Precision calculations for the LHCorFrom amplitudes to cross sectionsNigel GloverIPPP, Durham UniversityAmplitudes 2015, Zurich6 July 2015– p. 1

The Task for Experimental Particle Physics– p. 3

The Task for Theoretical Particle Physics– p. 4

Theoretical input✓✓vital in making precise perturbative predictions in quantum field theory, ingeneral, and in the Standard Model of particle physics, in particular.precise data enables information on new physics to be extracted indirectly(pre-discovery)– p. 5

Theoretical input✓not crucial for direct discovery!!– p. 6

Theoretical input✓but needed to interprete discovery as due to the production and decay of aStandard Model Scalar-like particle✓ H production cross section (σ) ✓ H branching ratio (BR)– p. 7

Why loops?✓Loop integrals play an intrinsic part in(a) the interpretation of experimental discoveries at the high energy frontier(b) extracting precise information from precision experiments(c) in making the case for the physics potential of future high energy facilities⇒ Importance of developments in Amplitudes that will be showcased at thisconference– p. 8

The challenge from the LHC✓✓✓✓Everything (signals, backgrounds, luminosity measurement) involves QCDStrong coupling is not small: α s (M Z ) ∼ 0.12 and running is important⇒ events have high multiplicity of hard partons⇒ each hard parton fragments into a cluster of collimated particles jet⇒ higher order perturbative corrections can be large⇒ theoretical uncertainties can be largeProcesses can involve multiple energy scales: e.g. p W T and M W⇒ may need resummation of large logarithmsParton/hadron transition introduces further issues, but for suitable (infrared safe)observables these effects can be minimised⇒ importance of infrared safe jet definition⇒ accurate modelling of underlying event, hadronisation, ...– p. 9

Cross Sections at the LHC– p. 10

Theoretical Frameworkp(P j )f p jj(ξ j P j )ˆσ ijQ 2p(P i )f p ii(ξ i P i )σ(Q 2 ) =∫ ∑i,jdˆσ ij (α s (µ R ),µ 2 R/Q 2 ,µ 2 F/Q 2 )⊗f p i (µ F)⊗f p j (µ F)[ ( )] 1+OQ 2✓ partonic cross sections dˆσ ij✓ running coupling α s (µ R )✓ parton distributions f i (x,µ F )✓✓renormalization/factorization scaleµ R , µ Fjet algorithm + parton shower + hadronisationmodel + underlying event + ...– p. 11

Theoretical Uncertainties- Missing Higher Order corrections (MHO)- truncation of the perturbative series- often estimated by scale variation - renormalisation/factorisation✓ systematically improvable by inclusion of higher orders- Uncertainties in input parameters- parton distributions- masses, e.g., m W , m h , [m t ]✓- couplings, e.g., α s (M Z )systematically improvable by better description of benchmark processes- Uncertainties in parton/hadron transition- fragmentation (parton shower)✓systematically improvable by matching/merging with higher orders- hadronisation (model)- underlying event (tunes)Goal: Reduce theory certainties by a factor of two compared to where we are now innext decade– p. 12

What is the hold up?Rough idea of complexity of process ∼ #Loops + #Legs (+ #Scales)- loop integrals are ultraviolet/infrareddivergent- complicated by extra mass/energyscales- loop integrals often unknown✓completely solved at NLO- real (tree) contributions are infrareddivergent- isolating divergences complicated✓completely solved at NLO- currently far from automation✓ mostly solved at NLOCurrent standard: NLO– p. 13

Anatomy of a NLO calculation✓✓one-loop 2 → 3 process✓ explicit infrared poles from loop integral✓ looks like 3 jets in final statetree-level 2 → 4 process✓ implicit poles from soft/collinear emission✓ looks like 3 or 4 jets in final state✓plus method for combining the infrared divergent parts✚ dipole subtraction Catani, Seymour; Dittmaier, Trocsanyi, Weinzierl, Phaf✚ residue subtraction Frixione, Kunszt, Signer✚ antenna subtraction Kosower; Campbell, Cullen, NG; Daleo, Gehrmann, Maitre✓ automated subtraction tools Gleisberg, Krauss (SHERPA); Hasegawa, Moch, Uwer(AutoDipole); Frederix, Gehrmann, Greiner (MadDipole); Seymour, Tevlin (TeVJet),Czakon, Papadopoulos, Worek (Helac/Phegas) and Frederix, Frixione, Maltoni, Stelzer(MadFKS)For a long time bottleneck was the one-loop amplitudes– p. 14

