Three-body interactions from the finite-volume QCD spectrum
Ruairí Brett, Chris Culver, Maxim Mai, Andrei Alexandru, Michael Döring, Frank X. Lee
TThree-body interactions from the finite-volume QCD spectrum
Ruair´ı Brett, ∗ Chris Culver, † Maxim Mai, ‡ Andrei Alexandru,
1, 3, § Michael D¨oring, ¶ and Frank X. Lee ∗∗ The George Washington University, Washington, DC 20052, USA Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom Department of Physics, University of Maryland, College Park, MD 20742, USA
We perform a fit of the finite-volume QCD spectrum of three pions at maximal isospin to constrainthe three-body force. We use the unitarity-based relativistic three-particle quantization condition,with the GWUQCD spectrum obtained at 315 MeV and 220 MeV pion mass in two-flavor QCD.For the heavier pion mass we find that the data is consistent with a constant contact term close tozero, whereas for the lighter mass we see a statistically significant energy dependence in tension withthe prediction of leading order ChPT. Our results also suggest that with enough three-body energylevels, the two-body amplitude could be constrained.
PACS numbers: 12.38.Gc, 14.40.-n, 13.75.Lb
I. INTRODUCTION
It is a long-term quest of nuclear physics to under-stand hadron interactions as they emerge from quarkand gluon dynamics. The main challenge lies in the factthat perturbation theory fails at low energies, becausethe interactions are strong. Lattice QCD (LQCD) offersa non-perturbative method which has quarks and glu-ons as fundamental degrees of freedom while keeping allsystematics under control. LQCD calculations are per-formed in a finite volume and in Euclidean time, leavingonly indirect methods to study real-time infinite-volumescattering. The relation between finite-volume spectrumand infinite-volume scattering amplitudes is called quan-tization condition, which has been known for two-hadronsystems since the pioneering work of L¨uscher [1]. Thelast decade has witnessed significant progress using thisapproach for a variety of interacting two-particle systems.Only recently has the quantization condition been ex-tended to the three-hadron sector.Recent theoretical advances [2–34] of the three-bodyformalism as well as related numerical studies [35–49] (seeRefs. [50, 51] for reviews) open a possibility for studyinghadronic processes that involve three-body interactionsfrom first principles. Many scattering channels of interestreceive contributions from three-particle states. For ex-ample in the meson sector the a (1260) resonance, seenexperimentally in τ decays, couples first to ρπ and σπ ,and then due to the instability of the ρ and σ mesonsto a final state with three pions. In the baryonic sectoran example where three-body states are relevant is theRoper resonance N (1440)1 / + which has both two- andthree-particle final states, as it decays to πN and ππN . ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] So far most of the effort in lattice QCD calculationsof three-hadron state energies has been concentrated onpure three-meson systems with maximal isospin (three pi-ons or three kaons) where relatively precise finite-volumespectra have been calculated [45, 52–57] and the formal-ism connecting it to infinite-volume amplitudes is betterunderstood [9, 16, 18, 19, 21, 40, 42–47, 58]. In the pi-oneering work by the NPLQCD collaboration in the 3 π and 3 K sectors [52, 53] threshold energy eigenvalues atdifferent pion masses were determined and a thresholdexpansion [59–61] was performed allowing for the firstdetermination of a three-body force.Finite-volume formalisms for excited states were devel-oped later. Among other recent developments [28, 47],there are two relativistic formalisms available for three-particle scattering usually referred to as relativistic finite-volume unitarity (FVU) [16] and relativistic effective fieldtheory (RFT) [9] approaches. The former uses unitarityof the S -matrix as a guiding principle, while the latterrelies on the re-summation of diagrams. A key ingredientto these formalisms is the parameterization of two andthree-body scattering amplitudes. The two-body inputcan be determined from experiment in combination withchiral extrapolations, but lattice QCD data, together withthe quantization conditions, can also be used.The first determination of the three-body force from theNPLQCD data [52] using such a formalism was achievedin Ref. [40] with the FVU framework. There, within theuncertainties of the lattice data, the three-body force wasfound to be zero. This study contains also the first pre-dictions for excited levels at different pion masses. Later,in Ref. [54] excited levels for different boosts and irre-ducible representations (irreps) were calculated that ledto a refined determination of the three-body force withthe RFT formalism [43]. In particular, the three-bodyforce was found to be finite, and even attempts to deter-mine its energy dependence could be made. Within theFVU formalism [44] the data of Ref. [54] were predictedusing only chiral extrapolations of two-body input up tonext-to-leading order (NLO) and assuming a vanishing a r X i v : . [ h e p - l a t ] J a n three-body force, resulting in a χ ≈ .
86 for the three-body sector and χ ≈ .
79 for the combined (correlated)two- and three-body sectors. Note also the predictionsof the data of Ref. [54] in Ref. [47] and in Ref. [54] itself,based on Ref. [62].The GWUQCD collaboration calculated the three- π + spectrum for different quark masses, box geometries, andboosts, mapping out a plethora of states and compar-ing also to FVU predictions that were made under theassumption of vanishing three-body forces [55]. The agree-ment found was fair with a noticeable tension betweenthe lattice data and predictions leading to a χ ≈ . π + spectrum at three different pion masses, in-cluding the physical point for the first time [56]. Theextraction of the three-body force with the RFT formal-ism was compared with the leading order (LO) ChiralPerturbation Theory (ChPT) prediction for the three-to-three process. Similar to Ref. [43], the three-body termwas found to be non-zero, its energy-independent partbeing in qualitative agreement with LO ChPT, but itsenergy-dependent part not.Recently, the three-body force was calculated andextracted by the Hadron Spectrum collaboration [45].The pion mass is relatively large in this calculation( m π ≈
390 MeV) and one could expect a sizable three-body force; yet, within uncertainties and different fitstrategies/parametrizations tried, the three-body term K was found to be compatible with zero.Obviously, the role of three-body forces is not settled. Inthis paper, we continue the investigation of the three-bodyforce with the FVU formalism. In particular, the men-tioned discrepancy of FVU predictions with GWUQCDdata ( χ ≈ .
68) [55] could be the result of a finitethree-body term. Therefore, we perform here a fit to thelattice data of Ref. [55] using the FVU formalism with spe-cial focus on the three-body contact term. In Section IIwe review the extraction of the three-pion finite-volumespectrum from the GWUQCD collaboration. Then inSection III, we review the three-body quantization condi-tion. We include technical implementation details such asparameterizations of the two- and three-body scatteringamplitudes, and establish a connection to a three-to-threecontact term. In Section IV we present the results ofvarious fit scenarios. Finally in Section V we discuss theimpact of the results.
II. FINITE-VOLUME QCD SPECTRUM
Here we review briefly the details of the calculationof the finite-volume spectrum performed by our collab-oration, referring the reader to Ref. [55] for additional
Label N t × N x,y × N z η a [fm] N cfg am π E × ×
24 1 .
00 0 . . E × ×
30 1 . − − . E × ×
48 2 . − − . E × ×
24 1 .
