Elastic Form Factors of Nucleon Excitations in Lattice QCD

The recent development of the parity-expanded variational analysis (PEVA) technique makes lattice QCD studies of the form factors of baryon excitations possible. In this paper, we present Sachs electromagnetic form factors for excitations of the proton and neutron. We study the two lowest lying odd-parity excitations and demonstrate that at heavier pion masses, these states are dominated by behaviour consistent with constituent quark models for the N(1535) and N(1650), respectively. At pion masses approaching the physical point, meson-baryon contributions are observed to play an increasing role. We also study the lowest-lying localised even-parity excitation, and find that its form factors are consistent with a radial excitation of the ground state nucleon. A comparison of the results from the PEVA analysis with those from a conventional variational analysis exposes the necessity of the PEVA approach in baryon excited-state studies.


I. INTRODUCTION
In Ref. [1], we presented a method for extracting the form factors of a baryonic state on the lattice using the PEVA technique, and established its effectiveness for accessing the structure of the ground-state nucleon. We now use this method (as summarised in Section II) to investigate the structure of the excitations of the proton and neutron observed in Lattice QCD.
On the lattice, we observe a tower of stable excitations of each ground state. These towers of stable finite-volume eigenstates are associated in a non-trivial manner with unstable finite-width resonances in nature. Understanding the structure of the states observed in Lattice QCD will enable predictions of the infinite-volume observables of nature via effective field theory techniques [2,3].
Investigating the structure of excited states in lattice QCD is recognised as an important frontier in the field. Progress has already been made in the meson sector, where the issues of parity mixing are not present [4,5].
Here we tackle the more challenging problem of calculating such quantities in the baryon sector.
While experimental measurements of resonance transition amplitudes have been made, it is much harder to measure elastic form factors in the resonance regime. This is because elastic form factors parameterise interactions where both the initial and final state are the same. To measure them for an (unstable) resonance, we need to first produce that resonance, and then probe it during the extremely short time window before it decays. On the other hand, the transition form factors parameterise the transformation of one state into another. We can probe a stable target such as a ground state proton and measure how it is excited into the unstable resonance of interest through an examination of its decay products.
It has been suggested that the magnetic dipole moment of the N * (1535) resonance could be measured through the γp → γηp process [6] using the Crystal Barrel/TAPS detector at ELSA or Crystal Ball @ MAMI, but this measurement has yet to be realised. The difficulty of measuring such quantities experimentally provides the opportunity for Lattice QCD to lead experiment and create new knowledge.
By using local three-quark operators on the lattice, both the CSSM [7,8] and the Hadron Spectrum Collaboration (HSC) [9,10] observe two low-lying odd-parity states in the resonance regimes of the N * (1535) and N * (1650). The local operators used only couple well to localised states. As a result, the states accessed by such analyses are expected to be either quark-model-like states dressed by a meson cloud, similar to the groundstate nucleon, or bound meson-baryon molecular states, such as the Λ(1405) [11]. Existing Hamiltonian effective field theory (HEFT) calculations [12] describe this energy spectrum as the finite volume spectrum of the N * (1535) resonance. However, given the presence of two resonances in this region, this is not necessarily the full picture, and a determination of the structure of these states on the lattice can give important insight. By comparing the structure of the observed lattice states to quark model predictions, we gain insight into the expected makeup of the states in HEFT. HEFT also holds the promise of linking our lattice results to experiment.
In this paper we present a determination of the Sachs electric and magnetic form factors for three localised spin-1 ⁄2 nucleon eigenstates on the lattice. Two of these states are negative-parity nucleon excitations, which we label N * 1 (or p * 1 for the proton excitation and n * 1 for the neutron excitation), and N * 2 (or the equivalent labels for the excited proton and neutron). The remaining eigenstate is a positive-parity excitation, and is denoted N ′ , p ′ , or n ′ . We compare the magnetic moments predicted from the negative-parity results to constituent quark model predictions for the magnetic moments of the N * (1535) and N * (1650) resonances [6,13]. Such quark model calculations can be extended to include effects from the pion cloud. We also compare our lattice results to two such extensions [13,14]. From these comparisons, we make connections to the basis states to be considered in HEFT [12].

II. PARITY EXPANDED VARIATIONAL ANALYSIS
The process of extracting finite-momentum matrix elements of baryonic excited states via the PEVA technique is presented in full in Ref [1]. We provide here a brief summary of this process to introduce the notation and concepts necessary to present our results.
We begin with a basis of n conventional spin-1 /2 operators {χ i (x)} that couple to the states of interest. Adopting the Pauli representation of the gamma matrices, we introduce the PEVA projector [15] Γ ±p ≡ 1 4 I + γ 4 I ± iγ 5 γ k p k , and construct a set of basis operators We note that we use a Euclidean metric δ µν , and hence there is no need to distinguish between contravariant and covariant indices.
We then seek an optimised set of operators φ α ±p (x) that each couple strongly to a single energy eigenstate α. These optimised operators are constructed as linear combinations of the basis operators. The optimum linear combinations are found by solving a generalised eigenvalue problem with G(p ; t + ∆t) and G(p ; t), where the correlation matrix with i and j ranging over both the primed and unprimed operators. This process is described in detail in Ref. [15]. Using the optimised operators, we can construct the eigenstate-projected two-point correlation function and the three point correlation functions where J (x) is some current operator, which is inserted with a momentum transfer q = p ′ − p. The consideration of G 3 − (J ; p ′ , p ; t 2 , t 1 ; α) (where the sink operator uses the opposite PEVA projector sign convention to the source operator) is required to optimise the extraction of the form factors for general kinematics. We note that it is sufficient to consider this change of projector for the sink operator alone, leaving the source operator as φ α +p (0) in all cases considered.
