Excited light and strange hadrons from the lattice with two Chirally Improved quarks
EExcited light and strange hadrons from the latticewith two Chirally Improved quarks
Georg P. Engel ∗ Università Milano-Bicocca, Italy andINFN, Sezione di Milano-Bicocca, ItalyE-mail: [email protected]
C.B. Lang
Institut für Physik, FB Theoretische Physik,Universität Graz, A–8010 Graz, AustriaE-mail: [email protected]
Daniel Mohler
Fermi National Accelerator Laboratory, Batavia, Illinois 60510-5011, USAE-mail: [email protected]
Andreas Schäfer
Institut für Theoretische Physik,Universität Regensburg, D–93040 Regensburg, GermanyE-mail: [email protected]
Results for excited light and strange hadrons from the lattice with two flavors of Chirally Improvedsea quarks are presented. We perform simulations at several values of the pion mass ranging from250 to 600 MeV and extrapolate to the physical pion mass. The variational method is appliedto extract excited energy levels but also to discuss the content of the states. Among others, weexplore the flavor singlet/octet content of Lambda states. In general, our results agree well withexperiment, in particular we confirm the Lambda(1405) and its dominant flavor singlet structure.
XV International Conference on Hadron Spectroscopy-Hadron 20134-8 November 2013Nara, Japan ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - ph ] N ov xcited light and strange hadrons from the lattice with two Chirally Improved quarks Georg P. Engel
1. Introduction
Our main knowledge about Strong Interactions lies in experimental data on hadron resonances [1],of which an ab-initio determination starting from QCD would be truly desirable. Although thereis noticeable progress in computing resonance properties on the lattice (for a review, see, e.g.,[2]), this is still a prohibitively difficult task for most resonances. To discuss an extensive list ofexcited hadrons, we consider the discrete spectrum of the Hamiltonian in a finite box. This discretespectrum becomes denser towards larger volumes, and the volume dependence is related to thephase shift of the resonance in the elastic region [3, 4]. However, in finite volume and also forunphysically heavy pion masses the decay channels are often closed or the related phase space issmall. Correspondingly, the energy levels in the finite system are close to the resonance peak, inparticular for narrow resonances. Hence, as a first approximation, the discrete energy levels canbe identified with the masses of corresponding resonances. We discuss all canonical channels ofisovector light and strange mesons and light and strange baryons and give results at the physicalpion mass which can be compared to experiment. A large basis of interpolators is considered,which, however, includes only quark–antiquark and 3-quark interpolators. In principle, the seaquarks should provide overlap of these interpolators with meson–meson and meson–baryon states.In practice, however, we cannot clearly identify such states. A possible explanation may be a weakoverlap with the used interpolators, indicating the need for more general interpolators for futurework. The results presented here have been published before in [5, 6, 7, 8, 9], for recent reviews onhadron spectroscopy on the lattice, see, e.g., [10, 11, 12].
2. Setup
While most lattice Dirac operators break chiral symmetry explicitly, a lattice version of the sym-metry can be formulated by choosing a particular discretization of the Dirac operator, which obeysthe non-linear “Ginsparg-Wilson" equation [13, 14]. We use the Chirally Improved (CI) fermionaction [15, 16], which is an approximate solution to this equation. For the gauge sector we use thetadpole-improved Lüscher-Weisz action [17]. The lattice spacing a is set at the physical pion massfor each value of the gauge coupling using a Sommer parameter r = .
48 fm, as discussed in [5].The lowest energy levels can be extracted considering correlation functions of hadronic interpola-tors and their exponential decay in euclidean time. The correlators are computed using Monte Carlosimulations with importance sampling for the gauge sector and the fermion determinant. To discussexcited states, the variational method is applied [18, 19]. One constructs several interpolators O i foreach set of quantum numbers and computes the cross-correlation matrix C i j ( t ) = (cid:104) O i ( t ) O j ( ) † (cid:105) . Itsgeneralized eigenvalue problem yields approximations to the eigenstates of the Hamiltonian. Theexponential decay of the eigenvalues is governed by the energy levels, and the eigenvectors tellabout the content of the states in terms of the lattice interpolators. We simulate two CI light seaquarks and consider a valence strange quark. We generate seven ensembles with pion masses in therange from 250 to 600 MeV, lattice spacings between 0.13 and 0.14 fm and lattices of size L ≈ . xcited light and strange hadrons from the lattice with two Chirally Improved quarks Georg P. Engel
3. Results m a ss [ G e V ] π -+ a ++ ρ -- π -+ a ++ b +- ρ -- T /E π -+ T /E a ++ T /E m a ss [ G e V ] K0 - K + K * - K + K - T /E K + T /E φ -- f T /E ++ Figure 1:
Energy levels for isovector light (left), strange and isoscalar (right) mesons in finite volume( L ≈ . σ , that ofthe experimental values by boxes of 1 σ . For spin 2 mesons, results for T and E are shown side by side.Open symbols denote a poor χ /d.o.f. of the chiral fits. Disconnected diagrams are neglected in the isoscalarchannels. m a ss [ G e V ] J P =1/2 + N ∆ Λ Σ Ξ Ω J P =3/2 + N ∆ Λ Σ Ξ Ω m a ss [ G e V ] J P =1/2 - N ∆ Λ Σ Ξ Ω J P =3/2 - N ∆ Λ Σ Ξ Ω Figure 2:
Like Fig. 1, but for positive (left) and negative parity (right) baryons.
