Phenomenology of TMD Fragmentation using the Winner-Take-All axis - PowerPoint PPT Presentation


 Phenomenology of 
 TMD Fragmentation 
 using the Winner-Take-All axis Lorenzo Zoppi Based on ongoing work with Andreas Papaefstathiou, Duff Neill, Wouter Waalewijn


 
 
 Outline 1. Framework 
 a) Ingredients: jets, hadron TMD, winner-take-all axis. 
 b) Technical details: factorization, resummation. 
 2. Results for e+e- collisions. Work in progress! 2

Jets “Jets live in-between theory and experiment” • In high-energy QCD collisions, final-state 
 hadrons are produced into collimated sprays. • Consequence of perturbative QCD being dominated by soft/collinear dynamics. • Rather than a unique object, there is a whole class of different jets. • Definition include radius, axis, and jet algorithm that prescribes how to build the jet from a collection of (observed/predicted) particles. • Different jets suit different physical applications. 3

Fragmenting-hadron TMD in Jets Inclusive Jet(hadron) production e + e − → Jet(h) X pp → Jet(h) X We study the differential distribution of hadron transverse momentum with respect to the jet axis. E J Jet energy E h ≡ z Hadron energy fraction E J zk ⊥ Hadron transverse momentum k ⊥ = E J sin( ϑ ) 4

Motivation • Jet shapes for pp collisions 
 Baumgart, Leibovich, Mehen, Rothstein ’14, 
 Bain, Dai, Hornig, Leibovich, Makris, Mehen ’16 
 Kaufmann, Mukherjee, Vogelsang ’15 
 Kang, Ringer, Vitev ’16 
 Dai, Kim, Leibovich ’16 • Complementary constraints on non-perturbative 
 Fragmentation Functions 
 Arleo, Fontannaz, Guillet, Nguyen ’13 
 Kaufmann, Mukherjee, W. Vogelsang ’15 • Jet quenching in heavy ion collisions 
 Chien, Vitev ’15 
 Chang, Quin ’16 
 Wang, Wei, Zhang ‘16 
 Kang, Ringer, Vitev ’17 5

Recoil-Free Jet axes ~ p soft + ~ p coll = 0 6

Winner-take-all axis Practical realization of recoil-free axis with a simple implementation. Run a recombination algorithm: (Define an distance measure between particles) 1. Find the two particles closest to each other; 2. If the distance is bigger than the jet radius, 
 Merge them; 
 ( E = E 1 + E 2 E 1 > E 2 ⇒ n ˆ = ˆ n 1 3. Otherwise, stop. Define remaining particles to be jets. By construction, the axis lie on a particle. 
 Soft particles are doomed to lose when recombined with collinear ones. 7

Factorization When the energy scales are separated, factorization follows (Soft Collinear Effective Theory) d � ( h ) ( E J , z h , ~ J ( ~ z h d ( h ) ( µ ) k ⊥ ) = H ( µ ) ⊗ k ⊥ , R, µ ) ⊗ dE J dz h dk 2 E J ⊥ Integrated FFs 
 Universal, thanks to 
 soft insensitivity. TMD Jet functions Depend on jet defintion (radius, algorithm). D. Neill, I.Scimemi, W. Waalewijn JHEP 1704 (2017) 020 8

Factorization When , a further factorization takes place. k ⊥ ⌧ E J R d � ( h ) ( E J , z h , ~ J ( ~ z h d ( h ) ( µ ) k ⊥ ) = H ( µ ) ⊗ k ⊥ , R, µ ) ⊗ dE J dz h dk 2 E J ⊥ d � ( h ) ( E J , z h , ~ z h C ( ~ z h d ( h ) ( µ ) k ⊥ ) = H ( µ ) ⊗ B ( R, µ ) ⊗ k ⊥ , µ ) ⊗ dE J dz h dk 2 E J ⊥ Boundary functions 
 Do not know about TMD. Matching coefficients 
 Do not know about jet size. 9


 Leading Log parton-shower picture Leading Log (LL) DGLAP equations can be derived imposing 
 strong angular ordering ϑ i +1 ⌧ ϑ i ϑ 0 = R This corresponds to studying the branching tree of 
 a parton shower. - Nodes of the branching tree splitting probability P(z) Final probability of hadron integrate over 
 with z energy fraction relevant branching history ϑ ' k ⊥ WTA: axis and hadron travel together until some splitting of order . 
 E J at each node, the axis follows the daughter particle with energy fraction . z > 1 2 After the splitting, further divisions do not alter in a significative way: standard DGLAP ϑ Before the splitting, the axis has to track the hadron: the function obeys a modified DGLAP Z 1 dz 0 ij ( z ) = P ( z ) θ ( z − 1 d ln µ = α s dD i ij ( z z 0 P 0 z 0 ) D j ( z 0 ) P 0 2) 2 π z 10

LL parton shower: broadening axis The same picture does not generalize to other recoil-free axis. 
 E.g. broadening axis A. Larkoski, D. Neill, J. Thaler ’14 2 E i sin ϑ i, ˆ Minimizes n X b = min E J 2 n ˆ i ∈ Jet Even in the planar case, the broadening axis always lie on a particle. The energy fractions on the left and on the right 1.0 contribute to determine which particle takes control of the axis. 
 Region I 0.8 z l + z L > 1 / 2 Axis goes leftwards ⇔ The non-planar case is further complicated by a non-trivial 
 0.6 interplay of angles and energy fractions. 
 z 0.4 The broadening axis does not necessarily lie on a particle. Region III 0.2 Region II Evolution of TMD fragmenting jet functions with respect to 
 the broadening axis cannot be cast in a DGLAP-like form. 0.0 0.5 0.6 0.7 0.8 0.9 1.0 x 11


