# Bounds on Gauge Bosons Coupled to Non-conserved Currents

Majid Ekhterachian<EMAIL_ADDRESS>Maryland Center for Fundamental Physics,
Department of Physics, University of Maryland, College Park, MD 20742.  Anson
Hook<EMAIL_ADDRESS>Maryland Center for Fundamental Physics, Department of
Physics, University of Maryland, College Park, MD 20742.  Soubhik Kumar
<EMAIL_ADDRESS>Berkeley Center for Theoretical Physics, Department of
Physics, University of California, Berkeley, CA 94720, USA  Theoretical
Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Yuhsin Tsai<EMAIL_ADDRESS>Department of Physics, University of Notre Dame, IN
46556, USA

###### Abstract

We discuss new bounds on vectors coupled to currents whose non-conservation is
due to mass terms, such as $U(1)_{L_{\mu}-L_{\tau}}$. Due to the emission of
many final state longitudinally polarized gauge bosons, inclusive rates grow
exponentially fast in energy, leading to constraints that are only
logarithmically dependent on the symmetry breaking mass term. This exponential
growth is unique to Stueckelberg theories and reverts back to polynomial
growth at energies above the mass of the radial mode. We present bounds coming
from the high transverse mass tail of mono-lepton+missing transverse energy
events at the LHC, which beat out cosmological bounds to place the strongest
limit on Stueckelberg $U(1)_{L_{\mu}-L_{\tau}}$ models for most masses below a
keV. We also discuss a stronger, but much more uncertain, bound coming from
the validity of perturbation theory at the LHC.

##  I Introduction

Recently, new light weakly coupled particles have increasingly become a focus
as either a mediator to a dark sector Boehm and Fayet (2004); Pospelov  _et
al._ (2008); Arkani-Hamed  _et al._ (2009), as dark matter itself Abbott and
Sikivie (1983); Dine and Fischler (1983); Preskill  _et al._ (1983); Arias
_et al._ (2012); Graham  _et al._ (2016), or to explain potential anomalies
Gninenko and Krasnikov (2001); Kahn  _et al._ (2008); Pospelov (2009); Tucker-
Smith and Yavin (2011); Batell  _et al._ (2011); Feng  _et al._ (2017). A
particularly well motivated candidate is a vector boson. The currents that
these vector bosons couple to can either be conserved, e.g. $U(1)_{B-L}$ with
Dirac neutrino masses, or they can be non-conserved, e.g. $U(1)_{L}$ or
$U(1)_{L_{\mu}-L_{\tau}}$.

In this letter we will continue a long line of research into the bounds that
can be placed on vector bosons coupled to non-conserved currents, see e.g.
Refs. Preskill (1991); Fayet (2006); Barger  _et al._ (2012); Karshenboim  _et
al._ (2014); Dror  _et al._ (2017a, b); Krnjaic  _et al._ (2020); Dror (2020).
As is well known, these models are non-renormalizable field theories and as a
result have amplitudes that grow with energy. Eventually these amplitudes grow
so large that tree level amplitudes violate unitarity at the energy scale
$\Lambda$, indicating that perturbation theory has broken down. Requiring that
new physics appears below the scale $\Lambda$ gives the unitarity bound.
Unitarity bounds have a long storied history, see e.g. Refs. Llewellyn Smith
(1973); Cornwall  _et al._ (1973); Joglekar (1974); Cornwall  _et al._ (1974),
and the famous application to the Higgs boson Lee  _et al._ (1977a, b);
Chanowitz and Gaillard (1985); Appelquist and Chanowitz (1987).

Typically non-conservation of the currents comes from either anomalies or mass
terms. In the first case, amplitudes involving longitudinal modes are enhanced
Preskill (1991). Later, Ref. Dror  _et al._ (2017a, b) showed that this
enhancement led to strong constraints on anomalous gauge theories. In the
second case, inclusive amplitudes were shown to exhibit exponential growth
Craig  _et al._ (2020). We will show that this exponential growth leads to
very strong constraints on these theories.

In this letter we will focus on the case of currents that are not conserved
due to mass terms. Specifying this starting point locks us into considering
the Stueckelberg limit of gauge theories111The Stueckelberg limit is when the
mass of the radial mode is taken to be heavier than the energy scale under
consideration. as including the radial mode renders mass terms gauge
invariant. A Stueckelberg theory coupled to a non-conserved current can have
inclusive rates that grow exponentially fast in energy due to multiparticle
emission Craig  _et al._ (2020). This exponential growth is unique to the
Stueckelberg limit and becomes polynomial at energies above the mass of the
radial mode.

The exponential growth of amplitudes in these models gives extremely strong
unitarity bounds Craig  _et al._ (2020). Redoing the unitarity bound using our
conventions, we find

$\Lambda\approx\frac{4\pi
m_{X}}{\sqrt{27}g_{X}}\log^{3/2}\left(\frac{m_{X}}{g_{X}m}\right),$ (1)

where $g_{X}$ and $m_{X}$ are respectively the coupling and the mass of the
gauge boson, and $m$ is the symmetry breaking mass term. To show how strong
this unitarity bound is, let us consider the Stueckelberg limit of
$U(1)_{L_{\mu}}$. Typically one assumes that the strongest unitarity bound on
this model comes from the fact that this current is anomalous. Redoing the
unitarity bound (for details see the Supplementary Material) using our
conventions leads to a slightly stronger result than the standard result
Preskill (1991)

$\Lambda_{a}\approx\frac{\sqrt{4\pi}m_{X}}{g_{X}}\frac{32\pi^{2}}{g^{2}},$ (2)

where $g$ is the $SU(2)_{W}$ gauge coupling. Comparing this with Eq. (1)
taking $m_{X}/g_{X}\simeq 1$ GeV, motivated by our later results, and $m\simeq
0.05$ eV, as the mass of the neutrino, we find that
$\Lambda\sim\Lambda_{a}/10$. This shows that despite the extremely small
neutrino mass, its non-zero value still gives a unitarity bound almost an
order of magnitude more stringent than the anomaly does! Thus before
considering moving to an anomaly free theory such as
$U(1)_{L_{\mu}-L_{\tau}}$, one should first UV complete the symmetry breaking
neutrino mass terms. In contrast, for a Higgsed $U(1)_{L_{\mu}}$ one can
naturally have the exact opposite scenario where $\Lambda_{a}\ll\Lambda$.

Unitarity in these models is restored by the inclusion of the radial mode.
However in these UV completions, the small fermion mass is the result of a
higher dimensional operator so that scattering of the radial mode has its own
unitarity bound. Constraints on the UV completion are fairly model dependent,
however since it is likely that the UV completion only couples to the SM via
the neutrino mass term, bounds on it are plausibly fairly weak. On the other
hand, for Stueckelberg gauge bosons we obtain robust, model-independent bounds
that do not rely on the dynamics of the radial mode. While these bounds can be
evaded by going away from the Stueckelberg limit by including the radial mode
at sufficiently low energies, the radial mode should then be taken into
account whenever processes at energies larger than its mass are studied.

In this letter we will focus on gauge bosons coupled to the lepton number
currents, which are only non-conserved because of the extremely small neutrino
masses. To find explicit bounds, we consider the total decay width of the $W$
boson and the high transverse mass tail of mono-lepton+MET (missing transverse
energy) events at the Large Hadron Collider (LHC). The bounds we obtain are
essentially independent of what the exact model under consideration is, thus
in the introduction we will only list our bounds on $U(1)_{L_{\mu}-L_{\tau}}$
with Dirac neutrino masses. A somewhat uncertain bound of $m_{X}/g_{X}\gtrsim
24\,\,\text{GeV}$ comes from requiring that physics is perturbative at the
LHC. We also find

$\frac{m_{X}}{g_{X}}>1.3\,\,\text{GeV};\qquad\frac{m_{X}}{g_{X}}>54\,\,\text{MeV}$
(3)

coming from mono-lepton+MET events at the LHC and the total decay width of the
W boson respectively. In the high mass limit, this is only a factor of $\sim$
100 weaker than the extremely precise measurements of the $g-2$ of the muon
Jegerlehner and Nyffeler (2009); Miller  _et al._ (2012). Meanwhile for
$m_{X}\lesssim 1$ keV, this is the strongest constraint on these models
beating out even cosmological bounds222If the gauge boson instead coupled to
$L_{e}$, the strongest bounds at low mass are from neutrino oscillations in
the earth Wise and Zhang (2018) and it is only for masses $m_{X}\lesssim
10^{-16}$ eV that our bounds win out..

