Matching of transverse momentum dependent distributions at twist-3

We derive the leading order matching of the quark generated polarized transverse momentum dependent (TMD) distributions onto the collinear functions at small values of the transverse distance. Starting from the very definition of the TMD operator and performing the light-cone operator product expansion up to twist-3 order, we evaluate each distribution directly in position space. We primarily consider the cases of Sivers, Boer-Mulders and worm-gear functions. The effects of the TMD process dependence on the matching are explicitly shown. We also discuss the moments of TMD distributions which can be relevant for lattice calculations.

technically simpler and more systematic in comparison to a direct evaluation and factorization of cross-sections, which is often used see e.g. [15][16][17].
The operators that define TMD distributions have a peculiar structure, which distinguishes them from more traditional parton distribution functions (PDFs) and fragmentation functions (FFs), distributions amplitudes and others. Namely, they include half-infinite light-like Wilson lines, and they are geometrically non-compact. Moreover, the direction of the Wilson lines depends on the underline process. This direction is future pointing for production mechanisms (such as fragmentation in semi-inclusive deep-inelastic scattering (SIDIS)), and past pointing for collision mechanisms (such as Drell-Yan process). It gives a superficial process dependence of the TMD distributions, in the form of sign-flip for P-odd distributions [18]. At the same time, the collinear distributions are perfectly independent on the process, therefore, any process dependence of TMD operator must reveal itself within OPE. We indeed observe this effect and demonstrate that it appears in the contributions specific for the TMD operators. Exactly these contributions give rise to the famous Efremov-Teryaev-Qiu-Sterman (ETQS) function [19][20][21][22]. Moreover, we have observed that these, process-dependent terms of OPE, also contribute to the P-even, and hence process-independent, distributions. Altogether, to our best knowledge, the analysis presented here is the first study of OPE for TMD operators beyond the twist-2 accuracy. As a final result, we obtain all leading order matching expressions for the TMDPDFs in the quark sector.
The definition framework and all relevant TMD distributions and operators are given in sec. II. The light-cone OPE for TMD operators up to linear in transverse positions terms is given in sec. III. The parameterization of relevant collinear distributions is presented in sec. IV. The assembling of OPE and its application to particular distributions is presented in sec. V. We also present by-product result for Mellin moments of worm-gear function in sec. VI. The final results are shown and commented in sec. VII.

II. DEFINITION OF TMD DISTRIBUTIONS
We outline our work for the transverse momentum dependent parton distribution functions (TMDPDFs) which, in the quark case, are defined by the matrix element [1,23,24]  Thus, any four-vector can be decomposed into the components where v + = (nv), v − = (nv), and v T is the transverse component orthogonal to the scattering plane (v T n) = (v Tn ) = 0. To specify the reference frame we state that v ± = (v 0 ± v 3 )/ √ 2. The transverse components play a special role in our consideration. The transverse subspace is projected out by the transverse part of the metric tensor g µν T = g µν − n µ p ν + p µ n ν (np) . (2.5) There are only two non-zero components, g 11 T = g 22 T = −1. In the following, we also need the transverse part of the Levi-Civita tensor where ǫ µνρσ is defined in the Bjorken convention (ǫ 0123 = −ǫ 0123 = 1). Consequently, we have ǫ 12 T = −ǫ 21 T = 1, which coincides with the definition [23,25,26], despite the opposite normalization of the four-dimension ǫ-tensor. The tensor ǫ µν T does not change sign when both indices are down, ǫ T µν = ǫ µν T , and ǫ µν T ǫ T,µρ = δ ν T, ρ . Since the transverse subspace is Euclidian, the scalar product transverse vectors is negative, v 2 T < 0. In the following, we use the bold font notation to designate the Euclidian scalar product of transverse vectors, i.e. b 2 = −b 2 > 0, when it is convenient.
The spin of the hadron is parameterized by the spin-vector S, The light-cone decomposition of the spin vector is where the helicity λ of the hadron is The vector s µ T is the transverse component of the spin, s 2 T = λ 2 − 1. With the help of ǫ T -tensor we can introduce another useful vectors and it impliess 2 T = s 2 T . The open spinor indices (ij) of the TMD operator in eq. (2.1) are to be contracted with different gamma-matrices, which we denote generically as Γ. The gamma-matrices that appear at the leading order of TMD factorization are where σ α+ T = g αβ T σ βγ n γ , and In the naive parton model interpretation, these gamma-structures are related to the observation of unpolarized (γ + ), longitudinally polarized (γ + γ 5 ) and transversely polarized (iσ α+ T γ 5 ) quarks within the hadron. Beyond the leading order factorization one expects that the power suppressed terms of TMD show also different gamma structures. However, currently, the TMD factorization theorem is not established beyond the leading order. Moreover, it is known that TMD distributions with different from (2.11) gamma-structure contain rapidity divergences that are not renormalized by the standard TMD soft factor [9].
For practical calculations it is convenient to write TMD distributions in momentum space as Fourier transform of distributions in position space in the usual manner where the scalar product (bp T ) is Euclidian. The decomposition in eq. (2.14-2.16) is then replaced by its analog it position space, This parameterization coincides 1 with the parameterization given in [28]. The explicit transformation rules for all these functions can be found in appendix A.

