arX

iv:0

906.

1958

v2 [

hep-

ph]

20

Aug

200

9

Edinburgh 2009/06

IFUM -941-FT

FREIBURG -PHENO -09/03

Precision determ ination ofelectroweak param eters

and the strange content ofthe proton

from neutrino deep{inelastic scattering

T he N N P D F C ollaboration:

Richard D.Ball1,LuigiDelDebbio1,Stefano Forte2,Alberto G u�anti3,Jos�e I.Latorre4,

Andrea Piccione2,Juan Rojo2 and M aria Ubiali1.

1 SchoolofPhysicsand Astronom y,University ofEdinburgh,

JCM B,KB,M ay�eld Rd,Edinburgh EH9 3JZ,Scotland2 Dipartim ento diFisica,Universit�a diM ilano and INFN,Sezione diM ilano,

Via Celoria 16,I-20133 M ilano,Italy3 Physikalisches Institut,Albert-Ludwigs-Universit�atFreiburg

Herm ann-Herder-Stra�e 3,D-79104 Freiburg i.B.,Germ any4 Departam entd’Estructura iConstituents de la M at�eria,Universitatde Barcelona,

Diagonal647,E-08028 Barcelona,Spain

Thispaper isdedicated to the m em ory ofW u-KiTung

A bstract:

W e use recent neutrino dim uon production data com bined with a globaldeep-inelastic

parton �tto constructa new parton set,NNPDF1.2,which includesa determ ination of

the strange and antistrange distributionsofthe nucleon.Theresultischaracterized by a

faithfulestim ation ofuncertaintiesthanksto the useofthe NNPDF m ethodology,and is

freeofm odelortheoreticalassum ptionsotherthan theuseofNLO perturbativeQ CD and

exact sum rules. Better controlofthe uncertainties ofthe strange and antistrange par-

ton distributionsallowsusto reassessthe determ ination ofelectroweak param etersfrom

the NuTeV dim uon data.W e perform a directdeterm ination ofthe jVcdjand jVcsjCK M

m atrix elem ents,obtaining centralvaluesin agreem entwith the currentglobalCK M �t:

speci�cally we �nd jVcdj= 0:244� 0:019 and jVcsj= 0:96� 0:07. O urresultfor jVcsjis

m ore precise than any previous direct determ ination. W e also reassess the uncertainty

on the NuTeV determ ination ofsin2�W through the Paschos-W olfenstein relation: we

�nd thatthevery large uncertaintiesin thestrangevalence m om entum fraction aresu�-

cientto bring the NuTeV resultinto com plete agreem entwith the resultsfrom precision

electroweak data.

1

C ontents

1 T he strange content ofthe nucleon 3

2 Experim entaldata 6

2.1 Data set,uncertaintiesand correlations . . . .. . .. . . .. . .. . . .. . 7

2.2 O bservables,kinem atic cutsand pseudo-data sam ple . . . .. . .. . . .. . 7

3 N euralnetw orks,parton distributions and physicalobservables 10

3.1 Param etrization ofthe strangePDF . .. . . .. . .. . . .. . .. . . .. . 10

3.2 Thedim uon physicalobservable. .. . .. . . .. . .. . . .. . .. . . .. . 12

3.3 Treatm entofthecharm m ass . . .. . .. . . .. . .. . . .. . .. . . .. . 12

3.4 NuclearCorrections .. . . .. . .. . .. . . .. . .. . . .. . .. . . .. . 14

4 R esults 16

4.1 TheNNPDF1.2 parton set:statisticalfeatures . . .. . . .. . .. . . .. . 16

4.2 TheNNPDF1.2 parton set:parton distributions . .. . . .. . .. . . .. . 21

4.3 Theoreticaluncertainties. . .. . .. . .. . . .. . .. . . .. . .. . . .. . 21

4.4 Determ ination ofthe strange distribution . . .. . .. . . .. . .. . . .. . 23

4.5 Com parison with experim entaldata . .. . . .. . .. . . .. . .. . . .. . 33

5 P recision determ ination ofelectrow eak param eters 35

5.1 Determ ination ofjVcsjand jVcdj. .. . .. . . .. . .. . . .. . .. . . .. . 35

5.2 PDF correctionsto the Paschos-W olfenstein ratio . .. . . .. . .. . . .. . 42

6 C onclusions and outlook 44

A K ernels for P hysicalO bservables 45

2

1 T he strange content ofthe nucleon

Thedeterm ination ofthestrange and antistrange quark distributionsofthenucleon isof

considerable phenom enologicalinterest,because m any �nalstatesin the standard m odel

and beyond coupledirectly to strangeness.A notableexam pleisthedeterm ination ofthe

electroweak m ixing angle by the NuTeV collaboration [1],which m ightprovide evidence

for physics beyond the standard m odel,and which is very sensitive [2]to the strange

contentofthenucleon.

Unfortunately,the bulk ofthe data which are used forparton determ ination,nam ely

neutral-current deep-inelastic scattering,have m inim alsensitivity to avour separation,

and nosensitivity atalltotheseparation ofquarkand antiquarkcontributions.Asaconse-

quence,untilvery recently in standard parton �tssuch asCTEQ 6.5[3]and M RST2006[4],

thestrangeand antistrangequark distributionswerenotdeterm ined directly:rather,they

wereassum ed to beequal,and then proportionalto thetotallightantiquark sea distribu-

tion. The only available attem pt at a determ ination ofthe strange and antistrange dis-

tributions[5]wasbased on a re-analysisofold (m ostly bubble-cham ber)charged-current

neutrino-nucleon scattering data:unfortunately,thequality oftheseold data wasinsu�-

cientfora reliable determ ination.

Thissituation haschanged recently,due to the availability ofa widersetofinclusive

neutrino deep-inelastic scattering data [6,7]and, m ore im portantly, of data for deep-

inelastic neutrino and anti-neutrino production ofcharm [8{10](\dim uon" data,hence-

forth), which is directly sensitive to the strange and antistrange parton distributions.

As a consequence,dedicated analyses ofthe strange quark distribution have been per-

form ed [11{14],and independentparam etrizations ofthe strange and antistrange distri-

butions are included in m ost recent parton �ts [15]. However,the standard m ethod of

parton determ ination used in allthese references,which is based on �tting the param -

eters ofa �xed functionalform ,is known to be hard to handle when the experim ents

are relatively unconstraining. Indeed,it is not uncom m on that the addition ofnew ex-

perim entalinform ation to a parton �t ofthis kind,actually leads to an increase rather

than a decrease ofuncertainty bands (see e.g.[16]),because the new data require the

use ofa m ore generalparam etrization. This ham pers a direct statisticalinterpretation

ofthe uncertainty bands on parton distributions obtained in this way: indeed,in som e

ofthese parton determ inations[15,17]experim entaluncertaintiesarein ated by suitable

\tolerance" criteria. Precision m easurem ents are thus very di�cult to obtain whenever

theresultsaresigni�cantly a�ected by parton uncertainties.Thisisclearly thecasein the

extraction ofthe electroweak m ixing angle from the NuTeV data ofRef.[1],and itcould

be m ore generally an issue for LHC observables which depend crucially on the strange

distribution,such asthe\standard candle" �Z =�W [17].

A m ethod ofparton determ ination which is free ofthese di�culties was developed

by us in a series ofpapers [18{20],and has led recently to the construction of a full

parton setbased on a �tto a globalsetofdeep-inelasticscattering data:NNPDF1.0 [21].

This m ethod is based on the use ofneuralnetworks for parton param etrization,and a

M onte Carlo m ethod supplem ented by a suitable training and stopping algorithm forthe

construction oftheparton �t.In thisapproach,parton distributionsaregiven asa M onte

Carlo sam plerepresenting theirprobability distributionsasinferred from thedata:so,for

instance,uncertaintiescan beobtained from thesam plebycom putingstandard deviations,

3

likelihood intervalsby determ ining frequency histogram s,and so on.

Itwasshown thatthism ethodologyislargelyfreeofbiasrelated toparton param etriza-

tion,and ithandlesin a satisfactory way incom pleteinform ation,contradictory data,and

theaddition ofnew datawithin asinglefram ework.In particular,in Ref.[21]itwasexplic-

itly veri�ed thatwhen data arerem oved by changing thekinem atic cuts,theuncertainty

bandswiden in such a way thatresultsbeforeand afterthecutsrem ain com patible,while

results outside the data region directly a�ected by the cuts rem ain stable. In Ref.[22]

it was further checked that the sam e behaviour is observed when the whole dataset is

altered,e.g.by rem oving alldata from oneorm oreexperim ents:a �tto a sm allerdataset

haswideruncertainties,butrem ainscom patible with the �tto the largerdataset.

That these stability properties ofthe NNPDF approach apply also to the way the

strangedistribution istreated wasshown in a dedicated study based on thesam em ethod-

ology [23],leading to theNNPDF1.1 parton set.In NNPDF1.1,thestrangeparton distri-

butionss� = s� �sareparam etrized by two independentneuralnetworks,instead ofbeing

taken to be proportionalto the light antiquark distribution as in NNPDF1.0. However,

the dataset is the sam e as for NNPDF1.0: so the s+ distribution is only very weakly

constrained,and the s� essentially unconstrained by the the data. Nevertheless,when

resultsofthis pairof�tsare com pared,they show rem arkable stability,despite the fact

thateach neuralnetwork isparam etrized by a very redundantsetofparam eters(the ad-

dition oftwo neuralnets results in the addition of74 extra free param eters in the �t).

Indeed,parton distributionswhich areuna�ected by theaddition ofindependentstrange

degreesoffreedom (such asthe gluon)are unchanged,and the only m arked e�ectofthe

independentparam etrization ofstrangenessisan increase,by abouta factor two,ofthe

uncertainty on thetotalvalence quark distribution (u � �u + d� �d+ s� �s ).Rem arkably,

statisticalanalysisoftheNNPDF1.0 setalonewasalready su�cientto show [21]thatthe

uncertainty on thiscom bination wasunderestim ated.

In this paper,by adding recent dim uon data to the globaldeep-inelastic scattering

dataset on which the NNPDF1.0 and NNPDF1.1 �ts were based,we construct a new

parton set,NNPDF1.2,which includes a determ ination ofthe strange and antistrange

distributions. Furtherm ore,we determ ine directly the jVcsjand jVcdjCK M m atrix ele-

m entswhich controlthestrength ofthecharged{currentcoupling to neutrinosin dim uon

production ofthestrangeand down quarksrespectively,and weuseourdeterm ination of

thestrangequark distribution to com putethecorrection to thePaschos-W olfenstein ratio

to beused in extractionsofthe electroweak m ixing angle.

W e �nd that the shape ofthe strange and antistrange distributions which are com -

patible with data are rather m ore general than those obtained in other recent stud-

ies [11{15,17]. O ur uncertainty on the ratio K S = [S+ ]=��U + �D

�ofstrange to light

sea m om enta is rather m ore asym m etric than hitherto assum ed: K S

�Q 2 = 20G eV 2

�=

0:71+ 0:19� 0:31

stat.Thism ay havenontrivialim plicationsforLHC observables,such astheZ=W

crosssection ratio m entioned above. Despite these increased uncertainties,we �nd that,

perhaps surprisingly,the dim uon data are su�cient to determ ine jV csj= 0:96 � 0:07tot.

This is one order ofm agnitude m ore precise than any other direct determ ination from

neutrino deep-inelastic scattering,and iscom parable to the currentPDG bestaverage of

direct determ inations from D m eson decays,(jVcsj= 1:04 � 0:06 [24]),though stilltwo

orders ofm agnitude worse than the results ofa globalCK M �t. The related CK M ele-

m ent jVcdjis also determ ined,jVcdj= 0:244 � 0:019tot,with a sim ilar accuracy to other

4

determ inationsfrom dim uon data.

W efurther�nd thatthes� �sdistribution,which m ustchangesign asafunction ofx in

orderforthetotalnucleon strangenessto vanish,can do so in a widevariety ofways,and

thatitssign atanygiven x isnotwelldeterm ined.Asaconsequence,theuncertainty in the

strangevalencem om entum fraction,and thusin thecorrection tothePaschos-W olfenstein

ratio,ism uch largerthan hitherto assum ed,and issuch thattheNuTeV m easurem entof

sin2�W isactually in com pleteagreem entwith determ inationsfrom precision electroweak

data once thisuncertainty istaken into account.

M any ofthe techniquesand tools thatwe use in thispaperare partofthe standard

NNPDF m ethodology,already described in detailin Refs.[19{21]and used there forthe

construction of the NNPDF1.0 parton set. Here we willfocus on the new aspects of

the NNPDF1.2 set,and then discuss our m ain results. Thus in Sect.2 we describe the

dim uon crosssection and itsavailableexperim entaldeterm inations,and in Sect.3 wewill

give itsexpression in perturbative Q CD and thusitsrelation to the strange distribution,

and discuss the way the strange and antistrange distributions are treated, as well as

som e speci�c theoreticalissues related to the treatm ent ofthis observable,such as the

treatm entofthe charm m assand ofnuclearcorrections. Fulldetailsofthe hard kernels

used to construct the physicalobservables are given in Appendix A. In Section 4 we

presentourdeterm ination ofthe strange and antistrange distributions,speci�cally their

shape and their contribution to the nucleon m om entum ,and com pare them to results

obtained by other groups. In Section 5 we willdiscuss in detailthe im plications ofour

resultsforprecision electroweak m easurem ents,and discussspeci�cally thedeterm ination

ofthe CK M m atrix elem entsjVcsjand jVcdjand the im pactofourresultson the NuTeV

determ ination oftheelectroweak m ixing angle.

5

x-410 -310 -210 -110 1

)2 (

GeV

2Q

1

10

210

310

410

NMC-pdNMCSLACBCDMSZEUSH1CHORUSFLH108NTVDMNZEUS-H2

Figure1:Experim entaldata in the(x;Q 2)planeused in theNNPDF1.2 analysisafterkinem atic

cuts.

