Structure of the Roper Resonance from Lattice QCD Constraints

# Structure of the Roper Resonance from Lattice QCD Constraints

## Abstract

Two different descriptions of the existing pion-nucleon scattering data in the region of the Roper resonance are constructed. Both descriptions fit the experimental data very well. In one scenario the resonance is the result of strong rescattering between coupled meson-baryon channels, while in the other scenario, the resonance has a large bare-baryon (or quark-model like) component. The predictions of these two scenarios are compared with the latest lattice QCD simulation results in this channel. Consideration of the finite volume spectra, the manner in which the states are excited from the vacuum in lattice QCD and the composition of the states in Hamiltonian effective field theory enable a discrimination of these two different descriptions. We find the second scenario is not consistent with lattice QCD results whereas the first agrees with the lattice QCD constraints. In this scenario, the mass of the quark-model like state is approximately 2 GeV and in the finite volume of the lattice is dressed to place the first radial excitation of the nucleon at 1.9 GeV. Within this description, the infinite-volume Roper resonance is best described as a resonance generated dynamically through strongly coupled meson-baryon channels.

###### pacs:
12.38.Gc Lattice QCD calculations 12.39.Fe Chiral Lagrangians
1

Since the discovery of the Roper resonance in1964 Roper (1964), its peculiar properties have challenged our understanding of the quark structure of hadrons and ultimately of quantum chromodynamics (QCD) itself Aznauryan et al. (2008); Joo et al. (2005); Weber (1990); Julia-Diaz and Riska (2006); Barquilla-Cano et al. (2007); Golli and Sirca (2008); Golli et al. (2009); Meissner and Durso (1984); Hajduk and Schwesinger (1984); Krehl et al. (2000); Schutz et al. (1998); Matsuyama et al. (2007); Kamano et al. (2010, 2013); Suzuki et al. (2010); Hernandez et al. (2002); Barnes and Close (1983); Golowich et al. (1983); Kisslinger and Li (1995). With the first negative-parity excitation of the nucleon, the , almost 600 MeV higher in mass, expectations – based upon the harmonic oscillator model which has enjoyed much success in treating hadron spectroscopy – suggest that the first positive-parity excited state should occur at a little over 2 GeV. Yet, empirically one finds the first positive-parity, spin-1/2 Roper resonance of the nucleon to have a mass of just 1.45 GeV, below the  Patrignani et al. (2016)!

To make matters worse, the first negative parity excitation of a strangeness -1 baryon, the famous , is lower in mass than both of these non-strange excited states of the nucleon  Patrignani et al. (2016). Fortunately, in this case there have recently been advances in our understanding, thanks to modern lattice QCD simulations of not only the mass of this state but the individual valence quark contributions to its electromagnetic form factors Hall et al. (2015, 2016). These simulations have also been supported by analysis involving an effective Hamiltonian Liu et al. (2016a), which allows a natural connection to be made between the results calculated on a finite lattice volume and the infinite volume of the real world Liu et al. (2017); Molina and Döring (2016). As a result of these studies, it is now clear that the is essentially an anti-kaon nucleon bound state with very little content corresponding to the sort of three-quark state anticipated in a typical quark model Hall et al. (2016).

In this Letter, we use similar techniques to those which proved so successful for the  Liu et al. (2016a) to investigate the nature of the Roper resonance. We first introduce the coupled channel scattering formalism Liu et al. (2016b) and show the two high quality fits obtained to existing data in the region of the Roper. In the first there is no significant three quark coupling, while in the second there is. These models produce rather different behaviour in the unobserved and channels and cannot be distinguished by experiment. We then use the same models on a finite volume to compute the spectrum one would expect to find in lattice QCD. Only the first scenario is consistent with recent lattice simulations, indicating the Roper resonance is also generated dynamically through the rescattering of coupled meson-baryon channels. The three-quark state anticipated in traditional quark models appears to lie nearer 2 GeV, which as explained earlier, is far more consistent with the mass of the observed .

In order to model the scattering data in the region of the Roper resonance and describe the observed inelasticity, we include three coupled channels, , and . In the rest frame, the Hamiltonian has the following form

 H=H0+HI, (1)

where the non-interacting Hamiltonian is

 H0 = ∑B0|B0⟩m0B⟨B0|+∑α∫d3→k (2) |α(→k)⟩[ωα1(→k)+ωα2(→k)]⟨α(→k)|.

Here denotes a bare baryon with mass , which may be thought of as a quark model state, designates the channel and () indicates the meson (baryon) state which constitutes channel . The energy .

