Finite-volume energy spectrum of the K − K − K − system
Andrei Alexandru, Ruairí Brett, Chris Culver, Michael Döring, Dehua Guo, Frank X. Lee, Maxim Mai
FFinite-volume energy spectrum of the K − K − K − system Andrei Alexandru,
1, 2, ∗ Ruair´ı Brett, † Chris Culver, ‡ Michael D¨oring,
1, 4, § Dehua Guo, ¶ Frank X Lee, ∗∗ and Maxim Mai †† The George Washington University, Washington, DC 20052, USA Department of Physics, University of Maryland, College Park, MD 20742, USA Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
The dynamics of multi-kaon systems are of relevance for several areas of nuclear physics. However,even the simplest systems, two and three kaons, are hard to prepare and study experimentally. Herewe show how to extract this information using first-principle lattice QCD results. We (1) extend therelativistic three-body quantization condition to the strangeness sector, predicting for the first timethe excited level finite-volume spectrum of three kaon systems at maximal isospin, and (2) present afirst lattice QCD calculation of the excited levels of this system in a finite box. We compare ourpredictions with the lattice results reported here and with previous ground state calculations andfind very good agreement.
PACS numbers: 12.38.Gc, 14.40.-n, 13.75.Lb
INTRODUCTION
In recent years hadron-hadron scattering informationfrom first-principles lattice QCD (LQCD) has becomesignificantly more accessible. Fostered by advances intheoretical and computational tools a large number ofhigh-precision studies have been performed in the mesonsector, see for example Refs. [1–30], and Ref. [31] for areview. Several research areas of nuclear physics bene-fit from these studies. For instance, the study of pion,kaon, and proton correlations in heavy ion collisions bythe ALICE@CERN collaboration [32] relies on the valueof the K − K − scattering length determined in a latticecalculation [33]. There are however only few results focus-ing on the strange sector [23, 33, 34], in contrast to thepion systems explored extensively by a number of collab-orations [6, 15, 20, 28, 34–40]. Furthermore, informationabout many- K − systems is relevant for the understand-ing of strange nuclear matter and its implications to theequation of state of neutron stars. In particular, it is wellknown that ultra dense environments (such as those inthe core of neutron stars) allow for an appearance of kaoncondensates [41–44], that can soften the equation of stateof neutron stars [42, 43, 45, 46]. Further details on theantikaon interaction with baryonic matter can be foundin reviews [47, 48].Today, the frontier of hadronic scattering in LQCDis in the scattering of three mesons. Pioneering latticecalculations have moved from the extraction of the groundstates of such systems [49–51] to the high-precision deter-mination of multiple excited three-hadron states [52–56].Significant progress has also been made in the develop-ment of formalisms relating the finite- and infinite-volumethree-hadron spectrum [57–97]. Applications of such ap-proaches to LQCD data have thus far been for three pionsystems in maximal isospin [54–56, 58, 62, 63].In this paper, we extend these methods to explore a new area: we present both the first determination of theexcited three-kaon finite-volume spectrum from LQCD,along with the first connection to infinite-volume scat-tering using the formalism of Refs. [29, 58, 63, 68, 72].The latter is extended to the three-flavor sector allowingfor chiral extrapolations along arbitrary M K ( M π ) tra-jectories using constraints from chiral symmetry. Suchimplementations are standard in the two-body sector [98–105], but not yet explored for the three-body systems.The present study closes this gap, using relativistic three-body formalism implementing two-body input from theinverse amplitude approach [106, 107]. FINITE-VOLUME SPECTRUM FROM LATTICEQCD
