$I=1/2$ $S$-wave and $P$-wave $K\pi$ scattering and the $\kappa$ and $K^*$ resonances from lattice QCD

We present a lattice-QCD determination of the elastic isospin-$1/2$ $S$-wave and $P$-wave $K\pi$ scattering amplitudes as a function of the center-of-mass energy using L\"uscher's method. We perform global fits of $K$-matrix parametrizations to the finite-volume energy spectra for all irreducible representations with total momenta up to $\sqrt{3}\frac{2\pi}{L}$; this includes irreps that mix the $S$- and $P$-waves. Several different parametrizations for the energy dependence of the $K$-matrix are considered. We also determine the positions of the nearest poles in the scattering amplitudes, which correspond to the broad $\kappa$ resonance in the $S$-wave and the narrow $K^*(892)$ resonance in the $P$-wave. Our calculations are performed with $2+1$ dynamical clover fermions for two different pion masses of $317.2(2.2)$ and $175.9(1.8)$ MeV. Our preferred $S$-wave parametrization is based on a conformal map and includes an Adler zero; for the $P$-wave we use a standard pole parametrization including Blatt-Weisskopf barrier factors. The $S$-wave $\kappa$-resonance pole positions are found to be $\left[0.84(14)- 0.338(55)\,i\right]\:{\rm GeV}$ at the heavier pion mass and $\left[0.662(88)- 0.368(44)\,i\right]\:{\rm GeV}$ at the lighter pion mass. The $P$-wave $K^*$-resonance pole positions are found to be $\left[0.8951(64)- 0.00249(21)\,i\right]\:{\rm GeV}$ at the heavier pion mass and $\left[0.8719(82)- 0.0132(11)\,i\right]\:{\rm GeV}$ at the lighter pion mass, which corresponds to couplings of $g_{K^* K\pi}=5.02(26)$ and $g_{K^* K\pi}=5.03(22)$, respectively.


