Pion decay constant in Dense Skyrmion Matter

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Pion decay constant in Dense Skyrmion Matter
Hee-Jung Lee ( Chungbuk Natl Univ.)
Collaborators Byung-Yoon Park (Chungnam Natl
Univ.) Mannque Rho (Saclay) Vicente Vento (Univ.
of Valencia) Dong-Pil Min (Seoul Natl Univ.)
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Contents
Overview SU(2) Skyrme Model Pion decay
constant in Skyrmion Matter Results
Summary and discussion
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Overview
Properties of Pion in free space
Properties of Pion in Nuclear Matter
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1. Chiral Perturbation Theory -
H. Yabu et al, PRD 50 (1994) 3549. -
V. Thorsson et al, NPA 589 (1995) 633.
- T. S. Park et al, PLB 409 (1997) 26.
- U. Meissner et al, Ann.Phys297(2002)27.
Fermi Gas Mean Field Approx.
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2. Optical Potential Theory - T. Waas et al,
PLB 405 (1997) - T. Yamazaki et al, PLB 418
(1998) S-wave pion- nucleon potential of
the deeply bound states in Pb nucleus
observed at GSI
Pion mass increases - a 7 at the
center Pb nucleus - a 3 in symmetric
nuclear matt.
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SU(2) Skymion Model Approach
- Baryons are appeared as static solitons of
the Skyrme Lagrangian. - The pion-matter
interactions are generated from the same
Skyrme Lagrangian as classical potentials. -
The results contain higher orders in density.
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SU(2) Skymion Model
1. The Model Lagrangian
2. Hedgehog Ansatz Single Baryon by topology
Baryon number density
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3. Skyrmion Matter Skyrmion Crystal - I.
Klevanov, NPB 262(1985) 133 - M. Kugler et
al, PRD 40 (1989) 3421.
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4. Pions in Skyrmion Matter The dynamics
of the pion in dense skyrmion matter. - S.
Saito et al, Prog. Theor. Phys. 75 (1986) 68.
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- Lagrangian
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Classical potentials
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- The Pion Effective Mass ( )
Pole equation of the Pion Propagator
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- The in medium decay constant ( )
PCAC Relation
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- Results
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At high den., the Chiral symm. is restored.
Trace (Scale) Anomaly in QCD
Introduce a scalar field into the Skyme theory

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QCD Trace Anomaly

• The classical QCD action is scale-invariant in
  the chiral limit under
• At quantum level, this invariance becomes broken
  even for the chiral limit.
• The dilatation current is not conserved

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Modified Skyrme Lagrangian - J. Ellis and
J. Lanik, PLB150 (1985) 289 - G. E. Brown and
M. Rho, PRL 66 (1991) 2720
The scalar field ( ) with unity scale dim. and
the potential energy generate the trace anomaly
in this effective Lagrangian -
Furnstahl, et al, PRC 52 (1995) 1368
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! If one interpret the scalar field as a pure
glueball, the gluon condensation gives the mass
of the scalar field as

• 1. Set A R. J. Furnstahl, et al, PRC 52 (1995)
  1368
• 2. Set B C. Song, et al, PRC 56 (1997) 2244
• 3. Set C M. S. Shifman, et al, NPB 147 (1979)
  385

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Energy per baryon
Potential Energy for the scalar field
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Energy per single skyrmion as a function of the
scalar field X for a given lattice size with the
parameter sets B.
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Phases of skyrmion matter
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Pions and Scalar in Skyrmion Matter
Fluctuations on top of the static skyrmion
crystal
Lagrangian for pions and the scalar field
Lagrangian for the static background skyrmion
matter
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Lagrangian for pions on the matter
Lagrangian for the scalar field on the matter
Lagrangian for the interaction between pions
and the scalar field on the matter
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Pion decay constants
External axial vector current
Axial vector current and vector current of SM
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Pion decay constants
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Pion velocity
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Pion propagator in medium


for massless pion
Pion velocity in medium
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Pion velocity in medium
H.-J. Lee, B.-Y. Park, M. Rho, V. Vento, Nucl.
Phys. A (2004)
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Relation between two kinds of constants
For massless pion
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Summary and ..

• Phase transition with respect to chiral symmetry
• happens with density in skyrmion matter.
• Mass and decay constants of pion decrease
• with density of skyrmion matter.
• 3. Velocity of pion decreases to minimum and
• returns the value in free space when chiral
  sym.
• is restored.
• 4. Relation in two kinds of constants for
  massive pion?
• This approach would be too simple to be a
• real physical situation.
• need to quantize skyrmions in skyrmion
  matter.
• 6. Heating skyrmion matter, what happens?

Thank you!!