Table Of Content

Nucleon and Roper electromagnetic elastic and transition form factors D.J. Wilson,1 I.C. Cloët,2 L. Chang,1 and C.D. Roberts1,3,4 1Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 2CSSM and CoEPP, School of Chemistry and Physics University of Adelaide, Adelaide SA 5005, Australia 3Institut fu¨r Kernphysik, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany 4Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616, USA We compute nucleon and Roper electromagnetic elastic and transition form factors using a Poincaré-covariant, symmetry-preserving treatment of a vector×vector contact-interaction. Ob- tained thereby, the electromagnetic interactions of baryons are typically described by hard form factors. In contrasting this behaviour with that produced by a momentum-dependent interaction, one achieves comparisons which highlight that elastic scattering and resonance electroproduction experimentsprobetheevolutionofthestronginteraction’srunningmassesandcouplingtoinfrared 2 momenta. For example, the existence, and location if so, of a zero in the ratio of nucleon Sachs 1 form factors are strongly influenced by the running of the dressed-quark mass. In our description 0 of the nucleon and its first excited state, diquark correlations are important. These composite and 2 fully-interacting correlations are instrumental in producing a zero in the Dirac form factor of the n proton’sd-quark;andindeterminingtheratioofd-to-uvalence-quarkdistributionsatx=1,aswe a show via a simple formula that expresses dv/uv(x = 1) in terms of the nucleon’s diquark content. J Thecontactinteraction producesafirstexcitationofthenucleonthatisconstitutedpredominantly 7 from axial-vector diquark correlations. This impacts greatly on the γ∗p→P11(1440) form factors, 1 our results for which are qualitatively in agreement with the trend of available data. Notably, our dressed-quark core contribution to F2∗(Q2) exhibits a zero at Q2 ≈0.5m2N. Faddeev equation ] treatmentsofahadron’sdressed-quarkcoreusuallyunderestimateitsmagneticproperties,hencewe h consider the effect produced by a dressed-quark anomalous electromagnetic moment. Its inclusion -t much improves agreement with experiment. On the domain 0<Q2 .2GeV2, meson-cloud effects l are conjectured to be important in making a realistic comparison between experiment and hadron c u structurecalculations. Wefindthatourcomputedhelicityamplitudesaresimilartothebareampli- n tudesinferredviacoupled-channelsanalysesoftheelectroproductionprocess. Thissupportsaview [ thatextanthadronstructurecalculations, whichtypicallyomitmeson-cloud effects,shoulddirectly becompared with thebare-masses, -couplings, etc., determined via coupled-channelsanalyses. 2 v PACSnumbers: 13.40.Gp;14.20.Dh;14.20.Gk;11.15.Tk 2 1 2 I. INTRODUCTION the N(1535)S , which is heavier than the Roper. Holo- 2 11 graphic models of QCD, viewed by some as a covariant . 2 generalisationofconstituent-quarkpotentialmodels,pre- Building a bridge between QCD and the observed 1 dict degeneracyof the (n,l)=(1,0) and (0,1) states [5]. 1 properties ofhadrons is one of the key problems in mod- Whilstithasbeenobservedthatconstituent-quarkmod- 1 ernscience. Theinternationalprogrammefocusedonthe els with Goldstone-boson exchange potentials can pro- : physicsofexcitednucleonsisclosetotheheartofthisef- v duce the observed level ordering [6], such a foundation i fort. Itaddressesthequestions: whichhadronstatesand X makes problematic a unified description of baryons and resonancesareproducedbyQCD,andhowaretheycon- mesons. r stituted? TheN∗ programthereforestandsalongsidethe a search for hybrid and exotic mesons as an integral part In order to correct the level ordering problem within of the search for an understanding of QCD. An example the potential model paradigm,other ideas have been ex- of the theory activity in this area is provided in Ref.[1]. plored. ThepossibilitythattheRoperissimplyahybrid baryonwith constituent-gluoncontentis difficult tosup- It is in this context that we consider the N(1440)P , 11 JP = (1/2)+ Roper resonance, whose discovery was report because the lightest such states occur with masses above1.8GeV[7]. Analternativeistoconsiderthepres- ported in 1964 [2]. In important respects the Roper ap- enceofexplicitconstituent-q¯qcomponentswithinbaryon pears to be a copy of the proton. However, its (Breit- bound-states [8]. Whilst not literally correct, such a Wigner) mass is 50% greater [3]. This feature has long picture may be interpreted as suggesting that πN final- presented a problem within the context of constituent- stateinteractionsmustplayanimportantroleinanyun- quark models formulated in terms of colour-spin poten- derstanding of the Roper. This perspective is common tials, which typically produce the following level ordering [4]: ground state, JP =(1/2)+ with radial quantum to modern coupled-channels treatments of baryon reso- nances[9–11],andfindssupportincontemporarynumer- number n = 0 and angular momentum l = 0; first excited state, JP = (1/2)− with (n,l) = (0,1); second icalsimulationsoflattice-QCD[12]andDyson-Schwinger excited state, JP =(1/2)+, with (n,l)=(1,0); etc. The equation (DSE) studies [13–15]. difficulty is that the lightest l = 1 baryon appears to be Given that an understanding of the Roper has long 2 p p − q q Γa TABLEI.