#  Phenomenological Motivation for Gravitational Positivity Bounds:
A Case Study of Dark Sector Physics

Toshifumi Noumi  Sota Sato  Junsei Tokuda  Department of Physics, Kobe
University, Kobe 657-8501, Japan

###### Abstract

Positivity bounds on scattering amplitudes provide a necessary condition for a
low-energy effective field theory to have a consistent ultraviolet completion.
Their extension to gravity theories has been studied in the past years aiming
at application to the swampland program, showing that positivity bounds hold
at least approximately even in the presence of gravity. An issue in this
context is how much negativity is allowed for a given scattering process. In
this paper we address importance of this rather technical issue by
demonstrating that it is relevant to physics within the scope of ongoing
experiments, especially in the context of dark sector physics. In particular,
we provide detailed analysis of dark photon scenarios as an illustrative
example. This motivates further studies on gravitational positivity bounds.

††preprint: KOBE-COSMO-22-5

##  I I. Introduction

The concept of effective field theory (EFT) and ultraviolet (UV) completion is
a key framework for connecting various scales in nature. When applied to
gravity, it will open up a road toward quantum gravity phenomenology. The
swampland program Vafa (2005) is aiming at this direction by exploring UV
constraints on gravitational EFTs and their implications for particle physics
and cosmology. See Brennan  _et al._ (2017); Palti (2019); van Beest  _et al._
(2021) for review articles.

Positivity bounds on scattering amplitudes is one of the interesting
approaches to such UV constraints on EFTs. While the bounds were originally
formulated in nongravitational theories with a mass gap Pham and Truong
(1985); Ananthanarayan  _et al._ (1995); Adams  _et al._ (2006), their
extension to gravity theories has been studied intensively, having application
to the swampland program in mind Hamada  _et al._ (2019); Bellazzini  _et al._
(2019); Alberte  _et al._ (2020); Tokuda  _et al._ (2020); Loges  _et al._
(2020a); Herrero-Valea  _et al._ (2021); Caron-Huot  _et al._ (2021); Alberte
_et al._ (2022); Bellazzini  _et al._ (2021); Chiang  _et al._ (2022). A
finding there is that positivity bounds hold at least approximately even in
gravity theories.

For concreteness, we consider an $s$-$u$ symmetric scattering amplitude
$\mathcal{M}(s,t)$ of the process $AB\to AB$ up to
$\mathcal{O}(M_{\text{Pl}}^{-2})$ and perform its infrared (IR) expansion as

$\displaystyle\mathcal{M}(s,t)$ $\displaystyle=$
$\displaystyle\sum_{n,m=0}^{\infty}\frac{c_{n,m}}{n!m!}\left(\frac{s-u}{2}\right)^{n}t^{m}$
(1) $\displaystyle+(s,t,u\text{-poles of graviton exchange})\,,$

where $(s,t,u)$ are the standard Mandelstam variables and they satisfy
$s+t+u=2(m_{A}^{2}+m_{B}^{2})$. Then, the positivity bound implies

$c_{2,0}\geq\frac{\lambda}{M_{\rm Pl}^{2}M^{2}}\quad(\lambda=\pm 1)\,,$ (2)

where the sign $\lambda$ and the scale $M$ depend on details of UV completion
of gravity. See the next section for the definitions of $\lambda$ and $M$ in
terms of gravitational Regge amplitudes at high energies and also the
assumptions used to derive the bound (2).

When the threshold energy $m_{\rm th}$, i.e., the mass of the lightest
intermediate on-shell state, is below the UV cutoff scale $\Lambda$ of the
EFT, it is convenient to define Bellazzini (2017); de Rham  _et al._ (2017)

$B^{(2)}(\Lambda):=c_{2,0}-\frac{4}{\pi}\int_{m_{\rm
th}^{2}}^{\Lambda^{2}}\mathrm{d}s\frac{\mathrm{Im}\,\mathcal{M}(s,0)}{(s-m_{A}^{2}-m_{B}^{2})^{3}}\,,$
(3)

which is calculable within the EFT. The bound (2) on $c_{2,0}$ is now
sharpened as

$B^{(2)}(\Lambda)\geq\frac{\lambda}{M_{\rm Pl}^{2}M^{2}}\,.$ (4)

