Positive Moments for Scattering Amplitudes

We find the complete set of conditions satisfied by the forward $2\to2$ scattering amplitude in unitarity and causal theories. These are based on an infinite set of energy dependent quantities -- the arcs -- which are dispersively expressed as moments of a positive measure defined at (arbitrarily) higher energies. We identify optimal finite subsets of constraints, suitable to bound Effective Field Theories (EFTs), at any finite order in the energy expansion. At tree-level arcs are in one-to-one correspondence with Wilson coefficients. We establish under which conditions this approximation applies, identifying seemingly viable EFTs where it never does. In all cases, we discuss the range of validity in both couplings and energy. We also extend our results to the case of small but finite~$t$. A consequence of our study is that EFTs in which the scattering amplitude in some regime grows in energy faster than $E^6$ cannot be UV-completed.

In this article we extend the positivity conditions on the forward amplitude to what appears to be a complete set. This is made possible by: i) writing the dispersion relations in terms of suitable energy dependent quantities in the EFT, the arcs, which are directly related to the Wilson coefficients at tree level and capture features of the RG evolution at the quantum level; ii) noticing that by unitarity the arcs correspond to the sequence of moments of a positive measure over a compact interval. The resulting setup precisely fulfils the hypotheses of Hausdorff's moment problem, whose known solution provides the set of necessary and sufficient positivity constraints on the forward amplitude. We also extend some of our results beyond the forward limit, using the classic result in Refs. [25,26] on the positivity of the t derivatives of the imaginary part of the amplitude.
Our results are related and partly overlap with those obtained by the geometric approach put forward in Ref. [10]. An important difference is that our method enables to identify optimal constraints that involve a finite number of arcs/Wilson coefficients only. This is the situation closer to questions of phenomenological interest.
The constraints we present are most effective in derivatively coupled theories, where the forward amplitude is finite in the massless limit. There, the arcs are single scale quantities that purely depend on the running couplings, and our constraints directly bound the RG flow (an aspect briefly touched in Refs. [4,17,18,20,21]). One well formulated hypotheses that can be tested in this context concerns the existence of symmetries -exact, accidental or weakly broken -which may appear in the lowenergy EFT. In particular, the non-linear transformations Typeset by REVT E X arXiv:2011.00037v2 [hep-th] 11 Oct 2021 with b s traceless, are symmetries in EFTs where interactions have many derivatives and, therefore, deliver soft amplitudes, i.e. amplitudes with a fast energy E growth. The case N = 0 is familiar: U (1) Goldstone bosons with 2 → 2 amplitudes M ∼ E 4 . The same behaviour arises for the EFT of massless particles with spin, like for the Euler-Heisenberg Lagrangian in QED, as well as the theories of Refs. [27][28][29]. The N = 1 case corresponds to Galileons [30], with M ∼ E 6 , and so on. We refer to theories with large exponent in M ∼ E 2n , n > 2, as supersoft [5,31]. Similarly, longitudinal polarizations in theories with massive spin-J particles have supersoft amplitudes with 2n ≥ 3J [20]. For example, approximate linear diffeomorphisms suppress the selfinteractions from the Einstein-Hilbert term, relative to the super-soft linearised (Riemann) 3 terms. Such a scenario for gravity is incompatible with tree-level UV completions [12]. In this work we will show that supersoftness in general cannot emerge, neither by structure nor by accident, from any reasonable, weakly or strongly-coupled UV completion. This paper is organized as follows. In section I, focussing on the forward limit, we construct the necessary general dispersion relations and derive optimal bounds on the arc variables. In sections II A and II B we apply our results, first assuming a weakly coupled UV completion and then lifting this assumption while focussing on the more specific case of an abelian Goldstone boson. In section III we consider a first foray of our methodology beyond the forward limit. Finally in section IV we summarize our results and offer an outlook.

I. ARCS AND THEIR CONSTRAINTS
We study the 2 → 2 scattering amplitude M of a single particle of mass m (including the limit m 2 → 0 if it exists). We discuss first the forward limit t → 0 in theories where it is finite, and define M(s) ≡ lim t→0 M(s, t); for concreteness we focus on the spin-0 case, but our results carry over to arbitrary spin and flavour structure [5]. Extensions to t = 0 are postponed to section III. We will assume the following analytic properties of M: • Crossing symmetry in s ↔ u with real analyticity, namely M(s) = M * (4m 2 − s * ).
• The singularities of M(s) in the complex splane consist solely of a unitarity cut at phys- ical energies s > 4m 2 , plus the crossing symmetric one at s < 0, see e.g. Ref. [32]. 1 • Unitarity, via the optical theorem, implies ImM(s) > 0.
In particular, we assume M(s)/s 2 → 0 as s → ∞. In the gapped case, validity of this condition is ensured by the Froissart bound [33,34].
Exploiting the analyticity of the amplitude in the upper half plane, we define the arc variables (or simply "arcs"), a n (ŝ) ≡ ŝ dŝ πiM whereŝ ≡ s−2m 2 is the crossing-symmetric variable, M(ŝ) ≡ M(s), andŝ represents a counterclockwise semicircular path in the upper half plane of radiusŝ, as shown in Fig. 1.
The Cauchy theorem implies that the integral over the closed contour C =ŝ + ∞+ l 1 + l 2 vanishes.
Thus, we deform the integral in Eq. (2) alongŝ into an integral along ∞+ l 1 + l 2 . For integer n, we can use crossing symmetry and real analyticity to relate the amplitude above the lefthand cut (path l 2 ) to the one above the righthand cut (path l 1 ), Due to the Froissart bound,M/ŝ 2 → 0, the integral along the semicircle of infinite radius vanishes for n ≥ 0. Thus, we can write Eq. (2) as a n (ŝ) = 2 π ∞ s dŝ ImM(ŝ ) s 2n+3 , n ≥ 0 .
