Extended Linear Sigma Model with (three-flavor) Baryons Phys. Rev. D 93 (2016) 034021 Lisa Olbrich 1 enyi 2 , Francesco Giacosa 1 , 3 , and Dirk H. Rischke 1 in collaboration with Mikl´ os Z´ et´ 1 Institute for Theoretical Physics, Goethe University, Frankfurt am Main, Germany 2 Wigner Research Center for Physics, Budapest, Hungary 3 Institute of Physics, Jan Kochanowski University, Kielce, Poland Chiral Group Meeting, May 30 th 2016
Introduction The Model Results Conclusions and Outlook Outline 1 Introduction 2 The Model 3 Results 4 Conclusions and Outlook Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook QCD - perturbative approach fails in low-energy regime Quantum Chromodynamics q ( x ) − 1 � � iγ µ D µ − m 2 Tr( G µν G µν ) L QCD = ¯ q ( x ) • only a few parameters • but it is not analytically solvable. 5 4.5 4 strong coupling constant 3.5 Perturbative approach fails 3 2.5 in the low-energy regime. 2 1.5 0 1000 2000 3000 4000 5000 energy scale [MeV] Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Extended Linear Sigma Model (meson part) Extended Linear Sigma Model �� � 2 � �� 2 � � � � � � ( D µ Φ) † D µ Φ) − m 2 Φ † Φ Φ † Φ Φ † Φ L meson = Tr 0 Tr − λ 1 Tr − λ 2 Tr �� m 2 − 1 � � 4 Tr ( L µν L µν + R µν R µν ) + Tr ( L µ L µ + R µ R µ ) 1 + ∆ 2 det Φ − det Φ † � 2 � � Φ + Φ † �� � + Tr H + c 1 + i g 2 Tr ( L µν [ L µ , L ν ]) + Tr ( R µν [ R µ , R ν ]) � � 2 + h 1 � � Tr ( L µ L µ + R µ R µ ) + h 2 Tr � � ( L µ Φ) † ( L µ Φ) + (Φ R µ ) † (Φ R µ ) Φ † Φ 2 Tr � Φ R µ Φ † L µ � + g 3 [Tr ( L µ L ν L µ L ν } + Tr { R µ R ν R µ R ν )] + 2 h 3 Tr + g 4 [Tr ( L µ L µ L ν L ν ) + Tr ( R µ R µ R ν R ν )] + g 5 Tr ( L µ L µ ) Tr ( R ν R ν ) + g 6 [Tr ( L µ L µ ) Tr ( L ν L ν ) + Tr ( R µ R µ ) Tr ( R ν R ν )] • exhibits the same symmetries as QCD • a lot more parameters • but good results already at tree level D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa and D. H. Rischke, Phys. Rev. D 87 (2013) 014011 S. Janowski, F. Giacosa and D. H. Rischke, Phys. Rev. D 90 (2014) 11, 114005 Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Outline 1 Introduction 2 The Model 3 Results 4 Conclusions and Outlook Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook The inclusion of baryons with strangeness L. Olbrich, M. Z´ et´ enyi, F. Giacosa, and D. H. Rischke Phys. Rev. D 93 , 034021 (2016) [arXiv:1511.05035 [hep-ph]] Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Quark-Diquark