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Do Quarks Really Form Diquark Clusters in the Nucleon?

arXiv:hep-ph/9302266v1 17 Feb 1993Do Quarks Really Form Diquark Clusters in the Nucleon?Derek B. LeinweberDepartment of Physics and Center for Theoretical PhysicsUniversity of Maryland, College Park, MD 20742AbstractA gauge invariant method for the investigation of scalar diquark clustering inthe nucleon ground state is presented. The method focuses on a comparisonof quark distributions in the nucleon with those in the ∆baryon resonance.Recent lattice QCD calculations of these quark distribution radii are analyzedin a search for evidence of scalar diquark clustering.

The analysis indicates thelattice results describe the negative squared charge radius of the neutron withlittle resort to hyperfine clustering between u-d-quark pairs. This result con-trasts both quark-diquark and nonrelativistic quark models where hyperfineattraction between u and d quarks in the nucleon is argued to play a signif-icant role.

Comparison of light quark distributions in Λ0 and Σ∗0 indicateonly a small reduction of the scalar diquark distribution radius relative to thevector diquark distribution. Current lattice QCD determinations of baryoncharge distributions do not support the concept of substantial u-d scalar di-quark clustering as an appropriate description of the internal structure of thenucleon.Typeset using REVTEX1

I. INTRODUCTIONThe one-gluon-exchange potential (OGEP) has been extensively used to describe thespin-dependent interactions of constituent quarks in low-energy phenomenology since itsinception [1].Among the earliest of OGEP successes is an explanation of the negativesquared charge radius of the neutron. Carlitz et al.

focused on the spin-dependent hyperfinerepulsion between the doubly represented d quarks in the neutron which naturally givesrise to a negative squared charge radius [2].Later analyses exploited the larger OGEPhyperfine attraction acting between constituent quark pairs in a scalar spin-0 state [3,4]. Inthese papers the attractive hyperfine force pulls the u-quark of the neutron into the centerwhile the two d-quarks are repelled to the periphery by the vector spin-1 repulsion of thehyperfine interaction.

These more quantitative analyses were also successful in accountingfor the neutron charge radius.The success of hyperfine interactions based on single-gluon exchange, led some authors tospeculate that the strong attractive hyperfine interaction pairs a u and d quark of the nucleoninto a scalar diquark [5] of considerable clustering [6–11]. It is often argued that point-likescalar diquarks naturally arise in QCD, or that scalar diquarks are energetically favored[12,13].

A factor in the popularity of diquark models is the numerical simplification of threebody problem to a two body problem [8,14,15]. Also the diquark model is not without along list of successful descriptions of low-energy hadronic phenomena.

For example, diquarkclustering can also explain the neutron charge radius.In this approach it is concludedthat the neutron has a scalar diquark core [16]. More recently it has been argued that aquark-diquark approximation of the three-quark structure of baryons is now available whichtakes into account the inner structure of baryons at low energies [8].

Still others argue thatdiquark models provide a natural forum for investigating deviations from SU(6)-spin-flavorsymmetry as low energy phenomena is not tractable from first principles [17].In this paper, a gauge invariant method for the examination of scalar diquark clustering inthe nucleon ground state is presented. Direct evidence indicating the absence of substantialdiquark clustering of u and d quarks in low-lying baryons will be presented.

Quark electriccharge distribution radii are calculated from first principles in lattice regularized QCD. Byexamining the quark distributions in octet baryons where quarks may form scalar diquarksand comparing these distributions with the relevant decuplet baryons where quarks arepredominantly paired in vector diquarks, one can determine the amount, if any, of scalardiquark clustering [18].

A central point of this paper is that it is a comparison of chargedistribution radii between octet and decuplet baryons (as opposed to within the baryonoctet) that reveals whether or not hyperfine interactions lead to scalar diquark clustering.Of course this is not the first paper to refute the diquark picture of low-energy baryons.As early as 1981, an analysis of πN partial widths suggested that diquark configurations donot contribute appreciably to the structure of low-lying resonances [19]. More recently it hasbeen argued that point-like scalar diquarks are unlikely in a relativistic bound-state quarkmodel [20] and that there is no clear indication of diquark clustering in a non-relativisticquark model of the nucleon [21].In some systems there is good reason to believe that some diquark clustering can occur.For example, in Ξ the two strange quarks are more localized due to their heavier mass [22].In high angular momentum systems, small attraction between quarks can lead to clustering2

[21]. However, the scalar diquark clustering argued to arise from the attractive part of thehyperfine interaction is not reproduced in the nonperturbative analysis presented here.In section II A, descriptions of the neutron charge radius in a quark-diquark model,nonrelativistic constituent quark model and lattice QCD calculations are compared.

Thepurpose of this section is to illustrate that the three different approaches lead to valuesfor the neutron to proton charge radius ratio which are consistent with each other andthat the lattice results are not inconsistent with the experimentally determined value. Theconclusion of this section is that reproduction of the neutron to proton charge radius ratioalone is insufficient to discriminate between the various quark distribution pictures.

SectionII B introduces the charge distributions in ∆to discriminate between these three differentdescriptions of quark distributions in the nucleon. Here the absence of significant scalardiquark clustering in the lattice results is discussed.

Section III explores charge distributionsin Λ0 and Σ∗0 hyperons.A similar analysis indicates the absence of substantial scalardiquark clustering in these hyperons. Section IV considers the possible sources of systematicuncertainty in the lattice calculations and how these uncertainties may affect the chargedistribution radii used in this analysis.

Finally the implications of the results are discussedand summarized in section V.II. SCALAR DIQUARK CLUSTERING IN THE NUCLEONA.

The Neutron Charge RadiusTo investigate the issue of scalar diquark clustering, we turn to the lattice investigationsof baryon electromagnetic structure of Ref. [22,23].

These lattice results are obtained in anumerical simulation of quenched QCD on a 24 × 12 × 12 × 24 lattice at β = 5.9 using Wil-son fermions. Twenty-eight gauge configurations are used in the analysis.

Charge radii areobtained from lattice calculations of electromagnetic form factors at ⃗q 2 = 0.16 GeV2. TheQ2 dependence of the form factors is taken as a dipole form.

However, monopole or linearforms yield similar radii which agree with the dipole results within statistical uncertainties.A third order, single elimination jackknife [24,25] is used to determine the statistical uncer-tainties in the lattice results. These uncertainties are indicated in parentheses describingthe uncertainty in the last digit(s) of the results.The lattice results have displayed some of the qualitative features expected from hyper-fine interactions.

Let us first focus on the mean square charge radius of the neutron asdescribed by the quark-diquark model [16], the nonrelativistic quark model [3,4] and thelattice simulations of Ref. [22,23].

The diquark description of quark distributions is as illus-trated in figure 1a. A u and d quark form a scalar diquark cluster whose radius in extremecases may be taken to be point like.

The second d quark has a larger charge distributionradius and provides the required negative charge at large radii surrounding a positive coreof charge +1/3.In the nonrelativistic quark model a similar scalar diquark clustering occurs between uand d quarks due to the attractive part of the hyperfine interaction. However this clusteringis relatively small compared to the clustering anticipated in diquark models.

The remainingd-quark is driven to the periphery of the neutron by the repulsive part of the hyperfine3

interaction. The clustering is averaged over the two d quarks of the neutron and the quarkdistributions may be described as in figure 1b.

An important point to note is that the hy-perfine interaction causes two alterations to the unperturbed wave function. The attractionbetween the u and d quarks causes the whole nucleon to shrink and the repulsion betweenthe two d quarks introduces the required asymmetry between the quark distributions toreproduce the neutron charge radius.In understanding the results of the lattice investigation the negative mean square chargeradius of the neutron was attributed to hyperfine repulsion between the doubly representedd quarks without resorting to any hyperfine attraction [22,26].

This argument is similar tothe original explanation provided by Carlitz et al. [2].

Of course it is the hyperfine attractionthat is argued to naturally give rise to scalar diquark clustering in the nucleon. The currentpicture of quark distributions within the neutron is described as in figure 1c.

The differencebetween figures 1b and 1c is the possible absence of any hyperfine attraction between uand d quarks. Of course the lattice results for the neutron charge radius alone may also beconsistent with either of the previous two descriptions [27].The presence of two degrees of freedom, namely the matter radius and the charge asym-metry, allow these three different descriptions of quark distributions within the neutron tolie in quantitative agreement when reproducing the neutron to proton charge radius ratio.Experimental measurements [28,29] of nucleon mean square charge radii produce a ratio of⟨r2 ⟩n⟨r2 ⟩p= −0.167(7) .

(1)The quark-diquark prediction of the mean square neutron to proton charge radius ratio is⟨r2 ⟩n⟨r2 ⟩p= −0.137(9) ,(2)when the parameter of the model is fixed by previous analyses [16]. Similarly the configu-ration mixing induced by hyperfine interactions in the nonrelativistic quark model [3] leadsto the result⟨r2 ⟩n⟨r2 ⟩p= −0.13 .

