Chiral Soliton Models and Nucleon Structure Functions
RReview
Chiral Soliton Models and Nucleon Structure Functions
Herbert Weigel Institute of Theoretical Physics, Physics Department, Stellenbosch University, Matieland 7602, South Africa;[email protected] Department of Mathematics, Kwame Nkrumah University of Science and Technology, Private Mail Bag, Kumasi,Ghana; [email protected] January 7, 2021 submitted to Symmetry
Abstract:
We outline and review the computations of polarized and unpolarized nucleon structurefunctions within the bosonized Nambu-Jona-Lasinio chiral soliton model. We focus on a consistentregularization prescription for the Dirac sea contribution and present numerical results from thatformulation. We also reflect on previous calculations on quark distributions in chiral quark solitonmodels and attempt to put them into perspective.
Keywords:
Chiral Quark Model, Regularization, Chiral Soliton, Hadron Tensor, Structure Functions
1. Introduction
In this mini-review we reflect on nucleon structure function calculations in chiral soliton models. Thisis an interesting topic not only because structure functions are of high empirical relevance but maybeeven more so conceptually as of how much information about the nucleon structure can be retrieved fromsoliton models. In this spirit, this paper to quite an extend is a proof of concept review.Solitons emerge in most non-linear field theories as classical solutions to the field equations. Thesesolutions have localized energy densities and can be attributed particle like properties. In the contextof strong interactions, that govern the structure of hadrons, solitons of meson field configurations areconsidered as baryons [1].Nucleon structure functions play an important role in deep inelastic scattering (DIS) that revealsthe parton substructure of hadrons. In DIS leptons interact with partons by the exchange of a virtualgauge particle. Here we will mainly consider electrons that exchange a virtual photon with either apion or a nucleon. The process is called deep inelastic as the produced hadrons are not detected. In acertain kinematical regime, the so-called Bjorken limit to be defined below, the DIS cross section can beparameterized as the product of the cross section for scattering off partons and distribution functions thatmeasure the probabilities to find these partons inside the hadron. This is the factorization scheme [2]. Inthis picture the structure functions are linear combinations of parton distribution functions.DIS can also be explored without direct reference to partons by writing the cross section in terms oflepton and hadron components. The latter is the hadron matrix element of a current-current correlatorand is parameterized by form factors. The structure functions are obtained from these form factors ina certain regime for the kinematic variables, again the Bjorken limit. The operator product expansionformally relates distribution and structure functions by expressing the hadron matrix elements of thecurrent-current correlator as matrix elements of bilocal and bilinear quark operators in the Bjorken limit.The microscopic theory for the structure of hadrons is quantum-chromo-dynamics (QCD) which is thenon-abelian gauge theory SU ( N C ) , where N C = Submitted to
Symmetry a r X i v : . [ h e p - ph ] J a n ersion January 7, 2021 submitted to Symmetry (perturbative) QCD only relates these functions at different energy scales and does so very successfully[3] within the DGLAP formalism [4], neither structure nor distribution functions can be computed fromfirst principles in QCD, except, maybe within the lattice formulation [5] . Hence model calculationsseem unavoidable for a theoretical approach to the structure functions that contain the information of thenon-perturbative nature of hadrons. In such models it may or may not be possible to relate structure anddistribution functions. For the quark model that we will employ, regularization stands in the way and weattempt to compute the structure functions directly from the current-current correlator.Though chiral (soliton) models for baryons have so far not been derived from QCD, there is ample ofmotivation to explore nucleon properties in chiral models. The soliton approach goes back to the Skyrmemodel [7] while the connection to QCD was later established by considering baryons in a generalizedversion of QCD with N C large [1]. Soon after those mainly combinatoric arguments for considering baryonsin an effective meson theory, static baryons properties were derived within the Skyrme model [8]. Thesoliton approach has ever since been very actively explored, cf. the reviews [9]. The point of departurefor most of these models is an effective meson theory that reflects the major symmetries of QCD onthe hadron level. In the low energy regime this is essentially the chiral symmetry with the pions aswould-be Goldstone bosons being the basic field degrees of freedom . Other mesons like ω and ρ werethen incorporated according to the rules of chiral symmetry. A major endeavor is to determine as manyas possible model parameters from mesons to gain a high predictive power in the soliton sector, i.e. forbaryon properties. Many of these properties have been reproduced in chiral soliton models to the accuracythat one expects from keeping the leading (and eventually next-to-leading) terms of a power expansion in N C when the actual value is N C = Another possibility is to apply QCD renormalization group equations to the empirical data at large energies and scale themdown to the point at which the probability interpretation becomes inconsistent [6]. On the other end, the heavy quark effective symmetry has also been combined with the soliton picture. This is outside the scopeof this review. The interested reader may trace relevant publications from Ref. [52] in the recent article [10].ersion January 7, 2021 submitted to
Symmetry that we only identify the symmetries of QCD when adopting this model to describe hadrons. At this stageof the project we will not identify the quark degrees of freedom with those of QCD, which, for example,means that the current quark mass is a free parameter. The theoretical framework has been derived alreadysome time ago [17] while the numerical results arising from costly simulations have only been obtainedrecently [18]. As mentioned above, a major motivation for the soliton picture arises from generalizingQCD to a non-abelian gauge theory with large N C . In this review we will make N C explicit in formulas,but actual calculations are performed with N C = a posteriori [19–27].We will comment on those approaches in Sect.7.The study of structure functions in soliton models has, to quite some extend, been triggered by theso-called proton spin puzzle [28]: Data on the polarized structure function suggested that, together withflavor symmetric relations, almost none of the nucleon spin was due to the spin of the quark constituents.This picture emerges from the non-relativistic quark model in which the nucleon spin equals the matrixelement of the axial singlet current. It is actually this matrix element that relates to the data and chiralsoliton models indeed yield a small value, some versions even predict zero [29]. See Ref. [30] for a recentreview on the present understanding of the proton spin structure.Earlier we have noted the relevance of structure and/or distribution functions for DIS. There areother regimes of relevance. Let us first make explicit the factorization theorem for the cross-section forelectron hadron scattering [2], σ e ( x , Q ) = ∑ a (cid:90) x d ξ f a ( ξ ) σ ea (cid:18) x ξ , Q (cid:19) . (1)Here f a ( ξ ) is the distribution function for parton a with momentum fraction ξ in the hadron and σ ea isthe (Born) cross-section for electron parton scattering. Note that the sum over a also includes differentdistributions for the same parton such as polarized and unpolarized. Furthermore x and Q are Lorentzinvariant kinematical variables that will be defined in Section 2. Essentially we are interested in the casewhere x is fixed but Q becomes large. The f a ( ξ ) are equally important for the Drell-Yan process in whichtwo hadrons ( A and B ) scatter into a lepton-antilepton pair and other hadrons. That pair originates from avirtual gauge boson that is produced by quarks ( q ) and antiquarks ( q ) within the hadrons. Without goinginto further detail this suggests that the scattering cross section is parameterized by the same distributionfunctions as DIS σ ∼ ∑ a (cid:90) x A d ξ q f q ( ξ q ) (cid:90) x B d ξ q f q ( ξ q ) σ (cid:48) ( ξ q , ξ q , Q ) , (2)where σ (cid:48) is the cross section for turning the quark-antiquark pair into a lepton-antileption pair by theexchange of a virtual gauge boson. For the detailed definition of the kinematic variables x A and x B for thetwo hadrons A and B we again refer to Ref. [2]. Here we will not pursue the Drell-Yan process any furtherbecause it is not related to a current-current correlation matrix element of a single hadron. However, wewill shortly come back to the Drell-Yan process in Section 7.The expansion, Eq. (1) indicates that different distributions f a ( ξ ) contribute with different inversepowers of Q to the total cross section through σ ea . Accordingly distributions are categorized by their twist which is extracted from the leading inverse power of Q in σ e . The definition of twist dwells in the operatorproduct expansion and relates to the dimensionality and spin of the operators in that expansion. Here it issufficient to mention that the leading contribution (as Q increases) has twist-2, distributions that contributelike 1/ Q to the total cross section have twist-3 etc. [31]. ersion January 7, 2021 submitted to Symmetry
The following section contains a brief recap of basic definitions in the context of structure functions.Section 3 describes the path from the self-interacting fermion theory to the bosonized chiral model togetherwith a review of the pion structure function calculation. This will be followed by the construction of thesoliton in that model in Section 4. We explain the soliton model calculation of structure in Section 5 anddiscuss the numerical results in Section 6. As mentioned, in Section 7 we will discuss related distributionfunction calculations in the chiral quark soliton model. Some concluding remarks are contained in Section8.
2. Framework of deep inelastic scattering
Deep inelastic scattering (DIS) is a major tool to explore the composition of the nucleon. In thisprocess electron scattering produces a virtual photon which then interacts with the charged components ofthe nucleon. To extract the structure functions, the scattering products need not be detected as they aresummed over in the final scattering cross section.The Feynman diagram to the right describesthe kinematical set-up, where k and k (cid:48) arethe momenta of the initial and final electrons,respectively, while p is the momentum of theincoming proton, typically taken in the rest frame.The set of final hadrons, X is not detected andsummed over, cf. Eq. (3). (cid:80)(cid:80)(cid:80)(cid:113)(cid:80)(cid:80)(cid:80) k (cid:16)(cid:16)(cid:16)(cid:49)(cid:16)(cid:16)(cid:16) k (cid:48) q = k − k (cid:48) (cid:115) (cid:45) p (cid:26)(cid:26)(cid:26)(cid:26)(cid:62)(cid:8)(cid:8)(cid:8)(cid:8)(cid:42)(cid:24)(cid:24)(cid:24)(cid:24)(cid:58)(cid:45)(cid:88)(cid:88)(cid:88)(cid:88)(cid:122)(cid:72)(cid:72)(cid:72)(cid:72)(cid:106) (cid:41) X The interaction vertex for the disintegration of the nucleon is the matrix element of the(electromagnetic) current J µ ( ξ ) . The cross-section contains the squared absolute value of this matrixelement and we sum/integrate over all final states subject to energy momentum conservation. This definesthe hadron tensor for electron nucleon scattering W µν ( p , q ; s ) = π ∑ X (cid:68) p , s (cid:12)(cid:12)(cid:12) J µ ( ) (cid:12)(cid:12)(cid:12) X (cid:69)(cid:68) X (cid:12)(cid:12)(cid:12) J † µ ( ) (cid:12)(cid:12)(cid:12) p , s (cid:69) ( π ) δ ( p + q − p X ) , (3)where s denotes the nucleon spin. The nucleon momentum is p and q = k − k (cid:48) is the momentum of thevirtual photon. As the interaction is inelastic we have q >
0. This, together with translational invariance,yields W µν ( p , q ; s ) = π (cid:90) d ξ e iq · ξ (cid:68) p , s (cid:12)(cid:12)(cid:12) [ J µ ( ξ ) , J † ν ( )] (cid:12)(cid:12)(cid:12) p , s (cid:69) . (4)The interaction is space-like and it is customary to introduce Q = − q > ν = p · qM N where M N is the nucleon mass. In the nucleon rest frame ν is the energy transferred from the electron to the virtualphoton. Most prominent is the Bjorken variable x = Q M N ν , (5) ersion January 7, 2021 submitted to Symmetry
Table 1.
Projection operators which extract the leading large Q components from the hadron tensor. Theprojectors given in the spin independent cases presume the contraction of W ρσ with S µνρσ = g µρ g νσ + g ρν g µσ − g µν g ρσ . The last row denotes the required spin orientation of the nucleon. f f g g T = g + g − g µν − xg µν i M N (cid:101) µνρσ q ρ p σ q · s − i M N (cid:101) µνρσ s ρ p σ spinindependent spinindependent (cid:126) s (cid:107) (cid:126) q (cid:126) s ⊥ (cid:126) q which in the parton model denotes the momentum fraction associated with a particular parton. Sinceon-shell p = M N , Q and x can be taken as the only dynamical Lorentz invariant variables so that thehadron tensor has the form factor decomposition W µν ( p , q ; s ) = (cid:18) − g µν + q µ q ν q (cid:19) M N W ( x , Q ) + (cid:18) p µ − q µ p · qq (cid:19) (cid:18) p ν − q ν p · qq (cid:19) M N W ( x , Q )+ i (cid:101) µνλσ q λ M N p · q (cid:18)(cid:104) G ( x , Q ) + G ( x , Q ) (cid:105) s σ − q · sq · p p σ G ( x , Q ) (cid:19) (6)for parity conserving processes like electromagnetic scattering of photons. The structure functions arethe form factors in the so-called Bjorken scaling limit that takes Q → ∞ with x fixed. For the spinindependent, unpolarized structure functions f ( x ) and f ( x ) that is M N W ( x , Q ) Bj −→ f ( x ) and p · qM N W ( x , Q ) Bj −→ f ( x ) . (7)For the spin dependent, polarized structure functions no further scaling is involved and G ( x , Q ) Bj −→ g ( x ) and G ( x , Q ) Bj −→ g ( x ) . (8)Contracting the hadron tensor with projectors listed in Tab. 1 extracts the pertinent structure functions. Forthe unpolarized structure functions these projectors directly lead to the Callan-Gross relation f = x f .Observe also that these projectors are to be combined with appropriate selections for the spin orientationof the nucleon state as indicated in the last row of Tab. 1. Even though we employ the Bjorken limit to theform factors, that leading expansion may still contribute with different (inverse) powers of Q to the totalcross section and thus the structure functions may be assigned different (leading) twist.Similarly to the commutator in the hadron tensor we consider the matrix element of the time-orderedcurrent-current product T µν ( p , q ; s ) = i (cid:90) d ξ e i q · ξ (cid:68) p , s (cid:12)(cid:12)(cid:12) T (cid:16) J µ ( ξ ) J † ν ( ) (cid:17) (cid:12)(cid:12)(cid:12) p , s (cid:69) = ( π ) ∑ X (cid:40) δ ( (cid:126) p X − (cid:126) q − (cid:126) p ) p X − q − p − i (cid:101) (cid:68) p , s (cid:12)(cid:12)(cid:12) J µ ( ) (cid:12)(cid:12)(cid:12) X (cid:69)(cid:68) X (cid:12)(cid:12)(cid:12) J † µ ( ) (cid:12)(cid:12)(cid:12) p , s (cid:69) + δ ( (cid:126) p X + (cid:126) q − (cid:126) p ) p X + q − p − i (cid:101) (cid:68) p , s (cid:12)(cid:12)(cid:12) J µ ( ) (cid:12)(cid:12)(cid:12) X (cid:69)(cid:68) X (cid:12)(cid:12)(cid:12) J † µ ( ) (cid:12)(cid:12)(cid:12) p , s (cid:69)(cid:41) . (9) ersion January 7, 2021 submitted to Symmetry
Cauchy’s principal value prescription x ± i (cid:101) = P (cid:16) x (cid:17) ∓ i πδ ( x ) shows that imaginary part of the first termis proportional to the hadron tensor as in Eq. (3) while the second term does not have an imaginary partfor the present kinematical set-up. Hence we have W µν ( p , q ; s ) = π Abs T µν , (10)where Abs stands for absorptive part. From the physics point of view, T µν is the forward amplitude fornucleon Compton scattering and the hadron tensor is its absorptive part.This paves the way towards computing the structure functions in the bosonized quark model. Theaction for that model is obtained from a functional integral of a self-interacting quark model. Within thatformulation matrix elements of time-ordered products are straightforward to compute. SubsequentlyCutkosky’s rules are applied to extract their absorptive parts.
