UPDATE FROM NNPDF

Nathan Hartland Nikhef / VU Amsterdam

DIS 2018 Kobe, 17/04/18

NNPDF

CURRENT GLOBAL FIT STATUS

Czakon, Heymes, Mitov [1511.00549], [1606.03350]

Boughezal et al, Gehrmann et al[1504.02131], [1507.02850]

Currie et al [1310.3993] [1611.01460]

Theoretical developments for NNPDF3.1➤ NNLO Results

tt

Inc. Jets➤ Fitted/intrinsic charm Ball et al [1510.02491], [1605.06515]

W/Z pT

Last release: NNPDF3.1[1706.00428]

➤ Broad dataset inc. LHC measurements

➤ Statistically validated methodology

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Figure 3.1: The NNPDF3.1 NNLO PDFs, evaluated at µ2 = 10 GeV2 (left) and µ2 = 104 GeV2 (right).

3.3 Parton distributions

We now inspect the baseline NNPDF3.1 parton distributions, and compare them to NNPDF3.0and to MMHT14 [7], CT14 [6] and ABMP16 [8]. The NNLO NNPDF3.1 PDFs are displayedin Fig. 3.1. It can be seen that although charm is now independently parametrized, it is stillknown more precisely than the strange PDF. The most precisely determined PDF over most ofthe experimentally accessible range of x is now the gluon, as will be discussed in more detailbelow.

In Fig. 3.2 we show the distance between the NNPDF3.1 and NNPDF3.0 PDFs. Accordingto the definition of the distance given in Ref. [98], d ' 1 corresponds to statistically equivalentsets. Comparing two sets with Nrep = 100 replicas, a distance of d ' 10 corresponds to adi↵erence of one-sigma in units of the corresponding variance, both for central values and forPDF uncertainties. For clarity only the distance between the total strangeness distributionss+ = s + s is shown, rather than the strange and antistrange separately. We find importantdi↵erences both at the level of central values and of PDF errors for all flavors and in the entirerange of x. The largest distance is found for charm, which is independently parametrized inNNPDF3.1, while it was not in NNPDF3.0. Aside from this, the most significant distances areseen in light quark distributions at large x and strangeness at medium x.

In Fig. 3.3 we compare the full set of NNPDF3.1 NNLO PDFs with NNPDF3.0. TheNNPDF3.1 gluon is slightly larger than its NNPDF3.0 counterpart in the x

⇠< 0.03 region, while

it becomes smaller at larger x, with significantly reduced PDF errors. The NNPDF3.1 lightquarks and strangeness are larger than 3.0 at intermediate x, with the largest deviation seenfor the strange and antidown PDFs, while at both small and large x there is good agreementbetween the two PDF determinations. The best-fit charm PDF of NNPDF3.1 is significantly

23

NEW DATA IN NNPDF3.1

4

New datasets in NNPDF3.1

Combined HERA inclusive data Run I+II quark singlet and gluon

D0 legacy W asymmetries Run II quark flavor separation

ATLAS inclusive W, Z rap 7 TeV 2011 strangeness

ATLAS inclusive jets 7 TeV 2011 large-x gluon

ATLAS low-mass Drell-Yan 7 TeV 2010+2011 small-x quarks

ATLAS Z pT 7,8 TeV 2011+2012 medium-x gluon and quarks

ATLAS and CMS tt differential 8 TeV 2012 large-x gluon

CMS Z (pT,y) 2D xsecs 8 TeV 2012 medium-x gluon and quarks

CMS Drell-Yan low+high mass 8 TeV 2012 small-x and large-x quarks

CMS W asymmetry 8 TeV 2012 quark flavor separation

CMS 2.76 TeV jets 2012 medium and large-x gluon

LHCb W,Z rapidity dists 7 TeV 2011 large-x quarks

LHCb W,Z rapidity dists 8 TeV 2012 large-x quarks

Measurement Data taking Motivation

Juan Rojo DIS2017, Birmingham, 04/04/2017(Table thanks to J. Rojo)

Measurement Data Taking Target PDF

NNPDF3.1 GLOBAL - PHENOMENOLOGY (GG)Significant reduction in uncertainties across the kinematic range

NNPDF3.1 GLOBAL - PHENOMENOLOGY (QQ)QQ Uncertainties decrease despite greater parametrisation freedom

Uncertainties often towards the percent levelThere are other sources of uncertainty to worry about

Neglects correlations with PDF fit parameters ( ) - important when experimental uncertainties are small

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DETERMINING ALPHAS FROM A GLOBAL FIT

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χ2

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NNPDF2.1 NNLO Global

Parabolic FitData

Figure 1: The χ2 as a function of αs using NNLO NNPDF2.1 parton distributions. The uncer-tainty shown is due to the finite size of the PDF replica sample used to compute the χ2 for eachvalue of αs. The curve is the result of a parabolic fit.

of data points is Ndat = 3357 [2], and at the minimum χ2/Ndat = 1.16. The error barshown on each point is the standard deviation determined using the bootstrap method asdescribed in Ref. [1]. A parabolic fit leads to

αNNLOs (MZ) = 0.1173 ± 0.0007stat ± 0.0001proc, (1)

where the statistical uncertainty corresponds to a shift by one from the minimum of theparabola, while the procedural uncertainty is propagated from the finite-size uncertaintyon the χ2 values, and it is clearly completely negligible. The χ2 per degree of freedom ofthe parabolic fit shown in Fig. 1 is χ2

par/Ndof = 1.1. This value remains unchanged if theouter two or four points are removed from the parabolic fit: this means that the behaviourof the curve shown is parabolic to within the accuracy of the individual points.

