Three-pion decays of the tau lepton, the \mathbf{a_{1}(1260)} properties, and the \bm{a_{1}\rho\pi} Lagrangian

# Three-pion decays of the tau lepton, the a1(1260) properties, and the a1ρπ Lagrangian

Martin Vojík Institute of Physics, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic    Peter Lichard Institute of Physics, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic Institute of Experimental and Applied Physics, Czech Technical University, Horská 3/a, 120 00 Prague, Czech Republic
July 15, 2019
###### Abstract

We show that the Lagrangian is a decisive element for obtaining a good phenomenological description of the three-pion decays of the lepton. We choose it in a two-component form with a flexible mixing parameter . In addition to the dominant intermediate states, the ones are included. When fitting the three-pion mass spectra, three data sets are explored: (1) ALEPH 2005 data, (2) ALEPH 2005 data, and (3) previous two sets combined and supplemented with the ARGUS 1993, OPAL 1997, and CLEO 2000 data. The corresponding confidence levels are (1) 28.3%, (2) 100%, and (3) 7.7%. After the inclusion of the resonance, the agreement of the model with data greatly improves and the confidence level reaches 100% for each of the three data sets. From the fit to all five experiments [data set (3)], the following parameters of the are obtained:  MeV,  MeV. The optimal value of the Lagrangian mixing parameter agrees with the value obtained recently from the annihilation into four pions.

###### pacs:
13.35.Dx,13.25.-k,14.40.Be

## I Introduction

The dominance of the meson, hereafter referred as , and its decay mode in the three-pion decays of the -lepton is firmly established experimentally pluto (); delco1986 (); markii1986 (); argus1986 (); mac (); cello (); argus1993 (); opal1995 (); opal1997 (); aleph1998 (); delphi (); cleo2000a (); cleo2000b (); cleo_prelim (); aleph2005 (); argus1996 (). A convincing demonstration was provided by the ARGUS Collaboration argus1993 (); argus1996 (), who compared the distribution of unlike- and like-sign two-pion masses. The intermediate state is the core of several models of the three-pion decays of the -lepton pham (); tornqvist1987 (); isgurtau (); kuhnsanta (); feindt (); kuhnmirkes (); poffenberger (); dpp (); drpp (); achasov (); cleo2000a (). It should be mentioned that the dominance in heavy lepton decays was proposed in 1971 tsai (), about five years before the -lepton was actually discovered.

The resonance, discovered almost fifty years ago goldhaber (), plays an important role in many phenomena of the nuclear and particle physics. Its properties have been studied in many processes, but even its basic parameters are not very well known. The values of the resonance mass determined from different processes or by different experimental groups often contradict one another. The same applies, even to a larger extent, to the width. Very little improvement has been achieved over the last thirty years, see Table 1.

The origin of those problems lies in the very nature of the resonance with its short lifetime and large width. The usual definition of the mass and usual procedures for its measurement are not applicable. The mass and width enter the formulas for experimentally accessible quantities via the assumed form of the resonance propagator, which generates a specific Breit-Wigner formula. Those formulas are further modified in different ways when modeling the dynamics of the processes in which the participates. As a result, we do not have a unique definition of the mass and width . In fact, every formula represents a specific definition of and . Given this, it is not surprising that different models yielded different results even when being applied to the same data. It would be natural to accept as the canonical parameters the results of a model that best describes a broad class of data on various processes and from various experiments. Unfortunately, we are not in such a situation yet.

The situation of the heavier meson states with is a little unclear and none of them has found its place in the Summary Table of the Review of Particle Properties pdg2008 (). The first indication of the state with mass of 1.65 GeV and width of 0.4 GeV appeared already in 1978 pernegr1978 (). The later experimental evidence, which comes mainly from hadronic reactions, is summarized in pdg2008 (), where this resonance is listed as and assigned the mass of  MeV and the width of  MeV. The three-pion decay of the -lepton is less convenient for studying the resonance (often denoted as in what follows) because of fundamental limitations due to the mass which is not big enough to provide sufficient phase space for three-pion final states with the needed invariant mass. Nevertheless, the DELPHI Collaboration delphi () performed the analysis of the Dalitz plots for different 3-pion mass ranges and observed an enhancement that “could be explained by a decay mode of the to a resonance of mass similar to or greater than the mass which then decays to three pions through the intermediate state of a pion plus a particle of mass 1.25 GeV or greater.” They interpreted this as an evidence for the . In our opinion, this observation need not signify the existence of the . It may also be a decay of the , produced with a larger than nominal mass, into and . A more convincing proof of the in the decay of the lepton comes from the CLEO Collaboration cleo2000a (). They showed that adding the term into the Breit-Wigner function improved significantly the agreement with the data.

