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Chiral Extrapolations of the ρ(770) Meson in Nf = 2 + 1 Lattice QCD Simulations

Several lattice QCD simulations of meson-meson scattering in p-wave and Isospin = 1 in Nf = 2 + 1 flavours have been carried out recently. Unitarized Chiral Perturbation Theory is used to perform extrapolations to the physical point. In contrast to previous findings on the analyses of Nf = 2 lattice data, where most of the data seems to be in agreement, some discrepancies are detected in the Nf = 2 + 1 lattice data analyses, which could be due to different masses of the strange quark, meson decay constants, initial constraints in the simulation, or other lattice artifacts. In addition, the low-energy constants are compared to the ones from a recent analysis of Nf = 2 lattice data.


Introduction
Unitarized Chiral Perturbation Theory (UChPT) is a nonperturbative method which combines the constraints from coupled-channel unitarity and chiral symmetry. In this approach, several resonances such as the ρ and σ mesons are identified with poles of the scattering amplitude in the complex-energy plane. Instead of expanding the scattering amplitude in powers of momenta, as it is usually done in Chiral Perturbation Theory, an expansion of its inverse is preferred. This so-called inverse amplitude method was introduced in Ref. [1] and extended in Refs. [2][3][4][5][6][7][8]. In this approach, the O(p 2 ) and O(p 4 ) chiral Lagrangians [9,10] provide the kernel of the scattering equation. While the lowest order term qualitatively leads to the generation of the scalar resonances, the O(p 4 ) chiral Lagrangian, which depends on the low energy constants, is responsible for the generatation the vector-meson resonances [11]. As a result, the UChPT model with O(p 2 ) and O(p 4 ) chiral Lagrangians dynamically generates the scalar-and vector-meson resonances and describes scattering data up to 1.2 GeV [11].
The UChPT model can be implemented in the finite volume by imposing that the scattering states can only have discrete momenta. This formalism, developed in Refs. [12][13][14][15] is equivalent to the Lüscher approach up to contributions kept in Ref. [13] that are exponentially suppressed with the cube volume. In Ref. [16] the method is generalized to the cases of moving frames and partial-wavemixing in coupled channels.
The above model was used to analyze the energy levels extracted from a lattice simulation of the GWU group of ππ scattering in p-wave in two flavours, see Ref. [17]. The result turned out to be in agreement with other N f = 2 lattice simulations. Later, this study was extended to other lattice data sets in N f = 2 [18]. Two common features were observed from the analyses: 1) The error ellipses of the low energy constants have a common overlap; 2) All extrapolations led to a consistently lower ρ mass (∼ 60 MeV) than its physical value. Furthermore, the N f = 2 simulation from the RQCD group at a pion mass close to the physical mass, 149 MeV, gives a ρ meson mass of around 710 MeV [19].
While it is possible that the reason for discrepancy lies in the scale settings or other lattice QCD artifacts, in Refs. [17,18], it was shown that the discrepancy can also be accomodated in terms of the strange quark. There, by adding the KK channel a posteriori in the analysis of the N f = 2 lattice data, it was observed that the ρ mass shifts upward in around 60 MeV, consistently for the different lattice data sets, being thus a likely reason for discrepancy.
In this work, we analyze the lattice data from simulations of the ρ meson phase shifts in N f = 2+1 flavours, which have been carried out in recent years [20][21][22][23][24][25][26][27][28][29]. The theoretical framework is the oneand two-coupled channel UChPT model of Refs. [17,18]. Chiral extrapolations to the physical point are provided for the various simulations. The values of the low energy constants (LECs) are discussed. Finally, the role of the strange quark in the N f = 2 + 1 lattice simulations is investigated.

