JHEP04(2018)118 Springer , April 5, 2018 April 11, 2018 April 20, 2018 : : : November 20, 2017 : Revised = 4 SYM theory at Accepted Published N Received ) of the hadronic tensor of deep 2 x, q ( Published for SISSA by . From type IIB superstring theory we 3 https://doi.org/10.1007/JHEP04(2018)118 5 F S [email protected] , perspective, the relevant part of the scattering amplitude comes 5 . 3 1711.06171 The Authors. c AdS-CFT Correspondence, Gauge-gravity correspondence

We calculate the structure function , [email protected] Instituto de La F´ısica Plata-UNLP-CONICET andFacultad Departamento de de Ciencias F´ısica, Exactas,Calle Universidad Nacional 49 de y La 115, Plata, C.C.E-mail: 67, (1900) La Plata,[email protected] Buenos Aires, Argentina Open Access Article funded by SCOAP Keywords: ArXiv ePrint: from the five-dimensional non-Abeliangravity obtained Chern-Simons from terms dimensional in reductionderive on the an SU(4) effective Lagrangian gaugedmation. describing super- the The four-point exponentiallyPomeron interaction small techniques. in regime the of local the approxi- Bjorken parameter is investigated using Abstract: inelastic scattering (DIS)strong of coupling charged and leptons atduality small from framework. values glueballs of This of theplitudes. is Bjorken From done parameter the in in AdS terms the gauge/string of theory type IIB superstring theory scattering am- Nicolas Kovensky, Gustavo Michalski and Martin Schvellinger DIS off glueballs fromchiral string anomaly theory: and the the role Chern-Simons of term the JHEP04(2018)118 . 26 17 3 = 4 F N and the ), x N ], related to 5  – 3 λ  . The compactification 14 5 S 7 is the coupling of the gauge × excited strings are produced, 13 5 9 λ YM g √ x 1 the process is well described in / 19 5 1 ]. In the calculation of the hadronic 8 –  x < 6 x 16 5 , where regime  S ], and also there are Kaluza-Klein modes. N x  6 at small λ ) – 3 2 λ √ 2 YM F / g – 1 – √ − = λ 12 = 4 SYM theory [ 10 leads to the maximally supersymmetric five-dimensional 5 N scattering amplitude S ) SYM theory with an IR cutoff has been proposed by Polchin- φ N + A ]. In the planar limit, at strong ’t Hooft coupling (1 → 1 = 4 SU( 1 φ + N 22 28 A ) SYM theory is dual to type IIB supergravity on AdS In the study of DIS in terms of the AdS/CFT correspondence there are a few distinct N amplitude 3.1 Chern-Simons interaction3.2 from the superstring The amplitude 4.1 Comments on the exponentially small- 2.1 Symmetric contributions 2.2 Antisymmetric contributions 1.1 Deep inelastic1.2 scattering in Yang-Mills Deep theories inelastic scattering and the gauge/string duality parametric regions which depend on’t the Hooft relation between coupling the of Bjorken thetheory. parameter gauge theory, For the parametricterms region of type where IIB 1 supergravity.therefore For it exp is ( necessary to consider type IIB superstring theory scattering amplitudes in the of type IIB supergravitysupergravity on with gauged SU(4)This symmetry dimensional [ reduction induces athe five-dimensional chiral Chern-Simons anomaly term in [ thetensor dual the chiral anomaly is reflected in the fact that it appears a structure function The holographic dual description ofglueballs deep in inelastic the scattering (DIS) ofski charged and leptons Strassler from in [ SU( 1 Introduction 5 Discussion A Conventions for the Killing vectorsB on Gamma matrix algebra in the three-point closed string scattering 4 Antisymmetric structure function 2 Heuristic effective Lagrangian from supergravity 3 Antisymmetric effective action from string theory Contents 1 Introduction JHEP04(2018)118 1 ). 2 x, q ( 3 The rele- F 2 limit. structure function N 1 F structure function of the , diffusion effects become x 2 27% (or better) for the first F . expansion. /N ). We consider DIS of charged leptons 2 x, q ), in the present work the interest is focused ( 2 3 – 2 – F x, q ( 2 F ]. ) SYM theory with an IR cutoff energy scale Λ, and corrections to DIS of charged leptons off glueballs at N 16 2 ) and /N 2 corrections have been considered. However, they have used an effective ] 1 x, q 2 9 ( = 4 SU( ]. 1 /N 9 F N of the pion [ ]. Then, by using these results for mesons a comparison with lattice 2 ]. These calculations have been extended to one-loop level type IIB F 13 ] also 1 15 – 10 11 ]. has been carried out, finding good agreement (within accuracy of 10% or ]. This is particularly important because it should hold for scalar and polarized 14 3 12 , limit and the limit in which the momentum transfer of the virtual photon is much -meson [ ρ 11 N While most of the investigations outlined above concern the calculation of the symmet- For scalar and polarized vector mesons new and very interesting developments have Moreover, the holographic dual description of DIS from flavor Dp-brane models has It is also possible to go beyond the tree-level approximation using type IIB supergravity. In the paper [ It could also holdLattice for QCD the results first of sub-leading the term first in moments the of 1 the pion and rho meson structure functions are presented 2 3 1 model given byamong a them scalar-vector in Lagrangian, comparisonhave which included with in has the our actual a paper possible very [ field small fluctuations number of of type modes IIBin supergravity and reference which interactions [ we ric structure functions on the antisymmetric structurefrom function glueballs in the describe it in terms of its string theory dual model. It is interesting to recall the origin of of the supergravity for the D3D7-brane system,to the finding tree-level an results, impressive now improvementthree fitting with moments lattice respect of QCD data within 1 been done in [ QCD data better) for an overall fittingpion, of and the (within first 21% three or moments better) of for the the first three moments of the the structure functions (foressential example information about those the similar internal structureto to of study the hadrons, other Callan-Gross which scattering can relation) processes.ing be provides In helpful connections in addition, among order universal different behavior confiningwe suggests find relativistic deep new quantum underly- Callan-Gross field type theories. relations for In the this antisymmetric work structure function been carried out verythat successfully. holographic Among dual the dynamical interestingtions mesons results, [ show it universal is properties worth forvector emphasizing the meson structure structure func- functionsvance for of QCD this itself, comes at from least the in fact the that the large discovery of properties such as relations among amplitude by considering scatteringall amplitudes possible of on-shell the states. incominglarge The and result outgoing of states thatlarger into than calculation the is IR very cutoff interesting,particle do intermediate not namely: states commute. (in the This terms indicates of that the in Cutkosky the rules) high give energy the limit leading two- contribution. important and Pomeron techniques can be used. Particularly in reference [ strong coupling have beentwo-hadron obtained, final which states. correspond to Cutkosky a rules DIS allow process us where to there calculate are the imaginary part of an holographic dual description. For exponentially small values of JHEP04(2018)118 . ] , 5 are  19 ) , (1.2) (1.3) (1.1) , the 5 . Con- , ]. This 18 ... , ) ABC ), where 6 z 17 AB f + , and also [ . δ 6 − ∗ C q 1 2  ABC 4 y A d E σ and ( p 4 ∂ A + ) = δ B o D ρ ) B z A ABC A T n d ABC B ν A − ∂ 0) in the T A if x , A m ( ( 2 ) are background fields, 4 A 1 4 / x δ ) SYM theory is [ ( CDE ≡ β f N 1 overall factor in the chiral ) ∂ 1 4 A,µ ∂x C mnopq − A ε and (1 + T α 2 ]. It can be easily seen by looking B 4 ∂ C σ 7 N ∂x T A ABC = 4 SU( A ρ d ( ∂ R-symmetry group. The field content T 2 N B ν µναβ π R ), where ε A i κ x 2) in the 96 ( it leads to the action for the SU(4) gauge  / , i.e. the µ 5 1 5 + ABC A,µ is the Levi-Civita tensor, , ∂ S S ∂x A ε – 3 – i d , which after dimensional reduction of type IIB ) 5 x 1 ]. A,mn ( 2 6 , which are normalized as Tr( A − µνρσ F µ π ]. Let us recall how this works. The starting point R = 1. Both the SU(4) gauged supergravity coupling ε 8 2 corrections only come from the Kaluza-Klein modes 48 – A xJ mn R 6 N 4 2 F d − ABC ] and a more complete AdS/CFT calculation was done by Bilal and R /N 7 2 SG 1 = i d g 1 − 4 2 > 5 − π ) 2 z 96 AdS ( N g C ) SYM theory has an SU(4) ρ − J = N ) symmetry corresponds to a gauge SU(4) symmetry in the AdS p y A (  are fixed in terms of the boundary theory R-current correlators which are R )) B ν x x κ J ( 5 ) ) ) can be rewritten as an operator equation µ d x symmetry group symbols defined by Tr( J After dimensional reduction on ( = 4 SU( µ 1.1 x, z Z R A µ ( is an integer and we set 4 D N symmetry is anomalous, i.e. it is broken at quantum level. The anomaly can be cal- A m ( κ ] = R A < J ]. 8 A as well as [ ρ are hermitian generators of SU(4) This was suggested by Witten [ d 4 ∂ 5 A ∂z SG S where g Chu [ arising from the dimensionalanomaly. reduction on fields The action of this supergravity contains aMoreover, Chern-Simons the term, thus AdS/CFT it is correspondence notanomaly gauge calculation of invariant. shows the the boundaryis matching theory type with [ IIB the supergravity chiral theory in it ten turns dimensions. out In that fact the if 1 one considers type IIB superstring This anomaly is reflectedboundary in the SU(4) bulk theorycorresponding in action a in the very bulk niceat is way, the not namely: gauge gauge invariant since sector [ supergravity the of on global the the action five-sphere leads in to AdS the maximal SU(4) gauged supergravity on AdS sidering the minimal coupling equation ( where the subindex minus indicatesone the which abnormal leads piece to of the thethe three-point chiral SU(4) function, anomaly i.e. [ the T culated exactly at one-loop level,three being external the points, connected corresponding by Feynmantriangle diagram three Feynman the diagram, chiral fermion one which is propagators. with of related the This to chiral is anomaly the the obtained three-point so-called perturbatively function. from The precise value theory. of the gauge theory includesthere six are real four scalars complex transforming Weyl spinorsR-symmetry in transforming group the in with representation the the fundamental chiralitySU(4) part representation of (0 the the antisymmetric structure functions which appear in the hadronic tensor in this gauge JHEP04(2018)118 ) ) ) 5 5 ), . 2 ]). 2 q N = 4 1.4 · N (1.4) 22 x, q , ( P N ) SYM 1 21 (2 F N / , while for β = 4 SU( , P /x 1 α  N q E σ ∝ A 3 = 4 SU( D ρ F µναβ A ε N B ν A ), which are sources for ], however this only contains a x, z CDE ( 20 f ). By matching equation ( A µ 1 4 x A ( 0 + A ) come with the same factor µ → J C σ values we obtain z 1.3 A x -channel Reggeized particle exchange, . Notice that for QCD in the case of ρ t lim λ ∂ 1 √ B ν 2 ) indicate the Chern-Simons term. Notice ≡ − A ) 1 ) is subleading in comparison with  ) 1.3 x for a scalar hadron of 2 ( µ 3 ) structure function [ /x A µ ∂ – 4 – 2 F x, q A ∂x (1 ( 3 x, q 1. In addition, the two-point R-symmetry current ( ∝ F 3 − 3 F µνρσ -channel is allowed. In the general Lorentz covariant = 0 since parity is preserved (see for instance [ symmetry currents 2 ε F coordinates, while Greek indices denote the boundary t 3 R ) structure functions. On the other hand, at larger values N 5 F 2 = ABC d x, q κ . This indicates that in terms of the Witten’s diagrams the ( 2 ) structure function. This tensor is not invariant under parity 2 2 π region, dominated by the F i κ π/N 96 x x, q ( = 3 = 4 ) and F A 2 )) SG ]. Parentheses in the action ( gauge field in the x ) it leads to g x, q 6 ( ( µ R 1 . Thus, this is the origin of the quantum chiral anomaly in the dual 1.2 J F ). µ 2 D ( ABC SU(4) d x, q ( ) SYM theory. From the Chern-Simons term above a three-point interaction in AdS 2 ⊂ F This work is organized as follows. In the Introduction we describe DIS in Yang- Our findings are interesting since, to our knowledge, this is the first result of the Now, let us explain the consequences of the Chern-Simons term for the calculation N There is a previous calculation of the 5 Mills theories and itsshow a description heuristic in derivation terms ofsymmetric the of contributions effective the as Lagrangians gauge/string well from supergravity, duality. as which antisymmetric includes In contributions. sectionheuristic five-dimensional 2 Then, approach we and in it section has been 3 done for we spin-1/2 hadrons. superstring theory calculation. Specifically, for small- the exponentially small- using Pomeron techniques wepure find electromagnetic interaction of the Bjorken parameter weand find that non-preserving parity structure function theory. We have