Holographic Odderon at TOTEM ?

,


I. INTRODUCTION
In the Regge limit, diffractive pp and pp scattering at large √ s is dominated by the Pomeron, with a smaller admixture from the Odderon [1], a tower of C-odd soft gluons.The Odderon is believed to be the C-odd partner of the C-even Pomeron.While the latter in its soft version dominates the diffractive pp cross section at high energy, the manifestation of the former is still being debated, although recent results from the TOTEM collaboration have claimed it [2].
Regge theory predicts the rise of the pp and pp elastic cross sections at collider energies.The rise follows from Regge poles.The dominant pole is the Pomeron with the largest intercept and positive signature.The Odderon carries a smaller intercept and negative signature.Negative signature Reggeons add in the pp channel, and subtract in the pp channel.
The hard QCD Pomeron at weak coupling, is a Reggeized BFKL ladder which resums the rapidity ordered collinear emissions.It is a C-even and Peven gluon ladder, which is identified with j-plane branch-points in the conformal limit.In the same regime, the Odderon is a Reggeized BKP ladder [3,4].It is a C-odd and P-odd exchange of gluons, with two nearby j-plane branch points.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 [5].
The purpose of this work is to explore the possible contribution of the Odderon in diffractive pp and pp elastic scattering, in light of the recently reported Odderon by the D / O and TOTEM collaborations [2].In gravity dual formulations, the general aspects of the Odderon were initially discussed in [5].Our approach will use the gravity dual formulation, with a repulsive wall to account for confinement.Only in this case, a full Reggeization with the famed Gribov diffusion is realized.For completeness, we note that the Odderon contribution in pp and pp scattering, was partially analyzed using an effective string theory in [6], and the AdS/CFT in the conformal limit in [7].
The outline of the paper is as follows: in section II we briefly review the salient features of the newly released pp and pp data by the TOTEM collaboration.In section III we will formulate the gravity dual description using the repulsive wall model.We will explicitly construct the bulk-to-bulk Pomeron and Odderon propagators, and show how the Reggeized forms of the Pomeron and Odderon emerge from pertinent analytical resummations.Unlike the soft wall, the repulsive wall reproduces the expected Gribov diffusion for the Reggeons on the boundary in the confining limit.The logarithmic diffusion in the conformal limit is shown to hold.In section IV we derive the holographic and Reggeized amplitudes for elastic pp and pp scattering with the associated total and differential cross sections.The rho and slope parameters are made explicit.In section V the holographic amplitudes are eikonalized.The Froissart bound is recovered at asymptotic energies.The eikonal results for the slope and rho parameters, as well as the elastic differential cross section are compared to the recently reported TOTEM data.Our conclusions are in section VI.More details regarding some aspects of the repulsive and soft wall as well as the conformal limit are given in a number of additional Appendices.

II. ODDERON IN pp AND pp
Diffractive pp and pp elastic scattering is dominated by Reggeized glueball exchanges as illustrated in Fig. 1.The Pomeron and Odderon carry positive and negative C-parity, respectively.In pQCD the Pomeron is a Reggeized exchange of a two-gluon ladder with rung induced by Lipatov vertices with positive parity 2 ++ , and the Odderon a Reggeized three-gluon ladder with negative parity 1 −− .Their contribution adds in the elastic pp scattering amplitude, and subtracts in pp, The difference between the two amplitudes at large √ s stems from the Odderon exchange, as it discriminates p from p by charge conjugation.The combined and large √ s elastic pp data from LHC and pp data from D / O, were recently analyzed by the TOTEM collaboration, with the results shown in Fig. 2.
The data for the differential cross section shows a clear diffractive dip and bump pattern for the pp channel, with a large bump-to-dip ratio that appears to be decreasing for larger √ s.In the pp scattering data it appears to be flat already at comparatively low √ s.Among the new and unexpected features of the elastic amplitude are: 1/ the persistent diffractive structure at low-t; 2/ the absence of additional secondary structures at large-t; 3/ the linear rise of the slope parameter.We now proceed to analyze (II.1) and their pertinent differential and total cross sections in the context of dual gravity.

