# Detecting the Féeton Fifth Force by Superconducting Josephson Junctions

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>Kavli IPMU
(WPI), UTIAS, University of Tokyo, Kashiwa, 277-8583, Japan  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

The intriguing $U(1)_{B-L}$ extension of the standard model predicts a fifth
force between particles carrying $B-L$ charges. The mediator is the $B-L$
gauge boson called Féeton. In this letter, we propose a novel experimental
design to detect the quantum phase difference caused by this fifth force using
a superconducting Josephson junction. We find that the experiment has the best
sensitivity to the gauge coupling when the gauge boson is within the mass
range of $10^{-2}\,$eV to $100\,$eV.

Introduction– The $B-L$ extension of the standard model (SM) is the most
attractive option beyond the SM. It predicts three right-handed neutrinos
(RHNs) to cancel the gauge anomalies. In addition, the high symmetry breaking
scale generates large Majorana masses for the RHNs, which is a key point for
the natural explanation of the observed small neutrino masses through the
seesaw mechanism Minkowski:1977sc ; Yanagida:1979as ; Yanagida:1979gs ;
GellMann:1980vs , as well as for the creation of the baryon asymmetry in the
present Universe via the leptogenesis Fukugita:1986hr ; Buchmuller:2005eh .
Although the $B-L$ breaking scale receives some limits Wilczek:1979hh , the
mass of the gauge boson (called as ”Féeton” Choi:2020kch ; Okada:2020evk ;
Lin:2022xbu ) is however unpredictable. The Féeton mass is determined by the
product of the $B-L$ gauge coupling constant $g_{B-L}$ and the breaking scale
$V_{B-L}$. The gauge coupling remains an unknown free parameter.

However, there exist several constraints on the Féeton gauge coupling
constant, as illustrated in Figure 3. For Féeton masses smaller than
$10^{-2}\,$eV, the gauge force exhibits long-range behavior and receives
strong constraints due to gravitational wave detection LIGOScientific:2021ffg
; Miller:2023kkd ; Frerick:2023xnf  and torsion-balance experiments
Wagner:2012ui ; Adelberger:2009zz . Meanwhile, the mass larger than $100\,$eV
is also severely restricted by star coolingHardy:2016kme , collider
ATLAS:2017fih ; CMS:2016cfx ; LHCb:2017trq ; FASER:2023tle  and fix target
experimentsGninenko:2016kpg ; LDMX:2018cma ; Chen:2022liu ; NA64:2022yly .
Notably, the constraints for the intermediate mass range are comparatively
weaker. The purpose of this paper is to propose a novel experiment to measure
the Féeton fifth force in this particular mass region.

The experimental setup proposed consists of a parallel array of Josephson
junctions. Because of the distance dependence of the $B-L$ gauge potential,
the two superconductors inside a junction feel different fifth force
strengths. Consequently, the two quantum superconducting states will evolve to
have a phase difference after a period of time. Such a phase difference can be
converted into a detectable charge current due to the Josephson effects. We
first give a general derivation of the phase difference induced by the fifth
force. Then, the experimental design and the predicted current are presented.
After considering all the backgrounds, we finally establish the projected
constraint on the Féeton coupling strength.

Phase Difference Induced by $B-L$ Gauge Interaction – In this paper, we
consider the simple extension of SM with a $U(1)_{B-L}$ gauge symmetry. The
detailed definition of the model is given in the Appendix.

The model leads to a new gauge force mediated by the $B-L$ gauge boson,
Féeton. In the non-relativistic regime, such a vector interaction can be
written in terms of the Yukawa type of potential function,

$V_{B-L}(r)=g_{B-L}^{2}\frac{Q_{B-L}e^{-m_{A^{\prime}}r}}{r}.$ (1)

The $Q_{B-L}$ is the product of the corresponding $B-L$ charges of two
interacting bodies.

