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The Virtual Photon Structure Functions and AdS/QCD Correspondence

We study the virtual photon structure functions from gauge/string duality. If the Bjorken variable x is not small, supergravity approximation becomes good in dual string theory. We calculate the virtual photon structure functions at large 't Hooft coupling in a moderate x-region and determine x-behavior of the structure functions. We also show that the Callan-Gross relation F_L=0 is satisfied to a good approximation in gravity calculation.


Introduction
The structure functions are important objects in QCD. For example, the nucleon structure functions control the cross section of deep inelastic scattering and they are related to the parton densities inside the initial hadrons. The nucleon structure functions however are defined by nucleon matrix elements of the electro-magnetic currents, so they cannot be calculated in the perturbative method. We can only calculate energy scale evolution by perturbation theory.
AdS/CFT correspondence relates N = 4 super Yang-Mills theory at large 't Hooft coupling in four dimensions to weakly-coupled string theory in AdS 5 × S 5 [1]. The authors of [2] have studied dual gravity description of nucleon structure functions and have calculated nucleon structure functions. This is called the hard-wall model. In this model, the AdS space has a cut-off at infrared region and conformal invariance is broken. This cut-off scale corresponds to the infrared mass scale of the gauge theory. At small coupling, probe photons scatter off partons inside hadrons. But at large 't Hooft coupling, the situation is completely different; probe photons scatter off entire hadrons and do not destroy hadronic states. Hadronic states are dual to (massless) string states in AdS space and we can calculate structure functions from supergravity interactions. Usually, the leading-twist operators have the leading contribution to OPE of currents correlators at small coupling. But at large 't Hooft coupling, these operators have large anomalous dimensions [3] and do not dominate in OPE.
In this article, we consider the virtual photon structure functions and study their property from bulk dynamics in AdS space. The photon structure functions are defined by the absorptive part of the four quark currents correlator. Naively, they can be obtained by replacing the initial hadronic states in hadronic tensors by the target photon states. So their properties are similar to nucleon structure functions; we can apply standard OPE technique and calculate the anomalous dimensions of twist-two operators and coefficient functions perturbatively. A crucial difference of the virtual photon structure functions from the nucleon structure functions is that one can determine x-behavior of the virtual photon structure functions perturbatively. But from theoretical view point, it is still interesting to consider QCD objects at strong coupling which are observed at small coupling [4].
photon whose momentum is q µ (p µ ) the probe (target) photon. If the invariant mass of the target photon is close to on-shell p 2 ≃ 0, then vector meson dominance is realized. This process is rather well described by vector meson-photon coupling than by photon-photon scattering. Dual gravity descriptions of such interactions are studied in [9], [11]. If the invariant mass of the target photon is far off-shell Λ 2 QCD ≪ p 2 , this process is described by virtual photon-virtual photon scattering.
We define the tensor T µναβ as where the J µ are quark currents which couple to the photons. The structure tensor of virtual photons is defined by the absorptive part of T µναβ .
This tensor has eight independent components. If the initial photon states are unpolarized, we average the target photon helicities and we obtain W µν can be decomposed into two structure functions F 1 (x, q 2 , p 2 ) and F 2 (x, q 2 , p 2 ) as . The structure functions F i (x, q 2 , p 2 ) are called the real photon structure functions in the region p 2 ≃ 0 and are called the virtual photon structure functions in the region Λ 2 QCD ≪ p 2 . In order to apply OPE for the product of the quark currents which couple to probe photons, the condition p 2 ≪ q 2 is required. This means that the distance between quark currents correlators which couple to the probe photons is much shorter than the others. So in the region Λ 2 QCD ≪ p 2 ≪ q 2 , we can calculate the virtual photon structure functions perturbatively at small coupling.
Inserting the final hadronic states X |X X| = 1, we can represent the structure tensor W µν as where we denote P X the momentum of the final state hadron X andJ µ is the Fourier transformation of J µ .

