Instanton effects in twist-3 generalized parton distributions

The instanton vacuum picture is used to study hadronic matrix elements of the twist-3 (dimension-4, spin-1) QCD operators measuring the quark spin density and spin-orbit correlations. The QCD operators are converted to effective operators in the low-energy effective theory emerging after chiral symmetry breaking, in a systematic approach based on the diluteness of the instanton medium and the $1/N_c$ expansion. The instanton fields induce spin-flavor-dependent"potential"terms in the effective operators, complementing the"kinetic"terms from the quark field momenta. As a result, the effective operators obey the same equation-of-motion relations as the original QCD operators. The spin-orbit correlations are qualitatively different from naive quark model expectations.


Introduction
The spontaneous breaking of chiral symmetry is fundamental to the emergence of hadrons from QCD.It generates most of the light hadron masses through the action of chromodynamic fields in the vacuum.It produces the pion as a quasi-massless excitation mediating long-range interactions between hadrons.The effective dynamics at large distances ∼ 1/M π is expressed by the chiral Lagrangian, describing the interaction of pions and their coupling to the massive hadrons.
Hadron structure is expressed in the matrix elements of composite QCD operators between hadronic states.At the level of the effective dynamics, these QCD operators can be represented by operators in the effective degrees of freedom.One can distinguish two types of QCD operators: (i) Conserved currents related to global symmetries, e.g. the vector and axial vector currents and the energy-momentum tensor.For such QCD operators the effective operators can be obtained from the response of the chiral Lagrangian to the symmetry transformations (Noether theorem).No dynamical information is needed beyond that contained in the chiral Lagrangian.(ii) QCD operators not related to symmetries.This covers the vast majority and includes the quark/gluon operators characterizing partonic structure, e.g. the local twist-2 spin-n and twist-3 spin-n operators determining the moments of the generalized parton distribution (GPDs).For such QCD operators the derivation of the effective operators requires dynamical information beyond what is in the chiral Lagrangian.
Numerous observations suggest that chiral symmetry breaking is caused by topological fluctuations of the gauge fields in the QCD vacuum (instantons); see Refs.[1,2] for a review.Lattice simulations of Euclidean (imaginary-time) QCD show that these fluctuations are localized with average size ρ ∼ 0.3 fm and occur at average distances R ∼ 1 fm, and that they involve strong semiclassical fields.These gauge fields induce lo-Email addresses: jykim@jlab.org(June-Young Kim), weiss@jlab.org(Christian Weiss) calized zero modes of the fermion fields with definite chirality, which delocalize to break chiral symmetry.The instanton vacuum is abstracted from these findings and describes the QCD gauge fields on the scale ρ−1 as a superposition of instantons, coupled to the fermions by the zero modes [1,2].The picture is endowed with a small parameter in the form of the packing fraction of instantons in the vacuum, κ ≡ π 2 ρ4 / R4 ≈ 0.1 (diluteness).Chiral symmetry breaking and hadronic correlation functions can be studied in a systematic expansion.It leads to a successful phenomenology of hadron structure, validated by many observations.Using techniques based on the 1/N c expansion (saddle point approximation, bosonization) one can construct the effective dynamics at the scale R, described by quarks with a dynamical mass M ∼ 0.3-0.4GeV, coupled to a chiral pion field [3,4,5].The chiral Lagrangian is obtained by integrating out the quarks and performing a gradient expansion in the pion field.
In the instanton vacuum one can derive the effective operators representing QCD operators in the effective theory after chiral symmetry breaking [4].The picture provides an explicit model of the nonperturbative gauge fields appearing in quarkgluon and pure gluon QCD operators.The effective operators can be derived systematically in an expansion in the packing fraction.The method has been used to analyze and predict the nucleon matrix elements of several higher-twist quark-gluon operators [6,7] and higher-dimensional gluon operators [8].It preserves operator relations following from QCD equations of motion [6], as well as low-energy theorems from the scale and U(1) A anomalies of QCD [4].
In this work we use the instanton vacuum to study the hadronic matrix elements of the twist-3 QCD operators measuring the quark spin density and spin-orbit correlations in hadrons.These operators appear in the theory of twist-3 generalized parton distributions (GPDs) and the phenomenology of nucleon spin structure.We derive the effective operators and find that parametrically large "potential" terms arise from the instanton gauge field in the covariant derivatives, complement-ing the "kinetic" terms from the quark field momenta.As a consequence, the effective operators obey the same equationof-motion relations as the original QCD operators, a very gratifying result.We discuss the quark spin-orbit correlations described by the instanton-based effective operators and show that they are qualitatively different from those obtained with quark model-based operators including only kinetic terms.

