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

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

Martin M. Block Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208    Loyal Durand Department of Physics, University of Wisconsin, Madison, WI 53706    Douglas W. McKay Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045
July 11, 2019
###### Abstract

We recently derived an explicit expression for the gluon distribution function in terms of the proton structure function in leading-order (LO) QCD by solving the LO DGLAP equation for the evolution of analytically, using a differential-equation method. We showed that accurate experimental knowledge of in a region of Bjorken and virtuality 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 quark structure function can be determined directly in terms of from the DGLAP gluon evolution equation. To illustrate the method and check the consistency of existing LO quark and gluon distributions, we used the published values of the LO quark distributions from the CTEQ5L and MRST2001LO analyses to form , and then solved analytically for . We find that the analytic and fitted gluon distributions from MRST2001LO agree well with each other for all x and , while those from CTEQ5L differ significantly from each other for large values, , at all . We conclude that the published CTEQ5L distributions are incompatible in this region. Using a non-singlet evolution equation, we obtain a sensitive test of quark distributions which holds in both LO and NLO perturbative QCD. We find in either case that the CTEQ5 quark distributions satisfy the tests numerically for small , but fail the tests for —their use could potentially lead to significant shifts in predictions of quantities sensitive to large . We encountered no problems with the MRST2001LO distributions or later CTEQ distributions. We suggest caution in the use of the CTEQ5 distributions.

###### pacs:
13.85.Hd,12.38.Bx,12.38.-t,13.60.Hb

## I Introduction

In a recent paper bdm1 (), we derived an explicit expression for the gluon distribution function in the proton in terms of the proton structure function for scattering. The result was obtained in leading-order (LO) QCD by solving the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation dglap () for the evolution of analytically, assuming massless quarks and using a differential-equation method. The method is model-independent and does not require individual quark distributions, so accurate experimental knowledge of in a region of Bjorken and virtuality is all that is needed to determine the gluon distribution in that region.

In the present paper, we extend the method using a Laplace-transform technique. We first re-derive the solution for in terms of

 Fγp2(x,Q2)=nf∑i=1e2ix[qi(x,Q2)+¯qi(x,Q2)] (1)

using our Laplace transform method. The same procedure can be used to derive from the singlet (S) quark distribution function

 FS(x,Q2)=nf∑i=1x[qi(x,Q2)+¯qi(x,Q2)] (2)

and other combinations of the quark distributions. We then obtain an analytic solution for in terms of by solving the LO gluon evolution equation exactly. These are the principal theoretical results in this paper.

To illustrate the method, we have used the LO CTEQ5L CTEQ5 () and MRST2001 MRST2001 () quark distributions to construct the proton and singlet structure functions as functions of and , solved analytically for the corresponding gluon distributions, and compared the results for with those given in the fits. We find that the analytic and published distributions agree to high accuracy for MRST2001 LO, but, to our surprise, that they are incompatible for CTEQ5L, differing significantly at large for all . We find the same situation when we use the evolution equations for the singlet quark distribution or for individual quarks to solve for . We have not encountered these difficulties in somewhat less-extensive calculations using the more recent CTEQ6L CTEQ6 () distributions, finding good agreement between the analytic and published gluon distributions.

To investigate the inconsistencies further, we have used the DGLAP evolution equation for the non-singlet (NS) distributions

 FNS ≡ nf∑i=1x[ui(x,Q2)+¯ui(x,Q2) (3) −di(x,Q2)−¯di(x,Q2)],

where the sum is over the up- and down-type quarks and antiquarks in each generation with both types active. As will be shown, can be used in both leading order (LO) and next-to-leading order (NLO) to construct sensitive tests of the quark distributions.

The tests based on are satisfied for the MRST2001 LO MRST2001 () and CTEQ6L CTEQ6 () quark distributions which also satisfied the gluon tests above to high accuracy. However, we find that the LO CTEQ5L and the NLO CTEQ5M quark distributions in the renormalization scheme fail to satisfy the NS evolution equation in the same regions at large where the gluons gave difficulty. This shows clearly that these published quark distributions are inconsistent at large .

