φ-meson lepto-production near threshold and the strangeness D-term
φφ -meson lepto-production near threshold and the strangeness D -term Yoshitaka Hatta
Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA andRIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA
Mark Strikman
Department of Physics, Penn State University, University Park PA 16802, USA (Dated: February 26, 2021)We present a model of exclusive φ -meson lepto-production ep → e (cid:48) p (cid:48) φ near threshold whichfeatures the strangeness gravitational form factors of the proton. We argue that the shape of thedifferential cross section dσ/dt is a sensitive probe of the strangeness D-term of the proton. I. INTRODUCTION
Exclusive lepto-production of vector mesons ep → e (cid:48) γ ∗ p → e (cid:48) p (cid:48) V is a versatile process that can address a widespectrum of key questions about the proton structure. Depending on the γ ∗ p center-of-mass energy W , photonvirtuality Q and meson species, the spacetime picture of the reaction looks different and requires different theoreticalframeworks. At high energy in the large- Q region, and for the longitudinally polarized virtual photon, a QCDfactorization theorem [1] dictates that the scattering amplitude can be written in terms of the generalized partondistribution (GPD), the meson distribution amplitude (DA) and the perturbatively calculable hard part. Even incases where such a rigorous treatment is not available, many phenomenologically successful models exist. Generallyspeaking, light vector mesons probe the up and down quark contents of the proton, whereas in the Regge-regime W → ∞ , or for heavy vector mesons such as quarkonia ( J/ψ, Υ , ... ), the process is primarily sensitive to the gluoniccontent of the proton.The φ -meson (a bound state of s ¯ s ) has a somewhat unique status in this context because its mass m φ = 1 .
02 GeV,being roughly equal to the proton mass m N = 0 .
94 GeV, is neither light nor heavy. In the literature, φ -productionis often discussed on a similar footing as quarkonium production. Namely, since the proton does not contain valence s -quarks, the φ -proton interaction proceeds mostly via gluon exchanges. However, the proton contains a small but non-negligible fraction of strange sea quarks already on nonperturbative scales of ∼ . It is then a nontrivial questionwhether the gluon exchanges, suppressed by the QCD coupling α s , always dominate over the s -quark exchanges.In this paper, we take a fresh look at the lepto-production of φ -mesons ep → e (cid:48) p (cid:48) φ in the high- Q region nearthreshold , namely when W is very low and barely enough to produce a φ -meson W (cid:38) m φ + m N = 1 .
96 GeV.Measurements in the threshold region have been performed at LEPS [2–4] for photo-production Q ≈ (cid:38) Q (cid:38) (cid:104) P (cid:48) | T µνg | P (cid:105) where T µνg is the gluon part of the QCDenergy momentum tensor [9–11] (see also [12, 13]). In lepto-production at high- Q , this connection can be cleanlyestablished by using the operator product expansion (OPE) [14]. The analysis [9] of J/ψ photo-production over awide range of W indicates that the t -dependence of these form factors (or that of the gluon GPDs at high energy W )is weaker than what one would expect from the Q -dependence of the electromagnetic form factors. This suggeststhat the gluon fields in the nucleon are more compact and localized than the charge distribution. Moreover, it hasbeen demonstrated [10, 11, 14] that the shape of the differential cross section dσ/dt is sensitive to the gluon D-term D g ( t ) which, after Fourier transforming to the coordinate