NNLL threshold resummation for the total top-pair production cross section

Transcription

1 NNLL threshold resummation for the total top-pair production cross section Christian Schwinn Univ. Freiburg (Based on M.Beneke, P.Falgari, S. Klein, CS, arxiv: [hep-ph] )

2 Introduction 1 Total cross section measurements at Tevatron and LHC σ Tevatron t t = σ TeV t t = σ t t/σ tt 7% NLO+NLL QCD: (D0) (CDF) (CMS) (ATLAS) (Nason, Dawson Ellis 88, Bonciani et al. 98) [pb] tt σ NLO QCD (pp) Approx. NNLO (pp) NLO QCD (pp) Approx. NNLO (pp) CDF D0 Single-lepton 179 ± 12 pb Dilepton ATLAS Preliminary pb All-hadronic 167 ± 81 pb Combined 177 CMS Preliminary, s=7 TeV pb CMS 2011 combination 166 ± 2 ± 11 ± 8 TOP (L= /fb) (val. ± stat. ± syst. ± lumi.) CMS e/ µ+jets+btag 164 ± 3 ± 12 ± 7 TOP (L= /fb) (val. ± stat. ± syst. ± lumi.) CMS dilepton (ee, µµ,eµ) 170 ± 4 ± 16 ± 8 TOP (L=1.1/fb) (val. ± stat. ± syst. ± lum) s [TeV] σ Tevatron t t = σ TeV t t = Need higher-order predictions! CMS all-hadronic 136 ± 20 ± 40 ± 8 TOP (L=1.1/fb) (val. ± stat. ± syst. ± lumi.) CMS dilepton ( µτ) 149 ± 24 ± 26 ± 9 TOP (L=1.1/fb) (val. ± stat. ± syst. ± lumi.) Approx. NNLO QCD, Aliev et al., Comput.Phys.Commun. 182 (2011) 1034 Approx. NNLO QCD, Kidonakis, Phys.Rev.D 82 (2010) Approx. NNLO QCD, Ahrens et al., JHEP 1009 (2010) 097 NLO QCD σ(tt) (pb)

3 Introduction 2 NNLO: in progress, several ingredients known ( Alex Mitov s talk) (Czakon et al.; Bonciani et al ; Körner et al , Anastasiou/Mert-Aybert 08;... ) Progress for soft gluon resummation: 1-loop hard functions for singlet/octet (Czakon/Mitov 08) massive 2-loop soft anomalous dimension (Becher/Neubert; Kidonakis; Mitov/Sterman/Sung; Beneke/Falgari/CS; Czakon Mitov/Sterman; Ferroglia et al 09) NNLO app. (Moch/Uwer(Langenfeld) 08/09, Beneke et al.; Ahrens et al. 09, Kidonakis 10) NNLL resummation (Ahrens et al. 10/11; Beneke et al. 11, Cacciari et al. 11)

4 Introduction 2 NNLO: in progress, several ingredients known ( Alex Mitov s talk) (Czakon et al.; Bonciani et al ; Körner et al , Anastasiou/Mert-Aybert 08;... ) Progress for soft gluon resummation: 1-loop hard functions for singlet/octet (Czakon/Mitov 08) massive 2-loop soft anomalous dimension (Becher/Neubert; Kidonakis; Mitov/Sterman/Sung; Beneke/Falgari/CS; Czakon Mitov/Sterman; Ferroglia et al 09) NNLO app. (Moch/Uwer(Langenfeld) 08/09, Beneke et al.; Ahrens et al. 09, Kidonakis 10) NNLL resummation (Ahrens et al. 10/11; Beneke et al. 11, Cacciari et al. 11) Further issues: EW corrections 2% (Bernreuther et al.; Kühn et al. 05/06) non-resonant corrections (Denner et al. Bevilacqua et al. 10) PDF uncertainties

5 Threshold resummation 3 Enhanced QCD corrections in threshold limit β = 1 4m 2 t /ŝ 0 Soft corrections: (Resummation in Mellin space: Sterman 87; Catani, Trentadue 89, Kidonakis,Sterman 97, Bonciani et al. 98,... ) α s log 2 (8β 2 ) α s log(8β 2 ) Coulomb gluon corrections (Fadin, Khoze 87; Peskin, Strassler 90, NRQCD,... ) α s 1 β Resummed cross section [ ˆσ pp σ (0) exp k=0 ( αs ln β g LL (α s ln β) + g NLL (α s ln β) + α s g NNLL (α s ln β) +... ) k {1 (LL,NLL); α s, β (NNLL);...} : β ]

6 Threshold resummation 4 ˆσ pp HH ŝ 4M 2 = R,i H i W R i J R (Beneke, Falgari, CS 10) Factorization into soft, hard and Coulomb functions Hard, soft and Coulomb functions: H i =, W i R =, J R = Soft radiation sees only total colour charge R of heavy particles (Singlet, octet,...)

