Analytic treatment of leading-order parton evolution equations: theory and tests

0.33 MB·English

by  Martin M. Block

Analytic treatment of leading-order parton evolution equations: theory and tests
Martin M. Block,1 Loyal Durand,2 and Douglas W. McKay3
1Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208
2Department of Physics, University of Wisconsin, Madison, WI 53706
3Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045
(Dated: January 13, 2009)
WerecentlyderivedanexplicitexpressionforthegluondistributionfunctionG(x,Q2)=xg(x,Q2)
in terms of the proton structure function Fγp(x,Q2) in leading-order (LO) QCD by solving the
2
LO DGLAP equation for the Q2 evolution of Fγp(x,Q2) analytically, using a differential-equation
2
method. We showed that accurate experimental knowledge of Fγp(x,Q2) in a region of Bjorken x
2
and virtuality Q2 is all that is needed to determine the gluon distribution in that region. We re-
derive and extend the results here using a Laplace-transform technique, and show that the singlet
9
0 quarkstructurefunctionFS(x,Q2)canbedetermineddirectlyintermsofGfromtheDGLAPgluon
evolution equation. To illustrate the method and check the consistency of existing LO quark and
0
gluon distributions, we used the published values of theLO quark distributions from theCTEQ5L
2
and MRST2001LO analyses to form F2γp(x,Q2), and then solved analytically for G(x,Q2). We
n findthattheanalytic andfitted gluon distributionsfrom MRST2001LO agree well with each other
a for all x and Q2, while those from CTEQ5L differ significantly from each other for large x values,
J x >∼ 0.03−0.05,atallQ2. WeconcludethatthepublishedCTEQ5Ldistributionsareincompatiblein
3 thisregion. Usinganon-singletevolutionequation,weobtainasensitivetestofquarkdistributions
1 whichholdsinbothLOandNLOperturbativeQCD.WefindineithercasethattheCTEQ5quark
distributions satisfy the tests numerically for small x, but fail the tests for x >∼ 0.03−0.05—their
]
h use could potentially lead to significant shifts in predictions of quantities sensitive to large x. We
p encountered no problems with the MRST2001LO distributions or later CTEQ distributions. We
- suggest caution in theuse of the CTEQ5 distributions.
p
e PACSnumbers: 13.85.Hd,12.38.Bx,12.38.-t,13.60.Hb
h
[

I. INTRODUCTION of G(x,Q2) by solving the LO gluon evolution equation
3
v exactly. Thesearetheprincipaltheoreticalresultsinthis
1 Inarecentpaper[1],we derivedanexplicitexpression paper.
0 for the gluon distribution function G(x,Q2)=xg(x,Q2) To illustrate the method, we have used the LO
2 in the proton in terms of the proton structure function CTEQ5L [3] and MRST2001 [4] quark distributions to
0 Fγp(x,Q2) for γ∗p scattering. The result was obtained construct the proton γ∗p and singlet structure functions
8. in2leading-order (LO) QCD by solving the Dokshitzer- as functions of x and Q2, solvedanalytically for the cor-
0 Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation [2] respondinggluondistributions,andcomparedtheresults
8 fortheQ2 evolutionofFγp(x,Q2)analytically,assuming forG(x,Q2)withthosegiveninthefits. Wefindthatthe
0 2
masslessquarksandusingadifferential-equationmethod. analytic and published distributions agree to high accu-
:
v The method is model-independent and does not require racy for MRST2001 LO, but, to our surprise, that they
Xi individual quark distributions, so accurate experimental are incompatible for CTEQ5L, differing significantly at
knowledge of Fγp(x,Q2) in a region of Bjorken x and large x for all Q2. We find the same situation when we
r 2
a virtualityQ2 is allthat is neededto determine the gluon use the evolution equations for the singlet quark distri-
distribution in that region. butionorforindividualquarkstosolveforG(x,Q2). We
In the present paper, we extend the method using a have not encountered these difficulties in somewhat less-
Laplace-transformtechnique. Wefirstre-derivethesolu- extensivecalculationsusingthemorerecentCTEQ6L[5]
tion for G(x,Q2) in terms of distributions, finding good agreement between the ana-
lytic and published gluon distributions.
nf Toinvestigatetheinconsistenciesfurther,wehaveused
Fγp(x,Q2)= e2x q (x,Q2)+q¯(x,Q2) (1)
2 i i i the DGLAP evolution equation for the non-singlet (NS)
Xi=1 (cid:2) (cid:3) distributions
using our Laplace transform method. The same proce-
dure can be used to derive G(x,Q2) from the singlet (S) nf
F x u (x,Q2)+u¯ (x,Q2)
quark distribution function NS ≡ i i
i=1
nf X (cid:2)d (x,Q2) d¯(x,Q2) , (3)
FS(x,Q2)= x qi(x,Q2)+q¯i(x,Q2) (2) − i − i
Xi=1 (cid:2) (cid:3) wherethesumisovertheup-anddown-type(cid:3)quarksand
and other combinations of the quark distributions. We antiquarksineachgenerationwithbothtypesactive. As
then obtain an analytic solution for F (x,Q2) in terms will be shown, F can be used in both leading order
S NS
2
(LO) and next-to-leading order (NLO) to construct sen- Since
sitive tests of the quark distributions.