obtainedin this five-dimensional in SU(4) two gauged different supergravity, ways: and then firstly from from a a first heuristic principles calculation type IIB hadronic tensor) this termproportional generates to a the tensortransformations, structure thus of a the non-conserving parity form SYM structure theory, function and appears at in smallorder values as of the the Bjorken parameter we find that this is of the same of the hadronic tensorThe of cubic a part scalar ofthe glueball the hadronic in Chern-Simons tensor, terms termU(1) of implies at that the small in gauge/string valuestensor the theory of decomposition holographic duality. the of calculation the Bjorken of current-current parameter, correlator (which the enters propagation the of definition of an the the boundary theory global SU(4) to equation ( correlator fixes leading contributions from both terms in the action ( AdS/CFT correspondence, obtaining the following equation where we havedimensional considered SU(4) gauged the supergravity, boundary values of the bulk gauge fields of the five- that Latin indices standgauge for theory coordinates. AdS The Chern-Simonssymbol term is proportional toSU( the SU(4) symmetric is derived, which leads to the three-point R-symmetry current correlator by using the exactly known [ JHEP04(2018)118 con- (1.5) (1.6) µν and the W q µν l and comment on the . x i 0 (b) P, h | scattered from a hadron with . This includes the derivation of µ 3 (0)] at small k ν F 3 1. The DIS regime corresponds to F ,J ) x ≤ , ( µ q x · J 2 [ | q ≤ P 2 P, h − – 5 – h x · = iq x x e . In particular, for scalar hadrons this decomposition 4 ν d Z ↔ i µ ≡ 0 involves soft processes, therefore it cannot be calculated in per- (a) µν hh µν regime. In section 5 we discuss our calculations and results. fixed. The hadronic tensor can be decomposed in symmetric and W as schematically shown in figure 1.a. The virtual photon carries four- W x is the leptonic tensor calculated from perturbative QED. In contrast, x µ µν P l . The associated differential cross section is proportional to the µ . Schematic pictures of DIS (a) and forward Compton scattering (b) processes. q , keeping 2 0 P h Figure 1  and 2 whose physical values belongq to the rangeantisymmetric 0 terms under to several identitiesof for the Lorentz hadronic covariantwhich tensor tensor. can structures As be multipliedBjorken a seen parameter by result, as the it functions so-called can of structure be the functions, written virtual as photon a momentum sum transfer h Time-reversal and translational invariance, hermicity restrictions and Ward identities lead momentum traction, where the hadronic tensor turbation theory. Its matrixof elements electromagnetic are currents defined between as the two-point initial functions and of final hadronic a states commutator with polarizations 1.1 Deep inelastic scattering inLet Yang-Mills theories us consider afour-momentum charged lepton with four-momentum carry out a derivation of thespecifically effective action leads to directly the from antisymmetric type structure IIBthe function superstring Chern-Simons theory interaction which from the4 superstring theory we scattering calculate amplitude. theexponentially In antisymmetric small- section structure function JHEP04(2018)118 N = 4 ) DIS. (1.8) (1.9) 2 q N · x, q ( P 2 .... F  ν ) + q 2 q 2 · x, q q . Also, dots indicate electromagnetic ( 2 P 3 q . . 2 for two of the Weyl F i / − 0  2 1 R-symmetry group under x ν q µν A and  σ P ]. Moreover, the procedure R T P, h 1 P x  ρ }| q . It is precisely in this context   5 Im µ (0) Λ. On the other hand, conformal q S π ν q µνρσ ∼ J 2 · ) q i ε = 2 x P M ( − µ µν A − ) J 2 { µ ˆ T P | background, with a constant dilaton and x, q  ( 5 (1.7) 2 – 6 – ) ,W S − 2 P, h F  h ) 2 x x × 2 · q 2 ) µν S x, q 5 2 iq ν ) SYM theory is dual to a particular solution of type ( q T x, q  · P N xe ( µν A µ x, q 4 1 P ( d Im F P 2 3 structure function is expected even for i W π F  Z 3 ) SYM theory. In order to break conformal invariance and σ ν i ) + F 2 q P 2 ) + N = 2 = 4 SU( q µ ρ ≡ 2 which can be omitted since after contraction with the leptonic q q 0 0) generator. This leads to charges µν x, q S N µ , µν ( hh − x, q q 2 for two complex scalars and the resulting gauge theory is anomaly 0 ]. In this work we use the three diagonal generators. It leads to a 1 ( W T / µνρσ , F µν 2 23 µν S i ε η / = 4 SU( µν 1 + η W  − N , = 0 [ = = 2 ) = / 2 333 d x, q ( µν The planar limit of Since there are contributions from soft processes to the DIS, the structure functions = diag(1 W 3 which the scalars andT the fermions arefermions and charged. charge 1 Thefree conventional since choice isnon-vanishing to Chern-Simons use term in the the dual supergravity description. planar limit of induce color confinementThen, the the standard hadron is procedure representedsymmetry by requires is a asymptotically to state recovered of in introduceof the mass an UV the limit, IR IR and at scaleprocess structure least is Λ. at are generated tree-level not by the details gauging important. a U(1) An subgroup analogue of the to SU(4) the virtual photon of the DIS IIB superstring theory, namely:units of AdS the fluxthat of the the holographic five-form field dualcan strength picture be through of extended DIS to was other developed string in theory [ dual models. In particular, we will focus on the The precise relation between the two tensors is given by the following two equations regimes DIS structure functions canDIS be is obtained related by to using therem. the forward The gauge/string Compton related theory scattering amplitude duality. (FCS) canexpectation process be value written through of in the terms two optical of electromagnetic theo- a currents tensor inside defined the by the hadron time-ordered as follows tensor they do notterm contribute would to not the be DISSYM differential included theory cross a if non-vanishing section. we had Notice, imposed that parity the conservation. third cannot However, be for obtained from pertubative SYM theory. Fortunately, in certain parametric The last line of thisterms equation proportional has to been rewritten in terms of leads to JHEP04(2018)118 = = 5 are ··· A (1.11) (1.10) ,..., m, n, (with = 1 A µ ··· J a, b, of the gauge theory, 6 . For example, for the λ φ 0. The relation between the characteristics of the . x → 2 5 limit the OPE is dominated z Ω d -channel graviton/gauge boson 2 t ]. DIS of charged leptons from , R → ∞ 23 and the metric 2 + , 2 YM q g
= 4 SYM theory is given by a solution R 20 2 ≡ 9 are the ten-dimensional indices, dz N s Λ. Since hadronic states at the boundary / + ) SYM theory has been described in detail ,..., ν N = 1 , the ’t Hooft coupling dx = 0 0 – 7 – µ z N is the string coupling. πλ , g dx ··· s 4 , with radius g 5 µν √ 3 are flat four-dimensional indices and η S = 4 SU( = ) corrections within this regime allow for the photon to M,N, × 2 2 2 0 2 N ,..., 5 z R α R /N = 0 = 2 ··· ds limit of µ, ν, N indices, is the string length and 5 2 s l . One-loop level (1 = region. Since the details of the IR are not important, we use an over-simplified 0 α → ∞ 15) in terms of the matter fields is given in [ z 4 are AdS ], in terms of the operator product expansion (OPE) of the two electromagnetic cur- The introduction of an IR scale Λ in the gauge theory corresponds to a cutoff in the On the one hand, one can see that for moderate values of The explicit expression for the full non-Abelian conserved current N We use the following conventions: 1 indices. 6 5 ,..., ,..., on glueballs created byKlein operators (KK) tower which associated are to dual the to ten-dimensional normalizable dilaton modes0 field in theS Kaluza- small deformation known as thesumed hard-wall to model, be exactly in valid whichare up the to dual the anti-de to point Sitter normalizableat description modes this is in point as- leads the to bulk, a by restriction imposing for Dirichlet the boundary dual conditions hadron mass. In this work, we will focus and the parameters of the string theory is given by where of type IIB supergravity on AdS In terms of thesethe coordinate number the of UV boundary color is degrees located of at freedom, regime of QCD. As weexchange will dominance see in in the detail, bulk. this is dual to1.2 the Deep inelasticThe scattering holographic and dual model the to gauge/string the duality planar limit of internal structure. This is relatedfor to the fact that particle creationstrike is a suppressed secondary in hadron the bulk frommuch smaller the values surrounding of cloud the ofby Bjorken hadrons. the parameter, protected in On the operators. the other This hand, is in for analogy with the Pomeron description of the Regge dimensions and the maingether contribution with to some the OPE specificthe is protected conserved given currents. operators by such double-trace as operators the to- energy-momentum tensorscattering are and somewhat differenttrace in operators comparison can with only QCD, create namely: or the annihilate relevant an double- entire hadron, not being able to probe its 1 glueballs in the large in [ rents inside the hadron.twist-two At operators. weak ’t However, Hooft at coupling large the coupling OPE these is operators dominated develop by large single-trace anomalous JHEP04(2018)118 On , at 1 , 7 (1.14) (1.12) (1.13) − ) ∆ ) qz ( 2 0 /q K , 2 2 3 regime the in- z R (Ω) 2 x / · ∆ 1 is some numerical iq Y − e i λ ∆ c ) q is order one, therefore  ·  0 ˜ z s z 0 n x ( α  i . x is its ten-dimensional coun- −  · , ) s iP 2 e /  ) = 4 1 1 = 0. The Bessel function of the 0) = 0 i λ R , z c q − ν − Λ → · x 1 x ]. In their conventions the spherical harmonic ( we see that n ≈ z , z  24 2 ν 0 / increases in the bulk, which indicates that x 1 ( (Ω) − z ∆ qz 1 λ πλ α Y 4 ) – 8 –  . √ 4) from the point of view of the five-dimensional , leading to a suppression factor (Λ m P z x ( For the ingoing field, the solution to the associated A = − /q -channel diagram in type IIB supergravity. In this 2 ,A 1 ] s and the corresponding boundary conditions are 8 s − ≡ ) 1 5 ∆ 2 ∼ 3 m 2 int qz J ∆(∆ R A ( z 2 2 1 z int ) diffusion effects in the radial direction become important (Ω) is a scalar spherical harmonic on the five-sphere. − x z 2 . · / ∆ R ˜ s 1 ,A iP Y qzK λ e x x · · in the bulk. − Λ iq 4 iq e P e m R µ µ √ A on the gauge field theory side. In this case the leading amplitude of exp( n n i c 2 / ]. ≤ ) vanishes exponentially as 1 ) = x − 0) = qz 25 λ , ( , z Ω) = 1 1 ν is not exponentially small. → x  K ( x , z, µ , z µ ν A is the four-dimensional Mandelstam variable and ˜ x > x x ( i ( s µ φ The gravity counterparts for the different parametric regimes described above from A The normalization condition is given in the appendix of [ In this work we use the convention is normalized over the unit five-sphere. 7 8 ∆ and the interaction cannotthe be Pomeron considered [ local. This region can be describedY in terms of on-shell propagator. In contrast,the type when IIB superstring theory dynamicsof becomes a relevant, and Reggeized consequently graviton theteraction mode exchange can dominates. be thought In of thededuced as exp( from local, thus flat-space it string canregion, theory be i.e. scattering described when in amplitudes. terms For of the an effective smallest action parametric terpart. Thus, the IIBrange 1 supergravity description ofthe the dual bulk FCS dynamics process correspondsparametric is regime to given the the by photon strikes an are the obtained entire from hadron. the Then, imaginary part the of DIS the structure two-point functions current correlator by considering the is the following parametric relation [ where second kind the interaction must occurleast at when the field theory viewpoint can be obtained by looking at the center-of-mass energy. There We can set the transversal polarization condition where in the lastnormalization expression constant we and have expandedthe near other hand, the the boundary. holographicmode dual of of a gauge the field virtualEinstein-Maxwell photon equations is on given AdS by a non-normalizable dimension ∆ has a KKtheory. mass Thus, the ten-dimensional field is given by incoming mode the solution corresponding to a state created by an operator with scaling JHEP04(2018)118 R A mn F (2.1) ) and (1.15) (1.16) n x ( m . B . Also, 2 , CS S /N . This amplitude (Ω) 2 + ) K π k  ( e is the dilaton and Y . d ··· φ 5 = 16 ∇ 4, can be written as + 2 5 2 de κ