The energy independent interaction Hamiltonian includes two parts, , where describes the vertex interaction between the bare particle and the two-particle channels

 g = ∑α,B0∫d3→k{|α(→k)⟩G†α,B0(k)⟨B0|+h.c.}, (3)

while the direct two-to-two particle interaction is defined by

 v=∑α,β∫d3→kd3→k′|α(→k)⟩VSα,β(k,k′)⟨β(→k′)|. (4)

For the vertex interaction between the bare baryon and the two-particle channels we choose:

 G2α,B0(k) = g2B0α4π2(kf)2lαu2α(k)ωα1(k), (5)

where the pion decay constant MeV and is the orbital angular momentum in channel . Here, since we are concerned with the Roper resonance, with isospin, angular momentum and parity, , is 1 for and , while it is 0 for . The regulating form factor, , takes the exponential form where is the regularization scale. For the direct two-to-two particle interaction, we introduce the separable potentials for the following five channels

 VSα,β(k,k′) = gSα,β¯Gα(k)√ωα1(k)¯Gβ(k′)√ωβ1(k′), (6)

where . The -matrices for two particle scattering are obtained by solving a three-dimensional reduction of the coupled-channel Bethe-Salpeter equations for each partial wave

 tα,β(k,k′;E) = Vα,β(k,k′;E)+∑γ∫q2dq× (7) Vα,γ(k,q;E)1E−ωγ1(q)−ωγ2(q)+iϵtγ,β(q,k′;E).

The coupled-channel potential is readily calculated from the interaction Hamiltonian

 Vα,β(k,k′) = ∑B0G†α,B0(k)Gβ,B0(k′)E−m0B+VSα,β(k,k′), (8)

with the normalization . The pole position of any bound state or resonance is obtained by searching for the poles of the -matrix in the complex plane.

In order to compare the predictions of this continuum model with the results of lattice QCD simulations, it is necessary to rewrite the problem on a finite volume. This procedure is by now well known and we refer to Refs. Liu et al. (2016a, 2017, b); Hall et al. (2013); Wu et al. (2014, 2016) for the details. By solving for the eigenstates of the Hamiltonian effective-field-theory (EFT), one obtains energy levels which can be compared with the energies found in lattice QCD simulations. The eigenvectors of the Hamiltonian system describe the composition of the eigenstates which can be compared with the interpolating fields used to excite the lattice QCD states.

We can also extend the formalism to unphysical pion masses. Using as a measure of the light quark masses, we consider the variation of the bare mass and -meson mass as

 m0B(m2π) = m0B|phy+α0B(m2π−m2π|phy), (9) m2σ(m2π) = m2σ|phy+α0σ(m2π−m2π|phy), (10)

where the slope parameter is constrained by lattice QCD data from the CSSM. In the large quark mass regime, where constituent quark degrees of freedom become relevant, one expects Cloet et al. (2002) . The nucleon and Delta masses away from the physical point are obtained via linear interpolation between the lattice QCD results.

By fitting the experimental data for scattering from 1200 MeV to 1800 MeV, we found two rather different models which appear equally acceptable in terms of the quality of the fit to existing data. The corresponding parameters are shown in Table 1. In Fig. 1, the phase shift, inelasticity and -matrix for the channel are shown for these two models. It is impossible to distinguish the merit of these two fits using the existing data. However, in the as yet unmeasured coupled channels there are considerable differences.

The main differences are that the coupling of in fit I is much larger than that in fit II, while the coupling of the bare state to and in fit I is much smaller than in fit II. This leads to two different pictures of the Roper. In fit I it is a resonance generated by strong rescattering in the meson-baryon channels. On the other hand, in fit II the rescattering is weaker and the observed resonance is dominated by coupling to an underlying bare, or quark-model like, state.

In light of the present experimental data, these two scenarios are both acceptable. However, as illustrated in Fig. 2, measurements of the scattering amplitudes in the coupled channels and , would enable us to distinguish between them.

Given the absence of the relevant experimental data, we now turn to the results provided by lattice QCD simulations, focussing on the recent work of Lang et al. Lang et al. (2017) and the CSSM Liu et al. (2016b). The former is particularly interesting as incorporated and five-quark non-local interpolating fields with the momenta of each hadron projected to provide excellent overlap with the low-lying scattering states of the spectrum. These are in addition to standard three-quark interpolating fields which have proved to favour localized states and miss the non-local scattering states Mahbub et al. (2012, 2013a, 2013b, 2014); Kiratidis et al. (2015, 2016) making the lattice spectrum incomplete. In reporting these results in Figs. 4 and 4 we have used solid(open) symbols to indicate states dominated by local(nonlocal) interpolating fields. This information can be used in addition to the standard analysis of the spectrum.

Just as insight into the composition of the lattice QCD states can be obtained from the eigenvectors of the lattice correlation matrix used to excite the states, Hamiltonian EFT also provides insight into their composition via the superposition of basis states for each eigenvector. In Figs. 4 and 4 we not only show the spectra calculated for Scenarios I and II but we also colour code the lines to indicate the amount of the bare-baryon (or three-quark) basis state in that particular eigenstate. The red, blue, green and orange lines indicate the states having the first, second, third and fourth largest bare-state contributions, respectively.

