CERN-TH-2020-095

An Analytical Toolkit for the S-matrix Bootstrap

Miguel Correia,a,b Amit Sever,a,c Alexander Zhiboedova

aCERN, Theoretical Physics Department, CH-1211 Geneva 23, Switzerland
bInstitute of Physics, Ecole Polytechnique Federal de Lausanne (EPFL) CH-1015
Lausanne, Switzerland
cSchool of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978,
Israel

Abstract

We revisit analytical methods for constraining the nonperturbative $S$-matrix
of unitary, relativistic, gapped theories in $d\geq 3$ spacetime dimensions.
We assume extended analyticity of the two-to-two scattering amplitude and use
it together with elastic unitarity to develop two natural expansions of the
amplitude. One is the threshold (non-relativistic) expansion and the other is
the large spin expansion. The two are related by the Froissart-Gribov
inversion formula. When combined with crossing and a local bound on the
discontinuity of the amplitude, this allows us to constrain scattering at
finite energy and spin in terms of the low-energy parameters measured in the
experiment. Finally, we discuss the modern numerical approach to the
$S$-matrix bootstrap and how it can be improved based on the results of our
analysis.

###### Contents

  1. 1 Introduction
    1. 1.1 Assumptions
    2. 1.2 Plan of the Paper
  2. 2 Amplitude Basics
    1. 2.1 Analyticity
    2. 2.2 Unitarity and Elastic Unitarity
    3. 2.3 Partial Wave Expansion
    4. 2.4 Froissart-Gribov Formula
    5. 2.5 Further Continuity Assumption
  3. 3 Analytic Continuation of Elastic Unitarity
    1. 3.1 Mandelstam Kernel
    2. 3.2 The Double Spectral Density and Crossing
    3. 3.3 Analytic Continuation in Spin
    4. 3.4 Analytic Continuation in $s$
    5. 3.5 Elastic Landau Curves
  4. 4 Positivity of $\rho$ and Multi-Particle Production
    1. 4.1 Positivity of The Double Spectral Density
    2. 4.2 The Aks Theorem and Necessity of Particle Production
    3. 4.3 Bounding Inelasticity
  5. 5 Threshold Expansion
    1. 5.1 Threshold Expansion of $f_{J}(s)$
    2. 5.2 Threshold Expansion of Discontinuity
    3. 5.3 Threshold Expansion for Double Spectral Density
    4. 5.4 Radius of Convergence
  6. 6 Large $J$ Expansion
    1. 6.1 Large Spin Expansion of $f_{J}(s)$
    2. 6.2 Large Spin Expansion of ${\rm Im}f_{J}(s)$
    3. 6.3 Threshold Expansion in the $J$-space
  7. 7 Finite $J$ and Finite $s$
    1. 7.1 A Toy Model Example
    2. 7.2 Elastic Unitarity and Coupling Maximization
  8. 8 Analytical Methods and Numerical Bootstrap
    1. 8.1 Review of the Numerical Framework of [15]
    2. 8.2 Why and What Should be Improved
    3. 8.3 How the Numerical Framework Can Be Improved
    4. 8.4 Mahoux-Martin Positivity
  9. 9 Comments on CFTs
  10. 10 Conclusions
  11. A Derivation of the Mandelstam Kernel
  12. B Useful Identities for Gegenbauer $P$\- and $Q$-functions
  13. C The $Q_{J}^{(d)}(z)$ Large $J$ Expansion
  14. D Gribov’s Theorem
  15. E Karplus Curves from the Mandelstam Equation
  16. F Threshold Expansion for Non-Integer $J$
  17. G Threshold expansion in $J$-space: Technical Details
    1. G.1 Higher Order Corrections to $\rho(s,t)$
    2. G.2 More General Integral
  18. H Keyhole Integrals in Odd $d$
    1. H.1 Partial Wave
    2. H.2 Double Spectral Density

##  1 Introduction

The idea of bootstrapping the $S$-matrix of a unitary, relativistic, gapped
theory in $d\geq 3$ was actively pursued in the 60’s. While many interesting
results have been derived [1, 4, 2, 3], no nonperturbative physical
$S$-matrices have been computed. The main reason being that no solid,
nonperturbative calculation scheme was ever put forward without relying on
some unreliable approximations. In addition to that, analytic properties of
the multi-point amplitudes were never fully understood. Moreover, even at the
level of the two-to-two scattering amplitude, the region of analyticity that
is usually assumed in the bootstrap analysis has not been rigorously
established.

While the problem of analytic properties of multi-point scattering amplitudes
is still widely open, the question of finding a good calculation scheme has
recently acquired an interesting twist with the development of the conformal
bootstrap [5, 6]. In this context, by exploring bounds on the OPE data in the
space of solutions to the CFT bootstrap equations, it was found that sometimes
the physical theories of interest saturate the bootstrap bounds [7, 8, 9] and
are in this sense solvable.111It is still an open question if the islands
observed in the exclusion plots shrink to zero size or not. Remarkably, a
similar phenomenon was observed for 2d $S$-matrices [10, 11, 12], where some
previously known two-dimensional scattering amplitudes of physical theories
were found to saturate the bootstrap bounds. We can loosely call such theories
bootstrap-solvable. A fundamental, open question in the $S$-matrix theory
program therefore is: Are there bootstrap-solvable $S$-matrices in $d\geq 3$?

A distinguishing feature characteristic to scattering in $d\geq 3$ dimensions
is a remarkable connection between scattering and particle production. Here,
by scattering we mean a non-trivial $2\to 2$ amplitude and by particle
production, we mean a non-zero $2\to n$ amplitude with $n>2$. While scattering
without particle production is a commonplace in $d=2$ [13], in $d\geq 3$ it is
widely believed that scattering implies production [14].222This result is
sometimes called the Aks theorem. However, since it relies on some unproven
assumptions that we discuss in detail below, its status is still not
completely solid. The underlying reason is an elegant interplay between
analyticity, elastic unitarity and crossing symmetry.

The simplest non-trivial S-matrix element is a $2\to 2$ connected scattering
amplitude $T(s,t)$ as a function of Mandelstam invariants $s$ and $t$. In a
gapped theory, this amplitude is subject to an exact non-linear equation
called elastic unitarity. This equation originates from the fact that when we
scatter the lightest particles in the theory at energies below the first
multi-particle threshold ($s_{0}$), only two particles can be produced in the
final state.333As such elastic unitarity is absent in theories with massless
particles. Similarly, there is no elastic unitarity in CFTs. As a result,
unitarity of the $S$-matrix becomes a non-linear equation satisfied by
$T(s,t)$ for $4m^{2}<s<s_{0}$. It is the purpose of the present paper to
revisit the implications of elastic unitarity, when combined with crossing and
analyticity, on the structure of the nonperturbative amplitude $T(s,t)$.

One motivation for our analysis is recent numerical investigations of higher-
dimensional scattering in [15, 16]. In these works elastic unitarity was not
imposed and it was observed that various bootstrap bounds tend to be saturated
by purely elastic functions. It is therefore an interesting, open question how
to efficiently implement elastic unitarity and particle production in the
current $S$-matrix bootstrap program. An obvious way to tackle the problem is
to include higher-point amplitudes in the bootstrap analysis explicitly.
However, due to unknown and complicated analytic properties of higher-
dimensional amplitudes it is not clear if it is feasible in $d\geq 3$. Another
possible way to make progress, which we will follow in the present paper, is
to focus on how to implement structures of the amplitude that are dictated by
elastic unitarity into the current numerical approach of [15, 16]. In this way
we hope to be able to zoom in closer on the physical higher-dimensional
$S$-matrices and, if we are lucky, maybe eventually solve them. Here, we
discuss various possibilities of doing that and will report the numerical
results in [17]. We start by laying out our assumptions that serve as the
basis for the further analytic study.

###  1.1 Assumptions

We assume that in the far past and in the far future states of the system are
described by a set of free particles. The Hilbert space therefore is taken to
be the Fock space of free particles.444To the best of our knowledge this
assumption, known as asymptotic completeness, does not follow from the non-
zero gap and Wightman axioms, see e.g. [18]. For simplicity we assume that the
spectrum contains a single scalar particle of mass $m$ together with its
multi-particle states. Under these conditions it has been rigorously proven
that the amplitude satisfies:

  1. 1.

Crossing symmetry: for scattering of identical particles we have

$\displaystyle T(s,t)=T(t,s)=T(u,t)\ ,\qquad\leavevmode\nobreak\ s+t+u=4m^{2}\
.$ (1.1)

  2. 2.

Real analyticity: for scattering of identical particles we have

$\displaystyle T(s^{*},t^{*})=T^{*}(s,t)\ .$ (1.2)

Our extra assumptions for the connected two-to-two scattering amplitude
$T(s,t)$, which have not been rigorously established, are the following:

  1. $3.$

Extended analyticity: $T(s,t)$ is an analytic function for complex $s$ and $t$
in some region ${\cal D}$, except for potential poles in $0<s<4m^{2}$ and a
cut starting at $s=4m^{2}$, as well as images of these singularities under
crossing. We will often assume that the region of analyticity ${\cal D}$
extends to the full complex $s$-plane ($t$-plane) for some finite region in
$t$-plane ($s$-plane), but many of our arguments can be adopted to the
situation when ${\cal D}$ is bounded in both variables simultaneously.555If
${\cal D}=\mathbb{C}^{2}$ one says that the function is maximally analytic.
Maximal analyticity is not necessary for the present paper. As usual in this
paper $T(s,t)$ stands for the analytic continuation of the amplitude from the
physical regime to the principle sheet – without going through the multi-
particle cuts.

  2. $4.$

Polynomial Boundedness: for fixed $t$

$\displaystyle|T(s,t)|<|s|^{J_{0}(t)}\ ,\qquad|s|\to\infty\
,\qquad(s,t)\in{\cal D}\ .$ (1.3)

The formula above assumes that ${\cal D}$ includes $s=\infty$ for fixed $t$.
More generally, we will assume that the amplitude is polynomially bounded on
the principal sheet (away from the bound states poles and multi-particle
thresholds).

We also have two extra technical assumptions:

  1. $5.$

Continuity: partial waves $f_{J}(s)$ are real analytic functions in the
elastic region $4m^{2}<s<s_{0}$.666This is closely related to what is called
absence of pathologies a-la A. Martin [19]. We discuss it in more detail in
section 2.5.

  2. $6.$

${\mathbb{Z}}_{2}$ symmetry and no bound states: for simplicity we assume no
bound states in the spectrum. We also assume that we have a single stable
particle of mass $m$ which is odd under ${\mathbb{Z}}_{2}$ symmetry. Therefore
only an even number of particles can be produced in the scattering of two
particles.

###  1.2 Plan of the Paper

In section 2 we review basics of the two-to-two scattering amplitude. We
consider scattering of identical scalar particles and briefly review known
analyticity results. We discuss unitarity and elastic unitarity in terms of
$T(s,t)$. We introduce partial wave expansion and the Froissart-Gribov
formula. We briefly discuss continuity properties of $T(s,t)$.

In section 3 we consider analytic continuation of elastic unitarity. We first
review the derivation of the Mandelstam equation for the double spectral
density $\rho(s,t)$, or equivalently double discontinuity of $T(s,t)$, based
on analytic continuation of elastic unitarity in one of the Mandelstam
variables ($s$ or $t$). We then discuss its relation to analytic continuation
of partial waves in spin $J$ via the Froissart-Gribov formula. We also
consider analytic continuation of elastic unitarity in energy $s$. Finally, we
exhibit the structure of the Landau-Karplus curves along which the double
spectral density develops a nontrivial support in the elastic region.

