Transversity generalized parton distributions for the deuteron
TTransversity generalized parton distributions for the deuteron
W. Cosyn and B. Pire Department of Physics and Astronomy, Ghent University, Proeftuinstraat 86, B9000 Ghent, Belgium Centre de Physique Th´eorique, ´Ecole Polytechnique, CNRS, 91128 Palaiseau, France (Dated: October 24, 2018)Transversity generalized parton distributions (GPDs) appear as scalar functions in the decompo-sition of off-forward quark-quark and gluon-gluon correlators with a parton helicity flip. For a spin1 hadron, we find 9 transversity GPDs for both quarks and gluons at leading twist 2. We studythese twist-2 chiral odd quark transversity GPDs for the deuteron in a light cone convolution model,based on the impulse approximation, and using the lowest Fock-space state for the deuteron.
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I. INTRODUCTION
The factorization of hard exclusive amplitudes in the generalized Bjorken regime [1, 2] as the convolution of gen-eralized parton distributions (GPDs) with perturbatively calculable coefficient functions allows to get access to the3 − dimensional structure of nucleons or nuclei through the extraction of the various quark and gluon GPDs. Theconnection between GPDs and parton-hadron helicity amplitudes allows an easy counting of twist-2 GPDs : thereare 2(2 J + 1) GPDs for each quark flavor (or for the gluon) in a nucleus of spin J . Half of these GPDs correspondto parton helicity non-flip, the other half - which are dubbed transversity GPDs - correspond to parton helicity flip.In the quark case, the helicity non-flip GPDs are chiral even, while the helicity flip GPDs are chiral-odd. The helicityflip and non-flip sectors evolve independently in the renormalization scale. Moreover, the quark and gluon sectors donot mix in the evolution of transversity GPDs.Nuclear GPDs [3–12] obey the same rules as nucleon GPDs and are accessible through coherent exclusive processeswhich may be isolated from incoherent processes where the target nucleus breaks during the hard interaction. Asthe simplest composite nucleus, the deuteron is a fascinating object to scrutinize in order to understand the QCDconfinement mechanism [13]. The study of hard reactions which allow to access its quark and gluon structure is atthe heart of the on-going physics program at Jefferson Lab (JLab) as well as the future electron-ion collider (EIC)program. The study of the deuteron GPDs should allow to understand more deeply the relation between the deuteronand nucleon structures. The spin 1 nature of the deuteron makes it a particularly rich object from the point of viewof building the spin from the constituent spins and orbital angular momenta.Contrarily to the nucleon GPDs which have been the subject of many works – both theoretically and experimentally– the study of deuteron GPDs is still in its infancy; its founding blocks are the definition of helicity non-flip quarkand gluon GPDs [14] and the calculation of deeply virtual Compton scattering (DVCS) and deep exclusive mesonproduction (DEMP) amplitudes [15, 16] in the coherent reactions on a deuteron. First results on coherent hardexclusive reactions have been obtained at JLab [17]. In the present paper, we study the transversity sector ofdeuteron twist-2 GPDs which was left aside up to now.The paper is organized as follows. The transversity GPDs of spin 1 hadrons are the objects of study in Sec. II:we start with introducing the necessary kinematic variables in Subsec. II A, list the general correlators and theirsymmetry properties in Subsec. II B, and subsequently introduce the transversity GPDs for spin 1 and commenton their properties for quarks (Subsec. II C) and gluons (Subsec. II D). In the following Section III, we outline theconvolution formalism for the deuteron, with kinematic variables defined in Subsec. III A, the deuteron light-frontwave function and chiral odd nucleon GPDs discussed in Subsecs. III B and III C, and finally the convolution modelis presented in Subsec. III D. Results obtained in the convolution formalism for transversity helicity amplitudes andGPDs in the quark sector are discussed in Sec. IV and sum rules from the first moments of the quark transversityGPDs are covered in Sec. V. Conclusions are stated in Sec. VI. The notation, sign and normalization conventions usedthroughout this article are summarized in Appendix A, while App. B contains a summary of the properties of parityand time reversal symmetries on the light front. The relations between transversity helicity amplitudes and GPDsfor spin 1 hadrons are listed in App. C, and a minimal convolution model used to obtain some analytical results isoutlined in App. D.We shall not deal with the phenomenology of these GPDs in this article, and leave this topic for further work. Sim-ilarly, the polynomiality properties of spin 1 GPDs and the connection between general moments and the generalizedform factors will be discussed elsewhere. At present, no parameterization for the nucleon gluon transversity GPDs isavailable [18]. Consequently, in this article we do not consider calculations in the convolution model for the deuterongluon transversity GPDs. a r X i v : . [ h e p - ph ] O c t I. TRANSVERSITY GPDS FOR SPIN 1 HADRONS
The central objects that define GPDs are Fourier transforms of gauge-invariant off-forward parton correlators,where the initial (final) hadron in the correlator matrix element has four-momentum p ( p (cid:48) ), light-front helicity λ ( λ (cid:48) )and mass M . For quarks these correlators take the form (cid:104) p (cid:48) λ (cid:48) | ¯ ψ ( − κn )Γ ψ ( κn ) | p λ (cid:105) , (1)with Γ a general Dirac structure, and the two quark fields are separated along a light-like four-vector n µ ( n = 0).In this work, we use the lightcone gauge ( nA ) = 0, so no explicit Wilson lines appear in the correlators. Similarcorrelators can also be introduced for gluons (see Subsec. II B). These objects encode long distance, strongly coupledQCD dynamics and can be diagramatically represented by the blob in Fig. 1. p p'k k' FIG. 1: Diagrammatic representation of an off-forward parton correlator.
A. Kinematical variables
We introduce the standard kinematic variables for these matrix elements, being the average hadron momentum P ,momentum transfer ∆, skewness ξ (which determines the longitudinal momentum transfer) and t : P = p + p (cid:48) , ∆ = p (cid:48) − p , t = ∆ ,ξ = − (∆ n )2( P n ) . (2)Depending on the skewness ξ , the momentum transfer squared t (which is negative) has a maximum value t = − M ξ − ξ , (3)and we can write t − t = − M ξ − ξ + 2( pp (cid:48) ) . (4)The four-vector 2 ξP + ∆ is orthogonal to n [((2 ξP + ∆) n ) = 0] and has norm(2 ξP + ∆) = − (1 − ξ )( t − t ) . (5)The following combination of kinematic variables occurs a lot in formulas in this work, so an extra dimensionlessvariable is defined: D ≡ (cid:112) ( t − t )(1 − ξ )2 M . (6)As we study parton correlators for spin 1 particles, we consider a basis of three polarization four-vectors, both for theinitial (unprimed four-vectors) and final (primed four-vectors) spin 1 hadron state [14], normalized to ( (cid:15) ∗ ( i ) (cid:15) ( i ) ) = − (cid:15) ( i ) p ) = ( (cid:15) (cid:48) ( i ) p (cid:48) ) = 0] : (cid:15) (0) µ = 1 M (cid:18) p µ − M ξ n µ ( P n ) (cid:19) ,(cid:15) (cid:48) (0) µ = 1 M (cid:18) p (cid:48) µ − M − ξ n µ ( P n ) (cid:19) ,(cid:15) (1) µ = − (cid:112) (1 − ξ )( t − t ) (cid:18) (1 + ξ ) p (cid:48) µ − (1 − ξ ) p µ − ξ ( t − t ) − t ξ n µ ( P n ) (cid:19) ,(cid:15) (cid:48) (1) µ = − (cid:112) (1 − ξ )( t − t ) (cid:18) (1 + ξ ) p (cid:48) µ − (1 − ξ ) p µ + ξ ( t − t ) + t ξ n µ ( P n ) (cid:19) ,(cid:15) (2) µ = (cid:15) (cid:48) (2) µ = 1 (cid:112) (1 − ξ )( t − t ) (cid:15) µναβ p (cid:48) ν p α n β ( P n ) . (7)We use (cid:15) (0) = (cid:15) (0) ,(cid:15) ( ± ) = ∓ e ± iφ ( (cid:15) (1) ± i(cid:15) (2) ) / √ , (8)as definite light-cone helicity polarization four-vectors for the initial hadron, and similar expressions for the primedpolarization four-vectors and the final hadron. In Eq. (8), φ is the azimuthal angle of the four-vector ∆ + 2 ξP . B. Correlators and symmetry properties
