Table Of ContentIPM School and Conference on Hadrn and Lepton Physics, Tehran, 15-20 May 2006 1
An approach to NNLO QCD analysis of non-singlet structure function
AliN.Khorramian
Physics Department, Semnan University, Semnan, Iran;
Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O.Box 19395-5531, Tehran, Iran
S.AtashbarTehrani
Physics Department, Persian Gulf University, Boushehr, Iran;
Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O.Box 19395-5531, Tehran, Iran
Weusethenext-to-next-to-leadingorder(NNLO)contributionstoanomalousdimensiongoverningtheevolution
of non-singlet quark distributions. We use the xF3 data of the CCFR collaboration to obtain some unknown
7 parameters which exist in the non-singlet quark distributions in the NNLO approximation. In the fitting
0 procedure, Bernsteinpolynomial method isused. Theresults ofvalence quarkdistributions inthe NNLO,are
0 ingoodagreementwiththeavailabletheoretical model.
2
n
a 1. Introduction 2. The theoretical background of the
J QCD analysis
1
1
Theglobalpartonanalysesofdeepinelasticscatter-
In the NNLO approximation the deep inelastic co-
2 ing(DIS)andtherelatedhardscatteringdataaregen- efficient functions are known and also the anomalous
v erallyperformedatNLOorder. Presentlythe next-to dimensions in n-Moment space are available at this
2 leading order (NLO) is the standard approximation
order[16, 17]. Since in this paper we want to calcu-
7 for most important processes in QCD.
late the non-singlet parton distribution functions in
1
0 The corresponding one- and two-loop split- the NNLO by using CCFR experimental data, we in-
1 ting functions have been known for a long time troducethemomentsofnon-singletstructurefunction
6 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The NNLO cor- up to three loops order.
0 rections should to be included in order to arrive at Let us define the Mellin moments for the νN struc-
h/ quantitatively reliable predictions for hard processes ture function xF3(x,Q2):
p occurring at present and future high-energy colliders.
- These corrections are so far known only for the 1
p structure functions in the deep-inelastic scattering n(Q2)= xn−1F3(x,Q2)dx. (1)
e M Z
[12, 13, 14, 15]. 0
h
:
v The theoretical expression for these moments obey
Recently much effort has been invested in com-
i the following renormalization group equation
X puting NNLO QCD corrections to a wide variety
r of partonic processes and therefore it is needed ∂ ∂
a to generate parton distributions also at NNLO so (cid:18)µ∂µ+β(As)∂A +γN(nS)(As)(cid:19)Mn(Q2/µ2,As(µ2))=0,
thatthetheorycanbeappliedinaconsistentmanner. s
(2)
whereA =α /(4π)istherenormalizationgroupcou-
s s
S.Mochandet al. [16,17]computedthe higheror- pling and is governedby the QCD β-function as
dercontributionsuptothree-loopssplittingfunctions
governing the evolution of unpolarized non-singlet ∂A
µ s =β(A )= 2 β Ai+2 . (3)
quark densities in perturbative QCD. ∂µ s − i s
Xi≥0
During the recent years the interest to use CCFR
data [18] for xF structure function in higher orders,
3 The solution of Eq.(3) in the NNLO is given by
based on orthogonal polynomial expansion method
has been increased [19, 20, 21, 22, 23, 24].
1 β1ln(lnQ2/Λ2 )
In this paper we determine the flavor non- ANNLO = MS +
s β lnQ2/Λ2 − β3(lnQ2/Λ2 )2
singlet parton distribution functions, xuv(x,Q2) and 0 MS 0 MS
xd (x,Q2), using the Bernstein polynomial approach 1
v [β2ln2(lnQ2/Λ2 )
up to NNLO level. This calculation is possible now, β5(lnQ2/Λ2 )3 1 MS −
0 MS
asthenon-singletanomalousdimensionsinn-Moment
β2ln(lnQ2/Λ2 )+β β β2].
space in three loops have been already introduced 1 MS 2 0− 1
[16, 17]. (4)
IPM-LHP06-19May
2 IPM School and Conference on Hadrn and Lepton Physics, Tehran, 15-20 May 2006
Notice that in the above the numerical expressions Innextsectionwewillintroducethefunctionalform
for β , β and β are of the valence quark distributions and we will param-
0 1 2
eterize these distributions at the scale of Q2.