We then analyze various implications of elastic unitarity:

  * •

In section 4 we discuss positivity properties of double spectral density
$\rho(s,t)$. We review the argument that scattering implies production in
$d\geq 3$. This result follows from the combination of crossing symmetry and
positivity of $\rho(s,t)$.

  * •

In section 5 we introduce the notion of the threshold expansion for partial
waves, as well as for the first and second discontinuities of the scattering
amplitude. The threshold expansion of partial waves and the discontinuity of
the scattering amplitude is a consequence of elastic unitarity and it
describes low-energy or non-relativistic scattering. The Mandelstam equation
then maps it to the expansion of double spectral density close to the boundary
of its nontrivial support, the so-called Kaprlus-Landau curve.

  * •

In section 6 we map the threshold expansion in the $t$-channel to the large
$J$ expansion of partial wave coefficients in the $s$-channel. It comes from
“inversion” of the threshold expansion via the Froissart-Gribov formula.
Turning to the ${1\over J}$ corrections we find that remarkably the
computations can be sometimes done exactly in ${1\over J}$.

  * •

In section 7 we turn the large $J$ results into finite $J$ predictions plus an
error estimate. This error estimate is deducted from a local bound on the
discontinuity of the amplitude in the region $s,t>4m^{2}$. No such rigorous
bound is known. We discuss natural error estimates and perform the finite
spin, finite energy computations in a simple toy model.

In section 8 we consider the modern numerical approach to the $S$-matrix
bootstrap following [15]. We discuss why and what should be improved in the
existing approach. We suggest several ways in which this approach can be
improved. In particular, we discuss various ways to implement elastic
unitarity numerically. In section 9 we briefly comment on the relation between
the present analysis and similar ideas in the conformal bootstrap. Finally, in
section 10 we conclude and present some future directions. Several appendices
contain technical details that should be helpful in understanding the details
of our arguments and calculations.

##  2 Amplitude Basics

In this section we briefly review the basic kinematics of the two-to-two
scattering of identical, scalar particles to set the conventions for the
further analysis. As usual we write the $S$-matrix as

$\hat{S}\equiv\hat{1}\hskip-4.55254pt1+i\,\hat{T}\ ,$ (2.1)

where $\hat{T}$ is zero in the theory of a free massive scalar. We are
interested in the matrix elements that describe two-to-two scattering

$\displaystyle S_{2,2}(p_{3},p_{4}|p_{1},p_{2})\equiv\langle
p_{3},p_{4}|\hat{S}|p_{2},p_{1}\rangle\ ,$ (2.2)

where the initial and final states are characterized by the on-shell momenta

$p^{2}={\vec{p}\,}^{2}-(p^{0})^{2}=-m^{2}\
,\qquad{\vec{p}\,}^{2}=\sum_{i=1}^{d-1}(p^{i})^{2}\ ,\qquad p^{0}>0\ .$ (2.3)

As in (2.1) we can separate the contribution of the disconnected and connected
parts of the $S$-matrix

$\displaystyle S_{2,2}(p_{3},p_{4}|p_{1},p_{2})$
$\displaystyle=S_{1,1}(p_{3}|p_{1})S_{1,1}(p_{4}|p_{2})+S_{1,1}(p_{4}|p_{1})S_{1,1}(p_{4}|p_{2})+S_{2,2}^{c}(p_{3},p_{4}|p_{1},p_{2}).$
(2.4)

The disconnected part is given by an overlap of the one particle states which
is uniquely fixed by Lorentz symmetry

$\displaystyle S_{1,1}(p|q)$
$\displaystyle=1\hskip-4.55254pt1_{1,1}(p|q)=2(2\pi)^{d-1}\sqrt{\vec{p}^{2}+m^{2}}\times\delta^{d-1}(\vec{p}-\vec{q})\
,$ (2.5)

The connected part is the main object of our interest

$\displaystyle S_{2,2}^{c}(p_{3},p_{4}|p_{1},p_{2})$ $\displaystyle\equiv
i\langle
p_{3},p_{4}|\hat{T}|p_{2},p_{1}\rangle=i(2\pi)^{d}\delta^{d}\left(p_{1}+p_{2}-p_{3}-p_{4}\right)T(s,t)\
.$ (2.6)

In the formula above we introduced Mandelstam invariants

$\displaystyle s$ $\displaystyle=-(p_{1}+p_{2})^{2}=4(m^{2}+\vec{p}^{2})\ ,$
$\displaystyle t$
$\displaystyle=-(p_{1}-p_{3})^{2}=-2\vec{p}^{2}(1-\cos\theta)\ ,$
$\displaystyle u$
$\displaystyle=-(p_{1}-p_{4})^{2}=-2\vec{p}^{2}(1+\cos\theta)\ ,$ (2.7)

where in the last equality we wrote their form in the center-of-mass frame,
$p_{1}=(m,\vec{p})$, $p_{2}=(m,-\vec{p})$. Here,

$\displaystyle\cos\theta=1+{2t\over s-4m^{2}}\ ,$ (2.8)

is cosine of the scattering angle. Only two of the Mandelstam invariant are
independent while the third is related to the other two through the relation

$\displaystyle s+t+u=4m^{2}\ .$ (2.9)

From above, the dimension of the amplitude is

$\displaystyle[T(s,t)]=m^{4-d}\ .$ (2.10)

###  2.1 Analyticity

In the discussion above scattering matrix elements were defined for physical
momenta that correspond to actual scattering. The starting point of the
$S$-matrix considerations is the statement that the physical matrix element
$T(s,t)$ is a boundary value of an analytic function of the relativistic
invariants regarded as complex variables

$\displaystyle T(s,t)=\lim_{\epsilon\to 0}T(s+i\epsilon,t)\ ,$ (2.11)

where we assumed $s$ and $t$ to be real and $-(s-4m^{2})<t<0$, $4m^{2}<s$.
This corresponds to scattering in the $s$-channel $1,2\to 3,4$. Using the
basic principles of QFT correlation functions, reduction formulas that relate
them to the $S$-matrix elements, and techniques of analytic completion one can
establish various analytic properties of the scattering amplitudes as
functions of complex $s$ and $t$, see e.g. [20, 21, 4] for a pedagogical
exposition.

The first result of this type is subtracted dispersion relations in $s$ for
fixed $-t_{0}<t\leq 0$ in the physical $s$-channel region [21, 22]. In this
case one has to understand analytic properties of $T(s,t)$ as a function of
complex $s$. For the case of $\pi^{0}\pi^{0}\to\pi^{0}\pi^{0}$ scattering, for
which our treatment applies directly, one can show that $T(s,t)$ is an
analytic function of $s$ with two cuts: $s>4m^{2}$ and $u>4m^{2}$, with
$t_{0}=28m^{2}$. See Table 1 of [21] for the processes for which a fixed-$t$
dispersion relation has been proven and the corresponding values of
$t_{0}$.777The cases for which a dispersion relation has not been proven,
baryon-baryon scattering for instance [21], still enjoy a domain of
analyticity that connects the $s$\- and $u$-channel cuts. Thus, the property
of crossing (see below) can still be established for these cases [23].

Another well-known property, originally due to Lehmann [24], concerns analytic
properties of $T(s,t)$ as a function of $t$ for fixed physical $s>4m^{2}$.
Lehmann showed that $T(s,\cos\theta)$ is analytic inside an ellipse, the so-
called Lehmann ellipse, in the $\cos\theta$ complex plane with foci at
$\cos\theta=\pm 1$ and semi-major axis $\cos\theta_{sL}>1$ which depends on
the details of the theory, energy and masses of particles, as well as the
scattering process. Lehmann also showed that the absorptive part or
discontinuity of the amplitude ${\rm Disc}_{s}T(s,\cos\theta)$ is analytic in
a larger ellipse, the so-called large Lehmann ellipse, with a semi-major axis
$\cos\theta_{LL}=2\cos^{2}\theta_{sL}-1$.

The third class of results concerns analyticity of $T(s,t)$ when both $s$ and
$t$ are complex. Bros, Epstein and Glaser [25] showed that any point
$(s,\cos\theta)$ in the physical region is surrounded by an analyticity
neighborhood whose precise form is not known in general, see e.g. [4] for
details. An explicit domain of simultaneous analyticity in both variables was
derived by Lehmann [26] for elastic processes that obey a fixed-$t$ dispersion
relation by continuing the Lehmann ellipse to complex $s$. For cases with
single variable dispersion relations in all three channels (as in
$\pi\pi\to\pi\pi$ scattering), Mandelstam [27] derived domains of the form
$|s\,t|<b$ for any complex $s$ and $t$ outside the single variable dispersion
relation cuts. For pion scattering the largest domain occurs for
$b=256m_{\pi}^{4}$. These domains have the drawback that when $s\to\infty$ the
Lehmann domain shrinks to the line $-t_{0}<t<0$ and the Mandelstam domain
shrinks to the point $t=0$.

Based on the results above, the analyticity domain was further enlarged using
unitarity by A. Martin [28]. The final result is that the amplitude is
analytic within $|t|<R$ and $s$ cut-plane. For scattering of identical
particles that we consider in the present paper $R=4m^{2}$. From this result
the standard bounds on high energy behavior of the amplitudes follow. It also
follows that for $|t|<4m^{2}$ the scattering amplitude admits fixed $t$
dispersion relations with at most two subtractions.

Finally, let us introduce the notion of maximal analyticity (or Mandelstam
analyticity) which states that the scattering amplitude is analytic in the
$(s,t)$ complex planes with only singularities on the principal sheet being
unitarity cuts $s,t,u>4m^{2}$ and bound state poles for $0<s,t,u<4m^{2}$. This
property is consistent, at least for the scattering of lightest particles in
the theory, with expectations from unitarity and analysis of perturbation
theory. However it is important to keep in mind that it has not been proven.
The original attempts to prove maximal analyticity in perturbation theory [29,
30] were later found to have a loophole [31] which, to the best of our
knowledge, has never been closed even for the scattering of lightest
particles. In this paper we freely assume analyticity beyond what has been
rigorously proven but we do not assume maximal analyticity.

There are two other properties that we will use. One is crossing

$\displaystyle T(s,t)=T(t,s)=T(u,t)\ ,$ (2.12)

which states that in particular that scattering in different channels is
described by different boundary values of a single analytic function. For the
two-to-two scattering it has been proven in [23]. Beyond the two-to-two
scattering only a partial progress has been achieved [32]. Another property is
real analyticity

$\displaystyle T(s^{*},t^{*})=T^{*}(s,t)\ .$ (2.13)

This was established within axiomatic quantum field theory in [33]. For the
$S$-matrix argument see e.g. [1].

Finally, let us also mention in a related context the result by A. Martin [34]
who showed that maximal analyticity in the form of the Mandelstam
representation together with knowledge of the double spectral density in the
elastic region fix the scattering amplitude completely. Similarly, knowledge
of the amplitude at fixed energy in the elastic region as a function of the
scattering angle is believed to fix it almost completely, see e.g. [35].