Virtual photon structure functions in AdS/QCD
In this section, we calculate the virtual photon structure functions from supergravity interaction. Scattering process is caused by gauged U(1) R-symmetry currents J µ . We treat the process to the lowest order in U(1) coupling constant and to all orders in strong coupling constant. Indices m, n, · · · denote the AdS 5 space and µ, ν, · · · denote four dimensional Minkowski space. Indices m, n, · · · are raised with the curved metric g mn and µ, ν, · · · are raised with η µν . We define q 2 = η µν q µ q ν and q = √ q 2 . In the hard-wall model, the AdS space has a infrared cut-off at r 0 = ΛR 2 and Λ corresponds to the infrared mass scale of the gauge theory. The metric of AdS 5 × W is where R = (4πg 2 N) 1/4 α ′ is the AdS radius and W is the five-dimensional internal space which has a U(1)-isometry. The final state hadron is dual to a string state in ten-dimensional space [2]. We consider this string state is given by the dilaton. The dilaton wave function Φ X is given by where c X is a dimensionless constant. J ∆−2 is a Bessel function and ∆ is related to the dilaton mass M 2 = ∆(∆ − 4)/R 2 in AdS 5 , and Y (Ω) is the wave function of dilaton in W direction which is normalized The Kaluza-Klein gauge field A m couples to the R-current at AdS boundary [6], [7]. The boundary condition of gauge field is imposed by where ǫ µ denotes the polarization vector of the gauge field in four dimensions. The bulk gauge field satisfies the Maxwell equation in the AdS 5 space. We take Lorentz gauge in five dimensions, then the nonnormalizable mode of gauge field are where K 0 , K 1 are modified Bessel functions of second type. Field strength tensors of A µ and A r are The insertion of R-symmetry currents excites the metric perturbation δg ma = A m (y, r)v a (Ω) with Killing vector v a (Ω) associated to a U(1)-isometry of W . The correlation function of two R-symmetry currents and a scalar corresponds to the supergravity interaction [8] From eq.(11), eq.(12) becomes where C = √ g W Y (Ω)v a v a . This factor does not affect x-behavior of F 1 (x, q 2 , p 2 ) and F 2 (x, q 2 , p 2 ), so we do not write it explicitly, then = c X Cs 1/4 Λ 1/2 − 2 (q · p)(ǫ(q) · ǫ(p)) − (q · ǫ(p))(p · ǫ(q)) ×qpA(x, q 2 , p 2 ) + 2 (ǫ(q) · ǫ(p))q 2 p 2 −(p · ǫ(q))(p · ǫ(p))q 2 − (q · ǫ(q))(q · ǫ(p))p 2 +(q · p)(q · ǫ(q))(p · ǫ(p)) B(x, q 2 , p 2 ) .
In the above equation, A(x, q 2 , p 2 ) and B(x, q 2 , p 2 ) are defined by We have to sum up the final-state hadrons to get the structure tensor W µν . In the hardwall model, the space has a cut-off at r 0 = ΛR 2 , so the zeros of the Bessel function eq.(7) are at M n = nπΛ and the on-shell condition of the dilaton becomes Using eq.(14) and eq.(16) and taking average for the polarization vectors of the target photon, we get the structure tensor eq.(5) Hence we have the following structure functions We stress that the condition p 2 ≪ q 2 is not used in the supergravity calculation of the structure functions. This is quite different from pQCD analysis in which the condition p 2 ≪ q 2 is necessary for applying OPE to the product of quark currents.
We analyze F i (x, q 2 , p 2 ), (i = 1, 2) in the kinematical region p 2 ≪ q 2 . K i (qR 2 /r) damps exponentially in small r region. So the region which contributes to integrals in A(x, q 2 , p 2 ) and B(x, q 2 , p 2 ) is qR 2 ≤ r.