space, can be interpreted as an internal force (or ‘pres-sure’) exerted by gluons inside the proton [15] (see however, [16]). The experimental determination of the parameter D g ( t = 0) is complementary to the ongoing effort [17–19] to extract the u, d -quark contributions to the D-term D u,d from the present and future deeply virtual Compton scattering (DVCS) experiments. Together they constitute thetotal D-term D (0) = D u (0) + D d (0) + D s (0) + D g (0) + · · · . (1)Only the sum is conserved (renormalization-group invariant) and represents a fundamental constant of the proton.The strangeness contribution to the D-term D s has received very little, if any, attention in the literature so far.However, a large- N c argument [20] suggests the approximate flavor independence of the D-term D u ≈ D d (nontrivialbecause u -quarks are more abundant than d -quarks in the proton) which by extension implies that D s could be a r X i v : . [ h e p - ph ] F e b comparable to D u,d at least in the flavor SU(3) symmetric limit. It is thus a potentially important piece of the sumrule (1) when the precision study of the D-term becomes possible in future. The purpose of this paper is to argue thatthe lepto-production of φ -mesons near threshold is governed by the strangeness gravitational form factors including D s . By doing so, we basically postulate that the φ -meson couples more strongly to the s ¯ s content of the proton thanto the gluon content at least near the threshold at high- Q , contrary to the prevailing view in the literature. Also,by using the local version of the OPE following Ref. [14], we do not work in the collinear factorization framework [1]whose applicability to the threshold region does not seem likely. In the appendix, we briefly discuss the connectionbetween the two approaches. Other approaches to φ -meson photo- or lepto-production near threshold can be foundin [21–23].In Section II, we collect general formulas for the cross section of lepto-production. In Section III, we describe ourmodel of the scattering amplitude inspired by the OPE and the strangeness gravitational form factors. The numericalresults for the differential cross section dσ/dt are presented in Section IV for the kinematics relevant to the JLab andthe future electron-ion collider (EIC). II. EXCLUSIVE φ -MESON LEPTOPRODUCTION Consider exclusive φ -meson production cross section in electron-proton scattering ep → e (cid:48) γ ∗ p → e (cid:48) p (cid:48) φ . The ep and γ ∗ p center-of-mass energies are denoted by s ep = ( (cid:96) + P ) and W = ( q + P ) , respectively. The outgoing φ hasmomentum k µ with k = m φ . The cross section can be written as dσdW dQ = α em π P · (cid:96) ) Q P cm (cid:90) dφ (cid:96) π L µν (cid:90) dt (cid:88) spin (cid:104) P | J µem ( − q ) | P (cid:48) φ (cid:105)(cid:104) P (cid:48) φ | J νem ( q ) | P (cid:105) , (2)where J em is the electromagnetic current operator, m N is the proton mass and P cm is the proton momentum in the γ ∗ p center-of-mass frame P cm = (cid:112) W − W ( m N − Q ) + ( m N + Q ) W . (3)For simplicity, we average over the outgoing lepton angle φ (cid:96) , but its dependence can be restored [24] if need arises.We can then write, in a frame where the virtual photon is in the + z direction, (cid:90) dφ (cid:96) π L µν = 2 Q − (cid:15) (cid:18) g µν ⊥ + (cid:15)ε µL ε νL (cid:19) (4)where g ij ⊥ = δ ij and ε µL ( q ) = 1 Q (cid:112) γ (cid:18) q µ + Q P · q P µ (cid:19) . (5)is the polarization vector of the longitudinal virtual photon. (cid:15) is the longitudinal-to-transverse photon flux ratio (cid:15) = 1 − y − y γ − y + y + y γ , − (cid:15) = 2 