On the unitarity and low energy expansion of the Coon amplitude

The Coon amplitude is a deformation of the Veneziano amplitude with logarithmic Regge trajectories and an accumulation point in the spectrum, which interpolates between string theory and field theory. With string theory, it is the only other solution to duality constraints explicitly known and it constitutes an important data point in the modern S-matrix bootstrap. Yet, its basics properties are essentially unknown. In this paper we fill this gap and derive the conditions of positivity and the low energy expansion of the amplitude. On the positivity side, we discover that the amplitude switches from a regime where it is positive in all dimensions to a regime with critical dimensions, that connects to the known $d = 26, 10$ when the deformation is removed. En passant, we find that the Veneziano amplitudes can be extended to massive scalars of masses up to $m^2 = 1/3$, where it has critical dimension 6.3. On the low-energy side, we compute the first few coupligs of the theory in terms of $q$-deformed analogues of the standard Riemann zeta values of the string expansion. We locate their location in the EFT-hedron, and find agreement with a recent conjecture that theories with accumulation points populate this space. We also discuss their relation to low spin dominance.

The Coon amplitude [1][2][3] is, together with the Veneziano amplitude, the only explicitly known fourpoint tree-level amplitude that describes an infinite exchange of higher-spin resonances which solves the duality constraints. It was discovered as a deformation of the Veneziano amplitude to non-linear Regge trajectories. The deformation is given in terms of a parameter q (0 ≤ q ≤ 1), which characterises a family of amplitudes defined by (in units α = 1) with σ = 1 + (s − m 2 )(q − 1), τ = 1 + (t − m 2 )(q − 1) (2) where s, t are Mandelstam variables (c.f. appendix). This amplitude describes the scattering of four identical scalars of mass m 2 . At q = 0, it reduces to a scalar theory, and at q = 1 it gives back the Veneziano model: Unlike for the Veneziano model, no worldsheet theory was found for the Coon amplitude, and to this day, its physical origin remains mysterious. In addition, and what concerns us in this paper, its basic properties; unitarity conditions and low-energy expansion, are essentially unknown.
In more recent times, the Coon amplitude was brought forward in the bootstrap analysis as an exception to the universality of linear Regge trajectories in [4], coming from the existence of an accumulation point in its spectrum, similar to that of the hydrogen atom, which allows the theory to evade the theorem of [4]. Related bootstrap constraints applied to the Wilson coefficients of effective field theories (EFTs) coming from unitarity, crossing and analyticity are known to impose bounds [5] that carve theory islands [6][7][8][9][10], and it appears that they are bigger than what is required to describe the basic theories of the world around us [11][12][13][14]. Even more interestingly, [14] recently conjectured that the space of gravitational EFTs is actually populated generically of theories with an accumulation point. Since the Coon amplitude has an accumulation point and connects continuously string theory and field theory, it provides an extremely interesting testing ground to investigate some aspects of these questions.
The main results of our analysis are as follows. Firstly, we map the positivity region of the amplitude in the (q, m 2 )-space, see fig. 2: for each point (q, m 2 ) we determine the maximal dimension in which no ghosts are exchanged as intermediate states. This generalises the known d = (10) 26 critical dimensions of (super)string theory for m 2 = (0)1. We discover a surprising regime of the amplitude where we can prove that it is ghost-free in all dimensions [15]. This goes against standard intuition that in high enough dimensions, string-like theories eventually cease to be unitary. In the other regime, assisted by analytical arguments, we determine numerically the positivity surface, which interpolates from infinite critical dimensions to the standard critical dimensions of string theory. Along the way, we also realised that the Coon and a fortiori Veneziano amplitudes can be extended to positive m 2 → 1/3, with corresponding critical dimension d 6.3 for the Veneziano amplitude [16].
Secondly, we compute some low energy couplings of the Coon amplitude in terms of q-polylogarithm values, which generalise the known zeta-values of the string lowenergy expansion. We compute explicitly the first coefficients g 2 (q), g 3 (q), g 4 (q) and map their location in the space of couplings, comparing to [6]. We also comment on the connection to the notion of low-spin dominance of [11].

THE COON AMPLITUDE
In this section, we review a few facts about the Coon amplitude. Note firstly that, as a function of s, t alone, as in (1), this amplitudes really describes bi-adjoint colorordered scalars and should be thought as being stripped of a color factor, in analogy with similar case in string theory [17]. Later, when we compare the low energy theory to [6], we consider the (s, t, u) symmetrized amplitude, given by summing over the (s, t), (t, u) and (s, u) channels, which corresponds to ordinary external scalars.
