decays in Chiral Perturbation Theory with Resonances
We have studied the decays within Chiral Perturbation Theory including resonances as explicit degrees of freedom. We have considered three different form factors according to treatment of final-state interactions. In increasing degree of soundness: Breit-Wigner, exponential resummation and dispersive representation. We find that although the first one fails in accounting for the data on the mode, the other two approaches provide good fits to them which are sensitive to the pole parameters, that are determined to be MeV and MeV. These values are competitive with the standard determination from decays. The corresponding predictions for the channel respect the current upper bound and hint to the discovery of this decay mode in the near future.
PACS : 13.35.Dx, 12.38.-t, 12.39.Fe, 11.15.Pg, 11.55.Bq
Keywords : Hadronic tau decays, Chiral Lagrangians, Dispersion relations, Analytic properties of S matrix.
Semileptonic tau decays represent a clean benchmark to study the hadronization properties of QCD due to the fact that half of the process is purely electroweak and, therefore,
free of uncertainties at the required precision Braaten:1991qm (); Braaten:1988hc (); Braaten:1988ea (); Braaten:1990ef (); Narison:1988ni (); Pich:1989pq (); Davier:2005xq (); Pich:2013?? ().
At the (semi-)inclusive level this allows to extract fundamental parameters of the Standard Model, most importantly the strong coupling Davier:2008sk (); Beneke:2008ad (); Pich:2011bb (); Boito:2012cr (). Tau decays containing Kaons have been split into the Cabibbo-allowed and suppressed decays Barate:1999hj (); Abbiendi:2004xa ()
rendering possible determinations of the quark-mixing matrix element Maltman:2008ib (); Antonelli:2013usa () and the mass of the strange quark Chetyrkin:1998ej (); Pich:1999hc (); Korner:2000wd (); Kambor:2000dj (); Chen:2001qf (); Gamiz:2002nu (); Gamiz:2004ar (); Baikov:2004tk (); Gamiz:2007qs () at high precision.
At the exclusive level, the largest contribution to the strange spectral function is given by the decays (). The corresponding differential decay width was measured by the ALEPH Barate:1999hj () and OPAL Abbiendi:2004xa () collaborations, and recently the B-factories BaBar Aubert:2007jh () and Belle Epifanov:2007rf () have published increased accuracy measurements. These high-quality data have motivated several refined studies of the related observables Jamin:2006tk (); Moussallam:2007qc (); Jamin:2008qg (); Boito:2008fq (); Boito:2010me () allowing for precise determinations of the pole parameters because this resonance gives the most of the contribution to the dominating vector form factor. These were also determined for the resonance and the relative interference of both states was characterized, although with much less precision than in the case of the mass and width.
In order to increase the knowledge of the strange spectral function, the decays have to be better understood (they add up to one third of the strange decay width), the and decays being also important for that purpose. The mode is also very sensitive to the resonance contribution and may be competitive with the decays in the extraction of its parameters. This is one of the motivations for our study of the decays in this article. We will tackle the analysis of the decays along the lines employed in other three-meson GomezDumm:2003ku (); Dumm:2009kj (); Dumm:2009va (); Dumm:2012vb () and one-meson radiative tau decays Guo:2010dv (); Roig:2013?? () elsewhere.
The decays were first measured by CLEO Bartelt:1996iv () and ALEPH Buskulic:1996qs () in the ’90s. Only very recently Belle Inami:2008ar () and BaBar delAmoSanchez:2010pc () managed to improve these measurements reducing the branching fraction to essentially half of the CLEO and ALEPH results and achieving a decrease of the error at the level of one order of magnitude. Belle Inami:2008ar () measured a branching ratio of and BaBar delAmoSanchez:2010pc () , which combined to give the PDG average Beringer:1900zz (). The related decay has not been detected yet, although an upper limit at the confidence level was placed by BaBar Lees:2012ks ().
