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Some recent research highlights from the CSSM B. G. Lasscock, J. Hedditch,

a

M. B. Parappillya , D. B. Leinwebera , A. G. Williamsa∗ ,

a

Special Research Centre for the Subatomic Structure of Matter, and Department of Physics, University of Adelaide, Adelaide SA 5005, Australia In this article we provide an overview of some of the recent research highlights from the CSSM. We present new results of our studies into the Θ+ pentaquark state from lattice QCD considering both spin-1/2 and spin-3/2 states. We continue our exploration of exotic states with a study of exotic meson states using hybrid meson interpolators with explicit gluonic degrees of freedom. To investigate scaling, we compare the behaviour of the quark mass function M (q 2 ), and wave-function renormalization function Z(q 2 ), in dynamical lattice QCD, for a variety of bare quark masses, on two lattices with lattice spacing of 0.090 and 0.125 fm, but similar physical volumes.

1. THE PENTAQUARK RESONANCE SIGNATURE IN LATTICE QCD We present the results of our search for the Θ+ pentaquark resonance, which has strangeness +1 and minimal quark content uudd¯ s. Since the spin, parity and isospin of the putative Θ+ remain open questions, we have explored [1–4] a wide space of quantum numbers using an exhaustive array of interpolating ﬁelds, including for the ﬁrst time a study of the Θ+ as a possible spin-3/2 state [2]. Key to this work is the formulation of a robust resonance signature in lattice QCD that can distinguish a resonance from possible scattering states. The volume dependence of the residue of the lowest lying state has been proposed as a way to identify a resonance [5,6]. Alternatively, hybrid boundary conditions, i.e. choosing a diﬀerent boundary condition for the u, d-quarks compared to the s-quark has been proposed in Refs. [7–9] to diﬀerentiate the resonance from a scattering state. Our method, which is complementary to these approaches, is to look for suﬃcient attraction between the constituents of the pentaquark state such that the mass of the pentaquark state is less than the energy of the corresponding free decay channel. We refer to this in [1,2] as the “standard lattice resonance signature” in lattice QCD because we have universally observed this behaviour at the quark masses that have been ∗ Speaker

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considered in studies of conventional baryons [10– 14]. 1.1. Simulation Parameters In our analysis we use a large 203 × 40 lattice. Using the mean-ﬁeld O(a2 )-improved LuscherWeisz plaquette plus rectangle action [15], the gauge conﬁgurations are generated via the Cabibbo-Marinari pseudoheat-bath algorithm with three diagonal SU(2) subgroups looped over twice. The lattice spacing is 0.128(2) fm, determined using the Sommer scale r0 = 0.49 fm. For the fermion propagators, we use the FLIC fermion action [16], an O(a)-improved fermion action with excellent scaling properties providing near continuum results at ﬁnite lattice spacing [17]. A ﬁxed boundary condition in the time direction is implemented by setting Ut (x, Nt ) = 0 ∀ x in the hopping terms of the fermion action, and periodic boundary conditions are imposed in the spatial directions. Gauge-invariant Gaussian smearing [18] in the spatial dimensions is applied at the fermion source at t = 8 to increase the overlap of the interpolators with the ground states. Eight quark masses are used, providing amπ = {0.540, 0.500, 0.453, 0.400, 0.345, 0.300, 0.238, 0.188}. For the pentaquark study we limited our calculations to the six largest quark masses to maintain a clear signal. The strange quark mass is taken to be the third largest (κ = 0.12885) quark mass. This κ provides a pseudoscalar mass

B.G. Lasscock et al. / Nuclear Physics B (Proc. Suppl.) 161 (2006) 248–255

of 697 MeV which compares well with the ex 2 − M 2 = 693 MeV, perimental value of 2MK π motivated by leading order chiral perturbation theory. The error analysis is performed by a third-order, single-elimination jackknife, with the χ2 per degree of freedom obtained via covariance matrix ﬁts. Further details of the fermion action and simulation parameters are provided in Refs. [16,17]. 1.2. Lattice resonance signature A test that has proved useful in studies of nucleon resonances [10–12] is a search for evidence of binding at quark masses near the physical regime. In Fig. 1 we show the spectrum of nucleon resonances [12]. The solid curve is the S-Wave N + π decay channel energy corresponding to the 1/2− and 3/2− states which clearly become bound, i.e. the mass of the resonance becomes less than its decay channel energy, at the quark masses shown. This is what we refer to as the standard lattice resonance signature of binding at quark masses near the physical regime. Note that the 3/2+ state also becomes bound as it decays to a PWave N + π (not shown), which due to ﬁnite volume eﬀects is at a higher energy than the S-Wave N + π.

