Natural Boundaries for Scattering Amplitudes

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Princeton, January 2023

Natural Boundaries

for Scattering Amplitudes

Sebastian Mizera

Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA

E-mail: smizera@ias.edu

Abstract: Singularities, such as poles and branch points, play a crucial role in investi-

gating the analytic properties of scattering amplitudes that inform new computational

techniques. In this note, we point out that scattering amplitudes can also have another

class of singularities called natural boundaries of analyticity. They create a barrier

beyond which analytic continuation cannot be performed. More concretely, we use

unitarity to show that 2

→

2 scattering amplitudes in theories with a mass gap can

have a natural boundary on the second sheet of the lightest threshold cut. There, an

inﬁnite number of ladder-type Landau singularities densely accumulates on the real

axis in the center-of-mass energy plane. We argue that natural boundaries are generic

features of higher-multiplicity scattering amplitudes in gapped theories.

ArXiv ePrint: 2210.11448

arXiv:2210.11448v2 [hep-th] 7 Jan 2023

Contents

1 Introduction 2

2 Background 6

2.1 Lacunary functions 6

2.2 Analytic continuation of elastic unitarity 7

2.3 Holomorphic unitarity cuts 11

2.4 Lehmann and Zimmermann ellipses 12

3 Natural boundaries on the second sheet 13

3.1 Unitarity and anomalous thresholds 13

3.2 Ladder-type Landau singularities 16

3.3 Natural boundaries 18

3.3.1 Fixed scattering angle 18

3.3.2 Fixed momentum transfer 20

4 Discussion 20

A Analyticity at ﬁxed momentum transfer 23

A.1 Bros–Epstein–Glaser analyticity 24

A.2 Toy model for analytic completion 26

B Analyticity at ﬁxed scattering angle 30

References 31

1 Introduction

Physical behavior of scattering amplitudes is encoded in their analytic properties. One

does not need to search hard to ﬁnd numerous examples where understanding of the

analytic structure lead to signiﬁcant improvements in computational techniques [

–

There is therefore great interest in further exploration of the analytic properties of

scattering amplitudes, especially those general enough to be applicable to realistic

processes in the Standard Model and beyond.

– 2 –

What kind of singularities are known to appear in scattering amplitudes? At tree-

level in perturbation theory, we only encounter poles of the type (

s−m2

)

−1

, together

with their multi-particle generalizations. At low loop level, direct computations give

square-root and logarithmic branch points of the kind

fa/2logbf

for some integers

a, b

and a function

of the kinematics [

–

]. It is also known that scattering amplitudes

can have branching of arbitrary order, for instance,

sα(t)

in the Regge limit (where

is large and

(

) is the Regge trajectory) [

] or

s

in dimensional regularization

around 4

−



dimensions [

]. Theories of quantum gravity are expected to produce

essential singularities of the form

e−s

in the high-energy limit

s→ ∞

at ﬁxed angle

[

]. Resummation of diagrams can lead to non-holonomic singularities of the

form (1

−fa/2logbf

)

−1

[

–

]. It is also known that scattering amplitudes can have

distributional support in special limits, such as deep inelastic scattering in QCD [

], or

accumulation curves of singularities in unphysical kinematics [

]. One may ask if the

list of possible singularities has been exhausted.

Mathematically, analytic functions can certainly have another feature called a

natural boundary of analyticity, formed by a dense set of singularities. As such, they

provide barriers to analytic continuation. Natural boundaries are no strangers to

physicists; for example, elliptic functions such as the Dedekind eta function

(

) have a

singularity at every rational

and hence a natural boundary on the real

-axis, see,

e.g., [

]. In Sec. 2.1 we review the basic mechanism behind such accumulations of

singularities. A natural question arises, whether scattering amplitudes can have natural

boundaries and where to ﬁnd them.1

To answer this question, we turn to the simplest non-analytic feature of the 2

→

S-matrix: the branch cut for the lightest-threshold exchange, starting at

= 4

, where

is the center-of-mass energy squared. We take the mass

to be strictly positive.

In four space-time dimensions, the threshold is square-root branched, meaning that it

divides the

-plane into two sheets glued by a branch cut, see Fig. 1.1. By convention,

they are called the ﬁrst (or physical ) and second sheet respectively. We will denote the

connected part of the S-matrix on the

-th sheet

(

s, z

), where

cos θ

is the cosine

of the scattering angle and is held ﬁxed (the analytic structure of the S-matrix at ﬁxed

momentum transfer

is less understood, see App. A). It is on the second sheet that we

will encounter a natural boundary, see Fig. 1.1 (right).

The physical S-matrix can be recovered by approaching the branch cut from the

In order to avoid circular arguments, we assume that the spectrum of particles in the theory itself

is not dense and it does not have accumulation points. In most of this work we will consider a theory

with a single scalar of mass

m >

0. Note that scattering amplitudes in superstring theory have a

natural boundary in the (non-holomorphic) coupling constant (see, e.g., [

]), but it is not a kinematic

singularity.