) must be included. abc ,..., = 1, the spherical harmonics A  mn ) F k n 1.16 = 0 x (Ω) are vector spherical harmonics, 1 4 ( ) ) . 1, the holographic dual description ], where non-forward Compton scat- k k ( − ( m a A m m, n ˜ 26 2 = 1, thus 2 Y ˜ B  and the Ramond-Ramond (RR) 4-form B ) ]. From the ten-dimensional perspective, , φ 5 R k 5 SO(6) representations, while R x 16 , m S X ma massless gauge fields arise as the following 2 ∂ [ h ≈ − ( 5 = 5 i.e. 1 2 A m SO(6). It can be gauged in order to construct a – 9 – B + ∼ . After diagonalization of the equations of motion mabc = 4 SYM R-symmetry group corresponds to the ≡ A a , with indices −R N A m K 5  dependence, and a kinematic factor A , a . At the lowest level, can be written as 5 0 5 5 α (Ω) AdS ) g k ( a − . The details of the five-dimensional reduction are given in Y 5 p ) S x n 5 x d ( Killing vectors ) k 5 ( m Z S 2 5 ]. However, for the holographic dual description of the antisymmetric B 1 1 κ 2 k X = is the Levi-Civita tensor density on = ]. The excitations of the graviton 1 label their corresponding SU(4) d ), which contains the 2 5 S ≥ ma regime [ abcde ) are vector fields in AdS h k  with one index in the AdS x m s, t, u ; x The supergravity action on AdS The effective Lagrangian may be obtained in a heuristic way by analyzing the five- In the holographic picture, the 0 ( α m ( mabc -channel. This heuristic method was discussed in [ ˜ In this section andis in the the non-Abelian following field one strength we associated set with the gauge fields, dimensional gauged supergravity diagramtering of at the tree level, photon-dilatont where to the photon-dilaton leading scat- tering diagram amplitudes for in dilatons the have high been calculated. energy limit is given by the it is necessary tostring theory go scattering beyond amplitudes, the whichG can supergravity be description. expressed asis Thus, the calculated product it in of requires order a to considering be pre-factor build calculated an after effective evaluating Lagrangian on from the which solutions the of hadronic the tensor fields can in AdS 2 Heuristic effective LagrangianIn from the supergravity high center-of-massof energy DIS regime, in the bulk is given by the exchange of excited strings states. In this situation The second contribution canconstruction be of ignored the effective in actionsmall- the that supergravity leads calculations to andstructure the functions also symmetric the structure in second functions the contribution in of the equation ( where B correspond to the associated with these modes,linear the combination fifteen AdS a where isometry group of the five-sphere,five-dimensional SU(4) gauged supergravity on AdS the corresponding gauge fieldsexpanded arise in as modes perturbationsreference on of [ some particular fields which are JHEP04(2018)118 ] ) ] 1 φA 29 (2.2) (2.3) (2.4) , AAh . , → 28 5 pq ), ( scattering F φA dual to the components pq φφh F 5 φA 3 m kl A , g ). → ) 4 1 0 1.3 − φA x, x lq , ( ) F 0 graviton. The process is x kp grav ( 5 F G A kl ] only the AdS pq T  g ) 26 0 kl g = A x, x kl , the scattering amplitude in terms mn ( g mn 2 3 Φ, thus neither the energy-momentum tensor T 5 mnkl − κ . Dots include the kinematic terms of the 2 G . ) √ nk x φ , T mn x g q ( + h ∂ 0 ml ∗ φ – 10 – g mn Φ φ T p + 0 → ∂ x ) nl 5 pq g g x d mk dual to the hadron, and a gauge field 5 is the Chern-Simons term defined in ( mn g d . Notice that in reference [ g φ  2 Z CS − 2 5 S ) = κ np 0 g = factors. mq x, x 5 is given by A g ( κ because they only contribute with a global constant. Given that the 9 + 5 S mnlk ) can be obtained. The Lagrangian arises from the nq graviton propagator in the high energy limit can be expressed as [ 2 ) being some function that is not relevant in the present case, while the g G 0 5 mp x, q g x, x ( ( 3 ). The last factor F = ( . Feynman diagram corresponding to the graviton exchange contribution to the DIS (FCS) grav φ G mn Aφφ In this section we calculate the heuristic Lagrangian obtained from the T We parameterize the perturbations as Φ 9 dilaton and gauge field energy-momentum tensors are given by nor the propagator have where the AdS with coming from the graviton couples to theof energy-momentum the tensor perturbations The idea is to calculate thestates four-point are scattering amplitude given where the byU(1) ingoing current, a and outgoing interacting dilaton throughoutschematically the shown exchange in figure ofof an the AdS field decomposition have been considered, thus ignoring the Kaluza-Klein modes from closed string amplitudes.effective Lagrangian from Then, which wefunction the will leading use contribution the towith same the the exchange techniques antisymmetric of structure to a calculate gauge field the 2.1 in five-dimensional gauged supergravity Symmetric on contributions AdS the Ricci scalar which includesfields the not relevant graviton for our analysis,and and ( also the interaction terms of the type ( scattering mediated by a graviton, and show that it coincides with the one calculated in [ Figure 2 process for small values of the Bjorken parameter JHEP04(2018)118 . 5 i 0 S x C | (2.8) (2.9) (2.5) (2.6) (2.7) (2.10) G | x h . Thus, in , G 10 i ]. , , ··· . 13 i ]. pq A + 4) 1 g T -dependence in the | pq ˜ lq u/ 5 coordinates and 0 g 4) ˜ t G F ) | 5 α 0 x ˜ u/ x S kp 0 ( ih α lq x included in the four-point | 0 limits. After integration, φ F − F l φ 2 ) Γ (1 + . In the DIS regime at strong 0 ∂ s T 5 Γ( → x
∗ ) ˜ h s (
4 ) to the imaginary part of the t ˜ φ u 4 x k kp ˜ t/ ˜ t, X 0 ∂ 2.2 ˜ t/ F 0 and ˜ 2 5 α 5 ˜ s, and )) α κ , 5 of equation (2.38) of [ , 0 . x ˜ t ) − ) ( ) 0 factor, it naturally appears when con- AdS α 0 s t 1 + ( ( φ g ( ' i ∼ 2 z t 2 z k G − A → ∞ ∂ ∇ x, x ∇ K| ) . ( 4) Γ T s p | 5 x 4) Γ + 0 + ( ˜ x s/ S x ad hoc 0 ∗ s t 5 ˜ s/ grav 2 2 0 ) correspond to the matrix elements φ α d ih – 11 – 0 l z z 0 G α − ∂ x , however the full the string theory pre-factor we | Z 1 = = x, x 5 G Γ( − 5 ( ) = | 5 5 S ˜ t 0 ) + ˜ t ˜ s x x x C ( Γ (1 + ( ∼ ih grav 2 5 A ) can be neglected with respect to the first one. However, φ 3 kl x 0 l κ G | T 64 ∂ α φ ) 2.9 )  grav 0 T 2 − x h ˜ G s ( ∗ 0 G . Nevertheless, at first order there is no x, x x  φ 5 ( X ) = k ˜ t ] we have corrected several mistakes in equation (82) of [ ˜ u ∂ x 13 ˜ t, X mnlk 2 ( ) all fields are evaluated at the same spacetime point, namely: we have 2 5 ˜ s, = Im h G , κ 0 2.7 ) × α x ( Sym ( eff can be thought of as function instead of a differential operator. Therefore ] we must multiply by the factor G S A ∼ 1 φ G mn T Note that in ( Since the derivation of this section is heuristic, in order to obtain the same action as In equation (2.38) of [ 10 this context included in the previous paragraphcoupling, depends the on both latter can beterm thought in of the as r.h.s. a ofthis number equation instead ( is of not an the operatoramplitude, since due case to the for the second fact that we only have to consider the imaginary part of In the graviton propagator where expressions ofNow, the for form thecurved solutions space that Mandelstam variables we act have as described second order in differential the operators defined introduction by the ten-dimensional is a constant coming from the reduction on built an effective four-point interaction.In the This supergravity is picture referred theof scattering as the amplitude the quantum can ultra-local mechanical be approximation. operator schematically written language in as terms scattering amplitude. Thus, the effective action turns out to be where the ten-dimensional solutions for the scalars depend on the While at this levelsidering of the derivation four-point this string isexchanging an a theory whole scattering tower amplitude. ofcause excited it It string leads leads states. to to This a the factor finite possibility is contribution particularly of relevant from be- equation ( in reference [ string theory scattering amplitude, where where we only writethis the expression leading matches terms the in index the structure of respectively. The contraction of these three tensors leads to JHEP04(2018)118 ). , see 11 1.3 (2.11) iQφ = 4 SYM = φ N a . In addition, ∂ a C K Q gauge fields, through ) the on-shell effective 3 m A 2.7 , gauge field which propagates in the (Ω) ) cannot be neglected. Y C m solution for each field and carrying out C A 2.9 5 ) derived in the last subsection gives the . x −Q 2.7 – 12 – (Ω) = Y a . However, we can see that the situation is different in ∂ 2 F C a K ]. Now, in order to investigate the leading order contribution and ) can be obtained, giving a contribution of the same order as 1 30 , 2 F 29 x, q , ( 3 20 [ F j s 1 these antisymmetric structure functions are sub-leading in comparison ' x . Feynman diagram corresponding to the gauge field exchange contribution to the FCS 1 regime, due to the Chern-Simons term present in the supergravity action ( regime where the last term in equation ( associated to the diagonal SU(4) generators. -channel, instead of the usual graviton exchange. Following the procedure of the t x space. This state couples to two dilatons in the bulk with coupling  C Since the incoming and outgoing states correspond to two In order to obtain the structure functions from equation ( 5 x K Note that the equation below differs from the conventional eigenvalue equation 11 appendix A. This is due to the convention of the generators of SU(4). there are the following eigenvalue equations for the spherical harmonics of the dilaton for structure function the symmetric ones. the Chern-Simons term theyAdS couple to another with the symmetric ones the From this term the antisymmetric structurein functions the arise when a gaugesubsection field 2.1, is we exchanged will derive an effective Lagrangian from which the glueball antisymmetric the exchange of gaugeleading fields. contribution to The thegives action symmetric no ( information structure about functionsexpect the for these structure antisymmetric the functions ones. to glueball.theory be In present at However, in the the it case electromagnetic DIS. of For QCD one would not 2.2 Antisymmetric contributions Up to now we haveprocess considered scales the as exchange ˜ ofto a spin-2 the field antisymmetric because DIS the structure amplitude of functions the at high energy it is necessary to consider the amplitude can beas considered local. mentioned in More the detailssmall- will introduction, be this given approximation in breaks section 4. down in Noteaction that the must exponentially be calculatedthe by integrals. inserting the AdS Figure 3 process for small values of the Bjorken parameter JHEP04(2018)118 , 1 ). a ∂  3 a 2.15 (2.13) (2.14) (2.15) (2.16) (2.12) x K . . = ) solutions. φ ) 3 contribute, 5 φ m K n ∂ 15 is the current , ∗ ∂ ) in the ∗ φ 33 ]. n 2 D φ d J 29 − , − x, q ∗ pre-factor, it can be ) ( ∗ φ and 3 φ . φ q m we obtain an effective F related to n R , ∂ 2 ∗ 338 ) 3 m 0 φ ∂ group. Then, the relevant s d φ φ ∂ ( x ad hoc A ( R ( can be obtained from type − . n B q D D 5 ∗ ) J 0 A φ ) p q 0 −Q ∂ . Only x, x A o C ( φ ∂ x, x ( A 33 ) = ( n 0 d 12 B p ∂ x gauge 2 5 ( CD mn A κ o n D G G ∂  ) becomes ) 2 mnopq A x ˜ s CD ∗ n ε ( δ G A 2.12 m  C C – 13 – m J ,J mn Q ∂ g 0 Im B q x C diagonal generators of SU(4) ) in section 4. ) has been included as an 5 A 2 ABC p ˜ u Q ) = d 0 ∂ 15 xd mnopq ˜ t, i 5 6 A o x, q K d ˜ s, ( A − x ε ABC x, x , 3 , which is expected when a gauge field is exchanged in the 5 n ( 0 d ˜ t Z F d ∂ 15 for the relevant case. α i 6 / 2 5 , and ( ) = CD mn κ 0 − Z G 8 x G ( = = 8 K = mnopq n D ε A C J ) 0 Asym eff ABC S d x, x ( i 6 = 3 and CD mn ) = G B is the current associated with the Chern-Simons term, while ) x ( x = m ( C m C m J A C J J Next step must be the evaluation of theAlthough at effective this Lagrangian point on the AdS Now, in order to obtain the antisymmetric structure function Then, the integrand of the amplitude ( Then, the supergravity amplitude becomes There is also a pure gauge component which does not contribute to this process [ 12 -channel. This is still a heuristic approach. In the next section we will show explicitly 3 Antisymmetric effective actionThe from Lagrangian string we theory haveof obtained the in five-dimensional the SU(4) previous gauged section supergravity from on the AdS Chern-Simons term directly from string theory.Lagrangian proportional Also, to by 1 t multiplying by an extrahow these ˜ factors emerge from closed superstring theory scattering amplitudes. where We present the calculation of understood from the fact that it appears in the four-point scattering amplitudes calculated tiply by the string theory factor. The effective Lagrangian is components of the symmetric symbol arewhich of are the related form to the regime we have to build an effectiveThen, Lagrangian similarly with the to tensor the structure symmetric of equation case ( described in the previous subsection, we must mul- As mentioned, the incoming gauge fieldswhich is correspond the to generator photons of one of the U(1) subgroups of the SU(4) Also, the gauge field propagator can be expressed as where associated with the dilaton. These currents are defined as JHEP04(2018)118 with (3.1) (3.2) ]. We 2 5 1 κ 2 .