(A)Computedquantitiesrequiredasinputforthe Ψa P = Ψb P Faddeevequation,obtained with αIR/π=0.93 and(in GeV) p q m=0.007, Λir =0.24, Λuv =0.905. (B) Nucleon and Roper masses,andassociatedunit-normalisedeigenvectors,obtained d p Γb therewith. (All dimensioned quantities are listed in GeV.) d M mqq0+ mqq1+ Eqq0+ Fqq0+ Eqq1+ Md1F/2 FIG.1. Poincaré covariantFaddeevequation,Eq.(B10),em- 0.368 0.776 1.056 4.354 0.499 1.3029 0.880 ployed herein to calculate baryon properties. Ψ in Eq.(B1) is the Faddeev amplitude for a baryon of total momentum mass (GeV) s a+ a0 a+ a0 1 1 2 2 P =pq+pd. It expresses therelative momentum correlation mN =1.14 0.88 -0.38 0.27 -0.065 0.046 between the dressed-quark and -diquarks within the baryon. m =1.72 -0.44 -0.030 0.021 0.73 -0.52 R TheshadedregiondemarcatesthekerneloftheFaddeevequa- tion, Sec.B, in which: the single line denotes the dressed- quark propagator, Sec.A1; Γ is the diquark Bethe-Salpeter amplitude, Sec.A4; and the double line is the diquark prop- properties of meson and baryon ground- and excited- agator, Eqs.(B4), (B9). states based on the symmetry-preserving treatment of asingle quark-quarkinteraction;namely, avector-vector contact-interaction. Thisprocedurehasalreadybeenap- eluded practitioners, it is unsurprising that this reso- plied to the spectrum of u,d-quark mesons and baryons nancehasbeenafocusoftheN∗ programmeatJefferson [13], and the electromagnetic properties of π- and ρ- Lab (JLab). Experiments at JLab [16–19] have enabled mesons, and their diquark partners [25–27]. These stud- anextractionofnucleon-to-Ropertransitionformfactors iesprovidethefoundationformuchofthatwhichfollows. and thereby exposed the first zero-crossing seen in any In Sec.II we present a brief overview of our frame- nucleon form factor or transition amplitude. Explaining work: both the Faddeev equation treatment of the nu- this new structure also presents a challenge for theory cleon and Roper dressed-quark cores, and the currents [20]. whichdescribethe interactionofaphotonwithabaryon Notwithstanding its history, an understanding of the composed from consistently-dressed constituents. Addi- Roperisperhapsnowbeginningtoemergethroughacon- tionalmaterialisexpressedinappendicesandreferredto structive interplay between dynamical coupled-channels as necessary. In Sec.III we describe the parameter-free models and hadron structure calculations, particularly calculation of nucleon elastic form factors within a DSE thosesymmetry-preservingstudies madeusingthe tower treatment of the contact interaction. Germane to our ofDyson-Schwingerequations[21–24]. One indicationof presentation are comparisons both with data and com- thisisfoundinpredictionsforthemassesofthebaryons’ putationsusingQCD-likemomentum-dependenceforthe dressed-quark-cores [13], which match the bare masses propagatorsand vertices. In addition, we use the elastic ofnucleonresonancesdeterminedbytheExcitedBaryon form factors to predict the ratio of valence-quark distri- AnalysisCenter(EBAC)[10]witharms-relativeerrorof bution functions at x=1. 14% and, in particular, agree with EBAC’s value for the We begin to describe our results for the Roper elastic bare-mass of the Roper resonance; viz. (in GeV), and nucleon-to-Roper transition form factors in Sec.IV. The description continues in Sec.V, with a considera- mQQQ =1.82 0.07 cf. mEBAC−bare =1.76 0.10. (1) Roper ± Roper ± tion of the impact on all form factors of a dressed-quark anomalous magnetic moment. In Sec.VI we explore the The DSE state is the first excitation of the ground-state effect of meson-cloud contributions to hadron structure nucleon whilst the EBAC bare state is the source for calculationsin the contextofthe γ∗p P (1440)helic- three distinct features in the πN-scattering P11 partial ity amplitudes, whichhavebeen analy→sed1u1sing coupled- wave, which migrate widely from the real-energy axis channels methods [28–31]. once meson-nucleon final-state interactions are enabled. Section VII is an epilogue. It is notable that the dressed-quarkcoreofthe nucleon’s parity partner is approximately 400MeV heavier than mQQQ and1.1GeVheavierthanthe coreofthe ground- Roper statenucleon,amagnitudecommensuratewithitsorigin II. ELECTROMAGNETIC CURRENTS in dynamical chiral symmetry breaking (DCSB) [13]. Herein we probe further into the possibility that πN We base our description of the dressed-quark-core of final-state interactionsplay a criticalrolein understand- the nucleon and Roper on solutions of a Faddeev equa- ing of the Roper, through a simultaneous computation tion, which is illustrated in Fig.1, and formulated and within the