0
β0 = 11 0.6667f , Byusingtheanomalousdimensionsinone,twoand
−
β = 102 12.6667f , three loops from [16] and inserting them in Eq. (9),
1
β = 1428−.50 279.611f +6.01852f2, (5) the moment of non-singlet structure function in the
2 − NNLO as a function of n and Q2 is available.
where f denotes the number of active flavors.
3. Parametrization of the parton
The solution of the renormalization group equa-
tionfornon-singletstructurefunctionxF canbepre- densities
3
sented in the following form [22]:
Inthissectionwewilldiscusshowwecandetermine
the parton distribution at the input scale of Q2 = 1
Mn(Q2) =exp As(Q2) γN(nS)(x)dx CN(nS)(As(Q2)) , GeV2. To start the parameterizations of the a0bove
Mn(Q20) (cid:20)−ZAs(Q20) β(x) (cid:21)CN(nS)(As(Q20()6)) mQ20enwtieonaessdumpaerttohne fdoilsltorwibinugtiofunnscattiotnhael fionrpmut scale of
where (Q2)isaphenomenologicalquantityrelated
Mn 0 xu (x,Q2)=N xa(1 x)b(1+c√x+dx), (11)
to the factorization scale dependent factor which we v 0 u −
will parameterize in next section. γNS is the anoma-
n
lous function and has a perturbative expansion as N
xd (x,Q2)= d(1 x)e xu (x,Q2). (12)
v 0 N − v 0
γNS(A )= γNS(n)Ai+1 . (7) u
n s i s
Xi≥0 Intheabovethexatermcontrolsthelow-xbehavior
partondensities, and (1 x)b,e large values of x. The
−
At the NNLO the expression for the coefficient func- remaining polynomial factor accounts for additional
tion CN(nS) can be presented as [25] medium-x values. Normalization constants Nu and
N are fixed by
d
C(n)(A )=1+C(1)(n)A +C(2)(n)A2 . (8)
NS s s s
1
With the corresponding expansion of the anomalous u (x)dx=2, (13)
v
dimensions,givenbyEq.(7),thesolutiontothethree Z0
loops evolution equation from Eq. (6), is as follows
1
NNLO(Q2)= As(Q2) γ0NS/2β0 Z0 dv(x)dx=1. (14)
Mn (cid:18)A (Q2)(cid:19) ×
s 0
Theabovenormalizationsareveryeffectivetocontrol
γ γNS β
1+[A (Q2) A (Q2)] 1 0 1 unknown parameters in Eqs. (11,12) via the fitting
(cid:26) s − s 0 (cid:18)2β1 − 2β0 (cid:19)β0 procedure. The five parameters with ΛNf=4 will
β2 γ γNS 2 be extracted by using the Bernstain poQlyCnDomials
+[A (Q2) A (Q2)]2 1 1 0
s − s 0 8β2 (cid:18)β − β (cid:19) approach.
0 1 0
1
+4[A2s(Q2)−A2s(Q20)] Using the valence quark distribution functions, the
momentsof u (x,Q2) andd (x,Q2)distributions can
1 β β2 β β v 0 v 0
γ 1γ + 1 − 2 0γNS be easily calculated. Now by inserting the Mellinmo-
(cid:18)β 2− β2 1 β3 0 (cid:19)(cid:27)
0 0 0 ments of uv and dv valence quark in the Eq.(10), the
1+C(1)As(Q2)+C(2)A2s(Q2) V(n,Q20). (9) function of V(n,Q20) involves some unknown parame-
(cid:16) (cid:17) ters.
Where V is the valence quark compositions as
V(n,Q2)=u (n,Q2)+d (n,Q2). (10) 4. Averaged structure functions
0 v 0 v 0
AsweseeinMellin-nspacethenon-singlet(NS)parts Although it is relatively easy to compute the nth
of structure function in the NNLO approximationfor momentfromthestructurefunctions,theinversepro-
example, i.e. NNLO(Q2), can be obtained from the cess is not obvious. To do this inversion, we adopt a
Mn
corresponding Wilson coefficients C(k) and the non- mathematically rigorous but easy method [26] to in-
singlet quark densities. vertthemomentsandretrievethestructurefunctions.