According to the Schrodinger equation, $i\partial_{t}\psi=\hat{H}\psi$, the
quantum phase $\phi(r,\tau)$ of any stationary state feeling the $B-L$ gauge
force will evolve as $i\phi(r,\tau)=i\int_{0}^{\tau}V_{B-L}(r)dt$. Since the
potential is distance-dependent, two states in different positions will
acquire a phase difference over a period of evolution. This quantum phase
effect can be detectable and utilized to probe this tiny Féeton fifth force.
The settings are illustrated in Fig. 1.

One first prepares two groups of particles in coherent states, $\ket{1}$ and
$\ket{2}$111They can be some coherent particles splitted into a superposition
of positions, atoms in Bose-Einstein condensation, Cooper pairs in
superconductor, and so on. Our work focuses on the case of Cooper pairs.. Both
states have a configuration with size $a$, $B-L$ charge $Q_{B-L}$, and are in
different space positions with a small separation of $\epsilon$. At the
distance of $d$ 222The Cooper pair has the same phase $\phi_{1}$ or $\phi_{2}$
as shown in Eq.(6) inside the superconductor 1 or 2, respectively. Thus the
distance $d$ means the average distance of the superconductor 1 from the
surface of the plate., there is a plate (a thick film) made of neutral atoms,
which has a thickness of $b$ and its length size is large enough to be
approximated as infinity. The above four lengths have the following hierarchy,
$b>d\gg\epsilon\sim a$. While the $B-L$ charges of protons and electrons
inside an atom cancel each other out, the neutrons contribute to a non-zero
$B-L$ charge. The neutron number density of the plate is $n$. With the
integration of contributions from the whole volume, the phase of $\ket{1}$
induced by $B-L$ gauge potential with a distance $d$ is,

$\displaystyle\phi_{f}(d)$ $\displaystyle=\tau
g_{B-L}^{2}Q_{B-L}n\int_{0}^{b}\int_{0}^{\infty}\frac{e^{-m_{A^{\prime}}\sqrt{(d+z)^{2}+r^{2}}}2\pi
r}{\sqrt{(d+z)^{2}+r^{2}}}drdz$ $\displaystyle=\tau
g_{B-L}^{2}Q_{B-L}n\frac{2\pi
e^{-m_{A^{\prime}}d}\left(1-e^{-m_{A^{\prime}}b}\right)}{m^{2}_{A^{\prime}}}.$
(2)

for a time period $\tau$. Here, index $f$ labels for the fifth force. For
$\ket{2}$, the distance argument is $d+\epsilon$. Because of the position
difference, the $B-L$ gauge interaction sourced from the plate induces a phase
difference $\Delta\phi_{f}$ between states $\ket{1}$ and $\ket{2}$ as,

$\Delta\phi_{f}\equiv\phi_{f}(d)-\phi_{f}(d+\epsilon),$ (3)

In the limit of tiny separation $\epsilon\ll d$ and long range force $d\ll
1/m_{A^{\prime}}$, $e^{-m_{A^{\prime}}(d+\epsilon)}\simeq
e^{-m_{A^{\prime}}d}(1-m_{A^{\prime}}\epsilon)\simeq(1-m_{A^{\prime}}\epsilon)$.
To maximize the signal, the thickness should be large enough that,
$e^{-m_{A^{\prime}}b}\sim 0$. Thus, the Eq. (3) becomes,

$\Delta\phi_{f}\simeq\frac{2\pi\tau
g_{B-L}^{2}Q_{B-L}n\epsilon}{m_{A^{\prime}}}.$ (4)

With the experimental materials determined, the phase difference can be
enhanced by adjusting the running time $\tau$ and the fifth force range
$1/m_{A^{\prime}}$. A longer force range makes the quantum states feel more
$B-L$ charges in plate, and a larger phase is accumulated over an extended
period of time.

Figure 1:  Two configurations of coherent states $\ket{1}$ and $\ket{2}$ with
sizes $a$ are in different positions with a separation of $\epsilon$. A large
plate with a thickness $b$ made of neutral atoms is placed at a distance of
$d$. It carries the neutron ($B-L$ charge) density $\rho$ and serves as a
potential source. The above four length parameters have the relationship as
$a\sim\epsilon\ll d<b$.