Belle’s paper Inami:2008ar () cites the few existing calculations of the decays based on Chiral Lagrangians Pich:1987qq (); Braaten:1989zn (); Li:1996md (); Aubrecht:1981cr () and concludes that ‘further detailed studies of the physical dynamics in decays with mesons are required’ (see also, e.g. Ref. Actis:2010gg ())222Very recently, the decays have been studied Kimura:2012 (). However, no satisfactory description of the data can be achieved in both decay channels simultaneously.. Our aim is to provide a more elaborated analysis which takes into account the advances in this field since the publication of the quoted references more than fifteen years ago. The considered decays are currently modeled in TAUOLA Jadach:1990mz (); Jadach:1993hs (), the standard Monte Carlo generator for tau lepton decays, relying on phase space. We would like to provide the library with Resonance Chiral Lagrangian-based currents Shekhovtsova:2012ra (); Nugent:2013hxa () that can describe well these decays for their analyses and for the characterization of the backgrounds they constitute to searches of rarer tau decays and new physics processes.
Our paper is organized as follows: the hadronic matrix element and the participating vector and scalar form factors are defined in section 2, where the differential decay distribution in terms of the latter is also given. These form factors are derived within Chiral Perturbation Theory () Weinberg:1978kz (); Gasser:1983yg (); Gasser:1984gg () including resonances () Ecker:1988te (); Ecker:1989yg () in section 3. Three different options according to treatment of final-state interactions in these form factors are discussed in section 4 and will be used in the remainder of the paper. In section 5, the decay observables are predicted based on the knowledge of the decays. These results are then improved in section 6 by fitting the BaBar and Belle data. We provide our predictions on the decays in section 7 and present our conclusions in section 8.
2 Matrix elements and decay width
We fix our conventions from the general parametrization of the scalar and vector matrix elements Gasser:1984ux ():
where . From eq. (1) one has
with , and . Instead of one can use defined through
and the normalization condition
which is obtained from
In terms of these form factors, the differential decay width reads
and Erler:2002mv () represents an electro-weak correction factor.
We have considered the mixing up to next-to-leading order in the combined expansion in , and Kaiser:1998ds (); Kaiser:2000gs () (see the next section for the introduction of the large- limit of QCD 'tHooft:1973jz (); 'tHooft:1974hx (); Witten:1979kh () applied to the light-flavoured mesons). In this way it is found that , , where Ambrosino:2006gk ().
The best access to is through semi-leptonic Kaon decay data. We will use the value Beringer:1900zz (); Antonelli:2010yf (). Eq. (8) makes manifest that the unknown strong-interaction dynamics is encoded in the tilded form factors, which will be subject of our analysis in the following section. We will see in particular that the use of instead of the untilded form factors yields more compact expressions that are symmetric under the exchange , see eqs.(12) and (16).
3 Scalar and vector form factors in with resonances
Although there is no analytic method to derive the form factors directly from the QCD Lagrangian, its symmetries are nevertheless useful to reduce the model dependence to a minimum and keep as many properties of the fundamental theory as possible.
Weinberg:1978kz (); Gasser:1983yg (); Gasser:1984gg (), the effective field theory of QCD at low energies, is built as an expansion in even powers of the ratio between the momenta or masses of the lightest pseudoscalar mesons over the chiral symmetry breaking scale, which is of the order of one GeV. As one approaches the energy region where new degrees of freedom -the lightest meson resonances- become active, ceases to provide a good description of the Physics (even including higher-order corrections Bijnens:1999sh (); Bijnens:1999hw (); Bijnens:2001bb ()) and these resonances must be incorporated to the action of the theory. This is done without any ad-hoc dynamical assumption by in the convenient antisymmetric tensor formalism that avoids the introduction of local terms at next-to-leading order in the chiral expansion since their contribution is recovered upon integrating the resonances out Ecker:1988te (); Ecker:1989yg (). The building of the Resonance Chiral Lagrangians is driven by the spontaneous symmetry breakdown of QCD realized in the meson sector, the discrete symmetries of the strong interaction and unitary symmetry for the resonance multiplets. The expansion parameter of the theory is the inverse of the number of colours of the gauge group, . Despite not being small in the real world, the fact that phenomenology supports this approach to QCD Manohar:1998xv (); Pich:2002xy () hints that the associated coefficients of the expansion are small enough to warrant a meaningful perturbative approach based on it. At leading order in this expansion there is an infinite number of radial excitations for each resonance with otherwise the same quantum numbers that are strictly stable and interact through local effective vertices only at tree level.