The finite-volume spectrum of hadronic states can bedirectly accessed by studying correlation functions inthe framework of LQCD. Here we review the procedurefor extracting the finite-volume spectrum of K − K − K − .The energy levels of hadrons in a finite volume can beextracted from the large time behavior of correlationfunctions consisting of interpolating operators, O i , whichcreate/annihilate the hadrons of interest, C ij ( t ) = D O i ( t ) O † j (0) E = X n h |O i | n i h n |O † j | i e − E n t . (1)If the operators are constructed to overlap with the states n of interest, we can extract the finite-volume energies E n . An important tool to allow the extraction of mul-tiple finite-volume energies is to perform a variationalanalysis on a matrix of correlation functions constructedfrom several operators. This is equivalent to solving ageneralized eigenvalue problem [108–110], and extractingthe finite-volume spectrum from the eigenvalues of thecorrelation matrix. Due to the precision with which we a r X i v : . [ h e p - l a t ] O c t Label N t × N a [fm] N cfg E × . aM π = 0 . af π = 0 . aM K = 0 . af K = 0 . E × . aM π = 0 . af π = 0 . aM K = 0 . af K = 0 . TABLE I. Details and results of the GWUQCD N f = 2ensembles used in this study. Here a is the lattice spacing, N cfg the number of Monte Carlo configurations for each ensemble,and aM π and aM K the pion and kaon masses, respectively.The errors in the parenthesis are stochastic. For the latticespacing we also include an estimate for the systematic errorof 2%. can measure the correlation functions, thermal effects dueto the finite temporal extent must be accounted for as inRef. [54].The overlap factor h n |O j | i is non-zero only if ouroperators and states n have the same quantum numbers.In a finite cubic volume, the rotational symmetry groupis reduced from SO (3) to O h . We have to thereforeconstruct our operators with definite quantum numbersaccording to the irreducible representations (irreps) of O h . An important consequence is that the irreps of O h mix different angular momentum from the infinite volume.The symmetry is further reduced if the system is studiedwith non-zero total momentum.To create operators which overlap with the three-kaonspectrum, we begin by constructing a single kaon inter-polator according to K − (Γ( p ) , t ) = ¯ u ( t )Γ( p ) s ( t ) , (2)where s, u are the quark fields, and the momentum matrixΓ( p ) = e i p · x γ projects the operator to definite momen-tum. Our three-kaon operators are now just a product ofthree single kaon operators. We project the three kaonoperators to irreps of the cubic group. To project to row λ of irrep Λ of group G , we evaluate O K K K = X g ∈ G U Λ λλ ( g )det( R ( g )) × K − ( R ( g ) p ) K − ( R ( g ) p ) K − ( R ( g ) p ) , (3)where p , p , p are the three-momenta of each kaon, R isthe three-dimensional rotation matrix associated with g ,and U is the representation matrix of g in irrep Λ.The GWUQCD ensembles are generated using twomass-degenerate light quarks ( N f = 2 QCD), usingthe nHYP-smeared clover action. Lattice parametersof the ensembles used here are listed in Table I. Details ★★ Physical point ���������������� ( �� ) ���������� ( �� ) ���� ���� ���� ���� ���� ���� ���� ���� - ��� - ��� - ��� - ��� - ��� - ��� - ��� FIG. 1. The I = 1 KK scattering lengths from NPLQCD [33]with statistical (red) and systematic (gray) error bars. Thechiral prediction at the physical point is indicated (green star),as well as predictions for different pion masses using f π , f K from NPLQCD [115] (solid blue line), or NLO extrapolating f π (blue dash-dotted line). of the ensemble generation, including some discussionon tuning the bare strange quark mass, can be foundin Refs. [18, 111]. The pion and kaon decay constants, f π , f K are determined using the procedure outlined inRef. [112]. The strange quark mass is tuned by settingthe ratio R = ( M K /M π ) to its physical value. For thevalence quarks appearing in (kaon) interpolating opera-tors, both light and strange (all-to-all) quark propagationis treated using the LapH method [113]. The all-to-allLapH propagators were computed using our optimizedinverters [114]. The lattice results and predictions are tab-ulated in the Supplementary Material. Jackknife samplesare provided as ancillary files with the arXiv submission. FINITE-VOLUME SPECTRUM FROMINFINITE-VOLUME PHYSICS