Equation (1) is the main result of this paper. It is interesting to compare this to theNLO result of Ref. [1], namely

αNLOs (MZ) = 0.1191 ± 0.0006stat ± 0.0001proc, (2)

which was obtained using exactly the same methodology, from PDFs in turn determinedusing the same methodology and the same data 1; the value of the χ2 per data point atthe minimum at NLO is also the same, namely χ2/Ndat = 1.16. The result appears tobe perturbatively quite stable. The small difference in statistical error between the NLO

1The number of datapoints for the NLO fit of Ref. [1] is Ndat = 3338 [13]: the slightly lower number ofdatapoints in the NLO case is due to the fact that the kinematic cuts on charm structure function datamust be more stringent at NLO in order to exclude data for which NNLO corrections are very large, asdiscussed in more detail in Ref. [2].

3

Figure 1.1. Comparison between the standard deviation of a pair of correlated variables (↵s, ✓) and theone-sigma range for the variable ↵s along the best-fit line of ✓. The best fit is denoted as (↵s, ✓) and theellipse is the one-sigma contour about it. The standard deviations on (↵s, ✓) are (�↵,�✓), while �old isthe one-sigma interval for ↵s with fixed ✓ = ✓.

correlated, especially if the semi-axes of the ellipse are of very di↵erent length, the procedureused in our previous work might lead to an underestimate of the uncertainty in ↵s. Hence thenew procedure becomes very relevant, now that some PDF uncertainties are rather small.

It turns out that the implementation of this simultaneous minimization within the NNPDFmethodology is nontrivial: it requires the development of a suitable generalization of the stan-dard NNPDF approach, which we call the correlated replica method. Using this strategy, ↵s

can be treated like any other quantity that depends on the PDFs. In particular, its central valueand uncertainty can be determined by performing statistics over a replica sample. This meansthat, for example, the uncertainty on ↵s is the standard deviation of an ensemble of ↵s values.As we shall see, this allows for a determination of ↵s with small experimental uncertainties, andnegligible methodological uncertainties. Having reduced very much the size of all other uncer-tainties, the problem of accurately estimating theoretical uncertainties becomes quite serious.This is specifically problematic in the case of missing higher-order uncertainties (MHOUs), forwhich no fully satisfactory method has been developed. Here we will conservatively estimate thetheoretical uncertainty due to missing higher order QCD corrections (N3LO and beyond) fromhalf the shift between the NLO and NNLO ↵s values.

This paper consists of two main parts. First, in Sect. 2 we present the correlated replicamethod used for the determination of ↵s, explain how it is used to estimate the associatedPDF uncertainties, and compare it with the method used in previous NNPDF determinations.Then, in Sect. 3 we present our determination of ↵s at NLO and NNLO together with a carefulassessment of all sources of uncertainty. Possible future developments are briefly outlined inSect. 4.

2 The correlated Monte Carlo replica method

As discussed in the introduction, the ↵s determination presented here di↵ers from our previousone [17,18] because now the value of ↵s and its uncertainty are determined from a correlated fittogether with the PDFs. After briefly summarizing the main aspects of the NNPDF methodologyand the way it was used to determine ↵s in Ref. [17, 18], we describe the main idea of the newmethod, and then discuss the details of its implementation. Only the salient aspects of theNNPDF methodology will be recalled here; the reader is referred to the original literature (seeRef. [16], of which we follow the notation, and references therein) and recent reviews [2, 21, 22]for a more detailed discussion.

3

Previous NNPDF determination Based on a scan of NNPDF2.1

Ideally one should minimise PDFs and simultaneously↵S

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Measure the of best fit PDF parameters as a function of ↵S

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[1802.03398]

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NNPDF FITS ARE EXPENSIVE

x5−10 4−10 3−10 2−10 1−10

2−1−

01

2

34

5

67

)2xg(x,QCurrent fit replicasCurrent fit mean value

error bandσCurrent fit 1Current fit 68% CL band

)2xg(x,Q

Monte Carlo uncertainties➤ PDFs are formed by ensembles:

Fits to independent, equallylikely samples (replicas) of the input dataset

➤ Each result requires 100/1000statistically independent analysis runs

�pp!X =X

i,j

Z 1

0dx1dx2 fi(x1, Q

2) fj(x2, Q2) �ij!X

�x1, x2, Q

2�

� =

nfX

i,j

nxX

↵,�

Wij↵� fi(x↵, Q20)fj(x� , Q

20)

Fitting a large dataset only possible making use of pre-computed tables

THE CORRELATED REPLICA METHODHow can we take into account PDF/ correlations in a ‘MC’ way?↵S

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Figure 3.1. The �2 profiles for each of the 379 c-replicas used for the NNLO determination of ↵s(mZ),Eq. (3.1). Each curve corresponds to an individual c-replica, and the color scale indicates the best-fit ↵s

value determined from the parabolic fit to that curve.

Figure 3.2. The probability distributions for the best-fit ↵(k)s values Eq. (2.6) at NNLO (left) and at

NLO (right). Each marker indicates the ↵(k)s value corresponding to each individual c-replica.

be supplemented by methodological and theoretical uncertainties, to be discussed in Sects. 3.2and 3.3 below.

