Parton distributions from nonlocal chiral SU(3) effective theory. I. Splitting functions

We present a new formulation of pseudoscalar meson loop corrections to nucleon parton distributions within a nonlocal covariant chiral effective field theory, including contributions from SU(3) octet and decuplet baryons. The nonlocal Lagrangian, constrained by requirements of local gauge invariance and Lorentz-invariant ultraviolet regularization, generates additional interactions associated with gauge links. We use these to compute the full set of proton $\to$ meson + baryon splitting functions, which in general contain on-shell and off-shell contributions, in addition to $\delta$-function terms at zero momentum, along with nonlocal contributions associated with the finite size of the proton. We illustrate the shapes of the various local and nonlocal functions numerically using a simple example of a dipole regulator.


I. INTRODUCTION
The important role played by chiral symmetry in hadron physics has been documented for many decades. Traditionally the purview of low-energy hadron and nuclear physics, more recently the relevance of chiral symmetry in QCD has become more prominent also in high-energy reactions, in which the quark and gluon (or parton) substructure of hadrons is manifest. One of the most striking expressions of the chiral symmetry and its approximate breaking is in the nonperturbative structure of the sea quark distributions of the nucleon [1,2]. In particular, the breaking of chiral SU(3) symmetry was anticipated [3] to generate unequal strange and (light) nonstrange sea quark distributions, and, even more dramatically, an excess ofd antiquarks overū. The latter was confirmed in proton-proton and proton-deuteron Drell-Yan experiments at CERN [4] and Fermilab [5], following earlier indirect indications from inclusive [6] and semi-inclusive [7] deep-inelastic scattering (DIS) data on proton and deuteron targets.
The observation of a larged −ū asymmetry has also served to motivate more challenging searches for other nonperturbative asymmetries, such as between strange and antistrange quarks in the proton, s −s [8][9][10], or between the helicity dependent light antiquark distributions, ∆d − ∆ū [11]. The phenomenological success in describing thed −ū asymmetry, in particular, in terms of nonperturbative models of the nucleon in which its peripheral structure is modeled by a pseudoscalar meson cloud suggested that signatures of chiral symmetry breaking may also be found in other types of parton distribution functions (PDFs) [8,[12][13][14][15][16][17][18].
While considerable experience has been accumulated with nonperturbative models, a challenge has been to compute the chiral symmetry breaking effects on the PDFs in a modelindependent way from QCD. An important step in establishing a direct connection with QCD was made with the observation [19] that the leading nonanalytic (LNA) behavior of moments of the nonsinglet PDFs, expanded in powers of the pion mass, m π , could be obtained from chiral effective field theory, which encodes the same chiral symmetry properties as present in QCD [20][21][22]. In addition to demonstrating how lattice QCD data on PDF moments and other observables simulated at unphysically large pion masses could be extrapolated to the physical point [23], the result [19] demonstrated unambiguously that a nonzero component ofd −ū arises as a direct consequence of the infrared structure of QCD.
The LNA behavior of the various contributions can be established model-independently by considering the infrared limit; however, the computation of the full amplitude requires specific choices for regularizing the divergences in the loop integrals. In the literature, regularization prescriptions such as transverse momentum cutoffs, Pauli-Villars and dimensional regularization have been used, as well as form factors or finite-range regulators. The latter take into account the finite size of hadrons, while the others are generally more suitable for theories that treat hadrons as pointlike.
In practice, the extended structure of the nucleon and other baryons does become important in many traditional hadronic physics applications. In nonrelativistic calculations, if the regulators are in three-dimensional momentum space, charge conservation, which is related to the time component of the current, is respected in the presence of form factors.
In relativistic calculations, on the other hand, the use of covariant or relativistic regulators often leads to explicit violation of charge conservation.