DSE framework of nucleon and Roper elastic describedinApps.A,B.TheFaddeevequationsarecom- formfactors,andtheformfactorsdescribingthenucleon- pleted by the quantities reported in Table IA, and our to-Roper transition. In so doing we add materially to a values for the nucleon and Roper masses and eigenvec- body of work that presents the unified analysis of many tors, the latter normalised to unity, are presented in 3 Q TABLEII.Row1: Resultscomputedhereinwiththecontact interaction, whose input is presented in Table I. Row 2: Re- sultsobtainedusingQCD-likemomentum-dependenceforthe P P dressed-quarkpropagatorsanddiquarkBethe-Salpeterampli- f Ψ Ψ i tudes in solving the Faddeev equation. Row 3: Values rep- f i resentative of experiment. Row 4: Contact interaction aug- mentedbyamodeldressed-quarkanomalouselectromagnetic moment (see Sec.V). P P f Ψ Ψ i f i r1pMN r2pMN r1nMN r2nMN κp κn contact 3.19 2.84 1.21 3.19 1.02 -0.92 Ref.[33] 3.76 2.82 0.59 3.14 1.67 -1.59 Ref.[34] 3.76 4.18 0.56 4.33 1.79 -1.91 Q contactQAMM 3.41 4.00 0.55 3.85 1.68 -1.24 P P f Ψ Ψ i f i transition form factors [Q γT =0, Eq.(A20)] axial vector scalar µ µ J∗(P ,P )=ieu¯ (P ) γTF (Q2) µ f i R f µ 1∗ 1 Q + (cid:2) σ Q F (Q2) u (P ). (3) µν ν 2∗ N i M +M R N (cid:21) FIG. 2. Interaction vertex which ensures a conserved cur- N.B. Electromagnetic current kinematics and the defini- rent for the elastic and transition form factors in Eqs.(2), tionofconstraint-independentformfactorsarediscussed (3). The single line represents the dressed-quark propagator, in Ref.[32], so that Eq.(2) may be viewed as a special S(p) in App.A1; thedoubleline, thediquarkpropagators in case of Eq.(3) which is simplified by the on-shell condi- Eqs.(B4)and(B9);andtheverticesaredescribedin App.C. tion u¯ (P )γ Qu (p )=0. B f B i From top to bottom, the diagrams describe the photon cou- With the co·ntact interaction described in App.A and pling: directlytothedressed-quark;toadiquark,inanelastic ourtreatmentoftheFaddeevequation,App.B,thereare scattering event; or inducing a transition between scalar and three contributions to the currents. They are illustrated axial-vector diquarks. inFig.2anddetailedinApp.C.Thecomputationofform factors is straightforward following the procedures out- lined in those appendices. Table IB. These masses are drawn from a unified spectrum of u,d-quark hadrons, obtained using a symmetry- III. NUCLEON ELASTIC preserving regularisationof a vector vector contact in- × teraction [13]. That study simultaneously correlates the masses of meson and baryon ground- and excited-states There are no free parameters in our computation of within a single framework. In comparison with rele- nucleon elastic form factors: all those associated with vant quantities, it produces a root-mean-square-relative- our treatment of the contact interaction are fixed in error/degree-of-freedom equal to 13%. The predictions Refs.[13, 27], see Table I. We report static properties in uniformlyoverestimatetheexperimentalvaluesofmeson Table II, and depict form factors for the proton in Fig.3 and baryonmasses [3]. Given that the employedtrunca- andthe neutroninFig.4. N.B. We usea Euclideanmet- tiondeliberatelyomitted meson-cloudeffects in the Fad- ric, App.E, and hence in elastic scattering one has deev kernel, this is a good outcome because inclusion of P2 = m2 =P2, Q2+2P Q=0, (4) such contributions acts to reduce the computed masses. f − B i i· As noted in the Introduction, Eq.(1), such effects are where m is the mass of the baryon involved. B particularly important for the Roper resonance. We are interested in three electromagnetic currents: thosedefiningthenucleonandRoperelasticformfactors A. Dirac and Pauli Form factors In our symmetry-preserving DSE-treatment of the JB(P ,P )=ieu¯ (P ) γ F (Q2) µ f i B f µ 1B contact interaction we construct a nucleon from di- 1 quarkswhoseBethe-Salpeteramplitudesaremomentum- + σ (cid:2)Q F (Q2) u (P ), (2) 2M µν ν 2B B i independent and dressed-quarks with a momentum- B (cid:21) independent mass-function, and arrive at a nucleon described by a momentum-independent Faddeev ampli- B = N, R and Q = P P ; and that expressing the tude. This last is the hallmark of a pointlike composite f i − 4 0 2 4 6 8 10 0 2 4 6 8 10 0 1 0.8 Lx -0.2 Lx 0.6 Hn H 1 p F 1 F 0.4 0 1 -0.4 0.2 0 0 1 -0.2 0.8 n p Κ L(cid:144)Κ 0.6 L(cid:144)x -0.4 x H H2p 0.4 F2n-0.6 F - -0.8 0.2 -1 0 0 2 4 6 8 10 0 2 4 6 8 10 x x FIG. 4. Neutron Dirac (upper panel) and Pauli (lower FIG. 3. Proton Dirac (upper panel) and Pauli (lower panel) form factors, as a function of x = Q2/m2 . Solid curve panel) form factors, as a function of x = Q2/m2N. Solid N curve – result obtained herein using the contact-interaction – result obtained herein using the contact-interaction and andhenceadressed-quarkmass-functionanddiquarkBethe- hence a dressed-quark mass-function and diquark Bethe- Salpeteramplitudesthataremomentum-independent;dashed Salpeteramplitudesthataremomentum-independent;dashed curve – result obtained in Ref.