The 379 c-replicas selected for the NNLO determination are shown in Fig 3.1. The colorscale of each curve indicates the best-fit ↵s value. It is apparent that the vast majority ofthe curves exhibit an approximately parabolic behaviour. The probability distributions of the

best-fit values ↵(k)s Eq. (2.6) which correspond to each c-replica, both at NLO and at NNLO,

are shown in Fig. 3.2, where the markers indicate the value of ↵(k)s for each specific c-replica.

These probability densities have been determined using the Kernel Density Estimate method,see [35]. We find that the probability distribution for ↵s (mZ) is both shifted to higher valuesand broadened when going from NNLO to NLO. The decrease of the best-fit value of ↵s (mZ)when going from NLO to NNLO has been repeatedly observed before, also in our previousdetermination [17,18], while the broadening is due to the poorer quality of the NLO fit.

The impact on the ↵s determination of any subset of the input data can be roughly assessedby studying its contribution to the total figure of merit. We have done this by determiningreplica by replica the corresponding partial �2

p for a process (or group of processes) p, definedas the figure of merit Eq. (2.3) with the summation over i, j now restricted to data whichbelong to the specific subset p. The ↵s fit procedure through the correlated replica method

9

Figure 3.1. The �2 profiles for each of the 379 c-replicas used for the NNLO determination of ↵s(mZ),Eq. (3.1). Each curve corresponds to an individual c-replica, and the color scale indicates the best-fit ↵s

value determined from the parabolic fit to that curve.

Figure 3.2. The probability distributions for the best-fit ↵(k)s values Eq. (2.6) at NNLO (left) and at

NLO (right). Each marker indicates the ↵(k)s value corresponding to each individual c-replica.

be supplemented by methodological and theoretical uncertainties, to be discussed in Sects. 3.2and 3.3 below.

The 379 c-replicas selected for the NNLO determination are shown in Fig 3.1. The colorscale of each curve indicates the best-fit ↵s value. It is apparent that the vast majority ofthe curves exhibit an approximately parabolic behaviour. The probability distributions of the

best-fit values ↵(k)s Eq. (2.6) which correspond to each c-replica, both at NLO and at NNLO,

are shown in Fig. 3.2, where the markers indicate the value of ↵(k)s for each specific c-replica.

These probability densities have been determined using the Kernel Density Estimate method,see [35]. We find that the probability distribution for ↵s (mZ) is both shifted to higher valuesand broadened when going from NNLO to NLO. The decrease of the best-fit value of ↵s (mZ)when going from NLO to NNLO has been repeatedly observed before, also in our previousdetermination [17,18], while the broadening is due to the poorer quality of the NLO fit.

The impact on the ↵s determination of any subset of the input data can be roughly assessedby studying its contribution to the total figure of merit. We have done this by determiningreplica by replica the corresponding partial �2

p for a process (or group of processes) p, definedas the figure of merit Eq. (2.3) with the summation over i, j now restricted to data whichbelong to the specific subset p. The ↵s fit procedure through the correlated replica method

9

•For each data sample (replica) perform a scan in

•Each replica has a preferred value of (the minimum of each parabola)

•These preferred values form a MC distribution

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RESULTS

0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.120�S(MZ)

Heavy: 68% CLLight: 95% CL

NNPDF3.1

NNPDF2.1

MMHT14

ABMP16

PDG2017

Figure 3.7. Comparison of the present NNLO determination of ↵s (mZ), Eq. (3.13), with the PDGaverage and with the previous ABMP16, MMHT14, and NNPDF2.1 results. For the NNPDF values, theinner (darker) error bar correspond to experimental uncertainties, while the outer (lighter) one indicatesthe sum in quadrature of experimental and theoretical uncertainties.

between PDFs and ↵s, while the theory uncertainty should be compared to Eq. (3.11) which isalso based on the CH method. We conclude that, in comparison to Ref. [18], the current resultis more precise, though with more conservatively estimated uncertainties.

In Fig. 3.7 we compare the NNLO result of Eq. (3.13) to our previous result [18], to the currentPDG average [3], and to two recent determinations obtained from simultaneous fit of PDFs and↵s (mZ), ABMP16 [41] and MMHT2014 [40]. We find good agreement with the PDG average aswell as with the MMHT14 and NNPDF2.1 determinations. It has been suggested [50, 51] thatthe lower ABMP16 value can be partly explained by the use of a fixed-flavour number schemefor the treatment of DIS data. In fact, it is interesting to observe that the current AMBP16value is higher than previous values of ↵s (mZ) obtained by the same group [52], from which itdi↵ers because of inclusion in Ref. [41] of LHC data described in an Nf = 5 scheme.

Interestingly, the ↵s (mZ) determination from the NNPDF3.1 fit is higher than any otherrecent determination from PDF fits. Inspection of Figs. 3.3 and 3.5 strongly suggests thatthis increase is driven by the high-precision LHC data, especially for gauge boson production(including the Z pT distribution) but also for top and jet production.

4 Summary and outlook

In this work we have presented a new determination of the strong coupling constant ↵s (mZ)jointly with a global determination of PDFs which, by relying on NNPDF3.1, for the first timeincludes a large amount of LHC data using exact NNLO theory in all cases. In comparison toa previous determination based on NNPDF2.1, our results exploit the new correlated replicamethod that is equivalent to the simultaneous fit of PDFs and ↵s. This new method thus fullyaccounts for the correlations between PDFs and ↵s in the determination of the best-fit value of↵s and of the associated uncertainty.

