Isospin-0 s-wave scattering length from twisted mass lattice QCD
We present results for the isospin-0 s-wave scattering length calculated with Osterwalder-Seiler valence quarks on Wilson twisted mass gauge configurations. We use three ensembles with unitary (valence) pion mass at its physical value (250 MeV), at 240 MeV (320 MeV) and at 330 MeV (400 MeV), respectively. By using the stochastic Laplacian Heaviside quark smearing method, all quark propagation diagrams contributing to the isospin-0 correlation function are computed with sufficient precision. The chiral extrapolation is performed to obtain the scattering length at the physical pion mass. Our result agrees reasonably well with various experimental measurements and theoretical predictions. Since we only use one lattice spacing, certain systematics uncertainties, especially those arising from unitary breaking, are not controlled in our result.
Quantum chromodynamics (QCD) is established as the fundamental theory of the strong interactions. QCD at low energies is largely determined by chiral symmetry, which is spontaneously broken. The effective theory of QCD at low energies is chiral perturbation theory (PT) Weinberg:1978kz (); Gasser:1983yg (); Gasser:1987ah (), representing a systematic expansion in the quark masses and momenta. Elastic scattering provides an ideal testing ground for the mechanism of chiral symmetry breaking. Since only the pions – the pseudo-Goldstone bosons of SU chiral symmetry – are involved, the expansion is expected to converge rapidly. In fact, the s-wave scattering length in the weakly repulsive isospin-2 channel can be reproduced by leading order (LO) PT Weinberg:1966kf () with a deviation of only 0.5% when compared to the results obtained from experiments combined with Roy equations Batley:2010zza ().
However, in the isospin-0 channel the situation is different: the interaction is attractive and much stronger than in the isospin-2 channel. The agreement between LO PT and experiments for the s-wave scattering length in the isospin-0 channel is much less impressive: they deviate by around 30% Weinberg:1966kf (); Batley:2010zza (); Pislak:2003sv (). Moreover, this channel accommodates the lowest QCD resonance – the mysterious or scalar meson. Although it plays a crucial role in some fundamental features of QCD, its existence was disputed for a long time. Only recently it was established unambiguously with dispersive analyses and new experimental data, see Ref. Pelaez:2015qba () for a review.
This makes a nonperturbative, ab initio computation of interaction properties in the isospin-0 channel directly from QCD highly desirable. Lattice QCD is the only available method to perform such a computation with controlled systematic uncertainties. Lüscher showed that the infinite volume scattering parameters can be related to the discrete spectrum of the eigenstates in a finite-volume box Luscher:1986pf (); Luscher:1990ux (). This allows one to compute scattering properties in lattice QCD, which is necessarily implemented in a finite volume and Euclidean space-time.
For the isospin-2 channel, many lattice results have become available. See Refs. Yagi:2011jn (); Fu:2013ffa (); Sasaki:2013vxa (); Helmes:2015gla () for the most recent ones. For the isospin-0 channel the situation is more complicated mainly due to the fermionic disconnected diagrams contributing here, which are challenging to compute in lattice QCD. To date there are only two full lattice QCD computations dedicated to this channel Fu:2013ffa (); Briceno:2016mjc (). In Ref. Fu:2013ffa (), the s-wave scattering length was computed for three unphysically large pion masses. An extrapolation to the physical point was performed to obtain the scattering length at physical pion mass. The authors of Ref. Briceno:2016mjc () on the other hand extracted many energy eigenstates in the corresponding channel and obtained the scattering amplitudes for two values of pion mass – 236 MeV and 391 MeV. The information about the meson is deduced from the pole structure in the scattering amplitudes at the two unphysical pion masses, respectively.