Josephson Junction Detector – We have seen that the $B-L$ gauge interaction
induces a phase difference between two groups of coherent states in different
space positions. The superconducting Josephson Junction (JJ) Josephson:1962zz
; Josephson:1974uf  can transfer such a phase difference to charge current
which can be detectable if it is sufficiently large.

A JJ is made of two separated superconductors and an insulator with width
$\epsilon$ in between. Once the temperature of the superconductor falls below
its critical temperature, the electrons inside the conductor form Cooper
pairs, becoming in a coherent state, and the resistance disappears. These
Cooper pair states in the two superconductors are the coherent states
$\ket{1}$ and $\ket{2}$ as we want. They can be described by order parameters
Landau:1950lwq ,

$\Psi_{1}=\sqrt{n_{1}}e^{i\phi_{1}}\,,\quad\Psi_{2}=\sqrt{n_{2}}e^{i\phi_{2}},$
(5)

analogous to a quantum mechanical wave function. Here, $n_{i}$ ($i=1,2$) is
the number density over the volume of Cooper pairs and $\phi_{i}$ is the
corresponding phase. For simplicity, we take the two superconductors the same
and $n_{1}=n_{2}$. From the semi-classical London theory, one can estimate the
number density through the London penetration length $\lambda$ and
$n=m_{e}/(e^{2}\lambda^{2})$ London:1935 . Usually, the penetration length
varies from $50\sim 500\,$nm for different types of materials. We take
$\lambda=100\,$nm so that the Cooper pair number density is
$n_{1}=n_{2}=2\times 10^{-20}\,$GeV3.

In the insulator region, there is a barrier potential $V$ higher than the
kinetic energy of electrons $E$. The wave functions can be parameterized as,

$\Psi(x)=C_{1}\cosh{x/\xi}+C_{2}\sinh{x/\xi},$ (6)

with $\xi=\sqrt{1/4m_{e}(V-E)}$. Taking Eq. (5) as boundary conditions, one
can solve,

$C_{1}=\frac{\sqrt{n_{1}}e^{i\phi_{1}}+\sqrt{n_{2}}e^{i\phi_{2}}}{2\cosh(\epsilon/2\xi)}\,,C_{2}=\frac{\sqrt{n_{1}}e^{i\phi_{1}}-\sqrt{n_{2}}e^{i\phi_{2}}}{2\sinh(\epsilon/2\xi)}.$
(7)

Although classically forbidden, the quantum tunneling effect induces a
current. The current density is generally $J=-(2e/m_{e}){\rm
Re}(\Psi^{*}i\nabla\Psi)$, and proportional to the phase difference as
Terry:1901 ,

$J=\frac{e\sqrt{n_{1}n_{2}}}{m_{e}\epsilon}\sin\Delta\phi\,,\quad\Delta\phi\equiv(\phi_{1}-\phi_{2}).$
(8)

First we locate the plate far away from the JJ. In the beginning, the JJ has
initial phases and the Josephson current is induced. However, the current can
be dissipated and the junction goes into its ground state $\Delta\phi=0$.
Then, we put the plate at a close distance $d$ to the JJ. The phase difference
$\Delta\phi$ is caused by the fifth force, which induces an extra electric JJ
current333We denote the Cooper pairs in the superconductors 1 and 2 by
$\Psi(x_{1})$ and $\Psi(x_{2})$, respectively. The Wilson line operator,
$\Psi(x_{2})\exp(i\int^{x_{2}}_{x_{1}}A)\Psi(x_{1})^{*}$, is gauge invariant.
Thus, the phase of this operator is gauge invariant. This shows that the phase
difference $\Delta\phi$ defined in the text is effectively gauge invariant in
our experimental setup.. Considering the size of superconductor $a^{2}$, which
usually taken as $a\simeq 10\,$nm, the current induced by the fifth force
$I_{f}$ is,

