Polynomiality sum rules for generalized parton distributions of spin-1 targets

We present the polynomiality sum rules for all leading-twist quark and gluon generalized parton distributions (GPDs) of spin-1 targets such as the deuteron nucleus. The sum rules connect the Mellin moments of these GPDs to polynomials in skewness parameter $\xi$, which contain generalized form factors (GFFs) as their coefficients. The decompositions of local currents in terms of generalized form factors for spin-1 targets are obtained as a byproduct of this derivation.

we discuss the counting of the number of GFFs that appear in the decomposition of the local operators found in Sec. III based on symmetries and selection rules. This counting provides an important check on the further results in this paper. Section V contains the main results of this paper, viz., the decomposition of the local operators in GFFs and the resulting polynomiality sum rules for the GPDs of spin-1 targets. We find agreement between our decompositions and the counting established in Sec. IV. Finally, conclusions are stated in Sec. VI. Appendix A summarizes the conventions we use in this work, and App. B outlines the differences between this work and a second gluon spin-1 GFF decomposition in the literature [19]. The connection with the gravitational form factors of spin-1 targets is left for a future study.

II. LEADING-TWIST GPDS OF SPIN-1 HADRONS
Quark and gluon GPDs are defined as Lorentz-invariant functions appearing in the decompositions of light cone correlators [1,2]. GPDs are classified by their collinear twist, which is equal to their dimension minus the projection of their spin onto the lightlike vector n defining the "plus" direction [53]. GPDs of higher twist1 make smaller contributions to cross sections for hard processes such as deeply virtual compton scattering (DVCS), and at high Q 2 the lowest-twist GPDs dominate the cross section. The lowest-twist GPDs are twist-2, and are often called leading-twist. The leading-twist GPDs are specifically what we consider here.
There are three leading-twist quark correlators, which are defined by the following off-forward matrix elements (conventions used in this work are summarized in App. A): e 2ix(Pn)κ p ′ , λ ′ |q(−nκ)/ n[−nκ, nκ]q(nκ)|p, λ , (1a) where is a Wilson line from x to y, with P signifying path-ordering. There are also three leading-twist gluon correlators, which are defined by: where the symmetrization operator S is defined in Eq. (A4). Here, and elsewhere in this work, the indices i and j signify transverse light cone coordinates. Each of these correlators has an additional dependence on the renormalization scale µ 2 , which we have not notated for brevity. The correlators defined in Eqs. (1,3) apply to any hadron, but the number of independent Lorentz structures this can be decomposed into, and thus the number of GPDs a hadron has, depends on the hadron's spin. Here, we give the decompositions of the light cone correlators for spin 1 specifically. The vector quark and gluon correlators have the following decomposition: while the axial vector quark and gluon correlators decompose as: where we use the shorthand ǫ xyzw = ǫ µνρσ x µ y ν z ρ w σ whenever x, y, z, w are four-vectors. These were first found in [26], though we define P to be half of the P used in ibid., producing some superficial differences in the formulas. The decompositions for the quark and gluon transversities for spin-1 hadrons were found later, in [28]. For the quark transversity, we have: and for the gluon transversity:

III. MELLIN MOMENTS OF BILOCAL OPERATORS
In this section, we consider the relationship between Mellin moments of generalized parton distributions and matrix elements of local currents. The Mellin moments of the GPDs defined in Sec. II can be found by evaluating the Mellin moments of the bilocal operators that define the correlators in Eqs. (1,3). The three quark operators of interest here are: while the gluon operators are: Like the correlators they produce through their off-forward matrix elements, these operators have an implicit dependence on a renormalization scale µ 2 .
The identity can be used in the Mellin transforms of the bilocal operators, and combined with the Leibniz product rule and chain rule to recast x s in terms of the gauge covariant derivative. The action of the covariant derivative depends on which representation of SU(3, C) the object being differentiated transforms under, and its actions on the field operators of interest are: If we define a two-sided covariant derivative using: (where we have absorbed the gauge link in order to make the notation less cumbersome), then we find that: where Γ is a generic Clifford matrix, and we have used G and G ′ to denote generic components of the gluon field strength or its dual. Lastly, if we define an auxilliary variable λ = 2(Pn)x, then one has: and the Mellin moments of generic quark and gluon operators become: where additional instances of n are pulled out of the operator, having come from the definitions of the light cone correlators. The Mellin moments of the bilocal operator have thus become local operators. It is worth noting that, since n µ 1 . . . n µ s is symmetric and traceless, we can additionally symmetrize and subtract the trace of the matrices between the quark or gluon fields in Eqs. (14)-that is, we can apply the operator S [see Eq. (A4)]-without changing the result. Moreover, since the actual matrices (quark case) or field operators (gluon case) are contracted with n at leading twist, the results for specific correlators can be symmetrized further. In particular, for quarks we find: Here, A is defined in Eq. (A5) and signifies explicit antisymmetrization of the indices denoted under it2. For the gluons, we have: These symmetrizations (and trace subtractions) serve two purposes: firstly, they refine the local operators to transform under irreducible representations of the Lorentz group, allowing straightforward classification and decomposition of the operators; and secondly, they ensure that none of the form factors in the decomposition of matrix elements of the local operators contain terms that will contract with n µ n µ 1 . . . n µ s to zero. This ensures that each of the generalized form factors we find is actually present in the Mellin moment of a GPD.
The local operators defined in Eqs. (15)- (16), like the bilocal operators we have derived them from, have an additional dependence on the renormalization scale µ 2 which we have not notated. Consequently, the form factors obtained from decomposing their matrix elements will generally have dependence on the renormalization scale as well. Electromagnetic form factors are a special exception to this rule.

IV. COUNTING GENERALIZED FORM FACTORS
In this section, we look at the decomposition of matrix elements of the local operators given in Eqs. (15) -(16) when sandwiched between initial and final state kets |p, λ and p ′ , λ ′ |, namely: for a hadron h, where λ(λ ′ ) is the lightfront helicity of the initial (final) state hadron. These matrix elements can be decomposed into a number of Lorentz structures containing the momenta P and ∆, and possibly Clifford matrices sandwiched between spinors (in the spin-half case) or polarization vectors (in the spin-one case). The Lorentz-invariant functions of the Lorentz scalar t = ∆ 2 multiplying these structures constitute the generalized form factors (GFFs) that we seek.
The number of GFFs that should appear in the decomposition of Eq. (17) can be determined by using the method outlined in Refs. [21,22]. This method involves looking at the crossed channel for hadron-antihadron production from the vacuum, and determining the number of form factors for this process by matching the J PC quantum numbers between the final state hh| and the operator O µν... . More specifically, the procedure is as follows: 2 Although σ µν is already antisymmetric, the sequence of covariant derivatives following is not, so this antisymmetrization following the denoted symmetrization is nontrivial. However, the additional antisymmetrizing terms this introduces into O ν µ µ 1 . . .µs σ contract with n µ n µ 1 . . . n µs to zero, so we are free to introduce them here. These terms are necessary for the local operator to transform under the n+2 2 , n 2 ⊕ n 2 , n+2 2 representation of the Lorentz group. See Ref. [54] for details.
1. For an operator O µν... , the possible J PC quantum numbers are determined using its transformation properties under the Lorentz group.
2. For each J PC quantum number, the total number of states (i.e., L values) that are possible in an hh system with this J PC are counted.
3. These counts for all possible J PC numbers are added up.
The number of form factors obtained for the crossed channel is equal to the number of GFFs for the original channel due to crossing symmetry.

