Abstract
We provide the complete decomposition of the local gaugeinvariant energymomentum tensor for spin1 hadrons, including nonconserved terms for the individual parton flavors and antisymmetric contributions originating from intrinsic spin. We state sum rules for the gravitational form factors appearing in this decomposition and provide relations for the mass decomposition, work balance, total and orbital angular momentum, mass radius, and inertia tensor. Generalizing earlier work, we derive relations between the total and orbital angular momentum and the Mellin moments of twist2 and 3 generalized parton distributions, accessible in hard exclusive processes with spin1 targets. Throughout the work, we comment on the unique features in these relations originating from the spin1 nature of the hadron, being absent in the lower spin cases.
Introduction
In recent years, a lot of attention has been given to the energymomentum tensor (EMT) as a fundamental object of study in hadronic physics and QCD, see e.g. [1, 2] and references therein. Hadronic matrix elements of the (local) EMT operator for quarks and gluons are parametrized in terms of gravitational form factors (GFFs),^{Footnote 1} just as hadronic matrix elements of the charge current operator are parametrized in terms of electromagnetic form factors. The GFFs encode properties that are of great interest, such as the hadronic mass and angular momentum sum rules and their spatial distributions [1, 4,5,6], and can shed light on novel properties of hadrons, such as the way that stress and shear forces are distributed within them [2, 4, 7, 8]. The topic places itself in the more general quest for a thorough understanding of the hadron structure, in which the spatial and momentum distributions of quarks and gluons play a crucial role.
Much of the recent literature on the hadron structure and QCD EMT has focused on either spin1 / 2 [2, 5, 6, 8,9,10] or spin0 [11,12,13] systems, see also [14,15,16,17,18] for recent lattice studies. In the former case, the proton has been the predominant object of study. This is natural, as nucleons are often understood as the primary building blocks of nuclear matter, and understanding the mass and spin decomposition of the proton is a necessary step in understanding the origin of most visible mass. In the latter case, the pion is of great interest not only for the unique role it plays in dynamical chiral symmetry breaking, which is believed to be the origin of the majority of hadron mass, but also because of its simplicity as a system and the ability to study its properties (including components of its EMT) on the lattice [19].
On the other hand, the hadronic physics community has in general dedicated little attention to the internal structure of hadrons of spin higher than 1/2. From a theoretical point of view, a full picture of higher spin hadrons and nuclei is desirable because it would serve in elucidating QCD dynamics: spin1 (and higher spin) systems carry information on nonnucleonic degrees of freedom, i.e. the dynamics beyond quarks and gluons confined within the individual nucleons [20].
Information on spin1 hadrons would allow us to thoroughly study such different parton contributions and dynamics in the spirit, for instance, of the theoretical calculations of the gravitational form factors for vector mesons in holographic QCD [21] and on the lattice [22]. Being nearly the only experimentally available hadronic spin1 target, the deuteron has attracted a fair amount of attention over the past decades. It is the simplest bound state of more than one nucleon and, therefore, it has been of prime importance to unravel the nature of the nuclear binding. On the other hand, its internal structure and dynamics are the ultimate effect of the interactions between the elementary constituents, and this makes the deuteron a promising avenue towards understanding how QCD produces the force that binds nucleons together in nuclei [23]. After the first measurement by the HERMES collaboration of a tensor polarized collinear structure function of the deuteron [24], the socalled \(b_1\) function defined in [25], it became clearer that going beyond the singlenucleon formulation is needed to describe the experimental data, especially in specific regions of the parton momentum range [26,27,28,29,30].
The same arguments hold for the study of the gluonic content of higher spin hadrons, which requires once again to account for additional gluon functions in momentum and coordinate space that are exclusive to tensor polarized structures and therefore related to spin1 or higher. This fact has stimulated a recent interest in the theoretical [31, 32] and lattice community [33, 34]. The deuteron is thus expected to play a major role in the 12 GeV program at Jefferson Lab (JLab) dedicated to spin1 targets [35].
As a fundamental entity encoding the spatial and mechanical properties of hadrons, the EMT of a spin1 system such as the deuteron contains much of this dynamical information that is of interest to the nuclear physics community. This information is encoded in GFFs familiar from the EMT of spin0 and spin1/2 systems, but also within a host of additional form factors novel to spin1 systems. This is analogous to the spin1 electromagnetic current containing one more form factor than the spin1/2 current, and has a similar origin. A spin1 system has an additional degree of freedom, which can manifest itself in higher multipole moments (in this case, a quadrupole moment) or a tensor polarization mode. A full understanding of spin1 systems requires a complete categorization of all the independent Lorentz structures that can appear in its EMT, and an elucidation of the physical significance of the GFFs that appear with these structures.
