Search

The spontaneous breakdown of an approximate symmetry implies that the spectrum of the theory contains approximately massless particles. The hidden symmetry very strongly constrains their masses. A numerical evaluation of these constraints on the lattice would allow a more precise determination of the quark mass ratios m_u:m_d:m_s and thereby reduce some of the uncertainties encountered in precision flavour physics., Comment: In memory of Harald Fritzsch, contribution to the book edited by G. Buchalla, D. L\"ust and Z.-Z. Xing

Introductory section for the volume 50 years of Quantum Chromodynamics (edited by F. Gross and E. Klempt, to be published in Eur. Phys. J. C), Comment: The present article represents a small part of the entire volume which is accessible at arXiv:2212.11107. arXiv admin note: substantial text overlap with arXiv:1211.6777

The Cottingham formula expresses the leading contribution of the electromagnetic interaction to the proton-neutron mass difference as an integral over the forward Compton amplitude. Since quarks and gluons reggeize, the dispersive representation of this amplitude requires a subtraction. We assume that the asymptotic behaviour is dominated by Reggeon exchange. This leads to a sum rule that expresses the subtraction function in terms of measurable quantities. The evaluation of this sum rule leads to $m_{QED}^{p-n}=0.58\pm 0.16\,\mbox{MeV}$., Comment: 26 pages, 9 figures. Update, matches the version published in EPJC

The Cottingham formula expresses the electromagnetic part of the mass of a particle in terms of the virtual Compton scattering amplitude. At large photon momenta, this amplitude is dominated by short distance singularities associated with operators of spin 0 and spin 2. In the difference between proton and neutron, chiral symmetry suppresses the spin 0 term. Although the angular integration removes the spin 2 singularities altogether, the various pieces occurring in the standard decomposition of the Cottingham formula do pick up such contributions. These approach asymptotics extremely slowly because the relevant Wilson coefficients only fall off logarithmically. We rewrite the formula in such a way that the leading spin 2 contributions are avoided ab initio. Using a sum rule that follows from Reggeon dominance, the numerical evaluation of the e.m. part of the mass difference between proton and neutron yields $m_{QED}^{p-n}=0.58\pm 0.16\,$MeV. The result indicates that the inelastic contributions are small compared to the elastic ones., Comment: 11 pages. A typo in Eq. (21) corrected

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle physics community. More specifically, we report on the determination of the light-quark masses, the form factor f+(0), arising in the semileptonic K -> pi transition at zero momentum transfer, as well as the decay constant ratio fK/fpi and its consequences for the CKM matrix elements Vus and Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)LxSU(2)R and SU(3)LxSU(3)R Chiral Perturbation Theory. We review the determination of the BK parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. The latter quantities are an addition compared to the previous review. For the heavy-quark sector, we provide results for mc and mb (also new compared to the previous review), as well as those for D- and B-meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. Finally, we review the status of lattice determinations of the strong coupling constant alpha_s., Comment: 383 pages, 32 figures, 160 tables, 795 references

Many of the quantities of interest at the precision frontier in particle physics require a good understanding of the strong interaction at low energies. The present talk reviews the theoretical framework used in this context. In particular, I draw attention to the fact that applications of effective field theory methods in the low energy domain involve two different aspects: dependence of the quantities of interest on the quark masses and dependence on the momenta. While the lattice approach gives an excellent handle on the low energy constants that govern the quark mass dependence, the most efficient tool to pin down the momentum dependence is dispersion theory. At the same time, the dispersive analysis enlarges the energy range where the effective theory applies. In the meson sector, the interplay of the various sources of information has led to a coherent framework that describes the low energy structure at remarkably high resolution. The understanding of the low energy properties in the baryon sector is less well developed. There is significant progress in the dispersive analysis of $\pi N$ scattering, for example, but it leads to puzzling conclusions concerning the pattern of SU(3) symmetry breaking in the baryon octet, which yet remain to be understood. Finally, I critically examine recent papers dealing with the Cottingham formula for the electromagnetic contribution to the mass difference between proton and neutron., Comment: Contribution to the proceedings of the Workshop on Chiral Dynamics, Pisa 2015. Extended the discussion of the sigma-term, to account for a new lattice result that appeared only after the closing of the workshop

