Monte Carlo 1. Introduction and Parton Showers

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Monte Carlo

1. Introduction and Parton Showers

Torbj¨orn Sj¨ostrand

Department of Astronomy and Theoretical Physics Lund University

S¨olvegatan 14A, SE-223 62 Lund, Sweden

TRIUMF, Vancouver, Canada, 4 July 2013

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Course Plan

Improve understanding of physics at the LHC Complementary to the “textbook” picture of particle physics, since event generators is close to how things work “in real life”.

Lecture 1 Introduction and generator survey Parton showers: final and initial

Lecture 2 Combining matrix elements and parton showers Multiparton interactions and other soft physics Hadronization

Conclusions

Tutorials Use PYTHIA to study aspects of Higgs physics Learn more:

A. Buckley et al., “General-purpose event generators for LHC physics”, Phys. Rep. 504 (2011) 145 [arXiv:1101.2599[hep-ph]];

also “PYTHIA 6.4 Physics and Manual”, JHEP05 (2006) 026

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A tour to Monte Carlo

. . . because Einstein was wrong: God does throw dice!

Quantum mechanics: amplitudes =⇒ probabilities

Anything that possibly can happen, will! (but more or less often) Event generators: trace evolution of event structure.

Random numbers ≈ quantum mechanical choices.

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The structure of an event – 1

Warning: schematic only, everything simplified, nothing to scale, . . .

p

p/p

Incoming beams: parton densities

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Beam remnants and other outgoing partons

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The structure of an event – 9

Everything is connected by colour confinement strings Recall! Not to scale: strings are of hadronic widths

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The structure of an event – 10

The strings fragment to produce primary hadrons

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The structure of an event – 11

Many hadrons are unstable and decay further

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The structure of an event – 12







These are the particles that hit the detector

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The Monte Carlo method

Want to generate events in as much detail as Mother Nature

=⇒ get average and fluctutations right

=⇒ make random choices, ∼ as in nature σfinal state= σhard processPtot,hard process→final state

(appropriately summed & integrated over non-distinguished final states) where P_tot= P_resP_ISRP_FSRP_MPIP_remnantsPhadronizationP_decays

with P_i =Q

jP_ij =Q

j

Q

kP_ijk = . . . in its turn

=⇒ divide and conquer

an event with n particles involves O(10n) random choices, (flavour, mass, momentum, spin, production vertex, lifetime, . . . )

LHC: ∼ 100 charged and ∼ 200 neutral (+ intermediate stages)

=⇒ several thousand choices (of O(100) different kinds)

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Event Generator Position

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Why generators?

Allow theoretical and experimental studies of complex multiparticle physics

Large flexibility in physical quantities that can be addressed Vehicle of ideology to disseminate ideas

from theorists to experimentalists Can be used to

predict event rates and topologies

⇒ can estimate feasibility simulate possible backgrounds

⇒ can devise analysis strategies study detector requirements

⇒ can optimize detector/trigger design study detector imperfections

⇒ can evaluate acceptance corrections

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The workhorses: what are the differences?

HERWIG, PYTHIA and SHERPA offer convenient frameworks for LHC physics studies, but with slightly different emphasis:

PYTHIA (successor to JETSET, begun in 1978):

• originated in hadronization studies: the Lund string

• leading in development of MPI for MB/UE

• pragmatic attitude to showers & matching HERWIG (successor to EARWIG, begun in 1984):

• originated in coherent-shower studies (angular ordering)

• cluster hadronization & underlying event pragmatic add-on

• large process library with spin correlations in decays

SHERPA (APACIC++/AMEGIC++, begun in 2000):

• own matrix-element calculator/generator

• extensive machinery for CKKW ME/PS matching

• hadronization & min-bias physics under development PYTHIA and HERWIG originally in Fortran, but now all in C++.

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MCnet

MCnet projects:

• PYTHIA (+ VINCIA)

• HERWIG

• SHERPA

• MadGraph

• Ariadne (+ DIPSY)

• Cedar (Rivet/Professor) Activities include

• summer schools

• short-term studentships

• graduate students

• postdocs

• meetings (open/closed)

training studentships

3-6 month fully funded studentships for current PhD students at one of the MCnet nodes. An excellent opportunity to really understand and improve the Monte Carlos you use!

www.montecarlonet.org for details go to:

Monte Carlo

London CERN

Karlsruhe Durham Lund

Application rounds every 3 months.

