Neutrino-nucleon DIS from Holographic QCD: PDFs of sea and valence quarks, form factors, and structure functions of the proton

We discuss unpolarized neutrino- and anti-neutrino-nucleon deep inelastic scattering (DIS) using a chiral doublet of baryonic sources with explicit symmetry breaking, in a slice of AdS$_5$ with both a hard and soft wall. We explicitly derive the direct and transition form factors for the vector and axial-vector currents for the holographic dual of a proton and neutron. We use them to derive the s-channel structure functions for neutrino and anti-neutrino scattering on a proton and neutron in bulk. The t-channel contributions stemming from the Pomeron and Reggeon exchanges are also evaluated explicitly. The pertinent even and odd structure functions in the limit of large and small parton momentum fraction $x$ are given. The results allow for the extraction of the nonperterbative parton distribution functions carried by the sea and valence quarks both at large-x and small-x regimes. Our holographic PDF sets compare well with LHAPDF and CTEQ PDF sets in the large-x and small-x regimes in the intermediate range of $Q^2<10~\rm{GeV^2}$.


I. INTRODUCTION
At extremely low-x, the measured nucleon structure functions on unpolarized nucleon targets, show a rapid growth of sea quarks and gluons at low-x [1,2]. Phenomenological arguments suggest that the growth saturates [3], a point supported by perturbative QCD arguments [4]. A central question is then: what is the primary mechanism for the growth of the sea quarks at low-x?
DIS in holography at moderate-x is different from weak coupling as it involves hadronic and not partonic constituents [5]. The large gauge coupling at the low renormalization point, causes the color charges to rapidly deplete their energy and momentum, making them visible to hard probes only through double trace operators. However, because the holographic limit enjoys approximate conformal symmetry, the structure functions and form factors exhibit various scaling laws including the parton-counting rules [6]. DIS scattering at low-x is partonic and fully saturated [7]. Recently, we have extended the notion of DIS scattering to nuclei as extremal RN-AdS black holes with an emerging Fermi surface [8].
In this paper we consider neutrino and anti-neutrino DIS scattering on an unpolarized nucleon described by a Dirac chiral doublet using holographic QCD. The scattering is chiefly due to neutral and charge exchanges. In QCD the charge exchange currents discriminate between partons and anti-partons. When combined with anti-neutrino DIS scattering, it allows for the separation * Electronic address: kmamo@anl.gov † Electronic address: ismail.zahed@stonybrook.edu between sea and valence partons. In holography, the partonic description holds only for very large x and very small x in the Regge limit. The purpose of the present study is to construct the even and odd parity leptonic structure functions for neutrino and anti-neutrino scattering, and use them to extract the sea parton distribution of the nucleon as a Dirac fermion in bulk.
The organization of the paper is as follows: in section II we briefly review the setting for the model with bulk chiral gauge fields and a doublet of Dirac fermions in a slice of AdS 5 , for both the soft and hard wall. The explicit breaking of chiral symmetry is enforced by the boundary value of a scalar bulk field in the bi-fundamental representation. In section III we derive the direct and transition form factors for the vector and axial-vector form factors for both the proton and neutron. In section IV, we briefly review the essentials of neutrino and anti-neutrino DIS scattering on unpolarized nucleon targets, and show how the valence and sea up and down parton distribution functions are related to the structure functions of W ± exchanges. The s-channel contributions to these structure functions are derived using the direct and transition form factors in holographic QCD. The t-channel contributions are also derived using the Pomeron and Reggeon exchanges in bulk. In section V, all the results are combined to extract the valence and sea partonic distributions at large and small x, with a detailed comparison to the LHAPDF and CTEQ PDFs. Our conclusions are in section V. A number of Appendices are added to support some of the derivations.

II. HOLOGRAPHIC MODEL
AdS/CFT duality maps a conformal and strongly coupled gauge theory in 1+3 dimensions at the boundary to a weakly coupled type-II supergravity in 1+9 dimensions in bulk, with a dilaton, an antisymmetric tensor field and additional odd (IIA) or even (IIB) forms. AdS 5 ×S 5 geometry emerges from solitonic and BPS charged D-brane supergravity solutions in bulk with the graviton and dilaton excitations dual to glueballs. The dual of the flavor excitations at the boundary, are obtained through probe Dp-branes and described by DBI and Chern-Simons effective actions.
A simple way to capture AdS/CFT duality in the nonconformal limit is to model it using a slice of AdS 5 with various bulk fields with assigned anomalous dimensions and pertinent boundary values, in the so-called bottomup approach which we will follow here. We consider AdS 5 both with a soft and hard wall, with a background metric g M N = (η µν , −1)R 2 /z 2 with the flat metric η µν = (1, −1, −1, −1) at the boundary. Confinement will be described by a background dilaton φ = κ 2 z 2 for mesons and φ =κ 2 z 2 for nucleons in the soft wall model. In the hard wall model, φ = 0 and confinement is enforced at z = z 0 .

A. Bulk vector mesons
The vector mesons fields in bulk are denoted by L µ , R µ . They are U(2) valued and described by the effective action [9,10] with F = dA − iA 2 in form notations and A = A a T a with T 0 = 3 2 1 2 and T i = 1 2 τ i . Note that the generators T a are fixed in such a way that the electromagnetic charge of the proton is 1, while that of the neutron is 0. The bulk U(2) vector field V = (R + L)/2 and axial-vector field A = (R − L)/2 reduce to the QCD flavor source fields at the boundary. The Chern-Simons contributions in (II.1) are The equation of motions for the bulk gauge fields follow by variation (D = d − iA) The coupling g 5 in (II.1) is fixed by the brane embeddings in bulk, with 1/g 2 5 ≡ 3N c /(12π 2 ) (D7-branes), and 1/g 2 5 ≡ (3 √ λ/2 5/2 π)N c /(12π 2 ) (D9-branes). When ignoring these embedding, the standard assignement is 1/g 2 5 ≡ N c /(12π 2 ) to match the vector 2-point correlation functions of QCD in the UV [13].