3. The Chiral Quark Model
We consider the simplest SU ( ) Nambu-Jona-Lasinio (NJL) model which contains a chirallysymmetric quartic fermion interaction in the scalar and pseudoscalar bilinears. In Minkowski spacethe Lagrangian reads [13], L NJL = q (cid:16) i/ ∂ − m (cid:17) q + G (cid:104) ( qq ) + ( q i γ (cid:126) τ q ) (cid:105) . (11)The field q ( x ) denotes a spinor with two flavors (up, u and down, d ). There are no color interactions buteach spinor has N C color components. Furthermore m and G are the current quark mass (average up anddown quark mass) and the dimensionful coupling constant, respectively. The symmetry transformationsare q −→ q + i (cid:126) (cid:101) · (cid:126) τ q for m ∝ and q −→ q + i γ (cid:126) (cid:101) · (cid:126) τ q for m = M that has a quadratic potential and couples linearly to the quark bilinears qq and q i γ (cid:126) τ q . Then thefermion part of the functional integral can be computed and its logarithm is an effective action whichbecomes a non-linear and non-local theory for M [12]. The entries of this matrix are identified withthe fields of the low-lying mesons. The model (by invention) breaks chiral symmetry dynamically forsufficiently large G and therefore the most import modes of M are the pseudoscalar pions ( π ± , π ). Themembers of this isospin triplet would be Goldstone bosons in the chiral limit characterized by m = γ , (via minimalsubstitution in L NJL ) and studying the decay π → γγ . On the other hand the real part is subjected tostandard regularization methods like proper-time [32] or Pauli-Villars [33]. Within the perturbative realm( i.e. zero soliton sector) one can even work with a sharp momentum cut-off [34].We would like to avoid the Wick-rotation because we want any imaginary part in our calculation ofthe hadron tensor being solely due to the absorptive components that we will extract via Cutkosky’s rules.There is indeed a procedure to identify the Minkowski space analogs of the real and imaginary parts of theEuclidian action [35]. To this end we define Dirac operatorsi D = i ∂ / − ( S + i γ P ) + v / + a / γ = : i D ( π ) + v / + a / γ ersion January 7, 2021 submitted to Symmetry i D = − i ∂ / − ( S − i γ P ) − v / + a / γ = : i D ( π ) − v / + a / γ , (12)where S = ( M + M † ) and P = ( M − M † ) . Furthermore v µ and a µ denote external (classical) sourcefields with respect which we will compute functional derivatives to explore correlation functions. Finallywe have also defined Dirac operators without those sources ( D ( π ) and D ( π ) ) for later use. Wick-rotating D produces the conjugate of the Wick-rotation of D so that Tr log [ DD ] corresponds to the real partof the Euclidian action while its imaginary part is associated with Tr log [ D ( D ) − ] . The introductionof D comes at a price. Some of the Ward identities derived from the standard Dirac operator D donot hold anymore and rather occur with opposite signs [17]. We will later cure that obstacle by aparticular calculational procedure to extract the polarized structure functions. This procedure is part of theregularization scheme. Even though the proper-time scheme has been very successfully applied for thesolitons of the NJL model, we do not implement it here. This scheme induces an exponential dependenceon the cut-off and it is unclear how to implement the Bjorken limit. Rather we adopt a version of thePauli-Villars scheme in which the cut-off essentially is additive to the quark mass and does not interferewith the Bjorken limit. With all these preliminaries we are now in a position to write down the effectiveaction for M : A NJL = A R + A I + G (cid:90) d x tr (cid:104) m ( M + M † ) − MM † (cid:105) (13) A R = − i N C ∑ i = c i Tr log (cid:104) − DD + Λ i − i (cid:101) (cid:105) , A I = − i N C (cid:104) − D ( D ) − − i (cid:101) (cid:105) .Here A R and A I are the Minkowski analogs of the real and imaginary parts of the Euclidian space effectiveaction. Furthermore ’Tr’ denotes the functional trace that includes space-time integration on top ofsumming over the discrete Dirac and flavor indexes. The Pauli-Villars regularization scheme requires c = Λ = ∑ i = c i = ∑ i = c i Λ i = Λ = Λ = Λ . For any quantity Q ( Λ ) that is subject to regularization we then have ∑ i = c i Q ( Λ i ) = Q ( ) − Q ( Λ ) + Λ Q (cid:48) ( Λ ) , (15)where the prime denotes the derivative with respect to the argument. For notational simplicity we willusually write the formulas as on the left hand, understanding that the right hand side is implemented inactual computations.To analyze the model we need to find the ground state solution, (cid:104) M (cid:105) . For symmetry reasons anynon-zero solution can only be a (real) constant that is proportional to the unit matrix. We thereforesubstitute (cid:104) M (cid:105) = m in the so-called gap equation12 G (cid:16) m − m (cid:17) = − N C m ∑ i = c i (cid:90) d k ( π ) (cid:104) − k + m + Λ i − i (cid:101) (cid:105) − (16) ersion January 7, 2021 submitted to Symmetry that arises from δ A NJL δ M =
0. For sufficiently large coupling G this equation has a solution with m (cid:29) m which obviously plays the role of a mass parameter when substituted for M into D (or D ). It is thereforecalled the constituent quark mass.Any non-trivial vacuum solution signals dynamical symmetry breaking and applying a symmetrytransformation onto that solution leads to (would-be) Goldstone bosons. In this case the relevanttransformation is chiral and the would-be Goldstone boson is the pseudoscalar iso-triplet pion (cid:126) π . Thisfield is most conveniently introduced via the non-linear realization M = mU = m exp (cid:104) i gm (cid:126) π · (cid:126) τ (cid:105) = m + i g (cid:126) π · (cid:126) τ + O ( (cid:126) π ) , (17)where U is the chiral field while g is the Yukawa coupling constant. In the next step, we expand theeffective action to quadratic order in the pion fields A NJL = g (cid:90) d p ( π ) (cid:126) (cid:101) π ( p ) · (cid:126) (cid:101) π ( − p ) (cid:20) N C q Π ( p ) − G m m (cid:21) + O (cid:16) (cid:126) π (cid:17) , (18)which has been written for the Fourier transform (cid:126) (cid:101) π ( p ) = (cid:82) d x e − i p · ξ (cid:126) π ( ξ ) . The quadratic contributioncontains the polarization function Π ( p ) = (cid:90) dx Π ( p , x ) with Π ( p , x ) = − i ∑ i = c i d k ( π ) (cid:104) − k − x ( − x ) p + m + Λ i − i (cid:101) (cid:105) − .(19)The factor in square brackets in Eq. (18) times g is the inverse pion propagator. Requiring this propagatorto have a pole at the physical pion mass enforces m = N C Gmm π Π ( m π ) . (20)Furthermore the residue of that pole should be one thereby relating the Yukawa coupling constant g toother model parameters, 1 g = N C ∂∂ m π (cid:104) m π Π ( m π ) (cid:105) . (21)We construct the axial current from the functional derivative with respect to the axial source a µ A µ ( ξ ) = δ A NJL δ a µ ( ξ ) (cid:12)(cid:12)(cid:12) v ν , a ν = .Expanding A µ ( ξ ) to linear order in (cid:126) (cid:101) π ( p ) yields the matrix element ( a and b are flavor labels) (cid:104) | A ( a ) µ ( ξ ) | (cid:101) π ( b ) ( p ) (cid:105) ! = δ ab f π ( p ) p µ e − i p · ξ from which we get the on-shell pion decay constant f π ( ) = f π = N c mg Π ( m π ) . Taking this togetherwith Eqs. (20) and (21) gives three equations for four model parameters ( g , Λ , G and m ) after insertingthe empirical data f π = m π = G , undetermined. We We expect that boson to be massless only when the original theory has an exact chiral symmetry, m ( ) = Symmetry employ the gap equation (16) to express that undetermined parameter as a function of the constituentquark mass m which we take as the sole variable from now on. After all, we have quite some intuitionabout m and expect it to be somewhere around 400MeV. This procedure is reflected by the first threecolumns of Tab. 2 in the proceeding Section. In this calculation the current quark mass is only about onethird of what is obtained within proper-time regularization scheme [15]. This significant difference againsuggests that quarks fields of the model are merely some effective degrees of freedom, with little or norelation to fundamental particles.To apprehend the nucleon structure function calculation let us have a short look at DIS off pionswhich is characterized by a single structure function, F ( x ) ,12 π Abs T µν ( p , q ) Bj −→ F ( x ) (cid:20) − g µν + q µ q ν q − q (cid:16) p µ − q µ x (cid:17) (cid:16) p ν − q ν x (cid:17)(cid:21) , (22)where the Bjorken limit defined after Eq. (6) has been indicated. In order to compute the Comptonamplitude (22) we calculate the time-ordered product T (cid:0) J µ ( ξ ) J ν ( ) (cid:1) = δ δ v µ ( ξ ) δ v ν ( ) A NJL (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v µ = (23)from the action, A NJL in Eq. (12) with a µ = v µ → v µ Q , where Q = diag ( − ) isthe quark charge matrix. In principle we would have to fully expand A NJL to quadratic order in both thephoton vector source, v µ and the pion field (cid:126) (cid:101) π ( p ) . Fortunately there is some simplification in expanding DD . Contributions to this product that are quadratic in either of the two fields add Feynman diagrams tothe Compton amplitude that depend only on one of the two momenta. These, kind of local diagrams, donot have an absorptive component. It is thus sufficient to consider − DD = ∂ + m + g γ [ ∂ /, (cid:126) π · (cid:126) τ ] − i ( ∂ / v / Q + v / Q ∂ / ) + i g γ [ (cid:126) π · (cid:126) τ , v / Q ] + . . . . (24)Even with this simplification, the expansion of the logarithm in A R ( A I does not contribute) has someun-wanted terms with the flavor trace tr [ (cid:126) π Q (cid:126) π Q ] that would lead to different structure functions for thecharged and un-charged pions. Fortunately these terms cancel in the Bjorken limit. Even when omittingterms which are suppressed in this limit or eventually do not contribute to the absorptive part, the pionCompton amplitude is still quite cumbersome to compute [17] (cid:90) d ξ e i q · ξ (cid:104) π ( p ) | δ δ v µ ( ξ ) δ v ν ( ) A NJL (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v µ = | π ( p ) (cid:105) = g N C ∑ i = c i (cid:90) d k ( π ) − k + m + Λ i − i (cid:101) (cid:2) − ( k − p ) + m + Λ i − i (cid:101) (cid:3) × (cid:40) − ( k − p ) + m + Λ i − ( k + q − p ) + m + Λ i − i (cid:101) tr ( p / γ µ q / γ ν + p / γ ν q / γ µ ) − − ( k − p ) + m + Λ i − ( k − q − p ) + m + Λ i − i (cid:101) tr ( p / γ µ q / γ ν + p / γ ν q / γ µ )+ m (cid:34) tr ([ k / − p / ] γ ν q / γ µ ) − ( k − q − p ) + m + Λ i − i (cid:101) − tr ([ k / − p / ] γ µ q / γ ν ) − ( k + q − p ) + m + Λ i − i (cid:101) (cid:35)(cid:41) . (25) ersion January 7, 2021 submitted to Symmetry
10 of 40
The last two terms are products of four propagators as expected from an expansion up to fourth order. Thefirst two terms only have three propagators and are represented by diagrams with a pion and a photon ata single vertex. This interaction stems from the last term in Eq. (24). As in Eq. (9) the absorptive part isextracted by putting all intermediate propagators on-shell according to Cutkosky’s rule1 − k + m + Λ i − i (cid:101) −→ − πδ ( k − m − Λ i ) − ( k ± q − p ) + m + Λ i − i (cid:101) −→ − i π q − δ (cid:0) q + ± ( k − p ) + (cid:1) . (26)In the second substitution we introduced light-cone coordinates (the full definition is given in Section 6.3)because they render the implementation of the Bjorken limit quite transparent: q − → ∞ and q + →− xp + = xm π √ (in the pion rest frame). These coordinates bring in the factor q − when extracting theabsorptive part. A posteriori this justifies the omission of all terms in Eq. (25) that did not contain a factor q /in the numerator . After taking the traces in color and spinor spaces the structure function can be read offfrom T + T
22 Bj −→ F ( x ) using, e.g. , γ q / γ −→ q − γ γ + γ = − q − γ + .We find F ( x ) = − ( N C g ) ∑ i = c i (cid:90) d k ( π ) πδ ( k − m − Λ i ) (cid:2) − ( k − p ) + m + Λ i − i (cid:101) (cid:3) × (cid:26) (cid:104) − ( k − p ) + m + Λ i (cid:105) (cid:2) δ ( k + − p + − q + ) − δ ( k + − p + + q + ) (cid:3) p + + m (cid:2) δ ( k + − p + + q + ) − δ ( k + − p + − q + ) (cid:3) ( k + − p + ) (cid:27) . (27)The δ -functions straightforwardly produce the k + integrals fixing this variable to either m π ( − x ) or m π ( + x ) . The integral over k − can then be computed via the δ -function in first numerator. The resultfrom these two integrals is F ( x ) = ( N C g ) ∑ i = c i (cid:90) d k ⊥ ( π ) (cid:40) M i ( x ) θ ( x ) θ ( − x ) x ( − x ) (cid:2) M i ( x ) − m π (cid:3) + (cid:0) x ←→ − x (cid:1)(cid:41) , (28)where M i ( x ) = x ( − x ) (cid:2) m + Λ i + k ⊥ (cid:3) . This expression for the pion structure function was earlierobtained using light-cone wave-functions [35,36]. In the chiral limit, m π =
0, this structure function is justa constant on the interval − ≤ x ≤
1. It is interesting to note that the light-cone coordinate momentum The full calculation also produces terms involving ( k / − p / ± q / ) in Eq. (25). With Eq. (26) it is obvious that they do not contributeto the absorptive part even though they have a finite Bjorken limit.ersion January 7, 2021 submitted to Symmetry
11 of 40 variables can also be integrated in the pion polarization function, Eq. (19) leading to the same k ⊥ integralallowing the compact expression F ( x ) = ( N C g ) ∂∂ p (cid:104) p Π ( p , x ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = m π . (29)At this point one important aspect has not been considered. As it stands, Eq. (29) is the pion structurefunction at the scale at which the NJL-model is supposed to approximate QCD. Stated otherwise,the structure function computed from Eq. (29) approximates the QCD result at a (presumably) lowrenormalization scale. To allow a comparison with data, the QCD evolution equations must be applied tothe model prediction. At that stage, the low renormalization scale enters as a new parameter that is tunedto optimize the agreement with the data at the higher energy scale of the experiments. This calculation hasbeen carried out in Ref. [37]. Here we will not further elaborate on QCD evolution but will get back to it inSection 6 in the context of the nucleon structure functions.The main lesson learned from this pion structure function study is that the calculation simplifiessignificantly when identifying the propagators that carry the momentum which is large in the Bjorken limitand ignoring the others (the many terms not shown in Eq. (25)) and/or simplifying them by approximatingthem with free quark propagators.