III. LIGHT-CONE EXPANSION FOR TMD OPERATOR
The small-b matching of TMD distribution to the integrated distributions is obtained by the operator product expansion (OPE) at small-b. The OPE is independent from the hadronic states and for this reason universal. Let us introduce a separate notation for the TMD operators. The operator that produces TMD distributions in the Drell-Yan case is where Γ represents the gamma-matrices of the leading set (2.11), and the half-infinite Wilson lines are defined as Here and in the following we also omit the T-ordering of the fields, since it is irrelevant in the tree order approximation. The operator that produces the TMD distributions for SIDIS reads Generally, the links which connect the end points of Wilson lines at a distant transverse plane must be added in both operators (for DY and for SIDIS). Here, we omit them for simplicity, assuming that some non-singular gauge (e.g. covariant gauge) is in use. In non-singular gauges the field nullifies at infinities, A µ (±∞n) = 0, and the contribution of distant gauge links vanish. The relation between the TMD distribution (2.1) and the TMD operator (3.1) is The light-cone expansion of the TMD operators corresponds to the expansion in the variable b. The OPE has a generic schematic form where C's are Wilson coefficient functions which depend on ln b 2 , O 's are light-cone operators, and the symbol * denotes some integral convolution between coefficient function and operators. Here, the superscripts tw2 and tw3 indicate the collinear twist, which in principle differs from the geometrical twist. We remind that the term collinear twist indicates the distributions which enter the same order of momentum expansion. It is not a well-defined quantum number, in contrast to the geometrical twist. The later is defined by "dimension-spin" value, and is a well-defined quantum number, in the sense that e.g. it conserves and does not mix under scaling transformations. As we will see the operators O tw3 are compositions of geometrical twist-2 and twist-3 operators. The coefficient functions are perturbatively calculable. In this work, we study the matching only at order α 0 s . At leading order in α s the quantum fields can be considered as classical fields, that satisfy QCD equations of motion. In this approximation, the small-b OPE is just the Taylor expansion at b = 0. Expanding up U to the linear order in b we obtain The leading term reads Therefore, we obtain that U Γ DY (z, 0) = U Γ DIS (z, 0), which is well known. The term linear in b µ is given by the derivative of the operator. Explicitly, it reads where ∂ T µ is the derivative with respect to the transverse components only. This expression can be written as where the covariant derivative and the field-strength tensor are defined as usual To obtain the expression (3.10) we have used the assumption that 2 A(−∞n) = 0, and the explicit expression for the total derivative of a Wilson line, 2 In singular gauges, one generally cannot expect the boundary condition A(±∞n) = 0, but A(±∞n, ∞) = 0. In this case the TMD operator receives the transverse link to corresponding infinity, which preserves the gauge invariance (for a discussion on the role of singular gauge see f.i. [29][30][31][32]). Therefore, transverse derivative operator ( is inserted at far-end of Wilson lines at ±n∞ + ∞ (compare to (3.10)), and as a result it also differentiates transverse links. Then the expansion formula (3.13) obtains an extra term where τ = τ b/|b|. The limit b → 0 is smooth and thus produces zero. In this way, the result in a singular gauge coincides with the result in a regular gauge. The similar consideration holds for SIDIS operators with replacement −∞n → +∞n.
