Higher-order QCD corrections to the Higgs-boson qt distribution Jun Gao Argonne National Laboratory June 5, 05 In collaboration with Radja Boughezal and Frank Petriello Radcor/Loopfest, UCLA 05
qq Motivation Transverse momentum distribution of the Higgs boson in gluon fusion: probe of standard 5 model EW dynamics, test of QCD factorization sum gg gq qq 500 700 00 300 500 700 00 300 500 700 00 300 500 700 σ i,j dσ i,j dp T [/GeV] 0 3 0 4 0 5 0 6 Operator O O O O O 3 O O 5 SM p T [GeV] s transverse momentum distributions for scalar coupling operators. The normalization factors gq qq sum (pb/gev) dσ/dp T. differentiate different origins of ggh effective couplings probe production mechanism through QCD color structures ij are ideal for testing QCD factorization in gluon fusion process gg H + X at LHC, m H = 5 GeV Grazzini et al, MRST00, σ = 39.4 pb ResBos, MRST00, step, σ = 36. pb ResBos, MRST00, smooth, σ = 36. pb Kulesza et al, CTEQ5M, σ = 35.0 pb Berger et al, CTEQ5M, σ = 37.0 pb MC@NLO, MRST00, σ = 3.4 pb PYTHIA 6.5, CTEQ5M, σ = 7.8 pb HERWIG 6.3, CTEQ5M, σ = 6.4 pb O 5 SM Operator O O O O O 3 O O 5 SM Harlander, Neumann, 308.5 500 700 00 300 500 700 00 300 500 700 00 300 500 700 00 300 Pseudoscalar Operator O ~ O ~ O ~ O ~ O ~ 3 O ~ O ~ 5 top loop p T [GeV] 0 0 0 40 60 80 (GeV) 00 Balazs et al., hep-ph/040305 g operators. FIG. : The Normalized normalization Higgsfactors ij are p T [GeV] transverse momentum distributions for scalar coupling operators. The normalizat given in table I. 500 700 00 300 500 700 00 300 500 700 00 300 500 700 p T
Experimental precision at LHC run is limited by statistics but could be largely improved at run or high luminosity LHC; current uncertainties on theoretical predictions are not small 4 [pb/gev] dσ / dp [pb/gev] 0 H T H T dσ / dp 0 HRES + HRES XH + XH XH = VBF XH + = VH VBF + + tth VH+ + bbh tth + bbh data, tot. data, unc. tot. unc. syst. unc. syst. unc. - - = 8 TeV, 0.3 fb s = 8 TeV, s 0.3 fb ATLAS ATLAS pp H pp H [/GeV] [/GeV] H /σ dσ / dp T H 0 T /σ dσ / dp 0 HRES + HRES XH + XH NNLOPS+PY8 NNLOPS+PY8 + XH + XH MG5_aMC@NLO+PY8 MG5_aMC@NLO+PY8 + XH + XH SHERPA SHERPA.. + XH.. + XH XH = VBF XH + = VH VBF + tth + VH+ + bbh tth + bbh data, tot. data, unc. tot. unc. syst. unc. syst. unc. - - = 8 TeV, 0.3 fb s = 8 TeV, s 0.3 fb ATLAS ATLAS pp H pp H 3 0 3 0 0 0 [pb] Ratio to HRes H dσ / d y 0 00 0 40 40 60 60 80 00 80 00 0 0 40 40 60 60 80 80 00 00 4 4 Ratio to HRes 0 0 0 00 0 40 40 60 60 80 80 00 00 0 0 40 40 60 60 80 80 00 00 40 [pb] 35 H dσ / d y 30 5 40 ATLAS ATLAS pp H pp H 35 HRES + XH XH XH ATLAS, syst. arxiv: unc. syst. unc. 504.05833 data, tot. data, unc. tot. unc. - s = 8 TeV, s = 8 0.3 TeV, fb0.3 fb 30 5 - HRES + XH ph T Ratio to HRes [GeV] ph [GeV] = VBF + = VH VBF + + tth VH+ + bbh tth + bbh T H /σ dσ / d y 3 0 0 0 0 40 40 60 60 80 00 80 00 0 0 40 40 60 60 80 80 00 00 Ratio to HRes 0 0 0 0 0 0 40 40 60 60 80 00 80 00 0 0 40 40 60 60 80 80 00 00.. HRES + HRES XH + XH H /σ dσ / d y ATLAS ATLAS ph T [GeV] ph [GeV] pp H pp H NNLOPS+PY8 NNLOPS+PY8 + XH + XH data, tot. data, unc. tot. unc. syst. unc. syst. unc. MG5_aMC@NLO+PY8 + XH + XH - - s = 8 TeV, s = 80.3 TeV, fb0.3 fb SHERPA SHERPA.. + XH.. + XH XH = VBF XH + = VH VBF + tth + VH+ + bbh tth + bbh T
