(c) – [boson, edge label=\( W^+\)] (d),
(d) – [anti fermion, edge label=\(\ell\)] (b),
(c) –  (e),
(f) –  (d)

(a) at (0,0) \(Z\);
(b) at (1,0);
(c) at (2,1.);
(d) at (2,-1.);
(e) at (3,1.) \( n_a\);
(f) at (3,-1.) \( n_b\);
(a) – [boson] (b),
(b) – [boson, edge label=\( W^-\)] (c),
(c) – [anti fermion, edge label=\(\ell\)] (d),
(d) – [boson, edge label=\( W^+\)] (b),
(c) –  (e),
(f) –  (d)

(a) at (0,0) \(Z\);
(b) at (1,0);
(c) at (2,1.);
(d) at (2,-1.);
(e) at (3,1.) \( n_a\);
(f) at (3,-1.) \( n_b\);
(a) – [boson] (b),
(b) – [boson, edge label=\( W^+ \)] (c),
(c) – [fermion, edge label=\(\ell\)] (d),
(d) – [boson, edge label=\( W^-\)] (b),
(c) –(e),
(f) –  (d)

(a) at (0,0) \(Z\);
(b) at (1,0);
(c) at (2,1.);
(d) at (2,-1.);
(e) at (3,1.) \( n_a\);
(f) at (3,-1.) \( n_b\);
(a) – [boson] (b),
(b) – [boson, edge label=\( Z\)] (c),
(c) – [anti fermion, edge label=\(n_i\)] (d),
(d) – [scalar, edge label=\( h\)] (b),
(c) –  (e),
(f) –  (d)

(a) at (0,0) \(Z\);
(b) at (1,0);
(c) at (2,1.);
(d) at (2,-1.);
(e) at (3,1.) \( n_a\);
(f) at (3,-1.) \( n_b\);
(a) – [boson] (b),
(b) – [scalar, edge label=\( h \)] (c),
(c) – [anti fermion, edge label=\(n_i\)] (d),
(d) – [boson, edge label=\( Z\)] (b),
(c) – (e),
(f) – (d)

Feynman diagrams contributing to the invisible $Z$ decays (in unitary gauge).

In Fig. <ref> we display the diagrams contributing to the invisible $Z$ decays, $Z \to n_a n_b$ (with $a=b$ or $a\neq b$).
Below we present the vector form factors contributing to the invisible $Z$-decays (we do not give the associated tensor and scalar form factors since the former are negligible and the latter vanish for on-shell decays); the superscript labels (a)-(f) refer to the topologies  depicted in Fig. <ref>.

\begin{eqnarray}
F_L^{V \, (a)} &=& \dfrac{g_w^3 }{16\pi^2 8 c_w^3 M_W^2 } \sum_{i,j}
\bigg\{
C_{ai}^* \bigg[C_{ij} \big[C_{jb} m_a m_i \bigg(c_w^2 m_b^2 (C_{22} +2  C_{12}+ C_{11})
+2  M_W^2 (C_{0}+C_{2}+ C_{1})
\nonumber\\ &\phantom{=}&
+ \left(m_b^2-m_j^2\right) c_w^2 (C_{2}+ C_{1})\bigg)
+C_{jb}^* m_a m_b m_i m_j c_w^2 \left( C_{22} +2  C_{12}+ C_{11}\right)\big]
\nonumber\\ &\phantom{=}&
+C_{ij}^* \big[C_{jb}^* m_a m_b\left( (D-2)c_w^2 C_{00}
- (D-2)M_W^2 (C_{0}+ 2 C_{1} +2 C_{2} +C_{11}+C_{22} + 2 C_{12})
- q^2 c_w^2 C_{12}\right)
\nonumber\\ &\phantom{=}&
+C_{jb} m_a m_j \left((D-2)c_w^2 C_{00}
-(D-2)  M_W^2 ( C_{0}+ C_{1}+ C_{2})- q^2 c_w^2 C_{12}
\right)\big]\bigg]
\nonumber\\ &\phantom{=}&
+C_{ai} \bigg[C_{ij}^* \big[C_{jb}^* m_b m_i
\left((D-2)c_w^2 C_{00}
-(D-2)M_W^2 (C_{0}+ C_{1}+ C_{2}) - q^2 c_w^2 C_{12}  \right)
\nonumber\\ &\phantom{=}&
+C_{jb} m_i m_j\left((D-2) c_w^2 C_{00} - (D-2) M_W^2 C_{0} - q^2 c_w^2 C_{12}\right)\big]
\nonumber\\ &\phantom{=}&
+C_{ij} \big[C_{jb}^* m_b m_j \left(
c_w^2  m_a^2 (C_{1} + C_{2} + C_{11} + C_{22} + 2 C_{12})
+2  M_W^2 (C_{0} +C_{1} + C_{2})
-m_i^2 c_w^2(C_{1} + C_{2}) \right)
\nonumber\\ &\phantom{=}&
+C_{jb} \bigg(\left(c_w^2 m_b^2+M_W^2 (D-2)\right) C_{11} m_a^2
-M_W^2 \left(2 B_0^{aZi}+2 B_0^{bZj}+(D-6) B_0^{qij}\right)
\nonumber\\ &\phantom{=}&
+\left(\left(m_a^2-m_i^2\right) \left(m_b^2-m_j^2\right) c_w^2+M_W^2 \left(2 m_a^2+2 m_b^2-2 m_i^2-2 m_j^2-(D+6) M_Z^2+2 q^2\right)\right) C_{0}
\nonumber\\ &\phantom{=}&
+\left(2 \left(m_a^2+3 m_b^2\right) M_W^2-c_w^2 \left(\left(m_j^2-2 m_b^2\right) m_a^2+m_b^2 m_i^2\right)\right) C_{2}
+m_b^2 \left(c_w^2 m_a^2+M_W^2 (D-2)\right) C_{22}
\nonumber\\ &\phantom{=}&
+\left(2 \left(3 m_a^2+m_b^2\right) M_W^2-c_w^2 \left(\left(m_j^2-2 m_b^2\right) m_a^2+m_b^2 m_i^2\right)\right) C_{1}
\nonumber\\ &\phantom{=}&
+\left(2 c_w^2 m_a^2 m_b^2+\left(m_a^2+m_b^2\right) M_W^2 (D-2)\right) C_{12}+2 M_W^2 (D-2) C_{00}\bigg)\big]\bigg]
\bigg\}\,,
\end{eqnarray}

\begin{eqnarray}
F_R^{V \, (a)} &=& \dfrac{g_w^3 }{16\pi^2 8 c_w^3 M_W^2 } \sum_{i,j}
\bigg\{
C_{ai} \bigg[C_{ij}^* \big[
-C_{jb} m_a m_b m_i m_j c_w^2\left(C_{11} + C_{22}+2  C_{12}\right)
\nonumber\\ &\phantom{=}&
+C_{jb}^* m_a m_i
\big(-c_w^2 m_b^2 (C_{1}+ C_{2}+ C_{22}+ C_{11}+2 C_{12} ) -2  M_W^2 (C_{0}+C_{1}+ C_{2})
+c_w^2 m_j^2(C_{1}+ C_{2})\big)
\big]
\nonumber\\ &\phantom{=}&
+C_{ij} \big[C_{jb} m_a m_b\left(q^2 c_w^2 C_{12} -(D-2)  C_{00} c_w^2
+  (D-2)M_W^2 (C_{0}+2C_{2}+2C_{1}+ C_{11} + C_{22} +2 C_{12})\right)
\nonumber\\ &\phantom{=}&
+C_{jb}^* m_a m_j\left( q^2 c_w^2 C_{12} -(D-2) c_w^2 C_{00} + (D-2) M_W^2 (C_{0}+C_{2}+ C_{1})\right)\big]\bigg]
\nonumber\\ &\phantom{=}&
+C_{ai}^* \bigg[C_{ij} \big[C_{jb} m_b m_i \left( q^2 C_{12} c_w^2-(D-2) c_w^2 C_{00} + (D-2) M_W^2 (C_{0}+ C_{2}+ C_{1})\right)
\nonumber\\ &\phantom{=}&
+C_{jb}^* m_i m_j\left( q^2 C_{12} c_w^2-(D-2)c_w^2 C_{00} +  (D-2) M_W^2C_{0}\right)\big]
\nonumber\\ &\phantom{=}&
+C_{ij}^* \big[C_{jb} m_b m_j \left(-c_w^2  m_a^2 (C_{1}+ C_{2}+ C_{11}+ C_{22} +2 C_{12}) -2  M_W^2 (C_{0}+C_{1}+ C_{2}) + c_w^2 m_i^2 (C_{1}+ C_{2}) \right)
\nonumber\\ &\phantom{=}&
+C_{jb}^* \big(M_W^2 \left(2 B_0^{aZi}+2 B_0^{bZj}+(D-6) B_0^{qij}\right) -\left(c_w^2 m_b^2+M_W^2 (D-2)\right) C_{11} m_a^2
\nonumber\\ &\phantom{=}&
+\left(M_W^2 \left(-2 m_a^2-2 m_b^2+2 m_i^2+2 m_j^2+(D-6) M_Z^2-2 q^2\right)-c_w^2 \left(m_a^2-m_i^2\right) \left(m_b^2-m_j^2\right)\right) C_{0}
\nonumber\\ &\phantom{=}&
+\left(c_w^2 \left(\left(m_j^2-2 m_b^2\right) m_a^2+m_b^2 m_i^2\right)-2 \left(m_a^2+3 m_b^2\right) M_W^2\right) C_{2}
\nonumber\\ &\phantom{=}&
-m_b^2 \left(c_w^2 m_a^2+M_W^2 (D-2)\right) C_{22}+\left(c_w^2 \left(\left(m_j^2-2 m_b^2\right) m_a^2+m_b^2 m_i^2\right)-2 \left(3 m_a^2+m_b^2\right) M_W^2\right) C_{1}
\nonumber\\ &\phantom{=}&
+\left(-2 c_w^2 m_a^2 m_b^2-(D-2) \left(m_a^2+m_b^2\right) M_W^2\right) C_{12}-2 (D-2) M_W^2 C_{00}\big)\big]\bigg]
\bigg\}\,,
\end{eqnarray}

with the Passarino-Veltman functions $C_{rs}= C_{rs}(m_a^2, q^2, m_b^2, M_Z^2, m_i^2, m_j^2)$.

\begin{eqnarray}
F_L^{V \, (b)} &=& \dfrac{g_w^3 }{16\pi^2 8 c_w^3 M_W^2 } \sum_{i,j}
\bigg\{
C_{ai}^* \bigg[C_{ij} \big[
C_{jb}^* m_a m_b m_i m_j c_w^2 \left(4 C_{0} +4  C_{1} +4  C_{2}  + C_{11}+ C_{22} +2  C_{12} \right)
\nonumber\\ &\phantom{=}&
m_a m_i c_w^2 \left(m_j^2 (2C_{0}+ C_{1} + C_{2} )
+m_b^2 (2C_{0} +3C_{1} + 3C_{2}+ C_{11} + C_{22} +2 C_{12} )\right)\big]
\nonumber\\ &\phantom{=}&
+C_{ij}^* \big[C_{jb}^*  m_a m_b c_w^2 \left( (D-2)C_{00}- q^2 C_{12}\right)
+C_{jb} m_a m_j c_w^2 \left( (D-2) C_{00}-q^2 C_{12}\right)\big]\bigg]
\nonumber\\ &\phantom{=}&
+C_{ai} \bigg[C_{ij} \big[C_{jb} c_w^2 \big(
m_a^2 m_b^2 (C_{0} +2C_{1} +2C_{2} + C_{11}+ C_{22} +2C_{12})
\nonumber\\ &\phantom{=}&
+ \left(m_i^2m_b^2+ m_a^2m_j^2\right) ( C_{0} + C_{1}+ C_{2} ) +m_i^2m_j^2 C_{0}\big)
\nonumber\\ &\phantom{=}&
+C_{jb}^* m_b m_j c_w^2\big( m_i^2(2 C_{0} +  C_{1}+  C_{2})    +m_a^2   (2C_{0} + 3C_{1}+3C_{2} + C_{11}+C_{22} +2C_{12}) \big)\big]
\nonumber\\ &\phantom{=}&
+C_{ij}^* \big[C_{jb}^* m_b m_i c_w^2\left((D-2) C_{00}- q^2 C_{12}\right)
+C_{jb} m_i m_j c_w^2  \left((D-2) C_{00}- q^2 C_{12}\right)\big] \bigg]
\bigg\}\,,
\end{eqnarray}

