21COE 出張報告 原子核理論研究室 Ｄ３　古城　徹

関空～シンガポール 約 6 時間 シンガポール～ロンドン 約 12 時間 ロンドン~ケンブリッジ 約 2 時間

(ニュートン、ダーウィン、ディケンズ , ... を輩出) ケンブリッジ ～ ３１のカレッジからなる大学町 トリニティカレッジ ～ニュートン輩出、ノーベル賞31人 (ニュートン、ダーウィン、ディケンズ , ... を輩出) 宿舎 キングスカレッジ

会議のテーマ 10-10m 10-15m カラーの閉じ込め 環境変化（有限温度、密度）に対するＱＣＤ真空の応答 量子色力学（ＱＣＤ）に関連するあらゆるトピックを扱い、 　　　ＱＣＤに対する総合的な理解を図る 10-10m 10-15m カラーの閉じ込め 環境変化（有限温度、密度）に対するＱＣＤ真空の応答 有限温度、密度における素励起の性質の変化

Topics 関心ある現象はいずれもＱＣＤの非摂動効果と関連 → 摂動論を超えた様々な手法を開発していく必要性 強相関 ＱＧＰ の現象論～重イオン衝突実験の現象論 (6 talks) 格子ＱＣＤの大規模数値解析 （ＱＣＤ相図、ＱＧＰ中における素励起スペクトル、 　ＱＧＰ流体の輸送係数～ずれ粘性、熱伝導度） (6 talks) モデル的アプローチ： (11 talks) ＱＣＤの対称性を反映した現象論的モデルの解析( PNJL, dual-GL model ) 次元を落とし簡単化したＱＣＤの解析　（ (2+1)-D Yang-Mills ) 内部対称性を落とし簡単化したＱＣＤの解析 ( SU(2) color QCD ) Holographic QCD, Gauge-Gravity duality ( AdS/CFT or AdS/QCD 対応） (20 talks) 超弦理論を介して量子効果が強いQCDを、双対な重力理論の古典計算で解析 →　真空中のハドロンの性質、ＱＧＰへの適用

Peristaltic modes of single vortex in U(1) and SU(3) gauge theories based on PRD75, 105015 (2007) Toru Kojo (Kyoto University) in collaboration with Hideo Suganuma (Kyoto University) Kyosuke Tsumura (Fuji film corporation) This work is supported by the Grand-in-Aid for the 21st Century COE. 「Exploring QCD 」 at Isaac Newton Institute, 2007. 8. 23

Contents I, Dual superconductor (brief review) I-1, Dual superconducting picture for string I-2, Dual Ginzburg Landau model II, Peristaltic modes (Main results) II-1, Static vortex solution and classification of vortex II-2, Fluctuation analysis ~ Peristaltic modes III, Summary and outlook

String picture of hadrons String picture of hadrons gives natural explanation for: Duality of the hadron reactions = s-channel t-channel string reaction Regge trajectories of hadrons constant string tension (hadron mass)2 angular momentum Linear potential between quarks lattice studies for QQ, 3Q potential universality of the string tension Creutz, PRL43, 553 (1979) T.T.Takahashi et al, PRL86, 18 (2001) The string picture may share important part of QCD.

Dual superconducting picture DSC picture connects the string picture and QCD. Abrikosov vortex in U(1) theory Color flux tube in QCD A.A.Abrikosov, Soviet Phys.JTEP 5, 1174(1957) Y.Nambu, PRD.122,4262(1974) ‘t Hooft , Nucl.Phys.B190.455(1981) B Mandelstam, Phys.Rep.C23.245(1976) electric Cooper-pair condensation dual magnetic monopole condensation squeeze magnetic field B E squeeze color electric flux z periodicity of the phase of Cooper-pair wave function quantization of the color electric flux (periodicity of the phase of monopole ) color confinement (static level) quantization of the total magnetic flux linear potential between quarks (topologically conserved)

Dynamics of color flux tube In most cases, we consider the moduli-dynamics of strings, i.e., rotation translation stringy vibration “Peristaltic” mode Instead, following the dual superconducting scenario, we focus on the dynamics of internal degrees of freedom, i.e., excitation of the flux tube with changing its thickness. When we consider short strings, this type of excitation becomes important instead of the stringy excitation because stringy excitation cost energy ~ π/L. (typical length ~ 1 fm for usual hadrons) simplification of the problem infinite length　（neglect the boundary） translational invariance along the vortex line, and cylindrical symmetry.

