Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion
description
Transcript of Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion
![Page 1: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/1.jpg)
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.)
![Page 2: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/2.jpg)
Introduction
and its soliton solution (Skyrmion)
We consider Skyrme theory (2-flavors),
,
.
The Skyrmion is not rotationally symmetric, and has free parameter ;
collective coordinate
![Page 3: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/3.jpg)
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?
![Page 4: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/4.jpg)
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?
![Page 5: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/5.jpg)
Deformation of spinning Skyrmions
lab frame body-fixed frame
spinning deformed Skyrmion static Skyrmion
. ..
![Page 6: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/6.jpg)
(1)
( 2 )
( 3)
. ..
Deformation of spinning Skyrmions
Particular combinations of A,B,C correspond to three modes of deformation.
C
2B
-A+2B+C
![Page 7: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/7.jpg)
. ..
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.)
![Page 8: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/8.jpg)
Requiring this to satisfy field theory EOM for constant , we get three differential equations for A,B,C.
For example, for ,
![Page 9: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/9.jpg)
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.
![Page 10: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/10.jpg)
Result 1. the shape of the baryons
original static Skyrmion
(at r=1 fm)
nucleondelta
![Page 11: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/11.jpg)
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
![Page 12: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/12.jpg)
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.
![Page 13: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/13.jpg)
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
![Page 14: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/14.jpg)
Back up Slides
![Page 15: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/15.jpg)
Exp. OursRigid
○
×
×
×
×
××
○
×
win: ○lose: ×
10%-20% relativistic corrections . Generally, the numerical values get worse.
numerical results for nucleon properties
![Page 16: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/16.jpg)
![Page 17: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/17.jpg)
1fm
![Page 18: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/18.jpg)
(1)
( 2 )
( 3)
C
2B
-A+2B+C
![Page 19: Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion](https://reader036.fdocuments.net/reader036/viewer/2022081520/568165bc550346895dd8baf6/html5/thumbnails/19.jpg)