In the past years, the bounds (2) and (4) have been used in the swampland
program and it turned out that the sign $\lambda$ and the scale $M$ are
crucial for deriving the swampland conditions. For example, the sign $\lambda$
is crucial in the context of the weak gravity conjecture Arkani-Hamed  _et
al._ (2007) for black holes, which implies that the charge-to-mass ratios of
extremal black holes are increased by higher derivative corrections. In
particular, negativity of $c_{2,0}$ for a certain helicity amplitudes of
light-by-light scattering implies violation of the conjecture. Hence, it is
crucial to understand for which scattering process and in what class of UV
completion of gravity, negativity of $c_{2,0}$ is allowed. See, e.g., Kats
_et al._ (2007); Cheung  _et al._ (2018); Hamada  _et al._ (2019); Bellazzini
_et al._ (2019); Jones and McPeak (2020); Loges  _et al._ (2020b); Cano  _et
al._ (2020a, b); Loges  _et al._ (2020a); Cremonini  _et al._ (2021); Arkani-
Hamed  _et al._ (2022); Cano  _et al._ (2022); Henriksson  _et al._ (2022);
Aalsma and Shiu (2022) for recent discussion.

The scale $M$ is also important to use the bound (4) for constraining the
particle spectrum in gravitational EFTs Cheung and Remmen (2014); Andriolo
_et al._ (2018); Chen  _et al._ (2019); Alberte  _et al._ (2020, 2021); Aoki
_et al._ (2021); Noumi and Tokuda (2021); Alberte  _et al._ (2022). As we
explain later, if the scale $M$ is at a UV scale $M\gtrsim\Lambda$, one may
derive nontrivial constraints generically. On the other hand, if the sign
$\lambda$ is negative and the scale $M$ is sufficiently low, the bounds are
trivially satisfied without giving any constraint on IR physics.
Interestingly, a recent paper Alberte  _et al._ (2022) pointed out that for a
certain helicity amplitude of graviton-photon scattering, the sign $\lambda$
has to be negative and also the scale $M$ has to be an IR scale for
compatibility with the sum rule. Since the sign $\lambda$ and the scale $M$
depend on each scattering process and details of its UV completion, it is of
great interests how generic this feature is beyond the graviton-photon
scattering111For example, if negativity of $c_{2,0}$ cannot be ruled out in
light-by-light scattering eventually, this provides a counterexample for the
black hole weak gravity conjecture. So far, there is no known counterexample
in string theory, so this issue would tell us if the conjecture is specific in
string theory or more generic in quantum gravity..

In this paper we address importance of this rather technical issue by
demonstrating that it is relevant to physics within the scope of ongoing
experiments, especially in the context of dark sector physics. In particular,
we study dark photon scenarios as an illustrative example.

##  II II. Gravitational Positivity Bounds

We begin by a brief review of positivity bounds in the presence of gravity. A
set of sufficient conditions for deriving the bounds (2) and (4) are the
following:

  1. (i)

Unitarity: The imaginary part of $\mathcal{M}$ is nonnegative,
$\mathrm{Im}\,\mathcal{M}(s\geq m_{\text{th}}^{2},0)\geq 0$.

  2. (ii)

Analyticity: $\mathcal{M}$ is analytic on the physical sheet of the complex
$s$-plane except for poles and discontinuities on the real axis demanded by
unitarity.

  3. (iii)

Mild behavior in the Regge limit: The high-energy behavior of
$\mathcal{M}(s,t)$ is sufficiently mild to satisfy
$\lim_{|s|\to\infty}|\mathcal{M}(s,t<0)/s^{2}|=0$.

  4. (iv)

Regge behavior: The $s^{2}$ bound (iii) is satisfied as a consequence of
Reggeization by an infinite tower of higher-spin states.

The properties (i)-(iii) are satisfied in the standard UV complete theory with
a mass gap. Also, they are satisfied by construction in perturbative string
theory (which emerged in the $S$-matrix theory context). Even though it is
nontrivial if these properties should hold in general gravity theories222See,
e.g., Häring and Zhiboedov (2022) for recent discussion on high-energy
behavior of gravitational scattering., we assume that (i)-(iii) hold for a
small negative $t<0$ up to $\mathcal{O}(M_{\text{Pl}}^{-2})$. By contrast, the
property (iv) is naturally derived from the properties (i)-(iii) in the
process with massless spin-2 exchange. For our purpose, it is convenient to
parameterize the imaginary part of the Regge amplitude up to
$\mathcal{O}(M_{\text{Pl}}^{-2})$ as

$\mathrm{Im}\,\mathcal{M}(s,t)\simeq
f(t)\left(s/M_{*}^{2}\right)^{\alpha(t)}+\cdots\,\,(s\geq M_{*}^{2})\,.$ (5)

Here $M_{*}$ is the Reggeization scale, which is well above the mass scale of
the higher-spin states, e.g., the string scale $M_{s}$ in perturbative string
theory. Also, we suppressed terms irrelevant to Reggeization of graviton
exchange.

