Excited state baryon spectroscopy from lattice QCD

Excited state baryon spectroscopy from lattice QCD

Robert G. Edwards edwards@jlab.org Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA    Jozef J. Dudek dudek@jlab.org Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA Department of Physics, Old Dominion University, Norfolk, VA 23529, USA    David G. Richards dgr@jlab.org Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA    Stephen J. Wallace stevewal@umd.edu Department of Physics, University of Maryland, College Park, MD 20742, USA
June 6, 2011
Abstract

We present a calculation of the Nucleon and Delta excited state spectrum on dynamical anisotropic clover lattices. A method for operator construction is introduced that allows for the reliable identification of the continuum spins of baryon states, overcoming the reduced symmetry of the cubic lattice. Using this method, we are able to determine a spectrum of single-particle states for spins up to and including , of both parities, the first time this has been achieved in a lattice calculation. We find a spectrum of states identifiable as admixtures of representations and a counting of levels that is consistent with the non-relativistic constituent quark model. This dense spectrum is incompatible with quark-diquark model solutions to the “missing resonance problem” and shows no signs of parity doubling of states.

for the Hadron Spectrum Collaboration

I Introduction

Explaining the excitation spectrum of baryons is core to our understanding of QCD in the low-energy regime, and if we truly understand QCD in the strong-coupling regime, we should be able to confront experimental spectroscopic data with first-principles calculations within QCD. The experimental investigation of the excited baryon spectrum has been a long-standing element of the hadronic-physics program. An important goal has been the search for so-called “missing resonances”, baryonic states predicted by the quark model based on three constituent quarks but which have not yet been observed experimentally; should such states not be found, it may indicate that the baryon spectrum can be modeled with fewer effective degrees of freedom, such as in quark-diquark models. In the past decade, there has been an extensive program to collect data on electromagnetic production of one and two mesons at Jefferson Lab, MIT-Bates, LEGS, MAMI, ELSA, and GRAAL. To analyse these data, and thereby refine our knowledge of the baryon spectrum, a variety of physics analysis models have been developed at Bonn, George Washington University, Jefferson Laboratory and Mainz.

The experimental efforts outlined above should be complemented by high-quality ab initio computations within lattice QCD. Historically, the calculation of the masses of the lowest-lying states, for both baryons and mesons, has been a benchmark calculation of this discretized, finite-volume computational approach, where the aim is well-understood control over the various systematic errors that enter into a calculation; for a recent review, see Hoelbling (2010). However, there is now increasing effort aimed at calculating the excited states of the theory, with several groups presenting investigations of the low-lying excited baryon spectrum, using a variety of discretizations, numbers of quark flavors, interpolating operators, and fitting methodologies Mahbub et al. (2010a, b); Engel et al. (2010); Mathur et al. (2005). Some aspects of these calculations remain unresolved and are the subject of intense effort, notably the ordering of the Roper resonance in the low-lying Nucleon spectrum.

A basis of baryon operators for states at rest, respecting the (cubic) symmetry of the lattice, was developed in Refs. Basak et al. (2005a, b), and subsequently used in calculations of the excited state Nucleon spectrum in both quenched QCDBasak et al. (2007), and with two dynamical quark flavorsBulava et al. (2009). In parallel, we studied Clover fermions on anisotropic latticesEdwards et al. (2008); Lin et al. (2009), with a finer temporal than spatial resolution, enabling the hadron correlation functions to be observed at short temporal distances and hence many energy levels to be extracted. Crucial to our determination of the spectrum has been the use of the variational method Michael (1985); Luscher and Wolff (1990); Blossier et al. (2009) with a large number of interpolating operators at both the source and the sink; we developed and used the “distillation” method, enabling the necessary correlation functions to be computed in an efficient manner. A recent calculation of the Nucleon, and excited-state spectrum demonstrated the efficacy of the methodBulava et al. (2010).

In this paper, we expand the above program of computations considerably, extending to baryons the spin-identification techniques developed for mesons in Refs. Dudek et al. (2009, 2010). We develop a new basis of interpolating operators with good total angular momentum, , in the continuum, which are then subduced to the various lattice irreducible representations (irreps). We find that the subduced operators retain a memory of their continuum antecedents to a remarkable degree. For example, hadron correlation functions between operators subduced from different continuum spins are suppressed relative to those subduced from the same , illustrating an approximate realization of rotational symmetry at the scale of hadrons. This allows us to determine reliably the spins of most of the (single-particle) states, which helps to delineate between the nearly degenerate energy levels we observe in the spectrum. We are thereby able to determine the highly excited spectrum, including spins up to , and to resolve the masses of the low-lying states with a statistical precision at or below 1%.