7 Threshold resummation 4 ˆσ pp HH ŝ 4M 2 = H i W R i J R (Beneke, Falgari, CS 10) Factorization into soft, hard and Coulomb functions R,i can include Coulomb resummation to all orders ((N)LO Coulomb-Green function: Fadin/Khoze 87; Beneke/Signer/Smirnov 99,... ) RGEs for hard and soft function (Becher, Neubert; Ferroglia et al.; Beneke/Falgari/CS; Czakon/Mitov/Sterman 09) Momentum space solution (Becher/Neubert 06) H(M, µ h ) f 1 (µ f )f 2 (µ f )H(M, µ f )J Rα W Rα (ω, µ f ) W R α ii (ω,µ s) µ h 2m t, Choice of µ s : RGE approach: fixed µ s (Becher, Neubert, Xu 07) Running scale µ s m t β 2 (Beneke, Falgari, Klein, CS 11) (frozen to µ s m t β 2 cut for β < β cut = 0.35 (Tevatron), 0.54 (LHC)) µ h µ f µ s

8 Total top-pair production cross-section 5 σ t t[pb] Tevatron LHC (7TeV) LHC (8TeV) LHC (14TeV) NLO NLL NNLO app NNLL (m t = GeV, (N)NLO MSTW08PDFs, 90% CL PDF+α s uncertainty) Different uncertainties for NNLL (added in quadrature above): Scale variation (µ f, µ h, µ C ): µ σ NNLL (TeV) = , µσ NNLL (LHC7) = Uncertainty in resummation procedure: (vary β cut by 20%, envelope of NNLO app, N3LO app, NNLL, ambiguity ŝ 2m t m t β 2 ) Res σ NNLL (TeV) = , Res σ +3.9 NNLL (LHC7) = 5.6 Const σ NNLL (TeV) = ±0.10, Const σ NNLL (LHC7) = ±4.7 Estimate of missing constant at O(α 2 s)

9 Total top-pair production cross-section 5 σ t t[pb] Tevatron LHC (7TeV) LHC (8TeV) LHC (14TeV) NLO NLL NNLO app NNLL (m t = GeV, (N)NLO MSTW08PDFs, 90% CL PDF+α s uncertainty) NNLL results include Coulomb effects beyond NNLO bound-state corrections below threshold (take Γ t 0 approximately into account) small effect: σ BS = 0.014pb (Tevatron), 0.67pb (LHC7), 3.1pb (LHC14) σ C = 0.052pb (Tevatron), 0.13pb (LHC7), 0.3pb (LHC14)

10 Comparison to other NNLO app /NNLL predictions 6 NNLL soft resummation (Cacciari/Czakon/Mangano/Mitov/Nason 11) (β-expansion, Mellin space, no Coulomb resummation, different α 2 s constant) Resummed differential cross sections Pair-invariant mass : dˆσ(t t) dm t t, (Ahrens/Ferroglia/Neubert/Pecjak/Yang 10) One-particle inclusive: dˆσ(t+x) dp T (Kidonakis 11; Ahrens et al. 11) (include some higher-order terms in β) Σpb 9 8 BFKS (m t GeV Kidonakis m t 173 GeV Ahrens et al. m t GeV Cacciari et al. m t GeV Tevatron Σpb LHC7 TeV BFKS (m t GeV Kidonakis (m t 173 GeV Ahrens et al. (m t GeV Cacciari et al. (m t GeV NLO NNLO NNLL CDF/D0 100 NLO NNLO NNLL Atlas/CMS

11 PDF uncertainties 7 Dependence on PDF and α s (Watt 12 using HATHOR (Aliev et al. 10)) (pb) tt σ NNLO (approx.) tt cross sections at the LHC ( CMS, L = fb -1 ATLAS, L = 0.7 fb s = 7 TeV) pole m t = GeV Vertical error bars Inner: PDF only Outer: PDF+α S 68% C.L. PDF MSTW08 NNPDF2.1 HERAPDF1.0 HERAPDF1.5 ABKM09 JR09 G. Watt (September 2011) Fix α s (M Z ) = α S (M ) Z (68% CL PDF/α s error + theory error) (Numbers from J. Piclum) Σ NNLL pb MSTW08 CT10 NN2.1 ABM11 JR09 ATLAS CMS m t GeV Α s M Z : NLO PDF :Α s M Z

