Veneziano Variations How Unique are String Amplitudes Cliﬀord Cheungaand Grant N. Remmenbc

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Veneziano Variations:

How Unique are String Amplitudes?

Cliﬀord Cheungaand Grant N. Remmenb,c

aWalter Burke Institute for Theoretical Physics,

California Institute of Technology, Pasadena, California 91125

bKavli Institute for Theoretical Physics,

University of California, Santa Barbara, CA 93106

cDepartment of Physics,

University of California, Santa Barbara, CA 93106

Abstract

String theory oﬀers an elegant and concrete realization of how to consistently couple

states of arbitrarily high spin. But how unique is this construction? In this paper we

derive a novel, multi-parameter family of four-point scattering amplitudes exhibiting

i) polynomially bounded high-energy behavior and ii) exchange of an inﬁnite tower

of high-spin modes, albeit with a ﬁnite number of states at each resonance. These

amplitudes take an inﬁnite-product form and, depending on parameters, exhibit mass

spectra that are either unbounded or bounded, thus corresponding to generalizations

of the Veneziano and Coon amplitudes, respectively. For the bounded case, masses

converge to an accumulation point, a peculiar feature seen in the Coon amplitude

but more recently understood to arise naturally in string theory [1]. Importantly, our

amplitudes contain free parameters allowing for the customization of the slope and

oﬀset of the spin-dependence in the Regge trajectory. We compute all partial waves

for this multi-parameter class of amplitudes and identify unitary regions of parameter

space. For the unbounded case, we apply similar methods to derive new deformations

of the Veneziano and Virasoro-Shapiro amplitudes.

e-mail: clifford.cheung@caltech.edu,remmen@kitp.ucsb.edu

arXiv:2210.12163v2 [hep-th] 30 Jan 2023

Contents

1 Introduction 3

2 Amplitude Construction 5

2.1 ProductAnsatz.................................. 5

2.2 Polynomial Residue Constraint . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 FinalAmplitude.................................. 10

3 Unitarity Bounds 16

3.1 AnalyticPositivity ................................ 16

3.2 NumericalPositivity ............................... 21

3.3 Additional Spectral Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Unbounded Variations 24

4.1 CyclicInvariant.................................. 25

4.2 PermutationInvariant .............................. 26

5 Discussion 27

A Analytic Partial Waves 28

1 Introduction

The amplitudes bootstrap exploits the serendipitous fact that many of the theories that

describe nature are actually ﬁxed by simple conditions imposed on the kinematic functions

that encode scattering. For example, while gauge theory and gravity can famously be derived

from high-minded geometric principles or gedankenexperiments, this is not our only recourse

for understanding their origins. Rather, an alternative path to discovering gauge theory and

gravity is to answer a concrete math question about the S-matrix: What is the space of

on-shell scattering amplitudes for massless spin-one and spin-two particles that are local,

Lorentz invariant, and mediate a long-range force?

It is then natural to ask, what is the analogous question in string theory? In broad terms,

string theory furnishes a working example of how gravity and quantum mechanics can be

rendered consistent. At a more technical level, it addresses the problem that high-energy,

ﬁxed-angle graviton scattering is ill behaved. Unitarizing this behavior requires the addition

of higher-spin degrees of freedom that must be included with care lest they exacerbate these

ultraviolet pathologies. Thus, string theory oﬀers an explicit answer to the concrete question

of how to consistently build an amplitude that exhibits the exchange of higher-spin modes

and is sensible at high energies. In his seminal work [2], Veneziano constructed precisely

such an amplitude,

Astring(s, t) = Γ(−α0−α0s)Γ(−α0−α0t)

Γ(−2α0−α0(s+t)) .(1)

In this paper, we revisit this line of inquiry, in particular asking to what extent string

amplitudes are the unique solutions to this particular math problem.

Concretely, we construct a multi-parameter space of Lorentz invariant, four-particle,

perturbative scattering amplitudes A(s, t)that describe an inﬁnite exchange of higher spins

while conforming to the following conditions:

i) Polynomial Boundedness. The high-energy behavior of the amplitude is polynomially

bounded. If this condition fails, causality and unitarity are generically violated [3–6].1

ii) Finite-Spin Exchange. Each pole in the amplitude describes the exchange of a ﬁnite

tower of spins. When this condition fails, the exchanged state is an inﬁnitely extended

object, thus undermining the locality of the theory.

Famously, string theory provides precisely a function A(s, t)that conforms to these criteria

in the form of Eq. (1). In this case, the amplitude vividly encodes the Regge spectrum of an

1Rigorous bounds exist for gapped theories, e.g., the Regge (A<s2) and Froissart (A<slogD−2s) bounds,

which apply to ﬁxed momentum transfer and scattering angle, respectively.

inﬁnite tower of stringy excitations, evenly spaced in mass-squared and with an ever-growing

collection of spins.

Perhaps less appreciated is the existence of yet another solution to the above constraints,

discovered—and then sadly forgotten by virtually all—in the remarkable work of Coon [7].

