String theory is our leading candidate for a unified and self-consistent formulation of quantum gravity. Its core assertion is exceedingly simple: perhaps the inventory of the universe is composed not of point particles, but rather of extended objects. This line of inquiry has sparked an extraordinary influx of new ideas across disciplines, including quantum field theory, phenomenology, cosmology, condensed matter theory, quantum information theory, and even mathematics.

Despite these successes, pressing questions remain. How unique is string theory? What is the minimal set of principles from which it follows? And most importantly, is it the only solution satisfying these criteria?

The amplitudes bootstrap is a versatile framework for interrogating these questions with mathematical precision. In this approach, one sculpts out a consistent space of theories by constraining an ansatz scattering amplitude with certain physical conditions. For example, imposing Lorentz invariance and factorization on the scattering of massless particles of spin one or two is sufficient to uniquely fix the dynamics, thus deriving gauge theory and gravity without the aid of an action [1–5].

In this paper, we show that this approach is even more powerful if we also assume a spectrum together with a short list of physical constraints that include crossing symmetry, bounded high-energy scaling, and finite spin for the exchanged states. Imposing these conditions, we derive a new infinite parameter class of amplitudes that subsumes the renowned Veneziano [6] and Coon [7] amplitudes as special cases. We then use analogous logic to derive generalizations of the Virasoro-Shapiro amplitude [8,9], which describes gravitational scattering.

We construct four-point, tree-level scattering amplitudes for massless scalars subject to a handful of simple physical criteria: (i)

Crossing symmetry. For external scalars exhibiting cyclic or full permutation invariance, the corresponding scattering amplitudes satisfy A(s,t)=A(t,s) and M(s,t)=M(t,s)=M(s,u)=⋯, respectively. Here A(s,t) and M(s,t) can be viewed as gauge theory and gravity amplitudes stripped of their kinematic prefactors, F4/u and R4, which are products of the linearized field strengths and curvatures encoding polarizations. For the rest of this section we focus on A(s,t), though our discussion generalizes straightforwardly to M(s,t).

The last and most nontrivial ingredient in our analysis is the spectrum of the theory, input via some sequence mn2. We will consider integer and q-integer spectra, leaving more general possibilities for future work.

To implement the bootstrap described above, we assume a dual resonant ansatz for an amplitude with polynomial residues. This leaves crossing symmetry as the final and most difficult condition to impose. As we will see, one can reduce this complex analysis problem in two variables to a simpler problem of a single variable by restricting t to a wisely chosen function of s.

To begin, let us consider an integer mass spectrum, mn2=n,(4)in which case the dual resonant form of the amplitude is A(s,t)=∑n=0∞Rn(t)n-s,(5)where Rn(t) is defined in Eq. (1). Here we again note that Eq. (5) only converges for t<0 or on poles in s.

Next, we study a special kinematic regime in which t is displaced from s by a positive integer k, t=s-k.(6)By inserting Eq. (6) into Eq. (5) and imposing crossing symmetry, we find that A(s,s-k)=A(s-k,s). This implies that ∑n=0∞Rn(s-k)n-s=∑n=0∞Rn(s)n+k-s, which upon a relabeling of terms becomes ∑n=k∞Rn(s-k)-Rn-k(s)n-s=-∑n=0k-1Rn(s-k)n-s.(7)The right-hand side comprises a finite collection of terms whose poles are at s=n<k. Since this expression is elsewhere regular, it has vanishing residues at n≥k. Demanding the same of the left-hand side yields Rn(n-k)=Rn-k(n),(8)for 1≤k≤n. Note that Eq. (8) should be handled with care since it is neither necessary nor sufficient for enforcing crossing symmetry for all s and t. It is not sufficient because it was derived for the special kinematic choice in Eq. (6), so crossing might hold on the support of that condition but not away from it. Meanwhile, it is not necessary because it originates from the infinite sum in Eq. (5), whose residues converge at poles in s for generic t but not necessarily for t satisfying Eq. (6). Nevertheless, for the present analysis we take Eq. (8) as an input assumption. Crucially, in all of our later examples, we find that the sums in Eqs. (5) and (7) indeed converge and can be trusted. Our resulting amplitudes also exhibit full crossing symmetry for generic s and t.

