Nucleon Excited State Wave Functions from Lattice QCD
AADP-13-28/T848
Nucleon Excited State Wave Functions from Lattice QCD
Dale S. Roberts, Waseem Kamleh, and Derek B. Leinweber
Special Research Centre for the Subatomic Structure of Matter,School of Chemistry and Physics, University of Adelaide, SA, 5005, Australia
We apply the eigenvectors from a variational analysis to successfully extract the wave functionsof even-parity excited states of the nucleon, including the Roper. We explore the first four states inthe spectrum excited by the standard nucleon interpolating field. We find that the states exhibit astructure qualitatively consistent with a constituent quark model, where the ground, first-, second-and third-excited states have 0, 1, 2, and 3 nodes in the radial wave function of the d -quark abouttwo u quarks at the origin. Moreover the radial amplitude of the probability distribution is similarto that predicted by constituent quark models. We present a detailed examination of the quark-mass dependence of the probability distributions for these states, searching for a nontrivial role forthe multi-particle components mixed in the finite-volume QCD eigenstates. Finally we examine thedependence of the d -quark probability distribution on the positions of the two u quarks. The resultsare fascinating, with the underlying S -wave orbitals governing the distributions even at rather large u -quark separations. PACS numbers: 12.38.Gc, 12.39.Jh, 14.20.Gk
I. INTRODUCTION
An examination of the wave function of a quark boundwithin a hadron provides deep insights into the underly-ing dynamics of the many-body theory of QCD. It en-ables a few-body projection of the underlying physicsthat can be connected with models, shedding light on theessential effective phenomena emerging from the complexdynamics of QCD.The hadron spectrum is the manifestation of the highlycomplex dynamics of QCD. It is an observable that isreadily accessible in collider experiments. While thequantum numbers of the states can be ascertained, prop-erties providing more insight into the structure of the res-onances often remain elusive to experiment. We aim toprovide some insight into the underlying dynamics gov-erning the structure of these states.In quantum field theory, a Schr¨odinger-like probabil-ity distribution can be constructed for bound states bytaking a simplified view of the full quantum field theorywave functional in the form of the Bethe-Salpeter wavefunction [1], herein referred to as simply the ‘wave func-tion’. Recent advances in the isolation of nucleon excitedstates through correlation-matrix based variational tech-niques in lattice QCD now enable the exploration of thestructure of these states and how these properties emergefrom the fundamental interactions of QCD.In this paper, we extend earlier results [2] focusingon the wave function of the Roper excitation [3] to thefour lowest-lying even-parity states excited by the stan-dard χ interpolating field which incorporates a scalardiquark construction. We examine the quark mass de-pendence of the probability distributions for these states.Here we search for a signature of multi-particle compo-nents mixed in the finite-volume QCD eigenstates at thetwo largest quark masses where the states sit close to the multi-particle thresholds. We also explore the de-pendence of the d -quark probability distribution on thepositions of the two u quarks along an axis through thecentre of the distribution.In presenting our results we make extensive use of iso-volume and surface plots of the probability distributionsfor the quarks. Such visualizations have already beenused to illustrate physical effects such as Lorentz con-traction [7, 8], the effect of external magnetic fields [9]and finite volume effects [2, 10], for example.Early explorations of these states were based on non-relativistic constituent quark models. The probabilitydistributions of quarks within hadrons were determinedusing a one-gluon-exchange potential augmented with aconfining form [4, 5]. These models have been the cor-nerstone of intuition of hadronic probability distributionsfor many decades. In this investigation, we will confrontthese early predictions for quark probability distributionsin excited states directly via Lattice QCD.In a relativistic gauge theory the concept of a hadronicwave function is not unique and the Bethe-Salpeterwave function underlying the probability distributionscan be defined in several different forms. For example,the gauge-invariant Bethe-Salpeter amplitude exploits astring of flux to connect the quarks annihilated at differ-ent spatial positions in a gauge invariant manner. Asthis leads to an explicit path dependence, an averageover the paths is desirable. Another approach consid-ers Bethe-Salpeter amplitudes in which the gauge degreeof freedom is fixed to a specific gauge. In lattice fieldtheory, Coulomb and Landau gauges are most commondue to their local gauge fixing procedure. Landau gaugeprovides distributions that compare favorably with con-stituent quark model predictions [2] and therefore we se-lect Landau gauge herein. a r X i v : . [ h e p - l a t ] N ov I. LATTICE TECHNIQUES
