aa r X i v : . [ h e p - l a t ] N ov Lattice calculation of κ meson Ziwen Fu ∗ Key Laboratory of Radiation Physics and Technology (Sichuan University) , Ministry of Education;Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610064, P. R. China.
We study the κ meson in 2 + 1 flavor QCD with sufficiently light u/d quarks. Using numericalsimulation we measure the point-to-point κ correlators in the “Asqtad” improved staggered fermionformulation. We analyze those correlators using the rooted staggered chiral perturbation theory(rS χ PT), particular attention is paid to the bubble contribution. After chiral extrapolation, weobtain the physical κ mass with 828 ±
97 MeV, which is within the recent experimental value 800 ∼ a ≈ .
12 fm.
PACS numbers: 12.38.Gc, 11.15.Ha, 12.40.Yx, 14.40.Df
I. INTRODUCTION
The so-called κ meson ( J P = 0 + ) is a scalar me-son with strangeness. In 2010, Particle Data Group(PDG) [1] lists the meson K ∗ (800) or κ with a very broadwidth (550 MeV). A recent analysis [2] gives its massabout 750 MeV. Moreover, the resonance of a scalarmeson is reported [2–5] to exist in the πK system with amass of the κ meson about 800 MeV.Until now, four lattice simulations of the κ mass havebeen reported. Prelovsek et al. reported a crude esti-mation of the κ mass as 1 . a mass [6]. Mathur et al. [7] studied the u ¯ s scalar me-son in the quenched approximation, and obtained thevalue of the κ mass to be 1 . ± .
12 GeV after remov-ing the fitted πη ′ ghost. With the dynamical N f = 2 seaquarks and a valence strange quark, the UKQCD Collab-oration [8] suggested the κ mass around 1000 − κ meson using the dynamical fermionfor the light u/d quark and the valence approximationfor the strange quark, which shows that the I = 1 / . κ mass to be about 1 . κ mass aslow as 826 ±
119 MeV in our previous study [12], however,here we neglected the taste-symmetry breaking and usedthe crude linear extrapolation. It is well known that, inthe staggered fermion formulation of lattice QCD, dueto the taste-symmetry breaking there exist many multi-hadron states with J P = 0 + which can proliferate be-tween the source and sink of the κ correlator. Of spe-cial interest for us are the two-pseudoscalar states (i.e.,bubble contribution) [13, 14]. With sufficiently light u quark, the κ meson propagator is dominated at large ∗ Electronic address: [email protected] time distances by these two-meson states. The bubblecontributions are significantly affected by the unphysicalapproximations which are often used in lattice simula-tions [13, 14]. They are expected to disappear in thecontinuum limit.In our previous work [15–18], we extended the analysisof Refs. [13, 14], examined the scalar meson correlators inlattice QCD with the inclusion of the disconnected dia-grams, and carried out a quantitative comparison of themeasured correlators and the predictions from rS χ PT.Despite the considerable complexity of the scalar mesonchannels with dozens of spectral components, the rS χ PTprovides a strict framework which permits the analysisof the scalar meson correlator precisely in terms of onlya small number of low-energy chiral couplings, which wemay determine through fits to the data. In this workwe extend the analyses of Refs. [15–18] to the κ me-son, treat the u quark as a valence approximation quark,while the valence strange quark mass is fixed to its phys-ical value [19] considering that the κ meson contains astrange quark and a light u quark, we perform a series ofnumerical simulations with MILC gauge configurationsin the presence of 2 + 1 flavors of Asqtad improved stag-gered dynamical sea quarks, generated by the MILC Col-laboration [20], and chirally extrapolate the mass of the κ meson to the physical π mass using the popular fourparameter fit with the inclusion of the chiral logarithms. II. PSEUDOSCALAR MESON TASTEMULTIPLETS
