Gluon PDF from Quark dressing in the Nucleon and Pion

Gluon dressing of the light quarks within hadrons is very strong and extremely important in that it dynamically generates most of the observable mass through the breaking of chiral symmetry. The quark and gluon parton densities, $q(x)$ and $g(x)$, are necessarily interrelated since any gluon emission and absorption process, especially dressing of a quark, contributes to $g(x)$ and modifies $q(x)$. Guided by long-established results for the parton-in-parton distributions from a strict 1-loop perturbative analysis of a quark target, we extend the non-perturbative QCD approach based on the Rainbow-Ladder truncation of the Dyson-Schwinger equations to describe the interrelated valence $q_{\rm v}(x)$ and the dressing-gluon $g(x)$ for a hadron at its intrinsic model scale. We employ the pion description from previous DSE work that accounted for the gluon-in-quark effect and introduce a simple model of the nucleon for exploratory purposes. We find typically \mbox{$\langle x \rangle_g \sim 0.20$} for both pion and nucleon at the model scale, and the valence quark helicity contributes 52\% of nucleon spin. We deduce both $q_{\rm v}(x)$ and $g(x)$ from 30 calculated Mellin moments, and after adopting existing data analysis results for $q_{\rm sea}(x)$, we find that NLO scale evolution produces $g(x)$ in good agreement with existing data analysis results for the pion at 1.3 GeV and the nucleon at 5 GeV$^2$. At the scale 2 GeV typical of lattice-QCD calculations, we obtain \mbox{$\langle x \rangle_g^{\rm N} = 0.42$} in good agreement with 0.38 from the average of recent lattice-QCD calculations.

Introduction: Recent progress in understanding the structure of hadrons is increasingly focussed on the separate roles of quarks and gluons. A consensus is starting to emerge [1,2] from experimental and theoretical work on integral properties such as the quark and gluon parton contributions to the nucleon spin, angular momentum, and lightcone momentum [3][4][5]. A er a long time restricted to low moments of parton distribution functions (PDFs), the la ice-regulated approach to QCD calculations has in recent years developed methods to obtain the momentum fraction x-dependence of PDFs, see e.g., Refs. [6,7].
Gluons have two very strong dynamical roles in hadron physics. Besides their role in binding quarks, perhaps the next most prominent role is the generation of around 95% of the mass of most light quark hadrons through the mechanism of dynamical breaking of chiral symmetry (DCSB). It is possible that one of these two roles dominates the gluon parton structure of such hadrons. La ice-QCD calculations of the gluon fraction of the nucleon spin and lightcone momentum are typically 35-50% at scale 2 GeV [4,5,8]. Any gluon emission and absorption process, including dressing of a quark, contributes to the gluon parton density g(x) and modi es the quark part q(x) [9]. e concept of a PDF of a parton in a parton is o en used in considerations of radiative processes that describe changes with resolving scale [10,11], pQCD issues within factorization [12,13], and explorations of the relation between 1-loop quark dressing and the consequent gluon-in-quark and quark-in-quark PDFs [9,14]. e la er work illustrates that a gluon dressing mechanism produces both x q < 1 and x g > 0 at any scale. It has recently been found that an assumption of zero gluon and sea distributions at low scales typical of models is incompatible with scale evolution and current data [15]. e application of the canonical QCD de nition [16] of g(x) (Eq. (2) below) to ob-tain the gluon-in-quark g(x) at the intrinsic scale of a nonperturbative hadronic model containing DCSB has not been made before.
We investigate this within the Dyson-Schwinger equation approach (DSE) to hadron physics which employs the ladderrainbow truncation of diagrams; an in nite subset of gluon emission and absorption processes are thereby included. is DSE-RL approach has proven to be very e cient for ground state masses, decay constants, and electromagnetic form factors [17][18][19][20]. It has been especially accurate for light quark pseudoscalar and vector mesons [21,22] because their properties are strongly dictated by the dynamical breaking of chiral symmetry and vector current conservation, which is built into the approach. It has been applied to pion, kaon and nucleon PDFs [23][24][25][26][27][28][29] mostly using the Ward Identity Ansatz to represent the relevant quark vertex for any PDF moment. As discussed later, this approximation is accurate only for the lowest (quark number) moment, and does not distinguish the gluon-in-quark and quark-in-quark PDFs. One DSE approach that does make this distinction has been applied to q(x) for the pion [30].
