1. VORTEX SOLUTIONS IN THE EXTENDED SKYRME FADDEEV MODEL In collaboration with Luiz Agostinho Ferreira, Pawel Klimas (IFSC/USP) Masahiro Hayasaka (TUS) Juha Jäykkä (Nordita) Kouichi Toda (TPU) NOBUYUKI SAWADO Tokyo University of Science, Japan sawado@ph.noda.tus.ac.jp arXiv:0908.3672, arXiv:1112.1085, arXiv:1209.6452, arXiv:1210.7523 At Miami 2012: A topical conference on elementary particles, astrophysics, and cosmology, 13-20 December, Fort Lauderdale, Florida 19 December, 2012

2. Objects of Yang-Mills theory (i)Gauge + Higgs composite models Abelian vortex (in U(1)) Abrikosov vortex, graphene, cosmic string, Brane world, etc. ‘tHooft Polyakov monopole GUT, Nucleon catalysis ( Callan-Rubakov effect), etc. The Skyrme-Faddeev Hopfions, vortices Glueball?, Abrikosov vortex?, Branes? (ii)Pure Yang-Mills theory Instantons In the Cho-Faddeev-Niemi-Shabanov decomposition Monopole loop Condensates in a dual superconductivity Confinement N.Fukui,et.al.,PRD86(2012)065020,``Magnetic monopole loops generated from two-instanton solutions: Jackiw-Nohl-Rebbi versus 't Hooft instanton”

3. Exotic structures of the vortex…… M.N.Chernodub and A..S. Nedelin, PRD81,125022(2010) ``Pipelike current-carrying vortices in two-component condensates’’ P.J.Pereira,L.F.Chibotaru, V.V.Moshchalkov, PRB84,144504 (2011) ``Vortex matter in mesoscopic two-gap superconductor square’’ Semi-local strings The Ginzburg-Landau equation

4. Summary We got the integrable and also the numerical solutions of the vortices in the extended Skyrme Faddeev Model. A special form of potential is introduced in order to stabilize and to obtian the integrable vortex solutions. We begin with the basic formulation.

5. Cho-Faddeev-Niemi-Shabanov (CFNS) decomposition electric magnetic remaining terms 2211 3×4 ― 6 = 6Degrees of freedom 6 L.D. Faddeev, A.J. Niemi, Phy.Rev.Lett.82 (1999) 1624, ``Partially dual variables in SU(2) Yang-Mills theory” ``renormalization group time’’ H. Gies, Phys. Rev. D63, 125023 (2001),``Wilsonian effective action for SU(2) Yang-Mills theory with Cho-Faddeev-Niemi -Shabanov decomposition The Gies lagrangian

6. Lagrangian (in Minkowski space) Sterographic project Static hamiltonian Positive definite for The integrability: the analytical vortex solutions The equation of the vortex

7. The zero curvature condition The equation becomes Traveling wave vortex The vortex solution in the integrable sector L.A.Ferreira, JHEP05(2009)001,``Exact vortex solutions in an extended Skyrme-Faddeev model” O.Alvarez,LAF,et.al,PPB529(1998)689,``A new approach to integrable theories in any dimension” One gets the infinite number of conserved quantity Additional constraint

8. The equation The solution has of the form: Ansatz

9. Derrick’s scaling argument G.H.Derrick, J.Math.Phys.5,1252 (1964), ``Comments on nonlinear wave equations as models for elementary particles’’ Scaling: Consider a model of scalar field: We need to introduce form of a potential to stabilize the solution.

10. The baby-skyrmion potential Plug into the equation it is written as Assume the zero curvature condition

11. Analytical solutions for n = 1, 2

12. The energy of the static/traveling wave vortex

13. The infinite number of conserved current Thus the current is always conserved: And the equation of motion is written as

14. The charge per unit length: The components:

15. Broken axisymmetry of the solution Symmetric: The baby-skyrmion exhibits a non-axisymmetric solution depending on a choice of potential I.Hen et. al, Nonlinearity 21 (2008) 399

16. A repulsive force between the core of the vortices might appear It might be similar with the force between the Abrikosov vortex. Erick J.Weinberg, PRD19,3008 (1979), ``Multivortex solutions of the Ginzburg-Landau equations” The vortex matter/lattice structure is observed.

17. Summary We got the integrable and the numerical solutions of the vortices in the extended Skyrme Faddeev Model. A special form of potential is introduced in order to stabilize and to obtain the integrable vortex solutions.

18. Thank you ！ Tanzan Jinja shrine,Japan, 16 Nov.,2012 Lago Mar Resort, USA, 17 Dec.,2012

20. The Skyrme-Faddeev model L.Faddeev, A.Niemi, Nature (London) 387, 58 (1997), ``Knots and particles’’ R.A.Battye, P.M.Sutcliffe, Phys.Rev.Lett.81,4798(1998) Lagrangian Static hamiltonian Positive definite for

21. Boundary conditions Coordinates: Hopfions(closed vortex) Hopf charge L.A.Ferreira, NS, et.al., JHEP11(2009)124, ``Static Hopfions in the extended Skyrme-Faddeev model” Axially symmetric ansatz Non-axisymmetric case: D.Foster, arXiv:1210.0926

22. (m, n) = (1, 1)(1, 2)(2, 1) (m, n) = (1, 3) (m, n) = (1, 4) (2, 2)(4, 1) Hopf charge density (3, 1)

23. corresponds to the zero curvature condition Dimensionless energy, Integrability The solution is close to the Integrable sector, but not exact.