Understanding the nucleon as a Borromean bound-state
aa r X i v : . [ nu c l - t h ] J un Understanding the nucleon as a Borromean bound-state
Jorge Segovia a , Craig D. Roberts b , Sebastian M. Schmidt c a Instituto Universitario de F´ısica Fundamental y Matem´aticas (IUFFyM), Universidad de Salamanca, E-37008 Salamanca, Spain b Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA c Institute for Advanced Simulation, Forschungszentrum J¨ulich and JARA, D-52425 J¨ulich, Germany
Abstract
Analyses of the three valence-quark bound-state problem in relativistic quantum field theory predict that the nucleon may beunderstood primarily as a Borromean bound-state, in which binding arises mainly from two separate e ff ects. One originates innon-Abelian facets of QCD that are expressed in the strong running coupling and generate confined but strongly-correlated colour-antitriplet diquark clusters in both the scalar-isoscalar and pseudovector-isotriplet channels. That attraction is magnified by quarkexchange associated with diquark breakup and reformation. Diquark clustering is driven by the same mechanism which dynamicallybreaks chiral symmetry in the Standard Model. It has numerous observable consequences, the complete elucidation of whichrequires a framework that also simultaneously expresses the running of the coupling and masses in the strong interaction. Plannedexperiments are capable of validating this picture. Keywords: confinement, continuum QCD, diquark clusters, dynamical chiral symmetry breaking, nucleon form factors
1. Introduction . The proton is the core of the hydrogen atom,lies at the heart of every nucleus, and has never been observedto decay; but it is nevertheless a composite object, whose prop-erties and interactions are determined by its valence-quark con-tent: u + u + d , i.e . two up ( u ) quarks and one down ( d ) quark.So far as is now known [1], bound-states seeded by two valence-quarks do not exist; and the only two-body composites are thoseassociated with a valence-quark and -antiquark, i.e . mesons.These features are supposed to derive from colour confinement.Suspected to emerge in QCD, confinement is an empirical real-ity; but there is no universally agreed theoretical understanding.Such observations lead one to a position from which the pro-ton may be viewed as a Borromean bound-state, viz . a systemconstituted from three bodies, no two of which can combine toproduce an independent, asymptotic two-body bound-state. InQCD the complete picture of the proton is more complicated,owing, in large part, to the loss of particle number conservationin quantum field theory and the concomitant frame- and scale-dependence of any Fock space expansion of the proton’s wavefunction [2–5]. Notwithstanding that, the Borromean analogyprovides an instructive perspective from which to consider bothquantum mechanical models and continuum treatments of thenucleon bound-state problem in QCD. It poses a crucial ques-tion: Whence binding between the valence quarks in the proton, i.e . what holds the proton together?In numerical simulations of lattice-regularised QCD (lQCD)that use static sources to represent the proton’s valence-quarks,a “Y-junction” flux-tube picture of nucleon structure is pro-duced, e.g . Ref. [6, 7]. This might be viewed as originating in Email addresses: [email protected] (Jorge Segovia), [email protected] (Craig D. Roberts), [email protected] (Sebastian M. Schmidt) the three-gluon vertex, which signals the non-Abelian characterof QCD and is the source of asymptotic freedom [8–10]. Suchresults and notions would suggest a key role for the three-gluonvertex in nucleon structure if they were equally valid in real-world QCD wherein light dynamical quarks are ubiquitous. Aswill become evident, however, they are not; and so a di ff erentexplanation of binding within the nucleon must be found.
