Study of hadron scattering at high-energy began at about early 1960s with the advent of the first high-energy accelerators. In this era, the main interest was on elastic scatterings, or in general, exclusive processes in which the kinematics of all particles in the final state is reconstructed. As long as we consider that each hadron is a fundamental object, to focus on the elastic scattering was natural because the elastic scattering is the simplest process.

The theoretical foundation was S-matrix theory, which requires several mathematical axioms for the scattering amplitude. The S-matrix theory led us to several representations / transformations of scattering amplitude, by using partial wave expansion and dispersion relations. The idea of complex angular momenta is one of the most important development, which is the consequence of the Watson-Sommerfeld representation of the scattering amplitude. The Regge theory was formulated basting on the complex angular momenta, which aims to gather and collect infinite number of (t-channel) hadronic resonances all together into a single expression as "amplitude of exchange of Regge trajectory". The Regge theory can describe the high-energy small-angle scattering data (with parameter fitting), by assuming poles (=Regge trajectories) in the complex angular momentum plane. The Veneziano amplitude, which is known as the scattering amplitude of the open string today, was discovered as an explicit realization of the idea of Regge theory, along with conjectured s-t duality. This was the born of the string theory and the idea that hadron is string. However, the Regge theory, Veneziano amplitude or the other string based amplitude remained merely models of the scattering amplitude and the fundamental theory of the hadrons was still unclear.

In the 1970s, the quantum chromodynamics (QCD) began to be understood as the fundamental theory of the hadrons. One of the main reasons was discovery of narrowresonance of J/ψ particle, which was understood as the bound state of charm and anticharm. On the other hand, people started to consider that the string models for hadrons are inappropriate itself, because the Veneziano amplitude differs from the experimental data of large angle (=large momentum transfer) scattering (hard scattering). The string theory predicts that the scattering amplitude falls down exponentially as a function of energy; this property seems to be inevitable because a string is not a pointlike object but has a finite size, so the large angle scattering is a rare event. However, the experimental data showed that the amplitude has a power-low behavior in terms of energy, which prefers the model that there are pointlike particles inside a hadron, as in the QCD.

Once it was noticed that quarks and gluons are elementary constituents inside the hadrons, the hadron processes which people were interested in changed. The elastic scatterings are complicated processes in the viewpoint of the QCD. Instead of such exclusive scatterings, the inclusive process became main interest such as total cross section of e+e- to hadrons or total cross section of e-p to hadrons, so-called deep inelastic scattering (DIS). This is because we can leave clean parton processes where QCD can be tested by the perturbation theory, by summing up all the final states and neglecting the information of individual hadrons.

Today, the QCD has been fully tested, and interests are shifting into enlarging processes in which perturbative QCD is available and unveiling non-perturbative properties of hadrons itself, such as spin structure of a hadron and spacial or momentum transverse distribution of partons in a hadron. The perturbative QCD has been established as a powerful tool for high-energy hadron scattering and is utilized to search for new physics at hadron colliders or the other high-energy experiments with parton distribution function (PDF) determined by experimental data. However, there are many high-energy hadron processes which require more complicated non-perturbative information of partons, for example, generalized parton distribution (GPD). Such parton information is taking attention not only it is needed to describe cross sections of hadron processes, but also it tells us properties of hadrons described above. To determine such non-perturbative parton information in hadrons, we need to develop theoretical understandings of non-perturbative properties, along with taking experimental data. In this context, the exclusive (elastic) scattering are taking attention again. Even today, Regge-theory based phenomenology is still useful for a variety of high-energy small-angle scatterings which perturbative QCD can not explain.

On the other hand, the string theory, which was born in order to explain hadron scattering but finally abandoned, has developed towards "theory of everything". The string theory includes D-brane as a Dirichlet boundary condition of open string, and it was noticed that D-brane can also be regarded as black hole (black brane). From this idea, AdS/CFT duality was conjectured, which states that a Yang-Mills (YM) theory corresponds to superstring theory on a warped geometry. AdS/CFT duality is an example of weak/strong correspondence, and therefore non-perturbative physics of the strongly coupled YM theory can be studied by the perturbation theory in the string theory.