On the one hand, the arcs as defined via the IR representation Eq. (2) are IR quantities that can be systematically computed as an expansion in powers of s in the domain of validity of the IR EFT s s max , with s max the cutoff. In the simple case of a tree-level amplitude in the forward limit,M(ŝ) = n=0 c 2nŝ 2n and the arcs match simply to Wilson coefficients, a n (ŝ) = c 2n+2 . Moreover, the scale dependence of arcs partly reflects the EFT RG flow, as we discuss in section II.
On the other hand, according to the UV representation in Eq. (3), arcs receive contributions from all microphysics scales up to the far UV. So, Eq. (2) ideally represents something measurable in our low energy experiment, while the representation in Eq. (3) requires knowledge of the theory at all scales. Nevertheless, in this form, Eq. (3) has interesting properties that we now discuss.

A. All Constraints
From Eq. (3) it follows that the arcs are positive, a n (ŝ) > 0 , since the imaginary part of the forward amplitude is positive. In fact, convoluting in (3) a positive function F , we have Clearly, to every function F , positive in (ŝ/ŝ ) 2 ∈ [0, 1], correspond inequality constraints relating arcs of different orders. Characterising the most general such function will allow us to address, Question 1: What is the complete set of constraints the arcs a n must satisfy?
To answer this question we will relate our problem to the theory of moments. The following change of variables simplifies the notation and defines a positive measure, so that we can writê s 2n+2 a n = 1 0 x n dµ(x) .
A sequence of dimensionless numbers, defined as in Eq. (7) with dµ positive, is called a sequence of moments. In fact, we have a one-parameter family of sequences because each moment is a function ofŝ. We comment on theŝ dependence below. We introduce the discrete derivatives (∆a) n =ŝ 2 a n+1 − a n , with higher order differences defined recursively, ∆ k = ∆(∆ k−1 ) and ∆ 0 a n = a n . For instance, (∆ 2 a) n = a n − 2ŝ 2 a n+1 +ŝ 4 a 2+n , etc. A sequence of moments necessarily satisfies are positive in the whole integration domain. The case k = 1 corresponds to Eq. (6). The converse is also true, as implied by the Hausdorff moment theorem: if a sequence satisfies then there exists a unique measure dµ such that Eq. (7) is satisfied. 2 This theorem, following from the fact that the functions in Eq. (9) -called Bernstein polynomials -are a basis of all positive functions in [0,1], provides an answer to our Question 1.
As mentioned above, both the arcs and their discrete derivatives depend explicitly on the scaleŝ, as captured by, 2 Arcs probing the theory at finite s are crucial for the mapping to moments on a compact interval (Hausdorff's problem). Instead, a sequence made of amplitude's residues at s → 0 (equivalent to arcs with vanishing radius) see e.g. [35], maps to a non-compact domain (Stieltjes half-moment problem), and the solution is not unique.
which is negative for all k, because of Eq. (10). Therefore, asŝ is increased, the arcs decrease -proportionally to ImM. This implies that, given an EFT, the constraints Eq. (10) for k ≥ 1 become more stringent as s increases. Conversely, if the conditions Eq. (10) are satisfied at one scaleŝ they are automatically satisfied at smaller scales. This behaviour will play an important role later on, when we discuss constraints on the arcs in specific EFTs.

B. Optimal Bounds for a Finite Set of Arcs
In practice we often focus on a finite number of arcs. For instance, in a typical EFT only the first few powers of s are phenomenologically interesting -at tree-level this corresponds to the first few arcs. Thus it is natural to ask, Here optimal means the projection of all constraints on the finite set. In the language of the previous section, we ask: assuming the components of a = {ŝ 2 a 0 , . . . ,ŝ 2+2N a N } are moments, what is the subspace A(N ) ⊂ R N on which a takes values?
Of course Eq. (10) still holds and, for k +n ≤ N , it involves only the first N arcs. The constraints with k + n ≥ N imply however additional conditions on the subset n ≤ N . In what follows we will show a simple procedure to extract this information.
Similarly to Question 1 -that implied finding the most general positive function in [0, 1] -Question 2 requires finding a parametrization for the most general polynomial p(x) = N i=1 α i x i of finite degree ≤ N , positive in x ∈ [0, 1]. Indeed, via Eq. (7), each such p(x) leads to a condition on arcs, One can prove that any such polynomial can be written as [36], type-4 (13) where q J,k (x)'s are non-zero real polynomials -not necessarily positive -such that p(x) is degree N , i.e. q J,k has at most degree d k , with d 1 = N/2 , d 2 = d 3 = (N − 1)/2 and d 4 = (N − 2)/2 , where k is the integer part of k ≥ 0. Since Eq. (13) is a sum over positive terms, it is sufficient to discuss them individually. Therefore we drop the index J and consider one by one generic polynomials q k of each type-k in Eq. (13), with arbitrary real coefficients α kj . We define the Hankel matrix (H N ) ij = a i+j+ , for i, j = 0, . . . , (N − )/2 , so that H N involves arcs up to a N for N − even and a N −1 for N − odd. For instance, Now, polynomials of type-1 imply, Since the vector {α 10 , α 11ŝ 2 , α 12ŝ 4 , . . . α 1,N/2ŝ N } is arbitrary, the following Hankel matrix must be positive definite, Following similar steps one finds that the positiveness of respectively. We refer to Eqs. (16)(17) as homogeneous and Eqs. (18,19) as inhomogeneous constraints. Since (13) is an arbitrary positive polynomial for x ∈ [0, 1], Eqs. (16)(17)(18)(19) represent the optimal constraints, providing an answer to Question 2. For instance, Eqs. (16)(17)(18)(19) with N = 2 define the A(2) region: a 0 a 1 a 1 a 2 0 , a 1 > 0 , a 0 >ŝ 2 a 1 , a 1 >ŝ 2 a 2 .
