The asymptotic behaviour of parton distributions at small and large x
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Abstract
It has been argued from the earliest days of quantum chromodynamics that at asymptotically small values of x the parton distribution functions (PDFs) of the proton behave as \(x^\alpha \), where the values of \(\alpha \) can be deduced from Regge theory, while at asymptotically large values of x the PDFs behave as \((1x)^\beta \), where the values of \(\beta \) can be deduced from the Brodsky–Farrar quark counting rules. We critically examine these claims by extracting the exponents \(\alpha \) and \(\beta \) from various global fits of parton distributions, analysing their scale dependence, and comparing their values to the naive expectations. We find that for valence distributions both Regge theory and counting rules are confirmed, at least within uncertainties, while for sea quarks and gluons the results are less conclusive. We also compare results from various PDF fits for the structure function ratio \(F_2^n/F_2^p\) at large x, and caution against unrealistic uncertainty estimates due to overconstrained parametrisations.
Keywords
Large Hadron Collider Gluon Distribution Valence Quark Regge Theory Nucleon Structure1 Introduction
An accurate determination of parton distribution functions (PDFs) is an essential building block for the precision physics program at the large hadron collider (LHC) [1, 2, 3, 4, 5]. Given current limitations in the understanding of nonperturbative quantum chromodynamics (QCD), such a determination is not achievable from first principles. Instead, PDFs are determined in a global fit to hardscattering experimental data [6, 7, 8, 9, 10, 11], using perturbative QCD to combine information from different processes and scales. In such an analysis, the bestfit values of the input PDF parametrisation are obtained by comparing the PDFdependent prediction of a suitable set of physical observables with their measured values, and then by minimising a figure of merit which quantifies the agreement between the two.
It should be emphasised that Eqs. (2)–(3) cannot be derived using perturbative QCD, but rather require other more general considerations. For instance, counting rules can be derived from Bloom–Gilman duality [22] or using AdS/QCD methods in nonperturbative QCD [23].^{1} The use of Eqs. (2)–(3) in the input PDF parametrisation, Eq. (1), could therefore lead to theoretical bias. For instance, as we will discuss below, perturbative QCD calculations predict a logarithmic, rather than a powerlike, growth of the PDFs at small x. Even if Eqs. (2)–(3) were a solid prediction from QCD (which they are not), they would not be particularly useful in the context of a global PDF analysis. First, it is unclear how small or large x should be in order for the power laws (2)–(3) to provide a reliably enough approximation of the underlying PDFs. Second, it is unclear at which values of \(Q^2\) Regge theory and Brodsky–Farrar quark counting rules should apply exactly. This is a serious limitation, given the nonnegligible PDF scale dependence around the input parametrisation scale \(Q^2\simeq Q_0^2\). In principle, the optimal values of \(Q^2\) should be chosen at the interface between perturbative and nonperturbative hadron dynamics, \(Q^2\simeq Q_0^2 = Q^2_\mathrm{in}\). It has been shown [25] that \(Q_\mathrm{in}^2\simeq 0.75\) GeV\(^2\) by matching the high and low\(Q^2\) behaviour of the strong coupling \(\alpha _s(Q^2)\) as predicted respectively by its renormalisation group equation in the \(\overline{\mathrm{MS}}\) scheme and its analytic form in the lightfront holographic approach.
The aim of this study is to present a methodology to quantify the effective asymptotic behaviour of PDFs at small and large values of x, and then apply it to compare recent global fits with various perturbative and nonperturbative QCD predictions. The paper is organised as follows. In Sect. 2 we introduce a definition of the effective PDF exponents, and we use them to quantify for which ranges of x and \(Q^2\), if any, PDFs exhibit a powerlaw behaviour of the form Eqs. (2)–(3). Once the asymptotic range has been determined, in Sect. 3 we investigate to which extent these exponents, as obtained from global PDF fits, are in agreement with the theoretical predictions of their values. In addition to Brodsky–Farrar quark counting rules, we will also compare the global fit predictions with other nonperturbative models of nucleon structure at large x. In principle, this comparison will allow us to discriminate among models, in the same way as was done for spindependent PDFs in Ref. [26].
