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Lattice study on πK scattering with moving wall source 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. The s-wave pion-kaon (πK) scattering lengths at zero momentum are calculated in lattice QCD with sufficiently light u/d quarksand strange quarkat its physical valueby thefinite size formula. Thelightquarkmassescorrespondtom =0.330−0.466GeV.Inthe“Asqtad”improvedstaggered π fermion formulation, we measure the πK four-point correlators for both isospin I = 1/2 and 3/2 channels,andanalyzethelatticesimulationdataatthenext-to-leadingorderinthecontinuumthree- flavor chiral perturbation theory, which enables us a simultaneous extrapolation of πK scattering lengthsat physicalpoint. Weadopt atechniquewith themovingwall sourceswithout gauge fixing to obtain the substantiable accuracy, moreover, for I = 1/2 channel, we employ the variational 2 method to isolate the contamination from the excited states. Extrapolating to the physical point 1 yields the scattering lengths as m a = −0.0505(19) and m a = 0.1827(37) for I = 3/2 and π 3/2 π 1/2 0 1/2 channels, respectively. Our simulation results for πK scattering lengths are in agreement with 2 the experimental reports and theoretical predictions, and can be comparable with other lattice n simulations. These simulations are carried out with MILC Nf = 2+1 flavor gauge configurations a at lattice spacing a≈0.15 fm. J 8 PACSnumbers: 12.38.Gc 1 I. INTRODUCTION was claimed as m a = 0.129 0.05. However, if ] π 3/2 − ∼ − t the scatteringhadrons containstrangequarks, χPT pre- a dictions usually suffer from considerable corrections due l Pion-kaon (πK) scattering at low energies is the sim- - to the chiral SU(3) flavor symmetry breaking, as com- p plest reactions including a strange quark, and it allows e for an explicit exploration of the three-flavor structure pared with the case of the ππ scattering. Therefore, a h lattice QCD calculation is needed to offer an alternative of the low-energy hadronic interactions, which is not di- [ important consistent check of the validity of χPT in the rectly probed in the ππ scattering. The measurement presence of the strange quarks. 2 of πK scattering lengths is one of the cleanest processes v and a decisive test for our understanding of the chiral 2 To date, four lattice studies of πK scattering length SU(3) symmetry breaking of the quantum chromody- 2 have been reported [10–13]. The first lattice calcula- namics (QCD). Inthe presentstudy, we will concentrate 4 tion of πK scattering length in I = 3/2 channel was 1 on the s-wave scattering lengths of πK system, which explored by Miao et al. [10] using the quenched ap- . have two isospin eigenchannels (I = 3/2,1/2) in the 0 isospin limit, and the low-energy interaction is repulsive proximation, and the value of mπa3/2 was found to 1 be 0.048. The first fully-dynamical calculation using for I = 3/2 channel, and attractive for I = 1/2 case, − 1 N =2+1 flavorsof the Asqtad-improved [14, 15] stag- f 1 respectively. gered sea quark [16, 17] was carried out [11] to calculate v: Experimentally, πK scattering lengths are obtained the I =3/2 scattering length for mπ =0.29 0.60 GeV, i through πK scattering phases using the Roy-Stainer and further indirectly evaluate the I = 1/−2 scattering X equations. The experiments at low energies are an im- lengthonthe basisof χPT.They obtainedasmallnega- r portant method in the study of the interactions among tive value of m a = 0.0574 for I =3/2 channel and a π 3/2 mesons [1–3], and these experiments have reported that a positive value of m a− =0.1725for I =1/2 channel, π 1/2 the s-wave scattering length (a0) in the I = 3/2 chan- respectively. Nagataetal. fulfilledfirstdirectlatticecal- nel, mπa3/2 has a smallnegative value, namely, 0.13 culation on I =1/2 channel [12] using the quenched ap- − ∼ 0.05. Moreover,the on-going experiments proposed by proximation. They investigated all quark diagrams con- − the DIRAC collaboration [4] to examine πK atoms will tributing to both isospin eigenstates,and found that the provide the direct measurements or constraints on πK scattering amplitudes can be expressed as the combina- scattering lengths. tions of only three diagrams in the isospin limit. This Atpresent,theorypredictsπK scatteringlengthswith work greatly inspires us to study πK scattering. How- a precision of about 10%, and it will be significantly im- ever, they did not observe the repulsive interaction even proved in the near future. Through the scalar form fac- for I =3/2 channel at their simulation points, and their tors in semi-leptonic pseudo-scalar-to-pseudo-scalar de- lattice calculations are relatively cheaper. Sasaki et al. cays, Flynn et al. [5] extracted the πK scattering length observed the correct repulsive interaction for I = 3/2 in the I = 1/2 channel as m a = +0.179(17)(14). channel and attractive for I = 1/2 case, and they ob- π 1/2 Three-flavorChiralPerturbationTheory(χPT)[6–9]has tained the scattering lengths of m a = 0.0500(68) π 3/2 − beenusedtopredictthescatteringlengthsinthestudyof and m a = 0.154(28) for the I = 3/2 and 1/2 chan- π 1/2 the low-energy πK scattering, and small negative value nels, respectively [13]. Moreover, to isolate the contam- 2 ination from the excited states, they construct a 2 2 A. πK four-point functions × matrix of the time correlation function and diagonalize it [13], this method will guide us to study πK scatter- Let us consider the πK scattering of one Nambu- ing for I = 1/2 channel in a correct manner. It should Goldstone pion and one Nambu-Goldstone kaon in the be stressed that, to reduce the computational cost, they Asqtad-improved staggered dynamical fermion formal- employed a technique with a fixed kaon sink operator ism. Using operators O (x ),O (x ) for pions at points π 1 π 3 for the calculation of πK scattering length for I = 1/2 x ,x ,andoperatorsO (x ),O (x )forkaonsatpoints 1 3 K 2 K 4 channel and then an exponential factor is introduced to x ,x ,respectively,withthe pionandkaoninterpolating 2 4 dropthe unnecessaryt-dependence appearingdue to the field operators defined by fixed kaon sink time [13]. In this work, we will improve this technique by using a “moving” wall source without (x,t) = d(x,t)γ u(x,t), π+ 5 gauge fixing where the exponential factor is not needed O −1 (x,t) = [u(x,t)γ u(x,t) d(x,t)γ d(x,t)], any more. Thus, there is no satisfactory direct lattice Oπ0 √2 5 − 5 calculation for I =1/2 channel until now. (x,t) = s(x,t)γ d(x,t), K0 5 In the presentstudy, we will use the MILC gaugecon- O (x,t) = s(x,t)γ u(x,t), (1) K+ 5 O figurationsgeneratedinthepresenceofN =2+1flavors f of the Asqtad improved[14, 15] staggereddynamicalsea we then represent the πK four-point functions as quarks[16,17]tostudythes-waveπK scatteringlengths for both I = 1/2 and 3/2 channels. Inspired by the ex- C (x ,x ,x ,x )= (x ) (x ) † (x ) †(x ) , πK 4 3 2 1 OK 4 Oπ 3 OK 2 Oπ 1 ploratory study of ππ scattering for I = 0 channel in (cid:10) ((cid:11)2) Ref. [18], we will adopt almost same technique but with where represents the expectation value of the path h···i moving kaon wall source operator without gauge fixing integral, which we evaluate using the lattice QCD simu- for I =1/2and3/2channelsto obtainthe reliableaccu- lations. racy. We calculated all the three diagrams categorized Aftersumming overspatialcoordinatesx , x , x and 1 2 3 in Ref. [12], and observed a clear signal of attraction x , we obtain the πK four-point function in the zero- 4 for I = 1/2 channel and that of repulsion for I = 3/2 momentum state, case. Moreover, for I = 1/2 channel, we employ the variationalmethodtoisolatethecontaminationfromthe C (t ,t ,t ,t )= C (x ,x ,x ,x ), πK 4 3 2 1 πK 4 3 2 1 excited states. Most of all, we only used the lattice sim- Xx Xx Xx Xx 1 2 3 4 ulation data of our measured πK scattering lengths for (3) bothisospineigenstatestosimultaneouslyextrapolateto- where x (x ,t ), x (x ,t ), x (x ,t ), and 1 1 1 2 2 2 3 3 3 wardthephysicalpointusingthecontinuumthree-flavor x (x ,t≡),andtstands≡forthetimediff≡erence,namely, 4 4 4 χPTatthe next-to-leadingorder. Ourlatticesimulation t ≡t t . 3 1 results of the scattering lengths for both isospin eigen- ≡To a−void the complicated Fierz rearrangement of the channelsareinaccordancewiththeexperimentalreports quarklines,we choosethe creationoperatorsatthe time and theoretical predictions, and can be comparable with slices which are different by one lattice time spacing as other lattice simulations. is suggested in Ref. [23], namely, we select t = 0,t = 1 2 This article is organized as follows. In Sec. II we de- 1,t = t, and t = t+1. In πK system, there are two 3 4 scribe the formalismforthe calculationofπK scattering isospineigenstates,namely,I =3/2andI =1/2,wecon- lengths including the Lu¨scher’s formula [19–21] and our struct the πK operators for these isospin eigenchannels computationaltechnique ofthe modifiedwallsourcesfor as [12] the measurement of πK four-point functions. In Sec. III wewillshowthesimulationparametersandourconcrete I=12(t) = 1 √2π+(t)K0(t+1) π0(t)K+(t+1) , lattice calculations. We will present our lattice simula- OπK √3n − o tion results in Sec. IV, and arriveat our conclusions and I=32(t) = π+(t)K+(t+1), (4) outlooks in Sec. V. OπK where (t) = (x,t), (t)= (x,t) II. METHOD OF MEASUREMENT OK0 OK0 OK+ OK+ Xx Xx (t) = (x,t), (t)= (x,t).(5) π0 π0 π+ π+ In this section, we will briefly review the formulas of O Xx O O Xx O the s-wave scattering length from two-particle energy in a finite box, with emphasis on the formulae for isospin If we assume that the u and d quarks have the same I = 1/2 πK system. Also we will present the detailed mass, only three diagrams contribute to πK scattering procedure for extracting the the energies of πK system. amplitudes[12]. Thequarklinediagramscontributingto Here we follow the original derivations and notations in πK four-point function denoted in Eq. (3) are displayed Refs. [12, 18, 22–24]. inFigure 1, labeling them as direct(D), crossed(C) and 3 Sasakiet al. solve this problem through the technique with a fixed kaon sink operator to reduce the computational cost[13]. Encouragedby the exploratoryworksof theππ scatteringatI =0channelinRefs.