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

2. 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

3. Abrikosov vortex in U(1) theory electric Cooper-pair condensation B squeeze magnetic field A.A.Abrikosov, Soviet Phys.JTEP 5, 1174(1957) B quantization of the total magnetic flux z periodicity of the phase of Cooper-pair wave function (topologically conserved) Dual superconducting picture DSC picture connects the string picture and QCD. color confinement (static level) linear potential between quarks Color flux tube in QCD dual magnetic monopole condensation squeeze color electric flux E Y.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 )

4. Dynamics of color flux tube In most cases, we consider the moduli-dynamics of strings, i.e., rotationtranslationstringy vibration simplification of the problem infinite length （ neglect the boundary ） translational invariance along the vortex line, and cylindrical symmetry. “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)

5. D.O.F for the dual string ‘t Hooft, Nucl.Phys.B190.455(1981) SU(3) gauge theory (QCD)U(1) 3 ×U(1) 8 gauge theory fix the gauge of the off diagonal elements (Abelian projection) 8-gluon 2 -gluon related to t 3,t 8 generators 6 -gluon related to other generators “monopoles” with U(1) 3 ×U(1) 8 magnetic charge remaining U(1) 3 ×U(1) 8 sym. “ photon ” “ charged matter ” topological configuration (electric 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

6. Ezawa-Iwazaki, PRD25(1982)2681 Maedan-Suzuki, PTP81(1989)229 Model – dual Ginzburg-Landau model After the Abelian gauge fixing, we get the D.O.F, especially magnetic monopole, necessary to construct the dual strings. dual photon fieldmonopole field same form as the Ginzburg-Landau type action 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

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

8. Excitation modes under the static vortex background neglect 3 rd and 4 th order terms of fluctuations because we focus on the case where the quantum fluctuation is not so strong: Consider only the axial symmetric fluctuation around the static vortex solution Euler-Lagrange equation at 2 nd 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:

9. eqs. for fluctuations in the radial direction dispersion relation: Peristaltic modes of the vortex “radial mass” propagation with “radial mass” m j ω 、ｋ ｚ conserved total color electric flux EZEZ monopole EZEZ EZEZ

10. Vortex-induced potential for fluctuations = M monopole 2 2 = M d-photon 2 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 κ 2 ) Only the radial direction of the potential is nontrivial.

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

12. 1 st excited state – wavefunction in the radial direction fluctuation of electric field large small monopole dual-photon Type-I Type-II BPS fluctuations of φ 、 A θ corporative oscillation ~ e ipr / r 1/2 oscillation ~ e ipr / r 1/2 → resonant scattering rr squeezed by monopole ( total flux is conserved to 0) (around)

13. Summary: excitation energy ~ 0.5 GeV κ 2 ~ 3 → Type-II resonant scattering type of vibrations appear. DGL parameters are taken to fit the QQ potential results. monopole self-coupling: λ ~ 25 dual-gauge coupling: g dual ~ 2.3 value of monopole cond.: v ~ 0.126 GeV flux-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:

14. Outlook and speculation: For the application to hot QCD: temperature Then, if the strength of effective 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.

15. 1.0 fm2.0 fm3.0 fm 0.25 GeV 2.0 GeV 1.0 GeV 1.5 GeV 0.1 GeV Type-II (DGL case) Type-I BPS Vortex – vortex “potential” per unit length ( for DGL, per 1 fm ) RR R R

16. String picture of hadrons String picture of hadrons gives natural explanation for: Duality of the hadron reactions == s-channel t-channelstring reaction Regge trajectories of hadrons constant string tension (hadron mass) 2 angular momentum The string picture may share important part of QCD. 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)

17. Static solution z G-L parameter minimize with B.C for finite energy vortex solution at vortex core, φ ＝ 0 in asymptotic region, sym. is restored gauge fixing: We search for the solution with cylindrical symmetry & topological charge = n. topological quantization (topological charge n)

18. Higgs photon mixing Importance of mixing BPS Type-II Type-I around BPS static vortex – Higgs bound state bound state of static vortex and - mixed state Same threshold leads the corporative behavior of Higgs & photon then, lowest excitation energy is considerably decreased. around the core, Higgs & photon are mixed