Expressions for the decomposition of the EMT for spin1 hadrons have appeared earlier in Refs. [21, 36, 37]. In the present work, we provide the complete EMT decomposition that also includes all nonconserved terms (appearing incompletely in [37]) and we study the properties of and the relations between the GFFs that parametrize the local operator for the EMT. More specifically, we derive new sum rules and present expressions for the mass and angular momentum decomposition of a spin1 hadron in terms of the new structures for quarks and gluons.
The results in this paper may be relevant for the experiments at JLab and a future EIC [38] and for the proposed fixedtarget projects @LHC [39], where different polarized hadrons and nuclei can be employed. Current data for spin1 tomography is rather scarce, with HERMES having measured deeply virtual Compton scattering (DVCS) on the deuteron with both unpolarized [40] and polarized targets [41]. In these measurements, hadrons were not detected in the final state, but simulations were used to select a sample of enhanced coherent deuteron contribution. More recently, Jefferson Lab has measured deeply virtual \(\pi ^0\) production on the deuteron [42] and a recent letter of intent allows for coherent deuteron DVCS measurements [43]. The latter should also be possible in Hall B using the ALERT detector [44]. Finally, generalized distribution amplitudes (GDAs) for the rhorho meson pair, accessible in the crossed reaction \(\gamma ^*\gamma \rightarrow \rho \rho \) [45,46,47], can be related to the rhomeson GFFs similarly to the pion case [12] and could also potentially be studied at Belle II.
This work is organized as follows. In Sect. 2, we give a full decomposition of the most general form that the EMT of a spin1 hadron can take. This section also contains sum rules that follow immediately from energymomentum conservation. In Sect. 3, we calculate the multipole moments of matrix elements of the spin1 EMT. Mass and angular momentum decompositions are derived in this section, along with additional sum rules and a workenergy balance relation. Section 4 explores the connections between the EMT and Mellin moments of twist2 and twist3 generalized parton distributions. Finally, in Sect. 5, we summarize our results. In addition, in Appendix A the form factors counting technique is reviewed, Appendix B and C contain additional information on Lorentz projectors and the polarization bilinears useful to obtain the parametrizations of the EMT, and Appendix D displays the covariant parametrization of the GPD correlator.
Decomposition of the energymomentum tensor
The goal of this section is to construct the most general possible parametrization for the EMT of an onshell, spin1 hadron in terms of GFFs. A variety of definitions exists for the EMT in QCD (for a review, see [1]), but here we work with the gaugeinvariant kinetic form of the QCD EMT \(T^{\mu \nu }=T^{\mu \nu }_q+T^{\mu \nu }_g\), where
with \( \overset{\leftrightarrow }{D^{\mu }} =(\overset{\rightarrow }{\partial ^{\mu }} \overset{\leftarrow }{\partial ^{\mu }}) 2igA^{\mu } \). Due to the presence of spin, the QCD EMT is in general not symmetric under exchange of its free Lorentz indices, with the entirety of the asymmetry in the quark contribution. The EMT \(T^{\mu \nu }\) is a conserved current, with the symmetric and antisymmetric components being separately conserved. Accordingly, we consider the general form of the EMT in two layers: the symmetric component of the EMT and the full asymmetric EMT.
Symmetric EMT
For a spin1 system, there are only six possible independent rank2 Lorentz structures that are symmetric, \(\mathsf {P}\)even, \(\mathsf {T}\)even, consistent with the hermiticity property, Lorentzcovariant, linear in each of the initial and final state polarization vectors, and conserved [21, 36]. This is fewer than the seven Lorentz structures that arise from the (1, 1) representation of the Lorentz group (see Appendix A), meaning one of those Lorentz structures is nonconserved. The conserved symmetric EMT takes the following form
where M is the hadron mass, \(P=(p'+p)/2\) is the average fourmomentum, \(t=\Delta ^2\) with \(\Delta =p'p\) is the fourmomentum transfer, and for each fourvector a, b one has \(a_{\{\mu } b_{\nu \}}=(a_\mu b_\nu +a_\nu b_\mu )/2\). Energy and momentum must be conserved in a closed system, so in this decomposition of the symmetric EMT a sum over all partons is understood. The partial EMT for quarks and gluons does not have to be conserved however, so there are three additional independent Lorentz structures that can appear
where \(a=q,g\). Summing over all partons, we should recover (2) which implies the following sum rules
Note that we have named the GFFs to agree with the conventions in [37], although we find an additional nonconserved pure trace GFF, in agreement with the counting in Appendix A. It should also be noted that unlike the total GFFs \(\mathcal {G}_i(t)\), the partial GFFs \(\mathcal {G}_i^a(t)\) are usually scale and scheme dependent.
