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Baryon Resonances and Pentaquarks on the Lattice ✩

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Baryon Resonances and Pentaquarks on the Lattice ✩ Nuclear Physics A 754 (2005) 248c–260c Baryon resonances and pentaquarks on the lattice ✩ F.X. Lee a,b,∗, C. Bennhold a a Center for Nuclear Studies, George Washington University, Washington, DC 20052, USA b Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USA Received 17 December 2004; accepted 22 December 2004 Available online 21 January 2005 Abstract We review recent progress in computing excited baryons and pentaquarks in lattice QCD. 2004 Elsevier B.V. All rights reserved. 1. QCD primer Quantum Chromodynamics (QCD) is widely accepted as the fundamental theory of the strong interaction. The QCD Lagrangian density can be written down simply in one line (in Euclidean space) 1 L = Tr F F µν +¯q γ µD + m q, (1) QCD 2 µν µ q where Fµν = ∂Aµ − ∂Aν + g[Aµ,Aν] is the gluon field strength tensor and Dµ = ∂µ + gAµ is the covariant derivative which provides the interaction between the gluon and quark terms. The action of QCD is the integral of the Lagrangian density over space– 4 time: SQCD = LQCD d x. QCD is a highly non-linear relativistic quantum field theory. It is well known that the theory has chiral symmetry in the mq = 0 limit and the symmetry is spontaneously broken in the vacuum. At high energies, it exhibits asymptotic freedom, ✩ Based on plenary talk by F.X. Lee at HYP2003, JLab. * Corresponding author. E-mail address: [email protected] (F.X. Lee). 0375-9474/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.12.072 F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c 249c while at low energies it has confinement. At the present, the only tool that provides a solu- tion to QCD with controlled systematic errors is lattice QCD which solves the theory on a discrete space–time lattice using numerical simulations. The basic building block for com- puting the spectrum is the fully-interacting quark propagator defined via the path integral = | [ ¯ | S(x,0) 0 T q(x)q(0) 0 − − − DADq Dq¯ [q(x)q(¯ )]e SQCD DAM 1 Me Sg = 0 = det − − , (2) DADq Dqe¯ SQCD DAdet Me Sg µ where M = γ Dµ + mq is the quark matrix. In the last step the quark fields are exactly integrated, resulting in an expression that resembles a statistical system with the weighting − factor det Me SG for which Monte-Carlo methods can be employed. In this sense the quark propagator is simply the expectation value of the inverse quark matrix S =M−1. The determinant det M proves costly to simulate so it is usually set to a constant, leading to savings of up to a factor of 100. This is called the quenched approximation which amounts to ignoring the quark–antiquark bubbles in the QCD vacuum. 2. Baryon resonances The rich structure of the excited baryon spectrum, as documented by the particle data group [1], provides a fertile ground for exploring the nature of quark–quark interactions. Most of the spectrum, however, is poorly known. Traditionally, quark models have led the way in making sense of the spectrum. But many puzzles remain. What is the nature of the roper resonance, and the Λ(1405)? How to explain the inverted ordering of the lowest-lying states which has the order of positive and negative-parity excitations inverted between N, ∆ and Λ channels? Two contrasting views have emerged about the nature of the hyperfine splittings in the baryons. One is from the constituent quark model [2,3] which c · c · has the interaction dominated by one-gluon-exchange type, i.e., color–spin λ1 λ2σ1 σ2. f · f · The other is based on Goldstone-boson-exchange [4] which has flavor–color λ1 λ2 σ1 σ2 as the dominant part. Lattice QCD is perhaps the most desirable tool to adjudicate the theoretical controversy surrounding these issues. Evidence from valence QCD [5] favors the flavor–color picture. Lattice QCD has evolved to the point that the best quenched calculation, of the ground- state hadron spectrum, shown in Fig. 1(left), has reproduced the observed values to within 7%, with the remaining discrepancy attributed to the quenched approximation. This bodes well for the exploration of the excited sectors of the spectrum, even in the quenched ap- proximation. 