Subleading Nc Improved Parton Showers
Simon Plätzer
DESY Theory Group
– in collaboration with Malin Sjödahl –
Physics: SP & M. Sjödahl, arXiv:1201.0260
Some technicalities: SP & M. Sjödahl, EPJ Plus 127 (2012) 26 (arXiv:1108.6180)
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
1 / 20
Motivation and general remarks.
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
2 / 20
Motivation and general remarks.
A yet unexplored country.
– How good is the large-Nc approximation?
– It can’t be that bad ... where to expect large effects?
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
2 / 20
Motivation and general remarks.
A yet unexplored country.
– How good is the large-Nc approximation?
– It can’t be that bad ... where to expect large effects?
A more precise input to fixed-order matching.
– NLO matching condition often only satisfied in large-Nc limit.
[independent approach for a single emission by Sherpa, arXiv:1111.1220]
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
2 / 20
Motivation and general remarks.
A yet unexplored country.
– How good is the large-Nc approximation?
– It can’t be that bad ... where to expect large effects?
A more precise input to fixed-order matching.
– NLO matching condition often only satisfied in large-Nc limit.
[independent approach for a single emission by Sherpa, arXiv:1111.1220]
Consider a simple approach, fitting well into existing MC generators.
[Matchbox and DipoleShower modules in Herwig++, SP & S. Gieseke arXiv:1109.6256]
We can’t (yet) do a full evolution at the level of amplitudes.
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
2 / 20
Outline.
– A brief recap of dipole factorization and showering.
– Color matrix element corrections for dipole showers.
– The Sudakov veto algorithm and other technicalities.
– Results and conclusions.
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
3 / 20
From dipole factorization to dipole showering.
Singly unresolved limits, i, j collinear or one of them soft:
[S. Catani & M. H. Seymour, Nucl. Phys. B485 (1997) 291]
|Mn+1 (..., pi , ..., pj , ..., pk , ...)|2 ≈
X
1
hMn (..., pij˜ , ..., pk̃ , ...)|Vij,k (pi , pj , pk )|Mn (..., pij˜ , ..., pk̃ , ...)i
2pi · pj
k6=i,j
i
Well established
subtraction scheme for
NLO calculations.
Use dipole factorization
to define a shower
algorithm →
Simon Plätzer (DESY Theory Group)
j
˜
ij
k̃
k
Subleading Nc Improved Parton Showers
4 / 20
From dipole factorization to dipole showering.
|Mn+1 (..., pi , ..., pj , ..., pk , ...)|2 ≈
X
1
hMn (..., pij˜ , ..., pk̃ , ...)|Vij,k (pi , pj , pk )|Mn (..., pij˜ , ..., pk̃ , ...)i
2pi · pj
k6=i,j
Vij,k = −Vij,k
→
Tij˜ · Tk
T2ij˜
Vij,k
˜ k colour connected)
δ(ij,
1 + δij˜
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
5 / 20
From dipole factorization to dipole showering.
|Mn+1 (..., pi , ..., pj , ..., pk , ...)|2 ≈
X
1
hMn (..., pij˜ , ..., pk̃ , ...)|Vij,k (pi , pj , pk )|Mn (..., pij˜ , ..., pk̃ , ...)i
2pi · pj
k6=i,j
2
dPij,k (p⊥
, z) =
2
2
Vij,k (p⊥
, z) dφn+1 (p⊥
, z)
1 + δij˜
dφn
˜ k colour connected)
×δ(ij,
Sensible for p⊥ ordering and improvements to initial state radiation.
[SP & S. Gieseke, JHEP 1101 (2011) 024]
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
5 / 20
Taking dipoles serious: Color matrix element corrections.
|Mn+1 (..., pi , ..., pj , ..., pk , ...)|2 ≈
X
1
hMn (..., pij˜ , ..., pk̃ , ...)|Vij,k (pi , pj , pk )|Mn (..., pij˜ , ..., pk̃ , ...)i
2pi · pj
k6=i,j
2
dPij,k (p⊥
, z) =
×
2
2
Vij,k (p⊥
, z) dφn+1 (p⊥
, z)
1 + δij˜
dφn
−1 hMn |Tij˜ · Tk |Mn i
|Mn |2
T2ij˜
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
6 / 20
Taking dipoles serious: Color matrix element corrections.
|Mn+1 (..., pi , ..., pj , ..., pk , ...)|2 ≈
X
1
hMn (..., pij˜ , ..., pk̃ , ...)|Vij,k (pi , pj , pk )|Mn (..., pij˜ , ..., pk̃ , ...)i
2pi · pj
k6=i,j
2
dPij,k (p⊥
, z) =
×
2
2
Vij,k (p⊥
, z) dφn+1 (p⊥
, z)
1 + δij˜
dφn
−1 hMn |Tij˜ · Tk |Mn i
|Mn |2
T2ij˜
Similar to matrix element corrections: ‘Color matrix element corrections’.