5 1
ThetestsbasedonF aresatisfiedfortheMRST2001 Fγp(x,Q2)= F (x,Q2)+ F (x,Q2), (9)
NS 2 18 S 6 NS
LO [4] and CTEQ6L [5] quark distributions which also
satisfiedthegluontestsabovetohighaccuracy. However, we should be able to derive G(x,Q2) consistently in LO
we find that the LO CTEQ5L and the NLO CTEQ5M
fromeitherofEqs.(5)or(6)providedEq.(7)issatisfied.
quarkdistributionsintheMSrenormalizationschemefail
WenotethatEqs.(6)and(7)continuetoholdatnext-
to satisfy the NS evolution equation in the same regions
to-leadingorder with modified quarkand gluonsplitting
at large x where the gluons gave difficulty. This shows
functions in Eqs.(4) and (6). These are givenin the MS
clearly that these published quark distributions are in- schemeinEqs.(4.7a),(4.7b)and(4.7c)ofFloratoset al.
consistent at large x.
[6].
The errorsin the quark distributions and their incom- To simplify the equations, we rewrite Eq.(5) as
patibilitieswithGcouldwellaffectpredictionsforexper-
imental quantities that are sensitive to large x. We con- 1 x dz
clude that the CTEQ5L distributions, though very con- FF(x,Q2)=x G(z,Q2)Kqg z z2 , (10)
venient because of the analytic parametrizations given Zx (cid:16) (cid:17)
[3], should not be used to make precise predictions at where we have introduced (x,Q2), which is defined
large x. as FF
−1
α
(x,Q2) = s e2 (x,Q2). (11)
II. PRELIMINARIES FF 4π i! F2
i
X
onWlyeofinrtshteinptrroodtouncestaruquctaunrteitfyuFnc2t(ixo,nQF2)γ,pw(xh,iQch2)d,ebpyends Squimarilkardliys,trwibeudtieofinneFF(Fx,SQ(x2),Qby2) in terms of the singlet
2 S
F2(x,Q2) ≡ ∂F∂2γlpn((xQ,2Q)2) FFS(x,Q2) = 2nf4απs −1F2(x,Q2), (12)
(cid:16) (cid:17)
αs 4Fγp(x,Q2)+16 Fγp(x,Q2)ln1−x and analogousquantities with appropriateprefactors for
−4π 2 3 2 x other structure functions such as FγZ , FZ , and FW∓.
(cid:26) (cid:20) 2(3) 2(3) 2(3)
1 Fγp(z,Q2) Fγp(x,Q2) dz Wenotealsothatifwewishtoevolveasinglemassless
+xZx (cid:18) 2 z − 2 x (cid:19)z−x(cid:21) Equqa.r(k4)dibstyriQbut=ionxqin(xL,OQ,2w)eancadn simpinlyErqep.(la1c1e) Fby2γpthine
8 1 x dz i i FF
x Fγp(z,Q2) 1+ . (4) appropriate
−3 2 z z2
Zx (cid:16) (cid:17) (cid:27) α −1
(x,Q2) = s (x,Q2), (13)
We define the corresponding quantities S(x,Q2) and FFi 4π Fi
F
(x,Q2) for the singlet and non-singlet quark distri- (cid:16) (cid:17)
NS
bFutions obtained by replacing Fγp by F or F . so that
2 S NS
The LO DGLAP evolution equations [2] for the 1 x dz
proton structure functions F2γp(x,Q2), FS(x,Q2), and FFi(x,Q2)=x G(z,Q2)Kqg z z2 , (14)
FNS(x,Q2)formasslessquarkscanbewrittencompactly Zx (cid:16) (cid:17)
in x space as with i=u,u¯,d,d¯,c,c¯,s,s¯.
Since the gluon content of Eq.(14) is independent of
α 1 x dz
(x,Q2) = s e2x G(z,Q2)K , (5) the type of quark, Eq.(14) has the important conse-
F2 4π Xi i Zx qg(cid:16)z(cid:17) z2 quence that, in LO,
FS(x,Q2) = 2nf4απsx 1G(z,Q2)Kqg xz zd2z, (6) FFi(x,Q2) =1, (15)
(x,Q2) 0, Zx (cid:16) (cid:17) (7) FFj(x,Q2)
NS
F ≡ forallxatafixedvirtualityQ2,andanychoiceofiandj.
wherethesumoverquarkchargesinEq.(5)includesboth As we will soon see, this turns out to be a very sensitive
quarks and antiquarks, and n in Eq. (6) is the number test of compatibility of quark distributions with the LO
f
of active quarks. vanishes only if both members of QCD evolution equations.
NS
F
a quark family are massless or can otherwise be treated An alternate form of this relation,which is readily ex-
as active. The LO g q splitting function is given by tendabletoNLO,followsfromtheevolutionequationfor
→ the non-singlet distribution F (x,Q2), (x,Q2) =
NS NS
K (x)=1 2x+2x2. (8) 0. Splitting the sums in Eq. (3) intoFsuFms over up-
qg
−
3
and down-type quarks, we find that the vanishing of or conversely, the inverse transform of a product to the
(x,Q2) implies that convolution of the original functions, giving
NS
FF
FFdtype(x,Q2) =1 (16) −1[g(s) h(s);v] = vGˆ(w)Hˆ(v w)dw
(x,Q2) L × −
FFutype Z0
v
for all x at any given virtuality Q2. This furnishes us = Hˆ(w)Gˆ(v w)dw. (24)
−
with an exactnumericaltest ofNLO perturbative QCD. Z0
We willlaterseethat,whilethe MRST2001andCTEQ6
Equation(23)allowsustowritetheLaplacetransform
partondistributionssatisfyourconstraintsforallQ2 and
of Eq.(10) as
x, the LO CTEQ5L and NLO MS CTEQ5M quark dis-
tributions satisfy these tests for small x, but fail badly f(s,Q2) = g(s,Q2) h(s), (25)
×
at large x.
