Recap: VINCIA Plug-in to PYTHIA 8 C++ (~20,000 lines) Giele, - - PowerPoint PPT Presentation

▶
recap vincia
SMART_READER_LITE
LIVE PREVIEW

Recap: VINCIA Plug-in to PYTHIA 8 C++ (~20,000 lines) Giele, - - PowerPoint PPT Presentation

Sep 12, 2022 240 likes •515 views

Recap: VINCIA Plug-in to PYTHIA 8 C++ (~20,000 lines) Giele, Kosower, Skands, PRD 78 (2008) 014026, PRD 84 (2011) 054003 Gehrmann-de Ridder, Ritzmann, Skands, PRD 85 (2012) 014013 i 1 1 j i I k Based on antenna factorization j I k K -

slide-1
SLIDE 1
  • P. S k a n d s

1 1 i j k I i j k I m+1 m+1 K K

Recap: VINCIA

1

Giele, Kosower, Skands, PRD 78 (2008) 014026, PRD 84 (2011) 054003 Gehrmann-de Ridder, Ritzmann, Skands, PRD 85 (2012) 014013

Plug-in to PYTHIA 8 C++ (~20,000 lines)

Based on antenna factorization

  • of Amplitudes (exact in both soft and collinear limits)
  • of Phase Space (LIPS : 2 on-shell → 3 on-shell partons, with (E,p) cons)

Evolution Scale

Infinite family of continuously deformable QE Special cases: transverse momentum, invariant mass, energy Improvements for hard 2→n: “smooth ordering” & LO matching

Radiation functions

Written as Laurent-series with arbitrary coefficients, anti Special cases for non-singular terms: Gehrmann-Glover, MIN, MAX + Massive antenna functions for massive fermions (c,b,t)

Kinematics maps

Formalism derived for infinitely deformable κ3→2 Special cases: ARIADNE, Kosower, + massive generalizations

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

⌦ ↵

0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(c)

2 ∗√ 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

pT mD Eg vincia.hepforge.org

slide-2
SLIDE 2
  • P. S k a n d s

One-Loop Corrections

Trivial Example (for notation): Z→qq First Order (~POWHEG)

2

Giele, Kosower, Skands, Phys.Rev. D78 (2008) 014026

Fixed Order: Exclusive 2-jet rate (2 and only 2 jets), at Q = Qhad

∗] = |M0 0 |2

1 + 2 Re[M0

0 M1 ∗]

|M0

0 |2

+ Z Q2

had

dΦant g2

s C Ag/q¯ q

!

Born Virtual Unresolved Real

¯ q = |M0 1 |2

|M0

0 |2

Markov Shower: Exclusive 2-jet rate (2 and only 2 jets), at Q = Qhad

|M0

0 |2 ∆(s, Q2 had) = |M0 0 |2

1 − Z s

Q2

had

dΦant g2

s C Ag/q¯ q + O(α2 s)

!

Born Sudakov Approximate Virtual + Unresolved Real

NLO Correction: Subtract and correct by difference Z s dΦant 2CF g2

s Ag/q¯ q = ↵s

2⇡ 2CF ✓ −2Iq¯

q(✏, µ2/m2 Z) + 19

4 ◆

2 Re[M0

0 M1 ∗]

|M0

0 |2

= ↵s 2⇡ 2CF

  • 2Iq¯

q(✏, µ2/m2 Z) − 4

  • |M 0

0 |2 →

⇣ 1 + αs π ⌘ |M 0

0 |2

IR Singularity Operator

)

Z0 → q¯ q

slide-3
SLIDE 3
  • P. S k a n d s

One-Loop Corrections

Getting Serious: second order

3

Ongoing work, with E. Laenen & L. Hartgring (NIKHEF)

QE mZ ∆qg(Q2

R, 0)

∆g¯

q(Q2 R, 0)