The remainder of the paper is organized as follows. In Section II we describe the parameters of the lattice gauge fields used in our calculation. The “distillation” method, and its application to the construction of baryon correlation functions is outlined in Section III. Section IV describes the procedure for constructing baryon interpolating operators with good continuum spin. Angular dependence transforming like orbital angular momenta is incorporated through covariant derivatives that transform as representations with and ; a detailed construction is provided in Appendix A. We also develop subduction matrices that allow the continuum operators to be subduced to irreducible representations of the cubic group; a derivation for half-integer spins up to is given in Appendix B, where tables of the subduction matrices are also given. Our implementation of the variational method is presented in Section V, and our procedure for determining the spins of the resulting lattice states is described in Section VI, which also shows tests of the approximate rotational invariance in the spectra. The stability of the resultant spectra with respect to changes in the analysis method is discussed in Section VII. We present our results in Section VIII, beginning with a detailed discussion of the spectrum at the heaviest of our quark masses before examining the quark mass dependence. Discussion of multi-particle states is given in Section IX. We summarize our findings and provide our onclusions in Section X.

Ii Gauge fields

volume
524 1.15 500 7 56
444 1.29 601 5 56
396 1.39 479 8 56
Table 1: The lattice data sets and propagators used in this paper. The light and strange quark mass as well as the baryon mass in temporal lattice units are shown. The pion mass, lattice size and number of configurations are listed, as well as the number of time-sources and the number of distillation vectors .

A major challenge in the extraction of the spectrum of excited states from a lattice calculation is that the correlation functions, or more specifically the principal correlators of the variational method that correspond to excited states, decay increasingly rapidly with Euclidean time as the energy of the state increases, whilst the noise behaves in the same manner as in the ground state. Hence the signal-to-noise ratio for excited-state correlators exhibits increasingly rapid degradation with Euclidean time with increasing energy. To ameliorate this problem we have adopted a dynamical anisotropic lattice formulation whereby the temporal extent is discretized with a finer lattice spacing than in the spatial directions; this approach avoids the computational cost that would come from reducing the spacing in all directions, and is core to our excited-state spectroscopy program. Improved gauge and fermion actions are used, with two mass-degenerate light dynamical quarks and one strange dynamical quark, of masses and respectively. Details of the formulation of the actions as well as the techniques used to determine the anisotropy parameters can be found in Refs. Edwards et al. (2008); Lin et al. (2009).

The lattices have a spatial lattice spacing with a temporal lattice spacing approximately times smaller, corresponding to a temporal scale GeV. In this work, results are presented for the light quark baryon spectrum at quark mass parameters and , and lattice sizes of with spatial extent fm. The bare strange quark mass is held fixed to its tuned value of ; some details of the lattices are provided in Table 1. The lattice scale, as quoted above, is determined by an extrapolation to the physical quark mass limit using the baryon mass (denoted by ). To facilitate comparisons of the spectrum at different quark masses, the ratio of hadron masses with the baryon mass obtained on the same ensemble is used to remove the explicit scale dependence, following Ref. Lin et al. (2009).

Iii Correlator construction using distillation

The determination of the excited baryon spectrum proceeds through the calculation of matrices of correlation functions between baryon creation and annihilation operators at time and respectively:

Inserting a complete set of eigenstates of the Hamiltonian, we have

(1)

where the sum is over all states that have the same quantum numbers as the interpolating operators . Note that in a finite volume, this yields a discrete set of energies.

Smearing is a well-established means of suppressing the short-distance modes that do not contribute significantly to the low-energy part of the spectrum, and in turn, allows for the construction of operators that couple predominantly to the low-lying states. A widely adopted version is Jacobi smearing, which uses the three-dimensional Laplacian,

where the gauge fields, may be constructed from an appropriate covariant gauge-field-smearing algorithm Morningstar and Peardon (2004). From this a simple smearing operator,

is subsequently applied to the quark fields ; for large , this approaches a Gaussian form . “Distillation”Peardon et al. (2009), the method we adopt, replaces the smearing function by an outer product over the low-lying eigenmodes of the Laplacian,

(2)

where the is the eigenvector of , ordered by the magnitude of the eigenvalue; the (volume-dependent) number of modes should be sufficient to sample the required angular structure at the hadronic scalePeardon et al. (2009); Dudek et al. (2010), but is small compared to the number of sites on a time slice. Thus distillation is a highly efficient way of computing hadron correlation functions.

