Jets in pp at NNLO João Pires Universita di Milano-Bicocca and Universita di Genova, INFN sezione di Genova HP.5-5 September 014 Florence, Italy based on: Second order QCD corrections to gluonic jet production at hadron colliders J. Currie, A. Gehrmann-De Ridder, T. Gehrmann, N. Glover, JP, S. Wells, arxiv:1407.5558 arxiv:1.99 JHEP 1401 (014) 1, arxiv:101.7 Phys.Rev.Lett. 1 (01) 16 Perturbative QCD description of jet data from LHC Run-I and Tevatron Run-II S.Carrazza, JP, arxiv:1407.701
Inclusive jet and dijet cross sections look at the production of jets of hadrons with large transverse energy in inclusive jet events pp j + X exclusive dijet events pp j cross sections measured as a function of the jet p T, rapidity y and dijet invariant mass m jj in double differential form (CMS-PAS-SMP-1-01) (ATLAS-CONF-01-01)
Inclusive jet cross section Motivation for NNLO experimental uncertainties at high-p T smaller than theoretical need pqcd predictions to NNLO accuracy
Inclusive jet cross section Motivation for NNLO experimental uncertainties at high-p T smaller than theoretical need pqcd predictions to NNLO accuracy collider jet data can be used to constrain parton distribution functions size of NNLO correction important for precise determination of PDF s inclusion of jet data in NNLO parton distribution fits requires NNLO corrections to jet cross sections
Inclusive jet cross section Motivation for NNLO experimental uncertainties at high-p T smaller than theoretical need pqcd predictions to NNLO accuracy collider jet data can be used to constrain parton distribution functions size of NNLO correction important for precise determination of PDF s inclusion of jet data in NNLO parton distribution fits requires NNLO corrections to jet cross sections α s determination from hadronic jet observables limited by theoretical uncertainty due to scale choice
inclusive jet and dijet cross sections State of the art: dijet production is completely known in NLO QCD [Ellis, Kunszt, Soper 9], [Giele, Glover, Kosower 94], [Nagy 0] NLO+Parton shower [Alioli, Hamilton, Nason, Oleari, Re 11] NLO EW corrections [Dittmaier, Huss, Speckner 1] approximate NNLO threshold corrections [Kidonakis, Owens 00], [Florian, Hinderer, Mukherjee, Ringer, Vogelsang 1] Goal: obtain the jet cross sections at NNLO exact accuracy in double differential form d σ dp T d y d σ dm jj dy
pp j at NNLO: gluonic contributions A (0) 6 (gg gggg) A(1) 5 (gg ggg) A() 4 (gg gg) [Berends, Giele 87], [Mangano, Parke, Xu 87], [Britto, Cachazo, Feng 06] [Bern, Dixon, Kosower 9] [Anastasiou, Glover, Oleari, Tejeda-Yeomans 01],[Bern, De Freitas, Dixon 0] dˆσ NNLO = dφ 4 dˆσ RR NNLO + dφ dˆσ RV NNLO + dφ dˆσ V V NNLO explicit infrared poles from loop integrations implicit poles in phase space regions for single and double unresolved gluon emission procedure to extract the infrared singularities and assemble all the parts in a parton-level generator differential cross sections kinematics of the final state intact to apply arbitrary phase space observable cuts
NNLO antenna subtraction dˆσ NNLO = + + dφ 4 dφ dφ ( ) dˆσ NNLO RR dˆσs NNLO ( ) dˆσ NNLO RV dˆσt NNLO ( ) dˆσ NNLO V V dˆσu NNLO dˆσ NNLO S : real radiation subtraction term for dˆσrr NNLO dˆσ NNLO T : one-loop virtual subtraction term for dˆσrv NNLO dˆσ NNLO U : two-loop virtual subtraction term for dˆσv V NNLO subtraction terms constructed using the antenna subtraction method at NNLO for hadron colliders presence of initial state partons to take into account contribution in each of the round brackets is finite, well behaved in the infrared singular regions and can be evaluated numerically
