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| Gross–Llewellyn Smith sum rule | |
|---|---|
| Name | Gross–Llewellyn Smith sum rule |
| Field | Particle physics |
| Introduced | 1969 |
| Authors | David J. Gross; C. H. Llewellyn Smith |
Gross–Llewellyn Smith sum rule The Gross–Llewellyn Smith sum rule is a relation in high-energy Deep inelastic scattering that connects the integral of the parity-violating structure function in neutrino–nucleon interactions to fundamental properties of quark content. It was formulated by David J. Gross and C. H. Llewellyn Smith and plays a central role in testing the predictions of Quantum Chromodynamics and the structure of the proton and neutron. The sum rule provides an experimental bridge between observed cross sections at facilities like CERN, Fermilab, and SLAC National Accelerator Laboratory and theoretical calculations involving renormalization and perturbative corrections.
The sum rule states that the first moment of the parity-violating structure function measured in charged-current neutrino scattering off an isoscalar target equals the number of valence quarks (three) up to calculable corrections. It forms part of a set of integral relations including the Adler sum rule and the Bjorken sum rule that have been used to validate aspects of Quantum Chromodynamics and the Parton model. Early theoretical context includes work by researchers at Princeton University, Harvard University, and Yale University applying operator product expansion techniques developed by figures such as Kenneth G. Wilson.
The derivation uses the Operator product expansion and current algebra for charged-current weak interactions mediated by the W boson in the framework of the Standard Model. One constructs moments of structure functions from the hadronic tensor in neutrino–nucleon scattering and relates them to matrix elements of local operators like the non-singlet axial and vector currents studied by Gerard 't Hooft and Steven Weinberg. Perturbative renormalization group methods such as those developed by David Gross and Frank Wilczek allow calculation of logarithmic scaling violations. The derivation highlights cancellations of sea-quark and gluon contributions in the non-singlet channel, connecting to concepts advanced by Richard Feynman in the Parton distribution function framework and by John C. Collins in factorization proofs.
Measurements have been performed using neutrino beams at experiments and laboratories including CERN experiments with the PS and SPS complexes, the NuTeV and CCFR experiments at Fermilab, and charged-current studies at SLAC National Accelerator Laboratory. Analyses combine cross-section data to form the integral over Bjorken-x and require corrections for target composition and radiative effects calculated with input from Particle Data Group conventions and PDFs provided by collaborations such as CTEQ, MSTW, and NNPDF. Results compare the measured first moment with the theoretical value of three, showing deviations accounted for by perturbative Quantum Chromodynamics corrections and higher-twist effects investigated by groups at DESY, Brookhaven National Laboratory, and KEK.
Perturbative Quantum Chromodynamics corrections reduce the leading-order prediction through calculable series in the strong coupling constant αs, with coefficients computed up to next-to-next-to-leading order by theorists at institutions like CERN and Brookhaven National Laboratory. Renormalization schemes developed by G. 't Hooft and others and computations using techniques by S. Weinberg and Eugene Wigner ensure scheme-independent predictions for measurable quantities. Non-perturbative higher-twist contributions, tied to multi-parton correlations and power-suppressed terms, have been evaluated using models inspired by Shifman–Vainshtein–Zakharov sum rules and lattice computations from groups at CERN and Fermi National Accelerator Laboratory.
The sum rule constrains the non-singlet combination of quark parton distribution functions and thus informs global PDF fits by collaborations such as CTEQ, MSTW, NNPDF, and HERAPDF. It provides a check on valence-quark normalization in the proton and neutron and influences interpretations of flavor asymmetries studied in experiments at CERN, Fermilab, and Jefferson Lab. By testing whether the integrated non-singlet distribution equals three after QCD corrections, the sum rule probes sea-quark dynamics, gluon radiation patterns described by the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi equations, and mechanisms explored in models by Isgur and Karl and in lattice QCD efforts from collaborations like QCDSF.
Proposed in 1969 by David J. Gross and C. H. Llewellyn Smith, the sum rule became a milestone for validating asymptotic freedom and the emerging Quantum Chromodynamics theory championed by Gross, Frank Wilczek, and David Politzer. Experimental programs at SLAC, CERN, and Fermilab tested the relation, motivating theoretical work on higher-order corrections from groups at CERN and Brookhaven National Laboratory and stimulating global PDF fitting efforts at CTEQ and MSTW. The interplay between theory and experiment around this sum rule contributed to awarding of the Nobel Prize in Physics to pioneers of QCD and cemented techniques now standard in analyses at the Large Hadron Collider and future facilities.
Category:Quantum chromodynamics Category:Deep inelastic scattering Category:Particle physics