Lattice Study of Scalar Mesons and Exotic Hadrons
Lattice Study of Scalar Mesons and Exotic Hadrons
Atsushi Nakamura in collaboration with Scalar Collaboration (T. Kunihiro, S. Muroya, C. Nonaka, M. Sekiguchi and A. Wada) and T. Saito RIISE, Hiroshima University, Higashi-Hiroshima, 739-8521, Japan
We report two lattice QCD simulation projects relating to multi-quark states: 1) full QCD simu- lations of scalar mesons, σ and κ using Wilson fermions, and 2) quench calculation of diquark force using Polyakov line correlations. We observe a low sigma mass, mπ
PACS numbers:
I. INTRODUCTION
This symposium focuses a very ambitious topics, i.e., “multiquark hadrons”. A question why hadrons are a q q q bound state of only three quarks or quark-antiquark sys- q q tem originated from the early days of the quark model, q q q q and now after the discovery of the penta-quarks[1], we should go a step further, i.e., what kind of multi-quark systems are available in QCD and what kind of dynamics works behind. Penta-quark ? Scalar meson ? In this workshop, Prof. Jaffe gave a beautiful lec- ture, where he described successfully the penta-quarks to- gether with other multiplets using the di-quark degrees of FIG. 1: Possible diquark contributions to the penta-quark freedom. Quark-quark system is color anti-triplet (anti- and scalar meson. symmetric) or sextet (symmetric), 3 × 3=3∗ +6 (1) × = + II. SCALAR MESONS
We may anticipate that, in the color anti-triplet chan- nel, the quark-quark interaction is attractive and strong The objective of SCALAR collaboration is to under- based on the perturbation [4] and the instanton induced stand scalar mesons in the framework of QCD. The con- model [5]. The penta-quark state, Θ+, may be a bound fidence level of the sigma meson has been increasing, and κ state of (ud)(ud)¯s,where(ud) stands for highly corre- an other scalar meson [8, 9], , has been reported by sev- lated u and d quark pairs.[6] Di-quark states play an eral experimental groups. In the modern hadron physics important role not only in multi-quark states, but also basedonQCD,thechiralsymmetryisanimportantin- in high energy reactions and color superconductivity at gredient and the sigma meson plays an essential role in high density.[3] There has been, however, no lattice cal- it together with the pion. culation to investigate the force between two quarks. In The existence of the sigma meson was obscure for the second part of this paper, we report our lattice sim- many years. The re-analyses of the π-π scattering phase ulation project of the q − q potential. shift have suggested a pole of the σ meson with I =0 PC ++ Even the scalar particle, σ, may be included in this and J =0 [10]. In this analysis, the chiral sym- framework. In Ref.[7], Alford and Jaffe calculated π − metry, analyticity, unitarity and crossing symmetry are π scattering amplitude in the quench approximation by taken into account. Contributions of the σ pole were ob- L¨uscher formula with neglecting diagrams (A) and (G) served in the decay processes such as D→ πππ[11] and in Fig.2, and found an evidence of the bound state, and Υ(3S) → Υππ [12]. In 1996 PDG(Particle Data Group), proposed a new picture of the scalar mesons as a (¯qq¯)(qq) “f0(400-1200) or σ” appeared below 1 GeV mass region, state. and “f0(600) or σ” in the 2002. In this paper, we report our approaches to the sigma If the sigma meson exists, it is natural to consider the meson and di-quark force. κ meson as a member of the nonet scalar states of chiral The indices a, fi and ν stand for color, flavor and Dirac indices, respectively. H(x)†|0 is a state whose quantum number is specified by the operator H. π mesons in the current lattice simulations are not suf- ficiently light and our sigma meson cannot decay into 2 π, i.e., its width is zero. Any other particle, e.g. ρ,∆and N ∗, also have zero width in lattice QCD calculations in the literature. In the case of the sigma meson, this flaw D A should be kept in mind, since the two-pion component may be important. There have been several attempts at lattice study of sigma mesons. To our knowledge, the first such calcu- lation was carried out by deTar and Kogut [15], where