# Thermal Relic Right-Handed Neutrino Dark Matter

Yu Cheng<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics and
Astronomy, Shanghai Jiao Tong University, China  Key Laboratory for Particle
Astrophysics and Cosmology (MOE) & Shanghai Key Laboratory for Particle
Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, China
Jie Sheng<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics
and Astronomy, Shanghai Jiao Tong University, China  Key Laboratory for
Particle Astrophysics and Cosmology (MOE) & Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai
200240, China  Tsutomu T. Yanagida<EMAIL_ADDRESS>Tsung-Dao Lee
Institute & School of Physics and Astronomy, Shanghai Jiao Tong University,
China  Key Laboratory for Particle Astrophysics and Cosmology (MOE) & Shanghai
Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong
University, Shanghai 200240, China

###### Abstract

It is known that two heavy Majorana right-handed neutrinos are sufficient to
generate the baryon asymmetry in the present universe. Thus, it is interesting
to identify the third right-handed neutrino $N$ with the dark matter. We
impose a new discrete symmetry $Z_{2}$ on this dark matter neutrino to
stabilize it. However, the $U(1)_{B-L}$ gauge boson $A^{\prime}$ couples to
the right-handed neutrino $N$. If the $B-L$ breaking scale $V_{B-L}$ is
sufficiently low, the dark matter neutrino $N$ can be in the thermal bath. We
find that the thermal relic $N$ can explain the dark matter abundance for the
$B-L$ breaking scale $V_{B-L}\sim O(10)$ TeV. After considering all the
constraints from the existing experiments, a narrow mass region of the thermal
produced right-handed neutrino dark matter $N$ is still surviving.

Introduction – The heavy Majorana right-handed neutrinos (RHN) $N_{i}$ are the
key ingredient for the leptogenesis Fukugita:1986hr ; Buchmuller:2005eh . It
has been pointed out Frampton:2002qc ; Endoh:2002wm  that two heavy Majorana
neutrinos $N_{1,2}$ are sufficient for generating the observed baryon
asymmetry in the present universe (through the leptogenesis) 111It is pointed
out King:1999mb  that single right-handed neutrino dominance has a natural
explanation for the large mixing angle in the neutrino sector.. This fact has
raised a new question, ’Why do we have three families?’ A simple answer to
this question is that one more RHN (called as $N$) is required to explain the
dark matter Kusenko:2010ik ; Ibe:2016yfo . Provided a natural extension of the
standard model (SM), that is, the $U(1)_{B-L}$ gauge symmetry, we need one
more family to cancel the gauge anomaly.

In the previous works, we have considered the non-thermal production of the
RHN DM $N$ Kusenko:2010ik ; Ibe:2016yfo , provided that the breaking scale of
the $U(1)_{B-L}$ gauge symmetry is extremely high such as a GUT scale. Thus,
it is very difficult to test such a model since the interaction between DM $N$
and the SM particles is extremely weak. However, the sufficient baryon
asymmetry can be produced at the Universe temperature of
$\mathcal{O}(10)\,$TeV if the two Majorana masses of $N_{1,2}$ are almost
degenerate Pilaftsis:1997jf ; Xing:2006ms . Therefore, it is interesting to
consider the $U(1)_{B-L}$ gauge symmetry is broken at a relatively low scale
around $\mathcal{O}(10)\,$TeV and the two Majorana masses for $N_{1,2}$ are
also of the same scale.

In this paper, we consider the thermal production of the RHN DM $N$ in the
early Universe assuming the $B-L$ breaking scale $V_{B-L}\simeq O(10)$ TeV to
explain the observed DM density 222The DM $N$ can be in the thermal bath even
when the $B-L$ breaking scale is very high if we introduce a new light scalar
boson $\phi$ couples to $N$ Cheng:2023hzw ; Cheng:2023dau .. After introducing
the model, we first show that such scenario can successfully explain the DM
density in the present Universe. Then, the constraints from all existing
experiments, including the fixed-target, collider, and DM direct detection
experiments, are discussed. The combined constraints from the direct detection
and collider exclude the mass parameter region $m_{N}\gtrsim 5$ GeV. The
direct $N$ production experiments at the fixed targets give us very strong
constraints on the lower mass range where $m_{N}<1$ GeV. Even though, a narrow
parameter region for $m_{N}\subset(10$ MeV, $5$ GeV$)$ still remains. Finally,
we discuss how to test our scenario in future experiments and give our
conclusions.

The RHN DM Model – In this paper, we consider a simple extension of the SM
which is based on the gauge symmetry $SU(3)\times SU(2)\times U(1)_{Y}\times
U(1)_{B-L}$. This simple extension is extremely attractive since we need to
add three RHNs to cancel the gauge anomalies. The presence of RHNs is a key
point of explaining the small neutrino masses via the seesaw mechanism
Minkowski:1977sc ; Yanagida:1979as ; Yanagida:1979gs ; GellMann:1980vs ;
Wilczek:1979hh  and of generating the observed baryon-number asymmetry in the
present universe through the leptogenesis Fukugita:1986hr ; Buchmuller:2005eh
.

A new Higgs field $\Phi$ is introduced to break the gauged $B-L$ symmetry
spontaneously, and its couplings to RHNs $N_{i}(i=1-3)$ generate large
Majorana masses for them through $\frac{1}{2}h_{i}\Phi N_{i}N_{i}$. The
related Lagrangian is,

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{i}{2}\bar{N}_{i}\gamma^{\mu}\partial_{\mu}N_{i}+\left(\lambda_{i\alpha}\bar{N}_{i}L_{\alpha}H-\frac{1}{2}M_{Ri}\bar{N}_{i}^{c}N_{i}+\text{
h.c. }\right)$
$\displaystyle-\frac{1}{2}g_{B-L}\bar{N}_{i}\gamma^{\mu}\gamma_{5}N_{i}A_{\mu}^{\prime}+g_{B-L}Q_{B-L}\bar{f}\gamma^{\mu}fA_{\mu}^{\prime}.$
(1)

Here, $L_{\alpha}$, $f$, and $H$ are the SM left-handed lepton doublets,
fermions, and Higgs boson. The index $i$ stands for the generation of RHN and
$\alpha$ for all the species of leptons. The vacuum expectation value of
$\Phi$ is the breaking scale of $B-L$ gauge symmetry
$\langle\Phi\rangle=V_{B-L}$. Thus, the RHN mass is $M_{Ri}=h_{i}V_{B-L}$. The
$B-L$ quantum numbers $Q_{B-L}$ in the present model for all particles are
shown in Tab. 1. The $q_{\alpha}$, $u_{R}$, $d_{R}$ and $e_{R}$ are the left-
handed quark doublets, right-handed up- and down-type quarks, and the right-
handed charged leptons, respectively.

Table 1: $B-L$ charge for different species Species | $q_{\alpha}$ | $u_{R}$ | $d_{R}$ | $L_{\alpha}$ | $e_{R}$ | $N_{i}$ | $\Phi$ | $H$
---|---|---|---|---|---|---|---|---
$Q_{B-L}$ | 1/3 | 1/3 | 1/3 | -1 | -1 | -1 | 2 | 0

As explained in the introduction, one of the right-handed neutrinos
$N_{i}(i=1-3)$ is the DM. We can choose the third $N_{3}$ to be this RHN DM
without losing the generality and hence $\lambda_{3\alpha}=0$. To guarantee
the stability of DM, we impose a discrete $Z_{2}$ symmetry. Only the RHN DM
$N$ is odd under the $Z_{2}$ parity while others are even. First of all, we
exclude the parameter region that, $g_{B-L}\sim\mathcal{O}(1)$. In such a
region, the right-handed neutrino $N$ is nothing but the so-called WIMP, and
it is most likely excluded by the direct detection experiments. Therefore, we
consider the small gauge coupling $g_{B-L}\ll 1$ region and hence the $B-L$
gauge boson $A^{\prime}$ is relatively light with mass $m_{A^{\prime}}\ll
V_{B-L}=\mathcal{O}(10)\,$TeV. On the other hand, we assume the new Higgs
boson $\Phi$ has a mass of the order of the $B-L$ breaking scale. The effects
of this Higgs boson at the freeze-out time of our DM $N$ can be neglected.
Thus, the physics at the decoupling time of the RHN DM is described by only
lighter particles, the DM $N$, the $U(1)_{B-L}$ gauge boson $A^{\prime}$, the
SM particles, and their interactions.

