21COE

6 12 ~ 2

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10-15m10-10m

Topics ( PNJL, dual-GL model ) (2+1)-D Yang-Mills ) ( SU(2) color QCD ) Holographic QCD, Gauge-Gravity duality ( AdS/CFT or AdS/QCD QCD(6 talks)(6 talks)(11 talks)(20 talks)

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

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

String picture of hadronsString picture of hadrons gives natural explanation for: The string picture may share important part of QCD.

Abrikosov vortex in U(1) theoryelectric Cooper-pair condensationBsqueezemagnetic field A.A.Abrikosov, Soviet Phys.JTEP 5, 1174(1957)Bquantization of the total magnetic fluxperiodicity of the phase of Cooper-pairwave function (topologically conserved)Dual superconducting pictureDSC picture connects the string picture and QCD. color confinement (static level)linear potential between quarksColor flux tube in QCDdualmagnetic monopolecondensationsqueeze color electric fluxEY.Nambu, PRD.122,4262(1974)t Hooft , Nucl.Phys.B190.455(1981)Mandelstam, Phys.Rep.C23.245(1976)quantization of the color electric flux(periodicity of the phase of monopole )

Dynamics of color flux tubeIn most cases, we consider the moduli-dynamics of strings, i.e., rotation translation stringy vibration

D.O.F for the dual string t Hooft, Nucl.Phys.B190.455(1981)SU(3) gauge theory (QCD)U(1)3U(1)8 gauge theoryfix the gauge of the off diagonal elements (Abelian projection)8-gluon2 -gluon related to t3 ,t8 generators6 -gluon related to other generatorsmonopoles with U(1)3U(1)8 magnetic charge remaining U(1)3U(1)8 sym. photon charged matter topological configuration(electric charge)

Ezawa-Iwazaki, PRD25(1982)2681 Maedan-Suzuki, PTP81(1989)229Model dual Ginzburg-Landau modelAfter 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?

Static solution (n = 1 vortex)G-L parameter:(under rescaled unit)

Excitation modes under the static vortex backgroundneglect 3rd and 4th order terms of fluctuations becausewe focus on the case where the quantum fluctuation is not so strong.Consider only the axial symmetric fluctuation around the static vortex solution

eqs. for fluctuations in the radial directiondispersion relation:Peristaltic modes of the vortexradial mass

Vortex-induced potential for fluctuationsOnly the radial direction of the potential is nontrivial.

Energy spectrum ( the effect of the diagonal potential )

1st excited state wavefunction in the radial directionfluctuation of electric fieldlargesmallType-IType-IIBPSfluctuations of Acorporativecorporativeoscillation ~ eipr / r1/2 oscillation ~ eipr / r1/2 resonant scattering rrsqueezed by monopolesqueezed by monopole( total flux is conserved to 0)( total flux is conserved to 0)(around)

Summary: excitation energy ~ 0.5 GeV2 ~ 3 Type-IIresonant scattering type of vibrations appear.DGL parameters are taken to fit the QQ potential results.monopole self-coupling: ~ 25dual-gauge coupling: gdual ~ 2.3value of monopole cond.: v ~ 0.126 GeVflux-tube in the vacuum: We found the characteristic discrete pole around BPS value of GL parameter. coherent vibration of Higgs and photon fields. For the application to QCD: We consider the vortex vibration with changing its thickness. For the general vortex case:

Outlook and speculation: For the application to hot QCD: Then, if the strength of the monopole self-interaction (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.

Type-II vortex Type-I realistic discrete pole BPS GL parameter

21COE

Vortex vortex potential per unit length ( for DGL, per 1 fm )

Thermodynamical Stabilityexact solution de Vega-Schaposnik, PRD14,1100(1976) no interaction between vorticesType-I vortices system is thermodynamically unstablevortex lattice with topological charge n=1( thermodynamically stablevortex-vortex interactionattractivevortex-vortex interactionrepulsiveBUsually, Type-I vortex is not considered, but we consider the external magnetic fieldsqueezed enough to generate only one vortex( Mnvortex mass with topological charge n )(at least in tree level)We study not only Type-II vortex but also Type-I vortexnot uniform

n > 1 vortex, classical profiles & potentials( = 1/2 case )n=1n=2n=3increasing ntotal magnetic flux ( =2n ) increasesCooper-pair around core is suppressedCooper-pair around core is suppressed& fluc. of Hz is enhancedprofilepotentiallarge potential around core for surface between Cooper-pair and magnetic flux shifts outwardmixing potential shifts outward

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

Vortex antivortex potential per unit length ( for DGL, per 1 fm )