2 The effective exponents
From Fig. 1 and Table 1 it is clear that both \(\alpha _{f_i}(x,Q^2)\) at \(x=10^{5}\) and \(\beta _{f_i}(x,Q^2)\) at \(x=0.9\) have converged to the fitted values of \(a_{f_i}\) and \(b_{f_i}\) within PDF uncertainties. In addition, by examining the x dependence of \(\alpha _{f_i}(x,Q^2)\) and \(\beta _{f_i}(x,Q^2)\), it is possible to identify the asymptotic regions in which they become roughly independent of x. Furthermore, since the definitions Eq. (4) may be applied at any value of \(Q^2\), we may use them to study the \(Q^2\) dependence of the effective exponents.
The effective exponents \(\alpha _{f_i}\) and \(\beta _{f_i}\) at \(Q^2=1\) GeV\(^2\) and \(x_a=10^{5}\) and \(x_b=0.9\) computed for the MSTW08 NLO PDF set with Eq. (4), compared to the corresponding fitted exponents \(a_i\) and \(b_i\)
\(f_{i}\)  \(\alpha _{f_i}(x_a,Q^2)\)  \(a_{f_i}\)  \(\beta _{f_i}(x_b,Q^2)\)  \(b_{f_i}\) 

\(u_V\)  \(+0.29\pm 0.01\)  \(+0.291^{+0.019}_{0.013}\)  \(+3.11\pm 0.04\)  \(+3.243^{+0.062}_{0.039}\) 
\(d_V\)  \(+1.02\pm 0.11\)  \(+0.968^{+0.110}_{0.110}\)  \(+5.67\pm 0.47\)  \(+5.944^{+0.510}_{0.530}\) 
g  \(0.30\pm 0.37\)  \(0.428^{+0.066}_{0.057}\)  \(+2.95\pm 0.39\)  \(+3.023^{+0.430}_{0.360}\) 

NNPDF3.0 PDFs are parametrised in the basis that diagonalises the DGLAP evolution equations [28]. The function \(\mathscr {F}(x,\{c_{f_i}\})\) is a multilayer feedforward neural network (also known as perceptron). The powerlaw term \(x^{a_{f_i}}(1x)^{b_{f_i}}\) in Eq. (1) is treated as a preprocessing factor that optimises the minimisation process: the exponents \(a_{f_i}\) and \(b_{f_i}\) are chosen for each Monte Carlo replica at random in a given range determined iteratively.

MMHT14 The PDFs parametrised are the valence distributions \(u_V\) and \(d_V\), the total sea S, the sea asymmetry \(\Delta _S=\bar{d}\bar{u}\), the total and valence strange distributions \(s^+=s+\bar{s}\) and \(s^=s\bar{s}\) and the gluon g. The function \(\mathscr {F}(x,\{c_{f_i}\})\) is taken to be a linear combination of Chebyshev polynomials. The exponents \(a_{f_i}\) and \(b_{f_i}\) are fitted, except for \(a_{s^+}=a_{S}\).

CT14 The PDFs parametrised are the valence distributions \(u_V\) and \(d_V\), the sea quark distributions \(\bar{u}\) and \(\bar{d}\), the total strangeness \(s^+\) and the gluon g. It is assumed that \(s=\bar{s}\). The function \(\mathscr {F}(x,\{c_{f_i}\})\) is a linear combination of Bernstein polynomials. The exponents \(a_{f_i}\) and \(b_{f_i}\) are parameters of the fit, but not all of them are free: specifically, it is assumed that \(b_{u_V}=b_{d_V}\), so that as \(x\rightarrow 1\) \(u_V(x,Q_0^2)/d_V(x,Q_0^2)\rightarrow k\), with k a constant, and that as \(x\rightarrow 0\) \(\bar{u}(x,Q_0^2)/\bar{d}(x,Q_0^2)\rightarrow 1\), which requires \(a_{\bar{u}}=a_{\bar{d}}\).