∆q¯

q(m2 Z, Q2 E)

ag/q¯

q

QR

dσq¯

q

Approximate → (1 + V0) |M0

1 |2 ∆2(m2 Z, Q2 1) ∆3(Q2 R1, Q2 had) ,

Fixed Order: Exclusive 3-jet rate (3 and only 3 jets), at Q = Qhad

V0 = αs/π

2→3 Evolution 3→4 Evolution 2→3 Evolution 3→4 Evolution

Exact → |M0

1 |2 + 2 Re[M0 1 M1∗ 1 ] +

Z Q2

had

dΦ2 dΦ1 |M0

2 |2

Born Virtual Unresolved Real

Markov Shower:

µR

slide-4
SLIDE 4
  • P. S k a n d s

Master Equation

NLO Correction: Subtract and correct by difference

4

Ongoing work, with E. Laenen & L. Hartgring (NIKHEF)

V1Z(q, g, ¯ q) = 2 Re[M0

1 M1⇤ 1 ]

|M0

1 |2

LC − ↵s ⇡ − ↵s 2⇡ ✓11NC − 2nF 6 ◆ ln ✓µ2

ME

µ2

PS

◆ + ↵sCA 2⇡ " − 2I(1)

qg (✏, µ2/sqg) − 2I(1) qg (✏, µ2/sg¯ q) + 34

3 # + ↵snF 2⇡ " − 2I(1)

qg,F (✏, µ2/sqg) − 2I(1) g¯ q,F (✏, µ2/sqg) − 1

# + ↵sCA 2⇡ " 8⇡2 Z m2

Z

Q2

1

dΦant Astd

g/q¯ q + 8⇡2

Z m2

Z

Q2

1

dΦant Ag/q¯

q

−

2

X

j=1

8⇡2 Z sj dΦant (1 − OEj) Astd

g/qg + 2

X

j=1

8⇡2 Z sj dΦant Ag/qg # + ↵snF 2⇡ " −

2

X

j=1

8⇡2 Z sj dΦant(1 − OSj) PAj Astd

¯ q/qg + 2

X

j=1

8⇡2 Z sj dΦant A¯

q/qg

−1 6 sqg − sg¯

q

sqg + sg¯

q

ln ✓sqg sg¯

q

◆ # , (72)

V0 µR Gluon Emission IR Singularity Gluon Splitting IR Singularity 2→3 Sudakov Logs 3→4 Sudakov Logs 3→4 Emit 3→4 Split

OEj = Gluon-Emission Ordering Function Q1 = 3-parton Resolution Scale OSj = Gluon-Splitting Ordering Function δA = LO Matching Terms (finite)

*)Note: here only Leading Color

δA2→3 δA3→4, Split δA3→4, Emit

ANLO = ALO (1+V1)

Standard Finite Terms Standard IR Singularities

slide-5
SLIDE 5
  • P. S k a n d s

Loop Corrections

5

(MC)2 : NLO Z → 2 → 3 Jets + Markov Shower

1.4 1.5 1.5 1.5 1.75 1.75 1.75 2

  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

lnHyijL lnHyjkL

QE=2pT HstrongL

µR = mZ ΛQCD = ΛMS αS(MZ) = 0.12

Ongoing work, with E. Laenen & L. Hartgring (NIKHEF)

Soft Antiquark-Collinear Quark-Collinear Hard Resolved Markov Evolution in: Transverse Momentum Size of NLO Correction:

  • ver 3-parton

Phase Space Parameters:

q(pi) ¯ q(pk) g(pj) yij = (pi · pj) M 2

Z

→ 0 when i||j & when Ej → 0 Scaled Invariants

slide-6
SLIDE 6
  • P. S k a n d s

Choice of µR

6

1.05 1.1 1.1 1.1 1.2 1.2 1.2 1.2

  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

lnHyijL lnHyjkL

QE=2pT HstrongL

1.4 1.5 1.5 1.5 1.75 1.75 1.75 2

  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

lnHyijL lnHyjkL

QE=2pT HstrongL

Markov Evolution in: Transverse Momentum, αS(MZ) = 0.12

µR = mZ ΛQCD = ΛMS µR = pTg ΛQCD = ΛCMW

Ongoing work, with E. Laenen & L. Hartgring (NIKHEF)