In Refs. [15, 16], we give a brief review of the rootedstaggered chiral perturbation theory with particular fo-cus on the tree-level pseudoscalar mass spectrum, andachieve the rooted version of the theory through the repli-cated theory [21].The tree-level masses of the pseudoscalar mesonsare [16, 22] M x,y,b = µ ( m x + m y ) + a ∆ b , (1)where x, y are two quark flavor contents which make up, b = 1 , ...,
16 are the taste, µ = m π / m q is the low-energychiral coupling constant of the point scalar current to thepseudoscalar field, and the term of a ∆ b comes from thetaste symmetry breaking. The m x and m y are the twovalence quark masses in the pseudoscalar meson, and m x is the light valence u/d quark mass by convention.In this work we investigate degenerate u and d quarks,treat u quark as a valence approximation quark, while thevalence strange quark mass is fixed to its physical value m s , thus it will be convenient to introduce the notations M Ub ≡ M π b = 2 µm x + a ∆ b M Sb ≡ M ss,b = 2 µm s + a ∆ b (2) M Kb ≡ M K b = µ ( m x + m s ) + a ∆ b , where M U is the mass of the Goldstone pion with twolight valence quark masses, M K is the mass of the Gold-stone kaon with one valence quark equal to the light va-lence quark and one at its physical mass m s , and M S isthe mass of a fictitious meson s ¯ s in a flavor nonsingletstate [23] with two valence quarks at physical mass m s .The isosinglet states ( η and η ′ ) are modified both bythe taste-singlet anomaly and by the two-trace (quark-line hairpin) taste-vector and taste-axial-vector operators[23, 24]. When the anomaly parameter m is large, weobtain the usual result M η,I = 13 M UI + 23 M SI , M η ′ ,I = O ( m ) . (3)In the taste-axial-vector sector we have M ηA = 12 [ M UA + M SA + 34 δ A − Z A ] M η ′ A = 12 [ M UA + M SA + 34 δ A + Z A ] (4) Z A = ( M SA − M UA ) − δ A M SA − M UA ) + 916 δ A , and likewise for V → A , where δ V = a δ ′ V is the hairpincoupling of a pair of taste-vector mesons, and δ A = a δ ′ A is the hairpin coupling of a pair of taste-axial mesons ( δ ′ V and δ ′ A are tree-level (LO) taste-violating hairpin param-eters [19]).In the taste-pseudoscalar and taste-tensor sectors, inwhich there is no mixing of the isosinglet states, themasses of the η b and η ′ b by definition are M η,b = M Ub ; M η ′ ,b = M Sb . (5)In Table I, we list the masses of the resulting taste multi-plets in lattice units for our chosen lattice ensemble withthe taste-breaking parameters δ A and δ V determined inRefs. [19, 23]. For Goldstone multiplet (taste P ), wemeasured their corresponding correlators and fitted themwith a single-exponential [19]. Then, using the tastesplittings in Refs. [19, 23], we calculated the masses ofother non-Goldstone taste multiplets. We do not needthe η ′ I masses in this work, hence, we do not list thesevalues in Table I. TABLE I: The mass spectrum of the pseudoscalar mesonfor MILC medium-coarse ( a = 0 .
12 fm) lattice ensemble with β = 6 . am ′ ud = 0 . am ′ s = 0 . am x taste( B ) aπ B aK B aη B aη ′ B .