y − y + y + y γ γ (6)where y ≡ P · qP · (cid:96) , γ ≡ x B m N Q = m N QP · q (7)The parameter γ accounts for the target mass correction which should be included in near-threshold production. Thevariable y can be eliminated in favor of s ep using the relation W = y ( s ep − m N ) + m N − Q . (8)In the hadronic part, let us define12 (cid:88) spin (cid:104) P | J µem ( − q ) | P (cid:48) φ (cid:105)(cid:104) P (cid:48) φ | J νem ( q ) | P (cid:105) ≡ − ( M µρ ) ∗ M ρν , (9)where the minus sign is from the vector meson polarization sum (cid:80) ε ρV ε ρ (cid:48) V = − g ρρ (cid:48) + k ρ k ρ (cid:48) m φ . (The amplitude M satisfiesthe conditions k ρ M ρµ = M ρµ q µ = 0.) Contracting with the lepton tensor (4), we get − (cid:18) g µν ⊥ + (cid:15)ε µL ε νL (cid:19) M ∗ ρµ M ρν = − (cid:18) g µν ⊥ + (cid:15) γ Q ( P · q ) P µ P ν (cid:19) M ∗ ρµ M ρν = (cid:32) g µν − (cid:0) + (cid:15) (cid:1) Q (1 + γ )( P · q ) P µ P ν (cid:33) M ∗ ρµ M ρν , (10)where we used g µν ⊥ = − g µν + ε µL ε νL − q µ q ν Q . (11)We thus arrive at dσdW dQ = α em π W ( W − m N )( P · (cid:96) ) Q (1 − (cid:15) ) (cid:90) dt dσdt , (12)with dσdt = α em W − m N ) W P cm (cid:32) g µν − (cid:0) + (cid:15) (cid:1) Q (1 + γ )( P · q ) P µ P ν (cid:33) M ∗ ρµ M ρν . (13)Note that the γ ∗ p cross section (13) depends on the ep center-of-mass energy s ep because (cid:15) depends on y , and y depends on s ep as in (8). Experimentalists at the JLab have measured dσdW dQ [5], and from the data they havereconstructed the differential cross section (13) and the total cross section σ ( W, Q ) = (cid:82) dt dσdt . In the next section wepresent a model for the scattering amplitude M . III. DESCRIPTION OF THE MODEL
Our model for the matrix element (cid:104) P (cid:48) φ ( k ) | J νem ( q ) | P (cid:105) has been inspired by the recently developed new approach tothe near-threshold production of heavy quarkonia such as J/ψ and Υ [14]. The main steps of [14] are summarized asfollows. One first relates the
J/ψ production amplitude (cid:104) P (cid:48) J/ψ ( k ) | J νem ( q ) | P (cid:105) to the correlation function of the charmcurrent operator J µc = ¯ cγ µ c (cid:90) dxdye ik · x − iq · y (cid:104) P (cid:48) | T { J µc ( x ) J νc ( y ) }| P (cid:105) , (14)in a slightly off-shell kinematics k (cid:54) = m J/ψ . One then performs the operator product expansion (OPE) in theregime Q (cid:29) | t | , m J/ψ and picks up gluon bilinear operators ∼ F F . The off-forward matrix elements (cid:104) P (cid:48) | F F | P (cid:105) areparameterized by the gluon gravitational form factors. These include the gluon momentum fraction A g (the secondmoment of the gluon PDF), the gluon D-term D g and the gluon condensate (trace anomaly) (cid:104) F µν F µν (cid:105) .When adapting this approach to φ -production, we recognize a few important differences. First, we need to keep s -quark bilinear operators ∼ ¯ ss rather than gluon bilinears. A quick way to estimate their relative importance in thepresent approach is to compare the momentum fractions A s +¯ s and α s π A g . Taking A s +¯ s ≈ . A g ≈ . α s = 0 . ∼ . s -quark contribution getsan additional factor ∼
2. Second, the condition Q (cid:29) | t | is more difficult to satisfy. For example, the momentumtransfer at the threshold is | t th | = m N ( m V + Q ) m N + m V . (15) In Ref. [14], the authors calculated the spin-averaged cross section g µν M ∗ ρµ M ρν . To directly compare with the lepto-production data,one should rather use the formula (10). This condition is needed in order to ensure that the large momentum Q does not flow into the nucleon vertex so that one can performthe JJ OPE.
