The Strong Interaction and LHC phenomenology

The Strong Interaction and LHC phenomenology Juan Rojo STFC Rutherford Fellow University of Oxford Theoretical Physics Graduate School course

Lecture 9: Parton Distributions and LHC phenomenology

QCD partons in the initial state As we saw in previous lecture, high-energy collisions involving hadrons in the inital state can be described within the naive parton model In the case of deep-inelastic scattering (one hadron in initial), we have Hadronic cross-section Parton Distribution Function (PDF): Probability of finding quark q in the proton with momentum fraction y Partonic cross-section We saw that in this case QCD collinear divergences do not cancel and need to be absorbed by a redefinition of the PDFs, which acquire a dependence on the factorization scale and that the dependence of the PDFs with this scale is purely perturbative, determined by the DGLAP evolution equations

Recap of DIS kinematics The kinematics of deep-inelastic scattering were defined as In the parton model, x Bj can be interpreted as the fraction of the proton momentum carried by the struck quark We also saw that PDFs obey certain valence and momentum sum rules, which are maintained by perturbative QCD evolution In this lecture we discuss how to determine the Parton Distributions of the proton from experimental data

PDF evolution equations Once parton distributions have been determined at a given scale, the so DGLAP (Dokshitzer-Gribov- Lipatov-Altarelli-Parisi) evolution equations can be used to evolve them to any other scale These equations are a set of N flav +1 coupled integro-differential equations A variety of numerical and semi-analytical methods exist for efficient solutions of the DGLAP equations Juan Rojo University of Oxford, 06/05/2014

Parton Distribution Functions (PDFs) Event rates at the LHC depend on the Parton Distribution Functions (PDFs) of the proton LHC Master Formula PDFs encode non-perturbative dynamics that determine the distribution of energy that quarks and gluons carry within the proton { Computation of PDFs from first principles not competitive PDFs: Determine PDFs via global QCD analysis Theory+Data+Methodology of hard-scattering data { Matrix elements: Theory Experimental data 7 6 5 4 3 2 1 0-1 -2-5 10 xg(x, Q = 2.0 GeV ) - s = 0.119-4 10 2-3 10 x 2 NNPDF2.3 NNLO CT10 NNLO MSTW2008 NNLO -2 10-1 10 1 QCD Theory PDFs Predictions @ LHC Statistical Methodology Parton Distributions: fundamental limit to theory predictions at LHC 6

Parton Distribution Functions Proton Parton Distributions determine the fraction of the proton energy that is carried by each of its constituents, quarks and gluons Gluon PDF, g(x 1,Q) (Bjorken) x 1 = E gluon / E proton Higgs Boson Gluon PDF, g(x 2,Q) Proton 1 / Q = resolution scale Higher energies ( large Q) -> Protons probed at small distances

Parton Distribution Functions Therefore, the main task here is to determine the Bjorken-x dependence of the PDFs from data Dependence on the momentum fraction is non-perturbative: needs to be extracted from experimental data Evolution in scale fixed by perturbative QCD dynamics In principle, one could use lattice QCD to determine this dependence with Bjorken-x, but current accuracy is far from the level of precision needed for LHC phenomenology We also need to provide a sound, statistically robust methodology, in order to avoid introducing any theoretical bias in the PDF determinations Also, it is of paramount importance to determine the associated PDF uncertainties, which can be of various origins Experimental uncertainties from the finite precision of data Theoretical uncertainties, like from missing higher perturbative orders Methodological uncertainties, like functional form bias

PDFs and LHC phenomenology 2) Very large PDF uncertainties (>100%) for new heavy particle production 1) PDFs fundamental limit for Higgs boson characterization in terms of couplings Supersymmetric QCD W mass Top Quark mass 3) PDFs dominant systematic for precision measurements, like W boson mass, that test internal consistency of the Standard Model

PDFs and LHC phenomenology 1) PDFs fundamental limit for Higgs boson characterization in terms of couplings Theory systematics (hatched areas) limiting factor for Higgs coupling extractions at the LHC