NNLO antenna subtraction universal factorisation of both colour ordered matrix elements and the (m+)- particle phase space colour connected unresolved particles i j l k M m+4 (..., i, j, k, l,...) J({p m+4 }) M m+ (..., I, L,...) J({p m+ }) X4 0 (i, j, k, l) I L i j l I L k momentum map {p i, p j, p k, p l } {p I, p L } enforces momentum conservation away from the unresolved limits phase-space factorisation dφ m+ (p a,..., p i, p j, p k, p l,..., p m+ ) = dφ m(p a,..., p I, p L,..., p m+ ) dφ Xijkl (p i, p j, p k, p l ) integrated antennae is the inclusive integral Xijkl 0 (s ijkl) = 1 C(ɛ) dφ Xijkl (p i, p j, p k, p l )X4 0 (i, j, k, l)
NNLO antenna subtraction Implementation checks pp j at NNLO: subtraction terms correctly approximate the matrix elements in all unresolved configurations of partons j, k dˆσ RR,RV NNLO {j,k},{j} 0 dˆσ S,T NNLO local (pointwise) analytic cancellation of all infrared explicit ɛ-poles when integrated subtraction terms are combined with one, two-loop matrix elements ( ) Poles dˆσ NNLO RV dˆσt NNLO = 0 ( ) Poles dˆσ NNLO V V dˆσu NNLO = 0 leading and subleading colour process independent NNLO subtraction scheme allows the computation of multiple differential distributions in a single program run
Jet production partonic channels Fraction of jets per initial state contribution LHC fraction of jets 1 0.9 0.8 0.7 0.6 s=8 TeV anti-k T R=0.7 MSTW008lo µ = µ = p R F T1 0.0 < y < 0. gg qg qq qq gg gg dominates at low p T 0.5 qg qg important in all p T regions 0.4 0. qq qq dominant at high p T 0. 0.1 Tevatron qg and q q dominant 0 (LHC 8 TeV) p (GeV) T Present results at NNLO for gg gg at leading colour gg gg at subleading colour fraction of jets 1 0.9 0.8 0.7 0.6 0.5 0.4 0. s=1.96 TeV anti-k T R=0.7 MSTW008lo µ = µ = p R F T1 0.0 < y < 0. gg qg qq qq q q gg at leading colour 0. 0.1 0 (Tevatron 1.96 TeV) p (GeV) T
Numerical setup (J.Currie, A. Gehrmann-De Ridder, T.Gehrmann, N. Glover, JP 1) pp collisions at s = 8 TeV jets identified with the anti-k T jet algorithm with resolution parameter R = 0.7 jets accepted at rapidities y < 4.4 leading jet with transverse momentum p T > 80 GeV subsequent jets required to have at least p T > 60 GeV MSTW008nnlo PDF for all fixed-order predictions dynamical factorization and renormalization scales equal to the leading jet p T (µ R = µ F = µ = p T 1 ) present results for full colour gg gg scattering and q q gg leading colour combined at NNLO
Inclusive jet p T distribution at NNLO (pb/gev) dσ/dp T 5 4 1-1 - - -4 6-5 -6 1.8 1.6 1.4 1. 1 s=8 TeV anti-k T R=0.7 MSTW008nnlo µ = µ = p R F T1 NLO/LO NNLO/NLO NNLO/LO qq->gg+x + gg->gg+x LO NLO NNLO p (GeV) T p (GeV) T all jets in an event are binned NNLO correction stabilizes the NLO k-factor growth with p T NNLO corrections 15 6% with respect to NLO