The residue crossing condition in Eq. (8) imposes n constraints on the n+1 free parameters in Eq. (1). Concretely, we eliminate λn,m for m<n in terms of λm≡λm,m, yielding the general solution, Rn(t)=∑m=0nλmm!t!(t-m)!n!(n-m)!,(9)where we analytically continue x! to Γ(x+1) at will. Physically, λm controls the exchange of spin m modes at levels n≥m. While Eq. (9) naively admits arbitrary λm, many choices render the sum in Eq. (5) nonconvergent. For example, this happens if we choose λm=δmm′ for some m′. Consequently, it is necessary to determine λm for which the expression in Eq. (5) actually converges.

Evaluating Eq. (9) for the very special choice λm=1m!, the Vandermonde identity implies that Rn(t)=(t+n)!t!n!=(t+nn). Inserting this residue into the dual resonant ansatz in Eq. (5) yields A(s,t)=∑n=0∞1n-s(t+nn)=Γ(-s)Γ(-t)Γ(-s-t),(10)which is the Veneziano amplitude. Here we obtained right-hand side by relating the sum to a hypergeometric function and applying the Gauss summation theorem.

Next, let us consider the more general parameter choice λm=r!(m+r)! for any real value of r. Here the ansatz residue in Eq. (9) becomes Rn(t)=(t+n+r)!r!(t+r)!(n+r)!, so Eq. (5) evaluates to A(s,t)=∑n=0∞1n-s(t+n+r)!r!(t+r)!(n+r)!=-1sF32[1,-s,1+t+r1-s,1+r;1].(11)Via a Thomae transformation [15], this expression equals A(s,t)=Γ(-s)Γ(-t)Γ(-s-t)F23[-s,-t,r-s-t,1+r;1],(12)which is the Veneziano amplitude times a manifestly crossing symmetric function of s and t that reduces to 1 at r=0. For arbitrary kinematics, Eq. (12) will be our definition of a new hypergeometric amplitude.

We can recast Eq. (12) in several illuminating forms. For nonzero real r, inserting the definition of the hypergeometric function yields an infinite sum over a special case of the generalized amplitudes derived in Ref. [13], A(s,t)=∑n=0∞1n!rr+nΓ(-s+n)Γ(-t+n)Γ(-s-t+n),(13)which also coincides with a particular choice of coefficients in the string-inspired ansatz in Ref. [16].

For nonnegative integer r, Eq. (12) is also equal to A(s,t)=(-1)rr!Γ(-s-r)Γ(-t-r)Γ(-s-t-r)+P2r-2(s,t)Q2r(s,t),(14)where P2r-2 and Q2r(s,t)=∏ℓ=1r(s+ℓ)(t+ℓ) are polynomials of degree 2r-2 and 2r, respectively. Here the first and second terms in Eq. (14) exhibit spurious poles in s and t that precisely cancel. Note that Eq. (14) can be analytically continued to negative integer r if we also divide λm and thus A(s,t) by an overall factor of r!. In this case the hypergeometric amplitude A(s,t) is given just by the first term in Eq. (14), which is also the basis amplitude comprising the string ansatz in Ref. [16].

Next, it will be instructive to study various limits of A(s,t). For example, for nonnegative integer r, the low-energy expansion of the amplitude is A(s,t)=-1s-1t+Hr,1+π26(s+t)+ζ(3)(s+t)2-Hr,1,2st+π4360(s+t)(4s2+4t2+st)-(π26Hr,2-Hr,1,3)st(s+t)+ζ(5)(s+t)2(s2+st+t2)+[π26Hr,3-(π26+Hr,2)ζ(3)-Hr,1,4]st(s+t)2+2r1+rF76[1,1,1,1,1,1,1-r2,2,2,2,2,2+r;−1]s2t2+⋯,(15)where we have defined the generalized harmonic number Hn,m=∑k=1nk-m and its moments Hn,m,k=∑ℓ=1nHℓ,mℓ-k [17]. Note that Eq. (15) apparently exhibits uniform transcendentality [20].

