Dense Baryonic Matter in the Hidden Local Symmetry Approach Yong-Liang Ma Department of Physics, Nagoya University. Talk APCTP-WCU Focus Program, APCTP, Pohang, Apr.14 – 24, 2013.

Baryonic APCTP-WCU Focus Program 2013/4/182 We studied baryon and baryonic matter using Skyrme model In Collaboration with: M. Harada, H. K. Lee, Y. Oh, B. Y. Park, M. Rho, G. –S. Yang References:  Hidden Local Symmetry and Infinite Tower of Vector Mesons for Baryons, Phys.Rev. D86 (2012)  Skyrmions with vector mesons in the hidden local symmetry approach, Phys. Rev. D87 (2013)  Baryonic Matter in the Hidden Local Symmetry Induced from Holographic QCD Models, arXiv: [hep-ph].  Dense Baryonic Matter in Hidden Local Symmetry Approach: Half-Skyrmions and Nucleon Mass, to appear.

2013/4/18 Baryonic APCTP-WCU Focus Program 3  A comprehensive summary of Skyrme model and our progress with HLS: Yongseok Oh’s talk on Monday  A review of the progress on FCC crystal matter: Byung-Yoon Park’s talk on Wendsday. Here I am going to talk: Dense Baryonic Matter in the Hidden Local Symmetry Approach: Half-Skyrmions and Nucleon Mass.

Outline I.Introduction II.Skyrmion matter from HLS III.Hadron properties with the FCC crystal background IV.Summary and Discussion 2013/4/18 Baryonic APCTP-WCU Focus Program 4

2013/4/18 Baryonic APCTP-WCU Focus Program 5  We included all the roles of ,  and  mesons using the HLS up to O(p 4 ) including the Wess- Zumino terms.  We use a general master formula to determine all the LECs self-consistently from a class of holographic QCD models.  The baryonic matter was studied by putting the solitons on the FCC crystals.  Hardron properties in medium were investigated by considering the meson fluctuations with respect to the classical solutions. I. Introduction

The hidden local symmetry Lagrangian to O(p 4 ) responsible for Skyrmion physics Parameters (17): f , g, a, y 1  y 9, z 4, z 5, c 1  c 3 Baryonic APCTP-WCU Focus Program 2013/4/186 M. Harada and K. Yamawaki, Phys. Rep. 381, 1 (2003).

HLS from hQCD models A z = 0 gauge Mode expansion 16 parameters in terms of 2 parameters: G YM, M KK Inputs: f , m . Baryonic APCTP-WCU Focus Program 2013/4/187

Baryonic APCTP-WCU Focus Program 8  - 3  interaction  -  -  interaction Minimal model

Baryonic APCTP-WCU Focus Program 2013/4/18 Lessons: Rho meson, attractive interaction; omega meson, repulsive interaction. Soliton mass and moment of inertia, parameter “a” independent. Chiral counting mechanism is valid. More degree of freedom is necessary to provide attractive interaction. Sigma meson ! 9

2013/4/18 Baryonic APCTP-WCU Focus Program 10 II. Skyrmion matter from HLS

2013/4/18 Baryonic APCTP-WCU Focus Program 11 A)no relative rotation. B)rotated by an angle π/2 wrt the axis parallel to the line joining the two skyrmions. C)the angle is π and the rotation axis is perpendicular to the line joining the two skyrmions. Park, Rho & Vento, Nucl. Phys. A736(2004)129.

2013/4/18 Baryonic APCTP-WCU Focus Program 12 M. Kugler, S. Shtrikman, Phys. Lett. B 208 (1988) 491; Phys. Rev. D 40 (1989) 3421.

2013/4/18 Baryonic APCTP-WCU Focus Program 13  In HLS( , ,  ), n 1/2  n 0. Comparing to HLS min ( , ,  ), O(p 4 ) terms and the other  -  -  interactions through hWZ terms makes n 1/2 larger by a factor 1.7. A noticeable improvement.  This higher n 1/2 may come from the fact that the size of the single skyrmion is smaller in the former and that the additional interactions in HLS( , ,  ) weaken the repulsive interactions from the omega.  Supported by the results from HLS( ,  ), where n 1/2  6n 0. The absence of  reduces the skyrmion size to almost half compared to that of HLS( , ,  ).  The weak dependence of E/B on the density in HLS( ,  ),is noticeable. Inclusion of the  mesons reduces the soliton mass a lot and almost saturates the Bogololny 's bound.  n min > n 1/2 for all three cases. The binding energy  150 MeV,  100 MeV and  50 MeV. As a complete theory of nuclear matter, n min should represent the nuclear matter ground state. That n min >n 0 and the binding energy at n min is larger than the empirical value signals that there is something missing in the present skyrmion crystal description for the EoS of nuclear matter.

2013/4/18 Baryonic APCTP-WCU Focus Program 14  The O(p 4 ) contributes always a attractive interaction.  (E/B) O(p2) ,  and (E/B) O(p2)  show distinctly different density dependence in the skyrmion phase and in the half-skyrmion phase. In the single skyrmion phase, (E/B) O(p2) ,  dominates over and (E/B) O(p2) , with their even increasing from  2 -  3 as the density increases whereas in the half-skyrmion phase, and (E/B) O(p2)  starts to decrease while (E/B) O(p2)  starts to increase. Then, at some density the repulsive interaction due to the  becomes dominant.

2013/4/18 Baryonic APCTP-WCU Focus Program 15 III. Hadron properties with the FCC crystal background Make substitution: The minimum energy solutions Fluctuating fields In-medium modification of meson properties:

2013/4/18 Baryonic APCTP-WCU Focus Program 16  The density dependence of f  * /f  shows the different behavior in the single skyrmion phase and in the half-skyrmion phase.  In the single skyrmion phase, f  * /f  decreases as the baryon number density increases.  On the other hand, in the half-skyrmion phase, it stays in a non-vanishing value around  Might indicate that, in the half-skyrmion phase, chiral symmetry is broken due to the multi-quark condsensate.

2013/4/18 Baryonic APCTP-WCU Focus Program 17 In-medium modification of the baryon properties:  Since in our HLS Lagrangian all the parameters are determined by the master formula, we can calculate the in-medium modification of all other parameters by using f  * and m  *. We shall assume that the master formula holds also in dense matter and further that the curvature function K 2 (z) does not change. The hQCD parameters M KK and G YM are modified as  All the LECs can be replaces with stared ones.  Baryon properties in-medium can be calculated.

2013/4/18 Baryonic APCTP-WCU Focus Program 18  The behaviour of the crystal size dependence of the soliton mass is the same as the behavior of f  *.  Support by M sol  f  * /e = f  * /g = f  *2 /m  *  N c which follows simply from the scaling of the nucleon mass in the N c limit as in matter-free space.  This a new result that has not been obtained in other models -- which is the most significant finding of this work -- that the nucleon mass which decreases with increasing density in the skyrmion phase, stops dropping at n 1/2 and stays constant in the half- skyrmion phase. In the case of HLS( , ,  )  0.6 M sol.

2013/4/18 Baryonic APCTP-WCU Focus Program 19  The crystal size dependence of the  -N mass difference strongly depends on the  meson.  In the calculation including  meson,  M decreases up to n 1/2 beyond which  M becomes nearly constant.  In the case of HLS( ,  ), in the low density,  M is unscaling, but near n 1/2, it scales up and after n 1/2  M becomes almost constant.

2013/4/18 Baryonic APCTP-WCU Focus Program 20  Skyrmion radius is strongly influenced by the presence of the  meson.  In the case of HLS( ,  ), the skyrmion radius is small and constant at varying density. This is simply because in the absence of , the skyrmion is almost point-like and remains stable with the crystal size.

IV. Summary and Discussion  Dense baryonic matter properties are studied using the HLS to O(p 4 ) ( , ,  ). We use a general master formula to determine all the LECs self-consistently from a class of holographic QCD (hQCD) models.  The hWZ terms in the HLS induced from the CS term in hQCD models are crucial for the inclusion of the omega meson effect in the baryon and baryonic matter properties.  The results clearly show that both the  meson attractive interaction and the  meson repulsive interaction affects on the Skyrmion properties.  Some of our results, such as n min and binding energy, are deviated from nature. Other meson such as sigma meson effects? Baryonic APCTP-WCU Focus Program 2013/4/1821

2013/4/18 Baryonic APCTP-WCU Focus Program 22

2013/4/18 Baryonic APCTP-WCU Focus Program 23 Further discussions:

2013/4/18 Baryonic APCTP-WCU Focus Program 24  The crystal-size (or density) dependence of f  is obtained with the O(p 2 ) terms of HLS. A complete calculation of this quantity up to O(p 4 ) should include.  Moreover, since we only consider fluctuations from the O(p 2 ), both  meson and  meson masses are the same as their vacuum values and independent of the density. In a complete calculation including the full O(p 4 ), these masses will change with density.  Similarly for the density dependence of the soliton properties, only the in-medium modified f  * is considered since the other input parameter m  is the same as its vacuum value in the present approximation. In a complete calculation up to O(p 4 ), these results might be changed.

2013/4/18 Baryonic APCTP-WCU Focus Program 25 Q1: How to include dilaton?  Chiral symmetry restoration at high density. f  *  0.  Nuclear binding energy and n min are away from nature. More d.o.f. Dilaton

2013/4/18 Baryonic APCTP-WCU Focus Program 26  Dilaton suppresses the soliton mass.  But, the suppression amplitude is too small within a reasonable choice of dilaon mass. f  = 240 MeV

2013/4/18 Baryonic APCTP-WCU Focus Program 27 FCC crystal  Binding energy does not change.  n 1/2 moves to the low density.  > 1 at large density.

2013/4/18 Baryonic APCTP-WCU Focus Program 28  Both f  * are m  * increased at high density From this model we cannot arrive at the physics we expect. How to include dilation in the present model?

2013/4/18 Baryonic APCTP-WCU Focus Program 29 Q2: Possible  or  meson mass scaling in HLS( , ,  )?  meson mass is not modified in-medium, either tree level or loop correction.  meson mass is modified at both tree level O(p 4 ) terms and loop level O(p 2 ) terms. U(2) in vacuum is broken to S(2)  U(1) in medium Baryon properties in medium: Master formula should be rederived from hQCD models  1 (z)   1  (z),  1  (z). hWZ terms are recombined.

Thanks ! Baryonic APCTP-WCU Focus Program 2013/4/1830