005 P 0 . . . . . . . . . . . . . . . . . . . − .
010 P 0 . . . . . . . . . . . . . . . . . . . − .
015 P 0 . . . . . . . . . . . . . . . . . . . − .
020 P 0 . . . . . . . . . . . . . . . . . . . − .
025 P 0 . . . . . . . . . . . . . . . . . . . − We should remind the readers that the data in Table Iare obtained with the valence strange quark mass fixedto its physical value. Here and below, we adopt the no-tation in Ref. [19], the primes on masses indicate thatthey are the dynamical quark masses used in the latticesimulations, not the physical masses m u , m d and m s . III. THE κ CORRELATOR FROM S χ PT In Refs. [15–18], using the language of the replicatrick [22, 25] and through matching the point-to-pointscalar correlators in chiral low energy effective theoryand staggered fermion QCD, we rederive the “bubble”contribution to the a channel of Ref. [14], and extendthe result to the σ channel. Here we further extend theseresults to the κ channel. A. Non-singlet κ correlator To simulate the correct number of quark species, weuse the fourth-root trick, which automatically performsthe transition from four tastes to one taste per flavor forstaggered fermion at all orders. We employ an interpo-lation operator with isospin I = 1 / J P = 0 + at thesource and sink, O ( x ) ≡ √ n r X a,g ¯ s ag ( x ) u ag ( x ) , (6)where g is the indices of the taste replica, n r is the num-ber of the taste replicas, a is the color indices, and weomit the Dirac-Spinor index. The time slice correlator C ( t ) for the κ meson can be evaluated by C ( t )= 1 n r X x ,a,b X g,g ′ (cid:10) ¯ s bg ′ ( x , t ) u bg ′ ( x , t ) ¯ u ag ( , s ag ( , (cid:11) , where , x are the spatial points of the κ state at source,sink, respectively. After performing Wick contractionsof fermion fields, and summing over the taste index [12],for the light u quark Dirac operator M u and the s quarkDirac operator M s , we obtain C ( t ) = X x ( − ) x D Tr[ M − u ( x , t ; 0 , M − † s ( x , t ; 0 , E , (7)where Tr is the trace over the color index, and x = ( x , t )is the lattice position.The mass of the κ meson can be reliably determined onthe lattice simulation. However, there exist many mul-tihadron states with J P = 0 + which can propagate be-tween the source and the sink. Of special interest forus is the intermediate state with two pseudoscalars P P which we refer to as the bubble contribution ( B ) [14].If the masses of P and P are small, the bubble termgives a considerable contribution to the κ correlator, andit should be included in the fit of the lattice correlator inEq. (7), namely, C ( t ) = Ae − m κ t + B ( t ) , (8)where we omit the unimportant contributions from theexcited states, the oscillating terms corresponding to aparticle with opposite parity, and other high-order mul-tihadron intermediate states. B. Coupling of a scalar current to pseudoscalar
Before we embark on the bubble contribution, herewe first derive the coupling of a point scalar current¯ s r ( x ) u r ( x ) to a pair of the pseudoscalar fields at thelowest energy order of the staggered chiral perturbationtheory (S χ PT), where the subscript r in the expression u r ( x ) is the index of the taste replica for a given quarkflavor u . The effective scalar current can be determined from the dependence of the lattice QCD