The following quark-quark correlators determine the leading twist-2 quark GPDs [1, 2, 19] V qλ (cid:48) λ = (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | ¯ ψ ( − κn )( γn ) ψ ( κn ) | p λ (cid:105) ,A qλ (cid:48) λ = (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | ¯ ψ ( − κn ) γ ( γn ) ψ ( κn ) | p λ (cid:105) ,T q iλ (cid:48) λ = (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | ¯ ψ ( − κn )( in µ σ µi ) ψ ( κn ) | p λ (cid:105) , (9)where i is a transverse index and transverse is relative to the light-like four-vectors n and ¯ n ( n = ¯ n = 0, n ¯ n =1).The decomposition for the first two (vector V qλ (cid:48) λ , axial vector A qλ (cid:48) λ ) was considered for spin 1 hadrons in Ref. [14]and determines the 9 chiral even quark GPDs for spin 1 (5 for V qλ (cid:48) λ , 4 for A qλ (cid:48) λ ). The decomposition of the tensorcorrelator T q iλ (cid:48) λ is given below (Subsec. II C) and determines 9 spin 1 chiral odd quark GPDs.Similarly, the following gluon-gluon correlators determine the leading twist-2 gluon GPDs [1, 2, 19]: V gλ (cid:48) λ = 2( P n ) (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | Tr (cid:2) n α G αµ ( − κn ) G βµ ( κn ) n β (cid:3) | p λ (cid:105) = 1( P n ) (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | Tr (cid:2) n α (cid:0) G αR ( − κn ) G βL ( κn ) + G αL ( − κn ) G βR ( κn ) (cid:1) n β (cid:3) | p λ (cid:105) ,A gλ (cid:48) λ = − i ( P n ) (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | Tr (cid:104) n α G αµ ( − κn ) (cid:101) G βµ ( κn ) n β (cid:105) | p λ (cid:105) = 1( P n ) (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | Tr (cid:2) n α (cid:0) G αR ( − κn ) G βL ( κn ) − G αL ( − κn ) G βR ( κn ) (cid:1) n β (cid:3) | p λ (cid:105) ,T g ijλ (cid:48) λ = − P n ) (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | Tr ˆ S (cid:2) n α G αi ( − κn ) n β G βj ( κn ) (cid:3) | p λ (cid:105) , (10) Our sign convention for the Levi-Civita tensor and other quantities is summarized in App. A i, j are transverse indices, the operator ˆ S implies symmetrisation and removal of trace, and transverse four-vector components a R/L are defined as a R = a x + ia y ,a L = a x − ia y . (11)Again, the decomposition of V qλ (cid:48) λ , A qλ (cid:48) λ for spin 1 hadrons has been discussed earlier [14] and the composition of thetensor correlator T g ijλ (cid:48) λ is given below in Subsec. II D.As T g RLλ (cid:48) λ = T g LRλ (cid:48) λ = 0, there remain two independent matrix elements for the tensor gluon-gluon correlator: T g RRλ (cid:48) λ = − P n ) (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | Tr (cid:2) n α G αR ( − κn ) n β G βR ( κn ) (cid:3) | p λ (cid:105) ,T g LLλ (cid:48) λ = − P n ) (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | Tr (cid:2) n α G αL ( − κn ) n β G βL ( κn ) (cid:3) | p λ (cid:105) . (12)Hermiticity and discrete light-front symmetries impose the following constraints on the correlators : • Hermiticity V ∗ λ (cid:48) λ (∆ , P, n ) = V λλ (cid:48) ( − ∆ , P, n ) ,A ∗ λ (cid:48) λ (∆ , P, n ) = A λλ (cid:48) ( − ∆ , P, n ) ,T q R/L ∗ λ (cid:48) λ (∆ , P, n ) = − T q L/Rλλ (cid:48) ( − ∆ , P, n ) ,T g RR ∗ λ (cid:48) λ (∆ , P, n ) = T g LLλλ (cid:48) ( − ∆ , P, n ) . (13) • Light-front parity P ⊥ V λ (cid:48) λ (∆ , P, n ) = V − λ (cid:48) − λ ( (cid:101) ∆ , (cid:101) P , ˜ n ) ,A λ (cid:48) λ (∆ , P, n ) = − A − λ (cid:48) − λ ( (cid:101) ∆ , (cid:101) P , ˜ n ) ,T q R/Lλ (cid:48) λ (∆ , P, n ) = − T q L/R − λ (cid:48) − λ ( (cid:101) ∆ , (cid:101) P , ˜ n ) ,T g RRλ (cid:48) λ (∆ , P, n ) = T g LL − λ (cid:48) − λ ( (cid:101) ∆ , (cid:101) P , ˜ n ) . (14) • Light-front time reversal T ⊥ V λ (cid:48) λ (∆ , P, n ) = ( − λ (cid:48) − λ V λλ (cid:48) ( − (cid:101) ∆ , (cid:101) P , ˜ n ) ,A λ (cid:48) λ (∆ , P, n ) = ( − λ (cid:48) − λ A λλ (cid:48) ( − (cid:101) ∆ , (cid:101) P , ˜ n ) ,T q R/Lλ (cid:48) λ (∆ , P, n ) = ( − λ (cid:48) − λ T q L/Rλλ (cid:48) ( − (cid:101) ∆ , (cid:101) P , ˜ n ) ,T g RRλ (cid:48) λ (∆ , P, n ) = ( − λ (cid:48) − λ T g LLλλ (cid:48) ( − (cid:101) ∆ , (cid:101) P , ˜ n ) . (15) • Finally, P ⊥ T ⊥ combined implies V λ (cid:48) λ (∆ , P, n ) = ( − λ (cid:48) − λ V − λ − λ (cid:48) ( − ∆ , P, n ) ,A λ (cid:48) λ (∆ , P, n ) = ( − λ (cid:48) − λ +1 A − λ − λ (cid:48) ( − ∆ , P, n ) ,T q R/Lλ (cid:48) λ (∆ , P, n ) = ( − λ (cid:48) − λ +1 T q R/L − λ − λ (cid:48) ( − ∆ , P, n ) ,T g RR/LLλ (cid:48) λ (∆ , P, n ) = ( − λ (cid:48) − λ T g RR/LL − λ − λ (cid:48) ( − ∆ , P, n ) . (16)where the notation (cid:101) P is defined in Eq. (A3). The properties of light-front parity and time reversal are summarized in App. B If the correlators in the following equations do not have a q or g superscript, the same relation is valid for both the quark-quark andgluon-gluon correlator. Transverse superscripts separated by a slash denote multiple possible values to be considered in sequence betweenthe left- and right-hand side. . Leading twist-2 quark transversity GPDs The leading twist-2 transversity quark GPDs are chiral odd and defined by matrix elements of the tensor correlator T q iλ (cid:48) λ . They are scalar functions depending on Lorentz invariants x, ξ, t multiplying all possible independent tensorstructures that appear in the decomposition of the correlator matrix element. These tensor structures are builtfrom the available four-vectors (cid:15), (cid:15) (cid:48) , n, P, ∆ and the decomposition has to obey the symmetry constraints given in theprevious subsection. We decompose the correlator as (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | ¯ ψ ( − κn )( in µ σ µi ) ψ ( κn ) | p λ (cid:105) = M ( (cid:15) (cid:48)∗ n ) (cid:15) i − (cid:15) (cid:48)∗ i ( (cid:15)n )2 √ P n ) H qT ( x, ξ, t )+ M (cid:20) P i ( (cid:15)n )( (cid:15) (cid:48)∗ n )2 √ P n ) − ( (cid:15)n ) (cid:15) (cid:48) i ∗ + (cid:15) i ( (cid:15) (cid:48)∗ n )2 √ P n ) (cid:21) H qT ( x, ξ, t )+ (cid:20) ( (cid:15) (cid:48)∗ n )∆ i − (cid:15) (cid:48) i ∗ (∆ n ) M ( P n ) ( (cid:15)P ) − ( (cid:15)n )∆ i − (cid:15) i (∆ n ) M ( P n ) ( (cid:15) (cid:48)∗ P ) (cid:21) H qT ( x, ξ, t )+ (cid:20) ( (cid:15) (cid:48)∗ n )∆ i − (cid:15) (cid:48) i ∗ (∆ n ) M ( P n ) ( (cid:15)P ) + ( (cid:15)n )∆ i − (cid:15) i (∆ n ) M ( P n ) ( (cid:15) (cid:48)∗ P ) (cid:21) H qT ( x, ξ, t )+ M (cid:20) ( (cid:15) (cid:48)∗ n )∆ i − (cid:15) (cid:48) i ∗ (∆ n )2 √ P n ) ( (cid:15)n ) + ( (cid:15)n )∆ i − (cid:15) i (∆ n )2 √ P n ) ( (cid:15) (cid:48)∗ n ) (cid:21) H qT ( x, ξ, t )+ (∆ i + 2 ξP i ) M ( (cid:15) (cid:48)∗ (cid:15) ) H qT ( x, ξ, t ) + (∆ i + 2 ξP i ) M ( (cid:15) (cid:48)∗ P )( (cid:15)P ) M H qT ( x, ξ, t )+ (cid:20) ( (cid:15) (cid:48)∗ n ) P i − (cid:15) (cid:48) i ∗ ( P n ) M ( P n ) ( (cid:15)P ) + ( (cid:15)n ) P i − (cid:15) i ( P n ) M ( P n ) ( (cid:15) (cid:48)∗ P ) (cid:21) H qT ( x, ξ, t )+ (cid:20) ( (cid:15) (cid:48)∗ n ) P i − (cid:15) (cid:48) i ∗ ( P n ) M ( P n ) ( (cid:15)P ) − ( (cid:15)n ) P i − (cid:15) i ( P n ) M ( P n ) ( (cid:15) (cid:48)∗ P ) (cid:21) H qT ( x, ξ, t ) . (17)All nine tensor structures are linearly independent, consequently so are the nine GPDs. This can be best seen byconsidering the transformation between the GPDs and helicity amplitudes (see App. C). Using the hermiticity, parityand time reversal constraints on the correlators written down in Sec. II B, we find the following properties of theGPDs: • All nine H qTi are real. • Even/odd behavior in skewness ξ : H qTi ( x, − ξ, t ) = H qTi ( x, ξ, t ) i ∈ { , , , , , } ,H qTi ( x, − ξ, t ) = − H qTi ( x, ξ, t ) i ∈ { , , } . (18) • Sum rules and form factors of local currents: Due to the odd nature of the GPD or the presence of n µ n ν / ( P n ) in the accompanying tensor, we have the following sum rules that equal zero (cid:90) − d x H qTi ( x, ξ, t ) = 0 i ∈ { , , , } . (19)The first moments of the other 5 GPDs give form factors of local tensor currents. • Forward limit: this corresponds to ∆ = 0 , ξ = 0 , ( (cid:15)P ) = ( (cid:15) (cid:48)∗ P ) = 0. The only GPD that does not decouple andis non-zero in this limit is H qT ( x, , h ( x ) defined in Ref. [20, 21]: h ( x ) = H qT ( x, , . (20)The correlators of Eq. (9) can be connected to parton-hadron scattering amplitudes in u -channel kinematics. Wecan thus write the helicity amplitudes of quark-hadron scattering A qλ (cid:48) µ (cid:48) ; λµ [with µ ( µ (cid:48) ) the light-front helicity of theoutgoing (incoming) parton line] as certain projections of Eq. (9) and one has for the chiral odd helicity amplitudes [19]: A qλ (cid:48) +; λ − = 12 T q Rλ (cid:48) λ , A qλ (cid:48) − ; λ + = − T q Lλ (cid:48) λ . (21)5lugging the explicit expressions of the polarization four-vectors of Eq. (7) in the decomposition of Eq. (17), we obtaina linear set of transformations between the nine independent helicity amplitudes A qλ (cid:48) +; λ − and the nine transversityGPDs H qTi . This set of equations and their inverse are listed in App. C. D. Leading twist-2 gluon transversity GPDs