Thermal Production – In this section, we discuss the thermal production of the
RHN DM $N$ and calculate its relic abundance. First of all, the $B-L$ gauge
boson $A^{\prime}$ can be in the thermal bath with the SM particles through
its decay and inverse decay, $A^{\prime}\leftrightarrow f+\bar{f}$. The
equilibrium condition requires $\Gamma_{A}\simeq
g_{B-L}^{2}m_{A^{\prime}}/12\pi\geq\mathcal{H}(m_{A^{\prime}})$ where
$\mathcal{H}(T)=1.66\sqrt{g_{*}}T^{2}/M_{PL}$ is the hubble constant and
$M_{PL}=1.22\times 10^{19}\,$GeV the planck mass. Since
$m_{A^{\prime}}=g_{B-L}\times V_{B-L}$, this condition reads to $g_{B-L}\geq
12\pi V_{B-L}/M_{PL}\simeq 10^{-13}$, which is easily satisfied.

Then, we consider possible thermal processes that make the RHN $N$ in the
thermal bath. The dominant processes are dependent of the mass difference
$\Delta$ between the $B-L$ gauge boson $A^{\prime}$ and the RHN $N$, defined
as $\Delta\equiv(m_{A^{\prime}}-m_{N})/m_{N}$. We shall discuss it in the
following three cases.

Figure 1:  The gradient red (blue) region in the upper (lower) panel
corresponds to the parameter space which gives the observed DM density for the
mass difference $0<\Delta<1$ ($\Delta<0$). The gray shaded region in the lower
panel stands for the CMB constraint while other colored shaded regions are
excluded by different experiments whose labels can be found in FIG. 2.

I. $m_{N}>m_{A^{\prime}}$ ($\Delta<0$): In this parameter region, there are
two processes rendering the RHN $N$ in the thermal bath with the SM particles.
First, the RHN DM can annihilate into gauge boson through $N+N\leftrightarrow
2A^{\prime}$ with a thermal averaged cross section,

$\langle\sigma v\rangle_{NN\rightarrow
2A^{\prime}}=\frac{g^{4}_{B-L}}{4\pi}\frac{m_{N}^{2}-m_{A^{\prime}}^{2}}{\left(m_{A^{\prime}}^{2}-2m_{N}^{2}\right)^{2}}\sqrt{1-\frac{m_{A^{\prime}}^{2}}{m_{N}^{2}}}.$
(2)

Second, it also annihilates into SM fermions $f$ and anti-fermion $\bar{f}$,
$N+N\leftrightarrow f+{\bar{f}}$, by exchanging the mediator $A^{\prime}$. The
thermal averaged cross section,

$\langle\sigma v\rangle_{NN\rightarrow
f\bar{f}}=\frac{N_{c}Q^{2}_{B-L}g^{4}_{\rm
B-L}(m_{f}^{2}+2m_{N}^{2})}{2x\pi(m_{A^{\prime}}^{2}-4m_{N}^{2})^{2}}\sqrt{1-\frac{m_{f}^{2}}{m_{N}^{2}}}$
(3)

is velocity dependent but with the same order of $g_{B-L}^{4}$.

The thermal evolution of the RHN yield $Y_{N}\equiv n_{N}/s$ (where $n_{N}$ is
the number density of RHN and $s=2\pi^{2}g_{*}T^{3}/45$ is the entropy density
of the Universe with degrees of freedom $g_{*}$) is then described by the
Boltzmann equation,

$\frac{dY_{N}}{dx}=-\frac{s\langle\sigma v\rangle_{\rm
ann}}{\mathcal{H}x}\left(Y_{N}^{2}-\left(Y_{N}^{\rm eq}\right)^{2}\right),$
(4)

where $\langle\sigma v\rangle_{\rm ann}$ is the total annihilation cross
section containing both channels. As the number density decreases, the
annihilation freezes out. The final yield can be solved analytically as,

$Y_{N}^{f}\approx\frac{3.79\left(g_{*}\right)^{-1/2}}{m_{N}M_{\rm
Pl}J_{f}},\quad J_{f}\equiv\int_{x_{f}}^{\infty}dx\frac{\left\langle\sigma
v\right\rangle_{\rm ann}}{x^{2}}.$ (5)

Here, $x_{f}$ is the freeze-out point. The DM relic density is related to
$Y^{f}_{N}$ as,

$\rho_{\chi}=m_{N}s_{0}Y^{f}_{N},\quad\Omega_{N}h^{2}=\frac{\rho_{N}}{\rho_{c}/h^{2}},$
(6)

where $s_{0}=2891.2$ cm-3 is the entropy density today and
$\rho_{c}=1.05\times 10^{-5}h^{2}$ GeV/cm3 is the critical density of the
Universe.

The parameter space which explains the observed DM density today
$\Omega_{N}h^{2}=0.12$ is shown as the gradient blue region of the lower panel
in Fig. 1. The mass range $m_{N}<10$ GeV has a very serious constraint from
the CMB observations Kawasaki:2021etm ; Cline:2013fm . Furthermore, other
parameter region is also excluded by the existing experimental data. More
details shall be discussed in the next section.

II. $2m_{N}>m_{A^{\prime}}>m_{N}$ ($0<\Delta<1$): The annihilation of RHN $N$
into gauge boson $A^{\prime}$ is kinetically suppressed because the gauge
boson becomes heavier. Thus, such a channel is neglected in Eq. (4), and the
freeze-out is determined only by the annihilation channel $N+N\leftrightarrow
f+\bar{f}$. As shown in the upper panel of Fig. 1, since the total
annihilation cross section does not change too much, the parameter space
(gradient red region) giving the observed DM relic density is similar to the
case that $\Delta<0$. Although there is no CMB constraint here for the
annihilation channel to gauge boson is forbidden, all the parameter space is
still excluded by experimental constraints.

III. $m_{A^{\prime}}>2m_{N}$ ($\Delta>1$): Once the gauge boson mass is larger
than two times of the RHN mass, the decay and inverse decay channel333 A
similar model was considered in Kaneta:2016vkq . However, the most important
process, the decay-inverse decay channel, is not taken into account in their
work. Hence, most of their parameter spaces for the thermal relic $N$ DM are
excluded at present, as we show in FIG.1 in this paper.
$A^{\prime}\leftrightarrow 2N$ dominants with a decay rate

$\Gamma_{A^{\prime}\rightarrow
2N}=\frac{g^{2}_{B-L}m_{A^{\prime}}}{24\pi}\left(1-\frac{4m_{N}^{2}}{m_{A^{\prime}}^{2}}\right)^{\frac{3}{2}}$
(7)

of order $g_{B-L}^{2}$. The Boltzmann equation in the early Universe becomes,

$\frac{dY_{N}}{dx}=-\frac{1}{\mathcal{H}x}\left[-\Gamma_{A^{\prime}\rightarrow
NN}\left(Y_{A^{\prime}}-\frac{Y_{N}^{2}Y^{\rm
eq}_{A^{\prime}}}{\left(Y_{N}^{\rm eq}\right)^{2}}\right)\right].$ (8)

The gauge boson $A^{\prime}$ keeps maintaining in equilibrium for its decay
rate is always larger than the Hubble constant in the concerning energy scale.
The freeze-out point $x_{f}$ can be obtained by the condition that the
coefficient of the collision term becomes order of one,

$\Gamma_{A^{\prime}\rightarrow NN}\frac{Y^{\rm eq}_{A^{\prime}}}{Y_{N}^{\rm
eq}}\simeq\mathcal{H}x_{f}.$ (9)

To match the observed DM density, the RHN mass has a relationship with the
coupling $g_{B-L}$ and mass difference $\Delta$ as Frumkin:2021zng ,

$m_{N}\simeq\left[\frac{g^{2}_{B-L}}{16\pi}\left(\frac{M_{PL}}{1.66\sqrt{g_{*}}}\right)\left(\frac{2^{3/2}\pi^{7/2}g_{*}T_{\rm
eq}}{60x_{f}^{1/2-3/\Delta}}\right)\right]^{\frac{1}{\Delta+1}}.$ (10)

Here, we applied the DM number density, $n_{N}(x_{f})\simeq\pi^{2}g_{*}T_{\rm
eq}m_{N}^{2}/30x_{f}^{2}$.

The corresponding parameter space is shown in Fig. 2 as the gradient red
padding area. The same as the standard freeze-out scenario, a larger DM mass
requires a larger coupling. Besides, for a fixed DM mass, the $B-L$ gauge
boson mass $m_{A^{\prime}}$ increases as the mass difference $\Delta$
increase. As a result, the number density of the gauge boson becomes smaller
and a larger coupling is needed to compensate for the decrease of the left-
handed side of Eq. (9). One can see that the DM with a mass $m_{N}\gtrsim 5$
GeV is excluded by DM direct detection and collider experiments. However,
there is still some regions surviving for $m_{N}\subset(10\,$MeV, $5\,$GeV$)$
with the vacuum-expectation value $V_{B-L}\subset(1\,$TeV, $5\,$TeV$)$.