###  2.2 Unitarity and Elastic Unitarity

In terms of the $T$-matrix, the unitarity of the $S$-matrix
$\hat{S}\cdot\hat{S}^{\dagger}=\hat{1}\hskip-4.55254pt1$ reads

$\displaystyle{1\over i}\langle
p_{3},p_{4}|\hat{T}-\hat{T}^{\dagger}|p_{2},p_{1}\rangle$
$\displaystyle=\langle
p_{3},p_{4}|\hat{T}\cdot\hat{T}^{\dagger}|p_{2},p_{1}\rangle$ (2.14)
$\displaystyle=\sum_{n=1}^{\infty}\int d\mu(q_{1},\dots,q_{2n})\langle
p_{3},p_{4}|\hat{T}|\\{q_{i}\\}_{i=1}^{2n}\rangle\,\langle\\{q_{i}\\}_{i=1}^{2n}|\hat{T}^{\dagger}|p_{2},p_{1}\rangle\
,$

where in the second line we have inserted a complete basis of asymptotic
states and the Lorentz invariant measure is

$d\mu(p_{1},\dots,p_{n})\equiv{1\over n!}\prod_{i=1}^{n}d\mu(p_{i})\ ,\qquad
d\mu(p)\equiv{1\over(2\pi)^{d-1}}\theta(p^{0})\,\delta(p^{2}+m^{2})\,d^{d}p\
.$ (2.15)

Due to the momentum conservation, a $2n$-particle intermediate state can only
contribute if $s>(2nm)^{2}$. Otherwise, there is no enough energy to create an
on-shell state with $2n$ particles. In particular, for $4m^{2}<s<16m^{2}$,
only two particle states are possible. Hence, in that regime we can replace
the sum in (2.14) by the first term and the equation closes on the $2\to 2$
transitions only. This is the elastic unitarity regime. Explicitly, we have

$\displaystyle 2T_{s}(s,t)={1\over 2}\int{d^{d-1}\vec{q^{\prime}}\
\over(2\pi)^{d-1}(2E_{\vec{q^{\prime}}})}\int{d^{d-1}\vec{q^{\prime\prime}}\over(2\pi)^{d-1}(2E_{\vec{q^{\prime\prime}}})}(2\pi)^{d}\delta^{d}(p_{1}+p_{2}-q^{\prime}-q^{\prime\prime})T^{(+)}(s,t^{\prime})T^{(-)}(s,t^{\prime\prime})\
,$ (2.16)

where we have introduced the notations

$\displaystyle T^{(\pm)}\equiv\lim_{\epsilon\to 0}T(s\pm i\epsilon,t)\ ,\qquad
T_{s}(s,t)={\rm Disc}_{s}T(s,t)$ $\displaystyle\equiv{1\over
2i}\left(T^{(+)}(s,t)-T^{(-)}(s,t)\right)\ ,$ (2.17)

and $t^{\prime}=-(\vec{p}_{1}-\vec{q^{\prime}})^{2}$,
$t^{\prime\prime}=-(\vec{q^{\prime\prime}}-\vec{p}_{4})^{2}$. In writing the
above we used real analyticity of the amplitude (2.13). Finally, the overall
factor of one half is a symmetry factor for identical bosons.

We will now reduce the integral to an integration over the two scattering
angles by performing all the kinematical integrations explicitly. For that
aim, we first go to the center of mass frame where
$\vec{q^{\prime}}=-\vec{q^{\prime\prime}}\equiv p\,\vec{n}$, where $\vec{n}$
is a unit $d-1$ vector and $p=\sqrt{s-4m^{2}}/2$. In these variables the
elastic unitarity constraint (2.16) becomes

$\displaystyle 2T_{s}(s,t)={p^{d-2}\over(2\pi)^{d-2}(2E_{p})^{2}}{E_{p}\over
4p}\int
d^{d-2}\Omega_{\vec{n}}\,T^{(+)}(s,t^{\prime})\,T^{(-)}(s,t^{\prime\prime})\
,$ (2.18)

where ${E_{p}\over 2p}={\sqrt{s}\over 2\sqrt{s-4m^{2}}}$ is the Jacobian
coming from the energy conservation delta-function. The integrand only depends
on the two scatttering angles

$\displaystyle
z^{\prime}=\cos\theta^{\prime}={\vec{p}_{1}\cdot\vec{n}\over|\vec{p}_{1}|}\qquad\text{and}\qquad
z^{\prime\prime}=\cos\theta^{\prime\prime}={\vec{p}_{3}\cdot\vec{n}\over|\vec{p}_{3}|}\
,$ (2.19)

in terms of which we can write the measure as

$\displaystyle\int
d^{d-2}\Omega_{\vec{n}}\equiv\int\limits_{-1}^{1}dz^{\prime}\int\limits_{-1}^{1}dz^{\prime\prime}\,{\cal
P}_{d}(z,z^{\prime},z^{\prime\prime})\qquad\text{where}\qquad
z=\cos\theta={\vec{p}_{1}\cdot\vec{p}_{3}\over|\vec{p}_{1}||\vec{p}_{3}|}=1+\frac{2t}{s-4m^{2}}\
,$ (2.20)

is the cosine of the external scattering angle. We find that (see appendix A
for more details)

$\displaystyle\mathcal{P}_{3}(z,z^{\prime},z^{\prime\prime})$
$\displaystyle=2\sqrt{1-z^{2}}\,\delta(1-z^{2}-z^{\prime 2}-z^{\prime\prime
2}+2zz^{\prime}z^{\prime\prime})\ ,$ (2.21)
$\displaystyle\mathcal{P}_{d>3}(z,z^{\prime},z^{\prime\prime})$
$\displaystyle={2\pi^{d-3\over 2}\over\Gamma\left({d-3\over
2}\right)}(1-z^{2})^{4-d\over 2}{\Theta(1-z^{2}-z^{\prime 2}-z^{\prime\prime
2}+2zz^{\prime}z^{\prime\prime})\over(1-z^{2}-z^{\prime 2}-z^{\prime\prime
2}+2zz^{\prime}z^{\prime\prime})^{5-d\over 2}}\ .$

Using that $E_{k}=\sqrt{s}/2$ , we can write (2.18) covariantly as

$\displaystyle\boxed{T_{s}(s,t)={(s-4m^{2})^{d-3\over 2}\over
8(4\pi)^{d-2}\sqrt{s}}\int\limits_{-1}^{1}dz^{\prime}\int\limits_{-1}^{1}dz^{\prime\prime}\,{\cal
P}_{d}(z,z^{\prime},z^{\prime\prime})\,T^{(+)}(s,t(z^{\prime}))\,T^{(-)}(s,t(z^{\prime\prime}))\
,\quad 4m^{2}\leq s\leq 16m^{2}}\ ,$ (2.22)

where $t(x)\equiv-(s-4m^{2})(1-x)/2$, not to be confused with the external
momentum transfer $t$, which is held fixed. The step/delta function in the
phase space integration kernel ${\cal P}_{d}(z,z^{\prime},z^{\prime\prime})$
has a simple geometrical origin, see figure 1.

Figure 1: In the elastic strip $4m^{2}<s<16m^{2}$ the discontinuity of the
amplitude comes from two intermediate particle exchange only. This result in
the exact elastic unitarity equation (2.22). The corresponding phase space
integration kernel ${\cal
P}_{d}(\cos\theta,\cos\theta^{\prime},\cos\theta^{\prime\prime})$ in (2.21) is
proportional to a step/delta function which has a simple geometrical origin.
In the center of mass frame we have three $(d-1)$-dimensional vectors,
$\vec{p}_{1}=-\vec{p}_{2}$, $\vec{p}_{3}=-\vec{p}_{4}$, and
$\vec{q^{\prime}}=-\vec{q^{\prime\prime}}$. The geometrical angles between
these three vectors, $\\{\theta,\theta^{\prime},\theta^{\prime\prime}\\}$ are
therefore restricted to the range $\theta_{1}+\theta_{2}\geq\theta_{3}$, where
$\theta_{1,2,3}$ are any permutation of
$\\{\theta,\theta^{\prime},\theta^{\prime\prime}\\}$.

For $s>16m^{2}$ and general $t$ the unitarity constraint involves scattering
elements with more than two particles. To get a constraint on the two particle
amplitude, we note that $\hat{T}\cdot\hat{T}^{\dagger}$ on the right hand side
of (2.14) is a positive semi-definite matrix. Hence, for any state $\Psi$ we
have that

$\langle\Psi|\hat{T}|\\{q_{i}\\}_{i=1}^{2n}\rangle\,\langle\\{q_{i}\\}_{i=1}^{2n}|\hat{T}^{\dagger}|\Psi\rangle=|\langle\Psi|\hat{T}|\\{q_{i}\\}_{i=1}^{2n}\rangle|^{2}\geq
0\ ,$ (2.23)

and hence, if we drop all the contributions with more than two particles in
(2.14) we get an inequality for the $2\to 2$ scattering matrix

$\displaystyle{1\over i}$ $\displaystyle\int
d\mu(p_{1},p_{1})\,d\mu(p_{3},p_{4})\,\psi(p_{1},p_{2})\,\psi^{*}(p_{3},p_{4})\times\langle
p_{3},p_{4}|\hat{T}-\hat{T}^{\dagger}|p_{2},p_{1}\rangle$ (2.24)
$\displaystyle\geq$ $\displaystyle\int
d\mu(p_{1},p_{1})\,d\mu(p_{3},p_{4})\,\psi(p_{1},p_{2})\,\psi^{*}(p_{3},p_{4})\times\int
d\mu(q_{1},q_{2})\,\langle p_{3},p_{4}|\hat{T}|q_{1},q_{2}\rangle\,\langle
q_{1},q_{2}|\hat{T}^{\dagger}|p_{2},p_{1}\rangle\geq 0\ .$

For example, if we pick a wave function that consists of two particles with a
specific momenta then we have the the amplitude in the forward limit where
$p_{3}=p_{1}$ and $p_{4}=p_{2}$. For this choice of wave function, the
unitarity constraint (2.24) becomes888The unitarity relation in the forward
limit is nothing but the optical theorem. In $d$ dimensions it takes the form
$\displaystyle T_{s}(s,0)={\rm
Im}[T(s,0)]=\sqrt{s(s-4m^{2})}\,\sigma_{tot}(s)\ ,$ (2.25) where
$\sigma_{tot}(s)$ is the total cross-section, of dimension
$[\sigma(s)]=L^{d-2}$.

$\displaystyle T_{s}(s,0)$ $\displaystyle\geq{(s-4m^{2})^{d-3\over 2}\over
8(4\pi)^{d-2}\sqrt{s}}\int\limits_{-1}^{1}dz^{\prime}\int\limits_{-1}^{1}dz^{\prime\prime}\,{\cal
P}_{d}(1,z^{\prime},z^{\prime\prime})\,T^{(+)}(s,t(z^{\prime}))\,T^{(-)}(s,t(z^{\prime\prime}))\
$ $\displaystyle\propto\int\limits_{-1}^{1}dz^{\prime}(1-z^{\prime
2})^{d-4\over 2}|T^{(+)}(s,t(z^{\prime}))|^{2}\,,\quad s\geq 16m^{2}\ ,$
(2.26)

where we used that
$\mathcal{P}_{d}(1,z^{\prime},z^{\prime\prime})\propto\delta(z^{\prime}-z^{\prime\prime})$,
see (B.7) for the precise formula. We also used that
$T^{(-)}(s,t(z^{\prime}))=\left(T^{(+)}(s,t(z^{\prime}))\right)^{*}$ for
$-1\leq z^{\prime}\leq 1$.