2 Experim entaldata

TheNNPDF1.2 parton determ ination isbased on thesam edata setused forNNPDF1.0,

supplem ented by data on deep-inelastic neutrino production ofcharm from NuTeV [8,25]

which giveusahandleon thestrangedistribution,whosedeterm ination isthem ain goalof

thispaper.W ealso add to thedata setsom erecently published m easurem entsofneutral

currentand charged currentdeep-inelastic crosssectionsby the ZEUS experim entbased

on HERA-IIdata [26,27].

An earlier m easurem ent of the dim uon cross section using the sam e detector (but

a di�erent beam -line) was perform ed by the CCFR collaboration [28]. This previous

m easurem entis signi�cantly less accurate and its com patibility with the NuTeV data is

debatable [8,14];we willnotinclude itin our�t. A recent m easurem entofthe dim uon

cross section has also been perform ed by the CHO RUS collaboration [9];unfortunately,

however,onlytheresultsofaleading{orderQ CD analysisofthisdatahavebeen published,

and notthecross-section data them selves,which thereforecannotbeused in ouranalysis.

Thetreatm entofexperim entaldatain thepresent�tfollowsRef.[21].In particular,all

inform ation on correlated system aticsisincluded in our�t,in thatthefullcovariancem a-

trix iscom puted including allavailable correlated uncertainties(including norm alization

uncertainties).

Below we give m ore explicit details ofthe new data and corresponding observables

6

E xperim ent Set N dat xm in xm ax Q2m in

Q2m ax

�tot (% ) F R ef.

ZE U S-H E R A -II

ZE U S06N C 90 (90) .005 0.65 200.0 30000.0 2.6 ~�N C ;e

�[26]

ZE U S06C C 37 (37) .015 0.65 280.0 30000.0 14 ~�C C ;e

�[27]

N U T E V D im uon

N uTeV D im uon � 45 (43) .0267 0.37 1.1 116.5 19 ~��;c

[25]

N uTeV D im uon �� 45 (41) .021 0.25 0.8 68.3 23 ~���;c

[25]

Total (including Tab. 1 of R ef. [21]) 4165 (3372)

Table 1:Furtherexperim entaldata included in thepresentanalysisin addition to thosegiven in

Table 1 ofRef.[21]. W e show the num ber ofpoints before (after) applying kinem atic cuts,the

kinem atic range,the average totaluncertainty after cuts and the observable which is m easured.

Di�erentsetswithin an experim entare correlated with each other,while data from di�erentex-

perim entsareuncorrelated.Thetotalnum berofdata pointsrefersto the fulldataset.

which have been included in the current�t.

2.1 D ata set,uncertainties and correlations

Thedatasetused forthepresent�tisobtained by supplem entingthedatasetused forthe

NNPDF1.0 �t,assum m arized in Table1ofRef.[21],with thedatasum m arized in Table 1

given here.A scatterplotofthe fulldata setisdisplayed in Figure 1.Note thatNuTeV

dim uon data overlap with the restof�xed targetexperim ents,providing inform ation on

the proton strangenessforx �> 10� 2.

Thecovariance m atrix iscom puted forallthedata included in the�t,asdiscussed in

Ref.[21]. The NuTeV dim uon data are a�ected by a com m on norm alization uncertainty

of2.1% [6]; eight correlated system atics; and a statisticaluncertainty. The statistical

uncertainty is around 15% for neutrino and around 25% for anti-neutrino data, while

correlated system atics are generally sm aller by a factor between three and �ve. This

dom inantstatisticaluncertainty isa�ected by abin by bin correlation duetotheunfolding

procedureused in extracting thedim uon crosssection from them easured observable.The

covariance m atrix which describesthese correlations isnotavailable. Itse�ect hasbeen

sum m arized in Ref.[25]by providing for each bin an \e�ective num ber of degrees of

freedom ",which providesthe expected value ofthe best-�t�2 to the given data bin,i.e.,

e�ectively,a rescaling for the statisticalerror. These rescaling factors can be as low as

30% ,and are typically around 50% ,indicating sizable correlations.

Rescaling ofstatisticalerrors in order to account for m issing correlations could bias

the �tin an unpredictableway and itisa dangerousprocedureifthe inform ation on the

covariance m atrix is lost. O n the other hand,only including correlations for the sub-

dom inantsystem aticerrorscould lead to an underestim ateoftherelativeim pactofthese

uncertainties. Hence,because the covariance m atrix ofthe NuTeV data isunfortunately

unavailable,the only consistent procedure for the treatm ent ofthese data is to add all

uncertaintiesin quadrature,and only considernorm alizationsascorrelated uncertainties.

Thisisthe procedurethatwe shallfollow.

2.2 O bservables,kinem atic cuts and pseudo-data sam ple

The set of observables considered in these �ts consists of the structure functions and

reduced cross-sectionsconsidered in Ref.[21]and sum m arized in Table1 ofthatreference,

7

E xperim ent ZE U S-H E R A -II N uTeV D im uon Totalfi

P E

»

D

F(art)

E

rep

–fl

dat

-4.2 � 10� 4

-5.4 � 10� 4

-2.3 � 10� 4

r

h

F(art)

i

0.999 0.999 0.999fi

P E

»

D

�(art)

E

rep

–fl

dat

6.5 � 10� 3 -2.6 � 10� 3 -6.1 � 10� 4

D

�(exp)

E

dat13.79% 21.23% 11.24%

D

�(art)

E

dat13.88% 21.17% 11.24%

r

h

�(art)

i

0.999 0.998 0.999D

�(exp)

E

dat0.287 0.034 0.146

D

�(art)

E

dat0.294 0.034 0.146

r

h

�(art)

i

0.994 0.978 0.996D

cov(exp)E

dat6.89 � 10� 4 0.169 1.61 � 10� 3

D

cov(art)

E

dat7.03 � 10

� 40.168 1.54 � 10

� 3

r

h

cov(art)

i

0.997 0.988 0.988

Table 2: Statisticalestim atorsforthe M onte Carlo arti�cialdata generation with N rep = 1000,

for experim ents not included in Ref.[21]. The de�nition ofthe statisticalestim atorsis given in

Appendix B of[20].ThefaithfulnessoftheM onteCarlo sam pling ofexperim entaldata isassessed

quantitatively by these estim ators.

supplem ented by the dim uon cross section. Neutrino dim uon production is induced by

charm production through charged currentinteractionsofneutrinoswith thetargetnuclei,

followed by the fragm entation ofthe charm quark into a charm ed hadron and its decay

into a m uon.Thecorresponding crosssection isgiven by

~��(��);c(x;y;Q 2)�1

E �

d2��(��);c

dxdy(x;y;Q 2)

=G 2FM N

2�(1+ Q 2=M 2W)2

"��

Y+ �2M 2

Nx2y2

Q 2� y

2

��

1+m 2

c

Q 2

�

+ y2

�

F�(��);c

2 (x;Q 2)

� y2F�(��);c

L(x;Q 2)� Y� xF

�(��);c

3 (x;Q 2)

#

; (1)

where

Q2 = 2M N E �xy; Y� = 1� (1� y)2: (2)

The charm production cross section is obtained from the published NuTeV neutrino

dim uon production crosssections[25]as

1

E �

d2��(��);c

dxdy(x;y;Q 2)=

1

hBr(D ! �)i� A (x;y;E�)

1

E �

d2��(��);2�

dxdy(x;y;Q 2); (3)

where hBr(D ! �)i is the average branching ratio ofcharm ed hadrons into m uons and

A (x;y;E �)isa bin-dependentexperim entalacceptance correction.Acceptancesare pro-

vided by the NuTeV collaboration,based on a leading-order m odel[29];next-to-leading

order acceptances [30](not publicly available) di�er by less than 3% from the leading-

orderones.Thebranching ratio used in the NuTeV analysis[10]com esfrom a reanalysis

ofthe em ulsion data ofthe FNAL E531 experim entand turnsoutto be hBr(D ! �)i=

0:099� 0:012,in agreem entwith otherdeterm inations[9,31].A sim ultaneousextraction of

thisparam eteralong with the determ ination ofstrangenessin Ref.[14]leadsto a sim ilar

8

result. In the determ ination ofthe dim uon crosssection,the branching ratio willbe set

equalto the centralvalue used in the NuTeV analysis [10]. The associated uncertainty

willthen beincluded in our�tasdiscussed in Section 3.2 below.

O urdata setisobtained by im posing on allthedata listed in Table 1 ofRef.[21]and

in Table 1 the sam e kinem aticalcutsasin NNPDF1.0,nam ely Q 2 > Q 2cut = 2 G eV 2 and

W 2 > 12:5 G eV 2.Afterthese cuts,84 outofthe 90 NuTeV dim uon data pointsare left.

Aftercuts,the totalnum berofdata pointsin the NNPDF1.2 analysisisN dat = 3372.

Errorpropagation from theexperim entaldata to the�tisperform ed through a M onte

Carlo procedure,described in detailin Ref.[21],by generating a setof1000 pseudo-data

replicas,whosefaithfulnesscan beveri�ed by studyingsuitablestatisticalestim ators.The

statisticalestim ators forthe new data setsincluded in the present�t,aswellasforthe

globaldata set,are sum m arized in Table 2.

9

3 N eural networks, parton distributions and physical ob-

servables

Physicalobservablesare determ ined from a setofPDFsgiven ata reference scale,which

arein turn param etrized in term sofneuralnetworks,accordingtotheform alism discussed

in detailin Sect.3-4ofRef.[21].Herewesum m arizethenew featuresofthisdeterm ination:

theuseofan independentparam etrization forthestrangeand antistrangedistribution and

itsconstruction in term sofneuralnetworks,thenew physicalobservablesused fordim uon

data,and som eissuesthatrequirereconsideringwhen dealingwith thisobservable,nam ely

the treatm entofthe charm m assand nuclearcorrections.

3.1 Param etrization ofthe strange PD F

In the NNPDF1.0 �tofRef.[21],parton distributionswere param etrized using �ve inde-

pendentneuralnetworks: fourindependentlinearcom binations ofthe two light avours

and anti- avours,and thegluon.Thestrangeand antistrangequark distributionswereas-

sum ed to begiven by s= �s= ���u + �d

�=2 with � = 0:5,and heavy quarksweregenerated

dynam ically,using a zero-m assvariable avournum berschem e (ZM -VFN).In thesubse-

quentNNPDF1.1 �t[23],two furtherneuralnetworkswereintroduced to param etrizethe

strange and antistrange quark distributions. Here,asin Ref.[23]we param etrize parton

distributionsin term sofseven independentneuralnetworks,aswe now discuss.

Theprim ary partonicquantitiesoutofwhich allphysicalobservablesarebuiltup are

the gluon,the singlet quark distribution,the totalvalence quark distribution,and ten

nonsinglet com binations ofthe valence (qi� �qi) or total(qi+ �qi) quark and antiquark

distribution for the i-th quark avor. These are constructed as in Ref.[21],to which

we referform ore details. The starting scale ischosen atthe charm threshold,where the

charm distributionsareassum ed to vanish,and therem ainingsix lightquark distributions

and thegluon distribution areparam etrized in term sofindependentneuralnetworks.The

possibility ofintroducing an intrinsiccharm distribution willnotbestudied in thepresent

�t,though there isno obstacle to including itin futurestudies.

The fourlightnon-strange distributionsand the gluon distribution are param etrized

in term sofneuralnetworksasin Ref.[21],by letting

f(x;Q 20)= A f (1� x)

m f x� nfNN f(x); (4)

where f(x;Q 20) is a linear com bination ofparton distributions,and NN f(x) is a m ulti-

layer feed-forward neuralnetwork with two interm ediate layers and architecture 2-5-3-1,

param etrized by 37 freeparam eters(weightsand thresholds).TheconstantsA f areeither

sim ply setto one,orelse used to enforce the valence and m om entum sum rules.

Thepreprocessingfunction (1� x)m f x� nf isincluded in ordertospeed up theconver-

genceofthe�t:theneuralnetwork only hasto �tthedeviation from thebehaviourofthe

preprocessingfunction,whoseexponentsarethus�xed to valueswhich absorb som eofthe

grossbehaviourofthefunction f(x;Q 20)withoutbiasing theresult(i.e.withoutim posing

a steep growth orfallwhich NN (x)would have trouble in reabsorbing).Independenceof

theresultson thechoiceofthepreprocessingexponentswasveri�ed in Ref.[21]by varying

them within a reasonable stability range.Thisstability range isidenti�ed in Ref.[21]by

10

PDF m n

�(x;Q 20) [2:7;3:3] [1:1;1:3]

g(x;Q 20) [3:7;4:3] [1:1;1:3]

T3(x;Q20) [2:7;3:3] [0:1;0:4]

V (x;Q 20) [2:7;3:3] [0:1;0:4]

� S(x;Q20) [2:7;3:3] [0;0:01]

s+ (x;Q20) [2:7;3:3] [1:1;1:3]

s� (x;Q20) [2:7;3:3] [0:1;0:4]

Table 3: The range ofvariation ofthe random ized preprocessing exponentsused in the present

NNPDF1.2 �t.

requiringthequality ofthe�tto beunchanged astheexponentsarevaried.A sm allresid-

ualdependenceon the preprocessing exponentswasfound in Ref.[21]forthe tripletand

totalvalencequark distributions.In orderto beableto disentangleaccurately thestrange

contribution itisim portantthatuncertaintieson alllightquark avoursareestim ated as

precisely aspossible:forthispurpose,in theNNPDF1.1 �tofRef.[23]and in thepresent

�tallpreprocessing exponentsare random ized:a di�erentvalue istaken foreach M onte

Carlo replica,uniform ly distributed within thestability range.

The choice oflinearcom binationsofthe two lightest avourswhich are param etrized

independently according to Eq.(4) is the sam e in the present �t as in NNPDF1.0. O n

top ofthem ,we add two independentneuralnetworks in the strange sector,in orderto

param etrize

s� (x;Q 2)� s(x;Q 2)� �s(x;Q 2) (5)

according to

s+ (x;Q 2

0) = (1� x)m

s+ x� n

s+ NN s+ (x); (6)

s� (x;Q 2

0) = (1� x)m

s� x� n

s� NN s� (x)� saux(x;Q20); (7)

where

saux(x;Q20)= A s�

�xrs� (1� x)

ts��: (8)

The exponents m ,n ofthe preprocessing functions are random ized as discussed above,

and theirrangesare also listed in Table 3.