B. Bulk fermionic doublet
The bulk Dirac fermion action in curved AdS 5 with minimal coupling to the left U(2) gauge fields is The left and right flavor field strengths are set to A L M N = L M N , A R M N = R M N . The Dirac or minimal fermionic coupling is standard, but the Pauli coupling is not. It can be shown to follow from the SUGRA action by reduction from the top-down approach. Without it, the neutron form factor vanishes. The nucleon doublet refers to (II.7) The nucleon fields in bulk form an iso-doublet p, n with 1, 2 referring to their ± = R, L chirality at the boundary [11].
They are dual to a boundary chiral doublet of baryonic sources with Ψ p1,2 ↔ O p,± and Ψ n1,2 ↔ O n,± with anomalous dimensions ±M = ±(∆ − 2) = ±(τ − 3/2). They map onto a non-normalizable solution or source, plus a normalizable or expectation value of the corresponding source at the boundary. This doubling is best seen in the top-down approach using a left and a right bulk filling brane. The fermionic potential V (z) =κ 2 z 2 will be used for both the soft and hard wall. Here e N A = zδ N A is the inverse vielbein. The components of the spin connection are ω µzν = −ω µνz = 1 z η µν , the Dirac gamma matrices Γ A = (γ µ , −iγ 5 ) are chosen in the chiral representation, and satisfy the flat anti-commutation relation {Γ A , Γ B } = 2η AB . The equation of motions for the bulk Dirac chiral doublet follows by variation with 1, 2 = R, L = ±. The inclusion of a fermionic potential which is set to the dilaton profile for simplicity, breaks conformal symmetry and garentees the Reggeization of the nucleon spectrum in bulk. (II.8) is supplemented with confining boundary conditions for the hard wall, and vanishing fields asymptotically for the soft wall. The solutions to (II.8) are briefly discussed in Appendix B.2. In short, for the soft wall and in the absence of a tachyon coupling in bulk, the spectrum Reggeizes with M 2 n = 4κ 2 (n + τ − 1) and the ground state proton, neutron states with n = 0 are degenerate. They follow from the mixed representation Ψ 1 (p, z) = ψ R (z)Ψ 0 R (p) + ψ L (z)Ψ 0 L (p) (II. 9) for n = 0, with ψ R (z ≈ 0) ≈ z τ +1/2 and ψ L (z ≈ 0) ≈ z τ +3/2 for the positive parity states 1 ≡ + at the boundary. Similar relations follow for the negative parity states 2 ≡ − at the boundary through the substitution ψ R,L ↔ ∓ψ L,R by parity. Note that the canonical dimension for the QCD baryonic sources is ∆ = 9/2 which would suggest a twist τ = 4. However, we expect non-vanishing anomalous dimensions to develop at strong coupling. We do not know of any reliable calculational scheme to assess them. We will assume ∆ and thus τ as a parameter with a twist τ = 3 to recover the hard scattering rules. The inclusion of additional twist contributions is discussed in [12].

C. Fermionic currents
For later use, it will be useful to define the bulk U(2) Dirac 1-form currents 10) and the bulk U(2) Pauli 2-form currents in terms of which the pertinent 1-and 2-form charged currents read respectively and

D. Bulk effective action
In terms of the charged and left 1-form gauge fields L ± N ≡ 1 √ 2 (L 1 N ∓ iL 2 N ), the bulk meson effective action (II.1) can be recast in the following form with the Chern-Simons contribution restriced to the charged left-currents through a neutral where we made use of the Abelian field strengths (II. 16) In terms of the 1-and 2-form currents, the bulk fermion effective action (II.5) now reads where we defined

E. Hard and soft wall models
We now detail the specifics of the hard and soft wall model with chiral symmetry breaking through the use of a flavor scalar in the bi-fundamental representation in a slab of AdS. These well motivated bottom-up models capture the essentials of the top-down approaches and make more transparent the essential aspects of holography. Their parameters are fixed by the brane embeddings and reduction in higher dimensions.

Hard wall with bi-fundamentals
To distinguish vector and axial-vector spectra and introduce a scalarX in the bi-fundamental representation of U (N f ) L × U (N f ) R with a scaling dimension ∆ = 3 and bulk action using the 1-form DX = dX − iA LX + iXA R . Note by replacing X → 2g 2 5 ×X, we can rewrite (II. 19), in a form similar to [14], as The equation of motion derived from the above action (II.20), for A = 0, has a background solution for the bifundamental scalar near the UV boundary (z → 0) given by Both the current mass matrix and the quark bilinear are diagonal, M q = m q 1 and Σ = σ1, with ..N f are the flavor indices. For the hard wall, the boundary condition for the U (N f ) gauge fields is F R,L (x, z 0 ) = 0.

Vectors:
The bulk vector field V M = (V µ , V 5 ) splits into a µ-transverse, µ-longitudinal and 5-contribution. The 5-longitudinal components of the vector field, mix through a Higgs-type effect. They can be decoupled by a pertinent choice of gauge. The transverse part of the vector field decouples, and its mode decomposition in terms of ψ n (z) yields the bulk equation subject to the confining conditions ψ n (0) = ψ n (z 0 ) = 0. The solutions are readily found as ψ n (z) ∼ zJ 1 (m n z) with the vector spectrum m n fixed by the zeros γ 0,n of the Bessel function J 0 (γ 0,n ) = 0, with normalization z0 0 dz √ g xx ψ n (z)ψ m (z) = δ nm (II. 23) and the completeness relation The first zero of J 0 or γ 0,1 = 2.40483, is identified with the rho meson state m 1=ρ = γ 0,1 /z 0 = 0.775 GeV which fixes the IR scale z 0 = 3.103/GeV. Note that asymptotically γ 0,n ≈ nπ with m n ≈ nπ/z 0 . The bulk-to-boundary vector propagator follows from (II.22) through the substitution m 2 n → −Q 2 with the boundary conditions V(Q, 0) = 1 and ∂ z V(Q, z 0 ) = 0 (confining). The solution can be obtained in closed form where we have also shown its mode decomposition with the decay constants g 5 F n = (−∂ z ψ n (z)/z) 0 .

Axials:
Similarly the bulk axial-vector field A M = (A µ , A 5 ) splits into a µ-transverse, µ-longitudinal and 5-contribution. The 5-longitudinal components of the axial-vector field, mix through a Higgs-type effect. They can be decoupled by a pertinent choice of gauge, at the expense of more coupling of the A 5 field with the tachyon field. For instance in the R-gauge, the A 5 field is identified with the pion field. These contributions will not be followed except when discussing the pion contribution to the direct and transitional axial-vector form factors below. The µ-transverse part of A µ is always decoupled. Its mode decomposition in terms ofψ n (z) yields the bulk equation Upon taking the chiral limit before the near-boundary limit (i.e., m q → 0 before z → 0), (II.26) reduces to with the same confining boundary conditionsψ n (0) = ψ n (z 0 ) = 0 and a similar normalization (II.23) and completeness relation (II. 24). Note that (II.27) can only be solved numerically.
However, we can find an equation that can be solved analytically, if we take the near-boundary limit first without taking the chiral limit (i.e., z → 0 with m q = 0) of (II.26) which reduces to (II.28) Definingm 2 n = m 2 n + 1 2 m 2 q , we can recast (II.28) in the form which maps onto the vector equation (II.22) with the replacement ofψ n (z) ↔ ψ n (z), andm n ↔ m n . In the near boundary or UV limit, the differences between the axial-vector and vector masses are only due to the explicit symmetry breaking effect with m q = 0, and vanishes in the chiral limit. In general however, the difference is largely due to the spontaneous breaking of chiral symmetry through Σ = σ1, as a numerical solution to (II.27) shows. For σz 3 0 1, a simple parametric estimate can be obtained using first order perturbation theory in (II.27) which reduces to the near boundary or UV limit result for σz 3 0 → 0. In the chiral limit (II.30) yields a chiral splitting between the axials and vectors m 2 n ≈ m 2 n + 0.38 solely due to the chiral condensate. Note that asymptotically, the linear mass splitting in the hard wall model vanishes, i.e.m n − m n ≈ σ 2 z 5 0 /n → 0 for n 1. Below and for simplicity, we will carry the analytical analysis using the near boundary limit, using the substitution The bulk-to-boundary axial-vector propagator follows from (II.29) through the substitution m 2 n → −Q 2 = −(Q 2 + 1 2 m 2 q ) with the boundary conditions A(Q, 0) = 1 and ∂ z A(Q, z 0 ) = 0 (confining). Note that A(Q, 0) = A(0, z) since A(0, z) = 1. The solution can be obtained in closed form as where we have also shown its mode decomposition with the decay constants g 5Fn = (−∂ zψn (z)/z) z=0 .

