Baryon parity doublets and chiralspin symmetry

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 chiralspin symmetries emerge in QCD. It is demonstrated here that the baryon parity-doublet Lagrangian is manifestly chiralspin-invariant. We construct nucleon interpolators with fixed chiralspin transformation properties that can be used in lattice studies at high T.


INTRODUCTION
A Dirac Lagrangian of a massless fermion field is chirally symmetric since the left-and right-handed components of the fermion field are decoupled, where The SU (2) L × SU (2) R chiral symmetry is a symmetry upon independent isospin rotations of the left-and righthanded components of fermions, where τ a are the isospin Pauli matrices and the angles θ a R and θ a L parameterize rotations of the right-and lefthanded components. The transformation (3) defines the (0, 1/2) ⊕ (1/2, 0) representation of the chiral group.
The mass term in the Lagrangian breaks explicitly the chiral symmetry since it couples the left-and righthanded fermions.
It is known for a long time that it is possible to construct a chirally symmetric Lagrangian for a massive fermion field provided that there are mass-degenerate fermions of opposite parity -parity doublets -that transform into each other upon a chiral transformation [1].
Consider a pair of the isodoublet fermion fields where the Dirac bispinors Ψ + and Ψ − have positive and negative parity, respectively. The parity doublet above is a spinor constructed from two Dirac bispinors and contains eight components. Note that there is in addition an isospin index which is suppressed. Given that the right-and left-handed fields are directly connected with the opposite parity fields the vectorial and axial parts of the chiral transformation law under the (0, 1/2) ⊕ (1/2, 0) representation of Here σ i is a Pauli matrix that acts in the space of the parity doublet. In the chiral transformation law (3) the axial rotation mixes the massless Dirac spinor ψ with γ 5 ψ , while in the chiral rotation of the parity doublet a mixing of two independent fields Ψ + and Ψ − is provided. Then the chiral-invariant Lagrangian of the free parity doublet is given as Note that this Lagrangian can be written in different equivalent forms [2,3]. The equivalence of the present form [1] with that one of ref. [3] was demonstrated in ref. [4] and the chiral transformation law (6) corresponds to the "mirror" assignment of ref. [3]. For the reader convenience we derive this equivalence in Appendix A. This Lagrangian can also be equivalently written in terms of the right-and left-handed fields of eq. (5), The latter form demonstrates that the right-and lefthanded degrees of freedom are completely decoupled and the Lagrangian is manifestly chiral-invariant. Since the latter Lagrangian is equivalent to the Lagrangian (7) it is also manifestly Lorentz-invariant (in order to avoid confusion note that Ψ L and Ψ R for the parity doublet are defined differently from the massless Dirac bispinor (2)). A crucial property of the Lagrangian (7-8) is that the fermions Ψ + and Ψ − are exactly degenerate and have a nonzero chiral-invariant mass m. This Lagrangian has a number of peculiarities. In particular, the diagonal axial charge of the fermions Ψ − and Ψ + vanishes, while the off-diagonal axial charge is 1.
Supplemented by the interactions with the pion and sigma-fields [2,3] this Lagrangian has been sometimes used in baryon spectroscopy [5] and as an effective model for chiral symmetry restoration scenario at high temperature or density where baryons with the nonzero mass do not vanish upon a chiral restoration [6].
Recently it was found on the lattice [7] that at temperatures above the critical one new symmetries emerge in QCD, SU (2) CS and SU (2N F ) [8,9], which implies a modification of the existing view on the nature of the strongly interacting matter at high temperatures [10]. In particular, the elementary objects at high temperatures are quarks with a definite chirality connected by the chromo-electric field, i.e. there are no free deconfined quarks.
These symmetries were observed earlier upon artificial truncation of the near-zero modes of the Dirac operator at zero temperature [11][12][13][14], for a recent theoretical development see [15].
metries are symmetries of the chromo-electric interaction in QCD. In addition to the chiral transformations the SU (2) CS and SU (2N F ) rotations mix the left-and right-handed components of quark fields. The chromomagnetic interaction as well as the quark kinetic term break these symmetries down to Given the observed new symmetries in high T QCD a question arises whether the parity doublet model could still be used as an effective description for the baryon-like objects at high T. Here we demonstrate that this model is actually manifestly SU (2) CS and SU (2N F ) invariant.
The su(2) algebra [Σ a , Σ b ] = 2iǫ abc Σ c is satisfied with any Euclidean gamma-matrix. U (1) A is a subgroup of SU (2) CS . The SU (2) CS transformations mix the left-and right-handed fermions. The free massless quark Lagrangian (1) does not have this symmetry.
An extension of the direct SU (2) CS ×SU (N F ) product leads to a SU (2N F ) group. This group contains the chiral symmetry of QCD SU (N F ) L × SU (N F ) R × U (1) A as a subgroup. Its transformations are given by where m = 1, 2, ..., (2N F ) 2 − 1 and the set of (2N F ) 2 − 1 generators is with τ being the flavour generators with the flavour index a and n = 1, 2, 3 is the SU The SU (2N F ) transformations mix both flavour and chirality. While the SU (2) CS and SU (2N F ) symmetries are not symmetries of the QCD Lagrangian as a whole, they are symmetries of the fermion charge operator and of the chromo-electric interaction in QCD. The chromomagnetic interaction as well as the quark kinetic term break these symmetries.

