Transversity generalized parton distributions in spin-3/2 particles

The definitions of the quark and gluon transversity generalized parton distributions (GPDs) in spin-3/2 particles are obtained in the light-cone gauge. It is found that they contain 16 independent components for each parton. Their even or odd property is found in terms of skewness variable, and the odd transversity GPDs vanish in the forward limit. There are 16 amplitudes with helicity flip of quark or gluon. We conclude that all the amplitudes, unpolarized, polarized, and transversity amplitudes, have a common factor $F(\zeta)$ carrying all the complex part. Here, the variable $\zeta$ is the helicity difference $\zeta=(\lambda' -\lambda)-(\mu' - \mu)$ for the amplitude $\mathcal{A}_{\lambda' \mu',\lambda\mu}$. This kinematical factor is associated with the transfer of the orbital angular momentum in the quark- and gluon-spin-3/2 particle scattering amplitudes. We also derive another main physical quantity, transversity distribution, from the transversity GPDs in the forward limit for the spin-3/2 particles.


Introduction
To investigate the construction of hadrons from quarks and gluons degrees of freedom, which are the elementary particles described by quantum chromodynamics (QCD), the notion of generalized parton distributions (GPDs) was introduced [1][2][3][4][5][6][7].The GPDs can provide a more detailed account of the internal structure of hadrons in terms of their elementary constituents than the parton distributions and form factors (FFs).Similar to the theoretical constructs, one of the related experiments utilized to investigate hadron structures is the deep inelastic scattering process, which indicates the parton distributions, including the unpolarized, polarized, and transversity distribution functions at the leading twist.We know that the GPDs depend on three variables: the longitudinal momentum fraction x, the square of the transferred momentum t, and the skewness variable ξ which is defined by the transferred longitudinal momentum, rather than just one variable x in the parton distribution functions (PDFs).The reaction that involves GPDs is the off-forward Compton scattering, which consists of at least one off-shell photon [7,8], including the deeply virtual Compton scattering (DVCS) process [3,5].Furthermore, the GPDs reduce to the PDFs in the forward limit.The spacelike GPDs are measured by the DVCS and meson-production processes at the Thomas Jefferson National Accelerator Facility (JLab) and in the CERN-AMBER project.The timelike GPDs are investigated by the two-photon process or the timelike Compton scattering at KEKB [9,10] and will be studied possibly at BESIII.In addition, there are future GPD projects at EICs [11,12], by hadron reactions at the Japan Proton Accelerator Research Complex (J-PARC) [13][14][15][16], and by neutrino reactions at Fermilab [17][18][19].
Corresponding to transverse momentum independent parton distributions [20], there are three types of GPDs, i) unpolarized GPD corresponding to the unpolarized parton distribution function, the probability of finding a parton with the longitudinal momentum fraction x in the unpolarized hadron; ii) polarized GPD corresponding to the polarized distribution function, the number density of a parton with the longitudinal momentum fraction x and spin parallel minus spin antiparallel to that of the polarized hadron; iii) transversity GPD corresponding to the transversity distribution function h 1 (x) [21,22] (called ∆ T (x) in Ref. [20], ∆(x) in Ref. [23] and δq(x) in Ref. [24]), which denotes the number density of a parton with the longitudinal momentum fraction x and polarization parallel to that of the hadron with transverse polarization minus the number density with antiparallel polarization.The transversity distribution function h 1 (x), which can be obtained from transversity GPDs in the forward limit as described above, has been studied fundamentally by Ralston et al. [25], Artru et al. [26], Jaffe et al. [27,28], Cortes et al. [29], Ji [30], and Barone et al. [31].Additionally, there is no logical reason to assume that the transversity parton distribution is significantly smaller than the other two distributions [31][32][33][34][35][36][37].For nearly two decades, the spin-1/2 and -1 quark transversity distributions have been measured by some experiments [38][39][40][41] and the gluon transversity is possibly measured by JLab [42], EIC (Electron-Ion Collider) [11,43,44], and EicC (Electron-Ion Collider in China) [12], and Fermilab [37,45,46].
Based on the parity and time reversal invariances, there are 2(2J + 1) 2 independent GPDs for each parton (either quark flavor or gluon) in a spin-J hadron at leading twist [8].These independent GPDs are discussed in Sec.3.5.4 of Ref. [8] and Ref. [47] for spin-3/2 hadrons, and it is also found by using the parity invariance in Eq. ( 7) of this paper.