ark and Gluon PDFs in the Nucleon: In the Bjorken kinematical limit, and at leading-twist, the unpolarized quark and gluon parton densities are de ned by the explicitly Poincaré-invariant matrix elements [16,[31][32][33] and Here n µ is the light-like longitudinal basis vector (given by (1, 0 T , −1) in the target rest frame), W (λ, A) is the Wilson line integral that restores gauge invariance to the non-local current, and G + µ (λn) = n ν G νµ (λn). At the same level, the helicity gluon PDF is given by where the dual tensor isG µν (z) = 1 2 µναβ G αβ (z). e quark helicity PDF is given by Eq. (1) with the replacement / n → / n γ 5 . As pointed out some time ago [16] parton momentum conservation is a consequence of the above formal de nitions, and is gauge-invariant, and scale-invariant. e sum of the unpolarized PDF momentum fractions from the above expressions gives where D + = n ν D ν is the lightcone projected covariant derivative, and q includes antiquarks and all relevant avors. e operator density on the RHS of Eq. (4) is proportional to the light-cone projection of the energy-momentum tensor density T ++ (x) = n µ T µν (x) n ν , where T µν (x) has the Belinfante improvement and is symmetric and gauge invariant [34]. e sum rule S x = P |T ++ (0)| P c /2(P + ) 2 = 1 then follows a er covariant normalization P |P = 2E δ 3 (P − P ).  14) for x m g using BSE-generated vertices as appropriate to each case. Bo om panel: An illustration of the BSE integral equation Eq. (13) for the vertex that carries the moment information of the PDF of the gluon-in-quark, as speci ed by the particular inhomogeneous term given in Eq. (15). e la er is the dressed generalization of the 1-loop counterpart in the bo om of Fig. 1.
ark target at 1-loop: We outline relevant partonic aspects of a single quark "hadronic" target, treated at 1-loop some time ago [9]. e previous de nitions employed Euclidean metric; from here on we adopt Euclidean metric as a prelude to the realistic numerical treatment of non-perturbative aspects using Rainbow-Ladder truncation which is de ned and well established in Euclidean metric. 1 From Eq. (1), the 1-loop result for x q , as displayed in Fig. 1, is given by where p is the quark target momentum, x = k · n/p · n, and for simplicity we have employed light-cone (LC) gauge to eliminate the Wilson line. Here with V = S(k) (−i n) S(k) and gluon momentum q = p − k. e spinor product is for a fully polarized quark and the normalization is such thatū 1 u 1 = 1 =ū 1 (−i n) u 1 . e notation e gauge-invariant quark helicity ∆ q is obtained from Eq. (6) by replacement of (−i n) by (−i n γ 5 ) throughout, and se ing k · n/p · n → 1.
From Eq. (2), the Wilson line makes no contribution to x g . At 1-loop, the gluon eld can only arise from dressing of the quark and, as displayed in Fig. 1, the gauge-invariant 1-loop result is where again x = k · n/p · n. Herê and q = p − k is the loop quark momentum. Integration by parts applied to the above con rms that x g where the rst term involves the Ward Identity vertex in terms of the 1-loop propagator S(p). Due to the conserved vector current, the rst (Ward Identity vertex) term is the unit quark number (thus xing Z 2 ) and the momentum sum is veri ed. e gauge-invariant 1-loop gluon helicity is given by Typical 1-loop results are where g 2 has been chosen to yield a magnitude for x (1) g typical of the non-perturbative results for the pion and nucleon to be discussed below. e integrals are regularized via the proper time method with Λ IR = 0.05 GeV and Λ UV = 20 GeV. e above quark target expressions are the 1-loop limit of the m = 1 PDF moments given by x m = u 1 Γ (m) (p) u 1 in terms of the appropriate dressed quark vertex that carrying information on the quark-in-quark or the gluon-in-quark PDF. is example illustrates that any gluon radiation, absorption or spli ing dynamics, especially dressing of a quark, generates linked contributions to both q(x) and g(x).
