The Strong Interaction and LHC phenomenology · The Strong Interaction and LHC phenomenology Juan...

The Strong Interactionand LHC phenomenology

Juan RojoSTFC Rutherford Fellow

University of Oxford

Theoretical Physics Graduate School course

Juan Rojo University of Oxford, 25/05/2014

Lecture 9:Parton Distributions

and LHC phenomenology


QCD partons in the initial stateAs 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

Partonic cross-section

Parton Distribution Function (PDF):Probability of finding quark q in the proton with momentum fraction y

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 kinematicsThe kinematics of deep-inelastic scattering were defined as

In the parton model, xBj 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


P a r t o n D i s t r i b u t i o n Function

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 Nflav +1 coupled integro-differential equations

A variety of numerical and semi-analytical methods exist for efficient solutions of the DGLAP equations

6

Parton Distribution Functions (PDFs)• Event rates at the LHC depend on the Parton Distribution Functions (PDFs) of the proton

• 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

Determine PDFs via global QCD analysis of hard-scattering data

PDFs Predictions @ LHCQCD Theory

Experimental data

Parton Distributions: fundamental limit to theory predictions at LHC

x-510 -410 -310 -210 -110 1

-2-1

01234

567

= 0.119s_) - 2 = 2.0 GeV2xg(x, Q

NNPDF2.3 NNLOCT10 NNLOMSTW2008 NNLO



= 0.119s_) - 2 = 2.0 GeV2xg(x, Q

Statistical Methodology

PDFs: Theory+Data+Methodology

{ {

Matrix elements: Theory

LHC Master Formula


Parton Distribution Functions

Higgs Boson

Gluon PDF, g(x1,Q)

Gluon PDF, g(x2,Q)

Proton

Proton

Proton

Parton Distributions determine the fraction of the proton energy that is carried by each of its constituents, quarks and gluons

(Bjorken) x1 = Egluon / Eproton

1 / Q = resolution scaleHigher energies ( large Q) -> Protons probed at small distances


Parton Distribution Functions Proton


Therefore, the main task here is to determine the Bjorken-x dependence of the PDFs from data

Evolution in scale fixed by perturbative QCD dynamicsDependence on the momentum fraction is non-perturbative: needs to be extracted from experimental data

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 phenomenology2) Very large PDF uncertainties (>100%) for new heavy particle production

Supersymmetric QCD1) PDFs fundamental limit for Higgs boson characterization in terms of couplings

3) PDFs dominant systematic for precision measurements, like W boson mass, that test internal consistency of the Standard Model

W mass

Top Quark mass


PDFs and LHC phenomenology1) PDFs fundamental limit for Higgs boson characterization in terms of couplings


Theory systematics (hatched areas) limiting factor for Higgs coupling extractions at the LHC

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PDF determinationPDF 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

QCD Theory

Methodology

Parton Distributions

Lepton-proton structure functions, heavy quark production, jet production, Drell-Yan pair production, electroweak bosons, isolated photons, ...

NNLO DGLAP evolution, NLO and NNLO hard scattering cross sections, heavy quark treatment, strong coupling, electroweak effects, hadronic production, PDF flavor separation,...

Ansatze for the x-dependence of PDFs q(x,Q0), propagation of experimental errors from data to PDFs, model uncertainties, minimization, definition of the figure of merit !2 , parallel computing, ....

LHC phenomenology, predictions for LHC processes, PDF and Higgs production, .....


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PDF determination


Our goal is to determine all the Parton Distributions (PDFs) of the proton from hard-scattering data, at some fixed input scale Q0 , 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

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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

fq (x) are smooth functions, should be flexible enough to parametrize any general underlying law

PDF parametrization in the MSTW08 set

14

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

15

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

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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

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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

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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

19

PDF evolution: heavy quark PDFs


The intrinsic component of heavy quark PDF is suppressed by terms O("qcd2 / mh2 ). 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

b(x)g(x)

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

20

PDF evolution: heavy quark PDFs


The intrinsic component of heavy quark PDF is suppressed by terms O("qcd2 / mh2 ). 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

21

Precision tests of the Factorization Theorem

Momentum Integral0.9 0.95 1 1.05 1.1 1.15 1.2 1.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8NNPDF2.1 LO*