where the vector n can be arbitrary. The segments of Wilson line between −∞ and τ cancel and we obtain In the case of SIDIS kinematics the Wilson lines point the future light-like infinity, and therefore, the same derivation gives Comparing the results for DY in eq. (3.13) and SIDIS in eq. (3.14) kinematics we observe that the first term is the same, while the last terms differ because of the limits of integration and a common sign. Therefore, already at this stage it is clear that the operator in the first term does not contribute to P-odd distributions (i.e. Sivers and Boer-Mulders functions) which is known to change sign in different kinematics. As expected, the non-compact (in the sense that it spans an infinite range of position space) TMD operator is expressed via a set of compact light-cone operators. The light-cone operators in eq. (3.13, 3.14) are not very well defined, in the sense, that they are of indefinite geometrical twist (more specifically, this issue concerns the first terms of eq. (3.13, 3.14)). At the next stage of the OPE we need to classify the contributions with respect to twist and decompose over independent components. These components are parameterized via parton distributions functions, which are universal and can be measured in different experiment.
As the key point here is the twist-expansion we provide some additional discussion. The standard approach to twist-decomposition of operators is to consider their local expansion. In the local expansion the contributions of different twists can be separated by the permutation algebra, and summed back to a non-local representation, see e.g. the detailed decomposition of similar operators in [33]. However, a much simpler approach consists in taking the operator directly in a non-local form [34,35]. In this approach, one starts with operators off the light-cone, and makes the twist-decomposition, and then perform the limit to the light-cone.
In principle, the procedure of twist-decomposition can be made at the level of operators, see e.g. [34]. However, practically it is involved, especially for tensor gamma-structure. The evaluation is significantly simpler instead in the terms of distributions, e.g. as it is done in ref. [35]. Here we are going to follow this second approach. In fact, the derivation presented in the following sections closely follows the procedure described in details in [35] for the case of meson distribution amplitudes. The difference in kinematics does not allow us to use the powerful method of conformal basis, but there is no principle difference in other aspects.
Prior to the parameterization and twist-decomposition let us prepare the operator for this procedure, and make its off-light-cone generalization. At our order of accuracy (twist-3) we do not need the most general form of the threepoint operators, since they are already of geometrical twist -3 and do not contain admixture with twist-2 operators. Therefore, the generalization should be done only for the two-point operators, and it can be simply achieved by the replacement zn µ → y µ with y 2 = 0. The result is conveniently re-written in the following form where we have used the formula for the stretch derivative of the Wilson line Note, that this expression is the same for DY and SIDIS operators. The last term of (3.15) is again pure twist-3 operator, and thus one can set it directly on the light-cone. Let us conclude this section with an intermediate summary of our main results. For convenience we introduce the generic notation for two-and three-point operators The expression for the first terms of small-b expansion for TMD operator reads (at leading order in α s ) The limit y → zn implies y 2 → 0 such that the light-like separation between fields is z.