The problem of preserving gauge invariance in theories with hadronic form factors can be formally alleviated by introducing nonlocal interactions into the gauge invariant local Lagrangian. A method for constructing nonlocal Lagrangians with gauge fields was described by Terning [30], based on the path-ordered exponential introduced by Wilson [31] and earlier by Bloch [32]. Variants of the method were subsequently used in phenomenological applications to strange vector form factors and other nucleon matrix elements by a number of authors [33][34][35]. The pion and σ meson properties have been studied by gauging nonlocal meson-quark interactions in relativistic quark models [36,37]. The nonlocal Lagrangian at the hadron level was also recently constructed and applied to electromagnetic form factors of the nucleon [38][39][40].
The presence of gauge links in the nonlocal Lagrangian connecting different spacetime coordinates generates additional diagrams which are needed to ensure the local gauge invariance of the theory. This guarantees that the proton and neutron charges, for example, are unaffected by meson loops, or that contributions to the strangeness in the nucleon from diagrams with intermediate state kaons and hyperons sum to zero. These basic features of the theory are not guaranteed for a local Lagrangian with a covariant regulator, but arise naturally in the nonlocal theory in which the Ward identities and charge conservation are automatically satisfied.
In this paper we describe how the nonlocal formulation of the chiral SU(3) effective theory can be used to derive the contributions from pseudoscalar meson loops to PDFs in the nucleon. We include both the SU(3) octet and decuplet baryon contributions, using a covariant regulator generated through the nonlocal Lagrangian that respects Lorentz and gauge symmetry. In the present paper we focus on the formalism and the derivation of the proton → baryon + meson splitting functions from the nonlocal chiral Lagrangian; a follow-up paper [41] will report on the results for the nucleon PDFs, computed through convolutions of the splitting functions and PDFs in the virtual mesons and baryons in the loops.
We begin by reviewing in Sec. II the familiar local effective Lagrangian in the standard chiral SU(3) effective field theory. The generalization of the effective Lagrangian to the nonlocal case is described in Sec. III, a procedure which allows the preservation of gauge invariance in the presence of covariant vertex functions for the nucleon-baryon-meson interaction. The main results for the proton → meson + baryon splitting functions are derived in Sec. IV for the full set of lowest order diagrams, including rainbow, bubble, tadpole and Kroll-Ruderman contributions, as well as additional terms that arise from the gauge links generated from the nonlocal interactions. Here we present the model independent results for the nonanalytic behavior of the moments of the splitting functions, and illustrate the relative shapes and magnitudes of the various functions using a simple example of a covariant dipole vertex form factor. Finally, in Sec. V we summarize our results and outline future applications of the new formalism.

II. LOCAL CHIRAL EFFECTIVE LAGRANGIAN
In this section we review the standard local chiral effective theory for mesons and baryons.
The lowest-order Lagrangian, consistent with chiral SU(3) L ×SU(3) R symmetry, describing the interaction of pseudoscalar mesons (φ) with octet (B) and decuplet (T µ ) baryons, is given by [42,43] where M B and M T are the octet and decuplet masses, D and F are the meson-octet baryon coupling constants, C and H are the meson-octet-decuplet and meson-decuplet-decuplet baryon couplings, respectively, f = 93 MeV is the pseudoscalar decay constant, and "h.c." denotes the Hermitian conjugate. The tensor ijk is the antisymmetric tensor in flavor space, and we define the tensors γ µν = 1 2 [γ µ , γ ν ] and γ µνα = 1 2 {γ µν , γ α } in terms of the Dirac γ-matrices. The octet-decuplet transition tensor operator Θ µν is defined as where Z is the decuplet off-shell parameter. The SU(3) baryon octet fields B ij include the nucleon N (= p, n), Λ, Σ ±,0 and Ξ −,0 fields, and are given by the matrix The baryon decuplet fields T ijk µ , which include the ∆, Σ * , Ξ * and Ω − fields, are represented by symmetric tensors with components In the meson sector, the operator U in Eq. (1) is defined in terms of the matrix of pseudoscalar fields φ, where φ includes the π, K and η mesons, The pseudoscalar mesons couple to the baryon fields through the vector and axial vector where υ a µ corresponds to an external vector field, and λ a (a = 1, . . . , 8) are the Gell-Mann matrices. The covariant derivatives of the octet and decuplet baryon fields in the chiral Lagrangian (1) are defined as [44,45] where υ 0 µ denotes an external singlet vector field, λ 0 is the unit matrix, and · · · denotes a trace in flavor space. For the covariant derivative of the decuplet field, we use the notation For the pseudoscalar meson fields, the covariant derivarive is written Expanding the Lagrangian (1) to leading order in the baryon and meson fields, the relevant interaction part for a meson and baryon coupling to a proton can be written explicitly as The terms involving the coupling H are not present because of the restriction to proton initial states. The current calculations below also do not involve the terms with the coupling H for the proton initial states.