ABM12 The PDFs parametrised are the valence distributions \(u_V\) and \(d_V\), the sea distributions \(\bar{u}\) and s, the sea asymmetry \(\Delta _S\) and the gluon g. It is assumed that \(s=\bar{s}\). The function \(\mathscr {F}(x,\{c_{f_i}\})\) has the form \(x^{P_{f_i}(x)}\), where \(P_{f_i}(x)\) is a function of x; for s, \(\mathscr {F}(x,\{c_{f_i}\}=1\). The exponents \(a_{f_i}\) and \(b_{f_i}\) are parameters of the fit, except for the condition \(a_{\Delta _S}=0.7\).

CJ15 The PDFs parametrised are the valence distributions \(u_V\) and \(d_V\), the light antiquark sea, \(\bar{u}+\bar{d}\), the light antiquark ratio \(\bar{d}/\bar{u}\), the total strangeness \(s^+\) and the gluon g. It is assumed that \(s=\bar{s}\). The function \(\mathscr {F}(x,\{ c_{f_i}\})\) is provided by the polynomial \((1+c_{f_i}^{(1)}\sqrt{x}+c_{f_i}^{(2)}x)\) for all the distributions except the light antiquark ratio and the total strangeness. Specifically, \(\bar{d}/\bar{u}\) is parametrised with a simple polynomial which ensures that as \(x\rightarrow 1\), \(\bar{d}/\bar{u}\rightarrow 1\), while it is assumed that \(s^+=\kappa (\bar{u}+\bar{d})\); \(c_{f_i}^{(1)}\), \(c_{f_i}^{(2)}\) and \(\kappa \) are parameters of the fit. A small admixture of \(u_V\) is added to \(d_V\) so that as \(x\rightarrow 1\) \(d_V/u_V\rightarrow k\), with k a constant.
Although the momentum distributions of strange and antistrange quarks are assumed to be identical in some of these PDF sets, it should be noted that a strange/antistrange asymmetry in the nucleon is predicted based on nonperturbative QCD models; see e.g. Ref. [29] and references therein. Strange and antistrange distributions may also be very different from each other in the polarised case, as it was shown in Ref. [29] based on a lightcone model of energetically favoured mesonbaryon fluctuations applied to the \(K^+\Lambda \). However, a study of a structured asymmetry in the momentum distributions of strange and antistrange quarks in a global QCD analysis is beyond the scope of this work, and has been addressed elsewhere [6, 7].
To the best of our knowledge, this is the first time that the onset of an asymptotic regime in the effective PDF exponents \(\alpha _{f_i}(x,Q^2)\) and \(\beta _{f_i}(x,Q^2)\) has been explicitly demonstrated. Remarkably, this onset takes place at x values close to the boundary between the data and extrapolation regions. Our results indicate that the three global PDF sets are broadly consistent among one other within uncertainties not only at the level of PDFs, but also at the level of their small and largex asymptotic behaviour. The main exceptions are \(u_V\) and \(d_V\) at small x, where the effective exponent of NNPDF3.0 is incompatible with those of CT14 and MMHT14. However, this is an extrapolation region where the Hessian approximation has some limitations and nonGaussian effects are large: indeed, if we compute with NNPDF3.0 the onesigma PDF interval as opposed to the \(68\,\%\) CL, the three sets become consistent.
