φ-meson lepto-production near threshold and the strangeness D-term

Exclusive lepto-production of vector mesons ep → e′γ∗p → e′p′V is a versatile process that can address a wide spectrum of key questions about the proton structure. Depending on the γ∗p center-of-mass energy W , photon virtuality Q and meson species, the spacetime picture of the reaction looks different and requires different theoretical frameworks. At high energy in the large-Q region, and for the longitudinally polarized virtual photon, a QCD factorization theorem [1] dictates that the scattering amplitude can be written in terms of the generalized parton distribution (GPD), the meson distribution amplitude (DA) and the perturbatively calculable hard part. Even in cases where such a rigorous treatment is not available, many phenomenologically successful models exist. Generally speaking, 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 gluonic content of the proton. The φ-meson (a bound state of ss̄) has a somewhat unique status in this context because its mass mφ = 1.02 GeV, being roughly equal to the proton mass mN = 0.94 GeV, is neither light nor heavy. In the literature, φ-production is 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 nonnegligible fraction of strange sea quarks already on nonperturbative scales of ∼ 1 GeV. It is then a nontrivial question whether the gluon exchanges, suppressed by the QCD coupling αs, always dominate over the s-quark exchanges. In this paper, we take a fresh look at the lepto-production of φ-mesons ep → e′p′φ in the high-Q region near threshold, namely when W is very low and barely enough to produce a φ-meson W & mφ + mN = 1.96 GeV. Measurements in the threshold region have been performed at LEPS [2–4] for photo-production Q ≈ 0 and at the Jefferson laboratory (JLab) for both photoand lepto-productions 4 & Q & 0 [5–7]. One of the main motivations of these experiments was to study the nature of gluon (or Pomeron) exchanges. Our motivation however is drastically different, so let us quickly provide relevant background information: It has been argued that the near-threshold photo-production of a heavy quarkonium (see [8] for an ongoing experiment) is sensitive to the gluon gravitational form factors 〈P ′|Tμν g |P 〉 where T g is the gluon part of the QCD energy momentum tensor [9–11] (see also [12, 13]). In lepto-production at high-Q, this connection can be cleanly established by using the operator product expansion (OPE) [14]. The analysis [9] of J/ψ photo-production over a wide 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 suggests that the gluon fields in the nucleon are more compact and localized than the charge distribution. Moreover, it has been demonstrated [10, 11, 14] that the shape of the differential cross section dσ/dt is sensitive to the gluon D-term Dg(t) which, after Fourier transforming to the coordinate space, can be interpreted as an internal force (or ‘pressure’) exerted by gluons inside the proton [15] (see however, [16]). The experimental determination of the parameter Dg(t = 0) is complementary to the ongoing effort [17–19] to extract the u, d-quark contributions to the D-term Du,d from the present and future deeply virtual Compton scattering (DVCS) experiments. Together they constitute the total D-term


I. INTRODUCTION
Exclusive lepto-production of vector mesons ep → e γ * p → e p V is a versatile process that can address a wide spectrum of key questions about the proton structure. Depending on the γ * p center-of-mass energy W , photon virtuality Q 2 and meson species, the spacetime picture of the reaction looks different and requires different theoretical frameworks. At high energy in the large-Q 2 region, and for the longitudinally polarized virtual photon, a QCD factorization theorem [1] dictates that the scattering amplitude can be written in terms of the generalized parton distribution (GPD), the meson distribution amplitude (DA) and the perturbatively calculable hard part. Even in cases where such a rigorous treatment is not available, many phenomenologically successful models exist. Generally speaking, 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 gluonic content of the proton.
