SKYRMIONS WITH VECTOR MESONS: SINGLE SKYRMION AND BARYONIC MATTER

Yongseok Oh (Kyungpook National University)

2013.6.26 SEMINAR @ BLTP, JINR

13년 6월 26일 수요일 CONTENTS

✴ Motivation ✴ Approaches in the Skyrme model • Conventional approaches / short review on the earlier works • HLS with hQCD • Lagrangian up to O(p4) • Role of vector mesons (ρ vs. ω) in Skyrmion properties • Nuclear matter ✴ Outlook

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13년 6월 26일 수요일 Collaborators:

研究室 クォーク・ハドロン理論研究室Mannque Rho Hyun Kyu Lee H Masayasu Harada Byung-Yoon Park ク・ハドロン多体系の解明を目指す「有限温度・有限密度 QCD」，および，「エキゾチックハドロン」です．これらの 研究対象に対して，現在世界各地で様々な実験が行われ， 世界各国の研究者が多数携わっています．そして，理論の 検証だけでなく，従来の理論からは予想もされなかった新 たな謎・知見を導く実験結果を提出している状況であり，Yong-Liang Ma Ghil-Seok Yang 理論側からの実験への物理提言が求められています．今後 も世界各地で数々の実験が計画されており，実験結果に裏 クォーク・グルーオン多体系の相図（理論的予想） 打ちされた理論の発展，そして理論側からの実験への提言 といったように，理論と実験の双方からQCDの解明がより 原田正康教授（前列中央） 野中千穂助教（前列左1番目） Ma, Yong-Liang いっそう進むことが期待されています． 特任助教（前列右1番目），大学院生と学部4年生 当研究室では，現在の実験から提出されている多くの謎 を解決すると共に，将来の実験に対して提言を行うべく， 験と密接な関連のもとに，有効模型，現象論的模型，格子 有効理論の枠組みでの新たな理論の開発とそれを用いた解 QCDなどの様々な手法を用い，これまでの理論では説明で 析，現象論的模型の枠組みでの新しい理論や模型の開発と きないデータに対して新たな理論を構築すると共に，将来 それを用いた解析，格子QCDを用いたQCD基礎論に基づく の実験に対しての提言を行うべく理論的解析を行っていま 解析，そして最近では超弦理論での双対性に基づく模型を 原田正康教授 す． 用いた解析などを行っています．これらの理論からハドロ ン物理の多角的な理解を目指します．国内だけでなく，ア （1）相転移とそれに伴う現象の解明 * 原田正康 教授 Masayasu Harada, Prof. メリカ，フランス，韓国などの研究者との国際的共同研究 ハドロン相からQGP相への相転移の兆候として，カイラ も積極的に行っています． 野中千穂 助教 Chiho Nonaka, Assist. Prof. ル対称性の回復に伴ってハドロンの性質が変化することが 以下に，当研究室での主な研究対象を紹介します． パイ中間子（メソン）と陽子（バリオン）の楕円フローの横運動量による変化． Ma, Yong-Liang 特任助教 Yong-Liang Ma, Research Assist. Prof. 期待されており，実験的シグナルを明らかにすることが重 実線と点線が理論的計算の結果．楕円フローが最大となる横運動量の値が 有限温度・有限密度QCD 要となっています．当研究室では，これらの相転移点近傍 ほぼ2：3（＝メソン：バリオン）と，構成クォークの数の比になってい の構造を，実験との関連のもとに解明すべく，理論的手法 ることを示し，RHICにおけるQGP生成を強く示唆するものである． 有限温度・有限密度QCDは，高温度・高密度状態での を開発しながら研究を進めています． クォーク・ハドロン多体系の相構造（図参照），QCD相転 研究室では，クォーク・グルーオンの力学を記述する 移機構やハドロンの性質の変化などを研究対象とするもの （2）重イオン衝突実験の現象論的解析 待されています．しかし，低温度・高密度状態において理 当基礎理論，量子色力学（QCD）の解明を目標として です． ハドロンの性質の変化，相転移現象の解明をめざし，一 論上の困難も多く，理解されていないことが数多く残され 研究を行なっています．ハドロン物理学の課題の一つに， この有限温度・有限密度QCDの解明のために，宇宙初期 連の重イオン衝突実験が行われてきました．特に近年では ています．当研究室では格子QCDそしてQCD有効模型を用 クォークやグルーオンが自由粒子のように振る舞う新しい の高温状態を実験室で再現すべく，一連の実験が行われて RHIC，そしてLHCといった高エネルギー重イオン衝突実 いて，この問題にアプローチしています． 物質相「クォーク・グルーオン プラズマ（QGP）」の物 います．2000年より稼働しているアメリカ・ブルックヘヴ 験が稼働し予想外の興味深い結果が次々と得られてきまし 性研究があります．この新しい物質相はQCDの漸近的自 ン国立研究所の重イオン衝突実験（RHIC実験）からこれ た．その結果の一つが強結合クォーク・グルーオン プラズ エキゾチックハドロン 由性（2004年ノーベル物理学賞）により予想されていたも までの理論では説明できない結果が次々と得られ，新しい －6 マ（QGP）です．これはQCD基礎論，QCD有効理論，実験 シグマ中間子は，質量生成機構と深く関係するクォーク のです．QGP状態はビッグバン後，10 秒後に存在したと 理論の整備が必要となっています．さらに2009年に稼働を とQCDを結びつける現象論的模型といった理論研究からの 2体（クォークと反クォーク）で構成される「シグナル粒子」 考えられ，QGP物理の理解は素粒子・原子核物理の重要な 開始したCERN/LHC実験からは昨年新しいデータが報告さ 実験結果理解の成功から導きだされたものです．当研究室 である可能性が高いものです．しかし，クォーク4体（クォー 課題であるだけでなく詳細な宇宙史の解明にもつながりま れてきました．数年後にはドイツ重イオン研究所（GSI）/ では，理論研究から実験結果に潜むQGPの普遍的な事実を ク2個と反クォーク2個）で構成されている可能性も指摘さ す．もう一つの課題として，質量生成機構を明らかにする FAIR実験，ドゥブナ合同原子核研究所（JINR）/NICA実験 探ることを目標にしています． れています．また，近年発見されたX（3872）等のチャー ことがあります．我々の質量の大部分は原子核を構成して が稼動する予定です．このように低エネルギーから高エネ ムクォークを含む新しい粒子もクォーク4体から構成され いる核子（陽子・中性子）で与えられます．核子を構成す ルギー領域までの豊富な実験結果が今まさに期待されると －30 （3）低温度・高密度領域 ている可能性が高いと考えられています． るクォークの質量はせいぜい20×10 kgと見積もられて きです．これらの重イオン衝突実験の理解にはQCD基礎論 －30 RHICそしてLHCでは高温度・低密度領域の物理が解明 当研究室では，これらのシグマ中間子やX（3872）等が， いて，クォーク3個を集めても陽子の質量1700×10 kgに だけでなく，実験状況の理解といった総合的な深い物理の されていくと期待されています．それに対し，相図上のも クォーク4体から構成されるエギゾチックハドロンである 遠く及びません．我々の質量の大部分は，QCDの強い相互 洞察力が必要です．つまり，物理現象をうまく捉えた現象 う一つのフロンティアが低温度・高密度領域です．この領 か，それとも従来のメソンと同じクォーク2体から構成さ 作用により引き起こされる「カイラル対称性の自発的破れ」 論的模型の構築も必要になってきます．当研究室では，実 References:（2008年ノーベル物理学賞）によって生成されていると考 域の物理の理解は，GSI/FAIR，JINR/NICA計画と関連し れているかを明らかにすべく，有効模型・格子QCDなどを えられており，その機構の解明が重要な課題となっていま ますし，中性子星などへの基本的な情報を与えることが期 用いて様々な角度から解析を行っています． す．さらに，Spring-8/LEPS実験でのペンタクォークΘ+や， + Y.-L.KEK/Belle実験でのZ Ma, Y.（4430）など，これまでの「メソン= Oh, G.-S. Yang, M. Harada, H.K. Lee, B.-Y. Park,院生への期待 and M. Rho, Hidden local symmetry and infinite tower of クォーク+反クォーク」，「バリオン=クォーク3体」という 構造では説明できないハドロンが発見されました．これら 院生には，研究室の歴史を作り上げる意気込みで，新しい理論を開発しながら研究を進めていってもらいた vectorの「エキゾチックハドロン」の解明も重要な課題です． mesons for baryons, Phys. Rev. D86, 074025 (2012)いと考えています．スタッフの行っている研究分野に限らず，自分で研究分野を開拓して行くことも奨励します． 以上の重要課題を念頭におき当研究室での主たる研究課 その際，スタッフも喜んで相談にのります．積極的な研究姿勢を期待しています．また，院生が，海外で開催 題は以下の2つです．高温度・高密度状態におけるクォー Y.-L. Ma, G.-S. Yang, Y. Oh, and M. Harada, Skyrmions withされる国際会議での発表や海外研究機関での滞在を通した国際的共同研究を行うことを奨励しています． vector mesons in the hidden local symmetry approach, Phys. http://hken.phys.nagoya-u.ac.jp ・ 昨年度まで，博士課程（前期課程）入学生の募集は，E研，H研，QG研共通で行われてきましたが，平成25 年度入学生の募集は，各研究室別々に行います． *連絡先 [email protected] FAX（052）789-2865 Rev. D87, 034023 (2013) ・ EHQG共通のM1，M2の写真は，QG研のページを参照してください． 教授：1／助教：1／特任助教：1／ DC：5／ MC：30（EHQG共通） Y.-L. Ma, M. Harada, H.K.野中千穂助教 Lee, Y. Oh, B.-Y. Park, and M. Rho, Dense baryonic matter in hidden local symmetry 8 9 approach: Half-Skyrmions and nucleon mass, arXiv:1304.5638 (submitted to Phys. Rev. D) 3

13년 6월 26일 수요일 MOTIVATION 1. 1. 2. 3. 4. (SHE) 2. 1. 2. 3. 4. 5. etc

Nuclear Physics g Hadron Physics g Nuclear Physics

4 13년 6월 26일 수요일 /12 IBS & RAON

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13년 6월 26일 수요일 6

13년 6월 26일 수요일 Modern problems in nuclear and elementary particle physics 13. 6. 26. 7:01

The 7th BLTP JINR-APCTP Joint Workshop "Modern problems in nuclear and elementary particle physics", July 14-19, 2013 General Info Russia, Irkutsk Region, Bolshiye Koty Organizing Committee

Program Committee

Scientific program

Sponsors

General information

The APCTP-BLTP JINR Joint Workshops were initiated to promote cooperation between the Asia Pacific Center for Theoretical Physics (APCTP) and the Bogoliubov Laboratory of Theoretical Physics (BLTP) of the Joint Institute for Nuclear Research (JINR). The first Joint Workshop was organized by BLTP JINR and was held at Dubna on June 18-23, 2007.

The second Joint Workshop was organized by the APCTP and was held at Pohang on April 20-24, 2008. Since then, it became tradition that each side hosts the workshop every two years. 7 It is, therefore, natural to have the third Joint Workshop at Dubna on May 28-30, 2009 that was 13년 6월 26일 수요일 organized by the BLTP. During the third workshop, there was extensive discussion on possible collaborations between the two countries in regard to the NICA (Nuclotron-based Ion Collider Accelerator) project in Russia, and to the Korean radioactive ion beam accelerator. We believe that positive atmosphere for collaborations between the two countries was achieved in regard to the future accelerator projects.

In 2010 the APCTP organized the meeting and the fourth Joint Workshop was held at APCTP, Pohang on July 24-26, 2010. In this workshop, about 25 talks were presented in various fields of nuclear physics which include hadron physics, heavy ion and rare isotope http://theor.jinr.ru/~APCTP_2013/main.html 1/2 SKYRME MODEL

1960s: T.H.R. Skyrme

Baryons are topological solitons within a nonlinear theory of pions.

2 f⇡ µ 1 2 = Tr @ U †@ U + Tr U †@ U, U †@ U L 4 µ 32e2 µ ⌫ ⇥ ⇤ f⇡ : pion decay constant e : Skyrme parameter

Topological soliton winding number = baryon number 1 B = ✏ Tr U †@ UU†@ UU†@ U µ 24⇡2 µ⌫↵ ⌫ ↵

T.H.R. Skyrme: Proc. Roy. Soc. (London) 260, 127 (1961), Nucl. Phys. 31, 556 (1962) 8

13년 6월 26일 수요일 REVIVAL

In large Nc, QCD ~ effective field theory of mesons and baryons may emerge as solitons in this theory.

E. Witten, 1980s

HEDGEHOG SOLUTION

R 1fm ⇠ f U =exp(iF (r)⌧ rˆ) M 146 B ⇡ 1.2GeV · sol ⇠ | | 2e ⇠ ✓ ◆ for B =1

M 1.23 12⇡2 B > 12⇡2 B sol ⇠ ⇥ | | | | in the Skyrme unit:f⇡/(2e)

Bogomolny bound

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13년 6월 26일 수요일 BARYON MASSES

To give correct quantum numbers

SU(2) collective coordinate quantization

U(t)=A(t)U0A†(t)

Mass formula: infinite tower of I = J 1 M = Msol + I(I + 1) : moment of inertia 2 I I3 15 M = M + ,M= M + N sol 8 sol 8 I I Adjust f! and e to reproduce the nucleon and Delta masses

f⇡ = 64.5MeV,e=5.45

Empirically, f⇡ = 93 MeV,e=5.85(?)

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13년 6월 26일 수요일 Skyrme model: results

Best-fitted results 554 G.S. Adkins et al. / Static properties of nucleons Radial Distance 7 0 2 4 6 8 40 I i i I I 3 Quantity Prediction Expt MN input 939 MeV M input 1232 MeV r2 1/2 0.59 fm 0.72 fm h iI=0 r2 1/2 0.92 fm 0.81 fm h iM,I=0 µp 1.87 2.79 µ 1.31 1.91 n µp/µn 1.43 1.46 | | l I I I I I I i [ I I I I [ I I I I I I I i i I i i .5 ~ ~.5 2 2.5 Radial Distance r (fermi) Fig. 1. Plot of F, the numerical solution of eq. (3). F appears in the Skyrme ansatz Uo(x) = exp [iF(r)x. ~]. The radial distance is measured in fm, and also in the dimensionless variable F= eF=r.

of any given A is not an eigenstate of spin and isospin. We need to treat A as a quantum mechanical variable, as a collective coordinate. The simplest way to do G.S. Adkins, C.R. Nappi, and E. Witten, Nucl.this is Phys.to write B228, the lagrangian 552 (1983) and all physical observables in terms of a time dependent A. We substitute U = A(t)UoA-l(t) in the lagrangian, where U0 is the A.D. Jackson and M. Rho, Phys. Rev. Lett.soliton 51, solution 751 (1983) and A(t) is an arbitrary time-dependent SU(2) matrix. This pro- cedure will allow us to write a hamiltonian which we will diagonalize. The eigenstates with the proper spin and isospin will correspond to the nucleon and delta. 11 So, substituting U = A(t)UoA-l(t) in (1), after a lengthy calculation, we get 13년 6월 26일 수요일 L = -M + ;t Tr [OoAOoA-1], (4)

where M is defined in (2) and )t =4zc(1/e3F=)A with

A = f ~sin2F[1 +4~F{'2+ --~)sin2F\] J d~. (5)

Numerically we find A = 50.9. The SU(2) matrix A can be written A = ao+ ia .,t, with ao2+a 2= 1. In terms of the a's (4) becomes

3 L=-M+2A ~ (di) 2. i=O Skyrme model for Nuclear Physics

Single Baryon Nuclear Matter

Topics Improvement of the model • Properties of single baryon • more degrees of freedom • Equation of State (mesons) • Phase transition • 1/Nc corrections • Application to nuclei • ChPT Approaches Extension to other hadrons • Modified Effective • SU(3) extension to hyperons Lagrangian • Heavy-quark baryons • Skyrmion Crystal • Hypernuclei & Exotic • Winding number n baryons solutions

Still there are many works to do 12

13년 6월 26일 수요일 LIGHT NUCLEI OF EVEN MASS NUMBER IN THE ... PHYSICAL REVIEW C 80,034323(2009)

TABLE IV. Energy levels of the quantized B 6 Skyrmion. = π 4 IJ Quantum state E ( 10" ) E (MeV) π

01+ 1, 0 0, 0 6.51.6 | α⊗| α 3+ 3, 0 0, 0 38.99.7 | α⊗| α 4" ( 4, 4 4, 4 ) 0, 0 73.518.3 | α"| " α ⊗ | α 5+ 5, 0 0, 0 97.324.2 | α⊗| α 5" ( 5, 4 5, 4 ) 0, 0 105.926.4 | α +| " α ⊗ | α 10+ 0, 0 1, 0 47.411.8 | α⊗| α 2+ 2, 2 1, 1 2, 2 1, 1 62.615.6 | α⊗| 2,α0+| 1", 0α⊗| " α 66.816.7 | α⊗| α 2" 2, 2 1, 1 2, 2 1, 1 73.118.2 | α⊗| " α +| " α⊗| α 3+ 3, 2 1, 1 3, 2 1, 1 82.020.4 | α⊗| α"| " α⊗| " α 3" 3, 2 1, 1 3, 2 1, 1 92.523.1 | α⊗| " α"| " α⊗| α 4+ 4, 2 1, 1 4, 2 1, 1 108.026.9 | α⊗| 4,α0+| 1", 0α⊗| " α 112.228.0 | α⊗| α 4" 4, 2 1, 1 4, 2 1, 1 118.529.5 | (α⊗4, 4| " 4α,+|4 )" α⊗1, 0| α 120.930.1 | α +| " α ⊗ | α 20" 0, 0 ( 2, 2 2, 2 )137.434.2 | α⊗ | α"| " α 1+ 1, 0 2, 0 148.637.0 | α⊗| α 1" 1, 0 ( 2, 2 2, 2 )143.935.9 | α⊗ | α +| " α 2+ 2, 2 2, 1 2, 2 2, 1 157.339.2 | α⊗| α"| " α⊗| " α 2" 2, 0 ( 2, 2 2, 2 )156.939.1 2, 2| α⊗2, 1| α"2, | 2" α 2, 1 167.841.8 | α⊗| " α"| " α⊗| α

2 where "33 U33V33 W .Itsallowedquantumstatesare correctly predicted to be a 1+ state (Fig. 3). We also find = " 33 listed in Table IV,togetherwiththeirenergylevelscomputed a3+ excited state, which is seen experimentally. However, using the inertia tensors in Appendix A2. we overpredict its excitation energy by roughly a factor of 5. The Skyrme model qualitatively reproduces the experimen- The model does not account for centrifugal stretching nor tal spectrum of lithium-6Nuclei and its isobars and predicts some the allowed decay of the 3+ state to an α-particle plus a π further states, including J 4", 5+,and5" excited states deuteron, which may be the reason for our overprediction. The of lithium-6 with isospin 0.= The ground state of lithium-6 is lowest allowed state with isospin 1 is a 0 state, which is seen Light nuclei in the Skyrme model (e.g., mass number 6) +

30.1MeV J=3π 28.2MeV I=2 29.1MeV J=2π Hydrogenπ6 26.1MeV J=4π 24.9MeV J=4π ,I=1 24.8MeV J=3π ,I=1 FIG. 3. (Color online) Energy Manton, Wood, PRD 74 (2006) level diagram for nuclei of mass 18.7MeV J=(1π ,2π ) 18.0MeV J=2π ,I=1 number 6. Mass splittings between REPARAMETRIZING THE SKYRME MODEL USING THE ... PHYSICAL REVIEW D 74, 125017 (2006) nuclei are adjusted to eliminate the collective coordinates, is just a constant, the static mass of the Skyrmion. proton/neutron mass difference and The quantized momenta corresponding to bi and ai become the body-fixed spin and isospin angular momenta remove Coulomb effects, as described L and K satisfying the su 2 commutation relations [2]. i i π " The usual space-fixed spin and isospin angular momenta Ji in Ref. [19]. and Ii are related to the body-fixed operators by T J D A L ;I D A K : (37) + π + i α⊗ π 2"ij j i α⊗ π 1"ij j 9.7MeV J=(2 , 1 , 0 ) We also have J2 L2, I2 K2. Thus the operators J, L, I and K form theα Lie algebraα of SO 4 SO 4 . π "J;L π "I;K Their action on A1 and A2 is given by 1 1 5.9MeV J=2+ ,I=1

J ;A A π ; I ;A π A ; (38) + ‰ i 2Šα2 2 i ‰ i 1Šα⊗2 i 1 FIG. 1. Baryon density isosurface (to scale) of the B 6 5.4MeV J=2 ,I=1 + α + 4.8MeV J=(2) ,I=1 Skyrmion. 4.1MeV J=0 ,I=1 + 1 1 3.6MeV J=0 ,I=1 + Li;A2 πiA2; Ki;A1 A1πi; (39) 3.1MeV J=0 ,I=1 ‰ Šα⊗2 ‰ Šα2 addition of a C axis which is orthogonal to the main C Heliumπ6 encouraging2 4results 2.2MeV J=3+ ,I=0 with all other commutators between momenta and coordi- symmetry axis. The group D4 is extended to D4d by nates zero. Berylliumπ6 including a reflection in a plane which contains the main + A basis for the Hilbert space of states is given by symmetry axis and bisects a pair of the C2 axes obtained by J=1 ,I=0 J; J3;L3 I; I3;K3 , with J J3, L3 J and I applying the C4 symmetry to the generating C2 axis. The Ij , K i I.j Concretely,i J;⊗ J ;L and I;I ;K ⊗ are optimized rational map, in a convenient orientation, is [6] 3 3  j 3 3i j 3 3i Lithiumπ6 Wigner functions of the Euler angles parametrizing A2 4 and A1 respectively. The ground states are the states with Battye,z Manton,a Sutcliffe, Wood, PRC 80 (2009) R z ‡ ;a0:16i: (41) the lowest values of J and I that are compatible with the FR π "αz2 az4 1 α constraints arising from the symmetries of the given π ‡ " 034323-5 Skyrmion. The allowed values of L3 and K3 are also con- The D4d symmetry can be seen by considering the expres- strained by the symmetries of the Skyrmion. In what sion for the baryon density in (14). The baryon density 13 follows, the arbitrary third components of the space and vanishes where the Wronskian W of the map R p=q, α isospace angular momenta J3 and I3 are omitted. given by Recall that physical states satisfy [7] 8 2 4 13년 6월 26일 수요일 W qp0 pq0 2z az 3a 1 z a ; (42) 2"in L 2"in K B α ⊗ α⊗ π ‡π ⊗ " ‡ " e
π e
π 1 π ; (40) j i α j i α π⊗ " j i vanishes. In addition to the nine zeros of W, the baryon so even B implies that the spin and isospin, J and I, are density also vanishes at z due to the factor of z2 in the integral, and odd B implies that they are half-integral. denominator of the rationalα 1 map. For general B>1,themomentsofinertiaarelargerfor The D subgroup is generated by two elements, a " rotations than for isorotations. Since these appear in the 4 rotation about the x1-axis, and a " rotation about the x1 denominator of the quantum Hamiltonian, the quantum x axis. The product of these is a "=2 rotation about theα states of lowest energy are those with minimal isospin, 2 x3-axis. The corresponding symmetries of the rational map and spin excitations of Skyrmions require less energy than are R z 1=R 1=z and R z 1=R i=z . Therefore, isospin excitations. In particular, for even B the ground the solutionπ "α is invariantπ " underπ "α⊗ the followingπ " two symme- state has zero isospin, but (because of the FR constraints) tries: not necessarily zero spin. These observations match what (i) a " rotation about the x1-axis combined with a " is seen experimentally for a large range of nuclei up to isorotation about the 1-axis, and baryon number about 40. (ii) a " rotation about the x1 x2 axis combined with a " isorotation about the 2-axis.α VI. QUANTIZATION OF THE B 6 SKYRMION T α For the first symmetry, we have n1 n2 1; 0; 0 , 1 α απ " The minimal-energy B 6 Skyrmion, shown in Fig. 1, and α1 α2 ". As required, R z n2 Rn1 1.We α α π ⊗ "α α has D symmetry, and isα well approximated by the ra- compute N 15 using the formula from the previous 4d α tional map ansatz. We recall that the dihedral group D4 is section, and deduce that ⊗FR 1. For the second sym- α⊗T 1 T obtained from C ,thecyclicgroupoforder4,bythe metry, we have n1 0; 1; 0 , n2 1; 1; 0 , and 4 απ " α p2 π " α1 α2 ". As required we have R z n Rn i. ⊗ 2 1 1 α α π ππ "α α Thanks to P.M. Sutcliffe, University of Durham, UK, for Again N 15, so ⊗FR 1. Therefore the FR con- providing the picture straints reduceα to α⊗

125017-5 VOLUME 86, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 APRIL 2001

those with little symmetry, B π 5, 8, 14, and 18, which

0.2 is very much as one would expect. The binding energy appears to increase to an asymptotic value of around 0.15–0.16. This is a clear consequence of the FB bound since it is linearly related to EπB. In fact, the value of EπB appears to have an asymptotic 0.15 value which is around 6%–7% above the FB bound, com- patible with the value obtained for a hexagonal lattice [13] which is the limit of an infinitely large fullerene (the ana- log of graphite in carbon chemistry). It is clear that an 0.1 infinitely large shell is physically unlikely and that there probably exists a value Bπ such that for B . Bπ the solu- tions no longer look like fullerene shells. In such a case the solutions are likely to begin to look more like portions cut 0.05 from the infinite Skyrme crystal [14] whose EπB is only NUCLEI 4% above the FB bound. Another possibility is an inter-