The Coon amplitude is a deformation of the Veneziano amplitude labeled by a single pa-

rameter q. The most notable attribute of the Coon amplitude is its hydrogen-like spectrum:

a discrete array of mass levels that converge to inﬁnite density at an accumulation point,

followed by a branch cut.2Recently, a number of works [8–10] have analyzed the Coon ampli-

tude from the perspective of positivity bounds derived from analytic dispersion relations. By

demanding that the spectral density be nonnegative—i.e., no ghosts—those authors mapped

out a putative consistent region for q.

The outline of the present work is as follows. In Sec. 2, we generalize the analysis of

Coon to derive a new multi-parameter family of amplitudes that are cyclic invariant on the

external legs and conform to the criteria described above. These amplitudes exhibit the

same mass spectrum of accumulation points as the Coon amplitude, but diﬀer markedly in

their distribution of spins at each level. Interestingly, they also exhibit customizable Regge

trajectories. We also describe the low- and high-energy limits of these amplitudes. We

emphasize that this multi-parameter class is not an exhaustive solution space but merely

oﬀers an existence proof of such deformations.3

Afterward, in Sec. 3 we derive constraints on this family of amplitudes coming from par-

tial wave unitarity. This enforces the nonnegativity of every coeﬃcient in the partial wave

decomposition of the amplitude on each residue. We derive both analytic and numerical pos-

itivity bounds and observe that there is a multi-parameter subspace of putatively consistent

amplitudes.

In Sec. 4 we then derive a new family of amplitudes obeying the above criteria but exhibit-

ing mass spectra that are free from accumulation points and thus unbounded. We generalize

the analysis of Sec. 2 to this case and present a multi-parameter space of amplitudes ex-

hibiting either cyclic or full permutation invariance on the external legs. Unfortunately, this

alternative class of theories appears to be disfavored by unitarity constraints.

Finally, we summarize our results and discuss promising future directions in Sec. 5.

2A reasonable worry is that an inﬁnite density of states is inconsistent with statistical mechanics because a

thermal ensemble will sample the inﬁnite reservoir of modes near the accumulation point. Similar logic would

imply tension with holographic bounds on degrees of freedom. However, recall that any such pathology

requires that the degenerate states be indistinguishable. If they are not—for example for the case of the

hydrogen atom and some recent string theoretic constructions [1]—then nothing is awry.

3In this work we restrict to the case of scalar external states. As is well known, however, the inclusion of spin

is achieved simply by multiplying scalar amplitudes by polarization-dependent prefactors. Unfortunately,

this operation generically worsens the high-energy and Regge behavior of the amplitudes.

2 Amplitude Construction

2.1 Product Ansatz

In this section we construct a family of amplitudes A(s, t)describing the scattering of ex-

ternal scalars subject to the constraints of polynomial boundedness and ﬁnite-spin exchange

described in Sec. 1. Here we assume cyclic invariance on the external legs, so A(s, t) = A(t, s)

is st-symmetric, like color-ordered amplitudes in gauge theory and open string theory.

If we view A(s, t)as a tree-level amplitude, then it exhibits simple poles on exchanged

resonances and can be written as a rational function of the form

A(s, t) = N(s, t)

∞

n=0(s−m2

n)(t−m2

.(2)

Here m2

ndescribes an arbitrary spectrum indexed by nonnegative integers n, properly ordered

such that m2

n< m2

n+1. Note that any state degeneracy at a given mass level is accounted for

by multiplicity factors in the numerator.

At this juncture, let us call attention to an important subtlety: inﬁnite products such

as ∞

n=0(s−m2

n)are actually ill deﬁned. In particular, convergence requires each factor in

the product to approach unity at large n, which is impossible for all s. To express the

amplitude in a convergent form, one needs to make a binary choice of whether the spectrum

is unbounded or bounded. For the latter case, m2

∞is ﬁnite and there is an accumulation

point in the spectrum. In the subsequent analysis we will assume that this is the case.

Note that precisely for this reason, none of our results will contradict those of Ref. [11],

whose analysis argued for the uniqueness of the Veneziano amplitude, but given the explicit

assumption that the spectrum is free from accumulation points.

With a spectrum bounded from above and below, it is convenient to deﬁne the auxiliary

kinematic variables σ, τ ,

s= (m2

0−m2

∞)σ+m2

∞

t= (m2

0−m2

∞)τ+m2

∞,(3)

which are related to the physical Mandelstam variables s, t by an aﬃne transformation. Here

the range 0≤σ≤1maps to m2

∞≥s≥m2

0, which scans through the spectrum of states. In

terms of the auxiliary kinematic variables, the amplitude takes the form

A(σ, τ) = N(σ, τ)

∞

n=0 1−f(n)

σ1−f(n)

τ,(4)

which exhibits an inﬁnite tower of simple poles located at

f(n) = m2

n−m2

∞

0−m2

∞

,(5)

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VenezianoVariations:HowUniqueareStringAmplitudes?CliordCheungaandGrantN.Remmenb;caWalterBurkeInstituteforTheoreticalPhysics,CaliforniaInstituteofTechnology,Pasadena,California91125bKavliInstituteforTheoreticalPhysics,UniversityofCalifornia,SantaBarbara,CA93106cDepartmentofPhysics,UniversityofCalifo...

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