The difference between the electromagnetic self-energies of proton and neutron can be calculated with the Cottingham formula, which expresses the self-energies as an integral over the electroproduction cross sections---provided the nucleon matrix elements of the current commutator do not contain a fixed pole. We show that, under the same proviso, the subtraction function occurring in the dispersive representation of the virtual Compton forward scattering amplitude is determined by the cross sections. The representation in particular leads to a parameter-free sum rule for the nucleon polarizabilities. We evaluate the sum rule for the difference between the electric polarizabilities of proton and neutron by means of the available parameterizations of the data and compare the result with experiment., Comment: 30 pages, 7 figures; The typos in Eqs. (10), (27), (56), (57), (63) and (B.46) are corrected. None of the results of the paper is affected

I briefly review the conceptual developments that led to the Standard Model and discuss some of its remarkable qualitative features. On the way, I draw attention to several puzzling aspects that are beyond the reach of our present understanding of the basic laws of physics., Comment: Contribution to 'Fifty Years of Quarks', H. Fritzsch and M. Gell-Mann, eds. (World Scientific, Singapore)

The problems encountered in the determination of m_u and m_d are discussed. While their sum is known quite well, the difference m_d-m_u, which measures the breaking of isospin symmetry in the QCD Lagrangian, is still subject to significant uncertainties. I focus on recent work based on the dispersive analysis of the decay eta -> 3 pi, which offers a good handle on isospin breaking, because in that transition, the contributions from the e.m. interaction are suppressed., Comment: Contribution to the proceedings of the workshop "Determination of the Fundamental Parameters of QCD", Nanyang Technological University, Singapore, March 2013

These lecture notes recall the conceptual developments which led from the discovery of the neutron to our present understanding of strong interaction physics., Comment: Lectures given at the International School of Subnuclear Physics Erice, Italy, 23 June to 2 July 2012

Low energy precision experiments provide significant tests of the laws of nature that can reveal physics beyond the Standard Model. A good theoretical understanding of the low energy properties of QCD is required for this purpose. The recent developments at the interface between lattice and effective field theory methods provide an excellent basis for pion physics already now, while the extension required to explore the low energy properties in the strange quark sector yet calls for further work, also in view of a better determination of the light quark masses. At the precision achieved in lattice determination of quark mass ratios, the e.m. self-energies of the mesons play an important role. I point out some unresolved issues occurring in this context and then summarize the present knowledge of the light quark masses., Comment: Contribution to the Proceedings of the Workshop on Chiral Dynamics, Bern, 2009

My talk was dedicated to the memory of Jan Stern. The brief account given below focuses on the progress achieved in the determination of the $\pi\pi$ S-wave scattering lengths, both experimentally and with light dynamical quarks on a lattice. In view of the excellent agreement, we can conclude that (a) the expansion in powers of the two lightest quark masses represents a very efficient tool for the analysis of the low energy structure of QCD and (b) the size of the energy gap of QCD is governed by the order parameter of lowest dimension, the quark condensate., Comment: Contribution to Proc. Confinement and the Hadron Spectrum VIII, Mainz, Germany, Sept. 1-6, 2008

The energy gap of QCD is now understood very well. There is no doubt that the expansion in powers of the two lightest quark masses does represent a very useful tool for the analysis of the low energy structure. Concerning the expansion in powers of m_s, however, the current situation leaves much to be desired. While some of the lattice results indicate, for instance, that the violations of the Okubo-Iizuka-Zweig rule in the quark condensate and in the decay constants are rather modest, others point in the opposite direction. I am confident that the dust will settle soon, so that the effective coupling constants that govern the dependence of the various quantities of physical interest on m_s can be determined, to next-to-next-to-leading order of the chiral expansion. The range of validity of ChPT can be extended by means of dispersive methods. The properties of the physical states occurring in the spectrum of QCD below KKbar threshold can reliably be investigated on this basis. In particular, as shown only rather recently, general principles of quantum field theory lead to an exact formula that expresses the mass and width of resonances in terms of observable quantities. The formula removes the ambiguities inherent in the analytic continuation from the real axis into the complex plane, which plagued previous determinations of the pole positions of broad resonances., Comment: Talks given at QCD08 and Confinement8