Now the twice-subtracted dispersion relation for $t<0$ reads

$\displaystyle c_{2,0}+\mathcal{O}(t)$ $\displaystyle=$
$\displaystyle\frac{2}{M_{\rm
Pl}^{2}t}+\frac{4}{\pi}\int_{M_{*}^{2}}^{\infty}\mathrm{d}s\frac{f(t)\left(s/M_{*}^{2}\right)^{\alpha(t)}}{(s+t/2-m_{A}^{2}-m_{B}^{2})^{3}}$
(6) $\displaystyle+\frac{4}{\pi}\int_{m_{\rm
th}^{2}}^{M_{*}^{2}}\mathrm{d}s\frac{\mathrm{Im}\,\mathcal{M}(s,t)}{(s+t/2-m_{A}^{2}-m_{B}^{2})^{3}}\,,$

where the singular graviton $1/t$-pole must be canceled with the second term
on the right-hand side (RHS). The cancellation fixes the value of $f(0)$.
Also, evaluating the $\mathcal{O}(t^{0})$ term gives Tokuda  _et al._ (2020)

$B^{(2)}(\Lambda)=\frac{4}{\pi}\int_{\Lambda^{2}}^{M_{*}^{2}}\mathrm{d}s\frac{\mathrm{Im}\,\mathcal{M}(s,0)}{(s-m_{A}^{2}-m_{B}^{2})^{3}}+\frac{\lambda}{M_{\rm
Pl}^{2}M^{2}}$ (7)

with $\lambda\,(=\pm 1)$ and $M\,(>0)$ defined by

$\frac{\lambda}{M^{2}}=-\left[\frac{2f^{\prime}(0)}{f(0)}-\frac{\alpha^{\prime\prime}(0)}{\alpha^{\prime}(0)}\right]\,.$
(8)

Then, $\mathrm{Im}\,\mathcal{M}(s,0)\geq 0$ implies the bounds (2) and (4).

##  III III. General consideration

We discuss general implications of the bound (4) for the dark sector physics.
For earlier discussions along this line of considerations, see, e.g., Andriolo
_et al._ (2018); Alberte  _et al._ (2022). Let us consider the following model
in which the Standard Model Lagrangian $\mathcal{L}_{\text{SM}}$ and the
Lagrangian of the dark sector $\mathcal{L}_{\text{DS}}$ are included in
addition to the Einstein-Hilbert term:

$\mathcal{L}=\frac{M_{\text{Pl}}^{2}}{2}R+\mathcal{L}_{\text{SM}}[A_{b},\cdots]+\mathcal{L}_{\text{DS}}[X,\cdots]\,.$
(9)

Here, $A_{b}$ and $X$ represent an ordinary photon and a hidden particle in
the dark sector, respectively. We define $\mathcal{L}_{\text{DS}}$ such that
direct interactions between the dark sector and the Standard Model, if exist,
are included in $\mathcal{L}_{\text{DS}}$. We assume that the matter sector
described by $\mathcal{L}_{\text{SM}}+\mathcal{L}_{\text{DS}}$ is
renormalizable and neglect nonrenormalizable terms for
simplicity333Nonrenormalizable terms suppressed by the cutoff scale $\Lambda$
would give $\mathcal{O}(\Lambda^{-4})$ contributions to $B^{(2)}(\Lambda)$.
This does not change the main point in this section.. We then discuss the
implication of (4) for the scattering amplitude $\mathcal{M}(s,t)$ of the
process $\gamma X\to\gamma X$. We will later consider the case
$X=\gamma^{\prime}$, the dark photon as an illustrative example.