To illustrate how distillation is applied to the construction of the baryon correlators, we specialize to the case of a positively charged isospin- baryon. A generic annihilation operator can be written

(3)

where and are and quark fields respectively, is a spatial operator, including possible displacements, acting on quark , are color indices, and are spin indices; encodes the spin structure of the operator, and is constructed so that the operator has the desired quantum numbers, as discussed in the next section. We now construct a baryon correlation function as

where the bar over and indicate these belong to the creation operator. Inserting the outer-product decomposition of from Eq. 2, we can express the correlation function as

(4)

where

encodes the choice of operator and

is the operator-independent “perambulator”, with the eigenvector indices, and the usual discretized Dirac operator. The perambulators play the role of the quark propagators between smeared sources and sinks. Once the perambulators have been computed, the factorization exhibited in Eq. (4) enables the correlators to be computed between any operators expressed through , including those with displaced quark fields, at both the source and the sink. This feature will be key to our use of the variational method and the subsequent extraction of a spin-identified baryon spectrum.

Iv Construction of baryon operators

The construction of a comprehensive basis of baryon interpolation operators is critical to the successful application of the variational method. The lowest-lying states in the spectrum can be captured with color-singlet local baryon operators, of the form given in Eq. (3)111With, in this case, being just ordinary Dirac gamma matrices.. Combinations of such fields can be formed so as to have definite quantum numbers, including definite symmetry properties on the cubic lattice. The introduction of a rotationally symmetric smearing operation, be it Jacobi smearing or distillation, in which the quark fields are replaced by quasi-local, smeared fields , does not change the symmetry properties of the interpolating operators. However, such local and quasi-local operators can only provide access to states with spins up to . In order to access higher spins in the spectrum, and to have a sufficient basis of operators to effectively apply the variational method, it is necessary to employ non-local operators.

A basis of baryon interpolating operators incorporating angular structure, respecting the cubic symmetry of the lattice, and able to access higher spins in the spectrum, was constructed in refs. Basak et al. (2005b, a), and these operators were employed in our earlier determinations of the spectrum. The identification of the continuum spins corresponding to the calculated energy levels remained challenging, however. To overcome this challenge, we adopt a different procedure for operator construction: we first derive a basis of operators in the continuum, with well-defined continuum spin quantum numbers, and then form the subduction of those operators into the irreducible representations of the octahedral group of the lattice.

iv.1 Continuum baryon interpolating operators

Baryons are color-singlet objects, and thus they involve totally anti-symmetric combinations of the color indices of the three valence quarks. Furthermore baryon interpolating operators have to be anti-symmetric under the exchange of any pair of quarks, which is automatically satisfied since they are constructed from anti-commuting Grassmann fields. Thus the remaining quark labels, namely those of flavor, spin and spatial structure, have to be in totally symmetric combinations.

We will construct our baryon interpolating operators from products of three quark fields. Before proceeding to classify our operators, we note that three objects can exist in four definite symmetry combinations: symmetric (), mixed-symmetric (), mixed antisymmetric () and totally antisymmetric ()222Mixed symmetry combinations are of definite symmetry under exchange of the first two objects.; projection operators that generate these combinations are specified in the Appendix, in Eqs. (18) and (19). We will write our baryon interpolating operators by applying projection operators that act on the flavor, spin and spatial labels of a generic three-quark operator, :

where are flavor, Dirac spin and spatial projection operators, respectively, and the subscripts and specify the symmetry combinations of flavor, Dirac spin and spatial labels. For each operator , we must combine the symmetry projection operators such that the resulting baryon operator is overall symmetric. The rules for combining symmetries of such direct products are given in Eq. (LABEL:eq:combine).

To illustrate the construction, we specialize to the case of local or quasi-local operators, where the spatial dependence is the same for each quark, and thereby symmetric. Thus we write this simplified interpolating operator as

Furthermore, we will only consider the case of two-component Pauli spin rather than four-component Dirac spin. In the standard convention, Pauli spin involves a label that can take two values: and . It is straightforward to extend the construction to Dirac spins using the Dirac-Pauli representation of Dirac matrices, as in Refs. Basak et al. (2005b, a). Dirac spins involve direct products of two Pauli spin representations: one for -spin and the other for -spin (intrinsic parity). The operator construction for Dirac spin is described in Appendix A.

The product rules of Eq. (LABEL:eq:combine) specify three ways to combine these flavor and spin projectors to yield an overall symmetric projector:

(5)
(6)
(7)

The four symmetric spin combinations simply correspond to the four states or operators of spin : , while the two mixed symmetric and two mixed antisymmetric combinations each correspond to states or operators of spin ; there is no antisymmetric combination of three objects taking only two values.

