YITP-23-57, KOBE-COSMO-23-05, UT-Komaba/23-2, IPMU23-0012, CTPU-PTC-23-13

Gravitational Positivity for Phenomenologists:

Dark Gauge Boson in the Swampland

Katsuki Aokia, Toshifumi Noumib,c, Ryo Saitod,e, Sota Satob,c,

Satoshi Shiraie, Junsei Tokudaf, Masahito Yamazakie,g

aCenter for Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto,
Japan

bGraduate School of Arts and Sciences, University of Tokyo,
Komaba, Meguro-ku, Tokyo 153-8902, Japan

cDepartment of Physics, Kobe University, Kobe 657-8501, Japan

dGraduate School of Sciences and Technology for Innovation,
Yamaguchi University, Yamaguchi 753-8512, Japan

eKavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo Institutes for Advanced Study,
University of Tokyo, Kashiwa 277-8583, Japan

fCenter for Theoretical Physics of the Universe,
Institute for Basic Science (IBS), Daejeon, 34126, Korea

gTrans-Scale Quantum Science Institute,
University of Tokyo, Tokyo 113-0033, Japan

###### Contents

  1. 1 Introduction
  2. 2 Gravitational Positivity Bounds
    1. 2.1 Positivity Bounds without Gravity
    2. 2.2 Positivity Bounds with Gravity
    3. 2.3 Generalizations for Realistic Theories
  3. 3 Practical Recipe for Gravitational Positivity
  4. 4 Positivity Bound on Dark U(1) Gauge Boson
    1. 4.1 Higgs Contribution
    2. 4.2 Fermion Contribution
    3. 4.3 Constraint on Dark Gauge Boson
  5. 5 Toward Constraints on Realistic Phenomenological Models
  6. 6 Conclusion and Discussion
  7. A Gravitational Contributions Beyond Graviton $t$-channel

##  1 Introduction

It is one of the outstanding questions in present-day physics to uncover the
origin and the identity of the dark matter (DM). More broadly we are
interested in the search for the dark sector containing DM, or any physics
beyond the Standard Model (BSM).

Traditionally, dark sectors have been analyzed in the framework of low-energy
Effective Field Theories (EFTs), where the effects of the dark sector have
been incorporated by small couplings of the dark sector to particles of the
Standard Model (SM). It is often the case that such couplings are severely
constrained by experiments/observations, leading to very small (absolute)
values of parameters. While certain fine-tunings are needed, the dominant
attitude has been that such small parameters are well-tolerated in the EFTs,
as long as such small parameters are technically natural [1]. This raises a
fundamental problem for dark-sector searches: while we can keep eliminating
parameter spaces for dark-sector-SM couplings, there is in principle no lower
bound on their size, and it seems that we can never exclude a “nightmare”
scenario where the dark sector interacts with the SM sector only through
gravitational interactions.

The situation is different, however, once we impose an extra condition that
the Infrared (IR) low-energy EFT has a consistent ultraviolet (UV) completion
with gravity. There has been mounting evidence recently that there are
necessary conditions for a low-energy EFT to have a consistent UV completion,
and such conditions are often called swampland constraints in the literature
[2, 3]. One way to obtain such constraints is to invoke the so-called
positivity constraints on scattering amplitudes at low energy. The positivity
constraints are well established in non-gravitational theories and their
application to the EFT of the Standard Model and cosmology has long been
discussed: see [4, 5, 6, 7] for early references on QCD. For more recent
developments, see e.g. [8], a review article [9] and references therein. The
advantage of this method is that we can derive an infinite set of inequalities
on the EFT couplings by minimal assumptions on the scattering amplitudes such
as unitarity, analyticity, and locality, without relying too much on the
details of the UV completion.

  1. 0.

Write down the (renormalizable) Lagrangian of your model, couple the model to
gravity, and consider a $2\to 2$ scattering process $AB\to AB$.

  2. 1.

Compute the amplitude ${\cal M}_{\text{non-grav}}$ of diagrams without
gravity:

$\displaystyle i{\cal M}_{\text{non-
grav}}(s,t)=\raisebox{-14.0pt}{\includegraphics{nongra.pdf} }+\cdots\,,$

and ${\cal M}_{\text{grav},t\text{-channel}}$ of graviton $t$-channel
diagrams:

$\displaystyle i{\cal
M}_{\text{grav},t\text{-channel}}(s,t)=\raisebox{-16.0pt}{\includegraphics{t-gra.pdf}}+\cdots\,.$

  3. 2.

Compute $B_{\text{non-grav}}$:

$\displaystyle B_{\text{non-
grav}}(\Lambda)=\frac{4}{\pi}\int_{\Lambda^{2}}^{\infty}ds\,\frac{{\rm
Im}\,\mathcal{M}_{\text{non-
grav}}(s,t=0)}{\left(s-m_{A}^{2}-m_{B}^{2}\right)^{3}}\,.$

  4. 3.

Compute series expansion of ${\cal M}_{\text{grav},t\text{-channel}}$ around
$t=0$ and $s=m_{A}^{2}+m_{B}^{2}$, and extract $B_{\text{grav}}$ from the
coefficient of $t^{0}(s-m_{A}^{2}-m_{B}^{2})^{2}$.

  5. 4.

Compute $B(\Lambda)=B_{\text{non-grav}}(\Lambda)+B_{\text{grav}}(\Lambda)$ and
check its positivity.

Figure 1: Workflow to get the gravitational positivity bound (scalar case).

It is challenging to derive positivity bounds in the presence of gravity
because scattering amplitudes of quantum gravity are less understood in
general. If one can derive such bounds, however, the bounds will be understood
as the swampland conditions on a given EFT. There are various attempts to
formulate positivity bounds in the presence of gravity: see e.g., [10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

In this paper, we discuss a particular version of positivity bounds with
gravitational effects included [14]. Interestingly, the effect of gravity
modifies the positivity bound, and in some cases strengthens it. While the
effects of gravity are suppressed by the powers of the Planck scale, gravity
can still change the conclusions dramatically since we are interested in small
couplings in dark sectors. Indeed, it has been known that if the gravitational
positivity bound is applied to the matter-matter scatterings at loop level in
$D=4$ dimensions, we obtain nontrivial constrains on the spectrum and
renormalizable couplings of light particles well below the Planck scale [24,
25, 26, 16, 27, 28, 29].

The gravitational positivity bound sometimes gives a lower bound on the size
of the dark sector couplings. When combined with upper bounds from
observations, one can then in principle exclude some dark sector models. The
gravitational positivity bound relies on some assumptions of the UV theory and
hence can be violated in observations. This, however, inevitably implies a
violation of some well-defined fundamental assumptions of the UV theories, and
hence we can derive sharp constraints of UV physics from the experiments of IR
physics.

It is an interesting question to systematically apply the gravitational
positivity bounds to various BSM models. It is fair to say, however, that a
systematic application of gravitational positivity bounds for BSM physics
remains mostly unexplored. Moreover, there are important unsolved problems
whose solutions are required for such analysis, as we will see below.

The goal of this paper is to make the gravitational positivity bound more
accessible to a broader audience of phenomenologists interested in BSM
physics. In Fig. 1, we show the procedures to obtain the gravitational
positivity bound for scalar fields. For this purpose, we list a concrete
recipe for deriving gravitational positivity bounds for a given EFT, and list
some fine prints in such discussions. We next illustrate the power and
limitations of the bounds in the concrete example of a massive dark
$\mathrm{U}(1)$ gauge boson (dark photon). We will find that the resulting
bound is impressively strong, however, for applications for realistic models,
there are some theoretical subtleties that need to be addressed.

The rest of this paper is organized as follows. In Sec. 2 we summarize the
gravitational positivity bound for non-experts. In Sec. 3 we present a
practical recipe for deriving the gravitational positivity bound. As a
concrete illustration of our procedure, we write down the gravitational
positivity bound for a massive $\mathrm{U}(1)$ gauge boson in Sec. 4. We will
find that the resulting bound is very strong and rules out many parameter
spaces. One should note, however, that there are important subtleties for
realistic model building, as we will discuss further in Sec. 5. The final
section (Sec. 6) is devoted to conclusions and discussions. We included an
appendix for gravitational contributions beyond graviton $t$-channel exchange.

##  2 Gravitational Positivity Bounds

In this section we summarize the gravitational positivity bounds of [14]. To
make the presentation friendly to non-experts, we outline the basic
assumptions, definitions of quantities to be computed, bounds, and their
interpretations, and leave their derivations to original references. We begin
with the non-gravitational case and then include gravity. We will close with
some remarks needed for applications to realistic phenomenological model
building.

###  2.1 Positivity Bounds without Gravity

Positivity bounds are formulated in terms of scattering amplitudes. For
technical simplicity, this subsection focuses on scalar scattering amplitudes
in gapped theories.

##### Assumptions.

Consider an $s$-$u$ symmetric scattering amplitude $\mathcal{M}(s,t)$ of
$AB\to AB$ type in a given low-energy EFT, where $s,t,u$ are the standard
Mandelstam variables that satisfy $s+t+u=2(m_{A}^{2}+m_{B}^{2})$, and $m_{A}$
and $m_{B}$ are the masses of the external particles $A$ and $B$,
respectively. We assume that the forward amplitude $\mathcal{M}(s,t=0)$
evaluated in the would-be UV complete theory satisfies the following
properties:

  1. 1.

Analyticity: The amplitude $\mathcal{M}(s,t=0)$ is analytic on the physical
sheet111There are in general unstable resonances, whose associated poles are
however located in other sheets in the complex $s$-plane. of the complex
$s$-plane except for poles and discontinuities on the real axis required by
unitarity.

  2. 2.

Unitarity: The imaginary part is non-negative, i.e., ${\rm
Im}\,\mathcal{M}(s,t=0)\geq 0$.

  3. 3.

$s^{2}$-boundedness: The amplitude is bounded by $s^{2}$ at high energy, i.e.,
$\displaystyle\lim_{|s|\to\infty}\mathcal{M}(s,0)/s^{2}=0$. This third
condition is guaranteed in local quantum field theories thanks to the
Froissart bound [30, 31] and the Phragmén-Lindelöf theorem.

##### Positivity bounds.

The aforementioned three properties imply an infinite set of consistency
relations among scattering amplitudes evaluated at the UV and IR. Let us
define

$\displaystyle a_{2n}:=\left[\frac{\partial^{2n}\mathcal{M}(s,t=0)}{\partial
s^{2n}}\right]_{s=m_{A}^{2}+m_{B}^{2}}\quad(n=1,2,\ldots)\,.$ (2.1)

Then, the following dispersion relation holds from the analyticity and
$s^{2}$-boundedness of the amplitude:

$\displaystyle a_{2n}=\frac{2\cdot(2n)!}{\pi}\int_{m_{\rm
th}^{2}}^{\infty}ds\,\frac{{\rm
Im}\,\mathcal{M}(s,t=0)}{(s-m_{A}^{2}-m_{B}^{2})^{2n+1}}\quad(n=1,2,\ldots)\,,$
(2.2)

where $m_{\rm th}$ is the threshold energy, i.e., the mass of the lightest
intermediate state. Note that the relation for $n=0$ does not follow under the
present assumptions, which requires a stronger assumption that the amplitude
is bounded by $s^{0}$ at high energy.

The EFT is defined with a UV cutoff which we denote by $\Lambda_{\rm UV}$. The
actual cutoff of the EFT, namely the scale of new physics, is unknown from the
low-energy perspective. We thus introduce a reference scale $\Lambda$ below
which the EFT is assumed to be valid, i.e., $\Lambda$ is assumed to satisfy
$\Lambda<\Lambda_{\rm UV}$.