These lecture notes concern recent developments in our understanding of the low energy properties of QCD. Significant progress has been made on the lattice and the beautiful experimental results on the Ke4 and K->3pi decays, as well as those on pionic atoms also confirm the results obtained on the basis of Chiral Perturbation Theory. There is an exception: one of the precision experiments on Kmu3 decay is in flat contradiction with the Callan-Treiman relation. If confirmed, this would indicate physics beyond the Standard Model: right-handed quark couplings of the W-boson, for instance. Furthermore, I discuss two examples where the estimates of the effective coupling constants based on saturation by resonances appear to fail. In the second part, the progress made in extending the range of validity of the effective theory with dispersive methods is reviewed. In particular, I draw attention to an exact formula, which expresses the mass and width of a resonance in terms of observable quantities. The formula removes the ambiguities inherent in the analytic continuation from the real axis into the complex plane, which plagued previous determinations of the pole positions associated with broad resonances. In particular, it can now be demonstrated that the lowest resonance of QCD carries the quantum numbers of the vacuum., Comment: Lectures given at the International School of Subnuclear Physics, Erice, 2007

The first part of this report reviews recent developments at the interface between lattice work on QCD with light dynamical quarks, effective field theory and low energy precision experiments. Then I discuss how dispersion theory can be used to analyze the low energy structure of the $\pi\pi$ scattering amplitude in a model independent manner. This leads to an exact formula for the mass and width of the lowest few resonances, in terms of observable quantities. As an application, I consider the pole position of the $\sigma$, paying particular to error propagation in the numerical analysis. The report is based on work done in collaboration with Irinel Caprini and Gilberto Colangelo., Comment: 10 pages, 4 figures, Contribution to the proceedings of the Workshop on Scalar Mesons and Related Topics, honouring the 70th birthday of Michael Scadron, Lisbon, Portugal, Feb. 11-16, 2008

Recent work in light flavour hadron physics is reviewed. In particular, I discuss the significance of the progress achieved with light dynamical quarks on the lattice for the effective low energy theory of QCD. Also, I draw attention to some puzzling results from NA48 and KTeV concerning the scalar form factor relevant for Kmu3 decay -- taken at face value, these indicate physics beyond the Standard Model., Comment: Contribution to the proceedings of the Rencontres de Moriond on QCD and Hadronic Interactions, La Thuile, Italy, March 2007; added references, minor changes in the text

Recent work in low energy pion physics is reviewed. One of the exciting new developments in this field is that simulations of QCD on a lattice now start providing information about the low energy structure of the continuum theory, for physical values of the quark masses. Although the various sources of systematic error yet need to be explored more thoroughly, the results obtained for the correlation function of the axial current with the quantum numbers of the pion already have important implications for the effective Lagrangian of QCD. The consequences for $\pi\pi$ scattering are discussed in some detail. The second part of the report briefly reviews recent developments in the dispersion theory of the scattering amplitude. One of the important results here is that the position of the lowest resonances of QCD can now be determined in a model independent manner and rather precisely. Beyond any doubt, the partial wave with $I=\ell=0$ contains a pole on the second sheet, not far from the threshold: the lowest resonance of QCD carries the quantum numbers of the vacuum., Comment: Contribution to the proceedings of the workshop "Chiral Dynamics, Theory & Experiment", Durham/Chapel Hill, NC, USA, September 2006

Recent developments in low energy pion physics are reviewed, emphasizing the strength of dispersion theory in this context. As an illustration of the method, I discuss some consequences of the forward dispersion relation obeyed by the isoscalar component of the scattering amplitude., Comment: Contribution to the proceedings of the workshop "Quark Confinement and the Hadron Spectrum VII", Ponta Delgada, Azores Islands, Portugal, September 2006