We find that the determination of ↵s (mZ) is considerably stabilized by the use of a wide set ofdi↵erent processes and data, and we provide evidence that a global simultaneous determination of↵s (mZ) and PDFs leads to a more stable and accurate result than the one obtained from subsetsof data. We thus obtain a value of ↵s (mZ) which is likely to be more precise and more accuratethan previous results based on similar techniques. We find that the LHC data consistently leadto an increase in the central value of ↵s (mZ), and observe good overall consistency between the

19

significant impact of LHC data in the more recent determination. In addition, the shift betweenNLO and NNLO PDFs is found to be smaller in NNPDF3.1 than in previous NNPDF sets [45],presumably because MHO terms pull in di↵erent directions and thus partly cancel each other toa greater extent in a more global fit. Indeed, we find a similar increase of perturbative stabilityof PDFs and of the associated ↵s (mZ) by repeating the analysis presented here for reduceddatasets [46]. Therefore, the reduction of the MHOU by a comparable factor in Eq. (3.8) incomparison to Ref. [18] is expected.

Nevertheless, the very small value of the MHOU at NNLO, Eq. (3.11), even smaller than thealready small experimental uncertainty Eq. (3.1), may seem rather too optimistic. In fact, onemay argue that the applicability of the CH method in this case is questionable, as it relies on theassumption of the existence of a common expansion Eq. (3.9). Indeed, the present determinationof ↵s (mZ) is obtained by combining several processes, each of which has its own perturbativeexpansion, and it was noted in Ref. [44] that the rescaling factor � can vary considerably betweenprocesses.

Taking into account these considerations, we prefer to adopt a more conservative estimate.Namely, we assume that the MHOU on the NNLO result is half the di↵erence between the NLOand NNLO results Eq. (3.8):

�↵ths = 0.0011 (0.9%) , (3.12)

about twice the size of the corresponding experimental uncertainty Eq. (3.1). Whereas this issurely a very crude estimate, we do not feel that any of the available methods can lead to amore reliable conclusion.

On top of the missing higher fixed-order QCD corrections, several other aspects of the the-ory used in the simultaneous determination of ↵s (mZ) and PDFs also lead to uncertainties.These include the values of the heavy quark masses, standard model parameters (specificallyCKM matrix elements and electroweak couplings), electroweak corrections, QCD resummationcorrections [47, 48], QCD power corrections, and nuclear corrections. Many of these uncertain-ties were assessed in the NNPDF3.1 PDF determination that we are relying upon [16], andfound to be smaller than PDF uncertainties. In particular, the dependence on the charm massin previous PDF determinations is removed in NNPDF3.1 thanks to the presence of an inde-pendently parametrized charm PDF [49], and electroweak corrections are carefully kept undercontrol thanks to the choice of suitable kinematic cuts. But PDF uncertainties mix with theexperimental uncertainty on ↵s (mZ), with which they are strongly correlated, and are in factindistinguishable from it, as discussed in Sect. 2.1, so the hierarchy of uncertainties on PDFsand ↵s (mZ) is the same. We conclude that we have evidence that most of these theoreticaluncertainties are sub-dominant in comparison to the experimental uncertainty Eq. (3.1), andthus even more so in comparison to the MHOU Eq. (3.12).

3.4 Final results and comparisons

We can now collect results. Combining the NNLO value and experimental uncertainty Eq. (3.1),the methodological uncertainty Eq. (3.7) and the theoretical uncertainty Eq. (3.12) we get

↵NNLOs (mZ) = 0.1185± 0.0005exp ± 0.0001meth ± 0.0011th = 0.1185± 0.0012 (1%) , (3.13)

where in the last step we have added all uncertainties in quadrature. For a comparison toother determinations, such as the PDG average, we recommend using only the experimentaluncertainty (the methodological uncertainty being negligible), which reflects the limitations ofour result and procedure, but not the limitation due to the fact that our result is obtained atNNLO. For precision phenomenology, however, we recommend use of the total uncertainty inorder to conservatively account for the MHOU.

This result can be compared to the previous one [18] based on NNPDF2.1, ↵NNLOs (mZ) =

0.1173±0.0007exp±0.0009th. In comparison to this older result, the central value of ↵s(mZ) hasincreased by �↵s = +0.0012 . As far as uncertainties are concerned, both the theoretical andexperimental uncertainties on this previous result are larger, if one compares like with like. Theexperimental uncertainty should actually be compared to Eq. (3.3) as it was obtained with thesame method. The uncertainty is somewhat underestimated because it neglects the correlation

18

[1802.03398]

THEORETICAL UNCERTAINTIES IN PDF FITSPDF uncertainties often represent only experimental and procedural factorsParametric uncertainties due to e.g strong coupling straightforward to handle

How can we estimate uncertainties due to missing higher orders?

At NLO - measure difference between fits at different perturbative orders

A more difficult question

FITS WITH SCALE VARIED THEORY

4−10 3−10 2−10 1−10 x 0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

) [re

f] 2

) / g

( x,

Q2

g ( x

, Q

Central scales

refµ=2

Rµ=

Fµ

/2refµ=

Rµ=

Fµ

NNPDF3.1 DIS-only NLO, Q = 100 GeV

THEORETICAL UNCERTAINTIES IN PDF FITSCan we build a theoretical ‘covariance matrix’?

