Bounds on scattering of neutral Goldstones

We study the space of $2\to 2$ scattering amplitudes of neutral Goldstone bosons in four space-time dimensions. We establish universal bounds on the first two non-universal Wilson coefficients of the low energy Effective Field Theory (EFT) for such particles. We reconstruct the analytic, crossing-symmetric, and unitary amplitudes saturating our bounds, and we study their physical content. We uncover non-perturbative Regge trajectories by continuing our numerical amplitudes to complex spins. We then explore the consequence of additional constraints arising when we impose the knowledge about the EFT up to the cut-off scale. In the process, we improve on some aspects of the numerical $S$-matrix bootstrap technology for massless particles.


Introduction
In the past decade there has been a surge of interest in studying general properties of scattering amplitudes stemming from the fundamental principles of causality, crossing, and unitarity.One powerful method for doing this is the numerical S-matrix bootstrap [1][2][3].It allows to determine model-independent bounds on physical observables and extract scattering amplitudes from them.
The case of massive particles has been widely studied in various space-time dimensions .Particles with spins have been studied in [35].Alternative approaches have been developed in [36][37][38][39], and progress has been made in establishing new rigorous results in [40,41].The case of massless particles has been investigated to a lesser extent, and the numerical algorithms are less effective.Nevertheless, some intriguing results have been obtained: bounds on the quark antiquark potential [42,43], universal constraints on pion low energy constants [44], no-go theorems for quantum gravity with supersymmetry [45,46], and universal bounds on photon scattering [47].
In this work, we apply the numerical bootstrap to a simple physical process: the 2 → 2 scattering amplitude of massless identical scalars in four space-time dimensions. 1 We focus on the class of IR safe amplitudes that can be described at low energies by a derivatively coupled Effective Field Theory (EFT) 2 Examples of such theories include neutral Goldstone bosons of the spontaneously broken U (1) symmetry, spontaneous conformal symmetry breaking, or co-dimension one defects.The real coefficients c n in the potential are called the Wilson coefficients.Their mass dimension reads as [c n ] = −n.
We can use the EFT Lagrangian density (1.1) to systematically compute the 2 → 2 scattering amplitude of Goldstones at each order in the small s expansion T (k) EFT ∼ O(s k ), where s is the Mandelstam variable describing the squared total energy of the process.The EFT expansion provides the most generic parametrization of the amplitude compatible with crossing symmetry, and unitarity.Since the Mandelstam variable s is dimensionful, the validity of the EFT expansion as an approximation of the amplitude itself depends on a scale, a priori unknown, called the EFT cutoff.For the moment, we will be agnostic about the cutoff.Instead, without loss of generality, we identify the EFT expansion with the small s 1 Stronger bounds can be obtained by including multi-particle constraints.This is an extremely hard problem in higher dimension.In 2d some progress has been made in [48]. 2 Here we have only included the kinetic term and terms which contain four fields ϕ and up to 8 derivatives.
-2 -asymptotic expansion of the nonperturbative scattering amplitude of Goldstones 3 T (s, t, u) EFT (s, t, u) + O(s N +1 ), (1.2) where the first terms are given by T EFT (s, t, u) = g 2 (s 2 + t 2 + u 2 ), T EFT (s, t, u) = g 3 stu, together with 4   T EFT (s, t, u) The s, t and u are the usual Mandelstam variables which obey the relation s + t + u = 0.The above expressions can be easily derived without any reference to the Lagrangian by using only crossing, and unitarity as shown in appendix B, where we provide the full expansion up to O(s 6 ) order.Let us stress that the logs in (1.4) contain the scale √ g 2 which defines our coupling g 4 .
Alternatively we could define the coupling g 4 (µ) with an arbitrary scale µ in the logs, namely we could use log(−s/µ 2 ) instead of log(−s √ g 2 ) in (1.4).Then the two observables are simply related as g 4 (µ) = g 4 − In perturbation theory this is simply a change in the renormalization scheme.
When working with the scattering of massive particles there is canonical choice of the physical units: setting the mass gap m = 1.In the case of massless particles there is no obvious choice.In the EFT framework the natural candidate is the cutoff scale.However, from a non-perturbative perspective, the cutoff is not a well defined concept.It is more natural to use the first dimensionful parameter g 2 in (1.2) to set the units.Notice that g 2 is a nice candidate because it is universal and notoriously positive [49].Two dimensionless couplings are then formed as ḡ3 ≡ g 3 . (1.5) For later purpose, we will always define dimensionless 'bar' variables in units of g 2 , for example the dimensionless Mandelstam variable reads s ≡ s √ g 2 . (1.6) The first goal of our paper is 3 To be more precise, the small s expansion is valid at fixed scattering angle θ.Since t = − s 2 (1 − cos θ), and u = − s 2 (1 + cos θ), then t, u are small.For simplicity of narration we also use the identification O(s n ) ∼ O (s n log m (−s)), where m ≥ 1. 4 The real coefficients gn introduced in the EFT expansion are simply related to the Wilson coefficients as g2 = 2c4, and g3 = 3c6.The relation between g4 and c4 is scheme-dependent.We work in the scheme in which the physical imaginary part is given by (1.4), this leads to the relation g4 = c8/4.
• To improve the numerical non-perturbative methods for massless particles.In particular to solve the issue with slow numerical convergence observed in [47].
What can we say a priori about the allowed values of {ḡ 3 , ḡ4 }?The first statement is that both couplings can take arbitrarily large values.To understand this, let us consider a situation in which all coefficients in the low energy expansion of the amplitude (1.2) are given by a simple formula of the form where k n are some order one real numbers, α is a dimensionless small parameter, and M an arbitrary scale.We can think of M as the lightest particle in a weakly coupled QFT which we integrated out, and α as the dimensionless coupling of this putative UV theory.We provide some explicit examples in appendix C. Plugging (1.7) into (1.5)we get (1.8) In the weak coupling limit α → 0 the observables (1.8) become infinitely large, and the logs in (1.4) become negligible.In this regime, we conjecture that it is possible to describe the bootstrap results in perturbation theory using tree-level models. 5Our conjecture makes a number of observable predictions that we verify using our numerics in section 2.3.The complementary scenario is when both ḡ3 , ḡ4 ∼ 1.This is the regime in which the non-perturbative bootstrap is most powerful.In this case the logarithms in (1.4) are not parametrically suppressed, and the bounds are non-trivial. 6In this regime the S-matrix bootstrap is superior to other methods like "positivity" because it bounds the real part of the scattering amplitude by employing non-linear unitarity constraints, see appendix A for details.For the explanation of this fact using a simple analytic model see [43].For the comparison between the bounds obtained using the S-matrix bootstrap and positivity in the case of massive particles see [24,25].
When talking about EFT approximation of a physical phenomenon, there is an important requirement to satisfy: the separation of scales between the IR physics and its UV completion.This separation is parametrized by the introduction of an additional dimensionful parameter, the cutoff of the EFT, that we denote by M .Recall, that g 2 was chosen to set the units.With the cutoff in mind, we can define a new dimensionless positive coupling ξ g 2 = ξM −4 . (1.9) In the example of the weakly coupled UV completion discussed above, we can compare this expression with (1.7).From this it follows that ξ = k 2 α is infinitesimally small for α → 0.
Contrarily, for strongly coupled UV completions we expect ξ ∼ 1.The effect of an additional gap due to the EFT cutoff on the bootstrap bounds in the case of massive particles has been discussed in [25].In [25], the authors proposed a simple algorithm to mimic the presence of the EFT cutoff in a non-perturbative amplitude by bounding its imaginary part at low energies s ≪ M 2 , and checked the consistency of the method with the EFT expectations.
Here, we employ a refined variation of that idea using the strategy inspired by [21] and we apply it to the case of massless scalars.
Imagine that an experimentalist tells us that the scattering amplitude for a massless process is well approximated by some experimental data for all s, −t ≤ M , and that the error of this approximation is given by a function err(s, t, u).This can be implemented by the following condition Problems of this type were originally studied in [21] in two dimensions in the case of form factors where instead of the experimental data the authors used the numerical data obtained using the Hamiltonian Truncation method [73]. 7In this paper we will explain how to implement (1.10) in four dimensions.Since we do not have any experimental or numerical data for the neutral Goldstone scattering we will use as a proxi of this data the EFT representation (1.2) up to O(s 4 ) order assuming is valid in the extended region s ∈ [0, M 2 ].The error function will be estimated as our ignorance of the O(s 5 ) order terms in this expansion.We refer to this type of study as model-dependent.Summarizing, we define the second goal of our paper as EFT (s, cos θ) ≤ err(s, cos θ). (1.11) The paper is organized as follows.In section 2 we present our numerical results on universal bounds and discuss their physical meaning in great detail.In section 3 we briefly summarize the technology of [3] and introduce our improvements for efficiently studying massless particles.In section 4 we introduce the machinery needed for obtaining modeldependent bounds defined by (1.11).We conclude in section 5. Some additional results are provided in appendices.We refer to them throughout the text in places where they become relevant.
The numerical data and a notebook to extract the amplitude and perform the fit analysis can be downloaded from the repository https://doi.org/10.5281/zenodo.8422615.