5 5 S ...N 1 ). String theory N F corresponds to the , 5 = Vol , obtaining the five- reduction are actually 0 limit with insertions ··· ...N 5 2 10 1 5 κ S + N h, h, φ, φ S → 5 ( Ansatz ˜ t  A 2 5 ...M 1 F ] it was pointed out that the ). ]. In this work we will mainly 2 M is the Hodge dual operator in ten 1 background. Firstly, we calculate . The 31 240 ∗ , 5 G ε , φ, φ 5 S . In the case of a graviton exchange, − − 3 27 F × ). In [ , √ ). The subtlety lies in the fact that, as , 5 A 2 5 Ansatz 10d [ 1 5! 2.1 F , where R 5 ( 5 =  ) for glueballs. A 5 2 ∗F G pre-factor can be straightforwardly obtained h, h, φ, φ − ...M ( = x, q – 14 – 1 ( √ 5 A 3 M x F ) F 10 ad hoc d ∗F ( Z for the graviton and the RR 4-form field perturbations. 1 2 10 , κ − ] 2 5 = Ansatz ...M 2 M ). Then, in subsection 3.2 we will derive the effective Lagrangian is the ten-dimensional metric. Recall that 2 a 1 , h d 5 M [ IIB sugra MN 10 F ). This means that we also have to consider a process with ingoing RR ∂ S ], the massless gauge fields that appear after the , G 5 2 . In this notation the five-form field strength and the self-duality condition F 1.16 3 = 5 ( π 5 A = ...M
1 5 M S Let us focus on the As a consistency check, these RR modes should be associated with the derivation of F compactification from the ten-dimensional type IIB supergravity solution generating 5 Type IIB supergravitydimensional action SU(4) can gauged supergravitylinearized be action equations consistently ( of reduced motionfrom on of other graviton fields, and which means four-form that field it excitations is are consistent decoupled to turn off all other fields and work together with the self-duality condition dimensions and Vol are written as The relevant ten-dimensional type IIB supergravity action is given by a three closed stringincoming closed scattering string amplitude states on onS flat a certain space-time, specific andthe then effective SU(4)-gauged we supergravity evaluatefollow on the the AdS first two references. contribution to the structure function 3.1 Chern-Simons interaction fromIn this the section superstring we derive amplitude thesupergravity structure from of type the IIB Chern-Simons string theory term on of the five-dimensional AdS gauged states, i.e. a scattering amplitude of the form the Chern-Simons term. Inamplitude the next section we willfrom show this how term. it This can Lagrangian be will obtained be from used the in section 4 for the calculation of the leading scattering amplitudes include allshould the ask possible is interchanged why modes.previous the section Then, cannot leading be a antisymmetric derived questionemphasized contributions from in one we [ found heuristicallylinear in combinations of the gravitonin and equation RR 4-form ( field modes. The precise relation is given from a firstclosed principles type derivation. IIB superstringcorresponding theory For to scattering that two amplitude dilatons purpose in andthe the we two gauge gauge have fields fields are to encodedstart calculate in from a metric the perturbations string four-point polarized theory in scattering a amplitude particular of way the [ form IIB superstring theory. Then, the JHEP04(2018)118 of . i (3.5) (3.6) (3.7) (3.8) (3.3) (3.4) e C K d B (in other Ansatz 5 K c S , ∇ i ) b A 3 ¯ z K ] , a , 3 4 34 z ∇ = 1. . ( 4 . 1) ...M , 1 − abcde , ...M  1 12345 MN M 1 5 ε − h M S ( NSNS abcde N (2) g N = ) given in the appendix A of + ε V F F √ ) 4 5 4 h B 2 abc S ¯ z A.8  Z MN 1 g , ...M 12345 C q defined from these Killing vectors, ...M 2 B m 1 G ε √ 1 . z A 5 ( A ) B p S as → = eA 2 1 MM MM ∼ A (1) a − F K o , d F ∂ ∇ 2 1 are the 15 Killing vectors on MN A n abcde ∇ − MN mabc ( G B RR a A MN V h m K , a ) G h abcde ∂ ,  1 – 15 –  10 B a ¯ − z 3 , iκ ≡ K √ 1 2 z abcde . The explicit computation of this integral leads to B m x mnopq ( ε 5 ) A − ε abc A 10 1 2  S 5 ) in the above equation, it is easy to see that the result term using Z d S − = , 5 2 ∼ g 5 ) = 1 2 3.3 S F √ − × is a pseudo-tensor on ma , h ( 5 RR h MN . For that we use equation ( = ] and references therein. In the case we are interested in, the V (2) 5 h A h abc AdS 4 i 34 F Z Z z – , ABC 2 abcde d ...M d  2 10 32 (1) 1 5 1 κ , it has been shown that only a particular linear combination of the 13 M 3 F 5 =1 ]. The form of the relevant perturbations is given by 3 ( i Y N and the covariant derivatives 2 S [ A F  Z 4 A m A ...M 1 A ∼ MM is the totally antisymmetric symbol such that F ε comes with an integral over G Now, the extension of this term to the curved spacetime background can be written as The starting point is the following flat-space three-point closed superstring theory 5 − For details see appendix B. 13 √ Thus, from the ten-dimensional pointAdS of view thethe five-dimensional Chern-Simons symmetric term symbol on the present work. By plugging the perturbations ( has the following structure and it corresponds tobe an obtained interaction by term perturbing in the the type IIB supergravity action which can found for example inRR [ modes correspond tomode self-dual is five-form the field graviton. strength The perturbations expression has while been the explicitly NSNS obtained in [ scattering amplitude where the vertex operators on the two-sphere and the corresponding conventions can be The Levi-Civita tensor is given by where words, the lowest vector spherical harmonics,the thus metric giving components). the usual Kaluza-Klein the volume form ical harmonics on fundamental modes of bothvector the modes graviton and the four-form field gives rise to the massless up to a numerical constant, and where the only with these perturbations. By expanding the fields in scalar, vector and tensor spher- JHEP04(2018)118 5 F (3.9) (3.13) (3.14) (3.11) (3.12) (3.10) in the of four and the . scattering 14 i GK φ . ) ) 4 φ , q ABC ¯ z = d φ . C , q ) + 4 φ ∂ J z N e A ( ( A ∂ 0) , φ ∂
ABC → . (0 B p d M NSNS N ( 4 i ) obtained in type IIB A φ V except for the fact that k ∂ R 2 ) o 3 4 ∂ + M 4 ¯ K z = ) presented in appendix A, A n k , , φ, φ 4 A 3 ∗ ...M A 5 z 1 φ ( m A.9 ) F is the kinematic factor M ...M q ∂ , , 1) 1 N 5 − K the ten-dimensional Mandelstam , M F F φ ∂ 1 ( e 4 3 GK ( − notation in the next section, where two of (2) k ( mnopq N A ∂ NSNS ε φ · e V F ...M C ) ) = 1 4 1 + 2 K  k ¯ z ]. The final result in the case where the (4) A 2 B d , ...M MM 2 1 = 0), and − 32 ) and inserting the explicit form of the , φ K → z F c ABC ( u  φ ) d (3) = ∇ 2 MM 1 2 (1) is obtained by writing the curved-space version + ˜ s + 3.12 i – 16 – A b − , φ u R ˜ t 4 , F ) ˜ 5 A 1 2 ˜ K u 4 S (2) 5 + a , = − φ ( 0 F s RR 3 ∇ ∗ and ˜ , V φ ) 2 → ) ˜ 1 u φ (1) q 5 k ˜ t ¯ z abcde ˜ s · F ,  ˜ s, ( 1 φ ∂ 1 2 10 scattering amplitude  , e z 0 k A κ ( ( 5 = 0 and ˜ 2 ) α ∂ φ S 1 2 ( 80 ) and the Killing vector identity ( 4 g −  k G − + − , √ B  d 1 2 limit (which is trivial for this particular = +  2.11 = A K 2 10 − ˜ t 2 3 ˜ t c ( RR κ s k K ) and constructing an effective four-point interaction Lagrangian that , V ∇ ) ˜ ˜ → s 4 h 20 ˜ ). This yields an integrand of the form u A + b i k regime and within the ultra-local approximation, we are interested in − , − φ 2 z · K 0 2 k 3.3 a x ) there are two relevant contributions to the = 1 d = + k + ∇ → u 2 4 =1 1 ˜ t φφ A i Y 2.16 5 − k ˜ s, F abcde 5 , Z 0  = F  α L ( s 0 limit. The scattering amplitude is given by the worldsheet correlation function G In this section we use the standard convention for string theory scattering amplitudes where all external A ∼  14 → ˜ current associated with dilaton come from states are ingoing. Wethe will states switch will back be to taken the to be outgoing. By using the relation ( in the ingoing/outgoing convention we see that both the symmetric symbol Finally, the full effectivetors action associated written with in the termsof expansion of the on the effective gauge actionperturbations fields corresponding ( and to the ( Killing vec- For the small- considering the small- we can take ˜ reproduces this scattering amplitude.Einstein frame turns The out Lagrangian to associated be with with ˜ variables ( Details of the computationRR-modes can correspond be to five-form foundrespond field in to strength [ the perturbations dilaton, and is the given NSNS-modes by cor- the product t vertex operators The results in thegrangian previous ( section indicateamplitude, that and in particularly orderNSNS we four-point to need closed calculate the string thesuperstring one scattering effective theory. coming amplitude La- from For small the values massless of RR-RR-NSNS- the Bjorken parameter we have to focus on the 3.2 The JHEP04(2018)118 ). (4.2) (4.3) (4.4) (4.1) ) the 2.15 ]. The µν 1 ). Also, W is similar to 2.13 . components 3 ) 5 F φ S q , ∂ ˜ t 0 ) for the glueball, ∗ 2 α 2 φ ) renders the Chern- ) presented in [ ). Also, let us recall , − m 2 x, q ( ∗ ( F 3.13 3 φ ) we predicted in section  . 1.13 q F µν ˜ s Asym C 0 . 4 ) and α 33 φ ∂ W (and the same for x 2.16 ∗ ( d 1 ∗ ν ]. The exact result is φ 3 p − 1 regime the holographic method n F m factors wherever it corresponds. and the four-dimensional one is µ ) and ( A 2 µν m / o asym n φ∂ s 1 R ∂   π iT 1 − ∗ − 2 2 δ 1.12 3 n λ s φ + A ) ˜ m , = =1 0 limit, the imaginary part of the pre- at small ˜ u ∞ ∂ m  X 2 ∗ , m 2  ∂ 3 µν φ 0 x sym 0 z → R ( F T 4 s ˜ t 0 m C → πα  µν Asym α = We recover ˜ t mnopq λ gA T = √ ≈ – 17 – ˜  s, √ µν 15 0 x ε , x − ˜ s 0 5 ]. T 0 e → 10 Im α 1 d ˜ t α d | ( ∗ ν  G ] n Z 2 Z  µ s 5 C n 35 S ) ˜ , Im g iQ ˜ u ≡ C 20 √ = ˜ t, Q Ω ˜ s, int 5 C , S 0 d Asym eff 33 α d which can be neglected. ( iS Z G 2 6 R  / − × i 1 . This means that we obtain the structure anticipated in equation ( − exc = λ D n ] the normalization of the fields is such that the interaction term between the dilaton Im 1 J ∼ 2 Asym eff S solutions we have to insert are given by ( /R is the second of the currents (involving dilatons) given in equation ( 0 factor combined with the second parentheses of equation ( 5 α C m J ABC As in the symmetric case, by taking the d Note that in [ 15 and the gauge field is in the regime under consideration andof up order to corrections from the radial and factor can be replaced by a sum over excited string states [ The AdS that the relation between the ten-dimensional invariant ˜ states, this on-shell action is given by where the last equality follows fromthe the one corresponding optical to theorem. the The symmetricstarting calculation structure point of functions is thederived effective from action first principles proposed in in section section 3. 2 Considering two from ingoing heuristic states arguments and and two outgoing consists in evaluatingprocess the and on-shell taking amplitude itssymmetric associated imaginary and with part. antisymmetric parts the Then,AdS/CFT as effective if dictionary supergravity implies we [ separate the hadronic tensor into its 4 Antisymmetric structure function In this section wefollowing obtain the the conventions of antisymmetric reference structureAs [ explained function in the introduction, in the the Simons current These results are in full2.3 using agreement heuristic with arguments. the effective Finally,this remember action paper that ( we for will the focus particular process on studied the in contribution proportional to where JHEP04(2018)118 ]  36 x (4.8) (4.9) (4.5) (4.6) (4.7) (4.10) (4.11)  , λ ) √ 2 − . e ) x, q , 2 ( ) we obtain the 2 . 1 q s F 1 x 0 2 x, q 1.7 ( . q α R σ 1 · 2 regime, but one can ∆ P I ρ