$I_{f}=I_{c}\sin\Delta\phi_{f}\,,\quad\text{where}\quad
I_{c}=\frac{en_{1}a^{2}}{m_{e}\epsilon}.$ (9)

For illustration, we propose the plate material to be Graphene made of carbon
atoms. From its mass density, $\rho_{C}=2.27$ g/cm3, one can estimate its
neutron number density $n=6.8\times 10^{23}$ cm-3. Inside a JJ, the insulator
usually has a width $\epsilon=1\,$nm and the Cooper-pair quantum states have a
$B-L$ charge $Q_{B-L}=-1\times 2=-2$ for two electrons. Assuming the running
time is one day, $\tau=86400\,$s, plate thickness $b=1\,$cm, the induced phase
difference Eq. (4), in the long-range-force limit, is,

$\Delta\phi_{f}=2\times
10^{-2}\times\left(\frac{g_{B-L}}{10^{-16}}\right)^{2}\left(\frac{\tau}{1\,\text{Day}}\right)\left(\frac{10^{-2}\,\text{eV}}{m_{A^{\prime}}}\right),$
(10)

which is hopeful of being detected everitt2011gravity . For a tiny phase
difference, $\sin\Delta\phi_{f}\simeq\Delta\phi_{f}$. The corresponding
Josephson current induced from the fifth force is,

$\displaystyle I_{f}$ $\displaystyle\simeq I_{c}\Delta\phi_{f}\simeq\frac{2\pi
e\tau g_{B-L}^{2}Q_{B-L}nn_{1}a^{2}b}{m_{e}}$ $\displaystyle=1\times
10^{-4}\,\text{A}\times\left(\frac{g_{B-L}}{10^{-16}}\right)^{2}\left(\frac{\tau}{1\,\text{Day}}\right)\left(\frac{10^{-2}\,\text{eV}}{m_{A^{\prime}}}\right).$
(11)

Again, with the experimental settings fixed, the signal is proportional to the
running time, force range, and gauge coupling. As indicated by Eq. (10), The
sine periodic variation of the current can even be observed for around one-
year running.

Figure 2:  The experimental design for detecting the $B-L$ gauge interaction.
The Josephson junction (with two superconductors as blue cube and one
insulator as yellow cube) is placed inside a conductor (gray shaded) with a
symmetric shape to avoid the Casimir backgrounds. The $B-L$ gauge potential
source at a distance of $d$ is made of Graphene.

Experimental Background and Projected Constraint – For the $B-L$ gauge boson
mass $m_{A^{\prime}}\lesssim 10^{-2}\,$eV, the gauge coupling $g_{B-L}$ has
already been strongly constrained from the fifth force detection experiments,
like torsion balance Wagner:2012ui ; Adelberger:2009zz , and gravitational
wave detection LIGOScientific:2021ffg ; Miller:2023kkd ; Frerick:2023xnf .
Meanwhile, the range $m_{A^{\prime}}\gtrsim 1\,$eV is also constrained by
stellar cooling effect Hardy:2016kme . Even in the mass range of
$10^{-3}\,$eV$\lesssim m_{A^{\prime}}\lesssim 1\,$eV, the limit has reached as
$g_{B-L}\lesssim 10^{-15}$. All the current constraints is shown as the color
shaded region in Fig. 3. Thus, the strength of Féeton fifth force is tiny and
the exclusion of experimental backgrounds becomes important. We shall discuss
the backgrounds, including the Casimir effect, gravitational force, and the
thermal noise, as follows.

