Testing Partonic Charge Symmetry at a High-Energy Electron Collider

0.36 MB·

by  T. J. Hobbs

Testing Partonic Charge Symmetry at a High-Energy Electron Collider
T.J. Hobbs,1 J. T. Londergan,1 D.P. Murdock,2 and A. W. Thomas3
1Department of Physics and Center for Exploration of Energy and Matter,
Indiana University, Bloomington, IN 47405, USA
2Department of Physics, Tennessee Technological University, Cookeville, TN 38505, USA
3CSSM, School of Chemistry and Physics, University of Adelaide, Adelaide, South Australia 5005, Australia
We examine the possibility that one could measure partonic charge symmetry violation (CSV)
by comparing neutrino or antineutrino production through charged-current reactions induced by
electrons or positrons at a possible electron collider at the LHC. We calculate the magnitude of
CSV that might be expected at such a facility. We show that this is likely to be a several percent
1 effect, substantially larger than the typical CSV effects expected for partonic reactions.
1
0 PACSnumbers: 12.15.+y.13.15.+g,24.85.+p
2

n
Charge symmetry is a very specific operation tron accelerator with protons or deuterons from
a
J involving isospin, which leads to the interchange the LHC [8]. In this paper we will show that such
ofprotonsandneutrons, orequivalentlytheinter- a facility (given the name LHeC) has the poten-
0
2 change of up and down quarks. The charge sym- tial to produce charge symmetry violating effects
metry operator P corresponds to a rotation of which areconsiderably largerthanthose expected
CS
] 180◦ about the 2 axis in isospin space, such that with other facilities. We will review the effect in
h
p P = eiπT2 , question, show the results of theoretical calcula-
- CS tions for the proposed CSV effects, and discuss
p PCS|u(cid:105) = −|d(cid:105); PCS|d(cid:105)=|u(cid:105). (1) why they ought to be expected to be relatively
e
h Itisofparticularimportancebecauseatlowener- large at energies accessible to an electron-ion col-
[ lider.
gies, where it has been studied extensively, charge
The reactions of interest are the charged cur-
1 symmetryisafarbettersymmetrythanisospinin
rent (CC) cross sections for electron and positron
v general, typically being respected to better than
3 deep inelastic scattering at energies in the range
1% [1, 2]. It is therefore natural to assume that
2 50-100 GeV on protons and deuterons at LHC
chargesymmetryisalsovalidatthepartoniclevel
9 energies, i.e., several TeV. These are important
3 and, indeed, almost all analyses of parton distri-
because they directly and unambiguously probe
. butionfunctions(PDFs)assumechargesymmetry,
1 the flavor structure of the proton PDFs in the va-
whether the assumption is stated or not. The im-
0
lence region. Consider the deep inelastic reaction
1 portance of charge symmetry violation in PDFs
(e−,ν ). Ahigh-energyelectronincidentonapro-
1 within the context of tests of the Standard Model e
: has recently been of considerable interest [3, 4]. tonproducesaneutrino. Theprocessresultsfrom
v a W− which is absorbed on quarks from the pro-
i To date, no violation of charge symmetry has
X ton, as shown schematically in Fig. 1; the final
been observed at the partonic level, although the
r one global analysis that did allow for CSV did hadronic state is not observed. The signature for
a
this process is disappearance of the electron, to-
find a preferred solution with a non-zero effect
gether with very large deposition of energy in the
– albeit with very large errors [5]. The current
hadronic sector.
upper limits are consistent with the validity of
The F structure function for the CC reaction
partonic charge symmetry in the range 5-10% [6]. 2
on a proton has the form
Theoreticalmodelsgenerallyproduceestimatesof
charge symmetry violation (CSV) in PDFs which FW−p(x)=2x[u(x)+c(x)+d(x)+s(x)]. (2)
for many observables give effects at roughly the 2
1% level [6, 7]. This presents a significant chal- These reactions will occur at extremely high ener-
lenge for experimentalists, first to observe effects giesandverylargeQ2. Therefore, anycorrections
of this magnitude and then to isolate the signal to the F structure functions in Eqs. (2) and (3)
2
from competing effects of similar size. arising from quark mixing matrices, quark masses
A new facility has recently been proposed that or higher-twist effects should be completely negli-
would collide electrons or positrons from an elec- gible.
reactions on protons initiated by electrons and by
positrons.
