Sigma resonance parameters from a lattice QCD simulation
Abstract
In this work we present the analysis of the energy spectrum from a recent twoflavor () lattice QCD calculation for pionpion scattering in the scalar, isoscalar channel (the meson). The lattice simulation was performed for two quark masses corresponding to a pion mass of and . The meson parameters are extracted using various parametrizations of the scattering amplitude. The results obtained from a chiral unitary parametrization are extrapolated to the physical point and read , where the uncertainties in the parentheses denote the stochastic and systematic ones. The behavior of the meson parameters with increasing pion mass is discussed as well.
XIV International Workshop on Hadron Physics,
1823 March 2018,
Florianópolis, SC, Brazil
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I Introduction
The is one of the resonances more discussed in the literature. Several phenomenological models have pointed out that it is relevant for nuclear forces. However, its properties have been subject to a long discussion due to a variety of models which tried to determine its mass and width from experimental data, often showing incompatible results. Even its existence was several times questioned and the Review of Particle Properties did not show it for twenty years. The particle is nowadays a wellestablished resonance whose pole properties have been determined more precisely from dispersion relations and coupledchannel approaches which rely upon chiral Lagrangians, analyticity, and unitarity relations Pelaez:2015qba ().
Recently, the GWU group has extracted the energy spectrum and phase shifts for pionpion scattering in the scalar, isoscalar channel, in a lattice QCD simulation for two different pion masses, and MeV. In this work, we present the analysis of these data by using several parameterizations and estimate the uncertainties due to the use of one or other. First, we consider a general expansion in an energyvariable conformally mapping the energy plane to the unit disk, similar to the analysis of Refs. Yndurain:2007qm (); Caprini:2008fc (). Second, we employ a model based on the chiral unitary approach (UChPT), used, e.g., in Refs. Hu:2017wli (); Hu:2016shf (); Guo:2016zos (). Subsequently, the UChPT amplitude is extrapolated to the physical point. Our final result, based on all lattice data presented here with and without the isovector channel data Guo:2016zos (), reads MeV and agrees with the result of the most recent analysis of experimental data Pelaez:2015qba () within the quoted region. Additionally, the study of the pion mass dependence of resonance mass and coupling to the channel is studied in a broader range of , as it was done in previous works Doring:2016bdr (); Hanhart:2008mx ().
Ii Analysis of the scattering amplitude
The discrete energy spectrum is obtained from the finite volume scattering amplitude,
(1) 
which depends explicitly on the form of the matrix Doring:2012eu (). For boxes with asymmetry in the direction and in the rest frame,
(2)  
and , being the momentum in the rest frame, , and is two pion loopfunction written in the conventional dimensional regularization form Guo:2018zss ().
The positions of poles in Eq. (1) determine the discrete energy spectrum on the lattice. The form of the function is not fixed by unitarity. In this work we use four versions of two different types of the matrix to gauge the systematic uncertainty tied to a particular choice.
We analyze data from different pion masses and channels individually and also simultaneously. The parameterizations, channels and pion masses employed and number of parameters (pars.) in the fits are:
 Individual fits, or MeV,

Conformal parameterizations, (2 pars.)

Chiral parameterization, and (4 pars.)
 Combined fits, for and MeV,

Chiral parameterization, (3 pars.)

