QCD AT THE LHC. Massimiliano Grazzini (INFN, Firenze) Italo-Hellenic School, june 2005

Transcription

1 QCD AT THE LHC Massimiliano Grazzini (INFN, Firenze) Italo-Hellenic School, june 25

2 Outline Introduction: Asymptotic freedom and the parton model Infrared divergences The factorization theorem DGLAP equations

3 QCD ad hadron colliders: factorization theorem Parton distributions QCD tools: Fixed order calculations: LO, NLO, NNLO Parton shower Montecarlos Corcella s lectures Resummed calculations Concluding remarks

4 Basics QCD Quantum - Chromo - Dynamics Quantum theory of colour: It is a non abelian gauge theory with SU(N c ) gauge group Completely specified given the number of colours N c L = 1 4 F a µνf µνa + N f f=1 ψ i f [iγ µ (D µ ) ij m f δ ij ] ψ j f (D µ ) ij = δ ij µ igt a ija a µ Covariant derivative i, j = 1...N c F a µν = µ A a ν ν A a µ + gf abc A b µa c ν Gluon field tensor Difference with QED: gluons are charged and interact among themselves

5 α i p β j Feynman rules i(/p + m) αβ p 2 m 2 + iɛ δij µ p ν a Gluon spin pol. tensor: gauge dependent b i p 2 + iɛ dµν (p) δ ab α i g µ β j ig(t a ) ij (γ µ ) αβ p 1 µ 2 a 2 p 2 gf a 1a 2 a 3 [g µ 1µ 2 (p 1 p 2 ) µ 3 p 3 +g µ 2µ 3 (p 2 p 3 ) µ 1 ] +g µ 3µ 1 (p 3 p 1 ) µ 2 µ 1 a 1 µ 3 a 3 µ 2 a 2 µ 3 a 3 p 2 p 3 p 1 µ 1 a 1 p 4 ig 2[ f ba 1a 2 f ba 3a 4 (g µ 1µ 3 g µ 2µ 4 g µ 1µ 4 g µ 2µ 3 ) ] +(2 3) + (2 4) µ 4 a 4

6 The spin polarization tensor is d µν (p) = λ ε µ (λ) (p)εν (λ) (p) and depends on the gauge choice d µν (p) = g µν + (1 α) pµ p ν p 2 + iɛ g µν + pµ n ν + p ν n µ p n n 2 covariant gauges pµ p ν (p n) 2 axial gauges In covariant gauges Lorentz invariance is manifest but ghosts must be included to cancel effect of unphysical gluon polarizations µ b a p b i p 2 + iɛ δab a c gf abc p µ In physical gauges (e.g. n µ A µ =, n arbitrary direction) only two tranverse polarizations are present more transparent physical picture: for lowest order or approximated calculations physical gauges make life simpler

7 Colour algebra The calculation of Feynman diagrams is similar to QED: just keep into account colour factors The explicit form of colour matrices is not important in practice [T a, T b ] = if abc T c (T a ) ij = t a ij fundamental (T a ) bc = if abc Useful relations: T r(t a t b ) = T R δ ab T R = 1/2 adjoint T r(t a ) = (t a t a ) il = C F δ il C F = N 2 c 1 2N c f adc f bdc = C A δ ab C A = N c Exercise: prove the above expressions for C F, C A (hint: t a, I form a basis for N c N c hermitian matrices)

8 Asymptotic freedom QCD is a renormalizable gauge theory Ultraviolet (UV) singularities appear in loop diagrams but they can be removed by the renormalization procedure Renormalization has two steps: Regularization: allows to make sense of divergent loop integrals Subtraction: redefine the coupling α S = g 2 /(4π) UV cut-off + = { Λ 2 cut α B 1 + α B β p 2 d 4 k } (k 2 ) 2 + O(α2 B) { } α B 1 + α B β (log Λ2 cut µ 2 + log µ2 p 2 ) + O(α B 2 ) } = α(µ 2 ) {1 + β α(µ 2 ) log µ2 p 2 + O(α B 2 ) coefficient of UV behaviour

