Monte Carlo Tools Frank Krauss Institute for Particle Physics - - PowerPoint PPT Presentation

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Monte Carlo Tools

Frank Krauss

Institute for Particle Physics Phenomenology Durham University

GGI, 24.&26.9.2007

• F. Krauss

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Topics of the lectures

1

Lecture 1: Tour through Event Generators

Hard physics simulation: Parton Level event generation Dressing the partons: Parton Showers Soft physics simulation: Hadronization Beyond factorization: Underlying Event 2

Lecture 2: Higher Orders in Monte Carlos

Some nomenclature: Anatomy of HO calculations Merging vs. Matching Thanks to the other Sherpas: T.Gleisberg, S.H¨

• che, S.Schumann, F.Siegert, M.Sch¨
• nherr, J.Winter;
• ther MC authors: S.Gieseke, K.Hamilton, L.Lonnblad, F.Maltoni, M.Mangano, P.Richardson,

M.Seymour, T.Sjostrand, B.Webber, . . . .

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Simulation’s paradigm

Basic strategy

Divide event into stages, separated by different scales.

Signal/background:

Exact matrix elements.

QCD-Bremsstrahlung:

Parton showers (also in initial state).

Multiple interactions:

Beyond factorization: Modeling.

Hadronization:

Non-perturbative QCD: Modeling.

Sketch of an event

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Today’s lecture: Event Generation in a Nutshell

Monte Carlo integration Parton level event generation Parton showers Multiple interactions Hadronization

• F. Krauss

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Monte Carlo integration

Convergence of numerical integration

Consider I =

1

• dxDf (

x). Convergence behavior crucial for numerical evaluations. For integration (N = number of evaluations of f ):

Trapezium rule ≃ 1/N2/D Simpson’s rule ≃ 1/N4/D Central limit theorem ≃ 1/ √ N.

Therefore: Use central limit theorem.

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Monte Carlo integration

Monte Carlo integration

Use random vectors xi − →: Evaluate estimate of the integral I rather than I. I(f ) = 1

N N

• i=1

f ( xi).

(This is the original meaning of Monte Carlo: Use random numbers for integration.)

Quality of estimate given by error estimator (variance) E(f )2 =

1 N−1 [I 2(f ) − I(f )2].

Name of the game: Minimize E(f ). Problem: Large fluctuations in integrand f Solution: Smart sampling methods

• F. Krauss

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Monte Carlo integration

Importance sampling

Basic idea: Put more samples in regions, where f largest = ⇒ improves convergence behavior (corresponds to a Jacobian transformation). Assume a function g( x) similar to f ( x);

• bviously then, f (

x)/g( x) is comparably smooth, hence E(f /g) is small.

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Monte Carlo integration

Stratified sampling

Basic idea: Decompose integral in M sub-integrals I(f ) =

M

• j=1

Ij(f ), E(f )2 =

M

• j=1

Ej(f )2 Then: Overall variance smallest, if “equally distributed”. = ⇒ Sample, where the fluctuations are. Divide interval in bins; adjust bin-size or weight per bin such that variance identical in all bins.

I = 0.637 ± 0.147/ √ N

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Monte Carlo integration

Example for stratified sampling: VEGAS

Assume m bins in each dimension of x. For each bin k in each dimension η ∈ [1, n] assume a weight (probability) α(η)

k

for xk to be in that bin. Condition(s) on the weights: α(η)

k

∈ [0, 1], Pm

k=1 α(η) k

= 1. For each bin in each dimension calculate I (η)

k

and E (η)

k

. Obviously, for all η, I = Pm

k=1I (η) k

, but error estimates different. In each dimensions, iterate and update the α(η)

k

; example for updating: α(η)

k

(rm new) ∝ α(η)

k

(rm old)

E(η) k Etot.(η)

!κ . Problem with this simple algorithm: Gets a hold only on fluctuations to binning axes.

• F. Krauss

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Monte Carlo integration

Multichannel sampling

Basic idea: Use a sum of functions gi( x) as Jacobian g( x). = ⇒ g( x) = N

i=1 αigi(

x); = ⇒ condition on weights like stratified sampling; (“Combination” of importance & stratified sampling).

Algorithm for one iteration: Select gi with probability αi → xj . Calculate total weight g( xj ) and partial weights gi ( xj ) Add f ( xj )/g( xj ) to total result and f ( xj )/gi ( xj ) to partial (channel-) results. After N sampling steps, update a-priori weights.