In this work we compute the scattering length of the isospin-0 channel in twisted mass lattice QCD Frezzotti:2000nk () and Lüscher’s finite volume method Luscher:1986pf (); Luscher:1990ux (). As discussed in Ref. Buchoff:2008hh (), the explicit isospin breaking of the twisted mass quark action makes it prohibitively complicated to study this channel with this action. To circumvent this complication we use a mixed action approach with Osterwalder-Seiler quarks Frezzotti:2004wz () in the valence sector, which preserves isospin symmetry. This approach introduces additional lattice artefacts due to unitarity breaking. These lattice artefacts are of and will vanish only in the continuum limit. In particular, due to isospin breaking in the sea there is possibly residual mixing with . Since we use only one value of lattice spacing, systematic uncertainties in our results are not fully controlled. Further calculations are needed to explicitly address these uncertainties. However, they are beyond the scope of this work.
This paper is organized as follows. The lattice setup is discussed in Sec. II. Lüscher’s finite volume method is introduced in Sec. III. In Sec. IV we present the computation of the finite volume spectrum of the isospin-0 system. The result for the scattering length is given in Sec. V. The last section is devoted to a brief summary and discussions.
Ii Lattice action
The results presented in this paper are based on the gauge configurations generated by the European Twisted Mass Collaboration (ETMC) with Wilson clover twisted mass quark action at maximal twist Frezzotti:2000nk (). The gauge action is the Iwasaki gauge action Iwasaki:1985we (). We use three ensembles with pion mass at the physical value, at 240 MeV and at 330 MeV, respectively. The lattice spacing is forflavor all three ensembles, as found in Ref. Abdel-Rehim:2015pwa () up to lattice artefacts. In Table 1 we list the three ensembles with the relevant input parameters, the lattice volume and the number of configurations. More details about the ensembles are presented in Ref. Abdel-Rehim:2015pwa ().
The sea quarks are described by the Wilson clover twisted mass action. The Dirac operator for the light quark doublet consists of the Wilson twisted mass Dirac operator Frezzotti:2000nk () combined with the clover term (in the so-called physical basis)
with , and the forward and backward lattice covariant derivatives. Here is the so-called Sheikoleslami-Wohlert improvement coefficient Sheikholeslami:1985ij () multiplying the clover term and , with the critical mass. is the average up/down (twisted) quark mass. is the lattice spacing and the Wilson parameter. acts on a flavor doublet spinor . In our case the clover term is not used for improvement but serves to significantly reduce the effects of isospin breaking Abdel-Rehim:2015pwa ().
The critical mass has been determined as described in Refs. Chiarappa:2006ae (); Baron:2010bv (). This guarantees automatic improvement Frezzotti:2003ni (), which is one of the main advantages of the Wilson twisted mass formulation of lattice QCD.
In the valence sector we introduce quarks in the so-called Osterwalder-Seiler (OS) discretization Frezzotti:2004wz (). The corresponding up and down single flavor lattice Dirac operators read
From this definition it is apparent that OS up and down quarks have explicit SU isospin symmetry if for both e.g. was used. The matching of OS to unitary actions is performed by matching the quark mass values, i.e. . The value of in the OS action can be shown to be identical to the unitary one and improvement stays valid Frezzotti:2004wz (). Moreover, we have shown in Ref. Ottnad:2015hva () that in such a mixed action approach disconnected contributions to and mesons can be computed and the results agree with the unitary ones Michael:2013gka () in the continuum limit. Therefore, this mixed action approach should works also in the case of scattering, where disconnected contributions can be expected to be less important, since OZI suppression is in place. However, there is a potential complication arising from the double poles in flavor-neutral meson propagators present in a quenched or partially quenched theory Bernard:1994zp (). The scalar correlators with disconnected diagrams suffer from unphysical contributions due to the double poles. The unphysical contributions to the and correlators have been studied in Refs. Aubin:2008wk (); Bardeen:2001jm (); Golterman:2005xa (); Bernard:1995ez (). In this work, we are not going to consider this problem since the formula of these unphysical contributions for our partially quenched approach is not available. Also, as will be presented in Sec. IV, our numerical results do not indicate large unphysical contributions. All the correlators we computed numerically are positive within the obtained statistics and are well described by a single exponential function of in a reasonably large time range. This would not be the case if there were large unphysical contributions as shown in Refs. Aubin:2008wk (); Bardeen:2001jm (); Golterman:2005xa (); Bernard:1995ez (). Nevertheless, one should keep in mind that the effects of the double poles may cause uncertainties that are not considered in our results.