Formalism
The two-channel scattering equation reads, in the UChPT model, as follows [11], where V 2 and V 4 are 2 × 2 matrices which contain the transition potentials derived from the O(p 2 ) and O(p 4 ) Lagrangians of the ChPT expansion [9,10]. With the channel ordering (ππ, KK) these potentials, projected on I = 1 and L = 1, V 2 and V 4 , are [17] − respectively. In the above equation, for the channel i, W is the centerof-mass energy, and m 1,2 refers to the masses of the mesons 1, 2 in the i channel. In the transition potential of Eq. (2), four distinct combinations of LECs are involved,l 1 ,l 2 , L 3 and L 5 , wherel 1 = 2L 4 + L 5 , andl 2 = 2L 1 − L 2 + L 3 . Note that in the present notation thel i are not the canonical SU(2) LECs but combinations of SU(3) LECs. In Eq. (1), G is a diagonal matrix whose elements are the two-meson loop functions, which are evaluated using dimensional regularization. The dependence on the subtraction constant in the case of the ρ meson can well be absorbed in the values of the LECs [18]. The meson decay constant dependence on the pion mass is taken from Ref. [6], where f π , f K are fitted in an analysis of combined lattice and experimental data.
The elements T i j of the scattering amplitude are related to S -matrix elements as follows, When the UChPT model is fitted to the experimental data for ππ and πK phase shifts, similarly as in Ref. [30], we obtain the values for the LECs in Eq. (2) given in the last row of Table 1. In what follows, we will refer to this fit as "Experimental".
The UChPT model can be adapted to the conditions of the lattice simulations, see Refs. [12][13][14][15][16][17]. While in Ref. [17] the eigenenergies were fitted, here we directly fit the extracted phase shifts. The correlation between energy W and phase shift δ(W) is taken into account, and also the correlations between eigenenergies themselves if available. We refer the reader to Ref. [31] for a more complete description of the fitting procedure.

Results
The lattice data for p-wave ππ phase shifts in I = 1 of Refs. [20-22, 24, 27, 28] are fitted by the UChPT model described in the previous section. We skip the analysis of the data from Ref. [29] because they have large uncertainties. Note that in the two-coupled-channel problem, see Eq. (2), there are now four parameters for the minimization problem. When performing the fits strong nontrivial correlations between the parameters appear and many different χ 2 minima are found. Then, it becomes necessary to restrict the model to study the parametrization and chiral behaviour. With that in mind, four different strategies (a), (b), (c), and (d) are pursued, which are described below.
Fit (a) -fixed L 3 , L 5 . This strategy, with two LECs L 3 and L 5 fixed to their values obtained by fitting phase shifts from experiment [17,18], is identical to the approach of Refs. [17,18]. The result for the extrapolated phase shifts at the physical point, in comparison to the lattice and experimental data, is shown in Fig. 1. Continuous lines represent the result from the two-channel SU(3) fits. Here one-channel SU(2) fits were also done, and the result is depicted with dashed lines. Both, one-and two-coupled-channel fits give very similar extrapolated phase shifts, which demonstrates that the explicit dynamics of the KK channel can be almost fully absorbed in the LECs of the SU(2) fit. A more detailed discussion on the SU(2) UChPT fits can be found in the Appendix of Ref. [31].
As shown in Table 1, see the χ 2 d.o.f , the lattice data are well described by the SU(3) UChPT model. The extrapolated phase shifts, as Fig. 1 shows, agree with the experiment for the Wilson15 and Bulava16 data. However, in the case of the Dudek13 [20] data, the extrapolated phase shift is at slightly lower energies than the experiment, for Aoki11 the extrapolation is at higher energies, and for the Alexandrou17 data, is much lighter than the one from experiment. In Fig. 2 (left), results for (m ρ , g) for different lattice data sets are presented. Empty and filled symbols stand for SU(2) and SU(3) analyses, respectively. The experimental point is indicated as "phys.". For the Bulava16, Wil-son15 and Dudek13 data, the values of m ρ and g in the SU(3) fits are close to the physical value, with larger discrepancies for the Alexandrou17 and Aoki11 data sets.
