QCD at the LHC

Vittorio Del DucaINFN LNF

Louvain 28-29-30 April 2009

Standard Model

The Standard Model has been a spectacular success,

weathering out all challenges

Past & Present:the LEP/SLD/Tevatron legacy

The comparison with the electroweak precision

measurements has not changed much in the last years

Precision tests of SM

Top quark Mass

Run I value: 178.0 ± 4.3 GeV

Run II value: 172.6 ± 0.8 ± 1.1 GeV

Tevatron

ICHEP 08

small effect on the quality of the SM fit and on the mH bounds

Precision tests of SM

Top quark Mass

Run I value: 178.0 ± 4.3 GeV

Run II value: 172.6 ± 0.8 ± 1.1 GeV

Tevatron

ICHEP 08

small effect on the quality of the SM fit and on the mH bounds

!m/m = 0.2 !"/" TH: δσ/σ = 9% Δm = 3 GeVAt the LHC the expected EXP error is Δm = 1 GeVso the TH cross section should be known at 3% level

Effects on global EW fits W-boson Mass March 08

tree level mW = mZ cos !W

W

t

b

W W W

H

W

one loopm2

W

!1! m2

W

m2Z

"=

!""2GF

(1 + !r)

!rt = ! 3!

16"

cos2 #W

sin4 #W

m2t

m2W

!rH =11!

48" sin2 #Wln

m2H

m2W

!mW ! m2

t!mW ! lnmH

δmt = 1 GeV ⇒ δmW(mt) = 6 MeV

so

ElectroWeak fits point to a light Higgs boson

use mt to estimate mH from EW corrections

as mt changes, large shifts in mH

mH = 87+36-27 GeVfrom EW fits

mH > 114.4 GeVfrom direct search at LEP

At 95% CLmH < 160 GeV from EW fitsmH < 190 GeV combined withdirect search at LEP

March 08

Effects on global EW fits Higgs boson Mass

... 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

TOTEM

ALICE : ion-ion,p-ion

ATLAS and CMS :general purpose

27 km LEP ring 1232 superconducting dipoles B=8.3 T

TOTEM (integrated with CMS):pp, cross-section, diffractive physics

LHCb : pp, B-physics, CP-violation

Here: ATLAS and CMS

• pp √s = 14 TeV Ldesign = 1034 cm-2 s-1 (after 2009)

Linitial ≤ few x 1033 cm-2 s-1 (until 2009)

• Heavy ions (e.g. Pb-Pb at √s ~ 1000 TeV)

LHC

Event rate

On tape

Level-1SM processes are backgrounds to New Physics signals

LHC is a QCD machine

L = 1034 cm-2 s-1 = 10-5 fb-1 s-1

design luminosity

integrated luminosity (per year)

L ≈ 100 fb-1 yr-1

With 1 fb-1 we shall get ...

jets (pT > 100 GeV)

final state events

109

jets (pT > 1 TeV) 104

W ! e! 2⋅107

Z ! e+e! 2⋅106

bb 5⋅1011

tt 8⋅105

107 (Tevatron)

overall # of events (2008)

106 (LEP)

109 (BaBar, Belle)

104 (Tevatron)

even at very low luminosity, LHC beats all the other accelerators

LHC: the next future

calibrate the detectors, and re-discover the SMi.e. measure known cross sections: jets,

understand the EWSB/find New-Physics signals(ranging from Z’ to leptons, to gluinos in SUSYdecay chains, to finding the Higgs boson)

constrain and model the New-Physics theories

W, Z, tt

in all the steps above (except probably Z’ to leptons)

precise QCD predictions play a crucial role

B production in the 90’s

discrepancy between Tevatron data and NLO prediction

Tales from the past

B cross section in collisions at 1.96 TeVpp

d!(pp ! HbX, Hb ! J/" X)/dpT (J/")

CDF hep-ex/0412071

total x-sect is 19.4 ± 0.3(stat)+2.1!1.9(syst) nbFONLL = NLO + NLL

Cacciari, Frixione, Mangano, Nason, Ridolfi 2003

use of updated fragmentation functions by Cacciari & Nasonand resummation

good agreement with data no New Physics

an unbroken Yang-Mills gauge field theory featuring asymptotic freedom and confinement

in non-perturbative regime (low Q2) many approaches: lattice, Regge theory, χ PT, large Nc, HQET

in perturbative regime (high Q2) QCD is a precision toolkit for exploring Higgs & BSM physics

LEP was an electroweak machine

Tevatron & LHC are QCD machines

QCD

Precise determination of

strong coupling constantparton distributionselectroweak parametersLHC parton luminosity

Precise prediction for

Higgs productionnew physics processestheir backgrounds

!s

QCD at LHC

Goal: to make theoretical predictions of signals and backgrounds as accurate as the LHC data

Strong interactions at high Q2 Parton modelPerturbative QCD

factorisation

universality of IR behaviour

cancellation of IR singularities

IR safe observables: inclusive rates

jets

event shapes

Factorisation

pb

pa

PB

PA

!X =!

a,b

"

1

0

dx1dx2 fa/A(x1, µ2

F ) fb/B(x2, µ2

F )

! !ab!X

#

x1, x2, {pµi }; "S(µ2

R), "(µ2

R),Q2

µ2R

,Q2

µ2F

$

!X =!

a,b

"

1

0

dx1dx2 fa/A(x1, µ2

F ) fb/B(x2, µ2

F )

! !ab!X

#

x1, x2, {pµi }; "S(µ2

R), "(µ2

R),Q2

µ2R

,Q2

µ2F

$

}X X = W, Z, H, QQ,high-ET jets, ...

! = C"nS(1 + c1"S + c2"

2

S + . . .)