Lagrangian andthe staggered chiral Lagrangian on the spurion field M ,where M is the staggered quark mass matrix. For n Kogut-Susskind (KS) flavors, M is a 4 nn r × nn r ma-trix. M = m uu I ⊗ I R m ud I ⊗ I R m us I ⊗ I R · · · m du I ⊗ I R m dd I ⊗ I R m ds I ⊗ I R · · · m su I ⊗ I R m sd I ⊗ I R m ss I ⊗ I R · · · ... ... ... . . . , (9)where I is a 4 × I R is the n r × n r unitreplica matrix. In short, M = m ⊗ I ⊗ I R , and m = m uu m ud m us · · · m du m dd m ds · · · m su m sd m ss · · · ... ... ... . . . (10)is the n × n quark mass matrix. Since the staggeredchiral lagrangian ( L SχP T ) is an effective equivalent La-grangian for L QCD in low energy limit, the effective cur-rent ¯ s r ( x ) u r ( x ) is obtained from¯ s r ( x ) u r ( x ) = − ∂ L SχP T ∂ M s r u r ( x ) , (11)where L SχP T = 18 f π Tr[ ∂ µ Σ ∂ µ Σ † ] − µf π Tr[ M † Σ + Σ † M ]is the staggered chiral Lagrangian [22]. We omit thehigh order terms and the terms that are independentof M , f π is the tree-level pion decay constant [22], µ is the constant with the dimension of the mass [26], andΣ = exp (cid:18) i Φ f π (cid:19) [22]. The field Φ = X b =1 T b ⊗ φ b isdescribed in terms of the mass eigenstate field φ b [22],where φ b is a 3 × φ bf r f ′ r ′ with flavor f, f ′ , the index of the tastereplica r, r ′ , and taste b which is given by generators T b = { ξ , iξ ξ µ , iξ µ ξ ν , ξ µ , ξ I } [22]. Hence, the field Φis 4 nn r × nn r pseudoscalar matrix in S χ PT [22], andthe subscripts u, s denote its valance flavor component.Therefore, Σ is also a 4 nn r × nn r matrix. The Tr is thefull 4 nn r × nn r trace. Then, the effective current is [26]¯ s r ( x ) u r ( x ) = µ Tr t [Φ( x ) ] s r u r , (12)where the notation Tr t stands for the trace over taste,and µ is the low-energy chiral coupling of the point scalarcurrent to the pseudoscalar field Φ( x ). C. Bubble contribution to the κ correlator In this subsection we compute the bubble contributionto κ correlator in Eq. (7) from two intermediate states.From the above discussion, the point scalar current canbe described in terms of the pseudoscalar field Φ by usingS χ PT [26][14],¯ s r ( x ) u r ( x ) = µ Tr t [Φ( x ) ] s r u r , ¯ u r ( x ) s r ( x ) = µ Tr t [Φ( x ) ] u r s r . (13)For concreteness, the bubble contribution to the κ cor-relator in a theory with n r tastes per flavor and threeflavors of KS dynamical sea quarks is [22] B SχP Tκ ( x )= µ n r n r X r,r ′ =1 ( h ¯ s r ( x ) u r ( x ) ¯ u r ′ (0) s r ′ (0) i Bubble ) = µ n r n r X r,r ′ =1 ( (cid:10) Tr t [Φ ] s r u r Tr t [Φ ] u r ′ s r ′ (cid:11)) , where the subscript κ specifies the bubble contribution for κ meson. If we consider this identityTr t (cid:0) T a T b (cid:1) = 4 δ ab , we arrive at B SχP Tκ ( x ) = µ n r X a =1 16 X b =1 N f X i =1 N f X j =1 n r X r,r ′ =1 n r X t,t ′ =1 h φ as r i t ( x ) φ ai t u r ( x ) φ bu r ′ j t ′ (0) φ bj t ′ s r ′ (0) i , where a, b, a ′ , b ′ are the taste indices, i, j are the flavorindices, and r, t, r ′ , t ′ are the indices of the taste replica.The Wick contractions result in the products of two prop-agators for the pseudoscalar fields. After summing overthe index of the taste replica, and considering that u, d quarks are degenerate, the bubble contribution can be ex-pressed in terms of the pseudoscalar propagators h φ b φ b i . B SχP Tκ ( x ) = n r µ X b =1 ( h φ bsd ( x ) φ bus (0) i conn h φ bss ( x ) φ bss (0) i conn + h φ bsu ( x ) φ bus (0) i conn h φ buu ( x ) φ buu (0) i conn + h φ bsu ( x ) φ bus (0) i conn h φ bss ( x ) φ bss (0) i conn ) + µ X b = I,V,A ( h φ bsu ( x ) φ bus (0) i conn h φ buu ( x ) φ buu (0) i disc + h φ bsu ( x ) φ bus (0) i conn h φ bss ( x ) φ bss (0) i disc +2 h φ bsu ( x ) φ bus (0) i conn h φ buu ( x ) φ bss (0) i disc ) . (14)The subscript conn and subscript disc stand for theconnected contribution and the disconnected contribu-tion, respectively. The propagators h φ b φ b i for the pseu-doscalar field φ b for various tastes b are intensivelystudied in Ref. [22]. The propagators for all tastes( I, V, A, T, P ) have the connected contributions, whileonly tastes