Let us define222In general, we can have
$m_{\text{th}}^{2}<m_{A}^{2}+m_{B}^{2}$ and branch cuts can run on the entire
real $s$-axis. The definitions (2.1) and (2.3) may become ill-defined. In such
a case, $B_{2n}(\Lambda)$ should be defined through “arcs” of [32, 33] (see
also [34]). $B_{2n}(\Lambda)$ defined in this way agree with (2.3) [or (2.7)]
when branch cuts do not run on the entire real $s$-axis, according to the
Cauchy integral theorem. [35, 36, 37]

$\displaystyle B_{2n}(\Lambda):=a_{2n}-\frac{2\cdot(2n)!}{\pi}\int_{m_{\rm
th}^{2}}^{\Lambda^{2}}ds\,\frac{{\rm
Im}\,\mathcal{M}(s,t=0)}{(s-m_{A}^{2}-m_{B}^{2})^{2n+1}}\quad(n=1,2,\ldots)\,.$
(2.3)

We emphasize that $B_{2n}(\Lambda)$ is defined in terms of the amplitude at
the IR scale below $\Lambda$, and therefore it is calculable within the EFT.
Then, the dispersion relation (2.2) implies

$\displaystyle
B_{2n}(\Lambda)=\frac{2\cdot(2n)!}{\pi}\int_{\Lambda^{2}}^{\infty}ds\,\frac{{\rm
Im}\,\mathcal{M}(s,t=0)}{(s-m_{A}^{2}-m_{B}^{2})^{2n+1}}\quad(n=1,2,\ldots)\,,$
(2.4)

which provides a consistency relation between the IR physics below $\Lambda$
(LHS) and the UV physics above $\Lambda$ (RHS). Furthermore, unitarity implies
that the RHS is non-negative, so that

$\displaystyle B_{2n}(\Lambda)\geq 0\quad\text{for all $\Lambda$ such that
$\Lambda<\Lambda_{\rm UV}$}\quad(n=1,2,\ldots)\,.$ (2.5)

These bounds are called the positivity bounds. Note that $B_{2n}(\Lambda)$ is
a monotonically non-increasing function of $\Lambda$ because the integrand in
(2.3) is positive. The typical behavior is shown in Fig. 2. When the bounds
(2.5) are satisfied at a certain scale, they also hold at any scale below it.
The positivity bound (2.5) is thus stronger for larger $\Lambda$.

##### Interpretations.

In a given EFT, one can compute the forward amplitude and evaluate
$B_{2n}(\Lambda)$ defined in Eq. (2.3) in terms of parameters of the EFT and a
scale $\Lambda$. The positivity bounds (2.5) then provide certain inequalities
among the EFT couplings and $\Lambda$, leading to two complementary
interpretations:

  1. 1.

Bounds on EFT couplings.
Suppose that the low-energy EFT describes a system below an energy scale $E$.
We can then identify $\Lambda$ with the energy scale of interest $E$ to find
the bounds on the EFT couplings. These are the necessary conditions for the
EFT to have a standard UV completion and be valid below the scale $E=\Lambda$.
(In here we are agnostic on the value of the cutoff scale $\Lambda_{\rm UV}$.)

  2. 2.

Bounds on the UV cutoff.
In some cases, one may want to assume certain values of the EFT couplings—for
example, coupling constants are already fixed by experiments or they need to
take certain values for phenomenological purposes (e.g., so that they can be
searched in a particular experiment). One can then ask up to which scale the
EFT can be valid, i.e., at which scale the new physics comes in. As shown in
Fig. 2, $B_{2n}(\Lambda)$ may become negative at some energy scale
$\Lambda_{*}$, in which case we can derive an upper bound $\Lambda_{\rm
UV}<\Lambda_{*}$ on the UV cutoff of the EFT.

The condition (2.5) is based on the assumptions on UV, but care must be taken
to ensure because we do not know the UV theory. If a violation of the
inequality (2.5) is observed in the experiment, it may indicate a violation of
the original assumptions. In this sense, we can also use the positivity bounds
(2.5) to probe UV from IR.

Figure 2: The expression (2.3) of $B_{2n}(\Lambda)$ is a monotonically non-
increasing function as a function of $\Lambda$, and crosses zero at the scale
$\Lambda=\Lambda_{*}$. For a well-defined UV completion, we need
$B_{2n}(\Lambda)\geq 0$, which defines an upper bound $\Lambda_{*}$ on the
scale $\Lambda$. Figure 3: Typical scenario: a UV completion of gravity is
achieved well below the Planck scale by an infinite tower of higher-spin
states (Regge tower). There may also exist new UV states beyond the EFT that
are described within QFT model building and that are not directly related to
the UV completion of gravity.

###  2.2 Positivity Bounds with Gravity

We next explain how the positivity bounds are extended to theories with
gravity. The main new ingredient is that the $t$-channel graviton exchange
contributes a new singularity to the low-energy scattering amplitudes as
$\sim-s^{2}/(M_{\rm Pl}^{2}t)$ ($M_{\rm Pl}\simeq 2.4\times
10^{18}\,\textrm{GeV}$ being the reduced Planck mass), invalidating the
definition (2.1) of $a_{2}$ and hence the positivity argument for
$B_{2}(\Lambda)$.333Note that the positivity argument for $B_{2n}$
($n=2,3,\ldots$) is unchanged. In the following, we focus on $B_{2}$. In this
subsection, we summarize how the positivity bounds are formulated in the
presence of gravity.

##### Assumptions.

The basic assumptions are the same as before, namely analyticity, unitarity,
and the $s^{2}$-boundedness
$\displaystyle\lim_{|s|\to\infty}\bigl{|}\mathcal{M}(s,t)/s^{2}\bigr{|}=0$ for
small negative $t$.

While the argument below can be applied to any UV complete theory with these
three properties, a typical candidate of such a UV completion is a
perturbative string theory with a particle spectrum as shown in Fig. 3.

##### Gravitational positivity bounds.

We define $a_{2}$ by subtracting the graviton $t$-pole from the amplitude and
taking the forward limit as444 For simplicity, we focus on the amplitude up to
$\mathcal{O}(M_{\rm Pl}^{-2})$ and only subtract the graviton $t$-pole as a
divergent term in the forward limit $t\to-0$ in the definition of $a_{2}$. See
[15, 20] for arguments with the $\mathcal{O}(M_{\rm Pl}^{-4})$ terms.

$\displaystyle a_{2}$
$\displaystyle:=\lim_{t\to-0}\left[\frac{\partial^{2}\mathcal{M}(s,t)}{\partial
s^{2}}+\frac{2}{M_{\rm Pl}^{2}t}\right]_{s=m_{A}^{2}+m_{B}^{2}}\,.$ (2.6)

With $a_{2}$ defined in this manner, we introduce

$\displaystyle B(\Lambda)$ $\displaystyle:=a_{2}-\frac{4}{\pi}\int_{m_{\rm
th}^{2}}^{\Lambda^{2}}ds\,\frac{{\rm
Im}\,\mathcal{M}(s,t=0)}{\left(s-m_{A}^{2}-m_{B}^{2}\right)^{3}}\,.$ (2.7)

Here and in what follows we suppress the subscript $2$ of $B_{2}(\Lambda)$ for
notational simplicity. The dispersion relation (2.4) is then modified as

$\displaystyle B(\Lambda)=\lim_{t\to-0}\left[\frac{2}{M_{\rm
Pl}^{2}t}+\frac{4}{\pi}\int_{\Lambda^{2}}^{\infty}ds\,\frac{{\rm
Im}\,\mathcal{M}(s,t)}{\left(s+\frac{t}{2}-m_{A}^{2}-m_{B}^{2}\right)^{3}}\right]\,.$
(2.8)

Due to the first term in the square bracket (which is negative for $t<0$), the
sign of $B(\Lambda)$ is undetermined. However, one can show that the
positivity of $B(\Lambda)$ holds at least approximately. The idea is that we
split the integral of the dispersion relation into the quantum gravity part
and the other,

$\displaystyle B(\Lambda)=\frac{4}{\pi}\int_{\Lambda^{2}}^{\Lambda_{\rm
QG}^{2}}ds\,\frac{{\rm
Im}\,\mathcal{M}(s,0)}{\left(s-m_{A}^{2}-m_{B}^{2}\right)^{3}}+\lim_{t\to-0}\left[\frac{2}{M_{\rm
Pl}^{2}t}+\frac{4}{\pi}\int_{\Lambda_{\rm QG}^{2}}^{\infty}ds\,\frac{{\rm
Im}\,\mathcal{M}(s,t)}{\left(s+\frac{t}{2}-m_{A}^{2}-m_{B}^{2}\right)^{3}}\right]\,.$
(2.9)

When we take the forward limit $t\to-0$, the l.h.s of (2.9) is finite because
it is defined by subtracting the $t$-pole term. Then, although the $t$-pole
term in the r.h.s diverges in the forward limit $t\to-0$, the sum in the
square bracket should be finite as a result of a cancellation with the quantum
gravity part:

$\displaystyle\lim_{t\to-0}\left[\frac{2}{M_{\rm
Pl}^{2}t}+\frac{4}{\pi}\int_{\Lambda_{\rm QG}^{2}}^{\infty}ds\frac{{\rm
Im}\,\mathcal{M}(s,t)}{\left(s+\frac{t}{2}-m_{A}^{2}-m_{B}^{2}\right)^{3}}\right]=:\frac{\sigma}{M_{\rm
Pl}^{2}M^{2}}\quad(\sigma=\pm 1)\,,$ (2.10)

where we defined the sign $\sigma$ and the mass scale $M$, which depend on
details of the cancellation mechanism. As a result, we find

$\displaystyle B(\Lambda)\geq\frac{\sigma}{M_{\rm Pl}^{2}M^{2}}\quad\text{for
all $\Lambda$ s.t.\ $\Lambda<\Lambda_{\rm UV}<\Lambda_{\rm QG}$ }\,,$ (2.11)

which we call the gravitational positivity bound.

The right-hand side of (2.11) is suppressed by powers of $M_{\rm Pl}$, and
hence disappears in the decoupling limit $M_{\rm Pl}\to\infty$, where the
gravitational positivity bound reduces to the positivity bound without
gravity.

The size of the right-hand side depends on both the scale $M$ and the sign
$\sigma$. While the derivation here works irrespective of the details of the
cancellation mechanism, the most plausible scenario is the Regge
behavior555See also [38, 39] for connections between Reggeization and
causality.: the gravitational amplitude is modified as ${\rm
Im}\mathcal{M}\sim s^{2+\alpha^{\prime}t+\cdots}$ with
$\alpha^{\prime}\sim\Lambda_{\rm QG}^{-2}$ by an infinite tower of higher-spin
states at $s\geq\Lambda_{\text{QG}}^{2}$ (see Fig. 3). The perturbative string
theory is a typical example of this scenario in which the quantum gravity
scale $\Lambda_{\text{QG}}$ corresponds to the string scale. The cancellation
of the $t$-pole can be explicitly shown under the assumption of Regge behavior
[14]. If we assume the gravitational Reggeization of the amplitude at high
energy, then a naive expectation is $M\sim\Lambda_{\rm QG}$. If IR physics
also plays a role in the Regge behavior, however, we can encounter a much
smaller value $M\ll\Lambda_{\rm QG}$. It is an interesting question to
systematically study the values of $M$ and $\sigma$ in string-theory
compactifications.

##### Interpretations.

In contrast to the non-gravitational positivity bound (2.5), the gravitational
positivity bound (2.11) contains not only the IR data $B(\Lambda)$ determined
by the EFT parameters up to the scale $\Lambda$, but also the UV data
$(M,\sigma)$ of gravitational Regge amplitudes (or, generically speaking,
quantum gravity amplitudes). Therefore, there are three possible
interpretations, depending on the context [28, 29, 16, 27, 17]:

  1. 1.

Quantum gravity constraints on IR physics.

If we specify/assume a quantum gravity scenario and the parameters
$(\sigma,M)$ of gravitational Regge amplitudes, we can think of the bounds
(2.11) as quantum gravity constraints on the low-energy EFT at a scale
$\Lambda=E$. Such constraints are useful to carve out the parameter space of
phenomenological models in the spirit of the Swampland Program.

  2. 2.

Constraints on the scale of new physics.

As in the non-gravitational case, we can interpret (2.11) as the bounds giving
the maximum cutoff $\Lambda_{*}$ when we assume the parameters of the EFT and
the UV data $(\sigma,M)$. When the bounds (2.11) are violated, one can try to
increase the cutoff of the EFT by adding new state(s) to EFT like Fig. 2. If
the bounds (2.11) are inevitably violated at $\Lambda>\Lambda_{*}$, this is a
sign of the necessity for a UV completion of gravity. In this case,
$\Lambda_{*}$ is not a maximum scale of just new physics, but the upper bound
on the quantum gravity scale $\Lambda_{\rm QG}$.