Universal bounds on Wilson coefficients
We investigate numerically the space of UV complete amplitudes of Goldstones in four dimensions.To this end, we use the numerical method introduced in [3] and [44].We briefly

Figure 1:
The allowed region for the parameters ḡ3 and ḡ4 defined in (1.5) lies above the orange line.review this method in section 3. There, we also introduce several crucial improvements of this method needed for studying massless particles efficiently.Below we summarize our physical results. 8y using this method we determine the lower bound on the coupling ḡ4 as a function of ḡ3 .This bound is given by the orange line in figure 1a.All values of the couplings ḡ3 and ḡ4 above the orange line are allowed.There is a nontrivial absolute minimum for ḡ4 min ḡ4 = 1.58 ± 0.02 (4π) 2  (2.1) attained for ḡ3 ≃ −0.54.The value of ḡ4 is scheme-dependent as it appears at the same order as the logs in the amplitude.We define it in terms of the physical amplitude as in equations (1.2) - (1.4).
Looking at figure 1a we observe that ḡ4 is unbounded from above, while ḡ3 is unbounded in both directions.We have already argued in section 1 how arbitrarily large values of the Wilson coefficients can be realized by a simple mechanism.The absence of these bounds can be proven theoretically by construction (in any number of space-time dimensions) as explained in appendix C. In figure 1b we present our bound for the ratio ḡ3 / √ ḡ4 . 9From figure 1b, we observe that the ratio ḡ3 / √ ḡ4 goes to a constant as |ḡ 3 | → ∞, and we can estimate the following bound on this ratio When ḡ3 = 0, our bound applies to the special case of the scattering of dilatons in N = 4 SYM on the Coulomb branch [74].By giving a vev to one of the six scalars, we induce the spontaneous symmetry breaking of both conformal and SU (4) R-symmetry.Fluctuations around the vev correspond to the dilaton, the remaining five real scalars correspond to the Goldstones of the unbroken Sp(4) symmetry.At low energies the amplitude with only dilatons matches the expansion in (1.2) up to order O(s 4 ) included, and the mixing with the other Goldstones happens at higher orders.The condition ḡ3 = 0 is consequence of SUSY and the soft theorems.By looking at our bound at ḡ3 = 0 point we report the following result (2.3)

Phenomenology of the boundary
At each point on the boundary of the allowed region, given by the orange line in figure 1, there is a unique amplitude which we can reconstruct numerically.The amplitudes on the boundary are called extremal.Our goal in this subsection is to understand the physics contained in these extremal amplitudes.
The simplest way to analyze a scattering process is to plot partial amplitudes T ℓ (s) (also called partial waves) for several values of angular momentum ℓ = 0, 2, 4, . ... Recall, that in the physical region the scattering amplitudes in d = 4 space-time dimensions is written in terms of the partial amplitudes as where P ℓ (x) is the Legendre polynomial.For more details, see appendix A.
In order to diagnose how the various degrees of freedom in the amplitude distribute in the spin channels, it is useful to look at the spin decomposition of some observables.A convenient choice is given by g 2 and its sum-rule representation, derived in appendix F. One has where σ tot is the total cross section.If we decompose the amplitude in the forward direction in partial waves inside the integral, we can define the following dimensionless coefficients that can be thought as branching ratios of the cross section in each spin channel [25], The values ḡℓ 2 , interpreted as a sort of partial cross sections, are thus a good indication of the spin content of the amplitude.In figure 2 we present the value of ḡℓ 2 using different colors for different spins as a function of ḡ3 that we use to parametrize the boundary.The sum rule (2.6) is already well approximated by the sum of the branching ratios ḡℓ 2 with few angular momenta ℓ = 0, 2, 4, 6.By looking at the figure we see that when ḡ3 ≫ 1 the ℓ = 0 ratio gives the most important contribution to the cross-section, suggesting that our bounds might have a simple description in terms of a perturbative QFT amplitude.On the other hand, when ḡ3 ≪ −1, the sum of the higher spin ratios dominate and we might envision a string-like amplitude.Close to the minimum value of ḡ4 where we do expect strongly coupled physics, the spins arrange in a non-trivial way.
We can refine our physical picture by studying the position of the spectrum of resonances as a function of ḡ3 .Consider partial waves Resonances are given by zeros s R of the partial waves S ℓ (s R ) = 0 in the upper half plane.Their position in the complex plane can be interpreted in terms of the mass and the width s R = (m + iΓ/2) 2 of the unstable particle [46].In figure 3 we plot the position of the lowest lying resonance in the ℓ = 0 partial wave for several values of ḡ3 .Connecting the dots we obtain a smooth continuous trajectory parametrized by ḡ3 .The nature of this zero changes as we follow it along the trajectory .When ḡ3 ≫ 1, in red, the zero shows up close to the real axis with a mass parametrically larger than its width m ≫ Γ.Then it can be described as a light weakly coupled scalar particle.On the opposite extreme, when ḡ3 ≪ −1, its interpretation is < l a t e x i t s h a 1 _ b a s e 6 4 = " P x 1 k W 8 5 n q g c + Figure 3: Trajectory of the lowest lying scalar zero as function of ḡ3 .
obscure as it goes close to the left cut region s < 0. Around the minimum value of ḡ4 , when ḡ3 ∼ −1, its real and imaginary part are of the same order and we can interpret it as a σ-like particle, as the famous scalar resonance in QCD.
Combining the information in figure 2 and figure 3, we can envision the existence of three distinct regions which will be investigated below, Region I (QCD-like) : Region II (string-like) : Region III (spin-0 exchange) : ḡ3 ≫ +1. (2.10)