This is the method of choice for parton level event generation!

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Monte Carlo integration

Selecting after sampling: Unweighting efficiency

Basic idea: Use hit-or-miss method; Generate x with integration method, compare actual f ( x) with maximal value during sampling; = ⇒ “Unweighted events”. Comments:

unweighting efficiency, weff = f ( xj )/fmax = number of trials for each event. Good measure for integration performance. Expect log10 weff ≈ 3 − 5 for good integration of multi-particle final states at tree-level. Maybe acceptable to use fmax,eff = Kfmax with K < 1. Problem: what to do with events where f ( xj )/fmax,eff > 1? Answer: Add int[f ( xj )/fmax,eff ] = k events and perform hit-or-miss on f ( xj )/fmax,eff − k.

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Monte Carlo integration

Particle physics example: Evaluation of cross sections

Simple example: t → bW + → b¯ lνl : |M|2 = 1 2 8πα sin2 θW !2 pt · pν pb · pl (p2

W − M2 W )2 + Γ2 W M2 W

Phase space integration (5-dim) Γ = 1 2mt 1 128π3 Z dp2

W

d2ΩW 4π d2Ω 4π 1 − p2

W

m2

t

! |M|2

Advantages

Throw 5 random numbers, construct four-momenta (= ⇒ full kinematics, “events”) Apply smearing and/or arbitrary cuts. Simply histogram any quantity of interest - no new calculation for each observable

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Parton level simulations

Stating the problem(s)

Multi-particle final states for signals & backgrounds. Need to evaluate dσN:

• cuts

N

• i=1

d3qi (2π)32Ei

• δ4
• p1 + p2 −
• i

qi

• |Mp1p2→N|2 .

Problem 1: Factorial growth of number of amplitudes. Problem 2: Complicated phase-space structure. Solutions: Numerical methods.

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Parton level simulations

Factorial growth: e+e− → q¯ q + ng

n #diags 1 1 2 2 8 3 48 4 384

1 2 3 4

Number of gluons

1 10 100 1000

Number of diagrams

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Parton level simulations

Basic ideas of efficient ME calculation

Need to evaluate |M|2 =

• i

Mi

• 2

Obvious: Traditional textbook methods (squaring, completeness relations, traces) fail = ⇒ result in proliferation of terms (MiM∗

j )

= ⇒ Better: Amplitudes are complex numbers, = ⇒ add them before squaring! Remember: spinors, gamma matrices have explicit form could be evaluated numerically (brute force) But: Rough method, lack of elegance, CPU-expensive

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Parton level simulations

Helicity method

Introduce basic helicity spinors (needs to “gauge”-vectors) Write everything as spinor products, e.g. ¯ u(p1, h1)u(p2, h2) = complex numbers. Also:

(p / + m) = ⇒

1 2

P

h

»„ 1 + m2

p2

« ¯ u(p, h)u(p, h) + „ 1 − m2

p2

« ¯ v(p, h)v(p, h) – (completeness relation)

Find other genuine expressions:

Y (p1, h1, p2, h2) := ¯ u(p1, h1)u(p2, h2) X(p1, h1, p2, h2, p3) := ¯ u(p1, h1)p /3u(p2, h2) Z(p1, h1, p2, h2; p3, h3, p4, h4) := ¯ u(p1, h1)γµu(p2, h2)¯ u(p3, h3)γµu(p4, h4) ,

all complex-valued functions of momenta & helicities.

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Parton level simulations

Taming the factorial growth in the helicity method

Reusing pieces: Calculate only once! Factoring out: Reduce number of multiplications! Implemented as a-posteriori manipulations of amplitudes.

• ;

• ;

• e

• +
• ;

• ;

• e

• +
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Parton level simulations

Recursion methods (off-shell)

Basic idea: Recursively build one-particle off-shell currents (various versions of this: Berends-Giele, Alpha etc.). “Classical” example: n-gluon amplitudes:

Start with two on-shell gluons, represented by their polarization vectors, hence the currents associated with them are Jν(k) = εν(k). Then the two-gluon current reads (no colors) Jµ(k = k1 + k2) =

ig3 (k1+k2)2 V µνρJν(k1)Jρ(k2).

From this, larger and larger currents can be built recursively. For quarks, the currents are given by spinors, and similar reasoning applies for the construction of the

• ne-particle off-shell currents.