In this paper, we investigate the electromagnetic properties of the proton and neutron by choosing the current operator J (x) to be the vector current. In particular, we use the O(a)-improved [16] conserved vector current used in Ref. [17], where r is the Wilson parameter, and j µ C (x) is the standard conserved vector current for the Wilson action.
This choice of current operator gives the matrix element where 2 is the squared fourmomentum with the conventional sign, and the invariant scalar functions F 1 (Q 2 ) and F 2 (Q 2 ) are respectively the Dirac and Pauli form factors. Here u α (p, s) is the spinor for the lattice eigenstate α moving with momentum p and spin s. As α is an eigenstate of the lattice Hamiltonian, this spinor takes the form of a conventional single particle spinor with the centre-of-momentum energy, E CM = E α (p) 2 − p 2 playing the role of mass.
The states considered in this work display an energymomentum relation consistent with a single particle dispersion relation [15]. As such, E CM ≈ m α throughout this work, but the techniques presented work more generally for states where E CM = m α .
To extract our desired signal from this spinor structure, we can take the spinor trace with some spinstructure projector Γ S . This trace is then called the spinor-projected three-point correlation function These spinor-projected correlation functions have a nontrivial time dependence, which can be removed by constructing the ratio [18] where Γ 4 = (I + γ 4 )/2 and Γ k = (I + γ 4 )(i γ 5 γ k )/2 form the basis for the spin projectors we use, and r µ and s µ are coefficients selected to determine the form factors. We can then define the reduced ratio, By investigating the r µ and s σ dependence of this ratio, we find that the clearest signals are given by where s is chosen such that p · s = 0 = p ′ · s , r is equal to q × s , and the sign ± in Eq. (10) is chosen such that 1 ± p · p ′ is maximised. This choice maximises the signal in the lattice determination of the correlation function ratios.
We can then find the Sachs electric and magnetic form factors, through appropriate linear combinations of R T ± and R S ∓ . We have shown how the PEVA technique can be applied to the calculation of elastic baryon form factors for arbitrary kinematics. We now proceed to investigate the Sachs electric and magnetic form factors of several excitations of the nucleon.

III. NEGATIVE PARITY EXCITATIONS
The results presented in this paper are calculated on the PACS-CS (2 + 1)-flavour full-QCD ensembles [19], made available through the ILDG [20]. These ensembles use a 32 3 × 64 lattice, and employ a renormalisationgroup improved Iwasaki gauge action with β = 1.90 and non-perturbatively O(a)-improved Wilson quarks, with C SW = 1.715. We use five ensembles, with stated pion masses from m π = 702 MeV to 156 MeV [19], and set the scale using the Sommer parameter with r 0 = 0.4921(64) fm [19]. More details of the individual ensembles are presented in Table I, including the squared pion masses in the Sommer scale. When fitting correlators, the χ 2 /dof is calculated with the full covariance matrix, and the χ 2 values of all fits are consistent with an appropriate χ 2 distribution.
The three heaviest pion masses available among these ensembles span m π = 411 MeV-702 MeV, a typical range for contemporary studies of baryon excitations. As such, these masses are appropriate for this world-first study of the electromagnetic structure of nucleon excitations. In presenting our discoveries, we will focus on the results at the lightest of these three masses, m π = 411 MeV. There are two even lighter masses, at m π = 156 MeV and 296 MeV. These approach the physical point, presenting a significant challenge in terms of gauge noise and computational cost, but offer the possibility of insight into important chiral physics.
For the variational analyses in this paper, we begin with the same eight-interpolator basis as in Ref. [1], in which we studied the electromagnetic form factors of the ground-state nucleon. This basis is formed from the conventional spin-1 ⁄2 nucleon interpolators with 16, 35, 100, or 200 sweeps of gauge-invariant Gaussian smearing [21] with a smearing fraction of α = 0.7, applied at the quark source and sinks in creating the propagators. For the PEVA analyses, this basis is expanded to sixteen operators as described in Section II. Before performing the Gaussian smearing, the gauge links to be used are smoothed by applying four sweeps of three-dimensional isotropic stout-link smearing [22] with ρ = 0.1. We study the first three excitations extracted by this basis, consisting of one positive-parity state and two negative-parity states. As we will see in the results presented below, the PEVA technique is very important in correctly extracting form factors of these excitations.
To extract the form factors, we fix the source at time slice N t /4 = 16 relative to a fixed boundary condition in time, and (utilising the sequential source technique [23]) invert through the current, fixing the current insertion at time slice 21. We choose time slice 21 by inspecting the two point correlation functions associated with each state and observing that excited-state contaminations in the eigenstate-projected correlators are suppressed by time Table I. Details of the gauge field ensembles used in this analysis. For each ensemble we list both the pion mass given in Ref. [19], with the lattice spacing set by hadronic inputs, and our determination of the the squared pion mass with the lattice spacing listed in the table, which is set by the Sommer parameter with r0 = 0.4921(64) fm [19]. slice 21. This is evaluated by fitting the effective mass in this region to a single state ansatz verifying that the full covariant χ 2 /dof is satisfactory. We then extract the form factors as outlined in Section II for every possible sink time and once again look for a plateau consistent with a single-state ansatz.

