In a leading color parton shower things are easy ...

Introduction Dipole Showers Results ˇ

Color corrections in parton showers

Malin Sjödahl

In collaboration with Simon Plätzer (Vienna) and Johan Thorén (Lund)

arXiv: 1808.00332 October 17, 2018

Malin Sjödahl

Introduction Dipole Showers Results ˇ

Section 1

Introduction

Introduction Dipole Showers Results ˇ

Parton shower basics

I A parton shower starts from a hard matrix element for some scattering process and dresses it up with additional radiation (mostly gluons)

Hard part

Radiation added by shower

Malin Sjödahl

Introduction Dipole Showers Results ˇ

In a leading color parton shower things are easy ...

I In standard parton showers where QCD is treated as if it had infinitely many colors, the colors are described by color lines and only color connected partons (sharing a line) can radiate coherently→ ∼ Npartondipoles

Introduction Dipole Showers Results ˇ

In real QCD with three colors

I For Nc= 3 radiation from almost any pair of partons can interfere→∼ Npartons² possibilities, suppressed by (1/Nc), 1/N_c²,...

Malin Sjödahl

Introduction Dipole Showers Results ˇ

Why investigate N

= 3 color corrections?

I Expect that color suppressed terms become very important for many partons

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

I Needed for exact matching of matrix elements to parton showers

I Needed for Nc = 3 hadronization

Introduction Dipole Showers Results ˇ

Section 2 Dipole Showers

Malin Sjödahl

Introduction Dipole Showers Results ˇ

Dipole Factorization

I Parton showers work under the approximation that the next parton to be emitted is soft or collinear to one of the

existing partons

I Dipole factorization gives, whenever i and j become collinear or one of them soft:

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

k6=i,j

1

2pi· p^jhMn(p˜ij, p˜k, ...)|Vij,k(pi, pj, pk)| Mn(p˜ij, p˜k, ...)i An emitter ˜ijsplits into two partons i and j, with the spectator ˜kabsorbing the momentum to keep all partons (before and after) on-shell. (Catani, Seymour

hep-ph/9605323)

ij˜ i

j

Color corrections in parton showers 8

Malin Sjödahl

In collaboration with Simon Plätzer (Vienna) and Johan Thorén (Lund)arXiv: 1808.00332

Introduction Dipole Showers Results ˇ

The spin averaged splitting kernel is

Vij,k(pi, pj, pk) =−8πα^sV_ij,k(pi, pj, pk)T˜ij· Tk

T²_˜ij

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_⊥/sijk − (1 + z)



Malin Sjödahl

Introduction Dipole Showers Results ˇ

Emission probability

For a leading Nc shower, the emission probability is dPij,k(p²_⊥, z) = Vij,k(p²_⊥, z)dφn+1(p²_⊥, z)

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

Including subleading emissions, instead gives dP_ij,k(p²_⊥, z) = Vij,k(p²_⊥, z)dφ_n+1(p²_⊥, z)

dφn ×−1 T²_˜

ij

hMⁿ|T^˜ij· T^˜k|Mⁿi

|M|²

Introduction Dipole Showers Results ˇ

Overall Picture

UsingHerwig’s dipole shower

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

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

T²_˜

ij

hMⁿ|T˜ij· T˜k|Mⁿi

|M|²

I Reweighting to encompass negative contributions

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

I Color structure is calculated usingColorFull(MS 1412.3967)

I Nc = 3 shower for a number of emissions, then standard leading Nc shower

Malin Sjödahl

Introduction Dipole Showers Results ˇ

Color structure

I A major challange is the SU(3) color structure of QCD

I The color structure can be decomposed in color bases

|Mⁿi =

d_n

X

α=1

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

and for this project we use trace bases

I ... but these standard “bases” are non-orthogonal and overcomplete, with a dimension scaling∼ (Ng+ N_q¯q)!→ (N_g+ N_q¯q)!²terms when squaring

I See next talk by Johan Thorén for better bases

Introduction Dipole Showers Results ˇ

New Features

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

I The g→ q¯q splitting

I Hadronic initial state, meaning initial state radiation

I Full compatibility with all of the additional functionality in Herwig 7.1. (So we can run any process now, in particular LHC events)

I Subsequent standard leading Ncshowering after the Nc = 3 shower

Malin Sjödahl

Dipole Showersˆ Results Current Status and Future Work

Section 3

Results

Dipole Showersˆ Results Current Status and Future Work

Full Color Shower Reaching Soft Scales

Since a limited number of Nc = 3 emissions are kept, up to 3 for LHC and 5 for LEP, we check the pT 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

I → We go far down in p^Tcompared to relevant jet scales, at LEP close to the hadronization scale

Malin Sjödahl

Dipole Showersˆ Results Current Status and Future Work

LEP Preliminary Results

For most e⁺e⁻observables we find small corrections, at the percent level. However, some observables (thrust, out-of-plane p_⊥, hemisphere masses, aplanarity, jet multiplicities 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 7

0.9 0.95 1.0 1.05

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

Ratio

Dipole Showersˆ Results Current Status and Future Work

LHC Preliminary Results

For LHC observables, corrections are typically of order a few percent, but some observables show corrections of 10− 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 Malin Sjödahl

Dipole Showersˆ Results Current Status and Future Work

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 only qg→ qg scattering and a 50 GeV analysis cut.