A. J PC of local operators
The J PC decompositions of the local operators of interest are independent of the hadron. These decompositions have been derived elsewhere [21][22][23][54][55][56], and here we only recall the results.  representation of the proper Lorentz group, but have opposite parity and charge conjugation quantum numbers, giving: In both cases, operators associated with the (s + 1)th Mellin moment of a quark GPD and the sth moment of a gluon GPD transform the same way under the Lorentz group, and will accordingly have the same number of generalized form factors. As a consequence these operators mix under QCD evolution. This will no longer be the case when we consider transversity operators. The transversity operator for the quark, defined in Eq. (15c), transforms under the s+1 representation of the proper Lorentz group. Both parity quantum numbers are available for each j value in the decomposition, giving two sequences of allowed J PC numbers: The helicity-flip operator for the gluon, defined in Eq. (16c), transforms under the s+4 2 , s 2 ⊕ s 2 , s+4 2 representation of the proper Lorentz group. As with the quark transversity, both parities contribute for each available j. We thus again get two sequences of J PC [23]: B. J PC counting and matching for spin-0 As explained above, the number of expected generalized form factors in the matrix element of a local operator is counted by matching the number of J PC states available to a hadron-antihadron state and the J PC decomposition of the local operator. This matching scheme has been performed for spin-half hadrons extensively elsewhere, so we will not repeat this for the spin-half case. We will look in detail at the bosonic cases spin-0 and spin-1.
First, we consider spin-0 as a simple case. The allowed J PC quantum numbers for a hadron-antihadron state are determined by the relations for two boson states: where J = |L − S|, . . . , L + S. Since S = 0 for a system of two spin-0 particles, one simply has J = L. The allowed states are given by the sequence We now proceed to match the J PC sequence arising from hadron-antihadron states with the J PC decomposition of the local operators of interest. We begin with the vector operators O Summing over all possible coincidences between these sequences, we get: where Θ(P) is 1 if the condition P within it is true, and 0 otherwise. If we reindex the sum using r = s + j + 1, we have: The number of generalized form factors for s = 0, 1, 2, 3, . . . should thus be 1, 2, 2, 3, . . ., which is exactly what is seen in existing literature on pion GFFs (see, e.g., [24]). Note that s ≥ 0 for quarks, and s ≥ 1 for gluons, since in the latter case we are taking the sth Mellin moment. We next consider the axial operators. From their decomposition, we found J PC = j (−) j+1 (−) s . There are no matches between this and the J PC of allowed hadron-antihadron states due to the mistmatch in parity. We thus reproduce the well-known result that the helicity-dependent GPDs of spin-0 particles identically vanish. The helicity-flip (transversity) GPDs, however, do not vanish for spin-0, so we have two more cases-the quark and gluon transversities-to consider.
For quark transversity operators, the cases J PC = j (−) j (−) s+1 and j (−) j+1 (−) s+1 must both be matched against J PC = j (−) j (−) j . Clearly, only the first of these sequences has matches. The limits are given by j ∈ {1, 2, . . . , s + 1}. Except for the lower limit in j being different, this looks identical to the helicity-independent case. Summing over all possible matches gives us: producing the sequence 1, 1, 2, 2, . . . for the number of GFFs when taking the (s + 1)th Mellin moment (s ≥ 0). This sequence agrees with what is seen in Ref. [25]. For gluon transversity operators, we consider O µναβµ 1 ...µ s gT , which means (in contrast to the other gluon cases) we are taking the (s + 1)th Mellin moments of the transversity GPDs. Doing so gives us the sequences Only the first of these sequences has matches with J PC = j (−) j (−) j . The number of matches we get is: The number of GFFs for the (s + 1)th moments of the gluon transversity GPDs thus follows the sequence 1, 1, 2, 2, . . . , which is the same as the number of transversity GFFs for the quarks. However, unlike in the non-helicity-flip cases, the moments of the quark and gluon operators are not offset from one-another. This curious fact was also noted for spin-half in Ref. [23].

C. J PC counting and matching for spin-1
The allowed J PC for a hadron-antihadron state consisting of two spin-1 particles are given by the relations (23) for two-boson states. The limits for J are given by J = |L − S|, . . . , L + S, and the possible values for S are given by S = 0, 1, 2. This gives us three sequences of J PC quantum numbers-one for each S value. We can codify rules for counting the number of states as follows.
For the S = 0 states, we replicate the spin-0 case. We have J = L, giving us a sequence J PC = j (−) j (−) j of allowed quantum numbers, with one L value for each J PC .
For S = 1, we have three sequences of states to count.
For S = 2, we have five sequences of states to count.
3. States for which j = L, which begin at L = 1 (for reasons also relating to forbidden states). These have J PC = j (−) j (−) j ( j ≥ 1). Adding these counts together, we have the following number of L values available in each J PC sequence:

States for which
where Θ(P) is defined to be 1 if the condition P is true, and 0 if P is false.  As an illustrative guide, we tabulate in Table I the allowed J PC quantum numbers for all three sequences up to J = 4. The L values are explicitly included, and one can confirm the formulas given in Eqs. (29) for J up to 4 by counting the L values in this table.
With the J PC sequences for spin-1 hadron-antihadron states in hand, we now proceed to count matches between these and the J PC decompositions of local operators.