Twist-3 QCD operators
We consider the twist-3 (dimension-4, spin-1) operators with [αβ] ≡ αβ − βα.Here is the QCD covariant derivative with gauge potential A cβ (x).ψ(x) is the quark field with N f light flavors; τ denotes a generic flavor matrix; both singlet (τ = 1) and non-singlets (τ = τ a ) will be discussed in following.The natural-parity operator Eq. ( 1) represents the antisymmetric part of the QCD energy-momentum tensor and measures the quark spin density in hadrons [9].The unnatural-parity operator Eq. ( 2) describes quark spin-orbit correlations [10].Using the QCD equations of motion, the operators can be expressed in terms of the twist-2 axial vector and vector current operators, where ∂ γ [...] denotes the total derivative.In Eq. ( 5) we have omitted operators proportional to the current quark masses.In our convention ϵ 0123 = 1, γ 5 ≡ −iγ 0 γ 1 γ 2 γ 3 [11].Equations ( 4) and (5) imply that the hadronic matrix elements of the operators ⟨p 2 |...|p 1 ⟩ are proportional to the 4-momentum transfer p 2 − p 1 and vanish in the forward limit.The scale dependence of the twist-3 operators can be inferred from Eqs. ( 4) and (5).The vector currents and the flavornonsinglet axial currents are conserved and scale-independent; the flavor-singlet axial current is subject to the U(1) A anomaly.
In the instanton vacuum the QCD gauge potential is represented by a sum of instanton and antiinstanton (I and Ī) fields in singular gauge, and the functional integral is performed over the collective coordinates (position, color orientation, size).Large instanton sizes are suppressed by instanton interactions, and an effective size distribution centered around ρ ∼ 0.3 fm is obtained.Fluctuations of the size are suppressed by 1/N c , and one can take ρ = ρ in leading order [12,4].
The quark field modes with Euclidean momenta p ≲ ρ−1 experience chiral symmetry breaking and can be described by an effective field theory.Each I( Ī) induces a multifermion vertex of the form of a flavor determinant (see Fig. 1a) where z is the I( Ī) position; the color orientation has been integrated over.We use the shorthand notation F(p) is the wave function of the I( Ī) fermionic zero mode with range p ≲ ρ−1 , normalized such that F(0) = 1 [4].The ground state of the interacting fermion system is constructed in the 1/N c expansion (saddle point approximation) [3,4].A non-trivial saddle point appears, characterized by a dynamical quark mass M of parametric order M 2 ∼ κ ρ−2 and numerical value M ∼ 0.3-0.4GeV.The functional integral is bosonized by introducing a chiral field (see Fig. 1b) where F π defines the normalization of the pion field.This converts the multifermion interaction to a Yukawa-type interaction of the quarks with the chiral field.At the saddle point, the effective action takes the simple form Hadronic correlation functions can be computed systematically in the 1/N c expansion [3,4,5].Baryon correlation functions are characterized by a non-trivial classical chiral field (soliton), giving rise to a rich structure [13].The applications of this model have been discussed extensively in the literature; the present study focuses on operators in the theory defined by Eq.( 10) and does not require details of the correlation functions.
In the same scheme of approximations one can integrate out the quark fields and derive the effective action of the chiral field, The dependence on the chiral field can be made explicit by gradient expansion in ∂U.The integrand in Eq. ( 12) is the chiral Lagrangian.The low-energy constants such as F 2 π are given by quark loop integrals in the effective theory of Eq. ( 10) and calculable in terms of the dynamical scales M and ρ.In this way chiral symmetry breaking by instantons quantitatively predicts the effective dynamics at the hadronic scale.

