#  On (Scalar QED) Gravitational Positivity Bounds

Yuta Hamada${}^{\diamond\star}\,$, Rinto Kuramochi${}^{\star}\,$, Gregory J.
Loges${}^{\diamond}\,$, Sota Nakajima${}^{\diamond}\,$
⋄Theory Center, IPNS, High Energy Accelerator Research Organization (KEK),
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
⋆Graduate University for Advanced Studies (Sokendai), 1-1 Oho, Tsukuba,
Ibaraki 305-0801, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study positivity bounds in the presence of gravity. We first review the
gravitational positivity bound at the tree-level, where it is known that a
certain amount of negativity is allowed for the coefficients of higher-
derivative operators. The size of these potentially negative contributions is
estimated for several tree-level, Reggeized gravitational amplitudes which are
unitary at high energies and feature the $t$-channel pole characteristic of
graviton exchange. We also argue for the form of the one-loop Regge amplitude
assuming that the branch cut structure associated with the exchange of the
graviton and higher-spin particles is reflected. We demonstrate how the one-
loop Regge amplitude appears by summing over Feynman diagrams. For our one-
loop amplitude proposal, the positivity bounds generically receive a finite
contribution from the Regge tower and do not lead to a parametrically small
bound on the cut-off scale of the low-energy EFT, consistent with recent
studies based on sum rules of the amplitude.

††preprint: KEK-TH-2492

##  1 Introduction

The space of low-energy EFTs is highly constrained by unitarity, locality, and
causality Adams:2006sv . Imposing these general requirements allows one to
bound, for example, the Wilson coefficients of higher-derivative operators and
rule out a whole swathe of theories as being inconsistent. For instance, the
optical theorem on the scattering amplitude in the forward limit leads to a
variety of positivity bounds on the Wilson coefficients. Recent developments
reveal that the infinite tower of higher-dimension operators must lie inside
the EFThedron Arkani-Hamed:2020blm ; Chiang:2021ziz ; Chiang:2022ltp .

The Swampland program Vafa:2005ui  (see Palti:2019 ; vanBeest:2021lhn ;
Grana:2021zvf ; Agmon:2022thq  for recent reviews) takes this one step further
by aiming to understand the nontrivial imprints that quantum gravity has on
low-energy physics. There are many conjectures such as the Weak Gravity
Conjecture Arkani-Hamed:2006emk , Distance Conjecture Ooguri:2006in , and so
forth. It is desirable to provide explanations for these conjectures.

It is natural to utilize the technique of the positivity bound to explain the
Swampland conjectures. This idea is particularly useful to demonstrate the
Weak Gravity Conjecture, which states the existence of the state where the
charge is larger than the mass.111In addition to the Weak Gravity Conjecture,
the S-matrix bootstrap has been used to obtain a lower bound on the
coefficient of the higher-derivative operators Guerrieri:2021ivu ;
Bern:2021ppb ; Bern:2022yes . The certain positivity condition on the four
derivative operators means that the Weak Gravity Conjecture is realized by the
nearly extremal black holes Kats:2006xp ; Cheung:2018cwt ; Hamada:2018dde ;
Bellazzini:2019xts ; Loges:2019jzs ; Goon:2019faz ; Jones:2019nev ;
Loges:2020trf ; Arkani-Hamed:2021ajd ; Cao:2022iqh . However, in contrast to
the non-gravitational case, it is difficult to obtain positivity bounds for
gravitational amplitudes. The main obstruction comes from the existence of the
massless graviton, which leads to the $t$-channel pole at $t=0$. Because of
this, one cannot directly take the forward limit, $t\to 0$, where the optical
theorem is applicable. Ref. Tokuda:2020mlf  bypasses this difficulty222Another
direction to study the gravitational positivity bound is to work with a fixed,
finite impact parameter Caron-Huot:2021rmr , which works for $d\geq 5$. by
assuming a Regge form for the amplitude at high energy, and shows that the
gravitational amplitude satisfies an approximate positivity bound.333See also
Ref. Hamada:2018dde  for the schematic idea of the approximate positivity
bound. The Regge amplitude means that, for $s\to\infty$ with fixed $t<0$, the
amplitude $\mathcal{A}$ satisfies
$\lim_{s\to\infty}\mathcal{A}/s^{2}=0$.444The argument based on the causality
Arkani-Hamed:2020blm  indicates
$\lim_{s\to\infty}\mathcal{A}/s^{2}=\text{(finite)}$, though it is hard to say
that the finite value is zero. The chaos bound Maldacena:2015waa ;
Chandorkar:2021viw  leads to the same conclusion. It is also argued that the
Regge boundedness for $d\geq 5$ follows from the reasonable assumptions in
Ref. Haring:2022cyf . This assumption is mainly motivated by the behavior of
the Virasoro-Shapiro or heterotic string amplitude in tree-level string
theory. Moreover, other amplitudes satisfying Regge boundedness and the IR
consistency conditions such as unitarity have been found Cheung:2022mkw ;
Geiser:2022exp .

Recently, the gravitational positivity bound at the one-loop level has been
used to speculate new Swampland conjectures. It was argued in Ref.
Alberte:2020bdz  that for scalar QED coupled to Einstein gravity, the
gravitational positivity bounds provide a nontrivial relationship between the
EFT cut-off $\Lambda$ and the scalar’s mass $m$ and charge $e$,

$\Lambda\lesssim\sqrt[\ ]{emM_{\text{Pl}}}\,,$ (1)

assuming that the massless graviton $t$-channel pole may be discarded.555The
bound is still valid after taking into account the effect of the $t$-channel
pole Tokuda:2020mlf ; Aoki:2021ckh  as long as we assume the Reggeized
amplitude motivated by the tree-level string theory. See Sec. 2.2 and Sec. 4.1
for the detail. Here $M_{\text{Pl}}$ is the reduced Planck mass. This scale is
much smaller than expected and is even parametrically smaller than that
provided by the magnetic Weak Gravity Conjecture Arkani-Hamed:2006emk ,

$\Lambda\lesssim eM_{\text{Pl}}\,,$ (2)

when the scalar is light, $m\ll eM_{\text{Pl}}$. A similar technique is used
to obtain the cut-off scale of the Standard Model Aoki:2021ckh  and dark
photon model Noumi:2022zht  as well as constraining the shape of the scalar
potential Noumi:2021uuv .

As stressed in Ref. Alberte:2020bdz , the bound (1) does not hold if the
graviton $t$-channel pole can not be discarded. This is roughly equivalent to
the question of whether we can neglect the contributions from the Regge states
or not. This is because, in addition to canceling the $t$-channel pole, the
Regge states can contribute a subleading $\mathcal{O}(t^{0}s^{2})$ term to the
amplitude which has a nontrivial effect on the positivity bounds in the low-
energy EFT Tokuda:2020mlf . Ref. Alberte:2021dnj  (and very recently
deRham:2022gfe  in the case of the graviton-graviton scattering) argues that
the contributions from the Regge states can not be neglected for the graviton-
photon scattering. Given an IR amplitude, sum rules indicate that the
amplitude from Regge states alters the argument which leads to Eq. (1).666See
also Ref. Herrero-Valea:2022lfd  for the study of the Regge amplitude
including the effect of the graviton loop. Ref. Tokuda:2020mlf  puts the bound
on the form of the Regge amplitude by using the finite energy sum rule and IR
part of null constraints. Moreover, in this paper, we provide a counterexample
of the supersymmetric version of Eq. (1) assuming the IR consistency and the
Regge boundedness of the amplitude. This is an independent argument that
graviton $t$ -channel pole can not be neglected in the case of photon-photon
amplitude.

If this is the case, then it is likely that the high energy Regge amplitude
contains the light mass parameter $m^{2}$ in the denominator of the
prefactor.777Another option is that the Regge tower is light, but we will
argue that this option is unlikely (at least for some cases) given our
counterexample. However, since it is expected that the Reggeization occurs due
to the effect of the heavy higher spin states, it is not obvious why the
amplitude with the light mass parameter appears. In this article, we
demonstrate that this can be understood by accounting for 1-loop effects on
the amplitude from $t$-channel exchange of Regge states.

We also study the properties of ‘deformed’ tree-level gravitational
amplitudes. A subleading $\mathcal{O}(t^{0}s^{2})$ term to the amplitude is
computed, which has an impact on the positivity bounds.

The remainder of this article is organized as follows. In Sec. 2 we review the
positivity bounds for $2$-to-$2$ photon scattering in non-gravitational
theories and how, when minimally coupled to Einstein gravity, the resulting
$t$-channel pole is canceled by Reggeization. In Sec. 3 we analyze the
properties of ‘deformed’ tree-level gravitational amplitudes which have been
recently proposed. These amplitudes satisfy the IR consistency conditions, and
we study the implications of these amplitudes on the positivity bound. In Sec.
4 we consider the subleading, 1-loop contributions to scalar QED and show that
the Regge states lead to a nonzero correction in the forward limit, $t\to 0$,
assuming that the branch cut structure associated with the exchange of the
graviton and higher spin particles is reflected in the Regge amplitude. We
conclude with a discussion in Sec. 5.

##  2 Positivity bounds at tree level

In this section we review how gravitational positivity bounds may be extracted
in the forward limit $t\to 0^{-}$ despite the presence of the $t$-channel pole
coming from graviton exchange. Reggeization of the gravitational amplitudes
provides a mechanism for canceling the $s^{2}/t$ term that appears in Einstein
gravity. After carefully subtracting such divergent terms, dispersive sum
rules allow one to derive positivity bounds on the remaining finite terms
which are calculable in a low-energy EFT.

We will focus on the fixed-$t$ scattering of two photons where all incoming
and outgoing particles have $+$ helicity. The amplitudes can be written in
terms of the usual Mandelstam variables $s,t,u$ which satisfy $s+t+u=0$. Later
in Sec. 4.3, for concreteness, we will consider scalar QED coupled to Einstein
gravity in 4d, for which the quadratic action is

$S=\int\mathrm{d}^{4}{x}\,\sqrt[\
]{-g}\left(\frac{M_{\text{Pl}}^{2}}{2}\,R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-D_{\mu}\phi^{\dagger}D^{\mu}\phi-m^{2}\phi^{\dagger}\phi\right)\,,$
(3)

where the mostly-plus notation is adopted. See Sec. A for more details on our
conventions and notation.

###  2.1 Review: (non-)gravitational positivity bounds

Let us begin by recalling the usual argument which allows one to derive
positivity bounds in EFTs, taking care of the difference between the non-
gravitational and gravitational amplitudes. Along the way we note the points
where the argument cannot immediately be adopted for theories coupled to
Einstein gravity.

