arXiv subjects
R. D. Ball
Publications and source records attributed to R. D. Ball.
The Challenge of Small x
We review the current understanding of the behaviour of inclusive cross sections at small x and large Q^2 in terms of Altarelli-Parisi evolution, the BFKL equation, and Regge theory, asking in particular to what extent they are mutually consistent.
Theoretical Analysis of Polarized Structure Functions
We review the analysis of polarized structure function data using perturbative QCD at next-to-leading order. We use the most recent experimental data to obtain updated results for polarized parton distributions, first moments and the strong coupling. We also discuss several theoretical issues involved in this analysis and in the interpretation of its results. Finally, we compare our results with other similar analyses in the recent literature.
Asymptotically Free Partons at High Energy
We describe the application of renormalization group improved perturbative QCD to inelastic lepton-hadron scattering at high center-of-mass energy but comparatively low photon virtuality. We construct a high energy factorization theorem which complements the mass factorization theorem used for processes with high virtualities. From it we derive a renormalization group equation which resums all large logarithms at high energy, thereby extending to this regime asymptotic freedom and thus the full range of perturbative computational techniques. We discuss the solution of this equation in various limits, and in particular show that the high energy behaviour of physical cross-sections is consistent with phenomenological expectations and unitarity bounds.
Structure Functions
We summarize recent developments in the understanding of nucleon structure. New data on F2, R and F2c, over a wide range of Q2 and x, are described. Conventional leading twist NLO perturbative QCD gives an excellent account of all the new data with Q2 above a GeV, leaving very little room for either higher twists or higher logarithms. We summarize the current status of NLO fits, and the determination of the gluon distribution and the strong coupling constant from structure function data. Finally we consider some of the theoretical issues raised by the new data.
Perturbative Evolution of Polarized Structure Functions
We review the perturbative evolution of the polarized structure functions g_1 and their associated parton distribution functions, with particular emphasis on the anomalous coupling of the first moment of the polarized gluon distribution. We also describe the small x behaviour of polarized parton distributions, contrasting it with that of the unpolarized distributions. We then explain how this theoretical analysis affects the extraction of the singlet axial charge from experimental data on g_1, and show that it may be possible to use such data to infer the existence of polarized gluons in the nucleon.
A Next-to-Leading Determination of the Singlet Axial Charge and the Polarized Gluon Content of the Nucleon
We perform a full next-to-leading analysis of the the available experimental data on the polarized structure function g_1 of the nucleon, and give a precise determination of its singlet axial charge together with a thorough assessment of the theoretical uncertainties. We find that the data are now sufficient to separately determine first moments of the polarized quark and gluon distributions, and show in particular that the gluon contribution is large and positive.
Determination of alpha_s from F_2^p at HERA
We compute the proton structure function F_2^p at small x and large Q^2 at next-to-leading order in alpha_s(Q^2), including summations of all leading and subleading logarithms of Q^2 and 1/x in a way consistent with momentum conservation. We perform a detailed comparison to the 1993 HERA data, and show that they may be used to determine alpha_s(M_Z^2)=0.120 pm 0.005(exp) pm 0.009(th). The theoretical error is dominated by the renormalization and factorization scheme ambiguities.
Momentum Conservation at Small x
We discuss how momentum conservation is implemented in perturbative computations based on expansions of anomalous dimensions appropriate at small $x$. We show that for any given choice of $F_2$ coefficient functions there always exists a factorization scheme where the gluon is defined in such a way that momentum is conserved at next to leading order.
Scheme Dependence at Small x
We discuss the evolution of F_2^p at small x, emphasizing the uncertainties related to expansion, fitting, renormalization and factorization scheme dependence. We find that perturbative extrapolation from the measured region down to smaller x and lower Q^2 may become strongly scheme dependent.
Next-to-Leading Order Structure Function Evolution at Small x and Large Q^2
We show that a unified approach to the perturbative evolution of structure functions which sums all logarithms of Q^2 and 1/x at leading and next-to-leading order yields results in full agreement with the 1993 HERA data for F_2. This makes it possible to determine alpha_s surprisingly accurately from these data alone.