The relevant effective Lagrangian for the lightest resonance nonets reads 333We comment on its extension to the infinite spectrum predicted in the limit in the paragraph below eq. (3).:
where all coupling constants are real, is the pion decay constant and we follow the conventions of Ref. Ecker:1988te (). Accordingly, stands for trace in flavour space, and , and are defined by
where (), and are matrices that contain light pseudoscalar fields, current quark masses and external left and right currents, respectively. The matrix () includes the lightest vector (scalar) meson multiplet 444In the limit of QCD the lightest scalar meson multiplet does not correspond to the one including the (or meson) Cirigliano:2003yq (), but rather to the one including the resonance., and stands for these resonances kinetic term. We note that resonances with other quantum numbers do not contribute to the considered processes (like the axial-vector and pseudoscalar resonances, which have the wrong parity).
The computation of the vector form factors yields
because and . We recall that the normalization of the vector form factor, , was pre-factored in eq. (8) together with .
The strangeness changing scalar form factors and associated S-wave scattering within have been investigated in a series of papers by Jamin, Oller and Pich Jamin:2000wn (); Jamin:2001zq (); Jamin:2006tk (); Jamin:2006tj () (see also Ref. Bernard:1990kw ()). The computation of the scalar form factors gives:
and can be written in terms of the , form factors computed in Ref.Jamin:2001zq ():
where, for the considered flavour indices, should correspond to the resonance. Besides (see the comment below equation (12)) it has also been used that
Indeed, using our conventions, the tilded scalar form factors become simply
The computation of the leading order amplitudes in the large- limit within demands, however, the inclusion of an infinite tower of resonances per set of quantum numbers 555We point out that there is no limitation in the Lagrangians in this respect. In particular, a second multiplet of resonances has been introduced in the literature SanzCillero:2002bs (); Mateu:2007tr () and bi- and tri-linear operators in resonance fields have been used GomezDumm:2003ku (); RuizFemenia:2003hm (); Cirigliano:2004ue (); Cirigliano:2005xn (); Cirigliano:2006hb (); Kampf:2011ty ().. Although the masses of the large- states depart slightly from the actually measured particles Masjuan:2007ay () only the second vector state, i.e. the resonance, will have some impact on the considered decays. Accordingly, we will replace the vector form factor in eq. (12) by
where the operators with couplings and are defined in analogy with the corresponding unprimed couplings in eq. (LABEL:lagrangian).
which yield the form factors
where Jamin:2006tk (); Jamin:2008qg (); Boito:2008fq (); Boito:2010me (). We note that we are disregarding the modifications introduced by the heavier resonance states to the relation (18) and to the definition of .
4 Different form factors according to treatment of final-state interactions
The form factors in eqs.(19) diverge when the exchanged resonance is on-mass shell and, consequently, cannot represent the underlying dynamics that may peak in the resonance region but does not certainly show a singular behaviour. This is solved by considering a next-to-leading order effect in the large- counting, as it is a non-vanishing resonance width 666Other corrections at this order are neglected. Phenomenology seems to support that this is the predominant contribution.. Moreover, since the participating resonances are not narrow, an energy-dependent width needs to be considered. A precise formalism-independent definition of the off-shell vector resonance width within has been given in Ref. GomezDumm:2000fz () and employed successfully in a variety of phenomenological studies. Its application to the resonance gives
where was defined in eq. (2). Several analyses of the SanzCillero:2002bs (); Pich:2001pj (); Dumm:2013zh () and Jamin:2008qg (); Boito:2008fq (); Boito:2010me () form factors where the and prevail respectively, have probed the energy-dependent width of these resonances with precision. Although the predicted width Guerrero:1997ku () turns to be quite accurate, it is not optimal to achieve a very precise description of the data and, instead, it is better to allow (as we will do in the remainder of the paper) the on-shell width to be a free parameter and write
where it has been taken into account that at the -scale the only absorptive cut is given by the elastic contribution.
At this point, different options for the inclusion of the resonances width arise. The most simple prescription is to replace by in eqs. (19). We shall call this option ‘dipole model’, or simply ‘Breit-Wigner (BW) model’. One should pay attention to the fact that analyticity of a quantum field theory imposes certain relations between the real and imaginary parts of the amplitudes. In particular, there is one between the real and imaginary part of the relevant two-point function. At the one-loop level its imaginary part is proportional to the meson width but the real part (which is neglected in this model) is non-vanishing. As a result, the Breit-Wigner treatment breaks analyticity at the leading non-trivial order.
