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The continuing challenge of quantum chromodynamics

31 March 2003

Thirty years after quantum chromodynamics was invented, tackling its difficulties continues to fascinate a host of theorists – and often proves a necessity for unravelling the limits of the Standard Model – as the 2002 DESY Theory Workshop revealed.

Quantum chromodynamics (QCD), the theory of the strong force, is a marvellous example of how the physical laws that describe a large variety of complex phenomena can be condensed into a very simple and elegant mathematical structure, known as non-abelian gauge theory. The fundamental equations can be written down in a single line, yet they describe how the nucleons acquire their masses from “nothing”, or how two nucleons smashed together at high energies disintegrate into dozens of new particles bundled into “jets” – the visible manifestations of the quarks and gluons. The fundamental equations are extremely hard to solve. At higher energies where the strong force weakens, the equations may be expanded in a perturbation series, where each new term demands more sophisticated analytical or numerical methods of computation. At energies of the order of the proton mass, the equations can only be solved by large-scale computers.

From September 24-27, 2002, approximately 130 high-energy physicists gathered in Hamburg at the annual DESY Theory Workshop to discuss their recent advances in the development of computational methods, and their successes (and sometimes failures) in comparing their calculations with experiments that continue to become more precise or to explore new phenomena. As emphasized by many talks at the workshop, these efforts go far beyond understanding how hadronic phenomena work. As the high-energy community gathers its resources to attack the fortress of the Standard Model, which has stood unconquered for the past 30 years, the strong interaction is a faithful, though not always loved, companion. Whether protons collide at the LHC to produce perhaps the Higgs boson or new particles, whether B mesons decay at SLAC and KEK to reveal the subtle asymmetry of matter and anti-matter, or whether the anomalous magnetic moment of the muon is measured to a part in a billion, an accurate computation of strong interaction effects will be required to ascertain finally a failure of the Standard Model.

Inside the proton

What is a proton? The answer is more difficult than just “three quarks”. In high-energy collisions the proton appears as a bunch of quarks and gluons collectively called partons. The (longitudinal) momentum distributions of these partons are fundamental input to the computation of any proton collision. James Stirling of Durham reviewed the current knowledge of parton distributions and concluded that the global fit is satisfactory. Methods are now being developed to assign reliable errors to these functions, which may soon be known with higher (“next-to-next-to-leading order”) theoretical accuracy. Closely related to the conventional parton distributions are the diffractive parton distributions, which give the probability of finding a parton in the proton under the additional condition that the proton stays intact in the collision. One of the surprising results of DESY’s HERA experiments is that this probability remains large, even at the highest momentum transfers. The physical interpretation of this was provided by John Collins of Penn State, who also emphasized that models of soft interactions in diffractive scattering should be taken as models for the corresponding parton distributions.

At high collision energies the number of partons with small momentum fraction x of the proton increases rapidly and the conventional, perturbative equations should break down. They should be replaced by an equation that sums logarithms in x, known as the BFKL equation. In the leading approximation, the solution to the BFKL equation overestimates the growth of high-energy cross sections. Victor Fadin of Novosibirsk discussed the progress made towards a next-to-leading approximation. With many parts now being completed, the calculation of the so-called photon impact factor is required before a comparison with experiments can be attempted. Whatever the result, at very small momentum fraction the growth of parton densities must stop. As explained by Alfred Mueller of Columbia, this occurs for quarks because the Pauli principle limits the number of fermions per phase space cell. For gluons, however, “saturation” already occurs classically when the density of gluons is so high that their combined field strength is non-perturbatively large. Mueller discussed the applicability of a classical description and estimates of the saturation scale during various stages of the collision process under the conditions at HERA and at Brookhaven’s RHIC collider.

The experimental verification of these phenomena at HERA remains ambiguous, according to Brian Foster of Bristol. He also showed an impressive amount of jet data – all in agreement with QCD computations – and demonstrated that the strong coupling constant can now be determined from electron-proton collisions with high accuracy. The HERA collider has become a veritable QCD factory, providing data over many orders of magnitude in momentum transfers and for many final states that probe different aspects of the strong interaction. Understanding the transition to soft, non-perturbative physics remains one of the most difficult challenges. This transition appears to be surprisingly smooth. Hans-Günther Dosch of Heidelberg showed that a simple model which views the QCD vacuum as an ensemble of Gaussian gauge field fluctuations, allows many features of soft hadronic interactions at high energy to be related to properties of the QCD vacuum.

