The quark-mass dependence of elastic $\pi K$ scattering from QCD

TL;DR

This study uses lattice QCD to analyze how the elastic πK scattering amplitudes depend on quark mass, revealing the evolution of resonances and bound states as pion and kaon masses vary.

Contribution

First lattice QCD determination of πK scattering amplitudes across multiple quark masses, providing insights into resonance behavior and analytic continuation limitations.

Findings

P-wave amplitude shows a pole evolving from bound state to resonance.

S-wave amplitude indicates a possible κ/K0*(700)-like resonance.

Lattice results highlight challenges in model-independent resonance extraction.

Abstract

We present a determination of the isospin-1/2 elastic $\pi K$ scattering amplitudes in S and P partial waves using lattice Quantum Chromodynamics. The amplitudes, constrained for a large number of real-valued energy points, are obtained as a function of light-quark mass, corresponding to four pion masses between 200 and 400 MeV. Below the first inelastic threshold, the P-wave scattering amplitude is dominated by a single pole singularity that evolves from being a stable bound-state at the highest quark mass into a narrow resonance that broadens as the pion and kaon masses are reduced. As in experiment, the S-wave amplitude does not exhibit an obviously resonant behavior, but instead shows a slow rise from threshold, which is not inconsistent with the presence of a $\kappa$/$K_0^\star(700)$-like resonance at the considered quark masses. As has been found in analyses of experimental scattering data, simple analytic continuations into the complex energy plane of precisely-determined lattice QCD amplitudes on the real energy axis are not sufficient to model-independently determine the existence and properties of this state. The spectra and amplitudes we present will serve as an input for increasingly elaborate amplitude analysis techniques that implement more of the analytic structure expected at complex energies.

Tables

Table 1. Table 1: A summary of the lattices used in this study, with spatial volume L 3 superscript 𝐿 3 L^{3} , temporal extent T 𝑇 T , and the masses of relevant stable hadrons. N cfgs subscript 𝑁 cfgs N_{\text{cfgs}} denotes the number of gauge configurations used and N vecs subscript 𝑁 vecs N_{\text{vecs}} is the number of distillation vectors  Peardon  et al. ( 2009 ) . N t src subscript 𝑁 subscript t src N_{\mathrm{t_{src}}} is the number of different timeslices used for source operators. Brackets denote the uncertainty on the final digit.

${\left(L/a_s\right)}^3$	$\times T/a_t$	$N_{\mathrm{cfgs}}$	$N_{\mathrm{vecs}}$	$N_{t_{\mathrm{src}}}$	$a_t{m}_{\pi }$	$a_t{m}_K$	$a_t{m}_{\eta }$	$a_t{m}_{\Omega }$	$\xi$	$m_{\pi }$/MeV
{${16}^3$, ${20}^3$, ${24}^3$}	$\times$ 128	{479,603,553}	{64,128,162}	2–8	0.06906(13)	0.09698(9)	0.10364(19)	0.2951(22)	3.444(6)	391
${24}^3$	$\times$ 256	309	162	4–8	0.05593(28)	0.09027(15)	0.09790(100)	0.2857(8)	3.456(9)	327
${24}^3$	$\times$ 256	400	162	4	0.04735(22)	0.08659(14)	0.09602(70)	0.2793(8)	3.455(6)	284
${32}^3$	$\times$ 256	485	384	2–4	0.03928(18)	0.08344(7)	0.09299(56)	0.2751(6)	3.453(6)	239

Table 2. Table 2: Reference amplitude fits corresponding to a linear-order K 𝐾 K -matrix with parameters γ 0 subscript 𝛾 0 \gamma_{0} , γ 1 subscript 𝛾 1 \gamma_{1} in S 𝑆 S -wave, and a Breit-Wigner in P 𝑃 P -wave with a Breit-Wigner mass parameter m R subscript 𝑚 𝑅 m_{R} and Breit-Wigner coupling g R subscript 𝑔 𝑅 g_{R} .

$m_{\pi }/$MeV	${\chi }^2/N_{\mathrm{dof}}$	${\gamma }_0$	${\gamma }_1$	corr.	$m_R$/MeV	$g_R$	corr.
239	$\frac{39.15}{28-4}=1.63$	$0.48\left(1\right)$	$2.29\left(19\right)$	-0.57	904(1)	4.81(7)	0.18
284	$\frac{20.13}{21-4}=1.18$	$0.90\left(15\right)$	$3.34\left(64\right)$	-0.33	911(2)	4.67(11)	0.01
327	$\frac{25.31}{18-4}=1.81$	$0.59\left(16\right)$	$3.14\left(90\right)$	-0.75	915(2)	5.20(16)	0.28
391	$\frac{51.36}{36-4}=1.60$	$0.77\left(24\right)$	$1.20\left(39\right)$	-0.91	936(1)	5.52(20)	-0.31

Table 3. Table 3: Scattering length and effective range parameters obtained from fitting the finite-volume spectra, corresponding to the bands in Fig.  3 . A three-parameter K 𝐾 K -matrix, as described in the text, has been fitted simultaneously in P 𝑃 P -wave.