I. INTRODUCTION
As the simplest two-meson system with unequal mass and carrying strangeness, the Kπ system plays an important role in particle and nuclear physics. A review of the early history of Kπ scattering and the associated resonances can be found in Ref. [1]. The Kπ system also occurs in heavy-meson weak decay processes that are used to search for physics beyond the Standard Model [2][3][4][5][6]. This includes multibody nonleptonic decays such as B → Kππ, in which large CP -violating effects have been observed and two-body resonant sub-structures are seen [7], and semileptonic decays such as B → Kπ + − , which currently provides hints for new fundamental physics [8][9][10][11][12][13][14][15].
The I = 1/2 S-wave scattering amplitude is observed to be elastic up to approximately 1.3 GeV [16,17]. The results for the elastic scattering phase shift slowly but monotonically increase and reach 60 • at approximately 1.1 GeV. The rise in the phase shift is likely due to a very wide resonance, the κ [also referred to as K * 0 (800), or, more recently, K * 0 (700)]. However, since the phase shift does not cross 90 • , the existence of the κ remains a topic of discussion [21]; the latest update of the Particle Data Group database [22] still lists the κ as "requires confirmation". Because the κ is such a wide resonance with total decay width Γ total ≈ 300 MeV, its description is more involved. The search for a proper description and explanation for this elusive resonance is a hot topic in hadronic physics. The basic idea is to construct a parametrization of the scattering amplitude and fit it to the experimental data; by analytically continuing the amplitude into the com-plex plane, one searches for a pole attributed to the κ resonance. The experimental data [16,17] have been described by effective lagrangians incorporating chiral symmetry [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] and models of meson interactions [38,39]. The κ was also studied in the large N c limit of QCD [40,41] and with the inverse amplitude method [42,43]. The authors of Ref. [44,45] used Roy-Steiner equations to determine the pole of the κ resonance. Less rigorous, but similar studies using dispersion relations [46,47] are able to consistently find the κ pole. The relations between chiral perturbation theory and dispersion relations were explored in Ref. [48]. Fits of various models [49] and Padé approximants [50] to the experimental data led to similar results. Furthermore, it was observed that a κ resonance is necessary to explain an enhancement [51,52] in other production channels in the experiments E791 [53] and BES [54,55].
In the P -wave, the I = 1/2 Kπ scattering amplitude at energies below the Kη threshold is well described by a simple Breit-Wigner form with a single resonance, the K * (892) 1 , as observed in various processes ranging from kaon beam experiments to τ decays and D-meson decays [22]. The total decay width of the K * (892) is approximately 50.8(0.9) MeV [22] with branching ratios to Kπ being 99.9%, to Kγ at the order of 10 −3 and less than 10 −5 to Kππ.
In lattice QCD, scattering amplitudes can be determined from finite-volume energy spectra using Lüscher's method [56][57][58][59][60][61][62][63][64][65]. For S-wave Kπ scattering, the early lattice QCD calculations focused on scattering lengths describing low-energy scattering. The first such calculation, published in 2004, was performed for I = 3/2 only and in the quenched approximation [66]. This was followed by a calculation in 2006 that included N f = 2 + 1 staggered sea quarks but employed a domain-wall valence action [67]; the authors determined the I = 3/2 Swave scattering length directly from the lattice and used chiral symmetry to extract also the I = 1/2 scattering length at several pion masses. The Swave scattering lengths have also been determined from scalar form factors for semileptonic decays [68]. Reference [69] contains the first direct lattice QCD calculation of the S-wave scattering length for both I = 1/2 and I = 3/2, albeit in the quenched approximation. The Kπ system was also investigated using a staggered action for both the valence and sea quarks in Refs. [70][71][72][73]. Note that the presence of extra non-degenerate fermion tastes when using a staggered action introduces complications for the Lüscher method that are not yet fully understood. More recent dynamic lattice QCD calculations of Kπ S-wave the scattering lengths employed valence Wilson fermions with either N f = 2 [74], N f = 2 + 1 [75] or N f = 2 + 1 + 1 dynamical Wilson fermions.
Early attempts to investigate the κ resonance on the lattice focused on the energy spectrum and involved searching for additional energy levels that could be associated with the resonance. Finitevolume energies were investigated for the κ system in Refs. [76] and Ref. [77]. In the latter reference, the quark-disconnected contributions were neglected. The authors later found that this leads to the wrong spectrum [74], as also discussed in Ref. [78] from a perturbative point of view. In the early 2010's it became clear that the κ does not behave like the typical resonance on the lattice. Using unitarized chiral perturbation theory models, Refs. [79][80][81][82][83][84] determined the finite-volume energies and investigated what can be expected in lattice QCD calculations.
To date, there have been few fully-fledged determinations of the energy dependence of Kπ scattering amplitudes with dynamical lattice QCD. The first such studies focused on the P -wave in the K * resonance region. In Ref. [85], N f = 2 + 1 staggered quarks were used to determine the K * phase shift from the rest frame spectra. A similar calculation with N f = 2 Wilson quarks included also moving frames [86] and determined the scattering phase shift and the K * width. The authors of Ref. [87] repeated the calculation for the ρ and the K * at an almost physical pion mass on two large N f = 2 gauge ensembles. A more comprehensive study was published in Refs. [88,89], where the authors calculated the scattering amplitudes in S-, P -and D-waves with I = 1/2 and I = 3/2 and determined their resonance content. They employed anisotropic gauge ensembles with N f = 2 + 1 Wilson fermions, similarly to Ref. [90]. Recently, Ref. [91] reported a calculation of I = 1/2 S-and P -wave scattering amplitudes at an unprecedented number of quark masses.
In the following, we present a new detailed analysis of I = 1/2 Kπ scattering using lattice QCD. This work provides further information on the interactions and resonances in this system, and also constitutes our first step toward a future lattice calculation of semileptonic form factors with Kπ final states based on the formalism developed in Ref. [92]. We simultaneously determine the energy dependence of both the S-wave and P -wave scattering amplitudes below the Kη threshold, and investigate several different parametrizations with and without an Adler zero for the S-wave amplitude. We determine the pole locations corresponding to the κ and K * resonances, and also present the K * Kπ couplings.
Our calculation is performed on two different gaugefield ensembles with N f = 2 + 1 dynamical clover fermions; the first has a lattice size of 32 3 ×96 with a spacing of a ≈ 0.114 fm and a pion mass of m π ≈ 317 MeV, while the second has a lattice size of 48 3 × 96 with a ≈ 0.088 fm and m π ≈ 176 MeV.
The paper is organized as follows: we begin by overviewing the continuum description of elastic Kπ scattering in Sec. II. In Sec. III, we briefly describe the lattice gauge field ensembles, while Sec. IV gives details on the construction of the interpolating operators and correlation matrices. Our spectrum results are shown in Sec. V, and the finite-volume methods for the determination of the scattering amplitudes are discussed in Sec. VI. We present our results for the energy dependence of the phase shifts and the pole locations in Sec. VII. Our conclusions, including a comparison with previous work, are given in Sec. VIII.