The contribution saux(x;Q20)in Eq.(7)is introduced in order to enforce the strange

valence sum rule:theconstantA s� is�xed by requiring

Z 1

0

dxs� (x)= 0; (9)

which givesthecondition

A s� =�(rs� + ts� + 2)

�(rs� + 1)�(ts� + 1)

Z 1

0

(1� x)m

s� x� n

s� NN s� (x)dx: (10)

Clearly,thesum rulesrequiress� to changesign atleastonce.Thisway ofim plem enting

the sum rule is designed in order to ensure that this crossing happens naturally in the

valenceregion,ratherthan in som econtrived way outsidethedata region wheretheshape

11

ofs� iscom pletely unconstrained. To thispurpose,the exponentsrs� ;ts� are chosen in

such a way thatsaux(x;Q20)peaksin the valence region,and that the sm allx and large

x behaviour ofs� (x;Q 20)are notcontrolled by the saux(x;Q

20) contribution. In practice

thelattercondition isenforced by requiring rs� � � ns� and ts� � m s� ,whiletheform er

is enforced by letting rs� = ts� =k,which sets the m axim um ofsaux(x;Q20) at x = 1

k+ 1.

W e then choose ts� = 3:5,and take k asa uniform ly distributed random num berin the

range k 2 [1;3].Theconsequencesofthisvery exibleim plem entation ofthestrangeness

valence sum rulewillbediscussed in Sect.4.4 below.

3.2 T he dim uon physicalobservable

TheNNPDF1.2 data set,displayed in Fig.1,containsdata forthesam esetofobservables

discussed in Ref.[21],with the addition ofthe dim uon cross section Eq.(1). The latter

isdeterm ined by the charm structure functionsF�(��);c

2 ,F�(��);c

Land xF

�(��);c

3 ,which in the

quark m odelare given by

F�;p;c

2 (x;Q 2)= xF�;p;c

3 (x;Q 2)= 2x�jVcdj

2d(x) + jVcsj

2s(x)+ jVcbj

2b(x)

�; (11)

F��;p;c

2 (x;Q 2)= � xF��;p;c

3 (x;Q 2)= 2x�jVcdj

2 �d(x) + jVcsj2 �s(x)+ jVcbj

2�b(x)�;(12)

with F�(��);c

L= 0. Fullexpressions for these structure functions in perturbative Q CD at

any scale in term softhe basisofPDFsused in our�tsare given in Appendix A.

Because they are not inclusive with respect to the �nalstate quark avour, these

structurefunctionsdependon CK M m atrixelem ents.Theseareextrem elywelldeterm ined

by current global�ts including unitarity constraints;for our global�ts we willuse the

currentbest-�tPDG [24]values:uncertaintieson them aretiny and willbeneglected.In

Section 4 we willthen study the quality ofour �t as the param eters jVcsjand jVcsjare

varied withoutthe unitarity constraint,and usethisto provide a directdeterm ination of

these param etersfrom the dim uon data.

Also,asalready discussed in Sect.2,thedim uon cross-section Eq.(3)dependson the

branching ratio hBr(D ! �)i. The uncertainty in this is actually rather signi�cant: in

previousanalyses[10,14]ofdim uon datathisturned outtobeoneofthedom inantsources

ofuncertainty.To take accountofthisuncertainty,the value ofthe branching ratio used

in the�thasbeen random ized aboutitscentralvalue,analogously to theprocedureused

forthepreprocessing exponents,with a G aussian distribution ofwidth equalto thestated

uncertainty hBr(D ! �)i= 0:099� 0:012 [10].

3.3 Treatm ent ofthe charm m ass

In the previous NNPDF1.0 and NNPDF1.1 parton determ inations,heavy quarks were

treated in azerom assvariable avournum ber(ZM -VFN)schem e,asdiscussed in Sect.3.4

ofRef.[21]. Contributionssuppressed by powers ofthe heavy quark m ass,i.e. oforder

m 2h=Q 2,which are neglected in thisschem e,have a sm allbutnotentirely negligible im -

pact[32],m ostlythrough theinitialcondition on thecharm distribution,which then a�ects

allotherPDFsdue to the m om entum sum rule.Forthe purposeofthe presentanalysis,

an im proved treatm entofthecharm m assisadvisable atleastfordim uon production,as

thedim uon crosssection m easurescharm production,and a sizablefraction oftheNuTeV

dim uon data are thusatscalesclose to the charm m ass.

12

0

5

10

15

20

25

30

35

0.02 0.05 0.1 0.2

Aν

* d2 σν,

c (x,y

,Eν)

/dx/

dy

x

Eν=88.3 GeV, y=0.324

Improved ZMZM

0

5

10

15

20

25

30

35

0.02 0.05 0.1 0.2

Aν

* d2 σν,

c (x,y

,Eν)

/dx/

dy

x

Eν=88.3 GeV, y=0.558

Improved ZMZM

0

5

10

15

20

25

30

35

0.02 0.05 0.1 0.2

Aν

* d2 σν,

c (x,y

,Eν)

/dx/

dy

x

Eν=88.3 GeV, y=0.771

Improved ZMZM

0

5

10

15

20

25

30

35

0.02 0.05 0.1 0.2

x

Eν=174.3 GeV, y=0.324

Improved ZMZM

0

5

10

15

20

25

30

35

0.02 0.05 0.1 0.2

x

Eν=174.3 GeV, y=0.558

Improved ZMZM

0

5

10

15

20

25

30

35

0.02 0.05 0.1 0.2

x

Eν=174.3 GeV, y=0.771

Improved ZMZM

0

5

10

15

20

25

30

35

0.02 0.05 0.1 0.2

x

Eν=247 GeV, y=0.324

Improved ZMZM

0

5

10

15

20

25

30

35

0.02 0.05 0.1 0.2

x

Eν=247 GeV, y=0.558

Improved ZMZM

0

5

10

15

20

25

30

35

0.02 0.05 0.1 0.2

x

Eν=247 GeV, y=0.771

Improved ZMZM

Figure 2: Com parison of the ZM and I-ZM com putations of the dim uon cross section

d2��(��);c=dxdy for typicalNuTeV kinem atics. Allcross sections in the plots are rescaled by a

factorA � = 100=G 2F M N E

2�. The neutrino kinem atic param eters(E �;y)are related to x and Q 2

by Eq.(2).Resultsforanti-neutrinosarevery sim ilar.

To thispurpose,we em ploy (forthe dim uon observable only)the im proved ZM -VFN

(I-ZM -VFN)schem e,proposed in Ref.[33]and discussed in detailin Ref.[34]. There,it

wasshown thatthebulk ofthecharm m asse�ectsnearthreshold can beaccounted forby

requiring thatthethreshold fortheinclusion ofheavy quarksin thesum over�nalstates

besetatitsphysicalvalueW 2 = m 2c,and thatthephase-spaceconstraintduetotheheavy

quark m ass be respected in convolution integrals. The latter requirem ent is in practice

im plem ented by replacing the Bjorken x variable by a rescaling variable �c de�ned as

�c � x

�

1+m 2

c

Q 2

�

: (13)

W hereas results obtained with this I-ZM -VFN schem e are in fair agreem ent with those

obtained with a fulltreatm entofthequark m ass(so-called generalm ass,orG M schem e),

and in ratherbetteragreem entwith thedata,they m ay lead toan excessivesuppression of

heavy quark production:to thispurpose,in Ref.[34]a one-param eterfam ily ofrescaling

variableshasbeen constructed,such thatthe agreem entwith the G M schem e can beop-

tim ized by tuning thisparam eter.Itturnsout,however,thatthesim plestchoiceEq.(13)

isactually very close to theoptim aloneforcharged currentdeep inelastic scattering.

Hence,in the presentanalysis we willuse the ZM -VFN for allinclusive observables,

butforthedim uon crosssection Eq.(3)wewillusetheI-ZM -VFN ofRef.[33].In practice,

13

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200

RA

σ

Point label

CHORUS data

Neutrino, CTEQ6.6, deFlorian-Sassot 03 NLOAnti-neutrino, CTEQ6.6, deFlorian-Sassot 03 NLO

Neutrino, CTEQ6.6, HKN07 NLOAnti-neutrino, CTEQ6.6, HKN07 NLO

0.9

0.95

1

1.05

1.1

1.15

1.2

3160 3170 3180 3190 3200 3210 3220 3230 3240 3250

RA

σ

Point label

NuTeV dimuon data

Neutrino, CTEQ6.6, deFlorian-Sassot 03 NLOAnti-neutrino, CTEQ6.6, deFlorian-Sassot 03 NLO

Neutrino, CTEQ6.6, HKN07 NLOAnti-neutrino, CTEQ6.6, HKN07 NLO

Figure3:Com parison ofthenuclearcorrectionsto thereduced crosssectionsforinclusiveCHO -

RUS data (left) and for NuTeV charm production data (right) from the de Florian-Sassot [35]

and HK N07 [36]param etrization.Thecorrection isshown forindividualexperim entaldata points,

versusthe pointlabel(arbitrary order).

thism eansthatwe willretain the fullm c dependence in Eq.(1),and in the expressions

forthe structurefunctionsF�;c

i Eq.(39)allconvolutionsare de�ned as

[C q]�x;Q

2�= �

�W

2� m

2c

�Z 1

�c

dy

yC�y;�s

�Q2��q

��c

y;Q

2

�

: (14)

Theim pactofthistreatm entofthecharm m assisshown in Fig.2,wherewecom pare

a NLO determ ination ofthedim uon crosssection Eq.(3)within theZM -VFN and I-ZM -

VFN schem es,based on ourpreviousNNPDF1.0 parton set.Thesuppression ofthecross

section atsm allx dueto�nitequarkm assisapparentfrom thisplot.Clearly,theinclusion

ofquark m asse�ectsonly in the determ ination ofthe dim uon crosssection,and then in

theI-ZM -VFN schem e,isan approxim ation.Thisapproxim ation willlead to a system atic

uncertainty in our determ ination ofthe strange PDFs and ofCK M m atrix elem ents in

thenextsections.W ewillestim ate thisuncertainty by com paring resultsobtained in the

ZM -VFN and I-ZM -VFN schem e:asthefullG M schem eisactually in between thesetwo,

this provides a rather conservative overestim ate ofthe associated uncertainty. W e will

then seethatthissystem aticuncertainty isactually sm allin com parison to thestatistical

uncertainty on strangenessand associated observables.

3.4 N uclear C orrections

Neutrino data areobtained from deep-inelastic scattering o� a nucleartarget:forNuTeV

essentially Fe,A N uTeV = 49:6 [25],and forCHO RUS (whose inclusive structure function

m easurem ents are also included in our data set) Pb,A chorus = 207,[7]. Therefore,a

suitable nuclear correction should be introduced in order to obtain from these data a

determ ination ofthePDFsoffreenucleons.

Nuclearcorrectionshave been determ ined by variousgroups[35{38],using m odelsof

nuclearstructure.Thecorrection

R A

�F�2(x;Q

2)��

F�;A

2 (x;Q 2)

AF�;p

2 (x;Q 2); (15)

14

x-210

-110]2

[GeV

2Q

1

10

210

]ν 2[F

AR

0.8

0.9

1

1.1

1.2

1.3

]ν[F

R

0.8

0.9

1

1.1

1.2

1.3

CHORUS inclusive

]ν[F

R

0.8

0.9

1

1.1

1.2

1.3

CHORUS inclusive

x-210

-110]2

[GeV

2Q

1

10

210

]ν 2[F

AR

0.8

0.9

1

1.1

1.2

1.3

]ν[F

R

0.8

0.9

1

1.1

1.2

1.3

NuTeV charm

]ν[F

R

0.8

0.9

1

1.1

1.2

1.3

NuTeV charm

Figure 4:Nuclearcorrectionsto the neutrino structure function F �2 forinclusive CHO RUS data

(left)and forNuTeV charm production data (right)from the K ulagin-Pettiparam etrization [37].

The correction isshown in the fullkinem atic region relevantforboth experim ents.

to the reduced cross sections ��(��)and ��(��);c,obtained using the param etrizations of

Refs.[35,36],aredisplayed in Figs.4 fortheexperim entalCHO RUS inclusiveand dim uon

NuTeV data.Itisapparentthatcorrectionsobtained using di�erentm odelscan besignif-

icantly di�erent,butthey areallquitesm all.Forthisreason,nuclearcorrectionswerenot

used in the NNPDF1.0 �t[21]. In the NNPDF1.2 �tpresented here we willnotinclude

nuclearcorrectionsin ourbaseline�t,but,in orderto determ inetheassociated system atic

uncertainty,wewillrepeatthe�twith thenuclearcorrectionscom puted using them odels

ofRefs.[35,36],which providecorrectionsto theparton distributions.Thedependenceof

thenuclearcorrection on thekinem aticvariablesisshown in Fig.4in thekinem aticregion

and for A values relevant for CHO RUS and NuTeV data,using the m odelofRef.[37],

which instead providesdirectly a correction to thestructurefunction.

15

�(0)

i;��(0)

i;g �(0)

i;T3�(0)

i;V T�(0)

i;� S�(0)

i;s+�(0)

i;s�N m axite r� N cop E sets N update

[10;1] [10;1] [1;0:1] [1;0:1] [1;0:1] [5;0:5] [1;0:1] 5000 1/3 120 3 10

Table 4:Param eterscontrolling the genetic algorithm m inim ization.Since we work with

N m ut = 2 thereare two entriesin each colum n forthevaluesof�(0).

4 R esults

In thissection wepresenttheNNPDF1.2 parton set.Afterdiscussing thegeneralfeatures

ofthe �t and its result,and com paring these to the previous NNPDF parton set, we

discussin detailthedeterm ination ofthestrangeand antistrangedistributions,which are

thenovelfeaturesofthis�t.W e�nally com pareresultsto experim entaldata,including a

com parison with therecent[26]determ ination ofthe F3 structurefunction.

4.1 T he N N PD F1.2 parton set: statisticalfeatures

W e have produced a set ofN rep = 1000 replicas ofseven PDFs,each determ ined as an

optim al�t to one of the M onte Carlo replicas obtained from the data set of Sect. 2.

