QCD at the LHC. Vittorio Del Duca INFN LNF

Transcription

1 QCD at the LHC Vittorio Del Duca INFN LNF Louvain April 2009

2 Standard Model Past & Present: the LEP/SLD/Tevatron legacy The Standard Model has been a spectacular success, weathering out all challenges The comparison with the electroweak precision measurements has not changed much in the last years

3 Precision tests of SM Top quark Mass Tevatron Run I value: ± 4.3 GeV Run II value: ± 0.8 ± 1.1 GeV ICHEP 08 small effect on the quality of the SM fit and on the mh bounds

4 Precision tests of SM Top quark Mass Tevatron Run I value: ± 4.3 GeV Run II value: ± 0.8 ± 1.1 GeV ICHEP 08 small effect on the quality of the SM fit and on the mh bounds δm/m =0.2 δσ/σ TH: δσ/σ = 9% Δm = 3 GeV At the LHC the expected EXP error is Δm = 1 GeV so the TH cross section should be known at 3% level

5 Effects on global EW fits March 08 W-boson Mass tree level one loop m 2 W t ( 1 m2 W m 2 Z m W = m Z cos θ W ) = πα 2GF (1 + r) H W W W W W b r t = 3α 16π r H = cos 2 θ W sin 4 θ W m 2 t m 2 W 11α 48π sin 2 θ W ln m2 H m 2 W so δm W m 2 t δm W ln m H δmt = 1 GeV δmw(mt) = 6 MeV ElectroWeak fits point to a light Higgs boson

6 Effects on global EW fits Higgs boson Mass March 08 mh > GeV from direct search at LEP mh = GeV from EW fits At 95% CL mh < 160 GeV from EW fits mh < 190 GeV combined with direct search at LEP use mt to estimate mh from EW corrections as mt changes, large shifts in mh

7 ... so, the Standard Model is very good, but that s not enough ElectroWeak Symmetry Breaking not tested neutrino masses and mixings dark matter baryogenesis??? foremost task of the LHC is to understand the EWSB: find the Higgs boson or whatever else is the cause of it

8 LHC pp s = 14 TeV L design = cm -2 s -1 (after 2009) L initial few x cm -2 s -1 (until 2009) Heavy ions (e.g. Pb-Pb at s ~ 1000 TeV) TOTEM (integrated with CMS): pp, cross-section, diffractive physics TOTEM ATLAS and CMS : general purpose 27 km LEP ring 1232 superconducting dipoles B=8.3 T Here: ATLAS and CMS ALICE : ion-ion, p-ion LHCb : pp, B-physics, CP-violation

9 LHC is a QCD machine SM processes are backgrounds to New Physics signals design luminosity L = cm -2 s -1 = 10-5 fb -1 s -1 integrated luminosity (per year) L 100 fb -1 yr -1

10 With 1 fb -1 we shall get... final state jets (pt > 100 GeV) events 10 9 overall # of events (2008) jets (pt > 1 TeV) 10 4 W eν Z e + e b b t t (Tevatron) 10 6 (LEP) 10 9 (BaBar, Belle) 10 4 (Tevatron) even at very low luminosity, LHC beats all the other accelerators

11 LHC: the next future calibrate the detectors, and re-discover the SM i.e. measure known cross sections: jets, W, Z, t t understand the EWSB/find New-Physics signals (ranging from Z to leptons, to gluinos in SUSY decay chains, to finding the Higgs boson) constrain and model the New-Physics theories in all the steps above (except probably Z to leptons) precise QCD predictions play a crucial role

12 Tales from the past B production in the 90 s discrepancy between Tevatron data and NLO prediction

13 B cross section in p p collisions at 1.96 TeV dσ(p p H b X, H b J/ψ X)/dp T (J/ψ) FONLL = NLO + NLL Cacciari, Frixione, Mangano, Nason, Ridolfi 2003 total x-sect is 19.4 ± 0.3(stat) (syst) nb CDF hep-ex/ use of updated fragmentation functions by Cacciari & Nason and resummation good agreement with data no New Physics

14 QCD an unbroken Yang-Mills gauge field theory featuring asymptotic freedom and confinement in non-perturbative regime (low Q 2 ) many approaches: lattice, Regge theory, χ PT, large Nc, HQET in perturbative regime (high Q 2 ) QCD is a precision toolkit for exploring Higgs & BSM physics LEP was an electroweak machine Tevatron & LHC are QCD machines

15 Precise determination of strong coupling constant parton distributions electroweak parameters LHC parton luminosity Precise prediction for QCD at LHC Higgs production new physics processes their backgrounds α s Goal: to make theoretical predictions of signals and backgrounds as accurate as the LHC data

16 Strong interactions at high Q 2 Parton model Perturbative QCD factorisation universality of IR behaviour cancellation of IR singularities IR safe observables: inclusive rates jets event shapes

17 computed in pqcd Factorisation extracted from data evolved through DGLAP is the separation between the short- and the long-range interactions P A p a }X X = W, Z, H, Q Q, high-e T jets,... ˆσ is known as a fixed-order expansion in α S P B p b ˆσ = Cα n S(1 + c 1 α S + c 2 α 2 S +...) c 1 = NLO c 2 = NNLO or as an all-order resummation ˆσ = CαS[1 n + (c 11 L + c 10 )α S +(c 22 L 2 + c 21 L + c 20 )αs ] where L = ln(m/q T ), ln(1 x), ln(1/x), ln(1 T ),... c 11,c 22 = LL c 10,c 21 = NLL c 20 = NNLL

18 Factorisation-breaking contributions underlying event power corrections (see Rick Field s studies at CDF) MC s and theory modelling of power corrections laid out and tested at LEP where they provide an accurate determination of models still need be tested in hadron collisions (see e.g. Tevatron studies at different s ) diffractive events double-parton scattering observed by Tevatron CDF in the inclusive sample p p γ + 3 jets potentially important at LHC σ D σ 2 S predicted breakdown in dijet production at N 3 LO Collins Qiu 2007 α S