[SP & M. Sjödahl, arXiv:1201.0260]
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
6 / 20
What amplitudes?
Choose a definite basis for color space:
|Mn i =
dn
X
α=1
cn,α |αn i
↔
Mn = (cn,1 , ..., cn,dn )T
|Mn i known for the hard process.
How to get |Mn+1 i to enter the next correction after emission?
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
7 / 20
What amplitudes?
Choose a definite basis for color space:
|Mn i =
dn
X
α=1
cn,α |αn i
↔
Mn = (cn,1 , ..., cn,dn )T
|Mn i known for the hard process.
How to get |Mn+1 i to enter the next correction after emission?
Observe that
and
|Mn |2 = M†n Sn Mn = Tr Sn × Mn M†n
†
hMn |Tij˜ · Tk̃ |Mn i = Tr Sn+1 × Tk̃,n Mn M†n Tij,n
˜
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
7 / 20
What amplitudes?
Choose a definite basis for color space:
|Mn i =
dn
X
α=1
cn,α |αn i
↔
Mn = (cn,1 , ..., cn,dn )T
|Mn i known for the hard process.
How to get |Mn+1 i to enter the next correction after emission?
Observe that
and
|Mn |2 = M†n Sn Mn = Tr Sn × Mn M†n
†
hMn |Tij˜ · Tk̃ |Mn i = Tr Sn+1 × Tk̃,n Mn M†n Tij,n
˜
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
7 / 20
What amplitudes?
Observe that
and
|Mn |2 = M†n Sn Mn = Tr Sn × Mn M†n
†
hMn |Tij˜ · Tk̃ |Mn i = Tr Sn+1 × Tk̃,n Mn M†n Tij,n
˜
Use an ‘amplitude matrix’ Mn as basic object:
Mn+1 = −
X X 4παs Vij,k (pi , pj , pk )
†
Tk̃,n Mn Tij,n
˜
2
pi · pj
T˜
i6=j k6=i,j
ij
where
Mnhard = Mnhard M†nhard
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
8 / 20
Technicalities.
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
9 / 20
Technicalities.
Two things to tackle:
– An efficient treatment of the color basis {|αi}.
[M. Sjödahl, in preparation]
– Splitting kernels which can become negative.
[SP & M. Sjödahl, EPJ Plus 127 (2012) 26]
Then hook into existing shower and sampling machinery.
[SP & S. Gieseke, arXiv:1109.6256], [SP, EPJ C72 (2012) 1929]
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
9 / 20
Technicalities: The color basis.
We use a trace basis.
The framework is general enough to cope with other choices.
Mapping to the basis after emission is simple:
k
i
l
j
i
=
k
l
j
i
l
k
j
−
more details in [M. Sjödahl, JHEP 0812 (2008) 083]
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
10 / 20
Technicalities: The Sudakov veto algorithm.
Evolve from Q → q in presence of an infrared cutoff µ. Driven by:
dSP (µ; q|Q)
= ∆P (µ|Q)δ(q − µ) + P(q)∆P (q|Q)θ(Q − q)θ(q − µ)
dq
with the Sudakov form factor
Z
∆P (q|Q) = exp −
Q
P(k)dk
q
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
11 / 20
Technicalities: The Sudakov veto algorithm.
Evolve from Q → q in presence of an infrared cutoff µ. Driven by:
dSP (µ; q|Q)
= ∆P (µ|Q)δ(q − µ) + P(q)∆P (q|Q)θ(Q − q)θ(q − µ)
dq
This is truly a probability density. General version of the Sudakov veto
algorithm proven to work for this case.
[SP & M. Sjödahl, EPJ Plus 127 (2012) 26]
Adaptive sampling methods are available: ExSample C++ library.