[18,23],similarly,wehandlethisproblembyevaluatingT quarkprop- agatorsonaL3 T lattice,eachpropagator,whichcorre- × spondstoawallsourceatthetimeslicet=0, ,T 1, ··· − is denoted by FIG. 1: Diagrams contributing to πK four-point functions. Dn′,n′′Gt(n′′)= δn′,(x,t), 0 t T 1, (6) ≤ ≤ − Short bars stand for wall sources. Open circles are sinks for Xn′′ Xx local pion or kaon operators. The thinkerlines represent the strange quark lines. where D is the quark matrix for the staggered Kogut- Susskind quark action. The combination of G (n) which t we apply for πK four-point functions is schematically rectangular (R), respectively. The direct and crosseddi- shown in Figure 1, where short bars stand for the posi- agramscanbeeasilyevaluatedbyconstructingthecorre- tion of wall source, open circles are sinks for local pion spondingfour-pointamplitudesforarbitraryvaluesofthe or kaon operators, and the thinker lines represent the timeslicest andt usingonlytwowallsourcesplacedat strange quark lines. Likely, the subscript t in the quark 3 4 the fixed time slices t and t . However, the rectangular propagator G represents the position of the wall source. 1 2 diagram(R) requiresanotherquarkpropagatorconnect- D, C, and R, are schematically displayed in Figure 1, ing the time slices t and t , which make the reliable and we can also expressed them in terms of the quark 3 4 evaluation of this diagram extremely difficult. propagatorsG, namely, CD(t ,t ,t ,t ) = ReTr G† (x ,t )G (x ,t )G† (x ,t )G (x ,t ) , 4 3 2 1 t1 3 3 t1 3 3 t2 4 4 t2 4 4 Xx Xx D h iE 3 4 CC(t ,t ,t ,t ) = ReTr G† (x ,t )G (x ,t )G† (x ,t )G (x ,t ) , 4 3 2 1 t1 3 3 t2 3 3 t2 4 4 t1 4 4 Xx Xx D h iE 3 4 CR(t ,t ,t ,t ) = ReTr G† (x ,t )G (x ,t )G† (x ,t )G (x ,t ) , (7) 4 3 2 1 t1 2 2 t4 2 2 t4 3 3 t1 3 3 Xx Xx D h iE 2 3 wheredaggersmeantheconjugationbytheeven-oddpar- combined to construct the physical correlationfunctions ity( 1)n forthestaggeredKogut-Susskindquarkaction, for πK states with definite isospin. As it is investigated − and Tr stands for the trace over the color index. The inRef.[12],intheisospinlimit,theπK correlationfunc- hermiticity properties of the propagator G are used to tion for I = 3/2 and 1/2 channels can be expressed as eliminate the factors of γ5. the combinations of three diagrams, namely, For πK rectangular diagram in Figure 1(c), it cre- ates the gauge-variant noise [18, 23]. One can reduce CI=12(t) I=21(t) I=21(0) =D+ 1N C 3N R, the noise by fixing gauge configurations to some gauge πK ≡DOπK |OπK E 2 f − 2 f I=3 I=3 I=3 ( e.g., Coulomb gauge), and select a special wall source CπK2(t)≡DOπK2(t)|OπK2(0)E=D−NfC, (8) to emit only the Nambu-Goldstone pion [25], however, the gauge non-invariantstates may contaminate the πK wheretheoperator I denotedinEq.(4)createsaπK four-pointfunction. Alternatively,we performthe gauge state with total isosOpiπnKI and the staggered-flavorfactor fieldaveragewithoutgaugefixingsincethegaugedepen- N isinsertedtocorrectfortheflavordegreesoffreedom f dentfluctuationsshouldneatlycanceloutamongthelat- of the Kogut-Susskind staggered fermion [24]. For the tice configurations. Besides these cancelations, the sum- pion and kaon operators it is most natural to choose the mation of the gauge-variant terms over the spatial sites one in the Nambu-Goldstone channel. This is the choice of the wall source further suppresses the gauge-variant for our current study. noise. In our current lattice simulation we found that To calculate the scattering lengths for hadron-hadron this method works pretty well. scatteringonthe lattice, orthe scatteringphase shifts in All three diagrams in Figure 1 are needed to be cal- general, one usually resorts to Lu¨scher’s formula which culated to study the πK scattering in both I =1/2 and relates the exact energy level of two hadron states in a I =3/2channels. Threetypesofthe propagatorscanbe finite boxto the scatteringphase shift in the continuum. 4 In the case of πK scattering, the s-wave πK scattering fermionsandcorrespondtothe contributionsfrominter- length in the continuum is defined by mediate states with opposite parity [27, 28]. The ellipsis suggests the contributions from excited states which are tanδ0(k) suppressed exponentially. a = lim . (9) 0 k→0 k Weshouldbearinmindthat,forthestaggeredKogut- Susskind quark action, there are further complications k is the magnitude of the center-of-mass scattering mo- in itself stemming from the non-degeneracyof pions and mentum which is related to the total energy EI of the πK kaonsintheGoldstoneandotherchannelsatafinite lat- πK system with isospin I in a finite box of size L by tice spacing. Briefly speaking, the contributions of non- Nambu-Goldstone pions and kaons in the intermediate EI = m2 +k2+ m2 +k2, (10) states is exponentially suppressed for large times due to πK π K q p their heavier masses compared to these of the Nambu- wherethemπ isthepionmass,andmK isthekaonmass. Goldstone pion and kaon [18, 23, 24]. Thus, we sup- We can rewrite Eq. (10) to an elegant form as pose that πK interpolator does not couple significantly to other πK tastes, and neglect this systematic errors. 1 m2 m2 2 In our concrete calculation, we calculated the pion k2 = EI + π− K m2. (11) 4(cid:18) πK EI (cid:19) − π mass mπ and kaon mass mK through the methods dis- πK cussedbytheMILC collaborationinRefs.[29,30]inour In the absence of the interactions between the π and K previousstudy[31]. Inthisworkweevaluatetotalenergy particles, k/tanδ (k) , and the energy levels occur EI of πK system with isospin I from Eq. (14). 0 → ∞ πK at momenta k = 2πn/L, (n is a integer), corresponding In the current study we also evaluate the energy shift to the single-particle modes in a cubic box. δ (k) is the δE =EI (m +m ) from the ratios 0 I πK − π K s-wavescattering phase shift, which canbe evaluated by the Lu¨scher’s finite size formula [19, 21], CX (0,1,t,t+1) RX(t)= πK , X =D,C, and R, C (0,t)C (1,t+1) π K tanδ (k) −1 √4π k2 (15) 0 = 1, , (12) (cid:18) k (cid:19) πL ·Z00(cid:18) (2π/L)2(cid:19) whereCπ(0,t)andCK(1,t+1)arethepionandkaontwo- pointfunctions,respectively. ConsideringEq.