z nuclear matter with three flavor extended linear sigma model - - PowerPoint PPT Presentation

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Mass Modifications of light mesons in nuclear matter with three flavor extended linear sigma model

Daiki Suenaga

(Central China Normal University) Phillip Lakaschus Dirk H. Rischke, Jurgen Schaffner-Bielich (Johann Wolfgang Goethe University)

2/20

・Mesons in nuclear matter

• 1. Introduction

related to… (Theoretically) Partial restoration of chiral symmetry, Change of axial anomaly, etc… (Experimentally) Modification of vector mesons experiments Mesic nuclei, etc… comprehensively within a chiral model in a unified way

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・Extended Linear Sigma Model (eLSM)

The extended Linear Sigma Model (eLSM) is a chiral model which can reproduce light meson properties successfully

• 1. Introduction

z

• D. Parganlija, P. Kovacs, Gy. Wolf, F. Giacosa, and D. H. Rischke;

PRD 87, 014011 (2013)

We applied the eLSM into cold nuclear matter reproducing saturation properties and studied mass changes of the mesons in nuclear matter comprehensively Includes 9 scalars, 9 pseudo-scalars9 vectors, and 9 axial-vectors (and 1 dilaton)

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・The eLSM

• 2. eLSM

(2 quark state) meson field is schematically written by

Φij ∼ ¯ qj

Rqi L

Lij

µ ∼ ¯

qj

Lγµqi L

Rij

µ ∼ ¯

qj

Rγµqi R

• chiral transformation laws and particle assignment

SU(3)L × SU(3)R

Φ → gLΦg†

R

Lµ → gLLµg†

L

Rµ → gRRµg†

R

z

• D. Parganlija, P. Kovacs, Gy. Wolf, F. Giacosa,

and D. H. Rischke; PRD 87, 014011 (2013)

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・Lagrangian

DµΦ = ∂µΦ − ig1(LµΦ − ΦRµ) − ieAµ[T3, Φ] Lµν = ∂µLν − ieAµ[T3, Lν] − ∂νLµ + ieAν[T3, Lµ]

Rµν = ∂µRν − ieAµ[T3, Rν] − ∂νRµ + ieAν[T3, Rµ]

The Lagrangian is given by

• 2. eLSM
• Axial anomaly

z

• D. Parganlija, P. Kovacs, Gy. Wolf, F. Giacosa,

and D. H. Rischke; PRD 87, 014011 (2013)

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・Input parameters

• 2. eLSM

z

• D. Parganlija, P. Kovacs, Gy. Wolf, F. Giacosa,

and D. H. Rischke; PRD 87, 014011 (2013)

7/20

• ・Input parameters
• 2. eLSM
• z
• D. Parganlija, P. Kovacs, Gy. Wolf, F. Giacosa,

and D. H. Rischke; PRD 87, 014011 (2013)

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・ fitting

χ2

C1 = m2

0 + λ1(φ2 N + φ2 S)

C2 = m2

1 + h1

2 (φ2

N + φ2 S)

z

• D. Parganlija, P. Kovacs, Gy. Wolf, F. Giacosa,

and D. H. Rischke; PRD 87, 014011 (2013)

• 2. eLSM

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・Notes

z

• D. Parganlija, P. Kovacs, Gy. Wolf, F. Giacosa,

and D. H. Rischke; PRD 87, 014011 (2013)

• 2. eLSM

λ1, h1

• λ1, h1

(f0)

f0

• mesons are not used as input since they include many uncertainties

10/20

We studied meson mass changes in nuclear matter without fixing scalar-isoscalar mesons (without fixing )

・Applying the eLSM into nuclear matter

• 3. eLSM in nuclear matter

λ1, h1

nucleon interactions

ψ1l(2r) → gLψ1l(2r) ψ1r(2l) → gRψ1r(2l)

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・Applying the eLSM into nuclear matter

nucleon interactions

ψ1l(2r) → gLψ1l(2r) ψ1r(2l) → gRψ1r(2l)

• terms are included by axial anomaly to fit

decay

k1, k2

• L. Olbrich, M. Zétényi, F. Giacosa and D. H. Rischke;

PRD 97, 014007 (2018)

N ∗(1535) → Nη

• 3. eLSM in nuclear matter

We studied meson mass changes in nuclear matter without fixing scalar-isoscalar mesons (without fixing )

λ1, h1

12/20

• mass formulae are given by

・Construction of nuclear matter

N(939), N ∗(1535)

m± = 1 2 2 4 s (G1 + G2)2φ2

N + 4

✓ M0 + k1 2 φ2

N

◆2 ⌥ (G2 G1)φN 3 5

The nuclear matter is constructed by fermion one loops with the eLSM and parity doublet model with the following input parameters

