A Euclidean bridge to the relativistic constituent quark model
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A Euclidean bridge to the relativistic constituent quark model
T. J. Hobbs ∗ and Mary Alberg , , Gerald A. Miller Department of Physics, University of Washington, Seattle, Washington 98195, USA Department of Physics, Seattle University, Seattle, Washington 98122, USA (Dated: October 16, 2018) bstract Background:
Knowledge of nucleon structure is today ever more of a precision science, withheightened theoretical and experimental activity expected in coming years. At the same time, apersistent gap lingers between theoretical approaches grounded in Euclidean methods ( e.g., latticeQCD, Dyson-Schwinger Equations [DSEs]) as opposed to traditional Minkowski field theories (suchas light-front constituent quark models).
Purpose:
Seeking to bridge these complementary worldviews, we explore the potential of a
Eu-clidean constituent quark model (ECQM). This formalism enables us to study the gluonic dressingof the quark-level axial-vector vertex, which we undertake as a test of the framework.
Method:
To access its indispensable elements with a minimum of inessential detail, we developour ECQM using the simplified quark+scalar diquark picture of the nucleon. We construct a hyper-spherical formalism involving polynomial expansions of diquark propagators to marry our ECQMwith the results of Bethe-Salpeter Equation (BSE) analyses, and constrain model parameters byfitting electromagnetic form factor data.
Results:
From this formalism, we define and compute a new quantity — the
Euclidean densityfunction (EDF) — an object that characterizes the nucleon’s various charge distributions as func-tions of the quark’s Euclidean momentum. Applying this technology and incorporating informationfrom BSE analyses, we find the dressing effect on the proton’s axial-singlet charge to be small inmagnitude and consistent with zero.
Conclusions:
The scalar quark + diquark ECQM is a step toward a realistic quark model inEuclidean space, and urges additional refinements. The small size we obtain for the impact ofthe dressed vertex on the axial-singlet charge suggests that models without this effect are on firmground to neglect it. ∗ [email protected] . INTRODUCTION Hadronic physics is presently at an important crossroads. On the one hand, with itsadvantageous representation of Minkowski field theory, light-front formalism [1–6] has madeimpressive gains in understanding the proton’s flavor and spin structure [7–10]. At much thesame time, techniques grounded in Euclidean field theory, such as Lattice QCD [11] and themethodology of Bethe-Salpeter Equations (BSEs) [12–16], continue to unfold an ever morerefined picture of the hadronic spectrum, as well as its various excitations and transitions.An effort to reconcile these two families of approaches is therefore more of a crying necessitythan ever before. The present analysis represents an initial step to bridge this enduringgap by formulating a
Euclidean constituent quark model (ECQM).To this end, we craft a simple model in Euclidean space which binds the constituentquark into the nucleon through the exchange of a scalar spectator diquark. While thequark-diquark approach itself is hardly new (such models have an established history in theanalyses of both the DIS sector [17–20] and elastic scattering [10]), our specific formulationof a Euclidean constituent quark model has not to our knowledge been previously attempted.Standard light-front theory [21, 22] extracts bound state properties ( e.g., elastic form fac-tors, inelastic structure functions) from overlaps of 3-dimensional light-front wave functions(LFWFs), which are themselves obtained by integrating a 4-dimensional Bethe-Salpeter am-plitude over the “minus” components of the internal momenta k − ≡ k − k ; these in turnprovide a means of relating the constituent quark model to form factors and GPDs [23–26].Despite the remarkable success of methods rooted in constituent quark models, an uncir-cuitous means of relating them to Euclidean approaches remains lacking, however. Thatis, although techniques for projecting, e.g., the pion’s Bethe-Salpeter amplitude onto theLF have been pioneered recently [27], a direct formulation of the quark model in Euclideanspace has not yet been put forth. The chief aim of the present article is to do precisely this,leading to the aforementioned ECQM. However, the implementation in Euclidean space re-quires techniques inspired by hyperspherical QED calculations [28–31], which we trace indetail in Sec. III below. Following angular integration of the resulting 4-dimensional ampli-tudes in Euclidean hyperspherical space, the formalism we develop outputs distributions forthe quark-level densities of the proton as functions of the intermediate quark’s Euclideanmomentum. These latter quantities we designate Euclidean density functions (EDFs), and3e carry out their evaluation in the sections below.In the present paper, we test our formalism by performing an analysis of the quarkhelicity share of the proton’s spin by evaluating the flavor-singlet axial charge as spelledout in later sections. The origin of the proton’s