4. Self-consistent soliton
The soliton is a static meson configuration that minimizes the bosonized action. To construct thisconfiguration we define a Dirac Hamiltonian h via the Dirac operators in Eq. (12)i D ( π ) = β ( i ∂ t − h ) and i D ( π ) = ( − i ∂ t − h ) β . (30)Its diagonalization h Ψ α = (cid:101) α Ψ α , (31)yields eigenvalues (cid:101) α and eigen-spinors Ψ α = ∑ β V αβ Ψ ( ) α as linear combinations of the free Dirac spinors Ψ ( ) α in a spherical basis.When constraining the meson configuration to the chiral circle, i.e. parameterizing M = mU withonly U being dynamical, the so-called hedgehog configuration [38] minimizes the action in the unit baryonnumber sector . This, together with (the assumption of) spherical symmetry suggest the ansatzh = (cid:126) α · (cid:126) p + β m U ( (cid:126) r ) where U ( (cid:126) r ) = exp [ iˆ r · (cid:126) τ γ Θ ( r )] . (32)The radial profile function Θ ( r ) is called the chiral angle. The hedgehog configuration, Eq. (32) is invariantunder so-called grand spin transformations that combine flavor and coordinate rotations. Accordingly, theDirac and flavor components of the eigenfunctions Ψ α are products of radial functions and grand spineigenfunctions. The latter are products of spherical harmonic functions, spinors and iso-spinors. Finaldiscretization is accomplished by imposing boundary conditions on the radial functions at a distance D much larger than typical extensions of the chiral angle [39]. Different boundary conditions are equivalentin the limit D → ∞ , however, at large but finite D a certain choice may be preferable depending on which We refer to the earlier review articles [15,16] for obstacles and their solutions for hedgehog configurations away from the chiralcircle.ersion January 7, 2021 submitted to
Symmetry
12 of 40 quantity is to be computed [40]. All possible boundary conditions require that there is no flux through thesphere at D .Once the structure of the spinors is established, particular profile functions can be considered. Forprofiles with Θ ( ) = − π and lim r → ∞ Θ ( r ) = (cid:101) v , eigen-spinor Ψ v ) in the grand spin zero channel. This level is referred to asthe valence quark level [15]: the wider the chiral angle, the more strongly bound is this distinct level. Its(explicit) occupation ensures unit baryon number.The functional trace in A R ( A I vanishes for static configurations) is computed as an integral over thetime interval T and a discrete sum over the basis levels defined by Eq. (31). In the limit T → ∞ the vacuumcontribution to the static energy is then extracted from A R → − TE vac . Collecting pieces, we obtain thetotal energy functional as [15,16] E tot [ Θ ] = N C [ + sign ( (cid:101) v )] (cid:101) v − N C ∑ i = c i ∑ α (cid:26)(cid:113) (cid:101) α + Λ i − (cid:113) (cid:101) ( ) α + Λ i (cid:27) + m π f π (cid:90) d r [ − cos ( Θ )] .(33)Here we have also subtracted the vacuum energy associated with the non-dynamical meson fieldconfiguration Θ ≡ N C as ascertained for baryon masses in QCD [1].The soliton profile is then obtained as the profile function Θ ( r ) that minimizes the total energy E tot self-consistently subject to the above mentioned boundary conditions on Θ ( r ) . The energy eigenvalues (cid:101) α are functionals of the chiral angle through the diagonalization in Eq. (31). Hence the minimization of E tot [ Θ ] involves δ(cid:101) α δ Θ ( r ) = m (cid:90) d r (cid:48) Ψ † α ( (cid:126) r (cid:48) ) β (cid:2) − sin Θ ( r (cid:48) ) + iˆ r (cid:48) · (cid:126) τ cos Θ ( r (cid:48) ) (cid:3) Ψ α ( (cid:126) r (cid:48) ) δ ( r − r (cid:48) ) ,by the chain rule. Self-consistency arises as the wave-functions in this functional derivative emerge fromdiagonalizing an operator that contains Θ ( r ) . Though this Hartree-type problem is quite elaborate, it hasbeen established some time ago [41] and ever been refined [15,16]. The two main contributions to E tot [ Θ ] act in opposite directions: the binding of the distinct level is attractive while the Dirac sea piece (partially)compensates for this reduction. As the binding of the valence level increases with the constituent quarkmass m , the soliton is kinematically stable against decaying into N C unbound quarks for m (cid:38) cf. Tab. 2.This soliton represents an object which has unit baryon number but neither good quantum numbersfor spin and flavor (isospin). Such quantum numbers are generated by canonically quantizing thetime-dependent collective coordinates A ( t ) that parameterize the spin-flavor orientation of the soliton via U ( (cid:126) r , t ) = A ( t ) U ( (cid:126) r ) A † ( t ) , (34)where U ( (cid:126) r ) is the self-consistent static configuration from Eq. (32). For a rigidly rotating soliton the Diracoperator becomes, after transforming to the flavor rotating frame [32],i D ( π ) = A β (cid:18) i ∂ t − (cid:126) Ω · (cid:126) τ − h (cid:19) A † and i D ( π ) = A (cid:18) − i ∂ t + (cid:126) Ω · (cid:126) τ − h (cid:19) β A † . (35) ersion January 7, 2021 submitted to Symmetry
13 of 40
Actual computations involve an expansion with respect to the angular velocities (cid:126) Ω that are defined by thattime derivative of the collective coordinates as A † ddt A = i2 (cid:126) Ω · (cid:126) τ . (36)According to the canonical quantization rules the angular velocities are replaced by the spin operator (cid:126) Ω −→ α (cid:126) J . (37)The constant of proportionality is the moment of inertia α = N C [ + sign ( (cid:101) v )] ∑ β (cid:54) = v |(cid:104) v | τ | β (cid:105)| (cid:101) β − (cid:101) v + N C ∑ α (cid:54) = β ∑ i = c i |(cid:104) α | τ | β (cid:105)| (cid:101) α − (cid:101) β (cid:101) α + (cid:101) α (cid:101) β + Λ i (cid:113) (cid:101) α + Λ i − (cid:101) β + (cid:101) α (cid:101) β + Λ i (cid:113) (cid:101) β + Λ i , (38)expressed by introducing eigenstates | α (cid:105) of h ; i.e. Ψ α ( (cid:126) r ) = (cid:104) (cid:126) r | α (cid:105) . The moment of inertia is O ( N C ) and isextracted as twice the constant of proportionality of the O ( (cid:126) Ω ) term in the Lagrange function ( A / T ). WithEq. (37) the expansion in (cid:126) Ω is thus equivalent to the one in N C . After quantizing the collective coordinatesthe Hamilton operator is that of a rigid rotor leading to the energy formula E ( j ) = E tot + α j ( j + ) , (39)with spin eigenvalues j = for the nucleon and j = for the ∆ -resonance. Note that this energyformula contains a piece linear in N C and one linear in N C . The contribution O ( N C ) , which is the vacuumpolarization energy from the meson fluctuations, is generally omitted in soliton models. There is no robustcalculation of this vacuum polarization energy because these models are not renormalizable. Estimatesindicate that the O ( N C ) component significantly reduces the energy [42]. Since this part does not dependon the baryon quantum numbers, it is customary to only consider mass differences, in particular, the ∆ -nucleon mass difference ∆ M = α . The results shown in Tab. 2 suggest that m ≈ | N (cid:105) is [8] (cid:104) N | D ab | N (cid:105) = − (cid:104) N | I a J b | N (cid:105) with D ab =
12 tr (cid:16) A † τ a A τ b (cid:17) . (40)Here I a and J b are iso- and spin operators, respectively. The above matrix element arises from the operatoridentity I a = − D ab J b which by itself reflects the invariance of the hedgehog configuration under combinedisospin and coordinate rotations.As an example for the computation of a static nucleon property we consider the vacuum contributionto the axial charge, g a , of the nucleon because in Section 5 it will be paradigmatic for how sum rulesfor structure functions emerge in this model and its treatment. In the first step we require the spatial ersion January 7, 2021 submitted to Symmetry
14 of 40 components of the axial current as a function of the collective coordinates A . This is achieved by expandingthe regularized action to leading order in the axial source a / with a = A NJL = − i N C ∑ i = c i Tr log (cid:110) β (cid:16) ∂ t + h (cid:17) β + β ( i ∂ t + h ) a / γ + a / γ ( − i ∂ t + h ) β + Λ i − i (cid:101) (cid:111) = − i N C ∑ i = c i Tr log (cid:110) ∂ t + h + { h , a / γ β } + Λ i − i (cid:101) (cid:111) . (41)The next simplification is that we only need the (space) integral of that current and therefore may take a / γ β = − (cid:126) a ( a ) · (cid:126) αγ τ a = − (cid:126) a ( a ) · (cid:126) Σ τ a with constant (cid:126) a ( a ) to compute ∂ A NJL ∂ (cid:126) a ( a ) (cid:12)(cid:12)(cid:12) (cid:126) a ( a ) = = i N C ∑ i = c i Tr (cid:26)(cid:16) ∂ t + h + Λ i − i (cid:101) (cid:17) − (cid:110) h , (cid:126) Σ τ a (cid:111)(cid:27) = i N C ∑ i = c i T (cid:90) d ω π Tr (cid:48) (cid:26)(cid:16) − ω + h + Λ i − i (cid:101) (cid:17) − (cid:110) h , (cid:126) Σ τ a (cid:111)(cid:27) . (42)As for any path integral, the limit T → ∞ extracts the vacuum (Dirac sea) component. In the next step wewant to evaluate the remaining trace Tr (cid:48) using the eigenvalues (cid:101) α and the eigenstates | α (cid:105) of h . Substitutingthe rotating hedgehog configuration from Eq. (34) and using the cyclic property of the trace yields ∂ A NJL ∂ (cid:126) a ( a ) (cid:12)(cid:12)(cid:12) (cid:126) a ( a ) = = i N C ∑ i = c i T (cid:90) d ω π ∑ α (cid:26)(cid:16) − ω + (cid:101) α + Λ i − i (cid:101) (cid:17) − (cid:101) α (cid:68) α (cid:12)(cid:12)(cid:12) (cid:126) Σ A † τ a A (cid:12)(cid:12)(cid:12) α (cid:69)(cid:27) = − N C TD ab ∑ i = c i ∑ α (cid:101) α (cid:113) (cid:101) α + Λ i (cid:68) α (cid:12)(cid:12)(cid:12) (cid:126) Σ τ b (cid:12)(cid:12)(cid:12) α (cid:69) , (43)where the frequency integral has been computed by contour integration. The vacuum contribution to theaxial charge is then obtained as the proton matrix element g ( s ) a = lim T → ∞ T (cid:42) P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ A NJL ∂ a ( ) z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) a ( ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P (cid:43) = N C ∑ i = c i ∑ α (cid:101) α (cid:113) (cid:101) α + Λ i (cid:104) α | Σ τ | α (cid:105) , (44)with spin projection J = + . In addition we have the contribution from the valence quark that we get viaa similar derivative after ”gauging” the valence level N C [ + sign ( (cid:101) v )] (cid:101) v −→ N C [ + sign ( (cid:101) v )] D ab (cid:68) v (cid:12)(cid:12)(cid:12) h + (cid:126) a ( a ) · (cid:126) Σ τ b (cid:12)(cid:12)(cid:12) v (cid:69) (45)so that g ( v ) a = − N C [ + sign ( (cid:101) v )] (cid:104) v | Σ τ | v (cid:105) . In total we have g a = g ( v ) a + g ( s ) a = − N C [ + sign ( (cid:101) v )] (cid:104) v | Σ τ | v (cid:105) + N C ∑ i = c i ∑ α (cid:101) α (cid:113) (cid:101) α + Λ i (cid:104) α | Σ τ | α (cid:105) . (46)It is illuminating to make the single cut-off regularization from Eq. (15) explicit g a = − N C [ + sign ( (cid:101) v )] (cid:104) v | Σ τ | v (cid:105) + N C ∑ α (cid:34) sign ( (cid:101) α ) − (cid:101) α (cid:101) α + Λ ( (cid:101) α + Λ ) (cid:35) (cid:104) α | Σ τ | α (cid:105) . (47) ersion January 7, 2021 submitted to Symmetry
15 of 40 m [ MeV ] m [ MeV ] Λ [ GeV ] E tot [ MeV ] α [ ] ∆ M [ MeV ] g a
350 7.9 0.77 1267 8.65 173 0.85400 8.4 0.74 1269 5.89 255 0.80450 8.5 0.73 1257 4.82 311 0.77
Table 2.
Model parameters and results. See the main text for their definitions.
The strongly bound valence level is also included in the sum over α . As the binding of that level isincreased, for example by increasing the constituent quark mass m in the self-consistent construction, thecorresponding energy eigenvalue eventually changes sign. The particular combination of valence and seacontributions ensures that g a is continuous as the terms with sign ( (cid:101) v ) cancel. This feature is universal forany quantity; there is no discontinuity as the sign of the valence energy eigenvalue changes . This is alsotrue for the energy, Eq. (33) and the moment of inertia, Eq. (38). Essentially this occurs by construction asthe prefactor [ + sign ( (cid:101) v )] is introduced to ensure unit baryon number B = [ + sign ( (cid:101) v )] − ∑ α sign ( (cid:101) α ) .Stated otherwise, when the valence level is so strongly bound that its energy eigenvalue is negative, thebaryon number is carried by the polarized Dirac sea (vacuum). This is an implicitly assumed feature oftopological chiral soliton models like the Skyrme model because the topological current is the leading termin the gradient expansion for the vacuum contribution of the baryon current in chiral quark models [43].In Tab. 2 we also list the model predictions for g a . They are about 30% below the empirical value of1.26 [44]. Note, however, that only the leading N C result is given. It has been asserted that, because of thetime-ordering prescription in the path integral for bosonization, subleading contributions can significantlyincrease the model prediction [16]. These contributions are, unfortunately, not without further problems.For example, they violate PCAC: In soliton models a partially conserved axial current (PCAC) results fromthe field equation for the soliton. This equation contains only the leading order in N C and any subleadingpiece in the axial current is not covered. Altering the field equation accordingly [45] does not produce astable soliton when the subleading Dirac sea contribution is properly included [16].
5. Hadron tensor for the nucleon as soliton
We now get to a central topic of this short review: extracting the nucleon structure functions fromthe hadron tensor in the soliton background while realizing regularization from the onset of the action,Eq. (13). Here we will consider mainly the example of the leading N C component of the longitudinalpolarized structure function, g . For this example we will also explain how sum rules are established inthe fully regularized formulation. For further details on other structure functions, that are obtained usingquite a similar procedure, we refer to to original literature [17,18].Similar to the pion structure function in Sect. 3 we start from the Compton tensor, Eq. (23). However,this time we have to account for the non-perturbative nature of the solitonic meson fields, and may notapproximate D ( π ) except for the N C expansion. As mentioned in that earlier Section, isospin violatingcontributions may arise that only cancel once the Bjorken limit is assumed. Can we anticipate this type of Taking the "chemical potential" to be zero is a choice anyhow. In analogy to g a the baryon number is obtained from a functional derivative with respect to constant v . The vacuum contributionstems from A I and is not regularized. As mentioned earlier A I is conditionally convergent in the sense that the sum over α mustbe taken over a symmetric interval. Early studies [46] omitted that part.ersion January 7, 2021 submitted to
Symmetry
16 of 40 (cid:116) (cid:116) Bj −→ (cid:116) (cid:116) Figure 1.