(20) This is illustrated in the left panel of Fig. 2. The first constraint in (20) implies a 0 a 2 > a 2 1 , and is saturated by the lowest parabola in Fig. 2; the fourth constraint in (20) is saturated by the upper line, the other constraints imply that the coordinates lie in the interval [0, 1]. For comparison, the green lines show the constraints obtained using Eq. (10) and arcs up to a 2 (solid), a 3 (dashed), and a 4 (dotted), and then projected onto the (s 2 a 1 /a 0 ,s 4 a 2 /a 0 ) plane: clearly these converge to the optimal result A(2).
Eqs. (16)(17)(18)(19) capture how the full constraint on the arc sequence is projected on the first N arcs. Of course we can consider the projection on any subset. In particular for a sequence a k , · · · , a N that starts at k = 0, Eqs. (16)(17)(18)(19) are generalised by the substitution → + k, e.g. Eqs. (16) becomes H k N 0. Eq. (6) belongs in this class.
To conclude this section we compare our results to those of Ref. [10], which -considering the forward amplitude -finds that consistent EFTs must satisfy the set of homogeneous Hankel matrix positivity constraints, Eqs. (16)(17). Indeed, the whole set of homogeneous constraints (i.e. for arbitrarily large N ) implies the ensemble of constraints in Eq. (10), 3 and thus by Hausdorff moment theorem is a necessary and sufficient set. However, when only a finite number N of arcs/Wilson coefficients is considered, as it is often the case, our equations Eqs. (16)(17)(18)(19) represent the optimal constraints. For instance, if we are interested in the allowed space for three arcs, as in Eq. (20), the first two homogenous conditions correspond to simple Hankel determinants and can be easily obtained with the methods of [10]. The latter two inhomogeneous conditions, instead, can only be obtained by considering infinite many homogeneous Hankel matrices.
In practice, this translation is complicated by the fact that arcs might receive contributions from (infinitely) many Wilson coefficients. In what follows we discuss under which circumstances this translation is possible: section II A discusses the tree-level approximation while section II B discusses sizeable quantum effects.

A. Tree Level
As a first application of our bounds, we focus on the forward limit in situations where we can consider just the IR tree level amplitude -we will discuss in the next section under which conditions this approximation holds. At low energy this takes a polynomial form 4 We compute the a n (ŝ) using the definition Eq. (2) and Eq. (22) and find, independently ofŝ. So the constraints of the previous sections can be read directly in terms of the coefficients appearing in the amplitude. From a practical point of view it is simpler to focus on a limited number of coefficients (rather than the complete series), so that Eqs. (16)(17)(18)(19) represent the relevant constraints. Of these, the homogenous constraints of Eqs. (16)(17) do not depend onŝ and thus represents properties of the UV theory that are intrinsic, i.e. independent of the overall scale of the dynamics. In particular they include Eq. (4), which implies that all coefficients c n be strictly positive [4].
On the other hand, the energy scaleŝ appears explicitly in Eqs. (18)(19) (this is translated in the normalisation of Figs. 2 beingŝ-dependent). Given an EFT, in the form of a set of c n satisfying Eqs. (16)(17), we can think of Eqs. (18)(19) as defining the highest possible cutoffŝ max where new dynamics must modify our EFT amplitude.
For the simple case of the first three coefficients, in addition to positivity of each of them, Eq. (20) and Eq. (23) imply For fixed values of the Wilson coefficients, asŝ increases, the arcs track trajectories in Fig. 2 (blue lines, for different values of c n ), that start forŝ → 0 at the origin and evolve along parabolae. In this case the cutoff must satisfyŝ 2 max < c 4 /c 6 . The simplest inhomogeneous constraints c 2ŝ 2 − c nŝ n > 0, suffice to rule out any, even approximate, supersoft behaviour, where the tree-level forward amplitude is dominated by the O(s n ) growth, n > 2. First of all, positivity already implies that supersoft symmetries Eq. (1) are never exact: they are always explicitly broken by a (possibly small) c 2 > 0. They could in principle have been appreciable in the regime of energyŝ (c 2 /c n ) 1 n−2 . However our bounds forbid that: super-soft theories cannot consistently be UV-completed at weak coupling, as the expansion in s must be strictly decreasing [19,20]. We will see in the next section that the same conclusion holds also beyond the weak coupling approximation.
As a concrete example, consider for instance interactions of the form (∂∂φ) 4 , which is invariant under the Galilean symmetry φ → φ + b + b µ x µ , and giving M(s) ∼ s 4 . Purely on the basis of symmetries, these terms could have naturally dominated the more relevant (∂φ) 4 interactions, which break the Galilean symmetry. This is however inconsistent with the first inequality in (24) which forces (∂∂φ) 4 to be subdominant.
Faster UV convergence. So far the bounds in this section do not depend on c 0 in (22). This changes if we assume that lim s→∞M (s) ≡ M ∞ is finite. This is the case, for instance, if the theory in the UV is described by a finite number of resonances in the treelevel approximation.
For a finite M ∞ we can extend the definition of arcs in Eq. (2) to n ≥ −1. Then we can repeat an analysis similar to the one that we did in the previous section for the subtracted amplitudeM − M ∞ , finding that the arcs define a sequence of positive moments {a −1 , a 0 , a 1 , . . . }.
Note in particular that a −1 =M(0) − M ∞ can be regarded as the difference of effective couplings defined by the value of the forward amplitude in the IR and in the UV respectively, according to the weak coupling intuition. Since the first of the homogeneous conditions on the arcs is now a −1 > 0, the IR forward amplitude is larger than the UV one. Moreover, a −1 a 1 − a 2 0 > 0 must be satisfied and can be regarded as an upper bound on c 2 2 /c 4 if the tree-level approximation of Eqs. (22,23) holds. An interesting case that we consider in detail in the next section is the theory of a single Goldstone boson for which c 0 = 0. If the UV completion is perturbative −M ∞ 16π 2 , then we have on the Goldstone theory. 5 The upper bound on c 2 2 /c 4 is in agreement with the expectation that RG running effects on c 4 -that we will see Eq. (28) -are expected to be small in a weakly coupled theory.