Figure 2:  The parameter space (red padding region) which gives the observed
DM density for the mass difference $\Delta>1$. The colored (green, blue,
black) shaded regions are excluded by different experiments. The blue lines
(solid, dashed, dash-dotted) from above to below correspond to $V_{\rm
B-L}=1,5,10\,$TeV.

Experimental Constraints – The RHN DM $N$, as well as a new massive gauge
boson $A^{\prime}$, is predicted in our model. Both of them have interactions
with SM particles and receive strong constraints from experiments. In this
section, we shall analyze the results from fixed-target, collider, and DM
direct detection experiments and show the surviving parameter space. All the
constraints are shown in Fig. 2.

I. Fixed-Target Experiments: The $B-L$ gauge boson $A^{\prime}$ can be
produced by the bremsstrahlung process in high-energy electron scattering off
heavy nucleus, $e^{-}+Z\rightarrow e^{-}+Z+A^{\prime}$. Its subsequent
invisible decay into a neutrino pair will cause missing energy events in
fixed-target experiments Gninenko:2016kpg ; LDMX:2018cma ; Chen:2022liu . With
no signal events found, NA64 experiment data NA64:2022yly  gives the stringent
constraints on the $B-L$ gauge coupling, $g_{B-L}<(2\times 10^{-5},10^{-4})$
according to the mass range $m_{A^{\prime}}\subset(10,100)$ MeV as shown in
the gray shaded region surrounded by solid curve.

The extra $B-L$ gauge symmetry also generates a non-standard $\nu-e$
interaction, which shall lead to some recoil excess in neutrino detectors.
Thus, the strongest constraint for the mass region
$m_{A^{\prime}}\subset(0.1,1)\,$GeV comes from $\nu-e$ scattering experiments,
CHARM II CHARM-II:1993phx , which is shown as the gray shaded region
surrounded by dashed curve.

II. Collider Search: Several searches for the $B-L$ gauge boson produced in
proton-proton collisions at LHC were also carried out by the ATLAS, CMS, LHCb
and FASER collaborations ATLAS:2017fih ; CMS:2016cfx ; LHCb:2017trq ;
FASER:2023tle . By searching for the new high-mass resonances phenomena in
dielectron and dimuon final states, the ATLAS collaboration gives the
constraint for $g_{B-L}<10^{-3}$ in $m_{A^{\prime}}\subset(100,1000)$ GeV
ATLAS:2017fih  as shown by the green shaded region surrounded by dashed curve.
An inclusive search for the gauge boson decay
$A^{\prime}\rightarrow\mu^{+}+\mu^{-}$ was performed by LHCb
experimentLHCb:2017trq ; Amrith:2018yfb . By searching for the displaced-
vertex signature for long-lived $A^{\prime}$ in $2m_{\mu}<m_{A^{\prime}}<350$
MeV and prompt-like $A^{\prime}$ decay in $350$ MeV $<m_{A^{\prime}}<70$ GeV,
the experiment constrains $g_{B-L}<10^{-4}$ as shown by the the green shaded
region surrounded by solid curve. The FASER experiment is designed to search
for long-lived particles travelling in the far-forward direction of the proton
beam. It excludes the parameter space where $g_{B-L}\subset(5\times
10^{-6},2\times 10^{-5})$ and $m_{A^{\prime}}\subset(15,40)$ MeV FASER:2023tle
as shown in the dark green shaded region surrounded by solid curve.

III. Direct Detection: The RHN itself can scatter with SM fermions by
exchanging a t-channel $B-L$ gauge boson. Thus, it is also constrained from
the DM direct detection experiments. The scattering is $p$-wave since the RHN
DM $N$ is Majorana type. Taking a general fermion target with mass $m_{f}$,
the leading order of the $p$-wave scattering cross section $\sigma_{S}$ is,

$\sigma_{S}=\frac{m_{N}^{2}m_{f}T_{N}(3m_{N}^{2}+2m_{N}m_{f}+m_{f}^{2})}{(m_{N}+m_{f})^{4}\pi}.$
(11)

Here, $T_{N}$ is the kinetic energy of initial DM. For the halo DM with
velocity $v\sim 10^{-3}$, the kinetic energy is of order $T_{N}\sim
10^{-6}m_{N}$. In the non-relativistic limit, the interaction between Majorana
RHN and SM fermion via a vector mediator contains both spin-independent and
spin-dependent contributionsFan:2010gt . The spin-independent part plays a
leading role and receives stronger constraints.

For the RHN DM with mass over $\mathcal{O}(1)$ GeV, the most sensitive
constraints are from the XenonXENON:2023cxc , LZLZ:2022lsv , and
PandaXPandaX-4T:2021bab  which are based on the DM-nuclear scattering. For the
coherent scattering of Xenon nuclear, the target mass is $m_{f}=131\,$GeV. As
shown in the blue shaded region surrounded by solid lines, the nuclear recoil
constraint is slightly weaker than collider ones. However, it fills the gap
around $(30,70)\,$GeV so that the possibility of $m_{N}\gtrsim 5\,$GeV is
totally excluded. On the other hand, one needs to consider the DM-electron
scattering ($m_{f}=m_{e}$) for the RHN DM with mass
$m_{N}\lesssim\mathcal{O}(1)$ GeV because of the kinetic threshold. In this
region, the most stringent constraint (blue shaded region surrounded by dashed
curve) is from SENSEISENSEI:2020dpa , DAMICDAMIC-M:2023gxo  and
DarksideDarkSide:2022knj . It is much weaker than that of fixed-target
experiments.

Combining all the constraints, one can clearly see that the parameter space
for $m_{A^{\prime}}<2m_{N}$ ($\Delta<1$), which prefers $g_{B-L}>10^{-3}$, has
already been excluded. The only remaining region is $m_{N}\subset(10\,$MeV,
$5\,$GeV$)$ for $\Delta>1$ and $g_{B-L}\lesssim 10^{-4}$ except for a small
region with $m_{N}<20$ MeV which is excluded by the FASER experiment.

Discussion and Conclusions – Why do we have three families for quarks and
leptons? This is one of the fundamental questions in particle physics. In this
paper, we assume a $U(1)_{B-L}$ gauge symmetry in addition to the SM symmetry,
$SU(3)\times SU(2)\times U(1)_{Y}$, since this provides us a theoretical
reason for the presence of an RHN $N_{i}$ in each family. The spontaneous
breaking of the $B-L$ gauge symmetry generates Majorana mass terms for the
RHNs $N_{i}~{}(i=1-3)$, which are key ingredients for the seesaw mechanism and
the leptogenesis. Thus, in this interesting framework, our question is
equivalent to the question of why nature needs three RHNs $N_{1,2,3}$.

It is known that we need at least two heavy Majorana neutrinos $N_{1,2}$ to
create the observed baryon-number asymmetry in the present Universe through
leptogenesis. In this paper, we investigate the possibility that the last
third RHN, which is called $N$, is required for the DM to produce the Universe
we can live in. Motivated by this anthropic selection Kusenko:2010ik ;
Ibe:2016yfo  of three families of quarks and leptons, we discuss the scenario
that the third RHN is the DM in the Universe.

It has been pointed out Pilaftsis:1997jf ; Xing:2006ms  that we can have a
successful low-energy scale leptogenesis at the temperature $T\sim 10$ TeV, if
the two Majorana RHNs $N_{1,2}$ have almost degenerate masses $M_{1}\simeq
M_{2}$. In this case, the $B-L$ breaking scale can be $V_{B-L}=O(10)$TeV.
Motivated by this observation, we consider thermal production of the RHN DM
$N$ in this paper and find the allowed parameter space, $m_{N}\subset(10$ MeV
$,5$ GeV$)$ and $g_{B-L}\simeq 10^{-4}$, consistent with the present DM
density and with all existing experiments 444The parameter region of
$m_{N}<10$ MeV has a tension with the success of BBN.. We show that the higher
mass region for $m_{N}\gtrsim 5$ GeV is already excluded by the DM direct
detection and the LHC experiments. The fixed-target experiments give us
stringent constraints for the lighter mass region, $m_{N}<\mathcal{O}(1)$ GeV.
Therefore, we expect that the high-luminosity LHC experiments will provide us
crucial tests of the present scenario for the RHN DM of the mass in the
surviving range and/or the $B-L$ gauge boson $A^{\prime}$ will be discovered
in future high intensity electron beam frontiers such as LDMX-style missing
momentum experiments Gninenko:2016kpg ; LDMX:2018cma ; Chen:2022liu  and Belle
II if the mass is in fact $m_{A^{\prime}}<\mathcal{O}(1)$ GeV.

