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 SAWADOTokyo 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

Objects of Yang-Mills theory

ℒ=𝛼 (𝜕𝜇❑�⃗�)2+𝛽 (𝜕𝜇

❑�⃗�×𝜕𝜈❑�⃗�)2+𝛾(𝜕𝜇

❑�⃗�)4

(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”

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

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

electric magnetic remaining terms

22 1 1

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”

t = ln k/ L ``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

``Magnetic symmetry’’

Lagrangian (in Minkowski space)

Sterographic project

Static hamiltonian

Positive definite for

The integrability: the analytical vortex solutions

The equation of the vortex

The zero curvature condition 𝜕𝜇𝑢𝜕𝜇𝑢=0

𝛽𝑒2=1 𝜕𝜇𝜕𝜇𝑢=0The equation becomes

or

or )(

𝑢=𝑣 (𝑧 )𝑤 (𝑦 )=𝑧𝑛𝑒𝑖𝑦𝑧=𝑥1+ 𝑖𝜖1𝑥2 , 𝑦=𝑥3−𝜖2𝑥

0

¿ (𝜌𝑎 )𝑛

𝑒𝑖 [𝜖1𝑛𝜑+𝑘 (𝑥3+𝜖2 𝑥0)]

Traveling wave vortex

The vortex solution in the integrable sectorL.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

(0)

The equation

The solution has of the form:

We have no solutions for

𝑢 (𝒓 , 𝑡 )=√ 1−𝑔 (𝑦 )𝑔 (𝑦 )𝑒𝑖(𝑛𝜑+𝜆𝑧+𝑘𝜏 )

𝑥𝜇=𝑟 0(𝜏 ,𝜌 cos𝜑 ,𝜌 sin𝜑 , 𝑧)

Vortex solutions in

Ansatz

= 0

and for

for

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.

The baby-skyrmion potential

Plug into the equation it is written as

and

Assume the zero curvature condition

0=( 𝛽𝑒2−1 ) 4𝑛3

𝑎4 {1+( 𝜌𝑎 )2𝑛}

− 3

[ (𝑛−1 )( 𝜌𝑎 )2𝑛−4

−(𝑛+1)( 𝜌𝑎 )4𝑛−4 ]

+2𝑟02𝜇2

𝑀 2 {1+(𝜌𝑎 )2𝑁}

−3

[(2+ 2𝑁 )( 𝜌𝑎 )4𝑁− 4

−(2− 2𝑁 )( 𝜌𝑎 )

2𝑁 −4 ]

with the potential: we assume

𝑢 (𝜌 ,𝜑 ,𝑧 ,𝑡 )=(𝜌𝑎 )𝑛

𝑒𝑖 [𝜖𝑛𝜑+𝑘 (𝑧+𝜏)]

𝑉 𝛼𝛽=𝜇2

2(1+𝑛3❑)𝛼(1−𝑛3❑)𝛾𝛼 ≥0𝛾>0

Analytical solutions for n = 1, 2

𝑎=|𝑛|4√𝑀2(𝛽𝑒2−1)𝑟 0

2𝜇2 and

𝑛=1 ,𝑘2=0.0 ,𝑟 0

2𝜇2

𝑀 2 =1.0 𝑛=2 ,𝑘2=0.0 ,𝑟02𝜇2

𝑀 2 =1.0

The energy per unit length of the traveling wave vortex with

The static energy per unit of length of the vortex with

The energy of the static/traveling wave vortex

𝐸𝑠𝑡𝑎𝑡𝑖𝑐=2𝜋+4 𝜋31

𝑎2(𝛽𝑒2−1)

The infinite number of conserved current

𝐽𝜇≔𝛿𝐺𝛿𝑢∗𝒦𝜇−

𝛿𝐺𝛿𝑢

𝒦𝜇∗ h𝑤 𝑒𝑟𝑒𝐺≔𝐺 (|𝑢|2 )

¿𝛿𝐺𝛿𝑢

¿

Thus the current is always conserved:

And the equation of motion is written as

𝜕𝜇𝒦𝜇−2𝑢∗ (1+|𝑢|2 )𝒦𝜇𝜕

𝜇𝑢=− 𝜇2

4¿¿

The zero curvature condition 𝒦𝜇𝜕𝜇𝑢=𝒦∗

𝜇𝜕𝜇𝑢∗=0 ,𝒦∗

𝜇𝜕𝜇𝑢=𝒦𝜇𝜕

𝜇𝑢∗

𝐽𝜌=0The transverse spatial structure of the polar component and the longitudinal component are a pipelike structure.

The charge per unit length:

𝑄=∫𝑑𝑥1𝑑𝑥2 𝐽 0=−8𝜋 𝑀 2𝑘𝑎2𝑟0 [𝑛6 1𝑎2 ( 𝛽𝑒2−1 )+ 1𝑛Γ (1+

1𝑛)Γ (1−

1𝑛)]

For

we get Noether current with

𝐽𝜇=−4 𝑖𝑀2 𝑢𝜕𝜇𝑢

∗−𝑢∗ 𝜕𝜇𝑢

(1+|𝑢|2 )2− 𝑖 8𝑒2

(𝛽𝑒2−1)2(𝜕𝜈𝑢𝜕

𝜈𝑢∗)(𝜕𝜇𝑢∗𝑢−𝑢∗𝜕𝜇𝑢)

¿¿¿

The components:

Broken axisymmetry of the solution

The energy density plot of for old-, and new-baby potentials

Old baby skyrmion potential

New baby skyrmion potential

　　　　　　　　　　　　 (old) 　　　　　　　　　　　　 newNonsymmetric: old

For the potential , the holomorphic solutions appear as a ground state!

Symmetric:

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

A sequence of the energy density plots of for the several for

the old-potential

𝛽 𝑒2=1.01 𝛽 𝑒2=1.1 𝛽 𝑒2=2.0 𝛽 𝑒2=20.0

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.

SummaryWe 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 toobtain the integrable vortex solutions.

OutlookWhat it the origin of the potential?How can I observe our solutions in Physics? For SC, we may introduce an external magnetic field

and see the structure change for the field. Geometrical patterns appear?

Our integrable solution thus carries an infinite number of conserved quantity.

The model (two gap model) hides a SU(2) structure even if it describes the U(1) like observation such as SC.

Thank you ！Tanzan Jinja shrine,Japan, 16 Nov.,2012

Lago Mar Resort, USA, 17 Dec.,2012

The Skyrme-Faddeev modelL.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

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

(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)

corresponds to the zero curvature condition

Dimensionless energy, Integrability

The solution is close to the Integrable sector, but not exact.

𝛽𝑒2