Dipole momenta and compositeness
of the $\tau$ lepton at Belle II

M. Fabbrichesia, and L. Marzolab

(a) INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy
(b) Laboratory of High-Energy and Computational Physics, NICPB, Rävala 10,
10143 Tallinn, Estonia

ABSTRACT

The large number of $\tau$ leptons available at the Belle II experiment makes
it possible to study their properties and the extent of their compositeness.
Our strategy relies on three observables defined in terms of elements of the
polarization density matrix of the produced $\tau$ pairs, which we obtain via
quantum tomography. The observables are able to explore values of the magnetic
dipole moment down to $6.3\times 10^{-4}$, which is two order of magnitude
better than the current experimental limit, constrain the electromagnetic
radius below $4.3\times 10^{-3}$ fm and exclude an electric dipole moment
larger than $1.7\times 10^{-17}$ e cm. The quoted limits are at the $95\%$
confidence level and are obtained for a benchmark integrated luminosity of 1
ab-1.

$\bowtie$

##  1 Introduction

T he abundant production of $\tau$-lepton pairs in
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ at Belle II [1] provides a promising
laboratory for a systematic analysis of their electromagnetic couplings. The
study is important because deviations of these parameters from their Standard
Model (SM) values would provide insights into the nature of underlying new
physics, implying for instance the presence of new particles or interactions.
Given the current data, the $\tau$ lepton —the heaviest among the leptons, as
the top quark among the quarks—could be the likeliest to show a behavior
departing from that described by the SM. In this paper we then investigate the
electric dipole moment, the anomalous magnetic moment and the possible
substructure of the $\tau$ lepton, represented by the size of its radius that
we define below.

The most general electromagnetic Lorentz-invariant coupling between the photon
and the $\tau$ lepton can be written as

$-ie\,\bar{\tau}\,\Gamma^{\mu}(q^{2})\,\tau\,A_{\mu}(q)=-ie\,\bar{\tau}\left[\gamma^{\mu}F_{1}(q^{2})+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m_{\tau}}F_{2}(q^{2})+\frac{\sigma^{\mu\nu}\gamma_{5}q_{\nu}}{2m_{\tau}}F_{3}(q^{2})\right]\tau\,A_{\mu}(q)\,,$
(1.1)

which defines the magnetic and electric dipole moments as

$a_{\tau}=F_{2}(0)\quad\mbox{and}\quad
d_{\tau}=\frac{e}{2m_{\tau}}F_{3}(0)\,,$ (1.2)

as well as the mean squared radius

$\langle\vec{r}^{\;2}\rangle=-6\left.\frac{dG_{E}}{d\vec{q}^{\;2}}\right|_{q^{2}=0}\,,$
(1.3)

with

$G_{E}(q^{2})=F_{1}(q^{2})+\frac{q^{2}}{4m_{\tau}^{2}}F_{2}(q^{2})\,.$ (1.4)

We take the form factors $F_{2,3}(q^{2})$ at the leading order for $q^{2}\to
0$, while we retain the first order in $q^{2}$ for the form factor
$F_{1}(q^{2})$.

Whereas the magnetic moments of lighter leptons are measured to an accuracy
that probes the respective radiative contributions expected within the SM,
that of the heavier $\tau$ lepton is still only known to a lower precision:
the current limits

$-0.052<a_{\tau}<0.013\quad\mbox{(95\%)
CL}\;\text{\cite[cite]{[\@@bibref{}{DELPHI:2003nah}{}{}]}}\,,$ (1.5)

are one order of magnitude above the theoretical value

$a_{\tau}^{\rm SM}=1.17721(5)\times
10^{-3}\,\text{\cite[cite]{[\@@bibref{}{Eidelman:2007sb}{}{}]}}.$ (1.6)

The electric dipole moment is constrained to be (with an integrated luminosity
of 833 fb-1)

$-0.185\times 10^{-16}<d_{\tau}<0.061\times 10^{-16}\;\mbox{e
cm}\quad\mbox{(95\%)
CL}\;\text{\cite[cite]{[\@@bibref{}{Belle:2021ybo}{}{}]}}$ (1.7)

assuming its value to be real. A non-vanishing value would signal the presence
of CP violation.

No direct limits on the electromagnetic radius of the $\tau$ lepton exist, but
there is a bound on the scale

$\Lambda_{\rm
C.I.}>7.9\;\mbox{TeV}\;\text{\cite[cite]{[\@@bibref{}{ALEPH:2006jhv}{}{}]}}$
(1.8)

for the related four-fermion contact interaction

$+\frac{2\pi}{\Lambda^{2}_{\rm
C.I.}}\,(\bar{e}_{L}\gamma^{\mu}e_{L})\,(\bar{\tau}_{L}\gamma_{\mu}\tau_{L})\,,$
(1.9)

constrained at the energy of about 200 GeV.

The form factors introduced in Eq. (1.1) translate, in the language of
effective field theory, into the $SU(2)\times U(1)$ invariant effective
operators for the $\tau$ lepton. The leading contributions come from the
following three higher-dimensional operators:

$\hat{O}_{1}=e\frac{c_{1}}{m_{\tau}^{2}}\bar{\tau}\gamma^{\mu}\tau
D^{\nu}F_{\mu\nu}\,,\quad\hat{O}_{2}=e\frac{c_{2}\,\upsilon}{2m_{\tau}^{2}}\bar{\tau}\sigma^{\mu\nu}\tau
F_{\mu\nu}\quad\mbox{and}\quad\hat{O}_{3}=e\frac{c_{3}\,\upsilon}{2m_{\tau}^{2}}\bar{\tau}\sigma^{\mu\nu}\gamma_{5}\tau
F_{\mu\nu}\,,$ (1.10)

where $D^{\nu}$ is the covariant derivative, $F_{\mu\nu}$ the electromagnetic
field strength tensor and $\upsilon=174$ GeV is the Higgs field vacuum
expectation value. In Eq. (1.10) the dimensionless Wilson coefficients $c_{i}$
are taken to be real. Left- and right-handed chiral fields enter
symmetrically. The operator $\hat{O}_{1}$ gives the leading $q^{2}$ dependence
of the form factor $F_{1}$, while $\hat{O}_{2,3}$ give the $q^{2}$-independent
term of the form factors $F_{2,3}$:

$F_{1}(q^{2})=1+c_{1}\frac{q^{2}}{m_{\tau}^{2}}+\ldots\qquad\mbox{and}\qquad
F_{2,3}(0)=2\,c_{2,3}\frac{\upsilon}{m_{\tau}}\,.$ (1.11)

The operators $\hat{O}_{2,3}$ are written for the sake of convenience with an
extra factor $\upsilon/m_{\tau}$, sourced after the electroweak symmetry
breaking by the dimension-six $SU(2)_{L}\times U(1)_{Y}$ gauge invariant
operators involving the Higgs field. Operators of higher dimensions can in
general contribute—they give further terms in the expansion of the form
factors—but their effect is suppressed.

Compared to the effective field theory approach, the form factors we use have
the advantage of directly answering a simple question: is the $\tau$ lepton a
point like particle or does it show a composite structure? The answer comes
without relaying on any assumption or UV model to select the relevant
operators out of the many present in the effective theory.

Our strategy to constrain the electromagnetic couplings in Eq. (1.1) exploits
the polarization density matrix, which can be experimentally reconstructed
through a procedure dubbed quantum tomography and gives a bird’s-eye view of
the possible observables available for a given process. The method has been
previously used to constrain physics beyond the SM affecting the top-quark [6,
7] and $\tau$ pair [7] production at the LHC, or yielding Higgs anomalous
couplings to $\tau$ leptons [8] and gauge bosons [9, 10, 11]. For the present
case, we use the entries of the polarization density matrix to define three
observables that provide the means to best constrain the parameters in Eq.
(1.1): one observable measures the entanglement [12] in the spin states of the
produced $\tau$ pairs, another is related to triple products involving one
momentum and the spin vectors of the $\tau$ leptons—and it is specific to the
CP-violating electric dipole moment, and the third is the total cross section.

We compare our results to the current experimental limits [13] and to recent
phenomenological estimates of the electric and magnetic dipole moments in [14]
and of the electric dipole moments in [15].

###  1.1 Events

We propose to probe the electromagnetic couplings of the $\tau$ lepton by
using the process $e^{+}e^{-}\to\tau^{+}\tau^{-}$ at the Belle-II experiment,
located at the SuperKEKB collider. The SuperKEKB collider delivers
$e^{+}e^{-}$ collisions at a center of mass energy of $\sqrt{s}=10.579$ GeV.
The Belle collaboration has published analyses of
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ production with data corresponding to an
integrated luminosity of up to $921$ fb-1 [16], equivalent to $841$ million
$\tau^{+}\tau^{-}$ events, and other 175 million events have now been added
with the Belle II dataset [17]. The aim of the SuperKEKB project is to collect
$50$ ab-1 of data [18, 1], corresponding to a dataset of about $50$ billion
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ events. Of these, we can use those involving
the $\tau^{-}\to\pi^{-}\nu_{\tau}$, $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ and
$\tau^{-}\to\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ decays to reconstruct the
polarization density matrix. The combination of these channels covers about
$21\%$ of $\tau$ pair decays.

We expect the dominant background to arise from mis-reconstructions of the
$\tau$ decay channel, affecting about $15\%$ of the
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ events at Belle II [17]. Backgrounds arising
from the process $e^{+}e^{-}\to\textrm{q}\bar{\textrm{q}}$ and from other
sources are negligible in comparison. Given their modest impact, we neglect
the effect of backgrounds in our study. It will have to be taken into account
when the analysis is performed on the actual data.

The detailed Monte Carlo simulation of the process
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ in [19] shows that a full quantum tomography
is achievable and provides an estimate of the uncertainties in the density
matrix that we are going to use in our analysis.