From Eq. (14) we can straightforwardly derive the behavior of the amplitude for high-energy fixed-angle scattering. In particular, sending |s|,|t|→∞, we obtain A(s,t)∼eB(s,t)+rst+⋯,(16)where B(s,t)=(s+t)log(s+t)-slogs-tlogt+⋯ and ellipses denote subleading contributions we will ignore. For nonnegative integer r, the first and second terms in Eq. (16) are trivially derived from the first and second terms in Eq. (14). In the physical region, -1≤cosθ=1+2ts≤1, we find that B(s,t)<0, so A(s,t)∼r/st falls off as a power law. In the unphysical region, t>0, we find that B(s,t)>0, in which case A(s,t)∼eB(s,t) exhibits the same behavior as the Veneziano amplitude, in accordance with general arguments [21].

Meanwhile, the Regge limit of s→∞ at fixed t is A(s,t)=sJ(s,t)+r(1+t)s+⋯,(17)where J(s,t)=t+⋯ and ellipses denote subleading terms. Here the first and second terms dominate depending on whether t>0 or t<0, respectively. Any spurious poles in t will cancel between terms, as in Eq. (14).

Rather incredibly, the hypergeometric amplitude in Eq. (12) has an integral representation that is tantalizingly reminiscent of the string worldsheet, A(s,t)=r∫01∫01dxdyx-s-1yr-1(1-xy)t(1-x)t+1.(18)Amazingly, the integral above is just the Koba-Nielsen formula for the five-point Veneziano amplitude [22], ∫01∫01dxdyx-s12-1y-s45-1(1-xy)s23+s34-s51(1-x)s23+1(1-y)s34+1,(19)evaluated at s12=s, s23=t, s34=s51=-1, and s45=-r. It is perhaps not so surprising that this object is a viable four-point amplitude. By setting s34, s45, s51 to constant values, we ensure that the resulting expression exhibits singularities only in s and t, while the s↔t symmetric choice of kinematics enforces crossing. For the case of r→0, the five-point Veneziano amplitude factorizes onto the massless pole at s45→0, so multiplying by r to remove the singularity yields the four-point Veneziano amplitude, as expected.

There is an infinite parameter space of amplitudes constructed from a weighted sum over the solutions above, λm=∫-∞∞dr μ(r)r!(m+r)!.(20)Given any choice of μ(r) for which the integral converges, the corresponding λm yield a consistent amplitude. The resulting object is a linear combination of our hypergeometric amplitudes, so its properties are straightforward to derive.

Next, consider a spectrum defined by q-deformed integers, mn2=[n]q=1-qn1-q,(21)where 0<q<1 [23]. The q-deformed integers reduce to the usual integers for q→1, allowing for a simple comparison to the results of the previous section.

To streamline our analysis, we define s=[σ]q and t=[τ]q in analogy with Eq. (21), where σ and τ are the natural kinematic variables given the spectrum. We also assume a dual resonant representation for the amplitude, A(σ,τ)=∑n=0∞Rn([τ]q)[n-σ]q,(22)where Rn([τ]q)=Rn(t) is the same quantity defined in Eq. (1). Furthermore, since 1[n-σ]q=1-(1-q)s[n]q-s, Eq. (22) exhibits simple poles in s at the points s=[n]q.

Next, let us consider special kinematics where σ and τ are offset by a nonnegative integer k, τ=σ-k.(23)Imposing crossing symmetry on the amplitude implies that A(σ,σ-k)=A(σ-k,σ), so ∑n=k∞Rn([σ-k]q)-Rn-k([σ]q)[n-σ]q=-∑n=0k-1Rn([σ-k]q)[n-σ]q.(24)As before, we are motivated to impose the crossing condition on the residues, Rn([n-k]q)=Rn-k([n]q)for  1≤k≤n.(25)Solving these equations for the ansatz in Eq. (1), we obtain the general solution, Rn([τ]q)=∑m=0nλmqm(m-1)2[m]q![τ]q![τ-m]q![n]q![n-m]q!,(26)which is the q-deformed generalization of Eq. (9).

