Analytic derivation of the next-to-leading order proton structure function $F_2^p(x, Q^2)$ based on the Laplace transformation

0.55 MB·

by  Hamzeh Khanpour

Analytic derivation of the next-to-leading order proton structure function Fp(x,Q2)
2
based on the Laplace transformation
Hamzeh Khanpour1,2,∗ Abolfazl Mirjalili3,† and S. Atashbar Tehrani4‡
(1)Department of Physics, University of Science and Technology of Mazandaran, P.O.Box 48518-78195, Behshahr, Iran
(2)School of Particles and Accelerators, Institute for Research in
Fundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran
(3)Physics Department, Yazd University, P.O.Box 89195-741, Yazd, Iran
(4)Independent researcher, P.O.Box 1149-8834413, Tehran, Iran
(Dated: September26, 2016)
Inthepresentarticle,ananalyticalsolutionbasedontheLaplacetransformationtechniqueforthe
6 DGLAPevolutionequationshasbeenpresentedatthenext-to-leadingorderaccuracyinperturbative
1 QCD. This technique is also has been applied to extract the analytical solution for the proton
0 structure function, Fp(x,Q2), in the Laplace s-space. We present the results for the separate
2
2 parton distributions for all parton species, including valance quark densities, the anti-quark and
p strange sea PDFs, and for the gluon distribution. We successfully compare the obtained parton
e distribution functions and theproton structurefunction with theresults from GJR08 [Eur. Phys. J
S C53(2008)355-366]andKKT12[J.Phys. G40(2013)045002]parametrizationmodelsaswellasthe
x-spaceresultsusingQCDnumcode. Ourcalculations showaverygood agreementwith theavailable
3
theoretical models as well as the DIS experimental data throughout the small and large values of
2
x. The usage of our analytical solution to extract the parton densities and the proton structure

function has been discussed in details to justify the analysis method considering the accuracy and
]
h speed of calculations. Overall, theaccuracy which we obtain from theanalytical solution using the
p inverse Laplace transforms technique found to be better than 1 part in 104−5. We also present a
- detailed QCD analysis of non-singlet structure functions using all available DIS data to perform
p
global QCD fits. In this regard we employ the Jacobi polynomials approach to convert the results
e
from Laplace s-space to Bjorken-x space. The extracted valence quark densities are also presented
h
and compared to the JR14, MMHT14, NNPDF and CJ15 PDFs sets. We evaluate the numerical effects
[
oftargetmasscorrections(TMCs)andhighertwistterms(HT)onvariousstructurefunctions,and

2 compare fitsto data with and without these corrections.
v
8 PACSnumbers: 12.39.-x,14.65.Bt,12.38.-t,12.38.Bx
0
5
3 CONTENTS PDF parametrizations 11
0 Data sets 11
.
1 I. Introduction 1 Statistical procedures 11
0 B. Target mass corrections (TMCs) 11
6 II. Theoretical formalism 2 C. Higher twist (HT) corrections 12
1
D. Results of QCD fit 13
:
v III. Singlet solution in Laplace space at the
i next-to-leading order approximation 3 VIII. Summary and conclusion 14
X
r IV. Non-singlet solution in Laplace space at the Acknowledgments 15
a
next-to-leading order approximation 5
Appendix A: The Laplace transforms of
V. Proton structure function F2p(x,Q2) in Laplace splitting functions at the NLO approximation 17
space 6
Appendix B: The Wilson Coefficients of singlet
VI. The results of Laplace transformation and non-singlet distributions in the Laplace s
technique 7 space at the NLO approximation 23
VII. Jacobi polynomials technique for the DIS
References 24
analysis 9
A. Method of the QCD analysis 11
I. INTRODUCTION
∗ [email protected] Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)
† [email protected] evolution equations [1–4] are a set of an integro-
‡ [email protected] differentialequationswhichcanbeusedtoevolvethepar-
2
tondistribution functions (PDFs) to anarbitraryenergy comparethe results with the DIS data usedin our PDFs
scale, Q2. The solutions of the DGLAP evolution equa- fit.
tionswillprovideusthegluon,valanceandseaquarkdis- The present paper organized as follows: In Section II,
tributions inside the nucleon. Consequently these equa- weshallprovideabriefdiscussionforthe theoreticalfor-
tions widely can be usedas fundamentaltools to extract malismoftheprotonstructurefunctionFp(x,Q2)atthe
2
the deep inelastic scattering (DIS) structure functions NLO approximation of QCD. A detailed formalism to
(SF) of proton, neutron and deuteron which enrich our establish an analysis method for the solution of DGLAP
current information about the structure of the hadrons. evolution using the repeated Laplace transforms for sin-
The standard procedure to obtain the x dependence of glet sector have been presented in Section. III. In Sec-
the gluonandquarkdistributions is to solvenumerically tion. IV, we also review the method of the analyticalso-
the DGLAP equations and compare the solutions with lution of DGLAP evolution equations based on Laplace
the data in order to fit the PDFs at some initial fac- transformationtechniques for non-singletsector. In Sec-
torization scale, typically less than the square of the c- tion. V, we utilize this method to calculate the proton
quark mass Q2 (m2 2 GeV2). The initial distribu- structure function Fp(x,Q2) by Laplace transformation.