19 Power corrections at Tevatron Ratio of inclusive jet cross sections at 630 and 1800 GeV M.L. Mangano KITP collider conf 2004 Bjorken-scaling variable x T = 2E T s In the ratio the dependence on the pdf s cancels dashes: theory prediction with no power corrections solid: best fit to data with free power-correction parameter Λ in the theory

20 Factorisation in diffraction?? diffraction in DIS double pomeron exchange in p p no proof of factorisation in diffractive events data do not support it

21 Parton distribution functions (PDF) just to get an idea of the PDF size, take some PDF fit

22 LHC kinematic reach LHC opens up a new kinematic range x range covered by HERA but Q 2 range must be provided by DGLAP evolution GeV physics is large x physics (valence quarks) at Tevatron, but smaller x physics (gluons & sea quarks) at the LHC rapidity distributions span widest x range Feynman x s for the production of a particle of mass M x 1,2 = M 14 TeV e±y

23 PDF evolution factorisation scale µ F is arbitrary cross section cannot depend on µ F µ F dσ dµ F =0 implies DGLAP equations V. Gribov L. Lipatov; Y. Dokshitzer G. Altarelli G. Parisi µ F df a (x, µ 2 F ) dµ F = P ab (x, α S (µ 2 F )) f b (x, µ 2 F )+O( 1 Q 2 ) µ F dˆσ ab (Q 2 /µ 2 F,α S(µ 2 F )) dµ F = P ac (x, α S (µ 2 F )) ˆσ cb (Q 2 /µ 2 F,α S (µ 2 F )) + O( 1 Q 2 ) P ab (x, α S (µ 2 F )) is calculable in pqcd

24 DGLAP evolution equations F ep 2,Fep L factorisation for the structure functions (e.g. ) F i (x, µ F 2 )=C ij q j + C ig g with the convolution [a b](x) C ij,c ig coefficient functions q i (x, µ F 2 ) g(x, µ F 2 ) PDF s 1 x ( ) dy x y a(y) b y DGLAP evolution equations ( ) ( d qi Pqi q d ln µ F 2 = j P qj g g P gqj P gg ) ( qj g ) perturbative series P ij α s P (0) ij + αsp 2 (1) ij + αsp 3 (2) ij anomalous dimension γ ij (N) = 1 0 dx x N 1 P ij (x)

25 HERA F 2 Bjorken-scaling violations large violations at small x small violations at large x

26 a list of PDF s general structure of the quark-quark splitting functions P qi q k = P q i q k =δ ik Pqq v + Pqq s P qi q k = P q i q k =δ ik Pq q v + Pq q s flavour non-singlet flavour asymmetry q ± ns,ik = q i ± q i (q k ± q k ) P ± ns d d ln µ 2 F ( qs g ) = = P v qq ± P v q q sum of valence distributions of all flavours n f qns v = (q r q r ) Pns v = Pqq v Pq q v + n f (Pqq s Pq q) s r=1 flavour singlet n f q s = (q i + q i ) i=1 with P qq = P + ns + n f (P s qq + P s qq) P qg = n f P qi g, P gq = P gqi ( ) Pqq P qg P gq P gg ( qs g )

27 PDF history leading order (or one-loop) anomalous dim/splitting functions Gross Wilczek 1973; Altarelli Parisi 1977 NLO (or two-loop) F 2,F L Bardeen Buras Duke Muta 1978 anomalous dim/splitting functions Curci Furmanski Petronzio 1980 NNLO (or three-loop) F 2,F L Zijlstra van Neerven 1992; Moch Vermaseren 1999 anomalous dim/splitting functions Moch Vermaseren Vogt 2004 the calculation of the three-loop anomalous dimension is the toughest calculation ever performed in perturbative QCD! one-loop two-loop γ (0) (0) ij /P ij γ (1) (1) ij /P ij 18 Feynman diagrams 350 Feynman diagrams three-loop γ (2) (2) ij /P ij 9607 Feynman diagrams 20 man-year-equivalents, 10 6 lines of dedicated algebra code

28 global fits MRST, MSTW: Martin et al. CTEQ: Pumplin et al. Alekhin (DIS data only) method χ 2 Perform fit by minimising to all data, including both statistical and systematic errors PDF global fits J. Stirling, KITP collider conf 2004 Q 2 0 Start evolution at some, where PDF s are parametrised with functional form, e.g. xf(x, Q 2 0) = (1 x) η (1 + ɛx γx)x δ Cut data at Q 2 >Q 2 min and at W 2 >Wmin 2 to avoid higher twist contamination Allow ū d as implied by E866 Drell-Yan asymmetry data accuracy NNLO evolution no prompt photon data included in the fits

29 PDF: recent developments more systematic treatment of uncertainties in global fits more accurate treatment of heavy flavours in the vicinity of the quark mass (few % effect on Drell-Yan at the LHC) MSTW, CTEQ, Alekhin neural network PDF s (NNPDF), based on unbiased priors (so far only DIS data, and only the non-singlet distributions) valence quark distribution note the larger uncertainty in NNPDF still far from global-fit analysis in NNPDF NNPDF Collaboration 08

30 Parton cross section 3 complementary approaches to ˆσ matrix-elem MC s fixed-order x-sect shower MC s final-state description hard-parton jets. Describes geometry, correlations,... limited access to final-state structure full information available at the hadron level higher-order effects: loop corrections hard to implement: must introduce negative probabilities straightforward to implement (when available) included as vertex corrections (Sudakov FF s) higher-order effects: hard emissions included, up to high orders (multijets) straightforward to implement (when available) approximate, incomplete phase space at large angles resummation of large logs? feasible (when available) unitarity implementation (i.e. correct shapes but not total rates) M.L. Mangano KITP collider conf 2004

