Towards Multijet Matching with Loops

Peter SkandsCERN / Fermilab

Towards Multijet Matching Towards Multijet Matching with Loopswith Loops

HP2.2, Buenos Aires, October 2007

The Second Talk of the Workshop

ME-to-PS matching in VINCIA - 2Peter Skands

Precision ChromodynamicsPrecision Chromodynamics► Monte Carlo problem

• Uncertainty on fixed orders and logs obscures clear view on hadronization and the underlying event

► So we just need …• An NNLO + NLO multileg + NLL

Monte Carlo, with uncertainty bands, please

► Then …• We could see hadronization and UE

clearly solid models

Energy Frontier

Inten

sity F

rontie

rThe Astro Guys

Precision Frontier

The Tevatron and LHC data will be all the energy frontier data we’ll have for a long while

Anno 2018


LL Shower Monte CarlosLL Shower Monte Carlos

► Evolution Operator, S• “Evolves” phase space point: X …

As a function of “time” t=1/Q Observable is evaluated on final configuration

• S unitary (as long as you never throw away or reweight an event) normalization of total (inclusive) σ unchanged (σLO, σNLO, σNNLO, σexp, …) Only shapes are predicted (i.e., also σ after shape-dependent cuts)

• Can expand S to any fixed order (for given observable) Can check agreement with ME Can do something about it if agreement less than perfect: reweight or add/subtract

► Arbitrary Process: X

Pure Shower (all orders)

O: Observable

{p} : momenta

wX = |MX|2 or K|MX|2

S : Evolution operator

Leading Order


““S” S” (for Shower)(for Shower)

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

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

NB: Will not focus on where Δ comes from here, just on how it expands

• = Generating function for parton shower Markov Chain

“X + nothing” “X+something”

A: splitting function


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

• The choice of evolution “time”

• The splitting functions (finite terms not fixed)

• The phase space map (“recoils”, dΦn+1/dΦn )

• The renormalization scheme (argument of αs)

• The infrared cutoff contour (hadronization cutoff)

Variations

Comprehensive uncertainty estimates (showers with

uncertainty bands)


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 (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, …)

• 3 different phase space maps Ariadne or Kosower “antenna” recoils, or Emitter + longitudinal Recoiler

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

a

b

r

VIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED ANTENNAEVIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED ANTENNAE

Giele, Kosower, PS : PRD78(2008)014026 + Les Houches ‘NLM’ 2007


Example: Jet RatesExample: Jet Rates► The unknown finite terms are important

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

• Uncertainty in hard region already at first order

• Cascade down to produce uncontrolled tower of subleading logs

αs(MZ)=0.137,

μR=pT,

pThad = 0.5 GeV

Varying finite terms only

with


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

• The choice of evolution “time”

• The splitting functions (finite terms not fixed)

• The phase space map (“recoils”, 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

• We’ve learned, however: some NLL effects can be (approximately) absorbed by judicious choices

E.g., (E,p) cons., coherence, using pT as scale in αs, with ΛMS ΛMC, … Effectively, precision is better than strict LL, but still not formally NLL

Variations

Comprehensive uncertainty estimates (showers with

uncertainty bands)

Clever choices fine (for process-independent things), can we do better? … + matching


Matching in a nutshellMatching in a nutshell► There are two fundamental approaches

• Additive

• Multiplicative

► Most current approaches based on addition, in one form or another

• Herwig (Seymour, 1995), but also CKKW, MLM, MC@NLO, ...

• In these approaches, you add event samples with different multiplicities Need separate ME samples for each multiplicity. Relative weights a priori unknown.

• The job is to construct weights for them, and modify/veto the showers off them, to avoid double counting of both logs and finite terms

► But you can also do it by multiplication• Pythia (Sjöstrand, 1987): modify only the shower

• All events start as Born + reweight at each step. Using the shower as a weighted phase space generator only works for showers with NO DEAD ZONES

• The job is to construct reweighting coefficients Complicated shower expansions only first order so far Generalized to include 1-loop first-order POWHEG

Seymour, Comput.Phys.Commun.90(1995)95

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

Norrbin, Sjöstrand : Nucl.Phys.B603(2001)297Massive Quarks

All combinations of colors and Lorentz structures


NLO with AdditionNLO with Addition► First Order Shower expansion

Unitarity of shower 3-parton real = ÷ 2-parton “virtual”

► 3-parton real correction (GGG + example finite terms; α, β)

► 2-parton virtual correction (same example)

PS

Finite terms cancel in 3-parton O

Finite terms cancel in 2-parton O (normalization)

Multiplication at this order A = |M3|2/|M2|2


► Herwig• In dead zone: Ai = 0 add events corresponding to unsubtracted |MX+1|

• Outside dead zone: reweighted à la Pythia Ai = |MX+1| no additive correction necessary

