S. Liuti · A. Courtoy · G. R. Goldstein J. O. Gonzalez Hernandez · A. Rajan
Defining the Observables for Quarks and Gluons Orbital
Angular Momentum Distributions
Received: 1 December 2014 / Accepted: 19 December 2014 / Published online: 21 May 2015 © Springer-Verlag Wien 2015
Abstract We present a critical discussion of the observables that have been recently put forth to describe
quarks and gluons orbital angular momentum distributions. Starting from a standard parameterization of the energy momentum tensor in QCD one can single out two forms of angular momentum, a so-called kinetic term, generally associated with the Ji decomposition, and a canonical term from the Jaffe Manohar decomposition. Orbital angular momentum has been connected to a Generalized Transverse Momentum Distribution (GTMD), for the canonical term, and to a twist three Generalized Parton Distribution for the kinetic term. We argue that while the latter appears as an azymuthal angular modulation in the longitudinal target spin asymmetry in deeply virtual Compton scattering, due to parity constraints, the GTMD associated with canonical angular momentum cannot be measured in a similar set of experiments.
1 Introduction
A major challenge for QCD theory is understanding the angular momentum or spin structure of the nucleon. An outstanding, yet unsolved question is in fact, determining a unique gauge invariant decomposition of the total quark and gluon angular momenta, Jq, and Jg, as defined from the QCD energy momentum tensor, into their respective spin and orbital components (see Refs. [1–3] for reviews). Out of the many possibilities for such a decomposition, emerge two fundamental forms, leading to the so-called kinetic Orbital Angular Momentum (OAM)—also known as Ji decomposition [4]—and the canonical orbital angular momentum—
S. Liuti (
B
)· A. RajanPhysics Department, University of Virginia, 382 McCormick Rd., Charlottesville, VA 22904, USA E-mail: sl4y@virginia.edu
A. Rajan
E-mail: ar5xc@virginia.edu A. Courtoy
IFPA, AGO Department, Université de Liège, Bât. B5, Sart Tilman, 4000 Liège, Belgium E-mail: aurore.courtoy@ulg.ac.be
G. R. Goldstein
Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA E-mail: gary.goldstein@tufts.edu
J. O. Gonzalez Hernandez
Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Torino, via P. Giuria, 1, 10125 Turin, Italy E-mail: joghdr@gmail.com
Jaffe Manohar [5] decomposition. The Ji decomposition reads [4], 1
2ΔΣ + L
q+ Jg= 1
2. (1)
Lq, the kinetic OAM, is at variance with the canonical OAM, Lqcan,g which is defined through the
decomposi-tion [5], 1 2 = 1 2ΔΣ + L q can+ ΔG + Lgcan. (2)
In the WW limit the distributions, Lq(x), and Lqcan, coincide, their differences depend on final state interactions
contained in this case in the genuine twist three terms. In particular, d x Lq(x) = 0, while
d x Lqcan(x) = 0.
By extending the Ji sum rule to twist three one obtains [7,8], d x x Gq2(x, 0, 0) = 1 2 − d x x(Hq(x, 0, 0) + Eq(x, 0, 0)) + d x Hq(x, 0, 0) (3) where Gq2is a specific twist three Generalized Parton Distribution (GPD) [8–11] (G2was renamed E2T in the full classification of GPDs given in Ref. [12], while Hq, Eq, and Hq are the twist two GPDs contributing to the observables for Deeply Virtual Compton Scattering (DVCS) processes introduced in [4] (see reviews in Refs. [13,14]. G2(x, 0, 0) can therefore be associated with the distribution of kinetic OAM, Lq(x).
On the other side, canonical OAM is constructed by parametrizing the unintegrated correlation function [5] in the following way [6,15–17],
Lqcan = p, Λ | d x−dx⊥iψ†(x × ∇)3ψ | p, Λ = − d xd2kT kT2 M2F14(x, 0, k 2 T, 0, 0), (4) F14is a specific GTMD—or an unintegrated over intrinsic-kT GPDs—appearing in the decomposition of the
vector component of the unintegrated quark–quark correlation function at twist two [12].
Equation (4) provides a plausible, intuituive identification which is inferred from the definition of canonical OAM originally suggested in [5]. Nevertheless, the fact that one can consider matching OAM onto experimental observables, only through a specific off-forward unintegrated parton distribution, or GTMD, entails various complications, from questions on both its factorizability and renormalizability in QCD, to the definition of a deeply virtual scattering process which could be sensitive to F14. Such complications are not present for the GPD, G2, although there exists no obvious, straightforward partonic interpretation of this twist three quantity. In either case it is necessary to identify processes where OAM can be observed directly. As shown in Ref. [18] this can be readily done for G2by connecting the helicity amplitudes combinations which contribute to DVCS at twist three [10,11] and that contain this structure function. to an observable, namely the sin 2φ modulation in the longitudinal Target Spin Asymmetry (TSA), Asin 2LUφ[18]. This term has already been mea-sured, and found to be quite substantial at HERMES [19] and CLAS [20]. It is also presently been analyzed at Jefferson Lab [21].