[SP, EPJ C72 (2012) 1929]
Also know how to combine several channels, P =
→ ‘competing processes’.
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
P
i
Pi
11 / 20
Technicalities: The Sudakov veto algorithm.
Evolve from Q → q in presence of an infrared cutoff µ. Driven by:
dSP (µ; q|Q)
= ∆P (µ|Q)δ(q − µ) + P(q)∆P (q|Q)θ(Q − q)θ(q − µ)
dq
What if P < 0?
Possible to solve, up to overall normalization.
P
Much simpler, if i Pi > 0 with some Pi < 0.
→ Interleave vetoing and competition.
– For each i propose candidates qi driven by Pi θ(Pi ).
– Select largest qi .
– Accept with probability
Simon Plätzer (DESY Theory Group)
P
i
Pi /
P
i
Pi θ(Pi ).
Subleading Nc Improved Parton Showers
12 / 20
Results: Warmup.
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
13 / 20
Results: Warmup.
Proof of concept: e + e − → jets.
No hadronization: check effects on the level of the shower.
Would anyway require rethinking, as manifestly a large-Nc picture.
Only gluon emission, as no colour correlations for g → qq̄.
Three types of approximations:
– Full: CF and correlations exact.
– Shower: CF exact, correlations large Nc .
– Strict large-Nc : Everything approximated, including CF ≈ CA /2
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
13 / 20
Results: Warmup.
What to expect, and what to check.
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
14 / 20
Results: Warmup.
What to expect, and what to check.
Look at first few color matrix element corrections, exact vs. large-Nc :
– No difference for two jets.
– Large-Nc reproduces shower implementation exactly for three jets.
– From four jets on, large-Nc sensitive to history:
match shower implementation if one sequence of dipoles dominates.
Check how well large-Nc reproduces the shower implementation:
differences at per-mille level.
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
14 / 20
Results: Warmup.
number of emissions
event fraction
1
full
shower
strict large-Nc
Results stable
when going up to
six emissions.
0.1
0.01
Full correlations
suppress radiation:
destructive
interferences.
x/full
DipoleShower + ColorFull
1.4
1.2
1
0.8
0.6
1
2
3
4
5
6
nemissions
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
15 / 20
Results.
Thrust, τ = 1 − T
full
shower
strict large-Nc
10
1
0.1
0.01
full correlations, n = 6
n=5
n=4
10
1
0.1
0.01
0.001
0.001
0.0001
0.0001
DipoleShower + ColorFull
1.2
1.1
1
0.9
0.8
x/n = 6
x/full
Thrust, τ = 1 − T
100
N −1 dN/dτ
N −1 dN/dτ
100
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
DipoleShower + ColorFull
1.2
1.1
1
0.9
0.8
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
τ
Simon Plätzer (DESY Theory Group)
τ
Subleading Nc Improved Parton Showers
16 / 20
Results.
average rapidity w.r.t. ~n3
Angle between softest jets
full
shower
strict large-Nc
0.8
N −1 dN/dhyi
N −1 dN/d cos α34
1
0.6
0.4
full
shower
strict large-Nc
1
0.1
0
1.2
1.1
1
0.9
0.8
DipoleShower + ColorFull
DipoleShower + ColorFull
x/full
x/full
0.2
-1
-0.5
0
0.5
1
1.2
1.1
1
0.9
0.8
0
Simon Plätzer (DESY Theory Group)
0.5
1
1.5
2
2.5
3
hyi
cos α34
Subleading Nc Improved Parton Showers
17 / 20
Conclusions and outlook.
First implementation of subleading Nc contributions in parton showers.
Small effects seen in e + e − → jets, except very special observables.
Solutions to technical issues, particularly negative splitting kernels.
→ Will serve as input for future work.
Larger effects expected for e.g. pp → jets.
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
18 / 20
Backup.
Simon Plätzer (DESY Theory Group)
Subleading Nc Improved Parton Showers
19 / 20
Large-Nc vs. shower implementation.
1
O
1
Nc2
V13
V13 +V23
4
1
V23
V13 +V23
1
2
3
O
1
Nc2
3
V13
V13 +V23
1
2
V23
V13 +V23
2
Simon Plätzer (DESY Theory Group)
2
Subleading Nc Improved Parton Showers
4
20 / 20