Hard wall without bi-fundamentals
Both the bulk-to-boundary vector and axial-vector propagators can be obtained in closed form in a variant of the Sakai-Sugimoto construction [15] using a hard wall model without the use of the bi-fundamental scalar field in bulk but with modified boundary conditions.

Axials:
Similarly, the axial vector spectrum follows with ψ n (z) ∼ zJ 1 (m n z) withm n = γ 1,n /z 0 . If z 0 is fixed by the rho mass then the ratio of the axial-to-rho meson mass ism 1 /M 1 = γ 1,1 /γ 0,1 = 1.593 which is consistent with the empirical ratio m A /m ρ = 1.587 as in the Sakai-Sugimoto construction. The higher excited modes fare less better empirically in both formulations.

Soft wall with bi-fundamentals
In the soft wall model with scalar bi-fundamentals, we replace (II.20) by [14] S[X] = d 5 x e −φ(z) √ g Tr |DX| 2 + 3|X| 2 (II. 35) with again the same background solution and boundary identification for the bi-fundamentals. Vectors: In this model, the bulk vector gauge field in terms of ψ n (z) yields the bulk equation The solutions are readily found as ψ n (z) = c nκ 2 z 2 L 1 n (κ 2 z 2 ), and m 2 n = 4κ 2 (n + 1) for n = 0, 1, ... with normalization coefficients c n = 2/n + 1 determined from the normalization condition (for the soft wall model with background dilaton φ =κ 2 z 2 ) For the soft wall model, the bulk-to-boundary vector propagator follows from (II.36) through the substitution m 2 n → −Q 2 with the boundary conditions V(Q, 0) = 1. The solution can be obtained in closed form as where we have also shown its mode decomposition with the decay constants g 5 F n = (−∂ z ψ n (z)/z) 0 , and defined a ≡ Q 2 /4κ 2 .

Axials:
For the soft wall model, the mode decomposition of the axial-vector gauge field in terms ofψ n (z) yields the bulk equation which, upon taking the chiral limit before the near-theboundary one (i.e., taking m q → 0 before z → 0), reduces with a similar normalization (II.37). Again, note that (II.40) can only be solved numerically. But, similar to the hard wall case, we can find an equation that can be solved analytically, for the soft wall model, if we take the near-boundary limit first without taking the chiral one (i.e., z → 0 with m q = 0) of (II.39) which reduces to Defining m 2 n =m 2 n − 1 2 m 2 q , we can rewrite (II.41) as which is essentially the same equation as the vector one (II.36) with the replacement ofψ n (z) ↔ ψ n (z), and M n ↔ m n .
For the soft wall model, the bulk-to-boundary axialvector propagator follows from (II.42) through the substitution m 2 n → −Q 2 = −(Q 2 + 1 2 m 2 q ) with the boundary conditionsṼ(Q, 0) = 1. Note thatṼ(Q, 0) =Ṽ(0, z) sincẽ V(0, z) = 1. The approximate solution follows as where we have also shown its mode decomposition with the decay constants g 5Fn = (−∂ zψn (z)/z) 0 , and defined a ≡Q 2 /4κ 2 . We emphasize that the exact form of (II.43) requires solving numerically (II.39) for the normalizable modes and using the mode decomposition (first line). Alternatively, one can solve (II.39) also numerically for the non-normalizable modes, after the substitutionm 2 n → −Q 2 . However, for DIS scattering which is the main thrust of this paper, this is not needed. Indeed, in the DIS regime with Qz 0 1, the near-boundary approximation giving (II.43) (second line) is sufficient. This will be assumed throughout.

III. DIRECT AND TRANSITION FORM FACTORS
The vector and axial-vector couplings to the Dirac fermion in bulk follow from the Witten diagrams. They involve both the Dirac and Pauli form factors. We note that neutrino scattering through the charged currents involve solely the charged left-currents. Here we construct both the direct and transition form factors needed for the vector and axial-vector currents and compare the direct ones to the most current data on the proton and neutron. We will use these form factors to construct the s-channel contributions for neutrino DIS scattering on nucleon targets.

A. Direct vector and axial form factors
The direct vector and axial form factors for the proton and neutron V, A + N (p) → N (p ), can be extracted from the boundary to bulk three point functions with pertinent LSZ reduction using for the chargeless form factor, and similarly for the charged currents and the electomagnetic currents  [16], with the baryonic decay constants defined and evaluated in [17] (and references therein). In our case they are tied to the bulk wave functions and given in (II.14-II. 15). The chargeless currentsJ at the boundary are identified with the quark (partonic) currents, with the quarks in the fundamental representation of U(N f = 2). They are sourced by the dual bulk vector fields V 0 µ (Q, z → 0) and V 3 µ (Q, z → 0) at the boundary, respectively. Similarly, the dual bulk axial vector fields A (0,3) µ (Q, z → 0) and at the boundary, are the dual of the quark currentsJ (0,3)µ A (0) and 2J ±µ A (0), respectively. We now proceed to evaluate the Abelian part, U (1)α V,A ⊂ U (2) of the Dirac and Pauli contributions to the direct vector or axial form factors of the proton and neutron (III.45). We will give a detailed account of the Dirac contribution to the direct parts of the ectromagnetic and axial-vector currents, setting up this way the various definitions and normalizations. The Pauli contributions will follow a similar reasoning and will be only quoted.

Direct vector form factor: Dirac
The Dirac contribution to the direct part of the electromagnetic current can be extracted from the bulk Dirac part of the action in (II. 17) in the soft wall model following the conventions for the interaction action used in [18]. Here µ (q) is the polarization of the EM probe. In the last equality in (III.51) we substituted the bulk gauge fields by (A.1.4), and the bulk fermionic currents in terms of the fermionic fields (II.15). The charge assignments are e nucleon = 1 for the proton, and e nucleon = 0 for the neutron.
We have also defined and the normalization (III.54) Here we have set Q 2 = −q 2 (space-like), and used the final state mass shell condition to identify n X = Q 2 (1/x − 1)/4κ 2 , with p 2 = M 2 0 = 4κ 2 (M + 1 2 ) = 8κ 2 for the initial nucleon state. The Dirac part of the electromagnetic current (III.47) can be extracted from (III.51) using which amounts to the Dirac or minimal contribution to the electromagnetic form factor (Q 2 < 0) or more explicitly in the soft wall model using the soft wall rho meson trajectory m 2 n = 4κ 2 (n + 1). Note that normalizing F EM (Dirac) 1 (0) = 1 for the proton, fixes 1 + O(N −2 c ) = 1. In other words, the 1/N c corrections to the EM form factors must vanish at Q = 0 due to the conservation of the electromagnetic current. Also note that (III.58) asymptotes a dipole form.