From the Lagrangian (1) one can also obtain the form of the electromagnetic current that couples to the external field υ a µ , For the SU(3) flavor singlet current coupling to the external field υ 0 µ , one has where again λ 0 is the unit matrix and · · · denotes a trace in flavor space.
The currents for a given quark flavor are then expressed as combinations of the SU (3) singlet and octet currents, where J µ 3 and J µ 8 are the a = 3 and 8 components of the octet current, respectively. Using Eqs. (14), (15) and (16), the currents J µ u , J µ d and J µ s can be written explicitly as where the terms involving the doubly-strange baryons Ξ 0,− and Ξ * 0,− and the triply-strange Ω − are not present because they cannot couple to the proton initial states.

III. NONLOCAL CHIRAL LAGRANGIAN
In this section we describe the generation of the nonlocal Lagrangian from the local meson-baryon Lagrangian in Sec. II. Evaluating the traces in Eq. (1) and introducing the minimal substitution for the electromagnetic field A µ , the local Lagrangian density can be rewritten more explicitly in the form where for the interaction part we show only those terms that contribute to a meson-baryon coupling to a proton, and we keep the dependence on the space-time coordinate x explicitly.
The covariant derivatives here are written so as to indicate the coordinate with respect to TABLE I. Coupling constants C Bφ , C T φ and C φφ † for the pBφ, pT φ and ppφφ † interactions, respectively, for the various allowed flavor channels.
which the derivative is taken, where e q B , e q T and e q φ are the quark flavor charges of the octet baryon B, decuplet baryon T and meson φ, respectively. For example, for the proton one has the charges e u p = 2e d p = 2, e s p = 0, while for the Σ + hyperon e u Σ + = 2e s Σ + = 2, e d Σ + = 0, and so forth. For the mesons, the flavor charges for the π + are e u π + = −e d π + = 1 but e q π 0 = 0 for all q, and for the K + these are e u K + = −e s K + = 1, e d K + = 0, and similarly for the charge conjugate states. These flavor charges may be read off from the currents given in Eqs. (17a)-(17c). The coefficients C Bφ in Eq. (18) depend on the coupling constants D, F and C, and are given explicitly in Table I for the processes discussed in this work.