II. PARAMETRIZATIONS OF THE SCATTERING AMPLITUDES
In this section we briefly review the K-matrix formalism describing 2 → 2 scattering [93], and then discuss the specific parametrizations we use to describe Kπ scattering with I(J P ) = 1/2(0 + ) and I(J P ) = 1/2(1 − ). In general, the multi-channel scattering matrix can be expressed as where T ( ) is the T -matrix (also known as the scattering amplitude), which depends on the invariant mass s of the system, and is the partial wave of the scattering process. From the unitarity of S ( ) one gets where we used that due to time-reversal invariance of the strong interaction the T -matrix is symmetric. Here, the indices i, j, ... label the scattering channels and s (i) thr denotes the threshold in channel i. Equivalently, That means that we can split the real and imaginary contributions to T ( ) −1 in the following way: where K ( ) has to be real and symmetric to ensure unitarity and time-reversal invariance. In order to incorporate the correct analytic structure from the Kπ threshold, we define the phasespace factor ρ, which is a diagonal matrix in channel space: Above, a and b label the two mesons undergoing elastic scattering in channel i. For example, at scattering energies above the Kη threshold we have two scattering channels, i = 0, 1, which correspond to the scattering of Kπ and Kη, respectively. However, the Kη channel is not relevant at the energies we consider here, and our spectra can be described by purely elastic Kπ scattering (i = 0 only). We omit the channel indices in the remainder of the paper.
Using the phase-space factor, we define the rescaled K-matrixK ( ) through The elastic scattering phase shift δ is related to the scattering amplitude as and to the K-matrix as We now proceed to the discussion of the parametrizations we use for the s dependence. For the P -wave, which is governed by the narrow K * resonance, we find that a simple one-pole K-matrix parametrization with Blatt-Weisskopf barrier factors describes the data well. For the S-wave, we also consider three additional parametrizations: the effective range expansion [94,95], Bugg's parametrization [96,97] that accounts for a zero in the scattering amplitude known as the Adler zero [98][99][100], and the conformal-map-based parametrization used in Ref. [50], which also has an Adler zero.

A. Chung's parametrization
Chung's parametrization is a raw K-matrix pole parametrization [93] combined with Blatt-Weisskopf barrier factors [101]. The latter describe a centrifugal barrier effect in the P -wave but are trivial for S-wave scattering. For both the S-and P -waves the K-matrix pole parametrization iŝ where α labels the resonances present and with Here, the γ ,α are the resonance couplings, B α (k, k α ) are the Blatt-Weisskopf barrier factors (defined further below), and the parameters Γ 0 ,α are related to the widths of the resonances. Since only the product γ 2 ,α Γ 0 ,α appears in Γ ,α ( √ s), we perform our fits in terms of new parameters g 0 ,α defined as Inserting Eq. (12) into Eq. (9) gives the final form The Blatt-Weisskopf factors are functions of k, the scattering momentum at the given s, and k α , the scattering momentum at s = m 2 α . The scattering momentum k is defined via which gives The Blatt-Weisskopf factors are equal to with R 1,α the characteristic range for = 1, which we also take to be a fit parameter. Since we include only one resonance in each partial wave, we omit the index α in the following and denote the fit parameters as m , g 0 (for = 0, 1), R 1 .