Direct axial form factor: Dirac
The direct axial form factors are derived using the same reasoning, with Wα µ A (Q 2 ) (III.45) following from the pertinent variation of the bulk Dirac action or more explicitly for the chargeless component, where the substitution (II.32) was used in the second line, i.e.
Similarly, for the charged components Here ± µ (q) is the polarization of the charged vectors. Following the normalization of the flavor generators T a in (II.1), the axial and electroweak charges are Finally, the isoscalar axial form factor can be extracted from (III.60) using which yields the minimal Dirac contribution to the isoscalar axial form factor as More explicitly, in the soft wall model we have following (III.58). (III.66) reduces to a dipole form factor asymptotically. Recall that the substitution (III.61) in the bulk-to-boundary axial-vector propagator is justified for Qz 0 1, as we noted in (II.43). For Qz 0 → 0, (III.65-III.66) give with the rightmost result corresponding to the leading perturbative correction as in (II.30) for σz 3 0 1. Using the soft wall parameters in [14] (model A) yields m 0 = 0.775 GeV andm 0 = 1.363 GeV, resulting in a large Dirac leading perturbative contribution to the isovector axial charge of the nucleon g 3(Dirac) A ≈ 1/3. The near-boundary approximation is not justified in this limit, with the exact but numerical bulk-to-boundary axial-vector propagator required.

Pion pole
Finally, we note that the axial-vector form factor is characterized by two invariant form factors F A (q 2 ) and H A (q 2 ) in general, which are defined as with q = p 2 − p 1 . In the chiral limit, the two form factors are tied by the conservation of the axial-vector current The absence of this contribution in (III.75) can be traced back to the A 5 field which we have ignored. As we noted above, the A 5 mixes with the tachyon X. When taken into proper consideration, this mixing after gauge fixing locks A 5 with the phase Π of the tachyon, i.e. X = X 0 e iΠ , which is identified with the pion field. Careful considerations in bulk yield precisely H A (q 2 ) as expected from current conservation in the chiral limit [20]. Here we can just reinstate this contribution by inspection, with the full pion pole q 2 → q 2 − m 2 π .

B. Transition form factors
In DIS scattering of neutrinos off nucleons, we will need the transition form factors of the left chargeless and charged currents. For that, we define the left vector transition form factor for the process V, L + p → X We first evaluate the U (1) L Dirac and Pauli contributions to the transition vector form fac-tor (III.69) and then generalize them to the correspond- The minimal Dirac interaction term between the bulk U (1) L gauge field L N and the bulk fermionic field Ψ 1 in the action is (III.70) following the canonical normalizations which makes the couplings and power counting manifest in Witten diagrams. The explicit form of (III.70) in terms of the bulk fermions is (III.72) The Dirac contribution to the U (1) L transition form factor reads explicitly For neutrino and anti-neuutrino scattering, the pertinent left-handed hadronic transition form factor is which can be evaluated using the Dirac and Pauli contributions to the fermionic action. To illustrate the normalizations, consider for simplicity the contribution due to the Dirac part, which yields the interaction vertex with the corresponding transition form factor (III.76)

Charged transition form factor: Pauli
The minimal Dirac bulk interaction does not contribute to the neutron transition form factor. This contribution arises from the U (1) L part of the Pauli interaction in bulk [19] The inclusion of this interaction is straightforward but tedious, with the final result for the charged currents and manifest current conservation, i.e. q µ W ±µν = 0 for on-shell spinors. The integrals are related to the general integrals of the type (III.53), namely with a = Q 2 /4κ 2 , and (III.82) We have made use of the recursive relation between the confluent hypergeometric functions together with their integral representation (III.84) and the property of the Laguerre polynomials L a n (w) = Γ(a + 1 + n) Γ(a + 1)Γ(n + 1) to evaluate the integrals.
The electromagnetic transition vector form factor W µ EM (Q 2 ) as defined through can be extracted from (III.51) with the inclusion of the Pauli contribution (III.77), using The result is The chargeless axial transition form factor W µ 0,3 (Q 2 ) as defined through follows the same reasoning, with the full result including the Dirac and Pauli contributions

IV. NEUTRINO AND ANTI-NEUTRINO DIS SCATTERING IN QCD
In QCD lepton nucleon scattering follows from the contraction of the leptonic tensor and hadronic tensor through the exchange of neutral currents carried by γ, Z and charged currents carried by W ± . Some useful insights on standard neutrino DIS scattering on a nucleon can be found in [21]. In this section we briefly review the key definitions and characteristics of this scattering as a prelude to the holographic analysis, which will make use of the transition form factors established above for the s-wave contributions to DIS.
A. Structure functions for ν, ν scattering An overall review of neutrino DIS scattering can be found in [21], so we will be brief in our presentation of the results for our ensuing analysis. For unpolarized nucleons, the hadronic tensor for neutrino (antineutrino) scattering can be organized under the strictures of Lorentz symmetry, parity and current conservation in terms of three invariant structure functions q µ , P µ are the 4-momenta of the virtual current and nucleon respectively. x = −q 2 /2P · q is the Bjorken parameter which is kinematically bounded 0 ≤ x ≤ 1. F 1,2 refer to the symmetric structure functions, while F 3 to the antisymmetric one. The formers are parity preserving, while the latter is not for neutrino and antineutrino probes. In the DIS limit with Q 2 = −q 2 P 2 and x fixed, the structure functions in QCD obey Bjorken scaling. In this limit, the parity even structure functions satisfy the Callan-Gross relation F 2 = 2xF 1 . We note that analyticity allows to relate the hadronic DIS tensor (IV.91) to the discontinuity of the forward Compton amplitude of a lepton on a nucleon To calculate the hadronic tensor (IV.91) we use the completeness of the hadronic spectrum f with the transition current matrix element (III.69), for excited states of squared mass M 2 For neutrino scattring the explicit form of the QCD quark (partonic) currents in (IV.92) are given by with e, e W the electric and electroweak charges, and sinθ W = e e W with the Weinberg angle θ W ≈ π 6 .
The imaginary part in (IV.92) receives contribution from the neutral γγ, ZZ, γZ as they mix, and the charged W ∓ W ± as they are conjugate.

B. Unpolarized parton distributions through charged currents
Neutrino and anti-neutrino scattering on a hadron through the charged currents yield very informative information on the parton content of a hadron in the DIS limit. At weak coupling, the parton model gives a very simple descriptive of the structure functions in terms of the partonic distribution functions of the hadron. Assuming two flavors for simplicity and isospin symmetry, the partonic model for νp → l − X through W + exchange gives The corresponding neutron structure functions follow by isospin symmetry. (IV.96-IV.97) can be inverted to give the unpolarized valence and sea parton distributions in the proton. We now proceed to evaluate these unpolarized partonic distributions using the holographic dual of neutrino scattering.