Treatment of color: Color-ordering the amplitudes = ⇒ C(1, ..., n) = Tr [T a1 . . . T an ], where T a are color matrices in fundamental representation. Problem: Need to sum over all allowed permutations.

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Parton level simulations

Integration methods: Multi-channeling

Basic idea: Translate Feynman diagrams into channels = ⇒ decays, s- and t-channel props as building blocks.

R.Kleiss and R.Pittau, Comput. Phys. Commun. 83 (1994) 141

Integration methods: “Democratic” methods

Rambo/Mambo: Flat & isotropic

R.Kleiss, W.J.Stirling and S.D.Ellis, Comput. Phys. Commun. 40 (1986) 359;

HAAG: Follows QCD antenna pattern

A.van Hameren and C.G.Papadopoulos, Eur. Phys. J. C 25 (2002) 563.

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Limitations of parton level simulation

Factorial growth

. . . persists due to the number of color configurations

(e.g. (n − 1)! permutations for n external gluons).

Solution: Sampling over colors, but correlations with phase space = ⇒ Best recipe not (yet) found. New scheme for color: color dressing

(C.Duhr, S.Hoche and F.Maltoni,JHEP 0608 (2006) 062)

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Limitations of parton level simulation

Factorial growth

Off-shell vs. on-shell recursion relations: Time [s] for the evaluation of 104 phase space points, sampled over helicities & color.

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Limitations of parton level simulation

Efficient phase space integration

Main problem: Adaptive multi-channel sampling translates “Feynman diagrams” into integration channels = ⇒ hence subject to growth. But it is practical only for 1000-10000 channels. Therefore: Need better sampling procedures = ⇒

• pen question with little activity.

(Private suspicion: Lack of glamour)

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Limitations of parton level simulation

General

Fixed order parton level (LO, NLO, . . . ) implies fixed multiplicity No control over potentially large logs (appear when two partons come close to each other). Parton level is parton level experimental definition of observables relies on hadrons. Therefore: Need hadron level event generators!

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Motivation: Why parton showers?

Some more refined reasons

Experimental definition of jets based on hadrons. But: Hadronization through phenomenological models

(need to be tuned to data).

Wanted: Universality of hadronization parameters

(independence of hard process important).

Link to fragmentation needed: Model softer radiation

(inner jet evolution).

Similar to PDFs (factorization) just the other way around

(fragmentation functions at low scale, parton shower connects high with low scale).

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Motivation: Why parton showers?

Common wisdom

Well-known: Accelerated charges radiate QED: Electrons (charged) emit photons Photons split into electron-positron pairs QCD: Quarks (colored) emit gluons Gluons split into quark pairs Difference: Gluons are colored (photons are not charged) Hence: Gluons emit gluons! Cascade of emissions: Parton shower

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Occurrence of large logarithms

e+e− → jets

Differential cross section:

dσee→3j dx1dx2 = σee→2j CF αs π x2

1 + x2 2

(1 − x1)(1 − x2)

Singular for x1,2 → 1. Rewrite with opening angle θqg and gluon energy fraction x3 = 2Eg/Ec.m.:

dσee→3j d cos θqg dx3 = σee→2j CF αs π " 2 sin2 θqg 1 + (1 − x3)2 x3 − x3 #

Singular for x3 → 0 (“soft”), sin θqg → 0 (“collinear”).

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Occurrence of large logarithms

Collinear singularities

Use

2d cos θqg sin2 θqg = d cos θqg 1 − cos θqg + d cos θqg 1 + cos θqg = d cos θqg 1 − cos θqg + d cos θ¯

qg

1 − cos θ¯

qg

≈ dθ2

qg

θ2

qg

+ dθ2

¯ qg

θ2

¯ qg

Independent evolution of two jets (q and ¯ q): dσee→3j ≈ σee→2j

• j∈{q,¯

q}

CFαs 2π dθ2

jg

θ2

jg

P(z) , where P(z) = 1+(1−z)2

z

(DGLAP splitting function)

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Occurrence of large logarithms

Expressing the collinear variable

Same form for any t ∝ θ2: Transverse momentum k2

⊥ ≈ z2(1 − z)2E 2θ2

Invariant mass q2 ≈ z(1 − z)E 2θ2 dθ2 θ2 ≈ dk2

⊥

k2

⊥

≈ dq2 q2

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Occurrence of large logarithms

Parton resolution

What is a parton? Collinear pair/soft parton recombine! Introduce resolution criterion k⊥ > Q0. Combine virtual contributions with unresolvable emissions: Cancels infrared divergences = ⇒ Finite at O(αs)

(Kinoshita-Lee-Nauenberg, Bloch-Nordsieck theorems)

Unitarity: Probabilities add up to one P(resolved) + P(unresolved) = 1. + =1.

