Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Theory of Deeply Inelastic Scattering

Johannes Blümlein

TL;DR

Deep-inelastic scattering (DIS) provides precision tests of Quantum Chromodynamics (QCD) by measuring structure functions and extracting the strong coupling constant $α_s(M_Z^2)$ and parton distribution functions (PDFs). The article surveys the theoretical framework, including the light-cone expansion, renormalization and factorization, heavy flavor corrections, and high-order Wilson coefficients and anomalous dimensions up to four loops, as well as QED/EW corrections, target-mass and higher-twist effects. It covers small-$x$ and large-$x$ resummations, sum rules, nuclear PDFs, and the solution of evolution equations in Mellin space, highlighting global NNLO analyses (ABM, MSTW, JR, NNPDF, CTEQ, HERAPDF) and their impact on collider phenomenology. The work underscores the importance of continued high-precision theory and data for DIS to sharpen tests of QCD and guide analyses at the LHC and future facilities such as the EIC and LHeC, especially for refining $α_s(M_Z^2)$ and PDFs.

Abstract

The nucleon structure functions probed in deep-inelastic scattering at large virtualities form an important tool to test Quantum Chromdynamics (QCD) through precision measurements of the strong coupling constant $α_s(M_Z^2)$ and the different parton distribution functions. The exact knowledge of these quantities is also of importance for all precision measurements at hadron colliders. During the last two decades very significant progress has been made in performing precision calculations. We review the theoretical status reached for both unpolarized and polarized lepton-hadron scattering based on perturbative QCD.

The Theory of Deeply Inelastic Scattering

TL;DR

Deep-inelastic scattering (DIS) provides precision tests of Quantum Chromodynamics (QCD) by measuring structure functions and extracting the strong coupling constant and parton distribution functions (PDFs). The article surveys the theoretical framework, including the light-cone expansion, renormalization and factorization, heavy flavor corrections, and high-order Wilson coefficients and anomalous dimensions up to four loops, as well as QED/EW corrections, target-mass and higher-twist effects. It covers small- and large- resummations, sum rules, nuclear PDFs, and the solution of evolution equations in Mellin space, highlighting global NNLO analyses (ABM, MSTW, JR, NNPDF, CTEQ, HERAPDF) and their impact on collider phenomenology. The work underscores the importance of continued high-precision theory and data for DIS to sharpen tests of QCD and guide analyses at the LHC and future facilities such as the EIC and LHeC, especially for refining and PDFs.

Abstract

The nucleon structure functions probed in deep-inelastic scattering at large virtualities form an important tool to test Quantum Chromdynamics (QCD) through precision measurements of the strong coupling constant and the different parton distribution functions. The exact knowledge of these quantities is also of importance for all precision measurements at hadron colliders. During the last two decades very significant progress has been made in performing precision calculations. We review the theoretical status reached for both unpolarized and polarized lepton-hadron scattering based on perturbative QCD.

Paper Structure

This paper contains 30 sections, 174 equations, 23 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The deep-inelastic process
  3. The Light-Cone Expansion
  4. Parton Models
  5. Renormalization and Factorization and Deep-Inelastic Structure Functions
  6. The strong coupling constant
  7. Anomalous Dimensions and Splitting Functions
  8. Leading Order
  9. Next-to-Leading Order
  10. 3-Loop Order
  11. 4-Loop Order
  12. Anomalous dimensions in the large $N_f$ limit
  13. Coefficient functions
  14. First Order
  15. Higher Orders
  16. ...and 15 more sections

Figures (23)

  • Figure 1: Charge distribution for the proton and the neutron implied by the form factors of Ref. Olson:1961zz, Figure 2(b); from Olson:1961zz, © (1961) by the American Physical Society.
  • Figure 2: Left: An early observation of scaling: $\nu W_2$ for the proton as a function of $-q^2$ for $W > 2 \,\hbox{GeV}$, at $x = 1/\omega$ = 0.25; Right: The Callan-Gross relation: $K_0 = F_2/(2xF_1) - 1$ vs $-q^2$. These results established the spin of the partons as $1/2$; from Kendall:1991np, © (1991) by the American Physical Society.
  • Figure 3: The function $\nu W_2$ plotted vs $|q^2|$ assuming $R = |q^2|/\nu^2$ for different fixed $\omega_W = (1/x + M_N^2/|q^2|)/(1+ 0.2/|q^2|)$; from Brasse:1972wk© (1972) by Elsevier Science.
  • Figure 5: Left: HERA combined NC $e^+p$ reduced cross section as a function of $Q^2$ for six $x$-bins compared to the separate H1 and ZEUS data input to the averaging procedure. The error bars indicate the total experimental uncertainty. The individual measurements are displaced horizontally for better visibility; Right : The combined data with the HERAPDF1.0 fit HERAPDF1.0 is superimposed. The bands represent the total uncertainty of the fit. Dashed lines are shown for $Q^2$ values not included in the QCD analysis; from herapdf:2009wt© (2009) Springer Verlag.
  • Figure 6: The data on $F_L$ versus $x$ obtained by the H1 collaboration Aaron:2010ry confronted with the 3-flavor scheme NNLO predictions based on the different parton distributions functions (PDFs). Solid line: ABM11 Alekhin:2012ig, dashes: JR09 JimenezDelgado:2008hf, dots: MSTW Martin:2009iq). The NLO predictions based on the 3-flavor NN21 PDFs Ball:2011mu are given for comparison (dashed dots). The value of $Q^2$ for the data points and the curves in the plot rises with $x$ in the range of $1.5 \div 45~{\rm GeV}^2$; from Alekhin:2012ig.
  • ...and 18 more figures