]>NUPHB14529S05503213(18)30358410.1016/j.nuclphysb.2018.12.018The Author(s)High Energy Physics – PhenomenologyFig. 1Left: R(5a) from Eq. (2.2) for all (t0a,t0′a) pairs on ensembles where multiple source times are employed. Right: tminplot from singleexponential fits (which ignore boundary effects) to the zeromomentum singlepion correlator over the range [tmin,tmax] for each t0 individually on the D200 ensemble. The pion mass extracted from the t0averaged correlator is shown by the error band.Fig. 1Fig. 2Top row: Comparison of K˜11−1(Ecm) (left) and δ1(Ecm) (right) between the C101 and D101 ensembles, which have the same parameters but different physical volumes. Bottom row: The same comparison for the N200 and N401 ensembles (white and gray markers respectively), which have (approximately) the same quark masses but different lattice spacings.Fig. 2Fig. 3Left: ratio of matrix elements for the O(a) counterterm over the leading matrix element for the D200 ensemble. Right: Thricesubtracted dispersive fit to lnQ3(s) on the D200 ensemble. The central 68% of bootstrap samples (thinned out for clarity) are shown for each individual point, together with the bestfit line.Fig. 3Fig. 4Form factor results shown with both the Gounaris–Sakurai (GS) parametrization and the thricesubtracted dispersive fit on the J303 ensemble (left) and D200 ensemble (right).Fig. 4Fig. 5Left: The twice and thricesubtracted dispersive fits on the D200 ensemble. Right: Comparison of form factor with thricesubtracted dispersive fits on both the N401 and N200 ensembles, which have similar quark masses and different lattice spacings.Fig. 5Fig. 6The resonance mass mρ (left) in physical units and the coupling gρππ (right) from Breit–Wigner fits to the scattering amplitude on all ensembles.Fig. 6Fig. 7K˜11−1 (left) and phase shift (right) on the C101 ensemble, together with the Breit–Wigner fit.Fig. 7Fig. 8Same as Fig. 7 for the D101 ensemble. One state in each of the A1+(3) and B2+(2) irreps which have very large errors have been removed from the plot.Fig. 8Fig. 9Top row: Same as Fig. 7 for the N401 ensemble. Bottom row: the timelike pion form factor (left) and the ratio employed in the thricesubtracted subtracted dispersive fit (right), which is also shown.Fig. 9Fig. 10Same as Fig. 9 for the N200 ensemble.Fig. 10Fig. 11Same as Fig. 9 for the D200 ensemble.Fig. 11Fig. 12Same as Fig. 9 for the J303 ensemble.Fig. 12Table 1Parameters of the CLS ensembles used in this work. The timelike pion form factor is not determined on the coarsest lattice spacing. After the ensemble ID in the first column, we list the gauge coupling, lattice spacing and dimensions, pseudoscalar meson masses, the separation between correlation function measurements in molecular dynamics units (MDU), the number of such measurements, and the minimum distance from an interpolator to a temporal boundary.Table 1IDβa [fm]L3 × Tmπ, mK [MeV]τmeas [MDU]Nconfmπtbnd
C1013.40.086483 × 96220, 47083002.5
D101643 × 12883032.3

N4013.460.076483 × 128280, 460162743.5

N2003.550.064483 × 128280, 46088543.0
D200643 × 128200, 48085582.1

J3033.70.050643 × 192260, 470163283.1
Table 2Parameters of the stochastic LapH implementation used in this work. (ρ,nρ) are the stout link smearing parameters, Nev the number of Laplacian eigenvectors, NR the number of independent noise sources, Nt0 the number of source times for fixed quark lines, and ND the total number of light quark Dirac matrix inversions per gauge configuration. Notation for the dilution scheme is explained in the text.Table 2ID(ρ,nρ)NevdilutionNRfixNRrelNt0ND
C101(0.1,20)392(TF,SF,LI16)F (TI8,SF,LI16)R6211408
D101928(TF,SF,LI16)F (TI8,SF,LI16)R6221792

N401(0.1,25)320(TF,SF,LI16)F (TI8,SF,LI16)R5221664

N200(0.1,36)192(TF,SF,LI8)F (TI8,SF,LI8)R522832
D200448(TF,SF,LI8)F (TI8,SF,LI8)R522832

J303(0.1,60)208(TF,SF,LI8)F (TI16,SF,LI8)R5231504
Table 3Finitevolume irreps Λ (second column) of the little group for various classes of total momenta Ptot = (2π/L)d (first column) employed here. The superscripts on the partial waves (ℓ) contributing to that irrep denote the number of multiple occurrences, while the ‘+’ indicates positive Gparity.Table 3dΛℓ
(0,0,0)T1u+1, 3, 52, …

(0,0,n)A1+1, 3, 52,…
E+1, 32, 53, …

(0,n,n)A1+1, 32, 53, …
B1+1, 32, 53 …
B2+1, 32, 53, …

(n,n,n)A1+1, 32, 52, …
E+1, 32, 54, …
Table 4Results of correlatedχ2 fits of the I = 1 elastic pwave ππ amplitude to Eq. (3.1). After the ensemble ID and the number of levels, the three subsequent columns show results from fits ignoring ℓ = 3 contributions. The remaining columns contain results from fits including the fwave contribution as described in the text.Table 4IDNlvlℓ=1 fitsℓ=1,3 fits
mρ/mπgρππχˆ2mρ/mπgρππmπ7a3×103χˆ2
D101433.366(15)6.19(10)2.513.370(15)6.23(10)−0.56(30)2.49
C101213.395(26)5.67(17)1.073.399(30)5.72(19)−0.18(26)1.11
N401192.717(16)5.84(12)1.642.721(16)5.88(13)−2.7(3.0)1.71
N200152.733(16)5.94(10)1.342.733(16)5.94(10)0.0(2.9)1.45
D200173.877(34)6.16(19)0.813.883(36)6.15(20)−0.61(94)0.84
J303183.089(25)6.30(17)0.753.096(25)6.32(17)−4.2(3.6)0.73
Table 5Results from twice and thricesubtracted dispersive fits (see Eq. (3.4)) to the form factor on each ensemble with the three finest lattice spacings. The large χˆ2 is due to the significant correlations between Ecm and lnQn.Table 5IDNlvln=2n=3
mπ2p1χˆ2mπ2p1mπ4p2χˆ2
N401180.1372(13)8.90.099(4)0.0144(7)5.5
N200150.1405(18)6.00.104(7)0.0143(12)3.6
D200170.0738(17)5.70.0674(16)0.0034(2)4.1
J303180.1129(15)2.70.101(4)0.0075(6)2.1
Table 6Results from the C101 ensemble.Table 6d2irrep.levelEcm/mπ(pcm/mπ)2(pcm/mπ)3cotδ1δ1
0T1u+03.148(23)1.477(37)2.99(40)31.0(2.5)
13.564(33)2.176(58)−3.34(42)136.1(4.6)

1A1+02.2864(61)0.3069(70)7.5(1.7)1.29(24)
13.318(32)1.752(53)1.18(31)63.0(5.0)
E+03.211(34)1.578(54)2.13(31)43.0(2.7)

2A1+02.4630(75)0.5165(93)6.4(1.0)3.32(45)
13.426(37)1.934(64)0.40(24)81.6(4.7)
23.821(26)2.650(50)−5.6(1.3)142.4(6.7)
B1+03.310(34)1.739(57)2.34(30)44.4(2.5)
13.783(22)2.579(42)−8.3(6.7)153(28)
B2+02.692(12)0.812(16)5.59(80)7.45(85)
13.409(40)1.905(68)−0.39(20)98.4(4.6)