The errors in the quark distributions and their incompatibilities with could well affect predictions for experimental quantities that are sensitive to large . We conclude that the CTEQ5L distributions, though very convenient because of the analytic parametrizations given CTEQ5 (), should not be used to make precise predictions at large .

## Ii Preliminaries

We first introduce a quantity , which depends only on the proton structure function , by

 F2(x,Q2) ≡ ∂Fγp2(x,Q2)∂ln(Q2) (4) −αs4π{4Fγp2(x,Q2)+163[Fγp2(x,Q2)ln1−xx +x∫1x(Fγp2(z,Q2)z−Fγp2(x,Q2)x)dzz−x] −83x∫1xFγp2(z,Q2)(1+xz)dzz2}.

We define the corresponding quantities and for the singlet and non-singlet quark distributions obtained by replacing by or .

The LO DGLAP evolution equations dglap () for the proton structure functions , , and for massless quarks can be written compactly in space as

 F2(x,Q2) = αs4π∑ie2ix∫1xG(z,Q2)Kqg(xz)dzz2, (5) FS(x,Q2) = 2nfαs4πx∫1xG(z,Q2)Kqg(xz)dzz2, (6) FNS(x,Q2) ≡ 0, (7)

where the sum over quark charges in Eq. (5) includes both quarks and antiquarks, and in Eq. (6) is the number of active quarks. vanishes only if both members of a quark family are massless or can otherwise be treated as active. The LO splitting function is given by

 Kqg(x)=1−2x+2x2. (8)

Since

 Fγp2(x,Q2)=518FS(x,Q2)+16FNS(x,Q2), (9)

we should be able to derive consistently in LO from either of Eqs. (5) or (6) provided Eq. (7) is satisfied.

We note that Eqs. (6) and (7) continue to hold at next-to-leading order with modified quark and gluon splitting functions in Eqs. (4) and (6). These are given in the scheme in Eqs. (4.7a), (4.7b) and (4.7c) of Floratos et al. Floratos ().

To simplify the equations, we rewrite Eq. (5) as

 FF(x,Q2)=x∫1xG(z,Q2)Kqg(xz)dzz2, (10)

where we have introduced , which is defined as

 FF(x,Q2) = (αs4π∑ie2i)−1F2(x,Q2). (11)

Similarly, we define in terms of the singlet quark distribution by

 FFS(x,Q2) = (2nfαs4π)−1F2(x,Q2), (12)

and analogous quantities with appropriate prefactors for other structure functions such as , , and .

We note also that if we wish to evolve a single massless quark distribution in LO, we can simply replace in Eq. (4) by and in Eq. (11) by the appropriate

 FFi(x,Q2) = (αs4π)−1Fi(x,Q2), (13)

so that

 FFi(x,Q2)=x∫1xG(z,Q2)Kqg(xz)dzz2, (14)

with .

Since the gluon content of Eq. (14) is independent of the type of quark, Eq. (14) has the important consequence that, in LO,

 FFi(x,Q2)FFj(x,Q2)=1, (15)

for all at a fixed virtuality , and any choice of and . As we will soon see, this turns out to be a very sensitive test of compatibility of quark distributions with the LO QCD evolution equations.

An alternate form of this relation, which is readily extendable to NLO, follows from the evolution equation for the non-singlet distribution , . Splitting the sums in Eq. (3) into sums over up- and down-type quarks, we find that the vanishing of implies that

 FFdtype(x,Q2)FFutype(x,Q2)=1 (16)

for all at any given virtuality . This furnishes us with an exact numerical test of NLO perturbative QCD. We will later see that, while the MRST2001 and CTEQ6 parton distributions satisfy our constraints for all and x, the LO CTEQ5L and NLO CTEQ5M quark distributions satisfy these tests for small , but fail badly at large .