Hadron PDFs: To generalize the above 1-loop quark vertex structures and apply them to valence quarks as o ered by a hadron, we employ the Rainbow-Ladder truncation of the DSE approach that has successfully described many hadron properties [17][18][19][20]. Without the Wilson line contribution, Eq. (1) applied to the nucleon in this DSE-RL approach produces where the trace is over Dirac and color indices, and 4 with Λ indicating the ultraviolet regularization mass scale. e quark vertex Γ q (p, x) is generated from the inhomogeneous term Z 2 (−i n) δ(x − p · n/P · n) via the Bethe-Salpeter integral equation [23,30].
e nucleon amplitude M f (p, P ) has the Dirac spinor structure of aqnucleon sca ering amplitude, and describes the probability amplitude for the target to present a dressed quark of avor f and momentum p that in turn yields a quark-in-quark with momentum fraction x to the hard DIS probe. e moments of the corresponding unpolarized quark PDFs are then where Γ (m) Ref. [30]. Our main emphasis is the nucleon and the present exploratory model for M f (p, P ) is explained later. e 1-loop version of the vertex Γ (m) q (p) in Eq. (12) has been used in the quark target example discussed earlier.
is Bethe-Salpeter equation for the vertex is where the DSE-RL kernel K µν (q) is given in Eq. (18) below, and the inhomogeneous term for the unpolarized quark PDFs is Γ (m) e moments of the unpolarized dressing gluon PDFs are where the 1-loop version of vertex Γ whereX g (p, k) is given by Eq. (8) except here the quark propagator is dressed. e corresponding helicity gluon PDF moments use the quark vertex Γ (m) ∆g (p) which is generated from the BSE with the inhomogeneous term with q = p − k. e combination g 2 /k 4 of the 1-loop formulas has been generalized to K g (k 2 ) to account for the nonperturbative dressing. Details are given below and in the Appendix. A er the BSE is solved for Γ (m) ∆g (p) it is found that the inhomogeneous term Eq. (16) is an excellent numerical approximation to the solution at the level of 10 −3 .
Interaction Kernels: In all cases, the dressed quark propagator S(p) is obtained as the solution of QCD's quark Dyson-Schwinger equation in Rainbow-Ladder truncation, which is where e standard DSE-RL interaction kernel [21,23,36] that generates quark propagators and BSE vertices and meson bound states is Hereα s (q 2 ) denotes a continuation of the 1-loop α s (q 2 ) to provide smooth non-singular coverage for the entire domain of q 2 . e rst term of Eq. (18) implements the infrared enhancement due to dressing e ects, while the second term, with F(q 2 ) = (1 − exp(−q 2 /(1 GeV 2 )))/q 2 , connects smoothly with the 1-loop renormalization group behavior of QCD. e DSE-RL kernel correlates a large amount of hadron physics [17][18][19][20].
For the vertex that generates the gluon PDF, the kernel K g (k 2 ) can be identi ed by a generalization of the procedure that de nes the standard BSE-RL kernel K RL (k 2 ) for the Bethe-Salpeter meson bound state equation which is linked by global symmetries to the dressed quark Dyson-Schwinger equation. For example see Ref. [37]. Here we are using the properties of multiplicative renormalizability to use the large renormalization scale dependence of propagators and vertex functions to produce their ultraviolet momentum dependence. e BSE-RL kernel for the interaction of 2 quark currents collects the ultraviolet 1-loop momentum dependence from , where γ m = 4/β 0 = 12/(33 − 2 N f ). In the present case there is an extra dressed gluon propagator, and hence the deep ultraviolet behavior is characterized by an extra factor Z 3 (q 2 , Λ 2 )/q 2 . We take the interaction kernel K g (q 2 ) to be the related form whereZ 3 (q 2 ) denotes a continuation of the corresponding 1-loop q 2 dependence a er the regularization mass scale Λ has been absorbed into the de nition of scale Λ QCD . Details and parameters are given in the Appendix.
Here  We employ the more realistic nucleon description in which the amplitude introduced in Eq. (11) has the form M f (p, P ) = D f (p c ) A(p, P ) where A(p, P ) is a Dirac scalar amplitude. is adopted form implements a number of realistic features including momentum dependence for quark number and polarization densities. We incorporate properties of the 3-quark description of a spin up proton that come from a generalization of the SU (6) spin and isospin state [38,39].
at is where χ c is the (antisymmetric) color singlet state, χ s (1; 2, 3) is a 3-quark Pauli spin state with a pair coupled to spin s and that coupled to the third quark to make S z = 1/2, and the φ i are the corresponding isospin states. e F α are symmetric spatial states F α = f α (p 1 ) f α (p 2 ) f α (p 3 ), with α = 0, 1 and A = 1 − E 12 − E 13 imposes overall antisymmetry. e standard SU(6) state corresponds to f 0 = f 1 in which case |Ψ is automatically antisymmetric without the need for operator A.
