# Measuring Contextual Relationships in Temporal Social Networks by Circle
Link

Cui Jing, Yiqing Zhang, and Xiang Li†

###### Abstract

Network science has released its talents in social network analysis based on
the information of static topologies. In reality social contacts are dynamic
and evolve concurrently in time. Nowadays they can be recorded by ubiquitous
information technologies, and generated into temporal social networks to
provide new sights in social reality mining. Here, we define _circle link_ to
measure contextual relationships in three empirical social temporal networks,
and find that the tendency of friends having frequent continuous interactions
with their common friend prefer to be close, which can be considered as the
extension of Granovetter’s hypothesis in temporal social networks. Finally, we
present a heuristic method based on circle link to predict relationships and
acquire acceptable results.

##  1 Introduction

In the past decades, we have witnessed fruitful and exciting advances in
studies of complex, large-scale social networks. Many algorithms and methods
have been brought up in predicting social relationships [1] and modeling
social network structure [2] based on local and global topological information
of network. Since the developments of ubiquitous digital technologies in
sensing, storage and communication, we have experienced the emergence of
large-scale human behaviour data with high temporal resolution. Consequently,
a new complex network concept, namely _Temporal Networks_ , has been
presented[3]; new methods using temporal information have been proposed for
social network analysis, such as closeness recognition[4] and interactive
patterns modeling[5]. A nascent interdisciplinary area, _temporal social
network_ , is coming to the stage.

Inspired by the work of Song et al.[6] about the predictability of human
mobility, Takaguchi et al.[7] measured the predictability of social contacts
in face-to-face networks applied the concepts of information entropy. They
declare that prior knowledge of social contact between the ego and his (or
her) current partner can decrease the uncertainty of social contact between
the ego and the next partner by a large percentage. Followed by their
conclusions, we define the contextual relationship between the current partner
and the next partner as _Circle Link_ , characterizing the potential social
circles. The empirical analyses of three temporal social networks show that
there is no universal memory mechanism in human contextual relationships, and
two vertices of dense edges have dense time-ordered edges with their common
neighbor, which can be considered as the extension of Granovetter’s hypothesis
in temporal social networks. At the last part of this article, we give a
heuristic method based on the definition of circle link for predicting
relationships with limited, local prior knowledge of social contacts.

##  2 Datasets

Three datasets are used in our research, where two of them are collected by
RFID technologies during the ACM Hypertext Conference in Torino and the
_INFECTIOUS: STAY AWAY_ art-science exhibition at the Science Gallery in
Dublin[8], respectively. The third one is derived from the Campus Wi-Fi login
records in Fudan University[9]. We refer these three datasets as HT, SG and WF
respectively, and use the data of three days in each dataset to generate the
corresponding temporal social networks, whose detailed properties are shown in
Table 1.

Table 1: Properties of all temporal social networks Dataset | Nodes | Records | Edges | Sparsity
---|---|---|---|---
HT | Day1 | 100 | 3460 | 946 | 5.93e-04
Day2 | 102 | 3510 | 1062 | 7.14e-04
Day3 | 97 | 2895 | 926 | 8.39e-04
SG | Day1 | 200 | 2684 | 714 | 7.92e-04
Day2 | 204 | 2770 | 739 | 7.61e-04
Day3 | 186 | 2467 | 615 | 7.39e-04
WF | Day1 | 1120 | 12833 | 10346 | 0.0120
Day2 | 2250 | 25772 | 21637 | 0.0067
Day3 | 1906 | 15798 | 13744 | 0.0057

Each network can be presented as a list of conversation events, consisting of
two participants, the start time and the duration. To analyze the contextual
relationships, we generate the event list of a specific ego as shown in the
left panel of Fig.1, ignoring the durations of events and being ordered by the
start time. The corresponding (static) weighted social network is aggregated
over the whole temporal social network, and the weight of edges is indicated
by the number of events between two persons (see the right panel of Fig. 1).

Figure 1: The illustrations of Circle Link

##  3 The Definition of Circle Link

We apply the definitions [7] of the uncorrelated entropy,
${H}_{i}^{1}=-\sum_{j\in{N}_{i}}{{P}_{i}(j)\log_{2}{{P}_{i}(j)}}$, and the
conditional entropy,
${H}_{i}^{2}=-\sum_{j\in{N}_{i}}{{P}_{i}(j)\sum_{l\in{N}_{i}}{{P}_{i}(l|j)\log_{2}{{P}_{i}(l|j)}}}$
in all temporal social networks (where ${N}_{i}$ is the set of ego _i_ ’s
partners/neighbors, ${P}_{i}(j)$ represents the historical probability that
ego _i_ contacts partner _j_ , and ${P}_{i}(l|j)$ represents the conditional
probability that ego _i_ contacts partner _l_ after a contact with partner
_j_). Fig. 2 shows that the conditional entropies in the datasets of SG and WF
are followed by normal distributions, in contrast to the results of [7].
Nevertheless, the conditional entropy is smaller than the corresponding
uncorrelated entropy in all networks, indicating prior knowledge of event with
the current partner can decrease the uncertainty of the event with the next
partner. Therefore we define _Circle Link_ to capture the potential social
relationship from such continuous contextual relationships. The weight of a
circle link is defined as the cumulative number of such continuous contextual
relationships and measures the potential degree. As shown in the left panel of
Fig.1, since the person $A$ has two continuous conversation events with his or
her two partners $B$ and $C$ at the time $t_{1},t_{2}$ and $t_{10},t_{11}$,
the weight of the circle link between $B$ and $C$ indicated by $cl_{A}(BC)$ is
$2$ (see the right panel of Fig.1). We can define the mean weight of each
circle link in network level as follows:

${W}_{CL}(i,j)\equiv\sum_{l\in\mathrm{V},l\neq i,j}{|{cl}_{l}(i,j)|}$ (1)

where $\mathrm{V}$ is the set of all nodes.

Figure 2: Distribution of entropies in three datasets

##  4 Results

###  4.1 Self Circle Link Phenomenon

The definition of circle link includes self-loop phenomenon. An individual can
be circle linked to himself or herself, like individual _D_ in the right panel
of Fig. 1. We define the following _Self Circle Rate_ (SCR) to represent the
percentage of self circle links among all circle links observed by the ego:

${\textbf{SCR}}_{i}\equiv\frac{|{cl}_{i}(jj)|}{|{cl}_{i}(lj)|},\quad l,j\in
N_{i},$ (2)

where $N_{i}$ is the set of ego $i$’s partners.

We apply the null hypothesis that an ego contacts his or her partners without
a memory mechanism, where $\textbf{SCR}^{null}_{i}=1/|N_{i}|$, and define the
ratio $m_{0}$ between $\textbf{SCR}_{i}$ and $\textbf{SCR}_{i}^{null}$
averaged on the whole network as follows:

$m_{0}\quad\equiv\sum_{i\in\mathrm{V}}\frac{\textbf{SCR}_{i}}{\textbf{SCR}_{i}^{null}}=\sum_{i\in\mathrm{V}}\frac{\textbf{SCR}_{i}}{1/N_{i}}$
(3)

where $\mathrm{V}$ is the set of all nodes. Table 2 shows that in the
Hypertext Conference, contacts among people have memory mechanism ($m_{0}>1$
for HT), different from contacts in the Gallery setting ($m_{0}\approx 1$ for
SG). Moreover, in the human indoor interaction network people repel to contact
with their current partner as the next one ($m_{0}\leq 0.65$ for WF),
suggesting that the model represented in [10] don’t capture the mechanisms of
all social contacts. Furthermore, $m_{0}$ is time invariant within a dataset,
indicating the memory or inverse-memory mechanism is only determined by the
contexts of social contacts.

Table 2: The rate $m_{0}$ in temporal social networks Dataset | Day1 | Day2 | Day3
---|---|---|---
HT | 1.8794 | 1.9596 | 1.5307
SG | 0.8308 | 0.7573 | 0.7497
WF | 0.6517 | 0.5587 | 0.3470

###  4.2 Strength and Clustering Coefficient of Social Ties Correlated with
Circle Link Weights

In the aggregated version of temporal social networks, the weights of edges
represent the strengths of social ties. Here we use the Pearson correlation
coefficients $\rho$ to characterize the correlation between the strengths of
social ties and temporal patterns. Table 3 shows that in all temporal social
networks, the Pearson correlation coefficients $\rho_{W_{L},W_{CL}}$ between
the weight of edges $W_{L}$ and the weight of circle links $W_{CL}$ have
relative high values $\rho_{W_{L},W_{CL}}>0.5$, indicating that two vertices
frequently continuously contact with their common neighbor have dense edges
between them. Furthermore, Fig. 3 shows that the weights of circle links are
inversely proportional to the corresponding link betweenness centralities,
indicating that the involving vertices are in the dense local network. These
are closely related to Granovetter’s hypothesis that states that in social
networks dense edges have on average higher weights [11]. However, in addition
to having higher weights, we find that dense edges are more commonly related
to “continuous group talk”, temporal patterns involving three individuals.

Furthermore, we apply the definition of edge clustering coefficient [12]
$CC_{L}(i,j)={n}_{C}(i,j)/{n}_{T}(i,j)$ , where ${n}_{C}(i,j)$ is the number
of common neighbors of individual _i_ and individual _j_ , ${n}_{T}(i,j)$ is
the total number of nodes neighbored individual _i_ or individual _j_. The
high edge clustering coefficient represents dense overlaps of the
corresponding two vertices’ neighborhoods. As shown in Table 3, the Pearson
correlation coefficients $\rho_{CC_{L},W_{CL}}$ between link clustering
coefficients and weights of circle links have relative low values
$\rho_{CC_{L},W_{CL}}<0.5$ in two face-to-face networks, but relative high
values $\rho_{CC_{L},W_{CL}}>0.5$ in human indoor interaction network,
indicating “continuous group talk” involving more than three individuals exits
in human indoor interaction network, but not in two face-to-face networks.

Table 3: Pearson correlation coefficient of weight of circle links and other network metrics | $\rho_{W_{L},W_{CL}}$ | $\rho_{CC_{L},W_{CL}}$
---|---|---
HT | Day1 | 0.7390 | 0.2851
Day2 | 0.7292 | 0.2382
Day3 | 0.5583 | 0.2248
SG | Day1 | 0.7031 | 0.4682
Day2 | 0.7170 | 0.3971
Day3 | 0.6993 | 0.4833
WF | Day1 | 0.5935 | 0.7532
Day2 | 0.5079 | 0.7745
Day3 | 0.5547 | 0.7701
Figure 3: Counter-relationship between $W_{CL}$ and $BC_{L}$

##  5 Relationship Prediction Method

Here we present a heuristic method based on circle link to predict
relationships in temporal social networks. We use the weights of circle links
$W_{CL}$ as the predictor of potential relationships, where $W_{CL}>0$
represents there exits a potential relationship and vice verse. The weights of
edges in weighted social networks $W_{L}$ are used to testify the predictor,
where $W_{L}>0$ indicates there exits a relationship in real, and vice verse.
Therefore, Table 4 shows four classifies of results in our prediction method.
_Precision_ ($TP/(TP+FP)$) and _recall_ ($TP/(TP+FN)$) are used to quantify
the exactness and completeness of our method.

Table 4: The classifies of results in relationship prediction method | $W_{L}>0$ | $W_{L}=0$
---|---|---
$W_{CL}>0$ | True Positive(TP) | False Positive(FP)
$W_{CL}=0$ | False Negative(FN) | True Negative(TN)

Table 5 shows that our method performs well in all temporal social networks,
giving the evidence that it is possible to observer the structure of a large-
scale social network by locating a few sensors and analyzing their temporal
interaction data. The well performance of our method can be intuitively
contributed to high positive correlation between the predictor $W_{CL}$ and
the tester $W_{L}$. Moreover, high _precision_ is also caused by high positive
correlation between the predictor $W_{CL}$ and the edge clustering coefficient
$CC_{L}$ when comparing Table 3 with Table 5. Another possible factor is the
clustering coefficient of the network. It has been testified in our previous
work [13] that the average temporal clustering coefficient of WF is larger
than those of face-to-face networks. We further calculated the following
Circle Rate (CR) of the weighted and boolean clustering coefficient ($WCC$ and
$CC$) of each node in the _WF_ dataset:

${\textbf{CR}}_{i}\equiv\frac{|{cl}_{l,j}\wedge l_{l,j}|}{|{cl}_{l.j}|},\quad
l,j\in{N}_{i}$ (4)

T-test results as shown in Table 6 give the evidence that high _precision_ is
possibly caused by high clustering phenomenon in temporal social networks.
Finally, the _recall_ of one face-to-face network (SG) is two times larger
than that of another face-to-face network (HT), which is because that people
are more temporally clustered in former network [8].

Table 5: The precision and recall of our method in all temporal social
networks*

Dataset | Precision | Recall
---|---|---
HT | Day1 | 0.5023 | 0.2650
Day2 | 0.4781 | 0.2374
Day3 | 0.4413 | 0.2563
SG | Day1 | 0.5322 | 0.5669
Day2 | 0.5501 | 0.5806
Day3 | 0.5681 | 0.6483
WF | Day1 | 0.8484 | 0.2537
Day2 | 0.7824 | 0.2491
Day3 | 0.7788 | 0.2896

*The precision and recall are averaged over all nodes.

Table 6: T-test results about the hypothesis that the $\textbf{CR}_{i}$ has a
larger mean than the $CC_{i}$ or the $WCC_{i}$ in _WF_ dataset*

p-value | $WeightedCC$ | $UnweightedCC$
---|---|---
Day1 | 2.7520e-07 | 2.7453e-10
Day2 | 0.0055 | 2.0769e-04
Day3 | 0.5311 | 0.1662

*The hypothesis is true when p-value is smaller than 0.005.

##  6 Conclusions

In this work, We defined a new term _Circle Link_ to measure the contextual
relationship of ego and help predict potential relationship between ego’s
partners. The empirical analyses confirmed that the memory mechanism is not
universal in all social contacts. Furthermore, the tendency of close friends
having frequent continuous interaction with their common friend can be seen as
an extension of Granovetter’s hypothesis to temporal social networks. Finally,
we presented a heuristic method of using contextual information to excavate
potential relationship within ego’s neighborhoods and discuss main influence
factors. We believe future amelioration of this method would help to implement
larger-scale data collection of temporal social networks with limited sensors.

## Acknowledgments

The authors acknowledged the SocioPatterns project for sharing their data on
human face-to-face proximity contact, the Informatization Office of Fudan
University for the WiFi Data collection. This work was partly supported by the
National Key Basic Research and Development Program (No.2010CB731403), the
NCET program (No.NCET-09-0317), and the National Natural Science Foundation
(No.61273223) of China.

## References

  * [1] L. Lü and T. Zhou.  Link prediction in complex networks: A survey.  Physica A, 390(6):1150–1170, 2011.
  * [2] D. J. Watts and S. H. Strogatz.  Collective dynamics of small-world networks.  Nature, 393(6684):440–442, 1998.
  * [3] P. Holme and J. Saramäki.  Temporal networks.  Phys. Rep., 519(3):97–125, 2012.
  * [4] N. Eagle, A. Pentland, and D. Lazer.  Inferring friendship network structure by using mobile phone data.  Proc. Natl. Acd. Sci. USA, 106(36):15274–15278, 2009.
  * [5] N. Perra, B. Gonçalves, R. Pastor-Satorras, and A. Vespignani.  Activity driven modeling of time varying networks.  Sci. Rep., 2, 2012.
  * [6] C. Song, Z. Qu, N. Blumm, and A.-L. Barabási.  Limits of predictability in human mobility.  Science, 327(5968):1018–1021, 2010.
  * [7] T. Takaguchi, M. Nakamura, N. Sato, and et al.  Predictability of conversation partners.  Phys. Rev. X, 1(1):011008, 2011.
  * [8] L. Isella, J. Stehlé, A. Barrat, and et al.  What’s in a crowd? analysis of face-to-face behavioral networks.  J. Theor. Biol., 271(1):166–180, 2011.
  * [9] Y.-Q. Zhang and X. Li.  Temporal dynamics and impact of event interactions in cyber-social populations.  Chaos, 23(1):013131, 2013.
  * [10] M. Karsai, K. Kaski, and J. Kertész.  Correlated dynamics in egocentric communication networks.  PloS ONE, 7(7):e40612, 2012.
  * [11] M. Granovetter.  The strength of weak ties.  Am. J. Sociol., 78(6):1360–1380, 1973.
  * [12] S. Pajevic and D. Plenz.  The organization of strong links in complex networks.  Nat. Phys., 8(5):429–436, 2012.
  * [13] J. Cui, Y.-Q. Zhang, and X. Li.  On the clustering coefficients of temporal networks and epidemic dynamics.  In Circuits and Systems (ISCAS), 2013 IEEE International Symposium on, pages 2299–2302. IEEE, 2013.