GFF counting for spin-1: vector operators
We first consider the number of GFFs arising from the (s + 1)th moment of the helicity-independent quark correlator, or the sth moment of the gluon correlator. The J PC decomposition of the relevant operator is J PC = j (−) j (−) s+1 , with j ∈ {0, 1, . . . , s + 1}. The J PC counts for the hadron-antihadron states are noted in Eqs. (29). Two of the sequences noted contribute, namely those from Eqs. (29a,29c). Counting the number of matches we get gives: which produces the sequence 3, 7, 8, 12, 13, . . . for the number of GFFs. This gives us a pattern of numbers which alternative increases by 4 and 1. For illustrative purposes, we provide in Table II

GFF counting for spin-1: axial vector operators
Next we consider the helicity-dependent correlators, namely the (s + 1)th moment of the quark correlator or the sth moment of the gluon correlator. The J PC sequence for the operator is j (−) j+1 (−) s . The hadron-antihadron states this can be matched with are those appearing in Eqs. (29b,29d). Counting the number of matches we have gives us: An explicit tabulation for matches for s up to 3 is included in Table III.

GFF counting for spin-1: quark transversity operators
For the quark helicity-flip form factors, two sequences of J PC are present in the decomposition of the opeator: j (−) j (−) s+1 and j (−) j+1 (−) s+1 . Since both parties are present, all of the sequences for hadron-antihadron states notated in Eqs. (29) contribute to the count. Counting all of the matches gives us: This follows the sequence 5, 8, 14, 17, 23, . . . for the number of GFFs, with the sequence alternatively increasing by 3 and 6. An explicit tabulation for matches for s up to 3 is included in Table IV.

GFF counting for spin-1: gluon transversity operators
Lastly, we look at helicity-flip correlators for the gluon. As in the spin-0 case, we consider the (s + 1)th Mellin moment, not offsetting this like we did with the non-flip moments. The J PC decomposition of the operators give us two sequences, namely j (−) j (−) s and j (−) j+1 (−) s , with j ∈ {2, 3, . . . , s + 2}. Since both parities are present in this decomposition, all of the sequences in the hadron-antihadron state spectrum notated in Eqs. (29) contribute to our counting. Counting the matches, we find: This produces the sequence 6, 9, 15, 18, 24, . . . for the number of gluon transversity GFFs. Like with the quark transversity GFFs, this sequences alternatively increases by 3 and 6. However, in contrast to the spin-0 and spin-half cases, the number of quark and gluon transversity GFFs do not coincide. Instead, for each value of s, there is one more gluon transversity GFF. An explicit tabulation for matches for s up to 3 is included in Table V.

V. RESULTS: GENERALIZED FORM FACTORS AND POLYNOMIALITY
In this section, we give explicit expressions for the matrix elements of the local operators that appear in Mellin moments of light cone correlators when the operators are sandwiched between kets for spin-1 hadrons. The spin-0 and spin-half cases have been considered elsewhere. We start by considering the operators with free indices, and then contract their decompositions with the appropriate number of n vectors and compare to the correlator decompositions to obtain polynomiality sum rules for the GPDs. Differences between our decompositions for local gluon currents and a second decomposition in the literature [19] are discussed in App. B.