Using the methods described in Refs. [30,[35][36][37][38][39][40], the nonlocal version of the local La-grangian (18) can be written as where the gauge link G q φ is introduced to preserve local gauge invariance, for the matter fields, and for the electromagnetic field, where θ(x) is an arbitrary function of the space-time coordi- The nonlocal Lagrangian density in Eq. (20) can be further decomposed by expanding the gauge link (21) in powers of the charge e q φ , where the O(e q φ ) term is and we have used a change of variables z µ → x µ +a µ t+b µ (1−t). This allows the Lagrangian L (nonloc) to be written as a sum of free and interacting parts, where to lowest order the latter consists of purely hadronic (L and respectively. For the δG q φ term in Eq. (24), which explicitly depends on the gauge link, the nonlocal interaction with the external gauge field yields the additional contribution to the Lagrangian density, For the nonlocal theory the quark current has two contributions: the usual electromagnetic current, J µ,q em , obtained with minimal substitution from Eq. (26), and an additional term obtained from the gauge link, respectively. Compared with Eqs. (13) and (17), the nonlocal interaction Lagrangian and currents in Eqs. (25)- (29) include the extra regulator function F (a). The local limit can be obtained by taking F (a) to be a δ-function, F (a) → δ (4) (a), which is equivalent to taking the form factor in momentum space to be unity. Since the Fourier transform of the δ-function in position space is a plane wave in momentum space, the value of the plane wave at the origin is unity. With the nonlocal interaction and current obtained here, in the next section we will discuss the splitting functions describing the interaction of the external field with the proton dressed by the pseudoscalar fields.

IV. SPLITTING FUNCTIONS
In this section we will derive the general expressions for the proton → pseudoscalar meson + baryon splitting functions for the full set of SU(3) octet and decuplet states. After giving the model independent results for arbitrary regulating function F (a), we derive explicit expressions for a specific choice of regulator in which the momentum dependence is given by a dipole shape.

A. Model independent results
The interaction of an external probe with a proton dressed by pseudovector mesons at leading order is given in Fig Fig. 1 will be relevant for the s −s asymmetry.

SU(3) octet intermediate states
Beginning with the meson rainbow diagram in Fig. 1(a), the vertex function for the nonlocal theory can be written as [46] Γ µ φB (2π where L (nonloc) had(B) is the part of the hadronic nonlocal Lagrangian (25) that depends on the octet baryon fields B. (Note also that we defined the vertex such that the quark flavor charge e q φ is included explicitly in the bare meson and baryon PDFs discussed in the next section.) Integrating over the space-time coordinates x µ , y µ and z µ , one has where the Dirac spinor u is normalized such thatūu = 1, and D φ and D B denote the propagator factors for the intermediate baryon and meson, respectively, where m φ and M B are for the meson and octet baryon masses. Defining the regulator in momentum space as the vertex operator becomes Taking the µ = + component of the integrand Γ µ φB , we define the splitting function f where k + = k 0 + k z and M is the nucleon mass. From Eq. (34) the splitting function for the meson rainbow diagram is then given by Similarly, the splitting functions for the baryon rainbow diagram of Fig. 1

(b) and the
Kroll-Ruderman (KR) diagram of Fig.1(c) can be expressed as and respectively.
As discussed in Sec. III, the current generated by the gauge link in Eq. (29) produces the additional diagrams in Fig. 1(d), 1(g) and 1(k). The amplitude for the Kroll-Ruderman additional diagram in Fig. 1(d) can be written as which after Wick contraction and integration over x µ , y µ and z µ , becomes Performing the integrations over the space-time coordinates a µ and b µ , the vertex can be further simplified to In analogy with the definition of the splitting function in Eq. (34), the splitting function for the nonlocal Kroll-Ruderman diagram in Fig. 1(d) induced by the gauge link can be written The main additional feature here compared with the splitting functions in the local theory is the dependence on the derivative of the hadronic form factor F on k − .
For the remaining meson tadpole and bubble diagrams in Fig. 1(e) and 1(f), the splitting functions are given by and where the coupling constant C φφ † is listed in Table I.
Finally, the vertex associated with the nonlocal tadpole diagram in Fig. 1(g), generated by the gauge link, is defined by and can be reduced to The splitting function for the nonlocal tadpole diagram is then given by T can be written

Decuplet intermediate states
where L (nonloc) had(T ) is the part of the hadronic nonlocal Lagrangian (25) that depends on the decuplet baryon fields T , and the operator Θ αβ is given in Eq. (2). Integrating over the space-time coordinates, one finds where the decuplet baryon propagator D T is the same as D B in Eq. (32b), but with M B replaced by decuplet baryon mass M T . The spin-3/2 projection operator P αβ , like the octetdecuplet vertex function Θ αβ , depends on the off-shell parameter Z, defined in Eq. (2).
However, as physical quantities do not depend on Z, it makes sense to simplify the form of the spin-3/2 propagator, and hence in our calculation we choose Z = 1/2, following Refs. [47,48], in which case the projector P αβ is written Note that for this choice one then has the operator Θ αβ = g µν − γ µ γ ν . Performing the integrations over the space-time coordinates a µ and b µ then gives The splitting function for the meson rainbow diagram with decuplet intermediate state is therefore given by Following similar procedures as for the octet baryon case, the splitting functions for the decuplet baryon rainbow diagram in Fig. 1(i) and the decuplet Kroll-Ruderman diagram in Fig. 1(j) can be written as and respectively. Finally, the splitting function for the nonlocal Kroll-Ruderman decuplet diagram in Fig. 1(k) induced by the gauge link is The set of functions f for the octet baryons, and f for the decuplet baryons, then represent the complete set of functions that describe the dressing at one loop of the interaction of an external current with the proton in the nonlocal meson-baryon field theory.

B. Covariant dipole form factor
To evaluate the splitting functions derived in the previous section requires a specific choice for the meson-baryon vertex form factor F (k). Consistency with Lorentz invariance restricts the form factor to in general be a function of the meson virtuality k 2 and the baryon virtuality (p − k) 2 . For convenience, we choose the regulator to have a simple dipole shape in k 2 with a cutoff parameter Λ [33,34], independent of the details of the baryon state, where D Λ = k 2 − Λ 2 + iε and we define Λ 2 ≡ Λ 2 − m 2 φ .

Octet splitting functions
With the dipole regulator in Eq. (56), after reduction of the γ matrices in Eq. (36) the splitting function for the meson rainbow diagram in Fig. 1(a) can be written as where the average mass M and mass difference ∆ are defined as It will be convenient to perform the d 4 k integration in terms of light-cone momentum components k ± = k 0 ± k z and transverse momentum k ⊥ . The first two terms in Eq. (57) have poles both on the upper and lower half-plane, so the integration over k − can be obtained using the residue of D B or D φ . For the third term, proportional to 1/D 2 φ , when k + = 0 both D φ and D Λ have poles on same half-plane, so the integral vanishes. On the other hand, when k + = 0 the integral becomes divergent. We can simplify this term using where we define The integration over k − in Eq. (59) can be written as [24,49] ∞ where µ is a momentum independent constant. After the k − integration, the splitting function for the meson rainbow diagram can be expressed as a sum of an on-shell term, The on-shell function is given by whereȳ = 1 − y, and we employed the shorthand notations [29] The δ-function contributions are nonzero only at y = 0, and arise from the local and nonlocal interactions. The local δ-function term is given by with The log Ω φ term in Eq. (65) gives rise to the leading nonanalytic contribution, which is independent of the regularization method, as we have verified using various methods, including Pauli-Villars, dimensional regularization or a hadronic form factor. In the limit when Λ → ∞, the second term in Eq. (65) ∼ Λ 2 /Ω Λ becomes a constant. Within dimensional regularization, the integral of a constant is defined to be zero, in which case the result coincides with that in Ref. [28], The nonlocal δ-function contribution, δf In the Λ → ∞ limit the first term in the integrand of δf (δ) φ vanishes, while the second term becomes a constant, independent of k ⊥ . In dimensional regularization the latter can again be taken to be zero. The local function f (δ) φ , on the other hand, retains a dependence on k ⊥ through the log Ω φ term, so that the splitting function for the rainbow diagram in Eq. (62) will reduce in this limit to the local splitting function. In the same limit, for the case φ = π and B = N , the integrand of Eq. (63) reduces to the familiar on-shell form found in the literature [1,50,51], for the specific dissociation p → π + n.