When Q (cid:29) m V , | t th | is a larger fraction of Q in φ -production m V = m φ ≈ m N than in J/ψ -production m V = m J/ψ ≈ m N . As one goes away (but not too far away) from the threshold, the region Q (cid:29) | t | does exist. In principle,our predictions are limited to such regions, though in practice they can be smoothly extrapolated to | t | > Q as longas | t | is not too small.We now perform the OPE. A simple calculation shows A µνs ≡ i (cid:90) d d re ir · q (cid:16) ¯ s (0) γ µ S (0 , − r ) γ ν s ( − r ) + ¯ s ( − r ) γ ν S ( − r, γ µ s (0) (cid:17) = − q − m s ) (cid:0) q g µα g νβ − g µα q ν q β − g νβ q µ q α + g µν q α q β (cid:1) T sαβ + · · ·→ − q − m s ) ( q · kg µα g νβ − g µα k ν q β − g νβ q µ k α + k α q β g µν ) T sαβ + · · · , (16)where S (0 , − r ) = (cid:90) d d q (2 π ) d e − iq · r i (/ q + m s ) q − m s , (17)is the s -quark propagator and T sαβ = i ¯ sγ ( α ←→ D β ) s. (18)is the s -quark contribution to the energy momentum tensor ( ←→ D ≡ D −←− D ). We neglect the s -quark mass m s wheneverit appears in the numerator. However, we keep it in the denominator to regularize the divergence q − m s ∼ Q + m s just in case we may want to extrapolate our results to smaller Q values in future applications. In the last line of(16), we have implemented minimal modifications to make A µνs transverse with respect to both q ν and k µ as requiredby gauge invariance. While this is ad hoc at the present level of discussion, we anticipate that total derivative/highertwist operators restore gauge invariance, similarly to what happens in deeply virtual Compton scattering (DVCS) [26]Of course, even after restricting ourselves to quark bilinears, there are other operators that can contribute to (16).Potentially important operators include the axial vector operator ¯ sγ µ γ s and the s -quark twist-two operators withhigher spins. The former is related to the (small) s -quark helicity contribution ∆ s to the proton spin in the forwardlimit. Its off-forward matrix element is basically unknown. The latter are discussed in the appendix where it isfound that, unlike in the quarkonium case [14], the twist-two, higher-spin operators are not negligible for the presentproblem. To mimic their effect, we introduce an overall phenomenological factor of 2.5 in (16).To evaluate the matrix element (cid:104) P (cid:48) |A µνs | P (cid:105) , we use the following parameterization of the gravitational form factors[27, 28] (cid:104) P (cid:48) | T αβs | P (cid:105) = ¯ u ( P (cid:48) ) (cid:20) A s ( t ) γ ( α ¯ P β ) + B s ( t ) ¯ P ( α iσ β ) λ ∆ λ m N + D s ( t ) ∆ α ∆ β − g αβ ∆ m N + ¯ C s ( t ) m N g αβ (cid:21) u ( P ) , (19)where ¯ P = P + P (cid:48) , ∆ µ = P (cid:48) µ − P µ and t = ∆ . D ( t ) / C ( t ) in the literature. We neglect B s following the empirical observation that the flavor-singlet B u + d is unusually small (see, e.g., [29]). We further set¯ C s = − A s assuming that the trace anomaly is insignificant in the strangeness sector. [However, this point may beimproved as was done for gluons in [11, 14].] For the remaining form factors, we employ the dipole and tripole ansatzesuggested by the perturbative counting rules at large- t [30, 31] A s ( t ) = A s (0)(1 − t/m A ) , D s ( t ) = D s (0)(1 − t/m D ) , (20)with A s (0) = A s +¯ s = 0 .
04 as mentioned above. We use the same effective masses m A = 1 .
13 GeV and m D = 0 . s -quarks in thenucleon are generated by the gluon splitting g → s ¯ s . The value D s (0) is our main object of interest, and is treatedhere as a free parameter. As mentioned in the introduction, even though s -quarks are much less abundant in the In [9] and more recently in [33], the authors fitted the φ -meson photo- and lepto-production data using the form dσ/dt ∝ A ( t ) ∝ / (1 − t/m A ) and found that the mass parameter m A is consistent with the J/ψ case. This partially supports our procedure. proton A s (cid:28) A u,d , a large- N c argument suggests that the D-terms are ‘flavor blind’ D s ∼ D u ≈ D d [20]. In theflavor SU(3) limit, and at asymptotically large scales, the relation ( C F = N c − N c = 4 / D s (0) ≈ C F D g (0) , (21)together with the recent lattice result for the gluon D-term D g (0) ≈ − . D s (0) ≈ − .
