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Quenched scalar meson correlator with Domain Wall Fermions

We study the singlet and non-singlet scalar-meson masses using domain wall fermions and the quenched approximation. The singlet mass is found to be smaller than the non-singlet mass and indicates that the lowest singlet meson state could be lighter than 1 GeV. The two-point functions for very small quark masses are compared with expectations from the small-volume chiral perturbation theory and the presence of fermionic zero modes.


Introduction
The nature of the lightest scalar mesons is not well understood. The observed σ resonance 0 ++ with m ≃ 480 MeV and Γ ≃ 320 MeV [1] can be either a manifestation of the ππ scattering or a distinct 0 ++ state. Recent lattice simulations explore whether the lightest singlet 0 ++ state is a qq state, a mixture ofqq and glueball or aq 2 q 2 state [2,3,4,5]. A dynamical simulation ofqq in Ref. [2] indicates m lat σ ≃ m lat ρ , while Ref. [3] finds m lat σ as low as m lat π indicating possible restoration of chiral symmetry at finite volume. In quenched studies,qq-glueball mixing renders the lowest scalar mass both below 1 GeV [3] and above 1 GeV [5].
The lightest observed non-singlet scalar states are a 0 (980) and a 0 (1450). Previous lattice simulations indicate that lowest scalar (qq) I=1 state is heavier than 1 GeV [3,5,6]. This opens the question on the nature of the a 0 (980), proposed to be aq 2 q 2 state in [4]. Dynamical simulation [7] observes an indication for the a 0 → η ′ π decay.
Those simulations were using Wilson [2,3,4,5] and Kogut-Susskind [7] fermions. We re-examine the light scalar spectrum using Domain Wall fermions (DWF), which allow us to study lighter quark masses. Good chiral properties of DWF are especially welcome in the study of the σ meson, which is intimately related to chiral symmetry breaking.
A light σ could play an important role as an in-termediate state in K→σ→ (ππ) I=0 , for example.

Lattice simulation
We use the quenched approximation together with DWF (L s = 16) and RG-improved gauge action (DBW2). This renders the chiral symmetry breaking parameter m res smaller than 1 MeV. Our lattice parameters are shown in Table 1.  Table 1: Simulation parameters, ranges of m π a and results for m 0 .
The disconnected quark diagram is simulated with the Kuramashi technique [8]. This technique is applied also for the connected diagram, so that a meson has a point source on every lattice site.

Connected diagram and quenching effects
The connected two-point function exhibits conventional exponential decay for m q > m s and gives us the a 0 mass consistent with [3,5].
The connected two-point function has a striking quenched effect for m q < m s [6] -it is negative at larger times! Fig. 1 shows that the negativity is more pronounced for smaller quark masses and smaller volumes. We also find that the correlation function is negative only for non-zero topological charge Q. This is consistent with the effects of fermionic zero modes, which are not suppressed in the quenched approximation and whose contributions are related by [9]  The negativity can be understood also within quenched chiral perturbation theory (QχP T ). The small-volume version of QχP T [10] respects the relation (1) and is discussed in the Section 5. It is applicable when pion wavelength λ π > L.
We extract m a0 using the large-volume version of QχP T where λ π < L [6]. The dominant negative contribution to a 0 (t)a † 0 (0) in this approach arises from the πη ′ intermediate state [6] (Fig.  2). This diagram dominates at larger t, where the e −ma 0 t term is sub-dominant. The qq −πη ′ coupling is m 2 π /m q . The finite volume is taken into account by replacing the integral over momenta with a finite sum. The value of the m 0 insertion is determined from the disconnected pseudoscalar diagram and is given in     δ = m 2 0 /(24π 2 f 2 π ) ≃ 0.14 ± 0.04 (f π ≃ 0.13 GeV) obtained from this m 0 is larger than δ obtained from m π [9] and this discrepancy is currently under study. We use m 2 0 from the hairpin to be consistent with m 2 1 determined in analogous way below. With this input, the diagram in Fig. 2 is predicted with no free parameters and agrees well with our data for different quark masses and volumes, as shown in Fig. 1. Surprisingly, the agreement is good also for the case of smallest m q and V , where effects from fermionic zero modes are thought to be important.
The complete QχP T description of the a 0 (t)a † 0 (0) correlator incorporates pseudoscalar and scalar fields [6], and is given by the sum of the a 0 -exchange diagram, the diagram in Fig. 2 and their chiral corrections (given in Fig. 8 of [6]). The sum of the diagrams describes our data very well and depends only on two free-parameters: the a 0 mass and its coupling √ 8f a0 m 2 π /m q to thē qq current 2 . On the lattice with the largest volume, which has smaller finite-size and quenching effects (Fig.1), we find f a0 ∼ 0.05 GeV insensitive to m q , while m a0 is shown in Fig. 3. Our result for m a0 is lower than those of [3,5] since we have taken into account the intermediate two-pseudoscalar states in the analysis of the two-point function 3 . Our result does not exclude

Disconnected diagram and σ meson
Our data for the disconnected two-point function is given in Fig. 4. We analyze it in terms of the meson diagram 4 in Fig. 5: The values of m a0 and f a0 are taken from the connected diagram above. We have checked also that m a0 , determined from the shape of the disconnected two-point function, is consistent with m a0 obtained from the connected two-point function. A one-parameter fit to m 2 1 gives m 2 1 a 2 = −0.11 ± 0.04 at m π a = 0.3003 ± 0.0019 for our largest lattice, consistent with results from the other two lattices. The sum of the diagrams with an arbitrary number of m 2 1 insertions in the dynamical theory gives the singlet scalar mass m 2 σ = m 2 a0 + m 2 1 . The sign of the disconnected diagram in our data has an important physical consequence as it implies that m 2 1 < 0 and m σ < m a0 . The resulting m σ is shown in Fig. 3. Fig. 6 shows the comparison between predictions of small-volume QχP T [10] and our data for two-point functions at fixed topology and very small m q . The dominance of the fermionic zero modes as V → 0 and m q → 0 (Eq. 1) is reflected in our data as well as the effective theory. Note that QχP T [10] gives a negative prediction for the pion correlator at this V for larger t, although a pion correlator with a point source and sink should be positive on every configuration by definition. This probably indicates that the exis somewhat higher than the physical value, so the chiral corrections are smaller. 4 We approximate m 2 1 to be a local coupling since the lightest glueball has a heavy mass of 1.6 GeV [5].

QχP T at small volume
pansion parameter 1/(Lf π ) in QχP T is not small enough at 8 3 ×24 with a −1 = 1.3 Gee and the comparison has to be performed at larger volume.

Conclusion
We examined the lightestqq singlet and nonsinglet scalar meson masses using Domain Wall fermions in the quenched approximation. Our conclusions on the scalar spectra are based on the lattice with the largest volume and are displayed in Fig. 3. The singlet is lighter than non-singlet -this is an inevitable consequence of the sign of the disconnected two-point function. Our results m a0 = 1.04±0.07 GeV and m σ = 0.9±0.1 GeV at the smallest quark mass indicate that the lightest singletqq state could be lighter than 1 GeV and can not rule out that the a 0 (980) is aqq state.