0 ≈ c ≈ 2
tionsforthegluonandquarkareusuallydeterminedina Weattemptadetailedcomparisonofournext-to-leading
globalQCDanalysisincludingawidevarietyofDISdata order results with recent results from the literature in
fromHERA[5–10]andCOMPASS[11],hadroncollisions Section. VI. We also discuss in details the usage of our
at Tevatron [12–15] and fixed-target experiments over a analyticalsolutiontojustifytheanalysismethodinterms
large range of x and Q2 as well as ν(ν¯)N xF data from of accuracy and speed. A completed comparison be-
3
CHORUS and NuTeV [16, 17], and also the data for the tween the obtained results and available DIS data are
longitudinal structure function F (x,Q2) [18]. Finally also presented in this section. The application of the
L
using the coupled integro-differential DGLAP evolution Laplace transformation techniques and Jacobi polyno-
equations one can find the PDFs at higher energy scale, mial expansion machinery at the next-to-leading order
Q2. For mostrecent studies onglobalQCD analysis,see was described in detail in Sec. VII. The method of the
for instance [19–26]. QCD analysis including the PDFs parametrization, sta-
tistical procedures and the data selection also presented
Some analytical solutions of the DGLAP evolution
in this section. The numerical effects of targetmass cor-
equations using the Laplace transform technique, ini-
tialized by Block et al., have been reported in recent rections(TMCs)andhighertwistterms(HT) onvarious
structure functions are also discussed. Finally, we give
years [27–37] with considerable phenomenological suc-
our summary and conclusions in Section. VIII. In Ap-
cess. In this paper, a detailed analysis has been per-
pendix A, we render the results for the different split-
formed, using the repeated Laplace transforms, in order
ting functions in the Laplace transformed, s space and
to find an analytical solutions of the DGLAP evolution
Appendix B includes the analytical expression for the
equationsatthenext-to-leadingorder(NLO)approxima-
Wilson coefficient functions of the singlet and gluon dis-
tions. Wealsoanalyticallycalculatetheindividualgluon,
tribution in s-space.
singlet and non-singlet quark distributions from the ini-
tial distributions inside the nucleon. We present our re-
sults for the valance quark distributions xu and xd ,
v v
II. THEORETICAL FORMALISM
the anti-quarkdistributionsx(d+u)andx∆=x(d u),
−
the strange sea distribution xs = xs and finally for the
gluondistributionxg. UsingtheLaplacetransformtech- The present DIS and hadron collider data provide the
nique,wealsoextracttheanalyticalsolutionsforthepro- best determination of quark and gluon distributions in
ton structure function Fp(x,Q2) as the sum of a flavour a wide range of x [7, 9, 10]. In this article we will be
2
singlet FS(x,Q2), Fg(x,Q2) and a flavour non-singlet concerned specifically with the proton structure func-
2 2
FNS(x,Q2) distributions. The obtained results indicate tion at the next-to-leading order accuracy in perturba-
2
anexcellentagreementwiththeDISdataaswellasthose tive QCD. In the common MS renormalization scheme
obtainedbyothermethodsuchasthe fittoF2p structure the F2(x,Q2)structurefunction, extractedfromthe DIS
function performed by KKT12 [20] and GJR08 [38]. epprocess,canbe writtenas the sumofa flavoursinglet
F (x,Q2), F (x,Q2) and a flavour non-singlet distri-
In the present work, we also demonstrate once more 2,S 2,g
butions F (x,Q2) in which we will have,
thecompatibilityoftheLaplacetransformtechniqueand 2,NS
the Jacobi polynomial expansion approach at the next- F (x,Q2) 1
to-leading order and extract the valence quark densities 2 = F2,S(x,Q2)+F2,g(x,Q2)+F2,NS(x,Q2)
x x
as well as the values of the parameter α (M2) from the
s Z =<(cid:0)e2 >C (x,Q2) q (x,Q2) (cid:1)
QCDfittotherecentDISdata. Theeffectoftargetmass 2,S ⊗ S
corrections (TMCs), which are important especially in +<e2 >C2,g(x,Q2) g(x,Q2)
⊗
thehigh-xandlow-Q2regions,andthecontributionfrom +C (x,Q2) q (x,Q2), (1)
2,NS NS
higher twist (HT) terms also considered in the analysis. ⊗
To quantify the size of these corrections,we evaluate the here g and q representthe gluon and quark distribution
i
structurefunctionsatnext-to-leadingorderinQCD,and functions respectively. The q stands for usual flavour
NS
3
nanond-qsingstleatndcofmorbitnhaetifloanv,oxuurv-si=ngxle(utq−uu¯a)r,kxddivst=ribxu(tdio−n,d¯) α 4(Qπ2)∂l∂nGQ2(x,Q2)=FS⊗ Pg0q + αs4(Qπ2)Pg1q (x,Q2)
S s (cid:18) (cid:19)
α (Q2)
Nf + G⊗ Pg0g + s4π Pg1g (x,Q2), (3)
xqS = x(qi+q¯i), (cid:18) (cid:19)
Xi=1 where αs(Q2) is the running coupling constant and the
splitting functions P0(x,α (Q2)) and P1(x,α (Q2)) are
where N denotes the number of active massless quark ij s ij s
f the Altarelli-Parisisplittingkernelsatoneandtwoloops
flavours. InEq.(1)the symboldenotestheconvolution
⊗ correction respectively as [4, 40, 41],
integralwhichturnsintoasimplemultiplicationinMellin
N-space and < e2 > represents the average squared α (Q2)
charge. The C2,S and C2,NS are the common next-to- Pij(x,αs(Q2))=PiLjO(x)+ s2π PiNjLO(x). (4)
leading order Wilson coefficient functions [39]. The ana-
In the evolution equations, we take the N = 4 for
lyticalexpressionfortheadditionalnext-to-leadingorder f
m2 < µ2 < m2 and N = 5 for m2 < µ2 < m2 and
gluoniccoefficientfunctionC2,g canbefoundinRef.[39]. c b f b t
adjust the QCD parameter Λ at each heavy quark mass
As we already mentioned the gluon and quark distribu-
tionfunctionsattheinitialstateQ2canbedeterminedby threshold,µ2 =m2c andm2b. Consequentlytherenormal-
0 ized coupling constant α (Q2) can be run continuously
fittothepreciseexperimentaldataoveralargenumerical s
rangeforxandQ2. Theindividualquarksandgluondis- when the Nf changes at the c and b mass threshold [42].