## Iii Solution of the LO evolution equation for the proton structure function Fγp2(x,Q2)

We now re-derive our analytic solution bdm1 () of Eq. (10) for the LO gluon distribution, using a new Laplace-transform method. Introducing the coordinate transformation

 v≡ln(1/x), (17)

we define functions , , and in -space by

 ^G(v,Q2) ≡ G(e−v,Q2) ^Kqg(v) ≡ Kqg(e−v) ^F(v,Q2) ≡ FF(e−v,Q2). (18)

Explicitly, from Eq. (8), we see that

 ^Kqg(v)=1−2e−v+2e−2v. (19)

Further, we can write

 ^F(v,Q2) = ∫v0^G(w,Q2)e−(v−w)^Kqg(v−w)dw (20) = ∫v0^G(w,Q2)^H(v−w)dw,

where

 ^H(v) ≡ e−v^Kqg(v) (21) = e−v−2e−2v+2e−3v.

We introduce the notation that the Laplace transform of a function is given by , where

 h(s)≡L[^H(v);s]=∫∞0^H(v)e−svdv, (22)

with the condition . The convolution theorem for Laplace transforms relates the transform of a convolution of functions and to the product of their transforms and , so that

 L[∫v0^G(w)^H(v−w)dw;s]=g(s)×h(s), (23)

or conversely, the inverse transform of a product to the convolution of the original functions, giving

 L−1[g(s)×h(s);v] = ∫v0^G(w)^H(v−w)dw (24) = ∫v0^H(w)^G(v−w)dw.

Equation (23) allows us to write the Laplace transform of Eq. (10) as

 f(s,Q2) = g(s,Q2)×h(s), (25)

where

 f(s,Q2)=L[^F(v,Q2);s] (26)

and and are the transforms of the functions and on the right-hand-side of Eq. (20). Solving Eq. (25) for , we find that

 g(s,Q2)=f(s,Q2)/h(s). (27)

We will generally not be able to calculate the inverse transform of explicitly, if only because is determined by a numerical integral of the experimentally-determined function . However, if we regard as the product of the two functions and and take the inverse Laplace transform using the convolution theorem in the form in Eq. (24) and the known inverse , we find that

 ^G(v,Q2) = L−1[f(s,Q2)×h−1(s);v] (28) = ∫v0^F(w,Q2)^J(v−w)dw

where is a new auxiliary function, defined by

 ^J(v)≡L−1[h−1(s);v]. (29)

The calculation of and the inverse Laplace transform of of are straightforward, and we find that

 ^J(v) = 3δ(v)+δ′(v) (30) −e−3v/2(6√7sin[√72v] +2cos[√72v]).

We therefore obtain an explicit solution for the gluon distribution in terms of the integral

 ^G(v,Q2) = ∫v0^F(w,Q2)^J(v−w)dw (31) = 3^F(v,Q2)+∂^F(v,Q2)∂v −∫v0^F(w,Q2)e−3(v−w)/2× (6√7sin[√72(v−w)] +2cos[√72(v−w)])dw.

Transforming Eq. (31) back into space, we find that the gluon probability distribution is given in terms of the function , assumed to be known, by

 G(x,Q2) = 3FF(x,Q2)−x∂FF(x,Q2)∂x (32) −∫1xFF(z,Q2)(xz)3/2× {6√7sin[√72lnzx]

It can be shown without difficulty that this expression for is equivalent to that which we derived in bdm1 (), where we first converted the evolution equation for into a differential equation in , and then solved that equation explicitly.