9 where we have defined the renormalized coupling ( ) α(µ 2 ) α B 1 + β α B log Λ2 cut µ 2 + O(αB) 2 Divergence desappeared but an arbitrary scale How does the coupling depend on the scale? µ is now present The theory does not predict the absolute value of the coupling but its dependence on the scale can be predicted dα S (µ 2 ) d log µ 2 = β α 2 S(µ 2 ) + O(α 3 S)!!"# $ it is one the renormalization group equations " % & % if for a given scale µ we have α S (µ 2 ) << 1 a perturbative solution of the renormalization group equation can be given " % '% " # % " #

10 In QED the dependence of the coupling on the scale has a simple physical interpretation The vacuum around a pointlike charge becomes polarized due to the emission of e + e pairs and produces a screening effect But: : the effective coupling decreases with the distance β < In QCD gluons are charged and provide a positive contribution to β!!! " " " "!! " " "! "! "!! "! " "! "! "! "! " "!!!! β = 11N c 2n F 12π > for n F < 16

11 Intuitively: gluons are charged and spread colour charge over larger distances: anti-screening effect ASYMPTOTIC FREEDOM In processes with large momentum transfer we can use perturbation theory even if we have not solved the full theory Nobel prize in Physics 24 D. Gross, H.D. Politzer, F. Wilczek Λ QCD scale at which the coupling becomes strong

12 Parton model Asymptotic freedom at large tranferred momenta hadrons behave as collections of free (weaklyinteracting) partons at small scales the interaction becomes strong but if we are not interested in the details of hadronic processes (consider inclusive enough final states) we can use the parton picture e + e annihilation produce a hard q q pair at scale Q (short time scale τ = 1/Q ) which travel far apart as free

13 Deep inelastic scattering e e Let us consider the process ep ex γ q Q 2 = q 2 = (k k ) 2 Λ 2 QCD If Q 2 < MZ 2 the cross section is dominated by one-photon exchange The photon acts as a probe of the proton structure p X Parton interactions in the proton characterized by a large time scale τ 1/Λ QCD with respect to τ hard = 1/Q! q Scattering is incoherent on the single partons zp σ(p) dzf(z) ˆσ(zp) p

14 The question: does the parton picture survive when radiative corrections are included?

15 e + e annihilation Consider the O(α S ) corrections to the total cross section Real: Kinematics: x i = 2p i Q/Q 2 = 2E i /Q energy fractions p 1 + p 2 + p 3 = Q x 1 + x 2 + x 3 = 2 (p 1 + p 3 ) 2 = (Q p 2 ) 2 = (1 x 2 )Q 2 Soft limit: x 3 Collinear limit: p 1 p 3 x 2 1 p 2 p 3 x 1 1

16 Virtual: Squaring the amplitudes real and virtual have similar structure but different kinematics real virtual Cut lines on shell

17 Real contribution: σ R = 1 Phase space Matrix element dx 1 dx 2 dx 3 δ(2 x 1 x 2 x 3 ) M real (x 1, x 2, x 3 ) 2 M real (x 1, x 2, x 3 ) 2 = σ C F α S 2π x x 2 2 (1 x 1 )(1 x 2 ) Singular when x 1, x 2 1 The singularity cannot be integrated! Let s look at its structure: Numerator: x x 2 2 = 1 + (1 x 3 ) 2 2(1 x 1 )(1 x 2 ) Denominator: 1 (1 x 1 )(1 x 2 ) = 1 2 x 1 x 2 Does not produce singularities ) ( 1 1 x x 2 = 1 x 3 ( 1 1 x x 2 ) The real contribution can be seen as a sum of independent terms divergent both in the soft and in the two collinear regions plus a finite remainder soft sing. collinear sing.

18 Q p 1 p 2 p 3 In the region M real 2 ( αs 2π ) P gq (x 3 ) = C F 1 + (1 x 3 ) 2 x 1 1 (p 2 p 3 ) x x 1 P qg (x 3 ) Universal q g splitting kernel Virtual contribution: p 1 1 dx 1 dx 2 δ(2 x 1 x 2 ) M virt dx 3 Q Taking the square the diagrams have the same structure of the real but different kinematics Differences in kinematics affect only finite parts The sign is opposite by unitarity The divergences in the real and virtual contributions cancel out and a finite result is recovered p 2

