Gravitational Positivity for Phenomenologists: Dark Gauge Boson in the Swampland

The gravitational positivity bound gives quantitative"swampland'' constraints on low-energy effective theories inside theories of quantum gravity. We give a comprehensive discussion of this bound for those interested in applications to phenomenological model building. We present a practical recipe for deriving the bound, and discuss subtleties relevant for realistic models. As an illustration, we study the positivity bound on the scattering of the massive gauge bosons in the Higgs/St\"{u}ckelberg mechanism. Under certain assumptions on gravitational amplitudes at high energy, we obtain a lower bound $m_{V} \gtrsim \Lambda_\mathrm{UV}^2 /g M_\mathrm{Pl}$ on the gauge boson mass $m_V$, where $g$ is the coupling constant of the gauge field, $M_\mathrm{Pl}$ is the reduced Planck mass and $\Lambda_\mathrm{UV}$ is the ultraviolet cutoff of the effective field theory. This bound can strongly constrain new physics models involving a massive gauge boson.


Introduction
It is one of the outstanding questions in present-day physics to uncover the origin and the identity of the dark matter (DM).More broadly we are interested in the search for the dark sector containing DM, or any physics beyond the Standard Model (BSM).
Traditionally, dark sectors have been analyzed in the framework of low-energy Effective Field Theories (EFTs), where the effects of the dark sector have been incorporated by small couplings of the dark sector to particles of the Standard Model (SM).It is often the case that such couplings are severely constrained by experiments/observations, leading to very small (absolute) values of parameters.While certain fine-tunings are needed, the dominant attitude has been that such small parameters are well-tolerated in the EFTs, as long as such small parameters are technically natural [1].This raises a fundamental problem for dark-sector searches: while we can keep eliminating parameter spaces for dark-sector-SM couplings, there is in principle no lower bound on their size, and it seems that we can never exclude a "nightmare" scenario where the dark sector interacts with the SM sector only through gravitational interactions.
The situation is different, however, once we impose an extra condition that the Infrared (IR) lowenergy EFT has a consistent ultraviolet (UV) completion with gravity.There has been mounting evidence recently that there are necessary conditions for a low-energy EFT to have a consistent UV completion, and such conditions are often called swampland constraints in the literature [2,3].One way to obtain such constraints is to invoke the so-called positivity constraints on scattering amplitudes at low energy.The positivity constraints are well established in non-gravitational theories and their application to the EFT of the Standard Model and cosmology has long been discussed: see [4][5][6][7] for early references on QCD.For more recent developments, see e.g.[8], a review article [9] and references therein.The advantage of this method is that we can derive an infinite set of 0. Write down the (renormalizable) Lagrangian of your model, couple the model to gravity, and consider a 2 → 2 scattering process AB → AB.
1. Compute the amplitude M non-grav of diagrams without gravity: and M grav,t-channel of graviton t-channel diagrams: 2. Compute B non-grav : 3. Compute series expansion of M grav,t-channel around t = 0 and s = m 2 A + m 2 B , and extract B grav from the coefficient of t 0 (s − m 2  A − m 2 B ) 2 .4. Compute B(Λ) = B non-grav (Λ) + B grav (Λ) and check its positivity.inequalities on the EFT couplings by minimal assumptions on the scattering amplitudes such as unitarity, analyticity, and locality, without relying too much on the details of the UV completion.
In this paper, we discuss a particular version of positivity bounds with gravitational effects included [14].Interestingly, the effect of gravity modifies the positivity bound, and in some cases strengthens it.While the effects of gravity are suppressed by the powers of the Planck scale, gravity can still change the conclusions dramatically since we are interested in small couplings in dark sectors.Indeed, it has been known that if the gravitational positivity bound is applied to the matter-matter scatterings at loop level in D = 4 dimensions, we obtain nontrivial constrains on the spectrum and renormalizable couplings of light particles well below the Planck scale [16,[24][25][26][27][28][29].
The gravitational positivity bound sometimes gives a lower bound on the size of the dark sector couplings.When combined with upper bounds from observations, one can then in principle exclude some dark sector models.The gravitational positivity bound relies on some assumptions of the UV theory and hence can be violated in observations.This, however, inevitably implies a violation of some well-defined assumptions of the UV theories, and hence we can derive sharp constraints of UV physics from the experiments of IR physics.
It is an interesting question to systematically apply the gravitational positivity bounds to various BSM models.It is fair to say, however, that a systematic application of gravitational positivity bounds for BSM physics remains mostly unexplored.Moreover, there are important unsolved problems whose solutions are required for such analysis, as we will see below.
The goal of this paper is to make the gravitational positivity bound more accessible to a broader audience of phenomenologists interested in BSM physics.In Fig. 1, we show the procedures to obtain the gravitational positivity bound for scalar fields.For this purpose, we list a concrete recipe for deriving gravitational positivity bounds for a given EFT, and list some fine prints in such discussions.We next illustrate the power and limitations of the bounds in the concrete example of a massive dark U(1) gauge boson (dark photon).We will find that the resulting bound is impressively strong, however, for applications for realistic models, there are some theoretical subtleties that need to be addressed.
The rest of this paper is organized as follows.In Sec. 2 we summarize the gravitational positivity bound for non-experts.In Sec. 3 we present a practical recipe for deriving the gravitational positivity bound.As a concrete illustration of our procedure, we write down the gravitational positivity bound for a massive U(1) gauge boson in Sec. 4. We will find that the resulting bound is very strong and rules out many parameter spaces.One should note, however, that there are important subtleties for realistic model building, as we will discuss further in Sec. 5.The final section (Sec.6) is devoted to conclusions and discussions.We included an appendix for gravitational contributions beyond graviton t-channel exchange.