spin in the angular momentum of itsQCD constituents is a problem that has bedeviled hadronic physics ever since the adventof the “spin crisis” in the late 1980s following the revelation [32, 33] of the European MuonCollaboration (EMC) concerning the small size of the proton’s integrated spin-dependentstructure function, R g p ( x ) dx = 0 . ± . ± . X q ∆ q + L q + J g , (1)and the contribution from the total quark helicity P q ∆ q is now understood to representapproximately one third of the total nucleon spin, and has been the focus of intense ex-perimental and theoretical effort [40–44]. Despite recent progress [10], obtaining this resultin the context of constituent quark models, including those formulated on the light-front,remains an elusive goal. For this reason, an assessment of the rˆole played by the exchange ofnonperturbative gluons in the setting of a constituent quark model could help weigh whetherthis effect substantially alters the spin decomposition of Eq. (1). To accomplish this, we usethe aforementioned hyperspherical ECQM to incorporate information from BSE analyses onthe quark’s dressed axial-vector vertex [45–48], ultimately finding a minimal effect.The remainder of the paper is organized as follows: Sec. II treats the standard covari-ant approach, with a description of the formalism needed to fit current data in the elasticelectromagnetic sector with the bare ECQM in Sec. II A, and a prediction of the proton’saxial-singlet charge in Sec. II B; Sec. III describes the hyperspherical formalism. Herein, thebasic properties of EDFs are introduced in Sec. III A, and the simplest nontrivial calculation— the EDF for the proton’s charge distribution — is given in Sec. III B. Having thus com-pletely determined the details of the bare hyperspherical ECQM, we use it to predict the4 ~ q ~ k ~ p ~ k ~ − k ~ p ~ ( a ) p ~ q ~ k ~ p ~ k ~ − k ~ p ~ = 0 ( b ) FIG. 1. (Color online) ( a ) The standard diagram responsible for the first nontrivial contributionto F , ( e q ). ( b ) The main graph for the quark contributions to the nucleon’s axial-singlet charge a . In both cases, solid internal lines represent the propagation of the interacting quark, while thedashed lines are for the scalar spectator diquark. The ovate blobs symbolize our prescription forthe momentum dependence of the nucleon-quark-diquark interaction as given by ϕ ( e k ) in Eq. (3). axial-singlet charge of the proton in Sec. III C, as well as the distribution of this axial chargeas a function of the struck quark Euclidean momentum e k . In Sec. IV we fold the latestnumerical estimates for the soft gluon dressing effect on the axial charge of an individualquark into our formalism, and draw our final conclusions in Sec. V. Lastly, select formulaeare postponed to Appendices A and B. II. THE BARE MODEL: ELECTROMAGNETIC STRUCTURE AND SPINA. Electromagnetic form factors
In the quark + scalar diquark picture, computing the Pauli and Dirac form factors F ( e q )and F ( e q ) as functions of the spacelike photon virtuality squared e q amounts to evaluatingthe leading triangle diagram in Fig. 1( a ), which here represents an amplitude formulated inEuclidean space. For this purpose, we take the propagators of the scalar diquark (of mass m D ) and quark (of mass m ) to be, respectively, D (cid:0) [ e p − e k ] (cid:1) = 1[ e p − e k ] + m D ,S (cid:0) e k (cid:1) = 1 i e k + m , (2)where we in general denote Euclidean 4-vectors as e v µ , and the main prescription-dependentingredient of the ECQM involves making a formal choice to characterize the binding of the5truck constituent quark into the nucleon. To accomplish this, it is necessary to stipulatea relativistic vertex factor for the momentum dependence of the nucleon-quark-diquarkinteraction, represented by the “blobs” appearing in both panels of Fig. 1. The systematicsinvolved in the implementation of such phenomenological vertex factors have been exploredin diverse contexts, including in models of nucleon structure [49–51] and nuclear scattering[52]; in the end, however, we select for simplicity a minimal choice consistent with Lorentzcovariance: a scalar function of the quark’s Euclidean 4-momentum e k with the general form ϕ ( e k ) ≡ g (cid:18) Λ e k + Λ (cid:19) . (3)Of course other analytic forms for the vertex function may also be used ( e.g., multipolesinvolving higher powers, or functions of the spectator diquark 4-momentum), but theseultimately lead to qualitatively similar results, and in practice we find use of Eq. (3) simplifiescalculations dramatically. For this reason, the remainder of the present analysis is carriedout using Eq. (3). In light of our choice for the nucleon-quark-diquark vertex function,the model parameters in our framework are thus the strength of the nucleon’s couplings toits internal quark/diquark degrees of freedom g (which acts as an overall normalization),the constituent masses of the quark and scalar diquark m and m D , respectively, and theultraviolet cutoff parameter Λ, all of which we take from fits in the electromagnetic sector.Namely, the form factors F , ( e q ) are extracted from the triangle diagram shown in Fig. 1,which gives the extended electromagnetic