N-je%ness Subtrac0ons

Transcription

1 -je%ness Subtrac0ons [arxiv: ] Maximilian Stahlhofen In collabora0on with Jonathan Gaunt, Frank ackmann, and Jonathan Walsh

2 IR Singulari0es Massless QCD: (example: e + e q q ) unitarity cut LO: M Maximilian Stahlhofen - JGU Uni Wien, Seminar - 2

3 IR Singulari0es Massless QCD: (example: e + e q q ) unitarity cut LO: M 2 k +... p 1 (p + k) 2 = 1 2 p k (1 cos ) k 0 0 so^ singularity collinear singularity IR singulari0es Maximilian Stahlhofen - JGU Uni Wien, Seminar - 3

4 IR Singulari0es Kinoshita-Lee-auenberg (KL) theorem: IR singulari0es cancel for physical (IR safe) observables. Implies sum over degenerate ini0al ( PDFs) and final states. Cancella0on between virtual and real quantum correc0ons at every order in perturba0on theory Maximilian Stahlhofen - JGU Uni Wien, Seminar - 4

5 IR Singulari0es Prac0cal problem: regulariza0on Virtual Dimensional Regulariza0on 1 n poles (Must at higher orders) Cancella0on fails Real PS integra0on can be (arbitrarily) complicated Dim. Reg. impossible! Aim: umerical integra0on Monte Carlo generator Maximilian Stahlhofen - JGU Uni Wien, Seminar - 5

6 IR Singulari0es Prac0cal problem: regulariza0on Virtual Real Dimensional Regulariza0on 1 n poles (Must at higher orders) Cancella0on Solu0on: IR subtrac0on fails PS integra0on can be (arbitrarily) complicated Dim. Reg. impossible! Aim: umerical integra0on Monte Carlo generator Maximilian Stahlhofen - JGU Uni Wien, Seminar - 6

7 Outline IR Subtrac0on/Slicing Methods he -je%ness Event Shape LO Ingredients Applica0ons/Results Possible Extensions Summary Maximilian Stahlhofen - JGU Uni Wien, Seminar - 7

8 IR Subtrac0on/Slicing Methods -jet cross sec0on (i.e. hard final state partons): [X = set of (Born-level) kinema0c variables the (differen0al) cross sec0on depends on.] LO(X) = d LO (X) -parton phase space LO(X) = d LO(X)+ V d LO(X) R +1 LO(X) = d LO(X)+ VV d LO(X)+ RV +1 d LO(X) RR Maximilian Stahlhofen - JGU Uni Wien, Seminar - 8

9 IR Subtrac0on/Slicing Methods -jet cross sec0on (i.e. hard final state partons): [X = set of (Born-level) kinema0c variables the (differen0al) cross sec0on depends on.] LO(X) = d LO (X) LO(X) = d LO(X)+ V d LO(X) R +1 LO(X) = d LO(X)+ VV d LO(X)+ RV +1 d LO(X) RR +2 2-loop diagrams 1-loop diagrams tree-level diagrams Maximilian Stahlhofen - JGU Uni Wien, Seminar - 9

10 IR Subtrac0on/Slicing Methods -jet cross sec0on (i.e. hard final state partons): [X = set of (Born-level) kinema0c variables the (differen0al) cross sec0on depends on.] LO(X) = d LO (X) LO(X) = d LO(X)+ V d LO(X) R +1 LO(X) = d LO(X)+ VV d LO(X)+ RV +1 d LO(X) RR +2 d # 4 + # 3 + # 2 + # +# d +1 # 2 + # +# Problem: Achieve cancella0on of IR poles! Z apple d +2 # Maximilian Stahlhofen - JGU Uni Wien, Seminar - 10

11 IR Subtrac0on/Slicing Methods Idea: introduce subtrac0on (counter-) terms LO(X) = d LO(X)+ V +1 d R LO(X) d R sing(x) + d sing(x) R +1 LO(X) = d LO(X)+ VV +1 d RV LO(X) d RV sing(x) + d sing(x) RV d RR LO(X) d RR sing(x) + d sing(x) RR +2 Local subtrac0ons: Construct such that they exactly match the d R,RV,RR in the singular limit point by point in PS ()LO are simple enough to be integrable in D dimensions d R,RV,RR sing Maximilian Stahlhofen - JGU Uni Wien, Seminar - 11

12 IR Subtrac0on/Slicing Methods Idea: introduce subtrac0on (counter-) terms LO(X) = d LO(X)+ V +1 d R LO(X) d R sing(x) + d sing(x) R +1 integrable in 4D integrable in D dim. LO(X) = d LO(X)+ VV +1 d RV LO(X) d RV sing(x) + d sing(x) RV Local subtrac0ons: d RR LO(X) integrable in 4D d RR sing(x) + d sing(x) RR +2 integrable in D dim. Construct such that they exactly match the d R,RV,RR in the singular limit point by point in PS ()LO are simple enough to be integrable in D dimensions d R,RV,RR sing Maximilian Stahlhofen - JGU Uni Wien, Seminar - 12

13 IR Subtrac0on/Slicing Methods Local subtrac0on schemes LO Catani-Seymour (Dipole) FKS agy-soper [Frixione, Kunszt, Signer, 96] [Catani, Seymour, 96] [agy, Soper, 07; Chung, Krämer, Robens, 10] FDU [Sborlini, Driencourt-Magnin, Hernandez-Pinto, Rodrigo, 16] ü Well established ü Arbitrary -jet processes ü Automated: HERWIG++, SHERPA, MUICH; MG5_aMC, POWHEG-BOX, WHIZARD; HELAC+DEDUCOR; MCFM, Maximilian Stahlhofen - JGU Uni Wien, Seminar - 13

14 IR Subtrac0on/Slicing Methods Local subtrac0on schemes LO very complicated, hard to automate, but successfully applied to several LHC processes, e.g.: Sector decomposi0on [Anastasiou, Melnikov, Petriello, 03] - pp H, V [Anastasiou, Melnikov, Petriello, 03-04] Sector-improved subtrac0on schemes (SRIPPER) - - pp pp t t [Czakon, Fiedler, Mitev, 13] H + j [Boughezal, Caola, Melnikov, Petriello, Schulze, 13-15] [Czakon, 10] Antenna subtrac0on [Gehrmann-De Ridder, Gehrmann, Glover, 05] - - e + e pp jj 3j (par0al)[gehrmann-de Ridder, Gehrmann, Glover, Pires 13] - [Chen, Gehrmann, Glover, Jaquier, 14-16] pp! H, Z + j [Gehrmann-De Ridder, Gehrmann, Glover, Huss, Morgan, 15] - pp t t (par0al) Colorful LO [Del Duca, Somogyi, rocsanyi, 05] - e + e 3j [Del Duca, Duhr, Somogyi, ramontano, rocsanyi, 16] [Gehrmann-De Ridder, Gehrmann, Glover, Heinrich 07] [Weinzierl, 08] [Abelof, Gehrmann-De Ridder, Maierhofer, Majer, Pozzorini, 11-15] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 14

15 IR Subtrac0on/Slicing Methods Alterna0ve method: Phase space Slicing (non-local subtrac0on) prototype: q - subtrac0on [Catani, Grazzini, 07] ransverse momentum q of color-neutral final state controls real emissions! q q >0 requires at least one non-singular real emission! Maximilian Stahlhofen - JGU Uni Wien, Seminar - 15

16 IR Subtrac0on/Slicing Methods q - subtrac0on [Catani, Grazzini, 07] LO(X) = 0 q cut dq d LO (X) dq + d LO (X) dq q cut dq diff. measurement (cuts) on Born-level kinema0cs Maximilian Stahlhofen - JGU Uni Wien, Seminar - 16

17 IR Subtrac0on/Slicing Methods q - subtrac0on [Catani, Grazzini, 07] LO(X) = diff. measurement (cuts) on Born-level kinema0cs 0 q cut dq d LO (X) dq + d LO (X) dq q cut dq q >0 1-jet LO process computable with LO tools Maximilian Stahlhofen - JGU Uni Wien, Seminar - 17

18 IR Subtrac0on/Slicing Methods q - subtrac0on [Catani, Grazzini, 07] LO(X) = 0 q cut dq d LO (X) dq + d LO (X) dq q cut dq diff. measurement (cuts) on Born-level kinema0cs expansion in small q d = H [B a B b S](q )+O q2 dq Q [Collins, Soper, Sterman 84] 2 known analy0cally to LO q >0 1-jet LO process computable with LO tools Maximilian Stahlhofen - JGU Uni Wien, Seminar - 18

19 IR Subtrac0on/Slicing Methods q - subtrac0on [Catani, Grazzini, 07] LO(X) = 0 q cut dq d LO (X) dq + d LO (X) dq q cut dq diff. measurement (cuts) on Born-level kinema0cs expansion in small q d = H [B a B b S](q )+O q2 dq Q [Collins, Soper, Sterman 84] 2 known analy0cally to LO q >0 1-jet LO process computable with LO tools Exploits exis0ng LO technology Easy to implement Large numerical cancella0ons as q cut Q ( computer intensive) Maximilian Stahlhofen - JGU Uni Wien, Seminar - 19

20 IR Subtrac0on/Slicing Methods q - subtrac0on [Catani, Grazzini, 07] LO(X) = 0 q cut dq d LO (X) dq + d LO (X) dq q cut dq Very successful for LHC processes with colorless (=0) final state: - H - W/Z - γγ - WH - ZH - ZZ - W + W - - Zγ/Wγ - WZ - HH [Catani, Grazzini, 07] [Catani, Cieri, Ferrera, de Florian, Grazzini, 09] [Catani, Cieri, Ferrera, de Florian, Grazzini, 12] [Ferrera, Grazzini, ramontano 11] [Ferrera, Grazzini, ramontano 15] [Cascioli et al., 14] [Grazzini, Kallweit, Rathlev 15] [Gehrmann et al., 14] [Grazzini, Kallweit, Pozzorini, Rathlev, Wiesemann, 16] [Grazzini, Kallweit, Rathlev, orre 14] [Grazzini, Kallweit, Rathlev 15] [Grazzini, Kallweit, Rathlev, Wiesemann, 16] [De Florian et al., 16] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 20

21 IR Subtrac0on/Slicing Methods q - subtrac0on [Catani, Grazzini, 07] LO(X) = 0 q cut dq d LO (X) dq +... but limited to colorless (=0) final states! V d LO (X) dq q cut dq transverse momentum of hard V+g final state (=1): q >0 even if, real emissions are in the collinear limit! g Maximilian Stahlhofen - JGU Uni Wien, Seminar - 21

22 IR Subtrac0on/Slicing Methods q - subtrac0on [Catani, Grazzini, 07] LO(X) = 0 q cut dq d LO (X) dq +... but limited to colorless (=0) final states! V d LO (X) dq q cut dq transverse momentum of hard V+g final state (=1): q >0 even if, real emissions are in the collinear limit! g need universal -jet resolu0on variable -je%ness Maximilian Stahlhofen - JGU Uni Wien, Seminar - 22

23 he -je%ness Event Shape Defini0on: final state jets = M k=1 min i 2 ˆq i p k [Stewart, ackmann, Waalewijn, 10] final state parton momenta ˆq i = q i jet direc0ons :, i = a, b, 1,..., Q i ˆq a,b fixed to the beam direc0ons! (jet momenta q i determined by any jet algorithm, Q i is arbitrary normaliza0on, e.g. Q i = Q or Q i = 2E i ) Maximilian Stahlhofen - JGU Uni Wien, Seminar - 23