PDF determination PDF determination is based on a global analysis of hard scattering data to extract, thanks to the factorization theorem, universal PDFs for LHC predictions Experimental data Lepton-proton structure functions, heavy quark production, jet production, Drell-Yan pair production, electroweak bosons, isolated photons,... QCD Theory NNLO DGLAP evolution, NLO and NNLO hard scattering cross sections, heavy quark treatment, strong coupling, electroweak effects, hadronic production, PDF flavor separation,... Methodology Ansatze for the x-dependence of PDFs q(x,q 0 ), propagation of experimental errors from data to PDFs, model uncertainties, minimization, definition of the figure of merit! 2, parallel computing,... Parton Distributions LHC phenomenology, predictions for LHC processes, PDF and Higgs production,... 11

PDF determination Our goal is to determine all the Parton Distributions (PDFs) of the proton from hard-scattering data, at some fixed input scale Q 0, then use DGLAP evolution to evaluate them in any other Q In principle we need to determine 13 PDFs, one for each quark and antiquark, and then the gluon: So the problem is essentially to determine 13 functions from a finite set of data points What information does QCD provide us with? PDFs should vanish when x=1 due to kinematic constraints PDFs need to satisfy the valence and momentum sum rules PDFs are not positive definite quantities beyond LO, though cross-sections should Heavy quark PDFs are generated radiatively from gluon splitting, no need of intrinsic heavy quark PDF In addition, experimental data has finite accuracy, so it is of paramount importance to estimate as well which are the PDF uncertainties for a given set of data and theory input 12

PDF determination With this information into account, we need to determine from data: The four normalization factors are determined by valence and momentum sum rules The (1-x) prefactor imposes the mass-shell kinematical constraint f q (x) are smooth functions, should be flexible enough to parametrize any general underlying law PDF parametrization in the MSTW08 set 13

PDF evolution Once PDFs are determined at some scale, typically O(few GeV), we can compute them at any other scale using the DGLAP evolution equations Low scale LHC scale 14

PDF evolution: high-energy limit In the limit when the momentum fraction carried by partons is very small, x << 1, it is possible to derive approximate analytical solutions to the DGLAP equations This is the DGLAP small-x limit, also known as small-x limit, because the invariant mass of the hadronic scattering is Therefore, for fixed four-momentum transfer Q, the small-x limit is the limit where the center of mass energy of the collision becomes very high In this limit, QCD becomes to first approximation a theory composed only by gluons, since splittings into gluons are strongly enhanced at small-x by the DGLAP evolution equations 15

PDF evolution: high-energy limit Let s consider DGLAP evolution equations for gluons only where in the small-x limit the gluon-gluon splitting function simplifies to It is advantageous to write the DGLAP evolution equations in Mellin (moment) space, Since it can be shown (exercise) that the DGLAP convolution factorizes in Mellin space and we find We have reduced an integro-differential equation to a simple ODE, which can be analytically solved 16

PDF evolution: high-energy limit The solution for the DGLAP evolution equations for gluons only in Mellin space in the small-x limit is Now we need to go back to momentum space, performing the inverse Mellin transform using the saddle point approximation (exercise) So the analytic solution for the gluon PDF in the small-x limit is 17

PDF evolution: high-energy limit Which can also be expressed as This equation shows that at small-x: The gluon PDF xg(x) rises steeply, and the more the larger the evolution lenght The initial shape of the gluon PDF is to a good extent washed up by the PDF evolution This steep gluon will feed down to quark PDFs via DGLAP mixing, and thus at small-x all PDFs rise steeply as x decreases 18

PDF evolution: heavy quark PDFs The intrinsic component of heavy quark PDF is suppressed by terms O(" qcd 2 / m h 2 ). Therefore, heavy quark PDFs are generated perturbatively, from gluon splitting For example, for the bottom PDF, at leading order, the DGLAP equations lead to (exercise) with the boundary condition that the bottom PDF vanishes below the bottom mass In a scheme where the bottom quark is massive, all bottoms arise from gluon splittings In a scheme where the bottom quark is massless, there is a bottom PDF in the proton as for all other quarks, that is constructed resumming gluon collinear splittings 19 g(x) b(x)

Precision tests of the Factorization Theorem Perturbative QCD requires that the momentum integral should be unity to all orders Is it possible to determine the value of the momentum integral from the global PDF analysis, rather than imposing it? Check in LO*, NLO* and NNLO* fits without setting M=1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 NNPDF2.1 LO* NNPDF2.1 NLO* NNPDF2.1 NNLO* 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 Momentum Integral Experimental data beautifully confirms the pqcd expectation Extremely non trivial test of the global analysis framework and the factorization hypotheses Good convergence of the QCD perturbative expansion 21 NNPDF Collaboration, arxiv:1107.2652

