Chiral-spin symmetry emergence in baryons and eigenmodes of the Dirac operator

Truncating the low-lying modes of the lattice Dirac operator results in an emergence of the chiral-spin symmetry $SU(2)_{CS}$ and its flavor extension $SU(2N_F)$ in hadrons. These are symmetries of the quark - chromo-electric interaction and include chiral symmetries as subgroups. Using as a tool the expansion of propagators into eigenmodes of the Dirac operator we here analytically study effects of a gap in the eigenmode spectrum on baryon correlators. We find that both $U(1)_A$ and $SU(2)_L \times SU(2)_R$ emerge automatically if there is a gap around zero. Emergence of larger $SU(2)_{CS}$ and $SU(4)$ symmetries requires in addition a microscopical dynamical input about the higher-lying modes and their symmetry structure.


I. INTRODUCTION
In a number of lattice spectroscopical studies with a chirally-invariant Dirac operator upon artificial truncation of the lowest modes of the Dirac operator [1,2] a large degeneracy was discovered in mesons [3][4][5] and baryons [6]. Corresponding symmetry groups, SU (2) CS and SU (2N F ) [7,8], turned out to be larger than the chiral symme- These symmetries emerge naturally, i.e. without any explicit truncation, in hot QCD above the pseudocritical temperature [9][10][11], where the near-zero modes of the Dirac operator are suppressed by temperature [17]. Consequently elementary objects in that range are not free quarks and gluons but rather chirally symmetric quarks bound i by the chromo-electric field into a color singlet objects, like a "string".
According to the Banks-Casher relation [12] the chiral symmetry breaking quark condensate is proportional to the density of the near-zero modes. A gap in the low lying Dirac eigenmode spectrum induces restoration of SU (N F ) L × SU (N F ) R symmetry. It was shown that it also induces restoration of U (1) A in the J = 0 mesons [13]. Analytical study of the J = 0 and J = 1 isovector meson propagators in terms of the eigenmodes of the Dirac operator revealed that all meson correlators that are connected by the U (1) A and/or SU (2) L ×SU (2) R transformations get necessarily degenerate if such a gap exists in the Dirac spectrum [14]. However, a possible emergence of SU (2) CS and of SU (2N F ) requires further dynamical properties encoded in certain matrix elements. Here we extend this analysis to baryons and show that the same conclusions remain valid in this case as well.

II. CHIRAL-SPIN SYMMETRY
The SU (2) CS chiral-spin transformations for quarks are given by where generators, defined in the Dirac spinor space are Here γ k , k = 1, 2, 3, 4, are hermitian Euclidean gamma-matrices, obeying the anticommutation relations Different k define four-dimensional representations that can be reduced into two-dimensional irreducible ones. The su (2) algebra is satisfied for any k in Eq. (2).
The fundamental vector of SU (2N F ) at N F = 2 is 3 The SU (2) CS and SU (2N F ) groups are not symmetries of the QCD Lagrangian as a whole.
In a given reference frame the quark-gluon interaction Lagrangian in Minkowski space can be splitted into temporal and spatial parts: Here D µ is a covariant derivative that includes interaction of the quark field ψ with the gluon field A µ , The temporal term includes an interaction of the color-octet charge densitȳ with the electric part of the gluonic gauge field. It is invariant under any unitary transformation acting in the Dirac and/or flavor spaces. In particular it is a singlet under SU (2) CS and SU (2N F ) groups. The spatial part consists of a quark kinetic term and interaction with the magnetic part of the gauge field. It breaks SU (2) CS and SU (2N F ). We conclude that interaction of electric and magnetic components of the gauge field with fermions can be distinguished by symmetry.
In order to discuss the notions "electric" and "magnetic" one needs to fix the reference frame. An invariant mass of the hadron is the rest frame energy. Consequently, to discuss physics of hadron mass generation it is natural to use the hadron rest frame.
In refs. [3][4][5] and [6] meson and baryon masses have been extracted from the asymptotic slope of the rest frame t-direction Euclidean correlator where O Γ (x, y, z, t) is an operator that creates a quark-antiquark pair for mesons or three quarks for baryons with fixed quantum numbers. Truncation of the near-zero modes of the Dirac operator resulted in emergence of the SU (2) CS and SU (2N F ) symmetries in hadrons.
This implies that a confining SU (2) CS -and SU (2N F )-symmetric quark-electric interaction is distributed among all modes of the Dirac operator. At the same time the quarkmagnetic interaction, that breaks both symmetries, is located only in the low-lying modes.
Consequently truncating the low-lying modes results in emergence of symmetries in the spectrum of hadrons.  A complete set of nucleon operators (J = 1/2, I = 1/2, P = ±1) with spin-zero diquarks consists of four operators [16] of the following form: where P ± = 1 2 (1 ± γ 4 ) is the parity projector. The matrices Γ  Table I.
Applying the U (1) A transformation on the given operator of Table I Below we present a set of nucleon operators that transform under irreducible representations of SU (2) CS , k = 4 [16]. These operators are linear combinations of the operators from the Table I: Explicitly these operators are: Here γ ± = 1 2 (1 ± γ 5 ) and B r (χ z ) is the nucleon interpolator in the irreducible representation of dimension r = 2χ+1 of SU (2) CS and with chiral-spin index χ z (z-projection of the chiralspin χ). In (14) the curly brackets {...} mean antisymmetrization between d b and u c quarks like in (12). Upon the chiral-spin transformation (1)-(2) with k = 4 only those nucleon operators are connected that belong to the same irreducible representation, as illustrated in