$m_{\pi }/$MeV	${\chi }^2/N_{\mathrm{dof}}$	$m_{\pi }a$	$m_{\pi }r$	corr.	$m$/MeV	$g$	${\gamma }_0$/GeV-2
239	$\frac{10.84}{15-4}=0.99$	$0.43\left(9\right)$	$-$	$-$	901(2)	0.367(39)	0.88(39)
	$\frac{36.58}{28-5}=1.59$	$0.46\left(3\right)$	$-0.72\left(18\right)$	0.95	902(1)	0.403(20)	0.29(9)
284	$\frac{18.79}{21-4}=1.11$	$0.91\left(8\right)$	$-$	$-$	908(2)	0.410(11)	0.30(8)
	$\frac{17.27}{21-5}=1.08$	$0.79\left(13\right)$	$-0.32\left(29\right)$	0.84	908(2)	0.411(12)	0.29(8)
327	$\frac{25.86}{18-4}=1.85$	$0.78\left(7\right)$	$-$	$-$	914(2)	0.412(13)	0.31(13)
	$\frac{24.70}{18-5}=1.90$	$0.65\left(14\right)$	$-1.10\left(110\right)$	0.95	915(2)	0.412(14)	0.08(25)
391	$\frac{31.90}{36-4}=1.00$	$1.29\left(10\right)$	$-$	$-$	933(1)	0.522(21)	0.55(13)
	$\frac{31.65}{36-5}=1.02$	$1.31\left(11\right)$	$0.20\left(41\right)$	0.43	933(1)	0.521(21)	0.56(15)

Table 4. Table 4: The parameterizations used in this study. χ 2 superscript 𝜒 2 \chi^{2} values in italics denote an amplitude that was rejected due to either a too large χ 2 / N dof superscript 𝜒 2 subscript 𝑁 dof \chi^{2}/N_{\mathrm{dof}} or nearby singularities not supported by the finite volume spectra.

$S$-amp	$P$-amp	$N_S$	$N_P$	$N_g$	$I_S$	$I_P$	$N_{\mathrm{pars}.}$	${\chi }^2/N_{\mathrm{dof}}$ for $m_{\pi }/\mathrm{MeV}$
							$S$+$P$	$\approx$ 391	$\approx$ 327	$\approx$ 286	$\approx$ 239
SL	BW	-	-	-	-$i\rho$	BW	1+2	$\frac{49.76}{36-3}=1.51$	$\frac{32.87}{18-3}=2.19$	$\frac{38.73}{21-3}=2.15$	$\frac{65.63}{28-3}=2.63$
ER	BW	-	-	-	-$i\rho$	BW	2+2	$\frac{49.72}{36-4}=1.55$	$\frac{24.79}{18-4}=1.77$	$\frac{33.50}{21-4}=1.97$	$\frac{45.98}{28-4}=1.92$
SL	Eq. 4	-	-	-	-$i\rho$	CM	1+3	$\frac{31.90}{36-4}=1.00$	$\frac{25.86}{18-4}=1.85$	$\frac{18.79}{21-4}=1.11$	$\frac{59.92}{28-4}=2.50$
ER	Eq. 4	-	-	-	-$i\rho$	CM	2+3	$\frac{31.65}{36-5}=1.02$	$\frac{24.70}{18-5}=1.90$	$\frac{17.27}{21-5}=1.08$	$\frac{36.58}{28-5}=1.59$
Eq. 2	BW	0	-	-	CM	BW	1+2	$\frac{62.75}{36-3}=1.90$	$\frac{59.08}{18-3}=3.94$	$\frac{108.3}{21-3}=6.01$	$\frac{241.47}{28-3}=9.66$
Eq. 2	BW	1	-	-	CM	BW	2+2	$\frac{51.36}{36-4}=1.60$	$\frac{25.31}{18-4}=1.81$	$\frac{20.13}{21-4}=1.18$	$\frac{39.15}{28-4}=1.63$
Eq. 2	Eq. 4	0	-	-	CM	CM	1+3	$\frac{42.21}{36-4}=1.32$	$\frac{31.20}{18-4}=2.23$	$\frac{70.80}{21-4}=4.16$	$\frac{239.23}{28-4}=9.97$
Eq. 2	Eq. 4	1	-	-	CM	CM	2+3	$\frac{31.93}{36-5}=1.03$	$\frac{24.54}{18-5}=1.89$	$\frac{12.50}{21-5}=0.78$	$\frac{37.70}{28-5}=1.64$
Eq. 2	Eq. 8	1	1	0	CM	CM	2+4	$\frac{28.74}{36-6}=0.96$	$\frac{23.09}{18-6}=1.92$	$\frac{12.48}{21-6}=0.83$	$\frac{37.70}{28-6}=1.71$
Eq. 2	Eq. 8	1	-	1	CM	CM	2+3	$\frac{33.00}{36-5}=1.06$	$\frac{24.64}{18-5}=1.90$	$\frac{12.48}{21-5}=0.78$	$\frac{37.70}{28-5}=1.64$
Eq. 2	Eq. 8	1	0	1	CM	CM	2+4	$\frac{31.93}{36-6}=1.06$	$\frac{24.54}{18-6}=2.04$	$\frac{12.48}{21-6}=0.83$	$\frac{37.70}{28-6}=1.71$
Eq. 5	BW	1	-	-	CM	BW	2+2	$\frac{47.88}{36-4}=1.50$	$\frac{25.20}{18-4}=1.80$	$\frac{39.83}{21-4}=2.34$	$\frac{53.67}{28-4}=2.24$
Eq. 5	Eq. 4	1	-	-	CM	CM	2+3	$\frac{31.20}{36-5}=1.01$	$\frac{25.09}{18-5}=1.93$	$\frac{20.94}{21-5}=1.31$	$\frac{42.40}{28-5}=1.84$
Eq. 6	BW	1(c),1(γ)	-	-	CM	BW	3+2	$\frac{40.76}{36-5}=1.31$	$\frac{24.44}{18-5}=1.88$	$\frac{19.89}{21-5}=1.24$	$\frac{37.01}{28-5}=1.61$
Eq. 6	Eq. 4	1(c),1(γ)	-	-	CM	CM	3+3	$\frac{28.27}{36-6}=0.94$	$\frac{24.34}{18-6}=2.03$	$\frac{12.49}{21-6}=0.83$	$\frac{32.32}{28-6}=1.47$
Eq. 7	BW	1	-	-	CM	BW	2+2	$\frac{51.30}{36-4}=1.60$	$\frac{24.48}{18-4}=1.75$	$\frac{25.89}{21-4}=1.52$	$\frac{38.60}{28-4}=1.61$
Eq. 7	Eq. 4	1	-	-	CM	CM	2+3	$\frac{31.93}{36-5}=1.03$	$\frac{24.39}{18-5}=1.88$	$\frac{13.73}{21-5}=0.86$	$\frac{32.41}{28-5}=1.41$
U$\chi$PT		-	-	-	$\sim$CM		4	$\frac{38.18}{31-4}=1.41$	$\frac{26.06}{18-4}=1.86$	$\frac{25.88}{21-4}=1.52$	$\frac{39.32}{28-4}=1.64$
U$\chi$PT		-	-	-	$\sim$CM		4	$\frac{49.29}{31-4}=1.64$	$\frac{24.41}{18-4}=1.74$	$\frac{25.26}{21-4}=1.49$	$\frac{39.66}{28-4}=1.65$

