Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion
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Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion. Toru KIKUCHI (Kyoto Univ.). Based on arXiv:1002.2464 ( Phys. Rev. D 82, 025017 ) arXiv:1008.3605. with Hiroyuki HATA (Kyoto Univ.) . Introduction. We consider Skyrme theory (2-flavors),. ,.

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Relativistic collective coordinate system of solitons and spinning skyrmion

Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion

Toru KIKUCHI

(Kyoto Univ.)

Based on

arXiv:1002.2464 (Phys. Rev. D 82, 025017)

arXiv:1008.3605

with Hiroyuki HATA (Kyoto Univ.)


Relativistic collective coordinate system of solitons and spinning skyrmion

Introduction Spinning Skyrmion

We consider Skyrme theory (2-flavors),

,

and its soliton solution(Skyrmion)

.

The Skyrmion is not rotationally symmetric, and has free parameter

;

collective coordinate


Relativistic collective coordinate system of solitons and spinning skyrmion

Skyrmions represent baryons. Spinning Skyrmion

The collective coordinate describes the d.o.f. of spins and isospins.

How do we extract its dynamics?

Rigid body approximation

[Adkins-Nappi-Witten, 83]

.

Substitute this into the action:


Relativistic collective coordinate system of solitons and spinning skyrmion

The necessity of the relativistic corrections Spinning Skyrmion

Large contribution of

the rotational energy

939MeV

1232MeV

8%

30%

Energy

0

Ω

nucleon

delta

High frequency

23

-1

velocity at r=1fm ~ light velocity

Ω ~ 10 s

The relativistic corrections seem to be important.

How do Skyrmions deform due to spinning motion?


Relativistic collective coordinate system of solitons and spinning skyrmion

Deformation of spinning Skyrmions Spinning Skyrmion

.

.

.

labframe

body-fixed frame

static Skyrmion

spinning deformed Skyrmion


Relativistic collective coordinate system of solitons and spinning skyrmion

Deformation of spinning Skyrmions Spinning Skyrmion

.

.

.

C

(3)

2B

Particular combinations of A,B,C correspond to

three modes of deformation.

(1)

(2)

-A+2B+C


Relativistic collective coordinate system of solitons and spinning skyrmion

Deformation of spinning Skyrmions Spinning Skyrmion

.

.

.

These are the most general terms that share several properties with the rigid body approximation.

ex.)

left and right constant SO(3) transformations on

rotations of field in real and iso space


Relativistic collective coordinate system of solitons and spinning skyrmion

Requiring this Spinning Skyrmion to satisfy field theory EOM for constant , we get three differential equations for A,B,C.

For example, for

,


Relativistic collective coordinate system of solitons and spinning skyrmion

Energy and isospin with corrections Spinning Skyrmion

To fix the parameters of the theory, take the data of

nucleon:

,

delta:

as inputs.

We are now ready to obtain the numerical results.


Relativistic collective coordinate system of solitons and spinning skyrmion

Result 1. the shape of the baryons Spinning Skyrmion

delta

nucleon

original static Skyrmion

(at r=1 fm)


Relativistic collective coordinate system of solitons and spinning skyrmion

Result 2. relativistic corrections to physical quantities Spinning Skyrmion

ours

rigid body

experiment

125MeV

108MeV

186MeV

0.59fm

0.68fm

0.81fm

1.04fm

0.94fm

1.17fm

0.85fm

0.95fm

0.82fm

2.79

1.65

1.97

・・・

・・・

・・・

・・・

The fundamental parameter of the theory becomes better.

However, most of the static properties of nucleon become worse.


Relativistic collective coordinate system of solitons and spinning skyrmion

A comment on the numerical results Spinning Skyrmion

Looking at the numerical ratio of each term of the energy,

nucleon

89

7

4

(%)

delta

14

18

68

it does not seem that these are good convergent series.

Conclusion

Relativistic corrections are important.

In fact, they are so large that our Ω-expansion is not a good one.


Relativistic collective coordinate system of solitons and spinning skyrmion

Summary Spinning Skyrmion

We calculated the leading relativistic corrections to the spinning Skyrmions.

⇒ The shape of the baryons

⇒ Relativistic corrections to various physical quantities

We found that the relativistic corrections are numerically important.

For more appropriate analysis of the spinning Skyrmion, a method beyond Ω-expansion is needed.


Relativistic collective coordinate system of solitons and spinning skyrmion

Back up Slides Spinning Skyrmion


Relativistic collective coordinate system of solitons and spinning skyrmion

numerical results for nucleon properties Spinning Skyrmion

win: ○

lose: ×

×

×

×

×

×

×

×

Exp.

Ours

Rigid

10%-20% relativistic corrections .

Generally, the numerical values get worse.


Relativistic collective coordinate system of solitons and spinning skyrmion

1fm Spinning Skyrmion


Relativistic collective coordinate system of solitons and spinning skyrmion

C Spinning Skyrmion

2B

(3)

(1)

(2)

-A+2B+C