Asymmetric EMT
When the constituents of a system possess intrinsic angular momentum, the EMT is in general expected to be asymmetric, see e.g. [1, 48, 49] for recent discussions. This is a simple consequence of the conservation of the generalized angular momentum \(\partial _\mu M^{\mu \alpha \beta }=0\), with \(M^{\mu \alpha \beta }=x^{\alpha } T^{\mu \beta }x^{\beta } T^{\mu \alpha }+S^{\mu \alpha \beta }\) and where \(S^{\mu \alpha \beta }\) is the intrinsic generalized angular momentum tensor, which when combined with the conservation of the EMT implies that \(T^{\alpha \beta }T^{\beta \alpha }=\partial _\mu S^{\mu \alpha \beta }\). In agreement with the counting in Appendix A, we find only two antisymmetric Lorentz structures satisfying all the constraints. The most general form of the EMT is therefore
where \(a_{[\mu } b_{\nu ]}=(a_\mu b_\nu a_\nu b_\mu )/2\). Since one of the two new tensors is nonconserved, energymomentum conservation imposes the additional sum rule
This is an interesting new feature of the spin1 target, since a spin0 target has no antisymmetric part and a spin1/2 target has only a conserved contribution. This has to do with the fact that the intrinsic generalized angular momentum tensor for a scalar field vanishes, \(S^{\mu \alpha \beta }_0=0\), and is completely antisymmetric for a Dirac field, as \( S^{\mu \alpha \beta }_{1/2} =\frac{1}{2}\,\epsilon ^{\mu \alpha \beta \lambda } \overline{\psi }\gamma _\lambda \gamma _5\psi \) with \(\epsilon _{\,0123}=+1\). In the case of a massive vector field \(V^{\mu }\), the intrinsic generalized angular momentum tensor reads \(S^{\mu \alpha \beta }_1=2F^{\mu [\alpha }V^{\beta ]}\), so that \(\partial _\alpha \partial _\mu S^{\mu \alpha \beta }_1\ne 0\), opening the possibility of having a nonvanishing intrinsic energy dipole moment beside intrinsic angular momentum [48].
Since we will be working with the kinetic form [1] of the QCD EMT, the QCD equations of motion imply that the antisymmetric part of the EMT can be expressed in terms of the axialvector current as follows [1, 50, 51]
The matrix elements of the axialvector current being parametrized as [52, 53]
with the notation \(\epsilon _{\mu \alpha \beta P}=\epsilon _{\mu \alpha \beta \lambda }P^{\lambda }\), we find from considering the matrix elements of (7)
and \(\mathcal {G}_{10}^g(t)=\mathcal {G}_{11}^g(t)=0\). Note that the vanishing of the antisymmetric part of \(T^g_{\mu \nu }\) has to do with the impossibility of writing down the gluon spin contribution in a form that is both local and gauge invariant [1]. We thus find that the antisymmetric part of the EMT for a spin1 hadron is conserved.^{Footnote 2}
Multipole moments of the energymomentum tensor
Much of the interesting information about a hadron’s mechanical properties that is contained in the EMT is encoded by the multipole moments of the EMT matrix elements. These include static observables such as mass, angular momentum, the inertia tensor, and so on, but additionally include information about how each of these decomposes into quark and gluon contributions. The decomposition of hadron mass and angular momentum into quark and gluon contributions – and the latter also into spin and orbital angular momentum components – has been a major focus of recent literature on the EMT. As with much of the other literature on the QCD EMT, this focus has been primarily directed towards spin1/2 systems (predominantly the proton) and spin0. In this section, we elaborate on the mechanical properties of spin1 hadrons encoded by the multipole moments of their EMT, including both properties that are analogous to the lesserspin cases and those that are new to spin1.