Note that the Ref.Smith:2024jve  shows our remaining parameter space has also
been excluded by the constraint from the invisible decay $B\rightarrow
K+A^{\prime}$. However, the definition of the $B-L$ charge has an ambiguity
due to the hypercharge gauge rotation. For $Q_{B-L}\rightarrow Q_{B-L}-2Y$,
the charge for $Q_{q_{\alpha}}=0$. Then, the above decay channel is
sufficiently suppressed even after the electroweak symmetry breaking, but the
$B-L$ charge of the right-handed neutrino $N$ is unchanged. Therefore, the
constraints in Smith:2024jve  are not applicable in our model with the new
$B-L$ charge and our conclusion in this paper remains the same.

## Acknowledgements

The authors thank Liang Tan for useful discussions. This work is supported by
the National Natural Science Foundation of China (12175134, 12375101,
12090060, 12090064, and 12247141), JSPS Grant-in-Aid for Scientific Research
Grants No. 19H05810, the SJTU Double First Class start-up fund No.
WF220442604, and World Premier International Research Center Initiative (WPI
Initiative), MEXT, Japan. T. T. Y. is an affiliated member of Kavli IPMU,
University of Tokyo.

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11footnotetext: This work has been submitted to the Central European Journal
of Economic Modelling and Econometrics and is under review process

# Has EU accession boosted patents performance in the EU-13? – A critical
evaluation using causal impact analysis with Bayesian structural time-series
models

Agnieszka<EMAIL_ADDRESS>Jan Kochanowski University of
Kielce, Kielce, Poland 22footnotemark: 2, Krzysztof
<EMAIL_ADDRESS>AGH University of Science and Technology,
Krakow, Poland

###### Abstract

Nowadays innovation is one of the main determinants of economic development.
Patents are a key measure of innovation output, as patent indicators reflect
the inventive performance of countries, technologies and firms. This paper
provides new insights on the causal effects of the enlargement of the European
Union (EU) by investigating the patents performance within the new EU member
states (EU-13). The empirical results based on data collected from the OECD
database from 1985 – 2017 and causal impact using a Bayesian structural time-
series model (proposed by Google) point towards a conclusion that joining the
EU has had a significant impact on patents performance in Romania, Estonia,
Poland, Czech Republic, Croatia and Lithuania, although in the latter two
countries the impact was negative. For the rest of the EU-13 countries there
is no significant effect on patent performance. Whether the EU accession
effect is significant or not, the EU-13 are far behind the EU-15 (countries
which entered the EU before 2004) in terms of patent performance. The majority
of patents (98.66%) are assigned to the EU-15, with just 1.34% of assignees
belonging to the EU-13.

Keywords: causal analysis, European Union, patents, innovations, Bayesian
structural time-series models

JEL Classification: O340, C11, C32, O52

##  1 Introduction

According to the European Patent Office [EPO, a] a patent is _“legal title
that gives inventors the right, for a limited period (usually 20 years), to
prevent others from making, using or selling their invention without their
permission in the countries for which the patent has been granted”_.

In an increasingly knowledge-driven economy, society invariably needs creative
or inventive ideas or concepts to improve existing features, add useful new
features to products or develop new products. European patent application
statistics from 2019 ([EPO, b]) show the top patent technology fields were
digital communication, medical technology and computer technology- all the
highly influential and fast-growing areas of technology.

In this rapidly changing environment patents are a long-term investment. The
patent applicant can commercialize the invention at any point during the time
of patent protection, either through developing products or services
incorporating the patented technology or by licensing it to others. This means
patents play an increasingly important role in innovation and economic
performance. There is evidence that patenting in general contributes
positively to financial performance. The number of patents is quantitative
output of technologically successful R&D activities and can be seen as one of
the key-components of a country’s innovation [Maresch et al., 2016].

In the literature (e.g. [Lee and Lee, 2020]), innovation is presented as the
most important factor in achieving economic and employment growth and one of
the main determinants of economic development in modern societies. Investing
in research is considered essential for achieving smart, sustainable and
inclusive growth and jobs in Europe. Investment protects R&D activities by
approval of patents rights and this can make the patent a risk reducing factor
for an investor. Given the strong connection between innovations and patents,
we may expect that less innovative countries will be lagging behind the highly
innovative ones in terms of total patent number.

Despite the efforts of European Union (EU) policy makers, disparities exist
between the EU-15 (countries which entered the EU before 2004) and EU-13
(countries that joined the EU in and after 2004) in various ways as
highlighted by many authors e.g. [Filippetti and Peyrache, 2013, Makkonen and
Mitze, 2016, Pazour et al., 2020]. Among other things, the EU is also
polarized in terms of innovations. According to [Hollanders, 2019] data from
the European Innovation Scoreboard suggest that the EU-13 lags behind the
EU-15. As showed by [Kleszcz, 2020], the headquarters of the EU’s top 1,000
R&D investors from the ICT sector are located in 16 countries, with an
overwhelming concentration in EU-15 countries. Furthermore, organisations
based in the EU-13 have benefited less from their participation in the
European Framework Programmes (FPs) than organisations from the EU-15 [Fresco
et al., 2015]. Most of the time, the EU-13 can be found at the lower end of
participation rankings concerning Horizon 2020 [Pazour et al., 2018]. The
majority of EU nations with smaller research budgets are former communist
countries in central and eastern Europe, which together with Cyprus and Malta
joined the EU after 2004 as emphasized by [Abbott and Schiermeier, 2019].

In recent years, there have been numerous academic and policy debates on the
delivery mechanisms of EU funds in member states. Accessing large amounts of
funding has often been seen as one of the main benefits of EU accession. It
was hoped that European funds like Structural Funds, direct payments in
agriculture, rural development and others would contribute towards reducing
disparities between regions. Economists and regional geographers have been
very interested in studying the effects of European transfers on national and
regional/local economies e.g. [Surubaru, 2021]. In various scientific
literature and reports analysis can be found assessing the impact of EU
membership on different aspects like political changes, economic reforms,
political values or foreign policies e.g. [van Houwelingen et al., 2021,
Nitoiu and Moga, 2021, Baas, 2020, Felbermayr et al., 2018]. Nevertheless,
there are different views on the EU and its influence on member countries.
Many of the Central and Eastern European countries surveyed positive views of
the political union, however on the other hand there are plenty of
Eurosceptics within the EU, as the UK Brexit referendum demonstrated. Hence it
is crucial to reliably and quantitatively assess how membership impacts member
states in many different aspects. Considering all aforementioned aspects of
disparities between EU-13 and EU-15 leads to a natural question of: Has EU-13
accession boosted the innovation of these countries? Since the question is
about the cause and an effect, a formal causal analysis is required to answer
it. However, causal impact analysis of EU enlargement on the EU-13 countries’
innovations could be identified in very few scientific publications. [Gyen,
2018] investigates the relationship between innovation and EU membership using
panel data on the firm level, while [Makkonen and Mitze, 2016] studied
scientific collaboration between ‘old’ and ‘new’ member states. Results from
the latter paper are a direct motivation for this research as they show that
the collaboration between the new and old member states has been affected by
EU enlargement.

International scientific co-publication is just one of the components of
innovation. The number of Patent Cooperation Treaty (PCT) patents applications
per billion GDP is another component of Summary Innovation Index [Hollanders,
2019]. In this paper, we used number of patents as one key measure of
innovation output. Despite the role of patents in innovation, no studies so
far have attempted to examine the dynamics of patents performance in the EU-13
over time, particularly including the effects of EU-13 accession on patents
performance. This study has tried to bridge that gap and analyse dynamical
causal impact of EU accession on patents performance.

Causal impact of EU accession on patents performance was assessed by utilizing
Bayesian structural time-series models. The method generalises the widely used
difference-in-differences approach to the time-series setting, by explicitly
modelling the counterfactual of a time series observed both before and after
the intervention. Hence, the research problem that guides the analysis is the
assessment of the question: Has EU membership significantly increased the
number of patents of the EU-13 countries.

Furthermore, the paper explores the total number of patents to highlight
disparities between EU-13 and EU-15 countries. The following research problems
are addressed in the paper:

  * •

To quantify the effects of accession of EU-13, in particular find causalities
if the EU-13 accession to EU has increased dynamics of patents performance in
the EU-13.

  * •

To analyse disparities between EU-13 and EU-15.