(3)Present lattice simulations calculate hadron properties at relatively heavy current quarkmasses due to the increasing computational demands encountered in inverting the fermionmatrix as the quarks become lighter. In Ref.

[22,23] charge radii are determined at threevalues of the Wilson hopping parameter κ = 0.152, 0.154, and 0.156 corresponding to currentquark masses ranging from the strange current quark mass ms through to ms/2. To makecontact with the physical world the lattice results are extrapolated, usually linearly as afunction of 1/κ which is proportional to the current quark mass or m2π, to the point atwhich the physical pion mass is reproduced.

Results from chiral perturbation theory [30–32]suggest that logarithmic terms divergent in the limit mπ →0 should be included in theanalytic structure of the nucleon charge radius extrapolation function, in addition to termslinear in m2π and other higher order terms finite in the chiral limit. However, the physics of4

light pions giving rise to the divergent logarithmic term is not included in either of the quarkmodels considered here. To allow a comparison with these models on a more equal footing,the lattice extrapolations of charge radii are done linearly in m2π, effectively subtractingthe nonanalytic contributions associated with light pion dressings from the charge radii.For linear extrapolations, the difference between extrapolations of quantities to the physicalpion mass as opposed to mπ = 0 are negligible.

In the following, radii are extrapolated toκcr = 0.159 8(2) where the pion mass vanishes.The neutron charge radius is sensitive to gluonic degrees of freedom and the lattice ratiohas a large statistical error associated with it. However, it is important to demonstrate thatthe lattice result is consistent with the previous analyses.

The lattice results indicate [23]⟨r2 ⟩n⟨r2 ⟩p= −0.11+0.10−0.14. (4)The qualitative features of the neutron charge radius anticipated by hyperfine interactionsare reproduced in each of the calculations considered.

Obviously, reproduction of the neutronto proton charge radius ratio alone is not sufficient to determine whether it is hyperfineattraction, repulsion, or a combination of the two that is actually responsible for giving riseto the neutron charge radius. If attraction plays a significant role then diquark-clusteringmodels may capture the essential features of the underlying quark-gluon dynamics.

On theother hand, if the lattice results indicate hyperfine attraction does not play a significantrole in determining the charge distributions then the realization of diquark clustering in thenucleon ground state is doubtful.B. Charge Distributions in ∆To discriminate between these three different descriptions of quark distributions withinthe nucleon we must turn to another system where the attractive part of the hyperfineinteraction does not play a significant role, i.e.

the ∆baryon resonance. By examiningthe changes in the quark distributions as the spin of the singly represented quark is setsymmetric with the doubly represented quarks we can investigate the relevance of scalardiquark clustering and search for the anticipated effects of one-gluon-exchange hyperfineinteractions on the quark distributions.Consider the quark distributions within the proton and how they will change in goingfrom p to ∆+.

Figure 2 illustrates the three previous scenarios for the proton as well as thequark distributions in ∆+. Hyperfine interactions are not expected to give rise to vectordiquark clustering.

Both the nonrelativistic quark model and the lattice results describethe ∆+ charge distribution as a sum of three equivalent quark distributions. In fact thesymmetry of the ∆three-point correlation function demands the quark distribution radii in∆to be equal under SU(2)-isospin symmetry.

This symmetry is manifest without resortingto actual calculations on the lattice [23].In the quark-diquark model the scalar diquark cluster is lost in ∆+ and the net positivecharge of the cluster in the nucleon moves to larger radii. For point-like diquarks the neteffect is huge, resulting in a much larger charge radius for ∆+ than for the proton.

Moreover,both u- and d-quark charge distributions swell in ∆due to the breakup of the (point-like)5

u-d-quark cluster. Unfortunately quantification of these statements does not appear to bepossible.

A comparison of p and ∆+ charge radii in a diquark model does not appear to havebeen considered. Some form factors of octet and decuplet baryons were recently examinedin the quark-diquark approximation [8], however electric form factors and charge radii for∆resonances were not reported.

To estimate a lower bound for the size of these anticipatedquark distribution swellings, we refer to the nonrelativistic quark model where the role ofhyperfine attraction plays a much weaker and less dramatic role.In the nonrelativistic quark model, both the u- and d-quark distributions become broaderas the attraction between u and d quarks is replaced by hyperfine repulsion in ∆. In themodel of Isgur-Karl-Koniuk [3] the predicted increase in the rms charge radius is [33,34]r∆rp= 1.28 ,(5)with the quark distributions experiencing a large swelling ofru∆rup= 1.33 ,rd∆rdp= 1.49 .

(6)In the scenario previously introduced for the lattice results, the dominant effect willbe the new hyperfine repulsion experienced by the d quark from the two u quarks in ∆+.Similarly the two u quarks already experiencing some hyperfine repulsion will feel additionalrepulsion from the single d quark. Hence the dominant effect will be the broadening of thenegatively charged d-quark distribution compensated by some broadening of the u-quarkdistribution.

In fact, the lattice results suggest that the ∆+ charge radius may actually besmaller than that of the protonr∆rp= 0.97(7) . (7)This result differs by 4 standard deviations from the prediction of the nonrelativistic quarkmodel.

The quark distribution radii indicate the dominant effect in the lattice results is thebroadening of the negatively charged d-quark distributionru∆rup= 1.01(9) ,rd∆rdp= 1.12(16) . (8)The striking difference between the lattice results and the two models considered hereis the absence of any significant change in the lattice u-quark distribution radius.

In bothmodels, the u-quark distribution was predicted to be broader in ∆, largely due to thedisappearance of scalar diquark clustering in going from the nucleon to ∆. The lattice resultsindicate that hyperfine attraction does not lead to substantial scalar diquark clustering inthe nucleon ground state [35].Figure 3 illustrates the extrapolation of quark distribution radii in p and ∆+ used toobtain the results of (8).

The u-quark distribution radii in p and ∆are nearly identicalfor each κ considered. Table I details the charge distribution radii for p and ∆and theirresiding quarks.6

III. SCALAR DIQUARK CLUSTERING IN Λ0Another place to search for evidence of scalar diquark clustering is in the charge dis-tribution radius of the light u and d quarks in Λ0.

In simple quark models these quarksare generally taken as a pure scalar diquark. In other words, in this approximation the Λ0magnetic moment is given by the intrinsic magnetic moment of the strange quark alone.The lattice results suggest that the u and d quarks do not form a pure scalar diquark andmay actually contribute to the Λ0 magnetic moment at the level of 10%.

However, for thepresent purpose we will ignore such effects.In the decuplet Σ∗0 hyperon state the u-d sector will be paired predominately as a vectordiquark. Hence with the assumption that the s quark plays a spectator role, comparisonof the light quark sector distribution radii in Λ0 and Σ∗0 will give some indication of therelevance of scalar diquark clustering.

If a scalar diquark is “dissolved” in going from Λ0 toΣ∗0, a broadening of the light quark distribution in Σ∗0 is expected. The ratio of the lightquark distributions from the lattice analyses [22,23] indicatesrlΣ∗0rlΛ= 1.04(10) ,(9)confirming the conclusions based on the N-∆quark distributions.

Once again substantialdiquark clustering is not seen.Figure 4 illustrates the extrapolation of quark distribution radii in Λ and Σ∗0 used toobtain the results of (9). The strange quark appears to play a satisfactory spectator role.Little change is seen between the combined light-quark distributions of Λ and Σ∗0 at eachvalue of κ.

Numerical values and statistical uncertainties for these radii are summarized intable II.IV. SYSTEMATIC UNCERTAINTIESThis comparison of charge distributions in octet and decuplet baryons has revealed re-markable differences between the lattice results and the anticipated role of hyperfine attrac-tion based on the OGEP.

Moreover, this analysis appears to render a diquark picture ofquark distributions in the nucleon ground state obsolete. As a result, it is important to con-sider the possible sources of systematic uncertainty in the lattice calculations.

Systematicuncertainties may have their origin in the quenched approximation, the extrapolation of thelattice results to the physical regime, finite volume effects or renormalization group scalingdeviations.It is worth noting that all of the lattice results presented to this point have involvedratios of lattice results. It is expected that the effects of these possible sources of systematicuncertainty will be suppressed in taking ratios.

For example, while the absolute values ofthe lattice predictions of magnetic moments [22] are small compared to the experimentallymeasured values, the lattice results predict ratios of the baryon magnetic moments to theproton moment as good as or better than models, which in most cases have parameterstuned to reproduce the experimental moments. A more detailed discussion of systematicuncertainties follows.7

A calculation of electromagnetic form factors in full QCD has not been done. However,at the present values of quark mass investigated on the lattice, hadron spectrum analysessuggest the dominant new physics in full QCD is a simple renormalization of the strongcoupling constant.

It is possible that over large distances the quenched approximation doesnot screen quark interactions as much as required by full QCD. However given the similarityof the quark distribution radii discussed in this paper it is unlikely that there would besignificant deviations from the results presented here.The extrapolation of the results to the physical regime is another source of concern.However, the arguments based on the extrapolated results are supported at each value ofquark mass investigated on the lattice as indicated in tables I and II.