For Scenario I, all of the lattice states dominated by local three-quark interpolating fields can be associated with a colored line. Similarly, all of the Hamiltonian states having the largest bare basis-state component, indicated in red in Fig. 4, have a nearby lattice QCD result. Thus, the predictions of Scenario I are consistent with lattice QCD.

On the other hand, Scenario II displays little correspondence to the lattice QCD results. Scenario II predicts a low-lying state with a large bare basis-state component of approximately 50%, approaching that for the ground state. Such a state would be easy to excite in lattice QCD with local three-quark operators. However this state is not seen in the simulations. The lattice state that is seen at light quark masses is actually excited by a scattering-state interpolating field. Indeed, Lang et al. Lang et al. (2017) only see this state when they include a non-local interpolating field.

In Scenario I, the lattice excitation at GeV is described by a Hamiltonian eigenstate dominated by the basis state. Fig. 5 shows the composition of the lowest ten Hamiltonian eigenstates as a function of pion mass for Scenario I. Near the physical mass the first eigenstate is dominated by basis states. Again, this is consistent with the work of Lang et al., who only find the lowest state when they include a non-local interpolating field. The second state is dominated by the channel but some mixing with is also apparent. The channel dominates the fifth excitation of the spectrum at light quark masses and this serves as a prediction for future lattice QCD calculations employing non-local momentum-projected scattering-state interpolators.

It is only for the seventh, eighth and ninth eigenstates that we find a significant bare basis-state contribution and this is precisely where the lattice QCD states excited by local three-quark operators reside. The ninth state has an extremely large bare-state component exceeding 50% and both the CSSM and Lang et al. observe lattice QCD eigenstates within one sigma of this state.

Next, we examine how the eigenstates evolve as the pion mass increases. Here one anticipates rescattering to become suppressed as the masses of the hadrons increase. In Fig. 5 we see that in scenario I the bare baryon content of the second and fourth eigenstates increases towards the upper end of the pion-mass range. Once again this is consistent with the lattice simulations as this is where the CSSM finds lower mass states in the spectrum with local three-quark operators. Overall the Hamiltonian eigenvectors obtained within Scenario I explain the lattice spectra very well.

In contrast, the content of the Hamilton eigenvectors in Scenario II is quite wrong. In particular, the lowest mass states are always dominated by the bare-baryon basis state and as noted before, this is inconsistent with the results of Lang et al.. Neither the CSSM nor the Cyprus collaboration Alexandrou et al. (2015) were able to identify such states in their lattice results.

In conclusion, the bare-baryon (or three-quark) basis state associated with the Roper resonance found in Nature lies at approximately 2 GeV. This large mass is required in order to provide a finite-volume spectrum consistent with lattice QCD. The result is also consistent with natural expectations from a harmonic oscillator spectrum. A lower bare basis-state mass leads to the prediction of quark-model like low-lying states dominated by a bare-state component. The absence of such states in the lattice QCD spectrum rules out this scenario.

In the preferred Scenario I with a 2 GeV bare-state mass, quark-model like states sit high in the spectrum. In the finite volume of the lattice, the bare state is dressed to produce states commencing at GeV for the lightest pion mass of 156 MeV. Indeed, the CSSM studied the three-quark wave function of this state and discovered it resembles the first radial excitation of the quark model Roberts et al. (2013, 2014).

With this new insight, the mystery of the low-lying Roper resonance may be nearing resolution. Evidence indicates the observed nucleon resonance at 1440 MeV is best described as the result of strong rescattering between coupled meson-baryon channels.

In working towards a definitive analysis there is ample scope for new data to further resolve the nature of this state. Better and more comprehensive experimental data on the channels coupled to the Roper resonance would be of significant interest. Further development of three-body channel contributions Hansen and Sharpe (2014, 2015) in effective field theory is desired. Similarly, a more comprehensive lattice QCD analysis of the complete nucleon spectrum in several lattice volumes would serve well to further expose the role of the coupled channels giving rise to the Roper resonance.

###### Acknowledgements.
Acknowledgements: We thank the PACS-CS Collaboration for making their flavor configurations available and the ongoing support of the ILDG. This research was undertaken with the assistance of the University of Adelaide’s Phoenix cluster and resources at the NCI National Facility in Canberra, Australia. NCI resources were provided through the National Computational Merit Allocation Scheme, supported by the Australian Government and the University of Adelaide Partner Share. This research is supported by the Australian Research Council through the ARC Centre of Excellence for Particle Physics at the Terascale (CE110001104) and through Grants No. DP151103101 (A.W.T.), DP150103164, DP120104627 and LE120100181 (D.B.L.).

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