The boundary. Which theories saturate Eqs. (16)(17)(18)(19)? Hausdorff theorem implies that the integration measure (the imaginary part of the UV forward amplitude) is uniquely determined if all arcs are known. If the measure has support on finitely many points, then a finite number of arcs suffices to determine the measure uniquely.
Physically, a measure that consists of P distinct delta functions, ImM , is realised in the tree-level approximation when integrating out heavy particles with P distinct masses M 1 < M 2 < . . . < M P and effective squared couplings g 2 k > 0. This situation corresponds to 6 a n = P k=1 These arcs lie at the boundary of the A(2P ) region (defined by Eqs. (16)(17)(18)(19) with N = 2P ), as we now show.
In the variable x ∈ [0, 1] of section I B, the measure dµ(x) consists of strictly positive delta functions located at x k =ŝ 2 /M 4 k for k = 1, . . . , P . In the EFT, i.e.ŝ < M 2 1 , there are thus P such delta's within (0, 1). Recalling Eq. (13), Hankel matrices of orderP correspond to the quadratic forms generated by integrating for a canonically normalized complex scalar field Φ. The tree-level forward scattering of the Goldstone bosons gives where m 2 h = λv 2 . Therefore, M∞ = −λ and cn = λ/m 2n h , so that c 2 2 /c 4 = λ, which saturates the bound (25). 6 For P → ∞, finiteness of a 0 requires that g 2 k /M 4n+4 k decays sufficiently fast as k → ∞. Considering a large but finite number of particles is therefore a good approximation. zeroes of qP −1 (x) coincide with the P delta's in the measure. This can only happen ifP −1 ≥ P , in which case the corresponding Hankel matrix has order > P : Hankel matrices of order ≤ P are strictly positive definite, while they are only positive semi-definite if their order is > P . Considering then Eqs. (16)(17)(18)(19), we conclude that, forŝ within the EFT, the arcs are in the interior of A(N ) for N < 2P and at the boundary of A(2P ).
This implies that -in weakly coupled theoriesthe measurements of the arcs in the IR allows to indirectly count the number P of resonances in the UV: P is just the smallestP for which det H 0 2P = 0. The same reasoning as above also implies that for s = M 2 1 , just at the edge of the EFT, the arcs are at the boundary of the A(2P − 1) region. Indeed, whenŝ approaches M 2 1 from below, x 1 =ŝ 2 /M 4 1 approaches the edge x = 1 of the integration region, and the contribution of the lightest resonance drops out when integrated against the type-3 polynomial (13). Then, forP = P the zeroes of qP −1 (x) can be chosen to coincide with the location of the remaining P − 1 heavier resonances, On the other hand, forP < P , qP −1 (x) has fewer zeroes than there are resonances. This implies that for N < 2P − 1 Eq. (18) is still strictly satisfied forŝ = M 2 1 and that s > M 2 1 is needed in order to violate it: N = 2P − 1 arcs allows an optimal estimate of the EFT cut-off M 1 , while for N < 2P − 1 the estimate is always suboptimal.

B. Beyond Tree Level
Under which circumstances is the tree-level formula, a n = c 2n+2 , a good approximation, and when do the conclusions of the previous section hold?
To answer this, we study an EFT including RG effects, restricting for simplicity to the massless case (whereŝ = s). Moreover, in order for the forward limit to be well-defined, we focus on the case of a derivatively coupled scalar, i.e. a Goldstone boson φ with symmetry φ → φ + b. Indeed, the explicit computation that we present below -up to two loops and O(s 6 ) -does not exhibit any IR divergence. Moreover, for the Goldstone theory, the tree-level amplitude is finite at t = 0, and divergences could originate only from collinear emissions (soft emission does not affect the total cross section and hence neither the imaginary part of the amplitude). Using collinear factorization (SCET, see e.g. [37]), it is easy to see that these are finite, as the positive powers of collinear momenta associated with the Goldstone derivative interactions always compensate putative divergences in collinear propagators. We therefore assume that the forward amplitude is well-defined.
In the upper half plane, up to O(s 6 ), the most general Goldstone boson amplitude starts at order O(s 2 ) and reads, where log is defined in the standard way, with the cut on the negative real semi-axis. Then we have 2 log(−is) = log(s) + log(−s) for Ims ≥ 0 and log(s)−log(−s) = iπ. For ease of notation we set the RG scale µ = 1, but the generic choice is reinstated through log(−is) → log(−is/µ 2 ), c n → c n (µ). An explicit calculation in the Goldstone case gives 8 where c 2,1 is the coefficient of s 2 t in the non-forward amplitude.
At O(s 6 ) in the energy-expansion, Eq. (27) does not receive any further correction at any number of loops. From (27), the first three arcs read, where ) .
