Study of the pion-mass dependence of ρ -meson properties in lattice QCD

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I. INTRODUCTION
One of the most significant challenges in hadron physics is to understand the internal composition of diverse hadrons.However, because of the non-perturbative nature of the strong interaction in the low energy regime, the structure of hadrons within Quantum Chromodynamics (QCD) has remained unsolvable analytically.In order to develop insight into hadron structure and guide experimental work, a wide variety of phenomenological models have been developed.This includes the constituent quark model [1], the MIT [2] and cloudy bag models [3,4] as well others based upon Schwinger-Dyson equations [5] and molecular [6] and hybrid [7] models.As it is often possible to adjust the parameters in these models in order to reproduce the limited experimental data, these models can typically not be distinguished solely on the basis of how well they describe experiments.
On the other hand, the approximate chiral symmetry of QCD means that the pion is a pseudo-Goldstone boson, with a much smaller mass, m π , than other hadrons.Because the mass of the pion squared is proportional to the quark mass over a wide range, it is reasonable to expand certain physical variables in terms of m π .For instance, the mass of a resonance R can be expressed perturbatively as follows: where {g i } is the set of free parameters of the model and Σ R is the self-energy term.The separation described in Eq. 1 differentiates between the known modelindependent coefficients of terms nonanalytic in the quark mass which are contained within the self-energy terms, and the unknown coefficients of terms analytic in the quark mass, coefficients that are constrained by fitting data.
In the real world, m π takes the fixed value µ π ≈ 138.5 MeV (for charged pions), and it is typically feasible to adjust the free parameters of a model, {g i }, to reproduce the correct experimental value, m R (µ π ).However, when we extend the model to unphysical pion masses these models may predict different values of m R versus m π .The dependence of various physical variables on m π offers a fresh perspective in exploring the structure of hadrons in the non-perturbative regime [8].As a result, it is of great significance to make measurements on m R at unphysical m π values.Lattice QCD (LQCD) is a well-established nonperturbative formulation of QCD, defined on a finite and discretized volume of four-dimensional Euclidean spacetime.Through simulation of the two-point Green functions of composite operators, one can obtain the finite volume spectrum of eigenvalues of the QCD Hamiltonian, with specific quantum numbers, as a function of m π .We stress that such results are genuine predictions of QCD, even though the light quark masses do not take their physical values.Moreover, the phase shift in the infinite volume can then be obtained through the wellknown Lüscher formula and its extensions [9][10][11][12][13][14].
LQCD has developed rapidly over the half century since Wilson's pioneering work was published in 1974 [15].With the substantial progress in simulation algorithms and tremendous advances in computing power, many LQCD collaborations [16,17] have extracted the finite volume spectra for various sets of quantum numbers, including the ρ meson, over a range of values of m π .
Experimentally, the ρ meson is identified as a broad peak around √ s = 770 MeV in the invariant mass distribution of the isovector P -wave of ππ scattering [18].It is often identified as a confined q q state, consistent with the constituent quark model.This picture is supported by several theoretical arguments, such as the the large-N c limit of QCD [19][20][21][22].Nevertheless, the sizable decay width, Γ ρ→ππ ≈ 140 MeV [18], signifies the ρ meson's strong coupling to the ππ channel.In other words, the observed peak structure results from the interaction be-tween a q q state, referred to as the bare ρ, and the ππ continuum at the hadronic level.Consequently, a comprehensive study of the ρ meson necessitates an exploration of the ππ scattering sector.
The Lüscher formula is the most practical way to relate lattice calculations to the elastic scattering phase shifts of two spinless particles.Therefore, when dealing with a system containing only one ππ channel, it is sufficient to utilize the standard Lüscher formula to relate the finite volume spectrum to the phase shifts.However, in our present study, we also aim to incorporate the ωπ channel, in order to assess its impact.While it is not an open channel, it does generate the leading non-analytic behaviour of the ρ mass as a function of m π .
While the Lüscher formalism can certainly include the additional ωπ channel, we note that there are other approaches which provide computational convenience with very little overhead in incorporating several two-particle channels.As an example, we mention that the unitarized chiral perturbation theory (χPT) can calculate the finite volume spectrum through the pole position of the T matrix defined in the finite volume as shown in Refs.[39,40].
Alternatively, Hamiltonian effective field theory (HEFT) also incorporates the Lüscher formalism and establishes a connection between the scattering process in infinite volume and the finite volume spectrum of the system [41].For multi-channel scattering, the advantages and practicality of HEFT have been demonstrated in studies of various resonances, including the Roper [42,43], the Λ(1405) [44], the N * (1535) [45] and the D * s (2317/2460) [46].Because the Hamiltonian operates within the Fock space, effectively describing interactions among various different channels, the HEFT approach has two important features.1) It provides insight into the composition of the eigenstates through the strength of various components of the eigenvectors.2) It also enables an examination of the quark-mass dependence of resonance properties in a single formalism, enabling this unified analysis.
Here, we consistently analyze the spectra provided by several different LQCD collaborations using the HEFT framework, drawing on results extracted in the rest frame [47], moving frame and elongated frames [48].Motivated by the physical picture mentioned, the Hamiltonian employed here is studied within a framework that involves a bare ρ, as well as ππ and ωπ channels.We obtain the bare ρ mass in various regularization schemes from the lattice energy levels and investigate its dependence on m π .We observe that the linear slope of the bare ρ mass with respect to m 2 π is minimally affected by scheme dependence.Furthermore, we investigate the composition of the ρ meson, using the eigenvector of the eigen-energy state closest to the physical ρ mass.Finally, we illustrate interesting discrepancies in contemporary lattice QCD calculations.
The paper is organized as follows.In Sec.II we provide an overview of the HEFT formulation and proceed to construct the finite volume Hamiltonian for the specific case under investigation.Section III presents the results of the numerical analysis and examines the dependence of various variables on m π .Finally, in Sec.IV, we draw the discussion to a close with a concise summary and a suggestions for further analysis.