Table 5. Table 5: The operators used in obtaining the energy levels shown in Fig.  5 . Numbers in square brackets denote the type of momentum, where p → = [ i ​ j ​ k ] = 2 ​ π L ​ ( i , j , k ) → 𝑝 delimited-[] 𝑖 𝑗 𝑘 2 𝜋 𝐿 𝑖 𝑗 𝑘 \vec{p}=[ijk]=\frac{2\pi}{L}(i,j,k) . Momentum directions are summed in constructing meson-meson-like operators as described in Ref.  Dudek  et al. ( 2012 ) . By q ​ q ¯ 𝑞 ¯ 𝑞 q\bar{q} we refer to constructions of the form ψ ¯ ​ Γ ​ D ​ … ​ D ​ ψ ¯ 𝜓 Γ 𝐷 … 𝐷 𝜓 \bar{\psi}\Gamma D...D\psi that resemble a single-meson structure. The operators used at the largest pion mass are listed in Ref.  Wilson  et al. ( 2015a ) . a t ​ m ℓ subscript 𝑎 𝑡 subscript 𝑚 ℓ a_{t}m_{\ell} and a t ​ m s subscript 𝑎 𝑡 subscript 𝑚 𝑠 a_{t}m_{s} appear in the lattice action from Refs.  Edwards  et al. ( 2008 ); Lin  et al. ( 2009 ) .