These functions are RG invariant by construction, and are the natural extensions of c 4,6 in the interacting theory. The dots in Eq. (29) denote higher powers of s, of which there are a priori infinitely many. If all such contributions were important, the EFT would have no predictive power. A meaningful EFT exists only under certain assumptions on the convergence of the series. To this aim, a first unavoidable assumption, is the perturbativity of the dimensionless couplings g 2 n ≡ c n (s)s n that control the IR loop expansion, This condition, which we assume throughout this work, implies that many of the higher order terms that (could) enter Eq. (29) must be small. For instance it implies that β 4 s 2 and the contribution ∝ s 4 c 4 c 2 /16π 2 in β 6 s 4 -see (28) -are subleading in a 0 . Nevertheless, Eq. (31) does not yet imply the validity of the tree-level approximation for the arcs. Indeed, the corrections to the tree level result a n = c 2n+2 , are controlled by a broader set of parameters given by the ratios We will refer to the situations where these parameters are small as strong perturbativity. For n < m these parameters grow in the IR and become smaller in the UV, and viceversa for n > m: in the standard RG parlance they respectively correspond to relevant and irrelevant deformations away from treelevel. For n = m, they capture instead the logarithmic RG running of Wilson coefficients (they measure the interaction strengths at the cutoff -see (25) for the weakly coupled case). For instance, in the sigma-model example of footnote 5, c n = λ/m 2n h and β n ∼ (λ 2 /16π 2 )/m 2n h are characterised by one scale and one coupling so that the ratios in Eq. (32) are of order (λ/16π 2 )(s/m 2 h ) n−m . Then, given λ/16π 2 1 and s/m 2 h < 1, for n ≥ m the parameters of Eq. (32) are always small (strongly perturbative). However, for n < m, they can be larger than unity at small enough energies, In fact, these quantum effects always dominate the far IR s → 0, where the arcs asymptote to For c 2 > 0, and given β 4 < 0, as implied by unitarity within the EFT these arcs fulfil all the constraints. What happens is that for s → 0, the arcs are fully dominated by the IR tail of the spectral density, which is positive and fully determined by the leading term ∝ c 2 2 in the 2 → 2 cross-section. The Hausdorff condition is then trivially satisfied and, as graphically represented in Fig. 2, all red trajectories flow to a common attractor as s → 0.
When higher energies are considered, predictivity is only retained if contributions above a certain finite positive power of s remain negligible, in particular when considering the arcs (29). In view of that, we will now discuss two scenarios for omitting higher order terms. We dub them Simplest EFT and Nextto-Simplest EFT . In the first case we assume that all irrelevant contributions to the arcs are negligible, namely β n s n c m (s)s m for n > m .
This is the standard situation in the context of EFTs. Instead, more complex scenarios arise by allowing irrelevant parameters to sizeably contribute to the arcs. In the Next-to-Simplest EFTs, we will allow β n s n ∼ c m (s)s m for some n > m , while all the other β coefficients still fulfil Eq. (35).
Simplest EFT. The IR relevant contribution to the arcs drastically modify the allowed region for the running Wilson coefficients c n (s). This is however not apparent in the first two arcs a 0 and a 1 . Indeed, for the first arc alone we have a 0 c 2 , since there are no possible relevant nor marginal perturbations that enter. Therefore the positivity constraint on a 0 gives c 2 > 0, equivalent to the tree-level case. Consider now the first two arcs. The size of β 4 /c 4 controls the logarithmic running in a 1 ≈ c 4 (s). The bounds of section I A, in particular the optimal bounds in Eqs. (16)(17)(18)(19), read in addition to c 2 > 0. Written in term of the running coefficient c 4 (s), Eqs. (37,38) have the same form as at tree-level constraints of Eq. (24). Notice in particular that c 4 (s) (the lowest running Wilson coefficient) cannot be negative. Moreover, the second inequality, together with perturbativity defined in Eq. (31), justifies our neglect of the term ∝ s 6 c 2 4 /16π 2 c 2 in β 8 s 8 /c 2 s 2 . In other words, it implies that these terms automatically respect strong perturbativity (35).
In a causal and unitary theory, violation of any of eqs. (37,38) must correspond to the failure of the hypothesis, Eq. (35). This implies that either the EFT is at its cutoff, or that it is non-standard, in the sense that irrelevant higher derivative terms must be included, as we will discuss below in an example.
Starting with the third arc, the distinction between arcs and running Wilson coefficients becomes appar- ent. Indeed, the third arc a 2 ≈ −β 4 /2s 2 + c 6 (s), 9 includes a relevant deformation from its tree-level expectation a 2 = c 6 . The bounds read, Given c 4 (s) and c 6 (s) at an energy s, the first expression is stronger and the second is weaker than the tree-level conditions in (24). The weaker condition implies that the determinant c 2 c 6 (s) − c 4 (s) 2 , and indeed even c 6 (s), can be negative. In fact the RG effects in Eq. (30) alone, already violate the naive tree-level bounds: in the far IR s → 0, RG evolution leads to c 6 (s) → β 6 log 2 s > 0, but makes the determinant negative, where we have used the explicit values for β 4 and β 6 from Eq. (28). Indeed it is now possible to have consistent EFTs where the determinant is so negative that, as energy increases, Eq. (40) is violated before Eq. (39): a radical difference w.r.t. the tree-level approximation, where only the inhomogeneous conditions depend on s. This is possible because quantum effects imply that arcs manifestly depend on the energy scale: Eq. (11) implies that all constraints become more stringent as s increases, so that both homogeneous and inhomogeneous conditions play now a role in defining the theory's regime of validity. This is illustrated in Fig. 3, where coloured regions correspond to Eqs. (39,40) and black dashed curves report the naive tree-level expectation.
We can therefore identify two interesting classes of theories. First, theories that possess a regime in which the parameters in Eq. (32) are small (in particular β 4 /c 4 1 and β 4 /c 6 s 2 1 in the case of three arcs) and can be approximated by the treelevel expressions. Weakly coupled theories where the EFT is obtained by integrating out massive particles at tree-level (like discussed on page 7), or at loop level (like for the Euler-Heisenberg Lagrangian), belong in this class. 10 In theories of this class, it is the inhomogeneous bounds (Eqs. (38,39) for three arcs) that are violated first, as s increases. This state of things is illustrated in Fig. 3, where we have chosen β 4 /c 4 (s) = 0.1, and where the region above the solid black lines correspond to varying sizes of β 4 /c 6 (s max )s 2 max , as indicated in the figure.
