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�
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)�
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