where
f(s,Q2)= [ˆ(v,Q2);s] (26)
III. SOLUTION OF THE LO EVOLUTION
LF
EQUATION FOR THE PROTON STRUCTURE
FUNCTION Fγp(x,Q2) andg(s,Q2)andh(s)arethetransformsofthefunctions
2 Gˆ and Hˆ on the right-hand-side of Eq. (20). Solving
Eq.(25) for g, we find that
We now re-derive our analytic solution [1] of Eq.(10)
for the LO gluon distribution, G(x,Q2) using a new g(s,Q2)=f(s,Q2)/h(s). (27)
Laplace-transform method. Introducing the coordinate
transformation We will generally not be able to calculate the inverse
transformofg(s,Q2)explicitly,ifonlybecausef(s,Q2)is
v ln(1/x), (17)
≡ determinedbyanumericalintegraloftheexperimentally-
we define functions Gˆ, Kˆqg, and Fˆ in v-space by dg(est,eQrm2)inaesdthfuenpcrtioodnucFtˆ(ovf,tQh2e)t.woHfouwnecvtieorn,siff(wse,Qr2e)gaanrdd
Gˆ(v,Q2) G(e−v,Q2) h−1(s) and take the inverse Laplace transform using the
≡ convolution theorem in the form in Eq. (24) and the
Kˆ (v) K (e−v)
qg ≡ qg known inverse −1[f(s,Q2);v]= ˆ(v,Q2), we find that
ˆ(v,Q2) (e−v,Q2). (18) L F
F ≡ FF Gˆ(v,Q2) = −1[f(s,Q2) h−1(s);v]
Explicitly, from Eq.(8), we see that L ×
v
= ˆ(w,Q2)Jˆ(v w)dw (28)
Kˆ (v)=1 2e−v+2e−2v. (19) F −
qg − Z0
Further, we can write where Jˆ(v) is a new auxiliary function, defined by
v
ˆ(v,Q2) = Gˆ(w,Q2)e−(v−w)Kˆ (v w)dw Jˆ(v) −1[h−1(s);v]. (29)
F qg − ≡L
Z0
v The calculation of h(s) and the inverse Laplace trans-
= Gˆ(w,Q2)Hˆ(v w)dw, (20) form of of h−1(s) are straightforward,and we find that
−
Z0
where Jˆ(v) = 3δ(v)+δ′(v)
Hˆ(v) ≡ e−vKˆqg(v) e−3v/2 6 sin √7v
= e−v 2e−2v+2e−3v. (21) − √7 " 2 #
−
√7
We introduce the notationthat the Laplace transform
+2cos v . (30)
of a function Hˆ(v) is given by h(s), where " 2 #!
∞
h(s) [Hˆ(v);s]= Hˆ(v)e−svdv, (22) We therefore obtain an explicit solution for the gluon
≡L distribution G(v,Q2) in terms of the integral
Z0
with the condition Hˆ(v) = 0 for v < 0. The convolu- v
Gˆ(v,Q2) = ˆ(w,Q2)Jˆ(v w)dw
tiontheoremforLaplacetransformsrelatesthetransform F −
of a convolution of functions Gˆ and Hˆ to the product of Z0
∂ ˆ(v,Q2)
their transforms g(s) and h(s), so that = 3ˆ(v,Q2)+ F
F ∂v
v v
Gˆ(w)Hˆ(v w)dw;s =g(s) h(s), (23) ˆ(w,Q2)e−3(v−w)/2
L − × − F ×
(cid:20)Z0 (cid:21) Z0
4
6 √7 determined by Fγp as shown in the preceding section.
sin (v w) 2
√7 " 2 − # Again, we simplify the notation by writing this equation
intermsofthe(rescaled)q gsplittingfunctionK (x)
gq
√7 and a quantity (x,Q2) a→s
+2cos (v w) dw. (31) GG
" 2 − #! 1 dz
(x,Q2) = x F (x,Q2)K (x/z) . (33)
GG S gq z2
Transforming Eq.(31) back into x space, we find that Zx
the gluon probability distribution G(x,Q2) is given in Here
terms of the function , assumed to be known, by 2
FF K (x) 2+x, (34)
gq
≡ x −
∂ (x,Q2)
G(x,Q2) = 3 (x,Q2) x FF and
FF − ∂x
3 α −1 ∂G(x,Q2)
1 x 3/2 (x,Q2) s
(z,Q2) GG ≡ 8 4π ∂lnQ2
−Zx FF (cid:16)z(cid:17) × (cid:26)(cid:16) (cid:17) 1
6 √7 z G(x,Q2) (33 2nf)+12ln(1 x)
sin ln − 3 − −
(√7 " 2 x# 1 (cid:18) z x x 2(cid:19) dz
12x G(z,Q2) 2+
√7 z dz − x − z − z z2
+2cos ln . (32) (cid:20)Zx (cid:18) (cid:16) (cid:17) (cid:19)
" 2 x#) z 1 z dz
+ G(z,Q2) G(x,Q2) , (35)
− z x z2
It can be shown without difficulty that this expression Zx (cid:18) − (cid:19) (cid:21)(cid:27)
(cid:0) (cid:1)
forn effectivelymasslessactivequarkflavors. Asbefore,
forGisequivalenttothatwhichwederivedin[1],where f
we first converted the evolution equation for Fγp(x,Q2) going to v space, we define the quantities
2
into a differential equation in v, and then solved that Fˆ (v,Q2) F (e−v,Q2),
S S
equation explicitly. ≡
ˆ(v,Q2) (e−v,Q2),
Even if we cannot integrate the last term of Eq.(32) G ≡ GG
analytically, the usual case, it can be integrated numer- Hˆ (v,Q2) e−vK (e−v)
gq gg
≡
ically, so that we always have an explicit solution which = e−v 2ev 2+e−v
can be evaluated to the numerical accuracy to which −
(x,Q2) is known. We emphasize that this solution = 2 2(cid:0)e−v+e−2v. (cid:1) (36)
−
fForFG is derived from F2γp on the assumption that the TheevolutionequationtheninvolvesaconvolutionofFˆS
quarks are either massless, or effectively so, a situation and Hˆ .