Two photon coupling to fermion loop. Thick lines are the full fermion propagators D ( π ) − (or D ( π ) − ) without any perturbation expansion. The thin line in the loop represents a free (massless) fermionpropagator, ∂ / − . Dashed lines denote Cutkosky cuts as discussed after Eq. (58). cancellations for the soliton configuration at an earlier stage and thus simplify the calculation (somewhat)?As a matter of fact the appearance of these terms is indeed an artifact of the simultaneous expansion inthe pion and photon fields, Eq. (24). We might equally well have expanded only in the photon field first(taking the charge matrix Q as part of v /, for simplicity) − Tr (cid:26)(cid:16) − D ( π ) D ( π ) + Λ i (cid:17) − (cid:20)(cid:16) D ( π ) v / + v / D ( π ) (cid:17) (cid:16) − D ( π ) D ( π ) + Λ i (cid:17) − (cid:16) D ( π ) v / + v / D ( π ) (cid:17)(cid:21)(cid:27) . (48)Here square brackets have been introduced to mark those factors that are sensitive to the large photonmomentum. Due to the cyclic properties of the trace this is merely a choice but it must contain all verticeswith v /. In momentum space the propagator inside the square brackets behaves like 1/ Q when assumingthe Bjorken limit. In particular this implies that (cid:104) . . . (cid:105) Bj −→ (cid:16) D ( π ) v / + v / D ( π ) (cid:17) (cid:16) − D ( π ) D ( π ) (cid:17) − (cid:16) D ( π ) v / + v / D ( π ) (cid:17) Bj −→ − D ( π ) v / (cid:16) D ( π ) (cid:17) − v / − v / (cid:16) D ( π ) (cid:17) − v / D ( π ) . (49)Terms with either the unit or the (cid:16) − D ( π ) D ( π ) (cid:17) − operators between two vector sources have been omittedbecause they will either not depend on the photon momentum, cf. the discussion before Eq. (24), or areadditionally suppressed by factors of Q . The above replacement tells us that in the Bjorken limit thepropagator through which the large photon momentum runs will not contain the cut-offs Λ i . In particularthere will be no contributions which behave like Q Λ i ; thereby the proper scaling behavior is manifest. Inother regularization schemes, like e.g. proper-time, wherein the cut-off is not additive to the loop momenta,the absence of such scaling violating contributions is not apparent. Previously, Eq. (48), we expandedthe operator in powers of the pion field leading to complicated three and four vertex quark loops. Nowwe see that the Bjorken limit enforces the cancellations among those diagrams that we observed for thepion structure function. The expression (49) simplifies even further by noting that the quark propagatorbetween the two photon insertions carries the large photon momentum and should hence be approximatedby the free massless propagator, (cid:104) . . . (cid:105) −→ D ( π ) v / ( ∂ / ) − v / − v / ( ∂ / ) − v / D ( π ) . (50)The transition from the expression (48) to (50) is illustrated in Fig. 1. Substituting this simplification intoEq. (48) leads to Tr (cid:26)(cid:20)(cid:16) D ( π ) (cid:17) − − (cid:16) D ( π ) (cid:17) − (cid:21) v / ( ∂ / ) − v / (cid:27) + reguarlization terms . (51) ersion January 7, 2021 submitted to Symmetry
17 of 40
Essentially we only include small and moderate momenta from the loop integrals for one of the twopropagators, keeping in mind that the sum of the momenta in the propagators is subject to the Bjorkenlimit. The integration regime in which that large momentum is distributed (approximately) equally amongthe two propagators does not contribute in the Bjorken limit [17].Having simplified the construction of the Compton tensor with the soliton background in the Bjorkenlimit we see that it will be sufficient to differentiate (bringing back the charge matrix Q ) A ( v ) Λ ,R = − i N C ∑ i = c i Tr (cid:26)(cid:16) − D ( π ) D ( π ) + Λ i (cid:17) − (cid:104) Q v / ( ∂ / ) − v / D ( π ) − D ( π ) ( v / ( ∂ / ) − v / ) Q (cid:105)(cid:27) (52)with respect to the vector sources. As already mentioned after Eq. (12) the operator D , which wasintroduced to accomplish regularization, produces an unconventional Ward identity because, in contrastto D , this γ -odd operator has a relative minus sign between the derivative operator i ∂ / and the axialvector source a / γ . To correct this regularization artifact in a way consistent with the Bjorken sum rule [47]for the nucleon axial charge, g a , this relative sign must also be reflected in the Dirac decomposition of ( v / ( ∂ / ) − v / ) = v µ ∂ ρ ∂ v ν ( γ µ γ ρ γ ν ) : γ µ γ ρ γ ν = S µρνσ γ σ − i (cid:101) µρνσ γ σ γ while ( γ µ γ ρ γ ν ) = S µρνσ γ σ + i (cid:101) µρνσ γ σ γ , (53)where S µρνσ = g µρ g νσ + g ρν g µσ − g µν g ρσ . We recall that the D model, which is not physical, has beensolely introduced as a device to allow for a regularization which maintains the anomaly structure of theunderlying theory by regularizing A R and A I differently. Hence it is not at all surprising that furtherspecification of this regularization prescription is demanded in order to formulate a fully consistent model.We stress that this issue is not specific to the Pauli-Villars scheme. All schemes that regularize the sum,log ( D ) + log ( D ) but not the difference, log ( D ) − log ( D ) will require the specification (53). Since onlythe polarized, i.e. spin dependent, structure functions are effected, this issue has not shown up whencomputing the pion structure function.For the imaginary part of the action the expression analogous to Eq. (52) reads A ( v ) Λ ,I = − i N C (cid:26)(cid:16) − D ( π ) D ( π ) (cid:17) − (cid:104) Q v / ( ∂ / ) − v / D ( π ) + D ( π ) ( v / ( ∂ / ) − v / ) Q (cid:105)(cid:27) = i N C (cid:26)(cid:16) D ( π ) (cid:17) − Q v / ( ∂ / ) − v / + (cid:16) D ( π ) (cid:17) − ( v / ( ∂ / ) − v / ) Q (cid:27) . (54)Again, it is understood that the large photon momentum runs only through the operators in squarebrackets in the first equation. Note that in the unregularized case ( Λ i ≡
0) the contributions associatedwith D would cancel in the sum of Eqs. (52) and (54) leaving A ( v ) = A ( v ) Λ ,R + A ( v ) Λ ,I = i N C (cid:26)(cid:16) D ( π ) (cid:17) − (cid:104) Q v / ( ∂ / ) − v / (cid:105)(cid:27) + regularization terms , (55)and the adjustment, Eq. (53) would not be efficacious. Expanding this expression to quadratic order inthe pseudoscalar field P produces the standard ‘handbag’ diagram with the propagators connecting thequark-pion and quark-photon vertices [36]. In particular, there are no isospin violating terms of the formtr ( Pv / Pv / ) .In the next step we will detail the calculation of the leading N C contribution from the polarizedvacuum (Dirac sea) to the nucleon structure functions. The contribution of the distinct valence level, willlater be added as for g a in Eqs. (45) and (46). For the NJL soliton model this valence quark contribution has ersion January 7, 2021 submitted to Symmetry
18 of 40 been thoroughly discussed in Refs. [48,49]. In other models, like the MIT bag model [50], the calculation isquite similar [51].The above discussion and definition of the structure functions (form factors) in the hadron tensor wasbased on translational invariance. To apply it to a soliton configuration we need to restore translationalinvariance. This is accomplished by introducing a collective coordinate, (cid:126) R , which describes the positionof a soliton (nucleon) [52] with its momentum (cid:126) p conjugate to this collective coordinate , i.e. (cid:104) (cid:126) R | (cid:126) p (cid:105) = √ E exp (cid:16) i (cid:126) R · (cid:126) p (cid:17) . Here E = (cid:113) (cid:126) p + M N denotes the nucleon energy. The Compton amplitude is thenobtained by taking the pertinent matrix element and averaging over the position of the soliton, T ab µν = M N (cid:90) d ξ (cid:90) d R e i q · ξ (cid:68) p , s (cid:12)(cid:12)(cid:12) T (cid:110) J a µ ( ξ − R ) J b † ν ( − R ) (cid:111) (cid:12)(cid:12)(cid:12) p , s (cid:69) = M N (cid:90) d ξ (cid:90) d ξ e i q · ( ξ − ξ ) (cid:68) s (cid:12)(cid:12)(cid:12) T (cid:110) J a µ ( ξ ) J b † ν ( ξ ) (cid:111) (cid:12)(cid:12)(cid:12) s (cid:69) . (56)Here we have made use of the fact that the initial and final nucleon states not only have identical momentabut are actually considered in the rest frame. For simplicity we will treat ξ and R as four-vectors notingthat their temporal components vanish, ξ = R =
0. The spin-isospin matrix elements will be evaluatedin the space of the collective coordinates A , which have been introduced in Eq. (35).To see how Cutkosky’s rule works in the soliton sector it instructive to briefly (and only formally)consider the leading N C contribution in the unregularized case T (cid:8) J µ ( ξ ) J ν ( ξ ) (cid:9) = i N C (cid:40) (cid:16) − D ( π ) (cid:17) − Q (cid:104) γ µ δ ( ˆ x − ξ ) ( ∂ / ) − γ ν δ ( ˆ x − ξ )+ γ ν δ ( ˆ x − ξ ) ( ∂ / ) − γ µ δ ( ˆ x − ξ ) (cid:105)(cid:41) . (57)Here ˆ x refers to the position operator. The above functional trace is computed by using a plane-wave basisfor the operator in the square brackets while the matrix elements of D ( π ) are evaluated employing theeigenfunctions Ψ α of the Dirac Hamiltonian (32): T µν ( q ) = − M N N C (cid:90) d ω π ∑ α (cid:90) d ξ (cid:90) d ξ (cid:90) d k ( π ) e i ξ ( q + k ) e − i ( (cid:126) ξ − (cid:126) ξ ) · ( (cid:126) q + (cid:126) k ) k + i (cid:101) × ω + (cid:101) α ω − (cid:101) α + i (cid:101) (cid:68) N (cid:12)(cid:12)(cid:12)(cid:40) Ψ α ( (cid:126) ξ ) Q A γ µ k / γ ν Ψ α ( (cid:126) ξ ) e i ξ ω − Ψ α ( (cid:126) ξ ) Q A γ ν k / γ ν Ψ α ( (cid:126) ξ ) e − i ξ ω (cid:41)(cid:12)(cid:12)(cid:12) N (cid:69) + O (cid:18) N C (cid:19) . (58)The dependence on the collective coordinates is contained in Q A = A † Q A . We clearly recognize the twopropagators, one in the massless plane wave basis and the other in the soliton background. Cutkosky’s This procedure is common to all soliton models when e.g. computing form factors [53].ersion January 7, 2021 submitted to
Symmetry
19 of 40 rule produces respective δ -functions − π i δ ( k ) and − π i δ ( ω − (cid:101) α ) . We perform the frequency integral,write t = ξ and employ the prescription from Eq. (53) so that the hadron tensor becomes W µν = M N N C ∑ α sign ( (cid:101) α ) (cid:90) dt (cid:90) d ξ (cid:90) d ξ (cid:90) d k ( π ) e i t ( q + k ) e − i ( (cid:126) ξ − (cid:126) ξ ) · ( (cid:126) q + (cid:126) k ) δ (cid:16) k (cid:17) k ρ × (cid:68) N (cid:12)(cid:12)(cid:12) S µρνσ (cid:40) Ψ α ( (cid:126) ξ ) Q A γ σ Ψ α ( (cid:126) ξ ) e i (cid:101) α t − Ψ α ( (cid:126) ξ ) Q A γ σ Ψ α ( (cid:126) ξ ) e − i (cid:101) α t (cid:41) − i (cid:101) µρνσ (cid:40) Ψ α ( (cid:126) ξ ) Q A γ σ γ Ψ α ( (cid:126) ξ ) e i (cid:101) α t + Ψ α ( (cid:126) ξ ) Q A γ σ γ Ψ α ( (cid:126) ξ ) e − i (cid:101) α t (cid:41)(cid:12)(cid:12)(cid:12) N (cid:69) + O (cid:18) N C (cid:19) . (59)In the above we have four contributions, two for each the unpolarized ( S µρνσ ) and polarized ( (cid:101) µρνσ )components. One of the two components propagates from ξ to ξ and the other in the opposite direction.Typically they are denoted particle and anti-particle distributions. Note, however, that in the present case (cid:101) α may have either sign so that both particle and anti-particles spinors contribute in all terms.In deriving Eq. (59) only the pole from ω = + (cid:101) α contributed. That will be different whenregularization is accounted for. We display the result without further derivation as the calculationfor the fully regularized scenario goes along the same lines as above T µν ( q ) = − M N N C (cid:90) d ω π ∑ α (cid:90) dt (cid:90) d ξ (cid:90) d ξ (cid:90) d k ( π ) e i ( q + k ) t e − i ( (cid:126) q + (cid:126) k ) · ( (cid:126) ξ − (cid:126) ξ ) k + i (cid:101) × (cid:68) N (cid:12)(cid:12)(cid:12)(cid:40) (cid:104) e i ω t Ψ † α ( (cid:126) ξ ) β Q A γ µ k / γ ν Ψ α ( (cid:126) ξ ) − e − i ω t Ψ † α ( (cid:126) ξ ) β Q A γ ν k / γ µ Ψ α ( (cid:126) ξ ) (cid:105) f + α ( ω ) (60) + (cid:104) e i ω t Ψ † α ( (cid:126) ξ ) Q A ( γ µ k / γ ν ) β Ψ α ( (cid:126) ξ ) − e − i ω t Ψ † α ( (cid:126) ξ ) Q A ( γ ν k / γ µ ) β Ψ α ( (cid:126) ξ ) (cid:105) f − α ( ω ) (cid:41)(cid:12)(cid:12)(cid:12) N (cid:69) + O (cid:18) N C (cid:19) ,with the spectral functions f ± α ( ω ) = ∑ i = c i ω ± (cid:101) α ω − (cid:101) α − Λ i + i (cid:101) ± ω ± (cid:101) α ω − (cid:101) α + i (cid:101) . (61)The first term in these spectral functions arises from the regularized real part, and the second from theunregularized imaginary part. Without regularization f + α ( ω ) ∼ ( ω + (cid:101) α ) ω − (cid:101) α + i (cid:101) and f − α ( ω ) ∼ πδ ( q + k ± ω ) which we subsequently useto integrate k . Then the δ -function for the absorptive part of the Compton amplitude is δ (( q ± ω ) − | (cid:126) k | ) .To perform the spatial integrals we define the Fourier transform of the single particle wave-functions as (cid:101) Ψ α ( (cid:126) p ) = (cid:90) d ξ π Ψ α ( (cid:126) ξ ) e i (cid:126) ξ · (cid:126) p (62) The single particle wave-functions are parity eigenfunctions so that spatial reflections can be compensated by factors of β .ersion January 7, 2021 submitted to Symmetry
20 of 40 and get W µν ( q ) = i M N N C π (cid:90) d ω π ∑ α (cid:90) d k × (cid:68) N (cid:12)(cid:12)(cid:12)(cid:40)(cid:104) (cid:101) Ψ † α ( (cid:126) q + (cid:126) k ) Q A βγ µ k / γ ν (cid:101) Ψ α ( (cid:126) q + (cid:126) k ) δ ( | (cid:126) k | − ( q + ω ) ) − (cid:101) Ψ † α ( (cid:126) q + (cid:126) k ) Q A γ ν k / γ µ β (cid:101) Ψ α ( (cid:126) q + (cid:126) k ) δ ( | (cid:126) k | − ( q − ω ) ) (cid:105) f + α ( ω ) (cid:12)(cid:12)(cid:12) p + (cid:104) (cid:101) Ψ † α ( (cid:126) q + (cid:126) k ) Q A ( γ µ k / γ ν ) β (cid:101) Ψ α ( (cid:126) q + (cid:126) k ) δ ( | (cid:126) k | − ( q + ω ) ) − (cid:101) Ψ † α ( (cid:126) q + (cid:126) k ) Q A β ( γ ν k / γ µ ) (cid:101) Ψ α ( (cid:126) q + (cid:126) k ) δ ( | (cid:126) k | − ( q − ω ) ) (cid:105) f − α ( ω ) (cid:12)(cid:12)(cid:12) p (cid:41)(cid:12)(cid:12)(cid:12) N (cid:69) , (63)where, again, we only wrote the leading N C term. An example for the pole extraction is (cid:32) ∑ i = c i ω − (cid:101) α − Λ i + i (cid:101) (cid:33) p = ∑ i = c i − i πω α [ δ ( ω + ω α ) + δ ( ω − ω α )] , with ω α = (cid:113) (cid:101) α + Λ i . (64)To get an expression that looks like a bilocal and bilinear distribution function we shift the integrationvariable (cid:126) p = (cid:126) q + (cid:126) k and recognize that the single particle wave-functions will have support only for small (cid:126) p ,as compared to the large momenta in (cid:126) q . This allows us to replace k / by − q / in the Bjorken limit (recall that k = − q ∓ ω from the t integral) for the Dirac matrices sandwiched between the spinors. Furthermore | (cid:126) k | − ( q ± ω ) = | (cid:126) p − (cid:126) q | − ( q ± ω ) = (cid:126) p − (cid:126) p · ˆ n | (cid:126) q | + | (cid:126) q | − ( q ± ω ) −→ − | (cid:126) q | [ (cid:126) p · ˆ n − ( M N x ∓ ω )] .Here ˆ n is the unit vector in the direction of the spatial photon momentum (cid:126) q . Then W µν ( q ) = i M N N C π (cid:90) d ω π ∑ α (cid:90) d p × (cid:68) N (cid:12)(cid:12)(cid:12)(cid:40)(cid:104) (cid:101) Ψ † α ( (cid:126) p ) Q A βγ µ n / γ ν (cid:101) Ψ α ( (cid:126) p ) δ ( (cid:126) p · ˆ n − ( M N x − ω )) − (cid:101) Ψ † α ( (cid:126) p ) Q A γ ν n / γ µ β (cid:101) Ψ α ( (cid:126) p ) δ ( (cid:126) p · ˆ n − ( M N x + ω )) (cid:105) f + α ( ω ) (cid:12)(cid:12)(cid:12) p + (cid:104) (cid:101) Ψ † α ( (cid:126) p ) Q A ( γ µ n / γ ν ) β (cid:101) Ψ α ( (cid:126) p ) δ ( (cid:126) p · ˆ n − ( M N x − ω )) − (cid:101) Ψ † α ( (cid:126) p ) Q A β ( γ ν n / γ µ ) (cid:101) Ψ α ( (cid:126) p ) δ ( (cid:126) p · ˆ n − ( M N x + ω )) (cid:105) f − α ( ω ) (cid:12)(cid:12)(cid:12) p (cid:41)(cid:12)(cid:12)(cid:12) N (cid:69) , (65)where n µ = (