We now in a position to briefly review the method
tributionsparameterizedwiththepre-determinedshapes
of extracting the parton distribution functions via an-
as a standard functional form. This function is given in
terms of x and a chosen value for the input scale Q2. alyticalsolutionofDGLAP evolutionequationusing the
0
The gluon distribution xg(x,Q2) is a far more difficult Laplace transformation technique. By considering the
0
variable changes ν ln(1/x) and w ln(1/z), one can
case for PDFs parameterizations to obtain precise infor-
≡ ≡
rewritetheevolutionequationspresentedinEqs.(2,3)in
mationduetothelittleconstraintprovidedbytherecent
terms of the convolution integrals and with respect to v
data [20, 25].
and τ variables as [27, 28],
In the following, we will present our analytic method
based on the newly-developed Laplace transform tech-
∂Fˆ
niquetodeterminethenon-singletFNS(x,Q2)andsinglet S(ν,τ)=
structure functions F (x,Q2) and G(x,Q2) using the in- ∂τ
aptutQd20i=str2ibGuetVio2n.sWFeNSuS0s(ext,hQe20K)K,TF1S20([x20,Q]a20n)daGnJdRG080([3x8,]Qin20)- ˆ0ν(cid:18)Kˆqq(ν−w) + α4s(πτ)Kˆq1q(ν−w)(cid:19)FˆS(w,τ)dw
putpartondistributionstodeterminetheindividualpar- ν α (τ)
tondistributionfunctionsatanarbitraryQ2 >Q20 which +ˆ Kˆqg(ν−w) + s4π Kˆq1g(ν−w) Gˆ(w,τ)dw
can be obtained, using the DGLAP evolution equations. 0 (cid:18) (cid:19)
(5)
Having the parton distribution functions and using the
inverse Laplace transform, one can extract the proton
structure function Fp(x,Q2) as a function of x at any ∂Gˆ
desired Q2 value. 2 ∂τ (ν,τ)=
ν α (τ)
Kˆ (ν w) + s Kˆ1 (ν w) Fˆ (w,τ)dw
ˆ gq − 4π gq − S
0 (cid:18) (cid:19)
III. SINGLET SOLUTION IN LAPLACE SPACE ν α (τ)
AT THE NEXT-TO-LEADING ORDER + Kˆ (ν w) + s Kˆ1 (ν w) Gˆ(w,τ)dw,
APPROXIMATION ˆ0 (cid:18) gg − 4π gg − (cid:19)
(6)
Forthemostimportanthighenergyprocessesthenext-
wheretheQ2 dependenceofaboveevolutionequationsis
to-leadingorderapproximationisthestandardonewhich
expressedentirelythoroughthevariableτ asτ(Q2,Q2)
we also consider it in our analysis. The DGLAP evolu- 0 ≡
tionequationscandescribethe perturbativeevolutionof 41π QQ22αs(Q′2)d lnQ′2. Note that we used the nota-
the singlet xqS(x,Q2) and gluon xg(x,Q2) distribution tion´ Fˆ0S(ν,τ) FS(e−ν,Q2) and Gˆ(ν,τ) G(e−ν,Q2).
functions. The coupled DGLAP evolution equations at The above con≡volution integrals show tha≡t using 1-loop
the next-to-leading order approximation, using the con- Kˆ0(ν) e−νP0(e−ν) and 2-loop Kˆ1(ν) e−νP1(e−ν)
volution symbol , can be written as [33, 34] keirjnels≡where tihje i and j are a comibjinat≡ion of qiujark q
⊗
or gluong, one canobtain the singletFˆ (ν,τ) and gluon
S
4π ∂FS (x,Q2)=F P0 + αs(Q2)P1 (x,Q2) Gˆ(ν,τ) sectors of distributions.
αs(Q2)∂lnQ2 S⊗(cid:18) qq 4π qq(cid:19) Defining the Laplace transforms f(s,τ)
≡
+ G P0 + αs(Q2)P1 (x,Q2), (2) L[FˆS(ν,τ);s] and g(s,τ) ≡ L[Gˆ(ν,τ);s] and using
⊗ qg 4π qg this fact that the Laplace transform of a convolution
(cid:18) (cid:19)
4
factors is simply the ordinary product of the Laplace −1[k (s,τ);ν] and the input distributions by Fˆ0(ν)
L ij S ≡
transform of the factors, which have been presented in −1[f0(s);ν] and Gˆ0(ν) −1[g0(s);ν]. Then the fol-
[27, 29], the Laplace transform of Eqs.(5, 6) convert to Llowingdecoupledsolution≡swLithrespecttoν andQ2vari-
an ordinary first order differential equations in Laplace ables and in terms of the convolutions integrals can be
space s with respect to τ variable. Therefore we will written as,
arrive at,
v
Fˆ (ν,Q2)= K (ν w,τ)Fˆ0(w)dw
∂f(s,τ)= ΦLO(s) + αs(τ)ΦNLO(s) f(s,τ) S ˆ0 FF − S
∂τ f 4π f ν
(cid:18) (cid:19) + K (ν w,τ)Gˆ (w)dw, (14)
+ ΘLO(s) + αs(τ)ΘNLO(s) g(s,τ), (7) ˆ0 FG − 0
f 4π f ν
(cid:18) (cid:19) Gˆ(ν,Q2)= K (ν w,τ)Gˆ (w)dw
ˆ GG − 0
0
∂g α (τ) ν
∂τ(s,τ)= ΦLgO(s) + s4π ΦNgLO(s) g(s,τ) +ˆ KGF(ν−w,τ)FˆS0(w)dw. (15)
(cid:18) (cid:19) 0
α (τ)
+ ΘLO(s) + s ΘNLO(s) f(s,τ), (8) Consideringtheν ln(1/x),onecanfinallyarriveatthe
g 4π g solutionsoftheDG≡LAPevolutionequationswithrespect
(cid:18) (cid:19)
to x and Q2 variables. As we mentioned earlier, the Q2
whose the leading-ordersplitting functions for the struc-
dependence of the distributions functions Fˆ (ν,Q2) and
turefunction F , presentedin[4,43]inMellin-space,are S
given by ΦLO 2and ΘLO at Laplace s space by: Gˆ(v,Q2) are specified by τ variable. Clearly the knowl-
(f,g) (f,g) edge on the initial distributions F0(x) and G0(x) at Q2
S 0
neededtoobtainedthedistributionsatanyarbitraryen-
8 1 1 ergy scale Q2
ΦLO =4 + +2(γ +ψ(s+1)) (,9)
f − 3 s+1 s+2 E Now we intend to extend our calculations to the next-
(cid:18) (cid:19)
to-leadingorderapproximationforgluonandsingletsec-
tors of unpolarized parton distributions. In this case, to
1 2 2
ΘLO =2N + , (10) decoupleandtosolveDGLAPevolutionsinEqs.(7,8)we
f f 1+s − 2+s 3+s need an extra Laplace transformationfromτ space to U
(cid:18) (cid:19)
space. The U will be a parameter in this new space. In
ΦLO =12 1 2 + 1 1 (γ +ψ(s+1)) the rest of the calculation, the αs(τ)/4π is replaced for
g s − 1+s 2+s − 3+s − E brevity by a(τ). Therefore the solution of the first order
(cid:18) (cid:19) differential equations in Eqs.(7, 8) can be converted to,
33 2N
f
+ − , (11)
3 UF(s,U)−f0(s)=ΦLfO(s)F(s,U)
and +ΦNLO(s) [a(τ)f(s,τ);U]
f L
8 2 2 1 +ΘLO(s) (s,U)+ΘNLO(s) [a(τ)g(s,τ);U],
ΘLO = + , (12) f G f L
g 3 s − 1+s 2+s (16)
(cid:18) (cid:19)
where the N is the number of active quark flavours,
f U (s,U) g0(s)=ΦLO(s) (s,U)
γE is the Euler’s constant and ψ is the digamma func- G − g G
tion. The next-to-leadingordersplitting functions ΦNLO +ΦNLO(s) [a(τ)g(s,τ);U]
(f,g) g L
and ΘNLO are too lengthy to be include here and we +ΘLO(s) (s,U)+ΘNLO(s) [a(τ)f(s,τ);U].