 
 
 
 
 Computing LL+NLO results • Focus on . e + e − → Jet ( hadron ) • Take the first Mellin moment of the cross section 
 Z 1 independent from set of Fragmentation Functions thanks to sum rule dz z d ( h ) ( µ ) = 1 0 • Compute every ingredient at its natural scale, evolve them up to the common scale (LL) 
 Z Q Z 1 d � N =1 n o e + e − → Jet ( h ) f ( ~ H e + e − ( µ ) ⊗ B ( R, µ ) ⊗ C ( ~ k ⊥ , µ ) ⊗ d ( h ) ( µ ) k ⊥ ) ≡ = dE J dz z dk ⊥ Emin 0 � k ⊥ � resum ln E J R G ( R, ~ k ⊥ , µ ) = J ( R, ~ k ⊥ , µ ) ⊗ d ( h ) ( µ ) resum ln( R ) • To extend the validity of the results to , match with fixed-order NLO results. k ⊥ ∼ E J R Z Q Z 1 n o f ( NLO ) ( ~ H e + e − ( µ ) ⊗ G ( R, ~ k ⊥ ) = k ⊥ , µ ) dE J dz z 0 Emin 12

Results

The angular distributions behave remarkably as a very definite power law Analytic Shower cutoff Jet radius causes discontinuity HW7, partons Analytic 14

Transverse momentum distributions ⇣ k ⊥ ⌘ J ( ~ k ⊥ , R, µ ) = A + B ln ✓ ( E J R − k ⊥ ) E J R E J,min < E J < Q 2 Gradually switches down the cross section 
 when approaching the jet boundary. The logarithmic term appears only at NLO: NLO is the first significative fixed order. 15

WTA vs recoil-sensitive axes Using traditional jet definitions, 
 Sudakov double logarithms arise as a consequence of soft/ collinear overlap. Insensitivity to soft radiation removes such double logs, resulting in the definite power law. Energy recombination scheme: p = p 1 + p 2 16

Jet shapes: quark vs gluon E J > 50 GeV, Q = 250, Tan R 2 = 0.4 10 5 1 Gluon Initiated 1 dG d θ ( θ ,R ) G ( R , R ) �������� Quark Initiated 0.50 0.10 0.05 0.005 0.010 0.050 0.100 0.500 θ Quark and gluon have different jet shapes. In addition, they are not a pure power law. Linearity of the transverse momentum/angular distributions is a non-trivial fact. 17

Subjets vs partons vs hadrons Probe of the factorization of jet boundary: Single hadrons and whole subjets yield the same when . ϑ > r Jet radius Shower cut off 18

Outlook • The Winner-Take-All scheme provides a framework to study hadron TMD without sensitivity to soft radiation. • NLO+LL results for differential distributions agree with Monte Carlo simulations and show a definite power-law behavior. Results of the simulations agree with the factorization framework. • The observables we considered could directly probe interesting physics. Investigate potential applications: 
 - Jet substructure for boosted objects. 
 - Probing Q/G plasma through jet quenching.

Outlook • The Winner-Take-All scheme provides a framework to study hadron TMD without sensitivity to soft radiation. • NLO+LL results for differential distributions agree with Monte Carlo simulations and show a definite power-law behavior. Results of the simulations agree with the factorization framework. • The observables we considered could directly probe interesting physics. Investigate potential applications: 
 - Jet substructure for boosted objects. 
 - Probing Q/G plasma through jet quenching. Thank you !

Backup slides

Jet matching coefficients: example J qq ( x, E J R, ~ k ⊥ , z, µ ) ⇣ 1 � 1 + z 2 = ↵ s C F 1 � 1 � � ✓ E J R − k ⊥ � (1 − x ) ✓ 2 − z � ⊥ /µ 2 � 2 ⇡ ⇡ µ 2 k 2 1 − z + 1 ⇣ E J R ⇣ ln(1 − x ) n h⇣ ⌘ ⌘ ⌘ i o − � 2 ( ~ 2(1 + x 2 ) k ⊥ ) � (1 − z ) + ln + + 1 − x 1 − x µ 1 − x + z − 1 1 ⇣ E J Rz ⇣ ln(1 − z ) �hn h⇣ ⌘ ⌘ ⌘ i i + � 2 ( ~ 2(1 + z 2 ) � k ⊥ ) � (1 − x ) ✓ + ln + + 1 − z 2 1 − z µ 1 − z + 21 + z 2 � 1 �h io⌘ � � + ✓ 2 − z 1 − z ln z (1 − z ) + (1 − z )

WTA axis - branching tree - details Z ϑ i − 1 h Z 1 n ∞ d ϑ i i X Y Y � � A ( z ) = δ (1 − z ) + dz i 2 P ( z i ) δ z − z j ϑ i 0 ϑ min n =0 I =1 j h Z 1 n ∞ 1 R i X Y Y n ! ln n � � � � = δ (1 − z ) + dz i 2 P ( z i ) δ z − z j ϑ min 0 n =1 i =1 j Z 1 � z dA dA dz 0 � A ( z 0 ) d ln µ = d ln R = z 0 2 P z 0 z 23