I, V and A have the disconnected contribu-tions [22]. h φ buu ( x ) φ buu (0) i V disc = − δ V ( k + M S V )( k + M U V )( k + M η V )( k + M η ′ V ) , h φ bss ( x ) φ bss (0) i V disc = − δ V ( k + M U V )( k + M S V )( k + M η V )( k + M η ′ V ) , h φ buu ( x ) φ bss (0) i V disc = − δ V ( k + M η V )( k + M η ′ V ) , h φ buu ( x ) φ buu (0) i I disc = − k + M S I ( k + M U I )( k + M ηI ) , h φ bss ( x ) φ bss (0) i I disc = − k + M U I ( k + M S I )( k + M ηI ) , h φ buu ( x ) φ bss (0) i I disc = −
43 1 k + M ηI . And likewise for the axial taste ( A ), we just require V → A . By plugging in the above mesonic propaga-tors (namely, carrying out the Wick contractions) andswitching to momentum space, Eq. (14) can be rewrittenas B SχP Tκ ( p )= µ X k ( n r X b =1 " k + p ) + M K b k + M U b + 1( k + p ) + M K b k + M S b − k + p ) + M K I k + M U I + 23 1( k + p ) + M K I k + M ηI − k + p ) + M K I k + M S I − δ V ( k + p ) + M K V k + M S V ( k + M M V )( k + M η V )( k + M η ′ V ) − δ A ( k + p ) + M K A k + M S A ( k + M U A )( k + M η A )( k + M η ′ A ) − δ V ( k + p ) + M K V k + M U V ( k + M S V )( k + M η V )( k + M η ′ V ) − δ A ( k + p ) + M K A k + M U A ( k + M S A )( k + M η A )( k + M η ′ A ) − δ V ( k + p ) + M K V k + M η V )( k + M η ′ V ) − δ A ( k + p ) + M K A k + M η A )( k + M η ′ A ) ) , (15)And likewise for the axial taste ( A ), we just require V → A . In Table I, we list all these pseudoscalar massesneeded for the current study.In the continuum limit, taste-symmetry is restored,namely, δ V = a δ ′ V → δ A = a δ ′ A →
0, then Eq. (15)reduces to B κ ( p )= µ X k ( n r X b =1 k + p ) + M K b k + M U b + X b =1 n r ( k + p ) + M K b k + M S b − k + p ) + M K I k + M U I + 23 1( k + p ) + M K I k + M ηI − k + p ) + M K I k + M S I ) . Here the total contribution from pairs of the states withmass M U and M S is proportional to(16 n r − , (16)which vanishes when n r = 1 /
4. The negative thresh-old has nicely canceled out the unphysical threshold KS .The surviving thresholds are the physical ηK . IV. SIMULATIONS AND RESULTS
We use the MILC lattices with 2 + 1 dynamical flavorsof the Asqtad-improved staggered dynamical fermions,the detailed description of the simulation parameters canbe found in Refs. [20, 23, 27]. We analyzed the κ correla-tors on the 0 .
12 fm MILC ensemble of 520 24 ×
64 gaugeconfigurations with bare quark masses am ′ ud = 0 . am ′ s = 0 .
05 and bare gauge coupling 10 /g = 6 . a − = 1 . +49 − GeV, which has aphysical volume approximately 2 . am s = 0 . u and d quarksare degenerate. In Table I, we list all the pseudoscalarmasses used in our fits with the exception of the masses M η A , M η ′ A , M η V , and M η ′ V . Those masses can vary withthe fit parameters δ A and δ V .For the light u quark Dirac operator M u and the s quark Dirac operator M s , we measure the point-to-pointquark-line connected correlator which is described byEq. (7). We use the conjugate gradient method (CG) toobtain the required matrix element of the inverse fermionmatrix M u and M s .In order to improve the statistics, we place the sourceon all the time slices t s = 0 , · · · , T −
1, therefore, weperform T = 64 inversions for each configuration and av-erage these correlators. Note that the time extent of ourlattices is more than twice the spatial extent. The ratherlarge effort to generate propagators allows us to evaluatethe correlators with high precision, which is importantto extract the desired κ masses reliably.Since the κ meson contains a strange s quark and alight u quark, we should treat the u quark as a valenceapproximation quark, while the valence strange quarkmass is fixed to its physical value [19]. The physical valueof the strange quark mass of the lattice ensemble used inthe present work has been precisely determined by MILCsimulations [20], namely, am s = 0 . a is thelattice spacing.The propagators of the κ meson are calculated with thesame configurations using five u valence quarks, namely,we choose am x = 0 . .
01, 0 . .
