deep inelastic scattering and global fits of parton distributions – … · 2011-06-03 · global...

Deep Inelastic Scattering and Global Fits of Parton Distributions

– lecture 4 –

Alberto AccardiHampton U. and Jefferson Lab

HUGS 2011

HUGS 2011 – Lecture [email protected] 2

Plan of the lectures Lecture 1 – Motivation

– Quarks, gluons, hadrons– Deep Inelastic Scattering (DIS)

Lecture 2 – Parton model– DIS revisited– Collinear factorization and Parton Distribution Functions (PDFs)– Limitations

Lecture 3 – The QCD factorization theorem– QCD factorization, universality of PDFs– DIS, Drell-Yan (DY) lepton pairs, W and Z production, hadronic jets

Lecture 4 – Global PDF fits– How to make a fit, and use its results– Fits as community service (e.g., measure PDFs, apply to LHC)– Fits as a tool to study hadron and nuclear structure


Lecture 4 – Global PDF fits

Global PDF fits– the basics– a selection of fine details– PDF uncertainties: experimental, theoretical

Examples and applications– Standard Model and beyond at the LHC– Impact of a proposed new accelerator: the EIC

– Studying hadron structure: the d/u ratio at x → 1


Global PDF fits


Global PDF fits

Problem:– we need a set of PDFs in order to calculate a particular

hard-scattering process (say, at LHC)

Solution:– Choose a data set for a set of different hard scattering processes – Generate PDFs using a parametrized functional form at initial scale

Q0; evolve them from Q0 to any Q using DGLAP evolution equations

– Use the PDF to compute the chosen hard scatterings– Repeatedly vary the parameters and evolve the PDFs again– Obtain an optimal fit to a set of data.

Modern PDF sets: CTEQ-TEA (CT10), CTEQ-JLab (CJ10), MSTW2008, NNPDF2.1, ABM11, JR, HERAPDF1.5


Global PDF fits as a tool

Test new theoretical ideas– e.g., are sea-quarks antisymmetric? Is there any “intrinsic” charm?

Phenomenology explorations– e.g., can CDF / HERA “excesses” be at all due to glue/quark

underestimate at large x?

Test / constrain models– e.g., by extrapolating d/u at x=1– Possibly, constrain nuclear corrections

Limitations– existing data– experimental errors– theoretical errors


Key points

Choice of data sets

Choice of kinematic cuts to perform calculations with confidence

Parametrized functional form for input PDFs at Q0

Definition of “optimal fit” – typically by a suitable choice of 2 function

Truncation of the perturbative series: – LO; NLO (state-of-the-art)– NNLO (fully available for DIS, DY – partially for other processes)

Treatment of errors– Experimental, statistical and systematic– Theoretical


Observables

Each observables involves a different linear combination, or product of PDFs:a diverse enough set of observables is needed for parton flavor separation– Some redundancy needed to cross-check data sets

Typical data sets used in global fits– Inclusive DIS– Vector boson production in p+p, p+D

– Hadronic jets, p+p or p+pbar: inclusive jets, +jet– neutrino DIS:

* use of nuclear targets require consideration of nuclear corrections to measure theproton / neutron PDFs; typically these induce large theoretical uncertainty, themore so for heavy nuclei. Fixed target DY is an exception: the probed x values in thenucleus are small enough to neglect corrections.

Need to establish a strategy to get to the particular PDFs one is interested in– Different groups make different choices

W§; Z0; DY lepton pairs

`§ + p; `¡ +D¤

º +A¤


Useful PDF properties - 1

The gluon dominates at low x and falls steeply as x increases

Symmetric sea quarks: anti-q and q comparable at low x (and anti-q fall off in x even faster than the gluons)

u and d dominate at large x with u > d ; at low x, u ≈ d


Useful PDF properties - 2

Gluon radiation – QCD evolution in Q2

– Gluon radiation causes parton momentum loss: • At large x, quarks and gluons shift to the left: PDFs get steeper

– Gluons create q, anti-q pairs, and g, g pairs:• At small x, quark and gluon PDFs increase, get steeper


Parametrization at Q0

In the beginning... first fits based on

Estimate by counting rules: = 2ns – 1 with ns = spectator quarks no.