  3. 3.

IR constraints on quantum gravity.

Once the parameters and the validity of the model are identified by
experiments at a scale $\Lambda=E$, we can use the bounds (2.11) to constrain
the parameters $(\sigma,M)$ of the gravitational Regge amplitudes required for
UV completion of gravity. Such constraints are useful as a necessary condition
for a quantum gravity theory to describe our real world.

As we mentioned in Sec. 2.1, the violation of original assumptions on UV
theory leads to the violation of (2.11). Differently from the non-
gravitational case, however, experimental tests of the violation of (2.11)
require knowledge of the UV data $(M,\sigma)$.

###  2.3 Generalizations for Realistic Theories

When one tries to apply the gravitational positivity bounds (2.5) to realistic
phenomenological models, the basic idea is the same as in the case of scalar
scatterings of the previous subsection. However, one encounters several new
subtleties to be carefully addressed, which we list in the following.

##### Spin.

The positivity bound can be extended to a scattering of particles with spin,
see [35, 40].666 For this purpose, it is useful to diagonalize the crossing
relation using the transversity formalism of [41]. For the use of this
formalism in the context of positivity bounds, see [40]. There are extra
technical complications, however, and here we highlight the issue of the
kinematic singularity, which is relevant for our discussion of gauge bosons in
Sec. 3.

Let us consider two-to-two scattering amplitudes of massive spin-1 particles
with mass $m_{V}$. The kinematic singularity is the singularity of the
scattering amplitude $\mathcal{M}$ from the pole $s=4m_{V}^{2}$ of the
scattering angle $\theta$ as in

$\displaystyle\cos\theta=1+\frac{2t}{s-4m_{V}^{2}}\ .$ (2.12)

The kinematic singularity disappears in the forward limit $t\to 0$ if the
scattering amplitude does not contain the massless $t$-pole. However, in our
setup, the kinematic singularity appears from $t$-channel graviton exchange
diagrams. We can deal with this singularity by defining a kinematic
singularity-free amplitude to be [40] (see also [42, 43]):

$\displaystyle\widetilde{\mathcal{M}}(s,t):=s^{2}\left(s-4m_{V}^{2}\right)^{2}\mathcal{M}(s,t)\
.$ (2.13)

We can then derive the positivity bound by essentially the same arguments, as
long as we replace the amplitude $\mathcal{M}$ by $\widetilde{\mathcal{M}}$.
In practice, this is the same as ignoring the contribution of kinematic
singularity to the $t$-channel graviton exchange diagrams.

##### Massless particles.

The general properties of the scattering amplitudes have been established only
in gapped systems, and not when massless particles are present. Even worse,
the traditional non-perturbative S-matrix does not exist for massless
particles, due to the issue of the IR divergence. To avoid this IR issue, we
estimate the contributions from IR divergent diagrams to the positivity bounds
by introducing an IR cutoff in Sec. 4. We find that they will be subdominant,
and hence we expect that the subtleties associated with the IR divergence will
not affect our main conclusion. However, this may not be the case in other
models, in which case an appropriate prescription for massless particles is
required to derive the bounds.

##### Unstable particles.

If a particle has a decay channel, such an unstable particle does not appear
in the asymptotic states and the standard scattering amplitudes cease to exist
for the particle [44]. However, unstable-particle scattering amplitudes can be
unambiguously defined by residues of higher-point amplitudes (see e.g. [45])
and their properties can be studied in the S-matrix theory.777There are still
IR divergences, which need to be addressed separately. Unitarity of the
S-matrix leads to certain constraints on the unstable-particle scattering
amplitudes, and in particular, there exists a positivity constraint on the
imaginary part [46], suggesting the existence of positivity bounds even from
scattering of unstable particles. In the present paper, however, we will only
focus on the scattering of stable particles and leave a precise treatment of
unstable particles for a future study.

##### Anomalous threshold singularities.

Even for stable particles, the analytic structure of a scattering of a heavier
particle is not as simple as that of a lighter particle. It is known that the
scattering of the heavier particle can have a new singularity called an
anomalous threshold, whose existence does not immediately follow from
unitarity. While general knowledge of anomalous threshold is still missing to
the best of our knowledge (see [47, 48] for recent discussions), we expect
that the anomalous thresholds appear only in the IR regime, i.e., they are
detectable and controlled within EFT. If this is indeed the case, the
anomalous thresholds will give additional technical complications but will not
spoil the success of positivity bounds, as long as these thresholds are taken
into account in the derivation of the dispersion relations.

##  3 Practical Recipe for Gravitational Positivity

We are now ready to discuss practical recipes for deriving gravitational
positivity bounds for EFTs. Instead of making the presentation fully general,
we focus on the case of $2\to 2$ scattering $VV\to VV$ of spin $1$ particle
$V$ with mass $m_{V}$. This will be the case relevant for the rest of this
paper.

The gravitational positivity bound is an inequality for the quantity
$B(\Lambda)$. When we wish to apply the gravitational positivity bound of the
previous section, we run into one practical problem: we can almost never
compute an exact scattering amplitude $\mathcal{M}(s,t)$ in an EFT. In fact,
if we know the exact expression for $\mathcal{M}(s,t)$, we can analytically
continue the expression to the whole complex plane, and we have effectively
solved for the quantum gravity already. The best we can do is to compute an
approximate expression $\mathcal{M}_{\rm EFT}(s,t)$ for $\mathcal{M}(s,t)$, by
including only a finite number of parameters out of an infinite number of
higher-dimensional operators. The differences between the two are small in the
regions of the validity of EFT (at scales $\Lambda\ll\Lambda_{\rm UV}$).
Hence, the computation of the approximate amplitude $\mathcal{M}_{\rm EFT}$ is
sufficient for evaluating the quantity $B(\Lambda)$ which only requires
information below $\Lambda$. In the following, we will discuss EFT scattering
amplitude $\mathcal{M}_{\rm EFT}$ and explain a practical way to calculate
$B(\Lambda)$ from the EFT amplitude $\mathcal{M}_{\rm EFT}$: for notational
simplicity we will simply denote this approximate amplitude as $\mathcal{M}$
in the rest of this section.

We consider the scattering amplitude of the gauge bosons $VV\to VV$.
Corresponding to polarizations of the gauge boson, we define the following
combinations of helicity amplitudes. For the scattering of transverse modes,

$\displaystyle\mathcal{M}_{TT}(s,t):=$
$\displaystyle\frac{1}{4}\left[\mathcal{M}\left(1^{+}2^{+}3^{+}4^{+}\right)+\mathcal{M}\left(1^{+}2^{-}3^{+}4^{-}\right)+\mathcal{M}\left(1^{-}2^{-}3^{-}4^{-}\right)+\mathcal{M}\left(1^{-}2^{+}3^{-}4^{+}\right)\right]\
,$ (3.1)

where superscripts $\pm$ denote helicities. For the scattering of the
transverse mode and the longitudinal mode,

$\displaystyle\mathcal{M}_{TL}(s,t):=\frac{1}{2}\left[\mathcal{M}\left(1^{+}2^{L}3^{+}4^{L}\right)+\mathcal{M}\left(1^{-}2^{L}3^{-}4^{L}\right)\right]\
,$ (3.2)

where superscripts $L$ denote longitudinal polarizations. For the scattering
of longitudinal modes,

$\displaystyle\mathcal{M}_{LL}(s,t):=\mathcal{M}\left(1^{L}2^{L}3^{L}4^{L}\right)\
.$ (3.3)

These amplitudes are defined such that they exhibit $s\leftrightarrow u$
symmetry (up to subtleties associated with the kinematical singularity
discussed around (2.13), which are not relevant to the following discussion).

We provide a step-by-step procedure for deriving the gravitational positivity
bound. The basic task is simple: we need to evaluate the cutoff-dependent
quantity $B(\Lambda)$ defined by a spin-$1$ generalization of (2.8):

$\displaystyle B(\Lambda):=a_{2}-\frac{4}{\pi}\int_{m_{\rm
th}^{2}}^{\Lambda^{2}}ds\,\frac{{\rm
Im}\,\widetilde{\mathcal{M}}(s,t=0)}{\left(s-2m_{V}^{2}\right)^{7}}\,,$ (3.4)

where $\widetilde{\mathcal{M}}$ is a kinematic singularity-free amplitude
defined in (2.13), $m_{\rm th}$ is the threshold energy, i.e., the mass of the
lightest intermediate state, and $a_{2}$ is basically defined as in (2.6), but
with gravitational corrections included (as we will soon see in (3.8)).

Step 1: Calculate $\mathcal{M}(s,t)$

We compute the $2\to 2$ scattering amplitude $\mathcal{M}(s,t)$ up to order
$\mathcal{O}\left(M_{\text{Pl}}^{-2}\right)$:

$\displaystyle\mathcal{M}(s,t)=\underbrace{\mathcal{M}_{\text{non-
grav}}(s,t)}_{\mathcal{O}\left(M_{\text{Pl}}^{0}\right)}+\underbrace{\mathcal{M}_{\text{grav}}(s,t)}_{\mathcal{O}(M_{\text{Pl}}^{-2})}+\underbrace{\vphantom{\mathcal{M}_{\text{grav}}}\cdots}_{\mathcal{O}(M_{\text{Pl}}^{-4})}\
.$ (3.5)

Here $\mathcal{M}_{\text{non-grav}}(s,t)$ is the contribution from diagrams
with no graviton exchange, while $\mathcal{M}_{\text{grav}}(s,t)$ is the
contribution from diagrams with one graviton exchange and is of the order
$\mathcal{O}\left(M_{\text{Pl}}^{-2}\right)$. The remaining terms are the
order of $\mathcal{O}\left(M_{\text{Pl}}^{-4}\right)$ and are neglected in the
rest of the analysis. Once we compute $\mathcal{M}$, it is straightforward to
obtain kinematic-singularity free amplitude $\widetilde{\mathcal{M}}$ (2.13),
both for non-gravitational and gravitational contributions. Note that such an
amplitude exists for each choice of helicity of external vector fields (as in
(3.1), (3.2), and (3.3)).

Step 2: Calculate $B_{\text{non-grav}}\left(\Lambda\right)$

We compute the non-gravitational contribution $B_{\text{non-grav}}(\Lambda)$
to $B(\Lambda)$. We assume that the EFT of interest is renormalizable in the
gravity decoupling limit $M_{\rm Pl}\to\infty$.

In such a case, the amplitude $\mathcal{M}_{\text{non-grav}}$ itself satisfies
the analyticity and the $s^{2}$-boundedness, which is the $s^{6}$-boundedness
in terms of $\mathcal{\widetilde{M}}_{\text{non-grav}}$, when the expression
is analytically continued throughout the complex plane, even outside the
regions of validity of the EFT. The non-gravitational part then satisfies the
twice-subtracted dispersion relation, giving a formula

$\displaystyle B_{\text{non-
grav}}(\Lambda)=\frac{4}{\pi}\int_{\Lambda^{2}}^{\infty}ds\,\frac{{\rm
Im}\,\widetilde{\mathcal{M}}_{\text{non-
grav}}(s,t=0)}{\left(s-2m_{V}^{2}\right)^{7}}\ .$ (3.6)

This formula is practically useful since we only need to compute the imaginary
part of the forward amplitude. This expression further simplifies in the limit
$\Lambda\gg m_{V}$:

$\displaystyle B_{\text{non-
grav}}(\Lambda)\simeq\frac{4}{\pi}\int_{\Lambda^{2}}^{\infty}ds\,\frac{{\rm
Im}\,\mathcal{M}_{\text{non-grav}}(s,t=0)}{\left(s-2m_{V}^{2}\right)^{3}}\;.$
(3.7)

In other words, in this limit, we can in practice neglect the subtleties
associated with the use of the kinematic-singularity-free amplitude
$\widetilde{\mathcal{M}}$, and use the original amplitude $\mathcal{M}$ for
the evaluation of $B_{\text{non-grav}}(\Lambda)$ in (3.7) (which coincides
(2.4) with $n=1$ and $m_{A}=m_{B}=m_{V}$).