D.O.F for the dual string SU(3) gauge theory (QCD) ‘t Hooft, Nucl.Phys.B190.455(1981) fix the gauge of the off diagonal elements (Abelian projection) SU(3) gauge theory (QCD) U(1)3×U(1)8 gauge theory 8-gluon remaining U(1)3×U(1)8 sym. 2 -gluon related to t3 ,t8 generators “ photon ” 6 -gluon related to other generators “ charged matter ” (electric charge) topological configuration “monopoles” with U(1)3×U(1)8 magnetic charge Merit: Further, we rewrite U(1)e×U(1)e photon part in terms of the U(1)m×U(1)m dual photon. Squeezing of electric flux can be described in the same way as the squeezing of magnetic flux (dual photon becomes massive). Dual-photon couples to the monopole current with the dual gauge coupling, strong coupling region can be treated as the weak coupling regime

Amemiya-Suganuma, PRD60(1999)114509 Model – dual Ginzburg-Landau model Ezawa-Iwazaki, PRD25(1982)2681 Maedan-Suzuki, PTP81(1989)229 After the Abelian gauge fixing, we get the D.O.F, especially magnetic monopole, necessary to construct the dual strings. Next question: Do monopoles really condense? Do the effects of off-diagonal gluon fluctuations make theory untractable? In low energy, QCD can be effectively described by “dual-photons” and monopoles (& quarks) degrees of freedom. lattice results monopole condenses ! off-diagonal gluons become heavy (~1.2 GeV) Amemiya-Suganuma, PRD60(1999)114509 dual photon field monopole field same form as the Ginzburg-Landau type action

Static solution (n = 1 vortex) = z = (under rescaled unit) G-L parameter: We search for the solution with cylindrical symmetry & topological charge = n. minimize static energy with B.C for finite energy vortex solution monopole color electric field energy density

Excitation modes under the static vortex background Consider only the axial symmetric fluctuation around the static vortex solution neglect 3rd and 4th order terms of fluctuations because we focus on the case where the quantum fluctuation is not so strong. Euler-Lagrange equation at 2nd order variation of the action Because of the translational (t, z) and rotational invariance of the static vortex background, eqs for (t, z) directions are easily solved axial symmetric fluctuations are completely decoupled from angular dependent modes. Remark:

conserved total color electric flux Peristaltic modes of the vortex eqs. for fluctuations in the radial direction dispersion relation: “radial mass” propagation 　with “radial mass” mj　 ω、ｋｚ conserved total color electric flux EZ monopole EZ

Vortex-induced potential for fluctuations Only the radial direction of the potential is nontrivial. 渦糸付近では対称性が回復　→　粒子の質量は軽くなる = Mmonopole2 2 = Md-photon2 energy “threshold” for continuum states V(r) for α (monopole) V(r) for β (dual photon) V(r) for α-β mixing ex）　Type-II case (independent of κ2)

Energy spectrum ( the effect of the diagonal potential ) Type-II monopole gauge field V(r) BPS Around BPS saturation, characteristic discrete pole appear as a result of monopole – dual photon corporative behavior monopole gauge field V(r) Type-I

1st excited state – wavefunction in the radial direction fluctuations of φ、 Aθ fluctuation of electric field small monopole dual-photon squeezed by monopole Type-I ( total flux is conserved to 0) (around) squeezed by monopole BPS ( total flux is conserved to 0) corporative oscillation ~ eipr / r1/2 Type-II → resonant scattering oscillation ~ eipr / r1/2 corporative large r r