unpolar distribu Theoretical predictions on SM Higgs boson qt distribution in different kinematic region:, non-perturbative region;, small-qt; 3, intermediate qt; 4, large qt; 5, Large-qT NP region small-qt Intermediate large-qt Large-qT ~GeV ~0 GeV 0 40 60 80 00 POWHEG+Pythia 8 Higgs qt distribution 4 ~m H ~m top
Resummation of small-qt logarithms (Log(qT/Q)) to all-order in QCD in impact parameter space, first developed by Collins, Soper and Sterman (CSS, 985) h h p p f f a b a b C C Catani et al., hep-ph/000884 c c. H S F. Recent CSS application on Higgs boson production: Florian et al., 0, NNLL+NLO [HqT/HRes] Wang et al., 0, NNLL+NLO [Resbos] [ dσf ] dq dq res. = a,b 0 dx dx db b J 0(bq ) f a/h (x,b 0/b ) f b/h (x,b 0/b ) 0 0 W F ab (x x s; Q, b). (3) Wab(s; F Q, b) = dz dz C ca (α S (b 0/b ),z ) C cb (α S (b 0/b ),z ) δ(q z z s) c 0 0 σc c F (Q, α S (Q )) S c (Q, b). (3) 5
Higgs qt resummation based on Soft-Collinear effective theory [note the different convention in counting of matching order] Y. Gao et al., 005, Idilbi et al, 005, Mantry, Petriello, 009 NNLL+NLO: Becher, Neubert, 00, Becher et al., 03 NNLL+NNLO: Chiu et al., 0, Neill et al., 05 NNLL: Echevarria et al., 05 Fixed-order calculations on Higgs qt spectrum, NLO/NNLO QCD in heavy top-quark limit, NLO QCD with approx. top mass dependence NLO: Florian et al., 999, Ravindran et al., 00, Glosser, Schmidt, 00 NNLO: Boughezal et al., 03, Chen et al., 04, Boughezal et al., 05 LO: Ellis et al., 988, Baur, Glover, 990 NLO (approx.): Dawson et al., 04 6
Framework small-qt factorization and resummation in SCET [following the scheme by Becher, Neubert, 00] dσ = σ 0 (µ) Ct (m t,µ) CS ( m H,µ) m H τs dy d q (π) d x e iq x B B µν B S c (ξ,x,µ) B cµν (ξ,x,µ) S(x,µ) Collinear anomaly and refactorization, qt<<m H ( ) dσ = σ 0 (µ) Ct (m t,µ) CS ( m H,µ) m H τs dy d q (π) d x e iq x ( x T m H B B S B b 0 ) Fgg (x T,µ) B µν g (ξ,x,µ) Bg ρσ (ξ,x,µ) 7
small-qt factorization and resummation in SCET [following the scheme by Becher, Neubert, 00] Matching on to standard collinear PDFs, qt>>λ QCD d σ dqt dy = σ 0(µ) Ct (m t,µ) CS ( m H,µ) dz ξ z i,j=g,q, q dz ξ z C gg ij (z,z,qt,m H,µ) φ i/p(ξ /z,µ) φ j/p (ξ /z,µ) C gg ij (z,z,qt,m H,µ)= 4π L = ln x T µ b 0 n=, d x e iq x ( x T m H b 0 ) Fgg (L,a s ) I (n) g i (z,l,a s ) I (n) g j (z,l,a s ). with qt>>λ QCD all colored objects can be calculated perturbatively; scale dependence are organized by RG equation with anomalous dimensions 8
Resummation of Sudakov logarithms is accomplished by evolving hard functions from canonical scales to a proper scale at which the expansion of remaining kernel C can be perturbatively controlled RG evolution of the hard function C t (m t,µ f )=β( α s (µ f )) /α s (µ f ) β ( α s (µ t ) ) /α s(µ t ) C t(m t,µ t ) [ C S ( m H iϵ,µ f)=exp S(µ h,µ f) a Γ (µ h,µ f) ln m H iϵ µ h proper scale (mu=<x T > - ) ~q*+qt, q*~8 GeV ] a γ S(µ h,µ f) C S ( m H iϵ,µ h) ( q = m H exp π ) Γ A 0 α s (q ) Improved expansion in momentum space, Lperp~/as / C gg ij (z,z,q T,m H,µ)= 0 n=, g A (η,l,a s )= [ ηl ]ϵ / dx T x T J 0 (x T q T )exp [ g A (η,l,a s ) ] Ī (n) g i (z,l,a s ) Ī(n) g j (z,l,a s ), [ a s ( Γ A 0 + ηβ 0 ) L 9 ] ϵ 0 +
