Resummation of the D Parameter. Andrew Larkoski Reed College work in progress with Andrea Banfi, Aja Procita

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1 Resummation of the D Parameter Andrew Larkoski Reed College work in progress with Andrea Banfi, Aja Procita UCLA, April 5, 2018

2 Legacy of LEP Precision Measurements Run:event 4093: 1000 Date T ime Ct rk(n= 39 Sump= 73.3) Ecal (N= 25 SumE= 32.6) Hcal (N=22 SumE= 22.6) Ebeam Evis 99.9 Emiss -8.6 Vtx ( -0.07, 0.06, -0.80) Muon ( N= 0 ) Se c V t x ( N= 3 ) Fde t ( N= 0 SumE= 0. 0 ) Bz=4.350 Thrust= Aplan= Oblat= Spher= Event Display from OPAL Y Z X 200. cm GeV Cent re of screen i s ( , , ) 2

3 Legacy of LEP Precision Measurements Run:event 4093: 1000 Date T ime Ct rk(n= 39 Sump= 73.3) Ecal (N= 25 SumE= 32.6) Hcal (N=22 SumE= 22.6) Ebeam Evis 99.9 Emiss -8.6 Vtx ( -0.07, 0.06, -0.80) Muon ( N= 0 ) Se c V t x ( N= 3 ) Fde t ( N= 0 SumE= 0. 0 ) Bz=4.350 Thrust= Aplan= Oblat= Spher= Event Display from OPAL Y X Thrust= Z Aplan= Oblat= Spher= Cent re of screen i s ( , , ) 200. cm GeV Event shape variables quantify energy flow distribution 3

4 Legacy of LEP Precision Measurements τ σ dσ dτ Thrust at N 3 LL + NNLO DELPHI ALEPH OPAL L3 SLD Fit at N3 LL for & theory scan error 0.30 Abbate, Fickinger, Hoang, Mateu, Stewart 2010 τ Cpar d dcpar SW GGGH C-parameter at NNLO ALEPH data GGGH LO NLO NNLO R 2 [0.5, 2] R 2 [0.5, 2] R 2 [0.5, 2] p Q2 =91.2GeV s (Q 2 )=0.118 GGGH C par Del Duca, Duhr, Kardos, Somogyi, Szőr, Trócsányi, Tulipánt 2016 SW Precision data/theory comparisons enable αs extraction 4

5 Legacy of LEP Precision Measurements Why thrust and C-parameter? NNLO: 2 3 at two-loops NNLL/N 3 LL: Small values restrict radiation e + q g e - q τ, C, B, first non-zero τ, C, B 1 Lots of tools! EERAD3, CoLoRFulNNLO, etc. Cross section factorizes Hard, Soft, Collinear functions 5

6 Legacy of LEP Precision Measurements LEP measured more than just τ and C! See: C 0: linear event codimension 2 D parameter at DELPHI, L3, OPAL 6 D 0: planar event codimension 1

7 Legacy of LEP Precision Measurements Challenges with D parameter: Fixed Order NNLO: 2 4 at two-loops Just starting to be calculated Badger, Brønnum-Hansen, Hartanto, Peraro 2017 Abreu, Febres Cordero, Ita, Page, Zeng x µ = 1 D parameter x µ = 0.1 State of the art NLO: 2 4 at one-loop : L3 data lines : NLO QCD D Nagy, Trócsányi

8 Legacy of LEP Precision Measurements Challenges with D parameter: Resummation NLL: D 0 has many regions 10 1 PT PT+NP σ -1 dσ/dd 0.1 Can have the same value of D State of the art NLL: Restrict to near-planar region D Banfi, Dokshitzer, Marchesini, Zanderighi 2001 y3 > 0.1 Not inclusive over final state

9 Legacy of LEP Precision Measurements Challenges with D parameter: Resummation 10 PT PT+NP σ -1 dσ/dd D Banfi, Dokshitzer, Marchesini, Zanderighi 2001 Inclusive over everything but D 9 y3 > 0.1

10 This Talk Remove all vestigial restrictions on the event Resum D parameter to NLL as proof-of-concept Approach inspired by developments in jet substructure Review: AJL, Moult, Nachman 2017 d dd = + 10

11 This Talk Method: Identify distinct configurations Resum Separately Sum together d dd = d di-jet dd + d tri-jet dd 11

12 Spherocity Tensor C and D are defined from the eigenvalues of the spherocity tensor = 1 X p i p i Q E i i C = 3( )= 3 Q 2 X i<j E i E j sin 2 ij D = = 27 Q 3 X i<j<k p i (p j p k ) 2 E i E j E k =1 12

13 Spherocity Tensor C = 3( )= 3 Q 2 X i<j E i E j sin 2 ij C (area of triangle) 2 E i E j ij D = = 27 Q 3 X i<j<k p i (p j p k ) 2 E i E j E k p i D (volume of parallelpiped) 2 13 p k p j