A) MZ

“Typical” Fixed-Order Choice

B) pT

= “Typical” Shower Choice

slide-7
SLIDE 7
  • P. S k a n d s

Choice of QEvol

7

Ongoing work, with E. Laenen & L. Hartgring (NIKHEF)

0.6 0.7 0.8 0.9 0.95 1.05 1.1 1.2 1.2 1.3 1.3 1.4 1.4 1.5 1.5 1.75 1.75 2 2

  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

lnHyijL lnHyjkL

QE=mD

1.05 1.1 1.1 1.1 1.2 1.2 1.2 1.2

  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

lnHyijL lnHyjkL

QE=2pT HstrongL

Markov Evolution in mD2 = 2min(sij,sjk) Markov Evolution in pTA2 = sijsjk/sijk

Missing Sudakov Suppression in Soft Region Too much Sudakov Suppression in Collinear Region Modest Corrections Everywhere

Parameters: αS(MZ) = 0.12, µR = pTA,

ΛQCD = ΛCMW

0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

pT

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

mD

slide-8
SLIDE 8
  • P. S k a n d s

1.05 1.1 1.2 1.3 1.3

  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

lnHyijL lnHyjkL

QE=2pT HstrongL

1.05 1.1 1.1

  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

lnHyijL lnHyjkL

QE=2pT HstrongL

Choice of Finite Terms

8

Ongoing work, with E. Laenen & L. Hartgring (NIKHEF)

Large finite terms → Small 3→4 Sudakov (much Sudakov Suppr)

Parameters: αS(MZ) = 0.12, µR = pTA, ΛQCD = ΛCMW

Small finite terms → Large 3→4 Sudakov (little Sudakov Suppr)

MIN Antennae: δA3→4 < 0 MAX Antennae: δA3→4 > 0

Note: this just for illustration. Matching to LO matrix elements fixes δA uniquely

slide-9
SLIDE 9

O u t l o o k

  • 1. Publish 3 papers (~ a couple of months: helicities, NLO multileg,

ISR)

  • 2. Apply these corrections to a broader class of processes,

including ISR → LHC phenomenology

  • 3. Automate correction procedure, via interfaces to one-loop

codes … (goes slightly beyond Binoth Accord; for LO corrections, we currently use own interface to modified MadGraph ME’s)

  • 4. Variations. No calculation is more precise than the reliability of its

uncertainty estimate → aim for full assessment of TH uncertainties.

  • 5. Recycle formalism for all-orders shower corrections?
slide-10
SLIDE 10
  • P. S k a n d s

Phase Space Contours

10 Mass-Ordering p⊥-ordering Energy-Ordering (m2

min)

( ⌦ m2↵

geometric)

( ⌦ m2↵

arithmetic)

Linear in y

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(a) Q2

E = m2 D = 2 min(yij, yjk)s 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(b) Q2

E = 2p⊥

√s = 2√yijyjks

0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(c) Q2

E = 2E∗√s = (yij + yjk)s

Quadratic in y

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(d) Q2

E = m4

D

s

= 4 min(y2

ij, y2 jk)s 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(e) Q2

E = 4p2 ⊥ = 4yijyjks 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(f) Q2

E = 4E∗2 = (yij + yjk)2s

Evolution Variables:

slide-11
SLIDE 11
  • P. S k a n d s

Consequences of Ordering

11 Mass-Ordering p⊥-ordering Energy-Ordering Linear in y

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(a) Q2

E = m2 D = 2 min(yij, yjk)s 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(b) Q2

E = 2p⊥

√s = 2√yijyjks

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(c) Q2

E = 2E∗√s = (yij + yjk)s

Quadratic in y

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(d) Q2

E = m4

D

s

= 4 min(y2

ij, y2 jk)s 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(e) Q2

E = 4p2 ⊥ = 4yijyjks 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij yjk

(f) Q2

E = 4E∗2 = (yij + yjk)2s

Ongoing work, with E. Laenen & L. Hartgring (NIKHEF)