Robust methods have been developed that allow theisolation and study of the states associated with these res-onances in Lattice QCD [2, 11–21]. In this study, we ap-ply the variational method [22, 23] to extract the groundstate and first three P excited states of the proton as-sociated with the Roper [3] and other higher-energy P states. We then combine this with lattice wave functiontechniques to calculate the probability distributions ofthese states at several quark masses and quark positions.We use the 2+1 flavour 32 ×
64 PACS-CS configurations[24] at a pion mass as low as 156 MeV.The wave function of a hadron is proportional to theparity-projected [25] two point Green’s function, G ± ij ( (cid:126)p, t ) = (cid:88) (cid:126)x e − i(cid:126)p · (cid:126)x tr ( γ ± (cid:104) Ω | T { χ i ( (cid:126)x, t ) ¯ χ j ( (cid:126) , } | Ω (cid:105) , (1)where χ i are the hadronic interpolating fields. In thecase of the proton the most commonly used interpolatoris given by χ ( x ) = (cid:15) abc ( u (cid:124) a ( x ) Cγ d b ( x ) ) u c ( x ) , (2)with the corresponding adjoint given by¯ χ (0) = (cid:15) abc ( ¯ d b (0) Cγ ¯ u (cid:124) a (0) ) ¯ u c (0) . (3)In order to construct the wave function, the quark fieldsin the annihilation operator are each given a spatial de-pendence, χ ( (cid:126)x, (cid:126)y, (cid:126)z, (cid:126)w ) = (cid:15) abc ( u (cid:124) a ( (cid:126)x + (cid:126)y ) Cγ d b ( (cid:126)x + (cid:126)z ) ) u c ( (cid:126)x + (cid:126)w ) , (4)while the creation operator remains local. This gen-eralizes G ( (cid:126)p, t ) to a wave function proportional to G ( (cid:126)p, t ; (cid:126)y, (cid:126)z, (cid:126)w ). In principle, we could allow each of thesecoordinates, (cid:126)y, (cid:126)z, (cid:126)w , to vary across the entire lattice,however, we can reduce the computational cost by takingadvantage of the hyper-cubic rotational and translationalsymmetries of the lattice. A near-complete description ofthe probability distribution of a particular quark withinthe proton can be formed by separating two of the quarksalong a fixed axis and calculating the third quark’s wavefunction for every lattice site. For this study, we will fo-cus on the probability distribution of the d quark fromEq. (4) with the u -quarks being separated along the x axis through the centre of the distribution, i.e. , χ ( (cid:126)x, (cid:126)d , (cid:126)z, (cid:126)d ; t ) = (5) (cid:15) abc ( u (cid:124) a ( (cid:126)x + (cid:126)d , t ) Cγ d b ( (cid:126)x + (cid:126)z, t ) ) u c ( (cid:126)x + (cid:126)d , t ) . where (cid:126)d i = ( d i , , d > d = − d for separationsacross an even number of lattice sites and d = − ( d −
1) for an odd separation. A symmetrised wave functionis presented by averaging the wave functions calculatedwith the interpolating field in Eq. (5) combined with the wave functions produced by the same interpolating fieldwhere d ↔ d .Landau gauge is a smooth gauge that preserves theLorentz invariance of the theory. While the size andshape of the wave function are gauge dependent, our se-lection of Landau gauge is supported by our results. Forexample, the ground state wave function of the d quark inthe proton is described accurately by the non-relativisticquark model using standard values for the constituentquark masses and string tension of the confining poten-tial [2]. Therefore this gauge provides a foundation for amore comprehensive wave function examination.To isolate energy eigenstates we use the correlationmatrix or variational method [22, 23]. As we are inter-ested in the wave functions for states at rest, we select (cid:126)p = 0 in Eq. (1). To ensure that the matrix elements areall ∼ O (1), each element of G ij ( t ) is normalized by thediagonal elements of G (0) as G ij ( t ) / ( (cid:112) G ii (0) (cid:112) G jj (0))(no sum on i or j ). Using an average of { U } + { U ∗ } configurations which have equal weight in the QCD ac-tion, our construction of the two-point functions is real[26, 27].A linear superposition of interpolators ¯ φ α = (cid:80) j ¯ χ j u αj creating state α provides the following recurrence relation G ij ( t + (cid:52) t ) u αj = e − m α (cid:52) t G ij ( t ) u αj , (6)from