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Tiivistelmä. Ryhmä on algebrallinen rakenne, joka koostuu joukosta ja sen alkioiden välisestä binäärisestä ja assosiatiivisesta operaatiosta. Mikäli joukko on äärellinen, puhutaan äärellisestä ryhmästä. Operaation binäärisyys tarkoittaa, että alkioiden välinen operaatio kuuluu aina edelleen joukkoon ja assosiatiivisuus tarkoittaa, että lopputulos ei riipu siitä, miten sulkeet on aseteltu alkioiden välisessä operaatiossa. Ryhmässä on aina olemassa neutraalialkio, jonka operaatio joukon alkioiden kanssa säilyttää alkiot muuttumattomina. Lisäksi ryhmässä jokaiselle alkiolle löytyy käänteisalkio, jonka operaatio alkion kanssa tuottaa ryhmän neutraalialkion. Ryhmäteoria on algebran osa-alue, joka tutkii ryhmiä, ryhmiin liittyviä käsitteitä ja näiden välisiä lainalaisuuksia. Eräs keskeinen ryhmiin liittyvä käsite on ryhmän aliryhmä. Aliryhmä on ryhmän osajoukko, joka on itse myös ryhmä. Mikäli aliryhmän kertaluku eli aliryhmän alkioiden lukumäärä on jonkin alkuluvun p potenssi, niin aliryhmää kutsutaan p-aliryhmäksi Sylowin lauseet ovat äärellisen ryhmäteorian keskeisimpiä tuloksia. Sylowin lauseet takaavat olemassaolon niin sanotuille Sylowin p-aliryhmille, jotka ovat inkluusion suhteen suurimpia mahdollisia ryhmän p-aliryhmiä. Sylowin p-aliryhmien lukumäärä on aina kongruentti luvun 1 kanssa alkuluvun p suhteen. Lisäksi kaikki Sylowin p-aliryhmät ovat aina toistensa konjugaatteja. Tämä tarkoittaa, että kaikki ryhmän Sylowin p-aliryhmät saadaan tuotettua operoimalla mitä tahansa Sylowin p-aliryhmää vasemmalta puolelta ryhmän alkiolla ja oikealta puolelta vastaavalla käänteisalkiolla. Alun perin lauseet todisti norjalainen matemaatikko Ludvig Sylow vuonna 1872, mutta lauseille on esitetty myöhemmin paljon vaihtoehtoisia todistuksia. Sylowin lauseet tarjoavat osittaisen käänteistuloksen Lagrangen lauseelle. Lagrangen lause kertoo, että ryhmän aliryhmän kertaluku jakaa aina ryhmän kertaluvun. Tämän lauseen käänteinen suunta ei ole yleisesti totta, mutta Sylowin lauseiden avulla voidaan osoittaa, että käänteinen suunta pätee osittain: Jos jokin alkuluvun potenssi jakaa ryhmän kertaluvun, niin tällöin ryhmällä on aliryhmä, jonka kertaluku on tämä alkuluvun potenssi. Sylowin lauseiden todistamiseen tarvitaan hyvä perusta ryhmäteoriasta, joka nojaa paljolti joukko-oppiin ja lukuteoriaan. Suurin osa tarvittavasta lukuteoriasta liittyy tavalla tai toisella kokonaislukujen jaollisuuteen. Joukko-opista olennaisimpia käsitteitä ovat kuvaukset ja relaatiot. Lisäksi joukko-opin perusteet ovat ryhmäteorian kannalta luonnollisesti olennaisessa osassa, sillä ryhmä on itsessään joukko, jossa on määritelty operaatio joukon alkioiden välille. Ryhmäteorian kulmakivenä on ryhmän määritelmä, jonka päälle rakennetaan uusia käsitteitä. Keskeisiä ryhmäteorian käsitteitä ovat esimerkiksi aliryhmä, tekijäryhmä ja homomorfismi. Näiden lisäksi tärkeä käsite Sylowin lauseiden todistamisessa on ryhmän toiminta, joka on eräänlainen operaatio ryhmän alkioiden ja jonkin joukon alkioiden välillä. Sylowin lauseet voidaan todistaa myös ilman ryhmän toimintaa käyttäen esimerkiksi induktiota.