The spectrum of the amplitude, depicted in fig. 1, can be immediately read off eq. 1. It has single poles located at σ = q n which correspond to [18] These poles build up an accumulation point as n → ∞, similar to the ionization threshold of the hydrogen atom for instance. At the accumulation point starts a cut, coming from the non-meromorphic factor ∼ q log(s) log(−t) log(q) . As we explain below, this factor is crucial to ensure polynomial residues, as was later realised [3]. Further analogies with atomic physics of such an analytic structure were drawn in [19] but never made precise [20]. This makes this amplitude depart from the strict tree-level, or meromorphic case studied in [4]. Similar non-analyticities were recently observed in holographic models for partonic interactions [21], in relation to the bending of hadron trajectories as t goes to the physical regime and it would be very interesting to understand this connection further. On a given pole s = m 2 n , the residue of the amplitude is given by a polynomial of degree n in t, corresponding to the exchange of particles of spins 0 to n and reads: On the pole, the non-meromorphic q-factor reduces to τ n and cancels an inverse factor τ −n coming from the infinite product [22]. This factor was missed in the original papers and in the more recent reference [23]. Interestingly, without it, the amplitude acquires some severe form of non-locality (non-polynomial residues), but its unitarity properties do not change much [24]. This illustrates an interesting aspect of this amplitude, maybe in relation to the arguments of [4, appendix C] : the presence of an accumulation point, which in itself did not look dramatic but yet allowed to escape the axioms of the theorem, seems linked to a subtle form of non-locality or non-meromorphicity which might explain its exotic nature. The Regge and fixed-angle regimes are straightforwardly extracted [1,25]. In the Regge regime, where s −t and t fixed, 1/σ vanishes and the amplitude behaves as where f (t) ∼ (1 − q n /τ ). Since q < 1, one sees that the amplitude is suppressed at physical negative t for m 2 ≥ 0. In particular, all bounds on couplings derived assuming standard twice-subtracted dispersion relations are expected to hold here, as they use a stronger condition of Regge boundedness A(s,t) s 2 → 0 at large s. We plot j(t) in appendix for a specific value of the mass, see fig 6. In the fixed angle regime, s −t fixed, both 1/σ and 1/τ vanish, therefore we get A q (s, t) ∼ e log(s)log(−t)/ log(q) ∼ s log(s cos(θ))/ log(q) . (9)

POSITIVITY OF THE AMPLITUDE
We will now present our results on the unitarity of the Coon amplitude, i.e. the conditions under which no negative norm states are present as intermediate states. These negative norm states or ghosts are characterised by negative residues on single poles. More precisely, near a resonance exchange in the s-channel, the amplitude takes the form As is standard and reviewed for instance in [4,8], Lorentz invariance implies that Res n (t) is a polynomial whose degree corresponds to the highest spin among the modes of mass m n being exchanged, and that it can be further decomposed into angular eigenfunctions in dimension d: Here cos(θ) = 1 + 2t s−4m 2 is the cosine of the scattering angle and each term in the sum corresponds to a different Gegenbauer polynomials (see appendix). The coefficients c n,J can be obtained using the orthogonality of the Gegenbauer polynomials, and unitarity demands that these should be positive, as a negative coefficient implies that the corresponding exchanged state has negative norm.
In the case of string theory, the no-ghost theorem [26,27] guarantees that such states decouple from all scattering amplitudes in for d ≤ 26 or 10. At the level of the four-point function (the Veneziano amplitude), a recent paper showed that residues should all be positively expandable on Gegenbauer polynomials [28] in d ≤ 6, (see also [29] for the states on the leading Regge trajectory in d = 4). It would still be desirable to be able to bridge the gap to d = 10 or 26 and maybe the q → 1 limit of the Coon amplitude could open an avenue to be combined with the techniques of these papers.
As regards Coon, an early study [30] did investigate the presence of ghosts in the amplitude in four dimensions. Some partial results were obtained, showing that some regions in the q, M 2 parameter space are ghost-free. While their (numerical) method finds ghosts in four dimensions, we do not, for any values of q. This is because we look at a different set-up : for them, the mass of the external particles M 2 was different from m 2 , the lowest mass of the amplitude, while for us, M 2 = m 2 .