###  2.3 Partial Wave Expansion

Unitarity of the $S$-matrix implies the non-linear integral relations that the
$2\to 2$ $T$-matrix has to satisfy, (2.22) and (2.24). To simplify these
complicated constraints we choose a wave function $\Psi$ that diagonalizes the
$T$-matrix and therefore also the integral kernel in (2.22), (2.24). This can
be done using the Lorentz symmetry of the problem. Namely, we decompose the
amplitude $T(s,t)$ in a complete basis of intermediate states which transform
in irreducible representations of the $SO(1,d-1)$ symmetry. These
representations are characterised by their energy and the little group
$SO(d-1)$ angular momentum in the center of mass frame, $E$ and $J$. For two
particle states the $SO(1,d-1)$ quantum numbers are enough to characterize the
states and we have

$\langle
p_{1},p_{2}|p,J,\vec{m}\rangle\propto\delta^{d}(p-p_{1}-p_{2})Y^{(d)}_{J,\vec{m}}(\hat{p}_{1})\
,$ (2.27)

where $p^{2}=E^{2}$, $Y^{(d)}_{J,\vec{m}}$ are the $d$-dimensional spherical
harmonics, and the energies dependant pre-factor will not be relevant for
us.999For $d=4$ the factor is
${\sqrt{E_{\vec{p}}\over|\vec{p}_{1}|E_{\vec{p}_{1}}E_{\vec{p}_{2}}}}$, see
[36] for details. We can now insert a complete basis to these states to
decompose the S-matrix element $\langle
p_{3},p_{4}|\hat{T}|p_{1},p_{2}\rangle$ in all possible spins. Since the
operator $\hat{T}$ is both, translation and $SO(1,d-1)$ invariant, due to the
Wigner-Eckart theorem we have that

$f_{J}(p^{2})\propto{\langle
p,J,\vec{m}|\hat{T}|p,J,\vec{m}\rangle\over\langle
p,J,\vec{m}|p,J,\vec{m}\rangle}\ ,$ (2.28)

where the convention-dependent proportionality factor is independent of the
energy and and the angular momentum $\vec{m}$. These functions are the so-
called partial wave coefficients, in terms of which the amplitude takes the
form

$T(s,t)=\sum_{J=0}^{\infty}n_{J}^{(d)}f_{J}(s)P_{J}^{(d)}\left(\cos\theta\right)\
,$ (2.29)

where the sum runs over all (even) spins and $n_{J}^{(d)}$ are convention-
dependent normalization factors. Here, $P_{J}^{(d)}(\cos\theta)$ are the
partial waves. They represents the angular dependence of the amplitude due to
the exchange of all the states with spin $J$. A simple way of determining
these functions is to go to the center of mass frame and act with the
$SO(d-1)$ quadratic Casimir on the two outgoing particle, while holding the
momentum of the two incoming particles fixed. This equation takes the form

$\left[(1-z^{2})^{\frac{4-d}{2}}\frac{d}{dz}(1-z^{2})^{\frac{d-2}{2}}\frac{d}{dz}+J(J+d-3)\right]P_{J}^{(d)}(z)=0\
,$ (2.30)

where $z=\cos\theta$ is cosine of the scattering angle (2.20). This second
order differential equation has two independent solutions. Spin $J$ unitary
representations are composed of states with angular momentum in the plane of
scattering ranging between $-J$ and $J$. Hence, the corresponding solution of
(2.30) is a degree $J$ polynomial of $\cos\theta$ that is given by

$P^{(d)}_{J}(z)={}_{2}F_{1}\left(-J,J+d-3,\frac{d-2}{2},\frac{1-z}{2}\right)\
.$ (2.31)

The partial wave coefficients can be extracted from the amplitude using the
orthogonality relation of these polynomials

$\displaystyle{1\over 2}\int\limits_{-1}^{1}dz\,(1-z^{2})^{{d-4\over
2}}P^{(d)}_{J}(z)P^{(d)}_{\tilde{J}}(z)={\delta_{J\tilde{J}}\over{\cal
N}_{d}\,n_{J}^{(d)}}\ .$ (2.32)

Here we have chosen the convention

${\cal N}_{d}={(16\pi)^{2-d\over 2}\over\Gamma\left({d-2\over 2}\right)}\
,\qquad n_{J}^{(d)}=\frac{(4\pi)^{d\over
2}(d+2J-3)\Gamma(d+J-3)}{\pi\,\Gamma\left({d-2\over 2}\right)\Gamma(J+1)}\ ,$
(2.33)

for which the unitarity constraint presented below takes a simple form. In
this convention we have

$f_{J}(s)={{\cal N}_{d}\over 2}\int\limits_{-1}^{1}dz\,(1-z^{2})^{d-4\over
2}P_{J}^{(d)}(z)\,T\left(s,t(z)\right)\ .$ (2.34)

Because the S-matrix is diagonal in the spin basis, so does the unitary
constraint. We consider first the elastic regime $4m^{2}<s<16m^{2}$ where this
constraints takes the form (2.22). Using (2.34), we project both sides to a
fixed spin $J$. On the left hand side we find the discontinuity of the partial
wave coefficient. Real analyticity (2.13) of $T(s,t)$ leads to real
analyticity of $f_{J}$

$\displaystyle f_{J}(s^{*})=f_{J}^{*}(s)\ .$ (2.35)

Hence, the discontinuity of the partial wave is equal to the imgionary part
${1\over 2i}\left(f_{J}(s+i\epsilon)-f_{J}(s-i\epsilon)\right)={\rm
Im}f_{J}(s)$. On the right hand side, it is useful to first represent the
kernel as a sum over partial waves of $z_{1}$, $z_{2}$ and $z$. Because this
kernel represents the angular integration in (2.18), its partial wave
decomposition must also be diagonal in spin. It takes the form (see appendix
C)

$\displaystyle{\cal P}_{d}(z,z^{\prime},z^{\prime\prime})=(4\pi)^{d-2}{\cal
N}_{d}^{2}\,(1-z^{\prime 2})^{d-4\over 2}(1-z^{\prime\prime 2})^{d-4\over
2}\sum_{J=0}^{\infty}n_{J}^{(d)}P_{J}^{(d)}(z)P_{J}^{(d)}(z^{\prime})P_{J}^{(d)}(z^{\prime\prime})\
.$ (2.36)

Using (2.34) for the three integrals and real analyticity (2.35), we arrive at
the elastic unitarity constraint

$\displaystyle\boxed{2{\rm Im}f_{J}(s)={(s-4m^{2})^{d-3\over
2}\over\sqrt{s}}|f_{J}(s)|^{2}}\ ,$ (2.37)

or equivalently

$|S_{J}(s)|=1\ ,\qquad\text{with}\qquad S_{J}(s)\equiv
1+i{(s-4m^{2})^{d-3\over 2}\over\sqrt{s}}f_{J}(s)\ .$ (2.38)

Here $1$ can be traced to back to $\hat{1}\hskip-4.55254pt1$ in (2.1). In this
way the trivial unitary $S$-matrix $\hat{S}=\hat{1}\hskip-4.55254pt1$ becomes
$S_{J}=1$ in the partial wave basis.

The solution to this is

$\displaystyle f_{J}(s)={\sqrt{s}\over(s-4m^{2})^{d-3\over
2}}i(1-e^{2i\delta_{J}(s)})\ ,$ (2.39)

with $\delta_{J}(s)$ being real for $4m^{2}<s<16m^{2}$ and is called the
scattering phase.

Similarly to the above, for $s>16m^{2}$ we chose $\psi(p_{1},p_{2})=\langle
p_{1},p_{2}|p,J,\vec{m}\rangle$ in (2.24). In that way we arrive at the same
equation, but with an inequality instead of an equality

$\displaystyle 2{\rm Im}f_{J}(s)\geq{(s-4m^{2})^{d-3\over
2}\over\sqrt{s}}|f_{J}(s)|^{2}\ .$ (2.40)

or equivalently, $|S_{J}(s)|\leq 1$, ${\rm Im}[\delta_{J}(s)]\geq 0$.

We close this section with a short discussion on the range of convergence of
the partial wave sum (2.29) for fixed physical $s$ as a function of
$\cos\theta$.101010 The convergence of the partial wave expansion can be seen
using Neumann’s argument [37]. Consider a function $f(z)$ analytic inside some
region ${\cal C}$ which includes the $[-1,1]$ interval. We can then write
$\displaystyle f(z)=\oint\limits_{\gamma}{dt\over 2\pi i}{f(t)\over
t-z}=\oint\limits_{\gamma}{dt\over 2\pi
i}\sum_{J=0}^{\infty}n_{J}^{(d)}P^{(d)}_{J}(z)\left(\mathcal{N}_{d}(t^{2}-1)^{d-4\over
2}Q^{(d)}_{J}(t)f(t)\right)=\sum_{J=0}^{\infty}n_{J}^{(d)}f_{J}P^{(d)}_{J}(z),$
(2.41) where $\gamma\in{\cal C}$ is some contour that wraps the interval
$[-1,1]$ counterclockwise and contains $z$ inside the integration contour. To
exchange the summation and integration, we also used that given $z$, ${1\over
t-z}=\mathcal{N}_{d}(t^{2}-1)^{d-4\over
2}\sum_{J=0}^{\infty}n_{J}^{(d)}P^{(d)}_{J}(z)Q^{(d)}_{J}(t)$ converges
uniformly in $t$ as long as $t$ is outside the ellipse with foci at $-1$ and
$1$ that passes through $z$. We also used the relation between
$Q_{J}^{(d)}(z)$ and $P_{J}^{(d)}(z)$ which will be explained below, see
(2.46). Therefore, the partial wave expansion (2.41) converges when $z$ is
inside the ellipse with foci at $-1$ and $1$ and $\in{\cal C}$. Note therefore
that the size of the domain of convergence of the partial wave expansion can
be less that the analyticity domain ${\cal C}$. It is a well-known fact that
the amplitude $T(s,\cos\theta)$ is analytic inside the small Lehmann-Martin
ellipse and its absorptive part, $T_{s}(s,\cos\theta)$ is analytic inside the
large Lehmann-Martin ellipse. These ellipses have foci at $\cos\theta=\pm 1$
and semi-major axis $z_{{\rm small}}$ and $z_{{\rm large}}$. Correspondingly,
inside these ellipses the sum (2.29) and its discontinuity converge.

In the case of scattering of identical lightest particles which is our main
interest we have

$\displaystyle z_{\rm small}=1+{8m^{2}\over s-4m^{2}}\ ,\qquad z_{\rm
large}=2z_{\rm small}^{2}-1=1+{32m^{4}\over(s-4m^{2})^{2}}\ .$ (2.42)

In section 4.1 we will see that extended analyticity, elastic unitarity and
crossing imply that the partial wave expansion converges in a larger region.

###  2.4 Froissart-Gribov Formula

The Froissart-Gribov formula is a representation of the partial wave
coefficients in terms of the discontinuity of the amplitude. It has multiple
applications and, in particular, it allows us to analytically continue partial
wave coefficients in spin. Correspondingly, in section 3.3 we will use the
Froissart-Gribov formula to analytically continue elastic unitarity in spin.
It will also allow us to better understand the analytic structure of the
amplitude in the Mandelstam invariants and relate the threshold expansion, see
section 5, to the large spin expansion, see section 6.