V. NEUTRINO AND ANTI-NEUTRINO DIS IN HOLOGRAPHY
We now consider neutrino DIS scattering on a nucleon as a Dirac fermion in bulk using holography, in the double limit of a large number of colors and strong gauge coupling λ = g 2 N c . Anti-neutrino DIS scattering follows from pertinent re-arrangements. DIS scattering on a nucleon as a bulk dilatino using holography was first addressed in [5] and later by others [18,26,27,30]. At large-x, DIS scattering using U (1) V probes follows from the direct and crossed Witten diagrams in bulk, and at small-x it follows from Pomeron exchange in the form of a Reggeized and warped close string exchange in bulk. We now review the analysis for a U (1) V current and then extend it to the electromagnetic vector U (1) EM V ⊂ U (2) V , and the left-handed U (1) ± L ⊂ U (2) L currents.

A. Structure functions: Baryonic exchange in the s-channel
For unpolarized U (1) L scattering, the hadronic tensor is with the spectral decomposition for the s-channel contributions The t-channel Reggeized contributions will be addressed below.

B. Dirac contribution
Using the relations we can make explicit the contracted hadronic tensor The additional normalization constant N L = (1/2g 2 5 ) × 2g 2 5 ×Ñ L compensates for the missing higher spin-j and O(g 0 5 ) corrections in the s-channel. Since s (u s )(p)(ū s )(p) = p + M , then the contracted hadronic tensor contributions from the s-channel are By approximating the sum by an integral in a continuous state, i.e., The same contraction applied to the canonical hadronic tensor decomposition (IV.91), for neutrino, with a transverse polarization · q = 0 (which we have already used in deriving (V.104), yields A comparison of (V.105) to (V.104), allows for the extraction of the s-channel baryonic contributions to the DIS structure functions The full left current vertex with the Pauli contribution is given in (III.78). A rerun of the preceding arguments yields the hadronic tensor for ν scattering in the untraced form (ν scattering follows from ν ↔ µ) where we have definedĨ LR (n x ) ≡ I ηL (n x )I ηR (n x ) + 4η 2 q 2 I RL (n x )I LR (n x ) and we have used + · q = 0, + * · q = 0, qνqμ νβμν = 0, and qνqμ μαµν = 0. Comparing (V.111) with the contraction of (IV.91), for neutrino ν, i.e., allows for the extraction of the structure functions where we have used P Xα = p α + q α , p α p β µναβ = 0, + · q = 0, + * · q = 0, and M 2 X = M 2 0 + q 2 (1 − 1/x). Strict bulk-to-boundary correspondence implies N + L = 1 in the double limit of a large number of colors N c and strong gauge t Hooft gauge coupling. Here we follow [18] and assume proportionality between the bulk and boundary structure functions with N + L an overall parameter that captures parts of the finite corrections to the strict double limit. It will be fixed by a point in the data. Similarly, we can find the structure functions for anti-neutrinoν scattering through W − exchange as DIS scattering at small-x is dominated by Reggeon and Pomeron exchanges. In this section, we first consider DIS scattering using U (1) L currents in the Pomeron regime and then generalize our results to the electromagnetic vector The Pomeron is dual to a close string exchange or graviton in the t-channel [22-24, 26, 27]. This is best obtained by recalling that the hadronic tentor ties to the forward scattering amplitude of a U (1) L current through where the Compton scattering amplitude T µν L is given by for massive graviton or glueball h µν exchange in the t-channel, with the explicit result given in (IV.18). The Pomeron contribution as a graviton exchange to the t-channel structure functions F L 1t,2t (x, Q) follow from for · q = 0. Inserting (IV.28) into (V.118) we obtain . More specifically, for the charged currents we have structure functions: Reggeon exchange in the t-channel Reggeon exchange is also a t-channel contribution stemming from a spin-1 exchange induced by the Chern-Simons contribution in (IV.91). The latter allows for the anomalous coupling of W − W + ω in bulk with ω µ a spin-1 flavor singlet U(1) gauge field in bulk (the analogue of the omega-meson). In principle, the Reggeized spin-1 exchange in bulk contributes to the unpolarized and parity odd structure function F 3 . A similar contribution was observed for the spin structure function in [27], following an earlier analysis in [26]. More specifically, the U(1) exchange of L 0 µ in bulk stems from (II.17) and (II.15), with the vertices LΨΨ : where we have defined In the last line in (V.122), we have used the fact that L + (q, z) = L − (q, z) = L(q, z).
Using the relation with W ν µν given by (IV.91), and A tot Lp→Lp (s, t) given by (III.32), we can extract the structure function F W ± 3 (x, Q 2 ) in the t-channel as

VI. HOLOGRAPHIC RESULTS AND COMPARISON TO DATA
(IV.96-IV.97) can be inverted to give the unpolarized valence and sea parton distributions in the proton, in terms of the pertinent holographic structure functions. The ones for the neutron follow by isospin symmetry. More specifically, we have for the unpolarized sea of the proton and for the unpolarized valence contribution of the proton  [19]), and soft-wall IR scaleκ = 0.350 GeV (fixed by the mass of proton and ρ meson in [19]).
The charged s-channel even-parity structure functions F W ± p  1 show the holographic Pomeron contribution at small-x for the xu and xd distributions of the proton at a resolution Q 2 = 6.5 GeV 2 , following from (VI.129) and (VI.130). The t Hooft coupling is λ = g 2 Y M N c = 9.533, the twist of proton is set to τ = 3 (fixed by the scaling of the electromagnetic form factor of proton in [19]), and the soft-wall IR scale is fixed toκ = 0.350 GeV (to reproduce the mass of the proton and ρ meson as in [19]). The green curve uses the normalization coefficient N ± Lt = 0.309. The blue-band corresponds to the normalization coefficients fixed between N ± Lt = 0.274 and N ± Lt = 0.375. Lt . The red points are the H1 and ZEUS data [33], and the orange points at low Q 2 are the data from E665 [34]. See text.
B. Fixing the charged normalization N ± Lt parameters Throughout we will use the same t Hooft coupling λ = g 2 Y M N c = 9.533 which is within the standard choice in the most holographic constructions. The charged normalizations N ± Lt are fixed between 0.274 and 0.375 and will be shown as a blue band in all results to follow. These normalizations are chosen for a best fit to the reduced non-charged (r,NC) and unpolarized deep inelastic e ± p scattering cross section σ ± r,N C at low-x. We recall, that σ ± r,N C is given by a linear combination of the generalised structure functions The overall structure functions,F 2 ,F L and xF 3 , are sums of structure functions, F X , F γZ X and F Z X , relating to photon exchange, photon-Z interference and Z exchange, respectively, and depend on the electroweak parameters asF Here v e and a e are the vector and axial-vector weak couplings of the electron to the Z boson, and κ Z ( . The values of sin 2 θ W = 0.23127 and M Z = 91.1876 GeV were used for the electroweak mixing angle and the Z-boson mass. In the quark-parton model (QPM) where the kinematic variable x Bj is equal to the fractional momentum of the struck quark, x, the structure functions are given in terms of the PDFs as In Fig. 2 we show the measured σ ± r,N C (x, Q, s) versus Q 2 in GeV 2 at √ s =318 GeV with increasing x-resolution. Fig. 2a follows from (VI.131) for x = 0.000016 using the small-x holographic PDF shown in Fig. 1. Fig. 2b shows the same for x = 0.00005. The middle green curve corresponds to N ± Lt = 0.309. Fig. 2c follows also from (VI.131) for x = 0.00008, with the middle green curve referring to N ± Lt = 0.314. Fig. 2d refers to x = 0.00013 with the middle-green curve referring to N ± Lt = 0.313. Fig. 2e follows again from (VI.131) at x = 0.0002 with the middle-green curve N ± Lt = 0.329. Fig. 2f refers to x = 0.00032 with the middle-green curve referring to N ± Lt = 0.313. Fig. 2f refers to x = 0.0008 with the middle-green curve referring to N ± Lt = 0.335. Fig. 2f refers to x = 0.0013 with the middle-green curve referring to N ± Lt = 0.359. The data from the H1 and ZEUS collaboration [33] are shown in red. The data shown in orange at very low Q 2 are from the E665 collaboration [34]. Note that we have reconstructed the orange data points for the cross section σ ± r,N C (x, Q, s) at √ s = 318 GeV from the E665 data set for the structure function F 2 (x, Q) at 0.0008 ≤ x ≤ 0.001 using [35] R(x, Q) = As noted above, the blue-band follows from the normalization range fixed by N ± Lt = 0.274 and N ± Lt = 0.375.