power counting and perturbative ingredients needed for the N3LL resummation QCD beta function to 4-loop [Ritbergen et al., 997], Cusp anomalous dimension to 4-loop [3-loop, Moch et al., 004], quark and gluon anomalous dimension to 3- loop [Becher, Neubert, 009], Ct and Cs to -loop [Ahrens et al, 009] at matching scales kernel of beam functions [scale independent terms] to -loop [Gehrmann et al, 0, 04], DGLAP splitting kernel to -loop [3-loop, Moch et al., 004], collinear anomaly [scale independent terms] to 3-loop [-loop, Becher, Neubert, 0] two unknown numbers Pade approx. for cusp at 4-loop, effects are small as in many other studies will vary 3-loop collinear anomaly (d3) in a reasonable range 0
Fixed-order matching (non-singular/power corrections) is important even in peak region for Higgs case, NNLO/N3LO [Glosser, Schmidt, 00, Boughezal et al., 05] [ dˆσ (fin.) Fab dq T ] f.o. = [ dˆσf ] ab dq T Long-distance effects are found to be small in peak region of the Higgs qt [Becher et al., 03, Echevarria et al., 05, Florian et al., 0] f.o. [ dˆσ (res.) Fab dq T ] f.o. ~ 0% in peak region We adopt the order matching, NLL+NLO, NNLL+NNLO, N3LL+N3LO [also require 3-loop [ DGLAP kernel]; ] will vary the canonical scale choices/hard functions in non-singular piece to further investigate the perturbative convergence 0.5 B g (n) (ξ,x T,µ)=f hadr (x T Λ NP ) B g (n)pert (ξ,x T,µ) f gauss hadr (x T Λ NP )=exp ( ) Λ NP x T f pole hadr (x T Λ NP )= +Λ NP x T dσ dq T pb GeV 0.3 0. NP 0 NP GeV NP GeV NP GeV NP GeV NP.0 GeV 0 5 0 5 0 q T GeV
Numerical results [preliminary] resummed component and scale variation [two matching scales +resummation scale, factorization scale], qt distribution no scale separation with scale separation. NLL. NLL.0 NNLL.0 NNLL d3 000 N3LL d3 000 N3LL 0 0 0 30 40 50 60 0 0 0 30 40 50 60 resummed cross sections can be divided into two separate scale invariant parts to have a better gauge on missing higher-order contributions [see Becher et al., 03]
resummed component and scale variation [two matching scales +resummation scale, factorization scale], intercept no scale separation with scale separation 0.5 NLL 0.5 NLL d Σ dp T pb GeV 0.0 5 d3 000 NNLL N3LL d Σ dp T pb GeV 0.0 5 d3 000 NNLL N3LL 0 0 5 0 5 0 0 0 5 0 5 0 resummed cross sections can be divided into two separate scale invariant parts to have a better gauge on missing higher-order contributions [see Becher et al., 03] 3
resummed component and variation with d3 [variation with cusp at 4-loop is negligible] qt distribution intercept. NNLL 0.5 NNLL.0 N3LL d3 3000 N3LL d3 000 d Σ dp T pb GeV 0.0 5 N3LL d3 3000 N3LL d3 000 0 0 0 30 40 50 60 0 0 5 0 5 0 F gg (L, α s )= d g n(l ) d q ( ) = dg 808 = C A C F C A 7 8ζ 3 4 7 T F n f ( αs ) n being the scale independent 4π part in collinear anomaly exponent, d=0, d=-95.8, d3=? 4
