Next-to-leading logarithmic processes in High Energy Jets...Next-to-leading logarithmic processes in...

Next-to-leading logarithmic processes in High EnergyJets

James A. Black,

with J. R. Andersen, H. Brooks and J. M. Smillie

Durham University

DIS Turin, April 10th 2019

Table of Contents

1 Introduction to HEJMRK limitFKL Contributions

2 Subleading ProcessesUnorderedExtremal qqCentral qq

3 Pure Jet Results

4 W+JetsComplications

James A. Black (IPPP) NLL in HEJ DIS2019 2 / 26

An Introduction to HEJ

High Energy Jets

A Partonic Monte Carlo Generator which aims to describe highmultiplicity events.

Provides perturbative predictions at LL accuracy (log(s/|t|)) withresummation of hard corrections to all orders.

Hard corrections are αs suppressed but phase space enhanced in thelarge invariant mass limit.

but we need a formalism...


Multi Regge Kinematic (MRK) Limit

The MRK Limit:

large s ; small PT ; strongly ordered jet rapidities (yj):

y1 � y2 � ...� yi � ..� yn−1 � yn

Some nice relations:

s2 ∼ −u2 → large

ti ∼ −p2⊥ji ∼ −p2⊥

log(

sij|tij |

)≈ |yj − yi |


FKL Contributions

FKL configurations are the leading contributions in the MRK limit.

(2→ n) amplitudes with strong rapidity ordering in final state

Internal jets (by rapidity) required to be gluons

Mediated by colour octet (gluon) t-channel exchange

√t1

√t2

√tn−1

pa

pn

p1

p2

p3

pn−1

pb Current

Current

Decreasin

gRapidity

Resum via effective LipatovVertices and the LipatovAnsatz.

For MRK kinematics, theseare leading power in powerexpansion. After integration,gives leading logarithmiccontribution.


HEJ Matrix element

=

1

4 (N2C − 1)

∥∥Sfafb→f1fn

∥∥2

·(g2s Kf1

1

t1

)·(g2s Kfn

1

tn−1

)

·n−2∏

i=1

(− g2

s CA

ti ti+1V µ(qi , qi+1)Vµ(qi , qi+1)

)

·n−1∏

j=1

exp[ω0(qj⊥)(yj+1 − yj)

]

Processes ⇔ currents, e.g. Sf1f2→f1Hf2 = jµ(p1, pa)V µνH (qj , qj+1)jν(pb, pn).

+ =


HEJ Matrix element

=

1

4 (N2C − 1)

∥∥Sfafb→f1fn

∥∥2

·(

g2s Kf1

1

t1

)·(

g2s Kfn

1

tn−1

)

·n−2∏

i=1

(− g2

s CA


)

·n−1∏

j=1


]


+ =


HEJ Matrix element

=

1

4 (N2C − 1)

∥∥Sfafb→f1fn

∥∥2

·(g2s Kf1

1

t1

)·(g2s Kfn

1

tn−1

)

·n−2∏

i=1

(− g2

s CA

titi+1V µ(qi , qi+1)Vµ(qi , qi+1)

)

·n−1∏

j=1


]


+ =


HEJ Matrix element

=

1

4 (N2C − 1)

∥∥Sfafb→f1fn

∥∥2

·(g2s Kf1

1

t1

)·(g2s Kfn

1

tn−1

)

·n−2∏

i=1

(− g2

s CA

ti ti+1Vµ(qi,qi+1)Vµ(qi,qi+1)

)

·n−1∏

j=1


]


+ =


HEJ Matrix element

=

1

4 (N2C − 1)

∥∥Sfafb→f1fn

∥∥2

·(g2s Kf1

1

t1

)·(g2s Kfn

1

tn−1

)

·n−2∏

i=1

(− g2

s CA


)

·n−1∏

j=1


]


+ =


Motivation: H+Jets FKL Only

0 1 2 3 4 5 6 7 8 9∆yfb

0.0

0.1

0.2

0.3

Rel

ativ

eco

ntr

ibu

tion 0

50

100

150

200

250

300

dσ/d

∆y f

b[f

b]

pp→ jjjH

Total rateAll-order componentFixed-order component

Resummed

FKL

Processes at FO

Unordered

Extremal qq

Other


Motivation: H+Jets Including Unordered

0 1 2 3 4 5 6 7 8 9∆yfb

0.0

0.1

0.2

0.3

Rel

ativ

eco

ntr

ibu

tion 0

50

100

150

200

250

dσ/d

∆y f

b[f

b]

pp→ jjjH


Resummed

FKL

Unordered

Processes at FO

Extremal qq

Other


Change due to Unordered

0 1 2 3 4 5 6 7 8∆yfb

−0.8

−0.6

−0.4

−0.2

0.0

0.2

Rel

ativ

eC

han

gein

dσ/d

∆y

fb


Points of Interest:

All-order componentincreased.