�+i = Oi(µR, µF )�Oi(2µR, 2µF ),

��i = Oi(µR, µF )�Oi(µR/2, µF /2)

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sha1_base64="e3MEgYazCO5d9fDWZjDqm2H5H5o=">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</latexit><latexit 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sha1_base64="e3MEgYazCO5d9fDWZjDqm2H5H5o=">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</latexit><latexit sha1_base64="e3MEgYazCO5d9fDWZjDqm2H5H5o=">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</latexit>

Consider shifts due to three-point scale variations

One can then construct a theory ‘covariance’

To be added to the experimental matrix

CovTh[Oi,Oj ] = �+i �

+j ���

i ��j

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CovTotal = CovTh[Oi,Oj ] + CovEx[Oi,Oj ]<latexit sha1_base64="TmuZqOYKzzzgCtebRGoVHIBAFxY=">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</latexit><latexit sha1_base64="TmuZqOYKzzzgCtebRGoVHIBAFxY=">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</latexit><latexit sha1_base64="TmuZqOYKzzzgCtebRGoVHIBAFxY=">AAACenichVHbSgMxEM2ut1pvVR9FCBZvKGW3Xdo+KBSK4JsVrBXaZcmmqY1mN0uSLZZlP8Jf880v8cUH0xuoWBwInDlnDjOZ8SNGpbKsd8NcWFxaXsmsZtfWNza3cts795LHApMm5oyLBx9JwmhImooqRh4iQVDgM9Lyn+sjvTUgQlIe3qlhRNwAPYa0RzFSmvJyr50Aqb4IkjofpF4yy+64QixN4SWco/fTtD1OMGLJTerR8+/ZkwvP5jivXv5xerm8VSiVKqWiDTVwnKpTmoBytQztgjWOPJhGw8u9dbocxwEJFWZIyrZtRcpNkFAUM5JmO7EkEcLP6JG0NQxRQKSbjFeXwkPNdGGPC/1CBcfsd0eCAimHga8rR1PK39qI/Etrx6pXdRMaRrEiIZ406sUMKg5Hd4BdKghWbKgBwoLqWSHuI4Gw0tfK6iXMfgrng/tiwbYK9q2Tr11M15EBe+AAnAAbVEANXIMGaAIMPox948g4Nj7NA/PUPJuUmsbUswt+hOl8AXyfxZA=</latexit><latexit sha1_base64="TmuZqOYKzzzgCtebRGoVHIBAFxY=">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</latexit>

Oi(µR, µF ), (0 < i < Ndat)<latexit sha1_base64="MGP3uRgz147yRmbY1ZA8EsTQhXk=">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</latexit><latexit sha1_base64="MGP3uRgz147yRmbY1ZA8EsTQhXk=">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</latexit><latexit sha1_base64="MGP3uRgz147yRmbY1ZA8EsTQhXk=">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</latexit><latexit sha1_base64="MGP3uRgz147yRmbY1ZA8EsTQhXk=">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</latexit>

For a set of predictions for an observable at a central scale

THEORY UNCERTAINTIES

Very preliminaryCombined covariance for DIS sets

In progress

How do these matrices influence the fit?

How do they compare to scale-varied fits?

PATH TO NNPDF 4.0 - NEW DATA

DIS HERA c. NOMAD CHORUS

JLAB S.Fs

F c2 , F

b2

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13TeV Standard CandlesInclusive jets

W/Z productionHigh-Mass DY

top-pair production

New processes2/3-jets

Prompt photonSingle topD-meson

Datasets/processes under consideration for NNPDF 4.0

NNPDF3.1 NNPDF3.1+ATLAS�

Fixed-target lepton DIS 1.207 1.203Fixed-target neutrino DIS 1.081 1.087HERA 1.166 1.169

Fixed-target Drell-Yan 1.241 1.242Collider Drell-Yan 1.356 1.346Top-quark pair production 1.065 1.049Inclusive jets 0.939 0.915Z pT 0.997 0.980

Total dataset 1.148 1.146

Table 4.5. Same as Table 4.4 now with individual experiments grouped into families of processes.

Figure 4.2. Left: comparison of the gluon PDF at Q = 100 GeV between the NNPDF3.1 andNNPDF3.1+ATLAS� fits, normalized to the central value of the former. Right: the correspondingrelative one-sigma PDF uncertainties in each case.

the kinematic coverage spanned by the ATLAS measurements shown in Fig. 2.1.The second is a downward shift of the gluon central value in the large-x region, by an

amount of up to two thirds of the PDF uncertainty. For instance at x ' 0.4 the gluon inNNPDF3.1+ATLAS� is about 4% smaller than in NNPDF3.1. Interestingly, the same trendwas observed when adding top-quark pair differential distributions to NNPDF3.0 [6]. The overallconsistency of the ATLAS direct photon data with the NNPDF3.1 dataset is highlighted by thefact that in the whole range of x the two fits are consistent within uncertainties.

In addition to the impact of the photon data on the gluon, it is important to determine if thenew data is consistent with the quark PDFs. In Fig. 4.3 we show the comparison of the quarkPDFs at Q = 100 GeV between the NNPDF3.1 and NNPDF3.1+ATLAS� fits. We find onlyrather small changes upon the addition of the photon data, both in terms of central values andof uncertainties, The exception is the charm PDF, which decreases in uncertainty across the fullx range, partly due to its relation to the gluon via perturbative evolution. We therefore concludethat the ATLAS data does not introduce tensions with the quark PDFs, and furthermore doesnot strongly impact the size of their respective uncertainties.