► CKKW and L-CKKW• At this order identical to Herwig, with “dead zone” for kT > kTcut introduced by hand

► MC@NLO• In dead zone: identical to Herwig

• Outside dead zone: AHerwig > |MX+1| wX+1 negative negative weights

► Pythia • Ai = |MX+1| over all of phase space no additive correction necessary

► Powheg• At this order identical to Pythia

no negative weights

HE

RW

IG T

YP

EP

YTH

IA T

YP

EMatching to X+1: Tree-levelMatching to X+1: Tree-level


Matching in VinciaMatching in Vincia► We are pursuing three strategies in parallel

• Addition (aka subtraction) Simplest, but has generic negative weights and hard to exponentiate corrections Guaranteed to fill all of phase space (unsubtracted ME in dead regions)

• Multiplication (aka reweighting) Complicated, so 1-loop matching difficult beyond first order, but has generic positive

weights and “automatically” exponentiates path to NLL Only fills phase space populated by shower: dead zones problematic

• Hybrid Trying to combine simple expansions with positive weights, full phase space, and

exponentiation

► Goal• Multi-leg “plug-and-play” NLO + “improved”-LL shower Monte Carlo

• Including uncertainty bands (exploring uncontrolled terms)

• Extension to NNLO + NLL ?


Second OrderSecond Order► Second Order Shower expansion for 4 partons (assuming first already matched)

min # of paths

AR pT + AR recoil

max # of paths

DZ

►Problem 1: dependence on evolution variable• Shower is ordered t4 integration only up to t3

• 2, 1, or 0 allowed “paths”

• Dead zone not good for multiplication QE = pT(i,j,k) = mijmjk/mijk

QE = pT

GGG

AVG

Vincia

AVG

Vincia

MAX

Vincia

MIN

QE = pT

Everyone’s usual

nightmare of a parton

shower

0

1

2

3


Second Order Second Order with Unordered Showerswith Unordered Showers

► For multiplication: allow power-suppressed “unordered” branchings

Vincia Uord

MIN

Vincia Uord

MAX

• Removes dead zone + better approx than fully unordered (Good initial guess better reweighting efficiency)

► Problem 2: leftover Subleading Logs • There are still unsubtractred subleading divergences in the ME

GGG Uord

AVG

Vincia Uord

AVG


Leftover LogsLeftover Logs► Most obvious for subtraction in Dead Zone

• ME completely unsubtracted in Dead Zone leftovers

► But also true in general: the shower is still formally LL everywhere• NLL leftovers are unavoidable

• Additional sources: Subleading color, Polarization

► Beat them or join them?• Beat them: not resummed

brute force regulate with Theta (or smooth) function ~ CKKW “matching scale”

• Join them: absorb leftovers systematically in shower resummationBut looks like we would need polarized NLL-NLC showers … !Could take some time … In the meantime, maybe we can cheat … (don’t stop matching)!

Note: more legs more logs, so ultimately will still need regulator. But try to postpone to NNLL level.


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

Sjöstrand-Bengtsson term2nd order matching term (with 1st order subtracted out)

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

Note: to maintain positivity for subleading colour, need to match across 4 events, 2 representing one color ordering, and 2 for the other ordering


General 2General 2ndnd Order Order (& NLL Matching)(& NLL Matching)

► Include unitary shower (S) and non-unitary “K-factor” (K) corrections

• S: branching probability modification, goes back into Sudakov resummed All logs should be here. Unitary does not modify normalization The simpler the better : will explicitly appear in 1-loop subtractions The simpler the better : will need to be evaluated once for every nested 24 branching (if NLL)

• K: event weight modification, does not go back into Sudakov not resummed Finite corrections can go here ( + regulated logs) Non-unitary changes normalization (“K” factors) Can be arbitrarily complicated: will not appear in 1-loop subtractions (?) Can be arbitrarily complicated: will only need to be evaluated once per event

► With this notation, • Addition/Subtraction: S = 1, K ≠ 1

• Multiplication/Reweighting: K = 1, S ≠ 1• Hybrid: S contains logs (kept as simple as possible), K contains the rest (stick complicated stuff here)


The ZThe Z3 1-loop term3 1-loop term► Second order matching term for 3 partons

► Additive (S=1) Ordinary NLO subtraction + shower leftovers

• Shower off w2(V)

• “Coherence” term: difference between 2- and 3-parton (power-suppressed) evolution above QE3. Explicit QE-dependence cancellation.

• δα: Difference between alpha used in shower (μ = pT) and alpha used for matching Explicit scale choice cancellation

• Integral over w4(R) in IR region still contains NLL divergences regulate

• Logs not resummed, so remaining (NLL) logs in w3(R)

also need to be regulated

► Multiplicative : S = (1+…) Modified NLO subtraction + shower leftovers

• A*S contains all logs from tree-level w4(R) finite.