2. DCSB and diquark correlations . Dynamical chiral sym-metry breaking (DCSB) is another of QCD’s emergent phenom-ena; and contemporary theory indicates that it is responsible formore than 98% of the visible mass in the Universe [11, 12]. Wejudge it probable that DCSB and confinement, defined via theviolation of reflection positivity by coloured Schwinger func-tions (see, e.g . Refs. [13–17] and citations thereof) have a com-mon origin in the Standard Model; but this does not meanthat DCSB and confinement must necessarily appear together.Models can readily be built that express one without the other, e.g . numerous constituent quark models express confinementthrough potentials that rise rapidly with interparticle separationbut nevertheless possess no ready definition of a chiral limit;and models of the Nambu–Jona-Lasinio type typically expressDCSB but not confinement.DCSB ensures the existence of nearly-massless pseudo-Goldstone modes (pions), each constituted from a valence-quark and -antiquark whose individual Lagrangian current-quark masses are <
1% of the proton mass [18]. In the presenceof these modes, no flux tube between a static colour source andsink can have a measurable existence. To verify this statement,consider such a tube being stretched between a source andsink. The potential energy accumulated within the tube mayincrease only until it reaches that required to produce a particle-antiparticle pair of the theory’s pseudo-Goldstone modes. Sim-
Preprint submitted to Physics Letters B 11 June 2015 lations of lQCD show [19, 20] that the flux tube then disap-pears instantaneously along its entire length, leaving two iso-lated colour-singlet systems. The length-scale associated withthis e ff ect in QCD is r σ ≃ (1 /
3) fm and hence if any such stringforms, it would dissolve well within hadron interiors.This discussion has exposed two corollaries of DCSB thatare crucial in determining the observable features of the Stan-dard Model. Another equally important consequence of DCSBis less well known. Namely, any interaction capable of creat-ing pseudo-Goldstone modes as bound-states of a light dressed-quark and -antiquark, and reproducing the measured value oftheir leptonic decay constants, will necessarily also generatestrong colour-antitriplet correlations between any two dressedquarks contained within a nucleon. Although a rigorous proofwithin QCD cannot be claimed, this assertion is based uponan accumulated body of evidence, gathered in two decades ofstudying two- and three-body bound-state problems in hadronphysics, e.g . Refs. [21–35]. No realistic counter examples areknown; and the existence of such diquark correlations is alsosupported by simulations of lQCD [36, 37].The properties of diquark correlations have been charted.Most importantly, diquarks are confined. However, this is nottrue if the leading-order (rainbow-ladder, RL [23, 38]) trunca-tion is used to define the associated scattering problem [26].Corrections to that simplest symmetry-preserving approxima-tion are critical in quark-quark channels: they eliminate bound-state poles from the quark-quark scattering matrix but preservethe strong correlations [23, 27, 28].Additionally, owing to properties of charge-conjugation, adiquark with spin-parity J P may be viewed as a partner tothe analogous J − P meson [21]. It follows that scalar, isospin-zero and pseudovector, isospin-one diquark correlations are thestrongest; and whilst no pole-mass exists, the following mass-scales, which express the strength and range of the correla-tion and are each bounded below by the partnered meson’smass, may be associated with these diquarks [21, 26, 36, 37]: m [ ud ] + ≈ . − . m { uu } + ≈ . − . m { dd } + = m { ud } + = m { uu } + in the isospin symmetric limit. Realisticdiquark correlations are also soft. They possess an electro-magnetic size that is bounded below by that of the analogousmesonic system, viz . [39, 40]: r [ ud ] + & r π , r { uu } + & r ρ , (1)with r { uu } + > r [ ud ] + . As in the meson sector, these scales are allset by that associated with DCSB.It is worth remarking here that in a dynamical theory basedon SU(2)-colour, diquarks are colour-singlets. They would thusexist as asymptotic states and form mass-degenerate multipletswith mesons composed from like-flavoured quarks. (Theseproperties are a manifestation of Pauli-G¨ursey symmetry [41,42].) Consequently, the [ ud ] + diquark would be massless in thepresence of DCSB, matching the pion, and the { ud } + diquarkwould be degenerate with the theory’s ρ -meson. Such identi-ties are lost in changing the gauge group to SU(3)-colour; butclear and instructive similarities between mesons and diquarksnevertheless remain, as we have described above. = a Ψ Pp q p d Γ b Γ − a p d p q b Ψ P q Figure 1: Poincar´e covariant Faddeev equation. Ψ is the Faddeev amplitudefor a baryon of total momentum P = p q + p d , where p q , d are, respectively,the momenta of the quark and diquark within the bound-state. The shaded areademarcates the Faddeev equation kernel: single line , dressed-quark propagator; Γ , diquark correlation amplitude; and double line , diquark propagator.
