Towards Jet Cross Sections at NNLO

Transcription

1 Towards Jet Cross Sections at HP.4, September, MPI Munich

2 Expectations at LHC Large production rates for Standard Model processes single jet inclusive and differential di-jet cross section will be measured to per cent accuracy Allow precision determinations strong coupling constant parton distributions Provided theory description is known to the same precision:!!"nb# proton - (anti)proton cross sections! jet (E T jet > "s/)! jet (E T jet > GeV)! jet (E jet T > "s/4)! Higgs (M H = GeV) GeV WJS9! tot! b! W! Z! t 5 GeV Tevatron LHC "s (TeV) J. Stirling events / sec for L = 33 cm - s -

3 Inclusive jet and dijet cross sections d y (pb/tev)!/dp d 5 T CMS Preliminary L = 4.7 fb s = 7 TeV anti-k T R =.7 4 y <.5 (" ) 3.5 < y <. (" ). < y <.5 (" ).5 < y <. (" ). < y <.5 (" ) - -5 NNPDF. µ = µ = p F R T NLO # NP..3 Jet p T (TeV) Data can be used to constrain parton distributions parton distribution fits currently include DIS structure functions and inclusive Drell-Yan cross sections Inclusion of jet data in parton distribution fits requires corrections to jet cross sections

4 calculations Require three principal ingredients (here: pp j) two-loop matrix elements explicit infrared poles from loop integral known for all massless processes one-loop matrix elements explicit infrared poles from loop integral and implicit poles from single real emission usually known from NLO calculations tree-level matrix elements implicit poles from double real emission known from LO calculations Infrared poles cancel in the sum Challenge: combine contributions into parton-level generator need method to extract implicit infrared poles 3

5 calculations Solutions sector decomposition: expansion in distributions, numerical integration (T. Binoth, G. Heinrich; C. Anastasiou, K. Melnikov, F. Petriello; M. Czakon) applied to corrections to Higgs and vector boson production (C. Anastasiou, K. Melnikov, F. Petriello) subtraction: add and subtract counter-terms: processindependent approximations in all unresolved limits, analytical integration several well-established methods at NLO q T subtraction at applied to Higgs and vector boson production, associated H+W production, diphoton production (S. Catani, M. Grazzini; with L. Cieri, G. Ferrera, D. de Florian, F. Tramontano) antenna subtraction at for jet observables in e + e - collisions (T. Gehrmann, N. Glover, AG) 4

6 methods: new developments sector decomposition combined with subtraction use sector decomposition to compute integrated subtraction terms numerically (M. Czakon; R. Boughezal, K. Melnikov, F. Petriello) applied to top quark pair production (P. Bärnreuther, M. Czakon, A. Mitov) non-linear mappings (C. Anastasiou, F. Herzog, A. Lazopoulos) applied to Higgs decay into bottom quarks (C. Anastasiou, F. Herzog, A. Lazopoulos) applied to Higgs production in botton fusion (S. Bühler, F. Herzog, A. Lazopoulos, R. Müller) This talk: antenna subtraction for jet observables in hadronic collisions 5

7 Subtraction Structure of m-jet cross section at hadron colliders dˆσ = with: + + dˆσ RR dˆσ S dφ m+ dˆσ RV dˆσ VS +dˆσ MF, dφ m+ dφ m dˆσ VV +dˆσ MF, Partonic contributions: dˆσ RR + dˆσ S + dφ m+ dˆσ RV Subtraction terms for double real radiation: Subtraction terms for one-loop real radiation: Mass factorization terms: dˆσ MF, dφ m+ dˆσ VS dˆσ VV dˆσ S dˆσ VS dˆσ MF, Challenge: construction and integration of subtraction terms 6

8 Antenna subtraction Subtraction terms constructed from antenna functions Antenna function contains all emission between two partons i j k I K i j k I NLO subtraction term dˆσ S NLO = m+ m+ dφ m+ (p,...,p m+ ; q) j Phase space factorization dφ m+ (p,...,p m+ ; q) =dφ m (p,..., p I, p K,...,p m+ ; q) dφ Xijk (p i,p j,p k ; p I + p K ) Integrated subtraction term X ijk = dφ Xijk X ijk K X ijk M m J (m) m (p,...,p I,p K,...,p m+ ) 7

9 Antenna functions Colour-ordered pair of hard partons (radiators) quark-antiquark pair quark-gluon pair gluon-gluon pair NLO (D. Kosower; J. Campbell, M. Cullen, N. Glover) Three-parton antenna: one unresolved parton X 3 (T. Gehrmann, N. Glover, AG) Four-parton antenna: two unresolved partons X 4 Three-parton antenna at one loop: X 3 Products of NLO antenna functions: X 3 X 3 Soft antenna function S 8

10 Antenna subtraction: incoming hadrons Three antenna types (NLO: A. Daleo,T. Gehrmann, D. Maitre) Final-final antenna Initial-final antenna Initial-initial antenna i j k I K i j k I K i j k I K i j k I K i k j I K i k j I K 9