the so-called disconnected diagram, or the OZI forbidden type diagram is discarded. The channel was called ‘va- lence’ sigma, σV . They measured the screening masses and observed that σV is much heavier than the π meson at the zero temperature, while σV and π degenerate as the temperature increases over Tc. Kim and Ohta calcu- C G lated the valence sigma mass with staggered fermions for the lattice spacing a =0.054 fm and lattice size 48a =2.6 fm [16]. They obtained mσ/mπ =1.4 ∼ 1.6byvarying FIG. 2: The four types of quark line contraction that con- mπ/mρ =0.65 ∼ 0.3. tribute to the pseudoscalar-pseudoscalar correlation function. Alford and Jaffe study diagrams D and C. [7] Lee and Weingarten [17] have stressed the importance of the mixing the scalar meson and glueballs and con- cluded that f0(1710) is the lightest scalar glueball dom- inant particle, while f0(1390) is composed of mainly the SU(3)⊗SU(3) symmetry. Recently, the κ with I =1/2is u and d quarkonium. As discussed in the introduction, reported with mass mκ ∼ 800 MeV [13, 14]. Alford and Jaffe analyzed the possibility that the sigma In order to establish the scalar meson spectroscopy as particle is an exotic state, i.e., qqq¯q¯ by a lattice QCD a sound and important piece of hadron physics, it is calculation [7]. very important now to investigate these scalar mesons All these calculations are in the quench approximation, by lattice QCD. Lattice QCD provides a first-principle i.e., the fermion determinant in Eq.(2) is dropped, which approach of hadron physics, and allows us to study non- corresponds to ignoring quark pair creation and annihi- perturbative aspects of quark-gluon dynamics. It is a rel- lation diagrams. McNeile and Michael observed that the ativistic formulation, and quarks are described as Dirac σ meson masses in the quench approximation and in the fermions. It is not a model, and apart from numerical full QCD simulation are very different [18]. They also limitations, there are no approximations. It is not a considered the mixing with the glueballs. They obtained bound state calculation: neither a potential model nor averysmallsigmamesonmass, even smaller than the π Bethe-Salpeter calculation. mass, in the full QCD case. Lattice QCD is usually formulated in the Euclidean path integral as There are two ongoing projects of σ meson spec- troscopy : Riken-Brookhaven-Columbia (RBC) collab- ¯ Z = DUDψ¯Dψe−SG−ψDψ = DU det De−SG , (2) oration [19] and Scalar collaboration [20, 21]. The two approaches are complementary. The RBC collaboration employs the domain wall fermions, which respect the chi- where SGis the gluon kinetic action, whose continuum 4 2 ral symmetry, but include a quench approximation, while limit is − d xTrF /4. We construct a state with a defi- µν Scalar collaboration uses Wilson fermions, which break nite quantum number and measure the decay in the chan- the chiral symmetry at a finite lattice spacing, but per- nel, forms the full QCD simulation. RBC reported their sim- −1 3 1 ¯ ulation at a =1.3GeVona16 × 32 lattice. They G x, y DUDψ¯DψH y H x † e−SG−ψDψ ( )=Z ( ) ( ) remedied the quench defect with the help of the chiral † −m|x−y| perturbation. They observed that the masses of the non- = H(y)H(x) →e (3) isosinglet scalar (a0) and the singlet scalar (σ)arealmost where H(x) is a hadron operator. For scalar mesons, degenerate when the quark mass is heavy (above s quark mass regions), and that as the quark mass decreases, the ¯af1 af2 a σ H(x)= Cf1f2 ψν ψν . (4) 0 mass remains almost constant, but the mass of the
a ν f1,f2 decreases. A. Propagators of Sigma meson