(8),wecan where the zeta function (1;q2) is denoted by write the amplitudes which project out the I =1/2 and 00 Z 3/2 isospin eigenstates as 1 1 (1;q2)= , (13) Z00 √4π nX∈Z3 n2−q2 RI=21(t) = RD(t)+ 12NfRC(t)− 32NfRR(t), R (t) = RD(t) N RC(t). (16) here q =kL/(2π) is no longer an integer, and 00(1;q2) I=32 − f Z can be efficiently calculated by the method described FollowingthediscussionsinRef.[24],wenowthencan in Ref. [26]. We also discussed this technique in Ap- extract the energy shift δE from the ratios pendix A, where we extend this discussion to the case I with the negative q2. In the case of the attractive inter- R (t)=Z e−δEIt+ , (17) action,k2 ontheboundstatehasanegativevalue,there- I I ··· fore k is pure imaginary, and δ0(k) is no longer physical where ZI stands for wave function factor, which is the scatteringphaseshift[13]. 00(1,q2),however,stillhave ratio of two amplitudes fromthe πK four-point function Z arealvalueevenforthiscase,hencetanδ0(k)/kobtained and the square of the pion two-point correlator and the by Eq. (12) is also real. If k2 is enough small, we can kaon two-point correlator, and the ellipsis indicates the | | consider tanδ0(k)/k as the physical scattering length at terms suppressed exponentially. In RI(t), some of the πk threshold [13]. fluctuations which contribute to both the two-point and The energy EI of πK system with isospin I can be four-pointcorrelationfunctions neatlycancelout,hence, πK obtainedfromπK four-pointfunctiondenotedinEq.(8) improvingthequalityoftheextractionoftheenergyshift with the large t. At large t these correlators will behave as compared with what we can obtain from an analysis as [27, 28] through the individual correlation functions [11]. ForI =3/2channel,wecanuseEq.(14)orEq.(17)to T CI (t) = Z cosh EI t + extract the energy shifts δE. We have numerically com- πK πK (cid:20) πK(cid:18) − 2(cid:19)(cid:21) pared the fitting values from two methods, and found T well agreement within statistical errors. In fact, using ( 1)tZ′ cosh EI′ t + . (14) − πK (cid:20) πK(cid:18) − 2(cid:19)(cid:21) ··· Eq. (17) to extract the energy shift δE has been exten- sivelyemployedforthestudyofπK scatteringatI =3/2 where EI is the energy of the lightest πK state with caseinRef.[11]. Hence,inthiswork,wewillonlypresent πK isospin I. The terms alternating in sign are a pecu- the energy shifts δE calculatedby Eq. (17), andthen its liarity of the Kogut-Susskind formulation of the lattice corresponding scattering lengths. 5 On the other hand, for I = 1/2 channel, the presence for the light u quark Dirac operatorM andthe s quark u of the kappa resonance is clearly shown in the low en- Dirac operator M , we obtain [31] s ergy [13], and therefore it should be necessary for us to separatethe groundstatecontributionfromthe contam- C (t))= ( )x Tr[M−1(x,t;0,0)M−1†(x,t;0,0)] , ination stemming from the excited states to achieve the κ Xx − D u s E reliablescatteringlengthasitisinvestigatedinRef.[13]. (20) For this purpose, we will construct a 2 2 correlation where Tr is the trace over the colorindex, and x=(x,t) × matrix of the time correlation function and diagonalize is the lattice position. it to extract the energy of the ground state. For the staggeredquarks,the meson propagatorshave the generic single-particle form, B. Correlation matrix (t)= A e−mit+ A′( 1)te−m′it+(t N t), C i i − → t− Xi Xi (21) For I = 1/2 channel, to separate the contamination wherethe oscillatingterms correspondto a particle with from the excited states, we construct a matrix of the oppositeparity. Forκmesoncorrelator,weconsideronly time correlation function, one mass with each parity in Eq. (21), namely, in our concrete calculation, our operator is the state with spin- 0 † (t) (0)0 0 † (t) (0)0 C(t)= h |OπK OπK | i h |OπK Oκ | i, tasteassignmentI⊗I anditsoscillatingtermwithγ0γ5⊗ 0 †(t) (0)0 0 †(t) (0)0 γ0γ5 [31]. Thus, the κ correlator was fit to the following  h |Oκ OπK | i h |Oκ Oκ | i  physical model, (18) where (t)isaninterpolatingoperatorforthe κmeson with zOerκo momentum, and πK(t) is an interpolating Cκ(t)=bκe−mκt+bKA(−)te−MKAt+(t→Nt−t), (22) O operator for πK system which is extensively discussed where b and b are two overlap factors. In Figure 6, in section IIA. The interpolating operator employed KA κ Oκ we clearly note this oscillating term. here is exactly the same as these in our previous stud- Weshouldbearinmindthat,forthestaggeredKogut- ies in Refs. [31–33], the notations adopted here are also Susskindquark action,our κ interpolating operatorcou- the same, but to make this paper self-contained, all the ples to various tastes as we examined the scalar a and necessary definitions will be also presented below. 0 σ mesons in Refs. [35, 36], where we investigated two- pseudoscalar intermediates states (namely, bubble contribution). In Ref. [32], we investigated the extracted κ 1. κ sector masses with and without bubble contribution for kappa correlators. Wefoundthatthereareonlyabout2%∼5% In our previous studies [31–33], we have presented differences in masses, although the bubble contributions a detailed procedure to measure kappa correlator are dominant in the κ correlators at large time region. 