The one-loop problemAny (massless) one-loop integral can be written as= ∑ i d i(D) + ∑ i c i(D) + ∑ i b i(D)M = ∑ d(D)boxes(D)+ ∑ c(D)triangles(D)+ ∑ b(D)bubbles(D)✓higher polygon contributions drop out✓ scalar loop integrals are known analytically around D = 4 Ellis, Zanderighi (08)✓need to compute the D-dimensional coefficients d(D) etc.The problem is complexity - the number of terms generated is too large to deal with,even with computer algebra systems, and there can be very large cancellations.– p. 15

Unitarity for one-loop diagramsSeveral important breakthroughs✓Sewing trees togetherBern, Dixon, Dunbar, Kosower (94)✓✓✓Freezing loop momenta with quadruple cutsOPP tensor reduction of integrandD-dimensional unitarityBritto, Cachazo, Feng (04)Ossola, Pittau, Papadopoulos (06)Giele, Kunzst, Melnikov (08)=⇒ automationHELAC/CutTools, Rocket, BlackHat+SHERPA, GoSam+SHERPA/MADGRAPH,NJet+SHERPA, MADLOOPS+MADGRAPH– p. 16

Numerical recursion for one-loop diagramsBreakthroughs on the “traditional" side✓ One-loop Berends-Giele recursion van Hameren (09)✓ Recursive construction of tensor numerator Cascioli, Maierhöfer, Pozzorini (11)=⇒ automationOpenLoops+SHERPA, RECOLA– p. 17

NLO - the new standard✓A lot of progress, and the “best" solution is still to emerge. In the meantime,there are public codes with NLO capability that could only be dreamed of a fewyears ago.– p. 18

NLO EW corrections✓✓Relevance and size of EW correctionsgeneric size O(α) ∼ O(α 2 s) suggests NLO EW ∼ NNLO QCDbut systematic enhancements possible, e.g.,✚ by photon emission, mass singular logs ∝ (α)ln(m l /Q) for bare leptons -important for measurement of W mass✚at high energies, EW Sudakov logs ∝ (α/sin 2 θ W )ln 2 (M W /Q)EW corrections to PDFs at hadron colliders✚photon PDF✓Instability of W and Z bosons✚✚realistic observables have to be defined via decay productsoff-shell effects ∼ O(Γ/M) ∼ O(α) are part of the NLO EW corrections? How to combine QCD and EW corrections in predictions?– p. 19

Mixed QCD - EW corrections✓ Tree contributions: O(α s α), O(α 2 )✓Loop contributions: O(α 2 sα)– p. 20

Example: W/Z+higher jet multiplicities at NLOKallweit, Lindert, Maierhoefer, Pozzorini, Schoenherr (15)– p. 21

NLO precision for event simulationFixed order calculations✓ Expansion in powers of the coupling constant✓ Correctly describes hard radiation pattern✓ Final states are described by single hard particles✓ NLO: up to two particles in a jet, NNLO: up to three..✓ Soft radiation poorly describedParton shower✓ Exponentiates multiple soft radiation (leading logarithms)✓ Describes multi-particle dynamics and jet substructure✓ Allows generation of full events (interface to hadronization)✓ Basis of multi-purpose generators (SHERPA, HERWIG, PYTHIA)✓ Fails to account for hard emissionsIdeally: combine virtues of both approachesShape: Real Radiation and Normalisation: Loops– p. 22