On the theoretical side, a radial excitation of the quark-antiquark system with a mass of 1.82 GeV appeared in a relativized quark model with chromodynamics of Godfrey and Isgur godfrey1985 (). Its decay width into the channel was calculated in the flux-tube-breaking model by Kokoski and Isgur kokoski1987 () with result  MeV (our estimate is based on their Table II). The seminal analysis of Barnes, Close, Page, and Swanson barnes1997 () has shown that the experimentally observed dominance of the D-wave over S-wave ves1995 (); adams1998 () excludes the hybrid meson nature of the and confirms it as a radial excitation of the quark-antiquark system.

A few meson states above the have been observed by a single group, mainly in the annihilation. They still need confirmation. For details, see pdg2008 ().

Another important ingredient that defines a particular model of the three-pion decay of the -lepton is, besides the propagator, the vertex. The models assembled by different authors are based on different vertexes. These are sometimes simply constructed as allowed combinations of the metric tensor and participating four-momenta. A more rigorous way lies in deriving them from the interaction Lagrangians among the axial, vector and pseudoscalar fields. Unfortunately, here the situation is unclear yet. Various theoretical concepts provide different effective Lagrangians theory (). This is probably the reason why the model builders preferred trivial Lagrangians or ad hoc vertexes. However, recent articles dpp (); drpp (); achasov () are different. Dumm, Pich, and Portolés dpp () got their Lagrangian from the resonance chiral theory. Their work was revised in the light of later developments in drpp (). Achasov and Kozhevnikov achasov () used the Generalized Hidden Local Symmetry model.

Several models of the three-pion decay of the tau lepton have been proposed. With some simplification one can say that each of them gives compatible results when applied to different sets of data, but the results of different models are incompatible. Also the agreement of many models with data (often verbally claimed as satisfactory) is poor when judged by usual statistical criteria. The most popular models were those of Isgur, Morningstar, and Reader (IMR) isgurtau () and of Kühn and Santamaria (KS) kuhnsanta (). Other models were much less successful in fitting the data. As an example we recall the results from argus1993 (), where the ARGUS Collaboration compared various models with their data. Using the ’s and the numbers of degrees of freedom (NDF) from their Table 4, we are getting the confidence level (C.L.) of for Bowler’s model bowler1986 () and 2.2% for the model of Ivanov, Osipov, and Volkov ivanov1991 (). The KS and IMR models look better with C.L. 10.7% and 79.0%, respectively. However, in a later article argus1995 () the ARGUS Collaboration used an enlarged set of data (integrated luminosity of 445 pb against 264 pb in argus1993 ()) and found that the KS model is rejected on a 7.4  level. The IMR model with parameters as given in isgurtau () was incompatible with the data on the same level argus1995 ().

Up to now, the best results have been obtained by the CLEO model cleo2000a () and by the model of Achasov and Kozhevnikov achasov (). The former obtained, when fitting the CLEO data cleo2000a (), C.L. of 54.6% without the resonance and 88.2% with it. The latter fitted the ALEPH data aleph2005 () assuming two heavier axial mesons and and got /NDF=79/102, which corresponds to C.L. of 95.6%. Unfortunately, each of those two successful models has been applied only to one data set.

The finding of an Lagrangian that leads to a satisfactory description of the three-pion production in the tau decays would have important consequences for other areas of the high energy and nuclear physics. For example, the resonance and its coupling to the system play important role in the evaluation of the dilepton and photon production rates from a hadronic fireball presumably created in the relativistic heavy ion collisions. The calculations performed so far, see, e.g., Refs. elmag (), have shown that the yield of electromagnetic signals strongly depends on the choice of the Lagrangian. Fixing its correct form is thus important for distinguishing the electromagnetic radiation of the Quark-Gluon Plasma (QGP) from the hadronic sources.

The outline for this paper is as follows. In Sec. II we describe our model and mention briefly its similarities and differences with other models. The experimental data used for testing our model and fixing its parameters are listed in Sec. III. Some details about our calculations and the results are presented in Sec. IV. We summarize our results and conclude in Sec. V. Two Appendixes contain technical details. The present work supersedes an earlier paper licvoj ().