In discussing dispersion relations we make the usual Regge-boundedness
assumption on the high-energy behavior of the amplitude:

$\lim_{\begin{subarray}{c}|s|\to\infty\\\ t<0\text{
fixed}\end{subarray}}\frac{\mathcal{A}(s,t)}{s^{2}}=0\,.$ (4)

For the non-gravitational case, this is guaranteed by the Froissart theorem
Froissart:1961ux ; Martin:1962rt  in a gapped theory. For the gravitational
case, this is of course violated by tree-level graviton exchanged, but
satisfied when the graviton is Reggeized, as we will consider below. We also
assume that $\mathcal{A}$ is crossing symmetric and holomorphic in $s$, except
for poles and cuts along the real axis.

$s$$\gamma$$\mathcal{C}_{\infty}$$\epsilon$$\mathcal{C}_{+}$$\mathcal{C}_{-}$
Figure 1: Analytic structure of $\mathcal{A}$ and the contour deformation of
$\gamma$ into $\mathcal{C}_{\infty}+\mathcal{C}_{+}+\mathcal{C}_{-}-\epsilon$.

Start with the identity

$\mathcal{A}(s,t)=s^{2}\oint_{\gamma}\frac{\mathrm{d}{s^{\prime}}}{2\pi
i}\,\frac{\mathcal{A}(s^{\prime},t)}{(s^{\prime}-s)s^{\prime 2}}\,,$ (5)

where $\gamma$ is any contour encircling $s$ within which $\mathcal{A}$ is
holomorphic. This integration contour can be deformed into a circle
$\mathcal{C}_{\infty}$ at $|s^{\prime}|\to\infty$, contours
$\mathcal{C}_{\pm}$ around any poles and branch cuts on the positive and
negative real axes, and a contour $\epsilon$ which encircles the pole at $s=0$
(see Fig. 1):

$\mathcal{A}(s,t)=s^{2}\left(\int_{\mathcal{C}_{\infty}}+\int_{\mathcal{C}_{+}}+\int_{\mathcal{C}_{-}}-\oint_{\epsilon}\right)\frac{\mathrm{d}{s^{\prime}}}{2\pi
i}\,\frac{\mathcal{A}(s^{\prime},t)}{(s^{\prime}-s)s^{\prime 2}}\,.$ (6)

The circle at infinity has vanishing contribution thanks to Eq. (4). In the
presence of gravity, $\mathcal{A}(s,t)$ may be expanded as

$\displaystyle\mathcal{A}(s,t)=\frac{a_{-1}(t)}{s}+a_{0}(t)+a_{1}(t)s+\cdots$
(7)

around the origin: note that the first term is absent without gravity. By
substituting this into Eq. (6), the pole at the origin gives rise to

$\displaystyle-s^{2}\oint_{\epsilon}\frac{\mathrm{d}{s^{\prime}}}{2\pi
i}\,\frac{\mathcal{A}(s^{\prime},t)}{(s^{\prime}-s)s^{\prime
2}}=\frac{a_{-1}(t)}{s}+a_{0}(t)+a_{1}(t)s\,.$ (8)

This means that the scattering amplitude $\mathcal{A}$ can be written as888See
Ref. Sinha:2020win  for manifestly crossing symmetric dispersion relations.

$\mathcal{A}(s,t,u)=\frac{a_{-1}}{s}+a_{0}+a_{1}s+\frac{s^{2}}{\pi}\int_{0}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}(s^{\prime},t,u^{\prime})}{s^{\prime
2}(s^{\prime}-s)}+\frac{u^{2}}{\pi}\int_{0}^{\infty}\mathrm{d}{u^{\prime}}\,\frac{\operatorname{Disc}_{u}\mathcal{A}(s^{\prime},t,u^{\prime})}{u^{\prime
2}(u^{\prime}-u)}\,,$ (9)

where the discontinuities $\operatorname{Disc}_{s}$ and
$\operatorname{Disc}_{u}$ on the real axis are defined by999The discontinuity
is related to the imaginary part by
$\operatorname{Disc}_{s}\mathcal{A}(s,t,u)=2i\operatorname{Im}\mathcal{A}(s+i\epsilon,t,u)$.

$\displaystyle 2i\operatorname{Disc}_{s}\mathcal{A}(s,t,u)$
$\displaystyle:=\mathcal{A}(s+i\epsilon,t,u)-\mathcal{A}(s-i\epsilon,t,u)\,,$
(10) $\displaystyle 2i\operatorname{Disc}_{u}\mathcal{A}(s,t,u)$
$\displaystyle:=\mathcal{A}(s,t,u+i\epsilon)-\mathcal{A}(s,t,u-i\epsilon)\,.$

By comparing $s^{2}$ terms around $s=0$ in (9), we obtain

$\partial_{s}^{2}\tilde{\mathcal{A}}(0,t,-t)=\frac{1}{\pi}\int_{0}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}(s^{\prime},t,u^{\prime})}{s^{\prime
3}}+\frac{1}{\pi}\int_{0}^{\infty}\mathrm{d}{u^{\prime}}\,\frac{\operatorname{Disc}_{u}\mathcal{A}(s^{\prime},t,u^{\prime})}{(u^{\prime}+t)^{3}}$
(11)

where we have defined

$\tilde{\mathcal{A}}:=\mathcal{A}-\frac{a_{-1}}{s}.$ (12)

This is an amplitude where the graviton $s$-channel pole is subtracted.

In the forward limit, the right-hand side of Eq. (11) must be non-negative by
unitarity. This immediately gives the following positivity bounds,

$\partial_{s}^{2}\mathcal{A}(0,0,0)\geq 0\,,$ (13)

which directly constrain the space of allowed EFTs. We can obtain stronger
bounds by extracting the portion of the integrals along the cuts which are
calculable at low energies. In order to do this, introduce $\Lambda$ as the
EFT cut-off and write

$\displaystyle\operatorname{Disc}_{s}\mathcal{A}(s^{\prime},t,u^{\prime})$
$\displaystyle=\operatorname{Disc}_{s}\mathcal{A}_{\text{IR}}(s^{\prime},t,u^{\prime})\,\theta(\Lambda^{2}-s)+\operatorname{Disc}_{s}\mathcal{A}_{\text{UV}}(s^{\prime},t,u^{\prime})\,\theta(s-\Lambda^{2})\,,$
(14)
$\displaystyle\operatorname{Disc}_{u}\mathcal{A}(s^{\prime},t,u^{\prime})$
$\displaystyle=\operatorname{Disc}_{u}\mathcal{A}_{\text{IR}}(s^{\prime},t,u^{\prime})\,\theta(\Lambda^{2}-s)+\operatorname{Disc}_{u}\mathcal{A}_{\text{UV}}(s^{\prime},t,u^{\prime})\,\theta(s-\Lambda^{2})\,.$

Eq. (11) now becomes

$\displaystyle\partial_{s}^{2}\tilde{\mathcal{A}}(0,t,-t)-\frac{2}{\pi}\int_{0}^{\Lambda^{2}}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}_{\text{IR}}(s^{\prime},t,u^{\prime})}{s^{\prime
3}}-\frac{2}{\pi}\int_{0}^{\Lambda^{2}}\mathrm{d}{u^{\prime}}\,\frac{\operatorname{Disc}_{u}\mathcal{A}_{\text{IR}}(s^{\prime},t,u^{\prime})}{(u^{\prime}+t)^{3}}$
(15)
$\displaystyle=\frac{2}{\pi}\int_{\Lambda^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}_{\text{UV}}(s^{\prime},t,u^{\prime})}{s^{\prime
3}}+\frac{2}{\pi}\int_{\Lambda^{2}}^{\infty}\mathrm{d}{u^{\prime}}\,\frac{\operatorname{Disc}_{u}\mathcal{A}_{\text{UV}}(s^{\prime},t,u^{\prime})}{(u^{\prime}+t)^{3}}\,,$

where again the right-hand side is non-negative by unitarity in the forward
limit. Without gravity there is no obstacle to taking $t\to 0$, in which case
the improved positivity bound reads Bellazzini:2016xrt ; deRham:2017imi ;
deRham:2017xox

$\partial_{s}^{2}\tilde{\mathcal{A}}(0,0,0)-\frac{2}{\pi}\int_{0}^{\Lambda^{2}}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}_{\text{IR}}(s^{\prime},0,u^{\prime})}{s^{\prime
3}}-\frac{2}{\pi}\int_{0}^{\Lambda^{2}}\mathrm{d}{u^{\prime}}\,\frac{\operatorname{Disc}_{u}\mathcal{A}_{\text{IR}}(s^{\prime},0,u^{\prime})}{u^{\prime
3}}\geq 0\,.$ (16)

By design everything here depends only on low-energy data and constrains the
space of allowed EFTs, e.g. through constraining the Wilson coefficients of
higher-derivative operators. One possibility is that the above equation
provides an upper bound on $\Lambda$, the scale where new physics must appear
to maintain unitarity.

###  2.2 Cancellation of the $\mathbf{t}$-channel pole in gravitational
amplitudes

For gravitational systems the argument which leads to Eq. (16) needs re-
evaluating; more care is needed when graviton exchange is included because of
the contribution

$\mathcal{A}_{\text{grav}}^{\text{tree}}\sim-\frac{s^{2}}{M_{\text{Pl}}^{2}t}$
(17)

which renders $\partial_{s}^{2}\tilde{\mathcal{A}}(0,0,0)$ ill-defined. Of
course this means that the integrals on the right-hand side of (11) must
diverge for $t\to 0^{-}$ as well. In the literatures, there are two ways to
deal with the graviton $t$-pole:

  1. 1.

Subtracting the graviton $t$-pole assuming that the amplitude is Reggeized at
the high energy Tokuda:2020mlf .

  2. 2.

Studying the gravitational positivity bound with a finite impact parameter
Caron-Huot:2021rmr ; Caron-Huot:2022ugt ; Caron-Huot:2022jli .