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1.3. Lattice techniques In lattice QCD we derive hadron masses from the 2-pt correlation function, exp(−i p · x) 0 |T χ(x) χ(0) ¯ | 0 . G(t, p) = x

(1) The time ordered product of ﬁelds shown above can be expressed in terms of quark propagators, allowing us to evaluate this function. Here the interpolating ﬁelds χ, ¯ (χ) create (annihilate) states with a particular set of quantum numbers. When studying baryon correlation functions, we must project out states of the appropriate spin and parity using the projection operator, Γ [12]. We extract the energy of the lowest lying state, with energy E0 , created by our interpolator using, G(t, p)

= t→∞

∝

trsp [Γ G(t, p)] exp (−E0 t) .

(2)

Since the contributions to the two-point function are exponentially suppressed at a rate proportional to the energy of the state, at zero momentum the mass of the lightest state, m0 , is obtained by ﬁtting a constant to the eﬀective mass, G(t, 0) = ln M eﬀ (t) G(t + 1, 0) t→∞

=

m0 .

(3)

Since we search for evidence that the resonance mass has become smaller than the free decay channel energy, i.e. we search for evidence of binding, it is useful to deﬁne an eﬀective mass splitting. For example, in an S-wave decay channel, ΔM eﬀ (t)

≡

eﬀ eﬀ M5q (t) − (MBeﬀ (t) + MM (t))

=

m5q − (mB + mM ) ,

t→∞

Figure 1. Summary of nucleon resonances [11,12], the solid curve is the mass of the S-Wave N + π decay channel corresponding to 1/2− , 3/2− parity states.

(4)

eﬀ where MBeﬀ (t) and MM (t) are the appropriate baryon and meson eﬀective masses for a speciﬁc channel. For a P -wave decay channel, the eﬀective masses are combined with the minimum nontrivial momentum on the lattice, 2π/L, to create the eﬀective energy, E eﬀ (t) = eﬀ (M (t))2 + (2π/L)2 , for each decay particle,

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where L is the lattice spatial extent. The advantage of this technique is that it measures a correlated mass diﬀerence, thereby suppressing the sensitivity to systematic uncertainties (such as using diﬀerent ﬁtting ranges). Moreover, correlations in the eﬀective masses can cancel, leading to a more accurate determination of the mass splitting. The two general types of pentaquark interpolating ﬁelds we consider are those based on an “N K” conﬁguration (either nK + or pK 0 ), and those based on a “diquark-diquark-¯ s” conﬁguration. Below is a summary of the interpolating ﬁelds we use that only couple to spin-1/2 states,

χP S

=

χSS

=

χN K

=

χN K

=

abc aef bgh (uT e Cdf )× (uT g Cγ5 dh )C s¯T c 1 √ abc (uT a Cγ5 db )× 2 (uT c Cγ5 de )C s¯T e 1 √ abc (uT a Cγ5 db )× 2 {uc (¯ se iγ5 de ) ∓ (u ↔ d)} 1 abc T a √ (u Cγ5 db )× 2 {ue (¯ se iγ5 dc ) ∓ (u ↔ d)} . (5)

For χN K and χN , the − and + corresponds to K the isospin I = 0 and 1 channels, while χP S and χSS access isoscalar and isovector states respectively. One can access spin- 32 states by replacing the spin-0 K-meson part of χN K with a spin-1 K ∗ vector meson operator, χμN K ∗

=

1 √ abc (uT a Cγ5 db )× 2 {uc (¯ se iγ μ de ) ∓ (u ↔ d)} ,

(6)

where the − and + corresponds to the isospin I = 0 and 1 channels, respectively. The ﬁeld χμN K ∗ transforms as a vector under the parity transformation, and has overlap with both spin1 3 2 and spin- 2 pentaquark states. Note that here we have extended the work of [2] including terms with μ = 1, 2 and we have enlarged the ensemble from 290 conﬁgurations in [2] to 397 conﬁgurations.