IV. COLLINEAR DISTRIBUTIONS
Evaluating the matrix elements of eq. (3.19, 3.20), and hence the matching of TMD distributions we meet with a number of collinear parton distributions. In this section we present the parameterization of two and three point parton distributions that appear in the final result. In fact, the functions that we find represent a complete set of geometrical twist 2 and 3 quark distributions. For the two-point functions we use the standard parameterization by [36]. For the three point functions there is not a commonly accepted parameterization, therefore, we introduce a parameterization inspiring in [37].

A. Parameterization of quark-quark correlators
The standard parameterization of light-cone quark-quark correlators is given [36] and reads 3) are non-zero for −1 < x < 1 and zero for |x| > 1 [38]. They can be represented by where q(x) andq(x) are the usual quark and anti-quark parton densities in the infinite momentum frame. A similar interpretation holds for g 1 and h 1 . At z → 0 the operators turn to local operators. The matrix elements of local operator can be parameterized in terms of the corresponding charges. This implies the existence of exact relations relations among the first moments of PDFs. In the present case the important relations are and they are another form of the Burkhard-Cottingham sum rule [39].
In order to proceed with the matching, we need also a parameterization of off light-cone collinear functions. In general, the parameterization of matrix elements off light-cone does not coincide with the parameterization of lightcone matrix elements, which is given in eq. (4.1, 4.2, 4.3). However, on and off light-cone parameterizations can be related each other order by order in the expansion over y 2 (where y is the distance between quark fields), see e.g. discussion in [40]. Such relations up to linear terms in y are presented in appendix B. Using the off-lightcone parameterization of eq. (B1, B2, B3) we derive the matrix elements of the first terms in the small-b OPE in eq. (3.19, 3.20). We find n α g µν T lim y→zn ∂ ∂y ν P, S|O γ α (y)|P, S = 0, (4.6) Moreover these expressions depend on the particular off-light-cone parameterization that is used. In any case, the functions g T and h L are not independent, and must be expressed in the terms of distributions with definite geometrical twist. Such a re-expression is also dependent on the parameterization. In the final result all (intermediate and offlight-cone) parameterization dependence cancels, and the result is uniquely defined using definite twist distributions.

B. Parameterization of quark-gluon-quark correlators
The parameterization of matrix elements of a three-point operator has the following general structure where the integration measure is [38] [dx] = dx 1 dx 2 dx 3 δ( In the rest of the paper we use the tilde notation for Fourier images of the functions In eq. (4.9) we have introduced a tensor t built of P µ , (single entry of)S µ , n µ , g µν T and ǫ µν T , and their scalar products, such that it preserves the permutation symmetry of indices on left-hand side, and it is invariant under rescaling z → αz. Such a tensor contains significant number of terms, which can be restricted by discrete symmetries, such as parity, time-reversal and charge-conjugation (which can be replaced by hermiticity due to CPT theorem). The parity invariance results into a relation among the terms of eq. (4.9) where the bar denotes the parity transformation of a vectorv µ = v µ , and η Γ,µ...

P
is the sign factor that appears in the parity transformation of the operator PT Γ P † = η Γ P T Γ . The time reversal transformation results into where η Γ,µ...

T
is the sign factor that appears in the time-reversal transformation of the operator T T Γ T † = η Γ T T Γ . In contrast to the two-point functions the time-reversal symmetry does not restrict the number of tensor structures t i , because the functions on left-and right-hand sides of eq. (4.13) are of different arguments. Additionally one has the hermiticity relation which gives where η H is sign of hermitian conjugation of the operator (T Γ ) † = η Γ H T Γ (here we expect that the tensors t are real). Together the time-reversal (4.13) and hermiticity (4.14) relations dictates the complex and symmetry properties of the functions F .
In general the number of tensors t is very large. However, for the current work we need only the tensors which are non-zero if open indices are transverse, and the rest of indices are contracted with n µ . In other words, we require the tensor structure of collinear twist-3. We find four such functions Here, the factors M are set to have dimensionless three-point PDFs T . The definition of distributions T and ∆T coincides 3 with the definition used in [37], up to a factor M . The comparison to ETQS 4 functions (here we compare to definitions in eq. (12) and eq. (21) of [41]) gives The distribution T are real dimensionless functions. According to eq. (4.13) they obey the following symmetry properties The Fourier transform of these distributions obey the same symmetry properties. These four functions are the only genuine twist-3 distributions in the quark sector. It appears very convenient to introduce the following integral combinations, The one-variable functions T (n) , ∆T (n) and δT (n) are in some aspects similar to the usual PDFs. For example, they have zero boundary conditions, In the following, we intensively use the functions in eq. (4.23-4.26), since they naturally arise and describe the wormgear functions and allow a simplification of formulas.