We focus on the first approach in this paper,101010As for the second approach,
the bound on the Wilson coefficients is obtained up to an infrared logarithm
Caron-Huot:2022ugt . The implications on the black hole weak gravity
conjecture are investigated in Ref. Henriksson:2022oeu . which was fleshed out
in Ref. Tokuda:2020mlf ; we assume that at energies well above the scale
$M_{\ast}$ the amplitude is Reggeized,

$\lim_{\begin{subarray}{c}|s|\gg M_{\ast}^{2}\\\ t<0\text{
fixed}\end{subarray}}\operatorname{Disc}_{s}\mathcal{A}(s,t)=f(t)\,\frac{s^{2+\alpha^{\prime}t}}{M_{\text{Pl}}^{2}M_{\ast}^{2\alpha^{\prime}t}}\approx
f_{\text{tree}}(t)\,\frac{s^{2+\alpha^{\prime}t}}{M_{\text{Pl}}^{2}M_{\ast}^{2\alpha^{\prime}t}}=:\operatorname{Disc}_{s}\mathcal{A}_{\text{Regge}}^{\text{tree}}(s,t)\,,$
(18)

where $\alpha^{\prime}$ provides the slope of the Regge trajectory, $f(t)$ is
a function of $t$ whcih has mass dimension $-2$. Here we have made the tree-
level approximation of a UV theory, $f\approx f_{\text{tree}}$. This is what
happens in string theory, for example, with $M_{\ast}^{2}\approx
M_{\text{s}}^{2}=\alpha^{\prime-1}$. Substituting this limiting expression
into the dispersive integral gives

$\int_{M_{\ast}^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}(s^{\prime},t)}{s^{\prime
3}}\approx\int_{M_{\ast}^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{u}\mathcal{A}(u^{\prime},t)}{(u^{\prime}+t)^{3}}\approx-\frac{f_{\text{tree}}(t)}{M_{\text{Pl}}^{2}\alpha^{\prime}t}=-\frac{f_{\text{tree}}(0)}{M_{\text{Pl}}^{2}\alpha^{\prime}t}-\frac{f_{\text{tree}}^{\prime}(0)}{M_{\text{Pl}}^{2}\alpha^{\prime}}+\cdots\,.$
(19)

As long as $f_{\text{tree}}(0)\sim\alpha^{\prime}$ is chosen appropriately,
this divergent UV contribution to the dispersion relation cancels the
$t$-channel pole in the IR. After the cancellation, the Regge contribution to
$\operatorname{Disc}\mathcal{A}$ need not be positive. For example, since at
tree-level in string theory the function $f_{\text{tree}}(t)$ only depends on
the dimensionful parameter $\alpha^{\prime}$, a certain amount of negativity
is allowed:

$\displaystyle\int_{M_{\ast}^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}(s^{\prime},t)}{s^{\prime
3}}\approx-\frac{1}{M_{\text{Pl}}^{2}t}-\mathcal{O}(1)\frac{\alpha^{\prime}}{M_{\text{Pl}}^{2}}+\mathcal{O}(t)\,.$
(20)

The size of this negative $\mathcal{O}(t^{0})$ term is crucial in discussing
the possibility that (nearly) extremal black holes satisfy the Weak Gravity
Conjecture Arkani-Hamed:2006emk ; Hamada:2018dde ; Loges:2019jzs ;
Loges:2020trf ; Arkani-Hamed:2021ajd .

By substituting the tree-level Regge amplitude into the dispersion relation,
we obtain

$\displaystyle\\!\\!\\!\partial_{s}^{2}\Delta\mathcal{A}(0,t,-t)-\frac{2}{\pi}\int_{0}^{\Lambda^{2}}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}_{\text{IR}}(s^{\prime},t,u^{\prime})}{s^{\prime
3}}-\frac{2}{\pi}\int_{0}^{\Lambda^{2}}\mathrm{d}{u^{\prime}}\,\frac{\operatorname{Disc}_{u}\mathcal{A}_{\text{IR}}(s^{\prime},t,u^{\prime})}{(u^{\prime}+t)^{3}}$
(21)
$\displaystyle=\frac{2}{\pi}\left(\int_{\Lambda^{2}}^{M_{*}^{2}}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}_{\text{UV}}(s^{\prime},t,u^{\prime})}{s^{\prime
3}}+\int_{\Lambda^{2}}^{M_{*}^{2}}\mathrm{d}{u^{\prime}}\,\frac{\operatorname{Disc}_{u}\mathcal{A}_{\text{UV}}(s^{\prime},t,u^{\prime})}{(u^{\prime}+t)^{3}}\right)$
$\displaystyle+\frac{2}{\pi}\left(\int^{\infty}_{M_{*}^{2}}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\Delta\mathcal{A}_{\text{UV}}(s^{\prime},t,u^{\prime})}{s^{\prime
3}}+\int^{\infty}_{M_{*}^{2}}\mathrm{d}{u^{\prime}}\,\frac{\operatorname{Disc}_{u}\Delta\mathcal{A}_{\text{UV}}(s^{\prime},t,u^{\prime})}{(u^{\prime}+t)^{3}}\right)-\frac{2f_{\text{tree}}^{\prime}(t)}{M_{\text{Pl}}^{2}\alpha^{\prime}},$

where

$\Delta\mathcal{A}:=\tilde{\mathcal{A}}-\mathcal{A}_{\text{grav}}^{\text{tree}}\,,\qquad\Delta\mathcal{A}_{UV}:=\mathcal{A}_{UV}-\mathcal{A}_{\text{Regge}}^{\text{tree}}\,.$
(22)

Everything is now manifestly finite for $t\to 0^{-}$. In addition, the second
line of Eq. (21) is positive in this limit by unitarity and in the third line
the terms in parentheses are negligible. The final term is Planck-suppressed,
but is negative. In cases where
$f_{\text{tree}}(t)=\alpha^{\prime}f_{\text{tree}}(\alpha^{\prime}t)$, the
last term can be estimated as

$\displaystyle-\frac{2f_{\text{tree}}^{\prime}(0)}{M_{\text{Pl}}^{2}\alpha^{\prime}}=-\mathcal{O}\left(\frac{\alpha^{\prime}}{M_{\text{Pl}}^{2}}\right)\,$
(23)

for $t\to 0$. The assumption here indicates that only the string scale appears
in $s\to\infty$, which looks reasonable, and this is indeed the case for the
Veneziano-Shapiro amplitude in string theory. To summarize, the approximate
gravitational positivity bound at tree level is

$\displaystyle\partial_{s}^{2}\Delta\mathcal{A}(0,0,0)-\frac{2}{\pi}\int_{0}^{\Lambda^{2}}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}_{\text{IR}}(s^{\prime},0,u^{\prime})}{s^{\prime
3}}-\frac{2}{\pi}\int_{0}^{\Lambda^{2}}\mathrm{d}{u^{\prime}}\,\frac{\operatorname{Disc}_{u}\mathcal{A}_{\text{IR}}(s^{\prime},0,u^{\prime})}{u^{\prime
3}}\geq-\mathcal{O}\left(\frac{\alpha^{\prime}}{M_{\text{Pl}}^{2}}\right).$
(24)

In Sec. 3 we will see two examples of Reggeized tree-level amplitudes for
which $f_{\text{tree}}^{\prime}(0)$ is calculable and allows for the slight
negativity argued for above. More generally, we argue in Sec. 4 that loop
effects will generate such a term.

##  3 Modified amplitudes

Before discussing one-loop amplitudes, we discuss two explicit examples of
Reggeized tree-level amplitudes where the negative $\mathcal{O}(t^{0}s^{2})$
contribution appears from the Regge states. Explicitly, we discuss the
deformed Virasoro-Shapiro amplitude identified recently in Arkani-
Hamed:2020blm  and the Coon amplitude Coon:1969yw ; Figueroa:2022onw ;
Geiser:2022exp ; Geiser:2022icl ; Chakravarty:2022vrp ; Maldacena:2022ckr ,
which is a $q$-deformation of the Veneziano amplitude (recently generalized in
Cheung:2022mkw ).

###  3.1 Virasoro-Shapiro amplitude

As a typical example of Regge-behaved amplitudes in the asymptotic limit, let
us consider the Virasoro-Shapiro amplitude which is identified with a 2-to-2
scattering of massless (gauge neutral) bosons in closed string theory. In Ref.
Tokuda:2020mlf , this amplitude has been studied in order to check the
cancellation of the $t$-channel graviton exchange and the violation of strict
positivity. In this subsection, we will discuss how the Virasoro-Shapiro
amplitude gives the $\mathcal{O}(t^{-1})$ term, but a $\mathcal{O}(t^{0})$
term does not appear (see Ref. Tokuda:2020mlf  for more detail). Next, we will
show that the deformation of the Virasoro-Shapiro amplitude considered in Ref.
Arkani-Hamed:2020blm , which maintains the power law for $s$ in the Regge
limit, leads to a negative $\mathcal{O}(t^{0})$ term.

The Virasoro-Shapiro amplitude is given by the following form:

$\displaystyle\mathcal{A}(s,t)=-K(s,t)\left.\frac{\Gamma(-\alpha(s))\Gamma(-\alpha(t))\Gamma(-\alpha(u))}{\Gamma(1+\alpha(s))\Gamma(1+\alpha(t))\Gamma(1+\alpha(u))}\right|_{u=-s-t}\,,$
(25)

where $K(s,t)$ is a kinematic factor and $\alpha(x)=\alpha^{\prime}x/4$. For
simplicity, we restrict our attention to the following form of $K(s,t)$:

$\displaystyle K(s,t)=P\left(s^{2}t^{2}+t^{2}u^{2}+s^{2}u^{2}\right)\,,$ (26)

where $P$ is a positive constant. By using the Stirling formula,
$\Gamma(z)\sim\sqrt[\ ]{2\pi}\,z^{z-\frac{1}{2}}e^{-z}$, one can easily see
that the asymptotic behavior of $\mathcal{A}(s,t)$ is

$\displaystyle\mathcal{A}(s,t)\sim
F_{\mathrm{VS}}(t)\alpha(s)^{2\alpha(t)+2}\,,$ (27)

where

$\displaystyle
F_{\mathrm{VS}}(t)=P\left(\frac{4}{\alpha^{\prime}}\right)^{4}e^{\pi
i\alpha(t)}\frac{\Gamma(-\alpha(t))}{\Gamma(1+\alpha(t))}\,.$ (28)

Note that $F_{\mathrm{VS}}(t)$ has poles on the real axis at $\alpha(t)=n$
with $n\geq 0$. The residue in the $t$-plane is

$\displaystyle\operatorname*{Res}_{\alpha(t)=n}\mathcal{A}(s,t)$
$\displaystyle=-K\left(s,\frac{4n}{\alpha^{\prime}}\right)\left(\frac{\Gamma(n+\alpha(s))}{n!\Gamma(1+\alpha(s))}\right)^{2}=-K\left(s,\frac{4n}{\alpha^{\prime}}\right)\frac{1}{\left(n!\right)^{2}}\prod_{j=1}^{n-1}\left(\alpha(s)+j\right)^{2}\,.$
(29)

It is clear that the residue at $\alpha(t)=n$ is a polynomial in $s$ of order
$2n+2$, and, in particular at $t=0$, the residue is

$\displaystyle-\frac{64Ps^{2}}{\alpha^{\prime 3}}\,.$ (30)

The pole expansion in the $t$\- and $u$-channel is thus given by

$\displaystyle\mathcal{A}(s,t)=-\frac{4}{\alpha^{\prime}}\sum_{n=0}^{\infty}\left(\frac{\left(\alpha(s)+1\right)_{n-1}}{n!}\right)^{2}\left[\frac{K\left(s,4n\alpha^{\prime-1}\right)}{t-4n\alpha^{\prime-1}}-\frac{K\left(s,-s-4n\alpha^{\prime-1}\right)}{t+s+4n\alpha^{\prime-1}}\right]\,,$
(31)

where we have used the Pochhammer symbol defined as

$\displaystyle(x)_{m}:=\frac{\Gamma(x+m)}{\Gamma(x)}=\prod_{j=0}^{m-1}(x+j)\,.$
(32)

