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) D h i ( x , µ ): ◮ Probability of hadron h resulting from parton π + 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 π − ρ + π 0 Today’s talk: subsets of jet particles → π + generalized fragmentation functions (GFFs) π 0 GFF F i ( x , µ ) describes distribution of observable x among some subset S of jet particles Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 2 / 20
Collinear Unsafe Observables z i = p T , i � Q i z κ p D � z 2 Jet Charge = T = i i p T , jet i ∈ jet i ∈ 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 � 1 µ d i ( x , µ ) = 1 d z α s ( µ ) � d µ D h P i → j , k ( z , α s ) D h j ( x / z , µ ) 2 π z x j , k D h q ( x ) Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 4 / 20
RG Evolution: Generalized Fragmentation Functions F i ( 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 d µ F i ( x , µ ) = 1 d z α s ( µ ) � � � P i → j , k ( z , α s ) d x 1 d x 2 F j ( x 1 , µ ) F k ( x 2 , µ ) 2 π j , k × δ ( x − ˆ x ( z , x 1 , x 2 )) F h q ( x 3 ) F h g ( x 5 ) F h g ( x 4 ) ¯ Jet Charge: ˆ x = z κ x 1 + (1 − z ) κ x 2 p D x = z 2 x 1 + (1 − z ) 2 x 2 F h q ( x 2 ) T : ˆ F h q ( x 1 ) F h g ( x 6 ) 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 p D T pp : µ = p T R , z i = p T , i e + e − : µ = ER , z i = E i ; p T , jet E jet envelopes: Pythia PS, Vincia PS, Dire PS; E , R combinations Evolution: κ = 1 Evolution: κ = 2 6 PS: µ = 100 GeV PS: µ = 100 GeV 100 GeV → 4 TeV 20 100 GeV → 4 TeV PS: µ = 4 TeV PS: µ = 4 TeV 4 15 Gluon GFF Gluon GFF F g F g Jet Charge All Particles 10 2 5 0 0 − 0 . 5 0 . 0 0 . 5 0 . 0 0 . 2 0 . 4 x x � Q i z κ � z 2 p D Jet Charge = T = i i i ∈ jet i ∈ jet 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 , x 1 , x 2 ) x p p 2 3 + + p E L p 1 4 z = E L + E R x 12 x 34 p 2 p 4 p 1 p 3 w 1 w 2 w 3 w 4 Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 7 / 20
Fractal Observables: Familiar Examples Weighted jet charge: ◮ w i = Q i ◮ ˆ x = z κ x 1 + (1 − z ) κ x 2 ◮ x = � i ∈ jet Q i z κ i p D T : ◮ w i = 1 ◮ ˆ x = z 2 x 1 + (1 − z ) 2 x 2 ◮ x = � i ∈ jet z 2 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 x = z κ x 1 + (1 − z ) κ x 2 + (4 z (1 − z )) κ/ 2 ˆ w i = 0 E 3 + E 4 E 1 + E 2 E 2 E 4 E 1 E 3 w 1 = 0 w 2 = 0 w 3 = 0 w 4 = 0 Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 9 / 20
Fractal Observables: Non-Associative Example x = z κ x 1 + (1 − z ) κ x 2 + (4 z (1 − z )) κ/ 2 ˆ w i = 0 E 3 + E 4 E 1 + E 2 �� κ/ 2 �� κ/ 2 � � � � E 1 E 1 E 3 E 3 x 1 = 4 1 − x 2 = 4 1 − E 1 + E 2 E 1 + E 2 E 3 + E 4 E 3 + E 4 E 2 E 4 E 1 E 3 Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 10 / 20
Fractal Observables: Non-Associative Example x = z κ x 1 + (1 − z ) κ x 2 + (4 z (1 − z )) κ/ 2 ˆ w i = 0 � κ � κ �� κ/ 2 � E 1 + E 2 � 1 − E 1 + E 2 � 4 E 1 + E 2 � 1 − E 1 + E 2 x = x 1 + x 2 + E jet E jet E jet E jet E 3 + E 4 E 1 + E 2 E 2 E 4 E 1 E 3 Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 11 / 20