Even if we cannot integrate the last term of Eq. (32) analytically, the usual case, it can be integrated numerically, so that we always have an explicit solution which can be evaluated to the numerical accuracy to which is known. We emphasize that this solution for is derived from on the assumption that the quarks are either massless, or effectively so, a situation that holds in general for , where mass effects due to the (massive) quark are negligible, or for a treatment of mass effects such as that used in CTEQ5 CTEQ5 () as discussed in Sec. V.111We will discuss mass effects in detail elsewhere.

Leading-order solutions for of exactly the same form follow from the evolution equations for the singlet structure function and the individual quark distribution functions, with replaced in Eq. (32) by or , as noted earlier. The latter can be used to obtain expressions for in terms of the structure functions for weak scattering processes, allowing the incorporation of other data sets.

## Iv Solution of the LO gluon evolution equation

To obtain an expression which relates the singlet structure function indirectly to experiment, we use the DGLAP equation for the evolution of the gluon distribution . This relates to which is determined by as shown in the preceding section. Again, we simplify the notation by writing this equation in terms of the (rescaled) splitting function and a quantity as

 GG(x,Q2) = x∫1xFS(x,Q2)Kgq(x/z)dzz2. (33)

Here

 Kgq(x)≡2x−2+x, (34)

and

 GG(x,Q2) ≡ 38{(αs4π)−1∂G(x,Q2)∂lnQ2 (35) − G(x,Q2)(13(33−2nf)+12ln(1−x)) − 12x[∫1xG(z,Q2)(zx−2+xz−(xz)2)dzz2 +∫1x(G(z,Q2)−G(x,Q2))(zz−x)dzz2]},

for effectively massless active quark flavors. As before, going to space, we define the quantities

 ^FS(v,Q2) ≡ FS(e−v,Q2), ^G(v,Q2) ≡ GG(e−v,Q2), ^Hgq(v,Q2) ≡ e−vKgg(e−v) (36) = e−v(2ev−2+e−v) = 2−2e−v+e−2v.

The evolution equation then involves a convolution of and .

Proceeding as before, we take the Laplace transform of both sides, solve for the transform of , invert the transform using the convolution theorem again, and find that

 ^FS(v,Q2) = ∂^G(v,Q2)∂v+∫v0^G(w,Q2)e−3(v−w)/2× (37) {6√7sin[√72(v−w)] −2cos[√72(v−w)]}dw.

Transforming back into space, is given by

 FS(x,Q2) = −x∂GG(x,Q2)∂x (38) {6√7sin[√72lnzx]

is a known function of the gluon distribution, , which, in turn, is a known function of the proton structure function that is measured in deep inelastic scattering. is therefore related back to through and the analysis in the preceding section, and could be used, in turn, to determine through Eq. (32) with replacing .

It is important to recognize that our method is essentially model-independent. When different parametrizations in and are used to fit experimental data, the LO gluon and distributions derived from the different fits must agree with each other to the accuracy of the fits in the regions in which the data exist. The results can only depend on the functional forms used to fit when they are extrapolated to and outside that region.

## V Numerical illustration of the method, and tests of LO gluon and quark distributions

To test of our methods, we have applied them to published parton distributions, using the quark distributions to construct and and then solving Eq. (32) to find . The methods work well, and reproduce the published gluons distributions except in the case of CTEQ5 CTEQ5 (). In that case, we found to our surprise that there are problems at high with the LO CTEQ5L distributions. While these are superseded by more recent CTEQ distributions CTEQ6 (); cteq6.5 (), they seem still to be used in some calculations, most likely because they have been parametrized analytically in a form convenient for calculation. Similar tests applied to the MRST2001 MRST2001 () and the CTEQ6L CTEQ6 () LO distributions uncovered no problems.

The results illustrate the sensitivity of the analytic methods in testing parton distributions, and may be of wider use for that purpose, as well as for deriving directly from data as originally proposed in bdm1 (). We present them here as a demonstration, and as a caution.