B. Effective-range expansion
The effective range expansion (ERE) to order k 2 for = 0 is given by [94,95] where a is the zero-energy scattering length and r 0 represents the effective range of the interaction. The actual fit parameters we use are C. Bugg's parametrization The author of Ref. [97] performed a fit to FOCUS and E791 data for the D → Kππ decay [102,103], using a modified version of the parametrization of Ref. [31] to accommodate the broad nature of the κ resonance. This is a K-matrix pole parametrization where the width is taken to have a zero of the form s − s A , intended to account for the prediction from chiral perturbation theory that the T -matrix has a zero at s = s A (the Adler zero), where [50] The parametrization enables the scattering amplitude to reproduce the experimental phase shift near the threshold much better. A model explanation for this form of the amplitude is discussed in Ref. [39]. We implement this parametrization by multiplying the K-matrix pole with an enveloping term of the form s − s A so that the T -matrix also becomes zero as predicted:K

D. Conformal map parametrization
The final parametrization we consider for the Swave is that of Ref. [50], which involves a power series in a new variable ω(s). The regions of analyticity in the complex-s plane are conformally mapped to the interior of the unit disk, while the elastic, inelastic, left-hand, and circular cuts are mapped to the circle |ω| = 1. The parametrization also includes an Adler zero and is given by with The conformal map is defined as where ∆ Kπ = m 2 K − m 2 π and y 0 ≡ y(s 0 ). The constant s 0 determines the maximum value of s for which the map is applicable, while the constant α determines the origin of the expansion. We set √ s 0 equal to the Kη threshold, using leading-order chiral perturbation theory to express m η in terms of m K and m π : We choose α = 1.3 for both ensembles so that the origin of the expansion is around the middle of the data points. We found it sufficient to expand to first order in ω.

III. GAUGE ENSEMBLES AND SINGLE-MESON ENERGIES
In this work, we utilize two different gauge-field ensembles, labeled C13 and D6, with parameters given in Table I. These ensembles use the tadpoleimproved tree-level Symanzik gluon action [104][105][106][107] and include 2+1 flavors of clover fermions [108,109]. The gauge links in the fermion action are Stout smeared [110] with a staple weight of ρ = 0.125. We use the same clover action also for the valence quarks. The lattice spacings, a, were determined using the Υ(2S) − Υ(1S) splitting [111,112] computed with improved lattice NRQCD [113].  When using Lüscher's method, the pion and kaon dispersion relations are needed to relate energies to scattering momenta. To study the dispersion relations on the lattice, we computed pion and kaon two-point correlation functions projected to different momenta. Fits of the dispersion relations on the D6 ensemble are shown in Figs. 1 and 2. We find that the data with | p| ≤ √ 3 · 2π/L are consistent with the relativistic continuum dispersion relation on both ensembles, and we therefore use this form in the further analysis.

IV. INTERPOLATING OPERATORS AND CORRELATION MATRIX CONSTRUCTION
To determine the Kπ scattering amplitudes, we use the Lüscher method; the first step of such a calculation is to determine the finite-volume spectra for different total momenta and irreducible repre- sentations. We determine the spectra in specific momentum frames and irreducible representations by calculating two-point correlation functions constructed from a set of interpolating operators.