PACS-CS
A particular concern in this analysis is the possibility of contamination by nearby multi-particle scattering states that have not been isolated by the current correlationmatrix analysis using local operators [7,8,24]. However, HEFT calculations indicate that the two odd-parity states observed on the lattice are the lowest states in the spectrum for the heaviest two quark masses considered, and scattering state contaminations are the order of 10 % or less for the three quark masses holding the focus of this investigation. In the following analysis, we demand that our two-and three-point correlators are consistent with a single-state ansatz within the Euclidean time regions considered. As one proceeds to precision calculations of these form factors, it will be essential to examine the scattering-state contributions to the form factors in detail.

A. GE for the first negative-parity excitation
Beginning with the lowest-lying negative-parity excitation observed in this study, we study how extractions of G E (Q 2 ) by both PEVA and a conventional analysis depend on the Euclidean time of the sink. In Fig. 1, we plot the connected contributions to G E (Q 2 ) from single quarks of unit charge for both quark flavours present in the nucleon interpolator. This plot is at m π = 702 MeV with the lowest-momentum kinematics of p = (0, 0, 0) and p ′ = (1, 0, 0). We see that the conventional extraction sits well below the PEVA extraction for all time slices between the current insertion and the point at which the signal is lost to noise. The conventional extraction also has a more significant time dependence than the PEVA extraction, forcing the conventional fit one time slice later. Both of these effects indicate that the conventional analysis is affected by opposite-parity contaminations, which are having a significant effect on the extracted form factor, introducing a systematic error of 12(4) % for the singly  represented quark flavour and 25(5) % for the doubly represented flavour. The lighter pion masses show a similar behaviour. The conventional analysis consistently has a plateau which starts later than the PEVA analysis and sits significantly lower. For example, at m π = 411 MeV, the magnitudes of the conventional plateaus with these low-momentum kinematics are systematically underestimated by 19(6) % and 26(6) % for the singly and doubly represented quark flavours respectively.
We can also consider changing the momenta to access different kinematics. By boosting the initial and final states while keeping the momentum transfer constant, we can access smaller values of Q 2 . We can also increase the three-momentum of the current insertion, giving access to larger values of Q 2 . For such kinematics at all masses we find that in general, the conventional plateaus are later in time and take smaller values than the PEVA plateaus.
These results indicate that the PEVA technique is critical to the correct extraction of the electric form factors of this nucleon excitation. The conventional analysis is contaminated by opposite-parity states, and when these states are removed by the PEVA technique it has a significant effect on the extracted form factor values. Hence, we now focus our attention only on the PEVA results for the remainder of this subsection.
In Fig. 2, we plot the Q 2 dependence of the electric form factor at m π = 411 MeV. The set of kinematics used to access the various Q 2 values is listed in Table II, and we exclude any fits for which there is no acceptable plateau, or the variational analysis fails. We see that the two quark flavours have very similar contributions to the electric form factor. They both agree well with a dipole ansatz with G 0 fixed to one, as we are working with single quarks of unit charge. These fits correspond to a RMS charge radius of 0.654 (20) fm for the doubly represented quark flavour and 0.670(26) fm for the singly represented quark flavour. These charge radii are similar to the charge radii of the individual quark sectors in the ground state examined in Ref. [1], (0.662 (12) fm for the doubly represented quark flavour and 0.633 (12) fm for the singly represented quark flavour). The doubly represented quark sector agrees to within one standard deviation. However, the singly represented quark sector in the excitation has a charge radius approximately 1.5 standard deviations larger than the ground state. We see similar behaviour for the other four masses. In all cases, the quark distributions are much smaller than the lattice length L ∼ 3 fm. The plots for these masses are omitted from this paper for the sake of brevity.
In order to compute the form factors of the first negative-parity excitation of the proton, G p * 1 E (Q 2 ), and neutron, G n * 1 E (Q 2 ), we need to take the correct linear combinations of the contributions from the doubly represented quark flavour and the singly represented quark flavour to reintroduce the multiplicity of the doubly represented quark and the physical charges of the up and down quarks. To this end we define In Fig. 3, we plot the nucleon electric form factors obtained by taking these combinations of the form factors at m π = 411 MeV. The form factor for the neutron excitation is close to zero, reflecting the similar charge radii of the individual quark flavours. By combining the dipole fits to the individual quark sectors in the same way as the data points one obtains a model for the Q 2 dependence of the electric form factors of the excited proton and neutron that includes full information from both quark sectors. If we do this for all five pion masses, we extract squared charge radii for the proton excitation ranging from 0.340(29) fm 2 to 0.470(54) fm 2 , increasing with decreasing pion mass. For the neutron excitation, the squared charge radii are close to or slightly below zero, Table II. Different kinematics used in our analysis to access a range of Q 2 values. The Q 2 value listed is for the first negativeparity nucleon excitation at the middle pion mass of mπ = 411 MeV. The statistical error listed for Q 2 comes from both the determination of the mass of the state and the conversion to physical units. In the so-called Breit frame kinematics, where the incoming and outgoing energies are equal, the correlated statistical errors from the mass cancel exactly, and as such the only source of errors is uncertainty in the lattice spacing used in converting to physical units. for example −0.033(24) fm 2 at m π = 296 MeV. As can be seen in Fig. 4, for each pion mass considered, the extracted squared charge radii for this first negative-parity excitation are consistent with the radii of the ground-state proton and neutron at the same mass, as obtained in Ref. [1]. This suggests that the size of this finite-volume state is similar to that of the ground-state nucleon. We see that the pion-mass dependence is fairly smooth, and has a clear trend to larger radii at lower pion masses. There is no hint of significant non-analytic behaviour in the quark-mass dependence, due to finitevolume suppression [25,26].