For the case of flavor, the flavor-symmetric combination yields the decuplet (). Hence Eq. (5), with , specifies the operators for the spin- . The mixed-symmetric combinations specify the octet (), so that correspond to the Nucleon. Thus we see that Eq. (7) specifies the operators for the spin- octet. Since there is no spin combination, this example cannot provide flavor-singlet () interpolating operators - angular structure through non-local behavior is required.

Covariant derivatives defined as in Ref. Basak et al. (2005a) are incorporated into the three-quark operators in order to obtain suitable representations that transform like orbital angular momentum. This is necessary in order to obtain operators with total angular momentum . First the covariant derivatives are combined in definite symmetries with respect to their action on the three quark fields. A single derivative is constructed from . There are two relevant symmetry combinations,

(8)

where the notation means that the derivative acts on the -th quark. There is no totally antisymmetric construction of one derivative and the symmetric combination is a total derivative that gives zero when applied to a baryon with zero momentum - it is omitted. Each derivative additionally has a direction index (suppressed above) for which we choose a circular basis such that they transform under rotations like components of a spin-1 object:

Totally symmetric baryon operators with one derivative are constructed by applying Eq. (8) to the mixed symmetry spin-flavor operators according to the rules given in Eq. (LABEL:eq:combine),

where superscripts in brackets indicate the number of derivatives in the operator.

Two derivative operators can be formed in definite three-quark symmetry combinations that transform like as follows,

(9)
(10)
(11)
(12)

The projection into comes from combining a pair of derivatives via their (suppressed) direction indices using an Clebsch-Gordan coefficient, i.e. as .

Although it is allowed, we omit the combination that couples to . It corresponds to the spatial Laplacian and vanishes at zero momentum. Several possibilities occur when totally symmetric baryon operators are formed using the rules of Eq. (LABEL:eq:combine) to combine the spatial derivatives and spins,

where no total derivatives are formed from these constructions. The angular momenta of spinors and derivatives are combined using the standard Clebsch-Gordan formula of in order to obtain operators with good in the continuum,

irreps,
Table 2: Continuum spins subduced into lattice irreps . There are two embeddings of in .
Singlet, 13 22 9
Octet, 28 48 20
Decuplet, 15 26 11
0 28 48 20
0 15 26 11
0 41 60 29
-1 43 74 31
-2 43 74 31
-3 15 26 11
Table 3: (left) Number of distinct operators, categorized by in each lattice irrep , using operators with up to two derivatives. (right) Number of distinct operators, categorized by isospin (I) and strangeness (S) in each lattice irrep , using operators with up to two derivatives. Each operator has “rows”. There are equal numbers of operators in positive and negative parity.

The scheme outlined in this section provides a classification of all baryon operators constructed from three quarks with either light or strange masses, up to two derivatives in their spatial structure and respecting classical continuum symmetries.

A naming convention for such operators is useful. A Nucleon operator, with spin and parity of the three quarks equal to , two derivatives coupled into and total spin and parity , is denoted as

(13)

where subscripts show that the flavor construction is mixed symmetry (), the spin construction also is mixed symmetry and the two derivatives are in a symmetric combination () as in Eq. (9). Direct products of these flavor, spin and space constructions yield an overall symmetric set of flavor, spin and spatial labels, as required.

A spin state like can be constructed in several ways because Dirac spinors are used and they have four components, two upper and two lower components. A superscript attached to the spin part indicates that the operator is the first of several embeddings with the same quantum numbers. Because of this, the basis set presented here is over-complete since operators featuring both derivatives and lower components are specified. However, this redundancy is intentional as we need a basis with multiple operators in each irrep for use with the variational method. If only Pauli spinors were used, we would have an classification of operators. That subset of our operators is referred to as “non-relativistic”.

We reiterate that the operator construction described here provides a basis set for spectrum calculations. We are not forcing the symmetries manifested in the operator basis into the spectrum results to be shown later. Rather, the dynamics of QCD will decide on the eigenstates, which can correspond to any linear superpositions of the operator basis which may exhibit quite different symmetries to those diagonalised by this basis.

iv.2 Subduction

In lattice QCD calculations the theory is discretized on a four-dimensional Euclidean grid. The full three-dimensional rotational symmetry that classifies energy eigenstates in the continuum is reduced to the symmetry group of a cube (the cubic symmetry group, or equivalently the octahedral group). Instead of the infinite number of irreducible representations labelled by spin and parity, , the double-cover cubic group for half-integer spin has only six irreducible representations (irreps): , , , , , , where parity is denoted by the (gerade) subscript for positive parity or the (ungerade) subscript for negative parity. The distribution of the various components of a spin- baryon into the lattice irreps is known as subduction, the result of which is shown in Table 2.