The more recent reference [23] which also studied the problem (which was unaware of [30]) is largely inconclusive and before the present work, nothing was known on the critical dimensions of the Coon amplitude. Our results are summarized in fig. 2. They show the existence of two regimes for q, distinguished critical value We find that, for q < q ∞ (m 2 ) the amplitude is ghostfree in all dimensions, which we demonstrate analytically. Then, for q > q ∞ (m 2 ), are critical dimensions exist and we numerically determined them, backed by an estimate of the envelope of the critical dimensions near q ∞ (m 2 ). Let us now give some details on this analysis.
Before describing the behaviour in q, we describe the range of m 2 . For masses m 2 < −1, the Veneziano amplitude is known to not be unitary (corresponding to intercepts greater than 1). For the Coon amplitude, a related statement holds : for m 2 < −1, there is a ghost at scalar mass-level 2 for q > 2 m 2 −1 . For q < 2 m 2 −1 , this state becomes a positive-norm state. We gathered solid numerical evidence that there exists a unitarity range similar to the one we describe below, for all masses m 2 → −∞. However, to keep a smooth q → 1 limit to string theory we restricted ourselves to m 2 ≥ −1, but the m 2 < −1 regime might contain some physics worth studying [31].
The upper bound is imposed on us by unitarity: for m 2 > 1 3 , the Coon amplitude has ghosts in all dimensions and for all values of q. One can check this explicitly by looking for example at the coefficient c 1,1 , which is given by Other coefficients are shown in the appendix in eq. (42) together with their q → 1 limit. Eq. (13) shows that for d ≥ 4 this state is a ghost if m 2 > 1 3 . This stems from the relation between cos θ and t, given by cos(θ) = 1+ 2t m 2 n −4m 2 , which becomes singular at the first resonance for m 2 = 1 3 and causes negative coefficients beyond this value. Therefore, we restrict ourselves to the range Interestingly, this upper bound seems to match the pion intercept fits of the massive endpoint string models of [32, eq. 4.2]. It would be interesting to investigate this question further.
We shall now prove that the Coon amplitude is unitary in all dimensions for 0 < q ≤ q ∞ (m 2 ). The proof uses the elementary fact that a polynomial with positive coefficients is positively expandable on Gegenbauer polynomials in all dimensions, as can be checked on eq. (36).
The starting point is to write a given residue as a function of x = cos(θ) = 1 + (15) where b n = (1 + 3m 2 (q − 1) − q n )/2, and The residue has all of its coefficients p n,k positive if and only if all of its roots are positive, thus we want to study the domain in the q, m 2 parameter space where x j,n ≤ 0 for all j and all n. Firstly, note that the roots are ordered: since q < 1, if j < j , then x j,n < x j ,n . Hence, a necessary and sufficient condition for all the roots to be positive (at fixed n) is that x n−1,n ≤ 0. Since x n−1,n increases monotonically with n, it is further enough to ensure that the limit n → ∞ of x n−1,n is negative. This amounts to setting q n → 0 in x n−1,n . For this to be negative, we need to have q and m 2 related by q < q ∞ (m 2 ) where q ∞ (m 2 ) is defined above in eq. (12).
We have now concluded that for q < q ∞ the residue is a polynomial with only positive coefficients, which implies that the amplitude is ghost-free in all dimensions in this region of parameter space.
Critical dimensions for q > q∞(m 2 ) In this range, ghosts are not excluded by our previous argument, and therefore we performed an extensive numerical study of the sign of the residues. We computed the Gegenbauer coefficients for the first 50 resonances and all spins for a grid of values of m 2 between −1 and 1 3 and varied q and d by small increments to obtain the critical dimension d(m 2 , q) for which all the coefficients become positive for each (m 2 , q), with accuracy 1/25 = 4%. This allowed us to map the surface in parameter space that separates the unitary and non-unitary regions, given by the green spline in fig. 2. In the q → 1 limit, the critical dimensions of the m 2 = −1 and m 2 = 0 models match the known values for the Veneziano and Neveu-Schwarz amplitudes (d = 26 and d = 10 respectively). Moreover, our results point towards a possible extremal unitary amplitude with m 2 = 1 3 and critical dimension d 6.3. Those three curves are plotted specifically in fig. 3. We provide some more details on the numerics in the appendix.
