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
z 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 E B squeeze magnetic field squeeze color electric flux 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)
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） “Peristaltic” mode translational invariance along the vortex line, and cylindrical symmetry. Dynamics of color flux tube In most cases, we consider the moduli-dynamics of strings, i.e., rotation translation stringy vibration
Further, we rewrite U(1)e×U(1)e photon part in terms of the U(1)m×U(1)mdual photon. Merit: 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 D.O.F for the dual string ‘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)8gauge theory 8-gluon remaining U(1)3×U(1)8 sym. 2 -gluon related to t3 ,t8 generators “ photon ” “ charged matter ” 6 -gluon related to other generators (electric charge) topological configuration “monopoles”withU(1)3×U(1)8magnetic charge
monopole condenses ! lattice results off-diagonal gluons become heavy (~1.2 GeV) Amemiya-Suganuma, PRD60(1999)114509 In low energy, QCD can be effectively described by “dual-photons” andmonopoles (& quarks) degrees of freedom. dual photon field monopole field same form as the Ginzburg-Landau type action 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?
= z We search for the solution with cylindrical symmetry & topological charge = n. minimize static energy with B.C for finite energy vortex solution monopole monopole color electric field color electric field energy density energy density Static solution (n = 1 vortex) G-L parameter: (under rescaled unit)
variation of the action Euler-Lagrange equation at 2nd order Remark: 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. 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:
conserved total color electric flux EZ ω、ｋｚ EZ propagation with “radial mass” mj EZ monopole Peristaltic modes of the vortex eqs. for fluctuations in the radial direction “radial mass” dispersion relation:
ex） Type-II case energy “threshold” for continuum states V(r) for α (monopole) 2 =Md-photon2 V(r) for β (dual photon) = Mmonopole2 (independent of κ2) V(r) for α-β mixing Vortex-induced potential for fluctuations Only the radial direction of the potential is nontrivial.
Type-II monopole V(r) gauge field Type-I gauge field V(r) monopole BPS Around BPS saturation, characteristic discrete pole appear as a result of monopole – dual photon corporative behavior Energy spectrum( the effect of the diagonal potential )
monopole dual-photon 1st excited state – wavefunction in the radialdirection fluctuation of electric field fluctuations of φ、 Aθ small squeezed by monopole Type-I ( total flux is conserved to 0) (around) squeezed by monopole ( total flux is conserved to 0) BPS corporative oscillation ~ eipr / r1/2 Type-II → resonant scattering oscillation ~ eipr / r1/2 corporative large r r
Summary: For the general vortex case: 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
= temperature Outlook and speculation: For the application to hot QCD: 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.
1.0 fm 2.0 fm 3.0 fm Type-II (DGL case) 2.0 GeV 0.25 GeV 1.5 GeV BPS R R R Type-I 1.0 GeV 0.1 GeV R Vortex – vortex “potential” per unit length ( for DGL, per 1 fm )
Duality of the hadron reactions = = s-channel t-channel string reaction Regge trajectories of hadrons (hadron mass)2 angular momentum constant string tension Linear potential between quarks universality of the string tension lattice studies for QQ, 3Q potential Creutz, PRL43, 553 (1979) T.T.Takahashi et al, PRL86, 18 (2001) String picture of hadrons String picture of hadrons gives natural explanation for: The string picture may share important part of QCD.
z We search for the solution with cylindrical symmetry & topological charge = n. gauge fixing: at vortex core, φ＝0 minimize with B.C for finite energy vortex solution in asymptotic region, sym. is restored Static solution G-L parameter topological quantization(topological charge n)
Higgs photon mixing bound state of static vortex and - mixed state Importance of mixing around the core, Higgs & photon are mixed static vortex – Higgs bound state Type-II BPS Type-I around BPS Same threshold leads the corporative behavior of Higgs & photon then, lowest excitation energy is considerably decreased.
ex） Type-II case V(r) for α (monopole) 2 V(r) for β (dual-photon) 0 V(r) for α-β mixing =Mmono2 The property of the potential in the radial direction energy “threshold” for continuum states V(r) potential induced by static vortex 2 =Md-photon2 (independent of κ2) “asymptotic” region ( r → ∞ ) “central” region ( r → 0 ) state below threshold state above threshold for all states α(r) →rm × const. α(r) →0 α(r) →eipr β(r) →rm× const. β(r) →0 α(r) →eipr (m ≧ 2 )
monopoles monopoles electric field electric field energy density energy density Static profile for Type-I & II κ＝δ/ξ(＝λ1/2/ e) : G-L parameter condensed matter DGL (QCD effective theory) penetration depth δ: ～ 500 A, 0.3 - 0.4 fm coherence length ξ: 25 – 104 A, 0.16 fm G-L parameter κ: 0.05－20, ~ 1.6 － 2.0 ex) pure metal ex) high Tc SC, metal with inpurity BPS =1/2 ξ δ (usually not considered) finite thickness string like
R R Abelian dominance off-diagonal gluon is heavy ( ~ 1.2 GeV ) Amemiya-Suganuma, PRD60(1999)114509 We will consider color flux tube linking specific charges. To discuss the color flux linking specific charges, we have only to consider this part. dual photon field monopole field same form as the Ginzburg-Landau type action Ginzburg-Landau action: Higgs (Cooper-pair) field photon field for Abrikosov vortex We have only to consider the GL-type action.