A er normalization of M f (p, P ) to quark numbers n f , the resulting quark polarization densities P f (p) are such that in the SU(6) limit the standard result (P u , P d ) = (4/3, −1/3) is recovered. e expression for A(p, P ) in terms of the f α (p) is given in the Appendix.
Results for Light-cone Momenta and Helicities: Models with parameters set rst by reproduction of scale-independent observables such as hadron masses and decays, do not have a naturally identi ed resolving scale µ 0 associated with the intrinsic PDFs. at scale can be determined by what is required to t, by DGLAP evolution upward, one or more empirical PDF x from global data analysis. e in nite subset of diagrams in a Rainbow-Ladder truncation has limited ability to accommodate parton spli ing and recombination processes that increase with resolving scale. e resulting DSE-RL µ 0 will be greater than Λ QCD and should be less than the QCD factorization scale used to factor cross sections into a perturbative sca ering mechanism and PDFs containing all nonperturbative physics at lower scales. We use the pion approach from Ref. [30] to set the infrared strength of the gluon-in-quark (hadron independent) kernel K g (q 2 ) of Eq. (19) so that under NLO DGLAP scale evolution [41] x g reproduces the JAM global analysis [35] at µ = 1.3 GeV. e pion model scale µ 0 = 0.78 GeV was previously determined by q v (x) under the non-singlet version of this evolution [30]. In the present singlet evolution case we require both x g and x sea be as close as possible to the JAM values. In some models [29], it has o en been assumed that suitable conditions at µ 0 are g(x) = q sea (x) = 0 and x qv = 1. As emphasized by Refs. [15,42] the choice of realistic DGLAP starting conditions at µ 0 (especially a non-zero gluon PDF) adds signi cant bene t to the quality of PDFs at µ > µ 0 . Here the reduced x π qv (µ 0 ) due to the established quark-in-quark e ect prevents adoption of the minimal boundary condition x π sea (µ 0 ) = 0; it would require taking x π g (µ 0 ) = 0.354 which is untenable because it is already greater than the JAM value at µ = 1.3 GeV and will only increase on evolution. e present model scale momentum fractions that minimize the RMS deviation from JAM values at µ = 1.3 GeV are displayed in Table I and the associated strength parameter of K g (q 2 ) is shown in Table IV. e present 2 x π uv (µ 0 ) is identical to earlier recent work [30] that also recognized the momentum carried by the gluon-in-quark e ect.
With K g (q 2 ) thus set, the employed nucleon amplitude then allows calculation of x N qv (µ 0 ) and x N g (µ 0 ), while x N sea (µ 0 ) is obtained from the sum rule. NLO evolution up to µ = √ 5 GeV and comparison with the NNPDF3.0 analysis [40] then identi es the nucleon model scale µ 0 = 0.56 GeV, and results are shown in Table III. e present x N uv+dv (µ 0 ) value is necessarily quite smaller than previous work [27] which ignored the gluon-in-quark e ect by using the convenience of the Ward Identity vertex Ansatz. is vertex Ansatz is correct only for the lowest Mellin moment (quark number) of quark PDFs; the 1-loop analysis for a quark target discussed earlier provides a simple illustration. A er NLO DGLAP evolution of the unpolarized n = 2 Mellin (momentum) moments to compare with data analysis, the results are shown in Table III. e RMS deviation of each set of 3 moments from the data analysis is typically 0.03 or less in each case.