Matrix Element improved Parton ShowerMEPS - mergingSeveral fixed order calculations of increasing multiplicity supplemented by PSCKKW: Catani, Krauss, Kuhn, Webber (01); MLM: Mangano– p. 23

Matrix Element improved Parton ShowerNLOPS - matchingOne fixed order calculation supplemented by PSMC@NLO: Frixione, Webber (02); POWHEG: Nason, Oleari (07)– p. 24

Matrix Element improved Parton ShowerMENLOPSSupplements core NLOPS with higher multiplicity MEPSHamilton, Nason; Hoeche, Krauss, Schoenherr, Siegert; Lonnblad, Prestel– p. 25

Matrix Element improved Parton ShowerMEPS@NLO (UNLOPS)Combines multiple NLOPSLavesson, Lonnblad; Hoeche, Krauss, Schoenherr, Siegert; Frederix, Frixione– p. 26

Reaching NNLOPS accuracyMINLOMultiscale improved NLO CKKW scale for Born piecesSudakov form factors for Born functions in POWHEGHamilton, Nason, ZanderighiExciting idea! starting from HJ@NLO+PS generate H rapidity distribution at NNLOHamilton, Nason, Oleari, Re, Zanderighi– p. 27

Motivation for more precise theoretical calculations✓✓✓Estimated signal strengths with largerLHC data setATL-PHYS-PUB-2013-014Theory uncertainty has big impact onmeasurementRevised wishlist of theoreticalpredictions for✚ Higgs processes✚ Processes with vector bosons✚Processes with heavy quarks orjets1405.1067ATLAS Simulation Preliminary-1-1s = 14 TeV: ∫ Ldt=300 fb ; ∫Ldt=3000 fbH→µµH→ττH→ZZH→WWH→ZγH→γγ(comb.)(incl.)(ttH-like)(VBF-like)(comb.)(VH-like)(ttH-like)(VBF-like)(ggF-like)(comb.)(VBF-like)(+1j)(+0j)(incl.)(comb.)(VH-like)(ttH-like)(VBF-like)(+1j)(+0j)→0.7→1.5→0.80 0.2 0.4∆µ/µ– p. 28

What NNLO might give you✓Reduced renormalisation scale dependence✓Event has more partons in the final state so perturbation theory can start toreconstruct the shower⇒ better matching of jet algorithm between theory and experiment✓LO NLO NNLOReduced power correction as higher perturbative powers of 1/ln(Q/Λ) mimicgenuine power corrections like 1/Q– p. 29

Motivation for NNLO✓Better description of transverse momentum of final state due to double radiationoff initial state✓✓✓✓LO NLO NNLOAt LO, final state has no transverse momentumSingle hard radiation gives final state transverse momentum, even if noadditional jetDouble radiation on one side, or single radiation of each incoming particlegives more complicated transverse momentum to final stateNNLO provides the first serious estimate of the error✓✓✓ and most importantly, the volume and quality of the LHC data!!– p. 30

Anatomy of a NNLO calculation e.g. pp → 2j✓✓✓double real radiation matrix elements dˆσ RRNNLO✓ implicit poles from double unresolved emissionsingle radiation one-loop matrix elements dˆσ RVNNLO✓ explicit infrared poles from loop integral✓ implicit poles from soft/collinear emissiontwo-loop matrix elements dˆσ V VNNLO✓ explicit infrared poles from loop integral✓ including square of one-loop amplitudedˆσ NNLO∼∫dΦ m+2dˆσ RRNNLO +∫dΦ m+1dˆσ RVNNLO +∫dΦ mdˆσ V VNNLO– p. 31

NNLO - amplitudes✓small number of two loop matrix elements known✓ 2 → 1: q¯q → V , gg → H, (q¯q → VH)✓ 2 → 2: massless parton scattering, e.g. gg → gg, q¯q → gg, etc✓2 → 2: processes with one offshell leg, e.g. q¯q → V +jet, gg → H+jet✓ 2 → 2: q¯q → t¯t, gg → t¯t known numerically Bärnreuther, Czakon, Mitov✓ 2 → 2: q¯q → VV, gg → VV new results in 2014 Cascioli et al✓ 2 → 3: gg → ggg first results in 2014 Badger, Frellesvig, Zhang?? Basis set of master integrals?? Efficient evaluation of master integrals?? Far from Automation✓ ✓Eager to have input from Amplitudes community– p. 32