3A1+02.5980(100)0.687(13)6.08(99)5.36(73)
12.999(14)1.249(21)2.5(1.3)29(11)
23.476(46)2.021(81)−1.12(16)111.4(3.9)
E+02.922(18)1.135(26)5.51(76)12.4(1.3)
13.570(56)2.186(100)−1.81(25)119.2(4.8)

4A1+02.710(15)0.836(21)5.9(1.5)7.4(1.5)
13.422(71)1.93(12)−0.44(43)99.3(9.7)
E+03.189(23)1.542(37)3.44(44)29.1(2.5)
13.593(48)2.228(87)−0.98(89)106(15)
Table 7Results from the D101 ensemble.Table 7d2irrep.levelEcm/mπ(pcm/mπ)2(pcm/mπ)3cotδ1δ1
0T1u+02.7894(88)0.945(12)4.23(43)12.3(1.0)
13.236(22)1.617(35)1.16(27)60.6(5.0)
23.646(30)2.324(55)−4.41(80)141.2(5.8)

1A1+02.1880(38)0.1968(42)4.6(1.6)1.09(30)
12.9555(99)1.184(15)4.42(47)16.2(1.5)
23.324(20)1.761(34)1.13(24)64.1(4.1)
33.564(24)2.175(42)−6.3(7.3)153(31)
43.674(20)2.374(36)−4.04(83)137.8(6.3)
E+02.912(12)1.121(17)4.32(47)15.4(1.3)
13.358(18)1.820(31)0.22(18)84.9(4.2)
23.740(26)2.496(49)−5.76(94)145.6(5.0)

2A1+02.3229(49)0.3490(57)4.7(1.2)2.49(55)
13.068(11)1.353(17)2.89(32)28.5(2.4)
23.297(19)1.717(32)0.15(19)86.2(4.8)
33.693(16)2.409(29)−2.6(1.2)125(13)
43.750(22)2.515(41)−9.0(4.2)156.0(8.7)
B1+02.988(13)1.233(19)3.55(30)21.1(1.2)
13.149(11)1.479(18)1.2(3.6)56(60)
23.535(30)2.124(54)−2.19(21)125.2(3.5)
B2+02.4392(51)0.4875(62)6.82(97)2.86(37)
13.114(11)1.424(18)2.73(24)31.9(2.0)
23.346(20)1.798(33)−0.03(20)90.8(4.8)
33.620(15)2.276(27)−13(12)165(14)
43.779(24)2.569(45)−12.2(4.3)161.4(6.1)

3A1+02.4411(78)0.4897(95)9.3(5.9)2.1(1.1)
12.6267(67)0.7250(88)4.2(2.5)8.4(4.2)
23.281(24)1.692(40)1.00(16)65.6(2.7)
33.742(19)2.501(35)−9.7(4.5)157.9(9.6)
43.878(22)2.760(44)−9.0(5.8)152.9(5.6)
53.938(28)2.876(55)−9.4(1.5)152.6(4.3)
E+02.6031(68)0.6940(89)6.04(69)5.47(55)
13.369(21)1.838(36)1.16(15)65.1(2.3)
23.697(16)2.417(30)−9(12)157(42)
33.862(16)2.728(31)2(10)69(65)

4A1+02.494(15)0.555(18)2.14(70)11.0(2.9)
13.171(13)1.514(21)2.34(18)38.5(1.7)
23.566(20)2.178(35)−1.62(23)116.7(3.6)
33.959(19)2.919(38)−8.0(1.9)148.2(6.5)
E+02.7979(94)0.957(13)4.39(68)12.0(1.6)
13.222(13)1.595(21)3.27(30)31.6(2.0)
23.406(16)1.900(28)−0.6(1.1)102(22)
33.638(19)2.309(34)−2.47(28)125.2(3.4)
43.980(21)2.959(43)−9.9(2.0)152.9(5.0)
Table 8Results from the N401 ensemble. The form factor is omitted for a single level in the Λ(d2)=A1+(4) irrep for which a plateau could not be identified.Table 8d2irrep.levelEcm/mπ(pcm/mπ)2(pcm/mπ)3cotδ1δ1Fπ
0T1u+02.714(22)0.841(30)0.281(64)70.0(3.3)17.05(69)
13.205(13)1.568(21)−5.18(57)159.2(2.4)3.518(53)

1A1+02.2248(24)0.2375(27)2.89(19)2.29(12)3.019(29)
12.798(22)0.957(31)−0.481(39)117.2(2.9)12.42(77)
E+02.775(20)0.925(28)0.269(48)73.2(2.2)13.92(99)

2A1+02.3597(55)0.3920(65)1.98(18)7.05(47)4.257(53)
12.836(23)1.010(33)−0.958(48)133.3(2.9)10.38(60)
B1+02.748(21)0.888(28)0.094(38)83.6(2.3)15.8(1.1)
B2+02.509(11)0.574(14)1.55(22)15.7(1.6)6.49(15)
12.907(22)1.113(33)−1.316(79)138.2(2.8)8.26(30)

3A1+02.4699(91)0.525(11)1.87(23)11.5(1.0)6.07(12)
12.806(15)0.969(20)−0.01(72)90(36)19.8(3.3)
22.950(34)1.176(51)−2.03(29)147.8(5.2)6.32(60)
E+02.627(31)0.725(41)0.96(26)32.7(4.6)11.26(98)
13.041(33)1.313(50)−2.99(38)153.3(4.1)5.78(21)

4A1+02.582(26)0.666(34)2.25(91)13.6(3.8)12.3(1.0)
12.834(23)1.008(32)−2.40(27)157.1(3.2)–
E+02.722(39)0.853(53)0.23(11)73.5(6.0)16.5(1.4)
13.194(16)1.550(26)−4.24(81)155.5(4.5)4.360(92)
Table 9Results from the N200 ensemble.Table 9d2irrep.levelEcm/mπ(pcm/mπ)2(pcm/mπ)3cotδ1δ1Fπ
0T1u+02.749(27)0.889(37)−0.077(41)95.3(3.1)15.3(1.1)

1A1+02.2654(24)0.2830(27)2.78(10)3.104(68)3.308(36)
12.874(22)1.065(32)−1.176(36)136.9(2.1)8.88(42)
E+02.756(35)0.898(48)−0.179(40)101.8(3.5)16.1(1.5)

2A1+02.4215(66)0.4659(80)2.22(16)8.14(36)4.94(12)
12.913(29)1.121(43)−2.059(96)150.0(2.6)7.04(48)
B1+02.676(51)0.791(68)−0.311(32)113.9(4.8)16.7(1.5)
B2+02.613(10)0.707(14)1.20(10)26.4(1.3)10.87(47)
13.089(24)1.385(37)−2.99(22)151.4(2.6)4.67(16)

3A1+02.482(18)0.540(22)1.26(17)17.5(1.2)7.29(21)
12.959(39)1.189(58)−2.51(52)152.7(6.8)5.85(45)
23.176(20)1.522(32)−5.2(1.4)160.3(5.0)3.60(51)
E+02.56(11)0.63(13)0.14(21)74(16)12.4(1.7)