We begin by illustrating the use of the analytic expression in Eq. (32) to derive from in the case of CTEQ5L CTEQ5 (). We take the published LO CTEQ5L quark distributions as our basic input, and use these distributions to calculate the proton structure function needed in in Eq. (11). Next, we solve Eq. (32) for the generated by this , labelling the solution , and compare the results with the published gluon distributions.

We follow the procedures of the CTEQ5 group in these calculations. The parton splitting functions used are those for massless quarks. Quark mass effects are included only in an approximate, -independent way, with and taken as zero for a massive quark when , and the quark treated as fully active when , with and included in the calculations of the structure functions and . The charge factors in Eqs.  (5) and (11), and the numbers of active quarks in Eqs. (6), (12), and later in Eq. (50), then change discontinuously at each threshold, but remain constant between thresholds. Despite these discontinuities, the quark and gluon distributions are continuous.222 The quark and gluon distributions are continuous solutions of the evolution equations, which are first-order differential equations in and integral equations in . The discontinuities on the right-had sides of Eqs. (5) and (6) as changes at a threshold are reflected in the final results by discontinuous changes in the derivatives of the structure functions on the left-hand sides of these equations. In particular, the heavy quark distributions and are identically zero for , then initially rise linearly from zero with increasing for . The contributions of and to , identically zero for , therefore jump to non-zero values for .

Because the mass effects do not depend on in this approach, the evolution equations retain the massless form between thresholds, and we can simply use the results derived above for massless quarks in our analysis, treating the distinct inter-threshold regions in separately.333Mass effects are treated the same way by CTEQ6 CTEQ6 (). The more accurate treatments of heavy-quark masses in MRST2001 MRST2001 () and the recent CTEQ analyses beginning with CTEQ6.5 cteq6.5 () involve -dependent effects. These will not affect the tests of the MRST2001 distributions given later using only the massless quarks.

In these calculations, we use the LO form of used in CTEQ5L alphacteq (),

 αs(Q2) = 4πβ0ln(Q2/Λ2), (39) β0 = 11−23nf, (40)

with and MeV for GeV, and MeV for 1.3 GeV GeV, and and MeV for GeV. These parameters give a continuous with CTEQ5 ().

The same methods can be used in LO to determine the gluon distributions generated by individual quark distributions, or by other combinations of the quarks, in particular, by combinations of the massless quarks and by the non-singlet distributions, as will be seen below.

Following these procedures, the gluon distribution we derive analytically from the calculated structure functions or , or from other quark combinations, should agree to the accuracy of the calculations with at all and .

The results of the calculations based on are shown in Fig. 1. The left-hand column compares (blue dashed curves) with (red solid curves) at , 20, and 100 GeV, (a), (b), and (c), respectively. All of the active, physically-relevant quarks are included in the input at each value of , with for (a) and (b) with active, and in (c) with now also active. For a scale comparison, we include the CTEQ5L up-quark distribution, .

The analytic and fitted  s do not agree well at large , with significant differences between them on the scale of and of the -quark distribution for . We do not believe that the calculations are reliable beyond : is very small , but the input and distributions and the individual integrals in are not, and the calculation in Eq. (32) involves large cancellations and becomes sensitive to small errors in the inputs.

The same discrepancies between the two  s are illustrated differently in the right-hand column of Fig. 1 where we plot the ratio

 rG=GAnalytic(x,Q2)/GCTEQ5L(x,Q2). (41)

In all cases, we find that is essentially constant and equal to one for all for small virtuality, and for all for large virtuality, showing good agreement between analytic and fitted gluon distributions at small . There are again significant deviations from one at larger , contrary to theoretical requirements.

We emphasize that the reason we find negative values of or for some is not that goes negative, but rather that goes negative. As can be seen from Eq. (4), this is a sensitive function of the difference between and the LO convolution integral of . In spite of the fact that the CTEQ5L quark distributions that go into are all positive, the resulting combination becomes negative for large . The appearance of negative values for the gluon distribution function suggests strongly that the differences between and in Fig. 1 result from problems with the input quark distributions, hence with , and are less likely to arise directly from problems with .