00 0 . . E × ×
28 1 . − − . E × ×
32 1 . − − . N f = 2 ensembles used in this study.Here η is elongation in the z -direction, a the lattice spacing, N cfg the number of Monte Carlo configurations. Ensembles E , E , E have a pion mass m π ≈
315 MeV, while E , E , E have a pion mass m π ≈
220 MeV. material. The ensembles employed use N f = 2 dynamicalfermions, with two sets of quark masses tuned so thatthe pions have masses of 315 MeV or 220 MeV. Thesedata sets were used to compute the two-pion spectrumin all three isospin channels [63–66], the three-pion I = 3channel [55] and the K − K − K − channel [57]. The quarkpropagators are estimated using LapH smearing [67] andan optimized code is employed to compute the requiredmatrix inversions [68]. The parameters describing theensembles in this study are listed in Table I. For detailsabout tuning the lattice spacing and other parameters werefer the reader to Ref. [69].For each pion mass, we calculated the I = 3 three-pion finite-volume spectrum on three different ensembles,which feature two different geometries. There is one cubicvolume and two volumes which are elongated in the z -direction, which is also the direction of the boost whencalculating the energies of moving states. The advantageof using elongated ensembles is an increase in energyresolution, since the momenta are quantized in units of2 π/L . For ensemble E , we are able to extract 16 energylevels below the inelastic threshold, compared to 3 in thecubic volume at the same pion mass. On the hand thenumerical cost of generating these ensembles is reducedsince the volume increases linearly with the elongation,as opposed to a cubic increase for symmetric boxes. Intotal we have 30 energy levels below 5 m π across thesix ensembles listed in Table I. They form the basis forour quantitative analysis of the three-body interaction inthis work. Precise values for the levels can be found inAppendix B of Ref. [55].Regardless of geometry, the finite volume introduces acritical change to the angular momentum quantum num-ber of the system of interest. In the infinite volume, thesymmetry group of rotations is SO (3), and the quantumnumber of the states are labelled by the angular momen-tum l , the irreducible representations (irreps) of the group.For a finite volume the symmetry group is reduced from SO (3) to O h in the case of cubic volumes, and D h in thecase of elongated volumes. If the system is at non-zerototal momentum the relevant symmetry group is C v ,provided the boost is along the direction of any spatialelongation, which is the case here. The rotation quantumnumber of states in a finite volume are thus labelled bythe irreps of the respective symmetry groups. Here wefocus only on states that overlap with l = 0, which willbe labelled by A u for symmetry groups O h and D h , orlabelled by A for the C v symmetry group. For a full dis-cussion on the relation between angular momentum l andirreps of the finite-volume symmetry groups in elongatedboxes we refer the reader to Ref. [70]. III. QUANTIZATION CONDITION
The dynamics of a (multi-)hadron system is accessedin lattice QCD by calculating correlation functions ona discretized Euclidean space-time in a finite volume.The so-called discretization effects are related to the fi-nite lattice spacing a . In principle, a continuum limit a → L , the fre-quently applied (also here) periodic boundary conditionsyield an infinitely large discrete set of allowed momenta S L = { (2 π/L ) n | n ∈ Z } . Unavoidably, using three-momenta from such a set changes the infinite-volumespectrum fundamentally, making it a discrete set of en-ergy eigenvalues. This holds for any finite L , such thata simple extrapolation (lim L →∞ ) is not useful, callingfor a non-trivial mapping between infinite- and finite-volume. Such mappings are referred to as quantizationconditions . An alternative possibility to use ordered dou-ble limit [71] techniques to extract (complex-valued) am-plitudes was explored in Ref. [11], with related techniquesin Refs. [48, 72, 73]. The goal of this section is to reiter-ate the form of the relativistic three-body quantizationcondition derived in Refs. [16, 40, 44, 55], unifying andsimplifying the nomenclature.As one of the currently available relativistic formu-lations of the three-body quantization condition (seeRefs. [50, 51] for reviews) the FVU approach derives fromthe relativistic unitary three-body formalism [74] in infi-nite volume. The formalism differs from the diagrammaticapproach followed in Ref. [9], but yields an equivalentfinite-volume spectrum given the same input, as shownusing time ordered perturbation theory in Ref. [34]. Alsoin the infinite volume both formalisms yield a unitarythree-body amplitude [75, 76]. Both formulations arecurrently being applied to a variety of calculations of sim-pler three-hadron systems such as 3 π + [40, 43–45, 55, 77]or 3 K − [57]. However, the practical implementation ofeach differs, favoring one or another formulation for agiven three-body system. See also Ref. [47] for a calcu- pP - p P - qq s σσ p q isobar/dimerspectatorin out FIG. 1. Schematic representation of a two-plus-one decompo-sition of the three-body system. P , p / q denote three-momentaof the three-body system and that of the in/outgoing specta-tors, respectively. Total three-body energy squared and theinvariant mass of the two-body systems are denoted by s and σ p / q , respectively. lation based on combination of a variational approachand the Faddeev method, including relativistic effects forthe pions and lattice spacing effects. In the following, wewill demonstrate and discuss an example of contrastingrepresentations of the three-body contact term. A. Design and implementation
We avoid discussing the derivation of the FVU formal-ism here [16, 40, 44, 55], but review the results and unifythe notation. At its core, the condition derives from a rel-ativistic unitary three-body scattering amplitude, whichresolves the three-body dynamics as a cluster of a two-body (also related to ‘isobar’ or ‘dimer’ notation) stateand a third particle – a ‘spectator’ [74] (see also Refs. [78–80] for recent progress in this direction). The kinematicnotation of such a configuration is depicted schematicallyin Fig. 1. Unitarity constrains the correct interplay be-tween the two- and three-body interactions accountingfor on-shell configurations, which are the source of singu-larities in the finite-volume correlator. The net result isthat in a finite volume √ s is the energy corresponding toan interacting three-body state when0 = det Q Lη ˜ P ( s ) := det h B ( s ) + C ( s ) − (1) E Lη (cid:0) K − ( s ) / (32 π ) + Σ Lη ˜ P ( s ) (cid:1)i . The determinant is taken over the S Lη × S Lη space,referring to the momentum of the in- and out-going spectator (third particle). Here, S Lη = { π/L ( n , n , n /η ) | ( n , n , n ) ∈ Z } refers to the mo-mentum configuration from elongated boxes used byGWUQCD and ˜ P is the total three-momentum of thesystem. The remaining building blocks of this condition( ∈ Mat S Lη ×S Lη ) are (cid:100) [ E Lη ] pq = δ pq L η p p + m π . (cid:100) [ B ( s )] − pq = − E p + q ( √ s − E p − E q − E p + q ) . This singular and non-diagonal matrix originates fromthe one-particle exchange contribution, and is a di-rect consequence of three-body unitarity in the infinitevolume. Here and in the following E x := p x + m π denotes the on-shell energy of a pion with a momentum x . (cid:100) [Σ Lη ˜ P ( s )] pq = δ pq σ p L η P ˜ k ∈S Lη J ˜ J E k ∗ ( σ p − E k ∗ ) . This singular, diagonal matrix accounts for theon-shell configurations in the two-body subsystemwith total energy squared σ p ( s ) = ( √ s − E p ) − p .The equation above is derived in the Appendixwith the final form given in Eq. (A29). Note thatthe summation is performed over ˜ k in the latticeframe, while the summands are expressed in termsof the k ∗ momenta in the two-body rest frame. Thecorresponding two-step boost readslatticeframe −→ three-bodyrest frame −→ two-bodyrest frame˜ k k (˜ k , ˜ P , s ) k ∗ ( k , p , s ) .The explicit formulas for the boosts and correspondingJacobians J and ˜ J are collected in Appendix A. (cid:100) [ C ( s )] pq . This regular and in general non-diagonal matrixparametrizes the isobar-spectator interaction and, thus,also the three-body contact interaction via a non-trivialmapping discussed in the Sec. III C. In both cases, thiscontribution is not fixed and needs to be obtainedfrom a fit to the lattice results as will be discussed inSec. III C and IV. Note that the normalization usedhere differs slightly from previous work [16, 74]. (cid:100) [ K − ( s )] pq = δ pq K − ( σ p ) . This regular, diagonal matrix parametrizes the dy-namics of the two-body sub-system. In that, it corre-sponds to the usual K -matrix as explained in detail inAppendix A 2. Similarly to the three-body term C ,this contribution needs to be determined from a fit tothe lattice eigenvalues. Such a fit can take two-body,three-body, or both data types into account as will bediscussed in Sec. IV.Before proceeding with the discussion of the two- andthree-body interaction terms K − and C , we note ageneric feature of the three-body quantization condition.In contrast to the well-established L¨uscher’s method inthe two-body case, the three-body quantization condi-tion in Eq. (1) emerges as a full-fledged determinantequation, even in the simplest case of identical particlesin S -wave. Thus, it prevents any one-to-one mappingbetween infinite- and finite-volume quantities. Instead,even for the simplest three-body systems, one must fixthe volume-independent quantities ( C and K − ) froma fit to lattice data and then use those to evaluate the K ( σ ) = + + + + . . . FIG. 2. Leading contributions to the K -matrix utilized forthe two-body input of the quantization condition in Eq. (1).Diagrams represent scattering amplitudes obtained from ChPTwith dots and squares denoting the leading and next-to-leadingchiral order vertices, respectively (tadpole contributions notshown). infinite-volume scattering amplitude. Note that this three-to-three amplitude provides only the three-body unitaryfinal state interaction of production processes; to actuallyobtain mass-spectra or Dalitz plots as, e.g., in Ref. [81],one needs additional information. B. Parametrization of the two-body input