##  II Unitarity bound

In this section, we derive the unitarity bound on these models, which was also
done in Ref. Craig  _et al._ (2020) using slightly different techniques333We
use a slightly different definition of the unitarity bound and we consider the
process $\nu+n\,\phi\rightarrow\nu+n\,\phi$ rather than
$\nu+\overline{\nu}\rightarrow n\,\phi$, which leads to stronger unitarity
bounds.. In order to demonstrate the exponential growth of amplitudes, we will
consider the toy scenario of a gauge boson coupled to only the left handed
piece of a Dirac fermion $\nu$, whose current is not conserved due to explicit
breaking by a small Dirac mass term $m_{\nu}$. Namely we consider a theory
with

$\mathcal{L}=-\frac{1}{4}F_{X}^{2}+i\overline{\nu}\left(\not{\partial}-ig_{X}\not{A}_{X}P_{L}\right)\nu-
m_{\nu}\overline{\nu}\nu+\frac{1}{2}m_{X}^{2}A_{X}^{2},$ (4)

where $F_{X}$ is the field strength of the gauge boson $A_{X}$. In the limit
of small $m_{\nu}$, the scale at which perturbation theory breaks down,
$\Lambda$, is much larger than the mass of the gauge bosons. In this limit,
things simplify as the high energy behavior of $A_{X}$ can be obtained from
the Goldstone boson equivalence theorem. Namely, the matrix element obeys
$\mathcal{M}(A_{X}^{L}+\cdots)\approx\mathcal{M}(\phi+\cdots)$, where
$A_{X}^{L}$ is a longitudinally polarized $A_{X}$ and $\phi$ is the Goldstone
boson.

We obtain the theory with Goldstone bosons by leaving unitarity gauge via a
chiral gauge transformation,
$\nu_{L}\rightarrow\exp(ig_{X}\phi/m_{X})\nu_{L}$. As the mass term is not
gauge invariant, it transforms into

$V=m_{\nu}\overline{\nu}e^{ig_{X}P_{L}\phi/m_{X}}\nu=\sum_{n}\overline{\nu}\frac{m_{\nu}}{n!}\left(\frac{ig_{X}P_{L}\phi}{m_{X}}\right)^{n}\nu.$
(5)

From this, it is clear that the theory is non-renormalizable and has a UV
cutoff. The Goldstone boson equivalence theorem lets us compute the
probability of emitting many longitudinal gauge bosons by calculating the much
simpler process of emitting many $\phi$ particles using the above interaction.

As one can see, considering processes involving $n$ $\phi$s gives a matrix
element $\sim m_{\nu}(g_{X}E/m_{X})^{n}$ that becomes more and more
insensitive to the small mass parameter $m_{\nu}$ as $n$ becomes larger and
larger. This causes the best unitarity bound to come from taking $n$ larger
and larger. However, in the large $n$ limit, the $1/n!$ coming from dealing
with identical final state particles penalizes taking $n$ too large. Thus the
optimal unitarity bound comes from taking an intermediate value of the number
of gauge bosons $n_{\rm opt}$.

A calculation done in the Supplementary Materials shows that the amplitude
$\hat{\mathbf{M}}$ of the process $\nu+n\,\phi\rightarrow\nu+n\,\phi$ is

$\displaystyle|\hat{\mathbf{M}}(\nu+n\,\phi$ $\displaystyle\rightarrow$
$\displaystyle\nu+n\,\phi)|=$ (6)
$\displaystyle\frac{g_{X}m_{\nu}}{2m_{X}(n+1)!n!(n-1)!}\left(\frac{g_{X}E}{4\pi
m_{X}}\right)^{2n-1}$

in the limit that $E\gg n\,m_{X}$. Unitarity requires that
$|\hat{\mathbf{M}}|<1$. In order to obtain the strongest bounds, we choose the
$n$ that maximizes Eq. (6) to obtain in the large $n$ limit

$\displaystyle|\hat{\mathbf{M}}(\nu+n_{\rm opt}\phi\rightarrow\nu+n_{\rm
opt}\phi)|\sim\frac{g_{X}m_{\nu}}{2m_{X}}e^{3\left(\frac{g_{X}E}{4\pi
m_{X}}\right)^{2/3}}.$ (7)

This maximum value of $|\hat{\mathbf{M}}|$ is obtained for $n_{\rm
opt}\approx(g_{X}E/4\pi m_{X})^{2/3}$. Requiring unitarity holds for Eq. (7)
gives the leading logarithmic behavior

$E=\Lambda\approx\frac{4\pi
m_{X}}{\sqrt{27}g_{X}}\log^{3/2}\left(\frac{m_{X}}{g_{X}m_{\nu}}\right).$ (8)

From this calculation we see the behavior claimed in the introduction.
Amplitudes have an exponential growth in energy and the strongest growth comes
from emitting multiple gauge bosons. In this calculation, we made the
approximation that the energy carried by each of the $A_{X}$ gauge bosons is
much larger than the mass when we utilized the Goldstone boson equivalence
theorem. Combining the expression for $n_{\rm opt}$ with Eq. (8) and the
requirement that $E\gg n_{\rm opt}\,m_{X}$, we find that our massless
approximation of the unitarity bound is valid when $g_{X}\lesssim
4\pi\sqrt{\log(m_{X}/g_{X}m_{\nu})/3}$. In the large log limit, the massless
limit is always a valid approximation.

##  III Models

We now briefly describe the models under consideration and set up some
notation. The results in the next section will be given in the 1-flavor
approximation so we will also discuss how to easily take into account the
standard 3-flavor set up. For ease of expression, in this section we will use
Weyl notation for fermions.

As mentioned before, we will be considering the Stueckelberg limit of
different $U(1)$ gauge theories. We first consider $U(1)_{L_{\mu}-L_{\tau}}$.
We assume Dirac neutrinos and that the right-handed neutrinos are neutral
under $L_{\mu}-L_{\tau}$. The flavor basis is related to the mass basis by
$\nu_{F}=U\nu_{M}$, where $\nu_{F}$ ($\nu_{M}$) are the flavor (mass) basis
left handed neutrinos and $U$ is the PMNS matrix. In the flavor basis the
neutrino mass term is $\nu^{c}M_{d}U^{\dagger}\nu_{F}$ where $M_{d}$ is a
diagonal matrix of the neutrino masses $m_{1,2,3}$. To leave unitary gauge,
the flavor-basis SM neutrinos are rotated by,

$\displaystyle\nu_{F}\rightarrow P\nu_{F}\quad
P=\text{diag}\left(1,e^{+ig_{X}\phi/m_{X}},e^{-ig_{X}\phi/m_{X}}\right).$ (9)

Thus the neutrino mass term involving the Goldstone bosons $\phi$ becomes

$\displaystyle\mathcal{L}^{D}_{\nu\,\rm mass}$ $\displaystyle=$
$\displaystyle\nu^{c}M_{d}U^{\dagger}P\nu_{F}+\rm{h.c.}$
$\displaystyle\supset$
$\displaystyle\sum_{n,j}\frac{1}{n!}\left(\frac{ig_{X}\phi}{m_{X}}\right)^{n}\nu^{c}_{j}M_{d,j}\left(U^{\dagger}_{j\mu}\nu_{\mu}+(-1)^{n}U^{\dagger}_{j\tau}\nu_{\tau}\right).$

From this, we see that any 1-flavor process involving $n$ high energy gauge
bosons can be converted into the 3-flavor result by replacing

$\displaystyle m_{\nu}^{2}\rightarrow\sum_{j=1}^{3}\left(|U_{\mu
j}|^{2}+|U_{\tau j}|^{2}\right)m_{j}^{2}.$ (10)

The other gauge theory we will consider is $U(1)_{L}$ with Majorana neutrino
masses. In this case, after leaving unitary gauge the mass term is

$\displaystyle\mathcal{L}^{M}_{\nu\,\rm mass}$ $\displaystyle=$ $\displaystyle
e^{2ig_{X}\phi/m_{X}}\nu^{T}_{M}M_{d}\nu_{M}+\rm{h.c.}$ (11)

From this, the 1-flavor results can be generalized using the substitution

$\displaystyle g_{X}\rightarrow 2\,g_{X},\qquad
m_{\nu}^{2}\rightarrow\sum_{l=e,\mu,\tau}\,\sum_{j=1}^{3}|U_{lj}|^{2}m_{j}^{2}.$
(12)

##  IV Constraints

We now present exclusions coming from the emission of many final state gauge
bosons. We will consider three different constraints coming from the total
decay width of the $W$ boson, the tail of high transverse mass mono-lepton+MET
events, and a rough estimate of when perturbativity breaks down at the LHC.
Our results will be presented in the simplified 1-flavor model. Eq. (10) and
Eq. (12) can be used to convert the results into the two specific models of
interest. In particular, in Fig. 1 and Fig. 2 we show the results after
incorporating the full 3-flavor set-up.