! = C"nS [1 + (c11L + c10)"S + (c22L

2 + c21L + c20)"2

S + . . .]

is known as a fixed-order expansion in! !S

c1 = NLO c2 = NNLO

or as an all-order resummation

L = ln(M/qT ), ln(1 ! x), ln(1/x), ln(1 ! T ), . . .wherec11, c22 = LL c10, c21 = NLL c20 = NNLL

is the separation betweenthe short- and the long-range interactions

extracted from dataevolved through DGLAP

computed in pQCD

Factorisation-breaking contributions

underlying event

power corrections

double-parton scattering

diffractive events

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

observed by Tevatron CDF in the inclusive sample

pp ! ! + 3 jets

potentially important at LHC !D ! !2

S

(see Rick Field’s studies at CDF)

!

s

predicted breakdown in dijet production at N3LO Collins Qiu 2007

Power corrections at TevatronRatio of inclusive jet cross sections at 630 and 1800 GeV

xT =2ET!

s

Bjorken-scaling variable

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-correctionparameter in the theory!

M.L. ManganoKITP collider conf 2004

Factorisation in diffraction ??

diffraction in DIS double pomeron exchange in pp

no proof of factorisation in diffractive eventsdata do not support it

just to get an ideaof the PDF size,take some PDF fit

Parton distribution functions (PDF)

LHC opens up a new kinematic range

LHC kinematic reach

Feynman x’s for the production of a particle of mass M x1,2 =M

14 TeVe±y

100-200 GeV physics is large x physics (valence quarks)at Tevatron, but smaller x physics(gluons & sea quarks) at the LHC

x range covered by HERAbut Q2 range must be provided by DGLAP evolution

rapidity distributions span widest x range

PDF evolution

factorisation scale is arbitraryµF

cross section cannot depend on µF

µF

d!

dµF

= 0

implies DGLAP equations

µF

dfa(x, µ2

F)

dµF

= Pab(x, !S(µ2

F )) ! fb(x, µ2

F ) + O(1

Q2)

µF

d!ab(Q2/µ2

F, "S(µ2

F))

dµF

= !Pac(x, "S(µ2

F )) " !cb(Q2/µ2

F , "S(µ2

F )) + O(1

Q2)

Pab(x, !S(µ2

F )) is calculable in pQCD

V. Gribov L. Lipatov; Y. DokshitzerG. Altarelli G. Parisi

d

d lnµ 2F

!

qi

g

"

=

!

PqiqjPqjg

PgqjPgg

"

!

!

qj

g

"

[a ! b](x) "

! 1

x

dy

ya(y) b

"

x

y

#

qi(x, µ2

F ) g(x, µ2

F )

factorisation for the structure functions (e.g. )

Fi(x, µ 2

F ) = Cij ! qj + Cig ! g

Fep2

, FepL

with the convolution

Cij , Cig coefficient functions

PDF’s

DGLAP evolution equations

perturbative series Pij ! !sP(0)ij + !

2sP

(1)ij + !

3sP

(2)ij

anomalous dimension !ij(N) = !

! 1

0

dx xN!1 Pij(x)

DGLAP evolution equations

HERA F2

Bjorken-scaling violations

large violations at small x

small violations at large x

d

d lnµ 2F

!

qs

g

"

=

!

Pqq Pqg

Pgq Pgg

"

!

!

qs

g

"

a list of PDF’sgeneral structure of the quark-quark splitting functions

Pqiqk= Pqiqk

=!ikPv

qq + Ps

qq

Pqiqk= Pqiqk

=!ikPv

qq + Ps

qq

flavour non-singlet

flavour asymmetryq±

ns,ik = qi ± qi ! (qk ± qk) P±

ns = Pv

qq ± Pv

qq

sum of valence distributions of all flavours

qv

ns =

nf!

r=1

(qr! q

r) P

vns = P

vqq ! P

vqq + nf (P s

qq ! Ps

qq)

flavour singlet

qs =

nf!

i=1

(qi + qi)

withPqq = P

+ns + nf (P s

qq + Ps

qq)

Pqg = nf Pqig , Pgq = Pgqi

PDF historyleading order (or one-loop)anomalous dim/splitting functions Gross Wilczek 1973; Altarelli Parisi 1977

NLO (or two-loop)

F2, FL Bardeen Buras Duke Muta 1978

anomalous dim/splitting functions Curci Furmanski Petronzio 1980

NNLO (or three-loop)

F2, FL 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

three-loop

!(0)ij /P (0)

ij

!(1)ij /P (1)

ij

!(2)ij /P (2)

ij

18 Feynman diagrams

350 Feynman diagrams

9607 Feynman diagrams

20 man-year-equivalents, lines of dedicated algebra code106

PDF global fitsJ. Stirling, KITP collider conf 2004

MRST, MSTW: Martin et al.

global fits

CTEQ: Pumplin et al.Alekhin (DIS data only)

methodPerform fit by minimising to all data,including both statistical and systematicerrors

!2

Start evolution at some , where PDF’s are parametrised with functional form, e.g.

Q2

0

xf(x, Q2

0) = (1 ! x)!(1 + !x0.5 + "x)x"

Cut data at and at to avoid higher twist contamination

Q2 > Q2

min W2

> W2

min

Allow as implied byE866 Drell-Yan asymmetry data

u != d

accuracyNNLO evolution

no prompt photon data included in the fits

PDF: recent developments

more accurate treatment of heavy flavours in the vicinity of the quark mass (few % effect on Drell-Yan at the LHC)

more systematic treatment of uncertainties in global fits

neural network PDF’s (NNPDF), based on unbiased priors(so far only DIS data, and only the non-singlet distributions)

MSTW, CTEQ, Alekhin

NNPDF Collaboration 08

note the larger uncertaintyin NNPDF

valence quark distribution

still far from global-fit analysisin NNPDF

matrix-elem MC’s fixed-order x-sect shower MC’s

final-state description

hard-parton jets.Describes geometry,

correlations, ...

limited access tofinal-statestructure

full information available at the hadron level

higher-order effects:loop corrections

hard to implement:must introduce

negative probabilities

straightforwardto implement

(when available)

included asvertex corrections

(Sudakov FF’s)

higher-order effects:hard emissions

included, up to highorders (multijets)

straightforwardto implement

(when available)

approximate,incomplete phase

space at large angles

resummation oflarge logs ? feasible

(when available)

unitarityimplementation

(i.e. correct shapesbut not total rates)

3 complementary approaches to !