which right and left eigenvalue equations are ob-tained [( G ( t )) − G ( t + (cid:52) t )] ij u αj = c α u αi , (7) v αi [ G ( t + (cid:52) t ) ( G ( t )) − ] ij = c α v αj , (8)with c α = e − m α (cid:52) t . The eigenvectors for state α , u αj and v αi , provide the eigenstate projected correlation function G α ± ( t ) ≡ v αi G ± ij ( t ) u αj , (9)with parity ± . The effective mass can then be calcu-lated from the projected two-point functions by m ( t ) =log ( G ( t ) /G ( t + 1)). While the effective mass is insensi-tive to a wide range of parameters [20], we follow Ref. [20]and select t to be 2 time slices after the source with∆ t = 2.Different interpolators exhibit different couplings tothe proton ground and excited states and hence can beused to construct a variational basis. The limited numberof local interpolators restricts the size of the operator ba-sis [11]. To remedy this, we exploit the smearing depen-dence of the coupling of states to one or more standardinterpolating operators in order to construct a larger vari-ational basis where the χ i and ¯ χ j from Eq. (1) containa smearing dependence. This method has been shown toallow access to states associated with resonances such asthe Roper [2, 20] and the Λ(1405) [28].The non-local sink operator used to construct the wavefunction is unable to be smeared, and hence the standardtechnique of Eq. (9) cannot be applied. However, Eq. (7)illustrates it is sufficient to isolate the state at the source2 IG. 1. The mass dependence of the four lowest-lying even-parity eigenstates excited by the χ interpolating field is com-pared with the S and P -wave non-interacting multi-particleenergy thresholds on the finite volume lattice. Plot symbolstrack the eigenvector associated with each state. using the right eigenvector. Thus, the probability distri-butions are calculated with each smeared source operatorand the right eigenvectors calculated from the standardvariational analysis are then applied in order to extractthe individual states of interest.Our focus on χ in this investigation follows fromthe results of Ref. [29], where the lowest-lying excita-tion of the nucleon was shown to be predominantly as-sociated with the χ interpolating field. The resultsfrom their 8 × χ and χ = (cid:15) abc ( u (cid:124) a ( x ) C d b ( x ) ) γ u c ( x ) revealed that χ plays amarginal role in exciting the Roper. The coefficientsof the Roper source eigenvector multiplying χ are nearzero. Further comparison with Ref. [29], identifies thethird state extracted herein as the fifth state of the twelvestates identified and the fourth state herein as the tenthstate. Figure 1 illustrates the quark mass dependence ofthese four states which will be examined in detail herein.The quark mass flow of these states tracked by their as-sociated eigenvectors [29] is not smooth and suggests thepresence of avoided level crossings as one transitions fromthe heaviest two quark masses to lightest three quarkmasses. At the two heaviest quark masses, it is seemslikely these states are dominated by multi-particle N π components, whereas at the lighter three quark masses,single particle components are more dominant. We willsearch for evidence of this in the wave functions of thesestates.In summary, the wave function for the d quark in state α having momentum (cid:126)p observed at Euclidean time t withthe u quarks at positions (cid:126)d and (cid:126)d is ψ αd ( (cid:126)p, t ; (cid:126)d , (cid:126)d ; (cid:126)z ) = (cid:88) (cid:126)x e − i(cid:126)p · (cid:126)x tr ( γ + 1) (10) (cid:104) Ω | T { χ ( (cid:126)x, (cid:126)d , (cid:126)z, (cid:126)d ; t ) ¯ χ j ( (cid:126) ,(cid:126) ,(cid:126) ,(cid:126)
0; 0) } | Ω (cid:105) u αj , where χ ( (cid:126)x, (cid:126)d , (cid:126)z, (cid:126)d ; t ) is given by Eq. (5).As discussed above, χ has the spin-flavour constructthat is most relevant to the excitation of the Roper fromthe QCD vacuum. As such, it is an ideal choice for reveal-ing the spatial distribution of quarks within the Roper.However, the selection of χ in Eq. (10) is not uniqueand other choices are possible. For example, the selec-tion of χ would reveal small contributions to the Roperwave function where vector diquark degrees of freedomare manifest. Similarly, D -wave contributions could beresolved through the consideration of a spin-3/2 isospin-1/2 interpolating field at the sink. Research exploringthese aspects of the wave functions is in progress. III. SIMULATION RESULTSA. Lattice Parameters
We use the 2+1 flavour 32 ×
64 configurations createdby the PACS-CS collaboration [24] constructed with theIwasaki gauge action [30] and the O ( a )-improved Wil-son action [31] with β = 1 .