Because the actuallattice calculations are done with u- and d-current-quark masses the order of the strange-current-quark mass, one might be concerned that hyperfine interactions such as those ofthe OGEP are suppressed in the lattice calculations to the point that one has no hope ofobserving diquark clustering. Fortunately, this is unlikely to be the case.To assess this issue more quantitatively, one can resort to the one-gluon-exchange hy-perfine interaction term which is inversely proportional to the product of quark masses.

Animportant point is the OGEP has some relevance only in phenomenology where constituentquark masses are used in the hyperfine term. While the current quark masses used in thisinvestigation are somewhat heavy, the corresponding constituent quark masses are not toodifferent from that used in phenomenology.Perhaps the easiest way to quantify this isto estimate the constituent quark mass to be one third the lattice proton mass.

At thelightest quark mass considered on the lattice, the constituent quark mass is approximately430 MeV which is not too different from the value obtained from the physical mass at 313MeV. If the hyperfine term of the one-gluon-exchange interaction is relevant, then the dif-ferences in the strength of the hyperfine interactions for these two quark masses are withina factor of 2.

For the intermediate quark mass value the strength is within a factor of 2.5.Hence, there is good reason to believe an extrapolation to the physical value is adequate.Moreover, significant mass splitting is observed between the nucleon and ∆on the latticewhere MN/M∆= 0.87(8). However, the change in the u-quark distribution radius remainsnegligible.

Perhaps it is also worth noting that single-gluon-exchange interaction strengthsbetween current quarks in this lattice QCD calculation exceed the strengths common to phe-nomenological analyses by an order of magnitude for the lightest quark mass investigatedon the lattice.It has been argued in the past that the finite volume of the lattice may affect the latticeresults, as surrounding periodic images of the baryon under study may restrict the baryon’ssize [22]. Note however, the u-quark distribution in ∆is not the broadest quark distributionseen on the lattice.

In addition, similar effects are seen for heavier quark masses where thequarks are more localized and less sensitive to the boundaries of the lattice. For a givenvalue of κ, the charge distribution radii calculated on the lattice tend to be similar in size.For this reason the finite volume of the lattice is more likely to affect the lattice results in aglobal manner.

For the case of a scalar u-d diquark cluster in p breaking up in ∆+, a finitevolume effect could not simultaneously accommodate a swelling of the d-quark distributionand yet restrict the u-quark distribution from following a similar swelling.The issue of deviations from asymptotic scaling is difficult to assess quantitatively sincea similar calculation at finer lattice spacings has not been done. Toussaint [36] has demon-8

strated that the nucleon to ρ mass ratio calculated at mπ/mρ = 0.6 is, to a good approxi-mation, independent of the value of β over a range of β = 5.7 to 6.3 for quenched Wilsonfermion calculations. Furthermore, Yoshie et al.

[37] have demonstrated that it is possible toreproduce the hadron spectrum in quenched QCD at β = 6.0 within statistical uncertaintiesof approximately 10% [36].The lattice prediction of the current investigation for the ∆/N mass ratio lies 12% or1.4 σ below the experimental ratio. One could adjust the parameters of the nonrelativisticquark model to reproduce the lattice N-∆splitting in which case the difference between thelattice and model predictions for baryon charge radii is reduced to 2σ.

However, the trendof the lattice results indicates that a doubling of the effects seen in the lattice results onlywidens the gap between the predictions of ∆+/p charge radii ratios and leaves the changein the u-quark radius in going from p to ∆+ at the order of 2%.V. DISCUSSION AND SUMMARYFaced with the discrepancies of these models which both lack mesonic degrees of freedomit is interesting to consider models such as the cloudy bag [38], or hedgehog models suchas the Skyrmion [39], hybrid (or little) bag [40] and the chiral-quark meson model [41].

Inhedgehog models, the proton and ∆+ radii are equal by construction. The difference incharge radii are 1/Nc suppressed and cannot be calculated using conventional semi-classicaltechniques.

It may be useful to note that the lattice results suggest that such higher ordereffects may be small.A calculation of the charge radii considered here in the cloudy bag model could providefurther insight to the importance of mesonic degrees of freedom in describing baryon chargedistributions. Since a large part of the neutron charge radius has its origin in the pion cloudof the cloudy bag model [42,43], it may be possible to circumvent the problems encounteredhere with hyperfine interactions.

Of course, to allow a direct comparison with the latticeresults and to avoid problems associated with open channel physics, the cloudy bag analysisshould be done with heavier pions. The ∆+ charge radius and internal quark distributionsprovide new additional information on the spin-dependence of quark interactions.Thetraditional models examined here do not reproduce the lattice results and a description ofthese phenomena remains an open challenge to models of QCD.Some have argued the existence of scalar diquarks based on large momentum transferphenomenology.

An analysis of scalar and vector diquarks at large momentum transfersin lattice QCD may provide useful insight. The results presented here may be of interestto those investigating large Q2 phenomena using nucleon ground state wave functions as amodel input representing the nonperturbative parts of the calculation [9,44].Finally, a few comments on the absence of significant quark clustering in the latticeresults relative to that anticipated by quark models are in order.

The first and most obvi-ous comment to make is that baryon charge distributions are sensitive to the long distancenonperturbative aspects of QCD. It should not be too surprising to find that hyperfineinteractions based on a single-gluon-exchange become virtually irrelevant in the nonpertur-bative regime.

It is well known that the one-gluon-exchange hyperfine interaction has somerelevance only when the quark masses are taken to be constituent quark masses. Exper-9

imental mass splittings indicate it is not the relevant interaction between current quarks.There is an infinite class of multiple-gluon-exchange diagrams to consider beyond the ex-change of a single gluon. Moreover, these diagrams are equally important since the theoryis nonperturbative.A second and possibly more interesting point concerns the spin of the singly representedquark in the nucleon.

From the lattice QCD analyses of baryon magnetic moments [22,23,45],it has become clear that the behavior of the singly represented quark in octet baryons isvery different from the predictions of SU(6)-spin-flavor symmetry. Briefly stated the latticeresults indicate that the proton moment is better described asµp = 43µu −16µd ,(10)where in constituent quark model language, the d quark has its net spin opposite that of theu quarks about half as much as suggested by SU(6).

In relation to scalar diquark clusteringthe probability of finding u and d quarks paired in a spin-0 state has been reduced. Onceagain the lattice results suggest that scalar diquark degrees of freedom do not provide anappropriate description of the internal quark structure of low-lying baryons.A gauge invariant method for the examination of scalar diquark clustering in the nucleonground state has been presented.

Results from lattice QCD describing the distributions ofquarks in the baryon octet and decuplet have been analyzed in a search for evidence of scalardiquark clustering. The results presented here contrast the predictions of the nonrelativisticquark model which has a relatively small diquark clustering compared to that demanded inquark-diquark models.

The lattice results do not support the concept of substantial diquarkclustering as an appropriate description of the internal structure of low-lying baryons.ACKNOWLEDGMENTSI wish to thank Zbigniew Dziembowski for stirring my interest in this subject. I also thankWojciech Broniowski, Tom Cohen, Manoj Banerjee, Nathan Isgur and Simon Capstick for anumber of interesting and helpful discussions.

Financial support from the U.S. Departmentof Energy under grant DE-FG05-87ER-40322 is gratefully acknowledged.10

REFERENCES[1] A. DeR´ujula, H. Georgi, and S. L. Glashow, Phys. Rev.

D 12, 147 (1975). [2] R. D. Carlitz, S. D. Ellis, and R. Savit, Phys.

Lett. B68, 433 (1977).

[3] N. Isgur, G. Karl, and R. Koniuk, Phys. Rev.

Lett. 41, 1269 (1978), ibid.

45, 1738 (E)(1980). [4] N. Isgur, G. Karl, and D. W. L. Sprung, Phys.

Rev. D 23, 163 (1981).

[5] D. B. Lichtenberg, Phys. Rev.

178, 2197 (1969). [6] K. S. Sateesh, Phys.

Rev. D 45, 866 (1992).

[7] M. Anselmino and E. Predazzi, Phys. Lett.

B254, 203 (1991). [8] G. V. Efimov, M. A. Ivanov, and V. E. Lyubovitskij, Z. Phys.

C 47, 583 (1990). [9] Z. Dziembowski and J. Franklin, Phys.

Rev. D 42, 905 (1990).

[10] H. G. Dosch, M. Jamin, and B. Stech, Z. Phys. C 42, 167 (1989).

[11] S. Fredriksson, Phys. Rev.

Lett. 52, 724 (1984).

[12] S. Fredriksson and T. Larsson, Phys. Rev.

D 28, 255 (1983). [13] S. Fredriksson, M. J¨andel, and T. Larsson, Z. Phys.

C 14, 35 (1982). [14] D. B. Lichtenberg, W. Namgung, E. Predazzi, and J. G. Wills, Phys.

Rev. Lett.