I. Casimir Effects: The quantum fluctuation of field will induce some
electromagnetic force between the surfaces of two macroscopic objects. This is
the so-called Casimir force 444Notice that the Van der Waals force can be
neglected for macroscopic objects Klimchitskaya:2015vra .. Both the surfaces
of superconductor and $B-L$ potential source are plates, the corresponding
Casimir potential is roughly $V\sim a^{2}/d^{3}$ Schwartz:2014sze , which is
larger than $B-L$ potential. One way to avoid such background is covering the
JJ symmetrically by a conductor with symmetric shape and grounding it, shown
as Fig. 2. The penetration length of electromagnetic wave, which is the so-
called skin depth of the conductor, is Jackson:1998nia

$\delta=\sqrt{\frac{2\rho}{\omega\mu}}.$ (12)

Here, $\rho$ is the resistivity and $\mu$ the magnetoconductivity of the
conductor. Taking copper as an example, $\mu\simeq 1$ and $\rho\simeq
1.7\times 10^{-8}$ $\Omega\cdot$m. The frequency of the Casimir
electromagnetic field $\omega$ is inversely proportional to the distance of
two surfaces, $\omega\gtrsim 1/d\simeq 10^{15}\,$Hz bruno:2002long . Thus, the
penetration length is roughly $\delta\simeq 10^{-8}\,$m. It means that as long
as the thickness of conductor is larger than this value, the electromagnetic
field outside the conductor is shielded Westphal:2020okx  and never affects
the JJ detector inside. Although the conductor itself also induces a Casimir
force on the detector, the potential is uniform and symmetric for both
superconductors and does not change their relative phase 555Recall that the
conductor also carries the $B-L$ charge. The symmetric setup can eliminate the
effects from it, too..

II. Gravitational Potential: The plate made of Graphene not only produces the
$B-L$ gauge potential, it is also a gravitational potential source. One carbon
atom has a $B-L$ charge $Q_{C}=6$ and a gravitational charge $m_{C}\simeq
12\,$GeV. Interacting with an electron, the gravitational force is the same
order of $B-L$ gauge force, $Gm_{C}m_{e}/r^{2}\simeq g_{B-L}^{2}\times
6/r^{2}$ ($G=6.8\times 10^{-39}\,$GeV-2 is the Newton constant), when
$g_{B-L}\simeq 10^{-20}$ for a simple estimation. Below that, the
gravitational force dominates. For the same settings, the quantum phase
induced by gravitational potential is,

$\phi_{g}(d)=\tau\times\int_{0}^{b}\varoiint_{S}\frac{2Gm_{e}\rho_{C}}{\sqrt{(d+z)^{2}+x^{2}+y^{2}}}dSdz,$
(13)

where we take the surface area of the plate $S=1\,$cm2. After running for a
day, the phase difference induced by gravity is
$\Delta\phi_{g}=\phi_{g}(d)-\phi_{g}(d+\epsilon)\simeq 10^{-9}$ for $d=1/$eV
and $b=1\,$cm, which is negligible comparing with the Eq. (10). Furthermore,
the gravity force from the Earth can be neglected by tuning the height of two
superconductors for equalizing the gravitational potentials in both sides.

III. Thermal Noise: The thermal noise produces an inevitable background
current, $I_{T}\approx ekT/\hbar\approx 10^{-7}(T/1K)A$ Perez:2023tld . In the
laboratory, researchers have achieved significantly low temperature
$T<0.1\,$mK in controlled environments Deppner:2021fks . However, the main
hurdle lies in replicating these conditions on a larger scale, both in terms
of space and time. For a conservative estimation, we take the environment
temperature $T=1\,$mK and the thermal current is $I_{T}=10^{-10}\,$A.

Figure 3:  The shaded regions are current constraints of $B-L$ gauge
interaction from Torsion-balance experiments (EP, ISL), stellar cooling bounds
from the Sun and Horizontal branch stars (HB). The red ($d=1\,$eV-1) and
orange ($d=0.1\,$eV-1) lines are the projected limits of our experimental
proposal. The different line types, dashed and solid, are for different
running times of one day and one month.

In summary, both the backgrounds from gravity and thermal noise are much
smaller than the expected signal. Therefore, the only condition for detecting
the signal is that the phase difference induced by the $B-L$ gauge force shall
accumulate to surpass the typical threshold, $\Delta\phi_{f}>10^{-3}$
everitt2011gravity . Such a requirement gives a projected constraint for the
$B-L$ gauge coupling as the colorful lines in Fig. 3.