V. LEADING MATCHING OF TMD DISTRIBUTIONS
In this section we assemble the result for the leading matching of TMD distributions up to terms linear in b. For this purpose we need to evaluate the matrix element of the operators in eq. (3.19, 3.20) using the parameterizations introduced in previous section. Here we should take into account the decomposition of collinear twist-3 distributions over the distributions with definite geometrical twist. In the following subsections we consider each gamma-structure individually, and discuss the features of its evaluation. For convenience we also collect the final results in sec. VII.

A. Vector operator
We start with the study of the vector operator, i.e. with Γ = γ + , in the DY kinematics. Taking the forward matrix element of the operator relation in eq. (3.19) we obtain where the contribution of the two-point correlator vanishes in accordance to eq. (4.6). The function T (z, vz −z) is symmetric in v due to the symmetry relation in eq. (4.19). Therefore, the anti-symmetric integral, which is the first in the square brackets of eq. (5.1), vanishes, Taking into account these simplifications we find In the case of SIDIS kinematic the operators are given by eq. (3.20). Applying the same procedure we find where we have used The only difference between Drell-Yan, eq. (5.4) and SIDIS, eq. (5.5) cases is the sign of the linear term. It corresponds to the famous process dependence of the Sivers function [18]. The TMD distribution is obtained by Fourier transformation over the light-cone distance, eq. (3.4). Performing it we obtain Here we have used, These expressions represent the leading matching of vector TMD distribution. Comparing it to the parameterization in eq. (2.18) we find the matching of individual functions. Naturally, the unpolarized TMDPDF matches the unpolarized PDF, . The Sivers function matching is process dependent and it reads Note, that the correction term is proportional to b 2 , and therefore, generically, contains twist-5 functions (and twist-4 functions for unpolarized distribution).
These expression, albeit in the different form, are well-known. In the two-point notation for ETQS function (4.18), the central value of three-point function T (−x, 0, x) corresponds to the diagonal value T q,F (x, x). Therefore, we can compare (5.10,5.11) to the expressions given in literature, where certain momentum space moments are calculated. Using the transformation rules presented in appendix A, one can check that Here the sign is given for the DY case, and should be changed for the SIDIS case. To our best understanding 5 these expression coincide with ones presented in [15,16,42,43].

B. Axial operator
Taking the forward matrix element of the operator in eq. (3.19) with Γ = γ + γ 5 , we obtain where we have used the parameterizations in eq. (4.2, 4.16) and the relation of eq. (4.7).
To proceed further we take the inverse Fourier transform. We have observed that these integrals naturally enter into the moments of the three-point functions, which are defined in eq. (4. 23-4.26). Moreover, it is convenient to present them as a Mellin convolution. Using these tricks we find The last integral in eq. (5.13) over the process-dependent term does not vanish, where we have used the assumption that the integrand goes to zero at infinity. The sign difference between these integrals, is compensated by the common sign difference in the operators for DY, eq. (3.19) and SIDIS, eq. (3.20) kinematics. Therefore, the contribution of seemingly process-dependent terms is the same for both operators. It is exactly compensated by the contribution of eq. (5.14), and thus the function ∆T (1) drops out of calculation.
Combining all together we obtain the same result for DY and SIDIS kinematics, which is Comparing to parameterizations in eq. (2.19) we find that the matching for the helicity TMD distribution g 1L (x, b) = g 1 (x) + O(b 2 ), and for the worm-gear-T distribution is The expression (5.19) is not the final one, because the function g T can be rewritten via functions of definite twist, 20) where ε + = 2m/M with m being the mass of a quark andȳ = 1 − y. The derivation of this decomposition is given in the appendix C 1. It is straightforward to check that it obeys the Burkhard-Cottingham sum rule eq. (4.5). Inserting the function g T into eq. (5.19) and using the associativity of Mellin transformation (see also (C6)) we obtain This is the final form of the matching of the worm-gear function to the twist-2 and twist-3 functions. The Mellin convolution, which is presented in eq. (5.21) by δ-function, can be explicitly integrated. It gives the following representation The obtained result can be compared to the first transverse momentum moments of TMD distribution derived in ref. [44] (see eqn. (47)), and agrees with it for x > 0.

