# Isospin-1/2 Dπ scattering and the lightest D_0^\ast resonance from lattice QCD

Luke Gayer, Nicolas Lang, Sinéad M. Ryan, David Tims, Christopher E. Thomas, David J. Wilson

PPrepared for submission to JHEP

Isospin-1/2 Dπ scattering and the lightest D ∗ resonance from lattice QCD Luke Gayer, a Nicolas Lang, a Sin´ead M. Ryan, a David Tims, a Christopher E. Thomas, b David J. Wilson b (for the Hadron Spectrum Collaboration) a School of Mathematics and Hamilton Mathematics Institute, Trinity College, Dublin 2, Ireland b Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences,University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:

Isospin-1/2 Dπ scattering amplitudes are computed using lattice QCD, work-ing in a single volume of approximately (3 . and with a light quark mass correspondingto m π ≈

239 MeV. The spectrum of the elastic Dπ energy region is computed yielding 20energy levels. Using the L¨uscher ﬁnite-volume quantisation condition these energies aretranslated into constraints on the inﬁnite volume scattering amplitudes. For the ﬁrst time,we ﬁnd a complex D ∗ resonance pole from lattice QCD, strongly coupled to the S -wave Dπ channel, with a mass m ≈ ≈

400 MeV. Combined with earlierwork investigating the D ∗ s , and D ∗ with heavier light quarks, similar couplings betweeneach of these scalar states and their relevant meson-meson scattering channels are deter-mined. The mass of the D ∗ is consistently found well below that of the D ∗ s , in contrast tothe currently reported experimental result. deceased a r X i v : . [ h e p - l a t ] F e b ontents t -matrix 114.2 Dπ with J P = 1 − and D ∗ π with J P = 1 + Dπ with J P = 0 + and J P = 1 − S -wave pole 215.2 P -wave pole 22 K -matrix 32C Simultaneously ﬁtting Dπ with J P = 0 + , − and D ∗ π with J P = 1 + Since their experimental discoveries in 2003, the lightest scalar charm-light D ∗ and charm-strange D ∗ s mesons have stimulated much theoretical activity. Within the quark-modelthey have a common P -wave q ¯ q construction and the origin of much of the mass diﬀerence isdue to the diﬀering light and strange quark masses. This results in predictions of two similarstates with the mass of the charm-light state below the charm-strange state. However, thestates have been observed experimentally at similar masses with vastly diﬀerent widths,– 1 –he D ∗ seen as a broad feature in Dπ amplitudes, in contrast with a narrow D ∗ s foundbelow DK threshold.The lightest experimentally observed scalar D ∗ resonance was ﬁrst reported by theBelle [1] and FOCUS [2] experiments with a mass in the range 2300-2400 MeV. Laterresults from BABAR [3] and a pair of studies by LHCb [4, 5] also found a broad resonanceat a similar mass, not so diﬀerent from quark potential models that predict a D ∗ around2400 MeV [6]. The D ∗ s (2317) was observed as a narrow peak in isospin-breaking D s π ﬁnalstates, with a mass below DK threshold. In contrast to the D ∗ the D ∗ s is reported some200 MeV below typical quark potential model predictions [7], perhaps even below the D ∗ .The vastly diﬀering widths, and how these aﬀect the masses, call for a deeper theoreticalanalysis of these states and the light quark mass dependence in the open charm sector [8].The limitations of quark potential models in scalar systems are understood. In partic-ular, since decay channels made of pairs of pseudoscalars open with no angular momentumbarrier they can have a signiﬁcant eﬀect, both when nearby but kinematically closed andwhen a strong coupling to a decay channel produces a large width. Many puzzling hadronicresonances arise close to thresholds, from the D ∗ and χ c (3872), to more recent observa-tions in exotic ﬂavour such as D − K + (with ﬂavour content ¯ c ¯ sdu ) [9, 10] and J/ψ J/ψ ﬁnalstates [11], and so a model-independent approach is required to explore fully the QCDdynamics.While experimental production of these hadrons proceeds through heavy meson decays,the simplest theoretical perspective is to observe them as part of scattering processes suchas Dπ → Dπ . Lattice QCD is a ﬁrst-principles approach that has been applied successfullyto a range of scattering processes. In the light meson sector properties of simple narrowresonances such as the ρ seen in P -wave ππ → ππ scattering have been computed [12–19].Scalar resonances have also been studied in elastic and coupled-channel systems includingthe σ [20], the f [21] and the a [22]. More recently, states involving spinning scatteringhadrons have been determined, such as the b [23], an exotic π [24], and several J P C = J −− resonances [25].In Ref. [26] Dπ , Dη , D s ¯ K scattering amplitudes were computed from 47 energy levelsin the ﬁrst coupled-channel calculation involving charm quarks using lattice QCD. A near-threshold scalar D ∗ bound-state was identiﬁed albeit with a heavier-than-physical lightquark mass corresponding to m π = 391 MeV. In this study, we compute Dπ scatteringamplitudes with m π = 239 MeV to investigate how the lightest D ∗ evolves with the lightquark mass. We have previously investigated several other channels at these two lightquark masses, beginning with the ρ [13, 14], followed by the σ [20] and πK [27, 28].Crucially for the present study, DK [29] was also determined at both masses and thoseresults combined with Ref. [26] allowed for a ﬁrst-principles comparison of the D ∗ s and D ∗ at m π = 391 MeV. In this case, the D ∗ was found signiﬁcantly below the D ∗ s . In thepresent study, we complete the quartet of calculations to better understand both systems.Charm-light systems are featured in several other studies using lattice QCD [30–36].Other than our earlier calculation [26], only Ref. [31] computed the I = 1 / Dπ scatteringamplitude. However, only two or three energy levels are obtained in a small volume withno dynamical strange quarks and so the physical implications are not clear.– 2 –he paper is organised as follows. Section 2 gives the lattice parameters and cal-culation details. In section 3 the ﬁnite volume spectra are presented and the scatteringamplitudes subsequently determined using L¨uscher’s formulation are described in section 4.The location of poles and their couplings is discussed in section 5. An interpretation andan analysis of the light quark mass dependence is given in section 6. A summary of thiswork and an outlook is given in section 7. Lattice QCD is a ﬁrst principles approach to QCD. Working in a ﬁnite cubic volume with aﬁnite lattice spacing, correlation functions are computed by numerically sampling the QCDpath integral in Euclidean spacetime. In a ﬁnite spatial volume, momentum is quantisedand thus only discrete spectra of scattering energies are accessible. However, the mappingbetween the discrete spectrum obtained in a ﬁnite volume and the inﬁnite volume scatteringamplitudes is known for two-hadron systems [37] and recent developments are reviewed inRef. [38].The results presented here are computed using a single lattice with (

L/a s ) × ( T /a t ) =32 × L is the spatial volume and T is the temporal extent, with a peri-odic boundary condition in space and an antiperiodic boundary condition in time. Ananisotropic lattice formulation is utilised in which the temporal lattice spacing, a t , is ﬁnerthan the spatial lattice spacing, a s , and ξ ≡ a s /a t ≈ .

5. To quote results in physical units,we compare the calculated Ω baryon mass to its physical value yielding a − t = 6 .

079 GeVand a spatial lattice spacing a s = 0 .

11 fm [28], corresponding to a physical spatial volumeof (3 . .A tree-level Symanzik-improved anisotropic action is used to describe the gauge sectorwhile for the fermion sector a tree-level, tadpole-improved Sheikholeslami-Wohlert actionwith N f = 2 + 1 dynamical quark ﬂavours and stout-smeared spatial gauge ﬁelds is used.This calculation uses 484 gauge conﬁgurations. The light quarks on this ensemble corre-spond to m π = 239 MeV while the heavier dynamical quark is tuned to approximate thestrange quark mass. The valence charm quark is simulated using the same action as forthe light and strange quarks with mass and anisotropy parameters tuned to simultaneouslyreproduce the physical η c mass and the pion anisotropy, determined from the dispersionrelations [36, 39]. To extract many states over a wide energy region, we follow the procedure used in Ref. [13].We compute matrices of correlation functions C ij ( t ) = (cid:104) |O i ( t ) O † j (0) | (cid:105) , (2.1)from large bases of interpolating operators O i that carry the quantum numbers of the I = 1 / Dπ system. The ﬁnite-volume eigenstates can then be obtained by solving ageneralised eigenvalue problem [40, 41] of the form C ij ( t ) v ( n ) j = λ n ( t, t ) C ij ( t ) v ( n ) j , (2.2)– 3 –here the correlation matrix C ij is diagonalised timeslice-by-timeslice to obtain the eigen-vectors v n , and the generalised eigenvalues, or principal correlators, λ n ( t, t ). The imple-mentation adopted here is outlined in Refs. [42, 43]. The energy spectrum is extracted froman analysis of the time dependence of the principal correlators. To account for possibleexcited state contaminations we ﬁt a sum of two exponentials, λ n ( t, t ) = (1 − A n ) e − E n ( t − t ) + A n e − E (cid:48) n ( t − t ) . (2.3)The parameter E n yields the energy of the state, while A n and E (cid:48) n are only used to stabilisethe ﬁt and play no further role in the subsequent analysis. A linear combination thatbest interpolates a given state n can then be constructed from the eigenvector v n throughΩ † n = (cid:80) i v ( n ) i O † i .The basis of operators in each irrep is constructed to achieve a good overlap with thestates we investigate. We use quark bilinears of the form ¯ ψ Γ D...ψ where Γ represents aproduct of γ -matrices, D is a gauge-covariant derivative and the ellipsis indicates that upto three derivatives are used. The inclusion of derivatives allows for the construction ofoperators with good overlap on to higher angular momenta states, more than typicallyobtained using only γ -matrices. Further details can be found in Ref. [43, 44].The operator bases also include meson-meson-like operators which take the form (cid:80) (cid:126)p + (cid:126)p = (cid:126)P C ( (cid:126)p , (cid:126)p )Ω † M ( (cid:126)p )Ω † M ( (cid:126)p ), where Ω † M i ( (cid:126)p i ) is a variationally-optimised operatorthat interpolates meson M i with lattice momentum (cid:126)p i . These operators are constructedfor each momentum from eigenvectors v n determined in variational analyses of π , K , η , D , D s and D ∗ mesons. Further details of the construction of these operators are given inRefs. [44, 45].The ﬁnite cubic volume breaks rotational symmetry so that eigenstates cannot belabelled by representations J of the orthogonal group. Instead they are categorised by theirreducible representations (irreps) of the cubic group O h when at rest [46], or by the littlegroup LG ( (cid:126)P ) at non-zero momentum [47]. The little groups correspond to crystallographicpoint groups with rotation and reﬂection symmetry determined by the direction of themomentum. The interpolating operators are projected into irreps of the correspondingsymmetry group. The subduction of continuum J P into lattice irreps relevant to thiscalculation is summarised in table 1.The operators used in constructing the correlation matrices are summarised in ap-pendix A. We use several constructions resembling a “single-meson” in each irrep, and weuse the non-interacting energies to guide the choice of appropriate “meson-meson” construc-tions. Non-interacting meson-meson energies are given by E = (cid:112) m + | (cid:126)p | + (cid:112) m + | (cid:126)p | for the scattering of mesons 1 and 2. We include all meson-meson constructions corre-sponding to a non-interacting level in the energy region below the Dππ threshold as wellas several additional operators that are only expected to produce levels at higher energiesto help ensure a reliable spectrum is obtained.Correlation functions are computed using the distillation framework [49], whereby thequark ﬁeld is projected into a low dimensional space spanned by the lowest N vec eigenvectorsof the gauge-covariant Laplacian ordered by eigenvalue. This projection suppresses highenergy modes, enhancing the overlap of the operators onto the states we investigate. At the– 4 – P Irrep J P ( (cid:126)P = (cid:126) Dπ J P [ N ] D ∗ π J P [ N ] Λ | λ | (˜ η ) ( (cid:126)P (cid:54) = (cid:126) A +1 + , 4 + + , ... ... T − − , 3 − − , ... ... E + + , 4 + + , ... ...[ n A (+) , 4 + , − , 2 + , ... ... E

1, 3 − , 2 + , ... + , ...[ nn A (+) , 2, 4 + , − , 2 +[2] , ... ... B , B

1, 3 − , 2 + , ... + , ...[ nnn ] A (+) , 3 + , − , 2 + , ... ... Table 1 . The lowest continuum J P and helicity λ , and corresponding Dπ (and similarly Dη and D s ¯ K ) and D ∗ π that subduce in each of the irrep and little groups used in this calculation. Overallmomentum is denoted by (cid:126)P = πL ( i, j, k ) = [ ijk ]. Bold J P denote a contribution that was used toobtain scattering information. [ N ] indicates the number of subductions when more than one arepresent in an irrep and ˜ η = P ( − J . “...” denotes higher partial wave contributions which are notconsidered in this calculation. For each J P several D ∗ π combinations can appear, we only makeuse of S +1 (cid:96) J = S and ignore P -wave and higher. For Dπ we consider up to J P = 2 + . Furtherdetails with more irreps and partial waves can be found in table 3 of Ref. [27] for Dπ (which followsthe same pattern as Kπ ), and tables 1, 5, 6 and 7 of Ref. [48] for D ∗ π . same time it provides an eﬃcient way to compute correlators and reuse them for diﬀerentoperator constructions. For this analysis we use N vec = 256 distillation vectors. In table 2 we summarise the stable hadron masses and the relevant thresholds for thiscalculation. These are obtained from the dispersion relation,( a t E ) = ( a t m ) + | (cid:126)d | (cid:18) πξ L/a s (cid:19) (2.4)where m is the hadron mass, (cid:126)d is a vector of integers. The anisotropy ξ obtained fromthe pion dispersion relation is ξ π = 3 . D is ξ D = 3 . ξ π including its uncertainty, to transform the moving-frame spectra to the rest-frame energies E cm . Both ξ π and ξ D are used subsequently to assess the uncertainties in the scatteringamplitudes. The mapping between the quantised ﬁnite-volume spectrum obtained from lattice QCD,and the inﬁnite volume scattering amplitudes is given by the L¨uscher quantisation condi-tion [37, 50, 51], and extensions thereof [52–60]. We use the form,det [ + i ρ ( s ) · t ( s ) · ( + i M ( s, L ))] = 0 , (2.5) The same number of vectors was used in a recent study of DK scattering in Ref. [29]; N vec = 384vectors were used in earlier studies of ππ [14, 20] and πK [28] scattering on this lattice. – 5 – t mπ K η D D s D ∗ a t E threshold Dπ Dππ Dη D s ¯ K D ∗ ππ Table 2 . Left: A summary of the stable hadron masses relevant for this calculation. Right:kinematic thresholds relevant for I = 1 / Dπ scattering. where s = E cm , ρ is a diagonal matrix of phase space factors, ρ ( s ) = 2 k ( s ) / √ s , t ( s ) is theinﬁnite volume t -matrix which is diagonal in partial waves, and is related to the scattering S -matrix through S = + 2 i √ ρ · t · √ ρ . The determinant is over partial waves and openchannels. The scattering momentum, k ( s ) = (cid:112) ( s − ( m D + m π ) ) ( s − ( m D − m π ) ) / (4 s ),and M ( s, L ) is a matrix of known functions that encode the eﬀect of the ﬁnite volume andmix partial waves. A more complete description including the subduction of this equationinto lattice irreps, which is relevant for pseudoscalar-pseudoscalar scattering, with thematrix indices exposed, is given in Ref. [27].The solutions of Eq. 2.5 for a given t ( s ) is the ﬁnite-volume spectrum for that speciﬁcscattering amplitude t ( s ). To determine the scattering amplitude from a spectrum wechoose to parameterise t ( s ) using several amplitudes that respect the unitarity of the S -matrix and are analytic except for cuts due to k ( s ) and poles. The free parameters in t ( s )are then minimised to best describe the spectrum obtained in the lattice calculation. Theenergies that solve Eq. 2.5 can be identiﬁed by numerically root-ﬁnding the determinant,or the eigenvalues of the matrix inside the determinant [61], as a function of s = E cm .These solutions are then used in a correlated χ ﬁt as deﬁned in Eq. 8 of Ref. [27]. Suitableparameterisations of t explored here include K -matrices, the eﬀective range expansion, aBreit-Wigner form, and unitarised chiral amplitudes.We note that Eq. 2.5 is valid only for two-hadron scattering processes, the choiceto work at heavier-than-physical pion masses raises three-hadron (and higher) thresholdsrelative to two-hadron thresholds. To go rigorously beyond three-hadron thresholds such as Dππ which appears at relatively low energies for the pion masses used here, an extension tothe theoretical framework is required [62]. The three-body quantisation condition has beenapplied recently to simple systems of three identical particles [63–68] while a very recentstudy has reported the generalisation of the quantisation condition to three non-identicalparticles [69]. While this is promising progress, a full consideration of the energy regionabove three-particle thresholds is beyond the scope of the current work.

