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Photon Structure Functions at Small X in Holographic Physics Letters B 751 (2015) 321–325 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Photon structure functions at small x in holographic QCD ∗ Akira Watanabe a, , Hsiang-nan Li a,b,c a Institute of Physics, Academia Sinica, Taipei, 115, Taiwan, ROC b Department of Physics, National Tsing-Hua University, Hsinchu, 300, Taiwan, ROC c Department of Physics, National Cheng-Kung University, Tainan, 701, Taiwan, ROC a r t i c l e i n f o a b s t r a c t Article history: We investigate the photon structure functions at small Bjorken variable x in the framework of the Received 20 February 2015 holographic QCD, assuming dominance of the Pomeron exchange. The quasi-real photon structure Received in revised form 9 October 2015 functions are expressed as convolution of the Brower–Polchinski–Strassler–Tan (BPST) Pomeron kernel Accepted 26 October 2015 and the known wave functions of the U(1) vector field in the five-dimensional AdS space, in which the Available online 29 October 2015 involved parameters in the BPST kernel have been fixed in previous studies of the nucleon structure Editor: J.-P. Blaizot functions. The predicted photon structure functions, as confronted with data, provide a clean test of the Keywords: BPST kernel. The agreement between theoretical predictions and data is demonstrated, which supports Deep inelastic scattering applications of holographic QCD to hadronic processes in the nonperturbative region. Our results are also Gauge/string correspondence consistent with those derived from the parton distribution functions of the photon proposed by Glück, Pomeron Reya, and Schienbein, implying realization of the vector meson dominance in the present model setup. Linear collider © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction photon can be regarded as a hadron rather than a pointlike object. Effective models are then appropriate for studies of the photon A photon is a fundamental particle, instead of a nonpertur- structure functions in this region. bative composite like hadrons. However, an energetic photon can The holographic QCD based on the AdS/CFT correspondence fluctuate into quark–antiquark pairs, which further radiate gluons, [12–14] is one of the effective models. The AdS/CFT correspon- in a hard scattering process. Hence, investigations on the pho- dence implies that one can analyze a gauge-theory process at ton structure have served as alternative tests of perturbative ap- strong coupling in the ordinary Minkowski space by means of proaches to QCD over recent decades. The authors in Refs. [1,2] the corresponding gravity theory at weak coupling in the higher first discussed the feasibility of the electron–photon deep inelas- dimensional AdS space. Since Maldacena first proposed this corre- tic scattering (DIS) and estimated its cross section, whose mea- spondence, its applications to QCD processes have been attempted surements, such as those conducted at the LEP, provided valuable intensively [15–22], in which the QCD scale QCD is introduced information of the photon structure functions. After Witten first and the conformal invariance is broken. In particular, phenomeno- evaluated the leading-order QCD corrections [3], many elaborated logical applications in hadron physics to, e.g., mass spectra, form higher-order calculations based on perturbative QCD had been per- factors, high-energy scattering processes, and so on, are quite suc- formed [4–10], and the next-to-next-to-leading-order results were cessful. available [11]. The consistency between experimental data and the- In this letter we will investigate the photon structure functions oretical results has strongly supported the reliability of the pertur- at small x, adopting the Pomeron exchange picture in the frame- bative calculations in QCD. The picture of the photon structure be- work of the holographic QCD. A Pomeron is considered as a color comes very different in the kinematic region with a small Bjorken singlet gluonic state composed of multi-gluon exchanges. It has variable x. A photon can fluctuate into vector mesons, so hadronic been known that cross sections for various high-energy scatter- contributions are not negligible to electron–photon DIS as x < 0.1 ing processes are well described in this picture [23]. A Pomeron in general. The hadronic component dominates as x < 0.01, and a in QCD corresponds to a reggeized graviton in the gravity theory in the five-dimensional AdS space. The authors in Ref. [24] evalu- ated the graviton exchange, and proposed the Brower–Polchinski– * Corresponding author. E-mail addresses: [email protected] (A. Watanabe), Strassler–Tan (BPST) Pomeron kernel. The elaborated analysis in [email protected] (H.