II. FORMALISM A. Hamiltonian model
The Hamiltonian in the center of mass frame of the interacting system is divided into two parts as follows, where H 0 is the non-interacting part, and H I is the interaction part.In this work, we include a bare ρ meson, which can be identified as a q q state, as well as two coupled channels, ππ and πω.In the infinite volume, characterized by SO(3) symmetry, it is most convenient to express the interaction in the JLS basis defined as [49] |α where C j1j2 (jm; m 1 m 2 ) is the Clebsch-Gordon (CG) coefficient of the SU(2) group, Y lm are the normalized spherical harmonics functions and | α; k * , σ 1 σ 2 ⟩ indicates the α = ππ or πω channels, with relative momentum k * and z-components of the spins of two particles, σ 1 and σ 2 , respectively.(For convenience, the quantities with an asterisk in this paper are all defined in the center of mass frame.)In addition, A α is the normalization factor that equals 1 √ 2 if α = ππ and otherwise is unity, m and σ are the z-components of the orbital angular momentum and total spin, respectively, S 1 and S 2 are the spins of the particles in channel α and J, M , ℓ and S are the total angular momentum, the z-component of the total angular momentum, orbital angular momentum, and total spin, respectively.The |α; k * , M ⟩ is normalized as, In general, there are interactions between the bare state and the two-particle channels as well as within and between the coupled channels.However, in this study, most of the energy levels and phase shifts are in the resonance region, so the dominant interaction is that between the bare state and two-particle channels.In addition, given the limited data concerning the energy levels from LQCD as well as experimental observables, we find that, in practice, the existing data can be described well without introducing channel-channel interactions.For this reason, the ππ − ππ, ππ − πω and πω − πω t/u-channel interactions are neglected.
For convenience, the JLS indices will be suppressed hereafter.
The free energy part of the Hamiltonian in this subspace, H 0 , is given by, where αi with α i = π or ω, m αi is for the mass of the particle in the α channel and m B ρ is the mass of the bare single-particle basis state.The interacting part, H I , is given within the model by, where the interaction term V α is independent of M , as a consequence of the Wigner-Eckart theorem, and given by where u ππ and u ωπ are the form factors parameterizing the internal structure of hadrons and ensuring the convergence of loop integrals.Here, the usual dipole form factors are used [50,51], where µ π = 138.5MeV is the physical mass of the pion.The scattering T-matrix, defined by can be obtained from the partial wave Lippmann-Schwinger equation [41,50,52], where V αβ comes from bare ρ exchange in the s-channel and is given by with V α defined in Eqs.(7) and (8).In the present case T ππ, ππ (p, q; E) can be obtained analytically where G(E) is the full propagator of the ρ meson defined by, with the self-energy The partial-wave phase shift, δ(E), for the P -wave ππ → ππ elastic scattering is then given by, where p = E 2 /4 − m 2 π is the on-shell momentum.The pole position of the ρ-resonance is located in the lower half plane of the unphysical Riemann sheet of the ππchannel but the first Riemann sheet of the ωπ-channel and determined by solving the equation

B. The Hamiltonian in finite volume
To obtain the energy levels in finite volume, we need to construct the finite volume Hamiltonian (FVH).Two major problems are encountered.Firstly, the correspondence between the Fock spaces spanned by the states with continuous and discrete momentum and secondly, the symmetry is reduced from the O(3) group to a finite subgroup, G, for the finite volume.As a result, J and M are no longer good quantum numbers.In Refs.[47,48], the standard formalism for the rest, moving and elongated frames were presented.Here we give a brief introduction to those aspects relevant to the present work.
To obtain the FVH in terms of the states with discrete momentum, one needs to make the following substitutions in Eqs.(5) and (6).First one sets because the bare ρ single-particle state does not change.However, it is very different for the two-particle state, and where V = ηL 3 is the volume of the box, with elongation factor η, and e n denotes a degenerate shell of the noninteracting Hamiltonian in the rest frame, because those states with the same e n share the same |k * (n)|.For example, in the rest frame of a cubic box, k * = 2π L n and hence e n = n 2 .However, for the general case, k * (n) and hence e n are not that simple.A detailed discussion of e n and k * (n) can be found in Ref. [48] and a summary is given in Appendix A. The finite volume basis vector, |α; e n , M ⟩, is given by an expression analogous to Eq. ( 3) with J = 1, ℓ = 1 and S = 0/1 for α = ππ/ωπ, respectively.Here {ê n } denotes the set of integer vectors with the same e n .
Note that the states defined in Eq. (3) with different values of JM ℓS are orthogonal, which is not the case in the finite volume since O(3) symmetry is broken.Thus, it is necessary to construct an orthogonal basis |α; e n , Γ, a⟩ furnishing an irreducible representation Γ of G.Such states take linear combinations of the basis states |α; e n , M ⟩ with reduction coefficients C Γ, G [48,53], ⟨α; e n , Γ, where Z Γ (e n ) is the normalization factor.In general there should be another index denoting the multiplicity of Γ, but in the present case that additional index is always 1 and hence it will be suppressed.The reduction coefficients relevant to the work reported here are shown in Appendix C. Similarly, satisfying With these well-defined orthogonal basis states |ρ B , Γ, a⟩ and |α; e n , Γ, a⟩, the FVH in the rest frame for a given irreducible representation, Γ, (note that because of the Wigner-Eckart theorem, the eigenvalue is independent of the "a" index, which is therefore suppressed) is given by |α; e n , Γ⟩ ⟨α; e n , Γ| Alternatively, these equations may be expressed in matrix form In principle the FVH is countably infinite-dimensional, while in practice it is found that the contribution of highenergy states to the low-lying eigenvalues of interest is negligible.Therefore, the matrix can be truncated by excluding the states with momentum higher than a certain value k * cut to obtain a finite-dimensional matrix.By comparing Eqs.(3),( 6), (24), (25) and (31), where the Jacobian, J α , just appears for a moving system, with the expression shown in Eq. (A2).

C. Fitting Formulas
In the present work, we fit the eigenvalues of the FVH, E H , to the lattice spectrum, E lat cm , with the usual least-χ 2 strategy.That is, we minimize the χ 2 defined as where C denotes the covariance matrix of the lattice spectrum.It should be noted that E lat cm is the spectrum that has been transformed into the rest frame.If a certain energy level, E lat n , is extracted from the composite operator with P ̸ = 0, it needs to be converted to E lat cm,n through In our model, there are five parameters, including the bare mass, m B ρ , two coupling constants g ρππ and g ωρπ , defined in Eqs.(7) and (8), and two cut-off parameters Λ ρππ and Λ ωρπ , defined in Eqs. ( 9) and (10), respectively.