### W Boson decay :

We will consider the partial decay width of the W boson into leptons and $n$
gauge bosons

$\Gamma_{n}(W^{-}\rightarrow L^{-}+\overline{\nu}+n\,A_{X}).$ (13)

The constraint we will impose is that the sum of these decay widths is less
than the total decay width

$\Gamma_{\text{BSM}}\equiv\sum_{n=2}^{\infty}\Gamma_{n}<\Gamma_{W}.$ (14)

For simplicity we will take $n\geq 2$ to avoid the soft divergence.

Using the Goldstone boson equivalence theorem, we find the result to leading
order in $m_{\nu}$ to be

$\Gamma_{\text{BSM}}=\frac{g_{2}^{2}m_{\nu}^{2}}{1536\pi
M_{W}}x^{4}\,_{2}F_{4}(\\{1,1\\},\\{2,3,3,5\\},x^{2})$ (15)

where $x=\frac{g_{X}M_{W}}{4\pi m_{X}}$ and $\,{}_{p}F_{q}$ is the generalized
hypergeometric function. In the limit of large argument, it simplifies as

$\qquad\,{}_{2}F_{4}(\\{1,1\\},\\{2,3,3,5\\},x^{2})\approx\frac{16\sqrt{3}}{\pi
x^{20/3}}e^{3x^{2/3}}$ (16)

showing the exponential growth of the amplitudes mentioned earlier.

Bounds are obtained by requiring that this decay width is smaller than the
total decay width of the W boson. For $U(1)_{L_{\mu}-L_{\tau}}$ ($U(1)_{L}$)
we find that $m_{X}/g_{X}>54$ MeV ($m_{X}/g_{X}>108$ MeV). Because of the
exponential, our results are insensitive to the details with which we
constrain the model. For example, requiring that this decay width is smaller
than $10^{-6}$ of the total decay width of the W boson (roughly 1 over the
number of W bosons produced at LEP) tightens the above
$U(1)_{L_{\mu}-L_{\tau}}$ bound to $76$ MeV.

Figure 1: Low mass constraints on a Stueckelberg $U(1)_{L_{\mu}-L_{\tau}}$
gauge boson with Dirac neutrino masses. The purple region shows the constraint
derived in Eq. (19) coming from the high transverse mass tail of mono-
lepton+MET events at the LHC. The dot-dashed purple line denotes the
constraint obtained from demanding the validity of perturbation theory at the
LHC, given by Eq. (20). Other constraints are from: $\Delta N_{\rm eff}$
during BBN through thermalized $A_{X}$ Huang  _et al._ (2018), black hole
superradiance (BHSR) instability Baryakhtar  _et al._ (2017), rare $K$ decays
Dror (2020), $\Delta N_{\rm eff}$ through $\nu\nu\rightarrow A_{X}A_{X}$ Dror
(2020); Huang  _et al._ (2018), constraints on $\nu$ decay through terrestrial
experiments Beacom and Bell (2002); Funcke  _et al._ (2020); Dror (2020). We
also update the cosmological constraints on $\nu$ decay as discussed in the
main text, based on the recent result of Ref. Barenboim  _et al._ (2020).
Given the $\nu$ decay bounds from cosmology are model-dependent and can be
relaxed, we show it via dashed lines. Figure 2: High mass constraints on a
Stueckelberg $U(1)_{L_{\mu}-L_{\tau}}$ gauge boson with Dirac neutrino masses.
The solid and dot-dashed purple lines are the same as in Fig. 1. Other
constraints are from: $\Delta N_{\rm eff}$ during BBN due to thermalization of
$A_{X}$ and entropy dump during its decay Huang  _et al._ (2018); Kamada and
Yu (2015); Escudero  _et al._ (2019) (solid: $T_{\rm RH}>4$ MeV, see e.g.
Hannestad (2004); de Salas  _et al._ (2015); dashed: $T_{\rm RH}>10$ MeV),
neutrino trident process at the CCFR experiment Mishra  _et al._ (1991);
Altmannshofer  _et al._ (2014), and a search for
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}A_{X},A_{X}\rightarrow\mu^{+}\mu^{-}$ at
BaBar Lees  _et al._ (2016). In the shaded green region, the muon $(g-2)$
anomaly Bennett  _et al._ (2006); Aoyama  _et al._ (2020) can be explained by
$A_{X}$ at 2$\sigma$ Baek  _et al._ (2001); Gninenko and Krasnikov (2001).

### mono-lepton+MET events :

In the next two sub-sections, we will discuss bounds that stem from the
breaking of perturbativity at the LHC. From the decay of the $W$ boson, we see
that we are faced with a theory which is becoming dominated by a large number
of final state particles. For black holes Banks and Fischler (1999), when the
cross section becomes dominated by a large number of final states, the cross
section for s-channel two-to-two scattering becomes highly suppressed. The
non-observation of this suppression leads to a constraint. As illustrated
below, it is plausible that a similar feature exists in our case when
perturbation theory breaks down

The case that we will consider is $pp\rightarrow W^{\star}\rightarrow l\nu$
via an off-shell $W$ boson. At high enough energies, the tree level off-shell
width of the $W$ boson becomes larger than the momentum flowing through the
propagator, suppressing the high transverse mass tail of mono-lepton+MET
events 444Perturbation theory has broken down when this occurs, so that our
calculation is only really an estimate of what happens in the full non-
perturbative theory.. We obtain a bound by requiring that this suppression is
small enough that the suppression was not observed in Ref. CMS (2021).

At high energies, the propagator of the $W$ boson is given by

$\frac{i}{s-M_{W}^{2}+\Sigma(s)}\qquad\text{Im}(\Sigma(s))=M_{W}\Gamma_{W}(s),$
(17)

where the earlier calculation of the decay width of the $W$ boson gives

$M_{W}\Gamma_{W}(s)=\frac{g_{2}^{2}m_{\nu}^{2}}{1536\pi}x^{4}\,_{2}F_{4}(\\{1,1\\},\\{2,3,3,5\\},x^{2})$
(18)

with $x=\frac{g_{X}\sqrt{s}}{4\pi m_{X}}$. When $s\ll M_{W}\Gamma_{W}(s)$,
mono-lepton+MET production is highly suppressed.

Ref. CMS (2021) observes the high $M_{T}$ tail of the mono-lepton+MET events
555The transverse mass $M_{T}$ is defined as $M_{T}=\sqrt{2p_{T}^{l}p_{T}^{\rm
miss}(1-\cos\Delta\phi)}$, where $\Delta\phi$ is the azimuthal angle between
$\vec{p}_{T}^{l}$ and $\vec{p}_{T}^{\rm miss}$. to be the expected SM result
when $M_{T}\lesssim 2$ TeV. Given $M_{T}\leq\sqrt{s}$ by definition, it
implies that at center of mass energies of at least 2 TeV, we do not see any
deviation from the SM expectation. The observed events dominantly come from
$pp\rightarrow W^{\star}\rightarrow l\nu$. The off-shell $W$ contribution
would be suppressed by at least a factor of 2 if
$\sqrt{2}s=M_{W}\Gamma_{W}(s)$ and would have been observed for any
$M_{T}\lesssim 2$ TeV. We obtain a bound by requiring that $\sqrt{2}s\gtrsim
M_{W}\Gamma_{W}(s)$ at 2 TeV. For $U(1)_{L_{\mu}-L_{\tau}}$ we find that

$\displaystyle m_{X}/g_{X}>1.3\leavevmode\nobreak\ \text{GeV}.$ (19)

For $U(1)_{L}$ with Majorana masses we find $m_{X}/g_{X}>2.6$ GeV.

### Calculability :

The next bound we place is that perturbation theory at the LHC is valid. This
is necessarily a somewhat fuzzy bound because the breakdown of perturbation
theory is not very well defined. For example, the unitarity bound derived in
Ref. Craig  _et al._ (2020) is weaker than ours by a factor of $\sim 3$
indicating that different techniques give $\mathcal{O}(1)$ different estimates
of when perturbation theory breaks down. However, it is undeniable that we can
calculate observables at the LHC and so perturbation theory must be valid.