B. GM for first negative-parity excitation
We now proceed to the magnetic form factor. In Fig. 5, we plot the plateaus at m π = 411 MeV with the lowestmomentum kinematics. Here we present results in terms of nuclear magnetons, µ N ≡ e 2m phys p , defined in terms of the physical proton mass, m phys p . While the conventional and PEVA plateaus for the doubly represented quark flavour are consistent, both in fit region and value, the conventional plateau for the singly represented quark flavour starts later and has a significantly more negative value than the PEVA plateau. We see a similar effect at all five pion masses and a variety of kinematics.
Having fit the form factor plateaus, we can investigate the Q 2 dependence of G M (Q 2 ). In Fig. 6, we plot the contributions to G M (Q 2 ) from both the singly represented quark flavour and the doubly represented quark flavour at m π = 411 MeV. Both quark flavours are consistent with a dipole fit. The Q 2 dependence is similar to that for G E (Q 2 ), and the same is true for the other pion masses considered.
As described in Section III A, we can take linear combinations of the individual quark flavour contributions to compute the magnetic form factors of the excited proton and neutron. We plot these combinations for m π = 411 MeV in Fig. 7. The squared magnetic radii given by are obtained via dipole fits to the quark sector contributions, which allow the value and slope of G M (Q 2 ) to be  extrapolated to Q 2 = 0. In obtaining hadronic magnetic radii, the quark sectors combine with additional weightings given by G M (0), that is  Figure 8. Quark-mass dependence of squared magnetic radii for the first negative-parity excitation of the proton.
For all five masses, we find that these squared magnetic radii mostly agree with the charge radii from G E (Q 2 ). In Fig. 8, we plot the pion-mass dependence of the squared magnetic radius obtained from quark-sector dipole fits to G M (Q 2 ) for the excited proton and neutron.
Returning to the individual quark sector results, we note that G M (Q 2 ) and G E (Q 2 ) have a similar Q 2 dependence over the range considered. In light of this, we hypothesise that G M (Q 2 ) and G E (Q 2 ) have the same Q 2 scaling in this region. If this hypothesis is valid, then the Since we are working with an improved conserved vector current, and single quarks of unit charge, G E (0) = 1 exactly, and G M (0) is the contribution of the quark flavour to the magnetic moment (up to scaling by the physical charge). Hence, the ratio is expected to provide a measure of the contribution to the magnetic moment from the given quark flavour. In Fig. 9, we plot this ratio at m π = 411 MeV as a function of Q 2 . We see that as expected, the ratio is approximately constant across the Q 2 range accessible by our kinematics. This holds true for all five pion masses considered in this work. This supports the underlying hypothesis that the Q 2 scaling of the contributions to G E (Q 2 ) and G M (Q 2 ) from each quark sector is the same, and hence suggests that µ eff is a good estimate for the magnetic moment of this state.
We take constant fits to µ eff at each quark mass, and plot their pion-mass dependence in Fig. 10. By taking linear combinations of these fits as described for G E (Q 2 ) and G M (Q 2 ) above, we obtain magnetic moment estimates for the excited proton and neutron, as plotted in Fig. 11. For the heaviest three pion masses, the effective magnetic moments show little pion mass dependence and have tight error bars. The lightest two pion masses have much larger errors, and we observe a discontinuity in G M (Q 2 ) at the the second lightest mass, appearing as a significantly smaller magnetic moment for both states. This suggests that there could be a change in the structure of this state at that mass. However, there is no  Figure 11. Quark-mass dependence of the magnetic moment of the first negative-parity excitations of the proton and neutron.
corresponding change in G E (Q 2 ). At the lightest mass, the magnetic moments appear to return to consistency with the values from the heavier masses. Hence, it is unclear whether the behaviour at the second lightest mass indicates a change in the nature of the state, the presence of significant scattering-state contamination, or is a result of increasing gauge noise at lighter pion masses. Further research will be required to determine which of these possibilities is realised.
In this section focusing on the first negative-parity excitation of the nucleon, we have demonstrated the importance of the PEVA technique in correctly extracting both the electric and magnetic form factors. From these, we derived charge radii, magnetic moments and magnetic radii.
While we regard the results for the three heaviest quark masses to be robust, we take some caution with the lightest quark mass results as they may be influenced by important unaccounted scattering-state contributions. Table III. Radii and magnetic moments of the ground state proton and neutron. Radii are obtained from from combinations of quark-sector dipole fits and magnetic moments are obtained from quark-sector ratios of GM (Q 2 ) to GE(Q 2 ).   In Table III, we present the charge radii, magnetic radii, and magnetic moments of the ground state proton and neutron. These measurements are obtained from the electric and magnetic form factors as described above. We provide these results here for easy comparison with the excited state results presented in this paper.
In Table IV, we present the same results for the first negative-parity excitation. We see that this state has radii similar to the ground state, but notably different magnetic moments. We find that at the heavier quark masses, these magnetic moments agree well with con-  stituent quark model predictions, as discussed below in Section III E.