Using Eqs. (2), (3) and (4) we can straightfor-
wardly show that the quantity R−(x) in (6) has
the form
x[−2s−(x)+δu−(x)−δd−(x)]
R−(x)= .
x[u+(x)+d+(x)+s+(x)+2c(x)]
(7)
Thus R−(x) is proportional to the valence quark
FIG.1: Schematicpictureofcharged-currentneutrino CSV parton distributions plus the strange quark
productioninDISinducedbyanelectrononaproton. asymmetry (the difference between the strange
and antistrange PDFs). Insofar as the strange
quark asymmetry exists, it should be large only
Wecanalsoconsiderthecorrespondingreaction
for positrons on protons, (e+,ν ). This reaction at quite small Bjorken x < 0.1, while theoretical
e
involves the absorption of a W+ on the proton, estimates of the valence CSV parton distributions
[9, 10] suggest that for Q2 ∼ 10 GeV2 they peak
with the resulting F structure function
2
at values x∼0.4.
FW+p(x)=2x[u(x)+c¯(x)+d(x)+s(x)]. (3) Consider a hypothetical collider with 50 GeV
2
electrons or positrons colliding with protons and
We can straightforwardly calculate the F struc-
2 deuterons of energy roughly 7 TeV. This would
ture functions (per nucleon) on the deuteron,
be similar to the possibilities if an electron col-
FW−D(x) = x[u+(x)+d+(x)+2c(x) lider were built at the LHC. Consider charged-
2
current reactions at such a facility with Q2 =105
+ 2s(x)−δd(x)−δu(x)] ;
GeV2. We know of two different mechanisms for
F2W+D(x) = x[u+(x)+d+(x)+2c¯(x) charge symmetry violation in parton distribution
+ 2s(x)−δd(x)−δu(x)] . (4) functions. The first arises from the radiation of a
photon by a quark. Such contributions are shown
In Eq. (4) we introduce combinations of quark
schematically in Fig. 2; they were first calculated
partondistributionfunctions(PDFs)thatareeven
by the MRST group [11] and Gluck et al. [12].
or odd under charge conjugation, and the CSV
These QED corrections are analogous to the cou-
PDFs
pling of gluons to quarks, except that photons do
q±(x) = x[q(x)±q¯(x)] ; not have the self-coupling terms possessed by glu-
δu(x) = up(x)−dn(x) ; ons. Inclusion of these ‘QED splitting’ terms will
produce charge symmetry violation in parton dis-
δd(x) = dp(x)−un(x) . (5)
tributionfunctionsbecauseoftheelectromagnetic
There are analogous relations to Eq. (5) for the (EM) coupling due to the different charges of up
antiquark CSV PDFs. For the remainder of this and down quarks.
letter we assume that c(x) = c(x). The distribu-
tionsq−(x), whichinvolvethedifferencesbetween
quarkandantiquarkPDFs(alternatively,theyare
the C-odd combinations of quark distributions),
are the valence parton distributions for a given
quark flavor.
Now define the following quantity,
FIG. 2: Schematic picture of quarks coupling to pho-
2(FW−D(x)−FW+D(x)) tons. This gives the origin of QED splitting that pro-
R−(x)≡ 2 2 . (6) duces CSV effects in parton distribution functions.