90, giving a lattice spacingof 0 . × χ operator with four smearinglevels; 16, 35, 100 and 200 sweeps [20] of gauge-invariantGaussian smearing [32]. These smearing levels corre-spond to smearing radii of 2.37, 3.50, 5.92 and 8.55 latticeunits or 0.215, 0.317, 0.537 and 0.775 fm respectively.The choice of variational parameters t = 2 , (cid:52) t = 2relative to the source position is ideal, resulting in theeffective mass plateaus of the states commencing at t = t = 2 as desired [2]. This indicates that the number ofstates contributing significantly to the correlation func-tions of the correlation matrix at t = 2 equals the di-mension of the correlation matrix. As such we examinethe wave functions of all four states with the caution thatthe fourth state is most susceptible to excited state con-tamination. In reporting the wave functions we selectthe mid point of the correlation matrix analysis at t = 3.The wave functions observed for all our states show anapproximate symmetry over the eight octants surround-ing the origin. To improve our statistics we average overthese eight octants when d = d = 0, and an averageover the four quadrants sharing an axis with the u quarkseparation at all other values of d and d .We fix to Landau gauge by maximizing the O ( a ) im-3roved gauge-fixing functional [33] F Imp = (cid:88) x,µ Re tr (cid:18) U µ ( x ) − u ( U µ ( x ) U ( x + ˆ µ ) + h . c . ) (cid:19) (11)using a Fourier transform accelerated algorithm [34].In carrying out our calculations, we average over theequally weighted { U } and { U ∗ } link configurations as animproved unbiased estimator [26]. The two-point func-tion is then perfectly real and the probability density isproportional to the square of the wave function. In thisanalysis, we choose to look at the zero-momentum prob-ability distributions. B. Wave Functions and Constituent Quark ModelPredictions
Figure 2 presents the wave functions for the first threestates at our lightest quark mass providing m π = 156MeV. In the excited states, the wave function changessign revealing a node structure consistent with 2 S and3 S excited state wave functions. To further explore thedetails of these wave functions, we construct a probabilitydensity from the square of the wave function and plot iton a logarithmic scale in Fig. 3.Our point of comparison with previous models of quarkprobability distributions comes from a non-relativisticconstituent quark model with a one-gluon-exchange mo-tivated Coulomb-plus-ramp potential. The spin depen-dence of the model is given in Ref. [5] and the radialSchrodinger equation is solved with boundary conditionsrelevant to the lattice data; i.e. the derivative of thewave function is set to vanish at a distance L x / √ σ =440 ±
40 MeV and optimize the constituent quark massto minimize the logarithmic difference between the quarkmodel and lattice QCD ground-state probability distri-butions illustrated in the left-hand column of Fig. 3. Wefind best fit results for √ σ = 400 MeV and the optimalconstituent quark masses range from 340 to 350 MeVover the range of PACS-CS quark masses available. Thequark mass dependence is more subtle than expected andmay be associated with the finite volume of the latticesuppressing changes in the wave function as the quarkmass is varied. At the lightest quark mass, just abovethose of Nature, the value of 340 MeV is in accord withthose traditionally used to describe the hadron spectrumor baryon magnetic moments.The lattice data for the first three states and all fivequark masses are compared with the constituent quarkmodel in Fig. 3. The wave functions are normalized to 1at the origin. As the quark model parameters are deter-mined using only the ground state probability distribu-tion, the probability densities illustrated for the excitedstates are predictions. FIG. 2. The wave function of the d quark in the proton aboutthe two u quarks fixed at the origin for the lightest quarkmass ensemble providing m π = 156 MeV. From top down,the plots correspond to the ground, first and second excitedstates observed in our lattice simulation. The wave functionchanges sign in the excited states and reveals a node structureconsistent with 1 s , 2 S and 3 S states. An examination of the left-hand column of Fig. 3 re-veals the subtle changes associated with the quark mass.The probability distribution of the heaviest ensemble fallsoff faster and requires a slightly heavier constituent quarkmass to fit the lattice results. This subtle mass depen-4
IG. 3. The probability distributions for the d quark about two u quarks fixed at the origin obtained in our lattice QCDcalculations (crosses) are compared with the quark model prediction (solid curve) for the ground (left column), first- (middlecolumn) and second- (right column) excited states. Quark masses range from the heaviest (top row) through to the lightest(bottom row). The ground state probability distribution of the quark model closely resembles the lattice data for all massesconsidered. The first excited states matches the lattice data well at small distances, but the node is placed further from thecentre of mass in the quark model, after which, the lattice data shows a distinct second peak, whereas the quark model risesto the boundary. It is interesting that the most significant difference is observed where long-distance physics associated withpion-cloud effects not included in the quark model are significant. For the third state the amplitudes of the shells between thenodes of the wave function are predicted well. d quark in the first excited state to that predicted by theconstituent quark model in the middle column of Fig. 3,we see a qualitative similarity but with important dif-ferences. The quark model predicts the behavior of thelattice wave function very well within the node and pre-dicts the position of the node rather well, particularly atthe lightest quark mass. However, the shape of the wavefunction tail is very poorly predicted, suggesting an im-portant role for degrees of freedom not contained withinthe quark model. For example, the long range pion tailof multi-particle components could alter the distributionof quarks within the state on the lattice. The poorestagreement is for the heaviest ensembles, where the baryonmass is in close proximity to the πN scattering threshold.Similar comments apply to the second excited stateillustrated in the right-hand column of Fig. 3. While thepositions of the nodes are predicted approximately, theamplitudes of the wave function between the nodes arevery accurately predicted by the quark model. Again thelargest discrepancies are for the heaviest states where thebaryon mass is in close proximity to the ππN scatteringthreshold. C. Quark Mass Dependence of the ProbabilityDistributions