02 and 0 . m x is the light valence u quark mass. In orderto obtain the physical mass of the κ meson, we then per-form extrapolation to the chiral (physical π mass, ob-tained from PDG) limit guided by chiral perturbationtheory. The correlators of the π, K meson and fictitiousmeson s ¯ s are also measured with the same configurationsfor calculating the pseudoscalar masses in Table I.Figure 1 shows the κ propagators with five differentlight valence u quark masses and their predicted bubblecontributions. For our chosen MILC configurations usedin the present study we obtain the positive predicted bub-ble contributions, since the small negative contributionin the bubble term is most likely outweighed by the pos- FIG. 1: The κ propagators for five valence u quarks. Overlaidon the data are their corresponding predicted bubble contri-butions. itive contribution as is discussed for the a correlators inRef. [14]. We add a constant 7 . e − κ propagatorsafter t ≥ C ( t ) = X i A i e − m i t + X i A ′ i ( − t e − m ′ i t + ( t → N t − t ) , (17)where the oscillating terms correspond to a particle withopposite parity. For the κ meson correlator, we consideronly one mass with each parity in the fits of Eq. (17),namely, in our concrete calculation, our operator is thestate with spin-taste assignment I ⊗ I and its oscillatingterm with spin-taste assignment γ γ ⊗ γ γ [16]. Fromthe aforementioned discussion, we must consider the bub-ble contribution. Therefore, all five κ correlators werethen fit to the following physical model C κ ( t ) = C meson κ ( t ) + B κ ( t ) , (18)here C meson κ ( t ) = b κ e − m κ t + b K A ( − ) t e − M KA t + ( t → N t − t ) , where the b K A and b κ are two overlap factors, and thebubble term B κ in the fitting function Eq (18) is given inmomentum space by Eq. (15). The time-Fourier trans-form of it yields B κ ( t ).This fitting model contains the explicit κ pole, togetherwith the corresponding negative-parity state K A and thebubble contribution. There are four fit parameters (i.e., FIG. 2: The κ masses as a function of the minimum timedistance. The effective mass plots will be a plateau in timerange 9 ≤ D min ≤ M κ , M K A , b K A , and b κ ) for each κ correlator with a givenvalence u quark mass m x . The bubble term B κ ( t ) was pa-rameterized by three low-energy couplings µ , δ A , and δ V .In our concrete fit, they were fixed to the values of theprevious MILC determinations [19]. The taste multipletmasses in the bubble terms were fixed as listed in Table I.The sum over intermediate momenta was cut off whenthe total energy of the two-body state exceeded 2 . /a orany momentum component exceeded π/ (4 a ). We deter-mined that such a cutoff gave an acceptable accuracy for t ≥
8. The lightest intermediate state in bubble term is πK . Therefore, this fitting model can remove a numberof unwanted πK states with different tastes and slightlydifferent energies.For am x = 0 . κ meson are shown in Figure 2. We find that the effec-tive κ mass suffers from large errors, especially in largerminimum time distance regions. To avoid possible largeerrors coming from the data at large minimum time dis-tance D min , we fit the effective mass of the κ meson onlyin the time range 9 ≤ D min ≤
11, where the effectivemasses are almost constant with small errors.In our fit, five κ propagators were fit using a minimumtime distance of 10 a . At this distance, the contaminationfrom the excited states is comparable to the statisticalerrors, we can neglect the systematic effect due to excitedstates, therefore we can extract the mass of the κ mesonefficiently.The fitted masses of the κ correlators are summarizedin Table II. The second block shows the masses of the κ meson in lattice units, and Column Four shows thetime range for the chosen fit. As a consistency check, wealso list the fitted masses of their corresponding negativeparity state K A in Column Three. We can note that thefitted values of the pseudoscalar meson K A masses areconsistent with our calculated values in Table I withinsmall errors. Column Five shows the number of degrees TABLE II: The summary of the results for the fitted κ masses. The second block shows the κ masses in lattice units.The third block shows the fitted K A masses. am x am κ aM K A Range χ / dof0 .
005 0 . . −
25 12 . / .
010 0 . . −
25 11 . / .
015 0 . . −
25 11 . / .