C. Results in the large-x regime
As we noted in the introduction, DIS scattering in holographic QCD at moderate values of partonic-x involves hadronic and not partonic constituents [5]. Indeed, in the large N c limit, the leading single trace twist contributions acquire large anomalous dimensions and are suppressed. The dominant contributions stem from double-trace operators with mesonic quantum numbers. Another way to see this is to note that the large gauge coupling at the low renormalization point, causes the color charges to undergo a rapid depletion into a cascade of even weaker charges, making them visible to hard probes only through double trace operators. At large-x, DIS scattering is almost off the entire hadron making the holographic approach pertinent. In this regime, the holographic limit enjoys approximate conformal symmetry, with the structure functions and form factors exhibiting various scaling laws including the parton-counting rules [5,6]. More specifically, we have for the sea contribution lim x→1 lim κ 2 with the even and odd structure functions given respectively by The large-x asymptotic of the Dirac+Pauli+Mixed structure functions following from (V.114), can be worked out in closed form. For the even-parity structure functions we have  Fig. 4, with the orange data points from JLAB [37]; b: same as in a for x = 0.85, with the red data points from the combined SLAC and BCDMS collaborations [35]. See text.
also in agreement with the recent results in [31] (see their Eq. 88, Eq 104 and Eq. 131). Note that eventhough the Pauli vertex contribution involves an additional vierbein in comparison to the Dirac one, hence an a priori extra suppression with the z-parameter and therefore Q 2 by duality, it is balanced by the extra z-derivative in the magnetic-like coupling F µz , causing both contributions to scale identically with large Q 2 at large-x. HereÑ ± L ≡ c(τ ) × N ± L with c(τ ) are undetermined coefficients, that can in principle be fixed rigorously in a specific model. As we noted earlier, strict bulk-to-boundary correspondence implies N ± L = 1 in the double and dual limit. Here we are assuming proportionality between the bulk and boundary structure functions with N ± L as overall parameters [18], that capture partially the finite corrections to the double limit. In the numerical analysis at large-x to follow, we fit only the two parametersÑ ± L which are defined as In terms of (VI.139), the structure functions (VI.136) simplify (VI.140) withκ = 0.350 GeV as fixed by the mass of the proton and ρ meson in [19].
(VI.140) scales as (1/Q 2 ) τ −1 asymptotically in agreement with the hard scaling laws expected from strong coupling [5], but vanishes as (1 − x) τ −2 at large x in contrast to (1 − x) 2τ −3 suggested in [36]. The large-x behavior at strong coupling follows from the observation that for the virtual photon with amplitude 1/Q to scatter off the nucleon as a Dirac fermion, the latter has to shrink to a size (1/Q) τ , with a scattering probability(1/Q 2 ) τ −1 . As a result, the structure function at large Q 2 but fixed s ∼ Q 2 (1 − x) scales as  Fig. 4; (c) F γZ 2 (x, Q) as given in (VI.132) using the large-x holographic PDFs shown in Fig. 4; (d) F Z 2 (x, Q) as given in (VI.132) using the large-x holographic PDFs shown in Fig. 4; (e) F γZ 3 (x, Q) as given in (VI.132) using the large-x holographic PDFs shown in Fig. 4; (g) F Z 3 (x, Q) as given in (VI.132) using the large-x holographic PDFs shown in Fig. 4.
To reproduce the hard scaling law asymptotically requires 2α + 6 − 4τ = 2 − 2τ or α = τ − 2, which is the large-x scaling in (VI.140). To recover Bjorken scaling for the structure function requires 2α + 6 − 4τ = 0 or α = 2τ − 3 which is the large-x scaling law suggested in [36]. We expect the latter to set in at very large Q 2 .
In Fig. 4a we show the behavior the valence distributions xu V (x, Q) in the upper blue-dark-band, and xd V (x, Q) in the lower green-light-band at Q 2 = 6.5 GeV 2 , following from (VI.135). In Fig. 4b we show the behavior of the sea distributions xu(x, Q) in the lower blue-dark-band, and xd(x, Q) in the upper green-light-band at the same Q 2 = 6.5 GeV 2 , following from (VI.134). The normalization coefficients delimiting the blue and green bands are determined by varyingÑ + L between 17.715 and 35.431, andÑ − L between 30.667 and 61.335. To assess the range of validity in parton-x of the holographic results at large-x, we re-asses the reduced neutral charge σ ± r,N C (x, Q, s) as given in (VI.131) solely in terms of the large-x holographic PDFs. The results are shown in Fig. 4 at low Q 2 for √ s = 5.565 GeV. The orange data points are from JLAB [37]. The red data points are from the combined SLAC and BCDMS collaborations [35] (see Figure 5.14 in [35]). Note that the F 2 (x, Q) data of [35] to σ ± r,N C (x, Q, s) where converted using (VI.133). These results show that the range of validity of the holographic results at large-x is limited to 0.75 ≤ x ≤ 1 (see also below). Our results for the reported DIS e ± p in Fig. 5 are consistent with the recent holographic results reported in [32] in the large-x regime (see Figure 5 in [32]).

D. Structure functions at intermediate Q 2
To investigate further the range of validity of the holographic PDFs at intermediate Q 2 and large-x, we show in Fig. 6a the holographic structure function F 2 (x) in (VI.132) versus x at Q 2 = 6.5 GeV 2 , evaluated using the results for the valence distributions shown in Fig. 4a. The red data points are from the combined SLAC and BCDMS collaborations in [35] for x = 0.65, x = 0.75, and x = 0.85. In Fig. 6b we show the differenceF 2 (x, Q)−F 2 (x, Q) versus x as given in (VI.132) for Q 2 = 6.5 GeV 2 using also the large-x holographic PDFs shown in Fig. 4. In Figs. 4c,d,e,f we show respectively, F γZ 2 (x, Q), F Z 2 (x, Q), F γZ 3 (x, Q) and F Z 3 (x, Q) versus large-x for Q 2 = 6.5 GeV 2 using also the large-x holographic PDFs in Fig. 4. Fig. 6a shows that the holographic results are compatible with the SLAC and BCDMS data [35] in the range 0.75 ≤ x ≤ 1 for Q 2 = 6.5 GeV 2 .