present(ft,gh)em in Appendix A. One can easily determine g F g L
(17)
thesenext-to-leadingordersplittingfunctionsinLaplace
s spaceusing the next-to-leadingorderresultsderivedin We can consider a very simple parameterizationfor a(τ)
Ref. [4, 40, 41]. The leading-order solution of the cou- as a(τ) = a0 . Generally to do a more precise calcula-
pled ordinary first order differential equations in Eqs.(7, tionatthe next-to-leadingorderapproximation,one can
8)intermsoftheinitialdistributionsarestraightforward. consider the following expression for the a(τ) as [28],
Considering the initial distributions for the gluon g0(s) a(τ) a +a e−b1τ. (18)
0 1
and singlet distributions f0(s) at input scale, Q2 = 2 ≈
0
GeV2, the evolved solutions in the Laplace s space are This expansion involves an excellent accurate to a few
parts in 104. Using a(τ) defined in above equation and
given by [27, 29],
the conventionswhichwerepresentedin[27,29], the fol-
f(s,τ)=kff(s,τ)f0(s) + kfg(s,τ)g0(s) lowing simplified notations for the splitting functions in
g(s,τ)=k (s,τ)g0(s) + k (s,τ)f0(s), (13) s space can be introduced by:
gg gf
Φ (s) ΦLO(s)+a ΦNLO(s),
The inverse Laplace transform of coefficients k’s in f,g ≡ f,g 0 f
above equations are defined as kernels K (ν,τ) Θ (s) ΘLO(s)+a ΘNLO(s). (19)
ij ≡ f,g ≡ g 0 g
5
Eqs.(16, 17) can be solved simultaneously to get the de- thepartondistributions. Forthenon-singletdistribution
sired coupled algebraicequations for singlet (s,U) and F (x,Q2), one can schematically write the logarithmic
NS
F
gluon (s,U) distributions and arrives at, derivativeofF asaconvolutionofnon-singletdistribu-
NS
G tion F (x,Q2) with the non-singlet splitting functions,
[U Φ (s)] (s,U) Θ (s) (s,U)= NS
− f F − f G pLO,NS and pNLO,NS [4, 40, 41]. Therefore the next-to-
qq qq
f0(s)+a1 ΦNfLO(s)F(s,U +b1)+ΘNfLO(s)G(s,U +b1) , leading order contributions for the FNS(x,Q2) can be
(20) written as:
(cid:2) (cid:3)
4π ∂F
Θg(s) (s,U)+[U Φg(s)] (s,U)= NS (x,Q2)
− F − G α (Q2)∂lnQ2
g0(s)+a ΘNLO(s) (s,U +b )+ΦNLO(s) (s,U +b ) . s
1(cid:2) g F 1 g G 1(2(cid:3)1) =FNS ⊗ pLqqO,NS+ αs4(Qπ2)pqNqLO,NS (x,Q2).
(cid:18) (cid:19)
The simplified solutions of above equations can be ob- (25)
tained by setting a = 0 in Eq.(18). For a(τ) = a , the
1 0
Eqs.(20, 21) lead us to, Again changing to the required variable, ν ln(1/x),
≡
andgoingtothe Laplacespaces,wearriveatthe simple
[U Φ (s)] (s,U) Θ (s) (s,U)=f0(s), (22)
− f F1 − f G1 solution as,
−Θg(s)F1(s,U)+[U −Φg(s)]G1(s,U)=g0(s).(23) ∂FˆNS(ν,τ)= ν pLO,NS(ν w) +
One can easily solve these equations and extract the ∂τ ˆ qq −
0
bFa1s(esd,Uon) tahnedinGp1u(st,qUu)ardkisstfr0ib(su)tiaonnds.g0T(hs)egrleusounltdsisctleriabruly- αs4(πτ)(cid:0)pqNqLO,NS(ν−w) FˆNS(w,τ)e−(ν−w)dw.
tion functions at Q2. Using the Laplace transform tech- (cid:19)
0 (26)
nique,itispossibletobackfromU spacetoτ spacewhich
is leading to the desired f(s,τ) and g(s,τ) expressions.