A knowledge of the radial distributions of quarks inside hadrons could lead to a better understanding of the QCD description of these hadrons and possibly suggest forms for phenomenological models. As a step in this direction, in an earlier work, the charge (vector) and matter (scalar) radial distributions of heavy-light mesons were measured in the quenched approximation on a 16^3x24 lattice with a lattice spacing of 'a' approx. 0.17 fm, and a hopping parameter corresponding to a light quark mass about that of the strange quark. Here several improvements are now made: 1) The configurations are generated using dynamical fermions with a approx 0.14 fm; 2) Many more gauge configurations areincluded; 3) The distributions at many off-axis, in addition to on-axis, points are measured; 4) The data analysis is much more complete. In particular, distributions involving excited states are extracted. The exponential decay of the charge and matter distributions can be described by mesons of mass 0.9 +- 0.1 and 1.5 +- 0.1 GeV respectively -- values that are consistent with those of vector and scalar $q\bar{q}$-states calculated directly with the same lattice parameters.

A knowledge of the radial distributions of quarks inside hadrons could lead to a better understanding of the QCD description of these hadrons and possibly suggest forms for phenomenological models. As a step in this direction, in an earlier work, the charge (vector) and matter (scalar) radial distributions of heavy-light mesons were measured in the quenched approximation on a 16^3x24 lattice with a lattice spacing of 'a' approx. 0.17 fm, and a hopping parameter corresponding to a light quark mass about that of the strange quark. Here several improvements are now made: 1) The configurations are generated using dynamical fermions with a approx 0.14 fm; 2) Many more gauge configurations areincluded; 3) The distributions at many off-axis, in addition to on-axis, points are measured; 4) The data analysis is much more complete. In particular, distributions involving excited states are extracted. The exponential decay of the charge and matter distributions can be described by mesons of mass 0.9 +- 0.1 and 1.5 +- 0.1 GeV respectively -- values that are consistent with those of vector and scalar $q\bar{q}$-states calculated directly with the same lattice parameters., 35 pages, 6 figures, Latex

We discuss the allowed decays of a hybrid meson in the heavy quark limit. We deduce that an important decay will be into a heavy quark non-hybrid state and a light quark meson, in other words, the de-excitation of an excited gluonic string by emission of a light quark-antiquark pair. We discuss the study of hadronic decays from the lattice in the heavy quark limit and apply this approach to explore the transitions from a spin-exotic hybrid to $\chi_b \eta$ and $\chi_b S$ where $S$ is a scalar meson. We obtain a signal for the transition emitting a scalar meson and we discuss the phenomenological implications., Comment: 18 pages, LATEX, 3 ps figures

We discuss the allowed decays of a hybrid meson in the heavy quark limit. We deduce that an important decay will be into a heavy quark non-hybrid state and a light quark meson, in other words, the de-excitation of an excited gluonic string by emission of a light quark-antiquark pair. We discuss the study of hadronic decays from the lattice in the heavy quark limit and apply this approach to explore the transitions from a spin-exotic hybrid to $\chi_b \eta$ and $\chi_b S$ where $S$ is a scalar meson. We obtain a signal for the transition emitting a scalar meson and we discuss the phenomenological implications.