We will use the unitarity bounds given in Eq. (8) as an order of magnitude
estimate for when perturbation theory breaks down. The highest center of mass
energy collision at the LHC was one that had a center of mass energy of $\sim
8$ TeV Sirunyan  _et al._ (2020). Requiring that LHC processes are calculable
at a center of mass energy of 8 TeV gives the constraint

$\displaystyle m_{X}/g_{X}\gtrsim 24\leavevmode\nobreak\ \text{GeV}$ (20)

for $U(1)_{L_{\mu}-L_{\tau}}$. As mentioned before, the breakdown of
perturbation theory is a somewhat nebulous concept and thus this bound should
be considered as only an estimate of where the bound coming from the breakdown
of perturbation theory must lie.

An astute reader will notice that the previous bounds are actually slightly
weaker than the unitarity bound found using their relevant energy scales. As
our previous bounds came from tree level calculations, they requires that
perturbation theory is valid. As such, strictly speaking, they only apply if
the stronger calculability constraint has misestimated the breakdown of
perturbation theory by a factor of few.

We show our constraints visually in Fig. 1 and Fig. 2 for
$U(1)_{L_{\mu}-L_{\tau}}$. We take normal ordering of the neutrino masses with
the lightest neutrino being massless. As explained in the text, the bounds are
not sensitive to the precise value of the mass. For other neutrino mixing
parameters, we use the results from Table 3 of Ref. Esteban  _et al._ (2020).

In Fig. 1 we have updated the cosmological bounds on these models coming from
the decay of the heavier neutrinos, which requires $\tau_{0}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}4\times 10^{5-6}\leavevmode\nobreak\ {\rm
sec}(m_{3}/50\leavevmode\nobreak\ {\rm meV})^{5}$ Barenboim  _et al._ (2020).
The two $\nu$ decay (cosmology) bounds represent the uncertainty in the bound
on the $\nu$ lifetime. After refining the calculation of the damping of
anisotropic stress due to neutrino decay and inverse decay, this new bound was
obtained and found to be several orders of magnitude weaker than previous work
(e.g., Archidiacono and Hannestad (2014); Escudero and Fairbairn (2019)). For
the $\Delta N_{\rm eff}$ constraints, we take $T_{\rm RH}<m_{\mu}$.
Additionally, in both figures we have neglected constraints coming from a
possible coupling to electrons as those bounds are model dependent and can be
avoided.

##  V UV completion

Unitarity in the emission of gauge bosons can be restored by introducing the
radial mode. To see the behavior of a UV completion, we consider a complex
scalar $\Phi$ with charge 1 and give the neutrinos a charge $q=g_{X}/g$ so
that the coupling of $A_{X}$ to neutrinos is given by $g_{X}$ while the
coupling of $A_{X}$ to $\Phi$ is given by $g$. The symmetry breaking mass term
comes from the higher dimensional operator

$V=\frac{y\Phi^{q}}{\Lambda^{\prime q}}HL\nu^{c}\qquad
m_{\nu}=\frac{y\,vf^{q}}{\Lambda^{\prime q}},$ (21)

which gives the standard neutrino mass term after $\Phi$ obtains a vev $f$.
The radial mode can be partially decoupled by taking $g\rightarrow 0$ and
$q,f\rightarrow\infty$ while holding $m_{X}=gf$ and $g_{X}=qg$ constant. This
limit attempts to decouple the radial mode $m_{\Phi}\lesssim f$ but at the
same time sends the symmetry breaking mass term to zero, $m_{\nu}\rightarrow
0$. Thus, when the neutrino mass is non-zero, the radial mode cannot be
decoupled.

Using Eq. (21), one can show that the unitarity bound $\Lambda$ satisfies

$\Lambda\approx\frac{4\pi
m_{X}}{\sqrt{27}g_{X}}\log^{3/2}\left(\frac{m_{X}}{g_{X}m_{\nu}}\right)>m_{\Phi}$
(22)

showing that unitarity did indeed predict the correct scale of new physics in
these models. In this UV theory, one can easily show that scattering involving
gauge bosons no longer grows. However, the price is that scattering involving
$\Phi$ does grow. Thus this UV completion will itself require a UV completion
at the scale $\Lambda^{\prime}$. As this particular higher dimensional
operator is identical to those seen in Froggatt–Nielsen models Froggatt and
Nielsen (1979), its UV completion proceeds along a manner completely analogous
to those models and will not be expounded upon here.

It is worth mentioning that while the UV completion presented above can easily
obtain $m_{\Phi}\gg 4\pi m_{X}/g_{X}$, it cannot saturate $\Lambda\sim
m_{\Phi}$. In this model, $m_{\Phi}\sim\sqrt{\lambda}f$ which can satisfy
$m_{\Phi}>m_{X}/g_{X}\sim f/q$ while perturbation theory is still under
control $\lambda q\lesssim 1$. However in this limit
$\Lambda\sim\sqrt{q}f>m_{\Phi}$, so that saturating the unitarity bound is not
possible. Even when one allows $\lambda q>1$ and instead uses semi-classical
methods Badel  _et al._ (2019), one still finds that it is impossible to
saturate the unitarity bound in this model.

The importance of multiparticle emission in Stueckelberg theories could have
been anticipated from this UV completion. In the UV completion, the symmetry
breaking mass term arises from a higher dimensional operator of dimension
$\sim q$, see Eq. (21). Discovering the bad high energy behavior of a higher
dimensional operator of dimension $\sim q$ requires $\sim q$ states. The IR
Stueckelberg theory should match the UV Higgs theory at the scale of the
radial mode. However, the IR Stueckelberg theory does not know which UV Higgs
theory to match onto. Thus all it can do is at a given energy scale $E$, match
onto the UV Higgs theory which has $m_{\Phi}=E+\epsilon$. As the energy scale
$E$ increases, the IR theory has to match onto different UV theories with
larger and larger $m_{\Phi}$ and hence larger and larger $q$. Thus when
scattering particles at higher and higher energies, the dominant final state
involves an ever increasing number of particles.

##  VI Conclusion

In this letter, we considered Stueckelberg gauge bosons coupled to non-
conserved currents broken by mass terms and calculated the bounds on these
models coming from mono-lepton+MET events at the LHC. For large gauge boson
masses, these constraints are only a bit weaker than even the strongest of
bounds, such as $g-2$ experiments. For most masses below a keV, these
constraints are the strongest bounds on these models.

The strength of these bounds comes from the exponential growth of inclusive
rates, a feature only present in the Stueckelberg limit Craig  _et al._
(2020). This growth is a double edged sword. On one hand, it allows one to use
very crude measurements to place extremely stringent constraints. On the other
hand, it does not benefit from precision measurements so that it is not easy
to improve on the constraints without access to a higher energy environment.
For example, a 100 TeV collider would improve upon the LHC bounds by about an
order of magnitude.

### Acknowledgments

We thank Simon Knapen and Maxim Pospelov for comments on the draft and Zhen
Liu for useful discussions. ME and AH were supported in part by the NSF grants
PHY-1914480, PHY-1914731, and by the Maryland Center for Fundamental Physics
(MCFP). SK was supported in part by the NSF grant PHY-1915314 and the U.S. DOE
Contract DE-AC02-05CH11231. YT was also supported in part by the NSF grant
PHY-2014165.

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Bounds on Gauge Bosons Coupled
to Non-conserved Currents

Supplementary Material
Majid Ekhterachian, Anson Hook, Soubhik Kumar, Yuhsin Tsai

This Supplementary Material contains additional calculations supporting the
results in the main text.