VOLUME 86, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 APRIL 2001 mediate state comprised of multiple shells [15], although all the known configurations of this kind have much larger confirmed by our results in all cases except B π 9 and 0 values of EπB. We have definitely shown that Bπ . 22 B π 13, that the polyhedronmulti--baryon-number0 associated 5 with the Skyrmion 10 15 Skyrmion 20 of charge B has a structure from the family of carbon and we believe that the connection between Skyrmions, cages for C4πB22". fullerenes, and rational maps will continue for much larger We had predicted in Ref. [10] that the Buckyball C60 configuration would be found for B π 17 and indeed values of B. an approximate0.15 rational map description was found in We acknowledge useful discussions with Conor Ref. [11]. Here, we see that such a solution is the mini- mum energy solution of the full nonlinear field equations Houghton and Nick Manton. Our research is funded by and in the rational map space. We see also that a large EPSRC (P.M. S.) and PPARC (R. A. B.). The parallel number of the other solutions have platonic symmetries which, from the mechanical point of view, implies the computations were performed at the National Cosmology structure packs well. It would appear, therefore, that such Supercomputing Centre in Cambridge. structures may be preferred over less symmetric ones in the minimization0.1 procedure. We should note, however, that this is not always the case and rational maps with platonic symmetries can easily be found, for example, at B π 9, which do not give minima [6]. The polyhedra found for B π 9 and 13 do not obey the [1] T. H. R. Skyrme, Proc. R. Soc. London A 260, 127 (1961). GEM rules nor are they of the fullerene type, since they contain four-valent links. They can, however, be related [2] E. Witten, Nucl. Phys. B223, 422 (1983). to a fullerene via the concept of symmetry enhancement, [3] G. S. Adkins, C. R. Nappi, and E. Witten, Nucl. Phys. as follows. A very common structure within the fullerene B228, 552 (1983). polyhedra is two0.05 pentagons separated by two hexagons. If the edge which is common to the two hexagons is shrunk to [4] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, and have zero length, the four polygons then form a C4 sym- R. E. Smalley, Nature (London) 318, 354 (1985). metric configuration containing a four-valent bond. For [5] P.W. Fowler and D. E. Manolopoulos, An Atlas of the case of B π 9, the polyhedron can be thought of as being created from a D2 symmetric fullerene by the action Fullerenes (Clarendon Press, Oxford, 1995). of two such operations, and in the B π 13 case six opera- [6] R. A. Battye and P.M. Sutcliffe, “Skyrmions, Fullerenes tions can be used to convert another D2 configuration into and Rational Maps” (to be published). one with O symmetry.0 Empirically, we see that each sym- metry enhancement operation0 appears 5 to be accompanied 10 15 20 [7] V.B. Kopeliovich and B. E. Stern, JETP Lett. 45, 203 by an equivalent one antipodally placed on the polyhedron, (1987). and single operations appear not to occur. We have computed the energies of the solutions which [8] J. J. M. Verbaarschot, Phys. Lett. B 195, 235 (1987). FIG. 2. On top the ionization energy IB plotted against B. No- [9] E. Braaten, S. Townsend, and L. Carson, Phys. Lett. B 235, are presentedtice in Table that I the using most the rational stable map solutions ansatz to are those with the most sym- create initial conditions which are then relaxed under the 147 (1990). metry, B π 4, 7, 17, while the least stable are those with little FIG. 1. The baryon density isosurfaces for the solutions which action of the full nonlinear field equations. It should be [10] R. A. Battye and P.M. Sutcliffe, Phys. Rev. Lett. 79, 363 we have identified as the minima for 7 # B # 22, and the noted that thesesymmetry valuesBattye, are (forB πB .5,Sutcliffe,1 8,) always 14, 18 a. little On less PRL the bottom 86 the(2001) binding 3989 energy associated polyhedral models. The isosurfaces correspond to than the correspondingper baryon valuesDE computedπB plotted solely against within theB. We see that for large B the (1997). π 0.035 and are presented to scale, whereas the polyhedra rational map ansatz. On the discrete grid the computed areB not to scale. binding energy appears to level out at around 0.15–0.16 as one [11] C. J. Houghton, N. S. Manton, and P.M. Sutcliffe, Nucl. value of the baryon number, Bdis, is less than the cor- responding integermightB expectsuggesting in that a simple the finite model difference of nuclei. Phys. B510, 507 (1998). of B π 9 and 13, the associated polyhedra are trivalent approximations used to compute the energy Edis will un- [12] P.J. M. van Laarhoven and E. H. L. Aarts, Simulated with 4πB 2 2" vertices [the geometric energy minimiza- derestimate the true energy. Moreover, in the initial con- Annealing: Theory and Applications (Kluwer Academic tion (GEM) rules] as predicted in Ref. [10], and for ditions one mustto impose remove the boundary a single condition Skyrmion,U π 1 at and the binding energy B $ 7 they are comprised of 12 pentagons and 2 B 2 7 the edge of the box. By using a wide range of different Publishers, Dordrecht, The Netherlands, 1987). 13년 6월 26일 수요일 π " per nucleon DEπB π E1 2 "EπBα, which is the energy hexagons. Such structures are common in a wide range grid sizes and spacing we have shown [6] that the value [13] R. A. Battye and P.M. Sutcliffe, Phys. Lett. B 416, 385 required to separate the solution into B well-separated of physical applications and have become a hot topic in of EdisαBdis can be computed accurately, and hence so can (1998). carbon chemistry where they correspond to shells with the true energySkyrmions using the formula dividedEB π B by3 πE thedisαB totaldis". baryon number. These [14] L. Castillejo, P.S. J. Jones, A. D. Jackson, J. J. M. carbon atoms placed at the vertices, the most famous We claim that our determinations of EdisαBdis are accurate being the icosahedrally symmetric Buckminsterfullerene in the absolutevalues sense to arewithin tabulated60.001 and that in the Table relative I and are plotted against B Verbaarschot, and A. Jackson, Nucl. Phys. A501, 801 C60 structure, which is also the traditional soccer ball values are probablyin Fig. even 2. more The accurate. ionization energy is largest for the most (1989). design. For this reason we refer to such solutions as being We havesymmetrical also computed solutions the ionizationB energy4, 7, and 17 and is least for [15] N. S. Manton and B. M. A. G. Piette, hep-th/0008110. of the fullerene type, with the prediction, spectacularly IB π EB21 1 E1 2 EB, which is the energy requiredπ

3991 3992 Single Baryon (with Pion Mass)

Adding the pion-mass term Adkins, Nappi, NPB 233 (1984) 1 = m2 f 2 (Tr(U) 2) Lpion 2 ⇡ ⇡ Quantity Prediction Prediction Expt (massless pion) (massive pion) MN input input 939 MeV M input input 1232 MeV f⇡ 64.5 MeV 54 MeV 93 MeV r2 1/2 0.59 fm 0.68 fm 0.72 fm h iI=0 r2 1/2 1.04 fm 0.88 fm h iI=1 1 r2 1/2 0.92 fm 0.95 fm 0.81 fm h iM,I=0 r2 1/2 1.04 fm 0.80 fm h iM,I=1 1 µp 1.87 1.97 2.79 µn 1.31 1.24 1.91 µ /µ 1.43 1.59 1.46 | p n|

13년 6월 26일 수요일 Why vector mesons?

Witten: QCD ~ weakly interacting mesons in large Nc • The lightest meson is the pion • The next low-lying mesons are vector mesons (ω and ρ)π " Stability of the soliton 2 f⇡ µ 1 2 = Tr @µU †@ U + Tr U †@µU, U †@⌫ U L 4 32e2 Skyrme terms ⇥ ⇤ • Without the Skyrme term, the soliton collapses. Derrick’s Theorem • Vector mesons can stabilize the soliton without the Skyrme term.

16

13년 6월 26일 수요일 Early Attempts to include VM

Including ! meson

= + + L Lpion L! Lint 2 2 f⇡ µ f⇡ 2 = Tr(@ U@ U †)+ m (Tr(U) 2) , Lpion 4 µ 2 ⇡ m2 1 = ! ! !µ ! !µ⌫ , = ! Bµ L! 2 µ 4 µ⌫ Lint µ

G.S. Adkins and C.R. Nappi, Phys. Lett. B137, 251 (1984)

Including ⇢ meson

= + + L Lpion L⇢ Lint µ ⌫ = ↵Tr(⇢ @ U †U@ U †) ⇢⇡⇡ interaction Lint µ⌫

G.S. Adkins, Phys. Rev. D33, 193 (1986)

17

13년 6월 26일 수요일 Early Attempts: results

Quantity Skyrme !⇢Expt (massive pion) MN input input input 939 MeV M input input input 1232 MeV f⇡ 54 MeV 62 MeV 52.4 MeV 93 MeV r2 1/2 0.68 fm 0.74 fm 0.70 fm 0.72 fm h iI=0 r2 1/2 1.04 fm 1.06 fm 1.08 fm 0.88 fm h iI=1 r2 1/2 0.95 fm 0.92 fm 0.98 fm 0.81 fm h iM,I=0 r2 1/2 1.04 fm 1.02 fm 1.06 fm 0.80 fm h iM,I=1 µp 1.97 2.34 2.16 2.79 µn 1.24 1.46 1.38 1.91 µ /µ 1.59 1.60 1.56 1.46 | p n| µI=0 0.365 0.44 0.39 0.44 µI=1 1.605 1.9 1.77 2.35

18

13년 6월 26일 수요일 SU(3) EXTENSION

Can we describe hyperons in the Skyrme model? Direct extension: SU(3) collective coordinate quantization

New approaches

exact diagonalization methods Yabu, Ando, NPB 301 (1988) Weigel et al., PRD 42 (1990) bound state model Callan, Klebanov, NPB 262 (1985)

13년 6월 26일 수요일 BOUND STATE MODEL

Starting point: flavor SU(3) symmetry is badly broken treat light flavors and strangeness on a different footing SU(3) SU(2) U(1) ! ⇥ Lagrangian = + L LSU(2) LK/K⇤ The soliton provides a background potential that traps K/K* (or heavy) mesons

Callan, Klebanov, NPB 262 (1985)

bound kaon

13년 6월 26일 수요일 BOUND STATE MODEL

Anomalous Lagrangian Pushes up the state of S =+1 states to the continuum → no bound state Pulls down the state of S = 1 states below the threshold → makes bound states g description of hyperons Renders two bound states with S = 1 the lowest state: p-wave → gives (+) parity ⇤(1116) 270 MeV energy difference excited state: s-wave → gives (-) parity⇤(1405) after quantization

13년 6월 26일 수요일 BOUND STATE MODEL

Experimental Data parity undetermined

negative parity

290 MeV

285 MeV positive parity

289 MeV

13년 6월 26일 수요일 BOUND STATE MODEL

Mass sum rules modification to GMO and equal spacing rule

3⇤+⌃ 2(N +⌅)=⌃⇤ (⌦ ⌅⇤) (⌦ ⌅⇤) (⌅⇤ ⌃⇤)=(⌅⇤ ⌃⇤) (⌃⇤ hyperfine relation 3 ⌃⇤ ⌃+ (⌃ ⇤)= N 2 The same relations hold for

+ + ⇤(1/2), ⌃(1/2), ⌃(3/2), ⌅(1/2 ), ⌅(3/2 ), ⌦(3/2)

13년 6월 26일 수요일 BOUND STATE MODEL

Best-fitted results based on the derived mass formula Particle Prediction (MeV) Expt N 939* N(939) Recently confirmed by COSY 1232* (1232) ⇤(1/2+) 1116* ⇤(1116) PRL 96 (2006) ⇤(1/2) 1405* ⇤(1405) + ⌃(1/2 ) 1164 ⌃(1193) BaBar : the spin-parity of ⌃(3/2+) 1385 ⌃(1385) Ξ(1690) is 1/2" ⌃(1/2) 1475 ⌃(1480)? ⌃(3/2) 1663 ⌃(1670) PRD 78 (2008) ⌅(1/2+) 1318* ⌅(1318) NRQM predicts 1/2+ ⌅(3/2+) 1539 ⌅(1530) ⌅(1/2) 1658 (1660) ⌅(1690)? puzzle in QM ⌅(1/2) 1616 (1614) ⌅(1620)? ⌅(3/2) 1820 ⌅(1820) Unique prediction of this model. ⌅(1/2+) 1932 ⌅(1950)? The Ξ(1620) should be there. ⌅(3/2+) 2120* ⌅(2120) still one-star resonance ⌦(3/2+) 1694 ⌦(1672) ⌦(1/2) 1837 ⌦(3/2) 1978 Ω’s would be discovered ⌦(1/2+) 2140 in future. ⌦(3/2+) 2282 ⌦(2250)? ⌦(3/2) 2604 YO, PRD 75 (2007)

13년 6월 26일 수요일 HEAVY QUARK BARYONS

Replace the strangeness by the heavy-flavor m /m m /m D ⇡ K ⇡ A dog wagging a tail?

large Nc vs. large mQ

(mQ >Msol) (mQ

13년 6월 26일 수요일 HEAVY QUARK BARYONS

Soliton-fixed frame Heavy-meson-fixed frame

300 MeV YO, B.Y. Park ZPA 359 (1997) fewer bound states

YO, B.Y. Park PRD 51 (1995)

13년 6월 26일 수요일 Vector Mesons

Systematic way to include vector mesons • Massive Yang-Mills approach Syracuse group • Hidden Local Symmetry Nagoya group • Equivalence of the two approaches

Skyrmions in the HLS • ρ meson stabilized model Y. Igarashi et al., Nucl. Phys. B259, 721 (1985) • ρ and ω meson stabilized model U.-G. Meissner, N. Kaiser, and W. Weise, Nucl. Phys. A466, 685 (1987)

• ρ, ω and a1 meson stabilized model N. Kaiser and U.-G. Meissner, Nucl. Phys. A519, 671 (1990) L. Zhang and N.C. Mukhopadhyay, Phys. Rev. D50, 4668 (1994)

27

13년 6월 26일 수요일 Recent Works for Skyrmions with Vector Mesons

Holographic QCD: infinite tower of vector mesons Solitons in hQCD

D.K. Hong, M. Rho, H.-U. Yee, and P. Yi, Phys. Rev. D76, 061901 (2007); JHEP 0709, 063 (2007) H. Hata, T. Sakai, S. Sugimoto, and S. Tamato, Prog. Theor. Phys. 117, 1157 (2007)

HLS Lagrangian O(p4) terms: M. Tanabashi, Phys. Lett. B316, 534 (1993) O(p4) terms & hQCD: M. Harada and K. Yamawaki, Phys. Rep. 381, 1 (2003)

Skyrmions in HLS with ρ meson up to O(p4) terms with hQCD K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev. D75, 086003 (2007) K. Nawa, A. Hosaka, and H. Suganuma, Phys. Rev. D79, 126005 (2009)

28

13년 6월 26일 수요일 Earlier works

O(p2) Lagrangian with HLS Hidden Symmetry

2 ⇠L,R(x) h(x)⇠L,R(x),h SU(2) f⇡ µ ! 2 = Tr(@ U@ U †)withU = ⇠† ⇠ µ L R V (x) ih(x)@ h†(x)+h(x)V (x)h†(x) L 4 µ ! µ µ

Covariant derivative: D ⇠ = @ ⇠ iV ⇠ µ L,R µ L,R µ L,R 1 ↵ˆµ = (Dµ⇠L⇠L† + Dµ⇠R⇠R† ) k 2i 1 Unitary gauge: ⇠L† = ⇠R = ⇠ ↵ˆµ = (Dµ⇠L⇠† Dµ⇠R⇠† ) ? 2i L R

HLS Lagrangian

= + a + m2 = ag2f 2 L LA LV Lkin V ⇡ 2 2 2 2 A = f⇡ Tr(↵ ˆ µ )= , V = f⇡ Tr(↵ ˆ µ ) 1 L ? L L k g⇢⇡⇡ = ag 1 2 2 kin = 2 Tr(Fµ⌫ ) L 2g a =2gives KSRF relation and the universality of ⇢ coupling

M. Bando, T. Kugo, and K. Yamawaki, Phys. Rep. 164, 217 (1988) 29

13년 6월 26일 수요일 As a , i.e., as m !1 V !1 ρ meson and the Skyrme term 2 V (↵µ Vµ) =0 L / k 1 where ↵µ = (@µ⇠L⇠L† + @µ⇠R⇠R† ) k 2i ) 1 2 Tr @ UU†,@ UU† = Lkin ! 32g2 µ ⌫ LSkyrme ⇥ ⇤ Skyrmion in the HLS with the ρ meson

726 Y. lgarashi et at / Stabilization of skyrmions

, , , ' 1 ' , '- , 1 , ' ' ' r ' ' . , i

M = (667 1575) MeV sol ⇠ ! for 1 a 4  

Msol = 1045 MeV for a =2

0 .5 1 1.5 2 radiat distance r (fro) Fig. 1. n = 1 solutions of F(r) and G(r) (dashed curves) for m~=0 (chiral limit) and a = 1, 2 and 4 fixing ag 2f 2 = m 2.

Y. Igarashi,' ' ' ' M.l ' Johmura,' -' • I i i A., Kobayashi,, I ' i , , ] H.I Otsu, T. Sato, and S. Sawada, Nucl. Phys. B259, 721 (1985) 3 m n =1 3 8 M eV 30

13년 6월 26일 수요일

a=4

0 .5 1 1.5 2 radial distance r (fml Fig. 2. n = 1 solutions of F(r) and G(F) (dashed curves) For m~ = 138 MeV and a = 1, 2 and 4 fixing ag2f ~ ~ m 2. The Lagrangian of O(p2) reads

2 µ 2 µ 1 µ" 1 2 2 † (2) = fπ Tr [ˆπ µπˆ ]+afπ Tr [ˆπ µπˆ ] 2 Tr [Vµ"V ]+ fπ mπ Tr U + U 2 (1.7). L π π " " π 2g 4 π π " The Lagrangian of O(p4) reads

y µ " µ " (4) = y1 Tr [ˆπ µπˆ πˆ "πˆ ]+y2 Tr [ˆπ µπˆ "πˆ πˆ ] L π π π π π π π π µ " µ " + y3 Tr [ˆπ µπˆ πˆ "πˆ ]+y4 Tr [ˆπ µπˆ "πˆ πˆ ] " " " " " " " " µ " µ " + y5 Tr [ˆπ µπˆ πˆ "πˆ ]+y6 Tr [ˆπ µπˆ "πˆ πˆ ] π π " " π π " " µ " µ " " µ + y7 Tr [ˆπ µπˆ πˆ "πˆ ]+y8 Tr [ˆπ µπˆ πˆ "πˆ ]+Tr [ˆπ µπˆ "πˆ πˆ ] " π π " π " π " π " π " µ " + y9 Tr [ˆπ µπˆ "πˆ πˆ ], α ⊗ (1.8) ρ and �π mesons" π " and

z µ " µ " (4) = iz4 Tr [Vµ"πˆ πˆ ]+iz5 Tr [V�µ" πˆ πˆ ]. (1.9) � meson: introducedL through HGSπ π like the meson" " The anomalous terms are written as Anomalous Lagrangian: source of the � meson 3 = gN (c c c )" Bµ Lan 8 c 1 π 2 π 3 µ g3N c (c + c )⊗µ"α⊗" tr (a ¯ ¯ ) π 32α2 1 2 µ " α ⊗ gN ig ig c c ⊗µ"α⊗ " tr (a v v )+ ‰ " tr (a a ) " tr ( a ) , π 8α2 3 π µ " α ⊗ 4 µ " α ⊗ π α ⊗ π 4 µ "α ⊗ ‰ (1.10) T. Fujiwara, T. Kugo, H. Terao, S. Uehara, K. Yamawaki, Determination of parameters The more complete form can be found in my previousProg. note. Theor. Phys., 73, 926 (1985) The radial wave functions for2 the fields2 at O(Nc) are defined as Minimal model: c = ,c = ,c =0 � 1 3 2 3 3 � B� term only Š =exp[iπ rˆF (r)/2], 3 Vector Dominance: c1 =1,c2 =0,c3 ·=1 No �� term "0 = W (r), " =0, G(r) Or fit them to known phenomenology 0 =0, α = (rˆ π ) . (1.11) gr "

ThenSee, we for have example, P. Jain, U.-G. Meissner, N. Kaiser, H. Weigel, N.C. Mukhopadhyay, etc

0 1 πˆ =0, ⊗ˆ = [f1(r)π + f2(r)rˆπ rˆ] π π π2 · gW(r) 1 F πˆ0 = , ⊗ˆ = G(r)+2sin2 (rˆ π ) , (1.12) U.-G. Meissner, N. Kaiser" π and2 W. Weise," Nucl.π2 Phys.r A466, 6852 (1987)" Š  31 where 13년 6월 26일 수요일 1 1 f (r)= sin F, f = F α sin F. (1.13) 1 r 2 π r

2 694 U.-G. Meissner et al. / Nucleons as Skyrme solitons

G(0) = -2, G(co)=O, (3.8a)

o’(0) = 0 ) o(oo)=O. (3.8b)

The boundary conditions on G(r) agree with the ones of Iqarashi et al. 15) whereas the boundary conditions on w(r) have already been discussed by Adkins and Nappi *“) in their work on theρ w-stabilized and skyrmion. � mesonsNote furthermore that eq. (3.7b) agrees exactly with the one given in ref. 15). It is evident that for large distances r, the pion field falls off exponentially, i.e.

. fXr)--e-mwr/r(l+ l/W&r) ) (l-,~) (3.9) with m, = 139 MeV. minimal model results with a = 2, f� = 93 MeV, g = 5.85 3.3. NUMERICAL SOLUTIONS

The set of Mcoupledsol =equations 1475 (3.7) MeV together with (3.2) and (3.8) constitutes a boundary value problem of ordinary differential equations that can be solved using

MINIMAL MODEL MESON PROFILES I I U.-G. Meissner et al. / Nucleons as Skyrme solitons 3

TABLE 1 3 2 Properties of the Skyrme soliton resulting from the lagrangians : I (2.11) or (2.19) with rr, p and w mesons k? 1 Minimal Complete Following model model ref. “) 0 MH [MeVl 1474 1465 1057

r H [f m l 0.5 0 0.48 0.27 -2 u For comparison, the results of the model of ref. 17) including ! pions and p mesons are also given. The parameters used are j -4 m, = 139 MeV, f, = 93 MeV, and g = 5.85. Here MH is the static soliton mass, and rt, the baryonic r.m.s. radius.

-t 15 Wer [fml refer to rH as the baryonic root mean square (r.m.s.) radius, since it measures Fig. 1. Meson profiles for the minimal model: the chiral angle F(r), and the vector meson profiles -G(r) and o(r) for g = 5.85, f, = 93 MeV, and m, = 138 MeV. Notice the different thescale forextension the o-meson. of the baryonic charge (baryon number dist~bution) of the soliton. The inclusion of the o-meson has two main effects: First, the skyrmion mass increases by -40% as compared to the model without w. Secondly, the soliton U.-G. Meissner, N. Kaiser and W. Weise,radius increases Nucl. Phys.by roughly A466, a factor 685 of (1987) 2. We will later demonstrate that a baryonic charge radius of rn - 0.5 fm leads to reasonable e~ectro~ug~ei~c charge radii of 32 protons and neutrons due to the inherent vector meson dominance. The role of the 13년 6월 26일 수요일 vector mesons is to increase the electromagnetic soliton radii to values typically between 0.8 and 1.0 fm.

4. Electromagnetic properties

4.1. PHOTON COUPLINGS TO MESONS

The electromagnetic interaction is added to the effective lagrangian (2.11) or (2.19) in the standard way, by gauging the subgroup generated by the electric charge

operator Q = j($+ TJ. For example, in the isovector sector, where the charge operator is QV= fan, the photon field A, is introduced in such a way that the covariant derivative (2.5) is replaced by

9&L,R = (8, -b& - P~CL)SL,R+tieSL,RT3AIL(X). (4.1) Evidently, the isovector photon is closely connected with the p-meson. Similarly, gauging the U(1) subgroup generated by the isoscalar charge Qs brings the isoscalar photon coupling into close relationship with the w meson. Let us discuss the characteristic couplings of isovector photons as implied by eq. (4.1) when inserted into the effective lagrangian, e.g. eq. (2.4). Again, the leading terms become particularly transparent in the weak field approxi- mation. We therefore return to eq. (2.7), recalling that there is a gauge parameter a which, if set a = 2, gives the proper relation (2.14) between the p-meson mass 710 U.-G. Meissner et ai. / N&eons as Skyme so&tons

TABLET Baryon properties; parameters as in table 1

Minimal Complete Experiment model model

0 [fm] 0.82 0.68 MA -MN DfeVl 359 437 293 MN [Mevl 1564 1575 939 r, - (r$)“2 [fm] 0.50 0.48 ~~~~~~~rE ” -0.220.92 -0.250.98 -0.119*0.0040.86*0.01

(rZ&‘” [fml 0.84 0.94 0.86 * 0.06 C&Y” [fml 0.85 0.93 0.88 * 0.07 fiP Ln.m.1 3.36 2.77 2.79 CL,[n.m.l -2.57 -1.84 -1.91 IP*/ELnI 1.31 1.51 1.46

(We show a comparison with data up to q2 = 0.6 GeV’, which is roughly the square of the vector meson mass; any further extrapolation would lead beyond the range of applicability of the model.) The complete model tends to produce charge and magnetic moment distributions which are slightly too large in size. The explicit treatment of p- and u-meson degrees of freedom is extremely important in achieving charge and magnetic radii close to the empirical ones, i.e. signifi~ntly larger than ρ and � mesons the radius (r$ri2== 0.5 fm of the baryon number density.