The φ-meson (a bound state of ss) 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-production is 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 nonnegligible fraction of strange sea quarks already on nonperturbative scales of ∼ 1 GeV 2 . It is then a nontrivial question whether 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 p φ in the high-Q 2 region near threshold, namely when W is very low and barely enough to produce a φ-meson W m φ + m N = 1.96 GeV. Measurements in the threshold region have been performed at LEPS [2][3][4] for photo-production Q 2 ≈ 0 and at the Jefferson laboratory (JLab) for both photo-and lepto-productions 4 Q 2 0 [5][6][7]. One of the main motivations of these experiments was to study the nature of gluon (or Pomeron) exchanges. Our motivation however is drastically different, so let us quickly provide relevant background information: It has been argued that the near-threshold photo-production of a heavy quarkonium (see [8] for an ongoing experiment) is sensitive to the gluon gravitational form factors P |T µν g |P where T µν g is the gluon part of the QCD energy momentum tensor [9][10][11] (see also [12,13]). In lepto-production at high-Q 2 , this connection can be cleanly established by using the operator product expansion (OPE) [14]. The analysis [9] of J/ψ photo-production over a wide 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 2 -dependence of the electromagnetic form factors. This suggests that the gluon fields in the nucleon are more compact and localized than the charge distribution. Moreover, it has been 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 'pressure') 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][18][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 the total D-term 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 (nontrivial because u-quarks are more abundant than d-quarks in the proton) which by extension implies that D s could be arXiv:2102.12631v1 [hep-ph] 25 Feb 2021 comparable to D u,d at least in the flavor SU(3) symmetric limit. It is thus a potentially important piece of the sum rule (1) when the precision study of the D-term becomes possible in future. The purpose of this paper is to argue that the 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 ss content of the proton than to the gluon content at least near the threshold at high-Q 2 , 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 connection between the two approaches. Other approaches to φ-meson photo-or lepto-production near threshold can be found in [21][22][23].
In Section II, we collect general formulas for the cross section of lepto-production. In Section III, we describe our model of the scattering amplitude inspired by the OPE and the strangeness gravitational form factors. The numerical results for the differential cross section dσ/dt are presented in Section IV for the kinematics relevant to the JLab and the future electron-ion collider (EIC).

II. EXCLUSIVE φ-MESON LEPTOPRODUCTION
Consider exclusive φ-meson production cross section in electron-proton scattering ep → e γ * p → e p φ. The ep and γ * p center-of-mass energies are denoted by s ep = ( + P ) 2 and W 2 = (q + P ) 2 , respectively. The outgoing φ has momentum k µ with k 2 = m 2 φ . The cross section can be written as 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 For simplicity, we average over the outgoing lepton angle φ , 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, is the polarization vector of the longitudinal virtual photon. is the longitudinal-to-transverse photon flux ratio The parameter γ accounts for the target mass correction which should be included in near-threshold production. The variable y can be eliminated in favor of s ep using the relation In the hadronic part, let us define where the minus sign is from the vector meson polarization sum (The amplitude M satisfies the conditions k ρ M ρµ = M ρµ q µ = 0.) Contracting with the lepton tensor (4), we get where we used We thus arrive at Note that the γ * p cross section (13) depends on the ep center-of-mass energy s ep because depends on y, and y depends on s ep as in (8). Experimentalists at the JLab have measured dσ dW dQ 2 [5], and from the data they have reconstructed the differential cross section (13) and the total cross section σ(W, Q 2 ) = dt dσ dt . In the next section we present a model for the scattering amplitude M.

III. DESCRIPTION OF THE MODEL
Our model for the matrix element P φ(k)|J ν em (q)|P has been inspired by the recently developed new approach to the near-threshold production of heavy quarkonia such as J/ψ and Υ [14]. The main steps of [14] are summarized as follows. One first relates the J/ψ production amplitude P J/ψ(k)|J ν em (q)|P to the correlation function of the charm current operator J µ c =cγ µ c in a slightly off-shell kinematics k 2 = m 2 J/ψ . One then performs the operator product expansion (OPE) in the regime Q 2 |t|, m 2 J/ψ and picks up gluon bilinear operators ∼ F F . The off-forward matrix elements P |F F |P are parameterized by the gluon gravitational form factors. These include the gluon momentum fraction A g (the second moment of the gluon PDF), the gluon D-term D g and the gluon condensate (trace anomaly) F µν F µν .
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 the present approach is to compare the momentum fractions A s+s and αs 2π A g . Taking A s+s ≈ 0.04, A g ≈ 0.4 [25] and α s = 0.2 ∼ 0.3 for example, we see that the strange sea quarks are more important than gluons, though not by a large margin. In the appendix, we give a slightly improved argument and show that the s-quark contribution gets an additional factor ∼ 2. Second, the condition Q 2 |t| is more difficult to satisfy. 2 For example, the momentum transfer at the threshold is When Q 2 m 2 V , |t th | is a larger fraction of Q 2 in φ-production m V = m φ ≈ m N than in J/ψ-production m V = m J/ψ ≈ 3m N . As one goes away (but not too far away) from the threshold, the region Q 2 |t| does exist. In principle, our predictions are limited to such regions, though in practice they can be smoothly extrapolated to |t| > Q 2 as long as |t| is not too small.