4A1+02.582(69)0.667(89)1.22(66)24.0(6.4)10.0(1.9)
E+02.778(96)0.93(13)−0.10(15)96(11)14.8(3.6)
Table 10Results from the D200 ensemble.Table 10d2irrep.levelEcm/mπ(pcm/mπ)2(pcm/mπ)3cotδ1δ1Fπ
0T1u+03.444(22)1.965(38)6.39(86)23.3(2.3)4.655(98)
13.980(75)2.96(15)−2.5(1.3)116(13)7.55(37)

1A1+02.3385(45)0.3672(53)10.1(1.3)1.26(14)1.656(16)
13.633(30)2.300(55)2.53(60)54.1(5.4)6.81(22)
E+03.548(32)2.147(57)4.64(68)34.1(2.8)5.81(18)

2A1+02.5386(65)0.6111(83)9.0(1.2)3.02(33)1.908(18)
13.748(40)2.512(75)1.05(47)75.3(5.6)7.77(27)
B1+03.637(26)2.308(48)4.32(41)39.1(1.9)6.81(21)
B2+02.851(11)1.033(16)8.5(1.3)7.06(94)2.369(31)
13.794(43)2.598(81)0.36(43)85.1(5.6)8.50(19)

3A1+02.701(11)0.824(14)11.2(2.5)3.83(74)2.056(21)
13.213(15)1.580(24)4.6(4.0)23(16)4.12(67)
23.886(91)2.78(18)−0.83(52)100.2(7.1)8.79(38)
E+03.128(19)1.446(29)8.7(1.6)11.4(1.7)3.137(40)
13.983(84)2.97(17)−1.79(62)109.3(7.6)8.13(64)

4A1+02.839(19)1.015(27)14.1(6.5)4.2(1.5)2.354(27)
E+03.499(37)2.061(64)8.4(3.2)19.3(5.4)4.94(16)
Table 11Results from the J303 ensemble.Table 11d2irrep.levelEcm/mπ(pcm/mπ)2(pcm/mπ)3cotδ1δ1Fπ
0T1u+03.044(28)1.317(42)0.50(11)71.6(3.0)11.36(30)

1A1+02.3048(48)0.3280(55)4.32(36)2.49(15)2.460(25)
13.211(29)1.578(47)−0.802(87)112.0(3.0)9.12(41)
E+03.082(43)1.374(66)0.32(14)78.6(4.0)11.51(46)

2A1+02.4890(79)0.5488(98)3.95(40)5.87(43)3.291(40)
13.236(48)1.617(77)−1.70(13)129.6(4.2)8.10(67)
B1+03.001(64)1.251(96)−0.05(13)91.9(5.4)11.63(64)
B2+02.755(27)0.898(38)2.66(65)17.7(3.0)5.55(12)
13.199(83)1.56(13)−3.48(83)150.8(8.5)7.9(1.2)

3A1+02.599(19)0.688(25)2.90(51)11.1(1.3)4.197(67)
13.165(36)1.505(57)−0.7(1.6)109(40)10.0(2.8)
23.30(11)1.71(18)−5.6(3.1)158(30)7.5(4.1)
E+02.936(28)1.155(42)1.92(34)32.9(3.3)9.12(41)
13.550(61)2.15(11)−5.22(85)148.9(5.8)3.89(18)