PDF fitting methodology

Neural Network PDFs A recent development in PDF fits is the use of robust unbiased interpolants to parametrize the nonperturbative QCD dynamics encoded in the PDFs from experimental data One example is the use of artificial neural networks to parametrize PDFs, promoted by the NNPDF Collaboartion NNPDF approach Perceptrons: Multi-layer Feed Forward Neural networks Traditional approach 24

Artificial Neural Networks Example 1: Pattern recognition. During the Yugoslavian wars, the NATO used ANNs to recognize hidden military vehicles A military aircraft is identified, despite being hidden below a commercial plane. Example 2: Marketing. A bank wants to offer a new credit card to their clients. Two possible strategies: 25 Contact all customers: slow and costly Contact 5% of the customers, train a ANN with their input (sex, income, loans) and their ourput (yes/no) and use the information to contact only clients likely to accept the offer Cost-effective method to improve marketing performance % of positive answers ANN based client selection Random client selection % of customers contacted

(Artificial) Neural Network PDFs Inspired by biological brain models, Artificial Neural Networks (ANNs) are mathematical algorithms widely used in a wide range of applications, from high energy physics to targeted marketing and finance forecasting. ANNs excel in same domains as their biological counterparts: pattern recognition, forecasting, classification,... from biology...... to high energy physics NNPDF approach to Parton Distributions ANNs provide universal unbiased interpolants to parametrize non-perturbative PDF dynamics Learn the underlying physical laws from experimental data using Genetic Algorithms No theory bias introduced in the PDF determination by the choice of ad-hoc functional forms NNPDF approach: one Artificial Neural Network per PDF, O(500) parameters total NB: ANNs redudant, PDFs identical if O(1000) parameters used Traditional approach: one simple polynomial per PDF, O(10-25) parameters total 26

Flexibility matters HERA-LHC benchmark studies Standard Approach NNPDF Approach PDF error PDF error Few Data Few Data Faithful extrapolation: PDF uncertainties blow up in regions with scarce data Crucial methodological ingredient for LHC searches at high masses 27

PDF Replica Neural Network Learning Each green curve corresponds to a gluon PDF Monte Carlo replica x g(x, Q 2 = 2 GeV 2 ) 28 x

Experimental constraints on PDFs

Experimental constraints on PDFs In global PDF analysis, a wide variety of experimental data needs to be used in order to constrain all relevant PDF flavor combinations in the widest possible range of Bjorken-x Typical dataset in a pre-lhc global PDF analysis Fixed-target Deep-inelastic scattering Neutral current and charged current Collider Deep-inelastic scattering Jet and Drell-Yan data from hadron colliders With LHC data, a much richer variety of processes has become available for PDF studies 30

Experimental constraints Traditional processes for PDF fits at hadron colliders are jet/dijet, Drell Yan and inclusive W,Z production The LHC is providing an impressive wealth of data here, already included in various PDF fits Inclusive jet production (ATLAS, CMS) W lepton asymmetry (ATLAS, CMS, LHCb) High mass Drell-Yan (ATLAS, CMS) Low mass Drell-Yan (ATLAS, CMS, LHCb): 31 DY rapidity Invariant mass Juan Rojo DIS2013, Marseille, 22/04/2013

Experimental constraints On top of traditional processes, like jets and W, Z production, a wide range of new processes that provide PDF information is now available at the LHC Top quarks: constrain large-x gluon W+charm: sensitivity to strangeness }Global arxiv:1303.7215 HERA+LHC CMS-SMP-12-002 high p T W and Z: gluon and on d/u ratio Isolated photons: complementary probe of the gluon, same x-range as for gg Higgs production d#(w + )/dp T / d#(w - )/dp T arxiv:1304.6754 arxiv:1202.1762 32 Juan Rojo DIS2013, Marseille, 22/04/2013

The strangeness conundrum In pre-lhc PDF fits, strangeness s(x,q) mostly constrained from DIS neutrino data W production in association with charm quarks provide a clean probe of the strange PDF at the LHC Measured by ATLAS and CMS with somewhat opposite (?) conclusions 33 But: different analysis techniques, kinematical cuts, selections, theory predictions used... Full differential distributions with covariance matrix Only meaningful comparison the results is provided by including both datasets in a global PDF analysis and determine the value of strange PDF which maximizes agreement with the two datasets All technical tools to carry this exercise available, see later in the talk