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

Positivity Bounds without Gravity
Positivity bounds are formulated in terms of scattering amplitudes.For technical simplicity, this subsection focuses on scalar scattering amplitudes in gapped theories.
Assumptions.Consider an s-u symmetric scattering amplitude M(s, t) of AB → AB type in a given low-energy EFT, where s, t, u are the standard Mandelstam variables that satisfy s , and m A and m B are the masses of the external particles A and B, respectively.We assume that the forward amplitude M(s, t = 0) evaluated in the would-be UV complete theory satisfies the following properties: s-plane except for poles and discontinuities on the real axis required by unitarity.
Positivity bounds.The aforementioned three properties imply an infinite set of consistency relations among scattering amplitudes evaluated at the UV and IR.Let us define Then, the following dispersion relation holds from the analyticity and s 2 -boundedness of the amplitude: where m th is the threshold energy, i.e., the mass of the lightest intermediate state.Note that the relation for n = 0 does not follow under the present assumptions, which requires a stronger assumption that the amplitude is bounded by s 0 at high energy.
The EFT is defined with a UV cutoff which we denote by Λ UV .The actual cutoff of the EFT, namely the scale of new physics, is unknown from the low-energy perspective.We thus introduce a reference scale Λ below which the EFT is assumed to be valid, i.e., Λ is assumed to satisfy Λ < Λ UV .
Let us define 2 [35-37] We emphasize that B 2n (Λ) is defined in terms of the amplitude at the IR scale below Λ, and therefore it is calculable within the EFT.Then, the dispersion relation (2.2) implies which provides a consistency relation between the IR physics below Λ (LHS) and the UV physics above Λ (RHS).Furthermore, unitarity implies that the RHS is non-negative, so that B 2n (Λ) ≥ 0 for all Λ such that Λ < Λ UV (n = 1, 2, . ..) . (2.5) These bounds are called the positivity bounds.Note that B 2n (Λ) is a monotonically non-increasing function of Λ because the integrand in (2.3) is positive.The typical behavior is shown in Fig. 2. When the bounds (2.5) are satisfied at a certain scale, they also hold at any scale below it.The positivity bound (2.5) is thus stronger for larger Λ.
2 In general, we can have m 2 th < m 2 A + m 2 B and branch cuts can run on the entire real s-axis.The definitions (2.1) and (2.3) may become ill-defined.In such a case, B 2n (Λ) should be defined through "arcs" of [32,33] (see also [34]).B 2n (Λ) defined in this way agree with (2.3) [or (2.7)] when branch cuts do not run on the entire real s-axis, according to the Cauchy integral theorem.Interpretations.In a given EFT, one can compute the forward amplitude and evaluate B 2n (Λ) defined in Eq. (2.3) in terms of parameters of the EFT and a scale Λ.The positivity bounds (2.5) then provide certain inequalities among the EFT couplings and Λ, leading to two complementary interpretations: 1. Bounds on EFT couplings.Suppose that the low-energy EFT describes a system below an energy scale E. We can then identify Λ with the energy scale of interest E to find the bounds on the EFT couplings.These are the necessary conditions for the EFT to have a standard UV completion and be valid below the scale E = Λ.(In here we are agnostic on the value of the cutoff scale Λ UV .) 2. Bounds on the UV cutoff.
In some cases, one may want to assume certain values of the EFT couplings-for example, coupling constants are already fixed by experiments or they need to take certain values for phenomenological purposes (e.g., so that they can be searched in a particular experiment).
One can then ask up to which scale the EFT can be valid, i.e., at which scale the new physics comes in.As shown in Fig. 2, B 2n (Λ) may become negative at some energy scale Λ * , in which case we can derive an upper bound Λ UV < Λ * on the UV cutoff of the EFT.
The condition (2.5) is based on the assumptions on UV, but care must be taken to ensure because we do not know the UV theory.If a violation of the inequality (2.5) is observed in the experiment, it may indicate a violation of the original assumptions.In this sense, we can also use the positivity bounds (2.5) to probe UV from IR.