III. GRAVITY DUAL DESCRIPTION
In the QCD dual gravity approach, the precursors to the Pomeron and the Odderon are identified with the graviton 2 ++ and the Kalb-Ramond 1 ±− fields, each to leading order sourced by the QCD boundary operators with all traces subtracted.The even spin-j trajectory for the Pomeron, and the odd spin-j trajectory for the Odderon are sourced, respectively, by the boundary operators with the proper symmetrization assumed.In particular, h j has conformal dimension ∆ h = 4 + j and twist τ h = 2 + j, and B j , C j have conformal dimension ∆ h = 6 + j and twist τ h = 5 + j, respectively.

A. Dual gravity
We now proceed to analyse the gravity dual description of the corresponding bulk fields, using the bottom-up approach with a repulsive wall.As we detail in Appendix A, the soft wall model fails to capture Gribov diffusive behaviour, necessary for offforward scattering.We will thus take a different path and linearize the bulk equations of motion in the dilaton, which resembles an additional potential term.To this end we write the metric as which is amenable to passing from the string frame to Einstein frame.We also introduced a constant prefactor a, to study the effects of a repulsive versus attractive dilaton, as well as the open versus closed string coupling.The repulsive dilaton bears much in common with the Nambu-Goto analysis of the Pomeron with a hard wall [9] (and references therein).Matching the parameters of type IIB supergravity (SUGRA) on AdS 5 to N = 4 Super Yang-Mills we get In the following we will set R = 1.

B. Reggeized even spin-j
The starting point for the Reggeization of the even spin-j exchange is the graviton with spin j = 2.We recall that the bulk equation of motion for the graviton follows from the linearization of the Einstein-Hilbert action.The traceless and transverse part of the metric fluctuation in physical gauge ϵ T T µν h j=2 , is amenable to a scalar equation with anomalous dimension with m 2 5 R 2 = 0.The closed string exchange resumming the even spin-j h j fields, can be sought as a graviton exchange with the anomalous dimension [10] ∆ g (j deformed by the closed string quantized mass spectrum In other words, the anomalous dimension (III.7) is with (III.10) Higher order corrections in 1/ √ λ to the Pomeron intercept, have been analyzed in [11].
We will seek the Reggeized bulk-to-bulk graviton propagator in terms of its Mellin transform with the even-signature contribution projected out via the Sommerfeld-Watson transform and the analytical continuation in spin j = 2 → j assumed.