0κ†(t)κ(0)0 . Tosimulatethe correctnumberofquark Thus, in the current study, we can reasonably assume h | | i species, we use the fourth-root trick [34], which auto- that the κ interpolator does not couple remarkably to matically performsthe transitionfromfourtastes to one other tastes, and ignore this systematic errors for the κ taste per flavor for staggered fermion at all orders. We sector [33]. employ an interpolation operator with isospin I = 1/2 and JP =0+ at the source and sink, 2. Off-diagonal sector 1 (x) s¯a(x)ua(x), (19) O ≡ √nr Xa,g g g Thecalculationsoftheoff-diagonalelementsincorrela- tion matrix C(t) in Eq. (18), namely, 0 † (t) (0)0 wbehreroefgthisetthaestiendriecpelsicoafs,thaeitsasttheerecpolloicra,inndriciesst,haenndumwe- panredvhio0u|Osκ†st(ut)dOyπfKor(0n)ona-rzeeerxoamctolymtehnetasaihnm|ReOeaπfs.K[t3h3e]Os,etκhineon|uori- omit the Dirac-Spinor index. The time slice correlator tations adopted here are also the same, but to make this fortheκmesoninthezeromomentumstatecanbeeval- paper self-contained, all the necessary definitions will be uated by also presented below. To avoid the complicated Fierz rearrangement of the 1 C (t)= s¯b (x,t)ub (x,t) u¯a(0,0)sa(0,0) , quarklines,we choosethe creationoperatorsatthe time κ n g′ g′ g g r xX,a,bXg,g′ (cid:10) (cid:11) slices which are different by one lattice time spacing as suggested in Ref. [23], namely, we select t = 0,t = 1, 1 2 where 0,x are the spatial points of the κ state at source and t =t for πK κ three-point function, and choose 3 → and sink, respectively. After performing Wick contrac- t = 0,t = t, and t = t+1 for κ πK three-point 1 2 3 → tionsoffermionfields,andsummingoverthetasteindex, function. 6 ure 2(a) and Figure 2(b), respectively, where short bars stand for the position of wall source, open circles are sinks for local pion or kaon operators, and the thinker lines represent the strange quark lines. Likely, the sub- scripttinthequarkpropagatorGrepresentstheposition of the wall source. The πK κ three-point function can be easily evalu- → ated by constructing the corresponding three-point am- plitudesforarbitraryvaluesofthetimeslicet usingonly FIG. 2: Diagrams contributing to πK → κ and κ → πK 3 twowallsourcesplacedatthe fixedtime slicest andt . three-pointfunctions. Shortbarsstandfor wall sources. The 1 2 However,thecalculationofκ πK three-pointfunction thinker lines represent the strange quark lines. (a) Quark → contractions of πK → κ, where open circle is sink for local is almostthe same difficult as that of the rectangulardi- kappa operator. (b) Quark contractions of κ → πK, where agram for πK four-point correlator function, since it re- open circle is sink for local pion operator. quires another quark propagator connecting time slices t and t . The κ πK and πK κ three-point func- 2 3 → → tions are schematically shown in Figure 2, and we can The quark line diagrams contributing to the κ πK also express them in terms of the quark propagators G, → and πK κ three-point function are plotted in Fig- namely, → C (t ,t ,t ) = ReTr[G (x ,t )G† (x ,t )G† (x ,t )] , πK→κ 3 2 1 t1 3 3 t2 3 3 t2 1 1 xX3,x1D E C (t ,t ,t ) = ReTr[G (x ,t )G† (x ,t )G† (x ,t )] . (23) κ→πK 3 2 1 t1 2 2 t3 2 2 t3 1 1 xX2,x1D E C. Extraction of energies λ (t,t ) (n=1,2) behaves as [33] n R Throughcalculatingthematrixofcorrelationfunction T C(t) denoted in Eq. (18), we can separate the ground λ (t,t ) = A cosh E t + n R n n state from first excited state in a clean way. It is very (cid:20)− (cid:18) − 2(cid:19)(cid:21) important to map out “avoided level crossings” between ( 1)tB cosh E′ t T , (25) the κ resonance and its decay products (namely, π and − n (cid:20)− n(cid:18) − 2(cid:19)(cid:21) K)in a finite box volume,because the firstexcited state is potentially close to the ground state. This makes the extractionofthegroundstateenergyunfeasibleifweonly for a large t, which mean 0 t < t T/2 to sup- R ≪ ≪ utilize a single exponential fit ansatz. Since we can not press the excited states and the unwanted thermal con- predict a priori whether our energy eigenvalues are near tributions. This equation explicitly contains an oscillat- to the resonance region or not, we find it always safe ing term. For the current study, we are only interested in practice to adopt the correlation matrix to analyze in eigenvalue λ (t,t ), here non-degenerate eigenvalues 1 R our lattice simulation data for isospin I = 1/2 channel. λ (t,t ) > λ (t,t ) are assumed. In practice, we found 1 R 2 R To extract the ground state, we follow the variational that the oscillating term in λ (t,t ) is not appreciable 1 R method [20]andconstructa ratioof correlationfunction for some t , we can also adopt following simple fitting R matrices as model [33], M(t,t )=C(t)C−1(t ), (24) R R λ (t,t )=Acosh( E(t T/2)), (26) 1 R − − with some reference time slice t [20], which is assumed R to be large enough such that the contributions to correlation matrix M(t,t ) from the excited states can be andthe difference between the fitting models of Eq.(25) R neglected, and the lowest two eigenstates dominate the and Eq. (26) is small. However, to make our extracted correlation function. The two lowest energy levels can ground energy E for isospin I = 1/2 channel always re- be extracted by a proper fit to two eigenvalues λ (t,t ) liable, in this work,we will present the ground energy E n R (n = 1,2) of matrix M(t,t ). Because here we work calculatedbyEq.