Positivity Bounds with Gravity
We next explain how the positivity bounds are extended to theories with gravity.The main new ingredient is that the t-channel graviton exchange contributes a new singularity to the low-energy UV states (if any) Figure 3: Typical scenario: a UV completion of gravity is achieved well below the Planck scale by an infinite tower of higher-spin states (Regge tower).There may also exist new UV states beyond the EFT that are described within QFT model building and that are not directly related to the UV completion of gravity.
scattering amplitudes as GeV being the reduced Planck mass), invalidating the definition (2.1) of a 2 and hence the positivity argument for B 2 (Λ). 3 In this subsection, we summarize how the positivity bounds are formulated in the presence of gravity.
Assumptions.The basic assumptions are the same as before, namely analyticity, unitarity, and the s 2 -boundedness lim |s|→∞ M(s, t)/s 2 = 0 for small negative t.
While the argument below can be applied to any UV complete theory with these three properties, a typical candidate of such a UV completion is a perturbative string theory with a particle spectrum as shown in Fig. 3.
Gravitational positivity bounds.We define a 2 by subtracting the graviton t-pole from the amplitude and taking the forward limit as4 With a 2 defined in this manner, we introduce Here and in what follows we suppress the subscript 2 of B 2 (Λ) for notational simplicity.The dispersion relation (2.4) is then modified as Due to the first term in the square bracket (which is negative for t < 0), the sign of B(Λ) is undetermined.However, one can show that the positivity of B(Λ) holds at least approximately.
The idea is that we split the integral of the dispersion relation into the quantum gravity part and the other, (2.9) When we take the forward limit t → −0, the l.h.s of (2.9) is finite because it is defined by subtracting the t-pole term.Then, although the t-pole term in the r.h.s diverges in the forward limit t → −0, the sum in the square bracket should be finite as a result of a cancellation with the quantum gravity part: where we defined the sign σ and the mass scale M , which depend on details of the cancellation mechanism.(The case σ = 0 corresponds to the exact cancellation.)As a result, we find which we call the gravitational positivity bound.
The right-hand side of (2.11) is suppressed by powers of M Pl , and hence disappears in the decoupling limit M Pl → ∞, where the gravitational positivity bound reduces to the positivity bound without gravity.
While the derivation here works irrespective of the details of the cancellation mechanism, the most plausible scenario is the Regge behavior5 : the gravitational amplitude is modified as QG by an infinite tower of higher-spin states at s ≥ Λ 2 QG (see Fig. 3).The perturbative string theory is a typical example of this scenario in which the quantum gravity scale Λ QG corresponds to the string scale.The cancellation of the t-pole can be explicitly shown under the assumption of Regge behavior [14].
Interpretations.In contrast to the non-gravitational positivity bound (2.5), the gravitational positivity bound (2.11) contains not only the IR data B(Λ) determined by the EFT parameters up to the scale Λ, but also the UV data (M, σ) of gravitational Regge amplitudes (or, generically speaking, quantum gravity amplitudes).Therefore, there are three possible interpretations, depending on the context [16,17,[27][28][29] As we mentioned in Sec.2.1, the violation of original assumptions on UV theory leads to the violation of (2.11).Differently from the non-gravitational case, however, experimental tests of the violation of (2.11) require knowledge of the UV data (M, σ).
Comments on (M, σ).As we explain more explicitly later, nontrivial constraints on the IR physics are obtained in the spirit of Interpretation 1 when Here, B grav is the gravitational part of B which will be defined in the next section.This condition is satisfied either when σ = 0, +1 or when M is sufficiently large, typically M ≫ m light with m light being the mass scale of light particles.This shows that details of the UV data (M, σ) are important in discussing phenomenological implications of the gravitational positivity [29].
Notably, recent studies proceed in the direction to carve out the parameter space of (M, σ) itself and sharpen the gravitational positivity bound (2.11).For scattering of identical scalars, the best bound obtained so far is schematically of the form [18,22] 7 (see also [21,23]), where m denotes the threshold energy, i.e., the energy of the lightest intermediate state.For tree-level amplitudes, this scale is essentially the mass of the lightest higher-spin particle (spin 2 or higher).For loop amplitudes, this corresponds to the lowest energy of the intermediate multi-particle states.Note that this bound is a necessary condition for the effective theory to have a standard UV completion, rather than a sufficient condition.Therefore, it is still a nontrivial question if there exists a consistent quantum gravity theory that accommodates a negative B(Λ),8 more specifically that with σ = −1 and M ∼ m light .Further studies in this direction are encouraged for phenomenological applications of the gravitational positivity.

