Introduction to Perturbative QCD—Foundation and Simple Applications

CTEQ Summer School, 2006Wu-Ki Tung

Pavel Nadolsky, Introduction to QCD and parton model

Module 2: Basic concepts of QCDBased on CTEQ School lectures by Wu-Ki Tung and George Sterman

Basic Elements of Quantum Chromodynamics (QCD)– a Non-abelian Gauge Field Theory with

SU(3) color Gauge Symmetry

Interaction vertices

Experimental Foundation of QCD I Long Distance Physics: Hadron Spectroscopy

and the Constituent Quark Model (1960’s)

Pseudo-scalar mesons Vector mesons

cf. PDG

Experimental Foundation of QCD I Long Distance Physics: Hadron Spectroscopy

and the Constituent Quark Model (1960’s)

“Octet” Baryons “Decuplet” baryons

Experimental Foundation of QCD II Short Distance Physics: Deep Inelastic Scattering, e+e- Annihilation, and the Parton Model (~1969-72)

Two-jet Events in e+e- Annihilation— Evidence for Quark – Anti-quark Production

A typical event in e+e- → hadron final state

Feynman diagram CM configuration

Partons are Point-like

Modern day “Rutherford Scattering”:In high energy inclusive e+e- annihilation and DIS, cross sections are hard (no form-factor like drop off for large Q)

First SLAC results on DIS (~ 1969)

Q2

elastic Sc. (FF)

DISσ / σpt

ECM (“Q”)

σ/σpt (pre-LEP)

Experimental Foundation of QCD II Short Distance Physics: Deep Inelastic Scattering, e+e- Annihilation, and the Parton Model (~1969-72)

Measuring the Spin of the Quarks

|spin ½

spin 0

•

• In DIS: FL ~ 0 (Callan-Gross) ⇔ σ ~ (1 + cosh2ψ) (cf. later)

Quarks are spin ½ fermions

Measuring the Spin of the Gluon

Very important question: (central to PQCD, and to this course)

How could the simple parton picture (with almost non-interacting partons) possibly hold in QCD (—a strongly interacting quantum gauge field theory)?

Answer: 3 distinctive Features of QCD

Asymptotic Freedom: A strongly interacting theory at long-distances (even confining) can become weakly interacting at short distances (due to scale dependence implied by the RGE).Infra-red Safety: There are classes of “infra-red safe” (IRS) quantities which are independent of long-distance physics, hence are calculable in PQCD.Factorization: There are an even wider class of physical quantities (inclusive cross sections) which can be factorized into long distance components (not calculable, but universal) & short-distance components (process-dependent, but infra-red safe, hence calculable).

The bulk of this course is devoted to exploring the ideas behind these features of QCD.

Analogies and correspondences (see later)

(Planck Scale)

Ultra-violet Renormalization and Asymptotic Freedom —the smallest time and shortest distances

t

x

What does renormalization do?Say, MS renormalization introduces a ren. scale μR . In principle, μR is arbitrary; in practice, μR is chosen ~ a physical scale Q, or √s .

Renormalization Group and the Running Coupling

The μ dependence of α(μ) is controlled by the renormalization group equation :

Solution of the RGE to 1-loop order sums leading quantum fluctuations to all orders of the fixed-coupling perturbative expansion.

β > 0 ⇒ α(μ) decreases as μ increases—QCD is asymptotically free.

How is αs(μ) measured in the variety of hadronic processes listed in the previous slide?

In general, how can one relate PQCD calculations (on leptons, quarks and gluons) to physical observables measured in the lepton-hadron world?

Answer: (i) IRS; and (ii) Factorization …

Order αs0 (LO) process and the Parton Model;

Order αs1 perturbative calculation

(NLO QCD correction):• Colinear and Soft Singularities;• Infra-red Safe (IRS) Physical Observables;• Factorization of (non-IRS) Physical

Observables into IRS (short-distance, calculable) and Universal (long-distance) pieces.

II: PQCD at Work: e+e- Annihilation

e+e- Annihilation into Hadrons:Leading Order in PQCD

Total cross section, normalized to point-like cross section:

Angular distribution:

qf2

e+e- Annihilation into Hadrons:Next to Leading Order (NLO) in PQCD

Differential cross section at the parton level:

Kinematics: Tree diagram(real gluon emission)

2

diverges when: xi → 1 (colinear)xi → 0 (soft)

Colinear and Soft Singularities

In both configurations, the virtual propagator line goes on mass shell: k2 → 0

x1 + x2 + x3 = 2

2

Colinear and Soft Singularities

In general (and for theory students):

cf. TASI lecture notes of Sterman

Introduction to the Parton Model and Pertrubative QCD

George Sterman, YITP, Stony Brook

CTEQ summer school, May 30, 2007

U. of Wisconsin, Madison

II. From the Parton Model to QCD

1. Color and QCD

2. Field Theory Essentials

3. Infrared Safety

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3. Infrared Safety

• To use perturbation theory, would like to choose µ ‘as largeas possible to make αs(µ) as small as possible.

• But how small is possible?

• A “typical” cross section, , define Q2 = s12 and xij =

sij/Q2,

σ

Q2

µ2, xij,

m2i

µ2, αs(µ), µ

=∞∑n=1

an

Q2

µ2, xij,

m2i

µ2

αns (µ)

with m2i all fixed masses – external, quark, gluon (=0!)