31 Parton shower MonteCarlo generators HERWIG B. Webber et al re-written as a C++ code (HERWIG++) PYTHIA T. Sjostrand 1994 SHERPA F. Krauss et al model parton showering and hadronisation

32 Matrix-element MonteCarlo generators several automated codes to yield large number of (up to 8-9) final-state partons can be straightforwardly interfaced to parton-shower MC s ideal to scout new territory large dependence on ren/fact scales example: Higgs (via gluon fusion) + 2 jets is αs 4 (Q 2 ) unreliable for precision calculations

33 Matrix-element MonteCarlo generators multi-parton LO generation: processes with many jets (or V/H bosons) ALPGEN M.L.Mangano M. Moretti F. Piccinini R. Pittau A. Polosa 2002 MADGRAPH/MADEVENT W.F. Long F. Maltoni T. Stelzer 1994/2003 COMPHEP A. Pukhov et al T. Ishikawa et al. K. Sato et al. 1992/2001 HELAC C. Papadopoulos et al processes with 6 final-state fermions PHASE E. Accomando A. Ballestrero E. Maina 2004 merged with parton showers all of the above, merged with HERWIG or PYTHIA

34 CKKW MonteCarlo interfaces S. Catani F. Krauss R. Kuhn B. Webber 2001 MLM L. Lonnblad 2002 M.L. Mangano 2005 procedures to interface parton subprocesses with a different number of final states to parton-shower MC s MC@NLO POWHEG S. Frixione B. Webber 2002 P. Nason 2004 procedures to interface NLO computations to parton-shower MC s Single top in MC@NLO Frixione Laenen Motylinski Webber 2005 at low pt, parton shower models collinear radiation at high pt, NLO models hard radiation

35 Leading Order Different methods (other than Feynman) to compute tree-level amplitudes Berends-Giele relations (1988) use recursively off-shell currents and go on-shell to get amplitudes = Cachazo Svrcek Witten (2004) compute helicity amplitudes by sewing MHV amplitudes (++ + ) Britto Cachazo Feng (2004) use on-shell recursive relations by shifting to complex momenta + = +

36 Leading Order CSW and BCF are ideal to obtain compact analytic amplitudes... but numerically Berends-Giele seems to be the fastest time [s] for 2 n gluon amplitudes for 10 4 PS points Duhr Höche Maltoni 2006 CO: colour ordered CD: colour dressed

37 Next to Leading Order Jet structure: final-state collinear radiation PDF evolution: initial-state collinear radiation Opening of new channels Reduced sensitivity to fictitious input scales: µ R,µ F predictive normalisation of observables first step toward precision measurements accurate estimate of signal and background for Higgs and new physics Matching with parton-shower MC s: MC@NLO POWHEG

38 Jet structure the jet non-trivial structure shows up first at NLO leading order NLO NNLO

39 pp t tz at NLO Lazopoulos McElmurry Melnikov Petriello 2008 dots: inclusive K factor dash: pt dependent K factor probes EW properties of top quark NLO/LO K factor = 1.35 Reduced theoretical error: 25-35% at LO; 11% at NLO

40 NLO cross sections: experimenter s wishlist 2005 Les Houches QCD, EW & Higgs working group hep-ph/

41 2007 Les Houches updated wishlist NLO multileg working group hep-ph/ private codes

42 High (EXP) demand for cross sections of Y + n jets Y = vector boson(s), Higgs, heavy quark(s),... Big TH community effort To compute the cross section of Y + n jets, we need: 1) tree-level amplitude for Y + (n+3) partons 2) one-loop amplitude for Y + (n+2) partons 3) a method to cancel the IR divergences and so to compute the cross section 3: until the mid 90 s, we did not have systematic methods to cancel the IR divergences 2: until , we did not have systematic methods to compute the one-loop amplitudes

43 NLO assembly kit example e + e 3 jets leading order M tree n 2 NLO real IR NLO virtual M tree n+1 2 M tree n 2 = dp S P split 2 ( A ɛ 2 + B ) ɛ d =4 2ɛ d d l 2(M loop n ) M tree n = ( A ɛ 2 + B ɛ ) M tree n 2 + fin.

44 NLO production rates Process-independent procedure devised in the 90 s slicing Giele Glover Kosower subtraction Frixione Kunszt Signer 1995; Nagy Trocsanyi 1996 dipole Catani Seymour 1996 antenna Kosower 1997; Campbell Cullen Glover 1998 σ = σ LO + σ NLO = dσm B J m + σ NLO m σ NLO = dσm+1j R m+1 + dσmj V m m+1 the 2 terms on the rhs are divergent in d=4 use universal IR structure to subtract divergences [ ] [ σ NLO = dσm+1j R m+1 dσ R,A m+1 J m + dσm V + m+1 the 2 terms on the rhs are finite in d=4 m m 1 ] dσ R,A m+1 J m

45 NLO history of final-state distributions e + e 3 jets K. Ellis Ross Terrano 1981 e + e 4 jets Bern et al.; Glover et al.; Nagy Trocsanyi e + e 4 fermions Denner Dittmaier Roth Wieders 2005 pp 1, 2 jets K. Ellis Sexton 1986 Giele Glover Kosower 1993 pp 3 jets pp V + 1 jet pp V + 2 jet Bern Dixon Kosower; Kunszt Signer Trocsanyi Nagy 2001 Giele Glover Kosower1993 Bern Dixon Kosower Weinzierl; Campbell Glover Miller Campbell K. Ellis 2003 pp V + 3 jet K. Ellis Melnikov Zanderighi; Berger et al. (BlackHat) 2009 pp Vb b Campbell K. Ellis 2003 pp VV Ohnemus Owens, Baur et al ; Dixon et al pp VV + jet Dittmaier Kallweit Uwer; Campbell K. Ellis Zanderighi 2007 pp V V V pp γγ pp γγ + 1 jet Lazopoulos Melnikov Petriello; Hankele Zeppenfeld 2007 Binoth Ossola Papadopoulos Petriello 2008 Bailey et al. 1992; Binoth et al Bern et al Del Duca Maltoni Nagy Trocsanyi 2003