Generalizations for Realistic Theories
When one tries to apply the gravitational positivity bounds (2.5) to realistic phenomenological models, the basic idea is the same as in the case of scalar scatterings of the previous subsection.However, one encounters several new subtleties to be carefully addressed, which we list in the following.
Spin.The positivity bound can be extended to a scattering of particles with spin, see [35,40]. 9here are extra technical complications, however, and here we highlight the issue of the kinematic singularity, which is relevant for our discussion of gauge bosons in Sec. 3.
Let us consider two-to-two scattering amplitudes of massive spin-1 particles with mass m V .The kinematic singularity is the singularity of the scattering amplitude M from the pole s = 4m 2 V of the scattering angle θ as in (2.13) The kinematic singularity disappears in the forward limit t → 0 if the scattering amplitude does not contain the massless t-pole.However, in our setup, the kinematic singularity appears from t-channel graviton exchange diagrams.We can deal with this singularity by defining a kinematic singularity-free amplitude to be [40] (see also [42,43]):10 We can then derive the positivity bound by essentially the same arguments, as long as we replace the amplitude M by M. In practice, this is the same as ignoring the contribution of kinematic singularity to the t-channel graviton exchange diagrams.
Massless particles.The general properties of the scattering amplitudes have been established only in gapped systems, and not when massless particles are present.Even worse, the traditional non-perturbative S-matrix does not exist for massless particles, due to the issue of the IR divergence.
To avoid this IR issue, we estimate the contributions from IR divergent diagrams to the positivity bounds by introducing an IR cutoff in Appendix A. We find that they will be subdominant, and hence we expect that the subtleties associated with the IR divergence will not affect our main conclusion.However, this may not be the case in other models, in which case an appropriate prescription for massless particles is required to derive the bounds.
Unstable particles.If a particle has a decay channel, such an unstable particle does not appear in the asymptotic states and the standard scattering amplitudes cease to exist for the particle [44].However, unstable-particle scattering amplitudes can be unambiguously defined by residues of higher-point amplitudes (see e.g.[45]) and their properties can be studied in the S-matrix theory. 11nitarity of the S-matrix leads to certain constraints on the unstable-particle scattering amplitudes, and in particular, there exists a positivity constraint on the imaginary part [46], suggesting the existence of positivity bounds even from scattering of unstable particles.In the present paper, however, we will only focus on the scattering of stable particles and leave a precise treatment of unstable particles for a future study.
Anomalous threshold singularities.Even for stable particles, the analytic structure of a scattering of a heavier particle is not as simple as that of a lighter particle.It is known that the scattering of the heavier particle can have a new singularity called an anomalous threshold, whose existence does not immediately follow from unitarity.While general knowledge of anomalous threshold is still missing to the best of our knowledge (see [47,48] for recent discussions), we expect that the anomalous thresholds appear only in the IR regime, i.e., they are detectable and controlled within EFT at least when the heavy particle is stable.If this is indeed the case, the anomalous thresholds will give additional technical complications but will not spoil the success of positivity bounds, as long as these thresholds are taken into account in the derivation of the dispersion relations.