2.1. Roper and S11 There exist a number of lattice studies of the excited baryon spectrum using a variety of actions [7–13]. The nucleon channel is the most-studied, focusing on two independent local fields: abc Ta b c χ1 = u Cγ5d u , (3) 250c F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c Fig. 1. Left: light hadron spectrum from quenched lattice QCD by the CP-PACS Collaboration [6]. Right: nucleon, 2 roper, and S11 masses as a function of mπ , using the standard nucleon interpolating field χ1. The insert is the ratio of roper to nucleon mass. The experimental values are indicated by the corresponding open symbols. abc Ta b c χ2 = u Cd γ5u . (4) χ1 is the standard nucleon operator, while χ2, which has a vanishing non-relativistic limit, is sometimes referred to as the ‘bad’ nucleon operator. Note that baryon interpolating fields couple to both positive- and negative-parity states, which can be separated by well- established parity-projection techniques. There are two problems facing these studies. First, they have not been able to probe the relevant low quark mass region while preserving chiral symmetry at finite lattice spacing (except Ref. [12] which uses the domain wall fermion). Since the controversy about the nature of roper hinges on chiral symmetry, it is essential to have a fermion action which explicitly exhibits the correct spontaneously broken chiral symmetry. Another difficulty of the calculation of the excited states in lattice QCD is that the conventional two-exponential fits are not reliable. Facing the uncertainty of the fitting procedure for the excited state, it has been suggested to use the non-standard nucleon interpolating field χ2, in the hope that it may have negligible overlap with the nucleon so that the roper state can be seen more readily. However, the lowest state calculated with this interpolation field (2.2 GeV) is much higher than the roper state. Employing the maximum entropy method allows one to study the nucleon and its radial excitation with the standard nucleon interpolation field [14]. However, with the pion mass at ∼600 MeV, the nucleon radial excitation is still too high (∼2 GeV). So the ordering of the nucleon, S11(1535) and the roper(1440) in these studies remains the same as that from quark models. In order to have a reliable prediction of the excited state in lattice QCD calculations by way of addressing the above mentioned difficulties, we aim to address both problems in our recent study [15]. First, we employ the overlap fermion [16] which has exact chi- ral symmetry even on a finite lattice, and on a large lattice which admits calculations with realistically small quark masses [17] to study the chiral region. Second, we implement con- strained curve fitting based on an empirical Bayes method. It has been advocated [18,19] recently as a powerful tool which utilizes more data points while better controlling the systematics. The details of our implementation are discussed in [20]. The lattice size we F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c 251c 3 use is 16 × 28 with the scale of a = 0.202(1) fm set from fπ . We consider a wide range of quark masses: 26 masses with the lowest mπ = 180 MeV (or mπ /mρ = 0.248), very close to the physical limit, and with 18 masses below the strange quark mass. We ana- lyzed 80 configurations. Our final result is shown in Fig. 1(right). We see that for heavy quarks (mπ 800 MeV), the roper, S11, and nucleon splittings are like those of the heavy quarkonium. When the quark mass becomes lighter, the roper and S11 have a tendency to coincide and cross over around mπ = 220 MeV. More statistics are needed to clarify this point. However, from the insert in the figure, we can see that the ratio of roper to nu- cleon has a smaller error (by ∼40%) than the roper mass itself. It shows that the roper is consistent with the experimental value near the physical pion mass. We use the form + + 2 C0 C1/2mπ C1mπ to extrapolate the roper and S11 to the physical pion mass. The re- sultant nucleon mass of 928(56) MeV, roper at 1462(157) MeV, and S11 at 1554(65) MeV are consistent with the experimental values. Our result confirms the notion that the order reversal between the roper and S11(1535) compared to the heavy quark system is caused by the flavor–spin interaction between the quarks due to Goldstone boson exchanges [4]. It serves to verify that the roper (1440) is a radial excitation of the nucleon with three valence quarks. It also casts doubts on the viability of using the non-standard interpolation field for the roper. We further support the notion that there is a transition from heavy quarks (where the SU(6) symmetry supplemented with color–spin interaction for the valence quarks gives a reasonable description) to light quarks (where the dynamics is dictated by chiral symme- try). It is suggested that this transition occurs at mπ ∼ 400 MeV for the nucleon. 2.2. The η ghost The above result is obtained only after the special effects of the so-called η ghost are removed. In full QCD, the η meson contributes to the proton via vacuum polarizations, as shown in Fig.
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