MARIE CURIE ACTIONS funded by:

Manchester Louvain Göttingen

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Other Relevant Software

Some examples (with apologies for many omissions):

Other event/shower generators: PhoJet, Ariadne, Dipsy, Cascade, Vincia Matrix-element generators: MadGraph/MadEvent, CompHep, CalcHep, Helac, Whizard, Sherpa, GoSam, aMC@NLO

Matrix element libraries: AlpGen, POWHEG BOX, MCFM, NLOjet++, VBFNLO, BlackHat, Rocket

Special BSM scenarios: Prospino, Charybdis, TrueNoir

Mass spectra and decays: SOFTSUSY, SPHENO, HDecay, SDecay Feynman rule generators: FeynRules

PDF libraries: LHAPDF

Resummed (p⊥) spectra: ResBos Approximate loops: LoopSim Jet finders: anti-k⊥and FastJet

Analysis packages: Rivet, Professor, MCPLOTS Detector simulation: GEANT, Delphes

Constraints (from cosmology etc): DarkSUSY, MicrOmegas

Standards: PDF identity codes, LHA, LHEF, SLHA, Binoth LHA, HepMC

Can be meaningfully combined and used for LHC physics!

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Putting it together

Standardized interfaces essential!

. . . but wide range of possible processes, some with special quirks.

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Multijets – the need for showers

Basic 2 → 2 process dressed up by bremsstrahlung!?

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Perturbative QCD

Order-by-order calculations: challenges more math than physics.

(courtesy Frank Krauss)

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Perturbative QCD

Order-by-order calculations: challenges more math than physics.

LO: solved for all practical applications.

NLO: in process of being automatized.

NNLO: the current calculational frontier.

Another bottleneck: efficient phase space sampling.

gg → H⁰ illustrates problems:

• Need high-precision calculations

• to search for BSM physics,

• but limited by poorly-understood slow convergence.

Perturbative calculations reliable for well separated jets, but . . .

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Divergences

Emission rate q → qg diverges when collinear: opening angle θqg→ 0 soft: gluon energy Eg → 0 Almost identical to e → eγ

but QCD is non-Abelian so additionally g → gg similarly divergent

αs(Q²) diverges for Q² → 0 (actually for Q² → Λ²_QCD)

Big probability for one emission =⇒ also big for several.

With ME’s need to calculate to high order and with many loops

=⇒ extremely demanding technically (not solved!), and involving big cancellations between positive and negative contributions.

Alternative approach: parton showers

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The Parton-Shower Approach

2 → n = (2 → 2) ⊕ ISR ⊕ FSR

FSR = Final-State Radiation = timelike shower Q_i²∼ m² > 0 decreasing

ISR = Initial-State Radiation = spacelike showers Q_i²∼ −m²> 0 increasing

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Why “time”like and “space”like?

Consider four-momentum conservation in a branching a → b c p⊥a= 0 ⇒ p_⊥c = −p⊥b

p+= E + pL ⇒ p_+a= p_+b+ p+c

p−= E − pL ⇒ p_−a= p−b+ p−c

Define p_+b = z p_+a, p_+c = (1 − z) p_+a Use p₊p−= E²− p_L² = m²+ p_⊥²

m²_a+ p²_⊥a p+a

= m²_b+ p_⊥b² z p+a

+ m²_c+ p²_⊥c (1 − z) p+a

⇒ m_a²= m_b²+ p²_⊥

z +m²_c+ p²_⊥ 1 − z = m²_b

z + m²_c

1 − z + p_⊥² z(1 − z) Final-state shower: mb= mc = 0 ⇒ m_a²= ^p

2

⊥

z(1−z) > 0 ⇒ timelike Initial-state shower: m_a = m_c = 0 ⇒ m²_b= −_1−z^p²^⊥ < 0 ⇒ spacelike

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Showers and cross sections

Shower evolution is viewed as a probabilistic process, which occurs with unit total probability:

the cross section is not directly affected However, more complicated than that

PDF evolution ≈ showers ⇒ enters in convoluted cross section, e.g. for 2 → 2 processes

σ = Z Z Z

dx1dx2dˆt fi(x1, Q²) fj(x2, Q²)dˆσ_ij dˆt Shower affects event shape

E.g. start from 2-jet event with p⊥1= p⊥2= 100 GeV.