Extending the analysis of Refs. Dudek et al. (2009, 2010) to baryons, we use “subduction” matrices for half-integer spins to project the continuum-based operators to irreducible representations of the octahedral group,

(14)

where is a lattice operator constructed as outlined in Appendix (A.3). As seen in Table 2, subduction of yields two occurrences of the irrep. Multiple occurrences of lattice irreps are the general rule for spins higher than . We denote by the occurrence of irrep in the subduction of spin . For example, the two occurrences of octahedral irrep in the subduction of are denoted as and . For each there is a subduction matrix in the values of and the rows of the irrep, , that maps the continuum spin operators to irreducible representations of the octahedral group. These subduction matrices are derived for half-integer spins up to in Appendix B and the resulting matrices are tabulated in Appendix B.

Classifying operators by symmetry, we have the total number of operators in each lattice irrep as shown in Table 3(left). Considered as a broken symmetry with an unbroken isospin symmetry, we have the number of operators shown in Table 3(right). The operator basis used in this work is constructed using flavor symmetry. Of course, the symmetry is broken in QCD, most notably by the mass of the strange quark. The operator basis allows the flavor symmetry breaking to be determined by the relative degree of overlap onto operators transforming in different representations of . This is comparable to what is done in quark models and other approaches, which helps to relate the lattice results to phenomenology.

With the construction of operators and their subduction to irreps of the cubic group outlined as above, we defer further details to the appendices. Appendix A provides details of the operator constructions, Appendix B derives the subduction matrices through an analysis of the quantum mechanics of continuum spin in the octahedral representation, and Appendix B provides the subduction matrices for half-integer spins.

V Correlator analysis

Figure 1: Principal correlator fits according to Eq. (17). For ten states of the Nucleon irrep for = 524 MeV, the plots show data and the fit for . Data used in the fit are shown in black, while points excluded from the fit are in grey.

The variational method for spectral extraction Michael (1985); Luscher and Wolff (1990); Blossier et al. (2009), which takes advantage of the multiplicity of operators within a given symmetry channel to find the best (in a variational sense) linear combination of operators for each state in the spectrum, is now in common usageBurch et al. (2009); Gattringer et al. (2008); Burch et al. (2006, 2004); Bulava et al. (2010). Our application of the method follows that developed in Refs. Dudek et al. (2008, 2009, 2010), and applied to the analysis of the excited meson spectrum, and we summarize it here.

The starting point is the system of generalized eigenvalue equations for the correlation matrix:

(15)

where , and where there is an orthogonality condition on the eigenvectors of different states (), . This orthogonality condition provides eigenvectors that distinguish clearly between nearly degenerate states, which would be difficult to distinguish by their time dependence alone. Equation (15) is solved for eigenvalues, , and eigenvectors, , independently on each timeslice, . Rather than ordering the states by the size of their eigenvalue, which can become uncertain because of the high level of degeneracy in the baryon spectrum, we associate states at different timeslices using the similarity of their eigenvectors. We choose a reference timeslice on which reference eigenvectors are defined, , and compare eigenvectors on other timeslices by finding the maximum value of which associates a state with a reference state . Using this procedure we observe essentially no “flipping” between states in either the principal correlators, , or the eigenvectors, , as functions of .

Any two-point correlation function on a finite spatial lattice can be expressed as a spectral decomposition

(16)

where we assume that , the temporal length of the box, so that the opposite-parity contributions arising from the other time ordering on the periodic lattice can be ignored. The “overlap factors”, are related to the eigenvectors by .

We obtain the masses from fitting the principal correlators, , which for large times should tend to . In practice we allow a second exponential in the fit form, and our fit function is

(17)

where the fit parameters are and . Typical fits for a set of excited states within an irrep are shown in Figure 1 where we plot the principal correlator with the dominant time-dependence due to state divided out. If a single exponential were to dominate the fit, such a plot would show a constant value of unity for all times. For the form of Eq. 17, the data would approach a constant at large times, and this is clearly satisfied for .

Empirically we find that the size of the second exponential term decreases rapidly as one increases . Further we find, in agreement with the perturbative analysis of Ref. Blossier et al. (2009) and with our earlier meson analysis, that for large values the extracted are larger than the value of , the largest “first” exponential mass extracted. At smaller values this is not necessarily true and is indicative of an insufficient number of states in Eq. (16) to describe completely. The values of and are not used elsewhere in the analysis.