Finally, in Fig. 4.4 we show the same comparison between theory predictions and experi-mental data as in Fig. 4.1 now for the NNPDF3.1 and NNPDF3.1+ATLAS� sets for the threerapidity bins of the ATLAS 8 TeV data included in the fit. We can see how in this case thepredictions obtained with NNPDF3.1+ATLAS� as an input move closer to the central valuesof the experimental data as compared to the NNPDF3.1 baseline, although by a small amount.These findings are consistent with the corresponding variations at the PDF level discussed inFigs. 4.2 and 4.3.

12

Impact of ATLAS prompt photon studied in J. Campbell et al [1802.03021]

PATH TO NNPDF 4.0 - METHODOLOGYAs our dataset gets larger, it’s important to assess our methodology

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A large dataset with a flexible parametrisation presents a complex optimisation problem

Our parametrisation is flexibleto minimise bias, but can it bemade more efficient?

Figure 2. Arc-lengths L for PDFs determined via the NGA and CMA-ES minimisers . Resultsare normalised to the NGA value, and uncertainties are given as asymmetric 68% confidenceintervals.

Figure 3. Gluon PDF ensemble distributions for the NGA (left) and the CMA-ES (right)minimisers. Each distribution is normalised to its central-value.

the results, further tests are needed. In Ref. [3] the consistency of the PDF uncertainties weredetermined by means of statistical closure tests. For the minimiser investigated here to providereliable fits of PDFs, it must be able to demonstrate closure on a pseudo-dataset. These testswe leave for a future work.

Acknowledgements

S. C. is supported by the HICCUP ERC Consolidator grant (614577) and by the EuropeanResearch Council under the European Union’s Horizon 2020 research and innovation programme(grant agreement n� 740006). N. H. is supported by an European Research Council StartingGrant “PDF4BSM”.

Fits are computationally expensiveCan modern optimisation tools help?

➤ Evolutionary Strategies

➤ Analytical gradients

SUMMARY

NNPDF3.1: released June 2017

➤ NNPDF3.1sx October 2017

➤ NNPDF3.1luxQED December 2017

➤ NNPDF January 2018 ↵S<latexit sha1_base64="s3mEFP6CJKApch0ET61O1O+fTuc=">AAAB73icdVDLSgNBEOyNrxhfUY9eBoPgadkNUSN4CHjxGNGYQLKE3slsMmT24cysEJb8hBcPinj1d7z5N05eoKIFDUVVN91dfiK40o7zaeWWlldW1/LrhY3Nre2d4u7enYpTSVmDxiKWLR8VEzxiDc21YK1EMgx9wZr+8HLiNx+YVDyObvUoYV6I/YgHnKI2UquDIhlg96ZbLDl2uVo9P3HIjFTcBTklru1MUYI56t3iR6cX0zRkkaYClWq7TqK9DKXmVLBxoZMqliAdYp+1DY0wZMrLpveOyZFReiSIpalIk6n6fSLDUKlR6JvOEPVA/fYm4l9eO9VB1ct4lKSaRXS2KEgF0TGZPE96XDKqxcgQpJKbWwkdoESqTUQFE8LiU/I/uSvbrmO715VS7WIeRx4O4BCOwYUzqMEV1KEBFAQ8wjO8WPfWk/Vqvc1ac9Z8Zh9+wHr/AnimkD0=</latexit><latexit sha1_base64="s3mEFP6CJKApch0ET61O1O+fTuc=">AAAB73icdVDLSgNBEOyNrxhfUY9eBoPgadkNUSN4CHjxGNGYQLKE3slsMmT24cysEJb8hBcPinj1d7z5N05eoKIFDUVVN91dfiK40o7zaeWWlldW1/LrhY3Nre2d4u7enYpTSVmDxiKWLR8VEzxiDc21YK1EMgx9wZr+8HLiNx+YVDyObvUoYV6I/YgHnKI2UquDIhlg96ZbLDl2uVo9P3HIjFTcBTklru1MUYI56t3iR6cX0zRkkaYClWq7TqK9DKXmVLBxoZMqliAdYp+1DY0wZMrLpveOyZFReiSIpalIk6n6fSLDUKlR6JvOEPVA/fYm4l9eO9VB1ct4lKSaRXS2KEgF0TGZPE96XDKqxcgQpJKbWwkdoESqTUQFE8LiU/I/uSvbrmO715VS7WIeRx4O4BCOwYUzqMEV1KEBFAQ8wjO8WPfWk/Vqvc1ac9Z8Zh9+wHr/AnimkD0=</latexit><latexit sha1_base64="s3mEFP6CJKApch0ET61O1O+fTuc=">AAAB73icdVDLSgNBEOyNrxhfUY9eBoPgadkNUSN4CHjxGNGYQLKE3slsMmT24cysEJb8hBcPinj1d7z5N05eoKIFDUVVN91dfiK40o7zaeWWlldW1/LrhY3Nre2d4u7enYpTSVmDxiKWLR8VEzxiDc21YK1EMgx9wZr+8HLiNx+YVDyObvUoYV6I/YgHnKI2UquDIhlg96ZbLDl2uVo9P3HIjFTcBTklru1MUYI56t3iR6cX0zRkkaYClWq7TqK9DKXmVLBxoZMqliAdYp+1DY0wZMrLpveOyZFReiSIpalIk6n6fSLDUKlR6JvOEPVA/fYm4l9eO9VB1ct4lKSaRXS2KEgF0TGZPE96XDKqxcgQpJKbWwkdoESqTUQFE8LiU/I/uSvbrmO715VS7WIeRx4O4BCOwYUzqMEV1KEBFAQ8wjO8WPfWk/Vqvc1ac9Z8Zh9+wHr/AnimkD0=</latexit><latexit sha1_base64="s3mEFP6CJKApch0ET61O1O+fTuc=">AAAB73icdVDLSgNBEOyNrxhfUY9eBoPgadkNUSN4CHjxGNGYQLKE3slsMmT24cysEJb8hBcPinj1d7z5N05eoKIFDUVVN91dfiK40o7zaeWWlldW1/LrhY3Nre2d4u7enYpTSVmDxiKWLR8VEzxiDc21YK1EMgx9wZr+8HLiNx+YVDyObvUoYV6I/YgHnKI2UquDIhlg96ZbLDl2uVo9P3HIjFTcBTklru1MUYI56t3iR6cX0zRkkaYClWq7TqK9DKXmVLBxoZMqliAdYp+1DY0wZMrLpveOyZFReiSIpalIk6n6fSLDUKlR6JvOEPVA/fYm4l9eO9VB1ct4lKSaRXS2KEgF0TGZPE96XDKqxcgQpJKbWwkdoESqTUQFE8LiU/I/uSvbrmO715VS7WIeRx4O4BCOwYUzqMEV1KEBFAQ8wjO8WPfWk/Vqvc1ac9Z8Zh9+wHr/AnimkD0=</latexit>