Double differential inclusive jet p T distribution at NNLO d y (pb/gev) T σ/dp d 16 15 1 11 9 7 5-1 - -5 s=8 TeV R=0.7 anti-k T MSTW008nnlo µ = µ = p R F T1 qq->gg+x+gg->gg+x 6 y <0. (x ) 0. 0.8 1..1.8.6 5 y <0.8 (x ) 4 y <1. (x ) y <.1 (x ) y <.8 (x ) 1 y <.6 (x ) 0 y <4.4 (x ) 1.8 1.6 1.4 1. 1 1.8 1.6 1.4 1. 1 1.8 1.6 1.4 1. 1 NLO/LO NNLO/NLO NNLO/LO NLO/LO NNLO/NLO NNLO/LO NLO/LO NNLO/NLO NNLO/LO double differential k-factors y <0. 0. y <0.8 0.8 y <1. NNLO prediction increases between 5% to 15% with respect to the NLO cross section p T p T p T (GeV) (GeV) (GeV) -7 p (GeV) T similar behaviour between the rapidity slices
Double differential exclusive dijet mass distribution at NNLO dσ /dmjj dy* (pb/gev) 15 1 9 6 1 - -6-9 -1 s=8 TeV qq->gg+x+gg->gg+x anti-k T R=0.7 MSTW008nnlo µ = µ = p R F T1 0 y*<0.5 (x ) 0.5 1.0 1.5.0.5.0.5 4.0 y*<1.0 (x ) 4 y*<1.5 (x ) 6 y*<.0 (x ) 8 y*<.5 (x ) y*<.0 (x ) 1 y*<.5 (x ) 14 y*<4.0 (x ) 16 y*<4.5 (x ) (GeV) m jj 1 0 1 0 1 0 NLO/LO NNLO/NLO NNLO/LO NLO/LO NNLO/NLO NNLO/LO NLO/LO NNLO/NLO NNLO/LO double differential k-factors y*<0.5 0.5 y*<1.0 1.0 y*<1.5 NNLO corrections up to 0% with respect to the NLO cross section similar behaviour between the y = 1/ y 1 y slices m jj m jj m jj (GeV) (GeV) (GeV)
Inclusive jet p T scale dependence (gg gg + X) dσ/dp (pb) T 90 80 70 60 s=8 TeV anti-k T R=0.7 µ = µ = µ R F 80 GeV < p < 97 GeV T LO (LO PDF+α s ) NLO (NLO PDF+α s ) LO (NNLO PDF+α s ) NLO (NNLO PDF+α s ) NNLO (NNLO PDF+α s ) 50 40 0 0 0-1 -1 1 4 5 6 7 8 9 µ/p T1 scale dependence study gluons only N F = 0 channel at leading colour dynamical scale choice: leading jet p T 1 flat scale dependence at NNLO
Threshold resummation approximation to exact NNLO Approximate NNLO results from an improved threshold calculation for the single jet inclusive production [de Florian, Hinderer, Mukherjee, Ringer, Vogelsang 1] pp j + X with the threshold limit given by s 4 = P X 0 near threshold phase space available for real-gluon emission is limited higher kth order coefficient functions dominated by large logarithmic corrections ( αsw k (k) log m ) ab (z) αk s, m k 1, z = s 4 z + s δ(z), 4th tower, O(z) (all-channels) (gluons only channel) - rapidity integrated
NNLO benchmark predictions for jet production S. Carrazza, JP, arxiv:1407.701 understand and characterise the validity of the NNLO threshold approximation by comparing it with the exact computation using the gg-channel µ R = µ F = p T for both predictions comparison performed differential in p T and rapidity following the exact experimental setups NNPDF nnlo as 0118 set for all fixed order predictions NLO benchmark curves green dashed curves black dashed curves blue dashed curves NLO-threshold gg-channel NLO-exact gg-channel NLO-exact all channels NNLO benchmark curves pink long-dashed curves NNLO-threshold gg-channel dσ thresh gg,nnlo /dσ gg,lo black long-dashed curves NNLO-exact gg-channel dσ exact gg,nnlo /dσ gg,lo
Tevatron CDF Run-II s=1.96 TeV S. Carrazza, JP, arxiv:1407.701 differences 15% at low-p T in the central regions in the forward region differences 40% for all p T regions
LHC ATLAS 0 s=7 TeV S. Carrazza, JP, arxiv:1407.701 differences large at small p T and increase with rapidity exact NNLO k-factor decreases with rapidity, NNLO threshold k-factor increases with rapidity