Our choice of is made using the “reconstruction” schemeDudek et al. (2008, 2010): the masses, , extracted from the fits to the principal correlators, and the extracted from the eigenvectors at a single time slice are used to reconstruct the correlator matrix using Eq. (16). This reconstructed matrix is then compared with the data for , with the degree of agreement indicating the acceptability of the spectral reconstruction. Adopting too small a value of leads to a poor reconstruction of the data for . In general, the reproduction is better as is increased until increased statistical noise prevents further improvement. The sensitivity of extracted spectral quantities to the value of used will be discussed in detail in section VII.1, but in short the energies of low-lying states are rather insensitive to and the reconstruction of the full correlator matrix usually is best when , but not too large.

From the spectral decomposition of the correlator, equation 16, it is clear that there should in fact be no time dependence in the eigenvectors. Because of states at higher energies than can be resolved with operators, there generally is a contribution to energies and ’s that decays more rapidly than the lowest mass state that contributes to a principal correlator. As for energies, we obtain “time-independent” overlap factors, from fits of , obtained from the eigenvectors, with a constant or a constant plus an exponential, in the spirit of the perturbative corrections outlined in Blossier et al. (2009)).

Vi Determining the spin of a state

vi.1 Motivation and procedure

Figure 2: The magnitude of matrix elements in a matrix of correlation functions, , at time-slice 5 is shown according to the scale at the lower right. The matrix is for the Nucleon irrep, with 28 operators, 16 operators and 4 operators.

A new method for identifying the spins of excited mesonic states was introduced in Refs. Dudek et al. (2009, 2010). In this work, we extend the method to identify spins of excited baryonic states. The explanation for the success of the new method is that there is an approximate realization of rotational invariance at the scale of hadrons in our correlation functions. There are two reasons for this claim. The first is that there are no dimension-five operators made of quark bilinears that respect the symmetries of lattice actions based on the Wilson formalism and that do not also transform trivially under the continuum group of spatial rotations. Thus, rotational symmetry breaking terms do not appear until in the action. This argument holds even though the action used in this work describes an anisotropic lattice. The second reason is that our baryon operators are constructed from low momentum filtered, smeared quark fields, where the smearing is designed to filter out fluctuations on small scales. Possible divergent mixing with lower dimensional operators is then expected to be suppressed. The baryon operators then are reasonably smooth on a size scale typical of baryons, which is fm. With the lattice spacing of fm, the breaking of rotational symmetry in the action or the baryon operators can be small: 0.015. Of course, these qualitative arguments should be backed up by evidence of approximate rotational invariance in explicit calculation.

The operators constructed in Section IV using subduction matrices transform irreducibly under the allowed cubic rotations, that is they faithfully respect the symmetries of the lattice. They also carry information about the continuum angular momentum, , from which they are subduced. To the extent that approximate rotational invariance is realized, we expect that an operator subduced from spin to overlap strongly only onto states of continuum spin , and have little overlap with states of different continuum spin. In fact this is clearly apparent even at the level of the correlator matrix as seen in Figure 2. Here the correlator matrix for the Nucleon is observed to be approximately block diagonal when the operators are ordered according to the spin from which they were subduced.

To identify the spin of a state, we use the operator “overlaps” for a given state extracted through the variational method presented in the previous section. In Figure 3 we show the overlaps for a set of low-lying states in the Nucleon and irreps of the calculation. The principal correlators for these states in are shown in Figure 1, and the corresponding mass spectrum is shown in Figure 9. The overlaps for a given state show a clear preference for overlap onto only operators of a single spin. While we show only a subset of the operators for clarity, the same pattern is observed for the full operator set.

Figure 3: Histograms of spectral overlaps, , are shown for a selection of eight operators (shown at the top using the naming convention of Eq. (13)). The 4 operators labelled with (color red) have subductions only to the irrep. The 3 operators labelled with (color green) and the 1 labelled with (color blue) have subductions to both and irreps. Each histogram is labelled by the value of mass of the state (in units of ) and has 8 vertical bars showing the relative values for each of the operators. The data are from the = 524 MeV ensemble and ’s are normalized so that the largest value across all states, for a given operator, is equal to . The lighter area at the head of each bar represents the one sigma statistical uncertainly. Note that for each state only one or two operators have a large relative value, and it is the same operators appearing across each of the irreps. Note also that nearly the same energies are obtained in and irreps for state subduced from one value.

The assignment of spin must hold for states with continuum spin subduced across multiple irreps. In the continuum our operators are of definite spin such that , and therefore from Eq. 14 the overlap of the subduced operator is . Only the spin states will contribute, and not any of the other spins present in the irrep . Inserting a complete set of hadronic states between the operators in the correlator and using the fact that the subduction coefficients form an orthogonal matrix (33), , we thus obtain terms in the correlator spectral decomposition proportional to for each we have subduced into, up to discretization uncertainties as described above. Hence, for example, a baryon created by a operator will have the same value in each of the irreps. This suggests that we compare the independently obtained -values in each irrep. In Figure 4 we show the extracted values for negative-parity states suspected of being spin across the and irreps, and of being spin across the , and irreps. As can be seen, there is good agreement of values in the different irreps that are subduced from spin , with only small deviations from exact equality.