Complex spin analysis of the amplitude minimizing ḡ4
In Region I, based on the previous analysis, we do expect to find amplitudes resembling QCD.
As a benchmark point we take the minimum value of ḡ4 .We study the spectrum of unstable resonances contained in the amplitude.Consider the phase shift δ ℓ (s) defined via The typical signature of a weakly coupled resonance is a rotation of the phase by π.Its mass can be estimated by looking at the energy at which the phase passes through π/2.The rotation of the phase can be associated to the presence of a zero of the partial wave in the complex plane close to the real axis.When the resonance is not weakly coupled it is more difficult to detect it by the phase shift analysis.The better way to study resonances is to inspect the complex plane as we do below.< l a t e x i t s h a 1 _ b a s e 6 4 = " S y q N L v M 5 I h 6 / E q q b U j f n S N B Y z 5 c = " > A    In figure 4 on the top panels we study δ 0 (s) and |S 0 (s)| in the complex s plane.We observe the presence of a broad resonance far away from the real axis.We call it σ resonance.In figure 4 on the bottom panels, we plot the phase shift δ 4 (s) and the absolute value of the spin four partial wave |S 4 (s)|.In this case, the δ 4 (s) clearly passes through π/2 around s ∼ 40, and it keeps growing at higher energies.In the complex plane we find two zeros: one close to the real axis, and a heavier one with large imaginary part.The former is interpreted as a weakly coupled spin four particle, the latter as a spin four σ-like resonance.Unitarity saturation in the spin four channel is not yet achieved by our numerics as it can be appreciated by looking at the solid blue line in the bottom-left panel.We know empirically that the lack of convergence does not affect the position of the resonances, but rather its width (the imaginary part) -see also the discussions in [7,14,46].
The theory of complex angular momenta [75] -see also [76] for a recent review -suggests that resonances with higher spins are different realizations of the same object, the Reggeon.Reggeons have a mass that continuously depend on the spin-ℓ parameter.Projecting the amplitude on various real spin channels we expect to follow the resonance along its trajectory, the famous Regge trajectory.The analytic continuation in spin of partial waves is performed through the Froissart-Gribov formula where Q ℓ (x) is the Legendre polynomial of the second kind.
We collect the various resonances in the Chew-Frautschi diagram for different spins ℓ ≤ 10 in figure 5 (top panel).The resulting distribution is suggestive.We clearly see two trajectories. 10The red dots belong to the leading trajectory, since it contains the particles that have the lowest mass for each spin; the blue dots to the sub-leading trajectory or more informally the daughter trajectory.The resonances in the leading trajectory are weakly coupled, and the linear universal behaviour emerges in our data [77] as we follow it to higher spins.The daughter particles are σ-like and deep in the complex plane, therefore the linearity is lost.In figure 5 (bottom left) we zoom in the low spin region where ℓ ≤ 2. We observe the crossing of the two trajectories around spin ℓ c ∼ 1.4.We indicate the lightest σ-like resonance of ℓ = 0 studied in figure 3 with a green dot.In the zoomed figure we appreciate how it does not belong to any trajectory, but rather is an isolated particle.This is in agreement with the fact that the analytic continuation in spin of our ansatz only converges for Re ℓ > ℓ 0 = 0.
In figure 5 (bottom right), we collect the position of the resonance in the complex plane.On the bottom left of the figure, we see how the leading trajectory moves and similarly for the sub-leading on the top part of the plot.In the complex plane s it becomes clear that the crossing of the two trajectories observed above is just the result of a projection.Moreover, we find a surprising behaviour between spins 2 ⪅ ℓ ⪅ 3. The corresponding arcs are highlighted with 'circles'.We believe the behaviour there is not physical, but rather consequence of poor convergence in that region of the complex-spin plane.We remind the reader that our numerics are affected by a systematic error due to truncation both in the size of the ansatz N max and the number of spin constraints imposed.The amplitude studied in this section gives the best approximation of the minimum value of ḡ4 at hand, but it is not necessarily well converged for all energies and spins.Indeed, the position of the resonance in those regions still has a strong dependence on N max .Analytic continuation of unitarity for complex spin is rigorously possible only in the elastic region.When the scattering is elastic it is possible to prove that |S ℓ | = 1 for Re ℓ > ℓ 0 .For the scattering of massless particles there is no such statement.However, since the extremal Do not trust ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲  2.0 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ < l a t e x i t s h a 1 _ b a s e 6 4 = " H 6  Re(s R ) < l a t e x i t s h a 1 _ b a s e 6 4 = " P x 1 k W 8 5 n q g c + Figure 5: Chew-Frautschi diagram for the amplitude that minimize ḡ4 .On the top panel, the leading (in red) and sub-leading (in blue) trajectories are presented up to spin 10.The spin 0 (green) resonance is not part of a trajectory.In this plot, we show various N max to express the convergence of the position of the zero and we use darker color for larger N max .On the bottom left panel, we zoom on the trajectories at low spin.The leading (red) is well converged in this region, this is not the case for the sub-leading (blue) and we expect it to move up .On the bottom right panel, we present the position of the resonance in the complex plane for both the leading (circle) and sub-leading (triangle) trajectory.In the circled region, we do not trust the position of the resonance.
amplitudes we study are mostly elastic at low energies and for all spins, it would be natural to expect that |S ℓ | ≈ 1.In appendix D, figure 16 we plot the absolute value of S ℓ on the real axis for for various spins in the neighboring of Re sR .We check that our expectations are realized when the Regge trajectories are smooth and monothonic, but fail for the problematic spins 2 ⪅ ℓ ⪅ 3, the circled arcs in figure 5.
We call ℓ(t) the parametric curve that describes the Regge trajectory as function of t.The highest intercept ℓ(0) among all Regge trajectories determines the asymptotic behaviour of the total cross section From figure 5, we can extract the intercept of the leading Regge trajectory to be ℓ(0) ≈ 0.75.However, due to the crossing phenomenon at spin ℓ c ≈ 1.4 the daughter trajectory becomes dominant at higher energies.Even though the numerical convergence is harder for the daughter trajectory, we do estimate that the leading intercept will have ℓ(0) ∼ 1, a feature typical of the pomeron trajectory. 11,12 Finally, we look at the total cross section as a function of s in figure 6. Curves with different colours correspond to different values of N max .At low energies we can compare the non-perturbative cross section with the EFT approximation in dashed black In this case, since the UV physics is strongly coupled, the cutoff of the EFT approximation is given by naive dimensional analysis and is expected to be around s ∼ 1.However, when s < 1 the EFT approximation is good for all angles, and this explains the nice agreement of the phase shifts in figure 4, where the EFT approximation is plotted in dashed red.The peak in the total cross-section happens at the mass of the lightest spin two resonance.
One could repeat a similar analysis to all amplitudes in the Region I.By simple inspection, we can generically say that all amplitudes in this region share the same qualitative features, although the details of the spectrum change.