The spin of the proton is 1/2, but how is it distributed over the various partons? A decade ago the “spin crisis” was proclaimed, after it was observed that the quarks carry only a fraction of the total spin. The talks by Elke-Caroline Aschenauer of DESY and Daniel Boer of Amsterdam highlighted that experimentalists and theorists still struggle to account for the remainder. For example, the gluon’s contribution to the spin remains largely unknown and its direct determination requires less inclusive measurements than polarised deep-inelastic scattering. Getting hold of orbital angular momentum is even harder and demands the introduction of new theoretical concepts (“generalized parton distributions”), which can be constrained by observing Compton scattering of virtual photons off protons.

Lattice calculations

Perturbative approximations are not adequate for ab initio calculations of hadron masses or, more generally, hadronic matrix elements, which are governed by strong-coupling physics. In these cases numerical simulation of QCD on a discrete space-time lattice provides the only systematic approach. Lattice QCD benefits greatly from the increasing speed of computers, where the scale of machines is currently set by TeraFlops (1012 operations per second). However, as emphasized by several speakers at this workshop, conceptual progress and the improvement of simulation algorithms play at least an equally important role.

Most calculations are still performed with a truncated version of QCD, which neglects quark-antiquark quantum fluctuations. Allowing quarks to fluctuate is costly, as discussed by Sinya Aoki of Tsukuba, and forces the use of smaller, coarser space-time lattices. Aoki showed that the computed hadron spectrum is in much better agreement with observations for dynamical quarks, but pointed to the need for better algorithms that would allow the simulation of light quarks with masses closer to their real values.

A different avenue was pursued by Hartmut Wittig of DESY, who reviewed the various methods to put massless (but still non-dynamical) quarks on the lattice. This became a real possibility a few years ago when it was discovered that QCD at finite lattice spacing has an exact symmetry that approaches the conventional chiral symmetry in the continuum limit. Wittig showed that the efforts to put this into practice are now bearing fruit for quantities such as the strange quark mass or the quark condensate, where the chiral behaviour is particularly important. Chiral symmetry is also important for kaon physics, where results from lattice calculations have a large impact on the interpretation of direct and indirect CP-violating effects. Indirect CP violation in kaon decay to two pions poses a particular challenge to lattice theorists, since the relevant matrix elements include final state interactions. Chris Sachrajda of Southampton described new ideas to extract these matrix elements by exploiting the finite size of the lattice.

Another impressive demonstration of progress in lattice gauge theory was given by Martin Lüscher of CERN. Using a new algorithm that allows the computation of large Wilson loops, he showed that the large-distance behaviour is consistent with the assumption that the low energy limit of SU(N) gauge theory is a bosonic string theory. Moreover, the perturbative regime joins smoothly to the string regime at a distance of about 0.5 fm.

Heavy quarks

Charm and bottom quarks are produced in large numbers at today’s high-energy colliders. The theory of single-inclusive heavy meson production and of quarkonium production was reviewed by Bernd Kniehl of Hamburg, who described the efforts to treat heavy quark mass effects correctly at all energy scales. He also concluded that, with the exception of polarisation measurements, the non-relativistic factorization approach to quarkonium production appears to be supported by existing data. A particularly interesting quarkonium system consists of a top-antitop pair. Although this system decays after little more than 10-25 s, the strong Coulomb force the quarks exert on each other leaves a visible enhancement in the energy dependence of the production cross section. This can be used (at an electron-positron collider) to determine the top quark mass to an accuracy of less than a permille. Thomas Teubner of CERN showed that many of the theoretical difficulties involved in the calculation of the threshold cross section have now been solved with non-relativistic effective field theory. Very similar calculations also determine the bottom and charm quark mass from quarkonium systems.

A complementary method uses inclusive heavy quark production in e+e collisions far above the production threshold. The corresponding hadronic spectral functions also provide an indispensable source of information for other fundamental constants, such as the strong coupling, the hadronic contribution to the electromagnetic coupling (at the scale of the Z mass) or the anomalous magnetic moment of the muon. The accuracy needed for these quantities is reflected in the development of sophisticated symbolic manipulation programs, which enable the computation of thousands of multi-loop Feynman diagrams. Matthias Steinhauser of Hamburg discussed recent advances, particularly in including quark mass effects and their impact on precision determinations of the coupling constants. Similar methods of algebraic reduction of Feynman integrals are now also being applied for jet physics, where many further difficulties come from the more complicated kinematics. The new frontier, stated Nigel Glover of Durham, is set by next-to-next-to-leading order calculations. He explained that while all the two-loop virtual effects are now completed, the construction of a usable Monte Carlo program that combines them with bremsstrahlung effects will probably require another few years of hard work.

