Exact Relations for Twist-3 Gluon Distribution and Fragmentation Functions from Operator Identities

We perform a systematic study on the twist-3 gluon distribution and fragmentation functions which appear in the collinear twist-3 factorization for hard inclusive processes. Three types of twist-3 distribution and fragmentation functions, i.e., intrinsic, kinematical and dynamical ones, which are necessary to describe all kinds of twist-3 cross sections, are related to each other by the operator identities based on the QCD equation of motion and the Lorentz invariance properties of the correlation functions. We derive the exact relations for all twist-3 gluonic distribution and fragmentation functions for a spin-1/2 hadron. Those relations allow one to express intrinsic and kinematical twist-3 gluon functions in terms of the twist-2 and dynamical twist-3 functions, which provides a basis for the renormalization of intrinsic and kinematical twist-3 functions. In addition, those model independent relations are crucial to guarantee gauge invariance and frame independence properties of the twist-3 cross sections.

Collinear twist-3 DFs and FFs can be in general classified into three types: intrinsic, kinematical and dynamical ones [59] . Although they all appear in the calculation of the twist-3 cross section formula, they are not independent from each other, but are related by QCD equation of motion (e.o.m.) and Lorentz invariance properties of the correlation functions. One of the present authors (Y.K.) performed a systematic study on the twist-3 quark DFs and FFs, and presented a complete set of those model independent relations, which are often called e.o.m. relations and the Lorentz invariance relations (LIR) [59]. These relations allow one to express the intrinsic and kinematical twist-3 DFs and FFs in terms of the twist-2 functions and the dynamical twist-3 functions. In addition, they play a critical role to guarantee the gauge invariance and frame independence of the twist-3 cross sections [29,59,60]. In this paper, we extend the study to gluonic twist-3 DFs and FFs for a spin-1/2 hadron [37-39, 61, 62] and derive all of those exact relations. For the twist-3 gluon DFs in the transversely polarized nucleon, which are relevant to SSAs in ep ↑ → eDX [27], p ↑ p → DX [11], p ↑ p → {γ, γ * }X [12] and also A LT for pp ↑ → DX [42], one of the present authors (S.Y.) derived such relations [63], while no such systematic studies exist for the twist-3 gluon FFs. There are several purely gluonic twist-3 FFs for a transversely polarized spin-1/2 hadron, so the present study is particularly important for the study of their contribution to the polarized hyperon production in the unpolarized proton-proton collision (pp → Λ ↑ X) [37,38] and SIDIS (ep → eΛ ↑ X), etc. Those exact relations for the gluonic DFs and FFs need to be taken into account in the derivation of the cross section and will be crucial to guarantee gauge invariance and the frame independence of the twist-3 cross sections as in the case of quark DFs and FFs.
The remainder of this paper is organized as follows: In section 2, we derive the relations among the twist-3 gluon DFs. After summarizing the complete set of purely gluonic distributions up to twist-3, we derive all the constraint relations among those functions. In section 3, we extend the study to the twist-3 gluon FFs. There are more number of twist-3 FFs compared to the twist-3 DFs due to the lack of a constraint from time reversal invariance. In particular, the dynamical FFs become complex, and the real and imaginary parts obey different constraint relations. Section 4 will be devoted to a brief summary.
2 Twist-3 gluon distributions 2.1 Intrinsic, kinematical and dynamical twist-3 gluon distributions We first summarize the definition of three types of purely gluonic distribution functions in the nucleon which has mass M , momentum P (P 2 = M 2 ) and the spin vector S (S 2 = −M 2 ). As usual, we introduce two light-like vectors p and n, which satisfy P µ = p µ + M 2 2 n µ and p · n = 1. Below we work in frame where p µ = P + g µ + and n µ = (1/p + )g µ − . The simplest collinear gluon distribution functions are defined from the lightcone correlation functions of gluon's field strengths F αν a with color index a in the nucleon state |P S as [61,62] where g αβ ⊥ = g αβ − p α n β − p β n α , and the transverse spin vector S µ ⊥ is defined as S µ = (S · n)p µ + (S · p)n µ + M S µ ⊥ . [0, λn] ≡ P exp{ig 0 λ dt A(tn) · n} is the gauge link which guarantees gauge invariance of the correlation function. Here and below we use the shorthand notation ǫ pnαβ ≡ ǫ νµαβ p ν n µ , etc. G(x) and ∆G(x) are, respectively, twist-2 unpolarized and helicity distributions and ∆G 3T (x) and ∆H 3T (x) are the intrinsic twist-3 distributions corresponding, respectively, to F +⊥ F +− and F +⊥ F ⊥⊥ correlators. Although ∆H 3T (x) drops from the correlator Φ αn,βn (x) which contribute to a cross section, we need the form (1) to derive a constraint relations among the twist-3 distributions. Each function in (1) has a support on |x| < 1.
The second type of the twist-3 gluon distributions are the kinematical ones which are defined as T (x) where · · · denotes twist-4 or higher. These three kinematical distributions ∆G (1) T (x), ∆G (1) T (x) and ∆H (1) T (x) can be also written as the k 2 ⊥ /M 2 -moment of the transverse momentum dependent (TMD) distributions. Note that the TMD distribution corresponding to ∆G (1) T (x) is naively Teven, while those for G (1) T (x) and ∆H (1) T (x) are naively T -odd. Each function in (2) has a support on |x| < 1.
The third type of distributions are the dynamical ones which are defined as the lightcone correlation functions of three field strengths ("F -type" distribution) [27,61] where f acb is the anti-symmetric structure constant for color SU(N) and · · · denotes twist-4 or higher. Here and below we often suppress the gauge link operators between the field strengths for simplicity. N (x 1 , x 2 ) satisfies the symmetry relation N ( and has a support on |x 1,2 | < 1 and |x 1 − x 2 | < 1. Replacing if abc by d abc (symmetric structure constants) in N αβγ F (x 1 , x 2 ), one can define another 3-gluon correlation functions. However, we shall not consider them, since they are not related to any other types of twist-3 gluon distributions. We call N ( where · · · denotes twist-4 or higher. It is easy to see that dx 1 N αβn D (x 1 , x) is reduced to Φ αn,βn (x), and the three distributions in the last three terms of (4) are thus related to those in (1) as Equations (1), (2), (3) and (4) define all necessary collinear twist-3 gluonic distribution functions in the collinear twist-3 formalism. Below we shall derive all constraint relations among those functions.