Effective operators from instantons
We now derive the effective operators for the twist-3 QCD operators Eqs. ( 1) and ( 2) in instanton vacuum, in the formulation of Refs.[4,6].To cover all cases of interest, we consider a general Euclidean operator with Γ α ≡ γ α or γ α γ 5 and unsymmetrized tensor indices α, β; (anti-) symmetrization will be performed in the expressions below.The Euclidean operator Eq. ( 13) is defined such that its spatial components coincide with those of the Minkowskian operators Eqs. ( 1) and ( 2) ).The gauge potential-dependent part of Eq. ( 13) is (see Fig. 1c) In leading order of the packing fraction, the gauge potential is evaluated in the field of a single I( Ī).For an I( Ī) centered at the origin and in standard color orientation, it is given by where (η ∓ ) c αβ ≡ ηc αβ , η c αβ are the 't Hooft symbols.The effective operator is constructed by substituting the I( Ī) gauge potential in Eq. ( 14), multiplying with the zero mode projector, integrating over color orientations, combining the I and Ī contributions, and bosonizing the vertex in leading order of 1/N c [4,6].We obtain (see Fig. 1d) where One sees that the gauge potential has been replaced by a color current of the quark fields coupling to the instanton zero mode, with a spin-flavor structure dictated by the symmetries of the instanton field.The effective operator Eq. ( 16) represents the QCD operator Eq. ( 14) in the bosonized effective theory of Eq.( 10) within the overall scheme of approximations (packing fraction, 1/N c expansion). 1he effective operator Eq. ( 16) can be inserted in correlation functions with hadronic currents to extract the hadronic matrix elements.In these correlation functions the momenta of the external quark fields (coupling to the hadronic currents) are of order p ∼ M ≪ ρ−1 , because the hadronic size is of order M −1 .Parametrically large contributions can arise only from loop diagrams in which the fields in the multifermion operator Eq. ( 16) are contracted among themselves and the internal momenta can extend up to l ∼ ρ−1 (see Fig. 1e).This permits further simplification and allows us to reduce the effective operator to a two-fermion operator of quarks in the chiral background field, similar to the interaction in Eq. (10).
The spatial variation of the chiral background field U(z) is on the scale M −1 ≪ ρ.The loop integral with momenta l ∼ ρ−1 is local on that scale.In computing the contractions of Eq. ( 16) we can therefore neglect the variation of the background field and set U = const., so that it becomes like a coupling constant.The contractions can then be computed in momentum representation, with the momentum assignments of Fig. 1e.The external momenta are p 1,2 ∼ M ≪ ρ−1 .Inside the loop, for the parametrically leading contribution we can neglect the mass term in the quark propagator with momentum l and replace it by the free propagator l • γ/l 2 .The momentum representation of the instanton field Eq.( 15) is where A(k) is a dimensionless scalar function of k ρ.The contraction of the operator Eq. ( 16) is obtained as The zero mode wave functions evaluated at the external momenta have been set to unity, F(p 1,2 ) = 1, because p 1,2 ≪ ρ−1 .
A parametrically large contribution arises from the integral in which the loop momenta in the numerator are projected as The factor l 2 cancels the denominator of the quark propagator and produces an integral that would be quadratically divergent if not for the internal zero mode wave function F(l).Making the projection, simplifying the products of gamma matrices, and dropping terms ∝ p 1,2 ∼ M, Eq. ( 18) becomes The final integral evaluates to This remarkable relation follows from the Dirac equation for the zero mode wave function in the instanton field in momentum representation [6].Eq. (20) reduces to which is independent of the instanton size ρ.This vertex represents the effect of the gauge potential in the operator Eq. ( 13) in the instanton vacuum.Altogether, including also the derivative term in Eq. ( 13), we find that the QCD operator Eq. ( 13) can be represented by the effective two-fermion operator This two-fermion operator gives the same result as the fourfermion operator Eq. ( 16) at the parametrically leading level when inserted in hadronic correlation functions with external momenta p 1,2 ∼ M ≪ ρ−1 .Here we have restored the position dependence of the background field U γ 5 (x) on the scale M ≪ ρ−1 .Equation ( 23) is the main result of present work and has numerous implications for hadron structure.We observe that the instanton induces a "potential" term in the effective operator, which accompanies the "kinetic" term of the quark field derivatives.The potential term is of the order of the dynamical quark mass term in the effective action Eq. ( 10) and therefore has a large effect on hadron structure.It is chirally odd and therefore involves the chiral background field (this is required by chiral symmetry).It has a specific spin/flavor dependence conditioned by the symmetries of the instanton field.Altogether, this highlights how instantons convert color dynamics to an effective spin-flavor dynamics.
The instanton effect depends on the spin or twist of the operator.In the twist-2 projection of Eq. ( 23) (symmetric traceless tensor) the potential term is absent.This happens because for Γ α = γ α the instanton-induced vertex in Eq. ( 23) involves γ α γ β and γ β γ α , which cannot produce a symmetric traceless tensor, the same happens for Γ α = γ α γ 5 .This generalizes to twist-2 spin-n operators [6].The twist-2 quark distributions in the instanton vacuum therefore can be calculated with the "kinetic" operators in leading order of the packing fraction.The consistency of this approximation is shown by the fact that the partonic sum rules (charge, momentum) are satisfied at this level [14].In the twist-3 projection of Eq. ( 23) (antisymmetric tensor) the potential term is present.In this case the products γ α γ β and γ β γ α in the instanton-induced vertex project on σ αβ and give rise to non-zero structures.Regarding the twist-4 projection (trace) of Eq. ( 23), see Footnote 1.