The ﬁnite volume spectra computed in this study are presented in Figs. 1 and 2. Asexplained above, these spectra are grouped according to the irreps of the cubic group and– 6 – . . . . . . . . a t E c m (a) [000] A +1

24 32 40 (b) [000] E +

24 32 40

DπD ∗ πDηD s ¯ KDππ

L/a s (c) [000] T − Figure 1 . Finite-volume spectra obtained in the at-rest [000] A +1 , [000] E + and [000] T − irreps.Dotted lines correspond to channel thresholds. Solid lines indicate non-interacting energy levelscorresponding to operators included in the simulation. Points with error bars represent the energylevels obtained from the variational analysis. Black points will be included in the subsequentscattering analysis while grey points will be excluded. little groups, labelled by [ (cid:126)d ]Λ ( P ) , where (cid:126)d indicates the direction of the overall momentumsuch that (cid:126)P = 2 π (cid:126)d/L . Parity P is only a valid quantum number at zero overall momentum.Energy levels used to constrain the scattering amplitudes are shown in black, while otherenergies extracted but not used are shown in grey. The cutoﬀ for energies used in thescattering analyses is Dππ threshold.We compute correlation functions on a single volume in this analysis, but we nonethe-less indicate the volume dependence of the non-interacting levels in these plots. Onlycertain continuum angular momenta subduce into a given irrep in the cubic volume asindicated in table 1, and higher partial waves are suppressed by a factor of k (cid:96) +1 nearthreshold. In Fig. 1, the at-rest irreps are shown. The Dπ S -wave ( (cid:96) = 0) only sub-duces into [000] A +1 in this ﬁgure. The Dπ P -wave ( (cid:96) = 1) subduces into [000] T − , whilethe lowest contribution for [000] E + is the Dπ D -wave ( (cid:96) = 2), as summarised in table 1.[000] E + will not be used in any of the ﬁts as discussed in section 4. The irreps shownin Fig. 2 have non-zero total momentum. The Dπ S -wave subduces into the top four ofthese ([100] A , [110] A , [111] A and [200] A ). The lower three irreps, [100] E , [110] B and[110] B predominantly contain the Dπ P -wave close to threshold and the D ∗ π S -wave atslightly higher energies and will be considered together with [000] T − , but separately fromthe Dπ S -wave. – 7 –ll irreps that have an (cid:96) = 1 contribution have a level far below threshold which maybe associated with a deeply bound D ∗ vector state that is stable at this heavier-than-physical light quark mass. In all irreps with an (cid:96) = 0 contribution we observe a levelaround Dπ threshold that is shifted downward with respect to the nearby non-interactinglevel. We also observe the appearance of what may be an extra level around a t E cm = 0 . S -wave interactions. In comparison, irreps having (cid:96) = 1 as thelowest partial wave contribution yield levels which are only marginally shifted away fromthe nearby non-interacting energies.Figure 3 shows the spectrum obtained for [000] A +1 and how this can vary when diﬀerenttypes of operator are removed. It is clear from the ﬁgure that neither the Dπ -like northe q ¯ q -like operators alone are enough to reliably compute the spectrum in this system.Using the q ¯ q operators alone produces a single level at a similar energy to the second levelfound when using a more complete basis, however the lowest level close to D [000] π [000] isabsent. Using only Dπ -like operators does not produce a single level consistent with thespectrum found when using the more complete basis. Adding the Dη -like and the D s ¯ K -likeoperators leaves the spectrum unchanged at lower energies and new levels arise close tothe non-interacting levels for D [000] η [000] and D s [000] ¯ K [000]. The lowest two levels usedin the scattering analyses are robust against any small changes in the operator basis, suchas removing higher lying operators and many individual q ¯ q -like operators.While qualitative indications of the interactions present can be obtained from lookingat the ﬁnite volume spectra, in order to gain a rigorous understanding the energies obtainedmust be related to inﬁnite volume scattering amplitudes, as we do in the next section.– 8 – . . . . . . . . a t E c m (a) [100] A

24 32 40 (b) [110] A

24 32 40 (c) [111] A

24 32 40

DπD ∗ πDηD s ¯ KDππ

L/a s (d) [200] A . . . . . . . . a t E c m (e) [100] E

24 32 40 (f ) [110] B

24 32 40

DπD ∗ πDηD s ¯ KDππ

L/a s (g) [110] B Figure 2 . As in Fig. 1, but for the moving-frame [100] A , [110] A , [111] A , [200] A irreps (top) and[100] E , [110] B and [110] B irreps (bottom). The dash-dotted curves indicate a non-interactinglevel for which no corresponding operator was included in the basis. – 9 – .350.360.370.380.390.400.41 Figure 3 . The spectra obtained in [000] A +1 when varying the operator basis. The operators in-cluded are marked below each column. The grey blocks show the 1 σ uncertainties of the energiesobtained from variational method. The lowest two energies from the column marked “all” corre-spond to the levels used in the scattering analyses. The histograms plotted next to each energy arethe magnitudes of the operator overlaps (cid:104) |O| n (cid:105) , normalised to their maximum contribution seenin any state, from each variational analysis. The solid lines indicate the non-interacting energies. – 10 – Scattering analyses

In this section, the spectra presented above are used to constrain the inﬁnite volume scat-tering amplitudes through the L¨uscher determinant condition given by Eq. 2.5. The goalis to extract the

Dπ S -wave. However due to the mixing of angular momenta introducedby the cubic volume it is necessary to consider other partial waves subducing into any ofthe irreps used in the analysis. As mentioned before, partial waves grow at threshold like k (cid:96) +1 which means that higher (cid:96) are suppressed close to threshold. In this calculation weonly ﬁnd eﬀects of any signiﬁcance from (cid:96) = 0 and (cid:96) = 1.At zero overall momentum [000] A +1 is the only irrep with an S -wave contribution,as is shown in table 1. The next higher partial wave that subduces in this irrep has (cid:96) = 4 and can be neglected. However there are not suﬃciently many energy levels toconstrain the amplitude from this irrep alone. In the A irreps at non-zero momentum Dπ can contribute with J P = 0 + , − , + , ... and thus it is necessary to consider these partialwaves simultaneously. The lowest D ∗ π non-interacting level in these irreps is D ∗ [100] π [100] in [110] A which lies well above the Dππ threshold, and the lowest contributing partialwave is D ∗ π in P -wave. Hence Dπ is the only relevant channel in these irreps within theenergy region we consider. In the moving frames [100] E and [110] B , irreps with nonzerototal momentum, J P = 1 − Dπ scattering is the leading partial wave. These irreps alsoinclude a contribution from D ∗ π in J P = 1 + and the lowest non-interacting level above Dπ threshold is D ∗ [100] π [000] . We therefore need to include D ∗ π in the t -matrix when makinguse of energy levels in these irreps.On the basis of these observations we initially perform two separate amplitude deter-minations: One using [000] T − , [100] E , and [110] B , to constrain the Dπ P -wave and toassess the D ∗ π S -wave contribution (section 4.2). The second determination uses [000] A +1 and the moving frame A irreps to constrain the Dπ S -wave and P -wave simultaneously(section 4.3). Treating energies separately simpliﬁes the Dπ S -wave analysis, while retain-ing constraints on the