Instead, one can try to devise a mechanism that keeps the complete complex two-point function. Ref. Guerrero:1997ku () used an Omnès resummation of final-state interactions in the vector form factor that was consistent with analyticity at next-to-leading order. The associated violations were small and consequently neglected in their study of the observables. This strategy was also exported to the decays of the in Refs. Jamin:2006tk (); Jamin:2008qg () where it yielded remarkable agreement with the data. We will call this approach to the vector form factor ‘the exponential parametrization’ (since it exponentiates the real part of the relevant loop function) and refer to it by the initials of the authors who studied the system along these lines, ‘JPP’.
A decade after, a construction that ensures analyticity of the vector form factor exactly was put forward in Ref. Boito:2008fq () and applied successfully to the study of the tau decays. It is a dispersive representation of the form factor where the input phaseshift, which resums the whole loop function in the denominator of eq. (19), is proportional to the ratio of the imaginary and real parts of this form factor. This method also succeeded in its application to the di-pion system Dumm:2013zh (), where it was rephrased in a way which makes chiral symmetry manifest at next-to-leading order. We will name this method ‘dispersive representation’ or ‘BEJ’, by the authors who pioneered it in the system.
We would like to stress that the Breit-Wigner model is consistent with only at leading order, while the exponential parametrization (JPP) and the dispersive representation (BEJ) reproduce the chiral limit results up to next-to-leading order and including the dominant contributions at the next order Guerrero:1998hd ().
In the dispersive approach to the study of the di-pion and Kaon-pion systems it was possible to achieve a unitary description in the elastic region that could be extended up to (the cut, which is phase-space and large- suppressed is safely neglected) and , respectively. Most devoted studies of these form factors neglect -in one way or another- inelasticities and coupled-channel effects beyond in them 777See, however, Ref. Moussallam:2007qc (), which includes coupled channels for the vector form factor., an approximation that seems to be supported by the impressive agreement with the data sought. However, this overlook of the problem seems to be questionable in the case of the decays where we are concerned with the first (second) inelastic cuts.
An advisable solution may come from the technology developed for the scalar form factors that were analyzed in a coupled channel approach in Refs. Jamin:2001zq (); Jamin:2001zr (); Jamin:2006tj () (for the strangeness-changing form factors) 888We will use these unitarized scalar form factors instead of the one in eq. (19) in the JPP and BEJ treatments (see above). and Guo:2012ym (); Guo:2012yt () (for the strangeness-conserving ones) unitarizing and (respectively) with explicit exchange of resonances Guo:2011pa (). However, given the large errors of the decay spectra measured by the BaBar delAmoSanchez:2010pc () and Belle Inami:2008ar () Collaborations and the absence of data on the channel we consider that it is not timely to perform such a cumbersome numerical analysis in the absence of enough experimental guidance 999One could complement this poorly known sector with the information from meson-meson scattering on the relevant channels GomezNicola:2001as (). Our research at next-to-leading order in the expansion treating consistently the mixing Kaiser:1998ds (); Kaiser:2000gs (); Escribano:2010wt () is in progress.. For this reason we have attempted to obviate the inherent inelasticity of the channels and tried an elastic description, where the form factor that defines the input phaseshift is given by eq. (19) with defined analogously to , i.e., neglecting the inelastic cuts. We anticipate that the accord with data supports this procedure until more precise measurements demand a better approximation.
Let us recapitulate the different alternatives for the treatment of final-state interactions that will be employed in sections 5-7 to study the decays. The relevant form factors will be obtained from eqs.(19) in each case by:
Dispersive representation (BEJ): A three-times subtracted dispersion relation will be used for the vector form factor. The input phaseshift will be defined using the vector form factor in eq. (19) with including only the cut and resumming also the real part of the loop function in the denominator. The unitarized scalar form factor will be used Jamin:2001zq (). More details can be found in appendix A.