picture Baryonic fields as quark-diquark states Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Quark-Diquark picture Baryonic fields as quark-diquark states u uds uus uud ([ d, s ] , [ s, u ] , [ u, d ]) = ˆ d dds uds udd � �� � s dss uss uds diquark � �� � quark Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Quark-Diquark picture Baryonic fields as quark-diquark states u uds uus uud ([ d, s ] , [ s, u ] , [ u, d ]) = ˆ d dds uds udd � �� � s dss uss uds diquark � �� � quark 6 + Σ 0 Λ Σ + p √ √ 2 6 − Σ 0 Λ Σ − ∼ n √ √ 2 Ξ − Ξ 0 − 2Λ √ 6 Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Quark-Diquark picture Baryonic fields as quark-diquark states u uds uus uud ([ d, s ] , [ s, u ] , [ u, d ]) = ˆ d dds uds udd � �� � s dss uss uds diquark � �� � quark 6 + Σ 0 Λ Σ + p √ √ 2 6 − Σ 0 Λ Σ − ∼ n √ √ 2 Ξ − Ξ 0 − 2Λ √ 6 � N 1 L ∼ D R q L , N 2 L ∼ D L q L N 1 R ∼ D R q R , N 2 R ∼ D L q R Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Mirror assignment Chiral transformation – mirror assignment C. E. DeTar and T. Kunihiro, Phys. Rev. D 39 (1989) 2805 Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Mirror assignment Chiral transformation – mirror assignment M 1 L ∼ D R γ µ ∂ µ q L M 2 L ∼ D L γ µ ∂ µ q L M 1 R ∼ D R γ µ ∂ µ q R M 2 R ∼ D L γ µ ∂ µ q R C. E. DeTar and T. Kunihiro, Phys. Rev. D 39 (1989) 2805 Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Mirror assignment Chiral transformation – mirror assignment Allows for chirally invariant mass terms within the Lagrangian. M 1 L ∼ D R γ µ ∂ µ q L M 2 L ∼ D L γ µ ∂ µ q L M 1 R ∼ D R γ µ ∂ µ q R M 2 R ∼ D L γ µ ∂ µ q R C. E. DeTar and T. Kunihiro, Phys. Rev. D 39 (1989) 2805 Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Octet baryons with three flavors Quark-diquark picture + mirror assignment → Four baryonic spin- 1 2 multiplets Λ(1116) Σ(1193) Ξ(1338) N ( 939 ) Λ(1600) Σ(1660) Ξ(1690) N ( 1440 ) Λ(1670) Σ(1620) Ξ(?) N ( 1535 ) Λ(1800) Σ(1750) Ξ(?) N ( 1650 ) Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Evaluation of the Lagrangian The Lagrangian ( N f = 3 ) L Nf =3 = Tr � ¯ N 1 L iγ µ D µ 2 L N 1 L + ¯ N 1 R iγ µ D µ 1 R N 1 R + ¯ N 2 L iγ µ D µ 1 L N 2 L + ¯ N 2 R iγ µ D µ � 2 R N 2 R + Tr � ¯ M 1 L iγ µ D µ M 1 R iγ µ D µ M 2 L iγ µ D µ M 2 R iγ µ D µ 4 R M 1 L + ¯ 3 L M 1 R + ¯ 3 R M 2 L + ¯ � 4 L M 2 R � � N 1 L Φ N 1 R + ¯ ¯ N 1 R Φ † N 1 L + ¯ N 2 L Φ N 2 R + ¯ N 2 R Φ † N 2 