• Any remaining logs in w3(V) cancel against NNLO NLL resummation if put back in S


VINCIA in Action: Jet RatesVINCIA in Action: Jet Rates

αs(MZ)=0.137,

μR=pT,

pThad = 0.5 GeV

Varying finite terms only

with

► The unknown finite terms are important• They are arbitrary (and in general process-dependent)

• Uncertainty in hard region already at first order

• Cascade down to produce uncontrolled tower of subleading logs


VINCIA in Action: LEPVINCIA in Action: LEP

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


The next big stepsThe next big steps► Z3 at one loop

• Opens multi-parton matching at 1 loop

• Required piece in NNLO Z matching

• Allows to get a fix on Sudakov terms generated by unordering

• Allows to get a fix on running coupling

► Work in progress• Write up complete framework for additive matching

NLO Z3 and NNLO matching within reach

• Derivations not yet finished for multiplicative matching … Complete NLL showers slightly further down the road

► Turn to the initial state, massive particles, other NLL effects


OverviewOverview► LL Shower Monte Carlos

• Constructing LL Showers: Uncertainties at LL

• The VINCIA Antenna Showers

► Matching• Multileg Matching 1: Additive (subtraction)

Simple subtraction terms Positive and Negative weights Subleading Logs not resummed need explicit regulators

• Multileg Matching 2: Multiplicative (reweighting) Positive weights Phase space coverage unordered showers (power-suppressed)

Exponentiated matching to 24: towards NLL showers Complicated subtraction terms

• Multileg Matching 3: Hybrid (subtraction + some reweighting) Best of both? Towards NNLO matching and beyond


DifferencesDifferences► Addition

• Weight(X+n) = ME(x+n) – Shower(X,X+1,X+2,…,X+n-1)

• Weight can have either sign negative weights (even at tree level)

• Special case 1: dead zones weight = ME Necessary in HERWIG Seymour’s 1995 paper Utilized in CKKW etc: force dead zones simpler matching, no negative weights

• Special case 2: shower function = ME(x+n)/ME(x+n-1) POWHEG: ensures Weight(X+n) = 0 and Weight(X+n-1) ~ KNLO * MELO

► Multiplication• Reweight(X+n) = ME(X+n) / Shower(X+n-1)

Physical matrix elements positive Reweight > 0

• Shower evolution is unitary Sudakov contains ME (as in Pythia, Powheg)

complicated subtractions beyond first order


min # of paths

AR pT + AR recoil

max # of paths

DZ

OrderingOrdering► Number of paths in 4-parton phase space

• Starting at 2-parton scale = 100 GeV

• X- and Y-axes = pT(0,1,2) and pT(1,2,3) So each (X,Y) bin contains many 4-parton PS points

• 10M 4-parton points generated with Rambo: test ordering

Mdaughter-dipole + AR recoil 3p-Thrust + AR recoil

D.Z.

D.Z.

0

1

2

3

pT(i,j,k) = mijmjk/mijk

Q2-ordering + AR recoil

pT = mijmjk/mijk

Mdd = min(mij,mjk)

M(1-T3) = min(mij,mjk,mik)

Q = max(mij,mjk)

AR pT + “longitudinal” recoil

3p-Thrust + “longitudinal” recoil


Ratio: Showers / ZRatio: Showers / Z4 ME4 ME

Vincia

AVG

GGG no C

AVGVincia

MAX

Vincia

MIN

QE = pT

C=0

Fini

te T

erm

s =

0 QE = pT

C=0

QE = T3

C=0

QE = pT

GGG

AVG

Vincia

AVG

Vincia

MAX

Vincia

MIN

QE = pT QE = T3

AVG MIN/MAX Alternative QE…

Everyone’s usual

nightmare of a parton

shower

(GGG/Vincia difference: Vincia only includes nestings of (23) that are ordered in the shower evolution variable)


Why NLO “multileg”?Why NLO “multileg”?► Including X at one loop “NLO” matching ?

= NLO only for distributions that are not a δ at LO (e.g., yX)

= LO for any distribution that “starts” at X+1 (e.g., pTX)

= “Improved” LL for any distribution that “starts” at X+2 (e.g., 2-jet rates)

Perturbative series still barely under control

► Combining MC@NLO with CKKW NLO + multi-leg tree ?= NLO only for distributions that are not a δ at LO

= LO for any distribution that “starts” at X+1, … X+N

= “improved” LL for any distribution that “starts” beyond X+N

► NLO N-jet precision can only be accessed by NLO multileg


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


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, if ordered, or from sij to 0 if unordered )

• 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.

Towards Multijet Matching with Loops

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