[PPT] - QCD Phenomenology at High Energy Bryan Webber CERN Academic PowerPoint Presentation

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QCD Phenomenology at High Energy

Bryan Webber CERN Academic Training Lectures 2008

Lecture 4: Jet Fragmentation and Hadron-Hadron Processes

Jet Fragmentation

❖ Fragmentation functions ❖ Small-x fragmentation ❖ Average multiplicity

Hadronization Models

❖ General ideas ❖ Cluster model ❖ String model

Hadron-Hadron Processes

❖ Parton-parton luminosities ❖ Lepton pair, jet and heavy quark production ❖ Higgs boson production

Survey of NLO Calculations for LHC

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Jet Fragmentation

Fragmentation functions F h

i (x, t) gives distribution of momentum fraction x for hadrons

f type h in a jet initiated by a parton of type i, produced in a hard process at scale t.
Like parton distributions in a hadron, Dh

i (x, t), these are factorizable quantities, in which

infrared divergences of PT can be factorized out and replaced by experimentally measured factor that contains all long-distance effects.

In e+e− annihilation, for example, the hard process is e+e− → q¯

q at scale equal to c.m. energy squared s; distribution of x = 2ph/√s is (for s ≪ M2

Z)

dσ dx = 3σ0 X

q

Q2

q

n F h

q (x, s) + F h ¯ q (x, s)

where σ0 is e+e− → µ+µ− cross section.
Fragmentation functions satisfy DGLAP evolution equation

t ∂ ∂tF h

i (x, t) =

X

j

Z 1

x

dz z αS 2πPji(z, αS)F h

j (x/z, t) .

Splitting functions Pji have perturbative expansions of the form Pji(z, αS) = P (0)

ji (z) + αS

2πP (1)

ji (z) + · · · 1

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Leading terms P (0)

ji (z) were given earlier. Notice that splitting function is Pji rather than

Pij since F h

j represents fragmentation of final parton j.

Solve DGLAP equation by taking moments as explained for DIS. As in that case, scaling

violation is clearly seen.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 1/σtot dσ/dx × c(flavour) 0.01 0.03 0.1 0.3 1 3 10 30 100 300

√s=91 GeV

107 107 107 107 L E P , S L C : a l l f l a v

u

r s 107 106 106 L E P , S L C : U p , D

w

n , S t r a n g e 106 105 105 L E P , S L C : C h a r m 105 104 104 L E P , S L C : B

t

t

m

104 103 103 103 L E P : G l u

n

(❍ 3-jet, ✶ FL,T)

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1

1 10 10 2 10 3 10

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x = 0.1 - 0.2 x = 0.2 - 0.3 x = 0.3 - 0.4 x = 0.4 - 0.5 x = 0.5 - 0.7 DELPHI ALEPH AMY TASSO 14 22 35 44 GeV CELLO MARKII

Q2(GeV2) 1/σtot (dσ/dx)

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Small-x fragmentation

Evolution of fragmentation functions at small x sensitive to moments near N = 1.

However, anomalous dimensions γ(0)

gq , γ(0) gg are not defined at N = 1: moment integrals

for N ≤ 1 are dominated by small z, where Pgi(z) diverges due to soft gluon emission.

At small z must take into account coherence effects. Recall evolution variable becomes

˜ t = E2[1 − cos θ], with angular ordering condition ˜ t′ < z2˜

t. Thus, redefining t as ˜

t, evolution equation in integrated form is Fi(x, t) = Fi(x, t0) + X

j

Z 1

x

dz z Z z2t

t0

dt′ t′ αS 2πPji(z)Fj(x/z, t′)

r in differential form

t ∂ ∂tFi(x, t) = X

j

Z 1

x

dz z αS 2πPji(z)Fj(x/z, z2t) .

Only difference from DGLAP equation is z-dependent scale on the right-hand side — not

important for most values of x but crucial at small x.

For simplicity, consider first αS fixed and neglect sum over j. Taking moments as usual,

t ∂ ∂t ˜ F (N, t) = αS 2π Z 1

x

dz zN−1P (z) ˜ F (N, z2t) .