[33], which employed QCD- curve – result obtained in Ref.[33], which employed QCD- like momentum-dependence for the dressed-quark propaga- like momentum-dependence for the dressed-quark propagators and diquark Bethe-Salpeter amplitudes in solving the tors and diquark Bethe-Salpeter amplitudes in solving the Faddeev equation; dot-dashed curve – a parametrisation of Faddeev equation; dot-dashed curve – a parametrisation of experimental data [34]. experimental data [34]. momentum-independent Faddeev amplitude suppresses particle and explains the hardness of the computed form quark orbital angular momentum, as may be seen from factors, which is evident in Figs.3, 4. the absence in Eqs.(B17) of a dependence on the rela- The hardness contrasts starkly with results ob- tive momentum. This explains the differences between tained from a momentum-dependent Faddeev ampli- the anomalous magnetic moments in Rows 1 and 2 of tudeproducedbydressed-quarkpropagatorsanddiquark Table II. Bethe-Salpeter amplitudes with QCD-like momentum- The differences between the anomalous moments in dependence; and with experiment. Evidence for a con- Rows2and3haveadifferentorigin;viz.,QCD’sdressed- nection between the momentum-dependence of each of quarks possess large momentum-dependent anomalous these elements and the behaviour of QCD’s β-function magnetic moments owing to dynamical chiral symmetry is accumulating; e.g., Refs.[25–27, 35–38]. The compar- breaking [39], and the discrepancy is resolved by incor- isonsinFigs.3,4addtothisevidence,inconnectionhere poratingthisphenomenon. Owingtothemomentumde- with readily accessible observables, and support a view pendence of these moments, the magnetic radii are also thatexperimentisasensitiveprobeoftherunningofthe affected, so that r , r in Row 2 are shifted markedly 2p 2n β-functiontoinfraredmomenta. Thisperspectivewillbe towardthevaluesinRow3. ThisisillustratedinRef.[40] reinforced by subsequent figures. and in Row 4, which is discussed further in Sec.V. Table II exposes another shortcoming in the descrip- In Fig.5 we depict a flavourdecomposition ofthe pro- tion of nucleons via a momentum-independent Faddeev ton’sDiracformfactor. Inneitherthedatanorthecalcu- amplitude; namely, the anomalous magnetic moments lations is the scaling behaviour anticipated from pertur- are far too small. In a Poincaré-covariant treatment, bativeQCDevidentonthemomentumdomaindepicted. the magnitude of the magnetic moment grows with in- This fact is emphasised by the zero in Fd , whose exis- 1p creasing quark orbital angular momentum. However, a tence is independent of the interaction. Its location is 5 0 2 4 3 0.6 p 2 u1 d 0.4 F p 2x (cid:144)Κp ,p 1 dF2 dF1 2x 0.2 2 x ì ì ì 0 0 0.6 0 2 4 6 8 x u 0.4 p Κ FIG. 5. Flavour separation of the proton’s Dirac form fac- (cid:144) tor, as a function of x=Q2/m2 : normalisation: Fu(0)=2, u2p N 1p F F1dp(0) = 1. Solid curve – u-quark obtained using the con- 2x 0.2 tact interaction; short-dashed curve – d-quark, contact interaction; dot-dashed curve – u-quark obtained from QCD- like momentum-dependence for the dressed-quark propaga- 0 tors and diquark Bethe-Salpeter amplitudes in the Faddeev 0 2 4 equation[33];andlong-dashed curve –d-quarkobtainedsim- ilarly. The data are from Refs.[42, 43]: u-quark, circles; and x d-quark, diamonds. The dotted curves are determined from theparametrisation of data in Ref.[44]. FIG. 6. Flavour separation of the proton’s Pauli form factor, as a function of x=Q2/m2 : d-quark, upper panel; and N u-quark, lower panel. Solid curve – result obtained using the contact interaction; dashed curve – obtained from QCD- like momentum-dependence for the dressed-quark propagators and diquark Bethe-Salpeter amplitudes in the Faddeev equation[33]; dotted curve –determinedfrom theparametrisation of data in Ref.[44]; and data from Refs.[42, 43, 45]. not, and the extrapolation of a modern parametrisation of data produces a zero which is coincident with that predicted by the QCD-based interaction [33, 41]. The B. Sachs form factors zero owes to the presence of diquark correlations in the nucleon. It has been found [33] that the proton’s singly- The lower panel of Fig.7 depicts the ratio of proton representedd-quarkismorelikelytobestruckinassocia- Sachs electric and magnetic form factors: tionwithanaxial-vectordiquarkcorrelationthanwitha scalar, and form factor contributions involving an axial- Q2 G (Q2)=F (Q2) F (Q2), (5a) vector diquark are soft. On the other hand, the doubly- Ep 1p − 4m2 2p N representedu-quarkispredominantlylinkedwith harder G (Q2)=F (Q2)+F (Q2). (5b) scalar-diquark contributions. This interference produces Mp 1p 2p the zero in the Dirac form factor of the d-quark in the Once again,the existence of a zero is independent of the proton. The location of the zero depends on the relative interaction upon which the Faddeev equation is based probability of finding 1+ and 0+ diquarks in the proton: but the location is not. That location is insensitive to withincreasingprobabilityforanaxial-vectordiquark,it the size of the diquark correlations [33]. movestosmaller-x–inRef.