x3−10 2−10 1−10 1

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Figure 3.1: The NNPDF3.1 NNLO PDFs, evaluated at µ2 = 10 GeV2 (left) and µ2 = 104 GeV2 (right).

3.3 Parton distributions

We now inspect the baseline NNPDF3.1 parton distributions, and compare them to NNPDF3.0and to MMHT14 [7], CT14 [6] and ABMP16 [8]. The NNLO NNPDF3.1 PDFs are displayedin Fig. 3.1. It can be seen that although charm is now independently parametrized, it is stillknown more precisely than the strange PDF. The most precisely determined PDF over most ofthe experimentally accessible range of x is now the gluon, as will be discussed in more detailbelow.

In Fig. 3.2 we show the distance between the NNPDF3.1 and NNPDF3.0 PDFs. Accordingto the definition of the distance given in Ref. [98], d ' 1 corresponds to statistically equivalentsets. Comparing two sets with Nrep = 100 replicas, a distance of d ' 10 corresponds to adi↵erence of one-sigma in units of the corresponding variance, both for central values and forPDF uncertainties. For clarity only the distance between the total strangeness distributionss+ = s + s is shown, rather than the strange and antistrange separately. We find importantdi↵erences both at the level of central values and of PDF errors for all flavors and in the entirerange of x. The largest distance is found for charm, which is independently parametrized inNNPDF3.1, while it was not in NNPDF3.0. Aside from this, the most significant distances areseen in light quark distributions at large x and strangeness at medium x.

In Fig. 3.3 we compare the full set of NNPDF3.1 NNLO PDFs with NNPDF3.0. TheNNPDF3.1 gluon is slightly larger than its NNPDF3.0 counterpart in the x

⇠< 0.03 region, while

it becomes smaller at larger x, with significantly reduced PDF errors. The NNPDF3.1 lightquarks and strangeness are larger than 3.0 at intermediate x, with the largest deviation seenfor the strange and antidown PDFs, while at both small and large x there is good agreementbetween the two PDF determinations. The best-fit charm PDF of NNPDF3.1 is significantly

23

The path to NNPDF4.0

➤ New data

➤ Theory uncertainties

➤ Methodology

Almost 40 new datasets due to be investigated

How can we best represent PDF uncertainties due to MHO corrections?

How can our fitting procedure keep pace with the dataset?

BACKUPS

INTRINSIC CHARM➤ The charm PDF is a borderline perturbative object

Most PDF fits assume that charm is generated perturbatively by evolution

Such an assumption can lead to a disproportionate influence of the charm mass

( GeV )XM10 210 310

Cha

rm -

Antic

harm

Lum

inos

ity

0.60.70.80.9

11.11.21.31.41.51.6

=1.47 GeVcm=1.33 GeVcm=1.61 GeVcm

NNPDF3 NLO, Fitted Charm, LHC 13 TeV

( GeV )XM10 210 310

Cha

rm -

Antic

harm

Lum

inos

ity

0.60.70.80.9

11.11.21.31.41.51.6

=1.47 GeVcm=1.33 GeVcm=1.61 GeVcm

NNPDF3 NLO, Perturbative Charm, LHC 13 TeV

[1605.06515]

Relaxing this assumption by fitting charm can stabilise results

NNPDF FITS ARE EXPENSIVE

x5−10 4−10 3−10 2−10 1−10

2−1−

01

2

34

5

67

)2xg(x,QCurrent fit replicasCurrent fit mean value

error bandσCurrent fit 1Current fit 68% CL band

)2xg(x,QSeveral procedural factors conspire to make NNPDF fits particularly demanding

Monte Carlo uncertainties➤ PDFs are formed by ensembles:

Each result requires 100/1000 statistically independent analysis runs

Neural Network parametrisation➤ Standard gradient descent is difficult:

Minimisation by Genetic Algorithm - typically 50,000 generations

�pp!X =X

i,j

Z 1

0dx1dx2 fi(x1, Q

2) fj(x2, Q2) �ij!X

�x1, x2, Q

2�

� =

nfX

i,j

nxX

↵,�

Wij↵� fi(x↵, Q20)fj(x� , Q

20)

Fitting a large dataset only possible making use of pre-computed tables

NNPDF FITS ARE EXPENSIVE

x5−10 4−10 3−10 2−10 1−10

2−1−

01

2

34

5

67

)2xg(x,QCurrent fit replicasCurrent fit mean value

error bandσCurrent fit 1Current fit 68% CL band

)2xg(x,QSeveral procedural factors conspire to make NNPDF fits particularly demanding

Monte Carlo uncertainties➤ PDFs are formed by ensembles:

Each result requires 100/1000 statistically independent analysis runs

Neural Network parametrisation➤ Standard gradient descent is difficult:

Minimisation by Genetic Algorithm - typically 50,000 generations

�pp!X =X

i,j

Z 1

0dx1dx2 fi(x1, Q

2) fj(x2, Q2) �ij!X

�x1, x2, Q

2�

� =

nfX

i,j

nxX

↵,�

Wij↵� fi(x↵, Q20)fj(x� , Q

20)

Fitting a large dataset only possible making use of pre-computed tables

NNPDF3.1 GLOBAL FIT RESULTS

( GeV )XM10 210 310

Glu

on -

Glu

on L

umin

osity

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2 NNPDF3.1

NNPDF3.0

LHC 13 TeV, NNLO ( GeV )XM10 210 310

Qua

rk -

Qua

rk L

umin

osity

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2 NNPDF3.1

NNPDF3.0

LHC 13 TeV, NNLO

�2

3.1 NNLO 3.0 NLO

HERA 1.16 1.14

ATLAS 1.09 1.37

CMS 1.06 1.20

LHCb 1.47 1.61

TOTAL (FC) 1.148 1.168

TOTAL (PC) 1.187 1.197

�2

Fit Quality

NNPDF3.1 GLOBAL FIT RESULTS - DATA VS METHODOLOGY

x 4−10 3−10 2−10 1−10

) [re

f] 2

) / g

( x,

Q2

g ( x

, Q

0.9

0.95

1

1.05

1.1

1.15 NNPDF3.1NNPDF3.1 (old dataset)NNPDF3.0

NNLO, Q = 100 GeV

NNPDF3.1 GLOBAL FIT RESULTS - DATA VS METHODOLOGY

x 4−10 3−10 2−10 1−10

) [re

f] 2

( x,

Q+

) / c

2 (

x, Q

+ c

0.8

0.9

1

1.1

1.2

1.3NNPDF3.1NNPDF3.1 (old dataset)NNPDF3.0

NNLO, Q = 100 GeV

)2 = 1.9 GeV2(x=0.023, QSR0.4 0.6 0.8 1 1.2 1.4

)d + u ) / ( s = ( s + SRNNPDF3.0

NNPDF3.1 global

NNPDF3.1 collider

NNPDF3.1 HERA+AWZ11

xFitter 2016

)d + u ) / ( s = ( s + SR

sR0 0.2 0.4 0.6 0.8 1 1.2 1.4

ABM12NNPDF3.0MMHT14CT14ATLAS-epWZ12

ATLAS-epWZ16exp uncertaintyexp+mod+par uncertaintyexp+mod+par+thy uncertainty

ATLAS, x=0.0232 = 1.9 GeV2Q

THE STRANGENESS PUZZLE

Tension in strangeness between global fits and xFitter persists in NN3.1

Rs(x,Q2) =

⇥s(x,Q2) + s(x,Q2)

⇤/⇥u(x,Q2) + d(x,Q2)

⇤

NNPDF3.1 Global NNPDF3.1 ColliderATLAS 2011 W/Z 2.14 1.55ATLAS 2010 W/Z 0.96 0.92NuTeV dimuon 0.82 26.5

Driven by disagreement between collider data and neutrino DIS

THE STRANGENESS PUZZLERs(x,Q

2) =⇥s(x,Q2) + s(x,Q2)

⇤/⇥u(x,Q2) + d(x,Q2)

⇤

x 4−10 3−10 2−10 1−10

)2 (

x, Q

SR

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6 NNPDF3.1 NNPDF3.1 colliderNNPDF3.0

NNLO, Q=100 GeV

INCLUSIVE JET DATA AT NNLOWhile the full NNLO calculation for inclusive jet production has been finalised, exact K-factors for several of the jet datasets were not available at time of publication

Therefore for NNLO fits, the jet data was included at NLO accuracy, but with an additional uncertainty determined by NLO scale variation

x 4−10 3−10 2−10 1−10

) [re

f] 2

) / g

( x,

Q2

g ( x

, Q

0.9

0.95

1

1.05

1.1

1.15 NNPDF3.1

NNPDF3.1 (NNLO jets)

NNPDF3.1 NNLO, Q = 100 GeV

x 4−10 3−10 2−10 1−10

) [re

f] 2

) / d

( x,

Q2

d ( x

, Q

0.9

0.95

1

1.05

1.1

1.15 NNPDF3.1

NNPDF3.1 (NNLO jets)

NNPDF3.1 NNLO, Q = 100 GeV

Reliability verified by comparison against fit with available NNLO corrections