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❖ Try solution of form F (N, t) ∝ tγ(N,αS). Then anomalous dimension γ(N, αS) must satisfy γ(N, αS) = αS 2π Z 1 zN−1+2γ(N,αS)P (z) . ❖ For N − 1 not small, we can neglect 2γ(N, αS) in exponent and obtain usual formula for anomalous dimension. For N ≃ 1, z → 0 region dominates, where Pgg(z) ≃ 2CA/z. Hence γgg(N, αS) = CAαS π 1 N − 1 + 2γgg(N, αS) = 1 4 "r (N − 1)2 + 8CAαS π − (N − 1) # = r CAαS 2π − 1 4(N − 1) + 1 32 s 2π CAαS (N − 1)2 + · · ·

To take account of running αS, write

˜ F (N, t) ∼ exp »Z t γgg(N, αS)dt′ t′ – ,

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and note that γgg(N, αS) should be γgg(N, αS(t′)). Use Z t γgg(N, αS(t′))dt′ t′ = Z αS(t) γgg(N, αS) β(αS) dαS , where β(αS) = −bα2

S + · · ·, to find

˜ F (N, t) ∼ exp " 1 b s 2CA παS − 1 4bαS (N − 1) + 1 48b s 2π CAα3

S

(N − 1)2 + · · · #

αS=αS(t)

.

In e+e− annihilation, scale t ∼ s and behaviour of ˜

F (N, s) near N = 1 determines form of small-x fragmentation functions. Keeping terms up to (N − 1)2 in exponent gives Gaussian function of N which transforms into Gaussian function of ξ ≡ ln(1/x): xF (x, s) ∝ exp » − 1 2σ2(ξ − ξp)2 – ,

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Width of distribution

σ = 1 24b s 2π CAα3

S(s)

!1

2

∝ (ln s)

3 4 .

1 2 3 4 5 6 7 8 1 2 3 4 5 6 ξ=ln(1/xp) 1/σ dσ/dξ

ZEUS* 10-20 GeV2 H1* 12-100 GeV2 ZEUS* 40-80 GeV2 ZEUS* 80-160 GeV2 H1* 100-8000 GeV2

DIS:

TASSO 22 GeV TASSO 35 GeV TASSO 44 GeV TOPAZ 58 GeV LEP 91 GeV LEP 133 GeV LEP 189 GeV LEP 206 GeV

e+e−: 6

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Peak position

ξp = 1 4bαS(s) ∼ 1 4 ln s 1 1.5 2 2.5 3 3.5 4 4.5 2 3 5 7 10 20 30 50 70 100 200 √s [GeV] ξp

ZEUS H1 DIS: BES TASSO MARK II TPC CELLO AMY ALEPH DELPHI L3 OPAL e+e−: MLLA QCD fit without coherence

Energy-dependence of the peak position ξp tests suppression of hadron production at small

x due to soft gluon coherence. Decrease at very small x is expected on kinematical grounds, but this would occur at particle energies proportional to their masses, i.e. at x ∝ m/√s, giving ξp ∼ 1

2 ln s. Thus purely kinematic suppression would give ξp increasing twice as

fast.

In p¯

p → dijets, √s is replaced by MJJ sin θ where MJJ is dijet mass and θ is jet cone angle.

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θ

Mjj=82 GeV

Mjj=105 GeV
Mjj=140 GeV
Mjj=183 GeV
Mjj=229 GeV
Mjj=293 GeV
Mjj=378 GeV
Mjj=488 GeV
Mjj=628 GeV
x=log( )

_ x 1

1

dN
dx

Nevent

CDF Preliminary

CDF Preliminary

Qeff = 256 13 MeV
+
MLLA Fit:

Fragmentation without color coherence L e a d i n g L

g

A p p r

x

i m a t i

n

(CDF Data only)

Mjj sinq (GeV/c

2 ) Peak position x

=ln(1/x
)

CDF M jj =80-630 GeV/c 2 , cone 0.47 CDF M jj =80-630 GeV/c 2 , cone 0.36 CDF M jj =80-630 GeV/c 2 , cone 0.28 e + e

and e

+ p Data

_

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Average Multiplicity

Mean number of hadrons is N = 1 moment of fragmentation function:

n(s) = Z 1 dx F (x, s) = ˜ F(1, s) ∼ exp 1 b s 2CA παS(s) ∼ exp s 2CA πb ln „ s Λ2 « (plus NLL corrections) in good agreement with data.