48,1653 (1982). [15] D. B. Lichtenberg, E. Predazzi, and J. G. Wills, Z. Phys.

C 17, 57 (1983). [16] Z. Dziembowski, W. J. Metzger, and R. T. Van de Walle, Z. Phys.

C 10, 231 (1981). [17] T. Uppal and R. C. Verma, Z. Phys.

C 52, 307 (1991). [18] Note that an individual quark flavor distribution radius is a gauge invariant quantity.The radius may be regarded as a baryon charge radius in which the remaining quarkflavors have zero electric charge.

[19] C. P. Forsyth and R. E. Cutkosky, Nucl. Phys.

B178, 35 (1981). [20] B. Ram and V. Kriss, Phys.

Rev. D 35, 400 (1987).

[21] S. Fleck, B. Silvestre-Brac, and J. M. Richard, Phys. Rev.

D 38, 1519 (1988). [22] D. B. Leinweber, R. M. Woloshyn, and T. Draper, Phys.

Rev. D 43, 1659 (1991).

[23] D. B. Leinweber, T. Draper, and R. M. Woloshyn, Phys. Rev.

D 46, 3067 (1992). [24] B. Efron, SIAM Rev.

21, 460 (1979). [25] S. Gottlieb, P. B. MacKenzie, H. B. Thacker, and D. Weingarten, Nucl.

Phys. B263,704 (1986).

[26] T. Draper, R. M. Woloshyn, and K. F. Liu, Phys. Lett.

B234, 121 (1990). [27] To this point we do not have sufficient information to exclude the possibility of a sig-nificant role for hyperfine attraction in the lattice results.

However, it will become clearthrough the analysis of ∆-baryons that hyperfine attraction does not play a significantrole in determining the quark charge distribution radii. [28] G. H¨ohler et al., Nucl.

Phys. B114, 505 (1976).

[29] O. Dumbrajs, Nucl. Phys.

B216, 277 (1983). [30] M. A.

B. B´eg and A. Zepeda, Phys. Rev.

D 6, 2912 (1972). [31] D. B. Leinweber and T. D. Cohen, Chiral Corrections to Lattice Calculations of ChargeRadii.

U. Maryland preprint 92-190. Accepted for publication in Phys.

Rev. D.[32] T. D. Cohen and D. B. Leinweber, The Pion Cloud in Quenched QCD.

U. Marylandpreprint 93-019. Recommended for publication in Comments on Nuclear and ParticlePhysics.11

[33] N. Isgur and G. Karl, Phys. Rev.

D 21, 3175 (1980). [34] Here we have taken θ∆= −θp, and φ∆= 0.

Equation (13) of Ref. [3] has been adjustedas described in the errata.

[35] In the preceding arguments, the focus in the lattice scenario has been on hyperfinerepulsion. However, the lattice results are also consistent with a mixture of hyperfineattraction and repulsion or even exclusive hyperfine attraction provided the clustering issmall.

The important point to draw from this examination of charge radii is that thereis no evidence of significant u-d-quark clustering in the nucleon ground state quarkdistributions. [36] D. Toussaint, The QCD Spectrum and Phase Diagram on the Lattice – 1991.

To appearin the Proceedings of the international symposium LATTICE-91, Tsukuba, Japan, Nov.1991, Nucl. Phys.

B (Proc. Suppl.

), U. Arizona preprint AZPH-TH/92-01. [37] T. Yoshie, Y. Iwasaki, and S. Saki, Quenched Hadron Spectrum on a 243 × 54 Lattice.Presented at the internation symposium LATTICE-91, Tsukuba, Japan, Nov. 1991 byT.

Yoshie. (unpublished).

[38] A. W. Thomas, in Advances in Nuclear Physics, Vol. 13, edited by J. Negle and E. Vogt(Plenum Press, New York, 1984).

[39] I. Zahed and G. E. Brown, Phys. Rep. 142, 1 (1986).

[40] L. Vepstas and A. D. Jackson, Phys. Rep. 187, 109 (1990).

[41] M. K. Banerjee, W. Broniowski, and T. D. Cohen, in Chiral Solitons, edited by K. F.Liu (World Scientific, Singapore, 1987). [42] S. Theberge, A. W. Thomas, and G. A. Miller, Phys.

Rev. D 22, 2838 (1980).

[43] F. Myhrer, Phys. Lett.

110B, 353 (1982). [44] H. Meyer and P. J. Mulders, Nucl.

Phys. A528, 589 (1991).

[45] D. B. Leinweber, Phys. Rev.

D 45, 252 (1992).12

TABLESTABLE I. Rms charge distribution radii normalized to unit charge in lattice units ⟨r2/a2 ⟩1/2.Baryonκ1 = 0.152κ2 = 0.154κ3 = 0.156κcr = 0.159 8(2)p3.70(13)4.04(18)4.39(33)5.06(48)∆+3.71(13)4.02(19)4.34(39)4.90(57)up3.69(12)4.00(18)4.28(38)4.86(57)u∆+3.71(13)4.02(19)4.34(39)4.90(57)dp3.63(14)3.86(21)3.99(66)4.39(82)d∆+3.71(13)4.02(19)4.34(39)4.90(57)TABLE II.Rms charge distribution radii normalized to unit charge in lattice units⟨r2/a2 ⟩1/2. lΛ indicates the combined light u and d quarks in Λ0 and similarly for Σ∗0.Baryonκ1 = 0.152κ2 = 0.154κ3 = 0.156κcr = 0.159 8(2)lΛ3.65(13)3.94(18)4.18(44)4.67(73)lΣ∗3.71(13)4.03(18)4.40(33)4.99(48)sΛ3.70(13)3.66(15)3.59(22)3.51(29)sΣ∗3.71(13)3.68(14)3.65(20)3.60(28)13

FIGURESFIG. 1.

Sketches of quark distribution radii in the neutron as described by a) the quark-diquarkmodel, b) the nonrelativistic quark model and c) the lattice investigations of baryon electromagneticform factors. The dashed lines are representative of the rms charge radius of the quark distributionsnormalized to unit charge.FIG.

2. Sketches of quark distribution radii in the proton as described by a) the quark-diquarkmodel, b) the nonrelativistic quark model and c) the lattice investigations of baryon electromag-netic form factors.d) An illustration of the quark distributions in ∆+.The dashed lines arerepresentative of the rms charge radius of the quark distributions normalized to unit charge.FIG.

3.Extrapolation of quark distribution radii in p and ∆calculated at the three (rightmost)values of κ to κcr where mπ vanishes. In this and the following figure, the lattice radii (r/a) havebeen scaled by a constant lattice spacing of a = 0.128 fm determined from the nucleon mass.For clairity, the u-quark radii are normalized to unit charge and d-quark radii are normalized tonegative unit charge.

The u-quark distribution radii in p and ∆are nearly identical at each valueof κ.FIG. 4.Extrapolation of quark distribution radii in Λ0 and Σ∗0.

The distribution radius ofthe combined light u and d quarks of Λ0 are denoted by lΛ and similarly for Σ∗0. The light-quarkradii are normalized to unit charge and strange-quark radii are normalized to negative unit charge.For each κ, lΛ ≃lΣ∗indicating the absence of significant scalar diquark clustering.14

SubmittedtoPhys.Rev.DDOE/ER/0-U.ofMD.PP#-00DoQuarksReallyFormDiquarkClustersintheNucleon?DerekB.LeinweberDepartmentofPhysicsandCenterforTheoreticalPhysicsUniversityofMaryland,CollegePark,MD0AbstractAgaugeinvariantmethodfortheinvestigationofscalardiquarkclusteringinthenucleongroundstateispresented.Themethodfocusesonacomparisonofquarkdistributionsinthenucleonwiththoseinthebaryonresonance.RecentlatticeQCDcalculationsofthesequarkdistributionradiiareanalyzedinasearchforevidenceofscalardiquarkclustering.Theanalysisindicatesthelatticeresultsdescribethenegativesquaredchargeradiusoftheneutronwithlittleresorttohyperneclusteringbetweenu-d-quarkpairs.Thisresultcon-trastsbothquark-diquarkandnonrelativisticquarkmodelswherehyperneattractionbetweenuanddquarksinthenucleonisarguedtoplayasignif-icantrole.Comparisonoflightquarkdistributionsin0and0indicateonlyasmallreductionofthescalardiquarkdistributionradiusrelativetothevectordiquarkdistribution.CurrentlatticeQCDdeterminationsofbaryonchargedistributionsdonotsupporttheconceptofsubstantialu-dscalardi-quarkclusteringasanappropriatedescriptionoftheinternalstructureofthenucleon.TypesetusingREVTEX