In the present work we utilize the three-body relativis-tic quantization condition (3bQC) derived in Ref. [72]extended later to higher irreps [68], elongations [54], andboosts [58]. For the I = 3 / S = − h B ( E ) + C ( E )+ E L (cid:0) K − ( E ) + ρ L ( E , P ) (cid:1) i Γ pq = 0 , (4)where E and P denote the center of mass energy andtotal three-momentum of the three-body system, respec-tively. Note that the implicit dependence on the latter issuppressed. The determinant is taken with respect to the ��������� - ��������������������� ( �� ) ���������� ( �� ) ���� ���� ���� ���� ���� ���� ������������������� ★★★ Physical point ���� ��� ���� ��� ���� ��� �������������������� ���� ���� ���� ���� ���� ���� ������������������� ���� ���� ���� ���� ���� ���� ������������������������� ���� ���� ���� ���� ���� ���� ���������������������������� ���� ���� ���� ���� ���� ���� ����������������������������
FIG. 2. Comparison of the predicted K − K − K − finite-volume spectrum to the results of LQCD calculations [51, 115] by theNPLQCD collaboration. Top and bottom row show projections to relevant irreps for P = and P = (1 , ,
0) cases, respectively.The M K ( M π ) trajectory is chosen as in the latter references, while the decay constants are determined from the NLO chiralextrapolations (dot-dashed line) or by setting them directly to the NPLQCD values (blue solid lines). In the top left figure, thered (gray) error bars represent the uncertainty quoted in Ref. [51] (including variation of scale setting). The insert in the A u plot shows the ground state data, predictions, and a prediction for the physical point. in/outgoing discrete lattice spectator momenta p / q afterprojecting the elements in parenthesis to an irrep Γ. Thenon-diagonal matrix B denotes the one-particle exchangeterm, while the diagonal matrix ρ L ( E , P ) represents thetwo-body self-energy term, see, e.g., appendix of Ref. [58]for explicit expressions. The propagation of the spectatoryields the factor [ E L ] pq = δ pq L p M π + p .The only unknown pieces of the quantization conditionare matrices K − ( E ) and C ( E ), encoding dynamics oftwo- (via the usual K -matrix) and three-body interac-tions, respectively. Since not many data is available yetfor the 3 K − -system and in analogy to the similar 3 π + system [63], we set the latter to zero. The two-body K -matrix is restricted to the dominant S -wave, noting thatdue to the nature of the 3bQC all relative partial wavesbetween the spectator and the two-body subsystem areincluded automatically by the one-particle exchange term B . Specifically, the K -matrix is chosen to match the in-verse amplitude method [102, 106, 107, 116, 117] – a verysuccessful description of two-meson scattering across wideenergy and meson mass ranges and all two-pseudoscalarmeson interaction channels [29], T ( s ) = ( T LO ( s )) T LO ( s ) − T NLO ( s ) = 1 K − ( s ) − ρ ( s ) . (5) Here, T (N)LO refers to the (next-to-)leading chiral orderscattering amplitudes [118], and ρ ( s ) denotes the usualfinite part of the two-body self-energy evaluated in di-mensional regularization. The K − K − amplitude to oneloop is obtained by using crossing symmetry for resultsof Ref. [106]. A summary of the relevant formulas is in-cluded in the Supplementary Material. In particular, thecorresponding K -matrix depends on { M π , M K , f π , f K } aswell as renormalized low-energy constants (LECs) { L ri } .The effect of the first set of parameters is more importantthan the latter for not too large meson masses, becausethe chiral series is ordered in powers of M /f . Thus,we fix the LECs to the results of the most recent globalfits to the lattice results [100] (discussion of older LECsis moved to the Supplementary Material), but explorevarious scenarios for the remaining inputs below.As a check we evaluate the scattering length M K a = T (4 M K ) at different meson massesand compare with the NPLQCD collaboration re-sults [33] along their ( M π , M K ) trajectory ( M π , M K ) ∈{ (293 , , (355 , , (493 , , (592 , } MeV. Forthe decay constants we compare two scenarios: ( S1 ) byextrapolating the pion decay constant using input at thephysical point and NLO chiral expressions [118] withLECs from Ref. [100] and ( S2 ) by using the meson decay ��������� - ��������������������� ( �� ) ���������� ( �� ) ���� ���� ���� ������������������� ★★ Physical point ���� ���� ���� ���������������� ���� ���� ���� �������������������
FIG. 3. Comparison of the predicted K − K − K − finite volume spectrum to the present lattice results (red data) for P = .The M K ( M π ) trajectory extends linearly from the physical point through the two shown lattice points. The insert in the A u plot shows the ground state data, predictions, and a prediction for the physical point. Note that the excited states in A u and E u are close to or beyond the πKKK threshold, but below the lowest relevant lattice threshold, for which two kaons necessarilyhave finite back-to-back momenta due to parity conservation. constants determined on the lattice [115]. These two sce-narios differ only by higher chiral orders and are employedas representatives of the systematic uncertainty of ourpredictions. The results are depicted in Fig. 1. They showthat the three-flavor formulation of the inverse amplitudeapproach (5) is a perfectly suitable parametrization of thetwo-body dynamics at unphysical meson masses. Higherorder terms yield sizable corrections above M π ≈ M phys π as expected.Before coming to the results on three-body spectra wepoint out the major difference between the 3bQC and itstwo-body equivalent. The 3bQC remains a determinantequation even for the simplest one-channel case. For afixed energy and momentum of the three-body system,the two-body input is required for a large kinematic range( s in Eq. (5)) due to the variable spectator momentum.Therefore, the two-body amplitude is often evaluatedfor subthreshold values of s . Various approaches to thisissue have been studied in the past [58, 72] and it wasfound that the obtained finite-volume spectra dependlittle on the subthreshold region. For the present case,we confirm this observation explicitly by varying the cutin the spectator momentum space in Eq. (4). As wechange this from the value used throughout this study, L | p max | = 2 π √
5, to L | p max | = 2 π √
11, the largest change( ∼ × − %) among all levels in the GWUQCD setup for M π = 315 MeV happens for the first excited level in A u .Similarly, we study the dependence on the subthreshold KK amplitude by replacing K − ( s ) with a real-valuedconstant at s = 3 M K and then at s = 3 . M K leading toa maximal change of any energy eigenvalue of 0.02% whichis a fraction of the smallest statistical uncertainty in theGWUQCD lattice data. The dependence of the resultson the use of modified IAM (mIAM) [119] instead of IAMleads to (cid:46) .
1% change of the scattering length and is, therefore, of similar size for the three-body ground stateenergy shift. In summary, these sources of systematicuncertainty are very small.
COMPARISON AND DISCUSSION
First, we turn to previous LQCD results, namely theground state A u (0) levels determined by the NPLQCDcollaboration [51, 115] in a cubic box of L = 2 . S1 and S2 are visualized by the lightblue band. We observe encouraging agreement betweenour predictions and the NPLQCD results. We also finda similar increase in the size of NNLO effects at higherpion masses, as observed in the two-body results. Goingbeyond the ground state level, we extend our predictionsto excited states, other irreps, and boosts (lower panel ofFig. 