ΓN ∗→Nη = 65 MeV mN = 939 MeV mN ∗ = 1535 MeV

gρNN = gV 2 = 2.5

[1] S. L. Zhu, Phys. Rev. C 59, 435 (1999)

[1]

ΣπN = 45 MeV E/A = −16 MeV

∂ ∂ρB (E/A − mvac

+ )|ρ0 = 0

ρ0 = 0.16 fm−3

• are remained as free parameters

M0, k1

• 3. eLSM in nuclear matter

m± → M0

(φN → 0)

hΦi0 = diag ✓φN 2 , φN 2 , φS p 2 ◆

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・Meson masses in nuclear matter

m∗2

X = Z∗2 X

⇣ ¯ m2

X + ΠX( ¯

mX,~ 0) ⌘

¯ mX Z∗

X

ΠX(q0, ~ q)

: mean field level : renormalization factor : one loop corrections

• 3. eLSM in nuclear matter

masses in nuclear matter to satisfy the gap equation in nuclear matter formula in the present analysis

• D. Suenaga, M. Harada, and S. Yasui PRC96 (2017)

× mean field

N, N ∗

+

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・Results

0.0 0.5 1.0 1.5 2.0 0.6 0.8 1.0 1.2 1.4 ρ/ρ0 Mass [GeV]

0.0 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ρ/ρ0 Mass [GeV]

spin 1 spin 0

π

f L

K η η0 a0

K∗

f H

ρ ω K∗ φ a1 K1

f1S M0 = 500 MeV ˜ k1 = 0 (˜

k1 = k1fπ)

• 3. eLSM in nuclear matter

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・Results

0.0 0.5 1.0 1.5 2.0 0.6 0.8 1.0 1.2 1.4 ρ/ρ0 Mass [GeV] 0.0 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ρ/ρ0 Mass [GeV]

˜ k1 = −0.4

• 3. eLSM in nuclear matter

spin 1 spin 0

π

f L

K η η0 a0

K∗

f H

ρ ω K∗ φ a1 K1

f1S M0 = 500 MeV

(˜ k1 = k1fπ)

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0.0 0.5 1.0 1.5 2.0 0.4 0.6 0.8 1.0 1.2 1.4 ρ/ρ0 Mass [GeV] 0.0 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ρ/ρ0 Mass [GeV]

M0 = 900 MeV

・Results

• 3. eLSM in nuclear matter

˜ k1 = 0 (˜

k1 = k1fπ)

spin 1 spin 0

π

f L

K η η0 a0

K∗

f H

ρ ω K∗ φ a1 K1

f1S

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˜ k1 = −0.5

0.0 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ρ/ρ0 Mass [GeV] 0.0 0.5 1.0 1.5 2.0 0.6 0.8 1.0 1.2 1.4 ρ/ρ0 Mass [GeV]

・Results

• 3. eLSM in nuclear matter

M0 = 900 MeV

(˜ k1 = k1fπ)

spin 1 spin 0

π

f L

K η η0 a0

K∗

f H

ρ ω K∗ φ a1 K1

f1S

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・Output parameters

The values of is small

• 3. eLSM in nuclear matter

˜ k1 = k1fπ

• |˜

k1| . 0.5

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The change of anomaly in nuclear matter is small

19/20

・Output parameters

The values of are large in contrast to larges counting

λ1, h1

Nc

• 3. eLSM in nuclear matter

ccordingly, the mass of becomes very small (What is happened if 4 quark state is included?)

f0L

20/20

・Conclusions

matter and studied changes of light meson masses comprehensively the axial anomaly is not changed at nuclear matter so much include four quark states

k1

• 4. Conclusions

21/20

Back up

22/20

・extended Linear Sigma Model (eLSM)

• g1

a1 π

L = −g1φN∂µπaaaµ

1 + m2 a1

2 aa

1µaaµ 1 + 1

2∂µπa∂πa + · · · = m2

a1

2 ✓ aa

1µ − g1φN

m2

a1

∂µπa ◆2 + 1 2 ✓ 1 − g2

1φ2 N

m2

a1

◆ ∂µπa∂πa + · · ·

=

a1

redefinition of

=

Renormalization factor

hΦi0 = diag

✓φN 2 , φN 2 , φS p 2 ◆

h0N = m2

0φN + λ1(φ2 N + φ2 S)φN + λ2

2 φ3

N

h0S = m2

0φS + λ1(φ2 N + φ2 S)φS + λ2φ3 S

23/20

・Mass formula

24/20

・Mass formula