that holds in general for Q2 4M2, where mass effects gq
due to the ith (massive) qua≫rk arei negligible, or for a ofPborothceseiddiensg,saoslvbeeffoorret,hewteratanksfeortmhefLaopflaFˆce,tinravnerstfotrhme
S S
treatment of mass effects such as that used in CTEQ5
transformusing the convolutiontheoremagain,and find
[3] as discussed in Sec. V.1
that
Leading-order solutions for G(x,Q2) of exactly the
∂ˆ(v,Q2) v
same form follow from the evolution equations for the Fˆ (v,Q2) = G + ˆ(w,Q2)e−3(v−w)/2
S
singlet structure function F and the individual quark ∂v G ×
S Z0
distribution functions, with (x,Q2) replaced in Eq.
(32)by (x,Q2)or (xF,QF2), asnotedearlier. The 6 sin √7(v w)
latter caFnFbSe used to oFbtFaiin expressions for G(x,Q2) in (√7 " 2 − #
terms of the structure functions for weak scattering pro- √7
cesses, allowing the incorporation of other data sets. 2cos (v w) dw. (37)
− " 2 − #)
Transforming back into x space, F (x,Q2) is given by
S
IV. SOLUTION OF THE LO GLUON
EVOLUTION EQUATION ∂ (x,Q2)
F (x,Q2) = x GG
S
− ∂x
Toobtainanexpressionwhichrelatesthesingletstruc- 1 x 3/2
+ (z,Q2)
ture function FS(x,Q2) indirectly to experiment, we use GG z ×
the DGLAP equation for the Q2 evolution of the gluon Zx (cid:16) (cid:17)
distribution G(x,Q2). This relates F to G which is 6 √7 z
S sin ln
(√7 " 2 x#
√7 z dz
2cos ln . (38)
1 Wewilldiscussmasseffects indetailelsewhere. − " 2 x#) z
5
(x,Q2) is a known function of the gluon distribu- of the structure functions Fγp and F . The charge fac-
tioGnG,G(x,Q2),which,inturn,isaknownfunctionofthe tors e2 in Eqs. (5) and2 (11), anSd the numbers of
i i
protonstructurefunctionFγp(x,Q2)thatismeasuredin active quarks n in Eqs. (6), (12), and later in Eq. (50),
2 f
deep inelastic scattering. F (x,Q2) is therefore related thenPchange discontinuously at each threshold, but re-
S
backto Fγp(x,Q2)throughG(x,Q2)and the analysisin mainconstantbetweenthresholds. Despite these discon-
2
the preceding section, and could be used, in turn, to de- tinuities, the quark and gluon distributions are continu-
termine G(x,Q2) through Eq. (32) with replacing ous.2
S
FF
. Becausethemasseffectsdonotdependonxinthisap-
FF
It is important to recognize that our method is essen- proach,the evolution equations retain the massless form
tially model-independent. When different parametriza- betweenthresholds,andwecansimplyusetheresultsde-
tions in x and Q2 are used to fit experimental Fγp data, rived above for massless quarks in our analysis, treating
2
the LO gluon and F distributions derived from the dif- the distinct inter-threshold regions in Q2 separately.3
S
ferent fits must agree with each other to the accuracy In these calculations, we use the LO form of α (Q2)
s
of the fits in the regions in which the data exist. The used in CTEQ5L [8],
results can only depend on the functional forms used to
fit Fγp(x,Q2) when they are extrapolated to x and Q2 4π
2 α (Q2) = , (39)
outside that region. s β ln(Q2/Λ2)
0
2
β = 11 n , (40)
0 f
− 3
V. NUMERICAL ILLUSTRATION OF THE
METHOD, AND TESTS OF LO GLUON AND withn =5andΛ =146MeVforQ>4.5GeV,n =4
f 5 f
QUARK DISTRIBUTIONS and Λ = 192 MeV for 1.3 GeV < Q 4.5 GeV, and
4
≤
n = 3 and Λ = 221 MeV for Q < 1.3 GeV. These pa-
f 3
To test of our methods, we have applied them to pub- rametersgive acontinuousαs(Q2) withαs(MZ2)=0.127
lishedpartondistributions,usingthequarkdistributions [3].
to construct Fγp and F and then solving Eq.(32) to The same methods can be used in LO to determine
2 S
find G. The methods work well, and reproduce the pub- thegluondistributionsG generatedbyindividualquark
i
lished gluons distributions except in the case of CTEQ5 distributions, or by other combinations of the quarks, in
[3]. In that case, we found to our surprise that there are particular,bycombinationsofthemasslessquarksu, d, s
problems at high x with the LO CTEQ5L distributions. and by the non-singlet distributions, as will be seen be-
WhilethesearesupersededbymorerecentCTEQdistri- low.
butions [5, 7], they seemstill to be used in some calcula- Following these procedures, the gluon distribution
tions, most likely because they have been parametrized G (x,Q2) we derive analytically from the calcu-
Analytic
analyticallyina formconvenientfor calculation. Similar latedstructure functions Fγp orF ,orfromotherquark
2 S
tests applied to the MRST2001 [4] and the CTEQ6L [5] combinations, should agree to the accuracy of the calcu-
LO distributions uncovered no problems. lations with G (x,Q2) at all x and Q2.