## Ii Model of the three–pion decays of the tau lepton

In this section we present our model, which will be used for fitting the three–pion mass spectra of the decays and . The basic information can be obtained by inspecting Figs. 1 and 2. In addition to the standard intermediate states we include also the states in which the couples to a pion and an (hereafter called ). The intermediate states improve the behavior of the differential decay width at small masses of the three-pion system and bring the difference between and decays. Their importance has been pointed out by the CLEO Collaboration cleo2000a ().

To show what is specific for our model, what differs it from other existing models of the three-pion decays of the tau lepton, we have to provide more information. This is done in the following subsections.

### ii.1 Phenomenological a1ρπ Lagrangian

The interaction Lagrangian among the , , and pion fields implies the form of the vertex in the Feynman diagrams. But sometimes a vertex is postulated that can hardly be related to any effective Lagrangian. In the literature, one can find several prescriptions for the vertex used in the calculation of the decay rate of the tau lepton into three pions and neutrino. The simplest one is , where index () couples to the () line. It can be derived from the interaction Lagrangian among the , , and fields without derivatives. It was used, e.g., in Ref. pham (). On the opposite pole of complexity is a two-component vertex used in the IMR model isgurtau (). Both its components are transversal both to the and four-momenta. The relative weight of the two components can vary, what gives the IMR model more flexibility. This is probably the main reason why this model sometimes fits the data a little better than the KS model kuhnsanta (), see, for example, opal1995 ().

To maintain both the flexibility and the correspondence with the effective field theory, we use a two-component Lagrangian of the interaction in the form

 La1ρπ=ga1ρπ√2(L1cosθ+L2sinθ), (1)

where

 L1 = Aμ⋅(Vμν×∂νP), L2 = Vμν⋅(∂μAν×P),

and . The isovectors , , and denote the operators of the , and fields, respectively.

Our Lagrangian differs from that derived by Wess and Zumino, see Eq. (67) in wz (), only by notation. We will consider the mixing angle a free parameter that has to be determined by fitting the experimental three-pion mass distribution. For each , the coupling constant can be determined from the decay width. The Lagrangian (1) implies the following vertex

 Xαμ = iga1ρπ√2{cosθ[pαρpμπ−(pπpρ)gαμ] − sinθ[pαρpμa1−(pa1pρ)gαμ]},

where ’s denote the four-momenta of the corresponding mesons (incoming , outgoing and ).

Lagrangian (1) has recently been used licjur (); jurlic () in a model of the electron–positron annihilation into four pions. Value of the mixing parameter was obtained by fitting the excitation function (dependence of the annihilation cross section on the invariant collision energy). From the channel the value of has been obtained licjur (). In jurlic (), a combined fit to and channels has provided the value of .

### ii.2 Other effective Lagrangians and their parameters

The Lagrangian describing the interaction of the triplet with strange pseudoscalar and vector mesons is not required when calculating the amplitudes of the three-pion decays of the , see Figs. 1 and 2. It is needed for evaluating the strange channel contribution to the total decay width of the . The latter enters the propagator discussed below. The Lagrangian is chosen in a form analogous to Eq. (1)

 La1K∗K=ga1K∗K√2(L′1cosθ+L′2sinθ),

with

 L′1 = ∂νK†AμK∗μν+H.c. , L′2 = K†∂μAνK∗μν+H.c.

Matrix notation is now used, in which

 K = (K+K0), K∗μ=(K∗+μK∗0μ), Aμ = ((a01)μ√2(a+1)μ√2(a−1)μ−(a01)μ),

and . As usual hokim (), a particle symbol denotes the field operator which annihilates that particle and creates its antiparticle. In the spirit of the SU(3) symmetry, we assume the same mixing angle as in the case (1). The coupling constant cannot be reliably extracted from the experimental data yet because of conflicting information about the branching fractions.111See pdg2008 () and the discussion on p. 253 in aleph2005 ().. We will therefore use the SU(3) symmetry relation

 g2a1K∗K=14g2a1ρπ. (2)

In order to evaluate the amplitudes of the Feynman diagrams depicted in Figs 1 and 2, we also need to specify the interaction Lagrangian among the , , and fields. We write it in the form

 La1σπ=g1(Aμ⋅∂μP)S+g2(Aμ⋅P)∂μS ,

where is the operator of the field. The Lorentz condition for the field implies that the amplitude of the decay is proportional to the difference

 ga1σπ=g1−g2. (3)

In the decay diagrams, where the off-mass-shell resonance is represented by its propagator (5), also the terms proportional to

 ha1σπ=g1+g2 (4)

The interaction Lagrangian between the and fields is given by

 Lσππ=gσππ(P⋅P)S .