#  The $\tau\to\nu_{\tau}\pi e^{+}e^{-}$ decay revisited

Adolfo Guevara  Departamento de Física Atómica, Molecular y Nuclear and
Instituto Carlos I de Física Teórica y Computacional Universidad de Granada,
E-18071 Granada, Spain.  Departament de Física Teòrica, IFIC, Universitat de
València - CSIC, Apt. Correus 22085, E-46071 València, Spain

###### Abstract

Abstract. The recent results from Belle of the
$\tau^{-}\to\nu_{\tau}\pi^{-}e^{+}e^{-}$ analysis has incentivated us to make
a reanalysis of a former work where the Structure Dependent parts are obtained
using Resonance Chiral Theory. Here we rely on the same theory accounting for
effects that explicitly break such symmetry. A motivation is the involved
$W\pi\gamma^{\star}$ vertex, utterly relevant in computing
$\tau^{-}\to\nu_{\tau}\pi^{-}$ radiative corrections.

###### pacs:

12.39.Fe, 12.38-t, 11.15.Pg

Resonance Chiral Theory, Chiral Lagrangians, hadronic tau decays, Belle
physics

##  1 Introduction

(This work is based on the research of ref. [1].) The increasing development
in the search for Beyond Standard Model Physics in flavor factories demands a
better control of backgrounds, specially in processes allowed in the Standard
Model with large suppressions. It has been shown that, to the precision
achieved in such experiments, important background may arise from radiative
corrections [2, 3, 4] and hadronic contamination [5]. In particular, $\tau$
decays provide a suitable scenario to search for BSM phenomena since they
involve low hadronic contamination compared to purely hadronic transitions and
since at experiments like Belle, a very large amount of them are created.
Therefore, such decays are suitable to look for BSM effects, such as Lepton
Flavor- Lepton Number and Lepton Universality-Violation, in semileptonic
$\tau$ decays. Some of the aforementioned effects could induce decays such as
$\tau^{-}\to\nu_{\tau}\pi^{+}\mu^{-}\mu^{-}$ or $\tau^{-}\to\mu^{-}\gamma$
that could be explored in Belle-II. In this work, we focus on an important
background of such processes, namely, the
$\tau^{-}\to\nu_{\tau}\pi^{-}\ell\overline{\ell}$. Furthermore, the relevance
of such decays relies on the effective vertex $W\gamma^{*}P$, which is of
utmost importance in the evaluation of radiative corrections to the
$\tau\to\nu_{\tau}P$ [6] decays as well as in determining high energy behavior
of the Transition Form Factor (TFF) of the $\pi$ meson, which is needed to
describe the most important parts of the Hadronic Vacuum Polarization
contribution to the anomalous magnetic moment of the $\mu$.

The recent measurement of the $\tau^{-}\to\nu_{\tau}\pi^{-}e^{+}e^{-}$ decays
in Belle [7] has confirmed our previous determination of the Branching
Fraction of such process [8], where we relied on Resonance Chiral Theory
(R$\chi$T) [9, 10] (which is $\chi$PT [11, 12, 13] expanded to include
resonances as active degrees of freedom) , where the lack of a pseudoscalar-
resonance exchange and some $\mathcal{O}(p^{6})$ contributions provide an
incomplete axial form factor in the sense of $VAP$ Green’s function analysis
[1, 14, 15]. This motivated us to revisit the analysis of such decays taking
into account the missing parts in the form factors by including the
pseudoscalar resonance exchange. These resonances, however, will necessarily
couple to a pseudo Goldstone boson and such coupling involves the mass of the
latter. Therefore, these resonances will contribute through terms that break
explicitly the Chiral Symmetry111Terms in the effective Lagrangian that depend
on the masses of the quarks (or equivalently on the masses of the pseudo
Goldstone bosons) will break chiral symmetry explicitly. Since the chiral
symmetry is spontanously broken $U(3)_{L}\otimes U(3)_{R}\to U(3)_{V}$,
considering quark-mass terms means that the $SU(3)_{V}$ subgroup gets
explicitly broken.. Thus, for the sake of consistency (to see how the $VVP$
Green’s function analysis [16, 17] relates to the vector form factor see
ref.[18]), all the leading-order contributions ($\mathcal{O}(m_{\pi}^{2})$) to
the remaining $U(3)_{V}$ breaking must be considered. Therefore, our analysis
will involve the missing $\mathcal{O}(p^{6})$ terms in ref. [8] as well as all
leading mass contributions to the
$\tau^{-}\to\nu_{\tau}\pi^{-}\ell\overline{\ell}$ decays. (Although we also
analyzed the $\tau^{-}\to\nu_{\tau}K^{-}\ell\overline{\ell}$ in ref. [1], here
we focus only in the $\pi$ decay-channel.)