2).New LQCD results are obtained in this work, includingfor the first time, excited K − K − K − energies, in multipleirreps. This provides an opportunity for a series of uniquetests of the predicted finite-volume spectra. Followingboth scenarios S1 and S2 along the chiral trajectory(see Table I) the predictions for the GWUQCD setupare shown in Fig. 3. The ground state is in excellentagreement with the predictions as was the case for theheavier pion mass results from NPLQCD. For the excited A u (0) levels the slight tension with the prediction couldbe some hint of the need for a non-zero three-body force.Of course, other possible sources for the discrepancy couldbe (i) the chiral prediction itself is not perfect, or (ii) thatthe partial quenching of the strange quark plays a role.This will be investigated in a future study.The E u (0) levels agree with the predictions well. Notethat this irrep is dominated by D -wave. Since the two-body interaction is typically smaller for higher partialwaves, the major contribution seems to come from theone-particle exchange term B , with no obvious need forcontact terms beyond that. In fact, this is very similarto the observed E − u (0) /A − u (0) pattern for the three-pionsystem noted in Ref. [58]. In both cases, the patternconfirms the dominance of the exchange contribution,which is a direct consequence of the S-matrix principle ofthree-body unitarity.In summary, we have traced a pathway for studyingmulti-kaon systems using lattice QCD. We presented thefirst LQCD calculation of excited three kaon states, inmultiple irreps, and at multiple pion masses. We havealso extended the relativistic three-body quantizationcondition to the strange sector, allowing for chiral ex-trapolations along arbitrary trajectories. We find thatthis extension consistently describes the data from twoindependent lattice calculations of multi-kaon systems. Inthe long run, this provides an avenue for extracting infor-mation relevant for strange resonances, kaon condensates,and heavy-ion collisions.This material is based upon work supported by theNational Science Foundation under Grant No. PHY-2012289 and by the U.S. Department of Energy un-der Award Number DE-SC0016582 (MD and MM) andDE-FG02-95ER40907 (AA,FXL,RB,CC). RB is also sup-ported in part by the U.S. Department of Energy andASCR, via a Jefferson Lab subcontract No. JSA-20-C0031.CC is supported by UK Research and Innovation grantMR/S015418/1. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] †† [email protected][1] X. Feng, K. Jansen, and D. B. Renner, Phys. Rev. D83 ,094505 (2011), arXiv:1011.5288 [hep-lat].[2] S. R. Beane, E. Chang, W. Detmold, H. W. Lin, T. C.Luu, K. Orginos, A. Parreno, M. J. Savage, A. Torok,and A. Walker-Loud (NPLQCD), Phys. Rev.
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Next-to-leading order chiral K − K − amplitude In the following we provide the explicit formulas for the scattering matrices of the two-body input. They rely onthe three-flavor chiral Lagrangian of the leading and next-to-leading chiral order [118]. We chose the formulationwith all decay constants replaced by the “physical” pion one, which is done consistently at the forth chiral order, i.e.,differences are of the order O ( p ). In practice, we obtain the K − K − → K − K − amplitude from the K + K − → K + K − amplitude quoted in Ref. [106] (see also Ref. [116] for the original calculation) by crossing symmetry which amounts toexchanging s ↔ u in the latter, T ( s, t, u ) = T LO ( s, t, u ) + T NLO ( s, t, u ) + . . . = (cid:20) M K − sf π (cid:21) LO + (cid:20) − µ K f π M K (cid:0) (cid:0) u + ut + t (cid:1) + 6 s − sM K − M K (cid:1) + µ π f π (cid:0) s − M K (cid:1) − u + 8 ut + 11 t + 8 sM K − M K M π + 9 (cid:0) u + t (cid:1) + 24 sM K − M K M K − M π ) ! + µ η f π M K − M π − s − (cid:0) u + t (cid:1) −
36 ( u + t ) M π + 8 M π M η + 9 (cid:0) u + t (cid:1) + 24 sM K − M K M π − M η ) ! + 4 f π (cid:0) L r ( s − M K ) + (2 L r + L r + L ) (cid:0) ( u − M K ) + ( t − M K ) (cid:1) − L r sM K − L r ( s − M K ) M π − L r − L r + L r ))) M K (cid:1) + 186 ut − s + 1032 sM K − M K f π π + 12 f π ( s − M K ) ¯ J KK ( s )+ 1288 f π (cid:18) (cid:0) u (2 u + t ) + 4 sM K − M K (cid:1) ¯ J KK ( u ) + 2(9 u − M K − M π − M η ) ¯ J πη ( u )3+ (9 u − M π − M η ) ¯ J ηη ( u ) + 3 (cid:0) u (cid:0) u + 4 t − M K (cid:1) − (cid:0) u + 2 t − M K (cid:1) M π (cid:1) ¯ J ππ ( u ) + ( u ↔ t ) (cid:19)(cid:21) NLO + . . . . (6)Here s, t, u denote the Mandelstam variables and ellipses denote the higher chiral orders not taken into account. Thetadpole integrals arising from, e.g., the wave function re-normalization procedure are denoted by µ i and read µ i = M i π f log M i µ , (7)where µ is the renormalization