vertex Γ µ (cid:0)e p ′ , e p (cid:1) of the nucleon as u ( e p ′ ) γ µ u ( e p ) −→ u ( e p ′ ) Γ µ (cid:0)e p ′ , e p (cid:1) u ( e p ) (4)= 1(2 π ) Z d e k u ( e p ′ ) i e k ′ + m ! γ µ (cid:18) i e k + m (cid:19) u ( e p ) ϕ ( e k ′ ) ϕ ( e k )[ e p − e k ] + m D ! , where e p (cid:0)e p ′ (cid:1) is the intial (final) proton 4-momentum, e k ′ = e k + e q , and e p ′ = e p + e q . Usingthe general form of the photon-proton vertex given by Eq. (A10) in App. A, we computethis latter amplitude using standard techniques [53, 54] involving Feynman parameters andmomentum shifts to obtain F ( e q ) = (cid:18) g Λ π (cid:19) Z dx Z − x dy Z − x − y dz Z − x − y − z dw (5) × (cid:20) (cid:21) + 2 N ( e q ) (cid:20) (cid:21) ! ,F ( e q ) = 2 (cid:18) g Λ π (cid:19) Z dx Z − x dy Z − x − y dz Z − x − y − z dw N ( z ) (cid:20) (cid:21) , (6)6n which N ( e q ) = (cid:0) m + zM (cid:1) − ( x + w ) (cid:0) − x − z − w (cid:1) e q , (7) N ( z ) = 2 M (1 − z ) (cid:0) m + zM (cid:1) , (8)∆ = ( x + w ) (cid:0) − x − z − w (cid:1) e q + ( x + y ) m + zm D − z (1 − z ) M + (1 − x − y − z ) Λ . (9)Above, M is the mass of the on-shell nucleon, and we have made use of the Euclidean GordonIdentity given by Eq. (A9) to decompose the amplitude of Eq. (4) into separate Pauli andDirac components `a la Eq. (A10).With these explicit expressions for F and F , it is simple to construct the familiar Sach’sparametrization of the nucleon’s electromagnetic form factors: G E ( e q ) ≡ F ( e q ) − e q M F ( e q ) ,G M ( e q ) ≡ F ( e q ) + F ( e q ) , (10)and we may determine the model parameters by fitting these expressions to experimentaldata on the proton. For this purpose, we treat the phenomenological parametrization ofKelly [55] as a proxy for the world’s experimental data and global fits thereof [56, 57],rather than preferencing individual sets; we may then minimize the numerical badness-of-fitmeasure χ ≡ n p n p X i =1 G E ( e q i ) − G p hen.E ( e q i ) G p hen.E ( e q i ) ! + G M ( e q i ) − G p hen.M ( e q i ) G p hen.M ( e q i ) ! . (11)We note that to ensure the numerical validity of the hyperspherical Euclidean space for-malism presented later in Sec. III, we in practice find it necessary to constrain the value ofthe diquark mass to be no less than that of the proton, m D ≥ M , while the other parametersare allowed to float freely over a broad range. This condition is a generic artifact of hyper-spherical techniques as applied to massive theories [30], and for QED can be circumventedwith an appropriate deformation of the integration contour in the complex e k plane. For theamplitudes under consideration here, however, such an approach meets further complicationsdue to the presence of quark denominators ∼ ( e k + m ) − , which can produce singularities inthe timelike region e k < e k is deformed; we therefore opt forthe simpler m D ≥ M condition in this initial study. We note of course that this procedure7 q fit to q < 1 GeV ~ ~ G E G M Kelly ( a ) q G M / ( m p G D ) G E / G D ~ ( b ) FIG. 2. (Color online) ( a ) A plot of the fitted electromagnetic form factors G E,M ( e q ), where weconstrain fits with the phenomenological parametrization of Kelly [55] for e q ≤ G E given in black and G M in red in both cases.( b ) A similar comparison, but in this case for the form factor ratios with respect to the well-knowndipole parametrization [55] G D ( e q ) ≡ (1 + e q (cid:14) Λ D ) − , where Λ D = 0 .
71 GeV . confers the added benefit of simulating the effects of a confining potential in the sense thatthe nucleon is thereby prohibited from decaying into its constituents ( m + m D ≥ M , for anychoice of m ).Also, for the sake of describing the nucleon axial-singlet charge (which is defined at e q = 0) we concentrate our fits at low photon virtualities, and hence only constrain themwith experimental information for e q ≤ G E and G M at 5 uniformly-chosen points in thedomain 0 ≤ e q ≤ i.e., n p = 5 in Eq. (11) above] results in the description plottedin Fig. 2, which corresponds to a χ per datum of 0 .
003 for the specific parameter valuesgiven in Table I. The numerical values of the fitting parameters imply a mass for the diquarkcomparable to that of the nucleon (consistent with Faddeev Equation studies, e.g.,
Ref. [58]),and a rather large constituent quark mass m ∼
600 MeV.In particular, the two panels of Fig 2 compare this fitted model to the parametrizationof Ref. [55] for the proton, both at the level of the separate form factors G E and G M themselves ( a ), as well as for the instructive ratios ( b ) with respect to the one-parameter8 m m D Λ g µ p ( µ N ) a M f M a G E and G M at low e q ≤ g and bare axial-singlet charge a determined in Sec. II are dimensionless, while the final two columns give the first moments of theelectric and axial-singlet quark charge EDFs M f and M a in GeV ; units elsewhere are in GeVunless otherwise noted. dipole approximation [55] G D ( e q ) ≡ (1 + e q (cid:14) Λ D ) − , with Λ D = 0 .