W e have used the genetic algorithm m inim ization and a cross-validation m ethod forthe

determ ination oftheoptim al�t,according to them ethod presented in Sect.4 ofRef.[21].

The param eters ofthe genetic algorithm are sum m arized in Table 4;they coincide with

those used in Ref.[21]forthe �vePDFsalready presentin that�t.

Thegeneralstatisticalfeaturesofour�nalparton setaresum m arized in Tables5-6,to

becom pared with thecorrespondingtables(Tables7-8)ofRef.[21],wherealltherelevant

quantitiesarede�ned (note thataverage uncertaintiesarenow given in percentage value,

whilethey were given asabsolute valuesin Ref.[21]).

Thestatisticalfeaturesofthe �tcan besum m arized asfollows:

� The generalfeatures ofthe total�t(Tab.5)are essentially indistinguishable from

thoseofRef.[21],and thecom m entswem adethen stillapply.Thesam eistruefor

the features ofthe �tto individualexperim ents (Tab.6) when these were already

included in the dataset ofRef.[21]. This stability upon the addition oftwo new

independentPDFs(thus74 extra free param eters)and a random ization ofthepre-

processing exponents supportsthe reliability ofthe resultsobtained in NNPDF1.0

forallPDFswhich weredeterm ined there.

� Thequality ofthe�tto thenew HERA IIdata iscom parableto thatto theBCDM S

data,and som ewhatworsethan thatofthe�tto otherHERA data.Thesenew data

m ostly probe the large x region,like BCDM S and unlike other HERA data (see

Fig.1),and are generally ratherprecise,also like BCDM S and unlike otherHERA

data (see Tab.1 and Tab.1 ofRef.[21]).Thissom ewhatlargervalue ofthe �2 for

largex high precision data,though com patiblewith statistical uctuationsand with

the theoreticalerror related to the use ofNLO perturbation theory,m ay suggest

som e m inordata incom patibility in thisregion.

� The �2 ofthe �tto dim uon data israthersm allerthan one. Thisisa consequence

ofthe fact that,as discussed in Sec.2.1,correlations have not been included for

16

�2tot 1.31

hE i 2.80

hE tri 2.75

hE vali 2.80

hTLi 1024�(exp)

�

dat11.0%

�(net)

�

dat4.0%

�(exp)

�

dat0.15

�(net)

�

dat0.32

cov(exp)

�

dat1:6 10� 3

cov(net)

�

dat6:1 10� 3

Table 5:Statisticalestim atorsforthe �nalPDF setwith N rep = 1000 forthe totaldata set.

E xperim ent �2tot

hE i

D

�(exp)

E

dat

D

�(net)

E

dat

D

�(exp)

E

dat

D

�(net)

E

dat

D

cov(exp )

E

dat

D

cov(net)

E

dat

SLA C 1.27 3.32 4.2% 2.6% 0.31 0.63 3:1 10� 5

2:7 10� 5

B C D M S 1.57 3.14 5.7% 4.5% 0.47 0.51 2:9 10� 5

1:0 10� 5

N M C 1.70 3.09 4.9% 2.3% 0.16 0.62 4:4 10� 4

3:8 10� 5

N M C -pd 1.46 3.12 1.7% 1.7% 3:3 10� 2

0.36 6:5 10� 6

6:0 10� 5

ZE U S 1.07 2.64 13% 3.9% 7:9 10� 2

0.26 1:5 10� 4

2:9 10� 5

H 1 1.03 2.52 12% 3.3% 2:7 10� 2

0.25 4:9 10� 2

2:7 10� 5

C H O R U S 1.37 2.88 15% 3.7% 9:4 10� 2

0.27 2:2 10� 3

3:8 10� 4

F LH 108 1.67 2.56 72% 5.7% 0.65 0.76 2:0 10� 2

2:5 10� 4

N uTeV D im uon 0.62 2.62 21% 22% 0.03 0.50 1:7 10� 3

1:7 10� 4

ZE U S-H E R A -II 1.51 2.90 14% 2.5% 0.29 0.34 6:9 10� 4

3:2 10� 5

Table6:Statisticalestim atorsforthe�nalPDF setwith N rep = 1000 forindividualexperim ents.

these data because the covariance m atrix isnotavailable.Theaverage value ofthe

�2 we obtain isin good agreem entwith thatexpected on the basisofthe\e�ective

num berofdegreesoffreedom " published in Ref.[25],and with other�tsto thesam e

data [14].

� The uncertainty ofthe �t to dim uon data,as m easured by the average standard

deviation h�i is very close to the uncertainty ofthe data,unlike that ofallother

data sets (re ected by the results for the total �t), where the �t uncertainty is

m uch sm aller than the data uncertainty (4% vs. 11% for the total�t). This is a

consequence ofthe factthatdim uon data have little redundancy,and are sensitive

to strangeness,to which otherdata are essentially insensitive;while allother data

have a very large redundancy,especially low-x HERA data which depend m ainly

on the quark singletand gluon.Thise�ectcan also be observed in the com parison

between experim entaldata and NNPDF1.2 predictionsofFig.15.

� Theaveragecorrelation isvery low forthedim uon data,becausetheonly correlated

system atics is norm alization. However,the �t to these data does display a corre-

lation ofthe sam e order ofm agnitude as for other data,re ecting the underlying

sm oothnessofparton distributions.

17

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

) 02 (

x, Q

Σx

0

1

2

3

4

5

6

7

8

9NNPDF1.2

NNPDF1.1

NNPDF1.0

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) 02 (

x, Q

Σx

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1NNPDF1.2

NNPDF1.1

NNPDF1.0

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

) 02xg

(x,

Q

-4

-2

0

2

4

6

8NNPDF1.2

NNPDF1.1

NNPDF1.0

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) 02xg

(x,

Q

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7NNPDF1.2

NNPDF1.1

NNPDF1.0

Figure5:Thesingletand gluon PDF atthestarting scaleQ 20 = 2 G eV 2,plotted versusx on a log

(left)orlinear(right)scale.ThePDFsfrom theprevioussetsNNPDF1.0 [21]and NNPDF1.1 [23]

are also shown forcom parison.Note thatwhile the PDFsfrom NNPDF1.2 and NNPDF1.0 have

been com puted with N rep = 1000,those ofNNPDF1.1 useN rep = 100 only.

18

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

) 02xV

(x,

Q

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6NNPDF1.2

NNPDF1.1

NNPDF1.0

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) 02xV

(x,

Q

0

0.2

0.4

0.6

0.8

1

1.2 NNPDF1.2

NNPDF1.1

NNPDF1.0

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

) 02

(x,

Q3

xT

0

0.1

0.2

0.3

0.4

0.5

0.6NNPDF1.2

NNPDF1.1

NNPDF1.0

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) 02

(x,

Q3

xT

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5NNPDF1.2

NNPDF1.1

NNPDF1.0

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

)02

(x,

QS

∆x

0

0.02

0.04

0.06

0.08

0.1

NNPDF1.2

NNPDF1.1

NNPDF1.0

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

)02

(x,

QS

∆x

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08NNPDF1.2

NNPDF1.1

NNPDF1.0

Figure6:Sam easFig.5,butforthe valenceand nonsingletPDFs.

19

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

) 02 (

x, Q

+xs

-1

-0.5

0

0.5

1

1.5

2NNPDF1.2

NNPDF1.1

NNPDF1.0

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) 02 (

x, Q

+xs

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3NNPDF1.2

NNPDF1.1

NNPDF1.0

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

)02

(x,

Q-

xs

-0.2

-0.15

-0.1

-0.05

-0

0.05

0.1

0.15

0.2NNPDF1.2

NNPDF1.1

NNPDF1.0

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

)02

(x,

Q-

xs

-0.15

-0.1

-0.05

0

0.05

0.1

0.15NNPDF1.2

NNPDF1.1

NNPDF1.0

Figure 7: Sam e as Fig.5,but for the strange sector PDFs. Note that in NNPDF1.0 s� were

assum ed to be respectively s+ (x;Q 20)=

1

2

��u + �d

�and s� (x;Q 2

0)= 0.

20

4.2 T he N N PD F1.2 parton set: parton distributions

TheNNPDF1.2 setofparton distributionsatthestarting scaleQ 20 = 2 G eV 2 isdisplayed

in Figs.5-7,and com pared to theprevioussetsNNPDF1.0 [21]and NNPDF1.1 [23].The

distances (de�ned as in Ref.[21]) between each pair of these three sets are shown in

Table 7.

ThegeneralfeaturesofthisPDF setand itscom parison to the previousNNPDF sets

are thefollowing

� In thesingletsector,thereisvery littledi�erencein centralvaluesand uncertainties

between theNNPDF1.2 and NNPDF1.0parton sets:thedistancebetween thesetsis

com patiblewith statistical uctuations.TheNNPDF1.1,which had an independent

param etrization forthestrangedistribution withoutany datatoconstrain itdisplays

an increasein theuncertainty ofthequark singletdueto thisunconstrained strange

contribution.

� The isospin tripletand the sea asym m etry are the sam e in allNNPDF setswithin

uctuations. The totalvalence hasthe sam e centralvalue in allsetswithin uctu-

ations,and the sam e uncertainty in the NNPDF1.2 and NNPDF1.1 sets,while the

uncertainty on itwassom ewhatunderestim ated in NNPDF1.0.Thisunderestim ate

ofthe NNPDF1.0 valence uncertainty was already singled out based on a statisti-

calstability analysis in Sect.5.4 ofRef.[21],where it was suggested thatit could

be cured by a random ization ofthe preprocessing exponents in Eq.(4). Thisran-

dom ization hasbeen im plem ented in NNPDF1.1 and NNPDF1.2,which indeed have

som ewhatlargervalenceuncertainty,com patiblewith each other.Thisisdespitethe

factthatthe strange contribution to the totalvalence isa�ected by a m uch larger

uncertainty in NNPDF1.1 than in NNPDF1.2.

� The centralvalue and uncertainty on the strange distributionsEq.(5)are com pat-

ible with those ofNNPDF1.1,where strangeness was independently param etrized

but essentially unconstrained by data, whereas they are incom patible with those

ofNNPDF1.0,where strangeness was determ ined by the assum ptionss+ (x;Q 20)=

12

��u + �d

�and s� (x;Q 2

0)= 0.Thism eansthatthissim ple assum ption,though per-

hapsnottoo faro�,is insu�cientto determ ine the strange distribution within its

stated accuracy. This conclusion was also reached recently in Ref.[13]. The un-

certainty on strangenessaswe determ ine ithere turnsoutto be ratherlargerthan

thatinduced by theNNPDF1.0 assum ption,butm uch sm allerthan thatobtained in

NNPDF1.1 in theabsenceofdim uon data.Itisthuspossibletodeterm inetheshape

ofs+ with reasonable accuracy. However,ourdeterm ination ofs� turnsoutto be

com patible with the NNPDF1.0 assum ption that s� (x;Q 20)= 0. W e shalldiscuss

the featuresofthe strange distribution in greaterdetailin Sect.4.4 below.

4.3 T heoreticaluncertainties

Asdiscussed in Sects.3.3-3.4,dim uon dataarepotentially sensitivetothetreatm entofthe

quark m ass,and neutrino data in generalare potentially sensitive to nuclearcorrections.

In order to explore this sensitivity, we have repeated the NNPDF1.2 �t using also for

21

N N P D F 1.2 vs. N N P D F 1.1

D ata E xtrapolation

�(x;Q20) 5 10

� 4� x � 0:1 10

� 5� x � 10

� 4

hd[q]i 2.7 1.2

hd[� ]i 3.1 1.8

g(x;Q20) 5 10

� 4� x � 0:1 10

� 5� x � 10

� 4

hd[q]i 2.4 2.0

hd[� ]i 1.3 1.4

T3(x;Q20) 0:05 � x � 0:75 10

� 3� x � 10

� 2

hd[q]i 1.5 0.9

hd[� ]i 1.1 1.2

V (x;Q20) 0:1 � x � 0:6 3 10

� 3� x � 3 10

� 2

hd[q]i 1.1 1.0

hd[� ]i 1.3 1.4

� S (x;Q20) 0:1 � x � 0:6 3 10

� 3� x � 3 10

� 2

hd[q]i 0.8 0.8

hd[� ]i 1.3 1.1

s+(x;Q

20) 5 10

� 4� x � 0:1 10

� 5� x � 10

� 4

hd[q]i 2.0 1.6

hd[� ]i 4.5 1.8

s� (x;Q 2

0) 0:1 � x � 0:6 3 10� 3

� x � 3 10� 2

hd[q]i 1.1 1.3

hd[� ]i 6.1 4.6

N N P D F 1.1 vs. N N P D F 1.0

D ata E xtrapolation

�(x;Q20) 5 10

� 4� x � 0:1 10

� 5� x � 10

� 4

hd[q]i 1.6 0.9

hd[� ]i 4.0 2.3

g(x;Q20) 5 10

� 4� x � 0:1 10

� 5� x � 10

� 4

hd[q]i 2.3 1.7

hd[� ]i 1.6 1.2

T3(x;Q20) 0:05 � x � 0:75 10

� 3� x � 10

� 2

hd[q]i 1.6 0.8

hd[� ]i 1.8 3.4

V (x;Q20) 0:1 � x � 0:6 3 10

� 3� x � 3 10

� 2

hd[q]i 1.8 1.7

hd[� ]i 5.3 5.2

� S (x;Q20) 0:1 � x � 0:6 3 10

� 3� x � 3 10

� 2

hd[q]i 1.2 1.0

hd[� ]i 1.6 1.1

s+(x;Q

20) 5 10

� 4� x � 0:1 10

� 5� x � 10

� 4

hd[q]i 1.0 1.0

hd[� ]i 5.4 2.3

s� (x;Q 2

0) 0:1 � x � 0:6 3 10� 3

� x � 3 10� 2

hd[q]i 1.1 1.3

hd[� ]i 7.4 4.6

N N P D F 1.2 vs. N N P D F 1.0

D ata E xtrapolation

�(x;Q 20) 5 10� 4

� x � 0:1 10� 5� x � 10� 4

hd[q]i 3.2 1.9

hd[� ]i 2.9 3.3

g(x;Q20) 5 10

� 4� x � 0:1 10

� 5� x � 10

� 4

hd[q]i 1.7 0.9

hd[� ]i 1.6 1.3

T3(x;Q20) 0:05 � x � 0:75 10� 3

� x � 10� 2

hd[q]i 1.1 1.0

hd[� ]i 2.0 3.2

V (x;Q20) 0:1 � x � 0:6 3 10

� 3� x � 3 10

� 2

hd[q]i 2.6 2.4

hd[� ]i 5.3 4.9

� S (x;Q20) 0:1 � x � 0:6 3 10� 3

� x � 3 10� 2

hd[q]i 1.4 0.9

hd[� ]i 1.5 1.2

s+(x;Q

20) 5 10

� 4� x � 0:1 10

� 5� x � 10

� 4

hd[q]i 6.2 3.7

hd[� ]i 5.7 3.8

s� (x;Q 2

0) 0:1 � x � 0:6 3 10� 3

� x � 3 10� 2

hd[q]i 1.3 1.2

hd[� ]i 6.8 6.5

Table7:Distancebetween theNNPDF1.0,NNPDF1.1 and NNPDF1.2 parton sets.Alldistances

arecom puted from a setofN rep = 100 replicas.

dim uon data the ZM -VFN schem e (as in Ref.[21]) instead ofthe im proved I-ZM -VFN

quark m asstreatm entdiscussed in Sect.3.3 and used forthe defaultNNPDF1.2 �t(the

ZM -VFN isused forallotherdata anyway).Thedistancesbetween resultsthusobtained

aredisplayed in Tab.8.Itisapparentthatthereisa certain changein thecentralvalueof

thestranges+ distribution in theregion ofthedata,oforderofaboutten,which,with 100

replicas,m eansthatthe centralvalue hasm oved by about1:4� in unitsofthe standard

deviation.Theuncertainty on s+ itself,and thecentralvalueofthesingletdistribution in

theregion ofthedata area�ected to a lesserextent,whileallotherPDFsareuna�ected.