A. Interpolating operators
We use two types of interpolating operators in this work. The single-hadron operators, built from local quark-antiquark bilinears, are constructed as follows: These operators have manifest isospin I = 1/2; the projection to irreducible representations of the lattice symmetry group is discussed further below. The two-hadron operators are constructed from products of pseudoscalar π and K operators with definite individual momenta, combined to I = 1/2 via SU (2) Clebsch-Gordan coefficients: where All quark fields in the single-hadron and multihadron operators are Wuppertal-smeared [114] with α Wup = 3.0 and N Wup = 20 on the C13 ensemble and N Wup = 55 on the D6 ensemble, using APEsmeared gauge links [115] with α APE = 2.5 and N APE = 25, 32 for C13, D6 in the smearing kernel.
Since the finite-volume box in which we perform our calculation reduces the symmetry with respect to the infinite volume, we project the operators to the irreducible representations that respect the symmetry of the lattice: Above, LG( P ) denotes the Little Group on the lattice for total momentum P (i.e., the subgroup of the cubic group that remains a symmetry for the given momentum), and χ Λ (R) are the characters for irrep Λ, which can be found for example in Ref. [116]. In the sum over p (with components being integer multiples of 2π/L), we fix the magnitudes | p 1 | = |R p | = | p | and | p 2 | = | P − R p |, and different choices for these magnitudes yield different operators in the same irrep. In the following we will denote the irrep-projected operators as O Λ, P A , where the operator index A counts the different internal structures within a given irrep, as detailed in Table II.

B. Wick contractions
The correlation matrix C Λ, P (t) for irreducible representation Λ of Little Group LG( P ) is obtained from the interpolators defined above as where t src is the source time and t snk is the sink time.
The correlation matrix elements are expressed in terms of quark propagators by performing the Wick contractions (i.e., by performing the path integral over the quark fields in a given gauge-field configuration). The resulting quark-flow diagrams are shown in Fig. 3 (for the case I = 1/2 considered here). The diagrams in Fig. 3 are obtained from point-toall propagators (labeled f ), sequential propagators (labeled seq) and stochastic timeslice-to-all propagators (labeled st), as in Ref. [117]. One summation over x at the source is eliminated using translational symmetry. In the following description of the different types of propagators we omit the smearing kernels for brevity, but we note that all propagators are smeared at source and sink with the parameters given in Sec. IV A. The point-to-all propagator for quark flavor q with full spin-color-dilution is given by The sequential propagator S seq follows from the solution of the lattice Dirac equation with right-hand side given by (39) restricted to the source time t src and dressed with a vertex, For the purpose of this calculation we require only Γ = γ 5 to realize the pseudoscalar pion or kaon interpolators and q = l/s for the second inversion of the light / strange Dirac operator. In addition we use timeslice-to-all stochastic propagators for the sink-to-sink quark propagation in diagram (e) in Fig. 3. These follow from solving the Dirac equation with a fully time-diluted stochastic source The space-spin-color components of η (tsrc) are independent, Z 2 × Z 2 -distributed random numbers with zero mean and unit variance. Finally, the all-source-to-all-sink quark propagation in diagrams (d) (lower fermion loop) and diagram (f) are realized with stochastic sourcetimeslice-to-all propagators from spin-diluted noise sources based on the one-end-trick [118], with components η ( x) a again Z 2 × Z 2 noise.

V. SPECTRUM RESULTS
We extracted the energy levels E Λ, P n from the correlation matrices using the generalized eigenvalue problem (GEVP) [119,120] where n labels the eigenpair. Here, t 0 is a reference timeslice whose variation does not affect noticeably the large-t behavior [117]. At large enough values of t and t 0 , the eigenvalues λ n (t, t 0 ) take the form of a single exponential λ n (t, t 0 ) = e −E Λ, P n (t−t0) . We can make some initial observations by looking at the effective energies shown in Fig. 4 for four different irreps of three different Little Groups. Dashed lines indicate the noninteracting energy levels of the Kπ system with the pion and kaon momenta | p 1 | and | p 2 | used for the corresponding operator basis. In the plot for irrep T 1u , note that the highest three energy levels are shifted upward relative to the noninteracting energies. These states overlap most strongly (in relative terms) with the states created by the corresponding multi-hadron operators. The lowest energy level is an extra energy level whose occurrence is related to the presence of a narrow resonance (the K * ). This state overlaps most strongly with the states created by the quark-antiquark operators. Similarly, on the top-right plot (Λ = A 1g , | P | 2 = 0), there is a downward shift with respect to the noninteracting energies from which we can expect an attractive interaction and a positive S-wave scattering phase shift.
With the absence of an extra energy level, one can not straightforwardly identify the presence of a resonance in the depicted energy range.
Our main results for the energies E Λ, P n are obtained directly from single-exponential fits to the generalized eigenvalues λ n (t, t 0 ) and are given in Ta  to the lattice-frame energy E Λ, P n through √ s We have chosen the fit ranges such that the contributions from higher excited states are negligible compared to statistical uncertainty. We ensured this by varying the lower bound of the fit range, t min , as shown in Figs. 5 and 6. In each case, the nominal value for t min /a is chosen such that ∆E fwd = E| tmin − E| tmin+a is consistent with zero.