FW−p(x)+FW+p(x)
2 2
The quantity R−(x) is given by the difference in The behavior of parton distributions with in-
theF structurefunctionspernucleonforelectron- creasing Q2 is given by the DGLAP evolution
2
deuteron and positron-deuteron CC reactions, di- equations [13–15]. We expand the DGLAP evolu-
videdbytheaverageF structurefunctionforCC tion equations to lowest order in both the strong
2
2
coupling α and the electromagnetic coupling α, distributionsevolve,tolowestorderinbothα and
S S
α. WithincreasingQ2,partonsradiategluonsand
∂q (x,µ2) α
i = S [P ⊗q +P ⊗g] photons which carry off momentum. Since the to-
∂lnµ2 2π qq i qg tal momentum fraction carried by quarks is given
α
+ 2πe2iP(cid:101)qq⊗qi ; basyQth2eisneccroenadsemsotmheenptaortfotnhedpisatrrtibountdioisntrfiubnucttiioonnss,
 
∂g(x,µ2) α (cid:88) will shift towards progressively smaller x values.
∂lnµ2 = 2πS  Pgq⊗qj +Pgg⊗g ComparisonofEqs.(11)and(12)showsthatthe
j radiationfromvalencequarkswillbegreaterthan
+ 2απe2iP(cid:101)qq⊗qi ; tchuarstbfreocmautsheetvoalloewnceestCoSrVdedriisntrαibutainodnsα.,Tvhailsenocce-
S
∂γ(x,µ2) α (cid:88) quark evolution contains contributions from both
= e2P ⊗q . (8)
∂lnµ2 2π j γq j gluon and photon radiation, whereas the valence
j
CSVdistributionhasonlyatermfromphotonra-
In Eq. (8), q (x,µ2) is the parton distribution for diation. ThissuggeststhatwithincreasingQ2 the
i
a given flavor i, g(x,µ2) is the gluon distribution valence parton distributions would experience a
andγ(x,µ2)isa“photonpartondistribution”[11]. larger shift to low x than will the valence CSV
In these equations the convolution is defined as distributions. We note that the quantity R−(x)
defined in Eq. (7) is proportional to the ratio of
(cid:90) 1 dy x valence CSV distributions to valence PDFs, at a
P ⊗q = P(y)q( ,µ2) , (9)
y y given x value. If the CSV valence distributions
x
are becoming larger relative to the valence PDFs
and the splitting functions are given by atlargeQ2,thenweexpectthequantityR−(x)to
growasQ2 increases; specificallywewouldexpect
P (y) P (y)
P(cid:101)qq(y) = qCq ; Pγq(y)= gCq ; the ratio to increase logarithmically with Q2.
F F Eq. (12), the QCD evolution equations for
P (y) = Pqg(y); P (y)=(cid:88)−2e2jδ(1−y) the valence CSV parton distributions, have been
qγ TR γγ 3 solvedbyGlu¨cket al. [12]andalsobytheMRST
j
group [11]; the two groups made slightly differ-
(10)
ent approximations for the initial conditions. We
We have analogous equations to Eq. (8) for an- stress that while the effect of photon radiation is
tiquarks. Taking the valence combinations from clear, it is far less obvious that the boundary con-
Eq.(5),weobtainforupanddownvalencequarks ditions imposed on the calculations are appropri-
ate. That is, we know of no rigorous proof that a
∂u−(x,µ2) α 2α lowscale,typicalofquarkmodels,istheappropri-
∂lnµ2 = 2πSPqq⊗u−+ 9πP(cid:101)qq⊗u− ; ateplacetosettheeffecttozero. Intheabsenceof
∂d−(x,µ2) α α a compelling theoretical derivation it is extremely
∂lnµ2 = 2πSPqq⊗d−+ 18πP(cid:101)qq⊗d− helpful to be able to test the idea experimentally.