46 NLO history of final-state distributions pp Q Q pp Q Q + 1 jet pp t tz pp t (+ W ) pp H + 1 jet pp H + 2 jets pp H + 3 jets pp HQ Q pp t tb b pp HQ Dawson K. Ellis Nason 1989 Mangano Nason Ridolfi 1992 Brandenburg Dittmaier Uwer Weinzierl Dittmaier Kallweit Uwer Lazopoulos McElmurry Melnikov Petriello 2008 Harris Laenen Phaf Sullivan Weinzierl 2002 Campbell K. Ellis Tramontano 2004; Cao Yuan et al W: Campbell Tramontano 2005 (GGF; mt ) C. Schmidt 1997 De Florian Grazzini Kunszt 1999 (GGF; mt ) Campbell K. Ellis Zanderighi 2006 (WBF) Campbell K. Ellis; Figy Oleari Zeppenfeld 2003 Ciccolini Denner Dittmaier 2007 (includes s channel) (WBF) Figy Hankele Zeppenfeld 2007 Beenakker et al. ; Dawson et al Bredenstein Denner Dittmaier Pozzorini 2008 (only qq annihilation) t: Maltoni Paul Stelzer Willenbrock 2001 b: Campbell, K. Ellis Maltoni Willenbrock 2002

47 ... summarising long time span to add one more jet to a cross section 2 2 and 2 3 processes: almost all computed and included into NLO packages 2 4 processes: very few computed e + e 4 fermions Denner Dittmaier Roth Wieders 2005 pp t tb b (only qq annihilation) Bredenstein Denner Dittmaier Pozzorini 2008 pp V + 3 jet 2 5 processes: none K. Ellis Melnikov Zanderighi; Berger et al. (BlackHat) 2009

48 J. Campbell, KITP collider conf 2004

49 J. Campbell, KITP collider conf 2004

50 Summary of NLO K factors NLO multileg working group hep-ph/ all: CTEQ6M at NLO K: CTEQ6L1 at LO K : CTEQ6M at LO gluon-initiated processes: large K factors more jets: smaller K factors

51 NLO progress: cross sections automated subtraction of IR divergences automating dipoles Gleisberg Krauss 2007 Frederix Gehrmann Greiner (MadGraph) 2008 Seymour Tevlin (TeVJet); Hasegawa Moch Uwer2008 dipoles proliferate with the spectators: may not be adapt for multi-jet environment Catani Seymour 1996 subtraction for multi-jet environment antenna new standard subtraction for each dipole, one must consider all possible spectators... result: in W + 2 jets (MCFM) 24 counterterm events for each real event Daleo Gehrmann Maitre 2006 Somogyi Trocsanyi 2006; Somogyi 2009

52 Dipoles To gain exact factorisation of phase space, modify momenta emitter momentum p µ ir = p µ i + pµ r y ir,n 1 y ir,n p µ n = p µ n = 1 1 y ir,n (p µ i + pµ r y ir,n Q µ irn ) spectator momentum 1 1 y ir,n p µ n y ir,n = 2p i p r Q 2 irn Catani Seymour 1996 momentum is conserved Q µ irn = pµ ir + pµ n = p µ i + pµ r + p µ n counterterm A M 0 m+1 2 = n i,r D ir,n (p 1,..., p m+1 )+...

53 New standard subtraction p µ ir = 1 1 α ir (p µ i + pµ r α ir Q µ ), p µ n = α ir = 1 2 [ y (ir)q y 2 (ir)q 4y ir ] Somogyi Trocsanyi α ir p µ n, n i, r y ir = 2p i p r Q 2 momentum is conserved p µ ir + n p µ n = p µ i + pµ r + n p µ n mapping à la Catani-Seymour, but all momenta but the unresolved ones are treated simultaneously like spectators i r 1 1 ĩr m+1 Q m+1 ĩr ir r Q m + 2 m + 2

54 2 1 one-loop amplitudes one-loop n-point amplitudes An are IR divergent IR divergences are universal Kunszt Signer Trocsanyi 1994; Catani 1998 IR finite terms are process dependent: many final-state particles many scales lengthy expressions An can be reduced to boxes, triangles and bubbles with rational coefficients A n = i 1 i 2 i 3 i 4 d i1 i 2 i 3 i 4 I D i 1 i 2 i 3 i 4 + i2 i1 i 1 i 2 i 3 c i1 i 2 i 3 I D i 1 i 2 i 3 + I: master integrals b, c, d: rational functions of kinematic variables higher polygons contribute only to O(ε) i3 i4 i1 i2 i 1 i 2 b i1 i 2 I D i 1 i 2 i3 i1 i2

55 NLO progress: unitarity method use unitarity cuts: A n = Cutkovsky rule Bern Dixon Dunbar Kosower 1994 d i1 i 2 i 3 i 4 Ii 4 1 i 2 i 3 i 4 + c i1 i 2 i 3 Ii 4 1 i 2 i 3 + b i1 i 2 Ii 4 1 i 2 + R n i 1 i 2 i 3 i 4 i 1 i 2 i 3 i 1 i 2 1 p 2 + i0 2πi δ+ (p 2 ) to factorise coefficients of An into products of tree amplitudes compute b, c, d with D=4: cut-constructible terms O(ε) part of b, c, d: rational term Rn Rn computable through on-shell recursion Berger Bern Dixon Forde Kosower 2006 generalized unitarity: quadruple cuts with complex momenta box coefficient di determined by product of 4 tree amplitudes Britto Cachazo Feng 2004 however, ci, bi still difficult to extract from triple and double cuts because terms already included in di must be subtracted