C. Tensor operator
The matrix element of the tensor operator in eq. (3.19) (i.e. with Γ = iσ α+ T γ 5 ) has a more complicated form where we have used the parameterizations of eq. (4.3, 4.17) and the relation (4.8). Its structure repeats the structure discussed during evaluations of the vector operator (for terms proportional to δT ǫ ) and axial operator (for terms proportional to δT g ). Therefore, we skip the discussion on the Fourier integrals and write down the final expression for the matching of transversally polarized TMD distribution. We obtain (compare to eq.(5.8, 5.7) and (5.18)) where the upper sign should be taken for the DY kinematics, and lower for the SIDIS kinematics. Comparing eq. (5.25) with the parameterization of eq. (2.20) we obtain the matching of individual TMD distributions. The transversity distribution is h 1 (x, b) = h 1 (x)+O(b 2 ). The Boer-Mulders functions depends on the underling process and reads The worm-gear L function is independent on the process and has the expression The pretzelosity distribution has no matching at this level of accuracy, despite the fact that the matrix element over free quarks is non-zero at b 2 → 0 [12]. It is expected that the first non-zero contribution to the pretzelosity is of twist-4. As in the case of the worm-gear T function, the expression for the worm-gear L function should be rewritten via a definite twist function. The derivation of function h L is given appendix C 2. It reads We note that there is no term proportional to the quark mass, since it cancels in the equations of motion. Consequently, the worm-gear L function is Let us point that the expressions for worm-gear L and worm-gear T functions has very similar structure, compare eq. (5.30) to eq. (5.21). The main difference is the factor y that appears in eq. (5.21). The integral over δ-function could be evaluated with the result This expression can be compared to the first transverse momentum moment of TMD distribution derived in ref. [44] (see eq.(49) in this reference). We find that our expression for x > 0 agrees with the result of ref. [44], apart from the quark mass contribution, which is absent in our expression but present in [44].
The calculation is done in momentum space. The result is given for the moments of distribution. † The calculation is done for the cross-section, with successive re-factorization into TMD distributions. ** The leading matching for the pretzelosity is currently unknown. We expect it to be of twist-4 level, see also discussion in [12]. where we omit the quark mass correction. The peculiar feature of this expressions is that functions with odd and even index n enter different moments independently. Such relations can be important for lattice studies of TMD distributions, where only Mellin moments of distributions can be evaluated. For example, in ref. [45] the lattice calculation of the first moment of g 1T is performed. It has been found that [45] ≈ 0.2. (6.5) The calculation has been done for the isovector combination of operators q = u − d. Here, the scales of TMD distributions are not clear since the translation rules between lattice scales and TMD evolution scales are not elaborated so far. Nonetheless, the evolution factors for both distributions are the same, and up to the first order of approximation the scale dependence of eq. (6.5) can be omitted. In ref. [45] it is shown that b-dependence of the ratio in eq. (6.5) is very weak. In particular, the value at b ≃ 0.46 practically coincides with eq. (6.5). This suggests that the small-b expansion is a good approximation. We can estimate the ratio in eq. (6.5) from our calculation. Using eq. (6.3) we find 1 + ∆T (2,1) 2f Assuming that the contribution of ∆T (2,1) is small, i.e. in the Wandzura-Wilczek approximation, we find this ratio is ∼ 0.13, which at this level of comparison is a good agreement.