I.INTRODUCTIONTheone-gluon-exchangepotential(OGEP)hasbeenextensivelyusedtodescribethespin-dependentinteractionsofconstituentquarksinlow-energyphenomenologysinceitsinception[].AmongtheearliestofOGEPsuccessesisanexplanationofthenegativesquaredchargeradiusoftheneutron.Carlitzetal.focusedonthespin-dependenthypernerepulsionbetweenthedoublyrepresenteddquarksintheneutronwhichnaturallygivesrisetoanegativesquaredchargeradius[].LateranalysesexploitedthelargerOGEPhyperneattractionactingbetweenconstituentquarkpairsinascalarspin-0state[,].Inthesepaperstheattractivehyperneforcepullstheu-quarkoftheneutronintothecenterwhilethetwod-quarksarerepelledtotheperipherybythevectorspin-repulsionofthehyperneinteraction.Thesemorequantitativeanalyseswerealsosuccessfulinaccountingfortheneutronchargeradius.Thesuccessofhyperneinteractionsbasedonsingle-gluonexchange,ledsomeauthorstospeculatethatthestrongattractivehyperneinteractionpairsauanddquarkofthenucleonintoascalardiquark[]ofconsiderableclustering[{].Itisoftenarguedthatpoint-likescalardiquarksnaturallyariseinQCD,orthatscalardiquarksareenergeticallyfavored[,].Afactorinthepopularityofdiquarkmodelsisthenumericalsimplicationofthreebodyproblemtoatwobodyproblem[,,].Alsothediquarkmodelisnotwithoutalonglistofsuccessfuldescriptionsoflow-energyhadronicphenomena.Forexample,diquarkclusteringcanalsoexplaintheneutronchargeradius.Inthisapproachitisconcludedthattheneutronhasascalardiquarkcore[].Morerecentlyithasbeenarguedthataquark-diquarkapproximationofthethree-quarkstructureofbaryonsisnowavailablewhichtakesintoaccounttheinnerstructureofbaryonsatlowenergies[].StillothersarguethatdiquarkmodelsprovideanaturalforumforinvestigatingdeviationsfromSU()-spin-avorsymmetryaslowenergyphenomenaisnottractablefromrstprinciples[].Inthispaper,agaugeinvariantmethodfortheexaminationofscalardiquarkclusteringinthenucleongroundstateispresented.Directevidenceindicatingtheabsenceofsubstantialdiquarkclusteringofuanddquarksinlow-lyingbaryonswillbepresented.QuarkelectricchargedistributionradiiarecalculatedfromrstprinciplesinlatticeregularizedQCD.Byexaminingthequarkdistributionsinoctetbaryonswherequarksmayformscalardiquarksandcomparingthesedistributionswiththerelevantdecupletbaryonswherequarksarepredominantlypairedinvectordiquarks,onecandeterminetheamount,ifany,ofscalardiquarkclustering[].Acentralpointofthispaperisthatitisacomparisonofchargedistributionradiibetweenoctetanddecupletbaryons(asopposedtowithinthebaryonoctet)thatrevealswhetherornothyperneinteractionsleadtoscalardiquarkclustering.Ofcoursethisisnottherstpapertorefutethediquarkpictureoflow-energybaryons.Asearlyas,ananalysisofNpartialwidthssuggestedthatdiquarkcongurationsdonotcontributeappreciablytothestructureoflow-lyingresonances[].Morerecentlyithasbeenarguedthatpoint-likescalardiquarksareunlikelyinarelativisticbound-statequarkmodel[0]andthatthereisnoclearindicationofdiquarkclusteringinanon-relativisticquarkmodelofthenucleon[].Insomesystemsthereisgoodreasontobelievethatsomediquarkclusteringcanoccur.Forexample,inthetwostrangequarksaremorelocalizedduetotheirheaviermass[].Inhighangularmomentumsystems,smallattractionbetweenquarkscanleadtoclustering

[].However,thescalardiquarkclusteringarguedtoarisefromtheattractivepartofthehyperneinteractionisnotreproducedinthenonperturbativeanalysispresentedhere.InsectionIIA,descriptionsoftheneutronchargeradiusinaquark-diquarkmodel,nonrelativisticconstituentquarkmodelandlatticeQCDcalculationsarecompared.Thepurposeofthissectionistoillustratethatthethreedierentapproachesleadtovaluesfortheneutrontoprotonchargeradiusratiowhichareconsistentwitheachotherandthatthelatticeresultsarenotinconsistentwiththeexperimentallydeterminedvalue.Theconclusionofthissectionisthatreproductionoftheneutrontoprotonchargeradiusratioaloneisinsucienttodiscriminatebetweenthevariousquarkdistributionpictures.SectionIIBintroducesthechargedistributionsintodiscriminatebetweenthesethreedierentdescriptionsofquarkdistributionsinthenucleon.Heretheabsenceofsignicantscalardiquarkclusteringinthelatticeresultsisdiscussed.SectionIIIexploreschargedistributionsin0and0hyperons.Asimilaranalysisindicatestheabsenceofsubstantialscalardiquarkclusteringinthesehyperons.SectionIVconsidersthepossiblesourcesofsystematicuncertaintyinthelatticecalculationsandhowtheseuncertaintiesmayaectthechargedistributionradiiusedinthisanalysis.FinallytheimplicationsoftheresultsarediscussedandsummarizedinsectionV.II.SCALARDIQUARKCLUSTERINGINTHENUCLEONA.TheNeutronChargeRadiusToinvestigatetheissueofscalardiquarkclustering,weturntothelatticeinvestigationsofbaryonelectromagneticstructureofRef.[,].TheselatticeresultsareobtainedinanumericalsimulationofquenchedQCDonalatticeat=:usingWil-sonfermions.Twenty-eightgaugecongurationsareusedintheanalysis.Chargeradiiareobtainedfromlatticecalculationsofelectromagneticformfactorsat~q=0:GeV.TheQdependenceoftheformfactorsistakenasadipoleform.However,monopoleorlinearformsyieldsimilarradiiwhichagreewiththedipoleresultswithinstatisticaluncertainties.Athirdorder,singleeliminationjackknife[,]isusedtodeterminethestatisticaluncer-taintiesinthelatticeresults.Theseuncertaintiesareindicatedinparenthesesdescribingtheuncertaintyinthelastdigit(s)oftheresults.Thelatticeresultshavedisplayedsomeofthequalitativefeaturesexpectedfromhyper-neinteractions.Letusrstfocusonthemeansquarechargeradiusoftheneutronasdescribedbythequark-diquarkmodel[],thenonrelativisticquarkmodel[,]andthelatticesimulationsofRef. [,].Thediquarkdescriptionofquarkdistributionsisasillus-tratedingurea.Auanddquarkformascalardiquarkclusterwhoseradiusinextremecasesmaybetakentobepointlike.Theseconddquarkhasalargerchargedistributionradiusandprovidestherequirednegativechargeatlargeradiisurroundingapositivecoreofcharge+/.Inthenonrelativisticquarkmodelasimilarscalardiquarkclusteringoccursbetweenuanddquarksduetotheattractivepartofthehyperneinteraction.Howeverthisclusteringisrelativelysmallcomparedtotheclusteringanticipatedindiquarkmodels.Theremainingd-quarkisdriventotheperipheryoftheneutronbytherepulsivepartofthehyperne

FIG..Sketchesofquarkdistributionradiiintheneutronasdescribedbya)thequark-diquarkmodel,b)thenonrelativisticquarkmodelandc)thelatticeinvestigationsofbaryonelectromagneticformfactors.Thedashedlinesarerepresentativeofthermschargeradiusofthequarkdistributionsnormalizedtounitcharge.interaction.Theclusteringisaveragedoverthetwodquarksoftheneutronandthequarkdistributionsmaybedescribedasingureb.Animportantpointtonoteisthatthehy-perneinteractioncausestwoalterationstotheunperturbedwavefunction.Theattractionbetweentheuanddquarkscausesthewholenucleontoshrinkandtherepulsionbetweenthetwodquarksintroducestherequiredasymmetrybetweenthequarkdistributionstoreproducetheneutronchargeradius.Inunderstandingtheresultsofthelatticeinvestigationthenegativemeansquarechargeradiusoftheneutronwasattributedtohypernerepulsionbetweenthedoublyrepresenteddquarkswithoutresortingtoanyhyperneattraction[,].ThisargumentissimilartotheoriginalexplanationprovidedbyCarlitzetal. [].Ofcourseitisthehyperneattractionthatisarguedtonaturallygiverisetoscalardiquarkclusteringinthenucleon.Thecurrentpictureofquarkdistributionswithintheneutronisdescribedasingurec.Thedierencebetweenguresbandcisthepossibleabsenceofanyhyperneattractionbetweenuanddquarks.Ofcoursethelatticeresultsfortheneutronchargeradiusalonemayalsobeconsistentwitheitheroftheprevioustwodescriptions[].Thepresenceoftwodegreesoffreedom,namelythematterradiusandthechargeasym-metry,allowthesethreedierentdescriptionsofquarkdistributionswithintheneutrontolieinquantitativeagreementwhenreproducingtheneutrontoprotonchargeradiusratio.Experimentalmeasurements[,]ofnucleonmeansquarechargeradiiproducearatioof