For the special choice of λm=qm(m+1)/2×1[m]q!, we use the q-Vandermonde identity to write the residue in Eq. (26) as Rn([τ]q)=[τ+n]q![τ]q![n]q!=(τ+nn)q, which is a q-deformed binomial distribution. Plugging this residue back into Eq. (22), we obtain A(σ,τ)=∑n=0∞1[n-σ]q(τ+nn)q.(27)Since Eq. (27) is literally the q-deformation of the Veneziano amplitude in Eq. (10), it is natural to conjecture that it is the Coon amplitude [7]. However, it is not. In fact, the sum in Eq. (27) does not even converge, so Eq. (25) is invalid, and s↔t crossing fails.

However, if we dress each term in the sum by hand with an additional factor of qτ(σ-n), we obtain A(σ,τ)=∑n=0∞qτ(σ-n)[n-σ]q(τ+nn)q=qστΓq(-σ)Γq(-τ)Γq(-σ-τ),(28)which is exactly the Coon amplitude [7,26,27].

Since the mysterious prefactor qτ(σ-n)=1 for σ=n, it does not affect the residues of the amplitude and contributes only contact interactions. Clearly, there is an infinite space of similar functions, but only the choice made above yields the Coon amplitude. We leave an investigation of other possible prefactors for future work.

Consider a more general parameter choice λm=qm(m+1)/2+rm×[r]q![m+r]q!, for which the residue is Rn([τ]q)=[τ+n+r]q![r]q![τ+r]q![n+r]q!. Again dressing each term in Eq. (22) with qτ(σ-n) and using the q-deformed Thomae transformation [28], we obtain A(σ,τ)=∑n=0∞qτ(σ-n)[n-σ]q[τ+n+r]q![r]q![τ+r]q![n+r]q!=qστΓq(-σ)Γq(-τ)Γq(-σ-τ)ϕ32[q-σ,q-τ,qrq-σ-τ,q1+r;q;q],(29)where the last factor is a basic hypergeometric function [29]. The object in Eq. (29) subsumes every amplitude we have discussed thus far, including hypergeometric (q=1), Coon (r=0), and Veneziano ({q,r}={1,0}).

In analogy with Eq. (20), we can also consider an arbitrary linear combination of the λm defined above, in which case the resulting amplitude is a corresponding linear combination of q-hypergeometric amplitudes that subsumes all of the amplitudes in Refs. [30,31].

There is a sizable parameter space, depicted in Fig. 1, for which our new amplitudes are consistent with unitarity. Following the analysis of Refs. [13,24,32–34], we consider external states with mass m02, which is accomplished by simply sending (s,t)→(s-m02,t-m02) in the q-hypergeometric amplitude in Eq. (29). The resulting amplitude depends on the set of parameters {q,r,m02}.

Expanding the residue of the pole at level n in partial waves, we obtain Rn(t)=∑ℓ=0nan,ℓGℓ(D)(cosθ), where cosθ=1+2ts-4m02 and Gℓ(D) denotes the D-dimensional Gegenbauer polynomials. Unitarity implies that an,ℓ≥0, thus sculpting out a consistent parameter region spanned by {q,r,m02,D} [35]. We find numerically that r is always larger than -1, from which we analytically find m02≤1/3 from an,n≥0. From an,n-1≥0, we then have the bound 2q-1-r≥3-q+m02(q-1), which reduces to r≥-(1+m02)/2 as q→1. We leave a full analysis of partial wave unitarity for future work.