For the baryon coupling rainbow diagram, Fig. 1(b), the splitting function in Eq. (37) can be reduced to f (rbw) Performing the k − integral, this can then be expressed as a sum of on-shell, local and nonlocal off-shell, and δ-function terms, Note that the on-shell splitting functions for the baryon and meson couplings are equivalent, while the δ-function contribution f is similar to that derived in Refs. [25,28], while the nonlocal off-shell term is given by In the Λ → ∞ limit, the nonlocal term behaves as Λ 8 /D 5 ΛB ∼ 1/Λ 2 , so vanishes, as expected. For the Kroll-Ruderman diagram in Fig. 1(c), the splitting function in Eq. (38) for the dipole regulator becomes which after the k − integration can be written in terms of the off-shell and δ-function terms, to be rearranged as After the k − integration, this reduces to a sum of the nonlocal off-shell and δ-function which generalizes the result in Ref. [28] to the nonlocal theory. Note that the local and nonlocal off-shell contributions f where f where the nonlocal function δf (δ) φ is given by Eq. (68). Finally, the splitting function for the nonlocal tadpole gauge link diagram in Fig. 1(g) from Eq. (47) with a dipole regulator is Combining Eqs. (79)-(81), one finds that the tadpole and bubble diagrams satisfy the generalized relation which confirms the gauge invariance of the nonlocal theory.

Decuplet splitting functions
Turning now to the splitting functions for the decuplet baryon intermediate states in Fig. 1(h)-1(k), the contribution from the rainbow diagram with coupling to the pseudoscalar meson in Eq. (52) for the covariant dipole form factor (56) is given by where the coupling constants C T φ for the decuplet intermediate states are listed in Table I, and the masses M T and ∆ T here are defined in analogy with Eq. (58), After performing the k − integration, the splitting function can be decomposed in terms of on-shell decuplet, end point, and local and nonlocal δ-function terms, As for the octet case, the first term in Eq. (85) is the on-shell splitting function for the meson rainbow with a decuplet spectator, where D φT and D ΛT are defined analogously to Eqs. (64), Since Λ 8 /D 4 ΛT → 1 in the Λ → ∞ limit, the decuplet on-shell function (86) reduces to the pointlike result found in Ref. [27].
The function f (on end) T in Eq. (85) is finite for finite values of Λ, but in the Λ → ∞ limit corresponds to the end point function in Ref. [27], with a singularity at y = 1. To see this, first note that D ΛT in Eq. (87b) can be written in the formȳD ΛT = −(X T +ȳ Ω Λ ), where X T = yΩ T − yȳM 2 and Ω T = k 2 ⊥ + M 2 T . In the Λ → ∞ limit, one can then write the factor At finite Λ, the term involving D 0 vanishes; however, care must be taken when evaluating this for Λ → ∞. ReplacingȳΩ Λ in the first and second terms in Eq. (89) by (−ȳD ΛT − X T ) and (−ȳD 0 − X 0 ), respectively, one obtains Since in the Λ → ∞ limit one hasȳD ΛT → −Λ 2 (ȳ + X T /Ω Λ ), the first term in parentheses in Eq. (90) can be written where we have taken Ω 0 Λ 2 . The right hand side of Eq. (91) has the properties that it vanishes ifȳ = 0, is divergent ifȳ = 0, and becomes log(X T /X 0 ) when integrated overȳ, so that it can be represented by a δ function, Similarly, for the 1/(ȳD ΛT ) n terms in Eq. (89) with n ≥ 2, one can write in the Λ → ∞ Since the same result is obtained when X T is replaced by X 0 , the 1/(yD ΛT ) n and 1/(yD 0 ) n terms cancel for n ≥ 2, and one obtains where µ is defined such that log(Ω T /µ 2 ) = log(Ω T /Ω 0 ) + 1. With this result, one can finally write the end point splitting function in the Λ → ∞ limit as This expression is identical to that for the end point term in Ref. [27], except for the k ⊥independent terms in (95). For dimensional regularization, however, these are again defined to be zero, so that the result does indeed match that in [27].