Depending on the system at hand and the research ob-jectives, various techniques in parametrizing the two-bodydynamics ( K − ) may be more or less advantageous. Forexample, a model based on ChPT was used in Ref. [40] tobridge the lattice results obtained at different pion mass.Another approach, related to an effective range expan-sion was employed in Ref. [43] allowing for a systematicextraction of threshold quantities.With the lattice data spanning over large energy rangesand two different pion masses we proceed here with thepath traced out in Ref. [40], and implement the modi-fied Inverse Amplitude Method [82–85] (mIAM) into thethree-body formalism. This is also motivated by our pre-vious applications to the isoscalar channel [65, 86] andmore recently cross-channel ( I = 0 , , ππ scattering)analysis [87] of GWUQCD lattice results [64–66] which isbased on this approach.A practical implementation of this requires equatingthe two-body scattering amplitude used in the infinitevolume analog of Eq. (1) to the mIAM amplitude [85].This procedure has been developed in Ref. [40], but is im-proved in the current study. In particular, the matchingprocedure is now exact and without the need of a redun-dant regulator. Details can be found in Appendices A 2and A 3 leading to K − ( σ ) = T ( σ ) − ¯ T ( σ ) + A ( σ )( T ( σ )) (2)(see Eq. (A17) there). Here, T Ln denotes the chiral ππ amplitude of order n projected to the isospin I = 2,angular momentum L = 0 partial wave. The barredsymbol denotes a reduced amplitude, where the s -channelone-loop diagram is subtracted. This algebraic re-shufflingensures that the above expression indeed contains onlyquantities which do not go on-shell in the physical region.Similarly, A ( σ ) is a function of the chiral amplitudes,introduced in Ref. [85] to improve the analytic behavior ofthe (infinite volume) scattering amplitude below threshold0 < √ σ < m π . In particular, adding A ( σ ) preventsthe two-body scattering amplitude ( ∼ / ( T − T )) fromdiverging around the Adler zero ( σ ⇔ T , ∞ ( σ ) = 0,where T , ∞ is the S -wave scattering amplitude). Recallthat the presence of the Adler zero itself is demanded bychiral symmetry. The diagrammatic representation of theleading contributions is given in Fig. 2, while formulasare provided for convenience in Appendix A 2.We emphasize that knowledge of the sub-threshold two-body dynamics is crucial to describing the three-bodydynamics with the quantization condition Eq. (1) sincefor some ˜ p , σ p ( ˜ p , ˜ P ,s ) < m π for ˜ p ∈ S Lη . The mIAMapproximates the left-hand cut by the next-to leadingchiral order. Thus, it is expected that at too low invariantmasses the latter model is a poor approximation of reality.To account for this, we have explored various possibilities,such as employing smooth form-factors, etc.. We foundthat fixing K − ( σ ) K − ( σ ) for σ < σ = 3 m π issufficient. Variating the matching point in a reasonablerange leads to uncertainties orders of magnitude smallerthan other effects, see Ref. [57]. Also, changes in sub-threshold behavior are supposed to be absorbed in three-body contact terms.Overall, the expression in Eq. (2) is a regular, volume-independent function of two-body energy σ , ensuring ininfinite volume an exact matching of the two-body dynam-ics of the mIAM approach. In that, Eq. (2) depends onfour renormalized low-energy constants { l r , l r , l r , l r } , seeRef. [88] for explicit formulas relating those to Lagrangianconstants. As it has been shown in previous studies of two-body lattice results [86, 89], l r and l r contribute weakly tothe two-body ππ dynamics. This is simply due to the factthat their appearance is solely rooted in the procedure ofreplacing the Lagrangian quantities – pion decay constantin the chiral limit and leading order pion mass – by theirphysical/lattice values. Thus, we simply fix the symmetrybreaking LECs { l r = 8 . · − , l r = 9 . · − } totheir typical values [87, 90] at the regularization scale of µ = 770 MeV. Another possibility is to directly use bothquantities as fit parameters as introduced in Ref. [91].This has an advantage of avoiding the scale setting discus-sion, which is beyond the current stage of three-hadronspectroscopy from QCD. C. Parametrization of the three-body input
The implementation of the three-body input into thequantization condition Eq. (1) is much less explored thanthat of the two-body. One of the goals of this paper is to improve on this, in exploring various ways of parametriz-ing three-body dynamics theoretically and in a practicalapplication to the lattice data.In the three-body finite-volume formalism (FVU) [16],the three-body dynamics is included via a regular (real-valued, infinite volume) function – the isobar-spectatorcontact term C ∼ C . This term is not fixed by uni-tarity and, other than demanded by data, there are noconstrains on its functional dependence with respect toenergies and masses. The RFT formalism [9] on the otherhand, relies on the diagrammatic counting of the on-shellconfigurations in derivation of the three-body quantiza-tion condition. Thus, it includes by construction a regularthree-body contact term ( K , df ), which again needs to bedetermined from a fit to the data.Interestingly, in the pioneering application to the lattice3 π + finite-volume spectrum [40], a simplistic fit to theNPLQCD data [52] led to c = (0 . ± . × − [MeV − ].Later, an analysis of more recent data at one pionmass [54] was performed within both RFT [43] andFVU [44]. In that, the FVU formalism with vanishingthree-body term C led to a χ ≈ . m π K , df = 270(160) wasobtained in the RFT approach by fitting two- and three-body lattice energies. The apparent discrepancy betweenthese results was raised in the community, but, so far, notresolved quantitatively.To find an algebraic connection between the three-to-three contact term ( ¯ T ) and the isobar-spectator coupling C we utilize the language of Ref. [16, 74] in matrixnotation for convenience. There, a fully connected three-body amplitude takes the form T = 13! h vτ T IS τ v i (3)for T IS , τ and v denoting isobar-spectator amplitude,isobar propagator and its coupling to asymptotically sta-ble states, respectively. Explicit definitions of the latterdo not matter for the derivation but can be found inRefs. [16, 74] and Appendix A 3. For simplicity, momen-tum and energy labels are omitted. Symmetrization overexternal momenta is taken into account by h ... i . For threedegenerate scalars in (relative) S -wave this leads to T = 32 vτ + v ( B + C ) vE − Lη τ v ( B + C ) vτ v . (4)Taking now the limit m π → ∞ for all intermediate pions Note that projection to shells (sets of momenta related by oc-tahedral symmetry transformation), was employed in this work,which is not part of the more general form of Eq. (1). There theemployed fit form was chosen as C = · c m π /m phys π . ¯ T = + + + . . . C K K �� �� �� �� ������������ p q s = [ m π ] m π T = × - × - × - × - × - × - × - �� �� �� �� ������������ p q s = [ m π ] m π T = × - × - × - × - × - × - × - �� �� �� �� ������������ p q s = [ m π ] m π T = × - × - × - × - × - × - × - FIG. 3.
Top:
Connection between the contact three-to-three ( ¯ T ) and isobar-spectator interaction ( C ). Two-body dynamicsis encoded in the K -matrix, which does not contribute to divergences. Bottom:
Visualisation of C (in MeV − ) from Eq. (6)for ¯ T from leading order ChPT at relevant values of total three-body energy √ s . In- and outgoing spectator momenta p and q are given by their index in the set (2 π ) / ( L ) { (0 , , , ..., ( ± , ± , } , ordered by magnitude. provides an expression equivalent to a contact term ( ¯ T )¯ T = 32 (cid:18) K − π (cid:19) − C − C E − Lη (cid:16) K − π (cid:17) − (cid:18) K − π (cid:19) − , (5)or equivalently C = (cid:18) K − π (cid:19) (cid:16)