We decompose diagrams up to $\mathcal{O}(M_{\text{Pl}}^{-2})$ into two parts:
the nongravitational diagrams in which gravitons are absent and those with a
graviton exchange. We refer to the contributions from the former diagrams and
the latter diagrams as $B^{(2)}_{\text{non-grav}}(\Lambda)$ and
$B^{(2)}_{\text{grav}}(\Lambda)$, respectively. Because the matter sector
itself is renormalizable, the nongravitational amplitude
$\mathcal{M}_{\text{non-grav}}$ satisfies the twice-subtracted dispersion
relation, leading to

$B^{(2)}_{\text{non-
grav}}(\Lambda)=\frac{4}{\pi}\int^{\infty}_{\Lambda^{2}}\mathrm{d}s\,\frac{\mathrm{Im}\,\mathcal{M}_{\text{non-
grav}}(s,0)}{(s-m_{X}^{2})^{3}}\,,$ (10)

where $m_{X}$ is the mass of the particle $X$.

By contrast, the GR amplitude does not satisfy the twice-subtracted dispersion
relation because of the nonrenormalizable nature of GR. A $t$-channel diagram
shown in FIG. 1 gives rise to the $\Lambda$-independent
$B^{(2)}_{\text{grav}}(\Lambda)\supset-\mathcal{O}(e^{2}M_{\text{Pl}}^{-2}m_{e}^{-2})$
while the corresponding $s,u$-channel diagrams give terms on the order of
$e^{2}M_{\text{Pl}}^{-2}\Lambda^{-2}$. Then, $B^{(2)}_{\text{grav}}(\Lambda)$
is given by a negative constant in a good approximation at $\Lambda\gg
m_{e}$:444Strictly speaking, loop corrections to the $XXh$-vertex could also
contribute to $B^{(2)}_{\text{grav}}$. The values of such contributions depend
on the details of $\mathcal{L}_{\text{DS}}$, the dark sector model. However,
as long as $X$ is a stable particle, the sign of such contributions would be
negative as far as we know. Hence, adding such terms will not change the
essence of the discussion.

$B^{(2)}_{\text{grav}}(\Lambda)=-\mathcal{O}\left(\frac{e^{2}}{M_{\text{Pl}}^{2}m_{e}^{2}}\right)\,.$
(11)

As a result, the bound (4) reads

$B^{(2)}_{\text{non-
grav}}(\Lambda)\geq\mathcal{O}\left(e^{2}M_{\text{Pl}}^{-2}m_{e}^{-2}\right)-\lambda
M_{\text{Pl}}^{-2}M^{-2}\,.$ (12)

The RHS depends on the unknown scale $M$ given in (8): a priori even the sign
of the RHS is undetermined. Implications of (12) crucially depend on the value
of $M$. If $M$ is given by an IR scale as $M\lesssim m_{e}/e$, the RHS of (12)
may become negative. In this case, the bound (12) is satisfied in any
renormalizable unitary matter theories coupled to gravity because of the
condition $B^{(2)}_{\text{non-grav}}(\Lambda)>0$, which follows from the
formula (10).

If $M$ is given by the scale of UV physics such as the mass of higher-spin
states and satisfies the condition $M\gg m_{e}/e$, we can then ignore the
second term on the RHS of (12), leading to

$\qquad\,\,\,B^{(2)}_{\text{non-
grav}}(\Lambda)\geq-B^{(2)}_{\text{grav}}(\Lambda)=\mathcal{O}\left(\frac{e^{2}}{M_{\text{Pl}}^{2}m_{e}^{2}}\right)\,.$
(13)

This gives a nontrivial bound on the EFT even though the matter sector is
renormalizable. In particular, this implies an upper bound on $\Lambda$ in
terms of parameters in the model such as coupling constants, because
$B^{(2)}_{\text{non-grav}}(\Lambda\to\infty)\to 0$ as explicitly seen from Eq.
(10).

Using the formula (10) and the relation $\mathrm{Im}\,\mathcal{M}(s,0)\simeq
s\sigma_{\gamma\text{-X}}(s)$ at $s\gg m^{2}_{X}$, (13) becomes555We can also
derive nontrivial bounds on processes with external Higgs bosons, if $M$ is at
a UV scale. See Alberte  _et al._ (2020); Noumi and Tokuda (2021) for
discussions of gravitational positivity bounds on scalar scatterings.