	$m_{\pi }\approx 239$ MeV	$m_{\pi }\approx 286$ MeV	$m_{\pi }\approx 327$ MeV
	$a_t{m}_{\ell }=-0.0860$, $a_t{m}_s=-0.0743$	$a_t{m}_{\ell }=-0.0856$, $a_t{m}_s=-0.0743$	$a_t{m}_{\ell }=-0.0850$, $a_t{m}_s=-0.0743$
	$L/a_s=32$, $T/a_t=256$	$L/a_s$=24, $T/a_t=256$	$L/a_s$=24, $T/a_t=256$
[000] $A_1^{+}$	$\pi \left[000\right]K\left[000\right]$ $\pi \left[100\right]K\left[100\right]$	$\pi \left[000\right]K\left[000\right]$ $\pi \left[100\right]K\left[100\right]$	$\pi \left[000\right]K\left[000\right]$ $\pi \left[100\right]K\left[100\right]$
	$\pi \left[110\right]K\left[110\right]$ $\pi \left[111\right]K\left[111\right]$
	$\eta \left[000\right]K\left[000\right]$ $\eta \left[100\right]K\left[100\right]$	$\eta \left[000\right]K\left[000\right]$	$\eta \left[000\right]K\left[000\right]$
	$7\times q\overline{q}$	$18\times q\overline{q}$	$5\times q\overline{q}$
[000] $T_1^{-}$	$\pi \left[100\right]K\left[100\right]$ $\pi \left[110\right]K\left[110\right]$	$\pi \left[100\right]K\left[100\right]$	$\pi \left[100\right]K\left[100\right]$
	$\eta \left[100\right]K\left[100\right]$
	$11\times q\overline{q}$	$44\times q\overline{q}$	$8\times q\overline{q}$
[000] $E^{+}$	$\pi \left[100\right]K\left[100\right]$ $\pi \left[110\right]K\left[110\right]$	$\pi \left[100\right]K\left[100\right]$	$\pi \left[100\right]K\left[100\right]$
	$\eta \left[100\right]K\left[100\right]$
	$22\times q\overline{q}$	$26\times q\overline{q}$	$16\times q\overline{q}$
[100] $A_1$	$\pi \left[000\right]K\left[100\right]$ $\pi \left[100\right]K\left[000\right]$	$\pi \left[000\right]K\left[100\right]$ $\pi \left[100\right]K\left[000\right]$	$\pi \left[000\right]K\left[100\right]$ $\pi \left[100\right]K\left[000\right]$
	$\pi \left[100\right]K\left[110\right]$ $\pi \left[110\right]K\left[100\right]$	$\pi \left[100\right]K\left[110\right]$ $\pi \left[110\right]K\left[100\right]$	$\pi \left[100\right]K\left[110\right]$ $\pi \left[110\right]K\left[100\right]$
	$\pi \left[110\right]K\left[111\right]$ $\pi \left[111\right]K\left[110\right]$
	$\pi \left[100\right]K\left[200\right]$ $\pi \left[200\right]K\left[100\right]$
	$\eta \left[100\right]K\left[000\right]$ $\eta \left[000\right]K\left[100\right]$	$\eta \left[100\right]K\left[000\right]$ $\eta \left[000\right]K\left[100\right]$	$\eta \left[100\right]K\left[000\right]$ $\eta \left[000\right]K\left[100\right]$
	$8\times q\overline{q}$	$14\times q\overline{q}$	$11\times q\overline{q}$
[110] $A_1$	$\pi \left[000\right]K\left[110\right]$ $\pi \left[100\right]K\left[100\right]$	$\pi \left[000\right]K\left[110\right]$ $\pi \left[100\right]K\left[100\right]$	$\pi \left[000\right]K\left[110\right]$ $\pi \left[100\right]K\left[100\right]$
	$\pi \left[110\right]K\left[000\right]$ $\pi \left[110\right]K\left[110\right]$	$\pi \left[110\right]K\left[000\right]$	$\pi \left[110\right]K\left[000\right]$
	$\pi \left[100\right]K\left[111\right]$ $\pi \left[111\right]K\left[100\right]$
	$\eta \left[110\right]K\left[000\right]$ $\eta \left[100\right]K\left[100\right]$	$\eta \left[110\right]K\left[000\right]$ $\eta \left[100\right]K\left[100\right]$	$\eta \left[110\right]K\left[000\right]$ $\eta \left[100\right]K\left[100\right]$
	$\eta \left[000\right]K\left[110\right]$	$\eta \left[000\right]K\left[110\right]$	$\eta \left[000\right]K\left[110\right]$
	$9\times q\overline{q}$	$24\times q\overline{q}$	$12\times q\overline{q}$
[111] $A_1$	$\pi \left[000\right]K\left[111\right]$ $\pi \left[100\right]K\left[110\right]$	$\pi \left[000\right]K\left[111\right]$ $\pi \left[100\right]K\left[110\right]$	$\pi \left[000\right]K\left[111\right]$ $\pi \left[100\right]K\left[110\right]$
	$\pi \left[110\right]K\left[100\right]$ $\pi \left[111\right]K\left[000\right]$	$\pi \left[110\right]K\left[100\right]$ $\pi \left[111\right]K\left[000\right]$	$\pi \left[110\right]K\left[100\right]$ $\pi \left[111\right]K\left[000\right]$
	$\eta \left[111\right]K\left[000\right]$ $\eta \left[110\right]K\left[100\right]$	$\eta \left[111\right]K\left[000\right]$ $\eta \left[000\right]K\left[111\right]$	$\eta \left[111\right]K\left[000\right]$ $\eta \left[000\right]K\left[111\right]$
	$\eta \left[100\right]K\left[110\right]$ $\eta \left[000\right]K\left[111\right]$
	$21\times q\overline{q}$	$12\times q\overline{q}$	$13\times q\overline{q}$
[200] $A_1$	$\pi \left[100\right]K\left[100\right]$ $\pi \left[000\right]K\left[200\right]$	$\pi \left[100\right]K\left[100\right]$ $\pi \left[000\right]K\left[200\right]$	$\pi \left[100\right]K\left[100\right]$ $\pi \left[000\right]K\left[200\right]$
	$\pi \left[110\right]K\left[110\right]$ $\pi \left[200\right]K\left[000\right]$	$\pi \left[110\right]K\left[110\right]$ $\pi \left[200\right]K\left[000\right]$	$\pi \left[110\right]K\left[110\right]$ $\pi \left[200\right]K\left[000\right]$
	$\eta \left[100\right]K\left[100\right]$	$\eta \left[100\right]K\left[100\right]$ $\eta \left[200\right]K\left[000\right]$	$\eta \left[100\right]K\left[100\right]$ $\eta \left[200\right]K\left[000\right]$
	$23\times q\overline{q}$	$9\times q\overline{q}$	$24\times q\overline{q}$
[100] $E_2$	$\pi \left[100\right]K\left[110\right]$ $\pi \left[110\right]K\left[100\right]$	$\pi \left[100\right]K\left[110\right]$ $\pi \left[110\right]K\left[100\right]$	$\pi \left[100\right]K\left[110\right]$ $\pi \left[110\right]K\left[100\right]$
	$\pi \left[110\right]K\left[111\right]$ $\pi \left[111\right]K\left[110\right]$
	$\eta \left[110\right]K\left[100\right]$ $\eta \left[100\right]K\left[110\right]$
	$33\times q\overline{q}$	$16\times q\overline{q}$	$39\times q\overline{q}$
[110] $B_1$	$\pi \left[100\right]K\left[100\right]$ $\pi \left[110\right]K\left[110\right]$	$\pi \left[100\right]K\left[100\right]$	$\pi \left[100\right]K\left[100\right]$
	$\pi \left[110\right]K\left[200\right]$ $\pi \left[200\right]K\left[110\right]$
	$\eta \left[100\right]K\left[100\right]$ $\eta \left[110\right]K\left[110\right]$	$\eta \left[100\right]K\left[100\right]$	$\eta \left[100\right]K\left[100\right]$
	$30\times q\overline{q}$	$16\times q\overline{q}$	$14\times q\overline{q}$
[110] $B_2$	$\pi \left[100\right]K\left[111\right]$ $\pi \left[111\right]K\left[100\right]$
	$\pi \left[110\right]K\left[110\right]$
	$28\times q\overline{q}$	$52\times q\overline{q}$	$16\times q\overline{q}$
[111] $E_2$	$\pi \left[100\right]K\left[110\right]$ $\pi \left[110\right]K\left[100\right]$	$\pi \left[100\right]K\left[110\right]$ $\pi \left[110\right]K\left[100\right]$	$\pi \left[100\right]K\left[110\right]$ $\pi \left[110\right]K\left[100\right]$
	$\eta \left[100\right]K\left[110\right]$ $\eta \left[110\right]K\left[100\right]$
	$33\times q\overline{q}$	$20\times q\overline{q}$	$16\times q\overline{q}$
[200] $E_2$	$\pi \left[110\right]K\left[110\right]$ $\pi \left[111\right]K\left[111\right]$	$\pi \left[110\right]K\left[110\right]$	$\pi \left[110\right]K\left[110\right]$
	$\eta \left[110\right]K\left[110\right]$
	$33\times q\overline{q}$	$16\times q\overline{q}$	$39\times q\overline{q}$