M.L. Mangano KITP collider conf 2004

Parton cross section

Parton shower MonteCarlo generators

HERWIG

PYTHIA

re-written as a C++ code (HERWIG++)

B. Webber et al. 1992

T. Sjostrand 1994

SHERPA F. Krauss et al. 2003

model parton showering and hadronisation

several automated codes to yieldlarge 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 scalesexample: Higgs (via gluon fusion) + 2 jets is αs4(Q2)

unreliable for precision calculations

Matrix-element MonteCarlo generators

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

COMPHEP A. Pukhov et al. 1999

GRACE/GR@PPA T. Ishikawa et al. K. Sato et al. 1992/2001

MADGRAPH/MADEVENT W.F. Long F. Maltoni T. Stelzer 1994/2003

HELAC C. Papadopoulos et al. 2000

merged with parton showers

all of the above, merged with HERWIG or PYTHIA

processes with 6 final-state fermions

PHASE E. Accomando A. Ballestrero E. Maina 2004

MonteCarlo interfaces

CKKW S. Catani F. Krauss R. Kuhn B. Webber 2001

MLM L. Lonnblad 2002 M.L. Mangano 2005

procedures to interface parton subprocesses witha different number of final states to parton-shower MC’s

MC@NLO S. Frixione B. Webber 2002

POWHEG P. Nason 2004

procedures to interface NLO computations to parton-shower MC’s

at low pT, parton showermodels collinear radiation

Frixione Laenen Motylinski Webber 2005

Single top in MC@NLO

at high pT, NLO models hard radiation

Leading Order

Different methods (other than Feynman) to compute tree-level amplitudes

use recursively off-shell currents and go on-shell to get amplitudes

Berends-Giele relations (1988)

Cachazo Svrcek Witten (2004)

compute helicity amplitudes by sewing MHV amplitudes (++⋅⋅⋅+−−)

Britto Cachazo Feng (2004)

use on-shell recursive relationsby shifting to complex momenta

=∑

∑

+−

+− +

−−

−

−

−−

++

+++

= ++ −−

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 104 PS points

Duhr Höche Maltoni 2006

CO: colour orderedCD: colour dressed

Next to Leading OrderJet structure: final-state collinear radiation

PDF evolution: initial-state collinear radiation

Opening of new channels

Reduced sensitivity to fictitious input scales: µR, µF

predictive normalisation of observables

first step toward precision measurementsaccurate estimate of signal and backgroundfor Higgs and new physics

Matching with parton-shower MC’s: MC@NLO POWHEG

Jet structurethe jet non-trivial structure shows up first at NLO

leading order

NLO

NNLO

p p! t t Z at NLO

Lazopoulos McElmurry Melnikov Petriello 2008

NLO/LO ≡ K factor = 1.35

Reduced theoretical error: 25-35% at LO; 11% at NLO

dots: inclusive K factor dash: pT dependent K factor

probes EW properties of top quark

NLO cross sections: experimenter’s wishlist

2005 Les Houches

QCD, EW & Higgs working group hep-ph/0604210

NLO multileg working group hep-ph/0803.04942007 Les Houches

updated wishlist

private codes

High (EXP) demand for cross sections of Y + n jetsY = 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) partons2) one-loop amplitude for Y + (n+2) partons3) 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 divergences2: until 2007-8, we did not have systematic methods to compute the one-loop amplitudes

NLO assembly kit

leading order

e+e!

! 3 jets

NLO virtual

NLO real

|Mtree

n|2

!

ddl 2(Mloopn )!Mtree

n =

"

A

!2+

B

!

#

|Mtreen |2 + fin.

= !

!

A

!2+

B

!

"

IR

d = 4 ! 2!

!

|Mtreen+1|

2 ! |Mtreen

|2 "

!dPS|Psplit|

2

example

NLO production rates Process-independent procedure devised in the 90’s

Giele Glover Kosower 1992-3

the 2 terms on the rhs are divergent in d=4

use universal IR structure to subtract divergences

the 2 terms on the rhs are finite in d=4

slicingsubtraction Frixione Kunszt Signer 1995; Nagy Trocsanyi 1996

dipole Catani Seymour 1996antenna Kosower 1997; Campbell Cullen Glover 1998

! = !LO

+ !NLO

=

!m

d!Bm Jm + !

NLO

!NLO

=

!m+1

d!Rm+1Jm+1 +

!m

d!Vm

Jm

!NLO

=

!

m+1

"

d!Rm+1Jm+1 ! d!

R,Am+1Jm

#

+

!

m

$

d!Vm

+

!

1

d!R,Am+1

%

Jm

NLO history of final-state distributionsK. Ellis Ross Terrano 1981e

+e!

! 3 jets

e+e!

! 4 jets Bern et al.; Glover et al.; Nagy Trocsanyi 1996-97

pp ! 1, 2 jets K. Ellis Sexton 1986 Giele Glover Kosower 1993

pp ! 3 jets Bern Dixon Kosower; Kunszt Signer Trocsanyi 1993-1995Nagy 2001

pp ! V + 1 jet Giele Glover Kosower1993

pp ! V + 2 jet Bern Dixon Kosower Weinzierl; Campbell Glover Miller 1996-97 Campbell K. Ellis 2003

pp ! V V Ohnemus Owens, Baur et al.1991-96; Dixon et al. 2000

pp ! V bb Campbell K. Ellis 2003

e+e!