(25),andthenitscorrespondings-wave R on the staggered fermions, and we can easily prove that scattering lengths. 7 III. LATTICE CALCULATION inversiontakesaboutone thousanditerationsduring the CG calculation. Therefore,alltogether we carryout 288 A. Simulation parameters inversionsonafullQCDconfiguration. Asshownfollow, this large number of inversions, performed on 450 configurations, provides the substantial statistics needed to WeusedtheMILClatticeswith2+1dynamicalflavors resolve the real parts of the I =1/2 and 3/2 amplitudes of the Asqtad-improved staggered dynamical fermions, with reliable accuracy. thedetaileddescriptionofthesimulationparameterscan Inthecalculationoftheoff-diagonalcorrelator,C (t), be foundinRefs.[16,17]. Onething we muststressthat 21 the quark line contractions results in a three-point dia- the MILC configurations are generated using the stag- gram. Since in this three-point diagram the pion field geredformulationof lattice fermions [37–39] with rooted and kaon field are located at the source time slice t , staggered sea quarks [30] which are hypercubic-smeared s t +1, respectively. We calculate the off-diagonal corre- (HYP-smeared)[40–43]. AsitwasshowninRefs.[44,45] s lator C (t) through thatHYP-smearinggaugelinkssignificantlyimprovesthe 21 chiral symmetry. C (t) = κ(t)(πK)†(0) 21 We analyzed πK four-point functions on the 0.15 fm = (cid:10)1 κ(t+t(cid:11))(πK)†(t ) , (28) MILC lattice ensemble of 450 163 48 gauge config- T s s urations with bare quark masses am× = 0.0097 and Xts (cid:10) (cid:11) ud ams = 0.0484 and bare gauge coupling 10/g2 = 6.572, where,again,wesumthecorrelatoroveralltimeslicests which has a physical volume approximately 2.5 fm. The and average it. As for the second off-diagonal correlator inverselatticespacinga−1 =1.358+−3153 GeV[16,17]. The C12(t), the pion field and kaon field are placed at the mass of the dynamical strange quark is near to its phys- sink time slices ts+t and t+ts+1, respectively, which ical value, and the masses of the u and d quarks are makethecomputationofC12(t)difficult. However,using degenerate. To avoid the contamination from pions and the relation C12(t) = C2∗1(t), we can obtain the matrix kaons propagating backward in time, periodic boundary element C12 for free. As it is studied in Ref. [46], since condition is applied to the three spatial directions while the sink and source operators are identical for a large in the temporal direction, Dirichlet boundary condition number of configurations, C(t) is a Hermitian matrix. is imposed, which reduce the original time extent of 48 The κ πK component agrees with πK κ within → → down to 24, moreover, it avoids the “fake effects” dis- the error,but the statisticalerrorsofthe matrixelement cussed in Ref. [13]. C12 should be larger than that of matrix element C21 for a large t. Therefore, in the following analyses we substitute matrix element C by the complex conjugate 12 B. Sources for isospin I =1/2 channel of matrix element C21, which is not only to save about 20% computation time, but also significantly to reduce statistical errors. To calculate the πK correlation functions, we use the For the κ correlator, C (t), we have measured the standard conjugate gradient method (CG) to obtain the 22 point-to-pointcorrelatorswith high precisionin our pre- required matrix element of the inverse fermion matrix. vious work [31]. Therefore, we can exploit the available The calculation of the correlation function for the rect- propagatorsto construct the κ-correlator angular diagrams naturally requires us to compute the propagators on all the time slices t = 0, ,T 1 of 1 both source and sink, which requiress the c·a·l·culat−ion of C22(t)= T κ†(t+ts)κ(ts) , (29) 48 separatepropagatorsin our lattice simulations. After Xts (cid:10) (cid:11) averaging the correlator over all 48 possible values, the where, also, we sum the correlator over all time slices ts statistics are greatlyimprovedsince we can put the pion and average it. sourceandkaonsourceatallpossibletimeslices,namely, One thing we must stress that, in the calculation of the correlator C (t) is calculated through the correlator (πK)(πK)† , we make our best-efforts to 11 h i reliablymeasuretherectangulardiagram,sincetheother C (t) = (πK)(t)(πK)†(0) two diagrams are relatively easy to evaluate. We found 11 D E that the rectangular diagram plays a major role in this 1 = (πK)(t+t )(πK)†(t ) , (27) correlator. Therefore, we get it properly for the πK sec- s s T Xts D E tor for isospin I =1/2 channel. Inthiswork,wealsomeasuretwo-pointcorrelatorsfor The best-effort to generate the propagators on all time pion and kaon, namely, slicesenablesustoobtainthecorrelatorswithhighpreci- G (t) = 0π†(t)π(0)0 , sion,whichisvitaltoextractthedesiredscatteringphase Gπ(t) = h0|K†(t)K(0|)i0 , (30) shifts reliably. K h | | i For each time slice, six fermion matrix inversions are where the G (t) is correlationfunction for the pion with π requiredcorrespondingtothe possible3colorchoicesfor zero momentum, and the G (t) is correlation function K the pion source and kaon source, respectively, and each for the kaon with zero momentum. 8 IV. SIMULATION RESULTS In our previous work [47], we have measured the pion and kaon point-to-point correlators. Using these correlators, we can precisely extract the pion mass (m ) and π kaon mass (m ) [47], which are summarized in Table I. K Using the same method discussed in Ref. [48] and the MILC code for calculating the pion decay constants f , π we precisely extract the pion decay constants f [49], π which are in agreement with the previous MILC deter- minationsatthissamelatticeensembleinRef.