One interesting observation from that study is that the scalar ghost sector seems define the unitarity surface. Below, we give a proof of this fact in the large d limit and an estimate of the critical dimensions for q near the critical line which matches the numerics, see fig. 3. Surprisingly, while our arguments fail when q goes closer to 1, we observe numerically that the scalar ghost criterion continues to hold. Proving this fact fully, maybe using the methods of [28], would allow to prove exactly the unitarity of Coon (up to finite numerical accuracy in d) and of Veneziano amplitude as a function of m 2 .
The argument about the scalar sector and the critical dimensions goes as follows. Consider the pole at σ = q n .
We want to relate the coefficients c n,L in the Gegenbauer decomposition of P n to the coefficients p n,k appearing in (15), which are given in terms of the roots x j,n by At fixed n, the c n,L coefficient only receives contributions from the p n,k coefficients with k ≥ L and with the same parity. Explicitly, for the scalar sector we have From the form of the roots x j,n one can see that the p L,n coefficients follow a well-defined pattern: when decreasing q starting from values where the amplitude is not unitary, p L,n become positive in decreasing order of spin. In particular, at some point, all the p n,k coefficients will have become positive except p n,0 .
The important point is that right before p n,0 turns positive (when x n−1,n becomes negative), c n,0 will become positive by continuity, as only the first term in (18) remains negative. Thus, at large d, c n,0 becomes positive when d = p n,2 −p n,0 Since p n,0 is small and negative, d is large as expected. As all the other c n,L coefficients do not receive contributions from p n,0 , we see that the coefficient c n,0 is the last one to become positive and thus the scalar ghosts are the ones defining the transition to the unitary regime in the large d limit.
Finally, similar arguments show that this ratio decreases as n → ∞ and therefore the limit curves enveloppes the region of unitarity of the amplitude. It turns out that the limit can be explicitly evaluated, by summing the infinite n limit of the double sum −p n,2 /p n,0 = − j1 =j2 1 xn,j 1 xn,j 2 . The resummed function is given in appendix, and is plotted in 3, and match nicely the numerical results at large d.

LOW ENERGY EXPANSION AND EFT-HEDRON
Recent times have witnessed a renewal of activity revolving around implications of dispersion relations, crossing symmetry and unitarity. Following the ideas of [5], various studies explored how the Wilson coefficients of weakly coupled EFTs are constrained [6][7][8][9][10]33] and live in some (sometimes small) regions of some positive region dubbed "EFThedron".
In this section, we compute the first few low energy couplings of the Coon amplitude. Because the Coon amplitude is well behaved at infinity, and respects analyticity and crossing, it admits dispersion relations and must fall in those positivity regions. We will see that the couplings indeed draw 1-dimensional varieties within those regions, parametrized by the value of q.
The first amplitude we consider is an (s, t, u) symmetric version of Coon, for external massless scalars (m 2 = 0) with no color indices: The (s, t, u) symmetry and momentum conservation s + t + u = 0 allow to expand at small s, t, u this function in terms of σ 2 = s 2 + t 2 + u 2 and σ 3 = stu, so that M q (s, t, u) = 1 s where the coefficients of this expansion are classically interpreted as low energy Wilson coefficients. A lengthy but straightforward explicit calculation gave us the first few coefficients, up to g 4 (q). Trivially, g 0 (q) = 1 − q. The next ones are given by functions related to q-zeta values, for instance g 2 (q) reads can be written in terms of q-deformed polylogarithms as defined for instance in [34], and whose q-zeta values are classically defined as q-values of those functions. Note that, compared to string theory, different orders of qtranscendentality appear to be mixed. The other couplings g 3 (q) and g 4 (q) are given in eqs. (37), (38). We also verified that when q → 1, they descend to the values given by the symmetrized sum A V (s, t) + A V (t, u) + A V (u, s): For generic EFTs, the allowed range of coefficients g 2 , g 3 , g 4 was determined in [6], fig. 8, in terms of dimensionless ratiosg 3 = g 3 M 2 /g 2 andg 4 = g 4 M 4 /g 2 with M 2 given by the scale of the first massive mode, which in our conventions is M 2 = [1] = 1. We show in fig. 4 the value of those ratios. They fall neatly in the domain determined in [6], albeit approaching tangentially the boundary at intermediate values of q.
One can also couple a massless scalar to a massive Coon amplitude, since 0 ≤ m 2 ≤ 1/3 are allowed. These amplitudes reduce to the extreme scalar case of [6] of a coupling the massless scalar to a single massive scalar of mass M 2 = m 2 and therefore accumulate to the upper right corner of their fig. 8. It is not surprising, and the same happens when coupling by a massless scalar to an amplitude made of massive Veneziano blocks.