Twist-3 effective operators
Using the master formula Eq. ( 23) we can now obtain the effective two-fermion operators for the twist-3 QCD operators Eqs. ( 1) and (2).It is convenient to present the effective operators in Minkowskian form, so that they can be compared directly to the original QCD operators.For the natural-parity operator Eq. ( 1) we obtain where we have used σ αβ U γ 5 = U γ 5 σ αβ .The potential term is proportional to the flavor commutator [τ, U γ 5 ] ≡ τU γ 5 − U γ 5 τ.
It is therefore zero in the flavor-singlet case τ = 1, which means that there is no instanton effect (or chiral field effect) on the total quark spin density.It is non-zero in the flavor-nonsinglet case τ 1, which means that there is an instanton effect on the individual flavor distributions of the quark spin.It is instructive to exhibit this effect in terms of the conventional pion field.Expanding the chiral field Eq. ( 11) in the pion field, the operator Eq. (25 One sees that the potential term couples the pion field to the isovector pseudotensor current of the quark field. For the unnatural-parity operator Eq. ( 2) we obtain Here the potential term is proportional to the flavor anticommutator {τ, U γ 5 } ≡ τU γ 5 + U γ 5 τ and affects both flavor singlet and nonsinglet quark spin-orbit correlations.

Effective equations of motion
In the effective theory obtained from instantons, the quark fields are subject to equations of motion in the chiral background field, governed by the effective action Eq.(10).These effective equations of motion imply certain relations between the operators in the effective theory.When applied to the twist-3 effective operators, the effective equations of motion reproduce the QCD operator relations Eqs. ( 4) and (5).In Minkowskian form, the equations of motion for quark fields in the effective theory Eq. ( 10) are Here we consider energies/momenta p ∼ M ≪ ρ−1 , which is sufficient for the typical fields in hadron structure.Using Eqs.(29), one can easily show that the Minkowskian twist-3 effective operator Eq. ( 25) obeys the relation where the operator on the R.H.S is the axial current in the effective theory.The relation holds for any τ (singlet and nonsinglet).The instanton-induced potential term in the effective operator is essential for bringing about Eq. (30); without it there would be a discrepancy of the order of the dynamical quark mass.The same is obtained for the unnatural-parity twist-3 effective operator Eq. ( 28).This remarkable result comes about because (i) in leading order of the packing fraction the gauge potential in the QCD operator is represented by a single instanton; (ii) the quark fields in the instanton background satisfy the QCD equations of motion (the zero mode is a solution to the Dirac equation in the instanton field).It attests to the consistency of the approximation scheme based on the packing fraction expansion (diluteness) and the 1/N c expansion (saddle point approximation).
An important practical consequence of the effective operator relation Eq. ( 30) is that the hadron structure results do not depend on which "version" of the operator is used.This problem afflicts dynamical models without a systematic derivation of the effective dynamics and the effective operators.