$\frac{4}{\pi}\int^{\infty}_{\Lambda^{2}}\mathrm{d}s\,\frac{\sigma_{\gamma\text{-X}}(s)}{s^{2}}\geq-B^{(2)}_{\text{grav}}=\mathcal{O}\left(\frac{e^{2}}{M_{\text{Pl}}^{2}m_{e}^{2}}\right)\,.$
(14)

Here $\sigma_{\gamma\text{-X}}(s)$ is a total cross section of the
$\gamma\text{-}X$ scattering at the center-of-mass energy $\sqrt{s}$.
Therefore, the bound (13) implies that a particle $X$ in the dark sector must
couple to photon to satisfy the lower bound (14) on $\sigma_{\gamma\text{-X}}$
at high energies $s\geq\Lambda^{2}$, meaning that the dark sector cannot be
too dark.666In the standard renormalizable theories, the cross section is
consistent with the Froissart-Martin bound. The integral in (14) may be then
dominated by its lower end. Assuming this, (14) implies a lower bound on
$\sigma_{\gamma\text{-X}}(\Lambda^{2})$ as
$\sigma_{\gamma\text{-X}}(\Lambda^{2})\gtrsim\mathcal{O}\left(\frac{e^{2}\Lambda^{2}}{M_{\text{Pl}}^{2}m_{e}^{2}}\right)\,.$
(15) This could also be useful for understanding the implication of the bound
(13) for given models in terms of the total cross section. This is reminiscent
of typical dark sector scenarios in extra dimensions, where decoupling of the
two sectors is realized by taking a large volume limit and so it cannot be
achieved as long as the volume is finite to make the four dimensional Planck
mass $M_{\rm Pl}$ finite. In the next section, we study implications of
(13)-(14) more concretely for dark photon models.

Figure 1: The diagram which gives the leading negative contribution to
$B^{(2)}_{\text{grav}}(\Lambda)$.

##  IV IV. Dark photons

The dark photon is the gauge boson $A_{a}^{\mu}$ of an extra $U(1)$ symmetry
in the dark sector to which we refer as $U(1)_{\text{d}}$ Holdom (1986) (see
Fabbrichesi  _et al._ (2020) for a review). The kinetic mixing of $A_{a}$ and
the ordinary photon $A_{b}$ is renormalizable, so that there is no reason to
exclude it. Indeed, it appears naturally if there exist heavy fields which are
charged under both of the electromagnetic $U(1)$ and the hidden
$U(1)_{\text{d}}$. Also, $A_{a}$ is massive in general. We then consider the
massive dark photon model described by (9) with

$\mathcal{L}_{\text{DS}}=-\frac{1}{4}(F_{a,\mu\nu})^{2}-\frac{1}{2}m_{A^{\prime}}^{2}(A_{a,\mu})^{2}-\frac{\epsilon}{2}F_{a,\mu\nu}F_{b}^{\mu\nu}+\mathcal{L}_{\text{hm}}\,,$
(16)

where $\mathcal{L}_{\text{hm}}$ denotes Lagrangian of matters in the dark
sector. The kinetic terms are diagonalized in terms of new fields
$(A,A^{\prime})$ which are defined by

$\displaystyle\begin{pmatrix}A_{a}\\\ A_{b}\\\ \end{pmatrix}$
$\displaystyle=\begin{pmatrix}1/\sqrt{1-\epsilon^{2}}&0\\\
-\epsilon/\sqrt{1-\epsilon^{2}}&1\\\ \end{pmatrix}\begin{pmatrix}A^{\prime}\\\
A\\\ \end{pmatrix}\,.$ (17)

The rotation (17) generates tiny couplings between $A^{\prime}$ and the
Standard Model: they are included in $\mathcal{L}_{\text{SM}}|_{A_{b}\to
A-\left(\epsilon/\sqrt{1-\epsilon^{2}}\right)A^{\prime}}$. By contrast,
couplings between $A$ and hidden matters are not induced by this rotation.
Below, we call $A$ and $A^{\prime}$ photon and the dark photon, respectively,
and analyze the photon-dark photon scattering
$\gamma\gamma^{\prime}\to\gamma\gamma^{\prime}$.