##  2 Methods

Q uantum tomography aims to fully determine the density matrix $\rho$ of a
quantum state. The $\tau$ leptons—whose spins are represented with two-level
quantum states, that is, qubits—act as their own polarimeters and the full
polarization density matrix can be reconstructed, within the inherent
uncertainties of the procedure, from the angular distribution of the $\tau$
decay products.

Polarizations are more difficult to measure than momenta and so the
reconstruction of the polarization density matrix from the data is
challenging. The main aim of this work is to show to what extent the
advantages of using the proposed method make the extra work in the
experimental analysis eventually worthwhile.

The density matrix describing the polarization state of a quantum system
composed by two fermions can be written as

$\rho=\frac{1}{4}\quantity[\mathbb{1}\otimes\mathbb{1}+\sum_{i}\operatorname{B}_{i}^{+}\,\quantity(\sigma_{i}\otimes\mathbb{1})+\sum_{j}\operatorname{B}_{j}^{-}\,\quantity(\mathbb{1}\otimes\sigma_{j})+\sum_{i,j}\operatorname{C}_{ij}\,\quantity(\sigma_{i}\otimes\sigma_{j})],$
(2.1)

where $i,j\in\\{n,r,k\\}$ and $\sigma_{i}$ are the Pauli matrices. The
decomposition refers to a right-handed orthonormal basis,
$\\{{\bf\hat{n}},{\bf\hat{r}},{\bf\hat{k}}\\}$ and the quantization axis for
the polarization is taken along ${\bf\hat{k}}$, so that
$\sigma_{k}\equiv\sigma_{3}$. In the fermion-pair center of mass frame we have

${\bf\hat{n}}=\frac{1}{\sin(\theta)}\quantity({\bf\hat{p}}\times{\bf\hat{k}}),\quad{\bf\hat{r}}=\frac{1}{\sin(\theta)}\quantity({\bf\hat{p}}-\cos(\theta){\bf\hat{k}})\,,$
(2.2)

where ${\bf\hat{k}}$ is the direction of the $\tau^{+}$ momentum and $\theta$
is the scattering angle satisfying ${\bf\hat{p}}\cdot{\bf\hat{k}}=\cos\theta$,
with ${\bf\hat{p}}$ being the direction of the incoming $e^{+}$.

The coefficients $\operatorname{B}^{\pm}_{i}$ in Eq. (2.1) give the
polarizations of the individual fermions, whereas the matrix
$\operatorname{C}_{ij}$ contains the spin correlations. By using the
properties $\mbox{Tr}\,(\sigma_{i}\sigma_{j})=2\delta_{ij}$ and
$\mbox{Tr}\,(\sigma_{i})=0$ we have:

$\operatorname{B}_{i}^{+}=\frac{\mbox{Tr}\,[\rho\quantity(\sigma_{i}\otimes\mathbb{1})]}{4}\,,\quad\operatorname{B}_{i}^{-}=\frac{\mbox{Tr}\,[\rho\quantity(\mathbb{1}\otimes\sigma_{i})]}{4}\,,\quad\operatorname{C}_{ij}=\frac{\mbox{Tr}\,[\rho\quantity(\sigma_{i}\otimes\sigma_{j})]}{4}\,,$
(2.3)

and, for the process at hand, it holds that $\operatorname{B}^{\pm}_{i}=0$.

The non-vanishing coefficients $\operatorname{C}_{ij}$ can be reconstructed in
the actual experiments by tracking the angular distribution of suitable
$\tau$-pair decay products. In particular, for events where each $\tau$ lepton
decays to a single pion and a neutrino, we have

$\frac{1}{\sigma}\,\frac{d\sigma}{d\cos\theta^{+}_{i}\,d\cos\theta^{-}_{j}}=\frac{1}{4}\,\left(1+\operatorname{C}_{ij}\,\cos\theta^{+}_{i}\,\cos\theta^{-}_{j}\right)\,,$
(2.4)

where $\cos\theta^{\pm}_{i}$ is the projections of the $\pi^{\pm}$ momentum
direction on the $\\{{\bf\hat{n}},{\bf\hat{r}},{\bf\hat{k}}\\}$ basis, as
computed in the rest frame of the decaying $\tau^{\pm}$. Crucial to the whole
procedure is the reconstruction of the neutrino kinematics. More details
pertaining to the experimental determination of the $\operatorname{C}_{ij}$
coefficients for different decay channels are presented, for instance, in Ref.
[19], which we closely follow.

Quantum tomography gives the coefficients $\operatorname{B}_{i}$ and
$\operatorname{C}_{ij}$ for the density matrix and there is a number of
observables that can be constructed with them. We consider the three that
provide the most stringent limits on the anomalous couplings:

  * •

The concurrence $\mathscr{C}[\rho]$ [12], which can be computed by means of
the particularly simple formula

$\mathscr{C}[\rho]=\frac{1}{2}\max\Big{[}0,\,|\operatorname{C}_{rr}+\operatorname{C}_{kk}|-(1+\operatorname{C}_{nn})\,,\,\sqrt{(\operatorname{C}_{rr}-\operatorname{C}_{kk})^{2}+4\operatorname{C}_{rn}^{2}}-|1-\operatorname{C}_{nn}|\Big{]}\,,$
(2.5)

because the $B_{i}^{\pm}=0$ and all off-diagonal elements but
$\operatorname{C}_{rn}$ vanish when the scattering angle is integrated over.
We use as observable the concurrence Eq. (2.5) computed from the
$\operatorname{C}$ matrix obtained by averaging over the angular distribution
of the $\tau$ leptons.

  * •

The second observable is just the total cross section:

$\sigma=\frac{1}{64\pi^{2}\,s}\int\mbox{d}\Omega\frac{|{\cal
M}|^{2}}{4}\sqrt{1-\frac{4m_{\tau}^{2}}{s}}\,,$ (2.6)

where we neglect the electron mass and $\sqrt{s}=10.579$ GeV. The spin-summed
squared amplitude, $|{\cal M}|^{2}$, is given by

$|{\cal
M}|^{2}=e^{4}\Bigg{\\{}\frac{s}{m_{\tau}^{2}}\left[\sin^{2}\theta\left(16\,c_{1}^{2}+\tilde{d}_{\tau}^{2}+a_{\tau}^{2}\right)+4\,c_{1}\left(\cos
2\theta+4\,a_{\tau}+3\right)\right]+2\,\frac{s^{2}}{m_{\tau}^{4}}\,c_{1}^{2}\Big{(}\cos
2\theta+3\Big{)}\\\
+4\left(8\,c_{1}\,\sin^{2}\theta+\left(\tilde{d}_{\tau}^{2}+a_{\tau}^{2}+1\right)\cos^{2}\theta-\tilde{d}_{\tau}^{2}+a_{\tau}^{2}+4\,a_{\tau}+1\right)+16\,\frac{m_{\tau}^{2}}{s}\sin^{2}\theta\Bigg{\\}}\,.$
(2.7)

  * •

The third observable, $\mathscr{C}_{odd}$, singles out the antisymmetric parts
of the density matrix. It is defined as

$\mathscr{C}_{odd}=\frac{1}{2}\,\sum_{\begin{subarray}{c}i,j\\\
i<j\end{subarray}}\Big{|}\operatorname{C}_{ij}-\operatorname{C}_{ji}\Big{|}\,,$
(2.8)

and contains only off-diagonal terms that change sign under transposition. The
observable encodes kinematical variables that can be written as the triple
products of momenta and spin vectors, for instance:

$\vec{k}\cdot\Big{(}\vec{s}_{\hat{n}}\times\vec{s}_{\hat{r}}\Big{)}\,,$ (2.9)

with $\vec{k}$ the momentum of one of the particles and $\vec{s}_{\hat{n}}$
and $\vec{s}_{\hat{r}}$ the projections of the spin vector along two
directions orthogonal to the momentum. In the present case there is only one
non-vanishing term coming from the elements $\operatorname{C}_{rn}$ and
$\operatorname{C}_{nr}$, which are equal in magnitude and with the opposite
sign. We build the observable $\mathscr{C}_{odd}$ from the $\operatorname{C}$
matrix obtained by averaging over the angular distribution of the $\tau$
leptons.

The integration over the angular distribution used for the operators
$\mathscr{C}[\rho]$ and $\mathscr{C}_{odd}$ is performed over the range
defined by $|\cos\theta|<0.4$ to optimize the quantum tomography procedure.

###  2.1 Uncertainties

To set limits on the parameters in Eq. (1.1) we need to know the uncertainties
of the operators we utilize to characterize the process
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ at Belle II. These have been estimated for a
benchmark luminosity of 1 ab-1 as follows. For the concurrence
$\mathscr{C}[\rho]$ and the CP odd operator $\mathscr{C}_{odd}$ we rescale the
corresponding uncertainties of $1.4\times 10^{-3}$ and $4.0\times 10^{-3}$
given in Ref. [19] (after averaging on the angular distribution of $\tau$
leptons) for a luminosity of 220 fb-1 and for a center of mass energy
$\sqrt{s}=10.579$ GeV. For the total cross section, instead, we rescale the
relative uncertainty of 0.3% on the integrated luminosity quoted in Ref. [20]
for 833 fb-1. All these uncertainties are at the $1\sigma$ confidence level
and contain (to different extents) also systematic errors.

##  3 Results

Figure 1:  Bounds on the electromagnetic couplings $a_{\tau}$,
$\tilde{d}_{\tau}=F_{3}(0)$ and $c_{1}$ obtained by means of Eq. (3.1) varying
two parameters at a time.

T he operators introduced in the previous section, generically denoted here as
$\mathscr{O}_{i}(a_{\tau},d_{\tau},c_{1})$, depend on the electromagnetic
couplings $a_{\tau}$, $d_{\tau}$ and $c_{1}$, and $\mathscr{O}_{i}(0,0,0)$ are
the values of the operators for the SM. To constrain the couplings, we
introduce a $\chi^{2}$ test set for a (68.3) 95% joint CL

$\sum_{i}\left[\frac{\mathscr{O}_{i}(a_{\tau},d_{\tau},c_{1})-\mathscr{O}_{i}(0,0,0)}{\sigma_{i}}\right]^{2}\leq(2.30)\;5.99\,,$
(3.1)

in which we set the uncertainties $\sigma_{i}$ for the operators
$\mathscr{O}_{i}$ at the values discussed in the previous section.