For $s\to\pm i\infty$, in particular, the leading contribution from
$t$-channel poles is

$\displaystyle\mathcal{A}(s,t)\sim-P\left(\frac{4}{\alpha^{\prime}}\right)^{5}\sum_{n=0}^{\infty}\frac{1}{(n!)^{2}}\frac{\alpha(s)^{2n+2}}{t-4n\alpha^{\prime-1}}\,.$
(33)

Note that this expression as an infinite sum coincides with the Regge behavior
(27) with $t$-channel poles expanded since $F_{\mathrm{VS}}(t)$ can be
expressed as

$\displaystyle
F_{\mathrm{VS}}(t)=-P\left(\frac{4}{\alpha^{\prime}}\right)^{5}\sum_{n=0}^{\infty}\frac{1}{(n!)^{2}}\frac{1}{t-4n\alpha^{\prime-1}}\,.$
(34)

By using (29) with $s\leftrightarrow t$ crossing symmetry, we find

$\displaystyle\operatorname{Disc}_{s}\mathcal{A}(s,t)$
$\displaystyle=-\sum_{n=0}^{\infty}\delta(\alpha(s)-n)\operatorname*{Res}_{\alpha(s)=n}\mathcal{A}(s,t)$
(35)
$\displaystyle=\frac{4}{\alpha^{\prime}}\sum_{n=0}^{\infty}K\left(\frac{4n}{\alpha^{\prime}},t\right)\left(\frac{\left(\alpha(t)+1\right)_{n-1}}{n!}\right)^{2}\delta(s-4n\alpha^{\prime-1})\,,$

and then the integral that we would like to evaluate is

$\displaystyle\int_{\Lambda^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}(s^{\prime},t)}{s^{\prime
3}}$ $\displaystyle=\frac{16P}{\alpha^{\prime
2}}\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{\left(\alpha(t)+1\right)_{n-1}}{(n-1)!}\right)^{2}+\mathcal{O}(t)$
(36) $\displaystyle=-\frac{32P}{\alpha^{\prime 3}t}+\mathcal{O}(t)\,,$

where we assume $1/\alpha^{\prime}>\Lambda^{2}$. The first term on the right-
hand side must ensure the cancellation of the graviton $t$-channel pole, and
hence we find

$\displaystyle\frac{32P}{\alpha^{\prime 3}}\sim M_{\text{Pl}}^{-2}\,.$ (37)

Note that we do not obtain any $\mathcal{O}(t^{0})$ contributions from the
Virasoro-Shapiro amplitude (25).

Let us next consider the following deformation:

$\displaystyle\mathcal{A}(s,t)\to\mathcal{A}(s,t)+\delta\mathcal{A}(s,t)\,,$
(38)

where $\delta\mathcal{A}$ is defined as

$\displaystyle\delta\mathcal{A}(s,t)=-\epsilon
K(s,t)\frac{\Gamma(1-\alpha(s))\Gamma(1-\alpha(t)\Gamma(1-\alpha(u))}{\Gamma(2+\alpha(s))\Gamma(2+\alpha(t))\Gamma(2+\alpha(u))}\,.$
(39)

The positivity of residues requires that the deformation parameter should be
bounded as $0\leq\epsilon\leq 1$ Arkani-Hamed:2020blm . Note that
$\delta\mathcal{A}$ can be expressed in terms of $\mathcal{A}$ as follows:

$\displaystyle\delta\mathcal{A}(s,t)=\epsilon\frac{\alpha(s)\alpha(t)\alpha(u)}{(1+\alpha(s))(1+\alpha(t))(1+\alpha(u))}\mathcal{A}(s,t)\,.$
(40)

It is hence straightforward to see the Regge behavior of
$\delta\mathcal{A}(s,t)$ is

$\displaystyle\delta\mathcal{A}(s,t)\sim\tilde{F}_{\mathrm{VS}}(t)\alpha(s)^{2\alpha(t)+2}\,,$
(41)

where

$\displaystyle\tilde{F}_{\mathrm{VS}}(t)=\epsilon\frac{\alpha(t)}{1+\alpha(t)}F_{\mathrm{VS}}(t)=\epsilon
P\left(\frac{4}{\alpha^{\prime}}\right)^{4}e^{\pi
i\alpha(t)}\frac{\Gamma(1-\alpha(t))}{\Gamma(2+\alpha(t))}\,.$ (42)

Note that $\delta\mathcal{A}$ has the same soft behavior as $\mathcal{A}$ in
the high energy region with $t<0$, but the first $t$-channel pole appears at
$\alpha(t)=1$ rather than at $t=0$, which corresponds to the exchange of a
massive particle with spin $4$. The residue of $\delta\mathcal{A}$ is

$\displaystyle\operatorname*{Res}_{\alpha(s)=n}\delta\mathcal{A}(s,t)=K\left(\frac{4n}{\alpha^{\prime}},t\right)\frac{\left(\alpha(t)\right)_{n-1}\left(\alpha(t)+2\right)_{n-1}}{(n-1)!(n+1)!}\,,$
(43)

and the integral is evaluated as

$\displaystyle\int_{\Lambda^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\delta\mathcal{A}(s^{\prime},t)}{s^{\prime
3}}$ $\displaystyle=-\frac{16\epsilon P}{\alpha^{\prime
2}}\sum_{n=0}^{\infty}\left(n+1\right)\frac{\left(\alpha(t)\right)_{n}\left(\alpha(t)+2\right)_{n}}{n!(n+2)!}+\mathcal{O}(t)$
(44) $\displaystyle=-\frac{8\epsilon P}{\alpha^{\prime
2}}{}_{2}F_{3}\left(2,\alpha(t),\alpha(t)+2;1,3;1\right)+\mathcal{O}(t)$
$\displaystyle=-\frac{8\epsilon P}{\alpha^{\prime 2}}+\mathcal{O}(t)\,.$

We can thus obtain the $\mathcal{O}(t^{0})$ negative term by deforming the
Virasoro-Shapiro amplitude by $\delta\mathcal{A}$ defined in (39). One can
also obtain the same result by using (36) and (41):

$\displaystyle\int_{\Lambda^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}(s^{\prime},t)}{s^{\prime
3}}$
$\displaystyle\sim\epsilon\frac{\alpha(t)}{1+\alpha(t)}\int_{\Lambda^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}(s^{\prime},t)}{s^{\prime
3}}$ (45) $\displaystyle=-\frac{8\epsilon P}{\alpha^{\prime
2}}+\mathcal{O}(t)\,.$

From (37) and the bound on $\epsilon$, we find the bound on this negative
contribution:

$\displaystyle\frac{8\epsilon P}{\alpha^{\prime
2}}\lesssim\frac{\alpha^{\prime}}{M_{\text{Pl}}^{2}}\,.$ (46)

This bound on the negativity is consistent with the recent paper Noumi:2022wwf
, in which constraints on Regge amplitudes are considered by using the finite-
energy sum rules and IR part of the null constraints.

###  3.2 Coon amplitude

The Coon amplitude Coon:1969yw  is a $q$-deformation of the Veneziano
amplitude which enjoys many of the same UV properties Figueroa:2022onw ;
Geiser:2022exp ; Geiser:2022icl ; Chakravarty:2022vrp  but has some unusual
features, most notably a spectrum with nonlinear Regge trajectories and an
accumulation point below which there are infinitely many states. Despite some
similarities with open string scattering in AdS Maldacena:2022ckr , to date
the Coon amplitude has no known worldsheet origin. Indeed, the unitarity of
the Coon amplitude has recently been called into question Jepsen:2023sia .
Nevertheless, we take it as a well-studied amplitude and use it as a building
block in a crossing-symmetric amplitude featuring both Regge behavior and a
$t$-channel pole characteristic of graviton exchange.

The Coon amplitude has the following product representation,111111The
Veneziano amplitude is recovered in the limit $q\to 1^{-}$.

$\mathcal{A}_{q}(s,t)=g^{2}(1-q)\exp\left(\frac{\log{\sigma}\log{\tau}}{\log{q}}\right)\prod_{n=0}^{\infty}\frac{(\sigma\tau-q^{n})(1-q^{n+1})}{(\sigma-q^{n})(\tau-q^{n})}\,,$
(47)

where $g$ is some coupling constant, $q\in(0,1)$ and $\sigma,\tau$ are related
to the usual Mandelstam variables as

$\sigma=1+(q-1)\Big{(}\frac{s}{\mu^{2}}-\delta\Big{)}\,,\qquad\tau=1+(q-1)\Big{(}\frac{t}{\mu^{2}}-\delta\Big{)}\,.$
(48)

The spectrum is quickly identified to be

$m_{n}^{2}=\mu^{2}(\delta+[n]_{q})\,,\qquad[n]_{q}:=\frac{1-q^{n}}{1-q}\,,$
(49)

with an accumulation point for large $n$:

$m_{n}^{2}\quad\xrightarrow{n\to\infty}\quad
m_{\ast}^{2}:=\mu^{2}\left(\delta+\frac{1}{1-q}\right)\,.$ (50)

There is a cut from the $\log{\sigma}$ factor which extends from the branch
point at $s=m_{\ast}^{2}$ out to infinity: see Fig. 2 for a sketch. Going
forward we take $\delta=0$ so that the spectrum contains massless states.

$\bm{\times}$$\bm{\times}$$\bm{\times}$$\bm{\times}$$\bm{\times}$$\bm{\times}$$\bm{\times}$$\bm{\times}$$\bm{\times}$$\bm{\times}$$\bm{\times}$$\bm{\times}$$\bm{\times}$$\mu^{2}$$m_{\ast}^{2}$
Figure 2: Schematic spectrum for the Coon amplitude with $\delta=0$. Massive
states begin at $m_{1}^{2}=\mu^{2}$ and have an accumulation point at
$m_{\ast}^{2}=\frac{\mu^{2}}{1-q}$ where a branch cut begins.

