Threshold photoproduction of $\eta_c$ and $\eta_b$ using holographic QCD

We discuss the possibility that threshold photoproduction of $\eta_{c,b}$ may be sensitive to the pseudovector $1^{+-}$ glueball exchange. We use the holographic construction to identify the pseudovector glueball with the Kalb-Ramond field, minimally coupled to bulk Dirac fermions. We derive the holographic C-odd form factor and its respective charge radius. Using the pertinent Witten diagrams, we derive and analyze the differential photoproduction cross section for $\eta_{c,b}$ in the threshold regime, including the interference from the dual bulk photon exchange with manifest vector dominance. The possibility of measuring this process at current and future electron facilities is discussed.


I. INTRODUCTION
At large center of mass energies, diffractive scattering of hadrons is dominated by Pomeron exchanges, Reggeized even gluon exchanges with even C-and P-assignments.Initial perturbative QCD (pQCD) arguments suggest that odd gluon exchanges in the form of Odderon exchanges with odd C-and P-assignments are also possible [1] (and references therein).The signature of this exchange maybe observed in the difference between the diffractive pp and pp cross sections, and the photoproduction or electroproduction of heavy pseudoscalar mesons.
Recently, the TOTEM collaboration at the LHC, has reported a difference between their extrapolated pp data at √ s = 1.96GeV [2], from the reported pp data by the D / O collaboration at Fermilab, at the same center of mass energy.Their analysis suggests that the difference is evidence for an Odderon.A number of recent analyses appear to also point in this direction [3] (and references therein).
At weak coupling, the hard Pomeron is a Reggeized Balitskii-Fadin-Kuraev-Lipatov (BFKL) ladder which resums the rapidity ordered C-even collinear emissions.In the conformal limit, it is is identified with the j-plane branch-points.By analogy, the Odderon is a Reggeized BKP ladder which resums the C-odd collinear emissions [4,5].At strong coupling in dual gravity, the Pomeron is identified with a Reggeized spin-j graviton, while the Odderon with a Reggeized spin-j Kalb-Ramond field [6].Further analyses of the gravity dual Odderon have been carried out in conformal geometries [7], and more recently in confining geometries [8] with a detailed comparison to the recent TOTEM data.
The purpose of this work is to explore the possible contribution of the C-odd gluonic exchange, in the diffractive photoproduction of charmed and bottom pseudoscalars η c,b near threshold, using dual gravity.This approach was recently applied to the description of photoproduction of charmonium near threshold at Jlab energies [9,10], with relative success in extracting the mass and scalar radii of the gluonic component of the nucleon [11].
In dual gravity, threshold charmoium photoproduction is dominated by the C-even and 2 ++ glueball exchange, with some admixture of 0 ++ glueball exchange for large skewness.Similarly, we expect that threshold photoproduction of charmed pseudoscalars to be dominated by C-odd 1 +− glueball exchanges, modulo the photon Primakoff exchange as depicted in Fig. 1.This process provides for a possible measure of the C-odd gluon charge radius.
The organization of the paper is as follows: in section II we outline the dual bulk action for the the photoproduction of heavy eta pseudoscalars.Following the initial suggestion in [6], the C-odd gluon exchange is identified with the Kalb-Ramond 2-form in the bulk, with coupling to the photon and pseudoscaars governed by the Chern-Simons term.In section III the dual photoproduction amplitudes are evaluated using the leading Witten diagrams.The C-odd bulk form factors are explicitly derived, and the corresponding charge radii derived.In section IV we detail the differential cross section for photopro-duction in the treshod region.Detail numerical results are presented at currently electron machines.Our conclusions are in section V. We have added a number of Appendices to detail some of the derivations in the main text.