C. GE for the second negative-parity excitation
We now proceed to examine the next negative-parity excitation observed in this study, N * 2 . In Fig. 12, we plot G E (Q 2 ) as a function of sink time for both quark flavours at m π = 702 MeV with the lowest-momentum kinematics. We see that the conventional extraction sits even further below the PEVA extraction than the first negative-parity excitation. While the PEVA fits start at time slice 24, the conventional fits are forced all the way out to time slice 28 and sit at only just above half of the values of the PEVA fits.
Moving on to the lighter pion masses, the discrepancy between the extracted form factors continues at m π = 570 MeV and 411 MeV, with the conventional analysis giving consistently lower values than PEVA. For example, Fig. 13 shows the plateaus at m π = 570 MeV.
At the lightest two pion masses, the signal gets significantly noisier, and the difference between the two techniques gets harder to distinguish. Increased statistics are required in order to clearly identify the effects of opposite-parity contaminations of this state at these masses. However, in principle, the enhancement of relativistic components of the baryon spinors at light quark  Figure 13. Quark-flavour contributions to GE(Q 2 ) for the second negative-parity excitation at mπ = 570 MeV for the lowest-momentum kinematics, providing Q 2 = 0.1458(44) GeV 2 . The conventions used in this plot are the same as in Fig. 12. The conventional fits have significantly lower values than the PEVA fits, and the plateau for the doubly represented quark flavour starts one time slice later than the corresponding PEVA plateau.
masses is expected to increase parity mixing in the conventional analysis.
In addition, in the tail of the two point correlation function at the lightest mass, contributions from lowlying states are evident from a χ 2 analysis of a single state ansatz. This effect was also seen in Ref. [27], where it was found that so long as a single-state ansatz is satisfied in the fit region, this contamination does not have a significant effect on the extracted mass. However, as shown in Ref. [1], contaminants that do not significantly perturb the extracted mass can still have a significant effect on the extracted form factors. As such, we must be cautious when interpreting results from this state at this mass. To gain a deeper insight into this state at this mass, multi-particle scattering operators will be necessary to properly isolate the low-lying scattering state.
Returning to the other ensembles, we find that the trends from the lowest-momentum kinematics continue for all other kinematics: for all masses for which the noise is sufficiently low, the conventional fits sit significantly lower than the PEVA fits.
Once again, these results clearly indicate that PEVA is critical to the correct extraction of the electric form factors of this nucleon excitation. The opposite-parity contaminations admitted by the conventional analysis lead to significant underestimation of the value of the electric form factor. Hence, we now focus our attention only on the PEVA results.
Plotting the acceptable plateaus as a function of Q 2 reveals that the contributions from the two quark flavours are once again very similar, and agree well with a dipole ansatz. For example, Fig. 14 shows dipole fits to  Figure 14. Quark-flavour contributions to GE(Q 2 ) for the second negative-parity excitation at mπ = 411 MeV. The curves are dipole fits to the form factor, with the y-intercept fixed to unity. They correspond to RMS charge radii of 0.679(38) fm for the doubly represented quark (up) and 0.715(31) fm for the singly represented quark (dp). We once again take the linear combinations discussed in Section III A to form the excited proton and neutron. For example, in Fig. 15, we plot the electric form factors obtained from the quark-sector combinations at m π = 411 MeV. At all five masses, the electric form factor for the second negative-parity excitation of the neutron is approximately zero. At the heaviest three pion masses, the linear combinations of the quark-sector dipole fits once again provide charge radii consistent with the ground-state nucleon at the same quark mass [1]. This suggests that this finitevolume excitation is also of similar size to the groundstate nucleon, as seen in Fig. 16, where we plot the pion-mass dependence of the charge radii. We see that the pion-mass dependence of the proton radius is fairly smooth at these heaviest masses, and has a clear trend to increasing charge radius at lower pion masses.
However, the results at the lightest two masses depart from this trend, with the proton radius for the second lightest mass sitting higher then the trend, and the lightest mass sitting low.

D. GM for the second negative-parity excitation
We now advance to the magnetic form factor of this state. In Fig. 17, we plot the heaviest pion mass of m π = 702 MeV and the lowest-momentum kinematics. While the plateau time-regions for the PEVA and conventional analysis are consistent, the values of those plateaus are very different, and in fact change ordering between the two extractions. We see a similar effect at m π = 296 MeV, 411 MeV, and 570 MeV, with similar inversions of the magnetic form factors between the two analyses. We see similar patterns for the other kinematics, with significantly different plateau values between the two analyses when the statistical noise is low enough to distinguish them.
Once again the PEVA technique is crucial to extracting the correct form factors. Hence, we focus only on the  PEVA results. Inspecting the Q 2 dependence of these form factors, we find that the contributions from both quark flavours agree well with a dipole ansatz. For example, Fig. 18 shows the form factors at m π = 411 MeV.
Here we have held the y-scale fixed to match previous plots, for ease of comparison. The most notable feature of these results is their small magnitude compared to the both the ground state and the excitation considered in Section III B. By taking linear combinations based on the multipli- city and charge of each quark flavour, as described in Section III A, we can obtain the magnetic form factors for the excited proton and neutron. Fig. 19 shows these combinations for m π = 411 MeV. The magnetic charge radii obtained by combining the quark-sector dipoles are consistent with the proton charge radii from G E (Q 2 ), although they often have very large errors due to the very small values of the magnetic form factors amplifying the effects of statistical fluctuations.
