Jet Fragmentation and Fractal Observables Ben Elder Massachusetts - - PowerPoint PPT Presentation

Jet Fragmentation and Fractal Observables

Ben Elder

Massachusetts Institute of Technology

July 17, 2017

Based on work with: Massimiliano Procura, Jesse Thaler, Wouter Waalewijn, and Kevin Zhou

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 1 / 20

Fragmentation Functions

Fragmentation function (FF) Dh

i (x, µ):

◮ Probability of hadron h resulting from parton

i, carrying momentum fraction x

◮ Non-perturbative (must be extracted from

data)

◮ Process independent ◮ Perturbative RG evolution

Jet substructure: typically don’t care about individual identified hadron Today’s talk: subsets of jet particles → generalized fragmentation functions (GFFs) GFF Fi(x, µ) describes distribution of

• bservable x among some subset S of jet

particles π+ ρ+ π− π0 π+ π0

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 2 / 20

Collinear Unsafe Observables

Jet Charge =

• i∈jet

Qizκ

i

pD

T =

• i∈jet

z2

i

zi = pT,i pT,jet

Phys.Rev. D93 (2016) no.5, 052003 following Krohn, Shwartz, Lin, Waalewijn:1209.2421

CMS Collab.-CMS-PAS-JME-13-002

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 3 / 20

RG Evolution: Standard Fragmentation Functions

Leading order evolution → DGLAP equations Follow evolution on one path → linear µ d dµDh

i (x, µ) = 1

2

• j,k

1

x

dz z αs(µ) π Pi→j,k(z, αs)Dh

j (x/z, µ)

Dh

q(x)

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 4 / 20

RG Evolution: Generalized Fragmentation Functions

Fi(x, µ) carries information about all particles in S Leading order evolution follows evolution along all paths → nonlinear NLO evolution involves 1 → 3 splittings µ d dµFi(x, µ) = 1 2

• j,k
• dz αs(µ)

π Pi→j,k(z, αs)

• dx1dx2 Fj(x1, µ)Fk(x2, µ)

× δ(x − ˆ x(z, x1, x2)) Fh

g (x5)

Fh

g (x6)

Fh

g (x4)

Fh

¯ q (x3)

Fh

q (x2)

Fh

q (x1)

Jet Charge: ˆ x = zκx1 + (1 − z)κx2 pD

T : ˆ

x = z2x1 + (1 − z)2x2

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 5 / 20

RG Evolution: Comparison to Parton Showers

Gluon GFF, Weighted Jet Charge and pD

T

pp : µ = pTR, zi = pT,i pT,jet ; e+e− : µ = ER, zi = Ei Ejet envelopes: Pythia PS, Vincia PS, Dire PS; E, R combinations

−0.5 0.0 0.5

x

2 4 6

Fg

Gluon GFF Jet Charge

Evolution: κ = 1

PS: µ = 100 GeV 100 GeV → 4 TeV PS: µ = 4 TeV

0.0 0.2 0.4

x

5 10 15 20

Fg

Gluon GFF All Particles

Evolution: κ = 2

PS: µ = 100 GeV 100 GeV → 4 TeV PS: µ = 4 TeV

Jet Charge =

• i∈jet

Qizκ

i

pD

T =

• i∈jet

z2

i

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 6 / 20

Fractal Observables

Construct observable with structure of (leading order) evolution equation Define clustering tree, final state jet particles = leaves of tree Assign weights to jet constituents (non-kinematic quantum numbers) Recursively combine from bottom to top of tree using recursion relation ˆ x(z, x1, x2) z = EL EL + ER

x x12 x34 p

1

+ p

2

p

3

+ p

4

w1 w2 w3 w4 p1 p2 p3 p4

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 7 / 20

Fractal Observables: Familiar Examples

Weighted jet charge:

◮ wi = Qi ◮ ˆ

x = zκx1 + (1 − z)κx2

◮ x =

i∈jet Qizκ i

pD

T :

◮ wi = 1 ◮ ˆ

x = z2x1 + (1 − z)2x2

◮ x =

i∈jet z2 i

These recursion relations are associative → independent of clustering tree

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 8 / 20

Fractal Observables: Non-Associative Example

ˆ x = zκx1 + (1 − z)κx2 + (4z(1 − z))κ/2 wi = 0 E1 + E2 E3 + E4 E1 E2 E3 E4

w1 = 0 w2 = 0 w3 = 0 w4 = 0

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 9 / 20

Fractal Observables: Non-Associative Example

ˆ x = zκx1 + (1 − z)κx2 + (4z(1 − z))κ/2 wi = 0

E1 + E2 E3 + E4 E1 E2 E3 E4 x1 =

• 4

E1 E1+E2

• 1 −

E1 E1+E2

κ/2 x2 =

• 4

E3 E3+E4

• 1 −

E3 E3+E4

κ/2

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 10 / 20

Fractal Observables: Non-Associative Example

ˆ x = zκx1 + (1 − z)κx2 + (4z(1 − z))κ/2 wi = 0 x = E1 + E2 Ejet κ x1+

• 1 − E1 + E2

Ejet κ x2+

• 4E1 + E2

Ejet

• 1 − E1 + E2

Ejet κ/2

E1 + E2 E3 + E4 E1 E2 E3 E4

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 11 / 20

Fractal Observables: Node Definition

x = xA + xB + xC =

• nodes
• ELER

E 2

jet

κ/2 κ = 2 = ⇒ x = 2(1 − pD

T )