FO component decreaseslinearly with yfb

Total rate mostlyunchanged.

Goal: Include more subleading processes within HEJ approximation.


Unordered Contributions

Pa P1Current

Puno

P4Pb

A gluon outside ofFKL rapidity orderingis known as anUnordered emission.

In HEJ this is modelled as a modified current. Where we now allow thatyuno ∼ y1 and y1 >> y2. (QMRK Limit)

MunoqQ→gqQ ∼

jµuno(pa, p1, puno)jµ(pb, p2)

t


Extremal qq

Pa P1

Current

P3

P2

Pb

The Extremal qqcase is an incominggluon splitting to qq.

In HEJ use a a modified current (related by crossing symmetry to Unocase) in the scattering.

Mqqqg→qQQ ∼

jµqq(pb, p2, p3)jµ(pa, p1)

t

There are 5 possible diagrams which contribute.


Extremal qq: Possibilities

Pa

Pb

P1

P2

P3

Pa

Pb

P1

P2

P3

Pa

Pb

P1

P2

P3

Pa

Pb

P1

P2

P3

Pa

Pb

P1

P2

P3


Central qq

Pa P1

P3

P2

Pb P4

In the case a Central qq pairis produced, we use aneffective vertex which fits theform:

Mqq→qQQq ∼⟨1|µ|a

⟩Xµν

⟨4|ν|b

⟩

t1t3

There are 7 possible diagrams which contribute.


Central qq

Pa

Pb

P1

P2

P3

P4

Pa

Pb

P1

P2

P3

P4

Pa

Pb

P1

P2

P3

P4

Pa

Pb

P1

P2

P3

P4

Pa

Pb

P1

P2

P3

P4

Pa

Pb

P1

P2

P3

P4

Pa

Pb

P1

P2

P3

P4


Scaling of the Matrix Elements

Higgs+3j: qQ → qgHQGluon Exchange

(FKL)Quark Exchange

(Unordered)Higgs Outside(Unordered)

0 2 4 6 8 10∆ = ∆yfb/2

10-17

10-16

10-15

10-14

10-13

10-12

|Mi|2/s

2

|M1|2/s2

|M2|2/s2

|M3|2/s2

0 2 4 6 8 10∆ = ∆yfb/2

10-12

10-11

10-10

10-9

s j1j 2×|M

i|2/s

2

sj1j2 × |M2|2/s2

sj1j2 × |M3|2/s2

In Multi Regge Theory:

|M| ∼ (sji jj )spin

Swapping propagator (gluon → quark) suppresses ME by (sji jj )1/2.


Reducing Dependence on Matching

Pa

Pb

P1

P2

P3

P4

Pa

Pb

P1

P2

P3

P4

Fixed Order

α4s

Add Resummation

(αs∆y )N

Inclusion within HEJ:

Previously: • 4j + 5j Fixed order result used directly in HEJ

Now: • 4j result reduced by virtual corrections. 5j result increased.• sum of 4j + 5j state is logarithmically controlled.


Reducing Dependence on Matching

Pa

Pb

P1

P2

P3

P4

Pa

Pb

P1

P2

P3

P4

Fixed Order

α4s

Add Resummation

(αs∆y )N

Inclusion within HEJ:

Previously: • 4j + 5j Fixed order result used directly in HEJNow: • 4j result reduced by virtual corrections. 5j result increased.

• sum of 4j + 5j state is logarithmically controlled.


Pure Jets: FKL Only

0 1 2 3 4 5 6

∆yfb

0.0

0.1

0.2

0.3

0.4

Rel

ativ

eco

ntr

ibu

tion

0.0

0.5

1.0

1.5

dσ/d

∆y f

b[µ

b]

pp→≥ 3j

TotalResummedFixed Order

Processes Resummed

FKL

Processes only at FO

Unordered

Extremal qq

Other

Nota Bene: we do not includeCentral qq since that only exists as asubleading process for 4j+


Pure Jets: All subleading processes

0 1 2 3 4 5 6

∆yfb

0.0

0.1

0.2

0.3

0.4

Rel

ativ

eco

ntr

ibu

tion

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

dσ/d

∆y f

b[µ

b]

pp→≥ 3j

TotalResummedFixed Order

Processes Resummed

FKL

Unordered

Extremal qq

Processes only at FO

Other

Nota Bene: we do not includeCentral qq since that only exists as asubleading process for 4j+


Change due to Subleading Pieces

0 1 2 3 4 5 6∆yfb

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

Rel

ativ

eC

han

gein

dσ/d

∆y

fb TotalResummedFixed order

Points of interest:

All-order component increased.