Practical Recipe for Gravitational Positivity
We are now ready to discuss practical recipes for deriving gravitational positivity bounds for EFTs.Instead of making the presentation fully general, we focus on the case of 2 → 2 scattering V V → V V of spin 1 particle V with mass m V .This will be the case relevant for the rest of this paper.
The gravitational positivity bound is an inequality for the quantity B(Λ).When we wish to apply the gravitational positivity bound of the previous section, we run into one practical problem: we can almost never compute an exact scattering amplitude M(s, t) in an EFT.In fact, if we know the exact expression for M(s, t), we can analytically continue the expression to the whole complex plane, and we have effectively solved for the quantum gravity already.The best we can do is to compute an approximate expression M EFT (s, t) for M(s, t), by including only a finite number of parameters out of an infinite number of higher-dimensional operators.The differences between the two are small in the regions of the validity of EFT (at scales Λ ≪ Λ UV ).Hence, the computation of the approximate amplitude M EFT is sufficient for evaluating the quantity B(Λ) which only requires information below Λ.In the following, we will discuss EFT scattering amplitude M EFT and explain a practical way to calculate B(Λ) from the EFT amplitude M EFT : for notational simplicity we will simply denote this approximate amplitude as M in the rest of this section.
We consider the scattering amplitude of the gauge bosons V V → V V .Corresponding to polarizations of the gauge boson, we define the following combinations of helicity amplitudes.For the scattering of transverse modes, where superscripts ± denote helicities.For the scattering of the transverse mode and the longitudinal mode, where superscripts L denote longitudinal polarizations.For the scattering of longitudinal modes, These amplitudes are defined such that they exhibit s ↔ u symmetry (up to subtleties associated with the kinematical singularity discussed around (2.14), which are not relevant to the following discussion).
We provide a step-by-step procedure for deriving the gravitational positivity bound.The basic task is simple: we need to evaluate the cutoff-dependent quantity B(Λ) defined by a spin-1 generalization of (2.8): where M is a kinematic singularity-free amplitude defined in (2.14), m th is the threshold energy, i.e., the mass of the lightest intermediate state, and a 2 is basically defined as in (2.6), but with gravitational corrections included (as we will soon see in (3.8)).
Step 1: Calculate M(s, t) We compute the 2 → 2 scattering amplitude M(s, t) up to order O M −2 Pl : Here M non-grav (s, t) is the contribution from diagrams with no graviton exchange, while M grav (s, t) is the contribution from diagrams with one graviton exchange and is of the order O M −2 Pl .The remaining terms are the order of O M −4 Pl and are neglected in the rest of the analysis.Once we compute M, it is straightforward to obtain kinematic-singularity free amplitude M (2.14), both for non-gravitational and gravitational contributions.Note that such an amplitude exists for each choice of helicity of external vector fields (as in (3.1), (3.2), and (3.3)).
Step 2: Calculate B non-grav (Λ) We compute the non-gravitational contribution B non-grav (Λ) to B(Λ).We assume that the EFT of interest is renormalizable in the gravity decoupling limit M Pl → ∞.
In such a case, the amplitude M non-grav itself satisfies the analyticity and the s 2 -boundedness, which is the s 6 -boundedness in terms of M non-grav , when the expression is analytically continued throughout the complex plane, even outside the regions of validity of the EFT.The non-gravitational part then satisfies the twice-subtracted dispersion relation, giving a formula This formula is practically useful since we only need to compute the imaginary part of the forward amplitude.This expression further simplifies in the limit Λ ≫ m V : In other words, in this limit, we can in practice neglect the subtleties associated with the use of the kinematic-singularity-free amplitude M, and use the original amplitude M for the evaluation of B non-grav (Λ) in (3.7) (which coincides (2.4) with n = 1 and m A = m B = m V ).
Step 3: Calculate B grav (Λ) We next compute the gravitational contribution B grav (Λ) to B(Λ).While this contains many different Feynman diagrams in general, we will argue in Appendix A that it is enough to consider the contribution from t-channel graviton exchange diagrams.In this case, the integral part of (2.7) is absent for this contribution because M grav,t-channel (s, t) does not have the s-channel or u-channel discontinuity at the non-gravitational one-loop order.Therefore the gravitational part of B (Λ) is simplified to a 2 defined in (2.6), evaluated for the gravity t-channel amplitude: When we assume Λ ≫ m V , we obtain an expression Step 4: Write down the inequality for total B(Λ) The constraints on EFT are obtained as the inequality (2.11) for the combined expression B(Λ) = B non-grav (Λ) + B grav (Λ).There are three interpretations of this inequality, as we discussed in Sec.2.2.

Positivity Bound on Dark U(1) Gauge Boson
A gauge interaction is one of the cornerstones of the quantum field theory (QFT).Gauge symmetries and their breakings are one of the most crucial ingredients of the Standard Model, and many BSM models also introduce new gauge symmetries.In particular, a light gauge boson is one of the candidates for the DM [49][50][51].
In this section, we work out the details of the gravitational positivity recipe for the dark U(1) gauge boson.Here we study the Abelian Higgs mechanism of the simplest gauge theory, the U(1) theory.We discuss theoretical constraints obtained by step-by-step computations spelled out in the previous section.

Higgs Contribution
In the Higgs mechanism, the gauge boson gets a mass when a charged scalar field Φ gets a non-zero vacuum expectation value (VEV) v.The renormalizable Lagrangian of the Higgs mechanism to generate a mass for a U(1) gauge boson is where 2) The gravitational interaction can be found by expanding the Lagrangian (4.1) in terms of the canonical gravitational field h µν as 3) The Higgs-gauge boson interaction (4.2) does not make the gauge boson V unstable since it has a Z 2 symmetry V ↔ −V .

Non-gravitational contributions
Fig. 4a show some examples of the non-gravitational Feynman diagrams involving the Higgs boson ϕ.From these (and other) diagrams we evaluate the non-gravitational part of B(Λ) to be Here • • • represents contributions suppressed by higher powers of Λ, which scale is assumed to be much greater than m ϕ and m V .We have also numerically estimated the two-loop contributions and confirm that these effects are smaller than the one-loop contributions in the parameter region of present interest.