The remaining δ-function terms in Eq. (85), namely, the local f . The local off-shell function is given by which in the Λ → ∞ limit reduces to , and is given by Using the relation in Eq. (94), one can show that in the Λ → ∞ limit this term is proportional to a δ function at y = 1, As for the octet case in Eq. (73), the decuplet splitting function also includes a nonlocal decuplet off-shell term, given by The presence of the 1/D 5 ΛT in the integrand of (105) ensures that in the Λ → ∞ limit the nonlocal function vanishes, δf For the δ-function contributions at y = 0, the local terms f As with the other nonlocal contributions, this term also vanishes in the Λ → ∞ limit.
The final diagram in Fig. 1 is that for the Kroll-Ruderman contribution with a decuplet intermediate state, Fig. 1(j). The splitting function corresponding to this diagram, after reducing the γ-matrices in Eq. (54), can be written After integrating over k − , the splitting function for the decuplet KR diagram can be expressed in terms of local and nonlocal off-shell and δ-function terms, each of which has been defined previously. Finally, the splitting function for the additional decuplet diagram induced by the gauge link, Fig. 1(k), is obtained from Eq. (55), With integration over k − , the splitting function for the nonlocal KR gauge link diagram can be simplified to a sum of nonlocal off-shell and δ-function contributions, This generalizes the result from Ref. [29] to nonlocal interactions in the presence of vertex functions parametrizing the extended nature of the proton.

C. Leading nonanalytic behavior
Having derived the complete set of splitting functions for the one-loop diagrams in Fig. 1 for the dissociation of a proton to a pseudoscalar meson (φ) and an SU (3) φ }, that vanish for pointlike particles. All of the diagrams in Fig. 1 are then represented by splitting functions that can be written as linear combinations of these basis functions.
Before presenting the numerical results for the splitting functions for the case of the covariant dipole form factor in Eq. (56), we first identify some features of the basis functions that do not depend on details of the regularization method, but are entirely determined by the infrared behavior of the chiral loops. Namely, expanding the lowest moments f i of the basis splitting functions, as a series in the pseudoscalar meson mass m φ , the coefficients of terms that are nonanalytic (NA) in m 2 φ (either odd powers of m φ or logarithms of m φ ) are determined by the low-energy properties of the nucleon and do not depend on the ultraviolet behavior of the functions [19][20][21][22][23]. In particular, the moments of the on-shell and off-shell functions f is given by where R = ∆ 2 − m 2 φ and R = m 2 φ − ∆ 2 . This agrees with the result found in Ref. [29] for strange octet contributions. In particular, for the latter case, when ∆ < m φ , the mass difference ∆ approaches zero first in the chiral limit, m φ → 0. The resulting LNA term is then simply 4m 2 φ log m 2 φ , consistent with Refs. [20][21][22][23][24]27]. For the case ∆ > m φ , expanding R as R = ∆ − m 2 φ /2∆ + O(m 4 φ ) one finds that the ∆ 2 log m 2 φ terms cancel, leaving behind the same LNA behavior ∼ m 2 φ log m 2 φ , but with a coefficient that is now 4 times smaller than for the ∆ < m φ case.
For the off-shell moment f (off) B , the NA contribution is The LNA behavior of the moment, f (δ) φ , of the δ-function term is These results generalize the LNA expressions given for hyperons and kaons in Ref. [29].
For the decuplet intermediate states, the NA term for the on-shell moment f For the case ∆ T < m φ , one finds in the ∆ T → 0 limit the LNA behavior 8 3 m 2 φ log m 2 φ . For ∆ T > m φ , one may again expand R T as , and note that the LNA term remains ∼ m 2 φ log m 2 φ due to a cancellation of the terms proportional to ∆ 2 T log m 2 φ , In both cases, therefore, the LNA term is given by m 2 φ log m 2 φ , although the the coefficient for ∆ T > m φ is 4 times smaller than that for ∆ T < m φ in the chiral limit.