Figure 4: Selected values across irreps are shown for states suspected of being (left panel) and (right panel), based on the = 524 MeV ensemble. The boxes at the top show the mass for the various states. There are two states of . The operators in the left panel, all projected onto , are left to right, , , , , , . Overlaps of these operators after subduction into and , agree well for each of the two states shown. The operators in the right panel, all projected onto , are left to right, , , , . Similarly, the operator overlaps for this state agree well across , , and .

These results demonstrate that the values of carefully constructed subduced operators can be used to identify the continuum spin of states extracted in explicit computation for the lattices and operators we have used.

We take the next step and use the identification of the components of the spin- baryon subduced across multiple irreps to make a best estimate of the mass of the state. The mass values determined from fits to principal correlators in each irrep differ slightly due to what we assumed to be discretization effects and, in principle, avoidable fitting variations (such as the fitting intervals). We follow Ref. Dudek et al. (2010) and perform a joint fit to the principal correlators with the mass being common. This method provides a numerical test that the state has been identified. We allow a differing second exponential in each principal correlator so that the fit parameters are and . These fits are typically very successful with correlated close to 1, suggesting again that the descretization effects are small. An example for the case of components identified in is shown in Figure 5. When we present our final, spin-assigned spectra it is the results of such fits that we show.

Figure 5: Fit to the three subduced principal correlators of lowest lying Nucleon using a common mass. Results are from the = 524 MeV ensemble. These levels correspond to the sixth excited , the seventh and the fourth . Plotted is . Grey points not included in the fit.

vi.2 Additional demonstration of approximate rotational invariance

Figure 6: Extracted Nucleon mass spectrum for various operator bases. (a) Full basis (dim = 48), (b) Only operators (dim = 28), (c) Only operators (dim = 16), (d) Only operators (dim = 4). Results are from the = 524 MeV ensemble. The dimensionality of the operator basis in each is shown in Table 10.

A further demonstration of approximate rotational invariance is based on the spectrum of energy levels. If indeed the mixing between states of different continuum spin is small, then the omission of such coupling should not much affect the excited state spectra. That proposition can be tested by extracting energies using all operators, and comparing them with the energies obtained from only operators subduced from a single value. If approximate rotational invariance were achieved in the spectrum, the energies would be nearly the same. As an example, we show results for the Nucleon irrep, in Fig. 6. The left column of the Fig. 6, labelled “all”, shows the lowest 12 energy levels obtained from matrices of correlation functions using the set of all 48 operators, and spin identified using the methods previously described. The states listed in Figure 3 correspond to a few of these 12 levels. The second column shows the lowest 6 levels resulting from the variational method when the operator basis is restricted to only those with continuum spin . Similarly, the third shows the lowest 4 levels when the operator basis is restricted to only those with continuum spin , and the last column are the lowest two levels when the basis is restricted to only operators.

The results are striking. We see that the masses of the levels in each of the restricted bases agree quite well with the results found in the full basis. The agreement is quite remarkable because one expects operators in the irrep that can couple with the ground state (the lowest state), will allow for the rapid decay of correlators down to the ground state as a function of time, . However, the higher-energy spin and states do not show such a decay; we obtain good plateaus in plots like those in Fig. 1. These results provide rather striking demonstration for the lack of significant rotational symmetry breaking in the spectrum.

Vii Stability of spectrum extraction

Figure 7: Extracted Nucleon mass spectrum as a function of . Horizontal bands to guide the eye. For clarity of presentation, only one of the highly excited and levels are shown.

In this section we consider to what extent the extracted spectrum changes as we vary details of the calculation, such as the metric timeslice, , used in the variational analysis, and the number of distillation vectors. We will use the Nucleon and the Delta irreps in the dataset to demonstrate our findings.

vii.1 Variational analysis and

Our fitting methodology was described in Section V where reconstruction of the correlator was used to guide us to an appropriate value of . As seen in Figure 7, for , the low-lying mass spectrum is quite stable with respect to variations of . This appears to be mostly due to the inclusion of a second exponential term in Eq. (17), which is able to absorb much of the effect of states outside the diagonalization space. The contribution of this second exponential typically falls rapidly with increasing both by having a smaller and a larger .

Overlaps, , can show more of a sensitivity to values being too low, as was found also in the analysis of mesons, Ref. Dudek et al. (2010).