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Hadronization Models

General ideas

Local parton-hadron duality

❖ Hadronization is long-distance process, involving small momentum transfers. Hence hadron-level flow of energy-momentum, flavour should follow parton level. ❖ Results on spectra and multiplicities support this.

Universal low-scale αS

❖ PT works well down to very low scales, Q ∼ 1 GeV. ❖ Assume αS(Q) defined (non-perturbatively) for all Q. ❖ Good description of heavy quark spectra, event shapes.

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Specific models

General ideas do not describe hadron formation. Main current models are cluster and string.
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Cluster (HERWIG)

❖ Non-perturbative g → q¯ q splitting after parton shower. ❖ Colour singlet q¯ q clusters have lower mass due to preconfinement property of parton shower.

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4

log10 Mass (GeV/c2) Frequency

50 GeV 500 GeV 5000 GeV 50000 GeV Singlet Random

❖ Clusters decay according to 2-hadron density of states. ❖ Few parameters: natural pT and heavy particle suppression ❖ Problems with massive clusters, baryons, heavy quarks

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String (PYTHIA)

❖ Uses string dynamics: colour string stretched between initial q¯ q breaks up into hadrons via q¯ q pair production. ❖ String gives linear confinement potential, area law for matrix elements. ❖ Gluons produced in shower give ‘kinks’ on string.

n

h hn-1

1 2

h h

q

q ..... x t

A

|M(q¯ q → h1 · · · hn)|2 ∝ e−bA ❖ Extra parameters for pT and heavy particle suppression. ❖ Some problems with baryons.

Both models describe e+e− data well . . .

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Jet rates and mean number of jets

OPAL (91 GeV)

Durham

2-jet 3-jet 4-jet 5-jet PYTHIA HERWIG ycut Jet Fraction

0.2 0.4 0.6 0.8 1 10

4

10

3

10

2

10

1

OPAL

Durham

91 GeV 133 GeV 177 GeV 197 GeV PYTHIA HERWIG ycut <N>

2 4 6 8 10 12 10

5

10

4

10

3

10

2

10

1

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Light quark and gluon fragmentation functions

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2

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1

1 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xE 1/NjetdNch/dxE

udsc Quark

OPAL

×10−1 ×10−2 ×10−3 ×10−4 DATA

Qjet =

6.4 GeV

Qjet = 13.4 GeV
Qjet = 24.0 GeV

⋆

√s/2 = 45.6 GeV

✷ ✷ ✷

Qjet = 46.5 GeV

√s/2 = 98.5 GeV

PYTHIA 6.1 HERWIG 6.2 ARIADNE 4.08

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7

10

6

10

5

10

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3

10

2

10

1

1 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xE 1/NjetdNch/dxE

Gluon

OPAL

×10−1 ×10−2 ×10−3 DATA

Qjet =

6.4 GeV

Qjet = 13.4 GeV
Qjet = 24.0 GeV

✷ ✷ ✷

Qjet = 48.5 GeV PYTHIA 6.1 HERWIG 6.2 ARIADNE 4.08

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Hadron-Hadron Processes

In hard hadron-hadron scattering, constituent partons from each incoming hadron interact

at short distance (large momentum transfer Q2).

j

Q µ µ i

For hadron momenta P1, P2 (S = 2P1 · P2), form of cross section is

σ(S) = X

i,j

Z dx1dx2Di(x1, µ)Dj(x2, µ)ˆ σij(ˆ s = x1x2S, αS(µ), Q/µ) where µ is factorization scale and ˆ σij is subprocess cross section for parton types i, j. ❖ Factorization scale is in principle arbitrary: it affects only what we call part of subprocess or part of initial-state evolution (parton shower). ❖ Rapidity of subprocess c.m. frame pµ = pµ

1 + pµ 2:

y ≡ 1

2 ln

h (p0 + p3)/(p0 − p3) i = 1

2 ln (x1/x2)

❖ Unlike e+e− or ep, we may have interaction between spectator partons, leading to soft underlying event and/or multiple hard scattering.

