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Baryon Parity Doublets and Chiral Spin Symmetry PHYSICAL REVIEW D 98, 014030 (2018) Baryon parity doublets and chiral spin symmetry M. Catillo and L. Ya. Glozman Institute of Physics, University of Graz, 8010 Graz, Austria (Received 24 April 2018; published 25 July 2018) The chirally symmetric baryon parity-doublet model can be used as an effective description of the baryon-like objects in the chirally symmetric phase of QCD. Recently it has been found that above the critical temperature, higher chiral spin symmetries emerge in QCD. It is demonstrated here that the baryon parity-doublet Lagrangian is manifestly chiral spin invariant. We construct nucleon interpolators with fixed chiral spin transformation properties that can be used in lattice studies at high T. DOI: 10.1103/PhysRevD.98.014030 I. BARYON PARITY DOUBLETS. fermions of opposite parity, parity doublets, that transform INTRODUCTION into each other upon a chiral transformation [1]. Consider a pair of the isodoublet fermion fields A Dirac Lagrangian of a massless fermion field is chirally symmetric since the left- and right-handed com- Ψ ponents of the fermion field are decoupled, Ψ ¼ þ ; ð4Þ Ψ− μ μ μ L iψγ¯ μ∂ ψ iψ¯ Lγμ∂ ψ L iψ¯ Rγμ∂ ψ R; 1 ¼ ¼ þ ð Þ Ψ Ψ where the Dirac bispinors þ and − have positive and negative parity, respectively. The parity doublet above is a where spinor constructed from two Dirac bispinors and contains eight components. Note that there is, in addition, an isospin 1 1 ψ R ¼ ð1 þ γ5Þψ; ψ L ¼ ð1 − γ5Þψ: ð2Þ index which is suppressed. Given that the right- and left- 2 2 handed fields are directly connected with the opposite parity fields The SUð2ÞL × SUð2ÞR chiral symmetry is a symmetry upon independent isospin rotations of the left- and right- 1 1 ΨR ¼ pffiffiffi ðΨ þ Ψ−Þ; ΨL ¼ pffiffiffi ðΨ − Ψ−Þ; ð5Þ handed components of fermions, 2 þ 2 þ a a a a θRτ θLτ the vectorial and axial parts of the chiral transformation ψ R → exp { ψ R; ψ L → exp { ψ L; ð3Þ 2 2 law under the ð0; 1=2Þ ⊕ ð1=2; 0Þ representation of SUð2ÞL × SUð2ÞR is a a where τ are the isospin Pauli matrices and the angles θR a a a and θ parametrize rotations of the right- and left-handed θVτ L Ψ → exp { ⊗ 1 Ψ; components. The transformation (3) defines the ð0; 1=2Þ ⊕ 2 ð1=2; 0Þ representation of the chiral group. θa τa Ψ → { A ⊗ σ Ψ: The mass term in the Lagrangian breaks explicitly the exp 2 1 ð6Þ chiral symmetry since it couples the left- and right-handed fermions. Here σi is a Pauli matrix that acts in the space of the parity It was known for a long time that it is possible to doublet. In the chiral transformation law (3) the axial construct a chirally symmetric Lagrangian for a massive rotation mixes the massless Dirac spinor ψ with γ5ψ, while fermion field provided that there are mass-degenerate in the chiral rotation of the parity doublet, a mixing of two Ψ Ψ independent fields þ and − is provided. Then the chiral- invariant Lagrangian of the free parity doublet is given as Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. ¯ μ ¯ Further distribution of this work must maintain attribution to L ¼ iΨγ ∂μΨ − mΨΨ the author(s) and the published article’s title, journal citation, μ μ 3 iΨ¯ γ ∂ Ψ iΨ¯ γ ∂ Ψ − mΨ¯ Ψ −mΨ¯ Ψ : and DOI. Funded by SCOAP . ¼ þ μ þ þ − μ − þ þ − − ð7Þ 2470-0010=2018=98(1)=014030(6) 014030-1 Published by the American Physical Society M. CATILLO and L. YA. GLOZMAN PHYS. REV. D 98, 014030 (2018) Note that this Lagrangian can be written in different II. CHIRAL SPIN SYMMETRY equivalent forms [2,3]. The equivalence of the present The SUð2ÞCS chiral spin transformations and generators form [1] with that one of Ref. [3] was demonstrated in [8,9], defined in the Dirac quark spinor space are Ref. [4] and the chiral transformation law (6) corresponds “ ” ’ to the mirror assignment of Ref. [3]. For the reader s εnΣn convenience we derive this equivalence in the Appendix. ψ → ψ 0 i ψ; ¼ exp 2 ð9Þ This Lagrangian can also be equivalently written in terms of the right- and left-handed fields of Eq. (5), n Σ ¼fγk; −iγ5γk; γ5g; ð10Þ L iΨ¯ γμ∂ Ψ iΨ¯ γμ∂ Ψ −mΨ¯ Ψ −mΨ¯ Ψ : ¼ L μ L þ R μ R L L R R ð8Þ n ¼ 1, 2, 3. Here γk, k ¼ 1, 2, 3, 4, is any Hermitian Euclidean gamma matrix, obeying the following anticom- The latter form demonstrates that the right- and left-handed mutation relations: degrees of freedom are completely decoupled and the γ γ γ γ 2δij; γ γ γ γ γ : Lagrangian is manifestly chiral invariant. Since the latter i j þ j i ¼ 5 ¼ 1 2 3 4 ð11Þ Lagrangian is equivalent