Equations

$$\displaystyle\det\Bigl{[}\bm{1}+i\rho(E)\,\bm{t}(E)\cdot\bigl{(}\bm{1}+i\bm{\mathcal{M}}(E,L)\bigr{)}\Bigr{]}=0,$$

$$t^{-1}(s)=K^{-1}(s)+I(s),K(s)$$

$$k_{\mathsf{cm}}(s)=\frac{1}{2s^{1/2}}{\left(s-(m_{\pi }+m_K{)}^2\right)}^{1/2}{\left(s-(m_K-m_{\pi }{)}^2\right)}^{1/2}.$$

$$t^{-1}(s)=(2k_{\mathsf{cm}}(s)\cdot K(s)\cdot 2k_{\mathsf{cm}}(s){)}^{-1}+I(s),K=\frac{g^2}{m^2-s}+{\gamma }_0,$$

$$K(s{)}^{-1}$$

$$K(s)$$

$$K(s{)}^{-1}$$

$$K(s)$$

Full text

††preprint: DAMTP-2019-13, JLAB-THY-19-2911

Introduction — $\pi K$ scattering has a long history, which mirrors closely the $\pi\pi$ case, with the $P$-wave containing a clearly visible narrow resonance, the $K^{\star}(892)$ which partners the $\rho$, while the $S$-wave features only a slow rise with energy. Much of our experimental knowledge is derived from the classic kaon beam experiments Estabrooks et al. (1978); *Aston:1986jb; *Aston:1987ey; *Aston:1987ir at SLAC where the dominance of pion exchange at small momentum transfers to proton targets was used to access an effective $\pi K$ initial state.

In a world where SU(3) flavor symmetry were exact, scattering amplitudes in isospin-$\frac{1}{2}$, isospin-1, and isospin-[math] would all appear in an octet and have a common resonant content in each partial-wave. Empirically, these channels show strikingly different behavior in $S$-wave, indicating a strong breaking of the SU(3) flavor symmetry. How the experimental observations evolve towards the SU(3) symmetric theory with varying quark mass is far from understood, and in this paper we will report on a study of this evolution in the kaon sector.

We compute the elastic scattering amplitudes for $\pi K$ in isospin-$\frac{1}{2}$ in $S$ and $P$ partial waves, using four values of the light-quark mass resulting in pion masses of approximately 239, 284, 329 & 391 MeV. We find clear evidence for the vector $K^{\star}$ state for all values of the quark masses, while the $S$-wave appears qualitatively similar to experiment with a broad enhancement seen across the elastic region.

Methods — We utilize lattice Quantum Chromodynamics (QCD) as the only first-principles, systematically-improvable and generally-applicable approach to QCD. The use of a discretized Euclidean spacetime of finite volume allows us to determine hadronic correlation functions via Monte-Carlo sampling of gauge-fields. The Euclidean time-dependence of these correlation functions is controlled by the discrete spectrum of eigenstates of QCD in the finite volume.

This spectrum can be used to constrain the infinite-volume scattering amplitudes via the Lüscher method  Luscher (1991); Rummukainen and Gottlieb (1995); Bedaque (2004); Kim et al. (2005); Fu (2012a); Leskovec and Prelovsek (2012); Gockeler et al. (2012); He et al. (2005); Lage et al. (2009); Bernard et al. (2011); Doring et al. (2011); Doring and Meissner (2012); Agadjanov et al. (2014); Doring et al. (2012); Hansen and Sharpe (2012); Briceno and Davoudi (2013); Guo et al. (2013) – for a recent review see Ref. Briceno et al. (2018a). Through use of multiple lattice volumes and consideration of frames moving with respect to the lattice, sufficiently many energy levels can be obtained to determine in detail the energy-dependence of scattering amplitude across a large energy region. Several previous studies have considered $\pi K$ scattering using lattice QCD Beane et al. (2006); Fu (2012b); Fu and Fu (2012); Lang et al. (2012); Prelovsek et al. (2013); Sasaki et al. (2014); Dudek et al. (2014); Wilson et al. (2015a); Bali et al. (2016); Brett et al. (2018); Helmes et al. (2018).

The cubic nature of the periodic spatial boundary of the lattice means that states are characterized by irreducible representations (irreps) of the cubic group, and of the relevant little groups when considering moving frames. The mismatch between the cubic symmetry and the continuous rotational symmetry of the infinite-volume theory means that the irreps contain an infinite number of mixed partial waves. Because near threshold only a relatively small number of low partial waves are expected to be significant, in practice only two or three amplitudes influence the spectrum in the energy region we will consider.

To extract scattering amplitudes from finite-volume spectra it is important to accurately obtain all of the energy levels in the region of interest, and in order to do this, we compute matrices of correlation functions using a basis of operators, and diagonalize to obtain several excited energy eigenstates Michael (1985); Blossier et al. (2009). A range of operators are considered, matching expectations of the kinds of finite-volume eigenstates in this case, consisting of $\bar{\psi}\Gamma{D}...{D}\psi$ constructions which resemble $q\bar{q}$ structures Dudek et al. (2010); Thomas et al. (2012), and meson-meson-like constructions Dudek et al. (2012). The meson-meson operators are built from products of variationally-optimized meson operators, themselves sums of many $\bar{\psi}\Gamma D...D\psi$ constructions with the flavor and spin-parity of the relevant hadron: $\pi$, $K$ or $\eta$ in this instance. The virtue of this method is that excited state contaminations from the single-meson object contained within the meson-meson object are greatly reduced, and signals may then be obtained at earlier Euclidean times where statistical noise is typically lower.