The values of the smallx effective exponent \(\alpha _{f_i}(x_a,Q^2)\) computed at \(Q^2=2\) GeV\(^2\) and \(Q^2=10\) GeV\(^2\) at \(x_a=10^{4}\), compared to the values of \(a_{f_i}\) predicted by Regge theory (and resummation of double logarithms). For the quark sea S and the gluon g we indicate the prediction of the soft Pomeron (and the NLLx perturbative result)
\(f_i\)  \(Q^2\)  \(\alpha _{f_i}(x_a,Q^2)\)  \(a_{f_i}\)  

(GeV\(^2\))  NNPDF3.0  CT14  MMHT14  ABM12  CJ15  
\(u_V\)  2.0  \(+0.48\pm 0.11\)  \(+0.72\pm 0.12\)  \(+0.65\pm 0.06\)  \(+0.76\pm 0.07\)  \(+0.61\pm 0.01\)  \(+0.5\) 
10.0  \(+0.46\pm 0.09\)  \(+0.66\pm 0.09\)  \(+0.61\pm 0.04\)  \(+0.70\pm 0.04\)  \(+0.60\pm 0.01\)  (0.63)  
\(d_V\)  2.0  \(+0.41\pm 0.11\)  \(+0.73\pm 0.12\)  \(+0.79\pm 0.06\)  \(+1.39\pm 0.10\)  \(+1.11\pm 0.03\)  \(+0.5\) 
10.0  \(+0.41\pm 0.11\)  \(+0.66\pm 0.07\)  \(+0.70\pm 0.04\)  \(+0.91\pm 0.08\)  \(+0.95\pm 0.05\)  (0.63)  
S  2.0  \(0.14\pm 0.06\)  \(0.15\pm 0.05\)  \(0.09\pm 0.04\)  \(0.16\pm 0.02\)  \(0.18\pm 0.03\)  \(0.08\) 
10.0  \(0.18\pm 0.04\)  \(0.20\pm 0.05\)  \(0.15\pm 0.04\)  \(0.19\pm 0.01\)  \(0.14\pm 0.02\)  (\(0.2\))  
g  2.0  \(0.16\pm 0.63\)  \(+0.06\pm 0.31\)  \(0.79\pm 0.43\)  \(+0.18\pm 0.10\)  \(+0.08\pm 0.03\)  \(0.08\) 
10.0  \(0.20\pm 0.46\)  \(0.15\pm 0.15\)  \(0.29\pm 0.09\)  \(0.15\pm 0.01\)  \(0.14\pm 0.01\)  (\(0.2\)) 
Same as Table 2 for the largex effective exponent \(\beta _{f_i}(x_b,Q^2)\) at \(x_b=0.9\) (for \(u_V\), \(d_V\) and g) and \(x_b=0.5\) (for S). The values of the exponent \(b_{f_i}\) predicted by Brodsky–Farrar quark counting rules are also shown
\(f_i\)  \(Q^2\)  \(\beta _{f_i}(x_b,Q^2)\)  \(b_{f_i}\)  

(GeV\(^2\))  NNPDF3.0  CT14  MMHT14  ABM12  CJ15  
\(u_V\)  2.0  \(+2.94\pm 0.52\)  \(+3.11\pm 0.28\)  \(+3.37\pm 0.07\)  \(+3.38\pm 0.06\)  \(+3.50\pm 0.01\)  \({\sim }3\) 
10.0  \(+3.30\pm 0.69\)  \(+3.38\pm 0.29\)  \(+3.62\pm 0.07\)  \(+3.61\pm 0.05\)  \(+3.78\pm 0.01\)  
\(d_V\)  2.0  \(+3.03\pm 1.96\)  \(+3.27\pm 0.37\)  \(+2.05\pm 0.59\)  \(+4.72\pm 0.43\)  \(+3.42\pm 0.06\)  \({\sim }3\) 
10.0  \(+3.23\pm 1.88\)  \(+3.52\pm 0.36\)  \(+2.29\pm 0.59\)  \(+4.92\pm 0.42\)  \(+3.68\pm 0.05\)  
S  2.0  \(+6.86 \pm 7.25\)  \(+6.41 \pm 1.22\)  \(+8.19 \pm 0.68\)  \(+8.16 \pm 0.38\)  \(+7.73 \pm 0.18\)  \({\sim }7\) 
10.0  \(+6.76 \pm 6.71\)  \(+6.91 \pm 1.14\)  \(+6.83 \pm 0.88\)  \(+8.51 \pm 0.38\)  \(+8.15 \pm 0.18\)  
g  2.0  \(+2.95\pm 1.25\)  \(+5.08\pm 2.18\)  \(+1.65\pm 0.23\)  \(+4.18\pm 0.06\)  \(+6.11\pm 0.33\)  \({\sim }5\) 
10.0  \(+3.25\pm 0.98\)  \(+5.13\pm 0.51\)  \(+2.24\pm 0.23\)  \(+4.44\pm 0.06\)  \(+4.91\pm 0.33\) 
3 Comparison with nonperturbative predictions
We now discuss how our findings compare with the expectations from Regge theory and the Brodsky–Farrar quark counting rules. In Tables 2, 3 we show the values of the effective exponents for the NNPDF3.0, CT14, MMHT14, ABM12 and CJ15 PDF sets, computed at \(x_a=10^{4}\) and \(x_b=0.9\) (\(x_b=0.5\) for S) at \(Q^2=2\) GeV\(^2\) and \(Q^2=10\) GeV\(^2\). We also include the values predicted by Regge theory and the Brodsky–Farrar quark counting rules.