1, ˆ n ) µ is a light-like vector. Eq. (65) is well suited for our numerical simulations in Section 6, inparticular when treating the δ -functions by averaging the directions of ˆ n [19]. However, the similarity withdistribution functions is more apparent when returning to coordinate space and writing the δ -functions asintegrals of exponential functions W ( s ) µν ( q ) = i M N N C (cid:90) d ω π ∑ α (cid:90) d ξ (cid:90) d λ π e i M n x λ ersion January 7, 2021 submitted to Symmetry
21 of 40 × (cid:68) N (cid:12)(cid:12)(cid:12)(cid:40)(cid:104) Ψ α ( (cid:126) ξ ) Q A γ µ n / γ ν Ψ α ( (cid:126) ξ + λ ˆ n ) e − i λω − Ψ α ( (cid:126) ξ ) Q A γ ν n / γ µ Ψ α ( (cid:126) ξ − λ ˆ n ) e i λω (cid:105) f + α ( ω ) (cid:12)(cid:12)(cid:12) p + (cid:104) Ψ α ( (cid:126) ξ ) Q A ( γ µ n / γ ν ) Ψ α ( (cid:126) ξ − λ ˆ n ) e − i λω − Ψ α ( (cid:126) ξ ) Q A ( γ ν n / γ µ ) Ψ α ( (cid:126) ξ + λ ˆ n ) e i λω (cid:105) f − α ( ω ) (cid:12)(cid:12)(cid:12) p (cid:41)(cid:12)(cid:12)(cid:12) N (cid:69) , (66)where we have added the superscript on W ( s ) µν ( q ) to clarify that Eq. (66) represents the vacuum (Dirac sea)component only. The valence component is most conveniently obtained by restricting the sum to α = vand omitting regularization W ( v ) µν ( q ) = i [ + sign ( (cid:101) v )] M N N C (cid:90) d ξ (cid:90) d λ π e i M n x λ (cid:68) N (cid:12)(cid:12)(cid:12)(cid:40)(cid:104) Ψ v ( (cid:126) ξ ) Q A γ ν n / γ µ Ψ v ( (cid:126) ξ − λ ˆ n ) e i λ(cid:101) v − Ψ v ( (cid:126) ξ ) Q A γ µ n / γ ν Ψ v ( (cid:126) ξ + λ ˆ n ) e − i λ(cid:101) v (cid:105)(cid:41)(cid:12)(cid:12)(cid:12) N (cid:69) . (67)Eqs. (66) and (67) indeed have the form of bilocal and bilinear quark distributions. However theseare the distributions for the quarks in the chiral model interacting self-consistently with the soliton. Sofar, no connection to distributions in QCD has been incorporated; our calculation is solely based on theelectromagnetic interaction within the chiral model. Several features needed consideration to arrive at anexpression of the form of distributions. Most importantly and, of course, not surprisingly the Bjorken limitwas implemented. In addition the one of the two propagators that occur in the Compton amplitude istaken to be that of a free massless fermion, while the other contains all information about the soliton thatresembles the nucleon. Again, this separation is an indirect consequence of the Bjorken limit. Furthermorewe made use of the fact that the (momentum space) quark wave-functions only have support at momentathat are tiny compared to the momentum of the exchanged virtual photon. Finally we stress that theappearance of single distribution functions in Eq. (66) is kind of deceptive as the spectral functions f ( ± ) α ( ω ) pick up more than a single pole.In Section 4 we have computed that axial charge, g a , of the nucleon. It is the prime example to seehow sum rules work in the presence of regularization. The Bjorken sum rule [47] relates that charge tothe x -integral of the isovector combination of longitudinal polarized nucleon structure functions g ( x ) for proton and neutron. These functions are obtained from the anti-symmetric component of the hadrontensor W ( s ,A ) µν = − M N N C (cid:101) µρνσ n ρ (cid:90) d ω π ∑ α (cid:90) d ξ (cid:90) d λ π e i M N x λ (cid:32) ∑ i = c i ω + (cid:101) α ω − (cid:101) α − Λ i + i (cid:101) (cid:33) p × (cid:68) N (cid:12)(cid:12)(cid:12) Ψ α ( (cid:126) ξ ) Q A γ σ γ Ψ α ( (cid:126) ξ + λ ˆ n ) e − i ωλ + Ψ α ( (cid:126) ξ ) Q A γ σ γ Ψ α ( (cid:126) ξ − λ ˆ n ) e i ωλ (cid:12)(cid:12)(cid:12) N (cid:69) . (68)The spectral function is fully regularized because it originates from f + α ( ω ) − f − α ( − ω ) = ∑ i = c i ω + (cid:101) α − ( − ω − (cid:101) α ) ω − (cid:101) α − Λ i + i (cid:101) + ω + (cid:101) α + ( − ω − (cid:101) α ) ω − (cid:101) α + i (cid:101) = ∑ i = c i ω + (cid:101) α ω − (cid:101) α − Λ i + i (cid:101) . ersion January 7, 2021 submitted to Symmetry
22 of 40
Here the prescription from Eq. (53) has had a major impact. Without this specification the relative signbetween the spectral functions would have been positive resulting in the spectral function ( ω + (cid:101) α ) / ( ω − (cid:101) α + i (cid:101) ) . In that case W ( s ,A ) µν would have to be associated with unregularized imaginary part of the actionwhich is not compatible with the sum rules. The reason is that the leading order (cid:16) in N C (cid:17) contribution tothe axial charges stems from the regularized real part of the action.Taking ˆ n = ˆ e and the projection operator given in Tab. 1 we find for the Dirac sea component of thelongitudinal polarized structure function g ( s ) ( x ) = − i M N N C (cid:68) N (cid:12)(cid:12)(cid:12) I (cid:12)(cid:12)(cid:12) N (cid:69) (cid:90) d ω π ∑ α (cid:90) d ξ (cid:90) d λ π e iM N x λ (cid:32) ∑ i = c i ω + (cid:101) α ω − (cid:101) α − Λ i + i (cid:101) (cid:33) p × (cid:104) Ψ † α ( (cid:126) ξ ) τ ( − α ) γ Ψ α ( ξ + λ ˆ e ) e − i ωλ + Ψ † α ( (cid:126) ξ ) τ ( − α ) γ Ψ α ( ξ − λ ˆ e ) e i ωλ (cid:105) , (69)where we have substituted the matrix element (40) of the collective coordinates, A , sandwiched betweennucleon states. To establish a sum rule we first note that 0 ≤ x < ∞ . The upper bound is not unity becausethe soliton breaks translational invariance. Eventually that will be accounted for by boosting the solitonto the infinite momentum frame [54], as will be discussed in Subsection 6.3. Furthermore the two termsin Eq. (69) are related by λ ↔ − λ which allows us to integrate only one of them but over − ∞ < x < ∞ thereby producing π M N δ ( λ ) . From parity conservation we have (cid:82) d ξ Ψ † α ( (cid:126) ξ ) τ γ Ψ α ( ξ ) = (cid:18) ω + (cid:101) α ω − (cid:101) α − Λ + i (cid:101) (cid:19) p = − i π(cid:101) α (cid:112) (cid:101) α + Λ (cid:20) δ (cid:18) ω + (cid:113) (cid:101) α + Λ (cid:19) + δ (cid:18) ω − (cid:113) (cid:101) α + Λ (cid:19)(cid:21) − i π (cid:20) δ (cid:18) ω + (cid:113) (cid:101) α + Λ (cid:19) − δ (cid:18) ω − (cid:113) (cid:101) α + Λ (cid:19)(cid:21) . (70)Because of δ ( λ ) there is no other dependence on ω in Eq. (69) and thus the second square bracket in Eq. (70)vanishes when integrating (cid:82) ∞ − ∞ dx g ( s ) ( x ) . Therefore the vacuum contribution to the Bjorken sum rule ( p and n are proton and neutron, respectively) (cid:90) ∞ dx (cid:16) g ( s ,p ) ( x ) − g ( s ,n ) ( x ) (cid:17) = g ( s ) A (71)is immediately verified from Eq. (44) after taking care of the isospin matrix elements of the nucleon.Adding the valence level component to this sum rule is a trivial simplification of the calculation leading toEq. (71).The above example for the verification of a sum rule is (almost) general. The symmetries under λ ↔ − λ extend the x integral over whole real axis, not only the positive half-line. That integral thenproduces δ ( λ ) which turns the bilocal matrix elements into the expectation values that occur in theexpressions for the static properties that occur in the particular sum rule. Then the sum rule is verified levelby level, i.e. separately for each term in ∑ α . The one exception is the momentum sum rule which involvesthe isoscalar component of the unpolarized structure function f ( x ) . When adapting the calculation of theBjorken sum rule to the unpolarized structure function f ( x ) , the integral (cid:82) dx x f ( x ) produces the fermion ersion January 7, 2021 submitted to Symmetry
23 of 40 part of the classical soliton energy in Eq. (33). However, there is an additional contribution proportionalto [ + sign ( (cid:101) v )] (cid:90) d ξ Ψ †v ( (cid:126) ξ ) (cid:126) α · (cid:126) ∂ Ψ v ( (cid:126) ξ ) − ∑ i = c i ∑ α (cid:101) α (cid:113) (cid:101) α + Λ i (cid:90) d ξ Ψ † α ( (cid:126) ξ ) (cid:126) α · (cid:126) ∂ Ψ α ( (cid:126) ξ ) and the sum rule is only verified when this piece vanishes. One shows that this is indeed the case byrecognizing that (cid:126) α · (cid:126) ∂ ∝ (cid:104) (cid:126) ξ · (cid:126) ∂ , h (cid:105) − m (cid:16) (cid:126) ξ · (cid:126) ∂ U ( ξ ) (cid:17) so that the matrix elements in that un-wanted contribution are those of the dilatation operator acting onthe soliton. In turn the above sum is the change in energy obtained when squeezing or stretching thesoliton infinitesimally. As the soliton minimizes the energy, this change must indeed be zero [17,19] . Wemust thus keep in mind that the momentum sum rule only works when summing all levels.For completeness (and an attempt to frighten the reader) we display the Bjorken limit of the hadrontensor including the next to leading order term for the expansion in N C , W ( s ) µν Bj −→ i M N N C (cid:90) d ω π ∑ α (cid:90) d ξ (cid:90) d λ π e i M N x λ (cid:68) N (cid:12)(cid:12)(cid:12) (72) × (cid:40)(cid:104) Ψ α ( (cid:126) ξ ) Q A γ µ n / γ ν Ψ α ( (cid:126) ξ + λ ˆ n ) e − i λω − Ψ α ( (cid:126) ξ ) Q A γ ν n / γ µ Ψ α ( (cid:126) ξ − λ ˆ n ) e i λω (cid:105) f + α ( ω ) (cid:12)(cid:12)(cid:12) p + (cid:104) Ψ α ( (cid:126) ξ ) Q A ( γ µ n / γ ν ) Ψ α ( (cid:126) ξ − λ ˆ n ) e − i λω − Ψ α ( (cid:126) ξ ) Q A ( γ ν n / γ µ ) Ψ α ( (cid:126) ξ + λ ˆ n ) e i λω (cid:105) f − α ( ω ) (cid:12)(cid:12)(cid:12) p + i λ (cid:104) Ψ α ( (cid:126) ξ ) (cid:126) τ · (cid:126) Ω Q A γ µ n / γ ν Ψ α ( (cid:126) ξ + λ ˆ n ) e − i λω + Ψ α ( (cid:126) ξ ) Q A (cid:126) τ · (cid:126) Ω γ ν n / γ µ Ψ α ( (cid:126) ξ − λ ˆ n ) e i λω (cid:105) f + α ( ω ) (cid:12)(cid:12)(cid:12) p + i λ (cid:104) Ψ α ( (cid:126) ξ ) (cid:126) τ · (cid:126) Ω Q A ( γ µ n / γ ν ) Ψ α ( (cid:126) ξ − λ ˆ n ) e − i λω + Ψ α ( (cid:126) ξ ) Q A (cid:126) τ · (cid:126) Ω ( γ ν n / γ µ ) Ψ α ( (cid:126) ξ + λ ˆ n ) e i λω (cid:105) f − α ( ω ) (cid:12)(cid:12)(cid:12) p + ∑ β (cid:104) α | (cid:126) τ · (cid:126) Ω | β (cid:105) (cid:32)(cid:104) Ψ β ( (cid:126) ξ ) Q A γ µ n / γ ν Ψ α ( (cid:126) ξ + λ ˆ n ) e − i λω − Ψ β ( (cid:126) ξ ) Q A γ ν n / γ µ Ψ α ( (cid:126) ξ − λ ˆ n ) e i λω (cid:105) g + αβ ( ω ) (cid:12)(cid:12)(cid:12) p + (cid:104) Ψ β ( (cid:126) ξ ) Q A ( γ µ n / γ ν ) Ψ α ( (cid:126) ξ − λ ˆ n ) e − i λω − Ψ β ( (cid:126) ξ ) Q A ( γ ν n / γ µ ) Ψ α ( (cid:126) ξ + λ ˆ n ) e i λω (cid:105) g − αβ ( ω ) (cid:12)(cid:12)(cid:12) p (cid:33)(cid:41)(cid:12)(cid:12)(cid:12) N (cid:69) ,with the spectral functions g ± αβ ( ω ) = ∑ i = c i ( ω ± (cid:101) α )( ω ± (cid:101) β ) + Λ i ( ω − (cid:101) α − Λ i + i (cid:101) )( ω − (cid:101) β − Λ i + i (cid:101) ) ± ( ω ± (cid:101) α )( ω ± (cid:101) β )( ω − (cid:101) α + i (cid:101) )( ω − (cid:101) β + i (cid:101) ) . (73) The factor x under the integral is written as a derivative with respect to λ . Integrating by parts and averaging over angles turnsthis into the expectation value of (cid:126) α · (cid:126) ∂ . There is a (numerically negligible) complication due to chiral symmetry breaking: for m π (cid:54) = O (cid:16) N C (cid:17) corrections, Eq. (39), which are not contained in this structure function. We also note that the sumrule actually yields E tot M N −
1, which does not vanish as we have defined the hadron tensor to contain the physical mass parameter.Nevertheless this sum rule is perfectly suited to test the numerical simulation.ersion January 7, 2021 submitted to
Symmetry
24 of 40
The subleading N C terms contain the angular velocity, Eq. (36). They arise from the expansions (aftertransforming | ω , β (cid:105) → A | ω , β (cid:105) ) (cid:104) ω , α | (cid:16) D ( π ) (cid:17) − | ω , β (cid:105) = δ αβ ω − (cid:101) α + ω − (cid:101) α (cid:104) α | (cid:126) τ · (cid:126) Ω | β (cid:105) ω − (cid:101) β + O (cid:16) (cid:126) Ω (cid:17) (74)and (cid:104) t , (cid:126) ξ | A ( ˆ t ) | ω , α (cid:105) = A ( t ) e − i ω t Ψ α ( (cid:126) ξ ) = A ( ) (cid:20) + i t (cid:126) τ · (cid:126) Ω (cid:21) e − i ω t Ψ α ( (cid:126) ξ ) + O (cid:16) (cid:126) Ω (cid:17) . (75)The explicit appearance of the time variable is treated in the context of the Fourier transform t e i q t = − i ∂∂ q e i q t while (in the nucleon rest frame) x = − − q + (cid:126) q M N q allows us to write ∂∂ q = ∂ x ∂ q ∂∂ x with ∂ x ∂ q = − M N − q xq
20 Bj −→ − M N .This clarifies that the factors of i λ in Eq. (72) originated from the explicit appearance of the time variablevia the derivative with respect to the Bjorken variable x .Again, Eq. (72) is the vacuum contribution. The valence part is most easily obtained by substitutingthe cranked valence level wave-function Ψ ( rot ) v ( (cid:126) r , t ) = (cid:40) Ψ v ( (cid:126) r ) + ∑ α (cid:54) = v Ψ α ( (cid:126) r ) (cid:104) α | (cid:126) τ · (cid:126) Ω | v (cid:105) (cid:101) v − (cid:101) α (cid:41) (76)into Eq. (67) and taking care of the bilocal dependence on time as in Eq. (75).In this chapter we have reviewed the formal derivation of the hadron tensor for a chiral quark solitonmodel starting from the electromagnetic coupling before bosonization and making ample use of theBjorken limit. We have detailed the case of the longitudinal polarized structure function to illuminate thecalculational principle and verify the relevant sum rule. Detailed formulas for other structure functionsare derived and presented in Refs. [17,18].