IV. SPECTRAL DECOMPOSITION
In this section we analyse the Euclidean nucleon propagators along t-direction upon truncation of the low-lying modes of the Dirac operator. We follow the procedure that was developed in Ref. [14] for a similar study of meson propagators. This approach is based on the spectral decomposition of the quark propagator in terms of the eigenmodes of the Dirac operator. The eigenmodes contain complete information about interaction of a quark with a gluonic field.
We work in Euclidean space-time and consider a hermitian massless Dirac operator D 0 ≡ iγ µ D µ . The eigenfunctions and eigenvalues of the Dirac operator are defined by the relation Because of {γ 5 , D 0 } = 0, the eigenvalues come in pairs with opposite signs (η n , −η n ) since In the following we will use the notation: η −n ≡ −η n . Here and in the rest of this work we assume that the Dirac operator D 0 does not have exact zero modes in its spectrum, which is equivalent to selecting gauge configurations with zero global topological charge.
The contribution of exact zero modes to observables vanishes in the thermodynamic limit.
The full Dirac operator for a quark field with mass m can be decomposed as where we used (15) and (16).
Now we consider baryon propagators and their decomposition using (17) for a theory with two mass degenerate quark flavours. A general baryon interpolator, see Eq. (14), can be written as for some choice of the coefficients c i ∈ C, in which whereΓ (i) 1 is given by a linear combination of products of Dirac matrices, Γ 2 is a generic product of gamma matrices and it satisfies the relation: The propagator associated with the operators O (i) (x) and O (j) (y), after the application of the Wick contractions is given by Furthermore we have called, e.g., D −1 u xaα|ya α = u xaαūya α A the quark propagator of the up quark between the space-time points x and y, with colour indices a and a , and Dirac indices α and α . In the case of two degenerate quark masses, then D −1 ≡ D −1 u = D −1 d . In absence of zero modes in the Dirac spectrum, the quark propagator D −1 can be expanded (see Ref. [14] and Eq. (17)) as where with h n ≡ h(m, η n ) = η n m 2 + η 2 n g n ≡ g(m, η n ) = m m 2 + η 2 n , n > 0. Substituting the Eq. (21) in the full propagator: we can express it in terms of h(m, η) and g(m, η), C(x, y) = n>0,k>0,l>0 g n g k g l S ggg (x, y) + g n g k h l S ggh (x, y) The functions S ggg (x, y),S ggh (x, y), S ghh (x, y) and S hhh (x, y) contain the information about the eigenfunctions of the Dirac operator and the structure of the baryon field under consideration.
Therefore the correlator C(x, y) has terms proportional to the g(m, η) function, like g n g k g l S ggg (x, y), g n g k h l S ggh (x, y) and g n h k h l S ghh (x, y), that we call g-terms, and terms proportional only to the h(m, η) function, that we call h-terms. A sketch of these two functions for different mass values is shown in Fig. 3.
In the chiral limit m → 0 the function g(m, η) approaches the delta-function π 2 δ(η). Hence a gap around zero in the spectrum of the Dirac operator will induce vanishing of the terms in Eq. (25) that contain at least one factor of g. In other words, all g-terms in Eq. (25) vanish in the chiral limit upon truncation of the near-zero modes of the Dirac operator.
The h(m, η) function is peaked at η = m and falls slower compared to the g(m, η) function at high eigenvalues η. Consequently while the h(m, η) function still suppresses higher eigenvalues η, making a small hole in the Dirac eigenspectrum will not necessarily lead to the vanishing of the h-term in Eq. (25) in the chiral limit unless some additional suppressing dynamical factors are contained in S hhh (x, y).
In the following we call nucleon operators g-equivalent if the difference of their propagators contains only g-terms.