56 NLO progress: OPP method Ossola Papadopoulos Pittau 2006 reduction of one-loop amplitude at integrand level A(q )= N(q) D 1 D m D i =(q + p i ) 2 m 2 i q = q + q q q =0 q lives in D dimensions; q lives in 4 dimensions ε part of numerator treated separately partial fraction the numerator into terms with 4, 3, 2, 1 denominator factors N(q) = (d i1 i 2 i 3 i 4 + d i1 i 2 i 3 i 4 ) i 1 i 2 i 3 i 4 + (c i1 i 2 i 3 + c i1 i 2 i 3 ) D i + i 1 i 2 i 3 i i 1 i 2 i 3 d, c, b vanish upon integration i i 1 i 2 i 3 i 4 D i for quadruple cuts, similar to BCF, but algorithmic: can be automated i 1 i 2 (b i1 i 2 + b i1 i 2 ) i i 1 i 2 D i reduced to problem of fitting d, c, b by evaluating N(q) at different values of q, e.g. singling out choices of q such that 4, 3, 2, 1 among all Di vanish, and then inverting the system. First find all possible 4-pt functions, then 3-pt functions, etc.

57 NLO progress: integer dimensions Giele Kunszt Melnikov 2008 in D dimensions the one-loop amplitude An can be reduced to pentagons, boxes, triangles and bubbles A n = e i1 i 2 i 3 i 4 i 5 Ii D 1 i 2 i 3 i 4 i 5 + d i1 i 2 i 3 i 4 Ii D 1 i 2 i 3 i 4 i 1 i 2 i 3 i 4 i 5 i 1 i 2 i 3 i 4 + c i1 i 2 i 3 Ii D 1 i 2 i 3 + b i1 i 2 Ii D 1 i 2 i 1 i 2 i 3 i 1 i 2 take Ds = # of spin states of internal particles in Ds dimensions, gluons have Ds-2 spin states, quarks have 2 (Ds-2)/2 spin states A (D,D s) n = d D q N (D s) (q) d 1 d n with D Ds dependence of A on Ds is linear N (D s) (q) =N 0 (q)+(d s 4)N 1 (q) - compute N0, N1 numerically separately through 2 different integer values of Ds - on Ds-dimension cuts, spin density matrix is well defined - choose basis of master integrals with no explicit D dependence in the coefficients - after reduction to master integrals, continue D to 4-2ε procedure can be completely automated

58 A Fortran 90 code based on OPP and integer dimensions Rocket Giele Zanderighi 2008 computes - up to one-loop 20-gluon amplitudes - one-loop W + 5-parton amplitudes - (leading-colour) NLO W + 3-jet cross section Giele Zanderighi 2008 K. Ellis Giele Kunszt Melnikov Zanderighi 2008 K. Ellis Melnikov Zanderighi 2009

59 W + 3-jet cross section at NLO K. Ellis Melnikov Zanderighi 2009 NLO very stable under scale variations K factor even smaller than in W + 2 jets

60 BlackHat Berger Bern Dixon Febres-Cordero Forde Ita Kosower Maitre 2008 A C++ code based on generalised unitarity, and on-shell recursion for the rational parts computes - one-loop 6-gluon (and MHV 7- and 8-gluon) amplitudes - (leading-colour) one-loop W + 5-parton amplitudes - (leading-colour) NLO W + 3-jet cross section BlackHat 2008 BlackHat 2009

61 Is NLO enough to describe data? Inclusive jet p T cross section at Tevatron good agreement between NLO and data over several orders of magnitude constrains the gluon distribution at high x

62 Drell-Yan Is NLO enough to describe data? W cross section at LHC with leptonic decay of the W MC@NLO NLO = O(2%) S. Frixione M.L. Mangano 2004 NNLO useless without spin correlations Precisely evaluated Drell-Yan W, Z cross sections could be used as ``standard candles to measure the parton luminosity at LHC

63 Drell-Yan W acceptances at LHC with leptonic decay of the W At LO, pe, mw/2 μ = mw/2, mw, 2mW NNLO corrections are large for pe, = 40, 50 GeV but are at the percent level for pe, = 20, 30 GeV K. Melnikov, F. Petriello 2006

64 Is NLO enough to describe data? Total cross section for inclusive Higgs production at LHC contour bands are lower µ R =2M H µ F = M H /2 upper µ R = M H /2 µ F =2M H scale uncertainty is about 10% NNLO prediction stabilises the perturbative series

65 Higgs production at NNLO a fully differential cross section: bin-integrated rapidity distribution, with a jet veto M H = 150 GeV (jet veto relevant in the C. Anastasiou K. Melnikov F. Petriello 2004 jet veto: require R =0.4 p j T <pveto T = 40 GeV for 2 partons R12 2 =(η 1 η 2 ) 2 +(φ 1 φ 2 ) 2 if R 12 >R p 1 T, p 2 T <p veto T if R 12 <R p 1 T + p 2 T <p veto T H W + W decay channel) K factor is much smaller for the vetoed x-sect than for the inclusive one: average p j T increases from NLO to NNLO: less x-sect passes the veto

66 Drell-Yan Z production at LHC Rapidity distribution for an on-shell Z boson 30%(15%) NLO increase wrt to LO at central Y s (at large Y s) NNLO decreases NLO by 1 2% scale variation: 30% at LO; 6% at NLO; less than 1% at NNLO C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003

67 Scale variations in Drell-Yan Z production µ R dashed: vary µ F only solid: vary and µ F together dotted: vary µ R only C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003

68 Drell-Yan W production at LHC Rapidity distribution for an on-shell W boson (left) W + boson (right) distributions are symmetric in Y NNLO scale variations are 1%(3%) at central (large) Y C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003

69 World average of α S (M Z ) α S (M Z )= ± S. Bethke hep-ex/ α S (M Z ) Rightmost 2 columns give the exclusive mean value of calculated without that measurement, and the number of std. dev. between this measurement and the respective excl. mean