PROTON CHARGE FORM FACTOR GE (q21 I

710 U.-G. Meissner et ai. / N&eons as Skyme so&tons

TABLET Baryon properties; parameters as in table 1

Minimal Complete Experiment model model

0 [fm] 0.82 0.68 MA -MN DfeVl 359 437 293 I 1 MN [Mevl 1564 1575 939 0 01 02 03 01, 05 t jZ[GeV21 r, - (r$)“2 [fm] 0.50 0.48 U.-G. Meissner et al. / Nucleons as Skyrme solirons 711 Fig. 7. The proton electric form factor. The solid line is the result of the minimal model. The result of ~~~~~~~rE ” -0.220.92 -0.250.98 -0.119*0.0040.86*0.01 the complete model is given by the long-dashed line. For comparison, the empirical dipole fit GE($) = NEUTRON CHARGE FORM FACTOR G; (q’, [ I- q2/0.71 GeV’]-’ is aisoI shown.I The data1 are Itaken from ref. 37). (rZ&‘” [fml 0.84 0.94 0.86 * 0.06 C&Y” [fml 0.85 0.93 0.88 * 0.07 0 08 fiP Ln.m.1 3.36 2.77 2.79 006 CL,[n.m.l -2.57 -1.84 -1.91 IP*/ELnI 1.31 1.51 1.46 0.04

0.02

(We show a comparison with data up to q2 = 0.6 GeV’, which is roughly the square 0 lq*l [GeV’l of the vector meson mass; any further extrapolation would lead beyond the range-002 I I I 1

of applicability of the model.) The complete model tends to produceFig. 8. Thecharge neutron electricand form factor. The solid line is the result of the minimal model, the long-dashed magnetic moment distributions which are slightly too large in size. The explicit line refers to the complete model. Data from 37). U.-G.treatment Meissner, of p- and N.u-meson Kaiser degrees and W. of Weise,freedom Nucl.is extremely Phys. important A466, 685in achieving (1987) charge and magnetic radii close to the empirical ones, i.e. signifi~ntly larger than 33 7 EON M~ET~ FORM FACTOR G~(~)/G~( ol 13년 6월 26일 수요일the radius (r$ri2== 0.5 fm of the baryon number density. -4 10

PROTON CHARGE FORM FACTOR GE (q21 I 05

a I I b 0 01 02 03 04 05 0 01 02 0.3 04 0.5 lq21[GeV21 lq*l[GeV*l

Fig. 9. (a) The proton magnetic form factor. The solid line is the result of the minimal model, the long-dashed line the one of the complete model. Data from 37), (b) The neutron magnetic form factor; notations otherwise as in fig. (a). Data from “1.

5. Axial properties

5.1. BASIC DEFINITIDNS

I 1 In this section, we discuss the axial properties of nucleons as they emerge in our 0 01 02 03 01, 05 t jZ[GeV21 model. We start with a brief summary of basic definitions. The most general form

Fig. 7. The proton electric form factor. The solid line is the result of the minimal model. The result of the complete model is given by the long-dashed line. For comparison, the empirical dipole fit GE($) = [ I- q2/0.71 GeV’]-’ is aiso shown. The data are taken from ref. 37). ρ, �, and a1 mesons

N. Kaiser and U.-G. Meissner, Axial vector meson Nucl. Phys. A519, 671 (1990) L. Zhang and N.C. Mukhopadhyay, U(x)=⇠L† (x)⇠M (x)⇠R(x) Phys. Rev. D50, 4668 (1994) 14 anomalousH. Forkel et al. terms / Skyrmions with vector mesons 465 cf. 6 independent terms in the ⇡⇢! system 1500 - Hard to control the parameters Esk WV)

Results1000 with- a = 2, f� = 93 MeV, g = g�/1.5 = 5.85, mV = 770 MeV

H. Forkel et al. / Skyrmions with vector mesons 465 Msol = 1002 MeV

1500 -

Esk WV)

1000 - 0 2 4 6 g, &a/l.5

Fig. 1. The behaviour of the skyrmion energy as the vector meson couplings and the masses go to zero. (a) g=g,JlS+O, (b) g+O, g,/1.5=5.85 (fixed), (c) g, + 0, g = 5.85 (fixed). In all cases the ratios g/m and gw/1.5m are kept constant at 5.85/770 MeV.

0 2 4 6 g, &a/l.5 g, the soliton energy becomes negative so that those vector mesons could eventually Fig. 1. The behaviour of the skyrmion energy as the vector meson couplings and the masses go to zero. destabilize the soliton. This is an artifact of the abelian approximation (a) g=g,JlS+O,to the (b) g+O, g,/1.5=5.85 (fixed), (c) g, + 0, g = 5.85 (fixed). In all cases the ratios g/m gauge-fieldH. Forkel, dynamics A.D. employedJackson, in andthat C.paper. Weiss, In the Nucl. full non-linearPhys. A526,problem 453 (1991)and gw/1.5m are kept constant at 5.85/770 MeV. 34 appropriate for the large-g domain, we find a positive and finite limit of theg, theenergy soliton energy becomes negative so that those vector mesons could eventually destabilize the soliton. This is an artifact of the abelian approximation to the 13년 6월 26일 수요일as g increases. When the ratio of g/m is fixed (5.85/770 MeV), this limiting energy is 941 MeV. (The corresponding limiting r.m.s. radius is 0.34 fm.) A finitegauge-field positive dynamics employed in that paper. In the full non-linear problem appropriate for the large-g domain, we find a positive and finite limit of the energy energy and r.m.s. radius are also obtained in the limit of large g for fixedas m.g increases. When the ratio of g/m is fixed (5.85/770 MeV), this limiting energy is 941 MeV. (The corresponding limiting r.m.s. radius is 0.34 fm.) A finite positive energy and r.m.s. radius are also obtained in the limit of large g for fixed m.

.4 b .4 b

rB cfrn) rB cfrn)

0 2 4 6 g* d1.5 Fig. 2. The behaviour of the r.m.s. baryon radius as the vector meson couplings and the masses go to 0 2 4 6 zero. Cases (a), (b) and (c) are the same as in fig. 1. g* d1.5 Fig. 2. The behaviour of the r.m.s. baryon radius as the vector meson couplings and the masses go to zero. Cases (a), (b) and (c) are the same as in fig. 1. ρ, �, and a1 mesons

The results are sensitive to the parameters.

L. Zhang and N.C. Mukhopadhyay, Phys. Rev. D50, 4668 (1994) 35

13년 6월 26일 수요일 Summary of the earlier works

1. a dependence • ambiguity in the value of a results in a large uncertainty in the soliton mass (in free space, a ~ 2 and at high temperature/density a ~ 1)

2. Higher order terms 4 2 • O(p ) etc are at O(Nc) like the O(p ) terms • More complicated form of the Lagrangian • Uncontrollably large number of low energy constants E.g. 6 anomalous terms for the ω meson at O(p2) 14 anomalous terms for the axial vector mesons at O(p2)

3. In this work, • O(p4) with ρ and ω mesons • Fix the couplings by using hQCD

36

13년 6월 26일 수요일 3

vector meson field Vµ is [8–10] Then one can construct the chiral Lagrangian up to O(p4) as g V = (π + " ) (5) 3 µ 2 µ µ HLS = (2) + (4) + anom, (7) vector meson field Vµ is [8–10] with Then one can construct the chiral Lagrangian up to L L L L 4 O(p ) as 0 + g "µ π2"µ Vµ = (πµ + "µ) (5) "µ = πµ " = 0 . (6) which is our working Lagrangian. Here, 2 · π2"π " π µ " µ " = + + , (7) with LHLS L(2) L(4) Lanom 3 1 0 + = f 2 Tr (ˆa aˆµ )+af 2 Tr aˆ aˆµ Tr (V V µ" ) , (8) "µ π2"µ (2) π µ π µ 2 µ" vector meson field Vµ is [8–10] Then one can constructL the chiral Lagrangian" " up to α α " 2g "µ = πµ " = 0 . (6) 4which is our working Lagrangian. Here, · π2"µπ "µ O(p ) as α ⊗ g π " " where f is the pion decay constant, a is the parameter of the HLS, g is the vector meson coupling constant, and the Vµ = (πµ + "µ) (5) π 2 field-strength tensor of the vector meson is HLS1 = (2) + (4) + anom, (7) with 2 µ 2 µ L L Lµ" L (2) = fπ Tr (ˆa µaˆ )+afπ Tr aˆ µaˆ 2 Tr (Vµ" V ) , Vµ" = αµV" α" Vµ i(8)[Vµ,V" ]. (9) L " " α α " 2g " " 0 + 4 "µ π2"µ In the mostα general⊗ form of the O(p )3 Lagrangian there are several terms that include two traces in the flavor space "µ = πµ " = . (6) which is our working Lagrangian. Here, where fπ is the pion decayπ constant,0 a is the parameter of the HLS, g is the vector meson coupling2 constant, and the · 2"µπ "µ such as the y10–y18 terms listed in Ref. [10]. These terms are suppressed by Nc compared to the other terms in the field-strength tensor ofπ the vector" meson" is 4 vector meson field Vµ is [8–10] Then one can constructLagrangian the and chiral are not Lagrangian considered up in to the present work. Then the O(p ) Lagrangian which we study in this paper is O(p4) as given by g Vµ" = αµV" α" Vµ i[Vµ,V" ]. (9) 2 µ 2 " µ " 1 µ" Vµ = (πµ + "µ) (5)(2) = fπ Tr (ˆa µaˆ )+afπ Tr aˆ µaˆ 2 Tr (Vµ" V ) , (8) 2In the most general formL of the O(p4) Lagrangian" " there areα severalα " 2g terms that include two traces in(4) the= flavor(4)y + space(4)z, (10) α ⊗ L L L 2 HLS = (2) + (4) + anom, (7) with wheresuchf asis the they10 pion–y18 decayterms constant, listed ina Ref.is the [10]. parameterThese of terms the HLS, are suppressedg is the vector by mesonNc compared coupling4 to constant, the other and terms the in the π L whereL L L4 field-strengthLagrangian andtensorHLS are of not the considered vector Lagrangian meson in is the present work. Then up the O(p to) Lagrangian O(p which we) study in this paper is 0 π + µ " µ " µ " µ " given"µ by 2"µ (4)y = y1Tr ⊗ˆ µ⊗ˆ ⊗ˆ " ⊗ˆ + y2Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ + y3Tr ⊗ˆ µ⊗ˆ ⊗ˆ " ⊗ˆ + y4Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ "µ = πµ " = 0 . (6) which is our working Lagrangian.L Here, " " " " " " " " α α α α α α α α · π2"π " Vµ" = αµV" α" Vµ i[Vµ,V" ]. (9) π µ " µ " " " µ ‰" µ ‰" "‰ µ ‰ = + + 4 (4) = (4)y + (4)z,+ y5Tr ⊗ˆ µ⊗ˆ ⊗ˆ " ⊗ˆ + y6Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ (10)+ y7Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ In theHGS most general(2) form(4) of the Oanom(p ) Lagrangian thereL areL severalL terms that include" " twoα tracesα in the flavor" " spaceα α " " α α L L L L 2 ‰ ‰ ‰ suchwhere as the y10–y18 terms listed in Ref. [10]. These terms are suppressed by Nc comparedµ to the" other terms in the" µ µ " µ µ 1 4 + y8 Tr ⊗ˆ µ⊗ˆ ⊗ˆ " ⊗ˆ +Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ + y9Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ , (11) Lagrangian and2 are not considered2 in the present work. Thenµ" the O(p ) Lagrangian" whichα we" studyα in this" paperα " is α " α " α (2) = fπ Tr (ˆa µaˆ )+afπ Tr aˆ µaˆ 2 Tr (Vµ" V ) , (8) givenL by " " µ α" α " 2g µ " Š µ µ "" ‰ µ " µ " ‰ ‰ (4)y = y1Tr ⊗ˆ µ⊗ˆ ⊗ˆ " ⊗ˆ + y2Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ (4)+zy=3Triz4Tr⊗ˆ µV⊗ˆµ"⊗⊗ˆˆ "⊗ˆ⊗ˆ ++izy45TrTr V⊗ˆµ"µ⊗⊗ˆˆ ⊗"ˆ⊗ˆ ⊗.ˆ (12) L " " "α " ⊗ " " " "L α α α" "α α αα α α α where f is the pion decay constant, a is the parameter of the HLS, g is the vector meson coupling constant, and the π µ ‰" (4) = (4)y + µ(4)‰"z, "‰‰ µ ‰ (10)‰ + y5Tr ⊗ˆ µ⊗ˆ ⊗ˆ " ⊗ˆ + yL6Tr ⊗ˆLµ⊗ˆ " ⊗ˆL ⊗ˆ + y7Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ field-strength tensor of the vector meson is " " α α " " α α " " α α where µ ‰" " µ ‰ µ ‰" Vµ+" y=8 αµTrV" ⊗ˆ αµ"⊗ˆVµ⊗ˆ "i[⊗ˆVµ,V+Tr" ]. ⊗ˆInµ⊗ theˆ " ⊗ presentˆ ⊗ˆ work,+ y9Tr we also⊗ˆ µ consider⊗ˆ " ⊗ˆ ⊗ˆ(9) the, anomalous par- In(11) order to study the properties of the soliton obtained µ "" " α "" α "ityµ hWZα" " termsα that areµ written"" α as" α µ " from the Lagrangian (7), we take the standard parame- (4)4 y = y1Tr ⊗ˆ Šµ⊗ˆ ⊗ˆµ" ⊗ˆ" + y2Tr‰ ⊗ˆ µ⊗ˆ µ" ⊗ˆ" ⊗ˆ + y3Tr‰ ⊗ˆ µ⊗ˆ ⊗ˆ " ⊗ˆ + y4Tr ⊗ˆ‰ µ⊗ˆ " ⊗ˆ ⊗ˆ In the most general form of the O(Lp )(4) Lagrangianz = iz4Tr" V thereµ"" ⊗ˆ" are⊗ˆ " several+ iz5Tr termsV"µ" that⊗ˆ" ⊗ˆ " include." two tracesα α inα theα flavor spaceα α α α terization(12) for the soliton configuration. For the pion field, L 2 "µ " ‰" α α µ ‰" "‰ µ 3 ‰ such as the y10–y18 terms listed in Ref. [10].+ y TheseTr ⊗ˆ terms⊗ˆ ⊗ˆ are⊗ˆ suppressed+ y Tr ⊗ˆ by⊗ˆNc⊗ˆcompared⊗ˆ + y Tr to the⊗ˆ other⊗ˆ ⊗ˆN terms⊗ˆc in the we use the standard hedgehog configuration, 5 µ "‰ 6 4 µ "‰ 7 µ =" c , (13) Lagrangian and are not considered in the present work." " Thenα α the O(p ) Lagrangian" " α α which weL studyanom" " in16 thisα 2α paperiL isi µ ‰" " µ ‰ µ ‰"i=1 F (r) given by + y8 Tr ⊗ˆ µ⊗ˆ ⊗ˆ " ⊗ˆ +Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ + y9Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ
, (11) ‰(r)=exp i" rˆ . (16) " α " α " αwhere" α " α " α · 2 In the present work,Š we also µ consider" the‰ anomalous µ " par- ‰ In order to study the‰ properties of the soliton obtained , - ity hWZ terms(4)z = iz that4Tr areVµ(4)" written⊗ˆ=⊗ˆ (4)+y as+iz5Tr(4)zV,µ" ⊗ˆ ⊗ˆ . from the Lagrangian (7),(10) we take the standard(12) parame- L L " L" L α α 3 3 The configuration of the vector mesons are written as [20] ‰ ‰ terization1 = i Tr for⊗ˆ theL⊗ˆR soliton⊗ˆR⊗ˆ configuration.L , For(14a) the pion field, where 3 L " Nc we use2 = thei Tr standard‡⊗ˆL⊗ˆR⊗ˆL hedgehog⊗ˆR , +17 configuration,terms (14b) G(r) anom = ci i, (13) L µ " L 16 2µ " L µ " µ " πµ = W (r) Š0µ, "0 =0, π = (rˆ " ) . (17) (4)y = y1Tr ⊗ˆ Inµ⊗ˆ the⊗ˆ present" ⊗ˆ work,+ y2Tr we⊗ˆ alsoµ⊗ˆ consider" ⊗ˆ ⊗iˆ=1 the+ anomalousy3Tr ⊗ˆ µ⊗ˆ par-⊗ˆ " ⊗ˆ +Iny order4Tr 3⊗ˆ to=Trµ study⊗ˆ "F‡⊗ˆV the(ˆ⊗ˆ⊗L properties⊗ˆR +⊗ˆR⊗ˆ ofL) the, soliton(14c) obtained gr α L " " " " " " " " α α α α L α α α α " F (r) ity hWZ terms that are written as
from the Lagrangian‰ (7),(r)=exp we takei the" r standardˆ . parame- (16) whereµ ‰" µ ‰" in the"‰ 1-formµ notation ‡ with ‰ +· 2 For the baryon number B = 1 solution, these wave func- + y5Tr ⊗ˆ µ⊗ˆ ⊗ˆ " ⊗ˆ + y6Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ + y7Tr ⊗ˆ µ⊗ˆ " ⊗ˆ terization⊗ˆ for the soliton configuration., For the- pion field, " " α α " "3 α α " " α α tions satisfy the following boundary conditions: µ ‰ N3c 3 µ ‰ weµ ‰ use the standard⊗ˆ =ˆ hedgehog⊗ ⊗ˆ , configuration, "1 = i Tr= ⊗ˆL⊗ˆR "⊗cˆR⊗ˆ,L , (13)(14a) " The configurationL of the vector mesons are written as [20] + y8 Tr ⊗ˆ µ⊗ˆ ⊗ˆ " ⊗ˆLLanom+Tr ⊗ˆ16 µ2⊗ˆ ""⊗ˆ i⊗ˆLi + y9Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ , α " " (11) " α " α " αi=1 " α " α " α ⊗ˆR =ˆ⊗ +ˆ⊗ , F (r) F (0) = ,F( )=0, Š 2‰= i Tr ‡⊗ˆL⊗ˆR
⊗ˆL⊗ˆR ,‰+ (14b) ‰ α " ⊗ µ " L µ " ‰(r)=exp i"2 rˆ . G(r) (16) G(0) = 2,G( )=0, (18) (4)z = iz4Tr Vwhereµ" ⊗ˆ ⊗ˆ + iz5Tr Vµ" ⊗ˆ ⊗ˆ . πµ = WFV(r)=Š0dVµ, "iV0 =0.(12)· , 2 π = (15)(rˆ " ) . (17) " ⊗ L " " 3 =Tr F‡α Vα(ˆ⊗L⊗ˆR +⊗ˆR⊗ˆL) , (14c) ", - gr α W ⊗(0) = 0,W( )=0. ‰ L ‰ " ⊗ = i Tr ⊗ˆ3 ⊗ˆ ⊗ˆ3 ⊗ˆ , (14a) The configuration of the vector mesons are written as [20] in the 1-form1 notation‡L withR R L + M. Harada and K. Yamawaki, Phys. Rep. 381, 1 (2003)Given the Lagrangian and the wave functions, it is now L " For the baryon number B = 1 solution, these wave37 func- 2 = i Tr ⊗ˆ ⊗ˆ ⊗ˆ ⊗ˆ , (14b) straightforward to derive the soliton mass Msol.The ‡ L R L R + tions satisfy the followingµ boundaryG(r) conditions: L 2 Another example of this kind is Tr πˆ Tr πˆ that generates In the present work, we also consider the anomalous⊗ˆL =ˆ par-⊗ ⊗ˆ , In order to study the propertiesπµ = W ( ofr) theŠ0µ, soliton"0 =0 obtained, π = µ (rˆ " ) . (17)explicit expression for the soliton mass is given in Ap- 13년 6월 26일 수요일 =Tr F (ˆ⊗ ⊗ˆ ⊗ˆ ⊗ˆ ) , (14c) π π ity hWZ terms that are written as 3 ‡ V L αR" + "RfromL the Lagrangianthe (7), mass we diπ takeerence the between standard the " and parame-α mesons.gr α L ⊗ˆR =ˆ⊗ +ˆ"⊗ , F (0) =π ,F" π "( )=0, pendix A. Minimizing the soliton mass then gives the α "terization for the soliton configuration. For the pion field, ⊗ in the 1-form3 notation‡ with 2 + For the baryon numberG(0)B == 12 solution,,G( these)=0 wave, func- (18) N FV = dV iV . (15) " ⊗ c we use the standard hedgehog configuration,W ⊗(0) = 0,W( )=0. anom = ci i, (13)" tions satisfy the following boundary conditions: L 16 2 L ⊗ˆL =ˆ⊗ ⊗ˆ , ⊗ i=1 α " " F (r)
⊗ˆR =ˆ⊗ +ˆ⊗ , ‰(r)=expGiveni" rˆ theF Lagrangian(0). = ,F and(16)( the)=0 wave, functions, it is now α " ⊗ where 2 straightforward· 2 G(0) = to2,G derive( the)=0 soliton, mass M(18)sol.The 2 FV = dV iV . µ (15) , - " ⊗ Another example of this kind" is Tr πˆ Tr πˆ µ that generates explicit expressionW ⊗(0) = 0,W for the( soliton)=0. mass is given in Ap- 3 3 Theπ configurationπ of the vector mesons are written as [20]⊗ 1 = i Tr ⊗ˆL⊗ˆRthe mass⊗ˆR⊗ˆL diπ, erence between(14a) the " andπ α" mesons.π " pendix A. Minimizing the soliton mass then gives the L " Given the Lagrangian and the wave functions, it is now 2 = i Tr ⊗ˆL⊗ˆR⊗ˆL⊗ˆR , (14b) L ‡ + straightforwardG(r to) derive the soliton mass Msol.The 2 µ π = W (r) Š , " =0, π = (rˆ " ) . (17) Another example of this kind is Tr πˆ Tr πˆ µµ that generates0µ 0 3 =Tr FV (ˆ⊗L⊗ˆR ⊗ˆR⊗ˆL) , (14c) π explicit expressiongr forα the soliton mass is given in Ap- L ‡ "+ π the mass diπerence between the " andπ α" mesons.π " pendix A. Minimizing the soliton mass then gives the in the 1-form notation‡ with + For the baryon number B = 1 solution, these wave func- tions satisfy the following boundary conditions: ⊗ˆL =ˆ⊗ ⊗ˆ , α " " ⊗ˆR =ˆ⊗ +ˆ⊗ , F (0) = ,F( )=0, α " ⊗ 2 G(0) = 2,G( )=0, (18) FV = dV iV . (15) " ⊗ " W ⊗(0) = 0,W( )=0. ⊗ Given the Lagrangian and the wave functions, it is now straightforward to derive the soliton mass Msol.The 2 µ Another example of this kind is Tr πˆ Tr πˆ µ that generates explicit expression for the soliton mass is given in Ap- π π the mass diπerence between the " andπ α" mesons.π " pendix A. Minimizing the soliton mass then gives the MA et al. PHYSICAL REVIEW D 87, 034023 (2013) 2 III. HIDDEN LOCAL SYMMETRY INDUCED π0 r;t A t π π" r π^ r^ π r^" r A t ; π "α π " g ⊗ 1π "‰ 2π "Š yπ " FROM HOLOGRAPHIC QCD ’ r A. Master formula !i r;t π " π r^ i: (21) π "α r π  " In this section, following Refs. [3,4], we provide a general master formula to determine the parameters of With these wave functions the moment of inertia can be the HLS Lagrangian by integrating out the infinite towers calculated, and its explicit expression is given in of vector and axial-vector mesons in a class of hQCD Appendix B. It is then straightforward to obtain the MA et al. models expressedPHYSICAL by the following REVIEW general D 87, 5D034023 action: (2013) Euler-Lagrange equations for the wave functions " r , 1π " " 0r , and ’ r by2 minimizing the moment of inertia, and III. HIDDEN LOCALDBI SYMMETRYCS INDUCED 2π r;t A t π π"1 r π^ r^ π r^"2 r Ay t ; S5 S5 S5 ; (29) theπ π results" "α areππ"" alsog ⊗ given π in"‰ Appendix B. Theπ "Š boundaryπ " FROM HOLOGRAPHICα ‰ QCD conditions imposed on the excited fields are where the 5D Dirac-Born-Infeld (DBI) part SDBI and the i ’ r i A. Master formula 5 ! r;t π " π r^ : (21) Chern-Simons part SCS are expressed as π "α r π  " In this section, following5 Refs. [3,4], we provide a " 0 " 0; 10 1 general master formula to1 determine the parameters of With these waveπ functions"α π1 the"α moment of inertia can be SDBI N G d4xdz K z Tr ⊗ " 0 " 0; the5 HLS Lagrangianc YM by integrating1 outF the⊗ F infinite towers calculated, and20 its explicit2 expression is given(22) in α ‡ 2 π " ⊗ Š π "α π1"α of vector and axial-vectorZ α mesons in a class of hQCD Appendix B.’ It0 is then’ straightforward0; to obtain the models expressedK z M2 byTr theF followingF ⊗z general; 5D action: (30) Euler-Lagrangeπ equations"α π1 for"α the wave functions " r , 2 KK ⊗z 1 HLS‰ π&" hQCD⊗ Š⊗ π " DBI CS and"2 "r1,r andand’ "r 2 byr minimizingat r 0 satisfy the moment the constraint of inertia, and S S S ; (29) π "π " π " π " α 5 5 5 the results are also given in Appendix B. The boundary CS αNc ‰ S5 2 w5 A : DBI (31) conditions imposed2" on1 0 the excited"2 0 fields2: are (23) where the 5D Dirac-Born-Infeldα 24α M4 (DBI)R π part" S5 and the π "‰ π "α CS Z  Chern-Simons part S5 are expressed as In the adiabatic" 0 collective" quantization0; scheme, the where the rescaled ’t Hooft coupling constant is defined as 10 1 3 1 baryon mass is givenπ "α by π1"α GDBIYM ‰= 108α and4 the field strength of the⊗ 5D gauge " 0 " 0; S5 3
NcGπ YM "d xdz K1 z Tr F ⊗ F 20 1. 25d action (22) field αAM is F MN @M‡A2 N π "@N⊗AM i AŠ M; AN . π "α π1"α Z α α ‡ ‡ ⊗ Š ’ 0 i i ’ 1 0; j j 1 Here, K1;2 z are the metric functions of z constrained by M M π ‰ " M DBIπ ‰ " ;CS (24) πK" z M2 Tr ⊗z ; (30) sol π "α Sπ15 "α= Ssol5 + S5 the gauge/gravity2 KK duality.F ⊗ ThezF gravity enters in the z de- α ‰ 2I α ‰ 2I ‰ π " ⊗ Š⊗ and "1 r and "2 r at r 0 satisfy the constraint pendence of the YM coupling, giving rise to the warping of π " π " α 4 where i and j are the isospin and the spin of the baryon, the space. In Eq.CS (31),NcM and R stand for the 4D S5 2 w5 A : (31) respectively. Then the2"1π0-N mass"2 0 difference2: reads (23) Minkowski space-time and z-coordinate4 space, respec- π "‰ π "α α 24α M R π " tively, and w A is the CS 5-form,Z  written as In the adiabatic collective quantization3 scheme, the where the rescaled5π " ’t Hooft coupling constant is defined as 3 i 1 πM Mπ MN : (25) GYM ‰= 108α and the2 field strength3 of the5 5D gauge baryon mass is given
by ‡ α 2I w5 A Tr AF A F A : (32) field3
Aπ "απ is F " @ ‰A2 @ ‡A10 i A ; A . M MNπ α M N ‡ N M ‡ ⊗" M NŠ The baryonic size ofi i a baryon1 shouldj bej computed1 by Here,Here, FK1;2 zdAare theiAA metricis functions the field of strengthz constrained of the by5D M Msol π ‰ " Msol π ‰ " ; (24) απ " ‰ the baryon-numberα current‰ 2I of theα Skyrmion.‰ 2I However, in gaugethe gauge/gravity field A duality.A dx⊗ TheA gravitydz. It enters should in the bez notedde- α ⊗ ‰ z order to intuitively see2. the induce effects of the the vector HLS mesons Lagrangian on thatpendence the from DBI of the S part YM5: integrate is coupling, of O ‰1 givingwhile out rise thethe to CSthe higher warping term is of ofmodes 4 thewhere Skyrmioni and j sizeare in theSKYRMIONS a isospin simple andway, the here WITH spin we of consider VECTOR the baryon, the MESONSOthe‰0 space. INwith THE the In ’tEq. HIDDEN Hooft (31), couplingMπ..."and constantR stand‰. for thePHYSICAL 4D REVIEW D 87, 034023 (2013) respectively. Then the π-N mass difference reads Minkowskiπ " space-time and z-coordinate space, respec- winding number and energy root-mean-squareinteg radii. The We should stress here that, as noted in Ref. [14], the root-mean-square (rms) radiusA x; of z the winding-numberAπ x; z cur- tively, and w5 A is the CS 5-form, written as K1Š z @z K2 z @z c n z αn c n z ; (34) ππ " ! 3 π " structure of theπ " action (29) is shared bothŠ by theπ top-" π " π "‰ α π " rent is defined by πM Mπ MN : (25) down Sakai-Sugimoto model2 i and3 the bottom-up1 5 models,
‡ α 2"I^ x c z "^ x w5 VA x Tr AF A F A : (32) π 0 πsuch as inπ π"α Refs. [25,26] as‰ 2 well as‡ the10 moose models in α 1=?2 π " π "⊗ kπ "⊗ π "‰ π " The baryonic2 1 size=2 of a1 baryon3 2 should0 be computed by Ref.Here, [11F], withdA the differenceiAA is appearing thewith field strengthα onlyn being in the of warpingthe then 5Dth eigenvalue (α0 0). By substituting r W d rr B r "^;π x c 1(26)z ; α ‰ (33) α the baryon-numberh i α current0 of the Skyrmion.π "⊗ k However,π " π " in factors.gauge field ThisA allowsA usdx to⊗ writeA downEq.dz. ( a It33 ‘‘master should) into thebe formula’’ noted action in Eq. (30), the HLS Lagrangian up πZ " α ⊗ ‰ z order to intuitively see the effects of the vector mesons on whichthat the applies DBI part to all is holographic of O ‰1 while models the4 and CS moose term is con- of where B0 r is the timewhere componentc n ofare the winding-number eigenfunctions satisfying the following to O p can be obtained. The explicit expressions for the the Skyrmion size in a simplef way,g here we consider the structionsO ‰0 with given the ’t appropriate Hooft couplingπ warping" constant factors.π ‰" . current thatπ " is explicitlyeigenvalue written as equation obtained from the action (30): LECs we need are derived as [3,4] winding number and energy root-mean-square radii. The πNow,We" should to induce stress the here HLS that, Lagrangian as noted in from Ref. the [4], action the root-mean-square (rms) radius of the winding-number cur- 1 (structure30), we use of the the action mode expansion(29) is shared of the both 5D by gauge the top- field rent is defined by0 2 B 2 2 F0sin F: (27) AdownM x; Sakai-Sugimoto z and integrate model out and all the bottom-up modes except models, the α‡2α r suchπ as in" Refs. [25,26] as well as the moose models1 in 2 21=2 pseudoscalar2 and2 the lowest-lying2 vector2 mesons, which 2 1=2 f N G M dzK z c_ z ; af N G M α c ; N G c ; r2 ⊗ 1 d3crr2YMB0 r KK ; 2 1=2 (26) 0 reducesRef. [11], withx;⊗ the z differenceto AcintegYMx; appearing z .KK In the1 onlyA 1x; in z the warping0 gauge,2 c YM 1 We define the energyW root-mean-squareα radius r E πas" π "‰ AM α M h z i g α h i 38 h i α π 0 π "" Z h i factors. Thisπ allows" us to writeπ " down4 a ‘‘masterπ "α formula’’ Z this2 implies2 the substitution [3,4] 2 2 0 y1 y N G1=2 1 c whichc applies;y to all holographicy modelsN G andc moose1 con-c ; where 13년 6월 26일 수요일B r2 is1=2 the time1 component1 32 2 of the winding-numberc YM 1 0 3 4 c YM 1 1 πr " E αŠd rrαŠMsol r ; hπ ⊗(28) Šstructions" i given appropriateአአwarping factors.h π ⊗ " i current thath i is explicitlyα πMysol written0 2y as yπ "" 2N G c 2 c3We2 ;y use the index M y ⊗;zy, with;y⊗ 0, 1, 2,2 3.N G c 1 c 1 c c 2 ; 5 Z 8 9 c YM 1 4Now,0 to induce6 theαπ HLS5 " Lagrangian7 α 7 from thec actionYM 1 1 1 0 α አአh (30As),i emphasized we use theα in mode Ref. Šπ [4 expansion],⊗ the procedure" of the of 5D‘‘integratingα gauge field out’’h π ⊗ "π ⊗ Š "i where Msol r is the0 soliton mass1 (energy)2 density given in adopted here is different from the ‘‘naive truncation’’ that 1 1 1 π " B F0sin F: (27) 2 A x; z and integrate out2 all the modes except the _ 2 2 Appendix A. α‡z4 2α22rN2 cGYM c 1 1 c 1 c 0 violates;zMπ the"5 chiral2 invariance.NcGYM c 1 1 c 1 ;c1 c 0 c 1 c 0 c 1 ; α h π ⊗ Š "ipseudoscalarአand the lowest-lyingh π ⊗ vector"i mesons, whichα ππ "2 ⊗ 6 Š 2α⊗⊗ 2 1=2 integ We define the energy root-mean-square radius1 r E 1as 1reduces A1 M x; z to AM x; z . In1 the Az x; z 0 gauge, _ 2 2 π " π " 4 _ 2π "α c2 c 0 c 1 ch0 i c 1 thisc implies1 the;c substitution3 [3,4] c 0 c 1 ; (35) α Š 21=2 ⊗ 6 ⊗ 2 ⊗ 2 α 2 2 1=2 1 ππ1 3 2 " 034023-4 α⊗⊗ ππ ⊗⊗ r E d rr Msol r ; (28) h i α M 0 π " 3 π sol Z " We use the index M ⊗;z , with ⊗ 0, 1, 2, 3. 4As emphasized in Ref.απ [4], the" procedureα of ‘‘integrating out’’ where Msol r is the solitonwhere massα (energy)is the smallest density given (nonzero) in adopted eigenvalue here is different of the ei- from the ‘‘naive truncation’’B. that The a independence Appendix Aπ ." 1 violates the chiral invariance. genvalue equation given in Eq. (34), and and are hi hhii In the phenomenological analysis, it is well known defined as that the HLS parameter a plays an important role [8–10]. 2 034023-4 With the leading Lagrangian at O p , the choice of A 1 dzK z A z ; A 1 dzA z (36) π " h i  1π " π " hh ii  π " a 2 reproduces the Kawarabayashi-Suzuki-Riazzudin- ZŠ1 ZŠ1 Fayyazudinα relation and the ‰ meson dominance in the for a function A z . Equation (35) provides the master 4 π " pion electromagnetic form factor. At O p , it was shown formula for the LECs in the HLS Lagrangian induced that the quantum correction enhances theπ infrared" value of from general hQCD models. Namely, one just needs to a, and, therefore, a good description of low-energy phe- plug the warping factor and the eigenfunctions of Eq. (34) nomenology can be achieved with the bare value of a being into Eq. (35) to obtain the values of the LECs. For example, 1=3 2 2 [10]. In the holographic approach, on the other hand, K1 z KŠ z and K2 z K z with K z 1 z the‡ parameter a is attributed to the normalization of the 5D correspondπ "α to theπ " Sakai-Sugimotoπ "α π model." π "α ⊗ wave function c 1 z , which cannot be determined from the In addition to the general hQCD models, we also con- homogeneous eigenvalueπ " equation (34). As a result, it turns sider the BPS model studied in Refs. [27,28] which is out [4] that the physical quantities are independent of the characterized by the flat space-time. In this case, instead 5 parameter a as far as the leading order in Nc is concerned. of solving the eigenvalue equation, the 5D gauge field is In order to explicitly see that any physical quantities expanded in terms of the Hermite function c n [27,28], calculated with the HLS Lagrangian induced from hQCD n 1 n 1 models are actually independent of the parameter a, we 1 Š z2=2 d Š z2 c n z πŠ " eŠ n 1 eŠ ; (37) start with Eq. (33). We first note that the vector-meson π "α n 1 dz Š n 1 !2 Š p⊗ mass and the pion decay constant are related by the relation π Š " q m2 ag2f2 , and m and f are fixed by their experimen- where n 1 and c corresponds to the wave function of ‰ α ⊗ ‰ ⊗ 1 tal values. Therefore, the HLS parameter a and the HLS the lowest-lying
vector meson. The wave function of the gauge coupling g are connected through ag2 m2 =f2 . Nambu-Goldstone pseudoscalar boson is expressed in α ‰ ⊗ terms of the error function erf z , Therefore, the g independence of physical quantities is π " equivalent to their a independence. 2 z 2 c z erf z e d : (38) To see the a independence explicitly, we define 0π "α π "αp⊗ Š 0 ~ Z c 1 z gc 1 z ; (39) In the following calculation, we use the Hermite function π "α π " ~ and the error function as wave functions of the vector mode so that the new function c 1 is normalized as and pseudoscalar mode, respectively. Then, the LECs of the HLS Lagrangian are determined by using the above 5When we include the loop corrections which can be regarded c 0 z and c 1 z in the master formulas in Eq. (35) with as a part of 1=Nc corrections, the a dependence becomes relevant K zπ " 1. π " to the physical quantities through the loop corrections. π "α