E. Comparison to the empirical CTEQ, LHAPDF and NNPDF data sets
We have also compared our holographic PDF sets both for the small-x and large-x regimes for the valence and sea distributions, to 292 PDF sets from the CTEQ and LHAPDF projects incorporated within the ManeParse Mathematica package [38]. The global comparison is displayed in Figs. 7 and Figs. 8 for Q 2 = 6.5 GeV 2 . Figs. 8a,c show the 292 PDF sets in multiple solid lines from CTEQ and LHAPDF data set [38], in comparison to our holographic PDF sets shown in blue-light-band for the up-distributions, and in green-light-band for the down-distributions. The holographic growth at intermediate-x overcomes these data sets, but joins with a smaller subset of these data at very low-x. Figs. 8b,d show that the holographic sea at large-x is relatively small but consistent overall with all the data sets in the expected applicability of our approach. A similar comparison for the holographic valence distributions is shown in Figs. 7a,c at low-x and Figs. 8b,d at large-x with consistency in both limits with the CTEQ and LHAPDF data set [38].
In Figs. 9 we comparre the holographic results for the sea distributions to the NNPDF collaboration PDF sets with error bars (shown in red band) from the LHAPDF projects incorporated within the ManeParse Mathematica package [38]. Figs. 9a,c show our results in blue-dark-band for the sea of up quarks, and in green-light-band for the sea of down quarks. Figs. 9b,d show also our results for the sea of up and down quarks at large-x. The growth of the holographic sea at low-x is larger in the intermediate range, but slower at very low-x. At large-x, our holographic results for the sea are relatively small but consistent with the reported data set. In Figs. 10 the valence up and down distributions are compared to the same data set with the same notations as for the sea. At low-x, the holographic results are consistent with zero for the valence up and down quark distributions. At larger-x, the holographic results are consistently larger for the up quark valence distribution in the region of validity of the holographic construction.  [38]; (c,d) Our holographic PDF sets (shown in green band) compared to 292 PDF sets (shown in multiple solid lines) from CTEQ and LHAPDF projects incorporated within the ManeParse Mathematica package [38].

VII. CONCLUSIONS
We presented a comprehensive holographic derivation of DIS neutrino-nucleon scattering process using AdS/CFT duality formulated in a slice of AdS 5 with bulk U(2) valued vector mesons to describe a tower of iso-vector and iso-axial particles, and a fermionic isodoublet to account for the even-and odd-parity proton and neutron excitations. To distinguish the vectors from the axials, chiral symmetry is broken either through a tachyon in the bi-fundamental representation both for the hard or soft wall model, or though boundary conditions for the hard wall. The Yukawa coupling of the bi-fundamental tachyon field between the even-odd bulk fermionic fields was ignored, since the DIS regime is mostly sensitive to the high lying part of the nuclon spectrum for which the effect is negligible.
Using Witten diagrams, we have derived the pertinent Dirac and Pauli fiorm factors for both the direct or diagonal currents and the transition or off-diagonal currents. The results for the direct vector form factors are consistent with those in [19], and for the transition form factors they are consistent with those in [18]. To our knowledge, the results for the direct and transition axial form factors are new. We use them for a holographic estimate of the axial coupling g A which is directly sensitive to the explicit and implicit breaking of chiral symmetry.
We have used the transition form factors for the left-currents, to explicitly construct the s-channel contributions to neutrino and anti-neutrino DIS scattering through the pertinent Witten diagram. This has led to the explicit identification of the s-channel holographic contributions to the even-and odd-parity structure functions for both W ± charged currents. The t-channel contributions stemming from the Reggeized Pomeron exchange is explicitly  [38]; (c,d) Our holographic PDF sets (shown in green band) compared to 292 PDF sets (shown in multiple solid lines) from CTEQ and LHAPDF projects incorporated within the ManeParse Mathematica package [38].
constructed and shown to dominate at low-x the even-parity structure functions. The Reggeon excchange through the bulk Chern-Simons interaction is shown to dominate the low-x odd-parity structure function.
For intermediate values of Bjorken-x, DIS scattering in holography is very different from QCD. The leading twist-2 operator in the OPE of the JJ-correlators acquire large anomalous dimension, and is dwarfed by the double trace operators which carry higher twists but are protected. In this regime, DIS scattering is off a poin-like hadron which is similar to scattering for x ∼ 1. When x 1, DIS scattering in QCD is dominated by Reggeon exchange and is dual to spin-2 or spin-1 Reggeized exchange in bulk. So lepton-nucleon scattering in holography allows to probe the partonic content of the nucleon in the two regimes of x ∼ 1 and x 1.
We have carried explicit calculations and comparison to the availabel data both for the structure functions and PDFs from LHAPDF and CTEQ, in both of these regimes. Our results show consistency for large-x for the intermediate range of Q 2 < 10 GeV 2 where scaling violations are still substantial. At low-x our results show a somehow larger growth at intermediate but low-x. The results at higher Q 2 -resolution can be obtained through standard QCD evolution.  [38]; (c,d) Our holographic PDF sets (shown in green band) compared to NNPDF collaboration PDF sets with error bars (shown in red band) from LHAPDF projects incorporated within the ManeParse Mathematica package [38].  [38]; (c,d) Our holographic PDF sets (shown in blue band) compared to NNPDF collaboration PDF sets with error bars (shown in red band) from LHAPDF projects incorporated within the ManeParse Mathematica package [38].

B. Bulk to boundary propagator
The bulk to boundary propagator (or the nonrenormalizable mode) of the bulk Dirac fermion is given by where the fermionic decay functions are defined as . (II.14) Note that the on-shell bulk Dirac fermions are just the residues at the poles p 2 − m 2 n in (II.13). The in-out baryonic states used in our DIS analysis throughout are where s i and s X label they in-out spin, and identified F R 0 = F L 0 = F N (p) (ground state) and F R n = F L n = F X (P X ) (excited state).

C. Yukawa coupling through the tachyon
The inclusion of a bi-fundamental tachyon field X(x, z) in bulk to lift the degeneracy between the vector and axialvector mesons, would also imply a Yukawa coupling between the even-odd bulk fermionic fields Ψ 1,2 which we have not considered here. More specifically [11] which would mix 1, 2 and lifts the degeneracy of the low-lying even and odd parity states in the nucleon sector, (Ψ 1 ± Ψ 2 )/ √ 2. A first order estimate in perturbation theory gives for the nucleon ground state with n = 0 with X 0 (z) given in (II.21). (II.16) through the expansion around the vev, X(x, z) ≈ X 0 (z)e iΠ(x,z) , would also generate a contribution to the pion-nucleon coupling and also the axial-charge of the direct and transition axial form factors [11]. Since our central interest is neutrino DIS scattering we can neglect this coupling and its effects on our results, as most of our analysis involves the behavior near the UV boundary where the effects of σ is negligible both in the nucleonic wavefunctions and the spectrum.