70 NNLO corrections may be relevant if the main source of uncertainty in extracting info from data is due to NLO theory: measurements NLO corrections are large: Higgs production from gluon fusion in hadron collisions α S NLO uncertainty bands are too large to test theory vs. data: b production in hadron collisions NLO is effectively leading order: energy distributions in jet cones

71 NNLO state of the art Drell-Yan W, Z production total cross section fully differential cross section Higgs production total cross section fully differential cross section e + e 3 jets event shapes, αs Hamberg van Neerven Matsuura 1990 Harlander Kilgore 2002 Melnikov Petriello 2006 Harlander Kilgore; Anastasiou Melnikov 2002 Ravindran Smith van Neerven 2003 Anastasiou Melnikov Petriello 2004 Catani de Florian Grazzini 2007 de Ridder Gehrmann Glover Heinrich 2007 Weinzierl 2008 α s (M 2 Z)= ± (stat) ± (exp) ± (had) ± (theo) TH uncertainty almost halved from NLO to NNLO no log resummation is included

72 e + e 3 jets de Ridder Gehrmann Glover Heinrich 2007 discrepancy found in logarithmic contributions to thrust distribution at N 3 LL by Becher Scwartz 2008 using SCET. Confirmed error in wide-angle soft emissions through independent NNLO calculation by Weinzierl 2008 any IR observable to NNLO with C O = N 2 c C lc O + C sc O + 1 N 2 c disagreement in O = α s 2π A O + C ssc O ( αs 2π N 2 c,n 0 c,n f N c ) 2 BO + + N f N c C nf O ( αs ) 3 CO 2π + N f C nfsc O N c + N 2 f C nfnf O traced back to soft counterterm missing in RR contribution Weinzierl 2008

73 NNLO cross sections Analytic integration first method Hamberg, van Neerven, Matsuura 1990 Anastasiou Dixon Melnikov Petriello 2003 flexible enough to include a limited class of acceptance cuts by modelling cuts as ``propagators Sector decomposition flexible enough to include any acceptance cuts Denner Roth 1996; Binoth Heinrich 2000 Anastasiou, Melnikov, Petriello 2004 cancellation of divergences is performed numerically can it handle many final-state partons? Subtraction process independent cancellation of divergences is semi-analytic can it be fully automatised? de Ridder Gehrmann Glover Heinrich 2007

74 e + e 3 jets NNLO assembly kit double virtual real-virtual double real

75 Two-loop matrix elements two-jet production qq qq,q q q q, q q gg, gg gg C. Anastasiou N. Glover C. Oleari M. Tejeda-Yeomans Z. Bern A. De Freitas L. Dixon 2002 photon-pair production q q γγ, gg γγ C. Anastasiou N. Glover M. Tejeda-Yeomans 2002 Z. Bern A. De Freitas L. Dixon 2002 e + e 3 jets γ q qg L. Garland T. Gehrmann N. Glover A. Koukoutsakis E. Remiddi 2002 V + 1 jet production T. Gehrmann E. Remiddi 2002 Drell-Yan V production R. Hamberg W. van Neerven T. Matsuura 1991 Higgs production q q Vg q q V gg H (in the m t limit) R. Harlander W. Kilgore; C. Anastasiou K. Melnikov 2002

76 Collinear and soft currents universal IR structure process-independent procedure universal collinear and soft currents 3-parton tree splitting functions Campbell Glover 1997; Catani Grazzini 1998; Frizzo Maltoni VDD 1999; Kosower parton one-loop splitting functions Bern Dixon Dunbar Kosower 1994; Bern Kilgore C. Schmidt VDD ; Kosower Uwer 1999; Catani Grazzini 1999; Kosower 2003 universal subtraction counterterms several ideas Kosower; Weinzierl; de Ridder Gehrmann Heinrich 2003 Frixione Grazzini 2004; Somogyi Trocsanyi VDD 2005 but devised only for e + e 3 jets de Ridder Gehrmann Glover 2005; Somogyi Trocsanyi VDD 2006

77 Why is NNLO subtraction so complicated? collinear operator back to NLO IR limits C ir M (0) m+2 (p i,p r,...) 2 1 (0) M m+1 (0)(p ir,...) ˆP f s i f r M m+1 (0)(p ir,...) ir soft operator S r M (0) m+2 (p r,...) 2 s ik M m+1 (0)(...) T i T k M m+1 (0)(...) s ir s rk counterterm r i r 1 2 C ir +S r M (0) m+2 (p i,p r,...) 2 performs double subtraction in overlapping regions

78 NLO overlapping divergences C ir S r can be used to cancel double subtraction C ir (S r C ir S r ) M (0) m+2 2 =0 S r (C ir C ir S r ) M (0) m+2 2 =0 the NLO counterterm A 1 M (0) m+2 2 = r i r 1 2 C ir + S r C ir S r M (0) m+2 (p i,p r,...) 2 i r has the same singular behaviour as SME, and is free of double subtractions C ir (1 A 1 ) M (0) m+1 2 =0 S r (1 A 1 ) M (0) m+1 2 =0 contains spurious singularities when parton s r becomes unresolved, but they are screened by J m

79 σ NNLO = m+2 NNLO subtraction dσ RR m+2j m+2 + m+1 dσ RV m+1j m+1 + the 3 terms on the rhs are divergent in d=4 use universal IR structure to subtract divergences m dσm VV J m

80 σ NNLO = m+2 NNLO subtraction dσ RR m+2j m+2 + m+1 dσ RV m+1j m+1 + the 3 terms on the rhs are divergent in d=4 use universal IR structure to subtract divergences [ ] σ NNLO = dσm+2j RR m+2 dσ RR,A 2 m+2 J m m+2 takes care of doubly-unresolved regions, but still divergent in singly-unresolved ones m dσm VV J m