VII. CONCLUSION
In this work we have evaluated the operator product expansion for the quark TMD operators up to linear in b terms. This order of expansion includes the majority of the polarized distributions. The summary of the matching relations is presented in table I. The main result of this study is the leading matching of Sivers, Boer-Mulders and worm gear function. We resume all the matching here for simplicity where the upper sign corresponds to the Drell-Yan operator, and lower sign corresponds to the SIDIS operator. The functions g 1 and h 1 are helicity and transversity PDFs. The functions T are collinear distributions of twist-3. Their definition is given in (4.15-4.17, 4.23-4.26).
The expressions presented here are only the leading order perturbative QCD terms. The sub-leading terms include the power corrections in b 2 and perturbative corrections. The perturbative corrections can be accumulated into the coefficient functions. For the distributions that match solely to twist-2 PDF these coefficient functions are already known at higher perturbative orders; next-to-next-to-leading order (NNLO) for unpolarized [10,11,17] and transversity [12] distributions) and at NLO for helicity distribution [9,46]. The polarized TMD distributions such as Sivers, Boer-Mulders, Collins and worm-gear functions, matches the twist-3 and twist-2 distributions. For these distributions, the coefficient functions are known only at LO and presented here altogether.
We find that the results obtained by us agree with expressions that we have found in the literature (as far as we can trace necessary definitions of various components). However, there are several essential differences. First of all, all known expressions are given in momentum space. Moreover, all of them are presented in terms of certain integrals of TMD distribution over p T . This fact complicates the comparison since such integrals are not well-defined within perturbation theory, and require some regularization procedure. In contrast, our calculation is done directly in the position space, and in this aspect, it represents a complete novelty. Another important distinctive fact of our work is that our calculation is based solely on the definition of TMD operators, whereas the majority of highertwist calculations are based on the evaluation of particular cross-sections with successive "reverse-engineering" into the TMD factorization formula. The only known example that we have found of a direct calculation is the one presented in ref. [44], where the leading matching for worm-gear functions is calculated. Finally, we consider all TMD distributions on the same foot and in the same framework which helps a consistent global understanding of these distributions.
Appendix C: Equation of motion and functions gT and hL The functions g T and h L are reducible, in the sense that they are compositions of geometrical twist -2 functions (g 1 and h 1 ) and geometrical twist -3 functions (T , ∆T and δT g ). The decomposition can be found with the help of the QCD equations of motion Dq = mq. The convenient technique is described in [40] using the example of distribution amplitudes. In the case of parton distributions the case is even simpler, since the total derivative contribution drops out, where dots represent any operator. In this appendix we present the derivation of functions g T and h L in details.

Function gT
As it is demonstrated in ref. [40] the relation between g T and definite twist functions is found with the help of the following operator relation q(y)γ µ γ 5 [y, −y]q(−y) = where m is the mass of quark. This is an exact operator relation, and is the consequence of QCD equations of motion [34]. Next, we evaluate the forward matrix element of equation (C1) using the parameterizations (B2, 4.15, 4.16). This operation transfers the variable y µ into the elementary function, and the derivative over y µ can be done. Next we take the limit y 2 → 0 as it is described in sec. B, and apply eq. (B11). After that procedure we obtain the vector equation which contains the functions of different twists. Its (2M s µ T )-component is where ε + = 2m/M . This equation relates collinear twist-3 function g T to the functions with geometrical twist 2 and 3.

Function hL
The convenient form of the equation of motions for the derivation of function h L as given in ref. [40], We note that the quark mass term is proportional to ε − = (mq − m q )/M , and thus, vanishes for quarks of the same flavor. Making the same steps as in the evaluation of the function g T , i.e. considering matrix element with parameterizations as in eq. (B3) and (4.17), taking derivative and limit y 2 → 0, we obtain The Fourier transform of this equation leads to the differential equation It is intriguing to observe that the expression for h L (C11) is structurally very similar to the expression for g T in eq. (C5).