Half of the total GPDs consist of the unpolarized and polarized GPDs that do not involve the helicity flip, while the remaining half is distributed to the transversity GPDs involving the helicity flip, i.e., there are 16 independent transversity GPDs for the spin-3/2 hadrons.The quark transversity GPDs are chiral-odd since the corresponding operator flips the quark chirality, in contrast to the chiral-even unpolarized and polarized GPDs.Moreover, the helicity flip distributions of the quark and gluon will mix in the evolution of the transversity GPDs.Many studies on the definition of GPDs [7,8,22,24,48] and the model calculations [49][50][51] have been conducted for the low spin (J ≤1) hadrons.
It should be stressed that the unpolarized and polarized GPDs of spin-3/2 particles have been decomposed, defined, and calculated in our recent papers [47,52].The priority of this work is to decompose and derive the definitions of the transversity GPDs for the spin-3/2 system.We expect that this series studies of GPDs for spin-3/2 particles give valuable references for possible future experiments at EIC, EicC, BESIII, KEK-B factory, and J-PARC.Moreover, the study of the gluon transversity distribution on ∆ isobar is also necessary to understand the non-nucleonic degrees of the hadrons in nuclei [23,[53][54][55].Experimentally, the timelike GPDs could be measured in principle at BESIII and KEK-B [10] even for spin-3/2 particles like e + e − → ∆ ∆ or Ω Ω.Furthermore, there are recent interests on the transition GPDs for N → ∆ at JLab and J-PARC [13,56], so that spin-3/2 GPD experimental project could become a realistic one in future.
The main content is concentrated in Sec. 2. Firstly, the definitions of transversity GPDs and their corresponding amplitudes, along with their symmetry properties, are explained and discussed.In the next two subsections, the specific components of the parton transversity GPDs respectively for the quark and gluon, and the interpretations of their symmetry properties are shown.The corresponding tensor form factors and tensor charges are obtained by the derived sum rules.Moreover, the amplitudes and the relations connecting transversity GPDs with transversity parton distributions are derived and elucidated.To summarize our work, we provide a short discussion in Sec. 3.

Conventions, correlators and symmetry properties
The GPDs are defined through matrix elements of the non-local parton operators.Following the conventions of Ji [7,8], the leading twist GPDs can be defined by the matrix element where O is the non-local operator at a light-like separation of the corresponding GPDs.The p (p ′ ) and λ (λ ′ ) respectively denote the momentum and helicity of the initial (final) state.In this work, the light-cone coordinate is employed and any four-vector v can be rewritten as The scalar product of any two four-vectors is u Moreover, the light-cone vector n = (0, 2, 0 ⊥ ) is needed and n 2 = 0.In addition, we use the same kinematical variables with our previous work [47], where k − ∆/2 (or k + ∆/2) is the initial (or final) parton momentum as represented in Fig. 1.Note that here we use ∆ instead of q in Ref. [47] to stand for the relative momentum.There are some other conventions in the following, like a In Ref. [47], the unpolarized (polarized) quark or gluon GPDs are defined and decomposed for a spin-3/2 system when we choose the non-local operator being O q = ψ/ nψ ( ψ/ nγ 5 ψ) or where G nµ = G ρµ n ρ in Eq. (1).In the present work, ψ(x) and G µν (x) are employed to respectively denote the quark field and gluon field strength tensor, and the corresponding dual field strength tensor is defined as Gρσ (x) = 1  2 ϵ ρσµν G µν (x).There is a Wilson line W − 1 2 z − , 1 2 z − along a light-like path between the two fields at positions of − 1 2 z − and 1 2 z − in the non-local operator O.The Wilson line is defined as [8] W where P stands for the path-ordering from b to a and A(x) represents the gluon field.In this work, the light-cone gauge A + = 0 for the gluon field is considered and then W − 1 2 z − , 1 2 z − = 1, so that the Wilson line does not appear in the operators.
Equation (1) with non-local operators of z) defines the quark and gluon transversity GPDs as and where i and j are the transverse indices, and the operator Ŝ in Eq. ( 5) stands for the removal of the trace and the symmetrization between i and j.As the transversity GPDs have different definitions from the helicity non-flip GPDs, the quark and gluon transversity GPDs have the different decompositions since gluon has the more index j.
One can also define the corresponding quark and gluon amplitudes [8] as where the label µ (µ ′ ) represents the emitted (re-absorbed) parton helicity.The amplitudes satisfy the constraints of [8,22] A from hermiticity, parity and time reversal transformations.Table 1 lists all the operators carrying the parton helicity at leading twist.Notice that the helicity flip operators on the quark have another forms, because of iσ +1 γ 5 = −iiσ +2 .Amplitudes (6) with µ ′ = µ contain 16 non-flip amplitudes in the spin-3/2 system, which have been explicitly given in Ref. [47].Analogously, the parity invariance restricts 16 independent quark or gluon transversity GPDs for the spin-3/2 particles when µ ′ ̸ = µ.Note that the helicity label ± on the partons of the quark stands for ± 1 2 and those of the gluon for ±1 according to the corresponding spin.More symmetry properties, such as hermiticity, light-front parity, and light-front time reversal, are also satisfied [22], and they do not provide any further constraints on the number of GPDs.