1. Ground-State Distribution
The mass dependence of the ground state probabilitydistribution for the d quark about the two u quarks fixedat the origin is illustrated in the two left-hand columns ofFig. 4. The plots are arranged from heaviest to lightestensembles with quark mass decreasing down the page.Although a Gaussian distribution is used to excite theground state from the vacuum, the well-known sharp-peaked shape associated with the Coulomb potentialis reproduced in the probability density for all quarkmasses. This is best observed in the left most columnwhere an isosurface reports the probability-density val-ues in the plane containing the two u quarks at the origin.Because the total probability density is normalised tounity in the spatial volume, the height of the peak dropsas the d quark becomes light and moves to larger dis-tances from the u quarks. The isosurface provides theclearest representation of the mass dependence of theground state.This gentle broadening of the distribution is also re-flected in the isovolume rendering of the projected groundstate probability density in the second column if Fig. 4.The isovolume has been cut into the plane containingthe u quarks at the origin. Probability-density valuesare depicted by a colour map similar to that used for theisosurface. The threshold for rendering the probabilitydistribution is 3 . × − , revealing a smooth sphere for the surface of the probability distribution.Finite volume effects do not appear to be significantin the probability densities of the ground state at any ofthe quark masses considered. This is in spite of the factthat the lightest ensemble has m π L = 2 .
2. First Excited State
Lattice results for the d -quark probability distributionabout the two u quarks at the origin in the first excitedstate of the proton are illustrated in the third and fourthcolumns of Fig. 4. In the light quark-mass regime, thisfirst excited state is associated with the Roper resonance.The darkened ring around the peak of the isosurface in-dicates a node in the probability distribution, consistentwith a 2 S radial excitation of the d quark. The node isbetter illustrated in the isovolume renderings where theprobability density drops below the rendering cutoff of3 . × − and leaves a void between the inner and outershells of the state.It is interesting that the narrowest distribution is seenat the heaviest quark masses, even though these stateshave energies coincident with the πN scattering thresh-old. Enforcing a colour singlet structure in annihilatingthe three spatially separated quarks prevents a direct ob-servation of the two-particle components contained in thedynamics governing the energy of the state. In this casethe multi-particle components only modify the three-quark distributions.The outer edge of the isovolume reveals interestingboundary effects which may be associated with the nec-essary finite-volume effects of multi-particle componentsmixed in the state. The deviation from spherical sym-metry in the outer shell will be reflected in the energyof the excited state observed in the finite-volume latticesimulation. At the lightest two quark masses, the distor-tion of the probability distribution is significant and willcorrespondingly influence the eigen-energy. Even with m π L = 4 . u αi describing the con-tributions of each of the source smearing levels to thestates α for the lightest quark-mass ensemble considered.For the ground state, all smeared sources contributepositively to the state. There is significant interplaybetween the smeared sources over the jackknife sub-ensembles giving rise to larger uncertainties for the pre-ferred operators. This is not the case for the excitedstates where particular superpositions of interpolatingfields are required to isolate the states.6 IG. 4. The dependence of the d -quark probability distribution on the masses of the quarks in the proton (two left-handcolumns) and its first excited state (two right-hand columns). The u quarks are fixed at the origin at the centre of the plot.The quark mass decreases from heaviest (top row) to lightest (bottom row). For each mass and state, the probability densityis normalised to unity over the spatial volume of the lattice. The isovolume threshold for rendering the probability distributionin the second and fourth columns is 3 . × − . IG. 5. The eigenvectors u αi describing the contributions ofeach of the source smearing levels to the states α for thelightest quark-mass ensemble considered. Indices i = 1 to 4correspond to 16, 35, 100 and 200 sweeps of gauge-invariantGaussian smearing. The superposition of positive and neg-ative Gaussian smearing levels is consistent with the nodalstructure recovered in the wave functions. For the first excited state, a single large width Gaus-sian contributes with a sign opposite to that of a narrowerGaussian, reflecting the wave function illustrated in thesecond plot of Fig. 2. The combination of sources creat-ing the second excited state has a similar pattern, with anarrow Gaussian contributing positively, an intermediateGaussian contributing negatively and a wide Gaussiancontributing positively, again reflecting the wave functionillustrated in Fig. 2 for this state. This sign alternatingstructure is also apparent for the fourth state suggestinga 3 S excitation for this state. We will examine this statefurther in the following.Turning our attention to the mass dependence of thenode we note the movement is somewhat unusual. Whilethere is a general trend of the node in the wave func-tion moving outwards as the quark mass decreases, thereis negligible movement in the node between the thirdand second lightest quark masses. We also note how thewidth of the void in the probability density increases withas the quarks become lighter.
3. Second Excitation
The probability distributions for the second excitationof the proton observed in this study are illustrated in thetwo left-hand columns of Fig. 6. Two nodes are evidentat all quark masses, consistent with a 3 S radial excitationfor the d quark. The first inner node is thin at the heaviermasses and difficult to see in the isovolume renderings.Finite volume effects are readily observed in the outer-most shell.For the heaviest mass, finite volume effects at the nodesare minimal. The nodes are spherical in shape and are largely unaffected by the boundary. Again there is atrend of the nodes moving further from the centre as thequarks become light.However, it is the middle quark mass considered thathas the broadest distribution. The quark mass flow of theeigenstate energies suggests avoided level crossings areimportant between the third and fourth heaviest quarkmasses. It may be a strong mixing with multi-particlestates that is giving rise to the broad distribution ofquarks at the middle quark mass.For the lightest two quark masses the outer node hastaken on a squared-off shape, having been distorted bythe boundary of the lattice. Again, this is an indica-tion that, even though the ground state wave functionpresents as spherical for this quark mass, this excitedstate is showing clear finite volume effects. Even at rela-tively modest quark masses, the wave functions of statesabove the decay thresholds show an important relation-ship with the finite volume of the lattice.
4. Third Excitation
The two right-hand columns of Fig. 6 illustrate themass dependence of the d -quark probability distributionfor the the highest excitation of the proton observed inour analysis. The presence of three nodes in the wavefunction is best observed at the heaviest and second light-est quarks masses.The inner-most node is easily observed in the surfaceplots. However it is very sharp and does not render inan obvious manner in the isovolume illustrations. Thesecond node is easily rendered and the third node is verybroad. To illustrate this node structure the outer-mostshell has become fragmented in the isovolume plots. Thefragments reveal the strong finite volume effects on thisstate.What is interesting is the manner in which the finitevolume effects on the outer shell change as a function ofquark mass. At the heaviest quark mass, the outer shellis strongest along the sides of the lattice. By the timeone encounters the lightest ensemble, the outer shell hasmoved to the corners as if there is no longer room for theouter shell along the sides of the 2.9 fm lattice. IV. QUARK SEPARATION
In order to investigate a more complete picture of thewave functions of the states isolated herein, we choose tofocus on the second-lightest quark mass ensemble provid-ing m π = 293 MeV and examine the dependence of the d quark probability distribution on the positions of thetwo u quarks composing the states. This mass providesthe best compromise between finite-volume effects, quarkmass and the ensemble size governing the signal qualityand associated uncertainties.8 IG. 6. The dependence of the d -quark probability distribution on the masses of the quarks in the proton for the second (twoleft-hand columns) and third (two right-hand columns) S -wave excited states of the proton observed herein. The u quarksare fixed at the origin at the centre of the plot. The quark mass decreases from heaviest (top row) to lightest (bottom row).For each mass and state, the probability density is normalised to unity over the spatial volume of the lattice. The isovolumethreshold for rendering the probability distribution in the second and fourth columns is 2 . × − and 3 . × − respectively.While the former renders the outer shell coherently, the latter better reveals the node structure of the 3 S distribution. IG. 7. The dependence of the d -quark probability distribution on the positions of the u quarks in the first even-parity excitedstate of the proton at the second lightest quark mass considered. The u quarks are fixed on the x -axis running from back rightthrough front left through the centre of the plot. The u quarks are fixed a distance of d and d from the origin located at thecentre. From left to right, the distance d = d − d increases, taking values 0, 1, 2 and 3 times the lattice spacing a = 0 . In continuing to investigate the wave function of the d -quark, we consider the separation of the u quarks alongthe x axis as described in Eq. (5). All integer separations, d = d − d , between zero and half the lattice extent inthe x direction ( i.e.