IR subtraction at NNLO✓The aim is to recast the NNLO cross section in the formdˆσ NNLO =[ ]dˆσ NNLO∫dΦ RR −dˆσ NNLOSm+2++[ ]dˆσ NNLO∫dΦ RV −dˆσ NNLOTm+1m[ ]∫dΦdˆσ NNLO V V −dˆσ NNLOUwhere the terms in each of the square brackets is finite, well behaved in theinfrared singular regions and can be evaluated numerically.– p. 33

NNLO - IR subtraction schemesWe do not have a fully general subtraction scheme as we have at NLOFive main methods:✚ Antenna subtraction Gehrmann, Gehrmann-De Ridder, NG (05)✚ q T subtraction Catani,Grazzini (07)✚ Colourful subtraction Del Duca, Somogyi, Tronsanyi✚ Stripper Czakon (10); Boughezal et al (11)✚ N-jettiness subtraction Boughezal, Focke, Liu, Petriello (15); Gaunt, Stahlhofen,Tackmann, Walsh (15)Each method has its advantages and disadvantages– p. 34

Higgs production at N3LO m t → ∞✓Aim to reduce the theoretical error for the inclusive Higgs cross section via gluonfusion to O(5%)✘✓✓In principle, need double box with top-quark loop! - currently not knownHiggs boson is lighter than the top-pair threshold1/m t corrections known to be small at NNLO⇒ Work in effective theory where top quark is integrated outL = L QCD,5 − 14v C 1HG a µνG aµν✓ Ingredients: Three-loop H+0 parton, Two-loop H+1 parton, One-loop H+2parton, Tree-level H+3 parton - all known as matrix elements for m t → ∞- key part is to extract the infrared singularities– p. 35

Higgs production at N3LO m t → ∞ˆσ ij (z)z= ˆσ SV δ ig δ jg +∞∑N=0ˆσ (N)ij (1−z) NAt N3LO,ˆσ SV = aδ(1−z)+5∑k=0[ ] log k (1−z)b k1−z+✓Plus-distributions produced by soft gluon emissions and already known adecade ago✓ a computed by Anastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Mistlberger (14)σ (N)ij =5∑k=0c (N)ijk logk (1−z)✓✓Describes subeading soft emissionsSingle emissions known exactly, but double and triple emissions known only asan expansion Anastasiou, Duhr, Dulat, Herzog, Mistlberger (14)– p. 36

Higgs cross section at N3LOAnastasiou, Duhr, Dulat, Herzog, Mistlberger– p. 37

Higgs cross section at N3LO– p. 38

pp → H + jet production at NNLO m t → ∞✓✓Key goal: Establish properties of the Higgs boson!experimental event selection according to number of jets✓ different backgrounds for different jet multiplicities✓ H+0 jet known at NNLOAnastasiou, Melnikov, Petriello; Catani, Grazzini✓✓H+n jets (n=1,2,3) known at NLOH+0 jet and H+1 jet samples of similar size✓NNLO H+1 jet crucial, particularly for WW channel✓ Three independent computations:✚ Stripper Boughezal, Caola, Melnikon, Petriello, Schulze✚ N-jettiness Boughezal, Focke, Giele, Liu, Petriello✚ Antenna (gluons only) Chen, Gehrmann, Jaquier, NG✓Fully differential and allows for arbitrary cuts on the final state– p. 39

pp → H + jet at NNLO m t → ∞✓✓large effects near partonic thresholdlarge K-factorσ NLO /σ LO ∼ 1.6σ NNLO /σ NLO ∼ 1.3✓ significantly reduced scale dependenceO(4%)– p. 40