II. BULK ACTION
We recall that the 1 +− Kalb-Ramond as a 2-form field, couples to the light flavor brane through the Chern-Simons term.For instance, for flavor D8 probe branes, with the 2-form F = 2πα ′ F + B, the sum of F = dA − iA 2 the flavor 2-form and the 1 +− Kalb-Ramond 2-form B. The tension of the D8 brane is denoted by T 8 .The light η field is usually identified with the singlet part of η, with N f = 1 + 2 (for a single heavy and two light flavor branes), while the U (1) gauge field with the space-time parts of F .We will assume that an analogous coupling carries to the heavy η c,b .The bulk action relevant to the photoproduction of η c,b reads for the Kalb-Ramond field B 2 and for the vector mesons, which will supply the relevant photon couplings through vector-meson-dominance (VMD).The coupling g CS is uniquely determined by the 5D Chern-Simons term to be g CS = Nc 24π 2 .The 5D Newton constant is given by g5 c and η P is the Pauli parameter, which will be fixed by matching the Pauli form factor to its experimental value.In the above equations the flavor trace will pick up the relevant charges for charmonia e c = 2/3 e or bottomonia e b = −1/3 e.Here H 3 = dB is the 3-form field strength of the 1 +− Kalb-Ramond field.The background is given by the AdS metric In the fermionic parts of the action we denote σ AB = i 2 Γ A , Γ B , with the gamma matrices given by Γ A = (γ µ , −iγ 5 ) and obeying the Clifford algebra Γ A , Γ B = 2η AB .The tetrads following from (II.4) are given by e M A = zδ M A .The positive and negative parity Dirac spinors follow from the mixed representation of (A.5) in Appendix A, to which we refer the interested reader for further details.The axial gauge field V M = (0, V µ ) is the projected spin-1 axial-field with the physical polarization ⟨0|V µ |V ; P ⟩ = ϵ µ (P ).The projection yields the 3 physical degrees of freedom out of the 6 gauge degrees of freedom in B, and guarantees the correct normalization for the ensuing kinetic term.We have included the sole coupling to a bulk Dirac fermion through its magnetic moment, as suggested by supergravity (SUGRA).In Appendix B we give the triple couplings 1 ±− ηγ in the Sakai-Sugimoto model for comparison.The dual field with boundary spin values Following [6], we make the boundary identifications with scalar and pseudoscalar gluonic operators with mass dimension ∆ = 6 which we interpret as C-odd twist-5 operators on the light front.In contrast, we note that in the context of pQCD, factorization arguments show that the leading contribution to the photoproduction of η c , in the large skewness limit, is a C-odd local twist-3 operator [12]  In dual gravity, threshold photoproduction of η c,b by exchange of a Kalb-Ramond field is illustrated by the Witten diagram in Fig. 2a.This process is very similar to the photoproduction of charmonium, with similar kinematics given the η c mass of 2.984 GeV, and the J/Ψ mass of 3.097 GeV.The essential differences stem from their quantum numbers: P-odd versus P-even couplings, with the former expected to be more suppressed.In light of this, and motivated by previous analyses [13][14][15][16], we have also included the tree level Witten diagram contribution stemming from the exchange of a bulk photon in Fig. 2b.

A. Dual photoproduction amplitude
Using (II.3), the Witten diagram in Fig. 2a yields the photoproduction amplitude for η c as with the bulk vertices where the field strength is now to be understood as F ρσ = iq ρ ϵ σ (q) − iq σ ϵ ρ (q) with ϵ µ (q) is the polarization of the external photon with momentum q.The massive spin-1 propagator in the mode-sum representation is given by Similarly, we obtain for the photon vertices in the bulk With the normalizable modes in (III.7)now substituted with their non-normalizable counterparts V(Q, z).
In the space-like region, we obtain for the Kalb-Ramond amplitude and for the photon amplitude analogously (III.16)