71 GeV — the latterserving to draw attention to subtleties in the form factors’ behavior at larger e q . In bothpanels also, solid curves represent the output of our fitted model, while dashed lines are theprediction of Ref. [55]. For the region of interest ( e q & G M , matching its qualitative dependence on e q quite closely; for G E , however, theagreement is somewhat weaker, as especially highlighted by the relatively steep downturn ofthe solid-black curve of Fig. 2( b ). At the same time, we adjudicate the better-than ∼ e q . . for G E and percent-level agreement for G M to be fullyadequate for our demonstration of the hyperspherical formalism here, which we pursue inthe following sections only for quantities defined in the real limit, e q = 0, including the axialcharge a . B. Axial-singlet charge
The total quark helicity contribution to the nucleon spin in Eq. (1) may be identifiedwith the matrix element for the axial-singlet charge of the proton [59], a = P q ∆ q , whichwe write explicitly as2 M e S µ a ≡ h e p, s | q γ µ γ q | e p, s i , e S µ ≡ M u ( e p ) γ µ γ u ( e p ) , (12)in which e S µ represents the nucleon’s Euclidean spin 4-vector, which obeys e S · e p = 0 and e S = −
1. For the non-pointlike proton basis states consistent with the bare quark + diquark9icture, the matrix element of Eq. (12) can be realized diagrammatically in a triangle graphakin to that which produced Eqs. (5) and (6) for the proton’s electromagnetic substructure— albeit with the appropriate ∼ γ µ γ operator entering at the axial current-quark vertex.This is shown explicitly in Fig. 1( b ), wherein e p ′ = e p , as is relevant for the axial-singletcharge defined at e q = 0. Using our established Euclidean conventions, this then gives theamplitude2 M e S µ a = 1(2 π ) Z d e k u ( e p ) (cid:18) i e k + m (cid:19) γ µ γ (cid:18) i e k + m (cid:19) u ( e p ) | ϕ ( e k ) | [ e p − e k ] + m D ! . (13)Thus we can follow a procedure similar to that used in the electromagnetic sector to computethe bare ( i.e., undressed ) quark + scalar diquark model prediction for the proton’s axial-singlet charge, keeping in mind that we will ultimately match our ECQM formalism to thestandard calculation in Sec. III C, constituting a vital test. We find2 M e S µ a = Γ(5) g Λ (2 π ) Z d e l ( e l + ∆ ) Z dx dy dz xy δ (cid:0) − [ x + y + z ] (cid:1) × u ( e p ) (cid:16) − i ( e l/ + z e p/ ) + m (cid:17) γ µ γ (cid:16) − i ( e l/ + z e p/ ) + m (cid:17) u ( e p ) ; (14)again using textbook [53] covariant methods, this can be manipulated to yield a = − (cid:18) g Λ π (cid:19) Z dy Z − y dz y (1 − y − z ) (cid:20) (cid:21) − m + zM ) (cid:20) (cid:21) ! , (15)where here the explicit expression for the denominator in terms of masses and Feynmanparameters is ∆ = (1 − y − z ) m + y Λ + z m D − z (1 − z ) M , (16)and we have implemented the shift e k µ → e l µ = e k µ − z e p µ , and made use of Eq. (A8). Thus,Eq. (15) is fully defined, and may be computed with the model parameters determined in theelectromagnetic sector — i.e., the values contained within the inner box of Table I. Insertingthese, we get a = 0 . II. HYPERSPHERICAL FORMALISMA. Euclidean density function
Here we introduce the framework necessary to obtain 4-dimensional Euclidean quark-leveldensities — for the proton’s electromagnetic charge in Sec. III B, and its axial-singlet chargein Sec. III C.Formally, we seek 4-dimensional densities dependent on the interacting quark’s Euclideanmomentum e k . Such quantities would be analogous to the squares of Bethe-Salpeter wavefunctions Ψ( k ; p ) from which LFWFs can be derived via the appropriate integral over R dk − at fixed LF time [21, 22] as described in Sec. I. Properly formulated, in our case these densityfunctions will allow the recovery of bulk properties of the nucleon from radial integrals inEuclidean space governed by the parameters of a constituent quark model. That is, thetotal nucleon charge and axial-singlet charge follow from the zeroth moment of the Euclideandensity functions (EDFs) f ( e k ) and a ( e k ), respectively: F ( e q = 0) = Z d e k f ( e k ) , (17) a = Z d e k a ( e k ) , (18)where the integrations over R d e k remain after summing over angles, and EDFs for othercharges may also be constructed. In fact, inasmuch as EDFs enjoy the proper support (inthis case, vanishing in the limit e k → ∞ ), their lowest moments in e k may also be computed: M n ¯ f ≡ Z d e k (cid:16)e k (cid:17) n f ( e k ) , (19) M n ¯ a ≡ Z d e k (cid:16)e k (cid:17) n a ( e k ) , (20)for which the choice ( n = 0) corresponds to the expressions in Eqs. (17) and (18), while thenontrivial first moments ( n = 1), corresponding to M ∼ h e k i , provide information on themean e k of the electromagnetic and axial-charge densities. We determine these explicitly inSecs. III B and III C below, and ultimately plot their associated integrands in Fig. 3.Pending this more detailed calculation, the proton’s charge EDF f ( e k ) may be describedto first approximation in the spirit of Feynman et al. [60], using a Euclideanized Gaussian11ave function ψ ( e k ) ∼ exp( − R e k (cid:14) F ( e q = 0) = 1(2 π ) Z d e k (cid:12)(cid:12) ψ ( e k ) (cid:12)(cid:12) = 1 → ψ ( e k ) = (cid:0) πR (cid:1) exp (cid:26) − R e k (cid:27) , (21)for which the dependence of the wave function on the quark momentum e k is governed purelyby the proton RMS radius, R ≡ h r p i / ≈ .
88 fm = 1 (cid:14) (0 .