23 ¯ T (cid:17) (cid:18) K − π (cid:19) + E − Lη (cid:0) ¯ T (cid:1) (cid:16) K − π (cid:17) , (6)where ¯ T denotes a real three-body contact term. Thisequation is schematically illustrated in Fig. 3 to the top.Note again that appearance of factors (32 π ) is caused bythe fact that we are working in the plane wave basis, seeAppendix A 2 for more details. This relation is of thesame form as the relation between the K , df and C term,which can be obtained from a matching of FVU and RFTformalisms, see for example Refs. [50, 51]. Also, as notedthere, it implies that in general an isotropic ¯ T leads toanisotropic C and vice versa.To expand on this further, we consider the followingexample. Chiral perturbation theory at leading chiralorder yields a three-to-three contact term of the form¯ T = 127 f π (cid:0) s − m π (cid:1) . (7) Using now Eq. (6) with the K -matrix from Eq. (2)we obtain a prediction for the isobar-spectator inter-action C ( √ s, p , q ), for the momenta belonging to thein/outgoing spectators. The resulting symmetric (in p , q )isobar-spectator contact interaction is depicted for severalvalues of total three-body energy 3 m π < √ s < m π inFig. 3. We observe that several orders of magnitude dif-ference between the overall scales of the ¯ T and C occursnaturally, connecting the results of Refs. [43, 44]. D. Three-body state energies
Given parameterizations of the two- and three-body in-teractions, the finite-volume spectrum can be determinedby searching for energies at which Eq. (1) is satisfied.To find the energies associated with a particular irrep Λof the symmetry group G , we first block-diagonalize thematrix Q Q = diag( Q Λ , Q Λ , . . . ) , ⇒ det Q = Y i det Q Λ i . (8)The determinant of Q Λ block can then be scanned sep-arately in s for the zeros which determine the energiesfinite-volume states for that irrep.To block-diaganolize the matrix Q we need to changeto an appropriate basis. The change of basis from theplane-wave basis in Eq. (1) can be done by constructingprojectors for the row λ of the irrep Λ of the group G : P Λ λ = d Λ | G | X g ∈ G Γ Λ λλ ( R ( g )) U ( R ( g )) (cid:124) , (9)where d Λ is the dimension of the irrep Λ, Γ Λ λλ ( R ) is therepresentation matrix of the group element g in row λ of the irrep Λ, R ( g ) is the rotation corresponding to g ,and U ( R ( g )) is the unitary transformation of R on theplane-wave space.We must truncate the plane-wave space S Lη to includeonly momenta below a sufficiently large threshold, p max ≈ p max . Inour case we found that there is a very mild dependence onthe choice of p max , for the energy levels predicted from thequantization condition, well below the stochastic errorsin the lattice data. By construction this truncated spaceis invariant under the symmetry transformations, sincethe momenta magnitudes are invariant under rotations.For each irrep Λ we project the entire plane-wave basisusing the projector above and the dimensionality of thisspace represents the number of independent multipletsassociated with Λ that appear in the selected plane-wavebasis. The restriction of Q to this subspace is Q Λ whichhas zeros corresponding to energies in the irrep Λ. IV. RESULTS
In this section we present the results for the three-bodyterms as extracted from fitting the finite-volume spectrum.For sake of clarity we will first discuss the extraction of thethree-body terms when the two-body dynamics is fixedby the mIAM parameterization with LECs determinedfrom fitting our lattice two-pion spectrum in all isospinchannels [44]. This model provides a good descriptionof our two-pion spectrum across all channels for the twoquark masses we studied. For the three-body terms weparameterize ¯ T , using no energy dependence: ¯ T ( s ) = t ,or ¯ T ( s ) = t + t s/m π , a linear function in s . Thesecond parametrization is consistent with the leadingorder prediction from ChPT in Eq. (7).The data points included in these fits are all three-bodyenergies below the inelastic threshold in the irreduciblerepresentations sensitive to the s-wave three-body terms.The relevant representations are A u for the states atrest and A for the moving states. The total numberof points is 12 (2, 3, and 7 for ensembles E , E , and E respectively) for the heavy quark mass and 7 (2, 2, and3 for ensembles E , E , and E respectively) for the lightquark mass. These energy levels are plotted with errorsin Fig. 4. FIG. 4. Finite-volume center-of-mass π + π + π + energies for m π = 315 MeV ( E , , ) and m π = 220 MeV ( E , , ). For eachpion mass there is one cubic box ( E , ) and two elongatedboxes ( E , , , ). Columns distinguish different irreps of therotational symmetry group containing energies below the in-elastic threshold, 5 m π (solid black line). The data pointsare the LQCD energy levels with error bars inside of the cir-cles. The dashed lines are the non-interacting energy levels.Boosted frames with non-zero total momentum are denoted bythe superscript [001] indicating a single unit of momentum inthe elongation ( z ) direction. The solid lines represent the pre-dicted central values of the spectrum from FVU, after fitting t , t , l r , and l r , for m π = 315 MeV (red) and m π = 220 MeV(blue) separately. - - - - - - FIG. 5. The three-body contact term ¯ T defined via Eq. (6) as a function of three-body energy √ s for m π = 315 MeV (left)and m π = 220 MeV (right). The data with error bars show the pointwise determination of ¯ T . The three lines correspond toconstant, linear and ChPT energy dependence. Dark bands indicate 1 σ uncertainties for fits of the three-body data using fixedmIAM LECs from Ref. [87]. Light bands indicate the 1 σ uncertainties from fits to the three-body data including l r and l r as fitparameters and priors from the cross-channel two-pion fits as described in Sec. IV. All fits to the three-body energy levels are performedseparately for the two quark masses since our parametriza-tion for the three-body term does not constrain the quarkmass dependence. We perform various fits, keeping someof the fit parameters fixed and varying others as shownin the Table III. Below we discuss these results.To get a sense whether these parametrizations are rea-sonable, we first extracted ¯ T ( s ) by fitting each individuallattice energy level alone. This offers a rough profile of¯ T as a function of energy. To stabilize these fits we fixthe LECs to the values extracted in our cross-channeltwo-pion study [44]. In Fig. 5 these results are representedby the data points, for the heavy and light quark masses.We note that while the errors are large the results sup-port ¯ T ( s ) being (weakly) dependent on energy. For theheavier quark mass the data is consistent with a constantparameterization for ¯ T , close to zero, whereas for thelighter mass we see a statistically significant fall in thedata with energy. Note that the magnitude of ¯ T in abso-lute terms is not that different, but the energy dependenceis more pronounced for the lighter mass. Additionally,for the light mass ¯ T is well constrained to be non-zeroaway from threshold. We also present in these plots theexpectation from the leading order ChPT, plotted with adashed line. At the light mass there is some tension withthe leading order ChPT prediction, in particular at higherenergies. Also, the energy dependent term has oppositesign to the ChPT result. This is similar to the tension inthe energy dependence of the RFT three-body term inRefs. [43, 56].The simultaneous fit to all the energy levels are indi-cated with color bands in Fig. 5. For these fits we allowedthe LECs to vary but constrained their variation usingpriors based on the probability distribution that was de-termined in our cross-channel two-pion data fits. Weprefer this strategy over the simultaneous fit of the two- and three- body energies, since it makes the results for thethree-body fit easier to analyze by isolating the two-bodycontribution. These priors are included by augmentingthe χ -function that is minimized by the fitter: χ = χ + δl i (Σ − ) ij δl j | {z } χ , (10)where δl i := l ri − ˆ l i , with ˆ l being the values of the LECs asdetermined in Ref. [44] and Σ ij their covariance matrix.Note that since the LECs variations tend to be stronglycorrelated, in order to include these priors properly weneed to consider the full correlation matrix Σ, not justthe diagonal terms associated with individual LECs errorestimates. In Ref. [44] we only reported the errors on theLECs. We include the estimate for the covariance matrixin Table II.The constant fit works reasonably well for the heavyquark mass ensembles and is consistent with the linear fitin s . For the lighter quark mass the linear and constant fitare quite different and the linear term is needed to describethe data well. The values extracted for the fit constantsand the χ value for the fit are listed in Table III.We also indicate in the figures with narrower bands thefit results for { t , t } when the LECs are not allowed tovary, corresponding to the second and third rows in Ta-ble III. We see that while the central values are almost thesame, the error bands are almost doubled when we allow TABLE II. Covariance matrix Σ ij for the LECs l r , l r asdetermined in Ref. [44].ˆ l i Σ ij / ( σ i σ j ) l r − . . l r . t · MeV t · MeV l r · l r · χ I =3 + χ χ . . − . + . .
39 0 2 . − . . . − . + . .
95 0 2 . − . .
1) +0 . − . + . .
76 0 2 . . . − . . .
08 0 .
14 2 . − . . . − . . .
41 1 .
52 2 . − . .
5) +0 . − . . .
33 0 .
10 2 . . . − . .
5) +5 . . t · MeV t · MeV l · l r · χ I =3 + χ χ . . − . + . .
78 0 4 . − . . . − . + . .
10 0 2 . . . − . − . + . .
41 0 2 . . . − . . .
80 3 .
79 4 . − . . . − . . .
50 0 .
30 2 . − . . − . . .
27 0 .
07 2 . . . − . .
8) +3 . .