034023-5 5

1 Here, = d +i is the field strength of the 5D gauge c = α˙ α2 , (35) field F= Adxµ +AA dz. It should be noted that the DBI 3 2 0 1 µ z ,, // partA is of AO(π1) whileA the CS term is of O(π0)withthe where π is the smallest (non-zero) eigenvalue of the ’t Hooft coupling constant π. 1 eigenvalue equation given in Eq. (34), and and We should stress here that as noted in Ref. [4], the are defined as α⊗ αα ⊗⊗ structure of the the action (29) is shared both by the top- down Sakai-Sugimoto model and the bottom-up models ⊗ A dzK (z)A(z), such as in Refs. [25, 26] as well as the moose models α ⊗ 1 in Ref. [11], with the diπerence appearing only in the αα⊗ ⊗ warping factors. This allows us to write down a “mas- A dzA(z) (36) αα ⊗⊗ ter formula” which applies to all holographic models and αα⊗ moose construction given appropriate warping factors. for a function A(z). Equation (35) provides the master Now to induce the HLS Lagrangian from the action formula for the LECs in the HLS Lagrangian induced (30), we use the mode expansion of the 5D gauge field from general hQCD models. Namely, one just needs to M (x, z) and integrate out all the modes except the pseu- plug the warping factor and the eigenfunctions of Eq. (34) A doscalar and the lowest lying vector mesons, which re- into Eq. (35) to obtain the values of the LECs. For integ 1/3 duces M (x, z)toA (x, z). In the Az(x, z) = 0 gauge, example, K (z)=K (z) and K (z)=K(z)with A M 1 α 2 this implies the following substitution [3, 4]:4 K(z)=1+z2 correspond to the Sakai-Sugimoto model. In addition to the general hQCD models, we also con- A (x, z) Ainteg(x, z) µ π µ sider the BPS model studied in Refs. [27, 28] which is =ˆ"µ (x)α0(z)+ "ˆµ (x)+Vµ(x) characterized by the flat space-time. In this case, instead " π of solving the eigenvalue equation, the 5D gauge field is +ˆ"µ (x)α1(z), (33) " π " expanded in terms of the Hermite function αn [27, 28], where αn are eigenfunctions satisfying the following { } n 1 n 1 eigenvalue equation obtained from the action (30), ( 1) α z2/2 d α z2 α (z)= " eα eα , (37) n n 1 dzn 1 1 (n 1)!2 α ‰ α 5 Kα (z)⊗z [K2(z)⊗zα (z)] = πnα (z), (34) " " 1 n n where n 10 and α1 corresponds to the wave function with πn being theDeterminationn-th eigenvalue (π0 = 0). By sub- of the lowestŠ lying of vector meson.couplings1 The wave function of stitutingHere, Eq.= (33)d into+i the actionis the infield Eq. strength (30), the of HLS the 5D gauge ˙ 2 F A µ AA the Nambu-Goldstonec3 pseudoscalar= α0α boson1 , is expressed in (35) field = µdx 4+ zdz. It should be noted that the DBI 2 LagrangianA upA to O(p )A can be obtained. The explicit terms of the error function,, erf(z), // expressionspart is forof O the(π LECs1) while we need the are CS derived term is as of [3,O 4](π0)withthe where π1 is the smallestz (non-zero) eigenvalue of the ’t Hooft coupling constant π. 2 2 "2 2 2 ˙ αeigenvalue0(z)=erf(z)= equatione givenα d‰. in Eq.(38) (34), and and fWeπ = shouldNcGYMM stressKK heredzK2( thatz) α0 as(z) noted, in Ref. [4], the ‰ are defined as α0 α⊗ αα ⊗⊗ structure2 of the2 theα action2 (29)⊗ is shared both by the top- af = NcG M π α , In the following calculation, we use the Hermit function downπ Sakai-SugimotoYM KK 1α 1 model⊗ and the bottom-up models ⊗ 1 2 and the error function as wave functionsA ofdzK the vector1(z)A(z), such2 = asNc inGYM Refs.α1 , [25, 26] as well as the moosemode models and pseudoscalar mode, respectively.α ⊗ Then, the g α ⊗ αα⊗ in Ref. [11], with the diπerence appearing2 onlyLECs in the of the HLS Lagrangian are determined by using y = y = N G 1+α α2 , ⊗ warping1 " factors.2 " c ThisYM allows1 " us0 to write down athe “mas- above α0(z) and α1(z) intoA the master formulasdzA(z) in (36) Eq. (35) with K(z) = 1. αα ⊗⊗ ter formula” which applies‰Š 2 to all2 holographic
models and αα⊗ y3 = y4 = NcGYM α1 (1 + α1) , moose" construction" given appropriate warping factors. for a function A(z). Equation (35) provides the master y =2y = y = 2N‰ G α2α2
, K1(z),K2(z) : metric functions Now5 to8 induce" 9 " thec HLSYM Lagrangian1 0 from the action formulaB. The fora independence the LECs in the HLS Lagrangian induced y = (y + y ) , 1/3 (30),6 we use5 the7 mode expansion‡ + of the 5D gauge field Kfrom1(z)= generalK hQCD(z),K models.2(z)=K Namely,(z) one just needs to " 2 My7(x,=2 zN)c andGYM integrateα1 (1 + α out1) 1+ allα the1 modesα0 , except theIn pseu- the phenomenologicalplug the warping analysis, factor2 it and is well the eigenfunctions known of Eq. (34) A " with K(z)=1+z doscalar and the lowest lying2 vector mesons, whichthat there- HLS parameter a plays an important role [8– z4 =2NcGYM ‡α1 1+α1 Š α0 , + into Eq. (35) to obtain the values of the LECs. For integ " 10]. With thein leading the Sakai-Sugimoto Lagrangian at O(p model21),/3 the choice of duces M (x, z)toA2 M (x, z). In the Az(x, z) = 0 gauge, example, K1(z)=Kα (z) and K2(z)=K(z)with z5 =A 2NcGYM‡ αŠ1 (1 + α1) , + this implies" the following substitution [3, 4]:4 a = 2 reproducesK(z)=1+ the Kawarabayashi-Suzuki-Riazzudin-z2 correspond to the Sakai-Sugimoto model. 1 1 1 Fayyazudin (KSRF) relation and the Š meson dominance c = α˙ α ‡α2 + α2 + , 1 0 1 0 integ 1 in the pion electromagneticIn addition form to the factor. general At hQCDO(p4), it models, we also con- Aµ(x, z) 2 Aµ 6 (x," z2) Two parameters ,, -π .// was shown thatsider the thequantum BPS correction model studied enhances in the Refs. in- [27, 28] which is ˙ 1 2 1 2 1 1 c2 = α0α1 =ˆ"αµ0 +(x)αα10+(z)+α1 +"ˆµ (x)+, Vµ(x)frared5 value ofcharacterizeda, and, therefore, by the a good flat space-time. description of In this case, instead "2 π 6 2 2 " ,, - .// low-energy phenomenologyof solvingKK the canmass eigenvalue be achieved equation, with the bare the 5D gauge field is +ˆ"µ (x)α1(z), (33) " π value" of a being 2 [10]. In the holographic approach, on 1 expanded in terms of the Hermite function αn [27, 28], Here, = d +i is the field strength of the 5D gauge ˙ 2 the other hand, the parameter‘t Hoofta is attributed coupling to the nor- F A µ AA wherec3 = αn αare0α1 eigenfunctions, satisfying the following(35) 4 n 1 n 1 field = µdx + zdz. It should be noted that the DBI As emphasized{ 2 in} Ref. [4], the procedure of “integrating out” malization of the 5D wave functionm =α 7761(z), MeV which cannot2 2 A A A eigenvalue,, equation// obtained from the action (30), ⇢( 1) α z /2 d α z part is of O(π1) while the CS term is of O(π0)withthe adopted here is diπerent from the “naive truncation” that vio- be determined fromα ( thez)= homogeneous" eigenvalueeα equa- eα , (37) n n 1 n 1 wherelatesπ the1 chiralis the invariance. smallest1 (non-zero) eigenvalue oftion the (34). As a result, it turns(fn⇡ out= [4]1)!2 92. that4 MeVα the‰ physical dz α 9 ’t Hooft coupling constant π. Kα (z)⊗z [K2(z)⊗zα (z)] = πnα (z), (34) " eigenvalue equation" 1 given in Eq. (34),n and n and We should stress here that as noted in Ref. [4], the a is still undetermined α⊗ αα ⊗⊗ 0 are defined as where n 1 and α1 corresponds to the wave function structure of the the action (29) is shared both by the top- with πn being the n-th eigenvalue (π0 = 0). By sub- Š TABLE I. Low energy constants of the HLSof the Lagrangian lowest lying at O vector(p4)with meson.a =2. The wave function of down Sakai-Sugimoto model and the bottom-up models stituting Eq. (33) into⊗ the action in Eq. (30), the HLS the Nambu-Goldstone pseudoscalar boson is expressed in Lagrangian upA to O(p4)dzK can1( bez)A obtained.(z), The explicit such as in Refs. [25, 26] as well as the moose models Model α y⊗ 1 y3 y5 y6 termsz of4 the errorz5 function erf(c1z), c2 c3 in Ref. [11], with the diπerence appearing only in the expressions for theα LECsα⊗ we need are derived as [3, 4] SS model 0.001096 ⊗ 0.002830 0.015917 +0.013712 0.010795 0.007325 +0.381653 z 0.129602 0.767374 π π π π π 2 warping factors. This allows us to write down a “mas- BPS model 0A.071910 dzA0.153511(z) 0.0122862 (36)0.196545 0.090338 0.130778 0.2069922 +3".031734 1.470210 2 παα ⊗⊗ 2 π π ˙ π απ0(z)=erf(zπ)= eα d‰. (38) ter formula” which applies to all holographic models and fπ = NcGYMαMα⊗KK dzK2(z) α0(z) , ‰ α0 39 moose construction given appropriate warping factors. for a function2 A(z). Equation2 α (35)2 provides⊗ the master afπ = NcGYMMKKπ1 α1 , In the following calculation, we use the Hermit function Now to induce the HLS Lagrangian from the action13년 6월 26일 수요일formula for the LECs in the HLSα ⊗ Lagrangian induced TABLE II.1 Skyrmion mass and size calculated in the HLS with theand SS theand error BPS models function with asa wave=2.Thesolitonmass functions of the vectorMsol and (30), we use the mode expansion of the 5D gauge field from general= hQCDN G models.α2 , Namely, one just needs to 2 2 2 µ the π-N mass2 di"cerenceYM π1M are in unit of MeV while r W andmoder andE are pseudoscalar in unit of fm. mode, The respectively. column of O(p Then,)+πµ theB is M (x, z) and integrate out all the modes except the pseu- plug the warpingg factorα and⊗ the eigenfunctions of2 Eq. (34)" α " α A “the minimal model” of Ref. [20] and that of O(p ) corresponds toLECs the model of the of HLS Ref. Lagrangian [19]. See the are text determined for more details. by using doscalar and the lowest lying vector mesons, which re- 2 2 π π into Eq. (35)y = toy obtain= N thecG values1+ ofα1 theα LECs., For integ 1 2 1/3 YM 0 the above α0(z) and α1(z) into2 the masterµ formulas2 in duces M (x, z)toAM (x, z). In the Az(x, z) = 0 gauge, example, K1HLS(z)="1(",Kα,α"π)HLS(z) and1(",Kα)HLS2(z)=" 1K("(z) )withBPS(", α, π)BPS(", α)BPS(") O(p )+πµB [20] O(p )[19] A 4 2 ‰ 2
Eq. (35) with K(z) = 1. this implies the following substitution [3, 4]: K(zM)=1+sol y z=correspond1184y = N toG the834አSakai-Sugimoto2 (1 + α )922, model.1162 577 672 1407 1026 3 " 4 " c YM 1 1 integ Inπ additionM to the448 general hQCD1707 models,1014 we also con- 456 4541 2613 259 1131 Aµ(x, z) A (x, z) ‰ 2 2
µ r2 y5 =20.433y8 = y9 = 20.247NcGYM α1α0.3090 , 0.415 0.164 0.225 0.540 0.278 π sider theW BPS model" studied" in Refs. [27, 28] which is B. The a independence " 2α =ˆ"µ (x)α0(z)+ "ˆµ (x)+Vµ(x) characterizedr E y6 = by0.608 the(y5 flat+ y7 space-time.) , 0.371 In this0.417 case, instead0.598 0.271 0.306 0.725 0.422 π " π" α " ‡ + of solving the eigenvalue equation, the 5D gauge2 field is +ˆ"µ (x)α1(z), (33) π y7 =2NcGYM α1 (1 + α1) 1+α1 α0 , In the phenomenological analysis, it is well known " π " " expanded in terms of the Hermite function2 αn [27, 28], z =2NcG α 1+α α , that the HLS parameter a plays an important role [8– where αn are eigenfunctions satisfying the following 4 YM ‡ 1 1 "Š 0 + 2 1.5 n 1 2 n 1 10]. With the leading Lagrangian at O(p ), the choice of { } ( 1) 2 d 2 eigenvalue equation obtained from the action (30), z5 = 2NcGYMα‡ αŠ1 (1 +z α/21) ,α+ z 3 αn(z)= " " eα eα , (37) a = 2 reproduces the Kawarabayashi-Suzuki-Riazzudin- n 1 n (r)1 HLS1(π) 1 (n 1)!2 α1 ‰ 1 dz1ξα1 Fayyazudin (KSRF) relation and the Š meson dominance Kα (z)⊗ [K (z)⊗ α (z)] = π α (z), (34) ˙ ‡ 2 2 + 1 z 2 z n n n c1 = "α0α1 α0 + α1 , HLS ( 4 " 1 2 6 " 2ξ (r) in the pion electromagnetic form factor. At1 πO,ρ()p ), it where n 10 and,, α corresponds- to the.//2 wave function 2 with π being the n-th eigenvalue (π = 0). By sub- Š 1 was shown that the quantum correctionHLS enhances(π,ρ,ω the) in- n 0 of the lowest lying˙ vector meson.1 2 1 The2 waveϕ1(r) × function 101 of F(r) 1 stituting Eq. (33) into the action in Eq. (30), the HLS c2 = α0α1 α0 + α1 + α1 + , frared value of a, and, therefore, a good description of the Nambu-Goldstone pseudoscalar"2 6 boson2 is expressed2 in 1 Lagrangian up to O(p4) can be obtained. The explicit 0.5 ,, - .// low-energy phenomenology can be achieved with the bare terms of the error function erf(z), value of a being 2 [10]. In the holographic approach, on expressions for the LECs we need are derived as [3, 4]  z the other hand, the parameter a is attributed to the nor- 2 2 2 0 4 " 2 2 ˙ As emphasizedα0(z)=erf( in Ref.z)= [4], the procedureeα d of‰. “integrating(38) out” malization of the 5D wave function α1(z), which cannot fπ = NcGYMMKK dzK2(z) α0(z) , 0 ‰ adopted here is diπerent from theα0 “naive truncation” that vio- be determined from the homogeneous eigenvalue equa- α ⊗ lates the chiral invariance. −1 G(r) af 2 = N G M 2 π α2 , In the following calculation, we use the Hermit function tion (34). As a result, it turns out [4] that the physical π c YM KK 1α 1⊗ 1 2 and the−0.5 error function as wave functions of the vector = NcG α , −2 g2 YMα 1⊗ mode and pseudoscalar mode, respectively. Then, the 2 LECs of0 the HLS Lagrangian1 are determined23 by using y = y = N G 1+α α2 , 0 1 23 1 " 2 " c YM 1 " 0 the above α0(z) and α1(z) intor (fm) the master formulas in r (fm) Eq. (35) with K(z) = 1. ‰Š 2 2 
y3 = y4 = NcGYM α1 (1 + α1) , " " FIG. 2. The excitations of the soliton profile, ⊗1(r)(solidline), FIG. 3. Comparison of the soliton wave functions F (r) 2 2 y =2y = y = 2N‰ G α α
, ⊗2(r)(dashedline),and (r)(dot-dashedline)calculatedin and G(r) in the three models, HLS ("), HLS (", α), and 5 8 " 9 " c YM 1 0 B. The a independence 1 1 the HLS1(", α, π)modelwitha = 2. Here, ⊗1(r)and⊗2(r) HLS1(", α, π), which are represented by the solid line, dashed y6 = (y5 + y7) , " ‡ + are dimensionless, while (r) is in unit of fm. lines, and dotted lines, respectively. y =2N G α (1 + α ) 1+α α2 , 7 c YM 1 1 1 " 0 In the phenomenological analysis, it is well known 2 that the HLS parameter a plays an important role [8– z4 =2NcGYM ‡α1 1+α1 Š α0 , + " 10]. With the leading Lagrangian at O(p2), the choice of 2 found that the inclusion of the π meson reduces and the HLS1(", π) models. However, as can be z5 = 2NcGYM‡ αŠ1 (1 + α1) , + " a = 2 reproducesthe soliton the mass. Kawarabayashi-Suzuki-Riazzudin- In the present work, the soli- seen by the dotted lines, inclusion of the α meson 1 2 1 2 1 Fayyazudin (KSRF) relation and the Š meson dominance c = α˙ α ‡α + α + , ton mass reduces from 922 MeV in the HLS1(") expands the wave functions. All these behaviors 1 0 1 0 1 in the pion electromagnetic form factor. At O(p4), it 2 6 " 2 to 834 MeV in the HLS (", π), which confirms the can be found in the rms sizes r2 and r2 ,, - .// was shown that the quantum correction1 enhances the in- W E 1 1 1 1 claim that the inclusion of the π meson makes the in Table II. Therefore, we concludeπ " that the ππ me-" c = α˙ α α2 + α2 + α + , frared value of a, and, therefore, a good description of 2 0 1 "2 0 6 1 2 1 2 Skyrmion closer to the BPS soliton. However, when son decreases the soliton massπ while the απmeson low-energy phenomenology can be achieved with the bare ,, - .// we include the α meson, the soliton mass increases value of a being 2 [10]. In the holographic approach, on increases it. the otherto 1184hand, MeV. the parameter This is ina is contrast attributed to the to the naive nor- ex- 4 pectation that including more vector mesons would 2. In the moment of inertia, or in the π-N mass dif- As emphasized in Ref. [4], the procedure of “integrating out” malization of the 5D wave function α1(z), which cannot adopted here is diπerent from the “naive truncation” that vio- be determineddecrease from the soliton the homogeneous mass. Since eigenvalue the α meson equa- inter- ference πM , through the collective quantization, lates the chiral invariance. tion (34).acts As with a result, the other it turns mesons out through [4] that the the hWZ physical terms, the role of the π and α mesons are the opposite to this observation shows the importance of the hWZ the case of the soliton mass. The mass di"erence terms in the Skyrmion phenomenology. The role of πM increases by the inclusion of the π meson, i.e., the α meson in the Skyrmion mass and size can also from 1014 MeV in the HLS1(") to 1707 MeV in be verified by comparing the soliton wave functions the HLS1(", π), which worsens the situation phe- shown in Fig. 3. This figure shows that the π me- nomenologically. Furthermore, in the nucleon and son shrinks the soliton wave functions, which can π masses, the rotational energy at O(1/Nc)iseven be seen by comparing the results from the HLS1(") larger than the soliton mass that is of O(Nc). This 7

