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 components 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.

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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. R The quark bispinor, divided into its R and L compo- ; ð16Þ L nents, is qR where R and L represent the upper and lower components q ; ¼ L ð23Þ of the right- and left-handed Dirac bispinors (2) q q u d 2 R where ¼ or . It transforms under the representation SU 2 ψ R ¼ ; ð17Þ of ð ÞCS. Therefore, the nucleon three-quark interpola- 0 tors fall into the representation 2 ⊗ 2 ⊗ 2 ¼ 21 ⊕ 22 ⊕ 4. Using the standard technique of Young tableaux for 0 representations of the permutation group ψ L ¼ : ð18Þ L

The SUð2ÞCS;k¼ 4 transformation (9) can then be rewritten as ð24Þ εnΣn εnσn R ψ → ψ 0 ¼ exp i ψ ¼ exp i : ð19Þ 2 2 L α β γ L R uL where indices , , denote either or , and, e.g., as is a The latter equation emphasizes that a reduction of the the left part of the up quark with the color index and the spin index s, we get representations (9) of SUð2ÞCS leads to a two-dimensional irreducible representation. Now we return to the parity-doublet Lagrangian. Given Eq. (5), the parity doublet (4) can be unitary transformed into a doublet Ψ ˜ R Ψ ¼ ; ð20Þ ΨL which is a two-component spinor composed of Dirac Ψ Ψ bispinors R and L (i.e., all together there are eight ð25Þ components). The Lagrangian (7) is obviously invariant upon the Hence, the nucleon fields in these irreducible representa- SUð2ÞCS rotation of the doublet (20), tions of SUð2ÞCS are given by n n 8 ΨR ε σ ΨR 1 L R R L L → i : < B −1=2 pffiffi u d − u d u ϵ f s; j; k ; exp ð21Þ 21 ð Þ¼ 2 ð as bj as bj Þ ck abc ð Þ ΨL 2 ΨL 2 ∶ 1 1 L R R L R : B 1=2 pffiffi u d − u d u ϵ f s; j; k ; 21 ð Þ¼ 2 ð as bj as bj Þ ck abc ð Þ It then follows that the parity-doublet Lagrangian is not only chirally invariant under the transformation (6), but is, ð26Þ

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8 qffiffi qffiffi qffiffi > B −1=2 2uL dL uR − 1uL dR uL − 1uR dL uL ϵ f s; j; k ; < 22 ð Þ¼ 3 as bj ck 6 as bj ck 6 as bj ck abc ð Þ 22∶ qffiffi qffiffi qffiffi ð27Þ > : B 1=2 1uL dR uR 1uR dL uR − 2uR dR uL ϵ f s; j; k ; 22 ð Þ¼ 6 as bj ck þ 6 as bj ck 3 as bj ck abc ð Þ

8 B −3=2 uL dL uL ϵ f s; j; k ; > 4ð Þ¼ as bj ck abc ð Þ > qffiffi > <> B −1=2 1 uL dL uR uL dR uL uR dL uL ϵ f s; j; k ; 4ð Þ¼ 3ð as bj ck þ as bj ck þ as bj ck Þ abc ð Þ 4∶ qffiffi ð28Þ > 1 R R L R L R L R R > B4ð1=2Þ¼ ðua d uc þ ua d uc þ ua d uc Þϵabcfðs; j; kÞ; > 3 s bj k s bj k s bj k :> B 3=2 uR dR uR ϵ f s; j; k ; 4ð Þ¼ as bj ck abc ð Þ

