Squark and top quark pair production at the LHC and Tevatron

Transcription

1 Squark and top quark pair production at the LHC and Tevatron M. Beneke (RWTH Aachen) 10th Hellenic School and Workshop on Elementary Particle Physics and Gravity Corfu, Greece, August 31, 2010 Outline Introduction: (s)particle pair production and resummation SCET and PNRQCD factorization Soft and Coulomb gluon resummation Top anti-top production Squark anti-squark production MB, P. Falgari, C. Schwinn, [hep-ph] and [hep-ph] (factorization, colour, anomalous dimension, squark production); MB, M. Czakon, P. Falgari, A. Mitov, C. Schwinn, arxiv: [hep-ph] (α2s log terms for tt ); MB, P. Falgari, S. Klein, C. Schwinn, proceedings of Loops & Legs 2010 and work in progress (top quark production) M. Beneke (RWTH Aachen) Corfu, 31 August / 23

2 Sparticle pair production Generic scenario (here supersymmetry): Production of coloured sparticles in pp collisions and cascade decay to the LSP. Sparticle production cross sections at NLO (Beenakker et al., 1997) M. Beneke (RWTH Aachen) Corfu, 31 August / 23

3 Radiative corrections Precision calculation of quantum fluctuations in perturbation theory; Feynman rules from L and visualization through diagrams; Loops... [Assume all couplings are small.] σ(g) = σ 0 + g 2 σ But perturbation theory breaks down when the new particles become too heavy, since this forces β to be small. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

4 Pair production and resummation σ = i,j ij (q q, qg, gg, e + e ) HH + X s 4 m 2 H dŝ L ij (ŝ/s, µ f ) ˆσ ij (ŝ, m H, m H; µ f ) β = 1 4 m 2 H /ŝ Non-perturbative despite small couplings: Soft gluon (photon) resummation, Sudakov logarithms: g 2 ln 2 β 1. Strong Coulomb force: g 2 /β 1 Sizeable decay widths of H, H : Γ H /m H g 2 Physical final states, Dyson resummation, non-resonant backgrounds Improve cross section predictions by (perturbative) resummations in the frameworks of (P)NRQCD, SCET, unstable particle EFT. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

5 Breakdown of perturbation theory Inhibited (soft) radiation Coulomb force Z Mβ A A 0 g 2 2 Z dω β 2 dθ Mλ ω λ θ A 0 g 2 ln 2 β Virtual correction has no kinematic restriction. Large effect when 2M ŝ, i.e. E = ŝ 2M Mβ 2 M. At LHC for M few hundred GeV. Gluon radiation. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

6 Breakdown of perturbation theory Inhibited (soft) radiation Coulomb force Z Mβ A A 0 g 2 2 Z dω β 2 dθ Mλ ω λ θ A 0 g 2 ln 2 β Virtual correction has no kinematic restriction. Large effect when 2M ŝ, i.e. E = ŝ 2M Mβ 2 M. At LHC for M few hundred GeV. Gluon radiation. A A 0 g 2 Z d 4 k M M 1 (2π) 4 ME + ( p + k) 2 ME + ( p + k) 2 k 2 A 0 g2 β For small E = Mβ 2, p Mβ the dominant quantum correction arises from low energy fluctuations with k 0 Mβ 2, k Mβ corresponding to the long-range QCD Coulomb interaction. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

7 Breakdown of perturbation theory Inhibited (soft) radiation Coulomb force Z Mβ A A 0 g 2 2 Z dω β 2 dθ Mλ ω λ θ A 0 g 2 ln 2 β Virtual correction has no kinematic restriction. Large effect when 2M ŝ, i.e. E = ŝ 2M Mβ 2 M. At LHC for M few hundred GeV. Gluon radiation. A A 0 g 2 Z d 4 k M M 1 (2π) 4 ME + ( p + k) 2 ME + ( p + k) 2 k 2 A 0 g2 β For small E = Mβ 2, p Mβ the dominant quantum correction arises from low energy fluctuations with k 0 Mβ 2, k Mβ corresponding to the long-range QCD Coulomb interaction. Perturbation theory breaks down due to the emergence of small scales Mβ, Mβ 2 M, ŝ. Sum the series of enhanced quantum fluctuations to all orders: j ff ( σ = σ 0 "1 + g 2 1 β, ln2,1 β + g 4 1 β 2, ln ) # 2,1 β, ln 4,3,2,1 β +... β M. Beneke (RWTH Aachen) Corfu, 31 August / 23