Quark transversity GPDs for a spin-3/2 particle
The quark transversity GPDs are defined by the matrix elements of transverse non-local quark-quark correlator in Eq. ( 4) as where The tensor function H qT,i,α ′ α (x, ξ, t) can be decomposed to 16 terms corresponding to only 16 independent tensor structures.Similar to our previous analysis [47], there are two ways to obtain the sufficiently tensor structures: 1) write down all the possible structures; 2) utilize the direct product between the transvesity spin-1/2 and unpolarized spin-1 structures or between the transversity spin-1 and unpolarized spin-1/2.We can then reduce the redundant tensor structures using some on-shell identities (seen also Appendix B).Consequently, the tensor function of H qT,i,α ′ α is decomposed at twist 2 by imposing Hermiticity, parity invariance, and time-reversal invariance as where the variables x, ξ, t in the quark transversity GPDs H qT i are omitted, and M is the mass of the spin-3/2 particle.
There is another equivalent definition of quark transversity GPDs using the matrix element of iσ ni γ 5 instead of iσ ni in Eq. ( 4).Due to the relation [24] iσ thus the equivalent definition is where which is given by HqT,i ′ ,α ′ α = iϵ 0i ′ i3 H qT,i,α ′ α .The quark transversity GPDs H qT i in Eq. ( 13) is the same as the corresponding ones in Eq. (10).
The relations of the amplitudes in Eq. ( 7) determines the even or odd property with respect to the variable of skewness ξ as where H qT 3,4,10,11,14,15 are odd and others are even in ξ.Clearly, all the odd GPDs go to zero when ξ = 0.It might be possibly insignificant but necessary to show the corresponding tensor FFs.In comparison with the unpolarized and polarized GPDs, one can consider the local quark-quark operator corresponding to quark transversity GPDs (4) as which give the ξ-independant tensor FFs.Since the first moments of GPDs in Eq. (10) give the tensor FFs in Eq. ( 15) by sum rules, the tensor structures corresponding to tensor FFs are contained in Eq. (10).In addition, the multi-n terms, like ∆ and so on, should not exist in the tensor structures corresponding to tensor FFs because there is only a position, µ or ν, permitted to place n.
Considering the transversity GPDs (10) and the terms with non-vanishing integration, we get the decomposition of Eq. ( 15) in terms of the tensor FFs as and the tensor FFs G qT i (t) are related with the transversity GPDs by the sum rules dx H qT j (x, ξ, t) = 0 with j = 3, 4, 9, 10, 11, 13 ∼ 16. ( There are three [24], five [22] and seven tensor FFs for the spin-1/2, -1 and -3/2 system, respectively.Since there there is only a nonzero form factor G qT 1 (0) − G qT 2 (0) describing the quark tensor charge.Beyond the Standard Model, the tensor charge is explained as the number of the electric dipole moment carried by the corresponding quark [57,58].
In addition, the quark amplitudes with helicity flip can be derived according to their definitions in Eq. ( 6).We use the same notations as in our previous work [47] |p The notation C represents the transfer of the transverse momentum between the initial and final hadron states, which induces that the hadron obtains the transfer of the orbital angular momentum that violates the helicity conservation, i.e. λ ′ − λ = µ ′ − µ for A λ ′ µ ′ ,λµ .In Eq. ( 18), the transversity indices x and y instead of 1 and 2 are used to avoid ambiguity.Moreover, C will be real when the azimuthal angle of the four-vector ∆ + 2ξP [8,22] is zero, i.e. (1 − ξ)p y = (1 + ξ)p ′y , and be zero in the forward limit.
Incorporating the amplitudes with helicity non-flip in Ref. [47], we find that any amplitude A λ ′ µ ′ ,λµ has a common factor where C ζ represents the ζ powers of C, θ(ζ) is the Heaviside step function with θ(0) = 1 2 and F (0) = 1.We conclude that F (ζ) has some properties, such as F (ζ ̸ = 0) = 0 in the forward limit and F * (ζ) = F (−ζ).Therefore, the amplitudes are real if the factor F (ζ) is excluded and we can define the real amplitude as and the relations in Eq. ( 7) becomes the real form, For the quark, the factors µ ′ and µ are µ ′ (µ) = ± 1 2 due to the quark spin.We believe that the low spin particles [22,24] have the analogous character 1 .The specific forms of the 16 quark amplitudes are 1 There might be a typo in (C15) of Ref. [22] according to this property.