Dπ P -wave.In this analysis, we only consider the region below the three-body

Dππ threshold at a t E cm ≈ . Dη threshold. No levels in this energy regionshow sensitivity to the inclusion of Dη operators. Eﬀects of higher partial waves with (cid:96) ≥ E + is an irrep where (cid:96) = 2 Dπ is the lowest partial wave. In this irrep, the lowest level isfound to be consistent with the lowest non-interacting level, D [100] π [100], at a t E cm =0 . ± . D -wave scattering phase δ = (0 . ± . ◦ . Sincethe phase must be zero at threshold it is reasonable to conclude that the Dπ D -wave isnegligibly small throughout the energy region used for scattering analyses. t -matrix We now introduce the t -matrix parameterisations used in this analysis. In elastic pseudoscalar-pseudoscalar scattering there is no coupling between partial waves in an inﬁnite volume, In the absence of a nearby resonance or bound-state pole. – 11 –he t -matrix is therefore diagonal in partial waves. In this case, a single partial wave ampli-tude can be described by the scattering phase shift δ (cid:96) ( E cm ) related to the t -matrix through t ( (cid:96) ) = ρ e iδ (cid:96) sin δ (cid:96) . Ultimately the results obtained should not be dependent on the intermediate parame-terisation used to described the t -matrix, and relying on only a single expression for t ( (cid:96) ) ( s )may introduce bias. We thus parameterise t ( (cid:96) ) ( s ) in a variety of ways. A ﬂexible parame-terisation respecting unitarity of the S -matrix is a K -matrix, which for elastic scatteringis given by ( t ( (cid:96) ) ) − ( s ) = 1(2 k ) (cid:96) K − ( s ) 1(2 k ) (cid:96) + I ( s ) , (4.1)for a partial wave (cid:96) , and K ( s ) that is real for real values of s . The factors (2 k ) − (cid:96) ensure theexpected threshold behaviour. Unitarity of the S -matrix is guaranteed if Im I ( s ) = − ρ ( s )above threshold. This places no constraint on Re I ( s ). One choice is to set Re I ( s ) to zeroabove threshold, giving I ( s ) = − iρ ( s ). Another option is to use the Chew-Mandelstamprescription [70], which uses the known Im I ( s ) to generate a non-zero Re I ( s ) through adispersion relation, whose explicit form is given in appendix B of Ref. [27]. To make thedispersion relation integral converge a subtraction at an arbitrary value of s is needed.A general expression for the amplitudes we use in sections 4.2 and 4.3 is K ( s ) = (cid:0) g (0) + g (1) s (cid:1) m − s + γ (0) + γ (1) s , (4.2)where g ( n ) , γ ( n ) and m are real free parameters that are obtained by the minimisationprocedure described in section 2.2. Various parameters may be ﬁxed to zero for diﬀerentapplications. When a K -matrix pole parameter is present, as in Eq. 4.2, when using aChew-Mandelstam phase space we subtract at the pole parameter, s = m .In section 4.4 we make use of variations of Eq. 4.2 and, additionally, of a ratio ofpolynomials, K − ( s ) = (cid:80) Nn =0 c n s n (cid:80) Mm =1 d m s m . (4.3)where c n and d m are real free parameters. Several of the low-order truncations of Eq. 4.3are algebraically identical to Eq. 4.2, however parameter correlations can diﬀer signiﬁcantly.One choice that often reduces correlations is the replacement s → ˆ s ≡ ( s − s thr . ) /s thr . . Whenusing Eq. 4.3 with a Chew-Mandelstam phase space, we choose to subtract at threshold, s = s thr . = ( m π + m D ) .In the case of single-channel elastic scattering, a common choice of amplitude is aneﬀective range expansion, given by k (cid:96) +1 cot δ (cid:96) = 1 a (cid:96) + 12 r (cid:96) k + P k + O ( k ) , (4.4) This is also true of the J P = 1 + D ∗ π amplitude we consider, assuming only S +1 (cid:96) J = S is relevantclose to threshold, neglecting S +1 (cid:96) J = D . Although referred to as matrices in general, in this calculation these are scalar equations for each partialwave amplitude. – 12 –here a (cid:96) and r (cid:96) are the scattering length and eﬀective range respectively, for partial wave (cid:96) . Another common parameterisation, that is appropriate for a single isolated resonance,is the relativistic Breit-Wigner parameterisation t ( (cid:96) ) ( s ) = 1 ρ ( s ) √ s Γ (cid:96) ( s ) m R − s − i √ s Γ (cid:96) ( s ) , (4.5)where the width is given by Γ (cid:96) ( s ) = g R π k (cid:96) +1 s m (cid:96) − R , which ensures the correct near-thresholdbehaviour, and m R , g R are free parameters.We also consider unitarised chiral amplitudes from Refs. [71–73]. Chiral perturbationtheory is an eﬀective ﬁeld theory (EFT) approach, derived from expanding a Lagrangian ofmeson degrees of freedom about the chiral ( m u , m d , m s →

0) and small-momentum limits.The number of terms grows order-by-order, and these come with unknown Wilson coeﬃ-cients not speciﬁed by the EFT that are estimated either from experimental data or froma ﬁrst-principles approach such as lattice QCD. In many cases of interest, when extrapo-lating these amplitudes away from threshold, they grow larger than permitted by unitarityof the S -matrix, necessitating unitarisation. This also enables resonance poles to be gen-erated, which can otherwise be diﬃcult to achieve through an expansion in momentum.Next-to-leading order is required for meson loops to appear, however, we choose to ﬁx thenext-to-leading order Wilson coeﬃcients to zero to reduce the number of free parameters.The amplitude used is similar to the K -matrices described above, and is shown to be al-gebraically identical to Eq. 4.3 with M = N = 2 and speciﬁc coeﬃcients c n and d m thatdepend on m D and m π , in Appendix B. In addition to the analyticity and unitarity (in s )shared with all the K -matrix amplitudes used, this amplitude also has the assumption ofchiral symmetry, that restricts the behaviour at threshold.We use a simple form that can be written as K − ( s ) = (cid:18) − π V J =0 (cid:19) − + α ( µ ) π + 2 π (cid:18) m D m π + m D log m D m π + log m π µ (cid:19) . (4.6)with a threshold-subtracted Chew-Mandelstam phase-space I ( s ), and V J =0 = − sF (cid:16) s − s ( m D + m π ) − ( m D − m π ) (cid:17) (4.7)corresponds to the S -wave projected leading-order elastic Dπ scattering amplitude. F and α ( µ ) are treated as free parameters, and the renormalisation scale µ is ﬁxed to a t µ = 0 . µ ≈ Dπ with J P = 1 − and D ∗ π with J P = 1 + We begin with a ﬁt of the [000] T − , [100] E , [110] B and [110] B spectrum. These irrepscontain Dπ in P -wave but do not have a Dπ S -wave contribution. There is a level farbelow threshold in T − that signals a J P = 1 − D ∗ bound state. The moving frame irrepscontain a contribution from D ∗ π in S -wave ( J P = 1 + ). To parameterise these two partial– 13 –aves, we use Eq. 4.2, once with a pole term in J P = 1 − Dπ , and again with a constantin J P = 1 + D ∗ π , and obtain γ (0) D ∗ π = (1 . ± . ± . g Dπ = (0 . ± . ± . m = (0 . ± . ± . · a − t . − . − . .

00 0 . . χ /N dof = . − = 1 . . (4.8)The ﬁrst uncertainty is obtained from the χ minimum, as is the matrix on the right thatshows the parameter correlation. The second uncertainty indicated is obtained by minimis-ing again several times, after varying in turn the π , D and D ∗ masses and anisotropy tothe maximum and minimum values within their 1 σ uncertainties, and taking the maximumdeviation. The phase shifts determined from this amplitude are shown in Fig. 4. The inner bandscorrespond to the parameters in Eq. 4.8 using the ﬁrst uncertainties and the correlations.The outer bands show the largest deviation determined by varying the mass and anisotropyvalues. The elastic

Dπ P -wave is small and in the next section we will also ﬁnd a similarsmall P -wave phase shift in the combined ﬁt of Dπ J P = 0 + and J P = 1 − . The D ∗ π contribution rises at threshold, perhaps indicating the tail of a higher D resonance. Someevidence for this can be read oﬀ from Fig. 2 where “extra” levels appear around a t E cm ≈ .

39. Additionally we include a simultaneous ﬁt of J P = 0 + , − Dπ and J P = 1 + D ∗ π inappendix C, where the resulting J = 1 amplitudes are very similar to those in Eq. 4.8 andFig. 4. Dπ with J P = 0 + and J P = 1 − We now determine the S and P -wave amplitudes simultaneously using 20 energy levelsbelow E = m D + 2 m π from the [000] A +1 , [000] T − , and the four moving-frame A irreps.We begin by ﬁtting a “reference” amplitude, which consists of an S -wave K -matrix witha pole term and a constant γ , and a P -wave with just a pole term. Both use a Chew-Mandelstam phase-space subtracted at s = m in each partial wave. After ﬁtting theseﬁve free parameters, we ﬁnd m = (0 . ± . ± . · a − t g = (0 . ± . ± . · a − t γ (0) = ( − . ± . ± . m = (0 . ± . ± . · a − t g = (0 . ± . ± . .

00 0 . − .

62 0 . − . . − .

85 0 .

17 0 . . − . − . . − . . χ /N dof = . − = 0 .

90 (4.9) For a given parameter x with central value ¯ x , x i values are obtained for each mass and each anisotropyvariation i , and uncertainties σ x i . Then the second uncertainty quoted is max i ( | ¯ x ± σ ¯ x | − | x i ± σ x i | ) wherethe two ± are changed simultaneously. In this case the variations considered are { m D → m D ± σ m D , m D ∗ → m D ∗ ± σ m D ∗ , m π → m π ± σ m π , ξ → ξ π + σ ξ π , ξ → ξ D − σ ξ D } , where the two anisotropy variations arethe largest possible deviations from the mean for ξ π . This procedure is repeated for the other minimahighlighted below. – 14 – .

35 0 .

37 0 . δ / ◦ a t E cm Dπ | thr D ∗ π | thr δ Dπl =1 δ D ∗ πl =0 Figure 4 . The phase shift for the

Dπ P -wave (blue) and D ∗ π S -wave (orange) amplitudes. Theinner band corresponds to the statistical uncertainties from the χ -minimum in Eq. 4.8. Theouter band shows the maximum possible deviation when varying the scattering particle masses andanisotropy within their uncertainties. The faded region begins at Dππ threshold indicating thehighest energy considered in this calculation. where the parameters with a subscript “1” describe the P -wave, and those without describethe S -wave. The meaning of the uncertainties are as described below Eq. 4.8.In Figs. 5 and 6, we show the spectrum obtained plotted with the solutions of theL¨uscher determinant condition, Eq. 2.5 with the parameters given in Eq. 4.9. The levelsbelow a t E ≈ .