CTEQ5L
The results illustrate the sensitivity of the analytic TheresultsofthecalculationsbasedonFγp areshown
2
methods in testing parton distributions, and may be of inFig.1. Theleft-handcolumncomparesG (blue
Analytic
wider use for that purpose, as well as for deriving G di- dashedcurves)with G (redsolidcurves)atQ2 =
CTEQ5L
rectlyfromdataasoriginallyproposedin[1]. Wepresent 5, 20, and 100 GeV2, (a), (b), and (c), respectively. All
them here as a demonstration, and as a caution.
We begin by illustrating the use of the analytic ex-
pression in Eq.(32) to derive G(x,Q2) from Fγp(x,Q2)
2
in the case of CTEQ5L [3]. We take the published LO 2 The quark and gluon distributions are continuous solutions of
CTEQ5L quark distributions as our basic input, and the evolution equations, which are first-order differential equa-
use these distributions to calculate the proton struc- tionsinlnQ2andintegralequationsinx. Thediscontinuitieson
ture function Fγp(x,Q2)= e2x[q (x,Q2)+q¯(x,Q2)] theright-hadsidesofEqs.(5)and(6)asnf changesatathresh-
2 i i i i oldarereflected inthe finalresults bydiscontinuous changes in
neededin (x,Q2)inEq.(11). Next,wesolveEq.(32) thederivativesofthestructurefunctions∂F/∂lnQ2 ontheleft-
for the GFgeFnerated by thisPFγp, labelling the solution hand sides of these equations. In particular, the heavy quark
2
GAnalytic(x,Q2), and compare the results with the pub- distributions xqi(x,Q2) and xq¯i(x,Q2) are identically zero for
lished gluon distributions. Q2 < Mi2, then initially rise linearly from zero with increasing
WefollowtheproceduresoftheCTEQ5groupinthese l∂n(xQq¯2i)fio/r∂Qln2Q≥2 tMoi2∂.FT/∂helncQon2,triibduentitoicnaslloyfz∂e(rxoqif)oir/∂Ql2nQ<2Mani2d,
calculations. The parton splitting functions used are thereforejumptonon-zerovaluesforQ2≥M2.
i
those for massless quarks. Quark mass effects are in- 3 Mass effects are treated the same way by CTEQ6 [5]. The
cluded only in an approximate,x-independent way,with more accurate treatments of heavy-quark masses inMRST2001
[4] and the recent CTEQ analyses beginning with CTEQ6.5 [7]
xq and xq¯ taken as zero for a massive quark i when
i i involve x-dependent effects. These will not affect the tests of
Q2 < M2, and the quark treated as fully active when
i theMRST2001distributionsgivenlaterusingonlythemassless
Q2 > Mi2, with qi and q¯i included in the calculations quarks.
6
(a) 1.2 2.0 ofGAnalyticorrGforsomex > 0.5isnot thatF2γp(x,Q2)
1.0 goes negative, but rather tha∼t (x,Q2) goes negative.
0.8 1.5 FF
0.6 As can be seen from Eq.(4), this is a sensitive function
G 0.4 rG1.0 of the difference between ∂Fγp(x,Q2)/∂ln(Q2) and the
0.2 0.5 LO convolution integral of F2γp(x,Q2). In spite of the
0.0 2
0.02.10 0.15 0.20 x0.30 0.50 00.0.001 0.0050.010 x 0.0500.100 fFaγcpt(txh,aQt2t)hearCeTaEllQp5oLsiqtiuvaer,kthdeistrreibsuulttiionngs cthomatbginoatiniotno
2
(b) 1.2 2.0 (x,Q2)becomesnegativeforlargex. Theappearance
1.0 FF
ofnegativevaluesforthegluondistributionfunctionsug-
0.8 1.5
G 00..46 rG1.0 gGests stronignlyFitgh.a1t trheesudltiffferroemncpesrobbeltewmesenwGithAntahlyetiicnapnudt
0.2 CTEQ5L
0.5 quark distributions, hence with G , and are less
0.0 Analytic
0.02.10 0.15 0.20 0.30 0.50 00.0.001 0.0050.010 0.0500.100 likely to arise directly from problems with GCTEQ5L.
x x We find very similar deviations if we use the singlet
(c) 1.2 2.0 structurefunctionF (x,Q2)inEq.(2)asinput,andthen
1.0 S
0.8 1.5 solve for G(x,Q2) using Eq. (32) with S replacing
0.6 . FF
G 0.4 rG1.0 FF
0.2 The discrepancies between the analytic and fitted Gs
0.5
0.0 are similar in magnitude to the differences between the
0.02.10 0.15 0.20 0.30 0.50 00.0.001 0.0050.010 0.0500.100 0.500 CTEQ5 and more recent CTEQ and MRST gluon dis-
x x
tributions in this region, and to the changes in the dis-
tributionsthatresultedfromtheadditionofinclusivejet
FIG. 1: Left -hand column: comparison of the G data to those analyses. Changes of this size are clearly
Analytic
(blue dashed curves) obtained by solving Eq.(32) using the significant.