The leading twist-2 transversity gluon GPDs are defined by matrix elements of the tensor correlator T g ijλ (cid:48) λ . Wedecompose this correlator as − P n ) (cid:90) dκ π e ixκ ( P n ) (cid:104) p (cid:48) λ (cid:48) | Tr ˆ S (cid:2) n α G αi ( − κn ) n β G βj ( κn ) (cid:3) | p λ (cid:105) = ˆ S (cid:26) (∆ i + 2 ξP i ) ( (cid:15) (cid:48)∗ n ) (cid:15) j − (cid:15) (cid:48) j ∗ ( (cid:15)n )( P n ) H gT ( x, ξ, t )+ (∆ i + 2 ξP i ) (cid:20) P j ( (cid:15)n )( (cid:15) (cid:48)∗ n )( P n ) − ( (cid:15)n ) (cid:15) (cid:48) j ∗ + (cid:15) j ( (cid:15) (cid:48)∗ n )( P n ) (cid:21) H gT ( x, ξ, t )+ (∆ i + 2 ξP i ) M (cid:20) ( (cid:15) (cid:48)∗ n )∆ j − (cid:15) (cid:48) j ∗ (∆ n ) M ( P n ) ( (cid:15)P ) − ( (cid:15)n )∆ j − (cid:15) j (∆ n ) M ( P n ) ( (cid:15) (cid:48)∗ P ) (cid:21) H gT ( x, ξ, t )+ (∆ i + 2 ξP i ) M (cid:20) ( (cid:15) (cid:48)∗ n )∆ j − (cid:15) (cid:48) j ∗ (∆ n ) M ( P n ) ( (cid:15)P ) + ( (cid:15)n )∆ j − (cid:15) j (∆ n ) M ( P n ) ( (cid:15) (cid:48)∗ P ) (cid:21) H gT ( x, ξ, t ) − (cid:20) ( (cid:15) (cid:48)∗ n ) P i − ( P n ) (cid:15) (cid:48) i ∗ ( P n ) (cid:21) (cid:20) ( (cid:15)n ) P j − ( P n ) (cid:15) j ( P n ) (cid:21) H gT ( x, ξ, t )+ (cid:20) ( (cid:15) (cid:48)∗ n )∆ i − (∆ n ) (cid:15) (cid:48) i ∗ P n ) (cid:21) (cid:20) ( (cid:15)n )∆ j − (∆ n ) (cid:15) j P n ) (cid:21) H gT ( x, ξ, t ) + (∆ i + 2 ξP i ) M (∆ j + 2 ξP j ) M ( (cid:15) (cid:48)∗ P )( (cid:15)P ) M H gT ( x, ξ, t )+ ∆ i + 2 ξP i M (cid:20) ( (cid:15) (cid:48)∗ n ) P j − (cid:15) (cid:48) j ∗ ( P n ) M ( P n ) ( (cid:15)P ) + ( (cid:15)n ) P j − (cid:15) j ( P n ) M ( P n ) ( (cid:15) (cid:48)∗ P ) (cid:21) H gT ( x, ξ, t )+ ∆ i + 2 ξP i M (cid:20) ( (cid:15) (cid:48)∗ n ) P j − (cid:15) (cid:48) j ∗ ( P n ) M ( P n ) ( (cid:15)P ) − ( (cid:15)n ) P j − (cid:15) j ( P n ) M ( P n ) ( (cid:15) (cid:48)∗ P ) (cid:21) H gT ( x, ξ, t ) (cid:27) . (22)The tensor structures that appear in the above equation are linearly independent. This is again best observed fromthe relations between the transversity GPDs and helicity amplitudes written out in App. C.Using the hermiticity, parity and time reversal constraints on the correlators written down in Sec. II B, we find thefollowing properties of the GPDs: • All nine GPDs are real and even in x . • Similarly as for the quark GPDs, the even or odd behavior in skewness ξ is as follows H gTi ( x, − ξ, t ) = H gTi ( x, ξ, t ) i ∈ { , , , , , } ,H gTi ( x, − ξ, t ) = − H gTi ( x, ξ, t ) i ∈ { , , } . (23) • Sum rules and form factors of local currents: Due to the odd nature of the GPD, we have the following sumrules that equal zero (cid:90) − d x H gTi ( x, ξ, t ) = 0 i ∈ { , , } , (24)the first moments of the remaining 6 GPDs give form factors of local tensor currents. • Forward limit: The only GPD that does not decouple and is non-zero is H gT ( x, , x ∆ defined in Ref. [22] [Eq. (1) within] or the collinear pdf xh T T ( x ) in Ref. [23] [Eq. (2.38)within]: H gT ( x, ,
0) = xh T T ( x ) . (25)This pdf is unique to the spin-1 case as a spin 1/2 hadron cannot compensate the gluon helicity flip.6he relation between helicity flip gluon-hadron helicity amplitudes A gλ (cid:48) +; λ − and the correlators of Eq. (10) is givenby [19] A gλ (cid:48) +; λ − = 12 T g RRλ (cid:48) λ , A gλ (cid:48) − ; λ + = 12 T g LLλ (cid:48) λ . (26)As for the quark sector, we can plug in the explicit expressions for the spin-1 polarization four-vectors and obtain thetransformation equations between the helicity amplitudes and the gluon transversity GPDs listed in App. C. III. DEUTERON CONVOLUTION MODEL: FORMALISM
In this section, we derive the expression of the spin 1 transversity GPDs for the case of the deuteron in the impulseapproximation (IA). In the IA, we consider the dominant
N N component of the deuteron depicted in the diagram ofFig. 2. The two quark lines in the correlators of Eqs. (9) are attached to the same nucleon and the second nucleon actsas a “spectator”. This is a standard first order approximation in the computation of partonic properties of nuclei [24–29]. The derivation presented here follows the approach used in Ref. [16]: the correlator T qR/Lλ (cid:48) λ for the deuteron isexpressed as a convolution of the deuteron light-front wave function with similar correlators for the nucleon. The latterare expressed through the four transversity GPDs of the nucleon. In the final step the correlators can be connectedto the transversity deuteron GPDs by inverting the relations between the complete set of helicity amplitudes definedby Eqs. (C1) – (C9) and the transversity spin 1 GPDs. A. Kinematics and conventions P D P' D p p' p Γ DNN Γ DNN k k'
FIG. 2: Diagrammatic depiction of the impulse approximation for the deuteron GPDs, considering the NN component of thedeuteron, for kinematics where x > ξ . As we will be dealing with kinematic variables on both the nuclear and nucleon level, we amend the notations ofSec. II A to differentiate clearly between the two. Using the four-momenta shown in Fig. 2, we introduce the followingkinematic variables: ¯ P D = 12 ( P D + P (cid:48) D ) , ¯ p = 12 ( p + p (cid:48) ) , ∆ = P (cid:48) D − P D = p (cid:48) − p ,ξ = − (∆ n )2( ¯ P D n ) , ξ N = − (∆ n )2(¯ p n ) , ¯ k = 12 ( k + k (cid:48) ) ,x = (¯ kn )( ¯ P D n ) , x N = (¯ kn )(¯ p n ) . (27)7e introduce light-front momentum fractions for the nucleons: α = 2 p nP D n , α = 2 p nP D n = 2 − α ,α (cid:48) = 2 p (cid:48) nP (cid:48) D n , α (cid:48) = 2 p nP (cid:48) D n = 2 − α (cid:48) , (28)and we have the following useful identities ( p n )( ¯ P D n ) = α (1 + ξ )2 , ( p (cid:48) n )( ¯ P D n ) = α (cid:48) (1 − ξ )2 ,α (1 + ξ ) = α (cid:48) (1 − ξ ) + 4 ξ ,ξ N = ξ α (1 + ξ ) − ξ ,x N = x α (1 + ξ ) − ξ . (29)The deuteron light-front wave function depends on the following dynamical variable, the three-momentum k d definedby k zd E k = α − , k ⊥ d = p ⊥ − α P ⊥ D , E k d = k d + m = m + ( k ⊥ d ) α α , (30)where m is the nucleon mass. The momentum k d corresponds to the relative momentum of the two on-shell nucleonsin the light-front boosted deuteron rest frame [30, 31]. The first two equations follow from the properties of light-frontboosts while the third equation can be obtained by equating ( k p + k n ) = 4 E k d = 2 m + 2( k n k p ), where k p , k n arethe on-shell nucleon momenta of the intermediate N N state.Finally, the phase space element of the active nucleon can be written as d Γ = dp +1 d p ⊥ (2 π ) p +1 = dα d p ⊥ (2 π ) α = (2 − α ) d k d (2 π ) E k d . (31) B. Deuteron light-front wave function
The deuteron light-front wave function [30–33] is given by the overlap of the deuteron single-particle state with theon-shell two-nucleon state, where all states are quantized on the light-front: (cid:104) N ( p , σ ); N ( p , σ ) | D ( P D , λ ) (cid:105) ≡ (2 π ) P + D δ ( p +1 + p +2 − P + D ) δ ( p ⊥ + p ⊥ − P ⊥ D )Ψ Dλ ( k d , σ , σ ) . (32)All involved momenta ( P D , p , p ) are on their mass shell, which means light-front energy (minus component ofmomentum) is not conserved in the transition D → N N . For the free two-nucleon state in the transition matrixelement of Eq. (32), an angular momentum decomposition can be performed in the light-front boosted deuteron restframe in a way very similar to the case of the non-relativistic deuteron wave function. The relative motion of the twonucleons in the deuteron rest frame can be projected on spherical harmonics and for the deuteron a radial S -wave( l = 0) and D -wave ( l = 2) can be coupled to the total spin S = 1 of the two nucleons to obtain total light-frontspin j = 1. The final form of the deuteron light-front wave function defined through Eq. (32) reflects this angulardecomposition: Ψ Dλ ( k , σ , σ ) = (cid:112) E k (cid:88) σ (cid:48) σ (cid:48) D σ σ (cid:48) [ R fc ( k µp /m )] D σ σ (cid:48) [ R fc ( k µn /m )]Φ Dλ ( k d , σ (cid:48) , σ (cid:48) ) , (33)with Φ Dλ ( k d , σ (cid:48) , σ (cid:48) ) = (cid:88) l =0 , λ l λ S (cid:104) lλ l λ S | λ (cid:105)(cid:104) σ (cid:48) σ (cid:48) | λ S (cid:105) Y λ l l (Ω k d ) φ l ( k ) , (34)8here the φ l ( k ) denote the radial components of the wave function and Y λ l l (Ω k ) are the spherical harmonics.The deuteron light-front wave function has two different features compared to the non-relativistic one that deservehighlighting. First, there is the appearance of two Melosh rotations D λ i λ (cid:48) i [ R fc ( k µi /m )] [34] in Eq. (33) that encoderelativistic spin effects arising from the quantization of particle states (and spin) on the light-front. Second, thedynamical variable that appears in the light-front wave function is the three-momentum k . In the calculationspresented in this article the radial wave functions φ l ( k ) are identified with those from non-relativistic wave functionparameterizations. We want to stress that this does not correspond with approximating the light-front wave functionwith the non-relativistic one given the differences pointed out above. This approach can be justified for momentaup to a few 100 MeV given the small binding energy of the deuteron. In Ref. [35], an explicit comparison betweenthe instant-form and front-form wave function for a two-particle bound state was carried out in a toy model. Theconnection between the non-relativistic instant form and light-front wave function as in Eq. (33) was found to holdfor (cid:15) B /M D < .