xF P m q x regime, one obtains the 3 2 2 ∆ + 2 generators, while this is integral over the Bessel 2∆ + 3 ≡ 2 Γ (∆ + 2) x / . ω 15 2 1 2 m µνρσ T ∆ ∆ + − ε I λ ∗ ν ) = 2 2 Γ n 2 π , ω and  µ . 2 πλ 2∆ + 3 | 2 ) n 8 (∆ + 2) 4 0 i x / x, q as 1 c m 1 T ( to be non-vanishing the hadron must Γ (∆ + 1) √ − 2 − ω 3  3 π ∆ λ Q | (2∆ + 2)(∆ + 2) qz (2∆ + 3)(∆ + 1) 4 F − ) √ 1  8 3 λ = 2  ω − 2 stands for the ( Ω √ q ∆ Λ ) = ω Q x ρ ), and working out the integration over the δ ], and ,F ) = − ω  ∆  1 ( ω  2 I 2 2 ( 1 4.2 2 q ∆ Λ 1 – 18 – ) = πλ ) is in agreement with the behavior found in [ K 2 R is not exponentially small, i.e. when 2∆+3 4  K s ω , 2 factor: for 0 1 ) x QI q √ 4.8 α I 1 x 2 x, q ω ) with the general decomposition ( Q 2 ( ( | 2 2∆+3  2 0 , while i 3 π c 4.6 F πλ C | ) = K 2 m 4 | 2 Ω ω 0 i Q X dω ω ρ c √ C | 2 4 = x, q π 33 Z ( 2∆+2 = d 3 1  (∆ + 1) = ˜ F s − 0 3 4 4 = ∆ α dω ω µν Ω asym  Q ρ Q 2 − 2 Z 2∆+3 W , q Λ ∗ ν 1 m = I n  ) = µ  2 ∆ 2 n δ 1 I x 2 case given by a dilatino mode. In the mentioned work, the antisymmet- / x, q m ( X 3 ) = F 2 is a dimensionless constant coming from the angular integral of the symmetric ) in the on-shell effective action ( Ω x, q 4.5 ρ . This sum can be expressed in terms of ( 2 1 Note that our result of equation ( Plugging the solutions for the gauge fields and the dilaton current together with equa- / F 1 − not necessary for the symmetric functions. for the spin-1 ric structure functionsextrapolate are the computed result in by considering the the exponentially ultra-local approximation. small- We find new Callan-Gross like relations that can be expressed as: One subtle difference arises frombe the charged under the U(1) groups associated to the were effective action, defined in equation (88) of [ Let us recall thatfollowing for symmetric the structure dilaton functions in the exp( Now, by comparison of equationantisymmetric ( structure function for the glueball where the chargefunctions is tion ( full ten-dimensional spacetime, we find λ which is well approximated by an integral for where the last factor can be ignored if JHEP04(2018)118 in = 2  π | u ) 0 (4.14) (4.15) (4.12) (4.13) z . More ( ∗ φ | φ ) structure 4 0 ν 2 z R does not include φ ∂  . µ x, q φ ( )) ∂  0 1 4 z F ( , 0 − φ 2 2 u p ) z  0 , and integrating. For ∂ | , 0 u − +4 is the Hamiltonian. ) ∗ ) z∂ 0 u − 2 z ) u φ u u ( z ∂ ( + ( . m ( φ λ τ | φ − 2 ∂ z u 2 √ 4 ( 4) P ∂ z δ 2 − =0 R t z − e 2 |   2 2 j ∆ / = 2) ( ˜ t/ +4 = λ λ 1 πτ 0 =0 j ]. 1 t z √ 2 α 1 λ | 2 √ , j 2 + 39  2 regime . Also the scalar solution z∂ -field part at ˜ t/ r – 5 ˜ s 2+ z 0 0 φ 4
S dependence in the exponential is known − + α = 0 λ x 2 α ˜ s 2 z∂ z √ t 2 0 37 , 2+ 2 ) ∂ , t  become important. This happens because − 0 α 0 3)
z 2 2 1 u ˜ s 5 z 0 2
0 25 F 2 − , ˜ | s α − − α R 0 u, u ∗ )] j n ( p u and the α 1 2 – 19 – = 2 13 qz K F , ( u ) )] 1 = 1 + (2 H u mn 2 ( qz factor was already present in the ultra-local approx- 2 )[ z ( ∂ ∆ 1 2 (1) A 2 qzK 0 2 s GF [ z = 2) = P . α 2.10 0 0 R − 0 λ qzK ) cannot be neglected. Thus, one must consider the full = z 1 2 , j [ dz √ j regime the ultra-local approximation does not hold due to + 4.4 z z dz ∆ dz dz du du t = 0 x 2 + 4. Then, the scattering amplitude can be written in terms of 2 0 limit takes the form Z is an arbitrary scale and there is no cut-off in the AdS spacetime. 2 z R Z Z Z , t 0 0 2 2 ν 2 2 2 → is a particular case of the Hodge-de Rham operator, defined more / / / α ref R R 1 1 1 = ˜ t z 2 Ω u, u 1 2 λ λ λ 2 ν ( ρ 0 Ω Ω Ω E K = 8 8 8 ). Note that the ˜ s ˜ πρ πρ πα πρ t 0 0 2 α α 16 = 0. ∆ = ≡ = = t 1 ) plays the role of time and F | ref = log( µν Sym τ z/z T with energies In the symmetric case, this leads to the interchange of a Pomeron. Let us start by ∗ ν This expression is valid after the angular integration on n u log( 16 µ n the scalar spherical harmonic.the Finally, definition notice of that the in ’t these Hooft expressions coupling we have absorbed a factor of 4 diffusion time. Thefield final part DIS of amplitude the isexample, effective obtained the Lagrangian part by at of evaluating the thefunction on-shell rest reads effective action of that the contributes to gauge- the where imation of the scattering amplitude.mass energy It for reflects a graviton the exchange.as appropriate The the scaling ( with diffusion the factor center-of- and the inverse of its coefficient gives the associated characteristic In the conformal case One can then diagonalize this operatorin in terms of its eigenfunctions,a which kernel are plane which waves in the generally by It can be− evaluated in terms of an auxiliary quantum mechanical problem where reviewing this in theconcretely, it conformal is limit. given by The the differential spin-2 operator Laplacian, and acts the on exponent reads We will set In the exponentially small- diffusion effects in thethe radial direction last of factor AdS indifferential operator equation of ( equation ( 4.1 Comments on the exponentially small- JHEP04(2018)118 . 3 x F and ) by 2 limit. (4.17) (4.19) (4.16) (4.18) (4.20) u λ x, q i ( 2 ) 2 0 F u ) is obtained factors since 2 + . τ u (  . λ z τ x, q are scalar factors .  2 ( √ and the resulting 2 2 1 z∂ and x − − F iνu e − . Also, the result for 17 s − ) , − 2∆ ˜ e τ 2 z , which naturally leads /x , ) ∂ 2 0 0 ++ dx e Λ)  2 q u i i h z As we have seen in section z ( ∞ − 2  ≈ η u u, u ( 18 ( Z ≈ , − + 2 λ s ]. iν ) F 1 2 ν ν √ π e | η 2 20 2 ) π +  √ dν 2 z spacetime to induce confinement ( 2 + ) + R 0 5 φ t erfc( | 0 u 2 2 ) = 2 − )) and inserting an extra factor of 2 η z iνu η − du u 0 e 2 ( z ˜ qz τe  8 α λ R τ ( R is formally recovered in the large π 2 erf( √ 2 1 2 R 1 2 2 1 √ − / K √ − 4 e ) = 1 0 h − u − ]. )) = 2 λ ) + – 20 – / ) = z − 1 πτ ( 39 u ) = 1 qz 2 + 3) = λ  ( -dependence is hidden in the u u ( η ( ( ν 1 x 2 0 x ) = 1 φ δ h r 0 ˜ τ K P (∆ λ , ( 2 0  erfc( du 0 2 √ 2 ⇒ ) z α , − R R 2 2 factor. A similar expression can be found for 2 / u, u τ q 1 2
1 ) and ( ˜ s π 1. These results are important to understand the holographic λ is introduced in the AdS 0 = 0 F + qz + 4˜ τ α − ( 0 < t 0 0 4˜ 2 )) = z z 1 2 ) | u z z ) √ ˜ τ K ( ] is similar to what we need for the dilaton case, we only outline the important 0 2 , 2 + u 0 α 0 limit the conformal kernel must be replaced by z 20 ( R ++ and we have defined the function u 2 h 1 2 q τ (2) 2 → A u, u = 2) = 2 = z ( t P / = ( 1 η z , j ˜ t − 0 ∂ )) = → 2 ) < F z α λ , consistent with energy-momentum conservation. Taking the reference value ( , the boundary condition on the Pomeron modes = 0 1 0 (1) 0 u A z ( z − , t P 0 = (4 (1) A = τ P u, u ( The process we have been analyzing is such that the leading contribution to the When the cut-off at ref Λ We are using light-cone coordinates.Since Also, the we analysis of consider [ only modes which are relevant in the high z leads to some very interesting results [ 17 18 K energy limit. steps. antisymmetric structure function comes from thewas exchange also of pointed a out Reggeized in the gauge2, DIS field. from this This dilatinos vector in reference mode [ implying interacts that with we the currents havewe instead to have of consider the the energy-momentum spin-one tensors, differential operator. Thus, in this case DIS process at highwritten energies. in terms In of fact,obtained at the the weak Pomeron structure coupling. kernel ofx Also, has the the amplitude a comparison at striking with strong the formal available coupling resemblance data with for DIS the at one small with Note that Therefore, in the where ˜ ditions on as eigenfunctions are replacing Of course, in thesein final this formulas regime the the four-dimensionalthe Mandelstam parametric invariant region is exp ( general steps of the above calculation remain valid, but one has to impose boundary con- that only depend onwith the corresponding the incoming curved solutions, metric.inserted and an all In identity contractions of the are theto made last the form appearance step of the we spin-2simply kernel. have by Now, written due multiplying to everything by the in optical a theorem terms 2 of where JHEP04(2018)118 , , ), Λ) , 0 4.15 (4.21) (4.24) (4.26) (4.22) (4.23) ρ ( i conformal 3 φ 2 ) F P 0 ρ ). However, ) + i ρ ( 2 . ). After diago- ) 2 λ 4.25 . 0 τ ρ ) 8 ρ 2 0 √ ∂ | ρ + ) − factors. Note that 0 ρ − + e ( z ρ ) ( λ τ ( τ 2 ρ ˜ τ 8 λ τ √ φ ∂ , 8 | √ − 2 2 ) grows more rapidly as − e / 2 and 0 ) 0− e ˜ z τ 2 s , ρ 8 / , x, q 1 πτ 2 2 ( R 2 λ / 3 0 + 3 = 4( ρ/ . F ( r 1 , ρ ) F 0 2 ρ Λ) = , + + ρ/ 0 . ρ ( 2 z ( ) ) ( 0 1 2 0 F ρ φ ρ can be split in the conformal − = 0 deformation − + + 3 e 3 ρ ρ 0 ( 2 ( F λ F z ) 1 2 λ 0 | τ 1 √ 8 ρ ,P √ + 2 ) − ) − e − − + ρ 1 e – 21 – ( λ . However, the conformal contribution is model qz
h λ 1 τ ( √ 0 zA ˜ s 8 2 √ 1 0 2 z ( / ) we can rewrite ∆ − α z − 1 K πτ 1 e ∂ ) conformal 2 λ 3
ref h ) since both the scaling with the center-of-mass and its ˜ s F (4.25) 0 qz 2 λ ) r ( / α 0 z/z 1 πτ = √ × 2 λ 3 K ρ, q (2 -dependence is contained in the ˜ ( 3 = 1) = F spacetime with no IR cut-off, plus the contribution from the / ) is still present. Now, let us consider the effect of introducing r z x A 2 ln( 1 3 z 5 , j P ( λ − q − 0 1 ρ √ 2 = 1) = = = 0 spacetime. The boundary condition on the Reggeized gauge field − ) = 1 u , j 5