CTTheEuQ-5qLuaFrk2γpd(ixst,rQib2u)tiaosnsinUpu(xt,,Qw2i)th=GxCuT(xE,QQ5L2)((rpedurcpulervdeost)-. x aSrinecferotmhethmeavjoarlecnocnetqriubaurtkiosnus atondFd2γ,p(wxe,Qha2v)eadtolnaregae
dashedcurves)areshowntogiveanindependentscaleatthe separatesetofcalculationsusingthesingletcombination
same x and Q2. (a) Q2 = 5 GeV2, nf = 4. (b) Q2 = 20 F = x(u+u¯+d+d¯) and Eq. (32) with re-
GeV2, nf = 4. (c) Q2 = 100 GeV2, nf = 5. Right-hand plSa,cuidng . No mass effects need to be incluFdeFdSi,nudthis
caoslu(am),n(:bt)h,e(cr)a.tTioherGra=tioGsAshnoaluytldic/eGquCaTlE1Q5fLoraatllthxeasnadmQe2Q. 2 case. ThFeFresults of the calculations, called Gud(x,Q2),
are shown in Fig. 2 for both CTEQ5L and MRST2001
LO.
Discrepancies between G and the G analyti-
of the active, physically-relevant quarks are included in CTEQ5L
theinputateachvalueofQ2,withnf =4for(a)and(b) cally reconstructed from FS,ud, here labeled Gud(x,Q2),
are again evident in the figure. The discrepancies have
with u, d, s, c active, and n = 5 in (c) with b now also
f
active. For a scale comparison, we include the CTEQ5L thesamepatternandareroughlythesamesizesasthose
up-quark distribution, U(x,Q2)=xu(x,Q2). seen in Fig. 1, suggesting that the problems originate
with the u and d distributions that drive the initial
TheanalyticandfittedGsdonotagreewellatlargex,
DGLAPevolutionatlargex. TheresultsforMRST2001
with significant differences between them on the scale of
Gandoftheu-quarkdistributionfor0.05<x<0.6. We LO show agreement between the calculated Gud and
do not believe that the calculations are r∼eliab∼le beyond GMRST2001 at the level of accuracy of the calculation,
which required the parametrization of numerical data
x 0.6: Gisverysmall,buttheinputuandddistribu-
≈ from [9].
tionsandthe individualintegralsin arenot,andthe
calculation in Eq. (32) involves largFeFcancellations and To exclude the possibility that the problems with
becomes sensitive to small errors in the inputs. CTEQ5 can be eliminated by going to NLO perturba-
The same discrepancies between the two Gs are il- tive QCD, we also looked at the evolution of the non-
lustrated differently in the right-hand column of Fig. 1 singlet structure function FNS(x,Q2), and at the gluon
where we plot the ratio distribution from the evolution of the singlet structure
function F (x,Q2), using the CTEQ5M NLO distribu-
S
r =G (x,Q2) G (x,Q2). (41) tions as input. We found a similar x-dependence for the
G Analytic CTEQ5L
gluon ratio obtained from the NLO singlet distribution,
In all cases, we find that r i(cid:14)s essentially constant and hence, continuing problems. The NS combination, for
G
equal to one for all x < 0.01 for small virtuality, and for both LO and NLO, is discussed in the next section, and
allx < 0.05forlargev∼irtuality,showinggoodagreement showsclearly that there areproblems with the CTEQ5L
betwe∼en analytic and fitted gluon distributions at small quark distributions.
x. There are again significant deviations from one at We note finally that, to reduce the possibility that
larger x, contrary to theoretical requirements. the discrepancies that we have encountered are a arti-
We emphasize that the reason we find negative values fact of the published parametrization of the CTEQ5L
7
(3) into sums over u- and d-type quarks, defined as
u x u(x,Q2)+u¯(x,Q2)+c(x,Q2)+c¯(x,Q2) ,(43)
type
≡
dtype x(cid:2)d(x,Q2)+d¯(x,Q2)+s(x,Q2)+s¯(x,Q2)(cid:3),(44)
≡
this relati(cid:2)on becomes (cid:3)
(x,Q2)= (x,Q2) (x,Q2)=0,,
NS utype dtype
FF FF −FF
(45)
wherethetwotermsaretobecalculatedseparatelyusing
the same NLO splitting functions as for the full F .
NS
Dividing by the u-type term, we obtain the ratio
(x,Q2)
dtype
r FF =1, (46)
NS ≡ (x,Q2)
utype
FF
a relation true for both LO and NLO when all the rele-
vant quarks are active.
To use this relation to check the consistency of the
quarkdistributions,weinserttheappropriateLOorNLO
quark and anti-quark distributions in Eqs. (43) and(44)
and use them to generate r (x,Q2) numerically using
NS
the LO or NLO splitting functions.
InthecaseoftheNLOCTEQ5MMSdistributions,we
FIG. 2: Plots of the n = 2 G (x,Q2) (blue dashed use
f ud
curves), G (x,Q2) (red curves), and the u-quark distri-
fitted 4π
bution U(x,Q2) = xu(x,Q2) (purple dot dashed curves) vs. α (Q2) =
x, for x > 0.1. Left-hand column: Gfitted = GCTEQ5L distri- s β0ln(Q2/Λ2) ×
butionsfor(a)Q2 =5GeV2,(b)Q2=20GeV2,(c)Q2=100 2β ln[ln(Q2/Λ2)]
GeV2. Right-handcolumn: Gfitted =GMRST2001 LOdistribu- 1− β21 ln(Q2/Λ2) , (47)
tions for (d) Q2 =5 GeV2, (e) Q2 =20 GeV2, (f) Q2 =100 (cid:20) 0 (cid:21)
GeV2. 2
β = 11 n (48)
0 f
− 3
19
β = 51 n , (49)
parton distributions [3], we have also compared them to 1 − 3 f
thecorrespondingnumericalCTEQ5Ldistributionsfrom
theDurhamwebsite[9]andconcludedthatthey arenu- withαs(Mz2)=0.118for nf =5, matchingto the α’s for
merically compatible. nf =4 and 3 at Q=4.5 and 1.3 GeV.