1) 7 . TABLE III. Fit results for ¯ T and LECs including I = 3 πππ energies only for m π = 315 MeV (left) and m π = 220 MeV (right).Bold font indicates parameters fixed to values from Ref. [87], others are left as free parameters of the fit. The final row in eachtable is for a fit using relaxed priors, as described in the text. the LECs to vary. This shows that even with small errorbars (few percent level) in the two-body parametrization,there is a large impact on the error of the three-bodyterms in this channel. This is partly due to the smallnessof three-body terms.In terms of fit quality, we note that for both masses, thefits for the linear form with varying LECs produce a χ perdegree of freedom around 2. This indicates that there is aslight tension between the data and the parametrizationused here. At this point it is not easy to determinewhether this is the result of the quality of the lattice dataor due to the lack of flexibility in the fit form used forthe three-body terms. We note that since the three-bodypredictions are sensitive to the two-body inputs, some ofthis tension might have as a source small discrepanciesin the two-body amplitudes used in the quantizationconditions. We note that when analyzing two-mesonenergy levels using mIAM framework, a similar level ofagreement between data and predictions was found [87].An interesting question is whether we can extract theLECs parametrizing the two-body interactions directlyfrom three-body energy levels. The three-body energydata set does not provide enough constraints to pin downboth the LECs and the three-body terms. We are howeverable to fit the LECs when the three-body terms are set tozero. Setting ¯ t , to zero makes sense, since their effect israther small. To stabilize the root finding routines used topredict the three-body energy levels as a function of theLECs, we constrained the region scanned for the LECs toa reasonable window, within one order of magnitude of thevalues determined from the two-body fits. Procedurallythis was accomplished using a set of relaxed priors. Weused a correlation matrix Σ relaxed = 30 × Σ, so that theequivalent error bands on the LECs were at the level of100%, in effect constraining only the order of magnitudeof the LECs.The results for these fits are included in the last rowsof Table III. We find that the values of the LECs are closeto the ones generated from the two-body fits, albeit withlarger error bars. This provides a good cross-check forthe formalism and suggests that with enough three-bodyenergy levels, we should be able to also constrain thetwo-body amplitudes. To put the results on the three-body force in perspective,we compare our determination of the three-body termwith those obtained in the literature [43, 56] in Fig. 6. Indoing so, the matching of corresponding three-body termscan be made on the level of scattering amplitudes applyingthe procedure discussed in Sec. III C. We note that thisyields an approximate identification K iso , , ’ t + 9¯ t )and K iso , , ’ t . We see reasonable agreement betweendifferent collaborations, not too different from the leadingorder ChPT prediction. This indicates the rapid progressmade in the community in mapping out the three-bodyforce. V. CONCLUSIONS
The field of three-body physics is rapidly advancing,fueled by progress on two fronts. On the one hand, pre-cise energy levels are being produced in LQCD for in-teracting systems such as three pions or kaons. On theother hand, formalisms that connect the finite-volumeQCD spectrum and infinite-volume three-body scatteringamplitude, called quantization conditions, are reachingmaturity. Such progress has allowed the possibility ofextracting quantitative information on the three-bodyforce from first principles.In this work we apply the FVU formalism to analyzethe spectrum obtained previously in Ref. [55]. We useda minimal parametrization for the three-body contactterm and constrain the parameters from fits to the spec-trum extracted using lattice QCD. We find that the heavyquark mass results are compatible with expectations fromleading order ChPT, but our lower mass results are intension with the predictions. Note that this is similarto other LQCD determinations of this term in the RFTframework [43, 56]. The three-body force terms are small,in broad agreement with other lattice QCD extractions.We also perform a fit to the three-body spectrum to con-strain the parameters of the two-body amplitude. Wefind that the results are in agreement with the valuesextracted from the two-body spectrum, indicating thatthe two-body amplitude can also be determined consis-tently from the three-body data. While this is expected0 - - - - - - FIG. 6. Three-body force as a function of the I = 2 scattering length. Results from this work for fixed and varied two-bodyinput is shown by filled red circles and squares, respectively. Expectations from leading order ChPT are shown by the gray lineand those of earlier (RFT) determinations in blue (ETMC [56]) and green (BRS [43]) symbols. The red circles are slightly offsetin the horizontal direction for legibility. The dashed vertical line shows the physical point. theoretically, it is an important feasibility check.This study serves as a benchmark for the fitting strategyand the tools developed to generate predictions for thefinite-volume three-body spectrum at maximal isospin. Inthis channel the effects of the three-body force are small,so to constrain it better we need more energy levels and/orbetter precision for the lattice data. This is likely to bedone in the near future. Another direction that is beingexplored is to study other three-body channels whereresonant amplitudes contribute. To this end, both latticeQCD data needs to be generated and the quantizationconditions have to be extended. ACKNOWLEDGMENTS
This material is based upon work supported by theNational Science Foundation under Grant No. PHY-2012289 and by the U.S. Department of Energy underAward Number DE-SC0016582 (MD and MM) and DE-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. The authors thank Fernando Romero-L´opez and Peter Bruns for useful discussions. [1] M. L¨uscher,
Two particle states on a torus and theirrelation to the scattering matrix , Nucl. Phys.
B354 (1991) 531.[2] S. Kreuzer and H.W. Hammer,
Efimov physics in a finitevolume , Phys. Lett.
B673 (2009) 260 [ ].[3] S. Kreuzer and H.W. Hammer,
On the modification ofthe Efimov spectrum in a finite cubic box , Eur. Phys. J.
A43 (2010) 229 [ ].[4] S. Bour, S. K¨onig, D. Lee, H.W. Hammer, andU.-G. Meissner,
Topological phases for bound statesmoving in a finite volume , Phys. Rev.
D84 (2011) 091503[ ].[5] S. Kreuzer and H.W. Grießhammer,
Three particles in afinite volume: The breakdown of spherical symmetry , Eur. Phys. J.
A48 (2012) 93 [ ].[6] K. Polejaeva and A. Rusetsky,
Three particles in a finitevolume , Eur. Phys. J.
A48 (2012) 67 [ ].[7] R.A. Brice˜no and Z. Davoudi,
Three-particle scatteringamplitudes from a finite volume formalism , Phys. Rev.
D87 (2013) 094507 [ ]. [8] U.-G. Meißner, G. R´ıos, and A. Rusetsky,
Spectrum ofthree-body bound states in a finite volume , Phys. Rev.Lett. (2015) 091602 [ ].[9] M.T. Hansen and S.R. Sharpe,
Relativistic,model-independent, three-particle quantization condition , Phys. Rev.
D90 (2014) 116003 [ ].[10] M.T. Hansen and S.R. Sharpe,
Expressing thethree-particle finite-volume spectrum in terms of thethree-to-three scattering amplitude , Phys. Rev.
D92 (2015) 114509 [ ].[11] D. Agadjanov, M. Doring, M. Mai, U.-G. Meißner, andA. Rusetsky,
The Optical Potential on the Lattice , JHEP (2016) 043 [ ].[12] P. Guo, One spatial dimensional finite volume three-bodyinteraction for a short-range potential , Phys. Rev.
D95 (2017) 054508 [ ].[13] M.T. Hansen and S.R. Sharpe,
Threshold expansion ofthe three-particle quantization condition , Phys. Rev.
D93 (2016) 096006 [ ].[14] R.A. Brice˜no, M.T. Hansen, and S.R. Sharpe,
Relatingthe finite-volume spectrum and the two-and-three-particle S matrix for relativistic systems of identical scalarparticles , Phys. Rev.
D95 (2017) 074510 [ ].[15] Y. Meng, C. Liu, U.-G. Meißner, and A. Rusetsky,
Three-particle bound states in a finite volume: unequalmasses and higher partial waves , Phys. Rev.
D98 (2018)014508 [ ].[16] M. Mai and M. D¨oring,
Three-body Unitarity in theFinite Volume , Eur. Phys. J.
A53 (2017) 240[ ].[17] P. Guo and V. Gasparian,
An solvable three-body modelin finite volume , Phys. Lett. B (2017) 441[ ].[18] H.-W. Hammer, J.-Y. Pang, and A. Rusetsky,
Three-particle quantization condition in a finite volume:1. The role of the three-particle force , JHEP (2017)109 [ ].[19] H.W. Hammer, J.Y. Pang, and A. Rusetsky, Threeparticle quantization condition in a finite volume: 2.general formalism and the analysis of data , JHEP (2017) 115 [ ].[20] P. Guo, M. D¨oring, and A.P. Szczepaniak, Variationalapproach to N -body interactions in finite volume , Phys.Rev.
D98 (2018) 094502 [ ].[21] M. D¨oring, H.W. Hammer, M. Mai, J.Y. Pang,§.A. Rusetsky, and J. Wu,
Three-body spectrum in afinite volume: the role of cubic symmetry , Phys. Rev.
D97 (2018) 114508 [ ].[22] P. Guo and T. Morris,
Multiple-particle interaction in(1+1)-dimensional lattice model , Phys. Rev.
D99 (2019)014501 [ ].[23] F. Romero-L´opez, A. Rusetsky, and C. Urbach,
Two-and three-body interactions in ϕ theory from latticesimulations , Eur. Phys. J.
C78 (2018) 846 [ ].[24] R.A. Brice˜no, M.T. Hansen, and S.R. Sharpe,
Three-particle systems with resonant subprocesses in afinite volume , Phys. Rev.
D99 (2019) 014516[ ].[25] P. Guo,
Propagation of particles on a torus , Phys. Lett.B (2020) 135370 [ ].[26] F. Romero-L´opez, S.R. Sharpe, T.D. Blanton,R.A. Brice˜no, and M.T. Hansen,
Numerical explorationof three relativistic particles in a finite volume includingtwo-particle resonances and bound states , JHEP (2019) 007 [ ].[27] S. Zhu and S. Tan, d -dimensional L¨uscher’s formula andthe near-threshold three-body states in a finite volume , .[28] F. Romero-L´opez, A. Rusetsky, N. Schlage, andC. Urbach, Relativistic N -particle energy shift in finitevolume , .[29] P. Guo, Modeling few-body resonances in finite volume , Phys. Rev. D (2020) 054514 [ ].[30] P. Guo,
Threshold expansion formula of N bosons in afinite volume from a variational approach , Phys. Rev. D (2020) 054512 [ ].[31] P. Guo and M. D¨oring,
Lattice model of heavy-lightthree-body system , Phys. Rev. D (2020) 034501[ ].[32] T.D. Blanton and S.R. Sharpe,
Relativistic three-particlequantization condition for nondegenerate scalars , .[33] T.D. Blanton and S.R. Sharpe, Alternative derivation ofthe relativistic three-particle quantization condition , Phys.Rev. D (2020) 054520 [ ]. [34] T.D. Blanton and S.R. Sharpe,
Equivalence of relativisticthree-particle quantization conditions , Phys. Rev. D (2020) 054515 [ ].[35] S. Kreuzer and H.W. Hammer,
The Triton in a finitevolume , Phys. Lett.