, t dρ dρ 0 ˜ s 0 z, q ( = 2 Z = 0 α A ρ, ρ 2 ρ ( space is deformed near , t P ( π 0 ) is given by K 12 5 2 Q × ρ, ρ is ( x, q Λ ( + . The formal expression obtained for ) = 3 0 K 2 A ρ F 0. Also, diffusion in It is difficult to obtain an analytical expression for the above integral ( x, q ( and → 3 F the contribution from the IRhow cut-off the is AdS model dependent,independent. in the sense that it is sensible to some approximations obtained from simplification of the scalar and gauge field external where The information about the together with the insertionThus, of the hard-wall spin-1 kernel instead of the conformal one. deformation induced by the IR cut-off: The explicit result is obtained by following the same steps that led us to equation ( The final form of theρ structure function in thisi.e. regime from is the given complete by complicated AdS integrals in However, the eigenfunctions areanalogous to modified the in onewe such above, obtain leading a to way an that identical modification this of condition the is kernel. actually Therefore, curvature correction change. Notex that this impliesa that cut-off in themodes AdS nalization of the relevant operator, we obtain a conformal kernel of the form The Regge slope is now 1 By introducing JHEP04(2018)118 z x = ⊂ R N (4.27) (4.28) -channel structure t . Its action 3 5 F ) is obtained 2 x, q , ( ) , 3 ], but it was taken 1 F  q/M dimensional reduction ( . Thus, there is a deep 2 1 5 = 4 SYM theory, which M S log ) ]. − N λ τ ABC . With this approximation, 0 2 ) has the power dependence √ d z 2 q/M 39 − ( ,  2 e QCD gauge boson such as in neutrino δ 2 25 x, q Λ log [ / 0 ( current, therefore allowing for the 1 πτ 1 3 λ M τ Z 2 λ q 2 ∼ √ F M ≈ − 1 x r or e ) λ λ 0 = 4 SYM theory and the emergence of 1 2 1  √ √ / z 2 = 4 SYM theories may share some generic 1 ( πτ electric W N 2 φ − λ = ) should be thought of as symmetrized in ) N ˜ s τ 0 r 1 λ qM α ,P 1 ( – 22 – √ λ 2 s  √  (and − 1 q 1 q ]. In that reference the wave-functions products were ) M s = ˜ s 0  − ˜ s 39 0 α 2 z λ ( α π  2 √ δ Q π 3 3 q 1 1 Q x ≈ ) comes from the chiral anomaly of ≈ ≈ ) 2 ) 2 , these gauge theories are essentially different. In particular, z, q x, q ( ( N x, q 3 A ) this implies ( F P 3 F is zero for the electromagnetic DIS, i.e. a charged lepton scattered from 4.28 3 F stands for some relevant mass scale, for example the proton mass. It should be can be gauged in order to describe the ). On the other hand, the fact that the chiral anomaly is related to the Chern- 2 M , and in ( = 4 SYM theory deformed by the introduction of the IR scale Λ. The reason for R 0 . From the string theory point of view this comes from the z In QCD x, q N ( 3 ABC features in the large- 4 SYM theory isSU(4) chiral. Theconstruction R-symmetry of current a associated bulkfunction with dual was the not photon global analyzed which in U(1) mediates the the original DIS calculation process. developed in The [ Feynman-Witten diagram of SU(4) gauged supergravity in the bulk. a hadron with exchange offunction a does virtual not photon, preserve duewhen considering to the an the parity interaction fact symmetry. mediated thatDIS. by However, this a Of though particular QCD course, and structure IR-deformed this would not be the case contains the Chern-Simons termconnection proportional between to the the symbol chiralF anomaly of the Simons term in thein bulk the is Bjorken variable reflected which in comes the from the fact propagation that of a gauge field in the the non-vanishing does not depend oncorrelation the function IR of deformation.d current This operators, anomaly can and beof is type seen proportional IIB from supergravity, to which the leads three-point the to the symmetric SU(4) gauged symbol supergravity on AdS 5 Discussion In this work wein describe the how dual the holographicin antisymmetric description structure function of DIS of charged leptons from glueballs at small- while the hard-wall contribution canin be the obtained context using of aand the similar kernel simplification. notation Note ˜ that the final expression for the conformal contribution takes the form approximated by Dirac delta functions determined by the relevant scales where directly related to the IR cut-off scale identified as Λ solutions lead to interesting results [ JHEP04(2018)118 5 1) x < -channel  s λ √ / OPE). On the the probability N ). First of all, we are sub-leading in JJ λ 3 √ F / . However, as seen by a 1 for glueballs. However, OPE in terms of the in- the twist of the hadronic 10 hadron is high enough so 2 T τ , the terms proportional to  F q ] from a heuristic viewpoint JJ x ]. Let us briefly recall it. Con- 1 35 1, the amplitude is dominated ,  and parent ]. , being 20 ) 1 1 41 λ boundary, and it therefore shrinks. F x < − √ τ 5 ) − 2  due to the form of the associated AdS /q is proportional to the metric warp factor, 2 2 1 / 4 1 F T − λ . This leads to a power law suppression in the – 23 – = 2 1, which implies that for moderate values of the /q 2  F ] in the context of the 1 D = . Thus, from a four-dimensional perspective the string 10 3 ˜ s /r 0 2 F α R . 2 = 4 SYM theory. A possibility to deal with a holographic = (string theory description) and on the right of the arrow we F ], in that case for instance scalar and polarized vector mesons z N x the probability that the photon strikes a parton is high, there- 40 and , massive string modes become relevant, therefore a string theory λ x 1 2 hadrons / F , being 10 T 2 ]. The scattering process in this context is the same as for the parity-preserving /R 2 35 values, being its size of order 1 r -channel diagram and the corresponding contributions to limit hadron production is suppressed, while for finite values of the four-dimensional hadronic state does not contain point-like partons. In the r s = become dominant in the OPE of two electromagnetic currents ( λ N 4 For the scalar glueball we have the following schematic behavior. On the left we A very interesting connection between the string theory and the partonic regimes was It is interesting to describe how our results for the supergravity regime (1 In the supergravity regime, i.e. when 2 T /N terplay between the anomalous dimensionsperturbative of QFT twist-two regime, operators, and whichnon-perturbative dominate the regime. for protected the operators, which are the leading onesconsider in small the values of that the virtual photonthat, strikes for a sufficiently hadron large surrounding virtual1 its photon momentum transfer other hand, for small fore the appropriate descriptionideas is were perturbative very quantum nicely field discussed theory in (QFT). [ Also, these enough scattering amplitude, which indeed goesscattered like state. (Λ Since thelarge whole string tunnelslarge from IR towards UV, this means that for sider a closed string, infour-dimensional an observer inertial the frame string itsi.e. tension tension is constant tension increases as theThis string implies approaches that the the AdS more efficient way for the string to scattered is to tunnel to large Bjorken parameter only supergravity modes takefor part smaller in the values dynamics. of analysis On is the other required. hand, proposed in the original paper by Polchinski and Strassler [ structure functions are related to ashould superstring notice theory that analysis invariable the (exp satisfies ( supergravity the regime condition the ten-dimensional Mandelstam have been described, as well as baryons as skyrmionsby [ the comparison with thefor symmetric polarized structure spin-1 functions solutions [ for the case of spin-1/2for hadrons. neutrino From DIS our within dual results it description is of not neutrino possiblemodel DIS to based is derive on to implications consider theIIA a D4D8anti-D8-branes superstring non-chiral constructed theory holographic by [ dual Sakai such and as Sugimoto the from type into account to some extent in related papers such as [ JHEP04(2018)118 , 1 ]. x 5 F (5.1) (5.2) (5.3) plays in 5 F . 3 F , -dependence for 2 . After the graviton ]) when reducing the x 3 2 − for moderate values of τ F ) 3 x F − (1 . This can be straightforwardly +1 N τ x 1 . − ) leads to a non-vanishing function of τ 2 2 −  2 F 2 N , respectively. However, this process only , = 0 since it is proportional to the Casimir -channel Chern-Simons contribution. This (supergravity description), q Λ ( t λ 0 1 x  √ F / limit it does not contribute to → – 24 – → → O regime the dominant contribution comes from a 1+2 λ N 1 λ λ − 1 1 √ x x √ √ 1 1 1 − 1 regime τ − − ], the presence of the cubic Chern-Simons interaction in regime the situation changes drastically because excited τ τ and  20 x   2 λ 2 2 2 2 2 q x < Λ √ q q / Λ Λ     2 2+2 . This is not the right place to look for 1 λ 1 1 x x x − µν x √ , and in the large ∝ ∝ ∝ / 2 W -power fall-off. In addition, the behavior of 19 2 3 1 2 /N F F F q . More specifically, in the symmetric case the relevant modes are given by 5 -channel Witten diagram which leads to contributions to the antisymmetric S s of the form 2 F As originally suggested in [ In the exponentially small- We omit further corrections from the Pomeron kernel. becomes sub-leading, i.e. it is suppressed by powers of 19 -channel Reggeized particle exchange. In the original description these modes belong to -channel tree-level Witten diagram. On the other hand, for the supergravity regime the RR vertex operators inis order due to to include thethe the role construction of that the the gaugetheory ten-dimensional fields on (described self-dual at five-form the field linear level strength in [ constructed an effective localproperties, four-point starting interaction from Lagrangian the byIn considering five-dimensional addition, symmetry SU(4) confirming gauged ourdirectly supergravity heuristic from Lagrangian results, the [ we analysis of haveThe a arrived four-point difference to type the with IIB same superstring the effective theory symmetric scattering action case amplitude. comes from the fact that one needs to consider the five-dimensional gauged supergravityof theory a is gauge field crucial,described exchange as the with it the corresponding leadsthe necessary scattering to one four-dimensional amplitude the hand, index possibility from after structure. describing two We the different have technique perspectives. in the well-known On symmetric case, we have and gives contributions to thethe structure hadronic tensor functions whichexchange, the characterize next-to-leading the order symmetric contributionis in part given terms by of of gauge center-of-mass fieldantisymmetric energy contributions exchange. scaling appear. As we have shown, it is in this context that the leading is suppressed by 1 strings must bet included as intermediatethe states. tower of The states dominant associated diagrams with are the given graviton. by This leads to the x seen from the factt that in thedominant small- part of the hadronic tensor implies the exchange of at least two on-shell particles, thus it Thus, while for theof 1 the scalar glueballkeeping under the the same Lorentz group, write down the behavior for larger values of JHEP04(2018)118 ) 2 ]. ]. 