D. Formulas for the mπ dependence
To extrapolate the results of the lattice calculations to the physical region, one needs to investigate the m πdependence of the properties of the ρ meson.This issue has been previously discussed in some studies [51,[54][55][56][57][58][59] and in this paper it will be studied within the framework of HEFT.As discussed in Ref. [55], in the framework of Chiral Perturbation Theory, the mass of the ρ is a function of m π of the form where m p ρ is related to the pole mass of the ρ, corresponding to the real part of the pole of the T-matrix in the complex plane.In general, m p ρ is different from the usual Breit-Winger mass, m BW ρ , which is defined as the real energy at which the phase shift is 90 degrees.For the present case, however, the difference is negligible.
As discussed in Refs.[51,55], the quark mass insertion at tree level only contributes to the m 2 π term in Eq. ( 38) up to O(m 4 π ), with the other two terms arising from pion-loop self energies.The m 3 π term comes from the ωπ loop, involving a vector-vector-pseudoscalar (V V P ) vertex, while both ππ and ωπ loops contribute to the log term.Within the present framework, the pole of the T-matrix is determined by Eq. ( 20), which tells us that where m p ρ − iΓ/2 is the complex pole position solved from Eq. (20).Comparing Eq. (38) and Eq. ( 39), the bare mass m B ρ is a quadratic function of m π at the leading order, In our analysis, this equation will be used to study the extracted m B ρ as function of m π .It is important to note that Eq. ( 40) is derived from the continuum field theory.In principle, to extrapolate the results obtained from lattice, the residual lattice artifacts should be estimated and removed.Since all actions are O(a)-improved, the effect of the finite lattice spacing can be estimated from O(a 2 ) as where ξ characterizes the rate at which it approaches the continuum limit and may vary from collaboration to collaboration as different fermion actions are used.

E. Model (in)dependence in HEFT
Understanding the model-dependent and modelindependent aspects of HEFT is important.As HEFT incorporates the Lüscher formalism [41,60], there are aspects of the calculation that share the same level of model independence as the Lüscher formalism itself.

Model independence
In particular, the Lüscher formalism provides a rigorous relationship between the finite volume energy spectrum and the scattering phase shifts and inelasticities of infinite-volume experiment.In HEFT, this relationship is mediated by a Hamiltonian.When the parameters of the Hamiltonian are sufficient to provide a high-quality description of lattice QCD results, then the associated scattering amplitudes are of high quality.The key is to have a sufficient number of tunable parameters to accurately describe the lattice QCD results.
In the baryon sector, high-quality lattice QCD results are scarce and HEFT is often fit to experimental data first.The HEFT formalism then describes the finitevolume dependence of the hadronic spectrum, indicating where future lattice QCD results will reside.
Fortunately, in the ρ-meson channel relevant to this analysis, several lattice QCD groups have resolved the finite-volume spectrum, taking care to assess the subtle shifts in the spectrum associated with avoided level crossings in the finite volume.This information is central to the Lüscher formalism and as such, is central to the HEFT analysis presented here.We will show excellent fits to the lattice QCD results such that HEFT provides rigorous predictions of the scattering observables with model independence at the level of the Lüscher formalism.
Of course, this model independence is restricted to the case of matched quark masses in finite-volume and infinite-volume.The Lüscher formalism provides no avenue for changing the quark mass.In other words, to make contact with experiment, the quark masses used in the lattice QCD simulations must be physical.
On the other hand, χPT is renowned for describing the quark mass dependence of hadron properties in a model-independent manner, provided one employs the truncated expansion in the power-counting regime, where higher-order terms not considered in the expansion are small by definition.Given that finite-volume HEFT reproduces finite-volume χPT in the perturbative limit by construction [60,61], it is reasonable to explore the extent to which this model independence persists in the full nonperturbative calculation of HEFT.
This was explored in Ref. [61].In the one channel case where a single particle basis state (e.g. a quark-model like ∆) couples to one two-particle channel (e.g.πN ), the independence of the results on the form of regularisation is reminiscent of that realised in χPT.Any change in the regulator is absorbed by the low-energy coefficients such that the renormalised coefficients are physical, independent of the renormalisation scheme.
However, in the more complicated two-channel case with a π∆ channel added, the same was not observed.The form of the Hamiltonian becomes constrained, describing experimental data accurately for only a limited range of parameters.The Hamiltonian becomes a model in this case, with regulator-function scales and shapes governed by the experimental data.The principles of chiral perturbation theory no longer apply in this nonperturbative calculation.However, for fit parameters that describe the data well, the model independence of the Lüscher formalism remains intact.The Hamiltonian is only mediary.

Quark mass variation
The consideration of variation of the quark masses away from the physical point provides further constraints on the Hamiltonian.In particular, lattice QCD results away from the physical point provide new constraints on the form of the Hamiltonian.In the two-channel case, the Hamiltonian becomes tightly constrained when considering experimental scattering data and lattice QCD results together.
With the Hamiltonian determined by one set of lattice results, one can then make predictions of the finitevolume spectrum considered by other lattice groups at different volumes and different quark masses.For the cases considered in the baryon spectrum the predictions of HEFT are in agreement with lattice QCD spectrum predictions.For example, in the ∆-channel HEFT successfully predicts the finite-volume spectrum of the CLS consortium [61,62].In the N ( 1 2 + ) channel, HEFT reproduces the lattice QCD results from Lang et al. [43,63].In the N ( 1 2 − ) channel, HEFT successfully predicts spectra from the CLS consortium [64,65], the HSC [64,66,67] and Lang & Verducci [64,68].Thus one concludes that the systematic errors of the HEFT approach to quarkmass variation are small on the scale of contemporary lattice QCD uncertainties.As the Hamiltonian is constrained by model-independent scattering data and lattice QCD results, we expect this success to be realised in the ρ-meson channel.
Variation in the quark mass is conducted in the same spirit as for χPT.The couplings are held constant and the hadron masses participating in the theory take values as determined in lattice QCD.The single-particle bare basis state acquires a quark mass dependence and this is done in the usual fashion by drawing on terms analytic in the quark mass.In most cases, lattice QCD results are only able to constrain a term linear in m 2 π , but on occasion, the data can demand a small m 4 π contribution.In the present analysis, we will see that the accuracy of contemporary lattice QCD results for the ρ-meson spectrum is sufficient to consider only a term linear in m 2 π .Even then we show that there are incompatibilities between the lattice QCD results.Contrary to the baryon sector, we show how a Hamiltonian constrained by the results of one group is incompatible with the results of other groups.Referring back to the model-independence of the Lüscher relation embedded within HEFT, the spectra of one group leads to scattering observables that are incompatible with the predictions of other groups.
The model independence associated with the movement of quark masses away from the physical point is largely governed by the distance one chooses to move from the physical quark-mass point.The HEFT approach is systematically improvable, reliant on highquality lattice QCD results to constrain the higherorder terms that one can introduce.In addition to the aforementioned analytic m 4 π term, one could also include higher-order interaction terms from the chiral Lagrangian.However, this increased level of precision is not yet demanded by contemporary lattice QCD results.