B694 (2011) 424 [ ].[36] L. Roca and E. Oset,
Scattering of unstable particles in afinite volume: the case of πρ scattering and the a (1260) resonance , Phys. Rev.
D85 (2012) 054507 [ ].[37] S. Bour, H.-W. Hammer, D. Lee, and U.-G. Meißner,
Benchmark calculations for elastic fermion-dimerscattering , Phys. Rev.
C86 (2012) 034003 [ ].[38] M. Jansen, H.W. Hammer, and Y. Jia,
Finite volumecorrections to the binding energy of the X(3872) , Phys.Rev.
D92 (2015) 114031 [ ].[39] P. Guo and V. Gasparian,
Numerical approach for finitevolume three-body interaction , Phys. Rev.
D97 (2018)014504 [ ].[40] M. Mai and M. D¨oring,
Finite-Volume Spectrum of π + π + and π + π + π + Systems , Phys. Rev. Lett. (2019)062503 [ ].[41] P. Klos, S. K¨onig, H.W. Hammer, J.E. Lynn, andA. Schwenk,
Signatures of few-body resonances in finitevolume , Phys. Rev.
C98 (2018) 034004 [ ].[42] R.A. Brice˜no, M.T. Hansen, and S.R. Sharpe,
Numericalstudy of the relativistic three-body quantization conditionin the isotropic approximation , Phys. Rev. D (2018)014506 [ ].[43] T.D. Blanton, F. Romero-L´opez, and S.R. Sharpe, I = 3 Three-Pion Scattering Amplitude from Lattice QCD , Phys. Rev. Lett. (2020) 032001 [ ].[44] M. Mai, M. D¨oring, C. Culver, and A. Alexandru,
Three-body unitarity versus finite-volume π + π + π + spectrum from lattice QCD , Phys. Rev. D (2020)054510 [ ].[45] M.T. Hansen, R.A. Brice˜no, R.G. Edwards,C.E. Thomas, and D.J. Wilson,
The energy-dependent π + π + π + scattering amplitude from QCD , Phys. Rev.Lett. (2021) 012001 [ ].[46] F. M¨uller, A. Rusetsky, and T. Yu,
Finite-volume energyshift of the three-pion ground state , .[47] P. Guo and B. Long, Multi- π + systems in a finitevolume , Phys. Rev. D (2020) 094510 [ ].[48] P. Guo and B. Long,
Visualizing resonances in finitevolume , Phys. Rev. D (2020) 074508 [ ].[49] J.-Y. Pang, J.-J. Wu, and L.-S. Geng,
DDK system infinite volume , Phys. Rev. D (2020) 114515[ ].[50] M.T. Hansen and S.R. Sharpe,
Lattice QCD andThree-particle Decays of Resonances , Ann. Rev. Nucl.Part. Sci. (2019) 65 [ ].[51] A. Rusetsky, Three particles on the lattice , PoS
LATTICE2019 (2019) 281 [ ].[52] W. Detmold, M.J. Savage, A. Torok, S.R. Beane,T.C. Luu, K. Orginos, et al.,
Multi-Pion States in LatticeQCD and the Charged-Pion Condensate , Phys. Rev.
D78 (2008) 014507 [ ].[53] W. Detmold, K. Orginos, M.J. Savage, andA. Walker-Loud,
Kaon Condensation with Lattice QCD , Phys. Rev. D (2008) 054514 [ ].[54] B. H¨orz and A. Hanlon, Two- and three-pionfinite-volume spectra at maximal isospin from latticeQCD , Phys. Rev. Lett. (2019) 142002 [ ].[55] C. Culver, M. Mai, R. Brett, A. Alexandru, andM. D¨oring,
Three pion spectrum in the I = 3 channel from lattice QCD , Phys. Rev. D (2020) 114507[ ].[56] M. Fischer, B. Kostrzewa, L. Liu, F. Romero-L´opez,M. Ueding, and C. Urbach,
Scattering of two and threephysical pions at maximal isospin from lattice QCD , .[57] A. Alexandru, R. Brett, C. Culver, M. D¨oring, D. Guo,F.X. Lee, et al., Finite-volume energy spectrum of the K − K − K − system , Phys. Rev. D (2020) 114523[ ].[58] F. M¨uller and A. Rusetsky,
On the three-particle analogof the Lellouch-L¨uscher formula , .[59] W. Detmold and M.J. Savage, The Energy of n IdenticalBosons in a Finite Volume at O ( L − ), Phys. Rev. D (2008) 057502 [ ].[60] S.R. Beane, W. Detmold, and M.J. Savage, n-BosonEnergies at Finite Volume and Three-Boson Interactions , Phys. Rev. D (2007) 074507 [ ].[61] S. Tan, Three-boson problem at low energy andimplications for dilute Bose-Einstein condensates , Phys.Rev. A (2008) 013636 [ ].[62] J.-Y. Pang, J.-J. Wu, H.W. Hammer, U.-G. Meißner,and A. Rusetsky, Energy shift of the three-particle systemin a finite volume , Phys. Rev.
D99 (2019) 074513[ ].[63] C. Pelissier and A. Alexandru,
Resonance parameters ofthe rho-meson from asymmetrical lattices , Phys. Rev.
D87 (2013) 014503 [ ].[64] D. Guo, A. Alexandru, R. Molina, and M. D¨oring,
Rhoresonance parameters from lattice QCD , Phys. Rev.
D94 (2016) 034501 [ ].[65] D. Guo, A. Alexandru, R. Molina, M. Mai, andM. D¨oring,
Extraction of isoscalar ππ phase-shifts fromlattice QCD , Phys. Rev.
D98 (2018) 014507[ ].[66] C. Culver, M. Mai, A. Alexandru, M. D¨oring, and F. Lee,
Pion scattering in the isospin I = 2 channel fromelongated lattices , Phys. Rev. D (2019) 034509[ ].[67]
Hadron Spectrum collaboration,
A Novel quark-fieldcreation operator construction for hadronic physics inlattice QCD , Phys. Rev.
D80 (2009) 054506 [ ].[68] A. Alexandru, C. Pelissier, B. Gamari, and F. Lee,
Multi-mass solvers for lattice QCD on GPUs , J. Comput.Phys. (2012) 1866 [ ].[69] H. Niyazi, A. Alexandru, F.X. Lee, and R. Brett,
Settingthe scale for nHYP fermions with the L¨uscher-Weiszgauge action , Phys. Rev. D (2020) 094506[ ].[70] F.X. Lee and A. Alexandru,
Scattering phase-shiftformulas for mesons and baryons in elongated boxes , Phys. Rev.
D96 (2017) 054508 [ ].[71] B.S. DeWitt,
Transition from discrete to continuousspectra , Phys. Rev. (1956) 1565.[72] M.T. Hansen, H.B. Meyer, and D. Robaina,
From deepinelastic scattering to heavy-flavor semileptonic decays:Total rates into multihadron final states from latticeQCD , Phys. Rev. D (2017) 094513 [ ].[73] R.A. Brice˜no, J.V. Guerrero, M.T. Hansen, andA. Sturzu, The role of boundary conditions in quantumcomputations of scattering observables , .[74] M. Mai, B. Hu, M. D¨oring, A. Pilloni, andA. Szczepaniak, Three-body Unitarity with IsobarsRevisited , Eur. Phys. J.
A53 (2017) 177 [ ]. [75] A. Jackura, S. Dawid, C. Fern´andez-Ram´ırez,V. Mathieu, M. Mikhasenko, A. Pilloni, et al.,
Equivalence of three-particle scattering formalisms , Phys.Rev. D (2019) 034508 [ ].[76] R.A. Brice˜no, M.T. Hansen, S.R. Sharpe, andA.P. Szczepaniak,
Unitarity of the infinite-volumethree-particle scattering amplitude arising from afinite-volume formalism , Phys. Rev. D (2019)054508 [ ].[77] T.D. Blanton, F. Romero-L´opez, and S.R. Sharpe,
Implementing the three-particle quantization conditionincluding higher partial waves , JHEP (2019) 106[ ].[78] JPAC collaboration,
Phenomenology of Relativistic → Reaction Amplitudes within the IsobarApproximation , Eur. Phys. J. C (2019) 56[ ].[79] A.W. Jackura, R.A. Brice˜no, S.M. Dawid, M.H.E. Islam,and C. McCarty, Solving relativistic three-body integralequations in the presence of bound states , .[80] M. Mikhasenko, Y. Wunderlich, A. Jackura, V. Mathieu,A. Pilloni, B. Ketzer, et al., Three-body scattering:Ladders and Resonances , JHEP (2019) 080[ ].[81] D. Sadasivan, M. Mai, H. Akdag, and M. D¨oring, Dalitzplots and lineshape of a (1260) from a relativisticthree-body unitary approach , Phys. Rev. D (2020)094018 [ ].[82] T.N. Truong,
Chiral Perturbation Theory and Final StateTheorem , Phys. Rev. Lett. (1988) 2526.[83] A. Dobado and J.R. Pelaez, The Inverse amplitudemethod in chiral perturbation theory , Phys. Rev.
