Relativistic Collective Coordinate System of Solitons and Spinning Skyrmion Toru KIKUCHI (Kyoto Univ.) Based on arXiv: ( Phys. Rev. D 82, ) arXiv: with 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 s 23 velocity at r=1fm ～ light velocity The necessity of the relativistic corrections Large contribution of the rotational energy ② Energy 0 Ω nucleondelta 939MeV1232MeV 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 framebody-fixed frame spinning deformed Skyrmion static Skyrmion...

(1) (２)(２) (３)(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) nucleon delta

Result 2. relativistic corrections to physical quantities The fundamental parameter of the theory becomes better. rigid body ours experiment 125MeV 108MeV186MeV However, most of the static properties of nucleon become worse. 0.68fm 0.59fm 0.81fm 1.04fm 1.17fm 0.94fm fm 0.85fm 0.82fm

A comment on the numerical results delta ：： nucleon 8974 ：： (%) 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

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Exp. Ours Rigid ○ × × × × × × ○ × win: ○ lose: × 10%-20% relativistic corrections. Generally, the numerical values get worse. numerical results for nucleon properties

1fm

(1) (２)(２) (３)(３) C 2B -A+2B+C