! 4 fermions Denner Dittmaier Roth Wieders 2005

pp ! V V + jet Dittmaier Kallweit Uwer; Campbell K. Ellis Zanderighi 2007

pp ! V V V Lazopoulos Melnikov Petriello; Hankele Zeppenfeld 2007Binoth Ossola Papadopoulos Petriello 2008

pp! V + 3 jet K. Ellis Melnikov Zanderighi; Berger et al. (BlackHat) 2009

pp ! !!

pp ! !! + 1 jetBailey et al. 1992; Binoth et al. 1999Bern et al. 1994 Del Duca Maltoni Nagy Trocsanyi 2003

NLO history of final-state distributionspp ! QQ

pp ! QQ + 1 jet

Dawson K. Ellis Nason 1989 Mangano Nason Ridolfi 1992

Brandenburg Dittmaier Uwer Weinzierl 2005-7Dittmaier Kallweit Uwer 2007-8

pp ! H + 1 jet

pp ! HQQ

(GGF; mt → ∞) C. Schmidt 1997 De Florian Grazzini Kunszt 1999

Beenakker et al. ; Dawson et al. 2001

(GGF; mt → ∞) Campbell K. Ellis Zanderighi 2006(WBF) Campbell K. Ellis; Figy Oleari Zeppenfeld 2003 Ciccolini Denner Dittmaier 2007 (includes s channel)

pp ! H + 2 jets

pp! t t b bBredenstein Denner Dittmaier Pozzorini 2008

(only qq annihilation)

Harris Laenen Phaf Sullivan Weinzierl 2002Campbell K. Ellis Tramontano 2004; Cao Yuan et al. 2004-5W: Campbell Tramontano 2005

pp! t (+ W )

t: Maltoni Paul Stelzer Willenbrock 2001b: Campbell, K. Ellis Maltoni Willenbrock 2002pp! H Q

(WBF) Figy Hankele Zeppenfeld 2007pp! H + 3 jets

Lazopoulos McElmurry Melnikov Petriello 2008pp! t t Z

... 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 computede+e!

! 4 fermions Denner Dittmaier Roth Wieders 2005

pp! t t b 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

J. Campbell, KITP collider conf 2004

J. Campbell, KITP collider conf 2004

Summary of NLO K factors

NLO multileg working group hep-ph/0803.0494

all: CTEQ6M at NLOK: CTEQ6L1 at LOK’: CTEQ6M at LO

gluon-initiated processes: large K factors

more jets: smaller K factors

NLO progress: cross sections

automated subtraction of IR divergencesautomating dipoles Gleisberg Krauss 2007

Frederix Gehrmann Greiner (MadGraph) 2008Seymour Tevlin (TeVJet); Hasegawa Moch Uwer2008

dipoles proliferate with the spectators:may not be adapt for multi-jet environment

subtraction for multi-jet environmentantennanew standard subtraction

Daleo Gehrmann Maitre 2006Somogyi Trocsanyi 2006; Somogyi 2009

for each dipole, one must considerall possible spectators ...result: in W + 2 jets (MCFM)24 counterterm events for each real event

Catani Seymour 1996

DipolesTo gain exact factorisation of phase space, modify momenta

pµir = pµ

i + pµr !

yir,n

1! yir,npµ

n

=1

1! yir,n(pµ

i + pµr ! yir,nQµ

irn)

pµn =

11! yir,n

pµn

Catani Seymour 1996

yir,n =2pi · pr

Q2irn

emitter momentum

spectator momentum

momentum is conserved Qµirn = pµ

ir + pµn = pµ

i + pµr + pµ

n

A |M0m+1|2 =

!

n !=i,r

Dir,n(p1, . . . , pm+1) + . . .

counterterm

New standard subtraction

pµir =

1

1 ! !ir

(pµi + pµ

r ! !irQµ) , pµ

n =1

1 ! !ir

pµn , n "= i, r

!ir =1

2

!

y(ir)Q !

"

y2(ir)Q ! 4yir

#

yir =2pi · pr

Q2

pµir +

!

n

pµn = p

µi + p

µr +

!

n

pµnmomentum is conserved

mapping à la Catani-Seymour, but all momenta but the unresolved ones are treated simultaneously like spectators

1

i

r

Q

!ir

!m + 2

m+1

···

· · ·

!Q

1

!ir

···

···

m+1

!m + 2

irr

Somogyi Trocsanyi 2006

one-loop amplitudes

↓ IR finite terms are process dependent: many final-state particles → many scales → lengthy expressions

Kunszt Signer Trocsanyi 1994; Catani 1998↑ IR divergences are universal

An can be reduced to boxes, triangles and bubbleswith rational coefficients

one-loop n-point amplitudes An are IR divergent

I: master integralsb, c, d: rational functions of kinematic variables

51

42

6

3

An =!

i1i2i3i4

di1i2i3i4 IDi1i2i3i4 +

!

i1i2i3

ci1i2i3 IDi1i2i3 +

!

i1i2

bi1i2 IDi1i2

i4

i3i2

i1 i3

i2

i1

i1 i2

higher polygons contribute only to O(ε)

use unitarity cuts: Cutkovsky rule

An =!

i1i2i3i4

di1i2i3i4 I4i1i2i3i4 +

!

i1i2i3

ci1i2i3 I4i1i2i3 +

!

i1i2

bi1i2 I4i1i2 + Rn

Bern Dixon Dunbar Kosower 1994

NLO progress: unitarity method 1

p2 + i0! 2!i "+(p2)

to factorise coefficients of An into products of tree amplitudes

compute b, c, d with D=4: cut-constructible termsO(ε) part of b, c, d: rational term Rn

Rn computable through on-shell recursionBerger Bern Dixon Forde Kosower 2006

generalized unitarity:quadruple cuts with complex momenta

box coefficient di determined byproduct of 4 tree amplitudes

however, ci, bi still difficult to extract from tripleand double cuts because terms already includedin di must be subtracted

Britto Cachazo Feng 2004

Ossola Papadopoulos Pittau 2006reduction of one-loop amplitude at integrand level

NLO progress: OPP method

for quadruple cuts, similar to BCF, but algorithmic: can be automated

A(q!) =N(q)

D1 · · · Dm

q’ lives in D dimensions; q lives in 4 dimensionsε part of numerator treated separately

Di = (q! + pi)2 !m2i q! = q + q q · q = 0

partial fraction the numerator into terms with 4, 3, 2, 1 denominator factors

N(q) =!

i1i2i3i4

(di1i2i3i4 + di1i2i3i4)"

i !=i1i2i3i4

Di

+!

i1i2i3

(ci1i2i3 + ci1i2i3)"

i !=i1i2i3

Di +!

i1i2

(bi1i2 + bi1i2)"

i !=i1i2

Di

d, c, b vanish upon integration

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 theninverting the system. First find all possible 4-pt functions, then 3-pt functions, etc.

