QCD Theory And Monte Carlo Tools - INDICO-FNAL (Indico)
QCD Theory and Monte Carlo ToolsJohn CampbellFermilab-CERN HCPSS1820-31 August 2018
References Useful resources for perturbative QCD — additional background and furtherreading for more advanced topics — are: QCD and Collider PhysicsR. K. Ellis, W. J. Stirling and B. R. WebberCambridge Monographs on Particle Physics, Nuclear Physics and Cosmology The Black Book of Quantum Chromodynamics: A Primer for the LHC EraJC, J. Huston, F. KraussOxford University Press Resource Letter: Quantum ChromodynamicsA. S. Kronfeld and C. QuiggarXiv: 1002.5032 [hep-ph], prepared for the American Journal of Physics2HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Outline Introduction to the theory of QCD- Lagrangian, color, Feynman rules, strong coupling QCD for hadron colliders- factorization, parton distribution functions, hard scattering Structure of QCD matrix elements- infrared singularities, real and virtual radiation Beyond leading order- techniques for NLO, NNLO and beyond Parton shower techniques- Sudakov factors, resolvable emissions, hadronization Modern event generators- merging, matching, hybrid schemes, precision matching3HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
10121011 It is no surprise that hadron collidersrequire an understanding of QCD.33 TeVHE LHC100 t106σ [pb] However, the level of sophisticationwe require is demonstrated by theinclusive cross-sections for finalstates that we are typicallyinterested in.14 TeVLHC105104(pT 50 GeV)103102WZ101γγ In order to test the SM (and modelsof new physics), we require aquantitative understanding of QCDand precise theoretical predictions. These lectures will describe theways in which we reach this goal.4HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools10321010(pT 50 GeV)gg H 100tttWWWZZZγγVBFttHWHZH10-1HH10-3110-1MCFM-210 -210-410-51010s [TeV]102Events / second @ 1033 cm2 s–1QCD: why we care8 TeVLHC
The challenge of QCDLQCD 5X1 a µ Fµ Fa q̄i (iDµ4HCPSS 2018 John Campbell QCD Theory and Monte Carlo Toolsflavorsµm)ij qj
QCD and color The Lagrangian looks a lot like the one for QED: a field strength termrepresenting the gluon field and a Dirac term for the quarks. However, it has one important difference: color.LQCD X1 aF µ Faµ q̄i (iDµ4µm)ij qjflavors Within the quark model, the additional quantum number of color was initiallyintroduced to accommodate the existence of the Δ baryon.- antisymmetry to satisfy Pauli exclusion principle carried by color- quarks and gluons carry color but observed hadrons are colorless The color degrees of freedom can also be directly probed in electron-positroncollisions, by comparing the production of hadrons and muons.6HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
“R-ratio” measurements7HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Color in the QCD Lagrangian: quarksX1 aµ LQCD F µ Fa q̄i (iDµ4µm)ij qjflavors The gauge principle — invariance under local gauge transformations —requires the introduction of the gauge-invariant derivative:(Dµ )ij @µij igs ta Aaµij When inserted in the Lagrangian this introduces interactions between quarksof color i and j, mediated by the gluon field Aaij. Strength of the interaction dependson the strong coupling (gs) anda matrix in color space (ta)that is related to a Gell-Mannmatrix (λa) by ta λa/2. These are Hermitian, traceless& satisfy commutation relation:8 a,b 2ifabccHCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
SU(3) The matrices ta are the generators of the group SU(3) in the fundamentalrepresentation: eight 3x3 matrices that satisfy: abt ,t ifabc tcwhere fabc form a set called theSU(3) structure constants. They’rereal numbers and are completelyantisymmetric in the indices. The matrices also obey a normalization condition:a bTr t t TRabwith TR 1/2 By inspection, we can also see that:Xa The quantity CF is called a Casimir.9HCPSS 2018 John Campbell QCD Theory and Monte Carlo Toolsta ta C F 14Nc2 1with CF 32Nc
Color in the QCD Lagrangian: gluonsX1 aµ LQCD F µ Fa q̄i (iDµ4µm)ij qjflavors The field strength tensor in the first term is fundamentally different from theQED case:aFµ a@ µ A a@ Aµb cgs fabc Aµ A The final term involves the SU(3) structure constants and two gluon fields. When inserted into the Lagrangian this leads to self-interactions betweengluons, involving three or four fields. To handle these gluon interactions we will need a second Casimir:Xfacd bcdf CAabc,d(can again check this by inspection)10HCPSS 2018 John Campbell QCD Theory and Monte Carlo Toolswith CA 3 Nc