Mass decomposition and balance equation
The mass decomposition and balance equation associated with a spin1 target are obtained in terms of the following properly normalized matrix element of the EMT [5, 8]
Using the covariant expression for the density matrix of a spin1 system [54, 55] (see also App. C for more details)
where the projector onto the subspace orthogonal to \(p^{\mu }\) is given by
and the covariant vector and tensor polarizations by
we find
The first two Lorentz structures do not depend on the spin and are indeed common to all targets. The last Lorentz structure is new. It is related to the target tensor polarization and therefore does not appear in the case of spin0 or spin1/2 targets. Because of Poincaré invariance, the forward matrix element of the total EMT has to assume the form^{Footnote 3}
from which we conclude that
using the constraints in Eq. (4).
The Lorentzinvariant coefficients in Eq. (15) can be interpreted in terms of proper internal energy and pressurevolume work [5, 8]. In the target rest frame, the partial internal energy is given by^{Footnote 4}
and the partial isotropic pressure–volume work by
The new feature of a spin1 target is the presence of a partial pressure–volume work anisotropy
associated with the tensor polarization. The mass decomposition then takes the form
and the balance equations read
Angular momentum decomposition
As explained in detail in [1, 56], higher spatial moments of the energymomentum distribution are ambiguous if defined naively as \(\langle \langle \int \text {d}^3r\,r^j T^a_{\mu \nu }(0,\vec {r})\rangle \rangle \). The reason for this is because information about the spatial distribution is lost in the forward limit \(\Delta \rightarrow 0\). Spatial distributions can only be defined in frames where no energy is transfered to the system \(\Delta ^0=\vec {P}\cdot \vec {\Delta }/P^0=0\). In this work, we will only consider the Breit frame \(\vec {P}=\vec {0}\) where threedimensional spatial distributions of the EMT are defined as [4, 6, 8]
with \(P^0=\sqrt{M^2+\frac{\vec {\Delta }^2}{4}}\). The dipole moment of the spatial distribution is then given by
Using the Breitframe expansion of the polarization fourvector bilinear derived in Appendix C, we find
Clearly, the only nonvanishing dipole moment is associated with the momentum distribution \(\langle T_a^{0k}\rangle (\vec {r})\) and is orthogonal to the vector polarization of the target \(\mathcal {S}^{\mu }=(0,\vec {s})\). It simply originates from the parton orbital angular momentum (OAM)^{Footnote 5}
In QCD, we then find that the parton total angular momentum (AM) is given in the target rest frame by
We naturally recover \(J^i_q=L^i_q+S^i_q\) with the quark spin contribution being given by
For gluons we simply have \(J^i_g=L^i_g\) because no local gaugeinvariant definition of the gluon spin does exist, see e.g. [1] for a recent detailed discussion.
Beside the term proportional to \(\mathcal {G}^a_5\) already obtained in [37], we also find a contribution from the \(\mathcal {G}^a_7\) GFF describing the nonconserved part of the EMT. Interestingly, such a contribution cannot appear for spin0 and 1/2 targets, since in these cases the nonconserved terms are necessarily of the form of a pure trace and hence decoupled from AM. Summing over all the partons, the \(\mathcal {G}_7\) contribution drops out according to Eq. (4) and we get the AM constraint
first derived in [21].
Mass radius and inertia tensor
Beside mass, two other important quantities characterizing the energy distribution can be defined, namely the mass radius and the inertia tensor. Both are expressed in terms of the following secondorder moments
The mass radius defined as \(R_M=\sqrt{\langle r^2\rangle }\) with
gives an idea of the spatial extension of the energy distribution. The inertia tensor [58, 59] defined as
allows one to determine the moment of inertia \(I_a^{\vec {n}} =I^{ij}_a n^in^j\) of the system about an arbitrary axis \(\vec {n}\) passing through the center of mass, which coincides in the Breit frame with the origin [48]. It is related to the mass quadrupole moment
which measures the deviation from a spherical distribution of the energy.
Using once more the Breitframe expansion of the polarization fourvector bilinear derived in Appendix C, we find
with
The squared mass radius, inertia tensor and mass quadrupole moment are then given by
As expected, the quadrupole moment in spin1 hadrons is different from zero due to the presence of the tensor polarization.