020 0 . . −
25 11 . / .
025 0 . . −
25 11 . / of freedom (dof) for the fit.In order to obtain the physical mass of the κ meson,we carry out the chiral extrapolation of the κ mass m κ tothe physical π mass using the popular three parameterfit with the inclusion of the next-to-next-to-leading order(NNLO) chiral logarithms. The general structure of thepion mass dependence of m κ can be written down as m κ = c + c m π + c m π + c m π ln( m π ) , (19)where c , c , c and c are the fitting parameters, and thefourth term is the NNLO chiral logarithms.We obtain the physical π mass from PDG [1] and use itas a chiral limit. In Figure 3, we show how physical value m κ is extracted, which gives χ / dof = 0 . /
1. The bluedashed line in Figure 3 is the linear extrapolation of themass of the κ meson to the physical pion mass m π . Thechirally extrapolated κ mass m κ = (828 ±
97) MeV, whichis in good accordance with the result in our previousstudy on a MILC medium coarse lattice ensemble [12].The cyan diamond in Figure 3 indicates this value.Using the fitting model in Ref. [23], we extract kaonmasses. In Figure 3, we display m κ , m K , m π , and m π + K in lattice units as a function of the pion mass m π . Weobserve that, as the valence quark mass increases, πK threshold grows faster than κ mass and, as a consequence, πK threshold is higher than κ mass for large pion mass(about am x ≥ . πK threshold is lower than κ meson for small pion mass (about am x ≤ . κ meson can decay on our lattice for small quarkmass, but for large quark mass, the decay κ → πK is nolonger allowed kinematically, which is in good agreementwith the results in Ref. [28].To understand the effects of these bubble contributions(or taste breakings), in the present study we also fittedour measured kappa correlators without bubble terms.The fitted results are tabulated in Table III. From Ta-ble II and Table III, we can clearly see that the bubbleterms contribute about 2% −
5% differences for the κ masses. V. SUMMARY AND OUTLOOK
In the present study we have extended the analyses ofthe scalar mesons in Refs. [15, 16], and derived the two-
FIG. 3: Characteristics of m κ , m K , m π and m π + K in latticeunits as a function of the pion mass. The chiral limit is ob-tained at the physical pion mass m π .TABLE III: The summary of the results for the fitted κ masses without bubble contributions. The second block showsthe κ masses in lattice units. The third block shows the fitted K A masses. am x am κ aM K A Range χ / dof0 .
005 0 . . −
25 12 . / .
010 0 . . −
25 11 . / .
015 0 . . −
25 10 . / .
020 0 . . −
25 10 . / .
025 0 . . −
25 10 . / pseudoscalar-meson “bubble” contribution to the κ cor-relator in the lowest order S χ PT. We used this physicalmodel to fit the lattice simulation data of the point-to-point scalar κ correlators for the MILC coarse ( a ≈ . u quark as a valence approximation quark, whilethe strange valence s quark mass is fixed to its physicalvalue, and chirally extrapolated the mass of the κ me-son to the physical pion mass. We achieved the physicalmass of κ meson with 828 ±
97 MeV, which is very closeto the recent experimental value 800 ∼
900 MeV. Prob-ably, it may be identified with the κ meson observed inexperiments.Most of all, we note that κ meson is heavier than πK threshold for enough small u quark mass. Therefore, itcan decay on our lattice for small quark mass. This pre-liminary lattice simulation will stimulate people to studythe decay mode κ → πK . We are beginning latticestudy of this decay channel with isospin representationof I = 1 / κ correlator in this work also provides a direct usefultest of the fourth-root recipe. The bubble term in S χ PTprovides a useful explanation of the lattice artifacts in-duced by the fourth-root approximation [15, 16]. The ar-tifacts include the thresholds at unphysical energies andthe thresholds with negative weights. These contribu-tions are clearly present in the κ channel in our QCD sim-ulation with the Asqtad action at a ≈ .