39 are in good agreement with the solutions of the L¨uscher determinantcondition.In Fig. 7, we show the phase shift of both the

Dπ S -wave and P -wave. The S -waveamplitude turns on rapidly at threshold and rises monotonically towards the edge of theelastic region, which in the ﬁnite volume produces statistically signiﬁcant energy shifts andperhaps an additional level, suggestive of a resonance. We defer the discussion of the polesand thus the resonance content of the t -matrix until after we have considered varying theform of the parameterisation. We now consider a range of parameterisations to explore the sensitivity to any particularchoice. We perform minimisations to two diﬀerent selections of energies. Motivated by thelack of volume dependence of the deeply bound levels due to the vector D ∗ state, and small P -wave phases found in the reference amplitude Eq. 4.9, we exclude the deeply bound levelseen in [000] T − and moving frame irreps. This results in 14 energy levels that are usedto obtain the amplitudes given in table 3, with only a constant K -matrix in P -wave. Theresults given in table 4 use the same 20 levels utilised for the reference amplitude, Eq. 4.9.– 15 – . . . . . . . . a t E c m (a) [000] A +1

24 32 40

DπD ∗ πDππ L/a s (b) [000] T − Figure 5 . Finite-volume spectra obtained in the at-rest A +1 and T − irreps, as in Fig. 1, plottedwith the solutions of the L¨uscher determinant condition using the reference parameterisation withthe parameters resulting from the χ -minimisation (orange points). In the region above threshold, all of the P -wave amplitudes produce phase shifts that areapproximately zero.In the following we present the minima found for a few key parameterisations thatare discussed further in section 6. Using the S -wave Breit-Wigner (parameterisation (q) oftable 4) gives the following parameter values m R = (0 . ± . ± . · a − t g R = (5 . ± . ± . m = (0 . ± . ± . · a − t g = (0 . ± . ± .

00 0 .

92 0 . − . .

00 0 . − . . − . . χ /N dof = 14 . / (20 −

4) = 0 .

91 (4.10)where the subscript “1” indicates the P -wave parameters, and the others are S -wave.Fitting the eﬀective range parameterisation (parameterisation (m) of table 4) gives a = (21 . ± . ± . · a t r = ( − . ± . ± . · a t m = (0 . ± . ± . · a − t g = (0 . ± . ± . .

00 0 .

90 0 . − . .

00 0 . − . . − . . χ /N dof = 14 . / (20 −

4) = 0 . . (4.11)– 16 – . . . . . . . . a t E c m (a) [100] A

24 32 40 (b) [110] A

24 32 40 (c) [111] A

24 32 40

DπD ∗ πDππ L/a s (d) [200] A Figure 6 . As Fig. 5, but for the moving frame A irreps. − .

35 0 .

37 0 . δ / ◦ a t E cm Dπ | thr D ∗ π | thr δ Dπl =0 δ Dπl =1 Figure 7 . Phase shift of the S -wave (red) and P -wave (blue) Dπ amplitudes. The inner linecorresponds to the reference parameterisation. The inner dark error band represents the statisticalerror from the χ -minimisation while the outer light error band additionally includes uncertaintiesfrom varying the input hadron masses and anisotropy within 1 σ . We plot this amplitude as a t k cot δ as a function of a t k compared to the reference ampli-tude Eq. 4.9 in Fig. 8. This shows the subthreshold constraint from [000] A +1 , [100] A and– 17 – Figure 8 . The S -wave eﬀective range parameterisation from Eq. 4.11 (red) and reference amplitudeEq. 4.9 (grey) plotted as a t k cot δ as a function of the momentum-squared a t k . The discrete pointsshow where the energies constrain the amplitudes by using Eq. 2.5 to obtain t level-by-level. Thebold square points are obtained from [000] A +1 , and the other points are from moving frame A irreps with P -wave and higher partial waves ﬁxed to zero. Both amplitudes are determined fromthe same spectra, as described in the text. [110] A . The amplitudes both describe the spectra well. However they do diﬀer withinuncertainties, as can be seen in the ﬁgure.For the unitarised chiral amplitude (parameterisation (s) of table 4) we obtain F = (0 . ± . ± . · a − t α ( µ ) = ( − . ± . ± . m = (0 . ± . ± . · a − t g = (0 . ± . ± . . − . − .

18 0 . .

00 0 . − . . − . . χ /N dof = 13 . / (20 −

4) = 0 . . The amplitudes in tables 3 and 4 are very similar at real energies. This is presented insection 6.1 as the outer red bands in the left panel of Fig. 10. However, they do diﬀer whencontinued to complex energies, which we investigate in the next section.– 18 – = 0 parameterisation (cid:96) = 1 parameterisation N pars χ /N dof K-matrix with a Chew-Mandelstam I ( s ) in both partial waves(ax) K = g m − s K = γ K = g m − s + γ (0) K = γ K = g m − s + γ (1) ˆ s K = γ K = ( g + g (1) s ) m − s K = γ K − = c (0) + c (1) ˆ s K = γ K − = c (0) + c (1) ˆ sc (2) ˆ s K = γ I ( s ) = − iρ ( s ) in both partial waves(gx) K = g m − s K = γ K = g m − s + γ (0) K = γ K = g m − s + γ (1) ˆ s K = γ K = ( g + g (1) s ) m − s K = γ . (kx) K − = c (0) + c (1) ˆ s K = γ K − = c (0) + c (1) ˆ sc (2) ˆ s K = γ k cot δ = 1 /a + r k K = γ k cot δ = 1 /a + r k + P , k K = γ . Breit-Wigner(ox) t = ρ m R Γ m R − s − im R Γ K = γ χ PT (px) t − = (cid:0) − π V J =0 (cid:1) − + 16 πG DR K = γ Table 3 . The parameterisations used that excluded the deeply-bound level around a t E cm = 0 . N pars indicates the number of free parameters in each parameterisation. An italicised χ /N dof valueindicates this ﬁt was not included in the amplitude ﬁgure and pole values due to an additional polefound on the physical sheet. – 19 – = 0 parameterisation (cid:96) = 1 parameterisation N pars χ /N dof K-matrix with Chew-Mandelstam I ( s ) in both partial wavesref. K = g m − s + γ (0) K = g m − s K = g m − s K = g m − s K = g m − s + γ (1) ˆ s K = g m − s K = ( g + g (1) s ) m − s K = g m − s K − = c (0) + c (1) ˆ s K = g m − s K − = c (0) + c (1) ˆ sc (2) ˆ s K = g m − s K = g m − s + γ (0) + γ (1) ˆ s K = g m − s . ∗ K-matrix with I ( s ) = − iρ ( s ) in both partial waves(g) K = g m − s + γ (0) K = g m − s K = g m − s K = g m − s K = ( g + g (1) s ) m − s K = g m − s K − = c (0) + c (1) ˆ s K = g m − s K − = c (0) + c (1) ˆ sc (2) ˆ s K = g m − s I ( s ) in S -wave, Eﬀective range in P -wave(l) K = g m − s + γ (0) k cot δ = 1 /a + r k I ( s ) in P -wave(m) k cot δ = 1 /a + r k K = g m − s k cot δ = 1 /a + r k + P , k K = g m − s . † Eﬀective range in both partial waves(o) k cot δ = 1 /a + r k k cot δ = 1 /a + r k k cot δ = 1 /a + r k + P , k k cot δ = 1 /a + r k . † Breit-Wigner in S -wave, K-matrix with Chew-Mandelstam I ( s ) in P -wave(q) t = ρ m R Γ m R − s − im R Γ K = g m − s χ PT (s) t − = (cid:0) − π V J =0 (cid:1) − + 16 πG DR K = g m − s † - these amplitudes were found to have physical sheet poles ∗ - this amplitude was found to have an additional resonance pole, as described in the text Table 4 . The parameterisations used that included the P -wave deeply bound level. N pars indicatesthe number of free parameters in each parameterisation. An italicised χ /N dof value indicates thisﬁt was not included in the amplitude ﬁgure and pole values, due to the presence of either physicalsheet poles, or additional resonance poles close to the left-hand cut. – 20 – Scattering amplitude poles

In this section we analyse the scattering amplitudes presented above for poles, by analyti-cally continuing to complex s = E cm . The amplitudes have been constrained only at realenergies, and when continuing to complex values it is possible that even apparently similaramplitudes diﬀer away from the real axis. However, if a nearby pole is present it is oftena universal feature across parameterisations that have similar shapes on the real axis. Byextracting the poles of the amplitudes we obtain the essence to compare among diﬀerentparameterisations, calculations, and experiment.Scattering amplitude poles unify bound-states and resonances in a single quantity thatprovides information about the spectral content of the channels under consideration. Inthe region of a pole, the t -matrix is dominated by a term t ∼ c / ( s − s ) where c is theresidue and s is the pole position. The factorised pole residue c gives a measure of thecoupling to the decay channel.The amplitudes we have used are analytic in s , except for cuts due to the cm momentum k ( s ) square-root function and poles. The s -channel cut leads to a multi-sheeted complex s plane, where each contributing channel doubles the number of sheets. Sheets can belabelled by the sign of the imaginary part of the momentum k i ( s ) for channel i . In thisanalysis we only consider a single channel and therefore the amplitudes as functions of s live on two sheets. The sheet with sgn(Im k ) = − k ) = +1 is called the physical sheet. The amplitudesutilised do not incorporate any eﬀects due to exchange processes that introduce additional(“left-hand”) cuts beginning at a t √ s = 0 . s .Causality restricts complex poles to occur only on the unphysical sheet, and withthe amplitudes we consider they will appear as complex-conjugate pairs. Bound statescorrespond to poles on the real axis below threshold on the physical sheet. Resonances arefound at complex energies, with √ s = m − i Γ / m is the mass and Γ is the width.We begin by investigating the S -wave amplitudes for poles. S -wave pole The reference amplitude, Eq. 4.9, has an S -wave pole on the unphysical sheet at a t √ s =(0 . ± . − i (0 . ± . S -wave poles form two clusters, one with − a t Im √ s ≈ .