For the case of a gravitational amplitude M(s,t) that vanishes as s→∞ at fixed t<0, we can compute the dispersion relation, M(s,t)=∮s′=sds′2πiM(s′,t)s′-s=∫−∞∞ds′πImM(s′,t)s′-s=∑n=0∞(1mn2-s+1mn2-u)Rn(t),(30)where we assume that the external states are massless, so s+t+u=0. Hence, M(s,t) can be expressed as an infinite sum over poles in the s and u channels.

Next, let us assume a linear spectrum as in Eq. (4), so that the dual resonant form of the amplitude is M(s,t)=∑n=0∞(1n-s+1n-u)Rn(t),(31)where R0(t)=ξ/t2 so that M(s,t)=ξstu+⋯ exhibits a long-range force at low energies. Since 1/stu is trivially crossing symmetric and dual resonant, it can be added or subtracted with impunity without violating our input assumptions. Hence, ξ is freely floating and cannot be related to any other parts of the amplitude by crossing.

For the residues at positive n, we define Rn(t)=∑m=02(n-1)κn,mtm,(32)so the modes exchanged at level n carry up to spin 2(n-1). For graviton scattering, M(s,t) is dressed with a polarization-dependent prefactor R4 that carries spin weight, in which case these modes carry up to spin 2n.

Next, we restrict to t=s-k, which implies that u=-2s+k. Since Eq. (31) is s↔u symmetric, we need only enforce crossing on s↔t. Equating M(s,s-k) to M(s-k,s) implies that ∑n=1∞Rn(s-k)n-s+Rn(s-k)n-k+2s=∑n=1∞Rn(s)n+k-s+Rn(s)n-k+2s. Relabeling the summation and equating the residue of each s-channel pole at s=n, we obtain Eq. (8). Meanwhile, equating the residue of each u-channel pole at s=-n-k2, we obtain Rn(-n+k2)=Rn(-n-k2),(33)where 1≤k≤n. Next, we constrain the ansatz in Eq. (32) with Eqs. (8) and (33), modulo any constraints involving ξ, since 1/stu can be freely added or subtracted from the amplitude as discussed above. We thus eliminate κn,m for m<2n in terms of the unfixed parameters κm≡κm,2m, yielding Rn(t)=∑m=0nκmm!(t+n+m)!(t+n)!(t-1)!(t-m-1)!(n-1)!(n-m-1)!,(34)where κm must be chosen so that the sum converges. For a q-integer spectrum, the above procedure yields no solutions, in accordance with general arguments [36].

Next, consider the parameter choice κm=1(m+1+r)!(m+1-r)!. In this case, the residue becomes Rn(t)=(t+n-1+r)!(t+r)!(n+r)!(t+n-1-r)!(t-r)!(n-r)!, which inserted back into the amplitude in Eq. (31) yields M(s,t)=Γ(-s)t2-r2F˜43[1,-s,t+r,t-r1-s,1+r,1-r;1]+s↔u.(35)By numerical evaluation, one can verify that Eq. (35) is not crossing symmetric for generic r. This is possible because the constraints in Eqs. (8) and (33) are not sufficient conditions for crossing. That said, for any integer r, Eq. (35) is in fact crossing symmetric, yielding a generalization of the Virasoro-Shapiro amplitude, M(s,t)=(-1)r+1Γ(-s+r)Γ(-t+r)Γ(-u+r)Γ(1+s+r)Γ(1+t+r)Γ(1+u+r),(36)which in fact forms a basis for the amplitudes described in the conclusions of Ref. [37].

Last but not least, we can actually take the principle of dual resonance even further, writing the Virasoro-Shapiro amplitude as a sum over s-channel poles alone, M(s,t)=∑n=0∞Rn(s,t)n-s,(37)where Rn(s,t)=(t+s-1)!t!s!(t+n-1)!t!n! depends on both s and t, and the sum is convergent for t<0.

The present work offers many lines of inquiry for future study. First and foremost, it would be interesting to bootstrap new amplitudes with different mass spectra. Second, there is the important question of whether our new amplitudes generalize to higher-point scattering. Finally, it would be interesting to perform a systematic analysis of the unitary regions of parameter space for our amplitudes.