III. DETAILS OF THE REGGEON EXCHANGE
The bulk gauge field L 0 µ (k, z) exchange contribution to the diffractive Compton scattering amplitude in the t-channel is given by We have defined p = (p 1 + p 2 )/2, t = −K 2 , V(Q, z) ≡ L(q = −Q 2 , z) as given in (A.1.8), and used the bulk-to-bulk gauge field propagator (A.1.9) with the substitutions q → k, Q → K, and V → L. We have also used the vertices in (V.121), and defined For z → 0, we can use (A.1.12) and simplify (III.1) as (K) is the form factor of the the nucleon due to L 0 µ . The Reggeization of the bulk spin-1 gauge field L 0 µ (k, z) exchange can be obtained, in a similar way to the Reggezation of the spin-2 graviton exchange, through the substitution followed by the summation over all spin-j meson exchanges using the Sommerfeld-Watson formula The contour C is to the left of all odd poles j = 1, 3, ... (in contrast to the Reggeized graviton where the contour is chosen to the left of the even poles), and requires the analytical continuation of the exchanged amplitudes to the complex j-plane. The spin-j normalized meson wavefunctions J L (m n (j), z) (III.6) are expressed in terms of the wavefunctions of massive scalar fieldsφ n (j, z) which are given, for the soft wall model, in terms of the generalized Laguerre polynomials asφ with w =κ 2 z 2 . The normalization coefficients are c n (j) = 2κ 2(∆(j)−1) Γ(n + 1) Γ(n + ∆(j) − 1) where, in the last line, we have used the fact that m 2 R 2 = −3. The spin-1 transverse bulk gauge field defined as zL 0 µ (m n , z) obeys the same bulk equation of motion as a bulk massive scalar fieldφ n (j = 1, z) with m 2 R 2 = −3 which is manifest in (V.10). We have also used the open string quantized mass spectrum m 2 for open strings attached to the D9 or D7-branes in bulk, and we have defined j 0 = 1 − 1/ √ λ. We now recall that the non-normalized bulk-to-boundary propagators of massive scalar fields are given in terms of Kummer's (confluent hypergeometric) function of the second kind, and their integral representations are (for space-like momenta k 2 = −K 2 ) after using the identity U (m, n; y) = y 1−n U (1 + m − n, 2 − n, y). Therefore, the bulk-to-bulk propagator of spin-j mesons 13) can be approximated at the boundary as (for space-like momenta k 2 = −K 2 ) where φ n (j, z → 0) = 1 zφ n (j, z → 0). We have defined the non-normalized bulk-to-boundary propagator of spin-j mesons (III.15) with the shift a + ∆(j) 2 → a + 1 + (∆(j) − 3) defined in such a way that the mass spectrum of massive scalar fields m 2 n = 4κ 2 (n + ∆(j=1) 2 ) and the mass spectrum of spin-1 gauge fields m 2 n = 4κ 2 (n + 1) match, i.e., we shift n + ∆(j=1) 2 → n + 1 + (∆(j = 1) − 3), giving the mass spectrum of spin-j mesons m 2 n (j) = 4κ 2 (n + 1 + (∆(j) − 3)) .
(III. 16) We have also used and the substitution φ n (j, (0) for the soft wall model. After the Reggeization, the scattering amplitude for the spin-j meson exchange becomes We have defined 1 2 (j + ∆(j) + 1) where G m,n p,q z a 1 , ..., a p b 1 , ..., b q is the Meijer G-function. We have used the identities to evaluate the integrals with ξ = Qz. The function F After summing over all contributions from the spin-j mesons, the total amplitude A tot Lp→Lp (s, t) is given by The contour C is at the rightmost of the branch-point of F LN 1 (j, K) and the leftmost of j = 1, 3, .... From (III.25), we determine the single Reggeon amplitude (total amplitude) in momentum space, after wrapping the j-plane contour C to the left, The imaginary part follows from the discontinuity of the Γ-function × Im 1 Γ(∆(j)) (III. 27) with the complex argument∆ (III.28) and j 0 = 1 − 1/ √ λ. For y → 0, we may approximate 1/Γ(iy) ≈ iy e iγy , with the Euler-Mascheroni constant γ = 0.55772.... The single Reggeon amplitude (total amplitude) in momentum space (III.25) can now be cast in block form We have sets ≡ s/κ 2 , andξ − π/2 = γ = 0.55772..... is Euler-Mascheroni constant. We note that the apparent pole in the Gamma-function at the Reggeon intercept, cancels out in the combination Γ(∆(j 0 ) − 2)V LLL (j 0 , Q, Q ).
In the block form (III.29), the spin-j integral I(j 0 , s) is similar to the spin-j integral in [24] (see Eq. 4.19), with the identifications , ξ(v) ↔ξ, and s ↔s. We then follow [24] to evaluate the spin-j integral by closing the j-contour appropriately. In the high energy limit √ λ/τ → 0 (τ ≡ logs), the single Reggeon contribution to the amplitude is We can rewrite the amplitude (III.31) as (III.34)

IV. DETAILS OF THE POMERON EXCHANGE
The transverse and traceless part of the graviton (η µν → η µν + h µν ) follows from the quadratic part of the Einstein-Hilbert action in de-Donder gauge, with Newton constant 16πG N = 8π 2 /N 2 c =g 2 5 = 2κ 2 . The massive glueball spectrum is determined by solving the equation of motion for h µν following from (IV.1), with for spin-2 glueballs m 2 n = 8κ 2 N (n + 1)g 5 f n = 2κ N (IV.2)

V. OPERATOR PRODUCT EXPANSION
The parton model emerges in QCD through a leading twist contribution to the structure functions. The twist expansion follows from the operator product expansion (OPE) of the JJ currents. We now illustrate this expansion to leading order for the charged current contributions in T-ordered product in (IV.92). Specifically, we have as x → 0 with S(x) = 2iγ · x/(2πx 2 ) 2 . With the help of the identity γ µ γ · xγ ν = (S µναβ + i µναβ γ 5 ) x α γ β (V.2) with the symmetric tensor S µναβ = η µα η νβ + η µβ ηνα − η µν η αβ (V. 3) in (V.1), the short distance contribution to the T-ordered product in (IV.92) is For unpolarized scattering, a comparison of (V.4) to (IV.92) suggests that the parity odd structure function F 3 can be identified with the antisymmetric tensor contribution, F 3 (x, Q 2 )P β ≈ (2e W ) 2 Im 1 x P | qγ β (1 − γ 5 )q |P (V.5) (V.5) is suggestive of a j = 1 exchange through the left singlet current, since s j=1 ↔ 1/x.
Although the twist-2 contribution to the OPE is real, it provides the relevant starting point for the Reggeization by summing over higher spin-j states in holography [26]. For that, we first note that the holographic dual of the singlet current form factor is P X | qγ β (1 − γ 5 )q |P = e N u N (P X )γ β (1 − γ 5 )u N (P ) The bulk-to-boundary propagator V(Q, z ) relates to the bulk-to-bulk propagator G(Q, z, z ) through lim z→0 2 z 2 G 1 (Q, z , z) = V(Q, z) (V.7) Using (V.7) into (V. 6) gives