The MRST group [11] did attempt an experi-
(11)
mental test of this method. Including QED radi-
For the valence CSV parton distributions, since ation in the DGLAP equations introduces a ‘pho-
δu−(x)=u−(x)−d−(x), from Eq. (11) we obtain ton parton distribution’ γ(x,Q2), which appears
p n
theevolutionequationsforthevalenceCSVPDFs, in Eq. (8). The MRST group attempted to iden-
to lowest order in α and α, tify this quantity in the process ep → eγX where
S
the final state e and γ are produced with equal
∂[δu−(x,µ2)] α and opposite large transverse momentum. This
∂lnµ2 ≈ 2π(e2u−e2d)P(cid:101)qq⊗u− ; process has been measured by the ZEUS Collabo-
√
∂[δd−(x,µ2)] α ration in ep collisions at s = 300 and 318 GeV
∂lnµ2 ≈ −2π(e2u−e2d)P(cid:101)qq⊗d− .(12) [16]. The observed cross sections were in reason-
able agreement with the MRST calculations but
Eq.(11)describeshowthevalencequarksevolve disagreed with calculations done using the Monte
with Q2 and Eq. (12) shows how the valence CSV Carlo simulations PYTHIA [17] and HERWIG
3
[18]. It would be useful to have other experimen- Valence CSV parton distributions arising from
tal tests of this method for including radiation of quark mass difference effects were calculated by
photonsbypartons,andtheexperimentsuggested Rodionov et al. [10]. They used bag model wave-
here could provide additional confirmation of this functions,includingtheeffectofquarkmassdiffer-
method. encesonthequarkwavefunctionsaswellasonthe
A second source of valence parton CSV arises di-quark and nucleon masses, using Eq. (13). The
naturally from the mass difference between the u Rodionov calculation preserved the correct sup-
and d quarks and may be calculated within light port and included the effect of transverse momen-
cone quark models. In such models the valence tumintheproton. Asinanyquarkmodelcalcula-
quark distribution can be expressed as [19–21] tion, the resulting leading twist PDFs are appro-
priate to a relatively low momentum scale (where
(cid:88) 1+γ0γ3
q (x,µ2) = M |(cid:104)X| ψ(0)|N(cid:105)|2 most of the momentum of the nucleon is carried
v 2
by valence quarks [21, 23]). In order to compare
X
× δ(M(1−x)−p+) . (13) with experimental data these PDFs are typically
X evolved up to Q2 = 10 GeV2. We subsequently
Eq.(13)denotestheprocesswhereavalencequark evolved these parton distributions to the higher
is removed from a nucleon |N(cid:105), and the result is Q2 values appropriate to an electron-ion collider
summedoverallfinalstates|X(cid:105). Thequantityp+ such as the LHeC.
X
istheenergyofthestatefollowingremovalofava- An alternate theoretical approach was due to
lence quark with momentum k. The quantity µ2 Sather [9], who investigated the expression for va-
represents the starting value for the Q2 evolution lence parton CSV distributions in a static quark
ofthepartondistribution. Eq.(13)isformallyex- picture. In such models the correct support is
actandprovidesanaturalstartingpointforcalcu- no longer guaranteed. In addition, Sather ne-
lations which preserve the correct support of the glected transverse quark momentum. By apply-
PDFs. ing Eq. (15) to Eq. (13) within this approxima-
Model quark wavefunctions are found to be tion scheme, Sather obtained an analytic approx-
nearlyinvariantunderthesmallmasschangestyp- imation relating valence quark CSV to deriva-
icalofCSV[10],soweconcentrateonthebreaking tives of the valence PDFs. The analytic approx-
of partonic charge symmetry associated with en- imation of Sather is appropriate only at Q2 val-
ergy shifts resulting from the u and d quark mass ues appropriate for quark model calculations, i.e.,
differences. Inparticular, weconsidertheeffectof Q2 ∼0.25−0.5 GeV2.
the n−p mass difference δM ≡ Mn −Mp = 1.3 We used the Sather prescription, differentiating
MeV, as well as the difference in diquark masses valencepartondistributionfunctionstoobtainva-
arising from the current quark mass difference be- lence CSV PDFs. For this purpose we used the
tweenupanddownquarks. Wedefinethequantity MRST2001 parton distributions [24] at the start-
ing scale, Q2 =1 GeV2. This is slightly too large
δm=m −m , (14) 0
(cid:101) dd uu a value of Q2 for the validity of Sather’s analytic
for which we have a robust estimate δm∼4 MeV approximation, but the resulting errors should be
(cid:101)
[22]. We determine CSV valence PDFs by calcu- small. We then inserted the resulting CSV PDFs
lating the variation of quark model parton distri- into the DGLAP evolution equations and evolved
butions from Eq. (13) with respect to these quan- to the Q2 appropriate for the electron collider ex-
tities, i.e., periments. The results were similar to those ob-
tainedusingtheCSVdistributionsofRodionovet
∂q ∂q
δq ≈ v δm+ v δM . (15) al. .