4A1+02.694(30)0.814(41)2.48(59)16.5(2.5)5.02(12)
13.281(73)1.69(12)−2.95(46)143.3(7.1)7.03(78)
E+03.049(64)1.324(98)0.37(24)76.3(6.9)11.94(54)
13.645(46)2.321(83)−27(38)173(84)2.598(91)
The I = 1 pion–pion scattering amplitude and timelike pion form factor from Nf = 2 + 1 lattice QCDChristianAndersenaJohnBulavaa⁎bulava@cp3.sdu.dkBenHörzbColinMorningstarcaCP3Origins, University of Southern Denmark, Campusvej 55, 5230 Odense M, DenmarkCP3OriginsUniversity of Southern DenmarkCampusvej 55Odense M5230DenmarkbPRISMA Cluster of Excellence and Institut für Kernphysik, University of Mainz, JohannJoachimBecherWeg 45, 55099 Mainz, GermanyPRISMA Cluster of ExcellenceInstitut für KernphysikUniversity of MainzJohannJoachimBecherWeg 45Mainz55099GermanycDepartment of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USADepartment of PhysicsCarnegie Mellon UniversityPittsburghPA15213USA⁎Corresponding author.Editor: HongJian HeAbstractThe elastic I=1 pwave ππ scattering amplitude is calculated together with the isovector timelike pion form factor using lattice QCD with Nf=2+1 dynamical quark flavors. Wilson clover ensembles generated by the Coordinated Lattice Simulations (CLS) initiative are employed at four lattice spacings down to a=0.05fm, several pion masses down to mπ=200MeV, and spatial volumes of extent L=3.1–5.5fm. The set of measurements on these ensembles, which is publicly available, enables an investigation of systematic errors due to the finite lattice spacing and spatial volume. The ππ scattering amplitude is fit on each ensemble by a Breit–Wigner resonance lineshape, while the form factor is described better by a thricesubtracted dispersion relation than the Gounaris–Sakurai parametrization.1IntroductionLattice QCD calculations of resonant twohadron scattering amplitudes have improved markedly in recent years thanks to algorithmic advances [1,2] and increased computing resources.11For recent reviews of the interplay between lattice QCD calculations and current computer architectures, see Refs. [3,4]. Many calculations of the elastic ππ amplitude in the vicinity of the ρ(770) exhibit sufficient statistical precision and energy resolution to determine the resonance parameters [5–15], while a few Kπ calculations similarly map out the K⁎(892) [16–19]. First coupled channel results have also appeared in Refs. [18,20–23] for the a0(980), f0(980), and Ds0⁎(2317) resonances. Resonant meson–meson amplitudes involving an external current have also been calculated in Refs. [10,24,25]. Compared to the meson–meson sector, calculations of resonant meson–baryon amplitudes are currently less advanced [26–28]. A recent review of lattice calculations of scattering amplitudes can be found in Ref. [29].The improvement in these calculations suggests that the quarkmass dependence of such amplitudes may be investigated quantitatively, providing valuable input to effective theories of lowlying hadron resonances as well as numbers at the physical point relevant for experiment. In order to obtain reliable results however, various systematic errors must be controlled. These include effects due to the finite lattice spacing and spatial volume inherent in lattice QCD simulations, as well as systematics in the calculation of finitevolume twohadron energies and matrix elements from which the amplitudes are determined.While lattice spacing effects are assessed in the usual way, the treatment of finite volume effects is more subtle. Since realtime scattering amplitudes cannot be naively calculated from Euclideantime lattice QCD simulations [30], the method proposed by Lüscher [31] is employed to infer twototwo hadron scattering amplitudes from shifts of finitevolume twohadron energies from their noninteracting values. This approach has been generalized to nonzero total momenta [32,33], nonzero spin [34–39], multiple coupled scattering channels [37,40,41], and amplitudes with an external current [42–47]. Extending this approach above threehadron thresholds has proven difficult and been applied to Monte Carlo lattice data only in a toy scalar field theory [48]. It is, however, under active development [49–60]. Recentlyproposed alternatives to this finitevolume formalism for total decay rates can be found in Refs. [61,62].The relation discussed above between finitevolume energy shifts and realtime twototwo scattering amplitudes can be written as [35](1.1)det[K˜−1(Ecm)−B(Λ,d)(Ecm)]=0 where Ecm is the finitevolume twohadron energy in the centerofmass frame, K˜ is proportional to the infinitevolume Kmatrix, and B is a known matrix encoding the effect of the finite volume. This determinant is block diagonalized so that Eq. (1.1) describes finitevolume energies in a single irreducible representation (irrep) Λ of the little group for a particular class of total momenta d=(L/2π)Ptot, where d is a vector of integers. The determinant is taken over total angular momentum (J), total spin (S), all coupled twohadron scattering channels, and an index enumerating the possibly multiple occurrences of a partial wave in irrep Λ.The determinant condition in Eq. (1.1) holds up to corrections which are exponentially suppressed in the spatial extent L. However, unlike finite volume corrections to singlehadron observables, the falloff of these residual exponential finite volume effects may in principle be different than mπ. The ‘rule of thumb’ mπL≳4 which is usually applied in singlehadron calculations to ensure that finitevolume effects are at the percent level must be reinvestigated in the context of scattering amplitudes. There exists therefore a hierarchy of finite volume effects: those described by Eq. (1.1) are polynomial in L−1 (and constitute the ‘signal’) while Eq. (1.1) holds only up to (unwanted) terms exponential in L.As a benchmark amplitude suitable for an investigation of these systematic effects, we consider here the elastic I=1 pwave pion–pion scattering amplitude relevant for the ρ(770). In order to extrapolate to the physical point and continuum, a range of pion masses mπ=200–280MeV and lattice spacings a=0.050–0.086fm are employed. Such extrapolations have been performed recently using chiral perturbation theory and its extensions [63–74] and treat all scattering data simultaneously. As these extrapolations are somewhat involved and have not yet treated cutoff effects, we present the determination of the amplitudes only in this work and leave extrapolation to the physical quark masses and continuum for the future. Nonetheless, in our results the residual finite volume and cutoff effects are evidently small compared to the statistical errors.We calculate scattering amplitudes over an energy range Ecm∈[2mπ,Emax]. Since Eq. (1.1) applies below n>2 hadron thresholds, Emax is reduced as mπ is lowered. Because of the chiral trajectory employed in this work (which is discussed in Sec. 2.1), for the heaviest quark masses the lowest inelastic threshold is K¯K, while the lowest n>2 hadron threshold is 4π. Although levels in the range 2mK<Ecm<4mπ could be treated using Eq. (1.1) with a coupledchannel Kmatrix, we nonetheless impose a restriction to elastic scattering, namely Emax=min(4mπ,2mK).In addition to this ππ scattering amplitude, we also calculate the I=1 timelike pion form factor, which encodes the coupling of an external (timelike) photon to two pions in an isovector configuration. Phenomenologically, it can be extracted from e+e−→hadrons and hadronic τdecays [75] and is of particular relevance for the hadronic vacuum polarization (HVP), a leading source of theoretical uncertainty in the anomalous magnetic moment of the muon (g−2)μ [76,77]. Using the optical theorem, the imaginary part of the HVP can be related to(1.2)Rhad(s)=σ(e+e−→hadrons)/4παem(s)23s, where σ(e+e−→hadrons) is the total cross section, αem the electromagnetic coupling, and s=Ecm2 the usual Mandelstam variable. In the elastic region Rhad is given by the twopion contribution(1.3)Rhad(s)=14(1−4mπ2s)32Fπ(s)2, which contains the timelike pion form factor Fπ(s). The phase of this form factor is fixed by Watson's theorem, so we are interested in the amplitude only here. Furthermore, we work in the isospin limit, where electromagnetic interactions are ignored and mu=md. Because of this, the elastic region persists up to either s=4mK2 or s=16mπ2.Although a precise determination of this form factor is phenomenologically desirable, there exists only the pioneering determination of Ref. [10] which employs a single lattice spacing, heavier quark masses, and a (single) smaller physical volume than this work. It is therefore imperative to also investigate lattice spacing and finite volume effects for this quantity, the former of which may be affected by renormalization and O(a)improvement of the electromagnetic current.In addition to its phenomenological impact, the timelike pion form factor is an important stepping stone toward more complicated resonance photoproduction amplitudes. Such amplitudes are relevant for ongoing and