Going to Laplace s space, we can obtain the first order
The complete solutions of Eqs.(20, 21) can be obtained
differential equations in Laplace space s with respect to
viaiterationprocesses. Theiterationcanbecontinuedto
τ variable for the non-singlet distributions f (s,τ),
NS
anyrequiredorderbutwewillrestrictourselfinwhichwe
get to a sufficient convergence of the solutions. Our re- ∂f α (τ)
NS(s,τ)= ΦLO+ s ΦNLO f (s,τ). (27)
sultsshowthatthesecondorderofiterationaresufficient ∂τ NS 4π NS,qq NS
(cid:18) (cid:19)
to get a reasonable convergence. Using the iterative so-
lution of Eqs.(20, 21) and the inverse Laplace transform The above equation has a very simplified solution,
technique to back fromU space to τ space,the following
f (s,τ)=eτΦNS(s)f0 (s), (28)
expressionsforthesingletandgluondistributionscanbe NS NS
obtained [27, 29, 35], whereΦ (s)iscontainingthenext-to-leadingordercon-
NS
f(s,τ)=k (a ,b ,s,τ)f0(s)+k (a ,b ,s,τ)g0(s), tributionsofthesplittingfunctionsatsspace,definedas
ff 1 1 fg 1 1
g(s,τ)=kgg(a1,b1,s,τ)g0(s)+kgf(a1,b1,s,τ)f0(s), Φ (s) ΦLO(s)+ τ2ΦNLO (s). (29)
NS ≡ NS τ NS,qq
(24)
The evaluation of ΦNLO (s)= e−νpNLO,NS(e−ν);s is
The analytical expressions for the next-to-leading order NS,qq L qq
straightforward but too lengthy to present here. The
approximationofcoefficients k , k ,k andk up to
ff fg gf gg (cid:2) (cid:3)
desiredstepsofiterationaregiveninAppendixB.Using analytical results for the unpolarized splitting functions
inthetransformedLaplacesspaceatthenext-to-leading
LaplaceinversioninEq.(24)fromstoν space,wecanar-
order approximation are given in Appendix A. The Q2
rivetothedecoupledsolutions(ν,τ)spaceastheresultof
convolution defined by the Eq.(14). The Q2 dependence dependenceoftheevolutionequationsisrepresentedbyτ
atleadingorderapproximationandbyτ atthe next-to-
of the solutions are performed by the τ variable and re- 2
leadingorderapproximationwhichthelatteronedefined
callingtheν ln(1/x),thesolutionscanbetransformed
≡ as [27, 29, 35],
back into the usual x space. Consequently, one can ob-
tainthe singletandgluondistributions asF (x,Q2)and
G(x,Q2) respectively. S τ 1 τα(τ′)dτ′ = ( 1 )2 Q2α2(Q′2) dlnQ′2.
2 ≡ 4π ˆ 4π ˆ s
0 Q2
0
(30)
IV. NON-SINGLET SOLUTION IN LAPLACE
SPACE AT THE NEXT-TO-LEADING ORDER Since all parts of the current analysis are done at the
APPROXIMATION next-to-leading order approximation, we should use the
τ variable as well. However for simplifying in notation,
2
Herewewishtoextendourcalculationstothenext-to- the τ variable is used insteadly through out the whole
leading orderapproximationforthe non-singletsectorof paper.
6
Similar to the singlet case, any non-singlet solution, parton types xq , obtained from GJR08 set of the free
i
F (x,Q2), can be obtained using the non-singletkernel parton distribution functions [38],
NS
KNS −1[eτΦNS(s);ν] which is defined by,
≡L u (s)=0.5889( B[4.7312,0.3444+s]
τ v
Fˆ (ν,τ)= K (ν w)Fˆ0 (w)dw. (31) 0.175 B[4.7312,0.8444+s]
NS ˆ NS − NS −
0
+ 17.997 B[4.7312,1.3444+s]) , (36)
Using again the appropriate change of variable, ν
≡
ln(1/x),thesolutionofEq.(31)canbeconvertedtousual
(x,Q2) space. dv(s)=0.2585( B[5.8682,0.2951+s]
1.0552 B[5.8682,0.7951+s]
−
+ 26.536 B[5.8682,1.2951+s]) ,
V. PROTON STRUCTURE FUNCTION Fp(x,Q2)
2 (37)
IN LAPLACE SPACE
d¯(s) u¯(s)=7.2874(B[19.756,1.2773+s]
We perform here an next-to-leading order analytical
−
analysis for the proton structure function Fp(x,Q2) us- 6.3187 B[19.756,1.7773+s]
2 −
ing Laplace transform technique. The result for singlet, + 18.306 B[19.756,2.2773+s]) , (38)
gluon and non-singlet parton distributions which we ob-
tained in previous sections are used to extract the nu- d¯(s)+u¯(s)=0.2295(B[9.8819, 0.1573+s]
cleon structure function. The next-to-leading order pro- −
tonstructurefunction Fp(x,Q2)for masslessquarkscan +0.8704 B[9.8819,0.3427+s]
2
be written as [1–4], + 8.2179 B[9.8819,0.8427+s]) , (39)
nf
F (x,Q2)= e2x(C (x,α ) [q (x,Q2)+q¯(x,Q2)] g(s)=1.3667 B[4.3258, 0.105+s], (40)
2 i q s ⊗ i i −
i=1
X whereBisthecommonEulerbetafunction. Thestrange
+C (x,α ) g(x,Q2)), (32)
g s ⊗ quark distribution function is assumed to be symmetric
where C and C are the next-to-leading order quarks ( xs = xs ) and it is proportional to the isoscalar light
q g
and gluon Wilson coefficients, and q , q¯ and g(x,Q2) quark sea which parameterized as
i i
are the quark, anti-quark and gluon distributions, re-
k
spectively. We exactly follow the method that we in- s(s)=s¯(s)= d¯(s)+u¯(s) , (41)
2
troduced before to solve the DGLAP evolution equa-
(cid:0) (cid:1)
tionsandanalyticallytodrivetheprotonstructurefunc- whereinpracticek isaconstantfixedtok =0.5[20,38].
