Low lying eigenmodes and meson propagator symmetries

In situations where the low lying eigenmodes of the Dirac operator are suppressed one observed degeneracies of some meson masses. Based on these results a hidden symmetry was conjectured, which is not a symmetry of the Lagrangian but emerges in the quantization process. We show here how the difference between classes of meson propagators is governed by the low modes and shrinks when they disappear.

Recently it was found that in certain situations a symmetry emerges that relates vector and scalar meson propagators but that is no symmetry of the action.That symmetry was observed in lattice QCD when low lying eigenmodes of the Dirac operator are suppressed either artificially by removing the eigenmodes from the quenched quark propagators [1][2][3][4][5] or naturally in the high temperature phase [6,7] either due to a gap 1 or because another rapid decrease towards zero eigenvalues.The symmetry group was called CS (chiral-spin) and has been suggested [12,13] to be SU (4) ⊃ SU (2) L × SU (2) R × U (1) A mixing the u-and d-quarks of a given chirality and also the leftand right-handed components.
Here we provide results to elucidate the observed symmetry.We find explicitly in an analytic calculation that some propagator identities emerge if the low lying eigenmodes of the Dirac operator are suppressed.We show that in that case propagators of different mesons become degenerate giving rise to the CS symmetry in part.For the other part of the symmetry further conditions have to be met.
Consider the eigenvalues of the hermitian Dirac operator.It is well known that the difference of the susceptibilities of, e.g., the propagator of the isovector scalar meson operator and of the isovector pseudoscalar operator are weighted by an eigenvalues density factor (on top of the generic eigenvalue density), that approaches a delta function in the zero mass limit.The approach is similar to the derivation of the Banks-Casher relation for the quark condensate [14].The difference between the scalar and pseudoscalar propagators and susceptibilities has been intensely studied earlier [15][16][17].We show here that this property applies to a large set of (scalar, pseudoscalar, vector and axial vector) meson propagator pairs and discuss the conditions for the CS symmetry.
We work in Euclidean space-time continuum and will briefly remind on the notation.The tool will be the spectral representation of the Dirac quark and the meson propagator.As there are sums over all (an infinite number of) eigenmodes we need some regularization (e.g., a finite volume lattice) and for this we rely on Fujikawa's approach [18][19][20], which we assume implicitly but will omit the actual derivation.
We choose hermitian γ-matrices γ † µ = γ µ and [γ µ , γ ν ] + = 2δ µν .The fermions are Grassmann fields and the Dirac action is real.The massless Euclidean Dirac operator D ≡ iγ µ D µ is hermitian with the eigensystem The dimension of the eigenvectors is n D n c n f (Dirac, color, flavor) at each point x ∈ R 4 and there are n D n c n f eigenvectors as functions of x.Only the Dirac index a is kept explicitly, the color-and flavor indices are implicit.The non-zero eigenvalues come in pairs as can be seen by multiplying with γ 5 : We use the notation η n ≡ η n with η −n = −η n and ψ (−n) = γ 5 ψ (n) .The eigenvalues are real and the eigenvectors form an orthonormal basis We formally regularize by point-splitting such that the Dirac operator becomes a matrix,

arXiv:1803.08693v3 [hep-ph] 25 May 2018
The indices for color and flavor are implicit.For simplicity we assume mass degenerate fermions.The Dirac operator has the spectral representation The fermion propagator then has the representation We assume that there are no exact zero modes with γ 5 ψ = ±ψ either because we are in the topological sector zero or because they have been removed.Anyhow they are suppressed in the thermodynamic limit.