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IPM School and Conference on Hadrn and Lepton Physics, Tehran, 15-20 May 2006 3
Themethodisbasedonthefactthatforagivenvalue with the available experimental data we can only use
of Q2, only a limited number of experimental points, the following 28 averages,including
covering a partial range of values of x are available.
The method devised to deal with this situation is to
take averages of the structure function weighted by F(exp)(Q2),F(exp)(Q2),F(exp)(Q2),..., F(exp)(Q2).
2,1 3,1 4,2 13,4
suitable polynomials. These are the Bernstein poly-
nomials which are defined by Anotherrestrictionweassumehere,istoignorethe
effects of moments with high order n which do not
Γ(n+2) stronglyconstrainthefits. Toobtaintheseexperimen-
B (x)= xk(1 x)n−k ; n k .
nk Γ(k+1)Γ(n k+1) − ≥ tal averages from CCFR data [18], we fit xF3(x,Q2)
− (15) for each bin in Q2 separately to the convenient phe-
Usingthebinomialexpansion,theaboveequationcan nomenologicalexpression
be written as
xF(phen) = xB(1 x)C . (20)
3
Γ(n+2)n−k ( 1)l A −
Bn,k(x)= Γ(k+1) l!(n −k l)! xk+l . (16) ThisformensureszerovaluesforxF3 atx=0,and
Xl=0 − − x = 1. In Table 1 we have presented the numerical
values of , and at Q2 =20,31.6,50.1,79.4,125.9
Now,wecancomparetheoreticalpredictionswiththe A B C
GeV2.
experimentalresultsfortheBernsteinaverages,which
are given by [27, 28]
Q2(GeV2)
A B C
1 20 4.742 0.636 3.376
F (Q2) dxB (x)F (x,Q2). (17)
nk ≡Z nk 3 31.6 5.473 0.694 3.659
0
50.1 5.679 0.698 3.839
Therefore, the integral Eq. (17) represents an aver-
79.4 4.508 0.567 3.757
age of the function F (x,Q2) in the region x¯
3 nk
1∆x x x¯ + 1∆x . The key point is, the va−l- 125.9 7.077 0.819 4.246
2 nk≤ ≤ n,k 2 nk
ues of F3 outside this interval have a small contribu- Table1: Numericalvaluesoffitting , ,
tion to the above integral, as Bnk(x) tends to zero parametersinEq.(20). A B C
very quickly. In order to ensure the equivalence of
the integralEq.(17)tothe sameintegralinthe range Using Eq. (20) with the fitted values of , and ,
A B C
txo1u=sex¯tnhke−no12r∆mxalnikzattoionx2fa=ctox¯rn,k +x212d∆xBxnk(,xw)einhathvee ofunnecctaionnsth.eSnomcoemspaumtpelFen(eekxxpp)e(rQim2)enintatleBrmersnostfeGinamavmera-
x1 nk
denominator of Eq. (17) which obRviously is not equal agesareplottedinFig.1inthehigherapproximations.
to 1. So it can be written: The errors in the F(exp)(Q2) correspond to allowing
nk
theCCFRdataforxF tovarywithintheexperimen-
Fnk(Q2)≡Rx¯x¯nnkk−+R21x¯12x¯∆n∆nkxkx−n+nk12k21∆∆dxxxnnkkBndkx(xB)nFk(3x(x),Q2) . (18) atafoannrladQelysrt2rsa≥iostr2ibsb0tyGaicraeasvVl,oe2iirn,drcitolnhurgsids[ie1nh3v8gao]s.ltuthWtheioeeenmhxatpehverreirtoiomuongfelhynstiiflmanalcpvlsluoyifrdsytetiendhmgrdeatasththiaec-
By a suitable choice of n, k we manage to adjust to olds.
the region where the averageis peaked around values Using Eq. (19), the 28 Bernstein averages F (Q2)
nk
which we have experimental data [18]. can be written in terms of odd and even moments.