monopoles d-photon 1st excited state – wavefunction in the radialdirection fluctuation of electric field Ez fluctuations of φ、 aθ small squeezed by monopoles Type-I ( total flux is conserved to 0) Ez Ez (around) squeezed by monopoles ( total flux is conserved to 0) BPS corporative Ez oscillation ~ eipr / r1/2 Type-II →long tail oscillation ~ eipr / r1/2 corporative large r r
B B B B B Thermodynamical Stability ( Mn：vortex mass with topological charge n ) vortex-vortex interaction vortex-vortex interaction exact solution de Vega-Schaposnik, PRD14,1100(1976) attractive repulsive 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, B not uniform 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= 1/2case ) 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 n=2 n=3 n = 2, 3 energy spectrum 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. The property as static vortex + photon excitation becomes strong. The threshold is unchanged, then continuum states behave like n =1 continuum states.
z Higgs HZ r r r ω、ｋｚ Summary for single Abelian vortex We have discussed the “peristaltic” modes of single vortex. We found, new discrete pole around κ2 ~ 1/2. This discrete pole is characterized by the corporative behavior of the Higgs and photon fields. As κ2 is increased, the low excitation modes change from the Higgs dominant modes to the photon dominant modes. Asnis increased, photon can excite more easily, and lowest excitation becomessofterone.
R Summary for single color electric flux We can directly apply the previous arguments to the color flux linking specific color charge. R profile in radial direction DGL gives κ2 ~ 3 - 4. → Type-II (mass of color electric flux per 1fm ~ 1.0 GeV/fm.) Only resonant scattering type of excitations appear. r excitation energy ~ 0.5 GeV. electric flux vibrate with long tail.
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 showonly the static profile, vortex- vortex potential, and vortex- anti vortex potential.
Bz same topological charge no vortex total magnetic flux is zero . Bz 1.0 fm 2.0 fm 3.0 fm 2.0 GeV 1.5 GeV R R 1.5 GeV sudden annihilation of the flux 1.0 GeV 0.5 GeV When d < 1.0 fm, our B.C, |ψ|２= 0 at the core is no more applicable. R R Vortex – antivortex “potential” per unit length ( for DGL, per 1 fm )
BPS Energy spectrum: New-type discrete pole V(r) Type-II monopoles dual photon field Type-I dual photon field monopoles Around BPS saturation, characteristic discrete pole appear as a result of monopoles – d-photon corporative behavior.
Sudden annihilation of fluxes (in DGL unit) 1.2 fm 1.0 fm Around d = 1.0 - 1.2 fm, the fluxes suddenly annihilate. This critical distance dcris related with the penetration depth δ dcr ~ (1.5 - 2.0) × 2δ This value seems to be considerably large.
Abelian projection ‘t Hooft, NPB190,455(1981) SU(3) gauge theory (QCD) U(1)×U(1) gauge theory (fix the gauge of the off diagonal elements) 2-gluon (“photon”) 8-gluon 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. mapping R3in physical space SU(2) variablesin internal space
Two vortices system ( static case ) field degrees of freedom： Reψ, Imψ, Ax, Ay (without cylindrical sym.) As the previous 1-vortex system, we first search for the static profiles which minimize the static energy. Step 1) Starting from the case where the distance between two vortices is large, adopt the product ansatz for initialB.C: ψ1+2＝ψ1 ψ2 /const. A1+2＝A1+ A2 (We need this B.C. only at the beginning of the calculation) 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. 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 .
Summary: We have discussed the “peristaltic” modes of single vortex. We found, new discrete pole around κ2 ~ 1/2. This discrete pole is characterized by the corporative behavior of the Cooper-pair and photon fields. To discuss thedynamics 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.
Abelian projection ‘t Hooft, NPB190,455(1981) SU(3) gauge theory (QCD) U(1)×U(1) gauge theory (fix the gauge of the off diagonal elements) 2-gluon (“photon”) 8-gluon 6-gluon (“charged matter”) Abelian monopole (topological object) Introduction Abrikosov vortex in U(1) theory Color confinement in QCD Y.Nambu, PRD.122,4262(1974) A.A.Abrikosov, Soviet Phys.JTEP 5, 1174(1957) ‘t Hooft , Nucl.Phys.B190.455(1981) Mandelstam, Phys.Rep.C23.245(1976) B Cooper-pair condensation magnetic monopole condensation dual Meissner effect dual Meissner effect squeezed color electric flux squeezed magnetic field B E
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
gauge fixing: Regge trajectories of hadrons (hadron mass)2 angular momentum constant string tension
Model – dual Ginzburg-Landau model Ezawa-Iwazaki, PRD25(1982)2681 Maedan-Suzuki, PTP81(1989)229 1) Abelian gauge fixing → U(1)2 monopole DOF naturally appear as topological objects. 2) include the auxiliary field U(1)mｘU(1)m dual photon B which couples with the monopole current. (Zwanziger, PRD3(19７0)880 ) 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) 4) integrate out the U(1)e ｘ U(1)e photon field A
z We search for the solution with cylindrical symmetry & topological charge = n. gauge fixing: monopoles monopoles electric field electric field energy density energy density Static solution G-L parameter Ansatz for topological charge n at vortex core, φ＝0 minimize static energy with B.C for finite energy vortex solution in asymptotic region, sym. is restored