19 Infrared divergences k Originate in theories with massless particles and are of two kinds Soft: the energy of a gluon vanishes p 1 (p + k) 2 = 1 2E p E k (1 cos θ) Collinear: two partons become parallel Physically a hard parton plus a soft gluon or two very close partons are indistinguishable They correspond to degenerate states hard parton hard parton +soft gluon 2 collinear partons Infrared divergences are a manifestation of long-distance effects

20 Even in QED we cannot separate an electron from an electron + a soft photon, or an electron from an electron plus a collinear photon Kinoshita-Lee-Naumberg theorem: if we limit ourself to considering quantities inclusive over initial and final states soft and collinear (degenerate) configurations infrared divergences cancel out The answer: intuitive parton picture survives to the computation of radiative corrections provided we consider inclusive enough processes NOTE: Phase space is flat Matrix elements are enhanced in soft and collinear regions The hadronic final state is typically formed by jets of collinear particles plus soft particles

21 In order to cancel the divergences it is not necessary to integrate over the full phase space According to the KLN theorem we can define a wide class of IR safe observables Leading Order (LO): σ LO = m M () ({p i }) 2 F m ({p i })dφ m Next-to-Leading Order (LO): Tree level matrix element σ NLO = m+1 dσ R + Measurement function m dσ V Phase space m number of partons at LO (e.g. 3 jet ) m = 3 Here one more parton Here same number of partons but one-loop matrix element To be sure that the presence of we should have: F does not spoil the cancellation F m+1 (...p i,...p j...) F m (...p i + p j...) if p i p j or p j

22 What about hadronization? We have seen that matrix elements are enhanced in soft and collinear regions: soft radiation fulfills the property of angular ordering: the angle at which soft gluons are emitted is continuosly reduced radiation is confined in collimated cones around the primary parton Hadronization is LOCAL: it does not spoil parton picture Moreover: the colour flow in the jet evolution is such that colour singlets are close in phase space

23 Hard processes with hadrons in the initial state DIS:! zp q Partonic cross section p Parton density Consider O(α S ) corrections to the partonic cross section Virtual: Real:

24 In the calculation we find infrared divergences Inclusive final state KLN cancellation of final state singularities However in the initial state we have only one parton and we cannot sum over degenerate states uncancelled collinear singularity α S (Q 2 ) 1 dθ 2 θ 2 Q 2 dk 2 T k 2 T Actually the collinear divergence would be regularized by a physical cutoff Q of the order of the typical hadronic scale The singularity implies the existence of long distance effects Multiple emission α n S log n Q 2 /Q 2 to be resummed Both problems are solved by the IN SHORT: Collinear singularities can be absorbed in bare parton densities FACTORIZATION THEOREM f (x) f(x, Q 2 )

25 Physical parton densities thus become scale dependent This is possible only if this redefinition is independent on the hard process Phase space: Propagator: d 3 k 2k k dk d cos θ dθ 2 1 (p k) 2 1 p k (1 cos θ) 1 θ 2 Vertex: /p/ɛ(k)(/p /k) θ in a physical gauge (helicity conservation in the quark gluon vertex!) dθ dθ2 θ 2 θθ θ2 θ 2 dθ 2 1 θ 2 θ not singular enough! In a physical gauge interferences can be neglected

26 General strategy: decompose diagrams in 2PI blocks (such that they cannot be disjoint by cutting only two lines) 2PI blocks are free of collinear singularities in a physical gauge 2PI = 2PI K T > µ F Introduce an arbitrary separation scale µ F Process dependent but finite 2PI K T < µ F.. Universal but divergent 2PI f () Reabsorb the divergent part in the redefinition of f(x) analogy with renormalization

27 σ(p, Q) = a 1 dzf a (z, µ 2 F )ˆσ a (zp, α S (Q 2 ); µ 2 F ) µ F Physical cross sections cannot depend on µ F µ F dependence must cancel between f a and ˆσ a The choice of µ F is arbitrary but if µ F is too different from the hard scale Q log Q 2 /µ 2 F reappear that spoil the perturbative expansion Introduction of an arbitrary scale Parton densities become scale dependent Such scale dependence is associated with the resummation of large collinear logs The scale dependence is perturbatively computable DGLAP equations