05 (orange markers in Fig. 9), and one slightly deeper in the complexplane with − a t Im √ s ≈ .

08 (blue). While both clusters correspond to amplitudes withperfectly acceptable χ /N dof values, the nearer cluster corresponds to three-parameter S -wave amplitudes, and the deeper cluster arises from amplitudes with two free parameters.In table 5, we compare 2, 3, and 4 parameter S -wave K -matrix ﬁts. The 3-parameter ﬁtcorresponds to the reference parameterisation, Eq. 4.9. The two-parameter ﬁt results in adeeper pole (amplitude (a)). The four-parameter ﬁt with a linear term γ (1) ˆ s (amplitude(f)) results in an amplitude with two poles, one around a t m ≈ .

29, far below threshold but– 21 –mp a t m a t g γ (0) γ (1) χ N dof Re( a t √ s ) -2Im( a t √ s ) a t | c | (a) 0.3916(42) 0.313(22) - - 0.90 0.3590(80) 0.0797(83) 0.381(33)ref. 0.4011(98) 0.419(83) -2.0(13) - 0.90 0.3592(35) 0.0512(95) 0.257(33)(f) 0.4222(92) 0.789(57) -8.6(16) -14.7(87) Table 5 . The result of varying the number of free parameters in the S -wave amplitude with a two-parameter P -wave as used in Eq. 4.9. “ref.” indicates the reference amplitude, Eq. 4.9. The ﬁnalamplitude (f) results in a parameterisation that produces two poles, one of them around a t m ≈ . close to the left-hand cut, and one similar to those found for the two and three-parameterﬁts. The χ /N dof increases suggesting that there is too much freedom. We choose toexclude parameterisations such as this that produce poles in the energy region of the leftcut. However these all have a pole consistent with the dotted region marked in Fig. 9, andso this choice does not aﬀect the ﬁnal result. We also exclude any parameterisation thatproduces physical sheet poles in S -wave.In table 5, the magnitude of the pole residue correlates with the magnitude of theimaginary part. This suggests how these amplitudes achieve very similar behaviours atreal energies despite having slightly diﬀerent pole positions. Although the data constrainthe real part of the pole position relatively well, some freedom remains in the imaginarypart. Nevertheless, the pole is present in all parameterisations and so appears to be anecessary feature to describe the lattice QCD spectra.Among the parameterisation variations we have implemented is a unitarised chiralamplitude, as mentioned above and described in detail in appendix B. This amplitude alsoproduces a pole that lies within the cluster of three-parameter S -wave amplitudes and ismarked by a black triangle in Fig. 9. The Breit-Wigner is also indicated (black square),which results in a pole with a larger imaginary part than many of the other amplitudes.The data demands a pole is present, however the scatter of pole positions shown inFig. 9 demonstrates that using any single parameterisation does not necessarily give areliable estimate of the uncertainties. Our ﬁnal value for the pole position and couplingtaking into account the statistical uncertainty from all parameterisations is a t √ s = (0 . ± . − i (0 . ± . a t c = (0 . ± .

13) exp iπ ( − . ± . , (5.2)this corresponds to the dotted area in Fig. 9. In physical units this corresponds to √ s = (cid:0) (2196 ± − i (425 ± (cid:1) MeV (5.3) c = ((1916 ± iπ ( − . ± . . (5.4) P -wave pole In J P = 1 − a deeply-bound D ∗ pole is found at a similar energy to the energy level farbelow threshold in all irreps where J P = 1 − subduces. Although from our amplitudes in– 22 – . − . − .

020 0 .

35 0 . a t I m √ s p o l e a t Re √ s poleK-matrix (reference)Breit-WignerEﬀective rangeuChiPT − . − . − . − .

10 0 . . . . a t I m c a t Re c Figure 9 . Poles on complex energy plane (left) and couplings (right). The black ﬁlled circle corre-sponds to the reference amplitude. Other amplitudes discussed in the text are shown with diﬀerentmarkers (see key). The coloured datapoints in both panels show the spread of poles (couplings)produced by the complete set of parameterisation variations (see tables 3 and 4). Orange crosshairscorrespond to three-parameter S -wave amplitudes, blue crosshairs to two-parameter ones. Thedotted rectangle encompasses the entire spread of the parameterisations including their statisticaluncertainties but excluding variations of mass or anisotropy for amplitudes other than the referenceparameterisation. table 4 a pole coupling can be extracted, the uncertainties are very large and the couplingdoes not appear to be particularly meaningful. The mass found is consistent with theresult considering only q ¯ q operators in table 2, suggesting little inﬂuence on this deeply-bound state from the Dπ operators. The experimental D ∗ is observed to be very narrow,found close to Dπ threshold. With the P -wave phase space opening relatively slowly, itwould likely require close-to-physical pion masses to observe signiﬁcant shifts away fromthe non-interacting Dπ energies, and thus determine a coupling to the Dπ decay channel.Across all parameterisations that include a P -wave pole term (see table 4) we obtain a t √ s = 0 . ± . . (5.5)This is consistent with the deeply bound state seen in irreps where J P = 1 − appears, asshown in Figs. 1 and 2. In physical units, the pole is located at √ s = (2006 . ± .

4) MeV . (5.6) We now interpret our results, ﬁrst comparing to earlier work at a light quark mass cor-responding to a larger pion mass, before considering the composition of the scalar state.– 23 –

150 2200 2250 2300 2350 2400

Figure 10 . The left panel shows the reference scattering amplitude at m π = 239 MeV (red)and 391 MeV (blue) plotted as ρ | t | with the energies that were used to constrain them shownbelow. The bold square points are from [000] A +1 , the other points are from moving frame irreps.The inner bands show the statistical uncertainty from the χ minimisation. The outer band for m π = 239 MeV includes variation over mass, anisotropy and parameterisations. The outer bandfor m π = 391 MeV includes variation over only mass and anisotropy; Ref. [26] found only a smalleﬀect from varying the parameterisation for this elastic system with a near-threshold bound state.The upper right panel shows the pole positions including the additional uncertainty found from thevariation over parameterisation, which is signiﬁcant for m π = 239 MeV. The pole at the lower pionmass is a resonance found on the unphysical sheet, and at the higher pion mass is a bound statefound on the physical sheet. The lower right panel shows the magnitudes of pole couplings to the Dπ channel. We also compare with studies of DK scattering at the same pion masses in the context of SU (3) ﬂavour symmetry. In Ref. [26], Dπ scattering was studied on three volumes with a light quark mass corre-sponding to m π = 391 MeV. A near-threshold bound state was found in S -wave with astrong coupling to the Dπ channel, that inﬂuenced a broad energy region. The referenceamplitude from that work , updated with an improved estimate of the D -meson mass fromRef. [29], results in Eq. 3.3 of Ref. [26]. – 24 – - - Figure 11 . The scattering amplitudes at m π = 239 and 391 MeV plotted as k cot δ as a function of k in scale-set units. Red shows the m π = 239 MeV amplitude and the m π = 391 MeV amplitudeis shown in blue. The points shown come from using the ﬁnite volume energies individually inthe L¨uscher determinant condition. The bold square points are obtained from irreps at rest, theother points are obtained from moving frame irreps. In both cases the P -wave does not have asigniﬁcant impact and was ﬁxed to zero where it appears in moving frames. The bound state massat m π = 391 MeV can be read oﬀ from the intersection of the blue curve with the dotted −| k | curveat negative k . m = (0 . ± . ± . · a − t . − . − .

39 0 . − .

06 0 . .

00 0 .

94 0 . − .

09 0 . .

00 0 . − .