227 GeV) [61]. Noting Eq. (B2),we conclude f WF1 ( e k ) = R e k exp n − R e k o , (22)a simple result to which we compare the model results of Secs. III B and III C below as aninstructive benchmark. Plotting the integrand 2 e k f WF1 ( e k ) of F (0) against e k in Fig. 3, theresulting distribution peaks predictably near e k & . R ,but then has a sharper momentum dependence at higher e k not found for the more realisticmodel calculations presented below; this fact alone highlights the necessity for the moredetailed hyperspherical treatment of nucleon spin structure outlined in Secs. III B–III C.Ultimately, in a utilitarian sense the EDFs of Eqs. (17) and (18) also permit an interfacewith the output of traditional Euclidean field-theoretic approaches, as emphasized in Sec. I.Whereas the formalism of Sec. II is adequate for the determination of the total protoncharge and helicity in the bare quark model, we ultimately wish to absorb the results ofBSE analyses into our ECQM to assess the gluon dressing effect. For this purpose, however,BSEs describe the impact of soft gluon exchange in the form of vertex functions of the quark’sEuclidean momentum, and there is no straightforward way to incorporate such quantitiesinto the bare calculation of Sec. II B, especially given the reliance of the latter upon shiftingloop momenta away from those given in Fig. 1( b ).On the other hand, given their status as vertex functions of the quark momentum, BSEresults may be incorporated directly into the integrated EDFs typified by Eq. (18) as quarkmomentum-dependent smearing functions f g ( e k ). It is precisely such a scheme that wepursue here for the quark helicity contribution to the nucleon spin, a . Thus, with theEDF a ( e k ) and the smearing function f g ( e k ) for the gluon-dressing effect in hand, one maycompute the impact of soft gluon exchange upon the total quark helicity contribution to theproton spin, leading to a corrected axial-singlet charge a ′ = Z d e k a ( e k ) f g ( e k ) , (23)12here in practice we identify the gluonic smearing function with the nonperturbative axial-vector vertex factor of BSE studies, f g ( e k ) = F R ( e k , e k ≫ f g ( e k ) = 1, in Eq. (23)simply recovers the bare ECQM calculation given by Eq. (18).We can in fact achieve the specifics of the general formalism described above, and thisamounts to the main result of the present paper. We derive the EDFs of Eqs. (17) and (18)by closely following the analogous calculation for the hadronic vacuum polarization effect inthe muon’s anomalous magnetic moment [28]; viz., we now evaluate Eq. (4) for e p ′ = e p inSec. III B and Eq. (13) in Sec. III C using a hyperspherical formalism originally adapted toQED [29–31]. B. Quark charge distribution
The hyperspherical formalism we describe below is of sufficient generality that it maybe deployed in the evaluation of various Euclidean momentum distributions. As an initialdemonstration, however, we highlight the calculation of the EDF for the proton’s electriccharge: i.e., the integrand leading to F ( e q = 0) of Eq. (17). As will be the case for thesubsequent determination of a ( e k ), we start at amplitude-level, in this case with Eq. (4),which at e q = 0 yields2 i e p µ F (0) = 1(2 π ) Z d ˜ k u ( e p ) (cid:18) i e k + m (cid:19) γ µ (cid:18) i e k + m (cid:19) u ( e p ) (cid:12)(cid:12) ϕ ( e k ) (cid:12)(cid:12) [ e p − e k ] + m D ! (24)= g Λ (2 π ) Z d e k u ( e p ) h − e k e k µ + (cid:16)e k + m (cid:17) γ µ − im { γ µ , e k } i u ( e p )( e k + m ) ( e k + Λ ) (cid:16) [ e p − e k ] + m D (cid:17) , (25)where we have again used Eq. (A10) for the general form of the electromagnetic vertex givenin App. A. To apply the hyperspherical formalism, we must express the numerator algebraleading to F (0) in terms of inner products. For this example, we achieve this by contractingboth sides of Eq. (25) with e p µ and using the identities of App. A, which brings us to theexpression F (0) = g Λ (2 π ) Z d e k e k + m − e p (cid:0)e p · e k (cid:1) + e p mM (cid:0)e p · e k (cid:1) ( e k + m ) ( e k + Λ ) (cid:16) [ e p − e k ] + m D (cid:17) . (26)13ore critically, rather than shifting away the term in the denominator ∼ ( e p · e k ) as in thestandard covariant calculations involving Feynman parameters [Eqs. (5) – (6) and (15)], weinstead make an expansion of the scalar diquark propagator:1[ e p − e k ] + m D = Z pk e p e k ∞ X n =0 (cid:16) Z pk (cid:17) n C n (cid:0) ˆ p · ˆ k (cid:1) , (27)where explicitly, Z pk ≡ e p e k (cid:16)e p + e k + m D − q ( e p + e k + m D ) − e p e k (cid:17) , (28)and we sometimes find it convenient to work in terms of the dimensionful object Z ≡ Z pk (cid:14)e p e k .In Eq. (27), the C n are Gegenbauer polynomials with the normalization and orthogonalityproperties described in App. B, and ˆ p is a unit vector in Euclidean space in the direction of e p µ .We can exploit these properties in App. B to perform the necessary angular integrations byfirst rendering the numerator of Eq. (26) in terms of a linear combination of the Gegenbauerpolynomials ( e p · e k ) = e p e k C (ˆ p · ˆ k ) , (29)( e p · e k ) = 14 e p e k (cid:16) C (ˆ p · ˆ k ) + C (ˆ p · ˆ k ) (cid:17) . (30)Inserting everything into Eq. (26) and using Eq. (B2) then results in F (0) = g Λ (2 π ) Z d e k e k Z ( e k + m ) ( e k + Λ ) Z d Ω ˆ k (cid:16) ∞ X n =0 (cid:0)e p e k Z (cid:1) n C n (ˆ p · ˆ k ) (cid:17) × − e k (cid:0) C (ˆ p · ˆ k ) + C (ˆ p · ˆ k ) (cid:1) + mM e p e p e k C (ˆ p · ˆ k ) + ( m + e k ) C (ˆ p · ˆ k ) ! ; (31)and we may use Eq. (B3) to evaluate the angular integral R Ω ˆ k . Before doing so, however, itis imperative to note that Eq. (31) is defined in general for spacelike 4-momenta (includingthe external nucleon 4-momentum e p ≥ e p = − M . By merit of our requirement that m D ≥ M , the integrationcontour e k ∈ [0 , ∞ ) remains unmenaced by branch points or singularities, and the nucleonmomentum may be straightforwardly continued to e p → iM . Doing so after evaluating theangular integrals, we finally obtain F (0) = (cid:18) g Λ π (cid:19) Z d e k e k Z ( e k + m ) ( e k + Λ ) e k M (cid:0)e k Z (cid:1) + mM e k Z + m ! , (32)14n which Z represents the analytic continuation of the rational function Z of Eq. (28), givenexplicitly by Z = − M e k (cid:18)e k + δ − q ( e k + δ ) + 4 M e k (cid:19) , (33)having defined the shorthand δ ≡ m D − M .It is notable also that the expression given in Eq. (32) constitutes an important checkof the hyperspherical formalism which we use in Sec. III C below for a , and one maystraightforwardly verify that it yields F (0) = 1 for the parameters of Table I. From it, wemay at last extract the Euclidean density function f ( e k ) for the proton’s quark-level chargethrough direct matching with Eq. (17), f ( e k ) = (cid:18) g Λ π (cid:19) e k Z ( e k + m ) ( e k + Λ ) e k M (cid:0)e k Z (cid:1) + mM e k Z + m ! ; (34)we plot this EDF in Fig. 3 alongside the analoguous quantity for the axial-singlet charge a ( e k ) derived in Sec. III C below.Having determined the quark-level EDF for the proton’s electric charge in Eq. (34), wemay use this result to evaluate higher moments of the charge distribution given in Eq. (19): M f = 0 . . (35)In this case, this value corresponds roughly to the center of the peak of the heavy-solidline in Fig. 3; more directly, we also plot the integrand over e k for the moment M f as thethin-solid line, multiplied by a factor of 2 for ease of comparison. C. Quark helicity
While the formalism in Sec. II B above was sufficient to determine the bare quark helicitycontribution to the proton axial-singlet charge a , we must ultimately interface our quark-diquark framework with the results of BSE analyses to estimate the gluon dressing effect asmentioned above. In this case, the BSE calculations we aim to incorporate are e k -dependentvertex factors as noted in Sec. III A, and thus we require an axial charge momentum distri-bution along the lines of Eq. (34) to evaluate Eq. (23).Hence, analogously to the calculation in Sec. III B, we now proceed by contracting both15 k, GeV ~ ~ ~
2k f (k )2k a (k )2k f (k ) ~ ~~ ~ FIG. 3. (Color online) A comparison of EDFs for the proton’s charge 2 e k f ( e k ) [Eq. (34), black-solid] and axial-singlet charge 2 e k a ( e k ) [Eq. (41), maroon-dashed] carried by the struck quark inthe scalar diquark ECQM as functions of its Euclidean momentum e k ; for illustration, we contrastthese with the result of using the Gaussian wave function, 2 e k f WF1 ( e k ) from Eq. (22) [red-dotted].The thin lines and associated shaded regions at bottom correspond to the integrands of thesedistributions’ first moments in e k , i.e., M ∼ h e k i of Eqs. (19) and (20). Note that these lattermoments have been rescaled by a factor of 2 for comparison. sides of Eq. (13) with the nucleon spin 4-vector e S µ to obtain2 M a = − g Λ (2 π ) Z d e k e S µ u ( e p ) (cid:16) − i e k + m (cid:17) γ µ γ (cid:16) − i e k + m (cid:17) u ( e p )( e k + m ) ( e k + Λ ) (cid:16) [ e p − e k ] + m D (cid:17) = − g Λ (2 π ) Z d e k M (cid:0) e S · e k ) + ( m − e k ) e S (cid:1) − m ( e p · e k )( e k + m ) ( e k + Λ ) (cid:16) [ e p − e k ] + m D (cid:17) , (36)and here we require an additional inner product:( e S · e k ) = 14 e S e k (cid:16) C ( ˆ S · ˆ k ) + C ( ˆ S · ˆ k ) (cid:17) . (37)Using this and Eq. (29) to re-write the inner products of Eq. (36) above, we incorporate thepolynomial expansion for (cid:0) [ e p − e k ] + m D (cid:1) − ; here this leads to16 = g Λ (2 π ) Z d e k e k Z ( e k + m ) ( e k + Λ ) Z d Ω ˆ k (cid:16) ∞ X n =0 (cid:0)e p e k Z (cid:1) n C n (ˆ p · ˆ k ) (cid:17) (38) × (cid:16) e k (cid:0) C ( ˆ S · ˆ k ) + C ( ˆ S · ˆ k ) (cid:1) − mM e p e k C (ˆ p · ˆ k ) + ( m − e k ) C ( ˆ S · ˆ k ) (cid:17) = (cid:18) g Λ π (cid:19) Z d e k e k Z ( e k + m ) ( e k + Λ ) × (cid:16) e k (cid:0)e p e k Z (cid:1) C ( ˆ S · ˆ p )3 − mM e p e k (cid:0)e p e k Z (cid:1) C (ˆ p · ˆ p )2 + ( m − e k C ( ˆ S · ˆ p ) (cid:17) . (39)As before, we analytically extend e p into the timelike region where it is on-shell, leading to a = (cid:18) g Λ π (cid:19) Z d e k e k Z ( e k + m ) ( e k + Λ ) − e k M (cid:0)e k Z (cid:1) + mM e k Z + m ! , (40)and Z is again given by the expression in Eq. (33). Lastly, we deduce the EDF appearingin Eq. (18) [and Eq. (23)] from Eq. (40) by simple matching, as had been done for ¯ f ( e k ): a ( e k ) = (cid:18) g Λ π (cid:19) e k Z ( e k + m ) ( e k + Λ ) − e k M (cid:0)e k Z (cid:1) + mM e k Z + m ! ; (41)in summary, we emphasize that to obtain Eqs. (36)–(41) we have contracted both sides ofthe first equation with e S µ and expanded the diquark propagator `a la Eq. (27).With these expressions, one may proceed to compute the bare quark contribution to theproton spin using the set of parameters determined from fits to the proton electromagneticform factors, given in Table I. Using these values in the conventional formalism of Sec. II Bthat led to Eq. (15), we found a = 0 .