28 DGLAP equations Q 2 k n ~x n p+k Tn z n Pn.. f(µ 2 F ) = f E(µ 2 F /Q 2 ) The important region is: Q 2 > k 2 T n >...k 2 T 1 > Q 2 n z 1 Q P 1 f () k 1 ~x 1 p+k T1 where the maximum power of log Q 2 /Q 2 is generated iterative structure f(x, Q 2 ) = f (x) + Q 2 Q 2 dk 2 T n k 2 T n 1 x dz n z n P n (α S (k 2 T n), z n )f(x/z n, k 2 T n) Taking derivative with respect to Q 2 Q 2 f(x, Q2 ) Q 2 = 1 x dz z P (α S(Q 2 ), z)f(x/z, Q 2 ) First order differential equation: can be solved if f is known at a reference scale Q

29 Probabilistic interpretation: a Q 2 a x Q 2 P ab (α S (Q 2 ), z) Q 2 x Q 2 fa = b P ab x/z fb Probability to find parton a in parton at scale Q 2 b P ab (α S, z) = α S 2π P () ab (z) + ( αs 2π P () ab and P (1) ab P (2) ab ) 2 P (1) ab (z) + ( αs 2π are known (now also!) P () ab ) 3 (2) P ab (z) +... Solving DGLAP equation using allows us to resum Leading Logarithmic (LL) contributions α n S log n Q 2 /Q 2 With we resum also Next-to-leading terms (NLL) P (1) ab Convolution α n S log n 1 Q 2 /Q 2 conservation of longitudinal momentum

30 [ 1 + z P qq () 2 (z) = C F + 3 ] δ(1 z) (1 z) distribution defined as 1 f(z) (1 z) + 1 f(z) f(1) 1 z [ P gg () z (z) = 2C A + 1 z (1 z) + z ( C A 2 3 T R n F ) δ(1 z) ] + z(1 z) [ ] 1 + (1 z) P gq () 2 (z) = C F z P () qg (z) = T R [ z 2 + (1 z) 2]

31 Exercise: compute P () qq Real: Work in axial gauge Use Sudakov parametrization: k µ = zp µ + k µ T + k2 k 2 T 2zp n nµ gauge vector (p k) µ = (1 z)p µ k µ T k 2 T 2(1 z)p n nµ Extract the leading contribution in 1/k 2 and check that P real qq (z) = C F 1 + z 2 1 z Virtual: must be of the form P virt qq = A δ(1 z) Can be obtained from real by using fermion number conservation 1 P qq (x)dx = integral of valence quark density cannot depend on Q 2

32 Solving DGLAP equation Consider for simplicity the case of non-singlet Q 2 f(x, Q2 ) Q 2 = Define Mellin moments f N = 1 f(x)x N 1 dx 1 x (f g) N = In Mellin space DGLAP becomes dz z P qq(α S (Q 2 ), z)f(x/z, Q 2 ) = = 1 1 (f g)(x) = 1 x (f g)(x) x N 1 dx dxx N 1 1 x dz z f(z)g(x z ) dz z f(z)g ( x z z N 1 t N 1 dt dzf(z) g(t) = f N g N Q 2 f N (Q 2 ) Q 2 = γ N,qq ( αs (Q 2 ) ) f N (Q 2 ) DGLAP equation is diagonal ) Anomalous dimension

33 Since: γ ab (N) = 1 The small x region corresponds to P ab (x)x N 1 small N, whereas x 1 selects large N γ gg 2C A N 1 N small γ aa 2C a log N as N Scaling violations are: Positive at small x Slightly negative at large x f(x,q 2 ) Q 2 Main effect of increase in is shift of partons from larger to smaller x Q 2 x

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35 QCD at hadron colliders In hadron collisions all phenomena are QCD related but we must distinguish between hard and soft processes Hard subprocess x 1 p 1 x 2 p 2 h 1 h 2 Production of low pt hadrons: most common events Soft underlying event Only hard scattering events can be controlled via the factorization theorem

36 h 1 x 1 a Q 2 σ(p 1, p 2 ; Q 2 ) = a,b Same parton densities measured in DIS! 1 dx 1 1 dx 2 f h1,a(x 1, µ 2 F ) f h2,b(x 2, µ 2 F ) h 2 x 2 b X ˆσ ab (x 1 p 1, x 2 p 2, α S (Q 2 ), µ 2 F ) Proof of factorization: similar to DIS: absorb initial state collinear divergences into a redefinition of parton distributions but... The proof that soft gluons do not spoil factorization is very difficult: soft gluons (large wavelenght) can transfer colour information between the two initial state hadrons Explicit calculations have shown that factorization breaking effects are present but are power suppressed in the high energy limit