[33]thescalar-diquarkprob- In order to assist in explaining the origin and location ability is 60%, whereas herein it is 78%. of a zero in the Sachs form factor ratio, in the top panel of Fig.7 we depict the ratio of Pauli and Dirac form factors: both the actual contact-interactionresult and that obtained when the Pauli form factor is artificially “soft- Weplottheflavourdecompositionoftheproton’sPauli ened;” viz., formfactorinFig.6. Onceagain,thecontact-interaction results are far too hard and the general trend of the F (Q2) data favours a Faddeev equation built from dressed- F2p(Q2)→ 1+Q2p2/(4m2 ). (6) quark propagators and diquark Bethe-Salpeter ampli- N tudes which are QCD-like in their momentum depen- AsobservedinRef.[46],asofteningoftheproton’sPauli dence. form factor has the effect of shifting the zero to larger 6 valuesofQ2. Infact,ifF becomessoftquicklyenough, 0 2 4 6 8 10 2p then the zero disappears completely. The Pauli form factor is a gauge of the distribution 1.6 of magnetisation within the proton. Ultimately, this magnetisation is carried by the dressed-quarks and in- p 1.2 1 fluenced by correlations amongst them, which are ex- F p pressed in the Faddeev wave-function. If the dressed- Κ 0.8 (cid:144) p quarksaredescribedbyamomentum-independentmass- 2 F function, then they behave as Dirac particles with con- 0.4 stant Dirac values for their magnetic moments and produce a hard Pauli form factor. Alternatively, suppose 0 that the dressed-quarkspossess a momentum-dependent 1 mass-function, which is large at infrared momenta but vanishes as their momentum increases. At small mo- 0.8 menta they will then behave as constituent-like parti- p M cles with a large magnetic moment, but their mass and G 0.6 à magneticmomentwilldroptowardzeroastheprobemo- (cid:144)p àá á ò mentumgrows. (N.B.Masslessfermionsdonotpossessa GE 0.4 à á ò measurablemagneticmoment[39].) Suchdressed-quarks p à Μ 0.2 will produce a proton Pauli form factor that is large for ò Q2 0butdropsrapidlyonthedomainoftransitionbe- 0 ∼ tween nonperturbative and perturbative QCD, to give a very small result at large-Q2. The precise form of the 0 2 4 6 8 10 Q2-dependence will depend on the evolving nature of Q2 HGeVL2 theangularmomentumcorrelationsbetweenthedressed- quarks. From this perspective, existence, and location if so,ofthezeroinµ G (Q2)/G (Q2)areafairlydirect FIG. 7. Upper panel: Normalised ratio of proton Pauli and p Ep Mp measureofthelocationandwidthofthetransitionregion Dirac form factors. Solid curve – contact interaction; long- dashed curve – result from Ref.[40], which employed QCD- between the nonperturbative and perturbative domains like momentum-dependence for the dressed-quark propaga- of QCD as expressed in the momentum-dependence of torsanddiquarkBethe-Salpeteramplitudes;long-dash-dotted the dressed-quark mass-function. curve – drawn from parametrisation of experimental data in Weexpectthatamass-functionwhichrapidlybecomes Ref.[34]; and dotted curve – softened contact-interaction re- partonic – namely, is very soft – will not produce a zero; sult,describedinconnectionwithEq.(6). Lower panel: Nor- have seen that a constantmass-function produces a zero malised ratio ofprotonSachselectricandmagneticform fac- at a small value of Q2, and know that a mass-function tors. Solidcurve andlong-dashed curve,asabove;dot-dashed which resemblesthat obtainedin the best availableDSE curve–linearfittodatainRefs.[50–54],constrainedtooneat studies[47,48]andvialattice-QCDsimulations[49],pro- Q2=0; short-dashed curve –[1,1]-Padéfit tothat data;and dotted curve – softened contact-interaction result, described duces a zero at a location that is consistent with extant in connection with Eq.(6). In addition, we have represented data. There is an opportunity here for very constructive a selection of data explicitly: filled-squares [51]; circles [53]; feedback between future experiments and theory. up-triangles [54]; and open-squares [55]. C. Valence-quark distributions at x=1 Q2 +2P Q = 0; i.e., one is dealing with elastic scat- · tering. Therefore, in the neighbourhood of x = 1 the At this point we would like to exploit a connection structurefunctionsaredeterminedbythetarget’selastic between the Q2 = 0 values of elastic form factors and form factors. The ratio in Eq.(7) expresses the relative the Bjorken-x = 1 values of the dimensionless structure probability of finding a d-quark carryingall the proton’s functions of deep inelastic scattering, Fn,p(x). Our first light-front momentum compared with that of a u-quark 2 remark is that the x=1 value of a structure function is doing the same or, equally, owing to invariance under invariant under the evolution equations [23]. Hence the evolution, the relative probability that a Q2 = 0 probe value of either scatters from a d-quark or a u-quark; viz., dv(x) , where dv(x) = 4FF22np((xx)) −1, (7) udv((xx)) = PP1pp,,ud. (8) uv(x)(cid:12)(cid:12)x→1 uv(x) 4− FF22np((xx)) v (cid:12)(cid:12)(cid:12)x→1 1 (cid:12) Plainly, in SU(6) const(cid:12)ituent-quark models, the right- (cid:12) is a scale-invariant feature of QCD and a discriminator hand-side of Eq.