Thus the charm m ass corrections displayed in Fig.2 have a sm allbut noticeable e�ect

on the determ ination ofthe totalstrange s+ distribution. O ur approxim ate treatm ent

willcorrespondingly be a source ofsystem atics,which we shalltake into account when

discussing quantitiesrelated to strangeness.

In orderto study thesensitivity to thenuclearcorrectionsdisplayed in Fig.3 wehave

22

ZM D e Florian-Sassot H K N 07

D ata Extrapolation D ata Extrap. D ata Extrap. .

�(x;Q 20) 5 10� 4

� x � 0:1 10� 5� x � 10� 4

hd[q]i 5.2 1.0 2.3 1.4 2.3 0.9

hd[�]i 2.5 1.6 1.5 1.2 1.2 1.1

g(x;Q 20) 5 10� 4

� x � 0:1 10� 5� x � 10� 4

hd[q]i 1.4 1.5 1.2 1.0 1.4 1.1

hd[�]i 1.8 1.5 1.2 1.2 1.2 1.4

T3(x;Q20) 0:05 � x � 0:75 10� 3

� x � 10� 2

hd[q]i 1.4 2.0 1.3 1.0 1.0 1.0

hd[�]i 2.9 0.9 1.4 1.5 1.1 1.1

V (x;Q 20) 0:1 � x � 0:6 3 10� 3

� x � 3 10� 2

hd[q]i 1.2 1.2 1.3 1.2 0.8 0.7

hd[�]i 1.5 1.1 1.3 1.5 1.3 0.9

� S (x;Q20) 0:1 � x � 0:6 3 10� 3

� x � 3 10� 2

hd[q]i 2.1 2.3 0.8 1.0 1.1 1.0

hd[�]i 1.1 1.1 1.2 1.3 1.0 1.3

s+ (x;Q 20) 5 10� 4

� x � 0:1 10� 5� x � 10� 4

hd[q]i 9.4 1.1 2.1 1.5 1.6 1.1

hd[�]i 3.4 1.6 1.5 1.0 1.5 1.0

s� (x;Q 20) 0:1 � x � 0:6 3 10� 3

� x � 3 10� 2

hd[q]i 0.9 0.9 1.0 1.1 1.3 1.1

hd[�]i 1.4 1.2 1.0 1.0 1.4 0.9

Table 8: Distancesbetween PDFscom puted from a setofN rep = 100 replicasfrom the default

NNPDF1.2set,and 100replicasobtainedusingaZM -VFN schem einsteadofthedefaultI-ZM -VFN

schem e ofSect.3.3,orintroducing nuclearcorrectionscom puted using the de Florian-Sassot[35]

and HK N07 [36]m odels.

repeated the NNPDF1.2 �twith allneutrino data corrected fornucleare�ectsaccording

to the m odels of de Florian-Sassot [35]and HK N07 [36]. The distances tabulated in

Tab.8 show thatthee�ectofnuclearcorrectionsisnegligible:�tswith orwithoutnuclear

correctionsdi�erby an am ountwhich iscom patible with statistical uctuations.

4.4 D eterm ination ofthe strange distribution

The determ ination ofthe strange and antistrange PDFs is problem atic because ofthe

scarcenessoftheexperim entalinform ation on thesequantities,which m akesitdi�cultto

separatethegenuineinform ation from theoreticalbias,asituation which ourm ethodology

is especially suited to dealwith. In previous parton �ts,a range ofpossible shapes of

the strange PDFswasexplored by assum ing di�erentfunctionalform sand studying the

variation ofresults[13].

Thes� (x;Q 20),s(x;Q

20)and �s(x;Q

20)strangePDFsEq.(5)areshown attheinputscale

in Fig.8,wheretheyarealsocom pared tothem ostrecentCTEQ 6.6[17]and M STW 08[15]

sets.W hereasthe CTEQ collaboration hasnotperform ed a fulldeterm ination ofthe s�

uncertainty band,a study ofthe dependence ofthe best-�t s� on assum ptions on its

functionalform was perform ed in Ref.[13]: severalofthe corresponding resultsare also

shown in Fig.8.Forgreaterclarity,in Fig.9 wealso plottheuncertaintieson thesePDFs.

In the data region x �> 0:03 alldeterm inations ofs� agree,however the NNPDF1.2

hasa m uch largeruncertainty than otherexisting determ inations.The origin ofthiscan

23

x-410 -310 -210 -110 1

) 02 (

x, Q

+xs

-0.2

0

0.2

0.4

0.6

0.8

1CTEQ6.6MSTW08

NNPDF1.0

NNPDF1.2

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) 02 (

x, Q

+xs

0

0.05

0.1

0.15

0.2

0.25

0.3CTEQ6.6MSTW08

NNPDF1.0

NNPDF1.2

x-410 -310 -210 -110 1

)02

(x,

Q-

xs

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04CTEQ6.5S-0CTEQ6.5S-1CTEQ6.5S-2MSTW08NNPDF1.0NNPDF1.2

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

)02

(x,

Q-

xs

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04CTEQ6.5S-0CTEQ6.5S-1CTEQ6.5S-2MSTW08NNPDF1.0NNPDF1.2

x-410 -310 -210 -110 1

) 02xs

(x,

Q

-0.2

0

0.2

0.4

0.6

0.8

1CTEQ6.6MSTW08

NNPDF1.0

NNPDF1.2

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) 02xs

(x,

Q

0

0.05

0.1

0.15

0.2

0.25

0.3CTEQ6.6MSTW08

NNPDF1.0

NNPDF1.2

x-410 -310 -210 -110 1

) 02 (

x, Q

sx

-0.2

0

0.2

0.4

0.6

0.8

1CTEQ6.6MSTW08

NNPDF1.0

NNPDF1.2

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) 02 (

x, Q

s x

0

0.05

0.1

0.15

0.2

0.25

0.3CTEQ6.6MSTW08

NNPDF1.0

NNPDF1.2

Figure8:From top to bottom ,the strangeC-even and C-odd com binationss+ (x;Q 20),s

� (x;Q 20)

Eq.(5)and thecorrespondingstranges(x;Q 20)and antistrange�s(x;Q

20)PDFs,plotted attheinput

scale versus x on a log (left) or linear(right) scale,com puted from the �nalset ofN rep = 1000

replicas. The NNPDF1.2 resultiscom pared to the M STW 08 [15]and CTEQ 6.6 [17]global�ts.

Fors� som eofthe resultsobtained from the CTEQ 6.5sstrangenessseries[13]arealso shown.24

x0.1 0.2 0.3 0.4 0.5 0.6 0.7

)2 0

(x,Q

+)

)/s

2 0(x

,Q+

( s

σ

-1.5

-1

-0.5

0

0.5

1

1.5 CTEQ6.6MSTW08

NNPDF1.0

NNPDF1.2

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

) )

2 0(x

,Q-

( s

σ

-0.03

-0.02

-0.01

0

0.01

0.02

0.03 MSTW08

NNPDF1.2

Figure9:Theuncertainty on thestrangePDFss� (x;Q 20)shown in Fig.8.Allbandscorrespond

toone�.Therelativeuncertainty isshown fors+ (left)and theabsoluteuncertainty fors� (right).

be understood by looking at Fig.10,where we display 25 random ly chosen replicas out

ofour fullset,and the m ean and standard deviation com puted from them : clearly,our

large uncertainty isa consequence ofthe great exibility a�orded by the neuralnetwork

param etrization.Thisisparticularly noticeablein thecaseofs� ,which m usthaveatleast

onenodebecauseofthesum ruleEq.(9):individualreplicascrossthex{axisin di�erent

places,with di�erentsign (from positivetonegativeorconversely),and som ereplicashave

m orethan one crossing.Itisinteresting to observethatthe\neck" in theuncertainty on

s� around x � 0:1 correspondsto the value ofx at which the crossing is m ostlikely to

occur. The role played by the valence sum rule Eq.(9)in determ ining these features of

thestrangenessasym m etry s� can beelucidated by repeating the�twithoutim posing it.

The results,displayed in Fig.11,show that even withoutthe sum rule constraint m any

replicasstillcrossthethe x{axis.

Another theoretical constraint which m ay help in reducing uncertainties is that of

positivity of cross sections. For instance, as in Ref.[21], in the determ ination of the

NNPDF1.2 PDF setwehaveim posed positivity ofthestructurefunction FL atlow x and

Q 2,which helpsin reducing the uncertainty ofthe gluon distribution atthe edge ofthe

HERA data region. In view ofthe factthat(see Fig.8)both s(x;Q 2o)and �s(x;Q 2

o)can

turn negative to within one sigm a forx �< 10� 2,and also in the large x �

> 0:2 region,one

m ay wonderwhetherim posingpositivity ofthedim uon crosssection m ightlikewisehelp in

reducingtheuncertainty on thestrangeand antistrangedistributions.In ordertotestthis,

in Fig.12 wedisplay thetotaldim uon crosssection,both attheinitialQ 2 = Q 20 = 2 G eV 2

and atthetypicalscaleoftheNuTeV dataQ 2 = 20G eV 2,com puted usingtheNNPDF1.2

PDFsofFig.8.Thecrosssection only becom essigni�cantly negative atlow Q 2 and very

low x �< 10� 5. For antineutrinos,it also becom e som ewhat negative at large x: at the

scale ofthe large x data Q 2�> 20 G eV 2 forx �

> 0:3. W e conclude thatthe constraintof

positivity only a�ects physicalobservablesquite farfrom the data region. W e have thus

notim posed thisconstraintin thecurrent�t.Itm ightbeworth im plem enting itin future

�tswhich includeDrell-Yan data,asthesecould furtherconstrain strangeness,especially

atlarge x.

Furtherconstraintscould bebased on theoreticalexpectations:forexam ple,one m ay

25

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1e-05 0.0001 0.001 0.01 0.1 1

s+(x

,Q2 =

2 G

eV2 )

x

NNPDF1.2, Nrep=25

Central value1-σ range

Individual replicas

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

s+(x

,Q2 =

2 G

eV2 )

x

NNPDF1.2, Nrep=25

Central value1-σ range

Individual replicas

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.0001 0.001 0.01 0.1 1

xs- (x

,Q2 =

2 G

eV2 )

x

NNPDF1.2, Nrep=25

Central value1-σ range

Individual replicas

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 0.2 0.4 0.6 0.8 1

xs- (x

,Q2 =

2 G

eV2 )

x

NNPDF1.2, Nrep=25

Central value1-σ range

Individual replicas

Figure10:A setofrandom ly chosen N rep = 25 replicasofthestrangePDFss+ (x;Q 20),s

� (x;Q 20)

outofthe fullsetofFig.8,and the PDFscom puted from them .

expect the strange PDF to be sm aller than the light quark valence PDFs;indeed,the

system atic im plem entation of theoretical or m odelconstraints in parton �ts has been

advocated e.g. in Ref.[39]. However,expectations based on m odelsofthe nucleon have

often turned out to be in disagreem ent with experim ent: for instance,in the polarized

case the strange distribution turnsoutto be unexpectedly large and in fact larger than

the up distribution (see e.g. Ref.[40]). To obtain reliable phenom enology,such as the

determ ination ofelectroweak param eterstobediscussed below,wepreferthereforetoonly

rely on exactconstraints,such asthe valence sum ruleorpositivity.

The features ofthe strange distributions which are m ost interesting for physics ap-

plications (as we shalldiscussin m ore detailin Section 5) are the m om entum fractions,

de�ned as�S��(Q 2)�

Z 1

0

dxxs� (x;Q 2); (16)

with sim ilar de�nitionsform om entsofotherPDF com binations,and in particulartheir

26

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.01 0.1 1

xs- (x

,Q2 =

2 G

eV2 )

x

Nrep=25

No strange SR

Central value1-σ range

Individual replicas

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 0.2 0.4 0.6 0.8 1

xs- (x

,Q2 =

2 G

eV2 )

x

Nrep=25

No strange SR

Central value1-σ range

Individual replicas

Figure 11:Sam e asthe lowerrow ofFig.10 when the sum rule Eq.(9)isnotim posed.