The operators $\mathscr{O}_{i}$ in Eq. (3.1) that we utilize to constrain
$a_{\tau}$ and $d_{\tau}$ are $\mathscr{C}[\rho]$ and $\mathscr{C}_{odd}$. We
employ instead $\mathscr{C}[\rho]$ and the cross section $\sigma$ to constrain
$a_{\tau}$ and $c_{1}$.

The bounds on each coupling can be extracted from Fig. 1 via marginalization
by assuming the parameters to be independent. The values obtained from the 95%
joint confidence interval are reported in Tab. 1, where they are compared to
the current experiment bounds.

PDG (2022) |  This work
---|---
$-1.9\times 10^{-17}\leq d_{\tau}\leq 6.1\times 10^{-18}$ e cm |  $|d_{\tau}|\leq 1.7\times 10^{-17}$ e cm
$-5.2\times 10^{-2}\leq a_{\tau}\leq 1.3\times 10^{-2}$ |  $|a_{\tau}|\leq 6.3\times 10^{-4}$
$\qquad\quad\Lambda_{\rm C.I.}\geq 7.9\;\text{TeV}$ |  $|c_{1}|\leq 9.5\times 10^{-5}\,,\quad\Lambda_{\rm C.I.}\geq 2.6\;\text{TeV}$
Table 1:  Bounds obtained on the single parameter after marginalization of the
95% joint confidence intervals for the electromagnetic couplings shown in Fig.
1, neglecting correlations. The values refer to a luminosity of 1 ab-1 at
Belle II; limits for higher luminosities can be (approximately) obtained by
rescaling these values by the square root of the ratio of the relative
luminosities. The current experimental limits are reported in the first
column. Figure 2:  Comparison between the power of entanglement (blue line)
and cross section (orange line) to constrain the electromagnetic couplings.
For the plot we use a common relative uncertainty of $0.1\%$. The green and
red areas denote the 95% and 68.3% one-parameter confidence level,
respectively.

The concurrence provides a limit 20% stronger than that given by the total
cross section in determining the anomalous magnetic moment, as shown in Fig.
2. Whether this improvement in the constraint justifies the extra work
required by quantum tomography depends on the details of the experimental
setup. Certainly, the method is not beneficial if the error in the cross
section measurement is much smaller than the corresponding uncertainty in the
determination of the $\tau$ lepton polarizations.

The radius of the $\tau$ lepton that can be explored is estimated from our
limit on $c_{1}$ and Eq. (1.4) as

$\langle\vec{r}^{\;2}\rangle=\frac{6}{m_{\tau}^{2}}\,\left(c_{1}+\frac{a_{\tau}}{4}\right)\,,$
(3.2)

which gives the root mean squared radius value of

$\sqrt{\langle\vec{r}^{\;2}\rangle}<5.1\times 10^{-3}\;\text{fm.}$ (3.3)

By comparison, the electron, which is believed to be a point-like particle,
has a limit on its root mean squared radius of about $10^{-5}$ fm [21] that
does not originate in radiative corrections.

##  4 Outlook

W e have outlined a strategy to constrain the electric and magnetic dipole
moments of the $\tau$ lepton, as well as its electromagnetic radius, via
quantum tomography.

In order to put our results into perspective, we compute the single-parameter
68.3% confidence intervals for $a_{\tau}$ and $d_{\tau}$ from Eq. (3.1) and
compare the resulting limits to those in the Literature. With a luminosity of
1 ab-1, Ref. [14] finds

$|d_{\tau}-d_{\tau}^{SM}|<1.44\times 10^{-18}\;\text{e
cm}\quad\text{and}\quad|a_{\tau}-a_{\tau}^{SM}|<1.24\times 10^{-4}$ (4.1)

at 68.3% CL. Our method instead yields

$|d_{\tau}|<6.7\times 10^{-18}\;\text{e cm}\quad\text{and}\quad
a_{\tau}<2.6\times 10^{-4}\,.$ (4.2)

The SM prediction for $d_{\tau}^{SM}$ is still negligible at this precision
level and so the first limit can be compared directly to ours. Differently,
the SM prediction for $a_{\tau}^{SM}$ is given in Eq. (1.6) and therefore the
limit in [14] is about one order of magnitude weaker than our result.

Ref. [15] quotes the following 68.3% confidence interval for the electric
dipole moment

$|d_{\tau}|<6.8\times 10^{-20}\;\text{e cm}\,,$ (4.3)

given for a luminosity of 50 ab-1. Rescaling the result we obtained to account
for the latter, we find

$|d_{\tau}|<9.4\times 10^{-19}\;\text{e cm}\,.$ (4.4)

It is not surprising that both the limits on the electric dipole moment
reported in Ref [15] and [14] are comparable to ours: these results are all
obtained by using an operator involving the triple product of two spin vectors
and one momentum, equivalent to the CP odd operator utilized in our analysis.
The limit in Eq. (4.3) is stronger because of the smaller uncertainty used,
which is obtained by combining in quadrature the yields of different decay
channels. Our result, in fact, is comparable to those obtained in Ref. [15]
for any single channel.

As for the bound obtained for the compositeness scale, see Tab. 1, one has to
bear in mind that the current experimental limit is obtained considering
energies of about 200 GeV, whereas ours is given for energies one order of
magnitude smaller, of about 10 GeV. Depending on the origin of the relevant
four-fermion interaction, the scaling of the related operator is at least
linear in the energy, implying that our result must be compared with the
approximated rescaled value $\Lambda_{\rm C.I.}\gtrsim 800\;\text{GeV}$ when
assessing the power of the method.

## Acknowledgements

We are happy to thank E. Gabrielli, C. Veelken and L. Zani for discussions.
L.M. is supported by the CoE grant TK202 “Universum”.

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e^{-\frac{\epsilon_{\ch{N2},\text{rot},\text{A}^{\prime},0,12}}{k_{B}T_{\text{ref}}}}}{Q_{\ch{N2},\text{rot},\text{A}^{\prime},0}}\text{
.}$ (71)

From (8) and (71), the respective effective process cross section
$\sigma_{\text{p},\text{av},0}^{10}$ is simply given by

$\sigma_{\text{p},\text{av},0}^{10}=\sigma_{\text{p},\text{av},0,12}^{10,12}\,\frac{g_{\ch{N2},\text{rot},\text{A}^{\prime},0,12}\cdot
e^{-\frac{\epsilon_{\ch{N2},\text{rot},\text{A}^{\prime},0,12}}{k_{B}T_{\text{ref}}}}}{Q_{\ch{N2},\text{rot},\text{A}^{\prime},0}}\text{
.}$ (72)

The quantity $Q_{\ch{N2},\text{rot},\text{A}^{\prime},0}$ corresponds to the
rotational partition function for
$\ch{N2}(\text{A}^{\prime}{}^{5}\Sigma_{\text{g}}^{+},0)$, which according to
its definition, is given by

$Q_{\ch{N2},\text{rot},\text{A}^{\prime},0}=\sum_{J=0}^{J_{\text{max}}}g_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}\cdot
e^{-\frac{\epsilon_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}}{k_{B}T_{\text{ref}}}}\text{
,}$ (73)

with $J_{\text{max}}$ being the maximum rotational quantum number that
$\ch{N2}(\text{A}^{\prime}{}^{5}\Sigma_{\text{g}}^{+})$ may assume. For
$J>J_{\text{max}}$ the nuclei are subjected to a purely repulsive potential,
which makes them depart from each other resulting in the dissociation of the
diatomic particle. Therefore, $J_{\text{max}}$ corresponds to the maximum
rotational quantum number $J$ for which the effective internuclear potential,
the so-called centrifugally corrected potential
$V_{J}(r)=V(r)+\hbar^{2}J(J+1)/(2\mu r^{2})$ remains with a well in its curve.
For higher $J$ values the well disappears, and the curve is transformed into
another with a slope which is non-positive throughout all of its extension.
The internuclear force, which corresponds to the symmetric value of the slope
and has a positive conventional signal in the direction of increasing
internuclear distances, is in turn transformed into a non-negative quantity,
imposing repulsiveness. One gets $J_{\text{max}}=115$, with the respective
centrifugally corrected potential curve $V_{J_{\text{max}}}(r)$ being
represented in Figure 16.

Figure 16: Centrifugally corrected potential curves $V_{J}(r)$ for $J=0,$
$40$, $80$ and $J_{\text{max}}=115$ obtained for
$\ch{N2}(\text{A}^{\prime}{}^{5}\Sigma_{\text{g}}^{+})$.

The sensible rotational energy
$\epsilon_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}$ corresponds to
$\epsilon_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}=B_{\ch{N2},\text{rot},\text{A}^{\prime},0}J(J+1)$,
with $B_{\ch{N2},\text{rot},\text{A}^{\prime},0}$ being the respective
spectroscopic vibronic-specific rotational function, usually labelled by
$B_{v}$. Since N2 is a homonuclear diatomic particle, the rotational degree of
degeneracy $g_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}$ corresponds to a
product between two contributions 120: one due to the nuclear spin,
$g_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}^{n}$, and another due to the
rotational quantum number, $2J+1$, i.e.
$g_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}=g_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}^{n}\cdot\left(2J+1\right)$.
The molecular term symbol ${}^{5}\Sigma_{\text{g}}^{+}$ associated with the
$A^{\prime}$ electronic level states that the rotational levels are symmetric
for even $J$, and antisymmetric for odd $J$ 120. Additionally, it is known
that the nitrogen nuclei follow Bose-Einstein statistics. The contribution of
the nuclear spin to the rotational degree of degeneracy of
$\ch{N2}(\text{A}^{\prime}{}^{5}\Sigma_{\text{g}}^{+},v,J)$ corresponds to 120

$g_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}^{n}=\begin{cases}(2I+1)(I+1),&\text{if
$J$ is even}\text{ ,}\\\ (2I+1)I,&\text{if $J$ is odd}\text{ ,}\end{cases}$
(74)

with $I$ being the so-called nuclei spin quantum number. For the case of the
nitrogen nuclei one has $I=1$ 120, meaning that
$g_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}^{n}=6$ for even $J$ and
$g_{\ch{N2},\text{rot},\text{A}^{\prime},0,J}^{n}=3$ for odd $J$.