I report on recent work done in collaboration with Irinel Caprini and Gilberto Colangelo. We observe that the Roy equations lead to a representation of the pion pion scattering amplitude that exclusively involves observable quantities, but is valid for complex values of s. At low energies, this representation is dominated by the contributions from the two subtraction constants, which are known to remarkable precision from the low energy theorems of chiral perturbation theory. Evaluating the remaining contributions on the basis of the available data, we demonstrate that the lowest resonance carries the quantum numbers of the vacuum and occurs in the vicinity of the threshold. Although the uncertainties in the data are substantial, the pole position can be calculated quite accurately, because it occurs in the region where the amplitude is dominated by the subtractions. The calculation neatly illustrates the fact that the dynamics of the Goldstone bosons is governed by the symmetries of QCD., Comment: Contribution to the proceedings of MESON 2006 (Krakow)

The scalar radius of the pion plays an important role in CHPT, because it is related to one of the basic effective coupling constants, viz. the one which controls the quark mass dependence of F_pi at one loop. In a recent paper, Yndurain derives a {\it robust lower bound} for this radius, which disagrees with earlier determinations. We show that such a bound does not exist: the "derivation" relies on an incorrect claim. Moreover, we discuss the physics of the form factors associated with the operators \ubar u, \dbar d and \sbar s and show that their structure in the vicinity of the K \Kbar threshold is quite different. Finally, we draw attention to the fact that the new data on the slope of the scalar K_l3 form factor confirm a recent, remarkably sharp theoretical prediction., Comment: 13 pages, 2 figures. v2: minor changes - version to appear on Phys. Lett. B. v3: Published version. Values of the measured K-pi radii corrected

In a recent paper, Pelaez and Yndurain evaluate some of the low energy observables of pi pi scattering and obtain flat disagreement with our earlier results. The authors work with unsubtracted dispersion relations, so that their results are very sensitive to the poorly known high energy behaviour of the scattering amplitude. They claim that the asymptotic representation we used is incorrect and propose an alternative one. We repeat their calculations on the basis of the standard, subtracted fixed-t dispersion relations, using their asymptotics. The outcome fully confirms our earlier findings. Moreover, we show that the Regge parametrization proposed by these authors for the region above 1.4 GeV violates crossing symmetry: Their ansatz is not consistent with the behaviour observed at low energies., Comment: Added more material, mostly in Sects. 7, 8 and 9, in support of the same conclusions. Latex, 28 pages, 3 figures

I draw attention to a recent breakthrough in the field of low energy pion physics: the consequences of the hidden symmetry of the QCD Hamiltonian have successfully been incorporated in the general dispersive framework for the pi-pi scattering amplitude, which is due to Roy. The meagre experimental information about the imaginary parts at and above 0.8 GeV suffices to unambiguously and accurately pin down the scattering amplitude at lower energies. The recent Brookhaven data on the reaction K -> pi pi e nu provide a significant test of the theory. They imply that the Gell-Mann-Oakes-Renner relation is approximately valid -- the bulk of the pion mass indeed originates in the quark condensate., Comment: Talk given in honour of Arnulfo Zepeda at the X Mexican School of Particles and Fields, Playa del Carmen, Mexico, 2002, 8 pp

The lectures review some of the basic concepts relevant for an understanding of the low energy properties of the strong interactions: chiral symmetry, spontaneous symmetry breakdown, Goldstone bosons, quark condensate. The effective field theory used to analyze the low energy structure is briefly sketched. As an illustration, I discuss the implications of the recent data on the decay $K\to \pi\pi e\nu$ for the magnitude of the quark condensate., Comment: Lectures given at the school of physics "Understanding the structure of hadrons", Prague, July 2001, 20 pp

The Standard Model prediction for the magnetic moment of the muon requires a determination of the electromagnetic form factor of the pion at high precision. It is shown that the recent progress in pi-pi scattering allows us to obtain an accurate representation of this form factor on the basis of the data on e+e- -> pi+ pi-. The same method also applies to the form factor of the weak vector current, where the data on the decay tau -> pi- pi0 nu are relevant. Unfortunately, however, the known sources of isospin breaking do not explain the difference between the two results. The discrepancy implies that the Standard Model prediction for the magnetic moment of the muon is currently subject to a large uncertainty. Talk given in May 2002, in honor of Arkady Vainshtein., Comment: Talk given at the Workshop "Continuous Advances in QCD 2002/Arkadyfest", Minneapolis, May 2002, 19 pp

