Glueball Regge trajectories Harvey Byron Meyer Lincoln College, Oxford Rudolf Peierls Centre for Theoretical Physics Department of Physics, University of Oxford arXiv:hep-lat/0508002v1 2 Aug 2005 Thesis submitted for the degree of Doctor of Philosophy at the University of Oxford Trinity term, 2004 · · Glueball Regge trajectories Harvey Byron Meyer Lincoln College Thesis submitted for the degree of Doctor of Philosophy at the University of Oxford Trinity Term, 2004 Abstract We investigate the spectrum of glueballs in SU(Nc) gauge theories. Our motivation is to deter- mine whether the states lie on straight Regge trajectories. It has been conjectured for a long time that glueballs are the physical states lying on the pomeron, the trajectory responsible for the slowly rising hadronic cross-sections at large centre-of-mass energy. After a review of Regge phenomenology, we show that string models of glueballs predict states to lie on linear trajectories with definite sequences of quantum numbers. We then move on to the lattice formulation of gauge theory. Because the lattice regularisation breaks rotational symmetry, there is an ambiguity in the assignment of the spin of lattice states. We develop numerical methods to resolve these ambiguities in the continuum limit, in particular how to extract high spin glueballs from the lattice. We also devise a multi-level algorithm that reduces the variance on Euclidean correlation functions from which glueball masses are extracted. In 2+1 dimensions, we determine the SU(2) spectrum up to spin 6, and relabel a previously published SU(3) spectrum with the correct spin quantum numbers. We find well-defined Regge trajectories, but the leading trajectory goes through the lightest scalar glueball and has an intercept close to (-1). In 3+1 dimensions, we perform a detailed survey of the SU(3) spectrum. A comparison to the low-lying SU(8) spectrum, that we also compute, indicates that these gauge theories are ‘close’ to SU( ). Although the spectrum is more complex than in two space dimensions, we can clearly identify∞ the leading trajectory: it goes through the lightest 2++ and 4++ states, has slope α′ =0.28(2) in units of the mesonic slope and intercept α0 =0.93(24), in remarkable agreement with phenomenological values. We conclude with some implications of these results. To Anne & R¨udiger Contents Acknowledgements iv 1 High energy hadronic reactions 1 1.1 Reggetheory ..................................... 1 1.2 Reggephenomenology .............................. ... 3 1.2.1 Thesoftpomeron ............................... 3 1.2.2 Thehardpomeron ............................... 5 1.2.3 Theodderon .................................. 6 1.3 Theperturbative-QCDpomeronandodderon . ....... 6 1.3.1 TheLow-Nussinovpomeron. .. 6 1.3.2 TheBFKLpomeron .............................. 7 1.3.3 Theperturbativeodderon . ... 8 1.4 Unitarisation ................................... ... 9 1.5 Non-perturbative models of the soft pomeron . .......... 11 1.6 Conclusion ...................................... 12 2 String models of glueballs 13 2.1 Twostringmodelsofglueballs . ...... 14 2.1.1 Theadjoint-stringmodel . ... 14 2.1.2 Theflux-tubemodel .............................. 15 2.2 Glueballspectrafromgravity . ....... 20 2.3 The flux-tube model from the Nambu-Goto action . ........ 21 2.3.1 Generalities.................................. 21 2.3.2 The vibrating closed string in 2+1 dimensions . ........ 22 2.3.3 The spinning and vibrating closed string in 3+1 dimensions........ 24 2.4 Conclusion ...................................... 26 3 Lattice gauge theory 27 3.1 Generalities .................................... ... 27 3.2 Thecontinuumlimit ............................... ... 29 3.3 Monte-Carlosimulations . ...... 30 3.3.1 Extracting the spectrum from two-point functions . .......... 32 3.3.2 Glueballspectrumcalculations . ...... 34 3.4 Outlook ......................................... 36 4 High-spin glueballs from the lattice 37 4.1 Introduction.................................... ... 37 4.2 Twomethodsofoperatorconstruction . ........ 38 4.2.1 Thematrixmethod .............................. 39 4.2.2 Thepath-findermethod . 43 4.2.3 A test for the operator construction methods . ........ 43 4.3 Highspinstatesonthelattice? . ....... 46 i 4.3.1 Latticevs. continuumsymmetrygroup . ..... 46 4.3.2 Twostrategies ................................. 48 4.4 ApplicationsofStrategyI . ...... 49 4.4.1 The 0− / 4− puzzle .............................. 50 4.4.2 Arecipefordataanalysis . ... 51 4.4.3 Results ..................................... 53 4.5 ApplicationsofStrategyII. ....... 55 4.5.1 Wave functions of the lightest A1 and A3 states............... 55 4.5.2 The 0− / 4− puzzlerevisited ......................... 