C. Reggeized odd spin-j
Similarly to the Reggeized even spin-j exchange, the starting point for the Reggeization of the odd spin-j exchanges that sums up to a closed string exchange, is the spin-1 C 2 ∼ ⋆dC gauge field.For that, we first recall that the bulk equation of motion for the rescaled gauge field zC, is the same as that of a scalar field with anomalous dimension [12] ∆ g (j = 1) = 2 + 4 + m 2 5 R 2 (III.12) with where we added m 2 k = k 2 , (4 + k) 2 , for the two branches of the Odderon [5].The latter branch identifies with the canonical dimensions of (III.2-III.3).The former branch is more sensitive to the presence of the adjoint scalars in the SUSY version.In this spirit, the closed string exchange resumming the odd spin-j C j gauge fields, can be sought as a vector exchange with the anomalous dimension deformed by the closed string quantized mass spectrum In other words, the anomalous dimension (III.13) is Higher order corrections in 1/ √ λ to the Odderon intercept, have been analyzed in [11].Similarly as for the graviton, the Reggeized Kalb-Ramond field is given by the odd-signature Sommerfeld-Watson transform (III.17) To conclude, the Reggeized bulk-to-bulk propagators that resum the even and odd trajectory are given by The analytically continued spin-2 and spin-0 are related by G 2 (j, t, z, z ′ ) = e −(j−2)(A(z)+A(z ′ )) G 0 (j, t, z, z ′ ) (III.19) with the warped scalar propagator obtained from and the pertinent Sturm-Liouville operator For the background (III.4) the scalar bulk-tobulk propagator obeys the Sturm-Liouville equation (III.20-III.21)with weights where After carrying out the coordinate transformations and field redefinitions laid out in Appendix A, and keeping in mind that the dilaton has been absorbed into the metric, the scalar bulk-to-bulk propagator is now seen to solve where we expanded the dilaton to linear order e 2av/3 = 1 + 2av/3.The independent homogeneous solutions to (III.23) are Whittaker functions with Kummer M (regular at v = 0) and Tricomi U (irregular with branch cut at v = 0) hypergeometric functions and The inhomogeneous solution to (III.23) is then with the normalization fixed by the Wronskian The cost of linearizing the dilaton is a shift in the conformal intercept resulting from the poles of the gamma function given by under the condition that 5aκ 2 R 2 = 2. From this identification it is apparent that only the repulsive dilaton with a > 0 gives the correct diffusive Regge behaviour, since the attractive dilaton with a < 0 would violate unitarity.As shown below and in Appendix A, this choice shifts the intercept slightly below the result from the soft wall or pure AdS, which is however to be expected since linearizing the dilaton introduces an additional potential term.We also find a pole with a higher intercept coming from the positive root which, however, will not be picked up by the contour integration of the Sommerfeld-Watson transform.
Reverting the rescalings and coordinate transformations, we obtain the resummed scalar bulk-tobulk propagator in the repulsive wall to be where we introduced the shorthand and similarly for U(z).The branch cut of is chosen to the left of the integration contour, along the negative real axis as in Fig. 3.In the large s/|t| limit, the integral is dominated by the saddle point.
For the Odderon the analytically continued spin-1 and spin-0 are related by with the rescaled scalar field (III.34) We recall that We expect higher order corrections in 1/ √ λ to the Odderon intercept in (III.35),following the strong coupling analysis in the conformal limit in [11].
The multiple branches of the Odderon, which are fixed from type IIB SUGRA, are given by m The structure of (III.35)shows that the k = 0 branch has either an intercept greater than one, or is super-diffusive.The branch with k = 1 has an intercept precisely at one, and is the leading branch cut that will be picked up by the saddle-point.Remarkably, a similar Odderon trajectory was noted in baryon-baryon scattering in AdS with D5 branes [7].
Performing the Sommerfeld-Watson transform via saddle-point approximation along the contour of Fig. 3, the bulk-to-bulk Odderon propagator is given by where which has a pole for the leading k = 1 branch.We note that, at large λ, (III.32) and (III.37)coincide for the k = 2 branch.The conformal limit is discussed in Appendix B

IV. SCATTERING AMPLITUDES
Elastic pp and pp scattering in the Regge limit with the exchange of Reggeons P, O are depicted in Fig. 1 using standard Feynman graphs.In dual gravity, the Feynman graphs are replaced by the Witten diagrams in Fig. 4. The Pomeron P is identified with a sum of massive even-spin glueballs, while the Odderon is identified with a sum of massive oddspin glueballs.The wavy-lines are the bulk-to-bulk G j± (j) propagators defined above.The bulk-toboundary Dirac fermions are represented by solid lines, and follow from the chiral Kaluza-Klein modes for the bulk Dirac fermions discussed in Appendix C.
The even and odd spin-j contributions to Fig. 1 are The even spin-j contribution to diffractive pp scattering in Fig. IV A is given by with the bulk Pomeron-nucleon vertex for the repulsive wall [12] where and the reduced even spin-j Reggeized graviton exchange with the Pomeron propagator given by (IV.43) and (IV.44) In the Regge limit the amplitude reduces to where and we used a = 4 for the closed string exchange and a = 1 for the open string fields as well as u(p 2 )γ µ u(p 1 ) = (p 1 + p 2 ) µ δ s1s2 .