12 fm. We findthat the “bubble” term must be included in a successfulspectral analysis of the κ correlator. The rS χ PT predictsfurther that these lattice artifacts vanish in the contin-uum limit, leaving only physical two-body thresholds. Itwould be nice to be able to investigate whether this ex-pectation is ruled out in lattice simulations at smallerlattice spacing.In this work we reported our preliminary results onone lattice ensemble, more physical one should be in thecontinuum limit. We are beginning a series of numericalsimulations with the MILC fine and super-fine lattice en-sembles. Furthermore, the kappa meson is a resonance,i.e. a state with a considerable width under strong in- teractions. In order to map out “avoided level crossings”between the resonance and its decay products in a finitebox volume proposed by L¨uscher [29], we are beginningto measure the πK scattering K + π → K + π channel andthe κ → K + π, K + π → κ cross-correlators in addition tothe κ → κ correlator [30–33]. Hopefully, we can reliablyobtain the resonance parameters of the κ resonance. Acknowledgments
This work is supported in part by Fundamen-tal Research Funds for the Central Universities(2010SCU23002) and by the Startup Grant from the In-stitute of Nuclear Science and Technology of SichuanUniversity. We would thank Carleton DeTar for kindlyproviding the MILC gauge configurations used for thiswork, and thank the MILC Collaboration for using theAsqtad lattice ensemble and MILC code. We are gratefulto Hou Qing for his support. The computations for thiswork were carried out at AMAX workstation, CENTOSworkstation, HP workstation in the Radiation PhysicsGroup of the Institute of Nuclear Science and Technol-ogy, Sichuan University. [1] Nakamura K et al. (Particle Data Group). J. Phys. G,2010, : 075021[2] Bugg D. Phys. Lett. B, 2006, : 471[3] Aitala M et al. Phys. Rev. Lett., 2002, : 121801[4] M. Ablikim et al. , Phys. Lett. B , 88 (2010).[5] Ablikim M et al. Phys. Lett. B, 2006, : 681[6] Prelovsek S, Dawson C, Izubuchi T et al. Phys. Rev. D,2004, : 094503[7] Mathur N, Alexandru A, Chen Y et al. Phys. Rev. D,2007, : 114505[8] McNeile C and Michael C. Phys. Rev. D, 2006, : 014508[9] Kunihiro T, Muroya S, Nakamura A et al. Phys. Rev. D,2004, : 034504[10] Kunihiro T, Muroya S, Nakamura A et al. Nucl. Phys.Proc. Suppl., 2004, : 242[11] Wada H, Kunihiro T, Muroya S et al. Phys. Lett. B, 2007, : 250[12] Fu Zi-wen et al. Lattice QCD study of the κ scalar meson.Chin. Phys. C (HEP & NP), to be published.[13] Prelovsek S. PoS, LAT2005, 2006, 085[14] Prelovsek S. Phys. Rev. D, 2006, : 014506[15] Fu Zi-Wen. Hybrid Meson Decay from the Lattice. UMI-32-34073, 2006, arXiv:1103.1541 [hep-lat].[16] Bernard C, DeTar C, Fu Z et al. Phys. Rev. D, 2007, : 094504[17] Fu Zi-Wen. Chin. Phys. Lett, 2011, (8): 081202[18] Fu Zi-Wen et al. Chin. Phys. C (HEP & NP), 2011, (10): 896[19] Aubin C et al. Phys. Rev. D, 2004, : 114501[20] Bazavov A et al. Rev. Mod. Phys., 2010, : 1349[21] Aubin C and Bernard C. Nucl. Phys. B, Proc. Suppl.,2004, : 182[22] Aubin C and Bernard C. Phys. Rev. D, 2003, : 034014[23] Aubin C et al. Phys. Rev. D, 2004, : 094505[24] Bernard C et al. Phys. Rev. D, 2001, : 054506[25] Damgaard P and Splittorff K. Phys. Rev. D, 2000, : 054509[26] Bardeen W, Duncan A, Eichten E et al. Phys. Rev. D,2001, : 014509[27] Bernard C et al. Phys. Rev. D, 2011, : 034503[28] Nebreda J and Pelaez J. Phys. Rev. D, 2010, : 054035[29] Lellouch L and L¨uscher M. Commun. Math. Phys, 2001,219