VI. LÜSCHER ANALYSIS
Assuming elasticity and neglecting exponential finite-volume effects, the energy levels of a twoparticle system with total momentum P in a cubic box with periodic boundary conditions are given by the solutions of the Lüscher quantization condition The object in parentheses is a matrix with indices m, m . The T -matrix introduced in Sec. II is diagonal, and for a single scattering channel as considered here, T ( ) denotes the scattering amplitude for partial wave . The amplitudes T ( ) depend only on the center-of-mass energy, or, equivalently, the scattering momentum k. The elements of M P for , ≤ 1 are given by [62] M P where the functions w m depend on the scattering momentum k, the box size L, and the total momentum P , Here, Z P m 1; (k L 2π ) 2 is the generalized zeta function and γ = E P / √ s is the Lorentz boost factor.
The matrix M P can be further simplified by taking into account the symmetries for a given Little Group LG( P ) and irreducible representation Λ [62]. Each irrep in principle contains infinitely many partial waves, the first few of which are listed in Table  II (since the K and π are both spinless, we have J = ). However, the contributions from higher partial waves are increasingly suppressed, and we neglect the contributions from ≥ 2 in this work. This then leads to the following quantization conditions, where we write the scattering amplitudes T (0) and T (1) in terms of the phase shifts δ 0 and δ 1 , respectively: 1, 1), LG = C 3v , Λ = E: Note that at nonzero momenta, the quantization conditions in the A 1 irreps depend on both the Swave and the P -wave phase shifts. This mixing between even and odd partial waves occurs because the reciprocal space of momenta in the unequal-mass Kπ system lacks inversion symmetry [62]. Traditionally, the Lüscher method has often been used to map individual energy levels on the lattice to individual phase shift values at the corresponding center-of-mass energies. However, this is no longer possible in the A 1 irreps with partial-wave mixing. Equation (55), for example, has the two unknowns δ 0 (k P ,A1 n ) and δ 1 (k P ,A1 n ), and it does not help to combine Eqs. (55) and (56) either, because the solutions of Eq. (56) occur at different values of the scattering momentum, k P ,E n . Since we want to use all irreps, we follow a different approach [121], in which we parametrize the energy dependence of the phase shifts δ 0 and δ 1 using the models discussed in Sec. II, and then perform a global fit of the model parameters for both the S-and P -waves to all energy levels by minimizing the following χ 2 function: is obtained from the parametrized scattering amplitudes using the Lüscher quantization conditions 2 . We fit 17 energy levels on the C13 ensemble and 26 levels on the D6 ensemble, as listed in Table III. In choosing these energy levels, we have stayed further below the Kη threshold, ∼ 0.95 √ s Kη , determined through Eq. (27) in order to safely avoid effects from the Kη threshold, the K (1410) resonance, or re-scattering from three particle channels [22]. In practice, we found it helpful to obtain initial guesses for the P -wave model parameters using an initial fit to only those irreps without S-wave contributions, followed by the full fit to all irreps. The results for the P -wave parameters from the full fits are consistent with the results from the reduced fits, but are about 10% more precise.
On each ensemble, we performed four different full fits that differ in the type of parametrization used for the S-wave amplitude: Chung's parametrization [Eq. (13)], effective-range expansion [Eq. (19)], Bugg's parametrization [Eq. (22)], and conformalmap parametrization [Eq. (24)]. The parametrization for the P -wave amplitude was always of the form given in Eq. (13). The best-fit parameters and χ 2 values of the full fits are given in Tables V and VI in the Appendix.