Let us introduce the Gegenbauer $Q$-functions. These are given by the second
linearly independent solution of the second order Casimir equation (2.30).
They are uniquely fixed by their asymptotic behavior

$\lim_{|z|\to\infty}Q^{(d)}_{J}(z)={c^{(d)}_{J}\over z^{J+d-3}}+\dots\ ,$
(2.43)

where $c^{(d)}_{J}$ is a normalization constant. The corresponding
$Q$-function is

$Q^{(d)}_{J}(z)={c_{J}^{(d)}\over
z^{J+d-3}}{}_{2}F_{1}\left(\frac{J+d-3}{2},\frac{J+d-2}{2},J+\frac{d-1}{2},{1\over
z^{2}}\right)\ .$ (2.44)

Our convention is

$c_{J}^{(d)}=\frac{\sqrt{\pi}\Gamma(J+1)\Gamma(\frac{d-2}{2})}{2^{J+1}\Gamma(J+\frac{d-1}{2})}\
.$ (2.45)

The $Q$-function has a cut running between $z=-1$ and $z=1$. The fact that
there are only two independent solutions to the Casimir equation means that
the discontinuity of $Q$ can be expressed in terms of $Q$ and $P$. The precise
relation takes the form

${\rm Disc}_{z}(z^{2}-1)^{{d-4\over 2}}Q^{(d)}_{J}(z)=-{\pi\over
2}(1-z^{2})^{{d-4\over 2}}P^{(d)}_{J}(z)\ ,\qquad z\in[-1,1]\ ,$ (2.46)

or equivalently (for integer $J$)

$\displaystyle Q^{(d)}_{J}(z)={1\over
2}\int\limits_{-1}^{1}dz^{\prime}\left({1-z^{\prime 2}\over
z^{2}-1}\right)^{d-4\over 2}{P^{(d)}_{J}(z^{\prime})\over z-z^{\prime}}\ .$
(2.47)

We can then plug (2.46) into the partial wave coefficient (2.34) as

$\displaystyle f_{J}(s)={\cal N}_{d}\oint\limits_{[-1,1]}{dz\over 2\pi
i}\left(z^{2}-1\right)^{{d-4\over 2}}Q^{(d)}_{J}(z)T(s,t(z))\ ,$ (2.48)

Figure 2: a. The partial wave projection integral (2.48) is a contour integral
(in blue) that circles around the cut of the $Q$-function, between $t=0$ and
$t=-(s-4m^{2})$, (in red). b. We partially open up the contour. Sometimes this
representation for partial waves is called the truncated Froissart-Gribov
formula. The advantage of this representation is that we only use a finite
amount of extended analyticity that has not been rigorously proven. c. We open
the contour all the way to infinity and arrive at the usual Froissart-Gribov
formula (2.49) with two integrations of the discontinuity of the amplitude
along the $t$-channel and $u$-channel cuts (in black).

where the integral is counterclockwise around the interval $z\in[-1,1]$. By
blowing up the contour, we get two integrals along the $t$\- and the
$u$-channel cuts, see figure 2

$f_{J}(s)={{\cal
N}_{d}\over\pi}\left[\int\limits_{z_{1}}^{\infty}\\!dz(z^{2}-1)^{{d-4\over
2}}Q^{(d)}_{J}(z)T_{t}(s,t(z))+\int\limits^{-z_{1}}_{-\infty}\\!dz(z^{2}-1)^{{d-4\over
2}}Q^{(d)}_{J}(z)T_{u}(s,u(z))\right]\ ,$ (2.49)

where

$z_{1}\equiv z|_{t=4m^{2}}=1+{8m^{2}\over s-4m^{2}}\ ,$ (2.50)

and we have assumed that $s>4m^{2}$, so that the $t$ channel cut runs from
$z_{1}=z_{1}>1$ to infinity. Here we have dropped the contributions of the
arcs at infinity. This is justified for large enough spin $J>J_{0}(s)$ using
(2.43), where $J_{0}(s)$ is the Regge intercept

$\displaystyle\lim_{|t|\to\infty}|T(s,t)|<|t|^{J_{0}(s)}\ .$ (2.51)

We can now use crossing to simplify (2.49). We change the integration variable
for the $u$-channel integral from $z$ to $-z$. Crossing symmetry implies that
$T_{u}\left(s,u(z)\right)=-T_{t}\left(s,t(-z)\right)$, where we have used that
$z(u)=-z$. Under this change of variables

$(z^{2}-1)^{{d-4\over 2}}\to(-1)^{d-4}(z^{2}-1)^{{d-4\over 2}}\ ,\qquad
Q^{(d)}_{J}(z)\to Q^{(d)}_{J}(-z)=(-1)^{J+3-d}Q^{(d)}_{J}(z)\ .$ (2.52)

We get that $f_{J}=0$ for odd $J$. For even $J$ we get

$\boxed{f_{J}(s)={2\,{\cal
N}_{d}\over\pi}\int\limits_{z_{1}}^{\infty}dz\,(z^{2}-1)^{{d-4\over
2}}Q^{(d)}_{J}(z)T_{t}(s,t(z))\ ,\qquad{\rm Re}J>J_{0}(s)}\ .$ (2.53)

As opposed to (2.34), the Froissart-Gribov representation of the partial waves
(2.53) is suitable for analytic continuation in $J$. It follows from the
Carlson theorem that this analytic continuation is the unique continuation
that does not grow too fast at large $J$. The Froissart-Gribov integral (2.53)
converges as long as ${\rm Re}J>J_{0}(s)$ thanks to (2.51) and (2.43).

This integral is written for $s>4m^{2}$. As $s$ approaches the threshold from
above $s-4m^{2}\to 0^{+}$, the lower end of the integral is pushed to
infinity, $z_{1}\to\infty$. To analyze $f_{J}(s)$ in this limit, it is useful
to use (2.43) and to switch back to an integral over $t$. In that way one
finds

$\displaystyle f_{J}(s)={2\,{\cal
N}_{d}\over\pi}c_{J}^{(d)}\left({s-4m^{2}\over
2}\right)^{J}\int\limits_{4m^{2}}^{\infty}{dt\over
t^{J+1}}T_{t}(4m^{2},t)\left(1+O\left((s-4m^{2})/t\right)\right)\ .$ (2.54)

This integral should be understood as follows. The large $t$ contribution is
finite because $|{\rm Im}_{t}T(4m^{4},t)|<|T(4m^{2},t)|<t^{J_{0}(4m^{2})}$ and
$J>J_{0}(4m^{2})$ by assumption. If the integrand diverges at some finite $t$,
and in particular as $t-4m^{2}\to 0^{+}$, then we should step back and write
it as a contour integral of $T(4m^{2},t)$ around the cut, which is manifestly
finite.

###  2.5 Further Continuity Assumption

Based on the standard QFT axioms, scattering amplitudes, cross sections and
partial waves are distributions rather than continuous functions. This leads
to various subtleties, sometimes known as pathologies a-la Martin [19]. For
example, we can imagine total cross-section having singularities which are
local in energy variable $s$, that are not detectable by the finite resolution
experiments. As such it is hard to exclude them based on physical grounds. One
way to produce such a singularity is to consider an infinite number of
resonances that accumulate on the real axis, see [19].

To the best of our knowledge there is no known, first principle argument that
can exclude these possibilities. One way to eliminate them is to simply assume
that various cross sections are continuous functions of energy $s$. We adopt
this practical approach add this to the list of our assumptions. More
precisely, we assume that boundary values of $T(s,t)$ (this includes both
single and double discontinuity) are continuous functions. It is common in the
literature to impose the condition that boundary values of $T(s,t)$ are
uniformly continuous or Hölder continuous, see e.g. [38], but we will not use
it in the present analysis.

Another related common assumption is regarding finiteness of scattering
lengths which are commonly measured in the experiments or using the lattice.
They are defined as follows, see (2.54),

$\displaystyle a_{J}=\lim_{s\to 4m^{2}}{m^{d-4}f_{J}(s)\over({s\over
4m^{2}}-1)^{J}}\geq 0\ ,\qquad J\geq 2\ .$ (2.55)

We will assume that the scattering lengths are finite for $J\geq 2$. Through
the Froissart-Gribov formula these are related to the assumption of finiteness
of the discontinuity $T_{t}(4m^{2},t)$ at $s=4m^{2}$ as well as convergence of
the $J=2$ Froissart-Gribov integral (2.54).

There is an interesting connection between the continuity of the amplitude and
macrocausality [39, 40]. Macrocausality is a set of statements about
scattering amplitudes when particles grouped according to space and time of
interactions and then moved away from each other by large translation. The
notion relevant for the continuity of scattering amplitudes is what is called
strong asymptotic causality in [39] and it has not been proved within the
field theory.

##  3 Analytic Continuation of Elastic Unitarity

The elastic unitarity relations (2.21) was derived for energies in the elastic
region, above the two particle threshold $4m^{2}<s<16m^{2}$, and for physical
kinematics $4m^{2}-s<t<0$. In this section we analytically continue this
relation in $t$, outside of the regime of real scattering angles. We also
consider the double discontinuity of the amplitude and the closely related
analytic continuation of elastic unitarity in spin.

###  3.1 Mandelstam Kernel

The dependence on the scattering angle $z=\cos\theta$ enters the right-hand
side of the elastic unitarity relation (2.22) through the kernels (2.21).
These kernels contain a delta or a step functions and are thus not suitable
for analytic continuation. To overcome this difficulty, we use the analyticity
of $T^{(\pm)}(s,t(z))$ inside the small Lehmann-Martin ellipse to express them
as a counterclockwise Cauchy integral around $[-1,1]$

$\displaystyle{\cal
T}^{(\pm)}(s,z^{\prime})=\oint\limits_{[-1,1]}{d\eta^{\prime}\over 2\pi
i}{{\cal T}^{(\pm)}(s,\eta^{\prime})\over\eta^{\prime}-z^{\prime}}\
,\qquad-1<z^{\prime}<1\ ,\qquad{\cal T}(s,z)\equiv T(s,t(z))\ .$ (3.1)

We can now exchange the order on integrations in (2.22) and perform the
$z^{\prime}$ and $z^{\prime\prime}$ integrals explicitly. In this way we
arrive at

$\displaystyle{\cal T}_{s}(s,z)$ $\displaystyle={(s-4m^{2})^{(d-3)/2}\over
8(4\pi)^{d-2}\sqrt{s}}\oint\limits_{[-1,1]}{d\eta^{\prime}\over 2\pi
i}\oint\limits_{[-1,1]}{d\eta^{\prime\prime}\over 2\pi i}\,{\cal
T}^{(+)}(s,\eta^{\prime}){\cal T}^{(-)}(s,\eta^{\prime\prime})\times
K_{d}(z,\eta^{\prime},\eta^{\prime\prime})\ ,$ (3.2)

where the new kernel is

$\displaystyle
K_{d}(z,\eta^{\prime},\eta^{\prime\prime})\equiv\int{d^{d-2}\Omega_{\vec{n}}\over(\eta^{\prime}-z^{\prime})(\eta^{\prime\prime}-z^{\prime\prime})}=\int\limits_{-1}^{1}dz^{\prime}\int\limits_{-1}^{1}dz^{\prime\prime}{{\cal
P}_{d}(z,z^{\prime},z^{\prime\prime})\over(\eta^{\prime}-z^{\prime})(\eta^{\prime\prime}-z^{\prime\prime})}\
.$ (3.3)

These integrals are evaluated in appendix A. For
$|\eta^{\prime}|,|\eta^{\prime\prime}|>1$ the result is

$\displaystyle K_{d=3}(z,\eta^{\prime},\eta^{\prime\prime})$
$\displaystyle={2\pi\over\eta_{+}-z}\left({\eta^{\prime}\over\sqrt{\eta^{\prime
2}-1}}+{\eta^{\prime\prime}\over\sqrt{\eta^{\prime\prime 2}-1}}\right)\ ,$
(3.4) $\displaystyle K_{d\geq 4}(z,\eta^{\prime},\eta^{\prime\prime})$
$\displaystyle={4\pi^{d-1\over 2}\over\Gamma({d-3\over
2})}\int\limits_{\eta_{+}}^{\infty}{d\eta\over\eta-z}{(\eta^{2}-1)^{4-d\over
2}\over(\eta-\eta_{+})^{5-d\over 2}(\eta-\eta_{-})^{5-d\over 2}}\ .$

where

$\displaystyle\eta_{\pm}(\eta^{\prime},\eta^{\prime\prime})\equiv\eta^{\prime}\eta^{\prime\prime}\pm\sqrt{\eta^{\prime
2}-1}\sqrt{\eta^{\prime\prime 2}-1}\ .$ (3.5)

The Mandelstam kernel for $|\eta^{\prime}|<1$ or $|\eta^{\prime\prime}|<1$ is
obtained from (3.4) by analytic continuation. Note that in (3.3) ${\cal
P}_{d}(z,z^{\prime},z^{\prime\prime})$ is not analytic in $z$, see (2.21). On
the other hand, the Mandelstam kernel
$K_{d}(z,\eta^{\prime},\eta^{\prime\prime})$ is analytic in $z$ and therefore
is suitable for analytic continuation.