002 (with (cid:15) B , M D the deuteron binding energy and mass), which holds for the deuteron case. C. Nucleon chiral odd quark GPDs
As the tensor correlator defining the nucleon chiral odd quark GPDs appears in the IA derivation, we brieflysummarize expressions for these in this subsection. We use the standard parametrization for the nucleon chiral oddquark GPDs introduced in Ref. [19]: (cid:90) dκ π e ix N κ (¯ p n ) (cid:104) p (cid:48) σ (cid:48) | ¯ ψ ( − κn )( in µ σ µi ) ψ ( κn ) | p σ (cid:105) = 12(¯ p n ) ¯ u ( p (cid:48) , σ (cid:48) ) (cid:20) H qT ( in µ σ µi ) + (cid:101) H qT (¯ p n )∆ i − (∆ n )¯ p i m + E qT ( γn )∆ i − (∆ n ) γ i m + (cid:101) E qT ( γn )¯ p i − (¯ p n ) γ i m (cid:21) u ( p , σ ) . (35)Substituting the standard light-front spinors [36], we list explicit expressions for the spinor bilinears multiplying theGPDs in the above expression. For ± and ∓ appearing in the following expressions the upper sign comes with the R component, the lower one with the L component. We have12(¯ p n ) ¯ u ( p (cid:48) , σ (cid:48) )( in µ σ µR/L ) u ( p , σ ) = − δ − σ (cid:48) ,σ (2 σ ∓ (cid:113) − ξ N , p n ) ¯ u ( p (cid:48) , σ (cid:48) ) (cid:34) (¯ p n )∆ R/L − (∆ n )¯ p R/L m (cid:35) u ( p , σ ) = δ σ (cid:48) ,σ √ t N − tm e ± iφ − δ − σ (cid:48) ,σ σ (cid:112) − ξ N ( t N − t ) m e (2 σ ± iφ , p n ) ¯ u ( p (cid:48) , σ (cid:48) ) (cid:20) ( γn )∆ R/L − (∆ n ) γ R/L m (cid:21) u ( p , σ ) = δ σ (cid:48) ,σ (1 ∓ σ ξ N ) √ t N − t m e ± iφ + δ − σ (cid:48) ,σ (2 σ ∓ ξ N (cid:112) − ξ N , p n ) ¯ u ( p (cid:48) , σ (cid:48) ) (cid:34) ( γn )¯ p R/L − (¯ p n ) γ R/L m (cid:35) u ( p , σ ) = ± δ σ (cid:48) ,σ σ (1 ∓ σ ξ N ) √ t N − t m e ± iφ − δ − σ (cid:48) ,σ (2 σ ∓ ξ N (cid:112) − ξ N , (36)where φ is the azimuthal angle of the four-vector ∆ + 2 ξ N ¯ p and t N = − m ξ N − ξ N . (37) D. Impulse approximation derivation
As the following derivation does not depend on the exact operator in the correlator, we leave it unspecified andcall it ˆ A . Consequently the equations below apply to any quark-quark or gluon-gluon GPD correlator written downin Subsec. II B.We start by inserting two complete sets of on-shell two-nucleon states in the correlator, use Eq. (32) to introducethe deuteron light-front wave functions and Eq. (A6) to evaluate the integrations over the spectator nucleon phase9pace elements: (cid:90) dκ π e ixκ ( ¯ P D n ) (cid:104) P (cid:48) D λ (cid:48) | ˆ A | P D , λ (cid:105) = (cid:88) N (cid:90) dp +1 d p ⊥ p +1 dp (cid:48) +1 d p (cid:48)⊥ p (cid:48) +1 dp +2 d p ⊥ p +2 P + D P (cid:48) + D δ + ⊥ ( P (cid:48) D − p (cid:48) − p ) × δ + ⊥ ( P D − p − p )Θ( α (1 + ξ ) − | x | − ξ ) (cid:2) Θ( ξ )Θ( α (1 + ξ ) − ξ ) + Θ( − ξ )Θ( α (1 + ξ )) (cid:3) × (cid:88) σ σ (cid:48) σ Ψ ∗ Dλ (cid:48) ( k (cid:48) d , σ (cid:48) , σ )Ψ Dλ ( k d , σ , σ ) (cid:90) dκ π e ix N κ (¯ p n ) (cid:104) p (cid:48) σ (cid:48) | ˆ A | p σ (cid:105) = (cid:88) N (cid:90) dα d p ⊥ α dα (cid:48) d p (cid:48)⊥ α (cid:48) P + D P (cid:48) + D p +2 δ ( − ∆ + − p +1 + p (cid:48) +1 ) δ ( − ∆ ⊥ − p ⊥ + p (cid:48)⊥ )Θ( α (1 + ξ ) − | x | − ξ ) × (cid:2) Θ( ξ )Θ( α (1 + ξ ) − ξ ) + Θ( − ξ )Θ( α (1 + ξ )) (cid:3) (cid:88) σ σ (cid:48) σ Ψ ∗ Dλ (cid:48) ( k (cid:48) d , σ (cid:48) , σ )Ψ Dλ ( k d , σ , σ ) (cid:90) dκ π e ix N κ (¯ p n ) (cid:104) p (cid:48) σ (cid:48) | ˆ A | p σ (cid:105) = (cid:88) N (cid:90) dα d p ⊥ α dα (cid:48) d p (cid:48)⊥ α (cid:48) − α δ ( α (cid:48) − α ξ − ξ + 4 ξ − ξ ) δ ( − ∆ ⊥ − p ⊥ + p (cid:48)⊥ )Θ( α (1 + ξ ) − | x | − ξ ) × (cid:2) Θ( ξ )Θ( α (1 + ξ ) − ξ ) + Θ( − ξ )Θ( α (1 + ξ )) (cid:3) (cid:88) σ σ (cid:48) σ Ψ ∗ Dλ (cid:48) ( k (cid:48) d , σ (cid:48) , σ )Ψ Dλ ( k d , σ , σ ) (cid:90) dκ π e ix N κ (¯ p n ) (cid:104) p (cid:48) σ (cid:48) | ˆ A | p σ (cid:105) = 2 (cid:88) N (cid:90) dα d k ⊥ d α (2 − α ) dα (cid:48) d k (cid:48) ⊥ d α (cid:48) δ ( α (cid:48) − α ξ − ξ + 4 ξ − ξ ) δ ( k (cid:48) ⊥ d − k ⊥ d − − α − ξ ∆ ⊥ − ξ − α − ξ ¯ P ⊥ D )Θ( α (1 + ξ ) − | x | − ξ ) × (cid:2) Θ( ξ )Θ( α (1 + ξ ) − ξ ) + Θ( − ξ )Θ( α (1 + ξ )) (cid:3) (cid:88) σ σ (cid:48) σ Ψ ∗ Dλ (cid:48) ( k (cid:48) d , σ (cid:48) , σ )Ψ Dλ ( k d , σ , σ ) (cid:90) dκ π e ix N κ (¯ p n ) (cid:104) p (cid:48) σ (cid:48) | ˆ A | p σ (cid:105) . (38)In the third step a factor 2 ¯ P D was brought into the Dirac delta function for the plus components. The sum N is overthe two possible active nucleons. The Heaviside functions originate from the requirement of positive light-front pluscomponents for the on-shell intermediate states. The | x | > | ξ | region gives the first Heaviside, the ERBL region theremaining ones.Up to here the derivation is valid for any correlator considered in Subec. II B. In the next step, we specialize to thecase of the twist-2 chiral odd quark GPDs. By taking ˆ A = ¯ ψ ( − κn )( in µ σ µR/L ) ψ ( κn ) in Eq. (38) and using Eqs. (35)and (36) we arrive at T R/Lλ (cid:48) λ = 4 (cid:90) dα d k ⊥ d α (2 − α ) dα (cid:48) d k (cid:48)⊥ d α (cid:48) δ ( α (cid:48) − α ξ − ξ + 4 ξ − ξ ) δ ( k (cid:48)⊥ d − k ⊥ d − − α − ξ ∆ ⊥ − ξ − α − ξ ¯ P ⊥ D ) × Θ( α (1 + ξ ) − | x | − ξ ) (cid:2) Θ( ξ )Θ( α (1 + ξ ) − ξ ) + Θ( − ξ )Θ( α (1 + ξ )) (cid:3)(cid:88) σ σ (cid:48) σ Ψ ∗ Dλ (cid:48) ( k (cid:48) d , σ (cid:48) , σ )Ψ Dλ ( k d , σ , σ ) . (cid:20) − δ − σ (cid:48) ,σ (2 σ ∓ (cid:113) − ξ N H IS T ( x N , ξ N , t )+ (cid:32) δ σ (cid:48) ,σ √ t − tm e ± iφ − δ − σ (cid:48) ,σ σ (cid:112) − ξ N ( t − t ) m e (2 σ ± iφ (cid:33) (cid:101) H IS T ( x N , ξ N , t )+ (cid:32) δ σ (cid:48) ,σ (1 ∓ σ ξ N ) √ t − t m e ± iφ + δ − σ (cid:48) ,σ (2 σ ∓ ξ N (cid:112) − ξ N (cid:33) E IS T ( x N , ξ N , t )+ (cid:32) ± δ σ (cid:48) ,σ σ (1 ∓ σ ξ N ) √ t − t m e ± iφ − δ − σ (cid:48) ,σ (2 σ ∓ ξ N (cid:112) − ξ N (cid:33) (cid:101) E IS T ( x N , ξ N , t ) (cid:35) , (39)where the nucleon GPDs are the isoscalar combinations X IS ( x N , ξ N , t ) = 12 (cid:0) X u ( x N , ξ N , t ) + X d ( x N , ξ N , t ) (cid:1) , (40)originating from the isoscalar nature of the deuteron np component considered here. Because of the non-conservationof the minus component in the D → N N vertex, the t appearing in the nucleon GPDs is in principle different10rom the t defined in the beginning [i.e. for the deuteron as defined in Eq. (2)]. Due to the small binding energy (cid:15) B of the deuteron, the difference between the two will go as (cid:15) B over some larger scale and can be neglected in afirst approximation. The deuteron transversity GPDs can be obtained from Eq. (39) by first calculating the helicityamplitudes [Eq. (21)] and subsequently using the results of App. C [Eqs. (C21) – (C29)] to compute the GPDs fromthe helicity amplitudes.Comparing our derivation with the one presented in Ref. [16], we notice the following differences. Equation numbersmentioned below refer to the ones in Ref. [16]: • Eq. (A2) is missing a factor 1/2 in the right-hand side so that the deuteron particle state is correctly normalized.As a consequence, Eq. (19) (and following) need an additional factor 1/4. • Eq. (29) should have a prefactor of π . There is a factor of 2 missing in the transition from Eq. (28) to (29)and a factor of 1/4 from the first bullet above. • The phase of Eq. (31) should read η λ = (2 λ ˜∆ x + i ˜∆ y ) / | ˜∆ ⊥ | . This can also be inferred from the helicityamplitudes written down in Eq. (61) of Ref. [1], where a factor e ± iφ is written in the A ∓ + , ± + amplitudes. IV. DEUTERON CONVOLUTION MODEL: RESULTS
In this section, we use Eq. (39) in combination with Eq. (21) and Eqs. (C21) – (C29) to compute the helicityamplitudes and transversity GPDs in the quark sector for the deuteron. For the chiral odd nucleon GPDs, we usethe parametrization of Goloskokov and Kroll (GK) [37], evaluated at a scale of µ = 2 GeV, and implemented in threedifferent models in Ref. [38] (the figures below use “model 2” therein, which has ˜ H T = H T , E T = ¯ E T − H T , ˜ E T = 0). In the forward limit of this parametrization the helicity pdfs enter [39]. For these we use the parametrizationof Ref. [40]. We use the AV18 parametrization of the deuteron wave function [41] unless otherwise noted. As fromthis section on we are only dealing with quark helicity amplitudes and GPDs, we omit the superscript q for thosequantities. We verified that the computed deuteron helicity amplitudes obey all the symmetry constraints listed inSubsec. II B, up to the numerical accuracy imposed on the integrations over the active nucleon phase space. − . . . . . . . . . × − A ++;+ − − . . . . . × − A − +; −− − . . . . . × − A − − . . . . . . . . . × − A − − . . . . . − . − . . × − A − +;0 − − . . . . . . . . . × − A ++;0 − − . . . . . − . . . . × − A −− − . . . . . − − × − A − +;+ − − . . . . . x − − × − A ++; −− S+DSDS+D no Mel.