The amplitude depends on the helicity configurations in general. For the
process with transversely polarized dark photon, we consider

$\displaystyle\mathcal{M}_{T}(s,t):=$
$\displaystyle\frac{1}{4}\bigl{[}\mathcal{M}(1^{+}2^{+}3^{+}4^{+})+\mathcal{M}(1^{+}2^{-}3^{+}4^{-})$
$\displaystyle+\mathcal{M}(1^{-}2^{-}3^{-}4^{-})+\mathcal{M}(1^{-}2^{+}3^{-}4^{+})\bigr{]}\,,$

where 1 and 2 (3 and 4) are the ingoing (outgoing) photon and dark photon,
respectively, and the superscript $\pm$ denotes the helicity. For the process
with longitudinal modes, we consider

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

where $L$ means the longitudinal polarization. For these amplitudes, the
crossing symmetry implies the $s\leftrightarrow u$ permutation invariance.

Below, we refer to $(B^{(2)}_{\text{non-grav}},\,B^{(2)}_{\text{grav}})$ for
$\mathcal{M}_{T}$ and $\mathcal{M}_{L}$ as $(B^{(2)}_{\text{T,non-
grav}},\,B^{(2)}_{\text{T,grav}})$ and $(B^{(2)}_{\text{L,non-
grav}},\,B^{(2)}_{\text{L,grav}})$, respectively.

###  IV.1 A simplest model

To begin with, we consider the simplest model with
$\mathcal{L}_{\text{hm}}=0$. In this case, the process
$\gamma\gamma^{\prime}\to\gamma\gamma^{\prime}$ is mediated only by the
Standard Model particles. For simplicity, we neglect the QCD sector for a
while. Since new particles coupled to photon below 1 TeV have not been found,
we choose $\Lambda\geq 1$ TeV.

Figure 2: The upper bounds (19) and (20) for $\Lambda=1$ TeV are plotted in
the $(m_{A^{\prime}},\epsilon)$-plane by the black dotted line and the white
line, respectively. We find that there is no parameter space compatible with
experiments and the bound (20). The figure is made based on the data given in
O’Hare (2020); Caputo  _et al._ (2021).

The form of interactions between $A^{\prime}$ and the Standard Model particles
are the same as those for $A$ except that the coupling strength is suppressed
by a factor $\epsilon/\sqrt{1-\epsilon^{2}}\simeq\epsilon$. As a result, the
calculations of $\mathcal{M}_{T}$ and $\mathcal{M}_{L}$ are similar to those
of the amplitude of the light-by-light scattering which has been done in Aoki
_et al._ (2021). As it is the case for the light-by-light scattering, in our
case, the dominant contributions to $B^{(2)}_{\text{non-grav}}(\Lambda)$ at
$\Lambda\geq 1\text{TeV}$ are given by the W-boson loop whereas the dominant
term in $B^{(2)}_{\text{grav}}$ arises from the electron, the lightest charged
particle in the Standard Model. The results are,

$\displaystyle B$ ${}^{(2)}_{\text{T,non-
grav}}(\Lambda)\simeq\frac{32\alpha^{2}\epsilon^{2}}{m_{W}^{2}\Lambda^{2}}\,,\quad
B^{(2)}_{\text{L,non-
grav}}(\Lambda)\simeq\frac{8\alpha^{2}\epsilon^{2}m_{A^{\prime}}^{2}}{\Lambda^{2}m_{W}^{4}}\,,$
(18) $\displaystyle B$ ${}^{(2)}_{\text{T,grav}}(\Lambda)\simeq
B^{(2)}_{\text{L,grav}}(\Lambda)\simeq-\frac{11\alpha}{180\pi
m_{e}^{2}M_{\text{Pl}}^{2}}\,,$

where $\alpha\simeq 1/137$ denotes the fine-structure constant. Then, (13)
implies a lower bound on $\epsilon$: from $\mathcal{M}_{T}$ we obtain

$\epsilon\geq\sqrt{\frac{11}{5760\pi\alpha}}\frac{m_{W}\Lambda}{m_{e}M_{\text{Pl}}}\simeq
1.9\times 10^{-12}\times\left(\frac{\Lambda}{1\,\mathrm{TeV}}\right)\,,$ (19)

while we obtain a stronger constraint from $\mathcal{M}_{L}$ as

$\displaystyle\epsilon$ $\displaystyle\geq$
$\displaystyle\sqrt{\frac{11}{1440\pi\alpha}}\frac{m_{W}^{2}\Lambda}{m_{A^{\prime}}m_{e}M_{\text{Pl}}}$
(20) $\displaystyle\simeq$ $\displaystyle 3.0\times
10^{-3}\times\left(\frac{\Lambda}{1\,\mathrm{TeV}}\right)\left(\frac{1\,\mathrm{keV}}{m_{A^{\prime}}}\right)\,.$