III. CHIRALSPIN SYMMETRY OF THE PARITY DOUBLETS
The chiralspin transformation (9) and generators (10) at k = 4 (the Euclidean γ 4 matrix coincides with the Minkowskian γ 0 matrix) can be presented in an equivalent form, as follows below. We will use the chiral representation of the γ-matrices. Then the SU (2) CS generators for the representation with k = 4 are Here 1 is the unit 2 × 2 matrix and the Pauli matrices σ i act in the space of spinors where R and L represent the upper and lower components of the right-and left-handed Dirac bispinors (2) The SU (2) CS , k = 4 transformation (9) can then be rewritten as (19) The latter equation emphasizes that a reduction of the representations (9) of SU (2) CS leads to a twodimensional irreducible representation. Now we return to the parity doublet Lagrangian. Given the Eq. (5) the parity doublet (4) can be unitary transformed into a doublet which is a two-component spinor composed of Dirac bispinors Ψ R and Ψ L (i.e., all together there are eight components). The Lagrangian (7) is obviously invariant upon the SU (2) CS rotation of the doublet (20), It then follows that the parity doublet Lagrangian is not only chirally invariant under the transformation (6), but is in addition SU (2) CS -and SU (2N F )-invariant with the generators of SU (2N F ) being

IV. CHIRALSPIN SYMMETRIC NUCLEON INTERPOLATORS
Here we construct nucleon three-quark interpolators that transform upon SU (2) CS representations (8-9) with k = 4. These interpolators should be used in lattice studies to establish the chiralspin symmetry and consequently to justify the parity doublet model as an effective description of the baryon-like objects at high temperatures.
Using the standard technique of Young tableaux for representations of the permutation group where indices α, β, γ denote either L or R, and e.g. u L as is the left-part of the up-quark with the color index a and the spin index s, we get Hence the nucleon fields in different irreducible representation of SU (2) CS are given by 4 : where ǫ abc is the Levi-Civita tensor in the color indices. f (s, j, k) is a function of the spin indices; but they are connected with the total angular momentum of the nucleon and at this step we do not perform any summation over s, j and k. We have denoted with B r (χ z ) the baryon field in the SU (2) CS irreducible representation of dim r = 2χ + 1 with the chiralspin χ and its projection χ z . Now we apply the Eqs. (26), (27) and (28) to nucleon interpolators with isospin 1/2, total angular momentum J = 1/2 with spin zero diquark. The structure of these nucleon interpolators is [14]: where P ± = 1 2 (1 ± γ 4 ). The Dirac structures of Γ  Table I.   TABLE I. List of Dirac structures for the N baryon fields with scalar or pseudoscalar diquarks, where I is the isospin, J P indicates spin and parity.
Cγ4 4 Now if we want to write the N (i) ± s as linear combination of the B r (χ z ) we need to find the proper expression for f (s, j, k) in the Eqs. (26), (27) and (28). At first we note that the diquark in Eq. (29) has spin zero and it does not carry any spin-index. Conversely since N (i) ± has spin 1/2, then it carries a spin index given by the quark u a . This means that in the Eqs. (26), (27) and (28) we need to sum over j and k, but not over s. C = iγ 2 γ 4 = σ 2 ⊗σ 3 , where the Pauli matrix σ 2 acts in the quark spin space. Therefore we can replace f (s, j, k) → (σ 2 ) jk in the Eqs. (26), (27) and (28) and sum over j and k but not over s. In this case B r (χ z ) will represent baryon interpolators with spin 1/2, in which the diquark has spin zero.
Finally we can rewrite the nucleon fields in the Eq. (29) for each i as where B r (χ z ) are and the brace brackets mean antisymmetrization in the flavour space, e.g.
The parity projector P ± in Eq. (30) can be replaced with P ± → 1/ √ 2, since in the computation of the correlators the presence of P ± is irrelevant.

V. CONCLUSIONS
In this paper we have demonstrated that a chirally symmetric parity doublet Lagrangian is not only chirally symmetric, but in addition is SU (2) CS and SU (2N F ) invariant. Then we have constructed nucleon three-quark interpolators that transform under irreducible representations of SU (2) CS . Such interpolators are required for lattice studies to establish the chiralspin symmetry.