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Occurrence of large logarithms

The Sudakov form factor

• Diff. probability for emission between q2 and q2 + dq2:

dP = αs

2π dq2 q2 1−Q2

0/q2

• Q2

0/q2

dzP(z) =: dq2

q2 ¯

P(q2) . No-emission probability ∆(Q2, q2) between Q2 and q2. Evolution equation for ∆: −d∆(Q2, q2)

dq2

= ∆(Q2, q2) P

dq2.

= ⇒ ∆(Q2, q2) = exp

• −

Q2

• q2

dk2 k2 ¯

P(k2)

• .
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Occurrence of large logarithms

Many emissions

Iterate emissions (jets) Maximal result for t1 > t2 > . . . tn: dσ ∝ σ0

Q2

• Q2

dt1 t1

t1

• Q2

dt2 t2 . . .

tn−1

• Q2

dtn tn ∝ logn Q2 Q2 How about Q2? Process-dependent!

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Occurrence of large logarithms

Ordering the emissions : Radiation pattern

q2

1 > q2 2 > q2 3, q2 1 > q′ 2 2

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Occurrence of large logarithms

Forward vs. backward evolution: Pictorially

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Occurrence of large logarithms

Use of DGLAP evolution

DGLAP evolution:

PDFs at (x, Q2) as function of PDFs at (x0, Q2

0 ).

Backward evolution:

start from hard scattering at (x, Q2) and work down in q2 and up in x.

Change in algorithm: ∆i(q2) = ⇒ ∆i(q2)/fi(xi, q2).

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Inclusion of quantum effects

Running coupling

Effect of summing up higher orders (loops): αs → αs(k2

⊥)

Much faster parton proliferation, especially for small k2

⊥.

Must avoid Landau pole: k2

⊥ > Q2 0 ≫ Λ2 QCD

= ⇒ Q2

0 = physical parameter.

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Inclusion of quantum effects

Soft logarithms : Angular ordering

Soft limit for single emission also universal Problem: Soft gluons come from all over (not collinear!) Quantum interference? Still independent evolution? Answer: Not quite independent.

Assume photon into e+e− at θee and photon off electron at θ Energy imbalance at vertex: kγ

⊥ ∼ zpθ, hence ∆E ∼ k2 ⊥/zp ∼ zpθ2.

Time for photon emission: ∆t ∼ 1/∆E. ee-separation: ∆b ∼ θee∆t > Λ/θ ∼ 1/(zpθ) Thus: θee/(zpθ2) > 1/(zpθ) = ⇒ θee > θ

• e

• Thus: Angular ordering takes care of soft limit.
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Inclusion of quantum effects

G.Marchesini and B.R.Webber, Nucl. Phys. B 238 (1984) 1.

Soft logarithms : Angular ordering

= ⇒ Gluons at large angle from combined color charge!

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Inclusion of quantum effects

Soft logarithms : Angular ordering

Experimental manifestation: ∆R of 2nd & 3rd jet in multi-jet events in pp-collisions

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Inclusion of quantum effects

Resummed jet rates in e+e− → hadrons

S.Catani et al. Phys. Lett. B269 (1991) 432

Use Durham jet measure (k⊥-type):

k2

⊥,ij = 2min(E 2 i , E 2 j )(1 − cos θij) > Q2 jet .