784 — a value which may also be recovered from thehyperspherical formalism as given by Eq. (40). Incidentally, this figure is in accord with themoment of the scalar diquark contribution to the quark helicity PDF obtained in a typicallight-front quark model (see Eqs. (61) and (62) of Ref. [10]):∆ q s = 13 (2 ∆ u − ∆ d ) ≈ .
75 ; (42)this latter expression assumed an SU ( ) ⊗ SU ( ) structure for the proton’s spin-flavor wavefunction.We point out as well that the axial-singlet EDF a ( e k ) given by Eq. (41) is not restrictedto be positive-definite, unlike the analogous electromagnetic charge EDF f ( e k ) of Eq. (34),17 ~ − k ~ L ~ k ~ L ~ ( a ) k ~ k ~ P ~ = 0 L ~ ( b ) FIG. 4. (Color online) ( a ) The diagram leading to the DSE for a quark of momentum e k dressedby a nonperturbative gluon carrying loop momentum e L . ( b ) The corresponding diagram for thequark axial-vector vertex BSE, responsible for the e k -dependent gluonic dressing correction to theaxial charge of an individual quark. which is related to the zeroth moments of traditional probabilistic quark density functions.In fact, for certain parameter combinations, a ( e k ) may experience substantial negativedownturns at larger spacelike quark momenta, e k ≥ a ( e k ) is instead dominated by a soft peak centered roughly at e k . . e k dependence appearing in Eqs. (34)and (41), the shapes of these distributions closely track each other, with f ( e k ) ≈ a ( e k ),particularly for e k ≪ m . Ultimately, we interpret this behavior as following from thecommon origin of both expressions in the diagrams of Fig. 1, which at e q = 0 differ only bythe appearance of γ .Moreover, for the higher ∼ h e k i moment of the axial-singlet EDF, we obtain the value M a = 0 . , (43)implying the proton’s distribution of axial-singlet charge is relatively softer than the chargedistribution [Eq. (35)] in the bare model. IV. GLUON DRESSING EFFECT
We now incorporate numerical estimates of the effect of dressing the quark-axial currentvertex with gluon exchange, which in principle may be determined from DSE-BSE analyses.Here, the relevant diagrams are displayed in Fig. 4, wherein panel ( a ) illustrates the dressed18ropagator responsible for QCD’s quark DSE, while panel ( b ) demonstrates the realizationof the BSE for the quark-level coupling of the axial-vector current dressed by soft gluonexchange. Naturally, the infrared momenta at which this effect is of interest demands the useof nonperturbative methods, and the standard procedure requires a prescription-dependenttruncation of the quark-gluon vertex (shown as the blobs in Fig. 4).In the context of BSE analyses [45–48], the dressed axial-vector vertex is representedby the structure Γ fg µ ( e K ; e P ), which is understood to connect an incoming quark of flavor g and momentum e K − = e K − (1 − η ) e P to an outgoing quark of flavor f and momentum e K + = e K + η e P ; here e P and e K represent the total and relative momentum of the quarkpair, and η is a dimensionless parameter upon which calculations cannot depend. Thus, forour purposes, we require the case e P = 0, such that e K + = e K − = e K ≡ e k , and we take thediagonal isospin-independent vertex f = g , as described in Ref. [48]. Then the structure ofthe quark-axial vector vertex of relevance here is simply u ( e k ) Γ µ ( e k ; 0) u ( e k ) = u ( e k ) γ h γ µ F R ( e k ; 0) + . . . i u ( e k ) , (44)and the ellipsis in Eq. (44) above represents additional contributions to the vertex that donot contribute in the present analysis. We therefore make the identification f g ( e k ) ≡ F R ( e k ; 0)mentioned in Sec. III A, and directly insert the numerical results reported in Ref. [48] tosmear the bare model axial charge as in Eq. (23).The behavior of f g ( e k ) depends crucially on the truncation scheme used to obtain the effec-tive quark-gluon vertices in the panels of Fig. 4. To get a sense for this source of prescriptiondependence, we compute the correction following from both schemes treated in Ref. [48] —the rainbow-ladder scheme (RL), and an ansatz based on a specific realization of dynamicalsymmetry breaking (DB), which we take numerically from Fig. 1 of Ref. [48]. Referringto these as f RL g ( e k ) (blue-dotted) and f DB g ( e k ) (red-dashed), we plot both dressing functionsagainst e k in Fig. 5( a ). Plainly, both truncation shemes predict a suppression of the quark’saxial charge for the lowest infrared momenta e k . . ∼ e k ∼ a ′ , 2 e k f RL g ( e k ) a ( e k ) (blue-dotted) and 2 e k f DB g ( e k ) a ( e k ) (red-dashed), in Fig. 5( b ) alongside the bare or “undressed” scenario, f g ( e k ) = 1 (black-solid).19 k, GeV f g (k ) ~ bare ( a ) DB RL ~ k, GeV ~ ( b ) bare DB RL FIG. 5. (Color online) ( a ) The gluon dressing function f g ( e k ) under several different scenarios:the perturbative limit f g ( e k ) = 1 (solid black), and f DB g ( e k ) (red-dashed) and f RL g ( e k ) (blue-dotted). ( b ) A plot of the integrand of Eq. (23) 2 e k a ( e k ) f g ( e k ) for several choices of the gluondressing function: f g ( e k ) = 1 (“ bare ,” shown in solid black), as well as the result of an improveddynamical chrial symmetry-breaking kernel in the BSE f DB g ( e k ) (“DB,” red-dashed line), and therainbow-ladder truncation method f RL g ( e k ) (“RL,” blue-dotted curve) of Ref. [45, 48]. From this, we find the net correction to the quark helicity contribution from gluon dressingto be (cid:18) a ′ a (cid:19) − − .