We make use of the distillation method Peardon et al. (2009) that allows all of the Wick contractions specified by QCD to be efficiently obtained. Anisotropic lattices, having a finer spacing in time ($a_{t}$) than space ($a_{s}=a_{t}\xi$), are used Edwards et al. (2008); Lin et al. (2009). Table 1 provides some details of these lattices – the heaviest and lightest pion-mass lattices have been used previously to study many other channels Dudek et al. (2012, 2013); Briceno et al. (2015); Briceño et al. (2016); Briceno et al. (2017); Dudek et al. (2016); Moir et al. (2016); Woss and Thomas (2016); Briceno et al. (2018b); Woss et al. (2018); Wilson et al. (2015b); Cheung et al. (2016), while the two intermediate pion-mass lattices are being used for the first time in this calculation. We have previously reported on $\pi K$ scattering on the 391 MeV lattice in refs. Dudek et al. (2014); Wilson et al. (2015a), and we make use of these same spectra again.

To quote results in physical units, the $\Omega$-baryon mass is used to set the scale via ${a_{t}^{-1}=\frac{m_{\Omega}^{\mathrm{phys.}}}{a_{t}m_{\Omega}^{\mathrm{latt.}}}}$, and in this paper, for all but the $m_{\pi}\approx 391$ MeV lattice, we use a new computation using 64 distillation vectors. For the $m_{\pi}\approx 239$ MeV lattice, this results in a more accurate value which supersedes that presented in Ref. Wilson et al. (2015b). A more complete description of the methods used to arrive at the lattice QCD spectra is presented in Ref. Wilson et al. (2015a).

Finite-volume spectra — We show a representative sample of the spectra obtained in Fig. 1, presenting two rest-frame spectra and one moving frame spectrum from the lightest pion mass considered.111Irreps are labeled as $[ijk]\Lambda^{P}$, where 3-momentum $\vec{k}=\frac{2\pi}{L}(i,j,k)=[ijk]$, $\Lambda$ is the irrep, and $P$ is the parity if $\vec{k}=\vec{0}$.,222We provide the remaining irreps and all of the spectra for the two new lattices, along with tables of operators used, in the supplemental material. Spectra on the heaviest pion-mass lattice are given in Ref. Wilson et al. (2015b). In the region below $\eta K$ threshold, the qualitative pattern is the same across all 4 lattices.,333Spectra from the smallest mass lattice can be compared to those shown in Ref. Brett et al. (2018), which uses a subset of the same gauge configurations, the same quark mass parameters, but a different method for computing correlation functions. In $[000]A_{1}^{+}$, $S$-wave interactions dominate and we observe large shifts in energy away from expectations in a theory without $\pi K$ interactions – there is an energy level below threshold, and another significantly below the next non-interacting energy, $\pi_{[100]}K_{[100]}$. At the $\eta K$ threshold a level appears that has significant overlap onto only the $\eta_{[000]}K_{[000]}$ operator, shown in green in Fig. 1, and such a level persists across all the light-quark masses considered.

The $[000]T_{1}^{-}$ irrep is dominated by $P$-wave with negligibly small contributions from $F$-wave and higher. An isolated level appears well below the lowest non-interacting energy, likely indicating the presence of a resonance which may be narrow given the relatively small shift of the next level up in energy.

The denser spectrum in $[100]A_{1}$ reflects the contribution of both $S$ and $P$-wave amplitudes. This commonly occurs in unequal mass systems in moving frames since parity is not a good quantum number.

We can estimate the size of $D$-wave scattering in the elastic region, which can have an impact in many moving frame irreps, by considering the $[000]E^{+}$ irrep. Here a level is obtained at $a_{t}E_{\mathsf{cm}}=0.1699(3)$ on the smallest mass lattice, coincident with the expected non-interacting energy for $\pi_{[100]}K_{[100]}$. This energy corresponds to a negligibly small phase shift of $\delta_{2}^{\pi K}=0.26(56)^{\circ}$, and similarly small values apply on the other lattices, such that we may neglect $D$-wave and higher partial-waves hereafter.

In our analysis, we choose to consider only energies below the first inelastic threshold, which depending on the irrep and pion mass, is either $\pi\pi K$ or $\eta K$. This results in 28, 21, 18 and 36 energy levels to constrain $S$ and $P$-wave scattering amplitudes on lattices with pion masses of 239, 284, 329 and 391 MeV respectively.

Analysis — The relationship between the discrete spectrum in a finite volume, $\{E_{n}(L)\}$, and the infinite-volume scattering matrix, $\bm{t}(E)$, is given by the solutions of Lüscher’s determinant condition Luscher (1991); Rummukainen and Gottlieb (1995); Bedaque (2004); Kim et al. (2005); Fu (2012a); Leskovec and Prelovsek (2012); Gockeler et al. (2012); He et al. (2005); Lage et al. (2009); Bernard et al. (2011); Agadjanov et al. (2014); Doring et al. (2012); Hansen and Sharpe (2012); Briceno and Davoudi (2013); Guo et al. (2013); Briceno (2014),

[TABLE]

where $\bm{\mathcal{M}}(E,L)$ is a matrix of known functions in the space of partial-waves for each irrep,444This only works for energies below three-particle thresholds. For on-going efforts to remove this restriction, see Refs. Briceño et al. (2019, 2017); Hansen and Sharpe (2014, 2019). and $\rho=\frac{2k_{\mathsf{cm}}}{E}$. To overcome the dependence of each energy level on $t(E)$ for multiple partial-waves, we parameterize the energy dependence of the scattering amplitudes using a variety of forms which respect unitarity and which have sufficient freedom to describe the spectra. For a given parameterization, the parameter values are found which upon solving the above determinant equation give finite-volume spectra that best describe the lattice spectra555minimizing the correlated $\chi^{2}$ presented in Ref. Wilson et al. (2015a). A representative example in which $S$-wave and $P$-wave are parameterized is shown by the orange points and curves in Fig. 1.

