Latest results from lattice QCD for the Roper resonance

In the excited baryon spectrum, of particular interest is the level order of the positive-parity excited nucleon, so-called the Roper resonance N (1440) and the negative-parity nucleon N(1535). It is worth mentioning that this pattern of the level order between positive and negative-parity excited states can be found universally in the ∆, Σ and flavor-octet Λ channels. It is also interesting to note that quark confining models such as either the harmonic oscillator quark model or the MIT bag model have some difficulty reproducing the correct level order, as eigenstates in each model alternate in parity.1) This wrong ordering problem does not seem to be easily alleviated. Hence, the first principle calculation, lattice QCD is demanded to resolve such a long-standing puzzle. In lattice QCD, computations of the hadron masses rely on the asymptotic behavior of the two-point hadron correlator in the large Euclidean time region as G(t) ∼ exp(−M0t). This approach mainly accesses the mass of the lowest-lying states, M0. As for the higher-lying states, we need more sophisticated analysis such as the matrix-correlator method.2), 3) The old calculation3) suggested the large mass of the first positive-parity excited nucleon and reported the possibility of missing the Roper resonance in lattice calculation. Recently, a first systematic calculation for the nucleon excited states in both parity channels showed that the wrong ordering between N ′ and N actually happens in the relatively heavy-quark mass region,1), 4) whereas the N state can be reproduced well in lattice QCD1), 5) (see Fig. 1). What one can see in this figure closely resembles the wrong ordering problem of the excited nucleon spectrum in the confining models. After this work, many lattice calculations confirmed this particular issue.6), 7), 8) The question arises whether or not lattice calculations fail to reproduce the Roper resonance. An answer to this question provides some perspective to understand the structure of the mysterious Roper resonance since above lattice calculations basically utilize the simple description of the Roper resonance as three valence quarks. However, we should not conclude anything from the previous lattice results without examining possible systematic errors. From the phenomenological point of view, the mass splitting between the ground state and the radial excited state, which has the same quantum number of the ground