We now perform the OPE. A simple calculation shows where is the s-quark propagator and is the s-quark contribution to the energy momentum tensor ( . We neglect the s-quark mass m s whenever it appears in the numerator. However, we keep it in the denominator to regularize the divergence s just in case we may want to extrapolate our results to smaller Q 2 values in future applications. In the last line of (16), we have implemented minimal modifications to make A µν s transverse with respect to both q ν and k µ as required by gauge invariance. While this is ad hoc at the present level of discussion, we anticipate that total derivative/higher twist 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 operatorsγ µ γ 5 s and the s-quark twist-two operators with higher spins. The former is related to the (small) s-quark helicity contribution ∆s to the proton spin in the forward limit. Its off-forward matrix element is basically unknown. The latter are discussed in the appendix where it is found that, unlike in the quarkonium case [14], the twist-two, higher-spin operators are not negligible for the present problem. To mimic their effect, we introduce an overall phenomenological factor of 2.5 in (16).
To evaluate the matrix element P |A µν s |P , we use the following parameterization of the gravitational form factors [27,28] whereP = P +P 2 , ∆ µ = P µ − P µ and t = ∆ 2 . D(t)/4 is often denoted by 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 = − 1 4 A s assuming that the trace anomaly is insignificant in the strangeness sector. [However, this point may be improved as was done for gluons in [11,14].] For the remaining form factors, we employ the dipole and tripole ansatze suggested by the perturbative counting rules at large-t [30,31] 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 = 0.76 GeV as for the gluon gravitational form factors from lattice QCD [32]. This is reasonable given that s-quarks in the nucleon are generated by the gluon splitting g → ss. 3 The value D s (0) is our main object of interest, and is treated here as a free parameter. As mentioned in the introduction, even though s-quarks are much less abundant in the proton A s A u,d , a large-N c argument suggests that the D-terms are 'flavor blind' D s ∼ D u ≈ D d [20]. 4 In the flavor SU(3) limit, and at asymptotically large scales, the relation (C F = together with the recent lattice result for the gluon D-term D g (0) ≈ −7.2 [32] gives D s (0) ≈ −1.35. We thus vary the parameter in the range 0 > D s (0) > −1.35, 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 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 P |A µν s |P 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 where e s = −1/3 and g γφ is the decay constant. There is actually an uncertainty of order unity in the overall normalization of the amplitude as mentioned in [14] (apart from the factor of 2.5 mentioned above). This can be fixed by 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 ≈ 3.5 GeV). The experimental data [5] have been taken in the range where t min is the kinematical lower limit of t which depends on Q 2 and W . Admittedly, even the maximal value Q 2 = 3.8 GeV 2 is not large from a perturbative QCD point of view. However, (23) is the only kinematical window where the lepto-production data exist. Our model actually provides smooth curves for observables in the above range of Q 2 . Fig. 1 shows dσ/dt at W = 2.5 GeV, Q 2 = 3.8 GeV 2 . We chose m s = 100 MeV for the current s-quark mass. 5 The four curves correspond to different values of the D-term, D s = 0, −0.4, −0.7, −1.3 in descending order. We see that if the D-term is large enough, it causes a flattening or even a bump in the |t|-distribution in the small-t region. This is due to the explicit factors of ∆ µ (t = ∆ 2 ) multiplying D s in (19) which tend to shift the peak of the t-distribution to larger values. Unfortunately, we cannot directly compare our result with the JLab data. The relevant plot, Fig. 18 of Ref. [5] is a mixture of data from different values of Q 2 in the range (23). For a meaningful comparison, dσ/dt should be plotted for a fixed (large) value of Q 2 , and there should be enough data points in the most interesting region |t| 1 GeV 2 . By the same reason, we cannot adjust the overall normalization of the amplitude mentioned at the end of the previous section. Incidentally, we note that in this kinematics the cross section is dominated by the contribution from the transversely polarized photon, namely, the part proportional to g µν ⊥ in (10). For illustration, in Fig. 2 we show the result with Q 2 = 20 GeV 2 and W = 2.5 GeV, having in mind the kinematics of the Electron-Ion Colliders (EICs) in the U.S. [34,35] and in China [36]. We chose √ s ep = 30 GeV for definiteness, but the dependence on s ep is very weak as it only enters the parameter in (13) and ≈ 1 in the present kinematics.