In the lower part of Table III we display the results for quark and gluon helicities. e valence quark helicity portion ∆Σ v of the nucleon spin is similar to recent LQCD results [43]. e complete quark helicity, ∆Σ = ∆u + + ∆d + with q + = q +q, requires the sea contribution which we do not produce in the present simple model. If instead the sea helicity is taken from the polarized PDF analysis of Ref. [44], then 2(∆ū + ∆d) ∼ −0.15 at a scale of 1 GeV. Together with the present valence result, this indicates ∆Σ ∼ 0.368, a value close to global PDF analyses [44,45]. e di ering scales of the components suggests caution, but the indications are promising. e gluon helicity portion of nucleon spin, 2∆G obtained here at the low model scale, is comparable to the value 0.166 obtained by a recent global polarized PDF analysis at starting scale 1 GeV [46]. A more comprehensive treatment of nucleon spin including quark and gluon orbital angular momentum is under investigation. e valence isovector axial charge g v A (µ 0 ) = ∆u v − ∆d v is not strictly scale invariant, unlike the physical g A = ∆u + − ∆d + . Under the common assumption of a avor symmetric sea, Table III would indicate g A ≈ g v A ≈ 0.864, signi cantly below the experimental value 1.26. It is typical for a relativistic model of dressed valence quarks to produce g v A ≤ 1.0, see e.g. Ref. [47]. However global analysis of polarized PDFs [44] indicates a sea contribution 2(∆ū − ∆d) ∼ 0.3 at a scale of 1 GeV; this addition to the present valence calculation yields 1.16 for g A . Again despite the di erent scales involved, this indication is promising. A treatment of the Dirac, avor and momentum structure of the nucleon amplitude that improves upon the present simple illustrative model could improve all helicity related quantities and such e ects are under investigation. e DSE-RL approach here compares well with the present LQCD consensus for x N g at 2 GeV [3,5,43,48,49] as follows: Here Ref. [3] Ref. [48] Ref. [43] Ref. [5] Ref. [ For the pion at scale 2 GeV, we obtain x π g = 0.343. Results for ark and Gluon PDFs: At µ 0 the PDF moments for m < 6 were obtained by numerical integration. For larger m numerical treatment of the integration over p in Eq. (12) and Eq. (14) is challenging because the vertices Γ (m) q/g (p) contain a factor (p · n/P · n) m . Nevertheless, the integrals are analytically convergent for arbitrarily large m because the result can only depend on scalar products of pairs of the external 4-vectors (P, n, ν, where ν is polarization). Since n · n = 0 the integrand factor (p · n) m has to partner with a factor (p · P ) m with m ≥ m which has to be generated by the integrand term M f (p, P ) as seen via a Taylor expansion of it about p · P = 0. e coe cients involve corresponding high order derivatives of M f (p, P ) and each increases the power of the p-dependent denominator. us the integral remains explicitly convergent with increasing m, with its domain of support shi ing steadily toward the UV such that for m > 6 only the ranges and power law fall-o of the ingredient factors are seen to be relevant. e integral can then be cast into standard Feynman representation for which the results are known in algebraic form. is is done by ing all elements including vertex amplitudes and propagators to quadratic form denominators respecting the ranges and power law indices.
We use moments up to m = 30 to clearly observe the asymptotic behavior c/m h+1 and so identify the end point behavior (1 − x) h . is produced h π q = 2 and h π g = 3 for the pion and h N q = 3 and h N g = 4 for the nucleon. ese exponents re ect the lowest non-zero derivative at the end point x = 1 which in turn re ects the UV limit Q 2 → ∞ implicit in the derivation of the asymptotically hard sca ering quark counting rules and the Drell-Yan West relation [50,51]. e gluon end point exponents are consistent with the gluon being sub-leading to its quark source.
To obtain the x dependence at model scale the moments are t to the moments of f ( x − x and B 4 (y) is the polynomial of degree 4 that uses the Bernstein basis rather than y n , namely is choice has proved quite e cient in global PDF analyses [52] because it signi cantly reduces overlaps in the x domains in uenced by the parameters a, b, c, d. e obtained values of the parameters for the model scale gluon PDFs are displayed in Table II. e resulting x dependence is displayed in Fig. 3. A er NLO DGLAP evolution 2 the resulting pion PDFs are displayed in Fig. 4 at scale µ = 1.3 GeV and compared with the results of the JAM global analysis [35]. e same procedures are applied to the nucleon, and after NLO DGLAP scale evolution to µ = √ 5 GeV, the PDFs are displayed in Fig. 5 in comparison with the data analysis results from NNPDF3.0 [40]. e quark sector relates to experiment somewhat be er than the earlier work [27] that employed a Faddeev equation description of the nucleon but did not account for the interrelated quark-in-quark and gluonin-quark e ects. A recent la ice calculation obtains isovector momentum x uv−dv = 0.16 at 2 GeV [55]. Our present result compares well to this and to the value 0.162 from the NNPDF3.0 analysis.