B. Chiral symmetry
We define chiral symmetry as the invariance of the massless Dirac operator.The transformation is where τ i are the generators of SU (2) f lavor and 1 f the unit matrix in flavor space.The kinetic term of the action is invariant, e.g., The chiral transformation commutes with the Euclidean Lorentz transformations O(4).
For the discussion it will be useful to split the four-Dirac-components of the eigenvectors into a pair , We choose a chiral basis for the Dirac matrices with and R, L having two components.
The CS transformations as a whole are no symmetry of the Dirac action.However, it has been observed, that CS is a symmetry of certain meson and baryon masses, if the low lying (quasi-zero) modes are absent [3][4][5]21].At zero temperature, with artificial removal of low lying modes in the valence sector, confinement seems to persist.Above the chiral temperature the zero modes are suppressed naturally.There are indications that some form of confinement persists as well [7].
The chromo-electric observables ψγ 4 ψ are symmetric under CS, the kinetic term of the action and the chromomagnetic terms ψγ k ψ (k = 1, 2, 3) are not.Removing the near-zero modes apparently restores the symmetry such that the influence of chromo-magnetism shrinks or disappears.One might conclude that confinement has its origin in the chromo-electric sector, which is symmetric under CS always [22].

III. MESON PROPAGATORS
We restrict ourselves to two mass degenerate quark flavors u and d.As already mentioned, we neglect exact zero modes, either because we are in that topological sector or because they have been removed.
We list the sink and source operator kernels.We also give sign factors s5 defined by Γγ5 = s5γ5Γ.FIG. 1.The weight functions g(η) and h(η) from ( 22) for typical values m equal to 0.02 (full), 0.06 (dashed) and 0.08 (dotted).
We study the propagators for mesons of type Ψ( τ ⊗Γ)Ψ and Ψ(1 f ⊗ Γ)Ψ.The Γ are listed in Table I; the choice has been motivated by the discussion of the CS symmetry in Ref. [12].We emphasize that our results are for the propagators themselves, not just the masses.
For the isotriplet we introduce the connected propagator for a given background gauge field A, where [. ..]A indicates Grassmann integration in an external field A. We relate source and sink by Γ snk = Γ † src ≡ s Γ Γ src , see Table I.For degenerate quark masses With the spectral representation for D −1 the meson propagator becomes The isoscalar propagators have also disconnected contributions proportional to Like other expressions used here, this has to be regularized (e.g., by lattice regularization) and there are standard tools to do this (e.g., [18][19][20]).The results and conclusions presented here are not affected.
We will find useful the identities where γ 5 Γγ 5 = s 5 Γ (see Table I).