We find very similar deviations if we use the singlet structure function in Eq. (2) as input, and then solve for using Eq. (32) with replacing .

The discrepancies between the analytic and fitted  s are similar in magnitude to the differences between the CTEQ5 and more recent CTEQ and MRST gluon distributions in this region, and to the changes in the distributions that resulted from the addition of inclusive jet data to those analyses. Changes of this size are clearly significant.

Since the major contributions to at large are from the valence quarks and , we have done a separate set of calculations using the singlet combination and Eq. (32) with replacing . No mass effects need to be included in this case. The results of the calculations, called , are shown in Fig. 2 for both CTEQ5L and MRST2001 LO.

Discrepancies between and the analytically reconstructed from , here labeled , are again evident in the figure. The discrepancies have the same pattern and are roughly the same sizes as those seen in Fig. 1, suggesting that the problems originate with the and distributions that drive the initial DGLAP evolution at large . The results for MRST2001 LO show agreement between the calculated and at the level of accuracy of the calculation, which required the parametrization of numerical data from MRSTweb ().

To exclude the possibility that the problems with CTEQ5 can be eliminated by going to NLO perturbative QCD, we also looked at the evolution of the non-singlet structure function , and at the gluon distribution from the evolution of the singlet structure function , using the CTEQ5M NLO distributions as input. We found a similar -dependence for the gluon ratio obtained from the NLO singlet distribution, hence, continuing problems. The NS combination, for both LO and NLO, is discussed in the next section, and shows clearly that there are problems with the CTEQ5L quark distributions.

We note finally that, to reduce the possibility that the discrepancies that we have encountered are a artifact of the published parametrization of the CTEQ5L parton distributions CTEQ5 (), we have also compared them to the corresponding numerical CTEQ5L distributions from the Durham web site MRSTweb () and concluded that they are numerically compatible.

Because the derived from either or fails to match well at large , we cannot expect the inverse problem, the analytic derivation of from to work well for the CTEQ5L distributions, and do not include such calculations here.

## Vi Tests of LO and NLO non-singlet distributions

For massless quarks in LO, the relation in Eq. (15) holds for any quark or anti-quark pair and ,

 FFi(x,Q2)FFj(x,Q2)=1. (42)

As discussed earlier, this can be generalized readily to NLO using the non-singlet evolution equation , Eq. (7) with the NLO NS splitting functions of Floratos et al. Floratos (). If we separate the sum in Eq. (3) into sums over - and -type quarks, defined as

 utype ≡ x[u(x,Q2)+¯u(x,Q2)+c(x,Q2)+¯c(x,Q2)], (43) dtype ≡ x[d(x,Q2)+¯d(x,Q2)+s(x,Q2)+¯s(x,Q2)], (44)

this relation becomes

 FFNS(x,Q2)=FFutype(x,Q2)−FFdtype(x,Q2)=0,, (45)

where the two terms are to be calculated separately using the same NLO splitting functions as for the full . Dividing by the -type term, we obtain the ratio

 rNS≡FFdtype(x,Q2)FFutype(x,Q2)=1, (46)

a relation true for both LO and NLO when all the relevant quarks are active.

To use this relation to check the consistency of the quark distributions, we insert the appropriate LO or NLO quark and anti-quark distributions in Eqs. (43) and(44) and use them to generate numerically using the LO or NLO splitting functions.

In the case of the NLO CTEQ5M distributions, we use

 αs(Q2) = 4πβ0ln(Q2/Λ2)× (47) [1−2β1β20ln[ln(Q2/Λ2)]ln(Q2/Λ2)], β0 = 11−23nf (48) β1 = 51−193nf, (49)

with for , matching to the ’s for and 3 at and 1.3 GeV.

In the left-hand column of Fig. 3, we plot vs. , for , 20, and 100 GeV and