ISR gives third jet, plus recoil to existing two, so p⊥1 = 110 GeV, p⊥2 = 90 GeV, p⊥1 = 20 GeV:

inclusive p⊥jet spectrum goes up hardest p⊥jet spectrum goes up

two-jets with both jets above some p⊥min comes down three-jet rate goes up

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Doublecounting

Do not doublecount: 2 → 2 = most virtual = shortest distance Conflict: theory derivations assume virtualities strongly ordered;

interesting physics often in regions where this is not true!

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Final-state radiation

Standard process e⁺e⁻→ qqg by two Feynman diagrams:

x_i = 2Ei

E_cm x₁+x₂+x₃= 2

dσ_ME

σ₀

=

_2π^α^s ⁴₃ _(1−x^x¹²^+x²²

1)(1−x₂)

dx

₁

dx

₂

Convenient (but arbitrary) subdivision to “split” radiation:

1

(1 − x₁)(1 − x₂)

(1 − x1) + (1 − x2)

x₃ = 1

(1 − x₂)x₃ + 1 (1 − x₁)x₃

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From matrix elements to parton showers

Rewrite for x2 → 1, i.e. q–g collinear limit:

1 − x₂ = m²₁₃ E_cm² = Q²

E_cm² ⇒ dx₂= dQ² E_cm² define z as fraction q retains

in branching q → qg

x₁ ≈ z ⇒ dx₁ ≈ dz x3 ≈ 1 − z

⇒ dP =dσ σ0

= α_s 2π

dx₂ (1 − x2)

4 3

x₂²+ x₁²

(1 − x1) dx1 ≈ α_s 2π

dQ² Q²

4 3

1 + z² 1 − z dz In limit x₁ → 1 same result, but for q → qg.

dQ²/Q² = dm²/m²: “mass (or collinear) singularity”

dz/(1 − z) = dω/ω “soft singularity”

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The DGLAP equations

Generalizes to

DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi)

dP_a→bc = α_s 2π

dQ²

Q² P_a→bc(z) dz P_q→qg = 4

3 1 + z²

1 − z

Pg→gg = 3(1 − z(1 − z))² z(1 − z) P_g→qq = n_f

2 (z²+ (1 − z)²) (n_f = no. of quark flavours) Universality: any matrix element reduces to DGLAP in collinear limit.

e.g. dσ(H⁰ → qqg)

dσ(H⁰ → qq) = dσ(Z⁰→ qqg)

dσ(Z⁰→ qq) in collinear limit

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The iterative structure

Generalizes to many consecutive emissions if strongly ordered, Q₁² Q₂² Q₃². . . (≈ time-ordered).

To cover “all” of phase space use DGLAP in whole region Q₁²> Q₂² > Q₃². . ..

Iteration gives final-state parton showers:

Need soft/collinear cuts to stay away from nonperturbative physics.

Details model-dependent, but around 1 GeV scale.

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The ordering variable

In the evolution with

dP_a→bc = αs

2π dQ²

Q² P_a→bc(z) dz

Q² orders the emissions (memory).

If Q²= m² is one possible evolution variable then Q⁰²= f (z)Q² is also allowed, since

d(Q⁰², z) d(Q², z)

=

∂Q⁰²

∂Q²

∂Q⁰²

∂z ∂z

∂Q²

∂z

=

f (z) f⁰(z)Q²

0 1

= f (z)

⇒ dP_a→bc = αs

2π

f (z)dQ²

f (z)Q² P_a→bc(z) dz = αs

2π dQ⁰²

Q⁰² P_a→bc(z) dz Q⁰²= E_a²θ_a→bc² ≈ m²/(z(1 − z)); angular-ordered shower Q⁰²= p_⊥² ≈ m²z(1 − z); transverse-momentum-ordered

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The Sudakov form factor – 1

Time evolution, conservation of total probability:

P(no emission) = 1 − P(emission).