D56 (1997) 3057 [ hep-ph/9604416 ].[84] A. G´omez Nicola, J.R. Pel´aez, and G. Rios,
The InverseAmplitude Method and Adler Zeros , Phys. Rev.
D77 (2008) 056006 [ ].[85] C. Hanhart, J.R. Pelaez, and G. Rios,
Quark massdependence of the rho and sigma from dispersionrelations and Chiral Perturbation Theory , Phys. Rev.Lett. (2008) 152001 [ ].[86] M. D¨oring, B. Hu, and M. Mai,
Chiral Extrapolation ofthe Sigma Resonance , Phys. Lett.
B782 (2018) 785[ ].[87] M. Mai, C. Culver, A. Alexandru, M. D¨oring, andF.X. Lee,
Cross-channel study of pion scattering fromlattice QCD , Phys. Rev. D (2019) 114514[ ].[88] J. Gasser and H. Leutwyler,
Chiral Perturbation Theoryto One Loop , Annals Phys. (1984) 142.[89] N.R. Acharya, F.-K. Guo, M. Mai, and U.-G. Meißner, θ -dependence of the lightest meson resonances in QCD , Phys. Rev. D (2015) 054023 [ ].[90] Flavour Lattice Averaging Group collaboration,
FLAG Review 2019: Flavour Lattice Averaging Group(FLAG) , Eur. Phys. J. C (2020) 113 [ ].[91] ETM collaboration,
The ρ -resonance with physical pionmass from N f = 2 lattice QCD , .[92] M. D¨oring, U.-G. Meißner, E. Oset, and A. Rusetsky, Scalar mesons moving in a finite volume and the role ofpartial wave mixing , Eur. Phys. J.
A48 (2012) 114[ ].[93] A. Gomez Nicola and J.R. Pelaez,
Meson mesonscattering within one loop chiral perturbation theory andits unitarization , Phys. Rev.
D65 (2002) 054009 [ hep-ph/0109056 ]. Appendix A: Two-body summations
To set a baseline, the previous approach to two-bodysummations, as used in Refs. [40, 44, 55], is reviewed.In Sec. A 2 the matching to ChPT is discussed motivat-ing a new implementation for evaluating the two-bodyparameterizations in Sec. A 3.
1. Previous two-body summation
In the following, the subscript “ ∞ ” is used to distin-guish infinite-volume quantities from their finite-volumecounterparts; as outlined in the main text, starred ( ∗ )three-momenta and energies are defined in the two-bodyrest frame; likewise, tilde (˜) indicates the lattice restframe, and momenta/energies without superscript aredefined in the three-body rest frame. Furthermore, p, q, l label incoming, outgoing, and intermediate spectator mo-menta, respectively; ˜ P is the momentum of the three-bodysystem and k ∗ labels the momentum of the pions from iso-bar decay, that are, of course, back-to-back, i.e., ( E k ∗ , k ∗ )and ( E k ∗ , − k ∗ ). The zero-components of all momentaare always taken on-shell, i.e., E l := l = p m π + l [74].The infinite-volume propagator τ ∞ in Eq. (4) ofRef. [16], adapted to current notation, reads τ − ∞ λ = σ − M λ − Σ ∞ , Σ ∞ = Z d k ∗ (2 π ) E k ∗ ( σ − σ + i(cid:15) ) , (A1)where σ := 4 E k ∗ and σ = s + m π − √ s E l is the in-variant two-body sub-energy squared (in the main text, σ is called σ p owing to the fact that p and q are usedto name spectator momenta). We refer to the term Σas “self energy” in the following although this formal-ism is a-priori not based on any diagrammatic expansion.Furthermore, M and λ in Eq. (A1) are fit parametersthat may be adjusted to two-body input; indeed, thetwo-to-two scattering T -matrix is simply T , ∞ = vSv = − v D v = − λτ ∞ λ (A2)with S and D from Ref. [74]. We also indicate the re-striction of the general vertex v ≡ v ( q i , q j ), that dependson the four-momenta of the decay pions q i , q j [74], tothe S -wave case λ ≡ λ ( σ ) that here depends only on σ . The vertex v depends on invariants formed by thethree available (isobar and decay) four-momenta. Thisensures three-body unitarity is maintained when v is eval-uated in different frames (three-body and two-body restframes) [81].In previous studies [40, 44, 55] a form factorfor regularization was included in the definition of v whichwe can drop in the current formulation as explained in Sec. A 3; for the time being we assume that the integralin Eq. (A1) is simply regularized by a cutoff (in Sec. A 3we construct convergent expressions).The finite-volume version of the isobar-spectator prop-agator is obtained from Eq. (A1) by imposing peri-odic boundary conditions in the lattice rest frame lead-ing to discretized momenta. For elongations η in z -directions, i.e., L x = L y = L , L z = ηL this im-plies a discrete set of allowed three-momenta ˜ k ∈S Lη := { π/L ( n , n , n /η ) | ( n , n , n ) ∈ Z } . Thefinite-volume propagator is diagonal in spectator momen-tum and reads τ − P Lη λ = σ − M λ − Σ ˜ P Lη Σ ˜ P Lη = 1 ηL X k ∗ ˜ J J E k ∗ σ − σ (A3)where k ∗ ≡ k ∗ ( k (˜ k ) , l (˜ l )), with the dependence on specta-tor momentum l explicitly denoted. The boost momentaare ˜ P from lattice to three-body rest frame, leading to theJacobian ˜ J , and − l from three-body rest frame to two-body rest frame, leading to the Jacobian J in Eq. (A3).To further discuss the kinematics, we consider the in-coming and outgoing pion momenta ˜ q j and ˜ p j , j = 1 , , π + system has then momentum ˜ P = ˜ q + ˜ q + ˜ q =˜ p + ˜ p + ˜ p . Momenta in the three-body rest frameare [92] l = ˜ l + (cid:20)(cid:18) E ˜ P √ s − (cid:19) ˜ l · ˜ P ˜ P − E ˜ l √ s (cid:21) ˜ P , (A4)and analogously for the other momenta k , p and q . InEq. (A4), E ˜ P = p s + ˜ P . The boost of Eq. (A4) leadsto the Jacobian˜ J = (cid:12)(cid:12)(cid:12)(cid:12) d l d ˜ l (cid:12)(cid:12)(cid:12)(cid:12) = E ˜ P √ s − ˜ l · ˜ P √ s E ˜ l . (A5)The isobar is not at rest in the three-body rest frame.Thus, an additional boost (by − l ) has to be performed forthe pertinent summation of momenta k ∗ in the self energyof the isobar of Eq. (A3). This is detailed in Eqs. (11, 12)of Ref. [16] and reads in the current notation k ∗ = k + l (cid:20) k · ll (cid:18) √ σ √ s − E l − (cid:19) + √ σ √ s − E l ) (cid:21) , (A6)leading to the Jacobian J = √ σ √ s − E l . (A7)5
2. Matching to ChPT
We use the Inverse Amplitude Method (IAM) [82, 83]for the isospin I = 2 ππ scattering amplitude, with a modi-fication for improved sub-threshold behavior (mIAM) [84].For this subsection, we denote the standard Mandelstamvariables by σ , t , u instead of s , t , u to avoid notation clashwith other parts of the paper. The starting point is givenby the ChPT result at leading (LO) and next-to-leadingorder (NLO), T I ( σ, t, u ) and T I ( σ, t, u ) , (A8)for I = 2 scattering. From now on we drop the index I for readability. The explicit expressions are given inRef. [88]. In S -wave ( L = 0), the unitary amplitude T L mIAM is written as T = ( T ) T − T + A , (A9)where A is constructed such that the Adler zero is at itsNLO position, i.e., A ( σ ) = T ( σ ) − (A10)( σ − σ A )( σ − σ ) σ − σ A (cid:0) T ( σ ) − T ( σ ) (cid:1) , where σ and σ A are zeroes of T and T − T , respectively,see Ref. [84] for more details.In Eqs. (A9) and (A10), T ( T ) is the LO (NLO)partial-wave contribution, obtained from the correspond-ing plane-wave expressions of Eq. (A8) according to T n ( σ ) = 132 N π Z − d x P T n ( σ, t, u ) , (A11)for n = 2 , x = cos θ where θ is the scattering angle.There is also a symmetry factor of N = 2 for identicalparticles, and P = 1 is the S -wave Legendre polynomial.In particular, T = 32 πT as the LO contribution is angleindependent.In a next step, we match this to a K-matrix formalismwhich is easier to implement into the three-body frame-work. For this we note that the NLO contribution to the ππ amplitude can be split as T ( σ, t, u ) = ¯ T ( σ, t, u ) + 12 ( T ( σ )) ¯ J ππ ( σ ) (A12)or T ( σ ) = ¯ T ( σ ) + 16 π ( T ( σ )) ¯ J ππ ( σ ) , (A13)where ¯ T and ¯ T are real for s > m π and only the ππ loop ¯ J ππ provides an imaginary part in the physical region.It reads [93] ¯ J ππ = 116 π (cid:20) σ log ˆ σ − σ + 1 (cid:21) , (A14) where ˆ σ = 2 k cm √ σ , k cm = r σ − m π . (A15)We can now determine the (real-valued) K -matrix byequating T = ( K − − ρ ) − (A16)with Eq. (A9). Using Eqs. (A13) this leads to K − = T − ¯ T + A ( T ) , ρ = 16 π ¯ J ππ , (A17)which, of course, only depends on σ . We obtain theplain-wave amplitude T mIAM from Eqs. (A16), (A11) andcan set it equal to the two-to-two scattering matrix ofEq. (A2), − ( λτ ∞ λ ) − = T − , ∞ = T − = K − π − ¯ J ππ . (A18)Indeed, the conventions of both T -matrices are identicalas an explicit evaluation of imaginary parts shows:Im T − , ∞ = Im Σ ∞ = − k cm π √ σ , Im T − = − π π Im ¯ J ππ = − k cm π √ σ . (A19)For the quantization condition, we also need the equivalentof the K -matrix of Eq. (A17) in the plane-wave basis.This explains the additional factor of 32 π in Eq. (1) ofthe main text using Eq. (A11).A final remark on symmetry factors is in order becausewe match our dispersive three-body framework to theFeynman-diagrammatic mIAM approach, and keepingtrack of these factors is important. The symmetry nor-malization of the two-body amplitude happens in thepartial-wave decomposition Eq.(A11) ( N = 2) such thatthe partial-wave amplitudes are connected to physicalobservables in the standard way; for example, the S -wave scattering length is a m π = T ( σ = 4 m π ) andthe phase shift reads tan δ = Im T Re T . When return-ing from the partial-wave T to the plane-wave T IAM in Eq. (A18) the corresponding factor 32 π contains the N = 2 of Eq. (A11), i.e., T IAM is not symmetry nor-malized. The same is true for T , ∞ : In the notationof Ref. [74], the matrix element for incoming pions atmomenta p , p and outgoing momenta q , q is given as h q , q | T | p , p i = 12! v ( q , q ) S ( σ ) v ( p , p ) (A20)= − λ ( σ ) τ ∞ ( σ ) λ ( σ ) = 12 T , ∞ , Strictly speaking, for a K -matrix formalism Re ¯ J ππ should beabsorbed in K , as well; somewhat sloppily, we still refer to K as“ K -matrix”. not contained in thedefinition of T , ∞ . Furthermore, note that v = 2˜ v wherethe factor of 2 accounts for the possibilities to connecta decay vertex ˜ v to two external pions [74]. In otherwords, v contains the same multiplicity as a Feynmanvertex would carry from the two possible contractionswith external fields.
3. Regularization-free, accelerated two-bodysummation
It is easy to see that Eq. (A3) diverges logarithmically.In the past [40, 44, 55] the same cutoff Λ was chosen forboth integration and summation in Eqs. (A1) and (A3)such that the real parts approximately match (indicatedwith “!”),Re Σ ∞ ( σ ) ! ≈ −
12 Re ¯ J ππ ( σ ) − d π (A21)which holds well over a large range of σ for the choice d = 0 .
86 and Λ = 42 m π [40]. Equation (A21) completedthe matching procedure.However, one can take a step back and question theform of the self energy in Eq. (A1). After all, its specificform was chosen to be able to match to time-orderedperturbation theory with explicit isobar fields [74], whichis not our goal here. In Ref. [74], different forms of theisobar-spectator are derived from its imaginary part givenin Eq. (A19). In particular, n -times subtracted dispersionrelations provide factors of ( σ/σ ) n in the integrand forthe self energy such that already with n = 1 one obtains aconvergent expression in the present case. As convergenceis drastically improved, no matching with hard cutoffs isneeded any more and the numerical summation can becut at much lower values, greatly improving the speedof fitting that was previously limited by the two-bodysummation. With the improvements discussed in thefollowing, typical speedups of a factor of 10 are achieved.Consider the twice-subtracted dispersion relation withsubtraction point arbitrarily chosen at σ = 0,Σ (2) ∞ ( σ ) = Σ (2) ∞ (0) + σ Σ (2) (0) + σ π ∞ Z m π d¯ σ Im Σ ∞ (¯ σ )¯ σ (¯ σ − σ − i(cid:15) )= − σ m π π + σ π ∞ Z d k ∗ k ∗ E k ∗ σ − σ + i(cid:15) , (A22)where σ = 4 E k ∗ and the second equal sign is obtainedwith the imaginary part from Eq. (A19), a variable substi-tution, and by matching Σ (2) ∞ and its derivative at σ = 0to − / J ππ from Eq. (A14). It can be easily shown that these two functions are identical for all σ ,Σ (2) ∞ ( σ ) = −
12 ¯ J ππ ( σ ) ≈ Σ ∞ ( σ ) + d π , (A23)where we have, for completeness, also quoted the rela-tion to the original expression according to Eq. (A21).The advantage of using the dispersion relation for thematching to ¯ J ππ is now apparent: the matching is exact,not approximate, it is independent of matching parame-ters (cut-off Λ and d ), and the corresponding integrationconverges quickly, as Eq. (A22) shows.By writingΣ (2) ∞ = − σ m π π + Z d k ∗ (2 π ) E k ∗ (cid:16) σσ (cid:17) σ − σ + i(cid:15) , (A24)the formal similarity with Eq. (A1), up to the factor( σ/σ ) , becomes apparent. By imposing periodic bound-ary conditions as in Sec. A 1, the finite-volume self energybecomesΣ (2)˜ P Lη = − σ m π π + 1 ηL X k ∗ ˜ J J E k ∗ (cid:16) σσ (cid:17) σ − σ . (A25)This expression may be used instead of Eq. (A3) as itconverges rapidly and provides exact matching to theChPT loop ¯ J ππ .The difference between Σ (2)˜ P Lη and Σ ˜ P Lη from Eq. (A3)is exponentially suppressed for all σ . Indeed, whenevera finite-volume pole arises at σ − σ = 0 (remember that σ is discrete in finite volume), we have ( σ/σ ) = 1, i.e.,the finite-volume poles in both expressions have the sameresidues. Furthermore, the difference is of the formΣ ˜ P Lη − Σ (2)˜ P Lη = A + A σ (A26)as an inspection of the arguments of the summationsshows: 1 σ − σ − (cid:16) σσ (cid:17) σ − σ = − σσ − σ . (A27)The exponentially suppressed difference can be sizable andbecomes, in general, larger for more subtractions in thedispersion relation; if one aims at regaining a particularform of the self energy, one can reduce the difference.To do this we solve explicitly for A , with the crucialadvantage that it has to be calculated only once for given L, η , three-body boost ˜ P , and two-body boost by − l . Thequantity A can then be recycled, e.g., in fitting, wherethe entire σ -dependence of Σ needs to be known. Inother words, one can still take advantage of the speedupprovided by Σ (2)˜ P Lη at the cost of having to determine A ,or A and A , once for every L , η , ˜ P , − l . The quantity7 A is easiest determined in the limit σ → −∞ . UsingEqs. (A22) and (A25), A = 14 π ∞ Z d k ∗ k ∗ σ E k ∗ − ηL X k ∗ ˜ JJσ E k ∗ , (A28)which is indeed exponentially suppressed.This concludes the derivation. Still, one can write theself energy, modified by the A -term, in the surprisinglysimple formΣ (1)˜ P Lη := Σ (2)˜ P Lη + A σ = 1 ηL X k ∗ ˜ J J E k ∗ σσ σ − σ . (A29)The factor σ/σ appears here linearly. Indeed, a closer inspection shows that we could have obtained Eq. (A29)directly by imposing periodic boundary conditions on aonce-subtracted dispersion relation, i.e., by starting thederivation withΣ (1) ∞ ( σ ) = Σ (1) ∞ (0) + σπ ∞ Z m π d¯ σ Im Σ ∞ (¯ σ )¯ σ (¯ σ − σ − i(cid:15) ) (A30)instead of Eq. (A22). Of course, by using directlyEq. (A29) one partially looses the speedup of the twice-subtracted results. But, at least, Eq. (A29) is still conver-gent in contrast to Eq. (A3). For the purpose of this study,we find the speed-up provided by Eq. (A29) sufficient andtrade the slight loss of speed for not having to calculate A separately for each L , η , ˜ P , − l . The exponentiallysuppressed term A0