Relations between D-and F -type DFs and QCD equation of motion
D-and F -type 3-gluon correlators in (4) and (3) are connected as where On the other hand, the correlator for the kinematical twist-3 (2) can be rewritten as Here and below we often suppress the color indices and gauge links for simplicity. For example One thus obtain the relation where N αβγ F (x, x) defines soft-gluon-pole functions. By comparing real and imaginary parts of both sides of (7), one obtains the following relations:

∆H
(1) The first two relations were derived in [63]. They show the D-type functions are determined by the F -type and kinematical functions. The last two relations (13) and (14) are the analogues of the relations for the quark distributions that show the k 2 ⊥ -moment of the "naively T -odd" TMD distribution functions, such as Sivers and Boer-Mulders functions, are proportional to the softgluon-pole (SGP) function of the F -type quark-gluon correlation function. It has been shown that the SGP functions N (x, x) and N (x, 0) contribute to SSAs for ep ↑ → eDX [27], p ↑ p → DX [11], p ↑ p → {γ, γ * }X [12] and p ↑ p → πX [16]. To the best of our knowledge, the relations (13) and (14) were not explicitly written in the literature.
To get further relations, we multiply g ⊥ βγ to (7), integrate over x 1 and use the relation D ⊥ β (λn)F nβ (λn) = −D n (λn)F np (λn) + gψ(λn)/ nt a ψ(λn) which follows from the QCD equation of motion (e.o.m.), (D µ F µα ) a = −gψγ α t a ψ. One then obtains where D g (x) is defined as as with the support on |x| < 1. The relation (15) is also new. From (7), one can obtain another relation involving ∆H 3T (x) as follows. We first write where we have used the relation d dλ [0, λn]F βµ (λn) = [0, λn]D n (λn)F βµ (λn) after integration by part. We then use the Bianchi identity Taking α, β and µ to be transverse, one arrives at the following relation Using the relations, (11) and (12), in this equation one eventually obtains This relation was derived here for the first time.
To summarize this subsection, we have obtained two relations (15) and (22) which relate the two intrinsic functions, ∆G 3T (x) and ∆H 3T (x), and one kinematical function, ∆G (1) T (x), to the dynamical functions. One needs another independent relation to express those three functions in terms of the dynamical functions.