v ∂(δm) (cid:101) ∂(δM)
(cid:101) Since the CSV effects arising from QED split-
From Eq. (15) the valence charge symmetry vi- ting effects and from quark mass differences are
olating parton distributions are obtained by tak- nearlyindependent,wehaveaddedthetwoeffects
ingvariationswithrespecttodiquarkandnucleon to produce a “net” CSV effect.
massesonvalencepartondistributionsfromquark The quark model estimates of valence parton
models. The resulting PDFs account for quark CSV can be compared with a recent lattice calcu-
and nucleon mass differences that lead to CSV ef- lationofvalencechargesymmetryviolatingparton
fects. distributions[25]. Thelatticecalculationprovides
4
striking confirmation of the quark model results Dimuon cross sections have been measured by
for parton CSV. The lattice calculation was car- the CCFR [29] and NuTeV [30] experiments.
ried out by considering small deviations from the From these reactions one can extract the quan-
SU(3) flavor-symmetric point where the strange tityxs−(x). Theseanalyseshavebeenundertaken
and light quarks masses are all equal. In that way by five groups: CTEQ [31]; Mason et al. [32];
they could estimate the effects of quark mass dif- the NNPDF Collaboration [33]; MSTW [34]; and
ferences on parton distributions. The lattice re- Alekhin, Kulagin and Petti [35]. We used the re-
sults gave estimates for the second moment of the sultsoftheanalysisofMasonet al. oftheNuTeV
valence CSV distributions neutrinoreactions[32]. Wemadeananalyticfitto
(cid:90) 1 thebest-fitresultfromMasonetal. corresponding
δU+ = xδu+(x)dx=−0.0023(6) , to Q2 =16 GeV2. The fit had the form
0
(cid:90) 1 xs−(x)=Axbexp(−cx)(x−0.004) . (18)
δD+ = xδd+(x)dx=+0.0020(3) . (16)
0
The resulting strange quark asymmetry was in-
The lattice results were obtained at a momentum
serted into the DGLAP evolution equation and
scale Q2 = 4 GeV2. Note that the results are
evolved to high Q2.
appropriate for the C-even combination of quarks
ratherthanthedesiredC-oddcombinationforva-
lence quarks, because the lattice calculations are
sensitive to the C-even combination. By com-
parison, the quark model valence quark calcula-
tions at a similar scale obtained δU− = −0.0014 0.005
and δD− = +0.0015 [10, 26]. Note that the lat-
tbioctehrtehsuelstigangraeneds wreiltahtivtheemqauganriktumdeodoeflthreessuelctosnidn x q(x) 0
moment of the valence CSV distributions. The
lattice results are 30-50% larger than the quark -0.005
model values. The differences may result from
the inclusion of singlet contributions in the lat-
0.2 0.4 x 0.6 0.8 1
tice calculations. The lattice results are also in
good agreement with the best value obtained for
valencequarkchargesymmetryviolationinaphe-
FIG. 3: [color online] Parton distributions that oc-
nomenological global fit to high energy data by
cur in the numerator of Eq. (7). Solid curve:
the MRST group [5]. However, the uncertainties xδu−(x); long-dashed curve: xδd−(x); short-dashed
in the lattice calculation are considerably smaller curve: xs−(x). The PDFs have been evolved to
than those from the global fit. Q2 =105 GeV2.
Thereisonefinaltermthatentersintothequan-
tity R−(x) of Eq. (7), namely the strange quark The parton distribution functions that occur in
momentum asymmetry [27, 28] the numerator of Eq. (7) are plotted in Fig. 3.
The solid curve is xδu−(x), the long-dashed curve
xs−(x)≡x[s(x)−s(x)] . (17)
is xδd−(x) and the short-dashed curve is xs−(x).