Model dependence
Now that the Hamiltonian has become a tightly constrained model, the eigenvectors describing the manner in which the non-interacting basis states come together to compose the eigenstates of the spectrum are model dependent.At the same time, there is little freedom in the model parameters of the Hamiltonian such that the predictions of the Hamiltonian are well defined.
With regard to the bare mass, there is interplay between the multi-particle channels included in the calculation, the regularisation scales considered and the associated bare mass.For example, as will be seen, upon introducing the additional ωπ channel, one observes an increase in the bare mass as some of this contribution is carried by the ωπ contribution.Thus the bare mass is defined only when the number of channels and their regularisation scales are fixed within the model.With the channels selected and preferred regularisation scales set, the bare mass becomes well-defined within the model.
While the bare mass is model dependent, we will show that the slope of the bare mass as a function of quark mass is insensitive to the number of two-particle channels considered.As such, this observation may be of assistance to those developing quark models.
Returning to the eigenvectors of the Hamiltonian, we emphasise that the parameters of the Hamiltonian model are well constrained, such that the predictions of the model are well defined.
The information contained in the Hamiltonian eigenvectors describing the basis-state composition of finitevolume energy eigenstates is analogous to the information contained within the eigenvectors of lattice QCD correlation matrices describing the linear combination of interpolating fields isolating energy eigenstates on the lattice.These too are model dependent, governed by the nature of the interpolating fields used to construct the correlation matrix.
What is remarkable is that with a suitable renormalisation scheme on the lattice (e.g.interpolators are normalised to set diagonal correlators equal to 1 at one slice after the source), the composition of the states drawn from the lattice correlation matrix is very similar to the description provided by HEFT [43,64].While both eigenvector sets are model dependent, their similarity does indeed provide some relevant insight into hadron structure.And because regularisation in the Hamiltonian is tightly constrained, one can begin to separate out the contributions of bare versus two-particle channels, something that is impossible in χPT.

Summary
In summary, there is a direct model-independent link between the finite-volume spectrum calculated at physical quark masses and the scattering observables of experiment.This model independence is founded on the Lüscher formalism embedded with HEFT.Similarly, variation of the quark masses away from the physical quark mass has systematic uncertainties that are small relative to contemporary lattice QCD spectral uncertainties.Finally, the Hamiltonian eigenvectors describing the basisstate composition of finite-volume energy eigenstates are model dependent.They are analogous to the interpolator dependent eigenvectors of lattice QCD correlation matrices describing the linear combination of interpolating fields isolating energy eigenstates on the lattice.The similarity displayed by these two different sets of eigenvectors suggests that they do indeed provide insight into hadron structure.

A. The LQCD Data
The finite volume spectra for the I = ℓ = 1 ππ sector with dynamical fermions at various pion masses have been provided by several LQCD collaborations over the past decade , including PACS-CS (2011, N f = 2+1) [24], HSC (2013, N f = 2+1) [27], HSC (2015, [34], and ETMC (2020, N f = 2 + 1(+1)) [28].Further details and energy levels are shown in Table I and the panels in Fig. 1, as well as all figures in Appendix D, where HEFT is fit to the various lattice data sets.

B. Three Fitting Schemes
In this work, our aim is to study the properties of the ρ meson by investigating the pion mass dependence of various relevant variables.In the HEFT framework, we have five free parameters which may be used to fit the lattice data with different pion masses.However, for each specific pion mass, there are only a few energy levels.
In addition, the ωπ contribution is considerably weaker than that of the ππ channel, since the threshold of ωπ is higher than the ρ mass.Consequently, we first adopt scheme A, wherein the interaction V ωπ is turned off, i.e., g ωρπ ≡ 0, while m B ρ , g ρππ , and Λ ρππ are treated as free fitting parameters.The finite volume spectra provided by various collaborations involving different π masses are each fit independently.Using scheme A, it is found that both g ρππ and Λ ωρπ show a very weak dependence on m π , while m B ρ is strongly dependent on m π , as anticipated earlier.
Building upon the results found using scheme A, in scheme B both g ρππ and Λ ρππ are fixed to be independent of m π , in accord with standard practice in chiral effective field theory.m B ρ is allowed to vary.As a result, this approach effectively combines spectra from various pion masses provided by different LQCD groups together in a unified analysis to constrain the variation of the bare mass, m B ρ , with m π .Finally, the contribution of the ωπ channel is examined in scheme C. In this case V ωπ is switched on, however, the two coupling constants, g ρππ and g ωρπ , as well as two cutoffs, Λ ρππ and Λ ωρπ , are taken to be independent of m π , while only m B ρ is permitted to vary in the fitting.These three schemes are summarized in Table II.

Results for Scheme A
Recall that in scheme A, the ρ − ωπ interaction is turned off and the lattice spectra are fit using three free parameters m B ρ , g ρππ , and Λ ρππ .The fitted spectra as a function of spatial extent, L, are shown by the blue curves in Fig. 1 for m π = 200 MeV as an illustration.All other fits are presented in Appendix D. The corresponding fitted parameter values are outlined in the columns dedicated to scheme A in Table III.In addition, the pion mass dependence of the three parameters, g ρππ , Λ ρππ , and m B ρ are shown in Fig. 2 and in the top panel of Fig. 3, respectively.From Fig. 2, it is found that the cut-off parameter Λ ρππ and the dimensionless coupling constant, g ρππ , both show a weak dependence on m π .In contrast, it is worth noting that for each m π , Λ ρππ exhibits a large uncertainty.Indeed, in some cases the upper uncertainty does not display in MINUIT2, which means that even for very large values of the cut-off, we can still find a reasonable fit for the energy levels.This observation suggests that the eigenvalue of the FVH around the region of the ρ mass is insensitive to Λ ρππ .This is consistent with the findings reported in Ref. [61], where the Λ-dependence of the eigenvalue and eigenvector are investigated in detail.A more common [27] dressed coupling constant, g BW ρππ , is where p is defined by 2 p2 + m 2 π = m BW ρ and Γ BW ρ→ππ is the partial width of the resonance in the Breit Wigner parameterizations.In our model, g BW ρππ depends on both the value of g ρππ and the form factor u ππ , which is determined by Λ ρππ .As a result, the slight m π -dependence seen in g ρππ and Λ ρππ clearly implies that the value of g BW ρππ , approximated to be around 6.0, remains independent of m π .This conclusion is consistent with the discussions presented in prior works, such as Refs.[29,31,33].
In contrast, the m π -dependence of m B ρ suffers a large fluctuation, as illustrated in the top section of Fig. (3).This behavior stems from the limited number of energy levels available for each pion mass, coupled with the considerable uncertainties associated with g ρππ and Λ ρππ .In order to reduce the uncertainty associated with the bare mass, one needs to find an appropriate method to combine the analysis of the various energy levels.Inspired by the weak dependence on m π , as well as the sizable uncertainties of g ρππ and Λ ρππ , we choose these two parameters to be constant for all pion masses and re-fit each spectrum using only one single parameter, m B ρ .