Recall that the transversity distribution function h 1 (x) can be measured in semi-inclusive deep inelastic scattering from transverse polarized targets in the scaling limit using the "Collins asymmetry" [59], where e q , f q 1 , ∆ 0 T D h q , and D h q are the quark charge, the unpolarized parton distribution function, the spin-dependent fragmentation function, and the spin-independent fragmentation function, respectively.

Gluon transversity GPDs
According to the corresponding definition (5), the gluon transversity GPDs at twist 2 can be obtained from where where the operator Ŝij implies the symmetrization and removal of trace between indices i and j, and the variables x, ξ, t in the gluon transversity GPDs H gT i are omitted too.Similarly, the relations in Eq. ( 7) constrains the even or odd behavior in terms of skewness ξ as where H gT 2,8,9,14,15 are odd and others are even in ξ.Odd GPDs vanish at ξ = 0. We also attempt to give the tensor FFs of the local current G µν (0)G ρσ (0) corresponding to the gluon transversity GPDs and The gluon transversity GPDs in Eq. ( 41) can determine the tensor FFs F µν,ρσ,α ′ α g by sum rules.Eliminating the n ρ n σ n δ and n ρ n σ n δ n ν terms since there are only two n's in the operator G ni (0)G jn (0) and considering the ξ symmetry ( 42), there remains 6 tensor FFs defined as and the sum rules collecting the tensor FFs and transversity GPDs are ( The forward limit tells that the local current G µν (0)G ρσ (0) contains a global symmetry and contributes to the unknown gluon tensor charge G gT 5 (0).Analogous to the quark amplitudes with helicity flip, the ones of gluon can be derived according to the definition in Eq. ( 6).There is also a common factor F (ζ) defined in Eq. ( 19) in the expressions of the gluon amplitudes.Different from the quark case, the spin-1 gluon takes µ ′ (µ) = ±1 instead of ± 1 2 .Using the same conventions in Eq. ( 18) and the definition of the real amplitude (20), we have the 16 gluon amplitudes with helicity flip as The other amplitudes A ′ g i+,j− can be obtained from the relation (21).In the forward limit, similar to the discussions on quark in Sec.2.2, there is no orbital contribution, i.e.
defined in Refs.[21,23] and ∆(x) has the similar interpretation with h 1 (x) in Eq. (38).Since (P •n)ū(p, λ ′ )u(p, λ) = M ū(p, λ ′ )/ nu(p, λ) in Eq. ( 41) is satisfied in the forward limit, H gT 5 (x, 0, 0) and H gT 11 (x, 0, 0) correspond to the same tensor structure and all the gluon transversity distributions are given by the spin-1 component.There is an obvious reason: the helicity flip of gluon leads to a change of two units of angular momentum, which is impossible for the spin-1/2 component.The minus sign in Eq. ( 62) is explained by the same reason as in Eq. ( 38).

Summary and discussion
In this work, the quark and gluon transversity GPDs of spin-3/2 particles are respectively derived and given for the first time.We find that there are 16 independent transversity GPDs for each parton, constrained by the parity invariance.The even or odd property with respect to ξ of the obtained transversity GPDs can help us to determine the corresponding tensor FFs that contain two tensor charges, where the quark tensor charge may describe the electric dipole moment from the quark beyond the Standard Model and the gluon tensor charge is unknown.We need more study for their physical meanings.There are seven tensor FFs for the local quark current and six for gluon from the sum rules connecting the transversity GPDs and the tensor FFs.
The amplitudes with helicity flip in terms of transversity GPDs are obtained.We conclude that all the amplitudes, including helicity non-flip and flip, have the common factor F (ζ), which carries the whole complex part and represents the transfer of the orbital angular momentum.In the forward limit, no orbital contribution, i.e.F (ζ ̸ = 0) = 0, constrains the only helicity conserved amplitudes, which give the corresponding parton distributions, to exist and the amplitudes with helicity flip give the transversity distributions of each parton.Following the Rarita-Schwinger spinor and our explanations, the transversity distribution of the parton in the spin-3/2 system is comprised of the spin-1/2 and spin-1 system, nevertheless the gluon transversity distribution only includes the spin-1 since the gluon helicity flip changes two units of angular momentum and it leads to the violation of the angular momentum conversation in the spin-1/2 system.