24 he -je%ness Event Shape Defini0on: = M k=1 min i 2 ˆq i p k [Stewart, ackmann, Waalewijn, 10] ˆq 1 ˆq b > 0 requires at least one non-singular real emission! [here: M=3, =1] ˆq a Maximilian Stahlhofen - JGU Uni Wien, Seminar - 24

25 he -je%ness Event Shape Defini0on: = M k=1 min i 2 ˆq i p k [Stewart, ackmann, Waalewijn, 10] ˆq 1 0 maximally singular limit: (+2) pencil-like jets [here: M=3, =1] ˆq b ˆq a Maximilian Stahlhofen - JGU Uni Wien, Seminar - 25

26 he -je%ness Event Shape Factoriza0on formula for small : d =r d Ĥ Ŝ B a B b i [Stewart, ackmann, Waalewijn, 10] J i + O Q Soft Jet 1 (FSR) Q E cm hard scale (ISR) Jet b Jet a (ISR) P a P b (FSR) Jet 2 Soft Maximilian Stahlhofen - JGU Uni Wien, Seminar - 26

27 he -je%ness Event Shape Factoriza0on formula for small : d =r d Ĥ Ŝ B a B b i [Stewart, ackmann, Waalewijn, 10] J i + O Q n n n n =0 factoriza0on n n n n Maximilian Stahlhofen - JGU Uni Wien, Seminar - 27

28 he -je%ness Event Shape Factoriza0on formula for small : d =r d Ĥ Ŝ B a B b i [Stewart, ackmann, Waalewijn, 10] J i + O Q Ĥ(,µ) hard func0on depends on Born PS matrix in color space B a (t a, x a,µ) beam func0on depends on virtuality t a and Bjorken variable x a of incoming parton Ŝ (k a,...,k, {ˆq i },µ) so^ func0on depends jet direc0ons ˆq i matrix in color space k j : so^ contribu0ons to from each jet J i (s i,µ) jet func0on depends on virtuality s i of outgoing parton Maximilian Stahlhofen - JGU Uni Wien, Seminar - 28

29 he -je%ness Event Shape Factoriza0on formula for small : d =r d Ĥ Ŝ B a B b Deriva0on in SCE i [Stewart, ackmann, Waalewijn, 10] J i + O Q n-collinear vector p µ = p nµ 2 + p+ n µ 2 + pµ (p, p +, p ) lightcone coordinates n 2 = n 2 = 0, n n = 2 SCE I degrees of freedom: n-collinear quark p µ (1, 2, )Q power coun0ng parameter n-collinear gluon p µ (1, 2, )Q 2 = /Q (ultra)so^ gluon p µ ( 2, 2, 2 )Q Maximilian Stahlhofen - JGU Uni Wien, Seminar - 29

30 he -je%ness Event Shape hard current Factoriza0on formula for small : d =r d Ĥ Ŝ B a B b Deriva0on in SCE Integrate out non-resonant dofs: n n n n n integrate out W n = n perms... n exp n n n n-collinear Wilson line g n P n A n i [Stewart, ackmann, Waalewijn, 10] J i + O Q eikonal propagator n factorize i n k n so^ Wilson line Maximilian Stahlhofen - JGU Uni Wien, Seminar - 30

31 he -je%ness Event Shape Factoriza0on formula for small : d =r d Ĥ Ŝ B a B b 1-loop calcula0on =1 Wilson coefficient [Stewart, ackmann, Waalewijn, 10] i J i + O Q matching purely virtual full QCD contribu0ons (finite parts) Maximilian Stahlhofen - JGU Uni Wien, Seminar - 31

32 he -je%ness Event Shape Factoriza0on formula for small : d =r d Ĥ Ŝ B a B b 1-loop calcula0on [Stewart, ackmann, Waalewijn, 10] i J i + O Q = Maximilian Stahlhofen - JGU Uni Wien, Seminar - 32