35. We thus varythe parameter in the range 0 > D s (0) > − .
35, with a particular interest in the possibility that | D s | is of order unity.Note that this makes φ -production rather special, compared to light or heavy meson productions. If A s (cid:28) A u,d,g but D s ∼ D u,d , the effect of the D-term will be particularly large in the strangeness sector.Finally, the proportionality constant between M µν and (cid:104) P (cid:48) |A µνs | P (cid:105) can be determined similarly to the J/ψ case(see Eqs. (48,49) of [14]). Using the φ -meson mass m φ =1.02 GeV and its leptonic decay width Γ e + e − = 1 .
27 keV,we find e s e m φ g γφ = 4 πe s α em m φ e + e − ≈ . , (22)where e s = − / g γφ is the decay constant. There is actually an uncertainty of order unity in the overallnormalization of the amplitude as mentioned in [14] (apart from the factor of 2.5 mentioned above). This can be fixedby fitting to the total cross section data. Then the shape of dσ/dt is the prediction of our model. IV. NUMERICAL RESULTS AND DISCUSSIONS
We now present our numerical results for the JLab kinematics with a 6 GeV electron beam ( √ s ep ≈ . . < Q < . , | t − t min | < . , < W < , (23)where t min is the kinematical lower limit of t which depends on Q and W . Admittedly, even the maximal value Q = 3 . is not large from a perturbative QCD point of view. However, (23) is the only kinematical windowwhere the lepto-production data exist. Our model actually provides smooth curves for observables in the above rangeof Q .Fig. 1 shows dσ/dt at W = 2 . Q = 3 . . We chose m s = 100 MeV for the current s -quark mass. Thefour curves correspond to different values of the D -term, D s = 0 , − . , − . , − . | t | -distribution in the small- t region. This isdue to the explicit factors of ∆ µ ( t = ∆ ) multiplying D s in (19) which tend to shift the peak of the t -distribution tolarger values. Unfortunately, we cannot directly compare our result with the JLab data. The relevant plot, Fig. 18 ofRef. [5] is a mixture of data from different values of Q in the range (23). For a meaningful comparison, dσ/dt shouldbe plotted for a fixed (large) value of Q , and there should be enough data points in the most interesting region | t | (cid:46) . By the same reason, we cannot adjust the overall normalization of the amplitude mentioned at the end of theprevious section. Incidentally, we note that in this kinematics the cross section is dominated by the contribution fromthe transversely polarized photon, namely, the part proportional to g µν ⊥ in (10).For illustration, in Fig. 2 we show the result with Q = 20 GeV and W = 2 . √ s ep = 30 GeV for definiteness,but the dependence on s ep is very weak as it only enters the parameter (cid:15) in (13) and (cid:15) ≈ ep cross section (12) strongly depends on s ep .) The contribution from the longitudinally polarizedphoton (the part proportional to (cid:15)ε µL ε νL in (10)) is now comparable to the transverse part. Again the impact of theD-term is noticeable, but the bump has almost disappeared and we only see a flattening of the curve in the extremecase D s = − .