FO component dramatically decreases with yfb

Total rate remains largely unchanged.


W+Jets at LL

In HEJ, W+Jets are usually calculated differently from Pure Jets by theuse of a modified current.

Pa P1

PW

+ Pa P1

PW

→ Pa P1

PW

Current

With the addition of the qq pairs we have additional places from which aW-Boson can be emitted.


W+Jets at NLL: Unordered

Complications to Unordered

No New Objects New Objects Required

Pa P1

PW

Current

Current

Puno

P4Pb

Pa P1

PW

Current

Puno

P4Pb


W+Jets at NLL: qq

Complications to qq

No New Objects New Objects Required

Pa P1

PW

Current

Current

P3

P2

Pb

PW

Pa P1

Current

P3

P2

Pb

PW

Pa P1

P3

P2

Pb P4

Current

Pa P1

P3

P2

Pb P4

PW


W+Jets at NLL: Extremal qq

Consider Process: qg → qQQW

AIM:

Factorise the t channel exchanges and the current scattering, resulting in anew effective current at either end of the FKL chain.

Need to find an amplitude for the process qg → qQQW of the form:

Mqg→qQQW ∼⟨1|µ|a

⟩Qµνρ (p2, pw , p3, pb) εν(pb)ε∗ρ(pw )

t1

Where Qµνρ is this effective current.


W+Jets at NLL: Central qq

Consider Process: qq → qQQWq

AIM:

Factorise Currents and effective qq vertex. As in the extremal qq

We therefore search for an expression of the form:

Mqq→qQQqW ∼⟨1|µ|a

⟩Xµν

⟨4|ν|b

⟩

t1t3


W+Jets Status Matrix Element Comparison

Pa P1

P3

P2

Pb P4

PW

↑-

-

↓

qq at fixed ∆y

0 2 4 6 8 100

1

2

1e 21

LOHEJ

0 2 4 6 8 100.0

2.5

5.0

||2 /

(256

5 s2 )

1e 18

LO*sqqHEJ*sqq

0 2 4 6 8 10

0.8

1.0

LO/LOHEJ/LO

Pa P1

P3

P2

Pb P4

PW

↑↑↓↓

qq at increasing ∆y

0 2 4 6 8 100.0

2.5

5.0

1e 22

LOHEJ

0 2 4 6 8 100.0

2.5

5.0

||2 /

(256

5 s2 )

1e 17

LO*sqqHEJ*sqq

0 2 4 6 8 10

0.8

1.0

LO/LOHEJ/LO


Conclusions and Further Considerations

Added resummation for Unordered, Extremal and Central qq

HEJ is now leading log accurate for all sub-leading processes

Verification process underway for W+Jets

Next steps for Next-to-Leading Log:

Virtual Corrections

HEJ2 recently had a public release!

https://hej.web.cern.ch/HEJ/


Backup slides


Lipatov Vertices

pA

pB

p1

p3

q +

pA

pB

p1

p3

p2

q1

q2

→

pA

pB

p1

p3

p2

q1

q2

V ρ(q1, q2) =− (q1 + q2)ρ

+pρA2

(q21

p2 · pA+

p2 · pBpA · pB

+p2 · p3pA · p3

)+ pA ↔ p1

− pρB2

(q22

p2 · pB+

p2 · pApB · pA

+p2 · p1pB · p1

)− pB ↔ p3.


Virtual Corrections

Lipatov Ansatz

One can obtain the virtual corrections in the MRK limit with the LipatovAnsatz, which is the following substitution within the analytic expressionfor the amplitudes:

1

ti→ 1

tiexp[α(qi )(yi−1 − yi )]

where

α = −g2CAΓ(1− ε)

(4π)2+ε2

ε

(q2µ2

)ε


Building Blocks

Pg

P2

P3

PW

JµV (p2, pA, pB , p3) =

(u2γ

ν(/p2 + /pA + /pB)γµu3

s2AB+

u2γµ(/p3 − /pA − /pB)γνu3

s3AB

)[uAγνuB ]


1a Contribution

Pa

Pb

P1

P2

P3

P4

Pa

Pb

P1

P2

P3

P4

Xµν1a =

gµνC1gwg4s

2√

2s23AB(s123AB)(pρ1)JVρ


4b Contribution

Pa

Pb

P1

P2

P3

P4

Pa

Pb

P1

P2

P3

P4

Xµν4b =

−gµνC1gwg4s

2√

2s23AB(s234AB)(pρ4)JVρ


3 Gluon Contribution

Pa

Pb

P1

P2

P3

P4

Xµν3g =

gwg2s T

geg ′T e23

2√

2 t1s23AB t3

[(q1)µ ηνρ + (q3)ν ηµρ−

(q1 + q3)ρ ηµν]JVρ(pa, pA, pB , p1)