81 σ NNLO = still contains m+2 NNLO subtraction dσ RR m+2j m+2 + m+1 dσ RV m+1j m+1 + the 3 terms on the rhs are divergent in d=4 use universal IR structure to subtract divergences [ ] σ NNLO = dσm+2j RR m+2 dσ RR,A 2 m+2 J m m+2 takes care of doubly-unresolved regions, but still divergent in singly-unresolved ones [ ] + dσm+1j RV m+1 dσ RV,A 1 m+1 J m 1/ɛ m+1 m dσm VV J m poles in regions away from 1-parton IR regions

82 σ NNLO = still contains m+2 NNLO subtraction dσ RR m+2j m+2 + m+1 dσ RV m+1j m+1 + the 3 terms on the rhs are divergent in d=4 use universal IR structure to subtract divergences [ ] σ NNLO = dσm+2j RR m+2 dσ RR,A 2 m+2 J m m dσm VV J m 1/ɛ poles in regions away from 1-parton IR regions [ ] + dσm VV + dσ RR,A 2 m+2 + dσ RV,A 1 m+1 J m m m+2 takes care of doubly-unresolved regions, but still divergent in singly-unresolved ones [ ] + dσm+1j RV m+1 dσ RV,A 1 m+1 J m m+1 2 1

83 2-step procedure construct subtraction terms that regularise the singularities of the SME in all unresolved parts of the phase space, avoiding multiple subtractions G. Somogyi Z. Trocsanyi VDD 2005 perform momentum mappings, such that the phase space factorises exactly over the unresolved momenta and such that it respects the structure of the cancellations among the subtraction terms G. Somogyi Z. Trocsanyi VDD 2006

84 A2 counterterm construct the 2-unresolved-parton counterterm using the IR currents regions: subtract the collinear limit of the soft one A 2 M (0) m+2 2 = { [ 1 6 C irs C ir;js S rs r s r i r,s j i,r,s + 1 ( CS ir;s C irs CS ir;s )] C ir;js CS ir;s 2 j i,r,s ( ) 12 [CS ir;s S rs + C irs S rs CS ir;s S rs i r,s + j i,r,s C ir;js ( 1 2 S rs CS ir;s S rs ) ]} M (0) m+2 2 G. Somogyi Z. Trocsanyi VDD 2005 performing double and triple subtractions in overlapping regions C irs (1 A 2 ) M (0) m+2 2 =0 C ir;js (1 A 2 ) M (0) m+2 2 =0 S rs (1 A 2 ) M (0) m+2 2 =0 CS ir;s (1 A 2 ) M (0) m+2 2 =0

85 Triple-collinear mapping p µ irs = 1 (p µ i 1 α + pµ r + p µ s α irs Q µ ), p µ n = irs [ y (irs)q α irs = 1 2 y 2 (irs)q 4y irs 1 p µ n, n i, r, s 1 α irs ] momentum conservation straightforward extension of NLO collinear mapping i r s p µ irs + p µ n = p µ i + pµ r + p µ s + n n phase space p µ n 1 1 m ĩrs Q m ĩrs irs r s Q m + 2 m + 2

86 NNLO counterterms σ NNLO {m+2} = σ NNLO {m+1} = σ NNLO = σ{m+2} NNLO + σ{m+1} NNLO + σ{m} NNLO m+2 [ [ dσ RR m+2 J m+2 dσ RR,A 2 m+2 J m m+1 dσ RR,A 1 m+2 J m+1 + dσ RR,A 12 m+2 J m ] {[ dσ RV m+1 + [ dσ RV,A 1 m ( ] dσ RR,A 1 m+2 J m+1 1 G. Somogyi Z. Trocsanyi VDD 2006 ) } dσ RR,A 1 A1 m+2 ]J m ε=0 d=4 σ NNLO {m} = m remainder is finite by KLN theorem { [ ] dσm VV + dσ RR,A 2 m+2 dσrr,a 12 m [ dσ RV,A 1 m+1 + ( 1 ) ]} dσ RR,A 1 A1 m+2 J m ε=0

87 Resummations coefficients of αs expansion of an observable Θ may contain logs in some kinematic regions, logs may become large, spoiling convergence if resummed to all orders, series convergence improves renormalisation of Θ is Θ (n) 0 (p i, Λ,g 0 )= n i=1 Z 1/2 i Θ0 independence of μ implies (Λ/µ, g(µ)) Θ (n) R (p i, µ, g(µ)), dθ (n) 0 dµ d ln Θ(n) R =0 d ln µ n = γ i (g(µ)) i=1 RG evolution resums α n s (µ 2 ) ln n (Q 2 /µ 2 ) into α n s (Q 2 ) example: e + e - hadrons, where Q 2 = s

88 Example: Higgs production from gluon fusion gluon density is rapidly increasing as x 0 Higgs production occurs near partonic threshold total energy of gluons in the final state is need to resum soft-gluon emission E X = ŝ m2 H 2m H 0 Higgs qt distribution Bozzi Catani de Florian Grazzini 2005 At small qt, NLO blows up

89 Resummation: Sudakov form factor Sudakov (quark) form factor as matrix element of EM current ( ) Q Γ µ (p 1,p 2 ; µ 2 2, ɛ) < 0 J µ (0) p 1,p 2 > = v(p 2 )γ µ u(p 1 ) Γ µ 2, α s(µ 2 ), ɛ obeys evolution equation Q 2 [ ( )] Q 2 Q 2 ln Γ µ 2, α s(µ 2 ), ɛ RG invariance requires µ dg dµ = 1 2 K is a counterterm; G is finite as ε 0 = µ dk dµ = γ K(α s (µ 2 )) γk is the cusp anomalous dimension solution is Γ ( Q 2, ɛ ) = exp { 1 2 Q 2 0 dξ 2 ξ 2 [ K ( α s (µ 2 ), ɛ ) ( )] Q 2 + G µ 2, α s(µ 2 ), ɛ Korchemsky Radyushkin 1987 [ G ( 1, ᾱ s (ξ 2, ɛ), ɛ ) 1 2 γ K (ᾱs (ξ 2, ɛ) ) ( )] } Q 2 ln ξ 2