Low spin dominance. The Coon amplitude A q (s, t), together with its Veneziano limit, exhibit a form of lowspin dominance, albeit weaker than that of [11]. It is a low spin dominance were not only the scalar state dominates the partial waves, but also the spin 1 state [35]. Let us explain how this comes about.
Following the conventions of [11], we Taylor expand the amplitude as A q (t, −s−t) = 1 s + 1 t + p<k a k,p s k−p t p . In order to match to [11], we look at the coefficients at level k = 2, 4, 6, which we compare to that of an amplitude given by a sum of a4,0 ≤ 6. The upper bound correspond to pure low-spin dominance, while the lower bound is pure spin-0 + spin-1 model. While for the coefficients a 2,1 /a 2,0 , the bounds are exactly satisfied, at k = 4 it can be seen that string theory lies a bit away from that, at (   a4,1 a4,0 , a4,2 a4,0 ) (2.9, 2.9). The relative accuracy of the model is explained by the fact that spin J exchanges come with q J(J+1)/2 , for which the linear approximation (spin 0 and 1) is a good approximation away from q = 1.

PERSPECTIVES
This study opens many perspectives, already mentioned in the text.
Firstly, it resonates very neatly with a conjecture of [14] that amplitudes with accumulation points populate the EFT-hedron of gravitational theories away from the small portion where usual theories seem to live. It would be very important to study this problem in more details, in relation with the Coon amplitude.
Secondly, it opens a way to attack the question of the positivity of the Veneziano amplitude recently studied in [28] thanks to its extra q-dependence. In particular, if one could prove our empirical observation that only the scalar ghost determines the critical dimension for values of q arbitrarily close to one, maybe using the techniques of [28], one could use the smoothness of the limit to prove positivity of the string theory four-point amplitude.
More generally, it would be of course essential to extend those results to the N -point function. Indeed, to make a statement about the unitarity of a theory (rather than the amplitude), in absence of a no-ghost theorem, one should indeed prove that ghosts decouple in all exchanges of all amplitudes. N -point functions were proposed as infinite sums in [2,36] and factorization was proven in [37,38].
It would also be interesting to study the Coon version of the Lovelace-Shapiro amplitude [39,40]. This amplitude, only recently understood from string theory [41], constitutes an interesting example where the N -point function shows defaults of unitarity.
Finally, the relation to q-zeta values might open an interesting avenue to produce worldsheet models for the Coon amplitude, and relate to various integral representation proposed in the literature [42,43].
Kinematics. Here we define our conventions for the article. In the centre-of-mass frame, we have where E is the center of mass energy, p the momentum transfer and θ the scattering angle. Note that the t ↔ u crossing simply sends θ → π − θ in the case of scattering of identical particles. The standard relations for the Mandelstam invariants read and cos(θ) = 1 + 2t Partial wave expansion. We follow the conventions of [4]. The polynomials P  (33) where G ( d−3 2 ) J (z) are known as Gegenbauer polynomials. They satisfy the following orthogonality relation: with normalization factors defined by (35) Positivity of a given monomial x n follows explicit expression [48]: .
for ρ positive integer. When n and J do not have the same parity modulo 2, vanishes. This condition on ρ and n, together with a factor of 2 are missing in [48].

Regge trajectories
For the interested reader, we show below how the Regge trajectories j(t)

Numerical study
We mapped the surface that separates the unitary and non-unitary regions in parameter space by a combination of two methods: To obtain the vertical part corresponding to the high d behaviour we started from a value q > q ∞ (m 2 ) where ghosts are present and decreased progressively q, while keeping m 2 and dimension d fixed until all the coefficients became positive. This allowed us to identify a d-dependent critical q, which we call q c (m 2 , d), that determines the boundary of the region of 3-dimensional parameter space q, m 2 , d in which the amplitude is unitary and approaches q ∞ (m 2 ) in the large d limit.
For the rest of the surface we used the fact that for each m 2 there exist a critical dimension below which the amplitude is ghost-free for all values of q. Fixing q and m 2 and starting from d lower than this critical dimension, we progressively increased d until we found the presence of ghosts, thus determining the critical surface.
In both cases we computed all the Gegenbauer coefficients for the first 50 resonances at each value of q with a numerical uncertainty on the critical dimension of 4%.