Similarly, the representation of ${\cal T}_{s}(s,z)$ in (3.2) is now suitable
for analytic continuation in $t$. That is because $t$ only enters through the
Mandelstam kernel that is manifestly analytic in $z$.

As for the kernel ${\cal P}_{J}^{(d)}$ (2.36), the Mandelstam kernel (3.3) is
also diagonal in spin. To represent it in angular momentum basis, we start
from the representation of ${\cal P}_{J}^{(d)}$ in (2.36) and plug it into the
definition (3.3). We then note that the $z^{\prime}$ and $z^{\prime\prime}$
integration has the effect of converting the partial waves
$P_{J}^{(d)}(z^{\prime})$ and $P_{J}^{(d)}(z^{\prime\prime})$ into
$Q_{J}^{(d)}(z^{\prime})$ and $Q_{J}^{(d)}(z^{\prime\prime})$ correspondingly,
(2.47). In that way we arrive at

$\displaystyle K_{d}(z,\eta^{\prime},\eta^{\prime\prime})=4(4\pi)^{d-2}{\cal
N}_{d}^{2}\,(\eta^{\prime 2}-1)^{d-4\over 2}(\eta^{\prime\prime
2}-1)^{d-4\over
2}\sum_{J=0}^{\infty}n_{J}^{(d)}P_{J}^{(d)}(z)Q_{J}^{(d)}(\eta^{\prime})Q_{J}^{(d)}(\eta^{\prime\prime})\
.$ (3.6)

###  3.2 The Double Spectral Density and Crossing

The combination of elastic unitary with crossing symmetry is very restrictive.
In the next sections we will explore some of its consequences at length. With
this aim in mind, we now represent elastic unitarity in a form that is more
suitable for imposing crossing. By taking another discontinuity of (3.2) with
respect to $t$, the left-hand side becomes the double discontinuity of the
amplitude111111Stated in words, the double spectral density $\rho(s,t)$ is
defined as a certain combination of boundary values of the analytic function
$T(s,t)$ unambiguously specified by $i\epsilon$ in the definition above.

$\displaystyle\rho(s,t)$ $\displaystyle\equiv-{1\over 4}\lim_{\epsilon\to
0}\left[T(s+i\epsilon,t+i\epsilon)-T(s-i\epsilon,t+i\epsilon)-T(s+i\epsilon,t-i\epsilon)+T(s-i\epsilon,t-i\epsilon)\right]$
$\displaystyle=\ \,{\rm Disc}_{t}{\rm Disc}_{s}T(s,t)={\rm Disc}_{s}{\rm
Disc}_{t}T(s,t)=\rho(t,s)\ .$ (3.7)

This crossing-symmetric function is known as the double spectral density.

By taking the $t$ discontinuity of (3.2) we arrive at

$\displaystyle\rho(s,t)$ $\displaystyle={(s-4m^{2})^{d-3\over 2}\over
8(4\pi)^{d-2}\sqrt{s}}\oint\limits_{[-1,1]}{d\eta^{\prime}\over 2\pi
i}\oint\limits_{[-1,1]}{d\eta^{\prime\prime}\over 2\pi i}\,{\cal
T}^{(+)}(s,\eta^{\prime}){\cal T}^{(-)}(s,\eta^{\prime\prime})\times{\rm
Disc}_{t}K_{d}(z,\eta^{\prime},\eta^{\prime\prime})\ ,$ (3.8)

where $4m^{2}<s<16m^{2}$. Outside of the elastic region there are additional
non-elastic contributions to the double discontinuity that will be considered
elsewhere. Next, we deform the $\eta^{\prime}$ and $\eta^{\prime\prime}$
integration to wrap the $t$ and $u$ channel cuts of ${\cal
T}^{(\pm)}(s,\eta)$. The discontinuity of the Mandelstam kernel in (3.8) is
analytic in $\eta^{\prime}$ and $\eta^{\prime\prime}$ along the deformation.
In that way, we end with real $\eta^{\prime}$ and $\eta^{\prime\prime}$ that
are positive on the $t$ channel cut and are negative on the $u$ channel cut.
For $\eta^{\prime}\eta^{\prime\prime}<-1$ the integral in the Mandelstam
kernel (3.4) starts at $\eta_{+}(\eta^{\prime},\eta^{\prime\prime})<-1$ and
can be chosen to run along the negative real $\eta$-axis. This choice make it
manifest that the discontinuity of the kernel for $z>1$ and
$\eta_{+}(\eta^{\prime},\eta^{\prime\prime})<-1$ is zero. We remain
with121212The result for $z<-1$ is obtained by analytic continuation and takes
the same form as (3.9) with ${\rm
Disc}_{z}K(-z,\eta^{\prime},\eta^{\prime\prime})$ instead of ${\rm
Disc}_{z}K(z,\eta^{\prime},\eta^{\prime\prime})$.

$\displaystyle\rho(s,t)={(s-4m^{2})^{d-3\over 2}\over
4\pi^{2}(4\pi)^{d-2}\sqrt{s}}\int\limits_{z_{1}}^{\infty}d\eta^{\prime}\int\limits_{z_{1}}^{\infty}d\eta^{\prime\prime}\,{\cal
T}_{t}^{(+)}(s,\eta^{\prime}){\cal T}_{t}^{(-)}(s,\eta^{\prime\prime})\,{\rm
Disc}_{z}K_{d}(z,\eta^{\prime},\eta^{\prime\prime})\ ,$ (3.9)

where we have mapped the $u$-channel cut to the $t$-channel cut using ${\cal
T}^{(\pm)}_{u}(s,\eta)={\color[rgb]{0,0,1}-\color[rgb]{0,0,0}}{\cal
T}^{(\pm)}_{t}(s,-\eta)$ and
$K(z,-\eta^{\prime},-\eta^{\prime\prime})=K(z,\eta^{\prime},\eta^{\prime\prime})$.
Here, the lower limit of integration is the point where the $t$ channel cut
starts (2.50).

Figure 3: The region of integration in equation (3.9). As $s$ or $t$
approaches the Landau curve from above, the integration region shrinks to
zero. As a result, the double spectral density vanishes below the Landau curve
$z=2z_{1}^{2}-1$.

The discontinuity of the kernel, for $\eta^{\prime}\eta^{\prime\prime}>0$ and
$z>1$ is given by

$\displaystyle{\rm Disc}_{z}K_{3}(z,\eta^{\prime},\eta^{\prime\prime})$
$\displaystyle=4\pi^{2}\delta(z-\eta_{+}){\sqrt{z^{2}-1}\over\eta_{+}-\eta_{-}}\
,$ $\displaystyle{\rm Disc}_{z}K_{d\geq
4}(z,\eta^{\prime},\eta^{\prime\prime})$ $\displaystyle={4\pi^{d+1\over
2}\over\Gamma({d-3\over 2})}\Theta(z-\eta_{+}){(z^{2}-1)^{4-d\over
2}\over(z-\eta_{-})^{5-d\over 2}(z-\eta_{+})^{5-d\over 2}}\geq 0\ .$ (3.10)

This formula was first derived by Mandelstam in $d=4$ [41]. In appendix A we
present the derivation for any dimension $d\geq 3$.

We now discuss the region of support of the double discontinuity $\rho(s,t)$.
Because the first discontinuity has only support for energies above the two
particle threshold, so does the double discontinuity. Looking at (3.9), it is
clear that the situation is more interesting. The discontinuities of the
amplitude only start at $\eta^{\prime},\eta^{\prime\prime}>z_{1}$. At the same
time, the Mandelstam kernel has a non-zero support only for
$z>\eta_{+}(\eta^{\prime},\eta^{\prime\prime})$. This constraint is an upper
bound on the $\eta^{\prime}$, $\eta^{\prime\prime}$ limits of integration, see
figure 3. Hence, the double spectral density in the elastic region is non-zero
only for

$\displaystyle z=1+{2t\over
s-4m^{2}}>\eta_{+}\left(z_{1},z_{1}\right)=1+{32m^{2}s\over(s-4m^{2})^{2}}\ .$
(3.11)

In terms of $s$ and $t$, the boundary of this region is known as the Landau or
Karplus curve and is given by (red in figure 4)

$\displaystyle t={16m^{2}\,s\over s-4m^{2}}\equiv t_{1}(s)\ .$ (3.12)

Figure 4: The double discontinuity of the amplitude $\rho(s,t)$ in the real
$(s,t)$ plane. In gray is the Steinmann shadow region where $\rho$ vanishes.
This region extends inside the elastic bands,
$\\{(4m^{2}<s<16m^{2},t),(s,4m^{2}<t<16m^{2})\\}$, and is bounded by the
Landau curve $t_{1}(s)={16m^{2}s\over s-4m^{2}}$ (in red) and its crossed
curve $t_{1}^{\text{cross}}(s)={4m^{2}s\over s-16m^{2}}$ (in blue). These two
curves extend out of the elastic bands, where there are additional multi-
particle contributions. They cross at $s=t=20m^{2}$.

As $t$ is increased above $t_{1}(s)$, the range of the $\eta^{\prime}$ and
$\eta^{\prime\prime}$ integration opens up. For example, for any value of
$t>t_{1}(s)$, the integral over $\eta^{\prime}$ is bounded in the range

$z_{1}\leq\eta^{\prime}\leq
z\eta^{\prime\prime}-\sqrt{(z^{2}-1)({\eta^{\prime\prime}}^{2}-1)}\leq
zz_{1}-\sqrt{(z^{2}-1)(z_{1}^{2}-1)}\ .$ (3.13)

Hence, as we increase $t$ inside the elastic region $4m^{2}<s<16m^{2}$ (where
(3.9) is valid), more and more channels of ${\cal
T}^{(+)}_{t}(s,\eta^{\prime})$ and ${\cal T}^{(-)}_{t}(s,\eta^{\prime\prime})$
kick in. Their corresponding contributions to $\rho(s,t)$ start at other
Landau curves in the $(s,t)$ plane that are above the elastic one (3.12), see
figure 4. In section 3.5 we present a minimal and complete set of Landau
curves in the elastic region $4m^{2}<s<16m^{2}$ that appears in the physical
$S$-matrices.