1 1 1 1 2 1 1

Number of antennae restricted by ordering condition

slide-12
SLIDE 12
  • P. S k a n d s

12

slide-13
SLIDE 13
  • P. S k a n d s

Idea:

Start from quasi-conformal all-orders structure (approximate) Impose exact higher orders as finite corrections Truncate at fixed scale (rather than fixed order) Bonus: low-scale partonic events → can be hadronized

Problems:

Traditional parton showers are history-dependent (non-Markovian) → Number of generated terms grows like 2N N! + Highly complicated expansions

Solution: (MC)2 : Monte-Carlo Markov Chain

Markovian Antenna Showers (VINCIA) → Number of generated terms grows like N + extremely simple expansions

Solution: (MC) 2

13

Markovian Antenna Shower:

After 2 branchings: 2 terms After 3 branchings: 3 terms After 4 branchings: 4 terms

Parton- (or Catani-Seymour) Shower:

After 2 branchings: 8 terms After 3 branchings: 48 terms After 4 branchings: 384 terms

“Higher-Order Corrections To Timelike Jets” GeeKS: Giele, Kosower, Skands, PRD 84 (2011) 054003

slide-14
SLIDE 14
  • P. S k a n d s

New: Markovian pQCD *

14

Legs Loops +0 +1 +2 +0 +1 +2 +3

|MF|2

Generate “shower” emission

|MF+1|2 LL ∼ X

i∈ant

ai |MF|2

Correct to Matrix Element Unitarity of Shower

P | | Virtual = − Z Real

Correct to Matrix Element

Z |MF|2 → |MF|2 + 2Re[M 1

FM 0 F] +

Z Real

The VINCIA Code

X

∈

ai → |MF+1|2 P ai|MF|2 ai →

Cutting Edge: Embedding virtual amplitudes = Next Perturbative Order → Precision Monte Carlos

PYTHIA 8

+

“Higher-Order Corrections To Timelike Jets” GeeKS: Giele, Kosower, Skands, PRD 84 (2011) 054003

*)pQCD : perturbative QCD

Start at Born level R e p e a t

slide-15
SLIDE 15
  • P. S k a n d s

Fixed Order: Recap

15

F @ LO

` (loops) 2

(2) (2)

1

. . .

1

(1) (1)

1

(1)

2

. . . (0) (0)

1

(0)

2

(0)

3

. . .

1 2 3

. . .

k (legs)

Max Born, 1882-1970 Nobel 1954

Leading Order

F @ NLO

` (loops) 2

(2) (2)

1

. . .

NLO for F + 0 → LO for F + 1

1

(1) (1)

1

(1)

2

. . . (0) (0)

1

(0)

2

(0)

3

. . .

1 2 3

. . .

k (legs)

Next-to-Leading Order

(from PS, Introduction to QCD, TASI 2012, arXiv:1207.2389)

Improve by computing quantum corrections, order by order

σNLO = σBorn + Z dΦF +1

  • M(0)

F +1

  • 2

+ Z dΦF 2Re h M(1)

F M(0)∗ F

i

→ 1/ϵ2 + 1/ϵ + Finite → -1/ϵ2 - 1/ϵ + Finite

= σBorn + Z dΦF+1 ⇣ |M(0)

F+1|2 − dσNLO S

⌘ | {z } Finite by Universality + Z dΦF 2Re[M(1)

F M(0)∗ F

] + Z dΦF+1 dσNLO

S

| {z } Finite by KLN .

The Subtraction Idea

Universal “Subtraction Terms” (will return to later)

slide-16
SLIDE 16
  • P. S k a n d s

Shower Types

16

HI IK KL H I K L Coll(I) Soft(IK)

Parton Shower (DGLAP)

aI aI + aK

Coherent Parton Shower (HERWIG [12,40], PYTHIA6 [11])

ΘIaI ΘIaI + ΘKaK

Global Dipole-Antenna (ARIADNE [17], GGG [36], WK [32], VINCIA)

aIK + aHI aIK

Sector Dipole-Antenna (LP [41], VINCIA)

ΘIKaIK + ΘHIaHI aIK

Partitioned-Dipole Shower (SK [23], NS [42], DTW [24], PYTHIA8 [38], SHERPA)

aI,K + aI,H aI,K + aK,I Figure 2: Schematic overview of how the full collinear singularity of parton I and the soft singularity

  • f the IK pair, respectively, originate in different shower types. (ΘI and ΘK represent angular vetos

with respect to partons I and K, respectively, and ΘIK represents a sector phase-space veto, see text.)