B. Dual form factors
The form factors are extracted from the 3-point functions with pertinent Lehmann-Symanzik-Zimmermann (LSZ) reduction.
For example, the Dirac form factor resulting from the current associated with the covariant derivative receivs contributions from ) . (III.18) Similar relations hold for the other 3-point amplitudes.The electromagnetic Dirac and Pauli form factors are thus given by where which follow from as previously obtained in [17].Note the appearance of an additional contribution to F 1 (Q) from the 5D Pauli term σ µz .The proton electromagnetic form factor normalizations are fixed by the charge F 1 (0) = 1 (Dirac) and magnetic moment given in units of the nuclear magneton where we used µ p /µ N = 2.7928.This fixes η P = 1.7928/C 3 (0) = 1.7928/4(τ − 1).Similarly, the Codd Kalb-Ramond or Odderon form factor is given by where we pulled out a facotr of 2M N to highlight the similarity with the electromagnetic Pauli form factor and p Fq is the regularized hypergeometric function.
The ensuing C-odd squared charge radius is (III.23) The C-odd form factor normalizations are fixed by the nucleon tensor charge (axial-Pauli) and the nucleon intrinsic spin (axial-Dirac).More specifically, the nucleon tensor charge is defined by the matrix element At a resolution of the order of the nucleon mass, lattice evaluation gives [18] δq = δu + δd ≈ 0.839 − 0.231 = 0.608 (III.25) The intrinsic spin of the nucleon is mostly due to the mixing with the gluons from the U (1) A anomaly The estimation from the QCD instanton vacuum gives Σ(0) = 0.3 at a resolution of about the nucleon mass [19], while lattice simulations give Σ(0) = 0.4, at a resolution of about twice the nucleon mass [20].Therefore, at the nucleon mass resolution, we set Using (III.23)we readily obtain the charge radius where γ E is the Euler-Mascheroni constant.For τ = 3 and κ γ = 0.3875 GeV we obtain For comparison, we note that the Odderon-nucleon coupling as a C-odd and un-Reggeized 3-gluon exchange in [13], is assumed monopole-like with unit normalization.Also in the eikonal dipole analysis at low-x [14], the Odderon-nucleon form factor is argued to be fixed by the leading twist quark generalized parton distribution (GPD), with a normalization to 1.In contrast, the Reggeized BKP Odderonnucleon form factor in [15] is relatively large, with even a rapid sign change at the origin.The form factors are displayed in Fig. 3 with for the open string sector and ϕ = κ 2 b z 2 = 4κ 2 z 2 for the closed string sector.We fix κ by the ρ meson pole in the (time-like) photon bulk-to-boundary propagator, as is required by VMD, giving (κ b , κ γ , κ N ) = (0.775, 0.3875, 0.3875) GeV.
(III.30)For moderate K 2 the dominant contribution on the light front stems from the F 1 contribution of the photon, in analogy to, but not as pronounced as, the Primakoff effect.This is due to the absence of the photon pole in VMD.At larger K 2 the C-odd contribution will dominate the differential cross section due to the kinematical nature of the coupling in (II.2), in agreement with pQCD calculations [14,16].

C. Threshold vertices
At threshold, the Odderon-η c,b γ vertices in the space-like region is given by (III.31) The pertinent LSZ reduction for the production of η c,b at the boundary results in a substitution rule for the bulk-to-boundary propagator hence reducing to a simple vertex factor (III.33) The absence of the dilaton/metric in the Chern-Simons term implies that (III.33) is divergent in the IR.Note that this is not the case if we were to use a slab geometry with a hard wall.With this in mind, to obtain an estimate for the coupling V Bηγ , we use the simple hard-wall cutoff obtained from (A.38) to obtain The vertex containing a single virtual and one real photon is given by As is the case for the Kalb-Ramond field, LSZ reduction picks out the pertinent normalizable mode and we obtain the same vertex as in (III.33) For the numerical analysis we fix the decay constant f c by the leading order decay rate from pQCD [21] with Q c the charm quark charge.From the experimental value Γ ηc→γγ = 5.376 × 10 −6 GeV [22] we obtain f ηc = 0.327 GeV, (III.37) where we used M ηc = 2.9839 GeV.The value for Γ η b →γγ is not reported.However, from heavy quark symmetry, it follows that which amounts to with M η b = 9.3897 GeV.