GFFs as moments of GPDs
The connection between the EMT and partonic distributions has long been a topic of consideration (see e.g. [60,61,62]). Generalized parton distributions (GPDs) in particular allow for GFFs to be extracted from their Mellin moments. Since GPDs parametrize the nonperturbative structure contributing to hard reactions such as deeply virtual compton scattering (DVCS) and virtual meson production, they are the most promising avenue for experimentally exploring the form factors appearing in the EMT decomposition.
Leadingtwist GPDs have been extensively studied for their polynomiality relations [62]. A specific case allows the second Mellin moments of helicityindependent twist2 GPDs to be related to the nontrace GFFs appearing in the symmetric component of the EMT. Such relations have been studied not only for spin0 [12, 63, 64] and spin1/2 [8, 61, 62], but also spin1 systems [21, 22, 37, 65].
Beyond leading twist, the Penttinen–Polyakov–Shuvaev–Strikman (PPSS) sum rule [66] relates the second moment of a twist3 GPD to the orbital angular momentum carried by quarks in spin1/2 hadrons, and it has also been shown [1] that twist4 GPDs contain information about the nonconserved GFF in a nucleon. In this section we will derive a spin1 analogue of the PPSS sum rule.
We proceed to derive sum rules for second Mellin moments of GPDs up to twist3.^{Footnote 6} Consider the quark and gluon vector correlators.^{Footnote 7}
where n is a lightlike fourvector and [y, z] denotes a straight Wilson line joining the spacetime points y and z. These correlators enter the description of deeply virtual Compton scattering and can be parametrized up to twist 3 as follows^{Footnote 8} [53, 68]
The first five terms (\(H^a_i\)) correspond to the twist2 GPDs, and the remaining nine (\(G^a_i\)) are purely twist3. In the quark sector, the twist3 GPDs satisfy the relation \(\int \mathrm {d}x\,G_i^q=0\) as a consequence of the charge current conservation. We suppressed the dependence of the GPDs on the parton longitudinal momentum x, longitudinal momentum transfer \(\xi =(\Delta n)/2(Pn)\), and squared momentum transfer \(t=\Delta ^2\) for conciseness of notation, and made use of the Sudakov decomposition of fourvectors \((n^2=\bar{n}^2=0,\, n\bar{n}=1)\)
The second Mellin moment of the lightfront string operators are related to the EMT up to twist 3 as follows
Taking the offforward matrix element on both sides allows us to relate seven of the GFFs to moments of leadingtwist vector GPDs [37, 65]. Comparing the Mellin moment of Eq. (40) with the decomposition in Eq. (2) in the symmetric frame \(P^{\mu }_T=0\), we find the following relations for quarks and gluons at twist 2
Since the GFFs appear as second Mellin moments of GPDs, they are special cases of generalized form factors, which correspond to arbitrary moments of GPDs. We note the following correspondence between the GFFs as defined in this work, and the \(s=2\) generalized form factors appearing in Ref. [65]:
We also find the following relations for quarks at twist 3
where we have used Eq. (9b).
Based on Eq. (27), we find that the total quark or gluon AM in a state with maximal vector polarization along the zdirection can be expressed in terms of twist2 GPDs as follows
which is nothing but Ji’s relation [61] for spin1 targets. Summing over quark and gluon contributions, we recover the spin sum rule derived in [21]
Unlike the case of spin1/2 targets [66, 69, 70], the quark OAM (26) requires not only a pure twist3 GPD but also a twist2 GPD
This twist2 GPD contribution is associated with the tensor polarization and is therefore absent in the case of spin1/2 targets.
Quark and gluon contributions to mass and pressurevolume work involve trace terms and hence twist4 GPDs. Only the partial pressurevolume work anisotropy can be related to a twist2 GPD
Summing over quarks and gluons, the mass sum rule and the balance equations imply the following constraints
These relations can be interpreted as statements of energymomentum conservation for collinear parton distribution functions (PDFs). Using the notation of [31, 71],^{Footnote 9} the unpolarized and tensorpolarized PDFs are given respectively by \(f_1^a(x) = H_1^a(x,0,0)\) and \(f^a_{1LL}(x) = H^a_5(x,0,0)\). In the case of quarks, they enter the deep inelastic structure functions \(F_1\) and \(b_1\) at leading order and leading twist as
The corresponding gluon PDFs mix with the quark ones and contribute to the DIS structure functions at higher order in the \(\alpha _s\) perturbative expansion. Thus Eq. (50) is a statement of the momentum sum rule for PDFs. The collinear structure functions for a scattering off a tensor polarized targets were first introduced in [20, 25, 72] and the separate contributions of quarks and gluons to Eq. (51) were previously discussed in [73, 74].