37 LHC parton kinematics Define rapidity y = 1 2 ln E + p z E p z x 1,2 = (M/14 TeV) exp(±y) Q = M M = 1 TeV At LO we have: 1 7 x 1 x 2 S = Q 2 x 1,2 = Q/ S e ±y Q 2 (GeV 2 ) M = 1 GeV M = 1 TeV Varying Q and y y = M = 1 GeV 6 Sensitivity to different x 1, x HERA fixed target x h 1 y x 1 x 2 Q 2 h 2

38 QCD tools We limit ourselves to considering hard scattering events: the starting point is the factorization theorem σ(p 1, p 2 ; Q 2 ) = a,b 1 dx 1 1 dx 2 f h1,a(x 1, µ 2 F ) f h2,b(x 2, µ 2 F ) ˆσ ab (x 1 p 1, x 2 p 2, α S (Q 2 ), µ 2 F ) Predictions at hadron colliders require good knowledge of: Parton distributions: Determined by global fits to different data sets Partonic hard cross section: computed with different methods

39 Fixed order calculations: truncate at a given order in LO, NLO, NNLO α S Resummed calculations: resum (where possible) large logarithmic contributions improving the reliability of fixedorder expansion: necessary when two different scales are present Parton shower simulations: use colour coherence to approximate multiparton QCD matrix elements quantum interferences approximated by angular ordering constraint See G. Corcella lectures

40 Parton distributions Determined by global fits to different data sets Standard procedure: Parametrize at input scale Q = 1 4 GeV xf(x, Q 2 ) = Ax α (1 x) β (1 + ɛ x + γx +...) Impose momentum sum rule: Evolve to desired Q 2 1 a Then fit to data to obtain the parameters Main groups: MRST, CTEQ Now also: Alekhin, ZEUS, NNPDF.. dxxf(x, Q 2 ) = 1 and compute physical observables

41 Typical processes: DIS: Fixed target: valence quark densities HERA: at small x f g, f sea Drell-Yan quark densities h 1 h 2 q q V pp collisions: sensitive to antiquarks and sea densities p p collisions: sensitive to flavour asymmetries of valence quarks

42 d(x)/ū(x) from FNAL E866/Nusea 8 GeV p + p and p + d µ + µ Obtain neutron pdfs from isospin symmetry: u d ū d σ pp 4 9 u(x 1)ū(x 2 ) d(x 1) d(x 2 ) x 1 x 2 σ pn 4 9 u(x 1) d(x 2 ) d(x 1)ū(x 2 ) Assuming σ pd σ pp + σ pn and using d(x) 4u(x) σ pd 2σ pp x1 x d(x 1 ) u(x 1 ) d(x 1 ) d(x 2 ) u(x 1 )ū(x 2 ) ( 1 + d(x ) 2 ) ū(x 2 ) σ pd 2σ pp 1 2 ( 1 + d 2 ū 2 )

43 W asymmetry in p p collisions u d = d W + in the proton d ū = u W in the proton If u in the proton is faster than d (u(x) > d(x)) W + (W ) produced mainly in p ( p) direction The W asymmetry is a measure of u(x 1 )d(x 2 ) d(x 1 )u(x 2 ) u(x 1 )d(x 2 ) + d(x 1 )u(x 2 ) In practice W lν A(y) = measure the charged lepton asymmetry dσ(w + ) dy dσ(w ) dy dσ(w + ) dy + dσ(w ) dy Charge Asymmetry CDF (11 pb -1 e+!) CTEQ-3M RESBOS MRS-R2 (DYRAD) MRS-R2 (DYRAD)(d/u Modified) MRST (DYRAD) CTEQ-3M DYRAD !Lepton Rapidity!