The role of the quark condensate for the low energy structure of QCD is discussed in some detail. In particular, the dependence of M_pi on m_u and m_d and the low energy theorems for the pi-pi scattering amplitude are reviewed. The new data on Ke4 decay beautifully confirm the standard picture, according to which the quark condensate is the leading order parameter of the spontaneously broken chiral symmetry., Comment: Talk given at the Workshop QCD@Work, Martina Franca, June 2001

We derive a representation for the pion nucleon scattering amplitude that is valid to the fourth order of the chiral expansion. To obtain the correct analytic structure of the singularities in the low energy region, we have performed the calculation in a relativistic framework (infrared regularization). The result can be written in terms of functions of a single variable. We study the corresponding dispersion relations and discuss the problems encountered in the straightforward nonrelativistic expansion of the infrared singularities. As an application, we evaluate the corrections to the Goldberger-Treiman relation and to the low energy theorem that relates the value of the amplitude at the Cheng-Dashen point to the \sigma-term. While chiral symmetry does govern the behaviour of the amplitude in the vicinity of this point, the representation for the scattering amplitude is not accurate enough to use it for an extrapolation of the experimental data to the subthreshold region. We propose to perform this extrapolation on the basis of a set of integral equations that interrelate the lowest partial waves and are analogous to the Roy equations for \pi\pi scattering., Comment: 97 pages (LaTeX), 16 figures. Two references added, correction in table one. Published version

We demonstrate that, together with the available experimental information, chiral symmetry determines the low energy behaviour of the $\pi\pi$ scattering amplitude to within very small uncertainties. In particular, the threshold parameters of the S-, P-, D- and F-waves are predicted, as well as the mass and width of the $\rho$ and of the broad bump in the S-wave. The implications for the coupling constants that occur in the effective Lagrangian beyond leading order and also show up in other processes, are discussed. Also, we analyze the dependence of various observables on the mass of the two lightest quarks in some detail, in view of the extrapolations required to reach the small physical masses on the lattice. The analysis relies on the standard hypothesis, according to which the quark condensate is the leading order parameter of the spontaneously broken symmetry. Our results provide the basis for an experimental test of this hypothesis, in particular in the framework of the ongoing DIRAC experiment: The prediction for the lifetime of the ground state of a $\pi^+\pi^-$ atom reads $\tau=(2.90\pm 0.10)10^{-15} sec$., Comment: 61 pages (LaTex), 11 figures

We show that, independently of the size of the quark condensate, chiral symmetry correlates the two S-wave \pi\pi scattering lengths. In view of this constraint, the new precision data on K_{e_4} decay allow a remarkably accurate determination of these quantities. The result confirms the hypothesis that the quark condensate is the leading order parameter., Comment: 4 pages (LaTex), 3 figures

The reasons why a considerable effort is made to resolve the low energy structure of QCD are discussed. The effective field theory used for this purpose is illustrated with the recent progress made in the predictions for pipi scattering and in understanding the low energy properties of the theory in the large Nc limit., Comment: Talk given at Chiral 2000, Jefferson Laboratory, Newport News, Virginia, July 2000, 15 pp

The sum rule determinations of m_s and the chiral perturbation theory results for the ratios m_u/m_d and m_s/m_d are reviewed and a method for the extrapolation of lattice data to the physical values of m_u and m_d is outlined., Comment: Talk given at Lattice 2000, Bangalore, India, Aug. 2000, 8 pp

In the first part of the talk, I discussed the nature of the effective theory that describes the properties of the pion-nucleon-interaction at low energies, using the static model as a starting point. In the second part, I then pointed out that the infrared singularities occurring in the pi-N scattering amplitude are stronger and of a more complex structure than those encountered in pi-pi scattering. A formulation of the effective theory that properly accounts for these was described and the results obtained thereby were illustrated with a few examples., Comment: Talk given at MENU99, Zuoz, Switzerland, 16-20 Aug 1999, $\pi N$Newslett. 15 (1999)45 pages