(Of course the ep cross section (12) strongly depends on s ep .) The contribution from the longitudinally polarized photon (the part proportional to ε µ L ε ν L in (10)) is now comparable to the transverse part. Again the impact of the D-term is noticeable, but the bump has almost disappeared and we only see a flattening of the curve in the extreme case D s = −1.3. The reason is simple. The cross section schematically has the form 4 The prediction |Du + D d | |Du − D d | from large-Nc QCD is supported by the lattice simulations [29]. Interestingly, and in contrast, the B-form factor is dominantly a flavor nonsinglet quantity |Bu − B d | |Bu + B d | as already mentioned. 5 While this choice is natural in the present framework, it leads to a too steep rise of the cross section σ(Q 2 ) as Q 2 is decreased towards 1 GeV 2 . Of course, our approach breaks down in this limit, but it is still possible to get a better Q 2 behavior in the low-Q 2 region by switching to the constituent s-quark mass ms = m φ /2 ≈ 500 MeV, or perhaps even ms → m φ as in the vector meson dominance (VMD) model. where f (t) is a low-order polynomial in t and a = 4, 5, 6. (See (20). The amplitude squared is a linear combination of A 2 s (t), A s (t)D s (t) and D 2 s (t).) The t-dependence of f (t) comes from the D-term and gamma matrix traces involving nucleon spinors (19). Clearly, f (t) can affect the shape of dσ/dt only when |t| < m 2 ∼ 1 GeV 2 . Beyond that, one simply has the power law dσ/dt ∼ 1/t c with c < 4. In Fig. 1, |t min | < 1 GeV, and this is why we see more interesting structures. As Q 2 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, the cross section is solely determined by the strangeness gravitational form factors, similarly to the J/ψ case where it is determined by the gluon counterparts [14]. Of particular interest is the value of D s , the strangeness contribution to the proton D-term. While D s is ignored in most literature, an argument based on the large-N c QCD suggests that it 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| < 1 GeV 2 region.
There are number of directions for improvement. As already mentioned, operators other than the energy momentum tensors 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 this needs to be checked. Also the renormalization group evolution of the form factors should be taken into account if in future one can measure this process over a broad range in Q 2 such as at the EICs in the U.S. and in China [34,36].
η is the skewness parameter and ξ is the analog of the Bjorken variable x B = −q 2 2P ·q in DIS. In the regime of our interest q 2 ∼ m 2 φ −q 2 = Q 2 , we can write where the second approximation is valid when Q 2 |t| = |∆ 2 |. In DIS or DVCS, one takes the scaling limit −q 2 → ∞ and 2P ·q → ∞ (and implicitly W 2 = (P + q) 2 → ∞) keeping the ratio 1 > ξ > 0 fixed. However, near the threshold where W 2 = (P + q) 2 is constrained to be close to (m N + m φ ) 2 , the scaling limit cannot be taken literally because x B and Q 2 are no longer independent (A3) In the limit Q 2 → ∞, we have that The partonic interpretation of scattering in this regime is rather peculiar. Normally one works in a frame in which the incoming and outgoing protons are fast-moving The incoming proton has the light-cone energy P + = (1 + η)P + , and it emits two partons with momentum fractions When η ≈ ξ ≈ 1, P + ≈ 2P + and the outgoing proton has vanishing light-cone energy P + ≈ 0. Moreover, the condition 2P · q ≈ Q 2 means q + ≈ −P + and (P + + q + )q − ∼ m 2 N . Therefore, the outgoing meson is not fast-moving in the minus direction q − ∼ O(m N ). Since the suppression of final state interactions due to large relative momenta is crucial for the proof of factorization [1], we suspect that the standard approach based on GPDs is not applicable for 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-quark contribution to the Compton form factor may be Taylor expanded as where H s is the s-quark GPD. The lowest moment is proportional to the gravitational form factor P |T ++ s |P that we keep, and higher moments give the form factors of the twist-two higher spin operators x n ∼ P |sγ + (D + ) n s|P . 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 2 ). (A8) The above integral proportional to If we only keep the first term in the Taylor expansion (corresponding to the energy momentum tensor), we get that is, 40% of the full result. This is in contrast to the J/ψ case [14] where one has instead the gluon GPD The first term in the Taylor expansion (corresponding to the gluon energy momentum tensor) . 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 are relatively innocuous in the gluon case. To cope with this, we introduce an overall factor 1/0.4 = 2.5 in the leading order result (16) as a model parameter. Note that the GPD H s contains a part related to the D-term [37], so this factor is common to both the A s and D s form factors.