Resonance a_1(1420) and the Three-Pion Decays of the Tauon

TL;DR

This study investigates the role of the a_1(1420) resonance in tau lepton decays to three pions, suggesting it may be the primary axial-vector resonance involved, based on recent data analysis.

Contribution

The paper provides evidence that the a_1(1420) resonance, rather than the traditionally assumed a_1(1640), is the main axial-vector resonance in tau three-pion decays, based on phenomenological modeling and data analysis.

Findings

Recent data favor a_1(1420) over a_1(1640)

a_1(1420) likely the main resonance in tau decays

More data needed for definitive conclusion

Abstract

The role of the a_1(1420) resonance in the three-pion decays of the tau lepton is investigated using a phenomenological model. For all data before 2008, roughly equal fit quality is achieved when the basic a_1(1260) resonance is supplemented with either a_1(1640) or a_1(1420). However, two recent and more precise data sets require resonances with masses that are not very far from that of a_1(1420). This suggests that the axial-vector resonance that accompanies the a_1(1260) in the three-pion decays of the tauon is a_1(1420), not a_1(1640), as believed up to now. More data are needed to demonstrate this definitely.

Tables

Table 1. Table 1: Confidence level (C.L.) of the fits to various data assuming various axial-vector recurrences a 1 ′ superscript subscript 𝑎 1 ′ a_{1}^{\prime} . Usual χ 2 superscript 𝜒 2 \chi^{2} and the number of degrees of freedom (NDF) are also shown.

		No $a_1^{\prime }$				$a_1^{\prime }\equiv a_1\left(1640\right)$				$a_1^{\prime }\equiv a_1\left(1420\right)$
Data		${\chi }^2$	NDF	C.L.(%)		${\chi }^2$	NDF	C.L.(%)		${\chi }^2$	NDF	C.L.(%)
Set A		$368.0$	462	100.0		$267.1$	460	100.0		$300.6$	460	100.0
ALEPH 13m		$31.2$	68	100.0		$30.3$	66	100.0		$30.5$	66	100.0
ALEPH 05c		$102.9$	110	67.0		$23.7$	108	100.0		$45.2$	108	100.0
ALEPH 13c		$103.6$	67	0.3		$73.6$	65	21.7		$74.3$	65	20.1
Nugent 09c		$552.8$	59	0.0		$194.1$	57	0.0		$78.8$	57	2.9
Belle 10c		$1607.1$	126	0.0		$845.4$	126	0.0		$572.5$	126	0.0

Table 2. Table 2: Searching for the a 1 ′ superscript subscript 𝑎 1 ′ a_{1}^{\prime} parameters. Only errors coming from Minuit minuit are shown, no attempt to estimate the systematic errors of the model has been made. Also the COMPASS compassprl results are shown for comparison.

Data	$m$ (MeV)	$\Gamma$ (MeV)	${\chi }^2$	NDF	C.L.(%)
Nugent 09c	$1375\pm 11$	$311\pm 8$	18.7	56	100.0
Belle 10c	$1376.7\pm 1.2$	$379.3\pm 6.3$	98.6	124	95.5
COMPASS	${1414}_{-13}^{+15}$	${153}_{-23}^{+8}$

Table 3. Table 3: Fit to the ALEPH 13c aleph13 data assuming the a 1 ′ superscript subscript 𝑎 1 ′ a_{1}^{\prime} parameters found from the Nugent 09c and Belle 10c data.

	${\chi }^2$	NDF	C.L.(%)
$a_1^{\prime }$ from Nugent 09c	65.1	65	47.3
$a_1^{\prime }$ from Belle 10c	65.9	65	44.6

Equations

$${\mathcal{L}}_{a_1\rho \pi }=\frac{g_{a_1\rho \pi }}{\sqrt{2}}\left({\mathcal{L}}_1\cos \theta +{\mathcal{L}}_2\sin \theta \right),$$

$$-i{G}_{a_1}^{\mu \nu }(p)=\frac{-g^{\mu \nu }+p^{\mu }{p}^{\nu }/m_{a_1}^2}{s-M_{a_1}^2(s)+i{m}_{a_1}{\Gamma }_{a_1}(s)}$$

$${\mathcal{L}}_{a_1\sigma \pi }=g_1\left({\mathbf{A}}^{\mu }\cdot {\partial }_{\mu }\mathbf{P}\right)S+g_2\left({\mathbf{A}}^{\mu }\cdot \mathbf{P}\right){\partial }_{\mu }S,$$

Full text

Recently, I have compared our model vojlic of the three-pion decays of the $\tau$ lepton to nineteen experimental data sets found in literature from 1986 to 2013. A detailed account will be presented elsewhere lichardprep .