5 Predictions for the decays
We note that eqs.(19) also hold for the form factors (see eq. (8) and comments below, as well). Therefore, in principle the knowledge of these form factors in the system can be transferred to the systems immediately, taking thus advantage of the larger statistics accumulated in the former and their sensitivity to the properties. This is certainly true in the case of the vector form factor in its assorted versions and in the scalar Breit-Wigner form factor. However, in the BEJ and JPP scalar form factor one has to bear in mind (see appendix A.3) that the () scalar form factors are obtained solving the coupled channel problem which breaks the universality of the form factors as a result of the unitarization procedure. As a consequence, our application of the form factors to the decays will provide a test of the unitarized results. Taking into account the explanations in Ref. Jamin:2001zq () about the difficult convergence of the three-channel problem (mainly because of the smallness of the contribution and its correlation with the channel) this verification is by no means trivial, specially regarding the channel, where the scalar contribution is expected to dominate the decay width.
In this way, we have predicted the branching ratio and differential decay width using the knowledge acquired in the decays. Explicitly:
In the dipole model, we have taken the , and mass and width from the PDG Beringer:1900zz () -since this compilation employs Breit-Wigner parametrizations to determine these parameters- and estimated the relative weight of them using (see discussion at the end of section 3) Ecker:1988te (). In this way, we have found .
In the JPP parametrization, we have used the best fit results of Ref. Jamin:2008qg () for the vector form factor. The scalar form factor has been obtained from the solutions (6.10) and (6.11) of Ref. Jamin:2001zq () 101010The relevant unitarized scalar form factors have been coded using tables kindly provided by Matthias Jamin.. The scalar form factors have also been treated alike in the BEJ approach.
In the BEJ representation, one would use the best fit results of Ref. Boito:2010me () to obtain our vector form factor. However, we have noticed the strong dependence on the actual particle masses of the slope form factor parameters, and . Ref. Boito:2010me () used the physical masses in their study of data. On the other hand we focus on the decays. Consequently, the masses should correspond now to instead of to . Noteworthy, both the and are lighter than the and and the corresponding small mass differences, given by isospin breaking, are big enough to demand for a corresponding change in the parameters. Accepting this, the ideal way to proceed would be to fit the BaBar data on decays Aubert:2007jh (). Unfortunately, these data are not publicly available yet. For this reason, we have decided to fit Belle data on the decay using the and masses throughout. The results can be found in table 1, where they are confronted to the best fit results of Ref. Boito:2008fq () 111111We display the results of this reference instead of those in Ref. Boito:2010me () because we are not using information from decays in this exercise. Differences are, nonetheless, tiny., both of them yield and are given for GeV, although the systematic error due to the choice of this energy scale is included in the error estimation. We will use the results in the central column of table 1 to give our predictions of the decays based on the results.
Parameter Best fit with fake masses Best fit Boito:2008fq () (MeV) (MeV) (GeV) (GeV) Table 1: Results for the fit to Belle data Epifanov:2007rf () with a three-times subtracted dispersion relation including two vector resonances in , according to eq. (19) and resumming the loop function in the denominator (see appendix A.2), as well as the scalar form factor Jamin:2001zq (). The middle column is obtained using the masses of the and mesons and the last column using the and masses actually corresponding to the data.
Proceeding this way we find the differential decay distributions for the three different approaches considered using eq. (8). This one is, in turn, related to the experimental data by using
We thank the Belle Collaboration for providing us with their data Inami:2008ar (). This was not possible in the case of the BaBar Collaboration delAmoSanchez:2010pc () because the person in charge of the analysis left the field and the data file was lost. We have, however, read the data points from the paper’s figures and included this effect in the errors. The number of events after background subtraction in each data set are (BaBar) and (Belle) and the corresponding bin widths are and MeV, respectively. In Fig.1 we show our predictions based on the system according to BW, JPP and BEJ. In this figure we have normalized the BaBar data to Belle’s using eq. (25). A look at the data shows some tension between both measurements and we notice a couple of strong oscillations of isolated Belle data points which do not seem to correspond to any dynamics but rather to an experimental issue or to underestimation of the systematic errors 121212We have also realized that the first two Belle data points, with non-vanishing entries, are below threshold, a fact which may indicate some problem in the calibration of the hadronic system energy or point to underestimation of the background.. In this plot there are also shown the corresponding one-sigma bands obtained neglecting correlations between the resonance parameters and also with respect to other sources of uncertainty, namely and , whose errors are also accounted for. The corresponding branching ratios are displayed in table 2, where the is also shown. We note that the error correlations corresponding to the fit results shown in table 1 have been taken into account in BEJ’s branching ratio of table 2.