Step 3: Calculate $B_{\text{grav}}\left(\Lambda\right)$

We next compute the gravitational contribution $B_{\text{grav}}(\Lambda)$ to
$B(\Lambda)$. While this contains many different Feynman diagrams in general,
we will argue in Appendix A that it is enough to consider the contribution
from $t$-channel graviton exchange diagrams. In this case, the integral part
of (2.7) is absent for this contribution because
$\widetilde{\mathcal{M}}_{\text{grav,$t$-channel}}(s,t)$ does not have the
$s$-channel or $u$-channel discontinuity at the non-gravitational one-loop
order. Therefore the gravitational part of $B\left(\Lambda\right)$ is
simplified to $a_{2}$ defined in (2.6), evaluated for the gravity $t$-channel
amplitude:

$\displaystyle B_{\text{grav}}\left(\Lambda\right)$ $\displaystyle\simeq
a_{2,\text{grav,$t$-channel}}:=\lim_{t\to-0}\left[\frac{2}{6!}\frac{\partial^{6}\mathcal{\widetilde{M}_{\text{grav,$t$-channel}}}(s,t)}{\partial
s^{6}}+\frac{2}{M_{\rm Pl}^{2}t}\right]_{s=2m_{V}^{2}}\,.$ (3.8)

When we assume $\Lambda\gg m_{V}$, we obtain an expression

$\displaystyle
B_{\text{grav}}\left(\Lambda\right)\,\simeq\lim_{t\to-0}\left[\frac{\partial^{2}\mathcal{M_{\text{grav,$t$-channel}}}(s,t)}{\partial
s^{2}}+\frac{2}{M_{\rm Pl}^{2}t}-\text{(kinematic
singularity)}\right]_{s=2m_{V}^{2}}\,.$ (3.9)

Step 4: Write down the inequality for total $B(\Lambda)$

The constraints on EFT are obtained as the inequality (2.11) for the combined
expression $B(\Lambda)=B_{\text{non-grav}}(\Lambda)+B_{\text{grav}}(\Lambda)$.
There are three interpretations of this inequality, as we discussed in Sec.
2.2.

##  4 Positivity Bound on Dark U(1) Gauge Boson

A gauge interaction is one of the cornerstones of the quantum field theory
(QFT). Gauge symmetries and their breakings are one of the most crucial
ingredients of the Standard Model, and many BSM models also introduce new
gauge symmetries. In particular, a light gauge boson is one of the candidates
for the DM [49, 50, 51].

In this section, we work out the details of the gravitational positivity
recipe for the dark $\mathrm{U}(1)$ gauge boson. Here we study the Abelian
Higgs mechanism of the simplest gauge theory, the $\mathrm{U}(1)$ theory. We
discuss theoretical constraints obtained by step-by-step computations spelled
out in the previous section.

###  4.1 Higgs Contribution

In the Higgs mechanism, the gauge boson gets a mass when a charged scalar
field $\Phi$ gets a non-zero vacuum expectation value (VEV) $v$. The
renormalizable Lagrangian of the Higgs mechanism to generate a mass for a
$\mathrm{U}(1)$ gauge boson is

$\displaystyle\mathcal{L}=|D_{\mu}\Phi|^{2}-\frac{\lambda}{4}(|\Phi|^{2}-v^{2})^{2}\,,$
(4.1)

where $D_{\mu}=\partial_{\mu}-ig_{\Phi}V_{\mu}$ is the covariant derivative
and $g_{\Phi}$ is a charge of $\Phi$ and $\lambda>0$. After
$\Phi=v+\phi/\sqrt{2}+iG$ develops the VEV, the Goldstone component $G$ is
absorbed into the longitudinal component of the gauge boson. There are a
massive gauge boson $V$ and a real scalar $\phi$ after the gauge symmetry
breaking, and the gauge boson gets a mass $m_{V}=\sqrt{2}g_{\Phi}v$ and a real
scalar $\phi$ has a mass $m_{\phi}=\sqrt{\lambda}v$. The interaction of the
Higgs and gauge bosons are given by

$\displaystyle\mathcal{L}\ni\left(\frac{g_{\Phi}^{2}}{2}\phi^{2}+g_{\Phi}m_{V}\phi\right)V^{\mu}V_{\mu}-\lambda\left(\frac{v}{2\sqrt{2}}\phi^{3}+\frac{1}{16}\phi^{4}\right)\,.$
(4.2)

The gravitational interaction can be found by expanding the Lagrangian (4.1)
in terms of the canonical gravitational field $h_{\mu\nu}$ as

$\displaystyle{\cal L}_{\rm grav}=-\frac{h_{\mu\nu}T^{\mu\nu}}{M_{\rm
Pl}}+{\cal O}(M_{\rm Pl}^{-2})\,;\quad
T^{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\partial(\sqrt{-g}{\cal
L})}{{\partial g_{\mu\nu}}}\,.$ (4.3)

The Higgs-gauge boson interaction (4.2) does not make the gauge boson $V$
unstable since it has a $\mathbb{Z}_{2}$ symmetry $V\leftrightarrow-V$.

####  4.1.1 Calculation of $B(\Lambda)$

(a) Non-gravitational Feynman diagrams. (b) $t$-channel graviton exchange
diagrams.

Figure 4: Some examples of the Higgs loop contribution to $B(\Lambda)$.

We calculate $B\left(\Lambda\right)$ at the one-loop order using the recipe
described in Sec. 3. In Figs. 4(a) and 4(b), we show some examples of the
gravitational and non-gravitational processes. The one-loop diagrams are
calculated by using the Mathematica packages FeynRules [52, 53, 54], FeynArts
[55], FeynCalc [56, 57, 58] and Package-X [59].

#### Non-gravitational contributions

Fig. 4(a) show some examples of the non-gravitational Feynman diagrams
involving the Higgs boson $\phi$. From these (and other) diagrams we evaluate
the non-gravitational part of $B(\Lambda)$ to be

$\displaystyle B^{TT}_{\text{non-grav}}$
$\displaystyle=\frac{g_{\Phi}^{4}}{4\pi^{2}\Lambda^{4}}+\frac{g_{\Phi}^{4}\left[m_{\phi}^{4}(12\log\left(\frac{m_{\phi}}{\Lambda}\right)+1)+m_{\phi}^{2}m_{V}^{2}(12\log\left(\frac{\Lambda^{3}m_{V}}{m_{\phi}^{4}}\right)-27)+24m_{V}^{4}\right]}{18\pi^{2}\Lambda^{6}m_{\phi}^{2}}+\cdots\,,$
(4.4) $\displaystyle B^{TL}_{\text{non-grav}}$
$\displaystyle=\frac{g_{\Phi}^{4}}{2\pi^{2}\Lambda^{4}}+\frac{g_{\Phi}^{4}\left[-3\left(m_{\phi}^{3}-2m_{\phi}m_{V}^{2}\right){}^{2}\log\left(\frac{m_{\phi}m_{V}}{\Lambda^{2}}\right)+17m_{\phi}^{4}m_{V}^{2}-20m_{\phi}^{2}m_{V}^{4}-8m_{\phi}^{6}+12m_{V}^{6}\right]}{9\pi^{2}\Lambda^{6}m_{\phi}^{2}m_{V}^{2}}+\cdots\,,$
(4.5) $\displaystyle B^{LL}_{\text{non-grav}}$
$\displaystyle=\frac{g_{\Phi}^{4}}{\pi^{2}\Lambda^{2}m_{V}^{2}}+\frac{g_{\Phi}^{4}\left(4m_{\phi}^{2}m_{V}^{2}\log\left(\frac{m_{V}^{2}}{\Lambda^{2}}\right)-8m_{\phi}^{2}m_{V}^{2}+5m_{\phi}^{4}+6m_{V}^{4}\right)}{8\pi^{2}\Lambda^{4}m_{V}^{4}}+\cdots\,.$
(4.6)

Here $\cdots$ represents contributions suppressed by higher powers of
$\Lambda$, which scale is assumed to be much greater than $m_{\phi}$ and
$m_{V}$. We have also numerically estimated the two-loop contributions and
confirm that these effects are smaller than the one-loop contributions in the
parameter region of present interest.

#### Gravitational contributions

As the next step, we compute the one-loop diagrams of the gravitational
contributions (Fig. 4(b)):

$\displaystyle B^{TT}_{\mathrm{grav}}$
$\displaystyle=\frac{-g_{\Phi}^{2}}{72\pi^{2}m_{\phi}^{2}M_{\rm
Pl}^{2}}\,g_{{TT}}\\!\left(\frac{m_{V}}{m_{\phi}}\right)\,,$ (4.7)
$\displaystyle B^{TL}_{\mathrm{grav}}$
$\displaystyle=\frac{-g_{\Phi}^{2}}{144\pi^{2}m_{V}^{2}M_{\rm
Pl}^{2}}\,g_{{TL}}\\!\left(\frac{m_{V}}{m_{\phi}}\right)\,,$ (4.8)
$\displaystyle B^{LL}_{\mathrm{grav}}$
$\displaystyle=\frac{-g_{\Phi}^{2}}{72\pi^{2}m_{V}^{2}M_{\rm
Pl}^{2}}\,g_{{LL}}\\!\left(\frac{m_{V}}{m_{\phi}}\right)\,,$ (4.9)

where the functions $g_{TT,TL,LL}$ are given by

$\displaystyle g_{{TT}}(x)$
$\displaystyle=\frac{2}{x^{8}\left(1-4x^{2}\right)^{2}}$ (4.10)
$\displaystyle\times\Bigg{[}-6\left(2x^{2}-1\right)\left(4x^{4}-5x^{2}+1\right)^{2}\log(x)+\left(200x^{8}-346x^{6}+230x^{4}-63x^{2}+6\right)x^{2}$
(4.11)
$\displaystyle-6\sqrt{\frac{1}{x^{2}}-4}\left(4x^{10}-32x^{8}+52x^{6}-35x^{4}+10x^{2}-1\right)x\log\left(\frac{1}{2}\left(\sqrt{\frac{1}{x^{2}}-4}+\frac{1}{x}\right)\right)\Bigg{]}\,,$
(4.12) $\displaystyle g_{{TL}}(x)$
$\displaystyle=\frac{-1}{x^{6}\left(1-4x^{2}\right)^{2}}$ (4.13)
$\displaystyle\times\Bigg{[}6\sqrt{\frac{1}{x^{2}}-4}x\left(51x^{2}+2\left(8x^{6}-92x^{4}+147x^{2}-93\right)x^{4}-5\right)\log\left(\frac{1}{2}\left(\sqrt{\frac{1}{x^{2}}-4}+\frac{1}{x}\right)\right)$
(4.14)
$\displaystyle+\left(4x^{2}-1\right)\left(-344x^{8}+416x^{6}-201x^{4}+30x^{2}+6\left(32x^{8}-128x^{6}+114x^{4}-41x^{2}+5\right)\log(x)\right)\Bigg{]}\,,$
(4.15) $\displaystyle g_{{LL}}(x)$
$\displaystyle=\frac{-244x^{6}+268x^{4}-123x^{2}+18}{x^{4}-4x^{6}}+\frac{6x\left(\frac{28x^{4}-19x^{2}+3}{x^{6}}-2\right)\left(\frac{1}{x^{2}}-2\right)^{2}\log\left(\frac{1}{2}\left(\sqrt{\frac{1}{x^{2}}-4}+\frac{1}{x}\right)\right)}{\left(\frac{1}{x^{2}}-4\right)^{3/2}}$
(4.16)
$\displaystyle+\frac{6\left(-4x^{6}+20x^{4}-13x^{2}+3\right)\log(x)}{x^{6}}\,.$
(4.17)

We plot the functions $g_{TT,TL,LL}$ in Fig. 5. These functions are real and
positive for any real values of $m_{V}$ and $m_{\phi}$, and we find that the
gravitational contributions from the Higgs to $B(\Lambda)$ are negative.

Figure 5: The functions $g_{{TT,TL,LL}}(x)$. These functions are real and
positive for $x>0$ and therefore the gravitational contributions are negative
for all the masses of the Higgs boson and the gauge boson.