Chiral parameterization, and (4 pars.).
For the conformal parameterizations, Eqs. (7) and (8), and (26) and (27) from Ref. Caprini:2008fc (), are used. These are named [cm1] and [cm2] respectively. The relation of in Ref. Caprini:2008fc () with the Kmatrix in Eq. (1), is , with . For the chiral parameterizations, we employ the chiral unitary approach described in Ref. Oller:1998hw (). We call them [chm1] and [chm2] for fits, and analyses, respectively. The data corresponding to the channel was analyzed previously in Ref. Guo:2016zos (). The same set of data is used in this work. The above parameterizations and details of the fits are described in detail in Refs. Guo:2016zos (); Guo:2018zss (). Below, we summarize our main results.
Iii Results
The results of the fits are collected in Table I of Ref. Guo:2018zss (), which all pass the Pearson’s test with the total lying inside of the 80% confidence interval. We discuss first the results from individual fits. The phase shifts are depicted in Fig. 1. There is a clear overlap of the fits with the data. Only for the heavy pion mass and energies deep below threshold there is some discrepancy that could be significant. Note again that the phaseshifts are not fitted directly, but rather the energies extracted from lattice QCD. There is a good agreement between the phase shifts from both conformal parameterizations used in the channel [cm1] and [cm2], that we are interested here, and also with the [chm2] chiral individual fits. Note that we used the conformal parameterizations with two parameters. Since the chiral model for alone has three parameters, uncertainties are larger if we only perform individual fits. Indeed, the chiral parameterization contains LECs which appear in both, and channels. Thus, the fit of both channels, and , constrains better the LECs used.
We perform an analytical continuation to the complex energy plane. On the second Riemann sheet of this plane we determine the position () and residuum () of the resonance pole. The results are collected in Table 1. In the three different parameterizations, and for individual fits at or MeV, there is a good agreement in the pole position, which lies above threshold, being a resonance in both cases.
For combined fits, the corresponding phaseshifts are depicted in Fig. 2. As shown in this picture, fitting the channel alone leads to larger uncertainties in the chiral unitary model, than in combination with the channel. Note that uncertainites are also larger in this channel for the raw lattice data. The best fit parameters are collected in Table I of Ref. Guo:2018zss (). For the lowenergy constants we observe agreement between the individual and combined fits results. The extrapolated phaseshifts to the physical point are depicted in Fig. 3, showing a good agreement with the experimental data below energies of MeV. The results for the pole properties are collected in the rows 58 of Table 1. The pole position and residuum at the physical mass is compatible in both chiral parameterizations, [chm1] and [chm2]. There is also agreement in the pole properties for the other pion masses. Note that, in the case of the heavy pion mass, the pole position is pushed towards the threshold in the combined fit because of the presence of the light pion mass, becoming a virtual state. The uncertainites for the pole properties at this mass are larger, partly because the state is near the transition to a virtual state, and partly because of the larger uncertainites in the raw lattice data.
MeV  MeV  MeV  
Parametrization  Fitted data  
cm1  –  –  –  –  –  –  
cm1  –  –  –  –  –  –  
cm2  –  –  –  –  –  –  
cm2  –  –  –  –  –  –  
chm1  
chm2  
chm2  
chm2  
Ref. Pelaez:2015qba ()  experimental  –  –  –  –  –  – 
With this in mind we make a prediction of the pole position and the corresponding coupling to the channel as a continuous function of the pion mass based on [chm2] fitted to both sets of lattice data, and , and both pion masses simultaneously. To remind the reader these sets are obtained from calculations at and . The result of the extrapolation is depicted in Fig. 4. With increasing pion mass, both poles for positive and negative travel in the 2on Riemann sheet towards the threshold, coupling more strongly to this channel. For , both poles meet at the real energy axis below threshold on the second Riemann sheet becoming virtual bound states. For higher pion mass, the poles evolve on the real axis towards and away from the threshold (), respectively. One of the pole reaches the twopion threshold at , where the coupling vanishes, and becomes a bound state for heavier pion masses, then, the coupling to two pions increases in this region monotonically. This behavior is remarkably similar to the one of Ref. Hanhart:2008mx ().
Iv Conclusions
To extract the parameters of the resonance, we have to use a parameterization that satisfies physical constraints, in particular unitarity, analyticity, and proper chiral behavior. To gauge the systematics associated with the choice of such parametrizations, we used two types of approaches (each in several variants): a generic one that makes no assumption about the underlying dynamics, and a chiral perturbation theory inspired one that allows to extrapolate the resonance parameters to different (i.e., physical) pion mass. The systematic errors associated with the choice of parametrization are about 10% for the pole position.
One of the strengths of the chiral parametrization is that it allows us to fit simultaneously both the and channel, for both pion masses. We find that the model describes the data well and that the results extracted from the simultaneous fit to both channels agree well with the channel fit results. We use the combined fit to extrapolate to the physical point and, based on the position of the pole in the complex energy plane, we find that . Here the first error is the stochastic error and the second one is the combined systematic error discussed above.
The extrapolation to the physical point agrees with the experimental phaseshifts and the pole mass and width of the is compatible with the result of recent analyses based on experimental data.
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