Results for Scheme B
The first task is to determine the constant values of g ρππ and Λ ρππ which will be used in the analysis of the data for all values of m π .To do that, we investigate m π = 236, 280, 322, 376, 386, 391 MeV cases, which have the highest values of the chi-squared per degree of freedom, χ2 (= χ 2 /dof), in scheme A. We present the distribution of the sum of the these χ2 with respect to some fixed g ρππ and Λ ρππ in Fig. 4. It is straightforward to find the preferred values of the coupling and cutoff from this distribution.It leads us to choose the values at the black point, where g ρππ = 7.07 and Λ ρππ = 890 FIG. 1. Spectra with mπ = 200 MeV provided by Bulava et al. [34] along with that calculated by HEFT using the fitting results for schemes A, B and C. The x-axis represents the spatial extent L in units of lattice spacing a, while the y-axis indicates the energy level.Tick marks on the y-axis are omitted for clarity.Text within the yellow box (n 2 , Γ) signifies spectrum extraction using operators in representation Γ and with total momentum Red points indicate the lattice spectrum provided by collaborations.Blue, orange dot-dashed and black dotted lines indicate the spectrum as the function of L calculated by HEFT using the fitting results of schemes A, B and C, respectively.The dashed gray lines indicate the non-interacting energy levels 2Eπ(k * ) and m B ρ (with m B ρ taken from Scheme A's fitting result for illustration).The turning points in the non-interacting energy levels are associated with energy crossings where the non-interacting two-particle energy becomes lower than the bare ρ-meson mass as L increases.MeV for N f = 2 + 1.On the other hand, it is found that for N f = 2, we should use slightly different values, namely g ρππ = 6.75 and Λ ρππ = 950 MeV [37,38].These values ensure the χ2 satisfies the condition χ2 ≲ 2 for all of these values of m π , as illustrated in Table III, in the column of χ2 for scheme B. For scheme B, the curves denoting the spectra as a function of the spatial extent, L, are shown as orange dot-dashed lines in the figures in Appendix D. The fitted m B ρ values are presented in Fig. 3, and the columns for scheme B in Table III.Comparing Fig. 3 It is apparent that there are substantial differences in the results from different LQCD groups.To address this issue, we examine the contribution of the coupling to the ωπ [51,55] channel, which is the closest two-particle channel in the present work and which, significantly, yields the leading non-analytic contribution to the ρ self-energy.

Results for Scheme C
Once V ωπ is turned on, two additional parameters, g ωρπ and Λ ωρπ , are involved.In addition, m ω also depends on m π .The m ω is not provided by most of the lattice QCD collaborations considered here.However, HSC reported a value of m ω at m π = 391 MeV [27].It provides an excellent opportunity to quantify the importance of quark-mass changes.They find a ρ-meson resonance position of a t m R = 0.15085 (18)(3) in the Briet-Wigner parameterization to be compared with their stable omega mass of a t m ω = 0.15678 (41).The difference/average is only 3.9% even at this very large pion mass.Since modern lattice QCD results are at lower quark masses, in our calculation for other collaborations, we just use the approximation m ω = m BW ρ .Since the threshold of ωπ is higher than the spectrum extracted, it is expected that the ωπ loop will shift the value of m B ρ but have a negligible influence on the resulting χ 2 .
Instead of allowing g ωρπ and Λ ωρπ to be two additional free parameters, we impose appropriate constraints to fix them.Two constraints are identified at the physical pion mass µ π : the decay width, Γ ω→3π , is primarily determined by the ω → ρπ → 3π mechanism (which is estimated to yield around 90% [70] of the width) and the P -wave phase shifts of ππ → ππ in the energy region around the ρ mass.
For simplification, we take the two cut-off, Λ ρππ and Λ ωρπ , to share the same value.Consequently, there are four undetermined parameters left, namely two coupling constants, one cut-off and one bare mass.Once g ρππ and Λ ρππ are fixed, the other parameters at µ π can also be determined from the decay width of ω → 3π and the Pwave phase shifts of ππ → ππ at the physical pion mass µ π .
The detailed procedure for parameter determination is presented in Appendix B. Here, we simply summarize the preferred values: g ρππ = 7.07 (6.75) and Λ ωρπ = Λ ρππ = 900 (980) MeV for N f = 2 + 1 (N f = 2) and g ωρπ = 18/GeV for either N f .It is worth mentioning that our value of g ωρπ is similar to that used in Ref. [51], 16/GeV.Additionally, g ρππ and Λ ωρπ are slightly shifted compared to those found using scheme B, because of the introduction of the ωπ channel.We then proceed to minimize the total χ 2 by fitting the value of m B ρ for each pion mass.
The fitted spectra as functions of the spatial extent, L, for scheme C are shown as black dotted lines in the figures of Appendix D, which illustrates only minor discrepancies from the orange dot-dashed lines found using scheme B, as expected.The fitted values of m B ρ are presented in the lower portion of Fig. 3 and are tabulated in the columns for scheme C in Table III.The preferred values of m B ρ are about 50 MeV higher than found in scheme B because of the additional self-energy term, Σ ωπ , as defined in Eq. (17).
Even after including the effect of the ωπ coupled channel, the m π -dependence of m B ρ shown in Fig. 3 is still scattered.It becomes apparent that the bare masses extracted from different lattice groups do not permit a consistent interpretation, which indicates the presence of intrinsic systematic differences between the lattice spectra provided by different collaborations.Such discrepancies lead us to consider the following issues that may influence the lattice results presented.These include: • Different residual lattice artefacts due to the different gauge and fermion actions considered.
• Varied scale-setting schemes employed by different collaborations.
• Different methods used to extract the finite volume spectra.
In the absence of systematic errors, the results provided by different collaborations should be consistent with each other after finite volume and lattice spacing artefacts are taken into account.Our HEFT approach enables one to account for the finite volume of the lattice, several values of the pion mass, as well as lattice spacing artefacts, all within a single formalism.As such, this is the first examination of the self consistency of world lattice QCD results for ππ scattering in ρ meson channel.