The strangeness conundrum In pre-lhc PDF fits, strangeness s(x,q) mostly constrained from DIS neutrino data W production in association with charm quarks provide a clean probe of the strange PDF at the LHC Measured by ATLAS and CMS with somewhat opposite (?) conclusions CMS: strange suppression in agreement with DIS data ATLAS: light quark sea symmetric preferred ( suppressed s(x) ) ( part. suppressed s(x) ) ( suppressed s(x) ) ( symmetric s(x) ) 34 But: different analysis techniques, kinematical cuts, selections, theory predictions used... Full differential distributions with covariance matrix Only meaningful comparison the results is provided by including both datasets in a global PDF analysis and determine the value of strange PDF which maximizes agreement with the two datasets All technical tools to carry this exercise available, see later in the talk

Top quarks as gluon luminometers The recent NNLO top quark cross section make top data the only LHC observable that is both directly sensitive to the gluon PDF and can be included consistently in a NNLO global analysis The precise 7 and 8 TeV LHC data can be used to discriminate between PDF sets and to reduce the PDF uncertainties on the poorly known large-x gluon The improved large-x gluon leads to more accurate theory predictions for BSM searches High mass Graviton Tail of the invariant tt mass distribution 36

Cross section ratios between LHC beam energies The staged increase of the LHC beam energy provides a new class of interesting precision observables: cross section ratios for different beam energies Can be computed with high precision due to correlation of theoretical errors at different energies Experimentally these ratios can also be measured accurately since many systematics, like luminosity or jet energy scale, cancel partially in the ratios These ratios allow stringent precision tests of the SM, in particular PDF discrimination Cross section ratios should thus be pursued as a novel approach to constrain PDF First implementation: measurement of jet cross section ratios by ATLAS between 7 and 2.76 TeV Reduced experimental and theory (scale) uncertainties, potentially can improve the sensitivity to PDFs of 7 TeV ATLAS jet data alone 37

Theoretical Developements in PDFs

QED corrections Photon-initiated diagrams are required for consistent electroweak calculations The DGLAP QCD equations can be modified with QED corrections, introducing a photon PDF NNPDF2.3 QED set is the only available QCD+QED PDF set with an independent determination of the photon PDF from DIS and LHC data Important for electroweak LHC phenomenology: W, Z searches, M W fits, WW production,... New public QCD+QED PDF evolution code available: APFEL pp -> l+l- NNPDF2.3QED pp -> W+W- 39

Electroweak corrections At present level of precision in QCD calculations, electroweak corrections become comparable if not larger Electroweak Sudakov logarithms grow with energy, more important at LHC 13 TeV Electroweak corrections affect the TeV scale phenomenology, both for New Physics searches in the highmass tails, Higgs characterization and precision SM measurements, such as PDF fits Electroweak corrections to high-pt jets @ LHC8 pp -> jet jet Therefore, including high-et data into global PDF fits requires inclusion of electroweak corrections More importantly, for consistency this requires also PDFs with electroweak corrections in the DGLAP evolution, that is, complement QCD and QED splitting functions with pure weak splittings and the W and Z PDFs in the proton Non trivial task: structure of EWK evolution equations very different from the QCD/QED ones 41

Going Beyond: PDFs at a 100 TeV collider Growing consensus that the next big machine more suitable to explore the energy frontier should be a 100 TeV hadron collider, possibly with also e+e- and ep operation modes The phenomenology of PDFs at such extreme energies is very rich: top quark PDFs, electroweak effects on PDFs and W/Z boson PDFs, ultra-low-x physics, BFKL dynamics, BSM physics with polarized PDFs,... First studies being now performed in the context of the CERN FCC working group BSM physics with polarized PDFs 42

Parton Distribution Functions In this lecture we have studied in more detail the determination of the Parton Distribution Functions of the proton, an essential ingredient for LHC phenomenology We have studied the theoretical constraints that exist on PDFs We have derived approximate analytical expressions for the gluon PDF, explaining which it grows so fast at small-x We have presented the global PDF analysis framework for PDF determination We have studied the methodology for PDF fitting, in particular the idea of using universal unbiased interpolants to parametrize the PDFs We have explored the experimental data, including LHC, that provides PDF constraints