E1 + E2 E3 + E4 E1 E2 E3 E4 xA =

• E1E2

E 2

jet

κ/2 xC =

• E3E4

E 2

jet

κ/2 xB =

• (E1+E2)(E3+E4)

E 2

jet

κ/2

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 12 / 20

Fractal Observables: Tree Dependence

Compute fractal observable on jet ensemble from Vincia anti-kT jets with R = 0.6 Recluster into fractal observable tree Non-associative recursion relation → tree dependence p = −1 → anti − kT p = 0 → C/A p = 1 → kT

5 10

x

0.00 0.25 0.50 0.75 1.00

Fg

Gluon GFF Vincia µ = 100 GeV

κ = 1

p = −1 p = 0 p = 1

1.0 1.5 2.0

x

2 4 6 8

Fg

Gluon GFF Vincia µ = 100 GeV

κ = 2

p = −1 p = 0 p = 1 2(1 − pD

T )

0.0 0.5 1.0

x

0.0 2.5 5.0 7.5 10.0

Fg

Gluon GFF Vincia µ = 100 GeV

κ = 4

p = −1 p = 0 p = 1

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 13 / 20

Fractal Observables: RG Evolution

Gluon GFF, C/A Trees

5 10

x

0.00 0.25 0.50 0.75 1.00

Fg

Gluon GFF p=0

κ = 1

PS: µ = 100 GeV 100 GeV → 4 TeV PS: µ = 4 TeV

1.5 2.0

x

5 10

Fg

Gluon GFF p=0

κ = 2

PS: µ = 100 GeV 100 GeV → 4 TeV PS: µ = 4 TeV

0.0 0.5 1.0 1.5

x

1 2 3 4

Fg

Gluon GFF p=0

κ = 4

PS: µ = 100 GeV 100 GeV → 4 TeV PS: µ = 4 TeV

ˆ x(z, x1, x2) = zκx1 + (1 − z)κx2 + (4z(1 − z))κ/2

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 14 / 20

Fractal Observables: RG Evolution

Gluon GFF, κ = 1

10 20

x

0.0 0.2 0.4 0.6

Fg

Gluon GFF κ = 1

anti-kT

PS: µ = 100 GeV 100 GeV → 4 TeV PS: µ = 4 TeV

5 10

x

0.00 0.25 0.50 0.75 1.00

Fg

Gluon GFF κ = 1

C/A

PS: µ = 100 GeV 100 GeV → 4 TeV PS: µ = 4 TeV

5 10

x

0.00 0.25 0.50 0.75 1.00

Fg

Gluon GFF κ = 1

kT

PS: µ = 100 GeV 100 GeV → 4 TeV PS: µ = 4 TeV

µ d dµFi(x, µ) = 1 2

• j,k
• dz αs(µ)

π Pi→j,k(z, αs)

• dx1dx2 Fj(x1, µ)Fk(x2, µ)

× δ(x − zx1 − (1 − z)x2 − (4z(1 − z))1/2)

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 15 / 20

Quark/Gluon Discrimination: Distributions

Multiple Partons, C/A Trees

2 4 6 8 10

x

0.0 0.2 0.4 0.6

Fi

Vincia µ = 4 TeV p = 0

κ = 1

Gluon GFF Quark GFF Down GFF Bottom GFF

1.0 1.5 2.0

x

2 4 6 8 10

Fi

Vincia µ = 4 TeV p = 0

κ = 2

Gluon GFF Quark GFF Down GFF Bottom GFF

0.0 0.5 1.0 1.5

x

0.0 0.5 1.0 1.5 2.0 2.5

Fi

Vincia µ = 4 TeV p = 0

κ = 4

Gluon GFF Quark GFF Down GFF Bottom GFF

better (more like multiplicity) ← pD

T →

worse (less like multiplicity)

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 16 / 20

Quark/Gluon Discrimination: ROC Curves

0.0 0.2 0.4 0.6 0.8 1.0

Quark Efficiency

0.0 0.2 0.4 0.6 0.8 1.0

Gluon Mistag Rate

Vincia, µ = 4 TeV p = 0

Node Product ROC Curves

κ = 1 κ = 2, (pD

T )

κ = 4

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 17 / 20

Soft-Drop Multiplicity

More about this in Chris Frye’s talk tomorrow at 4pm. Frye, Larkoski, Thaler, Zhou: ArXiv:1704.06266 C-unsafe in β → 0 limit Another application of GFFs Recursion relation: ˆ x(z, x1, x2) =          x2 0 ≤ z < zcut x2 + f (z) zcut ≤ z ≤ 1/2 x1 + f (z) 1/2 ≤ z ≤ 1 − zcut x1 1 − zcut < z ≤ 1

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 18 / 20

Conclusion

FFs → cross sections with single identified hadron GFFs → cross sections of fractal observables with subsets of final state particles GFFs are non-perturbative Nonlinear, DGLAP-like perturbative evolution Fractal observables at the LHC: jet charge and pD

T

Non-associative generalizations show promise for quark/gluon discrimination

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 19 / 20

Thank You

Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 20 / 20