Holographic QCD is an idea to study non-perturbative properties of hadron physics in the QCD by means of AdS/CFT duality. The gravity dual which is completely dual to the QCD is not known, but, several essential properties of the QCD can be implemented: confinement, breaking (eliminating) supersymmetry, adding flavor (quark), and chiral symmetry breaking, and so on. On such gravity duals, one can calculate low-energy observables in the strongly coupled gauge theory: spectra of hadrons, coupling constants among hadrons, and properties in finite temperature, and so on. So far, the holographic QCD is not good at quantitative precisions, but it is still a powerful tool which can derive several qualitative properties of hadrons and develop theoretical / conceptual understandings of non-perturbative properties fo hadrons.

Although the main interest of the holographic QCD has been low-energy hadron physics, nothing prevents us from using gravitational dual descriptions to study high-energy hadron scattering. As I depicted above, even in high-energy processes, we need non-perturbative information also, and AdS/CFT duality can be exploited for it. By using AdS/CFT, we can provide theoretical grounds for the traditional Regge theory and models based on string theory, and moreover, we can study how such traditional models should be modified, corresponding to the fact that the string theory is not on a flat (four) dimensional spacetime, but on a warped ten dimensional spacetime. Polchinski-Strassler (2001) showed that large-angle hadron scattering calculated in a gravity dual shows power-law behavior; the problem of traditional string model was cured. Brower-Polchinski-Strassler-Tan (BPST) (2006) formulated the Pomeron amplitude (= the leading trajectory in the Regge theory) in the gravity dual, which has both the property of the Regge theory and BFKL theory (=perturbatively formulated Pomeron). The BPST Pomeron captures the physics in the QCD because the transition between Regge theory and BFKL theory had been anticipated also in the QCD. However, potential power of gauge/string duality in hadron scattering is far from being fully exploited so far.

In this doctoral thesis, I deepen our understandings of 2 to 2 scatterings of hadron at high-energy (small Bjorken x) in holographic QCD by utilizing and extending BPST Pomeron. Especially, I study scattering of hadron h and virtual photon γ* ; this process is known as DIS and deeply virtual compton scattering (DVCS). From the cross sections of these processes, one can extract PDF and GPD. Although perturbative QCD can describe photonvirtuality q2-evolution of PDF and GPD, initial data of the evolution cannot be determined by perturbation theory. Such non-perturbative initial data for PDF can be obtained from DIS experiments, but GPD cannot be determined even from experimental data, without some theoretical modeling of non-perturbative physics. GPD describes parton distribution in the transverse directions and two parton correlation in a hadron in general, hence it is an interesting object on its own. I see that gravitational dual descriptions can determine how those non-perturbative scattering amplitudes depend on kinematical variables such as centerof-mass energy, momentum transfer, impact parameter and photon virtuality. Contrary to the study by Polchinski and Strassler (2002) which stated that the DIS in a gravity dual is completely different from DIS in the QCD, I show that qualitative properties of PDF/GPD of the QCD can be captured in a gravity dual by taking large but finite 1=x and q2. Gauge/gravity dual also tells us how to think about various theoretical ideas that various models of GPD have been based on.

Conceptual understandings of Regge theory are also developed. High energy behavior of elastic scattering amplitude A(s, t) is characterized by poles and their residues of its partial wave amplitude A(j, t) on the complex angular momentum j-plane; the poles and residues depend on momentum transfer t. The poles in the j-plane (=Regge trajectory) have often been assumed to depend linearly in t. Notable aspects for Regge theory and amplitude in string theory on a warped spacetime, however, include i) a single "Regge trajectory" of string theory on 10 dimensions gives rise to a Kaluza-Klein tower of infinite "Regge trajectories" on 4 dimensions, and ii) those trajectories do not remain linear for arbitrary negative t. The non-linear trajectories immediately result in non-Gaussian profile of GPD in the transverse directions, although Gaussian profile of GPD is often assumed in phenomenological analysis. We will also describe how the residues of the Regge poles are determined by holographic set-up, and also explain how the Kaluza-Klein tower of Regge trajectories organizes itself to become a single contribution with a form factor in momentum transfer t < 0.

An extra energy scale q2 is available in photon{hadron scattering, in addition to the center of mass energy W and confinement scale ∧. This extra parameter makes theoretical understanding of the non-perturbative amplitude interesting. The scattering amplitude is dominated by a contribution from a saddle point in the complex j-plane, not from a pole, for sufficiently large q ≫ ∧. The saddle point value j* depends on kinematical variables such as W, q and t. We find, by following this dependence of j*, instead of naively taking small x limit or large q2 limit, that observables characterizing scattering amplitude such as ln(1/q)-evolution parameter γ(eff)., ln(1/x)-evolution parameter λ(eff). and t-slope parameter B show qualitatively the same behavior in the strong coupling regime (gravity dual) as expected in perturbative QCD or observed in HERA DVCS experiment. As the saddle point value j* and the leading poles are both given by the kinematical variables of the scattering, crossover from the saddle-point phase to the leading pole phase may also be expected, when the photon virtuality decreases to a smaller value.