##  II Unitarity bounds - $n$-to-$n$ scattering

When discussing our unitarity bounds, we will follow the conventions and
discussions of Ref. Chang and Luty (2020); Abu-Ajamieh  _et al._ (2020) and
for simplicity we will be working in the limit of massless particles. Any S
matrix can be decomposed into $S=\mathbf{1}+iT$. The identity matrix describes
the situation where particles pass by without interacting while the transition
matrix $T$ describes nontrivial processes. The states we will be considering
have a continuous label $P$, the total momentum, and discrete labels $\alpha$.
These states are normalized as

$\langle P^{\prime},\alpha^{\prime}\mid
P,\alpha\rangle=(2\pi)^{4}\delta^{4}(P-P^{\prime})\delta_{\alpha\alpha^{\prime}}.$
(S1)

The amplitude $\hat{\mathbf{M}}$ is defined by

$\langle P^{\prime},\alpha^{\prime}\mid T\mid
P,\alpha\rangle=(2\pi)^{4}\delta^{4}(P-P^{\prime})\hat{\mathbf{M}}_{\alpha\alpha^{\prime}}\qquad\qquad\langle
P^{\prime},\alpha^{\prime}\mid S\mid
P,\alpha\rangle=(2\pi)^{4}\delta^{4}(P-P^{\prime})S_{\alpha\alpha^{\prime}}.$
(S2)

The main result that we will use is that
$|\hat{\mathbf{M}}_{\alpha\alpha^{\prime}}|\leq 1$ for all $\alpha$ and
$\alpha^{\prime}$ at tree level. When $\alpha\neq\alpha^{\prime}$, this
statement follows directly from the conservation of probability. For
$\alpha=\alpha^{\prime}$ we use unitarity

$1=\delta_{\alpha\alpha}=\sum_{\gamma}S^{\dagger}_{\alpha\gamma}S_{\gamma\alpha}=1-2\,\text{Im}\,\hat{\mathbf{M}}_{\alpha\alpha}+\sum_{\gamma}|\hat{\mathbf{M}}_{\gamma\alpha}|^{2}.$
(S3)

From this we have

$2\,\text{Im}\,\hat{\mathbf{M}}_{\alpha\alpha}=\sum_{\gamma}|\hat{\mathbf{M}}_{\gamma\alpha}|^{2}\geq|\hat{\mathbf{M}}_{\alpha\alpha}|^{2}$
(S4)

which can be massaged into the form
$1\geq|\text{Re}\,\hat{\mathbf{M}}_{\alpha\alpha}|^{2}+|\text{Im}\,\hat{\mathbf{M}}_{\alpha\alpha}-1|^{2}$.
From this we see that $|\text{Re}\,\hat{\mathbf{M}}_{\alpha\alpha}|\leq 1$.
Since $\hat{\mathbf{M}}_{\alpha\alpha}$ is real at tree level, we have
$|\hat{\mathbf{M}}_{\alpha\alpha}|\leq 1$.

We now have the unitarity bound
$|\hat{\mathbf{M}}_{\alpha\alpha^{\prime}}|\leq 1$ so we can apply it to the
theory described in Eq. (4). We will be considering the initial and final
states each with a neutrino $\nu$ and $n$ Goldstone bosons $\phi$. We define
our states as

$\mid P,n,\alpha\rangle=C_{n}\int
d^{4}xe^{-iPx}\phi^{(-)}(x)^{n}\nu_{\alpha}^{(-)}(x)\mid 0\rangle$ (S5)

where $C_{n}$ is a normalization constant, $(-)$ are the part of the fields
that contain the creation operators, and $\alpha$ is the spinor index of the
fermion. These states are normalized as

$\langle P^{\prime},n^{\prime},\dot{\alpha}\mid
P,n,\alpha\rangle=(2\pi)^{4}\delta^{4}(P-P^{\prime})\delta_{nn^{\prime}}\frac{\not{P}^{\alpha\dot{\alpha}}}{E}$
(S6)

where $E$ is the total energy. This normalization was chosen to reproduce Eq.
(S1) in the center of mass frame, which we will be using from here on. From
this, we have the normalization constant

$\frac{1}{|C_{n}|^{2}}=\frac{1}{2(n+1)(n-1)!}\left(\frac{E}{4\pi}\right)^{2n-1}.$
(S7)

We can finally calculate the amplitude of interest using Eq. (5) to give

$\displaystyle\langle P^{\prime},n,\alpha\mid(-i)\int
d^{4}x\overline{\nu}(x)\frac{m_{\nu}}{(2n)!}\left(\frac{ig_{X}P_{L}\phi(x)}{m_{X}}\right)^{2n}\nu(x)\mid
P,n,\alpha\rangle$ $\displaystyle=$
$\displaystyle(2\pi)^{4}\delta^{4}(P-P^{\prime})i\hat{\mathbf{M}}(\nu+n\phi\rightarrow\nu+n\phi)$
(S8) $\displaystyle|\hat{\mathbf{M}}(\nu+n\phi\rightarrow\nu+n\phi)|$
$\displaystyle=$
$\displaystyle\frac{g_{X}m_{\nu}}{2m_{X}(n+1)!n!(n-1)!}\left(\frac{g_{X}E}{4\pi
m_{X}}\right)^{2n-1}$ (S9)

Imposing $|\hat{\mathbf{M}}|\leq 1$ gives the results shown in the text. For
reference, the amplitude $\hat{\mathbf{M}}$ is related to the more familiar
matrix element $\mathcal{M}$ by normalization constants and phase space
integrals

$\displaystyle\hat{\mathbf{M}}_{\alpha\alpha^{\prime}}=C^{\star}_{\alpha}C_{\alpha^{\prime}}\int
d\Phi_{\alpha}d\Phi_{\alpha^{\prime}}\mathcal{M}_{\alpha\alpha^{\prime}}.$
(S10)

The phase space differentials $d\Phi_{\alpha}$ will be given below.

##  III Unitarity bounds on an anomalous $U(1)$

In this section we calculate the unitarity bounds on an anomalous gauge theory
in a manner analogous to what we did for $n$-to-$n$ scattering. We leave
unitarity gauge by a gauge transformation $\phi/m_{X}$. Because the theory is
anomalous, an anomaly term is added to the Lagrangian

$\mathcal{L}\supset\frac{g^{2}\mathcal{A}}{32\pi^{2}}\frac{g_{X}\phi}{m_{X}}W^{a}\tilde{W}^{a},$
(S11)

where
$\tilde{W}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}W_{\rho\sigma}$ and
$W^{a}$ are the gauge bosons with which $U(1)_{X}$ is anomalous and
$\mathcal{A}$ is the anomaly coefficient.

We will consider 2-to-2 scattering of $W^{1}$ gauge bosons via the Goldstone
boson $\phi$, which contains all of the leading high energy behavior in
Feynman-’t Hooft gauge. The largest amplitude occurs when all four gauge
bosons have the same helicity. A short calculation gives the amplitude

$\displaystyle|\hat{\mathbf{M}}(W^{1}+W^{1}\rightarrow W^{1}+W^{1})|$
$\displaystyle=$
$\displaystyle\frac{s}{4\pi}\left(\frac{g_{X}}{m_{X}}\frac{g^{2}\mathcal{A}}{32\pi^{2}}\right)^{2}$
(S12)

Requiring the unitarity bound $|\hat{\mathbf{M}}|<1$ be saturated at the
center of mass energy $\Lambda_{a}$, we arrive at the result

$\Lambda_{a}=\frac{\sqrt{4\pi}m_{X}}{g_{X}}\frac{32\pi^{2}}{g^{2}\mathcal{A}}.$
(S13)

##  IV Decay width of the $W$ boson

Here we compute the decay width of the $W$ boson into a lepton, a neutrino and
$n$ gauge bosons. The decay is dominated by the decay into longitudinal modes
which we calculate using the Goldstone boson equivalence theorem.

The relevant process is shown in Fig. S1.

Figure S1: The decay $W\rightarrow n\phi+l+\nu$.

We denote the four momenta of the $W$ boson, the outgoing lepton $l$, the
intermediate neutrino and the outgoing neutrino as $p_{W}$, $p_{l}$, $q$ and
$p_{\nu}$, respectively. We also denote the collective momenta of the $n$
Goldstone bosons $\phi$ as $p_{\phi}=p_{1}+p_{2}+\cdots+p_{n}$. We will ignore
the masses of $\phi,\nu$ and $l$ throughout our calculation, except those
appearing in the neutrino-Goldstone boson coupling. We will consider a single
neutrino flavor, and later generalize to the standard three-flavor structure.
The matrix element is

$\displaystyle
i\mathcal{M}=-\left(\frac{g_{2}}{2\sqrt{2}}\right)\bar{u}(p_{l})\gamma_{\alpha}(1-\gamma_{5})\epsilon^{\alpha}\frac{i\not{q}}{q^{2}}\kappa_{n}v(p_{\nu}),$
(S14)

where $\kappa_{n}$ denotes the coupling of neutrinos to $n$-Goldstone bosons,
obtained from Eq. (5). We see that $\kappa_{n}\propto\gamma_{5}$ for odd
values of $n$. However, it can be checked that $i\mathcal{M}$ is the same for
both even and odd values of $n$. The amplitude can be squared to give

$\displaystyle\frac{1}{3}\sum|\mathcal{M}|^{2}=\frac{2}{3}\left(\frac{g_{2}\kappa_{n}}{2\sqrt{2}}\right)^{2}\text{Tr}\left[\gamma^{\beta}\not{p}_{l}\gamma^{\alpha}\not{q}\not{p}_{\nu}\not{q}(1+\gamma_{5})\right]\frac{1}{q^{4}}\Pi_{\alpha\beta},$
(S15)

where
$\Pi_{\alpha\beta}=\left(-g_{\alpha\beta}+\frac{p_{W\alpha}p_{W\beta}}{M_{W}^{2}}\right)$.
The factor of $1/3$ comes from averaging over initial $W$ polarizations. After
some algebra this can be reduced to,

