1. -Nucleon Interaction and Nucleon Mass in Dense Baryonic Matter
Won-Gi Paeng, Hanyang University, Korea
in collaboration with Hyun Kyu Lee(HYU),
Mannque Rho(CEA Saclay & HYU), Chihiro Sasaki(FIAS)
arXiv: [nucl-th]
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2. Brief review on Dilaton-Limit Fixed Point
In hidden local symmetric theory with the nucleons, the authors of [1,2] found some point in which the -nucleon coupling is suppressed, that is, the  meson is decoupled from the nucleons. And, at that point, the nucleon mass becomes zero.
We call this point ``Dilaton-Limit Fixed Point’’ which is
estimated as the infrared fixed point in Renormalization Group Equations.
When we consider U(2) symmetry is conserved in medium, -nucleon coupling is suppressed as -nucleon coupling is suppressed. The suppression of -nucleon coupling causes the problem to the stability of dense baryonic matter. So, we would like to study what happen to - nucleon coupling in medium when we consider that U(2) symmetry is broken in medium so that -nucleon coupling is differently scaled from -nucleon coupling.
[1] C. Sasaki, H. K. Lee, W.-G. Paeng and M. Rho, Phys. Rev. D 84, (2011)
[2] W.-G. Paeng, H. K. Lee, M. Rho and C. Sasaki, Phys. Rev. D 85, (2012)
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3. Introduction to my model: Hidden Local Symmetry with Gglobal = [SU(2)L X SU(2)R]global chiral Symmetry and Hlocal = [SU(2) V X U(1)V]local hidden local symmetry
Nf = 2, in [SU(2)L X SU(2)R]global chiral Symmetry and [SU(2) V X U(1)V]local hidden local symmetry, we have the vector bosons(, ) as the gauge bosons of Hlocal and the pseudo scalar bosons() as the Nambu-Goldstone bosons associated with the spontaneous symmetry breaking of Gglobal.
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4. Introduction to my model: The meson Lagrangian in Hidden Local Symmetric model
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5. Introduction to my model: Parity doublet model
DeTar and Kunihiro suggested that the nucleon can have the mass, m0, which doesn’t break the chiral symmetry by introducing the parity partner of the nucleon and the mixing term between the nucleon and the parity partner.
The chiral invariant mass m0 doesn’t vanishes when we reach the chiral symmetry restored point, but the other part of the nucleon mass which is generated by the spontaneous chiral symmetry breaking vanishes.
Nucleon mass
where gL  SU(2)L and gR  SU(2)R.
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6. Introduction to my model: Parity doublet model
Nucleon mass
where gL  SU(2)L and gR  SU(2)R.
We can redefine the nucleon field as some combination of L, R, L, R, , which transforms as For example, we can define the physical nucleon field as
with
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7. Introduction to my model: The nucleon Lagrangian in Hidden Local Symmetric model
After diagonalizing the mass matrix, we arrive at the Lagrangian in the mass eigenstate.
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8. Introduction to my model: The nucleon Lagrangian in Hidden Local Symmetric model
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9. Introduction to my model: Hidden Local Symmetric Parity Doublet Model(PDHLS)
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10. Introduction to my model: Hidden Local Symmetric Parity Doublet Model(PDHLS)
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11. RGEs in PDHLS
In the standard assignment(m0 = 0), we have RGEs given by
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12. RGEs in PDHLS
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13. RGEs in PDHLS
In the standard assignment(m0 = 0), we have RGEs given by
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14. The diagrams for -nucleon coupling at one loop
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15. RGEs in PDHLS
In the standard assignment(m0 = 0), we have RGEs given by
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16. RGEs in PDHLS Dilaton-Limit fixed point(DLFP)
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17. RGEs in PDHLS
DLFP is infrared fixed point? Or ultraviolet fixed point?
DLFP cannot be the ultraviolet fixed point because
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18. RGEs in PDHLS
There can be tree-order 3 and -- couplings in the homogeneous Wess-Zumino term in the anormalous part of the HLS Lagrangian, which could contribute to RGE of gV, but at higher order.
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19. the possible interplay between the nucleon mass and the -nucleon coupling
Now, we examine in detail, with the help of the RGE of the nucleon mass, the possible interplay between the nucleon mass and the -nucleon coupling by studying how the -nucleon coupling is changed approaching to DLFP which assumed to be the infrared fixed point.
Case 1. -nucleon coupling is constant approaching to DLFP.
At a certain point, ‘A’ with density n(A),
Case 2. -nucleon coupling decreases very slowly approaching to DLFP, so keeping Fm be negative after the point A.
Case 3. -nucleon coupling decreases fast approaching to DLFP, but keeping Fm be positive after the point A.
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20. the possible interplay between the nucleon mass and the -nucleon coupling
DLFP
(=0)
(=)
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21. the possible interplay between the nucleon mass and the -nucleon coupling
In all case
The point A
DLFP
(=0)
(=)
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22. the possible interplay between the nucleon mass and the -nucleon coupling
constant
In all case
In case 1 and case 2,
The point A
DLFP
(=0)
(=)
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23. the possible interplay between the nucleon mass and the -nucleon coupling
constant
decreasing slowly
In all case
In case 1 and case 2,
The point A
DLFP
(=0)
(=)
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24. the possible interplay between the nucleon mass and the -nucleon coupling
constant
decreasing slowly
In all case
In case 1 and case 2,
In case 3,
The point A
DLFP
(=0)
(=)
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25. the possible interplay between the nucleon mass and the -nucleon coupling
constant
decreasing slowly
decreasing fast
In all case
In case 1 and case 2,
In case 3,
The point A
DLFP
(=0)
(=)
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26. Fixed Point
From the RGEs, we get the fixed point given as
It depends on the -nucleon coupling whether DLFP is the infrared fixed point or not. If the -nucleon coupling is scaled to be zero approaching to DLFP, DLFP will be evaluated as the infrared fixed point. And the scaling speed of the -nucleon coupling in medium will be different from the speed of - nucleon coupling.
From the RGEs, we find that RGEs shows the possible interplay between the nucleon mass and the -nucleon coupling. We will study this interplay by studying dilaton- implemented Hidden Local Symmetric Largrangian.
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27. Dilaton implemented Hidden Local Symmetric Lagrangian
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28. Dilaton implemented Hidden Local Symmetric Lagrangian
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29. Mean field calculation
  ,
a  30,
  0,
We do the mean field calculation in standard assignment because we didn’t yet do the calculation in mirror assignment. But, the interplay between the nucleon mass and the -nucleon coupling in mirror assignment will be similar with that in standard assignment.
In mean field calculation, we assume that the fluctuation of the fields are small, so we take its vacuum expectation value of the fields in the nuclear matter. In this calculation, we also consider a static, uniform matter so that all derivatives of the fields drop out and the spatial part of the fields become zero. The charge conservation of the nuclear matter is assumed, which makes the only third component of the iso-vector fields survive. Then, we have
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30. The interplay between the nucleon mass and the omega-nucleon coupling in mean field calculation*
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31. The interplay between the nucleon mass and the omega-nucleon coupling in mean field calculation*
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32. The interplay between the nucleon mass and the omega-nucleon coupling in mean field calculation*
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33. The interplay between the nucleon mass and the omega-nucleon coupling in mean field calculation*
when (gV*-1) = 0.
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34. The interplay between the nucleon mass and the omega-nucleon coupling in mean field calculation
which parameter sets are given to produce the binding energy of dense nuclear matter to be -16 MeV and the pressure to be zero at the normal nuclear density.
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35. Thank you
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36. Back up: Vacuum expectation value of the dilaton
From Coleman-Weinberg-type dilaton potential, we obtain the vacuum expectation value of  as or
by using the equation,
in matter-free space, and or
by using the equation,
in baryonic matter, where `*’ means medium effect and , T,  are the thermodynamic potential, the temperature and chemical potential of the baryonic matter.
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37. Back up: Vacuum expectation value of the dilaton
Case (1)
Case (2)
When the scale symmetry is restored,    0, the phase is changed from case (1) to case (2).
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38. Back up: At tree level
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39. Back up: At tree level
C. Sasaki, H. K. Lee, W.-G. Paeng, M. Rho (2011)
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40. Back up: At tree level
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41. Back up: At tree level
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42. Back up: At tree level
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43. Back up: one loop calculation
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44. Back up: one loop calculation
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45. Its application to the dense nuclear matter
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46. Its application to the dense nuclear matter
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47. Summary and discussion
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