– Valence quarks (qqq): ns = 2, = 3

– Gluons (qqqg): ns = 3, = 5

– Antiquarks (qqqqq): ns = 4, = 7

Estimate from Regge arguments, behavior of gluon radiation:

– Gluons and antiquarks: ≈ –1

– Valence quarks: ≈ –1/2

Overall normalization fixed by sum rules (momentum, charge conservation, ...)

fi(x) = Nix®i(1¡ x)¯i



With the large variety of precise data available today, needs more flexibility: multiply by a suitable function of x.

Examples for u, d quarks and gluons

– CTEQ6.1 – u,d,g:

– MSTW2008 – u,d:

g:

Caveats:– Choice of functional form or no. of free parameters can bias the results– Theoretical prejudices often built-in (e.g., CTEQ gluons can't go negative,

d/u ratio forced to either 0 or ∞ as x → 1 )

NNPDF: obtains “unbiased fits” by a neural-network parametrization, using a very large linear basis of functional forms

N xa1(1¡ x)a2ea3x£1 + ea4x

¤a5

N xa1(1¡ x)a2£1 + a3

px+ a4x

¤

N xa1(1¡ x)a2£1 + a3

px+ a4x

¤+N 0xb1(1¡ x)b2



Other points to keep in mind

– One should increase the number of parameters and the flexibilityof the parametrization until the data are well described

– Adding more parameters past that point simply results in ambiguities, false minima, unconstrained parameters, etc.

– May have to make some arbitrary decisions on parameter values that are not well constrained by the data

– A smaller numbers of parameters is not always better - it is the description of the data that counts.


“Optimal” fit

Needs a numerical measure of how good a fit is– choose a suitable 2 function

– vary parameters iteratively until 2 minimized

Simplest choice

– OK for 1 data set– And if data is statistically limited (errors not “too small”)

But nowadays we have– Several data sets for many observables– Correlated and uncorrelated errors– Overall normalization errors (due to, say, luminosity uncertainties)

D = exp.data = uncorrelated exp. errors T = calculation

Â2 =X

i

(Di ¡ Ti)2¾2i


“Optimal” fit

Normalization errors– assign a 2 penalty for normalization errors (different choices possible)

– Fit optimal normalization fN, compare to quoted one

Point-to-point systematic errors

– The data points Di are shifted by an amount reflecting the systematic

errors β with the shifts given the the sj parameters

– There is a quadratic penalty term for non-zero values of the shifts s

MSTW use a power 4


“Optimal” fit

Minimization of biases in treatment of normalizations– treat all errors on the same footing

Want to emphasize a given data set? use

– the weights wk and wN,k can be chosen to emphasize the contribution

of a given experiment or normalization to the total 2

[Ball et al., Nucl.Phys.B838:136,2010]


Order of perturbation theory

Lowest order in αs (LO) - easy to do, but– Hard scattering subprocesses do not depend on the factorization scale– May be missing large higher order corrections

Next-to-leading-order (NLO) - more complicated, but– Less dependent on scale choices since the PDFs and hard scattering

subprocesses both contain scale dependences which (partially) cancel– Some higher order corrections are now included

Next-to-next-to-leading-order (NNLO) - better, but– Splitting functions are known so NNLO evolution can be done– Some hard scattering subprocesses are known to NNLO (DIS, DY)

but not high-ET jets (yet)

NLO remains the state-of-the-art, but full NNLO analyses are coming


Order of perturbation theory

LO PDFs can be interpreted as probability distribution in x

At NLO, PDFs are defined to absorb IR and collinear divergences in the hard-scattering diagrams: they are no longer probabilities.

– Then, get rid of infinities by extracting PDFs from data (in this sense it is analogous to UV renormalization)

– Note: a “divergence” is defined differently in different subtration schemes (most commonly “modified minimal subtraction” MS, or DIS)

NLO PDFs are not “better” than LO PDFs – they are different objects:– you should use LO PDFs in LO calculations, NLO PDFs in NLO calcul'ns– ...and the same subtraction scheme, choice of scale

qLO(x) =

Zdz¡

2¼eixp

+z¡hpjÃ(z¡n)°+

2Ã(0)jpi

qNLO = qLO + fdivergencesg


PDF uncertainties

Experimental:

– uncertainties in measured data propagate into the fitted PDFs– can be quantified adapting statistical methods: “PDF error bands”– These PDF errors need to be interpreted with care

Theoretical:

– Several sources, cannot be quantified easily• Choice of data sets, kinematic cuts• Parametrization bias• Choice of 2 function• Truncation of pQCD series, heavy-quark scheme, scale choice• Higher-twist, target mass effects• Nuclear corrections• ...


