Exclusive Fashion Trends for Spring 2008

CERN October 2007

Exclusive FashionTrends for Spring 2008

Peter Skands

CERN & Fermilab

Exclusive Fashion - Trends for Spring 2008 - 2Peter Skands

OverviewOverview► Calculating collider observables

• Fixed order perturbation theory and beyond

• From inclusive to exclusive descriptions of the final state

► Uncertainties and ambiguities beyond fixed order

• The ingredients of a leading log parton shower

• A brief history of matching

• New creations: Fall 2007

► Designer showers, an example

• Exploring showers with antennae

• Some comments on matching at tree and 1-loop level

► Trends for Spring 2008?


Fixed Order (all orders)

“Experimental” distribution of observable O in production of X:

k : legs ℓ : loops {p} : momenta

Inclusive Fashion

High-dimensional problem (phase space)

d≥5 Monte Carlo integration

Principal virtues

1. Stochastic error O(N-1/2) independent of dimension

2. Full (perturbative) quantum treatment at each order

3. KLN theorem: finite answer at each (complete) order

Note 1: For k larger than a few, need to be quite clever in phase space sampling

Note 2: For ℓ > 0, need to be careful in arranging for real-virtual cancellations

“Monte Carlo”: N. Metropolis, first Monte Carlo calcultion on ENIAC (1948), basic idea goes back to Enrico Fermi


Exclusive Fashion

High-dimensional problem (phase space)

d≥5 Monte Carlo integration

+ Formulation of fragmentation as a “Markov Chain”:

1. Parton Showers:

iterative application of perturbatively calculable splitting kernels for n n+1 partons

2. Hadronization:

iteration of X X’ + hadron, according to phenomenological models (based on known properties of QCD, on lattice, and on fits to data).

A. A. Markov: Izvestiia Fiz.-Matem. Obsch. Kazan Univ., (2nd Ser.), 15(94):135 (1906)


Traditional GeneratorsTraditional Generators

► Generator philosophy:

• Improve Born-level perturbation theory, by including the ‘most significant’ corrections complete events

1. Parton Showers 2. Hadronisation3. The Underlying Event

1. Soft/Collinear Logarithms2. Power Corrections3. Higher Twist

roughlyroughly

(+ many other ingredients: resonance decays, beam remnants, Bose-Einstein, …)

Asking for fully exclusive events is asking for quite a lot …


Be wary of oraclesBe wary of oracles

PYTHIA Manual, Sjöstrand et al, JHEP 05(2006)026

Be even more wary if you are not told to be wary!

► We are really only operating at the first few orders (fixed + logs + twists + powers) of a full quantum expansion


Non-perturbativehadronisation, rearrangement de couleurs, restes de faisceau, fonctions de fragmentation non-perturbative, pion/proton, kaon/pion, ...

Soft Jets and Jet Structureémissions molles/collineaires (brems), événement sous-jacent (interactions multiples perturbatives 22 + … ?), brems jets semi-durs

Resonance Masses…

Hard Jet Tailhaut-pT jets à grande angle

& W

idths

+ Un-Physical Scales:+ Un-Physical Scales:

• QF , QR : Factorization(s) & Renormalization(s)

sInclusive

Exclusive

Hadron Decays

Collider Energy ScalesCollider Energy Scales


Problem 1: bremsstrahlung corrections are singular for soft/collinear configurations spoils fixed-order truncation

e+e- 3 jets

Beyond Fixed OrderBeyond Fixed Order


Beyond Fixed OrderBeyond Fixed Order

► Evolution Operator, S (as a function of “time” t=1/Q)

• “Evolves” phase space point: X …• Can include entire (interleaved) evolution, here focus on showers

• Observable is evaluated on final configuration

• S unitary (as long as you never throw away an event) normalization of total (inclusive) σ unchanged• Only shapes are affected (i.e., also σ after shape-dependent cuts)


Pure Shower (all orders)

wX : |MX|2

S : Evolution operator

{p} : momenta


Perturbative EvolutionPerturbative Evolution

► Evolution Operator, S (as a function of “time” t=1/Q)

• Defined in terms of Δ(t1,t2) – The integrated probability the system does not change state between t1 and t2 (Sudakov)


wX : |MX|2

S : Evolution operator

{p} : momenta

“X + nothing” “X+something”

A: splitting function

Analogous to nuclear decay:


Constructing LL ShowersConstructing LL Showers► The final answer will depend on:

• The choice of evolution variable

• The splitting functions (finite terms not fixed)

• The phase space map ( dΦn+1/dΦn )

• The renormalization scheme (argument of αs)

• The infrared cutoff contour (hadronization cutoff)

► They are all “unphysical”, in the same sense as QFactorizaton, etc.