Asymptotic bounds and tree level physics
When we take ḡ3 large, we expect the EFT approximation in equation (1.2) to be valid in the regime when To retain the relevant terms when s ≪ |ḡ 3 | −1 , it is convenient to define the rescaled variable ŝ = s|ḡ 3 | −1 .In this new regime, the amplitude can be approximated by 13T = 1 ḡ2 where the logs are suppressed by higher powers of ḡ3 .The large ḡ3 parameter plays the role of ℏ −1 in a putative effective action expansion, and the the amplitude is well approximated by the expansion of a tree level UV theory.Following this logic, we might expect the coefficient of the O(ŝ 4 ) term in (2.16) to be an order one number.The only possibility is that ḡ4 ∼ ḡ2 3 , and this prediction is precisely satisfied by the asymptotic behaviour of the boundary region in 1b.
In the ḡ3 → ∞ limit, we can apply the technology developed in [64] to determine bounds on the ratio ḡ3 √ ḡ4 .We obtain14 which is compatible with our asymptotic numerical estimate (2.2).The upper bound is saturated by a simple tree level massive scalar exchange [64], compatible with the spin zero dominance observed in figure 2, and the presence of a light weakly coupled scalar resonance in figure 3. The difference in the lower bound is likely due to the difficulty of the primal numerical algorithm to converge at large negative g 3 , therefore we expect the gap between the two methods to close.For the lower bound, there is no known function that saturates it.
Studying the extremal amplitudes that saturate the asymptotic bounds we can guess the tree level UV completion in the large ḡ3 limit.Moreover, using some simple unitarization model, we can estimate the behaviour of the amplitude away from the tree-level approximation for large, but finite ḡ3 .We report this analysis in the next paragraphs.

Region II (string-like)
In region II, the amplitude receives contribution from all spins starting at spin ℓ = 2 according to figure 2. Performing a partial wave analysis we observe the presence of weakly coupled massive resonances for all spins ℓ ≥ 2. We estimate the mass of these resonances and produce the corresponding Chew-Frautschi plot of the spin ℓ of the particle as a function of its mass squared m2 ≡ √ g 2 m 2 for each value of ḡ3 ≪ −1. 15 In figure 7 we show the resonances for ℓ ≤ 6 for two values of ḡ3 .All the points approximately lie on a single straight line, and we fit it with an ansatz of the form The coefficient α(0) is called the Regge intercept and α ′ the slope.The estimation of the coefficients α(0) and α ′ of the Regge trajectories for different values of ḡ3 is presented in figure 8. To match the tree level EFT description in (2.16), all mass parameters in the amplitude must scale as ḡ−1 3 at leading order.Indeed, we experimentally observe that16 We can conclude that in this region, the amplitudes exhibit a string-like behavior.In the limit ḡ3 → −∞, the amplitude at the boundary is described by a meromorphic string-like amplitude with the following properties • Spin ℓ contribution given by the limit ḡ3 ≪ −1 of figure 2. In particular, it has no spin 0 exchange.
• Leading Regge trajectory with intercept and slope given by (2.19)We are not aware of any string model which generates amplitudes compatible with these properties.For the recent works on string amplitudes, see [77,[79][80][81].
Region III (spin-0 exchange) In region III, the boundary precisely approaches the tree level positivity prediction lim (2.20) The positivity bound is saturated by a simple tree-level scalar exchange where λ is the dimensionless coupling and m the mass of the scalar exchanged.Expanding this amplitude around s = 0 keeping x ≡ cos θ fixed we obtain an expansion of the form (2.16).Matching the two expressions, upon identifying s in (2.21) with ŝ, we can express the low energy Wilson coefficients g 2 , g 3 in terms of the UV parameters λ, m The matching predicts also g 4 = λ 2 2m 8 , which is consistent with (2.20).In addition to the correct asymptotic behavior of the bound, this statement is supported by two additional observation.First, in this region, the sum-rule (2.6) is dominated by the spin-0 contribution as shown in figure 2. Second, in figure 9 we plot the real and imaginary part of the spin zero partial amplitude extracted on the boundary of figure 1 at ḡ3 = 1/2.Our result is depicted in red.In blue and orange we provide an estimate of this partial amplitude obtained by applying the Inverse Amplitude Method (IAM) to (2.21), and using the value for the mass and the coupling given by the matching conditions in (2.22) for ḡ3 = 1/2.We review the IAM in appendix E. The agreement is good in the region around the light resonance.At higher energies the amplitude is not trivial but contains broad higher spin resonance.Their mass does not scale with ḡ3 , so in units of the mass of the light resonance ŝ, they decouple.

S-matrix machinery for massless particles and its improvements
In this section we introduce and explain our improvements of the numerical S-matrix bootstrap machinery needed for studying massless particles.We start in subsection 3.1 by reviewing the standard machinery.Our improvements are then presented in subsections 3.2 and 3.3.

Machinery review
In order to study scattering amplitudes numerically in the generic scenario we use the machinery introduced in [3] combined with an extension of this machinery introduced in [44] needed for working with massless particle.It consists of writing the following ansatz Ansatz 1: where α abc are real coefficients.Crossing requires that they are fully symmetric in their indices.Due to the relation among the Mandelstam variables s + t + u = 0, not all terms are independent in the expansion (3.1), it is standard to require ∀(abc) ̸ = 0 : which removes the redundant terms.The ρ-variable is defined as where m is the mass of the external particle (in our case m = 0) and z 0 is an arbitrary parameter, numerical results do not depend on it.We choose z 0 = −1.The infinite sum in (3.1) is truncated in practice in such a way that max(a + b + c) ≤ N max .The numerics is ran for several values N max .This allows for an N max → ∞ extrapolation.The first term in the ansatz (3.1) has the square root branch point at s = 0, however the effective amplitude of Goldstones given by (1.2) -(1.4) has a log branch-point because of the log-terms in (1.4).The latter can never be represented by the former at finite N max .In order to deal with this issue we have added the additional term N(s, t, u) in (3.1).Let us explain how to obtain its explicit expression.Let us first denote the non-analytic part of the effective amplitude (1.2) -(1.4) (which contains the log-terms) by We can then also define where we the χ-variable is defined as and the H-function is given by The non-analytic expression (3.4) has an unlimited growth at large energies and, thus, violates unitarity.The term (3.5) instead remains finite at large energies.It is constructed in such a way that at low energies it reproduces (3.4), namely Next, we require that at low energies the ansatz (3.1) precisely reproduces the effective amplitude (1.2).Performing the expansion of (3.1) we simply get where k i (α, x) are some functions which depend on the coefficients of the ansatz and x ≡ cos θ, where θ is the scattering angle.These functions can be easily obtained in Mathematica, their form, however, is too large (and depends on N max ) in order to be written here.Comparing this expansion with (1.2) -(1.4) we conclude that After solving these constraints and plugging the solution back to (3.1) we obtain the ansatz which is fully compatible with the effective amplitude (1.2) -(1.4).
The ansatz (3.1) satisfies crossing and maximal analyticity by construction.The nonlinear unitarity formulated in the semi-definite positive way (A.12) is imposed numerically using SDPB [82,83].
The parameter g 2 defines the mass scale.In practice for performing the numerics we set it to some constant value, for example one could chose g 2 = 1.Our results do not depend on this choice since we always work with dimensionless quantities.It can happen, however, that at finite N max there is a preferred value of g 2 for which the numerics converges faster.In table 1 we show the lower bound on ḡ4 (keeping ḡ3 free) for different choices of g 2 .We see that the best bound is achieved for roughly g 2 = 500.When we construct bounds on ḡ4 as a function of ḡ3 the optimal value of g 2 changes with ḡ3 .As a result one needs to carefully adjust it in order to achieve the most optimal numerical convergence.