In comparing these expectations with the results from PDF fits, we need to choose a scale. Regge predictions are expected to hold only at low scales. For \(\alpha _{u_V}(x,Q^2)\) and \(\alpha _{d_V}(x,Q^2)\) this is not too much of a problem, since the scale dependence of nonsinglet distributions is quite weak (see Fig. 5). The values extracted from NNPDF3.0 are accordingly in good agreement with Regge expectations; those from the other global PDF fits are generally a little high (see Table 2). On the other hand, for \(\alpha _S(x,Q^2)\simeq \alpha _\Sigma (x,Q^2)\) and \(\alpha _g(x,Q^2)\), the scale dependence is rather strong (see Fig. 6), due to the double scaling behaviour. Making the comparison at low scales, we see reasonable agreement for the sea quarks with the Pomeron prediction, and also with the NLLx perturbative prediction. Uncertainties for the gluon intercept are inevitably large, so here the agreement is only qualitative. Note that for ABM12 and CJ15 the uncertainties are often substantially underestimated due to parametrisation constraints in the extrapolation region.
At large x, the Brodsky–Farrar quark counting rules predict that \(xf_i\sim (1x)^{2n_s1}\), where \(n_s\) is the minimum number of spectator partons. These are defined to be the partons that are not struck in the hardscattering process, since it is assumed that, in the limit \(x\rightarrow 1\), there can be no momentum left for any of the partons other than the struck parton. In a proton made of three quarks, one has for a valence quark, \(n_s=2\) and thus \(b_{u_V}=b_{d_V}=3\); for a gluon, \(n_s=3\) and \(b_g=5\); for a sea quark, \(n_s=4\) and \(b_S=7\); see Fig. 7. Note that the values of the exponents predicted by Brodsky–Farrar quark counting rules are different if the polarisation of the quark with respect to the polarisation of the parent hadron is retained [55]. This also affects the difference between up and down distributions. A detailed comparison between PDFs and quark counting rules in the polarised case was presented in Ref. [26]. Again it is unclear from the quark model argument at which scale these predictions are supposed to apply, but again we are fortunate that the scale dependence of largex PDFs is reasonably moderate (see Figs. 5, 6), and it is reasonable to make the comparison at a low scale [25].
The predictions \(b_{u_V}(x,Q^2)\) and \(b_{d_V}(x,Q^2)\) for the valence distributions are then in broad agreement with the effective exponents determined from most of the global PDF fits, though some deviations from Brodsky–Farrar quark counting rule expectations are observed for the MMHT14 down valence quarks: this seems to be a result of the oscillation noted already in Fig. 2. For the quark sea and the gluon, the success is again rather mixed, and only CT14 seems to provide results which agree with the prediction; for NNPDF3.0 the uncertainties on the quark sea are too large for the extraction to be meaningful, while the result for the gluon is a little low; for MMHT14 the result for the gluon is far too low, with a substantially underestimated uncertainty.