Traditional vs Coherent vs Global vs Sector vs Dipole

slide-17
SLIDE 17
  • P. S k a n d s

Global Antennae

17 ×

1 yijyjk 1 yij 1 yjk yjk yij yij yjk y2

jk

yij y2

ij

yjk

1 yij yjk q¯ q → qg¯ q ++ → + + + 1 ++ → + − + 1 −2 −2 1 1 2 +− → + + − 1 −2 1 +− → + − − 1 −2 1 qg → qgg ++ → + + + 1 −α + 1 2α − 2 ++ → + − + 1 −2 −3 1 3 −1 3 +− → + + − 1 −3 3 −1 +− → + − − 1 −2 −α + 1 1 2α − 2 gg → ggg ++ → + + + 1 −α + 1 −α + 1 2α − 2 2α − 2 ++ → + − + 1 −3 −3 3 3 −1 −1 3 1 1 +− → + + − 1 −α + 1 −3 2α − 2 3 −1 +− → + − − 1 −3 −α + 1 3 2α − 2 −1 qg → q¯ q0q0 ++ → + + −

1 2

++ → + − +

1 2

−1

1 2

+− → + + −

1 2

−1

1 2

+− → + − −

1 2

gg → g¯ qq ++ → + + −

1 2

++ → + − +

1 2

−1

1 2

+− → + + −

1 2

−1

1 2

+− → + − +

1 2

slide-18
SLIDE 18
  • P. S k a n d s

Sector Antennae

18 ¯ asct

j/IK(yij, yjk) = ¯

agl

j/IK(yij, yjk)

+ δIgδHKHk ( δHIHiδHIHj 1 + yjk + y2

jk

yij ! + δHIHj 1 yij(1 − yjk) − 1 + yjk + y2

jk

yij !) + δKgδHIHi ( δHIHjδHKHk 1 + yij + y2

ij

yjk ! + δHKHj 1 yjk(1 − yij) − 1 + yij + y2

ij

yjk !)

Sector j j radiated by i,k 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij sijsijk 1xk yjk sjksijk 1xi

Sector populated by IKijk

Sector k k radiated by j,i 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij sijsijk 1xk yjk sjksijk 1xi

Sector populated by JIjki

Sector i i radiated by k,j 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 yij sijsijk 1xk yjk sjksijk 1xi

Sector populated by KJkij

¯ agl

g/qg(pi, pj, pk) sjk→0

− → 1 sjk ✓ Pgg→G(z) − 2z 1 − z − z(1 − z) ◆

→ P(z) = Sum over two neigboring antennae

Global Sector

Only a single term in each phase space point → Full P(z) must be contained in every antenna Sector = Global + additional collinear terms (from “neighboring” antenna)

slide-19
SLIDE 19

P . Skands

) The Denominator v

19

In a traditional parton shower, you would face the following problem:

Existing parton showers are not really Markov Chains

Further evolution (restart scale) depends on which branching happened last → proliferation of terms

Number of histories contributing to nth branching ∝ 2nn!

~

+ + +

j = 2 → 4 terms j = 1 → 2 terms

(

~ +

Parton- (or Catani-Seymour) Shower:

After 2 branchings: 8 terms After 3 branchings: 48 terms After 4 branchings: 384 terms X

∈

ai → |MF+1|2 P ai|MF|2

(+ parton showers have complicated and/or frame-dependent phase-space mappings, especially at the multi-parton level)

slide-20
SLIDE 20

P . Skands

Matched Markovian Antenna Showers

+ Change “shower restart” to Markov criterion:

Given an n-parton configuration, “ordering” scale is Qord = min(QE1,QE2,...,QEn)