NNLO Higgs production via VBF✓✓Second largest source of Higgsbosonsdistinctive signature⇒ very useful for signal extractionand background suppression✓✓suppressed color exchange between quark lines gives rise to✚ little jet activity in central rapidity region✚✚✚scattered quarks → two forward tagging jetsHiggs decay products typically between tagging jetsmany Feynman diagrams suppressed by colour or kinematic considerationsNLO QCD corrections moderate and well under control (order 10% or less)– p. 41

NNLO Higgs production via VBFCacciari, Dreyer, Karlberg, Salam, Zanderighi✓✓NNLO QCD corrections are muchlarger in VBF setup than for inclusivecutsNNLO corrections appear to makejets softer, hence fewer events passthe VBF cut– p. 42

Improved precision for input parameters✓✓More precise measurements of strong couplingImproved parton distributions– p. 43

pp → 2 jets at NNLO✓✓One of key processes for perturbativeQCDCurrent experimental precisionO(5-10%) for jets from 200 GeV/c-1TeV/c(pb)dσ/dpT8070603×1090s=8 TeVR=0.7anti-k TMSTW2008nnloµ = µ = µR F80 GeV < p < 97 GeVTLO (NNLO PDF+α s )NLO (NNLO PDF+α s )NNLO (NNLO PDF+α s )✘- Need NNLO QCD and NLO EWOnly process currently included inglobal PDF fits that is not known atNNLO✓gg channel - leading colourCurrie, Gehrmann-De Ridder,Gehrmann, Pires, NG(pb)dσ/dpT5040302080703×1090s=8 TeVR=0.7anti-k TMSTW2008nnlo1µ = µ = µR F80 GeV < p < 97 GeVTLO (NNLO PDF+α s )µ/pT1NLO (NNLO PDF+α s )NNLO (NNLO PDF+α s )✓Scale variation much reduced for0.5 < µ/p T < 2.605040✓Size of corrections, and uncertainty,still depends on scale choice p T1 vp T .30201µ/pT– p. 44

Di-jet mass distribution (gluons only) at NNLOGehrmann-De Ridder, Gehrmann, Pires, NG; Currie, Gehrmann-De Ridder, Pires, NG✓✓NNLO corrections ∼25% wrt NLOsimilar behavior for different rapidityslices– p. 45

NNLO Z+1 jet productionGehrmann, Gehrmann-De Ridder, Huss, Morgan, NG✓✓✓✓✓An important background for beyondthe standard model searchesVery precise measurements can beobtained.Provides a fantastic testing groundfor precision QCD and electroweakcorrectionsUseful for detector calibration, jetenergy scale can be determined fromthe recoil of the jet against the Zboson.Useful process for PDF determinationInitial State σ (pb) % contributionqg 80.2 55.6q¯q 33.1 22.9¯qg 33.1 22.9gg -4.0 -2.7qq 1.8 1.2¯q¯q 0.1 0.1Total 144.3 100.0– p. 46

NNLO Z+1 jet productionGehrmann, Gehrmann-De Ridder, Huss, Morgan, NG✓Excellent convergence of NNLO in the jet p T distribution✓ Significant reduction in the scale uncertainty– p. 47

NNLO Z+1 jet productionGehrmann, Gehrmann-De Ridder, Huss, Morgan, NG✓ NNLO corrections uniform in rapidity, approximately 7%✓ Significant reduction in the scale uncertainty– p. 48

pp → ZZ at NNLOObserve✓✓✓Cascioli et alThe NNLO corrections increase theNLO result by an amount varyingfrom 11% to 17% as √ s goes from 7to 14 TeV.The loop-induced gluon fusioncontribution provides about 60% ofthe total NNLO effect.When going from NLO to NNLO thescale uncertainties do not decreaseand remain at the ±3% level.– p. 49

pp → W + W − at NNLOσ[pb]14012010080pp → W + W − +XATLASCMS✓✓Provides a handle on thedetermination of triple gaugecouplings, and possible new physicsSevere contamination of the W + Wcross section due to top-quark resonances604020gg → H → WW ∗added to all predictionsNNLO+ggNLO+ggNNLON+ggLONN+gg1.σ/σ NLO √7 8s[TeV] 13 14Gehrmann et al✓✓The NNLO QCD corrections increasethe NLO result by an amount varyingfrom 9% to 12% as √ s goes from 7to 14 TeV.Scale uncertainties at the ±3% level.– p. 50