In the limit of $m_{V}/m_{\phi}\to 0$ and $\to\infty$, the above expressions
reduce to

$\displaystyle B^{TT}_{\text{grav}}=$
$\displaystyle\begin{dcases}-\frac{g_{\Phi}^{2}}{72\pi^{2}m_{\phi}^{2}M_{\rm
Pl}^{2}}&(m_{V}/m_{\phi}\to 0)\,,\\\ -\frac{g_{\Phi}^{2}}{24\pi
m_{V}m_{\phi}M_{\rm Pl}^{2}}&(m_{V}/m_{\phi}\to\infty)\,,\end{dcases}$ (4.18)
$\displaystyle B^{TL}_{\text{grav}}=$
$\displaystyle\begin{dcases}-\frac{g_{\Phi}^{2}}{144\pi^{2}m_{V}^{2}M_{\rm
Pl}^{2}}&(m_{V}/m_{\phi}\to 0)\,,\\\ -\frac{g_{\Phi}^{2}}{24\pi
m_{V}m_{\phi}M_{\rm Pl}^{2}}&(m_{V}/m_{\phi}\to\infty)\,,\end{dcases}$ (4.19)
$\displaystyle B^{LL}_{\text{grav}}=$
$\displaystyle\begin{dcases}-\frac{g_{\Phi}^{2}}{72\pi^{2}m_{V}^{2}M_{\rm
Pl}^{2}}&(m_{V}/m_{\phi}\to 0)\,,\\\ -\frac{g_{\Phi}^{2}}{24\pi
m_{V}m_{\phi}M_{\rm Pl}^{2}}&(m_{V}/m_{\phi}\to\infty)\,.\end{dcases}$ (4.20)

###  4.2 Fermion Contribution

Next, let us introduce a dark $\mathrm{U}(1)$ charged Dirac fermion $\psi$ as
a matter field, whose interaction is given by

$\displaystyle\mathcal{L}_{\psi}=i\bar{\psi}{\ooalign{\hfil/\hfil\crcr$D$}}\psi-
m_{F}\bar{\psi}\psi\ ,$ (4.23)

where $D_{\mu}=\partial_{\mu}-ig_{F}V_{\mu}$ is the covariant derivative. Note
that this interaction can destabilize the gauge boson, by allowing
$V\to\psi\bar{\psi}$ decay if $m_{V}>2m_{F}$ and decay into gravitons.

If the mass of the charged particles is large $(m_{F}\gg\Lambda)$, these
charged particles are integrated out in the EFT and yield higher-derivative
corrections to the EFT action. These effects to $B(\Lambda)$ can be estimated
as $\mathcal{O}(g_{F}^{4}/m_{F}^{4})$ and we can ignore them compared to
contributions (4.4) from the Higgs field. Hence, our interest is the charged
particle within the regime of EFT ($m_{F}<\Lambda$).

####  4.2.1 Calculation of $B(\Lambda)$

(a) Diagram for $B_{\text{non-grav}}$. (b) Diagram for $B_{\text{grav}}$.

Figure 6: The fermion loop contributions to $B$.

We calculate the contributions of fermion-loop diagrams to $B(\Lambda)$ for
the dark photon scattering amplitudes at one-loop order.

#### Non-gravitational contributions

The diagram for a non-gravitational fermion loop is shown in Fig. 6(a). The
one-loop contributions are given by

$\displaystyle B_{\text{non-grav},F}^{TT}$
$\displaystyle=\frac{g_{F}^{4}\left(2\log\frac{\Lambda^{2}}{m_{F}^{2}}+1\right)}{4\pi^{2}\Lambda^{4}}+\cdots\,,$
(4.24) $\displaystyle B_{\text{non-grav},F}^{TL}$
$\displaystyle=\frac{4g_{F}^{4}m_{V}^{2}}{3\pi^{2}\Lambda^{6}}+\cdots\,,$
(4.25) $\displaystyle B_{\text{non-grav},F}^{LL}$
$\displaystyle=\frac{g_{F}^{4}m_{V}^{4}\left(4\log\frac{\Lambda^{2}}{m_{F}^{2}}-7\right)}{\pi^{2}\Lambda^{8}}+\cdots$
(4.26)

for $\Lambda\gg m_{V,F}$. These contributions of $LL$ and $TL$ vanish in the
limit $m_{V}/\Lambda\to 0$. Recall that $B_{\text{non-grav}}$ is determined by
the high-energy limit of the forward limit amplitude thanks to the dispersion
relation. The behaviors $B_{\text{non-grav},F}^{TL},B_{\text{non-
grav},F}^{LL}\to 0$ as $m_{V}/\Lambda\to 0$ can be understood by the
decoupling of the longitudinal sector in the high-energy limit. The
contributions of the charged particles to $B_{\text{non-grav}}^{TL}$ and
$B_{\text{non-grav}}^{LL}$ can be ignored in the limit $\Lambda\gg m_{V}$.

We thus focus on the $TT$ scattering. The charged spin-1/2 loops give positive
contributions to the non-gravitational diagram of the order $B_{\text{non-
grav},F}^{TT}\sim g_{F}^{4}/\Lambda^{4}$, which is of the same order as the
Higgs counterpart (4.4). Therefore, the inclusion of charged spin-1/2
particles does not drastically change the non-gravitational part. We reach a
similar conclusion even in the presence of charged spin-0 particles [16].

In contrast, as studied in [27, 29], charged spin-1 loops with the mass
$m_{\text{spin-1}}$ and the charge $g_{\text{spin-1}}$ lead to a different
asymptotic behavior $B_{\text{non-grav}}^{TT}\sim
g_{\text{spin-1}}^{4}/(m_{\text{spin-1}}^{2}\Lambda^{2})$ which dominates over
other contributions in the limit $\Lambda\gg m_{\text{spin-1}}$. One may then
wonder about contributions from higher-spin particles. The scaling in
$\Lambda$ is determined by the high-energy behavior of the imaginary part of
the amplitude in the forward limit, i.e., the total cross-section. Roughly,
spin-0, 1/2 loops give $\mathrm{Im}\,\mathcal{M}\propto s^{0}$ while spin-1
loops yield $\mathrm{Im}\,\mathcal{M}\propto s/m_{\text{spin-1}}^{2}$ in high-
energy limit. The faster growth in $s$ provides a larger contribution to
$B_{\text{non-grav}}(\Lambda)$. However, the asymptotic growth of the cross-
section in $s$ is bounded by the Froissart bound [30] as
$\mathrm{Im}\,\mathcal{M}<s\ln^{2}s$ in gapped theories, implying that the
spin-1 contribution almost saturates the bound. Higher-spin particles, if they
are described by gapped theories like QCD, may not give drastically different
contributions compared to spin-1 particles. For instance, in the case of
light-by-light scattering, the hadronic contribution can be estimated as
$\mathrm{Im}\,\mathcal{M}\propto s^{1.08}$ by employing the vector meson
dominance model [60, 61].

In short, if a charged spin-1 or higher-spin particle appears at $m$, the non-
gravitational contribution $B_{\text{non-grav}}^{TT}$ needs to be modified in
$\Lambda>m$. By contrast, spin-0 or spin-1/2 particles give additional
contributions but the qualitative behavior of $B_{\text{non-grav}}^{TT}$ is
the same.

#### Gravitational contributions

Figure 7: The functions $g_{{TT},F}(x)$ (red), $g_{{TL},F}(x)$ (green) and
$g_{{LL},F}(x)$ (blue). The solid and dashed lines represent the real and
imaginary parts, respectively.

Next, we discuss the gravitational parts (Fig. 6(b)), which are given by

$\displaystyle B_{\mathrm{grav},F}^{TT}$
$\displaystyle=-\frac{11g_{F}^{2}}{360\pi^{2}M_{\rm
Pl}^{2}m_{F}^{2}}\,g_{{TT},F}\left(\frac{m_{V}}{m_{F}}\right),$ (4.27)
$\displaystyle B_{\mathrm{grav},F}^{TL}$
$\displaystyle=-\frac{{11}g_{F}^{2}}{720\pi^{2}M_{\rm
Pl}^{2}m_{F}^{2}}\,g_{{TL},F}\left(\frac{m_{V}}{m_{F}}\right),$ (4.28)
$\displaystyle B_{\mathrm{grav},F}^{LL}$
$\displaystyle=-\frac{g_{F}^{2}m_{V}^{2}}{420\pi^{2}M_{\rm
Pl}^{2}m_{F}^{4}}\,g_{{LL},F}\left(\frac{m_{V}}{m_{F}}\right),$ (4.29)

where

$\displaystyle g_{{TT},F}(x)$
$\displaystyle=\frac{60\left(x^{2}-8\right)\left(x^{2}-2\right)\log\left(\frac{1}{2}x\left(\sqrt{x^{2}-4}-x\right)+1\right)-20x\sqrt{x^{2}-4}\left(2x^{4}-11x^{2}+24\right)}{11x^{5}\left(x^{2}-4\right)^{3/2}}\,,$
(4.30) $\displaystyle g_{{TL},F}(x)$
$\displaystyle=\frac{60\left(x^{4}-22x^{2}+56\right)\log\left(\frac{1}{2}x\left(\sqrt{x^{2}-4}-x\right)+1\right)-20x\sqrt{x^{2}-4}\left(x^{4}-19x^{2}+84\right)}{11x^{5}\left(x^{2}-4\right)^{3/2}}\,,$
(4.31) $\displaystyle g_{{LL},F}(x)$
$\displaystyle=\frac{70\left(\left(120-36x^{2}\right)\log\left(\frac{1}{2}x\left(\sqrt{x^{2}-4}-x\right)+1\right)+x\sqrt{x^{2}-4}\left(x^{4}+8x^{2}-60\right)\right)}{3x^{7}\left(x^{2}-4\right)^{3/2}}\,.$
(4.32)

We plot the functions $g_{AB,F}(x)~{}(A,B={T},{L})$ in Fig. 7.

Let us first discuss the limit $m_{V}\to 0$, which yields

$\displaystyle B_{\text{grav},F}^{TT}$
$\displaystyle=-\frac{11g_{F}^{2}}{360\pi^{2}M_{\text{pl}}^{2}m_{F}^{2}}\,,$
(4.33) $\displaystyle B_{\text{grav},F}^{TL}$
$\displaystyle=-\frac{11g_{F}^{2}}{720\pi^{2}M_{\text{pl}}^{2}m_{F}^{2}}\,,$
(4.34) $\displaystyle B_{\text{grav},F}^{LL}$
$\displaystyle=-\frac{g_{F}^{2}m_{V}^{2}}{420\pi^{2}M_{\text{pl}}^{2}m_{F}^{4}}\,.$
(4.35)

If the Higgs and the gauge boson are lighter than the charged particles
$(m_{\phi},m_{V}\ll m_{F})$, we can neglect these contributions to the
gravitational process $B_{\text{grav}}$ in comparison to the contributions of
the gauge boson sector. Note that the $LL$ part has an additional suppression
$m_{V}^{2}/m_{F}^{2}$ because the longitudinal mode should be decoupled from
the charged particle in the limit $m_{V}\to 0$.

By contrast, a peculiar behavior appears in a heavy mass range of the gauge
boson. The functions $g_{AB,F}$ are singular at $m_{V}=2m_{F}$ and become
complex numbers in $m_{V}>2m_{F}$. The threshold $m_{V}=2m_{F}$ corresponds to
the value at which the decay of the gauge boson to the charged particles
starts to be kinematically allowed. The singularity at $m_{V}=2m_{F}$ is
understood as an anomalous threshold. Let us consider the triangle diagram of
Fig. 6(b), which gives rise to the gravitational form factor of the gauge
boson. The triangle diagram has a normal threshold at $t=4m_{F}^{2}$. In
addition, the anomalous threshold comes up on the first sheet of the complex
$t$-plane in $m_{V}>\sqrt{2}m_{F}$ and the position of the singularity is at

$\displaystyle t=\frac{m_{V}^{2}}{m_{F}^{2}}(4m_{F}^{2}-m_{V}^{2})\,.$ (4.36)

Therefore, the form factor at $t=0$ is singular when $m_{V}=2m_{F}$,
generating the singularity of $B_{\text{grav},F}^{AB}$. In Fig. 7, we have
plotted the functions in the unstable range $m_{V}>2m_{F}$ as well. Recall
that $B(\Lambda)$ has to be real according to the dispersion relation (2.8) if
all the mentioned properties are satisfied. This implies that at least one of
the properties does not hold when the gauge boson decays. In fact, amplitudes
exhibit peculiar behaviors if the mass of the external particle is
extrapolated to the unstable region [47, 46]. The conventional positivity
bounds need to be modified for unstable particles.