Unique restart scale, independently of how it was produced

+ Matching: n! → n

Given an n-parton configuration, its phase space weight is: |Mn|2 : Unique weight, independently of how it was produced

20

Matched Markovian Antenna Shower:

After 2 branchings: 2 terms After 3 branchings: 3 terms After 4 branchings: 4 terms

Parton- (or Catani-Seymour) Shower:

After 2 branchings: 8 terms After 3 branchings: 48 terms After 4 branchings: 384 terms

+ Sector antennae → 1 term at any order (+ generic Lorentz- invariant and on-shell phase-space factorization)

Antenna showers: one term per parton pair

2nn! → n!

Larkosi, Peskin,Phys.Rev. D81 (2010) 054010 Lopez-Villarejo, Skands, JHEP 1111 (2011) 150 Giele, Kosower, Skands, PRD 84 (2011) 054003

slide-21
SLIDE 21

P . Skands

Approximations

21

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5 Fraction of Phase Space

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1

4 → Z

Vincia 1.025 + MadGraph 4.426

Strong Ordering 3 → Matched to Z GGG

PS

ψ

  • ord

D

m ARI

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5

  • 4
  • 3
  • 2
  • 1

10 1

5 → Z

Vincia 1.025 + MadGraph 4.426

Strong Ordering 3 → Matched to Z

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1

6 → Z

Vincia 1.025 + MadGraph 4.426

Strong Ordering 3 → Matched to Z

S T RO N G O R D E R I N G

Q: How well do showers do? Exp: Compare to data. Difficult to interpret; all-orders cocktail including hadronization, tuning, uncertainties, etc Th: Compare products of splitting functions to full tree-level matrix elements Plot distribution of Log10(PS/ME)

(fourth order) (third order) (second order) Dead Zone: 1-2% of phase space have no strongly ordered paths leading there*

*fine from strict LL point of view: those points correspond to “unordered” non-log-enhanced configurations

slide-22
SLIDE 22

P . Skands

2→4

Generate Branchings without imposing strong ordering

At each step, each dipole allowed to fill its entire phase space

Overcounting removed by matching + smooth ordering beyond matched multiplicities

22

⊥

2 Z

/m

2 T1

4p ln

  • 5
  • 4
  • 3
  • 2
  • 1

2 T1

/p

2 T2

p ln

  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6

q qgg → Z

VINCIA 1.025 ANT = DEF

AR

ψ KIN = (smooth)

T 2

ORD = p

>

4

<R

→ Ordered | 2nd | Unordered ← → Soft | 1st Branching | Hard ←

2 Z

/m

2 T1

4p ln

  • 5
  • 4
  • 3
  • 2
  • 1

2 T1

/p

2 T2

p ln

  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6

q qgg → Z

VINCIA 1.025 ANT = DEF

AR

ψ KIN = (strong)

T 2

ORD = p

>

4

<R

→ Ordered | 2nd | Unordered ← → Soft | 1st Branching | Hard ←

Dead Zone Smooth Ordering

= ˆ p2

⊥

ˆ p2

⊥ + p2 ⊥

PLL d parton triplets in = ˆ p2

⊥ last branching ⊥

+ p2

⊥

  • n triplets

current branching

slide-23
SLIDE 23

P . Skands

→ Better Approximations

23

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5 Fraction of Phase Space

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1

4 → Z

Vincia 1.025 + MadGraph 4.426

Smooth Ordering 3 → Matched to Z

AR

ψ GGG,

PS

ψ GGG,

KS

ψ GGG, (qg & gg)

AR

ψ ARI,

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5

  • 4
  • 3
  • 2
  • 1

10 1

5 → Z

Vincia 1.025 + MadGraph 4.426

Smooth Ordering 3 → Matched to Z

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1

6 → Z

Vincia 1.025 + MadGraph 4.426

Smooth Ordering 3 → Matched to Z

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5 Fraction of Phase Space

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1

4 → Z

Vincia 1.025 + MadGraph 4.426

Strong Ordering 3 → Matched to Z GGG

PS

ψ

  • ord

D

m ARI

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5

  • 4
  • 3
  • 2
  • 1

10 1

5 → Z

Vincia 1.025 + MadGraph 4.426

Strong Ordering 3 → Matched to Z

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1

6 → Z

Vincia 1.025 + MadGraph 4.426

Strong Ordering 3 → Matched to Z

S T RO N G O R D E R I N G S M O OT H M A R KOV

Distribution of Log10(PSLO/MELO) (inverse ~ matching coefficient)