44 Jets at Tevatron f g at large x MRST 22 and D jet data,! S (M Z )=.1197, " 2 = 85/82 pts 1.!"# # "#"! !!"""!#.5.5!!"""!# 1. 1.!!"""!# !!"""!# 2. 2.!!"""!# (Data - Theory) / Theory !"# # #"! !"# # #"! !"# # #"! 2. 1 Note: Strong interplay between possible new physics effects at large E T and extraction of the gluon E T (GeV)

45 Typical behaviour of parton densities in the proton x f(x,q) Overview of Parton Distribution Functions of the Proton CTEQ5M Q = 5 GeV Gluon / 15 d bar u bar s c u v d v (d bar -u bar ) * All densities vanish as x 1 At x Valence quarks vanish - Strong rise of the gluon, which becomes dominant - Also sea quarks increase driven by the gluon through g q q.8 x

46 In recent years a consistent effort has been devoted to the study of pdf uncertainties Experimental errors: - propagation of errors on parameters ( hessian method) - parameter scan ( lagrange-multiplier method) pdf with errors are now available

47 Overall picture is that quark densities are reasonably well constrained by DIS and DY data Luminosity function at TeV RunII.2 Luminosity function at LHC.1.1 Fractional Uncertainty #.1 G-G $ Q-Q --> W + (W - ) # Fractional Uncertainty ".1 # Q-Q --> W + G-G " ±.1# #!.1" " -.1 $ Q-Q --> "* (Z) -.1 # Q-Q --> W !s ^ (GeV) !s ^ (GeV) The gluon is instead more uncertain Note that: Also prompt photon production is sensitive to f g at large x but... Theoretical problems and possible inconsistencies between data sets suggested not to use it

48 What about theoretical errors? They are more difficult to quantify Hints: How to asses bias from initial parametrization? Alternative approach: Neural network pdfs S. Forte et al. Conservative partons: MRST23 Study stability with respect to cuts From 2 to 12 data points Select a data set where consistent predictions are available: Alekhin 23 DIS only at LO, NLO, NNLO Bottom line: more work is needed in this direction...

49 QCD calculations at LO Powerful techniques are available to compute multiparton matrix elements (ME): helicity amplitudes / colour - subamplitude decomposition direct ME generation from interaction Lagrangian These predictions are combined with automatic integration over multiparticle phase space Cross sections for up to 8 jet production can be computed Important because multijet final states can mimic signatures induced by physics beyond the standard model Contrary to standard MC event generator they correctly include large angle radiation Important to devise search strategies See G. Corcella lectures for a more complete description

50 When precision is an issue LO predictions are not sufficient LO calculations are affected by large uncertainties Physical quantities should be independent on renormalization and factorization scales but truncation of perturbative series at n-order induces a scale dependence of (n+1) order d!/de T (nb/gev) NLO LO.1 < " <.7 E T = 1 GeV CDF Data ! R /E T At LO the scale at which to evaluate is not even defined! α S At LO jets are partons: it is only including NLO corrections that we can distinguish among jet algorithms

51 QCD calculations at NLO Carried out over a period of 25 years Main difficulties: one has to consider virtual and real corrections They are affected by different kinds of singularities: UV singularities affect only virtual corrections: removed by renormalization Remember example in collisions e + e IR (soft and collinear) singularities: present in both virtual and real corrections use dimensional regularization (work in d = 4 2ɛ dimensions) Even if the relevant tree and loop amplitudes are known the integrals over unresolved partons cannot be performed analytically when arbitrary cuts are imposed!

52 An example: suppose you have to calculate Where the function F (x) is too complicated to perform the integral analytically I = lim ɛ The two terms are separately divergent NOTE: Phase space slicing: ( 1 dx x xɛ F (x) 1 ) ɛ F () x: energy of a gluon or angle between two partons Slice the integration region < x < δ, δ < x < 1 If we choose δ 1 we can approximate F (x) F () and write ( I lim ɛ F () = F () log δ + δ 1 δ ɛ dx x xɛ + 1 dx x F (x) δ virtual contribution but their sum is finite K. Fabricus, I. Schmitt, G. Kramer, G. Schierholz (1981) dx x xɛ F (x) 1 ɛ F () can be computed numerically The approximation relies on δ 1 but if δ is too small the two contributions are separately large )