hrinhrip=0:():()Thequark-diquarkpredictionofthemeansquareneutrontoprotonchargeradiusratioishrinhrip=0:();()whentheparameterofthemodelisxedbypreviousanalyses[].Similarlythecongu-rationmixinginducedbyhyperneinteractionsinthenonrelativisticquarkmodel[]leadstotheresulthrinhrip=0::()Presentlatticesimulationscalculatehadronpropertiesatrelativelyheavycurrentquarkmassesduetotheincreasingcomputationaldemandsencounteredininvertingthefermionmatrixasthequarksbecomelighter.InRef. [,]chargeradiiaredeterminedatthreevaluesoftheWilsonhoppingparameter=0:,0.,and0.correspondingtocurrentquarkmassesrangingfromthestrangecurrentquarkmassmsthroughtoms=.Tomakecontactwiththephysicalworldthelatticeresultsareextrapolated,usuallylinearlyasafunctionof=whichisproportionaltothecurrentquarkmassorm,tothepointatwhichthephysicalpionmassisreproduced.Resultsfromchiralperturbationtheory[0{]suggestthatlogarithmictermsdivergentinthelimitm!0shouldbeincludedintheanalyticstructureofthenucleonchargeradiusextrapolationfunction,inadditiontotermslinearinmandotherhigherordertermsniteinthechirallimit.However,thephysicsoflightpionsgivingrisetothedivergentlogarithmictermisnotincludedineitherofthequarkmodelsconsideredhere.Toallowacomparisonwiththesemodelsonamoreequalfooting,thelatticeextrapolationsofchargeradiiaredonelinearlyinm,eectivelysubtractingthenonanalyticcontributionsassociatedwithlightpiondressingsfromthechargeradii.Forlinearextrapolations,thedierencebetweenextrapolationsofquantitiestothephysicalpionmassasopposedtom=0arenegligible.Inthefollowing,radiiareextrapolatedtocr=0:()wherethepionmassvanishes.Theneutronchargeradiusissensitivetogluonicdegreesoffreedomandthelatticeratiohasalargestatisticalerrorassociatedwithit.However,itisimportanttodemonstratethatthelatticeresultisconsistentwiththepreviousanalyses.Thelatticeresultsindicate[]hrinhrip=0:+0:00::()Thequalitativefeaturesoftheneutronchargeradiusanticipatedbyhyperneinteractionsarereproducedineachofthecalculationsconsidered.Obviously,reproductionoftheneutrontoprotonchargeradiusratioaloneisnotsucienttodeterminewhetheritishyperneattraction,repulsion,oracombinationofthetwothatisactuallyresponsibleforgivingrisetotheneutronchargeradius.Ifattractionplaysasignicantrolethendiquark-clusteringmodelsmaycapturetheessentialfeaturesoftheunderlyingquark-gluondynamics.Ontheotherhand,ifthelatticeresultsindicatehyperneattractiondoesnotplayasignicantroleindeterminingthechargedistributionsthentherealizationofdiquarkclusteringinthenucleongroundstateisdoubtful.

B.ChargeDistributionsinTodiscriminatebetweenthesethreedierentdescriptionsofquarkdistributionswithinthenucleonwemustturntoanothersystemwheretheattractivepartofthehyperneinteractiondoesnotplayasignicantrole,i.e.thebaryonresonance.Byexaminingthechangesinthequarkdistributionsasthespinofthesinglyrepresentedquarkissetsymmetricwiththedoublyrepresentedquarkswecaninvestigatetherelevanceofscalardiquarkclusteringandsearchfortheanticipatedeectsofone-gluon-exchangehyperneinteractionsonthequarkdistributions.Considerthequarkdistributionswithintheprotonandhowtheywillchangeingoingfrompto+.Figureillustratesthethreepreviousscenariosfortheprotonaswellasthequarkdistributionsin+.Hyperneinteractionsarenotexpectedtogiverisetovectordiquarkclustering.Boththenonrelativisticquarkmodelandthelatticeresultsdescribethe+chargedistributionasasumofthreeequivalentquarkdistributions.Infactthesymmetryofthethree-pointcorrelationfunctiondemandsthequarkdistributionradiiintobeequalunderSU()-isospinsymmetry.Thissymmetryismanifestwithoutresortingtoactualcalculationsonthelattice[].Inthequark-diquarkmodelthescalardiquarkclusterislostin+andthenetpositivechargeoftheclusterinthenucleonmovestolargerradii.Forpoint-likediquarkstheneteectishuge,resultinginamuchlargerchargeradiusfor+thanfortheproton.Moreover,bothu-andd-quarkchargedistributionsswellinduetothebreakupofthe(point-like)u-d-quarkcluster.UnfortunatelyquanticationofthesestatementsdoesnotappeartobeFIG..Sketchesofquarkdistributionradiiintheprotonasdescribedbya)thequark-diquarkmodel,b)thenonrelativisticquarkmodelandc)thelatticeinvestigationsofbaryonelectromag-neticformfactors.d)Anillustrationofthequarkdistributionsin+.Thedashedlinesarerepresentativeofthermschargeradiusofthequarkdistributionsnormalizedtounitcharge.

possible.Acomparisonofpand+chargeradiiinadiquarkmodeldoesnotappeartohavebeenconsidered.Someformfactorsofoctetanddecupletbaryonswererecentlyexaminedinthequark-diquarkapproximation[],howeverelectricformfactorsandchargeradiiforresonanceswerenotreported.Toestimatealowerboundforthesizeoftheseanticipatedquarkdistributionswellings,werefertothenonrelativisticquarkmodelwheretheroleofhyperneattractionplaysamuchweakerandlessdramaticrole.Inthenonrelativisticquarkmodel,boththeu-andd-quarkdistributionsbecomebroaderastheattractionbetweenuanddquarksisreplacedbyhypernerepulsionin.InthemodelofIsgur-Karl-Koniuk[]thepredictedincreaseinthermschargeradiusis[,]rrp=:;()withthequarkdistributionsexperiencingalargeswellingofrurup=:;rdrdp=::()Inthescenariopreviouslyintroducedforthelatticeresults,thedominanteectwillbethenewhypernerepulsionexperiencedbythedquarkfromthetwouquarksin+.Similarlythetwouquarksalreadyexperiencingsomehypernerepulsionwillfeeladditionalrepulsionfromthesingledquark.Hencethedominanteectwillbethebroadeningofthenegativelychargedd-quarkdistributioncompensatedbysomebroadeningoftheu-quarkdistribution.Infact,thelatticeresultssuggestthatthe+chargeradiusmayactuallybesmallerthanthatoftheprotonrrp=0:():()Thisresultdiersbystandarddeviationsfromthepredictionofthenonrelativisticquarkmodel.Thequarkdistributionradiiindicatethedominanteectinthelatticeresultsisthebroadeningofthenegativelychargedd-quarkdistributionrurup=:0();rdrdp=:():()Thestrikingdierencebetweenthelatticeresultsandthetwomodelsconsideredhereistheabsenceofanysignicantchangeinthelatticeu-quarkdistributionradius.Inbothmodels,theu-quarkdistributionwaspredictedtobebroaderin,largelyduetothedisappearanceofscalardiquarkclusteringingoingfromthenucleonto.Thelatticeresultsindicatethathyperneattractiondoesnotleadtosubstantialscalardiquarkclusteringinthenucleongroundstate[].Figureillustratestheextrapolationofquarkdistributionradiiinpand+usedtoobtaintheresultsof().Theu-quarkdistributionradiiinpandarenearlyidenticalforeachconsidered.TableIdetailsthechargedistributionradiiforpandandtheirresidingquarks.

FIG..Extrapolationofquarkdistributionradiiinpandcalculatedatthethree(rightmost)valuesoftocrwheremvanishes.Inthisandthefollowinggure,thelatticeradii(r=a)havebeenscaledbyaconstantlatticespacingofa=0:fmdeterminedfromthenucleonmass.Forclairity,theu-quarkradiiarenormalizedtounitchargeandd-quarkradiiarenormalizedtonegativeunitcharge.Theu-quarkdistributionradiiinpandarenearlyidenticalateachvalueof.III.SCALARDIQUARKCLUSTERINGIN0Anotherplacetosearchforevidenceofscalardiquarkclusteringisinthechargedis-tributionradiusofthelightuanddquarksin0.Insimplequarkmodelsthesequarksaregenerallytakenasapurescalardiquark.Inotherwords,inthisapproximationthe0magneticmomentisgivenbytheintrinsicmagneticmomentofthestrangequarkalone.Thelatticeresultssuggestthattheuanddquarksdonotformapurescalardiquarkandmayactuallycontributetothe0magneticmomentatthelevelof0%.However,fortheTABLEI.Rmschargedistributionradiinormalizedtounitchargeinlatticeunitshr=ai=.Baryon=0:=0:=0:cr=0:()p.0().0().().0()+.().0().().0()up.().00().().()u+.().0().().0()dp.().().().()d+.().0(). ().0()

presentpurposewewillignoresucheects.Inthedecuplet0hyperonstatetheu-dsectorwillbepairedpredominatelyasavectordiquark.Hencewiththeassumptionthatthesquarkplaysaspectatorrole,comparisonofthelightquarksectordistributionradiiin0and0willgivesomeindicationoftherelevanceofscalardiquarkclustering.Ifascalardiquarkis\dissolved"ingoingfrom0to0,abroadeningofthelightquarkdistributionin0isexpected.Theratioofthelightquarkdistributionsfromthelatticeanalyses[,]indicatesrl0rl=:0(0);()conrmingtheconclusionsbasedontheN-quarkdistributions.Onceagainsubstantialdiquarkclusteringisnotseen.Figureillustratestheextrapolationofquarkdistributionradiiinand0usedtoobtaintheresultsof().Thestrangequarkappearstoplayasatisfactoryspectatorrole.Littlechangeisseenbetweenthecombinedlight-quarkdistributionsofand0ateachvalueof.NumericalvaluesandstatisticaluncertaintiesfortheseradiiaresummarizedintableII.FIG..Extrapolationofquarkdistributionradiiin0and0.Thedistributionradiusofthecombinedlightuanddquarksof0aredenotedbylandsimilarlyfor0.Thelight-quarkradiiarenormalizedtounitchargeandstrange-quarkradiiarenormalizedtonegativeunitcharge.Foreach,l'lindicatingtheabsenceofsignicantscalardiquarkclustering.