Conclusion
In this work, we found the most general form that the asymmetric, gaugeinvariant kinetic energymomentum tensor (EMT) of a spin1 hadron can take. Expressions were given for both the full EMT and the partial EMT due to a single parton type. We explored the physical meaning of the gravitational form factors appearing in this EMT, including sum rules imposed by conservation of momentum and angular momentum, the decomposition of spin1 hadron mass, and multipole moments of the EMT. We also explored connections between the gravitational form factors and other functions describing partonic structure, such as axial form factors and generalized parton distributions up to twist three.
The spin1 EMT was found to contain many more gravitational form factors than the corresponding spin0 or spin1/2 EMTs. A total of 11 form factors are present in the EMT decomposition, with 9 of these in the symmetric part and 2 in the antisymmetric part. Among them, 6 structures have no analogues in the lowerspin cases and are related to the presence of tensor polarization modes. They contribute to features new to spin1 hadrons such as a quadrupole moment and possibly a nonzero intrinsic energy dipole moment aside from the intrinsic angular momentum.
The structure of the spin1 EMT is rich, and there remains much to be explored. The pressure and shear force distributions encoded within it, and how these differ from the simpler spin0 case, are worthy of detailed study. It is also worth investigating how the EMT of a composite spin1 hadron compares to that of an elementary spin1 particle, such as a photon or one of the heavy electroweak gauge bosons. These topics will be the subject of future work, along with illustrative model calculations.
Experimentally, measurements of coherent hard exclusive processes with deuteron targets are possible at JLab and the future EIC with forward detectors. Extraction of the chiraleven vector GPDs from these measurements would then constrain the deuteron gravitational form factors through the Mellin moments of these GPDs. Similarly, extraction of GDAs for the rhorho meson pair from the crossed reaction \(\gamma ^*\gamma \rightarrow \rho \rho \) at Belle would constrain the rho meson gravitational form factors.
Note
Shortly after the present work was completed and made available, the independent work [75] appeared, dealing with the EMT of spin1 hadrons. The results are consistent with ours and the document contains an especially useful comparison between the different nomenclature for the GFFs present in the literature.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.]
Notes
We call the form factors “gravitational” since the EMT is usually understood as the source of gravitational interactions. However, we do not measure in practice these form factors through gravity and we do not know the exact form that a theory of quantum gravity will take. Thus, it is unclear whether gravitation sees the symmetric, Belinfanteimproved EMT as in general relativity or an asymmetric EMT as in EinsteinCartan theory [3]. Despite this, we refer to all form factors appearing in either the symmetric or asymmetric EMT as “gravitational” form factors.
We note that this conclusion holds for the gaugeinvariant kinetic EMT, but may not hold for the (nongaugeinvariant) canonical EMT. This is because the gluon contribution to the canonical EMT is asymmetric, and due to the fact that \({\partial _\alpha \partial _\mu S^{\mu \alpha \beta }_1\ne 0}\), the divergences of the symmetric and antisymmetric parts of the canonical gluon EMT are not expected to separately vanish.
This is consistent with the total fourmometum of the system being given by \(p^{\mu }=\langle \langle \int \text {d}^3r\, T^{0\mu }(0,\vec {r})\rangle \rangle \).
One can of course avoid having recourse to the rest frame and use instead the projectors \(p^{\alpha } p^{\beta }/M^2\) and \(P_{\alpha \beta }\) [5].
It is possible to consider up to twist4 for quarks and twist6 for gluons, but contributions beyond twist3 are less promising for future phenomenological studies.
For the complete covariant parametrization, see Appendix D.
Note that Ref. [31] uses \((Pn)=1\).
Interestingly, since \(\sigma ^{\mu \nu }=\frac{i}{2}[\gamma ^{\mu },\gamma ^{\nu }]\) we can interpret the Clifford algebra \(\{\gamma ^{\mu },\gamma ^{\nu }\}=2g^{\mu \nu }\mathbb I\) as the condition of vanishing quadrupole.
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Acknowledgements
We are grateful to Bernard Pire and Maxim Polyakov for useful discussions. The work of SC and CL was supported by the Agence Nationale de la Recherche under the projects No. ANR18ERC10002 and ANR16CE310019. AF was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, Contract No. DEAC0206CH11357 and an LDRD initiative at Argonne National Laboratory under Project No. 2017058N0.