We expect that the subtleties associated with the decay are negligible if the
particle is long-lived ($g_{F}\ll 1$). In other words, although the fermion
contribution to $B_{\rm grav}(\Lambda)$ is quite subtle in the mass range
$m_{V}>2m_{F}$, this subtle contribution can be smaller than the contribution
from the Higgs sector and could be simply negligible. In the following, we
shall adopt this optimistic expectation when discussing models in which the
gauge boson decays into other particles. The issue of unstable particles will
be studied elsewhere.

###  4.3 Constraint on Dark Gauge Boson

Table 1: Summary of the gravitational and non-gravitational contributions to $B$ for $m_{V}\ll m_{\phi,F}\ll\Lambda$ and $m_{\phi}\ll\sqrt{\Lambda m_{V}}$.  | Higgs loop | Fermion loop
---|---|---
| $B_{\text{non-grav}}$ | $B_{\text{grav}}$ | $B_{\text{non-grav}}$ | $B_{\text{grav}}$
$TT$ | $\dfrac{g_{\Phi}^{4}}{4\pi^{2}\Lambda^{4}}$ | $-\dfrac{g_{\Phi}^{2}}{72\pi^{2}M_{\rm Pl}^{2}m_{\phi}^{2}}$ | $\dfrac{g_{F}^{4}\left(2\log\frac{\Lambda^{2}}{m_{F}^{2}}+1\right)}{4\pi^{2}\Lambda^{4}}$ | $-\dfrac{11g_{F}^{2}}{360\pi^{2}M_{\rm Pl}^{2}m_{F}^{2}}$
$TL$ | $\dfrac{g_{\Phi}^{4}}{2\pi^{2}\Lambda^{4}}$ | $-\dfrac{g_{\Phi}^{2}}{144\pi^{2}M_{\rm Pl}^{2}m_{V}^{2}}$ | $\dfrac{4g_{F}^{4}m_{V}^{2}}{3\pi^{2}\Lambda^{6}}$ | $-\dfrac{11g_{F}^{2}}{720\pi^{2}M_{\rm Pl}^{2}m_{F}^{2}}$
$LL$ | $\dfrac{g_{\Phi}^{4}}{\pi^{2}\Lambda^{2}m_{V}^{2}}$ | $-\dfrac{g_{\Phi}^{2}}{72\pi^{2}M_{\rm Pl}^{2}m_{V}^{2}}$ | $\dfrac{g_{F}^{4}m_{V}^{4}\left(4\log\frac{\Lambda^{2}}{m_{F}^{2}}-7\right)}{\pi^{2}\Lambda^{8}}$ | $-\dfrac{g_{F}^{2}m_{V}^{2}}{420\pi^{2}M_{\rm Pl}^{2}m_{F}^{4}}$

In Table 1, we summarize our estimation of $B(\Lambda)$ for a light gauge
boson.

For simplicity of the presentation, we will in the following discuss the
inequality $B(\Lambda)>0$ for assumed $\Lambda$ (Interpretation 1 in Sec.
2.2). When we require $B(\Lambda)>0$ for each helicity process, we have non-
trivial conditions for the parameters of the dark $\mathrm{U}(1)$ model.

One should recall, however, that the actual bound is (2.11), which can deviate
from the simplified bound $B(\Lambda)>0$ by the effects of the RHS of (2.11);
the simplified bound coincides with the actual bound (2.11) only if we assume
that the scale $M$ is large enough so that we can neglect the RHS of (2.11).
Note that non-trivial constraints on the dark gauge boson exist as long as
$B_{\rm grav}-{\sigma}/(M_{\rm Pl}^{2}M^{2})<0$.

#### Abelian Higgs without matter fields

(a) (b)

Figure 8: The constraints on the dark gauge boson parameters in the Higgs
mechanism. (a): Lower limit of $g_{\Phi}$ as a function of $m_{V}$. The
positivity bound is satisfied in the region above the lines. (b): Contour of
the lower bound of $g_{\Phi}$ on $m_{V}-m_{\phi}$ plane from the positivity
bound. The dashed lines indicate the Higgs quartic coupling is large
$\lambda>\pi$. We choose $\Lambda=1$ TeV in both figures.

Let us first look at the case where there is no matter field and only gauge
and Higgs fields exist. For a very light gauge boson $m_{V}\ll
m_{\phi,F}\ll\Lambda$ and $m_{\phi}\ll\sqrt{\Lambda m_{V}}$, the following
conditions can be obtained from each helicity

$\displaystyle B^{TT}_{\text{non-
grav}}+B^{TT}_{\text{grav}}>0\quad\longrightarrow$ $\displaystyle\quad
m_{\phi}>\frac{\Lambda^{2}}{3\sqrt{2}g_{\phi}M_{\rm Pl}}\,,$ (4.37)
$\displaystyle B^{TL}_{\text{non-
grav}}+B^{TL}_{\text{grav}}>0\quad\longrightarrow$ $\displaystyle\quad
m_{V}>\frac{\Lambda^{2}}{6\sqrt{2}g_{\Phi}M_{\rm Pl}}\,,$ (4.38)
$\displaystyle B^{LL}_{\text{non-
grav}}+B^{LL}_{\text{grav}}>0\quad\longrightarrow$ $\displaystyle\quad
g_{\Phi}>\frac{\Lambda}{6\sqrt{2}M_{\rm Pl}}\,.$ (4.39)

In Fig. 8, we show the positivity conditions. Note that the higher terms of
$\Lambda$ for the non-gravitational contributions can be dominant if
$m_{\phi}\gtrsim\sqrt{m_{V}\Lambda}$ or $m_{\phi}\lesssim m_{V}^{2}/\Lambda$.

#### Stückelberg gauge boson with matter field

Figure 9: The constraints on the dark gauge boson parameters for the
Stückelberg $\mathrm{U}(1)$ gauge theory with a fermion field. The positivity
bound is satisfied in the region above the lines. We choose $\Lambda=1$ TeV.

Next, let us discuss the case where the Higgs field $\phi$ is absent in the
EFT, as in the case of the Stückelberg mechanism. We find

$\displaystyle B^{TT}_{\text{non-
grav}}+B^{TT}_{\text{grav}}>0\quad\longrightarrow$ $\displaystyle\quad
g_{F}\gtrsim 0.2\frac{\Lambda^{2}}{m_{F}M_{\rm Pl}\sqrt{\log(\Lambda
m_{F}^{-1})}}\,,$ (4.40) $\displaystyle B^{TL}_{\text{non-
grav}}+B^{TL}_{\text{grav}}>0\quad\longrightarrow$ $\displaystyle\quad
m_{V}\gtrsim 0.1\frac{\Lambda^{3}}{g_{F}m_{F}M_{\rm Pl}}\,,$ (4.41)
$\displaystyle B^{LL}_{\text{non-
grav}}+B^{LL}_{\text{grav}}>0\quad\longrightarrow$ $\displaystyle\quad
m_{V}\gtrsim 0.02\frac{\Lambda^{4}}{g_{F}m_{F}^{2}M_{\rm Pl}\sqrt{\log(\Lambda
m_{F}^{-1})}}\,.$ (4.42)

In Fig. 9, we show the positivity conditions on $m_{V}-g_{F}$ plane. Note that
for $m_{V}>2m_{F}$, delicate arguments are required and the positivity
conditions used previously cannot be applied as they are.

More generally, it is possible to incorporate both Higgs and fermions. The
helicity configurations involving longitudinal modes provide stronger
constraints. As discussed earlier, the Higgs contribution is dominant when
$g_{\Phi}\sim g_{F}$ and $m_{V}\ll m_{\phi,F}$, in which case the positivity
conditions will be similar to those of the Higgs scenarios.

#### Comparison with other swampland constraints for Stückelberg gauge bosons

Let us compare our swampland constraints with another swampland constraint on
the gauge boson mass [62] (see also [63]) motivated by the swampland distance
conjecture [3]. This bound states that the UV cutoff $\Lambda_{\rm UV}$ of an
EFT with a gauge boson should obey

$\displaystyle\Lambda_{\rm UV}\lesssim\textrm{min}((m_{V}M_{\rm
Pl}/g)^{1/2},g^{1/3}M_{\rm Pl})\,,$ (4.43)

where (as before) $m_{V}$ is the gauge boson mass and $g$ is the gauge
coupling constant. In Fig. 10 we compare our bounds and the bound (4.43) for
sample values of parameters.

(a) Light fermion (b) Heavy fermion

Figure 10: Comparison between the swampland constraints for Stückelberg gauge
bosons, the gravitational positivity bound (in red) and other swampland bounds
(4.43) (in blue) and the weak gravity conjectures (in green). We take the
cutoff scale $\Lambda$ is 1 GeV (dotted lines), 1 TeV (solid lines) and 1 PeV
(dashed lines). (a): Light fermion case $m_{F}=m_{V}$ and (b): Heavy fermion
case $m_{F}=0.1\Lambda$. In this plot, the lines are chopped in the parameter
space where $m_{V}>0.1\Lambda$.

One can see in Fig. 10 that overall our bound is stronger than the bound
(4.43). Let us quickly point out, however, that the two bounds are derived
under a different set of assumptions/arguments, and one needs to be careful in
any meaningful comparison.

The gravitational positivity bound is derived by general properties of
Reggeized scattering amplitudes of a weakly-coupled UV theory. The assumptions
are believed to be standard, but the bound has some ambiguities due to the
unknown UV mass scale/sign $(M,\sigma)$ in (2.10). By contrast, the bound
(4.43) is derived from a set of swampland conjectures. The $(m_{V}M_{\rm
Pl}/g)^{1/2}$ bound in (4.43) is derived by noting that the Stückelberg mass
is obtained when the gauge boson eats a fundamental axion, for which we can
apply an axionic version [63, 62] of the weak gravity conjecture [64]; the
$g^{1/3}M_{\rm Pl}$ bound in (4.43) is derived from a combination of the
(sub)lattice/tower weak gravity conjecture [65, 66] (see also [67, 25]) and
the species bound [68, 69, 70].

Let us also note that our bound applies to gauge boson masses both in the
Higgs mechanism and the Stückelberg mechanism, while the bound (4.43) applies
only to those in the Stückelberg mechanism. Our bound is more general in this
respect.

In Fig. 10 we also plotted the bound from the weak gravity conjecture [64],
which requires an existence of a particle with charge $g$ and mass $m$ such
that $\sqrt{2}g\geq m/M_{\rm Pl}$.

##  5 Toward Constraints on Realistic Phenomenological Models

The dark $\mathrm{U}(1)$ gauge sectors discussed so far have nothing to do
with our universe and are entirely theoretical constructs. However, any new
physics describing our universe must necessarily coexist with the Standard
Model. Since some important subtleties arise at this point, we will first
briefly discuss them and then discuss the specific model (B$-$L gauge model)
as a possible application of the gravitational positivity.

If we apply the discussion of gravitational positivity bounds to, for example,
the scattering of photons and dark gauge bosons, the dark sector necessarily
has interactions beyond those described by the Standard Model and gravity
[29]. In general, these interactions allow the dark gauge boson to decay into
photons and SM fermions and also decay into gravitons.

Figure 11: Implication to the $\mathrm{U}(1)_{\mathrm{B-L}}$ gauge boson from
the gravitational positivity bound. Here we adopt $\Lambda=1$ GeV. The
positivity constraint excludes the parameter region below the black lines. The
dotted lines indicate that the gauge boson mass is greater than neutrino
masses and our analysis cannot be directly applied. We also show several
experimental constraints from the fifth force searches [71, 72, 73, 74, 75,
76, 77, 78, 79], stellar cooling [80], cosmological constraints on the
neutrino properties [81, 82], and XENONnT constraint on the $e-\nu$ scattering
[83, 84]. We assume that right-handed neutrinos are heavy and not relevant to
the constraints.