Returning to the individual quark sector results with unit charge, and noting that the Q 2 dependence for G E (Q 2 ) and G M (Q 2 ) is similar, we once again take the ratio µ eff (Q 2 ) ≡ G M (Q 2 )/G E (Q 2 ). We find that this ratio is approximately flat for all five pion masses. For example, Fig. 20 shows the Q 2 dependence of the ratio at m π = 411 MeV. We can extract the contributions to the magnetic moment from both quark flavours from constant fits to this ratio. Fig. 21 shows the pion-mass dependence of these extracted magnetic moment contributions. It is remarkable that both quark flavours contribute with the same sign.
By taking the linear combinations discussed above, we can combine these individual quark flavour results to get the predicted magnetic moments for the second negative-parity excitations of the proton and neutron. In Fig. 22 we plot the dependence of these combinations on the squared pion mass. For the heaviest three pion masses, the effective magnetic moments show little pion mass dependence and have tight error bars. The magnetic moment of the proton excitation sits very close to zero, and the magnetic moment of the neutron excitation has a small but nonzero positive value. For the lightest two masses, the ordering of the two states flips, with the proton excitation taking on a more significant magnetic moment, and the neutron excitation dropping to be consistent with or below zero.
In summary, the PEVA technique is crucial for ex- tracting both the electric and magnetic form factors of the second negative-parity excitations of the proton and neutron. These extractions lead to a determination of the radii and magnetic moments of the proton and neutron excitations. The pion-mass dependence of the extracted magnetic form factors and magnetic moments provides evidence for an increased role for meson-baryon multiparticle contributions at light quark masses and a need to address possible scattering state contaminations in future correlation matrix studies.
In Table V, we present the charge radii, magnetic radii, and magnetic moments of the second negative-parity ex-  Table V. Radii and magnetic moments of the second negativeparity excitation of the proton and neutron. Radii are obtained from from combinations of quark-sector dipole fits and magnetic moments are obtained from quark-sector ratios of GM (Q 2 ) to GE(Q 2 ). At the lightest two pion masses, the form factor data was insufficient to properly constrain a dipole fit so we do not report magnetic radii at these masses. citation of the proton and neutron. We see that this state has similar radii to the ground state, but notably different magnetic moments. We find that at the heavier quark masses, these magnetic moments agree well with constituent quark model predictions, as discussed below.

E. Model comparison
In Section I, we introduced the two localised negativeparity excitations of the proton and neutron observed on the lattice by the CSSM and the HSC.
Here, we focus on the magnetic moments of these two states, as calculated in sections III B and III D. To date there have been no experimental determinations of the magnetic moments of these states, so we turn to theoretical models to interpret these results.
A HEFT description of these states would express them as a strongly interacting superposition of "bare" baryonic states and non-interacting meson-baryon states with definite relative momenta. Due to the localised nature of the interpolating operators used in this study, the only states the interpolators will have a significant overlap with are localised states with a significant barestate component. These bare states are associated with simple quark model states, so to understand our magnetic moment results, we turn to quark model predictions of the magnetic moments of the N * (1535) and N * (1650) resonances.
Our focus on the three heavier quark masses considered will be beneficial in comparing with quark models as it is this regime where constituent quark phenomenology is typically manifest.
We consider two constituent quark model (QM) predictions of the magnetic moments from Refs. [6,13], and two chiral constituent quark model (χQM) calculations which take the quark model calculations and include effects from the pion cloud [13,14].
In Fig. 23, we compare our magnetic moments extracted at m π = 702 MeV with these quark model predictions, which are calculated at the physical point. We can see that qualitatively, the results for the first negativeparity lattice excitation match up with the quark model N * (1535), and the second negative-parity lattice excitation with the quark model N * (1650). In fact, despite being at significantly different pion masses, the results are quantitatively very similar, with the lattice results sitting within the scatter of the model predictions for all states save the second negative-parity nucleon excitation, which sits slightly below all of the model predictions.
For comparison, we also plot lattice results produced using the conventional variational analysis. For these results, µ eff (Q 2 ) varies significantly for different kinematics, so rather than taking a constant fit across kinematics, we present only the result from the lowest-momentum (p = (0, 0, 0), p ′ = (1, 0, 0)) kinematics, which we expect to have the smallest opposite-parity contaminations. We see that the conventional results are significantly different to the PEVA results. In particular, the conventional extraction of the second negative-parity excitation is completely different to both the PEVA result and the quark model results. This once again demonstrates how critical the PEVA technique is to obtaining correct results.
This trend continues for m π = 570 MeV, and m π = 411 MeV, the latter shown in Fig. 24. Since the pionmass dependence of the magnetic moments between these three masses is quite small, the quantitative agreement remains good. This strongly suggests that these lattice states are dominated by two different bare states, corres- ponding to the two quark model states, at least at the three heaviest masses. This suggests that future HEFT studies of this resonance region should include two bare states, corresponding to the quark model N * (1535) and N * (1650) states.
Moving on to the second lightest mass, Fig. 25 shows that the lattice results depart from the model predictions at m π = 296 MeV. Again, an analysis of possible scattering-state contributions to the correlation functions is required to disentangle interesting meson-cloud effects from scattering-state effects. Fig. 25 also shows that there is still a significant disagreement between the conventional and PEVA results at m π = 296 MeV. It remains clear that the PEVA technique is playing an important role in addressing opposite parity contaminations.