In this contribution we address once more these issues, with the specific goal to provide additional support for pursuing experiments sensitive to both canonical and kinetic OAM. A more profound physical understanding of OAM may emerge only by defining a way to measure it. We discuss whether this goal can be met in either case, F14and G2, and which experimental setup would be required.
2 Parity Constraints
In Ref. [18] we demonstrated that there was a fundamental reason behind the claim that it was “not known how to extract Wigner distributions or GTMDs from experiments” [15], namely we explained how this inherent difficulty was a consequence of parity constraints on the helicity amplitudes which enter the general cross section formulation [22,23].
Differently from the Transverse Momentum Distributions (TMDs) and the Compton Form Factors (CFFs) which can be extracted from semi-inclusive and deeply virtual exclusive lepton nucleon scattering, GTMDs cannot be obtained from two body scattering processes. In particular, the DVCS process factors intoγ∗-quark elastic scattering and two body quark-proton scattering. In such a process, it is always possible to define a
Center-of-Mass (CoM) system where the two transverse momenta, kT andT cannot be independent from
one another (i.e. they belong to a single hadronic scattering plane).
To extract F14from experiment one first writes the helicity amplitudes for theγ∗p scattering process. The quark-proton scattering helicity amplitudes content of F14was identified as [18],
i ¯k
1
TΔ2− ¯k2TΔ1
M2 F14= A++,+++ A+−,+−− A−+,−+− A−−,−−, (5)
where ¯kT = (kT+ kT)/2, and we defined, AΛλ,Λλ,Λ(Λ) and λ(λ) being the proton and quark initial (final)
helicities, respectively. The helicity amplitudes obey the following parity relation,
A−Λ−λ,−Λ−λ = ηP(−1)Λ
−λ−Λ+λ
A∗Λλ,Λλ, (6)
ηP being the phase factor accounting for intrinsic parity and spin.
Therefore, for the F14contribution to the nucleon matrix elements to be non-zero, at least one pair of the helicity amplitudes must be imaginary, at variance with the other spin conserving structure functions. While for GPDs and TMDs this would be unphysical (parity violating) in the proton spin non flip case, for GTMDs, by allowing for relative phases among the amplitudes, the combination that forms F14can indeed be imaginary, as one simultaneously moves away from a collinear description, i.e. as ¯k andΔ are let to vary independently from one another. The specific combination of amplitudes giving rise to F14is therefore consistent with parity conservation so long as one gives up the idea of the quark proton scattering occurring in a single hadronic plane. We therefore here acknowledge that it is preferable to use an alternative choice of words to “parity odd” to describe this rather complicated situation.
This does not imply that F14cannot represent OAM. In fact, by observing that,
iσi j¯kT iΔT j = i j kΣk¯kT iΔT j = Σ · (¯kT × T) ≡ Σ3(¯kT × T)3, (7) one sees that because of the action ofΣ on the proton matrix elements, one has that iσi j¯kT iΔT j transforms in an opposite way to helicity, namely to the corresponding structures G14, in the GTMD sector, and H in the GPD sector [12,18].
Differently from helicity, which is promptly observable, the matrix element corresponding to F14is parity even. Although this is the source of the measurability issue for F14, it does not interfere with its identification with OAM which is also a parity even quantity.
Summarizing this part, to measure F14and G11and be consistent with the parity transformation properties in QCD one needs to define therefore an additional hadronic plane. Because F14has the kinematic factor for a longitudinally polarized target going to an unpolarized quark and spectator, it is clear that the hadronization process of the active quark will involve unpolarized functions. Also, as GTMDs depend on the momentum transfer, one has to consider exclusive processes, which rules out “dihadron” fragmentation functions. An exclusive process of the type:γ∗+ p → γ + π++ π−+ p will be required. The 4-momenta are set as
q+ p = q+ p1+ p2+ p. There are five invariants, s= (p + q)2, t = (p− p)2, s12= (p1+ p2)2, s13=
(p1+ p)2, t1= (q − p1)2. All other invariants can be written in terms of these. With this kinematical set of variables one can fix the kT of the incoming quark, as we will elaborate on in future work.