Summary and Outlook:
We extend the DSE-RL approach to enable the calculation of the gluon PDF a ributable to the dressing of quarks. Due to the strength of dynamical chiral symmetry breaking, this quark dressing mechanism is expected to produce most of the gluon PDF of light-quark hadrons. We obtain the interrelated quark and gluon parton momentum fractions using an approach based on the innite subset of diagrams implemented by the Rainbow-Ladder truncation of the Dyson-Schwinger equations applied to the pion and nucleon at their natural model scale. We nd the dressing gluon carries about 20% of the lightcone momentum fraction for both hadrons. From calculated moments x m g up to m = 30 we identify the dressing gluon-in-quark g(x) in the pion and nucleon. To enable NLO DGLAP evolution to higher scales to compare with existing data analysis, we employ the valence q π v (x) produced in previous work within this 2 e present exploratory model is not designed to produce the quark sea at model scale, so we employ the data analysis results from Ref. [53] for the pion at µ 0 = 0.632 GeV, and from Ref. [54] for the nucleon at µ 0 = 0.707 GeV and matched to the present x sea values. approach and calculate q N v (x) within the present exploratory nucleon model. e high x end point behaviors of g(x) are found to be (1 − x) h with h π = 3 and h N = 4; as expected on physical grounds these are 1 greater than the exponents of the corresponding q(x) which are the sources.
For this rst exploration of the gluon-in-quark PDF, we have used the triangle diagram in Landau gauge for q v (x) and thus have ignored the Wilson line contribution. As an estimate of its magnitude, we have tested a variation of the model scale boundary condition to start the upward evolution of the n = 2 Mellin moments for the pion. Use of x π qv + δ W and x π qs − δ W , with the gauge-invariant x m π g xed, shows that the new minimized RMS deviation from JAM momenta at 1.3 GeV can lower the previous 0.025-0.03 to 0.01 when δ W ∼ −0.035, with a slight increase of the favored model scale from 0.78 to 0.8 GeV. is suggests that the Wilson line e ect is about 3.5% for the lightcone momentum; this is much less important that other issues that need to be addressed. e present results for the pion add x m g and g(x) to the previously published quark results [30] of this parton-inparton approach. e dynamics of gluon exchange between di erent valence quarks is found to be down by a factor of 50 or more in its contribution to x g ; a future work will document this. e results here for the nucleon are new for all elements. e simple model for the nucleon amplitude used here for exploration produces results that are consistent with LQCD and experiment for unpolarized PDFs but are de cient in certain respects for gluon helicity. is is likely due to the simplicity of the presently employed modi ed SU(6) model nucleon amplitude. A generalized amplitude is under study. Improved QCD-based studies of the gluon PDF within hadrons will help prepare for experimental results from the anticipated Electron-Ion Collider [56].
Appendix: Form of interaction kernels: e interaction kernels in Eq. (18) and Eq. (19), which generate the quark vertices associated with the quark and gluon PDFs respectively, emploỹ α s (q 2 ) = π γ m 1 2 ln τ + 1 + q 2 /Λ 2 which extrapolates to the 1-loop coupling α s (q 2 ) in the ultraviolet. e second interaction kernel also employs which extrapolates to the ultraviolet q 2 behavior of the 1loop Landau gauge renormalization quantity Z 3 (q 2 , Λ 2 ). We use Λ QCD = 0.234 GeV. Apart from the xed quantities τ = e 2 − 1 and N f = 4, the parameters are given in Table IV.
Appendix: e Nucleon Model: For the amplitude A(p, P ) we employ the form A(p, P ) = N α α f α (p) f α (p) with the functions associated with the spin−1 and spin−0 terms of the underlying nucleon state in Eq. (21) having the form f α (p) = N α /((p − P/3) 2 + R 2 α ). Note that p − P/3 is the relative momentum of the active quark and spectator system, while the la er has momentum K = P − p. e ratio N 1 /N 0 replicates the relative infrared strength of the spin-1 and spin-0 qq correlations within the Faddeev amplitudes employed in Ref. [27], while N is determined by valence quark number. e parameters are displayed in Table IV.