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Properties of Pseudoscalar Flavor Singlet Mesons from Lattice

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Properties of Pseudoscalar Flavor Singlet Mesons from Lattice Properties of pseudoscalar flavor singlet mesons from lattice QCD DISSERTATION ZUR ERLANGUNG DES DOKTORGRADES (DR. RER. NAT.) DER MATHEMATISCH-NATURWISSENSCHAFTLICHEN FAKULTAT¨ DER RHEINISCHEN FRIEDRICH-WILHELMS UNIVERSITAT¨ BONN VORGELEGT VON KONSTANTIN OTTNAD AUS FREIBURG i.BR. BONN 2013 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn 1. Gutachter: Prof. Dr. Carsten Urbach 2. Gutachter: Prof. Dr. Ulf-G. Meißner Tag der Promotion: 05.02.2014 Erscheinungsjahr: 2014 Abstract The central topic of this work are masses and mixing parameters of the η–η′ system, which are in- vestigated within the framework of Wilson twisted mass lattice QCD, using gauge configurations provided by the European Twisted Mass Collaboration. We present the first calculation with Nf = 2 + 1 + 1 dynamical quark flavors performed at three different values of the lattice spacing and multiple values of the light quark mass, corresponding to charged pion masses ranging from 230 MeV to 500 MeV. Moreover, we use selected ensembles which differ only by the value of ∼ ∼ the strange quark mass while all other parameters are kept fixed in order to obtain information on the strange quark mass dependence of our observables. This allows us to carry out chiral and continuum extrapolations with well-controlled systematics for the mass of the η meson. Using the standard method, the statistical error for the η′ turns out significantly larger due to the large contributions of quark disconnected diagrams and autocorrelation effects. However, employing an improved analysis method based on an excited state subtraction in the connected pieces of the correlation function matrix, it becomes feasible to obtain a result for the η′ mass with controlled systematics as well. The values for both masses Mη = 551(8)stat(6)sys MeV and Mη′ = 1006(54)stat(38)sys(+64)ex MeV turn out to be in excellent agreement with experiment. Considering matrix elements in the quark-flavor basis, one expects the mixing in the η–η′ system to be described reasonably well by a single mixing angle φ and two decay constants fl, fs. The required accuracy of the matrix elements is again guaranteed by the aforementioned, improved analysis method, yielding a value of φ = 46.0(0.9)stat(2.7)sys◦ for the mixing angle extrapolated to the physical point. In addition we obtain results for the ratios fl/fPS = 0.859(07)stat (64)sys and fs/fK = 1.166(11)stat (31)sys . We find that our data is indeed described well by a single mixing angle, indicating that the η′ is mostly a flavor singlet state. Moreover, our results confirm that the charm quark does not contribute to any of the two states within errors. Apart from the flavor singlet sector, we also perform calculations of masses for the remaining light pseudoscalar octet mesons. Matching these masses to two-flavor Wilson chiral perturbation theory allows for a determination of the low energy constants W6′, W8′ and their linear combina- tion c which controls the a2 mass splitting between charged and neutral pion. We study 2 O the dependence of these low energy constants on the number of dynamical quark flavors and for different choices of the lattice action. Parts of this work have been previously published in journals [1–3] and conference proceedings [4–6]; see also the list included on the next page. iii iv List of publications [1] Konstantin Ottnad, Carsten Urbach, Chris Michael, and Siebren Reker. Eta and eta’ meson masses from Nf=2+1+1 twisted mass lattice QCD. PoS, LATTICE2011:336, 2011, 1111.3596. [2] Konstantin Ottnad et al. η and η′ mesons from Nf=2+1+1 twisted mass lattice QCD. JHEP, 1211:048, 2012, 1206.6719. [3] Krzysztof Cichy, Vincent Drach, Elena Garcia Ramos, Karl Jansen, Chris Michael, et. al. Properties of pseudoscalar flavour-singlet mesons from 2+1+1 twisted mass lattice QCD. PoS, LATTICE2012:151, 2012, 1211.4497. [4] Gregorio Herdoiza, Karl Jansen, Chris Michael, Konstantin Ottnad and Carsten Urbach. Determination of Low-Energy Constants of Wilson Chiral Perturbation Theory. JHEP, 1305:038, 2013, 1303.3516. [5] Chris Michael, Konstantin Ottnad and Carsten Urbach. η and η′ mixing from Lattice QCD. Phys. Rev. Lett. 111, 181602, 2013, 1310.1207. [6] Chris Michael, Konstantin Ottnad and Carsten Urbach. η and η′ masses and decay con- stants from lattice QCD with 2+1+1 quark flavours. 2013, 1311.5490. Contents Abstract iii Introduction 1 1 Theoretical Background 5 1.1 QuantumChromodynamics . 5 1.1.1 General properties and parameters . ...... 7 1.1.2 Chiralsymmetryandanomaly . 9 1.1.3 Spontaneous symmetry breaking and pseudoscalar mesons......... 11 1.1.4 Chiralperturbationtheory . .... 13 1.1.5 Mixing in the η–η′ system........................... 