3. Diquarks in the nucleon . The bulk of QCD’s particular fea-tures and nonperturbative phenomena can be traced to the evo-lution of the strong running coupling. Its unique characteristicsare primarily determined by the three-gluon vertex: the four-gluon vertex does not contribute dynamically at leading order inperturbative analyses of matrix elements; and nonperturbativecontinuum analyses of QCD’s gauge sector indicate that sat-isfactory agreement with gluon propagator results from lQCDsimulations is typically obtained without reference to dynam-ical contributions from the four-gluon vertex, e.g . Refs. [43–51]. The three-gluon vertex is therefore the dominant factor inproducing the class of renormalisation-group-invariant runninginteractions that have provided both successful descriptions ofand predictions for many hadron observables [52–56]. It is thisclass of interactions that generates the strong attraction betweentwo quarks which produces tight diquark correlations in analy-ses of the three valence-quark scattering problem.The existence of tight diquark correlations considerably sim-plifies analyses of the three valence-quark scattering problemand hence baryon bound states because it reduces that task tosolving a Poincar´e covariant Faddeev equation [22], depictedin Fig. 1. The three gluon vertex is not explicitly part of thebound-state kernel in this picture of the nucleon. Instead, onecapitalises on the fact that phase-space factors materially en-hance two-body interactions over n ≥ ff ect fine details of baryon structure, the dominant e ff ectof non-Abelian multi-gluon vertices is expressed in the forma-tion of diquark correlations. Such a nucleon is then a com-pound system whose properties and interactions are primarilydetermined by the quark + diquark structure evident in Fig. 1.It is important to highlight that both scalar-isoscalar andpseudovector-isotriplet diquark correlations feature within a nu-cleon. Any study that neglects pseudovector diquarks is unreal-istic because no self-consistent solution of the Faddeev equationin Fig. 1 can produce a nucleon constructed solely from a scalardiquark, e.g . pseudovector diquarks typically provide roughly150 MeV of attraction [32]. The relative probability of scalarversus pseudovector diquarks in a nucleon is a dynamical state-ment. Realistic computations predict a scalar diquark strengthof approximately 60% [29, 34, 35]. As will become clear, thisprediction can be tested by contemporary experiments.The quark + diquark structure of the nucleon is elucidated2 igure 2: Dominant piece in the nucleon’s eight-component Poincar´e-covariantFaddeev amplitude: s ( | p | , cos θ ). In the nucleon rest frame, this term describesthat piece of the quark-diquark relative momentum correlation which possesseszero intrinsic quark-diquark orbital angular momentum, i.e . L = p = P / − p q and cos θ = p · P / p p P . (The amplitude is normalisedsuch that its U Chebyshev moment is unity at | p | = in Fig. 2, which depicts the leading component of its Fad-deev amplitude: with the notation of Ref. [34], s ( | p | , cos θ ),computed using the Faddeev kernel described therein. Thisfunction describes a piece of the quark + scalar-diquark relativemomentum correlation. Notably, in this solution of a realis-tic Faddeev equation there is strong variation with respect toboth arguments. Support is concentrated in the forward di-rection, cos θ >
0, so that alignment of p and P is favoured;and the amplitude peaks at ( | p | ≃ M N / , cos θ = p q ≈ P / ≈ p d and hence the natural relative momentum iszero. In the antiparallel direction, cos θ <
0, support is con-centrated at | p | = i.e . p q ≈ P / p d ≈ P /
3. A realisticnucleon amplitude is evidently a complicated function; and sig-nificant structure is lost if simple interactions and / or truncationsare employed in building the Faddeev kernel, e.g . extant treat-ments of a momentum-independent quark-quark interaction – acontact interaction – produce a Faddeev amplitude that is alsomomentum independent [57, 58], a result exposed as unrealisticby Fig. 2 for any probe sensitive to the nucleon interior.A nucleon (and kindred baryons) described by Fig. 1 is a Bor-romean bound-state, the binding within which has two contri-butions. One part is expressed in the formation of tight diquarkcorrelations. That is augmented, however, by attraction gen-erated by the quark exchange depicted in the shaded area ofFig. 1. This exchange ensures that diquark correlations withinthe nucleon are fully dynamical: no quark holds a special placebecause each one participates in all diquarks to the fullest extentallowed by its quantum numbers. The continual rearrangementof the quarks guarantees, inter alia , that the nucleon’s dressed-quark wave function complies with Pauli statistics.It is impossible to overstate the importance of appreciatingthat these fully dynamical diquark correlations are vastly dif-ferent from the static, pointlike “diquarks” which featured in early attempts [59, 60] to understand the baryon spectrum andto explain the so-called missing resonance problem [61–63].Modern diquarks are soft, Eq. (1); and, as we shall explain,enforce certain distinct interaction patterns for the singly- anddoubly-represented valence-quarks within the proton. On theother hand, the number of states in the spectrum of baryonsobtained from the Faddeev equation in Fig. 1 [64] is similar tothat found in the three-constituent quark model, just as it is intoday’s lQCD calculations of this spectrum [65].