(8) is 1/2. On the other hand, when a between models. Next, when Bjorken-x is unity, then Poincaré-covariantFaddeev equation is employed to de- 7 0 2 4 6 8 10 TABLE III. Probabilities described after Eq.(9), from which 1 one may compute the evolution-invariant x = 1 value of the structurefunction ratio. 0.8 P1p,s P1p,a P1p,m udvv FF22np Lx 0.6 M=constant 0.78 0.22 0 0.18 0.41 Hp M(p2) 0.60 0.25 0.15 0.28 0.49 R, 1 0.4 F 0.2 scribe the nucleon, 0 1 Pp,d 2Pp,a+ 1Pp,m P1p,u = Pp,s3+11Pp,a3+12Pp,m, (9) p 0.8 1 1 3 1 3 1 R, Κ (cid:144) 0.6 L where we have used the notation of Ref.[33]. Namely, x H P1p,s = F1sp(Q2 = 0) is the contribution to the proton’s R,p 0.4 chargearising fromdiagrams with a scalardiquark com- 2 F ponentinboththeinitialandfinalstate: u[ud] γ u[ud]. 0.2 ⊗ ⊗ The diquark-photon interaction is far softer than the 0 quark-photon interaction and hence this diagram contributes solely to u at x = 1. Pp,a = Fa(Q2 = 0), 0 2 4 6 8 10 v 1 1p is the kindred axial-vector diquark contribution; viz., x 2d uu γ d uu +u ud γ u ud . At x = 1 { }⊗ ⊗ { } { }⊗ ⊗ { } this contributes twice as much to d as it does to u . v v FIG.8. Comparison ofcharged-Roperandproton Dirac(up- P1p,m = F1mp(Q2 = 0), is the contribution to the pro- perpanel)andPauli(lowerpanel)formfactors,asafunction ton’s charge arising from diagrams with a different di- of x = Q2/m2 : Solid curve – Roper; and dashed-curve – N quark component in the initial and final state. The ex- proton. All results obtained using the contact-interaction, istence of this contribution relies on the exchange of a andhenceadressed-quarkmass-functionanddiquarkBethe- quarkbetweenthediquarkcorrelationsandhenceitcon- Salpeter amplitudes that are momentum-independent. tributes twice as much to u as it does to d . If one uses v v the “static approximation” to the nucleon form factor, Eq.(B16), as with the contact-interaction herein, then IV. NUCLEON→ROPER TRANSITION AND Pp,m 0. ROPER ELASTIC 1 ≡ ItisplainfromEq.(9)thatd /u =0intheabsenceof v v axial-vector diquark correlations; i.e., in scalar-diquark- Acomputationofthenucleon-to-Ropertransitionform only models of the nucleon. Furthermore, Eq.(9) pro- factors must be performed in conjunction with that of duces d /u = 0.05, Fn/Fp = 0.30, using the case-II the Roper elastic form factors. They are connected via v v 2 2 solution in Ref.[56], which is fully consistent with Fig.5 orthonormalisation: the Roper is orthogonal to the nu- therein. cleon, which means F (Q2 = 0) = 0 for both the 1∗ Using the probabilitiesderivedfrom Table IB, one ob- chargedandneutralchannels;andthecanonicalnormali- tains the first row in Table III, whilst the second row sationoftheRoperFaddeevamplitudeisfixedbysetting is drawn from Ref.[33]. (Here we correct an error in F1R+(Q2 =0)=1. The transition is calculated with the Ref.[23],whichinadvertentlyinterchanged2 1ineval- kinematic arrangements: uatingthePp,a contribution.) BothrowsinT↔ableIIIare 1 Fconn/sFistpe=nt0w.4it5hdv0/.0u8v)=inf0e.r2r3e±d0r.e0c9en(9t0ly%vciaoncfiodnesnidceerlaetvioeln, Pf2 =−m2R, Pi2 =−m2N, m2R−m2N +2Pi·Q+Q2 =(100), 2 2 ± ofelectron-nucleusscatteringatx>1[57]. Ontheother fromthe transitioncurrentexpressedby the diagramsin hand, this is also true of the result obtained through a Fig.2, which are as explained in App.C except that the naive consideration of the isospin and helicity structure final baryon, Ψf, is the Roper resonance. These consid- of a proton’s light-front quark wave function at x 1, erations lead to the modifications described in App.D. ∼ which leads one to expect that d-quarks are five-times Note that in connection with all form factors involv- less likely than u-quarks to possess the same helicity as ingthe Roperresonance,weonly reportresultsobtained theprotontheycomprise;viz., d /u =0.2[58]. Plainly, with our symmetry-preserving treatment of the contact v v contemporaryexperiment-based analyses do not provide interaction. Thisisafirststep. Basedontheinformation a particularly discriminating constraint. Future experi- inSec.III,weanticipatethatamomentum-dependentin- ments with a tritium target could help [59]. teractionwillproduceRoper-relatedformfactorsthatare 8 0 2 4 6 8 10 TABLE IV. Row 1: Roper results computed herein with the contact interaction, whose input is presented in Table I. 0.4 Row 2: Related contact-interaction nucleon results repeated for ease of comparison. Rows 3, 4: Analogous results ob- L 0.2 tained with a model dressed-quark anomalous magnetic mo- Hx n ment,Sec.V. R, 0 1 F r1R+MN r2R+MN r1R0MN r2R0MN κR+ κR0 0 -0.2 1 Roper 2.96 2.66 0.81 3.19 0.61 -0.61 Nucleon 3.19 2.84 1.21 3.19 1.02 -0.92 -0.4 