\begin{eqnarray}
F_R^{V \, (b)} &=& \dfrac{g_w^3 }{16\pi^2 8 c_w^3 M_W^2 } \sum_{i,j}
\bigg\{
C_{ai} \bigg[C_{ij}^* \big[
-C_{jb} m_a m_b m_i m_j c_w^2 \left(4  C_{0}+4 C_{1} +4  C_{2} +C_{11} + C_{22} +2 C_{12}  \right)
\nonumber\\ &\phantom{=}&
-C_{jb}^* m_a m_i c_w^2 \left(m_j^2 (2C_{0}+C_{1}+ C_{2}) + m_b^2 (2C_{0} +3C_{1}+3C_{2} + 2 C_{12} + C_{11} +C_{22})\right)
\big]
\nonumber\\ &\phantom{=}&
+C_{ij} \big[C_{jb} m_a m_b c_w^2\left(q^2 C_{12}- (D-2)  C_{00}\right)+C_{jb}^* m_a m_j c_w^2\left( q^2 C_{12}- (D-2)  C_{00}\right)\big]\bigg]
\nonumber\\ &\phantom{=}&
+C_{ai}^* \bigg[C_{ij}^* \big[C_{jb}^*
- m_a^2 m_b^2 (C_{0} +2C_{1} +2C_{2}+C_{11} +C_{22}+ 2  C_{12})
\nonumber\\ &\phantom{=}&
-\left(m_j^2m_a^2 +m_b^2 m_i^2\right) (C_{0}+ C_{1} +C_{2}) - m_i^2m_j^2  C_{0}\big)
\nonumber\\ &\phantom{=}&
+C_{jb} m_b c_w^2\left(- m_i^2m_j   (2C_{0} +C_{1} +C_{2})
- m_a^2  m_j (2C_{0}+ 3C_{1} +3C_{2}+C_{11}+C_{22} +2 C_{12})
\right)\big]
\nonumber\\ &\phantom{=}&
+C_{ij} \big[C_{jb}  m_b m_i c_w^2\left(q^2 C_{12}- (D-2)  C_{00}\right)+C_{jb}^* m_i m_j c_w^2\left( q^2 C_{12}-(D-2)  C_{00}\right)\big]\bigg]
\bigg\}\,,
\end{eqnarray}

where $ C_{rs}= C_{rs}(m_a^2, q^2, m_b^2, M_H^2, m_i^2, m_j^2)$.

\begin{eqnarray}
F_L^{V \, (c)} &=& \dfrac{g_w^3 }{16\pi^2 2 c_w M_W^2 } \sum_{\rho}
\bigg\{
m_a m_b \mathcal{U}_{\rho a} \mathcal{U}_{\rho b}^* \bigg[
-2 (g_A+g_V) C_{00} +(g_A +g_V) B_0
\nonumber\\ &\phantom{=}&
-\left(g_A \left(m_b^2-m_\rho^2+(D-2) M_W^2\right)+g_V \left(m_b^2+m_\rho^2+(D-2) M_W^2\right)\right) C_{22}
\nonumber\\ &\phantom{=}&
-\left(g_A \left(m_a^2+m_b^2-2 m_\rho^2+2 (D-2) M_W^2\right)+g_V \left(m_a^2+m_b^2+2 m_\rho^2+2(D-2) M_W^2\right)\right)C_{12}
\nonumber\\ &\phantom{=}&
-\left(g_A \left(m_a^2-m_\rho^2+(D-2) M_W^2\right)+g_V \left(m_a^2+m_\rho^2+(D-2) M_W^2\right)\right) C_{11}
\nonumber\\ &\phantom{=}&
-(D-3) M_W^2 (g_A+g_V) C_{0}-2 (D-2) M_W^2 (g_A+g_V) (C_{1}+C_{2})
\bigg]
\nonumber\\ &\phantom{=}&
+\mathcal{U}_{\rho a}^* \mathcal{U}_{\rho b}
\bigg[ -2 M_W^2 (g_A+g_V) (B_0^{a}+B_0^{b})
\nonumber\\ &\phantom{=}&
+\big((g_A -g_V)m_\rho^2 - (g_A + g_V)(D-6) M_W^2 \big) B_0
\nonumber\\ &\phantom{=}&
+ m_a^2 \left(g_A \left(m_b^2-m_\rho^2+(D-2) M_W^2\right)+g_V \left(m_b^2+m_\rho^2+(D-2) M_W^2\right)\right) C_{11}
\nonumber\\ &\phantom{=}&
+m_b^2 \left(g_A \left(m_a^2-m_\rho^2+(D-2) M_W^2\right)+g_V \left(m_a^2+m_\rho^2+(D-2) M_W^2\right)\right) C_{22}
\nonumber\\ &\phantom{=}&
+\big(g_A [m_a^2 \left(2 m_b^2-m_\rho^2+(D-2) M_W^2\right)- m_b^2 \left(m_\rho^2-(D-2) M_W^2\right)]
\nonumber\\ &\phantom{=}&
+ g_V [m_a^2 \left(2 m_b^2+m_\rho^2+(D-2) M_W^2\right)+m_b^2 \left(m_\rho^2+(D-2) M_W^2\right)]\big)C_{12}
\nonumber\\ &\phantom{=}&
+2 \left(-g_A (m_\rho^2- (D-2) M_W^2) +g_V \left(m_\rho^2+(D-2) M_W^2\right)\right) C_{00}
\nonumber\\ &\phantom{=}&
+(g_A+g_V) \left(m_a^2 \left(2 m_b^2-m_\rho^2+2 M_W^2\right)-m_b^2 \left(m_\rho^2-6 M_W^2\right)\right) C_{2}
\nonumber\\ &\phantom{=}&
+(g_A+g_V) \left(m_a^2 \left(2 m_b^2-m_\rho^2+6 M_W^2\right)-m_b^2 \left(m_\rho^2-2 M_W^2\right)\right) C_{1}
\nonumber\\ &\phantom{=}&
+C_{0} \big( -g_A (D+1) m_\rho^2 M_W^2 +g_V (D-7) m_\rho^2 M_W^2
\nonumber\\ &\phantom{=}&
+(g_A+g_V) \big[
m_\rho^2(m_\rho^2-m_a^2 - m_b^2)
+ 2 M_W^2( m_a^2 + m_b^2 + q^2)
+ m_a^2 m_b^2
-(D-6) M_W^4 \big]\big)
\bigg]
\bigg\}\,, \nonumber \\
\end{eqnarray}

\begin{eqnarray}
F_R^{V \, (c)} &=& \dfrac{g_w^3 }{16\pi^2 2 c_w M_W^2 } \sum_{\rho}
\bigg\{
m_a m_b \mathcal{U}_{\rho a}^* \mathcal{U}_{\rho b}
\bigg[-(g_A +g_V) B_0
\nonumber\\ &\phantom{=}&
+\left(g_A \left(m_b^2-m_\rho^2+(D-2) M_W^2\right)+g_V \left(m_b^2+m_\rho^2+(D-2) M_W^2\right)\right) C_{22}
\nonumber\\ &\phantom{=}&
+\left(g_A \left(m_a^2+m_b^2-2 m_\rho^2+2 (D-2) M_W^2\right)+g_V \left(m_a^2+m_b^2+2 m_\rho^2+2 (D-2) M_W^2\right)\right)C_{12}
\nonumber\\ &\phantom{=}&
+\left(g_A \left(m_a^2-m_\rho^2+(D-2) M_W^2\right)+g_V \left(m_a^2+m_\rho^2+(D-2) M_W^2\right)\right) C_{11}
\nonumber\\ &\phantom{=}&
+(D-3) M_W^2 (g_A+g_V) C_{0}+2 (D-2) M_W^2 (g_A+g_V) (C_{1}+C_{2}) +2 (g_A+g_V) C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+\mathcal{U}_{\rho a} \mathcal{U}_{\rho b}^*
\bigg[
-m_a^2 \left(g_A \left(m_b^2-m_\rho^2+(D-2) M_W^2\right)+g_V \left(m_b^2+m_\rho^2+(D-2) M_W^2\right)\right) C_{11}
\nonumber\\ &\phantom{=}&
-m_b^2 \left(g_A \left(m_a^2-m_\rho^2+(D-2) M_W^2\right)+g_V \left(m_a^2+m_\rho^2+(D-2) M_W^2\right)\right) C_{22}
\nonumber\\ &\phantom{=}&
-\big(g_A m_a^2 \left(2 m_b^2-m_\rho^2+(D-2) M_W^2\right)-g_A m_b^2 \left(m_\rho^2-(D-2) M_W^2\right)
\nonumber\\ &\phantom{=}&
+g_V m_a^2 \left(2 m_b^2+m_\rho^2+(D-2) M_W^2\right)+g_V m_b^2 \left(m_\rho^2+(D-2) M_W^2\right)\big)C_{12}
\nonumber\\ &\phantom{=}&
+2 \left(g_A \left(m_\rho^2-(D-2) M_W^2\right)-g_V \left(m_\rho^2+(D-2) M_W^2\right)\right) C_{00}
\nonumber\\ &\phantom{=}&
-(g_A+g_V) \left(m_a^2 \left(2 m_b^2-m_\rho^2+2 M_W^2\right)-m_b^2 \left(m_\rho^2-6 M_W^2\right)\right) C_{2}
\nonumber\\ &\phantom{=}&
-(g_A+g_V) \left(m_a^2 \left(2 m_b^2-m_\rho^2+6 M_W^2\right)-m_b^2 \left(m_\rho^2-2 M_W^2\right)\right) C_{1}
\nonumber\\ &\phantom{=}&
+\big((g_A+g_V) \left(m_a^2 \left(-m_b^2+m_\rho^2-2 M_W^2\right)+m_b^2 \left(m_\rho^2-2 M_W^2\right)
-m_\rho^4 +(D-6) M_W^4 -2 M_W^2 q^2\right)
\nonumber\\ &\phantom{=}&
+g_A (D+1) m_\rho^2 M_W^2  -g_V (D-7) m_\rho^2 M_W^2
\big)C_{0}
\nonumber\\ &\phantom{=}&
+2 M_W^2 (g_A+g_V) (B_0^{a}+ B_0^{b})- (g_A-g_V) m_\rho^2 B_0
+(g_A+g_V) (D-6)  M_W^2 B_0
\bigg]
\bigg\}\,,
\end{eqnarray}

with the following Passarino-Veltman functions $C_{rs} = C_{rs}(m_a^2, q^2, m_b^2, M_W^2, m_\rho^2, m_\rho^2)$, $B_0 = B_0(q^2,m_\rho^2,m_\rho^2)$, $B_0^a = B_0(m_a^2,M_W^2,m_\rho^2)$ and $B_0^b = B_0(m_b^2,M_W^2,m_\rho^2)$.