B. Odd spin-j
The odd spin-j contribution to diffractive pp scattering is with the bulk Odderon-nucleon vertices corresponding to a Pauli and Dirac coupling, where and the reduced odd spin-j Reggeized spin-1 ex- where (IV.52) The vertices arise from the sources B M N and C M N of the boundary operators in (III.2) which are assumed to minimally couple to the chiral Dirac fermion current Ψσ AB Ψ in the bulk with The corresponding 1 −− fluctuations, which are tied through a topological mass term in type IIB SUGRA, are given by C µν and B µz [13].The latter already corresponds to a spin-1 exchange, while for C µν a projection onto the spin-1 content is obtained via utilizing the field strength The normalization is implied by the normalized kinetic term in the closed string SUGRA action, which we omit in the following.A similar reasoning holds for B µz .Note that the Pauli coupling vanishes in the forward limit, even with the normalization included, and is strongly suppressed in the Regge limit.The Dirac coupling does not and allows for the study of potential contributions of the Odderon as a spin-1 exchange in the forward region.Recall that the BKP Odderon is a Reggeized 4-vector ( 12 , 1 2 ) representation of the complexified SO (3,1).For completeness, we note that in [7] the Odderon is identified with an exchange of the B +z component of the Kalb-Ramond field in AdS 5 × S 5 .The self-dual 2form corresponds to the (0, 1) representation of the complexified SO(3,1), and is found to have non-zero coupling to the baryon vertex.
Similarly, after reducing the chiral Dirac spinors to 4D and utilizing the LSZ formula, we obtain for the Odderon (IV.54) In the Regge limit the amplitude is given by OΨΨ k α u(q 2 )σ αβ u(q 1 )P βν (p where we again used u(p 2 )γ µ u(p 1 ) = (p 1 + p 2 ) µ δ s1s2 for s → ∞.In particular we obtain for the forward amplitude OΨΨ For the computation of the total cross section, we recall that the signature factors are given by For the discussion of the analytical results we will drop the subleading piece in λ and only include it in the numerical analysis.In both cases the real part dwarfs the imaginary part at very strong coupling, but both imaginary parts are equal to 1.The total cross sections in pp and pp follow from the optical theorem and is given by OΨΨ e (j O (0)−j P (0))τ 1 − where we expanded for large s.The rho-parameters for both channels read with a constant asymptotic in the large rapidity limit.The high energy scattering data suggests ρ(s) ∼ 0.1 for √ s ≥ 100 GeV [14] (and references therein).Below we show that after the eikonal re-summation, the strong shadowing caused by the exchanged Reggeons will deplete (IV.60) to zero in the large rapidity limit, in qualitative agreement with the data.

D. Elastic differential cross-section
The spin averaged squared elastic amplitude is given by OΨΨ ) 2 e (j P (t)+j O (t))τ + 64e 2j O (t)τ g (1) where we note that the interference term is highly suppressed in the Regge limit but still leading compared to the g piece.Hence we will drop OΨΨ in our numerical analysis.The elastic differential cross sections for pp and pp follow from The elastic slopes are identical in the large rapidity limit The squared string length α ′ = 1 GeV −2 , is comparable to the canonical logarithmic value of the slope parameter [14] (and references therein) for both the Pomeron and the Odderon as closed string exchanges.