3. The reason is simple. The cross section schematically has the form dσdt ∼ f ( t )(1 − t/m ) a , (24) The prediction | D u + D d | (cid:29) | D u − D d | from large- N c QCD is supported by the lattice simulations [29]. Interestingly, and in contrast,the B -form factor is dominantly a flavor nonsinglet quantity | B u − B d | (cid:29) | B u + B d | as already mentioned. While this choice is natural in the present framework, it leads to a too steep rise of the cross section σ ( Q ) as Q is decreased towards1 GeV . Of course, our approach breaks down in this limit, but it is still possible to get a better Q behavior in the low- Q regionby switching to the constituent s -quark mass m s = m φ / ≈
500 MeV, or perhaps even m s → m φ as in the vector meson dominance(VMD) model. t σ dt FIG. 1: Differential cross section dσ/dt in units of nb/GeV as a function of | t | (in GeV ). W = 2 . Q = 3 . . The four curves correspond to D s = 0 , − . , − . , − . f ( t ) is a low-order polynomial in t and a = 4 , ,
6. (See (20). The amplitude squared is a linear combination of A s ( t ) , A s ( t ) D s ( t ) and D s ( t ).) The t -dependence of f ( t ) comes from the D-term and gamma matrix traces involvingnucleon spinors (19). Clearly, f ( t ) can affect the shape of dσ/dt only when | t | < m ∼ . Beyond that, onesimply has the power law dσ/dt ∼ /t c with c <
4. In Fig. 1, | t min | < Q gets larger, so does | t min | and the structure disappears.In conclusion, we have proposed a new model of φ -meson lepto-production near threshold. In our model, thecross section is solely determined by the strangeness gravitational form factors, similarly to the J/ψ case where it isdetermined by the gluon counterparts [14]. Of particular interest is the value of D s , the strangeness contribution tothe proton D-term. While D s is ignored in most literature, an argument based on the large- N c QCD suggests thatit may actually be comparable to D u,d [20]. If this is the case, we predict a flattening or possibly a bump in the t -distribution of dσ/dt in the small- t region. It is very interesting to test this scenario by re-analyzing the JLab data[5] or conducting new experiments focusing on the | t | < region.There are number of directions for improvement. As already mentioned, operators other than the energy momentumtensors should be included as much as possible, although this will unavoidably introduce more parameters in the model.We have argued in the appendix that the contribution from the twist-two, higher spin operators is small, but thisneeds to be checked. Also the renormalization group evolution of the form factors should be taken into account if infuture one can measure this process over a broad range in Q such as at the EICs in the U.S. and in China [34, 36]. Acknowledgments
Y. H. thanks the Yukawa Institute for Theoretical Physics for hospitality. This work is supported by the U.S.Department of Energy, Office of Science, Office of Nuclear Physics, under contracts No. DE- SC0012704, DE-SC-0002145 and DE-FG02-93ER40771. It is also supported in part by Laboratory Directed Research and Development(LDRD) funds from Brookhaven Science Associates.
Appendix A: Connection to the GPD approach
In this appendix, we argue how our OPE approach is connected to the usual light-cone approach in terms ofthe generalized parton distribution (GPD), see, e.g., [20] for a review. Consider doubly virtual Compton scattering(VVCS) p ( P ) γ ∗ ( q ) → p ( P (cid:48) ) γ ∗ ( q (cid:48) ) or deeply virtual meson production (DVMP) p ( P ) γ ∗ ( q ) → p ( P (cid:48) ) V ( q (cid:48) ) and introduce t σ dt FIG. 2: Differential cross section dσ/dt in units of pb/GeV as a function of | t | (in GeV ). W = 2 . Q = 20GeV , √ s ep = 30 GeV. The four curves correspond to D s = 0 , − . , − . , − . q = q + q (cid:48) , ¯ P = P + P (cid:48) , ∆ = P (cid:48) − P = q − q (cid:48) , ξ = − ¯ q P · ¯ q , η = − ∆ · ¯ q P · ¯ q . (A1) η is the skewness parameter and ξ is the analog of the Bjorken variable x B = − q P · q in DIS. In the regime of our interest q (cid:48) ∼ m φ (cid:28) − q = Q , we can write ηξ ≈