On the other hand, in theories in which the relevant perturbation β 4 /(c 6 (s)s 2 ) never becomes negligible, it is possible for c 2 , c 4 (s), c 6 (s) to lie at O(1) outside of the tree-level naive boundary Eq. (24). In the central and right plots of Fig. 3, these theories feature at all energy scales a value of c 2 c 6 (s)/c 4 (s) 2 significantly below its tree-level bound of 1, and include the option of a negative value, implying c 6 (s) < 0. All in all, even in the Simplest EFT scenario, quantum effects open a qualitatively new region in parameter space. In these theories, as energy increases, it is the homogeneous conditions Eq. (40) that are saturated first. This is illustrated by the red trajectories in Fig. 2 exiting from the lower parabola.
Next-to-Simplest EFT. We now study the simplest case in which there exist a sizeable effect associated with irrelevant parameters, as in Eq. (36)(i.e. a sizeable β n s n ∼ c m s m for some n > m). The Galileon limit, in which c 2,1 s c 2 , is an example of this. Since c 2,1 doesn't enter in the forward amplitude at tree-level, it is not directly bounded by our discussion in section II A (in section III we discuss the amplitude away from the forward limit, but we anticipate that a positive c 2,1 is unbounded by those arguments). In this limit, the part of β 6 involving c 2,1 , which we denoteβ 6 = −c 2 2,1 /(30(16π 2 )), can be potentially large and depart from strong perturbativity at sufficiently high energy within the EFT -though we still assume (strong) perturbativity for all higher coefficients.
Including also a 1 ≈ c 4 (s) +β 6 s 2 /2, we find the further conditions The first inequality implies that c 4 (s)s 2 must still be positive. It also implies another bound of the form of Eq. (42), that can be written explicitly as, and is stronger than the one implied by Eq. (42), for c 4 (s)s 2 < 2c 2 . The second inequality in Eq. (43) shows that c 4 (s)s 2 can now be larger than c 2 . However, compatibly with Eq. (42) it cannot exceed 2c 2 . Therefore the violation of the bound c 2 > c 4 (s)s 2 is only marginal, implying that supersoftness c 4 (s)s 2 c 2 remains forbidden. These results are summarised in Fig. 4.
If we further include information for the third arc, a 2 ≈ c 6 (s) (where c 6 (s) is dominated byβ 6 and we neglect the term ∝ β 4 /s 2 β 6 ), we have the inhomogeneous bound, So, a sizeable (negative) β 6 makes the lower bound on c 6 (s)s 4 /c 2 stronger, and shifts it towards larger values of c 4 (s)s 2 /c 2 . These effects, however, appear in a relatively uninteresting regime of the theory. Indeed, the quantum effects ∝ β 4 /s 2 discussed above were sizeable in the IR and allowed for theories that depart from the tree-level approximation, while still being valid over a relatively large energy regime. Instead, the effects discussed here, are controlled by the s 2β 6 term, that is sizeable only at high-energy, when the theory is already close to the cutoff. The couplings c 2,1 , c 4 and c 6 are all compatible with Galilean symmetry; c 4 and c 6 are exactly invariant while c 2,1 is a sort of Wess-Zumino-Witten (WZW) term [30,38]. Instead, c 2 violates the symmetry. Our analysis shows that c 2,1 can be at most a loop factor larger than c 4 , while the exactly invariant couplings c 4 and c 6 are directly limited by c 2 . In some sense, our constraints privilege the WZW couplings over the exactly symmetric ones.

III. BEYOND FORWARD
In this section we extend our methodology to the class of amplitudes that are also analytic in t in a finite region around t = 0. This trivially includes the case of tree level amplitudes generated by the exchange of massive states, but it also includes the general case of finite mass. Indeed, for fixed physical s ≥ 4m 2 the amplitude M(s, t) is analytic for complex cos θ ≡ 1 + 2t s−4m 2 inside the so called Lehmann ellipse, namely an ellipse with foci at cos θ = ±1 [25], see also e.g. [39].
For t finite and real, s ↔ u crossing symmetry and real analyticity dictate M(s, t) = M * (4m 2 −s * −t, t).  while a similar relation also holds in the u-channel. Then, unitarity of the partial waves together with positivity of the Legendre polynomials and their derivatives at t = 0, Eq. (47) imply [26] for all k and for s along the s-channel cut. The above additional positivity conditions can be exploited as we now discuss.
Considering the above properties we extend the definition of the arcs (2) to a n (ŝ, t) ≡ ŝ dŝ πiM where we recallŝ ≡ s − 2m 2 ,M(ŝ, t) = M(s, t) andŝ is now a contour with radiusŝ + t/2 centred at −t/2. The condition M(s, t) = M * (4m 2 −s * −t, t) ensures the reality of the arcs. Furthermore using thatM(ŝ, t)/s 2 → 0 for s → ∞ (as dictated by the analog of the Froissart bound at finite t [40]), the arcs can be expressed by a dispersive integral like in Eq. (3), 11 a n (ŝ, At this point we can take t-derivatives at t = 0 and use Eq. (48) to obtain positivity conditions and so on; where we have defined and a (0) n ≡ a n . Notice that while the arcs a n (ŝ, t) are defined for integer n and can be written purely in terms of IR data according to Eq. (49), the a (k) n (ŝ) of (51), are defined for half-integer n ≥ 0 and only through the UV representation in (52).
Notice also that it is in principle possible to build dispersion relations for other combinations that respect real analyticity, such as ∂tM − ∂ŝM/2. These also lead to Eq. (51).
Half-integer arcs a n+1/2 (n integer) and all a (k) m (m integer or half-integer for k > 0) are not calculable in the EFT because they are defined in terms of UV integrals (52), and have no IR counterpart like (49). On the contrary, a n and ∂ t a n , with integer n, can be computed directly within the IR. Our goal is therefore to understand the constraints on a n and ∂ t a n for arbitrary values of a (k) m and a n+1/2 , compatible with their being moments. In appendix A, we outline an analytic procedure to do so, from which the bounds below are derived, but that applies in principle to all N . For more t-derivatives this procedure becomes cumbersome: in Ref. [41] we propose a numerical technique, based on semi-definite programming, to extract the bounds efficiently.
