VORTEX SOLUTIONS IN THE EXTENDED SKYRME FADDEEV MODEL

VORTEX SOLUTIONS IN THE EXTENDED SKYRME FADDEEV MODEL NOBUYUKI SAWADO Tokyo University of Science, Japan sawado@ph. noda. tus. ac. jp In collaboration with 　　Luiz Agostinho Ferreira, Pawel Klimas (IFSC/USP) Masahiro Hayasaka (TUS) 　　　 Juha Jäykkä (Nordita) Kouichi Toda (TPU) 　　　ar. Xiv: 0908. 3672 , ar. Xiv: 1112. 1085, ar. Xiv: 1209. 6452, ar. Xiv: 1210. 7523 19 December, 2012 At Miami 2012: A topical conference on elementary particles, astrophysics, and cosmology, 13 -20 December, Fort Lauderdale, Florida

Objects of Yang-Mills theory (i) Gauge + Higgs composite models Abelian vortex (in U(1)) 　　　 Abrikosov vortex, graphene, cosmic string, Brane world, etc. ‘t. Hooft Polyakov monopole 　　　　　　GUT, Nucleon catalysis (Callan-Rubakov effect), etc. (ii) Pure Yang-Mills theory Instantons In the Cho-Faddeev-Niemi-Shabanov decomposition Monopole loop N. Fukui, et. al. , PRD 86(2012)065020, ``Magnetic monopole loops generated from two-instanton solutions: Jackiw-Nohl-Rebbi versus 't Hooft instanton” Condensates in a dual superconductivity 　　　　Confinement The Skyrme-Faddeev Hopfions, vortices Glueball? , Abrikosov vortex? , Branes?

Exotic structures of the vortex…… Semi-local strings M. N. Chernodub and A. . S. Nedelin, PRD 81, 125022(2010) ``Pipelike current-carrying vortices in two-component condensates’’ The Ginzburg-Landau equation P. J. Pereira, L. F. Chibotaru, V. V. Moshchalkov, PRB 84, 144504 (2011) ``Vortex matter in mesoscopic two-gap superconductor square’’

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.

Cho-Faddeev-Niemi-Shabanov (CFNS) decomposition L. D. Faddeev, A. J. Niemi, Phy. Rev. Lett. 82 (1999) 1624, ``Partially dual variables in SU(2) Yang-Mills theory” Degrees of freedom electric 2 magnetic 2 3× 4 ― 6 = 6 remaining terms 1 The Gies lagrangian 1 6 H. Gies, Phys. Rev. D 63, 125023 (2001), ``Wilsonian effective action for SU(2) Yang-Mills theory with Cho-Faddeev-Niemi -Shabanov decomposition ``renormalization group time’’

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

The vortex solution in the integrable sector L. A. Ferreira, JHEP 05(2009)001, ``Exact vortex solutions in an extended Skyrme-Faddeev model” O. Alvarez, LAF, et. al, PPB 529(1998)689, ``A new approach to integrable theories in any dimension” The zero curvature condition One gets the infinite number of conserved quantity Additional constraint The equation becomes Traveling wave vortex

Ansatz The equation The solution has of the form:

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

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

Analytical solutions for n = 1, 2

The energy of the static/traveling wave vortex

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

The components: The charge per unit length:

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

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, PRD 19, 3008 (1979), ``Multivortex solutions of the Ginzburg-Landau equations” The vortex matter/lattice structure is observed.

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.

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

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

Hopfions(closed vortex) Coordinates: L. A. Ferreira, NS, et. al. , JHEP 11(2009)124, ``Static Hopfions in the extended Skyrme-Faddeev model” Axially symmetric ansatz Boundary conditions Hopf charge Non-axisymmetric case: D. Foster, ar. Xiv: 1210. 0926

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

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