12 Prospects of m t -extraction 8 Fit m t -dependence of NNLL-cross-section: σ th t t (mpole t ) = m pole t! 4 c 0 + c 1 (m pole t 172.5) + c 2 (m pole t 172.5) 2 + c 3 (m pole t 172.5) 3 pb, Use ATLAS-CONF : c 0 = 166.5, c 1 = 1.15, c 2 = , c 3 = σ t t = ± 11.8pb Dependence on m MC t : σ exp t t (m MC t ) = ` m MC t pb Σ tt m t pb Σ th tt m t exp Σ tt mt maximize joined likelihood Z f(m t ) = f th (σ m t ) fexp(σ m t )dσ, m tgev with normalized Gaussians f th/exp m pole t = ( }{{} m MC t m pole t )GeV CMS result from σ t t = ± 18.4 pb: Approx. NNLO MSTW08NNLO m pole t / GeV m MS t / GeV Langenfeld et al. [7] Kidonakis [8] Ahrens et al. [9]

13 Summary 9 Threshold corrections log n β, 1 β n Factorization of soft and Coulomb corrections log β resummation from momentum space solution to RGEs combined Soft and Coulomb resummation possible NNLL resummation for t t small effects beyond NNLO (2 % Tevatron, 1 % LHC) theory uncertainty +4.2/ 6.5% Tevatron, ±4.5% LHC including resummation ambiguities (PDF+α s uncertainty larger) Outlook Mass determination from σ t t: m t ±5 GeV appears viable public program (in preparation w. J. Piclum) resummation for squarks/gluinos (NLL: Falgari/CS/Wever 12)

14 Bonus slides 10

15 Threshold behaviour 11 Top-pair production: two LO subprocesses: { q q t t : go t t : { 90% Tevatron 20 10% LHC % Tevatron 80 90% LHC7 14 Behaviour at production threshold ŝ 4m 2 t: (β = q q channel: colour octet, spin triplet ˆσ (8) q q = πβ 9m 2 t ˆσ (1) gg = 5πβ» 1 + α s 4π 1 4m2 t ŝ ) «2π 2 1 2N c β + 8C F log 2 8β 2 (32C F + 4N C ) log 8β gg channel: colour singlet/octet, spin singlet ˆσ (8) gg = 192m 2 t πβ 96m 2 t» h 1 + α s 4π 1 + α s 4π 2C F π 2 1β + 8N C log 2 8β 2 32N C log 8β 2 i π 2 2N C 1 β + 8N C log 2 8β 2 (32N C + 4N C ) log 8β 2 «+... Universal behaviour depending on initial/final colour states

16 Accuracy of threshold approximation 12 NLO sing : only log β, 1/β-terms NLO app : also constant terms (hard/soft function) Ambiguity: E = ŝ 2m t m t β 2 dσ qq pb dβ NLO NLO app Β NLO app E NLO sing Β NLO sing E Β dσ gg dβ pb NLO NLO app Β NLO app E NLO sing Β NLO sing E Β dσ dβ = 8βm2 q s(1 β 2 ) 2 L(β, µ f )ˆσ ( Tevatron, L: parton luminosity (MSTW08)) large ambiguities, but exact result covered: (µ f = µ r = m t = GeV) σ NLO,app (Tev) = pb σ NLO,app (LHC7) = pb σ NLO (Tev) = 6.50pb σ NLO (LHC7) = 150 pb

17 Coulomb Effects 13 Coulomb resummation in Coulomb Green function: (Fadin, Khoze 87; Peskin, Strassler 90) ˆσ t t(ŝ) R=1,8 σ 0,R (ŝ)im G R t t C (0, 0; E + iγ) Singlet channel, neglecting decay width: Im G 1 C (0, 0; E) = m 2 t πc F α s 4π ( e πc F α s mt E 1 ) 1 E > 0 n=1 δ(e E n)r n E < 0 4 Bound-state poles at E = ŝ 2m t m t β 2. E n = α2 sc 2 F m t 4n 2 smeared out by Γ t 0. dσ / dm [pb/gev] LHC 1 s = 14 TeV M [GeV] (Hagiwara et al. 08, Kiyo et al. 08)