$\displaystyle\frac{1}{3}\sum|\mathcal{M}|^{2}=\left(\frac{g_{2}\kappa_{n}}{2\sqrt{2}}\right)^{2}\frac{1}{3q^{4}}\left(8g^{\alpha\beta}\left[q^{2}(p_{l}\cdot
p_{\nu})-2(q\cdot p_{\nu})(q\cdot p_{l})\right]+32(q\cdot
p_{\nu})p_{l}^{\alpha}q^{\beta}-16q^{2}p_{l}^{\alpha}p_{\nu}^{\beta}\right)\Pi_{\alpha\beta}.$
(S16)

Let us now integrate over the phase space of the $n$ $\phi$ and the outgoing
neutrino. To this end, we will use the following identities involving $k-$body
phase space of massless particles Abu-Ajamieh  _et al._ (2020),

$\displaystyle\Phi_{k}(P)=\int
d\Phi_{k}(P)\equiv\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\frac{1}{2E_{1}}\cdots\frac{d^{3}p_{k}}{(2\pi)^{3}}\frac{1}{2E_{k}}(2\pi)^{4}\delta^{4}(p_{1}+\cdots+p_{k}-P)=\frac{1}{8\pi(k-1)!(k-2)!}\left(\frac{E}{4\pi}\right)^{2k-4},$
(S17) $\displaystyle\int
d\Phi_{k}(P)p_{1}^{\mu}=\frac{1}{k}\Phi_{k}(P)P^{\mu},$ (S18)
$\displaystyle\int d\Phi_{k}(P)p_{1}\cdot p_{2}=\frac{1}{2{k\choose
2}}\Phi_{k}(P)P^{2},$ (S19)

with $E=\sqrt{P^{2}}$ being the center of mass energy. Utilizing the above
identities and doing the contraction with $\Pi_{\alpha\beta}$ we get,

$\displaystyle\int
d\Phi_{n+1}(q)\left(\frac{1}{3}\sum|\mathcal{M}|^{2}\right)$
$\displaystyle=\frac{8}{3q^{2}}\left(\frac{g_{2}\kappa_{n}}{2\sqrt{2}}\right)^{2}\left[(p_{l}\cdot
q)+\frac{2(p_{W}\cdot q)(p_{l}\cdot
p_{W})}{M_{W}^{2}}\right]\frac{\Phi_{n+1}(q)}{(n+1)},$
$\displaystyle=\left(\frac{g_{2}\kappa_{n}}{2\sqrt{2}}\right)^{2}\frac{q^{2n-4}}{3\pi(n+1)!(n-1)!}\frac{1}{(4\pi)^{2n-2}}\left(3(p_{W}\cdot
p_{l})-\frac{2(p_{W}\cdot p_{l})^{2}}{M_{W}^{2}}\right).$ (S20)

As a final step, we go to the rest frame of the $W$ and integrate over the
lepton momenta to get,

$\displaystyle\Gamma(W\rightarrow
l+\nu+n\phi)=\frac{g_{2}^{2}M_{W}^{2n-1}\kappa_{n}^{2}}{(4\pi)^{2n}}\frac{1}{16\pi(n!)^{2}(n+2)!(n-1)}\quad\text{for}\leavevmode\nobreak\
n>1.$ (S21)

Here we have also multiplied by a factor of $1/n!$ to account for the $n$
identical $\phi$s in the final state. The total decay width of the $W$-boson
into a final state containing an arbitrary number of $\phi$s is then obtained
after summing over $n$,

$\displaystyle\Gamma_{\text{BSM}}\equiv\sum_{n>1}\Gamma(W\rightarrow
l+\nu+n\phi)=\frac{1}{16\pi\times
96}\frac{g_{2}^{2}m_{\nu}^{2}}{M_{W}}\left(\frac{M_{W}g_{X}}{4\pi
m_{X}}\right)^{4}{}_{2}F_{4}\left(\\{1,1\\},\\{2,3,3,5\\},\left(\frac{M_{W}g_{X}}{4\pi
m_{X}}\right)^{2}\right).$ (S22)

The $n=1$ case is special as there is a soft divergence leading to a log
enhancement of the form $\log\left(m_{W}/m_{X}\right)$. As this log is only
present for $n=1$, we conservatively neglect the $n=1$ contribution to the
decay width.


# High-efficiency single-photon source above the loss-tolerant threshold for
efficient linear optical quantum computing

Xing Ding  These authors contributed equally to this work.  Hefei National
Research Center for Physical Sciences at the Microscale and School of Physical
Sciences, University of Science and Technology of China, Hefei, Anhui 230026,
China  CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Hefei National Laboratory, University of Science and Technology of China,
Hefei, Anhui 230088, China  Yong-Peng Guo  These authors contributed equally
to this work.  Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei, Anhui 230026, China  CAS Center for Excellence in
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei, Anhui 230026, China  Hefei National Laboratory, University of
Science and Technology of China, Hefei, Anhui 230088, China  Mo-Chi Xu  These
authors contributed equally to this work.  Hefei National Research Center for
Physical Sciences at the Microscale and School of Physical Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, China  CAS
Center for Excellence in Quantum Information and Quantum Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei, Anhui
230088, China  Run-Ze Liu  These authors contributed equally to this work.
Hefei National Research Center for Physical Sciences at the Microscale and
School of Physical Sciences, University of Science and Technology of China,
Hefei, Anhui 230026, China  CAS Center for Excellence in Quantum Information
and Quantum Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China  Hefei National Laboratory, University of Science and
Technology of China, Hefei, Anhui 230088, China  Geng-Yan Zou  Hefei National
Research Center for Physical Sciences at the Microscale and School of Physical
Sciences, University of Science and Technology of China, Hefei, Anhui 230026,
China  CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Hefei National Laboratory, University of Science and Technology of China,
Hefei, Anhui 230088, China  Jun-Yi Zhao  Hefei National Research Center for
Physical Sciences at the Microscale and School of Physical Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, China  CAS
Center for Excellence in Quantum Information and Quantum Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei, Anhui
230088, China  Zhen-Xuan Ge  Hefei National Research Center for Physical
Sciences at the Microscale and School of Physical Sciences, University of
Science and Technology of China, Hefei, Anhui 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei, Anhui
230088, China  Qi-Hang Zhang  Hefei National Research Center for Physical
Sciences at the Microscale and School of Physical Sciences, University of
Science and Technology of China, Hefei, Anhui 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei, Anhui
230088, China  Hua-Liang Liu  Hefei National Research Center for Physical
Sciences at the Microscale and School of Physical Sciences, University of
Science and Technology of China, Hefei, Anhui 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei, Anhui
230088, China  Lin-Jun Wang  Center for Micro and Nanoscale Research and
Fabrication, University of Science and Technology of China, Hefei, Anhui
230026, China  Ming-Cheng Chen  Hefei National Research Center for Physical
Sciences at the Microscale and School of Physical Sciences, University of
Science and Technology of China, Hefei, Anhui 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei, Anhui
230088, China  Hui Wang  Hefei National Research Center for Physical Sciences
at the Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei, Anhui 230026, China  CAS Center for Excellence in
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei, Anhui 230026, China  Hefei National Laboratory, University of
Science and Technology of China, Hefei, Anhui 230088, China  Yu-Ming He  Hefei
National Research Center for Physical Sciences at the Microscale and School of
Physical Sciences, University of Science and Technology of China, Hefei, Anhui
230026, China  CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
China  Hefei National Laboratory, University of Science and Technology of
China, Hefei, Anhui 230088, China  Yong-Heng Huo  Hefei National Research
Center for Physical Sciences at the Microscale and School of Physical
Sciences, University of Science and Technology of China, Hefei, Anhui 230026,
China  CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Hefei National Laboratory, University of Science and Technology of China,
Hefei, Anhui 230088, China  Chao-Yang Lu  Hefei National Research Center for
Physical Sciences at the Microscale and School of Physical Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, China  CAS
Center for Excellence in Quantum Information and Quantum Physics, University
of Science and Technology of China, Hefei, Anhui 230026, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei, Anhui
230088, China  Jian-Wei Pan  Hefei National Research Center for Physical
Sciences at the Microscale and School of Physical Sciences, University of
Science and Technology of China, Hefei, Anhui 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, Anhui 230026, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei, Anhui
230088, China

###### Abstract

Photon loss is the biggest enemy for scalable photonic quantum information
processing. This problem can be tackled by using quantum error correction,
provided that the overall photon loss is below a threshold of 1/3. However,
all reported on-demand and indistinguishable single-photon sources still fall
short of this threshold. Here, by using tailor shaped laser pulse excitation
on a high-quantum efficiency single quantum dot deterministically coupled to a
tunable open microcavity, we demonstrate a high-performance source with a
single-photon purity of 0.9795(6), photon indistinguishability of 0.9856(13),
and an overall system efficiency of 0.712(18), simultaneously. This source for
the first time reaches the efficiency threshold for scalable photonic quantum
computing. With this source, we further demonstrate 1.89(14) dB intensity
squeezing, and consecutive 40-photon events with 1.67 mHz count rate.

