Chiral Structure of Baryon and Scalar Tetraquark Currents

We investigate chiral properties of local fields of baryons consisting of three quarks with flavor S U(3) symmetry. We construct explicitly independent local threequark fields belonging to definite Lorentz and flavor representations. We discuss some implications of the allowed chiral symmetry representations on physical quantities such as axial coupling constants and chiral invariant Lagrangians. We also systematically investigate chiral properties of local scalar tetraquark currents, and study their chiral transformation properties.


Introduction
As the chiral symmetry of QCD is spontaneously broken, S U(N f ) L ⊗ S U(N f ) R → S U(N f ) V (N f being the number of flavors), the observed hadrons are classified by the residual symmetry group representations of S U(N f ) V .The full chiral symmetry may then conveniently be represented by its non-linear realization and this broken symmetry plays a dynamical role in the presence of the Nambu-Goldstone bosons to dictate their interactions.
Yet, as pointed out by Weinberg [1,2], there are situations when it makes sense to consider algebraic aspects of chiral symmetry, i.e. the chiral multiplets of hadrons.Such hadrons may be classified in linear representations of the chiral symmetry group with some representation mixing.While the chiral representation can also be used as a theoretical probe for the internal structure of hadrons [3][4][5][6][7].
Motivated by these arguments, we perform a complete classification of baryon fields written as local products of three quarks according to chiral symmetry group S U(3) L ⊗ S U(3) R [8][9][10][11].We also systematically classified the scalar tetraquark currents and study their chiral transformation properties [12][13][14].

Baryon Currents
Local fields for baryons consisting of three quarks can be generally written as a e-mail: hxchen@buaa.edu.cnwhere a, b, c denote the color and A, B, C the flavor indices, C = iγ 2 γ 0 is the charge-conjugation operator, q A (x) = (u(x), d(x), s(x)) is the flavor triplet quark field at location x, and the superscript T represents the transpose of the Dirac indices only (the flavour and colour S U(3) indices are not transposed).The antisymmetric tensor in color space ϵ abc , ensures the baryons' being color singlets.
For local fields, the space-time coordinate x does nothing with our studies, and we shall omit it.
The three-quark fields may belong to one of several different Lorentz group representations which fact imposes certain constraints on possible chiral symmetry representations.This is due to the Pauli principle and can be explicitly verified by the method of Fierz transformations.As shown in Ref. [8], for Dirac fields without Lorentz index, there are one singlet field Λ and two octet fields N N 1 and N N 2 : For the Rarita-Schwinger fields with one Lorentz index, there are two non-vanishing independent fields (also independent of the previous three Dirac fields): where P 3/2 µν is the projection operator: For tensor fields with two antisymmetric Lorentz indices, there is only one non-vanishing independent field (also independent of the previous three Dirac fields and two Rarita-Schwinger fields): where We also perform chiral transformations and verify that Λ and N N 1 − N N 2 are together combined into one chiral multiplet ( 3, 3) ⊕ (3, 3); N N µ and ∆ P µ are together combined into another chiral multiplet (6, 3) ⊕ (3, 6); while N N 1 + N N 2 and ∆ P µν are transformed into themselves under chiral transformation, and they belong to chiral representations (8, 1) ⊕ (1, 8) and (10, 1) ⊕ (1, 10), respectively.
As simple applications of the present mathematical formalism, we can extract the (diagonal) axial coupling constants g A for these baryons, as shown in Table .1; while we can also construct all S U L (3)× S U R (3) chirally invariant non-derivative one-meson-baryon interactions [10,11].These baryon fields can be used in the Lattice QCD and QCD sum rule calculations [15][16][17].

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In this section we shall study the scalar tetraquark currents.Let us consider currents for the tetraquark ud s s having J P = 0 + .Here we consider only local currents.To write a current, Lorentz and color indices are contracted with suitable coefficients (L abcd µνρσ ) to provide necessary quantum numbers, where the sum over repeated indices (µ, ν, • • • for Dirac spinor indices, and a, b, • • • for color indices) is taken.Again, due to the Pauli principle, there are five independent currents which can be explicitly verified by the method of Fierz transformations [12]: ) .The similar results can be obtained for the general scalar tetraquark currents [13].
We also investigated the chiral structure of tetraquarks, and found all the chiral multiplets [14].Since these calculations for tetraquark are much more complicated than those for baryon, at the first stage we only considered the chiral (flavor) structure and other degrees of freedom remain undetermined.We concentrated on the tetraquarks belonging to the "non-exotic" [( 3, 3) ⊕ (3, 3)] and

A 3 =
( sa γ µ C sT b )(u T a Cγ µ d b ) , P 6 = ( sa C sT b )(u T a Cd b