FIG. 3: (Color online) Deuteron quark helicity amplitudes computed in the convolution formalism, at ξ = 0 . t = − .
25 GeV .Full blue curve includes the full deuteron wave function, dotted orange (dashed green) only includes the deuteron radial S -( D -)wave and the dashed-dotted red corve omits the Melosh rotations in the light-front deuteron wave function. . . . . . − × − H T − . . . . . H T − . . . . . − − − × H T − . . . . . × H T − . . . . . . . × H T − . . . . . − − H T − . . . . . − . − . − . . × H T S+DSDS+D no Mel. − . . . . . H T − . . . . . x − − − H T FIG. 4: (Color online) Deuteron quark transversity GPDs computed in the convolution formalism, at ξ = 0 . t = − .
25 GeV .Curves as in Fig. 3 . Figures 3 and 4 show the helicity amplitudes and quark transversity GPDs of the deuteron as a function of x (where − ≤ x ≤
1) , in kinematics ξ = 0 . t = − .
25 GeV . Next to the total result, Figs. 3 and 4 alsoshow the separate contributions to the helicity amplitudes and GPDs when only including the deuteron radial S -or D -wave. The remaining difference with the total result originates from S - D interference contributions. For thehelicity amplitudes, one observes that the deuteron helicity conserving ones (top row of Fig. 3) are dominated bythe pure S -wave contribution, whereas the ones with a helicity change for the deuteron receive sizeable contributionsfrom S - D interference terms. The effect of the Melosh rotations is generally smallest in amplitudes dominated by the S -wave contribution. Lastly, the two amplitudes with a complete deuteron helicity flip (Fig. 3 bottom row, middleand right panel) are identically zero for the S -wave as there is no orbital angular momentum available in the deuteronto compensate the change in helicities (two units for the deuteron, one unit for the quark).We compared calculations with the three slightly different implementations of the chiral odd nucleon GPD modelsused in Ref. [38]. The results proved to be rather insensitive to these choices as changes in the final deuteron GPDswere in the order of a few percent maximum. Similary, Figs. 5 and 6 show the use of three different deuteron wavefunctions in the calculation: the CD-Bonn [42] has a soft high-momentum tail, the WJC-1 [43] a hard one, and theAV18 [41] wave function is in-between. Consequently, the differences between the different parametrizations includedhere are largest at high p ⊥ or α close to its lower (0) and upper (2) bound. Both the amplitudes and GPDs arein general rather insensitive to the wave function details, even for the amplitudes that do not receive a pure S -wavecontribution, and which are dominated by high relative N N momenta in the convolution.Figures 7 and 8 show calculations at two values of momentum transfer. Helicity amplitudes without deuteron helicityflip shrink in size with higher momentum transfer. The amplitudes with a single helicity flip also become slightlysmaller but the effect is not as large. Finally, the amplitudes with a double helicity flip grow in size. This reflectsthe role angular momentum plays in these amplitudes, being supplied by the momentum transfer. The GPDs are ingeneral smaller at higher momentum transfer. H T has flipped sign, this is caused by the fact that H T is proportionalto the sum of helicity conserving and double helicity flip amplitudes (entering with different sign) [Eqs. (C26) and(C28)].Figures 9 and 10 show that most helicity amplitudes are dominated by the ¯ E T = 2 (cid:101) H T + E T nucleon chiral odd GPDfrom the GK parametrization. Only the A ++;0 − and A −− receive large contributions from H T . The dominanceof ¯ E T in most amplitudes is caused by its size on the one hand (which is larger than H T ) and the fact that both12 . . . . . × − A ++;+ − − . . . . . × − A − +; −− − . . . . . × − A − − . . . . . . . . . × − A − − . . . . . − . − . − . . × − A − +;0 − − . . . . . . . . . × − A ++;0 − − . . . . . − . . . × − A −− − . . . . . − − × − A − +;+ − − . . . . . x − − × − A ++; −− AV18CD-BonnWJC1
FIG. 5: (Color online) Deuteron quark helicity amplitudes computed in the convolution formalism, at ξ = 0 . t = − .
25 GeV with different deuteron wave functions. Deuteron wave functions are CD-Bonn [42], WJC-1 [43] and AV18 [41]. u and d quarks have same sign ¯ E T GPDs, whereas they have opposite for H T and thus are small for the isosingletcontribution entering in the convolution formula.Figs. 11 and 12 show the ξ dependence of the amplitudes and GPDs at a momentum transfer of t = − . .The deuteron helicity amplitudes with zero or two units of deuteron helicity flip decrease significantly with larger ξ ,while the ones with one unit of helicity flip are largest at intermediate values of ξ . For the GPDs, H T , H T decreasesignificantly with for larger ξ . V. SUM RULES IN THE DEUTERON CONVOLUTION PICTURE
In this section we focus on the quark transversity GPD sum rules of Eq. (19). Because of Lorentz invariance, theGPDs obey polynomiality properties that in particular predict that these first moments should be independent of thevalue of skewness. As we use a lowest order Fock space expansion in our convolution model, and this explicitly breaksLorentz invariance (no negative energy projections are included for instance), we investigate to which degree the ξ independence is violated in our convolution formalism. Fig. 13 depicts the results for the first moments of all thechiral odd quark GPDs at t = − . (which requires | ξ | < . ξ dependence, especially the GPDs H T , H T , H T and H T . Two of these ( H T , H T ) even should have zero first momentsaccording to Eq. (19). This could be seen as a requirement to include higher order contributions in the convolutionpicture, i.e. beyond the handbag diagram or including higher Fock states.To investigate this further, we look at the sum rules in a minimal convolution picture, detailed in App. D. Thisminimal convolution picture allows us to calculate the deuteron GPDs analytically. Looking at the final expressionsfor the deuteron GPDs listed in Eqs. D4, we see that only GPDs H T , H T , H T have a leading term O ( ξ ). InspectingEqs. D4, almost all GPDs have dominating terms proportional to D − (which is large for the deuteron kinematicsconsidered here) that go as higher powers of ξ , especially the GPDs that also show the largest ξ dependence in thefull convolution model. It is worth noting that the fact that H T = 0 in this minimal convolution is due to the lack ofa D -wave part in the deuteron wave function in this model and not a reflection of a sum rule.The violation of the sum rules thus is an inherent feature of all convolution models based on a Fock space expansion,even the simplest ones. One should thus blame their formulation for Lorentz invariance breaking. The contributionof higher Fock states is beyond the current scope of our study. One possible approach for the deuteron that respects13 . . . . . − × − H T − . . . . . . . . . H T − . . . . . − − − × H T − . . . . . × H T − . . . . . . . . . × H T − . . . . . − H T − . . . . . − . − . − . . × H T AV18CD-BonnWJC1 − . . . . . H T − . . . . . x − − − H T FIG. 6: (Color online) Deuteron quark transversity GPDs computed in the convolution formalism, at ξ = 0 . t = − .
25 GeV .Comparison between different wave functions. Lorentz invariance (and thus polynomiality of the GPDs) is the use of the covariant Bethe-Salpeter equation forthe deuteron. Current deuteron GPD implementations of the Bethe-Salpeter approach are limited to a contact
N N interaction [44], while the approach presented here allows the use of realistic deuteron wave function parametrizations.
VI. CONCLUSION
Our study completes the description of leading twist quark and and gluon GPDs in the deuteron, in a convolutionmodel based on the impulse approximation and using the lowest Fock space state for the deuteron in terms of nucleons.Although this picture is far from complete, it is a necessary starting point for the study of exclusive hard reactions inthe QCD collinear factorization framework. It will enable us to confront this framework to near future experimentalresults. We showed that the GPDs were not very model-sensitive to the nucleon nucleon potential as far as the impulseapproximation was used. However a richer structure as those involving a hidden color component [45] should lead toquite different GPDs, in particular in the gluonic sector.The transversity sector is remarkably quite difficult to access in hard reactions [46], mostly because of the chiral-oddcharacter of the quark transversity distributions. As far as transversity quark GPDs are concerned, the fact that theydo not contribute to the leading twist amplitude for the electroproduction of one meson [47, 48] lead to the study ofhigher twist [37, 49, 50] or quark mass sensitive [38, 51] contributions, and to the study of other reactions with moreparticles in the final state [39, 52–54]. The case for gluon transversity GPDs is rather different since they appear inthe leading twist DVCS [55] and timelike Compton scattering [56] amplitudes.We shall address the rich phenomenology of these reactions on the deuteron in future works, both for moderateenergy range of JLab [57] and for the very high energy range aimed at the EIC [58] and the LHeC [59] with deuteronbeams.
Acknowledgments
We acknowledge useful discussions with Adam Freese, C´edric Lorc´e, Claude Roiesnel, Lech Szymanowski andJakub Wagner. We thank Jakub Wagner for help with the numerical implementation of the chiral odd nucleon GPD14 . . . . . . . . . × − A ++;+ − − . . . . . × − A − +; −− − . . . . . × − A − − . . . . . . . . . × − A − − . . . . . − . − . − . . × − A − +;0 − − . . . . . . . . × − A ++;0 − − . . . . . − . − . . . × − A −− − . . . . . − . − . − . . × − A − +;+ − − . . . . . x − . . × − A ++; −− t = − .
25 GeV t = − .
50 GeV FIG. 7: (Color online) Deuteron quark helicity amplitudes computed in the convolution formalism, at ξ = 0 . t . parametrization. Appendix A: Conventions
This appendix summarizes the conventions and notations used throughout the text. We work with the followinglight-front conventions: • Light-front components and Levi-Civita tensor x ± = x ± x ,(cid:15) = 1 . (A1) • We use the transverse
R/L indices defined as a R = a x + ia y ,a L = a x − ia y , (A2)and for the action of light-front discrete symmetries we need the notation˜ a µ = ( a + , a − , − a , a ) . (A3)We have ˜ a R = − a L , ˜ a L = − a R . (A4)The product of two four-vectors can be written as a µ b µ = ( a + b − + a − b + − a R b L − a L b R ) . (A5)15 . . . . . − × − H T − . . . . . . . . . H T − . . . . . − − − × H T − . . . . . × H T − . . . . . . . . . × H T − . . . . . − . − . − . . H T − . . . . . − . − . − . . × H T t = − .