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Parton-Parton Luminosities

Useful to define the differential parton-parton luminosity dLij/dˆ

s dy and its integral dLij/dˆ s: dLij dˆ s dy = 1 S 1 1 + δij [Di(x1, µ)Dj(x2, µ) + (1 ↔ 2)] . Factor with Kronecker delta avoids double-counting when partons are identical.

We have dˆ

s dy = S dx1 dx2 and hence σ = X

i,j

Z dˆ s dy „ dLij dˆ s dy « ˆ σij(ˆ s) = X

i,j

Z dˆ s „dLij dˆ s « ˆ σij(ˆ s)

This can be used to estimate the production rate for subprocesses at LHC.

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Figure shows parton-parton luminosities at √s = 14 TeV for various parton combinations,

calculated using the CTEQ6.1 parton distribution functions and scale µ = √ ˆ

s. Widths of

curves estimate PDF uncertainties. Green = gg, Blue = gq + g¯ q + qg + ¯ qg, Red = q¯ q + ¯ qq (q = d + u + s + c + b).

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Lepton Pair Production

Inverse of e+e− → q¯

q is Drell-Yan process. At O(α0

S), mass distribution of lepton pair is

given by dˆ σ dM2(q¯ q → γ∗ → l+l−) = 4πα2 ˆ s 1 3Q2

q δ(M2 − ˆ

s) ❖ Factor of 1/3 = 1/N instead of 3 = N because of average over colours of incoming q. ❖ In higher orders vertex corrections (a) have M2 = ˆ s, gluon emission (b) and QCD Compton (c) diagrams give M2 < ˆ s.

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W± boson production is similar, except sensitive to different parton distributions, e.g.

u ¯ d → W + → l+νl

Transverse momentum of lepton pair, pT measures net transverse momentum of colliding

partons plus any intrinsic pT:

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Jet Production

Lowest-order subprocess for purely hadronic jet production is 2 → 2 scattering p1 + p2 →

p3 + p4 dˆ σ dΦ34 ≡ E3E4d6ˆ σ d3p3d3p4 = 1 32π2ˆ s X |M|2 δ4(p1 + p2 − p3 − p4) .

Many processes even at O(α2

S): 22

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Single-jet inclusive cross section obtained by integrating over one outgoing momentum:

Ed3ˆ σ d3p = d3ˆ σ d2pTdy − → 1 2πET d3ˆ σ dET dη = 1 16π2ˆ s X |M|2 δ(ˆ s + ˆ t + ˆ u) where (neglecting jet mass) ET ≡ E sin θ = |pT| , η ≡ − ln tan(θ/2) = y .

Jets can be defined by a modified version of kT algorithm discussed for e+e− in Lecture 2:

❖ For each final-state momentum pi and each pair of final-state momenta pi, pj, define kT i = ET i , kT ij = min{ET i, ET j}∆Rij/D where ∆Rij = p (ηi − ηj)2 + (φi − φj)2 and D = dimensionless parameter for angular size of jets (D = 0.5 − 1.0) ❖ If kT I is the smallest in the list of {kT i, kT ij}, define I as a jet and remove from list. ❖ If kT IJ is the smallest, combine I, J into one object K with pK = pI + pJ. ❖ Repeat until list is empty.

Use η rather than θ for invariance under longitudinal boosts: x1 → ax1, x2 → x2/a

gives ηi → ηi + ln a, so ηi − ηj is invariant.

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NLO predictions and data agree very well:

[nb/(GeV/c)]

JET T

dp

JET

/ dy σ

2

d

7

10

4

10

1

10

)

1

CDF data ( L = 1.0 fb Systematic uncertainties NLO: JETRAD CTEQ6.1M corrected to hadron level µ / 2 =

JET T

= max p

F

µ =

R

µ PDF uncertainties

|<0.7

JET

0.1<|y D=0.5

T

K |<0.7

JET

0.1<|y D=1.0

T

K

Data/Theory

0.5 1 1.5 2 [GeV/c]

JET T

p

200 400 600 HAD

C

1 1.5

Parton to hadron level correction Monte Carlo modeling uncertainties

[GeV/c]

JET T

p

200 400 600

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Rapidity dependence:

[GeV/c]

JET T

p

100 200 300 400 500 600 700

[nb/(GeV/c)]

JET T

dp

JET

/ dy σ

2

d

14

10

11

10

8

10

5

10

2

10 10

4

10

7

10

D=0.7

T

K )

1

CDF data ( L = 1.0 fb Systematic uncertainties NLO: JETRAD CTEQ6.1M corrected to hadron level µ / 2 =

JET T

= max p

F

µ =

R

µ PDF uncertainties

)

6

10 × |<2.1 (

JET

1.6<|y )

3

10 × |<1.6 (

JET

1.1<|y

|<1.1

JET

0.7<|y )

3

10 × |<0.7 (

JET

0.1<|y

)

6

10 × |<0.1 (

JET

|y

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Contribution of different parton combinations determined by subprocess cross sections and

parton distributions.