Remember prob. interpret. of Sudakov form factor:

R2(Qjet) = ˆ ∆q(Ec.m., Qjet) ˜2 R3(Qjet) = 2∆q(Ec.m., Qjet) · Z dq " αs (q)¯ Pq(Ec.m., q) ∆q(Ec.m., Qjet) ∆q(q, Qjet) ∆q(q, Qjet)∆g (q, Qjet) #

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Hadronization

Confinement

Consider dipoles in QED and QCD QED: QCD:

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Hadronization

Linear QCD potential in quarkonia

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Hadronization

Some experimental facts → naive parameterizations

In e+e− → hadrons: Limits p⊥, flat plateau in y. Try “smearing”: ρ(p2

⊥) ∼ exp(−p2 ⊥/σ2)

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Hadronization

Effect of naive parameterizations

Use parameterization to “guesstimate” hadronization effects:

E = Z Y dydp2

⊥ρ(p2 ⊥)p⊥ cosh y = λ sinh Y

P = Z Y dydp2

⊥ρ(p2 ⊥)p⊥ sinh y = λ(cosh Y − 1) ≈ E − λ

λ = Z dp2

⊥ρ(p2 ⊥)p⊥ = p⊥ .

Estimate λ ∼ 1/Rhad ≈ mhad, with mhad 0.1-1 GeV. Effect: Jet acquire non-perturbative mass ∼ 2λE (O(10GeV) for jets with energy O(100GeV)).

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Hadronization

Implementation of naive parameterizations

Feynman-Field independent fragmentation.

R.D.Field and R.P.Feynman, Nucl. Phys. B 136 (1978) 1

Recursively fragment q → q′+ had, where

Transverse momentum from (fitted) Gaussian; longitudinal momentum arbitrary (hence from measurements); flavor from symmetry arguments + measurements.

Problems: frame dependent, “last quark”, infrared safety, no direct link to perturbation theory, . . . .

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Hadronization

Yoyo-strings as model of mesons

B.Andersson, G.Gustafson, G.Ingelman and T.Sjostrand, Phys. Rept. 97 (1983) 31.

Light quarks connected by string: area law m2 ∝area. L=0 mesons only have ’yo-yo’ modes:

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Hadronization

Dynamical strings in e+e− → q¯ q

B.Andersson, G.Gustafson, G.Ingelman and T.Sjostrand, Phys. Rept. 97 (1983) 31.

Ignoring gluon radiation: Point-like source of string. Intense chromomagnetic field within string: More q¯ q pairs created by tunnelling. Analogy with QED (Schwinger mechanism): dP ∼ dxdt exp

• −πm2

q/κ

• , κ = “string tension”.
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Hadronization

Gluons in strings = kinks

B.Andersson, G.Gustafson, G.Ingelman and T.Sjostrand, Phys. Rept. 97 (1983) 31.

String model = well motivated model, constraints on fragmentation (Lorentz-invariance, left-right symmetry, . . . ) Gluon = kinks on string? Check by “string-effect” Infrared-safe, advantage: smooth matching with PS.

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Hadronization

Preconfinement

Underlying: Large Nc-limit (planar graphs). Follows evolution of color in parton showers: at the end of shower color singlets close in phase space. Mass of singlets: peaked at low scales ≈ Q2

0.

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Hadronization

Primordial cluster mass distribution

Starting point: Preconfinement; split gluons into q¯ q-pairs; adjacent pairs color connected, form colorless (white) clusters. Clusters (“≈ excited hadrons) decay into hadrons

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Hadronization

Cluster model

B.R.Webber, Nucl. Phys. B 238 (1984) 492.

Split gluons into q¯ q pairs, form singlet clusters: = ⇒ continuum of meson resonances. Decay heavy clusters into lighter ones; (here, many improvements to ensure leading hadron spectrum hard enough, overall effect: cluster model becomes more string-like); if light enough, clusters → hadrons. Naively: spin information washed out, decay determined through phase space only → heavy hadrons suppressed (baryon/strangeness suppression).

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Underlying Event

Multiple parton scattering?

Hadrons = extended objects! No guarantee for one scattering only. Running of αS = ⇒ preference for soft scattering.

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Underlying Event

Evidence for multiple parton scattering

Events with γ + 3 jets:

Cone jets, R = 0.7, ET > 5 GeV; |ηj| <1.3; “clean sample”: two softest jets with ET < 7 GeV;

σDPS = σγjσjj

σeff ,

σeff ≈ 14 ± 4 mb.

CDF collaboration, Phys. Rev. D56 (1997) 3811.

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Underlying Event

Definition(s)

• 1

Everything apart from the hard interaction including IS showers, FS showers, remnant hadronization.