The effective field theory relevant for the analysis of QCD at low energies is reviewed. The foundations of the method are discussed in some detail and a few illustrative examples are described., Comment: Contribution to the Festschrift in honour of B. L. Ioffe, 45 pages

We match the known chiral perturbation theory representation of the pi pi scattering amplitude to two loops with a phenomenological description that relies on the Roy equations. On this basis, the corrections to Weinberg's low energy theorems for the S-wave scattering lengths are worked out to second order in the expansion in powers of the quark masses. The resulting predictions, a00=0.220 \pm 0.005, a20=-0.0444 \pm 0.0010, contain remarkably small uncertainties and thus allow a very sensitive experimental test of the hypothesis that the quark condensate is the leading order parameter of the spontaneously broken chiral symmetry., Comment: 12 pages, one figure; a few more references added. Version to appear on Physics Letters B

The construction of the effective Lagrangian relevant for the mesonic sector of QCD in the large N_c limit meets with a few rather subtle problems. We thoroughly examine these and show that, if the variables of the effective theory are chosen suitably, the known large N_c counting rules of QCD can unambiguously be translated into corresponding counting rules for the effective coupling constants. As an application, we demonstrate that the Kaplan-Manohar transformation is in conflict with these rules and is suppressed to all orders in 1/N_c. The anomalous dimension of the axial singlet current generates an additional complication: The corresponding external field undergoes nonmultiplicative renormalization. As a consequence, the Wess-Zumino-Witten term, which accounts for the U(3)_R x U(3)_L anomalies in the framework of the effective theory, contains pieces that depend on the running scale of QCD. The effect only shows up at nonleading order in 1/N_c, but requires specific unnatural parity contributions in the effective Lagrangian that restore renormalization group invariance., Comment: 56 pages

We analyze the Roy equations for the lowest partial waves of elastic pi pi scattering and demonstrate that the two S-wave scattering lengths a_0^0 and a_0^2 are the essential parameters in the low energy region: Once these are known, the available experimental information determines the behaviour near threshold to within remarkably small uncertainties. An explicit numerical representation for the energy dependence of the S- and P-waves is given and it is shown that the threshold parameters of the D- and F-waves are also fixed very sharply in terms of a_0^0 and a^2_0. In agreement with earlier work, which is reviewed in some detail, we find that the Roy equations admit physically acceptable solutions only within a band of the (a_0^0,a_0^2) plane. We show that the data on the reactions e+e- -> pi pi and tau -> pi pi nu reduce the width of this band quite significantly. Furthermore, we discuss the relevance of the decay K -> pi pi e nu in restricting the allowed range of a_0^0, preparing the grounds for an analysis of the forthcoming precision data on this decay and on pionic atoms. We expect these to reduce the uncertainties in the two basic low energy parameters very substantially, so that a meaningful test of the chiral perturbation theory predictions will become possible., Comment: Latex, 84 pages, 17 figures

We show that in the presence of massive particles such as nucleons, the standard low energy expansion in powers of meson momenta and light quark masses in general only converges in part of the low energy region. The expansion of the scalar form factor $\sigma(t)$, for instance, breaks down in the vicinity of $t=4M_\pi^2$. In the language of heavy baryon chiral perturbation theory, the proper behaviour in the threshold region only results if the multiple internal line insertions generated by relativistic kinematics are summed up to all orders. We propose a method that yields a coherent representation throughout the low energy region while keeping Lorentz and chiral invariance explicit at all stages. The method is illustrated with a calculation of the nucleon mass and of the scalar form factor to order $p^4$., Comment: 66 pages, 12 postscript figures

In the large N_c limit, the variables required to analyze the low energy structure of QCD in the framework of an effective field theory necessarily include the degrees of freedom of the eta'. We evaluate the decay constants of the pseudoscalar nonet to one loop within this extended framework and show that, as a consequence of the anomalous dimension of the singlet axial current, some of the effective coupling constants depend on the running scale of QCD. The calculation relies on a simultaneous expansion in powers of momenta, quark masses and 1/N_c. Talk given at the Workshop on Nonperturbative Methods in Quantum Field Theory, NITP/CSSM, University of Adelaide, Australia, Feb. 1998., Comment: 14 pages

Talk given at QCD 97, Montpellier, July 1997. In the first part of the talk, I presented a review of the results for the light quark masses obtained on the basis of chiral perturbation theory. As this material is described elsewhere, the following notes only concern the second part, which dealt with the behaviour of the effective theory in the large N_c limit.