Analysis Reference K S

�Q 2 = 20G eV 2

�

NNPDF1.2 Thiswork 0:71+ 0:19� 0:31

M STW 08 [15] 0:56� 0:03

CTEQ 66 [17] 0:72� 0:05

AK P08 [14] 0:59� 0:08

Table 9: The relative strange m om entum fraction K S(Q2)Eq.(17),asdeterm ined from various

parton sets.Alluncertaintiescorrespond to 68% con�dencelevels.

ratio to thelightsea orrespectively lightvalence m om entum fractions:

K S(Q2) �

R10dx x s+

�x;Q 2

�

R10dx x

��u(x;Q 2)+ �d(x;Q 2)

� =[S+ ]

��U + �D

� ; (17)

R S(Q2) � 2

R10dxxs� (x;Q 2)

R10dxx(u� (x;Q 2)+ d� (x;Q 2))

= 2[S� ]

[U � + D � ]: (18)

In m any parton �ts,including theNNPDF1.0 �t,thesequantitiesaretaken to be�xed at

thestartingscale:thevalueoftherelativetotalstrangem om entum (som etim esalsocalled

strange suppression)is,since the earliestm easurem ents,taken to be [41]K S(Q20)� 0:5,

whilethe strange asym m etry isassum ed to vanish,i.e.R S(Q20)= 0.

Thevalueand uncertainty on thesequantitiescan bedeterm ined from theNNPDF1.2

setbyperform ingaveragesoverreplicaPDFs[21],which forK S and R S willnotnecessarily

coincidewith theratio ofaverage PDFs,becauseEqs.(17-18)arenotlinearin thePDFs.

In fact,because the denom inator in Eq.(17)can assum e rathersm allvalues,we expect

thatthe distribution ofvaluesofthe totalstrange fraction K S can be ratherasym m etric

and non-gaussian. The probability distribution of K S at Q 2 = 20 G eV 2 is shown in

Fig.13,and turnsoutto be indeed quite farfrom gaussian. Therefore,we com pute the

one-� uncertainty asacentral68% con�denceintegral,nam elyrequiringthetwooutertails

ofthe probability distribution (lighterblue region in Fig.13)to each correspond to 16%

probability,with thecentralvaluestillgiven by theaverage.Theresultwethusobtain for

27

0

10

20

30

40

50

60

70

0.001 0.01 0.1

Aν

* d2 σν,

c (x,y

,Eν)

/dx/

dy

x

NNPDF1.2, Q2 = 20 GeV2, y = 0.4

Neutrino - central valueNeutrino - 1-σ range

AntiNeutrino - central valueAntiNeutrino - 1-σ range

0

5

10

15

20

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Aν

* d2 σν,

c (x,y

,Eν)

/dx/

dy

x

NNPDF1.2, Q2 = 20 GeV2, y = 0.4

Neutrino - central valueNeutrino - 1-σ range

AntiNeutrino - central valueAntiNeutrino - 1-σ range

-10

0

10

20

30

40

50

0.001 0.01 0.1

Aν

* d2 σν,

c (x,y

,Eν)

/dx/

dy

x

NNPDF1.2, Q2 = 2 GeV2, y = 0.4

Neutrino - central valueNeutrino - 1-σ range

AntiNeutrino - central valueAntiNeutrino - 1-σ range

0

5

10

15

20

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Aν

* d2 σν,

c (x,y

,Eν)

/dx/

dy

x

NNPDF1.2, Q2 = 2 GeV2, y = 0.4

Neutrino - central valueNeutrino - 1-σ range

AntiNeutrino - central valueAntiNeutrino - 1-σ range

Figure 12:The totalneutrino and antineutrino dim uon crosssectionsatthe starting scaleQ 20 =

2 G eV 2 (lowerrow)and atthe \NuTeV" scale Q 20 = 20 G eV 2 (upperrow),plotted versusx on a

log (left)orlinear(right)scale.

28

]D + U ] / [ + = [ SSK0 0.5 1 1.5 2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

= 1000rep

, N2 = 20 GeV2NNPDF1.2, Q = 1000rep

, N2 = 20 GeV2NNPDF1.2, Q

Figure 13: Probability distribution ofK S atQ 2 = 20 G eV 2 com puted from the reference setof

N rep = 1000NNPDF1.2PDF replicas.Thecentralcross-hatched region correspondsto thecentral

68% con�denceinterval,K S

�Q 2 = 20G eV

2�= 0:71+ 0:19� 0:31

stat.

the expected K S and its uncertainty are shown in Table 9,along with the resultsfound

using other parton sets. The m edian ofthe probability distribution is equalto K m edS =

0:59,signi�cantly di�erentfrom the average because ofthe asym m etry. The NNPDF1.2

uncertainty ism uch largerthan thatfound in other�ts,forthe reasonsdiscussed above.

Notethat,however,allvaluesareessentially consistentwith thesim pleassum ption K S =

0:5 used in olderparton �ts.

In the case of the strange m om entum asym m etry R S Eq.(18) the denom inator is

�xed by knowledgeofthevalencecontentofthenucleon,which isknown quiteaccurately:

hence we expect the uncertainty to be sym m etric and dom inated by uncertainty ofthe

num erator.Indeed,theprobability distribution forR S,shown in Fig.14,turnsoutto be

approxim ately gaussian sothattheuncertainty com puted from thecentral68% con�dence

essentially coincides with the standard deviation ofthe distribution,while centralvalue

and uncertainty forR S areessentially proportionalto thoseofthestrangenessasym m etry

[S� ].Thislatterquantity hasbeen determ ined byvariousgroups,atvariousscales:several

oftheseresultsarecollected in Table10 and com pared to ourown.Resultsaregiven both

atthescaleatwhich they weredeterm ined,and then alsowhen evolved toacom m on scale,

exploiting thefactthatatNLO (though notatNNLO [43])[S� ]evolvesm ultiplicatively.

In thiscase,too,the NNPDF1.2 uncertainty ism uch largerthan thatobtained in other

�ts:whileforallothergroupsthereisan indication thata positivevalueof[S � ]isfavored

(allresultsbeing neverthelesscom patiblewith zero),thisindication loosesitssigni�cance

in ouranalysisdueto thevery large uncertainty.

In thenextsection we willsee that,surprisingly,even with such large uncertaintiesit

29

]- + D- ] / [ U- = 2 [ SS R-0.1 -0.05 0 0.05 0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

= 1000rep

, N2 = 20 GeV2NNPDF1.2, Q = 1000rep

, N2 = 20 GeV2NNPDF1.2, Q

Figure 14: Probability distribution ofR S atQ 2 = 20 G eV 2 com puted from the reference setof

N rep = 1000NNPDF1.2PDF replicas.Thecentralcross-hatched region correspondsto thecentral

68% con�denceinterval,R S

�Q 2 = 20G eV

2�= 0:006� 0:045stat.

Analysis Reference [S� ](Q 2)� 103 Q 2 [G eV 2] [S� ]�Q 2ref

= 20 G eV 2�� 103

NNPDF1.2 Thiswork 0:5� 8:6 20 0:5� 8:6

M STW 08 [15] 2:4� 2:0 1 1:7� 1:4

CTEQ 6.5s [13] 2:0� 1:8 1.69 1:6� 1:4

CTEQ 6.1s [11] 1:5� 1:5 1.69 1:2� 1:2

AK P08 [14] 1:0� 1:3 20 1:0� 1:3

NuTeV07 [10] 2:2� 1:3 16 2:2� 1:3

BPPZ03 [42] 1:8� 3:8 20 1:8� 3:8

Table10:Com parison ofvariousdeterm inationsofstrangenessm om entum asym m etry [S� ](Q 2)

Eq.(16).Alluncertaintiescorrespond to 68% con�dencelevels.Both thepublished valueisgiven,

and the value obtained evolving to Q 2ref = 20 G eV

2through NLO perturbativeevolution.

30

K S (m ean) R S

Reference 0:71+ 0:19� 0:31 (6� 45)� 10� 3

ZM -VFN 0:47+ 0:10� 0:20 (8� 39)� 10� 3

Nuclear-dFS03 0:74+ 0:21� 0:40 (12� 48)� 10� 3

Nuclear-HK N07 0:68+ 0:24� 0:29 (0� 40)� 10� 3

LO 0:61+ 0:33� 0:22 (1� 38)� 10� 3

No strange SR 0:62+ 0:20� 0:21 (17� 32)� 10� 3

Table 11: The strange relative totaland valence m om entum fractionsK S and R S,Eqs.(17,18),

atthescaleQ 2 = 20 G eV 2.The�rstrow givesthevaluecom puted from thereferenceNNPDF1.2

setofN rep = 1000 replicas,whiletheotherrowsgiveresultsfrom setsofN rep = 100 replicaseach

obtained from alternative �ts discussed in the text. Alluncertainties are one-� or 68% central

con�denceintervals.

ispossibleto exploitourdeterm ination ofK S and R S fora determ ination ofelectroweak

param eters.In view ofthis,itisusefulto also study possiblesourcesofsystem aticuncer-

tainty on these quantities.Possible signi�cantsourcesofsystem aticsarethe following:

� Heavy quark m asse�ects.Thetreatm entofheavy quark m asse�ectsentailsvarious

am biguitiesrelated totheprescription used todealwith subleadingterm s[33].In our

case,a furthersource ofsystem atics isdue to the fact thatthe charm quark m ass

is treated approxim ately, using the I-ZM -VFN schem e as discussed in Sect.3.3,

and then only for dim uon data. The corresponding uncertainty is conservatively

estim ated by repeating the �tin a pureZM -VFN schem e.

� Nuclear corrections. Their e�ect is estim ated by repeating the �t with CHO RUS

and NuTeV data corrected using thedeFlorian-Sassot[35]and HK N07 [36]m odels.

� HigherorderQ CD corrections.Thesearevery conservatively estim ated by repeating

the �tatLO .

Theresultsfrom K S and R S obtained in each ofthesecasesarecom pared in Table11

to thereference NNPDF1.2 result,allatthe scale Q 2 = 20 G eV 2.Itisapparentthatthe

e�ectofany ofthesesystem aticsisratherm oderate,even ifvery conservatively estim ated.

In the sam e table we also show the result ofa �t in which the sum rule Eq.(9) is not

im posed:even in thiscase theresultchangesvery little.

Estim ating the e�ect ofthe system atics from the sum in quadrature ofthe shift of

centralvaluesdueto the fourcentralrowsofTable 11 we get,atQ 2 = 20 G eV 2

K S = 0:71+ 0:19� 0:31

stat� 0:26syst; (19)

R S = 0:006� 0:045stat� 0:010syst: (20)

Thesystem aticson R S isthusnegligible,and m ostly dueto nucleare�ects.Thesystem -

aticson K S isnotquitenegligible,and alm ostentirely dueto thetreatm entoftheheavy

quark m ass:thisisan aspectofouranalysiswhich could beim proved in thefuturewithin

a m oreaccurate treatm entofquark m asse�ects.

31

0

5

10

15

20

25

0.02 0.05 0.1 0.2

Aν

* d2 σν,

c (x,y

,Eν)

/dx/

dy

x

Eν=88.3 GeV, y=0.324

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

Aν

* d2 σν,

c (x,y

,Eν)

/dx/

dy

x

Eν=88.3 GeV, y=0.558

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

Aν

* d2 σν,

c (x,y

,Eν)

/dx/

dy

x

Eν=88.3 GeV, y=0.771

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=174.3 GeV, y=0.324

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=174.3 GeV, y=0.558

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=174.3 GeV, y=0.771

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=247 GeV, y=0.324

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=247 GeV, y=0.558

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=247 GeV, y=0.771

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

Aν

* d2 σan

ti-ν,

c (x,y

,Eν)

/dx/

dy

x

Eν=77.9 GeV, y=0.349

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

Aν

* d2 σan

ti-ν,

c (x,y

,Eν)

/dx/

dy

x

Eν=77.9 GeV, y=0.579

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

Aν

* d2 σan

ti-ν,

c (x,y

,Eν)

/dx/

dy

x

Eν=77.9 GeV, y=0.776

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=143.7 GeV, y=0.349

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=143.7 GeV, y=0.579

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=143.7 GeV, y=0.776

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=226.8 GeV, y=0.349

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=226.8 GeV, y=0.579

NuTeV dataNNPDF1.2

0

5

10

15

20

25

0.02 0.05 0.1 0.2

x

Eν=226.8 GeV, y=0.776

NuTeV dataNNPDF1.2

Figure 15: Com parison between the NuTeV data and the NNPDF1.2 theoreticalpredictionsfor

neutrino (upper three rows) and anti-neutrino (lower three rows) dim uon production. Allcross

section in theplotsarerescaled by afactorA � =1

E �

102

G 2FM N E �

.Theneutrinokinem aticsparam eters

(E �;y)arerelated to x and Q2 by Eq.(2).Thesolid lineisthecentralNNPDF1.2 prediction and

the dashed linesthe 1-� interval.

32

x-110

)2

=500

0 G

eV2

(x, Q

Zγ 3xF

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6 NNPDF1.2

NNPDF1.0

ZEUS HERA-II

Figure 16: Com parison with the experim entaldeterm ination [26]ofthe interference structure

function xF Z

3 (x;Q 2)atQ 2 = 5000 G eV 2 with theNLO prediction obtained using theNNPDF1.0

and NNPDF1.2 parton sets.

4.5 C om parison w ith experim entaldata

The results obtained from a NLO com putation of the dim uon cross section from the

reference setofNNPDF1.2 parton distributionswith N rep = 1000 replicasare com pared

in Fig.15 to theNuTeV experim entaldata.Theagreem entisclearly excellentin allbins

except for the lowest Q 2 values (bottom left plot),where the approxim ate treatm ent of

the quark m assleadsto a deterioration in quality ofthe �t.

In Ref.[26],an extraction oftheinterferenceparity{violating structurefunction xF Z

3

evolved to a com m on scale Q 2 = 5000 G eV 2 wasalso presented.Thisextraction isbased

on data already included in our�t,so these data do notprovide any extra inform ation.