53 Subtraction method: Rewrite the same integral as ( 1 I = lim ɛ = 1 dx x dx x xɛ [F (x) F ()] + F () [F (x) F ()] The integration can now be done numerically Both methods require knowledge of F () General algorithms based on these two methods are now available R.K. Ellis, D.A.Ross, A.E.Terrano (1981) 1 dx x xɛ 1 ) ɛ F () The divergence cancels out! behaviour of the matrix element in the singular region W.T.Giele. E.W.N.Glover (1992) W.T.Giele. E.W.N.Glover, D.Kosower (1993) The capability to compute the relevant tree-level and one-loop amplitudes allowed to compute many reactions at NLO accuracy S.Frixione, Z.Kunszt, A. Signer (1995) S.Catani, M. Seymour (1996)

54 NLO history e + e 3 jets e + e 4 jets R.K. Ellis, D.A.Ross, A.E.Terrano (1981) Z. Nagy, Z. Trocsanyi (1997), L. Dixon, A. Signer (1997) pp 1, 2 jets pp 3 jets pp γγ pp γγ + 1 jet pp V + 1 jet pp V + 2 jet pp V V pp H + 1 jet D. de Florian, Z. Kunszt, MG (1999) pp HQ Q W. Beenakker et al.(21), S. Dawson et al. (21) pp Q Q pp Q Q + 1 jet Z. Kunszt and D.E. Soper (1992),W. Giele, N. Glover, D. Kosower (1993) Z. Nagy (21) B. Bailey et al. (1992), T. Binoth et al. (1999) V. del Duca et al. (23) W. Giele, N. Glover, D. Kosower (1993) J. Campbell, K. Ellis (23) J.Ohnemus, J. Owens (1991), Dixon et al. (1999) S. Dawson, K. Ellis, P. Nason, (1989), M.L. Mangano et al. (1992) A. Branderburg et al. (in progress)

55 Available NLO programs NLOJET++ Z. Nagy General pourpose C++ code to compute cross sections in DIS and hadron collisions e + e 4 jets Inclusive k algorithm p p 3 jets e + e ep ( 3 + 1) jets [nb/gev] d /dp T (1) LO NLO p T > 2GeV, <2 p T > 8GeV K(E T (1) ) E T (1) Q HS = p T /3.5<x R,F < p T (1) [GeV]

56 MCFM J.Campbell, K. Ellis General purpose fortran package for calculating many different processes involving, jets, vector bosons, Higgs and heavy quarks at hadron colliders p p V + 2 jets p p V + b b V = W, Z

57 WW,WZ,ZZ,Wγ,Zγ L.Dixon, Z.Kunszt,A.Signer, D. de Florian NLO codes to compute weak boson pair production at hadron colliders They include spin correlations and anomalous couplings EVENT2, DISENT M.Seymour Compute 3 jet observables in e + e and (2+1) jet in DIS DIPHOX/EPHOX P. Aurenche, T.Binoth, M. Fontannaz, J,. Guillet, G. Heinrich, E.Pilon, M.Verlen Compute p p γγ p p γ+ 1 jet

58 J. Campbell

59 J. Campbell

60 Why progress is slow? (From 3 to 4 jets in took almost 2 years!) e + e More legs implies more scales lengthy expressions Techniques to compute virtual corrections imply reduction of tensor to scalar integrals that involve large intermediate expressions and spurious singularities Work is currently being performed to overcome these difficulties A way out for the future: Z.Nagy, D.E.Soper (23) W. Giele, E.W.N. Glover (24), T. Binoth et al. (25) Isolate IR and UV singularities in the virtual contribution and compute the finite part numerically Fully numerical approach: combine real and virtual contributions to cancel IR singularities and compute their sum numerically (but no practical implementation so far)

61 The frontier: NNLO To extend the accuracy of fixed order calculations to the next perturbative order would help to better control processes where NLO predictions are affected by large uncertainties ( e.g. heavy quark and Higgs production at hadron colliders) In general: LO predictions give order of magnitude NLO corrections provide reliable estimate of cross sections and distributions NNLO corrections would give a better control of theoretical uncertainties Great progress in the last few years: Two-loop amplitudes needed for e + e 3 jets and pp 2 jets now computed Patter of IR cancellation to be understood

62 Three-loop anomalous dimensions LO anomalous dimensions 18 diagrams Gross, Wilczek (1973) Altarelli, Parisi (1977) NLO anomalous dimensions 35 diagrams Curci, Furmanski, Petronzio (198) NNLO corrections now completed 967 diagrams Moch, Vermaseren, Vogt (24) it is the most advanced QCD calculation ever performed: 2 man-year equivalents, several CPU years