33 he -je%ness Event Shape Factoriza0on formula for small : d =r d Ĥ Ŝ B a B b 1-loop calcula0on [Stewart, ackmann, Waalewijn, 10] i J i + O Q Maximilian Stahlhofen - JGU Uni Wien, Seminar - 33

34 he -je%ness Event Shape Factoriza0on formula for small : d =r d Ĥ Ŝ B a B b 1-loop calcula0on [Stewart, ackmann, Waalewijn, 10] i J i + O Q Maximilian Stahlhofen - JGU Uni Wien, Seminar - 34

35 LO Ingredients 2-loop hard func0on: purely virtual QCD correc0ons process-dependent some known, many not 2-loop beam func0on: ü quark ü gluon [Gaunt, MS, ackmann, 14] 2-loop jet func0on: ü quark ü gluon [Becher, eubert, 06] [Becher, Bell, 10] 2-loop so^ func0on: ü known analy0cally for =0 (e.g. Drell-Yan, H-produc0on, DIS, e + e jj) [Kelley, Schwartz, Schabinger, 11; Monni, Gehrmann, Luisoni, 11; Hornig, Lee, Stewart, Walsh, Zuberi, 11; Kang, Labun, Lee, 15] ü known numerically for =1 (e.g. H/W/Z+j produc0on) [Boughezal, Liu, Petriello, 15] ü shown how to obtain numerically for arbitrary from known 2-loop results + sector decomposi0on [Boughezal, Liu, Petriello, 15] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 35

36 LO Ingredients n-collinear momentum: p µ = p nµ 2 + p+ n µ 2 + pµ Generic quark beam func0on: B q = P n (p ) [ q n W n ](0) M / n 2 xp ( ˆp )[W nq n ](0) P n (p ) proton state collinear Wilson line collinear quark field Gluon beam func0on analogous: [W nq n ] 1 g [W n id n W n ] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 36

37 LO Ingredients n-collinear momentum: p µ = p nµ 2 + p+ n µ 2 + pµ Generic quark beam func0on: B q = P n (p ) [ q n W n ](0) M / n 2 ( ˆp )[W nq n ](0) P n (p ) ypical measurement operators: inclusive: M = (PDF) -je%ness: M = (t ˆp + ) inclusive p : M = (p ˆp ) ( MDPDF ) fully differen0al: M = (t ˆp + ) (p ˆp ) Maximilian Stahlhofen - JGU Uni Wien, Seminar - 37

38 LO Ingredients n-collinear momentum: p µ = p nµ 2 + p+ n µ 2 + pµ Generic quark beam func0on: B q = P n (p ) [ q n W n ](0) M / n 2 ( ˆp )[W nq n ](0) P n (p ) OPE: B i (t, x,µ) = virtuality j d I ij x, t,µ fj (,µ) + O 2 QCD t Wilson coefficient [Collins, Soper, Sterman; Stewart, ackmann, Waalewijn] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 38

39 LO Ingredients 2-loop qq-diagrams (axial gauge): + diagrams with gluons azached to Wilson lines (in Feynman gauge) Consider all possible cuts! Maximilian Stahlhofen - JGU Uni Wien, Seminar - 39

40 2-loop LO Ingredients qq-diagrams (axial gauge): [Gaunt, MS, ackmann, 14] I (2) qq I (2) q q Maximilian Stahlhofen - JGU Uni Wien, Seminar - 40

41 2-loop LO Ingredients qg-diagrams (axial gauge): [Gaunt, MS, ackmann, 14] I (2) qg Maximilian Stahlhofen - JGU Uni Wien, Seminar - 41

42 2-loop LO Ingredients gg-diagrams (axial gauge): [Gaunt, MS, ackmann, 14] I (2) gg +Wilson line diagrams and ghost loops (in Feynman gauge) Maximilian Stahlhofen - JGU Uni Wien, Seminar - 42