Masses computed with OS valence quarks differ from those computed with twisted mass valence quarks by lattice artefact of , in particular
For twisted clover fermions this difference is much reduced as compared to twisted mass fermions Abdel-Rehim:2015pwa (), however, the effect is still sizable. We use the OS pion mass in this paper, with the consequence that the pion mass value of the cA2.09.48 ensemble takes a value larger (around 250 MeV) than the physical one.
As a smearing scheme we use the stochastic Laplacian Heavyside (sLapH) method Peardon:2009gh (); Morningstar:2011ka () for our computation. The details of the sLapH parameter choices for a set of Wilson twisted mass ensembles are given in Ref. Helmes:2015gla (). The parameters for the ensembles used in this work are the same as those for ensembles with the corresponding lattice volume.
Iii Lüscher’s finite volume method
Lüscher’s finite volume method provides a direct relation between the energy eigenvalues of a two-particle system in a finite box and the scattering phase shift of the two particles in the infinite volume. Considering two particles with mass and in a cubic box of size , the energy of this system in the center-of-mass frame reads
where is the scattering momentum. For the following discussion, it is convenient to define a dimensionless variable via
which differs from an integer due to the interaction of the two particles.
The general form of Lüscher’s formula reads Luscher:1990ux ():
where is the phase shift of the partial wave with angular momentum , denotes an irreducible representation (irrep) of the cubic group. The matrix in the determinant is labeled by the pair , in which are the angular momenta subduced into the irrep and is an index indicating the occurrence of that in the irrep. The matrix element is a complicated function of but can be computed numerically.
In this work we are interested in the s-wave low energy scattering in the isospin-0 channel. Therefore, we will compute only the lowest energy level in the center-of-mass frame. In this case one should consider the irrep . Assuming that the effects of the partial waves with are negligible, the matrix in the determinant of Eq. 5 becomes one-dimensional and the equation reduces to
where is the Lüscher zeta-function which can be evaluated numerically for given value of . Using the effective range expansion of s-wave elastic scattering near threshold, we have
where is the scattering length and is the effective range parameter. Once the finite volume energy is determined from lattice QCD simulations, the scattering length can be calculated from the following relation
It should be pointed out that Lüscher’s formulas presented here, i.e. Eqs. 5 and 6, are for the scattering processes with . The phase shift in the continuum is only defined for positive . In the case of negative , one can introduce a phase which is related to by analytic continuation of Luscher:1990ux (). Only when there is a bound state at the corresponding energy, the phase has physical interpretation and its value equals to (modulo ) in the continuum and infinite volume limit. For the purpose of this paper – calculating the scattering length, we will only consider the lowest energy level in the center-of-mass frame. Since the interaction in the isospin-0 channel is attractive, this energy level is below the threshold, i.e. . For convenience, in the following we will always use the notation , which is understood as the analytically continued form for . Please note that Eq. 8 holds for both positive and negative as long as the modulus of is close to zero.
Iv Finite volume spectrum
In lattice QCD, the discrete spectra of hadronic states are extracted from the correlation functions of the interpolating operators that resemble the states. Due to the isospin symmetry breaking of the twisted mass quark action, it is difficult to investigate the isospin-0 channel directly in unitary twisted mass lattice QCD Buchoff:2008hh (). For this reason we use a mixed action approach with the OS action Frezzotti:2004wz () in the valence sector and choose in Eq. 2 for both up and down quarks, so that the isospin symmetry is guaranteed in the valence sector. In this section we describe our methodology to calculate the energies of the isospin-0 system.
iv.1 Computation of the correlation functions
We define the interpolating operator that represents the isospin-0 state in terms of OS valence quarks
with single pion operators summed over spatial coordinates to project to zero momentum
Here and represent the OS up and down quarks, respectively. With OS valence quarks all three pions are mass degenerate and will be denoted as .