However, it is interesting to com pare directly to it, because this structure function is

directly sensitive to the avour and valence/sea decom position ofPDFs (speci�cally to

strangeness),which isdi�culttoprobedirectly (seee.g.Ref.[21],appendixA.2).Because

the contribution ofxF Z

3 to the totalreduced cross{section issm alland only relevantin

a lim ited region ofphasespace,theagreem entbetween data and theory forthisquantity

could in principle be poorwithoutthis being signi�cantly re ected in the quality ofthe

global�t.

A com parison ofthese data with the NLO prediction obtained using the NNPDF1.0

and NNPDF1.2 parton setsisshown in Fig.16,and showsgood agreem ent:�2 = 1:53 for

NNPDF1.2,and �2 = 1:55 forNNPDF1.0,com parable to the value forotherdata in the

valence region (despite the fact that for the NNPDF1.0 �tneither the data ofRef.[26]

nor dim uon data were used). The widening ofthe uncertainty band when going from

NNPDF1.0 to NNPDF1.2 isa consequence ofthe sensitivity ofthisstructurefunction to

33

�(W + )Br`

W +! l+ �l

´

�(W � )Br`

W �! l+ �l

´

�(Z 0)Br`

Z 0! l+ l�

´

N N PD F 1.010 TeV 8.49 � 0.18 5.81 � 0.13 1.36 � 0.02

14 TeV 11.83 � 0.26 8.41 � 0.20 1.95 � 0.04

N N PD F 1.110 TeV 8.52 � 0.33 5.79 � 0.28 1.36 � 0.04

14 TeV 11.86 � 0.46 8.38 � 0.39 1.95 � 0.06

N N PD F 1.210 TeV 8.61 � 0.25 5.85 � 0.15 1.37 � 0.03

14 TeV 11.99 � 0.34 8.47 � 0.21 1.97 � 0.04

Table 12: Crosssectionsforgaugeboson production atthe LHC.Allquantitieshave been com -

puted atNLO using M CFM [44{47]and NNPDF partons.

valencecom binations,and strangenessin particular:very precisem easurem entsofitcould

greatly im prove avourseparation ofPDFs.

A detailed study of the phenom enological im plications of our reassessm ent of the

strangenessuncertainty forLHC observablesisbeyond the scope ofthiswork. However,

in Table 12 we collectthe totalcrosssection forW and Z production com puted atNLO

with M CFM [44{47]: results obtained with the NNPDF1.2 and NNPDF1.1 parton sets

arecom pared to thosefound using NNPDF1.0.already discussed in Ref.[21].Becauseof

theincreased uncertainty on thestrangedistribution,theuncertainty in thecrosssection

islargerin NNPDF1.1 and NNPDF1.2,though lessso in NNPDF1.2 dueto theconstraint

from dim uon data.

34

5 Precision determ ination ofelectroweak param eters

Neutrino DIS data, and especially dim uon data, can be used to perform direct m ea-

surem ents ofelectroweak param eters [48,49]. However the potentialprecision ofthese

m easurem entscan bespoiled by PDF uncertainties.Indeed,we have seen in Sect.4 that

the uncertainties we obtain on the strange distributions are quite large,typically larger

by alm ostoneorderofm agnitudethan those found in previousglobal�ts.

The CK M m atrix elem entscontrolthe strength ofthe coupling ofvariouspartonsto

neutrinosaccording to Eqs.(11,12).In spiteofthelargePDF uncertaintiesin thestrange

sector,weshallprovideherethem ostprecisedirectdeterm ination up to dateoftheCK M

m atrix elem ent jVcsjwithin a single experim ent. W e willalso provide a determ ination

of jVcdjwith an accuracy consistent with previous results from neutrino data. These

rem arkable resultsare possible because PDF uncertaintiesare free from param etrization

bias,thusthey m ay bedisentangled from the uncertainty on the physicalparam eters.

W e willthen turn to a study ofthe im pactofPDF uncertaintieson the extraction of

the electroweak m ixing angle sin2�W from the Paschos-W olfenstein ratio: we willshow

thatoncePDF uncertaintiesareproperly taken into account,theNuTeV m easurem entof

thisratio [1]isin fullagreem entwith thestandard m odelprediction.

5.1 D eterm ination ofjVcsjand jVcdj.

SincethepioneeringCDHS studies[41],neutrinoDIS hasbeen used asa m eansto directly

determ ine CK M m atrix elem ents: the parton{m odelexpressions for the neutrino and

anti-neutrino dim uon production Eqs.(11,12) provide two equations which relate two

experim entally m easurable crosssectionsto thetwo unknownsjVcdjand jVcsj.

However,these equations also contain as unknownsthe second m om ents ofthe light

quarkPDFs(thetotalcrosssection isproportionaltothesecond m om entofthePDF).The

standard lore [24,31,41]isthen thatifone assum esthatS� � 0,the linearcom bination

F�;c

2 � F��;c

2 only dependson thejVcdjand theu and d valencecom ponents,which arewell

m easured by otherexperim ents,so itcan beused to determ inejVcdj.O n the otherhand,

the orthogonalcom bination F�;c

2 + F��;c

2 dependson the jVcsj=jVcdjratio,butalso on K S

Eq.(17),and thusitcan only beused to determ inethecom bination jVcsjK S.Indeed,the

PDG [24]quotesa valueofjVcdj= 0:23� 0:11 obtained from theaverageneutrino dim uon

experim entsasthebestcurrentdirectdeterm ination.O nly thebound jVcsj� 0:74 at90%

con�dencelevel[31]wasquoted in previousPDG [50]editions,butthisisnow superseded

by a direct determ ination jVcsj= 1:04 � 0:06 from D decays (for a recent update,see

Ref.[51]).O fcourse,thevaluesobtained from thecurrentglobalCK M �ts[24,52,53]are

m uch m ore precise than these directdeterm inations(see Table 14 below).

In theNNPDF1.2 reference�t,jVcdj,jVcsj,and jVcbjareeach �xed to thecurrentPDG

value[24],obtained from theglobalCK M unitarity �t.W e now show that,thanksto the

factthatweare freeofbiasrelated to theparam etrization ofstrangeness,we can extract

both jVcsjand jVcdjfrom the �t. In orderto do this,we perform a scan over the values

ofjVcsjand jVcdjused in the �t,holding jVcbj�xed,butrelaxing the unitarity constraint

(in practice,becauseofitssm allness,theprecisevaluechosen forjVcbjisinconsequential).

The best{�tvalue and uncertainty forthe CK M param eters are then determ ined in the

standard way by m axim um likelihood from the �2 pro�le.

35

52

54

56

58

60

62

0.8 0.85 0.9 0.95 1 1.05 1.1

χ2

|Vcs|

NNPDF1.2, Nrep = 500, |Vcd|=0.2256

NuTeV DimuonParabolic fit (5 points)Parabolic fit (3 points)

Figure17:The�2 oftheNuTeV dim uon data asa function ofjVcsjwhen jVcdjiskept�xed atits

bestunitarity �tvalue.Thelong-dashed curveistheparabolic�tfrom which thecentralvalueand

one-� uncertainty Eq.(21) are obtained;the short-dashed curve is a parabolic �t to the central

and two outerpointsonly.

The�2 determ ined from asetofN datdata points uctuates,with astandard deviation

equalto ��2 =p2N dat. In orderto determ ine the �2 pro�le asthe underlying param e-

ters are varied,these uctuationsm ustbe kept undercontrol. W ithin our M onte Carlo

approach,this could be done by using the sam e setofdata replicas each tim e the �2 is

recom puted with di�erentvaluesofthe underlying param eters.Thism ighthoweverbias

theresultin arandom way dependingon theparticularsetofreplicaswhich hasbeen cho-

sen in the�rstplace.W epreferthusto vary random ly thesetofreplicaswhich isused for

di�erentparam etervalues: uctuationsarethen keptundercontrolby using a su�ciently

large setofreplicas,given the uctuation ofthe �2 com puted from a replica average has

a standard deviation equalto ��2=pN rep.Because only dim uon data aresensitive to the

CK M m atrix elem ents,we can determ ine their values from the dependence ofthe �2 of

the �tto these data only,rather than for that ofthe �tto the globaldataset. Because

we have (see Tab.1)84 dim uon data points,a setofN rep = 500 replicas is su�cientto

guarantee thatpoint-by-point uctuationsare sm allerthan �� 2 = 1.

First,wevary independently each ofthetwo CK M m atrix elem ents,keeping theother

�xed at its centralvalue in the CK M unitarity �t. The � 2 pro�le is com puted for �ve

equally spaced valuesoftheparam eterwhich isbeingvaried.Thevalueshavebeen chosen

on the basisofa prelim inary exploration ofthe space ofparam etersbased on �tswith a

sm allnum berofreplicas;they aredisplayed in Fig.19.Theensuing�2 pro�leisdisplayed

in Fig.17 forjVcsjand in Fig.18 forjVcdj.W eobservewell-de�ned m inim a in both cases.

36

48

50

52

54

56

58

60

62

0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28

χ2

|Vcd|

NNPDF1.2, Nrep = 500, |Vcs|=0.97334

NuTeV DimuonParabolic fit (5 points)Parabolic fit (3 points)

Figure18:The�2 oftheNuTeV dim uon data asa function ofjVcdjwhen jVcsjiskept�xed atits

bestunitarity �tvalue.Thelog-dashed curveistheparabolic�tfrom which thecentralvalueand

one-� uncertainty Eq.(22) are obtained;the short-dashed curve is a parabolic �t to the central

and two outerpointsonly.

37

jVcdj jVcsj

Statistical � 0:012 � 0:05

M asse�ects � 0:007 � 0:02

HigherorderQ CD � 0:010 � 0:03

Nuclearcorrections � 0:008 � 0:03

Totalsystem aticuncertainty � 0:014 � 0:05

Totaluncertainty � 0:019 � 0:07

Table 13: Sum m ary ofstatisticaland system atic uncertainties in the present determ ination of

jVcsjand jVcdj.

A parabolic �tleadsto

jVcsj = 0:93� 0:06; (21)

jVcdj = 0:248� 0:012; (22)

wheretheone-� uncertainty isobtained from thecondition �� 2 = 1.The�tisquitestable

upon the choice ofdi�erentsubsetsofthe �ve available points: ifitisrepeated by only

retaining thecentraland two outerpointsneitherthecentralvaluesnortheuncertainties

Eqs.(21-22)vary signi�cantly.Thiscon�rm sthatthenum berofreplicasused to com pute

the �2 issu�ciently large forthe resultnotto bebiased by statistical uctuations.Both

�tsare shown in Figs.17-18.

ThisshowsthateitherCK M m atrix elem entcan be determ ined from ourdata,with

com parable uncertainty,by taking the other�xed. W e can thusperform a sim ultaneous

determ ination ofboth these CK M m atrix elem ents. In order to im prove the accuracy

ofthisdeterm ination,we com pute the �2 atfourm ore pointsin the (jVcdj,jVcsj)plane,

denoted by squares in Fig.19. The �2 in these additionalpoints is com puted from a

sm allersetofN rep = 100 replicas.Theresultofthecom bined �tisthen

jVcsj = 0:96� 0:05; (23)

jVcdj = 0:244� 0:012: (24)

The uncertainties turn outto be alm ost identicalto the diagonaluncertainties,and the

correlation coe�cientisrelatively sm all� = 0:21,re ected in a m oderate shiftin central

valuesin com parison to the separate �tsEqs.(21-22). The location ofthe best-�tpoint

and one-� (�� 2 = 1) ellipse in the (jVcdj,jVcsj) plane for the best-�t �2 paraboloid is

shown in Fig.20.

Thisdeterm ination Eq.(24)isa�ected by the sam e system aticsthatwe exam ined in

Sect.4.3,nam ely,higher order Q CD corrections,treatm ent ofheavy quark e�ects and

m odeling ofnuclear corrections. In order to assess their im pact in the CK M elem ent

determ ination,we have repeated the determ ination ofeach ofthe two param etersasthe

other is kept �xed,Eqs.(21-22),by recom puting the �2 for a sm aller setofN rep = 100

replicasalongthepointsdenoted ascirclesin Fig.19,with each ofthesethreee�ectsvaried

in turn aswedid in Sect.4.3.W ethen taketheshiftin centralvalueasan estim ateofthe

corresponding uncertainty. The resultsare sum m arized in Table 13. Putting everything

38

0.8

0.85

0.9

0.95

1

1.05

1.1

0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28

|Vcs

|

|Vcd|

Figure 19: The grid ofpointsused in the determ ination ofthe CK M m atrix elem entsjVcsjand

jVcdj.O pen circlesdenotepointsused forthedeterm ination ofjVcdjEq.(22),and fullcirclespoints

used forthedeterm ination ofjVcsjEq.(21).Allpointsareused in thejointdeterm ination Eq.(24).

together,we �nd

jVcsj = 0:96� 0:07tot ; (25)

jVcdj = 0:244� 0:019tot: (26)

In Table 14 we com pare our �nalresults Eqs.(25-26) with the best CK M unitarity

�tresultsand with otherdirectdeterm inations. O urdeterm ination ofjVcdjisconsistent

with other direct determ inations,and ofcom parable accuracy,though one should bear

in m ind that previousdeterm inations from dim uon data were based on �tswith a �xed

functionalform ,and thussubjectto potentially largesystem aticsbias.O urdeterm ination

ofjVcsjis rather m ore accurate that any other direct determ ination from dim uon data,

m ore accurate than any single direct determ ination,and ofcom parable accuracy to the

PDF average ofdeterm inationsfrom D decays.

39

XX

CKM unit. fitCKM unit. fit

NNPDF1.2NNPDF1.2ÈVcsÈÈVcsÈ

ÈVcdÈÈVcdÈ0.22 0.23 0.24 0.25 0.26

0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02

Figure 20: Location ofthe best-�tpoint and one-� (statistical�� 2 = 1 uncertainty)ellipse in

the(jVcdj,jVcsj)planeforthebest-�t�2 paraboloid obtained from the�2 com puted atthepoints

displayed in Fig.19.The bestunitarity �tresult[24]isalso shown forcom parison.

40

Analysis Description Reference jVcsj

NNPDF1.2 D irectdeterm ination from globalPD F analysis Thiswork 0:96� 0:07tot

CDHS LO determ ination from �N ! �+��X [41] � 0:59 (90% C.L.)