Improvement 1
Let us consider the first derivative in t of the amplitude in the forward limit, namely ∂ t T (s, t = 0, u). (3.11) Up to O(s 5 ), (3.11) is finite in the series representation (1.2).It is reasonable to expect that (3.11) is finite at any order but we do not know of any general proof.From now on we assume that (3.11) is finite.This for instance allows to write the following sum-rule for the g 3 coefficient See appendix F for its derivation.Consider now the ansazt (3.1).Evaluating (3.11) using this ansatz we immediately see that (3.11) diverges.This is an unpleasant behaviour which prevents us for instance from using the sum-rule (3.12).In what follows we introduce a minor modification of the ansatz (3.1) which removes this divergence.
Let us define the following auxiliary variable Using it we can then define the following modified r-variable r 0 (z) ≡ 1, Using this variable we propose the new ansatz Ansatz 2: One can explicitly check that the ansatz (3.15) has finite value of (3.11).Below, we will check that this modification does not change the results of the bounds, see for example figure 13 for comparison of extrapolated bound.

Improvement 2
In order to numerically impose the unitarity condition (A.12) we need to project scattering amplitudes to partial amplitudes.The latter are labelled by the angular momentum ℓ = 0, 2, 4, . ... In theory we need to impose (A.12) for infinitely many values ℓ.In practice, however, this is impossible, and we impose (A.12) for a finite set of angular momenta ℓ = 0, 2, 4, . . ., L max .The parameter L max is the new player in the setup.In order to obtain physical results one needs to find the L max value high enough such that the numerical output stabilizes and remains independent of the change of L max .For some numerical problems (especially for massless particles) the output keeps changing with L max .In those situations one runs the numerics for several different values of L max and then performs the extrapolation L max → ∞.
In [47] it was observed that the numerics breaks down for large values of L max .This was a severe obstacle for performing the L max → ∞ extrapolation in [47] reliably.In this subsection we reproduce the same phenomenon in the case of neutral Goldstones using the ansatz (3.15) and propose a modified ansatz which solves this problem.Figure 10: The lower bound on ḡ4 using the ansatz 2. Both curves are constructed using the ansatz (3.15) with (3.16).In red curve with n max = 3, for large values of L max it starts changing rapidly.We refer to this region as the ramp.For the red curve, the ramp disappears and we are left with the stable bound.The bounds here are constructed with N max = 18.
Recall, that the ansatz (3.15) contains the log-terms via the expressions (3.4) and (3.7).We identified the ramp behaviour with the expression (3.7).It turned out that (3.7) is too restrictive.In order to add more freedom we can instead write where n max is a truncation parameter for the sum.The real coefficients γ abc are symmetric in the last two indices as required by crossing.We tune these coefficients in such a way that in the low energy expansion we get This ensures that the ansatz correctly reproduces the effective amplitude (1.2) at low energies.Due to this requirement (3.7) is the same as (3.16) but only up to O(s 3 ) terms.
The main result is given in figure 10.In this figure we compute the lower bound on ḡ4 (keeping ḡ3 free) as a function of L max using the ansatz (3.15) with a different amount of freedom in the log term N. We observe that for low n max the bound becomes abnormally strong.Increasing L max further we observe that the numerics breaks down (becomes unfeasible).This is clearly an unphysical and problematic behavior.The rapid increase of the red line will be referred to as the ramp. 17As we increase n max , the ramp is "pushed away" to higher L max and the extrapolation to infinite spin can be taken.For moderate values of L max the two curves are consistent with each other, however, for large L max the blue curve remains stable and does not exhibit any ramp behavior.Notice, that as the two curves are built with different n max , the freedom in ansatz used for the blue curve is bigger which explains why they do not overlap at intermediate spin.In the limit N max → ∞ this difference completely   Let us now focus on the case of N max = 20 and compare bounds with several different values of n max .We do it in figure 11.We can see that the ramp disappears in all these cases.In figure 12 we fix n max = 4 and scan instead over various values of N max .We see that the behaviour of the bound remains stable with L max for any N max (no ramp).Using the data presented in figure 12 we can perform the L max → ∞ extrapolation.We use several different fits in order to estimate the extrapolated value and its error.In the repository with our numerical data linked to this paper we provide further details about the fits used.In figure 13a we presented our L max → ∞ extrapolated bound (with error bars) on ḡ4 as a function of N max .
We have introduced several types of ansatz which at finite N max lead to consistent, but slightly different answers.Let us now perform the N max → ∞ extrapolation and show that different choices lead to the same result within the error bars.We will focus again on the ḡ4 minimization problem (keeping the ḡ3 value free).Our extrapolation for various ansatz choices is presented in figure 13a.The ḡ4 bound in the N max → ∞ limit for various ansatz choices is summarized in particular in figure 13b.

Model-dependent bounds
Let us now study the allowed space of amplitudes in the scenario in which we have an EFTinspired model.We impose that the effective field theory approximation of the amplitude is valid in the extended region s ∈ [0, M 2 ] for all angles.This is imposed by the condition (1.11).In practice we impose non-linear unitarity in the whole range of energies s ∈ [0, ∞].Thus, we can impose the condition (1.11)only on the imaginary part of the amplitude (the real part will be adjusted by the numerics accordingly), namely We emphasize here that the model is defined by a 'band' given by Im s T EFT (s, cos θ) ± err(s, cos θ) constraining the amplitude up to a cut-off scale M 2 .For example, experimental measure of the differential cross-section could be used to define the model.In that follow, we will describe how this can be implemented numerically in subsection 4.1 and present the result for a simple model in subsection 4.2.

Machinery for the model-dependent bounds
Here we explain how to construct numerically scattering amplitudes which obey crossing, maximal analyticity and unitarity at all energies and satisfy the condition (4.1) describing a model.Since we have two regions with s ≤ M 2 and s > M 2 it is natural to write the ansatz as a sum of two terms Here the first term takes care of the low energy behaviour of the amplitude, in other words it must approximately describe (1.2) in the extended region s ∈ [0, M 2 ].The second term takes care of the high energy behaviour and it is adjusted by the numerics in such a way that the full ansatz (4.2) obeys non-linear unitarity.Our proposal for the two terms in (4.2) reads as18 where N(s, t, u) is defined in (3.5) and the two types of the ρ-variable are defined as with r 1 defined by (3.14) using ρ 1 .Here z 0 < 0 and z ′ 0 ≤ 0 are real negative parameters which can be chosen at our will.In practice we choose z 0 = −1 and z ′ 0 = 0.The above variables are constructed in such a way that they contain a branch cut starting from s = 0 and s = M 2 respectively.
In order to impose unitarity (A.11) we choose a set of points using the Chebyshev grid.In practice we pick 100 points in the s ∈ [0, M 2 ] region and 200 points in the s > M 2 region.We impose non-linear unitarity at these 300 points and for angular momenta ℓ = 0, 2, . . ., 150.
In the region s ∈ [0, M 2 ] on top of the non-linear unitarity we also impose the condition (4.1).In practice this is done as follows.We pick a linear grid in the x ≡ cos θ variable and use 10 points in the interval x ∈ [0, 1]. 19We impose (4.1) for all possible combinations of 100 points in s (distributed with the Chebyshev grid) and the 10 points in x. 20 On top of it we also project the condition (4.1) into partial wave.We impose this condition only for spin-ℓ which lead to non-zero projection of the 'band' and the precise number of spin depend on the model.We impose the projected condition (4.1) for 100 points in s and each ℓ.
We observe that for the runs we have performed, one can choose a fixed size of ansatz T 1 ansatz (4.3) needed to reproduce the desired low energy behavior.Then an extrapolation can be performed in N 2 max which dictate the freedom of the high energy behavior.In practice, we chose N 1 max = 10, n max = 8 and converge in N 2 max .