PDF errors

Hessian method

T

“tolerance”


PDF errors

Tolerance T =

– Open a textbook, T=1 means 67% confidence level– But Hessian method works only if

• all data sets are statistically compatible• Exp. errors are Gaussian...• ...and have not been underestimated

(e.g., by neglect of a source of systematics)

– Correct this by a larger tolerance factor so that most data (90%, 67% of them) fall inside the PDF error band

• CTEQ6.1 used T=10, MRST used T=5• Nowadays a bit more refined procedure are adopted


PDF errors

Lagrange multipliers method

– Given an observable X, minimize a new function for fixed values of Lagrange multiplier

– Obtain a new set of parameters, Amin, and the pair

– Repeating for many variables, one obtains

– Chose a tolerance, read off the PDF error X


PDF errors

Monte-Carlo method

– Generate many replicas of the chosen data set– In each replica, randomize central data point within quoted errors– Make a fit for each replica– Obtain PDF errors from statistical analysis of all fit results

– This is adopted by the NNPDF collaboration, but is not limited to neural network based fits


Examples


Some recent PDFs


Some recent PDFs

HERAPDF1.0


Some recent PDFs


W and Z cross sections – a standard candle at the LHC

Partonic cross section for inclusive W,Z production well-known– These processes can be used to monitor the parton luminosity L (x)

– and reduce systematic uncertainty in other observables


Extra gauge boson W' at the LHC

DIS2011


Impact of a new accelerator – the EIC

Questions– What are the requirements in terms of energy, luminosity?– What physics do we expect to learn?– “Is it worthwhile building that accelerator?”

For example:– Is a DIS cross section measurement at the EIC going to improve the

PDF measurements?

This we can anwer with a global fit:– Generate pseudo-data – Include them in a global fit– Compare with old result



e+p collisions – using CTEQ-JLab fits [Accardi, Ent, Keppel]

– Pseudo data:



e+p collisions – using CTEQ-JLab fits [Accardi, Ent, Keppel]

– Sensible reduction in PDF error



e+A collisions – using NNPDF2.0 fits [Accardi, Guzey, Rojo]

– QCD fit to EIC pseudo-data for Pb only

– Assume energy scanL=4 fb-1 per energy setting0.04 < y < 0.8

– √s = 12, 17, 24, 32, 44 GeV(medium energy EIC – stage I)

– √s = 63, 88, 124 GeV(full energy EIC – stage II)



quarks gluons

e+A collisions – using NNPDF2.0 fits [Accardi, Guzey, Rojo]

– With only 1 nucleus target, impact comparable to present day world data:


Non perturbative structure of the proton – d/u ratio

Non-perturbative structure of the proton determines d/u as x → 1– 1 gluon exchange model: d/u → 0– Helicity conservation: d/u → 0.2– SU(6) quark model: d/u → 0.5

Standard parametrization forces d/u → 0 or ∞ – Needs a more flexible parametrization for the d-quark

so that d/u → c , fitted to data

Explore a range of nuclear correction for deuterium targets– Different Deuteron wave functions– A range of “off-shell” corrections

d0(x) = d(x) + cx®u(x)


Non perturbative structure of the proton – d/u ratio

PDF uncertainty band

– uses =1– With tolerance factor T=7,

comparable to nuclear uncertainty– Can be improved by new data

Nuclear uncertainty band– Using smallest and largest nuclear

corrections– Currently too large to pin down

d/u at x=1

– Needs theoretical improvement [Accardi et al., arXiv:1102.3686]

CTEQ-JLab fits


Lecture 4 - recap Global PDF fits

– the basics– a selection of fine details– PDF uncertainties: experimental, theoretical

– for more in-depth treatment, see [Owens] [Forte] [Tung] [Devenish]

Examples and applications– Standard Model and beyond at the LHC– Impact of a proposed new accelerator: the EIC

– Studying hadron structure: the d/u ratio at x → 1

deep inelastic scattering and global fits of parton distributions – … · 2011-06-03 · global...

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