Strange (antistrange) parton distributions can be As one might expect, the valence CSV distribu-
measured through opposite-sign dimuon produc- tions peak at a relatively large value x ∼ 0.2
tioninitiatedbyneutrinos(antineutrinos). Aneu- while the strange quark asymmetry peaks at an
trino undergoes a charged-current reaction, pro- extremely small x value. Note that due to valence
ducingaµ− andaW+,whichisabsorbedonans quark normalization, all of these quantities must
quarkproducingacharmquark. Thecharmquark have zero first moment, i.e., (cid:104)q(x)(cid:105) = 0, where
subsequently undergoes a semileptonic decay pro- q = [δu−,δd−,s−]. The strange quark asymme-
ducingaµ+ andansquark. Thecrosssectionfor try has zero first moment because the proton has
this process is proportional to the strange quark no net strangeness; the valence CSV distributions
distribution. Thecorrespondingreactioninitiated must have zero first moment because otherwise
byanantineutrinomeasurestheantistrangePDF. this would change the total number of valence
5
quarks in the neutron. So each of these curves tributions to valence PDFs should increase slowly
crosses zero at a small value of x (not shown in with Q2. Our best theoretical estimate of the ra-
Fig. 3). tio R−(x) from Eq. (7) at large x values is pre-
Another notable point is that the signs of these dicted to be rather large, of the order of several
quantities are such that (for values of x above the percent. For reasonably large values x > 0.1, the
crossover point for all of the parton distributions) ratio R−(x) is composed of relatively equal con-
all three contributions should add together in the tributionsfromvalencepartonCSVeffectsarising
numerator of Eq. (7). Fig. 4 shows the expected fromquarkmassdifferencesandfromQEDradia-
value of R−(x) vs. Bjorken x. The solid curve in tion. Thusthequantitativevaluesobtainedforthe
Fig. 4 includes only the QED splitting contribu- ratioR−(x)canprovideafurthercheckontheas-
tion to partonic CSV. The long-dashed curve in- sumptions made in determining charge symmetry
cludesbothQEDsplittingandquarkmasscontri- violation arising from QED radiation.
butions to valence quark CSV. The short-dashed In conclusion, a high energy electron/positron
curve is the result including all three terms in the collider whose beams interact with deuteron
numerator of Eq. (7), including also the contribu- beams from the LHC may produce the most
tion from strange quark asymmetry. promisingobservablewithwhichtosearchforpar-
tonic charge symmetry violating effects.
ThisworkwassupportedinpartbythetheU.S.
National Science Foundation grant NSF PHY-
0
0854805 (JTL) as well as by the Australian Re-
search Council (through an Australian Laure-
-0.02
ate Fellowship) and the University of Adelaide
(AWT).
x)-0.04
_R (
-0.06
-0.08 [1] E.M. Henley and G.A. Miller, Mesons in Nuclei,
0 0.2 0.4x 0.6 0.8 ed.M.RhoandD.H.Wilkinson,(North-Holland,
Amsterdam, 1979), p. 116.
[2] G.A. Miller, B.M.K. Nefkens and I. Slaus, Phys.
Rept. 194, 1 (1990).
FIG. 4: [color online] Contributions to the quantity
[3] W. Bentz, I. C. Cloet, J. T. Londergan
R−(x)vs.xfromEq.(7),wherethePDFsareevolved
et al., Phys. Lett. B693, 462-466 (2010).
to Q2 = 105 GeV2. Solid curve: contribution from
[arXiv:0908.3198 [nucl-th]].
QED splitting parton CSV term only; long-dashed
[4] J.T.Londergan,A.W.Thomas,Phys.Rev.D67,
curve: includes contribution also from quark mass
111901 (2003). [hep-ph/0303155].
CSV term; short-dashed curve: contribution from all
[5] A.D. Martin, R.G. Roberts, W.J. Stirling and
terms including strange quark asymmetry.
R.S. Thorne, Eur. Phys. J. C35, 325 (2004).