4. Nucleon current . The Poincar´e-covariant photon-nucleoninteraction current is: J µ ( K , Q ) = ie ¯ u ( P f ) " γ µ F ( Q ) + σ µν Q ν m N F ( Q ) u ( P i ) , (2)where P i ( P f ) is the momentum of the incoming (outgoing) nu-cleon; Q = P f − P i , K = ( P i + P f ) /
2: for elastic scattering, K · Q = K = − m N (1 + τ N ), τ N = Q / (4 m N ). The func-tions F , are, respectively, the Dirac and Pauli form factors: F (0) expresses the bound-state’s electric charge and F (0), itsanomalous magnetic moment, κ N = n , p . Notably, F ≡ G E = F − τ N F , G M = F + F .A nucleon described by the Faddeev equation in Fig. 1 is con-stituted from dressed-quarks, any two of which are always cor-related as either a scalar or pseudovector diquark. If this isa veracious description of Nature, then the presence of thesecorrelations must be evident in numerous empirical di ff erencesbetween the response of the bound-state’s doubly- and singly-represented quarks to any probe whose wavelength is smallenough to expose the diquarks’ nonpointlike character. Asso-ciating a monopole mass with the radii in Eqs. (1), it becomesapparent that this wavelength corresponds to momentum trans-fers Q & m ρ , where m ρ is the ρ -meson’s mass.In connection with electromagnetic probes, it is now pos-sible to check these predictions following the appearance ofhigh precision data on the neutron’s electric form factor outto Q = . [67]. The G nE data are significant largelybecause they can be combined with existing empirical informa-tion on G nM , G pE , M in order to produce a flavour separation ofthe proton’s Dirac and Pauli form factors [68, 69], i.e . a chartof the separate contributions of u - and d -quarks to the proton’sform factors. Supposing s -quark contributions are negligible,as seems the case [70], and assuming charge symmetry, then F u , = F p , + F n , , F d , = F n , + F p , . (3)In the future, nucleon-to-resonance transition form factorsmight be used similarly [71–73], in which event numerous newwindows on baryon structure would be opened.Evaluation of the nucleon currents is detailed in Ref. [34]and the results we describe herein are derived from that analy-sis, which provides a unified description of the electromagneticproperties of the nucleon, ∆ -baryon and Roper resonance [35].In what follows, it is important to note that the nucleon currentcan unambiguously be decomposed as follows: J µ ( K , Q ) = X k , l = ,..., J kl µ ( K , Q ) , (4)3 æææææææææææ æà à à àààà à à à ÷ ÷ ÷ ì ì ì Q @ GeV D Μ p G E p (cid:144) G M p Figure 3: Computed ratio of proton electric and magnetic form factors.Curves: solid (black) – full result, determined from the complete proton Fad-deev wave function and current; dot-dashed (red) – momentum-dependence ofscalar-diquark contribution [sum over k , l = , S -wave in the rest-frame [from J µ in Eq.(4)]; dotted (blue) – momentum-dependence of pseu-dovector diquark contribution [from the sum over k , l = , Q =
0. Data:circles (blue) [76]; squares (green) [77]; asterisks (brown) [78]; and diamonds(purple) [79]. where k , l , respectively, label the diquark component in thecomplete Faddeev wave function for the final and initial state.For example, J µ denotes that contribution to the current ob-tained when one selects for both the final and initial state ascalar diquark correlation with L = k , l = ,
5. Verifiable predictions of diquark pairing . Consider theratio of proton electric and magnetic form factors, R EM ( Q ) = µ p G E ( Q ) / G M ( Q ), µ p = G M (0). A series of experiments [74–79] has determined that R EM ( Q ) decreases almost linearly with Q and might become negative for Q & . Our first goalis to clarify the origin of this behaviour.A clear conclusion from Fig. 3 is that pseudovector diquarkcorrelations have little influence on the momentum dependenceof R EM ( Q ). Their contribution is indicated by the dotted (blue)curve, which was obtained by setting the scalar diquark com-ponent of the proton’s Faddeev amplitude to zero and renor-malising the result to unity at Q =