\begin{eqnarray}
F_L^{V \, (d)} &=& \dfrac{g_w^3 }{16\pi^2 4 c_w M_W^2 } \sum_{\rho}
\bigg\{
m_a m_b \mathcal{U}_{\rho a} \mathcal{U}_{\rho b}^* \bigg[2 c_{2w} m_\rho^2 C_{0}+2 c_{2w} C_{00}
\nonumber\\ &\phantom{=}&
+\left(c_{2w} \left(m_a^2+3 m_\rho^2\right)+2 c_w^2 (D-2) M_W^2\right) C_{1}
\nonumber\\ &\phantom{=}&
+\left(c_{2w} \left(m_b^2+3 m_\rho^2\right)+2 c_w^2 (D-2) M_W^2\right) C_{2}
\nonumber\\ &\phantom{=}&
+\left(c_{2w} \left(m_a^2+m_\rho^2\right)+2 c_w^2 (D-2) M_W^2\right) C_{11}
\nonumber\\ &\phantom{=}&
+\left(c_{2w} \left(m_b^2+m_\rho^2\right)+2 c_w^2 (D-2) M_W^2\right) C_{22}
\nonumber\\ &\phantom{=}&
+\left(c_{2w} \left(m_a^2+m_b^2+2 m_\rho^2\right)+4 c_w^2 (D-2) M_W^2\right)C_{12} \bigg]
\nonumber\\ &\phantom{=}&
+\mathcal{U}_{\rho a}^* \mathcal{U}_{\rho b}
\bigg[
-m_\rho^2 \left(c_{2w} \left(m_a^2+m_b^2\right)-4 c_w \left(c_w^2-1\right) M_W M_Z\right) C_{0}
\nonumber\\ &\phantom{=}&
-m_a^2 \left(c_{2w} \left(m_b^2+m_\rho^2\right)+2 c_w^2 D M_W^2\right) C_{11}
\nonumber\\ &\phantom{=}&
-m_b^2 \left(c_{2w} \left(m_a^2+m_\rho^2\right)+2 c_w^2 D M_W^2\right) C_{22}
\nonumber\\ &\phantom{=}&
- \left(c_{2w} \left(m_a^2 \left(2 m_b^2+m_\rho^2\right)+m_b^2 m_\rho^2\right) + 2 c_w^2 M_W^2 \left(D \left(m_a^2+m_b^2\right)-2 q^2\right)\right)C_{12}
\nonumber\\ &\phantom{=}&
-\left(2 c_{2w} m_\rho^2+8 c_w^2 (D-1) M_W^2\right) C_{00}
\nonumber\\ &\phantom{=}&
+\big(2 c_w M_W \left(c_w^2 M_Z \left(m_a^2+m_b^2\right)-c_w M_W \left(m_a^2+(D-1) m_b^2 -2 q^2\right)-M_Z \left(m_a^2+m_b^2\right)\right)
\nonumber\\ &\phantom{=}&
-c_{2w} \left(m_a^2  m_b^2  +m_a^2m_\rho^2 +2 m_b^2 m_\rho^2\right)\big)C_{2}
\nonumber\\ &\phantom{=}&
+\big(2 c_w M_W \left(c_w^2 M_Z \left(m_a^2+m_b^2\right)-c_w M_W \left((D-1) m_a^2 +m_b^2-2 q^2\right)-M_Z \left(m_a^2+m_b^2\right)\right)
\nonumber\\ &\phantom{=}&
-c_{2w} \left(m_a^2 m_b^2+2 m_a^2m_\rho^2+m_b^2 m_\rho^2\right)\big)C_{1}
\bigg]
\bigg\}\,,
\end{eqnarray}

\begin{eqnarray}
F_R^{V \, (d)} &=& \dfrac{g_w^3 }{16\pi^2 4 c_w M_W^2 } \sum_{\rho}
\bigg\{
m_a m_b \mathcal{U}_{\rho a}^* \mathcal{U}_{\rho b}
\bigg[-2 c_{2w} m_\rho^2 C_{0}-2 c_{2w} C_{00}
\nonumber\\ &\phantom{=}&
-\left(c_{2w} \left(m_a^2+3 m_\rho^2\right)+2 c_w^2 (D-2) M_W^2\right) C_{1}
\nonumber\\ &\phantom{=}&
-\left(c_{2w} \left(m_b^2+3 m_\rho^2\right)+2 c_w^2 (D-2) M_W^2\right) C_{2}
\nonumber\\ &\phantom{=}&
-\left(c_{2w} \left(m_a^2+m_\rho^2\right)+2 c_w^2 (D-2) M_W^2\right) C_{11}
\nonumber\\ &\phantom{=}&
-\left(c_{2w} \left(m_b^2+m_\rho^2\right)+2 c_w^2 (D-2) M_W^2\right) C_{22}
\nonumber\\ &\phantom{=}&
- \left(c_{2w} \left(m_a^2+m_b^2+2 m_\rho^2\right)+4 c_w^2 (D-2) M_W^2\right)C_{12}
\bigg]
\nonumber\\ &\phantom{=}&
+\mathcal{U}_{\rho a} \mathcal{U}_{\rho b}^*
\bigg[m_\rho^2 \left(c_{2w} \left(m_a^2+m_b^2\right)+4 c_w \left(c_w^2-1\right) M_W M_Z\right) C_{0}
\nonumber\\ &\phantom{=}&
+m_a^2 \left(c_{2w} \left(m_b^2+m_\rho^2\right)+2 c_w^2 D M_W^2\right) C_{11}
\nonumber\\ &\phantom{=}&
+m_b^2 \left(c_{2w} \left(m_a^2+m_\rho^2\right)+2 c_w^2 D M_W^2\right) C_{22}
\nonumber\\ &\phantom{=}&
+\left(c_{2w} \left(m_a^2 \left(2 m_b^2+m_\rho^2\right)+m_b^2 m_\rho^2\right)+2 c_w^2 M_W^2 \left(D \left(m_a^2+m_b^2\right)-2 q^2\right)\right)C_{12}
\nonumber\\ &\phantom{=}&
+\left(2 c_{2w} m_\rho^2+8 c_w^2 (D-1) M_W^2\right) C_{00}
\nonumber\\ &\phantom{=}&
+ \big(2 c_w M_W \left(c_w^2 M_Z \left(m_a^2+m_b^2\right)+c_w M_W \left(m_a^2+(D-1) m_b^2-2 q^2\right)-M_Z \left(m_a^2+m_b^2\right)\right)
\nonumber\\ &\phantom{=}&
+c_{2w} \left(m_a^2 m_b^2+m_a^2m_\rho^2+2 m_b^2 m_\rho^2\right)
\big)C_{2}
\nonumber\\ &\phantom{=}&
+ \big(2 c_w M_W \left(c_w^2 M_Z \left(m_a^2+m_b^2\right)+c_w M_W \left((D-1) m_a^2+m_b^2-2 q^2\right)-M_Z \left(m_a^2+m_b^2\right)\right)
\nonumber\\ &\phantom{=}&
+c_{2w} \left(m_a^2 m_b^2+2 m_a^2 m_\rho^2+m_b^2 m_\rho^2\right)
\big)C_{1}
\bigg]
\bigg\}\,,
\end{eqnarray}

in which $ C_{rs} = C_{rs}(m_a^2, q^2, m_b^2,  m_\rho^2, M_W^2, M_W^2)$ and $c_{2w} = \cos2\theta_w$.

\begin{eqnarray}
F_L^{V \, (e)} &=& \dfrac{g_w^3 }{16\pi^2 8 c_w^2 M_W^2 } \sum_{i}
\bigg\{
C_{ai}^*C_{ib}^* c_w m_a m_b \bigg[\left(m_i^2-m_b^2\right) (C_{2}+C_{22}) +\left(m_i^2-m_a^2\right) (C_{1}+ C_{11})
\nonumber\\ &\phantom{=}&
-\left(m_a^2+m_b^2-2 m_i^2\right) C_{12} -2 C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+C_{ai}^*C_{ib} c_w m_a m_i \bigg[\left(m_b^2-m_i^2\right) (C_{0}+  C_{2})
+\left(m_b^2-m_a^2\right) (C_{12}+  C_{11})
\nonumber\\ &\phantom{=}&
-\left(m_a^2-2 m_b^2+m_i^2\right) C_{1}
-2 C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+C_{ai}C_{ib}^* m_b m_i\bigg[
\left(c_w \left(m_i^2-m_a^2\right)+2 M_W M_Z\right) C_{0}+\left(c_w \left(m_i^2-m_b^2\right)-2 M_W M_Z\right) C_{2}
\nonumber\\ &\phantom{=}&
+\left(c_w \left(m_i^2-m_a^2\right)-2 M_W M_Z\right) C_{1}+c_w \left(m_a^2-m_b^2\right) (C_{22}+ C_{12}) -2 c_w C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
m_i^2 \left(c_w \left(m_b^2-m_a^2\right)+2 M_W M_Z\right) C_{0}+\left(c_w m_a^2 \left(m_b^2-m_i^2\right)-2 m_b^2 M_W M_Z\right) C_{2}
\nonumber\\ &\phantom{=}&
+\left(c_w m_a^2 \left(m_b^2-2 m_i^2\right)+c_w m_b^2 m_i^2-2 m_b^2 M_W M_Z\right) C_{1}+c_w m_a^2 \left(m_b^2-m_i^2\right) C_{11}
\nonumber\\ &\phantom{=}&
+c_w m_b^2 \left(m_a^2-m_i^2\right) C_{22}-c_w \left(m_a^2 \left(m_i^2-2 m_b^2\right)+m_b^2 m_i^2\right) C_{12}-2 c_w m_i^2 C_{00}
\bigg]
\bigg\}\,,
\end{eqnarray}

\begin{eqnarray}
F_R^{V \, (e)} &=& \dfrac{g_w^3 }{16\pi^2 8 c_w^2 M_W^2 } \sum_{i}
\bigg\{
C_{ai} C_{ib} c_w m_a m_b\bigg[
\left(m_b^2-m_i^2\right) (C_{2}+ C_{22})+\left(m_a^2-m_i^2\right) (C_{1} + C_{11})
\nonumber\\ &\phantom{=}&
+\left(m_a^2+m_b^2-2 m_i^2\right) C_{12}+2 C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+ C_{ai} C_{ib}^* c_w m_a m_i\bigg[
\left(m_i^2-m_b^2\right) (C_{0} + C_{2}) +\left(m_a^2-m_b^2\right) (C_{12}+ C_{11})
\nonumber\\ &\phantom{=}&
+\left(m_a^2-2 m_b^2+m_i^2\right) C_{1}+2 C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+ C_{ai}^* C_{ib} m_b m_i\bigg[
\left(c_w \left(m_a^2-m_i^2\right)-2 M_W M_Z\right) C_{0}+\left(c_w \left(m_b^2-m_i^2\right)+2 M_W M_Z\right) C_{2}
\nonumber\\ &\phantom{=}&
+\left(c_w \left(m_a^2-m_i^2\right)+2 M_W M_Z\right) C_{1}+c_w \left(m_b^2-m_a^2\right) (C_{22}+ C_{12}) +2 c_w C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+ C_{ai}^* C_{ib}^* \bigg[
m_i^2 \left(c_w \left(m_a^2-m_b^2\right)-2 M_W M_Z\right) C_{0}+\left(c_w m_a^2 \left(m_i^2-m_b^2\right)+2 m_b^2 M_W M_Z\right) C_{2}
\nonumber\\ &\phantom{=}&
+\left(2 m_b^2 M_W M_Z-c_w \left(m_a^2 \left(m_b^2-2 m_i^2\right)+m_b^2 m_i^2\right)\right) C_{1}+c_w m_a^2 \left(m_i^2-m_b^2\right) C_{11}
\nonumber\\ &\phantom{=}&
+c_w m_b^2 \left(m_i^2-m_a^2\right) C_{22}+c_w \left(m_a^2 \left(m_i^2-2 m_b^2\right)+m_b^2 m_i^2\right) C_{12}+2 c_w m_i^2 C_{00}
\bigg]
\bigg\}\,,
\end{eqnarray}

with $ C_{rs}=C_{rs}(m_a^2, q^2, m_b^2,  m_i^2, M_Z^2, M_H^2)$.