[6] J. T. Londergan, J. C. Peng, A. W. Thomas,
We see that for large x>0.2 the strange quark Rev. Mod. Phys. 82, 2009-2052 (2010).
contribution is essentially negligible. The pre- [arXiv:0907.2352 [hep-ph]].
dicted values of R−(x) are large; for x = 0.6 the [7] J. T. Londergan and A. W. Thomas, Prog. in
Part. Nucl. Phys. 41, 49 (1998).
calculated ratio is greater than 6%. This is quite
[8] For details see: http://www.lhec.org.uk
a sizeable result for partonic CSV terms, which
[9] E. Sather, Phys. Lett. B274, 433 (1992).
for most observables yield effects at the 1% level [10] E.N. Rodionov, A.W. Thomas and J.T. Londer-
or smaller [6]. This confirms our argument that, gan, Mod. Phys. Lett. A9, 1799 (1994).
whileboththevalencequarkandvalenceCSVdis- [11] A.D. Martin, R.G. Roberts, W.J. Stirling and
tributions shift to lower x values with increasing R.S. Thorne, Eur. Phys. J. C39, 155 (2005).
Q2, the CSV distributions experience a smaller [12] M. Glueck, P. Jimenez-Delgado and E. Reya,
Phys. Rev. Lett. 95, 022002 (2005).
shift(becausetolowestordertheCSVvalencedis-
[13] Y.L.Dokshitzer,Sov.Phys.JETP46,641(1977).
tributions only radiate photons while the valence
[14] V.N.GribovandL.N.Lipatov,Sov.J.Nucl.Phys.
parton PDFs radiate both gluons and photons), 15, 438 (1972).
andthusforagivenxtheratioofvalenceCSVdis- [15] G.AltarelliandG.Parisi,Nucl.Phys.B146,298
6
(1977). B558, 132 (2003); [hep-ph/0303155].
[16] S. Chekanov et al. (ZEUS Collaboration), [27] A. I. Signal, A. W. Thomas, Phys. Lett. B191,
Phys. Lett. B595, 86 (2004). 205 (1987).
[17] T. Sjostrand et al. , Comput. Phys. Commun. [28] A. W. Thomas, W. Melnitchouk, F. M. Stef-
135, 238 (2001). fens,Phys.Rev.Lett.85,2892-2894(2000).[hep-
[18] G. Marchesini et al. , Comput. Phys. Commun. ph/0005043].
67, 465 (1992). [29] A.O. Bazarko et al. (CCFR Collaboration), Z.
[19] R.L. Jaffe, Nucl. Phys. B229, 205 (1983). Phys. C65, 189 (1995).
[20] A.I. Signal and A.W. Thomas, Phys. Rev. D 40, [30] M. Goncharov et al. (NuTeV Collaboration),
2832 (1989). Phys. Rev. D64, 112006 (2001).
[21] A. W. Schreiber, A. I. Signal, A. W. Thomas, [31] H.L.Lai,P.M.Nadolsky,J.Pumplin,D.Stump,
Phys. Rev. D44, 2653-2662 (1991). W. K. Tung and C. P. Yuan, JHEP 0704, 089
[22] R.P. Bickerstaff and A.W. Thomas, J. Phys. G (2007).
15, 1523 (1989). [32] D. Mason et al., Phys. Rev. Lett. 99, 192001
[23] A.W.Schreiber,A.W.Thomas,J.T.Londergan, (2007).
Phys. Rev. D42, 2226-2236 (1990). [33] R. D. Ball et al. [NNPDF Collaboration], Nucl.
[24] A.D. Martin, R.G. Roberts, W.J. Stirling and Phys. B823, 195 (2009).
R.S. Thorne, Eur. Phys. J. C23, 73 (2002). [34] A. D. Martin, W. J. Stirling, R. S. Thorne and
[25] R. Horsley, Y. Nakamura, D. Pleiter, P.E.L. G. Watt, Eur. Phys. J. C63, 189 (2009).
Rakow, G. Schierholz. H. Stuben. A.W. Thomas, [35] S. Alekhin, S. Kulagin and R. Petti, Phys. Lett.
F. Winter, R.D. Young and J.M. Zanotti, B675, 433 (2009).
arXiv:1012.0215.
[26] J. T. Londergan and A. W. Thomas, Phys. Lett.
7