19. The property of the potential in the radial direction V(r) potential induced by static vortex = M mono 2 2 = M d-photon 2 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 0 2 (independent of κ 2) “central” region ( r → 0 ) “asymptotic” region ( r → ∞ ) α(r) → r m × const. β(r) → r m × const. α(r) → 0 β(r) → 0 α(r) → e ipr state below thresholdstate above threshold (m ≧ 2 ) for all states

20. electric field energy density monopoles electric field energy density Static profile for Type-I & II κ ＝ δ/ξ( ＝ λ 1/2 / e) : G-L parameter δ ξ pure metal ex) high Tc SC, metal with inpurity penetration depth δ: ～ 500 A, 0.3 - 0.4 fm ex) coherence length ξ: 25 – 10 4 A, 0.16 fm (usually not considered) condensed matter DGL (QCD effective theory) G-L parameter κ: 0.05 － 20, ~ 1.6 － 2.0 =1/2 BPS finite thickness string like

21. for Abrikosov vortex To discuss the color flux linking specific charges, we have only to consider this part. R R Abelian dominance off-diagonal gluon is heavy ( ~ 1.2 GeV ) Amemiya-Suganuma, PRD60(1999)114509 same form as the Ginzburg-Landau type action Higgs (Cooper-pair) field photon field Ginzburg-Landau action: We will consider color flux tube linking specific charges. dual photon fieldmonopole field We have only to consider the GL-type action.

22. 1 st excited state – wavefunction in the radial direction fluctuation of electric field E z large small monopoles d-photon Type-I Type-II BPS fluctuations of φ 、 a θ corporative oscillation ~ e ipr / r 1/2 oscillation ~ e ipr / r 1/2 → long tail rr squeezed by monopoles ( total flux is conserved to 0) (around) EzEz EzEz EzEz

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

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

25. n = 2, 3 energy spectrum n=2 n=3 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 lowest excitation becomes softer one giant vortex The threshold is unchanged, then continuum states behave like n =1 continuum states.

26. Summary for single Abelian vortex We have discussed the “ peristaltic ” modes of single vortex. We found, new discrete pole around κ 2 ~ 1/2. This discrete pole is characterized by the corporative behavior of the Higgs and photon fields. As κ 2 is increased, the low excitation modes change from the Higgs dominant modes to the photon dominant modes. As n is increased, photon can excite more easily, and lowest excitation becomes softer one. ω 、ｋ ｚ r z Higgs r HZHZ r

27. Summary for single color electric flux excitation energy ~ 0.5 GeV. (mass of color electric flux per 1fm ~ 1.0 GeV/fm.) r DGL gives κ 2 ~ 3 - 4. → Type-II Only resonant scattering type of excitations appear. profile in radial direction electric flux vibrate with long tail. R R We can directly apply the previous arguments to the color flux linking specific color charge.

28. 4, Calculation in 2 - D (Preliminary) Motivation: We would like to discuss: 1, excitation modes around 1-vortex without cylindrical symmetry. 2, the dynamics of the multi-vortices system, for example, vortex-vortex fusion into the giant vortex, the giant vortex fission to the small vortices, vortex - anti vortex annihilation and production, bearing in mind the future application to the hadron physics: ex) meson-meson reaction: scattering production of the resonance, especially exotic hadrons etc. In this talk, we show only the static profile, vortex- vortex potential, and vortex- anti vortex potential.

29. Vortex – antivortex “potential” per unit length ( for DGL, per 1 fm ) BzBz BzBz same topological charge no vortex total magnetic flux is zero. 1.0 fm2.0 fm3.0 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. R R R R

30. Energy spectrum: New-type discrete pole Type-II monopoles dual photon field V(r) dual photon field Type-I BPS Around BPS saturation, characteristic discrete pole appear as a result of monopoles – d-photon corporative behavior. monopoles

31. Sudden annihilation of fluxes (in DGL unit) 1.2 fm1.0 fm Around d = 1.0 - 1.2 fm, the fluxes suddenly annihilate. This critical distance d cr is related with the penetration depth δ d cr ~ (1.5 - 2.0) × 2δ This value seems to be considerably large.