\begin{eqnarray}
F_L^{V \, (f)} &=& \dfrac{g_w^3 }{16\pi^2 8 c_w^2 M_W^2 } \sum_{i}
\bigg\{
C_{ai}^* C_{ib}^* c_w m_a m_b \bigg[
\left(m_i^2-m_b^2\right) (C_{2}+  C_{22}) +\left(m_i^2-m_a^2\right) (C_{1}+ C_{11})
\nonumber\\ &\phantom{=}&
-\left(m_a^2+m_b^2-2 m_i^2\right) C_{12} -2 C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+ C_{ai}^* C_{ib} m_a m_i \bigg[
\left(c_w \left(m_i^2-m_b^2\right)+2 M_W M_Z\right) C_{0}+\left(c_w \left(m_i^2-m_b^2\right)-2 M_W M_Z\right) C_{2}
\nonumber\\ &\phantom{=}&
+\left(c_w \left(m_i^2-m_a^2\right)-2 M_W M_Z\right) C_{1}+c_w \left(m_b^2-m_a^2\right) (C_{12} + C_{11}) -2 c_w C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+ C_{ai} C_{ib}^* c_w m_b m_i \bigg[
\left(m_a^2-m_i^2\right) (C_{0} + C_{1})  +\left(m_a^2-m_b^2\right) (C_{22}+C_{12})
\nonumber\\ &\phantom{=}&
-\left(-2 m_a^2+m_b^2+m_i^2\right) C_{2} -2 C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+ C_{ai} C_{ib}\bigg[
m_i^2 \left(c_w \left(m_a^2-m_b^2\right)+2 M_W M_Z\right) C_{0}+\left(c_w m_b^2 \left(m_a^2-m_i^2\right)-2 m_a^2 M_W M_Z\right) C_{1}
\nonumber\\ &\phantom{=}&
+\left(c_w m_a^2 \left(m_b^2+m_i^2\right)-2 c_w m_b^2 m_i^2-2 m_a^2 M_W M_Z\right) C_{2}+c_w m_b^2 \left(m_a^2-m_i^2\right) C_{22}
\nonumber\\ &\phantom{=}&+c_w m_a^2 \left(m_b^2-m_i^2\right) C_{11}-c_w \left(m_a^2 \left(m_i^2-2 m_b^2\right)+m_b^2 m_i^2\right) C_{12}-2 c_w m_i^2 C_{00}
\bigg]
\bigg\}\,,
\end{eqnarray}

\begin{eqnarray}
F_R^{V \, (f)} &=& \dfrac{g_w^3 }{16\pi^2 8 c_w^2 M_W^2 } \sum_{i}
\bigg\{
C_{ai} C_{ib} c_w m_a m_b\bigg[
\left(m_b^2-m_i^2\right) (C_{2}+ C_{22}) +\left(m_a^2-m_i^2\right) (C_{1}+C_{11})
\nonumber\\ &\phantom{=}&
+\left(m_a^2+m_b^2-2 m_i^2\right) C_{12}  +2 C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+ C_{ai} C_{ib}^* m_a m_i\bigg[
\left(c_w \left(m_b^2-m_i^2\right)-2 M_W M_Z\right) C_{0}+\left(c_w \left(m_b^2-m_i^2\right)+2 M_W M_Z\right) C_{2}
\nonumber\\ &\phantom{=}&
+\left(c_w \left(m_a^2-m_i^2\right)+2 M_W M_Z\right) C_{1}+c_w \left(m_a^2-m_b^2\right) (C_{12}+ C_{11})+2 c_w C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+ C_{ai}^* C_{ib} c_w m_b m_i \bigg[
\left(m_i^2-m_a^2\right) (C_{0}+ C_{1})  +\left(m_b^2-m_a^2\right) (C_{22}+ C_{12})
\nonumber\\ &\phantom{=}&
+\left(-2 m_a^2+m_b^2+m_i^2\right) C_{2}+2 C_{00}
\bigg]
\nonumber\\ &\phantom{=}&
+ C_{ai}^* C_{ib}^*\bigg[
-m_i^2 \left(c_w \left(m_a^2-m_b^2\right)+2 M_W M_Z\right) C_{0}+\left(c_w m_b^2 \left(m_i^2-m_a^2\right)+2 m_a^2 M_W M_Z\right) C_{1}
\nonumber\\ &\phantom{=}&
+\left(-c_w m_a^2 \left(m_b^2+m_i^2\right)+2 c_w m_b^2 m_i^2+2 m_a^2 M_W M_Z\right) C_{2}+c_w m_b^2 \left(m_i^2-m_a^2\right) C_{22}
\nonumber\\ &\phantom{=}&
+c_w m_a^2 \left(m_i^2-m_b^2\right) C_{11}+c_w \left(m_a^2 \left(m_i^2-2 m_b^2\right)+m_b^2 m_i^2\right) C_{12}+2 c_w m_i^2 C_{00}
\bigg]
\bigg\}\,,
\end{eqnarray}

where $ C_{rs}=  C_{rs}(m_a^2, q^2, m_b^2, m_i^2, M_H^2, M_Z^2)$.

§ SM EXTENSIONS VIA STERILE FERMIONS AND NEUTRINO MASS GENERATION: FRAMEWORK AND CONSTRAINTS

In this section we describe the underlying framework
of our study: a well-motivated, low-scale mechanism of neutrino mass generation (relying on a minimal SM extension via sterile fermions).

§.§ The Inverse Seesaw

The type I seesaw (and its variants, as is the case of the Inverse Seesaw) are among the most minimal yet successful extensions of the SM accounting for a mechanism of neutrino mass generation. Moreover, it allows for a ”natural" (in the sense of 't Hooft [69]) explanation of the smallness of the the observed neutrino masses, as the latter vanish in the limit of lepton number symmetry restoration.

In the case of the ISS, two distinct species of sterile  fermions, $X$ and $\nu_R$, are added to the SM content; the Lagrangian encoding the mass terms for the neutral lepton sector can generically be written as

\begin{equation}\label{eq:ISS:lagrangian}
\mathcal L_\text{ISS}\, =\, -Y^D_{ij} \,\overline{L_i^c}\,\widetilde H \,\nu_{Rj}^c - M_R^{ij}\, \overline{\nu_{Ri}}\, X_j - \frac{1}{2}\mu_R^{ij}\, \overline{\nu_{Ri}^c}\,\nu_{Rj} - \frac{1}{2} \mu_X^{ij}\, \overline{X_i^c}\, X_j + \text{H.c.}\,,
\end{equation}

in which $\mu_X$ and $\mu_R$ are the only source of lepton number violation. After the Higgs boson acquires its vacuum expectation value, $v$, a Dirac mass term is generated, given by $m_D\equiv Y_D v$.
In the limit of vanishing $\mu_{X,R}$, lepton number is restored, and thus light neutrinos masses vanish. In this sense, assuming the hierarchy of scales $\mu_{X,R}\ll m_D\ll M_R$ is natural in the 't Hooft sense [70, 71].
Large deviations from unitarity of the PMNS can still be present, even in the massless neutrino limit, as $\eta$ is given by

\begin{equation}
\eta = \frac{1}{2}m_D^* \left(M_R^{-1}\right)^{\dagger}\left(M_R^{-1}\right)m_D^T\,,
\label{eq:def_eta_ISS}
\end{equation}

and thus it does not depend on $\mu_{X,R}$. Moreover, in the limit of approximate lepton number conservation, the term $\mu_R$ only contributes to light neutrino masses through loop effects [72]. Therefore, in the following we will not consider its contribution.

Although more minimal ISS realisations exist [41], here we will work in the (3,3) realisation, corresponding to the addition of
$n_R = n_X = 3$ generations of heavy neutral states.
The diagonalisation of the $9\times9$ mass matrix allows obtaining the full neutrino spectrum. In the approximate lepton number conserving limit in which $\mu_X \ll m_D\ll M_R$, one can derive an approximate expression for the masses of the light
(mostly active) neutrinos, given by

\begin{equation}
m_\nu \simeq  m_D \, \left( M_{R}^{-1} \right)^{T} \, \mu_X \, M_{R}^{-1} \, m_D^T\, \equiv U_\text{PMNS}^\ast\, m_\nu^\text{diag}\, U_\text{PMNS}^\dagger\,.
\label{eq:ISS:lightmasses}
\end{equation}

A detailed description of the relevant charged and neutral Lagrangian terms is provided in Appendix <ref>, and the associated vertices
in <ref>.

§.§ Relevant Lagrangian terms

In what follows we collect the most relevant terms in the Lagragian of the lepton sector (interactions with neutral and charged gauge bosons, Higgs and Goldstone bosons). The terms are presented in the physical lepton bases, and reflect the Majorana nature of the neutral leptons. (Some of the terms have been individually discussed in the main body of the manuscript, see Eqs. (<ref>, <ref>).)

\begin{align}\label{eq:lagrangian:WGHZ}
& \mathcal{L}_{W^\pm}\, =\, -\frac{g_w}{\sqrt{2}} \, W^-_\mu \,
\sum_{\alpha=1}^{3} \sum_{j=1}^{3 + n_S} \mathcal{U}_{\alpha j} \bar \ell_\alpha
\gamma^\mu P_L \nu_j \, + \, \text{H.c.}\,, \nonumber \\
& \mathcal{L}_{Z^0}^{\nu}\, = \,-\frac{g_w}{4 \cos \theta_w} \, Z_\mu \,
\sum_{i,j=1}^{3 + n_S} \bar \nu_i \gamma ^\mu \left(
P_L {C}_{ij} - P_R {C}_{ij}^* \right) \nu_j\,, \nonumber \\
& \mathcal{L}_{Z^0}^{\ell}\, = \,-\frac{g_w}{2 \cos \theta_w} \, Z_\mu \,
\sum_{\alpha=1}^{3}  \bar \ell_\alpha \gamma ^\mu \left(
{\bf C}_{V} - {\bf C}_{A} \gamma_5 \right) \ell_\alpha\,, \nonumber \\
& \mathcal{L}_{H^0}\, = \, -\frac{g_w}{4 M_W} \, H  \,
\sum_{i\ne j= 1}^{3 + n_S}    \bar \nu_i\,\left[{C}_{ij}\,\left(
P_L m_i + P_R m_j \right) +{C}_{ij}^\ast\left(
P_R m_i + P_L m_j \right) \right] \nu_j\ , \nonumber \\
& \mathcal{L}_{G^0}\, =\,\frac{i g_w}{4 M_W} \, G^0 \,
\sum_{i,j=1}^{3 + n_S}  \bar \nu_i \left[ {C}_{ij}
\left(P_R m_j  - P_L m_i  \right) + {C}_{ij}^\ast
\left(P_R m_i  - P_L m_j  \right)\right] \nu_j\,, \nonumber  \\
& \mathcal{L}_{G^\pm}\, =\, -\frac{g_w}{\sqrt{2} M_W} \, G^- \,
\sum_{\alpha=1}^{3}\sum_{j=1}^{3 + n_S} \mathcal{U}_{\alpha j}
\bar \ell_\alpha\left(
m_\alpha P_L - m_j P_R \right) \nu_j\, + \, \text{H.c.}\,.
\end{align}

Finally, we recall that ${C}_{ij} $ are defined as in Eq. (<ref>):

\begin{equation}
{C}_{ij} = \sum_{\rho = 1}^3
\mathcal{U}_{i\rho}^\dagger \,\mathcal{U}_{\rho j}^{\phantom{\dagger}}\:. \nonumber
\end{equation}

§.§ Feynman rules

Following the presentation of the Lagrangian terms, see Eq. (<ref>), we list in Table <ref> the Feynman rules for the vertices which were used in the computations carried out in this manuscript. Again notice that the neutral leptons are assumed to be of Majorana nature.

In the $Z$ and Higgs vertices, the arrows denote the momentum flow.
We note here that diagrams including at least one $Z n_i n_j$ or $H n_i n_j$ vertex have to be symmetrised (factor 2) due to the Majorana nature of the physical neutrinos.

(a) at (0,0) \(Z_\mu\);
(b) at (1,0);
(c) at (2,1.)\(n_i\);
(d) at (2,-1.)\(n_j\);
(a) – [boson] (b),
(c) – [momentum'=\( \)] (b),
(b) – [momentum'=\( \)] (d),

$= \, -i \dfrac{g_w}{4 c_w} \gamma_\mu \left(C_{ij}^* \,P_L -C_{ij} \,P_R \right)$

(a) at (0,0) \(H\);
(b) at (1,0);
(c) at (2,1.)\(n_i\);
(d) at (2,-1.)\(n_j\);
(a) – [scalar] (b),
(c) – [momentum'=\( \)] (b),
(b) – [momentum'=\( \)] (d),

$=\, -i\dfrac{g_w}{4 M_W} \left[C_{ij} \left( m_i\, P_L + m_j \,P_R \right)
+ C_{ij}^* \left( m_i\, P_R + m_j \,P_L \right) \right]$

(a) at (0,0) \(W_\mu^-\);
(b) at (1,0);
(c) at (2,1.)\(n_i\);
(d) at (2,-1.)\(\ell_\alpha^-\);
(a) – [boson] (b),
(c) – [fermion] (b),
(b) – [fermion] (d),

$=\, -i\dfrac{g_w}{\sqrt{2}} \,\mathcal{U}_{\alpha i}\, \gamma_\mu\, P_L$

(a) at (0,0) \(W_\mu^+\);
(b) at (1,0);
(c) at (2,1.)\(\ell_\alpha^+\);
(d) at (2,-1.)\(n_i\);
(a) – [boson] (b),
(c) – [fermion] (b),
(b) – [fermion] (d),

$=\, -i\dfrac{g_w}{\sqrt{2}} \,\mathcal{U}_{\alpha i}^*\, \gamma_\mu\, P_L$

(a) at (0,0) \(G^-\);
(b) at (1,0);
(c) at (2,1.)\(n_i\);
(d) at (2,-1.)\(\ell_\alpha^-\);
(a) – [scalar] (b),
(c) – [fermion] (b),
(b) – [fermion] (d),

$=\, i\dfrac{g_w}{\sqrt{2}M_W} \,\mathcal{U}_{\alpha i} \,\left(m_i\, P_R - m_\alpha\, P_L\right)$

(a) at (0,0) \(G^+\);
(b) at (1,0);
(c) at (2,1.)\(\ell_\alpha^+\);
(d) at (2,-1.)\(n_i\);
(a) – [scalar] (b),
(c) – [fermion] (b),
(b) – [fermion] (d),

$=\, i\dfrac{g_w}{\sqrt{2}M_W} \,\mathcal{U}_{\alpha i}^* \,\left(m_i \,P_L - m_\alpha \,P_R\right)$

Feynman rules for $W$, $Z$ and Higgs interactions (and associated Goldstone bosons) in SM extensions via Majorana sterile fermions.