14 0 . . − . − . .

00 0 . . g = (0 . ± . ± . · a − t γ = 13 . ± . ± . m = (0 . ± . ± . · a − t g = 1 . ± . ± . γ = ( − ± ± · a t χ /N dof = . − = 1 .

53 , (6.1)and the corresponding amplitude is plotted as the blue curve in Figs. 10 and 11. Thisamplitude contains an S -wave bound state pole at a t √ s = 0 . a t c Dπ = 0 . This is broadly in agreement with Ref. [26], with a slightly largerpole coupling. In scale-set units, this bound state remains approximately 2 ± S -wave and the same parameter-isation in P -wave as in Eq. 6.1 results in a poor χ /N dof . However, reducing the energyregion slightly to remove the 4 highest energy points, we ﬁnd a t m BW = 0 . g BW = 7 . χ /N dof = . − = 1 .

5. This results in a bound-state pole at Where the meaning of the ﬁrst and second uncertainties is as deﬁned below Eq. 4.8. – 25 –

200 2300 24000.20.40.60.81.0

Figure 12 . Breit-Wigner parameterisations compared to the reference K -matrix ﬁts at both lightquark masses - the numerical uncertainties shown include the statistical, and the mass, anisotropyvariations summed together. The complex t -matrix pole position is indicated with mass m = Re √ s ,and width Γ = − √ s . The pole coupling | c | to Dπ is also indicated. The ﬁt range used for theBreit-Wigner ﬁt is shortened relative to the K -matrix at m π = 391 MeV as described in the textand indicated in the plot by the faded region at the highest energies. a t √ s = 0 . a t c = 0 . Dπ channel, in bothcases the Breit-Wigner mass parameter bears little connection to the mass found from thecomplex pole. At m π = 239 MeV, the Breit-Wigner produces a broad pole that lies in thecluster of other two-parameter S -wave ﬁts, as indicated in Fig. 9.In Fig. 11 we plot the K -matrix amplitudes in Eq. 4.9 and Eq. 6.1 as k cot δ as a functionof k . This is the quantity in which the eﬀective range expansion is usually expressed, k cot δ = a + rk + ... . In both cases the scattering length has a large magnitude. Atthe larger pion mass a <

0, corresponding to a bound-state, seen as the intersection ofthe k cot δ curve and −| k | . The amplitude obtained at the smaller pion mass has a > I = 0, J = 0 ππ scattering [20], where a σ -like pole was found at the same masses, with a near-threshold bound state at the highermass that evolves into a resonance pole at the lower mass. This similarity is striking whenplotted as k cot δ : Fig. 11 is remarkably similar to the lower panel of Fig. 4 in Ref. [20].Conversely, S -wave πK in I = 1 / S -wave DK in I = 0 is found to have bound states at both – 26 –

39 3912100220023002400

Figure 13 . A summary of the real parts of the pole positions found in this analysis and Ref. [26]in S -wave Dπ scattering, and Ref. [29] in S -wave DK scattering. The D ∗ resonance pole found inthis study lies on the unphysical sheet with a large width. The other 3 poles are bound states. pion masses [29].One of the initial experimental surprises between the lightest D ∗ resonance with c ¯ l quark content, and the lightest D ∗ s resonance with c ¯ s quark content, was that the D ∗ s was found at a similar mass to the D ∗ . The experimental D ∗ was also found to be verybroad while the D ∗ s was found to be narrow. At m π = 391 MeV both states appear asbound states below their respective decay channels with the D ∗ lower in mass than the D ∗ s [26, 29]. At m π = 239 MeV, the D ∗ pole migrates deep into the complex plane while itsreal part stays close to Dπ threshold. Nevertheless, the “natural” mass ordering with the c ¯ l being lighter than the c ¯ s is retained. It should be noted that while in the m π = 391 MeVcalculation, the D ∗ s is more bound than in experiment, for m π = 239 MeV, the D ∗ s poleis found closer to threshold than it is in experiment [29]. We summarise the real parts ofthe pole positions as a function of m π in Fig. 13.The D ∗ computed in this study is below the reported mass for the experimentallyobserved D ∗ (2300), even though the light quark mass is larger than physical. This wasalso true in Ref. [26], which at a light quark mass corresponding to a higher pion massfound a bound-state just below Dπ threshold. Based on these two points, the real partof the complex resonance pole appears to move slowly with light quark mass, being found(77 ±

64) MeV above threshold at m π = 239 MeV and (2 ±

1) MeV below threshold at m π = 391 MeV. If this trend continues towards the physical light quark mass, then thecurrent estimate of the D ∗ mass from the experimental data appears a little too high.However given the large width it is possible that the experimental amplitudes are alsocompatible with a lower pole mass. Early suggestions this may be the case appeared inRef. [76]. During the preparation of this article, Ref. [77] appeared with a similar conclusioncomparing unitarised chiral amplitudes and LHCb data [78].In this study, we have not presented any amplitudes describing the coupled-channel– 27 –egion although a few energy levels were computed that extend above the elastic region. In[000] A +1 two levels were found coincident with the Dη and D s ¯ K non-interacting levels andwe concluded there were not suﬃciently many energy levels to make a reliable statementbased on this single volume alone. Additional eﬀects may be present from three-bodychannels such as Dππ and D ∗ ππ which is very close to D s ¯ K with m π = 239 MeV. InRef. [26], some of the levels around Dη and D s ¯ K showed only small shifts, qualitativelysimilar to what is seen here. However signiﬁcant eﬀects were observed in the scatteringamplitudes. The lightest D ∗ scalar resonance determined here is signiﬁcantly below where quark poten-tial models predict the lightest D ∗ state to arise [6]. Large contributions from meson-meson(molecular), and compact tetraquark components, in addition to quark-model q ¯ q compo-nents, have been proposed as possible explanations. However, these components are notorthogonal and any approach to distinguish them comes with a degree of model-dependence.One measure that is often claimed to distinguish between a fundamental or compositestate for near-threshold S -wave bound-states uses the ﬁeld renormalisation constant Z [79],where Z → Z → S -wave states like that found at m π = 391 MeV,with a near-threshold bound state that strongly inﬂuences the physical scattering region,this measure suggests the state is dominated by a molecular component. Fitting an eﬀectiverange parameterisation using the same 29 energy levels used for the Breit-Wigner resultsin a = − a t and r = − . a t and χ N dof = . − . Using Z = 1 − (cid:112) a/ ( a + 2 r ), thiscorresponds to Z ≈ . | X | = | − Z | (6.2)= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c dI ( s ) ds (cid:12)(cid:12)(cid:12) s → s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (6.3)from Refs. [83, 84], adapted to the deﬁnitions used here, where c is the pole residue, s is the pole position and I ( s ) is the Chew-Mandelstam function. For the reference param-eterisation Eq. 4.9, this results in | X | ≈ .

57, suggesting approximately equal importanceof both a fundamental q ¯ q contribution and a composite meson-meson contribution.An additional place where qualitative information arises in this calculation is in theoperator overlaps shown in Fig. 3. The operators used resemble q ¯ q and two-meson con-structions. The q ¯ q -like constructions alone produce a single level within the width of theresonance, similar to as was found in the case of the ρ resonance in Refs. [13, 14]. Using only Dπ operators does not produce a single level that is consistent with the spectrum whenusing all of the operators computed. Adding the q ¯ q operators to the Dπ -like constructionsresults in a consistent spectrum to that found when using all of the operators in the energyregion of the D ∗ . This is suggestive that both q ¯ q and Dπ components are necessary sinceboth the q ¯ q -like and Dπ -like operators appear essential to determine a reliable spectrum.– 28 – .3 SU(3) ﬂavour symmetry A useful perspective can be gained by considering the limit where the light and strangequark masses are set equal. An SU(3) ﬂavour symmetry then arises between the π , K andpart of the η that form the members of an SU(3) octet. Dπ and DK scattering are thenclosely related, as described in Section 6.4 of Ref. [29]. One key expectation is that thenumber of poles in each channel should not change as a function of the amount of SU(3)breaking. The pole couplings must be similarly related in the SU(3) limit, and it is possiblethat these only vary slowly with quark mass.When comparing the I = 1 / Dπ and I = 0 DK amplitudes at m π = 391 MeV, SU(3)symmetry is broken but only by a small amount ( m π /m K ≈ . I = 1 / D ∗ and I = 0 D ∗ s states are both near-threshold bound states, with the D ∗ s being more deeply bound than the D ∗ . In this calculation, working at m π = 239 MeV,SU(3) is more broken ( m π /m K ≈ .