The main contribution of this paper is assessment of patents performance for
EU-13 countries after accession to the EU using a new causal impact approach
with Bayesian inference. The method generalises the widely used difference-in-
differences approach commonly used in related work to the time-series setting
by explicitly modelling the counterfactual of a time series observed both
before and after the intervention.

The structure of this paper is as follows. In Section 2 we provide a the
review of the relevant literature on the patent analysis and different causal
relationships in the innovation field. In Section 3 we describe dataset with
methodology. In Section 4 we report and discuss our empirical results. Section
5 summarizes and concludes with policy recommendations.

##  2 Related work

Intellectual property law governs technological innovation. In the literature,
the field of innovations has been presented by [Lee and Lee, 2020, Szopik-
Depczyńska et al., 2018] as the most important factor to achieve economic and
employment growth. Patents are a key measure of innovation output, as patent
indicators reflect the inventive performance of countries, technologies and
firms. They are also used to track the level of diffusion of knowledge across
technology and internationalisation. Patent indicators can serve as a measure
of the output of R&D, its productivity, structure and the development of a
specific technology or industry.

As indicators, patents have the following advantages over other measures i) a
close link to invention, ii) they cover a broad range of technologies on which
there are sometimes few other sources of data, iii) the contents of patent
documents are a rich source of information, iv) patent data are readily
available from patent office [OECD, 2009]. A popular composite indicator for
measuring innovativeness at the national level is the Summary Innovation Index
(SII) annually published in a report: The European Innovation Scoreboard. One
of its indicators is the number of Patent Cooperation Treaty (PCT)
applications per billion GDP [Hollanders, 2019]. Another well-know composite
innovation indicator, which among other things contains quantifiable
information about patents (patent families filed in two or more offices;
Patent applications by origin; PCT international applications by origin), is
the Global Innovation Index, published by Cornell University, INSEAD and the
World Intellectual Property Organization [Dutta et al., 2020].

It is not only big organizations analysing patents. Independent scientific
researchers also often utilize patents as an indicator of country innovation
performance e.g. [Lee and Lee, 2020]. [Kim and Lee, 2015] moves the analysis
further by focusing on the different natures of national innovation systems in
East Asia and Latin America. The key conclusion is that it is not scientific
knowledge (academic articles) but technological knowledge (patents) that
matters for economic growth. Furthermore, generating scientific knowledge does
not automatically lead to the generation of technological knowledge. They
noticed that technological knowledge is primarily determined by corporate
research and development efforts, which used to be more lacking in Latin
American countries, compared to East Asia.

Companies invest in patents because, new inventions through efforts on R&D
activities get protected by approval of patents rights [Das, 2020]. The patent
system is one of a suite of policy levers that has been used to attempt to
bring the private returns captured by inventors closer to the social value of
their inventions. Patents aim to allow inventors to recoup the fixed costs of
their research investments by providing inventors with a temporary period of
market power [Williams, 2017]. A large amount of literature in economics,
management, finance, law, and related fields has developed over the past few
decades to investigate various aspects of the patent system.

Having said that, it is worth noting that there is a contradictory role of
patents in improving innovation found in the literature. On one hand, [Haber,
2016] summarized as follows “… the weight of the evidence supports the claim
of a positive causal relationship between the strength of patent rights and
innovation”. On the other hand, [Boldrin and Levine, 2013] presents
contrasting argument against patents “The case against patents can be
summarized briefly: there is no empirical evidence that they serve to increase
innovation and productivity…”. Investigations of the relationship between the
changes in patent activity and the amount of expenditure on R&D are not novel
e.g. [Griliches, 1990]. The conclusion is that the increase in the patent
activity of large enterprises can be achieved by increasing expenditure on
R&D.

[Sierotowicz, 2015] evaluated the efficiency of R&D expenditure (data obtained
from EUROSTAT) from the patent activity (data obtained from EPO) in EU-28
countries for the period 1999–2013. Among the 28 EU counties, 98.07% of
assignee patents granted by the EPO in that period were given to entities
belonging to the business enterprise sectors of the leading countries, which
included: Germany, followed by France, United Kingdom, Italy, Sweden,
Netherlands, Finland, Austria, Denmark and Spain, while the remaining 1.93% of
assignees belong to the remaining 18 countries of the EU. The author concluded
that the increase in total intramural expenditure on R&D activities in the
business enterprise sectors of the 10 leading EU countries caused the
increases in patenting activity of the sector in the long run. It was
presented that Germany has the highest value of 0.26 of patents granted by
enterprises per 1 million euros of the total intramural expenditure on R&D in
the business enterprise sector across the research period, while the lowest
value is Spain, at 0.03.

Empirical evidence on the link between firms’ R&D expenditure and patent
registrations in Spain was provided by [Altuzarra, 2019]. A bidirectional
causal relationship between R&D and patents was evidenced by the Granger
causality test in a panel of Spanish manufacturing firms for the period
1990–2013. A broader study of [Almeida and Teixeira, 2007] on whether
patenting negatively impacts R&D activity in a panel of 88 countries over an
eight-year period (1996–2003) found mixed support for the negativity of
patents on R&D investment. Accumulated patents positively impact on R&D
intensity for the set of less developed countries whereas no statistically
significant effect emerges in the case of higher developed converge clubs.
When restricting the highest developed convergence club down to countries with
a R&D intensity above 3%, the reversed negative causality arises,
corroborating the asymmetric impact of patents on R&D investment.

The review of existing studies covers different aspects of patents e.g.
interlinkages among R&D; patents and incomes in different countries or
comparative analysis between countries. Also, multiple reports point to the
issue of under-performance in many aspects by the EU-13 in comparison with the
EU-15 Member States. The effects of EU accession has been studied by many
authors, however, the particular aspect of innovations could be identified
only in a few scientific publications.

[Gyen, 2018] investigated the relationship between innovation and EU
membership using panel data at the business level by using a difference-in-
difference estimator considering access to the inner EU market as the
treatment. The findings are that there is a significant percentage point
decline in innovation efforts by firms in the new membership countries
relative to the change for the control group firms. [Makkonen and Mitze, 2016]
find that the most significant impact, in terms of co-publication intensities,
of the EU enlargement, has been the high increase in the level of scientific
collaboration that the EU-10 (new member states of the 2004 enlargement) and
EU-2 (new member states of the 2007 enlargement) have among each other.
Additionally, the collaboration between the new and old member states has been
affected by the EU enlargement. In both papers, author use causal model for
their analysis.

Similarly [Mtar and Belazreg, 2020] examined the causal relationship between
innovation, financial development, and economic growth using panel VAR
approach for 27 OECD countries over the period 2001–2016. Among others,
authors conclude that the relationship between innovation and economic growth
is complex and country-specific characteristics can play an important role in
fostering innovation and productivity. The paper concludes that governments
can play an important role in developing a legislative framework favoring the
development of innovation financing through the patent guarantee deposit.

Many additional factors have been listed in the literature, which could affect
innovation. Having said that, multiple findings and applications show that
despite the complicated nature of innovation, patents are one of the key
measurements of countries innovation performance. To the best of the author’s
knowledge, there are no other papers aiming at quantifying the causal effect
of EU accession on patents performance for specific EU countries.

Causal modeling techniques are not novel and are widely used to test causal
claims both within economics and in many other areas of social sciences e.g
[Woodward, 1995, Morgan and Winship, 2014]. The aforementioned papers use this
method frequently. Quasi-experimental methods, such as difference-in-
difference, regression discontinuity and other related methods, have had the
effect of overshadowing the role of economic theory in the specification of a
model.

Aforementioned Granger causality is based on autoregressive (AR) processes
applicable to problems of model identification, while the transfer entropy
method is an information-theoretic approach that does not need assumptions on
the structure of the process. It is based on the concept of Shannon-entropy
and is suitable for linear and nonlinear relations. Its key assumption is that
the sampled data should follow a well-defined probability distribution. Both
approaches are however essentially descriptive, as they are not based on a
structural modeling of the data generating process. This approach could be
found e.g.[Altuzarra, 2019]. However this method could have some limitation:
Granger-causality may not be sufficient in practice for counterfactual control
as identified by [Hoover, 2012].

The difference-in-difference (DiD) approach is a quantitative research design
for estimating causal relationships in quasi-experimental settings. It is
popular, for example, in empirical economics as well as other social sciences
and commonly applied when estimating the effects of certain policy
interventions or institutional changes that do not affect everybody at the
same time. This approach could be found in e.g [Makkonen and Mitze, 2016].
However, DiD is limited, for example:i) DiD is traditionally based on a static
regression model that assumes i.i.d. data despite the temporal aspect of the
data. When fit to serially correlated data, static models yield overoptimistic
inferences with too narrow uncertainty intervals; ii) most DiD analyses only
consider the point in time when the intervention happened, and no evolution of
the effect can be inferred; iii) Synthetic control construction can be
restricted in the case of time-series analysis.