1.4. Results In Fig. 2 we present a summary of our most relevant results from [1,2] in the isoscalar negative parity channel. In the I(J P ) = 0(1/2− ) channel the lowest energy non-interacting two-particle state is the N + K in S-wave. We ﬁnd excellent agreement in the mass of the I(J P ) = 0(1/2− ) state extracted with the N K and N K ∗ interpolators, in each case the mass extracted with the pentaquark interpolator is more massive than the lowest energy non-interacting two-particle state. − In the case of the I(J P ) = 0( 32 ) channel the lowest energy non-interacting two-particle state is the N + K ∗ in S-wave. Again the mass state extracted with the N K ∗ interpolator is more massive than the lowest energy non-interacting twoparticle state in this channel. Therefore we do not ﬁnd evidence of the standard lattice resonance signature in the spin-1/2 or spin-3/2 isoscalar negative parity channels. While this does not exclude the possible existence of the Θ+ in these channels, it has been shown that the I(J P ) = 0(1/2− ) and 0(3/2− ) in [5] and [9] respectively are scattering states, where the masses of these states are larger than their respective free decay channel energies because of ﬁnite volume eﬀects. We continue our analysis with the isoscalar positive parity channel. Once again we present our most relevant ﬁndings from [1,2] in Fig. 3. In this channel the lowest energy non-interacting two-particle state for both spin-1/2 and spin-3/2 states is the N + K in P-wave. We ﬁnd that + the mass of the I(J P ) = 0( 12 ) state extracted with the N K ∗ interpolator similar to the S-wave N ∗ +K or P-wave N +K ∗ energy, which is unfortuneatly approximately the same on our lattice. This suggests that the N K ∗ interpolator couples relatively weakly to the lower energy P-wave N + K state in this channel, which we found was also true for the N K interpolator studied in [1]. + The mass of the I(J P ) = 0( 12 ) state extracted with the P S interpolator is consistent with the energy of the non-interacting P-wave N + K twoparticle state. Most interestingly the mass of the + I(J P ) = 0( 32 ) state extracted with the N K ∗ interpolator becomes smaller than the energy of the non-interacting P-wave N + K two-particle state

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251

−

Figure 2. The masses of the I(J P ) = 0(1/2− ) (open circles) and I(J P ) = 0( 32 ) (closed circles) states − extracted with the N K ∗ interpolator and the mass of the I(J P ) = 0( 12 ) state (open squares) extracted with N K interpolator as a function of m2π . For comparison the S-Wave N + K mass (solid line), S-Wave N + K ∗ mass (dashed line) are shown. The data correspond to mπ 830, 770, 700, 616, 530 and 460 MeV. Some of the data points have been oﬀset horizontally for clarity. at the two smallest quark masses shown, i.e. we observe binding. We make a more accurate calculation the binding energy by calculating the mass splittings deﬁned in Eq. (4). The splitting between the mass + + of the I(J P ) = 0( 12 ) and I(J P ) = 0( 32 ) states extracted with the P S and N K ∗ interpolators respectively, and the energy of the P-wave N + K two-particle state is shown in Fig. 4. We ﬁnd that + the I(J P ) = 0( 32 ) state extracted with the N K ∗ interpolator is bound at the two smaller quark masses shown, which is evidence of the standard lattice signature. + The mass of the I(J P ) = 0( 12 ) state extracted with the P S interpolator is consistent with the energy of the P-wave N + K non-interacting twoparticle state to within errors. So we ﬁnd no evidence of binding but neither can we rule it out.

2. HYBRID AND EXOTIC FROM FLIC FERMIONS

MESONS

Exotic mesons have quantum numbers that cannot be described in terms of a simple quarkantiquark pair. In our study we access exotic meson states using hybrid interpolating ﬁelds, which have explicit gluonic degrees of freedom. The particle data group [19] reports two candidates for the 1−+ exotic at 1400 MeV and 1600 MeV. Further discussion of the experimental status of the 1−+ exotic can be found in Refs. [20,21] . We use the same lattice and simulation parameters in our study of the 1−+ exotic meson described in Sec. 1.1, however in this case we are able to probe the 1−+ at the two smallest quark masses which we were not able to use in our pentaquark study. To extract 1−+ exotic states we use local interpolating ﬁelds, coupling colour-octet quark bilinears to chromo-electric and chromo-magnetic ﬁelds. We consider four interpolating ﬁelds for

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+

+

Figure 3. The masses of the I(J P ) = 0( 12 ) (open circles) and I(J P ) = 0( 32 ) (closed circles) states + extracted with the N K ∗ interpolator and the mass of the I(J P ) = 0( 12 ) state extracted with the PS interpolator (open triangles), as a function of m2π . For comparison the the P-wave N+K, the S-Wave N + K ∗ and the P-Wave N + K ∗ are shown. The data correspond to a pion mass of 830, 770, 700,616 and 530 MeV. Some of the data points have been oﬀset horizontally for clarity. the 1−+ exotic, χ1 χ2 χ3 χ4

= = = =

q¯a γ4 Ejab q b ijkl q¯a γk Blab q b ijkl q¯a γ4 γk Blab q b jkl q¯a γ5 γ4 γk Elab q b .