6. Numerical results
The results discussed in this section are mostly taken from Ref. [18]. There are several steps until wecan perform a sensible comparison with experimental data. First we numerically simulate the analyticresults from the previous section. This produces structure functions that we call rest frame (RF) structurefunctions. We will present the results for the RF structure functions in the following two subsections.These structure functions have the unwanted feature that they do not vanish for | x | >
1. We will thereforebriefly describe a formalism [54] to boost the soliton to the infinite momentum frame (IMF). In the IMFthe structure functions indeed vanish for | x | >
1. That formalism is essentially adapted from a similarstudy [55] of the MIT bag model in D = +
1. This adoption is made possible as in the Bjorken limit itsuffices to restore Lorentz invariance in direction of the (large) photon momentum only. Once support ofthe structure functions is confined to | x | < ersion January 7, 2021 submitted to Symmetry
25 of 40 the spinors (and thus the sum rules). A typical simulation takes several CPU days/weeks on a standarddesktop PC. In related work [19,21] the expansion coefficients defined after Eq. (31) were directly used.Since they are discrete, some smoothing procedure was needed in that treatment of the momentum spacewave-functions.We will refrain from presenting lengthy formulas as, e.g. the extremely bulky expressions involvingthe Fourier transforms of the radial functions in Ψ α [18]. Rather we focus on explaining the treatmentof the pole terms without going into too much details. This treatment is non-trivial and interferes withregularization, the central topic of this review and therefore deserves closer consideration. Below wetherefore describe some key ingredients that are relevant for all our calculations.As in Ref. [19] we treat the Dirac δ -functions in Eq. (65) by averaging over ˆ n , that is, we replace these δ -functions by 14 π (cid:90) d Ω ˆ n δ ( E + (cid:126) p · (cid:126) n ) = | (cid:126) p | θ ( | (cid:126) p | − | E | ) (77)and generalizations thereof that contain additional factors of ˆ n under the solid angle integral. We thenneed to evaluate expressions like (in sums over single particle levels α , but omitting that index) (cid:90) d ω π (cid:90) d λ π ∑ i = c i (cid:32) ω + (cid:101)ω − (cid:101) − Λ i + i (cid:101) (cid:33) p (cid:90) d p (cid:101) Ψ † ( (cid:126) p ) (cid:101) Ψ ( (cid:126) p ) e i ( M N x − ˆ n · (cid:126) p ) λ e i ωλ . (78)Defining ω i = (cid:113) (cid:101) + Λ i we have (see also Eq. (64)) (cid:90) d ω π ∑ i = c i (cid:32) ω + (cid:101)ω − (cid:101) − Λ i + i (cid:101) (cid:33) p e i ωλ = − i2 ∑ i = c i | ω i | (cid:104) ( (cid:101) + ω i ) e i ω i λ + ( (cid:101) − ω i ) e − i ω i λ (cid:105) (79)and therefore (cid:90) d λ π e i ( M N x − ˆ n · (cid:126) p ) λ (cid:90) d ω π ∑ i = c i (cid:32) ω + (cid:101)ω − (cid:101) − Λ i + i (cid:101) (cid:33) p e i ωλ = − i2 ∑ i = c i | ω i | [( (cid:101) + ω i ) δ ( M N x − ˆ n · (cid:126) p + ω i ) + ( (cid:101) + ω i ) δ ( M N x − ˆ n · (cid:126) p − ω i )] −→ − i4 | (cid:126) p | ∑ i = c i | ω i | [( (cid:101) + ω i ) θ ( | (cid:126) p | − | M N x + ω i | ) + ( (cid:101) − ω i ) θ ( | (cid:126) p | − | M N x − ω i | )] , (80)where the arrow denotes the averaging procedure from Eq. (77). Note that, due to the step function, thecut-off also appears as the lower boundary of the momentum integral and we treat these boundariesaccording to the single cut-off prescription, Eq. (15) ∑ i = c i (cid:90) ∞ | M N x + ω i | pdp f ( p , ω i ) = (cid:90) ∞ | M N x + (cid:101) | pdp f ( p , (cid:101) ) − (cid:90) ∞ | M N x + √ (cid:101) + Λ | pdp f ( p , (cid:112) (cid:101) + Λ )+ Λ (cid:90) ∞ | M N x + √ (cid:101) + Λ | pdp ∂∂ Λ f ( p , (cid:112) (cid:101) + Λ ) Here f ( p , ω i ) contains angular matrix elements like (cid:82) d Ω (cid:126) p (cid:101) Ψ † ( (cid:126) p ) (cid:101) Ψ ( (cid:126) p ) or (cid:82) d Ω (cid:126) p (cid:101) Ψ † ( (cid:126) p ) (cid:126) α · (cid:126) p (cid:101) Ψ ( (cid:126) p ) multiplied by powers of ω i .ersion January 7, 2021 submitted to Symmetry
26 of 40 − Λ √ (cid:101) + Λ (cid:104) p f ( p , (cid:112) (cid:101) + Λ ) (cid:105) p = | M N x + √ (cid:101) + Λ | . (81) We will not present detailed formulas, except for some leading terms of the N C expansion. We referthe reader to Ref. [18] for more details (even though some factors of π were not written there). As anexample we present the expression for the isoscalar component of the unpolarized RF structure function [ f s ( x )] ∓ I = = M N N c π ∑ α ∑ i = c i (cid:90) ∞ | M N x ± α | p dp (cid:90) d Ω p (cid:40) ± (cid:101) Ψ † α ( (cid:126) p ) (cid:101) Ψ α ( (cid:126) p ) − (cid:101) α (cid:113) (cid:101) α + Λ i M N x ± α p (cid:101) Ψ † α ( (cid:126) p ) ˆ p · (cid:126) α (cid:101) Ψ α ( (cid:126) p ) (cid:41) , (82)where M N x ± α = M N x ± (cid:113) (cid:101) α + Λ i . (83)The superscripts ∓ denote the positive (negative) frequency components which are typically referred to asquark and antiquark distribution. They arise from the two poles (for a particular ω α ) of the δ -functionin Eq. (64) and materialize in the ± ω terms in Eq. (65). The total Dirac sea contribution to the isoscalarunpolarized structure function is the sum [ f s ( x )] I = = [ f s ( x )] − I = + [ f s ( x )] + I = . (84)On first sight it seems as if the first term under the integral in Eq. (82) would not be subject to regularization.This is not the case, as the momentum integral is computed according to Eq. (81). Since the isoscalarunpolarized structure functions are related to the classical energy of the soliton by the momentum sum rule,we must still subtract the analog of this calculation that is obtained by substituting spinor wave-functionsfor Θ =
0. We have numerically checked the sum rule and achieved agreement better than 1%. In viewof the many elaborate elements of the simulation, this is more than satisfactory. We get the valencecontribution from substituting Eq. (76) into the unregularized expression. This then adds [ f v1 ( x )] ∓ I = = − M N N c π [ + sign ( (cid:101) v )] (cid:90) ∞ M N | x ± v | p dp (cid:90) d Ω p (cid:40) ± (cid:101) Ψ †v ( (cid:126) p ) (cid:101) Ψ v ( (cid:126) p ) − M N x ± v p (cid:101) Ψ †v ( (cid:126) p ) ˆ p · (cid:126) α (cid:101) Ψ v ( (cid:126) p ) (cid:41) ,(85)to the positive and negative frequency components of the isoscalar unpolarized structure function. In thiscase there is no need to subtract the Θ = x . We consider this as an artifact of the Θ = N C and does not have any (artificial) Θ = ersion January 7, 2021 submitted to Symmetry
27 of 40 x -0.6-0.4-0.200.20.40.60.8 [f ] I = valence x -10-5051015 vacuum x -5051015 total Figure 2.
Model prediction (with m = x [f ] I = valence x -0.02-0.0100.010.02 vacuum x total Figure 3.
Same as Fig. 2 for the isovector unpolarized structure function. Observe the logarithmic scale forthe Bjorken variable x . In Fig. 4 we display the numerical results for the unpolarized structure function that enters theGottfried sum rule, i.e. f p ( x ) − f n ( x ) = x (cid:104) f p ( x ) − f n ( x ) (cid:105) , as the Callan-Gross relation holds in therest frame, cf. Tab. 1. At large x the vacuum contribution turns slightly negative. Though the valencecontribution is generally dominant, the small negative piece persists in the total contribution of thisstructure function. In Tab. 3, we compare our model prediction for the Gottfried sum rule, S G = (cid:90) ∞ dxx (cid:16) f p − f n (cid:17) , (86)for various constituent quark masses to that of the value extracted from data by the NM Collaboration [56].The agreement is astonishingly good. The integral is almost completely saturated by the valence levelcontribution.In contrast to the isoscalar unpolarized structure function, the isovector part does not undergoregularization. Such an alternating behavior between (un)regularized quantities is well-known for staticproperties [15,16] but it is interesting to see that it also holds for structure functions. Of course, that is aprediction of the formalism. ersion January 7, 2021 submitted to Symmetry
28 of 40 x f - f valence x -0.075-0.05-0.0250 vacuum x total Figure 4.
Model prediction of the unpolarized structure function f p ( x ) − f n ( x ) for the constituent quarkmass of m = Table 3.
The Gottfried sum rule for various values of m . The subscripts ’v’ and ’s’ denote the valence andvacuum contributions, respectively. The fourth column contains their sums. m [ MeV ] [ S G ] v [ S G ] s S G emp. value400 0.214 0.000156 0.214450 0.225 0.000248 0.225 0.235 ± For the polarized structure functions we will only list explicit formulas for the isovector longitudinalpiece which is leading in N C . Essentially this is the Fourier transform of Eq. (69). The vacuum contributionreads [ g s ( x )] ∓ I = = − M N N c π (cid:104) N | I | N (cid:105) ∑ α ∑ i = c i (cid:40) ∓ (cid:90) ∞ | M N x ± α | dp M N x ± α (cid:90) d Ω p (cid:101) Ψ † α ( (cid:126) p ) ˆ p · (cid:126) τγ (cid:101) Ψ α ( (cid:126) p ) (87) − (cid:101) α (cid:113) (cid:101) α + Λ i (cid:90) ∞ | M N x ± α | dp p (cid:34) A ± (cid:90) d Ω p (cid:101) Ψ † α ( (cid:126) p ) (cid:126) τ · (cid:126) σ (cid:101) Ψ α ( (cid:126) p ) + B ± (cid:90) d Ω p (cid:101) Ψ † α ( (cid:126) p ) ˆ p · (cid:126) τ ˆ p · (cid:126) σ (cid:101) Ψ α ( (cid:126) p ) (cid:35)(cid:41) ,where we have introduced the abbreviations, see also Eq. (83), A ± = p (cid:18) − ( M N x ± α ) p (cid:19) , B ± = p (cid:18) ( M N x ± α ) p − (cid:19) . (88)As before, the superscripts denote the positive and negative frequency components. The total Dirac seacontribution to g ( x ) again is the sum of the positive (+) and negative ( − ) frequency components. Thevalence quark contribution to the isovector longitudinal polarized structure function reads [ g v1 ( x )] ∓ I = = M N N c π [ + sign ( (cid:101) v )] (cid:104) N | I | N (cid:105) (cid:40) ∓ (cid:90) ∞ | M N x ± | dp M N x ± v (cid:90) d Ω p (cid:101) Ψ †v ( (cid:126) p ) ˆ p · (cid:126) τγ (cid:101) Ψ v ( (cid:126) p ) − (cid:90) ∞ | M N x ± v | dp p (cid:34) A ± (cid:90) d Ω p (cid:101) Ψ †v ( (cid:126) p ) (cid:126) τ · (cid:126) σ (cid:101) Ψ v ( (cid:126) p ) + B ± (cid:90) d Ω p (cid:101) Ψ †v ( (cid:126) p ) ˆ p · (cid:126) τ ˆ p · (cid:126) σ (cid:101) Ψ v ( (cid:126) p ) (cid:35)(cid:41) . (89) ersion January 7, 2021 submitted to Symmetry
29 of 40 x [ g ] I = valence x vacuum x total Figure 5.
Model prediction ( m = Table 4.
Axial isovector and isoscalar charges for various values of the constituent quark mass m fromintegrating the longitudinal structure functions. Subscripts are as in Tab. 3. Data in parenthesis give thenumerical results as obtained from the coordinate space representation, cf. Eq. (47) and Tab. 2. m [ MeV ] [ g A ] v [ g A ] s g A emp. value [ g A ] v [ g A ] s g A emp. value400 0.734 0.065 0.799 (0.800) 1.2601 0.344 0.0016 0.345 (0.350)450 0.715 0.051 0.766 (0.765) ± ± The numerical results are shown in Fig. 5. The isoscalar counterpart is subleading in N C and we display atypical model prediction in Fig. 6. We have already discussed the Bjorken sum rule for the isovector piece.Its verification serves as a test for the accuracy of the numerical simulation. The isoscalar combinationalso has a sum rule which gives the matrix element of the isoscalar axial current Ψ γ µ γ Ψ . As mentioned,its empirical determination has triggered much of the research on structure functions. Our results forboth sum rules are shown in Tab. 4. We note that the isoscalar axial charge is significantly less than one inagreement with phenomenology of the proton spin puzzle [28]. x [ g ] I = valence x -0.0200.020.04 vacuum x total Figure 6.
Same as Fig. 5 for the isoscalar longitudinal polarized structure functions. ersion January 7, 2021 submitted to
Symmetry
30 of 40 x -0.0100.010.020.03 [ g ] I = valence x vacuum x total Figure 7.
Model prediction of the isoscalar structure function, g , for the constituent quark mass of m = According to the projectors listed in Tab. 1 the transverse polarized structure function g T ( x ) hasmatrix elements similar to those above. From its computation we subsequently identify g ( x ) = g ( x ) − g T ( x ) (90)for both the isoscalar and isoscalar combinations. Typical results are shown in Figs. 7 and 8. Here it occursthat the vacuum piece dominates. However, that is mainly a consequence of cancellations for the valencecontribution via Eq. (90). It is customary to introduce light-cone coordinates x ± = (cid:0) x ± ˆ n · (cid:126) x (cid:1) / √ q − → ∞ and x = − q + p + .As discussed before, the fermion propagator is free and massless in the Bjorken limit. Massless fermionshave the singular function (cid:8) Ψ ( ξ ) , Ψ ( ) (cid:9) = π ∂ / δ ( ξ ) (cid:101) ( ξ ) . This can be used to turn the current-current x -0.0200.020.040.06 [ g ] I = valence x -0.02-0.0100.010.02 vacuum x -0.0200.020.040.06 total Figure 8.