Our proposal gives the strongest exclusion of the coupling $g_{B-L}\gtrsim
10^{-17}$ for Féeton mass $10^{-2}\,$eV $<m_{A^{\prime}}<1\,$eV by assuming no
signals detected. The limit becomes weaker when $m_{A^{\prime}}\gtrsim 1/d$
since the Yukawa potential is exponentially suppressed by distance. Thus, a
larger source distance $d=0.1\,$eV-1 (orange lines) has a wider detectable
range than the smaller case $d=1\,$eV-1 (red lines). On the other hand, in the
long range force limit $1/b<m_{A^{\prime}}<1/d$, the phase difference, or the
constraint, is independent of $d$ and is inversely proportional to
$m_{A^{\prime}}$ as indicated by Eq. (4). In addition, a longer running time
naturally yields stronger experimental constraints. If $\tau$ increase by a
factor of $30$ from $1\,$day (solid) to $1\,$month (dashed), the limit
stronger by a factor of $\sqrt{30}$ correspondingly.

Discussion and Conclusions – In this paper, we propose a novel experimental
design that utilizes a superconducting Josephson junction to detect the fifth
force mediated by $B-L$ gauge boson. The Josephson junction consists of two
superconductors at different distances from the $B-L$ gauge potential source.
As a result, a potential difference arises between the two Cooper-pairs-
coherent states in the superconductors, leading to the generation of a phase
difference over time. Due to the Josephson effects, this phase difference can
be converted into an electric current and detected. By extending the running
time, the signal can be significantly amplified, enabling more precise
observation of the fifth force.

After eliminating the background caused by Casimir effects, the experiment has
the strongest sensitivity to $g_{B-L}\subset(10^{-18},10^{-15})$, for the
gauge boson mass $m_{A^{\prime}}\subset(10^{-2}\,$eV$,100\,$eV). To achieve
this result, a reasonable running time of $\tau\sim 1\,$Day was considered.
This result is not limited to the $B-L$ extension and can be generalized to
other fifth force searches. We hope that this experimental proposal can become
a reality in the near future.

## Acknowledgements

The authors thank Jiadu Lin, Zixuan Dai, Prof. Qingdong Jiang, and Prof. Shao-
Feng Ge for useful discussions. And T.T.Y. thanks Kazuya Yonekura for a very
helpful discussion on the gauge invariant description of the system. 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.

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##  Appendix A Model for the $B-L$ extension

In this paper, we consider a simple model that extends the SM with a $B-L$
gauge group. The related Lagrangian is,

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{i}{2}\bar{N}_{i}\gamma^{\mu}\partial_{\mu}N_{i}+\left(\lambda_{i\alpha}\bar{N}_{i}L_{\alpha}H-\frac{1}{2}M_{Ri}\bar{N}_{i}^{c}N_{i}+\text{
h.c. }\right)$
$\displaystyle-\frac{1}{2}g_{B-L}\bar{N}_{i}\gamma^{\mu}\gamma_{5}N_{i}A_{\mu}^{\prime}+g_{B-L}q_{B-L}\bar{f}\gamma^{\mu}fA_{\mu}^{\prime}.$
(14)

Here, $L_{\alpha}$, $f$, $H$ and $N$ are the SM left-handed lepton doublets,
fermions, Higgs boson and RHNs. The index $i$ stands for the generation of RHN
and $\alpha$ for all the species of leptons. A new Higgs field $\Phi$ is
introduced to break the gauged $B-L$ symmetry spontaneously and the vacuum
expectation value of $\Phi$ is the breaking scale of $B-L$ gauge symmetry
$\langle\Phi\rangle=V_{B-L}$. Its couplings to RHNs $N_{i}(i=1-3)$ generate
large Majorana masses for them through $\frac{1}{2}h_{i}\Phi N_{i}N_{i}$.
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.

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

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. Regarding composite states, a proton or neutron is made
up of three quarks with $B-L$ charge $+1$. Furthermore, the $B-L$ charge of an
atom equals to the number of neutrons, since the charges of electrons always
cancel out that of the protons.