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Hidden local symmetry and infinite tower of vector mesons for baryonsYang, Ghil-Seok( ) Recent progress in hadron physics -From hadrons to quark and gluon- 2013 (Feb. 18-22, Yonsei Univ.)

Department of Physics&CHEP (Center for High Energy Physics) Kyungpook Nat'l University in collaboration with Yongseok Oh (Kyungpook Natl Univ.) Yong-Liang Ma (Nagoya Univ., Japan) Masayasu Harada (Nagoya Univ., Japan) Hyun Kyu Lee (Hanyang Univ.) Byung-Yoon Park (Chungnam Natl Univ.) Mannque Rho (CEA Saclay, France & Hanyang Univ.)

Motivation & Soliton Picture / Vector mesons

HLS Lagrangian up to O(p4) Soliton mass & M = m mN

Results : Three modelsHLS(, , ) modelHLS(, ) modelHLS() model

Summary

OutlineReferences: Y.-L. Ma, Y. Oh, G.-S. Yang, M. Harada, H.K. Lee, B.-Y. Park, M. Rho, Hidden local symmetry and infinite tower of vector mesons for baryons, Phys.Rev.D 86, 074025 (2012) [arXiv:1206.5460]

Y.-L. Ma, G.-S. Yang, Y. Oh, M. Harada, Skyrmions with vector mesons in the hidden local symmetry approach, Phys.Rev.D 87, 034023 (2013) [arXiv:1209.3554]3Motivation & Soliton PictureDense baryonic matter Studies for nucleon structure, compact stars, and so onPossible approach With a chiral Lagrangian, unify both elementary baryons and multi-baryons system * Skyrme model : (Skyrmion) (multi-Skyrmions)

Brown, Rho, The Multifaceted Skyrmions (Book) H.-J.Lee, B.-Y.Park, D.-P.Min, M.Rho, and V.Vento, Nucl.Phys.A723,427(2003) - single baryon is generated as a Skyrmion- multi-Skyrmions can be put on the crystal lattice to simulate many-body system and dense matterSkyrme model1960s: T.H.R. Skyrme

Baryons are topological solitons within a nonlinear theory of pions.

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

Skyrme (1961) Baryons are solitons in the non-linear sigma model t Hooft (1974) In large-Nc limit, QCD becomes equivalent to EFT of mesonsWitten (1979) Baryons may emerge as solitons in large-Nc theory of mesons

Hedgehog solutionSUf(2) collective coordinate quantization & Mass formulae

M = m mN To give correct quantum numbersMass formulae : infinite tower of I =JAdjust f and e to reproduce the nucleon and Delta masses

f = 64.5 MeV, e = 5.45Empirically, f = 93 MeV, e = 5.85(?)

G.S. Adkins, C.R. Nappi, and E. Witten, Nucl. Phys. B228, 552 (1983)A.D. Jackson and M. Rho, Phys. Rev. Lett. 51, 751 (1983)Best-fitted results from Skyrme model Hidden Local Symmetry(HLS) As energy scale goes up, infinite number of local symmetries appear HLS: corresponding gauge fields infinite vector & axial-vector mesons

Skyrme model for Nuclear Physicssingle baryonnuclear matterImprovement of the modelmore degrees of freedom (mesons)1/Nc correctionsTopicsProperties of single baryonEquation of statePhase transitionApplication to nucleus Why vector mesons ?Witten: QCD ~ weakly interacting mesons in large Nc The lightest meson is . The next low-lying mesons are vector mesons (, ).

Stability of the soliton

Without the Skyrme term, the soliton collapses [Derricks theorem] However, vector mesons can stabilize the soliton without the Skyrme term

Skyrmions with HLS and hQCD - meson stabilized model : Igarashi et al.(1985) - and mesons stabilized model : Meissner, Kaiser, Weise(1987) - , and a1 mesons stabilized model : Kaiser, Meissner(1990), Zhang, Mukhopadhyay(1994) - hQCD : Y. Kim / D.K.Hong, M.Rho, H.-U.Yee, and P.Yi (2007) - O(p4) : Tanabashi (1993), Harada, Yamawaki (hQCD, 2003), Nawa, Suganuma, Hosaka, Kojo (2007,2009)

Skyrme termEarly Attempts to include VM

Early Attempts : Results

m2= a g 2 f 2Status of the Skyrme model with HLS - Hidden Local Symmetry (HLS) free parameter: a dependence normally taken as (hadronic medium) 1 a 2 (free space) Ex) Msol within a -meson stabilized model (Igarashi et al, Nucl.Phys.B259,1985) : Msol = (667~1575)MeV for 1 a 4, Msol = 1045MeV for a =2 ambiguity of the value of a results in a large uncertainty of the soliton mass

In this work, 1. Introduce holographic QCD (hQCD) : Integrating out of the tower of vector mesons except , O (p4) with and mesons2. All LECs are fixed by only two phenomenological inputs in hQCD3. Skyrmion properties and roles of vector mesons - Difficulties for systematic studies from higher order terms 1) In HLS, higher order terms such as O(p4) are at O(Nc) like the O(p2) terms 2) More complicated form of the Lagrangian due to the higher order terms 3) Uncontrollably large number of low energy constants Ex) 6 anomalous terms of the mesons at O(p2), 14 anomalous terms for the axial vector mesons at O(p2) HLS Lagrangian up to O(p4)

where

wherehomogenous Wess-Zumino term ()17 parameters !Soliton mass in HLS up to O(p4)

Low energy constants of the HLS Lagrangian at O(p4) with a=2hQCD modelsSS (Sakai-Sugimoto) modelBPS (Bogomolnyi-Prasad-Sommerfeld) model

Merit of this work:Precise set of parameter-free calculation that have not been done previously in the field. (first complete and parameter-free soliton cal. with vector mesons up to O(p4) )

17 parameters ! but they can be fixed by using two values (f, m) Comparison with Skyrme LOriginal Skyrme LAfter integrating out VM in HLS

e=5.45

e=7.31 : SS modele=10.02 : BPS modelSince I ~ 1/e3, large e small I large M = m - mN effective Skyrme parameterResults : Three ModelsHLS(, , ) model : full O(p4) Lagrangian with hWZ terms HLS(, ) model : without hWZ terms, the meson decouples

HLS() model : integrates out VMs same as the LSkyrme but e is fixed by the HLSComparison of the three models

meson : shrink the soliton wave function (Msol ) meson : expand the soliton wave function (Msol ) * interacts with other mesons through hWZ terms Msol 1184 MeV, (emp.: 867 MeV)M = m - mN 448 MeV, (emp.: 292 MeV) in HLS(, , ) model : improved Msol than minimal model of HLS up to O(p2) Skyrmion mass and size calculated in the HLS with the SS and BPS modelsinclusion ofMsolM WEThe role of the and in M is opposite to the case of Msol

Without meson, M of O(1/Nc) > Msol of O(Nc)

a independence of the Skyrme propertiesSummaryThe first step in series of studies made to arrive at a description of dense baryonic matter relevant for the physics of nuclear structure or compact star in unified scheme in which both single baryon and multi-baryon are treated on the same footing. (The first complete and parameter-free soliton calculation with VMs up to O(p4) )

The role of mesonreduction of the soliton mass: from 922 MeV to 834 MeVincrease of the -N mass difference: from 1014 MeV to 1707 MeVshrink the soliton profile: from 0.417 fm to 0.371 fm

The role of meson increase of the soliton mass: from 834 MeV to 1184 MeVdecrease of the -N mass difference: from 1707 MeV to 448 MeVexpand the soliton profile: from 0.371 fm to 0.608 fm

Without mesonM of O(1/Nc) > Msol of O(Nc)

The independence of aDirect consequence from hQCDTheoretical Nuclear and Hadron PhysicsDepartment of PhysicsKyungpook National UniversityProf. Yongseok Oh & visiting Prof. KochelevHiroaki Kohyama : Dimensional regularization in NJL model (Inagaki, Kimura @ Hiroshima) Ghil-Seok Yang : Nuclear structure by shell model & Hypernuclei by EFT (Otsuka@CNS, Suzuki@Hihon & Ando@Daegu) Myunghwan Mun : Nuclear fission/fusion for SHEs (YM Kim@RISP, Antonenko@BLTP) Hana Gil : Hypernuclei (Hiyama@RIKEN) Two freshmen for master courseTheoretical Nuclear and Hadron PhysicsDepartment of PhysicsKyungpook National UniversityProf. Yongseok Oh & visiting Prof. KochelevHiroaki Kohyama : Dimensional regularization in NJL model (Inagaki, Kimura @ Hiroshima) Ghil-Seok Yang : Nuclear structure by shell model & Hypernuclei by EFT (Otsuka@CNS, Suzuki@Hihon & Ando@Daegu) Myunghwan Mun : Nuclear fission/fusion for SHEs (YM Kim@RISP, Antonenko@BLTP) Hana Gil : Hypernuclei (Hiyama@RIKEN) Two freshmen for master course