At low energies, $s,t,u\ll\mu^{2}$, one finds

$\mathcal{A}_{q}(s,t)\approx-g^{2}\mu^{2}\left(\frac{1}{s}+\frac{1}{t}\right)\,,$
(51)

so that the crossing-symmetric combination

$\tilde{\mathcal{A}}_{q}(s,t)=(s^{2}+t^{2}+u^{2})\big{[}\mathcal{A}_{q}(s,t)+\mathcal{A}_{q}(t,u)+\mathcal{A}_{q}(u,s)\big{]}$
(52)

at low energies features the $t$-channel pole characteristic of tree-level
graviton exchange:

$\tilde{\mathcal{A}}_{q}(s,t)\approx
g^{2}\mu^{2}\,\frac{(s^{2}+t^{2}+u^{2})^{2}}{stu}\quad\xrightarrow{t\to
0}\quad-4g^{2}\mu^{2}\,\frac{s^{2}}{t}\,.$ (53)

In the Regge limit the Coon amplitude takes the form

$\mathcal{A}_{q}(s,t)\quad\longrightarrow\quad
g^{2}(1-q)\sigma^{\frac{\log{\tau}}{\log{q}}}\prod_{n=0}^{\infty}\frac{1-q^{n+1}}{1-\frac{q^{n}}{\tau}}\approx
f_{q}(t)\left(-\frac{s}{m_{\ast}^{2}}\right)^{\frac{\log{\tau}}{\log{q}}}$
(54)

and so

$\tilde{\mathcal{A}}_{q}(s,t)\quad\longrightarrow\quad\tilde{f}_{q}(t)\left(-\frac{s}{m_{\ast}^{2}}\right)^{2+\frac{\log{\tau}}{\log{q}}}\,,$
(55)

which marginally satisfies Eq. (4) for $t<0$ ($\tau>1$) since $\log{q}$ is
negative. The contour integral around the positive real-$s$ axis splits into a
sum over poles and an integral along the branch cut:

$\int_{\mu^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}_{q}(s^{\prime},t)}{s^{\prime
3}}=-\pi\sum_{k=1}^{\infty}\frac{1}{m_{k}^{6}}\operatorname*{Res}_{s=m_{k}^{2}}\mathcal{A}_{q}(s,t)+\int_{m_{\ast}^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}_{q}(s^{\prime},t)}{s^{\prime
3}}\,.$ (56)

The residues are polynomial in $t$,

$\operatorname*{Res}_{s=m_{k}^{2}}\mathcal{A}_{q}(s,t)=g^{2}\mu^{2}q^{k}\prod_{n=0}^{k-1}\frac{\tau-q^{n-k}}{1-q^{n-k}}\quad\xrightarrow{t\to
0}\quad g^{2}\mu^{2}q^{k}\,,$ (57)

and the sum over $k$ gives a finite contribution for $t\to 0$. In contrast,
the integral over the branch cut diverges for $t\to 0$ and cancels the
$t$-channel pole. The range $s>m_{\ast}^{2}$ corresponds to $\sigma<0$, so we
have

$\displaystyle\operatorname{Disc}_{s}\mathcal{A}_{q}(s,t)$
$\displaystyle=-g^{2}(1-q)(-\sigma)^{\frac{\log\tau}{\log
q}}\sin{\left(\pi\,\frac{\log\tau}{\log
q}\right)}\prod_{n=0}^{\infty}\frac{(\sigma\tau-q^{n})(1-q^{n+1})}{(\sigma-q^{n})(\tau-q^{n})}$
$\displaystyle=g^{2}(1-q)(-\sigma)^{\frac{\log\tau}{\log
q}}\sin{\left(\pi\,\frac{\log\tau}{\log
q}\right)}\frac{(\sigma\tau-1)}{(\sigma-1)\frac{t}{\mu^{2}}}\prod_{n=1}^{\infty}\frac{(\sigma\tau-q^{n})(1-q^{n+1})}{(\sigma-q^{n})(\tau-q^{n})}$
$\displaystyle=-\pi g^{2}(-\sigma)^{-\frac{1-q}{\log
q}\frac{t}{\mu^{2}}}\frac{1-q}{\log q}\big{[}1+\mathcal{O}(t)\big{]}$ (58)

and thus

$\frac{\operatorname{Disc}_{s}\tilde{\mathcal{A}}_{q}(s^{\prime},t)}{(s^{\prime})^{3}}\approx-2\pi
g^{2}\mu^{2}\left(\frac{s^{\prime}}{\mu^{2}}\right)^{-1-\frac{1-q}{\log{q}}\frac{t}{\mu^{2}}}\frac{1-q}{\log
q}\big{[}1+\mathcal{O}(t)\big{]}$ (59)

gives

$\displaystyle\int_{m_{\ast}^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\tilde{\mathcal{A}}_{q}(s^{\prime},t)}{(s^{\prime})^{3}}$
$\displaystyle\approx-2\pi g^{2}\mu^{-2}\frac{1-q}{\log
q}\big{[}1+\mathcal{O}(t)\big{]}\int_{m_{\ast}^{2}}^{\infty}\mathrm{d}{s^{\prime}}\,\left(\frac{s^{\prime}}{\mu^{2}}\right)^{-1-\frac{1-q}{\log{q}}\frac{t}{\mu^{2}}}$
(60) $\displaystyle=-\frac{2\pi
g^{2}\mu^{2}}{t}\big{[}1+\mathcal{O}(t)\big{]}$

This leading $1/t$ behavior exactly cancels the $t$-channel pole of Eq. (53)
in the dispersion relations. Here the details of the cancellation are somewhat
different, however, since the sum over poles does not generate a $1/t$
contribution despite there being an infinite number of states. Knowing the
contributions from poles and the branch cut must exactly compensate for the
massless state, the $\mathcal{O}(t^{0})$ can be shown to be nonzero,

$4g^{2}\mu^{2}\left(-\frac{1}{t}+\frac{1-q}{\mu^{2}}+\mathcal{O}(t)\right)\,,$
(61)

providing another example where a Reggeized amplitude leaves a finite
contribution after canceling the $1/t$ divergence. We emphasize, however, that
although $\tilde{\mathcal{A}}_{q}(s,t)$ as introduced in Eq. (52) has the
features we argue for generally, it is not known if this amplitude arises in a
fully-consistent gravitational theory.

##  4 Positivity bounds at one loop

In this section, we first see how the positivity bound at the one-loop level
leads to an apparent puzzle that the cut-off scale of the theory is extremely
low, under the assumption that the graviton $t$-channel pole is neglected.
Then, we propose how the Reggeization extends to one-loop level.

###  4.1 Small cut-off scale

Suppose that the approximate gravitational positivity bound (24) at tree-level
is correct even at the one-loop level. Then, Ref. Alberte:2020bdz  found that
the left-hand side of (24) is computed as

$\displaystyle\partial_{s}^{2}\Delta\mathcal{A}(0,0,0)-\frac{2}{\pi}\int_{0}^{\Lambda^{2}}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}_{s}\mathcal{A}_{\text{IR}}(s^{\prime},0,u^{\prime})}{s^{\prime
3}}-\frac{2}{\pi}\int_{0}^{\Lambda^{2}}\mathrm{d}{u^{\prime}}\,\frac{\operatorname{Disc}_{u}\mathcal{A}_{\text{IR}}(s^{\prime},0,u^{\prime})}{u^{\prime
3}}$
$\displaystyle\approx\frac{e^{4}}{4\pi^{2}\Lambda^{4}}-\frac{e^{2}}{180\pi^{2}m^{2}M_{\text{Pl}}^{2}}\,.$
(62)

Note that the second term in the second line corresponds to a gravitational
diagram while the first term corresponds to a non-gravitational diagram. Then,
using (24), we obtain

$\displaystyle\frac{e^{4}}{4\pi^{2}\Lambda^{4}}-\frac{e^{2}}{180\pi^{2}m^{2}M_{\text{Pl}}^{2}}\geq-\mathcal{O}\left(\frac{\alpha^{\prime}}{M_{\text{Pl}}^{2}}\right),$
(63)

which leads to the upper bound on $\Lambda$:

$\Lambda\lesssim\sqrt[\ ]{emM_{\text{Pl}}}\,,\qquad\quad\text{for $m^{2}\ll
e^{2}\alpha^{\prime-1}$}\,.$ (64)

Similar results are reported for the Standard Model Aoki:2021ckh  and the dark
photon model Noumi:2022zht . In particular, using this argument, it is argued
that almost all parameter regions of the dark photon model are excluded
Noumi:2022zht .

Note that even if a higher-dimensional term is added to quadratic action (3),
there is no change in the positivity bound (64). For example, we can consider
the additional term with mass dimension six,

$S_{6}=\int\mathrm{d}^{4}{x}\,\sqrt[\
]{-g}\frac{c}{\Lambda^{2}}\phi^{\dagger}\phi F_{\mu\nu}F^{\mu\nu}\,,$ (65)

where $c$ is a constant. A new vertex (Fig. 3) and diagrams arise from Eq.
(65), but we can confirm that their contributions to the left-hand side of
(24) cancel out.

Figure 3: A star vertex in the right diagram is a new vertex from Eq. (65). We
consider new diagrams obtained by replacing the left diagram with the right
one.

Originally, the bound (64) is derived by assuming that the graviton
$t$-channel pole can be discarded, which is almost equivalent to the
approximately gravitational positivity bound (24) at tree-level. After that,
it turns out that in the case of graviton-photon Alberte:2021dnj  and
graviton-graviton deRham:2022gfe  scatterings, the $t$-channel pole can not be
neglected. This implies that the one-loop Regge amplitude cancels the negative
contribution corresponding to the graviton exchange.

In the following, as an independent argument for the importance of the
contribution from the Regge states, we provide a counterexample of Eq. (64) in
string theory, assuming that the inequality similar to Eq. (64) holds in the
case of 4d $\mathcal{N}=2$ theory. Let us take type IIB theory compactified on
Calabi-Yau threefold.121212We can also consider the IIA mirror dual. The
number of supersymmetry is reduced from $32$ to $8$, and then the resultant 4d
theory has $\mathcal{N}=2$ supersymmetry. The theory has a $h^{2,1}$ number of
$U(1)$ gauge symmetries corresponding to the dimensional reduction of 10d
Ramond-Ramond 4-form field, where $h^{2,1}$ is the Hodge number of the Calabi-
Yau threefold. On top of that, there are $h^{1,1}+1$ number of neutral
hypermultiplets, which are not important in our context. A part of the moduli
space is parameterized by the VEV of the scalar partner of $U(1)$ gauge field.
In general, this moduli space has a conifold singularity, where the $D3$-brane
wrapping on the internal cycle becomes massless Strominger:1995cz . In the 4d
language, this massless particle is a 4d $\mathcal{N}=2$ hypermultiplet
charged under $U(1)$, since $D3$-brane is charged under Ramond-Ramond 4-form
in 10d. By choosing the VEV slightly away from the conifold point, we have an
electron (charged hypermultiplet) with an arbitrarily small mass. In contrast,
the mass of the other heavy states, which is interpreted as the cutoff scale,
is finite.131313When the distance to the singularity is finite, only a finite
number of the state becomes massless. On the other hand, an infinite number of
states become massless for an infinite distance, as dictated by the Distance
Conjecture. This is a counterexample to the Eq. (64) because the right-hand
side of the inequality becomes arbitrarily small. Therefore, we arrive at the
conclusion that the Regge amplitude cancels the negative contribution even in
the case of photon-photon scattering. This counterexample also shows that the
negative contribution would not be canceled by modifying the Regge slope
$2+\alpha^{\prime}t$, but by the modification of the prefactor $f(t)$ (cf. Eq.
(18)) at the loop level. This is because the tower of higher spin states does
not become massless at the conifold point. This is a novel point compared with
the argument based on sum rules.