L − g N Tr � � M 1 L Φ † M 1 R + ¯ ¯ M 1 R Φ M 1 L + ¯ M 2 L Φ † M 2 R + ¯ − g M Tr M 2 R Φ M 2 L − m 0 , 1 Tr � ¯ N 1 L M 1 R + ¯ M 1 R N 1 L + ¯ N 2 R M 2 L + ¯ � M 2 L N 2 R − m 0 , 2 Tr � ¯ N 1 R M 1 L + ¯ M 1 L N 1 R + ¯ N 2 L M 2 R + ¯ � M 2 R N 2 L � N 2 L Φ N 1 R Φ † � � N 2 R Φ † N 1 L Φ † � N 1 R Φ † N 2 L Φ + ¯ ¯ − κ ′ N 1 L Φ N 2 R Φ + ¯ ¯ − κ 1 Tr 1 Tr � M 2 L Φ † M 1 R Φ † � � M 2 R Φ M 1 L Φ † � M 1 R Φ M 2 L Φ + ¯ ¯ − κ ′ M 1 L Φ † M 2 R Φ + ¯ ¯ − κ 2 Tr 2 Tr Tr � ¯ � � N 2 R Φ † � � N 1 L Φ † �� ¯ N 1 L Φ � − ǫ 1 Tr { N 2 R Φ } + Tr Tr Tr � ¯ � � M 2 L Φ † � � M 1 R Φ † �� ¯ M 1 R Φ � − ǫ 2 Tr { M 2 L Φ } + Tr Tr Tr � ¯ Φ † Φ � − ǫ 3 Tr � N 1 L M 1 R + ¯ M 1 R N 1 L + ¯ N 2 R M 2 L + ¯ � M 2 L N 2 R Tr � ¯ − ǫ 4 Tr � Φ † Φ � N 1 R M 1 L + ¯ M 1 L N 1 R + ¯ N 2 L M 2 R + ¯ � M 2 R N 2 L L. Olbrich, M. Z´ et´ enyi, F. Giacosa, and D. H. Rischke Phys. Rev. D 93 , 034021 (2016) Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
Introduction The Model Results Conclusions and Outlook Evaluation of the Lagrangian The Lagrangian ( N f = 3 ) L Nf =3 = Tr � ¯ N 1 L iγ µ D µ 2 L N 1 L + ¯ N 1 R iγ µ D µ 1 R N 1 R + ¯ N 2 L iγ µ D µ 1 L N 2 L + ¯ N 2 R iγ µ D µ � 2 R N 2 R + Tr � ¯ M 1 L iγ µ D µ M 1 R iγ µ D µ M 2 L iγ µ D µ M 2 R iγ µ D µ 4 R M 1 L + ¯ 3 L M 1 R + ¯ 3 R M 2 L + ¯ � 4 L M 2 R � � N 1 L Φ N 1 R + ¯ ¯ N 1 R Φ † N 1 L + ¯ N 2 L Φ N 2 R + ¯ N 2 R Φ † N 2 L − g N Tr � � M 1 L Φ † M 1 R + ¯ ¯ M 1 R Φ M 1 L + ¯ M 2 L Φ † M 2 R + ¯ − g M Tr M 2 R Φ M 2 L − m 0 , 1 Tr � ¯ N 1 L M 1 R + ¯ M 1 R N 1 L + ¯ N 2 R M 2 L + ¯ � M 2 L N 2 R − m 0 , 2 Tr � ¯ N 1 R M 1 L + ¯ M 1 L N 1 R + ¯ N 2 L M 2 R + ¯ � M 2 R N 2 L � N 2 L Φ N 1 R Φ † � � N 2 R Φ † N 1 L Φ † � N 1 R Φ † N 2 L Φ + ¯ ¯ − κ ′ N 1 L Φ N 2 R Φ + ¯ ¯ − κ 1 Tr 1 Tr � M 2 L Φ † M 1 R Φ † � � M 2 R Φ M 1 L Φ † � M 1 R Φ M 2 L Φ + ¯ ¯ − κ ′ M 1 L Φ † M 2 R Φ + ¯ ¯ − κ 2 Tr 2 Tr Tr � ¯ � � N 2 R Φ † � � N 1 L Φ † �� ¯ N 1 L Φ � − ǫ 1 Tr { N 2 R Φ } + Tr Tr Tr � ¯ � � M 2 L Φ † � � M 1 R Φ † �� ¯ M 1 R Φ � − ǫ 2 Tr { M 2 L Φ } + Tr Tr Tr � ¯ Φ † Φ � − ǫ 3 Tr � N 1 L M 1 R + ¯ M 1 R N 1 L + ¯ N 2 R M 2 L + ¯ � M 2 L N 2 R Tr � ¯ − ǫ 4 Tr � Φ † Φ � N 1 R M 1 L + ¯ M 1 L N 1 R + ¯ N 2 L M 2 R + ¯ � M 2 R N 2 L L. Olbrich, M. Z´ et´ enyi, F. Giacosa, and D. H. Rischke Phys. Rev. D 93 , 034021 (2016) Extended Linear Sigma Model with (three-flavor) Baryons Lisa Olbrich
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