+ K (z) Tr (F F µz)+Tr F˜ F˜µz , the Skyrme parameter e becomes the α meson coupling, 2 µz µz so that we have e = g 6, which is close to the empirical " π " α⊗(51) value e =5.45 that is determined from the π-N mass di"erence. In the HLS Lagrangian up to O(p4), however, 4 where FMN is the field strength of the SU(2) gauge field we have additional contributions from the pure O(p ) ˜ ASU(2) and FMN stands for that of the U(1) gauge field terms that lead to the Skyrme term. Explicitly, after ˜ AU(1). This explicitly shows that, without the CS term, integrating out the α meson, the e"ective Lagrangian is when the 5D model is de-constructed from the 4D model, obtained as only the kinetic and mass terms of the iso-scalar vector 2 µ ChPT = f" Tr ‰ µ‰ meson π are allowed. This conclusion can be explicitly L " " verified by using a specific hQCD model such as the SS -1 z4. y1 y2 2 + π Tr ‰ µ, ‰ α model. 2g2 π 2 π 4 " " As can be read from Appendex A, the contribution + , y1 + y2 2 - . from the kinetic and the mass terms of the π meson to + Tr ‰ µ, ‰ α , (56) 4 " " the soliton mass is where [ , ] is the commutator/ and 0, is the anticom- { } π 1 2 2 2 2 2 mutator. The second terms is the Skyrme term and we M =4" dr r ag f W + W π , (52) sol π2 " can read the e"ective Skyrme parameter e as ‰ Š ‡ 
which gives the equation of motion of W as 1 1 z4 y1 y2 = π . (57) 2e2 2g2 π 2 π 4 2 2 2 W ππ = ag f"W W π (53) π r Since the gauge/gravity duality implies that y1 = y2, the last term of Eq. (56) vanishes. Using Eq. (57)π and in the absence of the CS term. By making use of the the analytic expressions for the LECs given in Eq. (35), 7 partial integration with the boundary conditions given the Skyrme parameter is written as π in Eq. (18), Msol can be calculated as 1 NcGYM µz µz the Skyrme= parameter(1 Še2)2becomes, (58) the α meson coupling, π 1 2 2 2 2˜ ˜ 2e2 2 π 0 + KM2sol(z=4) "Trdr (FµzFr ag)+Trf"W FWµzπ FW ππ W , π2 π r π so that we have e =1 g 6,2 which is close to the empirical ‰ Š + , ‡ With the experimental values of the two inputs f" and =0π, " α⊗(54) " (51) m⊗,value we obtaine the=5 Skyrme.45 that parameter is determinede as from the π-N mass because of the equation of motion of Eq. (53). Therefore, di"erence. In the HLS Lagrangian up to O(p4), however, e 7.31. (59) 4 where FMN is thein the field absence strength of the CS ofterm, the the SU(2)π field decouples gauge from field we have additional" contributions from the pure O(p ) the other fields and M π vanishes. This is consistent with ˜ sol in the SS model, while in the flat space-time case, i.e., ASU(2) and FMNthestands earlier studies for on that the Skyrmions of the U(1) stabilized gauge by vector field terms that lead to the Skyrme term. Explicitly, after for the BPS soliton model, we obtain ˜ mesons [20]. As can be seen from the equation of motion AU(1). This explicitly shows that, without the CS term, integrating out the α meson, the e"ective Lagrangian is of W (r) given in Appendex A, the hWZ terms provide e 10.02. (60) when the 5D modelthe source is de-constructed terms of the π meson from field. the Therefore, 4D model, in the obtained as " absence of the hWZ terms, the π field decouples and does These values are larger than the empirical value of the only the kinetic and mass terms of the iso-scalar vector 2 µ not contribute to the soliton formation. Skyrme parameter=e =5f .Tr45 because‰ ‰ of the contributions ChPT " µ 4 meson π are allowed. This conclusion can be explicitly from theLy1, y2, and z4 terms that" are" of O(p ). Comparison with theSince Skyrme the moment of inertia Lagrangianis proportional to 1/e3 verified by using a specific hQCD model such as the SS -1 I z4. y1 y2 2 D. The eπective Skyrme parameter e in the Skyrme model,+ with a larger value of e,wehaveaπ Tr ‰ µ, ‰ α model. 2g2 π 2 π 4 " " smaller moment of inertia,+ which results in a larger mass, As can be readFinally from we Appendex estimate the Skyrme A, the parameter contributione in the splitting between the π yand+ they nucleon as is verified 2 - . 7 numerically in the next Section.1 2 from the kineticoriginal and Skyrme the mass model terms by integrating of the outπ themeson isovector to + Tr ‰ µ, ‰ α , (56) Original Skyrme Lagrangian 4 " " the soliton massα meson is from the HLS. The original Skymre model La- + K (z) Tr (F F µz)+Tr F˜ F˜µz , thegrangian Skyrme reads parameter e becomes the α meson coupling, / 0 2 µz µz so that we have e = g 6, which is close to the empirical whereIV. NUMERICAL [ , ] is the RESULTS commutator FOR THE and , is the anticom- 2 SKYRMION π " α⊗ f" 1 µ " 1 2 { } π (51) value =e =5Tr.45⊗2 thatU⊗ U2 is† determined2+ 2 Tr ⊗ fromUU2 † the, ⊗ UUπ-†N ,mass mutator. The second terms is the Skyrme term and we M =4" Skdr r µ ag f W 2+ Wµπ ,α (52) sol diL"erence.4 In2 the HLS Lagrangian" 32e up to O(p4), however, π 
- .(55) 4 Incan this section, read the we present e"ective the results Skyrme of numerical parameter cal- e as where FMN is the field strength of the SU(2) gauge field ‰we haveŠ additional contributions2 from the‡ pure O(p ) ˜ where the chiral field U is U = and the first term is the culations on the Skyrmion properties in the framework ASU(2) and FMN stands for that of the U(1) gauge field terms that lead to the Skyrme term.
Explicitly, after which gives thenonlinear equationAfter sigma ofintegrating model motion Lagrangian of outW thatas VM can bein written HLS as of the HLS discussed in the1 previous1 section.z4 They HLS1 y2 A˜ . This explicitly shows that, without the CS term, integrating2 µ out the α meson, the e"ective Lagrangian is 4 = . (57) U(1) f"Tr(‰ µ‰ ), where ‰ µ is defined as‰ ˆ µ without the Lagrangian up to O(p ) in Eq.2 (7) which2 is considered inπ when the 5D model is de-constructed from the 4D model, obtainedvector" field. as" In the earlier" analyses [29]" with the HLS the present calculation contains2e 172g parameters,π 2 π namely,4 only the kinetic and mass terms of the iso-scalar vector 2 2 2 2 LagrangianW ππ ==ag upf 2 Tr tof O‰W(p ‰),µ it isW knownπ that the Skyrme(53) f", a, g, yi (i =1,...9), z4, z5, and c1,2,3.Wedeter- meson π are allowed. This conclusion can be explicitly termChPT can be obtained" " µ from the α meson kinetic energy mineSince all these the LECs gauge/gravity through hQCD models, duality which implies are that y = y , L " π" r 1 π 2 verified by using a specific hQCD model such as the SS term in the limit of-1 infinitez4. α mesony1 y2 mass. In this case,2 characterizedthe last by term the warping of Eq. factor (56)K(z vanishes.) and the wave Using Eq. (57) and + π Tr ‰ µ, ‰ α e 7.31 model. 2g2 π 2 π 4 " " in the absence of the CS+ term. By making, use of the the analytic expressions for the LECs given in Eq. (35), As can be read from Appendex A, the contribution - . ' partial integration with they1 + boundaryy2 conditions2 given in the SS model from the kinetic and the mass terms of the π meson to + Tr ‰ µ, ‰ α , (56) the Skyrme parameter is written as π 4 " " the soliton mass is in Eq. (18), Msol can be calculated as where [ , ] is the commutator/ and 0, is the anticom- { } 1 NcGYM 2 2 π 1 2 2 2 2 2 mutator. The second terms is the Skyrme term and we = (1 Š ) , (58) Msol =4" dr r ag f"W + W π π, (52) 1 2 2 2 2 2 0 π2 M =4" drcan read ther e"agectivef SkyrmeW parameterW We as W 2e 2 π ‰ Š ‡sol " π ππ 
π2 π r π 1 2 which gives the equation of motion of W as ‰ Š + 1 1 z4 y1 y2 , ‡ With the experimental values of the two inputs f and = π . (57) " =0, 2 2 (54) 2e 2g π 2 π 4 m⊗, we obtain the Skyrme parameter e as 2 2 2 W ππ = ag f"W W π (53) π r because of the equationSince the gauge/gravity of motion duality of Eq. implies (53). that Therefore,y1 = y2, the last term of Eq. (56) vanishes. Using Eq. (57)π and e 7.31. (59) in the absence of the CS term. By makingin the use absence of the ofthe the analytic CS expressions term, the forπ thefield LECs decouples given in Eq. from (35), " partial integration with the boundary conditions given the Skyrmeπ parameter is written as π the other fields and Msol vanishes. This is consistent with in Eq. (18), Msol can be calculated as in the SS model, while in the flat space-time case, i.e., the earlier studies on the Skyrmions1 N G stabilized by vector = c YM (1 Š2)2 , (58) for the BPS soliton model, we obtain 40 π 1 2 2 2 2 mesons [20]. As can be seen2e2 from2 the equationπ 0 of motion Msol =4" dr r ag f"W W π W ππ W π2 π r π 1 2 ‰ Š + of W (,r)13년 6월 26일 수요일 given‡ With in Appendex the experimental A, the values hWZ of the terms two inputs providef" and e 10.02. (60) =0, (54) the source termsm⊗, of we the obtainπ themeson Skyrme field. parameter Therefore,e as in the " because of the equation of motion of Eq. (53). Therefore, These values are larger than the empirical value of the absence of the hWZ terms, the π fielde 7.31 decouples. and does(59) in the absence of the CS term, the π field decouples from " Skyrme parameter e =5.45 because of the contributions the other fields and M π vanishes. This isnot consistent contribute with to the soliton formation. sol in the SS model, while in the flat space-time case, i.e., 4 the earlier studies on the Skyrmions stabilized by vector from the y , y , and z terms that are of O(p ). for the BPS soliton model, we obtain 1 2 4 mesons [20]. As can be seen from the equation of motion Since the moment of inertia is proportional to 1/e3 of W (r) given in Appendex A, the hWZ terms provideD. The eπective Skyrmee 10 parameter.02. e (60) in the Skyrme model, with a largerI value of e,wehavea the source terms of the π meson field. Therefore, in the " absence of the hWZ terms, the π field decouples and does These values are larger than the empirical value of the smaller moment of inertia, which results in a larger mass not contribute to the soliton formation. Skyrme parameter e =5.45 because of the contributions splitting between the π and the nucleon as is verified Finally we estimate the Skyrme parameter e4 in the from the y1, y2, and z4 terms that are of O(p ). original SkyrmeSince model the by moment integrating of inertia outis proportional the isovector to 1/e3 numerically in the next Section. I D. The eπective Skyrme parameterα mesone from thein the HLS. Skyrme The model, original with a larger Skymre value modelof e,wehavea La- grangian reads smaller moment of inertia, which results in a larger mass Finally we estimate the Skyrme parameter e in the splitting between the π and the nucleon as is verified IV. NUMERICAL RESULTS FOR THE numerically in the next Section. original Skyrme model by integrating out the isovector2 SKYRMION α meson from the HLS. The original Skymre modelf La-" µ 1 2 = Tr ⊗ U⊗ U † + Tr ⊗ UU†, ⊗ UU† , grangian reads Sk µ 2 µ α L 4 IV. NUMERICAL32e RESULTS FOR THE 2 SKYRMION (55) In this section, we present the results of numerical cal- f" µ 1 2 
- . = Tr ⊗ U⊗ U † + Tr ⊗ UU†, ⊗ UU† , 2 LSk 4 µ 32e2 µ whereα the chiral field U is U = and the first term is the culations on the Skyrmion properties in the framework nonlinear(55) sigma modelIn this section, Lagrangian we present that the resultscan be of written numerical as cal- 
2 - . of the HLS discussed in the previous section. The HLS where the chiral field U is U = and the first2 term is theµ culations on the Skyrmion properties in the framework 4 f"Tr(‰ µ‰ ), where ‰ µ is defined as‰ ˆ µ without the Lagrangian up to O(p ) in Eq. (7) which is considered in nonlinear sigma model Lagrangian that can be written" as" of the HLS" discussed in the previous" section. The HLS 2 µ 4 f"Tr(‰ µ‰ ), where ‰ µ is defined as‰ ˆvectorµ without field. the InLagrangian the earlier up to O analyses(p ) in Eq. [29] (7) which with is considered the HLS in the present calculation contains 17 parameters, namely, vector" field." In the earlier" analyses [29]Lagrangian" with the HLS upthe to presentO(p2), calculation it is known contains that 17 parameters, the Skyrme namely, f , a, g, y (i =1,...9), z , z , and c .Wedeter- 2 " i 4 5 1,2,3 Lagrangian up to O(p ), it is known that the Skyrme f", a, g, yi (i =1,...9), z4, z5, and c1,2,3.Wedeter- term can be obtained from the α mesonterm kinetic can energy be obtainedmine all these from LECs the throughα meson hQCD kinetic models, energywhich are mine all these LECs through hQCD models, which are term in the limit of infinite α meson mass.term In this in case,the limitcharacterized of infinite by theα meson warping mass. factor K In(z) thisand the case, wave characterized by the warping factor K(z) and the wave Three models

• HLS(π, ρ, ⍵) model: full O(p4) Lagrangian with hWZ terms • HLS(π, ρ) model: without hQZ terms, the ⍵ meson decouples • HLS(π) model: integrates out VMs same as the Skyrme Lagrangian but e is fixed by the HLS

41

13년 6월 26일 수요일 33 vector meson field V is [8–10] Then one can construct the chiral Lagrangian up to vector meson field µV is [8–10] Then one can construct the chiral Lagrangian up to µ O(p4) as g O(p4) as Vµ = (gπµ + "µ) (5) Vµ =2 (πµ + "µ) (5) 2 HLS = (2) + (4) + anom, (7) with L HLS =L (2) +L (4) +L anom, (7) with L L L L 0 π + "µ 0 π2"µ + "µ = πµ " = "µ 20"µ . (6) which is our working Lagrangian. Here, "µ = π·µ " = π2"µπ "µ 0 . (6) which is our working Lagrangian. Here, · π π2"π " " " π µ " µ "

1 2 µ 2 µ 1 µ" (2) = fπ Tr2 (ˆa µaˆ )+µ afπ Tr2 aˆ µaˆ µ 2 Tr (Vµ" V µ)", (8) L (2) = fπ Tr (ˆ"a µ"aˆ )+afπ Tr αaˆ µαaˆ " 2g 2 Tr (Vµ" V ) , (8) L " " α α ⊗α " 2g where f is the pion decay constant, a is the parameter of theα HLS, ⊗g is the vector meson coupling constant, and the whereπf is the pion decay constant, a is the parameter of the HLS, g is the vector meson coupling constant, and the field-strengthπ tensor of the vector meson is field-strength tensor of the vector meson is