The coupling constant could be estimated from the data on the mass and width e791sigma (); cleosigma (). But because this constant enters the amplitudes of the three-pion decays of the taon multiplied by or , which are both unknown, it does not have much sense.

### ii.3 Propagator of the a1 resonance

We choose an analytically correct form tornqvist1987 (); isgurtau () of the propagator featuring the running mass and the energy-dependent total width in the denominator

 −iGμνa1(p)=−gμν+pμpν/m2a1s−M2a1(s)+ima1Γa1(s) . (5)

The following conditions should hold

 M2a1(m2a1)=m2a1, (6) dM2a1ds(m2a1)=0, (7) Γa1(m2a1)=Γa1, (8)

where and are the nominal mass and width of the resonance, respectively. The denominator in (5) is the boundary value of a function analytic in the complex -plane () with a cut running along the real axis from the three-pion threshold to infinity. The running mass squared can therefore be obtained from a once-subtracted dispersion relation222In Refs. tornqvist1987 (); cleo2000a () an unsubtracted dispersion relation was used. with as input

 M2a1(s)=M2a1(0)−sπP∫∞9m2πma1Γa1(s′)s′(s′−s)ds′ . (9)

Symbol P denotes the Cauchy principal value. We have chosen the subtraction point at , instead of as in isgurtau (). The advantage is that the integrand in (9) contains just one singular point instead of two, what makes the evaluation more stable and much faster. The disadvantage is that the condition (6) is not satisfied automatically and must be recalculated if or any parameter inside changes.

The following decays are considered when calculating :

 a1 → ρ+π→3π (10) a1 → ¯K∗K,K∗¯K→K¯Kπ, (11) a1 → σ+π→3π, (12)

where the mass of the decaying is taken to be . We neglect the interference between the amplitudes of (10) and (12) despite the identical final states. We argue that the decay proceeds in the S and D orbital momentum states, whereas in the P state. This argument is not entirely watertight because neither (10) nor (12) satisfies the conditions for being factorized as a two-step process aps (). Channel (11) is described by four Feynman diagrams, two of them have identical final states (e.g., in the case of .) We checked that the interference can be safely neglected in this case.

The Lagrangian between the vector () and pseudoscalar (, ) fields is chosen in a standard form with coupling constant . Moreover, the empirical widths pdg2008 () of the and provide the ratio

 g2K∗Kπg2ρππ=0.883±0.035, (13)

where the error has been enlarged to absorb the difference between the charged and neutral . Value (13) is a little higher than the SU(3) value of . Relations (2) and (13) enable us to express the product of coupling constants squared acting in (11) as a multiple of

 G2=g2a1ρπg2ρππ, (14)

which determines the partial decay width of (10). We introduce the ratio

 x=g2a1K∗Kg2K∗KπG2, (15)

the value of which is given by multiplying (2) by (13).

As we have already mentioned, there is no way of getting the product from the data, as the partial decay width of (12) is unknown. We therefore proceed in another way. We define the parameter

 y=ga1σπgσππG. (16)

If we insert the parameters and into the formula for the total decay width, it becomes proportional to . So does the derivative of the running mass squared (9). When the condition (7) is applied, can be canceled. With known , the condition (7) thus becomes an equation for the unknown . As we neglect the possible interference between the Feynman diagrams containing the with those containing the , the sign of is not essential and we choose . The dimension of is (energy) because the two Lagrangians that describe the decay via have together three derivatives, while those via just one.

An important note concerns the hadron vertexes. The effective Lagrangian approach takes hadrons as elementary quanta of the corresponding fields, ignoring thus their internal structure. As a consequence, the interaction strength is overestimated at higher momentum transfers. To describe the interaction among participating mesons more realistically, we explore the chromoelectric flux-tube breaking model of Kokoski and Isgur kokoski1987 (), as it was done already in the IMR model isgurtau (). Each strong interaction vertex is modified by the factor

 F(q)=exp{−q212β2}, (17)

where is the three-momentum magnitude of a daughter meson in the rest frame of the parent one (virtual masses are taken in the intermediate states). In the original paper kokoski1987 (), the value  GeV/ was established. We will use this value in all our calculations, as we found that moving from it did not bring statistically significant improvement of the agreement of our model with data. A cutoff similar to (17) was used in the model by CLEO Collaboration cleo2000a (). Their parameter corresponds to .

During the course of development of our model we tried various versions of the propagators, from the most primitive one with the constant mass and width to the most sophisticated and best physically justified one (5). The best fit to data has been provided by the latter.