NLO progress: integer dimensions Giele Kunszt Melnikov 2008

An =!

i1i2i3i4i5

ei1i2i3i4i5 IDi1i2i3i4i5 +

!

i1i2i3i4

di1i2i3i4 IDi1i2i3i4

+!

i1i2i3

ci1i2i3 IDi1i2i3 +

!

i1i2

bi1i2 IDi1i2

in D dimensions the one-loop amplitude An can be reduced to pentagons, boxes, triangles and bubbles

take Ds = # of spin states of internal particlesin Ds dimensions, gluons have Ds-2 spin states, quarks have 2(Ds-2)/2 spin states

A(D,Ds)n =

!dDq

N (Ds)(q)d1 · · · dn

with D ≤ Ds

dependence of A on Ds is linear N (Ds)(q) = N0(q) + (Ds ! 4)N1(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

RocketGiele Zanderighi 2008

A Fortran 90 code based on OPPand integer dimensions

computes- up to one-loop 20-gluon amplitudes- one-loop W + 5-parton amplitudes- (leading-colour) NLO W + 3-jet cross section

Giele Zanderighi 2008K. Ellis Giele Kunszt Melnikov Zanderighi 2008

K. Ellis Melnikov Zanderighi 2009

W + 3-jet cross section at NLOK. Ellis Melnikov Zanderighi 2009

NLO very stable under scale variations

K factor even smaller than in W + 2 jets

BlackHatA C++ code based on generalised unitarity,and on-shell recursion for the rational parts

Berger Bern Dixon Febres-Cordero Forde Ita Kosower Maitre 2008

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 2008BlackHat 2009

Inclusive jet cross section at Tevatron

good agreement between NLO and data over several orders ofmagnitude

constrains the gluondistribution at high x

Is NLO enough to describe data ?

pT

Is NLO enough to describe data ?Drell-Yan cross section at LHC with leptonic decay of the W W

|MC@NLO ! NLO| = O(2%)

NNLO useless without spin correlations

S. Frixione M.L. Mangano 2004

Precisely evaluated Drell-Yan cross sections could be used as ``standard candles’’ to measure the parton luminosity at LHC

W, Z

Drell-Yan W acceptances at LHC with leptonic decay of the W

K. Melnikov, F. Petriello 2006

μ = mW/2, mW, 2mW

NNLO corrections are large for pe,⊥ = 40, 50 GeVbut are at the percent level for pe,⊥ = 20, 30 GeV

At LO, pe,⊥ ≤ mW/2

Is NLO enough to describe data ?

Total cross section for inclusive Higgs production at LHC

µR = 2MH µF = MH/2

lowercontour bands are

upperµR = MH/2 µF = 2MH

scale uncertaintyis about 10%

NNLO prediction stabilises the perturbative series

Higgs production at NNLOa fully differential cross section:bin-integrated rapidity distribution, with a jet veto

jet veto: requireR = 0.4

for 2 partons

|pjT | < p

vetoT = 40 GeV

|p1

T |, |p2

T | < pveto

T

|p1

T + p2

T | < pveto

T

R2

12 = (!1 ! !2)2 + ("1 ! "2)

2

if

if

R12 > R

R12 < R

MH = 150 GeV (jet veto relevant in the decay channel)H ! W+W

!

K factor is much smaller for the vetoed x-sect than for the inclusive one:average increases from NLO to NNLO: less x-sect passes the veto|pj

T |

C. Anastasiou K. Melnikov F. Petriello 2004

Drell-Yan production at LHCZ

Rapidity distribution foran on-shell bosonZ

NLO increase wrt to LO at central Y’s (at large Y’s)30%(15%)NNLO decreases NLO by 1 ! 2%

scale variation: ! 30% at LO; ! 6% at NLO; less than at NNLO1%

C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003

Scale variations in Drell-Yan productionZ

solid: vary and together

dashed: vary only

dotted: vary only

µR

µR

µF

µF

C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003

Drell-Yan production at LHCW

Rapidity distributionfor an on-shell boson (left) boson (right) W

+

W!

C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003

distributions are symmetric in Y

NNLO scale variations are at central (large) 1%(3%) Y

World average of !S(MZ)

S. Bethke hep-ex/0606035

!S(MZ) = 0.1189 ± 0.0010

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

!S(MZ)

NNLO corrections may be relevant ifthe main source of uncertainty in extracting info from data is due to NLO theory: measurements!S

NLO corrections are large: Higgs production from gluon fusion in hadron collisions

NLO uncertainty bands are too large to testtheory vs. data: b production in hadron collisions

NLO is effectively leading order:energy distributions in jet cones

NNLO state of the art Drell-Yan W, Z production total cross section Hamberg van Neerven Matsuura 1990

Harlander Kilgore 2002

Higgs productiontotal cross section

fully differential cross section

Harlander Kilgore; Anastasiou Melnikov 2002Ravindran Smith van Neerven 2003

Anastasiou Melnikov Petriello 2004Catani de Florian Grazzini 2007

e+e!

! 3 jetsde Ridder Gehrmann Glover Heinrich 2007Weinzierl 2008

Melnikov Petriello 2006fully differential cross section

event shapes, αs

!s(M2Z) = 0.1240± 0.0008 (stat)± 0.0010 (exp)± 0.0011 (had)± 0.0029 (theo)

TH uncertainty almost halved from NLO to NNLOno log resummation is included

discrepancy found in logarithmic contributions to thrust distribution at N3LL by Becher Scwartz 2008 using SCET. Confirmed error in wide-angle soft emissions through independent NNLO calculation by Weinzierl 2008

e+e!