The energy of the isospin-0 state can be computed from the exponential decay in time of the correlation function
where is the temporal lattice extend. The four diagrams contributing to this correlation function, namely the direct connected diagram , the cross diagram , the box diagram and the vacuum diagram , are depicted in Fig. 1 (a)-(d). The correlation function can be expressed in terms of all relevant diagrams as
and the contributions from individual diagrams and are plotted in Fig. 2 for the three ensembles.
Even though we have full SU isospin symmetry in the valence sector when using OS valence quarks as described above, we have to consider effects of unitarity breaking. This may in particular happen due to the vacuum diagram . There is no symmetry available to prevent this diagram to couple for instance to intermediate states of unitary neutral pions (the neutral pion has vacuum quantum numbers in maximally twisted mass lattice QCD), since parity is not a good quantum number for our action. Since the neutral pion is the lightest meson in the spectrum with Wilson twisted mass fermions at finite value of the lattice spacing, the appearance of such states with (and maybe ) will dominate the large Euclidean time behavior of the correlation function , if the overlap of the used interpolating operators with these states is nonzero. In order to resolve this mixing, we build a matrix of correlation functions
with labeling the operator and the unitary neutral pion operator
where and are the (unitary) Wilson clover twisted mass up and down quarks. We use to distinguish it from OS down quark in Eq. 10. The twisted mass up quark coincides with the OS up quark with our matching scheme of the OS to the unitary action.
The computation of the disconnected diagrams, e.g. Fig. 1 (c), (d), (f), and (g), requires the quark propagator from a time slice to the same time for every . This has been a challenge in lattice QCD for decades. By using the stochastic LapH quark smearing method Peardon:2009gh (); Morningstar:2011ka (), we have all-to-all propagators available. The disconnected diagrams can be computed directly from those propagators.
We can reduce lattice artefacts in the vacuum diagrams following the ideas worked out in Ref. Ottnad:2015hva (). In the continuum limit the difference between () quarks corresponding to and () quarks corresponding to vanishes Frezzotti:2004wz (). Therefore, we can write
where are the OS quark propagators and we have used . This shows that the imaginary part of the loop needed in the contraction of the vacuum diagram is a pure lattice artefact and we will drop it in the computation. The same argument holds for the vacuum diagrams shown in Fig. 1 (f) and (g).
iv.2 Determination of the energies
Due to the finite temporal extend of the lattice, the correlation functions of multiparticle operators are polluted by so-called thermal states Feng:2009ij (). In the case of interest here, there is a constant contribution to of the form
which vanishes in the infinite volume limit . However, at finite -values it will dominate the correlation function at large Euclidean time. To remove this artefact we define a shifted correlation matrix
The new matrix is then free of any constant pollution from the thermal states. The value of can be adjusted for optimal results. We take in our analysis. Note that the shifting procedure also subtracts any constants stemming from vacuum expectation values.
The energy levels can then be obtained by solving the generalized eigenvalue problem (GEVP) Michael:1982gb ()
The eigenvalues are expected to have the following time dependence
The -like behavior comes from the shifting of the correlation functions in Eq. 15. The energies are then obtained by fitting the above functional form to the eigenvalues in the range where the effective energy shows a plateau. The value of should be chosen such that the correlation function at is dominated by the states we are interested in. We tried various values in the range of 1 to 7 and found negligible differences in the energies. In the following we set . With the two operators used here, we obtain two energy levels, one of which is far below the other one. The lower one agrees with the unitary neutral pion mass, while the higher one is close to . Hence, we identify the higher one to be the isospin-0 state. In principle, multi neutral and charged unitary pion states could also appear in the spectrum. To resolve these states, more operators need to be included. We have tried so and merely found increased statistical errors of the state. Therefore, we cannot finally exclude possible contaminations from such states.
As an example, the effective energies of the two eigenvalues for ensemble cA2.09.48 are shown in Fig. 3 (a). The fitted energies and fit ranges are indicated by the grey bands in the plot.