For instance, for N = 0 we find the constraint on arcs t-derivatives to be ∂ t a 0 > − 3 2 a0 s , a condition that appears in different form already in Ref. [8].
For N = 1 our conditions constrain the space {∂ t a 0 , ∂ t a 1 } as, which we illustrate in Fig. 5 for negative ∂ t a 0 (for ∂ t a 0 > 0, the allowed region is unbounded: the distance between the upper and lower boundaries of the projection on the {ŝ 2 a 1 /a 0 ,ŝ 3 ∂ t a 1 /a 0 } plane increases withŝ∂ t a 0 ). These conditions are more stringent than just ∂ t a 0 > − 3 2 a0 s (which corresponds to the left boundary of the box in Fig. 5), as shown also in Fig. 4.
We illustrate an application of these bounds to the Wilson coefficients at tree-level 12 Eq. (54) reads at t = 0, with t → 0 before taking m → 0, is finite. Therefore, for statements up to O(s 4 t), the use of the tree-level expressions is justified. We postpone a more refined discussion of these loop effects to Ref. [41]. where we have identified c n ≡ c n,0 to match the notation of the previous sections. Eq. (56) implies that c 2,1 can be negative but not arbitrarily so, as it is limited in magnitude by √ c 4 c 2 .
To further illustrate the constraining power of Eq. (56), we can for instance use it to test the consistency of a theory where the amplitude is dominated by ∝ E 10 terms for sufficiently large E within the EFT domain of validity. By Lorentz invariance the only option is M stu(s 2 + t 2 + u 2 ), corresponding to a c 4,1 s 4 t term dominating c 2 s 2 , c 2,1 s 2 t and c 4 s 4 . This hierarchy appears natural, since it is protected by one of the approximate symmetries in Eq. (1), see [31]. Such example is not constrained by our tree-level forward bounds in section II A, simply because the largest contribution to the amplitude vanishes at t = 0. However, Eq. (56) provides upper and lower bounds on c 4,1ŝ 5 , controlled by more relevant Wilson coefficients, see Fig. 5 and its caption. In the physical region |t| < s, these bounds imply that c 4,1 s 4 t can never dominate the terms with lower powers of E. In other words, supersoft amplitudes that vanish in the forward limit and have more powers of energy than c 2,1 s 2 t, are excluded by our bounds.
The same arguments lead to constraints on higher t-derivatives of arcs. The structure is always the same: ∂ k t a 0 are bound from below, but can be arbitrarily large when positive. Instead ∂ k t a n , n > 0 are bound from below and from above.
At tree level, this means that c n,k , with n > 2

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

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

< l a t e x i t s h a 1 _ b a s e 6 4 = " z t N x K s 8 H w W k 0 L V E s o D x Y + C 4 c + q U = " > A A A B + n i c b V C 7 T s N A E D z z D O E V o K Q 5 E S F R o M i O I k E Z Q U M Z J I I j J V a 0 v m z C K e e H 7 t Z I k Z W P o I W K E t H y M x T 8 C 7 Z x A Q l T j W Z 2 t b P j x 0 o a s u 1 P a 2 V 1 b X 1 j s 7 J V 3 d 7 Z 3 d u v H R z e m y j R A r s i U p H u + W B Q y R C 7 J E l h L 9 Y I g a / Q 9 a f X u e 8 + o j Y y C u 9 o F q M X w C S U Y y m A M s k V w 7 R 5 3 p w P a 3 W 7 Y R f g y 8 Q p S Z 2 V 6 A x r X 4 N R J J I A Q x I K j O k 7 d k x e C p q k U D i v D h K D M Y g p T L C f 0 R A C N F 5 a x J 3 z 0 8 Q A R T x G z a X i h Y i / N 1 I I j J k F f j Y Z A D 2 Y R S 8 X / / P 6 C Y 0 v v V S G c U I Y i v w Q S Y X F I S O 0 z H p A P p I a i S B P j l y G X I A G I t S S g x C Z m G T F V L M + n M X v l 4 n b b D i t h u P c t u r t q 7 K a C j t m J + y M O e y C t d k N 6 7
A u E 2 z K n t g z e 7 F S 6 9 V 6 s 9 5 / R l e s c u e I /  area encompasses coefficients which are not independent because of crossing symmetry in the single-flavour case. c 0 is unconstrained, c 2 is constrained to be positive, c 2,1 is constrained to be larger than a combination of the other coefficients. All other coefficients are bounded below and above.
for any k, fulfil two-sided bounds. Moreover, in the single flavour case considered here, crossing symmetry implies that c 2,k with k ≥ 2 are related to c 2+k,0 (for k even) or c 1+k,1 (for k odd), which are already bounded from above from our constraints (grey area in table I). For instance, the amplitude M(s, t) ∝ (s 2 + t 2 + u 2 ) 2 implies c 2,2 = 3c 4 , and c 4 is bounded from above. In other words, only c 2 and c 2,1 are unbounded from above by tree-level arguments; all other coefficients are instead bounded from below and above. We illustrate this in table I. As pointed out in Ref. [10], Eq. (48) contains more information than just positivity, which is what we have exploited so far. Indeed, the ∂ k t ImM(s, t)| t=0 (and their integrals a (k) n (ŝ)) are not merely positive, but are the sum of positive parameters with known coefficients: (s − 4m 2 ) k ∂ k t P (cos θ)| t=0 = ( + k)!/( − k)!k!. This implies more bounds involving at least three a (k) n (ŝ), with k + n = constant [10]. Because they involve at least three different tderivatives, and because they are saturated by the crossing symmetry condition in the simplest cases, these bounds didn't play a role in our discussion of supersoftness; for a relevant application see [42]. In Ref. [41] we will show how these bounds emerge in the language of moments and discuss quantitatively the impact of IR divergences.