For a heavy-light meson with a static heavy quark, we can explore the light quark distribution. The charge and matter radial distributions of these heavy-light mesons are measured on a 16^3 * 24 lattice at beta=5.7 and a hopping parameter corresponding to a light quark mass about that of the strange quark. Both distributions can be well fitted up to 4 lattice spacings (r approx 0.7 fm) with the exponential form w_i^2(r), where w_i(r)=A exp(-r/r_i). For the charge(c) and matter(m) distributions r_c approx 0.32(2) fm and r_m approx 0.24(2) fm. We also discuss the normalisation of the total charge and matter integrated over all space, finding 1.30(5) and 0.4(1) respectively., 31 pages including 7 ps figures

In nuclear many-body problems the short-range correlation between two nucleons is well described by the corresponding correlation in the {two}-body problem. Therefore, as a first step in any attempt at an analogous description of many-quark systems, it is necessary to know the two-quark correlation. With this in mind, we study the light quark distribution in a heavy-light meson with a static heavy quark. The charge and matter radial distributions of these heavy-light mesons are measured on a lattice with a light quark mass about that of the strange quark. Both distributions can be well fitted upto r approx 0.7 fm with the exponential form w_i^2(r), where w_i(r)=A exp(-r/r_i). For the charge(c) and matter(m) distributions r_c approx 0.32(2) fm and r_m \approx 0.24(2) fm. We also discuss the normalisation of the total charge (defined to be unity in the continuum limit) and matter integrated over all space, finding 1.30(5) and 0.4(1) respectively for a lattice spacing approx 0.17 fm., Comment: 8 pages, 3 ps figures

A brief review is given of attempts to understand the energies of four-quark systems calculated on a lattice in terms of nuclear-physics-inspired many-body techniques involving interquark potentials. Results are given for the next stage of this study where the wavefunctions of heavy-light mesons are also calculated on a lattice., 3 pages, one figure. Contribution to Confinement IV held in Vienna, 2000

String breaking is a non-perturbative long-distance feature of QCD that is involved in for example meson decays. A mixing analysis of lattice operators at zero temperature gives the level splitting and mixing energy between the broken and unbroken string states., 4 pages, 1 eps figure. Talk given at Confinement 2000 in Osaka, Japan

The crossing from a static quark-antiquark system to a system of two static-light mesons when the separation of the static quarks is increased is calculated in zero-temperature lattice QCD. The mixing of these two states is extracted from the lattice operators. We also discuss the breaking of an excited string of a hybrid meson., 4 pages, 2 eps figures

The potential between static-light mesons forming a meson-meson or a meson-antimeson system is calculated in quenched and unquenched SU(3) gauge theory. We use the Sheikholeslami-Wohlert action and statistical estimators of light quark propagators with maximal variance reduction. The dependence of the potentials on the light quark spin and isospin and the effect of meson exchange is investigated. Our main motivation is exploration of bound states of two mesons and string breaking. The latter also involves the two-quark potential and the correlation between two-quark and two-meson states., Contribution to LATTICE99 (QCD spectrum). 3 pages, 4 eps figures

The potential between two heavy-light mesons as a function of the heavy quark separation is calculated in quenched SU(3) lattice QCD. We study the case of heavy-light mesons with a static heavy quark and light quarks of mass close to the strange quark mass. We explore the case of light quarks with the same and with different flavours, classified according to the light quark isospin. We evaluate the appropriate light quark exchange contributions and explore the spin-dependence of the interaction. Comparison is made with meson exchange., Comment: 31 pages, 10 figures; error bars in Fig. 2 corrected (now smaller)

The energies of four-quark states are calculated for geometries in which the quarks are situated on the corners of a series of tetrahedra and also for geometries that correspond to gradually distorting these tetrahedra into a plane. The interest in tetrahedra arises because they are composed of {\bf three } degenerate partitions of the four quarks into two two-quark colour singlets. This is an extension of earlier work showing that geometries with {\bf two} degenerate partitions (e.g.\ squares) experience a large binding energy. It is now found that even larger binding energies do not result, but that for the tetrahedra the ground and first excited states become degenerate in energy. The calculation is carried out using SU(2) for static quarks in the quenched approximation with $\beta=2.4$ on a $16^3\times 32$ lattice. The results are analysed using the correlation matrix between different euclidean times and the implications of these results are discussed for a model based on two-quark potentials., Comment: Original Raw PS file replace by a tarred, compressed and uuencoded PS file