To further improve our results we adopt a method to remove the excited state contaminations Neff:2001zr (), which we have recently used successfully to study and mesons Jansen:2008wv (); Michael:2013gka (). It is based on the assumption that vacuum diagrams are only sizeable for low lying states, but negligible for higher excited states. Of course, the validity of this assumption must be checked in the Monte-Carlo data. In our case we know there is a very sizable disconnected contribution to the unitary neutral pion, which represents a pure lattice artefact Dimopoulos:2009qv (). A similar contribution has not been found to any other unitary correlation function. For the correlation function there are indications that the disconnected contribution is already small by itself Fu:2013ffa ().
The connected contractions contributing to are computed with sufficient precision, so we can reliably determine the ground states in the connected correlators and subtract the excited state contributions. We then build a correlation matrix from the subtracted connected and the original disconnected correlators. If disconnected contributions were relevant only for the ground states, one should find – after diagonalizing – a plateau for both states from small values of on. Note that with this procedure only the small behavior of the correlation functions is altered.
To be more specific, the connected contributions to the correlation function is given by
We fit the functional form Eq. 17 to the shifted correlator . After obtaining the parameters and from the fit, the connected correlator is reconstructed as
in which the excited state contaminations are subtracted. We repeat the fit to the data of for many different fit ranges. The expectation values of the fit parameters are computed as the weighted median over these many fits Helmes:2015gla (). By doing this, the systematics arising from different fit ranges is expected to be taken into account. The full correlator is then given by , where is the shifted vacuum correlator . The same procedure is performed for the unitary correlation function.
In Fig. 3 (b), we present the effective energies of the two eigenvalues of the subtracted correlator matrix for the same ensemble as in Fig. 3 (a). Clearly a plateau appears at much earlier -value compared to Fig. 3 (a), while the fitted energies agree very well. Therefore, we use this procedure – which allows us to determine in particular the interacting energy with much higher accuracy – for the results presented here.
The effective energies after removing the exited states for the ensembles and are shown in Fig. 4. In Table 2, we collect the values of and the unitary mass obtained from the procedure described above. The unitary charged pion mass and the OS pion mass are added for convenience.
In order to estimate possible artefacts from the mixing with the unitary , we also determined the energy by fitting to only the single correlator , without including the operator for the unitary neutral pion. The values are given in the last column of Table 2. One can see that the mean value of is slightly lower than for all three ensembles, but they agree very well with each other considering the statistical error. Keeping in mind that mixing is purely a lattice artefact, the agreement between and indicates that we are not likely to suffer severe lattice artefact here. This can be also seen in the small mass splitting between the unitary charged and neutral pions.
v.1 Scattering length
The scattering momentum is calculated from Eq. 3 with the energies and the OS pion masses listed in Table 2. Then the scattering length can be obtained from Eq. 8. Considering the relatively strong interaction in the isospin-0 channel, one has to investigate the contribution of and higher order terms in the effective range expansion. Since we only have one energy level for each pion mass, we are not able to determine the effective range with our lattice simulations. We rely on the values determined from PT Gasser:1983yg (). See Appendix A for the details of the values used in our analysis.
The values of , and in lattice units for all three ensembles are given in Table 3. For the ensembles and the scattering momentum is small due to the large volume. Therefore, the contribution of is expected to be small. As visible from Table 3, the value of is indeed less than of for these two ensembles. We compute the scattering length from Eq. 8 by ignoring the term, which is well justified for the ensembles and . The values of for these two ensembles are also given in Table 3. The first error is the statistical error and the second error comes from the uncertainty of the effective range .