11 ( ) Initial-initial antenna functions are crossings of final-final antennae: four-parton case ) quark-antiquark ( antennae ( ) ( ) A 4 A ) ) ( ( 4( q,ĝ,g,q, A 4 ( q, g, ĝ,q, A 4 ( q,g,g, q ), A ( ( ) 4 q,ĝ,ĝ,q ) Ã 4 Ã ) 4( q,ĝ,g,q, Ã 4( q, ) g, g, q, Ã ( ) 4 q,ĝ,ĝ,q B4 B4( q, q, q, q ), B4( q,q, q, q ), B4 ( q, q, q, q ) C4 C4( q, q,q,q ), C4 )) (( ( q, q, q,q, C 4 q, q, q,q ) (, C ) ) ( 4 q,q, q, q ) ) ( ( ( ) ) ( ) quark-gluon antennae D4 D4 ) ) ( ) ( ) ( q,ĝ,g,g ) ( ( ) ) ( ), D 4 ( q,g,ĝ,g, D 4 q,ĝ,ĝ,g, D 4 q,ĝ,g,ĝ E4 E4( q, q, q,g ), E4( q,q, q, ĝ ), E4( q, q, q,g ), E4( q, q, q, ĝ ), Ẽ4 Ẽ4( q, q, q,g ), Ẽ 4( q,q, q, ĝ ), 4( Ẽ q, q, q,g ) (, Ẽ 4 q, q, q, ĝ ) gluon-gluon antennae F4 F4 (ĝ,ĝ,g,g ) (ĝ,g,ĝ,g ), F 4 G 4 G ) (ĝ, 4(ĝ, q,q,g, G ) 4 q, q,g, G (ĝ, ) ( 4 q, q,ĝ, G ) 4 g, q, q,g (ĝ, ) (ĝ, ) ( ) q,q,g, G 4 q, q,ĝ, G 4 g, q, q,g G 4 G 4 H4 H4( q, q,q, q ), H4 ( q, q, q, q )

12 Integrated antenna functions Analytical integration over unresolved part of phase space only phase space integrals reduced to masters (C. Anastasiou, K. Melnikov) Final-final:, one scale: q 4 tree level (4 master integrals) 3 one loop (3 master integrals) Initial-final: q k + k + k 3 (+k 4 ) q + p k + k (+k 3 ) (A. Daleo, T. Gehrmann, G. Luisoni, AG) 3 tree level (9 master integrals) one loop (6 master integrals), two scales: q, x Initial-initial: p, three scales: q + p q + k (+k ), x, x 3 tree level ( master integrals) (R. Boughezal, M. Ritzmann, AG; T. Gehrmann, M. Ritzmann, AG) one loop (5 master integrals) (T. Gehrmann, P.F. Monni)

13 Integrated initial-initial antennae Tree-level antenna functions X 4 Kinematics: p + p q + k j + k k Phase space factorization (A. Daleo, T. Gehrmann, D. Maitre) dφ m+ (k,...,k m+ ; p,p )=dφ m ( k,..., k i, k l,..., k m+ ; x p,x p ) s s j s k ˆx = s s s j s k ˆx = s δ(x ˆx ) δ(x ˆx )[dk j ][dk k ] dx dx s s j s k s j s k + s jk s s j s k s s j s k s j s k + s jk s s j s k Fix x,x by imposing collinear limits; Lorentz boost to frame with x p + x p q ; q = q Integration: 3 particle phase space with x,x fixed

14 Integrated initial-initial antennae Integration of tree-level antenna functions X 4 Express phase space integrals as masters with x,x fixed Distinguish Hard region x,x : transcendentality Collinear regions x =, x or x,x =: transcendentality 3 Soft region x =x =: transcendentality 4 Determine master integrals from differential equations in x,x Antenna functions with secondary fermion pair: masters (R. Boughezal, M. Ritzmann, AG) Full set of antennae now completed: contains masters (T. Gehrmann, M. Ritzmann, AG) Integrated initial-initial antennae all known: X 3,X 3,X 4 3

15 Jet production at Double real radiation at for Contributions from all tree-level 4 processes Test case: gg gggg pp j (N. Glover, J. Pires) dσ R = N αs N born dφ4 (p 3,...,p 6 ; p,p ) π A 4! 6(ˆ g, ˆ g,i g,j g,k g,l g )J (4) (p i,...,p l ) + 4! + 4! P (i,j,k,l) (3,4,5,6) P (i,j,k,l) (3,4,5,6) P C (i,j,k,l) (3,4,5,6) A 6(ˆ g,i g, ˆ g,j g,k g,l g )J (4) (p i,...,p l ) A 6(ˆ g,i g,j g, ˆ g,k g,l g )J (4) (p i,...,p l ) three topologies according to initial state gluon positions 4