3 Impact Parameter Parton Distribution Functions
The recent developments in [6–8,15,17,18] allow us to see how both canonical and kinetic OAM are described by integrals over different parton distributions in impact parameter space. In both cases one can see how a net OAM is generated by quark distributions that are displaced from the origin in the transverse plane. For F14, the displacement is originated as follows,
F.T. −i¯k1Δ2− ¯k2Δ1 M2 F14(x, 0, ¯kT, T) = − 1 M2 i j T ¯k i T ∂ ∂bjF14(x, 0, ¯k T, b), (8)
where, Ti j = −+i j, and,
F14(x, 0, ¯kT, b) = d2ΔT (2π)2 e −ib·TF 14(x, 0, ¯kT, T), (9)
u by (fm) bx (fm) -0.4 0.0 0.4 -0.4 0.0 0.4 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 u by (fm) bx (fm) kT d bx (fm) |kT| = 0.3 GeV x = 0.3 (i/M 2 ) F.T. [ ( k1Δ2 - k2Δ1 )F14 ] -0.4 0.0 0.4 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 d bx (fm) |kT| = 0.3 GeV x = 0.3 (i/M 2 ) F.T. [ ( k1Δ2 - k2Δ1 )F14 ] kT
Fig. 1 Fourier transform of the correlator components defining F14, Eq. (8) for x= 0.3, k1= 0.3 GeV, and k2= 0. The u and d
quarks contributions are represented on the LHS and RHS, respectively (adapted from Ref. [25])
u by (fm) bx (fm) -0.4 0.0 0.4 -0.4 0.0 0.4 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 d bx (fm) x = 0.3 F.T. ( i Δ1 G2 ) -0.4 0.0 0.4 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10
Fig. 2 Fourier transform of the correlator components defining E2T, Eq. (10) at x= 0.3, integrated over kT. The u and d quarks
contributions are represented on the LHS and RHS, respectively (adapted from Ref. [25]
Notice that ¯kT needs to be kept at a fixed value in order to see this displacement, i.e. an integration over ¯kT
gives a zero result. This configuration corresponds to OAM generated through circular motion in the x− y plane [15].
For the configuration corresponding to the unintegrated G2/E2T(for more details on the specific GTMDs see Refs. [12,18]), the distortion in the transverse plane is described by the Fourier transform of the distribution,
F.T. −i i j T Δj M E2T(x, 0, ¯kT, T) = −1 M i j T ∂ ∂bj E2T(x, 0, ¯kT, b) (i, j = 1, 2), (10)
where we used the notation of [12]. In this case kT can be parallel toT, so Eq. (10) gives a non zero result
when integrated over kT, and OAM points in the direction orthogonal to both SL andT, consistently with
the representation given in Ref. [24].
In Fig.1we show the distributions in the transverse plane corresponding to Eq. (8) for both the u and d quarks, at the kinematics defined by: x = 0.3, ξ = 0, ¯k1= 0.3 GeV, and ¯k2 = 0. In Fig.2we show Eq. (10) integrated over ¯kT. All functions were obtained in the reggeized quark-diquark picture of Ref. [26,27].
4 Conclusions
In conclusion, we have analyzed the issue of observability of both canonical and kinetic OAM. Canonical OAM can be identified with the second moment in kT of the GTMD F14, a formal proof that such quantity
is related to OAM having been given in Ref. [6]. The observability of OAM is however hampered by the fact that F14, and an analogously a GTMD in the axial-vector sector, G11, cannot be connected to any of the GPDs and/or TMDs, thus making it challenging to define physical processes which would be sensitive to these quantities. In spite of the fact that non-zero results for F14can be obtained from either models, or by direct calculations on the lattice, a process that selects this quantity has not been yet identified. The physical content of the models, whether these are “perturbative” or arise from “effective field theories” cannot be taken as a proof of the existence of an observable. However, we notice that each of the models explored so far giving a non-zero result for F14, carry some remnant of confinement, while only one model calculation that clearly does not have confinement—the quark-target model—(as an “ensemble of free quarks”, with no gluon across the vertices) gives zero for F14(see analogous calculation in Ref. [28] ]on g2). We take this as an indication that the gauge link structure of F14plays a fundamental role, as already suggested in [17] and that, looking at future studies, its connection with the final state interactions implicitly present in the twist three definition of OAM through G2/E2T will give key information on the nature of OAM.
Finally, we reiterate that in our analysis, while reinforcing the use of the LF, we give a physical motivation for the fact that F14has not yet been associated to any observable, that goes beyond simply stating the issue [12,15]. Our explanation is founded on the transformation properties of the unintegrated correlator under parity which do not allow for the specific combination of helicity amplitudes generating F14to be observed in any given single hadronic plane. This prompts the derivation of an extension of the “master formula” used so far to describe both semi-inclusive and exclusive lepton-proton scattering [22]. At the same time we point out that the transformation property of the matrix element associated with F14is a distinct issue that should not be confused with the observability of the quantity through its decomposition in quark-proton helicity amplitudes. In this respect, F14is consistent with the transformation under parity of OAM.
Acknowledgments S.L. benefitted from the many lively discussions in particular with Ben Bakker, Stan Brodsky, Michael
Engelhardt, Paul Hoyer, Cheung Ji, Cedric Lorcé, Andreas Metz and Barbara Pasquini, in the congenial atmosphere provided by the workshop organizers. This work was funded in part by the Belgian Fund F.R.S.-FNRS via the contract of Chargée de recherches (A.C.), and by U.S. D.O.E. Grant DE-FG02-01ER4120 (S.L., A.R.). A.R. is grateful to the Université de Liège for partial funding.
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