Color Matrix Element Corrections in Herwig

Color Matrix Element Corrections in Herwig

Malin Sjodahl

→ Simon Pl¨atzer?

Collaboration of Simon Pl¨atzer, Malin Sjodahl and Johan Thor´en

Work submitted to JHEP, arXiv 1808.00332 August 28, 2018

Section 1

Motivation

Why investigate color matrix element corrections?

Effects of order1/N_c² can be comparable to other

uncertainties, and 1/Ncsuppression is present if there are two or more qq-pairs in the process

The colored initial state and the higher energy at the LHC gives rise to many colored partons and hence many color suppressed terms

For a leading Nc shower, the number of color connected pairs grows roughly as Npartons, but the number of pairs of colored partons grows as N_partons² → expect larger effects at LHC Needed for exact (N)NLO matching

A step towards a full color shower, including virtual color rearranging gluon exchanges

Section 2

Dipole Showers

Dipole Factorization

Dipole factorization gives, wheneveri and j become collinear or one of them soft:

|Mn+1(..., pi, ..., pj, ..., pk, ...)|² = X

k6=i,j

1

2p_i· pjhMn(p_ij˜, p_˜k, ...)|Vij,k(p_i, p_j, p_k)| Mn(p_ij˜, p_k˜, ...)i An emitter ˜ij splits into two partons i and j, with the spectator ˜k absorbing the momentum to keep all partons (before and after) on-shell. (Catani, Seymour hep-ph/9605323)

ij˜ i

j

k

Dipole Factorization

The spin averaged dipole insertion operator is

V_ij,k(pi, pj, p_k) =−8παsV_ij,k(pi, pj, p_k)Tij˜ · Tk

T²_˜

ij

(1)

Where, for example, for a final-final dipole configuration, we have Vq→qg,k(pi, pj, pk) = CF

 2(1− z)

(1− z)²+ p²_⊥/s_ijk − (1 + z)

 (2)

Emission probability

For a leadingNc shower, the emission probability is dP_ij,k(p²_⊥, z) = V_ij,k(p²_⊥, z)dφn+1(p²_⊥, z)

dφ_n ×δ( ˜ij, ˜k color connected) 1 + δ_{ij g˜}

(3) Including subleading emissions, instead gives

dP_ij,k(p²_⊥, z) = V_ij,k(p²_⊥, z)dφn+1(p²_⊥, z)

dφn ×−1

T²_˜

ij

hMn|Tij˜ · Tk˜|Mni

|M|²

(4)

e⁻

e⁺ γ

q

¯ q

g

q

¯ q

g

Section 3

Color Matrix Element Corrections

Overall Picture

UsingHerwig’s dipole shower

Instead of only allowing color connected emitter-spectator pairs to radiate, all possible pairs can radiate

All pairs may radiate in proportion to (for the first emission) ωⁿ_ik= −1

T²_ij˜

hMn|Tij˜ · T˜k|Mni

|M|² (5)

Reweighting to encompass negative contributions

The full color structure is evolved to be able to evaluate the above factor for the next emission

Color structure is calculated using ColorFull(MS 1412.3967) Nc= 3 shower for a number of emissions, then standard leading N_c shower

Density Operator

We can write the amplitude as a vector in some basis (trace, multiplet, etc.),

|Mni =

dn

X

α=1

c_n,α|αni ↔ Mn= (c_n,1, ..., c_n,dn)^T (6)

and construct a “density operator”Mn=MnM^†n, that we evolve by

Mn+1=−X

i6=j

X

k6=i,j

4παs

pi· p^j

V_ij,k(p_i, p_j, p_k) T²_˜

ij

Tk,n˜ MnT_˜^†

ij,n (7) where

V_ij,k = T²_ij˜p_i· pk

pj· pk

. (8)

This allows us to calculate the color matrix element corrections.