###  4.2 Reggeization at the loop level

In the previous section, we first argue that the bound (64) follows from the
gravitational positivity bound (24). Then, we argue that the bound contradicts
with the well-established string theory vacua. This indicates that the
derivation of Eq. (64) in Sec. 2.2 should be revisited.

In the following, we argue that the Regge amplitude at the loop level changes
the argument leading to Eq. (64). In Eq. (18), we have used $f\approx
f_{\text{tree}}$. Instead, if we keep the quantum correction
$f=f_{\text{tree}}+f_{\text{loop}}$, then the term

$\displaystyle-\frac{2f_{\text{loop}}^{\prime}(t)}{M_{\text{Pl}}^{2}\alpha^{\prime}}$
(66)

is added to the right-hand side of Eq. (21). By comparing this with Eq. (62),
we observe that if the term (66) cancels the negative contribution
corresponding to the graviton exchange, then it helps with positivity. This is
a resolution of the contradiction in the previous section. Naively, the
cancellation between Eq. (66) and $-e^{2}/(180\pi^{2}m^{2}M_{\text{Pl}}^{2})$
is unlikely because the UV Regge amplitude should “know" the IR mass scale
$m$. However, here we propose the generalization of the Regge amplitude to the
one-loop level, which realizes the cancellation naturally.

At the tree level, we have assumed the Reggeization

$\displaystyle\mathcal{A}\to
F_{\mathrm{tree}}(t)\frac{s^{2+\alpha^{\prime}t}}{M_{\text{Pl}}^{2}M_{*}^{2\alpha^{\prime}t}}$
(67)

for fixed $t$ and large $s$. Here $F_{\mathrm{tree}}$ has mass dimension $-2$,
and the relation with $f_{\text{tree}}$ in Eq. (18) is
$F_{\mathrm{tree}}\sin(\alpha^{\prime}t)=f_{\text{tree}}$. This is expected to
happen by the exchange of the higher spin particles. Given this
interpretation, the important requirement to the behavior of
$F_{\mathrm{tree}}(t)$ is that

  * •

$F_{\mathrm{tree}}(t)$ must have poles at $t=n\,\alpha^{\prime-1}$ where
$n=0,1,\ldots$. Therefore, $F_{\mathrm{tree}}(t)$ is written as

$\displaystyle
F_{\mathrm{tree}}(t)=\frac{f_{0}}{t}+\frac{f_{1}}{t-\alpha^{\prime-1}}+\frac{f_{2}}{t-2\alpha^{\prime-1}}+\cdots\,,$
(68)

where $f_{i}$ is a constant. Since each residue of the pole must be positive,
we obtain $f_{i}>0$. The presence of the pole at $t=0$ is crucial to maintain
the positivity bound at the tree-level. Because of the poles, the amplitude at
$t=n\,\alpha^{\prime-1}$ is dominated by the diagram corresponding to the
exchange of the particle of the mass $n\,\alpha^{\prime-1}$.

Next, we have to impose an extra requirement at the one-loop level. We denote
the amplitude as

$\displaystyle\mathcal{A}\to\big{[}F_{\mathrm{tree}}(t)+F_{\mathrm{loop}}(t)\big{]}\frac{s^{2+\alpha^{\prime}t}}{M_{\text{Pl}}^{2}M_{*}^{2\alpha^{\prime}t}}\,.$
(69)

At the one-loop level, contributing diagrams include those with a loop of
light particles and tree-level exchange of higher spin particles (see Sec.
4.3). Importantly, these diagrams have a branch cut starting from $t=4m^{2}$
corresponding to particle production.

  * •

At the one-loop level, the branch cut must appear at $t=4m^{2}$.
$F_{\mathrm{loop}}$ is written as

$\displaystyle
F_{\mathrm{loop}}(t)=\tilde{f}_{0}L_{0}(t,m^{2})+\frac{\tilde{f}_{1}\alpha^{\prime-1}}{t-\alpha^{\prime-1}}L_{1}(t,m^{2})+\frac{\tilde{f}_{2}\alpha^{\prime-1}}{t-2\alpha^{\prime-1}}L_{2}(t,m^{2})+\cdots\,,$
(70)

where $L_{i}(t,m^{2})$ is a loop function that contains the branch cut
starting from $t=4m^{2}$ (see Sec. 4.3 for the detail), and has an expansion

$\displaystyle
L_{i}(t,m^{2})=\sum_{j=1}^{\infty}\frac{l_{j}^{(i)}}{m^{2}}\left(\frac{t}{m^{2}}\right)^{j-1}\,,$
(71)

up to $\log(m^{2})$. Here $l_{j}$ is a dimensionless constant. On top of the
branch cut, there exist poles at $t=k\alpha^{\prime}$ where $k=1,2,\ldots$.

Now we discuss the implications of the Regge amplitude on the dispersion
relation. The contribution from Eq. (70) is

$\displaystyle\int^{\infty}_{M_{*}^{2}}\mathrm{d}{s^{\prime}}\,\frac{\operatorname{Disc}\mathcal{A}(s^{\prime},t)}{s^{\prime
3}}\sim\frac{\sin(\alpha^{\prime}t)}{\alpha^{\prime}t}\left(\tilde{f}_{0}L_{0}(t,m^{2})+\frac{\tilde{f}_{1}\,t}{t-\alpha^{\prime-1}}L_{1}(t,m^{2})+\frac{\tilde{f}_{2}\,t}{t-2\alpha^{\prime-1}}L_{2}(t,m^{2})+\cdots\right).$
(72)

For $t\to 0$, by choosing $\tilde{f}_{i}=\mathcal{O}(\alpha^{\prime})$
appropriately, the inequality (21) contains a finite contribution from the
one-loop diagrams with higher-spin spins exchanged. In particular, the
positivity bounds for scalar QED can be satisfied without requiring an
unusually low cut-off.

###  4.3 An argument for the form of $F_{\mathrm{loop}}$

Figure 4: The one-loop diagrams corresponding to the $t$-channel exchange of
the graviton and the higher spin particles. The solid, wavy, double solid, and
double wavy lines correspond to the scalar, graviton, photon, and higher spin
particles, respectively. These produce the branch cut starting from
$t=4m^{2}$.

In this section we provide an explanation for Eq. (70) by estimating the
$t$-channel exchange diagrams of Fig. 4, where the double line is either the
graviton or the higher spin particles (see also the caption). We show that
this class of diagrams leads to the contribution where the light mass
parameter $m^{2}$ shows up in the denominator. In App. B, we argue that the
other diagrams do not exhibit this feature by the power counting argument. The
loop integral in Fig. 4 is (see App. A for the notation)

$\displaystyle\frac{g_{1}g_{2}}{M^{2L-2}}\int\frac{\mathrm{d}^{4}{p}}{(2\pi)^{4}}\frac{1}{p^{2}-m^{2}}\frac{1}{(p+k_{1})^{2}-m^{2}}\frac{1}{(p+k_{1}+k_{3})^{2}-m^{2}}(2p+k_{1})^{\mu}(2p+2k_{1}+k_{3})^{\nu}$
(73) $\displaystyle\qquad\qquad\times(p^{\rho_{1}}\cdots
p^{\rho_{n_{1}}})\bigg{(}(p+k_{1}+k_{3})^{\sigma_{1}}\cdots(p+k_{1}+k_{3})^{\sigma_{n_{2}}}\bigg{)}\,,$

where the second line comes from the interaction between $\phi$ and higher
spin-$L$ particle $\Phi_{\alpha_{1}\cdots\alpha_{L}}$,

$\displaystyle\frac{g_{1}}{M^{L-1}}\Phi_{\alpha_{1}\cdots\alpha_{L}}\partial^{\alpha_{1}}\cdots\partial^{\alpha_{i}}\phi\partial^{\alpha_{i+1}}\cdots\partial^{\alpha_{L}}\phi\,,$
(74)

and $n_{1}+n_{2}=L$ with $M$ and $g_{1}$ being the mass and coupling of
$\Phi_{\alpha_{1}\cdots\alpha_{L}}$. Note that the coupling between the photon
and $\Phi_{\alpha_{1}\cdots\alpha_{L}}$ is schematically given by

$\displaystyle\frac{g_{2}}{M^{L-1}}\Phi_{\beta_{1}\cdots\beta_{L}}\partial^{\beta_{3}\cdots\beta_{m}}F^{\beta_{1}\mu}\partial^{\beta_{m+1}\cdots\beta_{L}}{F^{\beta_{2}}}_{\mu},$
(75)

where $g_{2}$ is the coupling constant. By introducing Feynman parameters, we
obtain

$\displaystyle\frac{g_{1}g_{2}}{M^{2L-2}}\int\frac{\mathrm{d}^{4}{l}}{8\pi^{4}}\int_{0}^{1}\mathrm{d}{y}\int_{0}^{1-y}\mathrm{d}{x}\,\frac{(2l+(1-2x-2y)k_{1}-2yk_{3})^{\mu}(2l+2(1-x-y)k_{1}+(1-2y)k_{3})^{\nu}}{[l^{2}-m^{2}+y(1-x-y)t]^{3}}$
$\displaystyle\times\bigg{(}(l-(x+y)k_{1}-yk_{3})^{\rho_{1}}\cdots(l-(x+y)k_{1}-yk_{3})^{\rho_{n_{1}}}\bigg{)}$
$\displaystyle\times\bigg{(}(l+(1-x-y)k_{1}+(1-y)k_{3})^{\sigma_{1}}\cdots(l+(1-x-y)k_{1}+(1-y)k_{3})^{\sigma_{n_{2}}}\bigg{)}$
(76)

where $l=p+(x+y)k_{1}+yk_{3}$.