To avoid bias, we consider a wide selection of scattering amplitude parameterizations that fall into four familiar categories: effective-range expansions, Breit-Wigners, $K$-matrices, as given in Ref. Wilson et al. (2015a) in Eqs. 9-13, and unitarized chiral perturbation theory (U$\chi$PT) Truong (1988); Dobado et al. (1990); Gomez Nicola and Pelaez (2002); Gomez Nicola et al. (2008). The $K$-matrix features the most flexibility, and we opt to use the Chew-Mandelstam phase-space in which a logarithm is generated from the known imaginary part from unitarity Wilson et al. (2015a). Our $K$-matrix forms respect $s$-channel unitarity, but do not include any features from scattering in the cross-channels (no “left-hand cuts”).

The U$\chi$PT amplitudes share the logarithm mentioned above associated with the $s$-channel cut, but they also contain perturbative features associated with the cross-channels. All the masses considered here are far from the chiral SU(3) symmetric point about which these amplitudes are expanded. The amplitudes would break unitarity without a unitarization step which, although not unique, results in U$\chi$PT amplitudes that respect unitarity perturbatively. We choose to apply the $\mathcal{O}(p^{4})$ SU(3) amplitudes specifically because they have been used in studying the pion mass dependence of $\pi K$ scattering in Ref. Nebreda and Pelaez. (2010).

The $S$ and $P$-wave phase-shifts of all considered amplitude parameterizations which can describe the finite-volume spectra with $\chi^{2}/N_{\mathrm{dof}}$ below $2.0$ are plotted in Fig. 2 – there are 14-17 per pion mass, and a complete list can be found in the supplemental material. The central curves are from a four parameter fit with a Breit-Wigner in $P$-wave and a two-parameter $K$-matrix, linear in $s=E_{\mathsf{cm}}^{2}$, in $S$-wave. This same choice is used to produce the orange curves in Fig. 1. Very little variation is seen between parameterizations – the amplitudes are well-determined and there is little sensitivity to the precise form used. As the pion mass reduces we see a clear trend towards the experimental phase-shifts. The striking difference in the $P$-wave amplitude between the $m_{\pi}\approx 391$ MeV and 327 MeV lattices is caused by the $K^{\star}$ changing from a bound state below $\pi K$ threshold to a resonant $K^{\star}$ above threshold.

The $S$-wave amplitude is presented in a different manner in Figure 3, via $k_{\mathsf{cm}}\,\cot\delta_{0}$, the quantity which has an effective range expansion $\frac{1}{a}+\tfrac{1}{2}rk_{\mathsf{cm}}^{2}+\ldots$, where $a$ is the scattering length and $r$ is the effective range. The discrete points shown reflect the $S$-wave for a fixed $P$-wave three-parameter $K$-matrix amplitude, with the uncertainty including a sampling of several amplitudes, while the curves show scattering-length and effective-range amplitudes with the same three-parameter $K$-matrix in $P$-wave. It is clear that for the three largest pion masses, the amplitude over the whole elastic region is acceptably well described by just a scattering-length, while at the smallest mass an additional effective-range term is required. A clear trend of decreasing $m_{\pi}a$ is observed with decreasing light-quark mass, which is qualitatively consistent with leading order chiral perturbation theory. Fig. 3 shows little evidence for a large effective range parameter that might signal the presence of a narrow resonance, nor for any strong enhancement below threshold that would be suggestive of important effects from an Adler zero in $t(E)$ Adler (1965a, b).

Poles — The singularity content of a scattering amplitude, considered as a function of complex energy, is closely connected to its spectroscopic content, with the presence of a pole, $t\sim c^{2}/\left(s_{0}-s\right)$ on an unphysical Riemann sheet, typically interpreted as being the most rigorous signal for a resonance. The pole position can be related to the mass $m_{R}$ and width $\Gamma_{R}$, $\sqrt{s_{0}}=m_{R}\pm i\Gamma_{R}/2$, and the residue gives access to the coupling, $c$.

All the $P$-wave amplitudes that we found were able to describe our finite-volume spectra feature a single pole close to the real axis that we summarize in Fig. 4. For the heaviest pion mass we considered, the pole is on the real axis, corresponding to a stable bound state, but otherwise it is off the real axis, corresponding to an unstable resonance. A smooth evolution is seen with an approximately flat effective coupling $|c|/|k|$ as a function of the pion mass. The scatter due to parameterization choice is observed to be quite modest, comparable to the size of the statistical uncertainty.

The $S$-wave amplitude presented in Figures 2 and 3 is superficially very simple: there is a rising phase shift, usually attributed to an attractive system, but no sharp features that signal the presence of a nearby pole or other singularity. This suggests that if any resonance pole is influencing this behavior, it must lie far into the complex plane. In order to determine such distant poles, it is necessary to consider the features of partial-wave amplitudes at complex $s$ which arise due to known properties of scattering, like crossing symmetry and unitarity. In elastic $\pi K$ scattering, the complex plane contains three cuts Mandelstam (1958); Oller and Oset (1999); *Cherry:2000ut; *Buettiker:2003pp; *Zheng:2003rw; *Zhou:2006wm; *DescotesGenon:2006uk; *Pelaez:2018qny: in addition to the $s$-channel unitarity cut, which is correctly handled in the finite-volume formalism, unitarity in the cross-channels leads to a circular cut at $|s|=(m_{K}^{2}-m_{\pi}^{2})$ and a left-hand cut that spans $-\infty<s<(m_{K}-m_{\pi})^{2}$. If these cuts are as close to the elastic scattering region as any hypothetical resonance pole, their effect must be accounted for if the pole is to be accurately determined. Of the amplitudes applied here, only U$\chi$PT has any contributions from the cross-channels, and the degree to which they are correctly handled has been debated Boglione and Pennington (1997); Nieves et al. (2002); Cavalcante and Sa Borges (2002); Gomez Nicola et al. (2008). When the U$\chi$PT amplitudes have their parameter freedom constrained by the finite-volume spectra presented above, a complex pole is found with a real energy around $m_{\pi}+m_{K}$ and a large imaginary part, not dissimilar to the experimental $\kappa$ resonance. In addition, many of the $K$-matrix forms we implement, which lack any explicit left-hand cut behavior, also feature poles at similar energies; however, some do not and many have other nearby poles. Even with precise information about the amplitude for real energies, the analytic continuation required to reach any pole is sufficiently large that a unique result is not found.