25 GeV t = − .
50 GeV − . . . . . H T − . . . . . x − − − H T FIG. 8: (Color online) Deuteron quark transversity GPDs computed in the convolution formalism, at ξ = 0 . t . • Single-particle state normalization of light-front helicity states (cid:104) p (cid:48) λ (cid:48) | p λ (cid:105) = (2 π ) p + δ λλ (cid:48) δ + ⊥ ( p − p (cid:48) ) . (A6) • Creation and annihilation operators are normalized through[ a λp , a λ (cid:48) † p (cid:48) ] ± = (2 π ) δ λλ (cid:48) δ + ⊥ ( p − p (cid:48) ) . (A7) • The last two equations imply | p λ (cid:105) = (cid:112) p + a † λp | (cid:105) . (A8) • The Dirac field in light-front quantization becomes ψ ( x ) = (cid:88) λ = ± (cid:90) dk + ⊥ √ k + (2 π ) (cid:104) a λk u ( k, λ ) e − ikx + b † λk v ( k, λ ) e ikx (cid:105) , (A9)with u ( k, λ ) , v ( k, λ ) the standard light-front spinors [36]. • The gluon field (with an implicit summation over a color index and SU (3) generators implied) A µ ( x ) = (cid:88) λ = ± (cid:90) dk + ⊥ √ k + (2 π ) (cid:104) a λk (cid:15) µ ( k, λ ) e − ikx + a † λk (cid:15) µ ∗ ( k, λ ) e ikx (cid:105) , (A10)where the polarization four-vectors are (cid:15) µ ( k, +) = (cid:104) + − − √ k R k + − √ − i √ (cid:105) ,(cid:15) µ ( k, − ) = (cid:104) + − √ k L k + √ − i √ (cid:105) . (A11)16 . . . . . . . . . × − A ++;+ − − . . . . . × − A − +; −− − . . . . . × − A − − . . . . . . . . . × − A − − . . . . . − . − . − . . × − A − +;0 − − . . . . . . . . × − A ++;0 − − . . . . . − × − A −− − . . . . . − − × − A − +;+ − − . . . . . x − − × − A ++; −− full¯ E T only FIG. 9: (Color online) Deuteron quark helicity amplitudes computed in the convolution formalism, at ξ = 0 . t = − .
25 GeV .Dashed curve is a calculation only considering the ¯ E T nucleon GPD. Finally, the field strength and dual field strength are G µν ( x ) = ∂ µ A ν ( x ) − ∂ ν A µ ( x ) − ig [ A µ ( x ) , A ν ( x )] (cid:101) G µν ( x ) = − (cid:15) µνρσ G ρσ . (A12) Appendix B: Light-front discrete symmetries
Light-front discrete symmetries were first considered in Ref. [60] and are discussed in several other instances of theliterature with slightly different forms of the operators between them (see for instance in Refs. [61–63]). We followthe definitions used in Refs. [62, 63] as the combination of light-front parity and time reversal with the standardcharge conjugation is consistent with the instant form
CP T . To our knowledge, the action of these light-front discretesymmetry operators on single-particle states and quark and gluon fields has not been summarized in detail or inthe case it has been written out [62], the intermediary formulas contain a number of errors and inconsistencies. Wetherefore include a summary here as a pedagogical appendix.
1. Light-front parity
We can introduce the light-front parity symmetry transformation by its action on a coordinate four-vector:Λ( P ⊥ ) : x µ (cid:55)→ ˜ x µ = ( x + , x − , − x , x ) . (B1)As an operator there are a few possible choices to implement this transformation. These differ in an overall sign ofthe phase in the exponential, but do not yield differences when considering the action of P ⊥ on correlator matrixelements. We choose P ⊥ = e − iπJ P = e − i π J e iπJ e i π J P , (B2)17 . . . . . − . . H T − . . . . . . . . . H T − . . . . . − − − × H T − . . . . . − × H T − . . . . . . . . . × H T − . . . . . − . − . − . . H T − . . . . . − . − . − . . × H T − . . . . . − H T − . . . . . x − − − H T full¯ E T only FIG. 10: (Color online) Deuteron quark transversity GPDs computed in the convolution formalism, at ξ = 0 . t = − .
25 GeV .Dashed curve is a calculation only considering the ¯ E T nucleon GPD. with P the standard instant form parity operator. We first consider the massive case. In the rest frame, acting with P ⊥ on a massive single-particle state with spin j yields using Eq. (B2) P ⊥ | j m (cid:105) = e − iπj η | j m (cid:105) , (B3)where η is the intrinsic parity of the particle. For light-front helicity states (defined with the standard light-frontboosts) and using the commutation relations of the Lorentz group algebra, we obtain P ⊥ | p λ (cid:105) = e − iπj η | ˜ p − λ (cid:105) . (B4)Light-front parity thus flips the light-front helicity of the particle and transforms its momentum. For the creationand annihiliation operators we obtain P ⊥ a † λp P †⊥ = η a e − iπj a †− λ ˜ p , P ⊥ a λp P †⊥ = η ∗ a e iπj a − λ ˜ p (B5)For the Dirac field, we have P ⊥ ψ ( x ) P †⊥ = (cid:88) λ = ± (cid:90) d ˜ k + ⊥ (cid:112) k + (2 π ) (cid:104) η ∗ a e iπj a − λ ˜ k u ( k, λ ) e − i ˜ k ˜ x + η b e − iπj b †− λ ˜ k v ( k, λ ) e i ˜ k ˜ x (cid:105) . (B6)The light-front spinors have γ γ u (˜ k, − λ ) = u ( k, λ ) ,γ γ v (˜ k, − λ ) = v ( k, λ ) , (B7) The normalization of particle states and fields is given in App. A. . . . . . × − A ++;+ − − . . . . . × − A − +; −− − . . . . . . . . × − A − − . . . . . . . × − A − − . . . . . − . − . . × − A − +;0 − − . . . . . . . × − A ++;0 − − . . . . . − . − . . × − A −− − . . . . . − − − × − A − +;+ − − . . . . . x − − − × − A ++; −− ξ = 0 . ξ = 0 . ξ = 0 . ξ = 0 . FIG. 11: (Color online) Deuteron quark helicity amplitudes computed in the convolution formalism, at various ξ for t = − . . and when requiring η b = − η ∗ a as in the instant form case, we have P ⊥ ψ ( x ) P †⊥ = η ∗ a e iπj γ γ ψ (˜ x ) , P ⊥ ψ † ( x ) P †⊥ = η a e − iπj ψ † (˜ x ) γ γ . (B8)When considering the transformation under P ⊥ of the quark-quark correlators introduced in Eq. (9), the phases e ± iπj and intrinsic parities do not contribute as they cancel between the initial and final state and the two quark fields.For massless states, the little group is characterized by a reference state with momentum along the z -axis ¯ k andtwo possible J eigenvalues (if parity is a good symmetry). For this reference state one has P ⊥ | ¯ k λ (cid:105) = η | ¯ k − λ (cid:105) . (B9)For a massless light-front helicity state with general momentum, this yields P ⊥ | p λ (cid:105) = η | ˜ p − λ (cid:105) , (B10)so as in the massive case momentum transforms and light-front helicity flips. Creation and annihilation operatorstransform as in Eq. (B5) but without the phase factor. For the gluon field this yields P ⊥ A µ ( x ) P †⊥ = (cid:88) λ = ± (cid:90) d ˜ k + ⊥ (cid:112) k + (2 π ) (cid:104) η ∗ a a − λ ˜ k (cid:15) µ ( k, λ ) e − i ˜ k ˜ x + η a a †− λ ˜ k (cid:15) µ ∗ ( k, λ ) e i ˜ k ˜ x (cid:105) . (B11)The polarization four-vectors of Eq. (A11) have (cid:15) µ ( k, ± ) = ˜ (cid:15) µ (˜ k, ∓ ) , (B12)and with η a real we have for the gluon field and field strength P ⊥ A µ ( x ) P †⊥ = η a (cid:101) A µ (˜ x ) , P ⊥ G µν ( x ) P †⊥ = η a ¯ G µν (˜ x ) , (B13)19 . . . . . − × − H T − . . . . . . . . . H T − . . . . . − . − . − . . × H T − . . . . . − . . . × H T − . . . . . H T − . . . . . − − − × − H T − . . . . . − . − . − . . × H T ξ = 0 . ξ = 0 . ξ = 0 . ξ = 0 . − . . . . . − H T − . . . . . x − − H T FIG. 12: (Color online) Deuteron quark transversity GPDs computed in the convolution formalism, at various ξ for t = − . . .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 .