Result using a model for the IR amplitude
Here we define a simple model based on the EFT expansion of the amplitude.The error function can be defined as follows.In appendix B we compute the imaginary part of T (5) EFT term exactly.We can equate the error function to this term.However, in order to take into account corrections due to higher order terms T (n) EFT with n ≥ 6 we replace ḡ3 by ϵ and use ϵ as an extra parameter.We find then the 'band' of the model by This model is thus defined by two dimensionless parameters, namely ξ defined in (1.9) and ϵ defined in (4.7).In what follows we will focus our attention on the ξ = 1 case.This choice favors the class of amplitudes with strongly coupled UV completions.See the discussion below (1.9) for the explanation.Regarding ϵ, we impose the condition (4.1) for two different values, namely ϵ = 1/2 and ϵ = 1.The resulting bound is given in figure 14.As one can see the lower bound agrees with the universal bound in figure 1a.Indeed, the lower inequality of the extended EFT condition in (4.1) is never active, in agreement with the fact that the amplitudes in Region I admit a strongly coupled UV completion, and that in some sense are extremal.
19 For a t − u symmetric amplitude, the amplitude is symmetric in x → −x and we could only consider the half interval x ∈ [0, 1].In general, one could choose a grid in the full interval x ∈ [−1, +1]. 20In practice, we observe that convergence in the size of the grid in x is fast and 10 points were a safe choice. - < l a t e x i t s h a 1 _ b a s e 6 4 = " o Z q 8 n 0 w L e y v l P a Y Z R x t R w Y b w + S n 9 n 1 z v u f 6 B 6 1 3 u l 8 p n 0 z j y Z I t s k 1 3 i k y N S J h e k Q q q E E 0 n u y S N 5 c m 6 c B + f Z e Z m 0 5 p z p z C b 5 B u f 1 A 6 K 3 k F 4 = < / l a t e x i t > A < l a t e x i t s h a 1 _ b a s e 6 4 = " L z 6 / 9 s D U q x r g 0        It is interesting to see how the EFT amplitude is UV completed along the boundary.In figure 15, we show the imaginary part of the amplitude in the forward limit at two pairs ot points for the case ϵ = 1.For s < M 2 , the constraint imposed (4.1) is clearly satisfied and is saturated along the upper boundary.In contrast, for s > M 2 , the imaginary part is unitarised differently depending on the point on the boundary.

Conclusions and outlook
In this paper we initiated the exploration of the space of 2 → 2 scattering amplitudes of massless neutral Goldstone bosons in four dimensions.We numerically determined the allowed region of the first two non-universal Wilson coefficients appearing in the 2 → 2 scattering, dubbed ḡ3 and ḡ4 both in the universal scenario, and by imposing a low energy model for the amplitude.
In the universal scenario we used the numerical approach of [3,44] where an analytic ansatz is written for the amplitude and unitarity is imposed numerically.In this paper we proposed two new improvements of the ansatz in section 3. The first improvement allows us to use the usual Bootstrap ansatz to match the fixed-s, small t behaviour of the amplitude predicted by the EFT expansion in (1.3).The second removes the ramp instability for large number of angular momentum constraints L max discovered in [47].We presented the main result in the universal scenario in figure 1.
On the boundary given by the orange line in figure 1 we can reconstruct the amplitudes numerically.Analyzing the branching ratios of the cross section in various spin channels and the behaviour of the lightest spin zero resonance, we identified three different regions along the boundary.The two asymptotic regions correspond respectively to weakly coupled string amplitudes with approximately linear Regge trajectories, and weakly coupled scalar exchanges.In between these two regimes, the amplitude resembles a strongly coupled theory like QCD.For the special point that minimizes ḡ4 , we performed for the first time a thorough analysis of the spectrum of resonances continuing the scattering amplitude to complex spins.We could interpolate the first two Regge trajectories by following the resonances in spin, and extract reliably their intercepts without fitting.We observed a curious crossing phenomenon among the two trajectories.
The amplitudes we extracted on the boundary for large values of ḡ3 and ḡ4 are not fully converged for the finite size ansatz we used.This is rather unsatisfactory.In the future, one could attempt to solve this issue by centering the ρ-variables at multiple different points.One of the realizations of this idea called the wavelet ansatz was advocated for in [25].Numerical convergence is controlled also by the number of spin constraints that should be taken to be very large.However, computing the partial wave projection with high precision and for very large spin is numerically expensive.In [46], the authors introduced the notion of unitarity in the sky, and showed how dramatically improves the convergence in the spin cutoff.We believe that by combining the improvements introduced in this paper with both the ideas mentioned in this paragraph might pave the way to more efficient numerical studies of massless scattering amplitudes.
The universal bounds determined in this paper apply to the dilatons of N = 4 SYM on the Coulomb branch.It would be interesting to consider also the other massless Goldstones of the R-symmetry breaking, and study the mixed system of dilatons and Goldstones.Another possible direction is to generalize the Bootstrap of neutral Goldstones to other dimensions.In three dimensions, we could study the low energy dynamics of M 2 branes, using maximal supersymmetry to obtain the bounds on the Wilson coefficients from the scattering of neutral massless scalars.On the other hand, in higher dimensions such bounds could be used to constrain the Wilson coefficients of the effective action of various strings compactifications and to test the consistency of the available non-perturbative results with the principles of analyticity, crossing, and unitarity.
In the model-dependent scenario we assumed that the amplitude is described by an EFTinspired model at energies below a chosen cut-off scale M .In section 4.1 we proposed a concrete numerical implementation of this idea inspired by [21].The main result in the modeldependent scenario is given in figure 14.This method is not limited to massless particles.One could use this approach also for the scattering of physical pions, and improve the results obtained in [7] by injecting the available experimental or lattice data as low energy constraints into the bootstrap setup.
As briefly reviewed in the end of appendix C the Wilson coefficients of the dilaton scattering in six dimesions is related to the difference of the UV and IR a-anomalies denoted by ∆a.In general, as it is well known, there are no bounds on ∆a in 6d from the 2 → 2 scattering of dilatons.However, adding an extra assumption/knowledge about their IR behaviour up to some energy scale one can apply our model-dependent scenario to get some bounds on ∆a.In QFTs with explicit conformal symmetry breaking one can introduce a massless scalar probe field (also often called the dilaton) which carries information about ∆a.No new bounds can be constructed on ∆a in this case using our technique.obtain the full amplitude which obeys (A.11).Let us write this full amplitude in the following form T full (s, t, u) = T tree (s, t, u) + N (s, t, u).