Quantum computers can solve certain hard problems beyond the reach of any
classical computer. Recent noisy intermediate-scale quantum (NISQ) processors
implementing Gaussian boson sampling with photons [1] or random circuit
sampling with superconducting qubits [2] have poised for the technological
singularity called quantum computational advantage, which raised serious
challenges to the extended Church-Turing Thesis [3]. However, without
effective quantum error correction, the NISQ processing fidelity decreases
exponentially when scaling up the size. Building fault-tolerant universal
quantum computers represents the most important challenge ahead [4].

Single photons are fast-flying, can be operated at room temperature, and have
very weak interactions with the environment. However, the single photons can
be easily lost either owing to inefficient photon sources and detectors, or in
the lossy photonic circuits. This represents arguably the biggest enemy of
universal photonic quantum computing.

The celebrated threshold theorem [5] states that if the physical error rate is
below a certain threshold, arbitrarily long calculations at arbitrarily low
error rates, i.e., fault-tolerant quantum computing, are possible through
quantum error correction. To tackle the problem of photon loss, various
quantum error correction codes have been developed [6, 7]. In particular, in
the measurement-based model of quantum computation, Varnava, Browne, and
Rudolph [8] showed that, if the product of single-photon source and detection
efficiency is above 2/3, then linear optical quantum computing is possible.
Such an extremely relaxed photon loss threshold for the primary error
mechanisms in the photonic system has motivated extensive efforts in
developing increasingly high-efficiency quantum light sources.

Over the past 50 years, tremendous progress has been made in the performance
of the quantum light source, from the pioneering entangled photons used for
Bell test [9] to the quantum computing experiments recently with tens of
indistinguishable photons [1, 10, 11, 12]. In particular, spontaneous
parametric down-conversion or four-wave mixing can provide heralded single
photons based on the probabilistically generated two-photon pairs, which have
been engineered to have simultaneously near-unity collection efficiency and
indistinguishability [13, 14]. To overcome its probabilistic, spatial or
temporal multiplexing has been attempted, however, the best source efficiency
achieved at a cost of high multi-photon noise ($g^{2}(0)=0.27$) still cannot
exceed the threshold [15].

Quantum dots (QDs) are in principle deterministic single-photon emitters.
Pulsed resonant excitation to the single QDs embedded in finely tuned
microcavities such as micropillars [16, 17], bullseye [17], and open cavity
[18] have created single photons with increasingly high efficiency and
indistinguishability. However, the best reported end-to-end single-photon
source efficiency is 0.57 [18], still below the loss-tolerant threshold. In
this letter, we report a single-photon source, for the first time, reaches the
loss tolerant efficiency threshold for scalable photonic quantum computing,
with a system efficiency of 0.712(18), while maintaining a single-photon
purity of 0.9795(6) and indistinguishability of 0.9856(13).

Figure 1: The design of the QDs in a fully tunable open-cavity setup. The top
concave cavity mirror is mounted on a stack of three piezo nano-positioners.
The bottom part, including 30 pairs DBR and QD layer, is mounted on a stack of
three nano-positioners (XYZ) and two tilt stages ($\Theta,\Phi$). The
$\lambda$-thickness InAs QD layer is grown on top of a 30 pairs AlAs/GaAs DBR.
The center wavelength of the bottom DBR is 890 nm. The top surface of the QD
layer is coated with 50 nm thickness Au/Ti grids, which facilitates the
location of the selected QDs with high quantum efficiency. This setup provides
the in-situ ability to align the cavity with selected QDs. The inset shows a
numerical simulation of the radiation from a dipole in an open-cavity
structure. The numerically simulated $Q$-factor is $\sim$9000 and the Purcell
factor is 18. The probability that single photons can transmit through the top
of the cavity is $\eta_{top}=0.939$.

To optimize the single QD-microcavity coupling, we use a tunable open cavity
[18], as shown in Fig. 1(a), in a plane-concave Fabry-Perot cavity
configuration. The advantage of the open microcavity is that it provides full
degrees of freedom to maximize the coupling between selected QDs with high
quantum efficiencies and the microcavity spectrally and spatially. The open
cavity consists of a 5.5 pairs SiO2/Ta2O5 concave distributed Bragg reflector
(DBR) layer in the top, and in the bottom a $\lambda$-thickness QD membrane,
which is directly grown on the 30 pairs AlAs/GaAs planar DBR layer by
molecular beam epitaxy. The top cavity mirror is fabricated using FIB etching
concave surface on a SiO2 substrate. The radius of curvature and diameter of
the concave mirror is 15 $\mu$m and 6.3 $\mu$m respectively. The reflectivity
of the top mirror is $\sim$98.6%, while the reflectivity of the bottom mirror
is $\sim$99.97%.

The QD-open microcavity system is cooled down to 4 K in a closed-cycle
cryostat. Note that while the open cavity brings the key advantage of full
tunability, it comes with an extreme technical challenge of high stability,
because even a 1 nm drift of the cavity length would cause a shift of the
cavity’s resonance frequency and a drop of 78% of the maximum Purcell factor.
We made various efforts to overcome the environmental vibrations, especially
that from the cold head. The cryostat is mounted on a passive vibration damper
to isolate vibrations from the environment.

Figure 2: High-efficiency single-photon source (SPS) under shape-optimized
pulse excitation. (a) Excitation scheme of a polarized single-photon source in
a mode-splitting cavity. (b) Deterministic generation of polarized single
photons under a series of pulse excitations with different pulse widths. (c)
The corresponding single-photon purity under different pulse width
excitations. (d) The RF photon counts as a function of the square root of
excitation laser power when the pulse width is 69 GHz. At the $\pi$ pulse, the
end-to-end single-photon counts per second reach a maximum of $(14.28\pm
0.01)$ million, under 25.38 MHz repetition rate laser excitation, which
corresponds to a system efficiency of 0.712(18).

To maximize the brightness of the single-photon emitters, we first pre-select
QDs with high quantum efficiencies by removing the top concave cavity and
comparing the $\pi$-pulse resonance fluorescence (RF) of hundreds of bare QDs.
The positions of the brightest candidates are recorded. Then, the top mirror
returns, forming a fully tunable open microcavity that allows in-situ coupling
of the cavity spatially and spectrally.

The basic excitation protocol follows the resonant excitation of a QD in a
degenerate mode-splitting cavity [17]. The mode-splitting here is induced by
the birefringence of the GaAs material, and the magnitude of mode-splitting
can be manipulated by applying strains in the GaAs material [19]. Compared
with geometrically asymmetric cavities, birefringence-induced mode-splitting
does not induce far-field optical mode deformation, which is favorable for
higher single-mode fiber collection efficiency. Figure 2(a) shows the
experimental excitation scheme used in practice. The measured $Q$-factor of
each mode is $\sim$8400 and splitting between two modes is $\sim$83 GHz, which
gives the ratio of the cavity splitting ($\Delta\omega$) to the cavity
linewidth ($\delta\omega$) of 2.07. A singly negatively charged QD emitting at
a wavelength of 884.5 nm with high quantum efficiency $\eta_{qd}$ is coupled
to the cavity’s H-polarized mode. Then we can excite the QD with a V-polarized
laser pulse and collect the H-polarized single photons.

It is important to note that, in previous experiment [17], due to the
frequency mismatch between the V-polarized excitation laser and the
H-polarized cavity mode, only a small fraction of the laser can actually be
used to excite the QD. The frequency of the partial excitation laser may be
far detuned with the QD, leading to a reduction of the $\pi$ pulse single-
photon counts. Compared to low $Q$ resonator structures like bullseye cavity,
this problem is more severe in high $Q$ resonators such as micro-pillar and
open-cavity.