16 lattice units) are considered.Figure 7 illustrates the probability distribution of the d quark for u quarks separated by 0, 1, 2 and 3 latticesteps in the first excited state associated with the Roperresonance. The most notable feature is the rapid reduc-tion in the overlap of the interpolator with the state asthe two u quarks are moved away from the origin. Whilesome broadening of the distribution peak is apparent, it isclear that using a normalization suitable for zero u quarkseparation is not effective for illustrating the probabilitydistribution at large u quark separations.To better illustrate the underlying shape of the wavefunctions, the probability distributions are normalised tokeep the maximum value of the probability density con-stant. For small u quark separations, the centre peakheight of the distribution is held constant, but for largerseparations the maximum value can be elsewhere in thedistribution.Figure 8 presents the d -quark probability distributionsfor u -quark separations of d = 0, 2, 4, 6 and 8 times thelattice spacing a = 0 . u -quark separations of 10, 12, 14 and16 times the lattice spacing. Each column correspondsto a different state with the ground, first-, second-, andthird-excitations illustrated from left to right. The twosmall spheres above the isosurface indicate the positionsof the two u quarks.
1. Ground-State Distribution
Focusing first on the ground state, on separation of the u quarks the probability distribution of the d quark formsa single broad peak. The structure is slightly roundeduntil d = d − d = 12 a = 1 .
09 fm, with small peaks atthe u -quark positions. At a separation of d = 13 a = 1 . d/a = 14. Theseresults are similar to the earlier quenched wave functionresults of Refs. [9, 35].
2. First Excited-State Distribution
As the u quarks are separated in the first excited stateassociated with the Roper, the central peak of the d -quark distribution broadens in a manner similar to thatfor the ground state. However by d = 4 a = 0 .
36 fmstrength in the wave function is seen to move from thecentre into the outer shell of the 2 S state. This transitioncontinues to d = 10 a = 0 .
91 fm where the u quarksare still well inside the original node position of the 2 S distribution. At this point, the central peak has beensuppressed entirely leaving a hole inside the ring or shellin three dimensions. In other words, the node of the wavefunction has shrunk to the origin. It’s interesting how thering-like probability density is enhanced in the directionperpendicular to the separation of the u quarks.At d = 11 a = 1 .
00 fm, small peaks form in the proba-bility distributions at the positions of the u quarks reveal-ing the first onset of scalar diquark clustering similar tothat in the second row of Fig. 9. At d = 12 a = 1 .
09 fm,the u quarks are still within the node of the original wavefunction. but the radius of the outer shell of the wavefunction illustrated by the ring in the probability densityhas reduced slightly. At d = 14 a = 1 .
27 fm, the u quarkssit in the node of the original wave function and there islittle memory of the original 2 S structure. Only a slightswelling at the centre of the distribution remains. Thecentral probability density reduces at d = 15 a = 1 .
36 fmsuch that scalar-diquark clustering dominates the prob-ability distribution at d = 16 a = 1 .
45 fm.10
IG. 8. The dependence of the d -quark probability distribution on the positions of the u quarks in the proton and its excitedstates. From left to right, the columns correspond to the ground, first, second and third S -wave excitations. The u quarks arefixed on the x -axis running from back right through front left through the centre of the plot. The u quarks are fixed a distanceof d and d = − d from the origin located at the centre. The distance between the quarks, d = d − d , increases from thetop row through to the bottom row, taking values 0, 2, 4, 6 and 8 times the lattice spacing a = 0 . IG. 9. The dependence of the d -quark probability distribution on the positions of the u quarks in the proton and its excitedstates. From left to right, the columns correspond to the ground, first, second and third S -wave excitations. The u quarks arefixed on the x -axis running from back right through front left through the centre of the plot. The u quarks are fixed a distanceof d and d = − d from the origin located at the centre. The distance between the quarks, d = d − d , increases from thetop row through to the bottom row, taking values 10, 12, 14, and 16 times the lattice spacing a = 0 . . Second Excitation For the second excited state observed herein, we againsee a shift of the probability density from the central peakto the next shell of the original 3 S -like wave function. At d = 6 a = 0 .
54 fm in the fourth row of Fig. 8, a similarenhancement in the first shell is observed as for the Roperat d = 8 a = 0 .
73 fm with strength in the probabilitydensity enhanced in the direction perpendicular to theseparation of the u quarks.The radius of the first shell about the central peakof the original distribution shrinks as the u quarks arepulled apart and at d = 8 a = 0 .
73 fm corresponding tothe bottom row of Fig. 8, the u quarks are now in the firstshell where four peaks are apparent. The original firstnode has shrunk to the centre and may have emerged,centered about each of the u quarks. Further evidenceof this process is discussed in the analysis of the thirdexcitation below. The second node of the original wavefunction now surrounds the four peaks.The u quarks approach the position of the second nodeof the wave function at d = 10 a = .
91 fm. The node isstill apparent in the front and back of the distribution,orthogonal to the u -quark separation axis. The peaks inthe probability distribution are still associated with thefirst shell surrounding the central peak of the originaldistribution with zero u -quark separation.By d = 12 a = 1 .