From the Lagrangian to calculations To use this Lagrangian we need to be able to calculate scattering amplitudesand ultimately cross-sections. The main toolbox for collider physics is perturbative QCD:- expand the Lagrangian about the free (non-interacting) case in powers of the coupling- interactions correspond to at least one power of the coupling- represent amplitudes as Feynman diagrams, with rules read off from the Lagrangian In the free case (gs 0) there are only propagators. These are easily read off from two-point interactions in the Lagrangian, thatgive the inverse propagator, after making the momentum-space replacement(c.f. Fourier expansion): @µ ! ipµ E.g. quarks: q̄i (i@µjimom. p11µm)(pµp2ij qj ! q̄i (pµµ m)m2HCPSS 2018 John Campbell QCD Theory and Monte Carlo Toolsijµm)ij qjNote: inverse of gluon propagatorrequires extra gauge-fixing term
QCD derivative)(Dµ )ij @µ12ijTriple-gluonvertex (fieldstrength tensor,one derivative) igs taaAµ ijHCPSS 2018 John Campbell QCD Theory and Monte Carlo ToolsaFµ @µ Aa Four-gluonvertex (fieldstrength tensor,no derivatives)@ Aaµgs fabc Abµ Ac
Color factors in action As an example of how these work, consider additional gluon emission from ahard quark or gluon. Looking at just the color matrices (i.e. ignoring kinematics) we can find theeffect on gluon emission probabilities from just the Feynman rules so far:2Pemit (q) X atijatij a,jPemit (g) Xa atij tji CFa,j(sum over colors)2 X(Hermitian)(Casimir)fabc fabc CAb,c(sum over colors)(Casimir) This gives rise to the expectation that gluon jets radiate more copiously thanquark jets.13HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Color factors and cross-sections In calculations of cross-sectionsthese sums over color factorsare ubiquitous (and can bearduous to handle, for verymany colored particles). The Casimirs of SU(3), CA andCF, are intrinsic to the theoryof QCD. Their values have beentested, e.g. in measurementsof jet event shapes at LEP.- A neat demonstration of themanifestation of group theoryin physical observables.14HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
The strong coupling The effects of the quantum field theory vacuum — populated by short-lived,virtual quark and gluon pairs — affects the strength of the coupling. Trying to measure the coupling of an individual quark by probing with a gluonis only possible at sufficiently high energy. At lower energies the probe will instead only resolve a cloud of virtualparticles that partially screen the coupling. This dependence on the energy scale μ is encoded in the beta function,which governs the running of the strong coupling:@ s (µ)@ s (µ)( s ) µ @µ@(log µ) It can be computed perturbativelyby considering exactly thesevacuum fluctuation diagrams,e.g. one power of αs:15HCPSS 2018 John Campbell QCD Theory and Monte Carlo Toolsgs2with s 4
Running the strong coupling At this order the result is:b0 s2( s ) .11CA 2nfb0 6 With quark loops alone (c.f. QED) the result would be a positive betafunction; in contrast, in QCD the gluon loops make it negative. Can now solve for the coupling relative to some other scale Q:@ s (µ) @(log µ)b0 s (µ)2 )1 s (µ) b0 [log µ)]µ Qµ Q s (Q) ) s (µ) 1 s (Q)b0 log(µ/Q) Note that this diverges at the scale Λ Q when the denominator vanishes. This condition gives an alternate expression for the running:1 s (µ) b0 log(µ/ )16Λ is the QCD scale; it represents theposition of the Landau pole in QCDHCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Consequences and tests1 s (µ) b0 log(µ/ ) Strong coupling decreasesat high energy:asymptotic freedom. Perturbation theory requiressufficiently high energy,unreliable close to Λ. Measured value of thestrong coupling values of Λ around250 MeV.17HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Protons and partons We now have to understand how to apply QCD in the era of hadron colliders. To do so, we have to understand how to apply a theory of quarks and gluonsto the protons found in the beams. The appropriate formalism iscalled collinear factorization. It divides the problem into: soft physics, corresponding tothe probability of finding,within a proton, a parton witha given momentum fraction x. hard physics, the subsequentscattering between theincident quarks and gluons. Strictly only proven in specialcases: Drell-Yan and deepinelastic scattering (DIS).18HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Parton distribution functions (pdfs) Depend on the momentum fraction (xa) and the factorization scale (μF), thatis implicit in the separation into soft and hard scales: Interpret as a probability must integrate over fraction xa (and xb) In the simplest, non-interacting, picture onemight assume the proton consists of just thethree valence quarks. With no quark preferredabove others one would get: By construction, these satisfy the momentum sum rule: A more sophisticated guess would be to imagine elastic interactions betweenthe quarks, “rubber bands” holding them together- only effect would be to smear out the δ-function, smoothing the sharp peak at x 1/3.19HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
