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:1008.3605with Hiroyuki HATA

(Kyoto Univ.)

Introduction

and its soliton solution (Skyrmion)

We consider Skyrme theory (2-flavors),

,

.

The Skyrmion is not rotationally symmetric, and has free parameter ;

collective coordinate

Substitute this into the action:

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

Rigid body approximation [Adkins-Nappi-Witten, 83]

.

How do we extract its dynamics?

Ω ～ 10 s23 -1 velocity at r=1fm ～ light velocity

The necessity of the relativistic corrections

Large contribution of the rotational energy

②

Energy

0 Ωnucleon delta

939MeV 1232MeV

8% 30%

①

High frequency

The relativistic corrections seem to be important. How do Skyrmions deform due to spinning motion?

Deformation of spinning Skyrmions

lab frame body-fixed frame

spinning deformed Skyrmion static Skyrmion

. ..

(1)

( ２ )

( ３)

. ..

Deformation of spinning Skyrmions

Particular combinations of A,B,C correspond to three modes of deformation.

C

2B

-A+2B+C

. ..

left and right constant SO(3) transformations on

Deformation of spinning Skyrmions

rotations of field in real and iso space

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

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

For example, for ,

Energy and isospin with corrections

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

nucleon:delta:

as inputs.

,

We are now ready to obtain the numerical results.

Result 1. the shape of the baryons

original static Skyrmion

(at r=1 fm)

nucleondelta

Result 2. relativistic corrections to physical quantities

The fundamental parameter of the theory becomes better.

rigid body ours experiment

125MeV 108MeV 186MeV

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

0.68fm0.59fm 0.81fm

1.04fm1.17fm 0.94fm

1.971.65 2.79

・・・

0.95fm0.85fm・・・・・・ ・・・

0.82fm

A comment on the numerical results

delta 68 14 18： ：nucleon 89 7 4： ： (%)

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

Conclusion

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

it does not seem that these are good convergent series.

Summary

We calculated the leading relativistic corrections to the spinning Skyrmions.

We found that the relativistic corrections are numerically important.

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

・

・

・

⇒ The shape of the baryons ⇒ Relativistic corrections to various physical quantities

Back up Slides

Exp. OursRigid

○

×

×

×

×

××

○

×

win: ○lose: ×

10%-20% relativistic corrections . Generally, the numerical values get worse.

numerical results for nucleon properties

1fm

(1)

( ２ )

( ３)

C

2B

-A+2B+C