Figure 1. Feynman diagrams of the SI contributions.

##  2 Amplitudes

There are two types of contributions to the decay amplitude, the first is a
Structure Independent (SI), which stems from Inner Bremsstrahlung, that is,
the virtual photon is radiated by the $\tau$ and $\pi$ meson where the latter
is taken as a point-like particle, therefore, the correspondig matrix element
can be obtained using scalar QED. The second is Structure Dependent (SD),
which means that the virtual photon can resolve the internal structure of the
$\pi$, this means that the matrix element of the weak and the electromagnetic
quark currents cannot be factorized into a matrix element of the weak current
times a matrix element of the electromagnetic one. This is where one needs to
rely on an Effective Field Theory. Separating the left weak current into the
vector and axial currents we can express the decay amplitude as follows

$\displaystyle\mathcal{M}_{IB}$
$\displaystyle=-iG_{F}V_{ud}fm_{\tau}\frac{e^{2}}{k^{2}}J_{\ell}^{\nu}\times$
$\displaystyle\overline{u}_{\nu_{\tau}}(1+\gamma_{5})\left[\frac{2p_{\nu}}{2p\cdot
k+k^{2}}+\frac{2{p_{\tau}}_{\nu}-\not{k}\gamma_{\nu}}{-2p_{\tau}\cdot
k+k^{2}}\right]u_{\tau},$ (1a) $\displaystyle\mathcal{M}_{V}$
$\displaystyle=-G_{F}V_{ud}\frac{e^{2}}{k^{2}}J_{\ell}^{\nu}J_{\tau}^{\mu}F_{V}(W^{2},k^{2})\varepsilon_{\mu\nu\alpha\beta}k^{\alpha}p^{\beta},$
(1b) $\displaystyle\mathcal{M}_{A}$
$\displaystyle=iG_{F}V_{ud}\frac{e^{2}}{k^{2}}J_{\ell}^{\nu}J_{\tau}^{\mu}\times$
$\displaystyle\left\\{F_{A}(W^{2},k^{2})\left[(W^{2}+k^{2}-m_{\pi}^{2})g_{\mu\nu}-2k_{\mu}p_{\nu}\right]\frac{}{}\right.$
$\displaystyle\left.-\frac{}{}A_{2}(W^{2},k^{2})k^{2}g_{\mu\nu}+A_{4}(W^{2},k^{2})k^{2}(p+k)_{\mu}p_{\nu}\right\\},$
(1c)

where $J_{\ell}^{\nu}=\overline{u}(p_{-})\gamma^{\nu}v(p_{+})$ and
$J_{\tau}^{\mu}=\overline{u}(q)(1+\gamma_{5})\gamma^{\mu}u(p_{\tau})$ are the
lepton electromagnetic and $\tau$ weak charged currents, respectively. The
definition of momenta is given in Fig. 1 Also, $W^{2}=(p_{\tau}-q)^{2}$ and
$k=p_{+}+p_{-}$. It is in the Form Factors of eq. (1b) and (1c) that the
hadronic dynamics that cannot be described by means of the underlying theory
(QCD) is encoded. These are the expressions that we will compute using
R$\chi$T.

##  3 Structure dependent part

Figure 2. Feynman diagrams of the vector SD contributions.

Using R$\chi$T we find that the Feynman diagrams that contribute to the vector
form factor are shown in Figure 2, while those contributing to the axial form
factors are given in Figure 3. We refer the reader to ref. [1] for the
operator basis used. The expression for the vector form factor gives

$F_{V}(W^{2},k^{2})=\frac{1}{3f}\left\\{-\frac{N_{C}\frac{}{}}{8\pi^{2}}+64m_{\pi}^{2}C_{7}^{W\star}\hskip
60.27759pt\right.$
$-8C_{22}^{W}(W^{2}+k^{2})+\frac{4{F_{V}^{ud}}^{2}}{M_{\rho}^{2}-W^{2}}\frac{d_{3}(W^{2}+k^{2})+d_{123}^{\star}m_{\pi}^{2}}{M_{\omega}^{2}-k^{2}}$
$+\frac{2\sqrt{2}F_{V}^{ud}}{M_{V}}\frac{c_{1256}W^{2}-c_{1235}^{\star}m_{\pi}^{2}-c_{125}k^{2}}{M_{\rho}^{2}-W^{2}}$
$\left.\hskip
17.22217pt+\frac{2\sqrt{2}F_{V}^{ud}}{M_{V}}\frac{c_{1256}k^{2}-c_{1235}^{\star}m_{\pi}^{2}-c_{125}W^{2}}{M_{\omega}^{2}-k^{2}}\right\\},$
(2)

which is the same result as for the $\pi^{0}$-TFF (accounting for factors of 2
from Bose symmetry) with the leading-order chiral symmetry breaking terms
[18].

Figure 3. Feynman diagrams of the axial SD contributions.