VII. RESULTS FOR THE PHASE SHIFTS AND POLE POSITIONS
The phase-shift curves obtained from the four different fits and the two ensembles are presented in Fig. 7. In addition, we determined the positions of the closest T -matrix poles in the complex √ s plane, which are associated with the κ and K * resonances.
The pole positions are shown in Fig. 8 and are listed in Table IV. All poles are located on the second Riemann sheet. In the following, we discuss our observations separately for the S-and P -waves.

A. S-wave scattering
The S-wave phase-shift curves from the four different parametrizations are in reasonable agreement with each other, given the uncertainties. We observe that the phase shifts remain below 80 • in the energy region considered. Even though there is little model dependence in the phase-shift curves for real-valued √ s, the positions of the resulting poles of the scattering amplitude vary widely between the different parametrizations. Moreover, some of the parametrizations lead to a much stronger dependence on the pion mass than others: • The ERE parametrization [Eq. (19)] yields a pole at [1.11(6) − 0.38 (34) i] GeV on the C13 ensemble and at [0.33(23) − 0.35 (22) i] GeV on the D6 ensemble. This change is significantly larger than expected from the relatively minor change in the quark masses. Note that the ERE is meant to describe the behavior near the threshold and likely becomes unreliable in the upper range of our energy region.
• The poles for Chung's K-matrix parametrization without an Adler zero [Eq. (13)] appear above 1.1 GeV -in the region where we do not have data points to fully constrain the amplitude -with only mild dependence on the quark masses. What may be happening is that this   fit is sensing the K * 0 (1430) resonance located not that far away [22].
• For the conformal-map-based parametrization [Eq. (24)] and Bugg's parametrization [Eq. (22)], which both include an Adler zero, the poles appear near [0.8 − 0.3 i] GeV consistently for the two pion masses, and consistently for the two parametrizations. On the lower-pion-mass ensemble in particular, these parametrizations also yield much smaller statistical uncertainties for the pole locations.
In summary, the poles of the S-wave amplitude are significantly more stable for parametrizations incorporating an Adler zero. Because the conformal-map parametrization describes the data on the lower-pion-mass ensemble (D6) better than Bugg's parametrization, we choose the conformalmap parametrization as our nominal parametrization.
In Fig 9, we compare the phase-shift curves from this parametrization to experimental results from Ref. [17]. We see that the S-wave phase-shift curves approach the experimental data as the pion mass is lowered toward its physical value. The pion mass of the D6 ensemble is nearly physical, and the resulting curve is very close to the data. The behavior of the curve as a function of the pion mass seen here is also consistent with the lattice results in Ref. [91] at a higher pion mass.

B. P -wave scattering
The results presented in Figs. 7, 8 and Table IV were all obtained with Chung's Kmatrix parametrization, Eq. (13), for the P -wave. While we explored other P -wave parametrizations that do not include Blatt-Weisskopf barrier factors, we found little variation (an explicit comparison can be found for I = 1 ππ scattering in Ref. [117], which also showed no significant variation). Furthermore, the P -wave phase-shift curves and pole locations do not significantly depend on the choice of the S-wave parametrization, which confirms that partial-wave mixing between = 0 and = 1 is under good control. We choose the same combination of parametrizations as above (Chung's parametrization for the P -wave combined with the conformal map for the S wave) for our nominal results for the P wave.
A comparison of the phase-shift curves for the two different pion masses with experimental data [17] is shown in Fig. 9. A clear resonance shape is observed for both pion masses. In this channel, the resonance width Γ K * →Kπ = −2 Im( √ s R ), where √ s R is the location of the pole, depends strongly on the pion mass due to the large change in available phase space. We find while the value in nature is 50.8(0.9) MeV [22]. Consequently, even at the close-to-physical pion mass of the D6 ensemble, the phase shift curve is still noticeably steeper than in nature. In this situation, it is more appropriate to consider the K * Kπ coupling g K * Kπ , which can be obtained from the decay width through where k * is the scattering momentum for √ s = Re( √ s R ). This gives g D6 K * Kπ = 5.03 (22).
These values are consistent with each other, and also consistent with similar calculations [87,91]. Our results are slightly below the experimental value of g K * Kπ = 5.603 (4). A comparison of our results for m K * and g K * Kπ with previous lattice results is shown in Fig. 10.