§.§ Constraints on HNL extensions of the SM

In the phenomenological analysis whose results are summarised in Sections <ref> and <ref>, we have taken into account numerous constraints, which were applied to the present ISS (3,3) realisation.
In addition to ensuring that such a mechanism of neutrino mass generation does comply with neutrino oscillation data, we have further imposed several experimental limits, including
EW precision observables, universality bounds from tau and meson decays, cLFV bounds, among others.
Finally, we also took into account perturbative unitarity constraints for the heavy sterile states (including
tree-level decay widths of heavy $N$ from the channels $N_i\to \ell_\alpha W$, $N_i\to N_j Z$ and $N_i \to N_j H$ where $N_j$ can also be light).

Oscillation data The first constrain on any SM extension aiming at accounting for neutrino masses and mixings (in particular via the addition of HNL) is that of reproducing the measured neutrino oscillation parameters. From the latest NuFIT 5.1 global fit results [58], these parameters are given in Table <ref>. As already stated in the main text (see Section <ref>), we vary the light neutrino mass in the range $m_0\in \left[ 10^{-10},\, 10^{-3}\right]$ eV and fix oscillation data to their best-fit values as summarised in Table <ref>, assuming normal ordering for the light neutrino spectrum, while varying the CP phase within its range. Note that none of these assumptions have any impact in our results nor on the constraints.

Normal ordering     Inverted ordering

$\sin^2\theta_{12}$      $0.304^{+0.013}_{-0.012}$     $0.304^{+0.012}_{-0.012}$

$\sin^2\theta_{23}$     $0.573^{+0.018}_{-0.023}$     $0.578^{+0.017}_{-0.021}$

$\sin^2\theta_{13}$     $0.02220^{+0.00068}_{-0.00062}$     $0.02238^{+0.00064}_{-0.00062}$

$\Delta m_{21}^2/10^{-5}\,\mathrm{eV}$     $7.42^{+0.21}_{-0.20}$     $7.42^{+0.21}_{-0.20}$

$\Delta m_{3\ell}^2/10^{-3}\,\mathrm{eV}$     $2.515^{+0.028}_{-0.028}$     $-2.498^{+0.028}_{-0.029}$

Global fit results obtained by NuFIT 5.1 [58] for neutrino mixing data, without the inclusion of the atmospheric sample from Super-Kamiokande. For normal ordering we have $\Delta m_{3\ell}^2\equiv\Delta m_{31}^2>0$, while in the case of inverted ordering one finds $\Delta m_{3\ell}^2\equiv \Delta m_{32}^2<0$.

Universality bounds from tau-lepton and light meson decays
The non-unitarity of the PMNS mixing matrix impacts several low-energy observables, which can then place strong constraints on the active-sterile mixings.
Among them, we have the universality ratios from kaon and pion leptonic decays, defined as [34, 32]

\begin{equation}
\Delta r_P\equiv \frac{\Gamma\left(P^+\rightarrow e\nu\right)}{\Gamma\left(P^+\rightarrow \mu\nu\right)}-1\quad \left(P=\pi,K\right),
\end{equation}

as well as $\tau$ decays (also sensitive to the modification of the $W\ell \nu$ vertex), with the associated observable, $R_{\tau}$, given by

\begin{equation}
R_{\tau}\equiv \frac{\Gamma\left(\tau \rightarrow \mu \nu \nu\right)}{\Gamma\left(\tau \rightarrow e \nu \nu\right)}.
\end{equation}

Charged lepton flavour violation
There are several cLFV transitions and decays placing strong constraints on neutrino mass models upon the appearance of flavour violation. Although here we will mostly focus on LFUV (while aiming at exploring regimes with suppressed cLFV), it is interesting to consider the interplay between the two sets of observables in order to discuss future prospects. Particularly relevant are cLFV three-body  decays, $\mu - e$ conversion in atoms, as well as cLFV $Z$ boson decays.
The expressions for the cLFV observables (in the context of SM extensions via HNL) can be found, for instance, in [12, 11, 24, 29, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 19, 20, 21, 22, 25]
The current bounds and future sensitivities are collected in Table <ref>.

Observable     Current bound     Future sensitivity

$\text{BR}(\mu\to e \gamma)$
$<4.2\times 10^{-13}$   (MEG [73])
$6\times 10^{-14}$   (MEG II [74])

$\text{BR}(\tau \to e \gamma)$
$<3.3\times 10^{-8}$   (BaBar [75])
$3\times10^{-9}$   (Belle II [76])

$\text{BR}(\tau \to \mu \gamma)$
$ <4.4\times 10^{-8}$   (BaBar [75])
$10^{-9}$   (Belle II [76])

$\text{BR}(\mu \to 3 e)$
$<1.0\times 10^{-12}$   (SINDRUM [77])
$10^{-15(-16)}$   (Mu3e [78])

$\text{BR}(\tau \to 3 e)$
$<2.7\times 10^{-8}$   (Belle [79])
$5\times10^{-10}$   (Belle II [76])

$\text{BR}(\tau \to 3 \mu )$
$<3.3\times 10^{-8}$   (Belle [79])
$5\times10^{-10}$   (Belle II [76])

$5\times 10^{-11}$  (FCC-ee [80])

$\text{BR}(\tau^- \to e^-\mu^+\mu^-)$
$<2.7\times 10^{-8}$   (Belle [79])
$5\times10^{-10}$   (Belle II [76])

$\text{BR}(\tau^- \to \mu^-e^+e^-)$
$<1.8\times 10^{-8}$   (Belle [79])
$5\times10^{-10}$   (Belle II [76])

$\text{BR}(\tau^- \to e^-\mu^+e^-)$
$<1.5\times 10^{-8}$   (Belle [79])
$3\times10^{-10}$   (Belle II [76])

$\text{BR}(\tau^- \to \mu^-e^+\mu^-)$
$<1.7\times 10^{-8}$   (Belle [79])
$4\times10^{-10}$   (Belle II [76])

$\text{CR}(\mu- e, \text{N})$
$<7 \times 10^{-13}$   (Au, SINDRUM [64])
$10^{-14}$    (SiC, DeeMe [81])

$2.6\times 10^{-17}$    (Al, COMET [82, 83, 84])

$8 \times 10^{-17}$    (Al, Mu2e [85])

$\mathrm{BR}(Z\to e^\pm\mu^\mp)$       $< 4.2\times 10^{-7}$  (ATLAS [86])       $\mathcal O (10^{-10})$  (FCC-ee [80])

$\mathrm{BR}(Z\to e^\pm\tau^\mp)$       $< 4.1\times 10^{-6}$  (ATLAS [87])       $\mathcal O (10^{-10})$  (FCC-ee [80])

$\mathrm{BR}(Z\to \mu^\pm\tau^\mp)$       $< 5.3\times 10^{-6}$  (ATLAS [87])       $\mathcal O (10^{-10})$  (FCC-ee [80])

Current experimental bounds and future sensitivities on relevant cLFV observables. The quoted limits are given at $90\%\:\mathrm{C.L.}$ (Belle II sensitivities correspond to an integrated luminosity of $50\:\mathrm{ab}^{-1}$.)

EW precision tests
The presence of the HNL leads to a shift on the EW oblique parameters $S$, $T$ and $U$ [55] from the SM predictions. The dominant contribution from heavy neutrinos is found for the $T$ parameter [43]. Even if present bounds on $T$ are not currently competitive to constrain the explored parameter space, notice that prospective sensitivities of FCC-ee (improving current constraints by about two orders of magnitude [62]) will place very strong bounds on the size of the deviations of $T$ from its SM expectation.

Moreover, muon decays are affected by the presence of the heavy sterile states. Given that we use $G_{\mu}$ - the measurement of the Fermi constant - as an input parameter, we must take into account the non-unitarity effect of HNL already at tree level. In particular, the muon decay rate is now given by

\begin{equation}
\Gamma_{\mu}= \frac{m_{\mu}^5 G_\mu^2}{192\pi^3} =\frac{m_{\mu}^5 G_F^2}{192\pi^3}\sum_{i,j=1}^3|\mathcal{U}_{\mu i}|^2|\mathcal{U}_{ej}|^2\simeq \frac{m_{\mu}^5G_F^2}{192\pi^3}\left(1-2\eta_{\mu\mu}-2\eta_{ee}\right)\,,
\end{equation}

from which we identify $G_{\mu}\equiv G_F\left(1-\eta_{\mu\mu}-\eta_{ee}\right)$, assuming $\eta_{\alpha\alpha}\ll1$.

Perturbative unitarity
We can also impose bounds on the HNL parameter space by taking into account perturbative unitarity constraints [88, 89, 90, 91, 92], which restricts the heavy states decay width to comply with

\begin{equation}
\frac{\Gamma(N_i)}{m_{N_i}}<\frac{1}{2},\quad \text{for}\quad i\geq4\,.
\end{equation}

For heavy neutrinos as considered here, the main contributions to the decay rate will be given by the two-body decays into a SM boson and a lepton. At tree level, these are given by

\begin{eqnarray}
\Gamma\left(N_i\rightarrow Wl_{\alpha}\right)&=&\frac{g_w^2}{64\pi}\left|\mathcal{U}_{\alpha i}\right|^2\frac{\lambda^{1/2}(m_i^2,M_W^2,m_{\ell_{\alpha}}^2)}{m_i}\left\lbrace 1+\frac{m_{\ell_{\alpha}}-2M_W^2}{m_i^2}+\frac{(m_i^2-m_{\ell_{\alpha}}^2)^2}{m_i^2M_W^2}\right\rbrace,\\
\Gamma\left(N_i\rightarrow ZN_j\right)&=&\frac{g_w^2}{128c_w^2\pi}\left|C_{ij}\right|^2\frac{\lambda^{1/2}(m_i^2,M_Z^2,m_{j}^2)}{m_i}\left\lbrace 1+\frac{m_{j}-2M_Z^2}{m_i^2}+\frac{(m_i^2-m_{j}^2)^2}{m_i^2M_Z^2}\right\rbrace,\\
\Gamma\left(N_i\rightarrow HN_j\right)&=&\frac{g_w^2}{128\pi}\left|C_{ij}\right|^2\frac{m_i^2}{M_W^2}\frac{\lambda^{1/2}(m_i^2,M_H^2,m_j^2)}{m_i}\left\lbrace\left(1+\frac{m_j^2-M_H^2}{m_i^2}\right)\left(1+\frac{m_j^2}{m_i^2}\right)+4\frac{m_j^2}{m_i^2}\right\rbrace, \nonumber
\\
\end{eqnarray}

where $\lambda(a,b,c)=\left[a-(\sqrt{b}-\sqrt{c})^2\right]\left[a-(\sqrt{b}+\sqrt{c})^2\right]$ is the Källén function.

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#  High Voltage Generation by Fiber-Coupled Pulsed Laser for a Simple Receiver
Circuit Structure ††thanks: This material is based upon work supported by the
National Science Foundation under Grant No. 1808489.