Constraint relations from nonlocal operator product expansion
Here we derive a relation from the nonlocal version of the operator product expansion (OPE) for general correlation functions not necessarily on the lightcone. The method was originally developed in [45,66], and have been frequently used for the twist-3 distributions [24,45,63,67,68], the twist-3 fragmentation functions [59,69], and the distribution amplitudes for hard exclusive processes [66,70,71], etc. This method is equivalent to OPE, and incorporates all the constraints from Lorentz invariance property of the correlation functions. Here we apply this method to the twist-3 gluon distribution functions to derive constraint relations.
We start from the following operator identity: In the left hand side (l.h.s.) of this equation, one should first make y non-lightlike, and take the lightcone limit y µ → y − g µ − after taking the derivative. From translational invariance, we have another identity, We take the expectation value of (23) by |P S , and use (24) to eliminate the first term in the right hand side (r.h.s.). We then obtain From this equation, one obtains the identity ∂ ∂y β P S|F αy (−y)[−y, y]F βy (y)|P S = P S|F α µ (−y)[−y, y]F µy (y)|P S + 2 P S|F yα (−y)gψ(y)/ yt a ψ(y)|P S where we used the QCD e.o.m., D β (y)F βµ (y) = −gψ(y)γ µ t a ψ(y), in the second term of r.h.s. In order to get a relation among the twist-3 distributions from (26), one needs inverse Fourier transform of (1), (3) and (17). In particular, to calculate l.h.s. and the first term in the r.h.s. of (26), one has to use the following form: In taking the derivative of (27) with respect to y β , one should use the form S µ ⊥ = S µ − S·y p·y p µ and g µν ⊥ = g µν − p µ y ν +p ν y µ p·y , keep all components of y µ with y 2 = 0 and then take y µ → g µ − y − limit. With this procedure, we have eventually obtained the following relation: This relation is independent from (15) and (22), and the three relations (15), (22) and (28) allow one to solve ∆G 3T (x), ∆H 3T (x) and ∆G (1) T (x) in terms of ∆G(x) and the dynamical functions. Here we comment on the relations obtained from operator identities other than (23). One can derive a constraint relation by considering the following correlation function: We found that this correlator simply gives the relation that is obtained from (15) and (22), which supplies a good consistency check. We also found that the operator identity for the correlator with F βν = 1 2 ǫ βνρτ F ρτ gives the same relation as (28), which also serves to confirm our result.
It is interesting to compare our approach and that in [63]. The authors of [63] analyzed the correlator (30) to express ∆ G 3T (x) and ∆ G (1) T (x) in terms of ∆G(x) and the dynamical twist-3 distributions. They started from the identity The second term in the r.h.s. can be rewritten further to be expressed in terms of the F -type functions. In our approach, the l.h.s. and the first term in the r.h.s. are calculated by using (27) and are expressed in terms of the intrinsic distributions. In this method, ∆H 3T (x) does not survive in the l.h.s., while it does appear in the first term of the r.h.s. This procedure leads to the same relation as (28). As for the method of [63], they treated the l.h.s. of (31) in the same way as ours (although they did not refer to the presence of ∆H 3T term). On the other hand, they analyzed the first term in the r.h.s. of (31) in a different way. They did not use the form (27), but rewrote it directly in terms of the F -type functions. Therefore they could obtain the constraint relation among the twist-3 distribution functions without recourse to ∆H 3T (x) contribution at any stage. As we will see in the next subsection, our results for ∆G 3T (x) and ∆G (1) T (x) agree with those in [63]. Our approach can also supply the expression for ∆H 3T (x). (See next subsection.)