tion at the next-to-leading order approximation first in The proton structure function Fp(x,Q2) in Laplace s
2
Laplace s space and then in Bjorken x space. As we al- space,uptothenext-to-leadingorderapproximation,can
ready mentioned, only the initial knowledge for singlet be written as
F0(x), gluonG0(x) andnon-singletF0 (x) distributions
S NS p,light(s,τ)= S(s,τ)+ G(s,τ)+ NS(s,τ),(42)
arerequiredtosolvetheDGLAPevolutionequationsvia F2 F2 F2 F2
Laplace transform technique. where the flavour singlet S and gluon G contribution
For our numericalinvestigation,we use the KKT12[20] read F2 F2
and GJR08 [38] parton distribution functions at Q2 = 2
0 4 1 1
GeV2. The valance quarkdistributions xuv andxdv, the S(s,τ)= 2u¯(s,τ)+ 2d¯(s,τ)+ 2s¯(s,τ)
anti-quark distributions x(d + u) and x∆ = x(d u), F2 9 9 9
(cid:18) (cid:19)
the strange sea distribution xs = xs and the gluon−dis- 1+ τ C(1)(s) , (43)
tribution xg of KKT12 and GJR08 models are generically × 4π q
parameterizedviathefollowingstandardfunctionalfrom, (cid:16) (cid:17)
2 τ
xq =aqxbq(1−x)cq(1+dqxfq +eqx), (33) F2G(s,τ)= 9g(s,τ) 4πCg(1)(s) . (44)
subject to the constraints that 1u dx=2, 1d dx= (cid:16) (cid:17)
0 v 0 v Finally the non-singlet contribution for three active
1, and the total momentum sum´ rule ´
(light) flavours is given by
1
x[u +d +2(u¯+d¯+s¯)+g] dx=1. (34) 4 1 τ
ˆ v v NS(s,τ)= u (s,τ)+ d (s,τ) 1+ C(1)(s) ,
0 F2 9 v 9 v 4π q
After changingto the variableν ln(1/x)andusingthe (cid:18) (cid:19)(cid:16) (cid:17)
Laplace transform q(s)= [e−νq≡(e−ν);s], one can easily (45)
L
obtained Eq.(33) in Laplace s space, (1) (1)
where the C (s) and C (s) are the common next-
q g
q(s)=aq(B[1+cq,bq+s]+eq B[1+cq,1+bq+s] to-leading order approximation of Wilson coefficients
+dq B[1+cq,bq+fq+s]) . (35) functions, derived in Laplace s space by cq(s) =
We use the following standard parameterizations in [e−νcq(e−ν);s] and cg(s)= [e−νcg(e−ν);s],
L L
Laplace s space at the input scale Q2=2 GeV2 for all
0
7
C(1)(s)=
q
2π2 2 6 2 4
C 9 + + +
F − − 3 − (1+s)2 1+s − (2+s)2 2+s
(cid:18)
2(γ +ψ(s+2)) 2(γ +ψ(s+3))
E E
3(γ +ψ(s+1))+ + +
E
1+s 2+s
1
π2+6(γ +ψ(s+1))2 6ψ′(s+1) +4ψ′(s+1) ,
E
3 −
(cid:19)
(cid:0) (cid:1) (46)
C(1)(s)=
g
2 2 4 16 4 16
f + +
(1+s)2 − 1+s − (2+s)2 2+s (3+s)2 − 3+s−
(cid:18)
2(γ +ψ(s+2)) 4(γ +ψ(s+3)) 4(γ +ψ(s+4))
E E E
+ .
1+s 2+s − 3+s
(cid:19)
(47)
Once again the Q2 dependence of proton structure startedtheirevolutionatQ2 =2GeV2,weusedF0 ,F0
0 NS S
function in Eq.(42) is performed by τ(Q2,Q2) and G0 constructed from their values at Q2 in Eq.(33).
0 ≡ 0
1 Q2α (Q′2)d lnQ′2. The final desired solution of The results for the evolved non-singlet distributions are
t4hπe´Qp20rotson structure function in Bjorken x space, depicted in Fig. 1. To do double check and indicate the
sufficient precision of our analysis, we have also used
Fp,light(x,Q2), are readily found using the inverse
2 the QCD evolution package, QCDnum [42] and link it to
Laplace transform and the appropriate change of vari-
LHAPDF [23] package for the GJR08 PDFs which directly
ables.
render us the parton densities in x-space. As can be
Thenext-to-leadingordercontributionofheavyquark,
seen from the related figures, a good agreementbetween
Fic,b(x,Q2), to the proton structure function can be cal- our results and the other ones exist. It indicates the
culated in the fixed flavour number scheme (FFNS) ap- evolution to work well beyond the charm quark mass
proach[20,44–50]andwillyieldthetotalstructurefunc- threshold, Q2 > Q2 ( m2 = 2 GeV2). In this figure
tions as Fp,total(x,Q2) = Fp,light(x,Q2)+Fheavy(x,Q2) thestraightlinerepr0ese≈ntstchesolution,resultedfromthe
2 2 i
where the Fp,light(x,Q2) refers to the common u, d, s Laplacetransformtechniqueandtheredcirclesrepresent
2
(anti) quarks and gluon initiated contributions, and the valance quark distributions from GJR08 global QCD
Fheavy(x,Q2)=Fc(x,Q2)+Fb(x,Q2)arethecharmand analysis. The dashed line indicates the results, arising
i 2 2
bottom quarks structure functions. We should mention out from QCDnum evolution package. One can conclude
that for the Fp,total only its light contribution is derived that the agreement, over the large span of 0 < x < 1, is
2
quite striking. The accuracy of the present analysis has
by Laplace transform technique. Its heavy contribution
been investigated and is typically better than about 1
isresultedfromtheusualMellintransformtechnique. In
the present analysis we use the GJR08 values for m = part in 105 at small and large value of Bjorken-x for the
c
up-valencequarkdistributionxu . Forthe down-valence
1.30 GeV and m = 4.20 GeV which slightly differ from v
b
KKT12 default values of m = 1.41 GeV and m = 4.50 quark distribution xdv, disagreements between our cal-
c b
culation and the GJR08 results are less than 1–2% for
GeV.
0<x<0.2.