non-singular component in conventional scheme, canonical mu=mh, no RG improvement [restrict to NNLO here]... NLO ns NNLO d3 000 NLL NLO sin.0 NNLO ns.0 NNLO sin.0 NNLL NNLO sin dσ dp T pb GeV NNLL dσ dp T pb GeV NNLO ns dσ dp T pb GeV N3LL N3LO sin NNLO ( ) 0 5 0 5 0 0 0 30 40 50 60 0 0 30 40 50 60 q T dσ (n) sin dq T α n s ln n ( q T m H ) q T dσ (n) ns dq T α n s q T m H ln n ( q T m ) H from left to right: vanishing of non-singular piece when qt->0; non-singular piece dominate in FO when qt~60 GeV; turn-off of resummation at high qt 5
non-singular component in conventional/modified scheme, canonical mu=mh, no/with RG improvement [restrict to NNLO here] naive(ns) alternative(ns) naive, mu m H NLO NNLO alternative, mu m H NLO NNLO 0 0 0 30 40 50 60 0 0 0 30 40 50 60 non-singular contributions are sensitive to the matching scheme used including choice of the canonical scale; uncertainties reduced from NLO to NNLO; N3LO must needed to match the precision in N3LL resummation 6
non-singular component in conventional/modified scheme, canonical mu=mh/q*+qt, no RG improvement [restrict to NNLO here] mu=mh(ns) mu=q*+qt(ns) naive, mu m H NLO NNLO naive, mu q q T NLO NNLO 0 0 0 30 40 50 60 0 0 0 30 40 50 60 non-singular contributions are sensitive to the matching scheme used including choice of the canonical scale; uncertainties reduced from NLO to NNLO; N3LO must needed to match the precision in N3LL resummation 7
non-singular component in conventional/modified scheme, mu=mh/ q*+qt, no/with RG improvement [restrict to NNLO here] NLO(ns) NNLO(ns) NLO m H nav m H alt q q T nav q q T alt NNLO m H nav m H alt q q T nav q q T alt 0 0 0 30 40 50 60 0 0 0 30 40 50 60 non-singular contributions are sensitive to the matching scheme used including choice of the canonical scale; uncertainties reduced from NLO to NNLO; N3LO must needed to match the precision in N3LL resummation 8
uncertainties in the combined spectrum, for NNLL+NNLO and N3LL +N3LO [preliminary].0.8 Ratio NNLL NNLO NNLL NNLO.0.8 only shown for 0-60 GeV, and mat. scheme yet Ratio d3 000 N3LL N3LO.6.4..0 NNLL NNLO3 NNLL NNLO4.6.4..0 0 0 0 30 40 50 60 0 0 30 40 50 60 with N3LL+N3LO accuracy we expect to bring down the perturbative uncertainties to within 0% from peak region to large qt (not include uncertainties from unknown d3) 9
Summary We report progress in a N3LL+N3LO calculation of qt spectrum of the SM Higgs boson in gluon fusion production based on SCET N3LL resummation reduce the scale uncertainties significantly and the resummation show good convergence Non-singular component (FO matching) is important for predictions even around the peak region; N3LO contributions are needed With all available perturbative inputs a reduction of perturbative uncertainties to within 0% are expected for the spectrum from smallqt to ~mh 0
non-singular component in conventional/modified scheme, canonical mu=mh/q*+qt, with RG improvement [restrict to NNLO here] mu=mh(ns) mu=q*+qt(ns) alternative, mu m H NLO NNLO alternative, mu q q T NLO NNLO 0 0 0 30 40 50 60 0 0 0 30 40 50 60 non-singular contributions are sensitive to the matching scheme used including choice of the canonical scale; uncertainties reduced from NLO to NNLO; N3LO must needed to match the precision in N3LL resummation