RoperQAMM 3.29 3.90 0.22 3.46 1.75 -1.20 NucleonQAMM 3.41 4.00 0.55 3.85 1.68 -1.24 0 -0.2 n R, similarforQ2 .0.5GeV2butsofteratlargermomentum (cid:144)Κ -0.4 L scales. x H R,n -0.6 2 F - -0.8 A. Roper Faddeev amplitude -1 TheFaddeevamplitudefortheRoperresonanceinTa- 0 2 4 6 8 10 bleIB,whoseoriginisexplainedinApps.B,D,contrasts x strikingly with that of the nucleon and suggests a fas- cinating new possibility for the structure of the Roper’s FIG. 9. Comparison of neutral-Roper and neutron Dirac dressed-quark core. To explain this remark, we focus (upper panel) and Pauli (lower panel) form factors, as a first on the nucleon, whose Faddeev amplitude describes function of x = Q2/m2 : Solid curve – neutral-Roper; a ground-state that is dominated by its scalar diquark N and dashed-curve – neutron. All results obtained using the component (78%). The axial-vectorcomponent is signif- contact-interaction,andhenceadressed-quarkmass-function icantly smaller but nevertheless important. This heavy and diquarkBethe-Salpeter amplitudes that are momentum- weightingofthescalardiquarkcomponentpersistsinso- independent. lutions obtained with more sophisticated Faddeev equation kernels (see, e.g., Table 2 in Ref.[33]). From a per- spectiveprovidedbythenucleon’sparitypartnerandthe diquark correlation. It is important to check whether radial excitation of that state, in which the scalar and this outcome survives with a Faddeev equation kernel axial-vector diquark probabilities are [15] 51%-49% and built from a momentum-dependent interaction. 43%-57%, respectively, the scalar diquark component of the ground-state nucleonactually appears to be unnatu- rally large. B. Roper elastic One can nevertheless understand the structure of the nucleon. As with so much else, the composition of the The Roper mass and Faddeev amplitude in Table IB nucleon is intimately connected with dynamical chiral produce the radii and anomalous magnetic moments in symmetry breaking. In a two-color version of QCD, the Table IV and the elastic form factors depicted in Figs.8, scalardiquarkisaGoldstonemode,justlikethepion[60]. 9. Notwithstandingthemarkedlydifferentinternalstruc- (This is a long-knownresult of Pauli-Gu¨rseysymmetry.) ture, the Roper elastic form factors are similar to those A memory of this persists in the three-color theory and of the nucleon, both in magnitude and Q2-evolution. isevidentinmanyways. Amongstthem,throughalarge The exception is the Dirac form factor of the neutral value of the canonically normalized Bethe-Salpeter am- Roper, which exhibits a zero at Q2 3m2 . This be- ≃ N plitude and hence a strong quark+quark diquark cou- haviour derives from a constructive interference between − pling within the nucleon. (A qualitatively identical ef- Diagrams2and3inFig.2that,withincreasingQ2,sums fect explains the large value of the πN coupling con- to overwhelm the always-negative contribution from Di- stant.) Thereisnosuchenhancementmechanismassoci- agram 1. As Q2 increases, the dominant contributions ated with the axial-vector diquark. Therefore the scalar expressed by Diagrams 2 and 3 are associated with a diquark dominates the nucleon. photon scattering from the positively-charged [ud] and With the Faddeev equation treatment described ud correlations, whereas Diagram 1 is alone in mea- { } herein,theeffectontheRoperisdramatic: orthogonality suring only a negative charge; i.e., that of the d-quark. of the ground- and excited-states forces the Roper to be Ultimately, therefore, suppression of the scalar-diquark constituted almost entirely (81%) from the axial-vector component in the Roper is responsible for the zero in 9 0 2 4 6 0 2 4 6 L x 0.4 H 0.4 p ® p R 0 ® 1 R F Κ 0 10 -0.4 L(cid:144)x H Lx, -0.8 p® -0.4 H R p 2 ® F R -1.2 - 2 -0.8 F 0 -1.6 1 0.12 0.2 à à ò 0.1 ò 0 ò Lx 0.08 Lx øø Hp® 0.06 ø à Hp® -0.2 àòà R R 1 à 2 F 0.04 ò F à -0.4 0.02ø 0 à -0.6 0 2 4 6 0 2 4 6 x x FIG. 10. Upper panel – F1∗ (solid and dot-dashed with FIG. 11. Comparison between F2∗(x) computed using the dressed-quark anomalous magnetic moment, Sec.V) and F2∗ framework described herein and available data [17–19], with (dashed and dotted with dressed-quark anomalous magnetic x=Q2/m2N. Upperpanel –normalisedtounityatx=0;and moment) as a function of x = Q2/m2 , computed using lower panel, as computed. In both panels the dashed curve N the framework described herein. Lower panel – Computed wascomputedwithamodelforthedressed-quarkanomalous form of F1∗(x) compared with available data [17–19]. The electromagnetic moment, Sec.V. The squares, triangles and squares, triangles and stars are preliminary results [61] from stars arepreliminary resultsfrom asimulation ofNf =2+1 a simulation of Nf = 2 + 1 lattice-QCD at, respectively, lattice-QCD at, respectively, m2π/m2πexpt. ≃8, 10, 40 [61]. m2/m2 ≃8, 10, 40. π πexpt. 2+1 results also support the presence of a zero in F . 