Fractal Observables: Node Definition � κ/ 2 � E L E R � ⇒ x = 2(1 − p D x = x A + x B + x C = κ = 2 = T ) E 2 jet nodes � κ/ 2 � ( E 1 + E 2 )( E 3 + E 4 ) x B = E 2 jet E 3 + E 4 E 1 + E 2 � κ/ 2 � κ/ 2 � � E 1 E 2 E 3 E 4 x A = x C = E 2 E 2 jet jet E 2 E 4 E 1 E 3 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 p = − 1 → anti − k T anti- k T jets with R = 0 . 6 p = 0 → C/A Recluster into fractal observable tree p = 1 → k T Non-associative recursion relation → tree dependence κ = 1 κ = 2 κ = 4 8 10 . 0 Gluon GFF Gluon GFF Gluon GFF 1 . 00 p = − 1 p = − 1 Vincia Vincia Vincia p = 0 p = 0 µ = 100 GeV µ = 100 GeV µ = 100 GeV 6 7 . 5 p = 1 p = 1 0 . 75 p = − 1 p = 0 F g F g F g 4 5 . 0 p = 1 0 . 50 2(1 − p D T ) 2 2 . 5 0 . 25 0 . 00 0 0 . 0 0 5 10 1 . 0 1 . 5 2 . 0 0 . 0 0 . 5 1 . 0 x x x Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 13 / 20
Fractal Observables: RG Evolution Gluon GFF, C/A Trees κ = 1 κ = 2 κ = 4 4 PS: µ = 100 GeV PS: µ = 100 GeV PS: µ = 100 GeV 1 . 00 100 GeV → 4 TeV 100 GeV → 4 TeV 100 GeV → 4 TeV 10 3 PS: µ = 4 TeV PS: µ = 4 TeV PS: µ = 4 TeV Gluon GFF Gluon GFF Gluon GFF 0 . 75 p=0 p=0 p=0 F g F g F g 2 0 . 50 5 1 0 . 25 0 0 0 . 00 0 5 10 1 . 5 2 . 0 0 . 0 0 . 5 1 . 0 1 . 5 x x x x ( z , x 1 , x 2 ) = z κ x 1 + (1 − z ) κ x 2 + (4 z (1 − z )) κ/ 2 ˆ Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 14 / 20
Fractal Observables: RG Evolution Gluon GFF, κ = 1 anti- k T k T C/A 0 . 6 PS: µ = 100 GeV PS: µ = 100 GeV PS: µ = 100 GeV 1 . 00 1 . 00 100 GeV → 4 TeV 100 GeV → 4 TeV 100 GeV → 4 TeV PS: µ = 4 TeV PS: µ = 4 TeV PS: µ = 4 TeV 0 . 4 Gluon GFF Gluon GFF Gluon GFF 0 . 75 0 . 75 κ = 1 κ = 1 κ = 1 F g F g F g 0 . 50 0 . 50 0 . 2 0 . 25 0 . 25 0 . 0 0 . 00 0 . 00 0 10 20 0 5 10 0 5 10 x x x µ d d µ F i ( x , µ ) = 1 � d z α s ( µ ) � � P i → j , k ( z , α s ) d x 1 d x 2 F j ( x 1 , µ ) F k ( x 2 , µ ) 2 π j , k × δ ( x − zx 1 − (1 − z ) x 2 − (4 z (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 κ = 1 κ = 2 κ = 4 2 . 5 Gluon GFF Gluon GFF Gluon GFF 0 . 6 10 � Quark � GFF � Quark � GFF � Quark � GFF Down GFF Down GFF 2 . 0 Down GFF 8 Bottom GFF Bottom GFF Bottom GFF 0 . 4 1 . 5 F i F i F i Vincia Vincia Vincia 6 µ = 4 TeV µ = 4 TeV µ = 4 TeV p = 0 p = 0 1 . 0 p = 0 4 0 . 2 0 . 5 2 0 . 0 0 . 0 0 2 4 6 8 10 1 . 0 1 . 5 2 . 0 0 . 0 0 . 5 1 . 0 1 . 5 x x x ← p D better (more like multiplicity) T → worse (less like multiplicity) Ben Elder (MIT) Jet Fragmentation and Fractal Observables July 17, 2017 16 / 20
Quark/Gluon Discrimination: ROC Curves Node Product ROC Curves 1 . 0 Vincia , µ = 4 TeV p = 0 0 . 8 κ = 1 Gluon Mistag Rate κ = 2 , ( p D T ) 0 . 6 κ = 4 0 . 4 0 . 2 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 Quark Efficiency 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. C-unsafe in β → 0 limit Another application of GFFs Recursion relation: x 2 0 ≤ z < z cut x 2 + f ( z ) z cut ≤ z ≤ 1 / 2 x ( z , x 1 , x 2 ) = ˆ x 1 + f ( z ) 1 / 2 ≤ z ≤ 1 − z cut x 1 1 − z cut < z ≤ 1 Frye, Larkoski, Thaler, Zhou: ArXiv:1704.06266 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 p D 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
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