The results presented in this section provide new insight into the structure of the negative-parity nucleon excitations observed in Lattice QCD. At the heavier pion masses considered, the two negative-parity excitations agree well with quark-model descriptions of the N * (1535) and N * (1650). This gives significant insight into the nature of these finite-volume eigenstates. Coupled with the observed single-particle dispersion relations seen in Ref. [15], the results indicate that these states are localised and behave like three-quark constituent-quarkmodel states. Indeed, the electric form factors indicate these states are similar in size to the ground states. These insights provide an important guide for future HEFT calculations.

IV. POSITIVE PARITY EXCITATION
We now move to the positive parity sector, studying the first localised positive-parity excitation of the nucleon observed on the lattice. This state sits at an effective mass of approximately 2 GeV for all five pion masses, well above 1.43(2) GeV, the mass of the Roper resonance observed in nature [28]. This has long been a puzzle for the particle physics community, but recent HEFT results indicate that the Roper resonance is dynamically generated from meson-baryon scattering states [3], and hence the lattice spectrum in this energy region has poor overlap with local three-quark operators. This means that the lattice state studied here is likely associated with the N * (1710), N * (1880), and/or N * (2100) resonances.

A. Electric form factor
We plot the dependence of G E (Q 2 ) on the Euclidean sink time at m π = 702 MeV in Figs. 26 and 27. The form factor values extracted from the PEVA and conventional analyses for each sink time look very similar, and this is reflected in the fits. The conventional and PEVA extractions both have clear plateaus over the same range of sink times, and these plateaus have consistent values.
This trend continues for lighter pion masses: the PEVA and conventional analyses have the same fit ranges and consistent fit values. This is also true for all kinematics for which we are able to find acceptable plateaus. This suggests there are no significant effects from oppositeparity contaminations on G E (Q 2 ) for this state, at least at this level of statistics.
Focusing on the PEVA results, in Fig. 28, we plot the Q 2 dependence of the electric form factor for the two valence quark flavours at m π = 411 MeV. We see that the two quark flavours make very similar contributions to the electric form factor and agree well with a dipole fit corresponding to charge radii of 0.88(4) fm for the doubly represented quark flavour and 0.89(5) fm for the singly represented quark flavour. This is significantly larger than the ground-state nucleon. Similar behaviour is seen   Figure 27. Contributions to GE(Q 2 ) from the singly represented quark flavour for the first positive-parity nucleon excitation. The pion mass, kinematics and plotting convention are the same as in Fig. 26 above. Both fits are from time slice 24-28. for the other four masses. As above, we take linear combinations of the individual quark flavour contributions, including the charges of the quark flavours and their multiplicity, to get the electric form factors for the first positive-parity excitations of the proton and neutron. In Fig. 29, we plot these combinations at m π = 411 MeV. At this and the other four masses, we find that the electric form factor for the neutron excitation is approximately zero.
For the heaviest three masses, combining the quarksector dipole fits in the same way as the lattice res- ults gives squared charge radii for the proton ranging from 0.67(7) fm 2 to 0.75(7) fm 2 , increasing with decreasing pion mass. These radii are all significantly larger than the charge radius of the ground-state proton at the corresponding mass, as can be seen in Fig. 30. This indicates that the second positive-parity excitation is a larger state than the ground state proton, at least at these pion masses.
At the lightest two masses, the central values of the radii are smaller, but the statistical errors are large enough that they are consistent with the radii at the heavier masses (as well as being consistent or almost consistent with the ground-state proton radii at the corresponding masses). In Fig. 30, we plot the pion-mass dependence of charge radii extracted from combinations of the quark-sector dipole fits to the electric form factor. We clearly see here this different behaviour at the lightest two masses, which appears to indicate a partial loss of signal, though it could hint at this state also changing structure at these masses. Repeating the calculations with increased statistics at these masses should allow these results to be more clearly resolved, and any physics concealed within these larger error bars to be revealed.

B. Magnetic form factor
Having investigated the electric form factor for this state, we now consider the magnetic form factor. In Fig. 31 we plot the Euclidean sink-time dependence of the extracted form factors at m π = 702 MeV, with the lowest-momentum kinematics. We see that the form factors and plateaus for both analyses are very similar, and there is no evidence for opposite-parity contamination of this state. We see similar results for the other masses and kinematics, with no clear differences between the conventional and PEVA plateaus. For example, Fig. 32 shows this behaviour at m π = 411 MeV with the same lowest-momentum kinematics. This suggests that, like G E (Q 2 ), G M (Q 2 ) for the first positiveparity excitation is not affected by opposite parity excitations, at least at this level of statistics.
Focusing on the PEVA results, we plot the Q 2 dependence of the plateau fits for the two valence quark flavours at m π = 411 MeV in Fig. 33. We see that both quark flavours agree well with a dipole ansatz. This is also true for the two heavier pion masses, and the two lighter masses are also consistent, though they are too noisy to significantly constrain the fit.
As discussed above, we can take combinations of the quark flavour contributions to form the excitations of the proton and neutron. In Fig. 34, we plot these combinations at m π = 411 MeV. By combining the dipole fits to the quark-sector results in the same way, we obtain magnetic radii that are consistent with the corresponding excited proton charge radius. This can be seen in Fig. 35, in which we plot the pion-mass dependence of the squared magnetic radii obtained from the linear combinations of the dipole fits to the form factors. These plots show fairly consistent results for the heavier three masses, with some pion-mass dependence. We omit results for the lightest two masses because the form factor data is insufficient to constrain dipole fits at these masses.