16 Antenna subtraction for gg gg at Double real radiation: gg gggg (N. Glover, J.Pires) Subtraction terms involve only gluon-gluon antennae in all three configurations (initial-initial, initial-final, final-final) F4 for colour-connected double unresolved limits F3 F3 for oversubtracted single unresolved limits and colour unconnected double unresolved limits F3 S for large-angle soft gluon radiation for single unresolved limits F 3 %! " # $ antenna subtraction terms constructed, implemented and tested in all unresolved limits 5

17 Integrated antennae combine with either real-virtual (m+ partons) or double virtual (m partons) channel Integrated double real subtraction terms dφ m+ dσ S = one-particle integrals dσ S, two-particle integrals dσ S, contains contains dφ m+ dσ S, + dφ m Integrated antennae depend on momentum fractions x,x of initial state partons dσ S, F 3 M m+ F 3 F 3 M m S F 3 M m F 4 M m F 3 F 3 M m 6

18 Jet production at Real-virtual radiation at for Contributions from all one-loop 3 processes Test case: gg ggg pp j (N. Glover, J. Pires, AG) dˆσ RV =N αs N born dφ3 (p 3,...,p 5 ; p,p ) π 3! + 3! P (i,j,k) (3,4,5) P (i,j,k) (3,4,5) A 5(ˆ g, ˆ g,i g,j g,k g )J (3) (p i,p j,p k ) A 5(ˆ g,i g, ˆ g,j g,k g )J (3) (p i,p j,p k ) two topologies according to initial state gluon positions one-loop matrix elements contain explicit infrared poles 7

19 Real-virtual subtraction for gg gg Single unresolved limit of one-loop amplitudes (Z. Bern, L.D. Dixon, D. Dunbar, D. Kosower) j unresolved Loop m+ Accordingly: Split tree X ijk Split tree Loop m + Split loop Tree m Split loop X ijk i j I i j k I i j k k K I K I m+ m+ K m+ K dσ VS,a =N dφ m+(p 3,...,p m+3 ; p,p ) j +N dφ m+ (p 3,...,p m+3 ; p,p ) j X 3 M () m+ J (m) m (p 3,...,p m+ ) X 3 M () m+ J (m) m (p 3,...,p m+ ) 8

20 Real-virtual subtraction for gg gg Structure of subtraction term approaches in all single unresolved limits dσ VS,a dσ VS,b dσ RV dσ VS =dσ VS,a +dσvs,b removes oversubtraction of explicit and implicit poles dσ VS,b = N dφ m+(p 3,...,p m+3 ; p,p ) ik X 3 (s ik ) X 3 M m+ J (m) m such that: Poles dˆσ RV dˆσ VS dˆσ S, dˆσmf, = strong check on explicit pole cancellation in real-virtual channel 9

21 Real-virtual subtraction for gg gg Check of the subtraction terms choose scaling parameter x for each limit generate phase space trajectories into each limit require reconstruction of two hard jets compute ratio (matrix element)/(subtraction term): Example: soft limit : i k #phase space points = 7 outside the plot outside the plot outside the plot s ij s X= -5 X= -6 X= -7 Ratio approaches unity in all unresolved limits Strong check on implementation of subtraction terms M RV /S term j

22 Jet production at Double-virtual radiation at for pp j Contributions from all one-loop processes Test case: gg gg (N. Glover, J. Pires, T.Gehrmann, AG) dˆσ VV =N αs N born π! +! P (i,j) (3,4) P (i,j) (3,4) dφ3 (p 3,p 4 ; p,p ) two topologies according to initial state gluon positions contains (two-loop*tree) and (one-loop) explicit infrared poles up to /e 4 from loop integrals A 4(ˆ g, ˆ g,i g,j g )J () (p i,p j ) A 4(ˆ g,i g, ˆ g,j g )J () (p i,p j )

23 Double virtual channel Explicit poles of double virtual contribution cancel with integrated double real subtraction terms F4 M m F3 F3 M m integrated one-loop subtraction terms F3 M m dσ S, dσ VS of the form of the form F 3 M m l mass factorization counter terms dσ MF, In all three configurations (final-final, initial-final, initial-initial)

24 Double virtual channel For purely gluonic contributions to pp j, we obtain Poles dˆσ VV + dˆσ S, + dˆσ VS +dˆσ MF, = Highly non-trivial check of analytic cancellation of infrared singularities between double-real, real-virtual and doublevirtual corrections Proof of principle for antenna subtraction method applied to hadronic collisions 3

25 Conclusions antenna subtraction method generalized to hadronic collisions completed analytic integration of all antenna functions for one or two partons in the initial state: full set of integrated antennae now available in all configurations Proof-of-principle implementation for to pp j contribution subtraction terms in double real and real virtual channel constructed and implemented observe point-wise convergence for matrix element/subtraction term Double virtual channel observe analytical cancellation of all infrared poles Parton level generator : In progress 4