All the issues of DiD are fixed by Google’s ([Brodersen et al., 2015])
proposal called _CausalImpact_. The method generalises DiD to the time-series
setting by using Bayesian structural time-series models to construct a
counterfactual effect estimator. Multiple advantages of the methods described
in section 3 make the method particularly suited to the needs of this paper.

##  3 Materials and methods

In this section we present formal methods of assessment if EU membership
change patents performance for new members (EU-13). The method from [Brodersen
et al., 2015] was used in the paper as a primary causal analysis method. All
causal impact calculations were performed with Google’s _CausalImpact_ R
package.

Patent information crucial for the analysis can be obtained from different
sources. Starting from publicly available databases like EPO, OECD or Google
Patents to the more advanced commercial ones like Derwent, PATSTAT or Global
Patent Index. The analysis reported in this paper is based on a dataset from
the Organisation for Economic Co-operation and Development (OECD).

###  3.1 Dataset

We used the OECD database [OECD.Stat, 2021] where patents were counted
according to the inventor’s country of residence. When a patent was invented
by several inventors from different countries the respective contributions of
each country was accounted for in order to eliminate multiple counting of such
patents (hence fractional counts existed in the dataset). The research
concerns the number of yearly patents in 28 EU countries: CY- Cyprus, MT-
Malta, LV- Latvia, LT- Lithuania, EE- Estonia, HR- Croatia, BG- Bulgaria, SK-
Slovak Republic, RO- Romania, LU- Luxembourg, SI- Slovenia, PT- Portugal, EL-
Greece, CZ- Czech Republic, HU- Hungary, PL- Poland, IE- Ireland, DK- Denmark,
ES- Spain, BE- Belgium, FI- Finland, AT- Austria, SE- Sweden, NL- Netherlands,
IT- Italy, UK- United Kingdom, FR- France, DE- Germany. The analysis is
carried out on patents collected over the years 1985–2017. A whole group of
countries, namely the so-called New Member States that joined the European
Union in 2004 (CZ, EE, HU, LV, LT, PL, SK, SI, MT and CY) 2007 (BG, RO) and
2013 (HR) will be collectively referred to as the EU-13 with EU-10, EU-2, EU-1
denoting the subgroups respectively. Old Member States countries (AT, BE, DE,
DK, EL, ES, FI, FR, IE, IT, LU, NL, PT, SE, UK) we will refer to as the EU-15.
For each EU-13 country the year of EU accession divides the patent’s dataset
for that particular country into two time-series refereed to as _before
accession_ and _after accession_.

###  3.2 Causal impact

Causal modelling enables reasoning about the cause and the effect in contrast
to correlation models where only the association can be reasoned about [Pearl,
2009]. The causal impact of a treatment is the difference between the observed
value of the response and the value that would have been obtained under the
alternative treatment, that is, the effect of treatment on the treated. In
this paper the response variable is a time series $y_{t}$, so the causal
effect of interest is the difference between the observed series and the
series that would have been observed had the intervention not taken place. In
the research reported here, the intervention is the EU accession and the
response variables represent yearly patents assigned to the country. As the
global political environment is not an isolated experimental environment it is
not possible to measure the counterfactual response. Thus we applied a method
proposed by [Brodersen et al., 2015] that uses time-series prediction to
estimate the counterfactuals.

A time-series model fitted to the observations from before intervention can
predict what would have happened if the country had not accessed the EU. The
uncertainty of the prediction is handled by using Bayesian structural time-
series models. This allows estimation of the statistical significance of the
causal impact. Since prediction intervals often grow fast as the number of
predicted steps increases, a simple time series model is insufficient to
conduct a meaningful analysis. Time-series observations from the countries not
affected by the intervention (yet correlated with the analyzed time series)
are crucial for the model to capture global trends. In this paper we used
patent time-series for the EU-15 to reduce prediction uncertainty for the new
EU countries. Such a decision is motivated by research of [Makkonen and Mitze,
2016], where EU-15 formed a control group in a DiD method. The next section
describes mathematical details of the time-series model used for prediction.

###  3.3 Linear Gaussian State Space Model

Linear Gaussian State Space Model (LG-SSM) is a general family of time series
models. [Murphy, 2012] defines it as the following linear dynamical system:

$\displaystyle\bm{z}_{t}$
$\displaystyle=\bm{F}_{t}\bm{z}_{t-1}+\bm{R}_{t}\bm{\varepsilon}_{t},\quad\bm{\varepsilon}_{t}\sim\mathcal{N}(\bm{0},\bm{Q}_{t}),$
(1) $\displaystyle y_{t}$
$\displaystyle=H_{t}^{T}\bm{z}_{t}+\delta_{t},\quad\delta_{t}\sim\mathcal{N}(0,\sigma_{y}^{2}),$
(2)

where $\bm{z}_{0}\sim\mathcal{N}(\bm{b}_{0},\bm{Q}_{0})$ is the
$n$-dimensional initial hidden state and $x_{t}\in\mathbb{R}$, $t\geq 0$, is
the observed time series. Throughout, $\mathcal{N}(\cdot,\cdot)$ denotes a
multivariate Gaussian distribution and the noises
$(\bm{\varepsilon}_{t},\delta_{t})$ are independent across time $t$. The state
dynamics are parameterized by the transition matrix
$\bm{F}_{t}\in\mathbb{R}^{n\times n}$, control matrix
$\bm{R}_{t}\in\mathbb{R}^{n\times q}$, and covariance matrix
$\bm{Q}\in\mathbb{R}^{q\times q}$. The observations $x_{t}$ are noisy linear
projections of the states $\bm{z}_{t}$ and further parameterized by the
observation vector $H\in\mathbb{R}^{n}$ and the observation noise variance
$\sigma_{y}^{2}$. Many classical time series models can be represented as an
LG-SSM, this makes it a popular choice for time series forecasting [Harvey,
1990]. In this paper we focus on two main components: local level and linear
regression.

Mathematically a local level model is defined as LG-SSM with a real hidden
state, namely, a current level $l_{t}$. It evolves as:

$\displaystyle y_{t}^{l}$
$\displaystyle=l_{t}+\epsilon_{t}^{y}\quad\epsilon_{t}^{y}\sim\mathcal{N}(0,\sigma_{y}^{2})$
(3) $\displaystyle l_{t}$
$\displaystyle=l_{t-1}+\epsilon_{t}^{l}\quad\epsilon_{t}^{l}\sim\mathcal{N}(0,\sigma_{l}^{2})$
(4)

where $\epsilon_{t}^{y}$ and $\epsilon_{t}^{l}$ are independent. The future
depends only on the past observations of the time-series.

Often time series can be explained by another time series and in addition to
its past. In LG-SSM it is possible to use external observations (time-series)
$\bm{x}_{t}$ in the form of linear regression. A _static linear regression_ is
obtained by setting $H_{t}=\bm{\beta}^{T}\bm{x}_{t}$ and $z_{t}^{r}=1$, where
$\bm{x}_{t}$ is a vector of covariates (in our case the EU-15 with additional
column of ones) and $\bm{\beta}$ is vector of regression coefficients. Since
the sum of LG-SSMs is also an LG-SSM the simple components (local level and
linear regression) can be summed to form structural time series model:
[Murphy, 2012]:

$\displaystyle y_{t}$
$\displaystyle=l_{t}+\bm{\beta}^{T}\bm{x}_{t}+\epsilon_{t}^{y},$ (5)

Where $l_{t}$ is the local level (4). In terms of the general representation
from (1) we obtain a block-diagonal transition and control matrices and
concatenated observation vectors performing the summation of local level and
regression components in (5). Given the model parameters, the state
$\bm{z}_{T}$ at time of intervention and the contemporaneous predictor
variables after intervention $\bm{x}_{t\geq T}$, the system dynamics (1)–(2)
enable forecasting for $y_{t\geq T}$. Since the model is linear and Gaussian,
the forecast has a normal distribution. If the patent data were just counts, a
hierarchical model with Poisson distribution for the response variable would
be required. Having said that we emphasize that the patents in our dataset are
fractional so we use vanilla LG-SSM.

###  3.4 Bayesian inference

In order to fit LG-SSM to the patent data of interest, we follow a Bayesian
approach that captures uncertainty via a posterior distribution as deeply
described by [Greenberg, 2012, van de Schoot et al., 2021]. For Bayesian
inference in LG-SSM a joint prior distribution is to be specified for the
initial states and the unknown model parameters.