Conclusions antenna subtraction method generalised for the calculation of NNLO QCD corrections for exclusive collider observables with partons in the initial-state explicit ɛ-poles in the matrix elements are analytically cancelled by the ɛ-poles in the subtraction terms non-trivial check of analytic cancellation of infrared singularities between double-real, real-virtual and double-virtual corrections successful inclusion of subleading colour contributions at NNLO with the antenna subtraction method first exact results for gg gg + X and q q gg + X at NNLO perfomed comparison between exact NNLO results and approximate NNLO results from threshold resummation in the gg-channel Future work: largest differences arise at low-p T for central rapidities and all p T at large rapidities differences are smaller at the Tevatron than at the LHC 7 TeV include remaining channels involving the quark contributions qg channel - most important at the LHC leading colour N F pieces qq channel - important at high p T
Back-up slides
Singly unresolved limits 180 160 140 Single collinear limit for gg gggg #PS points=000 517 outside the plot 5160 outside the plot 5159 outside the plot x=- -8 x=- - x=- -1 x=s 1i /s # events 0 80 60 40 0 Single collinear limits: q qg generate phase space points with s jk or s 1i small Distributions with a broader shape due to: 0 0.9 0.9 0.94 0.96 0.98 1 1.0 1.04 1.06 1.08 1.1 R g gg angular correlations in matrix elements and antenna functions when an initial/final state gluon splits into two gluons not accounted for by the subtraction term non-locality of subtraction term
Angular terms subtraction angular terms vanish after averaging over the azimuthal angle not-relevant for reproducing the correct 1/ɛ-poles in virtual contributions 1 π 1 π dφ (p l k ) = 0, dφ (p l k ) = k π 0 π 0 p p l n p l p n Θ F 0 (i, j, z, k ) A cos(φ + α) combine phase space points related to each other by a rotation of the system of unresolved partons {p i, p j } {p i, p j } p i φ P p j p µ i = zpµ + k µ k z n µ p n, pµ j = (1 z)pµ k µ k 1 z n µ p n, with p i p j = k z(1 z), p = n = k.p = k.n = 0
0 000 9000 8000 Single collinear limit for gg gggg #PS points=000 7 outside the plot 7 outside the plot 0 outside the plot x= -8 x= - x= -1 x=s jk /s 000 8000 Single collinear limit for gg gggg #PS points=000 8 outside the plot 0 outside the plot 0 outside the plot x=- -8 x=- - x=- -1 x=s 1i /s 7000 6000 6000 # events 5000 # events 4000 4000 000 000 000 00 0.9 0.9 0.94 0.96 0.98 1 1.0 1.04 1.06 1.08 1.1 R g gg jk 0 0.9 0.9 0.94 0.96 0.98 1 1.0 1.04 1.06 1.08 1.1 R g gg j 1 i 1 i l l k locality of the subtraction method achieved by combining -phase space points in both collinear limits combining phase space points largely cancels angular dependent terms generalizable to multiple collinear emission subtract correlations systematically at the phase space generation level
QCD cross sections at subleading color beyond NLO (J.Currie, A. Gehrmann-De Ridder, N. Glover, JP 1) subleading colour matrix elements have incoherent interferences, gluon scattering M 0 6 = g 8 N 4 (N 1) σ S 6 /Z 6 [ A (0) 6 (σ) + ( ) ] N A0 6 (σ) A 0 6 (σ ) + A 0 6 (σ ) + A 0 6 (σ ) one-loop five parton matrix elements, gluon scattering ( ) R M 0 5M 1 [ 5 = g 8 N 4 (N 1) R A 0 5 (σ)a1 5(σ) + 1 ] N A 0 5 (σ)a1 5,1(σ ) σ S 5 /Z 5 σ, σ, σ have no common neighbouring partons with σ no single, double or triple collinear singularities at subleading colour subtract divergences associated with single and double soft gluons only which at subleading colour map completely to the tree-level single soft gluon current X 0 tree level three parton antenna