The values of vibronic-specific effective process cross sections
$\sigma_{\text{p},\text{av},0}^{10}$ obtained through expression (72) are
presented in Table 2.

Table 2: Vibronic-specific effective process cross sections
$\sigma_{\text{p},\text{av},0}^{10}$ computed through expression (72) using
the rovibronic-specific effective process cross sections
$\sigma_{\text{p},\text{av},0,12}^{10,12}$ issued by Ottinger et al. 116.

M He Ne Ar Kr Xe H2 N2 NO O2
$\sigma_{\text{p},\text{av},0}^{10}[10^{-2}\text{\AA}^{2}]$ $2.953$ $4.430$
$8.122$ $11.075$ $14.767$ $6.645$ $5.907$ $12.552$ $4.430$

Since no data are available for the case $\ch{M}=\ch{N}$, it was decided to
make a study about the dependence of the effective process cross section on
the atomic collision partner - in a similar way to what was done in the two
first studied V-E processes - and from it obtain the respective data. It was
indeed found that the effective process cross section
$\sigma_{\text{p},\text{av},0}^{10}$ increased exponentially with the hard-
sphere diameter $d_{\text{M}}$ of the atomic collision partners, although the
same cannot be said about the molecular collision partners. By fitting an
exponential curve to the data points
$(d_{\text{M}},\sigma_{\text{p},\text{av},0}^{10})$ for the atomic collision
partners, the value of the effective process cross section for $\ch{M}=\ch{N}$
was obtained. The respective results are depicted by Figure 17. The data
values for the effective process cross sections deviate from the fit ones from
$-8.6\%$ to $10.6\%$. A value of
$\sigma_{\text{p},\text{av},0}^{10}=6.863\times 10^{-2}\text{\AA}^{2}$ was
obtained for the case $\ch{M}=\ch{N}$.

Figure 17: Upper plot: exponential curve fitted to the data
$(d_{\text{M}},\sigma_{\text{p},\text{av},0}^{10})$ for process (66) involving
solely the atomic collision partners. Lower plot: relative deviations of the
data,
$\delta=\left(\sigma_{\text{p},\text{av},0}^{10}-\sigma_{\text{p},\text{av},0}^{10,\text{fit}}\right)/\sigma_{\text{p},\text{av},0}^{10,\text{fit}}$.
Atomic collision partners: ; molecular collision partners: ; nitrogen atom N:
.

The dependence of the effective process cross section
$\sigma_{\text{p},\text{av},0}^{10}$ on the temperature was assumed to be the
one referred by Parmenter et al. 53, 54, and therefore, the respective forward
rate coefficient may be mathematically expressed by

$k_{0}^{10}(T_{\text{tr}_{\text{h}}})={\sigma^{\prime}}_{p,\text{av},0}^{\,10,\text{ref}}\,\sqrt{\frac{8k_{B}T_{\text{tr}_{\text{h}}}}{\pi\mu}}\,e^{\frac{\varepsilon}{k_{B}T_{\text{tr}_{\text{h}}}}}\text{
,}$ (75)

with
${\sigma^{\prime}}_{p,\text{av},0}^{\,10,\text{ref}}=\sigma_{p,\text{eff},0}^{10,\text{ref}}\,e^{-\frac{\varepsilon}{k_{B}T_{\text{ref}}}}$,
where $\sigma_{p,\text{eff},0}^{10,\text{ref}}$ is the effective process cross
section evaluated at the reference temperature $T_{\text{ref}}$ of Ottinger et
al. 116 \- the object of study considered above, and ambiguously labelled as
$\sigma_{\text{p},\text{av},0}^{10}$. Note that relation (75) may be
conveniently expressed through a modified Arrhenius function.

##  5 Conclusions

In this work a extensive set of vibronic energy levels for N2, N2+, N and N+
was built using the most up-to-date (to the limits of knowledge of the present
authors) data available in the literature. A near complete database of
vibronic-specific kinetic processes involving these species was congregated in
which special caution was taken to ensure physical consistency up to the
highest temperature values. The database comprises chemical processes such as
dissociation, ionisation, dissociative recombination and charge exchange, as
well as non-chemical processes, i.e. excitation and de-excitation of the
vibronic energy levels of the particles either due to collisions or
spontaneous emission of radiation. Note, however, that the term “near
complete” was herein used to describe the current state of the database, since
it misses collision-induced transitions between the high electronic energy
levels of the molecular particles, the bound-bound radiative processes of
absorption and induced emission, spontaneous emission processes from some very
high vibronic levels of the molecular particles (due to a lack of numerically
and experimentally obtained Einstein coefficients), bound-free radiative
processes (photodissociation and photoionisation), and free-free radiative
processes (bremsstrahlung). Efforts in modelling these should be taken in the
future.

This work focused almost entirely on two types of processes: the vibrational
and the vibronic transitions of the molecular particles by collisions with
heavy particles. Regarding vibrational transition processes, the Forced
Harmonic Oscillator model was used in place of the most commonly considered
one, the Schwartz-Slawsky-Herzfeld model, to compute the respective rate
coefficients since in contrast with the latter, it is physically consistent at
the high temperature values attained in atmospheric entries. The obtained
vibrational transition rate coefficients were, in some part, indirectly
validated, as thermal dissociation rate coefficients of
$\ch{N2}(\text{X}{}^{1}\Sigma_{\text{g}}^{+})$ by collisions with N2 and N
which depend on these transition rates were calibrated using state-of-the-art
experimental results. An agreement between $-59.9$ and $8.9\,\%$, and between
$-80.9$ and $-36.1\,\%$ were obtained for the former and latter interactions,
respectively. Furthermore, the values were compared with the most recent
Quasi-Classical Trajectory model calculations deviating by a maximum of
$56.5\,\%$. Note that although these deviations seem to be significant, they
should be regarded as reasonable, since the deviations are evaluated within an
extensive set of temperatures (varying in many thousands of kelvins) for which
the rate coefficients suffer changes of several orders of magnitude.

The authors tried to increase the degree of fidelity of the current models for
vibronic transitions by collisions with heavy particles by employing the so-
called exponential gap law for the effective process cross section. This takes
into account two parameters that are specific to the set of electronic levels
- the characteristic cross section $\sigma_{0}$ and characteristic energy
$E_{0}$ \- and a dependence on the energy defect $\Delta E$. However, the
computed effective process cross sections differed from the experimental ones
at room temperature by as much as one order of magnitude, which evidences that
other models with a even greater complexity should be tried. Furthermore,
since the available experimental data that was used for calibration was only
with respect to the low electronic levels of N2, it was not possible to model
transitions between the highest ones, as well as transitions between the
vibronic levels of N2+.

The herein developed database of kinetic processes was employed in Euler
unidimensional simulations of several shots of the $62^{\text{nd}}$ campaign
of the Electric Arc Shock Tube. The respective results are reported in a
companion paper  CITE COMPANION PAPER.

We would like to express gratitude to Vasco Guerra from the N-PRiME group for
his literature recommendations which were important for the conception of the
3 section. We also want to thank Dr. Annarita Laricchiuta from CNR Bari for
her clarifications and providing us her values for the branching ratios
associated with the process of non-dissociative ionisation of nitrogen
molecules by electron impact, as well as the used database for the vibrational
energies of the nitrogen molecule in its ground electronic level. We thank Dr.
Steven Guberman from the Institute for Scientific Research for his noteworthy
discussion about the valid temperature range for which his numerical data may
be used in the modelling of rates of dissociative recombination of molecular
nitrogen ions. We also acknowledge the clarifications provided by Dr. Julien
Annaloro from the National Centre for Space Studies in France about his
lumping procedure with respect to the electronic levels and Einstein
coefficients for spontaneous emission issued by NIST.

This work has been partially supported by the Portuguese Science Foundation
FCT, under Projects UIDB/50010/2020 and UIDP/50010/2020.

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## TOC Graphic

##  Appendix A Appendix A: Potential curves of N2 and N2+ in their different
electronic levels

Internuclear potential curves $V(r)$ for the electronic levels
X${}^{1}\Sigma_{\text{g}}^{+}$, A${}^{3}\Sigma_{\text{u}}^{+}$,
B${}^{3}\Pi_{\text{g}}$, W${}^{3}\Delta_{\text{u}}$,
B${}^{\prime}{}^{3}\Sigma_{\text{u}}^{-}$,
a${}^{\prime}{}^{1}\Sigma_{\text{u}}^{-}$, a${}^{1}\Pi_{\text{g}}$,
w${}^{1}\Delta_{\text{u}}$, A${}^{\prime}{}^{5}\Sigma_{\text{g}}^{+}$,
C${}^{3}\Pi_{\text{u}}$, b${}^{1}\Pi_{\text{u}}$,
c${}_{3}{}^{1}\Pi_{\text{u}}$, c${}_{4}^{\prime}{}^{1}\Sigma_{\text{u}}^{+}$,
b${}^{\prime}{}^{1}\Sigma_{\text{u}}^{+}$ and o${}_{3}{}^{1}\Pi_{\text{u}}$ of
the nitrogen molecule N2 were obtained. All curves were constructed by
application of the Rydberg 69, 70-Klein 71-Rees 72 method and posterior
extrapolation, as described in the 3 section. Values of the parameters
$Y_{ij}$ of the Dunham expansion that describes the rovibrational energy of
the molecular particles,

$T_{vJ}=\sum_{i,j=0}^{\infty}Y_{ij}\left(v+\frac{1}{2}\right)^{i}\left[J\left(J+1\right)\right]^{j}\text{
,}$ (76)

and are input variables of the RKR method, were required. Table 3 shows such
values, being taken from the literature. The electronically corrected (i.e.
with the sensible electronic energy summed) internuclear potential curves
$V_{e}(r)=V(r)+T_{e}$ obtained for each of the electronic levels of molecular
nitrogen N2 are depicted in Figure 18. All of the dissociation products shown
in Figure 18 were taken from the work of Lofthus and Krupenie 121 with the
exception of the one associated with the electronic level
A${}^{\prime}{}^{5}\Sigma_{\text{g}}^{+}$, which was taken from the work of
Partridge et al. 122.