In summary it appears that variational fitting is reliable provided is “large enough”. Using two-exponential fits in principal correlators we observe relatively small dependence of masses, but more significant dependence for the values which we require for spin-identification.

Figure 8: Extracted the Delta mass spectrum as a function of number of distillation vectors in the dataset. For , spin identification is lost for the highest and levels.

vii.2 Number of distillation vectors

The results presented so far are based on the analysis of correlators computed on lattices using 56 distillation vectors. We consider how the determination of the spectrum varies if one reduces the number of distillation vectors and thus reduces the computational cost of the calculation. This is particularly important given that, as shown in Peardon et al. (2009), to get the same smearing operator on larger volumes one must scale up the number of distillation vectors by a factor equal to the ratio of spatial volumes. To scale up to a lattice this would require vectors which is not currently a realizable number without using stochastic estimation Morningstar et al. (2011).

In figure 8 we show the low-lying part of the extracted Delta spectrum on the lattice as a function of the number of distillation vectors used in the correlator construction. It is clear that the spectrum is reasonably stable for but that the spectrum quality degrades rapidly for fewer vectors. In particular, the methods for spin identification fail for the highest state as well as the first state.

The need for a large numbers of distillation vectors has been discussed in Ref. Dudek et al. (2010) for the case of isovector mesons. The conclusion drawn is that to describe high spin hadrons having large orbital angular momentum, one needs to include sufficient vectors to sample the rapid angular dependence of the wavefunction over the typical size of a hadron. The results for baryons presented here are consistent with these observations.

In summary one is limited as to how few distillation vectors can be used if one requires reliable extraction of high-spin states. The results shown here suggest 32 distillation vectors on a lattice is the minimum, so at least 64 distillation vectors on a lattice are likely to be required.

Viii Results

Our results are obtained on lattices with pion masses between 396 and 524 MeV. More complete details of the number of configurations, time sources and distillation vectors used are given in Table 1. The full basis of operators in each irrep listed in Table 3 is used for the variational method construction.

viii.1 MeV results

Using the variational method outlined in Section V, the spectrum of energies in each lattice irrep for the Nucleon and for the Delta are shown in Figures 9 and 10, respectively. By applying the spin-identification procedure described above, we can subsequently associate continuum labels to each of these states, as labelled on the figures. For the remainder of this paper, we will therefore label the energies by their assigned continuum spins; these are shown for the lattices at in Figure 11, with the energies obtained from a joint fit to the principal correlators as described in Section V.

Figure 9: Extracted Nucleon spectra by irrep for . Colors are black (), red (), green (), blue (). Masses are shown in ratios of the baryon mass.
Figure 10: Extracted Delta spectra by irrep for . Colors are black (), red (), green (), blue (). Masses are shown in ratios of the baryon mass.
Figure 11: Spin-identified spectrum of Nucleons and Deltas from the lattices at , in units of the calculated mass.
Figure 12: Spin-identified spectrum of Nucleons and Deltas from the lattices at , in units of the calculated mass.

There are several notable features in these spectra. As we will discuss, the patterns of states have a good correspondence with single-hadron states as classified by symmetry. The numbers of low-lying states in each are similar to the numbers obtained in the non-relativistic quark model which is a particular realisation of the symmetry above (e.g., Isgur and Karl (1978, 1979)). For the purposes of these comparisons, it is helpful to introduce a spectroscopic notation: , where is the Nucleon or the Delta , is the Dirac spin, , , ,…denotes the combined angular momentum of the derivatives, , , or is the permutational symmetry of the derivative, and is the total angular momentum and parity. This notation also is used in Table 4, which we discuss now.

Nucleon ()
2
2
1
4
5
3
1
Delta ()
1
1
0
2
3
2
1
Table 4: Subset of the operator basis classified by multiplets and total spin and parity . The entries are the orbital angular momentum structures outlined in Section IV and Appendix A that contribute within each . The operators listed here are all from the first embedding in Dirac spin in Table 6, and correspond to only upper components (referred to as “non-relativistic”). A spectroscopic notation of is used, where or , is the Dirac spin, , , ,…is the combined angular momentum of the derivatives, , , or is the permutational symmetry of the derivatives, and is the total angular momentum and parity. In Sections IV-VI, an operator notation like was used which in this spectroscopic notation would be . Dimensions and parities of representations are listed in column 4 for the doublet spin states of column 2, and in column 5 for the quartet spin states of column 3. The number, , of operators for each is listed in the rightmost column. This same spectroscopic notation and classification of spatial structure is also used for comparisons with models where represents the orbital angular momentum.

In the negative parity spectrum, there is a pattern of five low-lying levels, consisting of two