peter skands theoretical physics, fermilab eugene, february 2009 higher order aspects of parton...

Peter SkandsTheoretical Physics, FermilabPeter SkandsTheoretical Physics, Fermilab

Eugene, February 2009

Higher Order Aspects of Parton ShowersHigher Order Aspects of Parton Showers

A Guide to Hadron Collisions - 2Peter Skands

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 k+ℓ > 0, need to be careful in arranging for real-virtual cancellations

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


► Naively, brems suppressed by αs ~ 0.1

• Truncate at fixed order = LO, NLO, …

• However, if ME >> 1 can’t truncate!

► Example: SUSY pair production at 14 TeV, with MSUSY ~ 600 GeV

• Conclusion: 100 GeV can be “soft” at the LHC Matrix Element (fixed order) expansion breaks completely down at 50 GeV With decay jets of order 50 GeV, this is important to understand and control

Bremsstrahlung Example: SUSY @ LHCBremsstrahlung Example: SUSY @ LHC

FIXED ORDER pQCD

inclusive X + 1 “jet”

inclusive X + 2 “jets”

LHC - sps1a - m~600 GeV Plehn, Rainwater, PS PLB645(2007)217

(Computed with SUSY-MadGraph)

Cross section for 1 or more 50-GeV jets larger than total σ, obviously non-sensical


Beyond Fixed Order 1Beyond Fixed Order 1► dσX = …

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

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


► But it’s not a parton shower, 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) + ...

Just an approximation of a sum of trees no real-virtual cancellations

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

KLN guarantees that sing{int(real)} = ÷ sing{virtual} approximate virtual = int(real)

dσX

α sab

saisibdσX+1 dσ

X+2

dσX+2

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

“DLA”


Beyond Fixed Order 2Beyond Fixed Order 2► dσX = …

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



+ Unitarisation: σtot = int(dσX)

σX;excl = 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 ~ s and Qend = Qhad

► σX;excl = int(dσX) - int(dσX+1,2,3,…;excl) = int(dσX) EXP[ - int(dσX+1 / dσX ) ]

► σX;tot = Sum (σX+0,1,2,3,…;excl ) = int(dσX)

dσX

α sab

saisibdσX+1 dσ

X+2

dσX+2

Given a jet definition, an

event has either 0, 1, 2, or … jets

“DLA”


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► In the previous slide, you saw many dependencies on things not

traditionally found in matrix-element calculations:

► 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 (vertex-by-vertex argument of αs)

• The infrared cutoff contour (hadronization cutoff)

Variations

Comprehensive uncertainty estimates (showers with uncertainty bands)

Matching to MEs (& NnLL?)

Reduced Dependence (systematic reduction of uncertainty)


A (complete idiot’s) Solution?A (complete idiot’s) Solution?► Combine different starting multiplicites

inclusive sample?

► In practice – Combine

1. [X]ME + showering

2. [X + 1 jet]ME + showering

3. …

► Doesn’t work

• [X] + shower is inclusive

• [X+1] + shower is also inclusive

X inclusiveX inclusive

X+1 inclusiveX+1 inclusive

X+2 inclusiveX+2 inclusive ≠X exclusiveX exclusive

X+1 exclusiveX+1 exclusive

X+2 inclusiveX+2 inclusive

Run generator for X (+ shower)

Run generator for X+1 (+ shower)

Run generator for … (+ shower)

Combine everything into one sample

What you get

What you want

Overlapping “bins” One sample


The Matching ProblemThe Matching Problem► [X]ME + shower already contains sing{ [X + n jet]ME }

• So we really just missed the non-LL bits, not the entire ME!

• Adding full [X + n jet]ME is overkill LL singular terms are double-counted

► Solution 1: work out the difference and correct by that amount add “shower-subtracted” matrix elements

• Correction events with weights : wn = [X + n jet]ME – Shower{wn-1,2,3,..}

• I call these matching approaches “additive”

► Solution 2: work out the ratio between PS and ME multiply shower kernels by that ratio (< 1 if shower is an overestimate)

• Correction factor on n’th emission Pn = [X + n jet]ME / Shower{[X+n-1 jet]ME}

• I call these matching approaches “multiplicative”


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, ...