Solution for intrinsic and kinematical DFs in terms of twist-2 and dynamical twist-3 DFs
As we found in previous subsections, eqs. (15), (22) and (28) constitute a complete set of the independent relations among the twist-3 intrinsic, kinematical and dynamical DFs. Here we provide a solution for the intrinsic and kinematical functions in terms of the twist-2 and dynamical twist-3 DFs. Taking the sum of (15) and (22), we obtain Inserting this into (28) to eliminate ∆H 3T (x), we have This equation can be integrated to give Combining this result and (15), one obtains the expression for ∆G (1) T (x) as ∆G (1) The result in (34) and (35) agrees with that in [63]. Insertion of (34) into (32) gives the expression for ∆H 3T (x) as This result is new. As shown in (34), (35) and (36), the intrinsic and kinematical twist-3 gluonic distributions are completely determined by ∆G(x) (often called Wandzura-Wilczek contribution) and the F -type purely gluonic correlation function N (x 1 , x 2 ) and the quark-gluon correlation function G F (x 1 , x 2 ). Since these relations are model independent exact relations, they need to be satisfied in phenomenological applications. These relations also provide a basis for the renormalization of the intrinsic and the kinematical twist-3 distributions. The evolution equation for N (x 1 , x 2 ) and G F (x 1 , x 2 ) have already been derived in [55]. The above relations (34), (35) and (36) shows it also determines the scale dependence of ∆G 3T (x), ∆G (1) T and ∆H 3T (x).

Intrinsic, kinematical and dynamical twist-3 gluon fragmentation functions
In this section we extend our analysis in the previous section to the twist-3 gluon fragmentation function (FFs). We consider FFs for a spin-1/2 baryon with mass M h , four momentum P h (P 2 h = M 2 h ), and the spin vector S (S 2 = −M 2 h ). In the twist-3 accuracy, we can treat P h as lightlike and introduce another lightlike vector w by the relation P h · w = 1. We again work in a frame where P µ h = P + h g µ + and w µ = g µ − /P + h . Transverse spin vector for the baryon S µ ⊥ is normalized as S 2 ⊥ = −1. Similarly to (1), the gluon's collinear FFs can be defined from the following fragmentation matrix elements [62]: where N = 3 is the number colors for SU (N ) and + · · · denotes twist-4 or higher. All functions in (37) are defined as real. Note that the last two terms drop in the correlator ∼ F wν F wµ , but we need this general correlator to derive relations among the twist-3 gluonic FFs. G(z) and ∆ G(z) are, respectively, twist-2 unpolarized and helicity FFs, and other 4 functions ∆ G 3T (z), ∆ G 3T (z), ∆ H 3T (z) and ∆ H 3T (z) are intrinsic twist-3 FFs. Compared with the distribution functions, the number of twist-3 FFs is doubled due to the absence of the constraint from time reversal invariance, i.e., "naively T -odd" FFs ∆ G 3T (z) and ∆ H 3T (z) survive in addition to "naively T -even" ∆ G 3T (z) and ∆ H 3T (z). Each function in (37) has a support on 0 < z < 1. The second type of gluon's FFs are the kinematical FFs, which are defined by where · · · denotes twist-4 or higher. These three kinematical FFs G T (z) can also be written as the k 2 ⊥ /M 2 h -moment of the TMD FFs as in (2) for the distribution functions. Each function has a support on 0 < z < 1.
The third type of twist-3 FFs are the dynamical ones which are defined as the three gluon correlation function [37][38][39]: where the color indices of the field strength are contracted by the anti-symmetric structure constant if abc and the presence of appropriate gauge links similar to (37) is implied to guarantee gauge invariance of the FFs. There are two independent F -type FFs N 1 1 z 1 , 1 z 2 and N 2 1 z 1 , 1 z 2 which are in general complex, meaning that the number of independent F -type FFs is four times more than the distribution case. z 2 have a support on 1 z 2 > 1 and 1 z 2 > 1 z 1 > 0. Replacing if abc by the symmetric structure constants d abc , one can define another F -type FFs. Although they appear in a certain cross section, e.g., pp → Λ ↑ X [37,38], they are not related to other twist-3 FFs. We therefore do not consider those FFs hereafter.
One can also define another set of twist-3 FFs by the replacement of gF wγ (µw) → D γ (µw) in (39), which gives where gauge links are suppressed for simplicity. D 1,2,3 1 z 1 , 1 z 2 are also complex functions, and are called D-type FFs. Functions in the last two lines are related to those in (37): From the relation it is easy to see Finally we introduce another dynamical FFs defined by where the spinor indices i, j are shown explicitly. These two functions D F T and G F T are, in general, complex functions with their naively "T -even" real part and the "T -odd" imaginary part. They have a support on 1 z 1 > 0, 1 z 2 < 0 and 1 z 1 − 1 z 2 > 1. As we will see below, constraint relations for the twist-3 gluonic FFs involve these F -type quark-gluon correlation functions through QCD e.o.m. We collectively call the functions in (39) and (43) dynamical twist-3 FFs.