The quantum numbers of the σ meson are I =0and J PC =0++; we adopt the σ meson operator of u¯(x)u(x)+d¯(x)d(x) σˆ(x) ≡ √ , (5) 2 where u and d indicate corresponding quark spinors, and σ σ FIG. 3: Quark diagrams for the scalar meson, , in lattice we suppress the color and Dirac indices. The meson simulations of full QCD. propagator is written as † Gσ(y,x)=Tˆσ(y)ˆσ(x) 1 u¯(y)u(y)+d¯(y)d(y) B. Numerical simulations = DUDu¯DuDd¯Dd √ Z 2 † In this project, we use only standard well-established u¯(x)u(x)+d¯(x)d(x) × √ techniques for numerical calculations, and want to see the 2 outcome. We employ Wilson fermions and the plaquette ¯ × e−SG−uDu¯ −dDd. (6) gauge action. Full QCD simulations were done by the Hybrid Monte Carlo (HMC) algorithm. By integrating over u,¯u, d and d¯ fields, the σ meson CP-PACS performed a very large-scale simulation of propagator leads to light meson spectroscopy in the full QCD calculation [22]. G y,x − D−1 x, y D−1 y,x We use here the same values of the simulation parame- σ( )= Tr ( ) ( ) β . κ . . . −1 −1 ters, i.e., =48and =01846, 0 1874, 0 1891, ex- +2TrD (y,y)TrD (x, x) cept lattice size; our lattice, 83 × 16, is smaller. We − 2TrD−1(y,y)TrD−1(x, x) employ the point source and sink; smaller lattice size − D−1 x, y D−1 y,x leads to larger mass due to a higher state mixture, = Tr ( ) ( ) Gσ = Cexp(−mt)+C exp(−m t)+···.Inthepoint +2(σ(y) −σ(y))(σ(x) −σ(x))(7) source/sink case, C, C , ··· are positive. In other words, where σ(x) ≡ TrD−1(x, x). “Tr” represents a summation our mass values on the small size lattice should be consid- over color and Dirac spinor indices. In Eq.(7), D−1’s are ered as the upper limit. We have checked that the values u and d quark propagators. Here we assumed that the u of mπ and mρ are consistent with those of CP-PACS. and d quark propagators are equivalent because u and d quark masses are almost the same.
From Eq.(7), we see that the σ propagator consists -2 10 of two terms. The first term corresponds to the con- nected diagram, i.e.,a¯qq-type meson which is the first ) term in Fig.3. The second term is the “disconnected” τ −1 -4 diagram; it is the correlation of σ(x)=TrD (x, x)at G( 10 x y π two points and corresponding to the second term in ρ Fig.3. The term “disconnected” is not appropriate since σ -6 the corresponding matrix element is, of course, not fac- 10 torized; quark lines are connected by gluon interactions. Nevertheless, we use this jargon in the following. 02468τ The quantum number of the σ meson (I =0and J P =0+) is the same as that of the vacuum, and the π ρ σ κ . vacuum expectation value of the σ operator, σ(x),does FIG. 4: Propagators of , and for =01891. not vanish. Therefore, the contribution of σ(x) should σ be subtracted from the operator. See Eq.(7). It is very difficult to evaluate the disconnected part of Note that since the full QCD simulation includes the the propagator, since we must calculate TrD−1(x, x)for q − q¯ pair creation/annihilation, not only the first two x Z qqqq qqqqqq ··· all lattice sites .Weusedthe 2 noise method to calcu- diagrams in Fig.3 but also ¯ ¯ ,¯¯ ¯ , appear at late the disconnected diagrams and the subtraction terms intermediate states. σ κ+ H x saua, of the vacuum . Each of these terms is of the order of For , the operator is ( )= a ν ¯ν ν and we ten, and (σ −σ)(σ −σ) becomes less than 10−4,as have only the connected diagram, shown in Fig.5. Therefore, high accuracy is required for −1 −1 the calculation. One thousand random Z2 numbers are Gκ(y,x)=−TrDs (x, y)D (y,x), (8) generated. Our numerical results show that the values of −1 where Ds is the s quark propagator. the first and the second terms in Eq.(7) are of the same order. Therefore, in order to obtain the signal correctly 2.0 3.0 mρa κ as the difference between these terms, high-precision nu- 2.5 s =0.1840 mσa 1.5 2 mκa merical simulations and careful analyses are required. We ()mπa 2.0 mΚ*a mΚa have investigated the relationship between the amount of 1.0 1.5 Z2 noise and the achieved accuracy in Ref.[20]. Gauge 1.0 0.5 configurations were created by HMC in the SX5 vector 0.5 supercomputer, and most disconnected propagator cal- 0.0 0.0 5.2 5.3 5.4 5.5 5.2 5.3 5.4 culations by the Z2 noise method were mainly performed 1/κ 1/κ on the SR8000 parallel machine at KEK. π ρ σ κ . 2 The propagators of , and for =01891 are FIG. 6: Left : mρ, mσ and mπ in the lattice unit as functions shown in Fig.4. The connected and disconnected parts of the inverse hopping parameter. Right : mκ, mK∗ and of the σ propagator are shown in Fig.5. It is difficult to mK in the lattice unit as functions of the inverse hopping obtain σ propagator at large τ since the precision of our parameter. The s quark hopping parameter is κs =0.1840. calculation is limited to O(G(τ)) ∼ 10−4.