Because the G (x,Q2) derived from either Intheleft-handcolumnofFig. 3,weplotrNS vs. x,for
Analytic
Fγp(x,Q2) or F (x,Q2) fails to match C (x,Q2) Q2 = 5, 20, and 100 GeV2 and nf = 4. The solid (red)
2 S CTEQ5L
curveswerecalculatedusingthe LOCTEQ5Lquarkdis-
well at large x, we cannot expect the inverse problem,
tributions as input, and the dashed (blue) curves, the
the analyticderivationofF fromGtoworkwellforthe
S
NLO CTEQ5Mquark distributions in the MS renormal-
CTEQ5L distributions, and do not include such calcula-
izationscheme. Weseethatthepatternsareverysimilar
tions here.
for all Q2, with ratios r 1 in absolute normalization
NS
forallx < 0.01,andwithv≈erylargedeviationsoccurring
at large ∼x. This result, independent of the value of Q2,
VI. TESTS OF LO AND NLO NON-SINGLET
shows explicitly that there are problems with the quark
DISTRIBUTIONS
distributionsatlargex: theNSevolutionequationisnot
satisfied in either leading or next-to-leading order.
For massless quarks in LO, the relation in Eq.(15)
To look at the size of the deviations in a more quanti-
holds for any quark or anti-quark pair i and j,
tative way,we introduce the notionof a LO “non-singlet
gluon” distribution G (x,Q2), defined by the equation
(x,Q2) NS
i
FF =1. (42)
(x,Q2) α 1 x dz
FFj (x,Q2)=2n sx G (z,Q2)K .(50)
FNS f4π NS qg z z2
As discussed earlier, this can be generalized read- Zx (cid:16) (cid:17)
ily to NLO using the non-singlet evolution equation G = 0 if (x,Q2) is a solution of the non-singlet
NS NS
(x,Q2)=0,Eq.(7)withtheNLONSsplittingfunc- DGLAPequaFtionforanevennumberofeffectivelymass-
NS
F
tions ofFloratoset al. [6]. Ifwe separatethe sumin Eq. lessquarks. Further,G isdeterminedmainlybytheu
NS
8
vanish indicates an inconsistency of the u and d distri-
butions as solutions of the DGLAP evolution equations,
and absolute magnitudes of the ratio near 1 would indi-
cate that the integrated discrepancies are large.
Asseenfromtheright-handcolumninFig. 3,theratio
r is zero for small x, but has large absolute values
NS,G
for x > 0.1 for Q2 = 5, 20, and 100 GeV2, Figs. 3(d),
(e), an∼d (f) respectively. In particular, around x 0.3,
∼
the ratio is about unity, indicating a serious discrepancy
near the maxima of the valence-quark distributions.
Thepatterninxofthediscrepanciesbetweenthegluon
distributions found analytically from either Fγp or F
2 S
and the distributions published by the CTEQ5 group
closelyfollowsthepatternseeninthedepartureof
NS
FF
from zero. For example, and the differences be-
NS
FF
tween the analytic and fitted gluon distributions all be-
comenegativeorpositiveinthesameregions. Thiswould
seem to indicate that the quark distributions at large x
(and, in particular, the dominant u and d valence distri-
butions) are the origin of the difficulty, the combination
oftheirshapes andQ2 dependence notbeingcompatible
with the DGLAP evolution equations, either in LO or
in NLO in the MS renormalizationscheme.
FIG. 3: Left-hand column: the LO and NLO non-singlet
VII. CONCLUSIONS
d-type to u-type ratio of Eq. (46) for n = 4 and Q2 = 5,
f
20, and 100 GeV2, (a), (b), and (c), respectively. The ratio
shouldbe1atallxandQ2. Thesolid(red)curvesareLO,cal- 1. We have first developed a powerful method for
culated from Eq.(43), Eq.(44), and Eq.(16) using CTEQ5L the analytic solution of the LO DGLAP evolu-
data for quark distributions . The dashed (blue) curves are tion equations based on Laplace transforms. This
NLOcalculatedfortheMSCTEQ5Mquarkdistributions[3]. method allows us to determine G(x,Q2) directly
Right-handcolumn: rNS,G =GNS(x)/GCTEQ5L(x)vs. x,for from Fγp(x,Q2) or other measurable structure
Q2 = 5, 20, and 100 GeV2, (d), (e), and (f), respectively. function2s such as FγZ , FZ , and FW∓, provid-
Thenon-singlet“gluon”distributionGNS(x)calculatedfrom 2(3) 2(3) 2(3)
ing close and simple connections to experiment for
u and d quarks is defined in the text in Eq.(50) and is ex-
pected to be zero. massless or effectively massless quarks.