! 3 jets de Ridder Gehrmann Glover Heinrich 2007

CO = N2c Clc

O + CscO +

1N2

c

CsscO + NfNc Cnf

O +Nf

NcCnfscO + N2

f CnfnfO

O =!s

2"AO +

!!s

2"

"2BO +

!!s

2"

"3CO

any IR observable to NNLO

with

disagreement in N2c , N0

c , NfNc

traced back to soft counterterm missing in RR contributionWeinzierl 2008

NNLO cross sectionsAnalytic integration

Sector decomposition

Hamberg, van Neerven, Matsuura 1990Anastasiou Dixon Melnikov Petriello 2003

first method

flexible enough to include a limited class of acceptance cutsby modelling cuts as ``propagators’’

cancellation of divergences is performed numerically

Denner Roth 1996; Binoth Heinrich 2000Anastasiou, Melnikov, Petriello 2004

flexible enough to include any acceptance cuts

can it handle many final-state partons ?

Subtractionprocess independent

cancellation of divergences is semi-analyticcan it be fully automatised ?

de Ridder Gehrmann Glover Heinrich 2007

NNLO assembly kit e+e!

! 3 jets

double virtual

real-virtual

double real

Two-loop matrix elements

two-jet production qq!! qq

!, qq ! qq, qq ! gg, gg ! gg

C. Anastasiou N. Glover C. Oleari M. Tejeda-Yeomans 2000-01

Z. Bern A. De Freitas L. Dixon 2002

photon-pair production qq ! !!, gg ! !!

C. Anastasiou N. Glover M. Tejeda-Yeomans 2002Z. Bern A. De Freitas L. Dixon 2002

e+e!

! 3 jets

L. Garland T. Gehrmann N. Glover A. Koukoutsakis E. Remiddi 2002

!!! qqg

V + 1 jet productionT. Gehrmann E. Remiddi 2002

Drell-Yan productionV

R. Hamberg W. van Neerven T. Matsuura 1991

Higgs productionR. Harlander W. Kilgore; C. Anastasiou K. Melnikov 2002

qq ! V

qq ! V g

gg ! H (in the limit)mt ! "

universal collinear and soft currents

3-parton tree splitting functions

Campbell Glover 1997; Catani Grazzini 1998; Frizzo Maltoni VDD 1999; Kosower 2002

universal IR structure

Collinear and soft currents process-independent procedure

2-parton one-loop splitting functions

Bern Dixon Dunbar Kosower 1994; Bern Kilgore C. Schmidt VDD 1998-99;Kosower Uwer 1999; Catani Grazzini 1999; Kosower 2003

universal subtraction countertermsseveral ideas Kosower; Weinzierl; de Ridder Gehrmann Heinrich 2003

Frixione Grazzini 2004; Somogyi Trocsanyi VDD 2005

but devised only for e+e!

! 3 jetsde Ridder Gehrmann Glover 2005; Somogyi Trocsanyi VDD 2006

back to NLO IR limits

collinear operatorCir|M

(0)m+2(pi, pr, . . .)|

2 !1

sir"Mm+1(0)(pir, . . .)|P

(0)fifr

|Mm+1(0)(pir, . . .)#

soft operator

Sr|M(0)m+2(pr, . . .)|

2 !sik

sirsrk

"Mm+1(0)(. . .)|Ti · Tk|Mm+1(0)(. . .)#

counterterm!

r

"

#

!

i!=r

1

2Cir + Sr

$

% |M(0)m+2(pi, pr, . . .)|

2

performs double subtraction in overlapping regions

Why is NNLO subtraction so complicated ?

A1|M(0)m+2|

2 =!

r

"

#

!

i!=r

1

2Cir +

$

%Sr !!

i!=r

CirSr

&

'

(

) |M(0)m+2(pi, pr, . . .)|

2

the NLO counterterm

contains spurious singularities when parton becomes unresolved, but they are screened by

s != r

Jm

Cir (Sr ! CirSr) |M(0)m+2|

2 = 0

can be used to cancel double subtractionCirSr

Sr (Cir ! CirSr) |M(0)m+2|

2 = 0

NLO overlapping divergences

has the same singular behaviour as SME, and is free of double subtractions

Cir (1 ! A1) |M(0)m+1|

2 = 0 Sr (1 ! A1) |M(0)m+1|

2 = 0

NNLO subtraction

!NNLO

=

!m+2

d!RRm+2Jm+2 +

!m+1

d!RVm+1Jm+1 +

!m

d!VVm

Jm

the 3 terms on the rhs are divergent in d=4use universal IR structure to subtract divergences

NNLO subtraction

!NNLO

=

!m+2

d!RRm+2Jm+2 +

!m+1

d!RVm+1Jm+1 +

!m

d!VVm

Jm

the 3 terms on the rhs are divergent in d=4use universal IR structure to subtract divergences

!NNLO

=

!

m+2

"

d!RRm+2Jm+2 ! d!

RR,A2

m+2 Jm

#

takes care of doubly-unresolved regions,but still divergent in singly-unresolved ones

NNLO subtraction

!NNLO

=

!m+2

d!RRm+2Jm+2 +

!m+1

d!RVm+1Jm+1 +

!m

d!VVm

Jm

the 3 terms on the rhs are divergent in d=4use universal IR structure to subtract divergences

!NNLO

=

!

m+2

"

d!RRm+2Jm+2 ! d!

RR,A2

m+2 Jm

#

takes care of doubly-unresolved regions,but still divergent in singly-unresolved ones

+

!

m+1

"

d!RVm+1Jm+1 ! d!

RV,A1

m+1 Jm

#

still contains poles in regions away from 1-parton IR regions1/!

NNLO subtraction

!NNLO

=

!m+2

d!RRm+2Jm+2 +

!m+1

d!RVm+1Jm+1 +

!m

d!VVm

Jm

the 3 terms on the rhs are divergent in d=4use universal IR structure to subtract divergences

+

!

m

"

d!VVm

+

!