Vµ" = αµV" α" Vµ i[Vµ,V" ]. (9) Vµ" = αµV" α" Vµ" i[Vµ,V" ]. (9) " " In the most general form of the O(p4) Lagrangian there are several terms that include two traces in the flavor space In the most general form of the O(p4) Lagrangian there are several terms that include two traces in the flavor space such as the y –y terms listed in Ref. [10].2 These terms are suppressed by N compared to the other terms in the4 such as the10y –18y terms listed in Ref. [10].2 These terms are suppressed by Nc compared to the other terms in the Lagrangian and10 are18 not considered in the present work. Then the O(p4) Lagrangianc which we study in this paper is Lagrangian and are not considered in the present work. Then the O(p4) Lagrangian which we study in this paper is given by coupled equations of motion for the wave functions F (r), root mean square (rms) radius of the winding number given by W (r), and G(r). These are also given in Appendix A. current is defined by = + , (10)The classical configuration of the soliton obtained (4) = (4)y + (4)z , (10) 1/2 L (4) L (4)y L (4)z above should be quantized to describe physical baryons 1/2 " L L L 4 r2 = d3rr2B0(r) , (26) where of definite spin and isospin. Here, we follow the standard W where ‰ Š 0 collective quantization method [24], which transforms the π" α µ " µ " µ " µ " coupled equations of motion for the wave functions F (0r), root mean square (rms) radius of the winding number (4)y = y1Tr ⊗ˆ µ⊗ˆ ⊗µˆ " ⊗ˆ " + y2Tr ⊗ˆ µ⊗ˆ " ⊗ˆ µ⊗ˆ " + y3Tr ⊗ˆ µ⊗ˆ ⊗µˆ " ⊗ˆ " + y4Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗µˆ " chiral field and the vector meson field as where B (r) is the time component of the winding num- L (4)y = y1Tr "⊗ˆ µ"⊗ˆ "⊗ˆ ""⊗ˆ + y2Tr "⊗ˆ µ"⊗ˆ ""⊗ˆ "⊗ˆ + y3Tr ⊗αˆ µ⊗αˆ ⊗αˆ " ⊗αˆ + y4Tr ⊗αˆ µ⊗αˆ " ⊗αˆ ⊗αˆ W (r), and G(r). These are also given in Appendix A. current is defined by L " " " " " " " " α α α α α α ber current that is explicitly written as µ ‰" µ ‰" α "‰ µ α ‰ 4 The classical configuration of the soliton obtained + y5Tr ⊗ˆ µ⊗ˆ ⊗µˆ " ⊗ˆ ‰"+ y6Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆµ ‰" + y7Tr ⊗ˆ µ⊗ˆ " ⊗ˆ ⊗ˆ"‰ µ ‰ π(r) π(r,t)=A(t) π(r)A†(t), 1/2 + y5Tr "⊗ˆ µ"⊗ˆ α⊗ˆ "α⊗ˆ + y6Tr "⊗ˆ µ"⊗ˆ "α⊗ˆ α⊗ˆ + y7Tr ⊗"ˆ µ⊗"ˆ " ⊗αˆ ⊗αˆ π 0 1 1/2 2 " above should be quantized to describe physical baryons B = 2 F π sin F. 3 2 0 (27) " " α ‰α " " α ‰α " " α ‰α V (r) V (r,t)=A(t) V (r)A†(t), (19) 2r2 = d rr B (r) , (26) + y Tr ⊗ˆ ⊗ˆµ⊗ˆ ⊗ˆ" +Tr ⊗ˆ ⊗ˆ ⊗ˆ" ⊗ˆµ + y Tr ⊗ˆ ⊗ˆ ⊗ˆµ ⊗ˆ" , (11) coupledµ of equations definiteµ spin of motion and isospin.µ for the Here, wave we functions follow theF ( standardr), root mean square 2‰‰r (rms)ŠW radius of the winding number 8 µ µ " ‰" µ " " µ ‰ 9 µ " µ ‰" π π"0 α + y8 Tr "⊗ˆ µα⊗ˆ "⊗ˆ "α⊗ˆ +Tr "⊗ˆ µα⊗ˆ ""⊗ˆ α⊗ˆ + y9Tr ⊗"ˆ µ⊗αˆ " ⊗"ˆ ⊗αˆ , (11) collective quantization method [24], which transforms the 1/2 " " α " α " " α α 4 W (r), and G(r). These are also given in Appendix A. current is defined by 2 Š µ " α ‰ µ " ‰α " ‰ where A(t) is a time-dependent SU(2) matrix. We de- We define the energywhere B root0(r) mean is the square time component radius r E of theas winding num- (4)z = iz4Tr VŠµ" ⊗ˆ ⊗µˆ " + iz5Tr ‰Vµ" ⊗ˆ ⊗ˆµ ". ‰ ‰ (12) Thechiral classical field configurationand the vector of meson the field soliton as obtained ‰ Š L = iz Tr V "⊗ˆ "⊗ˆ + iz Tr V ⊗ˆ α⊗ˆ . (12)fine the angular velocity π of the collective coordinate 1/2 (4)z 4 µ" 5 µ"α ber current that is explicitly" 1 written/2 as L " " α α rotationcoupled asabove shouldequations be quantizedof motion to for describe the wave physical functions baryonsF (r), root mean1 square2 1/"2 (rms) radius3 2 of0 the winding number ‰ ‰ π(r) π(r,t)=A(t) π(r)A†(t), 2 1/2 r =3 2 d rr B (r) , (26) ‰ ‰ r = W d rr Msol(r)1 , (28) Soliton WaveWof(r definite), and G spin( rFunctions). and These isospin.π are also Here, given we follow in Appendix the standard A. currentE is defined‰ Š by 00 2 ‰ Š Msol 0 Bπ" = F π sinα F. (27) collectiveiπ quantizationπ VAµ†((rt))" methodAV(µt)(.r,t [24],)= whichA(t) Vµ transforms(20)(r)A†(t), the (19) π " 2‰2αr2 The classical configurationπ0 of the soliton obtained 0 In the present work, we also consider the anomalous par- In order to study the properties of the soliton obtained chiral field· and" the vector meson field as where M where(r) isB the(r) soliton is the mass time component(energy) density of the given winding1/2 num- In the present work, we also consider the anomalous par- In order to study the properties of the soliton obtained above shouldwhere beA(t quantized) is a time-dependent to describe SU(2) physical matrix. baryons We de-sol We define2 the1/2 energy root" 3 mean2 0 square radius r2 1/2 as ity hWZ terms that are written as from the Lagrangian (7), we take the standard parame-Under the rotation (19), the space component of the α in Appendixber A. current thatr isW explicitly= writtend rr B as(r) , ‰ Š(26)E ity hWZ terms that are written as from the Lagrangian (7), we take the standard parame- of definitefine spin the and angular isospin. velocity Here,π weof follow thei collective the standard coordinate ‰ Š 0 terization for the soliton configuration. For the pion field,field and the time componentπ(r) π( ofr,t the)=⊗Afield,(t) π(r i.e.,)A†α(t),and π" 1 α 1/2 3 terization for the soliton configuration. For the pion field,0 collectiverotation quantization asπ method [24], which transforms the 10/2 1 " 2 N 3 we use the standard hedgehogClassical configuration, Solution ⊗ , get excited. TheCollective most general formsQuantization for the vector- 0 2B = F π sin3 F.2 (27) c chiral field andVµ(r the) vectorVµ(r,t meson)=A(t field) Vµ(r as)A†(t), (19) where B (r) isr theE time= 2‰ component2r2 d rr ofM thesol( windingr) , num-(28) anom = N2c ci i, (13) we use the standard hedgehog configuration, meson excitations are writtenπ as [20] III. HIDDEN LOCAL‰ Š SYMMETRYMsol 0 INDUCED L anom =16 cLi i, (13) iπ π A†(t)" A(t). (20)ber current that is explicitlyπ " written as α L 16 2i=1 L F (r) where A(t) is a time-dependent· " SU(2)0 matrix. We de- WeFROM define HOLOGRAPHIC the energy root mean QCD square radius r2 1/2 as
i=1 ‰(r)=exp i" rˆ F (r) . (16) π(r) π(r,t)=A(t) π(r)A†(t), where M (r) is the soliton mass (energy) densityE given
0 2 sol 1 ‰ Š ‰(r)=exp i"· rˆ 2 . (16)⊗ (r,t)=fineA( thet) angular[π π ππ( velocityr)+πˆ rπˆ πofrˆ theπ (r collective)] A†(t), coordinate 0 2 where Under the1 rotation (19), the2 space component of the α in AppendixB A. = F π sin F. 1/2 (27) where , · 2 - g Vµ(·r) Vµ(r,t·)=A· (t) Vµ(r)A†(t), (19)i 2 2 , - rotationfield as and the time component of the ⊗ field, i.e., α and A.2 1 Master/2 formula1 2"‰ r3 2 3 3 π r = d rr Msol(r) , (28) 1 = i Tr ⊗ˆL⊗ˆR ⊗ˆR⊗ˆL , (14a) The configuration of the vector mesons are written as [20] i (r)0 i E 3 3 The configuration!µ of= theW vector(r) 0µ mesons, are written as [20] ⊗ , get excited. The most general forms for the vector- ‰ Š Msol 0 2 1/2 L 1 = i Tr ⊗ˆL⊗ˆR" ⊗ˆR⊗ˆL , (14a) α (r,twhere)= A((tπ) is arˆ) time-dependent, SU(2) matrix.(21) We de- We define the energyπ root" mean square radiusα r as L " r mesonα excitationsiπ π areA† written(t)"0A( ast). [20] (20)In this section,III. following HIDDEN Refs. [3, LOCAL 4], we provide SYMMETRY a gen- INDUCEDE 2 = i Tr‡⊗ˆL⊗ˆR⊗ˆL⊗ˆR , + (14b) G(rG) (r) fine the angular velocity· "π of the collective coordinate ‰ Š L 2 = i Tr ⊗ˆL⊗ˆR⊗ˆL⊗ˆR , (14b) where Msol(r) isFROM the soliton HOLOGRAPHIC mass (energy) density QCD given ‡ + πµ = W (r) Š0µ⇢, 0 =0"0 =0, ⇢, =π = G(rˆ(r)(rˆ⌧ ) " ) . (17) eral master formula to determine the parameters of the1/2 3L=Tr FV (ˆ⊗ ⊗ˆ ⊗ˆ ⊗ˆ ) , (14c) With theserotationUnder wave the as functions rotation the (19),2 moment the space of inertia component can of the α 1/2 1 " ‡ L R + R L πµ = W (r) Š0µ, "0 =0, πgr= gr ⇥(rˆα " ) . (17) 0 HLS Lagrangianin Appendix by integrating2 A. out the infinite3 2 towers of L 3 =Tr F‡ V (ˆ⊗L⊗ˆR" +⊗ˆR⊗ˆL) , (14c) gr α be calculated and⊗ its(r,t explicit)=A( expressiont) [π π π1 is(r given)+πˆ inrˆ π Ap-rˆ π2(ri )] A†(t), r E = d rr Msol(r) , (28) L " field and the time componentg · of the ⊗ field,· i.e.,· α and ‰ Š Msol 0 vector and axial-vector mesons inA. a class Master0 of hQCD formula models in the 1-form notation‡ with + For the baryon number B = 1 solution, these wave func-pendix B.⊗ It, is get then excited. straightforwardi Theπ π mostA to general†(t obtain)"0A( formst the). Euler- for the vector-(20) π " α in the 1-form notation‡ with + For the baryonBoundary number B Conditions= 1 solution, these wave func- i ·(r) " i expressed by the following general 5D action: tions satisfy the following boundary conditions: Lagrangemeson equations excitationsα ( forr,t)= the waveare written( functions,π rˆ) as, [20]π1(r), π2(r), (21)whereIII. HIDDENMsol(r) is LOCAL the soliton SYMMETRY mass (energy) INDUCED density given ⊗ˆL =ˆ⊗ ⊗ˆ , tions satisfy the following boundary conditions: r α In this section,DBI followingCS Refs. [3, 4], we provide a gen- α " and (rUnder) by minimizing the rotation the (19), moment the of space inertia, component and the of the α in AppendixS5FROM= A.S HOLOGRAPHIC+ S , QCD(29) ⊗ˆL =ˆ⊗" ⊗ˆ , i 5 5 ⊗ˆ =ˆ⊗ +ˆα "⊗ ," F (0) = ,F( )=0, results arefield also0 and given the in time Appendix2 component B. The of boundary the ⊗ field, con- i.e., α and eral master formula to determine the parameters of the R ⊗ (rWith,t)=A these(t) [ waveπ π functionsπ (r)+πˆ therˆ π momentrˆ π (r)] ofA ( inertiat), can DBI ⊗ˆR =ˆα⊗ +ˆ"⊗ , F (0) = ,F⊗( )=0, 0 Boundary Conditions1 2 †where the 5D Dirac-Born-InfeldHLS Lagrangian (DBI)by integrating part S outand the the infinite towers of α "2 G(0) = 2,G( ⊗)=0, (18)ditions imposed⊗ , get excited.be on calculated the excited Theg and most· fields its general are explicit· forms expression· for the is given vector- in Ap- 5 FV = dV iV . (15) G(0) =" 2,G⊗( )=0, (18) CSA. Master formula 2 W ⊗(0) = 0,W( )=0. Chern-Simons (CS)vector part andS axial-vector5 are expressed mesons as in a class of hQCD models FV = dV" iV . (15) " ⊗ meson excitationspendix B.(r) It are is thenwritteni straightforward as [20] to obtain the Euler- III. HIDDEN LOCAL SYMMETRY INDUCED " W ⊗(0) = 0,W⊗( )=0. i ⊗ α (rLagrange,t)=π1π (0) = equationsπ(1π( )=0rˆ) , for, the wave functions, π ((21)r), π (r), expressedFROM by the HOLOGRAPHIC following general 5D QCD action: Given the Lagrangian and the wave functions, it is now r ⊗α 1 DBI2 In this section,4 following1 Refs. [3, 4], weµπ provide a gen- 0 andππ (0)(r) =2 byπ ( minimizing)=0, the moment of inertia, andS5 the= NcGYM d xdz K1(z)TrDBI [ µπ CS ] Given the Lagrangian and the wave functions, it is now ⊗ (r,t)=2A(t) [2π π π (r)+πˆ rˆ π rˆ π (r)] A†(t), 2S5 = S5 F+ SF5 , (29) straightforward to derive the soliton mass Msol.The ⊗ 1 2 eral master formula to determine the parameters of the µ FOR B=1 Soliton Withresults these wave areg also functions· given in the Appendix moment· · B. of The inertia boundary can con- " ⊗ 2 Another example of this kind is Tr πˆ Tr πˆ that generates straightforward to derive the soliton mass Msol.The (0) = ( )=0, (22) A. Master formula µ µ explicit expression for the soliton mass is given in Ap- HLS Lagrangian by integrating out the infinite towersDBI of 2 Another example of this kind is Tr ππˆ Tr ππˆ that generates be calculatedditions imposed and its⊗ explicit on the excited expression fields is are given in Ap- where the 5D Dirac-Born-Infeld2 µz (DBI) part S5 and the the mass diπerence between the " and α mesons.µ explicit expression for the soliton mass is given in Ap- i (r) i + K2(z)MKKTr [ µzCS ] , (30) π π pendix A. Minimizing the soliton mass then gives the vector and axial-vector mesons in a class of hQCD models π " π " and π (r)αpendix and(r,tπ )=( B.r) atIt isr = then(π 0 satisfy straightforwardrˆ) , the constraint, to obtain the Euler-(21) Chern-Simons (CS) partFS5F are expressed as the mass diπerence between the " andπ α" mesons.π " pendix A. Minimizing the soliton mass then gives the 1 2 r α expressedIn this by section, the following following general Refs. 5D [3, 4], action: we provide a gen- Lagrange equations for theπ1π (0) wave = π1 functions,( )=02, π1(r), π2(r), CS 2 Nc ⊗ DBI 4 1 µπ I S5J =eral master2 formulaw5(A). to determineDBI CS the parameters(31) of the Withand these(r)2 byπ wave1(0) minimizing + functionsπ2(0) =π the 2(0). the moment = π moment( )=0 of inertia, of,(23) inertia and the can S54 = NcSGYM= S d xdz+ S , K1(z)Tr [ µπ(29) ] Mbaryon(I,J)=2π Msol2 + = Msol + 24‰ M R 5 5 5 2 F F beresults calculated are also and given its explicit in Appendix expression B.⊗ The2 is boundary given in con- Ap- 2 HLS Lagrangian" α by integrating" ⊗ out the infinite towers of (0) = ( )=0, where the(22)where rescaled the ’t 5D Hooft Dirac-Born-Infeld coupling constant (DBI) is part definedSDBI and the In thependixditions adiabatic B. imposed It collective is then on straightforward the quantization excited fields scheme,⊗ to are obtainI the the Euler- Ivector and axial-vector mesons in a2 class of5 hQCDµz models 3 +CSK2(z)MKKTr [ µz ] , (30) baryon mass is given by as GYM Chern-SimonsexpressedŠ/(108‰ ) by and (CS)the the following part fieldS5 strength generalare expressed of 5D the action:F 5D as F Lagrangeand equationsπ1(r) and forπ2( ther) at waver = 0 functions, satisfy theπ1 constraint,(r), π2(r), "3 ππ (0) = π ( )=0, gauge field M is MN = "M N "N M i[ M , N ]. and (r) by minimizing1 the1 moment of inertia, and the A F CS NcA DBIA 1 CSA A i(i + 1) ⊗j(j + 1) Here, K (z) areDBI theS metric= functionsS54= S5 w of+(zAS)constrained.5 , µπ (29)(31) ππ (0) =2ππ (0)( +)=0π (0), = 2. (23)1,2 S5 = 5NcGYM 2 d xdz 5 K1(z)Tr [ µπ ] resultsM = M aresol + also given2= inM Appendixsol12+ B.2 The boundary(24) con- 42 24‰ M 4 R 2 F F 2 ⊗ 2 by the gauge/gravity duality. The" gravityα⊗ enters in the z DBI ditions imposed onI the (0) excited = ( fields)=0I , are (22) where the 5D Dirac-Born-Infeld" (DBI) part S5 and the In the adiabatic collective quantization scheme,dependence the of thewhere YM the coupling rescaled giving ’t HooftCS rise coupling to the warp- constant is defined 13년 6월 26일 수요일 ⊗ Chern-Simons (CS) part S 2are expressedµz as where i and j are isospin and spin of the baryon. Then + K4 32(z)5MKKTr [ µz ] , (30) baryon mass is given by ing of the space.as G InYM Eq. (31),Š/(108M‰ and) andR thestand fieldF forF strength the of the 5D the π-N massand π1 di(r")erence and π2 reads(πr1π)(0) at r ==π1 0( satisfy)=0 the, constraint, "3 four-dimensionalgauge Minkowski field space-timeM is MN and= "Mz1-coordinateN "N M i[ M , N ]. ⊗ DBI Nc 4 µπ ππ (0) = π i((i +)=0 1) , j(j +space, 1) respectively,S5CS = and= NcwG(AYMA) isdF thewxdz(A CS). 5-formAK1 written(z)TrA [ µ π A(31)] A 22π (0) +2π (0) = 2. (23) Here,5 K1,22(5z) are the5 metric functions2 ofFz constrainedF M = M1 sol + 32⊗ = Msol + (24) 24‰ 4 πM Mπ MN = 2. (25) 2 as by the gauge/gravityM "R duality.⊗ The gravity enters in the z " (0) = 2( )=0, (22) " α ⊗I I wheredependence the rescaled of ’t the Hooft YM coupling coupling2 constant giving rise isµz to defined the warp- In the adiabatic collectiveI quantization scheme, the 2 i +3 K2(z1)MKK5 Tr [ µz ] , (30) where i and j are isospin and spin of the baryon. Thenw (A)=Tr + 3 . 4 F(32)F The baryonicandbaryonπ1(r size) mass and of isπ a2( givenbaryonr) at byr should= 0 satisfy be computed the constraint, by as5 GYMing ofŠ the/(108 space.‰ ) and In Eq. the (31), fieldM strengthand R ofstand the 5D for the the π-N mass di"erence reads " AF 2A F 10A gaugefour-dimensional field3π Ncis Minkowski= " space-timeα" andi[ z-coordinate, ]. the baryon number current of the Skyrmion. However, in SCS = M MN wM(A)N. N M M N (31) 2π (0) + π (0) = 2. (23) space,5 A respectively,2 F and5 wA(A ) is theA CS A 5-formA written order to intuitively see the e"ects1 i(i of+ the 1)2 vector mesonsj(j +3 on 1) Here, K1,2(z)24 are‰ theM 4 metricR functions5 of z constrained M = Msol + π M= MsolM+ = . (24) (25) as " α the Skyrmion size in a simple way,M2 here" π we considerN 22 the by the gauge/gravity duality. The gravity enters in the z In the adiabatic collectiveI quantizationII scheme,3 We the use thedependencewhere index M the=( rescaled ofµ, the z)with YM ’tµ couplingHooft=0, 1, 2, coupling3. giving rise constant to the is warp- defined winding number and energy root mean square radii. The 3 2 i 3 1 5 where i and j are isospin and spin of the baryon. Then as G Š/w(1085(A)=Tr‰ ) and the+4 field strength of. the 5D(32) baryon massThe is baryonic given by size of a baryon should be computed bying ofYM the space. In Eq. (31),AFM 2andA FR stand10A for the the π-N mass di"erence reads gauge field"3 is π= " " i[ α , ]. the baryon number current of the Skyrmion. However, infour-dimensionalAM MinkowskiFMN space-timeM AN andN AMz-coordinate AM AN order to intuitivelyi(i + see 1) the e"ects ofj( thej + vector 1) mesons onspace,Here, respectively,K1,2(z) are and thew metric(A) is functions the CS 5-form of z constrained written M = Msol + = Msol +3 (24) 5 the SkyrmionπM sizeM2 inπ a simpleMN = way,. here2 we consider(25) theasby the gauge/gravity duality. The gravity enters in the z " I 2 I 3 We use the index M =(µ, z)withµ =0, 1, 2, 3. winding number and energy rootI mean square radii. Thedependence of the YM coupling giving rise to the warp- where i and j are isospin and spin of the baryon. Then 2 i 3 4 1 5 ing of thew5( space.A)=Tr In Eq.+ (31), M and R stand. (32) for the the πThe-N baryonicmass di" sizeerence of a reads baryon should be computed by AF 2A F 10A the baryon number current of the Skyrmion. However, in four-dimensional Minkowskiπ space-time andα z-coordinate order to intuitively see the e"ects of the3 vector mesons on space, respectively, and w5(A) is the CS 5-form written the Skyrmion sizeπM in aM simpleπ M way,N = here. we consider(25) the as " 2 3 winding number and energy root meanI square radii. The We use the index M =(µ, z)withµi=0, 1, 2, 3. 1 w (A)=Tr 2 + 3 5 . (32) The baryonic size of a baryon should be computed by 5 AF 2A F 10A the baryon number current of the Skyrmion. However, in π α order to intuitively see the e"ects of the vector mesons on the Skyrmion size in a simple way, here we consider the winding number and energy root mean square radii. The 3 We use the index M =(µ, z)withµ =0, 1, 2, 3. 13

understanding of the equation of state for compact-star with Variety of Flavors” from MEXT. Y.-L.M. was sup- matter as shown in Ref. [40]. As suggested in Ref. [5], ported in part by the National Science Foundation of a reliable treatment will require low-mass scalar degrees China (NSFC) under Grant No. 10905060. The work of freedom which will figure at subleading order in Nc. of M.H. was supported in part by the Grant-in-Aid for Such work is in progress and will be reported elsewhere. Nagoya University Global COE Program “Quest for Fun- damental Principles in the Universe: from Particles to the Solar System and the Cosmos” from MEXT, the ACKNOWLEDGMENTS JSPS Grant-in-Aid for Scientific Research (S) π 22224003, (c) π 24540266. Two of us (G.-S.Y. and Y.O.) were We are grateful to M. Rho for valuable discussions and supported in part by the Basic Science Research Pro- critical reading of the manuscript. Fruitful discussions gram through the National Research Foundation of Ko- with H. K. Lee and B.-Y. Park are also gratefully ac- rea (NRF) funded by the Ministry of Education, Science knowledged. The work of Y.-L.M. and M.H. was sup- and Technology (Grant No. 2010-0009381). The research ported in part by Grant-in-Aid for Scientific Research of Y.O. was supported by Kyungpook National Univer- on Innovative Areas (No. 2104) “Quest on New Hadrons sity Research Fund, 2012.

Appendix A: The soliton mass and the equations of motion for F (r), W (r), and G(r)

Using the wave functions definedSoliton in Eqs. (16) and (17) and mass the Lagrangian in Eq. (7), the soliton mass in the HLS up to O(p4) is obtained as

Msol =4" dr M(2)(r)+M(4)(r)+Manom(r) , (A1) π where M , M , and M are from , "+ , and , respectively.α Their explicit forms are (2) (4) anom L(2) L(4)y L(4)z Lanom 2 2 2 2 2 2 2 2 fπ 2 2 2 ag fπ 2 2 2 2 F W π r Gπ G 2 M (r)= F π r +2sin F W r + af G +2sin + + (G + 2) , (A2) (2) 2 π 2 π 2 π 2 g2 2g2r2 ‰ Š 2 ⊗ 2 2 2 2 2 2 2 r 2 2 2 r 2 2 4 2 r g W 1 2 F M (r)= y F π + sin F y F π F π sin F y G +2sin (4) π 1 8 r2 π 2 8 π r2 π 3 2 2 π r2 2 ‰ Š ‰ Š  ‰ Š
2 2 2 2 2 2 2 2 2 g W r g W 1 2 F y5 2 2 2 g W 1 2 F y G +2sin + r F π +2sin F G +2sin π 4 2 4 π r2 2 4 2 π r2 2 ‡ ‰ Š +  ‰ Š
⊗ 2 2 2 2 2 2 2 y7 sin F 2 F g W r 2 2 2 F π 2 F + y8 2 G +2sin + y9 F π + 2 sin F + G +2sin π 2 r 2 ‡ 8 r 4 2 + , - ‰ Š ‰ Š ‰ Š 2 2 sin F z5 2 F + z GπF π sin F + G(G + 2) + G(G + 2) G +2sin , (A3) 4 2r2 2r2 2 . / ‰ Š 2 2 2 F M (r)=α F πW sin F + α WFπ G +2sin anom 1 2 2 ‰ Š 2 F α G(G + 2)WFπ +2sinF WGπ W π G +2sin , (A4) π 3 π 2 . 0 ‰ Š1/ where 3gN gN gN α = c (c c ) , α = c (c + c ) , α = c c . (A5) 1 16"2 1 π 2 2 16"2 1 2 3 16"2 3 The Euler-Lagrange equations for F (r) and G(r) are obtained as

F ππ + Gππ = , A1 A2 B Gππ + F ππ = , (A6) A3 A4 D where 43

2 2 3 2 2 2 = f r (y + y ) r F π (y y )sin F 13년 6월 26일 수요일 A1 π π 2 1 2 π 1 π 2 MA et al. PHYSICAL REVIEW D 87, 034023 (2013)

MA et al. 2 2 2 PHYSICAL2 REVIEWF 2 D287,2 034023 (2013) 3 sin 2F 2 B 2fπrF0 fπ sin 2F 2afπ sin F G 2sin fπmπr sin F y1 y2 rF0 y1 y2 F0 π" F α α π α 2" α sin 2F α⊗ α α⊗ " 2 B 2f2 rF f2 sin 2F 2af2 sin F G 2sin 2sin 2F f2 m2 r2 sin F y y rF 3 Fy y 2 F 2 F 3 π" π 0 α π α π α y 2 αsinπ2Fπ y yα⊗g21Wα2 G2 2sin0 α⊗2 1 sin" F2 y2 G0 2sin 2 sin F π " 1 r"2 α⊗ 3 α 4 α 2 " 3 r2 α 2 sin 2F F 2 πF 3 " π " y sin 2F y y g2W2 G 2sin 2 sing2FrW y G 2sin 2 singF2W2 F 1 " 1 r2 α⊗ 3 α 4 α y 2 y " W3 r2 rWαF y2 y sin 2F y y G 2sin 2 G sin FF F π "⊗ 5 α" 9 2 ⊗ απ 0 0 α⊗ 5 α" 9 4 α⊗ 5 " 9 α 2 0 α 2 0 0 2 2 2 π "π " g rW g W 2 3 2 F 1 2 y5 y9 W rW0 F0 y5 y9 sin 2sinF 2F y52 F y9 G sin2sinF G0 2 F sin FF0 F0 sin 2F 2 F 2 y5 4 G 2sin y5 G2 2sin 2 2y8 y7 G 2sin "⊗ α ⊗ α α⊗ α " 2r2 αα⊗ "2 "π α r2 α"π α 2 α⊗ "" 2r2 α 2 2 3 π " π 2 " π " sin 2F 2 F sin F 2 F sin 3F sinF 2F sin 2F 2 F G G 2 F y5 2 G 2sin y5 2 G 2y2sin y 2y8G y2sin7 2 2 Gz 2sin G G 2 z ⊗ α G 2sin 2 sin F " MA2ret al. α 2 " r α 8 72 α⊗2 " PHYSICAL 2r REVIEW4α D2 87, 0340232 (2013) 5 2 π " π α⊗ " " r π α 2" απ 2r ⊗ " α α r π α 2" sin 3F F sin 2FF G G 2 F 2 sinF 2F F 2f2 rF f2 sin 2F 2af2 2 sin F G 2sin 2 2 f2 m2 r2 sin F y y rF 32 y y 2 F 2 2 2y8B y7 π 0 πG 2sin π z4 "1sinG GFWπ 0π2 "2zW5 0 ⊗G1 α2sin2 0 G 12sin22"2WGsin0 0 GF 2sin α⊗ "π" r2 α α α 2 απ α"2r2 2"⊗α α " α α⊗ rα2 α⊗2α "" 22 α 2 π " π 3 π " " π " sin 2F 2 2 2 2 F 2 2 F y1 sin F y3 y4 g WF 2G 2sin sin F y3 G F2sin sin FF F 2 " r2 α⊗ α 2 α 2 " r2 2α 2 2 2 2 "1sin FW0 "2W0 G 2sin π 2""23WGG" 0GG 2 2sinW0π 2 G "2sin G0W 2 cos F G 2sin W0 2sin FW0 ; (A11) " " g2rW α 2 "α g2Wα2 ⊗ αα α 2π α F 2" 1 α π α 2" α ⊗ y y π W rW F "y y sinπ 2F y y G" 2sin 2 G sin FF F "⊗ 5 α 9 2 ⊗ α 0 0 α⊗ 5 α 9 4 α⊗ 5 " 9 α 2 0 α 2 0 0 2 F π 2 F "π 2 " "3 G G sin2 W 2F0 2 G F 2sin2 sin 3GF 0W 2 cosF F G 2sinsin 2F W0 2sinF 2FW0 ; (A11) α ⊗ αy αG 2sin α2 y 2 G α2sin 2 2yαFy 12 G α2sin 2 g2W2 1 F 2 F α " 5 2r2 απ 2 " 5 r"2 α2 2 2 α⊗π 8 2" 7 2r2 " α 2 ⊗ 2 2 2 π " D π ag fπ G " 2sin 2 Gπ G 1 G" 2 y3g 2 G 2sin G 2sin 3 π α 2 α r ⊗ α ⊗ α α 2 " r α 2 α 2 sin F 2 F sin 2F π G G" 2 2 F α π " ⊗π " 2y8 y7 G 2sin z4 G G4 22 z5 ⊗ α G 2sin2 sin F 2 2 2 2 g W 2 2 2 F g 2 2 F g F α⊗ " F r π1 α " α 2r ⊗ α αg W r 21 π α " F2 2 2 F 2 2 2 2 2 2 F 2 y4 2 G F2sin y5 2F0 2 sin F G2 2sin 2y8 y7 2 sin F G 2sin D ag fπ G 2sin2 2 G G 12 G α2 y23g α2 22 G" 2sin4 α r G 2sin α 2 α⊗ " 2r α 2 π α"1sin FW20 α"2rW0 G⊗ 2sinα ⊗ α2" α2WG0 G π2sin2 " r " α π 2 α "π 2 " π " π " "" π α 2" " 2π αα 2" π 2 " ⊗π 2 " 2 4 2 2 g 2 2 F g 22 g 2 g 2 F g W F g 2 F 2y F G 2sin2 F F z2 cos FFg z sin F G F1 z G 1 G 2sin " G G 2 2W0 2 G 2sin 2 G0W 92 cos2 F0 G 2sin W2 0 2sin 4FW0 ; 0 4 2 2 (A11) 2 5 2 y G3 2sin y F2 α sin4 F G α 2sin2 2 " 2y2 y αsin 2Fr G 2sin⊗ α α 2r ⊗ α α 2 α 4 2 α αα⊗ α 2α "π α5 4 0" α rα2 π πα α " 2α" α⊗ 8 ⊗" 7 2r2 α 2 α π " π " π "π 2 "F 2 π2 F 2 " 2 F 2 2 G G 22 G 2sin "22g WF0 G 2sin "3g 2W0 sin F WF0 G 2sin : (A12) g 2 2 F g 2 g 2 2 2 2 g 2 F y F 2G2 2sin 2 F 1z cos FFα ⊗z α2 g sinW Fα1 G 1 2 2 Fαz G 2 F1αG 2sin2 α " α 2 9 D 0ag fπ G 2sin 4G G 1 G 02 y34g 2 ⊗π G 2sin" 5 G2 2sinπ " α π "⊗ α 4 α 22 "r2 2 α 2r 2 r2⊗ α α2 2r ⊗ α2 α 2 π π π α "" α ⊗ α ⊗ α α α " π α " ⊗π αα " π " g4W2 F g2 The equation2 of motionF of W reads g2 F y G 2sin2 F2 y 2 F 2 sin 2F G2 F2sin 2 22y y sin 2F G 2sin 2 2 F G G α2 4 G2 2sinα 2 ""52g4 WF0 α0 rG22 2sin α 2"3α⊗g 28 W" 0 7sin 2r2Fg2WWF2 α01G 22sin F 2 : (A12)g2W 2 α ⊗ α απ 2 "α π α "π 22 2 α " 2 " π α " 2 2 2 2 ⊗π2 " 2W00 π W0 2 ag fπ"W y3 α y2 4 g W π2 G 2sin "⊗ y5 y9 F0 2 sin F g 2 2 F g π"2 r gα 2 α⊗ αg 2 "r2 F α 2 "⊗ α 4 αr y9 F0 G 2sin z4 cos FF0 z4 sin F G 1 z5 G 1α G 2sin π " ⊗ π " α 4 α 2 " 2 α 2r2 ⊗ α α 2r2 ⊗ α α 2 The equation of motionπ of W reads" sin2F F α F π2 " " F F F 0 2 3 F 2 2 2 2 22 2 "1 2 F0 "2 G 2sin 2 2 G 2G 2sin F F0 2cosF G 2sin F0 4sinFG0 : (A13) 2 G G 2 G 2sin "g2gWWF0"G1 2sinr2 ""3Fg 2r22W0 sinαF WF20gGWα2sinr2 ⊗ 2:α α(A12) α α 2 α W W agα 2f⊗ 2 αW ⊗πy α y g22"Wα π α G 22sin" α 2 α π y " y π" α F α2 2"⊗ sin2F π " ⊗ 00 π"r 0 α π α⊗ 3 α 4 2 "r2 α 2 "⊗ 5 α 9 4 0 αr2 The equation of motion of W readsα π " ⊗ π " 2 This evidently shows that the hWZ terms, i.e., the ci terms, are the source terms of W r . sin F 2 F0 F 2 g2W"2 1 F 2 g2W 2 F ⊗ W W ag2f2 W y y 2g2W 3 G2 2sin2 2y y F 2 sin2F 2 "1 00 2 F0 0 "2 2π G 32sin4 2 2G 2G 2sin F5 F90 2cos0 F G2 2sin F0 4sinFG0 : (A13) " rπ"r "α r α⊗α α 2 αα2r "r⊗ π α α2" ⊗"⊗ α α4 π αr α " 2 α sin2F π F "F 2 " α πF " ⊗ 0 2 APPENDIX3 2 B: MOMENT2 OF INERTIA2 AND EQUATIONS OF MOTION OF THE EXCITED FIELDS "1 2 F0 "2 2 G 2sin 2 G 2G 2sin F F0 2cosF G 2sin F0 4sinFG0 : (A13) This evidently shows" r that the" hWZr π α terms,2" i.e.,α r theα⊗ c αterms,α are theα sourceπ termsα of2"Wαr . ⊗ When thei collective rotation is introduced, the⊗ Lagrangian can be written as This evidently shows that the hWZ terms, i.e., the c terms, are the source terms of W r . i ⊗