A. Vector operators
For the vector operator towers, we can write the following decomposition where the number of factors ∆ is related to the T-even or odd nature of the accompanying tensor, and the last term contains 2 terms comparable to the D-term for the spin 1/2 case. Polarization fourvectors of the initial (final) hadron are denoted by ǫ (ǫ ′ * ), where the helicity index λ (λ ′ ) is implicit in both these vectors and the bra and ket.
The number of GFFs appearing at each value of s in Eq. (34) can be counted as follows: each each odd value of s, one gets an additional D s+1,i , E s+1,i , F s+1 and G s+1 , so the count increases by 4. At each even value, an additional A s+1,i , B s+1,i , and C s+1,i appear, but the D-term-like GFFs F s+1 and G s+1 drop out, so the count increases by 1. Only 3 GFFs are non-zero at s = 0namely, A 1,0 , B 1,0 , and C 1,0 , so the count starts at 3, the sequence goes as 3, 7, 8, 12, 13, . . . , and can be written s + 3 1 + ⌊ s+1 2 ⌋ . This agrees with the number derived through J PC matching.
Combining the decomposition of Eq. (34) with that of Eq. (4) gives the following sum rules for moments of quark GPDs: Note that H 5,1 = 0, which is related to the Kumano-Close sum rule [57]. The electromagnetic structure function b 1 (x) appearing in the sum rule, which states that ∫ 1 0 b 1 (x) dx = 0, is related to the GPD H 5 at leading order and leading twist by: where x ∈ [0, 1] is the support region for b 1 . The Kumano-Close sum rule follows from H q 5,1 = 0 if H q 5 vanishes in the forward limit at negative x values. This sum rule can be violated if the sea carries tensor polarization [57], but H q 5,1 = 0 is an inviolate consequence of Lorentz symmetry.
The tower of local operators arising from the vector gluon correlator is related to the tower from the quark operator we just explored, but with the value of s offset. Specifically, the (s + 1)th moment of the vector quark correlator has the same Lorentz transformation properties and quantum numbers as the sth moment of the vector gluon correlator, where s ≥ 1 in this context since there is not a zeroth Mellin moment. This tower thus has a familiar decomposition: where we have an extra factor of 2 relative to the quark case, by convention, and where we have chosen to label the second index as µ s rather than ν in order to make the correspondence with the quark decomposition clearer. If we compare this to the definitions of the helicity-independent gluon GPDs, we get the following polynomiality relations for odd values of s: where the reflection symmetry around x = 0 of gluon GPDs was used to reduce the integration range to [0, 1], and where the GPD moments for even s are zero because of this same symmetry.

B. Axial vector operators
For the axial vector operator towers, we can write the following decomposition: where the number of factors ∆ is related to the T-even or odd nature of the accompanying remaining tensor. The number of GFFs for any value of s is 2(s + 1), which is equal to the number derived through J PC matching.
Combining the decomposition of Eq. (39) with that of Eq. (5) gives the following sum rules for moments of helicity-dependent quark GPDs: and we see that H q 3,1 = 0 and H q 4,1 = 0.
As in the vector case, the tower of local axial vector gluon operators matches up with the axial vector quark operators, with s offset to (s − 1). We obtain the following decomposition for this gluon tower: which leads us to the following sum rules for even values of s: with the moments vanishing for odd s since the helicity-dependent gluon GPDs are odd under reflection about x = 0.

C. Quark transversity operators
For the quark tensor operator towers, we can write the following decomposition of quark transversity GPDs: where we note that H

D. Gluon transversity operators
We lastly look at the tower of gluon transversity operators, for which we can write the following decomposition: Finally, this gives us the following relations for the moments of the gluon transversity GPDs, with s ≥ 0 and s even: where H

VI. CONCLUSION
In this work, we obtained polynomiality sum rules for spin-1 targets. This was accomplished by decomposing off-forward matrix elements of the local currents that appear in Mellin moments of bilocal operators. Thus, as a byproduct of this derivation, we have also obtained the decomposition of said local currents into independent generalized form factors. The method of Ji & Lebed [21] was used to count the number of independent generalized form factors that should appear in the decomposition of each local current, and we find agreement with our results.
In principle, such work could be extended to systems with greater spin. There exist spin-3/2 nuclei such as Lithium-7 which may be of future experimental interest, such as in studies of the polarized EMC effect. On an alternative route, the meaning of the generalized form factors appearing in the Mellin moments of GPDs for spin-1 systems warrants more in-depth exploration. The form factors appearing in the second Mellin moment of the helicity-independent GPDs appear in the Lorentz-covariant decomposition of the energy-momentum tensor, and encode properties of great interest such as the distribution of mass, angular momentum, and forces (including shear and pressure forces) inside the hadron.
One curious feature of spin-1 targets, which contrasts with spin-0 and spin-1/2 targets, is the appearance of two independent "D-like" terms, one each in the second Mellin moment of H 1 and H 3 . Two form factors may be necessary to describe the distribution of forces inside a hadron with more complicated structure, including three helicity states and a quadrupole moment. This, and other aspects of the spin-1 energy-momentum tensor, will be the focus of a future work.
In comparison with our decomposition, [DPS] are missing the following GFFs: A g s+1,s , B g s+1,s , C g s+1,s (s even). For the axial vector operators, we compare [DPS] Appendix B with our Eq. (41). This results in the following correspondences: