Precise determination of the and pole parameters from a dispersive data analysis
We use our latest dispersive analysis of scattering data and the very recent experimental results to obtain the mass, width and couplings of the two lightest scalar-isoscalar resonances. These parameters are defined from their associated poles in the complex plane. The analytic continuation to the complex plane is made in a model independent way by means of once and twice subtracted dispersion relations for the partial waves, without any other theoretical assumption. We find the pole at MeV and that of the at MeV, whereas their respective couplings to two pions are GeV and GeV.
The or sigma and resonances are of great interest in several fields of Physics. First, the two pion exchange in the scalar-isoscalar channel, I=0, J=0, where these resonances appear, plays a key role in Nuclear Physics, where the nucleon-nucleon attractive interaction has been for long Johnson:1955zz () modeled by the exchange of a “sigma” resonance. Second, this channel is also relevant for the QCD non-Abelian nature, since it is where the lightest glueball is expected to appear. However, the glueball identification is complicated by its possible mixing into different states, like the , and heavier resonances, which may be mesons, tetraquarks, molecules, or most likely a mixture of them all. Actually, most of the controversy around these resonances comes from the identification of scalar multiplets —see the Review of Particle Physics (PDG) “Note on Scalar Mesons”PDG (). Third, the , being the lightest hadronic resonance with vacuum quantum numbers, plays a relevant role in many models of QCD spontaneous chiral symmetry breaking. Furthermore, this state is of interest in order to understand why, despite being so light and strongly coupled to pions, it plays such a small role, if any, in the saturation Ecker:1988te () of the low energy constants of Chiral Perturbation Theory (ChPT). Moreover, the position of this pole could be setting the limit of applicability of the chiral expansion. Finally, this state is of interest for Electroweak Physics due to its many similarities —but even more by its many differences—with the Higgs mechanism now under scrutiny at LHC.
Still, the properties of these resonances are the subject of an intense debate. Let us recall that the was listed in the PDG as “not well established” until 1974, removed in 1976, and listed back in 1996. This was due to its width being comparable to its mass, so that it barely propagates and becomes a broad enhancement in the traditional, and often contradictory, scattering analyses, extracted from experiments, using different models affected by large systematic uncertainties. After 2000 these resonances have been observed in decays of heavier mesons, with well defined initial states and very different systematics from scattering, which led the PDG to consider, in 2002, the as “well established”, but keeping until today a too conservative estimate of: “Mass: 400 to 1200 MeV” and “Width: 600 to 1000 MeV”. For the the situation is not much better, with an estimated width “from 40 to 100 MeV”. However, not all the uncertainty comes from experiment. The shape of these resonances varies from process to process and that is why their masses and widths are quoted from their process independent pole positions, defined as . But many models do not implement rigorous analytic continuations and lead to incorrect determinations when poles are deep in the complex plane or close to threshold cuts, as it happens with the and the , respectively. Actually, this is one of the main causes of the huge PDG uncertainties PDG ().
This model dependence can be avoided by using dispersive techniques, which follow from causality and crossing, and provide integral relations and a rigorous analytic continuation of the amplitude in terms of its imaginary part in the physical region, which can be obtained from data. For example, dispersion relations combined with ChPT determine the pole at Dobado:1996ps () or Zhou:2004ms (). We focus here on dispersive analyses, but other approaches yield similar values sigma (); Oller:1998hw ()—see Table 1 and Klempt:2007cp () for a review and references.
|Pelaez:2010fj () (fit D)||3.5|
Generically, the main difficulty lies in the calculation of the left cut integral, which in Dobado:1996ps (); Zhou:2004ms () was just approximated. This left cut is due to crossing symmetry and can be incorporated rigorously in a set of infinite coupled equations written long ago by Roy Roy:1971tc () ( see also roy70 () for applications and references). Recently, Roy equations have been used to study low energy scattering Ananthanarayan:2000ht (), sometimes combined with ChPT Colangelo:2001df (), or also to test ChPT DescotesGenon:2001tn (), as well as to solve old data ambiguities Kaminski:2002pe (). Most recently Caprini:2005zr (), the and poles were shown to lie within the applicability region of Roy eqs. Since data were not reliable and to improve accuracy, Roy eqs. were supplemented by ChPT predictions in Caprini:2005zr (), to yield : , without using data below 800 MeV on S and P waves. In that work an pole is also found at . Note that, generically, scattering data around 900 MeV tend to produce a narrower Oller:2003vf (); Oller:1998hw (); Caprini:2005zr () than that seen in production processes or the PDG estimate. In Table 2 we list some other recent determinations of the parameters.
Our aim in this work is to provide a precise and model independent simultaneous determination of the and parameters from data alone, profiting from two relevant results developed over the last half year: On the one hand, the final analysis of decays by the NA48/2 Collaboration ULTIMONA48 (), which provides reliable and precise scattering phases below the mass of the kaon. On the other hand, a set of Roy-like eqs.—called GKPY eqs. and developed by our group GarciaMartin:2011cn ()— which is much more stringent in the resonant region than standard Roy eqs. The reason is that, in order to avoid divergences, dispersion relations are weighted at low energy with “subtractions”, but then amplitudes are only determined up to a polynomial, whose coefficients depend on threshold parameters. Since Roy eqs. have two subtractions they have an polynomial term multiplied by the isospin 2 scalar scattering length, whose large uncertainty thus grows markedly in the and region. In contrast, the GKPY eqs. have just one subtraction and their output, even without using ChPT predictions at all, provides GarciaMartin:2011cn () a very precise description of scattering data, discarding a long-standing conflict concerning the inelasticity—and to a lesser extent the phase shift—right above the region.