47) and the lightest states diﬀer signiﬁcantly. The ( c ¯ s ) D ∗ s remains bound, although less than with the heavier light quark, since the scattering D and K mesons both contain light quarks. As we have seen, the ( c ¯ l ) D ∗ becomes a resonancewith a pole deep in the complex plane.The poles are found to be strongly-coupled to the relevant meson-meson channels.The D ∗ s at both masses and the resonant D ∗ studied here have couplings consistent with | c | ≈ D ∗ found at the larger pion mass has a slightly smallercoupling. However it is found incredibly close to Dπ threshold and this may have an eﬀectas the pole transitions from a bound state to resonance.The results obtained here and in Refs. [26, 29] are qualitatively consistent with theexpectation that these poles and couplings only change slowly as a function of the amountof SU (3) breaking. In this analysis we have presented the ﬁrst computation of a D ∗ resonance pole fromlattice QCD. Working at m π = 239 MeV on a single lattice with a spatial volume ofapproximately (3 . , 20 energy levels were obtained that constrain the inﬁnite volumescattering amplitudes. Signiﬁcant deviations are observed from the spectrum expected inthe absence of interactions, that correspond to a rapid increase in the S -wave Dπ scatteringamplitude near threshold. Several amplitude parameterisations are considered, and theseall produce a single nearby resonance pole with a mass m = (2196 ±

64) MeV and awidth Γ = (425 ± ±

64) MeV above Dπ threshold, but deep in thecomplex plane. Much of the uncertainty on the width arises from considering a range ofparameterisations.This calculation has been performed at a heavier-than-physical pion mass, and alongwith an earlier calculation that found a bound state at a heavier mass [26], likely indicatesa slow decrease in the D ∗ pole mass with decreasing pion mass. At both pion masses, the D ∗ pole is found to couple strongly to the Dπ channel. At m π = 391 MeV, the pole isbound but its eﬀect is felt over a broad energy region. At m π = 239 MeV, a similar largecoupling is found but the pole migrates deep into the complex plane, with a signiﬁcant– 29 –nﬂuence over the whole S -wave elastic scattering energy region. The D ∗ pole computedhere has a smaller mass than the currently reported experimental values for the lightest D ∗ resonance [7]. Given the large width, the experimentally-determined amplitudes mayalso be compatible with a smaller mass [77, 78].The D ∗ s was also computed at both light quark masses and was found to be signiﬁcantlyheavier than the D ∗ in both cases [29]. The pole couplings to the relevant thresholds areall found to be large, which may account for some of the puzzling diﬀerences highlighted byearly experimental studies of these systems. In particular, the diﬀerence in widths betweenthe D ∗ s and D ∗ can be understood from the vastly diﬀerent phase space available in eachcase, since the couplings computed are similar. The poles themselves are found to have a“natural” ordering with the c ¯ l lighter than the c ¯ s .This calculation completes a quartet of analyses of the D ∗ and D ∗ s systems at twolight quark masses. The puzzle of a broad D ∗ more massive than the narrow D ∗ s foundin experiment is not present for light quark masses corresponding to m π = 239 MeV and391 MeV where the D ∗ pole is found consistently lower in mass than the D ∗ s . Further-more, the couplings of the D ∗ and D ∗ s poles to Dπ and DK respectively are compatible,suggestive of a common origin. Using a ﬁrst-principles approach to QCD, with externalinputs only to ﬁx quark masses, these analyses thus point to a possible resolution of thepuzzling experimental masses and widths. Acknowledgments

Chroma [85],

QUDA [86, 87],

QPhiX [88], and

QOPQDP

AppendicesA Operator Lists

In tables 6 and 7 we summarise the operators used in this study. A +1 [000] A [100] A [110] A [111] A [200] D [000] π [000] D [000] π [100] D [000] π [110] D [000] π [111] D [100] π [100] D [100] π [100] D [100] π [000] D [100] π [100] D [100] π [110] D [110] π [110] D [110] π [110] D [100] π [110] D [110] π [000] D [110] π [100] D [200] π [000] D [111] π [111] D [100] π [200] D [110] π [110] D [111] π [000] D [210] π [100] D [000] η [000] D [110] π [100] D [111] π [100] D [211] π [100] D [200] η [000] D [100] η [100] D [110] π [111] D [210] π [100] D ∗ [110] π [100] D s [000] ¯ K [000] D [111] π [110] D ∗ [100] π [100] D [111] η [000] D [200] π [100] D ∗ [111] π [100] D s [111] ¯ K [000] D [210] π [110] D [110] η [000] D [000] η [100] D s [110] ¯ K [000] D [100] η [000] D s [000] ¯ K [100] D s [100] ¯ K [000] × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψ Table 6 . Operators used in the variational analyses for irreps featuring S -wave Dπ as the lowestsubduced partial wave. Subscripts indicate momentum types. Γ represents some monomial of γ matrices and derivatives. – 31 – − [000] E [100] B [110] B [110] D [100] π [100] D [100] π [110] D [100] π [100] D [100] π [111] D [110] π [110] D [110] π [100] D [110] π [110] D [110] π [110] D ∗ [100] π [100] D ∗ [000] π [100] D [210] π [100] D [111] π [100] D ∗ [100] π [000] D ∗ [100] π [100] D ∗ [000] π [110] D ∗ [110] π [000] D ∗ [100] π [100] { } D ∗ [110] π [000] D ∗ [111] π [100] × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψ × ¯ ψ Γ ψ Table 7 . As table 6, but for operators used in irreps without an S -wave Dπ subduction. Thenumber in curly parentheses indicates the number of operators of this momentum combination.This arises due to the D ∗ appearing in both [100] A and [100] E that when combined with a pionin [100] A results in two linearly independent operators in [110] B . B Relation between unitarised chiral amplitudes and the K -matrix There have been several applications of unitarised chiral amplitudes to heavy-light systemssuch as Dπ and DK [71–73, 91, 92]. We use the amplitude deﬁnitions from a recentexample [73], that considered the coupled-channel Dπ, Dη, D s ¯ K spectra from ref. [26]. Inthis implementation, we only consider the elastic Dπ channel and ﬁt only to the spectrapresented above with no other inputs.Taking the “loop function” G DR as deﬁned in Eq. 14 of ref. [73] and the Chew-Mandelstam function I ( s ) as deﬁned in Appendix B of ref. [27], it is straightforward toshow that they diﬀer only by normalisation and terms independent of s . If the subtractionpoint of the Chew-Mandelstam function is chosen as threshold and I ( s thr . ) = 0 then therelation is,16 π G DR ( s, m , m ) = I ( s ) + α ( µ ) π + 2 π (cid:18) m m + m log m m + log m µ (cid:19) (B.1)where m = m π and m = m D are the scattering particle masses, α ( µ ) and µ appear in G DR , where µ is an energy scale taken to be 1 GeV and α ( µ ) ≈ − . V J =0 is the S -wave projected elastic Dπ scattering amplitude, then in the deﬁnitionsused throughout this paper this can be written as, K − ( s ) = (cid:18) − π V J =0 (cid:19) − + α ( µ ) π + 2 π (cid:18) m m + m log m m + log m µ (cid:19) . (B.2)Using only the leading order expression for V J =0 , V J =0 = C LO sF (cid:0) s − s ( m + m ) − ( m − m ) (cid:1) (B.3)where F ≈ f π , and C LO = − I = 1 / Dπ scattering [73]. Equation B.2 can thenbe written as a ratio of polynomials up to O ( s ) as in Eq. 4.3. For simplicity, we donot consider higher order terms and allow F and α ( µ ) to ﬂoat such that the spectra are– 32 –ell described. We have veriﬁed that the next-to-leading order term does not make alarge change to the amplitude using typical values for the next-to-leading order Wilsoncoeﬃcients given in the literature, when applied at m π ≈

239 MeV.

C Simultaneously ﬁtting Dπ with J P = 0 + , − and D ∗ π with J P = 1 + As an additional check, we also perform a simultaneous ﬁt to all the black points shown inFigs. 1 and 2, except [000] E + . We use Dπ amplitudes with J P = 0 + , − , and the J P = 1 + D ∗ π in a relative S -wave, neglecting any D ∗ π D -wave dynamical mixing. The S and P -waves are parameterised as Eq. 4.9, and the D ∗ π + wave is parameterised as in Eq. 4.8.After minimising the parameters to best describe the spectra, we obtain m = (0 . ± . ± . · a − t g = (0 . ± . ± . · a − t γ (0) = ( − . ± . ± . m = (0 . ± . ± . · a − t g = (0 . ± . ± . · a − t γ (0) [ D ∗ π ] = (1 . ± . ± . .

00 0 . − .

92 0 .

37 0 . − . . − .

97 0 .

35 0 . − . . − . − .

66 0 . .

00 0 . − . . − . . χ /N dof = 21 . / (29 −

6) = 0 . . (C.1)The corresponding phase shifts are shown in Fig. 14, where it can be seen that the Dπ phase shifts are very similar to those shown in Fig. 7. A positive phase shift is found inthe D ∗ π , similar to that shown in Fig. 4. − .

35 0 .

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