04% (DB scheme) , (45)= +2 .
98% (RL scheme) . (46)The magnitude of the effect from gluon dressing is therefore quite small, and in the presentanalysis, actually consistent with zero in the sense that depending upon the choice of trun-cation scheme, one may obtain a modest enhancement (RL) or tiny suppression (DB) ofthe proton’s total quark helicity. The smallness of the effect can be understood from themomentum dependence shown in Fig. 5( b ), in which the interplay of the shapes of f g ( e k )and a ( e k ) are such that the axial-singlet charge is slightly suppressed at low e k and enhancedat higher e k . These two effects largely cancel, however, in the integral over e k involved in thecomputation of a ′ according to Eq. (23), such that a ′ ≈ a , and we conclude the dressingeffect in a to be minimal. 20 . CONCLUSION In this paper we have proposed a model in Euclidean space formulated in terms of con-stituent quark degrees of freedom. The essential products of the resulting ECQM technologyare density functions of the quark’s Euclidean momentum (the EDFs) obtained from hy-perspherical angular integrations of 4-dimensional amplitudes. The special value of thesederived quantities is their ability to recover nucleon charges through integrals over the in-ternal momenta of their constituent quarks, a fact that empowered us to couple them topredictions of other Euclidean analyses — in this case, BSEs.Thus, having introduced this formalism, we tested it preliminarily by computing boththe nucleon’s quark charge density, as well as its axial-singlet charge. For the latter, thistest assumed the form of an assessment of the impact of BSE calculations for the dressedquark axial-vector vertex. There are of course various sources of model dependence on theside of both our ECQM for the nucleon-quark interaction and of the BSE analyses. Despitethese sources of model-dependence, we find the effect of the gluon dressing to be small —at most a several percent correction to the total quark helicity in the bare ECQM.Naturally, the analysis presented here is essentially exploratory, and if anything, suggeststhe need for further refinements. For instance, the scalar diquark picture alone cannotrealistically approximate the nucleon’s full spin structure as evidenced by the large valuewe obtain for the bare axial-singlet charge ( a = 0 . shape obtained for a ( e k ) shown in Fig. 5 for the present scalar diquark ECQMwould hold also for amplitudes involving spin-1 exchanges, so that the essential details ofsuch a calculation would resemble our presentation here. That being the case, our ultimateconclusion is unlikely to change: models in which bare constituent quarks carry the greatpredominance of the total quark helicity are on robust footing.Similarly, it should be noted that other possible considerations have not been treated sys-tematically, including the momentum dependence of the constituent quark’s dynamical mass,the implementation of which would require a self-consistent scheme not typical of the fittedconstituent quark model presented here. Such issues, as well as continued improvementsto the Euclidean hyperspherical formalism and BSEs for the axial-vertex dressing functions21ill be of enormous value in extending the current state-of-the-art regarding quark helicity,the nucleon spin problem, and Euclidean modeling of nucleon structure. VI. ACKNOWLEDGEMENTS
We thank Ian Clo¨et, Javier Men´endez, Brian Tiburzi, Andre Walker-Loud, and XilinZhang for helpful exchanges. The work of TJH and GAM was supported by the U.S. Depart-ment of Energy Office of Science, Office of Basic Energy Sciences program under Award Num-ber DE-FG02-97ER-41014. The work of MA was supported under NSF Grant No. 1516105.
Appendix A: Euclidean space conventions
We proceed using the Minkowski ↔ Euclidean transcription dictionary as outlined in, e.g.,
Refs. [15, 16], wherein 4-momenta and Dirac matrices transform according to k = ik , k j = − k j ,γ = γ , γ j = iγ j ; j ∈ { , , } . (A1)The Dirac algebra in this setting is then specified by n γ µ , γ ν o = 2 δ µν , (A2)such that the Euclidean inner product for any two 4-vectors e a µ , e b µ is e a · e b ≡ X µ e a µ e b µ = e a e b + · · · + e a e b , (A3)and, by extension, e p/ ≡ γ e p + · · · + γ e p . (A4)We also note the definition γ = − γ γ γ γ . (A5)We may give explicit expressions for the Euclidean Dirac spinors, which we obtain followingthe conventional Wick rotation as u λ ( p ) = p M + p χ λσ · p M + p χ λ → u λ ( e p ) = p M + i e p χ λ − σ · e p M + i e p χ λ , (A6)22here the helicity states χ [ λ = ↑↓ ] = , are proportional to the standard eigen-vectors of σ . These spinors are endowed with the typical normalization, u u = 2 M, u ( e p ) γ µ u ( e p ) = 2 i e p µ , (A7)and obey the Dirac Equation u ( e p ′ )( i e p/ ′ + M ) = ( i e p/ + M ) u ( e p ) = 0 . (A8)Moreover, in Euclidean space, the Gordon Identity assumes the slightly altered form u ( e p ′ ) γ µ u ( e p ) = 12 M u ( e p ′ ) n − i e P µ + σ µν e q ν o u ( e p ) , (A9)where we have defined e P µ ≡ e p ′ µ + e p µ and σ µν ≡ ( i/ γ µ , γ ν ]. By similar logic, we obtain thegeneral form for the extended electromagnetic vertex of the proton, u ( e p ′ ) Γ µ (cid:0)e p ′ , e p (cid:1) u ( e p ) = u ( e p ′ ) (cid:26) F ( e q ) γ µ + F ( e q ) σ µν e q ν M (cid:27) u ( e p ) . (A10) Appendix B: Hyperspherical formalism
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