2

d!RR,A2

m+2 +

!

1

d!RV,A1

m+1

#

Jm

!NNLO

=

!

m+2

"

d!RRm+2Jm+2 ! d!

RR,A2

m+2 Jm

#

takes care of doubly-unresolved regions,but still divergent in singly-unresolved ones

+

!

m+1

"

d!RVm+1Jm+1 ! d!

RV,A1

m+1 Jm

#

still contains poles in regions away from 1-parton IR regions1/!

2-step procedure

G. Somogyi Z. Trocsanyi VDD 2005

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 2006

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

A2 countertermconstruct the 2-unresolved-parton counterterm using the IR currents

Cirs (1 ! A2) |M(0)m+2|

2 = 0

Cir;js (1 ! A2) |M(0)m+2|

2 = 0

Srs (1 ! A2) |M(0)m+2|

2 = 0

CSir;s (1 ! A2) |M(0)m+2|

2 = 0

performing double and triple subtractions in overlapping regions

Constructing d!RR,A2

m+2

! Use IR factorization formulae valid in the doubly-unresolved regions Catani-Grazzini, Campbell-Glover

! To account for double subtraction in overlappingregions: subtract the collinear limit of the soft one

A2|M(0)m+2|

2 =!

r

!

s !=r

"!

i !=r,s

#1

6Cirs +

!

j !=i,r,s

1

8Cir;js +

1

2Srs

+1

2

$

CSir;s ! CirsCSir;s !!

j !=i,r,s

Cir;jsCSir;s

%&

!!

i !=r,s

#

CSir;sSrs + Cirs

$1

2Srs ! CSir;sSrs

%

+!

j !=i,r,s

Cir;js

$1

2Srs ! CSir;sSrs

% &'

|M(0)m+2|

2

Easy to check:

Cirs(1!A2)|M(0)m+2|

2 = 0

Cir;js(1!A2)|M(0)m+2|

2 = 0

CSir;s(1!A2)|M(0)m+2|

2 = 0

Srs(1!A2)|M(0)m+2|

2 = 0

Zoltán Trócsányi: Matching of Singly- and Doubly-Unresolved Limits of Tree-level QCD Squared Matrix Elements ETH Zürich, March 8, 2005 – p.16/22G. Somogyi Z. Trocsanyi VDD 2005

Triple-collinear mappingpµ

irs =1

1 ! !irs

(pµi + pµ

r + pµs ! !irsQ

µ) , pµn =

1

1 ! !irs

pµn , n "= i, r, s

!irs =1

2

!

y(irs)Q !

"

y2(irs)Q ! 4yirs

#

pµirs +

!

n

pµn = p

µi + p

µr + p

µs +

!

n

pµnmomentum conservation

straightforward extension of NLO collinear mapping

···

· · ·

m

Q

1

!m + 2

!irs

ir

s

···

···

m

!m + 2

1

!irs ! irs

r

sQ

phase space

The subtraction scheme at NNLO

!NNLO = !NNLO{m+2} + !NNLO

{m+1} + !NNLO{m}

!NNLO{m+2} =

!

m+2

"

d!RRm+2 Jm+2 ! d!RR,A2

m+2 Jm

!d!RR,A1

m+2 Jm+1 + d!RR,A12

m+2 Jm

#

d=4

!NNLO{m+1} =

!

m+1

"

d!RVm+1 Jm+1 ! d!RV,A1

m+1 Jm

+

!

1

$

d!RR,A1

m+2 Jm+1 ! d!RR,A12

m+2 Jm

%#

d=4

!NNLO{m} =

!

m

"

d!VVm +

!

2

d!RR,A2

m+2 +

!

1

d!RV,A1

m+1

#

d=4

Jm

has to be finitein the doubly-unresolvedregions!

RestrictsA1 and A12

severely

Zoltán Trócsányi: Matching of Singly- and Doubly-Unresolved Limits of Tree-level QCD Squared Matrix Elements ETH Zürich, March 8, 2005 – p.13/22

!NNLO{m+1} =

!

m+1

"#

d!RVm+1 +

!

1

d!RR,A1

m+2

$

Jm+1

!

#

d!RV,A1

m+1 +

%

!

1

d!RR,A1

m+2

&

A1

$

Jm

'

!=0

remainder is finite by KLN theorem

!NNLO{m} =

!

m

"

d!VVm

+

!

2

#

d!RR,A2

m+2 !d!RR,A12

m+2

$

+

!

1

#

d!RV,A1

m+1 +

%

!

1

d!RR,A1

m+2

&

A1

$'

!=0Jm

NNLO counterterms

G. Somogyi Z. Trocsanyi VDD 2006

Resummationscoefficients 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

!(n)0 (pi,", g0) =

n!

i=1

Z1/2i ("/µ, g(µ)) !(n)

R (pi, µ, g(µ)),

Θ0 independence of μ implies

d!(n)0

dµ= 0 ! d ln !(n)

R

d lnµ= "

n!

i=1

!i(g(µ))

renormalisation of Θ is

RG evolution resums !ns (µ2) lnn(Q2/µ2) into !n

s (Q2)

example: e+e- → hadrons, where Q2 = s

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 EX =s!m2

H

2mH" 0

need to resum soft-gluon emission

Higgs qT distribution Bozzi Catani de Florian Grazzini 2005

At small qT, NLO blows up

!µ(p1, p2;µ2, !) !< 0|Jµ(0)|p1, p2 >= v(p2)"µu(p1) !!

Q2

µ2, #s(µ2), !

"

Resummation: Sudakov form factorSudakov (quark) form factor as matrix element of EM current

obeys evolution equation

Q2 !

!Q2ln

!!