From the diagrams of Figure 3 we find the expressions

$F_{A}(W^{2},k^{2})=\frac{F_{V}^{ud}}{2f}\frac{F_{V}^{ud}-2G_{V}-m_{\pi}^{2}\frac{4\sqrt{2}d_{m}}{M_{\pi^{\prime}}^{2}}(\lambda_{1}^{PV}+2\lambda_{2}^{PV})}{M_{\rho}^{2}-k^{2}}$
$-\frac{F_{A}}{2f}\frac{F_{A}-2m_{\pi}^{2}\frac{4\sqrt{2}d_{m}}{M_{\pi^{\prime}}^{2}}\lambda_{1}^{PA}}{M_{a_{1}}^{2}-W^{2}}$
$\hskip
43.05542pt+\frac{\sqrt{2}}{f}\frac{F_{A}F_{V}^{ud}}{M_{a_{1}}^{2}-W^{2}}\frac{\lambda_{0}^{\star}m_{\pi}^{2}-\lambda^{\prime}k^{2}-\lambda^{\prime\prime}W^{2}}{M_{\rho}^{2}-k^{2}},$
(3)

$A_{2}(W^{2},k^{2})=\frac{2}{f}\left(G_{V}+\frac{2\sqrt{2}m_{\pi}^{2}d_{m}}{M_{\pi^{\prime}}^{2}}\lambda_{1}^{PV}\right.\hskip
43.05542pt$ $\hskip
43.05542pt\left.+\frac{\sqrt{2}F_{A}}{M_{a_{1}}^{2}-W^{2}}W^{2}(\lambda^{\prime}+\lambda^{\prime\prime})\right)\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}},$
(4)

$A_{4}(W^{2},k^{2})=\frac{2}{f}\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}\left[\frac{G_{V}}{W^{2}-m_{\pi}^{2}}\right.\hskip
51.6665pt$ $\hskip
47.36096pt\left.+\frac{2\sqrt{2}d_{m}m_{\pi}^{2}\lambda_{1}^{PV}}{M_{\pi}^{2}\left(W^{2}-m_{\pi}^{2}\right)}+\frac{\sqrt{2}F_{A}(\lambda^{\prime}+\lambda^{\prime\prime})}{M_{a_{1}}^{2}-W^{2}}\right].$
(5)

Taking the chiral limit ($m_{\pi}\to 0$), it can be seen that $A_{2}$ and
$A_{4}$ recover their linear dependency [14, 15, 19].

Imposing the high-energy beahviour of QCD form factors allows us to constrain
the parameters of the model. The behavior of the vector form factor in the
limits $\lim_{\lambda\to\infty}F_{V}(\lambda W^{2},0)$ and
$\lim_{\lambda\to\infty}F_{V}(\lambda W^{2},\lambda k^{2})$ [20, 21] give the
constraints (see ref. [18])

  * •

$F_{V}(W^{2},k^{2})$, $\mathcal{O}(m_{P}^{0})$:

$\displaystyle C_{22}^{W}$ $\displaystyle=$ $\displaystyle 0\,,$ (6)
$\displaystyle c_{125}$ $\displaystyle=$ $\displaystyle 0\,,$ (7)
$\displaystyle c_{1256}$ $\displaystyle=$
$\displaystyle-\frac{N_{C}M_{V}}{32\sqrt{2}\pi^{2}F_{V}}\,,$ (8)
$\displaystyle d_{3}$ $\displaystyle=$
$\displaystyle-\frac{N_{C}M_{V}^{2}}{64\pi^{2}F_{V}^{2}}\,.$ (9)

  * •

$F_{V}(W^{2},k^{2})$, $\mathcal{O}(m_{P}^{2})$:

$\displaystyle\lambda_{V}$ $\displaystyle=$
$\displaystyle-\frac{64\pi^{2}F_{V}}{N_{C}}C_{7}^{W*}\,,$ (10) $\displaystyle
c_{1235}^{*}$ $\displaystyle=$
$\displaystyle\frac{N_{C}M_{V}e^{V}_{m}}{8\sqrt{2}\pi^{2}F_{V}}+\frac{N_{C}M_{V}^{3}\lambda_{V}}{4\sqrt{2}\pi^{2}F_{V}^{2}}.$
(11)

We also use the relations fron VVP Green’s Function, for the sake of
predictability. Also, we use the values for some parameters of the fit done in
ref. [18]. These are

$c_{125}=c_{1235}=0,\hskip
8.61108ptc_{1256}=-\frac{N_{C}M_{V}}{32\sqrt{2}\pi^{2}F_{V}},$
$\kappa_{5}^{P}=0,\hskip
8.61108ptd_{3}=-\frac{N_{C}M_{V}^{2}}{64\pi^{2}F_{V}^{2}}+\frac{F^{2}}{8F_{V}^{2}}+\frac{4\sqrt{2}d_{m}\kappa_{3}^{PV}}{F_{V}},$
$C_{7}^{W}=C_{22}^{W}=0,\hskip 12.91663ptd_{123}=\frac{F^{2}}{8F_{V}^{2}}.$
(12)

However, if we impose that the axial form factors vanish in the same limits as
the vector form factors we obtain no constraint since they already have the
correct asymptotic behavior. Instead, we use the VAP Green’s Function behavior
[14, 15, 19], which gives the following relations