observed in our correlation functions [22]. Therefore we identify this state as the 1−+ exotic meson. We extrapolate to the chiral limit by ﬁtting the function, (7)

The interpolating ﬁelds which couple large-large and small-small spinor components (i.e χ2 and χ3 ) provide the strongest signal for the 1−+ state. In Fig. 5 we present the 1−+ exotic mass extracted with the χ2 interpolator, the a1 η decay channel energy, at each quark mass, and the π1 (1600) experimental candidate. At the four largest quark masses the standard lattice resonance signature of binding at quark masses near the physical regime is observed. In the approach to the large quark mass limit, quark counting rules imply that the mass splitting should become larger, which we observe. Further we are satisﬁed that at the lighter quark masses the χ2 interpolator is not accessing the a1 η decay channel as no evidence of negative metric contributions is

m1−+ = a0 + a2 m2π + a4 m4π ,

(8)

−+

mass in Fig. 5 and recover a mass of to the 1 1.74(25) GeV which is consistent within errors of the π1 (1600) experimental candidate. Systematic errors associated with the naive extrapolation are estimated to be ±50 MeV [22]. 3. SCALING BEHAVIOUR OF THE QUARK PROPAGATOR IN FULL QCD We study the scaling behavior of the quark propagator on two lattices with similar physical volumes in Landau gauge with 2+1 ﬂavors of dynamical quarks [23,24]. We use conﬁgurations generated with an improved staggered (“Asqtad”) action by the MILC collaboration. The calculations are performed on a 283 × 96 lattice with

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B.G. Lasscock et al. / Nuclear Physics B (Proc. Suppl.) 161 (2006) 248–255

+

+

Figure 4. The splitting between the I(J P ) = 0( 12 ) (open circles) and I(J P ) = 0( 32 ) (closed circles) states extracted with the P S and N K ∗ interpolators and the energy of the P-wave N +K non-interacting two-particle state. For comparison we include the Θ+ (1540) mass minus the energy of free P-wave N+K energy adjusted for the ﬁnite volume eﬀects of our lattice (star). The data correspond to a pion mass of 830, 770, 700,616 and 530 MeV. Some of the data points have been oﬀset horizontally for clarity.

lattice spacing of a = 0.090 fm and on a 203 × 64 lattice with lattice spacing of a = 0.125 fm. In Fig. 6, we show the quark mass function M (q 2 ), and wave-function renormalization function Z(q 2 ) calculated on the ﬁner lattice (light sea-quark mass and valence quark mass equal) with the light quark mass set to m = 27.1 MeV. This is compared with Z(q 2 ) and M (q 2 ) calculated on the coarser lattice by a simple linear interpolation from the four diﬀerent data sets such that the running masses are the same at q 2 = 3.0 GeV. Our results show that there is no signiﬁcant diﬀerence in the wave-function renormalization function and quark mass function on the two sets of lattices. Therefore the scaling behavior is good for the lattice spacing of a ≤ 0.125 fm. Figure 5. The 1−+ mass extracted with the χ2 interpolator (closed triangles), the a1 η decay channel (open triangles) and the π1 (1600) experimental candidate (square).

4. CONCLUSIONS We have presented an overview of recent research highlights from the CSSM Lattice Collaboration. In conclusion, we have completed a comprehensive search for the Θ+ pentaquark and the

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1−+ exotic meson in lattice QCD, and studied the scaling behaviour of the full QCD mass and wavefunction renormalisation functions. We ﬁnd evidence of the standard lattice signature in the spin-3/2 isoscalar positive parity channel, making it a promising candidate for future research. An enlarged ensemble and an analysis of the volume dependence of this signal is required to determine if this signature implies the existence of the Θ+ pentaquark. With the χ2 hybrid meson interpolator we extract a state which is a candidate for the 1−+ exotic meson. Extrapolation to the chiral limit yields, for the ﬁrst time, a mass that is consistent with the π1 (1600) experimental candidate. We have presented the ﬁrst scaling analysis of the full QCD quark propagator, wavefunction renormalisation and mass functions. We ﬁnd that the Z(q 2 ) and M (q 2 ) functions on lattices with similar volumes and lattice spacing a = 0.090 and a = 0.125 fm are in good agreement. Therefore good scaling behaviour is found for a ≤ 0.125 fm. REFERENCES

Figure 6. Comparison of wave-function renormalization function Z(q 2 ) and mass function M (q 2 ) for two diﬀerent lattices. The triangles corresponds to the quark propagator at mass m = 27.1 MeV from the 283 ×96 lattice, with lattice spacing a = 0.090 fm. The open circles are the data from the 203 × 64 lattice with lattice spacing a = 0.125 fm.

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