09 fm the quarks have moved beyondthe second node. The radius of node has reduced and canbe seen in the dark-blue regions at the centre of the plot.At d = 13 a = 1 .
18 fm the second node has collapsed tothe origin and explains the strong separation of the twopeaks observed at d = 14 a = 1 .
27 fm in the third row ofFig. 9. Even at the largest quark separations examined,the node structure of this state is apparent, suppressingthe probability density between the two peaks once againgoverned by scalar-diquark dynamics.
4. Third Excitation
The third excitation displays a wonderfully complexstructure that mirrors the transitions observed for thefirst and second excitations for the first few separations.For example at d = 4 a = 0 .
36 fm, one can see the en-hancement of the first shell in a direction orthogonal tothe u quark separation axis.At d = 6 a = 0 .
54 fm, a four-peak structure emergesas the u quarks enter the first shell of the wave function.Remarkably, a new nodal structure has emerged. Uponshrinking to the origin, the original first node emergedsurrounding each of the peaks at the u quark positions.This node now cuts through the first shell strength ofthe underlying 4 S configuration and divides what wouldnormally be a ring shape into four peaks.By d = 10 a = 0 .
91 fm the third node surrounds allsignificant structure in the distribution. The second node has shrunk to surround the small peak in the centre andthe first surrounds the u quark peaks.At d = 12 a = 1 .
09 fm the third node now surroundsboth of the major peaks and the fore and aft humps nearthe centre. The first node continues to surround eachof the peaks at the u quark positions, The emergence ofa second node around each of the u quarks is becomingapparent at the left- and right-hand edges of the plot.At d = 14 a = 1 .
27 fm the third node maintains a cir-cular structure centred about the origin and cuts throughthe ring like structures forming around each of the u quarks. The rings clearly reveal the shifting of the firstand second nodes to surround each of the u quarks.At the largest u -quark separation of d = 16 a = 1 .
45 fmthe third node has shrunk further to just touch the insideedges of the rings which have formed though the first andsecond nodes shrinking to the origin and emerging aroundthe two u quarks. V. CONCLUSIONS
In this first study of the quark probability distributionwithin excited states of the nucleon, we have shown thatall the states accessed in our correlation matrix analysisdisplay the node structure associated with radial excita-tions of the quarks. For example the first excited stateassociated with the Roper resonance displays a node inthe d quark wave function consistent with a radial exci-tation of the d quark. The second and third excitationsdisplay two and three nodes respectively.It is beautiful to observe the emergence of these cor-ner stones of quantum mechanics from the complex manybody theory of quantum field theory. The few-body pro-jection of the underlying physics can be connected withmodels, shedding light on the essential effective phenom-ena emerging from the complex dynamics of QCD.On comparing these probability distributions to thosepredicted by the constituent quark model, we find goodqualitative similarity with interesting differences. Thecore of the states is described very well by the model andthe amplitudes of the S -wave shells between the nodesare predicted very accurately by the constituent quarkmodel. The discovery of a node structure provides a deepunderstanding of the success of the smeared-source/sinkcorrelation matrix methods of Ref. [20].Finite volume effects are shown to be particularly sig-nificant for the excited states explored herein at relativelylight quark mass. As these excited states have a multi-particle component, the interplay between the lattice vol-ume, the wave function and the associated energy are keyto extracting the resonance parameters of the states.Fascinating structure in the d -quark probability distri-butions of the nucleon excited states is revealed when sep-arating the u quarks from the origin. As the u quarks areseparated the original node structure of the wave func-tion shrinks in size. For example, the Roper reveals aringed structure in the surface plots corresponding to an13mpty shell in three dimensions as the node collapses tothe origin. The second excited state reveals a four-peakedstructure at mid-range quark separations. At large sep-arations these states all display diquark clustering withthe d quark most likely found near one of the u quarks.The third state reveals the most exotic structure withnew nodes centred about the u quarks appearing afterthe original nodes collapsed to the origin.Future calculations will explore the structure of thesestates in more detail, examining the effect of the in-troduction of isospin-1/2 spin-3/2 interpolating fields[36, 37] to reveal the role of D -wave contributions. Whileour use of improved actions suppresses lattice discretiza-tion errors, ultimately simulations will be done at a va-riety of lattice spacings directly at the physical quarkmasses. An analysis of finite volume effects will also beinteresting to further reveal the interplay between the fi-nite volume of the lattice, the structure of the states andthe associated energy of the states; thus connecting the lattice QCD simulation results to the resonance physicsof Nature. ACKNOWLEDGEMENTS
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