TABLEII.Rmschargedistributionradiinormalizedtounitchargeinlatticeunitshr=ai=.lindicatesthecombinedlightuanddquarksin0andsimilarlyfor0.Baryon=0:=0:=0:cr=0:()l.().().().()l.().0().0().()s.0().().().()s.().(). (0).0()IV.SYSTEMATICUNCERTAINTIESThiscomparisonofchargedistributionsinoctetanddecupletbaryonshasrevealedre-markabledierencesbetweenthelatticeresultsandtheanticipatedroleofhyperneattrac-tionbasedontheOGEP.Moreover,thisanalysisappearstorenderadiquarkpictureofquarkdistributionsinthenucleongroundstateobsolete.Asaresult,itisimportanttocon-siderthepossiblesourcesofsystematicuncertaintyinthelatticecalculations.Systematicuncertaintiesmayhavetheirorigininthequenchedapproximation,theextrapolationofthelatticeresultstothephysicalregime,nitevolumeeectsorrenormalizationgroupscalingdeviations.Itisworthnotingthatallofthelatticeresultspresentedtothispointhaveinvolvedratiosoflatticeresults.Itisexpectedthattheeectsofthesepossiblesourcesofsystematicuncertaintywillbesuppressedintakingratios.Forexample,whiletheabsolutevaluesofthelatticepredictionsofmagneticmoments[]aresmallcomparedtotheexperimentallymeasuredvalues,thelatticeresultspredictratiosofthebaryonmagneticmomentstotheprotonmomentasgoodasorbetterthanmodels,whichinmostcaseshaveparameterstunedtoreproducetheexperimentalmoments.Amoredetaileddiscussionofsystematicuncertaintiesfollows.AcalculationofelectromagneticformfactorsinfullQCDhasnotbeendone.However,atthepresentvaluesofquarkmassinvestigatedonthelattice,hadronspectrumanalysessuggestthedominantnewphysicsinfullQCDisasimplerenormalizationofthestrongcouplingconstant.ItispossiblethatoverlargedistancesthequenchedapproximationdoesnotscreenquarkinteractionsasmuchasrequiredbyfullQCD.Howevergiventhesimilarityofthequarkdistributionradiidiscussedinthispaperitisunlikelythattherewouldbesignicantdeviationsfromtheresultspresentedhere.Theextrapolationoftheresultstothephysicalregimeisanothersourceofconcern.However,theargumentsbasedontheextrapolatedresultsaresupportedateachvalueofquarkmassinvestigatedonthelatticeasindicatedintablesIandII.Becausetheactuallatticecalculationsaredonewithu-andd-current-quarkmassestheorderofthestrange-current-quarkmass,onemightbeconcernedthathyperneinteractionssuchasthoseoftheOGEParesuppressedinthelatticecalculationstothepointthatonehasnohopeofobservingdiquarkclustering.Fortunately,thisisunlikelytobethecase.Toassessthisissuemorequantitatively,onecanresorttotheone-gluon-exchangehy-perneinteractiontermwhichisinverselyproportionaltotheproductofquarkmasses.AnimportantpointistheOGEPhassomerelevanceonlyinphenomenologywhereconstituent0

quarkmassesareusedinthehyperneterm.Whilethecurrentquarkmassesusedinthisinvestigationaresomewhatheavy,thecorrespondingconstituentquarkmassesarenottoodierentfromthatusedinphenomenology.Perhapstheeasiestwaytoquantifythisistoestimatetheconstituentquarkmasstobeonethirdthelatticeprotonmass.Atthelightestquarkmassconsideredonthelattice,theconstituentquarkmassisapproximately0MeVwhichisnottoodierentfromthevalueobtainedfromthephysicalmassatMeV.Ifthehypernetermoftheone-gluon-exchangeinteractionisrelevant,thenthedif-ferencesinthestrengthofthehyperneinteractionsforthesetwoquarkmassesarewithinafactorof.Fortheintermediatequarkmassvaluethestrengthiswithinafactorof..Hence,thereisgoodreasontobelieveanextrapolationtothephysicalvalueisadequate.Moreover,signicantmasssplittingisobservedbetweenthenucleonandonthelatticewhereMN=M=0:().However,thechangeintheu-quarkdistributionradiusremainsnegligible.Perhapsitisalsoworthnotingthatsingle-gluon-exchangeinteractionstrengthsbetweencurrentquarksinthislatticeQCDcalculationexceedthestrengthscommontophe-nomenologicalanalysesbyanorderofmagnitudeforthelightestquarkmassinvestigatedonthelattice.Ithasbeenarguedinthepastthatthenitevolumeofthelatticemayaectthelatticeresults,assurroundingperiodicimagesofthebaryonunderstudymayrestrictthebaryon'ssize[].Notehowever,theu-quarkdistributioninisnotthebroadestquarkdistributionseenonthelattice.Inaddition,similareectsareseenforheavierquarkmasseswherethequarksaremorelocalizedandlesssensitivetotheboundariesofthelattice.Foragivenvalueof,thechargedistributionradiicalculatedonthelatticetendtobesimilarinsize.Forthisreasonthenitevolumeofthelatticeismorelikelytoaectthelatticeresultsinaglobalmanner.Forthecaseofascalaru-ddiquarkclusterinpbreakingupin+,anitevolumeeectcouldnotsimultaneouslyaccommodateaswellingofthed-quarkdistributionandyetrestricttheu-quarkdistributionfromfollowingasimilarswelling.Theissueofdeviationsfromasymptoticscalingisdiculttoassessquantitativelysinceasimilarcalculationatnerlatticespacingshasnotbeendone.Toussaint[]hasdemon-stratedthatthenucleontomassratiocalculatedatm=m=0:is,toagoodapproxi-mation,independentofthevalueofoverarangeof=:to.forquenchedWilsonfermioncalculations.Furthermore,Yoshieetal. []havedemonstratedthatitispossibletoreproducethehadronspectruminquenchedQCDat=:0withinstatisticaluncertaintiesofapproximately0%[].Thelatticepredictionofthecurrentinvestigationforthe=Nmassratiolies%or.belowtheexperimentalratio.OnecouldadjusttheparametersofthenonrelativisticquarkmodeltoreproducethelatticeN-splittinginwhichcasethedierencebetweenthelatticeandmodelpredictionsforbaryonchargeradiiisreducedto.However,thetrendofthelatticeresultsindicatesthatadoublingoftheeectsseeninthelatticeresultsonlywidensthegapbetweenthepredictionsof+=pchargeradiiratiosandleavesthechangeintheu-quarkradiusingoingfrompto+attheorderof%.

