QCD ai colliders adronici. Vittorio Del Duca INFN Torino

Transcription

1 QCD ai colliders adronici Vittorio Del Duca INFN Torino Cortona 26 maggio 2005

2 QCD Premio Nobel 2004! an unbroken Yang-Mills gauge field theory featuring asymptotic freedom confinement in non-perturbative regime (low ) many approaches: lattice, Regge theory, χ PT, large N c, HQET in perturbative regime (high ) QCD is a precision toolkit for exploring Higgs & BSM physics Q 2 LEP was an electroweak machine Q 2 Tevatron & LHC are QCD machines

3 Precision QCD Precise determination of strong coupling constant parton distributions electroweak parameters LHC parton luminosity α s Precise prediction for Higgs production new physics processes their backgrounds

4 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

5 Factorisation 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

6 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 ) double-parton scattering observed by Tevatron CDF in the inclusive sample p p γ + 3 jets potentially important at LHC σ D σs 2 diffractive events α S

7 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

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

9 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

10 Matrix-element MonteCarlo generators multi-parton generation: processes with many jets (or W/Z/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 SHERPA F. Krauss et al. 2003

11 HERWIG Shower MonteCarlo generators B. Webber et al being re-written as a C++ code (HERWIG++) PYTHIA T. Sjostrand 1994 CKKW and more S. Catani F. Krauss R. Kuhn B. Webber 2001 a procedure to interface parton subprocesses with a different number of final states to parton showers MC@NLO S. Frixione B. Webber 2002 a procedure to interface NLO computations to shower MC s

12 NLO features 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

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

14 Desired NLO cross sections

15 NLO history of final-state distributions e + e 3 jets K. Ellis, D. Ross, A. Terrano 1981 e + e 4 jets Z. Bern et al., N. Glover et al., Z. Nagy Z.Trocsanyi pp 1, 2 jets K. Ellis J. Sexton 1986, W. Giele N. Glover D. Kosower 1993 pp 3 jets Z. Bern et al., Z. Kunszt et al , Z. Nagy 2001 pp γγ B. Bailey et al 1992, T. Binoth et al 1999 pp γγ + 1 jet Z. Bern et al. 1994, V. Del Duca et al pp V + 1 jet W. Giele N. Glover & D. Kosower1993 pp V + 2 jet Bern et al., Glover et al , K. Ellis & Campbell 2003 pp V b b K. Ellis & J. Campbell 2003 pp V V Ohnemus & Owens, Baur et al , Dixon et al pp Q Q pp Q Q + 1 jet pp H + 1 jet pp HQ Q Dawson K. Ellis Nason 1989, Mangano Nason Ridolfi 1992 A. Brandenburg et al. 2005? C. Schmidt 1997, D. De Florian M. Grazzini Z. Kunszt 1999 W. Beenakker et al. ; S. Dawson et al e + e 4 fermions Denner Dittmaier Roth Wieders 2005

16 J. Campbell, KITP collider conf 2004

17 J. Campbell, KITP collider conf 2004

18 e + e 3 jets NLO assembly kit 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.

19 NLO production rates Process-independent procedure devised in the 90 s slicing Giele Glover & Kosower subtraction dipole Catani & Seymour antenna Kosower; Campbell Cullen & Glover σ = σ LO + σ NLO = dσm B J m + σ NLO m σ NLO = dσm+1j R m+1 + dσmj V m Frixione Kunszt & Signer; Nagy & Trocsanyi 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

20 Observable (jet) functions vanishes when one parton becomes soft or collinear to another one J m Jm(p1,..., pm) 0, if pi pj 0 dσ B m is integrable over 1-parton IR phase space vanishes when two partons become simultaneously soft and/or collinear J m+1 J m+1(p 1,..., p m+1) 0, if p i p j and p k p l 0 (i k) R and V are integrable over 2-parton IR phase space observables are IR safe J n+1 (p 1,.., p j = λq,.., p n+1 ) J n (p 1,..., p n+1 ) if λ 0 J n+1 (p 1,.., p i,.., p j,.., p n+1 ) J n (p 1,.., p,.., p n+1 ) if p i zp, p j (1 z)p for all n m

21 NLO complications loop integrals are involved and process-dependent more particles many scales lenghty analytic expressions even though it is known how to compute loop integrals with 2 n particles no one-loop amplitudes with n > 4 have been computed (either analytically or numerically) no fully numeric methods yet for hadron collisions counterterms are subtracted analytically

22 Is NLO enough to describe data? 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) NLO + NLL good agreement with data (with use of updated FF s by Cacciari & Nason) The CDF value in the inset was preliminary. The published value is (CDF hep-ex/ ) 19.4 ± 0.3(stat) (syst) nb Cacciari, Frixione, Mangano, Nason, Ridolfi 2003

23 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

24 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

25 Summary of α S (M Z ) S. Bethke hep-ex/ world average of α S (M Z ) using MS and NNLO results only α S (M Z ) = ± (cf α S (M Z ) = ± outcome almost identical because new entries wrt LEP jet shape observables and 4-jet rate, and HERA jet rates and shape variables - are NLO ) filled symbols are NNLO results

26 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 in short, NNLO is relevant where NLO fails to do its job