Color Matrix Element Corrections

Evolving the density operator, we can calculate the color matrix element corrections for any number of emissions

ω_ikⁿ = −1 T²_˜

ij

Tr



Sn+1× T˜k,nMnT_˜^†

ij,n



Tr(S_n× Mn) (9)

Note thatω_ikⁿ can be negative, this is included through the weighted Sudakov algorithm(Bellm, SP, Richardson, Siodmok, Webster, 1605.08256)

This initially resulted in very large weights. Modifications to the weighted Sudakov veto algorithm drastically reduced the weights.

New Features

Compared to our previouse⁺e⁻ results(SP, MS 1206.0180), we have added

The g→ q¯q splitting

Hadronic initial state, meaning initial state radiation Full compatibility with all of the additional functionality in Herwig7.1. (So we can run any process now, in particular LHC events)

Subsequent standard leading Nc showering after theNc= 3 shower

Section 4

Preliminary Results

N

= 3 Shower Reaching Soft Scales

Since a limited number ofNc= 3 emissions are kept, up to 3 for LHC and 5 for LEP, we check thep_T of the last corrected emission

5th Nc=3 emission

0 1 2 3 4p⟂

0.00 0.05 0.10 0.15

LEP

3rd Nc=3 emission

5 10 15 20 25 p⟂

0.01 0.02 0.03 0.04 0.05 0.06

LHC 50 GeV

→ We go far down in pT compared to relevant jet scales, at LEP close to the hadronization scale

→ We expect convergence of most standard hard observables (this is also confirmed by allowing fewerN_c= 3 emissions)

LEP Preliminary Results

For moste⁺e⁻ observables we find small corrections, at the percent level. However, some observables (thrust, out-of-planep⊥, hemisphere masses, aplanarity, jet multiplicites for many jets) are corrected by∼ 5%.

Leading Nc 1 Nc=3 emission 3 Nc=3 emissions 5 Nc=3 emissions

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

Jets

2 3 4 5 6 7 8

0.9 0.95 1.0 1.05

Jets

Ratio

Leading Nc 1 Nc=3 emission 3 Nc=3 emissions 5 Nc=3 emissions

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 1 10¹

Aplanarity

1/σdσ/dA

0 0.05 0.1 0.15 0.2

0.9 0.95 1.0 1.05

A

Ratio

Figure:Number of jets withE > 5GeV, and aplanarity

LHC Preliminary Results

For LHC observables, corrections are typically of order a few percent, but some observables show corrections of10− 20%

Leading Nc 1 Nc=3 emission 2 Nc=3 emissions 3 Nc=3 emissions 10⁵

10⁶

Rapidity of first jet

dσ/dy(jet1)[pb]

-4 -2 0 2 4

0.8 0.9 1.0 1.1 1.2

y(jet 1)

Ratio

Leading Nc 1 Nc=3 emission 2 Nc=3 emissions 3 Nc=3 emissions 10⁴

10⁵ 10⁶

Rapidity of second jet

dσ/dy(jet2)[pb]

-4 -2 0 2 4

0.8 0.9 1.0 1.1 1.2

y(jet 2)

Ratio

Figure:Rapidity of hardest and second hardest jetusing a 50GeV analysis cut

LHC Preliminary Results

If we could study quark-gluon scattering, we would find large corrections

Leading Nc 3 Nc=3 emissions

10² 10³ 10⁴ 10⁵ 10⁶

Rapidity of first jet

dσ/dy(jet1)[pb]

-4 -2 0 2 4

0.8 0.9 1.0 1.1 1.2

y(jet 1)

Ratio

Leading Nc 3 Nc=3 emissions

10¹ 10² 10³ 10⁴ 10⁵

Rapidity of second jet

dσ/dy(jet2)[pb]

-4 -2 0 2 4

0.8 0.9 1.0 1.1 1.2

y(jet 2)

Ratio

Figure:Rapidity distribution of the hardest and second hardest jet while considering onlyqg→ qg scattering and a 50 GeV analysis cut.

... but we cannot

LHC Preliminary Results

Requiring one forward (quark dominated) and one central (gluon dominated) jet we find sizable corrections for many observables

Leading Nc 3 Nc=3 em.