The available experimental information is consistent with the Weinberg predictions for the threshold parameters of pi-pi-scattering. The data, however, only test those relations that are insensitive to the quark masses m_u and m_d. Recent theoretical progress leads to remarkably sharp predictions for the behaviour of the S- and P-wave phase shifts in the elastic region -- one of the very few low energy phenomena in QCD, where theory is ahead of experiment and where new precision data would subject the theory to a crucial test (Talk given at the DAFCE workshop, Frascati, November 1996).

The implications of the hidden, spontaneously broken symmetry for the properties of the sound waves of a solid are analyzed. Although the discussion does not go beyond standard wisdom, it presents some of the known results from a different perspective. In particular, I argue that, as a consequence of the hidden symmetry, the equations of motion for a sound wave necessarily contain nonlinear terms, describing phonon-phonon scattering and emphasize the analogy with the low energy theorems for pion-pion scattering., Comment: 12 pages, Latex file

Lecture Notes, Summer School on Effective Theories and Fundamental Interactions, Erice, 1996. The application of effective field theory methods to the low energy structure of QCD is discussed. I emphasize the universal structure of the corresponding effective Lagrangian and then discuss the physics of the effective coupling constants. The implications of the effective theory for the masses of the light quarks are analyzed in some detail., Comment: 30 pages, 3 postscript figures

Lecture Notes, Summer School on Masses of Fundamental Particles, Cargese, 1996. In the first part, some qualitative aspects the mass pattern are discussed, concerning the breaking of isospin and eightfold way symmetries. The second part deals with the chiral perturbation theory results for the masses of the pseudoscalar octet, which lead to very stringent constraints for the relative size of the light quark masses., Comment: 16 pages, 3 postscript figures

The paper collects the various pieces of information concerning the relative size of m_u, m_d and m_s. A coherent picture results, which constrains the mass ratios to a rather narrow range: m_u/m_d = 0.553 \pm 0.043, m_s/m_d = 18.9 \pm 0.8., Comment: 11 pages, 2 figures

Talk given at the Conference on Fundamental Interactions of Elementary Particles, ITEP, Moscow, Oct. 1995. The paper reviews the current status of knowledge concerning m_u, m_d and m_s. Qualitative aspects of the resulting picture for the breaking of isospin and eightfold way symmetries are discussed. At a more quantitative level, the review focuses on the chiral perturbation theory results for the masses of the Goldstone bosons. The corresponding bounds and estimates for the ratios m_u/m_d and m_s/m_d are described in some detail., Comment: 23 pages, 3 figures

We demonstrate that the decay eta -> 3 pi represents a sensitive probe for the breaking of chiral symmetry by the quark masses. The transition amplitude is proportional to the mass ratio (m_d^2-m_u^2)/(m_s^2-m^2). The factor of proportionality is calculated by means of dispersion relations, using chiral perturbation theory to determine the subtraction constants. The theoretical uncertainties in the result are shown to be remarkably small, so that eta-decay may be used to accurately measure this ratio of quark masses., Comment: 14 pages, 1 figure

Taken by itself, the interference with the etaprime appears to strongly affect the amplitude of the transition eta->3 pi. I point out that this effect is fictitious and also occurs in the mass spectrum of the pseudoscalars. Chiral symmetry implies that the same combination of effective coupling constants which determines the small deviations from the Gell-Mann-Okubo formula also specifies the symmetry breaking effects in the decay amplitude and thus ensures that these are small., Comment: 8 pages, Latex

The corrections to the current algebra mass formulae for the pseudoscalar mesons are analyzed by means of a simultaneous exansion in powers of the light quark masses and powers of 1/N. A set of mass formulae is derived, including an inequality, which leads to bounds for ratios of quark masses., Comment: 11 pages, Latex