-n. Li). Ref. [25–30] has indicated that applications of this kernel to the http://dx.doi.org/10.1016/j.physletb.2015.10.069 0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 322 A. Watanabe, H.-n. Li / Physics Letters B 751 (2015) 321–325 and suggests the application of the Pomeron exchange picture to this process. The photon structure functions in Eq. (1) are given in the five- dimensional AdS space by [27,28], 2 3/2 2 γ 2 αg0ρ Q (i) 2 2 F (x, Q ) = dzdz P (z, Q )P24(z , P ) i 32π 5/2 13 2 × (zz )Im[χc(W , z, z )], (3) Fig. 1. Deeply inelastic electron–photon scattering process, where the bubble repre- with i = 2, L, where the adjustable parameters g2 and ρ con- sents a parton density of the target photon. 0 trol the magnitude and the energy dependence of the structure functions, respectively. The imaginary part of the BPST Pomeron nucleon structure functions at small x led to results well consis- kernel [24,28] tent with data [31–33]. √ Following the same formalism, we will express the photon 2 (1−ρ)τ −[(log2 z/z )/ρτ] Im[χc(W , z, z )]=e e / τ , (4) structure functions at small x as convolution of the BPST kernel with the wave functions of the incident and target photons in with the conformal invariant the 5th dimension. Most parameters in the BPST kernel have been = 2 fixed in the previous studies of the nucleon structure functions [32, τ log(ρzz W /2), (5) 33]. The known five-dimensional U(1) vector field will be adopted arises from the single-Pomeron/graviton exchange. for the photon wave functions, which couple to leptons at the ul- The overlap functions P13(z) and P24(z ), describing the density traviolet (UV) boundary. That is, we do not need approximations distributions of the incident and target particles in the AdS space, or models to describe the incident and target particles here, in respectively, obey the normalization condition dzPij(z) = 1in contrast to the case of the nucleon DIS. Therefore, our study of the on-shell case. They depend on the particle virtualities Q 2 and the photon structure functions involves only a single free param- 2 (L) 2 P in the present study with off-shell photons. P (z, Q ) de- eter, which appears as an overall coefficient for the cross section. 13 notes the wave function of the longitudinally polarized photon, The model dependence is then reduced, and the predicted pho- 2 (2) 2 ton structure functions, as confronted with data, serve as a clean and P24(z , P ) takes the form the same as P13 (z, Q ). The mass- check of the validity of the BPST kernel. It will be demonstrated less 5D U(1) vector field, satisfying the Maxwell equation in the γ 2 five-dimensional AdS background spacetime, can be identified as that our results for the photon structure function F2 (x, Q ) are in agreement with the LEP data, though available data at small x the physical photon at the UV boundary. We adopt the wave func- are limited. In addition, our results are also consistent with those tion of this field for the overlap functions, which are then written as [36] obtained from the parton distribution functions (PDFs) of the pho- ton proposed by Glück, Reya, and Schienbein [34]. Because they (2) P (z, Q 2) = Q 2z K 2(Qz) + K 2(Qz) , (6) included the hadronic component in the photon PDFs, which dom- 13 0 1 inates at small x, this consistency may imply realization of the (L) 2 = 2 2 P13 (z, Q ) Q zK0 (Qz). (7) vector meson dominance in the present setup. Our predictions for the photon structure functions at very low x can be tested at fu- Both K0(Qz) and K1(Qz), the modified Bessel functions of the ture linear colliders. second kind, diverge at the origin and vanish at large Qz, consis- tent with the fact that a photon is a non-normalizable mode, and 2. Kinematics and model setup can penetrate into the larger z, i.e., infrared (IR) region at low Q . Namely, Eq. (6) exhibits the feature that a quasi-real photon prop- The electron–photon DIS eγ → eX is schematically shown in agates into the bulk of the AdS space, while keeping a substantial Fig. 1, where q and p are the four-momenta of the probe and tar- component in the UV region even at small x. −5 −2 2 get photons, respectively, and W represents the invariant mass of In the considered region with 10 ≤ x ≤ 10 and 1GeV ≤ 2 2 the hadronic final state. The associated differential cross section is Q ≤ 10 GeV , nonperturbative hadronic contribution to the struc- γ 2 expressed in terms of the two structure functions F2 (x, Q ) and ture functions may become important.
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