QCD effects in pdfs In fact, the valence quarks inside the proton will emit gluons (that can furthersplit into quark-antiquark pairs). These emissions will tend to be soft with respect to the original quark,meaning that the additional sea partons will be more likely to be found atsmall values of x. In fact, to a fair approximation:λ 1 (gluons, sea quarks)λ -1/2 (valence quarks) Effect of QCD interactions:- pdfs increase at small x- valence peak shifts to lower x 0.1and broadens (due to emission)20HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Probing pdfs Since they represent truly soft, non-perturbative, physics the pdfs cannot becalculated from first principles. However, the factorization procedure is based on the fact that they areuniversal: independent of the hard scattering and the rest of the collision.- therefore they can be extracted from experimental data. Deep inelastic scattering in electron-proton collisions, historically at HERA, isan ideal environment for this.- pdf enters only in part of the initial state.- the rest is well-known QED. This process is called “deep” due tothe fact that the probing photon is ofvery high virtuality:Q2 q21 GeV This is the scale of the pdf that is probed.21HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Pdf evolution Although they are essentially non-perturbative objects, their evolution — thedependence on the probing scale — depends on the emission of quarks andgluons and is calculable in perturbative QCD. Just like the strong coupling, the pdfs obey (coupled) evolution equations.At first order these take the form: Called the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation. The kernels of this evolution equation, the quantities Pab, are called splittingfunctions (more on these later).ba They represent the parton splitting:c (implicit)22HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
QCD-improved parton model Taking account of thisevolution results in theQCD-improved partonmodel. It gives rise to so-calledscaling violation, which isclearly visible inexperimental data. See for example thecombination of HERA data(from experiments H1 andZEUS) taken over theperiod 1994-2000.23HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Pdf fitting: general strategy Since the Q2 evolution of the pdfs is known, the traditional approach is toparametrize them at some reference scale, typically Q0 1-2 GeV. Typically starting ansatz is:with a smooth function P and free parameters A1, A2, Perform a global fit to available data, using DGLAP equation to evolve thepdfs to the appropriate scale first. Lots of room for interpretation:- choice of input data sets (especially in cases of conflict)- order of perturbation theory (in theory predictions and DGLAP evolution)- input parametrization and other theoretical prejudice (e.g. always positive or not). Global fitting industry: continuous improvements to the fitting procedure andtheoretical input. Main groups are CTEQ, MSTW/MMHT and NNPDF. NNPDF has a different approach to starting ansatz, instead using a sample ofpdf replicas generated by neural network to try to avoid parametrization bias.24HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Typical data setsNNPDF2.3 xed-larget,TevatronQ2 / M2 / p2 [ GeV2 36pbCMS-WEASY-840PBLHCB-W-36pbdeep inelasticscatteringATLAS-JETS-10101 -5102510-4HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools-310x10-210-11
LHC parton kinematicsPdf requirements ) x1,2M y p es High-mass or high-rapidityparticle production may beoutside fit range and sufferfrom larger pdf uncertainties.810x1,2 (M/14 TeV) exp( y)Q MM 10 TeV7106M 1 TeV105104M 100 GeV1022Q (GeV ) Simplest case: productionof a single particle withmass M and rapidity y. Kinematics:psp1 x 1(1, 0, 0, 1)p2sp2 x 2(1, 0, 0, 1)2pf M (cosh y, 0, 0, sinh y)910310y 642046210M 10 PSS 2018 John Campbell QCD Theory and Monte Carlo Tools-210-110010
Uncertainties and consistency The associated pdf uncertainties typically cover the spread between differentfitting groups, at least in the well-constrained region 50 - 500 GeV.- uncertainties on cross-sections at the level of 2–4% (important for modern precision!) Beyond that, differences begin to emerge and uncertainties are O(10%).- prescriptions for combining them to capture the spread exist, e.g. PDF4LHC.uncertainty for gg Xuncertainty for qq XLHC 13 TeV, NNLO, αS(MZ) 0.1181.3Gluon - Gluon Luminosity1.251.21.151.21.151.1PDF4LHC15 50.810271.25Quark - Antiquark LuminosityPDF4LHC15 priorCT14NNPDF3.0MMHT14LHC 13 TeV, NNLO, αS(MZ) 0.118210 M ( GeV )X310HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools0.90.8510102 M ( GeV )X103
Example pdfsCT14 NNLOCT14 NNLO0.80.8g!x,Q"#5g!x,Q"#5x f!x,Q" at Q 2 GeV0.6x f!x,Q" at Q 100 .010.20.030.10.31xQ 2 GeV Near starting scale for theevolution. u, d still peaked near x 1/3.28HCPSS 2018 John Campbell QCD Theory and Monte Carlo Toolsd!baru!bars!bar0.0.0010.0030.010.030.10.31xQ 100 GeV Typical LHC kinematics. u,d flattened, less important. gluon dominant for x 0.1.