All the data required to obtain the internuclear potential curves $V(r)$ for
the different electronic levels of the nitrogen molecular ion N2+ are found in
Table 4. The considered electronic levels for this species were
X${}^{2}\Sigma_{\text{g}}^{+}$, A${}^{2}\Pi_{\text{u}}$,
B${}^{2}\Sigma_{\text{u}}^{+}$, D${}^{2}\Pi_{\text{g}}$ and
C${}^{2}\Sigma_{\text{u}}^{+}$. The resultant electronically corrected
internuclear potential curves $V_{e}(r)$ are depicted in Figure 19. All of the
dissociation products shown in Figure 19 were taken from the work of Lofthus
and Krupenie 121.

Figure 18: Electronically corrected internuclear potential curves
$V_{e}(r)=V(r)+T_{e}$ for the different electronic levels of the nitrogen
molecule N_2. The terms N$\left({}^{4}\text{S}_{\text{u}}\right)$ \+
N$\left({}^{4}\text{S}_{\text{u}}\right)$,
N$\left({}^{4}\text{S}_{\text{u}}\right)$ \+
N$\left({}^{2}\text{D}_{\text{u}}\right)$,
N$\left({}^{4}\text{S}_{\text{u}}\right)$ \+
N$\left({}^{2}\text{P}_{\text{u}}\right)$,
N$\left({}^{2}\text{D}_{\text{u}}\right)$ \+
N$\left({}^{2}\text{D}_{\text{u}}\right)$ and
N$\left({}^{2}\text{D}_{\text{u}}\right)$ \+
N$\left({}^{2}\text{P}_{\text{u}}\right)$ represent the dissociation products
of the nitrogen molecule in the electronic levels associated with the
immediately below potential curves.

Table 3:  Spectroscopic constants $T_{e}$, $D_{e}$ and $r_{e}$, maximum
vibrational quantum number $v_{\text{max}}$ for which the respective Dunham
expansion is valid, Dunham parameters $Y_{i0}$ with $i=0,\,1,\,...,\,6$ and
$Y_{i1}$ with $i=0,\,1,\,...,\,4$ and the shape of the extrapolated long-range
part of the potential $V_{\text{lr}}$ of the different electronic levels of
molecular nitrogen N2.

$e$ |  X${}^{1}\Sigma_{\text{g}}^{+}$ |  A${}^{3}\Sigma_{\text{u}}^{+}$ |  B${}^{3}\Pi_{\text{g}}$ |  W${}^{3}\Delta_{\text{u}}$ |  B${}^{\prime}{}^{3}\Sigma_{\text{u}}^{-}$ |  a${}^{\prime}{}^{1}\Sigma_{\text{u}}^{-}$ |  a${}^{1}\Pi_{\text{g}}$ |  w${}^{1}\Delta_{\text{u}}$ |  A${}^{\prime}{}^{5}\Sigma_{\text{g}}^{+}$ |  C${}^{3}\Pi_{\text{u}}$ |  b${}^{1}\Pi_{\text{u}}$ |  c${}_{3}{}^{1}\Pi_{\text{u}}$ |  c${}_{4}^{\prime}{}^{1}\Sigma_{\text{u}}^{+}$ |  b${}^{\prime}{}^{1}\Sigma_{\text{u}}^{+}$ |  o${}_{3}{}^{1}\Pi_{\text{u}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Ref. for $T_{e}$ |  123 (a ) |  123 (a ) |  123 (a ) |  124 (a ) |  124 (a ) |  124 (a ) |  124 (a ) |  124 (a ) | 125 |  123 (a ) |  123 (a ) |  123 (a ) |  123 (a ) |  123 (a ) |  123 (a )
$T_{e}$ [cm-1] | $0.00$ | $50203.66$ | $59618.77$ | $59805.85$ | $66271.53$ | $68152.79$ | $69283.34$ | $72097.31$ | $75990.03$ | $89137.92$ | $101660.02$ | $104217.77$ | $104418.37$ | $105215.38$ | $105873.81$
Ref. for $D_{e}$ |  123 (b ) |  123 (b ) |  123 (b ) |  121 (b ) |  121 (b ) |  121 (b ) |  121 (b ) |  121 (b ) | 122 |  123 (b ) |  123 (b ) |  123 (b ) |  123 (b ) |  123 (b ) |  123 (b )
$D_{e}$ [cm-1] | $79886.67$ | $29686.89$ | $39499.33$ | $39308.13$ | $42456.54$ | $50186.28$ | $49055.63$ | $46241.86$ | $3450.00$ | $9980.02$ | $16677.65$ | $47196.90$ | $87946.60$ | $24185.10$ | $88858.57$
Ref. for $r_{e}$ | 126 | 126 | 126 | 127 | 121 | 121 | 121 | 121 | 76 | 126 | 126 | 126 | 126 | 126 | 126
$r_{e}$ [Å] | $1.09768$ | $1.28660$ | $1.21260$ | $1.28$ | $1.27838$ | $1.27542$ | $1.22025$ | $1.26883$ | $1.60840$ | $1.14860$ | $1.28400$ | $1.11630$ | $1.10800$ | $1.44390$ | $1.17840$
Ref. for $v_{\text{max}}$ | 123 | 123 | 123 | 124 | 124 | 124 | 124 | 124 |  This work (c ) | 123 | 123 | 123 | 123 | 123 | 123
$v_{\text{max}}$ | $15$ | $16$ | $21$ | $11$ | $17$ | $18$ | $14$ | $5$ | $5$ | $4$ | $19$ | $4$ | $8$ | $28$ | $4$
Ref. for $Y_{\text{i0}}$ | 123 | 123 | 123 | 124 | 124 | 124 | 124 | 124 |  (c ) | 123 | 123 | 123 | 123 | 123 | 123
$Y_{00}$ [cm-1] | $7.30(-2)$ | $-1.89(-1)$ | $-8.30(-2)$ | — | $1.38(-1)$ | $1.67(-1)$ | $1.20(-2)$ | $1.05(-1)$ | $-2.50(-2)$ | $-1.79(0)$ | $6.53(0)$ | $3.90(0)$ | $7.27(-1)$ | $2.89(-1)$ | $4.24(0)$
$Y_{10}$ [cm-1] | $2.36(3)$ | $1.46(3)$ | $1.73(3)$ | $1.51(3)$ | $1.52(3)$ | $1.53(3)$ | $1.69(3)$ | $1.56(3)$ | $7.31(2)$ | $2.05(3)$ | $6.42(2)$ | $2.20(3)$ | $2.17(3)$ | $7.59(2)$ | $1.97(3)$
$Y_{20}$ [cm-1] | $-1.43(1)$ | $-1.40(1)$ | $-1.44(1)$ | $-1.26(1)$ | $-1.22(1)$ | $-1.21(1)$ | $-1.39(1)$ | $-1.20(1)$ | $-1.09(1)$ | $-2.89(1)$ | $2.17(1)$ | $-2.56(1)$ | $-1.33(1)$ | $-3.40(0)$ | $-1.09(1)$
$Y_{30}$ [cm-1] | $-3.31(-3)$ | $2.40(-2)$ | $-3.30(-3)$ | $3.09(-2)$ | $4.19(-2)$ | $4.13(-2)$ | $7.94(-3)$ | $4.54(-2)$ | $-2.25(0)$ | $2.25(0)$ | $-1.41(0)$ | — | $-2.94(-1)$ | $1.78(-2)$ | —
$Y_{40}$ [cm-1] | $-1.95(-4)$ | $2.56(-3)$ | $-7.90(-4)$ | $-7.10(-4)$ | $-7.30(-4)$ | $-2.9(-4)$ | $2.9(-4)$ | — | — | $-5.51(-1)$ | $2.29(-2)$ | — | — | $-1.78(-3)$ | —
$Y_{50}$ [cm-1] | — | — | $4.20(-5)$ | — | — | — | — | — | — | — | — | — | — | — | —
$Y_{60}$ [cm-1] | — | — | $1.68(-6)$ | — | — | — | — | — | — | — | — | — | — | — | —
Ref. for $Y_{\text{i1}}$ | 123 | 123 | 123 | 124 | 124 | 124 | 124 | 124 | 76 | 123 | 123 | 123 | 123 | 123 | 123
$Y_{01}$ [cm-1] | $2.00(0)$ | $1.45(0)$ | $1.64(0)$ | $1.47(0)$ | $1.47(0)$ | $1.48(0)$ | $1.62(0)$ | $1.50(0)$ | $9.31(-1)$ | $1.83(0)$ | $1.39(0)$ | $1.98(0)$ | $1.93(0)$ | $1.16(0)$ | $1.73(0)$
$Y_{11}$ [cm-1] | $-1.73(-2)$ | $-1.75(-2)$ | $-1.79(-2)$ | $-1.70(-2)$ | $-1.67(-2)$ | $-1.66(-2)$ | $-1.79(-2)$ | $-1.63(-2)$ | $-1.71(-2)$ | $-2.40(-2)$ | $-1.42(-2)$ | $-3.80(-2)$ | $-1.96(-2)$ | $-1.04(-2)$ | $-2.75(-2)$
$Y_{21}$ [cm-1] | $-3.01(-5)$ | $-1.40(-4)$ | $-1.00(-4)$ | $-1.01(-5)$ | $1.84(-5)$ | $2.41(-5)$ | $-2.93(-5)$ | — | — | $1.90(-3)$ | $-5.21(-4)$ | — | — | $3.90(-4)$ | —
$Y_{31}$ [cm-1] | $-6.93(-8)$ | — | $5.00(-6)$ | $3.30(-7)$ | $-4.50(-7)$ | — | — | — | — | $-6.00(-4)$ | — | — | — | $-1.73(-5)$ | —
$Y_{41}$ [cm-1] | — | — | $2.10(-7)$ | — | — | — | — | — | — | — | — | — | — | — | —
$V_{\text{lr}}$ | $V_{\text{HH}}$ | $V_{\text{ER}}$ | $V_{\text{ER}}$ | $V_{\text{ER}}$ | $V_{\text{ER}}$ | $V_{\text{ER}}$ | $V_{\text{ER}}$ | $V_{\text{ER}}$ | $V_{\text{HH}}$ | $V_{\text{HH}}$ | $V_{\text{ER}}$ |  (d ) |  (d ) | $V_{\text{ER}}$ |  (d )

  * •

The number between parenthesis in the $Y_{i0}$ and $Y_{i1}$ cells correspond
to the orders of magnitude of these quantities.