When investigating the presence of the suspected radial recurrence of the , denoted as , we supplement the propagator (5) with the term

 −iGμνa′1(p)=α−gμν+pμpν/m2a′1s−m2a′1+ima′1Γa′1(s) , (18)

where is a complex parameter. We assume that the energy dependent total decay width exhibits the same energy behavior as that of and write

 Γa′1(s)=Γa1(s)Γa1(m2a′1)Γa′1 ,

where is the assumed width of the resonance.

What concerns the relation to the previous models, our propagator is closest to that used by the CLEO Collaboration cleo2000a (). If we ignored the momentum dependent terms in the numerators of (5) and (18), we would recover their Breit-Wigner function.

### ii.4 Propagators of the ρ, K∗, and σ resonances

In order to calculate the amplitudes of the taon’s three-pion decays we need also the and propagators. They play a role also in decays (10) and (12). In addition, the evaluation of the decay width (11) requires the knowledge of the propagator.

We choose the propagator of both the charged and neutral rho resonances in the form

 −iGμνρ(p)=−gμν+pμpν/m2ρs−M2ρ(s)+imρΓρ(s) , (19)

which uses the running mass squared and the energy dependent total width from Ref. running (). The denominator of propagator (19) is an analytic function in the -plane with a cut running from to infinity, as required by general principles. The real function is calculated from using a once-subtracted dispersion relation, which guarantees that the condition is satisfied. The condition

 dM2ρds(m2ρ)=0

is not fulfilled automatically and serves as a check that all important contributions to the total -meson width have properly been taken into account. They include, in addition to the basic two-pion decay channel, the , , , and , which get open as the resonance goes above its nominal mass. The structure of the participating mesons is taken into account by means of the Kokoski-Isgur form factor (17).

The running mass description of the propagator running () differs from other approaches that appeared in the literature GS (); VW (); melikhov04 (). Gounaris and Sakurai GS () considered only the two-pion contribution to the total width of the resonance and ignored structure effects. The result is a simple analytic formula, the main reason why their approach is so popular. Vaughn and Wali VW () took into account the strong form factor, but again ignored higher decay channels. Melikhov, Nachtmann, Nikonov, and Paulus melikhov04 () included the and channels, but did not consider the strong form factors. The running mass formalism running () takes into account both the higher decay channels and the structure effects.

The propagator of the resonance is required only for the calculation of the decay rate (11), which contributes to the total decay width of the resonance. It does not act in the three-pion decay of the lepton. It is chosen in a simpler form, with the constant mass and energy dependent decay width

 −iGμνK∗(p)=−gμν+pμpν/m2K∗s−m2K∗+imK∗ΓK∗(s) . (20)

The decay width includes only the contribution from the channel and is normalized to the nominal width at . The corresponding formula, taking into account also the Kokoski-Isgur form factor (17), is

 ΓK∗(s)=m2K∗s[q(s)q(m2K∗)]3F(q(s))F(q(m2K∗))ΓK∗,

where is the momentum of a daughter particle in the rest frame of the parent with the mass . The is a narrow resonance and we experienced numerical instabilities when calculating integrals containing the square of (20). To get rid of problems, we have used the procedure described in Appendix B.

Also for the propagator we use the form with fixed mass and energy dependent width

 −iGσ(p)=1s−m2σ+imσΓσ(s) ,

where includes only the contribution from the two-pion decay channel and is equal to

 Γσ(s)=m2σs√s−4m2πm2σ−4m2πF(q(s))F(q(m2σ)) Γσ.

As the current Review of Particle Physics pdg2008 () is not very specific about the mass and width, we rely on the mutually compatible values obtained by the Fermilab E791 Collaboration e791sigma () and the CLEO Collaboration cleosigma (), who both analyzed the mesons decays. The results (in MeV) of E791 are , , whereas those of CLEO are , . We adopt the weighted averages  MeV and  MeV.

## Iii Experimental data

We will compare the calculated three–pion mass distribution in the and decays with the outcome of the five experiments.

(1) The ARGUS Collaboration argus1993 () used the ARGUS detector at the DORIS II storage ring at DESY and studied the decay. Their background and acceptance corrected three pion mass distribution is given in twenty-eight bins within the mass range 0.425–1.775 GeV.