• Generically, the an depend logarithmically on their arguments,so a choice of large µ results in large logs of m2

i/µ2.

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• But if we could find quantities that depend on m′is only

through powers, (mi/µ)p, p > 0, the large-µ limit wouldexist.

σ

Q2

µ2, xij,

m2i

µ2, αs(Q), µ

= σ

Q

µ, xij,

m2i

µ2, αs(µ), µ

=∞∑n=1

an

Q

µ, xij

αns (µ) + O

m2i

µ2

p

• Such quantities are called infrared (IR) safe.

• Measure σ → solve for αs. Allows observation of the runningcoupling.

• Most pQCD is the computation of IR safe quantities.

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• To analyze diagrams, we generally think of m → 0 limit inm/Q.

• Generic source of IR (soft and collinear) logarithms:

p

αp

• IR logs come from degenerate states:Uncertainty principle ∆E → 0 ⇔ ∆t → ∞.

• For soft emission and collinear splitting it’s “never too late”.But these processes don’t change the flow of energy . . . Theproblem is asking for particle content.

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• For IR safety, sum over degenerate final states in perturbationtheory, and see what the sum is. This requires us to introduceanother regularization, this time for IR behavior.

• The IR regulated theory is like QCD at short distances, butis better-behaved at long distances.

IR-regulated QCD not the same as QCD except for IR safequantities.

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• See how it works for the total e+e− annihilation cross sectionto order αs. Lowest order is 2 → 2, σ

(0)2 ≡ σ0, σ3 starts at

order αs.

– Gluon mass regularization: 1/k2 → 1/(k2 −mG)2

σ(mG)3 = σ0

4

3

αs

π

2 ln2 Q

mg− 3 lnQmg −

π2

6+

5

2

σ(mG)2 = σ0

1 −4

3

αs

π

2 ln2 Q

mg− 3 ln

Q

mg−π2

6+

7

4

which gives

σtot = σ(mG)2 + σ

(mG)3 = σ0

1 +αs

π

– Pretty simple! (Cancellation of virtual (σ2) and real (σ3) gluon diagrams.)

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– Dimensional regularization: change the area of the sphere

from 4πR2 to (4π)(1−ε) Γ(1−ε)Γ(2(1−ε))R

2−2ε with ε = 2−D/2in D dimensions.

σ(ε)3 = σ0

4

3

αs

π

(1 − ε)2

(3 − 2ε)Γ(2 − 2ε)

4πµ2

Q2

ε

×1

ε2−

3

2ε−π2

2+

19

4

σ(ε)2 = σ0 [1 −

4

3

αs

π

(1 − ε)2

(3 − 2ε)Γ(2 − 2ε)

4πµ2

Q2

ε

×

1

ε2−

3

2ε−π2

2+ 4

]

which gives again

σtot = σ(mG)2 + σ

(mG)3 = σ0

1 +αs

π

• This illustrates IR Safety: σ2 and σ3 depend on regulator,but their sum does not.

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• Generalized IR safety: sum over all states with the sameflow of energy into the final state. Introduce IR safe weight“e(pi)”

dσ

de=

∑n

∫PS(n) |M(pi)|2δ (e(pi) − w)

with

e(. . . pi . . . pj−1, αpi, pj+1 . . .) =

e(. . . (1 + α)pi . . . pj−1, pj+1 . . .)

• Neglect long times in the initial state for the moment andsee how this works in e+e− annihilation: event shapes andjet cross sections.

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• “Seeing” Quarks and Gluons With Jet Cross Sections

• Simplest example: cone jets in e+e− annihilation!

"Q

• Intuition: eliminating long-time behavior ⇔ recognize theimpossibility of resolving collinear splitting/recombination ofmassless particles

• No factors Q/m or ln(Q/m) Infrared Safety.

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• In this case,

σ2J(Q, δ, ε) =3

8σ0(1 + cos2 θ)

×1 −

4αs

π

4 ln δ ln ε+ 3 ln δ +π2

3+

5

2

• Perfect for QCD: asymptotic freedom → dαs(Q)/dQ < 0.

• No unique jet definition. ↔ Each event a sum of possiblehistories.

• Relation to quarks and gluons always approximate but correc-tions to the approximation computable.

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• The general form of a jet cross section:

σjet = σ0∞∑n=0

cn(yi, N,CF )αns (Q)

• Choices for yi: δ, Ωjet, T, ycut, . . .

• δ, cone size; Ω, jet direction

• Shape Variable, e.g. thrust (T = 1 for “back-to-back” jets

T =1

smaxn

∑i|n · ~pi|

• ycut Cluster Algorithm: yij > ycut,

yij = 2minE2i , E

2j

(1 − cos θij

)

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Summarize: what makes a cross section infrared safe?

• Independence of long-time interactions:

p

αp

More specifically: should depend on only the flow of energyinto the final state. This implies independence of collinearre-arrangements and soft parton emisssion.

But if we prepare one or two particles in the initial state (as inDIS or proton-proton scattering), we will always be sensitiveto long time behavior inside these particles. The parton modelsuggests what to do: factorize. This is the subject of Part III.

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