1 10¹ 10² 10³ 10⁴

Rapidity of second jet

dσ/dy(jet2)[pb]

-4 -2 0 2 4

0.85 0.9 0.95 1.0 1.051.1

y(jet 2)

Ratio

Leading Nc 3 Nc=3 emissions

10² 10³

Azimuthal separation between jets

dσ/d∆φ(jet1,jet3)[pb]

0 0.5 1 1.5 2 2.5 3

0.85 0.9 0.95 1.0 1.051.1

∆φ(jet 1, jet 3)

Ratio

Figure:Rapidity and∆φ1,3 for the central/forward case

(400< M12<600 GeV, 3.8 <|y¹+ y2| < 5.2, 1.5 < |y²− y¹| < 3.5)

LHC Preliminary Results

We have compared to LHC data for a wide range of observables.

In general we find small corrections and no overall visible change in data description.

b b b b b b b b b b b b b

Data

b

Leading Nc 1 Nc=3 emission 2 Nc=3 emissions 3 Nc=3 emissions

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Central Transv. Thrust, 90 GeV<p^{jet 1}_⊥ <125 GeV,√ s=7 TeV

1/NdN/dln(1−TC)

-12 -10 -8 -6 -4 -2

0.6 0.8 1 1.2 1.4

ln(1−T_C)

MC/Data b b b b b b b b b b b b b

Data

b

Leading Nc 1 Nc=3 emission 2 Nc=3 emissions 3 Nc=3 emissions

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0.4Central Transv. Minor, 90 GeV<p^{jet 1}_⊥ <125 GeV,√ s=7 TeV

1/NdN/dln(Tm,C)

-6 -5 -4 -3 -2 -1

0.6 0.8 1 1.2 1.4

ln(T_m,C)

MC/Data

Figure:Central transverse thrust and thrust minor for√s = 7GeV,CMS 1102.0068TC= maxnˆT

P

i|p_⊥,i·ˆnT| P

ip⊥,i , Tm,C =

P

i|p_⊥,i׈nT| P

ip⊥,i for jeti, with η < 1.3

Conclusion, Hard Perturbative Region

In the hard perturbative region:

We have considered a wide range of observables at LEP and LHC and compared to data

Overall the data description does not change

As long as soft scales/observables with very many jets are not considered, the matrix element correction type of corrections are accurately described by correcting the first few emissions In general, percent level corrections are found at LEP, for some observables (thrust, out-of-plane p⊥, hemisphere masses, aplanarity, jet multiplicities for many jets) effects of around 5%

At the LHC, corrections are often a few percent, for some observables (mostly rapidity) corrections around 10-20%

Going Soft/Very Many Colored Partons

For soft QCD, where we cannot expect reliable results due to the need of more color suppressed terms, resummation, hadronization and MPI, we find larger corrections in many cases, (jet resolution scales, cluster masses in Herwig, number of very soft jets at LEP, charged multiplicity distribution, individual hadron multiplicities), indicating that subleadingN_ceffects probably play an important role for soft(ish) QCD

Leading 1 sub em 2 sub em 3 sub em 4 sub em 5 sub em 10−1

1 10¹ 10² 103 10⁴ 105

First Cluster Mass

1 101 102

0.50.6 0.70.8 0.91.0 1.11.2 1.31.4

mcluster[GeV]

Ratio

Leading Nc 1 Nc=3 emission 3 Nc=3 emissions 5 Nc=3 emissions 10⁻⁵

10⁻⁴ 10−3 10−2 10⁻¹ Jets

2 4 6 8 10 12 14

0.50.6 0.70.8 0.91.0 1.11.2 1.31.4

Jets

Ratio bbbbbbbbbbbbbbbbbbbbbbbbbb

Data

b

Leading Nc 3 Nc=3 emissions 5 Nc=3 emissions

10−5 10−4 10−3 10−2 10−1

Charged multiplicity distribution

1/NdN/dNch

10 20 30 40 50

0.50.6 0.70.8 0.91.0 1.11.2 1.31.4

Nch

MC/Data

Figure:Examples of large corrections: first clustermass in Herwig, number of jets at LEP using a 2 GeV energy cut, charged multiplicity distribution

Conclusion, Soft Region

In the soft region/region of many colored partons:

In this region, we cannot claim accurate results, however, we often find large corrections of several ten percent This affects the state going into the hadronization

meaning that we can expect a significant effect on the tune Subleading N_c effects can therefore be hidden in the tune Need to retune

Section 5

Current Status and Future Work

Current Status and Future Work

We can run the Nc= 3 parton shower for any LEP or LHC process

Tuning should be performed before a reliable comparison to standard showers can be done

We still miss virtual corrections, which rearrange the color structure without any real emissions. These are important for gap-survival observables.