To overcome this problem, we use a 4f optical pulse shaping system to shape
the laser pulse. The shaping system consists of two reflective diffraction
gratings, two identical lenses, and one slit. The groove density of the
gratings is 1200g/mm, and the focal length of the lenses is 1830 mm. The first
grating is used to transform frequency components into spatial ones, and the
second grating is for an inverse process. According to the detailed
description of the excitation scheme in supplementary material [20], with a
narrower excitation pulse width in frequency, the intra-cavity pulse would
resonantly excite the QD more effectively. Thus we set a width-tunable slit at
the Fourier plane to filter out unwanted frequency components. As shown in fig
2(b), by resonantly exciting the selected QD with different pulse widths, from
$\sim$96 GHz to $\sim$46 GHz, a set of clear Rabi oscillations are observed.
These oscillations illustrate an improvement in the $\pi$ pulse single-photon
counts, as well as a reduction in the power required for their excitation when
utilizing narrower pulse widths.

It should also be noted that narrowing the pulse width in frequency will
broaden the pulse length in the time domain, which can cause re-excitation of
the QD and reduce the single-photon purity. In fig 2(c), we show the measured
second-order correlations of the single photons under different excitation
pulse widths. By comparing the purity and brightness of the single photons, we
can figure out that 69 GHz is the most optimized pulse width to excite our
selected QD in the open cavity. Under this shape-optimized pulse excitation, a
single-photon source with minimum excitation power, near unity single-photon
purity, and maximum single-photon counts can be achieved simultaneously. The
shaped pulse excitation generates 13% more single photons at the $\pi$ pulse,
in comparison to the unmodified pulse excitation with a pulse width of 96 GHz.

Having optimized the shaping system, we proceeded to measure the efficiency of
our single-photon system. To avoid the dead time zone (30ns) of the
superconducting nanowire single-photon detector (SNSPD), we utilized an
amplitude electro-optic modulator to decrease the repetition rate of the
excitation laser to 25.38 MHz, which is one-third of the original frequency of
76.13 MHz. The single photons collected at the single-mode fiber end are
directly connected to the SNSPD, whose detection efficiency is 0.79(2). In
Figure 2(d), we present the corrected RF photon counts plotted against the
square root of the excitation laser power. Using a $\pi$ pulse power of 10.24
nW, we observed a $\pi$ pulse single-photon count rate of approximately 14.28
million counts per second, resulting in a system efficiency of 0.712(18). This
is the highest achieved single-photon source’s system efficiency of all
reported physical systems [20].

Figure 3: System efficiency characterization of the single-photon source. (a)
The single-photon stream with a time bin of 1.0us. 1000 bins of single-photon
counts are recorded. (b) The corresponding normalized histogram. The intensity
fluctuation of single photons is shown as deep blue dots and fitted with a
binomial function. The green shadow shows the shot-noise-limited source with
the same photon count rate. The directly measured intensity squeezing of the
single-photon source compared with the shot-noise limit is
$\sigma_{SPS}/\sigma_{SN}=0.65(2)$ under $\pi$ pulse excitation. The grey
shadow shows the corrected intensity squeezing value at the first lens is
$3.92\pm 0.18$ dB. (c) Statistics of consecutive n-photon events within 10
minutes, up to 40 photons in a row can be acquired, the line indicates an
exponential decay fit.

An important feature of high-efficiency single-photon source can be manifested
by intensity squeezing [21], i.e., the reduction of intensity fluctuation.
Such an observation was only possible very recently [22], after 20 years
development of QD single photons, highlighting the importance of improving the
system efficiency of single-photon sources. The standard deviation of counts
in a single-photon source is reduced by an intensity squeezing factor of
$\sqrt{(1-\rho)}$ compared to the shot noise, so the value of intensity
squeezing is a natural benchmark to characterize the overall efficiency of
single-photon sources. Here $\rho$ is the overall efficiency, including both
the collection efficiency and the detection efficiency. A time-correlated
single-photon counting (TCSPC) system is used to record the arrival time of
each photon for further analysis. Figure 3(a) shows the real-time detected
single photons at $\pi$ pulse with a time bin of 1.0 $\mu$s, the corresponding
histogram is shown in Fig 3(b). The standard deviation of single-photon counts
(blue) in Fig 3(b) is 2.43(8), which shows sub-shot-noise intensity
fluctuation with an intensity squeezing of 1.89(14) dB.

We analyzed the consecutive single-photon events to reveal the overall
efficiency. Figure 3(c) illustrates the number of consecutively detected
n-photon streams. The integration time is 10 minutes, and a maximum of
40-photon events are observed at a count rate of 1.67 mHz. The event rate of
consecutive $n$-photons should be $\rho^{n}\times R_{laser}$, here $R_{laser}$
is the repetition rate of the pulse laser. By fitting the exponential decay,
the overall efficiency $\rho$ was determined to be $0.5652(7)$. Taking into
account the detection efficiency of the used detector is $0.79(2)$, the system
efficiency of the single-photon source is $0.715(18)$, which again confirms
the efficiency measurement.

Figure 4: Characterization of the pulsed RF single photons. (a) The intensity
correlation histogram of the single photons under $\pi$ pulse excitation
measured with an HBT setup. The second-order correlation at zero time delay is
$g^{2}(0)=0.0205(6)$. (b) Quantum interference between two single photons. The
input two photons are $\pi$ pulse excited and prepared in cross and parallel
polarization. The raw HOM visibility is 0.928(1). (c) Single-photon
indistinguishability as a function of emission time separation. The corrected
photon indistinguishability drops very slightly, from 0.9856(13) at 13.1 ns to
0.959(2) at $\sim$2.67 $\mu$s.

In addition to measuring system efficiency, we conducted comprehensive
characterizations of the single-photon source by evaluating its purity and
indistinguishability. The single-photon purity is characterized with a Hanbury
Brown and Twiss (HBT) setup [23] and presented in Fig 4(a), with
$g^{2}(0)=0.0205(6)$. The data are directly measured without any further
filtering. The excitation laser is found to be the primary source of residual
multi-photon events. The photons’ indistinguishability is measured using a
non-postselective two-photon Hong-Ou-Mandel interferometer (HOM) [24]. Figure
4(b) shows the histograms of the normalized two-photon counts for cross and
parallel polarization respectively. The raw two-photon quantum interference
visibility is 0.928(1). After correction with the residual multi-photon
probability of $g^{2}(0)=0.0205(6)$ and an unbalanced beam splitter split
ratio of 45:55, the corrected single-photon indistinguishability is
0.9856(13).

To further prove that the single-photon source is ready for scale-up, it is
imperative to verify that that the single photons are nearly transform-limited
[25, 26]. Here we test the photon indistinguishability with different photon
time delays using a delay line adjustable HOM interferometer [25]. As shown in
Fig 4(c), the corrected HOM visibility is 0.9856(13), 0.985(1), 0.970(2), and
0.959(1) for the time delays of 13.1 ns, 670 ns, 1.31 $\mu$s and 2.67 $\mu$s,
respectively. Compared with the 13.1 ns delay, the single-photon
indistinguishability with 2.67 $\mu$s delay only drops less than 3%,
indicating that the emitted single photons are close to the transform limit.
Such high photon indistinguishability with long-time delay guarantees the
performance in scalable photonic applications.

In conclusion, by resonant excitation of a QD embedded in an open-cavity, we
have realized a deterministic single-photon source with single-photon purity
of 0.9795(6), indistinguishability of 0.9856(13), and system efficiency of
0.712(18) simultaneously, which for the first time exceeds the required
efficiency threshold for scalable photonic quantum computing [8]. Such a high-
efficiency source can be immediately used for applications such as boson
sampling, photonic cluster states generation, quantum communication, etc.
Furthermore, this QD-open cavity system can be readily extended to the strong
coupling regime by increasing the coupling rate, thereby facilitating the
implementation of photon-photon quantum gates. With its exceptional
performance and significant potential for various applications, our work marks
a critical step towards scalable quantum technologies reliant on single
photons.

###### Acknowledgements.

Our work is supported by the National Natural Science Foundation of China
(Grant No. 12012422), the National Key R&D Program of China (Grant No.
2019YFA0308700), the Chinese Academy of Sciences, the Anhui Initiative in
Quantum Information Technologies (Grant No. AHY060000), the Science and
Technology Commission of Shanghai Municipality (Grant No. 2019SHZDZX01), the
Innovation Program for Quantum Science and Technology (Grant No.
2021ZD0301400, 2021ZD0300204) and China Postdoctoral Science Foundation (Grant
No. 2021M703102).

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