We study the isovector light and strange meson channels J = , , C -parities and the light and strange baryon channels J = / , / Λ ( ) . In Fig. 3 (rhs) we show the corresponding resultsfor the individual ensembles in this channel. In some cases our results suggest the existence ofyet experimentally unobserved resonance states. However, some obtained energy levels mismatchexperiment. The first excitation in the nucleon ( J P = / + ) channel, e.g., lies considerably higherthan the Roper resonance. A possible interpretation is a weak overlap of our interpolators withthe physical state. Another possible source of deviation from experiment are finite-volume effects.Fig. 3 shows our results for a selection of states after extrapolation to infinite volume. In generalour results in the infinite volume limit compare very well with experiment. For the baryons, weanalyze the flavor content by identifying the singlet/octet/decuplet contributions. For example,3 xcited light and strange hadrons from the lattice with two Chirally Improved quarks Georg P. Engel m a ss [ G e V ] J P =1/2 + N Λ Σ Ξ J P =3/2 + ∆ Λ Σ J P =1/2 - Λ J P =3/2 - ΛΞ Ω m π [GeV ] m a ss [ G e V ] Λ - (L~2.2 fm) Figure 3:
Lhs: Like Fig. 1, but for hadrons in the infinite volume limit. Rhs: Energy levels for the baryonchannel Λ ( J P = / − ) in a finite box of linear size L ≈ . t -101-10102 [Singlet χ ]03 [Singlet χ ]10 [Singlet χ ]18 [Singlet χ ]26 [Octet χ ]27 [Octet χ ]34 [Octet χ ]42 [Octet χ ] E i g e nv ec t o r c o m pon e n t s Λ (1/2 − ) : ground state (A66) Λ (1/2 − ) : 1 st excit. (A66) t -101-101 02 [Singlet]09 [Octet]10 [Octet]16 [Octet] E i g e nv ec t o r c o m pon e n t s Λ (3/2 + ) : ground state (A66) Λ (3/2 + ) : 1 st excit. (A66) Figure 4:
Lhs: Eigenvectors for the ground state and the first excitation for m π ≈
255 MeV, for the baryonchannel Λ ( J P = / − ). The ground state Λ ( ) is dominated by flavor singlet, the first excitation by octetinterpolators. Rhs: Same as left hand side, but for J P = / + . We find the first excitation to be dominatedby flavor singlet interpolators, which vanish exactly for point-like interpolators due to Fierz-identities. we find a dominance of singlet interpolators (mixing of 15-20% with octet) for Λ ( ) , and adominance of octet interpolators of the first excitation. The corresponding eigenvectors are shownin Fig. 4. In the Λ ( J P = / + ) channel, the first excitation appears to be dominated by flavor singletinterpolators. Such interpolators vanish exactly for point-like quark-fields due to Fierz-identities.Using different quark-smearing widths to side-step these identities, we are able to construct non-vanishing interpolators nevertheless. To conclude, we remark that our ab-initio determination ofthe excited hadron spectrum agrees well with experiment, and provides also further insights, suchas yet unknown states, and also the content of the states in terms of lattice interpolators.
4. Acknowledgments
We thank E. Gamiz, Ch. Gattringer, L. Y. Glozman, M. Limmer, W. Plessas, H. Sanchis-Alepuz,M. Schröck and V. Verduci for valuable discussions. The calculations have been performed atthe Leibniz-Rechenzentrum Munich and on clusters at the University of Graz. We thank these4 xcited light and strange hadrons from the lattice with two Chirally Improved quarks
Georg P. Engel institutions. G.P.E. and A.S. acknowledge support by the DFG project SFB/TR-55. G.P.E. wassupported by the MIUR–PRIN 20093BM-NPR. Fermilab is operated by Fermi Research Alliance,LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy.
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