  * a

This reference only reports the value for $\Delta
T_{e0}:=T_{e0}-T_{\text{X}0}$, in which $T_{e0}$ is the quantity
$T_{ev}=T_{e}+G_{v}$ with $v=0$ for the electronic level $e$, and
$T_{\text{X}0}$ is the homologous quantity for the ground electronic level X.
Therefore, and sticking with the introduced notation, one can obtain $T_{e}$
by using the relation $T_{e}=\Delta T_{e0}+G_{0}^{X}-G_{0}^{e}$.

  * b

This reference only reports the value for the dissociation energy
$D_{0}:=D_{e}-G_{0}$, and therefore the potential well depth needs to be
computed using the relation $D_{e}=D_{0}+G_{0}$.

  * c

The values for $Y_{i0}$, with $i=0,\,1,\,...,\,3$ for the case
$e=\text{A}^{\prime}{}^{5}\Sigma_{\text{g}}^{+}$ were obtained by fitting the
vibrational energies values $G_{v}$, with $v=0,\,1,\,...,\,5$, taken from the
literature, considering as fitting function a Dunham expansion of third order,
i.e. $G_{v}=\sum_{i=0}^{3}Y_{i0}\left(v+\frac{1}{2}\right)^{i}$, with $Y_{00}$
approximated by 128
$\frac{B_{e}}{4}+\frac{\alpha_{e}\omega_{e}}{12B_{e}}+\frac{\left(\alpha_{e}\omega_{e}\right)^{2}}{144B_{e}^{3}}-\frac{\omega_{e}x_{e}}{4}$.
Note that in this relation, $B_{e}=Y_{01}$, $\alpha_{e}=-Y_{11}$,
$\omega_{e}=Y_{10}$, and $\omega_{e}x_{e}=-Y_{20}$. The $G_{0}$ value was
taken from the work of Huber and Vervloet 129, and the other $G_{v}$ values
were taken from the work of Hochlaf et al. 76 (note that in this article the
values appear subtracted by $G_{0}$).

  * d

Since it is known that the potential curve for this electronic level does not
have a conventional shape like the $V_{\text{HH}}(r)$ or $V_{\text{ER}}(r)$
curves, and its dissociation products were not already determined, no attempt
to extrapolate the RKR potential was performed. Therefore, it was assumed that
the dissociation of the molecule in this electronic level does not occur, and
only the vibrational levels restrained to RKR determined part may be accessed.

Figure 19: Electronically corrected internuclear potential curves
$V_{e}(r)=V(r)+T_{e}$ for the different electronic levels of the nitrogen
molecular ion N_2+. The terms N$\left({}^{4}\text{S}_{\text{u}}\right)$ \+
N+$\left({}^{3}\text{P}\right)$ and N$\left({}^{2}\text{D}_{\text{u}}\right)$
\+ N+$\left({}^{3}\text{P}\right)$ represent the dissociation products of the
nitrogen molecular ion in the electronic levels associated with the
immediately below potential curves.

Table 4:  Spectroscopic constants $T_{e}$, $D_{e}$ and $r_{e}$, maximum
vibrational quantum number $v_{\text{max}}$ for which the respective Dunham
expansion is valid, Dunham parameters $Y_{i0}$ with $i=0,\,1,\,...,\,7$ and
$Y_{i1}$ with $i=0,\,1,\,...,\,3$ and the shape of the extrapolated long-range
part of the potential $V_{\text{lr}}$ of the different electronic levels of
nitrogen molecular ion N2+.

$e$ |  X${}^{2}\Sigma_{\text{g}}^{+}$ |  A${}^{2}\Pi_{\text{u}}$ |  B${}^{2}\Sigma_{\text{u}}^{+}$ |  D${}^{2}\Pi_{\text{g}}$ |  C${}^{2}\Sigma_{\text{u}}^{+}$
---|---|---|---|---|---
Ref. for $T_{e}$ |  123 (a ) |  123 (a ) |  123 (a ) |  121 (b ) |  123 (a )
$T_{e}$ [cm-1] |  $0.00$ |  $9167.34$ |  $25462.43$ |  $52814.06$ |  $64610.79$
Ref. for $D_{e}$ |  123 (c ) |  123 (c ) |  123 (c ) |  121 (c ) |  123 (c )
$D_{e}$ [cm-1] |  $71368.43$ |  $62201.49$ |  $45903.08$ |  $18557.87$ |  $25988.34$
Ref. for $r_{e}$ |  126 |  126 |  126 |  121 |  126
$r_{e}$ [Å] |  $1.11642$ |  $1.17490$ |  $1.07400$ |  $1.47082$ |  $1.26200$
Ref. for $v_{\text{max}}$ |  123 |  123 |  123 |  — |  123
$v_{\text{max}}$ |  $21$ |  $27$ |  $8$ |  $5$ |  $6$
Ref. for $Y_{\text{i0}}$ |  123 |  123 |  123 |  121 |  123
$Y_{00}$ [cm-1] |  $-1.20(-1)$ |  $1.70(-2)$ |  $-1.43(0)$ |  $1.67(-1)$ |  $-1.80(0)$
$Y_{10}$ [cm-1] |  $2.21(3)$ |  $1.90(3)$ |  $2.42(3)$ |  $9.12(2)$ |  $2.07(3)$
$Y_{20}$ [cm-1] |  $-1.62(1)$ |  $-1.51(1)$ |  $-2.41(1)$ |  $-1.26(1)$ |  $-8.30(0)$
$Y_{30}$ [cm-1] |  $4.00(-3)$ |  $1.12(-2)$ |  $-3.00(-1)$ |  $5.55(-2)$ |  $-6.30(-1)$
$Y_{40}$ [cm-1] |  $-6.10(-3)$ |  $-2.70(-4)$ |  $-6.67(-2)$ |  — |  $1.30(-2)$
$Y_{50}$ [cm-1] |  $3.90(-4)$ |  — |  — |  — |  —
$Y_{60}$ [cm-1] |  $-1.40(-5)$ |  — |  — |  — |  —
$Y_{70}$ [cm-1] |  $2.00(-7)$ |  — |  — |  — |  —
Ref. for $Y_{\text{i1}}$ |  123 |  123 |  123 |  121 |  123
$Y_{01}$ [cm-1] |  $1.93(0)$ |  $1.74(0)$ |  $2.09(0)$ |  $1.11(0)$ |  $1.51(0)$
$Y_{11}$ [cm-1] |  $-1.88(-2)$ |  $-1.87(-2)$ |  $-2.12(-2)$ |  $-2.00(-2)$ |  $1.00(-3)$
$Y_{21}$ [cm-1] |  $-6.77(-5)$ |  $-6.00(-5)$ |  $-5.00(-4)$ |  — |  $-1.50(-3)$
$Y_{31}$ [cm-1] |  $-2.32(-6)$ |  $-1.10(-6)$ |  $-8.80(-5)$ |  — |  $6.00(-5)$
$V_{\text{lr}}$ |  $V_{\text{ER}}$ |  $V_{\text{ER}}$ |  $V_{\text{ER}}$ |  $V_{\text{ER}}$ |  $V_{\text{HH}}$

  * •

The number between parenthesis in the $Y_{i0}$ and $Y_{i1}$ cells correspond
to the orders of magnitude of these quantities.

  * a

This reference only reports the value for $\Delta
T_{e0}:=T_{e0}-T_{\text{X}0}$, in which $T_{e0}$ is the quantity
$T_{ev}=T_{e}+G_{v}$ with $v=0$ for the electronic level $e$, and
$T_{\text{X}0}$ is the homologous quantity for the ground electronic level X.
Therefore, and sticking with the introduced notation, one can obtain $T_{e}$
by using the relation $T_{e}=\Delta T_{e0}+G_{0}^{X}-G_{0}^{e}$.

  * b

This reference only reports the value for
$\Delta_{\text{n}}T_{e0}:=T_{e0}-T_{\text{n},\text{X}0}$, in which $T_{e0}$ is
the quantity $T_{ev}=T_{e}+G_{v}$ with $v=0$ for the electronic level $e$, and
$T_{\text{n},\text{X}0}$ is the homologous quantity for the ground electronic
level X of the counterpart neutral species. Therefore, and sticking with the
introduced notation, one can obtain $T_{e}$ by using the relation
$T_{e}=\Delta_{\text{n}}T_{e0}-\Delta_{\text{n}}T_{\text{X}0}+G_{0}^{X}-G_{0}^{e}$.

  * c

This reference only reports the value for the dissociation energy
$D_{0}:=D_{e}-G_{0}$, and therefore the potential well depth needs to be
computed using the relation $D_{e}=D_{0}+G_{0}$.