It can be seen that the BW model gives a too low decay width and that the function shape is not followed by this prediction, as indicated by the high value of the that is obtained. On the contrary, the JPP and BEJ predictions yield curves that compare quite well with the data already. Moreover, the corresponding branching fractions are in accord with the PDG value within errors. Altogether, this explains the goodness of the , which is . Besides, we notice that the error bands are wider in the dispersive representation than in the exponential parametrization, which may be explained by the larger number of parameters entering the former and the more complicated correlations between them that were neglected in obtaining Fig. 1 and the JPP result in table 2.
From these results we conclude that quite likely the BW model is a too rough approach to the problem unless our reference values for and the resonance parameters were a bad approximation. We will check this in the next section. On the contrary, the predictions discussed above hint that JPP and BEJ are appropriate for the analysis of data that we will pursue next.
|Dipole Model (BW)|
6 Fits to the BaBar and Belle data
We have considered different fits to the data. In full generality we have assessed that the data is not sensitive either to the low-energy region or to the peak region. This is not surprising, since the threshold for production opens around MeV which is some MeV larger than , a characteristic energy scale for the region of dominance. This implies first that the fits are unstable under floating and (which affects all three approaches) and second that the slopes of the vector form factor, which encode the physics immediately above threshold, can not be fitted with data (this only concerns BEJ). We have considered consequently fits varying only the mass and width and and sticking to the reference values discussed in the previous section for the remaining parameters in every approach.
Our best fit results for the branching ratios are written in table 3, where the corresponding can also be read. These are obtained with the best fit parameter values shown in table 4, which can be compared to the reference values, which were used to obtain the predictions in the previous section, that are recalled in table 5. The corresponding decay distributions with one-sigma error bands attached are plotted in Fig. 2.
These results show that the BW model does not really provide a good approximation to the underlying physics for any value of its parameters and should be discarded. Oppositely, JPP and BEJ are able to yield quite good fits to the data with values of the around one. This suggests that the simplified treatment of final state interactions in BW, which misses the real part of the two-meson rescatterings and violates analyticity by construction, is responsible for the failure.
Fitting alone is able to improve the quality of both approaches by . The fitted values are consistent with the reference ones (see table 5): in the case of BEJ at one sigma, being the differences in JPP slightly larger than that only. This is satisfactory because both the and the decays are sensitive to the interplay between the first two vector resonances and contradictory results would have casted some doubts on autoconsistency.
When the parameters are also fitted the results improve by in JPP and by in BEJ. This represents a reduction of the by in JPP and by in BEJ. It should be noted that the three-parameter fits do not yield to physical results in BW. Specifically, mass and width tend to the values and happens to be one order of magnitude larger than the determinations in the literature. Therefore we discard this result. We also notice that although the branching ratios of both JPP and BEJ (which have been obtained taking into account the parameter fit correlations) are in agreement with the PDG value, the JPP branching ratios tend to be closer to its lower limit while BEJ is nearer to the upper one. It can be observed that the deviations of the three-parameter best fit values with respect to the default ones lie within errors in BEJ, as it so happens with in JPP. However, there are small tensions between the reference and best fit values of and in JPP.
These results are plotted in Fig. 2. Although the BW curve has improved with respect to Fig. 1 and seems to agree well with the data in the higher-energy half of the spectrum, it fails completely at lower energies. On the contrary, JPP and BEJ provide good quality fits to data which are satisfactory along the whole phase space. We note that JPP goes slightly below BEJ and its error band is again narrower possibly due to having less parameters. BEJ errors include the systematics associated to changes in which is slightly enhanced with respect to the case.
Despite the vector form factor giving the dominant contribution to the decay width, the scalar form factor is not negligible and gives of the branching fraction in the JPP and BEJ cases. In the BW model this contribution is .
|Dipole Model (BW) (Fit )|
|Dipole Model (BW) (Fit , ,||Unphysical result||-|
|JPP (Fit )|
|JPP (Fit , ,|
|BEJ (Fit )|
|BEJ (Fit , ,|
|Dipole Model (BW)||JPP||BEJ|
|Dipole Model (BW)||JPP||BEJ|