scale and f is the meson decay constant in the chiral limit. The latter is determineddynamically, from employed sets of LECs and { M π , M K , f π , f K } (see discussion below).The finite parts of the meson-meson loop integrals ¯ J.. ( s ) are given by¯ J P Q ( s ) = 132 π (cid:16) ∆( M P , M Q ) s − Σ( M P , M Q )∆( M P , M Q ) (cid:17) log M Q M P (8) − ν ( s, M P , M Q ) s (cid:16) log ∆( M P , M Q ) + s + ν ( s, M P , M Q )∆( M P , M Q ) − s − ν ( s, M P , M Q ) − log ∆( M P , M Q ) − s + ν ( s, M P , M Q )∆( M P , M Q ) + s − ν ( s, M P , M Q ) (cid:17)! , where ν ( s, M P , M Q ) = p ( s − ( M P + M Q ) )( s − ( M P − M Q ) ), ∆( M P , M Q ) = M P − M Q and Σ( M P , M Q ) = M P + M Q . The corresponding formula for equal masses simplifies to¯ J P P ( s ) = 116 π σ ( s ) log σ ( s ) − σ ( s ) + 1 ! with σ ( s ) = q − M P /s , (9)and resembles the self energy part ρ ( s ) = 16 π ¯ J KK ( s ) used in the main part of the manuscript.For the purpose of the present work, these formulas are projected to the S -wave. This implies a factor N = 2 for thetwo identical kaons [106]. The amplitude is subsequently back-transformed to the (two-body) plane-wave basis asrequired by the form of the quantization condition in Eq. (4).There are three mass relations and three relations for the decay constants (see, e.g., Refs. [106, 118]) setting thesephysical quantities in relation to the tree level quantities f , M π , M K , and M η at NLO (we also replace M η usingthe Gell-Mann–Okubo relation). For the chiral predictions, we follow two strategies, S1 : The tree level quantities aredetermined at the physical point from M π , M K , M η , f π , f K , f η . Then, f π is obtained by replacing tree level massesby the meson masses on the lattice in the relation f π = f (cid:20) − µ π − µ K + 4 M π f ( L r + L r ) + 8 M K f L r (cid:21) . (10)For strategy S2 : The tree level quantity f is directly inferred from the available lattice information, in the presentcases (GWUQCD and NPLQCD) M π , M K , f π , and f K . We consider this method more reliable as one source ofextrapolation uncertainty is removed.We replace also the η -mass in Eq. (6) using the Gell-Mann–Okubo relation. All discussed replacements only leadto O ( p ) effects and are, thus, consistent to the chosen chiral order. Note also that the scattering amplitudes areregularization ( µ ) independent, which implies that the LECs are only defined at the given scale for which we choosethe same value of µ = 770 MeV as in Ref. [100] to be able to use their LECs for predictions. Note that we cannot usethe LECs of Ref. [102] because they correspond to chiral amplitudes formulated in terms of f π , f K , f η as explained inRef. [117]. We emphasize that effects due to change of the renormalization scale have been studied thoroughly in thetwo-flavour case in Ref. [29], where they have been found subdominant to the other effects, e.g., due to pseudo-scalarmasses, decay constants and LECs.While the sets of available SU(2) LECs produce very consistent predictions of the three-pion spectrum [54, 58], theeight SU(3) LECs are less well determined; we chose the LECs of Ref. [100] (fit 4) for this study because the pertinentfit includes lattice data; if one uses the older LECs of Ref. [106] obtained from only fits to experimental data, thescattering length turns out to be about 25% smaller than the one shown in Fig. 1, in contradiction with the NPLQCDresults [115]; likewise, the three-body ground state energy shift gets about 25% smaller, in contradiction with thelattice data. Lattice energies
Here we tabulate the lattice energy levels extracted from the ensembles in Table I in the main text. For the 315 MeVensemble ( E ), triple exponential fits were performed as in Ref. [54]. Due to an increase in the noise on the 220 MeVensemble ( E ), only single exponential fits were performed. The energies are tabulated in Table II. The final columnshows the energies predicted from SU(3) IAM using lattice decay constants and masses from Table I of the main text. Ensemble Irrep E lat /M K E pred /M K E A u . . . . E u . . E A u . . . . E u . ..