Designer Showers and Subtracted Matrix Elements

Freiburg, Apr 16 2008

Designer Showers and Subtracted Matrix Elements

Peter Skands

CERN & Fermilab

Time-Like Showers and Matching with Antennae - 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 parton shower

• A brief history of matching

• New creations: Fall 2007

► A New Approach

• Time-Like Showers Based on Dipole-Antennae

• Some hopefully good news

• VINCIA status and plans


► Main Tool: Matrix Elements calculated in fixed-order perturbative quantum field theory

• Example:

QQuantumuantumCChromohromoDDynamicsynamics

Reality is more complicated

High-transverse momentum interaction


Fixed Order (all orders)

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

k : legs ℓ : loops {p} : momenta

Monte Carlo at Fixed OrderMonte Carlo at Fixed Order

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


Parton ShowersParton Showers

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)

S: Evolution operator. Generates event, starting from {p}X


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. All of the above (+ more?)

roughlyroughly

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

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


Non-perturbativehadronisation, colour reconnections, beam remnants, non-perturbative fragmentation functions, pion/proton ratio, kaon/pion ratio, ...

Soft Jets and Jet StructureSoft/collinear radiation (brems), underlying event (multiple perturbative 22 interactions + … ?), semi-hard brems jets, …

Resonance Masses…

Hard Jet TailHigh-pT jets at large angles

& W

idths

sInclusive

Exclusive

Hadron Decays

Collider Energy ScalesCollider Energy Scales

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

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

• QE : Evolution(s)


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

e+e- 3 jets

Beyond Fixed OrderBeyond Fixed Order


Diagrammatical Explanation 1Diagrammatical Explanation 1► dσX = …

► dσX+1 ~ dσX g2 2 sab /(sa1s1b) dsa1ds1b

► dσX+2 ~ dσX+1 g2 2 sab/(sa2s2b) dsa2ds2b


► But it’s not yet an “evolution”

• What’s the total cross section we would calculate from this?

• σX;tot = int(dσX) + int(dσX+1) + int(dσX+2) + ...

Probability not conserved, events “multiply” with nasty singularities! Just an approximation of a sum of trees.

But wait, what happened to the virtual corrections? KLN?

dσX

α sab

saisibdσX+1 dσ

X+2

dσX+2

This is an approximation of inifinite-order tree-level cross sections

“DLA”


Diagrammatical Explanation 2Diagrammatical Explanation 2► dσX = …

► dσX+1 ~ dσX g2 2 sab /(sa1s1b) dsa1ds1b



+ Unitarisation: σtot = int(dσX)

σX;PS = σX - σX+1 - σX+2 - …

► Interpretation: the structure evolves! (example: X = 2-jets)• Take a jet algorithm, with resolution measure “Q”, apply it to your events

• At a very crude resolution, you find that everything is 2-jets

• At finer resolutions some 2-jets migrate 3-jets = σX+1(Q) = σX;incl– σX;excl(Q)

• Later, some 3-jets migrate further, etc σX+n(Q) = σX;incl– ∑σX+m<n;excl(Q)• This evolution takes place between two scales, Q in and Qfin = QF;ME and Qhad

► σX;PS = int(dσX) - int(dσX+1) - int(dσX+2) + ...

= int(dσX) EXP[ - int(α 2 sab /(sa1s1b) dsa1 ds1b ) ]

dσX

α sab

saisibdσX+1 dσ

X+2

dσX+2

Given a jet definition, an

event has either 0, 1, 2, or … jets

“DLA”


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 (σLO, σNLO, σNNLO, σexp, …)

• Only shapes are predicted (i.e., also σ after shape-dependent cuts)

Fixed Order (all orders)

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


{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., (E,p) cons., “angular ordering”, using pT as scale in αs, with ΛMS ΛMC, …

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. implementations)

► 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• Winter, Krauss (Dec 2007) : Dipole-antenna showers

• (incl. implementation in SHERPA.) Similar to ARIADNE, but more antenna-like for ISR

• Nagy, Soper (Jun 2007 + Jan 2008): Quantum showers subleading color, polarization (so far no implementation)

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

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

• Giele, Kosower, PS (Jul 2007): Antenna subtraction• VINCIA + an extension of that I will present here for the first time


Some Holy GrailsSome Holy Grails► Matching to first order + (N)LL ~ done

• 1st order : MC@NLO, POWHEG, PYTHIA, HERWIG

• Multi-leg tree-level: CKKW, MLM, … (but still large uncertainties)

► Simultaneous 1-loop and multi-leg matching

• 1st order : NLO (Born) + LO (Born + m) + (N)LL (Born + ∞)

• 2nd order : NLO (Born+1) + LO (Born + m) + (N)LL (Born + ∞)

► Showers that systematically resum higher logs

• (N)LL NLL NNLL … ?

• (N)LC NLC … ?

► Solving any of these would be highly desirable

• Solve all of them ?

• NNLO (Born) + LO (Born + m) + (N)NLL + string-fragmentation

• + reliable uncertainty bands


Parton ShowersParton Showers► The final answer depends on:

• The choice of evolution variable

• The splitting functions (finite/subleading 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, at LO, NLO, NNLO, …

• 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:

• 3 different shower evolution variables:• pT-ordering (= ARIADNE ~ PYTHIA 8)

• Dipole-mass-ordering (~ but not = PYTHIA 6, SHERPA)

• Thrust-ordering (3-parton Thrust)

• 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 + Les Houches 2007


Dipole-Antenna ShowersDipole-Antenna Showers► Dipole branching and phase space

Giele, Kosower, PS : hep-ph/0707.3652

( Most of this talk, including matching by antenna subtraction, should be applicable to ARIADNE and the SHERPA dipole-shower as well)


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

► Generalize to arbitrary double 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

Frederix, Giele, Kosower, PS : Les Houches NLM, arxiv:0803.0494


ComparisonComparisonFrederix, Giele, Kosower, PS : Les Houches ‘NLM’, arxiv:0803.0494


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 don’t tune!)