2 / / 30 1 42 (5.4) (5.5) , − λ ( and the 39 , O q = 4 SYM 36 sub-leading even when it N 1 [ ) leads to two actually grows 3 i.e. 3 ∼ F 5.5 F is not sub-leading χ From these one can ] obtaining a similar 3 F 43 20 structure function calculated 3 F does not vanish even for scalar 3 F . λ , 1 √ ]. ) 2 ]. Thus, it leads to a different shift. The − 43 1 region 25  , , φ, φ x 1 5 1 x region as in the symmetric case. However, F  , x – 25 – 5 ∼ F ) ( 0 will fulfil the Froissart bound. In order to restore x . The result we show in equation ( A ( 3 x → F x -channel Laplacian is now associated to spin-one fields as t . The saturation regime is reached when χ expansion. In the high energy limit these contributions are 2 ] one can consider a composite object built out of /N 1 does. Secondly, the last term of the exponent shows that an 2 F for extremely small values of the Bjorken parameter. 2 F 1 exchange since there are some subtleties that should be taken into account [ ≈ A very interesting question is what is the relation of the In the symmetric structure functions, at some point, the fast rising of the single- Focusing on the dependence in the Bjorken parameter, the precise calculation of the j In this context, one needs both the imaginary and the real part of the kernel. 20 dependence on the Bjorken’t parameter, Hooft the coupling, virtual but photonexpected replacing momentum the since transfer scaling in dimension thephenomenological of case implications the of are hadron glueballs developed state in they by [ both its twist, are as the same. Certain very remarkable in the present paper, i.e.baryons. using the Following string [ theorygluinos, analysis, which with might the corresponding bebaryon state. ones interpreted for We have as carried out a the corresponding holographic calculations representation in [ of a dual spin-1 inclusion of Pomeron propagators,construct build the from eikonal the phase PomeronWe kernel. think that similar features wouldin take place this for paper. the However, antisymmetric one contributionsthe should studied be cautious in performing the eikonal approximation for unitarity, it is necessary tocontributions consider in the contribution the fromdominated 1 loop by diagrams, multi-Pomeron exchange.readily As it generalized is to known, includeformula the these resumes formalism the diagrams used full by class above using of can ladder the be diagrams, eikonal where the notation. exchanged particles The lead eikonal to the opposed to the spin-2 operatorparticular considered value in is [ of thefaster same than sign, but it is smaller, which meansPomeron that exchange results when was subleading for largerinteresting values conclusions. of Firstly, inhadrons. the Furthermore, small- the firstsince it term grows in as theshift exponent appears implies that inthe the differential exponentially operator in small- the amplitude leads to This means that DIS of a charged lepton from a scalar has a non-zero antisymmetric case we find that the relevant scattering amplitude is of the form as suggested by the analysis of section 3.1. two dilaton and two graviton perturbations (with specific polarizations), whereas in the JHEP04(2018)118 . a 0 v the N also T 3 F , species of T 2 N spacetime will . On the other 2 5 N compactification of , at any 1 1 , while at high S S T isometries. This affects × 5 3 ], by considering the planar ) corrections from type IIB R W 3 0 ], where the spectrum of type 15 α , = 4 SYM at finite temperature ( 44 13 O N – ] it has been shown that the planar 5 7 , in terms of the Killing vectors 11 S ] using the black-hole embedding. In a v 47 µ [ . In [ A 2 1 S , reflecting the contribution of N = 2 × N 3 µa – 26 – is in the high-temperature phase, i.e. unconfined, h R = 1 SYM theory [ T N a five-dimensional Einstein manifold, as pointed out has a confining phase at low ]. It corresponds to 5 1 and on S 46 , at any W 1 7 1 × it has been found that, in a deconfined phase of a strongly ]. Also, mild IR deformations of the AdS S S 3 2 × 45 S × F 3 3 S , being R 5 and on W 1 ]. , following the same scaling arguments which hold for the calculations T 2 F × 48 N , 5 ] it has been obtained the corresponding 47 are imposed for fermions, while scalars get mass at one-loop level, rendering 48 1 conifold background for Killing vectors. S = 4 SYM plasma, they scale with 1 , 5 1 N S T . For instance, one in principle could carry out very similar calculation in the as considered by Witten [ × 5 ] the calculation will only be modified by the effects of In the cases of A very interesting situation can be analyzed by studying an As a general remark, whenever one considers the holographic dual calculation of DIS 5 5 1 W . Notice that supersymmetry is completely broken since anti-periodic boundary condi- A Conventions for the KillingIn vectors this on appendix weand describe the the explicit relation between the SU(4) Gell-Mann matrices We thank Carlos a N´u˜nezfor work critical has been reading supported on byICET) this the and the National manuscript National Scientific and Agency Research(ANPCyT-FONCyT) for Council comments. Grant the PICT-2015-1525, of Promotion and Argentina of the This (CON- PIP-UE Science CONICETde and Grant Technology nueva B´usqueda of f´ısica. Argentina should scale with carried out in [ Acknowledgments gluons. The same scaling occurs for all the structurecoupled functions. addition, in [ superstring theory. One expects that for the YM theory on YM theory at finite limit of the YMtheory theory becomes on unconfined, thereforehand, its the free YM theory energy on istherefore proportional its to free energy always scales with AdS T tions along a (non-supersymmetric) Yang-Mills theory. In the planar limit Witten has considered the IIB supergravity is knownonly [ change the overall normalizationNotice constant that coming in from all these thein cases, AdS-field the as wave-functions. fundamental well representation as of for the probeslimit gauge flavor of group Dp-branes [ the emulating matter SYM fields This (or is YM) expected theory, since one we only always consider obtains the structure confining functions phase scaling of as these theories. processes in AdS in [ the parametrization of theof graviphoton, AdS JHEP04(2018)118 (A.6) (A.4) (A.5) (A.1) (A.2) (A.3) 3) it is symbol ,
≥ , ). ABC [35] N      d K A.5 3 − . + 2 ) symbol appears for } C [26] . Now, let us consider ] 1 0 00 0 1 00 0 0 10 0 0 0 K ,T . ij ABC , [      B d − C K T 6 AB T which appears in the anti- , { δ 1 √ [14] 6 A 7 . The Killing vectors are the 2 T 6 K 4 traceless hermitian matrices. R ABC 6 3 ABC = × if d 3 π ,..., 15 i √ − 2 components. The only non-vanishing ] = = 2 Tr( = 1 ,T B , where for example C = ) = . }      5 = 4 SYM theory, for the antisymmetric 33 8 ,T
d S ABC 15 A K ( 21 6. T N 2 0 [35] [ , i, j ab , d √ − i ↔ g K . For example, the precise mapping for the / ∂ , whose generator is generally associated with 8 6 ,..., ]) b B j , – 27 – R + C x 1 0 00 1 0 00 0 0 0 0 0 0 K = 1 matrices satisfy the orthonormality condition and = 1 = 4 SYM theory, and the DIS of AB a ,T − A [26]      δ ,T 15 T B j , A K 1 2 3 K N ∂ T
; SU(4) 5 i [ 33 1 √ A S d x + A 2 g [26] T ⊂ T ) = into the Euclidean space R { = √ K B [14] R = ] 5 T Ω 8 ij Tr ( + S K [ 5 A i 3 and d − 2 T K ,T √ 5 [14] − / S 6 K      Tr( i Z electromagnetic = √ i = 1 = 1 0 0 = 2 338 ABC 15 d f − 3 K 0 0 00 0 0 0 0 0 1 0 0 0 K are the completely antisymmetric structure constants. For SU( ↔      generators is ]. In gauge/gravity duality applications, the electromagnetic current in general ↔ are Cartesian coordinates on R 8 1 2 15 3 T – i T T is the R-symmetry group of ABC 6 = x f ]. Thus, in the R 3 T On the other hand, in terms of the gauge/gravity duality the R-symmetry is realized The Lie algebra of SU(4) describes the full set of 4 20 The normalization has a minus sign due to the imaginary unit included in equation ( [ 21 3 The resulting Killing vectors are normalized as diagonal context, one has a differentthe basis given canonical by embedding the 15 of rotation Killing generators vectors where T structure functions we are onlycomponents interested in are the as the isometry group of the five-sphere, SO(6), which is isomorphic to SU(4). In this SU(4) example in the anomalyappears of in the three-point front function oftheory of the [ the Chern-Simons R-currents. interactionis The in modeled the by action gauging of a the U(1) dual gravitational where also useful to considercommutations. a In completely terms symmetric of traces symbol of the generators, these objects are given by are the only diagonalthe elements. commutation The relations The canonical basis given is by JHEP04(2018)118 (B.1) (B.2) (A.7) (A.8) (A.9) , , which ]. In the [35] DB C K 27 , D 2) C B
i/ C
T M , ] Γ M in terms of the five- 0 = ( is non-vanishing for , Γ kk 3 . 1 C [ e F − BA AC C e K K ABC , AC d d K C ∂ C −C −C
d ] (which of course only takes 33 0 ∂ N , d = = = . 0 jj Γ B e [ c C c C B kk 0 K K is proportional to the eigenvalue K AB AB T A AB b b 3 ) jj
C
3 0 ∂ ∂ ζ 5 raises and lowers the indices of the ii ] M M F 0 A a ε M M ABC Γ C ii [ (Γ a Γ d ··· K 1 ≡ C i K 2 R 4 M ] 0 ζ Γ abcde kk = C ε ][ abcde 0 Tr B ε d 5 Ω jj 5 ][ M K Ω – 28 – 0 d c 5 1 MN ··· ii 5 , ∂ [ d 1 h ˜ S d 5 (5) A C M b . , Z 5)-plane. This means that S 10 A , F Z K κ B
6 a [35] i 2 A 6 = R ∂ M i