future experiments which photoproduce resonances. An additional step in this direction is the πγ→ππ amplitude studied using lattice QCD in Refs. [24,25]. However, the timelike pion form factor calculated here does not require disconnected flavorsinglet Wick contractions, which are ignored in Refs. [24,25].Preliminary work toward the results reported here is found in Ref. [78]. The remainder of this paper is organized as follows. For completeness we review the gauge field ensembles, methods for calculating finitevolume twopion energies and matrix elements, and their relation to infinitevolume scattering amplitudes in Sec. 2. Results are given in Sec. 3 and conclusions in Sec. 4.2Lattice QCD methodsThe subset of CLS ensembles used in this work is discussed in Sec. 2.1 and application of the stochastic LapH method for alltoall quark propagation in Sec. 2.2. The analysis strategy used to extract the required finite volume energies and matrix elements from temporal correlation functions is contained in Sec. 2.3, while the relation between finitevolume quantities and infinitevolume scattering amplitudes is given in Sec. 2.4.2.1Gauge field ensemblesThe ensembles of gauge field configurations employed here are from the Coordinated Lattice Simulations (CLS) initiative and are presented in Refs. [79,80]. They employ the treelevel improved Lüscher–Weisz gauge action [81] and nonperturbatively O(a)improved Wilson fermions [82]. Open boundary conditions [83] are implemented in the temporal direction. Although these boundary conditions were adopted to reduce autocorrelation times of the global topological charge, they also influence finitetemporalextent effects in temporal correlation functions.Contributions to twohadron correlation functions where the hadrons propagate in opposite temporal directions, which for identical particles and zero total momentum yield a constant in time [84–87], are present with periodic boundary conditions but absent in this setup. Therefore, for large temporal extent T and if both interpolators are far from the boundaries, all twopoint correlation functions with open temporal boundary conditions have the form(2.1)limT→∞t0,(T−tf)→∞CT(t0,tf)=C(tf−t0)×{1+O(e−E0tbnd)}, where CT(t0,tf)=〈O(tf)O¯(t0)〉T is the correlator with open boundaries of extent T, C(t)=〈O(t)O¯(0)〉 the correlator in the T→∞ limit, E0 the lightest state with vacuum quantum numbers and tbnd=min(t0,T−tf) the minimal distance from an interpolator to the temporal boundaries. Since presumably E0≈2mπ, if mπtbnd≳2 then the exponential corrections in Eq. (2.1) are parametrically similar to exponentially suppressed finitevolume effects in singlehadron energies.While correlation functions are affected by temporal boundary conditions, the transfer matrix (and therefore also the spectrum) is unaffected. Although having an interpolating operator near a temporal boundary does not change its quantum numbers, we are after excited states and employ generalized eigenvalue methods requiring hermitian correlation matrices. The source and sink interpolators therefore must not be significantly affected by the temporal boundaries in order to maintain hermiticity. To this end, we always choose a minimum distance to the boundary (tbnd) of at least mπtbnd≳2. As in Ref. [6], our resulting insensitivity to finiteT effects can be demonstrated using the singlepion correlation function. Fitting this correlation function to a single exponential ignores contributions from the temporal boundaries. Fits of this type on a single ensemble are shown in Fig. 1, where the fitted energy is shown to be insensitive to the source interpolator position t0. Insensitivity to t0 in our most precisely determined correlation function suggests that temporal boundary effects may be neglected in subsequent fits.Another measure of finiteT effects is the ratio(2.2)Rt0,t0′(t)=CT(t0,t+t0)CT(t0′,t+t0′) which under the asymptotic assumptions of Eq. (2.1) receives corrections to unity of O(e−E0tbnd). R(5a) is also shown in Fig. 1 for various (t0,t0′) pairs on all ensembles with multiple source times. This ratio shows significant deviations from unity for mπtbnd≲3, despite no observable difference in the fitted energies. While the deviation of Rt0,t0′(t) from unity in the singlepion correlator suggests that averaging over source times may affect the hermiticity of correlation matrices, such deviations are not visible in twopion correlation functions. For the D200 ensemble the ratio with (t0/a,t0′/a)=(32,52), shown for the pion in Fig. 1 left panel as the leftmost point, is consistent with unity for the singleρ meson correlator at rest and the twopion correlator (each with a single unit of momentum) in the same channel.While we omit a complete discussion of algorithmic details used in configuration generation, some aspects are relevant for the analysis of correlation functions measured on these ensembles. The CLS ensembles employ twistedmass reweighting [88] for the degenerate light quark doublet and use the RHMC algorithm [89] for the strange quark determinant. One reweighting factor (W0) is used to change the light quark action to clover Wilson, while another (W1) corrects for the RHMC approximation. Efficient evaluation of these reweighting factors is discussed in Ref. [79]. Measurements of primary observables must be multiplied by the corresponding reweighting factors on each configuration according to(2.3)〈A〉=〈AW〉W〈W〉W where W=W0W1 and 〈…〉W denotes an ensemble average with respect to the simulated action. The denominator of Eq. (2.3) must also be taken into account in any resampling procedure used to estimate statistical errors or covariances. If both the twisted mass parameter and the range and degree of the rational approximation are chosen appropriately, these reweighting factors are typically close to unity. However, we observe anomalously large fluctuations in the reweighted zeromomentum singlepion correlator for a single source time on each of the C101 and D101 ensembles. These ensembles have the lightest quark mass at the coarsest lattice spacing and large fluctuations may indicate an inefficient choice of the simulated action. Data on these two source times are removed from the final analysis and are not included in Table 1.There are several possibilities for the trajectory of ms as ml=mu=md is lowered toward its physical value. Quark masses on the CLS ensembles employed here are tuned to lie on a chiral trajectory with trMq=const., where Mq is the bare quark mass matrix Mq=diag(mu,md,ms), in order to reduce the quark mass dependence of certain renormalized quantities. An additional chiral trajectory in which ms=const. is presented in Ref. [80]. While it is interesting to investigate the quarkmass dependence of scattering amplitudes on both chiral trajectories, we present results on the trMq=const. trajectory only.As discussed in Ref. [90], fixing the trace of the bare mass matrix is not equivalent to fixing the sum of renormalized quark masses. There a Taylor expansion is employed to slightly shift the quark masses in order to satisfy ϕ4=8t0(mK2+12mπ2)=const., which is not performed here. At the coarsest lattice spacing, imposing trMq=const. results in deviations of less than 5% of trMR/(trMR)symm (where the trace in the denominator is evaluated at the symmetric point mu=md=ms) from unity at the lightest pion masses considered in Ref. [90]. This small deviation from our desired chiral trajectory presumably has little effect on the observables considered here.Properties of the CLS ensembles used in this work are given in Table 1, which also contains τmeas, the separation in molecular dynamics units (MDU) between our measurements of hadronic correlation functions, and tbnd, the minimum distance from an interpolator to a temporal boundary. The timelike pion form factor is not determined on the coarsest lattice spacing. A more precise scale determination can be found in Tab. 3 of Ref. [90], while pseudoscalar meson masses and decay constants can be found in Tab. 2 of that work.As discussed above, open temporal boundary conditions are employed to decrease the integrated autocorrelation time of the global topological charge. However, there is still a significant amount of autocorrelation present in some observables on the CLS ensembles. A method to estimate statistical errors in the presence of large autocorrelations is outlined in Refs. [91,92]. This involves propagating the errors linearly, a method which may not be suitable for our purposes given the nonlinear nature of the Bmatrix elements given in Eq. (1.1) and discussed further in Sec. 2.3. Therefore, we simply ‘bin’ our correlator measurements and employ the bootstrap procedure with NB=800 bootstrap samples. Although no statistically significant autocorrelations are observed in any of our correlation functions, the largest integrated autocorrelation times (τint) measured on these ensembles span the range τint≈30–150 for β=3.4–3.7, respectively [79].2.2Correlation