Gravitational contributions
As the next step, we compute the one-loop diagrams of the gravitational contributions (Fig. 4b): The functions g T T,T L,LL (x).These functions are real and positive for x > 0 and therefore the gravitational contributions are negative for all the masses of the Higgs boson and the gauge boson.
where the functions g T T,T L,LL are given by We plot the functions g T T,T L,LL in Fig. 5.These functions are real and positive for any real values of m V and m ϕ , and we find that the gravitational contributions from the Higgs to B(Λ) are negative.

Non-gravitational contributions
The diagram for a non-gravitational fermion loop is shown in Fig. 6a.The one-loop contributions are given by for Λ ≫ m V,F .These contributions of LL and T L vanish in the limit m V /Λ → 0. Recall that B non-grav is determined by the high-energy limit of the forward limit amplitude thanks to the dispersion relation.The behaviors B T L non-grav,F , B LL non-grav,F → 0 as m V /Λ → 0 can be understood by the decoupling of the longitudinal sector in the high-energy limit.The contributions of the charged particles to B T L non-grav and B LL non-grav can be ignored in the limit Λ ≫ m V .We thus focus on the T T scattering.The charged spin-1/2 loops give positive contributions to the non-gravitational diagram of the order B T T non-grav,F ∼ g 4 F /Λ 4 , which is of the same order as the Higgs counterpart (4.4).Therefore, the inclusion of charged spin-1/2 particles does not drastically change the non-gravitational part.We reach a similar conclusion even in the presence of charged spin-0 particles [16].
In contrast, as studied in [27,29], charged spin-1 loops with the mass m spin-1 and the charge g spin-1 lead to a different asymptotic behavior B T T non-grav ∼ g 4 spin-1 /(m 2 spin-1 Λ 2 ) which dominates over other contributions in the limit Λ ≫ m spin-1 .One may then wonder about contributions from higherspin particles.The scaling in Λ is determined by the high-energy behavior of the imaginary part of the amplitude in the forward limit, i.e., the total cross-section.Roughly, spin-0, 1/2 loops give Im M ∝ s 0 while spin-1 loops yield Im M ∝ s/m 2 spin-1 in high-energy limit.The faster growth in s provides a larger contribution to B non-grav (Λ).However, the asymptotic growth of the cross-section in s is bounded by the Froissart bound [30] as Im M < s ln 2 s in gapped theories, implying that the spin-1 contribution almost saturates the bound.Higher-spin particles, if they are described by gapped theories like QCD, may not give drastically different contributions compared to spin-1 particles.For instance, in the case of light-by-light scattering, the hadronic contribution can be estimated as Im M ∝ s 1.08 by employing the vector meson dominance model [60,61].
In short, if a charged spin-1 or higher-spin particle appears at m, the non-gravitational contribution B T T non-grav needs to be modified in Λ > m.By contrast, spin-0 or spin-1/2 particles give additional contributions but the qualitative behavior of B T T non-grav is the same.

Gravitational contributions
Next, we discuss the gravitational parts (Fig. 6b), which are given by where We plot the functions g AB,F (x) (A, B = T, L) in Fig. 7.
Let us first discuss the limit m V → 0, which yields If the Higgs and the gauge boson are lighter than the charged particles (m ϕ , m V ≪ m F ), we can neglect these contributions to the gravitational process B grav in comparison to the contributions of the gauge boson sector.Note that the LL part has an additional suppression m 2 V /m 2 F because the longitudinal mode should decouple from the charged particle in the limit m V → 0. However, since the longitudinal mode does not decouple from gravity, the T L mode does not have such a suppression factor in the diagram where the longitudinal mode attaches to the graviton line.
By contrast, a peculiar behavior appears in a heavy mass range of the gauge boson.The functions g AB,F are singular at m V = 2m F and become complex numbers in m V > 2m F .The threshold m V = 2m F corresponds to the value at which the decay of the gauge boson to the charged particles starts to be kinematically allowed.The singularity at m V = 2m F is understood as an anomalous threshold.Let us consider the triangle diagram of Fig. 6b, which gives rise to the gravitational form factor of the gauge boson.The triangle diagram has a normal threshold at t = 4m 2 F .In addition, the anomalous threshold comes up on the first sheet of the complex t-plane in m V > √ 2m F and the position of the singularity is at Therefore, the form factor at t = 0 is singular when m V = 2m F , generating the singularity of B AB grav,F .In Fig. 7, we have plotted the functions in the unstable range m V > 2m F as well.Recall that B(Λ) has to be real according to the dispersion relation (2.8) if all the mentioned properties are satisfied.This implies that at least one of the properties does not hold when the gauge boson decays.In fact, amplitudes exhibit peculiar behaviors if the mass of the external particle is extrapolated to the unstable region [46,47].The conventional positivity bounds need to be modified for unstable particles.
We expect that the subtleties associated with the decay are negligible if the particle is long-lived (g F ≪ 1).In other words, although the fermion contribution to B grav (Λ) is quite subtle in the mass range m V > 2m F , this subtle contribution can be smaller than the contribution from the Higgs sector and could be simply negligible.In the following, we shall adopt this optimistic expectation when discussing models in which the gauge boson decays into other particles.The issue of unstable particles will be studied elsewhere.