Quarks dominate at large ET since this selects large x1,2:

ˆ s = x1x2S > 4E2

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Heavy Quark Production

Lowest-order subprocesses for heavy quark production are (a) light quark-antiquark

annihilation (10% at LHC) and (b) gluon-gluon fusion (90% at LHC)

NLO top quark cross section = 840±30(scale)±20(pdf) pb at LHC

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Standard Model Higgs Boson Production

Lowest-order subprocesses for Higgs boson production at hadron colliders:

(a) Gluon-gluon fusion (via top loop) (b) Vector boson fusion (c) Associated production with W, Z boson (d) Associated production with t¯ t.

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NLO Higgs cross sections

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Discovery decay channels depend on Higgs mass

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Status of NLO Calculations for LHC

2 → 2 parton processes — all available, e.g. in MCFM (CaEl∗)
2 → 3 parton processes

Final State Authors∗ Comments 3 jets KuSiTr,BerDixKo,GiKi,Na Public code available V + 2 jets ElCa,CaGlMi Public code available V b ¯ b ElCa Massless b quarks V b ¯ b ReFeWa Massive b quarks H+ 2 jets FiOlZep Vector boson fusion H+ 2 jets CaElZa Gluon fusion V V + 2 jets JaOlZep Vector boson fusion γγ jet deFKu,DelMalNaTr,BiGuMah t¯ tH, b¯ bH ReDaWaOr,BeeDitKrPlSpZer t¯ t jet DitUwWe HHH PlRa,BiKarKauRu W W jet DiKalUw ZZZ LaMePe

∗Beenakker,Bern,Binoth,Campbell,Dawson,deFlorian,DelDuca,Dittmaier,Dixon,Ellis,FebresCordero,Figy,

Giele,Glover,Guillet,Jager,Kallweit,Karg,Kauer,Kilgore,Kramer,Kosower,Kunszt,Lazopoulos,Mahmoudi, Maltoni,Melnikov,Miller,Nagy,Oleari,Orr,Petriello,Plehn,Plumper,Rauch,Reina,Ruckl,Signer,Spira, Troscsanyi,Uwer,Wackeroth,Weinzierl,Zanderighi,Zeppenfeld,Zerwas

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Les Houches 2005 wish list of “feasible” NLO calculations

Final State Relevance Progress? V V jet t¯ tH, new physics V V = γγ, W W V V V SUSY trilepton ZZZ V V b¯ b VBF→ H → V V , t¯ tH, new physics V V + 2 jets VBF→ H → V V V + 3 jets various new physics signatures t¯ t + 2 jets t¯ tH t¯ t b¯ b t¯ tH

Les Houches 2007: W W W , b¯

bb¯ b, 4 jets added.

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Summary of Lecture 4

Jet fragmentation functions also obey DGLAP evolution equations.

❖ Scaling violation seen in e+e−. ❖ Small-x fragmentation sensitive to coherence effects. ❖ Gaussian peak in ln(1/x), peak position shows coherence. ❖ Average hadron multiplicity predicted.

Hadronization models needed for simulation of full final states.

❖ General ideas describe spectra and event shapes. ➞ Local parton-hadron duality. ➞ Universal low-scale αS. ❖ Specific models needed for hadron distributions. ➞ String model (PYTHIA). ➞ Cluster model (HERWIG).

In hadron-hadron processes, factorization permits cross section calculations.

❖ Parton-parton luminosities important: uncertainties ∼ 10 − 20%. ❖ Lepton-pair, jet, top and Higgs production reliably predicted (NLO or NNLO). ❖ All 2 → 2 and many 2 → 3 subprocesses predicted to NLO.

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