A. Correlators of N (i) operators
Now we apply results of the previous section to correlators of nucleon operators from Table I. The details of the expansion in g-terms and h-term of the nucleon propagators are given in Appendix A.
In Fig. 4 we show how the difference of two correlators (25) calculated with any two operators from Table I is expressed via the ggg, ggh, ghh and hhh terms. We see from Fig. 4 that all nucleons connected by U (1) A and/or SU (2) A transformations, see Fig. 1, are g-equivalent, see for details Appendices B and C. Consequently a gap in the low-lying spectrum of the Dirac operator results (in the chiral limit) in degeneracy of all correlators obtained with operators connected by dashed red and/or blue arrows in Fig. 1. We conclude that a gap in the Dirac spectrum implies necessarily restoration of both U (1) A and SU (2) L × SU (2) R symmetries in nucleons. It is similar to the results for meson correlators obtained in [14].
Such degeneracies of the nucleon correlators have been observed on the lattice in Ref. [6]. However, the observations of Ref. [6] went essentially further than simply U (1) A and SU (2) L × SU (2) R restoration. It was noticed that a larger symmetries SU (2) CS and SU (4) emerge in baryon masses upon low-modes truncation.
From the analytical side we can now conclude the following. Comparing Fig. 4 with where the sum x,y,z is over the all space.
However summation over all spatial points x, y, z does not convert an h-connection between the N (2) and N (3) operators in Fig. 4 into a g-connection. We do not get further g-equivalence as compared the ones indicated in Fig. 4. The presence of a gap in the Dirac spectrum does not automatically imply emergence of SU (2) CS and SU (4) symmetries in the rest-frame correlators, which were observed in lattice study upon the low-mode truncation [6]. This implies that some additional suppressing microscopic dynamics should be contained in S hhh matrix elements. Regarding the SU (2) L × SU (2) R , each operator from Fig. 2 is a linear combination of positive and negative parity operators (12). Different operators (12) belong to different irreducible representations of the parity-chiral group, as was discussed above, so no definite representation of SU (2) L × SU (2) R can be ascribed to the operators (14). Now we apply a spectral decomposition of Sec. IV to the propagators built with the baryon operators (14) C(x, y) r,χz = B r (χ z )(x)B r (χ z )(y) .
We find that the difference between two generic propagators C(x, y) r,χz and C(x, y) r ,χ z , always contains hhh-terms. This means that a gap in the spectrum of the Dirac operator does not yet automatically imply emergence of the SU (2) CS and SU (4)  Consequently we analyze now baryon correlators in the rest frame, i.e. we consider the correlators C(t) r,χz = x,y,z where the sum x,y,z is over the all space.
(30) in (28) we get where in the third line we used Eq. (30). Since we are averaging over all possible quark fields we can remove the label P in the last line of Eq. (31) (because parity is a symmetry of the QCD action and the measure in the average · is parity-invariant).
Eq. (31) tells us that for a given irreducible representation r of SU (2) CS we have C(t) r,χz − C(t) r,−χz = 0, for all χ z . Hence in the rest frame the correlators for the baryons within the doublet 2 1 and 2 2 representations are equal. This is a general statement, irrespective whether there is or there is not a gap in the spectrum of the Dirac operator. This fact does not mean, however, that the SU (2) CS symmetry is manifest in the rest-frame correlators, because in the representation 4 the correlators with χ z = ±1/2 are not equal to the correlators with A presence of a gap in the Dirac spectrum does not automatically make the correlators with χ z = ±1/2 and with χ z = ±3/2 g-equivalent. The emergence of SU (2) CS requires some additional suppression of matrix elements with higher-lying modes as was discussed in the previous subsection.  Table I. It is given by The last line of Eq. (A1) can be written as the following sum, − (same terms as above with α ↔ γ and a ↔ c ) . (A2) Using (22) we can rewrite the coefficients in front of the eigenfunction products in (A2) in terms of g n and h n , i.e.
moreover other coefficients can be found exploiting that f −n = −f * n , see Eq. (22). Therefore by linearity of (A2), we can get the expression of C(N (i) ± ) in terms proportional to g n g k g l , g n g k h l , g n h k h l and h n h k h l .
where we used that γ 4 P ± = ±P ± , (P + ) ω + (P − ) ω = δ ω and that Γ for all values of i, see Table I. We expand the quark propagator according to (A2) and use ψ (−n) = γ 5 ψ (n) and γ 5 Γ Using (22) the coefficients in front of the eigenfunction products can be written as Hence in the difference of nucleon propagators with opposite parity contains no terms proportional to hhh as indicated in Fig. 4.
Appendix C: C(N (1) ± ) and C(N In order to prove that also the propagators C(N (i) ± ) and C(N (i+1) ± ) for i = 1, 3 are gequivalent we notice that from Table I we Using (22) we can rewrite the coefficients in front of the eigenfunction products in terms of h n and g n , namely f n f k f l − f −n f k f −l = −2(h n g k g l + i h n h k g l + g n g k h l + i g n h k h l ), f n f k f −l − f −n f k f l = −2(h n g k g l + i h n h k g l − g n g k h l − i g n h k h l ), f n f −k f l − f −n f −k f −l = −2(h n g k g l − i h n h k g l + g n g k h l − i g n h k h l ), f n f −k f −l − f −n f −k f l = −2(h n g k g l − i h n h k g l − g n g k h l + i g n h k h l ). ± ) are only proportional to ggh and ghh terms, as indicated in Fig. 4.