function constructionSince we employ twopion interpolators in which each pion is projected to definite momentum, quark propagators between all space–time points are required. We employ the stochastic LapH method to estimate such alltoall propagators and efficiently construct correlation functions [2]. Based on Ref. [1], this method endeavors to make alltoall propagators tractable by considering quark propagation between a lowdimensional subspace defined by the lowest Nev modes of the threedimensional gaugecovariant Laplace operator, hereafter referred to as the ‘LapH subspace’. This projection is a form of quark smearing, with an approximately Gaussian spatial profile and width controlled by the Nevth eigenvalue. In order to maintain a constant width, Nev must be scaled proportionally to the spatial volume.This smearing procedure enables more efficient stochastic estimation schemes by employing noisy combinations of Laplacian eigenvectors. It was determined in Ref. [2] that (at least for the range of spatial volumes considered there) with a moderate level of dilution the quality of the stochastic estimator remains constant as the volume is increased while maintaining a fixed number of dilution projectors. This work further demonstrates that the quality of the stochastic LapH estimator does not degrade for even larger volumes. Without significantly increasing the number of dilution projectors, we obtain precise results for scattering amplitudes with stochastic LapH on spatial volumes up to V=(5.5fm)3.In the stochastic LapH framework, NR stochastic sources {ρr} are introduced in time, spin, and Laplacian eigenvector indices. These sources are diluted by specifying Ndil complete orthogonal dilution projectors {Pb} so that an unbiased estimator of the smearedsmeared alltoall quark propagator is furnished by(2.4)Q(y,x)≈1NR∑r=1NR∑b=1Ndilφrb(y)ϱrb†(x), where ϱrb=VsPbρr is the smeared stochastic source, φrb=SQϱrb the smeared sink, Q the quark propagator, and S=VsVs† the smearing operator which projects onto the LapH subspace. To date, only schemes where dilution in each of these indices is done independently have been employed. A common strategy is to interlace n dilution projectors uniformly (denoted In) in the index in question. The ‘full’ dilution limit (denoted ‘F’) is recovered if n is equal to the total dimension of the index. Full specification of a dilution scheme therefore specifies a prescription in each of time, spin, and Laplacian eigenvector space. For example (TF,SF,LI8) refers to full dilution in time and spin, and eight dilution projectors interlaced uniformly among the Laplacian eigenvectors.As discussed in Ref. [2], it is typically beneficial to employ different dilution schemes for ‘fixed’ quark propagators (denoted by the subscript ‘F’), where x0≠y0 and ‘relative’ quark propagators (denoted ‘R’) where x0=y0. We therefore employ either full or interlace dilution in time, full dilution in spin, and interlace dilution in eigenvector space. The dilution scheme and other parameters of the stochastic LapH algorithm employed here are given in Table 2.Table 2 also contains information on the LapH subspace and thus the smearing operator S applied to quark fields in our interpolating operators. Before calculating eigenvectors, the gauge link field entering the covariant 3D Laplace operator is stout smeared [93]. The stout smearing parameters (ρ,nρ) together with the number of retained eigenvectors Nev therefore define our smearing scheme. We maintain an approximately constant physical linksmearing radius (rlink/a)2=ρnρ by tuning nρ appropriately. The quark smearing procedure is defined by retaining all eigenvectors with eigenvalue λ≲(aσs)2, where σs=1GeV. As the physical volume (V) is increased the number of eigenvectors must be scaled as Nev∝V. The stout smearing parameters and Nev are given in Table 2.We employ interpolating operators with light quarks only, since we calculate elastic pion–pion scattering amplitudes. The number of required light quark Dirac matrix inversions per configuration, denoted ND, is also given in Table 2. Our treatment of alltoall propagators enables us to efficiently evaluate all required Wick contractions involving twopion and singleρ interpolators, which are enumerated in Ref. [2]. An unbiased estimator results only if each quark line in a diagram employs independent stochastic sources. As discussed in Ref. [94], in each diagram we typically average over some number of multiple noise ‘orderings’, i.e. different permutations of the NR available quark lines.The correlation functions used in pion–pion scattering require smeared quark fields only. However, correlation functions for the timelike pion form factor contain the unsmeared vector current operator. These current correlation functions are easily constructed in the stochastic LapH framework although they employ quark fields which are not projected onto the LapH subspace.As done in Ref. [95], by exploiting γ5hermiticity it can be ensured that quark fields in the vector current bilinear are always unsmeared sinks ϕrb=Qϱrb. This motivates the construction of ‘current sinks’ defined as(2.5)Jrb;r′b′(d,Λ)(t)=∑x,yϕrb†(x)Γ(d,Λ)(x,y)ϕr′b′(y), where t=x0=y0 and Γ(d,Λ) denotes projection onto an irreducible representation (irrep) Λ of the little group of total momentum d. J(d,Λ)(t) has two noise/dilution indices and can therefore be employed in the correlation function construction procedure of Ref. [2] exactly as a smeared ρmeson interpolator.As suggested by Eq. (2.4), in order to save disk space the quark sinks are typically projected onto the LapH subspace before they are written to disk. However, the current functions of Eq. (2.5) must be constructed from unprojected sinks. For fixed quark lines the {ϕrb} must be kept in memory until calculation of J(d,Λ)(t) is complete. After construction of the current functions, the quark sinks are smeared and written to disk in the usual way.The calculation of the Laplacian eigenvectors is performed using a variant of the thick restarted Lanczos method [96], which entails global reorthogonalizations of the Krylov subspace. These reorthogonalizations scale poorly with Nev, so that as L is increased calculation of the Laplacian eigenvectors will eventually dominate the computational cost. However, for the L≲5.5fm volumes considered here the Dirac matrix inversions are still most computationally intensive.We perform these Dirac matrix inversions using the efficient DFL_SAP_GCR solver in the openQCD software suite.22http://luscher.web.cern.ch/luscher/openQCD/. In summation, our workflow consists of three main tasks: (1) Dirac matrix inversion, (2) hadron source/sink construction, and (3) formation of correlation functions. Due to their large storage footprint, the Laplacian eigenvectors are computed first in task 1 and not saved to disk. They are then recomputed during task 2, which is implemented entirely in 3D. Task 3 then no longer requires any latticewide objects and is simply tensor contraction of noisedilution indices.These different tasks are typically performed on different computer architectures, but a rough breakdown of the relative cost is 70–80% for the Dirac matrix inversions, 20–26% for task 2, and 1–5% for task 3. In total, 1–3% of the total for these three tasks is spent on calculating Laplacian eigenvectors.2.3Finitevolume energies and matrix elementsWe consider all elastic energy levels in isovector irreps where the JPG=1−+ partial wave is the leading contribution up to total momentum d2≤4, which are tabulated in Table 3. To calculate the energies, we follow the procedure of Refs. [6,16], which is outlined below.Outside the resonance region, interacting finitevolume twopion energies are close to their noninteracting values, while for levels with Ecm near mρ these gaps are larger. To exploit the small differences outside the resonance region and to treat all energies in a unified manner, we employ the ratio fits described in Ref. [6]. Using this method, we construct ratios(2.6)Rn(t)=Cˆn(t)Cπ(p12,t)Cπ(p22,t),Cˆn=(vn(t0,td),C(t)vn(t0,td)) where (p1,p2) are momenta of the constituent pions in the nearest noninteracting level and Cπ(p2,t) is a singlepion correlation function with momentum p2. The vector vn(t0,td) is a generalized eigenvector of the correlator matrix C(t) solving the generalized eigenvalue problem (GEVP) C(td)vn=λnC(t0)vn [97,98]. The {vn} are used to define the diagonal correlators Cˆn(t) between operators with optimal overlap onto the nth level, and are determined for a single (t0,td) only. The fitted energies vary little as these diagonalization times, as well as the operator basis, are varied.The difference ΔEn between an energy and its closest noninteracting ππ counterpart is extracted directly using singleexponential fits to the ratio in Eq. (2.6). Alternatively, the interacting energy may be obtained from single or twoexponential fits to Cˆn directly. All of these correlatedχ2 fits are