V. EIKONAL ELASTIC SCATTERING
The single Pomeron and Reggeon exchanges violate unitarity at large √ s.A simple remedy is to resum the s-channel exchanges or a process known as eikonalization, which is dominant in the Regge limit.With this in mind and following [15], we write the holographic total cross section for the Reggeons as where ψ ij (j R , z) are the holographic wavefunctions for the in-out states in the Regge limit, with the metric factors e −3A(z) (Pomeron) and e −2A(z) , e −A(z) (Odderon) for each vertex implicit.
Here χ(j R , s, b ⊥ , z, z ′ ) is the eikonal phase-shift for the Reggeon R = P, O with j ± = 2, 1 for the Pomeron and Odderon, respectively.In the following we will use the shorthand χ R ≡ χ(j R , s, b ⊥ , z, z ′ ).More specifically, from (III.31-III.36)we have The eikonalized elastic amplitudes for pp and pp scattering in impact parameter space are by analogy with the scattering amplitude in quantum mechanics, with 1 2 χ ± ≡ 1 2 (χ P ± χ O ) playing the role of the phase shifts.Note that when reverted to momentum space, the eikonalized elastic amplitudes are To evaluate (V.70) we write with R = P, O and V P (s, t, z, z ′ ), V O (s, t, z, z ′ ) the unaveraged versions of the vertex and kinematical factors defined in (IV.45) and (IV.55), respectively.Defining a ± = a P ± a O , we continue to carry out the angular integration to get The exchange of sum and integration follows from the absolute convergence of the series and the integrability of the ensuing function.For numerical evaluations, we will mostly use (V.70), for faster nu-merical convergence.

A. Eikonalized cross sections
In the forward limit, the remaining sum in (V.72) can be carried out analytically with the result where Γ(a, b) is the incomplete Gamma function.
For fixed z, z ′ , the forward scattering amplitudes are then where the incomplete Gamma function is seen to be suppressed exponentially, in the large rapidity limit.In terms of (V.74), the total cross sections (V.66) are where the averaging is over the in-out states using for each of the Reggeon exchanged R = P, O.The averaging procedure in (V.75) amounts to replacing a ± by Hence, the total cross sections are or, more explicitly, which is seen to asymptote the Froissart bound in the large rapidity limit in conformity with unitarity.The bound is fixed by the Pomeron intercept, which is larger than the highest Odderon intercept for k = 1.
The difference between the pp and pp cross sec-tions, is exponentially small at large rapidities The pp cross section is smaller than the pp cross section for the Odderon branch m 2 k = k 2 → 0, which yields the Odderon trajectory with intercept j O (0) = 1 + 1 2 √ λ , which is, however, not picked up by the contour in Fig. 3.This situation is reversed for the higher k > 1 contributions and also the branch m 2 k = (4 + k) 2 , which are Odderons with intercepts below 1.

B. rho and B parameters
The corresponding rho-parameters following from the eikonal amplitudes, are now given by In particular, at large rapidity which asymptote 0. This is seen to follow from the strong shadowing brought about by the eikonal resummation.
To proceed numerically, we fix the strong 't Hooft coupling by setting the soft Pomeron intercept to the Luscher contribution from the Nambu-Goto string time-like [9] 2 which is close to the soft phenomenological Donnachie-Landshoff Pomeron intercept [16].Remarkably 1 6 is precisely the entanglement entropy of free bosons (string bits) with a fixed (open) boundary in 1+3-dimensions.Further, due to the intercept of the Odderon being below 1, as enforced by the signature factor f − (λ), the dependence on the Odderon coupling g OΨΨ is very subtle.We thus refrain from a fit of g (2) OΨΨ to the data and instead use different input values to obtain an upper bound on the coupling, which will later serve as input for the differential cross section.
In Fig. 5 we show the total cross section for pp and pp for a global fit to the pp, pp total cross sections, rho parameter and B(s) data with √ s ≥ 1 TeV [17][18][19][20][21][22][23][24].The parameters are listed in Tab.I.The fit results confirm the very subtle dependence on the Odderon coupling.For the plots we fixed g (2) OΨΨ = 15, which minimized the mean-squares error and simultaneously showed the best numerical behaviour in the rho parameter, which is only poorly constrained by the data in the considered energy regime.In addition to the parameters α ′ , g PΨΨ we introduced an overall scale factor N σ for the total cross section and N ρ for the rho parameter.A more meaningful fit will follow from the differential cross sections below.
OΨΨ α ′ (GeV OΨΨ .See text. In Fig. 6b we show the empirical fit for the elastic slope parameter in (IV.64).Note that the slope parameter can be recast as a measure of the rms radius where the averaging is understood using the Tmatrix (V.91) below.Recall that for a Gaussian T-matrix e We have also introduced an overall scale N ρ in the numerical analysis of the ρ parameter, as listed in Tab.I.In 6a we show the fit results for the ρ parameter in (V.83).