Error ellipses for the (l 1 ,l 2 ) parameters, in the SU(3) analyses are shown in Fig. 3 (left). We note that the error ellipses do not have an as clearly common overlap region as in the analysis of N f = 2 data of Ref. [18]. In particular, the ellipses of Aoki11 and Alexandrou17 are separated from the bulk of other N f = 2 + 1 lattice data and the N f = 2 result from Lang11. In the analysis of the N f = 2 + 1 lattice data of Ref. [27] (Fu16), where we found some discrepancies with other sets of lattice data for similar pion masses, and also unusually large differences with the experimental data when the extrapolations are done, is discussed in Appendix of Ref. [31].
Fit (b) -free L 3 , L 5 . Before going into detail we want to make here a remark. The LECs that appear in the UChPT model are not only constraint by data in the p-wave, Isospin = 1, strangeness = 0 channel, but also in other channels, Isospin = 0, strangeness = −1, etc. This implies that by fitting with only data in the ρ channel can lead (and leads) to multiple solutions. Because of this, one can always choose one solution which has an extrapolation to the physical point compatible with the experiment (this would be almost equivalent to fitting the lattice data in combination with the Mπ =236MeV Wilson15 500 600 700 800 900 1000 1100  experimental data). However by no means this is a good prediction of the extrapolation of the lattice data. In order to have a good prediction of the lattice data at the physical point, more data are needed, as data for several pion masses, or in several channels, pion decay constants used etc. Otherwise it is convenient to impose extra constraints in the LECs, as for example it is done in strategy (a). This strategy will be useful to show that when allowing that the different lattice data sets have extrapolated phase shifts compatible with experiment, quite different values of the LECs are obtained for some of those, which is not possible. We choose a minimum in which the predicted χ 2 evaluated with the experimental phase shifts is small, i.e., a minimum with excellent chiral extrapolation, and compare the resulting values of L 3 and L 5 with strategy (a) for consistency. All extrapolated phase shifts for the different lattice data sets agree very well with experiment (not shown here, see Ref. [31] for further details) and the χ 2 d.o.f obtained is close to one for most of the lattice data sets, see Table 1. However, there are clear discrepances in the LECs for some of the lattice data sets. In Fig. 3 (right), the error ellipses for the L 3 , L 5 parameters obtained from minimization  [20-22, 24, 28]. The superscript (f) indicates parameters held fixed at their experimental values.
In the last two rows, the results from Ref. [17] and from the fit to the experimental data [30] are shown.    [20-22, 24, 28] (all N f = 2 + 1). The thick blue line shows the ellipse for strategy (d). The red dash-dotted ellipse (Lang11) represents the uncertainties from the analysis of the N f = 2 data of Ref. [33]; see Ref. [18]. The star stands for the result obtained in the fit to the experimental data (very small uncertainties, not shown). Right: Left: Error ellipses (68% confidence) for L 3 and L 5 in the minimization strategy (b) compared to the values obtained from the experimental fit (star).   [21,22,24,28] are highlighted in red. The extrapolation to the physical point in comparison to the experimental data (blue) [32] is shown in the bottom figure. Right: Density histogram from resampling obtained in minimization (c). Ellipses in white show the result obtained in Ref. [18] for the fit of the N f = 2 data of Ref. [33] (Lang11) using the SU(2) UChPT model. There, L 3 and L 5 were held fixed to their experimental values such that the corresponding ellipses degenerate to lines in the figure. The stars indicate the LECs obtained from experimental phase shifts. The diagonal elements show histograms for the distributions ofl 1 ,l 2 and L 3 .
(b) are shown. All the ellipses are close to the values of L 3 and L 5 obtained from the experimental fit, except for the Dudek13 and Alexandrou17 cases shown in the inset. In the latter cases, a good chiral extrapolation can only be achieved at the cost of very different values of the LECs.