##  Appendix B Appendix B: The kinetic processes database

Appendices B, B, B and B present the database of kinetic processes developed
in this work. In the case of Appendices B and B which are with respect to the
regarded collisional processes, a codename for the type of process was defined
to simplify the description of the process. It consists of three terms
separated by hyphens. If the process solely corresponds to a transition in the
internal energy modes of the collision partners, the first term of the
codename is defined by a set of capital letters each one representing the
involved internal energy modes (“V” from “vibrational” or/and “E” from
“electronic”). If the process corresponds to a bond breaking or/and forming of
the internal structure, the first term is defined by a set of capital letters,
each one representing a type of bond breaking or forming (“I” from
“ionisation”, “R” from “recombination”, “D” from “dissociation” or “A” from
“association”). The second and third terms correspond to labels of minuscule
letters representing the type of the collision partners (“h” from “heavy
species”, “m” from “molecular particle”, “a” from “atomic particle”, or “e”
from “electron”).

Collisional processes due to heavy particle impact for which forward rate
constants were obtained. The symbol after the reference in the column
“Reference” represents the physical quantity which was extracted from it:
process cross section (if $\sigma_{p}$), average process cross section (if
$\sigma_{p,\text{av}}$) or forward rate constant (if $k_{f}$). Type Chemical
equation Remarks Reference V-m-h
$\ch{N_{2}}\left(e,v\right)\ch{+M}\ch{<=>N2}\left(e,v^{\prime}\right)\ch{+M}$
$e\in\\{\text{X},\text{A},\text{B},\text{W},\text{B}^{\prime},\text{a}^{\prime},\text{a},\text{w},\text{A}^{\prime},\text{C},\text{b},\text{c}_{3},\text{c}^{\prime}_{4},\text{b}^{\prime},\text{o}_{3}\\}$,
$\forall\,v$, $\forall\,v^{\prime}>v$ and
$\ch{M}\in\\{\ch{N2},\ch{N2+},\ch{N},\ch{N+}\\}$ This work (a ) V-m-h
$\ch{N_{2}+}\left(e,v\right)\ch{+M}\ch{<=>N2+}\left(e,v^{\prime}\right)\ch{+M}$
$e\in\\{\text{X},\text{A},\text{B},\text{D},\text{C}\\}$, $\forall\,v$,
$\forall\,v^{\prime}>v$ and $\ch{M}\in\\{\ch{N2},\ch{N2+},\ch{N},\ch{N+}\\}$
This work (a ) VE-m-a
$\ch{N2}\left(\text{A},v\right)\ch{+N}({}^{4}\text{S}_{\text{u}})\ch{<=>N2}\left(\text{X},v^{\prime}\right)\ch{+N}({}^{2}\text{P}_{\text{u}})$
$\forall\,v,$ and $\forall\,v^{\prime}$ 111-$k_{f}$ VE-m-a
$\ch{N2}\left(\text{A},v\right)\ch{+N}({}^{4}\text{S}_{\text{u}})\ch{<=>N2}\left(\text{B},v^{\prime}\right)\ch{+N}({}^{4}\text{S}_{\text{u}})$
$\forall\,v,$ and $\forall\,v^{\prime}$ 47-$\sigma_{p,\text{av}}$ VE-m-a
$\ch{N2}\left(\text{W},v\right)\ch{+N}({}^{4}\text{S}_{\text{u}})\ch{<=>N2}\left(\text{B},v^{\prime}\right)\ch{+N}({}^{4}\text{S}_{\text{u}})$
$\forall\,v,$ and $\forall\,v^{\prime}$ 47-$\sigma_{p,\text{av}}$ VE-m-a
$\ch{N2}\left(\text{A}^{\prime},0\right)\ch{+N}({}^{4}\text{S}_{\text{u}})\ch{<=>N2}\left(\text{B},10\right)\ch{+N}({}^{4}\text{S}_{\text{u}})$
— 116-$\sigma_{p,\text{av}}$ VE-m-m
$\ch{N2}\left(\text{A},v_{1}\right)\ch{+N2}\left(\text{X},v_{2}\right)\ch{<=>N2}\left(\text{X},v^{\prime}_{1}\right)\ch{+N2}\left(\text{X},v^{\prime}_{2}\right)$
$\forall\,v_{1}$, $\forall\,v_{2}$, $\forall\,v^{\prime}_{1}$ and
$\forall\,\,v_{2}^{\prime}$ 108-$k_{f}$ VE-m-m
$\ch{N2}\left(\text{A},v_{1}\right)\ch{+N2}\left(\text{X},v_{2}\right)\ch{<=>N2}\left(\text{B},v^{\prime}_{1}\right)\ch{+N2}\left(\text{X},v^{\prime}_{2}\right)$
$\forall\,v_{1}$, $\forall\,v_{2}$, $\forall\,v^{\prime}_{1}$ and
$\forall\,\,v_{2}^{\prime}$ 47-$\sigma_{p,\text{av}}$ VE-m-m
$\ch{N2}\left(\text{A},v_{1}\right)\ch{+N2}\left(\text{A},v_{2}\right)\ch{<=>N2}\left(\text{B},v^{\prime}_{1}\right)\ch{+N2}\left(\text{X},v^{\prime}_{2}\right)$
$\forall\,v_{1}$, $\forall\,v_{2}$, $\forall\,v^{\prime}_{1}$ and
$\forall\,\,v_{2}^{\prime}$ 104-$k_{f}$ VE-m-m
$\ch{N2}\left(\text{A},v_{1}\right)\ch{+N2}\left(\text{A},v_{2}\right)\ch{<=>N2}\left(\text{C},v^{\prime}_{1}\right)\ch{+N2}\left(\text{X},v^{\prime}_{2}\right)$
$\forall\,v_{1}$, $\forall\,v_{2}$, $\forall\,v^{\prime}_{1}$ and
$\forall\,\,v_{2}^{\prime}$ 105-$k_{f}$ VE-m-m
$\ch{N2}\left(\text{W},v_{1}\right)\ch{+N2}\left(\text{X},v_{2}\right)\ch{<=>N2}\left(\text{B},v^{\prime}_{1}\right)\ch{+N2}\left(\text{X},v^{\prime}_{2}\right)$
$\forall\,v_{1}$, $\forall\,v_{2}$, $\forall\,v^{\prime}_{1}$ and
$\forall\,\,v_{2}^{\prime}$ 47-$\sigma_{p,\text{av}}$ VE-m-m
$\ch{N2}\left(\text{A}^{\prime},0\right)\ch{+N2}\left(\text{X},0\right)\ch{<=>N2}\left(\text{B},10\right)\ch{+N2}\left(\text{X},0\right)$
— 116-$\sigma_{p,\text{av}}$ E-a-h
$\ch{N}\left(e\right)\ch{+M<=>N}\left(e^{\prime}\right)\ch{+M}$ $\forall e$,
$\forall e^{\prime}>e$ and $\text{M}\in\\{\ch{N},\ch{N2}\\}$ 31-$k_{f}$ E-a-h
$\ch{N+}\left(e\right)\ch{+M<=>N+}\left(e^{\prime}\right)\ch{+M}$ $\forall e$,
$\forall e^{\prime}>e$ and $\text{M}\in\\{\ch{N},\ch{N2}\\}$ 31-$k_{f}$ D-m-h
$\ch{N2}\left(e,v\right)\ch{+M}\ch{<=>N}\left(e^{\prime}_{1}\right)\ch{+N}\left(e^{\prime}_{2}\right)\ch{+M}$
$e\in\\{\text{X},\text{A},\text{B},\text{W},\text{B}^{\prime},\text{a}^{\prime},\text{a},\text{w},\text{A}^{\prime},\text{C},\text{b},\text{b}^{\prime}\\}$,
$\forall\,v$ and $\ch{M}\in\\{\ch{N2},\ch{N2+},\ch{N},\ch{N+}\\}$ This work (a
) D-m-h
$\ch{N2+}\left(e,v\right)\ch{+M}\ch{<=>N}\left(e^{\prime}_{1}\right)\ch{+N+}\left(e^{\prime}_{2}\right)\ch{+M}$
$e\in\\{\text{X},\text{A},\text{B},\text{D},\text{C}\\}$, $\forall\,v$ and
$\ch{M}\in\\{\ch{N2},\ch{N2+},\ch{N},\ch{N+}\\}$ This work (a ) I-a-h
$\ch{N}\left(e\right)\ch{+M<=>N+}\left({}^{3}\text{P}\right)\ch{+M+e-}$
$\forall e$ and $\text{M}\in\\{\ch{N},\ch{N2}\\}$ 31-$k_{f}$ IR-m-a
$\ch{N2}\left(\text{X},v\right)\ch{+N+}({}^{3}\text{P})\ch{<=>N2+}\left(\text{X},v^{\prime}\right)\ch{+N}({}^{4}\text{S}_{\text{u}})$
VRP on $v$ and $v^{\prime}$ from case $v=0$ and $\sum_{v^{\prime}}$
130-$\sigma_{p}$ a Note that although the respective chemical equation does
not show any possible transition in the vibrational level (or even
dissociation) of the second collision partner (if it is a molecular particle),
such possibility is implicit.