(2) The OPAL Collaboration published their results on the three-pion-mass squared distribution in the charged-pion channel in two papers. The first of them opal1995 () was based on the data collected with the OPAL detector at the CERN Large Electron-Positron Collider (LEP) during 1992 and 1993. We used it in our recent publication licvoj (). In this work we explore the updated version opal1997 (), in which also the data of 1994 were included. The three-pion-mass-squared plot is corrected for background and efficiency and consists of twenty-three bins with much smaller statistical errors than in opal1995 ().

(3) The -lepton decay into three pions and neutrino was also investigated by the CLEO Collaboration at the Cornell Electron Storage Ring (CESR). Their results on the all-charged-pions channel still exist only in a preliminary form cleo_prelim (). We can therefore use only the data on the channel cleo2000a (). The background-subtracted, efficiency corrected three- mass spectrum is given in 47 bins.

(4,5) The ALEPH Collaboration at CERN LEP have published an article summarizing their results about the branching ratios and spectral functions of the decays aleph2005 (). It is based on the data collected with the ALEPH detector during 1991-1995 but processed by an improved method. We use the tables of the corrected three-pion mass squared spectra both in the and decays, which are publicly accessible at the website alephweb (). The all-charged pion spectrum contains 116 bins 0.025 GeV wide starting at 0.225 GeV. In the two-neutral pion case the spectrum starts at 0.2 GeV, but we discard the bin centered at 0.2375 GeV with a zero value and, comparing to its neighbors, an unrealistically small error. We are thus left again with 116 bins. In both cases we ignore the correlation matrices among the errors in different bins and add statistical and systematic errors linearly.

## Iv Calculations and results

To be sure that our results are free of programming errors, we have written two independent computer codes, one in C++ (M.V.), another in Fortran 95 (P.L.) and debugged them until they produced identical results.

We found that the parity-violating term in the decay amplitude influences the three-pion-mass distribution only negligibly and have not considered it any longer in our calculations. The decay amplitude in (22) then depends only on relativistic invariants , which are symmetric against transformation . Using this symmetry when calculating the innermost integral in (22) enables us to speed up the computing by a factor of two.

In the diagrams with the in the intermediate states, depicted in Figs. 1 and 2, also the product plays a role (for definitions, see Sec. II.2). We introduce additional free parameter

 z=ha1σπga1σπ, (21)

which allows us to express that unknown product as a multiple of , which is determined by the method described in Sec. II.3. The parameter itself will be obtained by minimalization of when fitting the experimental three-pion mass distributions in the three-pion decays of the lepton.

To be closer to experimental conditions, we do not calculate the (unnormalized) differential decay rate at certain values of the three-pion mass or its square , but its averages over the experimentally given bins in argus1993 (); cleo2000a () or opal1997 (); aleph2005 ().

When the contribution of the to the propagator is not considered, the calculated differential decay rates, and thus also the evaluated from them and data, depend on the following four parameters: (1) the nominal mass , (2) the nominal width , (3) the Lagrangian mixing parameter , and (4) the off-mass-shell coupling constant ratio defined by Eq. (21). Quantity (16) is not an extra parameter, condition (7) determines it as an implicit function of and .

The ARGUS, OPAL, and CLEO experiments present the three-pion mass spectra in the acceptance corrected number of events. Both ALEPH spectra are normalized to the integrated branching fractions. As the outcome of our model is not normalized (the coupling of the meson to the boson is not fixed by meson dominance vavd ()), we opt to compare just shape of the mass distribution and introduce five multiplicative constants. The values of them are obtained by minimizing the individual ’s for each experiment while keeping the common parameters (, , , and ) fixed.

To get a quick insight into the dependence of the quality of the fit on the Lagrangian mixing parameter , we first fix the basic parameters at the “standard” values, frequently used in theoretical considerations, namely,  GeV/ and  GeV. We also set and calculate the ratio of the usual to the number of experimental points for each of the five data sets as a function of . The results are shown in Fig. 3.

It is clear that the choice of the correct Lagrangian is of the utmost importance for obtaining a good agreement with the data. The fact that all five experiments point to the same narrow region in is extremely important. In addition, this region overlaps with the interval based on the results of the model licjur (); jurlic () of the electron-positron annihilation into four pions built around the same Lagrangian (1). It indicates the soundness both of the present model and of the annihilation model.

The region is not shown in Fig. 3 because for those values of it is impossible to satisfy condition (7) by procedure described in Sec. II.3. The square of parameter , defined by Eq. (16), acquires negative, i.e. unphysical, values, which mean the negative branching ratio of decay (12).