IV. SUMMARY AND OUTLOOK
In this paper we introduced a set of energy dependent quantities, the arcs a n (s) of Eq. (2), that conveniently encode the constraints of causality, uni-tarity and crossing on the forward 2 → 2 scattering amplitude. A dispersion relation, Eq. (3), allows to express the arcs as the moments of a positive measure in the range [s, ∞). Hausdorff's moment theorem then establishes the set of necessary and sufficient conditions the arcs must satisfy, given the positivity of the measure. Both the arcs and the constraints are infinite sets. However we derive the projection of the full set of constraints on the subsets of the lowest arcs, a 0 , . . . , a N for any N . These are expressed by Eqs. (16)(17)(18)(19) and fall into two classes, inhomogeneous and homogeneous, according to their explicit, or only implicit, dependence on s. These projections are interesting within EFTs, because the lowest a n 's encode the effects of the correspondingly lowest Wilson coefficients.
Our result is particularly relevant to determine the acceptable range of validity of EFTs in both couplings and energy. Concerning the latter and as implied by Eq. (11), the energy dependence of the arcs leads to stronger bounds on the EFT parameters as the energy is increased. Satisfaction of the positivity constraints at a certain s, automatically implies satisfaction at lower s but not at higher s. When the parameters of a given EFT violate the constraints above a certain scale, the only option to respect positivity is the breakdown of the EFT description at or below that scale.
In the idealized limit where the tree level approximation holds exactly, the arc series is in one to one correspondence with the series of Wilson coefficients in the expansion of the forward amplitude. The energy dependence of the arcs arises at the quantum level from two sources: the RG evolution of the Wilson coefficients and collinear radiation from the initial state. The latter induce effects that typically go like powers of ln s/m 2 , and thus diverge in the massless limit, corresponding to the IR divergence of the total cross section. There are however situations, in particular when the interactions are purely derivative, where these IR divergences are absent and the arcs energy dependence is purely controlled by RG evolution. It is this simpler situation that we have mostly considered for illustrative purposes, focussing on the theory of one abelian Goldstone boson. We could have similarly considered the case of massless vectors or fermions, where gauge invariance or supersymmetry mandate derivative interactions.
The constraints are conveniently described by grouping EFTs into two broad classes. The first are EFTs that emerge from a weakly coupled UV completion, either at tree level or from loops. Here, for s not too much below the physical EFT cut-off, the arcs are reliably approximated by the Wilson coefficients at tree level. Eqs. (16)(17)(18)(19) then translate directly into sharp constraints on the Wilson coefficients. In particular, the energy dependent inhomogeneous constraints dictate the strict convergence of the s expansion of the forward amplitude M(s) and rule out the possibility for supersoft EFTs, where M(s) grows faster than s 2 . Indeed the energy dependent constraints, through their violation, also allow to infer the maximal cut-off of the EFT. As we discussed, even in these weakly coupled EFTs the tree level approximation for the arcs breaks down at sufficiently low s because of mixing at the quantum level with more relevant Wilson coefficients. Consequently in the far IR the allowed region for the running Wilson coefficients differs significantly with respect to the near cut-off region. This fact is also directly related to the existence of the second class of EFTs, for which the tree level approximation for the arcs is never realized in their domain of validity. Positivity still implies strict constraints in couplings space and in particular rules out supersoft EFTs, at least under the simplest assumptions we could check.
In this second class of theories the homogeneous constraints, now energy dependent because of quantum effects, can control the maximal allowed UV cut-off.
Our study mostly concerned the forward amplitude, but in section III we extend it to t = 0. We studied the arc first t-derivative; in derivatively coupled theories, this too is free of IR-divergences. The bounds we obtain complete our analysis into the realm of theories with suppressed forward amplitude. In particular they show that M(s) can be dominated over a limited range of energies by ∝ s 2 t behaviour, but that anything softer is forbidden. More bounds than those presented here can be derived using the explicit form of Legendre polynomials (in addition to positivity of their derivatives) [10], but always involve at least two t-derivatives. These involve quantities that are IR divergent in the m → 0 limit, and goes beyond the scope of the present study (see comments further down).
Our investigation could be furthered in a number of ways. One could be to try and connect to the S-matrix bootstrap [43][44][45]. The latter approach exploits the full 2 → 2 unitarity equation, schematically 2ImT > |T | 2 , while our analytical bounds purely exploit positivity ImT > 0. Besides trying to implement full unitarity one could perhaps use an ansatz for the 2 → 2 amplitude similar to the S-matrix bootstrap approach of [43].
An obvious way to extend our work would be to more systematically study other instances of derivatively coupled theories. In particular one could consider cases involving states of different helicity and with a flavor structure. Here it would be interesting to consider the forward amplitude for superpositions of helicity and flavor. A less obvious one would be to consider non derivatively coupled EFTs where the cross section is affected by collinear divergences. In this case the bounds on the Wilson coefficients at some scale s will seemingly have a dependence, to be determined, on the mass m providing the IR regulation. Alternatively one could treat the mass m itself as the RG scale and work with s ∼ m 2 .
In this way, for N integer, the constraints Eqs. (16)(17)(18)(19) on h N (involving both integer and half-integer arcs) can be written in terms of constraints on H N (integer arcs only for integer) and H +1/2 N (halfinteger arcs only): where we defined ∆H N ≡ H N −1/2 −ŝH +1/2 N and used Schur's complement. The lower label of Hankel matrices is implicit and corresponds to N .
Together with the positivity bounds for a (1) n , involving a (1) n up to N − 1/2 , they define the space of allowed arc derivatives, once we eliminate all the a (1) n .