In this Letter I report on one interesting byproduct of my study that is related to the $a_{1}(1420)$ resonance, which was discovered in 2013 by the COMPASS Collaboration at CERN compassprl . They performed a careful partial wave analysis of the $\pi^{-}\pi^{-}\pi^{+}$ system produced in diffractive dissociation of 190 GeV$/c$ pions off a stationary hydrogen target and then used the resonance-model fit to the spin density matrix to find the resonance parameters.

I first recapitulate the main features of our model vojlic . Its physical content is depicted by the meson dominance vavd diagrams in Figs. 1 and 2.

The interaction among the $a_{1}(1260)$ (denoted simply as $a_{1}$ in what follows), $\rho$ and $\pi$ fields is described by a two-component Lagrangian

[TABLE]

where ${\mathcal{L}}_{1}={\mathbf{A}}^{\mu}\cdot\left({\mathbf{V}}_{\mu\nu}\times\partial^{\nu}{\mathbf{P}}\right)$, ${\mathcal{L}}_{2}={\mathbf{V}}_{\mu\nu}\cdot\left(\partial^{\mu}{\mathbf{A}}^{\nu}\times{\mathbf{P}}\right)$, and ${\mathbf{V}}_{\mu\nu}=\partial_{\mu}{\mathbf{V}}_{\nu}-\partial_{\nu}{\mathbf{V}}_{\mu}$. The isovectors ${\mathbf{A}}^{\mu}$, ${\mathbf{V}}_{\mu}$, and $\mathbf{P}$ denote the operators of the $a_{1}$, $\rho$ and $\pi$ fields, respectively. Although one can find several flaws in this Lagrangian from the QCD point of view, its soundness for low-energy phenomenology is supported by the fact that the same Lagrangian provided a good description of the four-pion production in the $e^{+}e^{-}$ annihilation licjur , even for the same value of the mixing parameter $\sin\theta$, as shown in vojlic .

The $a_{1}$ propagator

[TABLE]

features running mass $M_{a_{1}}(s)$ given by a once-subtracted dispersion relation with the energy-dependent total width $\Gamma_{a_{1}}(s)$ as input. The following contributions to $\Gamma_{a_{1}}(s)$ are considered: $a_{1}\rightarrow\rho+\pi\rightarrow 3\pi$, $a_{1}\rightarrow\bar{K^{*}}K,K^{*}\bar{K}\rightarrow K\bar{K}\pi,$ and $a_{1}\rightarrow\sigma+\pi\rightarrow 3\pi$ and the following conditions are satisfied: $M^{2}_{a_{1}}(m_{a_{1}}^{2})=m_{a_{1}}^{2}$, $\Gamma_{a_{1}}(m_{a_{1}}^{2})=\Gamma_{a_{1}}$.

When an additional axial-vector resonance $a_{1}^{\prime}$ is included, the same Lagrangian (1) is used with the same mixing parameter $\sin\theta$. Instead of (2), a simpler $a_{1}^{\prime}$ propagator is chosen, with running mass replaced by the nominal $a_{1}^{\prime}$ mass and with the energy dependent total width given by $\Gamma_{a_{1}^{\prime}}(s)=k\Gamma_{a_{1}}(s),$ with constant $k$ guaranteeing that the condition $\Gamma_{a_{1}^{\prime}}(m^{2}_{a_{1}^{\prime}})=\Gamma_{a_{1}^{\prime}}$ is fulfilled.

The propagator of the $\rho(770)$ resonance is taken in a variable-width, running-mass form runningrho .

The interaction Lagrangian among the $a_{1}$, $\pi$, and $\sigma$ fields is written in the form

[TABLE]

where $S$ is the operator of the $\sigma$ field. For the $\sigma$ propagator we use a simple form with the fixed mass and energy dependent width.

All strong interaction vertexes are modified by the form factor $F(q)=\exp\{-q^{2}/(12\beta^{2})\}$, where $q$ is the three-momentum magnitude of a daughter meson in the rest frame of the parent meson. It is taken from the chromoelectric flux-tube breaking model of Kokoski and Isgur kokoski1987 , together with their value $\beta=0.4$ GeV/$c$.