performed over some time range [tmin,tmax], the variation of which should not affect the fitted energies for asymptotically large t. Energies obtained from ratio, single, and twoexponential fits all typically depend little on tmax, while ratio fits typically exhibit a reduced dependence on tmin as well. However, the excited state contamination in ratio fits may be nonmonotonic leading to ‘bumps’ in tmin plots. As an important consistency check, we check agreement of the energies obtained from these three types of fits, different (t0,td) combinations, and GEVP operator sets.Our fit ranges are chosen conservatively so that the systematic errors discussed above due to the GEVP and fit ranges are smaller than the statistical ones. This is demonstrated using extensive comparisons of tminplots for different fit types, (t0,td) choices, and GEVP operator bases, similar to Refs. [6,16]. Bootstrap resamples of all reweighted correlation functions are publicly available in HDF5 format,33https://doi.org/10.5281/zenodo.1341045. as is a python Juypter notebook44https://github.com/ebatz/jupan. which performs the entire analysis chain. This tool not only provides an interface to view systematics related to our choices of fitting procedure, fitting ranges, and GEVP, but also enables direct access to all results at each step. Generally, in physical units we take (t0,td)≈(0.5,0.9)fm, tmin=0.7–1.3fm, and tmax=2–2.6fm.In addition to determining the energies, on the three finest lattice spacings we calculate matrix elements of the electromagnetic current(2.7)jμem=23u¯γμu−13d¯γμd+… where the ellipsis denotes contributions from heavier quarks. For the vacuumtoππ matrix elements considered in this work, we require insertions of the isovector component and a dimensionfive counterterm required to implement O(a)improvement(2.8)Vμa=ψ¯γμτa2ψ,∂˜νTμνa=i∂˜νψ¯σμντa2ψ, where ψ=(u,d)T, τa the usual Pauli matrices in isospin space, σμν=i2[γμ,γν], and ∂˜μ the symmetrized lattice derivative. The isovector index a is taken to be maximal and henceforth omitted.The determination of the timelike pion formfactor then employs linear combinations(2.9)V(Λ,d)=∑μbμ(Λ,d)VR,μ,∑μbμ(Λ,d)⁎bμ(Λ,d)=1 where the coefficients bμ(Λ,d) project the current onto (a row of) irrep Λ and spatial momentum d. The vector current bilinear appearing in Eq. (2.9) has been renormalized and O(a)improved nonperturbatively according to(2.10)(VR)μ=ZV(1+abVml+ab‾VtrMq)(VI)μ,(VI)μ=Vμ+acV∂˜νTμν, where the renormalization and improvement coefficients ZV, bV, b‾V, and cV are functions of the gauge coupling only in this massindependent scheme.We take Z˜V=ZV(1+abVml+ab‾VtrMq) and cV from the nonperturbative determination of Ref. [99]. An alternative determination of the nonsinglet current renormalization constants for this lattice discretization is found in Ref. [100]. Another preliminary nonperturbative determination of ZV can be found in Ref. [101], while nonperturbative determinations of bV and b‾V are performed in Refs. [102,103].Operationally, we calculate current correlation functions using Eq. (2.5) for both the dimension four and five operators in Eq. (2.8) projected onto definite momentum d and irrep Λ. These current correlation functions, which are vectors in the GEVP index, are given as(2.11)D(Λ,d)(t−t0)=〈J(Λ,d)(t)O¯(Λ,d)(t0)〉, where J denotes either of the operators in Eq. (2.8) projected according to Eq. (2.9) and Oˆ(Λ,d) is an interpolator for irrep (Λ,d).To extract the finite volume matrix elements 〈0Vˆ(Λ,d)Λdn〉, we calculate the current correlation functions (defined in Eq. (2.11)) containing each the two operators in Eq. (2.8). These are used to form optimized current correlation functions using the GEVP eigenvectors(2.12)Dˆn(t)=(D(t),vn), where the inner product is taken over the GEVP index. Using these optimized current correlators, we then construct three ratios which plateau to the desired matrix elements asymptotically for large t (up to GEVP systematics)(2.13)Rn(1)(t)=Dˆn(t)Cˆn(t)e−Ent,Rn(2)(t)=Dˆn(t)Ane−Ent,Rn(3)(t)=Dˆn(t)AnCˆn(t), where An and En are determined previously from the ratio fits to Cˆn(t). The final matrix elements are then obtained from a plateau average of these ratios over a range [tmin,tmax].Each of these ratios possesses different excited state contamination. In analogy with the determination of the energies discussed above, consistency of the matrix elements using different ratios, (t0,td), and GEVP bases provides a stringent check in their determination. GEVP corrections to An have a different form than those of the energies [104], and our choices of fit ranges are optimized with the energies in mind. For this reason we take Rn(1)(t) in Eq. (2.13) as the best estimate of the matrix elements. Nonetheless, all three ratios are typically consistent. Data illustrating the tmindependence of these ratios and the comparisons mentioned above may also be found in the HDF5 files and Jupyter notebook. After determining the matrix elements for each current operator, we combine them to form the renormalized combination in Eq. (2.10).2.4Amplitudes from finite volume energies and matrix elementsFirst we determine the elastic I=1 pwave pion–pion scattering amplitude using the finitevolume energies discussed in Sec. 2.3. This amplitude is obtained from the determinant condition introduced for general twototwo scattering in Eq. (1.1). However, considerable simplification occurs for elastic scattering between spinless identical particles. Here the Kmatrix is diagonal in orbital angular momentum ℓ with trivial structure in the irrep occurrence index nocc. The box matrix B, which encodes the effect of the finite periodic spatial volume, mixes different orbital angular momenta and is dense in nocc.All the irreps used in this work are given in Table 3 together with the pattern of partial wave mixing induced by the infinitedimensional Bmatrix. Explicit expressions for all Bmatrix elements up to ℓ≤6 are given in Ref. [35]. Several offdiagonal Bmatrix elements vanish for identical particles, preventing partial wave mixing between even and odd ℓ. If contributions from ℓ≥3 partial waves are neglected, this simplification provides a onetoone correspondence between energies in the irreps of Table 3 and(2.14)K˜11−1(Ecm)=(qcmmπ)3cotδ1(Ecm), where qcm is the center of mass momentum and δ1 the I=1 ππ phase shift. This approximation is justified by the nearthreshold suppression of higher partial waves and to test it we perform global fits including fwave contributions. Refs. [9,35] perform similar fits and find such contributions negligible.We turn now to the determination of the timelike pion form factor Fπ(Ecm) using the finitevolume matrix elements calculated according to Sec. 2.4. The relations employed here for zerototwo matrix elements are given in Refs. [10,43,46] which are based on the seminal work of Ref. [47].We first define the angle ϕ1(d,Λ) using B11(d,Λ)=(qcm/mπ)3cotϕ1(d,Λ) where B is from Eq. (1.1). This pseudophase is used together with the physical phase shift to relate the finite and infinitevolume matrix elements(2.15)Fπ(Ecm)2=gΛ(γ)(qcm∂δ1∂qcm+u∂ϕ1(d,Λ)∂u)3πEcm22qcm5L3〈0V(d,Λ)dΛn〉2, where qcm is the magnitude of the centerofmass (three) momentum, u2=L2qcm2/(2π)2, and(2.16)gΛ(γ)={γ−1,Λ=A1+γ,otherwise where γ=E/Ecm. The infinite volume matrix elements are therefore obtained from their finite volume counterparts using the multiplicative Lellouch–Lüscher–Meyer (LLM) factor shown on the r.h.s. of Eq. (2.15).As is evident from Eq. (2.15), determination of Fπ(Ecm) requires not only the finitevolume matrix element 〈0V(d,Λ)dΛn〉 but also the derivative of δ1. This derivative is obtained from a parametrization of the phase shift points described above and covariances between all data are treated explicitly using the bootstrap procedure. Parametrization of δ1(Ecm) and Fπ(Ecm) is discussed in Sec. 3.3ResultsWe first present results for the elastic ππ scattering amplitude. As discussed in Sec. 2.4, if ℓ≥3 contributions to Eq. (1.1) are neglected there is a onetoone correspondence between finitevolume energies and K˜11(Ecm) defined in Eq. (2.14). This energy dependence is parametrized by a Breit–Wigner shape(3.1)K˜11−1(Ecm)=(mρ2mπ2−Ecm2mπ2)6πEcmgρππ2mπ involving two free parameters gρππ2 and mρ2/mπ2. A correlatedχ2 fit of all points is performed according to the ‘determinant residual’ method of Ref. [35] with μ=10, although without ℓ≥3 contributions the K˜−1 and Bmatrices are onedimensional so that the determinant is trivial. Results for these fit parameters, which are both constrained to be positive, as well as the χ2 per degree of freedom, χˆ2=χ2/Nd.o.f, are given in Table 4 for each of the ensembles employed here.The influence of ℓ≥3 partial waves is assessed by enlarging the determinant condition of Eq. (1.1) to include the ℓ=3 contributions noted in Table 3. For this fit the fwave is parametrized by an unconstrained constant K˜33−1(Ecm)=−(mπ7a3)−1 yielding results which are also shown in Table 4. We see