In the more distant future, an update of hadronization models to an Nc= 3 final state would be and interesting research task

Thank you! I hope you could hear me...

Section 6

Backup Slides

Weight distribution

Leading Nc 1 Nc=3 emission 2 Nc=3 emissions 3 Nc=3 emissions 4 Nc=3 emissions 5 Nc=3 emissions

-10 -5 0 5 10

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 1

Weight distribution

w N(w)/Ntot

Leading Nc 1 Nc=3 emission 2 Nc=3 emissions 3 Nc=3 emissions

-10 -5 0 5 10

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 1

Weight distribution

w N(w)/Ntot

Figure:Weight distribution fore⁺e⁻ (left) and pp collisions (right) depending on the number ofNc= 3 emissions allowed. All generated events are used in these plots, i.e., no further analysis cut is applied.

N

= 3 Shower Reaching Soft Scales

4th Nc=3 emission 5th Nc=3 emission 6th Nc=3 emission

0 1 2 3 4p⟂

0.00 0.05 0.10 0.15 0.20

LEP

3rd Nc=3 emission 4th Nc=3 emission

5 10 15 20 25 p⟂

0.02 0.04 0.06 0.08

LHC 50 GeV

More LEP Observables

bbbbbbbbb b b b b b b b b

Data

b

Leading Nc 3 Nc=3 emissions 5 Nc=3 emissions

10⁻² 10−1 1 10¹

102Out-of-plane p⊥in GeV w.r.t. thrust axes

Ndσ/dpout⊥

0 0.5 1 1.5 2 2.5 3 3.5

0.50.6 0.70.8 0.91.0 1.11.2 1.31.4

p^out⊥/ GeV

MC/Data b b b b b b b b b

Data

b

Leading Nc 3 Nc=3 emissions 5 Nc=3 emissions

10−1 1 10¹

Light hemisphere masses, M²l/Evis²

Ndσ/dM

2 l/E 2 vis

0 0.02 0.04 0.06 0.08 0.1

0.50.6 0.70.8 0.91.0 1.11.2 1.31.4

M²l/E²vis

MC/Data bbbbbbbbbbbbbbbbbbbbbbbbbb

Data

b

Leading Nc 3 Nc=3 emissions 5 Nc=3 emissions

10⁻⁵ 10−4 10−3 10⁻² 10−1

Charged multiplicity distribution

1/NdN/dNch

10 20 30 40 50

0.50.6 0.70.8 0.91.0 1.11.2 1.31.4

Nch

MC/Data

Figure:Out-of-planep⊥ w.r.t. the thrust and thrust major axes (left), light hemisphere mass (middle) and fraction of events containingNch

charged particles. DELPHI, ALEPH

Top at LHC

b b b b b b b b b b b b b b b b b b

Data

b

Leading Nc 3 Nc=3 emissions

0.7 0.75 0.8 0.85 0.9 0.95 1.0

Gap fraction vs. Q0for veto region:|y| <0.8 fgap

50 100 150 200 250 300

0.94 0.96 0.98 1.0 1.02 1.04

Q0[GeV]

MC/Data bbbbb b b b b b b b b b b b b b

Data

b

Leading Nc 3 Nc=3 emissions

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1.0

Gap fraction vs. Qsumfor veto region:|y| <0.8 fgap

50 100 150 200 250 300 350 400

0.94 0.96 0.98 1.0 1.02 1.04

Qsum[GeV]