-2 10 and therefore these two scalar meson masses can be dif- ferent. The κ and the valence σ (connected part) have -4 10 s
) the same structure in their propagators, but quark mass τ is heavier than those of u and d. Therefore mκ >mconn, G(
-6 m 10 where conn is a mass corresponding to the connected σ Connected part, V . Part The calculations reported here have limitations: -8 10 Disconnected Part masses of u and d quarks are not light enough, simu- lations are done at a strong coupling region and the ac- 02468 τ curacy of the disconnected part is not sufficient. These defects will, however, be gradually overcome. In lattice FIG. 5: Propagators of the connected and disconnected dia- calculation, once someone establishes the scale of simu- gram of σ for κ =0.1891. lation required to obtain meaningful results, many im- proved and large-scale works succeed. Therefore in a few years, the lattice study of scalar mesons will pro- From our results, we evaluate the critical value of the vide important and fundamental information to deepen κ . a hopping parameter c =0195(3) and lattice space = our understanding of hadron physics. 0.207(9) fm (CP-PACS has obtained κc =0.19286(14) and a =0.197(2) fm). Figure 6 (left) shows masses of ρ, σ π /κ m /m and as functions of 1 . We find that σ ρ at III. POTENTIAL IN DIQUARK SYSTEM the chiral limit is 0.33 ± 0.09. Our results indicate the σ m
3 3 exp(−βFs(R)) = TrL(x 1)TrL(x 2) + TrL(x 1)L(x 2) 4 4 0.4 T/T =5.61 = , c T/Tc=3.04 3 3 0.2 Symmetric T/T =2.02 exp(−βFas(R)) = TrL(x 1)TrL(x 2)− TrL(x 1)L(x 2) c 2 2 = (15) V/T 0 F where s(as) are the difference of the free energy with -0.2 and without the static color symmetric (anti-symmetric) Anti-symmetric x 1 x 2 R |x 1 − x 2| β /T quarks located at and , = and =1 . -0.4 The formula (15) is gauge dependent, and therefore we need the gauge fixing. We employ the stochastic gauge 0 0.5 1 1.5 2 RT fixing quantization (SGFQ) with the Lorentz-type gauge fixing proposed by Zwanziger[25]: FIG. 7: The behavior of qq free energy in symmetric and τ . dAa δS anti-symmetric channel at Langevin step width ∆ =003. µ − 1 Dab A ∂ Ab ηa, = a + µ ( ) ν ν + µ (16) dτ δAµ α ab a In order to investigate the temperature dependence of where α, Dµ and ηµ correspond to a gauge parameter, a covariant derivative and Gaussian random noise, re- these free energies in more detail, we define an effective spectively. The lattice formulation and a more detailed force as explanation of this algorithm can be found in Refs. [26– f R, T ( ) − 1 F R − F R . 28]. T 2 = T 2 ( ( +1) ( )) (17) In the quenched approximation, the system has Z3 symmetry. The gauge action is invariant under the trans- The temperature dependence of the effective force 2 formation, U4(x) → zU4(x)orz U4(x) on a time-slice strength is shown in Fig. 8. As the temperature in- hyperplane, where z =exp(2πi/3). In this transforma- creases, the effective force strength in both channels de- tion, both TrL(x 1)TrL(x 2)andTrL(x 1)L(x 2)getafactor creases. z2 or z4. After infinite sweeps, Eqs.(15) vanishes since In the leading order perturbation (LOP), coefficient 1+z2 + z4 = 0. Therefore we restrict our simulations in of the color exchange terms in the symmetric and anti- 1 ∗ 2 the region −π/3 < Phase L<π/3. symmetric channels are Cqq[6] = + 3 and Cqq[3 ]=− 3 . 0.4 ∆τ = 0.03 1 10 ∆τ = 0.02 T/T =5.61 c 0.2 ∆τ = 0.0125 2 Symmetric T/Tc=3.04 0 -f/T 10 T/Tc=2.02 V/T 0 2 f/T -1 10 -0.2 Anti-symmetric Symmetric Anti-symmetric -2 10 -0.4
-3 10 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 RT RT RT
FIG. 10: Langevin step width ∆τ dependence of singlet and FIG. 8: The temperature dependence of the effective force octet potentials at T/Tc =3.04. in symmetric and anti-symmetric channels at Langevin step width ∆τ =0.03.