VIII. CONCLUSIONS
We have obtained precise results for the I = 1/2 S-and P -wave Kπ scattering phase shifts as functions of the center-of-mass energy up to 1.1 GeV, for quark masses corresponding to m π ≈ 176 MeV and m π ≈ 317 MeV. We also determined the positions of the closest poles in the scattering amplitudes, which we identify with the K * 0 (700) (also referred to as κ) and K * (892) resonances.
For the S-wave amplitude, we investigated several different parametrizations proposed in the literature, some including an Adler zero and some without it. All parametrizations considered, including the effective-range expansion that is similar to the widely used LASS parametrization [17], describe the phase shifts well for real √ s in the energy region considered. However, we found that the pole positions are stable only for those parametrizations that include an Adler zero. Using a conformal-mapbased parametrization with an Adler zero, we found the poles in the S-wave scattering amplitude at  9. Phase shift results, using the conformal-map parametrization for the S-wave and Chung's parametrization for the P -wave, compared to experimental results from Ref. [17]. The fitted parameters with the corresponding covariances used to produce these plots are included in the ancillary files as D6 fit parameters.dat and C13 fit parameters.dat.
Despite the unphysical pion masses and the lack of continuum extrapolations, these results are consistent with the κ pole position extracted from experiments as reported by the Particle Data Group [22].
Earlier lattice calculations at a heavier pion mass of m π ≈ 390 MeV performed by the Hadron Spectrum Collaboration [88,89] found the κ as a bound state. More recently, the same collaboration reported results for a wider range of pion masses down to approximately 200 MeV [91]. Investigating a large number of parametrizations, the authors did not find a sufficiently unique result to report numerically. For the parametrizations inspired by unitarized chiral perturbation theory, they did, however, find a κ pole with a real part near the Kπ threshold and a large imaginary part. This is consistent with our findings for parametrizations that include the Adler zero.
In the vector channel, our results for the K * mass and width have high statistical precision. Since the K * width depends strongly on the pion mass through kinematic effects, it is more appropriate to consider the K * Kπ coupling. Our results for this coupling and for the K * mass are compared with previous lattice results [74,85,87,88,90,91] in Fig. 10. Note that the calculations were performed with different numbers of flavors, different gluon and fermion discretizations, and with different procedures to set the lattice scale; none of the calculations included a continuum extrapolation. Keeping in mind these caveats, we note that our results for both g K * Kπ and m K * agree well with previous calculations, except for the higher mass obtained by Fu et al. using a staggered fermion action [85]. Apart from this outlier, the results for m K * show only very mild quark-mass dependence, while g K * Kπ has no discernible quark-mass dependence, similar to g ρππ [117]. Furthermore, the results from N f = 2 + 1 and N f = 2 ensembles appear to follow common staight lines.
The calculations performed here can also be used in future lattice determinations of 1 → 2 transition matrix elements of external currents with the same Kπ states. The scattering amplitudes are needed to map the finite-volume matrix elements to infinite-volume matrix elements via the formalism of Ref. [92], as has already been done for πγ * → ππ [122,123]. Such a calculation will be particularly important for rare B → Kπ + − decays [8][9][10][11][12][13][14][15]124].