0. As apparent from thedot-dashed (red) curve, the evolution of R EM ( Q ) with Q isprimarily determined by the proton’s scalar diquark component.In this component, the valence d -quark is sequestered inside thesoft scalar diquark correlation so that the only objects within thenucleon which can participate in a hard scattering event are thevalence u -quarks. (Any interaction with the d -quark attracts a1 / Q suppression because it is always locked into a correlationdescribed by a meson-like form factor [39].)It is known from Ref. [55] that scattering from the proton’svalence u -quarks is responsible for the momentum dependenceof R EM ( Q ). However, the dashed (green) curve in Fig. 3 re-veals something more, i.e . components of the nucleon associ-ated with quark-diquark orbital angular momentum L = æææææææææææææ ææ ææ æ æ æ æ æà à à à à = Q (cid:144) M N @ x F p D (cid:144) F p æ ææææææ æ æ æ æ æ æ - = Q (cid:144) M N F n (cid:144) @ x F n D Figure 4:
Upper panel . Proton ratio R ( x ) = xF ( x ) / F ( x ), x = Q / M N .Curves: solid (black) – full result, determined from the complete proton Fad-deev wave function and current; dot-dashed (red) – momentum-dependence ofthe scalar-diquark contribution; dashed (green) – momentum-dependence ofthat component of the scalar diquark contribution to the proton’s Faddeev wavefunction which is purely S -wave in the rest-frame; dotted (blue) – momentum-dependence of the pseudovector diquark contribution. Lower panel . Neutronratio R n ( x ) = F n ( x ) / [ xF n ( x )]. Curve legend as in the upper panel. The data inboth panels are drawn from Refs. [67, 68, 80–84]. tably, the presence of such components is an inescapable con-sequence of the self-consistent solution of a realistic Poincar´e-covariant Faddeev equation for the nucleon. The visible impacton R EM ( Q ) is primarily driven by a marked reduction in F p anda lesser e ff ect on F p when the L = ff ect can be understood once it is recalled that a Gordonidentity may be used to re-express the γ µ term in Eq. (2) as asum of two equally important terms, viz . a convection current,as appears in the nonrelativistic case, and a spin current, whichleads to a gyromagnetic ratio of two for a pointlike fermion.It must also be noted that the presence of diquark correlationsand the use of a Poincar´e covariant framework is insu ffi cient toexplain the data in Fig. (3). It is possible to incorporate bothbut still fail in this comparison, e.g . Faddeev equation studiesbased on a quark-quark contact interaction always generate azero in the neighbourhood Q ≃ M N [57, 58], and are thusruled-out by the data. As explained in Refs. [57, 85], the flawin those studies is the contact interaction itself, which generatesa momentum-independent dressed-quark mass. The existenceand location of a zero in R EM ( Q ) are a measure of nonpertur-bative features of the quark-quark interaction, with particular4ensitivity to the running of the dressed-quark mass [85].It is natural now to consider the proton ratio: R ( x ) = xF ( x ) / F ( x ), x = Q / M N , drawn in the upper panel of Fig. 4.As with R EM , the momentum dependence of R ( x ) is princi-pally determined by the scalar diquark component of the proton.Moreover, the rest-frame L = L , ff ect is manifest in analyses of the N → ∆ transition [33, 34].The lower panel of Fig. 4 displays an analogous ratio forthe neutron: R n ( x ) = F n ( x ) / [ xF n ( x )]. Here the curve ob-tained in the absence of pseudovector diquarks does not resem-ble the data, despite the fact that both the scalar-diquark-onlyand pseudovector-diquark-only curves are finite at x =