αs(MZ)=0.137,

μPS=pT,

pThad = 0.5 GeV

Varying finite terms only

with

(huge variation with μPS from pure LL point of view, but NLL tells you using pT at LL (N)LL. Formalize that.)


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 (= correction events to be added)

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



Matching by Reweighted ShowersMatching by Reweighted Showers

► Go back to original shower definition

► Possible to modify S to expand to the “correct” matrix elements ?


wX : |MX|2


{p} : momenta

Sjöstrand, Bengtsson : Nucl.Phys.B289(1987)810; Phys.Lett.B185(1987)435

Norrbin, Sjöstrand : Nucl.Phys.B603(2001)2971st order: yes

Generate an over-estimating (trial) branching

Reweight it by vetoing it with the probability

But 2nd and beyond difficult due to lack of clean PS expansion

w>0 as long as |M|2 > 0


Towards NNLO + NLLTowards NNLO + NLL► Basic idea: extend reweigthing to 2nd order

• 23 tree-level antennae NLO

• 23 one-loop + 24 tree-level antennae NNLO

► And exponentiate it

• Exponentiating 23 (dipole-antenna showers) (N)LL

• Complete NNLO captures the singularity structure up to (N)NLL

• So a shower incorporating all these pieces exactly should be able to reach NLL resummation, with a good approximation to NNLL; + exact matching up to NNLO should be possible

• Start small, do it for leading-color first, included the qqbar 24 antennae, A04 , B0

4 .

• Gives exact matching of Z4 since these happen to be the exact matrix elements for that process.

• Still missing the remaining 24 functions, matching to the running coupling in one-loop 23, and inclusion of next-to-leading color

• Full one-loop 23 matching (i.e., the finite terms for Z decay)


224 Matching 4 Matching by reweightingby reweighting

► Starting point:

• LL shower w/ large coupling and large finite terms to generate “trial” branchings (“sufficiently” large to over-estimate the full ME).

• Accept branching [i] with a probability

► Each point in 4-parton phase space then receives a contribution

• Also need to take into account ordering cancellation of dependence

1st order matching term (à la Sjöstrand-Bengtsson) 2nd order matching term (with 1st order subtracted)

(If you think this looks deceptively easy, you are right)


223 one-loop Matching 3 one-loop Matching by reweightingby reweighting

► Unitarity of the shower effective 2nd order 3-parton term contains

• An integral over A04 over the 34 phase space below the 3-parton evolution

scale (all the way from QE3 to 0)

• An integral over the 23 antenna function above the 3-parton evolution scale (from MZ to QE3)

• (These two combine to give the an evolution-dependence, canceled by matching to the actual 3-parton 1-loop ME)

• A term coming from the expansion of the 23 αs(μPS)

• Combine with 34 evolution to cancel scale dependence

• A term coming from a tree-level branching off the one-loop 2-parton correction.

► It then becomes a matter of collecting these pieces and subtracting them off, e.g., A1

3 .

• After cancellation of divergences, an integral over the shower-subtracted A04

remains Numerical? No need to exponentiate must be evaluated once per event. The other pieces (except αs) are already in the code.


Tree-level 2Tree-level 23 + 23 + 24 in Action4 in Action► 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)

αs(MZ)=0.137,

μR=pT,

pThad = 0.5 GeV

Varying finite terms only

with


LEP ComparisonsLEP Comparisons

Still with αs(MZ)=0.137 : THE big thing remaining …


What to do next?What to do next?► Further shower studies

• Include the remaining 4-parton antenna functions

• Measuring, rather than tuning, hadronization?

► Go further with one-loop matching

• Include exact running coupling from 3-parton one-loop• + Exponentiate

• Include full 3-parton one-loop (i.e., including finite terms) Shower Monte Carlo at NNLO + NLL

► Extend to the initial state

• The Krauss-Winter shower looks close; we would concentrate on the uncertainties and matching.

► Extend to massive particles

• Massive antenna functions, phase space, and evolution (+matching?)


Extra MaterialExtra Material► Number of partons and number of quarks

• Nq shows interesting dependence on ordering variable

Frederix, Giele, Kosower, PS : Les Houches Proc., in preparation


TThe he BBottom ottom LLine ine

The S matrix is expressible as a series in gi, gin/Qm, gi

n/xm, gin/mm, gi

n/fπm

, …

To do precision physics:

Solve more of QCD

Combine approximations which work in different regions: matching

Control it

Good to have comprehensive understanding of uncertainties

Even better to have a way to systematically improve

Non-perturbative effects

don’t care whether we know how to calculate them

FO DGLAP

BFKL

HQET

χPT


MatchingMatchingPure Shower (all orders)

wX : |MX|2


{p} : momenta

“X + nothing” “X+something”

A: splitting function

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



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

Frederix, Giele, Kosower, PS : Les Houches NLM, arxiv:0803.0494


Example: Z decaysExample: Z decays► Why is the dependence on the evolution variable so small?

• Conventional wisdom: evolution variable has huge effect• Cf. coherent vs non-coherent parton showers, mass-ordered vs pT-ordered, etc.

► Dipole-Antenna showers resum radiation off pairs of partons interference between 2 partons included in radiation function

• If radiation function = dipole formula intrinsically coherent •

• Remaining dependence on evolution variable much milder than for conventional showers

► The main uncertainty in this case lies in the choice of radiation function away from the collinear and soft regions

dipole-antenna showers under the hood …

Gustafson, PLB175(1986)453

Documents

Designer Showers and Subtracted Matrix Elements