From Fig. 8, we see that in the region in which the valence quarks are constrained by experimental data, i.e. \(x\lesssim 0.5\), the predictions for both ratios from all the PDF sets are in reasonable agreement with each other within uncertainties, as might be expected. For \(x\gtrsim 0.5\), the mutual consistency of PDF sets deteriorates rapidly, and a wide range of different behaviours is observed. This is a consequence of the reduced experimental information in this region: different PDF collaborations extrapolate to large x using different assumptions. For those sets with very weak assumptions on the PDF behaviour at large x, namely NNPDF3.0 and MMHT14, the uncertainties on the ratios expand rapidly, and at very large x there is no predictive power at all. For the two sets which assume that \(d_V/u_V\rightarrow k\) at large x, namely CT14 and CJ15, uncertainties are inevitably much reduced and a value of k is predicted. ABM12 is different again, in that they find as a result of their fit that \(b_{d_V}>b_{u_V}\) at more than two standard deviations (see Table 3), so that \(d_V/u_V\rightarrow 0\) as \(x\rightarrow 1\), and an unrealistically small uncertainty band in a region where there are actually no data.
It follows that all the various model predictions displayed in Fig. 8 are compatible with the NNPDF3.0 and MMHT14 predictions, while ABM12 confirms the Chiral Quark Model but appears to rule out all the others. The CT14 and CJ15 sets favour values of k in the region \(0\lesssim k \lesssim 0.25\), thus disfavouring the SU(6) prediction but unable to discriminate between the others. The preference for smaller values of k results in effect from a linear extrapolation of the downwards trend in the data region \(x\lesssim 0.5\). Not all the predictions respect the Nachtmann bound, Eq. (16).
4 Conclusions and outlook
In summary, in this work we have introduced a novel methodology to determine quantitatively the effective asymptotic behaviour of parton distributions, valid for any value of x and \(Q^2\). For the first time, we have unambiguously identified the ranges in x and \(Q^2\) where the asymptotic regime sets in, allowing us to compare in detail perturbative and nonperturbative QCD predictions at large and small x with the results of modern global PDF fits.
Concerning the smallx region, we have found broad agreement between the results from PDF fits and the predictions from Regge theory for the behaviour of the valence quark distributions. For the singlet and gluon distributions, the agreement with Regge predictions is still only qualitative, due in part to the substantial scale dependence, as well as the limited experimental information available in that region. On the other hand, the perturbative QCD double asymptotic scaling predictions are in excellent agreement with the results of PDFs fits over a wide range of \(Q^2\).
Concerning the largex region, we have found that the predictions of the Brodsky–Farrar counting rules for the behaviour of the valence quark distributions are in broad agreement with the global fit results, within PDF uncertainties. For the sea and gluon distributions uncertainties are much larger, and the agreement is only qualitative. The scale dependence of the effective exponents based on global PDF fits is in excellent agreement with the perturbative QCD expectation from the cusp anomalous dimension in a wide range of \(Q^2\). We have also compared the ratios \(d_V(x,Q^2)/u_V(x,Q^2)\) and \(F_2^n(x,Q^2)/F_2^p(x,Q^2)\) among PDF fits and with nonperturbative models of nucleon structure, but found that the interpretation of this comparison depends significantly on the assumptions built into the PDF parametrisation, to the extent that it is impossible at present to draw any firm conclusions.
We therefore conclude that, while the ancient wisdom of Regge theory and the Brodsky–Farrar counting rules seems to have some degree of truth, particularly in the valence quark sector, they are no substitute for the precise empirical PDF determinations provided by global analysis, and when used as constraints may lead to unrealistically accurate predictions in kinematic regions where there is no experimental data. Global PDF fits will always be hampered to some extent by the lack of data to constrain PDFs in extrapolation regions, and new measurements from the LHC and other facilities, such as JLab, are required to shed more light on the asymptotic behaviour of parton distributions at small and large x. The methodology presented in this work should find applications in future comparisons between different global PDF fits, and between PDF fits and nonperturbative models of nucleon structure.
Footnotes
Notes
Acknowledgments
J.R. is supported by an STFC Rutherford Fellowship ST/K005227/1 and by an European Research Council Starting Grant “PDF4BSM”. The work of J.R. and E.R.N. is supported by a STFC Rutherford Grant ST/M003787/1.
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