2

Remnant-remnant interactions, soft and/or hard. = ⇒ Lesson: hard to define

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Underlying event

Model: Multiple parton interactions

To understand the origin of MPS, realize that σhard(p⊥,min) =

s/4

• p2

⊥,min

dp2

⊥

dσ(p2

⊥)

dp2

⊥

> σpp,total for low p⊥,min. Here:

dσ(p2 ⊥) dp2 ⊥

=

1

R dx1dx2dˆ tf (x1, q2)f (x2, q2) d ˆ

σ2→2 dp2 ⊥

δ “ 1 − ˆ

tˆ u ˆ s

” (f (x, q2) =PDF, ˆ σ2→2 =parton-parton x-sec)

σhard(p⊥,min)/σpp,total ≥ 1 Depends strongly on cut-off p⊥,min (Energy-dependent)!

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Orientation MC integration Matrix elements Parton showers Hadronization Underlying Event Upshot

Underlying event

Old Pythia model: Algorithm, simplified

T.Sjostrand and M.van Zijl, Phys. Rev. D 36 (1987) 2019.

Start with hard interaction, at scale Q2

hard.

Select a new scale p2

⊥

(according to f =

dσ2→2(p2 ⊥) dp2 ⊥

with p2

⊥ ∈ [p2 ⊥,min, Q2])

Rescale proton momentum

(“proton-parton = proton with reduced energy”).

Repeat until below p2

⊥,min.

May add impact-parameter dependence, showers, etc.. Treat intrinsic k⊥ of partons (→ parameter) Model proton remnants (→ parameter)

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Orientation MC integration Matrix elements Parton showers Hadronization Underlying Event Upshot

Underlying Event

In the following: Data from CDF, PRD 65 (2002) 092002, plots partially from C.Buttar

Observables

• ∆φ

∆φ ∆φ ∆φ

• η

η η η

• η

η η η

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Orientation MC integration Matrix elements Parton showers Hadronization Underlying Event Upshot

Underlying event

Hard component in transverse region

• η

η η η

• η

η η η

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Orientation MC integration Matrix elements Parton showers Hadronization Underlying Event Upshot

Underlying event

Energy extrapolation

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Orientation MC integration Matrix elements Parton showers Hadronization Underlying Event Upshot

Underlying event

General facts on current models

No first-principles approach for underlying event:

Multiple-parton interactions: beyond factorization Factorization (simplified) = no process-dependence in use of PDFs.

Models usually based on xsecs in collinear factorization: dσ/dp⊥ ∝ p4−8

⊥

= ⇒ strong dependence on cut-off pmin

⊥ .

“Regularization”: dσ/dp⊥ ∝ (p2

⊥ + p2 0)2−4, also in αS.

Model for scaling behavior of pmin

⊥ (s) ∝ pmin ⊥ (s0)(s/s0)λ, λ =?

Two Pythia tunes: λ = 0.16, λ = 0.25.

Herwig model similar to old Pythia and SHERPA New Pythia model: Correlate parton interactions with showers, more parameters.

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Summary so far

1

Hard MEs:

Theoretically very well understood, realm of perturbation theory. Fully automated tools at tree-level available, 2 → 6 no problem at all. Obstacle(s) for higher multiplicities: factorial growth, phase space integration. NLO calculations much more involved, no fully automated tool, only libraries for specific processes (MCFM, NLOJET++), typically up to 2 → 3. NNLO only for a small number of processes.

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Orientation MC integration Matrix elements Parton showers Hadronization Underlying Event Upshot

Summary so far

1

Parton showers:

Theoretically well understood, still in realm of perturbation theory, but beyond fixed order. Consistent treatment of leading logs in soft/collinear limit, formally equivalent formulations lead to different results because of non-trivial choices (evolution parameter, etc.).

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Orientation MC integration Matrix elements Parton showers Hadronization Underlying Event Upshot

Summary so far

1

Hadronization

Various phenomenological models; different levels of sophistication, different number of parameters; tuned to LEP data, overall agreement satisfying; validity for hadron data not quite clear - differences possible (beam remnant fragmentation not in LEP).

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Orientation MC integration Matrix elements Parton showers Hadronization Underlying Event Upshot

Summary so far

1

Underlying event

Various definitions for this phenomenon. Theoretically not understood, in fact: beyond theory understanding (breaks factorization); models typically based on collinear factorization and semi-independent multi-parton scattering = ⇒ very naive; models highly parameter-dependent, leading to large differences in predictions; connection to minimum bias, diffraction etc.? even unclear: good observables to distinguish models.

• F. Krauss

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