43 2-loop LO Ingredients gq-diagrams (axial gauge): [Gaunt, MS, ackmann, 14] I (2) gq Maximilian Stahlhofen - JGU Uni Wien, Seminar - 43

44 2-loop I (2) ij (t, z,µ) = 1 µ 2 L t ( i 0 )2 3 µ 2 ij (1 z) µ 2 L t i 3 2 µ i B ij (1 z)+3p (0) ij (z) + 1 µ 2 L 1 LO Ingredients t i µ 2 1 ( i 0) ( i B0 )2 4 L n (x) spli%ng func0on i B0 ij (1 z) ln n (x) x i 0 I (1) ij (z) 2( i B0 + 0 )P (0) ij (z) µ 2 L 0 t µ 2 ( i 0) i 0 B0 i 2 12 k i B1 2 P (0) ik (z) zp (0) kj (z) ij (1 z) i P(0) ij (z) ( i B )I (1) ij (z) (t) 4I (2) ij (z) ew! k I (1) ik (z) zp (0) kj (z) + 4P (1) ij (z) [Gaunt, MS, ackmann, 14] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 44

45 LO Ingredients 2-loop hard func0on: purely virtual QCD correc0ons process-dependent some known, many not 2-loop beam func0on: ü quark ü gluon [Gaunt, MS, ackmann, 14] 2-loop jet func0on: ü quark ü gluon [Becher, eubert, 06] [Becher, Bell, 10] 2-loop so^ func0on: ü known analy0cally for =0 (e.g. Drell-Yan, H-produc0on, DIS, e + e jj) [Kelley, Schwartz, Schabinger, 11; Monni, Gehrmann, Luisoni, 11; Hornig, Lee, Stewart, Walsh, Zuberi, 11; Kang, Labun, Lee, 15] ü known numerically for =1 (e.g. H/W/Z+j produc0on) [Boughezal, Liu, Petriello, 15] ü shown how to obtain numerically for arbitrary from known 2-loop results + sector decomposi0on [Boughezal, Liu, Petriello, 15] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 45

46 2-loop hard func0on: purely virtual QCD correc0ons process-dependent some known, many not LO Ingredients 2-loop beam func0on: ü quark ü gluon 2-loop jet func0on: With these ingredients we can compute ü quark ü gluon [Becher, eubert, 06] [Gaunt, MS, ackmann, 14] sing [Becher, Bell, 10] [Becher, Bell, 10] LO (X) = d LO (X) d + O cut 0 d Q 2-loop so^ func0on:...for a given 2-loop hard func0on ü known analy0cally for =0 (e.g. Drell-Yan, H-produc0on, DIS, e + e jj) [Kelley, Schwartz, Schabinger, 11; Monni, Gehrmann, Luisoni, 11; Hornig, Lee, Stewart, Walsh, Zuberi, 11; Kang, Labun, Lee, 15] ü known numerically for =1 (e.g. H/W/Z+j produc0on) [Boughezal, Liu, Petriello, 15] [Becher, eubert, 06] ü shown how to obtain numerically for arbitrary from known 2-loop results + sector decomposi0on cut [Boughezal, Liu, Petriello, 15] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 46

47 -je%ness slicing recipe Applica0ons/Results 1 Compute LO +1 jet cross sec0on with partonic event generator: d LO (X) d = cut d d LO (X) ( > cut ) +1 - for each event determine jet direc0ons by mapping parton- onto jet-momenta ˆq i [Boughezal, Focke, Liu, Petriello, 15] [Gaunt, MS, ackmann, Walsh, 15] (1) - differences in the mapping procedure (jet algorithm) cut /Q [Jouzenus, Stewart, ackmann, Waalewijn, 11] - compute for each event and select events with > cut 2 Compute singular cross sec0on using SCE factoriza0on formula (input: 2-loop virtuals) sing LO (X) = 0 cut d d LO (X) d (2) 3 Add (1) + (2) and check cut independence! Maximilian Stahlhofen - JGU Uni Wien, Seminar - 47