18 Threshold expansion at O(α 2 s) 14 All threshold enhanced O(α 2 s) terms (Beneke, Czakon, Falgari, Mitov, CS 09 Implemented in HATHOR, Aliev et.al. 10) Pure soft corrections: (also Moch/Uwer+Langenfeld (08/09)) σ (2) s α 2 s(c (2) LL ln4 β + c (2) NLL ln3 β + c (2) NNLL,2 ln2 β + c (2) ln β NNLL,1 ) }{{} 2-loop γ H,s Potential corrections: 2nd Coulomb, NLO potentials σ p (2) α 2 s ( c C β 2 β (c(2) + C,0 c(2) C,1 log β) + c(2) ln β n-c ) }{{} spin-dependent (using Beneke, Signer, Smirnov 99, Czarnecki/Melnikov 97/01) mixed Coulomb/soft, hard corrections: σ (2) p sh α s β α s(c (1) ln LL β2 + c (1) NLLln β + c + H(1) ) }{{} process dependent

19 Comparison to other NNLO app /NNLL predictions 15 σ t t[pb] Tevatron LHC (7TeV) NNLO app NNLO app +NNLL (Beneke, Klein, Falgari, CS 11) NLO+NNLL N (Cacciari et al. 11) Ambiguities in momentum space results: fixed soft scale: σ NNLL = 7.08 pb (TeV); pb (LHC 7TeV) in N-space results (Cacciari et al. 11) set O(α 2 s) constant in β expansion to zero: σ N NNLL = 6.97 pb (TeV); don t include 1/N suppressed terms: σ N NNLL = 6.84 pb (TeV); pb (LHC 7TeV) pb (LHC 7TeV)

20 Comparison to other NNLO app /NNLL predictions 16 Pair invariant mass cross sections (Kidonakis,Sterman 97, Ahrens et al. 10) [ ] ( ) dˆσ(t t) log n (1 z) 1 z, PIM SCET : log z ds 4 dm t t 1 z +, z = M2 t t ŝ One particle inclusive cross sections: (Laenen et al. 98, Ahrens et al. 11) [ ] ( ) dˆσ(t + X) log n (s 4 /m 2 ) ; s 4 = p 2 X s m2 t, 1PI SCET : log s 4 / m 2 + s αs corrections (pb) s = 7 TeV gg-channel µ f = m t 50 0 Exact 1PI SCET 1PI 1PI 0 SCET -25 PIM 1PI SCET PIM PIM SCET PIM β β corrections (pb) α 2 s s = 7 TeV gg-channel µ f = m t

21 Approximate N 3 LO results 17 Expand NNLL to O(α 3 s), e.g. σ (3) qq,nnll = log6 β log 5 β log 4 β log 3 β + 1 β ( log β log 2 β log 3 β log 4 β ) log2 β log β 145 { + log β 1,2, 1/β, C (3)} +scale dep. β 2 }{{} not known exactly N 3 LO A : keep all terms, including k-dependence and constants N 3 LO B : only keep terms known exactly dσ qq pb dβ NNLOapp N3LOAkh 2,ks 1 N3LOB dσ gg pb dβ NNLOapp N3LOAkh 2,ks 1 N3LOB Β Β

22 Resummation: ambiguities 18 Soft scale choice in momentum-space resummation RGE approach: fixed µ s, vary by (Becher, Neubert, Xu 07) Running scale with cutoff (Beneke, Falgari, Klein, CS 11) m t βcut 2 µ s, forβ < β cut m t β 2, forβ > β cut O(α 2 s) constant in threshold expansion ) σ (2) tt (7TeV) = [1.70 Estimate ( C (2) qq ( C (2) gg, ) ( (2) )] C gg,1 pb 1000 C 2 C 2 1: σ (2) tt ± 5 pb Kinematic ambiguity log β log(( ŝ 2m t )/m t ) : σ tt 1 pb

23 Resummation: ambiguities 19 Running scale: Introduce β cut allow for different implementations β < β cut : NNLL (µ s = k s m t βcut 2 ) with/without constant at O(α2 s) β > β cut : NNLL (µ s = k s m t β 2 ); NNLO approx ; NNNL 3 (A/B) ΣA,B,Βcu t Βcut Choose β cut so that not too sensitive to ambiguities for β 1 breakdown of perturbation theory for β 0 (E.g. LHC7: µ s = 2m t β 2, β cut = 0.54 µ s > 100 GeV)