"Q2

µ2, "s(µ2), #

#$=

12

!K

%"s(µ2), #

&+ G

"Q2

µ2, "s(µ2), #

#$

RG invariance requires

µdG

dµ= !µ

dK

dµ= !K("s(µ2))

γK is the cusp anomalous dimension

K is a counterterm; G is finite as ε→ 0

Korchemsky Radyushkin 1987

!!Q2, !

"= exp

#12

$ !Q2

0

d"2

"2

%G

!!1, #s("2, !), !

"! 1

2$K

!#s("2, !)

"ln

&!Q2

"2

'()solution is

cusp anomalous dimensionloop expansion of the cusp anomalous dimension

K =!

6718! !2

"CA !

109

TF Nf

!(i)K = 2Ci

"s(µ2)#

+ KCi

!"s(µ2)

#

"2

+ · · ·

with

Factorisation of a multi-leg amplitude

Mueller 1981Sen 1983Botts Sterman 1987Kidonakis Oderda Sterman 1998Catani 1998Tejeda-Yeomans Sterman 2002Kosower 2003Aybat Dixon Sterman 2006Becher Neubert 2009Gardi Magnea 2009

to avoid double counting of soft-collinear region (IR double poles), Ji removes eikonal part from Ji, which is already in SJi/Ji contains only single collinear poles

MN (pi/µ, !) =!

L

SNL("i · "j , !) HL

"2pi · pj

µ2,(2pi · ni)2

n2i µ

2

# $

i

Ji

"(2pi · ni)2

n2i µ

2, !

#

Ji

"2("i · ni)2

n2i

, !

#

pi = !iQ0/!

2 value of Q0 is immaterial in S, J

N = 4 SUSY in the planar limit

colour-wise, the planar limit is trivial:can absorb S into Ji

each slice is square rootof Sudakov form factor

Mn =n!

i=1

"M[gg!1]

#si,i+1

µ2, !s, "

$%1/2

hn({pi}, µ2, !s, ")

β fn = 0 ⇒ coupling runs only through dimension

ln!!

"Q2

µ2, !s(µ2), "

#$= !1

2

!%

n=1

"!s(µ2)

#

#n "!Q2

µ2

#"n!&

$(n)K

2n2"2+

G(n)(")n"

'

⇒ IR structure of N = 4 SUSY amplitudes

Sudakov form factor has simple solution

!s(µ2)µ2! = !s("2)"2!

- introduce auxiliary vector ni (ni2 ≠ 0) to separate collinear region

- define a jet using a Wilson line along ni

Jet definition

u(p) J

!(2p · n)2

n2µ2, !

"=< p|"(0)!n(0,!")|0 >

!n(!2, !1) = P exp

!ig

" !2

!1

d!n · A(!n)

#

Wilson line

partonic jet

eikonal jet J!

2(! · n)2

n2, "

"=< 0|!!(!, 0)!n(0,"!)|0 >

Eikonal jet & cusp anomalous dimension

eikonal jet does not depend on any kinematic scale

single poles carry (! · n)2/n2 dependence,

thus violate classical rescaling symmetry wrt β ⇒ cusp anomalous dim

δj is a constant

double poles and kinematic dependence of single poles are controlled by cusp γK, as for quark form factor

J!

2(! · n)2

n2, "

"= exp

#14

$ µ2

0

d#2

#2

%$Ji

&%s(#2, ")

'! 1

2&K

&%s(#2, ")

'ln

!2(! · n)2µ2

n2#2

"()

Soft function S(cN )ijklSNL

!!a · !b, "s(µ2), #

"

=#

i!j!k!l!

< 0|!k,k!

!!2(0,!)!i,i!

!1(!, 0)!j,j!

!3(0,!)!l,l!

!!4(!, 0)|0 > (cL)i!j!k!l!

matrix evolution equation

µd

dµSJL

!!a · !b, "s(µ2), #

"

= !#

N

[!S ]JN

!!a · !b, "s(µ2), #

"SNL

!!a · !b, "s(µ2), #

"

formal solution

S!!a · !b, "s(µ2), #

"= P exp

#!1

2

$ µ2

0

d$2

$2!S

!!a · !b, "s(µ2), #

"%

ΓS soft anomalous dimension

!S =!!

n=1

!(n)S

"!s(µ2)

"

#n

Soft anomalous dimensionfor an amplitude with an arbitrary # of legs

!(2)S =

K

2!(1)S Aybat Dixon Sterman 2006

K is 2-loop coefficient of cusp anomalous dimension

ΓS has cusp singularities like γJ

Reduced soft function

S does not have cusp singularities ⇒ must respect rescaling βi → κi βi

S depends only on !ij =("i · "j)2

2("i · ni)2

n2i

2("j · nj)2

n2jfactorisation becomes

MN (pi/µ, !) =!

L

SNL("ij , !) HL

"2pi · pj

µ2,(2pi · ni)2

n2i µ

2

# $

i

Ji

"(2pi · ni)2

n2i µ

2, !

#

Dixon Magnea Sterman 2008

S has only single poles due to large-angle soft emissions

SJL (!ij , ") =SJL (#i · #j , ")

n!

i=1

Ji

"2(#i · ni)2

n2i

, "

#

Reduced soft anomalous dimension

!

j !=i

!

! ln "ij!S("ij , #s) =

14$(i)

K (#s)

evolution equation for reduced soft anomalous dimension

Becher Neubert; Gardi Magnea 2009

solution

with

!(i)K = Ci!K("s) !K = 2

"s(µ2)#

+ K

!"s(µ2)

#

"2

+ K(2)

!"s(µ2)

#

"3

+ · · ·

only 2-eikonal-line correlations

!S(!ij , "s) = !18#K("s)

!

i !=j

ln(!ij)Ti · Tj +12

$S("s)n!

i=1

Ci

generalises 2-loop solution

Are there any 3(or more)-line correlations ?Gardi Magnea say maybe; Becher Neubert say no

ConclusionsQCD is an extensively developed and testedgauge theory

a lot of progress in the last few years in

MonteCarlo generators

NNLO computations

better and better approximations of signal andbackground for Higgs and New Physics

NLO computations with many jets

resummations