C. Elastic differential cross section
Away from the forward limit, the spin averaged elastic differential cross sections in the eikonal approximation, are with the Reggeon amplitudes for fixed impact parameter and z, z ′ given in (V.72), and the averaging carried using the weight (V.76).More specifically, with the rightmost result following from the large rapidity limit.Note that the spin-tracing is trivial in the eikonal limit.It follows that the slope parameter (IV.64) is unchanged after the eikonal resummation, at asymptotic rapidities.
To understand the diffractive nature of the elastic differential cross section, we can recast (V.88) in the form with t = −q 2 ⊥ .For large √ s, the Froissart bound (V.81) is reached in the eikonal approximation.In this limit, we may approximate the Fourier inverse of the T-matrix in (V.90) by a black disc with a growing radius with rapidity b(s) = 2α ′ (j P (0) − 1)τ (V.92) for both pp and pp.
In Fig. 7a we show the behavior of T-matrix versus b ⊥ for fixed √ s, with vertices set to 1.The blackdisc limit is reached for √ s = 1 PeV.The parameters used are those listed in Tab.II for g OΨΨ = 15.They follow from a global fit to the empirical differential cross sections from TOTEM [25][26][27][28][29] and D / O [30].Note that the ratio τ / √ λ, which we argued to be large to carry out the integrals via saddle point approximation above, is between 4.2 and 5.3 for the data sets used in the fits.
In Fig. 7b we show the behavior of the differential cross section versus rapidity χ = lns, for b ⊥ = 0.8, 1, 1.2 fm.The dashed line refers to the saturation line set by the condition For b ⊥ = 0.8, 1, 1.2 fm the crossing takes place in the rapidity range χ S = 14 − 20, in agreement with a recent estimate using the standard Nambu-Goto string [31] (see their Fig. 2).Inserting (V.91) into (V.90)yields asymptotically sat. with the expected diffraction oscillations.Note that in the black-disc limit, the first minimum in (V.95) corresponds to The diffractive minimum shifts down with increasing √ s, a pattern that is consistent with the reported measurements in Fig. 8a.We can readily check that (V.95) recovers the Froissart bound.Indeed, the total cross sections σ ± (s) are tied to the elastic differential cross sections by where the rightmost result follows asymptotically, since the rho-parameters in (V.84) vanish in this limit.Recall that in the the black-disc limit, the elastic and inelastic cross sections are equal to πb 2 (s).(V.97) is in agreement with (V.81).In Fig. 8a we show the fit results for the holographic eikonalized elastic differential pp cross section (V.90) at center of mass energies of √ s = 2.76, 7, 8, 13 TeV [25][26][27][28][29], as well as a linear extrapolation of the fit parameters from Tab. II with weighted errors to √ s = 1.96TeV.The extrapolation is in qualitative agreement with Fig. 2. The diffractive tail is very well reproduced for the scattering data with √ s ≥ 7 TeV.For the data sets above √ s = 7 TeV, the model parameters seem to be converging to common values.The bump-dip region is not well pronounced, and with relatively large errors in the scattering data sets from TOTEM at √ s = 2.76 TeV.We performed the same fit with different input values for g OΨΨ and found that, due to the low intercept, the dependence on this coupling is even less pronounced as for the forward quantities in Tab.I.
Our result for pp at √ s = 1.96TeV are shown in Fig. 8b, and compared to the results reported by the D / O collaboration.We note that the diffractive peak is almost absent in this channel and at this center of mass energy.In Fig. 9 we show the extrapolations of the pp differential cross section to √ s = 2.76, 7, 8, 13 TeV, with the diffractive pattern still visible.Data at these higher energies will be welcome.
The agreement of our holographic results for pp and pp elastic cross sections at large center of mass energies with the reported TOTEM data, does not support the contributions of both a Pomeron and Odderon.Which is, however, mostly due to not being able to fix the Odderon intercept at 1.At large center of mass energies, the elastic pp and pp amplitudes receive contributions from C-even Pomeron and C-odd Odderon exchanges.In dual gravity, these exchanges follow from the Reggeization of the graviton and the Kalb-Ramond bulk fields [5].