As for the ensemble cA2.60.32, the value of is rather large compared to . This indicates that the effective range expansion up to might not be a good approximation and the term might not be negligible. Since the contribution of is unclear, we refrain from giving the scattering length for this ensemble. There are two possible reasons for the invalidity of the effective range expansion. First, due to the relatively small volume of the ensemble cA2.60.32, the value of is not small enough to make the expansion converge rapidly. Second, at the pion mass around 400 MeV, which is the OS pion mass of the ensemble cA2.60.32, there might be virtual or bound state poles appearing in the isospin-0 scattering amplitude Albaladejo:2012te (); Pelaez:2010fj (); Hanhart:2008mx (); hanhart:priv (); Bernard:2010fp (); Doring:2011vk (). The appearance of such states will give a very large scattering length – positively (negatively) large if it was a virtual (bound) state. Hence, the leading order in the effective range expansion, i.e. , becomes very small compared to the higher orders. In this case the NLO term can contribute significantly compared to the LO term even when is small. Assuming that the contribution of term is not bigger than the term, we can qualitatively estimate the scattering length for this ensemble to be a very large positive number, which features a virtual state. However, we do not exclude the possibility of a bound state because we do not include single meson operators when we compute the matrix of correlation functions. Including these operators might change the resulting spectrum and thus the scattering length.
v.2 Chiral extrapolation
In order to obtain the scattering length at the physical pion mass, one needs to perform a chiral extrapolation. scattering has been studied extensively in PT in the literature Weinberg:1966kf (); Gasser:1983yg (); Bijnens:1997vq (); Bijnens:1995yn (); Colangelo:2001df (). Since we only have two data points, we fit the NLO PT formula, which contains one free parameter, to our data. When expressed in terms of and computed from lattice simulations, the formula reads Fu:2013ffa ()
where is a combination of the low energy coefficients ’s :
In this expression, the renormalization scale is set to be the physical pion decay constant . Note that we work in the normalization with . By using the formula in Eq. 20, we have ignored the effects of unitarity breaking. In principle, we should use mixed action PT to perform the chiral extrapolation. The PT for the mixed action with twisted mass sea quarks and OS valence quarks is presented in Ref. WalkerLoud:2009nf (). However, using the two data points at one value of lattice spacing, we are not able to implement the mixed action PT formula. With this caveat in mind, we proceed with our analysis.
The OS pion decay constant has not been determined by ETMC yet. We compute for the three ensembles used in this paper. The values of are collected in the last column of Table 3. The details of their computation are presented in Appendix B. We recall that the OS pion mass values are larger than their unitary counterpart, see Table 2, such that our lowest OS pion mass value is at around 250 MeV.
The method we are applying here is valid only in the elastic region. Therefore, the pion mass values must be small enough to be below threshold where the meson becomes stable. This threshold is not known exactly, but results obtained with the 1-loop inverse amplitude method Hanhart:2008mx () (see also Refs. Pelaez:2010fj (); Hanhart:2014ssa (); Albaladejo:2012te ()) suggest that should be safe hanhart:priv (). Our two data points are obtained at pion mass around 250 MeV and 320 MeV respectively, both are below this threshold. Furthermore, the pion mass value should also be small enough to make the chiral expansion valid. To be safe, we perform the chiral extrapolation using only the data point with the lower pion mass ( 250 MeV). The results of this extrapolation are given in Table 4 as fit-1 and illustrated in Fig. 5. The results of the fit with the two data points, which are given in Table 4 as fit-2, agree with fit-1 within errors. We take the difference as an estimate of the systematics arising from chiral extrapolation. This leads to our final result for the scattering length:
We remark here that the extrapolation is strongly constrained since must vanish in the chiral limit. This explains the small statistical uncertainty on the value extrapolated to the physical point.
We compare our result in Table 5 to other results available in the literature. Our result for is lower, but within errors still compatible with most recent experimental, lattice and Roy equations results. Due to our comparably low value for , the value for is also relatively low. This is a direct consequence of the NLO PT description we are using.