C. Diquark force at zero-temperature
Data presented here are very encouraging. By fixing the gauge, we can investigate the quark-quark potential. 0 It is very interesting if we can extend the analysis to -0.1 zero temperature. An obstacle is that we measure here -0.2 the correlation between the Polyakov loops, which vanish -0.3 V /V at zero temperature. Wilson loops are not proper for -0.4 s as studying qq potential. -0.5 -0.6 T/T = 5.61 An interesting operator was proposed in Ref.[32], c which is correlations of finite length Polyakov loops: -0.7 T/Tc = 3.04 T/T = 2.02 -0.8 c Gqq¯ R L x ,t n L x ,t n †, -0.9 n ( )= ( 1 ; ) ( 2 ; ) (18) -1 0 0.5 1 1.5 RT where R ≡| x1 − x2|,and
L( x, t; n) ≡ U4( x, t)U4( x, t +1)···U4( x, t + n). (19) FIG. 9: The temperature dependence of the ratio Vs/Vas at Langevin step width ∆τ =0.03. The dash-dot line stands for In the Coulomb gauge, the operator L( x, t; n)hasthe ∗ the expected ratio, Cqq[6]/Cqq[3 ]=−0.5 at short distances. following remnant gauge symmetry in time slices,
L( x, t; n) → ω(t)†L( x, t; n)ω(t + n) (20)
Once we carry out the Coulomb gauge fixing, the corre- We compare the force strength in symmetric and anti- lation, (18), is similar to the Wilson loop, and it behaves symmetric channel and define the ratio, Fs/Fas,which as ∗ approaches the value Cqq[6]/Cqq[3 ]=−0.5 at short dis- qq¯ −σ2Rn Gn (R) ∼ e , (21) tances. Fig.9 shows the ratios at T/Tc =2.02, 3.04, 5.61. As the temperature increases the ratios at short distances in the confinement phase, where σ is the string tension. are comparable with the expected values; however, as the We extend this operator to the di-quark system. temperature decreases, it deviates from −0.5. qq G (R)=L( x1,t; n)L( x2,t; n). (22) We here check the ∆τ dependence. In Fig. 10, Vs and n Vas are shown for several values of ∆τ. They depend This qq correlation vanishes due to the remnant symme- little on ∆τ =0.0125 ∼ 0.03. try (20). Therefore we further a temporal gauge fixing in Two gauge dependent channels, qq symmetric and the Coulomb gauge, which can be realized by maximizing anti-symmetric, are well controlled in the framework of the stochastic quantization with Lorentz type gauge fix- ReT rU4( x, t). (23) ing term. As a result we find that the anti-symmetric free energy indicates the attractive force while the symmetric Under the transformation, U4(x) → zU4(x)or 2 one the repulsive force. z U4(x), on a time-slice hyperplane between t and t + n, 2 4 L( x1,t; n)L( x2,t; n) get a factgor z or z . Therefore Based on the formula, we are now preparing the cal- qq after infinite sweeps, Eq.(22) vanishes and we should re- culation of Gn (R) to study the diquark force in the con- strict our simulations in the region −π/3 < Phase L< finement phase. π/3.
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