I also develop the method to study skewness parameter of GPD, which relates the virtuality-difference of initial and final photon. With the full skewness dependence included in this analysis, it is also possible to use the result of this study to bridge a gap between data in such scattering processes at non-zero skewness and the transverse profile of partons in a hadron, which is encoded by the generalized parton distribution functions at zero skewness.

For this study, I found that the BPST formalism is insufficient and we have to extend it in a number of points. First, hadron matrix elements of total derivative operators are irrelevant for the h-γ(*) scattering with zero skewness (like DIS), but they do contribute to the skewed scattering amplitude. This contribution needs to be implemented in the language of gravity dual. Secondly, Pomeron/Reggeon propagators have been treated as if it were for a scalar field in BPST Pomeron, but they correspond to exchange of stringy states with non-zero (arbitrary high) spins; for the study of scattering with non-zero skewness, the polarization of higher spin state propagator should also be treated with care. Finally, this also means that infinitely many gauge degrees of freedom in string theory (which extends the general coordinate transformation of the graviton) need to be dealt with properly.

本論文は6章からなり、第1章は導入、第2章では光子ハドロン散乱の基本事項のまとめと一般化パートン分布の定義が与えられ、第3章でゲージ/弦対応に基づくホログラフィー的ゲージ理論の構成について述べられ、第4章では本論文に直接関連する先行研究に基づくセットアップ、特にポメロン振幅の重力双対による記述が説明されている。第5章が本論文の主要結果であり、前章の枠組みを深非弾性散乱および深仮想コンプトン散乱へ応用し、パートン分布に関する諸量を導いている。第6章では、まとめと議論が与えられている。

本論文の主たる目標は、ハドロン内のパートン分布に関する情報をホログラフィー的手法によって非摂動的効果も含めて得ることにある。特に、摂動論や、あるいは非摂動的手法としてこれまで強力であった格子理論では、扱えないような運動量領域における情報を引き出すことにある。しかしながら、現実の量子色力学に双対な重力背景は未だ知られておらず、そのままでは絵に描いた餅とならざるを得ない。そこで、論文提出者は、Brower-Polchinski-Strassler- Tan (以下BPSTと略) による2006年の研究に目を付けた。BPSTでは、運動方程式の解ではないが、十分に物理的性質を反映した重力背景を用いて、ハドロン散乱振幅を解析し、背景の構造の詳細にあまり依らず、ポメロン軌跡の寄与をホログラフィー的に定式化することに成功していた。これは、高次元の曲がった時空における弦の散乱振幅を用いることで、現実の4次元時空におけるハドロン散乱振幅を導出するというアイデアである。

論文提出者は渡利泰山氏とともに、この先行研究の枠組みを発展させて、仮想光子とハドロンの2体散乱振幅を詳細に調べ、深非弾性散乱や深仮想コンプトン散乱に対応する領域での振る舞いから、パートン分布関数や一般化パートン分布のある種のパラメータ依存性を決定することに成功した。これは、摂動論では解析できない強結合の情報を含んでいるものであり、重要な結果である。もちろん現実の量子色力学に近づくには、まだまだ詰めていかなければならない点があるものの、これまでより広い運動量領域に対して、一貫した枠組みで結果を導いているなど、注目すべき成果であると評価できる。特に、t-スロープパラメータに関しては、本研究において初めて導出された結果であり、その意義は大きい。

また、本研究においてなされたような研究方向性、すなわち弦理論のような理論的整合性をもつ手法にきちんと基づいて、現実の物理量、特に実験と最終的には比較し得るような過程に対する諸量の導出に果敢に挑んでいく姿勢は、現時点では十分確立した方法とは言えないまでも、今後両者を橋渡ししていく先駆的な研究として今後の発展に寄与すると考えられる。

なお、本論文第5章の内容は、渡利泰山氏との共同研究に基づくものであるが、論文提出者が主体となって解析を行ったもので、論文提出者の寄与が十分であると判断する。以上により、審査員一同は、本論文をもって論文提出者に、博士(理学)の学位を授与できると認める。