11 + ∆ Q ≈ , η ≈ x B − x B (cid:16) − ∆ Q (cid:17) ≈ x B − x B , (A2)where the second approximation is valid when Q (cid:29) | t | = | ∆ | . In DIS or DVCS, one takes the scaling limit − ¯ q → ∞ and 2 ¯ P · ¯ q → ∞ (and implicitly W = ( P + q ) → ∞ ) keeping the ratio 1 > ξ > W = ( P + q ) is constrained to be close to ( m N + m φ ) , the scaling limit cannot be taken literally because x B and Q are no longer independent x B = Q P · q = Q W + Q − m N ≈ Q Q + 2 m φ m N + m φ . (A3)In the limit Q → ∞ , we have that x B ≈ ξ ≈ η ≈ . (A4)The partonic interpretation of scattering in this regime is rather peculiar. Normally one works in a frame in whichthe incoming and outgoing protons are fast-moving ξ ≈ η ≈ P + − P (cid:48) + P + + P (cid:48) + . (A5)The incoming proton has the light-cone energy P + = (1 + η ) ¯ P + , and it emits two partons with momentum fractions η + x η , η − x η . (A6)When η ≈ ξ ≈ P + ≈ P + and the outgoing proton has vanishing light-cone energy P (cid:48) + ≈
0. Moreover, thecondition 2 P · q ≈ Q means q + ≈ − P + and ( P + + q + ) q − ∼ m N . Therefore, the outgoing meson is not fast-movingin the minus direction q − ∼ O ( m N ). Since the suppression of final state interactions due to large relative momentais crucial for the proof of factorization [1], we suspect that the standard approach based on GPDs is not applicablefor near-threshold production, at least in its original form.Nevertheless, we can make a rough connection to the present approach as follows. When ξ ≈
1, the s -quarkcontribution to the Compton form factor may be Taylor expanded as A s ( ξ, η ) ∼ (cid:90) − dx (cid:18) ξ − x − i(cid:15) − ξ + x − i(cid:15) (cid:19) H s ( x, η, t ) ≈ (cid:90) − dx ( x + x + · · · ) H s ( x, , t ) , (A7)where H s is the s -quark GPD. The lowest moment is proportional to the gravitational form factor (cid:104) P (cid:48) | T ++ s | P (cid:105) thatwe keep, and higher moments give the form factors of the twist-two higher spin operators (cid:104) x n (cid:105) ∼ (cid:104) P (cid:48) | ¯ sγ + ( D + ) n s | P (cid:105) .To estimate the impact of the latter, let us substitute the asymptotic form at large renormalization scales [20] H s ( x, η = 1 , t ) ∝ x (1 − x ) . (A8)The above integral proportional to (cid:90) dx x (1 − x )1 − x = 13 . (A9)If we only keep the first term in the Taylor expansion (corresponding to the energy momentum tensor), we get (cid:90) dxx (1 − x ) = 215 , (A10)that is, 40% of the full result. This is in contrast to the J/ψ case [14] where one has instead the gluon GPD (cid:90) dx H g ( x, η = 1 , t )1 − x ∼ (cid:90) dx (1 − x ) − x = 23 . (A11)The first term in the Taylor expansion (corresponding to the gluon energy momentum tensor) (cid:90) dx (1 − x ) = 815 . (A12)accounts for 80% of the total. The origin of this difference is easy to understand. The s -quark GPD H s vanishes at x = 0 because the s and ¯ s sea quarks are symmetric H ¯ s ( x ) = − H s ( − x ) = H s ( x ). On the other hand, the gluon GPD H g is peaked at x = 0 so higher moments in x are numerically more suppressed.We thus conclude that the twist-two, higher spin operators are not negligible in the s -quark case, although they arerelatively innocuous in the gluon case. To cope with this, we introduce an overall factor 1 / . . H s contains a part related to the D-term [37], so thisfactor is common to both the A s and D s form factors. [1] J. C. Collins, L. Frankfurt and M. Strikman, Phys. Rev. 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