In this case, the dark gauge boson has a finite decay width, and hence care
must be taken when applying the positivity constraint. Despite those caveats,
the decay rate of light and weakly interacting gauge bosons is extremely
small, and we expect no major practical problems.

As an example, we consider the $\mathrm{U}(1)_{\mathrm{B-L}}$ extension of the
Standard Model. We assume the gauge charge of the B$-$L breaking Higgs is same
as that of the SM leptons. In Fig. 11, we show the positivity bound on the
$\mathrm{U}(1)_{\mathrm{B-L}}$ model and current experimental constraints.
Here we take $\Lambda=1$ GeV. In this plot, we assume that the B$-$L Higgs
contributions are dominated over the SM fermions and neglect the SM
contributions. The dashed lines in the figure show the parameter region where
the dark gauge boson can decay into neutrinos. The stringent bound from the
positivity constraint comes from TL scattering mode. Note that the constraint
is more severe for the higher cutoff, as seen in the TL constraint (4.38). The
gravitational positivity bound requires the larger gauge coupling for smaller
masses and has strong tensions with the experimental searches.

##  6 Conclusion and Discussion

In this paper, we discussed the practical procedure for deriving gravitational
positivity bounds. We illustrated the procedure for dark gauge bosons, and
obtained the lower limit of the dark photon mass.

There are two complementary comments regarding our results. First and
foremost, our bound is very strong—in Fig. 11 most of the parameter regions
unexplored by experiments are already excluded by the positivity bound. This
is a clear demonstration of the power of the gravitational positivity bound,
and we believe that the gravitational positivity bound can give rather strong
constraints for many EFTs in BSM physics. It is a fascinating question to
explore this point further.

Despite this optimism, we should also be well aware of the fine prints and
limitations of our results, and we should be careful in interpreting the
results. Our bound is derived from several assumptions of UV physics as
spelled out in Sec. 3, which can be violated in some UV scenarios. In
particular, the assumption $B_{\rm grav}-\sigma/(M_{\rm Pl}^{2}M^{2})<0$ is
crucial to obtain non-trivial constraints on IR physics (Interpretation 1 in
Sec. 2.2). This inequality is satisfied if either $\sigma=+1$ or
$M\sim\Lambda_{\rm QG}$, while the validity of the inequality requires
detailed knowledge of the gravitational Regge behavior if $\sigma=-1$ and $M$
is an IR scale. Conversely, if the violation of the inequality $B(\Lambda)>0$
is experimentally verified, it can be regarded as an IR constraint on quantum
gravity (Interpretation 3). Moreover, there are other subtleties when we wish
to apply the bounds to realistic models, as discussed in Sec. 5. We expect in
general that the dark photon interacts with the Standard Model via kinetic
mixings with the Standard-Model gauge bosons. This means that there are large
non-gravitational amplitudes from QCD that can have a significant impact on
the dark photon constraint. Another issue is the stability of the dark photon.
The dark photon is expected to decay into Standard Model fermions and photons
once it interacts with the SM sector. In this case, the dark photon has a
finite decay width, and care must be taken when applying the positivity
constraint, even though we may have no major practical problems when the decay
rate is sufficiently small. It requires careful theoretical analysis to fully
address these details, which we hope will be discussed in future works.

## Acknowledgements

This work was partly performed during our mutual visits to Kobe University,
Kyoto University (Yukawa Institute), University of Tokyo (Kavli IPMU), and
Yamaguchi University. We would like to thank the institutes for their
hospitality. M.Y. is grateful to the Kavli Institute for Theoretical Physics
in Santa Barbara for hospitality during the Integrable22 workshop.

The Feynman diagrams in this paper were drawn with the help of TikZ-FeynHand
[85, 86].

The work of KA was supported in part by Grant-in-Aid from the Scientific
Research Fund of the Japan Society for the Promotion of Science, No. 20K14468.
TN was supported in part by JSPS KAKENHI Grant No. 20H01902 and No. 22H01220,
and MEXT KAKENHI Grant No. 21H05184 and No. 23H04007. This work of RS is
supported in part by JSPS Grant-in-Aid for Scientific Research 20H05860 and
No. 23H01171. S. Sato is supported in part by JST SPRING Grant No. JPMJFS2126.
The work of S. Shirai is supported by Grant-in-Aid for Scientific Research
from the Ministry of Education, Culture, Sports, Science, and Technology
(MEXT), Japan, 18K13535, 20H01895, 20H05860 and 21H00067, and by World Premier
International Research Center Initiative (WPI), MEXT, Japan. The work of JT
was supported by IBS under the project code, IBS-R018-D1. The work of MY is
supported in part by the JSPS Grant-in-Aid for Scientific Research (No.
19H00689, 19K03820, 20H05860, 23H01168) and by JST, Japan (PRESTO Grant No.
JPMJPR225A and Moonshot R&D Grant No. JPMJMS2061).

##  Appendix A Gravitational Contributions Beyond Graviton $t$-channel

Figure 12: An example of a gravitational contribution beyond $t$-channel
exchange.

The gravitational contribution $B_{\text{grav}}(\Lambda)$ to $B(\Lambda)$ can
be divided into $B_{\text{grav,$t$-channel}}(\Lambda)$ from the graviton
$t$-channel exchange and the remaining part $B_{\text{grav,others}}(\Lambda)$
(see Fig. 12). In this Appendix we show that we can neglect
$B_{\text{grav,others}}(\Lambda)$ in comparison with
$B_{\text{grav,$t$-channel}}(\Lambda)$.

Calculation of $B_{\text{grav,others}}(\Lambda)$ is challenging due to the
presence of IR-divergent diagrams in
$\widetilde{\mathcal{M}}_{\text{grav,others}}(s,t)$. While it is more
desirable to define IR-finite amplitudes using the dressed state formalism, it
is beyond the scope of this work and will be pursued in future research. To
regularize this IR divergence, instead, we introduce an IR regulator
$\mu_{\mathrm{IR}}$ by deforming the graviton propagator as

$\displaystyle\frac{iP_{\mu\nu\rho\sigma}}{q^{2}+i\epsilon}\rightarrow\frac{iP_{\mu\nu\rho\sigma}}{q^{2}-\mu_{\mathrm{IR}}^{2}+i\epsilon}\
,$ (A.1)

where $P_{\mu\nu\rho\sigma}$ is defined by
$P_{\mu\nu\rho\sigma}=\left(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\eta_{\mu\nu}\eta_{\rho\sigma}\right)/2$
in the harmonic gauge. Using the deformed graviton propagator, we can confirm
that $|\widetilde{\mathcal{M}}_{\text{grav,others}}(s,t)|$ is bounded by
$\mathcal{O}(s^{6})$ as $|s|\to\infty$. Along with the analyticity of
$\widetilde{\mathcal{M}}_{\text{grav,others}}$, this makes it possible to
derive the relation

$\displaystyle B_{\text{grav, others}}(\Lambda)$
$\displaystyle=\frac{4}{\pi}\int_{\Lambda^{2}}^{\infty}ds\,\frac{{\rm
Im}\,\widetilde{\mathcal{M}}_{\text{grav,
others}}(s,t=0)}{\left(s-2m_{V}^{2}\right)^{7}}\ .$ (A.2)

Moreover, since
$\textrm{Im}\,\widetilde{\mathcal{M}}_{\text{grav,others}}(s,t)$ is also
bounded by $\mathcal{O}(s^{6})$, we obtain

$\displaystyle B_{\text{grav, others}}(\Lambda)$
$\displaystyle=\mathcal{O}\left(\frac{1}{M_{\text{Pl}}^{2}\Lambda^{2}}\right)\
.$ (A.3)

This is much smaller compared to the contribution from the graviton
$t$-channel exchange diagrams and can be safely neglected.