D. Extrapolation in mπ
In the last section, we present the outcomes of our fitting approach applied to the finite volume spectra provided by various collaborations for a wide range of values of m π .With these results, we now investigate the m π -dependence of the properties of the ρ meson and extrapolate them into the physical region.In scheme A, our investigations reveal that both g ρππ and Λ ρππ display little variation as m π varies.In the spirit of chiral effective field theory, the couplings and regulator parameters are held fixed.Thus, we could concentrate on the m π -dependence of the bare ρ mass, m B ρ , using Eq. ( 40).It is possible to extrapolate the fitting results of m B ρ in schemes B and C, but not in scheme A, since there the values of m B ρ are correlated with g ρππ and Λ ρππ .In principle, it is natural to consider putting all the values of m B ρ (m π ) together and performing a global fit to make full use of the lattice data.However, from the two lower figures in Fig. 3, it is hard to extract useful information, since the data show large inconsistent variations.The possible reasons have been discussed in the previous section.
For example, with reference to the discussion about the lattice spacing effect in Eq. ( 41), the coefficient ξ are different for each LQCD group in principle.To confirm this, we applied Eq. ( 41) with a single value of ξ and TABLE IV.Extrapolation results are summarized for schemes B and C. For each collaboration, the results of B and C are given in the first and second row, respectively.The second and third column present the coefficient c0 and c1 defined in Eq. (40).The fourth column presents the extrapolated m B ρ at the physical pion mass.The fifth column presents the pole mass, defined by Eq. (20).found a large χ 2 .Also, there is a huge difference between the values of observables calculated using the parameters extrapolated to physical m π and those measured in experiments.This suggests that we should make the extrapolation of the data to the physical point collaboration by collaboration.
Because for each group, there are only a limited number of values of m π and the lattice spacing does not change a lot, the lattice spacing term can be absorbed and we just use Eq.(40) to perform the extrapolations.Furthermore, we have two free parameters in Eq. ( 40), thus only the data of collaborations having no less than two m π points are analyzed.The fitting and extrapolation results are shown in Fig. 5 and Table IV.
As shown in Fig. 5, there are five collaborations having no less than two different m π points.For each collaboration, the points show a good linear relation between m B ρ and m 2 π , whether the ωπ loop is included or not.The only notable exception is one point with a large uncertainty from ETMC.With c 0 and c 1 determined, we can obtain the bare mass of the ρ at the physical pion mass.Subsequently, we can get m p ρ by solving Eq. ( 20).The results are listed in the last column of Table IV.
For MILC, HSC and Bulava et al., even though their values of c 0 and c 1 are quite different, the extrapolated m p ρ all agree with the experimental value.However, for ETMC and Guo et al., they are about 30 MeV higher and 50 MeV lower compared to the experimental value, respectively.The relatively high m p ρ obtained by ETMC is not so surprising, since in their previous work [31] a higher value of m ρ compared to the others was also reported.The lower m p ρ extracted from Guo et al. also agrees with their own result, presented in Ref. [32], which possibly results from using N f = 2. Clearly, the physical ETMC 0.06 0.08 0.10 0.12 0.  39) using m B ρ = c0 + c1 m 2 π with c0 and c1 being the extrapolation coefficients for scheme B presented in Table IV.Red points also represent Re Σππ but using m B ρ from fitting results of scheme B presented in Table III.
ρ masses obtained here all indicate the consistency between our method and previous work, while in addition we provide detailed information on the m π -dependence of m B ρ .