[17]. Here weusedallthe631latticeconfigurationsofthisensemble. We also recapitulated these fitted values in Table I. TABLE I: Summary of the pion masses, kaon masses and the pion decay constants. The third and fourth blocks show pionmassesandkaonmassesinlatticeunits,respectively,and FIG. 3: (color online). Individual amplitude ratios RX(t) Column five shows the pion decay constants in lattice units. of the πK four-point function calculated through the mov- Thesecondblockgivespionmasses inGeV,wheretheerrors ing wall source without gauge fixing as the functions of t for areestimatedfromboththeerroronthelatticespacingaand am = 0.0097. Direct diagram (magenta diamonds) shifted thestatistical errors in Column three. x by 0.8, crossed diagram (red octagons) and rectangular dia- am m (GeV) am am af gram (bluesquares). x π π K π 0.00970 0.334(6) 0.2459(2) 0.3962(2) 0.12136(29) 0.01067 0.350(6) 0.2575(2) 0.3996(2) 0.12264(34) 0.01261 0.379(7) 0.2789(2) 0.4066(2) 0.12425(27) tot 20,whichsuggestsanattractiveforcebetweenthe ∼ 0.01358 0.392(7) 0.2890(2) 0.4101(2) 0.12482(32) pionandkaoninI =1/2channel. Furthermore,themag- 0.01455 0.406(7) 0.2987(2) 0.4134(2) 0.12600(26) nitudeoftheslopeissimilartothatofthecrossedampli- tude but with opposite sign. These features are what we 0.01940 0.466(8) 0.3430(2) 0.4300(2) 0.12979(27) eagerly expected from the theoretical predictions [6, 24]. We can observe that the crossed and rectangular amplitudes have the same value at t=0, and the close values forsmallt. BecauseouranalyticalexpressionsinEq.(7) A. Diagrams D,C, and R forthe twoamplitudes coincideatt=0,they shouldbe- have similarly until the asymptotic πK state is reached. The πK four-pointfunctions are calculatedwith same latticeconfigurationsusingsixuvalencequarks,namely, In Figure 4, we display the ratio R (t) projected onto I am = 0.0097, 0.01067, 0.01261, 0.01358, 0.01455 and the isospin I = 1/2 and 3/2 channels for am = 0.0097, x x 0.0194,wherem isthelightvalenceuquarkmass. They whicharedenoted inEq.(16). Adecreaseofthe ratioof x all have the same strange sea quark mass am =0.0426, R (t)indicatesapositiveenergyshiftandhenceare- s I=3/2 which is fixed at its physical value [17]. pulsiveinteractionforI =3/2channel,whileanincrease In Figure 3 the individual ratios, which are defined in ofR (t)suggestsanegativeenergyshiftandhencean I=1/2 Eq. (15) corresponding to the diagrams in Figure 1, RX attractionforI =1/2channel. Adipatt=3forI =1/2 (X = D,C and R) are displayed as the functions of t channel can be clearly observed [23]. The systematically for am = 0.0097. We can note that diagram D makes oscillatingbehaviorforI =1/2channelinthelargetime x thebiggestcontribution,thendiagramC,anddiagramR region is also clearly observed, which is a typical char- makes the smallest contribution. The calculation of the acteristic of the Kogut-Susskind formulation of lattice amplitudes for the rectangular diagram stands for our fermions and corresponds to the contributions from the principal work. Clear signals observed up to t = 20 for intermediate states of the opposite parity [27, 28], this the rectangular amplitude demonstrate that the method alsoclearlyindicatesthe existenceofthecontaminations ofthe movingwallsourcewithoutgaugefixingusedhere from other states rather than the pion-kaon scattering is practically applicable. state [13]. Therefore,to isolate the potential contamina- The values of the direct amplitude RD are quite close tions, we will use the variational method [20] to analyze to unity, indicating that the interaction in this channel the lattice simulation data. As for I =3/2 channel, this is veryweak. The crossedamplitude, on the other hand, oscillating characteristic is not appreciable, we will use increases linearly, which implies a repulsion in I = 3/2 the traditional method, namely, using Eq. (17) to com- channel. After an initial increase up to t 4, the rect- pute the energy shift δE and then calculate the corre- ∼ angular amplitude exhibits a roughly linear decrease up sponding scattering length. 9 where we can watch the fitted functional form as compared with the lattice simulation data for I =3/2 channel. For the other five light u valence quarks, we obtain the similar results, therefore we do not show these ratio R (t) plots here. The fitted values of the energy I shifts, δE in lattice units and wave function factor Z I I for I = 3/2 channel are summarized in Table II. The wavefunctionZ factorsarepretty closetounity andthe χ2/dof is quite small for I =3/2 channel, indicating the values of the extracted scattering lengths are substan- tially reliable. TABLE II: Summary of the lattice simulation results for the energy shifts in lattice units for I = 3/2 channel. The third blockshowstheenergyshiftsinthelatticeunit,Columnfour shows the wave function factors Z, Column five shows the timerangeforthechosenfit,andColumnsixshowsthenum- ber of degrees of freedom (dof) for the fit. All errors are FIG. 4: (color online). RI(t) for πK four-point function at calculated from jackknife. zero momenta calculated by the moving wall source without gaugefixingasthefunctionsoftforamx =0.0097. Solidline Isospin amx aδE Z Range χ2/dof in I =3/2 is exponential fits for 7≤t≤16. 