Model prediction of the isovector polarized structure functions, g , frame for the constituentquark mass of m = ersion January 7, 2021 submitted to Symmetry
31 of 40 correlator in the hadron tensor into a matrix element of bilocal bilinear fermion operators [58] (These arefundamental fermion operators, not the eigenfunctions of h in Eq. (32).) f ( x ) = x π (cid:90) d ξ − e − i p + ξ − (cid:104) N | Ψ ( ξ ) γ + Q Ψ ( ) − Ψ ( ) γ + Q Ψ ( ξ ) | N (cid:105) ξ + = ξ ⊥ = .This singles out the coordinate along the photon momentum as the most relevant variable. We will seeshortly that this is indeed realized in the IMF, which also has ξ + = p n , the matrixelements of bilocal bilinear quark operators can be shown to be non-zero only when p + n − ( − x ) p + = x = n and the nucleonare negligible small and/or p + becomes very large. The limit of large p + defines the IMF. It is thereforesuggestive to consider the soliton model structure functions in the IMF as well. To boost the system to theIMF, the collective coordinate method of Eq. (56) for any local object Γ must be extended to Γ ( (cid:126) ξ , ξ ) −→ S ( Λ ) Γ ( (cid:126) ξ (cid:48) − (cid:126) R (cid:48) , ξ (cid:48) ) S − ( Λ ) where ξ (cid:48) µ = (cid:16) Λ − (cid:17) µ ν ξ ν . (91)Here Λ parameterizes a Lorentz transformation and S ( Λ ) is the corresponding generator for Γ .Subsequently, the collective coordinates (cid:126) R are averaged as in Eq. (56).A Lorentz boost with rapidity Ω along the light cone transforms the RF coordinates as p + −→ M N √ Ω and p − −→ M N √ − Ω (92)while the transverse components are left unchanged. The transformation to the IMF is thus characterizedby Ω → ∞ which also implies that Λ − singles out ξ (cid:48)− so that ξ (cid:48) + →
0. In Ref. [54] this transformationwas applied together with the collective coordinate average for bilocal bilinear quark composites like thosein Eq. (59). Essentially that study adapted a two-dimensional MIT bag model calculation [55] to the solitonmodel by ignoring effects on the transverse coordinates as Lorentz covariance is only restored along ˆ n .The result is a simple transformation prescription for the structure functions: f IMF ( x ) = Θ ( − x ) − x f RF ( − ln ( − x )) , (93)where f RF is any of the structure functions like that in Eq. (69) which are obtained from the hadron tensorin the RF according to the calculations in the previous section. Obviously this prescription ensures that thetransformed structure functions have support only in the kinematically allowed interval 0 ≤ x ≤
1. Thusthe structure function f IMF ( x ) is a suitable input for the DGLAP evolution program. In what follows wewill omit the label IMF for the boosted structure functions. Of course, we wish to compare our model predictions with data. In this section we describe theremaining step with focus on the polarized structure functions. All model results presented in thisSubsection have been obtained for the constitutent quark mass m = µ = Q which is thus an adjustable hidden parameter in the ersion January 7, 2021 submitted to Symmetry
32 of 40 approach and can be thought of as the identification scale with QCD. This low mass scale is different fromthe high energy scales, Q at which DIS data are available. To compare with the DIS data, we adopt theleading order Altarelli-Parisi (DGLAP) equations [4] for parton distributions to evolve the model structurefunctions. To apply this formalism we, unfortunately, have to identify the model structure functions withQCD distribution functions of quarks since the chiral model is not renormalizable and does not have arenormalization group equation to sum the leading logs .Let h ( I = ) ( x , t ) be the isovector combination of any twist-2 distribution with t = ln (cid:18) Q Λ QCD (cid:19) . Thechange in momentum scale is governed by the differential equation dh ( I = ) ( x , t ) dt = α s ( t ) π C R ( F ) (cid:90) x dyy P qq ( y ) h ( I = ) (cid:18) xy , t (cid:19) : = α s ( t ) π C R ( F ) P qq ⊗ h ( I = ) ( x , t ) . (94)Here α s ( t ) = πβ t , is the running coupling constant of QCD, in which β = N C − N f and C R ( F ) = N f − N f are combinatoric factors in the QCD renormalization group equation for N f flavors. Most importantly P qq ( y ) is the splitting function that describes the probability of a quark emitting a gluon and a quark withmomentum fraction y . This splitting function and those for the isoscalar combination to be discussedbelow are given in Ref. [4]. The right-hand-side of Eq. (94) serves as the definition of the evolution product" ⊗ ". As initial condition, h ( I = ) ( x , t ( µ )) , to integrate this differential equation we take the distributionsidentified from the boosted structure functions in the IMF. The endpoint of integration is the scale Q atwhich data from experiment are available. We attempt to tune µ to optimize the agreement with thesedata and take the very same identification scale for all evolution calculations.The isoscalar combinations, h ( I = ) ( x , t ) are more complicated. By the pure nature of the quantumnumbers h ( I = ) ( x , t ) mixes with the gluon distribution g ( x , t ) and the evolution equations are coupleddifferential equations dh ( I = ) ( x , t ) dt = α s ( t ) π C R ( F ) (cid:104) P qq ⊗ h ( I = ) ( x , t ) + P qg ⊗ g ( x , t ) (cid:105) dg ( x , t ) dt = α s ( t ) π C R ( F ) (cid:104) P gq ⊗ h ( I = ) ( x , t ) + P gg ⊗ g ( x , t ) (cid:105) . (95)The only sensible identification of the gluon distribution g ( x , t ) is to have it vanish at µ , otherwise sumrules would be violated. This is again an unavoidable (and undesirable) identification of QCD degrees offreedom.We are now in the position to confront the model prediction with data from experiment. For thelongitudinal polarized structure function of the proton this is done in the left panel of Fig. 9. We chose µ = and get a reasonable (though not perfect) match with the data after evolving the boostedstructure function to the scale of the experiment, Q = . Any further fine-tuning of µ has onlymarginal effects. The predictions are obviously in the right ballpark, but deviations clearly emerge indetail. Surprisingly, the RF result appears to match data best. This is an indication that the boost formalismoveremphasizes the low x regime. For the neutron data are available in terms of the helium structurefunction [61] g He1 ( x ) ≈ P n g n ( x ) + P p g p ( x ) − (cid:104) g p ( x ) − g n ( x ) (cid:105) , (96) In Ref. [61] direct neutron data are only given as the ratio g n ( x ) / F ( x ) .ersion January 7, 2021 submitted to Symmetry
33 of 40 x g E143DGLAPIMFRF x -0.04-0.0200.020.040.06 g He RFIMFDGLAPE=4.74GeV, JLABE=5.89GeV, JLAB
Figure 9.
Model prediction for the longitudinal polarized proton structure functions. Left panel: g p ( x ) ; right panel: g He1 ( x ) . These functions are "DGLAP” evolved from µ = to Q = afterbeing projected to the IMF. Data are from Refs. [59,60] for the proton and from Ref. [61] for helium. In thelatter case E refers to the electron energy. x -0.500.511.5 g , E1437.0 , E143DGLAPIMFRF Figure 10.
Model prediction for the polarized proton structure functions g p ( x ) . This function is "DGLAP”evolved from µ = to Q = after being projected to the IMF. Data are from Ref [64]. with P n ≈ P p ≈ − x we find thestructure function to be small and positive while for moderate x the observed negative trough is presentbut somewhat too strong.The evolution of the transverse polarized structure functions is even more complicated because g ( x , t ) is the sum of two terms. One has twist-2 [62] g WW ( x , t ) = − g ( x , t ) + (cid:90) dy y g ( y , t ) (97)and the remainder, g ( x , t ) = g ( x , t ) − g WW ( x , t ) is associated with twist-3. The twist-2 part undergoesthe DGLAP evolution described above. For the twist-3 piece we extract Mellin moments M j ( Q ) = (cid:90) dx x j − g ( x , t ) (98)that scale as M j ( Q ) M j ( µ ) = (cid:20) ln ( µ ) ln ( Q ) (cid:21) γ j − β . (99)So far, only the leading large N C terms of γ j − are known [63]. At the initial scale µ we disentanglethe twist components, evolve them separately to Q , invert the Mellin transformation, and put the twocomponents back together to build g ( x , t ) . The result of this procedure for the proton channel is comparedto available data in Fig. 10. Our estimate produces the main structure seen experimentally: g p ( x , t ) isnegative and small in magnitude at large x and increases substantially as x decreases. ersion January 7, 2021 submitted to Symmetry
34 of 40 x -0.2-0.100.10.2 f - f NMCDGLAPIMF
Figure 11.
Model prediction ( m = µ = to Q = after transformation to the IMF. Data are from Ref. [56]. Twist-3 by itself is interesting as data have been recently reported [61] for the second moment d ( n ) ( Q ) = (cid:90) dx x g ( n ) ( x , t ) (100)in the neutron channel at two different transferred momenta: d ( n ) ( ) = ( − ± ) × − and d ( n ) ( ) = ( − ± ) × − (we added the reported errors in quadrature). Our modelcalculations for m = − × − and − × − , respectively. While the lower Q result matches the observed value, the higher one differs by about three standard deviations. The resultsindicates that the large N C approximation to evolve g requires improvement.Finally we comment on the isovector unpolarized structure function that is compared to data inFig. 11, see also Figs. 3 and 4. Though the negative contribution to f from the Dirac vacuum, cf. Fig. 3,around x = x to obtain f and (ii)when transformed to the IMF because of the Jacobian factor 1/ ( − x ) thereby worsening the agreementwith the experimental data from NMC [56]. To some extend, this dilutes the perfect agreement betweenthe model prediction and data for the Gottfried sum rule, Eq. (86), discussed earlier . Under that integralthe model result arises from cancellations not seen in the empirical structure function [56].
7. Related approaches
One of the major obstacles when computing structure functions within chiral quark soliton modelsis the consistent implementation of the regularization prescription. Various approaches have beenundertaken. The numerical results do not differ significantly as the dominant contribution to the structurefunctions arises from the explicitly occupied valence level (in particular when m (cid:46) m = N C effects and the separation of isoscalar and -vectorcomponents without encountering complicated expressions like those in Eqs. (72) and (73). The results ersion January 7, 2021 submitted to Symmetry
35 of 40 from Sect. 6 show that this is indeed a reasonable approximation for the polarized structure functions;maybe to a lesser extend for the unpolarized structure functions.From Eq. (66) we see that the regularized hadron tensor at leading order in N C is a sum of fourterms, while the unregularized version only has two. Similarly, when acting with the projection operatorsfrom Tab. 1 to extract a certain structure function, the spectral functions f ( ± ) α ( ω ) combine to reduce thenumber of terms that contribute to the hadron tensor to two as well. We have seen that explicitly for thelongitudinal polarized structure function g ( x ) in Eq. (69). These two terms are formally distributionfunctions that take the fermions forward and backward in space time along the direction of the virtualphoton momentum. The (formal) appearance of such distributions is general to all fermion models in theBjorken limit. It is therefore suggestive to consider distribution functions in such models regardless ofwhether or not other peculiarities in the model, like regularization, require more detailed consideration.In this context the authors of Ref. [19] derived two equivalent expressions for unregularized quarkdistribution functions (coefficients adjusted to comply with Eq. (62)) D ( ) ( x ) = π N C M N ∑ α ,occ. (cid:90) d p (cid:102) Ψ α ( (cid:126) p ) n / Γ (cid:101) Ψ α ( (cid:126) p ) δ ( p + (cid:101) α − M N x ) D ( ) ( x ) = − π N C M N ∑ α ,non − occ. (cid:90) d p (cid:101) Ψ α ( (cid:126) p ) n / Γ (cid:101) Ψ α ( (cid:126) p ) δ ( p + (cid:101) α − M N x ) , (101)in the large N C limit. Obviously these expression combine to D ( x ) = (cid:104) D ( ) ( x ) + D ( ) ( x ) (cid:105) = − π N C M N ∑ α sign ( (cid:101) α ) (cid:90) d p (cid:102) Ψ α ( (cid:126) p ) n / Γ (cid:101) Ψ α ( (cid:126) p ) δ ( p + (cid:101) α − M N x ) , (102)for the Dirac sea contribution which, by definition, has the negative energy levels occupied (occ) andthe positive energy levels empty (non-occ). Similarly anti-quark distribution functions are obtained with D ( x ) = − D ( − x ) . Using D ( x ) , D ( x ) and suitable linear combinations of the spin flavor matrices Γ theauthors would then compute the structure functions. Considering, for example, the unregularized versionof of the first term within the square brackets in Eq. (69) and noticing that [ ω + (cid:101) α ] δ ( ω − (cid:101) α ) = sign ( (cid:101) α ) δ ( ω − (cid:101) α ) (103)we observe the very same structure as in D ( x ) . In the notation of Ref. [19] the second term in Eq. (69)represents the antiquark distribution D ( x ) that must be added to complete the structure function. Notunexpectedly, without regularization these approaches are thus equivalent. Ref. [19] performs a twostep regularization for the distributions, first a smoothing function is multiplied in the level sum inEq. (102) with a scale E max . Then the calculation is repeated with a second, larger constituent quark massand the difference is extrapolated to E max → ∞ . That second constituent quark mass conceptually is aPauli-Villars mass, M PV . Its numerical value is determined from the pion decay constant f π as follows:compute the unregularized polarization functions, Eq. (19), that enter f π for both m and M PV , multiplyboth polarization functions by m and M PV , respectively and tune M PV such that the difference is f π /4 N C ,with f π = . Among other questions Equivalent expressions arise from trace identities.
E.g. , ∑ α (cid:101) α = (cid:101) v > (cid:101) v + ∑ α | (cid:101) α | = ∑ α ,occ. (cid:101) α . Whetheror not such identities hold depends on the particular regularization prescription. The caption to Fig. 1 in the second of Ref. [19] suggests that the valence level contribution would also undergo this Pauli-Villarstype subtraction. If correctly interpreted, that seems in contradiction to unit baryon number.ersion January 7, 2021 submitted to
Symmetry
36 of 40 one might ask why should the second calculation have the same smoothing scale E max ; and if different,what is the effect? We also note that a single subtraction does not produce a finite gap equation , Eq. (16),and further obstacles may occur away from the chiral limit when quadratic divergences may occur. Inthe onset we have distinguished between regularized and non-regularized parts in the action, Eq. (13).Any kind of a posteriori regularization faces the dilemma that such a distinction is difficult to implement.There are combinations of distributions that are ultraviolet finite even without regularization. Must theynevertheless undergo regularization? In this context refer to the discussion on the Gottfried sum rule inSection 6. We also note that the restriction to the leading N C terms does not distinguish between isoscalarand -vector components.Most of those early distribution function calculations did not attempt the DGLAP evolution, rathercompared the results with empirical distributions at a low renormalization point, that result from applyingthe inverse of the DGLAP evolution to data [6].In Refs. [20,21] similar calculations for the N C corrections to the unpolarized distributions have beenperformed while Ref. [22] discusses the polarized distributions with subleading N C terms included andalso implements the DGLAP evolution program. Similar to our calculations those authors observe that theDirac sea contribution to the polarized structure functions is almost negligibly small.The extension to three light flavors has also been addressed. These studies were first performed in thevalence level only approximation for the hadron tensor [65] and soon after by formulating distributionsincorporating the a posteriori regularization [27] with a Pauli-Villars mass as described above. Technicallythe main difference is that the collective coordinates are from SU ( ) and that there are eight instead ofthree angular velocities. Furthermore flavor symmetry breaking must be included because the strangequark mass (represented by the pseudoscalar kaon) is much larger than that of the up and down quarks.Of course, that extension allows a closer look at strangeness in the nucleon. In this regard the numericalresults of that model calculation agree with data [66], at least qualitatively.Once the identification of distribution functions is accepted, other processes than DIS, that in QCDare described by various bilocal bilinear quark operators, can also be explored within chiral quark solitonmodels. Let us mention two examples. Transversity distributions complete the description of the nucleonspin [67] and are relevant for the Drell-Yan process [68]. There are two of them which are similar tothe two polarized structure functions: the transverse h T ( x ) and longitudinal h L ( x ) . In the language ofdistributions the relevant bilocal bilinear quark operators are similar to those for the polarized ones,except for different Dirac matrices. Again, transverse and longitudinal refers to the alignment of spin andexternal momentum. In the soliton model these distributions were first estimated in the large N C limitand valence level only approximation [23]. Subsequently N C corrections were included [24] and finallyDirac sea contributions were considered in Refs. [22,25]. Transversity distributions have sum rules withtensor charges, (cid:104) N | Σ ( τ ) | N (cid:105) . These charges can be directly computed in the chiral soliton model withoutany ambiguity from regularizing the Dirac sea component. That component was found to be very small[24] suggesting that the valence level only approximation is reliable for these distributions. This was laterconfirmed by the computation with the a posteriori regularization prescription [22]. The twist-3 distribution e ( x ) , which has a sum rule with the π N − σ -term, has been considered in Refs. [26]. Again, this distributionis not a structure function accessible in DIS but can be extracted from pion photoproduction [69]. Eventhough the relevant bilocal bilinear quark operator is as simple as Ψ ( )( τ ) Ψ ( λ n ) , the actual computationis quite intricate because of a potential δ -function behavior of the isoscalar combination at x =
0. The sum In Ref. [19] this problem is bypassed by postulating a non-zero constituent quark mass in D ( π ) and define the model by thatoperator.ersion January 7, 2021 submitted to Symmetry
37 of 40 rule is only satisfied with the inclusion of such a behavior [70]. The model calculation of Ref. [26] indeedconfirms that singular structure.Let us also briefly comment on the historical development. Ref. [19] mentions that some preliminaryresults on structure functions had been ”announced” in Ref. [71]. But that reference only states that thesecalculations are in progress pointing to [19]. So it seems fair to state that the first results for structurefunctions in a soliton model were presented in Ref. [48] according to the journal received dates thoughthere was some delay of the actual publication.We have seen that, modulo regularization, the matrix elements to be computed are formally the sameas if the bilocal bilinear quark operators were directly transferred from QCD to the model. That is, ratherthan merely taking the NJL model as one for some of the QCD symmetries, it is considered a model forQCD degrees of freedom. This is a frequently adopted point of view, not only for the NJL model, but also, e.g. the MIT bag model [50], in particular in the context of structure functions [51]. Furthermore it opensthe door to explore quark distributions others than those parameterizing (electromagnetic) DIS.