$\displaystyle\lambda_{0}=\frac{F^{2}}{4\sqrt{2}F_{V}F_{A}},\hskip
8.61108ptd_{m}\lambda_{2}^{PV}=\frac{3F^{2}+2F_{A}^{2}-2F_{V}^{2}}{16\sqrt{2}F_{V}},\hskip
4.30554pt$
$\displaystyle\lambda^{\prime\prime}=-\frac{F^{2}+F_{A}^{2}-2F_{V}G_{V}}{2\sqrt{2}F_{V}F_{A}},\hskip
8.61108ptd_{m}\lambda_{1}^{PV}=-\frac{F^{2}}{4\sqrt{2}F_{V}},$
$\displaystyle\lambda^{\prime}=\frac{F^{2}+F_{A}^{2}}{2\sqrt{2}F_{V}F_{A}},\hskip
8.61108ptd_{m}\lambda_{1}^{PA}=\frac{F^{2}}{16\sqrt{2}F_{A}}\,.\hskip
47.36096pt$ (13)

##  4 Fit

Although the spectra of the invariant $m_{\pi e^{+}e^{-}}$ mass shows the
typical $\rho(1450)-\rho(1700)$ interference (see [1, 7]), involving heavier
copies of the $\rho$ meson implies considering more unconstrained parameters,
wich cannot be well contrained using the data due to large errors. It is,
therefore, expected that some relations stemming from high-energy QCD or
Green’s Functions will not be fulfilled. Thus, we do not rely on Weinberg’s
sum rules.

Comparing the branching fraction data from Belle with the expected signal
events distribution, we obtain an estimation for the deconvolution of signal
from the detector, which we do not know. This is taken as a systematic
uncertainty, which is comparable to the one reported by Belle. Also, the use
of incomplete expressions for the axial form factors by the Belle
collaboration to obtain the branching ratios and invariant mass spectrum [1,
7] can lead to biased estimations. We, therefore, choose to fit the total
branching fraction to the data instead of computing it from the partial width
expression. Thus, we fit $F_{V},F_{A},\lambda^{\star}_{0}$ and $\mathcal{B}$,
the branching ratio.

The resolution of the data is not precise enough in order to obtain a
determined set of parameters that minimize the $\chi^{2}$ of the fit.
Therefore, we fix one of the parameters and fit the other three. We fix first
$F_{A}=130$ MeV, and then $\lambda_{0}^{\star}=102\times 10^{-3}$. The former
is in agreement with the high energy relation of ref. [22], while the latter
is the estimation of $\lambda_{0}^{\star}$ neglecting chiral symmetry breaking
effects [23]. A third fit fixing $F_{V}=\sqrt{3}F$ (its correct high energy
value [22]) was done, however, it furnishes a very poor fit. The fits obtained
are shown in Table 1. As can be seen from set 2 in this Table, $F_{V}$ is far
closer to $\sqrt{2}F$ than to $\sqrt{3}F$, which reiforces the decision of
neglecting the Weinberg’s sum rules. The spectra obtained from the number of
events spectrum for each parameter set along with the experimental data is
shown in Figure 4.

Figure 4. Normalized invariant mass spectra obtained with the two sets of
parameters obtained from fitting to the Belle data. The purple line
corresponds to the data with $F_{A}$ fixed, while the green one stands for
that with $\lambda_{0}^{\star}$ fixed. The blue data corresponds to
measurements of $\tau^{-}$ decays, which show best agreement with our model.
[7]

##  5 Predictions

Using the phase space configuration of ref. [1] and the parameters of both
sets we obtained the $m_{\pi e^{+}e^{-}}$ and $m_{e^{+}e^{-}}$ invariant mass
spectra and the total branching ratio. We computed the
$\mathcal{B}(\tau^{-}\to\nu_{\tau}\pi^{-}e^{+}e^{-})$ implementing the
kinematical cut used by Belle to obtain their measurement of the branching
ratio, $m_{\pi e^{+}e^{-}}\geq 1.05$ GeV. The difference between our
prediction and Belle’s measurement is taken as a systematic uncertainty due to
unfolding. Thus, taking into account the correlation of the fit, we produced
2400 points using a Gaussian distribution in the parameter space to integrate
the branching fraction. We define our prediction of the branching ratio with
uncertainty for each decay channel to be the union of the intervals given by
the central values with their errors for the prediction thrown by each
parameter set, this is, the union of the intervals of Table 2; this is given
in Table 1. We also give the $m_{\pi^{-}e^{+}e^{-}}$ invariant mass spectrum
in Figure 5 and the $m_{e^{+}e^{-}}$ in Figure 6.

Figure 5. Invariant mass spectra $m_{\pi^{-}e^{+}e^{-}}$ for $P=\pi$, the
thickness represents the error band obtained from the difference between the
two sets.

Figure 6. Invariant mass spectra $m_{\pi^{-}e^{+}e^{-}}$, the thickness
represents the error band obtained from the difference between the two sets.
The green line is the prediction of ref. [8].

##  6 Conclusions

We have given an improved description of the decay amplitude from ref. [8].
This involves a more accurate theoretical description by including flavor
breaking terms as well as predictions with reduced uncertainty. As found in
the VVP Green’s function, the inclusion of pseudoscalar resonances in the form
factors is needed to give compatible expressions with the VAP Green’s Function
analyses [14, 15, 19]. A reasonably good fit $\chi^{2}/dof\approx 1.2$ was
obtained. However, a better set of data for the $m_{\pi^{-}e^{+}e^{-}}$ and
data for the $m_{e^{+}e^{-}}$ invariant mass spectra would allow to resolve
the dynamics of the heavier copies of the $\rho$ meson involved in these
decays. The fact that the fits favor a value $F_{V}=\sqrt{2}F$ shows that
dynamics from such heavier copies are indeed involved, making thus necessary
to improve the experimental analyses to obtain more precise data and the need
for a $m_{e^{+}e^{-}}$ spectra.