8 All-order systematics of soft-gluon and Coulomb resummation Expansion of the partonic cross section ˆσ(β) = ˆσ (0) X «αs k h i exp ln β g 0 (α s ln β) + g 1 (α s ln β) + α sg 2 (α s ln β) +... β k=0 {z } {z } {z } (LL) (NLL) (NNLL) n 1 (LL,NLL); α s, β (NNLL); α 2 o s, αsβ, β2 (NNNLL);... Counting: α s/β, α s ln β 1 [Note: Non-relativistic logs can also be cast into the above form.] In fixed order [Note: NNLL can be α s/β[coulomb] β[sub-leading soft] α s ln 2 β beyond the standard soft gluon approximation!] j ff ( 1 LL α s β, ln2 β ; α 2 1 s β 2, ln ) 2 β β, ln4 β ;..., j ff NLL α s ln β; α 2 ln β s β, ln3 β ;..., NNLL α s n1, β ln 2,1 o j ff β ; α 2 1 s β, ln2,1 β, β ln 4,3 β ;..., M. Beneke (RWTH Aachen) Corfu, 31 August / 23

9 Recent work Diagrammatic resummation methods developed since late 1980s (DY: Sterman, 1987; Catani, Trentadue, 1989; Two particles/jets: Kidonakis, Sterman, 1997; Bonciani et al, 1998) Resurge of recent interest due to advent of LHC and effective field theory resummation method. Towards NNLL accuracy. t t, total cross section Moch, Uwer, 2008; Cacciari, Frixione, Mangano, Nason, Ridolfi, 2008; Kidonakis, Vogt, 2008; Langenfeld, Moch, Uwer, 2009; MB, Falgari, Klein, Schwinn,2010; Ahrens, Ferroglia, Neubert, Pecjak, Li, 2010 sparticle pairs, total cross section Kulesza, Motyka, 2008/2009; Langenfeld, Moch, 2009; Beenakker, Brensing, Laenen, Krämer, Kulesza, Niessen, 2009/2010; MB, Falgari, Schwinn, 2009/2010 Invariant mass distributions Hagiwara, Sumino, Yokoya, 2008; Kiyo, Kühn, Moch, Steinhauser, Uwer, 2008 (t t threshold) Hagiwara, Yokoya, 2009 (sparticle pair threshold) Ahrens, Ferroglia, Neubert, Pecjak, Li, 2010 (t t, resummation for large z 1) Present talk: towards joint NNLL resummation of soft and Coulomb gluons. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

10 Why threshold expansion? Strictly valid for high masses 2 m H s had (heavy sparticles). Certainly not for tops at LHC7. Invariant mass distribution peaks at 380 GeV, corresponding to β 0.4, but the average β is larger. No large logs. Assume that threshold expansion provides a good approximation for the integral over all β. Works reasonable well for gg at LO and NLO, less well for q q, and probably better at NNLO, because the average is dominated by smaller β as the order increases. [Note: Multiplying the expanded relative NLO correction by the exact tree improves the approximation.] Tevatron LHC7 LHC gg LHC14 NLO NLOapprox β gg,nlo LO NLOsing NLO NLO sing Β NLO approx Cross section in pb, MSTW2008nnlo PDFs. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

11 Effective field theory Deal with scales one by one. Integrate out high-frequency fluctuations (short-distance) above some scale µ f. L(g i ) L eff (g i,eff ; µ f ) [µ f cancels in observables.] Changing µ f in small steps and integrating to the low scale sums ln β terms in perturbation theory to all orders. [ Renormalization group ] L eff describes physics at small scales (long wavelength) accurately and simpler. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

12 Relevant modes In heavy particle pair production and annihilation, integrate out short-distance modes with k M and larger. Several low-energy [more precisely, virtuality k 2 = k 2 0 k 2 ] modes are relevant. Mode Scaling Virtuality soft k µ Mβ 2 M 2 β 4 collinear k + M, k Mβ 2, k Mβ M 2 β 2 massless only anti-collinear k M, k + Mβ 2, k Mβ M 2 β 2 massless only potential k 0 Mβ 2, k Mβ M 2 β 2 massive only semi-soft k µ Mβ M 2 β 2 k ± = k 0 ± k 3 n ± k. All in centre-of-mass with collision-axis the 3-axis. Soft-collinear and non-relativistic effective field theory Not enough: due to several low-energy modes, multiple fields must address decoupling ( soft-collinear and soft-potential factorization ) M. Beneke (RWTH Aachen) Corfu, 31 August / 23