0.00970 0.00621(53) 0.9880(59) 7−16 0.0536/8 0.01067 0.00615(52) 0.9893(58) 7−16 0.0395/8 I = 3 0.01261 0.00602(50) 0.9914(56) 7−16 0.0226/8 2 B. Fitting analyses for I =3/2 channel 0.01358 0.00595(49) 0.9923(55) 7−16 0.0176/8 0.01455 0.00589(48) 0.9930(54) 7−16 0.0142/8 AccordingtoourdiscussionsinSectionII,inthiswork, 0.01940 0.00561(45) 0.9958(50) 7−16 0.0067/8 wewillmakeuseofEq.(17)toextracttheenergyshiftδE for I = 3/2 channel. Then we insert these energy shifts into the Eqs. (9) and (12) to obtain the corresponding Now we can insert these energy shifts in Table II into s-wave scattering lengths. Therefore, properly extract- theEqs.(9)and(12)toobtainthecorrespondings-wave ing the energy shifts is a crucial step to our final results scattering lengths. The center-of-mass scattering mo- in this paper. A convincing way to analyze our lattice mentum k2 in GeV calculated by Eq. (11), from which simulationdata is withthe “effective energyshift” plots, we can easily estimate its statistcal errors. Once we ob- a variant of the effective mass plots, where the propa- tain the values of k2, the s-wavescattering lengths a0 in gators were fit with varying minimum fitting distances lattice units can be obtained through Eqs. (9) and (12). D , and with the maximum distance D either at All of these values are summarized in Table III. Here we min max the midpointofthe lattice orwherethe fractionalstatis- utilizepionmassesandkaonmassesgiveninTableI.The tical errors exceeded about 20% for two successive time errors of the center-of-mass scattering momentum k and slices. For each valence quark m , the effective energy the s-wave scattering lengths come from the statistical x shiftplotsasafunctionofminimumfittingdistanceDmin errors of the energy shifts energies δE, pion mass mπ for I = 3/2 channel are shown in Figure 5. The central and kaon mass mK. value anduncertaintyofeachparameterwasdetermined by the jackknife procedure over the ensemble of gauge TABLEIII:Summaryofthelatticesimulationresultsofthes- configurations. wavescattering lengths for I =3/2 channel. The third block The energy shifts aδE of πK system for I = 3/2 shows the center-of-mass scattering momentum k2 in GeV, channel are extracted from the “effective energy shift” Column four shows the s-wave scattering lengths in lattice plots, and the energy shifts were selected by looking for units, and Column fiveshows the pion mass times scattering a combinationof a “plateau”in the energy as a function lengths. of the minimum distance D , and a good confidence level (namely, χ2) for the fitm.inWe found that the effec- Isospin amx k2[GeV2] a0 mπa0 tive energy shifts for I = 3/2 channel have only relative 0.00970 0.00350(27) −0.558(55) −0.137(13) small errorswithin broadminimum time distance region 0.01067 0.00357(28) −0.569(55) −0.146(14) 5≤Dmin ≤10 and are taken to be quite reliable. I = 32 0.01261 0.00366(30) −0.582(55) −0.162(15) We utilize the exponential physical fitting model in 0.01358 0.00374(30) −0.593(56) −0.171(16) Eq. (17) to extract the desired energy shifts for I = 3/2 0.01455 0.00379(34) −0.600(60) −0.179(18) channel. InFigure4wedisplaytheratioRI(t)projected 0.01940 0.00396(37) −0.624(62) −0.214(21) onto the I = 1/2 and 3/2 channels for am = 0.0097, x 10 FIG. 5: (color online). The effective πK energy shift plots, aδE as the functions of the minimum fitting distance Dmin in the fit for I =3/2 channel. The effective πK energy shift plots for I =3/2 channelhave only relative small errors within a broad minimum distance region 5≤Dmin≤10. C. Fitting analyses for I =1/2 channel In Figure 6, we show the real parts of the diagonal components (πK πK and κ κ) and the real part → → of the off-diagonal component πK κ of the correla- → tion function C(t) denoted in Eq. (18). Since C(t) is a Hermitian matrix, we will substitute the off-diagonal component κ πK by πK κ to reduce statistical → → errors in the following analyses. We calculate two eigenvalues λ (t,t ) (n = 1,2) for n R the matrix M(t,t ) in Eq. (24) with the reference time R t =7. Inthis work,we areonlyinterestedin the eigen- R value λ (t,t ) 1. In Figure 7 we plot our lattice results 1 R for λ (t,t ) for each valence quark m in a logarithmic 1 R x scaleasthefunctionsofttogetherwithacorrelatedfitto theasymptoticformgiveninEq.(25). Fromthesefitswe then extract the energies that will be used to determine FIG.6: (coloronline). Realpartsofthediagonalcomponents the s-wave scattering lengths. (πK →πK and κ→κ) and the real part of the off-diagonal To extractthe energiesreliably,we musttake twoma- component πK → κ of the time correlation function C(t). jor sources of the systematic errors into consideration. Occasionalpointswithnegativecentralvaluesforthediagonal One arises from the excited states which affect the cor- component κ → κ and the off-diagonal component πK → κ are not plotted. relator in low time slice region. The other one stems from the thermal contributions which distort the corre- latorinhightimesliceregion. Bydenotingafittingrange [t ,t ] and varying the values of the t and t , min max min max we can control these systematic errors. In our concrete fitting, we take t to be t +1 and increase the refer- min R 1 Inourprevious study[33],wehavepreliminarilyexaminedthe ence time slice tR to suppress the excited state contam- behaviorofλ2(t,tR). inations. Moreover, we select tmax to be sufficiently far