therefore that there is little dependence on including ℓ≥3.Interested readers may perform further fits using App. A, where energies and phase shift points for all ensembles (neglecting ℓ≥3) are tabulated and plotted, or the Jupyter notebook described in Sec. 2.3 where bootstrap samples of all energy levels are available. However, it is worth comparing some of the ensembles in this data set here. An explicit check of finite volume effects using the C101 and D101 ensembles, which have the same parameters but different volumes, is shown in Fig. 2. That figure also shows a comparison between the N401 and N200 ensembles, which have (approximately) the same quark masses but different lattice spacing. It is thus evident on these ensembles that both effects are not visible within our statistical errors. Finally, all results for mρ and gρππ in shown in Fig. 6, where they are converted to physical units using the scale determined in Ref. [90].We turn now to results for the I=1 timelike pion form factor, which are determined according to Sec. 2.3. Results for the form factor from Eq. (2.15) are also tabulated in App. A. As discussed in Sec. 2.3, we employ a nonperturbative determination [99] of cV multiplying the dimensionfive counter term in Eq. (2.10). Apart from the one from Ref. [99], there is the preliminary determination of Ref. [101] which obtains values larger in magnitude using a different improvement condition. If cV is nonnegligible, the relative magnitude of the leading order matrix elements to this counterterm is of interest. Their ratio is shown in Fig. 3. Given its 5–15% size, a larger cV could indicate larger cutoff effects in the form factor than we observe using cV(g0) from Ref. [99], which is at the fewpercent level.We now turn to parametrization of the form factor. Ref. [10] employs the Gounaris–Sakurai parametrization [105](3.2)FπGS(s)=f0qcm2h(s)−qρ2h(mρ)+b(qcm2−qρ2)−qcm3si,b=−h(mρ)−24πgρππ2−2qρ2mρh′(mρ),f0=−mπ2π−qρ2h(mρ)−bmρ24,h(s)=2πqcmsln(s+2qcm2mπ), where the notation is from Ref. [106] and qρ is the centerofmass momentum at the resonance energy. This parametrization depends only on mρ and gρππ, and therefore describes the form factor with no additional free parameters.Additional parametrizations are suggested by unitarity constraints. In the elastic approximation, the form factor satisfies the nsubtracted dispersion relation(3.3)Fπ(s)=∑k=0n−1skk!dkdskFπ(0)+snπ∫4mπ2∞dzzntanδ1(z)ReFπ(z)z−s−iϵ. This dispersion relation has the Omnès–Muskhelishvili solution [107,108] of(3.4)Fπ(s)=Qn(s)exp{snπ∫4mπ2∞dzznδ1(z)z−s−iϵ}=Qn(s)Ωn[δ1](s),lnQn(s)=∑k=1n−1pksk, where due to charge conservation the k=0 term vanishes, and we have defined the Omnès function Ωn[δ1](s). The constants pk are fit parameters and proportional to logarithmic derivatives of Fπ. Given the Breit–Wigner parametrization for the phase shift, the twicesubtracted dispersion relation (n=2) has a single additional parameter (p1) and the thricesubtracted (n=3) two additional parameters, p1 and p2. These parameters appear in lnQn(s), while Ωn[δ1](s) depends only on the Breit–Wigner parametrization of δ1.For these fits, we isolate lnQn(s) by constructing ln(Fπ/Ωn[δ1]) and fit it to the appropriate polynomial. An example of a thricesubtracted fit is shown in Fig. 3, while the twice and thrice subtracted fits are compared in Fig. 5 for the D200 ensemble. Finally, the Gounaris–Sakurai parametrization is compared to the thricesubtracted fit on the J303 and D200 ensembles in Fig. 4.The results from these fits are compared in Table 5. The large χˆ2 is due to the significant correlation between the horizontal and vertical errors, which is visible in Fig. 3. Fits with four subtractions (n=4) do not significantly reduce χˆ2. Results from the N401 and N200 ensembles, which have similar quark masses but different lattice spacings, are shown in Fig. 5 together with thricesubtracted fits. Agreement between these two ensembles indicates that cutoff effects are also under control in the form factor.4ConclusionsThis work presents an Nf=2+1 calculation of the I=1 elastic pwave ππ scattering phase shift and timelike pion form factor which addresses systematic errors due to the finite lattice spacing, mixing of higher partial waves, and residual (exponential) finite volume effects.While we do not perform continuum and chiral extrapolations here, our data can be used for future such extrapolations. Chiral extrapolations of lattice scattering data using unitarized extensions of chiral effective theory [65,67–74] have been performed, although to date cutoff effects have not been considered. Nonetheless, it is evident in our data that cutoff effects in both the scattering amplitude (shown in Fig. 2) and the timelike pion form factor (Fig. 5) are small with respect to our statistical errors. The coarser lattice spacing in this comparison is a=0.075fm. Furthermore, for the scattering amplitude we also check explicitly (in Fig. 2) that finite volume effects are also insignificant at our coarsest lattice spacing. The two volumes used here have mπL=4.6 and 6.1.As discussed in Sec. 1, complete extrapolations of the energy dependence of amplitudes are left for future work. However as a necessary ingredient to determine the form factor, we use Breit–Wigner fits to model the energy dependence of δ1(Ecm). A summary of the fit results for the resonance mass and coupling are shown in Fig. 6. The CLS ensembles employed here adjust ms as ml is lowered to its physical value such that trMq=const. is fixed. This is to be contrasted with the more common strategy of fixing ms to its physical value for all ml such as the recent Nf=2+1 results in Ref. [7] which employs rooted staggered fermions. Compared with the more standard trajectory, it appears that the slope of mρ is somewhat flatter here. Although not shown in this work, recent summaries of existing results for the ρresonance parameters are found in Refs. [5,6].In addition to future fits and extrapolations, our results for the timelike pion form factor can be used to extend the vector–vector correlator as described in Ref. [76] and implemented in Ref. [109] without reliance on experiment. This may significantly improve lattice determinations of hadronic vacuum polarization contribution to anomalous magnetic moment of the muon, aμHVP.Finally, the computational effort expended on the CLS lattices for this work can be largely reused for other twototwo amplitude calculations. First work in this direction for Nπ scattering has already appeared in Ref. [26]. The set of ensembles used here will also be augmented by several others at lighter quark masses, include one with L=6.5fm at the physical point, which is presented in Ref. [110].AcknowledgementsWe gratefully acknowledge communications with H. Meyer, J. R. Pelaez, C. Hanhart, M. T. Hansen, S. Schaefer, M. Dalla Brida, and F. Erben. Special thanks to F. Knechtli for helping us secure computer resources used for this project and V. Koch for contributions at an early stage. BH was supported by Science Foundation Ireland under Grant No. 11/RFP/PHY3218. CJM acknowledges support from the U.S. NSF under award PHY1613449 and through TeraGrid/XSEDE resources provided by TACC, SDSC, and NICS under grant number TGMCA07S017. Computing facilities were provided by the Danish eInfrastructure Cooperation (DeIC) National HPC Centre at the University of Southern Denmark. The authors wish to acknowledge the DJEI/DES/SFI/HEA Irish Centre for HighEnd Computing (ICHEC) for the provision of computational facilities and support. We are grateful to our CLS colleagues for sharing the gauge field configurations on which this work is based. We acknowledge PRACE for awarding us access to resource FERMI based in Italy at CINECA, Bologna and to resource SuperMUC based in Germany at LRZ, Munich. Furthermore, this work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID s384. We are grateful for the support received by the computer centers. The authors gratefully acknowledge the Gauss Centre for Supercomputing (GCS) for providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS share of the supercomputer JUQUEEN at Jülich Supercomputing Centre (JSC). GCS is the alliance of the three national supercomputing centres HLRS (Universität Stuttgart), JSC (Forschungszentrum Jülich), and LRZ (Bayerische Akademie der Wissenschaften), funded by the German Federal Ministry of Education and Research (BMBF) and the German State Ministries for Research of BadenWürttemberg (MWK), Bayern (StMWFK) and NordrheinWestfalen (MIWF). The USQCD QDP++ library [111] was used in designing the software used for the calculations reported here.Appendix AFinite volume energies and scattering amplitudesIn this appendix, we tabulate and plot the finitevolume energies, scattering amplitude, and timelike pion form factor for all ensembles in Table 1. The scattering amplitude and form factor tabulated here employ the truncation to ℓ≤1. 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