Returning to the individual quark sector results and noting that once again the electric and magnetic form factors have a similar Q 2 dependence, we take the ratio µ eff (Q 2 ) ≡ G M (Q 2 )/G E (Q 2 ). In Fig. 36, we plot this ratio as a function of Q 2 for m π = 411 MeV. We find that the ratio is once again very flat in Q 2 , supporting our hypothesis that the form factors have the same Q 2 scaling in this region, and the validity of µ eff as an estimate of the magnetic moment.
In Fig. 37, we plot the pion-mass dependence of µ eff for individual quarks of unit charge. We can once again take combinations of the individual quark-flavour contributions to get the excited proton and neutron. In Fig. 38, we plot the pion-mass dependence of these combinations.
We see that the excited-state magnetic moments agree well with the ground-state magnetic moments, particularly at the heaviest quark masses, where the agreement    is impressive. These results are in accord with a simple 2S constituent-quark-model state.
In this section, we have shown that the first positiveparity excitation of the nucleon has no obvious oppositeparity contaminations. However, variational analysis techniques in general have given good access to this state at several pion masses. This has allowed us to ascertain that these states have a larger radius than the groundstate nucleon, but have very similar magnetic moments. This is consistent with these states being dominated by a radial 2S excitation of the ground-state nucleon as seen in Refs. [29,30].
In Table VI, we present the charge radii, magnetic  Figure 35. Quark-mass dependence of squared magnetic radii of the first positive-parity excitation of the proton and neutron from quark-sector dipole fits to GM (Q 2 ). At the lightest two pion masses, the form factor data was insufficient to properly constrain a dipole fit so we do not report magnetic radii at these masses.
radii, and magnetic moments of the positive-parity excitation of the proton and neutron. We see that this state has consistently larger radii than the ground state, but similar magnetic moments.

V. CONCLUSION
We have presented the first implementation of the PEVA technique for matrix elements of exited baryons moving with nontrivial momentum. We have considered a variety of momentum frames to access a range of Q 2 values, with some approaching 0. We have shown that the PEVA technique is critical to correctly extracting the form factors of proton and neutron excitations on the lattice. Such extractions give us insight into the structure of the states seen on the lattice.
We find the size of the two lowest-lying negative parity excitations to be similar to the ground state nucleons. However the lowest-lying positive parity excitation we access is significantly larger suggesting a radial excitation as observed in the wave function analysis of Roberts et  Table VI. Radii and magnetic moments of the positive-parity excitation of the proton and neutron. Radii are obtained from from combinations of quark-sector dipole fits and magnetic moments are obtained from quark-sector ratios of GM (Q 2 ) to GE(Q 2 ). At the lightest two pion masses, the form factor data was insufficient to properly constrain a dipole fit so we do not report magnetic radii at these masses. al. [29,30]. At the heaviest three pion masses considered, the first negative-parity excitation observed is consistent with the N * (1535) of the quark model, as predicted by HEFT. Similarly, the second negative-parity excitation has the properties of the quark model N * (1650). This suggests that future HEFT studies should explore the incorporation of two bare basis states associated with the two different localised states observed herein.
At the lightest two pion masses, we observe a rearrangement in the structure of the second negative-parity excitation. This is evident in both a significant shift in the magnetic moments of the excited proton and neutron, and significant curvature in the pion-mass dependence of the electric form factor. A description of this state as a molecular bound state of KΣ dressed by KΛ, ηN and πN is an intriguing possibility, analogous to the description of the odd-parity Λ(1405) excitation as a molecular bound state of KN dressed by πΣ [11,31]. The proximity of the non-interacting KΣ to the effective energy of the observed lattice state is suggestive. For example, at the lightest mass m N * 2 = 1.65(7) GeV is consistent with the two-particle threshold of m K + m Σ = 1.72(3) GeV.
Further statistics will be required to pin down the exact behaviour at the lightest pion masses and refine this picture. In addition, the inclusion of multi-hadron operators will improve overlap with these states, and would allow for the investigation of other excitations in this region, providing more comprehensive inputs for a detailed HEFT study of the nature of these states.
The positive-parity excitation observed in this study is difficult to isolate, and has proven quite noisy in previous lattice studies. To extract its form factors is a remarkable outcome, which can only be attained through the cancellation of statistical fluctuations enabled by the combination of a conserved current and an appropriately selected correlator ratio. This state has a charge radius approximately 30 % larger than the ground state, but it has nearly identical magnetic moments. This is consistent with the state being a radial excitation of the ground-state nucleon as seen in Refs. [29,30].
In this paper we presented the first calculations of the elastic form factors of lattice nucleon excitations from the first principals of QCD. Substantial differences between the form factors calculated in a conventional variational analysis and those calculated with the PEVA approach show the PEVA technique to be central to progress in understanding the structure of these excited states. We anticipate its broad adoption in future baryon excited state studies.
In future analyses, it will be important to perform higher-statistics studies incorporating both single and multi-particle interpolating operators. This will be a challenging endeavour, owing to the computational cost of estimating the loop propagators that are necessary to compute meson-baryon scattering-state operators. However, such studies will allow for the lattice determination of the form factors of the multi-particle dominated excitations at light quark masses, and will allow for finitevolume effective field theory studies to make a robust connection between these states and the infinite-volume resonances of nature.