Summary: For the general vortex case: For the application to QCD: We consider the vortex vibration with changing its thickness. We found the characteristic discrete pole around BPS value of GL parameter. → coherent vibration of Higgs and photon fields. For the application to QCD: flux-tube in the vacuum: DGL parameters are taken to fit the QQ potential results. κ2 ~ 3 → Type-II monopole self-coupling: λ ~ 25 dual-gauge coupling: gdual ~ 2.3 resonant scattering type of vibrations appear. value of monopole cond.: v ~ 0.126 GeV excitation energy ~ 0.5 GeV

Outlook and speculation: For the application to hot QCD: temperature Then, if the strength of the monopole self-interaction λ(T) becomes weak, = becomes weak, and the property of color-electric flux approach to the Type-I vortex. The monopole - dual photon coherent vibration can appear in non-zero temperture.

質疑応答（要約） Ｑ，円柱対称な励起よりも歪んだ形の振動の方が重要では？ Ａ，Type-II の場合はvortex は表面を減らす方がエネルギー的には得する　　 　　ので円柱対称なモードだけで十分でしょう。Type-I については逆なので 　　今の解析からはあまり確定的なことは言えません。 Ｑ，実際のハドロンでこのような形で励起スペクトルは見えるのか？ Ａ，今の解析では大分簡単化した場合を扱っているので、実際のハドロン 　　 物理への応用を行なうにはもう少しrealistic な状況設定にする必要が 　　 あると思ってます。これについてはいずれやろうと思っています。 Ｑ，実際の超伝導体ではどうか？ Ａ，今回見つけたdiscrete pole が出るのは BPS 付近の GL parameter の場合ですが、対応する材料があれば見えると思います。

まとめ 今回の海外出張は、物理関係者のみならず、 外国の人と沢山話す機会があり、非常に良い経験となった。 →　英語の練習は普段からもっとしておくべきだったが。。 質疑応答に関しては、もっと経験を積む必要性を痛感した。 会議中のトークに刺激を受け、 　　いくつか研究テーマの候補を思いつくことができた。 最後に、このような機会を与えて下さった 21COE 外国旅費補助委員会の方々に感謝したいと思います。

Vortex – vortex “potential” per unit length ( for DGL, per 1 fm ) 1.0 fm 2.0 fm 3.0 fm 0.25 GeV 2.0 GeV 1.0 GeV 1.5 GeV 0.1 GeV Type-II (DGL case) Type-I BPS R

Thermodynamical Stability ( Mn：vortex mass with topological charge n ) vortex-vortex interaction exact solution de Vega-Schaposnik, PRD14,1100(1976) vortex-vortex interaction attractive repulsive B B B no interaction between vortices vortex lattice with topological charge n=1 ( thermodynamically stable） Type-I vortices system is thermodynamically unstable (at least in tree level) Usually, Type-I vortex is not considered, not uniform B but we consider the external magnetic field squeezed enough to generate only one vortex We study not only Type-II vortex but also Type-I vortex

n > 1 vortex, classical profiles & potentials (κ２ = 1/2 case ) n=1 n=2 n=3 profile increasing n total magnetic flux ( =2πn ) increases Cooper-pair around core is suppressed potential Cooper-pair around core is suppressed large potential around core for φ & fluc. of Hz is enhanced “surface” between Cooper-pair and magnetic flux shifts outward “mixing” potential shifts outward

n = 2, 3 energy spectrum n n=2 n=3 giant vortex the lowest excitation becomes softer one The topological defect of Cooper-pair condensation is enlarged, then photon can easily excite around the core. n The property as static vortex + photon excitation becomes strong. The threshold is unchanged, then continuum states behave like n =1 continuum states.

Vortex – antivortex “potential” per unit length ( for DGL, per 1 fm ) R Bz same topological charge no vortex total magnetic flux is zero . 1.0 fm 2.0 fm 3.0 fm 2.0 GeV 1.0 GeV 0.5 GeV 1.5 GeV sudden annihilation of the flux When d < 1.0 fm, our B.C, |ψ|２= 0 at the core is no more applicable.