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Appendices
Appendix A: Form factor counting
Let us consider the total number of GFFs we may expect to appear in the decomposition of the spin1 EMT. This will depend on the EMT definition we use. First, let us consider the most general possible rank2 tensor that is even under charge conjugation. It decomposes into several representations of the Lorentz group, namely (0, 0), \((1,0)\oplus (0,1)\), and (1, 1), which are respectively 1, 6, and 9dimensional representations. These representations have \(J^{PC}\) quantum number decompositions given in Table 1.
With these \(J^{PC}\) numbers, we can use the method of Ji & Lebed to count form factors [76].
We consider a crossed channel matrix element \(\langle h\bar{h}\hat{\mathcal {O}}0\rangle \), and write out all the possible \(J^{PC}(L)\) quantum numbers that \(h\bar{h}\) can have up to \(J=2\). We then count the matches between these \(J^{PC}\) and those in our hypothetical rank2 tensor. Such lists can be found in Table 2.
With these lists in hand, we can count the number of form factors that are contributing to the general rank2 tensor by each of the representations it decomposes into. The numbers are given in Table 3.
Each of the rows in Table 3 tells us something meaningful about form factor counts.

The first row gives us trace terms, which can be thought of as “nonconserving” terms. There is one such term for both spin0 and spin1/2, and they are associated with the \(\bar{C}(t)\) GFFs. For spin1, there are two nonconserving terms of this kind.

The second row tells us the number of GFFs appearing in the antisymmetric part of the EMT.

The third row tells us the number of GFFs in the decomposition of a traceless symmetric rank2 tensor, and they correspond to the number of GFFs that are known to appear in the second Mellin moments of twist2 vector GPDs.
Appendix B: Parametrization with Lorentz projectors
The matrix elements of the most general local, gaugeinvariant EMT for a massive spin1 target can be written as
where the effective vertex \(T_{\mu \nu ,\alpha \beta }(P,\Delta )\) is a Lorentz tensor constructed out of the invariant tensors \(g_{\mu \nu }\) and \(\epsilon _{\mu \nu \alpha \beta }\), and the available fourvectors P and \(\Delta \). Because of the onshell relations \((p^{\prime }\epsilon ^{\prime *})=(p\epsilon )=0\), we choose to discard terms involving \(P_\alpha \) or \(P_\beta \) as they are not independent from those involving \(\Delta _\alpha \) or \(\Delta _\beta \). Finally, discrete symmetries impose further constraints on the effective vertex. Invariance under parity and time reversal implies that
where a bar over a fourvector or a Lorentz index stands for the paritytransformed object \(\bar{a}^{\mu }=a^{\bar{\mu }}=(a^0,\varvec{a})\). The hermiticity property leads to the last constraint
It follows from these constraints that the effective vertex is real, and that its symmetric part under the exchange \(\alpha \leftrightarrow \beta \) involves only even powers of \(\Delta \), whereas the antisymmetric part involves only odd powers of \(\Delta \). Note also that the LeviCivita tensor need not be considered since it is the only object with negative intrinsic parity and since the product \(\epsilon _{\mu \nu \alpha \beta }\epsilon _{\rho \sigma \tau \lambda }\) can be rewritten in terms of the metric only.
Interestingly, we can find a complete parametrization of the effective vertex in terms of the Lorentz projectors onto (0, 0), \((1,0)\oplus (0,1)\), and (1, 1) representations
We recognize in particular the generators of Lorentz transformations \((M^{\mu \nu })^{\alpha \beta }=2i(D^{\mu \nu })^{\alpha \beta }\) associated with the vector representation \((\frac{1}{2},\frac{1}{2})\). Taking into account all the constraints discussed above, we find that the most general effective vertex can be written as a linear combination of the following eleven independent Lorentz structures (omitting the \(\alpha ,\beta \) indices for convenience)
At first sight, it seems that these are at odds with the counting in Appendix A, since we found three structures involving the projector onto the (0, 0) representation and six involving the projector onto the (1, 1) representation. In fact, the counting in Appendix A corresponds to the projections applied on the pair of indices \(\mu \nu \). The structures corresponding to the (0, 0) part of the effective vertex are then the only two pure trace terms \(I_{\mu \nu }\) and \(g_{\mu \nu }Q_{\Delta \Delta }\). The structures corresponding to the (1, 1) part of the effective vertex are those obtained from removing the trace of the seven other symmetric terms \(P_\mu I_{\nu P}\), \(\Delta _\mu I_{\nu \Delta }\), \(Q_{\mu \nu }\), \(\Delta _\mu Q_{\nu \Delta }\), \(\Delta _\nu Q_{\mu \Delta }\), \(P_\mu P_\nu Q_{\Delta \Delta }\), and \(\Delta _\mu \Delta _\nu Q_{\Delta \Delta }\).