to the Lagrangian (7) it is also manifestly Lorentz invariant [in order to avoid confusion Different k defines different four-dimensional representa- 2 note that ΨL and ΨR for the parity doublet are defined tions that can be reduced into irreducible ones of dim ¼ . a b abc c differently from the massless Dirac bispinor (2)]. The suð2Þ algebra ½Σ ; Σ ¼2iϵ Σ is satisfied with any A crucial property of the Lagrangian (7)–(8) is that the Euclidean gamma matrix. Ψ Ψ U 1 is a subgroup of SU 2 . The SU 2 trans- fermions þ and − are exactly degenerate and have a ð ÞA ð ÞCS ð ÞCS nonzero chiral-invariant mass m. This Lagrangian has a formations mix the left- and right-handed fermions. The number of peculiarities. In particular, the diagonal axial free massless quark Lagrangian (1) does not have this Ψ Ψ symmetry. charge of the fermions − and þ vanishes, while the off- diagonal axial charge is 1. An extension of the direct SUð2ÞCS × SUðNFÞ product Supplemented by the interactions with the pion and leads to a SUð2NFÞ group. This group contains the chiral sigma fields [2,3] this Lagrangian has been sometimes used symmetry of QCD SUðNFÞL × SUðNFÞR × Uð1ÞA as a in baryon spectroscopy [5] and as an effective model for the subgroup. Its transformations are given by chiral symmetry restoration scenario at high temperature or ϵmTm density where baryons with the nonzero mass do not vanish ψ → ψ 0 ¼ exp i ψ; ð12Þ upon a chiral restoration [6]. 2 Recently it was found on the lattice [7] that at temper- 2 2 atures above the critical one new symmetries emerge in where m ¼ 1; 2; …; ð2NFÞ − 1 and the set of ð2NFÞ − 1 QCD, SUð2ÞCS, and SUð2NFÞ [8,9], which implies a generators is modification of the existing view on the nature of the a n a n strongly interacting matter at high temperatures [10].In fðτ ⊗ 1DÞ; ð1F ⊗ Σ Þ; ðτ ⊗ Σ Þg ð13Þ particular, the elementary objects at high temperatures are quarks with a definite chirality connected by the chromo- with τ being the flavor generators with the flavor index a electric field; i.e., there are no free deconfined quarks. and n ¼ 1, 2, 3 is the SUð2ÞCS index. These symmetries were observed earlier upon artificial The fundamental vector of SUð2NFÞ at NF ¼ 2 is truncation of the near-zero modes of the Dirac operator at 0 1 – u zero temperature [11 14]; for a recent theoretical develop- R SU 2 ⊃ U 1 SU 2N ⊃ ment see [15]. The ð ÞCS ð ÞA and ð FÞ B u C B L C SUðNFÞL × SUðNFÞR × Uð1ÞA symmetries are sym- ψ ¼ B C: ð14Þ @ d A metries of the chromo-electric interaction in QCD. In R SU 2 d addition to the chiral transformations, the ð ÞCS and L SUð2NFÞ rotations mix the left- and right-handed compo- nents of quark fields. The chromo-magnetic interaction as The SUð2NFÞ transformations mix both flavor and well as the quark kinetic term break these symmetries down chirality. to SUðNFÞL × SUðNFÞR × Uð1ÞA. While the SUð2ÞCS and SUð2NFÞ symmetries are not Given the observed new symmetries in high T QCD, a symmetries of the QCD Lagrangian as a whole, they are question arises of whether the parity-doublet model could symmetries of the fermion charge operator and of the still be used as an effective description for the baryon-like chromo-electric interaction in QCD. The chromo-magnetic objects at high T. Here we demonstrate that this model is interaction as well as the quark kinetic term break these actually manifestly SUð2ÞCS and SUð2NFÞ invariant. symmetries. 014030-2 BARYON PARITY DOUBLETS AND CHIRAL SPIN SYMMETRY PHYS. REV. D 98, 014030 (2018) III. CHIRAL SPIN SYMMETRY OF THE in addition, SUð2ÞCS and SUð2NFÞ invariant with the PARITY DOUBLETS generators of SUð2NFÞ being The chiral spin transformation (9) and generators (10) fðτa ⊗ 1Þ; ð1 ⊗ σnÞ; ðτa ⊗ σnÞg: ð22Þ at k ¼ 4 (the Euclidean γ4 matrix coincides with the Minkowskian γ0 matrix) can be presented in an equivalent form, as follows below. We will use the chiral representa- IV. CHIRAL SPIN SYMMETRIC NUCLEON INTERPOLATORS tion of the γ matrices. Then the SUð2ÞCS generators for the k 4 representation with ¼ are Here we construct nucleon three-quark interpolators that transform upon SUð2ÞCS representations (8)–(9) Σn 1 ⊗ σ1; 1 ⊗ σ2; 1 ⊗ σ3 : ¼f g ð15Þ with k ¼ 4. These interpolators should be used in lattice studies to establish the chiral spin symmetry and i Here 1 is the unit 2 × 2 matrix and the Pauli matrices σ act consequently to justify the parity-doublet model as an in the space of spinors effective description of the baryon-like objects at high temperatures.
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