These bounds (19) and (20) are plotted in FIG. 2. The bound (20) also shows
the existence of a lower bound on $m_{A^{\prime}}$ for given $\epsilon$ and
$\Lambda$ (this bound applies to massive dark photons only, so massless dark
photons are free from this constraint). The origin of the lower bound on
$m_{A^{\prime}}$ is the suppression factor $(m_{A^{\prime}}/m_{W})^{2}$
contained in $B^{(2)}_{\text{L,non-grav}}$. This factor appears because
longitudinal modes of dark photons are decoupled from W-bosons in the limit
$m_{A^{\prime}}\to 0$.

We have neglected QCD contributions so far. Since the QCD scale is lower than
the electroweak scale, it may give relevant contributions to
$B^{(2)}_{\text{non-grav}}$. Indeed, QCD effects, particularly the Pomeron
exchange, provide the leading contribution to $B^{(2)}_{\text{non-grav}}$ in
the Standard Model analysis of Aoki  _et al._ (2021). If we extend the
analysis to transversely polarized dark photons simply by multiplying the
factor $\epsilon$ to the coupling between photon and Pomeron, the bound (19)
is slightly relaxed as $\epsilon\gtrsim 10^{-13}$ for $\Lambda\sim 1$ TeV.

It would be interesting to investigate how much the bound (20) is relaxed when
QCD is taken into account. In general, the QCD contributions to
$\mathcal{M}_{L}$ will be suppressed by the dark photon mass $m_{A^{\prime}}$
at least as long the mass is the Stückelberg mass. We thus expect that the
existence of a lower bound on $m_{A^{\prime}}$ like (20) is a robust result,
leaving further studies for future work.

###  IV.2 UV completion with bi-charged particles

Are models with a tiny $\epsilon$ inconsistent with the bound (13)? This is
not necessarily the case. We generically expect the existence of bi-charged
particles which generate the kinetic mixing via loops. If such particles are
light enough, implications of the bound (13) change. To see this, let us set
$\epsilon=0$ and instead add to the model two bi-charged vector bosons
$(V^{\mu}_{1},V^{\mu}_{2})$ with the same charges
$(q_{\text{em}},q_{\text{em}})$ under $U(1)_{\text{em}}$ but the opposite
charges $(q_{\text{d}},-q_{\text{d}})$ under $U(1)_{\text{d}}$. We choose
$q_{\text{em}}=q_{\text{d}}=1$ and write the $U(1)_{\text{d}}$ gauge coupling
as $e^{\prime}$ below. We assume that their masses $m_{V_{1}}$ and $m_{V_{2}}$
are approximately identical: $m_{V}:=m_{V_{1}}\simeq m_{V_{2}}$. With this
choice, these particles generate a tiny kinetic mixing suppressed by a factor
$|m_{V_{1}}^{2}-m_{V_{2}}^{2}|/m_{V}^{2}\ll 1$ so that the model can be
consistent with the current observations. Since bi-charged particles are
coupled to photon with coupling strength $e$, they must be sufficiently heavy
to be consistent with experiments: we then impose $m_{V}\gtrsim 1\text{TeV}\gg
m_{e}$.

In this model, $B^{(2)}_{\text{T,nongrav}}(\Lambda)$ and
$B^{(2)}_{\text{L,nongrav}}(\Lambda)$ at $\Lambda\gg m_{V}$ are evaluated up
to one-loop level as

$\displaystyle B$ ${}^{(2)}_{\text{T,non-
grav}}(\Lambda)\simeq\frac{4e^{2}e^{\prime
2}}{\pi^{2}m_{V}^{2}\Lambda^{2}}\,,$ (21) $\displaystyle B$
${}^{(2)}_{\text{L,non-grav}}(\Lambda)\simeq\frac{e^{2}e^{\prime
2}m_{A^{\prime}}^{2}}{\pi^{2}m_{V}^{4}\Lambda^{2}}\,.$ (22)

Bi-charged particles also contribute to $B^{(2)}_{\text{grav}}$, but it is
negligible compared to the electron contribution as long as $m_{V}\gg m_{e}$
and $e^{\prime}\lesssim 1$. The bound (13) is 777Note that, even if we add bi-
charged particles with spin smaller than one, the bound will not be relaxed
comparing with (19) because its contributions to $B^{(2)}_{\text{T,non-grav}}$
and $B^{(2)}_{\text{L,non-grav}}$ are proportional to $\Lambda^{-4}$ and
$m_{A^{\prime}}^{2}\Lambda^{-6}$, respectively.