We have used a bottom-up approach gravity dual construction with a repulsive wall, with bulk gravitons, vector and Dirac fields.The vector fields, arising from the form fields B 2 and C 2 , are allowed to couple through a Pauli coupling as expected for antisymmetric fields.We recall that the BKP Odderon in QCD, is vector-like, with a non-vanishing forward coupling, as supported by our computation.
Using the leading Witten diagrams in dual gravity, we have explicitly derived the diffractive scat-tering amplitudes for both pp and pp scattering in the Regge limit.The graviton-nucleon coupling is tensorial (one coupling), while the Odderon-nucleon coupling is vectorial (two couplings).The resulting amplitudes were eikonalized by resumming the leading Pomeron and Odderon contributions.
The model is characterized by the standard dual gravity parameters α ′ , λ, g 5 and κ the size of the confining and repulsive wall.We further introduced κ N to account for the different mass scale involved for the nucleon trajectory.These parameters are in principle fixed by the nucleon mass and even 2 ++ and odd 1 −− glueball trajectories, but we have elected to trade them for the Pomeron and Odderon couplings and fixed them by the scattering data.
The eikonalized holographic results for the forward pp and pp slope and rho parameters are in relatively good agreement with the world data, including the recently reported data from TOTEM.Remarkably, the differential cross sections for elastic pp scattering in the range √ −t < 1GeV, are well reproduced for all reported center of mass energies including the most recent TOTEM result at √ s = 13 TeV, with a very weak dependence on the Odderon coupling.The agreement with the data seems to be better with larger √ s.The first diffractive oscillations (dip-bump) are reproduced.In our model these diffractive patterns require only a strong Pomeron exchange.
To the order in 1/ √ λ considered and due to the pole in f − (λ), we are unable to fix the Odderon intercept at 1, which would in principle allow for a stronger dependence of the amplitude on the Odd-eron coupling.Hence, at high energies, its contribution is obscured by the Pomeron.The spurious pole at an intercept of 1 appears to be independent of the chosen background, as shown in (III.32),(III.37)and (B.9-B.10).This may be removed by retaining further higher order corrections in 1/ √ λ, using the strong coupling analysis in [11].
In contrast, our eikonalized holographic differential cross section for pp appears to reproduce the empirical results quite well for √ −t < 1 GeV and already at the center of mass energy of √ s = 1.96TeV.Our holographic analysis in the pp channel exhibits the same diffractive pattern as in the pp channel.This is chiefly due to the absence of a cancellation between the Pomeron and Odderon exchange contributions with an intercept below 1.Therefore, we conclude that an underlying holographic Odderon exchange is not present in the currently reported TOTEM data.
In addition, the empirical TOTEM data for diffractive pp scattering indicate that the proton saturates for rapidities in the range 14 < χ S < 20, in overall agreement with a recent estimate using the Nambu-Goto string in 4-dimensions [31].The lower bound translates to parton-x saturation for x S > 10 −6 in deep inelastic scattering, a challenging range for the upcoming electron-ion colliders.
Finally, the questions we raised in the introduction can now be answered in the context of dual gravity: 1/ the persistent diffractive structure at low-t in the pp data is due to the exchange of mostly eikonalized Pomerons with strong shadowing; 2/ The absence of secondary structure at large-t is due to strong shadowing; 3/ The linear rise of the forward slope measures is a measure of the rise of the Pomeron slope which is intimately related to the rise of the 2 ++ glueball Regge trajectory.and take the limit z → 0. Which coincides with the bulk-to-boundary propagator quoted in [10] once the z 2 term from G 0 is taken into account.
In light of the preceeding calculations to evaluate the Sommerfeld-Watson transform via saddle point we rewrite the Whittaker function involving the Tricomi U to better display its singular character.This is best seen by rewriting it in terms of Kummer M functions, using the identity The singular part in v is subleading in the saddle-point approximation and hence will be dropped in the following.Reverting the rescalings and coordinate transformations, we arrive again at the symmetric spin-j bulk-to-bulk propagator where we introduced the shorthand