E. Discussion and Exploration
In this section, we make some remarks concerning the numerical results.In scheme A, we first found that the coupling constants g ρππ and cut-off Λ ρππ are both weakly dependent on m π .This conclusion supports our previous study of the baryon resonances, Λ(1405), N * (1535) and N * (1440) (Roper), where we only considered the m π dependence of the masses of various hadrons but not the couplings and cut-off.Furthermore, the large uncertainty found for the cut-off is also acceptable, because in principle the physical observables should not be so sensitive to it.
In Eq. ( 38), there are many nonlinear terms in m 2 π , arising from the self-energy part.In the right column of Fig. 6, we show the value of the Re Σ ππ at the pole position as a function of m 2 π .Given m B ρ , Re Σ ππ is calculated by m p ρ − m B ρ as in Eq. ( 39), where m p ρ is obtained from Eq. (20).Scalars in the figure are calculated by using the values of m B ρ from the fitting results of scheme B presented in Table III, while the lines with error bands use m B ρ = c 0 +c 1 m 2 π , with c 0 and c 1 being the coefficients for scheme B presented in Table IV.
Clearly both these scalars and lines in Fig. 6 exhibit a nonlinear behavior.However, the difference of Re Σ ππ at different m π are just around 10 MeV when m π varies over the range 140 to 400 MeV.Therefore, there will also be an approximately linear relation between m p ρ and m 2 π , as shown in the left column of Fig. 6, where the Breit-Wigner masses provided by the three collaborations are also shown.This is the reason why the Breit-Wigner ρ mass could be described well by a linear function of m 2 π in Ref. [33].
Things are similar even when the ωπ channel is introduced.From Table IV one can see that m p ρ is nearly unchanged while m B ρ changes a lot when V ωπ turns on.Nonetheless, m B ρ still exhibits a linear relation to m 2 π as shown in Fig. 5, where the two lines are almost parallel for each LQCD group.Also, it is apparent that the contribution from the ωπ loop only makes a significant change in the value of c 0 , but just slightly modifies the slope, c 1 .These facts suggest that Re Σ ωπ is also weakly dependent on m π and its effect can be effectively absorbed into c 0 .
In summary, the slow variation of the contributions from ππ and ωπ loops significantly affects c 0 , while only slightly influencing the slope, c 1 .Therefore, it makes little sense to talk about m B ρ solely based on experimental results, as the fitted m B ρ strongly depends on how the hadronic loops are estimated.It is the slope c 1 that contains more useful, less model dependent, physical information concerning the structure of the ρ meson, which can only be extracted from the LQCD data at unphysical m π .Furthermore, in principle, on the theoretical side the slope c 1 can be calculated at the quark-level in various models.Thus, with the help of c 1 the relevant models could be distinguished.This is quite a good example of the idea that the data extracted at unphysical values of The linear relation between m B ρ and m 2 π is consistent with the assumption that |ρ B ⟩ is a pure q q state.Additionally, the contribution from hadron loops to m p ρ accounts for only approximately 20% of the total mass for the optimal value of the regulator parameter.Consequently, we can also conclude that the bare ρ plays the most important role in the structure of the observed ρ meson.To confirm this, we pick out several eigenstates whose energy is close to the physical ρ mass and look at their composition, a feature which is characteristic of HEFT.The eigenstates of the FVH are the counterpart in the finite volume of continuous scattering states in the infinite volume [43].Therefore, obviously, it is expected that the eigenstate whose energy is closest to the resonance region is the counterpart of the ρ.For illustration, we pick five eigenstates from the spectrum generated by HSC(2015) with m π = 236 MeV and calculate the probabilities of the bare state, ππ and ωπ components.The results are presented in Table V.It is found that the component ρ B has a probability around 75%.That is, it is definitely dominant.
With the parameters listed in Table III, we can predict the ππ → ππ P -wave phase shift at unphysical masses.Here we focus on the three collaborations whose values of m p ρ at the physical m π agree with the experimental value.In Fig. 7, we show the phase shifts calculated for four different values of m π .In the first row, at the physical point, the phase shifts of the three collaborations are nearly the same.This is expected since their values of m B ρ at physical m π are consistent.They are in good agreement with the experimental data, except for the region away from resonance, where the ππ−ππ t/u-channel interaction may not be negligible.
In other rows we predict the phase shifts of the three collaborations at three unphysical values of m π .Since the values of c 0 and c 1 obtained by analysis of the data from these three collaborations are very different, their phase shifts are not consistent.Especially, for m π = 400 MeV, we predict that the typical line shape would disappear in the phase shifts of ππ scattering for Bulava et al., since it would be a bound state of ππ.
The most serious problem found in this work is that the two coefficients, c 0 and c 1 , differ a lot for different LQCD groups.This unexpected variation arises from dif-ferences in the lattice QCD simulations, the interpolating fields considered in constructing correlation matrices, the analysis methods applied to the correlation matrices and finally the scale-setting schemes.
The use of different O(a)-improved actions gives rise to different O(a 2 ) errors such that the coefficient of a 2 differs for each set of lattice QCD results.It will be important to have two or more different lattice spacings available in high-quality sets to enable a determination and elimination of this lattice artefact.
State isolation is key to measuring the subtle shifts of the finite-volume energies from their non-interacting energies, vital to measuring a phase shift.In obtaining consistency across lattice collaborations, modern projected correlator methods should be adopted to reduce excited state contamination in the extracted energy eigenvalues.Moreover, all nonlocal two particle momentum-projected interpolating fields participating in the resonance region need to be considered and mixed with the single particle operators to ensure multiple eigenstates are not participating in the state-projected correlation functions of the correlation matrix.In the present case, single particle ρ interpolations are to be mixed, not only with ππ correlations, but also with πω correlators.Indeed, Fig. 1 and the figures in Appendix.D, illustrating the fits of HEFT to contemporary lattice QCD results, raise concerns about multi-state contamination in the analysis.
Finally, the choice of scale setting scheme is of vital importance when attempting to describe QCD properties away from the physical point.While all schemes are designed to extrapolate to the physical point, the manner in which they move away from the physical point is different.The Sommer scale is designed to be physical.Maintaining physics at the scale relevant to the charmonium spectrum, it naturally includes changes in the renormalization of the strong coupling constant due to changes in the sea quark masses.However, other schemes where the quark mass has no effect on the coupling constant are possible, provided the only goal is to get to the physical point.In light of the plethora of scale setting schemes currently proposed, we encourage the consideration of how well a proposed scheme is suited to learning the properties of QCD in a universe with different quark masses.

IV. SUMMARY AND OUTLOOK
In this work, we collected finite volume spectra for the I = ℓ = 1 ππ sector provided by LQCD collaborations over the past decade.These spectra were fit in a consistent manner within the framework of Hamiltonian Effective field theory.The basic states included in the Hamiltonian were a bare ρ state and the ππ and ωπ coupled channels.In this framework, we successfully fit the finite volume spectra in the rest frame, moving frame and the elongated box, and complemented this with experimental data for the ππ → ππ P -wave phase shifts.
We employed three schemes to fit the energy levels obtained at various pion masses.Through scheme A, we found that g ρππ and Λ ρππ exhibit a weak dependence on m π .In the following scheme B, we set these two parameters as constant in order to obtain the m π -dependence of the bare ρ basis state, m B ρ (m π ).In scheme C, where the ωρπ vertex was included with additional constraints from experimental data, we again extracted m B ρ (m π ).Finally, we used the linear relation between m B ρ and m 2 π , as shown in Eq. ( 38), to perform an extrapolation.Because the relationship between m B ρ and m 2 π was highly dependent on the LQCD group whose data we used, we were unable to fit all m B ρ simultaneously, and resorted to extrapolating collaboration by collaboration.
Based upon the extrapolations of data from the five LQCD groups, it was found that for each collaboration, m B ρ (m π ) could be described well by Eq. ( 40).This supports the hypothesis that the single-particle bare ρ meson plays an important role in forming a physical peak in the P -wave ππ scattering.The pole mass m p ρ defined by Eq. ( 39) are also calculated.When extrapolated to physical pion mass µ π , the m p ρ for MILC, Bulava et al. and HSC agree with the experimental measurements.We can also predict the observables such as the phase shift for each collaboration at some certain unphysical m π .For each group, the pole mass m p ρ and the coefficients c 1 in Eq. ( 40) were stable with and without the ωπ loop contribution, while the bare mass had a shift of around 50 MeV.Unfortunately, from the current LQCD data, the extracted value of c 1 is dependent on the lattice collaboration whose data is used.This indicates the important discrepancies in the results of today's lattice QCD calculation among different collaborations due to the possible reasons discussed previously.In the future it would obviously be ideal to have lattice data for the ρ meson at different pion masses with all systematic effects carefully controlled in order to resolve the m π -dependence of the ρ meson.