In the following, we concentrate on the term which contains the light mass
parameter $m^{2}$ in the denominator for $t\to 0$. From a power-counting
argument, this corresponds to the terms where the numerator does not contain
$l$:

$\displaystyle\frac{g_{1}g_{2}}{M^{2L-2}}\int\frac{\mathrm{d}^{4}{l}}{8\pi^{4}}\int_{0}^{1}\mathrm{d}{y}\int_{0}^{1-y}\mathrm{d}{x}\,\frac{((1-2x-2y)k_{1}-2yk_{3})^{\mu}(2(1-x-y)k_{1}+(1-2y)k_{3})^{\nu}}{[l^{2}-m^{2}+y(1-x-y)t]^{3}}$
$\displaystyle\times\bigg{(}(-(x+y)k_{1}-yk_{3})^{\rho_{1}}\cdots(-(x+y)k_{1}-yk_{3})^{\rho_{n_{1}}}\bigg{)}$
$\displaystyle\times\bigg{(}((1-x-y)k_{1}+(1-y)k_{3})^{\sigma_{1}}\cdots((1-x-y)k_{1}+(1-y)k_{3})^{\sigma_{n_{2}}}\bigg{)}.$
(77)

The structure of the cut becomes clear by integrating over $l$, $x$ and $y$.
We obtain the following expression:

$\displaystyle\int_{0}^{1}\mathrm{d}{y}\int_{0}^{1-y}\mathrm{d}{x}\int\frac{\mathrm{d}^{4}{l}}{8\pi^{4}}\frac{1}{[l^{2}-m^{2}+y(1-x-y)t]^{3}}$
$\displaystyle\qquad=-i\int_{0}^{1}\mathrm{d}{y}\int_{0}^{1-y}\mathrm{d}{x}\int\frac{\mathrm{d}^{d}{l_{E}}}{8\pi^{4}}\frac{1}{[l_{E}^{2}+m^{2}-y(1-x-y)t]^{3}}$
$\displaystyle\qquad=-i\int_{0}^{1}\mathrm{d}{y}\int_{0}^{1-y}\mathrm{d}{x}\,\frac{1}{16\pi^{2}}\frac{1}{m^{2}-y(1-x-y)t}$
$\displaystyle\qquad=-\frac{i}{t}\;\mathrm{Li}_{2}\left(\frac{2\sqrt[\
]{t}}{\sqrt[\ ]{t}-\sqrt[\
]{t-4m^{2}}}\right)-\frac{i}{t}\;\mathrm{Li}_{2}\left(\frac{2\sqrt[\
]{t}}{\sqrt[\ ]{t}+\sqrt[\ ]{t-4m^{2}}}\right)\,,$ (78)

where $\mathrm{Li}$ is the polylogarithm function.141414Note that Eq. (78) is
finite at $t=4m^{2}$. Here for simplicity, we ignore $x$ and $y$ dependence in
the numerator, but the structure of the branch cut does not change. The Eq.
(78) has a small-$t$ expansion of the form (71)

$\displaystyle-\frac{i}{t}\;\mathrm{Li}_{2}\left(\frac{2\sqrt[\ ]{t}}{\sqrt[\
]{t}-\sqrt[\
]{t-4m^{2}}}\right)-\frac{i}{t}\;\mathrm{Li}_{2}\left(\frac{2\sqrt[\
]{t}}{\sqrt[\ ]{t}+\sqrt[\ ]{t-4m^{2}}}\right)$ (79)
$\displaystyle=-i\left(\frac{1}{2m^{2}}+\frac{t}{24m^{4}}+\frac{t^{2}}{180m^{6}}+\cdots\right)\,.$

We write Eq. (77) in the following schematic way:

$\displaystyle\eqref{eq:integral2}=(k_{1}+\\#k_{3})^{\mu}(k_{1}+\\#k_{3})^{\nu}\cdots(k_{1}+\\#k_{3})^{\sigma_{n_{2}}}F_{L-2}(t,m^{2})\,,$
(80)

where $F_{L-2}(t,m^{2})$ is defined by

$\displaystyle F_{L-2}(t,m^{2})=$
$\displaystyle(-1)^{n_{1}}\int\frac{\mathrm{d}^{4}{l}}{8\pi^{4}}\int_{0}^{1}\mathrm{d}{y}\int_{0}^{1-y}\mathrm{d}{x}\,\frac{2(1-x-y)(1-2x-2y)(x+y)^{n_{1}}(1-x-y)^{n_{2}}}{[l^{2}-m^{2}+y(1-x-y)t]^{3}}.$
(81)

The indices $\mu$ and $\nu$ are contracted with the external polarizations
$\epsilon_{1}^{\mu}$ and $\epsilon_{3}^{\nu}$. This gives a factor

$\displaystyle(k_{1}+\\#k_{3})^{\mu}(k_{1}+\\#k_{3})^{\nu}\epsilon_{1\mu}\epsilon_{3\nu}\sim\frac{tu}{s}\to
t\,,\quad\text{for $s\to\infty$}$ (82)

Other indexes are contracted with $k_{2},k_{4},\epsilon_{2}$ or
$\epsilon_{4}$. For large $s$, this contribution behaves as $s^{L}$.
Multiplying the propagator of the higher spin particle, we obtain

$\displaystyle\text{Figure~{}\ref{fig:diagram}}\;=\frac{g_{1}g_{2}}{M^{2L-2}}\frac{s^{L}t}{t-M^{2}}F_{L-2}(t,m^{2})$
(83)

for large $s$. Suppose that the mass $M$ and the spin $L$ of the particle are
given by

$M^{2}=k\alpha^{\prime-1}\,,\qquad L=2+k\,,\qquad k=0,1,2,\ldots\,,$ (84)

as usual in the graviton Regge tower. Then, the amplitude is

$\displaystyle\frac{g_{1}g_{2}}{M^{2L-2}}\frac{s^{2+k}t}{t-k\alpha^{\prime-1}}F_{k}(t,m^{2})\,.$
(85)

When we take $t\to k\alpha^{\prime-1}$, we expect that the contribution from
the exchange of spin $2+k$ particle dominates. This reproduces Eq. (70).

We would like to explain an idea why the summation of the $t$-channel diagram
leads to the Reggeized amplitude. At the tree-level, the $t$-channel amplitude
is

$\displaystyle\frac{s^{2}}{M_{\text{Pl}}^{2}t}+g_{2}^{2}\frac{s^{3}\alpha^{\prime
2}}{t-\alpha^{\prime-1}}+g_{2}^{2}\frac{s^{4}\alpha^{\prime
3}}{t-2\alpha^{\prime-1}}+\cdots\,.$ (86)

After an appropriate analytic continuation, we obtain
$s^{2+\alpha^{\prime}t}/(M_{\text{Pl}}^{2}t)$ for fixed $t$.151515See Eq. (31)
and App. C for the detail. Similarly, at the one-loop level, the amplitude is

$\displaystyle\left(\frac{s^{2}}{M_{\text{Pl}}^{2}}+g_{1}g_{2}\frac{s^{3}t\,\alpha^{\prime
2}}{t-\alpha^{\prime-1}}+g_{1}g_{2}\frac{s^{4}t\,\alpha^{\prime
3}}{t-2\alpha^{\prime-1}}+\cdots\right)F(t,m^{2})\,,$ (87)

from Eq. (85). Here we have assumed that all $F_{k}$ are the same order of
magnitude, and have defined $F:=F_{k}$. Since the expression in the
parenthesis is the same form as the tree-level $t$-channel amplitude, it is
natural to expect that this is also Reggeized as
$s^{2+\alpha^{\prime}t}/M_{\text{Pl}}^{2}$ for fixed $t$. In fact, the formal
summation of the $t$-channel diagram leads to

$\displaystyle\frac{s^{2}}{M_{\text{Pl}}^{2}}+g_{1}g_{2}\sum_{n=1}^{\infty}\frac{s^{2}t\,\alpha^{\prime}}{t-n\alpha^{\prime-1}}(\alpha^{\prime}s)^{n}=\frac{s^{2}}{M_{\text{Pl}}^{2}}-g_{1}g_{2}\left(\alpha^{\prime}s\right)^{3}t\,\Phi(\alpha^{\prime}s,1,1-\alpha^{\prime}t)\,,$
(88)

where $\Phi$ is the Lerch zeta function. By expanding around $t=0$, we obtain

$\displaystyle-\left(\alpha^{\prime}s\right)^{3}t\,\Phi(\alpha^{\prime}s,1,1-\alpha^{\prime}t)=-\alpha^{\prime}s^{2}\sum_{n=1}^{\infty}\left(\alpha^{\prime}t\right)^{n}\mathrm{Li}_{n}(\alpha^{\prime}s)\to\alpha^{\prime}s^{2}\sum_{n=1}^{\infty}\frac{\left(\alpha^{\prime}t\log(\alpha^{\prime}s)\right)^{n}}{n!}\,,$
(89)

for $\alpha^{\prime}s\gg 1$. Here we have used

$\displaystyle\mathrm{Li}_{n}(\alpha^{\prime}s)=-\frac{\left(\log(\alpha^{\prime}s)\right)^{n}}{n!}+\mathcal{O}\left(\left(\log(\alpha^{\prime}s)\right)^{n-1}\right),$
$\displaystyle\text{for $\alpha^{\prime}s\gg 1$}\,.$ (90)

Now, assuming that $g_{1}g_{2}\alpha^{\prime}=M_{\text{Pl}}^{-2}$, we obtain

$\displaystyle\frac{s^{2}}{M_{\text{Pl}}^{2}}+g_{1}g_{2}\sum_{n=1}^{\infty}\frac{s^{2}t\,\alpha^{\prime}}{t-n\alpha^{\prime-1}}(s\alpha^{\prime})^{n}$
$\displaystyle\to\frac{s^{2}}{M_{\text{Pl}}^{2}}+g_{1}g_{2}\alpha^{\prime}s^{2}\sum_{n=1}^{\infty}\frac{\left(\alpha^{\prime}t\log(\alpha^{\prime}s)\right)^{n}}{n!}$
$\displaystyle=\frac{s^{2}}{M_{\text{Pl}}^{2}}e^{\alpha^{\prime}t\log(\alpha^{\prime}s)}=\frac{s^{2+\alpha^{\prime}t}}{M_{\text{Pl}}^{2}}\alpha^{\prime\alpha^{\prime}t}\,.$
(91)

As a result, we obtain

$\displaystyle\frac{s^{2+\alpha^{\prime}t}}{M_{\text{Pl}}^{2}}\alpha^{\prime\alpha^{\prime}t}F(t,m^{2})\,.$
(92)

This is nothing but our proposal (70) for small $t$. Although we are not able
to compute the numerical factor of each diagram,161616In general,
$n$-dependent coefficient appears in the summation. Moreover, the couplings
$g_{1,2}$ can depend on $n$. It is interesting to study the condition to
realize the Reggeized amplitude. this illustrates the idea of how Reggeization
occurs, and how the parametrically large prefactor appears.

##  5 Discussion

In this paper, we have studied the gravitational positivity bound at the one-
loop level. We first reviewed the gravitational positivity bound at the tree-
level, where the $t$-channel pole corresponding to the graviton exchange can
be canceled by assuming Regge behavior for the amplitude at high energy. Next,
we saw two examples of unitary, Reggeized gravitational amplitudes for which
the potentially negative finite contribution to the positivity bounds can be
found. Finally, we moved to the features of Reggeized amplitudes at one loop.
It is known that the one-loop EFT amplitude leads to the parametrically large
negative contribution to the positivity bound. We argued for a form of the
Reggeized amplitude at the one-loop level based on the analytic structure of
the $t$-channel exchange diagram of the graviton and the higher spin particles
in the Regge tower. We also provide an argument that the graviton $t$-channel
pole can not be neglected by showing a counterexample to Eq. (1) in string
theory. These results are consistent with Refs. Alberte:2021dnj ; Herrero-
Valea:2022lfd ; deRham:2022gfe .