V.DISCUSSIONANDSUMMARYFacedwiththediscrepanciesofthesemodelswhichbothlackmesonicdegreesoffreedomitisinterestingtoconsidermodelssuchasthecloudybag[],orhedgehogmodelssuchastheSkyrmion[],hybrid(orlittle)bag[0]andthechiral-quarkmesonmodel[].Inhedgehogmodels,theprotonand+radiiareequalbyconstruction.Thedierenceinchargeradiiare=Ncsuppressedandcannotbecalculatedusingconventionalsemi-classicaltechniques.Itmaybeusefultonotethatthelatticeresultssuggestthatsuchhigherordereectsmaybesmall.Acalculationofthechargeradiiconsideredhereinthecloudybagmodelcouldprovidefurtherinsighttotheimportanceofmesonicdegreesoffreedomindescribingbaryonchargedistributions.Sincealargepartoftheneutronchargeradiushasitsorigininthepioncloudofthecloudybagmodel[,],itmaybepossibletocircumventtheproblemsencounteredherewithhyperneinteractions.Ofcourse,toallowadirectcomparisonwiththelatticeresultsandtoavoidproblemsassociatedwithopenchannelphysics,thecloudybaganalysisshouldbedonewithheavierpions.The+chargeradiusandinternalquarkdistributionsprovidenewadditionalinformationonthespin-dependenceofquarkinteractions.ThetraditionalmodelsexaminedheredonotreproducethelatticeresultsandadescriptionofthesephenomenaremainsanopenchallengetomodelsofQCD.Somehavearguedtheexistenceofscalardiquarksbasedonlargemomentumtransferphenomenology.AnanalysisofscalarandvectordiquarksatlargemomentumtransfersinlatticeQCDmayprovideusefulinsight.TheresultspresentedheremaybeofinteresttothoseinvestigatinglargeQphenomenausingnucleongroundstatewavefunctionsasamodelinputrepresentingthenonperturbativepartsofthecalculation[,].Finally,afewcommentsontheabsenceofsignicantquarkclusteringinthelatticeresultsrelativetothatanticipatedbyquarkmodelsareinorder.Therstandmostobvi-ouscommenttomakeisthatbaryonchargedistributionsaresensitivetothelongdistancenonperturbativeaspectsofQCD.Itshouldnotbetoosurprisingtondthathyperneinteractionsbasedonasingle-gluon-exchangebecomevirtuallyirrelevantinthenonpertur-bativeregime.Itiswellknownthattheone-gluon-exchangehyperneinteractionhassomerelevanceonlywhenthequarkmassesaretakentobeconstituentquarkmasses.Exper-imentalmasssplittingsindicateitisnottherelevantinteractionbetweencurrentquarks.Thereisaninniteclassofmultiple-gluon-exchangediagramstoconsiderbeyondtheex-changeofasinglegluon.Moreover,thesediagramsareequallyimportantsincethetheoryisnonperturbative.Asecondandpossiblymoreinterestingpointconcernsthespinofthesinglyrepresentedquarkinthenucleon.FromthelatticeQCDanalysesofbaryonmagneticmoments[,,],ithasbecomeclearthatthebehaviorofthesinglyrepresentedquarkinoctetbaryonsisverydierentfromthepredictionsofSU()-spin-avorsymmetry.Brieystatedthelatticeresultsindicatethattheprotonmomentisbetterdescribedasp=ud;(0)whereinconstituentquarkmodellanguage,thedquarkhasitsnetspinoppositethatoftheuquarksabouthalfasmuchassuggestedbySU().Inrelationtoscalardiquarkclustering

theprobabilityofndinguanddquarkspairedinaspin-0statehasbeenreduced.Onceagainthelatticeresultssuggestthatscalardiquarkdegreesoffreedomdonotprovideanappropriatedescriptionoftheinternalquarkstructureoflow-lyingbaryons.Agaugeinvariantmethodfortheexaminationofscalardiquarkclusteringinthenucleongroundstatehasbeenpresented.ResultsfromlatticeQCDdescribingthedistributionsofquarksinthebaryonoctetanddecuplethavebeenanalyzedinasearchforevidenceofscalardiquarkclustering.Theresultspresentedherecontrastthepredictionsofthenonrelativisticquarkmodelwhichhasarelativelysmalldiquarkclusteringcomparedtothatdemandedinquark-diquarkmodels.Thelatticeresultsdonotsupporttheconceptofsubstantialdiquarkclusteringasanappropriatedescriptionoftheinternalstructureoflow-lyingbaryons.ACKNOWLEDGMENTSIwishtothankZbigniewDziembowskiforstirringmyinterestinthissubject.IalsothankWojciechBroniowski,TomCohen,ManojBanerjee,NathanIsgurandSimonCapstickforanumberofinterestingandhelpfuldiscussions.FinancialsupportfromtheU.S.DepartmentofEnergyundergrantDE-FG0-ER-0isgratefullyacknowledged.

REFERENCES[]A.DeRujula,H.Georgi,andS.L.Glashow,Phys.Rev.D,().[]R.D.Carlitz,S.D.Ellis,andR.Savit,Phys.Lett.B,().[]N.Isgur,G.Karl,andR.Koniuk,Phys.Rev.Lett.,(),ibid.,(E)(0).[]N.Isgur,G.Karl,andD.W.L.Sprung,Phys.Rev.D,().[]D.B.Lichtenberg,Phys.Rev.,().[]K.S.Sateesh,Phys.Rev.D,().[]M.AnselminoandE.Predazzi,Phys.Lett.B,0().[]G.V.Emov,M.A.Ivanov,andV.E.Lyubovitskij,Z.Phys.C,(0).[]Z.DziembowskiandJ.Franklin,Phys.Rev.D,0(0).[0]H.G.Dosch,M.Jamin,andB.Stech,Z.Phys.C,().[]S.Fredriksson,Phys.Rev.Lett.,().[]S.FredrikssonandT.Larsson,Phys.Rev.D,().[]S.Fredriksson,M.Jandel,andT.Larsson,Z.Phys.C,().[]D.B.Lichtenberg,W.Namgung,E.Predazzi,andJ.G.Wills,Phys.Rev.Lett.,().[]D.B.Lichtenberg,E.Predazzi,andJ.G.Wills,Z.Phys.C,().[]Z.Dziembowski,W.J.Metzger,andR.T.VandeWalle,Z.Phys.C0,().[]T.UppalandR.C.Verma,Z.Phys.C,0().[]Notethatanindividualquarkavordistributionradiusisagaugeinvariantquantity.Theradiusmayberegardedasabaryonchargeradiusinwhichtheremainingquarkavorshavezeroelectriccharge.[]C.P.ForsythandR.E.Cutkosky,Nucl.Phys.B,().[0]B.RamandV.Kriss,Phys.Rev.D,00().[]S.Fleck,B.Silvestre-Brac,andJ.M.Richard,Phys.Rev.D,().[]D.B.Leinweber,R.M.Woloshyn,andT.Draper,Phys.Rev.D,().[]D.B.Leinweber,T.Draper,andR.M.Woloshyn,Phys.Rev.D,0().[]B.Efron,SIAMRev.,0().[]S.Gottlieb,P.B.MacKenzie,H.B.Thacker,andD.Weingarten,Nucl.Phys.B,0().[]T.Draper,R.M.Woloshyn,andK.F.Liu,Phys.Lett.B,(0).[]Tothispointwedonothavesucientinformationtoexcludethepossibilityofasig-nicantroleforhyperneattractioninthelatticeresults.However,itwillbecomeclearthroughtheanalysisof-baryonsthathyperneattractiondoesnotplayasignicantroleindeterminingthequarkchargedistributionradii.[]G.Hohleretal.,Nucl.Phys.B,0().[]O.Dumbrajs,Nucl.Phys.B,().[0]M.A.B.BegandA.Zepeda,Phys.Rev.D,().[]D.B.LeinweberandT.D.Cohen,ChiralCorrectionstoLatticeCalculationsofChargeRadii.U.Marylandpreprint-0.AcceptedforpublicationinPhys.Rev.D.[]T.D.CohenandD.B.Leinweber,ThePionCloudinQuenchedQCD.U.Marylandpreprint-0.AcceptedforpublicationinCommentsonNuclearandParticlePhysics. []N.IsgurandG.Karl,Phys.Rev.D,(0).

[]Herewehavetaken=p,and=0.Equation()ofRef.[]hasbeenadjustedasdescribedintheerrata.[]Intheprecedingarguments,thefocusinthelatticescenariohasbeenonhypernerepulsion.However,thelatticeresultsarealsoconsistentwithamixtureofhyperneattractionandrepulsionorevenexclusivehyperneattractionprovidedtheclusteringissmall.Theimportantpointtodrawfromthisexaminationofchargeradiiisthatthereisnoevidenceofsignicantu-d-quarkclusteringinthenucleongroundstatequarkdistributions.[]D.Toussaint,TheQCDSpectrumandPhaseDiagramontheLattice{.ToappearintheProceedingsoftheinternationalsymposiumLATTICE-,Tsukuba,Japan,Nov.,Nucl.Phys.B(Proc.Suppl.),U.ArizonapreprintAZPH-TH/-0.[]T.Yoshie,Y.Iwasaki,andS.Saki,QuenchedHadronSpectrumonaLattice.PresentedattheinternationsymposiumLATTICE-,Tsukuba,Japan,Nov.byT.Yoshie.(unpublished).[]A.W.Thomas,inAdvancesinNuclearPhysics,Vol.,editedbyJ.NegleandE.Vogt(PlenumPress,NewYork,).[]I.ZahedandG.E.Brown,Phys.Rep.,().[0]L.VepstasandA.D.Jackson,Phys.Rep.,0(0).[]M.K.Banerjee,W.Broniowski,andT.D.Cohen,inChiralSolitons,editedbyK.F.Liu(WorldScientic,Singapore,).[]S.Theberge,A.W.Thomas,andG.A.Miller,Phys.Rev.D,(0).[]F.Myhrer,Phys.Lett.0B,().[]H.MeyerandP.J.Mulders,Nucl.Phys.A,(). []D.B.Leinweber,Phys.Rev.D,().

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