14 Spherocity Tensor Simultaneous measurement of C and D isolate configurations! Three parametric relations in D 0 limit: Region 1: Large Areas, Small Volumes 3 D C Region 2: Small Areas, Smaller Volumes D C 2 1 Region 3: Small Areas, Small Volumes D C

15 Spherocity Tensor Simultaneous measurement of C and D isolate configurations! End goal: d dd = Z dc apple d I dc dd + d II dc dd + d III dc dd Each of these cross sections are easy to calculate!!! 15

16 Region 1 Resummation C = 3( )! D = = C 2 Region considered by Banfi, et al., 2001 D C 1 Only large logs are of D/C 2 ~ D Additional soft/collinear emissions do not affect event plane 16

17 Region 1 Resummation Factorization: d I dx 1 dx 2 dd = H(x 1,x 2 )J 1 (D) J 2 (D) J 3 (D) S 123 (D) S 123 (D) J i (D) C =6 (1 x 1)(1 x 2 )(1 x 3 ) x 1 x 2 x 3 D C 1 H(x 1,x 2 ) Resummation by RG evolution from hard scale to soft and jet scales 17

18 Region 2 Resummation C = 3( )! 3 2 D = ! C 2 Novel configuration D C 1 Large logs of both C and D/C 2 Additional soft/collinear emissions do not affect event plane 18

19 Region 2 Resummation Factorization: d II dx 1 dx 2 dd = H(Q2 )H 2!3 (x 1,x 2 )J 1 (D) J 2 (D) J 3 (D) S 123 (D) S 123 (D) J i (D) Just a re-factorization of the hard function from region 1 D C 1 H(Q 2 )H 2!3 (x 1,x 2 ) Now, there is the 2 2 hard function H and the splitting function H2 3 19

20 Combining Regions 1 and 2 Option 1: Just additively match d I&II dx 1 dx 2 dd = d I dx 1 dx 2 dd + d II dx 1 dx 2 dd d I!II dx 1 dx 2 dd Exact to any logarithmic accuracy 20

21 Combining Regions 1 and 2 Option 2: Set scales to smoothly interpolate d I&II dx 1 dx 2 dd = H(Q2 )H 2!3 (x 1,x 2 )J 1 (D) J 2 (D) J 3 (D) S 123 (D) When C ~ 1, H(Q 2 ) 1 When C 1, H2 3 region 2 value 21

22 d I&II µ H2!3 = Combining Regions 1 and 2 Option 2: Set scales to smoothly interpolate dx 1 dx 2 dd = H(Q2 )H 2!3 (x 1,x 2 )J 1 (D) J 2 (D) J 3 (D) S 123 (D) 8 >< >: 6 1+ A 1 A 1 (2 x 1 x 2 )C 6 (2 x1 x 2 )C (2 x 1 x 2 )C 1 6 Q, C >C 1. 1/2 Q, C <C 1, Exact to NLL accuracy May not hold at higher orders 22

23 Region 3 Resummation C = 3( )! 3( ) D = ! C 2 Novel configuration D C 2 Only large logs are of C 2 ~ D Additional soft/collinear emissions can affect event plane 23

24 Region 3 Resummation Want for Factorization: d III dc dd = H(Q2 )J 1 (C, D) J 2 (C, D) S 12 (C, D) Issues: D is not additive; convolution doesn t work Because emissions that set D contribute to C, need to keep track of all emissions May be related to recent work on non-global logarithms Caron-Huot 2015; AJL, Moult, Nachman 2015; Becher, Neubert, Shao 2016; Ángeles Martínez, De Angelis, Forshaw, Plätzer, Seymour

25 Region 3 Resummation Redux Throw out factorization! d dc dd = d dc d (C) dd p(c, D) =p(c)p(d C) Exploit conditional probabilities to make progress d dc d (C) dd Calculable to high fixed and resummed order Has all of the subtleties of this region 25

26 Region 3 Resummation Redux To make progress, we calculate what we can Exact to NLL: d NLL dc dd = d NLL dc d (C) NLL dd Approximate NLL: d NLL dc dd d NLL dc d (C) 2 s dd Misses some logs at NLL Think it gets all logs to αs n log n+1 26

27 Region 3 Resummation Redux Need to restrict to D ~ C 2 " d III dc dd d NLL d (C) 2 s d (C) 2 s,d C2 dc dd dd # Then, can additively match with regions 1 and 2 Advantages Gets a result Systematically improvable Downsides No formal logarithmic accuracy Requires new definition of formal accuracy? 27

28 Complete Result Now just combine and integrate over C! Maximum value of C when D 0 d dd = Z 3/4 p 4 3 D dc apple d I&II dc dd + d III dc dd From constraint in region 3 C 3(λ2+λ3) D 27 λ2λ3 All overlaps have been subtracted Accomplishes fully inclusive resummation of the D parameter! 28

29 Conclusions The D parameter is one of the legacy precision measurements from LEP Due to its complicated event selection, only near-planar region had been resummed Inclusive resummation accomplished secondary measurement that simplifies event configuration Expect comparisons to LEP data soon! 29