Collisional processes due to electron impact for which forward rate constants
were obtained. The symbol after the reference in the column “Reference”
represents the physical quantity which was extracted from it: process cross
section (if $\sigma_{p}$), average process cross section (if
$\sigma_{p,\text{av}}$) or forward rate constant (if $k_{f}$). Type Chemical
equation Remarks Reference V-m-e
$\ch{N_{2}}\left(\text{X}{}^{1}\Sigma_{\text{g}}^{+},\,v\right)\ch{+e-<=>N_{2}}\left(\text{X}{}^{1}\Sigma_{\text{g}}^{+},\,v^{\prime}\right)\ch{+e-}$
$\forall v$ and $\forall\,v^{\prime}>v$, ADV 61-$\sigma_{p}$ (from 131) VE-m-e
$\ch{N2}\left(\text{X},v\right)\ch{+e-}\ch{<=>N2}\left(e^{\prime},v^{\prime}\right)\ch{+e-}$
$e^{\prime}\in\\{\text{A},\text{B},\text{W},\text{B}^{\prime},\text{a}^{\prime},\text{a},\text{w},\text{C}\\}$,
VRP on $v$ and $v^{\prime}$ from case $v=0$ and $\sum_{v^{\prime}}$
132-$\sigma_{p}$ $e^{\prime}\in\\{\text{c}_{3},\text{o}_{3}\\}$, VRP on $v$
and $v^{\prime}$ from case $v=0$ and $\sum_{v^{\prime}}$ 133-$\sigma_{p}$
$e^{\prime}\in\\{\text{b},\text{c}^{\prime}_{4},\text{b}^{\prime}\\}$, VRP on
$v$ and $v^{\prime}$ from case $v=0$ and $\sum_{v^{\prime}}$ 134-$\sigma_{p}$
$e^{\prime}=\text{A}^{\prime}$, Assumption of same reference values as for
$e^{\prime}=\text{A}$, VRP on $v$ and $v^{\prime}$ from case $v=0$ and
$\sum_{v^{\prime}}$ — VE-m-e
$\ch{N2+}\left(\text{X},v\right)\ch{+e-}\ch{<=>N2+}\left(e^{\prime},v^{\prime}\right)\ch{+e-}$
$e^{\prime}=\text{B}$, $v=0$ and $v^{\prime}=0$, Remainder of $v$ and
$v^{\prime}$: VRP from case $v=0$ and $v^{\prime}=0$ 135-$\sigma_{p}$
$e^{\prime}\in\\{\text{A},\text{D},\text{C}\\}$, Assumption of same reference
values as for $e^{\prime}=\text{B}$, $v=0$ and $v^{\prime}=0$, Remainder of
$v$ and $v^{\prime}$: VRP from case $v=0$ and $v^{\prime}=0$ — E-a-e
$\ch{N}\left(e\right)\ch{+e-<=>N}\left(e^{\prime}\right)\ch{+e-}$
$(e,e^{\prime})\in\\{({}^{4}\text{S}_{\text{u}},{}^{2}\text{D}_{\text{u}}),({}^{4}\text{S}_{\text{u}},{}^{2}\text{P}_{\text{u}}),({}^{2}\text{D}_{\text{u}},{}^{2}\text{P}_{\text{u}})\\}$
84-$\sigma_{p}$ Remainder of $(e,e^{\prime})$, with $e^{\prime}>e$ 89-$k_{f}$
E-a-e $\ch{N+}\left(e\right)\ch{+e-<=>N+}\left(e^{\prime}\right)\ch{+e-}$
$\forall e$ and $\forall e^{\prime}>e$ 89-$k_{f}$ D-m-e
$\ch{N_{2}}\left(\text{X}{}^{1}\Sigma_{\text{g}}^{+},\,v\right)\ch{+e-<=>N}\left(e_{1}^{\prime}\right)\ch{+N}\left(e_{2}^{\prime}\right)\ch{+e-}$
$\forall\,v$,
$(e_{1}^{\prime},e_{2}^{\prime})=({}^{4}\text{S}_{\text{u}},{}^{4}\text{S}_{\text{u}})$,
ADV 61-$\sigma_{p}$ (from 131) $\forall\,v$,
$(e_{1}^{\prime},e_{2}^{\prime})=({}^{4}\text{S}_{\text{u}},{}^{2}\text{D}_{\text{u}})$,
ADV 58-$\sigma_{p}$ (from 131) DR-m-e
$\ch{N2+}\left(\text{X}{}^{2}\Sigma_{\text{g}}^{+},\,v\right)\ch{+e^{-}<=>N}\left(e_{1}^{\prime}\right)\ch{+N}\left(e_{2}^{\prime}\right)$
$v\in\\{0,2\\}$ and
$(e^{\prime}_{1},e^{\prime}_{2})\in\\{({}^{4}\text{S}_{\text{u}},{}^{2}\text{D}_{\text{u}}),({}^{4}\text{S}_{\text{u}},{}^{2}\text{P}_{\text{u}}),({}^{2}\text{D}_{\text{u}},{}^{2}\text{D}_{\text{u}})\\}$,
136-$k_{f}$ $v\in\\{1,3,4\\}$ and
$(e^{\prime}_{1},e^{\prime}_{2})\in\\{({}^{4}\text{S}_{\text{u}},{}^{2}\text{D}_{\text{u}}),({}^{4}\text{S}_{\text{u}},{}^{2}\text{P}_{\text{u}}),({}^{2}\text{D}_{\text{u}},{}^{2}\text{D}_{\text{u}}),({}^{2}\text{D}_{\text{u}},{}^{2}\text{P}_{\text{u}})\\}$,
Remainder of $v$ with
$(e^{\prime}_{1},e^{\prime}_{2})\in\\{({}^{4}\text{S}_{\text{u}},{}^{2}\text{D}_{\text{u}}),({}^{4}\text{S}_{\text{u}},{}^{2}\text{P}_{\text{u}}),({}^{2}\text{D}_{\text{u}},{}^{2}\text{D}_{\text{u}}),({}^{2}\text{D}_{\text{u}},{}^{2}\text{P}_{\text{u}})\\}$:
VRP from case $v=4$ I-m-e
$\ch{N_{2}}\left(\text{X}{}^{1}\Sigma_{\text{g}}^{+},\,v\right)\ch{+e^{-}<=>N_{2}^{+}}\left(e^{\prime},v^{\prime}\right)\ch{+2e^{-}}$
$\forall\,v$ and $e^{\prime}\in\\{\text{X},\text{A},\text{B}\\}$, ADV, VRP on
$v^{\prime}$ from case $\sum_{v^{\prime}}$ 60-$\sigma_{p}$ (from 131) I-a-e
$\ch{N}\left(e\right)\ch{+e-<=>N+}\left({}^{3}\text{P}\right)\ch{+2e-}$
$e={}^{4}\text{S}_{\text{u}}$ 85-$\sigma_{p}$
$e\in\\{{}^{2}\text{D}_{\text{u}},{}^{2}\text{P}_{\text{u}}\\}$
86-$\sigma_{p}$ Remainder of $e$ 89-$k_{f}$

Molecular spontaneous emission processes for which Einstein coefficients were
obtained. The symbol after the reference in the column “Reference” represents
the quantity which was extracted from it: Einstein coefficient (if $A$) or sum
of the electronic-vibrational transition moments (if $\sum R_{e}^{2}$).
Species Electronic system $e$ \- $e^{\prime}$
$(v_{\text{max}},v^{\prime}_{\text{max}})$ Reference N2 Vegard-Kaplan
A${}^{3}\Sigma_{\text{u}}^{+}$ \- X${}^{1}\Sigma_{\text{g}}^{+}$ $(21,21)$ 137
\- $A$ (from 138) First positive B${}^{3}\Pi_{\text{g}}$ \-
A${}^{3}\Sigma_{\text{u}}^{+}$ $(21,21)$ 92 \- $A$ Wu-Benesch
W${}^{3}\Delta_{\text{u}}$ \- B${}^{3}\Pi_{\text{g}}$ $(21,17)$ 137 \- $A$
(from 138) IR afterglow B${}^{\prime}{}^{3}\Sigma_{\text{u}}^{-}$ \-
B${}^{3}\Pi_{\text{g}}$ $(21,21)$ 137 \- $A$ (from 138) Lyman-Birge-Hopfield
a${}^{1}\Pi_{\text{g}}$ \- X${}^{1}\Sigma_{\text{g}}^{+}$ $(21,21)$ 137 \- $A$
(from 138) Second positive C${}^{3}\Pi_{\text{u}}$ \- B${}^{3}\Pi_{\text{g}}$
$(4,21)$ 92 \- $A$ Birge-Hopfield I b${}^{1}\Pi_{\text{u}}$ \-
X${}^{1}\Sigma_{\text{g}}^{+}$ $(24,60)$ 139 \- $\sum R_{e}^{2}$ Worley-
Jenkins c${}_{3}{}^{1}\Pi_{\text{u}}$ \- X${}^{1}\Sigma_{\text{g}}^{+}$
$(11,60)$ 139 \- $\sum R_{e}^{2}$ Carroll-Yoshino
c${}_{4}^{\prime}{}^{1}\Sigma_{\text{g}}^{+}$ \-
X${}^{1}\Sigma_{\text{g}}^{+}$ $(11,60)$ 139 \- $\sum R_{e}^{2}$ Birge-
Hopfield II b${}^{\prime}{}^{1}\Sigma_{\text{u}}^{+}$ \-
X${}^{1}\Sigma_{\text{g}}^{+}$ $(46,60)$ 139 \- $\sum R_{e}^{2}$ Worley
o${}_{3}{}^{1}\Pi_{\text{u}}$ \- X${}^{1}\Sigma_{\text{g}}^{+}$ $(21,60)$ 139
\- $\sum R_{e}^{2}$ N2+ Meinel A${}^{2}\Pi_{\text{u}}$ \-
X${}^{2}\Sigma_{\text{g}}^{+}$ $(27,27)$ 137 \- $A$ (from 138) First negative
B${}^{2}\Sigma_{\text{u}}^{+}$ \- X${}^{2}\Sigma_{\text{g}}^{+}$ $(12,21)$ 92
\- $A$ Second negative C${}^{2}\Sigma_{\text{u}}^{+}$ \-
X${}^{2}\Sigma_{\text{g}}^{+}$ $(6,27)$ 137 \- $A$ (from 138)

Atomic spontaneous emission processes for which Einstein coefficients were
computed. Species Number of processes Reference N 279 (a ) NIST64 N+ 276 (a )
NIST64 a As a reminder to the reader: representative Einstein coefficients
were computed considering the lumping procedure performed on the split
electronic levels.