In general, the positions of the Landau curves look somewhat technical.
However, they are all kinematic and therefore have a geometrical origin. For
example, the constraint (3.11) takes a simple form in terms of the integral
scattering angles $\eta^{\prime}=\cosh\theta^{\prime}$,
$\eta^{\prime\prime}=\cosh\theta^{\prime\prime}$, and is given by

$\theta^{\prime}+\theta^{\prime\prime}\geq\theta\ .$ (3.14)

In physical kinematics, this constraint follows from a simple geometrical
consideration that is described in figure 1. The analytic continuation to the
non-physical kinetatical regime of positive $s$ and $t$ effect the range of
the angles, but leaves this geometrical constraint unchanged. At the technical
level, this is because the kinematical constraint only involves the cosines of
the scattering angles.

We end this section with a comment regarding the region below the Landau
curve, where the double discontinuity vanishes (the gray region in figure 4).
For the two-to-two scattering, existence of this region is a direct
consequence of elastic unitarity continued to $s,t>4m^{2}$. The precise shape
of the region depends on the details of the unitarity kernel $K_{d}$, as well
as on the analytic structure of the amplitude that enters into the elastic
unitarity relation. A similar phenomenon occurs in the higher-point amplitudes
as well. In this case one considers double discontinuity in the so-called
overlapping channels, see e.g. [42] for a detailed definition. This time it is
possible to consider double discontinuity for physical kinematics directly, as
opposed to the two-to-two case which requires continuation in one of the
Mandelstam invariants. It then follows that for physical kinematics the double
discontinuity vanishes for the overlapping channels. These are known as the
Steinmann relations [43, 44], and it is again a direct consequence of the
multi-particle unitarity. Steinmann relations are useful for constraining the
analytic structure of amplitudes with six and more particles, see [45] for a
recent discussion. Based on the two-to-two case one can try to analytically
continue the relevant unitarity relations to the unphysical values of the
relevant kinematical invariants, and find the extended region where the double
discontinuity vanishes. It would be interesting to understand the precise
shape of this region for the multi-particle case. This would require analytic
continuation of multiparticle unitarity kernels, as well as relevant
scattering amplitudes, analogous to the one made above.

###  3.3 Analytic Continuation in Spin

We will now argue that the elastic unitarity relation (2.37) holds in the
complex $J$ plane, provided that the partial wave coefficients are
analytically continued using the Froissart-Gribov representation (2.53). For
that aim, we first rewrite the elastic unitarity condition (2.37) with the
$i\epsilon$ prescription explicitly

$\displaystyle f_{J}(s+i\epsilon)-f_{J}(s-i\epsilon)=i{(s-4m^{2})^{d-3\over
2}\over\sqrt{s}}f_{J}(s+i\epsilon)f_{J}(s-i\epsilon)\ ,\qquad
4m^{2}<s<16m^{2}\ .$ (3.15)

In our previous discussions $f_{J}(s)\equiv f_{J}(s+i\epsilon)$. For integer
$J$ and real $s$ in that range, real analyticity (2.35) leads to (2.37). The
form (3.15) is however more suitable for analytic continuation.

Originally, (3.15) was derived for $J$ being integer and even, however using
the Froissart-Gribov representation we can continue partial waves in spin $J$

$\displaystyle f_{J}(s\pm i\epsilon)$ $\displaystyle={2\,{\cal
N}_{d}\over\pi}\int\limits_{z_{1}}^{\infty}dz\,(z^{2}-1)^{{d-4\over
2}}Q^{(d)}_{J}(z){\cal T}_{t}^{(\pm)}(s,z)\ ,\qquad{\rm Re}[J]>{\rm
Re}[J_{0}(s)]\ .$ (3.16)

This representation can in principle be continued to the whole complex $J$
plane, going beyond the ${\rm Re}[J]>{\rm Re}[J_{0}(s)]$ region. However, the
continued partial waves are not guaranteed to coincide with the physical ones
for $J<J_{0}$. Moreover, it is clear from (2.13) and the reality properties of
$Q_{J}^{(d)}(z)$ (3.16), that real analyticity of partial waves (2.35) is only
guaranteed to hold for real $J>J_{0}$.

We can then use (3.16) to separately analytically continue the left and right
hand sides of (3.15). It follows from the Carlson theorem that these two
analytic continuations have to agree in the whole complex $J$ plane. Namely,
the two analytic continuations agree for real integer $J>J_{0}$ and do not
grow too fast at large $|J|$, as can be seen from the large $J$ exponential
decay of $f_{J}(s)$, see section 6.1 for more details,

$f_{J}(s)\sim\left({\sqrt{s}+2m\over\sqrt{s}-2m}\right)^{-J}\ .$ (3.17)

Let us relate the analytic continuation in $J$ of (3.15) to the Mandelstam
equation (3.9). To get the former from the latter we integrate (3.9) with
$\int_{2z_{1}^{2}-1}^{\infty}dz(z^{2}-1)^{{d-4\over 2}}Q_{J}(z)$, where
$Q_{J}(z)$ are given in (2.47). We then use the following identity (for
derivation see appendix B)

$\displaystyle\int\limits_{\eta_{+}}^{\infty}dz\,(z^{2}-1)^{{d-4\over
2}}Q_{J}(z){\rm
Disc}_{z}K_{d}(z,\eta^{\prime},\eta^{\prime\prime})={4\pi^{d/2}\over\Gamma({d-2\over
2})}\left[(\eta^{\prime 2}-1)^{{d-4\over
2}}Q_{J}^{(d)}(\eta^{\prime})\right]\left[(\eta^{\prime\prime 2}-1)^{{d-4\over
2}}Q_{J}^{(d)}(\eta^{\prime\prime})\right]\ ,$ (3.18)

which is valid for ${\rm Re}[J]>-1$, $d\geq 3$ and $|\eta_{1}|,|\eta_{2}|>1$.
In practice, we will use this identity in (3.9), where $\eta^{\prime}$ and
$\eta^{\prime\prime}$ are real and positive.

Finally, we can write

$\displaystyle f_{J}(s+i\epsilon)-f_{J}(s-i\epsilon)$ $\displaystyle=2i{2{\cal
N}_{d}\over\pi}\\!\\!\\!\int\limits_{2z_{1}^{2}-1}^{\infty}\\!\\!\\!\\!dz(z^{2}-1)^{{d-4\over
2}}Q^{(d)}_{J}(z)\rho(s,t(z))\ ,\qquad{\rm Re}[J]>{\rm Re}[J_{0}(s)]\ .$
(3.19)

By combining (3.18) with (3.19), it is easy to check that (3.9) becomes
precisely (3.15).

Let us also mention that analytically continued elastic unitarity constrain
the possible Regge limit behavior of the amplitude. In particular, the leading
Regge singularity of the amplitude in the elastic region $4m^{2}<s<16m^{2}$
cannot we a pole located at some real Regge spin $\tilde{J}_{0}(s)$. We review
the derivation of this and a slightly more general result, known as Gribov’s
theorem, in appendix D.

###  3.4 Analytic Continuation in $s$

As $s$ is increased above $16m^{2}$ there are new multi-particle cuts
contributions to ${\rm Im}f_{J}(s)$ that are not captured by (2.37). Still, if
we denote by $f^{\circlearrowleft}_{J}(s)$ the partial wave that was
analytically continues on the second sheet through the elastic cut
$4m^{4}<s<16m^{2}$, then the equation

$\displaystyle f_{J}(s)-f_{J}^{\circlearrowleft}(s)=i{(s-4m^{2})^{d-3\over
2}\over\sqrt{s}}f_{J}(s)f_{J}^{\circlearrowleft}(s)$ (3.20)

holds true when analytically continued away from the elastic strip in the full
multi-sheet complex $s$ plane. In that sense, elastic unitarity can be
analytically continued in $s$.

Note that there is a difference in the nature of the two-particle cut between
odd and even dimensions. Due to the power of $(s-4m^{2})$ in (2.38) we get
that

$S_{J}^{\circlearrowleft}(s)\equiv 1+i{(s-4m^{2})^{d-3\over
2}\over\sqrt{s}}f_{J}^{\circlearrowleft}(s)\times\left\\{\begin{array}[]{ll}-1&d\text{-even}\\\
+1&d\text{-odd}\end{array}\right.\ .$ (3.21)

Correspondingly, for even dimension the continued elastic unitarity condition
(3.20) becomes

$\displaystyle S_{J}(s)S_{J}^{\circlearrowleft}(s)=1\ ,\qquad d\text{-even}\
.$ (3.22)

This relation implies that the elastic cut $4m^{2}<s<16m^{2}$ describes a two-
sheeted Riemann surface in even $d$. Indeed, continuing (3.22) around the
elastic cut again we conclude that
$S_{J}^{2\times\circlearrowleft}(s)=S_{J}(s)$. In odd $d$, because of the sign
difference in (3.21), we do not have (3.22) and the elastic cut is not two-
sheeted. Similar conclusion holds for $T(s,t)$. As we will see below, the
nature of the elastic unitarity cut is the one of a simple square-root for
even $d$ and is infinitely-sheeted for odd $d$. It immediately follows from
(3.22) that a pole of $S_{J}^{\circlearrowleft}(s)$ (resonance) corresponds to
a zero of $S_{J}(s)$ in even $d$. Similarly, in odd $d$, a pole of
$S_{J}^{\circlearrowleft}(s)$ corresponds to a zero of
$1-i{(s-4m^{2})^{d-3\over 2}\over\sqrt{s}}f_{J}(s)$ on the principal sheet.

###  3.5 Elastic Landau Curves

In this section we further study the kinematical properties of the double
spectral density in the elastic strip. In general, discontinuities of the
amplitude result from the exchange of intermediate on-shell states.
Correspondingly, the double discontinuity of the amplitude receives its
support from intermediate on-shell particles exchanged in both channels. As we
have seen in section 3.2, the first double discontinuity in the elastic strip
starts at the leading Landau curve (3.12). It comes from the exchange of two
particle in the $s$-channel and four particles in the $t$-channel. As we
increase $s$ or $t$, more and more processes become accessible. The curves in
the $s-t$ plane where they start to contribute are higher Landau curves. We
now derive an infinite set of higher Landau curves in the elastic strip that
are required by elastic unitarity.

To each of these Landau curves one can associate a Landau diagram. In the
elastic strip, all these diagrams have a simple iterative structure that is
plotted in figure 5. Correspondingly, the shape of the curve has a relatively
simple geometrical origin, (for the case of the first Landau curve, see figure
1 and the discussion around (3.14)). Here instead, we follow a shortcut and
derive their shape directly from the Froissart-Gribov representation and the
elastic unitarity constraint.

Figure 5: a. The Landau diagrams that contribute to the double spectral
density in the elastic strip $4m^{2}<s<16m^{2}$ and $4m^{2}<t$. In this
kinematical regime there can be only two particles in the $s$-channel. Hence,
the corresponding Landau diagrams have a simple structure of iterative two-
particle exchange in the $s$-channel and in between, any number of particles
exchange in the $t$-channel. b. Analogously to figure 1, the corresponding
Landau curves originate from a simple geometric constraint on physical
kinematics, see discussion after (3.14).