• At strict LL, any choice is equally good

• However, 20 years of parton showers have taught us: many NLL effects can be (approximately) absorbed by judicious choices

• Effectively, precision is much better than strict LL, but still not formally NLL

• E.g., story of “angular ordering”, using pT as scale in αs, …

Clever choices good for process-independent things, but what about the process-dependent bits? … + matching


MatchingMatching

► Traditional Approach: take the showers you have, expand them to 1st order, and fix them up

• Sjöstrand (1987): Introduce re-weighting factor on first emission 1st order tree-level matrix element (ME) (+ further showering)

• Seymour (1995): + where shower is “dead”, add separate events from 1st order tree-level ME, re-weighted by “Sudakov-like factor” (+ further showering)

• Frixione & Webber (2002): Subtract 1st order expansion from 1st order tree and 1-loop ME add remainder ME correction events (+ further showering)

► Multi-leg Approaches (Tree level only):

• Catani, Krauss, Kuhn, Webber (2001): Substantial generalization of Seymour’s approach, to multiple emissions, slicing phase space into “hard” M.E. ; “soft” P.S.

• Mangano (?): pragmatic approach to slicing: after showering, match jets to partons, reject events that look “double counted”

A brief history of conceptual breakthroughs in widespread use today:


New Creations: Fall 2007New Creations: Fall 2007► Showers designed specifically for matching

• Nagy, Soper (2006): Catani-Seymour showers• Dinsdale, Ternick, Weinzierl (Sep 2007) & Schumann, Krauss (Sep 2007): implementations

• Giele, Kosower, PS (Jul 2007): Antenna showers • (incl. implementation)

► Other new showers: partially designed for matching• Sjöstrand (Oct 2007): New interleaved evolution of FSR/ISR/UE

• Official release of Pythia8 last week

• Webber et al (HERWIG++): Improved angular ordered showers

• Nagy, Soper (Jun 2007): Quantum showers subleading color, polarization (implementation in 2008?)

► New matching proposals• Nason (2004): Positive-weight variant of MC@NLO

• Frixione, Nason, Oleari (Sep 2007): Implementation: POWHEG

• Giele, Kosower, PS (Jul 2007): Antenna generalization of MC@NLO• VINCIA

For more details PhenClub. Thursday, 11 am, 1-1-025:01 Nov : Sjöstrand, Richardson, PS:

Modern Showers (Pythia8 (+ Vincia?), Herwig++)


Towards Improved GeneratorsTowards Improved Generators► The final answer will depend on:

• The choice of evolution variable

• The splitting functions (finite terms not fixed)

• The phase space map ( dΦn+1/dΦn )

• The renormalization scheme (argument of αs)

• The infrared cutoff contour (hadronization cutoff)

► Step 1, Quantify uncertainty: vary all of these (within reasonable limits)

► Step 2, Systematically improve: Understand the importance of each and how it is canceled by

• Matching to fixed order matrix elements

• Higher logarithms, subleading color, etc, are included

► Step 3, Write a generator: Make the above explicit (while still tractable) in a Markov Chain context matched parton shower MC algorithm


Gustafson, PLB175(1986)453; Lönnblad (ARIADNE), CPC71(1992)15.Azimov, Dokshitzer, Khoze, Troyan, PLB165B(1985)147 Kosower PRD57(1998)5410; Campbell,Cullen,Glover EPJC9(1999)245

VINCIAVINCIA

► Based on Dipole-Antennae• Shower off color-connected pairs of partons

• Plug-in to PYTHIA 8.1 (C++)

► So far:

• Final-state QCD cascades (massless quarks)

• 2 different shower evolution variables:• pT-ordering (~ ARIADNE, PYTHIA 8)

• Mass-ordering (~ PYTHIA 6, SHERPA)

• For each: an infinite family of antenna functions • Laurent series in branching invariants with arbitrary finite terms

• Shower cutoff contour: independent of evolution variable IR factorization “universal”

• Several different choices for αs (evolution scale, pT, mother antenna mass, 2-loop, …)

• Phase space mappings: 2 different choices implemented • Antenna-like (ARIADNE angle) or Parton-shower-like: Emitter + longitudinal Recoiler

Dipoles (=Antennae, not CS) – a dual description of QCD

a

b

r

VIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED ANTENNAE

Giele, Kosower, PS : hep-ph/0707.3652


Example: Z decaysExample: Z decays► Dependence on evolution variable



Example: Z decaysExample: Z decays► VINCIA and PYTHIA8 (using identical settings to the max extent possible)

αs(pT),

pThad = 0.5 GeV

αs(mZ) = 0.137

Nf = 2

Note: the default Vincia antenna functions reproduce the Z3 parton matrix element;