13 Effective Lagrangians Soft-collinear effective theory for the colliding partons [here interactions involving collinear fermions (quarks)] (Bauer, Fleming, Pirjol, Stewart, 2000/2001; MB, Chapovsky, Diehl, Feldmann 2002)! 1 n/+ L = ξc in D c + g sn A s(x ) + id/ c id/ c in +D c 2 ξc + ξ c x µ nν Wc gfs µν (x )W n/ + c 2 ξc + qsw c id/ cξ c ξ ci D/ c W cq s(x ) +... Z! 0 id c = i + g sa c(x), W c(x) P exp ig s ds n +A c(x + sn +) Potential non-relativistic effective theory for the produced heavy squarks and gluinos (Pineda, Soto, 1997; Brambilla, Pineda, Soto, Vairo, 1999; MB, Signer, Smirnov, 1998/1999) L PNRQCD = ψ i 0 + g sa 0 s (t, 0 ) + Z + d 3 h r ψ T (R)a i ψ ( r ) 2 + iγ H 2m H 2 αs r! ψ + ψ id 0 s + 2 ψ T (R )a ψ (0) 2m H + iγ! H ψ 2 g sψ (x) x E(t, 0 )ψ(x) g sψ (x) x E(t, 0 )ψ (x) +... M. Beneke (RWTH Aachen) Corfu, 31 August / 23

14 Soft-collinear-potential decoupling Factorization of soft, collinear and Coulomb gluons is non-trivial [but well-known for soft-collinear], since soft gluons attach to and between Coulomb ladders and collinear radiation. Kinematics: [k ] c [k 0 ] p k µ s Also Coulomb exchange carries colour structure T A R TA R. Field redefinitions ξ c(x) = S n (3) (x ) ξ c (0) (x), ψ(x) = S v (R) (x 0 ) ψ (0) (x) etc. Basis of irreducible colour representations R α. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

15 Soft-collinear-potential decoupling Factorization of soft, collinear and Coulomb gluons is non-trivial [but well-known for soft-collinear], since soft gluons attach to and between Coulomb ladders and collinear radiation. Kinematics: [k ] c [k 0 ] p k µ s Also Coulomb exchange carries colour structure T A R TA R. Field redefinitions ξ c(x) = S n (3) (x ) ξ c (0) (x), ψ(x) = S v (R) (x 0 ) ψ (0) (x) etc. Basis of irreducible colour representations R α. Factorization formula (MB, Falgari, Schwinn, 2009) ˆσ(β, µ) = X a X i,i H a ii Z (M, µ) dω X R α J a R α (E ω 2 ) Wa,Rα ii (ω, µ). [Sum over a includes sub-leading powers in β. At NNLL there is factorization dependence on µ also in the Coulomb function.] M. Beneke (RWTH Aachen) Corfu, 31 August / 23

16 Resummation (1/3) ˆσ(β, µ) = X i Z H i (M, µ) dω X R α J Rα (E ω 2 ) WRα i (ω, µ). H i (M, µ) Short-distance production of heavy particle pair Sums (g 2 ln 2 β) n by renormalization group evolution in µ from M to Mβ 2. W Rα i (ω, µ) Soft function for the production of a single particle in irrep R α Ŵ Rα (z, µ) 0 T[SRα {aα,bβ} v,βκ S n,jb S 2 n,ib ](z)t[s n,a1 is n,a2 js Rα 1 v,κα ](0) 0 J Rα Potential function related to a correlation function of non-relativistic fields in PNRQCD. Sums Coulomb-exchange (g 2 /β) n to all orders for HH in irrep R α. At LL and NLL J Rα (E) Im " (2m red) 2 4π (s E» 1 + α s( D Rα ) 2m red 2 ln 8 m «rede µ 2 + ψ 1 αs( D Rα ) «)# 2 p E/(2m red ) M. Beneke (RWTH Aachen) Corfu, 31 August / 23