The NA contribution to the moment of the decuplet off-shell function f (off) T is given by The decuplet δ-function moment does not have an LNA term, but has contributions at higher order in m π , The decuplet results for the total LNA behavior coincide with those for the π∆ intermediate states in Ref. [27], arising from the f In this section we explore the features of the meson-baryon splitting functions for the various octet and decuplet contributions that are nonzero at y > 0, for a finite dipole cutoff parameter Λ in Eq. (56). For illustration, we consider the nucleon and lightest Λ hyperon states for the octet baryons, and the ∆ and Σ * for the decuplet states. Unless otherwise indicated, we will use a typical value for the cutoff mass of Λ = 1 GeV.
In Fig. 2 we show the basis splitting functions for the on-shell f former vanishing at y = 0 and the latter increasing in magnitude as y → 0.
Interestingly, both the on-shell and off-shell end point functions at Λ = 1 GeV peak at rather small values of y, while formally they become δ-functions at y = 1 for Λ → ∞. The dramatic change in the shape of the end point functions with increasing Λ is illustrated in The pattern of cancelations between the various contributions from the basis functions to particular diagrams in Fig. 1 is further explored in Fig. 5, which shows the decomposition of the splitting function for the nucleon-coupling rainbow diagram, f (rbw) N π . For the case of the covariant dipole form factor with Λ = 1 GeV, Fig. 5(a), one observes very strong cancelation between the positive on-shell and negative off-shell contributions, with the total closely resembling the purely nonlocal off-shell function δf (off) . At first sight this may be perplexing, if one interprets the result to suggest that the total nucleon-coupling rainbow function may be very small in the pointlike limit, where δf (off) vanishes. In practice, however, the on-shell and off-shell functions vary differently with Λ, so that the degree of cancelation depends on the cutoff. This is illustrated in Fig. 5(b), which shows the decomposition of f (rbw) N π for the case of a local theory with a Pauli-Villars regulator, which preserves the necessary symmetries of the theory [28,29]. In this case there is no nonlocal contribution, and the total is given by the sum of the on-shell and off-shell terms. For the on-shell splitting function f while for the off-shell splitting function f (off) N the regulator is given by In order to compare the shapes more directly, we choose the Pauli-Villars regulator to give the same total momentum y = Since the contributions from the loop diagrams are ultraviolet divergent, care must be taken to ensure that the integrals are regularized in a way that preserves the underlying symmetries of the effective theory, such as gauge invariance, Lorentz invariance, and chiral symmetry. A common approach adopted in the literature involves the use of local inter-actions with regulators that explicitly depend on the 3-momentum of the meson. While this does take into account the extended nature of hadrons and renders finite results, this approach is in practice ad hoc and destroys the local gauge and Lorentz invariance of the theory.
The virtue of the nonlocal formulation, on the other hand, is that it allows the use of a 4-dimensional regulator while preserving all the necessary symmetries. In this case the regulator is generated directly from the nonlocal Lagrangian, and gives rise to additional diagrams that appear from the expansion of the gauge link [see Fig. 1 The results derived here will serve as a basis for future applications of the formalism to computing meson loop contributions to parton distributions in the nucleon. Within the effective theory, these can be computed by matching twist-two quark level and effective hadronic level operators, which leads to a convolution representation for the PDFs, where f j (y) are the meson-baryon splitting functions, and q v j is the valence distribution for the quark flavor q in the hadronic configuration j. In a forthcoming paper [41], we will use this formalism to study flavor asymmetries in the nucleon generated through meson loops, such as in the light antiquark sea (d −ū) or for strange quarks (s −s), consistently within the 4-dimensional chiral effective theory framework.