Constraint on Dark Gauge Boson
In Table 1, we summarize our estimation of B(Λ) for a light gauge boson.We apply the gravitational positivity bound (2.11) to these amplitudes: (4.30) For simplicity of the presentation, in the following, we will discuss the constraints on the dark gauge boson parameters for σ = 0 with a fixed value of Λ (Interpretation 1 in Sec.2.2).One should recall, however, that the actual bound is (4.30).The bound becomes stronger for σ = +1 and weaker for σ = −1.We will later see more quantitatively how the σ/(M 2 Pl M 2 ) term changes the bound in Fig. 12.

Abelian Higgs without matter fields
Let us first look at the case where there is no matter field and only gauge and Higgs fields exist.For a very light gauge boson m V ≪ m ϕ,F ≪ Λ and m ϕ ≪ √ Λm V , the following conditions can be obtained from each helicity In Fig. 8, we show the positivity conditions.Note that the higher terms of Λ for the non-gravitational contributions can be dominant if

Stückelberg gauge boson with matter field
Next, let us discuss the case where the Higgs field ϕ is absent in the EFT, as in the case of the Stückelberg mechanism.We find In Fig. 9, we show the positivity conditions on m V − g F plane.Note that for m V > 2m F , delicate arguments are required and the positivity conditions used previously cannot be applied as they are.More generally, it is possible to incorporate both Higgs and fermions.The helicity configurations involving longitudinal modes provide stronger constraints.As discussed earlier, the Higgs contribution is dominant when g Φ ∼ g F and m V ≪ m ϕ,F , in which case the positivity conditions will be similar to those of the Higgs scenarios.

Comparison with other swampland constraints for Stückelberg gauge bosons
Let us compare our swampland constraints with another swampland constraint on the gauge boson mass [62] (see also [63]) motivated by the swampland distance conjecture [3].This bound states that the UV cutoff Λ UV of an EFT with a gauge boson should obey where (as before) m V is the gauge boson mass and g is the gauge coupling constant.In Fig. 10 we compare our bounds and the bound (4.37) for sample values of parameters.One can see in Fig. 10 that overall our bound is stronger than the bound (4.37).Let us quickly point out, however, that the two bounds are derived under a different set of assumptions/arguments, and one needs to be careful in any meaningful comparison.

WGC Axionic WGC Positivity
The gravitational positivity bound is derived by general properties of Reggeized scattering amplitudes of a weakly-coupled UV theory.The assumptions are believed to be standard, but the bound has some ambiguities due to the unknown UV mass scale/sign (M, σ) in (2.10).By contrast, the bound (4.37) is derived from a set of swampland conjectures.The (m V M Pl /g) 1/2 bound in (4.37) is derived by noting that the Stückelberg mass is obtained when the gauge boson eats a fundamental axion, for which we can apply an axionic version [62,63] of the weak gravity conjecture [64]; the g 1/3 M Pl bound in (4.37) is derived from a combination of the (sub)lattice/tower weak gravity conjecture [65,66] (see also [25,67]) and the species bound [68][69][70].
Let us also note that our bound applies to gauge boson masses both in the Higgs mechanism and the Stückelberg mechanism, while the bound (4.37) applies only to those in the Stückelberg mechanism.Our bound is more general in this respect.
In Fig. 10 we also plotted the bound from the weak gravity conjecture [64], which requires an existence of a particle with charge g and mass m such that √ 2g ≥ m/M Pl .