The difference between two meson propagators depends on
• the generic distribution density of eigenvalues ρ(m, η) (which depends on the gauge configuration and the Dirac operator), • the values of the overlap matrix elements ψ † Γψ (which are bounded from above due to the orthogonality and normalization of the eigenvectors), and • a weight function discussed below.
The generic distribution of eigenvalues ρ(m, η) is needed only for the small eigenvalues, where it vanishes fast enough or there is even a gap, the relevant cases of this study.The bulk behavior is inconspicuous [23].
We focus here on the third factor.As derived in the Appendix the functions (see Fig. turn up in the sums over eigenvalues in the next section.They are essential for the argumentation.Both functions give large weight to contributions from small η.The function g is peaked at η = 0 and approaches (π/2)δ(η) for small masses m → 0; for large η it falls like 1/η 2 .Propagator differences weighted by g vanish for small m if there is a gap in the density ρ(m, η) at low lying eigenmodes, i.e. if there are no eigenvalues below some value, or if the density vanishes fast enough for η → 0. Propagators that differ only by terms proportional to g will be called g-equivalent.
The function h is peaked at η = m and falls like 1/η for large η.Compared to g this behaviour may not suppress the higher modes enough, depending on the Dirac structure.Propagators that differ also by terms proportional to h will be called h-equivalent.For these the existence of a gap at low eigenvalues is not sufficient to obtain propagator agreement and more conditions have to be met.
In the next section we discuss the main results.The full derivations can be found in the appendix.The resulting equivalences between the meson propagators are listed in Table II and  For Γ ∈ {1, γ k , γ 4 , γ k γ j , γ k γ 4 } the difference between meson isovector propagators is y,d .These propagator pairs are g-equivalent.
For small masses m → 0 the functions g emphasize the contributions of small eigenvalues.If the eigenvalue density ρ(m, η) vanishes at small eigenvalues, then the propagator difference vanishes as well and axial symmetry is restored.The factors with eigenvectors are bounded (the eigenvectors are normalized).
The integral over x, y and sum over a and the other, hidden indices gives the susceptibility.For Γ = 1 the second term in (23) vanishes due to orthogonality.The first term gives δ nk .The susceptibility difference (the U A (1) susceptibility) then is for the eigenvalue density ρ(m, η) (cf.[15], the discussion in [17] and [8][9][10][11]16]).This term vanishes if there is a gap in ρ(m, η) or if the density vanishes fast enough2 for η → 0. The susceptibilities for the other difference pairs of Table I are also g-equivalent.
Compared with g-equivalence the now discussed type is more subtle with factors h(m, η), needing further bounds or eigenmode properties to find propagator agreement.
To see this we use the chiral basis of Sect.II B.
The disconnected contributions for propagators with Γ ∈ {γ k , γ 4 , γ k γ 5 , γ 4 γ 5 } have terms with factors like where σ is a 2 × 2 matrix (i.e., a sub-block of Γ; for the complete expression see App.C).The prefactor again favors low eigenmodes for small m.Therefore the disconnected contributions become much smaller if the low modes are suppressed in the generic density.
Even if the low modes are absent, however, the higher modes still contribute to the difference more than in the g-equivalent case since h(m, η) decreases slower with η than g(m, η).The quality of the agreement then depends on the matrix elements in (26) and not only on the eigenvalue density.This is discussed in the subsequent section.
If the high modes contribution can be neglected the isoscalar propagator agrees with the isovector propagator for the listed Γ.Considering the results for the connected propagators this implies also agreement of the isoscalar propagator pairs (γ k , γ k γ 5 ) and (γ 4 , γ 4 γ 5 ).

2.
Γ vs. Γγ4 Finally let us consider the connected propagator pairs for (Γ, Γγ 4 ) for Γ ∈ {1, γ k , γ 5 , γ k γ j , γ k γ 5 }; these are central for the CS symmetry.The propagator differences are sums of two types of terms and The first term becomes negligible if the fermion mass is small and if there is a gap in ρ(m, η) at small η.In the second term, unlike the connected propagators discussed in Sect.III A 1, all four types R † σ L, L † σ R, R † σ R, L † σ L enter the propagator difference multiplying h.When there are no eigenvalues below some |η| < η 0 or the generic density vanishes fast enough towards η = 0 the propagator difference is dominated by the terms with h.The factors L † σ R, etc., encode the dynamics of QCD.
There are a few observations that may shed some light: TABLE II.Related meson propagators; For g-equivalent propagators the differences vanish in the massless limit if there are no low lying modes.Further assumptions are necessary for h-equivalence.
-The h-terms vanish for chiral eigenmodes of the form (R, 0) or (0, L) or will be suppressed for almost chiral eigenmodes (where either |R| |L| or |R| |L|).However, such behaviour is expected mainly for the low lying modes which are truncated or suppressed anyhow in the situation of relevance here.
-The mesons with Γ = γ 4 or γ 4 γ 5 have only terms R † L and L † R; now R † and L correspond to the FIG. 2. The equivalence relations between the corresponding meson propagators for vector mesons are shown.The arrows symbolize the entries in Table II and the equivalence type as discussed in the text is shown.The arrangement of the operators follows [12] for better comparison; the left-hand column indicates the chiral structure [24].
FIG. 3. The equivalence relations between the corresponding meson propagators for scalar mesons are shown.The hequivalences have been omitted (although they are listed in Table II) as they relate to unphysical states, The left-hand column indicates the chiral structure [24].
same helicity which cannot add up to zero.The states cannot be physical scalars [12].For this reason we omit these states in Fig. 3.
-There is numerical evidence [4] indicating that the scalar propagators show less agreement than the vector propagators.This is a hint that the vector matrix elements ψ (n) † γ j ψ (k) are smaller than the scalar ones ψ (n) † ψ (k) .