Uncrossed Contributions

Pa

Pb

P1

P2

P3

P4

Xµνuncross =

〈A|σ|B〉(pA + pB)2

u2

[γσ(/p2 + /pA + /pB)γν(/p3 + /p4 − /pb)γµ

(s2AB)(tint2)+

γν(/pa − /p1 − /p2)γσ(/p3 + /p4 − /pb)γµ

(tint1)(tint2)+

γν(/pa − /p1 − /p2)γµ(/p3 + /pA + /pB)γσ

(tint1)(s3AB)

]u3


Crossed Contributions

Pa

Pb

P1

P2

P3

P4

Xµνcross =

〈A|σ|B〉(pA + pB)2

u2

[γσ(/p2 + /pA + /pB)γν(/pa − /p1 − /p3)γµ

(s2AB)(tint3)+

γν(/p2 − /p4 − /pb)γσ(/pa − /p1 − /p3)γµ

(tint3)(tint4)+

γν(/p2 + /p4 − /pb)γµ(/p3 + /pA + /pB)γσ

(tint4)(s3AB)

]u3


Matching with Fixed Order

σresum,match2j =

∑f1,f2

∑m

m∏j=1

∫ pBj⊥=∞

pBj⊥=0

d2pBj⊥

(2π)3

∫ dyBj

2

(2π)4 δ(2)

m∑k=1

pBk⊥

· xBa fa(xBa ,Q

Ba ) xBb fb(x

Bb ,Q

Bb )|MB |

2

(sB )2

· |MtreeHEJ|

−2(2π)−4+3m 2m

(sB )2

xBa fa,f1 (xBa ,Q

Ba ) xB

bfb,f2 (x

Bb,QB

b)

·∞∑n=2

∫ p1⊥=∞

p1⊥=.9pj,⊥

d2p1⊥

(2π)3

∫ pn⊥=∞

pn⊥=.9pj,⊥

d2pn⊥

(2π)3

n−1∏i=2

∫ pi⊥=∞

pi⊥=λ

d2pi⊥

(2π)3(2π)4 δ(2)

n∑k=1

pk⊥

· Ty

n∏i=1

(∫dyi

2

)Oe

mj

m−1∏l=1

δ(2)(pBJl⊥

− jl⊥)

m∏l=1

δ(yBJl− yJl )

O2j ({pi})

· xafa,f1 (xa,Qa) xb fb,f2 (xb,Qb)|Mf1 f2→f1g···gf2

HEJ ({pi})|2

s2.

arxiv:1805.04446


https://arxiv.org/abs/1805.04446

Matching with Fixed Order

σresum,match2j =

∑f1,f2

∑m

m∏j=1

∫ pBj⊥=∞

pBj⊥=0

d2pBj⊥

(2π)3

∫ dyBj

2

(2π)4 δ(2)

m∑k=1

pBk⊥

· xBa fa(xBa ,Q

Ba ) xBb fb(x

Bb ,Q

Bb )|MB |

2

(sB )2

· |MtreeHEJ|

−2(2π)−4+3m 2m

(sB )2

xBa fa,f1 (xBa ,Q

Ba ) xB

bfb,f2 (x

Bb,QB

b)

·∞∑n=2

∫ p1⊥=∞

p1⊥=.9pj,⊥

d2p1⊥

(2π)3

∫ pn⊥=∞

pn⊥=.9pj,⊥

d2pn⊥

(2π)3

n−1∏i=2

∫ pi⊥=∞

pi⊥=λ

d2pi⊥

(2π)3(2π)4 δ(2)

n∑k=1

pk⊥

· Ty

n∏i=1

(∫dyi

2

)Oe

mj

m−1∏l=1

δ(2)(pBJl⊥

− jl⊥)

m∏l=1

δ(yBJl− yJl )

O2j ({pi})

· xafa,f1 (xa,Qa) xb fb,f2 (xb,Qb)|Mf1 f2→f1g···gf2

HEJ ({pi})|2

s2.

Fixed Order

Overlap HEJ

arxiv:1805.04446


https://arxiv.org/abs/1805.04446

Next-to-leading logarithmic processes in High Energy Jets...Next-to-leading logarithmic processes in...

Documents

Transcript of Next-to-leading logarithmic processes in High Energy Jets...Next-to-leading logarithmic processes in...