55 4.5.3 Wave functions of the lightest E states.................... 56 4.5.4 The masses and Fourier coefficients in the continuum limit ........ 57 4.6 Conclusion ...................................... 57 5 Multi-level algorithms 61 5.1 Locality&multi-levelalgorithms . ......... 62 5.2 Afirstapplication ................................ ... 64 5.3 A2-levelversionofthealgorithm . ........ 66 5.4 Thealgorithmanditsparameters. ....... 67 5.5 Optimisationprocedure&performance . ........ 69 5.6 Glueball calculation in 3+1 dimensions . .......... 72 5.7 Conclusion ...................................... 72 6 GlueballReggetrajectoriesin2+1dimensions 80 6.1 High-energy scatteringin 2+1dimensions . .......... 81 6.1.1 Reggetheorypredictions. .... 81 6.1.2 QCD2 athighenergies............................. 81 6.2 The SU(2)spectrum.................................. 84 6.2.1 Operatorsandspinidentification . ...... 84 6.2.2 Results ..................................... 84 6.3 SU(Nc > 2)....................................... 86 6.4 Physicaldiscussion .............................. ..... 87 6.4.1 The glueball spectrum in a Chew-Frautschi plot . ........ 87 6.4.2 Comparisontoglueballmodels . .... 88 6.4.3 Implications for high-energy reactions . ......... 89 6.5 Conclusion ...................................... 89 7 GlueballReggetrajectoriesin3+1dimensions 94 7.1 Latticetechnology ............................... .... 94 7.1.1 Operatorconstruction . ... 94 7.1.2 Spin identification on a cubic lattice . ....... 96 7.2 ThequenchedQCDspectrumfromthelattice . ....... 100 7.2.1 Spinidentificationinaction . ..... 100 7.2.2 Bi-torelon states and other finite-volume effects . ........... 101 7.2.3 Scatteringstates&decays. .... 103 7.2.4 Continuum extrapolation and polarisation-averaged spectrum . 105 7.3 The lattice glueball spectrum at large Nc ...................... 107 7.4 Physicaldiscussion .............................. ..... 107 7.4.1 Implications for high-energy reactions . ......... 109 8 Conclusion 126 Appendices 130 ii A The symmetry group of the lattice 130 A.1 Irreducible representations of the square group . ............. 130 A.2 Parity and rotations in continuous (2+1) dimensions . ............. 130 A.3 Irreducible representations of the cubic group . ............. 131 B Numerical Recipes for Lattice Gauge Theory 133 B.1 SU(N)updatealgorithms............................... 133 B.1.1 TheCabibbo-Marinarialgorithm . ..... 133 B.1.2 TheKennedy-Pendletonalgorithm . ..... 134 B.1.3 SU(2)over-relaxation . .. .. .. .. .. .. .. .. .. .. .. .. 135 B.1.4 The unitarisation to SU(N)matrices .................... 135 B.2 Thegeneralisedeigenvalueproblem . ......... 137 B.3 Jacknifeerroranalysis . ...... 137 B.4 Correlatedcoshfits............................... .... 137 C Regge theory in 2+1 dimensions 139 D Regge trajectories in the Isgur-Paton model 141 D.1 Thephononictrajectory . ..... 141 D.2 Theorbitaltrajectory . .. .. .. .. .. .. .. .. .. .. .. .. ..... 142 iii Acknowledgements First and foremost I would like to thank my supervisor Dr. Michael Teper for sharing his experi- ence and providing guidance and support. Most of the ideas and methods presented in this thesis resulted from the enjoyable discussions we had throughout the course of my D. Phil. I would also like to express my gratitude to every member of the Rudolf Peierls Centre for Theoretical Physics, for creating a friendly environment and making the time I spent in the Centre a very rewarding experience; I have profited greatly from discussions with many of its members. In particular, Dr. Biagio Lucini and Dr. Urs Wenger often provided me with useful suggestions concerning physics and programming issues. The numerical calculations presented in this thesis were performed partly on Compaq Alpha workstations, partly on ‘Hydra’, the cluster of 80 Xeon processors of the Theoretical Physics Centre. These machines were partly funded by EPSRC and PPARC grants. I warmly thank Lory Rice and Jonathan Patterson for their efficient help and support. I am indebted to Prof. Martin L¨uscher for introducing me to the subject of lattice gauge theory, and to Prof. Kari Rummukainen, who introduced me to the art of Monte-Carlo simulations and continued offering his advice by email. I would also like to use this opportunity to extend my gratitude to Prof. Mikhail Shaposhnikov who helped me a lot in obtaining a position as a graduate student at Oxford University and whose lasting influence has accompanied me over the past three years. Last but not least I thank my family, especially my parents, Anne and R¨udiger Walter Meyer, for their sustained and unconditional support throughout my studies of physics. This work was supported by the Berrow Scholarship of Lincoln College, the ORS Award Scheme (UK) and the Bourse de Perfectionnement