CCFR NLO determ ination from �N ! �+��X [28,31] � 0:74 (90% C.L.)

PDG 08 Averagesofdeterm inationsfrom D decays [24] 1:04� 0:06

Hocker Averagesofdeterm inationsfrom �N ! �+��X [54] 1:04� 0:16

DELPHI D irectm easurem entfrom W+! c�s decays [55] 0:94

+ 0:32� 0:26 � 0:13

PDG 08 CK M unitarity �t [24] 0:97334� 0:00023

Analysis Description Reference jVcdj

NNPDF1.2 D irectdeterm ination from globalPD F analysis Thiswork 0:244� 0:019tot

CDHS LO determ ination from �N ! �+��X [41] 0:24� 0:03

CCFR NLO determ ination from �N ! �+��X [31] 0:232+ 0:017� 0:019

PDG 08 Averagesofdirectdeterm inationsfrom �N ! �+��X [24] 0:230� 0:011

PDG 08 Average ofdeterm inationsfrom D ! K =�l� decays [24] 0:218� 0:023

PDG 08 CK M unitarity �t [24] 0:2256� 0:0010

Table 14: Com parison ofthe present determ ination ofthe CK M m atrix elem ents jVcsj(upper

table)and jVcdj(lowertable)with otheravailable directm easurem ents,averagesand CK M con-

strained �ts.

41

5.2 PD F corrections to the Paschos-W olfenstein ratio

Thesuccessfuldeterm ination oftheCK M m atrix elem entswhich controlcharged current

scattering suggests that we m ight use our parton set for a reliable reassessm ent ofthe

determ ination of the coupling which controls neutralcurrent neutrino DIS.As is well

known [56],this coupling depends on the electroweak m ixing angle,which can thus be

extracted from itsexperim entalm easurem ent.Speci�cally,in the parton m odelone has

R PW ��(�N ! �X )� �(��N ! ��X )

�(�N ! ‘X )� �(��N ! �‘X )

=1

2� sin2�W +

�([U � ]� [D � ])+ ([C � ]� [S� ])

[Q � ]

1

6

�3� 7sin2�W

��

; (27)

where�W istheelectroweak m ixing angle,[S� ]isthestrangevalencem om entum fraction

Eq.(16),[U � ],[D � ]and [C � ]thevalencem om entum fractionsofotherquark avors,and

[Q � ]� ([U � ]+ [D � ])=2.

Therecentexperim entaldeterm ination [1]

sin2�W

���N uTeV

= 0:2277� 0:0014stat� 0:0009sys = 0:2277� 0:0017tot ; (28)

isobtained using Eq.(27)undertheassum ption thatforan isoscalarnucleon target[U � ]-

[D � ]= [C � ]= [S� ]= 0,so the term in square bracketsin Eq.(27)vanishes. O fcourse,the

NuTeV iron targetisnotexactly isoscalar;however,the corresponding correction can be

com puted [1]with sm alluncertainty [2].TheresultEq.(28)disagreesatthethree-� level

with the valuedeterm ined in globalprecision electroweak �ts,such as[57,58]

sin2�W

���EW �t

= 0:2223� 0:0003 : (29)

Possibleexplanationsforthisincludenucleare�ects,electroweak corrections,Q CD correc-

tions,and physicsbeyond thestandard m odel[2](seee.g.[59]foran updated listofrefer-

ences).However,onem ay also [2]question thevalidity oftheassum ption ofthevanishing

ofthe contribution in square bracketsin Eq.(27). The possibility that[U � ]� [D � ]6= 0

even foran isoscalartargetdueto isospin violation induced by Q ED evolution e�ectswas

discussed in Ref.[60]:itcould easily explain abouta third oftheobserved discrepancy.

In our�t,isospin sym m etry isassum ed,and furtherm ore [C � ]= 0. W e are then left

with the correction

�ssin2�W = � R S

1

6

�3� 7sin2�W

�; (30)

with R S de�ned in Eq.(18).Using thevalueofR S Eq.(20),obtained atthetypicalscale

Q 2 = 20 G eV 2 oftheNuTeV data (and whosescaledependenceisvery sm allanyway [43])

we obtain

�ssin2�W = � 0:001� 0:011PD Fs� 0:002th; (31)

wherethetheoreticaluncertainty com esfrom thee�ectsdiscussed abovein Sect.4.3,and

it is not to be confused with the experim entalsystem atics in the NuTeV m easurem ent

Eq.(28).

Even neglecting these theoretical uncertainties (which we estim ated very conserva-

tively),the additionalPDF uncertainty due to strangenessalone isthusabouttwice the

42

0.215

0.22

0.225

0.23

0.235

0.24

0.245si

n2 θ W

Determinations of the weak mixing angle sin2θW

NuTeV01 NuTeV01 Global EW fit+ NNPDF1.2 [S-]

Figure21:Com parison between theNuTeV determ ination ofsin2 �W ,Eq.(28),theresultfrom the

globalelectroweak �t,Eq.(29),and the NuTeV resultafterthe correction due to the uncertainty

on S� Eq.(32).

observed discrepancy in sin2�W . W e m ust conclude therefore that the apparent incon-

sistency between theNuTeV m easurem entand theglobalelectroweak �tdisappearsonce

the uncertainty on the strange distribution isproperly taken into account. Applying the

correction Eq.(31)the NuTeV resultbecom es

sin2�W

���N uTeV

= 0:2263� 0:0014stat� 0:0009sys� 0:0107PD Fs: (32)

W erecom m end thatthecorrected resultEq.(32)beused,forinstancein globalelectroweak

�ts.Thiscorrected resultiscom pared graphically in Fig.21 to theoriginalNuTeV result

Eq.(28)and the resultfrom theglobalelectroweak �tEq.(29).

43

6 C onclusions and outlook

W ehavepresented an upgradeoftheNNPDF1.0 parton set,which now includesan inde-

pendentparam etrization forthe strange distributions,and the inclusion ofdim uon data

which constrain them .Besidesbeingan interm ediatestep towardsafully global�tinclud-

ing hadronic data,ourresultsare interesting asa testofthe NNPDF m ethodology,asa

state-of-theartdeterm ination ofthestrangePDFs,and asa determ ination ofelectroweak

param eters.

W ehaveshown thattheNNPDF approach hasno di�culty in dealing with situations

whereexperim entalinform ation isscarce and only provideslooseconstraintson theform

of parton distributions. W ithin our approach, this does not require the introduction

oftheoreticalassum ptions or constraints in order to obtain stable results. W e can thus

providereliableestim atesofuncertainties,freeofbiasinduced by theoreticalassum ptions.

W e have obtained a determ ination of the strange m om entum fraction and of the

strangeness valence com ponent, which, though in agreem ent with previous determ ina-

tions,turn outto be a�ected by uncertainties which are sizably larger than those found

by othergroups.

Nevertheless,we have shown that,with the uncertainty on the strange PDF carefully

estim ated,the dim uon data can be used to provide a good determ ination ofthe CK M

m atrix elem ents jVcdjand jVcsj. In particular, our determ ination of jVcsjis the m ost

accurate ever obtained from neutrino deep-inelastic scattering data,and it is also m ore

accuratethan anyindividualdirectdeterm ination from D decays.W ehavealsoshown that

once PDF uncertaintiesare estim ated reliably,the value ofthe electroweak m ixing angle

extracted from NuTeV inclusive data isin agreem entwith standard m odelexpectations.

Them ain defectofourresultsisthatthey arestillbased on an approxim atetreatm ent

ofthecharm m ass.W ithin thecontextofthepresentwork,theonly signi�cantim plication

ofthis is a slight increase in the system atic uncertainty on our determ ination ofjVcdj.

However,thisalso entailsa furthersm allbutnon-negligiblesystem aticuncertainty in our

determ ination ofPDFs[21].

Itwillbeinterestingtostudy theim plicationsforLHC observablesofthisreassessm ent

ofthe uncertainty on the strange distribution. The NNPDF1.2 release is available from

the webpage ofthe NNPDF Collaboration http://sophia.ecm.ub.es/nnpdf/.

A cknow ledgm ents

Thiswork waspartly supported by grantsPRIN-2006 (Italy),M EC FIS2004-05639-C02-

01 (Spain)and by the European network HEPTO O LS undercontract M RTN-CT-2006-

035505.L.D.D.isfunded byan STFC Advanced Fellowship and M .U.byaSUPA graduate

studentship. W e acknowledge discussions with S.Alekhin,P.Nadolsky,P.Nason and

A.Vicini.W e are especially gratefulto D.M ason forproviding uswith the NuTeV data

and acceptances, to F.O lness for inform ation on NLO acceptances, to R.Sassot and

R.Pettiforproviding uswith theirnuclearPDF setsand to A.Tapperand K .Nagano for

help with theHERA-IIdata.J.R.acknowledgesthehospitality oftheCERN TH Division

wherepartofthiswork wascom pleted.

44

A K ernels for PhysicalO bservables

In thisappendixweexpand thephysicalobservablesfordim uonproduction in theevolution

basisofthe PDFs,and derive expressionsforthe kernels,in the sam e way and using the

sam e notation as in Appendix A ofRef.[21]. Allconvolutions m ay be perform ed either

in the ZM -VFNS orin theI-ZM -VFN schem e,asdiscussed in Sect.3.3.

Thecross-section forcharm production in neutrino scattering o� an isoscalarnucleon

isgiven by Eq.(1),which we write as

e��(��);c= �[eY+ F

�(��);c

2 � y2F�(��;c)

L� Y� xF

�(��;c)

3 ]; (33)

where

� =G 2F M N

2�(1+ Q 2=M 2W)2; eY+ =

�

Y+ �2M 2

N x2y2

Q 2� y

2

��

1+m 2

c

Q 2

�

+ y2: (34)

Taking into accounta possiblenon-isoscalarcom ponentofthenucleartargetby de�n-

ing � � 1� 2Z=A,in the quark m odelwe have

F�;c

2 = F�;c

L= xF

�;c

3 = x�jVcdj

2((1+ �)u + (1� �)d) + 2jVcsj2s+ 2jVcbj

2b�; (35)

F��;c

2 = F��;c

L= � xF

��;c

3 = x�jVcdj

2((1+ �)�u + (1� �)�d) + 2jVcsj2�s+ 2jVcbj

2�b�;(36)

where allexplicit dependence on x and Q 2 has been dropped. In term s of the PDF

evolution eigenstateswethen have

F�(��);c

2 = F�(��);c

L= � xF

�(��);c

3 = x�16w0(�� V )+ 1

2�w3(T3 � V3)

+ 16w8(T8 � V8)+

112w15(T15 � V15)+

120w24(T24 � V24)+

130w0(T35 � V35)

;(37)

where the + (-) sign corresponds to neutrino (anti-neutrino) scattering, and the CK M

factorsare

w0 � jVcdj2 + jVcsj

2 + jVcbj2; w3 � jVcdj

2; w8 � jVcdj

2� 2jVcsj

2;

w15 � jVcdj2 + jVcsj

2; w24 � jVcdj

2 + jVcsj2� 4jVcbj

2: (38)

Unitarity ofthe CK M m atrix is im posed setting w0 = 1;in the CK M determ ination in

Sec.5.2 itishoweverleftunconstrained.Below bthreshold Vcb = 0,so w0 = w15 = w24.

In perturbativeQ CD thecharm production neutrino structurefunctionsthustakethe

form

F�(��);c

i = Csi;q

16w0�+ C i;g

1nfw0g� C

si;q

16w0V + Ci;q

�12�w3(T3 � V3)

+ 16w8(T8 � V8)+

112w15(T15 � V15)+

120w24(T24 � V24)+

130w0(T35 � V35)

;(39)

F�(��);c

3 = � Cs3;q

1

6w0�+ C

si;q

1

6w0V � Ci;q

�1

2�w3(T3 � V3)

+ 16w8(T8 � V8)+

112w15(T15 � V15)+

120w24(T24 � V24)+

130w0(T35 � V35)

;(40)

where i= 2;L and nf is the num ber ofactive avours. W e can thus write the charm

production neutrino cross-sectionsas

e��(��);c= � xfK�(��);c

� �0 + K

�(��);cg g0 � K

�(��);c

V V0

+ K�(��);c

+ (12�w3T3;0 +

16w8T8;0 +

112w15T15;0)

� K�(��);c

� (1

2�w3V3;0 +

16w8V8;0 +

112w15V15;0)g; (41)

45

wherein M ellin space the kernelsare

K�(��);c

�= (eY+ C

s2;q � y

2CsL;q + Y� C

s3;q)(

16w0�

qq

S+ 1

20w24�

24;q

S+ 1

30w0�

35;q

S)

+ (eY+ C2;g � y2CL;g)

1nfw0�

gq

S; (42)

K�(��);cg = (eY+ C

s2;q � y

2CsL;q + Y� C

s3;q)(

16w0�

qg

S+ 1

20w24�

24;g

S+ 1

30w0�

35;g

S)

+ (eY+ C2;g � y2CL;g)

1nfw0�

gg

S; (43)

K�(��);c

V= (eY+ C

s2;q � y

2CsL;q + Y� C

s3;q)(

1

6w0�

vN S +

1

20w24�

24N S +

1

30w0�

35N S); (44)

K�(��);c

+ = (eY+ C2;q � y2CL;q + Y� C3;q)�

+N S; (45)

K�(��);c

�= (eY+ C2;q � y

2CL;q + Y� C3;q)�

�

N S: (46)

Below the bthreshold thesingletkernelssim plify to

K�(��);c

�= 1

4w15

�(eY+ C

s2;q � y

2CsL;q + Y� C

s3;q)�

qq

S+ (eY+ C2;g � y

2CL;g)�

gq

S

�; (47)

K�(��);cg = 1

4w15

�(eY+ C

s2;q � y

2CsL;q + Y� C

s3;q)�

qg

S+ (eY+ C2;g � y

2CL;g)�

gg

S

�; (48)

K�(��);c

V= 1

4w15(eY+ C

s2;q � y

2CsL;q + Y� C

s3;q)�

vN S: (49)

46

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49