(B.2)
Here the tree level amplitude in the small energy expansion is given by (B.We solve this equation iteratively order by order in small energy expansion by simply requiring non-linear unitarity (A.11). 21Off course, this procedure stops at the order in which particle production appears as emphasize in (B.3).
Leading order computation Let us start by focusing on the very first term in (B.1), namely we simply consider the following tree level amplitude According to (A.2) we can compute the partial amplitudes.Only ℓ = 0 and ℓ = 2 components are non-zero.They read where the kinematic coefficients read as We can obtain the leading order of the right-hand side in (B.3) by (B.6), we get the following result then The function f (s|t, u) has the following small energy expansion and the leading term in this expansion has just been found precisely and reads This is precisely the result quoted in (1.4).
The case of 6d As another example let us also write the effective amplitude in 6d.It reads EFT (s, t, u) + T EFT (s, t, u) + T EFT (s, t, u) + T EFT (s, t, u) + O(s 6 ), (B.18) where we have EFT = g 3 stu, T EFT (s, t, u) = g 5 stu s Comparing this expression with (1.2), we conclude that for the first amplitude we have Analogously we can obtain the low energy coefficients for the second amplitude, they read T stu pole : Plugging the results (C.5) and (C.6) into the definition of dimensionless couplings (1.5) we obtain the following expressions We summarize the above result in a compact form in table 2. For generic values of the coupling λ the amplitudes (C.1) and (C.2) do not satisfy the requirement of the non-linear unitarity.However, in the extreme weakly coupling limit λ → 0 this problem is alleviated and unitarity is reduced to the positivity of the residues.In this sense, both amplitudes satisfied "weak coupling" unitarity [64].
Let us now discuss if any bounds on the parameter ḡ3 can be constructed.From (C.7) we observe that the amplitude T spin 0 allows for infinitely large values of ḡ3 in any number of dimensions, namely This means that there always exists a weakly coupled theory with a very large positive value of ḡ3 , thus by construction no upper bound exists for this quantity.Analogously we can conclude that there is no lower bound on ḡ3 in d > 2 space-time dimensions.This follows from the ḡ3 expression for the T stu pole amplitude in (C.8), namely (C.10) The latter equality holds in d > 2 and can be straightforwardly checked by using the definition (C.3).Also notice, that from the definition of γ(d) (C.3), it follows that g 2 ≥ 0 for the amplitude T stu pole as required by positivity.Rerunning these arguments for the ḡ4 parameter we observe that no upper bound on ḡ4 can be constructed.However, no statement can be made about the lower bound, which is consistent with our numerical result presented in figure 1.
It is interesting to discuss the absence of bounds on ḡ3 in the context of the 6d a-theorem.In [84] the authors used the massless probe field (often ambiguously called the dilaton) to measure the difference between the UV and the IR a-anomaly.This difference is denoted by ∆a.By using the Weyl anomaly matching they obtained the low energy behaviour of the scattering amplitudes of the probe field.Their result is given by equations (3.18) and (3.19) in [84].We can connect their notation with ours by comparing equations (3.18) and (3.19) in [84] with (1.2).We find that

E Inverse amplitude method
Using elastic unitarity it is possible to unitarize tree-level amplitudes in the s-channel.One method for doing this is called the Inverse Amplitude Method (IAM) (see for example [85,86]).The main idea is to assume elastic unitarity which implies . (E. 3) The partial amplitude (E.3) is clearly a pure phase and, thus, obeys elastic unitarity.The real function α ℓ (s) is called the seed.Its explicit form depends on the model under consideration.We refer to the partial amplitude (E.3) as the amplitudes obtained with the IAM 1.
In the case of massless particles, we know that the imaginary part arises from the log(−s) term.Using this information we can, thus, make another choice for solving (E.In figure 9 we have plotted the spin zero partial amplitudes obtained using IAM1 and IAM2 for the three-level amplitude (2.21).Both IAM1 and IAM2 expressions have a similar behaviour.The advantage of IAM 2 is that it naturally reproduces the correct low energy behaviour (1.2) up to O(s 4 ) order.
F Sum-rules for the coefficients g n The coefficients g n are defined in (1.2).There are relation which relate the value of these coefficients to certain integral over the imaginary part of the interacting part of the scattering amplitude.We refer to this relations as the sum-rules.The most efficient way to derive this sum-rules is by using the technology of [69].There the authors introduced the notion of arcs a n (s, t).They are defined as The arcs (F.1) are related to the g n coefficients.Indeed, by consider a contour C s of small radius, we can use T EFT in (F.1).For example, the first arc is given by a 0 (s, t) = 2g 2 − g 3 t + O(s 2 ) (F.3) The two simplest sum-rules follows directly and reads Using the positivity constraint (A.14) one immediately concludes that g 2 ≥ 0. No similar statement can be made about g 3 .The above sum-rule for g 3 exists only if the first derivative in t in the forward limit, namely ∂ t Im T (s, t = 0), is finite.This not required by basic axioms.However, it is true in all example we encounter.It would be interesting to understand if this statement can be proven.From the perspective of the arc (F.3), this is equivalent to the statement that the O(s 2 ) satisfy the same property.
It is very useful to decompose the amplitudes in (F.4) and (F.5) into partial amplitudes according to (A.6).We can then define the following coefficients

G Large energy constraints
In this appendix we study the large energy behaviour s → ∞ of the simplest ansatz (3.1) reviewed in section 3.1.Let us write this ansatz here again for convenience The term N(s, t, u) contains the logs.Its precise form depends on the dimension.In what follows we will completely ignore N(s, t, u) for simplicity.However, once N(s, t, u) is known the discussion below can be straightforwardly adopted in order to take it into account.The ρ-variables were defined in (3.3).In the s → ∞ limit the ρ-variables have the following expansions where x is defined as the cosine of the scattering angle θ, see (1.5).Plugging these into the ansatz (G.1) we can write the following expansions (G.4) The coefficient A 0 is real instead the coefficients A n (x) with n ≥ 1 have both real and imaginary parts.
The expansion (G.3) can then be used to evaluate the large energy behaviour of the partial amplitude related to the scattering amplitude via (A.2).Let us write this expression here for convenience The vertical dashed line indicates the position of the absolute minimum of the bound.We plot the ratio: ḡ3 / √ ḡ4 vs ḡ3 .Dashed lines indicate the approximate asymptotes of the bound.

Figure 2 :
Figure 2: The numerical values of the branching ratio of the cross section ḡℓ 2 defined in (2.6) of the amplitudes on the lower boundary in figure 1 as a function of ḡ3 .The colors represent different values of the angular momentum ℓ = 0, 2, 4, 6.
t e x i t s h a 1 _ b a s e 6 4 = " H 6 J k I + 2 b D l W C b T S T t 0 8 m D m R g y h / o o b F 4 q 4 9 U P c + T d O 2 y y 0 9 c C F w z n 3 c u 8 9 f i K 4 A s v 6 N p a W V 1 b X 1 k s b 5 c 2 t 7 Z 1 d c 2 + / r e J U U t a i s Y h l 1 y e K C R 6 x F n A Q r J t I R k J f s I 4 / u p r 4 n Q c m F Y + j e 8 g S 5 o Z k E P G A U w J a 8 s y K A + w R 8 p t w X H N 8 I r H y 7 o 4 9 s 2 r V r S n w I r E L U k U F m p 7 5 5 f R j m o Y s A i q I U j 3 b S s D a w g g 3 h 6 1 P 6 P z k r u 9 6 u y 4 4 r p b 3 9 U R w z Z J 1 s k C 3 i E Z / s k U N y R O p E k F t y T x 7 J k 3 P n P D j P z s t n 6 5 g z m l k l P + C 8 f g B o o 5 U 0 < / l a t e x i t > Re 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " M 4 D O r c M x 1 C J V r U D i N I L M b W k 6 o K c = " > A A A B + H i c d V D L S g M x F M 3 4 r P X R q k s 3 w S K 4 G j L t 9 L E s u n F Z 0 T 6 g L U M m z b S h m c y Q Z I Q 6 7 Z e 4 c a G I W z / F n X 9 j + h B U 9 E D g c M 6 9 3 J P j x 5 w p j d C H t b a + s b m 1 n d n J 7 u 7 t H + T y h 0 c t F S W S 0 C a J e C Q 7 P l a U M 0 G b m m l O O 7 G k O P Q 5 b f v j y 7 n f v q N S s U j c 6 k l

3 .
s 1 Y 7 x + A H r L d P e u e T o w = = < / l a t e x i t > × 10 -5 < l a t e x i t s h a 1 _ b a s e 6 4 = " N d n a + j z 8 T 1 b D 7 B I D 7 k 6 c = < / l a t e x i t > |S 4 |

Figure 4 :
Figure 4: On the left, we plot the phase shifts for both spins ℓ = 0, 4 in red, and the absolute value |S ℓ | of the corresponding partial waves in blue.Red dashed is the one-loop EFT approximation expected to be reliable up to the scale s ∼ 1.On the right, we plot the absolute value |S ℓ | in the complex s plane.There we observe the presence of zeros that we interpret as resonances.