Multiplicativeness, with T_i = (i /n)T , 0 ≤ i ≤ n:

P_no(0 ≤ t < T ) = lim

n→∞

n−1

Y

i =0

P_no(T_i ≤ t < T_{i +1})

= lim

n→∞

n−1

Y

i =0

(1 − Pem(Ti ≤ t < T_{i +1}))

= exp − lim

n→∞

n−1

X

i =0

P_em(Ti ≤ t < T_{i +1})

!

= exp

− Z _T

0

dPem(t) dt dt

=⇒ dPfirst(T ) = dPem(T )exp

− Z T

0

dPem(t) dt dt

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The Sudakov form factor – 2

Expanded, with Q ∼ 1/t (Heisenberg) dP_a→bc = α_s

2π dQ²

Q² P_a→bc(z) dz

× exp



−X

b,c

Z Q_max² Q²

dQ⁰² Q⁰²

Z α_s

2πP_a→bc(z⁰) dz⁰





where the exponent is (one definition of) the Sudakov form factor A given parton can only branch once, i.e. if it did not already do so Note thatP

b,c

R R dP_a→bc ≡ 1 ⇒ convenient for Monte Carlo (≡ 1 if extended over whole phase space, else possibly nothing happens before you reach Q0 ≈ 1 GeV).

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The Sudakov form factor – 3

Sudakov regulates singularity for first emission . . .

. . . but in limit of repeated soft emissions q → qg (but no g → gg) one obtains the same inclusive Q emission spectrum as for ME, i.e. divergent ME spectrum

⇐⇒ infinite number of PS emissions

More complicated in reality:

energy-momentum conservation effects big since αs big, so hard emissions frequent

g → gg branchings leads to accelerated multiplication of partons

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Coherence

QED: Chudakov effect (mid-fifties)

QCD: colour coherence for soft gluon emission

solved by • requiring emission angles to be decreasing or • requiring transverse momenta to be decreasing

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Common Showering Algorithms

Standard shower language with a → bc successive branchings:

HERWIG: Q² ≈ E²(1 − cos θ) ≈ E²θ²/2 old PYTHIA: Q²= m² (+ brute-force coherence) Newer ARIADNE picture of dipole emission ab → cde:

is the basis for most current-day algorithms (HERWIG excepted)

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Parton Distribution Functions

Hadrons are composite, with time-dependent structure:

f_i(x , Q²) = number density of partons i at momentum fraction x and probing scale Q².

Linguistics (example):

F₂(x , Q²) = X

i

e_i²xf_i(x , Q²) structure function parton distributions

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PDF evolution

Initial conditions at small Q₀² unknown: nonperturbative.

Resolution dependence perturbative, by DGLAP:

DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi) df_b(x , Q²)

d(ln Q²) =X

a

Z 1 x

dz

z fa(y , Q²) αs

2πP_a→bc

z = x

y

DGLAP already introduced for (final-state) showers:

dP_a→bc = αs

2π dQ²

Q² P_a→bc(z) dz Same equation, but different context:

dP_a→bc is probability for the individual parton to branch; while df_b(x , Q²) describes how the ensemble of partons evolve by the branchings of individual partons as above.

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Initial-State Shower Basics

• Parton cascades in p are continuously born and recombined.

• Structure at Q is resolved at a time t ∼ 1/Q before collision.

• A hard scattering at Q² probes fluctuations up to that scale.

• A hard scattering inhibits full recombination of the cascade.

• Convenient reinterpretation:

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Forwards vs. backwards evolution

Event generation could be addressed by forwards evolution:

pick a complete partonic set at low Q0 and evolve, consider collisions at different Q² and pick by σ of those.