Pythia8 includes matching to Z3

Beyond the 3rd parton, Pythia’s radiation function is slightly larger, and its kinematics and hadronization cutoff contour are also slightly different


Dipole-Antenna FunctionsDipole-Antenna Functions► Starting point: “GGG” antenna functions, e.g.,

► Generalize to arbitrary Laurent series:

Can make shower systematically “softer” or “harder”

• Will see later how this variation is explicitly canceled by matching

quantification of uncertainty

quantification of improvement by matching

yar = sar / si

si = invariant mass of i’th dipole-antenna


Gehrmann-De Ridder, Gehrmann, Glover, JHEP 09 (2005) 056

Singular parts fixed, finite terms arbitrary


Quantifying MatchingQuantifying Matching► The unknown finite terms are a major source of uncertainty

• DGLAP has some, GGG have others, ARIADNE has yet others, etc…

• They are arbitrary (and in general process-dependent)

Using αs(MZ)=0.137, μR=1/4mdipole, pThad = 0.5 GeV


MatchingMatching


Matched shower (including simultaneous tree- and 1-loop matching for any number of legs)

Tree-level “real” matching term for X+k

Loop-level “virtual+unresolved” matching term for X+k




Tree-level matching to X+1Tree-level matching to X+11. Expand parton shower to 1st order (real radiation term)

2. Matrix Element (Tree-level X+1 ; above thad)

Matching Term:

variations in finite terms (or dead regions) in Ai canceled (at this order)

• (If A too hard, correction can become negative negative weights)

Inverse phase space map ~ clustering



Soft Standard Hard

Matched Soft Standard Matched Hard

Phase Space PopulationPhase Space Population

Positive correction Negative correction


Quantifying MatchingQuantifying Matching► The unknown finite terms are a major source of uncertainty

• DGLAP has some, GGG have others, ARIADNE has yet others, etc…

• They are arbitrary (and in general process-dependent)



1-loop matching to X1-loop matching to X► NLO “virtual term” from parton shower (= expanded Sudakov: exp=1 - … )

► Matrix Elements (unresolved real plus genuine virtual)

► Matching condition same as before (almost):

► You can choose anything for Ai (different subtraction schemes) as long as you use the same one for the shower

Tree-level matching just corresponds to using zero• (This time, too small A correction negative)



Note about “NLO” matchingNote about “NLO” matching► Shower off virtual matching term uncanceled O(α2) contribution

to 3-jet observables (only canceled by 1-loop 3-parton matching)

► While normalization is improved, shapes are not (shape still LO)


Tree-Level Matching “NLO” Matching


What happened?What happened?► Brand new code, so bear in mind the green guy

► Naïve conclusion: tree-level matching “better” than NLO?

• No, first remember that the shapes we look at are not “NLO” • E.g., 1-T appears at O(α) below 1-T=2/3, and at O(α2) above • An “NLO” thrust calculation would have to include at least 1-loop corrections to Z3• (The same is true for a lot of other distributions)

• So: both calculations are LO/LL from the point of view of 1-T

► What is the difference then?

• Tree-level Z3 is the same (LO)

• The O(α) corrections to Z3, however, are different• The first non-trivial corrections to the shape!• So there should be a large residual uncertainty the 1-loop matching is “honest”

• The real question: why did the tree-level matching not tell us?• I haven’t completely understood it yet … but speculate it’s to do with detailed balance • In tree-level matching, unitarity Virtual = - Real cancellations. Broken at 1 loop• + everything was normalized to unity, but tree-level different norms


What to do next?What to do next?► Go further with tree-level matching

• Demonstrate it beyond first order (include H,Z 4 partons)

• Automated tree-level matching (w. Rikkert Frederix (MadGraph) + …?)

► Go further with one-loop matching

• Demonstrate it beyond first order (include 1-loop H,Z 3 partons)• Should start to see cancellation of ordering variable and renormalization scale

• Should start to see stabilization of shapes as well as normalizations

► Extend the formalism to the initial state

► Extend to massive particles

• Massive antenna functions, phase space, and evolution


Summary: Trends for 2008Summary: Trends for 2008►Designer showers

• Does matching get easier?

• Can matching be extended deeper into the perturbative series ?• A practical demonstration combining 1-loop and multi-leg matching ?

• A practical demonstration of 1-loop matching beyond first order ?

• Unexplored territory beyond first few orders, leading NC, (N)LL

►Generators in 2008: theorists are learning C++

• Enter PYTHIA-8 (SHERPA & HERWIG++ already there)

►What more is needed for high precision at LHC ?• Need improvements beyond showers: e.g., higher twist / UE / … ?

• Is hadronization systematically improvable?

• + even the most super-duper Monte Carlo is useless without constraints on its remaining uncertainties! (“validation”)

Documents

Exclusive Fashion Trends for Spring 2008