Dipole Showersˆ Results Current Status and Future Work

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 1.05 1.1

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 1.05 1.1

∆φ(jet 1, jet 3)

Ratio

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

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

Malin Sjödahl

Dipole Showersˆ Results Current Status and Future Work

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−TC)

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(Tm,C)

MC/Data

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

P

i|p_⊥,i·ˆnT|

Pp , Tm,C=

P

i|p_⊥,i׈nT|

Pp for jet i, with

Dipole Showersˆ Results Current Status and Future Work

Conclusion, Hard Perturbative Region

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

I Overall the data description does not change

I 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

I 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%

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

Malin Sjödahl

Dipole Showersˆ Results Current Status and Future Work

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 subleading Nc

effects probably play an important role for soft(ish) QCD

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,

Color corrections in parton showers 22

Malin Sjödahl

In collaboration with Simon Plätzer (Vienna) and Johan Thorén (Lund)arXiv: 1808.00332

Dipole Showersˆ Results Current Status and Future Work

Conclusion, Soft Region

In the soft region/region of many colored partons:

I In this region, we cannot claim accurate results, however,

I we often find large corrections of several ten percent

I This affects the state going into the hadronization

I meaning that we can expect a significant effect on the tune

I Subleading Nc effects can therefore be hidden in the tune

I Need to retune

Malin Sjödahl

Dipole Showersˆ Results Current Status and Future Work

Section 4

Current Status and Future Work

Dipole Showersˆ Results Current Status and Future Work

Current Status and Future Work

I We have a fully functional Nc = 3 parton shower for any LEP or LHC process

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

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

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

Thank you!

Malin Sjödahl

Section 5

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 for e⁺e⁻(left) and pp collisions (right) depending on the number of Nc= 3 emissions allowed. All generated events are used in these plots, i.e., no further analysis cut is applied.

Malin Sjödahl

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-plane p⊥w.r.t. the thrust and thrust major axes (left), light hemisphere mass (middle) and fraction of events containing Nch

charged particles.DELPHI, ALEPH

Malin Sjödahl

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 with p⊥ above Q0

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

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(η^3−φ2)(η²3^|−η2), using (left) an underlying 2→ 2 hard process and (right) an underlying 2 → 3 hard process.

CMS 1102.0068

Malin Sjödahl

Density Operator

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

|Mⁿi =

d_n

X

α=1

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

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

M_n+1=−X

i6=j

X

k6=i,j

4παs

pi· p^j

V_ij,k(pi, pj, pk) T²_˜

ij

T_˜k,nM_nT_˜^†

ij,n (2)

where

V_ij,k = T²_˜ijpi· p^k

pj· p^k. (3)

Color Matrix Element Corrections

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

ωⁿ_ik= −1 T²_˜

ij

Tr

S_n+1× T˜k,nMnT_˜^†

ij,n



Tr(Sn× Mⁿ) (4)

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

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

Malin Sjödahl

Standard veto algorithm

Standard veto algorithm: we want to generate a scale q and additional splitting variables x (e.g. z andφ) according to a distribution dSP.

dSP(µ, xµ|q, x|Q)

= dqd^dx(∆P(µ|Q)δ(q − µ)δ(x − x^µ)

+P(q, x)θ(Q− q)θ(q − µ)∆P(q|Q)) Where∆Pis 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 (change P→ R in the above eqs.).

Standard veto algorithm

Standard veto algorithm: we want to generate a scale q and additional splitting variables x (e.g. z andφ) according to a distribution dSP.

dSP(µ, xµ|q, x|Q)

= dqd^dx(∆P(µ|Q)δ(q − µ)δ(x − x^µ)

+P(q, x)θ(Q− q)θ(q − µ)∆P(q|Q)) Where∆Pis 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 (change P→ R in the above eqs.).

Where we require R(q, x)Malin Sjödahl≥ 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. Generate q and x according to SR(µ, x_µ|q, x|k).

2. If q= 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, set k= q and go back to 1.

Weighted veto algorithm

Introduce an acceptance probability 0≤ (q, x|k, y) < 1 and a weightω. Set k = Q, ω = 1.

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

2. If q= 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.

Malin Sjödahl

Example of 1 /N

suppressed terms

Leading color structure:

2

=

= TR = T_R²(N_c²− 1) ∝ Nc².

Example of 1 /N

suppressed terms

Leading color structure:

2

∝ Nc². Interference term:

   ∗

=

= TR −T_R

Nc

= 0− TR²

N_c²− 1 Nc ∝ N^c.

Malin Sjödahl

Example of 1 /N

suppressed terms

   ∗

=

= TR

| {z }

∝N²_c

−TR

Nc

| {z }

∝N_c²