0. Ap-parently, something more than orbital angular momentum and arunning quark mass is important in understanding and explain-ing the behaviour of nucleon electromagnetic form factors; andwhatever it is must distinguish between isospin partners. Thiscould have been anticipated from Ref. [29] through a compari-son of Figs. 6 and 13 therein: whilst so-called precocious scal-ing was evident in R ( x ), this was not the case for R n ( x ). Theadditional feature, of course, is the presence of scalar and pseu-dovector diquark correlations, which have di ff erent impacts onthe doubly and singly represented valence-quarks.Figure 5 displays the proton’s flavour separated Dirac andPauli form factors. The salient features of the data are: the d -quark contribution to F p is far smaller than the u -quark con-tribution; F d /κ d > F u /κ u on x < x >
2; and in both cases the d -quark contribution falls dra-matically on x > u -quark contribution remainsroughly constant. Our calculations are in semi-quantitativeagreement with the empirical data. They reproduce the qual-itative behaviour and also predict a zero in F d at x ≃ u [ ud ]; namely, a u -quark in tandem with a [ ud ] scalar correlation, which pro-duces 62% of the proton’s normalisation [88]. If this were thesole component, then photon– d -quark interactions within theproton would receive a 1 / x suppression on x >
1, because the d -quark is sequestered in a soft correlation, whereas a spectator u -quark is always available to participate in a hard interaction.At large x = Q / M N , therefore, scalar diquark dominance leadsone to expect F d ∼ F u / x . Available data are consistent with thisprediction but measurements at x > æææææææææ æ æ æ æààààààààà à à à à x F d , x F u æææææææææ æ æ æ æààààààààà à à à à = Q (cid:144) M N Κ d - x F d , Κ u - x F u Figure 5:
Upper panel . Flavour separation of the proton’s Dirac form factoras a function of x = Q / M N . Curves: solid – u -quark; and dashed d -quarkcontribution. Data: circles – u -quark; and squares – d -quark. Lower panel .Same for Pauli form factor. Data: Refs. [67, 68, 80–84]. mation. Furthermore, as first remarked in Refs. [89, 90], scalardiquark correlations cannot be the entire explanation becausethey alone cannot produce a zero in F d .Consider the images in Fig. 6, which expose the relativestrength of scalar and pseudovector correlations in the flavourseparated form factors. The upper panel shows that whilst thescalar diquark component of the proton is the dominant deter-mining feature of F u , i.e . in connection with the doubly rep-resented valence-quark, the pseudovector component neverthe-less plays a measurable role.In the case of F d (lower panel, Fig. 6) the pseudovector cor-relation provides the leading contribution. The proton’s pseu-dovector component appears in two combinations: u { ud } and d { uu } . The latter involves a hard d -quark and is twice as proba-ble as the former (isospin Clebsch-Gordon algebra). The pres-ence of pseudovector diquarks in the proton therefore guaran-tees that valence d -quarks will always be available to partici-pate in a hard scattering event. F d possesses a zero becauseso does each of its separated contributions. (This is evident inRef. [29], discussion of Fig. 3, lower-right panel.) The locationof the predicted zero therefore depends on the strength of inter-ference with the scalar diquark part of the proton. Hence, likethe ratios of valence-quark parton distribution functions at largeBjorken- x [87, 91], the location of the zero in F d is a measureof the relative probability of finding pseudovector and scalar di-quarks in the proton: with all other things held equal, the zeromoves toward x = F d would be definitive evidence that the “preco-5 ææææææ ææ æ æ æ æ x F u æææææææ ææ æ æ æ æ = Q (cid:144) M N x F d Figure 6:
Upper panel . u -quark contribution to the proton’s Dirac form factoras a function of x = Q / M N . Curves: solid (black) – complete contribution;dot-dashed (red) – scalar-diquark contribution; dotted (blue) – pseudovectordiquark contribution. Lower panel . d -quark contribution to the proton’s Diracform factor. Curve legend same as upper panel. Data: Refs. [67, 68, 80–84]. cious scaling” of R ( x ) is accidental, existing only on a narrowdomain because of fortuitous cancellations amongst the manyscattering diagrams involved in expressing the current of a pro-ton comprised from tight quark-quark correlations.In Fig. 7 we draw analogous figures for the proton’s flavour-separated Pauli form factor. Plainly, F u is far more sensitive tointerference between scalar and pseudovector diquark correla-tions than F u . On the other hand, F d , exhibit similar patternsof interplay between scalar and pseudovector diquarks.The information contained in Figs. 5 – 7 provides clear evi-dence in support of the notion that many features in the mea-sured behaviour of nucleon electromagnetic form factors areprimarily determined by the presence of strong diquark cor-relations in the nucleon. Importantly, whilst inclusion of a“pion cloud” can potentially improve quantitative agreementwith data, it does not qualitatively a ff ect the salient features ofthe form factors [58, 92].
6. Summary . We explained how the emergent phenomenonof dynamical chiral symmetry breaking ensures that Poincar´ecovariant analyses of the three valence-quark scattering prob-lem in continuum quantum field theory yield a picture of thenucleon as a Borromean bound-state, in which binding arisesprimarily through the sum of two separate contributions. Oneinvolves aspects of the non-Abelian character of QCD thatare expressed in the strong running coupling and generatetight, dynamical colour-antitriplet quark-quark correlations inthe scalar-isoscalar and pseudovector-isotriplet channels. Thisattraction is magnified by quark exchange associated with di- æææææææ ææ æ æ æ æ Κ u - x F u æææææææ ææ æ æ æ æ = Q (cid:144) M N Κ d - x F d Figure 7:
Upper panel . u -quark contribution to the proton’s Pauli form factoras a function of x = Q / M N . Curves: solid (black) – complete contribution;dot-dashed (red) – scalar-diquark contribution; dotted (blue) – pseudovectordiquark contribution. Lower panel . d -quark contribution to the proton’s Pauliform factor. Curve legend same as upper panel. Data: Refs. [67, 68, 80–84]. quark breakup and reformation, which is required in order toensure that each valence-quark participates in all diquark corre-lations to the complete extent allowed by its quantum numbers.Combining these e ff ects, one arrives at a properly anti-symmetrised Faddeev wave function for the nucleon and ispositioned to compute a wide range of observables. Capi-talising on this, we illustrated and emphasised that numer-ous empirical consequences derive from: Poincar´e covariance,which demands the presence of dressed-quark orbital angularmomentum in the nucleon; the behaviour of the strong run-ning coupling as expressed, for instance, in the momentum-dependence of the dressed-quark mass; and the existence ofstrong electromagnetically-active scalar and pseudovector di-quark correlations within the nucleon, which ensure markeddi ff erences between properties associated with doubly- andsingly-represented valence-quarks. Planned experiments aretherefore capable of validating the proposed picture of the nu-cleon and placing tight constraints, e.g . on the rate at whichdressed-quarks shed their clothing and transform into partons,and the relative probability of finding scalar and pseudovectordiquarks within the nucleon. Acknowledgements . We are grateful for insightful commentsfrom D. Binosi, I. C. Clo¨et, R. Gothe, T.-S. H. Lee, V. Mokeev,J. Papavassiliou, S.-X. Qin, T. Sato and S.-S. Xu. J. Segovia ac-knowledges financial support from a postdoctoral IUFFyM con-tract at
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