ 4 MeV ~ 7 MeV ~ 135

MeV

~1.5 GeV ~ 5 GeV ~ 178 GeV


Local Symmetry vector gluons in the Adjoint

Representation, the 8

• Gauge Transformation coupling

• Quarks have both a flavor and a color index (and spin) - ,f aq

• Color singlet states (1) with no indices - just what we wanted!!

- mesons

- baryons

a aq q

abc a b cq q q

Documents

Quantum Chromodynamics, Colliders & Jetsstaff.washington.edu/sdellis/QCD08Lect_1.pdfQuantum Chromodynamics, Colliders & Jets Stephen D. Ellis University of Washington Maria Laach September