$\displaystyle\Lambda$ $\displaystyle\leq$ $\displaystyle
24\sqrt{\frac{5}{11}}\frac{m_{e}}{m_{V}}e^{\prime}M_{\text{Pl}}$ (23)
$\displaystyle\simeq$ $\displaystyle 2.0\times 10^{13}\,\,\text{GeV}\times
e^{\prime}\left(\frac{1\,\text{TeV}}{m_{V}}\right)$

for $\mathcal{M}_{T}$, while for $\mathcal{M}_{L}$ it reads

$\displaystyle\Lambda$ $\displaystyle\leq$ $\displaystyle
12\sqrt{\frac{5}{11}}\frac{m_{e}m_{A^{\prime}}}{m_{V}^{2}}e^{\prime}M_{\text{Pl}}$
(24) $\displaystyle\simeq$ $\displaystyle 10\,\,\text{TeV}\times
e^{\prime}\left(\frac{m_{A^{\prime}}}{1\,\text{keV}}\right)\left(\frac{1\,\text{TeV}}{m_{V}}\right)^{2}\,.$

Eq. (23) shows that, by adding sufficiently light bi-charged spin-one
particles appropriately, we can make massless dark photon models with a tiny
kinetic mixing $\epsilon$ consistent with (13) even for large cutoff
$\Lambda\gg 1\,\text{TeV}$. When the dark photon is massive
$m_{A^{\prime}}>0$, the scattering of longitudinal modes gives a stronger
constraint (24). The constraint implies $m_{A^{\prime}}\gtrsim e^{\prime}$ keV
because of the condition $\Lambda,m_{V}\gtrsim 1$ TeV. As discussed below
(20), this is due to the suppression of couplings between longitudinal modes
and matters at small $m_{A^{\prime}}$.

We conclude that dark photon cannot be too light, at least in models we
discussed. It would be interesting to explore models that accommodate a very
light dark photon without violating the bound (13), leaving it for future
work.

##  V V. Conclusion

In this paper we addressed importance of determining the sign $\lambda$ and
the scale $M$ in the gravitational positivity bound (4). If $\lambda$ is
negative and $M$ is as small as the electron mass scale, the bound is
trivially satisfied. By contrast, when $M$ is sufficiently large, the bound
gives a nontrivial constraint on IR physics even if the sign $\lambda$ is
negative. This gives a general bound (14) on the total cross section of the
photon-dark particle scattering, which implies that the dark sector cannot be
too dark.

We also provided more detailed analysis of dark photon scenarios. A general
finding is that the scattering of longitudinal modes of dark photons gives a
stronger constraint than that of transverse modes. In particular, dark photons
cannot be too light. It would be interesting to explore its possible
connection to the swampland argument on photon masses Reece (2019). Besides,
we observed that kinetic mixing with massive dark photons favors UV completion
by bi-charged vector bosons whose masses are bounded by (24), which is
reminiscent of gauge unification.

We stress that the bounds obtained under the assumption $M\gg m_{e}/e$ are
within the scope of ongoing dark photon searches. Therefore, a rather
technical issue about the scale $M$ is important not only theoretically, but
also from the phenomenological perspective. This motivates further studies on
gravitational positivity bounds in four dimensional spacetime, especially
about in which scattering processes the scale $M$ is at IR/UV.

###### Acknowledgements.

#### Acknowledgments.

We would like to thank Yoshihiko Abe, Katsuki Aoki, Kimihiro Nomura, Ryo
Saito, Satoshi Shirai and Masahito Yamazaki for useful discussion. T.N. is
supported in part by JSPS KAKENHI Grant No. 20H01902 and No. 22H01220, and
MEXT KAKENHI Grant No. 21H00075, No. 21H05184 and No. 21H05462. S.S. is
supported in part by JST SPRING Grant No. JPMJFS2126. J.T. is supported in
part by JSPS KAKENHI Grant No. 20J00912 and No. 21K13922.

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