QCD and the Standard Model

Many processes that would otherwise provide clean probes of fundamental interactions are ultimately sensitive to QCD through quantum fluctuations. One particularly well known example is the flavour-changing neutral current process B → Xsγ, reviewed by Mikolaj Misiak of Warsaw, where strong interaction effects double the predicted branching fraction. Experiment and theory currently agree, but to what precision can one compute strong interaction effects? Misiak explained how quark mass renormalization prescriptions influence the prediction, but concluded that the dominant uncertainties can still be reduced by perturbative calculations. They would however be very difficult. The discussion was continued with a review of exclusive heavy meson decays, where the problem of hadronization is even more direct. Understanding decays such as B → ππ, which can now be studied in detail at the B-factories, is crucial in order to ascertain the (in)consistency of the Kobayashi-Maskawa mechanism for CP violation in the quark sector. Gerhard Buchalla of Munich reported progress in applying QCD factorization methods to exclusive B decays, which have led to new insights into dynamical details of these reactions.

The anomalous magnetic moment of the muon has remained a hot topic since the announcement of the result by Brookhaven in 2001 (CERN Courier April 2001 p4 and September 2002 p8). The experimental value, precise to 0.7 ppm, is not quite in agreement with the theoretical result, but whether the discrepancy is the first signal of a breakdown of the Standard Model is a matter of debate. The blame could once more be on the strong interaction. The current status of theoretical calculations was presented by Eduardo de Rafael of Marseille and Fred Jegerlehner of DESY. A controversy on the sign of the so-called light-by-light scattering contribution, a tiny but relevant quantum effect has now been settled, bringing the prediction in better agreement with the data. However, the size of the effect itself remains quite uncertain. Another important development concerns hadronic photon vacuum polarisation effects, which must be determined from low-energy data, in particular around the ρ meson resonance. This year has seen new results from CMD-2 and from an analysis of τ-decays at LEP. Unfortunately, the two do not agree, the difference being more than twice the estimated error. Depending on the input, the theoretical muon anomalous magnetic moment is now 1 to 3 standard deviations smaller than Brookhaven’s experimental result.

QCD also matters in the production of new particles, foremost the Higgs boson. Michael Krämer of Edinburgh reviewed higher order calculations of Higgs, Higgs with top-antitop, and supersymmetric particle production. All these processes are now under good theoretical control. Krämer emphasized, however, that the calculation of signal processes must be accompanied by an equally detailed understanding of backgrounds.

Extreme conditions

The investigation of quark or nuclear matter under extreme conditions of temperature and density has a long history, with possible applications to neutron stars and quark-to-nuclear matter phase transitions in the early universe. With the advent of heavy-ion collisions, most recently (and on-going) at Brookhaven’s RHIC collider, these phenomena are now subject to terrestrial explorations. An interpretation of the first RHIC results was given by Miklos Gyulassy of Columbia, who described the geometric and saturation effects that appear in the collisions of large nuclei. Some of these effects are clearly seen in the data. He also explained how the pattern of energy loss should reveal information about the matter density in the collision region. While the dynamics of a nuclear collision is extremely complicated, the thermodynamics of strong matter is amenable to simulations in lattice QCD. The critical temperature and energy density at the phase transition are now rather well determined, says Edwin Laermann of Bielefeld, at least in the approximation that all quarks are massless. The influence of the strange quark mass on the phase diagram is a very interesting question. Recent theoretical developments concern lattice simulations at finite chemical potential. The difficulty lies in numerical cancellations that occur for a complex action. Laermann explained that it is now possible to investigate small chemical potentials using expansions, reweighting methods or analytic continuation from imaginary chemical potentials.

The phase diagram in the direction of chemical potential was illustrated in the concluding talk by Krishna Rajagopal of MIT, who showed that gluon exchange makes the Fermi surface unstable, rendering dense quark matter a BCS-like colour superconductor. Many more phenomena can occur, depending on the number of quark flavours or the strange quark mass, such as a condensation of colour and flavour quanta in an intertwined pattern. The workshop concluded with the tantalizing speculation that quark matter could actually be crystalline, and a review of the possibilities of detecting this phenomenon in supernovae explosions or pulsar quakes.

The plenary talks were preceded by introductory lectures on Deep Inelastic Scattering and Jets (Keith Ellis of Fermilab), Lattice QCD (Karl Jansen of NIC and DESY), Non-perturbative Methods (Andreas Ringwald of DESY) and Finite-temperature Field Theory (Dietrich Bödeker of Bielefeld), which were very well received both by students and experts. The interest of the community in strong interaction physics was also reflected by around 35 parallel session talks given by young researchers from different countries.

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