10 0 . ξ − . − . − . − . − . − . . . R d x H T i ( x , ξ , t ) H T H T H T H T H T H T H T H T H T FIG. 13: (Color online) First moments of the chiral odd quark GPDs at t = − . as a function of skewness ξ . Dashedcurves are the GPDs that have zero sum rules. The moderate ξ -dependence is a consequence of Lorentz symmetry breakingdue to lowest order approximation of the convolution picture. where ¯ G µν (˜ x ) = G µν (˜ x ) for an even number of indices 1, and with a minus sign for an odd number. As in thequark case, the intrinsic parities and phases e ± iπj cancel in the light-front parity transformation of the gluon-gluoncorrelators of Eq. (10). 20 . Time reversal We can introduce the time reversal symmetry transformation by its action on a coordinate four-vector :Λ( T ⊥ ) : x µ (cid:55)→ − ˜ x µ = ( − x + , − x − , x , − x ) . (B14)Because of the anti-unitarity of T ⊥ momenta transform as p µ (cid:55)→ ˜ p µ = ( p + , p − , − p , p ) . (B15)As with P ⊥ there are several choices to write T ⊥ on the operator level, with no difference at the level of transformationof correlator matrix elements. We take T ⊥ = e − iπJ T = e − i π J e iπJ e i π J T , (B16)where T is the standard instant form time reversal operator. For massive particles, we have in the rest frame for aspin j particle using Eq. (B16) T ⊥ | j m (cid:105) = e − iπm ˜ η | j m (cid:105) , (B17)with ˜ η a phase. For light-front helicity states we obtain T ⊥ | p λ (cid:105) = e − iπλ ˜ η | ˜ p λ (cid:105) . (B18)Consequently light-front time reveral does not flip light-front helicity, but momentum transforms. For the creationand annihiliation operators we obtain T ⊥ a † λp T †⊥ = ˜ η a e − iπλ a † λ ˜ p , T ⊥ a λp T †⊥ = ˜ η ∗ a e iπλ a λ ˜ p , (B19)and for the Dirac field one has T ⊥ ψ ( x ) T †⊥ = (cid:88) λ = ± (cid:90) d ˜ k + ⊥ (cid:112) k + (2 π ) (cid:104) ˜ η ∗ a e iπλ a λ ˜ k u ∗ ( k, λ ) e − i ˜ k ( − ˜ x ) + ˜ η b e − iπλ b † λ ˜ k v ∗ ( k, λ ) e i ˜ k ( − ˜ x ) (cid:105) . (B20)The light-front spinors have − γ γ u (˜ k, λ ) = e iπλ u ∗ ( k, λ ) , − γ γ v (˜ k, λ ) = e − iπλ v ∗ ( k, λ ) , (B21)and when requiring ˜ η b = ˜ η ∗ a as in the instant form case, we arrive at T ⊥ ψ ( x ) T †⊥ = ˜ η ∗ a ( − γ γ ) ψ ( − ˜ x ) , T ⊥ ψ † ( x ) T †⊥ = ˜ η a ψ † ( − ˜ x )( − γ γ ) . (B22)In the transformation under T ⊥ of the quark-quark correlators of Eq. (9), all the phases cancel, but there remains an( − λ (cid:48) − λ factor originating from from the transformation of the initial and final state [Eq. (B18)].For the massless case, we have for the reference state T ⊥ | ¯ k λ (cid:105) = ˜ η | ¯ k λ (cid:105) , (B23)and for the massless light-front helicity states with general momentum p T ⊥ | p λ (cid:105) = ˜ η | ˜ p λ (cid:105) , (B24)Also in the massless case, light-front time reversal conserves light-front helicity and momentum is transformed. Cre-ation and annihilation operators transform as in Eq. (B19) but without the phase factor. For the transformation ofthe gluon field, we arrive at T ⊥ A µ ( x ) T †⊥ = (cid:88) λ (cid:90) d ˜ k + ⊥ (cid:112) k + (2 π ) (cid:104) ˜ η ∗ a a λ ˜ k (cid:15) µ ∗ ( k, λ ) e − i ˜ k ( − ˜ x ) + ˜ η a a † λ ˜ k (cid:15) µ ( k, λ ) e i ˜ k ( − ˜ x ) (cid:105) . (B25)21he polarization four-vectors of Eq. (7) have (cid:15) µ ∗ ( k, ± ) = − ˜ (cid:15) µ (˜ k, ± ) , (B26)and with ˜ η a real we have for the gluon field and field strength T ⊥ A µ ( x ) T †⊥ = − ˜ η a (cid:101) A µ ( − ˜ x ) , T ⊥ G µν ( x ) T †⊥ = − ˜ η a ¯ G µν ( − ˜ x ) . (B27)When considering the transformation of the gluon-gluon correlators of Eq. (10) with T ⊥ , the phases drop out but afactor ( − λ (cid:48) − λ remains from the transformation of the initial/final state. Appendix C: Explicit relations between transversity GPDs and helicity amplitudes
In the quark sector, the helicity amplitudes A qλ (cid:48) +; λ − can be written as a function of the 9 transversity GPDs usingEq. (21): A q ++;+ − = − e iφ D (cid:20) ξ − ξ ( H qT − H qT ) + H qT + D − ξ ) H qT + 12(1 − ξ ) ( H qT − H qT ) (cid:21) (C1) A q − = e iφ D (cid:20) − √ H qT + ξH qT ) + 2 ξ − ξ H qT + 2 D − ξ H qT + 1 √ H qT + 2 D − − ξ − ξ H qT + D − ξ (1 − ξ ) H qT − ξ − ξ H qT − D − ξ H qT (cid:21) (C2) A q − = e iφ √ (cid:20) (1 − ξ )2 √ H qT − H qT ) − (cid:18) D − ξ − ξ − ξ (cid:19) H qT + (cid:18) D ξ − − ξ − ξ − ξ (cid:19) H qT + ξ √ − ξ ) H qT − D ξ H qT − D (1 + ξ )(1 − ξ ) (cid:0) D − ξ (cid:1) H qT + (cid:18) − D − ξ ) + ξ − ξ (cid:19) H qT + (cid:18) (3 − ξ ) D − ξ ) − ξ − ξ (cid:19) H qT (cid:21) (C3) A q ++;0 − = D √ − ξ ) (cid:20) − H qT + H qT + 2 H qT + D + ξ − ξ H qT + 12 ( H qT − H qT ) (cid:21) (C4) A q − +;+ − = e iφ D − ξ (cid:20) ξ ( H qT − ξH qT ) + D H qT + ( ξH qT − H qT ) (cid:21) (C5) A q ++; −− = e − iφ D − ξ ) H qT (C6)The other three helicity amplitudes A qλ (cid:48) +; λ − can also be obtained by applying Eq. (16) to the ones obtained above: A q − +; −− = − e iφ D (cid:20) ξ ξ ( H qT + H qT ) + H qT + D − ξ ) H qT − ξ ) ( H qT + H qT ) (cid:21) (C7) A q − +;0 − = e iφ √ (cid:20) (1 + ξ )2 √ H qT + H qT ) + (cid:18) D
11 + ξ − ξ − ξ (cid:19) H qT − (cid:18) D ξ + 11 − ξ + 2 ξ − ξ (cid:19) H qT − ξ √ ξ ) H qT − D − ξ H qT − D (1 − ξ )(1 − ξ ) (cid:0) D + ξ (cid:1) H qT + (cid:18) D ξ ) + ξ − ξ (cid:19) H qT + (cid:18) (3 + ξ ) D − ξ ) + ξ − ξ (cid:19) H qT (cid:21) (C8)22 q −− = D √ ξ ) (cid:20) H qT + H qT + 2 H qT + D − ξ − ξ H qT −
12 ( H qT + H qT ) (cid:21) (C9)The determinant of the matrix relating the helicity amplitudes and the GPDs in the above equations yieldsDet q = − / e iφ D , (C10)which shows that all tensor structures appearing in Eq. (9) are linearly independent away from the forward limit.For the gluon helicity amplitudes we obtain largely similar expressions as the tensors that are used in the decom-position are very similar. The main differences are (i) the right-hand side of all equations is multiplied with an extra e iφ D factor compared to the quark helicity amplitudes and (ii) there are differences for the factors multiplying the H qT and H qT GPDs as a different tensor structure was used: A g ++;+ − = − e iφ D (cid:20) ξ − ξ ( H gT − H gT ) + D − ξ H gT + 11 − ξ ( H gT − H gT ) (cid:21) (C11) A g − = e iφ D (cid:20) − √ H gT + ξH gT ) + 2 ξ − ξ H gT + 2 D − ξ H gT + 14 ( H gT + H gT ) + D − ξ (1 − ξ ) H gT − ξ − ξ H gT − D − ξ H gT (cid:21) (C12) A g − = e iφ √ D (cid:20) − ξ √ H gT − H gT ) − (cid:18) D − ξ − ξ − ξ (cid:19) H gT + (cid:18) D ξ − − ξ − ξ − ξ (cid:19) H gT −
12 ( H gT − ξH gT ) − D (1 + ξ )(1 − ξ ) (cid:0) D − ξ (cid:1) H gT + (cid:18) − D − ξ ) + ξ − ξ (cid:19) H gT + (cid:18) (3 − ξ ) D − ξ ) − ξ − ξ (cid:19) H gT (cid:21) (C13) A g ++;0 − = e iφ √ D − ξ (cid:20) − H gT + H gT + D + ξ − ξ H gT + 12 ( H gT − H gT ) (cid:21) (C14) A g − +;+ − = 4 ξD − ξ ( H gT − ξH gT ) + e iφ ( H gT − ξ H gT ) + e iφ D − ξ (cid:104) D H gT + 2( ξH gT − H gT ) (cid:105) (C15) A g ++; −− = D − ξ H gT (C16)The other three helicity amplitudes A gλ (cid:48) +; λ − can be obtained by using Eq. (16): A g − +; −− = − e iφ D (cid:20) ξ ξ ( H gT + H gT ) + D − ξ H gT −
11 + ξ ( H gT + H gT ) (cid:21) (C17) A g − +;0 − = e iφ √ D (cid:20) ξ √ H gT + H gT ) + (cid:18) D
11 + ξ − ξ − ξ (cid:19) H gT − (cid:18) D ξ + 11 − ξ + 2 ξ − ξ (cid:19) H gT −
12 ( H gT + ξH gT ) − D (1 − ξ )(1 − ξ ) (cid:0) D + ξ (cid:1) H gT + (cid:18) D ξ ) + ξ − ξ (cid:19) H gT + (cid:18) (3 + ξ ) D − ξ ) + ξ − ξ (cid:19) H gT (cid:21) (C18)23 g −− = e iφ √ D ξ (cid:20) H gT + H gT + D − ξ − ξ H gT −