63 NNLO results Drell-Yan, W, Z production - Total cross section - Rapidity distribution Hamberg, Matsuura, Van Neerven (199) Dixon et al. (23) Higgs production - Total cross section - Fully differential cross section S. Catani et al. (21), R.Harlander and W. Kilgore (21), C.Anastasiou et al. (22),W. Van Neerven et al. (23) C.Anastasiou, K.Melnikov, F.Petrello (24) e + e 3 jets in progress A. & T. Gehrmann, E.W.N. Glover

64 Resummation Example: consider the production of a vector boson at small transverse momentum Two-scale problem: q T Q = M W The recoiling gluon is forced to be either soft or collinear to one of the incoming partons q q W Real radiation strongly inhibited: KLN cancellation still at work but large logarithmic contributions of the form α n S log 2n Q 2 /qt 2 appear spoiling the perturbative expansion LO: NLO: dσ dq T + dσ dq T as q T

65 Single-gluon contribution in the small q T limit q α S 2π = α S 2π = α S 2π dω ω dq 2 T 1 Q 2 q 2 T 1 dθ 2 ( δ 2 θ 2 (q T q T 1 ) δ 2 (q T ) ) 1 + z 2 C F 1 z dq 2 T 1 q 2 T 1 Real Virtual ( δ 2 (q T q T 1 ) δ 2 (q T ) ) C F 1 qt 1 /Q ( δ 2 (q T q T 1 ) δ 2 (q T ) ) ( A (1) log Q2 qq splitting kernel q 2 T 1 dz ( 2 1 z q ) + B (1) +... ) (1 + z) q T 1 q T W q T 1 = θe g ω = E g /Q z = 1 ω A (1) = C F Write delta functions in b-space Neglect O(q T 1 /Q) terms B (1) = 3 2 C F = α S 2π d 2 b (2π) 2 eibq T Q 2 dq 2 T 1 q 2 T 1 ( e ibq T 1 1 ) ( A (1) log Q2 q 2 T 1 + B (1) ) Universal coefficients Performing irrelevant azimuthal integration α S 2π d 2 b (2π) 2 eibq T Q 2 dq 2 T 1 q 2 T 1 ( ) ( J (q T 1 b) 1 A (1) log Q2 qt B (1) )

66 More gluons: in b-space kinematics fulfill exact factorization δ 2 (q T q T 1... q T n ) e ib(q T q T 1...q T n ) = e ibq T i e ibq T i Adding a 1/n! symmetry factor one can show that the single gluon contribution exponentiates { d 2 b Q 2 (2π) 2 eibq T exp dq 2 T 1 q 2 T 1 (J (bq T 1 ) 1) α S 2π (A (1) log Q2 q 2 T 1 + B (1) ) } Replacing α S α S (q 2 T 1) we can control subleading effects The resummed cross section is now finite as q T This is what we observe in the data! Note: This resummation is approximately performed by MC event generators

67 In general: even if KLN theorem guarantees the cancellation of IR singularities, soft-gluon effects can still be large when real and virtual contributions are kinematically unbalanced If y is the variable that becomes small in this region (e.g. y = q T /Q ) double logarithmic contributions of the form αsl n 2n ( L log y ) appear (one soft and one collinear log) α S L 2 find an improved perturbative expansion when α 2 SL 4... α n SL 2n α L αsl αsl n 2n 1 S A similar (but physically different) example of resummation is the running of α S (UV origin): one log for each power of Leading Logs (LL) Next to Leading Logs (NLL) Resummation is possible if both kinematics and dynamics do factorize Resummed result must then be matched to fixed order to obtain a result which is as good as the fixed order but much better as y Note: α S α S (Q 2 ) = α S L 2 1 α S (µ 2 ) 1 + α S β log Q 2 /µ 2

68 Concluding remarks More than 3 years after discovery of asymptotic freedom and the advent of QCD QCD is a lively field with still room for fresh progress In the last few years it is turning to precision physics: this is what we need! The LHC start is approaching and most of the events will be QCD background we want to have them under control! A lot of exciting work is sti! ahead: room for you to contribute!