Fit (c) -Combined data M π < 320 MeV. Data from all lattice simulations with M π smaller or equal to 320 MeV are fitted simultaneously, while L 5 is kept fixed to the value from the fit to the experimental data. In this case the number of minima found in strategy (b) is significantly reduced and we are able to pin down the values of the other low-energy constants with much higher precision. The obtained low-energy constants are shown in Table 1. Phase shifts are depicted in Fig. 4 (left). The UChPT model describes the lattice data consistently, except for the Aoki11 data (M π = 300 MeV). These data appear in tension to the others within the present model. The chiral extrapolation to the physical point postdicts the experimental phase shifts rather well (lower left figure). The corresponding values of (m ρ , g) are shown in Fig. 2 (right) as red hexagon. The value of m ρ is compatible with the physical one. To show the correlations between the parameters, density histograms obtained from resampling are displayed in Fig. 4 (right) for different pairs of low-energy constants. The ellipse show the results from the analysis of Ref. [18] of the N f = 2 data from Lang et al. [33]. The star stands for the result from the experimental fit. The uncertainty region (light areas) obtained from fit (c) does not overlap with the result from the N f = 2 analysis, which is also closer to the experiment.
Fit (d) -combined data Hadron Spectrum. The Hadron Spectrum Collaboration provides phase shifts for the ρ meson at two different pion masses of 236 and 391 MeV [20,21]. We fit these data simultanously with L 3 and L 5 fixed as in minimization (a). The quality of the fit is indeed very good, with a χ 2 d.o.f = 1.1, although the extrapolated ρ mass, ∼ 750 MeV, turns out to be slightly lower than the experimental value, see Fig. 2 (left). The LECs obtained in this strategy are given in Table 1. They are remarkably close to the values obtained in N f = 2 fits [17,18]. Notice thatl 1 which is related to the ρ mass [17] has been reduced a factor 2-3 from the result obtained for these sets in minimization With the values of the LECs obtained in this strategy, the pion mass dependence of the ρ is plotted in Fig. 5, in comparison with that from the analysis of Ref. [17], and with the lattice data. The N f = 2+1 Alexandrou17 ρ mass (orange filled square at M π = 317 MeV) is below the prediction of fit (d) and, in fact, very close to the N f = 2 GWU result at similar mass. The N f = 2 + 1 ρ masses of the Fu16 data [27] are larger than the predictions of fit (d) which could be related to a rather different value of the kaon mass or other problems (see discussion in Appendix).
To simulate the absence of the strange quark one can remove the KK channel from the result of fit (d). The red dashed band is obtained predicting consistently lighter ρ masses that are indeed compatible with most of the N f = 2 data shown in the figure (Lang11 [33], CP-PACS07 [34], GWU16 [17], RQCD16 [19]).
In Ref. [17], the KK channel was included after fitting the N f = 2 GWU16 data to simulate the missing strange quark of the lattice results. The outcome, including systematic model uncertainties, is shown with the blue solid band in Fig. 5. A substantial shift of the ρ mass was observed, this time to larger values. The outcome is in agreement with the physical ρ mass and, marginally, with fit (d).

Conclusions
The present analysis shows that UChPT can be used to parametrize the light-quark (or pion mass) dependence of the phase-shifts in the rho-channel well. In most of the N f = 2 + 1 simulations, the extrapolation to the physical point of the mass is close to the experimental value. Still, the analysis reveals inconsistencies between some of the N f = 2 + 1 data. Particularly, the Alexandrou17 (where the extrapolated ρ mass is significantly lower) and Fu16 (unknown value of the kaon mass) data sets, produce inconsistent results with other sets within our model. We also found that the LECs from a combined fit of the Hadron Spectrum Collaboration data for two different pion masses are in agreement with the ones of N f = 2 UChPT analyses, and that, when the KK channel is removed, the value of the mass of the ρ shifted downwards, in good agreement with most of the N f = 2 simulations and with the results found in Refs. [17,18]. This further shows that the disagreement between the extrapolation of the N f = 2 lattice results and the experiment is not an artifact of the extrapolation. As a further research direction, a combined fit to lattice data in other partial waves and, e.g., other