LHC cross sections at NNLO✓✓✓✓✓Past few years, there has been an accelerating progress in computing NNLOcorrections to important LHC processes✚ H+Jet, H+2 Jet (VBF), H+W, H+Z, H+H, ...✚ W+Jet, Z+Jet, Di-Jet, Di-Gamma, WW, ZZ, Z+Gamma, single t, t¯t, ...Mostly of 2 → 2 variety based on amplitudes computed a while agoSeveral IR subtraction technologies availableWhenever gluons are involved, the NNLO contributions can be largeAll show a reduced scale variation– p. 51

Accuracy and Precision (A. David)– p. 52

Accuracy and Precision (A. David)– p. 53

Accuracy and Precision (A. David)– p. 54

Accuracy and Precision (A. David)– p. 55

Accuracy and Precision (A. David)– p. 56

Estimating uncertainties of MHO✓ Consider a generic observable O (e.g. σ H )whereO(Q) ∼ O k (Q,µ)+∆ k (Q,µ)O k (Q,µ) ≡k∑c n (Q,µ)α s (µ) n ,∆ k (Q,µ) ≡∑···c n (Q,µ)α s (µ) nn=0n=k+1✓ Usual procedure is to use scale variations to estimate ∆ k ,∆ k (Q,µ) ∼ max( [O k Q, µ ) ],O k (Q,rµ)r∼ α s (µ) k+1where µ is chosen to be a typical scale of the problem and typically r = 2.Choice of µ and r = 2 is convention– p. 57

Theoretical uncertainty on σ HForte, Isgro, VitaWarning: Scale variation may not give an accurate estimate of the uncertainty inthe cross section!!– p. 58

Going beyond scale uncertainties✓ Series acceleration David, Passarino✓sequence transformations gives estimates of some of the unknown terms inseriesEstimate coefficients using information on the singularity structure of the Mellinspace cross section coming from all order resummationBall et al- large N (soft gluon, Sudakov)- small N (high energy, BFKL)✓ Bayesian estimate of unknown coefficients Cacciari, Houdeau✓make the assumption that all the coefficients c n share a (process dependent)upper bound ¯c > 0 leading to density functions f(c n |¯c) and f(ln¯c)recent refinement of methodBagnaschi, Cacciari, Guffanti, JennichesAccepting that scale variation does not give reliable error estimate, can predictthe part of the N 3 LO cross section coming from scale variations.Pressure is building to better estimate MHO– p. 59

Theoretical uncertainty on σ H revisitedBagnaschi, Cacciari, Guffanti, JennichesUncertainty in Modified CH approach larger but more realistic!!– p. 60

Where are we now?✓✓Witnessed a revolution that has established NLO as the new standard- previously impossible calculations now achieved- very high level of automation for numerical code- standardisation of interfaces - linkage of one-loop and real radiationproviders- take up by experimental communitySubstantial progress in NNLO in past couple of years- several different approaches for isolating IR singularities- several new calculations available- codes typically require significant CPU resource– p. 61

Where are we going?✓NNLO automation?- as we gain analytical and numerical experience with NNLO calculations, canwe benefit from (some of) the developments at NLO, and the improvedunderstanding of amplitudes- automation of two-loop contributions?- automation of infrared subtraction terms?- standardisation of interfaces - linkage to one-loop and real radiationproviders?- interface with experimental communityNext few years:✓✓✓Les Houches wishlist to focus theory attentionNew high precision calculations that will appear such as, e.g. N3LO σ H , couldreduce Missing Higher Order uncertainty by a factor of twoNNLO will emerge as standard for benchmark processes such as dijetproduction leading to improved pdfs etc. could reduce theory uncertainty due toinputs by a factor of two– p. 62

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