This technique has the advantage of showing the connection with lower spin targets. For spin0 targets we have \(I_{\mu \nu }=\frac{1}{4} g_{\mu \nu }\), \(D_{\mu \nu }=0\) and \(Q_{\mu \nu }=0\), and for spin1 / 2 targets we have \(I_{\mu \nu }=\frac{1}{4} g_{\mu \nu }\mathbb I\), \(D_{\mu \nu }=i\sigma _{\mu \nu }\) and \(Q_{\mu \nu }=0\).^{Footnote 10} According to (B5) this leads respectively to 3 and 5 GFFs, in agreement with Appendix A.
Appendix C: Polarization fourvector bilinears
Matrix elements of spin1 targets can be written in terms of polarization fourvector bilinears contracted with some Lorentz tensors
In the forward limit, the polarization fourvector bilinear reduces to the covariant density matrix which can be written as [54, 55]
where the vector and tensor polarization operators read
and are orthogonal to \(p^{\mu }\) as required by the onshell relation \((p\epsilon )=0\). Clearly, the covariant vector and tensor polarizations can be taken so as to satisfy the same properties as the associated operators
They carry the dependence on the polarizations \(\lambda \) and \(\lambda '\), which has been omitted for convenience.
The generators of Lorentz transformations in the \((\frac{1}{2},\frac{1}{2})\) representation being given by
the unit, vector and tensor polarization operators take the explicit form
where \(P_{\mu \nu }\) is the projector onto the subspace orthogonal to \(p^{\mu }\) given by
All these operators are orthogonal to each other, so that the normalization, covariant vector and tensor polarizations can simply be obtained as follows
Denoting the restframe fourmomentum as \(k^{\mu }=Mg^{\mu 0}\), the general polarization fourvector bilinear can be written as
where the standard canonical rotationless Lorentz boost tensor reads
or in a more covariant form [79, 80]
Denoting for convenience the restframe covariant vector and tensor polarizations as
we find that the general polarization fourvector bilinear can be expressed as
In the forward limit, we recover the standard expression for the covariant density matrix
where
As noticed in the case of Dirac bilinears [81], in the forward limit \(\Delta =0\) the dependence on the restframe fourmomentum \(k^{\mu }\) can be absorbed into the pdependent covariant polarization. However, in the general offforward case \(\Delta \ne 0\) this is not possible anymore and the kdependence remains explicit. This dependence is unavoidable in a relativistic theory and comes from the fact that, because of the noncommutativity of boosts, canonical polarization of a massive particle has to be defined from a boost of the restframe polarization to the frame of interest [82].
In the Breit frame \(\vec {P}=\vec {0}\), we find that the general polarization fourvector bilinear reduces to
Appendix D: Covariant parametrization of Generalized Parton Distributions
Just like for local operators, hadronic matrix elements of nonlocal operators can be parametrized in a covariant form if one includes in the list of available fourvectors the direction of nonlocality, namely the lightlike fourvector n in the case of parton distributions like GPDs. The most general parametrization of the quark GPD correlator (39a39b), that respects the constraints imposed by hermiticity, parity and timereversal, reads
with
and \(F_i = F_i(x,\xi ,t)\). Since \(\mathcal {V}^{\mu , \alpha \beta }(P,\Delta ,n)\) must be invariant under a rescaling of the lightlike direction \(n\mapsto \alpha n\), factors of M / (Pn) appear whenever necessary. Additional factors of the hadron mass M have also been included to keep GPDs dimensionless. The relation between \(F_i\) and the standard GPD basis \(H_i,G_i\) of (40) can easily be obtained by projection onto the twist2 and 3 parts.
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Cosyn, W., Cotogno, S., Freese, A. et al. The energymomentum tensor of spin1 hadrons: formalism. Eur. Phys. J. C 79, 476 (2019). https://doi.org/10.1140/epjc/s1005201969813
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DOI: https://doi.org/10.1140/epjc/s1005201969813