As a future direction, we may consider a string theory setup to explicitly
check the form of the amplitude at the one-loop of the matter fields. Ideally,
we want to analyze the amplitude in the Calabi-Yau compactification near the
conifold point, but this is hard. Instead, we may consider the
compactifications on the flat background. For instance, we may take the non-
supersymmetric $\operatorname{SO}(16)\times\operatorname{SO}(16)$ heterotic
string and compactify it on $T^{6}$.171717We can consider the same setup for
the supersymmetric heterotic string theory. In 10d there are fermions whose
representation is $({\bf 16},{\bf 16})+({\bf 128},{\bf 1})+({\bf 1},{\bf
128})$ under $\operatorname{SO}(16)\times\operatorname{SO}(16)$. These will
become Dirac fermions in 4d after compactification. By turning on a VEV for
the Wilson lines, the gauge symmetry is broken from
$\operatorname{SO}(16)\times\operatorname{SO}(16)$ to
$\operatorname{U}(1)^{16}$.181818In addition, there are KK and winding
$\operatorname{U}(1)$s, but these are not important here. Then the $({\bf
16},{\bf 16})$ fermions have charge

$\big{(}\underbrace{1,0,\ldots,0}_{\operatorname{U}(1)^{8}},\underbrace{1,0,\ldots,0}_{\operatorname{U}(1)^{8}}\big{)}$
(93)

(and its permutations) under $\operatorname{U}(1)^{16}$. Similarly, $({\bf
128},{\bf 1})$ has charge

$\big{(}\underbrace{\pm\tfrac{1}{2},\ldots,\pm\tfrac{1}{2}}_{\operatorname{U}(1)^{8}},\underbrace{0,\ldots,0}_{\operatorname{U}(1)^{8}}\big{)}\,.$
(94)

These fermions receive masses from the Wilson line, and the masses are chosen
to be arbitrary values. This setup is close to the non-supersymmetric QED
coupled with light matters, and loops of these fermions may give a negative
contribution to the coefficient of $s^{2}$. On the other hand, the typical
mass of the higher-spin particles is always the string scale, independent of
the VEV of the Wilson line. It is an interesting task to compute the one-loop
string amplitude at $\mathcal{O}(M_{\text{Pl}}^{-2})$ in this setup to check
the details of our proposal in Sec. (4.2).

### Acknowledgments

The work of Y.H. and G.L. is supported in part by MEXT Leading Initiative for
Excellent Young Researchers Grant Number JPMXS0320210099. The work of S.N. is
supported in part by JSPS KAKENHI Grant Number 21J15497.

##  Appendix A Notation

We follow the notation in Ref. Alberte:2020bdz . We consider the scattering
$\gamma_{1}\gamma_{2}\to\gamma_{3}\gamma_{4}$. The momenta $k^{\mu}_{1,2,3,4}$
are parametrized as (all-ingoing notation)

$\displaystyle k^{\mu}_{1}=(k,0,0,k)\,,$ $\displaystyle
k^{\mu}_{2}=(k,0,0,-k)\,,$ $\displaystyle
k^{\mu}_{3}=-(k,k\sin\theta,0,k\cos\theta)\,,$ $\displaystyle
k^{\mu}_{4}=-(k,-k\sin\theta,0,-k\cos\theta)$ (95)

We define $\epsilon^{\mu}_{1,2,3,4}$ as a polarization vector of the external
photon. The $\pm$ polarization corresponds to

$\displaystyle\epsilon^{\mu}_{1}(\pm)=\frac{1}{\sqrt[\ ]{2}}(0,1,\pm 1,0)\,,$
$\displaystyle\epsilon^{\mu}_{2}(\pm)=\frac{1}{\sqrt[\ ]{2}}(0,-1,\pm 1,0)\,,$
$\displaystyle\epsilon^{\mu}_{3}(\pm)=\frac{1}{\sqrt[\ ]{2}}(0,\cos\theta,\pm
i,-\sin\theta)\,,$ $\displaystyle\epsilon^{\mu}_{4}(\pm)=\frac{1}{\sqrt[\
]{2}}(0,\cos\theta,\pm i,\pm\sin\theta)\,.$ (96)

In all-ingoing notation, the amplitude is written as

$\displaystyle\mathcal{A}(h_{1},h_{2},h_{3},h_{4};k_{1},k_{2},k_{3},k_{4})=\epsilon_{1}^{\mu}(h_{1})\epsilon_{2}^{\nu}(h_{2})\epsilon_{3}^{\alpha}(h_{3})\epsilon_{4}^{\beta}(h_{4})\mathcal{A}^{\mu\nu\alpha\beta}(k_{1},k_{2},k_{3},k_{4})\,,$
(97)

where $h_{1,2,3,4}=\pm 1$ is the helicity of the external photons in all-
incoming notation. $\epsilon_{ij}$ is defined as

$\displaystyle\epsilon_{ij}:=\epsilon_{i}\cdot\epsilon_{j}$ (98)

Explicitly, we obtain

$\displaystyle\epsilon_{12}=-\frac{1}{2}-\frac{h_{1}h_{2}}{2}\,,$
$\displaystyle\epsilon_{13}=-\frac{h_{1}h_{3}}{2}+\frac{1}{2}+\frac{t}{s}\,,$
$\displaystyle\epsilon_{14}=-\frac{h_{1}h_{4}}{2}-\frac{1}{2}-\frac{t}{s}\,,$
$\displaystyle\epsilon_{34}=-\frac{1}{2}-\frac{h_{3}h_{4}}{2}\,,$
$\displaystyle\epsilon_{24}=-\frac{h_{2}h_{4}}{2}+\frac{1}{2}+\frac{t}{s}\,,$
$\displaystyle\epsilon_{23}=-\frac{h_{2}h_{3}}{2}-\frac{1}{2}-\frac{t}{s}\,,$
(99)

The inner products between external momenta and polarizations are

$\displaystyle
k_{1}\cdot\epsilon_{3}=k_{3}\cdot\epsilon_{1}=k_{2}\cdot\epsilon_{4}=k_{4}\cdot\epsilon_{2}=-\frac{\sqrt[\
]{tu}}{\sqrt[\ ]{2s}}\,,$ $\displaystyle
k_{1}\cdot\epsilon_{4}=k_{4}\cdot\epsilon_{1}=k_{2}\cdot\epsilon_{3}=k_{3}\cdot\epsilon_{2}=\frac{\sqrt[\
]{tu}}{\sqrt[\ ]{2s}}\,,$ $\displaystyle\mathrm{(others)}=0\,.$ (100)

The inner products among external momenta are

$\displaystyle k_{1}\cdot k_{2}=k_{3}\cdot k_{4}=s/2\,,$ $\displaystyle
k_{1}\cdot k_{3}=k_{2}\cdot k_{4}=t/2\,,$ $\displaystyle k_{1}\cdot
k_{4}=k_{2}\cdot k_{3}=u/2\,,$ $\displaystyle
k_{1}^{2}=k_{2}^{2}=k_{3}^{2}=0\,.$ (101)

where $(s,t,u)$ are the Mandelstam variables:

$\displaystyle s=(k_{1}+k_{2})^{2}=2k_{1}\cdot k_{2},$ $\displaystyle
t=(k_{1}+k_{3})^{2}=2k_{1}\cdot k_{3},$ $\displaystyle
u=(k_{1}+k_{4})^{2}=2k_{1}\cdot k_{4}\,.$ (102)

##  Appendix B Unimportant diagrams

Figure 5: The diagrams whose contribution does not contain the light mass
parameter $m^{2}$ in the denominator for $t\to 0$.

In the main text, we have considered the diagrams in Fig. 4. However, there
are other diagrams in Fig. 5, which are potentially relevant. These diagrams
are not relevant to us because these contributions do not contain the light
mass parameter $m^{2}$ in the denominator for $t\to 0$. This can be seen from
the power counting of the loop integral. All the diagrams in Figs. 5 contain
two scalar propagators, which scale as $p^{-4}$, where $p$ is the loop
integral. Therefore, after integrating over $p$, we get $\log(m^{2})$ at most.
The derivative interaction at the vertex does not change the conclusion
because it changes the scaling of the integrand as $p^{-4+n}$ where $n\geq 0$.

##  Appendix C Reggeization at the tree-level

In this appendix, we show how the tree-level amplitude is Reggeized starting
from the pole expansion (31), along the line with the end of Sec. 4.191919Of
course, we know the full expression (25). It is easy to show the Regge
behavior from Eq. (25). Nevertheless, it is instructive to see how the same
result emerges from the pole expansion. In the following, we assume the limit
$s\to\pm i\infty$.202020The argument is not valid for other directions in
$s$-plane. For large $s$, Eq. (31) is written as

$\displaystyle\mathcal{A}(s,t)$
$\displaystyle\sim-\frac{4Ps^{4}}{\alpha^{\prime}}\sum_{n=0}^{\infty}\frac{1}{(n!)^{2}}\left(\frac{\alpha^{\prime}s}{4}\right)^{2n-2}\left(\frac{1}{t-4n\alpha^{\prime-1}}-\frac{1}{t+s+4n\alpha^{\prime-1}}\right)$
$\displaystyle=-\frac{64Ps^{2}}{\alpha^{\prime
3}}\left[\left(\frac{1}{t}-\frac{1}{t+s}\right)+\sum_{n=1}^{\infty}\frac{1}{(n!)^{2}}\left(\frac{\alpha^{\prime}s}{4}\right)^{2n}\left(\frac{1}{t-4n\alpha^{\prime-1}}-\frac{1}{t+s+4n\alpha^{\prime-1}}\right)\right]\,.$
(103)

We expand the second term around $t=0$, and then perform the summation from
$n=1$ to $n=\infty$. By keeping the leading term for $s\to\infty$ in each
order of $t$, we obtain

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{(n!)^{2}}\left(\frac{\alpha^{\prime}s}{4}\right)^{2n}\left(\frac{1}{t-4n\alpha^{\prime-1}}-\frac{1}{t+s+4n\alpha^{\prime-1}}\right)\sim\frac{\alpha^{\prime}}{4}\sum_{m=1}^{\infty}\frac{1}{m!}\left(\frac{\alpha^{\prime}t}{4}\right)^{m-1}\left(\log\left(\frac{\alpha^{\prime
2}s^{2}}{16}\right)\right)^{m}\,.$ (104)

By substituting this into Eq. (103), we obtain the Reggeized amplitude:

$\displaystyle\mathcal{A}(s,t)$
$\displaystyle\sim-\frac{16Ps^{2}}{\alpha^{\prime
2}}\sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{\alpha^{\prime}t}{4}\right)^{m-1}\left(\log\left(\frac{\alpha^{\prime
2}s^{2}}{16}\right)\right)^{m}+\frac{64Ps^{2}}{\alpha^{\prime 3}(t+s)}$
$\displaystyle=-\frac{64Ps^{2}}{\alpha^{\prime 3}}\left(\frac{\alpha^{\prime
2}s^{2}}{16}\right)^{\alpha^{\prime}t/4}+\frac{64Ps^{2}}{\alpha^{\prime
3}(t+s)}\,.$ (105)

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