2∗ F at Q2 >0. In Fig.12 we display the separate contributions from 1R0 eachdiagramrepresentedbythecurrentinFig.2. Whilst Diagram 1 with a scalar diquark bystander is plainly C. Transition dominant,asignificantcontributionisalsoreceivedfrom Diagram 2 with a photon probing the structure of the axial-vectordiquarkcorrelations. Theformfactorisneg- In Figs.10, 11 we depict the charged-Roper proton → ative at Q2 = 0 owing to orthogonality, which produces transition form factors computed using our treatment of s s < 0, and passes through zero because of the zero the contact interaction. The calculated form factors un- R N derestimate the data on the domain 0 < Q2 < 3GeV2 intheRoper’sFaddeevamplitude,whichischaracteristic of a radial excitation. andareveryprobablytoohard. Bothofthesedefectsare natural given that we have: deliberately omitted effects Figure 13 depicts the neutral-Roper neutrontransi- associatedwithamesoncloudinthe Faddeevkerneland tion form factors. Each possesses a ze→ro at Q2 ≃ 3m2N; the current; and used a contact interaction. the Dirac form factor is an order-of-magnitude smaller than its analogue in the charged-Roper transition; and On the other hand, the results are qualitatively in regarding F cf. F , in the neighbourhood of agreementwiththe trendapparentinavailabledata and 2R0→n 2R+→p reproduce the zero in F (Q2) at Q2 0.5m2 without Q2 = 0 the similar magnitude but opposite sign is con- 2∗ ≃ N sistent with available data [3]. fine tuning. These are meaningful successes given that they are features derived only from that which we consider to be the Roper’s dressed-quark core. As shown in the figures, lattice-QCD results are also V. ANOMALOUS MAGNETIC MOMENTS available for these form factors [61]. They have roughly the same magnitude as the experimental data. In con- It is noticeable from the lower panel of Fig.11 that trasttoearliersimulationsofquenched-QCD,theseN = the magnitude of F (Q2 = 0) is underestimated in our f 2∗ 10 0 2 4 6 0.04 Lx s H 0.6 m p ® a R agr 0 0F1 0.3 di 1 - L, 0 x Lx -0.04 Hn H ® -0.3 p ® R R F1 F2 -0.08 00 -0.6 0 2 4 6 1 L x x H p 0.8 ® R FIG. 12. Separation of F2∗(x) into contributions from differ- F2 0.4 ent diagrams, with x = Q2/m2 : solid – photon on u-quark 0 N 1 with scalar diquark spectator; dashed – photon on scalar di- , 0 quark with u-quark spectator; dot-dashed – photon on axial- Lx H vectordiquarkwithquarkspectator;dotted –photon-induced n -0.4 ® transition between scalar and axial-vector diquarks with u- R 2 quark spectator. N.B. Owing to Eq.(C5), there is no contri- F -0.8 0 bution involving an axial-vector diquarkspectator. 1 0 2 4 6 x framework: 0.1 cf. experiment [17], 0.56 0.02. A similar but s−maller deficit is apparent−in our±computed FIG. 13. Upper panel – F1R0→n (solid) as a function of x= nucleon anomalous electromagnetic moments, Table II. Q2/m2N compared with F1R+→p (dashed), computed using theframework described herein. Lower panel – Analogue for In this connection it is interesting to explore the effect produced by the dressed-quark anomalous electromag- F2R0→n. netic moment, which is produced by DCSB [39] and is known to have a material impact on the nucleons’ Pauli from the transition form factors in Eq.(3): form factors [40]. A (Q2)=c(Q2) F (Q2)+F (Q2) , (11a) 1 1∗ 2∗ To this end we modified the quark-photoncoupling as 2 describedinApp.C6andrecomputedalltheformfactors S (Q2)= qCM(cid:2)Sc(Q2) F (Q2)m(cid:3)R+mN described above. Some results for the nucleon are sum- 21 − √2 − 1∗ Q2 (cid:20) marisedinthelastrowofTableII:ineachcase,inclusion F (Q2) 2∗ of the dressed-quark anomalous magnetic moment pro- + , (11b) m +m duces a significant improvement in the comparison with R N(cid:21) data. A similar comparison is made for the Roper in with Table IV. Results for the Roper proton transition form factor c(Q2)= παQ2− 21 , q = Q2−Q2+, (12) are included in Figs.10, 1→1. Inclusion of a dressed-quark (cid:20)mRmNK(cid:21) CMS q2mR anomalouselectromagneticmomenthasapronouncedef- whereQ2 =Q2+(m m )2,K =(m2 m2 )/(2m ), fect on F2∗, which moves the result a little closer to ex- and α is±QED’s fine sRtr±uctuNre constant.R− N R periment: F (Q2 = 0) = 0.1 0.16 cf. experiment 2∗ − → − In addition to our own computation, Fig.14 displays [17] 0.56 0.02. It does not, however,compensate suf- − ± results obtained using a light-front constituent-quark ficiently for the absence of meson-cloud effects. model [62], which employed a constituent-quark mass of 0.22GeV and identical momentum-space harmonic os- cillator wave functions for both the nucleon and Roper (width= 0.38GeV) but with a zero introduced for the Roper, whose location was fixed by an orthogonality VI. MESON CLOUD condition. The quark mass is smaller than the DCSB- induced value we determined from the gap equation(see Table I) but a more significant difference is the choice of In Fig.14 we draw the helicity amplitudes for the spin-flavour wave functions for the nucleon and Roper. γ∗p P (1440) transition. They may be computed In Ref.[62] they are simple SU(6) O(3) S-wave states 11 → ×