To begin with, we evaluate F 1 (x, q 2 , p 2 ). We show later that the dominant contribution to F i (x, q 2 , p 2 ) comes from A(x, q 2 , p 2 ) and we can neglect the second term in eq.(18) for p 2 ≪ q 2 .
When p 2 ≪ q 2 is satisfied, we can also use the leading behavior K 1 (pR 2 /r) ≃ r/pR 2 . Then, with an approximation 1/Λ ∼ ∞, this integrals can be performed and we get where 2 F 1 (a, b; c; z) is a hypergeometric function. From the second line to the third line we use the identity 2 F 1 (a, b; c; z) = (1 − z) −a 2 F 1 (a, c − b; c; z/(z − 1)). Then F 1 becomes One can see that F 1 (x, q 2 , p 2 ) increases with x in the small x-region and decreases near x max = 1/(1 + p 2 q 2 ). The structure functions should vanish at x max and actually eq.(21) satisfies this requirement. From eq.(21), we have found that F 1 (x, q 2 , p 2 ) scales as F 1 (x) in the limit p 2 /q 2 → 0. This is a crucial difference between the photon structure functions and the nucleon structure functions in gravity calculation. If we define ∆ i as the conformal dimension of the dilaton or the dilatino which is dual to the initial state hadron, the momentum dependence of the nucleon structure functions always behave (Λ/q) 2∆ i −2 [2]. Therefore the nucleon structure functions do not have the scaling property in general ∆ i . The nucleon structure functions have the scaling property when the conformal dimension is special value ∆ i = 1 [11].
Next, we evaluate F 2 . We transform w = qR 2 /r in eq.(15), B(x, q 2 , p 2 ) can be written as When the condition p 2 ≪ q 2 is satisfied, we can use the asymptotic form K 0 (wp/q) ≃ log(2q/wp). Then where we used log x = lim ǫ→0 (x ǫ − 1)/ǫ. Therefore B(x, q 2 , p 2 ) 2 is a subleading contribution to F 2 . From the expression for F 2 , we calculate the longitudinal structure function defined as F L = F 2 − xF 1 .
From the above equation, in the limit p 2 /q 2 → 0, one can see that the Callan-Gross relation F L = 0 holds in gravity calculation. In QCD, the Callan-Gross relation implies the partons involved in the scattering process are spin 1/2 particles. This relation is weakly broken by higher order corrections in real QCD. In our calculation, the Callan-Gross relation is weakly broken by the effect of gauge field in radial direction.

Summary and discussions
In this article, we have calculated the virtual photon structure functions in the hard-wall model. The structure functions F i (x, q 2 , p 2 ) (i = 1, 2) we obtained possess the following properties which we have in QCD.
1) F i 's increase at small x-region, decrease at large x-region and vanish at x = x max .
2) F L is small compared to F i 's. It goes to zero when the limit p 2 ≪ q 2 as q 2 → ∞. It seems peculiar that the Callan-Gross relation holds in gravity calculation, although there exists no spin 1/2 particles in our case. But as mentioned in [9], the interaction ΦF 2 in eq.(12) is the low-energy effective action describing a higgs decay into two gammas through a heavy quark triangle loop [13]. Since the heavy quark is a spin 1/2 particle , this might be a reason why the Callan-Gross relation holds in our calculation.
We simply comment some extension of the virtual photon structure functions at strong coupling. In the hard-wall model, mass square of dilatons and radial numbers exhibit quadratic relation, but actually relations mass square between radial number is linear (Regge trajectory). A background which reproduces Regge trajectory is proposed in [10]. This is called the soft-wall model. Deep inelastic scattering in the soft-wall model is studied and the leading order results in soft wall model is same as the one in the hard-wall model [12]. We treat the virtual photon structure functions in not so small x-region. At small x-region, we have to take into account excited string states [2], [14]. Then, we cannot use the insertion of dilaton states and have to treat four currents correlator directly. The analysis would become much more complicated. It is interesting to investigate the virtual photon structure functions in such situations.