Scale Dependence and Small-x Behaviour of Polarized Parton Distributions
We discuss perturbative evolution of the polarized structure function g_1 in the (x,Q^2) plane, with special regard to the small-x region. We determine g_1 in terms of polarized quark and gluon distributions using coefficient functions to order alpha_s. At small x g_1 then displays substantial scale dependence, which necessarily implies a corresponding scale dependence in the large-x region. This scale dependence has significant consequences for the extraction of the first moment from the experimental data, reducing its value while increasing the error. Conversely, the scale dependence may be used to constrain the size of the polarized gluon distribution.
Summation of Leading Logarithms at Small x
We show how perturbation theory may be reorganized to give splitting functions which include order by order convergent sums of all leading logarithms of $x$. This gives a leading twist evolution equation for parton distributions which sums all leading logarithms of $x$ and $Q^2$, allowing stable perturbative evolution down to arbitrarily small values of $x$. Perturbative evolution then generates the double scaling rise of $F_2$ observed at HERA, while in the formal limit $x\to 0$ at fixed $Q^2$ the Lipatov $x^{-λ}$ behaviour is eventually reproduced. We are thus able to explain why leading order perturbation theory works so well in the HERA region.
Scheme Independence and the Exact Renormalization Group
We compute critical exponents in a $Z_2$ symmetric scalar field theory in three dimensions, using Wilson's exact renormalization group equations expanded in powers of derivatives. A nontrivial relation between these exponents is confirmed explicitly at the first two orders in the derivative expansion. At leading order all our results are cutoff independent, while at next-to-leading order they are not, and the determination of critical exponents becomes ambiguous. We discuss the possible ways in which this scheme ambiguity might be resolved.
The Rise in F_2^p at HERA
We show that the rise in $F_2^p$ at small $x$ and large $Q^2$ seen at HERA is indeed the non-Regge double asymptotic scaling behaviour expected from the perturbative emission of strongly ordered hard gluons. An alternative explanation, in which there is no strong ordering, and a new hard Reggeon is generated, is also tried but found wanting: its theoretical short-comings are betrayed by its failure to properly account for the HERA data.
Calculating F_2^p at small x and large Q^2
We show that the double asymptotic scaling of the HERA structure function data is consistent with pre-HERA data at larger $x$, soft pomeron behaviour at small $x$ and a sensible starting scale $Q_0$. We can thus actually calculate $F_2^p$ at small $x$ and large $Q^2$ by evolving up perturbatively at two loops, without any fitting.
A Direct Test of Perturbative QCD at Small x
We show that recent data from HERA on the proton structure function $F_2$ at small $x$ and large $Q^2$ provide a direct confirmation of the double asymptotic scaling prediction of perturbative QCD. A linear rise of $\ln F_2$ with the scaling variable $σ$ is observed throughout the kinematic region probed at HERA, and the measured slope is in excellent agreement with the QCD prediction. This provides a direct determination of the leading coefficient of the beta function. At large values of the scaling variable $ρ$ the data display a small but statistically significant scaling violation.
Double Asymptotic Scaling at HERA
Perturbative QCD predicts that at sufficiently large $Q^2$ and small $x$ nucleon structure functions should exhibit scaling in the two variables $\sqrt{\ln\smallfrac{1}{x}\ln\ln Q^2}$ and $\sqrt{\ln\smallfrac{1}{x}\big/\ln\ln Q^2}$, provided only that the small-$x$ behaviour of the input to the perturbative QCD evolution is sufficiently soft. We derive these asymptotic results by writing the gluonic Altarelli--Parisi equation at small $x$ as a two--dimensional wave equation, which propagates the gluon distribution from its boundaries into the asymptotic region. We then show that the existing experimental data on $F_2^p(x,Q^2)$ from HERA provide a remarkable confirmation of both of these scaling predictions. The so--called `hard' pomeron, which does not scale, may thus be excluded by more than three standard deviations, at least in the presently accessible kinematical regime. We propose that existing and future data from HERA should be binned in the two scaling variables, in order to search for scaling violations.