8. Conclusions
The standard model for elementary particles is a gauge theory for leptons, quarks and gauge bosons.To make contact with the world of mesons and baryons, knowledge about their composition in terms ofquarks (and gluons) is inevitable. The binding of the fundamental constituents to mesons and baryons,known as color-confinement in QCD, is a non-perturbative effect. Distribution functions that combineto structure functions parameterize this non-perturbative composition of mesons and baryons. Thesestructure functions cannot be computed from first principles in QCD but are either extracted from empiricaldata, computed in lattice simulations or obtained from some model estimates. The chiral soliton model isone of the many popular and successful models for baryons. Here meson fields are the model degrees offreedom and solitons are (static) solutions to the respective, non-linear fields equations.The calculation of nucleon structure functions in chiral soliton models has been a long issue.Traditional soliton models like the Skyrme model and its extensions by incorporating vector mesonsin addition to the pions face the problem of only representing local quark bilinear combinations. On theother hand, models that carry through the bosonization, that transforms the quark into a meson theory, areplagued by the need for regularization. Here we have reviewed a method that takes regularization seriouslyfrom the initial formulation of the action for the quark model rather than empirically implementingregularization for distribution functions that linearly combine to structure functions. The formal relationbetween structure functions and quark distributions is no longer obvious when regularization is required.We stress that this formulation only identifies chiral symmetries of QCD with no statement on howthe model and QCD quarks relate. Yet all the sum rules that relate integrals of the structure functionsto observables like hadron masses, isospin etc. and that are commonly derived from the probabilityinterpretation of distribution functions are also valid in this approach. The project should thus beconsidered more like a proof of concept rather than attempting precise predictions for the structure functions.The point of departure is a self-interacting chirally symmetric quark model. It is particularlyformulated to make feasible the full process of bosonization. At each step of this calculation regularizationis carefully traced resulting in consistently regularized structure functions. The model is defined suchthat only one part of the bosonized action is regularized in order to maintain the chiral anomaly. Henceit is suggestive that only some of the structure functions will be subject to regularization. The treatmentreviewed here is constructive in the sense that it dictates for which structure function regularizationmust be implemented and for which this is not the case. This goes beyond analyzing whether or not theparticular structure function is ultraviolet convergent. The method is also predictive in case the structure ersion January 7, 2021 submitted to
Symmetry
38 of 40 function does not have a sum rule that is related to a static property with an established regularizationprescription.Regularization, of course, only concerns the vacuum (Dirac sea) contribution to any observable. Inaddition there is always the contribution from the valence level (strongly) bound by the self-consistentsoliton. This level contribution must be included to deal with a unit baryon number object. We haveactually seen that this level contribution is dominant for almost all structure functions except the isoscalarunpolarized combination. For this combination we see a strong enhancement at small Bjorken x . This wasalso seen in the numerical simulation of Ref. [19], though not quite as drastic as in our case. We recallthat the unpolarized isoscalar structure function has a sum rule with the energy, which in soliton modelsis the classical soliton energy. The standard definition of this energy subtracts the zero soliton vacuumcounterpart to get a finite result for the soliton energy and therefore this structure function should undergoan analog subtraction. It is important to note that this energy subtraction has no dynamic effect, i.e. itdoes not enter the field equation for the soliton. Any additional (finite) subtraction would be possible ina renormalizable theory. Hence this piece is not without ambiguity. Of course, it is very suggestive tosubtract the zero soliton energy to determine the binding of the soliton. But that is only a (regularization)condition for the integrated structure function. Formally, however, the subtraction is obtained from adifferent action functional. The result that the zero soliton vacuum structure function is not a constant iskind of surprising as it suggests that the trivial vacuum has structure. One may also speculate whether thisunexpected result is related to the numerical treatment of discretizing wave-functions with box boundaryconditions. In close proximity to the boundary, completeness of the wave-functions is not guaranteed [72].We are currently exploring this speculation.The biggest conceptual problem unsolved so far is the fact that the computed structure functions havesupport for | x | > x > | x | ≤
1. We have adopted a procedure from the D = + x regime and also interfere with the rotational N C corrections but it maintains thesum rules. Other approaches, that merely omit the | x | > x = x regime, this ad hoc approach violates the sum rules, at least formally.A possible extension of the approach reviewed here would be the consideration of inelastic scatteringwith neutrino induced interactions. That would bring in a generalization of the Compton tensor thatwould also include couplings to the axial current as governed by the weak component of the standardmodel and thus form factors and structure functions that are not disallowed by parity conservation. Author Contributions:
The authors mutually agree that their contributions warrant co-authorship. That must beenough information for the public readership!
Funding:
H. W. is supported in part by the National Research Foundation of South Africa (NRF) by grant 109497.
Conflicts of Interest:
The authors declare no conflict of interest.
References
1. E. Witten, Nucl. Phys. B (1979) 57.2. J. C. Collins, D. E. Soper and G. F. Sterman, Adv. Ser. Direct. High Energy Phys. (1989) 1.3. H. Abramowicz et al. [H1 and ZEUS], Eur. Phys. J. C (2015) 580. ersion January 7, 2021 submitted to Symmetry
39 of 40
4. V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. (1972) 438; G. Altarelli and G. Parisi, Nucl. Phys. B (1977) 298; Y. L. Dokshitzer, Sov. Phys. JETP (1977) 641.5. H. W. Lin et al. , Prog. Part. Nucl. Phys. (2018), 107; M. Constantinou, et al. [arXiv:2006.08636 [hep-ph]].6. M. Gluck, E. Reya and A. Vogt, Z. Phys. C (1995) 433.7. T. H. R. Skyrme, Proc. Roy. Soc. Lond. A (1961) 127.8. G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B (1983) 552.9. G. Holzwarth and B. Schwesinger, Rept. Prog. Phys. (1986) 825; I. Zahed and G. E. Brown, Phys. Rept. (1986) 1; U. G. Meissner, Phys. Rept. (1988), 213. B. Schwesinger, H. Weigel, G. Holzwarth and A. Hayashi,Phys. Rept. (1989) 173; H. Weigel, Lect. Notes Phys. (2008) 1.10. Y. Liu, M. A. Nowak and I. Zahed, Phys. Rev. D (2019) 126023.11. M. Chemtob, Nucl. Phys. A (1987) 613.12. D. Ebert and H. Reinhardt, Nucl. Phys. B (1986) 188.13. Y. Nambu and G. Jona-Lasinio, Phys. Rev. (1961) 345.14. M. Wakamatsu and H. Yoshiki, Nucl. Phys. A (1991) 561.15. R. Alkofer, H. Reinhardt and H. Weigel, Phys. Rept. (1996), 139.16. C. V. Christov et al. , Prog. Part. Nucl. Phys. (1996) 91.17. H. Weigel, E. Ruiz Arriola and L. P. Gamberg, Nucl. Phys. B (1999) 383.18. I. Takyi and H. Weigel, Eur. J. Phys. A (2019) 128; I.Takyi, Ph.D. thesis, Stellenbosch University (2019), https://scholar.sun.ac.za/handle/10019.1/106075 .19. D. Diakonov, V. Y. Petrov, P. Pobylitsa, M. V. Polyakov and C. Weiss, Nucl. Phys. B (1996) 341, Phys. Rev. D (1997) 4069.20. P. V. Pobylitsa, M. V. Polyakov, K. Goeke, T. Watabe and C. Weiss, Phys. Rev. D (1999) 034024.21. M. Wakamatsu and T. Kubota, Phys. Rev. D (1998) 5755.22. M. Wakamatsu and T. Kubota, Phys. Rev. D (1999) 034020.23. P. V. Pobylitsa and M. V. Polyakov, Phys. Lett. B (1996) 350.24. L. P. Gamberg, H. Reinhardt and H. Weigel, Phys. Rev. D (1998) 054014.25. P. Schweitzer et al. , Phys. Rev. D (2001) 034013.26. Y. Ohnishi and M. Wakamatsu, Phys. Rev. D (2004) 114002; P. Schweitzer, Phys. Rev. D (2003), 11401027. M. Wakamatsu, Phys. Rev. D (2003) 034005; Phys. Rev. D (2003), 034006; Phys. Rev. D (2014) 034005.28. J. Ashman et al. [European Muon Coll.], Phys. Lett. B (1988) 364.29. S. J. Brodsky, J. R. Ellis and M. Karliner, Phys. Lett. B (1988) 309; R. Johnson, N. W. Park, J. Schechter, V. Soniand H. Weigel, Phys. Rev. D (1990) 2998.30. A. Deur, S. J. Brodsky and G. F. De Téramond, Rept. Prog. Phys. (2019) 076201.31. R. L. Jaffe, [arXiv:hep-ph/9602236 [hep-ph]].32. H. Reinhardt, Nucl. Phys. A (1989) 825.33. M. Jaminon, P. Stassart and G. Ripka, Phys. Lett. B (1989) 191.34. S. P. Klevansky, Rev. Mod. Phys. (1992) 649; U. Vogl and W. Weise, Prog. Part. Nucl. Phys. (1991), 195.35. R. M. Davidson and E. Ruiz Arriola, Phys. Lett. B (1995) 163; E. Ruiz Arriola and L. L. Salcedo, Nucl. Phys.A (1995) 703.36. T. Frederico and G. A. Miller, Phys. Rev. D (1994) 210.37. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D (2002) 094016.38. W. Pauli, Meson Theory of Nuclear Forces . Interscience Publishes, Inc., New York, 1946.39. S. Kahana and G. Ripka, Nucl. Phys. A (1984) 462.40. R. Alkofer, H. Reinhardt, J. Schlienz and H. Weigel, Z. Phys. A (1996) 181.41. H. Reinhardt and R. Wunsch, Phys. Lett. B (1988) 577; T. Meissner, F. Grummer and K. Goeke, Phys. Lett. B (1989) 296; R. Alkofer, Phys. Lett. B (1990) 310.42. F. Meier and H. Walliser, Phys. Rept. (1997) 383; H. Weigel, R. Alkofer and H. Reinhardt, Nucl. Phys. A (1995) 484.43. R. Alkofer, H. Reinhardt, H. Weigel and U. Zückert, Phys. Rev. Lett. (1992) 1874. ersion January 7, 2021 submitted to Symmetry
40 of 40
44. R. M. Barnett et al. [Particle Data Group], Phys. Rev. D (1996) 1.45. R. Alkofer and H. Weigel, Phys. Lett. B (1993) 1.46. M. Wakamatsu and T. Watabe, Phys. Lett. B (1993) 184.47. J. D. Bjorken, Phys. Rev. (1966) 1467; Phys. Rev. D (1970) 1376.48. H. Weigel, L. P. Gamberg and H. Reinhardt, Phys. Lett. B (1997) 287.49. H. Weigel, L. P. Gamberg and H. Reinhardt, Phys. Rev. D (1997) 6910.50. A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf, Phys. Rev. D (1974) 3471; A. Chodos,R. L. Jaffe, K. Johnson and C. B. Thorn, Phys. Rev. D (1974) 2599; A. W. Thomas, Adv. Nucl. Phys. (1984) 1.51. R. L. Jaffe, Phys. Rev. D (1975) 1953; A. I. Signal and A. W. Thomas, Phys. Lett. B (1988) 481; V. Sanjoseand V. Vento, Phys. Lett. B (1989) 15; A. W. Schreiber, A. I. Signal and A. W. Thomas, Phys. Rev. D (1991)2653.52. J. L. Gervais, A. Jevicki and B. Sakita, Phys. Rept. (1976) 281.53. E. Braaten, S. M. Tse and C. Willcox, Phys. Rev. D (1986) 1482.54. L. P. Gamberg, H. Reinhardt and H. Weigel, Int. J. Mod. Phys. A (1998) 5519.55. R. L. Jaffe, Annals Phys. (1981) 32.56. M. Arneodo et al. [New Muon], Phys. Rev. D (1994) 1.57. V. Y. Alexakhin et al. [COMPASS], Phys. Lett. B (2007) 8.58. R. L. Jaffe, in Relativistic Dynamics and Quark Nuclear Physics , proceedings of the Workshop, Los Alamos, NewMexico, edited by M. B. Johnson and A. Picklesimer (Wiley, New York, 1986).59. K. Abe et al. [E143], Phys. Rev. Lett. (1995) 346.60. K. Abe et al. [E143], Phys. Rev. D (1998) 112003.61. J EFFERSON L AB H ALL
A collaboration, Phys. Rev. D (2016) 052003.62. S. Wandzura and F. Wilczek, Phys. Lett. (1977) 195.63. R. L. Jaffe, Comments Nucl. Part. Phys. (1990) 239; X. D. Ji and C. h. Chou, Phys. Rev. D (1990) 3637; R. L.Jaffe and X.-D. Ji, Phys. Rev. D (1991) 724.64. K. Abe et al. [E143], Phys. Rev. Lett. (1996) 587.65. O. Schroeder, H. Reinhardt and H. Weigel, Nucl. Phys. A (1999) 174.66. A. O. Bazarko et al. [CCFR], Z. Phys. C (1995) 189; V. Barone, C. Pascaud and F. Zomer, Eur. Phys. J. C (2000) 243.67. J. P. Ralston and D. E. Soper, Nucl. Phys. B (1979) 109; R. L. Jaffe, X. M. Jin and J. Tang, Phys. Rev. Lett. (1998) 1166.68. R. L. Jaffe and X. D. Ji, Nucl. Phys. B (1992) 527.69. H. Avakian et al. [CLAS], Phys. Rev. D (2004) 112004; A. Airapetian et al. [HERMES], Phys. Rev. Lett. (2000) 4047; Phys. Rev. D (2001), 09710170. M. Burkardt and Y. Koike, Nucl. Phys. B (2002) 311.71. D. Diakonov, From Instantons to Nucleon Structure proceedings of the int. symposium on