Summary — We have extracted $S$ and $P$-wave elastic isospin-$\frac{1}{2}$ $\pi K$ scattering amplitudes from lattice QCD spectra using Lüscher’s formalism at four light-quark masses, corresponding to pion masses between 239 and 391 MeV. The resulting amplitudes show a smooth evolution towards experimental data as the quark mass approaches its physical value. Continuing the $P$-wave amplitude to complex values of the energy, the lowest-lying vector $K^{\star}$ resonance appears as a pole singularity in a way consistent with the canonical picture of a compact quark-antiquark state that acquires a decay width by coupling to the meson-meson continuum. The $S$-wave amplitudes are well-determined for real energies, however the analytic continuation into the complex plane does not yield a unique result that we can interpret in terms of the $\kappa$ pole. Along with our previous study of the $\sigma$ Briceno et al. (2017), this provides motivation for future analyses that incorporate now-standard lattice QCD analysis techniques, namely Lüscher-like analysis of finite-volume spectra, and those in use in the amplitude analysis community, e.g. Roy-Steiner equations, which account for the known singularities due to cross-channel physics. In the current case, an input to such a calculation would be information about $\pi\pi\to K\overline{K}$ in several partial waves, which can be obtained in a lattice calculation using the generalization of the Lüscher formalism for coupled-channels Briceno et al. (2018b); Wilson et al. (2015b).

In closing, we believe this poses a timely challenge for the lattice QCD and amplitude-analysis communities to address jointly. In so doing, we will not only be able to acquire a detailed picture of the mysterious $\sigma$ and $\kappa$ resonances, but an understanding of the breaking of SU(3) flavor symmetry and thus the origin and nature of these resonances.

Supplemental material

.1 Spectra

The spectra obtained in this study are shown in Fig 5.

.2 Highlighted parameterizations

In the text we give three pairs of parameterizations special attention – a “reference” parameterization that features in Figs. 1 and 2 is given by a linear polynomial in $s=E_{\mathsf{cm}}^{2}$,

[TABLE]

where $\gamma_{i}$ are free parameters and $s_{\mathrm{thr.}}=(m_{\pi}+m_{K})^{2}$ for the $S$-wave amplitude, and a Breit-Wigner in $P$-wave given by Eq. 10 from Ref. Wilson et al. (2015a). The Chew-Mandelstam phase space $I$ from appendix B of Ref. Wilson et al. (2015a) is used in $S$-wave with the subtraction point fixed at $\pi K$ threshold. This has a logarithmic real part for real $s$ above threshold, and better properties when analytically continued compared to using the simpler phase space $I=-i\rho=-i2k_{\mathsf{cm}}/\sqrt{s}$, where

[TABLE]

The numerical results of using this pair of parameterizations across the four light-quark masses are given in Table 2.

The other two pairs of parameterizations highlighted are scattering lengths and effective ranges in $S$-wave, defined by $k_{\mathsf{cm}}^{2\ell+1}\cot\delta_{\ell}=\frac{1}{a_{\ell}}+\frac{1}{2}r_{\ell}k^{2}_{\mathsf{cm}}+\mathcal{O}(k^{4}_{\mathsf{cm}})$, with $\ell=0$, and a three-parameter “pole plus constant” $K$-matrix parameterization in $P$-wave,

[TABLE]

with a Chew-Mandelstam phase space $I(s)$, which is a specific case of Eq. 11 and Eq. 12 in Ref. Wilson et al. (2015a). We make use of these two pairs of amplitudes in producing Fig. 3. The effective range $S$-wave amplitudes do not have the logarithmic term of the Chew-Mandelstam phase space $I(s)$ and thus are relatively badly behaved deep in the complex plane.

.3 Explicit list of parameterizations

The amplitudes described above are used, and additionally the following,

[TABLE]

where $\gamma_{i}$, $c_{i}$ are free parameters. In Eq. 7, which is just a special case of Eq. 6, $s_{A}$ is fixed at the Adler zero position from leading order chiral perturbation theory. We also use the effective range expansion as described above. In Table 4 we refer to the leading order effective range $1/a_{\ell}$ as SL and the linear order effective range version as ER.

In $P$-wave we make use of a Breit-Wigner parameterization and $K$-matrix parameterizations,

[TABLE]

We summarize the results of fitting spectra using these amplitudes in Table 4. Several of the amplitudes used did not perform well at every value of light-quark masses – these are marked in italics. Some amplitudes also produced unphysical features such as nearby physical sheet singularities – several of these were also excluded and are marked in italics. Many more parameterisations were tested, we have chosen to show only parameterisations which were successful on at least one of the lattices.

The columns $I_{S}$ and $I_{P}$ indicate the phase spaces used – CM corresponds to Chew-Mandelstam, $-i\rho$ is the simple phase described above, BW indicates the Breit-Wigner formula was used. Up to arbitrary subtractions, the U$\chi$PT amplitudes have the same logarithm as the Chew-Mandelstam phase space, indicated by $\sim$CM.

.4 Operator lists

We use the operators listed in Table 5 in obtaining the energy levels shown in Fig. 5.