ACKNOWLEDGMENTS
We acknowledge useful discussions and valuable comments from Xu Feng, Feng-Kun Guo, Chuan Liu, Liuming Liu, Guang-Juan Wang, Mao-Jun Yan, Yi-Bo Yang, Zhi Yang, Bing-Song Zou.This work was partly supported by the National Natural Science Foundation of China (NSFC) under Grants Nos.12175239 and 12221005 (J.J.W), and by the National Key R&D Program of China under Contract No. 2020YFA0406400 (J.J.W), and by the Chinese Academy of Sciences under Grant No. YSBR-101 (J.J.W).This research was also undertaken with assistance of resources from the National Computational Infrastructure(NCI), provided through the National Computational Merit Allocation Scheme.This work was supported by the Australian Research Council through Discovery Projects DP190102215 and DP210103706 (DBL) and DP230101791 (AWT).Y. L. is supported by the Excellence Hub project "Unraveling the 3D parton structure of the nucleon with lattice QCD (3D-nucleon)" id EXCELLENCE/0421/0043 co-financed by the European Regional Development Fund and the Republic of Cyprus through the Research and Innovation Foundation.Y. L. further acknowledges computing time granted on Piz Daint at Centro Svizzero di Calcolo Scientifico (CSCS) via the project with id s1174, JUWELS Booster at the Jülich Supercomputing Centre (JSC) via the project with id pines, and Cyclone at the Cyprus institute (CYI) via the project with ids P061, P146 and pro22a10951.since we have boosted the T-matrix element into ρ rest frame and W (E; k) ≡ (E − E π (k)) 2 − k 2 .Therefore, the spin-averaged decay width is given by Γ where dΦ 3 is the Lorentz-invariant three-body phase space element defined in [18].It is convenient to take dΦ 3 ∝ dm 12 dΩ 3 dΩ * 1 and ∝ dm  where ϵ is the polarization vector, u ππ and V ππ are the form factor and potential defined in Eq.( 9) and Eq.( 7), respectively.p * 1/2 are the four-vectors of π, P and P ′ are the four-vectors of ω and ρ, respectively.We do not introduce a form factor for V ω→ρπ since there is no loop integral related to it.Following that in Ref. [70], the interference term is given by, With these ingredients Γ ω→ρπ→3π can then be calculated.
In Sec.(III C) we found that when V ωπ = 0, g ρππ and Λ ρππ can be fixed at 7.07 and 890 MeV, respectively.It is expected that the introduction of the ωρπ vertex would slightly shift these parameters, we try to take g ρππ and Λ ρππ at 7.40 and 900 MeV, respectively.Besides, we assume that Λ ωρπ = Λ ρππ .The remaining two parameters m B ρ and g ωρπ are constrained by Γ ω→3π and δ ℓ=1 ππ→ππ .
If we adopt g ωρπ = 18/GeV, which is close to that in Ref. [51], and m B ρ = 870 MeV, it is found that Γ ω→ρπ→3π = 7.12 MeV (B13) while the experimental value of the partial decay width Γ ex ω→3π = 7.74 (13) MeV.The 10% discrepancy is accounted for by the neglected direct interaction between the |ω⟩ and the |3π⟩ channel [70].Furthermore, the phase shift defined by Eq.( 19) can also be obtained and shown in Fig. 8 The approximate consistence between the theoretical and experimental results justify the values adopted.Furthermore, it is interesting that the extrapolated m B ρ of HSC, MILC and Bulava et al. in scheme C are nearly the same as that adopted here.12. Same as in Fig. 1 but for the ETMC collaboration [28].

FIG. 2 .
FIG. 2. mπ-dependence of gρππ and Λρππ in scheme A. The dashed gray vertical line indicates the physical mπ.The dashed red horizontal lines indicate the value that will be fixed for N f = 2 + 1 in scheme B.
(A) with (B), one finds that the uncertainties in m B ρ are significantly reduced, albeit with slight shifts in the central values.The m π -dependence of m B ρ remains unclear.In particular, the values of m B ρ at m π = 301, 302, 315, 316 and 322 MeV extracted from MILC, ETMC, Guo et al., C. Alexandru et al., and ETMC respectively, significantly fluctuate between 840 and 920 MeV.

FIG. 3 .
FIG. 3. mπ-dependence of m B ρ in schemes A, B and C from top to bottom.The dashed gray vertical line indicates the physical mπ value.

FIG. 5 .
FIG. 5. mπ-dependence and extrapolation of m B ρ for each collaboration.Red and blue points indicate the fitting results of m B ρ in schemes B and C, respectively.Gray bands represent the quadratic function m B ρ = c0 + c1 m 2 π with uncertainty, where c0 and c1 for both schemes are given in TableIV.For convenience the lower bound of the m 2 π -axis is set as the physical value µ 2 π .

FIG. 6 .
FIG.6.mπ-dependence of m p ρ and the real part of the selfenergy Σππ.Left column: m p ρ defined by Eq.(20) using parameters for schemes B and C, and mρ obtained from the Breit-Wigner parameterization of the phase shift provided in each paper.Right column: real part of the self-energy Σππ.Blue lines with gray bands represent the Re Σππ = m p ρ (m B ρ ) − m B ρ as in Eq.(39) using m B ρ = c0 + c1 m 2 π with c0 and c1 being the extrapolation coefficients for scheme B presented in TableIV.Red points also represent Re Σππ but using m B ρ from fitting results of scheme B presented in Table III.

FIG. 13 .
FIG.13.Same as in Fig.1but for the ETMC collaboration[28].The turning point of the black dotted line for mπ = 322 MeV is due to an avoided level crossing when the ω π channel is included.

FIG. 16 .
FIG.16.Same as in Fig.1but for Guo et al.[32].Note that there is an additional quantity η denoting the elongation factor in the yellow box compared to the others.

TABLE I .
Details of the spectra by different collaborations.Columns, from left to right, show collaboration and year of spectra, pion mass (mπ), number of flavor (N f ), lattice size (L) and spacing (a) in fm, employed gauge and fermion actions, number of energy levels (N lvl ), and energy level extraction method.Besides, the spectra provided by Guo et al. are extracted in a elongated box with factorη = 1, 7 6 , 4 3 for mπ = 226 MeV, and η = 1, 1.25, 2 for mπ = 315 MeV.COLLAB.(Year)mπ(MeV) N f L(fm) a(fm)

TABLE III .
[69]ing results of schemes A, B, and C provided by the MINUIT2 program[69].In the columns for scheme A, we show asymmetric uncertainties, since in this scheme the upper and lower uncertainties are quite different.The question mark means that some upper uncertainties are not provided by MINUIT2 since even for a very large Λρππ the energy levels can still be fit well.For scheme B, gρππ, and Λρππ are fixed at 7.07 (6.85) and 890 (950)MeV for N f = 2 + 1 (2), respectively.For scheme C, gωρπ and Λωρπ are fixed at 18/GeV and 900 MeV, while gρππ and Λρππ are fixed at 7.07 (6.85) and 900 (980) MeV for N f = 2 + 1 (2), respectively.χ2 represents the reduced chi-square, i.e, χ 2 /dof with dof = N lvl − 3 for scheme A and N lvl − 1 for schemes B and C.

Table IV .
For convenience the lower bound of the m 2 π -axis is set as the physical value µ 2 π .