K 3 3 − R Γ π T M 3 3 2 = IIB Γ π AB 1 , abcde ζ 4 − ε BC C A = of the spherical harmonic that defines the dilaton solution with δ C − ) becomes an integral expression for = Dirac indices. We have the following properties 0 (3) closed = = = [35] A.7 B kk A ABC 0 1 Q d AB − BC jj DB 0
C C 2) ] as follows ii and / 1 1), one has the identity ε M D 34 − AB Γ C  A C ) = (1 M ], appendix B. The conjugation matrix C (Γ 33 Q (RR,RR,NSNS), involving two RR-vertex operators and one NS-vertex operator. is given by , being C AC (3) B ζ basis, equation ( C 33 A A d amplitude }
Note that in this language the relevant combination is We can also write the additional identity A M K Γ Now, we have to calculatereference the [ trace of twelvegamma gamma matrices. matrices. We The follow corresponding the indices notation in of the definition of the gamma matrices are tude This is given in [ where B Gamma matrix algebra in the three-point closedThe starting string point is scattering the flat-space three-point type IIB superstring theory scattering ampli- means that our finalQ result ≡ for the structurerespect function rotations onhadrons the charged internal with (3 respect to familiar notation. which is usefull in the computation of the effective action in subsection 3.2. dimensional Levi-Civita symbol andgiven the by Killing vectors (together with their derivatives) This allows one to rewrite the Chern-Simons term in the action in the more non-zero values which leads to{ the expression for the Chern-Simons interaction given in [ Defining a new symmetric symbol as JHEP04(2018)118 , ]. (B.3) (B.4) (B.5) JHEP Nucl. = 2 ] . SPIRE , ,
IN N N , ][ Γ .  5 2 ]. (1998) 253 1 N − DB 2 Γ ·F C 1 is traceless. supergravity ··· D F SPIRE 1 C h ]. IN
N 1 = 5 MN Γ ][ ]. M g arXiv:1601.01627 D M ··· [ 5 − Γ hep-th/9804058 supergravity in five-dimensions SPIRE 5 [
M IN SPIRE = 8 corrections to deep inelastic M Γ N 2 [ Correlation functions in the Γ IN = 8 N . [ /N ·F N C ··· 1 AC 1 B M (2016) 113 3 The mass spectrum of chiral (1999) 96 S M −C F B Γ hep-th/9907106 04 Gauged Adv. Theor. Math. Phys. = + A (1986) 598 [ , Gauged Compact and noncompact gauged supergravity R B (1985) 389 N 3 Tr Towards A 5 − B 546  N ]. ·F JHEP = T ··· – 29 – , B 272
1 M D 32 2 5 3 N BC F M (1999) 181 F SPIRE S 5 ··· 5 1 IN ]. M  Deep inelastic scattering and gauge/string duality AB M ··· ][ Nucl. Phys. 1 Γ R , ), which permits any use, distribution and reproduction in . B 562 2 M 1 MN  Nucl. Phys. Phys. Rev. h ]. F SPIRE , , IN 5 10 A note on the chiral anomaly in the AdS/CFT correspondence and [ S 1 MN κ , h i SPIRE 15 1 10 IN (1985) 268 − BA κ − CC-BY 4.0 ][ i Nucl. Phys. 2 = , −C hep-th/0209211 correspondence − This article is distributed under the terms of the Creative Commons [ (1985) 460 = = +1 B 154 Anti-de Sitter space and holography d 1 (3) closed − AB ]. ]. Therefore, one obtains the following contractions: supergravity on A C B 259 correction 32 (3) closed /AdS d 2 A (2003) 012 = 10 SPIRE /N IN hep-th/9802150 1 scattering from the gauge/gravity duality [ theories in five-dimensions CFT [ D Phys. Lett. Phys. 05 Next we can calculate the last trace using similar arguments as in appendix A of Then, the scattering amplitude becomes A. Bilal and C.-S. Chu, D. Jorrin, N. Kovensky and M. Schvellinger, M. L.J. G¨unaydin, Romans and N.P. Warner, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, E. Witten, M. Gunaydin, L.J. Romans and N.P. Warner, M. Pernici, K. Pilch and P. van Nieuwenhuizen, J. Polchinski and M.J. Strassler, H.J. Kim, L.J. Romans and P. van Nieuwenhuizen, [8] [9] [5] [6] [7] [3] [4] [1] [2] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. reference [ where omitted indices are contracted. The last term vanishes since matrices Other useful properties are listed below also we use the fact that for any two matrices which are themselves products of gamma JHEP04(2018)118 ] , ]. ] , ] D SPIRE (1997) 5 332 ] IN from = 4 B 562 ][ x N (1971) 980 D 56 ]. Phys. Rev. Lett. and arXiv:1406.7576 , D 3 ]. [ 5 arXiv:1609.01202 Phys. Rept. [ , SPIRE hep-th/9204004 Nucl. Phys. ]. AdS [ , IN Phys. Rev. SPIRE hep-th/9802203 (1997) 399] , ][ +1 [ IN ]. d arXiv:1311.2601 ][ SPIRE Phys. Rev. [ ]. ]. SYM: where is spin at strong 281 , IN (2016) 003 (2014) 075018 ][ (1992) 75 SPIRE The Pomeron and gauge/string ]. 12 = 4 IN Conformal symmetry and differential SPIRE SPIRE (1998) 153 ][ N 218 IN IN D 90 supergravity on (2014) 166 SPIRE ][ ][ JHEP ]. IN , = 8 01 ][ B 532 Erratum ibid. arXiv:1506.01385 ]. N [ Deep inelastic scattering off scalar mesons in [ Hadron structure functions at small Deep inelastic scattering cross sections from the arXiv:1507.07942 Deep inelastic scattering from holographic Deep inelastic scattering structure functions of The theory and phenomenology of polarized deep SPIRE [ On Phys. Rev. IN JHEP Consistent type IIB reductions to maximal arXiv:1112.1459 – 30 – , Annals Phys. SPIRE [ ][ ]. , 1, IN (1995) 1 ]. Polarized DIS in arXiv:1412.6509 ≥ ][ [ Nucl. Phys. -brane system ]. f 7 , Hard scattering and gauge/string duality arXiv:0912.4333 arXiv:0905.2493 SPIRE N 261 (2015) 009 D [ [ (2015) 065004 IN 3 hep-th/0603115 SPIRE [ (2012) 103 ][ D IN 12 SPIRE Nonforward Compton scattering in AdS/CFT correspondence ]. Structure functions in deep inelastic scattering from gauge/string ][ IN 02 D 92 (2015) 001 Spin physics and polarized structure functions ][ hep-th/0607237 Photon and dilepton production in supersymmetric Yang-Mills plasma JHEP SPIRE (2009) 007 05 [ Phys. Rept. , IN hep-th/0109174 , (2007) 005 Graviton and gauge boson propagators in AdS JHEP 08 (2010) 035008 ][ [ , 12 Pion and rho structure functions from lattice QCD Conformal symmetry and three-point functions JHEP Phys. Rev. , hep-th/9902042 , D 81 JHEP [ (2006) 015 hep-ph/9810270 , expansion from the ]. ]. ]. ]. ]. [ JHEP 12 , hep-lat/9703014 /N [ 1 (2002) 031601 SPIRE SPIRE SPIRE SPIRE SPIRE hep-ph/9501369 IN IN IN IN IN Phys. Rev. supergravity (1999) 330 88 duality inelastic scattering [ (2000) 1 JHEP regularization of the three gluon[ vertex coupling? superconformal Yang-Mills theory [ [ gauge/string duality the [ string theory 2743 duality beyond single-hadron final[ states spin-one hadrons holographic spin-1 hadrons with A. Baguet, O. Hohm and H. Samtleben, E. D’Hoker et al., J. Polchinski and M.J. Strassler, R.C. Brower, J. Polchinski, M.J. Strassler and C.-I. Tan, J.-H. Gao and B.-W. Xiao, B. Lampe and E. Reya, S. Caron-Huot et al., Y. Hatta, T. Ueda and B.-W. Xiao, M. Anselmino, A. Efremov and E. Leader, E.J. Schreier, D.Z. Freedman, G. Grignani, K. Johnson and N. Rius, D. Jorrin, M. Schvellinger and N. Kovensky, S. Ferrara, C. Fronsdal and A. Zaffaroni, E. Koile, N. Kovensky and M. Schvellinger, C. Best et al., E. Koile, N. Kovensky and M. Schvellinger, E. Koile, S. Macaluso and M. Schvellinger, E. Koile, S. Macaluso and M. Schvellinger, J.-H. Gao and Z.-G. Mou, [27] [28] [24] [25] [26] [22] [23] [20] [21] [18] [19] [16] [17] [13] [14] [15] [11] [12] [10] JHEP04(2018)118 ] ]. ]. , ]. ]. JHEP , ]. ] , SPIRE SPIRE Phys. SPIRE 114 IN IN , IN B 475 ][ ][ ]. SPIRE ][ SPIRE IN reduction of IN ][ ][ SYM plasma at Prog. Theor. hep-th/0611122 SPIRE , [ IN SO(6) = 4 ][ ]. Nucl. Phys. ]. Some tree-level string N , arXiv:1312.4703 ]. ]. [ Prog. Theor. Phys. SPIRE (2007) 019 arXiv:0908.2301 arXiv:0710.2148 SPIRE , arXiv:0912.4704 (2000) 066001 IN [ [ [ IN SPIRE hep-th/0003103 08 Consistent ][ SPIRE ][ [ IN Regge limit of R-current IN Eikonal approximation in arXiv:1507.02172 ][ D 61 ][ (2014) 408 Spectrum of type IIB supergravity on [ ]. JHEP arXiv:1007.2259 , [ (2010) 153 (2008) 026 String-gauge dual description of deep (2010) 012 ]. ]. SPIRE (2000) 275 B 884 Deep inelastic scattering from polarized 01 04 IN , in preparation. (2015) 010 Phys. Rev. ][ B 830 SPIRE , hep-th/9807080 ]. SPIRE On the Pomeron at large ’t Hooft coupling On the eikonal approximation in AdS space IN [ 12 IN (2010) 051 JHEP arXiv:0904.2870 B 586 Sphere-level Ramond-Ramond couplings in – 31 – JHEP ][ ]. , [ ][ hep-th/9803131 , ]. arXiv:0710.5297 Deep inelastic scattering at strong coupling from Deep inelastic scattering off a [ 11 [ SPIRE Nucl. Phys. SCFT’s IN Holographic current correlators at finite coupling and JHEP , SPIRE , ][ SPIRE Nucl. Phys. IN (1998) 199 = 1 Superstring scattering from D-branes JHEP IN , Superconformal field theory on three-branes at a Calabi-Yau ][ Saturation in deep inelastic scattering from AdS/CFT , N Nucl. Phys. ][ from string theory x Polarized deep inelastic and elastic scattering from gauge/string (1998) 505 ]. , More on a holographic dual of QCD Low energy hadron physics in holographic QCD (2008) 063 arXiv:0804.1562 x 5 (2009) 015025 2 [ S B 536 01 hep-th/0412141 arXiv:0707.2408 [ SPIRE [ D 80 IN ][ JHEP arXiv:0710.4378 , [ hep-th/0507073 hep-th/9603194 Nucl. Phys. [ (2008) 096010 Anti-de Sitter space, thermal phase transition and confinement in gauge theories (2005) 843 1: Predictions on [ , (2009) 050 1 hadrons at small- ]. ]. T Phys. Rev. 2 03 , 113 / × D 78 1 5 (2009) 092 SPIRE SPIRE hep-th/9905226 IN IN Adv. Theor. Math. Phys. strong coupling scattering off a supersymmetric plasma singularity AdS [ (2005) 1083 Rev. spin- inelastic scattering at small- Phys. gauge/string duality: the saturation line JHEP 03 amplitudes in the NSR formalism duality Ramond-Neveu-Schwarz formalism [ (1996) 193 correlators in AdS supergravity AdS/CFT: from shock waves to[ four-point functions type IIB supergravity on Y. Hatta, E. Iancu and A.H. Mueller, B. Hassanain and M. Schvellinger, I.R. Klebanov and E. Witten, A. Ceresole, G. Dall’Agata, R. D’Auria and S. Ferrara, E. Witten, L. Cornalba and M.S. Costa, N. Kovensky, G. Michalski and M. Schvellinger, R.C. Brower, M. Djuric, I. Sarcevic and C.-I. Tan, T. Sakai and S. Sugimoto, T. Sakai and S. Sugimoto, R.C. Brower, M.J. Strassler and C.-I. Tan, R.C. Brower, M.J. Strassler and C.-I. Tan, K. Becker, M. Becker, I.V. Melnikov, D. Robbins and A.B.J.-H. Royston, Gao and B.-W. Xiao, Y. Hatta, E. Iancu and A.H. Mueller, H.R. Bakhtiarizadeh and M.R. Garousi, M.R. Garousi and R.C. Myers, L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, M. Cvetiˇc,H. L¨u,C.N. Pope, A. Sadrzadeh and T.A. Tran, J. Bartels, J. Kotanski, A.M. Mischler and V. Schomerus, [47] [48] [44] [45] [46] [42] [43] [39] [40] [41] [37] [38] [34] [35] [36] [32] [33] [30] [31] [29]