where ϵabc is the Levi-Civita tensor in the color indices. In this case BrðχzÞ will represent baryon interpolators with fðs; j; kÞ is a function of the spin indices; but they are spin 1=2, in which the diquark has spin zero. connected with the total angular momentum of the nucleon Finally, we can rewrite the nucleon fields in Eq. (29) for and at this step we do not perform any summation over s, j, each i as and k. We have denoted with BrðχzÞ the baryon field in the ð1Þ SUð2ÞCS irreducible representation of dim r ¼ 2χ þ 1 with N −P B −3=2 B 3=2 ¼ ½ð 4ð Þ 4ð ÞÞ the chiral spin χ and its projection χz. rffiffiffi 1 Now we apply Eqs. (26)–(28) to nucleon interpolators þ ðB2 ð1=2Þ ∓ B2 ð−1=2ÞÞ 1=2 J 1=2 6 2 2 with isospin , the total angular momentum ¼ rffiffiffi with spin zero diquark. The structure of these nucleon 1 þ ðB2 ð1=2Þ ∓ B2 ð−1=2ÞÞ interpolators is [14] 2 1 1 rffiffiffi 1 B 1=2 B −1=2 ; NðiÞ ϵ P ΓðiÞu uTΓðiÞd − dTΓðiÞu ; þ 3ð 4ð Þ 4ð ÞÞ ¼ abc 1 að b 2 c b 2 cÞ ð29Þ Nð2Þ P − B −3=2 B 3=2 ¼ ð 4ð Þ 4ð ÞÞ 1 ðiÞ where P ¼ ð1 γ4Þ. The Dirac structures of Γ and rffiffiffi 2 1 1 ðiÞ Γ are reported in Table I. þ ðB2 ð1=2Þ ∓ B2 ð−1=2ÞÞ 2 6 2 2 NðiÞ Now if we want to write the s as linear combination rffiffiffi of the BrðχzÞ we need to find the proper expression for 1 þ ðB4ð1=2ÞB4ð−1=2ÞÞ fðs; j; kÞ in Eqs. (26)–(28). At first we note that the diquark 3 rffiffiffi in Eq. (29) has spin zero and it does not carry any spin 1 i Nð Þ 1=2 þ ðB2 ð1=2Þ ∓ B2 ð−1=2ÞÞ ; index. Conversely, since has spin , then it carries a 2 1 1 spin index given by the quark ua. This means that in rffiffiffi 1 Eqs. (26)–(28) we need to sum over j and k, but not over s. Nð3Þ −iP B −1=2 ∓ B 1=2 ¼ ð 22 ð Þ 22 ð ÞÞ C ¼ iγ2γ4 ¼ σ2 ⊗ σ3, where the Pauli matrix σ2 acts in the 6 quark spin space. Therefore, we can replace fðs; j; kÞ → 2 þ pffiffiffi ðB4ð−1=2ÞB4ð1=2ÞÞ ðσ2Þjk in Eqs. (26)–(28) and sum over j and k but not over s. 3 rffiffiffi 1 B −1=2 ∓ B 1=2 ; TABLE I. List of Dirac structures for the N baryon fields with þ ð 21 ð Þ 21 ð ÞÞ P 2 scalar or pseudoscalar diquarks, where I is the isospin and J rffiffiffi indicates spin and parity. ð4Þ 3 N ¼ iP − ðB2 ð−1=2Þ ∓ B2 ð1=2ÞÞ 2 2 2 P ðiÞ ðiÞ I;J Γ1 Γ2 i rffiffiffi 1 1 Cγ 5 1 þ ðB2 ð−1=2Þ ∓ B2 ð1=2ÞÞ ; ð30Þ γ C 2 1 1 ðiÞ 1 1 5 2 N ð2 ; 2 Þ i1 Cγ5γ4 3 iγ5 Cγ4 4 where BrðχzÞ are