InFig.2,theresultsforseaquarkandsingletdistribu-
VI. THE RESULTS OF LAPLACE tions have been shown and compared with the next-to-
TRANSFORMATION TECHNIQUE leading order analysis of GJR08 model as well as QCDnum
evolutionpackage. Howeverpeoplearereportingthesin-
In this section, we shall present our results that have glet solution rather than the individual distribution for
been obtained for the parton distribution functions and sea quarks, but following the technique which was in-
proton structure function Fp(x,Q2) using the Laplace troduced in [41, 51], it is possible to present separately
2
transformation technique to find an analytical solution the see quark distributions. The analytical solution for
for the DGLAP evolution equations. We obtain the the gluon distribution, G(x,Q2) = xg(x,Q2), has also
valance quark distributions, xu (x,Q2) and xd (x,Q2), been shown. All distributions are obtained from Eq.(24)
v v
usingEq.(31)andcomparethemwiththenext-to-leading in (s,τ) space and then converted to the (x, Q2) space,
order GJR08 results. Since the GJR08 collaboration using the convolution integrals in Eq.(14). The results
8
0.7 3 3
0.6 2Q)2.52 GQQJCCRDD0NN8UUMM 2Q)2.25 QQ22==1200 GGeeVV22
0.5 QCDNUM u(x,1.5 LLaappllaaccee d(x,1.5 NLO
2) Laplace +x 1 +x 1
Q 0.4 GJR08 xu0.5 xd0.5
x,
(v0.3 0.000010.001 0.01 0.1 1 0.000010.001 0.01 0.1 1
u x x
x
0.2
30 8
0.1 25 Q2=10GeV2
2)20 Q2=20GeV2 2)6
0 Q Q
0 0.2 0.4 0.6 0.8 1 x,15 x,4
x g(10 Σ(
x x2
5
0.7 0 0
0.00010.001 0.01 0.1 1 0.00010.001 0.01 0.1 1
x x
0.6
Figure 2: (Color online) Sea quarksand singlet
0.5 NLO
distributions in comparison with thenext-to-leading order
)
2Q 0.4 Q2=9.795 GeV2 results of GJR08 model. Thegluon distribution has also been
x, shown. The solid-line correspond toQ2 = 10 GeV2 and the
(v0.3 dashed-line correspond to Q2 = 20 GeV2. The results from
d
theQCD evolution package, QCDnum, havealso been
x
0.2 presented (dash-dottedand dash-dotted-dottedlines).
0.1
0 1.5
0 0.2 0.4 0.6 0.8 1
x NLO
GJR08
Figure 1: (Color online) Ourresults for thenon-singlet 2 2
Q =10 GeV Laplac
distribution, xuv(x,Q2) and xdv(x,Q2), using Eq.(31) and
its comparison with theglobal QCD analysis of GJR08. The 1 Q2=20 GeV2 Laplace
x-spaceresults from theQCD evolution package, QCDnum, 2) Q2=10 GeV2 QCDNUM
havealso been presented (dashed line). Q
x, Q2=20 GeV2 QCDNUM
(
s
x
0.5
indicatebysolidlinearecorrespondingtoQ2 =10GeV2
and the ones with dashed line to Q2 = 20 GeV2 respec-
tively. Thestrangeseadistributionxs=xs anditscom-
parison with the next-to-leading order results of GJR08
0
model is also shown in Fig. 3 at Q2 = 10 GeV2 and Q2 0.0001 0.001 0.01 0.1 1
= 20 GeV2. This figure indicate that the obtained re- x
sults from present analysis based on Laplace transform
Figure 3: (Color online) The strange sea distribution
techniqueareingoodagreementswiththeonesobtained
xs=xs in comparison with thenext-to-leadingorder results
by global QCD analysis of GJR08 for the parton distri- of GJR08 model. The solid-line correspond toQ2 = 10 GeV2
bution functions and also the obtained results from the and thedashed-linecorrespond to Q2 = 20 GeV2. The
QCDevolutionpackage,QCDnum. Onecanconcludefrom results from theQCD evolution package, QCDnum, havealso
Figs. 2 and 3 that the agreements between our results been presented (dash-dotted and dash-dotted-dottedlines).
and GJR08 global analysis are excellent over the entire
rangeofmomentumfraction-xandthevirtualityQ2. We
foundslightlydisagreementsbetweenx-spaceresultscal-
culated from QCDnum package and GJR08 analysis which next-to-leading order results from KKT12 global QCD
is 1.5–2% for all parton species except for the gluon dis- analysisanddepictedinFig.4. Inthisfigureouranalyt-
tribution. It is clear from mentioned plots that, overthe ical solution based on the Laplace transform technique
enormous Q2 and x span, our analytic solutions are is have been presented for sea and singlet distributions as
satisfactory agreements with the GJR08 analysis. well as for the gluon distribution G(x,Q2) = xg(x,Q2)
A detailed comparison has also been shown with the at Q2 = 100 GeV2. The analytical solution is arising
9
4 4 2
NLO
)3 Laplace )3 Q2=100 GeV2 NLO Laplace
2Q KKT12 2Q 1.5 Q2=9.795 GeV2 GJR08
x, x, E665
xu(2 xd(2 2Q)
+ +
xu1 xd1 (x, 1
P
2
F
0 0
0.00010.001 0.01 0.1 1 0.00010.001 0.01 0.1 1 0.5
x x
50
0
0.5 0.0001 0.001 0.01 0.1 1
40 x
2Q)30 2Q) 0.4 Figure 5: (Color online) The next-to-leadingorder
g(x,20 q(x,v0.3 Fp,atoptpalr(oxx,iQm2a)t,ioans aoffuthnecttiootnalofprxotaotnQs2tr=uc9tu.7r9e5fuGnecVti2o.nT, he
x x 0.2 2
input distributions are obtained from GJR08 model [38].
10 0.1 Herethe straight line represents ourresult, using the
Laplace transform technique,and thered circles represent
0.000010.001 0.01 0.1 1 0.000010.001 0.01 0.1 1 the proton structurefunction arising from GJR08 global
x x QCD analysis. A comparison with theE665 experimental
data [52] havealso been done.
Figure 4: (Color online) Sea quarks,gluon and non-singlet
distributions and its comparison with theresults from
next-to-leadingorderKKT12 global QCD analysis at Q2 =
100 GeV2.
order proton structure function, Fp(x,Q2), and its good
2
agreement with other theoretical models as well as ex-
perimental data, one can evaluate the parton distribu-
out from Eq.(24) which is relating to the KKT12 initial tions functions at the input scale Q20 by performing a
distributions at Q2 = 2 GeV2. global QCD fit to the all available and up-to-date DIS
0
The resultsofanalyticalsolutionsforallpartondistri- and hadron collision data, using the Jacobi polynomials
bution functions clearly show significant agreement over approach. We plantopresentourdetailedQCDanalysis
awiderangeofxandQ2 variables. Theonlyseriousdis- based on the analytical calculation in the next section.
agreementswhichwefoundbetweenourcalculationsand
theKKT12resultsareforxu+xu¯andxd+xd¯distributions,
which is to be smaller than 2–2.5% at 0.01<x<0.1. VII. JACOBI POLYNOMIALS TECHNIQUE
As a numerical illustration for our analytical ap- FOR THE DIS ANALYSIS
proaches at the next-to-leading order approximation of
the total proton structure function, Fp(x,Q2), we com- Globalanalysis of deep-inelastic scattering(DIS) data
2
pareourresultswiththeGJR08protonstructurefunction in the frameworkofQCD, providesone with new knowl-
and depict them in Figs. 5 and 6. A comparison with edge of hadron physics and serves as a test of reliability
E665dataatfixed-targetexperiments [52]andH1inclu- of our theoretical understanding of the hard scattering
sivedeepinelasticneutralcurrentdata[8]havealsobeen of leptons and hadrons. Various QCD analyses, both
shown there. The results for the total proton structure for polarized and un-polarized case, can be constructed
functionFp,total(x,Q2)havebeenpresentedasafunction using all available data from Fixed-target experiments,
2
of x (both for large and small x) at Q2 = 9.795 and 25 DIS data and the precise data from hadron colliders.