17 Resummation (2/3) Resummation of enhanced quantum corrections via renormalization group equations. [Follow the SCET formalism developed for DY production by (Becher, Neubert, Xu, 2007). Early work on resummation via RGEs for Wilson lines (Korchemsky, Marchesini, 1992; MB, Braun, 1995) ] NNLL (NLL) soft-gluon resummation needs the 3-loop (2-loop) cusp anomalous dimension and 2-loop (1-loop) soft anomalous dimension in d d ln µ «Ŵ Rα i (L) = (Γ r cusp + Γr cusp )L 2γRα W,i Ŵ Rα i (L), Two-loop soft anomalous dimension at threshold (MB, Falgari, Schwinn, 2009) γ Rα W,i = γ Rα H,s + γr s + γr s is a sum over single particle terms that satisfy Casimir scaling (at least up to two loops) γ Rα H,s = C Rα γ H,s with! ζ 3 γ (1) H,s = C A 9 2π T Fn f, M. Beneke (RWTH Aachen) Corfu, 31 August / 23

18 Resummation (3/3) Parton scattering: pp HH + X ˆσ res ˆσ (0) pp (ŝ, µ pp f ) = (ŝ, µ f ) X «h i (M, µ h ) U Rα 2M 2η i (M, µ h, µ s, µ f ) β µ i,rα s s Rα i ( η, µ e 2γ E η Z s) dω J Rα (E ω 2 ) «ω 2η Γ(2η) 0 ω µ s U Rα i = exp[4s(µ h, µ s) 2a V i (µ h, µ s) + 2a φ,r (µ s, µ f ) + 2a φ,r (µ s, µ f )] 4M 2! 2aΓ (µ h,µs) µ 2 h s Rα i (ρ, µ) = Z 0 dωe sω W Rα i (ω, µ) [a(µ 1, µ 2 ) denote integrated anomalous dimensions] Non-relativistic log summation must be added separately relevant from NNLL. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

19 Squark-antisquark production (1/4) Born production processes q i (k 1 ) q j (k 2 ) q σ1 k(p 1 ) q σ2 l (p 2) g(k 1 )g(k 2 ) q σ1 i(p 1 ) q σ2 j (p 2), Must be separated into colour-singlet and colour-octet production. q q g q q, q q g q q g g g q q, g g q q, q g g q q q g, g q q M. Beneke (RWTH Aachen) Corfu, 31 August / 23

20 Squark-antisquark production (2/4) Results at NLL for pp squark+antisquark + X at s = 14 TeV NLL = Tree C and W, 1-loop anomalous dim. + LO Coulomb Green function + matching to NLO fixed order (from Beenakker et al, 1997; Prospino code)) Size of corrections beyond NLO Σ Σ NLO NLL NLL nobs NLL s h C NLL s h m GeV q ( s = 14 TeV; m g /m q = 1.25) Resummation correction significantly larger than in other implementations (Kulesza, Motyka; Beenakker et al.), because: Coulomb scale is lower Interference of soft and Coulomb correction included (to black) Bound state contributions included (to purple) Resummation is 10% (mass-dependent) effect at the natural scale. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

21 Squark-antisquark production (3/4) σ(p p q q)(pb), s = 1.96 TeV (Tevatron) m q [GeV] LO NLO NLL K NLL % % % % % σ(pp q q)(pb), s = 7 TeV (LHC) m q [GeV] LO NLO NLL K NLL % % % % % % m g = 1.25m q. The error estimates are obtained by varying the various factorization scales but do not include PDF errors. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

22 Squark-antisquark production (4/4) Results at NLL for pp squark+antisquark + X at s = 14 TeV (LHC) Scale dependence ( theoretical uncertainty) at LO, NLO, NLL Σ pb NLO NLL 0.1 Μ f m q LO Σ pb NLO NLL Μ f 5.0 m q LO (pp collisions at s = 14 TeV; m g /m q = 1.25; m q = 1 TeV [left] and p p collisions, m q = 400 GeV [right] Green band: variation of all factorization scales.) Theoretical uncertainty is reduced. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

23 Top-quark pair production (1/3) Total hadronic t t cross section with NLL resummation and singular terms at NNLO. Differences to previous results: NLL resummation with SCET (rather than Mellin N-space) formalism Correct singular terms in β at O(α 2 s ) (MB, Czakon,Falgari, Mitov, Schwinn) Numerically, the differences are at the few percent level. Complete NNLL in progress. Example: gg [t t] 8 X ˆσ(β) (2) αs gg [t t] 8 X = ˆσ(β)(0) 4π log 4 β ln «2» π 4 27β 2 + π ( 2 32 ln β + β 9 «log 3 β + «32 ln 2 ln β ln ln π ln ln π 2 ln ln ζ(3) π π ln «log 2 β! log β ) M. Beneke (RWTH Aachen) Corfu, 31 August / 23