014030-4 BARYON PARITY DOUBLETS AND CHIRAL SPIN SYMMETRY PHYS. REV. D 98, 014030 (2018) 1 1−γ5 T 1−γ5 The cross-correlation matrix for interpolators BrðχzÞ and B2 ð−1=2Þ¼ϵabc pffiffiffi γ4ua db C uc 1 2 2 2 Br ðχ0zÞ is 0 T 1−γ5 † u d Cγ u ; CðtÞχ;χ ;χ ;χ ¼h0jBrðχz; tÞBr ðχ0z;0Þ j0i: ð32Þ þ a b 4 2 c 0 z 0z 0 1 1−γ5 T 1þγ5 The SUð2ÞCS restoration requires that the cross-correlators B2 ð1=2Þ¼ϵabc pffiffiffi ua db C uc 1 2 2 2 of operators from different representations of SUð2ÞCS vanish and the cross-correlators of operators within a given T 1þγ5 þγ4ua db Cγ4 uc ; representation of SUð2ÞCS that are diagonal in indices χz 2 χ rffiffiffi and 0z coincide while the off-diagonal vanish. In this case, C t 1−γ5 2 T 1þγ5 the diagonal correlators ð ÞNðiÞ of nucleon interpolators B2 ð−1=2Þ¼ϵabc − ua db Cγ4 uc 2 2 3 2 NðiÞ rffiffiffi from Table I are given as 1 T 1−γ5 − ua db Cγ4 uc 2 1 1 6 2 C t i C t C t C t ð ÞNð Þ ¼ ð Þ3=2 þ ð Þ1=22 þ ð Þ1=21 rffiffiffi 3 12 4 1 1−γ γ u dTC 5 u ; for i ¼ 1; 2; 3 þ 6 4 a b 2 c rffiffiffi 1 3 C t 4 C t C t : 1−γ 1 1 γ ð ÞNð Þ ¼ ð Þ1=22 þ ð Þ1=21 ð33Þ 5 T þ 5 4 4 B2 ð1=2Þ¼ϵabc ua d C uc 2 2 6 b 2 rffiffiffi C t C t χ χ 3=2 Here ð Þ3=2 is a correlator ð Þχ;χ0;χz;χ0z with ¼ 0¼ 1 1 γ5 T þ and with any χz ¼ χ0z, and similar for CðtÞ1=2 and CðtÞ1=2 . − γ4ua db Cγ4 uc 1 2 6 2 Hence, in the SU 2 -symmetric regime all correlators rffiffiffi ð ÞCS C t i i 1 2 1−γ5 ð ÞNð Þ with ¼ , 2, 3 should be degenerate. Such a − γ u dTCγ u ; 3 4 a b 4 2 c degeneracy was indeed observed at zero temperature upon 1−γ 1−γ truncation of the near-zero modes of the Dirac operator B −3=2 −ϵ 5 u dTC 5 u ; 4ð Þ¼ abc 2 a b 2 c in Ref. [14]. rffiffiffi 1 1−γ 1 γ B −1=2 ϵ 5 −u dTCγ þ 5 u V. CONCLUSIONS 4ð Þ¼ 3 abc 2 a b 4 2 c In this paper we have demonstrated that a chirally 1−γ T 5 symmetric parity-doublet Lagrangian is not only chirally þua db Cγ4 uc 2 symmetric, but in addition is SU 2 and SU 2NF ð ÞCS ð Þ 1−γ invariant. Then we have constructed nucleon three-quark −γ u dTC 5 u ; 4 a b 2 c interpolators that transform under irreducible representa- rffiffiffi tions of SUð2ÞCS. Such interpolators are required for lattice 1 1−γ 1−γ B 1=2 ϵ 5 γ u dTCγ 5 u studies to establish the chiral spin symmetry. 4ð Þ¼ 3 abc 2 4 a b 4 2 c 1 γ ACKNOWLEDGMENTS −γ u dTCγ þ 5 u 4 a b 4 2 c This work is supported by the Austrian Science 1 γ Fund FWF through Grants No. DK W1203-N16 and u dTC þ 5 u ; þ a b 2 c No. P26627-N27. The authors thank Christian Rohrhofer 1−γ 1 γ for cross-checking of all equations of Sec. IV. B 3=2 ϵ 5 γ u dTC þ 5 u ; 4ð Þ¼ abc 2 4 a b 2 c ð31Þ APPENDIX CHIRAL TRANSFORMATION OF and the brace brackets mean antisymmetrization in the THE PARITY-DOUBLET LAGRANGIAN T T T flavor space, e.g., fd Cucg¼d Cuc − u Cdc. The parity b b b pffiffiffi We begin with the Lagrangian (2.36) of [3] with two P P → 1= 2 projector in Eq. (30) can be replaced with , Dirac fermions since in the computation of the correlators the presence of P μ μ is irrelevant. L ¼ iψ¯ 1γμ∂ ψ 1 þ iψ¯ 2γμ∂ ψ 2 − mðψ¯ 1ψ 2 þ ψ¯ 2ψ 1Þ; ðA1Þ NðiÞ Hence, we have rewritten the nucleon interpolators in terms of interpolators that transform according to the which is invariant under chiral transformation with the irreducible representations of SUð2ÞCS. “mirror assignment”:

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ψ 1 → exp ðiαγ5Þψ 1; ψ 2 → exp ð−iαγ5Þψ 2: ðA2Þ Extending the Uð1ÞA mirror transformation (A2) to the axial part of the mirror SUð2ÞL × SUð2ÞR transformation From this we can obtain the chiral transformation of the doublet (4) Ψ αaτa Ψ ¼ þ ; ðA3Þ ψ → i γ5 ψ ; Ψ 1 exp 2 1 − αaτa where ψ → −i γ5 ψ ; 2 exp 2 2 ðA6Þ 1 Ψ ¼ pffiffiffi ðψ 1 þ ψ 2Þ; þ 2 1 Ψ− ¼ pffiffiffi ðγ5ψ 1 − γ5ψ 2Þ: ðA4Þ SU 2 SU 2 2 we obtain the axial part of the ð ÞL × ð ÞR transformation of the parity doublet (6): Then upon the “mirror” transformation (A2) the parity doublet (A3) transforms as Ψ Ψ Ψ αaτa Ψ þ → exp ðiασ1Þ þ : ðA5Þ þ → exp i ⊗ σ1 þ : ðA7Þ Ψ− Ψ− Ψ− 2 Ψ−

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