IV. CONCLUSIONS
Here we studied the rôle of low lying eigenmodes of the Dirac operator in meson propagators.The study is motivated by lattice QCD calculations where it was found that the differences between meson propagators of a large class disappear if the low lying (i.e., close to zero) modes of the Dirac operator are suppressed.The mass degeneracies have been observed when the low modes were truncated explicitly [3][4][5]12] or dynamically suppressed at large temperature [7].3There are two qualitatively different kinds of relations.Those with a weight factor g(m, η) we call g-equivalent.
Meson propagators that are g-equivalent (Fig.s 2 and 3 and table II) approach each other for small quarkmass, if there is a low-eigenvalue suppression or gap in the generic eigenvalue density.These equivalences, when realized, restore the axial symmetries SU (2) A and U (1) A .
The second type called h-equivalence needs further constraints in order to provide vanishing propagator differences.The weight factor h(m, η) is also peaked at small η but does not suppress the higher modes as efficient.In that case the quality of agreement depends on the overlap of eigenvectors.
We find: • The connected (isovector) propagators P c (Γ) and P c (iΓγ 5 ) for Γ ∈ {1, γ k , γ 4 , γ k γ j , γ k γ 4 } differ only by g-type terms.If there is a low mode suppression the propagators of a pair agree with each other for m → 0. The susceptibilities of the connected (isovector) propagators P c (Γ) inherit the gequivalence property.
• For some isoscalar mesons (see Sect.III A 2) the propagators' disconnected contributions are g-type terms.For these mesons the isoscalar and isovector propagators agree in the massless limit if there is a suppression of low eigenvalues.
In summary the axial symmetries between the meson propagators and susceptibilities are recovered for decreasing quark mass upon suppression of low lying eigenmodes in the eigenvalue distribution.A similar behaviour for the observed γ 4 symmetry requires in addition small overlap of the higher eigenmodes.The emerging agreement between the meson propagators explains numerical lattice QCD results for meson mass degeneracies.Based on the meson mass pattern the symmetries CS and SU (4) were conjectured [12].These may have far-reaching consequences [22].
Here we used relations like (20) , (22) and The propagator is where we replaced in (A3) s Γ by s Γγ5 and Γ by Γγ 5 and utilized (20).
The difference between the propagators becomes in all cases Appendix B: Disconnected terms These terms are responsible for the difference between isovector and isoscalar propagators and have the form where we used (20) in the 2nd step.Equivalent derivation for the 2nd sum leads to For Γ ∈ {1, γ k γ j , γ k γ 4 , i γ 5 , i γ k γ j γ 5 , i γ k γ 4 γ 5 } we find s 5 = 1 and giving (25).
For Γ ∈ {γ k , γ 4 , γ k γ 5 , γ 4 γ 5 } we find s 5 = −1 and which is discussed below in App. C. Then in all cases we find the form (σ depends on the actual Γ and is proportional to a Pauli matrix) ×(R (k) † (y) σ L (k) (y) + L (k) † (y) σ R (k) (y)) , In all terms upper components couple to lower ones.If the overlap is small (e.g., is the eigenmodes are close to chiral) then this contribution is small and the isovector and isoscalar propagators for that Γ are similar.
Again we find a term with g(m, η k )g(m, η n ) which vanishes if the small modes disappear.The term with h(m, η k )h(m, η n ) has significant contributions from low modes which disappear with them.It decays, however, slower for increasing η.If the modes above the gap are close to chiral, this term becomes small as well.In that case we are left with the g-type terms and the propagators are g-equivalent.

Appendix C :
More disconnected termsThis concerns the disconnected terms (B5) in Sect.III B 1. We consider the disconnected contribution to propagators for Γ ∈ {γ k , γ 4 , γ k γ 5 , γ 4 γ 5 } discussed at the end of App.B. Since the functions h(m, η) have slower decay towards larger η we have a closer look at the matrix elements.In the chiral basis of (13) the matrices Γ have the