GeV2. It is seen that our analytical solutions based on For further literature on various PDFs models, we refer
the inverse Laplace transform technique at the NLO ap- the reader to review articles [19, 53–63]. The kinematics
proximationfortheprotonstructurefunctionoverawide spannedbyeachDISdatasetusedinourfitaredescribed
rangeofxandQ2 valuesarecorrespondingwellwiththe in subsections VIIA.
experimental data and the QCD analysis performed by We shall focus here on the non-singlet (NS) structure
GJR08 analysis. One can conclude that, in spite of small functions, FNS(x,Q2), with their corresponding Laplace
2
disagreement for the parton densities, we found a satis- s-spacemoments NS(s,Q2)inordertoperformaQCD
M
factory agreement for the protonstructure function over analysis of deep inelastic scattering data up to the next-
a wide range of x and Q2. The overall agreement found to-leadingorder(NLO).Basedonapopularparametriza-
to be 1 part in 105. tionforthepartondistributionfunctions(PDFs), weap-
Based on our obtained results for the next-to-leading plytheJacobipolynomialformalism. Weconsiderawide
10
3 whered= p+n. Intheregionofx 0.3forthedifference
NLO of proton 2p and deuteron d d≤ata, we use:
M2 M2
Laplace
2.5
GJR08
E665 MN2S(s,τ)≡2(Mp2−Md2)(s,τ)
2 H1 1 2
= (u d )(s)+ (u¯ d¯)(s)
2Q) Q2=25 GeV2 (cid:18)3 v− v 3 − (cid:19)×
x, 1.5 1+ τ C(1)(s) eτΦNS(s) (50)
p( 4π q
F2 (cid:16) (cid:17)
Since sea quarks can not be neglected for x smaller
1
than about 0.3, in our calculation we suppose the d¯ u¯
−
distribution from JR14 [57] at Q2 = 2 GeV2,
0
0.5
x(d¯ u¯)(x,Q2)=37.0x2.2(1 x)19.2(1+2.1√x), (51)
− 0 −
As we mentioned at the beginning of this section, the
0
0.0001 0.001 0.01 0.1 1 method in which we have employed is using the Jacobi
x
polynomials expansion of the structure functions. The
detailed of the Jacobi polynomials approach are pre-
Figure 6: (Color online) Thenext-to-leadingorder
approximation of the total proton structurefunction, sented in details in our previous work [64]. Here we out-
Fp,total(x,Q2), as a function of x at Q2 = 25 GeV2. The line a brief review of this method. According to this ap-
2
input distributions are obtained from GJR08 model [38]. proach, using the Jacobi polynomials moments an(Q2),
Here thestraight line represents ourresult, using the one can reconstruct the structure function as,
Laplace transform technique,and thered circles represent
the proton structure function arising from GJR08 global
QCD analysis. The square and up-triangle signs represent Nmax
xf(x,Q2)=xβ(1 x)α a (Q2)Θα,β(x), (52)
thetheE665 experimental data [52] and H1 inclusive deep − n n
inelastic neutral current data [8], respectively. n=0
X
where N is the number of polynomials and Θα,β(x)
max n
are the Jacobi polynomials of order n,
rangeofDISdatacorrespondingthemomentumtransfer
from low Q2 & 2GeV2 to high Q2 30000GeV2 where n
0 ∼ Θα,β(x)= c(n)(α,β)xj, (53)
the approach reasonably still works. In this section, we n j
first give an introductory description of the Jacobi poly- Xj=0
nomials approach, as the method of our QCD analysis
in which c(n)(α,β) are the coefficients that expressed
for the non-singlet(NS) structurefunctions andthe pro- j
through Γ-functions and satisfy the orthogonality rela-
cedure of the QCD fit to the data.
tion with the weight wα,β =xβ(1 x)α as follows
In the common MS factorization scheme, one can ob- −
tainedtherelevantF structurefunctionuptoNLOfrom
2
combinationofnon-singlet,flavoursingletandgluoncon- 1
dxxβ(1 x)αΘα,β(x)Θα,β(x)=δ . (54)
tributions of Eqs.(43–45). ˆ − m n mn
0
At Laplace s-space, the combinations of parton densi-
Using above equations, we can relate the proton,
tiesatthevalenceregionx 0.3fortheprotonstructure
function p in NLO can b≥e written as: neutron and non-singlet structure functions with their
M2 Laplace s-space moments,
4 1
p(s,τ)= u (s)+ d (s)
M2 (cid:18)9 v 9 v (cid:19)× Nmax
1+ 4τπCq(1)(s) eτΦNS(s) (48) F2p,d,NS(x,Q2)=xβ(1−x)α n=0 Θαn,β(x)
X
(cid:16) (cid:17) n
Inthe aboveregion,the combinationsofpartondensi- c(n)(α,β) p,d,NS(s=j+1,Q2),
tiesforthedeuteronstructurefunction darealsogiven × j M2
M2 j=0
by, X
(55)
Md2(s,τ)= 158(uv(s)+dv(s))×, wsphaecreewMhic2ph,dp,NrSes(sen,Qte2d)inarEeqst.h(e48m–5o0m)efonrtsthienprLoatpolna,cneeus--
τ tron and non-singlet structure functions. Here the Q2-
1+ 4πCq(1)(s) eτΦNS(s), (49) dependence of the structure functions will be provided
(cid:16) (cid:17)