If we now use these GKPY dispersion relations to continue analytically that amplitude, we find:
Let us describe next the whole approach in detail and provide determinations for other quantities of interest, like their couplings and the parameters, as well as other checks of our calculations from Roy eqs.
Ours is what is traditionally called an “energy-dependent” analysis of scattering and decay data pipidata (); Rosselet:1976pu ()—in particular the latest results from NA48/2 ULTIMONA48 (). Our procedure, described in a series of works Kaminski:2006qe (); GarciaMartin:2011cn () was first to obtain a simple set of unconstrained fits to these data (UFD) for each partial wave separately up to 1420 MeV, and Regge fits above that energy. Next we obtained constrained fits to data (CFD) by varying the UFD parameters in order to satisfy within uncertainties two crossing sum rules, a complete set of Forward Dispersion Relations as well as Roy and GKPY eqs., while simultaneously describing data. The details for all CFD waves can be found in GarciaMartin:2011cn (), but since we are now interested in the scalar isoscalar partial wave , we show in Fig. 1 the resulting phase shift. It should be noticed that the CFD result is indistinguishable to the eye from the UFD, except in the 900 to 1000 MeV region, which we also show in detail and is essential for the determination of the parameters. Note that both the UFD and CFD describe the data in that region, but the GKPY dispersion relations require the CFD phase to lie somewhat higher than the UFD one. This is relevant since it yields a wider , correcting the above mentioned tendency to obtain a too narrow from unconstrained fits to scattering data alone. In the inner top panel, we show the good description of the latest NA48/2 data on decays, which are responsible for the small uncertainties in our input parametrization and constrain our subtraction constants. As seen in Fig. 1, the inelasticity shows a “dip” structure above 1 GeV required by the GKPY eqs. GarciaMartin:2011cn (), which disfavors the alternative “non-dip” solution. Having this long-standing “dip” versus “no-dip” controversy Au:1986vs () settled GarciaMartin:2011cn () is very relevant for a precise determination.
The interest of this CFD parametrization is that, while describing the data, it satisfies within uncertainties Roy and GKPY relations up to their applicability range, namely 1100 MeV, which includes the region. In addition, the three Forward Dispersion Relations are satisfied up to 1420 MeV. In Fig. 2 we show the fulfillment of the S0 wave Roy and GKPY eqs. and how, as explained above, the uncertainty in the Roy eq. is much larger than for the GKPY eq. in the resonance region. The latter will allow us now to obtain a precise determination of the and poles from data alone, i.e. without using ChPT predictions.
Hence, we now feed our CFD parameterizations as input for the GKPY and Roy eqs., which provide a model independent analytic continuation to the complex plane, and determine the position and residues of the second Riemann sheet poles. It has been shown Caprini:2005zr () that the and poles lie well within the domain of validity of Roy equations, given by the constraint that the values which are integrated to obtain the partial wave representation at a given should be contained within a Lehmann-Martin ellipse. These are conditions on the analytic extension of the partial wave expansion, unrelated to the number of subtractions in the dispersion relation, and equally apply to GKPY eqs.
Thus, in Table 3 we show the , and poles resulting from the use of the CFD parametrization inside Roy or GKPY eqs. We consider that our best results are those coming from GKPY eqs. since their uncertainties are smaller, although, of course, both results are compatible.
Several remarks are in order. First, statistical uncertainties are calculated using a MonteCarlo Gaussian sampling of the CFD parameters with 7000 samples distributed within 3 standard deviations. A systematic uncertainty due to the different charged and neutral kaon masses is relevant for the due to the existence of two thresholds separated by roughly 8 MeV, which we have treated as a single threshold at MeV. In order to estimate this systematic uncertainty, we have refitted the UFD and CFD sets to the extreme cases of using or instead of . As it could be expected, the only significant variation is for the —actually, only for its half width, which changes by MeV for GKPY eqs, and MeV for Roy eqs. The changes by roughly 1 MeV and the barely notices the change—less than 0.1 MeV. The effect on residues is smaller than that of rounding the numbers. We have added all these uncertainties in quadrature to the statistical ones. Second, both the mass and width of the are compatible with those in ref.Caprini:2005zr () within one standard deviation. Since we are not using ChPT and ref.Caprini:2005zr () did not use data below 800 MeV, this is a remarkable check of the agreement between ChPT and low energy data. Third, the width is no longer so narrow—as it happens in typical scattering analyses— and we find MeV, very compatible with results from production processes. The mass overlaps within one standard deviation with the PDG estimate. These results show that the effect of the too narrow pole and the use of further theoretical input like ChPT do not affect significantly the resulting parameters.
In Table 3 we also provide for each resonance its coupling to two pions, defined from its pole residue as:
where . This residue is relevant for models of the spectroscopic nature of these particles, particularly for the sigmacoupling (), which are beyond the pure data analysis scope of this work. Differences between previous values of these couplings can be seen in Tables 1 and 2.
In summary, using a recently developed dispersive formalism, which is especially accurate in the resonance region, we have been able to determine, in a model independent way, the , poles and couplings from data with no further theoretical input. We hope this works helps clarifying the somewhat controversial situation regarding the parameters of these resonances.
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