90 cusp anomalous dimension loop expansion of the cusp anomalous dimension γ (i) K =2C i α s (µ 2 ) π ( αs (µ 2 ) 2 ) + KC i + π with K = ( ) ζ 2 C A 10 9 T F N f

91 Factorisation of a multi-leg amplitude Mueller 1981 Sen 1983 Botts Sterman 1987 Kidonakis Oderda Sterman 1998 Catani 1998 Tejeda-Yeomans Sterman 2002 Kosower 2003 Aybat Dixon Sterman 2006 Becher Neubert 2009 Gardi Magnea 2009 M N (p i /µ, ɛ) = L p i = β i Q 0 / 2 ( 2pi p j S NL (β i β j, ɛ) H L µ 2, (2p i n i ) 2 ) n 2 i µ2 i value of Q0 is immaterial in S, J J i ( (2pi n i ) 2 n 2 i µ2 J i ( 2(βi n i ) 2 n 2 i ), ɛ ), ɛ to avoid double counting of soft-collinear region (IR double poles), Ji removes eikonal part from Ji, which is already in S Ji/Ji contains only single collinear poles

92 N = 4 SUSY in the planar limit colour-wise, the planar limit is trivial: can absorb S into Ji each slice is square root of Sudakov form factor M n = n i=1 [ ( )] 1/2 M [gg 1] si,i+1 µ 2, α s, ɛ h n ({p i },µ 2, α s, ɛ) β fn = 0 coupling runs only through dimension ᾱ s (µ 2 )µ 2ɛ =ᾱ s (λ 2 )λ 2ɛ Sudakov form factor has simple solution [ ( )] Q 2 ln Γ µ 2, α s(µ 2 ), ɛ = 1 2 ( αs (µ 2 ) n ( ) ) Q 2 nɛ [ ] γ (n) K π µ 2 2n 2 ɛ 2 + G(n) (ɛ) nɛ n=1 IR structure of N = 4 SUSY amplitudes

93 Jet definition - introduce auxiliary vector ni (ni 2 0) to separate collinear region - define a jet using a Wilson line along ni partonic jet ū(p) J ( (2p n) 2 n 2 µ 2 ), ɛ = <p ψ(0)φ n (0, ) 0 > Wilson line eikonal jet J Φ n (λ 2, λ 1 )=P exp ( 2(β n) 2 n 2 ), ɛ [ ig λ2 λ 1 dλn A(λn) = < 0 Φ β (, 0)Φ n (0, ) 0 > ]

94 Eikonal jet & cusp anomalous dimension eikonal jet does not depend on any kinematic scale J ( 2(β n) 2 single poles carry (β n) 2 /n 2 dependence, thus violate classical rescaling symmetry wrt β cusp anomalous dim n 2 ), ɛ = exp { 1 4 µ 2 0 dλ 2 λ 2 [ δ Ji ( αs (λ 2, ɛ) ) 1 2 γ K (ᾱs (λ 2, ɛ) ) ( 2(β n) 2 µ 2 )] } ln n 2 λ 2 δj is a constant double poles and kinematic dependence of single poles are controlled by cusp γk, as for quark form factor

95 Soft function S (c N ) ijkl S NL ( βa β b, α s (µ 2 ), ɛ ) < 0 Φ k,k β 2 (0, )Φ i,i i j k l = matrix evolution equation µ d dµ S ( JL βa β b, α s (µ 2 ), ɛ ) = N β 1 (, 0)Φ j,j [Γ S ] JN ( βa β b, α s (µ 2 ), ɛ ) S NL ( βa β b, α s (µ 2 ), ɛ ) β 3 (0, )Φ l,l β 4 (, 0) 0 > (c L ) i j k l ΓS soft anomalous dimension formal solution { S ( β a β b, α s (µ 2 ), ɛ ) = P exp 1 2 µ 2 0 dλ 2 λ 2 Γ S ( βa β b, α s (µ 2 ), ɛ )} Γ S = n=1 Γ (n) S ( αs (µ 2 ) π ) n

96 Soft anomalous dimension for an amplitude with an arbitrary # of legs Γ (2) S = K 2 Γ(1) S Aybat Dixon Sterman 2006 K is 2-loop coefficient of cusp anomalous dimension ΓS has cusp singularities like γj

97 Reduced soft function S JL (ρ ij, ɛ) = n i=1 S JL (β i β j, ɛ) J i ( 2(βi n i ) 2 n 2 i ), ɛ Dixon Magnea Sterman 2008 S does not have cusp singularities must respect rescaling βi κi βi S depends only on ρ ij = factorisation becomes M N (p i /µ, ɛ) = L (β i β j ) 2 2(β i n i ) 2 2(β j n j ) 2 n 2 i ( 2pi p j S NL (ρ ij, ɛ) H L µ 2, (2p i n i ) 2 ) ( (2pi n i ) 2 ) n 2 J i i µ2 n 2, ɛ i i µ2 n 2 j S has only single poles due to large-angle soft emissions

98 Reduced soft anomalous dimension evolution equation for reduced soft anomalous dimension j i ln ρ ij Γ S(ρ ij, α s )= 1 4 γ(i) K (α s) Becher Neubert; Gardi Magnea 2009 solution Γ S(ρ ij, α s )= 1 8 ˆγ K(α s ) i j with γ (i) K = C iˆγ K (α s ) ˆγ K =2 α s(µ 2 ) π only 2-eikonal-line correlations generalises 2-loop solution ln(ρ ij ) T i T j ˆδ S(α s ) + K Are there any 3(or more)-line correlations? Gardi Magnea say maybe; Becher Neubert say no n i=1 C i ( αs (µ 2 ) 2 ( ) + K (2) αs (µ 2 ) 3 ) + π π

99 Conclusions QCD is an extensively developed and tested gauge theory a lot of progress in the last few years in MonteCarlo generators NLO computations with many jets NNLO computations resummations better and better approximations of signal and background for Higgs and New Physics