Summary so far Have illustrated how the QCD Lagrangiancan be translated into Feynman rules, withan emphasis on the special role of color.LQCD X1 a µ Fµ Fa q̄i (iDµ4µm)ij qjflavors Have discussed the strongcoupling and the idea ofcollinear factorization forhadron collisions and theintroduction of pdfs. Will now spend some time onthe calculation of thehard scattering process.29HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Hard scattering calculations First we have to break down the partonic cross-section we identified into afew constituent parts:ˆab!n (µF , µR ) Incoming partonic flux: Transition amplitude (or matrix element) squared: Integrated over the available n-parton phase-space element, d𝚽n.Lorentz-invariant phase-spaceelement for each final state particle30HCPSS 2018 John Campbell QCD Theory and Monte Carlo Toolsensure overall fourmomentum conservation
W-production Consider one of the simplest-possiblehadron-collider processes, which is primarilymediated by up-anti-down annihilation. Application of the (mostly EW) Feynman rules gives the matrix element:weak couplingCKM elementquark spinorstrivial color factorDirac algebra(LH current)polarizationvector Squaring and summing over spins and colors is an exercise in Dirac algebra:color sumaveraged over initialcolors and spins31HCPSS 2018 John Campbell QCD Theory and Monte Carlo ToolsQ p1 p2
Partonic cross-section Putting the ingredients together we have:where Recalling our earlier kinematics we also haveŝ xu xd syW1xu log2xd so that we can perform the convenient change of variable:32HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
Final result The maximum rapidity is constrained by x 1 to be: This is the lowest order (tree-level) result for the inclusive cross-section.- the result for W- is obtained by interchanging u and anti-d quarks. In this form we immediately see that the rapidity distribution of the W-bosonis entirely defined by the (quark) pdfs.33HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
W rapidity distribution: Tevatron vs. LHC Tevatron: valence quarks in protons drive production of W to positive rapidityand anti-protons favor W- at negative rapidity.- asymmetry is used to constrain high-x valence quark pdfs(although indirectly, through diluted lepton asymmetry) LHC: no preferred direction and sea quarks play an important role;impact of valence quarks still evident in wider plateau for W .34HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
W rapidity at the LHC and beyond As energy of collisions increases, so does accessible range of W rapidities. The value of x required to produce a W boson decreases, leading to moreimportant role for sea quarks. Eventually sea quarks dominate and, at central rapidities, W and W- crosssections become similar.35HCPSS 2018 John Campbell QCD Theory and Monte Carlo Tools
The main toolbox for collider physics is perturbative QCD: - expand the Lagrangian about the free (non-interacting) case in powers of the coupling - interactions correspond to at least one power of the coupling - represent amplitudes as Feyn
BBSR QCD, Collider physics, neutrino physics, and dark matter. NISER, BBSR QCD VECC, Kolkata QCD IIT, Bombay QCD and Physics beyond standard model SINP, Kolkata QCD Bose Inst. Kolkata QCD AMU, Aligrah QCD IIT Madras QFT IIT Delhi QCD
The Markov Chain Monte Carlo Revolution Persi Diaconis Abstract The use of simulation for high dimensional intractable computations has revolutionized applied math-ematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through micro-local analysis. 1 IntroductionCited by: 343Page Count: 24File Size: 775KBAuthor: Persi DiaconisExplore furtherA simple introduction to Markov Chain Monte–Carlo .link.springer.comHidden Markov Models - Tutorial And Examplewww.tutorialandexample.comA Gentle Introduction to Markov Chain Monte Carlo for .machinelearningmastery.comMarkov Chain Monte Carlo Lecture Noteswww.stat.umn.eduA Zero-Math Introduction to Markov Chain Monte Carlo .towardsdatascience.comRecommended to you b
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