Sanghyeon Park and Juan Rivas-Davila
Stanford University
{spark15<EMAIL_ADDRESS>
###### Abstract

Almost all high-voltage dc generation for low-power applications is done by
either electrostatic machines or voltage multipliers. Electrostatic machines
use mechanically moving parts to transfer charge and energy from the low-
voltage side to the high-voltage side. Voltage multipliers use capacitive and
inductive networks to achieve the same purpose of energy transfer. Considering
the pros and cons inherent in those mechanical, capacitive, and inductive
energy transfer, a new means of energy transfer may provide a superior design
of a high voltage dc generator. Here we investigate the design of high voltage
generators based on optical power transfer. Optical power delivered via fiber-
optic cable allows extensive input-to-output dc insulation and spatial
separation. These characteristics lead to advantages with respect to ease of
insulation, ease of electromagnetic shielding, and scalability. We
experimentally validate the idea by building and testing a 5.5 kV dc generator
module solely powered by a 20 kHz pulsed laser, and cascading three of those
modules to obtain 14.7 kV dc output voltage. We then discuss possible
improvements to the circuit design to make it useful for real-world
applications. Finally, we demonstrate an optically powered electroadhesion
gripper to show the practicality of the proposed high voltage generator.

##  I Introduction

High voltage power supplies with an output in the kilovolts range are
essential for numerous scientific instruments that need high voltage dc bias.
Examples include avalanche photo diodes (500 V) [1], mass spectrometers (10 kV
and higher) [2], and various photon and particle detectors for spacecrafts
(from 150 V to 26 kV) [3]. Most of those applications draw only tens of
microamperes of current or even less from the power supply. Such an extremely
low power requirement enables some unique ways to generate the necessary dc
bias voltage. This paper explores a high voltage dc generation technique using
a pulsed laser delivered through a fiber optic cable.

A voltage multiplier is arguably the most popular technique to generate a high
voltage dc, appearing in many recent papers [4, 5, 6]. The multiplier consists
of two major parts. The first is a number of cascaded rectifiers made of
diodes and capacitors. The other is a mechanism to block dc voltage and pass
ac voltage from the input terminal to each of the rectifiers. There is little
variation in the rectifier structure; it always consists of many half-wave
rectifiers (two serialized diodes) connected in series, and sometimes in
parallel as well to accommodate multi-phase inputs. On the other hand, the dc-
blocking mechanism exhibits a great variation with different types of
components and their arrangements.

The dc-blocking ac-coupling devices in multipliers are mostly either a
capacitor network or a combination of capacitor and transformer networks. The
capacitor network realizes capacitive coupling between the input terminal and
rectifiers by presenting a low impedance at the frequency of operation. Famous
topologies based on capacitor networks include Cockcroft-Walton [7, 8] and
Dickson [9, 10]. The transformer network realizes inductive coupling by using
multi-winding transformers of high interwinding breakdown voltage. Examples of
using the combination of both capacitor and transformer networks are a 300 kV
generator developed by Enge [11], a 160 kV generator by Mao et al. [12], and a
100 kV generator by Pokryvailo et al. [13].

Interestingly, the task of blocking dc and coupling ac through a high-voltage
insulation barrier is similar to transferring power across a distance,
commonly known as wireless power transfer (WPT). Capacitive and inductive
couplings are indeed two most common approaches for near-field (nonradiative)
power transfer. But there is another category of WPT techniques called far-
field (radiative) power transfer which uses visible light and microwaves as a
means of power transmission. Considering those parallels between high voltage
dc generation and WPT, it is natural to ask whether the idea of optically
generating a high voltage is useful, and if so, how.

Several studies, mostly from the field of power management integrated
circuits, have shown the feasibility of high voltage generation by optical
power transfer [14, 15, 16]. Lee et al. [14] achieved an open-circuit voltage
of 150 V using an integrated circuit of 100 series-connected photovoltaic (PV)
cells. Similarly, Ortega et al. [15] achieved an open-circuit voltage of 103 V
using 169 PV cells. Rentmeister et al. [16] used 196 series-connected PV cells
to produce a 125 V open-circuit voltage.

Our work differs from the existing literature by not relying on a massive
number of PV cells for high output voltage. A resonant operation combined with
a step-up transformer provides enough voltage gain, allowing us to build a
high voltage generator using only low-cost commercially available parts and
assembly service. Considering the often prohibitive cost of custom-built high
voltage integrated circuits, our work is attractive in that it makes available
optically powered high-voltage generators to a wider range of research
projects and applications.

In this paper, we firstly review existing techniques for high voltage dc
generation in section II. Then we discuss viable configurations of light-based
high voltage generators in section III. The purpose of these two sections is
to put the proposed light-based scheme into perspective by comparing it with
other generator designs of different working principles. We then proceed to
implement and test the proposed voltage multiplier, of which the experimental
results are given in section IV. Also presented in the section are the
practical limits we experienced during our design process and a projection on
achievable performance. To demonstrate the practicality of the proposed
system, we show in section V an application of it to an electroadhesive
gripping device. We conclude the paper in section VI with a couple of
questions worth exploring for future research.

##  II Review of Popular Techniques for High Voltage Generation

This section reviews some of the most popular types of high voltage dc
generators. Designs are grouped based on how the input power from the low-
voltage side is transferred through an insulation barrier to the high-voltage
side. Their working principles and associated pros and cons make each of the
designs suitable for different applications with varying degrees of power,
noise, size, space, and reliability requirements.

###  II-A Mechanical coupling

The first to be discussed is a group of devices in which the power is conveyed
mechanically from low- to high-voltage sides. Commonly called electrostatic
generators, those devices are the first to appear in the history of high
voltage dc generators, dating back as far as 1663 [17]. Just to name some of
the most famous designs we refer to Kelvin water dropper [18], Wimshurst
influence machine [19], Van de Graaff generator [20], Felici’s rotating
cylinder [21], and a varying capacitance machine [22, 23].

Figure 1: The working principle of electrostatic generators, which generate a
high-voltage dc by using mechanical energy coupling.

Fig. 1 depicts the working principle of electrostatic generators. Firstly, an
insulated object is loaded with some amount of static charge (marked with
\small1⃝ in Fig. 1). Methods for charging the insulated object include the
triboelectric effect, the electrostatic induction, an electret, a corona
discharge, and a direct contact with a voltage source such as a battery.

Secondly, the object is moved across the insulation barrier toward the high-
voltage side mechanically (marked with \small2⃝). As the object is pulled away
from the low-voltage side, the mechanical energy is converted into electric
potential energy of the charge. Popular methods to actuate the moving part
include a rotor, a circulating belt, and a vibrator.

Lastly, after the charge has gained enough electric potential energy, it is
dumped to a filter capacitor (marked with \small3⃝). The role of the capacitor
is to smooth the output voltage ripple and, in the case of pulsed
applications, to accumulate enough energy for a high-current discharge. Once
this step is completed, the moving part is returned to the low-voltage side
and the process repeats.

After more than a century of study, electrostatic generator is still a subject
of active research. This largely owes to the advances in the field of micro-
electromechanical systems (MEMS) and the microfluidics technology, combined
with the interest in developing a miniature energy harvesting generator to
power small portable devices. Recent works based on microfluidics technology
include a chip-based Kelvin dropper [24], a ballistic Kelvin dropper [25], and
a mercury-droplet-based influence machine [26]. Recent works based on MEMS
technology include a varying capacitance machine [27, 28] and an influence
machine [29],

The last to mention is a peeling-tape-based X-ray source[30, 31]. The
operation is based on triboluminescence, a phenomenon that can be understood
by the same principle discussed in this section: the peeling of the tape
creates and separates a static charge (\small1⃝ and \small2⃝ of Fig. 1,
respectively); this charge immediately jumps across the vacuum insulation gap
and its high-energy recombination generates X-ray.

###  II-B Capacitive and inductive coupling

Figure 2: The structure of voltage multipliers, which transfer the energy from
low- to high-voltage side via capacitive and/or inductive coupling.

Capacitive and inductive coupling is extensively used for building a high
voltage dc generator. Fig. 2 shows a generalized structure of such system. It
consists of an ac voltage source, a capacitive and/or transformer network, an
$n$-stage voltage multiplier (i.e., $n$ half-wave rectifiers in series), and a
filter capacitor network.

The ac-coupling network’s role is to transfer the ac voltage $v_{ac}$ from the
low-voltage side to the high-voltage side through the insulation barrier.
Ideally, the coupling network presents zero impedance at the switching
frequency, thus the ac voltages $v_{ac,1},\dots,v_{ac,n}$ are equal in
amplitude to the input ac voltage $v_{ac}$. The coupling network may be built
with capacitors (e.g., [7, 8, 9, 10]), transformers (e.g., [32, 33, 34]), or
both (e.g., [11, 12, 13]). A step-up transformer with a Cockcroft-Walton
multiplier is a popular combination and often appear in literature. Different
capacitor-transformer structures result in different dc voltage stress that
each of the components need to withstand, their necessary capacitance and
inductance values, as well as the output impedance and power of the
multiplier. The specifics of the coupling network’s structure are determined
based on the required performance, limits on the total space and weight,
availability of high voltage parts, etc.

The filter capacitor network provides dc voltages denoted by
$v_{dc,1},\dots,v_{dc,n}$ by smoothing the half-wave rectifiers’ output. The
capacitors in the network are connected in either series or parallel, or
sometimes a mixture of both. Because a voltage multiplier is often built with
only capacitors and diodes, many multiplier topologies are named based on
their capacitor network structures. Cockcroft-Walton (or Greinacher) type [8,
7] is when all the capacitors are serialized in both the ac-coupling capacitor
network and the filter capacitor network. Dickson type [9, 10] is when all the
capacitors are in parallel. A mixture of series and parallel capacitors can
also be found in literature, for example, [35, 36].

##  III High Voltage by Optical Power Transfer

Compared to high voltage generators based on mechanical coupling
(electrostatic generators) and capacitive-inductive coupling (voltage
multipliers), investigations on optical power transfer for high voltage
generation are relatively lacking. Optical power-based systems have many
potential benefits which are to be discussed in this section. The biggest
downside of optical power transfer is the low power efficiency and
consequently a low available output power. Therefore, this scheme should be
used only for applications that need high voltage but not much power.

Here, we focus on a system that uses fiber-coupled laser to transfer power.
The reason is twofold: First, a laser of sufficiently high output power and
high intensity is readily available on the market. Second, using a fiber-
coupled laser allows us to configure the experiment in a flexible manner
without having to fix the relative location of the light source and the
receiver.

###  III-A System powered by a continuous-wave laser

Using fiber-coupled laser to deliver power to low-voltage electronics has been
investigated by several authors [37, 38, 39, 40]. One can easily conceive an
approach where the load electronics in those works are replaced with a dc-to-
dc high voltage generator. Fig. 3 shows the configuration of such system.
Using continuous-wave laser allows one to operate the PV cell at its maximum
power point. This potentially leads to a high efficiency as well as high
available power.

Figure 3: The configuration of a high voltage generator based on fiber-coupled
continuous-wave laser.

If a sufficiently large number of PV cells are in series, the voltage produced
from the PV array can be high enough such that the dc-to-dc voltage boost is
not necessary. Such is the case in works by Lee et al. [14] and Ortega et al.
[15] where authors integrated over a hundred of PV cells on a silicon die to
produce an open-circuit voltage of 100 V and higher.

The work by Rentmeister et al. [16] is perhaps the closest to the strategy
described in Fig. 3. The authors firstly produced 125 V using 196 on-chip PV
cells, then used an off-chip Dickson voltage multiplier to generate over 600 V
dc output. The authors made it clear that their method can achieve 1.5 kV
output voltage and even higher by simply increasing the number of PV cells in
the array.

One disadvantage of a continuous-wave based high-voltage generation is the
necessity for an dc-to-ac converter (inverter). Unless special care is taken
such as implementing the inverter circuit on a low-power integrated circuit
[16], the dc-to-ac conversion may overburden the limited power budget of a
solar cell.

Also, using a PV array necessitates balancing the optical power among
individual cells so as to maximize the output power from the array. Spreading
the light evenly over multiple cells may add to the difficulty of
implementation, especially when the light is from a fiber-optic cable which is
effectively a point light source. Moreover, such a massively integrated PV
array is not readily available on the market; at this moment it is only
available to those who can afford the time and cost for development of a
custom integrated circuit.