##  7 Acknowledgments

We are indebted to Denis Epifanov and Yifan Jin for leading the Belle analysis
of these decays. We thank Pablo Sánchez Puertas his useful comments on short
distance constraints and Juan José Sanz Cillero for useful comments on finding
the optimal parameter-space basis for the fitting procedure. This work was
supported partly by the Spanish MINECO and European FEDER funds (grant
FIS2017-85053-C2-1-P) and Junta de Andalucía (grant FQM-225), partly by the
Generalitat Valenciana (grant Prometeo/2017/053) and by Cátedras Marcos
Moshinsky (Fundación Marcos Moshinsky).

| set 1 | set 2
---|---|---
|  $F_{A}=130$ MeV | $\lambda_{0}^{\star}=102\times 10^{-3}$
$F_{A}$ | 130 MeV | ($122\pm 0$) MeV
$F_{V}$ | (135.5$\pm 1.1$) MeV | ($137.4\pm 1.6$) MeV
$\lambda_{0}^{\star}$ | $(384\pm 0)\times 10^{-3}$ | 102$\times 10^{-3}$
$\mathcal{B}$ | $(6.01\pm 0)\times 10^{-6}$ | $(6.36\pm 0.12)\times 10^{-6}$
$\chi^{2}/dof$ | 31.1/26 | 31.4/26
Table 1: Our best fit results for $F_{A}$ , $F_{V}$, $\lambda_{0}^{*}$ and the branching ratio. For the fit results shown on the left (right) columns we fix $F_{A}=130$ MeV ($\lambda_{0}^{\star}=102\times 10^{-3}$), respectively. A $0$ error means that the fit uncertainty in the parameter is negligible with respect to its central value. | set 1 | set 2 | IB
---|---|---|---
$\mathcal{B}(\tau^{-}\to\nu_{\tau}\pi^{-}e^{+}e^{-})$ | $(2.38\pm 0.28\pm 0.11)\cdot 10^{-5}$ | $(2.45\pm 0.45\pm 0.04)\cdot 10^{-5}$ | $1.457(5)\cdot 10^{-5}$
Table 2: Full branching ratios accounting for both (dominant) systematic and statistical uncertainties. In the right column we show the SI contribution with the error arising from numerical integration of the differential decay width.  $P,\ell$ | $\mathcal{B}(\tau^{-}\to\nu_{\tau}P^{-}\ell\overline{\ell})$
---|---
$\pi,e$ | $(2.41\pm 0.40\pm 0.12)\cdot 10^{-5}$
Table 3: Branching ratios for the different decay channels. The central value
is the mean of the union of intervals given in both columns of Table 2, the
first error covers the width of such union of ranges and the second error is
the quadratic mean of statistical uncertainties in Table 2.

## References

  * $1$. A. Guevara, G. López Castro and P. Roig, [arXiv:2111.09994 [hep-ph]].
  * $2$. K. Kampf, J. Novotný and P. Sánchez-Puertas, Phys. Rev. D 97 (2018) no.5, 056010.
  * $3$. T. Husek, K. Kampf, S. Leupold and J. Novotny, Phys. Rev. D 97 (2018) no.9, 096013.
  * $4$. A. Guevara, G. López-Castro and P. Roig, Phys. Rev. D 95 (2017) no.5, 054015.
  * $5$. A. Guevara, G. López Castro, P. Roig and S. L. Tostado, Phys. Rev. D 92 (2015) no.5, 054035.
  * $6$. Z. H. Guo and P. Roig, Phys. Rev. D 82 (2010), 113016
  * $7$. Y. Jin et al. [Belle], Phys. Rev. D 100 (2019) no.7, 071101.
  * $8$. A. Guevara, G. López Castro and P. Roig, Phys. Rev. D 88 (2013) no.3, 033007 [arXiv:1306.1732 [hep-ph]].
  * $9$. G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B 321 (1989), 311-342.
  * $10$. G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Phys. Lett. B 223 (1989), 425-432.
  * $11$. S. Weinberg, Physica A 96 (1979) no.1-2, 327-340.
  * $12$. J. Gasser and H. Leutwyler, Annals Phys. 158 (1984), 142.
  * $13$. J. Gasser and H. Leutwyler, Nucl. Phys. B 250 (1985), 465-516.
  * $14$. V. Cirigliano, G. Ecker, M. Eidemüller, A. Pich and J. Portolés, Phys. Lett. B 596 (2004), 96-106.
  * $15$. V. Cirigliano, G. Ecker, M. Eidemüller, R. Kaiser, A. Pich and J. Portolés, Nucl. Phys. B 753 (2006), 139-177.
  * $16$. P. D. Ruiz-FemenÃa, A. Pich and J. PortolÃ©s, JHEP 07 (2003), 003.
  * $17$. K. Kampf and J. Novotny, Phys. Rev. D 84 (2011), 014036
  * $18$. A. Guevara, P. Roig and J. J. Sanz-Cillero, JHEP 06 (2018), 160 [arXiv:1803.08099 [hep-ph]].
  * $19$. M. Knecht and A. Nyffeler, Eur. Phys. J. C 21 (2001), 659-678.
  * $20$. G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22 (1980), 2157.
  * $21$. S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31 (1973), 1153-1156.
  * $22$. P. Roig and J. J. Sanz Cillero, Phys. Lett. B 733 (2014), 158-163.
  * $23$. J. A. Miranda and P. Roig, Phys. Rev. D 102 (2020), 114017.