APPENDIX B: MOMENT OF INERTIA AND EQUATIONS OF MOTION OFL THEM EXCITEDI Tr A_ A FIELDS_ ; (B1) APPENDIX B: MOMENT OF INERTIA AND EQUATIONS OF MOTION OF THE EXCITEDπ" FIELDSsol α ⊗ y When the collectiveWhen the collective rotation rotation is introduced, is introduced, the the Lagrangian Lagrangian can canbe written be written as Moment as of Inertia where the moment of inertia I is summarized as

_ _ L Msol I Tr AAy ; _ _ (B1) L π" Msolα I⊗ Tr AAy ; I 4π dr I 2 r I 4 r I anom(B1)r : (B2) π" α ⊗ π ‰ ⊗ ⊗ α ⊗ ⊗ α ⊗ Š where the moment of inertia I is summarizedSKYRMIONS as WITH VECTOR MESONS IN THE HIDDEN ... SKYRMIONSZ WITH VECTORPHYSICAL MESONS REVIEW IN THE HIDDEN D 87, 034023... (2013) PHYSICAL REVIEW D 87, 034023 (2013) 2 1 F 2 1 1 2’2 where the moment of inertia is summarized as 2 2 2 2 2 2 2 2 2 2 2 2 I The contributions2 from 1 , , and I 2Farer 2 representedfπ1r sin F af byπr1 "1 r"2,2’ 2 "r1 , and2sin agr ,fπ respectively.’ ’0 We 2 2 2 L 2 2 L2 4 y L 24 z Lanom2 π "α3 2 2⊗ 32 Iπ 2 2⊗ "I⊗4 2I anom 6 6 ⊗ r2 I 4πI 2 rdr I 2 frπr sinI 4Fr afI⊗ anom πr r⊗ " :1 α"2 ⊗ 2 "1 2sin (B2)ag fπ’ π ⊗’ 0⊗ 2⊗ "⊗ α ⊗ ⊗ " α furtherπ π write"α‰ 3⊗ ⊗ α ⊗ ⊗ ⊗ α3 ⊗ Šπ ⊗ " ⊗ 2 r62 6 ⊗4 r 2 Z π " α ⊗ 3" 2 2" " "" 2 G2 α" 1 " " 1 G2 2G 2 "2: (B4) 2 ⊗ 3g2 π 10 ⊗ 10 20 ⊗ 20 "⊗3g2 π 1 "π 1 ⊗ 2 "⊗3g2 π ⊗ ⊗ " 2 4π dr r r 2 r 2 4r :2 2 2 (B2)2 The contributions from L 2 , L 4 y I L 4 z, and LanomIare2 represented3"0 I 24" by0 "I0 2 r"I0,anomI 4 r , andG "I1anom1r ," respectively.1 "2 1 We G 2G 2 " : (B4) ⊗ ⊗ ⊗ ⊗ 2 1 ⊗ 1 2⊗ 2 ⊗ 2 2 2 α π ‰⊗ 3g⊗ π α⊗ ⊗ α⊗⊗ "⊗⊗3⊗g Š π EachI⊗ "π4 term⊗ of Iy4 iIisy calculated"⊗3g π asziI⊗z : ⊗ " (B3) further write π " i i Z ⊗ π i α i 1 2 2 2 2 2 Each term of I 4 is calculated as X X I y r r sin F F0 sin F ; (B5) The contributions from , , and π " yare representedz : by r , r , and (B3)r , respectively.1 π "α We3 ⊗ r2 L 2 L 4 y L 4 z IL4 anom iI yi iI zi I 2 I 4 I anom " α ⊗ π α ⊗ ⊗ α ⊗ The momenti of inertiai from is⊗ obtained ⊗ 1 ⊗ as ⊗ 2 ⊗ further write X X L 2 r r2sin 2F F 2 sin 2F ; (B5) I⊗y1 0 2 1 2 2 2 π "α 3 ⊗ r I y2 r r sin FF0 ; (B6) The moment of inertia from L 2 is obtained as " α π "α3 ⊗ y z : (B3) I 4 iI yi iI zi 1 1 4 F 2 2 F F ⊗ 2 2 2 I r g2’2 g2W2 G 2sin 2 g2W’ G 2sin 2 " 2sin 2 π i α i I y r r sin FF0 ; y3 2 (B6) 1 2 π "α3 034023-14π "α 12 π r " ⊗ 2α ⊗ ⊗ 3 " ⊗ 2α" 2α X X 2 2 034023-14 1 2 2 2 1 2 F 2 2 F r g W G 2sin "1 "2 2 "1 2sin ; (B7) The moment of inertia from L is obtained as 1 4 F 2 2 ⊗ π2 F 3 " ⊗ F2α ⊗ππ ⊗ " ⊗ " 2α ⊗ 2 r g2’2 g2W2 G 2sin 2 g2W’ G 2sin 2 " 2sin 2 ⊗ I y3 2 1 π "α 12 r ⊗ 2 ⊗ 3 r2 ⊗ 2 F 2 2 1 F F π " α I⊗ r g2W"2 " " 2 2α"" 2sin 2 α g2W’ g2W’ 8 G 2sin 2 " 2sin 2 1 1 F 2 y4 π "α 2 π 1 ⊗ 2" ⊗F 21 2 12 ⊗ 2 1 2 r2g2W2 G 2sin 2 " " 2 2 π" 2sin 2 " ; α ⊗ π (B7) " α" α⊗ 1 2 1 2 2 2 ⊗ 2 3 ⊗ 2 π ⊗ " ⊗1 2 F g2’ 2 π " α ⊗π G" 2sin 2α ⊗ "1 "2 ; (B8) 034023-14 ⊗ 3 " ⊗ 2α π r ⊗π ⊗ " ⊗ 2 2 r 2 2 2 2 F 1 2 2 2 F 2 22 F 2 2 2 r g W " " 2 " 2sin g W’ g W’1 2 82 G2 2 2sin "2 F 2sinr 2 2 2 2 F 2 g ’ I y4 1 2 1 I r sin F r g W 2 G 2sin 1 F sin F 2 " 2sin " " ; (B9) π "α 2 π ⊗ " ⊗ 2 12 y5 ⊗ 2 20 2 1 1 2 2 π " α ⊗ ππ"α6 π " " ⊗ α" 2α ⊗ 12 " α⊗⊗ r απ " 2α ⊗π ⊗ " 2r ⊗ 1 F 2 g2’2 G 2sin 2 " " 2 ; 1 (B8)g’ 2 2 1 2 r sin 2F rgW ; (B10) ⊗ 3 ⊗ 2 r ⊗π ⊗ " I y6 " α π ⊗ π "α6 " 2r α

2 2 2 1 g’2 2 2 F F 1 2 2 2 2 2 F r 2 2 2 I 2 Fr sin 2F rgW2 g ’ 4 G 2sin 2 " 2sin 2 ; (B11) I r sin F r g W 2 G 2sin F sin F 2 " 2siny7 6 " " 2r ; (B9) 2 1 2 y5 0 2 1 π "α 1 π" 2 α 2⊗ " ⊗ α" α⊗ π "α6 π " ⊗ 2α ⊗ 12 " ⊗ r απ " 2α ⊗π ⊗ " 2r ⊗ 1 g’ 2 F F I r sin 2F rgW 4 G 2sin 2 " 2sin 2 ; (B12) 1 g’ 2 y8 π "α3 2r ⊗ 2 1 2 I r sin 2F rgW ; π" α " (B10) α" α⊗ y6 π "α6 2r " r2 α r2 F 2 r2 2 1 2 I r g2W2sin 2F F 2 " 2sin 2 F 2 sin 2F " " 2 g2’2 F 2 sin 2F ; (B13) y9 π "α 6 ⊗ 6 0 1 2 12 0 r2 π 1 ⊗ 2" ⊗ 24 0 ⊗ r2 2 " α " α " α 1 2 g’ 2 F 2 F I r sin F rgW and4 G 2sin " 2sin ; (B11) 44 y7 π "α6 2r ⊗ ⊗ 2 1 2 π" α " α" α⊗ 2 2 I r sin 2F G 1 " " r2 sin FF " ; (B14) 13년 6월 26일 수요일 z4 1 2 0 10 2 π "α3 ‰ π "⊗ Š 3 1 2 g’ 2 F 2 F I y r sin F rgW 4 G 2sin "1 2sin ; (B12) 8 π "α3 2r ⊗ 2 2 F2 F π" α " α" 2 α⊗ 2 I z5 r G 2sin G 1 "1 "2 "2 "1 "2 G 1 "1 "2 "1 2sin : (B15) π "α 3 " ⊗ 2α ‰ π " Šπ ⊗ "⊗‰ π "⊗ Š" 2α‰ r2 r2 F 2 r2 The2 hWZ terms give 1 2 r g2W2sin 2F F 2 " 2sin 2 F 2 sin 2F " " 2 g2’2 F 2 sin 2F ; (B13) I y9 0 1 0 2 1 2 0 2 π "α 6 ⊗ 6 2 12 r π gN⊗c " ⊗ 24 2 gN⊗c r 2 F 2 F " α " I anom rα 2 c1 c2 ’F0sin F" 2 c1 c2 ’Fα0 G 2sin "1 2sin π "α8π π " 24π π ⊗ " " ⊗ 2α" 2α and gN F c c ’F G" G " 2" ’ sin F " G ’ sin F G " 4sin 2 : (B16) ⊗ 24π2 3 0π 1 1 2"⊗ π 10 0"⊗ 0 1 ⊗ 2 2 2 2 2 " α‰ I z r sin F G 1 "1 "2 r sin FF0"10 ; (B14) 4 π "α3 ‰ π "⊗Then theŠ equations3 of motion for "1, "2, and ’ are obtained as

2 2 F 2 F I z5 r G 2sin G 1 "1 "2 "2 "1 "2 G 1 "1 "2 "1 2sin 034023-15: (B15) π "α 3 " ⊗ 2α ‰ π " Šπ ⊗ "⊗‰ π "⊗ Š" 2α‰ The hWZ terms give

gNc 2 gNc 2 F 2 F I anom r 2 c1 c2 ’F0sin F 2 c1 c2 ’F0 G 2sin "1 2sin π "α8π π " 24π π ⊗ " " ⊗ 2α" 2α gNc 2 F 2 c3 ’F0 G"1 G "1 2"2 ’ sin F "10 G0 ’0 sin F G "1 4sin : (B16) ⊗ 24π π "⊗ π "⊗ " ⊗ 2α‰

Then the equations of motion for "1, "2, and ’ are obtained as

034023-15 9

TABLE I. Low energy constants of the HLS Lagrangian at O(p4)witha =2.

Model y1 y3 y5 y6 z4 z5 c1 c2 c3 SS model 0.001096 0.002830 0.015917 +0.013712 0.010795 0.007325 +0.381653 0.129602 0.767374 BPS model π0.071910 π0.153511 π0.012286 0.196545 0.090338 π0.130778 0.206992 +3π .031734 1.470210 π π π π π π

TABLE II. Skyrmion mass and size calculated in the HLS with the SS and BPS models with a =2.ThesolitonmassMsol and 2 2 2 µ the π-N mass di"erence πM are in unit of MeV while r and r are in unit of fm. The column of O(p )+π B is " αW " αE µ “the minimal model” of Ref. [20] and that of O(p2) corresponds to the model of Ref. [19]. See the text for more details. π π 2 µ 2 HLS1(", α, π)HLS1(", α)HLS1(") BPS(", α, π)BPS(", α)BPS(") O(p )+πµB [20] O(p )[19] Msol 1184 834 922 1162 577 672 1407 1026 πM 448 1707 1014 456 4541 2613 259 1131 r2 0.433 0.247 0.309 0.415 0.164 0.225 0.540 0.278 " αW Solutions r2 0.608 0.371 0.417 0.598 0.271 0.306 0.725 0.422 π" αE 8 π 1.5 functions π and π . Then all the LECs are obtained HLS(0 π1 , ρ, ⍵) model 3 3 through the master formulas given in Eq. (35), which F(r) (r) HLS1(π) ξ1 contain the mass scale MKK, the ’t Hooft coupling " (or G(r) HLS (π,ρ) 1 ξ (r) 2 1 GYM), and the integrals of the warping factor K(z) and 2 W(r) 2 (r) HLS1(π,ρ,ω) the wave functions π0 and π1. In the present work, we ϕ × 10 F(r) consider two hQCD models, the SS model and the BPS 1 1 model. 0.5 In hQCD models, MKK and GYM are free parameters. In the present work, we fix them by using the empirical 0 0 values of fπ and m": 0 −1 −1 G(r) m" = 775.49 MeV, f = 92.4 MeV. (61) −0.5 π −2 −2 Then we have complete information to calculate all LECs 0 1 23 0 1 23 0 1 23 of the HLS Lagrangian through the master formulas. r (fm) r (fm) r (fm) Note, however, that the master formulas can determine only the product of ag2 = m2/f 2, and, therefore, a or FIG. 1. The soliton wave functions obtained in the " π FIG. 2. The excitations of the soliton profile, ⊗1(r)(solidline), FIG. 3. Comparison of the soliton wave functions F (r) g remains unfixed. However, as discussed in the previ- HLS1(π, ", α)modelwitha =2. F (r), G(r), and W (r) are ⊗2(r)(dashedline),and (r)(dot-dashedline)calculatedin and G(r) in the three models, HLS ("), HLS (", α), and ous section, the physical quantities are independent of given by the solid, dashed, and dot-dashed lines, respectively. 1 1 the HLS1(", α, π)modelwitha = 2. Here, ⊗1(r)and⊗2(r) HLS (", α, π), which are represented by the solid line, dashed a (or g). To be specific, we will first work with a = 2, Here, F (r)andG(r) are dimensionless, but W (r)isinunit 1 of 1/fm. are dimensionless, while (r) is in unit of fm. lines, and dotted lines, respectively. which is widely used in the model of the O(p2) HLS La- grangian [9, 10], and then examine how each components of the soliton mass and the moment of inertia behave as The values of the LECs obtained with a = 2 are given in the value of the HLS parameter a is varied. This verifies found that the inclusion of the π meson reduces and the HLS1(", π) models. However, as can be the first row of Table. I. numerically how the a independence comes about. the soliton mass. In the present work, the soli- seen by the dotted lines, inclusion of the α meson Equipped with the numerical values of the LECs given In the present work, we consider three versions of the ton mass reduces from 922 MeV in the HLS1(") expands the wave functions. All these behaviors in Table I, the equations of motion for the soliton wave 2 2 HLS model induced from each hQCD model. The first to 834 MeV in the HLS1(", π), which confirms the can be found in the rms sizes r and r function and for the soliton excitations can be solved nu- π "W π "E version is the model that includes the pion, α meson, and claim that the inclusion of the π meson makes the in Table II. Therefore, we conclude that the π me- merically, which allows us to calculate the soliton mass ⊗ meson. The second one is the model without the hWZ Skyrmion closer to the BPS soliton. However, when son decreases the soliton massπ while the απmeson and the moment of inertia. The main results of the terms, i.e., the model that includes the pion and the α we include the α meson, the soliton mass increases increases it. present work are summarized in the columns of HLS meson. The third one is obtained by integrating out the α 1 to 1184 MeV. This is in contrast to the naive ex- of Table II. The results of two models with the HLS of meson in the second version of the model. Therefore, this O(p2) are also presented for comparison. pectation that including more vector mesons would 2. In the moment of inertia, or in the π-N mass dif- corresponds to the original Skyrme model but with the The obtained soliton wave functions for the decrease the soliton mass. Since the α meson inter- ference πM , through the collective quantization, Skyrme parameter determined by the hQCD model. In acts with the other mesons through the hWZ terms, the role of the π and α mesons are the opposite to this section, we will examine the three versions of the SS HLS1( , α, ⊗) model are shown in Fig. 1 and Fig. 2. This model results in the soliton mass M 1184 MeV this observation shows the importance of the hWZ the case of the soliton mass. The mass di"erence model and of the BPS model. The obtained results will sol π 2 and the moment of inertia 0.661 fm that leads terms in the Skyrmion phenomenology. The role of πM increases by the inclusion of the π meson, i.e., be compared with those of the O(p ) models, such as the I π to πM 448 MeV. These numbers should be com- the α meson in the Skyrmion mass and size can also from 1014 MeV in the HLS1(") to 1707 MeV in α-stabilized model of Ref. [19] and “the minimal model” π be verified by comparing the soliton wave functions the HLS (", π), which worsens the situation phe- of Ref. [20] that includes the ⊗ meson in a minimal way. pared with the empirical values, Msol = 867 MeV 1 and πM 292 MeV. Compared with the widely used shown in Fig. 3. This figure shows that the π me- nomenologically. Furthermore, in the nucleon and π 2 45 “minimal model” of the HLS up to O(p ) [20, 32, 33], son shrinks the soliton wave functions, which can π masses, the rotational energy at O(1/Nc)iseven this shows that the HLS1( , α, ⊗)modelimprovesthe be seen by comparing the results from the HLS (") larger than the soliton mass that is of O(N ). This A. Skyrmion in the HLS induced from the 1 c soliton mass. 13년 6월 26일 수요일Sakai-Sugimoto model We then consider the HLS1( , α) model that is con- structed from the HLS Lagrangian without the hWZ In this subsection, we first consider the Sakai-Sugimoto terms. In other words, we set c1 = c2 = c3 = 0 and model [30, 31] to determine the LECs of the HLS La- remove the ⊗ meson mass term and its kinetic energy grangian. This model is characterized by the following term in the HLS1( , α, ⊗) model to obtain the HLS1( , α) warping factor: model. Therefore, this model is very similar to the model studied in Refs. [16, 34]. In addition to the soliton 2 1/3 K1(z)= 1+z π , mass, however, we also calculate the moment of iner- 2 tia which was not given in Refs. [16, 34]. And then we K2(z)=1+π z . " (62) finally consider the HLS1( ) model that is defined with Since MKK and GYM are determined by fπ and m", all the Lagrangian (56) with the Skyrme parameter given in LECs except a or g can be determined. This will be Eq. (59). All the results are summarized in Table II and called HLS1 model [5]. As we discussed above, we take here are several comments made in order. the commonly used value a = 2 as a typical example and then we will test the results by varying the value of a. 1. As claimed in the literature [16, 27, 28, 34], we 9

TABLE I. Low energy constants of the HLS Lagrangian at O(p4)witha =2.