Leading Order, Leading Color, Flat phase-space scan, over all of phase space (no matching scale) No dead zone

slide-24
SLIDE 24

P . Skands

+ Matching (+ full colour)

24

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5 Fraction of Phase Space

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1

5 → Z

Vincia 1.025 + MadGraph 4.426

Color-summed (NLC) 4 → Matched to Z

AR

ψ GGG,

PS

ψ GGG,

KS

ψ GGG, (qg & gg)

AR

ψ ARI,

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1

6 → Z

Vincia 1.025 + MadGraph 4.426

Color-summed (NLC) 5 → Matched to Z

Remaining matching corrections are small

(fourth order) (third order)

M AT C H E D M A R KOV

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5 Fraction of Phase Space

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1

4 → Z

Vincia 1.025 + MadGraph 4.426

Smooth Ordering 3 → Matched to Z

AR

ψ GGG,

PS

ψ GGG,

KS

ψ GGG, (qg & gg)

AR

ψ ARI,

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5

  • 4
  • 3
  • 2
  • 1

10 1

5 → Z

Vincia 1.025 + MadGraph 4.426

Smooth Ordering 3 → Matched to Z

(PS/ME)

10

log

  • 2
  • 1.5
  • 1
  • 0.5

0.5

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1

6 → Z

Vincia 1.025 + MadGraph 4.426

Smooth Ordering 3 → Matched to Z

S M O OT H M A R KOV

→ A very good all-orders starting point

slide-25
SLIDE 25
  • P. S k a n d s

IR Singularity Operators

25

I(1)

q¯ q

  • ✏, µ2/sq¯

q

  • = −

e✏ 2Γ (1 − ✏)  1 ✏2 + 3 2✏

  • Re

✓ − µ2 sq¯

q

◆✏ I(1)

qg

  • ✏, µ2/sqg
  • = −

e✏ 2Γ (1 − ✏)  1 ✏2 + 5 3✏

  • Re

✓ − µ2 sqg ◆✏ I(1)

qg,F

  • ✏, µ2/sqg
  • =

e✏ 2Γ (1 − ✏) 1 6✏ Re ✓ − µ2 sqg ◆✏

A0

3(1q, 3g, 2¯ q) =

1 s123 s13 s23 + s23 s13 + 2s12s123 s13s23

  • q¯

q → qg¯ q antenna function

Poles

  • A0

3(s123)

  • = −2I(1)

q¯ q (, s123)

Finite

  • A0

3(s123)

  • = 19

4 .

Integrated antenna Singularity Operators for qg→qgg for qg→qq’q’

Gehrmann, Gehrmann-de Ridder, Glover, JHEP 0509 (2005) 056

X0

ijk = Sijk,IK

|M0

ijk|2

|M0

IK|2 2

X 0

ijk(sijk) =

  • 8π2 (4π)− eγ

dΦXijk X0

ijk.

s performed analytically in d dimensions to ma

slide-26
SLIDE 26
  • P. S k a n d s

Loop Corrections

26

1.2 1.3 1.3 1.4 1.4 1.5 1.5 1.75 1.75 2 2

  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

lnHyijL lnHyjkL

QE=mD

1.3 1.3 1.3 1.3 1.3 1.3 1.4 1.4 1.5 1.5

  • 8
  • 6
  • 4
  • 2
  • 8
  • 6
  • 4
  • 2

lnHyijL lnHyjkL

QE=2pT HstrongL

The choice of evolution variable (Q)

Variation with µR = mD = 2 min(sij,sjk) Parameters: αS(MZ) = 0.12, ΛQCD = ΛCMW