The model without $a_{1}^{\prime}$ contains six adjustable parameters: (i) a multiplicative constant ensuring the correct normalization to the data, which absorbs the product of all coupling constants from the $a_{1}\rightarrow\rho$ branch, (ii) $a_{1}$ mass, (iii) $a_{1}$ width, (iv) ${a_{1}\rho\pi}$ Lagrangian mixing parameter $\sin\theta$, (v-vi) two different products of the coupling constants from the $a_{1}\rightarrow\sigma$ branch scaled by the product of coupling constants from the $a_{1}\rightarrow\rho$ branch.

If an $a_{1}^{\prime}$ is included, a complex constant enters, which multiplies the $a_{1}^{\prime}$ propagator before adding it to the $a_{1}$ propagator. The number of adjustable parameters is thus eight. If the mass and width of $a_{1}^{\prime}$ are allowed to vary, then the number of adjustable parameters rises to ten. Finally, if a combined set of $N$ individual data sets is fitted, then the number of adjustable parameters increases by $N\!-\!1$ because every data set requires its own normalization constant.

When we were completing our model vojlic , the existing data on the three-pion decays of the tauon were not precise enough to consider the $a_{1}(1640)$ mass and width as free parameters. Therefore, we used their values from the 2008 Review of Particle Physics pdg2008 , which are still used today. As I will show, the situation has changed with the advent of two more precise sets of data. The first, which appeared in the PhD Thesis of I. M. Nugent nugent09 , published in 2009, may be regarded as a very preliminary version of the BaBar data note1 . The other set contains very precise data from the Belle Collaboration at the KEKB collider, published in 2010 belle10 .

An important indicator of the soundness of our model is that the parameters required for a good fit to various data sets are very similar. It is therefore possible to obtain a satisfactory fit for several data sets combined together, as shown in vojlic .

The original model vojlic has been slightly modified: (i) the bin width in the modified ALEPH data aleph13 varies from point to point in the high mass part of the spectrum, which required a corresponding change in the software; (ii) following the advice of Dr. David Bugg bugg , I introduced an option to mimic the Adler zero adler in the $\sigma\rightarrow\pi+\pi$ amplitude; (iii) in the original model vojlic , one of the two $a_{1}\sigma\pi$ coupling constants was fixed by requiring the vanishing derivative of the $a_{1}$ running mass at the resonance point. I have released this connection and introduced another free parameter (already included in the list above).

After I learned of the discovery of the $a_{1}(1420)$ resonance by the COMPASS Collaboration compassprl , I made all the calculations with both options, namely with $a_{1}^{\prime}\equiv a_{1}(1640)$ and $a_{1}^{\prime}\equiv a_{1}(1420)$. The results presented here were evaluated with the $a_{1}(1420)$ mass of 1411.8 MeV and width of 158 MeV in accordance with a recent COMPASS submission to arXiv krinner .

Concerning the influence of the presence or choice of a particular $a_{1}^{\prime}$ on the agreement of our model with data, the experimental data can be divided into three categories.

In the first category, there are four data sets note2 (CELLO 90c cello90 , CELLO 90m cello90 , Nugent 09c nugent09 , Belle 10c belle10 ) which our model is not able to describe (confidence level, C.L., was less than 10%) whether the $a_{1}^{\prime}$ is included or not.

Then, there is a group of eight data sets (Argus 93c argus93 , OPAL 95c opal95 , OPAL 97c opal97 , ALEPH 98m aleph98 , OPAL 99c opal99 , OPAL 99m opal99 , ALEPH 05m aleph05 , ALEPH 13m aleph13 ), five of them with C.L. of 100%, with which our model agrees, even if the $a_{1}^{\prime}$ is not considered.

The last category includes seven experiments (Mark II 86c markii86 , MAC 87c mac87 , MAC 87m mac87 , ALEPH 98c aleph98 , CLEO 00m cleo00 , ALEPH 05c aleph05 , ALEPH 13c aleph13 ) in which the inclusion of the $a_{1}^{\prime}$ in the model improves its agreement with data (for three of them, the C.L. then reached more than 98%). The quality of the fit depends only marginally on the $a_{1}^{\prime}$ species.

The examples of data sets from each category are shown in Table 1. The results of five selected individual data sets are accompanied by the result of a simultaneous fit to six data sets (ARGUS 93c, OPAL 99c, OPAL 99m, CLEO 00m, ALEPH 05c, ALEPH 05m), which is denoted as Set A. The first two lines (Set A and ALEPH 13m) show examples of a good fit achieved without the help of $a_{1}(1640)$ or $a_{1}(1420)$. If either of them is included, the $\chi^{2}$ decreases, but there is no influence on the already perfect confidence level.