– 3 –

4m2

9m2

3m2

Figure 1.1

. Two sheets of the lightest threshold cut at

s >

, in the

-plane at ﬁxed

on which

(

s, z

) and

(

s, z

) are deﬁned respectively.

Left:

First sheet has the

-channel

normal-threshold cuts extending to the right, while the

t, u

-cuts run to the left. The physical

region

s >

is approached from the upper half-plane (orange). Two paths of analytic

continuation (red and blue) go through the elastic region (4

m2,

) and end up on the second

sheet.

Right:

Second sheet has a natural boundary (thick black line) extending throughout

and arising from a dense accumulation of ladder-type Landau singularities. Additional

singularities, such as bound-state poles in the lower half-plane might exist, but we illustrate

them only schematically. The physical region (gray) is approached from the lower half-plane,

from which two paths of analytic continuation (red and blue) emerge. The S-matrix cannot

be analytically continued past the natural boundary.

upper half-plane on the ﬁrst sheet or lower half-plane on the second. We would like to

emphasize that the second sheet is arguably the one carrying more physically-interesting

analytic features. For instance, it contains the resonance poles of unstable particles,

resulting in the Breit–Wigner peaks observable in particle colliders. For this reason,

we believe that properties of

(

s, z

) deserve to be understood at a deeper level. For

example, one of the questions we raise is whether the existence of the natural boundary

can result in quantiﬁable eﬀects on the physical region. Another motivation is to learn

to what extent the properties of

(

s, z

) can be used as an input in the non-perturbative

S-matrix bootstrap [24].

A connection between the two sheets is provided by unitarity, which embodies

the physical principle of probability conservation, or—more speciﬁcally—its analytic

continuation [25–35]. To be concrete, in four dimensions the Ti’s are related by

T1(s, z)−T2(s, z) = √4m2−s

√sZ

P >0

dz1dz2

pP(z;z1, z2)T1(s, z1)T2(s, z2),(1.1)

where

= 1

−z2−z2

1−z2

+ 2

zz1z2

. The left-hand side is the analytic continuation

of the imaginary part of the amplitude and the right-hand side corresponds to the

continuation of unitarity cuts. As such,

(1.1)

provides a consistency condition for

, or at least some of its properties, are known. One can write a formal solution for

– 4 –

θt

θu

θt

Figure 1.2

. The class of ladder-type Landau singularities building up the natural boundary.

Every edge is an on-shell state in the theory. The resulting singularity contributes at the

speciﬁc angle given by

(1.2)

, and more generally by

(3.11)

, with

θu

θt

. Each of the

rungs exchanges ki>1 particles.

in terms of holomorphic unitarity cuts [36]. We review these aspects in Sec. 2.2-2.3.

Using unitarity, we will show that

(

s, z

) has to have an inﬁnite number of

singularities at the non-perturbative level, corresponding to intermediate states going

on-shell and being interpretable as classical scattering trajectories in complexiﬁed space-

time. In perturbation theory, these are known as anomalous thresholds or Landau

singularities [37–39].

To demonstrate the existence of a natural boundary, we consider speciﬁc singularities

of the ladder-type, see Fig. 1.2. They correspond to scattering of two particles with

mediation points, at which

particles are being exchanged. We show that such

a singularity develops whenever the scattering angle reaches the value

θ∗

k1,k2,...,kn

which in the simplest case reads

θ∗

k1,k2,...,kn=

i=1

θt

ki(s) with θt

ki(s) = arccos 1 + 2k2

im2

s−4m2.(1.2)

Each

θt

(

) is the speciﬁc scattering angle across the interaction point with

states.

In order to ﬁnd the singular angle

, one simply adds them up modulo 2

. A more

general case is derived in Sec. 3.2. Note that positions of these singularities in the

Mandelstam invariants (

s, t

) would be extremely diﬃcult to describe: it is working with

the scattering angles that allows us to write such a compact expression for any

. In

contrast with the perturbation-theory Landau analysis, which only gives a necessary

condition for singularities, unitarity is a much stronger constraint which also guarantees

suﬃciency, up to possible cancellations.

Natural boundaries arise from a dense accumulation of such singularities as

n→ ∞

even if we consider a small subset with all

’s equal to

, i.e., equal-rung ladders. In

Sec. 3.3, we analyze positions of these singularities in the

-plane for ﬁxed-

and ﬁxed-

– 5 –

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Princeton,January2023NaturalBoundariesforScatteringAmplitudesSebastianMizeraInstituteforAdvancedStudy,EinsteinDrive,Princeton,NJ08540,USAE-mail:smizera@ias.eduAbstract:Singularities,suchaspolesandbranchpoints,playacrucialroleininvesti-gatingtheanalyticpropertiesofscatteringamplitudesthatinformnewcom...

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