The prior for the local level diffusion scale ($\sigma_{l}$) is inverse Gamma
interpreted as a scaled inverse-$\chi^{2}$ distribution (as described by
[Gelman et al., 2013]) with standard deviation $0.1\hat{\sigma}_{y}$ and 32
degrees of freedom, where $\hat{\sigma}_{y}$ is standard deviation of the time
series. The prior is truncated to maximum value at $\hat{\sigma}_{y}$. Initial
state prior is Normal with mean equal to the first observation $y_{0}$ of the
time series and scale equal to $\hat{\sigma}_{y}$. For the regression we used
Zellner’s g-prior with spike-and-slab. The prior for $\bm{\beta}$ and
$\sigma_{y}$ is commonly expressed in terms of expected model size (in our
case equal to 3), expected $R^{2}$ (0.8) and degrees of freedom (50). Spike-
and-slab prior enforces sparsity of $\bm{\beta}$, i.e. most of the values will
be exactly 0. Such a prior acts as a Bayesian model selection algorithm and
allows the addition of multiple covariates in the model without overfitting.
Only the most important covariates get nonzero coefficient thus they are
present in the model. All the other variables are multiplied by zero so they
are not part of the model. The hyperparameters for these distributions are set
using heuristics from _CausalImpact_ package that aim to provide default prior
distributions. In order to sample from the posterior distribution we apply
Monte Carlo Markov Chain (MCMC) [Murphy, 2012].

##  4 Results

As Figure 1 shows, the number of patents (as a sum in years 1985-2017) among
the EU member states varies greatly between countries. For better
visualization we present EU-13 and EU-15 in logarithmic scale (maximum number
of patents for country from EU-15 (Germany) was above 645,912 while in EU-13
(Poland) above 5,000). The greatest number of patents from EU-15 was assigned
to Germany, followed by France, United Kingdom, Italy and the Netherlands. The
highest number of patents from the EU-13 was assigned to Poland, Hungary and
Czech Republic. Also, the fraction of countries from the EU-13 (when counting
the total number of patents) was negligible compared to the EU-15. In total,
98.66% (1,462,149.9) of patents during the analyzed period were allocated in
fifteen EU-15 countries while in the EU-13 countries only 1.34% (19,783.4)
were allocated. This difference represents a huge dominance of the EU-15 in a
number of patents. Even accounting for the fact that countries from the EU-13
account for approximately 20% of the EU population, it is still a negligible
amount. Interestingly from a visual perspective (considering two periods
before and after accession), we could observe that the fraction of patents
after accession is higher (points above diagonal line in Figure 1) in the
EU-13, while for the EU-15 there was no big change (corresponding points lay
nearly diagonal line).

Figure 1: Number of patents before and after accession with the percentage of
increase for a given country. The diagonal line marks zero change due to
accession.

Simple analysis of total number of patents before accession and after
accession in terms of percentage increase reveals the highest increase of
patents occurred in EU-13 countries suggesting that EU accession affected
patents performance in those countries see Figure 1. The decrease was noticed
(points lying below the diagonal line) for Bulgaria (-17) and United Kingdom
(-5.4). The result for Croatia is incomparable because of an uneven period
1985- 2012 versus 2013-2017. The lowest increase was observed for Germany
(6.8), Italy (8.5) and France (9.8). Whereas the number of patents increased
over the two periods especially in Estonia (+530), Poland (+512) and Lithuania
(+500). In general the highest increase of patents performance during the
period considered was in the EU-13 and the lowest in EU-15 countries. However,
from this visual examination, it cannot be inferred with statistical precision
if the growth can be explained by the overall increasing trend over the last
decades or whether joining the EU caused the changes.

Thus, we apply a formal statistical causal modeling approach which allows us
to analyze whether EU membership has led to a statistically significant growth
in the number of patents in EU-13 countries compared to the counterfactual
scenario that each country had not accessioned the EU. We rely on the patent
dynamics before accession and the number of patents in the EU-15 that received
no treatment, for obtaining accurate predictions for the counterfactual
scenario. Summaries of the analysis obtained for 20,000 MCMC samples is
collected in Table 1.

The rows with _Stat_ Average (Avg.) refer to the average patents (over the
time period) during the post-intervention period. The rows with _Stat_
Cumulative (Cum.) represent the sum of the individual yearly observations. The
interpretation of our results for a given country e.g. Poland (which could be
generalized for the rest of the countries- accordingly to their values) is as
follows. In the post-intervention period, the response variable had an average
value of approx. 312.2 as reported in column _Actual_. By contrast, in the
absence of an intervention, we would have expected an average response of
101.1 (see: column _Val_). The 95% interval of this counterfactual prediction
is [22.4 - 228.6 ] (see: columns _lower_ and _upper_ in _Prediction_ group).
Subtracting this prediction from the observed response yields an estimate of
the causal effect that the intervention had on the response variable. The same
interpretation applies to entries in a row (_Cum_). In relative terms, the
response variable showed an increase of +212% (see column _Val_ in _Relative_
group). The 95% interval of this percentage is [+84%, +290%] (respectively
columns _lower_ and _upper_ in _Relative_ group ) means that the positive
effect observed during the intervention period is statistically significant.
Note that this increase is the causal effect relative to a predicted
counterfactual value, not to be confused with the percentage from Figure 1.

Statistical significance of the analysis can be assessed from the last column
_p_. The effect observed during the intervention period is statistically
significant and unlikely to be due to random fluctuations if the probability
of obtaining this effect by chance is very small (Bayesian one-sided tail-area
probability _p_ ). For countries where $p<0.05$ (posterior prob. of a causal
effect equal $1-p>=95\%$) the causal effect can be considered statistically
significant, otherwise the effect may be spurious and would generally not be
considered statistically significant. A significant causal impact on patents
performance after accession can be observed in Romania, Estonia, Poland, Czech
Republic, Croatia and Lithuania. However, as results from analysis show for
Croatia and Lithuania, joining the EU caused a negative impact for patents
performance. For the remaining countries: Hungary, Latvia, Cyprus, Slovakia,
Slovenia, Bulgaria and Malta joining the EU did not statistically
significantly affect patents performance. An interesting result is observed
for Latvia where the percentage change increase (see Figure 1) could
misleadingly suggest significant influence after accession (above 300%
increase after accession) however, model prediction classified this increase
as statistically not significant. It means that this increase came from trends
before accession, so such growth is not influenced by EU Membership.

Table 1: Summaries of causal impact analysis | Prediction$\pm$95% CI | Relative $\pm$95% CI |
---|---|---|---
| Stat | Actual | Val | lower | upper | Val | lower | upper | p
RO | Avg | $54.0$ | $29.3$ | $20.6$ | $38.5$ | $85\%$ | $53\%$ | $114\%$ | $0.000$
RO | Cum | $540.0$ | $292.6$ | $205.6$ | $385.2$ | $85\%$ | $53\%$ | $114\%$ | $0.000$
EE | Avg | $30.4$ | $13.9$ | $7.1$ | $23.5$ | $118\%$ | $50\%$ | $168\%$ | $0.000$
EE | Cum | $394.9$ | $180.9$ | $91.7$ | $305.1$ | $118\%$ | $50\%$ | $168\%$ | $0.000$
PL | Avg | $312.2$ | $100.1$ | $22.4$ | $228.6$ | $212\%$ | $84\%$ | $290\%$ | $0.001$
PL | Cum | $4,058.5$ | $1,301.0$ | $291.7$ | $2,971.3$ | $212\%$ | $84\%$ | $290\%$ | $0.001$
CZ | Avg | $229.6$ | $126.3$ | $20.5$ | $223.1$ | $82\%$ | $5\%$ | $166\%$ | $0.018$
CZ | Cum | $2,984.7$ | $1,641.5$ | $266.9$ | $2,900.6$ | $82\%$ | $5\%$ | $166\%$ | $0.018$
HR | Avg | $19.2$ | $27.9$ | $18.0$ | $36.7$ | $-31\%$ | $-63\%$ | $5\%$ | $0.040$
HR | Cum | $77.0$ | $111.8$ | $71.8$ | $147.0$ | $-31\%$ | $-63\%$ | $5\%$ | $0.040$
LT | Avg | $20.1$ | $33.3$ | $17.3$ | $46.2$ | $-40\%$ | $-78\%$ | $9\%$ | $0.047$
LT | Cum | $261.7$ | $432.8$ | $224.7$ | $601.2$ | $-40\%$ | $-78\%$ | $9\%$ | $0.047$
HU | Avg | $183.8$ | $96.1$ | $-105.0$ | $224.1$ | $91\%$ | $-42\%$ | $300\%$ | $0.071$
HU | Cum | $2,388.9$ | $1,249.8$ | $-1,365.3$ | $2,913.9$ | $91\%$ | $-42\%$ | $300\%$ | $0.071$
LV | Avg | $14.1$ | $9.5$ | $4.3$ | $18.2$ | $48\%$ | $-43\%$ | $102\%$ | $0.115$
LV | Cum | $183.3$ | $123.8$ | $56.5$ | $236.7$ | $48\%$ | $-43\%$ | $102\%$ | $0.115$
CY | Avg | $8.8$ | $6.3$ | $1.7$ | $11.0$ | $40\%$ | $-35\%$ | $113\%$ | $0.133$
CY | Cum | $114.5$ | $81.8$ | $21.8$ | $142.9$ | $40\%$ | $-35\%$ | $113\%$ | $0.133$
SK | Avg | $43.3$ | $33.3$ | $15.7$ | $55.5$ | $30\%$ | $-37\%$ | $83\%$ | $0.217$
SK | Cum | $562.9$ | $433.0$ | $204.7$ | $721.1$ | $30\%$ | $-37\%$ | $83\%$ | $0.217$
SI | Avg | $101.1$ | $93.9$ | $53.8$ | $139.4$ | $8\%$ | $-41\%$ | $50\%$ | $0.353$
SI | Cum | $1,314.5$ | $1,221.0$ | $699.9$ | $1,812.2$ | $8\%$ | $-41\%$ | $50\%$ | $0.353$
BG | Avg | $27.4$ | $24.8$ | $-7.0$ | $72.1$ | $11\%$ | $-181\%$ | $139\%$ | $0.389$
BG | Cum | $273.7$ | $247.7$ | $-70.3$ | $721.2$ | $11\%$ | $-181\%$ | $139\%$ | $0.389$
MT | Avg | $9.7$ | $9.6$ | $4.2$ | $16.4$ | $1\%$ | $-70\%$ | $57\%$ | $0.445$
MT | Cum | $125.9$ | $124.9$ | $54.0$ | $213.2$ | $1\%$ | $-70\%$ | $57\%$ | $0.445$