24 Top-quark pair production (2/3) LHC running at s = 7 TeV and plans to accumulate R L ( ) pb 1 in Important milestone: re-discovery of the top quark in Europe. Expect top anti-top pairs per 100 pb 1 Production mechansim gg, q q, gq, g q t t + X Compare NLO to NLL resummed plus all enhanced terms at second order O(α 2 s ). (NLO from Nason, Dawson, Ellis, 1988) M. Beneke (RWTH Aachen) Corfu, 31 August / 23

25 Top-quark pair production (2/3) LHC running at s = 7 TeV and plans to accumulate R L ( ) pb 1 in Important milestone: re-discovery of the top quark in Europe. Expect top anti-top pairs per 100 pb 1 Production mechansim gg, q q, gq, g q t t + X Compare NLO to NLL resummed plus all enhanced terms at second order O(α 2 s ). (NLO from Nason, Dawson, Ellis, 1988) (MB, Falgari, Schwinn, Klein, 2010) [using MSTW2008NNLO PDFs] M. Beneke (RWTH Aachen) Corfu, 31 August / 23

26 Top-quark pair production (2/3) LHC running at s = 7 TeV and plans to accumulate R L ( ) pb 1 in Important milestone: re-discovery of the top quark in Europe. Expect top anti-top pairs per 100 pb 1 Production mechansim gg, q q, gq, g q t t + X Compare NLO to NLL resummed plus all enhanced terms at second order O(α 2 s ). (NLO from Nason, Dawson, Ellis, 1988) Small scale dependence after resummation Parton distribution uncertainty dominates (MB, Falgari, Schwinn, Klein, 2010) [using MSTW2008NNLO PDFs] [Other recent work on top anti-top pair production beyond NLO: Moch, Uwer, 2008; Cacciari, Frixione, Mangano, Nason, Ridolfi, 2008; Kidonakis, Vogt, 2008; Langenfeld, Moch, Uwer, 2009; Ahrens, Ferroglia, Neubert, Pecjak, Li, 2010; Hagiwara, Sumino, Yokoya, 2008; Kiyo, Kühn, Moch, Steinhauser, Uwer, 2008 ] M. Beneke (RWTH Aachen) Corfu, 31 August / 23

27 Top-quark pair production (3/3) All results preliminary. Tevatron LHC7 LHC10 LHC NLO MSTW ABKM NLL+NLO MSTW ABKM NNLO app MSTW ABKM NNLO app + NLL MSTW ABKM NNLL work in progress Cross section in pb. Error includes scale variation µ i /2... 2µ i for all µ i (first number) and PDF error (second error). (MB, Falgari, Klein, Schwinn, in progress) M. Beneke (RWTH Aachen) Corfu, 31 August / 23

28 Conclusion 1) Joint resummation possible due to factorization of soft and Coulomb gluon effects (in SCET NRQCD) Leading soft function diagonal to all orders in a simple colour basis. 2-loop anomalous dimension at threshold determined. 2) Top pair cross section at threshold known at O(α 2 s ) at threshold up to the constant term. For any coloured heavy particles, given the one-loop hard matching coefficients. 3) Soft gluon resummation at NNLL for total cross section possible. For complete NNLL combine with non-relativstic log resummation. Phenomenological analysis forth-coming. M. Beneke (RWTH Aachen) Corfu, 31 August / 23

29 Conclusion 1) Joint resummation possible due to factorization of soft and Coulomb gluon effects (in SCET NRQCD) Leading soft function diagonal to all orders in a simple colour basis. 2-loop anomalous dimension at threshold determined. 2) Top pair cross section at threshold known at O(α 2 s ) at threshold up to the constant term. For any coloured heavy particles, given the one-loop hard matching coefficients. 3) Soft gluon resummation at NNLL for total cross section possible. For complete NNLL combine with non-relativstic log resummation. Phenomenological analysis forth-coming. 4) Ready for LHC data and discoveries M. Beneke (RWTH Aachen) Corfu, 31 August / 23