12 ( H gT + H gT ) (cid:21) (C19)The determinant of the above set of equations yieldsDet g = − e iφ D , (C20)which is again non-zero away from the forward limit.For completeness, we also list the inverse relations for both quarks and gluons as these are used to obtain thedeuteron GPDs from the helicity amplitudes calculated in the convolution formalism.For the quark GPDs we have H qT = (cid:34) √ e − iφ ξD (1 − ξ ) (cid:0) A q ++;+ − − A q − +; −− (cid:1) + 2 e − iφ (cid:18) − ξ A q − + 11 + ξ A q − +;0 − (cid:19) + 21 + ξ A q ++;0 − + 21 − ξ A q −− + 2 √ D (1 − ξ ) (cid:0) e − iφ A q − +;+ − − e iφ A q ++; −− (cid:1)(cid:35) , (C21) H qT = (cid:34) √ e − iφ D (1 + ξ ) (cid:18) D − ξ − ξ (cid:19) A q ++;+ − − √ e − iφ D (1 − ξ ) (cid:18) D ξ + ξ (cid:19) A q − +; −− + 2 √ e − iφ ξ (1 − ξ ) D A q − − e − iφ − ξ (cid:0) A q − − A q − +;0 − (cid:1) + 2 (cid:18) ξ ) + 2 ξ D (1 − ξ )(1 + ξ ) (cid:19) A q ++;0 − − (cid:18) − ξ ) + 2 ξ D (1 − ξ )(1 − ξ ) (cid:19) A q −− + √ e − iφ ξD (1 − ξ ) A q − +;+ − − √ e iφ ξD (1 − ξ ) (cid:0) ξ + D (3 + ξ ) (cid:1) A q ++; −− (cid:35) , (C22) H qT = (cid:20) − e − iφ D (cid:18) − ξ ξ A q ++;+ − − ξ − ξ A q − +; −− (cid:19) − √ D (cid:18) − ξ ξ A q ++;0 − − ξ − ξ A q −− (cid:19) + 2 e iφ ξD (1 − ξ ) A q ++; −− (cid:21) , (C23) H qT = (cid:20) e − iφ D (cid:18)
11 + ξ A q ++;+ − + 11 − ξ A q − +; −− (cid:19) + 1 √ D (cid:18) − ξ ξ A q ++;0 − + 1 + ξ − ξ A q −− (cid:19) + e − iφ D A q − +;+ − − e iφ D (cid:18) D − ξ − ξ (cid:19) A q ++; −− (cid:21) , (C24) H qT = 1 √ (cid:20) − e − iφ D (cid:18) ξ ) (cid:18)
12 + D − ξ (cid:19) A q ++;+ − + 1(1 − ξ ) (cid:18)
12 + D ξ (cid:19) A q − +; −− (cid:19) + e − iφ (1 − ξ ) D A q − + e − iφ √ − ξ ) (cid:0) A q − + A q − +;0 − (cid:1) − √ ξ ) (cid:18) − ξD (1 − ξ ) (cid:19) A q ++;0 − − √ − ξ ) (cid:18) ξD (1 + ξ ) (cid:19) A q −− − e − iφ D (1 − ξ ) A q − +;+ − − e iφ (cid:0) D (3 ξ + 1) + 4 ξ (cid:1) D (1 − ξ ) A q ++; −− (cid:35) , (C25) H qT = − D (cid:2) e − iφ (cid:0) A q ++;+ − + A q − +; −− (cid:1) + e − iφ A q − +;+ − + e iφ A q ++; −− (cid:3) , (C26) H qT = 2 e iφ (1 − ξ ) D A q ++; −− , (C27) H qT = 1 D (cid:34) − e − iφ (cid:18) − ξ ξ A q ++;+ − − ξ − ξ A q − +; −− (cid:19) + √ ξD (cid:18) − ξ ξ A q ++;0 − + 1 + ξ − ξ A q − +;0 − (cid:19) + 4 e iφ ξ D (1 − ξ ) A q ++; −− (cid:35) , (C28) H qT = − D (cid:34) e − iφ ξ (cid:18)
11 + ξ A q ++;+ − − − ξ A q − +; −− (cid:19) + √ ξD (cid:18) − ξ ξ A q ++;0 − − ξ − ξ A q − +;0 − (cid:19) + e − iφ A q − +;+ − − e iφ (cid:18) ξ D (1 − ξ ) (cid:19) A q ++; −− (cid:21) . (C29)24or the gluon GPDs we have H gT = (cid:34) √ e − iφ ξD (1 − ξ ) (cid:0) A g ++;+ − − A g − +; −− (cid:1) + e − iφ D (cid:18) − ξ A g − + 11 + ξ A g − +;0 − (cid:19) + e − iφ D (cid:18)
11 + ξ A g ++;0 − + 11 − ξ A g −− (cid:19) + √ − ξ ) (cid:0) e − iφ A g − +;+ − − A g ++; −− (cid:1)(cid:35) , (C30) H gT = (cid:34) − √ e − iφ D (cid:18) ξ ) ( ξ − D ) A g ++;+ − + 1(1 − ξ ) ( ξ + D ) A g − +; −− (cid:19) + 2 √ e − iφ ξD (1 − ξ ) A g − − e − iφ (cid:18)
11 + ξ A g − − − ξ A g − +;0 − (cid:19) + e − iφ D (1 − ξ ) (cid:18) ξ + D (1 − ξ ) ξ A g ++;0 − − ξ + D (1 + ξ ) − ξ A g −− (cid:19) − √ ξ (cid:0) D (1 + ξ ) + 2 ξ (cid:1) D (1 − ξ ) A g ++; −− (cid:35) , (C31) H gT = (cid:20) − e − iφ D (cid:18) − ξ ξ A g ++;+ − − ξ − ξ A g − +; −− (cid:19) − e − iφ √ D (cid:18) − ξ ξ A g ++;0 − − ξ − ξ A g −− (cid:19) + ξD (1 − ξ ) A g ++; −− (cid:21) , (C32) H gT = (cid:20) e − iφ D (cid:18)
11 + ξ A g ++;+ − + 11 − ξ A g − +; −− (cid:19) + e − iφ √ D (cid:18) − ξ ξ A g ++;0 − + 1 + ξ − ξ A g −− (cid:19) + e − iφ D A g − +;+ − − e iφ D (cid:18) D − ξ − ξ (cid:19) A g ++; −− (cid:21) , (C33) H gT = (cid:20) e − iφ D (1 − ξ ) (cid:18) D (1 − ξ )(1 + 2 ξ ) + 2 ξ (1 − ξ )(1 + ξ ) A g ++;+ − + D (1 + ξ )(1 − ξ ) − ξ (1 − ξ )(1 − ξ ) A g − +; −− + 2 ξ A g − (cid:19) + √ e − iφ ξ D (1 − ξ ) (cid:0) A g − + A g − +;0 − (cid:1) − √ e − iφ D (1 − ξ ) (cid:18) D (1 − ξ ) − ξ ξ A g ++;0 − + D (1 + ξ ) + 2 ξ − ξ A g −− (cid:19) + e − iφ − ξ A g − +;+ − + ( D (1 − ξ ) − ξ )( D + 4 ξ ) D (1 − ξ ) A g ++; −− (cid:21) , (C34) H gT = (cid:20) − e − iφ D (1 − ξ ) (cid:18) D (1 − ξ ) − ξ (1 − ξ )(1 + ξ ) A g ++;+ − + D (1 + ξ ) + 2 ξ (1 − ξ )(1 − ξ ) A g − +; −− + 2 A g − (cid:19) + √ e − iφ D (1 − ξ ) (cid:0) A g − + A g − +;0 − (cid:1) − √ e − iφ ξ D (1 − ξ ) (cid:18) D (1 − ξ ) − ξ ξ A g ++;0 − + D (1 + ξ ) + 2 ξ − ξ A g −− (cid:19) + e − iφ − ξ A g − +;+ − + ( D (1 − ξ ) − ξ )( D (1 + ξ ) + 2 ξ ) D (1 − ξ ) A g ++; −− (cid:21) , (C35) H gT = 1 − ξ D A g ++; −− , (C36) H gT = 1 D (cid:20) e − iφ (cid:18) − − ξ ξ A g ++;+ − + 1 + ξ − ξ A g − +; −− (cid:19) + e − iφ √ D (cid:18) − ξ ξ A g ++;0 − + 1 + ξ − ξ A g −− (cid:19) + 2 ξ D (1 − ξ ) A g ++; −− (cid:21) , (C37) H gT = 1 D (cid:20) e − iφ (cid:18) − ξ ξ A g ++;+ − + 1 + ξ − ξ A g − +; −− (cid:19) + e − iφ √ D (cid:18) − − ξ ξ A g ++;0 − + 1 + ξ − ξ A g −− (cid:19) − (cid:18) ξ D (1 − ξ ) (cid:19) A g ++; −− (cid:21) . (C38) Appendix D: Minimal convolution model for the deuteron
In this appendix, we outline a minimal convolution model for the deuteron GPDs. The model allows to calculatethe transversity GPDs analytically and to check certain trends seen in the full convolution model.The minimal model starts from the following assumptions:25
We only include the nucleon chiral odd GPD ¯ E T and put all others equal to zero. Figs. 9 and 10 show that thisis a reasonable starting point. • We do not include a D -wave component in the deuteron wave function. • We do not consider a spatial wave function for the S -wave. This means we only include the nucleon spin sums(through Clebsch-Gordan coefficients) and consider the following symmetric kinematics in the convolution: P ⊥ = 0 , ∆ y = 0 ,φ = 0 ,α = 1 + ξ , α (cid:48) = 1 − ξ ,k x ⊥ = − ∆ x , k y ⊥ = 0 ,k (cid:48) x ⊥ = ∆ x , k (cid:48) y ⊥ = 0 ,ξ N = 2 ξ ξ , x N = 2 x ξ . (D1)With the choice of this kinematics the symmetry constraints of Subsec. II B are still obeyed.In this minimal convolution model, we obtain for the nucleon helicity amplitudes (cid:90) d x N A N ++;+ − ( x N , ξ N , t ) = (1 − ξ N ) √ t N − t m F ( t ) , (cid:90) d x N A N − +; −− ( x N , ξ N , t ) = (1 + ξ N ) √ t N − t m F ( t ) , (cid:90) d x N A N ++; −− ( x N , ξ N , t ) = − ξ N (cid:112) − ξ N F ( t ) , (cid:90) d x N A N − +;+ − ( x N , ξ N , t ) = 0 , (D2)where F ( t ) = (cid:82) d x N ¯ E T ( x N , ξ N , t ).Using Eq. (39) in the minimal version, we obtain for the deuteron helicity amplitudes (cid:90) d x A q ++;+ − ( x, ξ, t ) = (1 − ξ ) √ t N − t m F ( t ) , (cid:90) d x A q − +; −− ( x, ξ, t ) = (1 + ξ ) √ t N − t m F ( t ) , (cid:90) d x A q − ( x, ξ, t ) = (1 + ξ ) √ t N − t m F ( t ) , (cid:90) d x A q − ( x, ξ, t ) = (cid:90) d x A q − +;0 − ( x, ξ, t ) = 0 , (cid:90) d x A q ++;0 − ( x, ξ, t ) = (cid:90) d x A q −− ( x, ξ, t ) = − √ ξ − ξ F ( t ) , (cid:90) d x A q ++; −− ( x, ξ, t ) = (cid:90) d x A q − +;+ − ( x, ξ, t ) = 0 . (D3)Note that the first moments of A − ( x, ξ, t ) and A − +0 − ( x, ξ, t ) are zero because the first moment of A − ++ − N ( x N , ξ N , t ) is zero (which is the only one contributing to those on the nucleon level), and the first moments of A ++ −− ( x, ξ, t ) and A − ++ − ( x, ξ, t ) are zero because we did not include a D -wave in the deuteron wave function.26inally, using Eqs. (C21) to (C29), we obtain for the chiral odd quark GPDs (cid:90) d x H T ( x, ξ, t ) = − √ ξ (1 − ξ ) (cid:32)(cid:115) (1 − ξ )( t N − t )( t − t ) M m + 1 (cid:33) F ( t ) , (cid:90) d x H T ( x, ξ, t ) = − √ ξ (1 − ξ ) (cid:18) ξ D (cid:19) (cid:18) D (3 + ξ ) √ t N − t m + 8 ξ − ξ (cid:19) F ( t ) , (cid:90) d x H T ( x, ξ, t ) = 4 ξ (1 + ξ )(1 − ξ ) (cid:115) ( t N − t )( t − t )(1 − ξ ) Mm F ( t ) − ξ D (1 − ξ ) F ( t ) , (cid:90) d x H T ( x, ξ, t ) = 2 (1 + 3 ξ )(1 − ξ ) (cid:115) ( t N − t )( t − t )(1 − ξ ) Mm F ( t ) − ξ ξ D (1 − ξ ) F ( t ) , (cid:90) d x H T ( x, ξ, t ) = 16 √ ξ D (1 − ξ ) F ( t ) + 8 √ ξ (1 + ξ )(1 − ξ ) F ( t ) − (cid:112) (1 − ξ )( t − t )( t N − t ) √ M m (1 + 3 ξ )(1 − ξ ) F ( t ) − (cid:115) t N − t )( t − t )(1 − ξ ) M m ξ (3 + ξ )(1 − ξ ) F ( t ) , (cid:90) d x H T ( x, ξ, t ) = − (1 + ξ ) (cid:115) ( t N − t )( t − t )(1 − ξ ) Mm F ( t ) , (cid:90) d x H T ( x, ξ, t ) = 0 , (cid:90) d x H T ( x, ξ, t ) = 16 ξ ξ − ξ M m (cid:115) t N − t ( t − t )(1 − ξ ) F ( t ) + 16 ξ (1 + ξ ) D (1 − ξ ) F ( t ) , (cid:90) d x H T ( x, ξ, t ) = 8 ξ ξ − ξ M m (cid:115) t N − t ( t − t )(1 − ξ ) F ( t ) − ξ D (1 − ξ ) F ( t ) . (D4) [1] M. Diehl, Phys. Rept. , 41 (2003), hep-ph/0307382.[2] A. V. Belitsky and A. V. Radyushkin, Phys. Rept. , 1 (2005), hep-ph/0504030.[3] S. Fucini, S. Scopetta, and M. Viviani, Phys. Rev. C98 , 015203 (2018), 1805.05877.[4] R. Dupr´e and S. Scopetta, Eur. Phys. J.
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