A detailed time evolution of the inferred causal impact can be observed in
Figures 3 and 4 for both groups of countries: with respectively significant
and insignificant effect of EU accession on patents performance. Each plot
consist of three separate charts:

original)

The observed time series (solid line) and fitted model’s forecast
counterfactuals (dashed line) with 95% credible interval (shaded area).

pointwise)

The point-wise causal effect with 95% credible interval, as estimated by the
model. This is the difference between the observed outcome and the predicted
outcome.

cumulative)

The cumulative effect with 95% credible interval – the total number of patent
due to EU accession.

The vertical line represents EU accession year (intervention) for the
particular country (2004/2007/2013). Further information that can be obtained
from the plots include: (original) the detailed time distribution (trajectory)
of EU-13 countries. (pointwise) The difference between observed data and
counterfactual predictions is the inferred causal impact of the intervention.
A key characteristic of the inferred impact series is the progressive widening
of the posterior intervals (shaded area). This effect emerges naturally from
the model structure that predictions should become increasingly uncertain as
we look further and further into the future. (cumulative) Another way of
visualizing posterior inferences is by means of a cumulative impact plot. It
shows, for each year, the summed effect up to that year. If the 95% credible
interval of the cumulative impact crosses the zero-line after the
intervention, then at that point we would no longer declare a significant
overall effect.

The same analysis performed for combined EU-13 countries shows significant
positive impact of accession patent performance. All EU-13 countries in total
have produced on average over 7,000 patents that can be attributed to the
accession. However, surprisingly 7 of the EU-13 countries have not
significantly benefited from the EU accession.

An interesting by-product of the causal analysis reported in this paper is a
discovery of international relations in patent time series. Since coefficients
in the linear regression are sparse due to spike-and-slab prior, only the most
important covariates get nonzero coefficient. In the model, the covariates for
each of the EU-13 country are the patents for the EU-15 countries. Let us
consider two countries $c\in\text{EU-15}$ and $c^{\prime}\in\text{EU-13}$. If
the coefficient $\beta_{c}$ is zero, it means that patent activity of country
$c$ does not explain the trend observed for country $c^{\prime}$. Contrary, a
nonzero value of $\beta_{c}$ indicates that both countries have a similar
dynamic of patents. The similarity may originate from different socio-
political interactions whose analysis is far beyond the scope of this paper.
We only point out the existence of those pan-European interactions

Figure 2: Fraction of nonzero regression coefficients

In Figure 2 we visualized what fraction of 20,000 MCMC samples resulted in
nonzero coefficient (estimate of slab probability). The $x$ axis represents
covariates ($c$), while $y$ axis corresponds to the new EU members
($c^{\prime}$). Interestingly Luxembourg and Sweden participate very weakly in
the model, while countries like Italy or Germany are frequently selected by
the Bayesian model selection.

On the other hand, Hungary’s patents are mostly explained solely by the time
series for the Netherlands. Similarly, the patent activity of Romania is
mostly explained by data from Spain. A deeper analysis of such relations could
be helpful for policymakers in designing targeted research funds aimed at
equalizing cooperation among the EU.

Figure 3: Significant causal impact

(a) Romania (b) Estonia

(c) Poland (d) Czech Republic

(e) Croatia (f) Lithuania

Figure 4: Insignificant causal impact

(a) Hungary (b) Latvia

(c) Cyprus (d) Slovak Republic

(e) Slovenia (f) Bulgaria

Figure 4: Insignificant causal impact (cont) (a) Malta

##  5 Conclusion

One of the most important initiatives defined by the EU in its Europe 2020
Strategy was to create an innovation-friendly environment that supports the
generation, emergence, and diffusion of innovations. Taking it into account,
it is worth highlighting that patents play an increasingly important role in
innovation and economic performance. In an increasingly knowledge-driven
economy, society invariably needs creative or inventive ideas or concepts to
improve existing features, add useful new features to products or develop new
ones. Patents are widely used as an indicator of how much innovation is taking
place in a given industry. Patents are also one output of technologically
successful R&D activities and a long-term investment. Patenting has
experienced a sizeable boom in the last two decades in many fields. Despite
the importance of innovation in the knowledge-driven economy it is also worth
highlighting that significant disparities of innovation occur between EU
member states, in particular between EU-15 and EU-13. In terms of Summary
Innovation Index, the majority of EU-13 are below the EU average. Our analysis
also clearly showed that the total number of patents in the years 1985-2017
among the EU member states varies greatly between countries - up to few orders
of magnitude. From the analysis a manifestation of the Pareto principle is
observed i.e. most patents are issued by only a few countries. Accordingly,
the greatest number of patents was assigned in Germany, followed by France,
United Kingdom and Italy. Although, EU Membership could bring benefits for
member states (e.g. boosted trade in goods or non-tariff trade cost) there are
different views on the EU and its influence on member countries. Hence, it is
crucial to have a reliable and quantitative assessment of how much the
membership influences member states. Influence of EU enlargement with causal
analysis on innovations could be identified in very few scientific
publications. In our research we have examined the dynamics of patents
performance in the EU-13 over time, in particular the impact of EU-13
accession on their patents performance. The proposed approach, based on causal
impact using a Bayesian structural time-series model, indicates a conclusion
that joining the EU has brought a statistically significant impact on patents
performance in Romania, Estonia, Poland, Czech Republic, Croatia and
Lithuania. Having said that, it is worth noting that in Croatia and Lithuania
the effects were negative. For the remaining countries: Hungary, Latvia,
Cyprus, Slovakia, Slovenia, Bulgaria and Malta we did not find any evidence
that joining to the EU has significantly affected their patents performance.

To achieve a high patents performance, countries need an innovation system
based on a research system, human resources as well as finance and support. EU
member states contribute to the Union’s common budget, but member states also
receive money back in various forms. However, EU-13 are less effective in
getting research funding (e.g. Horizon 2020 or European Research Council)
which could also affect patenting. The presented results could be a valuable
input for a policy-oriented discussion on innovation, particularly related to
patents performance in each EU country. The limitation of our analysis is the
dataset because data were available only until 2017. A more enhanced database
could enable a causal impact model of the changes of patents cooperation
between EU countries after and before accession. This could provide further
useful information. Nevertheless, the presented approach which utilised
Google’s Causal Impact could be widely used for the assessment of many aspects
of EU-13 accession or any other political event. Results from our analysis
could be a valuable source of assessment of European enlargement on patents
performance and provide further helpful information for policymakers in
reforming government subsidies concerning innovations in the New Member States
of the EU.

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