32. SU(3) gauge theory (QCD)U(1)×U(1) gauge theory Abelian projection ‘t Hooft, NPB190,455(1981) (fix the gauge of the off diagonal elements) 8-gluon 2-gluon (“photon”) 6-gluon (“charged matter”) Abelian monopole (topological object) Abelian projection and monopoles Usually magnetic monopole does not appear in U(1) gauge theory, but if theory includes SU(N) ( N>1) gauge fields, their specific topological configuration constructs U(1) point like singularity as a topological object. R 3 in physical space SU(2) variables in internal space mapping

33. Two vortices system ( static case ) field degrees of freedom ： Reψ, Imψ, Ax, Ay (without cylindrical sym.) Starting from the case where the distance between two vortices is large, adopt the product ansatz for initial B.C: ψ 1+2 ＝ ψ 1 ψ 2 /const.A 1+2 ＝ A 1 + A 2 Step 1) As the previous 1-vortex system, we first search for the static profiles which minimize the static energy. Step 2) Fix the |ψ| ２ = 0 at the vortices cores, and minimize the static energy with checking that the total magnetic flux is quantized appropriately. (We need this B.C. only at the beginning of the calculation) Step 3)After convergence, change the distance of the vortex core. Step 4)Adopt the previous profile as I.C. and return to the step 2. Then we acquire the static profile and the potential between two vortices.

34. Summary: We have discussed the “peristaltic” modes of single vortex. We found, new discrete pole around κ 2 ~ 1/2. This discrete pole is characterized by the corporative behavior of the Cooper-pair and photon fields. To discuss the dynamics of color flux, we need more careful treatments to retain the confinement property. We have also discussed the potential between vortices as a preparation for the dynamics of multi-vortices system. Future work: Non axial symmetric excitation of single vortex. Dynamics of two vortices. Careful treatment of color flux with projection.

35. Introduction Abrikosov vortex in U(1) theory Cooper-pair condensation B Meissner effect squeezed magnetic field A.A.Abrikosov, Soviet Phys.JTEP 5, 1174(1957) Color confinement in QCD dual B magnetic monopole condensation dual Meissner effect squeezed color electric flux E Y.Nambu, PRD.122,4262(1974) ‘t Hooft, Nucl.Phys.B190.455(1981) Mandelstam, Phys.Rep.C23.245(1976) SU(3) gauge theory (QCD)U(1)×U(1) gauge theory Abelian projection ‘t Hooft, NPB190,455(1981) (fix the gauge of the off diagonal elements) 8-gluon 2-gluon (“photon”) 6-gluon (“charged matter”) Abelian monopole (topological object)

36. Static U(1) vortex solution (n=1 case) magnetic field penetrates with small kinetic energy magnetic field is strongly squeezed by Higgs field finite thickness string

37. Regge trajectories of hadrons constant string tension (hadron mass) 2 angular momentum gauge fixing:

38. 1) Abelian gauge fixing → U(1) 2 monopole DOF naturally appear as topological objects. 2) include the auxiliary field U(1) m ｘ U(1) m dual photon B which couples with the monopole current. ( Zwanziger, PRD3(19 ７ 0)880 ) 4) integrate out the U(1) e ｘ U(1) e photon field A 3) sum up the monopole trajectory → leading order ： kinetic term of monopole field correction ： monopole self interaction ( Bardakci, Samuel, PRD18(1978)2849 ) (this is included phenomenologically) Ezawa-Iwazaki, PRD25(1982)2681 Maedan-Suzuki, PTP81(1989)229 Model – dual Ginzburg-Landau model

39. Static solution z G-L parameter minimize static energy with B.C for finite energy vortex solution at vortex core, φ ＝ 0 in asymptotic region, sym. is restored gauge fixing: We search for the solution with cylindrical symmetry & topological charge = n. Ansatz for topological charge n monopoles electric field energy density electric field energy density monopoles