Vi Discussion and Summary
In this paper, the isospin-0 scattering is studied with Lüscher’s finite volume formalism in twisted mass lattice QCD. We use a mixed action approach with the OS action in the valence sector to circumvent the complications arising from isospin symmetry breaking in the twisted mass quark action. The stochastic LapH quark smearing method is used to compute all-to-all quark propagators, which are required to compute the quark disconnected diagrams contributing to the isospin-0 correlation function. The lowest energy level in the rest frame is extracted for three ensembles with three different pion mass values. The scattering length is computed with Lüscher’s formula for the two ensembles with the lowest pion mass values. For the third ensemble with the largest pion mass value the scattering length cannot be determined reliably. In the computation of the scattering length we include the term in the effective range expansion using values for the effective range, which we compute using PT. The chiral extrapolation of is performed using NLO PT. Extrapolated to the physical value of , our result is , which is compatible within errors with the newer experimental and theoretical determinations available in the literature.
Since we work at a single lattice spacing value only, we cannot estimate lattice artefacts in our result. In particular, we cannot exclude that our result is affected by residual systematic uncertainties stemming from unitarity breaking, which will vanish in the continuum limit. Moreover, the control over higher order contributions from PT is rather limited. We cannot exclude that such contributions are sizable.
For these reasons a future study should include several lattice spacing values and ideally ensembles at the physical point. In order to avoid isospin breaking and unitarity breaking effects, we will repeat this computation with an action without isospin breaking.
Acknowledgements.We thank the members of ETMC for the most enjoyable collaboration. The computer time for this project was made available to us by the John von Neumann-Institute for Computing (NIC) on the Jureca and Juqueen systems in Jülich. We thank A. Rusetsky and Zhi-Hui Guo for very useful discussions and R. Briceño for valuable comments. We are grateful to Ulf-G. Meißner for carefully reading this manuscript and helpful comments. This project was funded by the DFG as a project in the Sino-German CRC110. S. B. has received funding from the Horizon 2020 research and innovation program of the European Commission under the Marie Sklodowska-Curie programme GrantNo. 642069. This work was granted access to the HPC resources IDRIS under the allocation 52271 made by GENCI. The open source software packages tmLQCD Jansen:2009xp (), Lemon Deuzeman:2011wz (), DDAMG Alexandrou:2016izb () and R R:2005 () have been used.
Appendix A Effective range from Pt
In order to investigate the contribution of the term in the effective range expansion, we need to know the value of the effective range . As explained in Section V.1, we estimate from PT.
In Ref. Gasser:1983yg (), the chiral expansion of the threshold parameter to NLO is given as
where is the renormalization scale, , and are the renormalized, quark mass independent couplings. In this expression, we have replaced the low energy parameters and with their (lattice) physical values and using the NLO chiral expansions of and , which are given in the same reference. Please also note that our convention of (130 MeV) is different from the (92.4 MeV) used in Ref. Gasser:1983yg ().
The effective range is related to as . In order to avoid the uncertainty in lattice scale setting, we write in lattice units as a function of the dimensionless parameters and :
Here the renormalization scale is set to be the physical pion decay constant . To write the formula as a function of , we have replaced with . The corrections due to this replacement appear at next-to-next-to-leading order.
We take the values of the scale independent couplings , and determined in Ref. Bijnens:2014lea ():
From the relations between and
with , and , we calculate the values of at :
The effective range is calculated with the ’s in Eq. 26 and the values of and in Table 2 and Table 3. The results of for the three ensembles are presented in Table 6. The errors are estimated from the errors of ’s and the statistical uncertainties of and .
Appendix B Determination of the OS values
The chiral extrapolation of the scattering length is most conveniently performed in . For this reason we need to compute also the OS pion decay constant. It is given by the following relation Constantinou:2010gr ()
with the (OS valence quark) axial vector component . The corresponding renormalization constant has been determined in Ref. Alexandrou:2015sea () for the action and -value used here. It reads
The matrix element can be determined from suitable correlation functions. We used the operator
together with from Eq. 10 to build a correlation matrix. For the matrix element we need local operators, hence, we cannot use sLapH. Instead, we performed dedicated inversions with local operators and the one-end-trick Boucaud:2008xu (). Since the off-diagonal correlators have a -like behaviour, we perform a constrained fit to this correlator matrix to determine the ground state energy and the corresponding matrix element at large Euclidean times.
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