IV. DIFFERENTIAL CROSS SECTION
The differential cross section is obtained by averaging over the initial state spins and polarizations and by summing over the final state spins (IV.40) The cross sections are then obtain from with respectively, the C-even F O , photon F γ , and mixed F Oγ contributions with zero mixing between the tensor contributions.

A. Estimate of g Bψ
The overall magnitude of the cross section (IV.41) hinges considerably on the value of the bulk coupling g Bψ .For an estimate, we can take the eikonal limit where the B-exchange as a closed string is exchanged between two open string dipoles.As a result, the coupling is of order g Bψ ∼ √ g s , with the string coupling g s = λ/4πN c (pure AdS geometry).For λ ∼ 10, we obtain g Bψ ∼ 0.5.Alternatively, if we use the identification (II.5) for the boundary operator, then the near forward Codd gluonic matrix element in the QCD instanton vacuum is about [23] Here f (qρ) is the induced form factor by an instanton of size ρ.For the kinematical range of interest in Fig 10, ρ |t min | ∼ 1, and f (q min ρ) ∼ 1.Indeed, for a "dense instanton ensemble" [24], the instanton packing fraction is κ I+ Ī ∼ 0.7 with a mean instanton size ρ ∼ 1 3 fm.It follows that a simple estimate for the dual coupling is g Bψ ∼ κ 2 I+ Ī ∼ 0.5, in agreement with the string estimate.For a "dilute instanton ensemble" [25], the packing fraction is κ I+ Ī ∼ 0.1, with a weaker dual coupling g Bψ ∼ κ 2 I+ Ī ∼ 0.01.The suppression of the gluons compared to the quarks in a topologically active vacuum, at the resolution 1/ρ, is similar to the suppression factor noted for the gluons in comparison to the quarks in the nucleon spin budget [19].
It would be very useful to carry a lattice simulation check for the QCD instanton vacuum estimate (IV.44).

B. Numerical results
With this in mind, the total cross section for threshold production of η c with κ ′ s fixed to the mass spectra, and g Bψ = {1, 0.5} is σ(W = 4.3 GeV) = {10.3,2.76} pb.It is sensitive to the overall value of g Bψ which we estimated above.In pQCD it corresponds to the fraction of gluons contributing to the nucleon tensor charge as measured by the quarks in (III.24).In the numerical results to follow, all the holographic results will be quoted for g Bψ = {1, 0.5}.
In Fig. 4a we show the differential cross section for W = 4.3 GeV versus the threshold-t, with the Pwave photon contributions (dotted-green: Pauli and solid-green: Dirac), the Odderon contribution (solidred) and the sum total (solid-black).At this center of mass energy and modulo the value of g Bψ , the differential cross section is dominated by the Odderon exchange near threshold, but is rapidly overtaken by the P-wave photon exchange.In Fig. 4b we compare our results for the differential cross section, to the recent estimate using the Primakoff photon exchange estimate (open-blue-dots) in [16].The holographic result is substantially larger.
In Fig. 5a we show the same differential cross section for η c production at the center of mass energy W = 10 GeV.The total cross section at this energy is σ(W = 10 GeV) = {202, 50} pb.Again, the Pwave photon contributions (dotted-green: Pauli and solid-green: Dirac) are compared to the Odderon contribution (solid-red) and the sum total (solidblack).At this energy, the Odderon contribution is dominant throughtout the threshold region.In Fig. 5a the holographic results are compared to the results obtained using the eikonalized dipole approximation for the Odderon in [14].The holographic results for the P-wave photon exchange (solid-green), the Odderon (solid-red) and total (solid-black), are compared to Odderon (red-triangle), photon (greendiamond) and total (black-diamond) in [14].The sum total of the differential cross section are about comparable at t = t min with the eikonalized results falling off much faster, although there is a substantial difference in the respective contributions, with no crossing in the holographic case in this kinematical range.The difference in the photon contribution stems from the VMD nature of the holographic photon exchange in the bulk in comparison to the simple Primakoff exchange used in [14] which dwarfs the Odderon contribution at threshold.In Fig. 6b we show the differential cross section at W = 50 GeV, the relevant kinematical range for the future electron-ion-collider (EIC).The integrated cross sections is given by σ(W = 50 GeV) = {242, 59} pb.
At much higher center of mass energy, say W = 300 GeV, the integration interval becomes very large since t min ∼ 0 and −t max ∼ W 2 and the integrated cross section starts to diverge.Although the Reggeization may start to be important in this kinematical range, we show our un-Reggeized C-odd bulk Odderon exchange in Fig. 7a with the P-wave photon exchange (solid-green: Dirac and dottedgreen: Pauli), the Odderon exchange (solid-red) and the sum total (solid-black).In Fig. 7b the holographic results are compared to the the estimate for photon-exchange (open-blue-dots) in [16], and the Odderon model exchange (green-triangle) from [27] and the Odderon model exchange (orange-diamond) from [26].
In Fig. 8a we show the same differential cross section for photoproduction of η b at W = 11 GeV, with the P-wave photon exchange (solid-green: Dirac and dashed-green: Pauli), the Odderon exchange (solidred) and the the sum total (solid-black).The unseparated contributions are shown in Fig. 8b for the photon (solid-green), Odderon (solid-red) and sum (solid-black).For η b production the integrated cross sections are σ(W = 11 GeV) = {0.002,0.001} pb and σ(W = 22 GeV) = {1.20,0.29} pb For η b , the photon contribution crosses the Odderon contribution twice in the threshold region, underlying the sensitivity to the unfixed overall g Bψ parameter.