Relations between D-and F -type FFs and QCD equation of motion
The gluon FFs introduced in (37)- (40) are not independent, but are related by various operator identities. Using the identity (6), we find D-type and F -type FFs are related as An important difference of this relation from the similar one for the distribution function (7) is that the correlator for the kinematical FFs appear directly as the coefficient of δ-function. This is because F -type FFs become 0 at z 1 = z 2 due to the support property as shown in [64,65]. From (44), we have These relations show D 1,2,3 by contracting (44) with g ⊥ βγ as where D F T (z) is defined from the dynamical FFs in (43) as , and it has a support on 0 < z < 1. Real and imaginary part of (48), respectively, reads and The relation (51) is the FF version of (15). We can also derive another relation from (44). Following a similar step from (19) to (20), we obtain the following relation.
Using (46) and (47) in the r.h.s. of this equation and comparing real and imaginary parts of both sides, one obtains the following two relations.
The second one is the FF version of (22) for the distribution function.
To summarize this section, we have derived two independent relations among the intrinsic, kinematical and dynamical functions, (51) and (54), for the "T -even" sector, and two independent ones (50) and (53) for the "T -odd" sector. One needs another independent relation for the former and two more relations for the latter.

Constraint relations from nonlocal operator product expansion
In this subsection, we will derive the relations among the twist-3 gluonic FFs, employing the method used in section 2.3. To this end, we consider operator identities for the correlation functions away from the lightcone which become the fragmentation matrix element in the lightlike limit. We need to calculate a matrix element like for y 2 = 0 and take the y µ → δ µ − y − limit after differentiation. To calculate (55), we use the following operator identities: In (55)-(58), we have explicitly written gauge links and color indices. Below we will suppress them for brevity. Inserting (56) and (57) into (55), and using (58) to eliminate the term containing 0|D ρ (−y)F αν (−y)|hX , one obtains ∂ ∂y ρ 0|F αν (−y)|hX hX|F βµ (y)|0 This equation is the starting point of our analysis in this section. Constraint relations for the twist-3 FFs can be obtained by expressing each term of (59) in terms of the FFs defined in Sec. 3.1. To calculate the l.h.s. of (59), we need the Fourier inversion of (37) for non-lightlike separation (y 2 = 0), which can be written as In calculating the derivative of the l.h.s. of (59), one need to use S µ ⊥ = S µ − S·y P h ·y P µ h and g µν ⊥ = g µν − 1 P h ·y (P µ h y ν + P ν h y µ ) in (60). This way the l.h.s. of (59) can be written in terms of the intrinsic FFs in (60). Likewise the second and the third terms in the r.h.s. of (59) can be easily expressed by using the dynamical FFs in (39). In order to express the first term in the r.h.s. of (59) in terms of the dynamical FFs, we introduce two particular contractions with respect to the Lorentz indices which allows use of the QCD e.o.m. F µα a (y) ← − D µ (y) = −gψ(y)γ α t a ψ(y).