MC/Data

Figure:Fraction of events having no additional jet withp⊥ aboveQ0

within a rapidity interval|y| < 0.8 (left) and fraction of events where the scalar sum of transverse momenta within|y| < 0.8 does not exceed Q^sum (right) fortt events at√s = 7 TeV.ATLAS 1203.5015

QCD “Coherence” observable

b b b b b b b b b b b b b b b b b b

Data

b

Leading Nc 1 Nc=3 emission 2 Nc=3 emissions 3 Nc=3 emissions

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

CMS,√_s=7 TeV, central jet 2–3 correlation,|η2| <0.8 Fη2(β)

0.5 1 1.5 2 2.5 3

0.50.6 0.70.8 0.91.0 1.11.2 1.31.4

β

MC/Data b b b b b b b b b b b b b b b b b b

Data

b

Leading Nc 1 Nc=3 emission 2 Nc=3 emissions

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

CMS,√_s=7 TeV, central jet 2–3 correlation,|η2| <0.8 Fη2(β)

0.5 1 1.5 2 2.5 3

0.50.6 0.70.8 0.91.0 1.11.2 1.31.4

β

MC/Data

Figure:The angleβ, tan β = sign^|φ(η³2^−φ)(η²3^|−η2), using (left) an underlying 2→ 2 hard process and (right) an underlying 2 → 3 hard process. CMS 1102.0068

Standard veto algorithm

Standard veto algorithm: we want to generate a scaleq and additional splitting variablesx (e.g. z and φ) according to a distribution dSP.

dS_P(µ, x_µ|q, x|Q)

= dqd^dx (∆_P(µ|Q)δ(q − µ)δ(x − xµ)

+P (q, x)θ(Q− q)θ(q − µ)∆P(q|Q)) Where∆P is the Sudakov form factor,

∆P(q|Q) = exp



− Z Q

q

dk Z

d^dzP (k, z)



To do this we use an overestimate of the distribution (with nicer analytical properties)dSR (changeP → R in the above eqs.).

Where we requireR(q, x)≥ P (q, x) for all q, x.

Standard veto algorithm

Standard veto algorithm: we want to generate a scaleq and additional splitting variablesx (e.g. z and φ) according to a distribution dSP.

dS_P(µ, x_µ|q, x|Q)

= dqd^dx (∆_P(µ|Q)δ(q − µ)δ(x − xµ)

+P (q, x)θ(Q− q)θ(q − µ)∆_P(q|Q)) Where∆P is the Sudakov form factor,

∆P(q|Q) = exp



− Z Q

q

dk Z

d^dzP (k, z)



To do this we use an overestimate of the distribution (with nicer analytical properties)dSR (changeP → R in the above eqs.).

Where we requireR(q, x)≥ P (q, x) for all q, x.

Standard veto algorithm

P (q, x) > 0 and R(q, x)≥ P (q, x). Set k = Q

1 Generateq and x according to SR(µ, xµ|q, x|k).

2 Ifq = µ, there is no emission above the cutoff scale.

3 Else, accept the emission with the probability P (q, x)

R(q, x).

4 If the emission was vetoed, setk = q and go back to 1.

Weighted veto algorithm

Introduce an acceptance probability0≤ (q, x|k, y) < 1 and a weightω. Set k = Q, ω = 1.

1 Generateq and x according to S_R(µ, x_µ|q, x|k).

2 Ifq = µ, there is no emission above the cutoff scale.

3 Accept the emission with the probability(q, x|k, y), update the weight

ω→ ω × 1

 ×P R

4 Otherwise update the weight to ω → ω × 1

1−  ×

 1−P

R



and start over at 1 with k = q.

Example of 1/N

suppressed terms

Leading color structure:

2

=

= T_R = T_R²(N_c²− 1) ∝ Nc².

Example of 1/N

suppressed terms

Leading color structure:

2

∝ Nc². Interference term:

   ∗

=

= T_R −T_R

N_c

= 0− TR²

N_c²− 1 N_c ∝ N^c.

Example of 1/N

suppressed terms

   ∗

=

= TR

| {z }

∝N_c²

−TR

Nc

| {z }

∝N_c²