In the next step we allow all four parameters to vary and use the CERN computer library program Minuit of James and Roos minuit () for finding their values that minimize for the three data sets defined in Sec. III. The results are summarized in Table 2. As always, the assessment of errors of the

parameters is a difficult task. We combined the errors provided by Minuit, which reflect the errors of experimental data, with our estimates of the errors induced by the uncertainties of the input parameters (the mass, various coupling constants). The agreement of our model with the ALEPH data is perfect (confidence level 100%), the agreement with two other data sets is satisfactory by measures usually accepted in the high energy physics (/NDF ). The values of the mass obtained from the three data sets are mutually compatible, as well as those of the width. What is especially remarkable are the values of the Lagrangian mixing parameter . Not only their small errors () and mutual consistence, but also a perfect agreement with the values obtained from the analyzes of the annihilation into four pions. Parameter characterizes the part of the Lagrangian that acts only for virtual and cannot be compared with anything yet (the annihilation model licjur (); jurlic () did not consider the intermediate states).

Now we add the contribution (18) to the propagator (5). Not to increase the number of free parameters too much, we fix the mass and width of the at the PDG 2008 values 1647 MeV and 254 MeV, respectively. The same approach was used by the CLEO Collaboration cleo2000a (), just their values were a little different (1700 MeV and 300 MeV). The number of the free parameters thus increases by two [the real and imaginary parts of , see (18)]. The results of the minimalization procedure are shown in Table 3 for all three data sets.

The comparison of Tables 2 and 3 shows that the addition of the resonance to the propagator greatly improves the agreement of the model with data in all cases. For the all-charged-pions ALEPH data aleph2005 (), the drops from 119.1 to 30.7 and the confidence level rockets from 28.25% to 100%. The improvement of the confidence level is even more substantial for the third data set, where the total is a sum of the individual ’s for the ARGUS, OPAL, CLEO, ALEPH , and ALEPH data. The mass and width of the as well as other two free parameters ( and ) are very stable against the inclusion of . Their new values (Table 3) differ only very little from the corresponding old ones (Table 2). Also the values obtained from different data sets are mutually compatible. This is true also for two new parameters Re  and Im .

The calculated three-pion-mass distribution is compared to the data of ALEPH Collaboration aleph2005 () in Fig. 4.

We have just learned that the resonance greatly improves the agreement with data. It is therefore a little surprising that there is no bump or shoulder corresponding to this resonance visible in Fig. 4. To investigate this conundrum we calculate the model distribution in three cases: (1) both and terms in the propagator (this is the curve presented already in Fig. 4); (2) only the term 5 in the propagator; (3) only the term (18). The model parameters in all three cases are identical. They are taken from the ALEPH row of Table 3. The findings, see Fig. 5, show that the underlying mechanism leading to the agreement with data is somewhat surprising. The final distribution is a result of the destructive interference between the dominant amplitude containing the propagator (5) and the amplitude containing the propagator (18).

Similar analysis performed for the decay, see Fig. 6, leads to the same conclusion.

## V Summary and conclusions

The main message of this study is that the form of the Lagrangian is the decisive factor for achieving a good model description of the three-pion decays of the tau lepton. This is again illustrated in Fig. 7 where the total , which is calculated as a sum of the individual ’s from the five experiments, is divided by the total number of experimental points and plotted as a function of the Lagrangian mixing parameter . Two different cases are considered: (1) only included in the propagator, (2) both and included. In contrast to Fig. 3, the other parameters are fixed at their optimal values taken from the appropriate tables (Tabs. 2 and 3). Even if the curve (2) is shifted a little toward smaller values of , the minima of both curves fall to the interval found in the model of the annihilation into four pions licjur (); jurlic ().

Our further finding, even not documented in this work in detail, concerns the form of the propagator. We have found that the running mass form (5), suggested and already used in several papers tornqvist1987 (); isgurtau (); cleo2000a (), provides a better fit to the taon three-pion decay data than simpler forms with a constant mass and a constant or energy dependent total decay width.

The running mass squared is given by the dispersion relation. In this work we have chosen a once-subtracted version (9). As the input for the dispersion relation, the energy dependent total decay width of the for all above the three-pion threshold is required (9). We approximated it as a sum of the decay widths to the three pion final states (via the and intermediate states) and the final states (via + c.c.). A typical behavior of the energy dependent total width

is shown in Fig. 8. The hump centered around  GeV develops as the mass of the two-pion subsystem falls predominantly first on the ascending and then on the descending side of the rho propagator. We ignored the channels , , and , which have been seen in the decays pdg2008 () and which open at higher .333The inclusion of them would bring additional free parameters, what we wanted to avoid. This is probably the reason why the running mass behaves wildly, see Fig. 9, and does not have a nice plateau around the nominal mass, as it did in the case of the running ().