Conformal limit
In the confining case, the bulk-to-bulk propagator is given by where a = ∆g(j)− t 2 and b = ∆ g (j) − 1.As κ → 0 we have t = t/8κ 2 → ∞ and we can rewrite where in our case x = |t|z 2 /4.For small κ we thus obtain The Regge trajectory can again be resummed by means of a Sommerfeld-Watson transform In the Regge limit the Bessel functions become triv-ial at the saddle point and hence we recover the With this in mind, the Sommerfeld-Watson transform (III.18)becomes We will evaluate (B.7) again via saddle point approximation.The branch cut of Γ(2 − ∆ g (j)) = Γ(iy) ≈ e −iγ E y /iy at ∆ g (j) − 2 = −iy = is chosen to the left of the integration contour, along the negative real axis.In the large s/|t| limit, the integral is dominated by the saddle point for fixed but large rapidity χ, with j P/O given by (III.10),(III.16) and where The signature factor for the k = 2 branch again coincides with the Pomeron signature factor.As is the case for the repulsive wall, due to the pole the intercept cannot be fixed at 1 with m 2 k = 0.In the conformal limit the diffusion is logarithmic in z, with the latter identified with the size of the transverse dipoles or string bits composing the exchanged Pomeron/Odderon.

Alternative derivation: forward region
Alternatively and in the conformal limit with κ → 0 and t = 0, the ensuing scalar propagator is a superposition of conformal plane waves with the shifted Pomeron, j P , and Odderon, j O , intercepts.The shift is caused by the diffusion in rapidity in warped AdS, with a diffusion constant D = 1 2 √ λ , in agreement with the original analysis in [5,32].

Alternative derivation: Finite impact parameter
The forward bulk-to-bulk part propagator can also be obtained from its explicit form for a finite impact parameter.For that, consider the bulk equation We note that S j = 2 √ λ(j − j O ) develops a branch point at j = j O as seen in Fig. 3. Using the Sommerfeld-Watson formula (III.18),we have with the contour integral to the right of the branch point j O , and summing over the odd poles j = 1, 3, ....In the forward limit with t = q 2 = 0, we can switch b ⊥ → ξ with the measure 1 − e −iπ(j−1) sin(π(j − 1)) (α ′ szz ′ ) j−1 e (2−∆g(j))ξ 4π (B.17 with ξ 0 = |logz/z ′ |.In the double limit of large rapidities τ = log α ′ szz ′ ≫ 1 and strong coupling √ λ ≫ 1, the j-integration can be evaluated in leading order in √ λ/τ ≪ 1, following the original arguments in [33], with the result

FIG. 3 .
FIG.3.Complex j-plane structure for the odd spinj bulk-to-bulk propagator.The poles give rise to the vector glueball spectrum and the cut gives rise to the Odderon.

FIG. 5 .
FIG.5.Total cross sections for pp and pp scattering, with the parameters given in Tab.I.
FIG. 7. a: the real part of the pp T-matrix approaches a step function at large rapidities and large λ; b: the differential pp cross section in (V.93) crosses the saturation dashed-line at large rapidities and fixed b ⊥ for pp.

s = 1 .FIG. 8
FIG.8.a: Results for the differential pp cross section (V.90), together with a linear extrapolation of the results to √ s = 1.96TeV, with a comparison to the corresponding TOTEM data and extrapolation[25][26][27][28][29]; b: Results for the differential pp cross section (V.90), compared to the data from D / O[30].

d 2 b
⊥ = 2πzz ′ sinhξ dξ and deform the contour to the left along the cut C L , to obtain G 1 (s, 0, z, z ′ ) =