Sets ALEPH 05c and ALEPH 13c are examples of data that are fitted better when any $a_{1}^{\prime}$ is included. For ALEPH 13c, the confidence level rose from 0.3 to 21.7 per cent [20.1 per cent] if $a_{1}(1640)$ [$a_{1}(1420)$] is chosen. For ALEPH 05c, confidence level even reaches 100 per cent for either choice.The last two rows show that our model is unable to fit Nugent 09c and Belle 10c data even if $a_{1}^{\prime}$ is considered.

In any case, it is encouraging that with the Nugent 09c data, replacing the $a_{1}(1640)$ with $a_{1}(1420)$ leads to a dramatic drop in the $\chi^{2}$ and some increase in the confidence level, from $\chi^{2}/$NDF= 194.1/57 (C.L.=0%) to $\chi^{2}/$NDF=78.8/57 (C.L.=2.9%). This has prompted me to allow both the $a_{1}^{\prime}$ mass and width vary. As a result, the confidence level climbs to 100%, based on $\chi^{2}/$NDF= 18.7/55. Corresponding values of the $a_{1}^{\prime}$ mass and width are shown in Table 2, together with their Minuit minuit errors. A comparison of the experimental and calculated three-pion mass spectra appears in Fig. 3.

Repeating the procedure with the Belle 10c data, I have again achieved a very good fit (C.L.=95.5%) for the parameters displayed in Table 2. The $a_{1}^{\prime}$ mass is almost identical to that of the Nugent 09c data; though the width is higher. The model curve goes perfectly through the experimental points, which have very small errors. See Fig. 4.

It may seem disturbing that the resonances that have been found in the Nugent 09c and Belle 10c data by minimizing $\chi^{2}$ are not visible in Figs. 3 and 4 as bumps or shoulders. The reason is that the interference between the $a_{1}(1260)$ and $a_{1}^{\prime}$ is destructive, as discussed in vojlic . See Figs. 5 and 6 there.

The situation with the ALEPH 13c data is special. An attempt to obtain the optimal $a_{1}^{\prime}$ parameters by minimizing $\chi^{2}$ was unsuccessful. The minimization procedure strayed into an unphysical region. But when the $a_{1}^{\prime}$ parameters extracted from the Nugent 09c or Belle 10c data shown in Table 2 were used, the resulting confidence level became much better than that with the $a_{1}(1640)$ or $a_{1}(1420)$. Compare Table 3 with ALEPH 13c row in Table 1.

It is tempting to assume that the resonance extracted from the Nugent 09c and Belle 10c data, which is also preferred by the ALEPH 13c data, is identical to the resonance $a_{1}(1420)$ observed by the COMPASS Collaboration. It would be surprising to have two different resonances so close in mass.

However, when we consult Table 2 and compare the parameters of the $a_{1}^{\prime}$ resonance from the study of the three-pion decays of the tauon with those found by the COMPASS Collaboration, we find a serious discrepancy in the decay widths. The $a_{1}(1420)$ decay width is about half of those from the Nugent 09c and Belle 10c data. A possible reason for this discrepancy may be a different parametrization of the resonance width. The COMPASS Collaboration used the width modified by the Blatt–Weisskopf centrifugal-barrier factors in their relativistic Breit–Wigner formulas haas . On the other hand, in our model the $a_{1}^{\prime}$ width is obtained by rescaling the width of $a_{1}(1260)$.

The COMPASS Collaboration has shown that the $a_{1}(1420)$ decays into pion and $f_{0}(980)$. The main drawback of our model is that we consider the $\sigma\equiv f_{0}(500)$ instead. However, at this stage, it does not make much sense to begin a labor intensive modification of our model by allowing different scalar resonances for $a_{1}(1260)$ and $a_{1}(1420)$. Our model in its present form shows perfect agreement with both the Nugent 09c and Belle 10c data. Clearly, more data are needed that would falsify our model and force it to discriminate between the $\sigma$ and $f_{0}(980)$.

To conclude, the phenomenological analysis of the Nugent nugent09 and Belle belle10 data offers a serious hint that the excited axial-vector resonance which plays a role in the three-pion decays of the tauon is the $a_{1}(1420)$, not $a_{1}(1640)$.