If instead a single PV cell is used, the output voltage from the cell is
around 0.6 V. This voltage is insufficient for many electronic circuits and
power transistors which demand at least 0.7 V and often higher. There are some
relaxation oscillators and integrated circuit solutions that work on a supply
voltage of 0.6 V or less, for example, Joule thief oscillator [41] and an
energy harvesting chip [42]. However, they still suffer from a too low
available power or output voltage.

###  III-B System powered by a pulsed laser

(a)

(b)

Figure 4: The configuration of a high voltage generator based on fiber-coupled
pulsed laser. (a) Schematic. (b) Light intensity versus time plot. $f$ and $D$
denote the switching frequency and duty cycle of the pulsed laser,
respectively.

In order to overcome the discussed shortcomings of a system based on
continuous-wave laser, we propose a scheme based on a pulsed laser. Fig. 4
illustrates the proposed arrangement. The laser diode is driven on and off
repeatedly with a set frequency $f$ and a duty cycle $D$ (Fig. 4b) and
consequently an ac voltage is produced from the PV cell.

The advantages of the proposed approach is as follows. First, the receiving
end does not need an inverter and thus the circuit design is simplified.
Whereas a continuous-wave system needs to perform dc-to-ac conversion followed
by ac-to-dc conversion, the proposed pulsed system’s receiving end only needs
to perform ac-to-dc conversion. As a result, any overhead associated with the
inverter circuit is avoided.

Second, when multiple high-voltage circuits are cascaded, it is easy to
stagger the phases of pulsed lasers driving multiple modules and have the
ripples cancel each other. Ideally, when $n$ modules are cascaded, one can
reduce the ripple ratio by $n$ times or more using this phase interleaving
technique. Previous works [43, 13, 44, 45] have investigated the idea of
reducing the output voltage ripple by segmenting a voltage multiplier and
driving each segment at staggered phases.

Fig. 5 illustrates a conceptual ripple reduction when a multiplier is
segmented into three parts and their switching phases are staggered by 120∘ to
each other. The output voltage $v_{out}$ is the sum of each ac-to-dc circuit’s
output voltages $v_{1}$, $v_{2}$, and $v_{3}$. Compared to the case without
interleaving where $v_{1}$, $v_{2}$, and $v_{3}$ waveforms overlap, the phase-
interleaved scheme reduces the ripple amplitude of $v_{out}$ by more than
threefold. A more general in-depth study of the interleaving principle can be
found in [46].

Figure 5: Conceptual illustration of ripple reduction by phase interleaving.
Segmenting the circuit into three parts and driving them at 120∘ out of phase
reduces the ripple amplitude of $v_{out}$ by more than three times.

The proposed pulsed-laser system can easily achieve the ripple reduction by
phase interleaving because all the laser diode drivers are placed on the low-
voltage side and share the same ground potential. In contrast, a high voltage
generator consisting of continuous-wave-driven modules need means of
communication between modules, otherwise the peak-to-peak ripple voltage will
be that of the worst possible case because of frequency mismatches between
oscillators.

At the same time, this high voltage generation method has several limitations.
The first is a relatively low power efficiency, because the PV cell cannot
operate at its maximum power point like the continuous-wave system. The second
is a relatively low power level, partly because of the low efficiency and
partly because the power is transferred only for fraction of time. Another
limitation of the proposed method is that the ac voltage amplitude from the PV
cell is inherently restricted to the turn-on voltage of the intrinsic p-n
junction diode.

###  III-C Comparison with other power transfer methods

Energy coupling through an insulation barrier by a pulsed laser enables
extensive electrical and physical isolation between the low-voltage input side
and the high-voltage output side. This feature leads to following advantages
that do not exist in systems that rely on mechanical or capacitor-inductive
energy couplings only.

First, optical energy coupling can be used to break down a long voltage
multiplier into smaller segments and thus prevent an excessive output voltage
drop. For a conventional $n$-stage Cockcroft-Walton voltage multiplier, the
deviation of the actual output voltage from the ideal value is often called an
output voltage drop and denoted by $\Delta V$. This $\Delta V$ is almost
proportional to $n^{3}$. By breaking down the $n$-stage multiplier into $m$
segments of $(n/m)$-stage multipliers, and powering each of those multipliers
via optical power transfer, one can reduce $\Delta V$ by almost $m^{2}$ times.

More specifically, the formula for $\Delta V$ of an $n$-stage Cockcroft-Walton
multiplier is

$\Delta
V=\frac{i_{out}}{fC}\left[\frac{2}{3}n^{3}+\frac{1}{2}n^{2}-\frac{1}{6}n\right]$
(1)

where $i_{out}$ is the average output dc current, $f$ is the ac input
frequency, and $C$ is the capacitance value of coupling capacitors. (The
derivation of this formula can be found in many academic papers, e.g., [8, 47]
and textbooks [48, 49, 50].) If this multiplier is broken into $m$ segments in
series, the new voltage drop $\Delta V^{\prime}$ is (assuming $m$ is a factor
of $n$)

$\displaystyle\Delta V^{\prime}$
$\displaystyle=\frac{i_{out}}{fC}\left[\frac{2}{3}\left(\frac{n}{m}\right)^{3}+\frac{1}{2}\left(\frac{n}{m}\right)^{2}-\frac{1}{6}\left(\frac{n}{m}\right)\right]\times
m$ (2)
$\displaystyle=\frac{i_{out}}{fC}\left[\frac{2}{3}\left(\frac{n^{3}}{m^{2}}\right)+\frac{1}{2}\left(\frac{n^{2}}{m}\right)-\frac{1}{6}n\right].$
(3)

When $n$ is sufficiently large, $\Delta V^{\prime}$ is almost $m^{2}$ times
smaller than $\Delta V$.

The second advantage of building a multiplier based on optical power transfer
is that it is easy to separate the high voltage part physically from the rest
of the system. Power from a fiber-coupled laser is easily routed losslessly
via long and pliable fiber optic cables. As a result, rectifiers on the
receiving end of the cable can be arranged in any way the designer sees fit.
The physical and electrical isolation of the high voltage part can be taken
advantage for better protection of low voltage circuits against arc discharge
events.

The spatial separation also allows better protection of the load side from
electromagnetic noise, which is important for sensitive scientific
applications that need a very low noise environment. In contrast, voltage
multipliers based on transformer networks need careful design and placement of
magnetic cores to minimize flux leakage. Also, because magnetic cores are
often rigid, brittle, and difficult to machine, they impose constraints on the
placements of transformers and the circuits surrounding them.

##  IV Prototype Circuit Driven by Pulsed Laser

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: The prototype high voltage dc generator powered by fiber-coupled
laser. (a) Schematic. (b) Implemented assembly of laser and a PV cell. A
visible laser (638 nm) is used in this photo only for the purpose of showing
the laser-illuminated area. All subsequent experiments were conducted with 808
nm laser. (c) Voltage multipliers built on a flexible circuit board, before
folding (left) and after (right). (d) Completed high voltage generator module
after assembly. (e) Size comparison with a US quarter coin. (f) Experimental
setup.

Fig. 6 shows the implemented light-powered high voltage generator. As
described in Fig. 6a, the circuit consists of a PV cell, a 1:100 step-up
transformer, and a 84-stage bipolar Cockcroft-Walton voltage multiplier. The
laser is pulsed at frequency $f$ and as a result the output voltage $V_{out}$
appears across the load resistor $R_{load}$. The multiplier is built on a
flexible circuit board as shown in Fig. 6c. This is so that we can fold the
multiplier into a small box-shaped volume and thus realize a compact overall
power supply design, as can be seen in Fig. 6d. Table I lists the parts used
in this implementation.

TABLE I: Component list of the prototype high voltage dc generator in Fig. 6a. Name | Part number, Manufacturer | Description
---|---|---
laser | C3743, CivilLaser | 808 nm, 2 W
PV cell | KXOB25-14X1F, IXYS | monoX-Si
$T$ | LPR6235-752S, Coilcraft | 1:100 turns ratio
$D$ | BAS16HLP-7, Diodes Inc. | 100 V, 215 mA
$C$ |  GRM155R62A104KE14D, Murata | 0.1 $\mathrm{\SIUnitSymbolMicro}$F, X5R

The equivalent circuit model of Fig. 6a is shown in Fig. 7. Pulsed laser
shining on the solar cell generates a relatively small positive current
$i_{pv}$. This current by itself may not be large enough for a sufficient ac
voltage amplitude $V_{pri}$.

To maximize $V_{pri}$, we take advantage of the parallel $LC$ circuit
consisting of the transformer’s magnetizing inductance $L_{tr}$, the PV cell’s
junction capacitance $C_{pv}$, and the voltage multiplier’s input capacitance.
We match the frequency of the pulsed laser (hence the frequency of the input
current $i_{pv}$) to the resonant frequency of the $LC$ circuit. In this way
the resonant ac current dominates the total current circulating in the
circuit, resulting in an ac voltage $V_{pri}$ much larger than without
resonance.

Figure 7: The equivalent circuit model of the prototype high voltage dc
generator in Fig. 6a.

###  IV-A Equivalent circuit model and characterization

We firstly characterize the implemented power supply in Fig. 6a using the
equivalent circuit model of Fig. 7. The first step is to determine the PV
cell’s circuit model. The input current $i_{pv}$ is equal to 98 % of the
number of incident photons per second according to the datasheet ([51]; the
external quantum efficiency for 808 nm wavelength is 98 %). The diode $D_{pv}$
is a silicon p-n junction diode with a forward voltage of around 0.6 V.

To figure out the PV cell’s capacitance $C_{pv}$, we temporarily disconnect
the voltage multiplier from the transformer to make a parallel $LC$ tank. This
$LC$ tank consists of $C_{pv}$ and the transformer’s magnetizing inductance
$L_{tr}$ of 7.5 $\mathrm{\SIUnitSymbolMicro}$H. We then shine a pulsed laser
light at 50 % duty cycle on the PV cell while changing the laser’s pulsing
frequency. The $LC$ tank is found to resonate at 74 kHz, from which $C_{pv}$
is calculated as 0.62 $\mathrm{\SIUnitSymbolMicro}$F.

To measure the parasitic series resistance $(R_{pcb}+R_{tr})$, we firstly use
pulsed laser to cause resonance at 74 kHz, then turn off the laser and observe
the oscillating voltage across the PV cell. The voltage decays exponentially
with the time constant of 96.2 $\mathrm{\SIUnitSymbolMicro}$s, from which
$(R_{pcb}+R_{tr})$ is found to be 0.16 $\Omega$.

###  IV-B Test results: single module

We test the performance of the circuit in Fig. 6 using a range of load
resistances and laser pulsing frequencies. Throughout the test, the laser
pulse’s duty cycle is fixed to 50 % and its average output power to 1.2 W
(i.e., 2.4 W peak power).

For voltage measurements we use a high voltage differential probe, Rigol
RP1100D, which has an input resistance of 100 M$\Omega$. This input resistance
of the probe is too low for the load resistance range we want to use, from 100
M$\Omega$ up to 2100 M$\Omega$. Connecting the probe in parallel with a load
resistance would cause significant loading effect.

To circumvent this issue, we connect multiple 100 M$\Omega$ resistors in
series to the differential probe and use the resistor–probe–resistor chain as
the load resistor for the experiment. This resistor chain doubles as an
$n$-to-1 voltage divider, which adequately scales down the voltage and
prevents the possible overvoltage damage to the probe. The use of voltage
dividers is the reason for discrepancies between the apparent and actual
voltage scales in the oscilloscope screen captures in the following sections.

Results from preliminary experiments reveal that the output voltage $V_{out}$
peaks at about 12 kHz to 20 kHz pulsing frequency $f$. We therefore vary the
frequency from 10 kHz to 22 kHz in 2-kHz increments. The load resistance
$R_{load}$ is varied from 100 M$\Omega$ to 700 M$\Omega$ so that sufficient
data are obtained for estimation of the open-circuit output voltage.