Another important ingredient of our model are the intermediate states. On one side, they enter the calculation of the total decay width of the resonance, which is necessary for constructing the running mass propagator (5). On the other side, they contribute to the decay rates of the three-pion decays of the tau lepton, Figs. 1 and 2.

To investigate the role of the intermediate states in the evaluation of the decay width, we split the distribution depicted in Fig. 6 by the full curve into its and components. The result is shown in Fig. 10.

It is obvious that the intermediate states play a unique role in describing the behavior of the differential decay width at small three-pion masses. What is a little suspicious, is the large magnitude at the intermediate masses. To see whether it is reasonable or not, we integrate the distributions to get the branching ratio

 B=Γ(τ−→ντπ−σ→ντπ−π0π0)Γ(τ−→ντπ−π0π0)

with the result . This number is more than twice higher than the experimental value of obtained by the CLEO Collaboration (Ref. cleo2000a (), Table III).

The source of this obvious deficiency of our model is the following: The important parameter , which regulates the rate of the transition, was obtained from the condition that the derivative of the running mass at the nominal-mass point should vanish (7). But the absence of higher decay channels, which influence the values of the running mass at all , may modify the resulting value of significantly. The larger than correct may mimic the missing channels.

A new feature of our work, which, to our knowledge, has not appeared in the literature yet, is that we fit the data from several experiments simultaneously. We intend to continue in this approach and include not only the data concerning the three-pion decays of the tau lepton, but also the experimental results from other weak, electromagnetic, and perhaps also strong interaction processes. The natural candidate is the electron-positron annihilation into four pions, for which a model based on the same Lagrangian as here has already been built. The value of the Lagrangian mixing parameter we have obtained here perfectly agrees with values obtained from the annihilation into four charged pions licjur () and from the combined fit to both annihilation channels jurlic ().

To summarize:
(1) We have shown that the right form of the Lagrangian is extremely important for obtaining a good agreement with data. We have obtained an unprecedented confidence level of 100% for all three sets of data we considered. The optimal value of the Lagrangian mixing parameter perfectly agrees with the value obtained from the annihilation into four pions.
(2) Our confirmation of the existence of the resonance with the mass and width compatible with the PDG pdg2008 () values is based on the increase of the confidence level from 7.7% to 100% after the has been included.
(3) We have explained why the resonance, which is important for getting a good agreement with data, is not visible in the three-pion-mass spectrum as a bump or shoulder.
(4) From the common fit to the data from five experiments we have obtained the following results:

Mass of the  MeV;

Width of the  MeV;

Lagrangian mixing parameter .

###### Acknowledgements.
One of us (P. L.) is indebted to J. Kapusta for discussions many years ago that triggered this investigation. This work was supported by the Czech Ministry of Education, Youth and Sports under contracts LC07050 and MSM6840770029.

## Appendix A Differential decay rate formula

We use the following formula for the differential decay rate in the invariant three-particle mass in a four-body decay :

 dΓdW = |p1|16(2π)6m2a∫W−m2m3+m4 dm34|p∗2| |p′3| (22) × ∫1−1dcosθ∗2∫1−1dcosθ′3∫2π0dφ′3|M|2.

The asterisk denotes the (2,3,4) rest frame, the prime the (3,4) rest frame. is the mass of the system consisting of particles 3 and 4, and are its energy and momentum, respectively, in the (2,3,4) rest frame. In the rest frame of the parent particle the momentum of particle 1 points along the negative -axis. In the (2,3,4) rest frame, the momentum of particle 2 lies in the (xz) plane.

## Appendix B Integrating over a narrow peak

Let us assume that we need to evaluate an integral over an interval that includes a narrow resonance peak

 Q=∫s2s1f(s)(s−M2(s))2+m2Γ2(s) ds , (23)

where is a slowly varying function. Further, let the two functions in the denominator satisfy conditions and . If then the integrand is rapidly varying function of and a numerical quadrature of very high order is required to get reliable results. After introducing a new variable by substitution , where , , , and , the integral (23) becomes

 Q=a2−a12mγ∫1−1(s−m2)2+m2γ2(s−M2(s))2+m2Γ2(s)f(s) dξ ,

which can be safely evaluated using, e.g., the Gauss-Legendre quadrature. We apply this method for calculating the integrals containing the square of the propagator (20). In that case .

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