We can also determine G(x,Q2) analytically from
the singlet quark distribution F (x,Q2) if the
S
and d distributions at large x, so measures the accuracy quark distributions are known, or F (x,Q2) from
S
of those distributions. G(x,Q2)whenthe latterisknown. Thesetofrela-
The normalization on the right-hand side of Eq.(50) tions provide consistency checks on LO quark and
hasbeenchosensothatthisequationisthe formalcoun- gluon distributions obtained in other ways. These
terpart of the singlet equation Eq.(7). are the principal theoretical results of the paper.
The ratio
2. Asanillustrationofourmethods,wehaveusedthe
GNS(x,Q2) analyticsolutionsto the evolutionequationsto ob-
r , (51)
NS,G ≡ G(x,Q2) taintestsoftheconsistencyofpublishedquarkand
gluon distributions. In particular, we compare the
shouldvanishforsolutionsoftheDGLAPequations,and gluon distributions determined analytically from
anydeviationsarescaledtothesizeoftheactual(singlet) structure functions Fγp(x,Q2) and F (x,Q2) cal-
2 S
gluon distribution GCTEQ5L. culatedfromthe quarkdistributionsdeterminedin
Weconsiderspecificallythenon-singletcombinationof various analyses, to the gluon distributions given
massless u and d quarks, by the same analyses. We have found no prob-
lems for analytic and fitted gluon distributions for
F (x,Q2) x u(x,Q2)+u¯(x,Q2)
NS,ud theMRST2001LOorCTEQ6Lsolutions,whilethe
≡
(cid:2)d(x,Q2) d¯(x,Q2)](cid:3), (52) analytic and fitted gluon distributions for CTEQ5,
− −
though consistent at small x, show small, but sig-
and calculate r for that combination, shown in the nificant, deviations in the large x region, indicat-
NS,G
right-hand column of Fig. 3. Any failure of the ratio to ingthatthe quarkandgluonsdistributionsarenot
9
completely consistent [10]. 4. We suggest that groups that calculate parton dis-
tributions numerically from the coupled DGLAP
3. A further analysis using the non-singlet structure
equations construct the appropriate analytic solu-
function F (x,Q2) and both the LO CTEQ5L
NS tions to test the consistency of their results.
andtheNLOCTEQ5Mquarkdistributionsshowed
definitively that there are problems with those dis-
Wenote,finally,thatweareworkingonanextensionof
tributions at high x. The quark distributions are
our analyticLO solutionfor G(x,Q2) to include massive
not consistent with either LO (CTEQ5L) or NLO
candbquarks,usingthemethodsof[7],andontheNLO
perturbative QCD in the MS scheme (CTEQ5M)
gluon solution.
in the sense that they do not satisfy the appropri-
ate non-singlet relations for x > 0.05 even though
they satisfy them very well at∼small x. The dis-
Acknowledgments
crepancies,ofunknownorigin,arelargeforx 0.3,
∼
andcouldhaveseriousconsequencesforpredictions
of processes sensitive to that x region. We con- The authors would like to thank the Aspen Center for
clude that the CTEQ5 distributions [3] should not Physics for its hospitality during the time much of this
beusedtomakesuchpredictions. Again,wefound work was done.
no problems with the MRST2001 distributions [4] D.W.M. receives support from DOE Grant No. DE-
or the more recent CTEQ6 distributions [5, 7]. FG02-04ER41308.
[1] M. M. Block, L. Durand,and D. W.McKay, Phys.Rev. [hep-ph/0110215].TheMRST2001distributionswereob-
D 77 , 094003 (2008) [arXiv:0710.3212 [hep-ph]]. tained using the Durham parton distribution generator
[2] V.N.Gribov andL. N.Lipatov,Sov.J. Nucl.Phys. 15, at http://durpdg.dur.ac.uk/hepdata/pdf3.html.
438(1972);G.AltarelliandG.Parisi,Nucl.Phys.B126, [5] J. Pumplin, D.R. Stump, J. Huston, H.L. Lai,
298 (1977); Yu.L.Dokshitzer, Sov.Phys.JETP 46, 641 P. Nadolsky, W.K. Tung, JHEP 02, 07:012 (2002)
(1977). [hep-ph/0201195].
[3] CTEQ Collaboration, H. L. Lai et al., Eur. Phys. [6] E. G. Floratos, C. Kounnas and R. Lacaze, Nucl. Phys.
J. C12, 375 (2000) [hep-ph/9903282]. The LO B192, 417 (1981).
CTEQ5L parton distributions were calculated us- [7] CTEQCollaboration,W.K.Tung,H.L.Lai,A.Belyaev,
ing the Mathematica package CTEQ5L of the CTEQ J. Pumplin, D. Stump, and C.-P. Yuan, J. High Energy
group, http://www.phys.psu.edu∼cteq/Mathematica, Phys. 0702:053, (2007) [hep-ph/0611254].
and independently using the CTEQ5 distributions [8] arXiv:hep-ph/0512167 v4, J. Pumplin et al. (2006);
from the Durham parton distribution generator at http://www.phys.psu.edu∼cteq/CTEQ5Table.
http://durpdg.dur.ac.uk/hepdata/pdf3.html. The pa- [9] http://durpdg.dur.ac.uk/hepdata/mrs.html.
rametersfortheLOαs(Q2)aregivenintheintroductory [10] There were some hints that early CTEQ pdf’s might
notes in the CTEQ5L Fortran program on the CTEQ be inconsistent with those derived using other evolution
web site, http://www.phys.psu.edu∼cteq/. codes; see J. Blumlein et al., hep-ph/9609400 (1996).
[4] A.D. Martin, R.G. Roberts, W.J. Stirling and R.S.
Thorne, MRST2001, Eur. Phys. J. C23, 73 (2002)