(a)

(b)

(c)

(d)

Figure 8: Summary of the test results for a single high-voltage module. (a)
Heat map of the measured $V_{out}$ values in kilovolts for the entire
$R_{load}$ and $f$ ranges. (b) Waveforms at the point of highest measured
output power, $R_{load}$ of 100 M$\Omega$ and $f$ of 12 kHz. (c) Waveforms at
the point of highest measured output voltage, $R_{load}$ of 700 M$\Omega$ and
$f$ of 20 kHz. $V_{out}$ is measured via a 7:1 voltage divider, hence the
seven times discrepancy between the apparent and actual voltage scales. (d)
$I$–$V$ curve consisting of data points with the highest output voltage for
each load resistance value.

Fig. 8 summarizes the test results. Fig. 8a shows measured $V_{out}$ values in
kilovolts for the entire $R_{load}$ and $f$ ranges. Two trends stand out in
the plot. First, the output voltage $V_{out}$ increases with $R_{load}$. This
is because a higher load resistance causes a low average output current, which
results in a less output voltage drop as described by (1). Second, as
$R_{load}$ increases, the frequency $f$ that maximizes $V_{out}$ also
increases. This positive shift in the resonant frequency is because a higher
$R_{load}$ leads to a larger $V_{out}$ as previously explained, and the larger
$V_{out}$ means a larger ac voltage swing across each of the rectifying
diodes. Because a p-n junction diode capacitance decreases exponentially with
the reverse bias voltage, a higher ac voltage swing means a smaller effective
capacitance presented by the diodes. As a result, the sum of all the diode
junction capacitances decreases, thus the voltage multiplier’s input
capacitance decreases, and consequently the resonant frequency $f$ increases.

Within the data of Fig. 8a, $R_{load}$ of 100 M$\Omega$ and $f$ of 12 kHz
yields the highest output power 68.6 mW, resulting in the light-to-electricity
power conversion efficiency of 5.7 % (Fig. 8b). In addition, $R_{load}$ and
$f$ of 700 M$\Omega$ and 20 kHz yields $V_{out}$ of 5.49 kV which is the
highest output voltage measured from this single high-voltage module (Fig.
8c).

When the circuit operates at the resonance, the measured voltage swing of
$V_{pri}$ is always limited to approximately $\pm 0.5$ V, and consequently the
voltage swing of $V_{sec}$, about $\pm 50$ V throughout the experiment. Two of
the examples can be seen in Fig. 8b and Fig. 8c. The reason is probably that
the positive voltage swing of $V_{pri}$ is clipped by the PV cell’s intrinsic
diode $D_{pv}$.

Selecting a data point of the highest output voltage for each load resistance
gives the $I$–$V$ curve of Fig. 8d. From the fitted curve the open-circuit
voltage is estimated to be 6.49 kV although its actual value is likely to be
higher considering the rate of increase in $V_{out}$ at low values of the
output current $i_{out}$. The measured output resistance of 155 M$\Omega$ is
roughly in line with theoretical calculations [48, 49, 50].

###  IV-C Test results: three modules cascaded

(a)

(b)

(c)

(d)

(e)

Figure 9: The prototype laser-powered high voltage dc generator consisting of
three modules cascaded. (a) Schematic. (b) High-voltage modules, before
stacking. (c) Finished high-voltage modules (front). (d) Finished high-voltage
modules (back). (e) Experimental setup.

We proceed to cascade three high-voltage modules in series and thereby produce
an output voltage three times that of a single module. The purpose is to
demonstrate that cascading multiple units of the proposed circuit allows one
to obtain a higher output voltage without performance degradation usually seen
in other high voltage generation methods. Fig. 9 shows the implemented circuit
and the experimental setup.

(a)

(b)

(c)

Figure 10: Test results for three high-voltage modules cascaded. (a) Heat map
of the measured high voltage output $V_{out,3}$ in kilovolts. (b) The ratio of
three-module output voltage $V_{out,3}$ to the single module output voltage
$V_{out}$. (c) One of the occasional voltage spikes measured when $V_{out,3}$
was larger than 12 kV. $V_{out}$ is measured via a 18:1 voltage divider, hence
the 18 times discrepancy between the apparent and actual voltage scales.

Fig. 10a shows the test results. Fig. 10a is the heat map of the measured
output voltage $V_{out,3}$. As one can see from Fig. 10b plotting
$(V_{out,3}/V_{out})$, for all frequency and load conditions $V_{out,3}$
values have almost tripled compared to the output voltages $V_{out}$ from a
single module shown in Fig. 8a. The voltage increase is slightly more than
three-fold at a low voltage ($V_{out,3}\lessapprox 12$ kV) because newly built
modules produced a higher voltage then the one tested in section IV-B. On the
other hand, the voltage increase is reduced to less than three-fold as the
output voltage becomes higher ($V_{out,3}\gtrapprox 12$ kV), presumably
because of corona discharge that onsets at around 6 kV ground-referenced
potential. To corroborate our claim, a hissing sound from the circuit was
evident when the output voltage exceeded 10 kV. Also, occasional voltage
spikes such as Fig. 10c were continually measured at the output, suggesting
partial discharges (localized electrical discharge) occurring within the
circuit.

###  IV-D Potential improvements and future opportunities

In designing and implementing the light-powered high voltage generator, we
encountered several practical difficulties. First, there was no off-the-shelf
PV cell readily available to match the area irradiated by the laser. As can be
seen in Fig. 6b, the PV cell we used is in a long rectangular shape whereas
the laser-illuminated area is only a small circular patch at the center. As a
result, more than two third of the PV cell junction area contributes nothing
but instead increases the capacitance $C_{pv}$ and the current through
$D_{pv}$, both of which increases the power loss. Second, although using a PV
array instead of a single cell to increase the ac voltage swing would have
helped, such design was not attempted due to the difficulty of evenly
distributing the laser light over multiple cells. Third, related to first and
second, low voltage swing from the PV cell led to the use of a high turns-
ratio transformer. Because of this high turns ratio, the capacitance of the
voltage multiplier presented a large value when reflected to the transformer’s
primary side. This effect, combined with the presence of extra large $C_{pv}$,
lowered the resonant frequency of the circuit and thus resulted in the huge
output resistance as seen in Fig. 8d [48, 49, 50].

In view of our findings, we list the following possible modifications to
improve the performance of the circuit and bring it closer to becoming a
competitive solution for high voltage applications. The first is to customize
the PV cell’s shape so that the unused junction area is minimized for a
smaller capacitance $C_{pv}$. The second is to use a multi-cell array with a
means of distributing light so that a high-amplitude ac voltage is produced.
Then the transformer turns ratio can be lowered. The first and the second
approaches combined will increase the resonant frequency of the circuit and,
therefore, increase the available output power from the multiplier. Needless
to say, using a higher-power light source with the wavelength closer to the
absorption edge of the PV cell will further improve the performance of the
circuit.

##  V Application Example on Electroadhesion

This section shows a possible application of the proposed technique. We
demonstrate a simple gripper that grips and moves an object by using an
electroadhesive pad [52, 53] powered by the developed high voltage generator.

The electroadhesive pad, shown in Fig. 11a, consists of interdigitated
electrodes laid on and insulated by a thin polyamide film. When a voltage of
several kilovolts is applied to the electrodes while the pad is in contact
with a material surface, an opposite static charge is induced on the surface.
The resulting electrostatic attraction between the pad and the material
enables gripping and material handling. Particularly in the case of our
electroadhesive pad, the adhesive force under an ideal condition is measured
to be over 100 gram-force at the applied voltage of over 2 kV, as plotted in
Fig. 11b. (The data are taken from our previous work in which we used the same
pad for a similar type of demonstration [54].)

(a)

(b)

Figure 11: The electroadhesive pad used for the demonstration, measuring 10
cm$\times$10 cm in size. (a) Photograph. (b) Adhesive force versus applied
voltage measured under an ideal condition (taken from our previous work [54]).

Let us assume a scenario where one wants to design a gripper that has an
electroadhesive pad mounted on tip of an actuation mechanism such as a robot
arm. In this scenario, the designer wants to electrically isolate the high
voltage generator and the pad so that no single-point insulation failure leads
to dangerous electrical discharge through either the user or the handled
object. This requirement for electrical safety is comparable to the Class-II
standard (also known as double-insulation) determined for consumer appliances
by International Electrotechnical Commission (IEC) [55].

For electrical isolation, the most common and straightforward approach is to
use an isolation transformer as in Fig. 12a. The transformer should not only
be able to withstand several kilovolts of dc potential difference between the
primary and secondary windings, but also should have a low inter-winding
capacitance to prevent the ac component of the discharge current from flowing
across the windings.

(a)

(b)

Figure 12: Structure comparison of electroadhesive gripper designs. (a)
Approach using an isolation transformer. (b) Approach using the proposed high
voltage generator.

The potentially large size and weight of the isolation transformer can be
avoided by using the proposed light-powered high voltage generator as
described in Fig. 12b. The fiber optic cable provides a mechanically flexible
power coupling with minimal stray capacitance and high insulation voltage.
This scenario is where the proposed high voltage generator can be superior
over conventional solutions, because the power consumption of the
electroadhesive pad is minuscule (Measured current is well below 10 nA at 3 kV
applied voltage.) and the means of electrical insulation dominates the total
system build cost.

Figure 13: The implemented electroadhesive gripper.

Fig. 13 is the implemented gripper. We attach a wooden stick to the
electroadhesive pad perpendicularly to use it as a handle. The high voltage
module is taped to the wooden stick, and the output nodes are wired to the
pad’s electrodes.

(a)

(b)

(c)

Figure 14: Demonstration of the electroadhesive gripper lifting an envelope
weighing 12 grams. (a) Experimental setup. (b) Gripper lifted when the laser
diode is powered off. (c) Gripper lifted when the laser diode is powered on.
Notice the envelope attached to the pad and lifted.

Fig. 14 demonstrates the operation of the electroadhesive gripper. The pad on
the gripper is pressed against a regular postal envelope weighing 12 grams.
When the laser diode remains powered off (Fig. 14b), the gripper does not lift
the envelope. When the diode is turned on (Fig. 14c), the high voltage
generator energizes the electroadhesive pad, inducing electrostatic force on
the pad and lifting the envelope. All the while, the fiber optic cable
mechanically decouples the laser diode from the gripper, thereby minimally
restricting the movement of the gripper.

##  VI Conclusion

In this paper, we investigated the feasibility of high voltage generation by
optical power transfer. We reviewed the existing approaches for high voltage
dc generation and classified them by how the input-to-output dc isolation is
achieved. From that perspective, electrostatic machines are based on
mechanical input-to-output energy coupling, and voltage multipliers are based
on capacitive and/or inductive coupling. The approach subject to our study was
optical energy coupling, that is, using fiber-coupled laser and a PV cell to
deliver power from the low-voltage input side to the high-voltage output side.
The optical scheme’s high input-to-output electrical insulation and spatial
separation are helpful for cascading multiple circuits in series for a higher
output voltage. Different from other optically powered high voltage generators
using continuous-wave light and an integrated PV array, our design used pulsed
laser and discrete components for simple construction.

We experimentally demonstrated the operation of the proposed pulsed-laser
powered high voltage generator. A PV cell converts a laser pulse train to a
low-voltage ac voltage, which is amplified to become a high-voltage ac via a
step-up transformer and finally becomes a high-voltage dc via a diode-
capacitor voltage multiplier. We used a light source of 1.2 W average output
at 10-to-22 kHz pulsing frequencies to power the circuit and measured 5.5 kV
dc output from a single module. From the subsequent experiment we powered
three cascaded modules using three laser diodes and measured 14.7 kV at the
output.

As a concluding remark, we list a few questions to suggest future directions
for improved light-based high voltage generation suitable for real-world
applications. First, higher switching frequencies: For applications that
demand a low ripple output, can a light-powered circuit be built to operate at
high-enough frequencies so as to make a separate output filter capacitor
unnecessary? Second, ripple-free dc generation: Can a ripple-free high voltage
dc be obtained by massively integrated PV arrays with no switching elements?
Third, system optimization: For a given dc output voltage, how can one
determine the optimal proportion of voltage gains by a PV array, a
transformer, and a multiplier? Lastly, system miniaturization: Other than the
transformer, can all the components in the high voltage module be integrated
into a monolithic circuit for compact system arrangement? By addressing those
questions we hope that future studies will identify design spaces where light-
based systems are a superior solution in terms of performance, build cost, and
reliability.

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