27 NNLO state of the art Drell-Yan W, Z total cross section production Hamberg, van Neerven, Matsuura 1990 Harlander, Kilgore 2002 rapidity distribution Anastasiou et al Higgs production total cross section fully differential cross section e + e 3 jets Harlander, Kilgore; Anastasiou, Melnikov 2002 Anastasiou, Melnikov, Petriello 2004 the and terms the Gehrmanns, Glover C 3 F 1/N 2 c

28 NNLO cross sections Sector decomposition Denner Roth 1996; Binoth Heinrich 2000 Anastasiou, Melnikov, Petriello 2003 the only method which, so far, yields useful NNLO cross sections cancellation of divergences is performed numerically process dependent Subtraction process independent cancellation of divergences is semi-analytic

29 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

30 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

31 Higgs production at LHC 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

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

33 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 V g q q V gg H (in the m t limit) R. Harlander W. Kilgore; C. Anastasiou K. Melnikov 2002

34 NNLO cross sections universal IR structure process-independent procedure universal collinear and soft currents 3-parton tree splitting functions J. Campbell N. Glover 1997; S. Catani M. Grazzini 1998; A. Frizzo F. Maltoni VDD 1999; D. Kosower parton one-loop splitting functions Z. Bern W. Kilgore C. Schmidt VDD ; D. Kosower P. Uwer 1999; D. Kosower 2003 universal subtraction counterterms several ideas and works in progress D. Kosower; S. Weinzierl; the Gehrmanns & G. Heinrich 2003 S. Frixione M. Grazzini 2004; G. Somogyi Z. Trocsanyi VDD 2005 but completely figured out only for e + e 3 jets the Gehrmanns & N. Glover 2005

35 10 9 LHC parton kinematics J. Stirling 10 8 x 1,2 = (M/14 TeV) exp(±y) Q = M M = 10 TeV M = 1 TeV Q 2 (GeV 2 ) M = 100 GeV y = M = 10 GeV HERA fixed target x

36 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

37 Parton distribution functions (PDF) factorisation for the structure functions (e.g. F ep 2, F ep ) L 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)

38 PDF s general structure of the quark-quark splitting functions P qi q k = P qi q k =δ ik Pqq v + Pqq s P qi q k = P qi 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 )

39 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

40 Numerical examples exact NNLO results, estimates from fixed moments and leading small- x term

41 HERA F 2 Bjorken-scaling violations H1, ZEUS: ongoing fits for PDF s; so far NNLO not included

42 global fits MRST: Martin Roberts Stirling Thorne 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 NLO evolution and fixed moments of NNLO no prompt photon data included in the fits

43 New formal developments BCF recursion relation for n-gluon tree amplitudes Britto Cachazo Feng 2004 ditto + Witten 2005 K 1,k = k i=1 p i A k+1, A n k+1 are on-shell tree amplitudes Hatted momenta are shifted by a complex amount

44 Momentum shift introduce spinors in chiral representation represent momenta through spinors shift momenta ˆλ 1 = λ 1 K 2 1,k n K 1,k 1 λ n ˆ λ 1 = λ 1 ˆλ n = λ n ˆ λ n = λ n + K 2 1,k n K 1,k 1 λ 1 to conserve momentum K 1,k also shifted by a quantity λ n λ1 BCF recursion relations also for gluon amplitudes with quarks Luo Wen 2005 massive particles Badger Glover Khoze Svrcek 2005 special one-loop gluon amplitudes Bern Dixon Kosower 2005

45 6-gluon amplitude 220 Feynman diagrams use fixed helicity, colour decomposition, symmetries Mangano Parke Xu 1988 use BCF relation simpler (although at the cost of spurious singularities) 2 (6 + 1) 5 7-gluon amplitude compactness of formulae even more striking BCF confirms results found analysing IR behaviour of N=4 SUSY one-loop amplitudes Bern Dixon Kosower VDD 2004

46 Twistor space A half-fourier transform of spinor space λȧ = i µȧ µȧ = i λȧ twistor space is 4-dimensional (λ 1, λ 2,, µ 1, µ 2 ) MHV tree amplitudes ij n1 except for momentum conservation (λ a, λȧ) depend only on δ( i p i ) = for each leg λ i d 4 x exp(ix i spinors, λ i λi ) introduce differential operator F ijkȧ = ij λȧk + jk λȧi + ki λȧj i, j, k have collinear support if amp A is annihilated by F ijkȧ n-gluon amplitudes in twistor space

47 MHV rules for gluon amplitudes continue MHV amplitudes off shell A tree,mhv n = ij n1 with η 2 = 0 sew vertices with scalar propagators = Cachazo Svrcek Witten 2004 ij 4 η n 1 η results independent of η ; agree with Feynman MHV rules can be extended to fermions Georgiou Glover Khoze; Wu Zhu 2004 Higgs bosons + gluons Badger Dixon Glover Khoze 2004 vector bosons Bern Forde Kosower Mastrolia 2004 massive fermions Schwinn Weinzierl 2005

48 Conclusions QCD is an extensively developed and tested gauge theory a lot of progress in the last 4-5 years in MonteCarlo generators NLO cross sections with one more jet NNLO computations better and better approximations of signal and background for Higgs and New Physics new formal developments: QCD as a string theory in twistor space novel ways of computing (analytically) tree multi-parton amplitudes and (N=4) loop amplitudes E. Witten 2003