## References

  * [1] G. ’t Hooft, _Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking_ , _NATO Sci. Ser. B_ 59 (1980) 135.
  * [2] C. Vafa, _The String landscape and the swampland_ , hep-th/0509212.
  * [3] H. Ooguri and C. Vafa, _On the Geometry of the String Landscape and the Swampland_ , _Nucl. Phys. B_ 766 (2007) 21 [hep-th/0605264].
  * [4] T.N. Pham and T.N. Truong, _Evaluation of the Derivative Quartic Terms of the Meson Chiral Lagrangian From Forward Dispersion Relation_ , _Phys. Rev. D_ 31 (1985) 3027.
  * [5] M.R. Pennington and J. Portoles, _The Chiral Lagrangian parameters, l1, l2, are determined by the rho resonance_ , _Phys. Lett. B_ 344 (1995) 399 [hep-ph/9409426].
  * [6] B. Ananthanarayan, D. Toublan and G. Wanders, _Consistency of the chiral pion pion scattering amplitudes with axiomatic constraints_ , _Phys. Rev. D_ 51 (1995) 1093 [hep-ph/9410302].
  * [7] J. Comellas, J.I. Latorre and J. Taron, _Constraints on chiral perturbation theory parameters from QCD inequalities_ , _Phys. Lett. B_ 360 (1995) 109 [hep-ph/9507258].
  * [8] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, _Causality, analyticity and an IR obstruction to UV completion_ , _JHEP_ 10 (2006) 014 [hep-th/0602178].
  * [9] C. de Rham, S. Kundu, M. Reece, A.J. Tolley and S.-Y. Zhou, _Snowmass White Paper: UV Constraints on IR Physics_ , in _Snowmass 2021_ , 3, 2022 [2203.06805].
  * [10] Y. Hamada, T. Noumi and G. Shiu, _Weak Gravity Conjecture from Unitarity and Causality_ , _Phys. Rev. Lett._ 123 (2019) 051601 [1810.03637].
  * [11] B. Bellazzini, M. Lewandowski and J. Serra, _Positivity of Amplitudes, Weak Gravity Conjecture, and Modified Gravity_ , _Phys. Rev. Lett._ 123 (2019) 251103 [1902.03250].
  * [12] G.J. Loges, T. Noumi and G. Shiu, _Duality and Supersymmetry Constraints on the Weak Gravity Conjecture_ , _JHEP_ 11 (2020) 008 [2006.06696].
  * [13] L. Alberte, C. de Rham, S. Jaitly and A.J. Tolley, _Positivity Bounds and the Massless Spin-2 Pole_ , _Phys. Rev. D_ 102 (2020) 125023 [2007.12667].
  * [14] J. Tokuda, K. Aoki and S. Hirano, _Gravitational positivity bounds_ , _JHEP_ 11 (2020) 054 [2007.15009].
  * [15] M. Herrero-Valea, R. Santos-Garcia and A. Tokareva, _Massless positivity in graviton exchange_ , _Phys. Rev. D_ 104 (2021) 085022 [2011.11652].
  * [16] L. Alberte, C. de Rham, S. Jaitly and A.J. Tolley, _QED positivity bounds_ , _Phys. Rev. D_ 103 (2021) 125020 [2012.05798].
  * [17] L. Alberte, C. de Rham, S. Jaitly and A.J. Tolley, _Reverse Bootstrapping: IR Lessons for UV Physics_ , _Phys. Rev. Lett._ 128 (2022) 051602 [2111.09226].
  * [18] S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, _Sharp boundaries for the swampland_ , _JHEP_ 07 (2021) 110 [2102.08951].
  * [19] S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin, _Causality constraints on corrections to Einstein gravity_ , 2201.06602.
  * [20] M. Herrero-Valea, A.S. Koshelev and A. Tokareva, _UV graviton scattering and positivity bounds from IR dispersion relations_ , _Phys. Rev. D_ 106 (2022) 105002 [2205.13332].
  * [21] C. de Rham, S. Jaitly and A.J. Tolley, _Constraints on Regge behaviour from IR physics_ , 2212.04975.
  * [22] T. Noumi and J. Tokuda, _Finite energy sum rules for gravitational Regge amplitudes_ , 2212.08001.
  * [23] Y. Hamada, R. Kuramochi, G.J. Loges and S. Nakajima, _On (Scalar QED) Gravitational Positivity Bounds_ , 2301.01999.
  * [24] C. Cheung and G.N. Remmen, _Infrared Consistency and the Weak Gravity Conjecture_ , _JHEP_ 12 (2014) 087 [1407.7865].
  * [25] S. Andriolo, D. Junghans, T. Noumi and G. Shiu, _A Tower Weak Gravity Conjecture from Infrared Consistency_ , _Fortsch. Phys._ 66 (2018) 1800020 [1802.04287].
  * [26] W.-M. Chen, Y.-T. Huang, T. Noumi and C. Wen, _Unitarity bounds on charged/neutral state mass ratios_ , _Phys. Rev. D_ 100 (2019) 025016 [1901.11480].
  * [27] K. Aoki, T.Q. Loc, T. Noumi and J. Tokuda, _Is the Standard Model in the Swampland? Consistency Requirements from Gravitational Scattering_ , _Phys. Rev. Lett._ 127 (2021) 091602 [2104.09682].
  * [28] T. Noumi and J. Tokuda, _Gravitational positivity bounds on scalar potentials_ , _Phys. Rev. D_ 104 (2021) 066022 [2105.01436].
  * [29] T. Noumi, S. Sato and J. Tokuda, _Phenomenological Motivation for Gravitational Positivity Bounds: A Case Study of Dark Sector Physics_ , 2205.12835.
  * [30] M. Froissart, _Asymptotic behavior and subtractions in the Mandelstam representation_ , _Phys. Rev._ 123 (1961) 1053.
  * [31] A. Martin, _Unitarity and high-energy behavior of scattering amplitudes_ , _Phys. Rev._ 129 (1963) 1432.
  * [32] B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva, _Positive moments for scattering amplitudes_ , _Phys. Rev. D_ 104 (2021) 036006 [2011.00037].
  * [33] B. Bellazzini, M. Riembau and F. Riva, _IR side of positivity bounds_ , _Phys. Rev. D_ 106 (2022) 105008 [2112.12561].
  * [34] N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang, _The EFT-Hedron_ , _JHEP_ 05 (2021) 259 [2012.15849].
  * [35] B. Bellazzini, _Softness and amplitudes’ positivity for spinning particles_ , _JHEP_ 02 (2017) 034 [1605.06111].
  * [36] C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, _Positivity bounds for scalar field theories_ , _Phys. Rev. D_ 96 (2017) 081702 [1702.06134].
  * [37] C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, _Massive Galileon Positivity Bounds_ , _JHEP_ 09 (2017) 072 [1702.08577].
  * [38] X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, _Causality Constraints on Corrections to the Graviton Three-Point Coupling_ , _JHEP_ 02 (2016) 020 [1407.5597].
  * [39] G. D’Appollonio, P. Di Vecchia, R. Russo and G. Veneziano, _Regge behavior saves String Theory from causality violations_ , _JHEP_ 05 (2015) 144 [1502.01254].
  * [40] C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, _UV complete me: Positivity Bounds for Particles with Spin_ , _JHEP_ 03 (2018) 011 [1706.02712].
  * [41] A. Kotanski, _Transversity Amplitudes and Their Application to the Study of Collisons of Particle with Spin_ , _Acta Phys. Pol. B_ 1 (1970) 45.
  * [42] L.-L.C. Wang, _General Method of Constructing Helicity Amplitudes Free from Kinematic Singularities and Zeros_ , _Phys. Rev._ 142 (1966) 1187.
  * [43] G. Cohen-Tannoudji, A. Morel and H. Navelet, _Kinematical singularities, crossing matrix and kinematical constraints for two-body helicity amplitudes_ , _Annals Phys._ 46 (1968) 239.
  * [44] M.J.G. Veltman, _Unitarity and causality in a renormalizable field theory with unstable particles_ , _Physica_ 29 (1963) 186.
  * [45] R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, _The analytic S-matrix_ , Cambridge Univ. Press, Cambridge (1966).
  * [46] K. Aoki, _Unitarity and unstable-particle scattering amplitudes_ , _Phys. Rev. D_ 107 (2023) 065017 [2212.05670].
  * [47] H.S. Hannesdottir and S. Mizera, _What is the i $\varepsilon$ for the S-matrix?_, SpringerBriefs in Physics, Springer (1, 2023), 10.1007/978-3-031-18258-7, [2204.02988].
  * [48] M. Correia, _Nonperturbative Anomalous Thresholds_ , 2212.06157.
  * [49] A.E. Nelson and J. Scholtz, _Dark Light, Dark Matter and the Misalignment Mechanism_ , _Phys. Rev. D_ 84 (2011) 103501 [1105.2812].
  * [50] P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel, J. Redondo and A. Ringwald, _WISPy Cold Dark Matter_ , _JCAP_ 06 (2012) 013 [1201.5902].
  * [51] P.W. Graham, J. Mardon and S. Rajendran, _Vector Dark Matter from Inflationary Fluctuations_ , _Phys. Rev. D_ 93 (2016) 103520 [1504.02102].
  * [52] N.D. Christensen, P. de Aquino, C. Degrande, C. Duhr, B. Fuks, M. Herquet et al., _A Comprehensive approach to new physics simulations_ , _Eur. Phys. J. C_ 71 (2011) 1541 [0906.2474].
  * [53] N.D. Christensen and C. Duhr, _FeynRules - Feynman rules made easy_ , _Comput. Phys. Commun._ 180 (2009) 1614 [0806.4194].
  * [54] A. Alloul, N.D. Christensen, C. Degrande, C. Duhr and B. Fuks, _FeynRules 2.0 - A complete toolbox for tree-level phenomenology_ , _Comput. Phys. Commun._ 185 (2014) 2250 [1310.1921].
  * [55] T. Hahn, _Generating Feynman diagrams and amplitudes with FeynArts 3_ , _Comput. Phys. Commun._ 140 (2001) 418 [hep-ph/0012260].
  * [56] R. Mertig, M. Bohm and A. Denner, _FEYN CALC: Computer algebraic calculation of Feynman amplitudes_ , _Comput. Phys. Commun._ 64 (1991) 345.
  * [57] V. Shtabovenko, R. Mertig and F. Orellana, _FeynCalc 9.3: New features and improvements_ , _Comput. Phys. Commun._ 256 (2020) 107478 [2001.04407].
  * [58] V. Shtabovenko, R. Mertig and F. Orellana, _New Developments in FeynCalc 9.0_ , _Comput. Phys. Commun._ 207 (2016) 432 [1601.01167].
  * [59] H.H. Patel, _Package-X: A Mathematica package for the analytic calculation of one-loop integrals_ , _Comput. Phys. Commun._ 197 (2015) 276 [1503.01469].
  * [60] A. Donnachie and P.V. Landshoff, _Total cross-sections_ , _Phys. Lett. B_ 296 (1992) 227 [hep-ph/9209205].
  * [61] M. Klusek, W. Schafer and A. Szczurek, _Exclusive production of rho0 rho0 pairs in gamma gamma collisions at RHIC_ , _Phys. Lett. B_ 674 (2009) 92 [0902.1689].
  * [62] M. Reece, _Photon Masses in the Landscape and the Swampland_ , _JHEP_ 07 (2019) 181 [1808.09966].
  * [63] A. Hebecker and P. Soler, _The Weak Gravity Conjecture and the Axionic Black Hole Paradox_ , _JHEP_ 09 (2017) 036 [1702.06130].
  * [64] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, _The String landscape, black holes and gravity as the weakest force_ , _JHEP_ 06 (2007) 060 [hep-th/0601001].
  * [65] B. Heidenreich, M. Reece and T. Rudelius, _Evidence for a sublattice weak gravity conjecture_ , _JHEP_ 08 (2017) 025 [1606.08437].
  * [66] B. Heidenreich, M. Reece and T. Rudelius, _The Weak Gravity Conjecture and Emergence from an Ultraviolet Cutoff_ , _Eur. Phys. J. C_ 78 (2018) 337 [1712.01868].
  * [67] M. Montero, G. Shiu and P. Soler, _The Weak Gravity Conjecture in three dimensions_ , _JHEP_ 10 (2016) 159 [1606.08438].
  * [68] N. Arkani-Hamed, S. Dimopoulos and S. Kachru, _Predictive landscapes and new physics at a TeV_ , hep-th/0501082.
  * [69] G. Dvali, _Black Holes and Large N Species Solution to the Hierarchy Problem_ , _Fortsch. Phys._ 58 (2010) 528 [0706.2050].
  * [70] G. Dvali and M. Redi, _Black Hole Bound on the Number of Species and Quantum Gravity at LHC_ , _Phys. Rev. D_ 77 (2008) 045027 [0710.4344].
  * [71] J.K. Hoskins, R.D. Newman, R. Spero and J. Schultz, _Experimental tests of the gravitational inverse square law for mass separations from 2-cm to 105-cm_ , _Phys. Rev. D_ 32 (1985) 3084.
  * [72] G.L. Smith, C.D. Hoyle, J.H. Gundlach, E.G. Adelberger, B.R. Heckel and H.E. Swanson, _Short range tests of the equivalence principle_ , _Phys. Rev. D_ 61 (2000) 022001.
  * [73] D.J. Kapner, T.S. Cook, E.G. Adelberger, J.H. Gundlach, B.R. Heckel, C.D. Hoyle et al., _Tests of the gravitational inverse-square law below the dark-energy length scale_ , _Phys. Rev. Lett._ 98 (2007) 021101 [hep-ph/0611184].
  * [74] T.A. Wagner, S. Schlamminger, J.H. Gundlach and E.G. Adelberger, _Torsion-balance tests of the weak equivalence principle_ , _Class. Quant. Grav._ 29 (2012) 184002 [1207.2442].
  * [75] J.G. Lee, E.G. Adelberger, T.S. Cook, S.M. Fleischer and B.R. Heckel, _New Test of the Gravitational $1/r^{2}$ Law at Separations down to 52 $\mu$m_, _Phys. Rev. Lett._ 124 (2020) 101101 [2002.11761].
  * [76] S.-Q. Yang, B.-F. Zhan, Q.-L. Wang, C.-G. Shao, L.-C. Tu, W.-H. Tan et al., _Test of the Gravitational Inverse Square Law at Millimeter Ranges_ , _Phys. Rev. Lett._ 108 (2012) 081101.
  * [77] W.-H. Tan et al., _Improvement for Testing the Gravitational Inverse-Square Law at the Submillimeter Range_ , _Phys. Rev. Lett._ 124 (2020) 051301.
  * [78] Y.J. Chen, W.K. Tham, D.E. Krause, D. Lopez, E. Fischbach and R.S. Decca, _Stronger Limits on Hypothetical Yukawa Interactions in the 30–8000 nm Range_ , _Phys. Rev. Lett._ 116 (2016) 221102 [1410.7267].
  * [79] MICROSCOPE collaboration, _MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle_ , _Phys. Rev. Lett._ 129 (2022) 121102 [2209.15487].
  * [80] E. Hardy and R. Lasenby, _Stellar cooling bounds on new light particles: plasma mixing effects_ , _JHEP_ 02 (2017) 033 [1611.05852].
  * [81] M. Escudero and S.J. Witte, _A CMB search for the neutrino mass mechanism and its relation to the Hubble tension_ , _Eur. Phys. J. C_ 80 (2020) 294 [1909.04044].
  * [82] M. Ibe, S. Kobayashi, Y. Nakayama and S. Shirai, _Cosmological Constraint on Vector Mediator of Neutrino-Electron Interaction in light of XENON1T Excess_ , _JHEP_ 12 (2020) 004 [2007.16105].
  * [83] XENON collaboration, _Search for New Physics in Electronic Recoil Data from XENONnT_ , _Phys. Rev. Lett._ 129 (2022) 161805 [2207.11330].
  * [84] S.K. A., A. Majumdar, D.K. Papoulias, H. Prajapati and R. Srivastava, _Implications of first LZ and XENONnT results: A comparative study of neutrino properties and light mediators_ , _Phys. Lett. B_ 839 (2023) 137742 [2208.06415].
  * [85] J. Ellis, _TikZ-Feynman: Feynman diagrams with TikZ_ , _Comput. Phys. Commun._ 210 (2017) 103 [1601.05437].
  * [86] M. Dohse, _TikZ-FeynHand: Basic User Guide_ , 1802.00689.