19 1.2 LatticeQCD ...................................... 24 1.2.1 QCDinEuclideanspacetime . 24 1.2.2 Lattice regularization . .... 27 1.2.3 Latticeactions ................................ 29 1.2.4 Symmetriesonthelattice . 32 1.3 Wilson twisted mass formulation . ....... 34 1.3.1 Lightsector................................... 34 1.3.2 Propertiesandsymmetries . 37 1.3.3 Heavysector .................................. 39 1.3.4 Symmetriesintheheavysector . .... 42 1.3.5 Wilson chiral perturbation theory . ...... 43 2 Spectroscopy and analysis methods 47 2.1 Interpolatingoperators. ....... 47 2.1.1 Pionsector ................................... 49 2.1.2 Kaonsector................................... 50 2.1.3 Flavor singlet sector . 52 2.2 Correlation functions and extraction of observables . ............... 53 2.2.1 Calculation of correlation functions . ......... 57 2.2.2 Generalized eigenvalue problem . ...... 60 2.2.3 Observables and renormalization . ...... 63 2.2.4 Scale setting and extrapolations . ...... 66 v vi CONTENTS 2.3 Dataanalysis .................................... 67 2.3.1 Autocorrelation............................... 68 2.3.2 χ2-fitting .................................... 69 2.3.3 Bootstrapmethod ............................... 70 3 Analysis and results 71 3.1 Lattice action and parameters . ....... 72 3.2 Masses and flavor non-singlet decay constants . ........... 74 3.2.1 Chargedpionandkaon ............................ 74 3.2.2 Flavor singlet and neutral pion masses . ....... 76 3.3 Chiral extrapolations and scaling artifacts for η,η′ ................. 81 3.3.1 Strange quark mass dependence and scaling test . ........ 81 3.3.2 Extrapolation to the physical point . ....... 84 3.4 Removal of excited states for η, η′ .......................... 87 3.4.1 Scaling behavior and strange quark mass dependence . .......... 90 3.4.2 Extrapolation to the physical point . ....... 94 3.4.3 Discussionofresults . 97 3.5 η, η′-mixing....................................... 100 3.5.1 Chiralextrapolations. .... 102 3.5.2 Discussionofresults . 106 3.6 Determination of low-energy constants of WχPT .................. 108 3.6.1 Nf = 2 + 1 + 1 WtmLQCD with Iwasaki gauge action . 109 3.6.2 Nf = 2 WtmLQCD with tree-level Symanzik improved gauge action . 113 3.6.3 Discussionofresults . 116 Summary and outlook 119 A Conventions and definitions 125 A.1 Indexconventions................................ .... 125 A.2 Paulimatrices................................... 125 A.3 Gell-Mannmatrices ............................... 126 A.4 Diracmatrices ................................... 126 B Correlation function matrices 127 B.1 Correlation functions for the neutral pion . ........... 128 B.2 Correlation functions for η, η′ ............................. 128 C Fitting details 131 C.1 ParametersforGEVP ............................... 131 C.2 Parametersforfactorizingfit . ....... 134 Bibliography 149 Introduction The rise of modern theoretical physics is very closely tied to the concept of symmetries. In fact, quantum field theories which are based on so-called local gauge symmetries have been proven to be tremendously successful in describing elementary particle interactions and provide the theoretical foundation for the present standard model of particle physics. Besides local symmetries which are essential to the construction of quantum field theories, there are further global symmetries. These include discrete symmetries such as parity ( ), time-reversal ( ) or P T charge conjugation ( ), as well as continuous ones like chiral symmetry. However, while gauge C symmetries are required to be exact for theoretical reasons, many other symmetries are actually broken in nature, leading to a wide range of intriguing phenomena. In principle, there are several ways of breaking symmetries. Firstly, there is explicit symmetry breaking which refers to a situation where a symmetry is violated (or only realized to some approximation) due to terms in the Lagrangian that do not respect the symmetry in question, while the remaining parts of the Lagrangian are indeed invariant. For instance, the presence of non-vanishing quark masses explicitly violates chiral symmetry in quantum chromodynamics (QCD), hence it is only realized as an approximate symmetry. Secondly, there is also the possibility of spontaneous symmetry breaking, which refers to a situation where the Lagrangian itself is invariant under a given symmetry, but the system develops a non-trivial vacuum expectation value, hence breaking the symmetry of the ground state. An example for such a mechanism is found in chiral symmetry, which is spontaneously broken by a non-vanishing quark condensate1. Another kind of symmetry breaking is again related to strong interactions and concerns only a specific subgroup of chiral symmetry, the so-called
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