# Presentation: 638370

## Higher Order Aspects of Parton Showers

## Monte Carlo at Fixed Order

**High-dimensional problem (phase space)**
- d≥5  Monte Carlo integration

- Principal virtues
- Stochastic error O(N-1/2)  independent of dimension
- Full (perturbative) quantum treatment at each order
- (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

- A Guide to Hadron Collisions

## 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

- 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 @ LHC

- A Guide to Hadron Collisions

## Beyond Fixed Order 1

- dσX =  ...
- dσX+1 ~ dσX g2 2 sab /(sa1s1b) dsa1ds1b
- dσX+2 ~ dσX+1 g2 2 sab/(sa2s2b) dsa2ds2b
- dσX+3 ~ dσX+2 g2 2 sab/(sa3s3b) dsa3ds3b

- 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
- saisib

- dσX+1

- dσX+2

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

- “DLA”

- A Guide to Hadron Collisions

## Beyond Fixed Order 2

- dσX =  ...
- dσX+1 ~ dσX g2 2 sab /(sa1s1b) dsa1ds1b
- dσX+2 ~ dσX+1 g2 2 sab/(sa2s2b) dsa2ds2b
- dσX+3 ~ dσX+2 g2 2 sab/(sa3s3b) dsa3ds3b
**+** 	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, Qin ~ **s** and Qend = **Q****had**
- 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
- saisib

- dσX+1

- dσX+2

- Given a jet definition, an event has *either*  0, 1, 2, or ... jets

- “DLA”

- A Guide to Hadron Collisions

## LL 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
*w**X** = |M**X**|**2  *or*  K|M**X**|**2** *
*S* : Evolution operator

**Leading Order**

- A Guide to Hadron Collisions

## “S” (for Shower)

*“**S” **(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

- A: splitting function

- A Guide to Hadron Collisions

## Constructing 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 Guide to Hadron Collisions

## A (complete idiot’s) Solution?

- Combine different starting multiplicites
-  *inclusive *sample?
- In practice – Combine
- [X]ME + showering
- [X + 1 jet]ME + showering
- ...
- Doesn’t work
- [X] + shower is inclusive
- [X+1] + shower is also 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

- A Guide to Hadron Collisions

## The 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”**

- A Guide to Hadron Collisions

## Matching 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

- A Guide to Hadron Collisions

## 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

- In dead zone: *A**i** = 0*  add events corresponding to unsubtracted *|M*X+1*|*
- Outside dead zone: reweighted à la Pythia  *A**i** = **|M*X+1*|*
-  no additive correction necessary
- CKKW and L-CKKW
- At this order identical to Herwig, with “dead zone” for *k**T** > k**T*cut introduced by hand
- MC@NLO
- In dead zone: identical to Herwig
- Outside dead zone: *A**Herwig** >** ** **|M*X+1*|*   *w*X+1 negative  negative weights
- Pythia
*A**i** = **|M*X+1*|* over all of phase space  no additive correction necessary
- Powheg
- At this order identical to Pythia
-  no negative weights

- HERWIG TYPE

- PYTHIA TYPE

- Matching to X+1: Tree-level

- A Guide to Hadron Collisions

## VINCIA

- 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

- 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, *p**T*, *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

- A Guide to Hadron Collisions

## Ordering

- kT

- m2

- pT
- (Ariadne)

- mant

- 1-T

- collinear

- Phase Space for 23

- Partitioned-Dipole

- Dipole-Antenna

- Eg

- Angle

- soft

- A Guide to Hadron Collisions

## Second Order

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

- 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

- A Guide to Hadron Collisions

## Second OrderAVERAGEs of Over/Under-counting

- Second OrderAVERAGEs of Over/Under-counting

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

- Define over/under-counting ratio:    PStree / MEtree

- NB: AVERAGE of R4 distribution

- A Guide to Hadron Collisions

## Second OrderEXTREMA of Over/Under-counting

- Second OrderEXTREMA of Over/Under-counting

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

- Define over/under-counting ratio:    PStree / MEtree

- NB: EXTREMA of R4 distribution (100M points)

- A Guide to Hadron Collisions

## (Stupid Choices)

- A Guide to Hadron Collisions

## Dependence on Finite Terms

- Antenna/Dipole/Splitting functions are ambiguous by finite terms

- A Guide to Hadron Collisions

## The 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

- A Guide to Hadron Collisions

## Matching 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 ?

- A Guide to Hadron Collisions

## NLO with Addition

- First Order Shower expansion

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

- 3-parton real correction (*A**3** = |M**3**|**2**/|M**2**|**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 )

- A Guide to Hadron Collisions

## Matching at Higher Orders  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 resummation
- But 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.

- A Guide to Hadron Collisions

## Z4 Matching by multiplication

- Z4 Matching by 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

- A Guide to Hadron Collisions

## The Z3 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

- A Guide to Hadron Collisions

## General 2nd Order (& NLL Matching)

- General 2nd Order (& 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 24 branching (if NLL)

**Addition/Subtraction**: 	S = 1, 	K ≠ 1
**Multiplication/Reweighting:** 	S ≠ 1 	K = 1
**Hybrid:** 			S = logs 	K = the rest

- A Guide to Hadron Collisions

## 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

- 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 Action

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

- A Guide to Hadron Collisions

## 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

- 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 Action

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

- A Guide to Hadron Collisions

## Can vary 
evolution variable, kinematics maps, radiation functions, renormalization choice, matching strategy (here just varying splitting functions)
After 2nd order matching
Non-pert part can be precisely constrained.
(will need 2nd order logs as well for full variation)

- Can vary
- evolution variable, kinematics maps, radiation functions, renormalization choice, matching strategy (here just varying splitting functions)
- After 2nd order matching
- Non-pert part can be precisely constrained.
- (will need 2nd order logs as well for full variation)

- VINCIA in Action

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

- A Guide to Hadron Collisions

## The next big steps

- Z3 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 Z3 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?

- A Guide to Hadron Collisions