Inefficient:

1 have to evolve and check for all potential collisions, but 99.9. . . % inert

2 impossible (or at least very complicated) to steer the production, e.g. of a narrow resonance (Higgs)

Backwards evolution is viable and ∼equivalent alternative:

start at hard interaction and trace what happened “before”

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Backwards evolution master formula

Monte Carlo approach, based on conditional probability : recast df_b(x , Q²)

dt =X

a

Z ₁

x

dz

z f_a(x⁰, Q²) αs

2πP_a→bc(z) with t = ln(Q²/Λ²) andz = x /x⁰ to

dP_b= df_b

f_b = |dt|X

a

Z

dz x⁰fa(x⁰, t) xf_b(x , t)

αs

2πP_a→bc(z) then solve for decreasing t, i.e. backwards in time,

starting at high Q² and moving towards lower, with Sudakov form factor exp(−R dP_b)

Webber: can be recast by noting that total change of PDF at x is difference between gain by branchings from higher x and loss by branchings to lower x .

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Coherence in spacelike showers

with Q²= −m²= spacelike virtuality kinematics only:

Q₃²> z1Q₁², Q₅²> z3Q₃², . . . i.e. Q_i² need not even be ordered

coherence of leading collinear singularities:

Q₅²> Q₃²> Q₁², i.e. Q² ordered

coherence of leading soft singularities (more messy):

E₃θ₄> E₁θ₂, i.e. z₁θ₄> θ₂

z 1: E₁θ₂≈ p_⊥2² ≈ Q₃², E₃θ₄≈ p_⊥4² ≈ Q₅² i.e. reduces to Q² ordering as above z ≈ 1: θ4> θ2, i.e. angular ordering of soft gluons

=⇒ reduced phase space

(47)

Evolution procedures

DGLAP: Dokshitzer–Gribov–Lipatov–Altarelli–Parisi

evolution towards larger Q²and (implicitly) towards smaller x BFKL: Balitsky–Fadin–Kuraev–Lipatov

evolution towards smaller x (with small, unordered Q²) CCFM: Ciafaloni–Catani–Fiorani–Marchesini

interpolation of DGLAP and BFKL GLR: Gribov–Levin–Ryskin

nonlinear equation in dense-packing (saturation) region, where partons recombine, not only branch

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Did we reach BFKL regime?

Study events with ≥ 2 jets as a function of their y separation.

Ratio of the inclusive to exclusive dijet cross sections:

Δy|

|

0 1 2 3 4 5 6 7 8 9

inclR

1 1.5 2 2.5 3 3.5 4 4.5 5

2010 data PYTHIA6 Z2 PYTHIA8 4C HERWIG++ UE-7000-EE-3 HEJ + ARIADNE CASCADE

= 7 TeV s CMS, pp,

dijets > 35 GeV pT

|y| < 4.7

Azimuthal decorrelation:

No strong indications for BFKL/CCFM behaviour onset so far!

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Initial- vs. final-state showers

Both controlled by same evolution equations dP_a→bc = αs

2π dQ²

Q² P_a→bc(z) dz · (Sudakov) but

Final-state showers:

Q² timelike (∼ m²)

decreasing E , m², θ both daughters m² ≥ 0 physics relatively simple

⇒ “minor” variations:

Q², shower vs. dipole, . . .

Initial-state showers:

Q² spacelike (≈ −m²)

decreasing E , increasing Q², θ one daughter m² ≥ 0, one m² < 0 physics more complicated

⇒ more formalisms:

DGLAP, BFKL, CCFM, GLR, . . .

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Combining FSR with ISR

Separate processing of ISR and FSR misses interference (∼ colour dipoles)

ISR+FSR add coherently in regions of colour flow and destructively else in “normal” shower by azimuthal anisotropies automatic in dipole (by proper boosts)

(51)

Coherence tests

Current-day generators for pseudorapidity of third jet:

and past incoherent:

Coherence tests – 1

old normal showers with/without ' reweighting:

⌘

₃

: pseudorapidity of third jet

↵: angle of third jet around second jet

Torbj¨orn Sj¨ostrand Monte Carlo 1 slide 51/1

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Summary and Outlook

A multitude of physics mechanisms at play in pp collisions.

Event generators separate problem into manageable chunks.

Random numbers ≈ quantum mechanical choices.

Often need to combine several software packages.

Matrix element calculations at core of process selection.

Parton shower offers convenient alternative to HO ME’s.

Unitarity by Sudakov form factor.

Tutorial today: begin using PYTHIA; Higgs production as example.

Tomorrow:

Combining matrix elements and parton showers.

Multiparton interactions and other soft physics.

Hadronization.

Conclusions.