10 < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 e t 9 h y 4
t e x i t s h a 1 _ b a s e 6 4 = " y h b 7 B e P w D C Q 5 N Z < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " H 6 J k I + 2 b D l W 2 j y y p s Q 3 M W X l 0 n j r O x e l J 2 7 8 2 L l O o s j R 4 7 I M S k R l 1 y S C r k l V V I n n C T k m b y S N + v J e r H e r Y 9 5 6 4 q V z R T I H 1 i f P x R L l G M = < / l a t e x i t > Re(s R ) < l a t e x i t s h a 1 _ b a s e 6 4 = " H 6 J k I + 2 b D l W 2 j y y p s Q 3 M W X l 0 n j r O x e l J 2 7 8 2 L l O o s j R 4 7 I M S k R l 1 y S C r k l V V I n n C T k m b y S N + v J e r H e r Y 9 5 6 4 q V z R T I H 1 i f P x R L l G M = < / l a t e x i t >

Figure 6 :
Figure6: Cross section σ tot = s −1 Im s T (s, t = 0) for the amplitude that minimize ḡ4 .The dashed black line is the EFT expansion in(2.14).The different colors is the cross section of the amplitude with different N max .

Figure 7 :
Figure 7: The masses of the lightest resonances plotted versus their spin.They lie on a line.We interpret this line as a Regge trajectory.

Figure 8 :
Figure 8: Estimation of the coefficients α(0) and α ′ of the Regge trajectories defined in (2.18) for different values of ḡ3 .

Figure 9 :
Figure9: Real and imaginary part of the spin zero partial amplitude S 0 , obtained on the boundary of figure1at ḡ3 = 1/2.Our numerical result is depicted in red.The blue and orange lines provide the rough estimates obtained using the inverse amplitude method reviewed in appendix E.

Figure 11 :Figure 12 :
Figure 11: Lower bound on ḡ4 for the two types of the ansatz (3.1) and (3.15) and various values of n max in (3.16).Here we focus only on N max = 20.

Figure 13 :
Figure 13: Extrapolation of the lower bound on ḡ4 in different scenarios.In the left plot we show the numerical data by dots and the extrapolation by the solid line.In the right plot we show the extrapolated bound for different cases.Solid lines show the error bars.The results are fully consistent among each other within the error bars.
e y t h P W p p g x t R H k b w u e n 5 H 9 S O 3 T 9 Y 9 e 7 O i q W y r M 4 c r A L e 3 A A P p x C C S 6 h A l V g I O E e H u H J u X E e n G f n Z d o 6 5 8 x m d u A b n N c P p D y Q X w = = < / l a t e x i t > B < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 d F v e O r 9 L H 2 g D 9 W u a 0 a Q 0

m 7 a 8 Q
w 5 C e 8 e F D E q 7 / j z b + x s w i u D w o e 7 1 V R V S 9 M p D D o e e / O 3 P z C 4 t J y b i W / u r a + s V n Y 2 q 6 Z O N U c q j y W s W 6 E z I A U C q o o U E I j 0 c C i U E I 9 H J y P / f o t a C N i d Y 3 D B I K I 9 Z T o C s 7 Q S o 0 W w h 1 m F 6 N 2 o e S 5 / p n v H 3 v 0 N / F d b 4 I S m a H S L r y 1 O j F P I 1 D I

l a t e x i t > DFigure 14 :
Figure 14: The allowed region of parameters is enclosed inside the islands.Two colors represent two different values of the parameter ϵ introduced in (4.7).The two values we use are ϵ = 1/2 and ϵ = 1.For building this plot we also chose ξ = 1 defined in (1.9).The orange line indicate the bound of figure 1.In figure 15, we plotted the imaginary part of the amplitude at the points A,B,C,D.
t e x i t s h a 1 _ b a s e 6 4 = " G S Z a h s r T O n e I 5 A v i I I A 3 1 f 7 c 6 H I 2 6 x b J b c W c g y 8 T L S B k y 1 L r F r 0 4 v Y k n I F T J J j W l 7 b o x + S j U K J v m k 0 E k M j y k b 0 Q F v W 6 p o y I 2 f z q 6 e k G O r 9 E g / 0 r b s + p n 6 e y K l o T H j M L C d I c W h W f S m 4 n 9 e O 8 H + p Z 8 K F S f I F Z s v 6 i e S Y E S m E Z C e 0 J y h H 4 h l d 4 c x 6 d F + f d + Z i 3 5 p x s 5 h D + w P n 8 A Y p c k m 4 = < / l a t e x i t > Point A < l a t e x i t s h a 1 _ b a s e 6 4 = " C / y 5 s w j c A d f A U E w s o U x z e 6 v L R l R T h j a o g g 3 B X 3 5 5 l T T P K / 5 l x b u 9 K F d L W R x 5 c k J K 5 I z 4 5 I p U y Q 2 p k w Z h R J N n 8 k r e n E f n x X l 3 P h a t O S e b O S Z / 4 H z + A I 1 m k n A = < / l a t e x i t > Point C < l a t e x i t s h a 1 _ b a s e 6 4 = " t B 5 w W 2 m f U 2 d 5 E i d o F S + U

Figure 15 :
Figure 15: Examples of UV completion of the amplitude at different point labeled by A,B,C,D in the boundary of figure14for ϵ = 1.We picked pairs of points close to each extremity and plotted the imaginary part of the amplitude in the forward limit at the upper/lower boundary.In gray, we represented the band chosen using (4.6) and (4.7).

12 )Figure 16 :
Figure 16: Absolute value of the partial amplitude S ℓ for complex spin on the real axis.The data are represented in blue.|S ℓ | = 1 is plotted in orange for reference.The position of the leading resonance of the leading trajectory is indicated by a black dashed line.

7 )480π 2
Since we work with the scattering of identical particles ℓ = 0, 2, 4, . . .Comparing these to (F.4) and (F.5) we conclude that to obtain from the explicit expression (A.7).For completeness let us also write the sum-rule for the coefficient g 4 .It reads as in d = 4 and β = 0 in d ≥ 5.In order to derive this expression we compute the arc a 1 explicitly in the forward limit to obtaing 4 + 21β log (s √ g 2 ) + O (s

Table 1 :
Lower bound on ḡ4 for different choices of g 2 .The best bound is achieved at g 2 = 500.Here we use N max = 20 and L max = 80.
1) and the function N (s, t, u) represents all the loop corrections.Loop computation is a hard task.Luckily there is a simpler way to determine the function N (s, t, u) as we show in this appendix.

Table 2 :
The low energy expansion coefficients for the amplitudes (C.1) and (C.2).
′ (s ′ + t)] n+1 , (F.1)where C s is the circle centered at −t/2 and of radius s + t/2 and exclude the real axis.For n ≥ 0 we can deform the contour and drop the arc at infinity provided that the amplitude admit 2 subtractions.22ImT(s ′ , t) [s ′ (s ′ + t)] n+1 .(F.2)