Model y1 y3 y5 y6 z4 z5 c1 c2 c3 SS model 0.001096 0.002830 0.015917 +0.013712 0.010795 0.007325 +0.381653 0.129602 0.767374 BPS model π0.071910 π0.153511 π0.012286 0.196545 0.090338 π0.130778 0.206992 +3π .031734 1.470210 π π π π π π

TABLE II. Skyrmion mass and size calculated in the HLS with the SS and BPS models with a =2.ThesolitonmassMsol and 2 2 2 µ the π-N mass di"erence πM are in unit of MeV while r and r are in unit of fm. The column of O(p )+π B is " αW " αE µ “the minimal model” of Ref. [20] and that of O(p2) corresponds to the model of Ref. [19]. See the text for more details. π π 2 µ 2 HLS1(", α, π)HLS1(", α)HLS1(") BPS(", α, π)BPS(", α)BPS(") O(p )+πµB [20] O(p )[19] Msol 1184 834 922 1162 577 672 1407 1026 πM 448 1707 1014 456 4541 2613 259 1131 r2 0.433 0.247 0.309 0.415 0.164 0.225 0.540 0.278 " αW Solutions r2 0.608 0.371 0.417 0.598 0.271 0.306 0.725 0.422 π" αE 8 π 9 1.5 functions π and π . Then all the LECs are obtained HLS(0 π1 , ρ, ⍵) model 3 3 through the master formulas given in Eq. (35), which F(r) (r) HLS1(π) ξ1 contain the massTABLE scale M I.KK Low, the energy ’t Hooft constants coupling of" the(or HLS Lagrangian at O(p4)witha =2. G(r) HLS (π,ρ) 1 ξ (r) 2 1 GYM), and the integrals of the warping factor K(z) and 2 W(r) 2 (r) HLS1(π,ρ,ω) Model the wavey functions1 π0 yand3 π1. In they5 present work,y6 we z4 z5 c1 c2 c3 ϕ × 10 F(r) SS modelconsider0.001096 two hQCD0 models,.002830 the SS0.015917 model and +0 the.013712 BPS 0.0107951 0.007325 +0.381653 0.129602 0.767374 1 BPS modelmodel.π0.071910 π0.153511 π0.012286 0.196545 0.090338 π0.130778 0.206992 +3π .031734 1.470210 0.5 π π π π π π In hQCD models, MKK and GYM are free parameters. In the present work, we fix them by using the empirical 0 0 values of fπ and m": 0 TABLE II. Skyrmion mass and size calculated in the HLS with the SS and−1 BPS models with a =2.ThesolitonmassMsol and 2 2 2 µ −1 G(r) the π-N mass di"erence πMmare" = in 775 unit.49 of MeV MeV, while r W and r E are in unit of fm. The column of O(p )+πµB is 2 " α " α “the minimal model” of Ref.f [20]= and 92.4 that MeV of. O(p ) corresponds(61) to the model of Ref. [19]. See the text for more details. −0.5 π π π −2 −2 HLS (", α, π)HLS(", α)HLS(") BPS(", α, π)BPS(", α)BPS(") O(p2)+π Bµ [20] O(p2)[19] Then we1 have complete1 information to1 calculate all LECs 0 1 23µ 0 1 23 M 1184 834 922 1162 577 672 1407 1026 0 1 23 sol of the HLS Lagrangian through the master formulas. r (fm) r (fm) r (fm) πM Note, however,448 that the1707 master formulas1014 can determine456 4541 2613 259 1131 r2 only the0.433 product of ag0.2472 = m2/f 2,0.309 and, therefore,0.415a or FIG.0.164 1. The 0.225soliton wave functions0.540 obtained in0.278 the W " π FIG. 2. The excitations of the soliton profile, ⊗1(r)(solidline), FIG. 3. Comparison of the soliton wave functions F (r) " 2α g remains unfixed. However, as discussed in the previ- HLS1(π, ", α)modelwitha =2. F (r), G(r), and W (r) are r E 0.608 0.371 0.417 0.598 0.271 0.306 0.725 0.422 ⊗2(r)(dashedline),and (r)(dot-dashedline)calculatedin and G(r) in the three models, HLS ("), HLS (", α), and π" α ous section,Comparison the physical quantities of arethe independent three of modelsgiven by the solid, dashed, and dot-dashed lines, respectively. 1 1 10the HLS1(", α, π)modelwitha = 2. Here, ⊗1(r)and⊗2(r) HLS (", α, π), which are represented by the solid line, dashed π a (or g). To be specific, we will first work with a = 2, Here, F (r)andG(r) are dimensionless, but W (r)isinunit 1 of 1/fm. are dimensionless, while (r) is in unit of fm. lines, and dotted lines, respectively. which is widely used in the model of the O(p2) HLS La- 1.5 1.5 Soliton Mass Moment of Inertia grangian [9, 10], and then examine how each components 2000 1 of the soliton mass and the moment of inertia behave as 3 2 (r) The values of the LECs obtained with aHLS= 21 are(π) given in found that the inclusion of theHLSπ meson(π,ρ) reduces O(p ) hWZ the value of the HLS parameter aξ1is varied. This verifies 1 and the HLS1(", π) models. However, as can be 4 sum 0.8 the first row of Table. I. HLS (π,ρ) the soliton mass. In the present work, the soli- seen1500 by the dotted lines,O(p inclusion) of the α meson 1 numerically how the a independenceξ (r) comes about. 1 1 HLS1(π,ρ,ω) 2 Equipped2 with the numerical values of the LECs given In the present work, we consider three versions of the ton mass reduces from 922 MeV in the HLS1(") expands the wave functions. All these behaviors ϕ (r) × 10 in Table I, the equations of motion forHLS the soliton1(π,ρ,ω wave) ξ (r) 2 2 0.6 HLS model induced from each hQCD model. The first F(r) to 834 MeV1 in the HLS1(", π), which confirms the can be found in the rms sizes r and r function and for the soliton excitations can be solved nu- 1000 π "W π "E version is the model that includes the pion, α meson, and 1 claim that the inclusion of the π meson makes the in Table II. Therefore, we conclude that the π me- 0.5 merically, which allows us to calculate the soliton mass 0.5 0.4 ⊗ meson. The second one is the model without the hWZ Skyrmion closer to the BPS soliton. However, when son decreases the soliton massπ while the απmeson (MeV) and the moment of inertia. The main results of the (fm) I

terms, i.e., the model that includes the pion and the α we include the α meson, the soliton mass increases increasessol 500 it.

present work are summarized in the columns of HLS M 0.2 meson. The third one is obtained by integrating out the α 0 1 to 1184 MeV. This is in contrast to the naive ex- of Table II. The results of two models with the HLS of 0 meson in the second version of the model. Therefore, this 0 O(p2) are also presented for comparison. pectation that including more vector mesons would 2. In the moment of inertia, or in the π-N mass dif- corresponds to the original Skyrme model but with the 0 0 −The1 obtainedG(r) soliton wave functions for the decrease the soliton mass. Since the α meson inter- ference πM , through the collective quantization, Skyrme parameter determined by the hQCD model. In acts with the other(r) mesons through the hWZ terms, the role of the π and α mesons are the opposite to HLS1( , α, ⊗) model are shown in Fig. 1 and Fig. 2. ξ2 −0.2 this section, we will examine the three versions of the SS −0.5this observation shows the importance of the hWZ the−500 case of the soliton mass. The mass di"erence −0.5 This model results in the soliton mass Msol 1184 MeV model and of the BPS model. The obtained results will −2 π 123123 2 and the moment of inertia 0.661 fm that leads terms in the Skyrmion phenomenology. The role of πM increases by the inclusion of the π meson, i.e., be compared with those of the O(p ) models, such as the I π a a 0 1 23to π 448 MeV. These numbers should be com- the0 α meson in the1 Skyrmion mass23 and size can also from 1014 MeV in the HLS1(") to 1707 MeV in α-stabilized model of Ref. [19] and “the minimal model” 0M π 1 23 r (fm) be verified by comparing r (fm) the soliton wave functions the HLS1(", π), which worsens the situation phe- of Ref. [20] that includes the ⊗ meson in a minimal way. pared with the empirical r values,(fm) Msol = 867 MeV FIG. 5. Dependence of the soliton mass and the moment of and πM 292 MeV. Compared with the widely used shown in Fig. 3. This figure shows that the π me- nomenologically. Furthermore, in the nucleon and inertia on the HLS parameter a in the HLS1(", α, ⊗)model. “minimalπ model” of the HLS up to O(p2) [20, 32, 33], son shrinks the soliton wave functions, which can45 π masses, the rotational energy at O(1/N )iseven FIG. 2. The excitations of the soliton profile, ⊗1(r)(solidline), FIG. 3. Comparison of the soliton wave functions F (r) FIG. 4. Same as Fig. 3 but for π1(r)andπ2(r). Dotted, dashed, and dot-dashed lines are thec contributions this shows that the HLS1( , α, ⊗)modelimprovesthe be seen by comparing the results from the HLS (") larger than2 the soliton4 mass that is of O(N ). This ⊗2(r)(dashedline),andA. Skyrmion (r)(dot-dashedline)calculatedin in the HLS induced from the and G(r) in the three models, HLS ("), HLS (", α), and 1 from the O(p ), O(p ), and hWZ terms. Solidc lines are their soliton mass. 1 1 the HLS1(13년 6월 26일 수요일", α, π)modelwithSakai-Sugimotoa = 2. Here, model⊗1(r)and⊗2(r) HLS (", α, π), which are represented by the solid line, dashed sums. 1We then consider the HLS ( , α) model that is con- are dimensionless, while (r) is in unit of fm. lines, and dotted lines, respectively.1 raises a serious problem to the validity of the col- structed from the HLS Lagrangian without the hWZ In this subsection, we first consider the Sakai-Sugimoto lective quantization method in these models. How- terms. In other words, we set c1 = c2 = c3 = 0 and model [30, 31] to determine the LECs of the HLS La- remove the ⊗ meson mass term and its kinetic energy ever, inclusion of the π meson reduces πM and This evidently shows that the role of the π me- grangian. This model is characterized by the following found that the inclusion of the π meson reduces termand in the the HLS HLS1( 1,(α",,⊗π)) model models. to obtain However, the HLS as1( can, α) be the rotational energy appreciably. The compar- son may be even more addressed in nuclear matter. thewarping soliton factor: mass. In the present work, the soli- model.seen Therefore, by the dotted this model lines, is very inclusion similar of to the the modelα meson ison of the soliton wave functions obtained from Therefore, it is highly desirable to investigate the ton mass reduces from 922 MeV in1/3 the HLS (") studied in Refs. [16, 34]. In addition to the soliton the O(1/Nc) rotational energy can be found in role of the π meson in more detail in Skyrmion 2 π 1 expands the wave functions. All these behaviors K1(z)= 1+z , mass, however, we also calculate the moment2 of iner- 2 to 834 MeV in the HLS1(", π), which confirms the can be found in the rms sizes r and r Fig. 4. This shows that "1,2(r) are expanded in the matter. Our analysis shows that the three com- 2 tia which was not given in Refs. [16, 34]. AndW then we E claim that the inclusionK2(z)=1+ ofπ thezπ. "meson makes the(62) π " π " HLS1(α, ⊗, π) model than in the HLS1(α, ⊗)model. ponents of the Skyrmion mass represented by the finallyin Table consider II. the Therefore, HLS ( )model we conclude that is defined that the withπ me- Skyrmion closer to the BPS soliton. However, when son decreases the soliton1 massπ while the απmeson This leads to a larger value for the moment of in- dotted, dashed, and dot-dashed lines in Fig. 5 have Since MKK and GYM are determined by fπ and m", all the Lagrangian (56) with the Skyrme parameter given in we includeLECs except the αameson,or g can the be soliton determined. mass increases This will be Eq.increases (59). All the it. results are summarized in Table II and ertia in the HLS1(α, ⊗, π) model, which leads to a very di"erent behavior with a,buttheirsumisin- smaller π through the relation given in Eq. (25). to 1184called MeV. HLS1 Thismodel is [5]. in contrastAs we discussed to the above, naive we ex- take here are several comments made in order. M dependent of the parameter a. Similar conclusions pectationthe commonly that including used value morea = vector 2 as a typical mesons example would and 2. In the moment of inertia, or in the π-N mass dif- can be drawn from the decomposition of the mo- 3. The contributions from each term of the La- decreasethen wethe will soliton test mass.the results Since by the varyingα meson the value inter- of a. 1.ferenceAs claimedπ , in through the literature the collective [16, 27, 28, quantization, 34], we ment of inertia as well. Here, we found a slight M grangian (7) to the soliton mass and the moment acts with the other mesons through the hWZ terms, the role of the π and α mesons are the opposite to dependence on a, which might be related to the ap- of inertia are also analyzed as functions of the HLS this observation shows the importance of the hWZ the case of the soliton mass. The mass di"erence proximate method adopted by the collective quanti- parameter a. The results are summarized in Fig. 5. terms in the Skyrmion phenomenology. The role of π increases by the inclusion of the π meson, i.e., zation method. More detailed studies on the quan- M The contributions from the O(p2)terms,O(p4) the α meson in the Skyrmion mass and size can also from 1014 MeV in the HLS (") to 1707 MeV in tization method is, therefore, desirable. 1 terms, and the hWZ terms are represented by the be verified by comparing the soliton wave functions the HLS (", π), which worsens the situation phe- 1 dotted, dashed, and dot-dashed lines, respectively, shown in Fig. 3. This figure shows that the π me- nomenologically. Furthermore, in the nucleon and All these observations show the importance of the π while the solid lines are their sums. We first verify son shrinks the soliton wave functions, which can π masses, the rotational energy at O(1/N )iseven meson in Skyrmions. The π meson increases the soli- c that the contribution from the O(p2)termstothe be seen by comparing the results from the HLS1(") larger than the soliton mass that is of O(Nc). This ton mass and decreases the moment of inertia, which is soliton mass increases with a. On the contrary, the exactly the opposite to the role of the ⊗ meson. Further- 4 contribution from the O(p ) terms has a negative more, only when the π meson is included, the rotational slope with a and its magnitude is smaller than the energy is smaller than the soliton mass and thus the stan- 2 6 O(p )terms, which shows that this order counting dard collective quantization can be justified. is reasonable in the Skyrmion mass and size. How- ever, the contribution from the hWZ terms that are connected to the π meson is highly nontrivial. In B. Skyrmion in the HLS induced from the BPS particular, its contribution is stable as a becomes model smaller while the O(p2) contribution decreases. As a result, when a 1, which corresponds to the It was claimed in Refs. [27, 28] that the BPS Skyrmion, value in nuclear mediumπ 7 [10], the contribution of i.e., the soliton in the flat space 5D YM action, has the the hWZ terms is close to that of the O(p2)terms. potentially important feature in the Skyrmion structure. In this subsection, we determine the LECs of the HLS with the flat space 5D YM action, which we call the BPS 6 The magnitude of the O(p4)contributionisabout15%ofthe model. To investigate the warping factor e"ect we con- O(p2)contributionata =2. sider a gauge theory in flat 5D Minkowski space-time. In 7 1/Nc corrections are expected to be highly important in medium, the sense of large ’t Hooft parameter expansion, the flat so this feature should be taken with caution. space-time means that we are going to consider the O( ) 9

TABLE I. Low energy constants of the HLS Lagrangian at O(p4)witha =2.

Model y1 y3 y5 y6 z4 z5 c1 c2 c3 SS model 0.001096 0.002830 0.015917 +0.013712 0.010795 0.007325 +0.381653 0.129602 0.767374 BPS model π0.071910 π0.153511 π0.012286 Results0.196545 0.090338 π0.130778 0.206992 +3π .031734 1.470210 π π π π π π

TABLE II. Skyrmion mass and size calculated in the HLS with the SS and BPS models with a =2.ThesolitonmassMsol and 2 2 2 µ the π-N mass di"erence πM are in unit of MeV while r and r are in unit of fm. The column of O(p )+π B is " αW " αE µ “the minimal model” of Ref. [20] and that of O(p2) corresponds to the model of Ref. [19]. See the text for more details. π π 2 µ 2 HLS1(", α, π)HLS1(", α)HLS1(") BPS(", α, π)BPS(", α)BPS(") O(p )+πµB [20] O(p )[19] Msol 1184 834 922 1162 577 672 1407 1026 πM 448 1707 1014 456 4541 2613 259 1131 r2 0.433 0.247 0.309 0.415 0.164 0.225 0.540 0.278 " αW r2 0.608 0.371 0.417 0.598 0.271 0.306 0.725 0.422 π" αE π M M MN 1.5 ⌘ 3 10 (r) HLS1(π) ξ1 HLS ( 1 ξ (r) 1 π,ρ) 1.5 2 2 Soliton Mass Moment of Inertia 2000 1 a independence of the ϕ (r) × 10 HLS1(π,ρ,ω) F(r) 2 HLS O(p ) 1(π,ρ) 1 hWZ Skyrmion0.5 properties 4 sum 0.8 1500 O(p ) 1 HLS1(π,ρ,ω) (r) 0 0.6 ξ1 0 1000 0.5 0.4 −1 G(r) (MeV) (fm) I

sol 500

−0.5 M 0.2 0 −2 0 0 0 1 23 0 1 23 (r) r (fm) r (fm) ξ2 −0.2 −0.5 −500 FIG. 2. The excitations of the soliton profile, ⊗1(r)(solidline), FIG. 3.123 Comparison of the123 soliton wave functions F (r) ⊗ (r)(dashedline),and (r)(dot-dashedline)calculatedin a a 02 1 23and G(r) in the three models, HLS1("), HLS1(", α), and the HLS (", α, π)modelwitha = 2. Here, ⊗ (r)and⊗ (r) 1 r (fm) 1 2 HLS1(", α, π), which are represented by the solid line, dashed are dimensionless, while (r) is in unit of fm. FIG. 5.lines, Dependence and dotted of lines, the respectively.soliton mass and the moment of 46 inertia on the HLS parameter a in the HLS1(", α, ⊗)model. FIG. 4. Same as Fig. 3 but for π (r)andπ (r). 13년 6월 26일 수요일 1 2 Dotted, dashed, and dot-dashed lines are the contributions from the O(p2), O(p4), and hWZ terms. Solid lines are their found that the inclusion of the π meson reduces sums. and the HLS1(", π) models. However, as can be raises athe serious soliton problem mass. to In the the validity present of work, the col- the soli- seen by the dotted lines, inclusion of the α meson lective quantizationton mass reduces method from in these922 MeV models. in the How- HLS1(") expands the wave functions. All these behaviors 2 2 to 834 MeV in the HLS1(", π), which confirms the can be found in the rms sizes r and r ever, inclusion of the π meson reduces πM and This evidently shows that the role ofW the π me- E claim that the inclusion of the π meson makes the in Table II. Therefore, we concludeπ " that the ππ me-" the rotational energy appreciably. The compar- son may be even more addressed inπ nuclear matter.π ison ofSkyrmion the soliton closer wave to the functions BPS soliton. obtained However, from when Therefore,son decreases it is highly the soliton desirable mass to while investigate the α themeson we include the α meson, the soliton mass increases the O(1/N ) rotational energy can be found in role ofincreases the π it.meson in more detail in Skyrmion to 1184c MeV. This is in contrast to the naive ex- Fig. 4. This shows that " (r) are expanded in the matter. Our analysis shows that the three com- pectation that including1,2 more vector mesons would 2. In the moment of inertia, or in the π-N mass dif- HLS1(α, ⊗, π) model than in the HLS1(α, ⊗)model. ponents of the Skyrmion mass represented by the decrease the soliton mass. Since the α meson inter- ference πM , through the collective quantization, This leadsacts to with a larger the other value mesons for the through moment the hWZ of in- terms, dotted,the dashed, role of the andπ dot-dashedand α mesons lines are in the Fig. opposite 5 have to ertia inthis the observation HLS1(α, ⊗, π shows) model, the importance which leads of to the a hWZ very dithe"erent case of behavior the soliton with mass.a,buttheirsumisin- The mass di"erence smaller πM through the relation given in Eq. (25). dependent of the parameter a. Similar conclusions terms in the Skyrmion phenomenology. The role of πM increases by the inclusion of the π meson, i.e., can be drawn from the decomposition of the mo- the α meson in the Skyrmion mass and size can also from 1014 MeV in the HLS1(") to 1707 MeV in 3. The contributions from each term of the La- ment of inertia as well. Here, we found a slight grangianbe (7) verified to the by soliton comparing mass the and soliton the wave moment functions the HLS1(", π), which worsens the situation phe- shown in Fig. 3. This figure shows that the π me- dependencenomenologically. on a, which Furthermore, might be related in the to nucleon the ap- and of inertia are also analyzed as functions of the HLS son shrinks the soliton wave functions, which can proximateπ masses, method the adopted rotational by energy the collective at O(1/N quanti-)iseven parameter a. The results are summarized in Fig. 5. c be seen by comparing the results2 from the HLS4 1(") zationlarger method. than More the soliton detailed mass studies that is on of theO(N quan-c). This The contributions from the O(p )terms,O(p ) tization method is, therefore, desirable. terms, and the hWZ terms are represented by the dotted, dashed, and dot-dashed lines, respectively, All these observations show the importance of the π while the solid lines are their sums. We first verify meson in Skyrmions. The π meson increases the soli- 2 that the contribution from the O(p )termstothe ton mass and decreases the moment of inertia, which is soliton mass increases with a. On the contrary, the exactly the opposite to the role of the ⊗ meson. Further- 4 contribution from the O(p ) terms has a negative more, only when the π meson is included, the rotational slope with a and its magnitude is smaller than the energy is smaller than the soliton mass and thus the stan- 2 6 O(p )terms, which shows that this order counting dard collective quantization can be justified. is reasonable in the Skyrmion mass and size. How- ever, the contribution from the hWZ terms that are connected to the π meson is highly nontrivial. In B. Skyrmion in the HLS induced from the BPS particular, its contribution is stable as a becomes model smaller while the O(p2) contribution decreases. As a result, when a 1, which corresponds to the It was claimed in Refs. [27, 28] that the BPS Skyrmion, value in nuclear mediumπ 7 [10], the contribution of i.e., the soliton in the flat space 5D YM action, has the the hWZ terms is close to that of the O(p2)terms. potentially important feature in the Skyrmion structure. In this subsection, we determine the LECs of the HLS with the flat space 5D YM action, which we call the BPS 6 The magnitude of the O(p4)contributionisabout15%ofthe model. To investigate the warping factor e"ect we con- O(p2)contributionata =2. sider a gauge theory in flat 5D Minkowski space-time. In 7 1/Nc corrections are expected to be highly important in medium, the sense of large ’t Hooft parameter expansion, the flat so this feature should be taken with caution. space-time means that we are going to consider the O( ) Discussions

1. The role of ρ meson • reduction of the soliton mass: from 922 MeV to 834 MeV • increase of the α-N mass difference: from 1014 MeV to 1707 MeV • shrink the soliton profile: from 0.417 fm to 0.371 fm 2. The role of ⍵ meson • increase of the soliton mass: from 834 MeV to 1184 MeV • decrease of the α-N mass difference: from 1707 MeV to 448 MeV • expand the soliton profile: from 0.371 fm to 0.608 fm 3. Without ⍵ meson • the α-N mass difference of O(1/Nc) > the soliton mass of O(Nc) 4. The independence of a • Direct consequence from hQCD

47

13년 6월 26일 수요일 Nuclear Matter: Skyrme Crystal

Skyrme Crystal (FCC) z I. Klebanov, Nucl. Phys. B262, 133 (1985) 0 0 z M. Kugler et al., Phys. Lett. B208, 491 (1988) 0 0 H.-J. Lee, B.-Y. Park, D.-P. Min, M. Rho, and V. x Vento, Nucl. Phys. A723, 427 (2003) y y x 0 0 y z 0 0 x Half-Skyrmion Phase

increasing density

48

13년 6월 26일 수요일 Skyrme Crystal

3 n0 • Skyrmion number density n = 1/(2L ) - • normal nuclear density n0 : 0.17 fm 3 HLSmin(,,) 1600 HLS (,) corresponds to L ∼ 1.43 fm HLS (,,)

(MeV) n ~ 2n0 1200

E/B binding energy per baryon 800 ~150 MeV, ~100 MeV, ~50 MeV (too big!) 1

0.8

0.6

0.4 transition to the half-Skyrmion phase

0.2 <σ> = 0

0 2.5 2 1.5 1 0.5 L (fm) n ~ n0

49

13년 6월 26일 수요일 2 1500 O (p ) , 4 1200 O (p ) , 900 (MeV) 600

E/B 300

0

300 2.5 2 1.5 1 0.5 L (fm)

50

13년 6월 26일 수요일 Change of Meson Properties

51

13년 6월 26일 수요일 1

0.9 1200

0.8 1000 f ⁄ 0.7

(MeV) 800 f sol

0.6 M 600 HLSmin (,,) HLSmin (,,) 0.5 HLS (,) HLS (,) HLS (,,) 400 HLS (,,) 0.4 2.5 2 1.5 1 0.5 2.5 2 1.5 1 0.5 L (fm) L (fm)

52

13년 6월 26일 수요일 HLS 1.2 min(,,) HLS (,) HLS (,,) 0.9 W 2 r 0.6

0.3

0 1.2

0.9 E 2 r

0.6

0.3

0 2.5 2 1.5 1 0.5 L (fm)

53

13년 6월 26일 수요일 Summary

1. The nontrivial role of the ⊗ meson 2. The presence of topological change from Skyrmions to half-Skyrmions at slightly above the normal nuclear density of higher density 3. Problems: minimal energy occurs at too high density with too high binding energy 4. The structure may be robust, but the numbers? 1/Nc corrections: Casimir energy

F. Meier and H. Walliser,438 Phys. Rep.F. 289,Meier, 383H. Walliser (1997) / Physics Reports 289 (1997) 383-448

Table 4.1 Tree and one-loop contribution to various quantities for parameter set A (e = 4.25, 9~- 0)

Tree One-loop ~ Exp.

M (MeV) 1629 -683 946 939 (MeV) 54 -22 32 45 4- 7 {r2) s (fin 2) 1.0 +0.3 1.3 1.6±0.3 9A 0.91 --0.25 0.66 1.26 (r2)a (fin 2) 0.45 --0.04 0.41 0 a9 +°'Is "--- --0.08 (r2) s (fm 2) 0.62 -0.11 0.51 0.59 /t v 1.62 +0.62 2.24 2.35 (r2) v (fill)2 0.77 --0.13 0.64 0.73 (10 -4 fm 3) 17.8 --8.0 9.8 9.5 ± 5

Table 4.2 Tree and one-loop contribution to various quantities for parameter set B (e = 4.5, 9~.o- 1.0) 54

13년 6월 26일 수요일 Tree One-loop ~ Exp. M (MeV) 1599 -646 953 939 cr (MeV) 58 -14 44 45 ± 7 (r2) s (fm 2) 1.1 +0.3 1.4 1.6 4- 0.3 9A 0.96 --0.15 0.81 1.26 (r2)A (fro 2) 0.44 -0.01 0.43 0/19+°18 .... --0.08 (r2) s (fm 2) 0.64 -0.09 0.55 0.59 /~v 1.69 +0.88 2.57 2.35 (r2) v (fro 2) 0.76 -0.17 0.59 0.73 (10 -4 fm 3) 19.4 -5.3 14.1 9.5 4- 5

Table 4.3 Tree and one-loop contribution to various quantities for parameter set C (e = 5.845, g~)= 2.2)

Tree One-loop ~ Exp.

M (MeV) 1490 -539 951 939 t7 (MeV) 67 +13 80 45 i7 (r2) s (fm 2) 1.1 +0.4 1.5 1.6 4- 0.3 9A 1.03 --. 11 0.92 1.26 (r2)A (fm 2) 0.40 +0.16 0.56 0.42+°0.J088 (r2)Se (fm 2) 0.68 --0.06 0.62 0.59 #v 1.75 +1.38 3.13 2.35 (r2) v (fm 2) 0.72 -0.19 0.53 0.73 (10-4 fm 3) 21.8 +5.4 27.2 9.5±5

meson species included in the model. This is also obvious from Tables 4.1-4.3, where the tree level mass is shown to be in the 1500-1630 MeV range for the three models. • With values between 54 and 67 MeV against experimental 45 MeV, the o- term is also over- estimated in tree for all parameter combinations under consideration here. This is not a common feature of all Skyrme-type models and depends on our usage of the full ChO 4 lagrangian Outlook

1. The role of vector mesons • previous works: more VMs lead to the Bogomolny bound • the inclusion of the ρ meson confirms it • but, the ⍵ meson has the opposite role: important from both the theoretical and phenomenological views 2. Issues 0 • next order corrections: O(Nc ) pion fluctuation • next order terms in the HLS: in Nc and in p 3. Final goal • few-nucleon systems ⇒ semi-empirical mass formula? • nuclear matter, Skyrmion crystal • equation of state, nuclear symmetric energy

55

13년 6월 26일 수요일 πank y" very much

for y"r aαention

13년 6월 26일 수요일