NNPDF2.1 NLO*

NNPDF2.1 NNLO*

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

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

NNPDF Collaboration, arxiv:1107.2652


PDF fitting methodology


23

(Artificial) Neural Network PDFsInspired 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 approachJuan Rojo University of Oxford, 25/05/2014

24

Neural Network PDFsA recent development in PDF fits is the use of robust unbiased interpolants to parametrize the non-perturbative QCD dynamics encoded in the PDFs from experimental dataOne example is the use of artificial neural networks to parametrize PDFs, promoted by the NNPDF Collaboartion

NNPDF approachJuan Rojo University of Oxford, 25/05/2014

Perceptrons: Multi-layer Feed Forward Neural networks

NNPDF approach

Traditional approach

25

Artificial Neural Networks

% of customers contacted

% o

f pos

itive

ans

wer

s

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:

Contact all customers: slow and costlyContact 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

Random client selection

ANN based client se

lection


26

(Artificial) Neural Network PDFsInspired 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

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 totalNB: ANNs redudant, PDFs identical if O(1000) parameters used

NNPDF approach

Traditional approach: one simple polynomial per PDF, O(10-25) parameters total

NNPDF approach to Parton Distributions


27

Flexibility matters

Standard Approach NNPDF Approach

PDF errorPDF error

Few DataFew Data

HERA-LHC benchmark studies

Faithful extrapolation: PDF uncertainties blow up in regions with scarce data Crucial methodological ingredient for LHC searches at high masses


28

x g

(x,

Q2

= 2

Ge

V2)

x

PDF Replica Neural Network LearningEach green curve corresponds to a gluon PDF Monte Carlo replica


Experimental constraints on PDFs


30

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 currentand charged current

ColliderDeep-inelastic scattering

Jet and Drell-Yandata from hadroncolliders

With LHC data, a much richer variety of processes has become available for PDF studies

31

Experimental constraintsTraditional 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

Juan Rojo DIS2013, Marseille, 22/04/2013

Inclusive jet production (ATLAS, CMS) W lepton asymmetry (ATLAS, CMS, LHCb)

High mass Drell-Yan (ATLAS, CMS) Low mass Drell-Yan (ATLAS, CMS, LHCb):

arxiv:1202.1762

Invariant massDY rapidity

32

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

Juan Rojo DIS2013, Marseille, 22/04/2013

Top quarks: constrain large-x gluon W+charm: sensitivity to strangeness

high pT W and Z: gluon and on d/u ratio Isolated photons: complementary probe of the gluon, same x-range as for gg Higgs production

}Global

HERA+LHC

CMS-SMP-12-002

d#(W+)/dpT / d#(W-)/dpT

arxiv:1303.7215

arxiv:1202.1762

arxiv:1202.1762

arxiv:1304.6754

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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

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

( symmetric s(x) )

34

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

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

CMS: strange suppression in agreement with DIS data ATLAS: light quark sea symmetric preferred

( symmetric s(x) )

( symmetric s(x) )

( suppressed s(x) )

( suppressed s(x) )

( part. suppressed s(x) )

35

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

arxiv:1202.1762Juan Rojo University of Oxford, 25/05/2014

At the LHC, gluon-gluon contribution is almost 90% of the total cross-section

36

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

arxiv:1202.1762

The improved large-x gluon leads to more accurate theory predictions for BSM searches

High mass Graviton Tail of the invariant tt mass distribution

37

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


Theoretical Developements in PDFs


39

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, MW fits, WW production, .... New public QCD+QED PDF evolution code available: APFEL

pp -> l+l-

NNPDF2.3QED

pp -> W+W-


40

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, MW fits, WW production, .... New public QCD+QED PDF evolution code available: APFEL


41

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

S. Dittmaier

Electroweak corrections affect the TeV scale phenomenology, both for New Physics searches in the high-mass tails, Higgs characterization and precision SM measurements, such as PDF fits

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

Electroweak corrections to high-pT jets @ LHC8

pp -> jet jet

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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 withpolarized PDFs

Parton Distribution Functions

P a r t o n D i s t r i b u t i o n Function

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


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