Toward Constraints on Realistic Phenomenological Models
The dark U(1) gauge sectors discussed so far have nothing to do with our universe and are entirely theoretical constructs.However, any new physics describing our universe must necessarily coexist with the Standard Model.Since some important subtleties arise at this point, we will first briefly discuss them and then discuss the specific model (B−L gauge model) as a possible application of the gravitational positivity.
If we apply the discussion of gravitational positivity bounds to, for example, the scattering of photons and dark gauge bosons, the dark sector necessarily has interactions beyond those described by the Standard Model and gravity [29].In general, these interactions allow the dark gauge boson to decay into photons and SM fermions and also decay into gravitons.
In this case, the dark gauge boson has a finite decay width, and hence care must be taken when applying the positivity constraint.Despite those caveats, the decay rate of light and weakly interacting gauge bosons is extremely small, and we expect no major practical problems.
As an example, we consider the U(1) B−L extension of the Standard Model.We assume the gauge charge of the B−L breaking Higgs is the same as that of the SM leptons.In Fig. 11, we show the positivity bound (4.30) on the U(1) B−L model and current experimental constraints.Here we take σ = 0 and Λ = 1 GeV.In this plot, we assume that the B−L Higgs contributions are dominated over the SM fermions and neglect the SM contributions.The dashed lines in the figure show the parameter region where the dark gauge boson can decay into neutrinos.The stringent bound from the positivity constraint comes from TL scattering mode.Note that the constraint is more severe for the higher cutoff, as seen in the TL constraint (4.32).The gravitational positivity bound requires the larger gauge coupling for smaller masses and has strong tensions with the experimental searches.

Conclusion and Discussion
In this paper, we discussed the practical procedure for deriving gravitational positivity bounds.We illustrated the procedure for dark gauge bosons, and discussed implications of the bounds.
There are two complementary comments regarding our results.First and foremost, our bound is very strong-in Fig. 11 most of the parameter regions unexplored by experiments are already excluded by the positivity bound.This is a clear demonstration of the power of the gravitational

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

Figure 2 :
Figure 2: The expression (2.3) of B 2n (Λ) is a monotonically non-increasing function as a function of Λ, and crosses zero at the scale Λ = Λ * .For a well-defined UV completion, we need B 2n (Λ) ≥ 0, which defines an upper bound Λ * on the scale Λ.

Figure 4 :
Figure 4: Some examples of the Higgs loop contribution to B(Λ).

2 +
the covariant derivative and g Φ is a charge of Φ and λ > 0. After Φ = v + ϕ/ √ iG develops the VEV, the Goldstone component G is absorbed into the longitudinal component of the gauge boson.There are a massive gauge boson V and a real scalar ϕ after the gauge symmetry breaking, and the gauge boson gets a mass m V = √ 2g Φ v and a real scalar ϕ has a mass m ϕ = √ λv.The interaction of the Higgs and gauge bosons are given by

Figure 7 :
Figure7: The functions g T T ,F (x) (red), g T L,F (x) (green) and g LL,F (x) (blue).The solid and dashed lines represent the real and imaginary parts, respectively.

Figure 9 :
Figure9: The constraints on the dark gauge boson parameters for the Stückelberg U(1) gauge theory with a fermion field.The positivity bound is satisfied in the region above the lines.We choose Λ = 1 TeV.
b) Heavy fermion

Figure 10 :
Figure 10: Comparison between the swampland constraints for Stückelberg gauge bosons, the gravitational positivity bound (in red) and other swampland bounds (4.37) (in blue) and the weak gravity conjectures (in green).We take the cutoff scale Λ is 1 GeV (dotted lines), 1 TeV (solid lines) and 1 PeV (dashed lines).(a): Light fermion case m F = m V and (b): Heavy fermion case m F = 0.1Λ.In this plot, the lines are chopped in the parameter space where m V > 0.1Λ.
]: 1.Quantum gravity constraints on IR physics.If we specify/assume a quantum gravity scenario and the parameters (σ, M ) of gravitational Regge amplitudes, we can think of the bounds (2.11) as quantum gravity constraints on the low-energy EFT at a scale Λ = E.Such constraints are useful to carve out the parameter space of phenomenological models in the spirit of the Swampland Program 6 .2. Constraints on the scale of new physics.As in the non-gravitational case, we can interpret (2.11) as the bounds giving the maximum cutoff Λ * when we assume the parameters of the EFT and the UV data (σ, M ).When the bounds (2.11) are violated, one can try to increase the cutoff of the EFT by adding new state(s) to EFT like Fig. 2. If the bounds (2.11) are inevitably violated at Λ > Λ * , this is a sign of the necessity for a UV completion of gravity.In this case, Λ * is not a maximum scale of just new physics, but the upper bound on the quantum gravity scale Λ QG .3. IR constraints on quantum gravity.Once the parameters and the validity of the model are identified by experiments at a scale Λ = E, we can use the bounds (2.11) to constrain the parameters (σ, M ) of the gravitational Regge amplitudes required for UV completion of gravity.Such constraints are useful as a necessary condition for a quantum gravity theory to describe our real world.

Table 1 :
Summary of the gravitational and non-gravitational contributions to B for m V ≪ m ϕ,F ≪ Λ and m ϕ ≪ √ Λm V .