Sum
comparison to η c is shown in (f).The black data points are from GlueX [28].The magenta and green data points are from SLAC [29] and Cornell [30], respectively.The holographic result from [9] uses the normalization constant N = 4.637 ± 3.131, and replaces A(t) by A(t) + η 2 D(t) in Eq.VIII.58 of [9], where η is the skewness parameter given explicitly by Eq.III.33 in [10].We have used the holographic gluonic gravitational form factors A(t) and D(t) = 4C(t) extracted by the J/Ψ − 007 collaboration at JLab [11].Note that we have ignored the D-term for the total cross section (f).Also note that the upper limit in the shaded region corresponds to N = 4.637 + 3.131 = 7.768 used by the J/Ψ − 007 collaboration at JLab [11].

V. CONCLUSIONS
We have analyzed the differential and integrated cross sections for photoproduction of heavy pseudoscalars η c,b in the threshold regime using dual gravity.In this limit, we have suggested that the dominant contribution stems from the exchange of a Kalb-Ramond B 2 -field in the bulk, which is the a: Holographic differential cross section for threshold photoproduction of ηc at W = 300 GeV, with the P-wave photon exchange (solid-green: Dirac and dotted-green: Pauli), the Odderon exchange (solid-red with g Bψ = {1, 0.5}) and their coherent sum (solid-black with g Bψ = {1, 0.5}); b: Holographic differential cross section as in a, compared to the Primakoff photon-exchange (blue-open-circles) from [16], the Odderon model (orange-diamonds) from [26] and the Odderon model (green-triangles) from [27].
dual of a 1 +− glueball.The glueballs are sourced by a twist-5 boundary operator, which we have argued to be tied to the tensor coupling of Dirac fermions in the bulk as dual to nucleons, modulo an overall constant g Bψ not fixed by holography.This gluon mediated constant was estimated to be small, using the QCD instanton vacuum at low resolution.
A possible measure of the diffractive gluon mediated photoproduction of heavy mesons near threshold would bring an important insight on the C-odd gluonic mass content of the proton, at about the nucleon mass resolution.It would also be an impor-tant precursor for the elusive Odderon, expected to set in at higher energies through Reggeization of the C-odd glueballs.
Near threshold, the η c,b photoproduction cross sections through diffractive 1 +− glueballs are shown to be very sensitive to the value of this coupling g BΨ .This notwithstanding, we have found that the ensuing diffractive differential cross sections overtake the P-wave photon mediated differential cross sections for g BΨ = 0.5 − 1, as suggested by both a string estimate, and an estimate using the (dense) QCD instanton vacuum for the dual boundary operator.The production is depleted by almost two orders of magnitude, say for g Bψ = 0.01 using the (dilute) QCD instanton vacuum for the dual boundary operator.These observations hold for the current electron facility at JLab with W ∼ 5 GeV, and the future electron facility at the EIC with W ∼ 50 GeV.(f) FIG. 9. Holographic differential and total cross sections for threshold photoproduction of J/Ψ (blue-shaded) from [9], and the present results for ηc (black-shaded with g Bψ = {1, 0.5}): (a) W = 4.58 GeV, (b) W = 4.30 GeV, (c) W = 10 GeV, (d) W = 50 GeV, (e) W = 300 GeV; and holographic total cross section for threshold photoproduction of J/Ψ (f).The black data points are from GlueX [28].The magenta and green data points are from SLAC [29] and Cornell [30], respectively.