48 Applica0ons/Results LO Z and H rapidity (as proof of principle): used 0 ( beam thrust ) - slicing LO Z/H+j computed with MCFM [Campbell, Ellis, 02, 10] - - [Gaunt, MS, ackmann, Walsh 15] Good agreement with exis0ng codes VRAP and HLO (per-mill level!) [Anastasiou, Dixon, Melnikov, Petriello 03, 04] [Catani, Grazzini, 07, Grazzini 08] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 48

49 Applica0ons/Results LO Z and H rapidity (as proof of principle): d nons d ln := d LO d ln d sing d ln 2 s ln , = /Q Check - - cut 0 independence: [Gaunt, MS, ackmann, Walsh 15] cut 0 cut 0 Observe convergence of nonsingular terms toward zero Maximilian Stahlhofen - JGU Uni Wien, Seminar - 49

50 Applica0ons/Results LO Figure 5. he dependence of the LO coefficient on the slicing para cut cut are presented using the -jettiness ( ) (circle) and Q -slicing (trian methods. he dashed lines correspond to the errors associated with the fitting Compare 0-je%ness to q slicing in pp! : Δσ (δ ) [ ] contribution, which scales as [Campbell, [42, 74], Ellis, Li, Williams, 16] 1 s CF n 2n cut n LO cut LO LO > )/ log +.. e 5. he dependence of the LO coefficient on the slicing parameter cut(. Results n! Q esented using the -jettiness (δ cut=τ cut ) (circle) and Q -slicing (triangles) ( cut Qcut ) for the -jettiness calculation. In this equation Q is an appropriate hard errors associated with the fitting procedure. δ to ds. he dashed lines correspond the = here by the transverse momentum of the photons. A similar analysis for LO cut 5. he dependence of the LO coefficient on the slicing parameter. Results the result [75], cut cut cut nted using the -jettiness ) (circle) and Q -slicing (triangles) ( Qcut ) ( 1 2 s CF n 2n Qcut n bution, which scales as [42, 74], LO cut LO he dashed lines correspond to the errors associated with the fitting procedure. (Q > Q )/ log + n n! Q cut n LO - ( > cut )/ LO 1 s CF 2n (3.2) log +.expects.. herefore one similar computational effort for the two methods w n! Q cut cut δ [ ] and Q are related by [74], tion, which scales as [42, 74], e -jettiness calculation. In this equation Q is an appropriate hard scale that is given n cut p2 cut cut 1 C n s F LOmomentum y the transverse yields ' Q ( > cut )/ oflothe photons. A similar log2n analysis +..for. Q -slicing(3.2). For Q n! Q Q sult [75], log cancella0ons similarlyusing large For -jettiness calculation. In this equation Q is an appropriate scale is given n Q =hard 40 onethat therefore expects the LO calculation Qcut cutgev Q 1 2 C n s F LO cut LO 2n cut (Q > Q )/ log +... (3.3) asanalysis expensivefor asq the one with similar = computa0onal GeV. the transverse momentum yields effort! of the photons. A similar -slicing n! Q cut Figure 5 shows the dependence for the LO coefficient, lt [75], fore one expects similar computational effort for the two methods when the values of n It is clear cutthat the dependence is much more pronounced than at LO Q 1 2 C n s F LO cut LO 2n for nd Qcut are related by [74], accuracy requires value of cut around GeV or Q (Q computed with log +... LOslicing, (3.3)aautomated LO 0-jet processes -je%ness in MCFM: >Q )/ n! p Q about 0.02 GeV. Once again power corrections are less significant for Q cut 2 cut Q pp! H, pp! Z, pp! W, pp! HZ, HW,! is comparable (+decays) for the two time topp achieve equivalent accuracy in both m re one expects similar computational methods when the! values of pp ' effort. computing (3.4) Q Q [Boughezal, Ellis, Focke, Liu, Petriello, Williams, 16] line with thecampbell, scaling expected fromgiele, Eq. (3.4). he LO results repor Qcut are related by [74], cutthe obtained from asymptotic! 0 results obtained by a fit to the cu p = 40 GeV one therefore expects the LO calculation using Q = 0.04 GeV to be Maximilian Stahlhofen JGU Uni Wien, Seminar - 50 Qcut cut is represented by the solid red line in figure 5. We observe that for valu cut ensive as the one with = ' GeV.. (3.4)