For instance:
SubstitutingEq.(16)inEq.(17),itfollowsthatthe
averagesofF withB (x)asweightfunctionscanbe F (Q2)=6 (2,Q2) (3,Q2) ,
3 nk 2,1
M −M
obtained in terms of odd and even moments, . (cid:0) (cid:1)
.
. (21)
(n k)!Γ(n+2)
F = −
nk Γ(k+1)Γ(n k+1) × TheunknownparametersaccordingtoEqs.(11,12)
n−k ( 1)l− will be a,b,c,d,e and ΛNQCfD. Thus, there are 6 pa-
− ((k+l)+1,Q2). rametersfor each orderto be simultaneously fitted to
Xl=0 l!(n−k−l)! M the experimentalFnk(Q2)averages. Using the CERN
(19) subroutine MINUIT [30], we defined a global χ2 for
allthe experimentaldatapointsandfoundanaccept-
WecanonlyincludeaBernsteinaverage,F ,ifwe able fit with minimum χ2/dof = 74.772/134= 0.558
nk
have experimental points covering the whole range in the NNLO case with the standard error of order
[x¯ 1∆x ,x¯ + 1∆x ] [29]. This means that 10−3. The best fit is indicated by some sample
nk − 2 nk nk 2 nk
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2.5
NNLO
0.8 xu (x)
F v
2 62
NNLO Fit
Fnk F 0.6 MA0R5ST
1.5 52 Q2=1 GeV2
0
F
42 0.4
1
1
0.2
Fnk0.75 F105
0
F116 10-4 10-3 10-2 10-1 x
0.5
F
85
0 50 100
Q2 xd (x)
0.5 v
NNLO Fit
Figure 1: NNLO fit to Bernstein averages of xF3. 0.4 MA0R5ST
0.3
curves in the Fig. 1. The fitting parameters and the
minimumχ2 valuesineachorderarelistedinTable2. 0.2
0.1
From Eqs.(11,12), we are able now to determine
the xu and xd at the scale of Q2 in higher order
v v 0 0
corrections. In Fig. 2 we have plotted the NLO and 10-4 10-3 10-2 10-1 x
NNLO approximation results of xu and xd at the
v v
input scale Q2 = 1.0 GeV2 (solid line) compared Figure 2: The parton densities xuv and xdv at the input
to the results0obtained from NNLO analysis (left scale Q20 =1.0GeV2 (solid line) compared to results
obtained from NNLOanalysis by MRST(dashed line)
panels) and NLO analysis (right panels) by MRST
[31] and A05(dashed-dotted line)[32].
(dashed-dotted line) [31] and A05(dashed line)[32].
5. x dependent of valence quark
densities
NNLO
N 5.134 In the previous section we parameterized the non-
u
a 0.830 singlet parton distribution functions at input scale of
Q2 = 1 GeV2 in the NNLO approximations by us-
b 3.724 0
ing Bernstein averages method. To obtain the non-
c 0.040
singlet parton distribution functions in x-space and
d 1.449 for Q2 > Q2 GeV2 we need to use the Q2-evolution
0
N 3.348 inn-space. Toobtainthex-dependenceofpartondis-
d
tributions from the n dependent exact analytical so-
e 1.460 −
lutions in the Mellin-moment space, one has to per-
Λ(Q4C)D,MeV 230 formanumericalintegralinordertoinverttheMellin-
χ2/ndf 0.558 transformation[33]
Table2: ParametervaluesoftheNNLOnon-singlet 1 ∞
fk(x,Q2) = dw
QCDfitatQ20 =1GeV2. π Z0 ×
Im[eiφx−c−weiφM (n=c+weiφ,Q2)],
k
All of the non-singlet parton distribution functions (22)
in moment space for any order are now available, so
we can use the inverse Mellin technics to obtain the where the contour of the integration lies on the right
Q2 evolutionofvalancequarkdistributionswhichwill of all singularities of M (n = c + weiφ,Q2) in the
k
be done in the next section. complex n-plane. For all practical purposes one may
IPM-LHP06-19May
IPM School and Conference on Hadrn and Lepton Physics, Tehran, 15-20 May 2006 5
choose c 1,φ=135◦ and an upper limit of integra- References
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