Quark spin-orbit correlations
Our findings have consequences for the phenomenology of quark spin-orbit correlations in the nucleon described by the matrix elements of the twist-3 operator Eq. ( 2) [10].A detailed study of the nucleon matrix elements in the mean-field picture based on the instanton-induced effective theory and the 1/N c expansion will be presented elsewhere.Here we limit ourselves to two limiting cases of this picture, the "quark model" and "chiral soliton" limits (their relation to the general mean-field picture is discussed e.g. in Ref. [15]).They illustrate the impact of the instanton-induced potential in the effective operator and provide a baseline for further detailed studies.
In the quark model limit, we consider the matrix element of the effective operator Eq. ( 28) in a weakly bound system of massive quarks (U γ 5 ≡ 1).The essential point can be seen by taking the operator Eq. ( 28) with τ = 1 and U γ 5 ≡ 1, and computing the expectation value in a free quark state with Minkowskian 4-momentum p and p 2 = M 2 , Dirac spinor wave function u satisfying ( / p − M)u = 0, and polarization vector s α ≡ ūγ α γ 5 u.The matrix element of the kinetic term is The matrix element of the potential term is obtained as The matrix element of the complete effective operator is then this could also be inferred from the fact that the complete effective operator is a total derivative.One sees that the instantoninduced potential term cancels the trivial result in the free state arising from the kinetic term and makes the the spin-orbit correlation a pure interaction effect.In the "chiral soliton" limit, the quark fields are integrated out, and we consider the nucleon as a soliton of the chiral Lagrangian Eq. ( 12).The effective operator corresponding to this situation is obtained from Eq. (28) by averaging over the quark fields in the background of the chiral fields, The average can be computed using the Green function of the quark field in the background of the chiral field (loop diagram).The dependence on the chiral field can be made explicit by gradient expansion in chiral field, similar to the effective action Eq. ( 12); the technique is described in Refs.[13,14].For example, the operator for the isovector vector current in the chiral theory is obtained as (here this result can also be obtained by applying the symmetry transformation to the leading-order term of the chiral Lagrangian Eq. ( 12) [16].For the isovector twist-3 operator Eq. ( 35) (τ = τ a ), the gradient expansion gives which is the total derivative of the isovector current of the chiral field and agrees with the equation-of-motion relation Eq. (5).In this case we observe that the leading-order result in ∂U arises from the kinetic term in the operator Eq. (28); the contributions from the potential term come in only at higher orders.For the isoscalar twist-3 operator Eq. ( 35) (τ = 1), the gradient expansion has to be carried out including terms of order ∂U 4 , as the isoscalar vector current of the chiral field is of order ∂U 3 [16]; the expressions will be presented elsewhere.

Summary and extensions
Instantons induce potential terms in the twist-3 quark operators measuring the quark spin density and spin-orbit correlations in hadrons.The potential terms are proportional to the dynamical quark mass emerging from chiral symmetry breaking, are independent of the instanton size, and have a distinctive spin-flavor structure.They ensure that the effective operators obey the same equation-of-motion relations as the original QCD operators.The potential terms qualitatively change the spin-orbit correlations in hadrons compared to estimates with the kinetic terms only.
Our results provide the basis for a systematic analysis of twist-3 GPDs in the instanton vacuum.The present study has focused on a special case of twist-3 operators that are related to well-known twist-2 operators through the equation-of-motion, which has allowed us to test the consistency of the approximations based on the instanton packing fraction and the 1/N c expansion.Future studies can address cases where such simple relations are not available and one has to rely on the approximations to organize the dynamics.
The methods developed here can be extended to several other cases of interest: (i) higher moments of the chirally-even twist-3 GPDs; (ii) chirally-odd twist-3 GPDs; (iii) matrix elements of twist-4 operators.
The present study employs the minimal description of the QCD vacuum in terms of instanton-type fluctuations, which induce fermionic zero modes and cause chiral symmetry breaking.Recent work has explored a more extended description including other types of topological fluctuations (so-called molecules and zig-zag paths), which do not possess zero modes but contribute to observables such as Wilson lines; see Refs.[17,18] and other articles in the series.Extending the effective operator approach to this more general framework would be an interesting problem for further study.

Figure 1 :
Figure 1: Instanton vacuum and effective operator formalism.(a) Multifermion interactions in instanton vacuum.(b) Bosonization of multifermion interaction.Dashed lines denote the chiral background field U(x).(c) QCD quark-gluon operator.(d) Effective operator induced by instanton.(e) Correlation function of effective operator with quark fields of momenta p 1,2 ∼ M. The parametrically large contribution arises from the contractions shown here.