The normalizable modes are given by ϕ n (z) = c n κzL n (κ 2 z 2 ) (A.17 where where Γ is the antisymmetric part of the Christoffel symbol, the Kalb-Ramond field can be viewed as a source for torsion, which was first observed in [41].The couplings arising from H 3 in the covariant derivative, and in particular ∆ (1) , in (D.1) for the 1 +− field V σ in the main text reduce to where we note that the H Zµν Γ Zµν coupling vanishes after the reduction to the 4D spinor is carried out.This means that also the B µz fluctuation corresponding to 1 −− does not couple through this term.
However the fluctuations B µν , C µZ and B µZ , C µν form the physical 1 ±− states and we obtain from F 3 Γ (3) in ∆ (2) for the 1 (D.7) Note that both couplings have the correct 5D parity.After the spin sums, the resulting squared matrix elements are highly suppressed at low K 2 .Other couplings yield the nucleon axial-tensor charge (u L (p 2 ) − u R (p 2 ))σ µν u(p 1 ) = u(p 2 )γ 5 σ µν u(p 1 ), (D.8) up to a factor √ K 2 , which is again suppressed in the near-forward regime.

10 )
The bulk coupling to η c,b is also governed by the Chern-Simons term in (II.1) via the substitution B M N → F M N .Note that there are no metric and dilaton factors in the coupling of B 2 and A µ to γ − η c,b in (III.7)since this interaction is purely governed by the Chern-Simons term in (II.1).The baryon couplings on the other hand are governed by the Dirac-Born-Infeld (DBI) part of the action and only receive 1/N c corrections from the Chern-Simons term.The photon couplings follow analogously with κ b replaced by κ.For z ′ → 0 and t = −K 2 we can use (A.35) in (III.7) to get δS EM Dirac δϵ µ (III.17) with e N = e for the proton and e N = 0 for the neutron, F N (p) = ⟨0| O N (0) |N (p)⟩ the nucleon source constant and

K 2 [
FIG. 3. C-even and C-odd nucleon form factors in the approximation κN = κγ with the normalization fixed by the charge, magnetic moment and (III.27).
t min/max fixed by the kinematics of the process, which are detailed in Appendix C.Carrying out the polarization and fermion spin sums we arrive at Photon Photon

FIG. 10 . 2 Ψ 1 , 2 2 (
FIG. 10.Minimal and maximal transverse momentum transfer tmin, tmax in the physical region for ηc (a) and η b (b) versus W = √ s.The photon momentum is taken to be at the optical point q 2 = −Q 2 = 0 and the hadron masses are given by MN = 0.938 GeV, mη c = 2.984 GeV and Mη b = 9.399 GeV.

D. 3 )
and Γ A... the antisymmetrized product of gamma matrices.By introducing the chiral spin-connectionω (±)AB M = ω AB M ± 1 4 • 2! e N A e OB H M N O , (D.4) which is amenable to a spin connection with torsion ω AB M = ω AB M + e A N e BO ΓN M O , (D.5)