51 Applica0ons/Results More (impressive!) results for LHC: used 1 - slicing LO W+jj computed with MCFM[Campbell, Ellis, 02, 10] [Boughezal, Focke, Liu, Petriello, 15] W boson p leading jet p cut 1 independence checked in the range cut GeV (per-mill level!) Maximilian Stahlhofen - JGU Uni Wien, Seminar - 51

52 Applica0ons/Results More (impressive!) results for LHC: used 1 - slicing LO H+jj computed with MCFM [Campbell, Ellis, 02, 10] [Boughezal, Focke, Giele, Liu, Petriello, 15] Higgs p leading jet p 1 cut cut independence checked in the range GeV (per-mill level!) Agreement with results from sector-improved subtrac0ons at per-mill level! Maximilian Stahlhofen - JGU Uni Wien, Seminar - 52

53 Applica0ons/Results More (impressive!) results for LHC: used 1 - slicing LO Z+jj computed with MCFM [Campbell, Ellis, 02, 10] [Boughezal, Campbell, Ellis, Focke, Giele, Liu, Petriello, 15] Z boson p leading jet p cut 1 independence checked in the range cut GeV (per-mill level!) Maximilian Stahlhofen - JGU UM Seminar - 53

54 Possible Extensions Single-differen0al -je%ness subtrac0on: (X) = 0 o d d sing (X) d + d d (X) d d sing (X) d ( < o ) + O Q - point-by-point subtrac0on in - expect bezer numerical convergence cut feasible (?) - requires PS map to disentangle from rest of PS integra0on Mul0-differen0al -je%ness subtrac0ons: - differen0al in i variable ( invariant mass of each jet/beam sector) - differen0al in both, and q [Jain, Procura, Waalewijn, 12] [Gaunt, MS 14] -... [Procura, Waalewijn, Zeune 15] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 54

55 Possible Extensions Include sub-leading power correc0ons: (X) = 0 cut d d sing (X) d + d (X) d cut d - Could improve accuracy/performance of slicing method Include quark masses: + O cut Q Systema0cally calculable in SCE - if m q Q, treat heavy partons similar to color-neutral final states, but more complicated so^ func0on [Li, Wang 16] - if m q Q, use massive SCE requires modified defini0on of! [Pietrulewicz, Gritschacher, Hoang, Jemos, Mateu, 14] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 55

56 Summary ew LO subtrac0on method: -je%ness slicing (X) = 0 cut d d sing (X) d + d (X) d cut d + O cut Q ( > cut ) from LO tools sing ( < cut ) from SCE, all universal ingredients known Impressive LO results: pp W/Z/H + j Possible extensions: more-differen0al subtrac0on, quark masses, power correc0ons Maximilian Stahlhofen - JGU Uni Wien, Seminar - 56

57 Back-Up [Gaunt, MS, ackmann, Walsh 15] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 57

58 Back-Up [Boughezal, Campbell, Ellis, Focke, Giele, Liu, Petriello, 15] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 58

59 Back-Up [Boughezal, Focke, Giele, Liu, Petriello, 15] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 59

60 Back-Up [Gaunt, MS, ackmann, Walsh 15] Maximilian Stahlhofen - JGU Uni Wien, Seminar - 60

61 Back-Up [Gaunt, MS, ackmann, Walsh 15] ( = ) ( = ) Maximilian Stahlhofen - JGU Uni Wien, Seminar - 61