• 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)297

Massive Quarks

All combinations of colors and Lorentz structures


► 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

YT

HIA

TY

PE

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


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:

• Choice of evolution time: pT-ordering

Dipole-mass-ordering

Thrust-ordering

• Splitting functions QCD singular terms + arbitrary finite terms (Taylor series)

• Phase space map Antenna-like or Parton-shower-like

• Renormalization scheme ( μR = {evolution scale, pT, s, 2-loop, …} )

• Infrared cutoff contour (hadronization cutoff) Same options as for evolution time, but independent of time universal choice

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


OrderingOrdering

kT m2

pT

(Ariadne)

mant 1-T

collinear

Phase Space for 23

Par

titio

ned

-Dip

ole

Dip

ole-

Ant

enn

a

Eg Anglesoft


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”

• 0 = Dead Zone : not good for reweighting QE = pT(i,j,k) = mijmjk/mijk

0

1

2

3


Second OrderSecond OrderAVERAGEs of Over/Under-countingAVERAGEs of Over/Under-counting

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

Define over/under-counting ratio: PStree / MEtree

0

1

2

3

NB

: AV

ER

AG

E o

f R

4 di

stri

buti

on


Second OrderSecond OrderEXTREMA of Over/Under-countingEXTREMA of Over/Under-counting

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

Define over/under-counting ratio: PStree / MEtree

0

1

2

3

NB

: EX

TR

EM

A o

f R

4 di

stri

buti

on (1

00M

poi

nts)


(Stupid Choices)(Stupid Choices)


Dependence on Finite TermsDependence on Finite Terms► Antenna/Dipole/Splitting functions are ambiguous by finite terms


The Right ChoiceThe Right Choice► Current Vincia without matching, but with “improved” antenna

functions (including suppressed unordered branchings)

• Removes dead zone + still better approx than virt-ordered (Good initial guess better reweighting efficiency)

► Problem 2: leftover Subleading Logs after matching

• There are still unsubtractred subleading divergences in the ME


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

• Addition (aka subtraction) Simplest & guaranteed to fill all of phase space (unsubtracted ME in dead regions)

But has generic negative weights and hard to exponentiate corrections

• Multiplication (aka reweighting) Guaranteed positive weights & “automatically” exponentiates path to NLL

Complicated, so 1-loop matching difficult beyond first order.

Only fills phase space populated by shower: dead zones problematic

• Hybrid Combine: simple expansions, full phase space, positive weights, and

exponentiation?

► Goal

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

• Including uncertainty bands (exploring uncontrolled terms)

• Extension to NNLO + NLL ?


NLO with AdditionNLO with Addition► First Order Shower expansion

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

► 3-parton real correction (A3 = |M3|2/|M2|2 + 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 α, β = 0 (POWHEG )


Matching at Higher Orders Matching at Higher Orders Leftover Subleading Logs Leftover Subleading Logs

► 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 … do it by exponentiated matching

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


ZZ4 Matching 4 Matching by multiplicationby multiplication

► 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 term

2nd 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


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


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

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

• K: event weight modification (special case: add/subtract events) Non-unitary changes normalization (“K” factors)

Non-unitary does not modify Sudakov not resummed

Finite corrections can go here ( + regulated logs)

Only needs to be evaluated once per event

• S: branching probability modification Unitary does not modify normalization

Unitary modifies Sudakov resummed

All logs should be here

Needs to be evaluated once for every nested 24 branching (if NLL)

• Addition/Subtraction: S = 1, K ≠ 1

• Multiplication/Reweighting: S ≠ 1 K = 1

• Hybrid: S = logs K = the rest


► Can vary • evolution variable, kinematics maps,

radiation functions, renormalization choice, matching strategy (here just varying splitting functions)

► At Pure LL, • can definitely see a non-perturbative

correction, but hard to precisely constrain it

VINCIA in ActionVINCIA in Action

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




► At Pure LL, • can definitely see a non-perturbative

correction, but hard to precisely constrain it






► After 2nd order matching Non-pert part can be precisely

constrained.(will need 2nd order logs as well for full variation)




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

• Opens multi-parton matching at 1 loop

• Required piece for NNLO matching

• If matching can be exponentiated, opens NLL showers

► Work in progress

• Write up complete framework for additive matching NLO Z3 and NNLO matching within reach

• Finish complete framework multiplicative matching … Complete NLL showers slightly further down the road

► Then…

• Initial state, masses, polarization, subleading color, unstable particles, …

► Also interesting that we can take more differentials than just δμR

• Something to be learned here even for estimating fixed-order uncertainties?

peter skands theoretical physics, fermilab eugene, february 2009 higher order aspects of parton...

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int d x int d x

production of x

s ab s a1 s

s ab s a2 s

s ab s a3 s

int d x xexcl

jet inclusive x

b ds a1 ds