The starting point is the kinematical structure of the discontinuity and the
double discontinuity of the amplitude as function of one of the Mandelstam
invariants

$\displaystyle{\rm Disc}_{t}T(s,t)=\qquad
T_{t}(s,t)=\sum_{n=1}^{\infty}\Theta(t-(2nm)^{2})\,T_{t}^{2\to 2n}(s,t)\
,\qquad 4m^{2}-t<s<0\ ,$ (3.23) $\displaystyle{\rm Disc}_{s}{\rm
Disc}_{t}T(s,t)=\rho(s,t)=\sum_{i}\Theta(t-t^{(i)}(s))\,\rho^{(i)}(s,t)\
,\qquad\qquad\ 4m^{2}<s,t\ ,$ (3.24)

where the functions $t^{(i)}(s,t)$ are the Landau curves we are after. For
that aim, we first have to analytically continue the discontinuities of the
amplitudes, $T_{t}^{2\to 2n}(s,t)$, to the regime of positive $s$ and $t$. As
we do so, they develop new thresholds. Because $T_{t}(s,t)$ is a real analytic
function, these new thresholds must coincide with the Landau curves
$t^{(i)}(s)$. What is important for us here is that the multi-particle
thresholds that are manifest in (3.23) are also present in the non-physical
kinematical regime.

To derive the functional shape of the Landau curves in the elastic strip we
impose the consistency of (3.24) and (analytically continued) (3.23) with
elastic unitarity. This can be done by either plugging them into the
Mandelstam equation or by imposing elastic unitarity at the level of the
partial waves. Here we follow the latter strategy and in appendix E we present
the former, both leading to the same result.

To impose consistency of (3.23) and (3.24) with the partial waves elastic
unitarity, we plug them into the Froissart-Gribov projection (2.53), that we
quote here for convenience

${\rm Re}f_{J}(s)+i\,{\rm Im}f_{J}(s)={2\,{\cal
N}_{d}\over\pi}\int\limits_{z_{1}}^{\infty}dz\,(z^{2}-1)^{{d-4\over
2}}Q^{(d)}_{J}(z)\left[{\rm Re}\,T_{t}(s,t(z))+i\rho(s,t(z))\right]\ .$ (3.25)

Next, we take the large $J$ limit of (3.25). Using the large $J$ decay of the
$Q$ functions that can be schematically written as, see section 6.1,

$Q_{J}^{(d)}(z)\sim\lambda^{-J}(z)\ ,\qquad\lambda(z)\equiv
z+\sqrt{z^{2}-1}=e^{\theta}\ ,\qquad z>1\ ,$ (3.26)

as well as the two-particle step function in (3.23), we conclude that the
leading large $J$ behavior of ${\rm Re}\,f_{J}(s)$ is

${\rm Re}f_{J}(s)\sim\lambda^{-J}(z_{1})\ .$ (3.27)

Similarly to (3.27), the $n$-particle threshold of $T_{t}(s,t)$ and the $i$’th
Landau curve threshold of $\rho(s,t)$ in (3.23) result in contributions to
${\rm Re}\,f_{J}(s)$ and ${\rm Im}\,f_{J}(s)$ that start at large $J$ as

${\rm Re}f_{J}(s)\sim\lambda^{-J}(z_{n})\
,\qquad\operatorname{Im}f_{J}(s)\sim\left.\lambda^{-J}\left(z\right)\right|_{t=t^{(i)}(s)}\
,$ (3.28)

where $z_{n}\equiv 1+{8n^{2}m^{2}\over s-4m^{2}}$. Importantly, (3.26)
receives only $1/J$-power corrections and no nonperturbative exponential
corrections, see appendix C. Hence, $\operatorname{Im}f_{J}(s)$ does not
receive any exponential large $J$ behavior other than the ones that result
from Landau curves thresholds (3.28). Similarly, the exponential large $J$
behavior of ${\rm Re}f_{J}(s)$ can only come from thresholds of $T_{t}(s,t)$
in (3.25), but not from nonperturbative terms in the large $J$ expansion of
$Q_{J}^{(d)}(z)$.

We now plug these set of large $J$ exponential behaviors into elastic
unitarity and derive the minimal set of Landau curves, $t^{(i)}(s)$, that are
required to close the equation, together with new thresholds of $T_{t}(s,t)$
that are not present in physical kinematics.

Elastic unitarity (2.37) can be written in the following schematic form

$\displaystyle\mathrm{Im}f_{J}(s)\propto[\mathrm{Re}f_{J}(s)]^{2}+[\mathrm{Im}f_{J}(s)]^{2}\
,$ (3.29)

where we omitted the pre-factor because it is irrelevant for the present
discussion. First, we see that the multi-particle threshold exponents of ${\rm
Re}f_{J}(s)$ in (3.28) result in the following exponents in
$\operatorname{Im}f_{J}(s)$

$\displaystyle\operatorname{Im}f_{J}(s)\sim\left[\lambda(z_{n})\lambda(z_{m})\right]^{-J}\
,\qquad n,\,m\geq 1\ .$ (3.30)

Using (3.28), these then lead to the Landau curves

$\left.\lambda\left(z\right)\right|_{t=t^{(n,m)}(s)}=\lambda(z_{n})\lambda(z_{m})\
.$ (3.31)

For example, the leading Landau curve (3.12) corresponds to the case where
$m=n=1$. This behavior must then also exist in ${\rm Re}f_{J}(s)$. To see
that, recall that $T_{t}(s,t)$ is a real analytic function of $t$ for fixed
$s>4m^{2}$. Hence, a threshold in the imaginary part of $T_{t}(s,t)$, namely
in $\rho(s,t)$, must be accompanied by a corresponding threshold in ${\rm
Re}\,T_{t}(s,t)$.

Having established the presence of the quadratic in $\lambda(z_{n})$ terms in
the large $J$ expansion of ${\rm Re}f_{J}(s)$ and $\operatorname{Im}f_{J}(s)$,
we now come back to elastic unitarity (3.29). The presence of these terms
together with the linear one (3.28), now induces higher powers of
$\lambda(z_{n})$ in the large $J$ expansion of $\operatorname{Im}f_{J}(s)$
and, via real analyticity, of ${\rm Re}f_{J}(s)$ as well. These take the
general form

$\displaystyle\mathrm{Re}f_{J}(s)$
$\displaystyle\sim\left[\lambda(z_{n_{1}})\cdots\lambda(z_{n_{L}})\right]^{-J}\
,\qquad n_{i}\geq 1\ ,\quad L\geq 1\ ,$ (3.32)
$\displaystyle\mathrm{Im}f_{J}(s)$
$\displaystyle\sim\left[\lambda(z_{n_{1}})\cdots\lambda(z_{n_{L}})\right]^{-J}\
,\qquad n_{i}\geq 1\ ,\quad L\geq 2\ .$ (3.33)

This large $J$ structure implies existence of an infinite set of Landau curves
labeled by a set of integers $\\{N_{1},N_{2},N_{3},\dots\\}$ and given by the
following equation

$\displaystyle\lambda\left(z(t_{\\{N_{1},N_{2},N_{3},\dots\\}})\right)=\lambda^{N_{1}}(z_{1})\,\lambda^{N_{2}}(z_{2})\,\lambda^{N_{3}}(z_{3})\cdots\
.$ (3.34)

In terms of the scattering angle, for which $z=\cosh\theta$ and
$\lambda=e^{\theta}$, these Landau curves take the form

$\displaystyle\theta(z)=N_{1}\theta_{1}+N_{2}\theta_{2}+N_{3}\theta_{3}+\dots\
,$ (3.35)

with $\theta_{n}=\arccos(z_{n})$. This form is the generalization of (3.14)
for all the elastic Landau curves. In a direct analogy with figure 1, it also
has a simple geometrical origin that is discussed in figure 5.b. Using that
$z=1+{2t\over s-4m^{2}}$, we can translate (3.34) into polynomial equations in
$s$ and $t$ with real coefficients.131313In this form they are familiar in the
study of perturbative Feynman integrals, see e.g. [1] for more details. For
example, we have

$\displaystyle t_{\\{2,0,\dots\\}}={16m^{2}s\over s-4m^{2}}\ ,\qquad
t_{\\{3,0,\dots\\}}={36m^{2}(s+{4m^{2}\over 3})^{2}\over(s-4m^{2})^{2}}\
,\qquad t_{\\{4,0,\dots\\}}={64m^{2}s(s+4m^{2})^{2}\over(s-4m^{2})^{3}}\ .$
(3.36)

The Landau curves have the following asymptotic behavior

$\lim_{s\to
4m^{2}}t_{\\{N_{1},N_{2},\dots\\}}(s)\sim(s-4m^{2})^{1-\sum_{j=1}^{\infty}N_{j}}\
,\qquad\lim_{s\to\infty}t_{\\{N_{1},N_{2},\dots\\}}(s)=\Big{(}2m\sum_{j=1}^{\infty}jN_{j}\Big{)}^{2}\
.$ (3.37)

These asymptotes are precisely the $t$-channel normal thresholds (3.23). This
is evident from the Landau diagram interpretation of figure 5 and it would be
interesting to see if it can be established rigorously.141414For the case of
two particles in the final state it was done by Mandelstam and we reviewed it
in section 3.1. For the multi-particle case some limited results have been
obtained [46, 47, 48, 49].

##  4 Positivity of $\rho$ and Multi-Particle Production

Drawing an analogy with the conformal bootstrap, one can expect that combining
crossing symmetry of $\rho(s,t)$ with some sort of positivity property leads
to nontrivial constraints on scattering amplitudes.

In this section we show that there is indeed a finite region in the elastic
strips of the $(s,t)$-plane, where $\rho(s,t)$ is positive, as noted in [50].
In subsection 4.2 we discuss a direct consequence of this positivity and
crossing – the necessity of multi-particle production $T_{2\to 2n}$ for any
$n$.

###  4.1 Positivity of The Double Spectral Density

Figure 6: The regime of positivity of the double spectral density $\rho(s,t)$
in the elastic strip $4m^{2}<t<16m^{2}$, (dashed region). This is the region
above the first Karplus curve $s_{1}(t)$ (in blue) and below the curve
$t=4m^{2}{\sqrt{s}+2m\over\sqrt{s}-6m}$ (in green). There is an identical
positive region in the crossed strip of $4m^{2}<s<16m^{2}$, see (4.4).

To establish positivity, we assume that the integral in the Mandelstam
equation for the double spectral density (3.9) converges and study under what
conditions the integrand is positive.

Consider first the discontinuity of the Mandelstam kernel ${\rm
Disc}_{z}K_{d}(z,\eta^{\prime},\eta^{\prime\prime})$ given in (3.2). This
function is strictly positive for $z>1$, which is the case for any
$s,t>4m^{2}$. Next we turn our attention to ${\cal
T}_{t}^{(\pm)}(s,\eta^{\prime})$. It is clear that ${\cal
T}_{t}(s,\eta^{\prime})$ is non-negative in the region where the partial wave
expansion converges151515In this region we can drop the $(\pm)$ subscript of
${\cal T}_{t}^{(\pm)}(s,\eta^{\prime})$ and ${\cal
T}_{t}^{(\pm)}(s,\eta^{\prime\prime})$ in (3.9) because both functions are
equal.

$\displaystyle{\cal
T}_{t}(s,\eta^{\prime})=T_{t}(s,t^{\prime})=\sum_{J}n_{J}^{(d)}\,{\rm
Im}f_{J}(t^{\prime})P_{J}^{(d)}\left(1+{2s\over t^{\prime}-4m^{2}}\right)\
,\qquad t^{\prime}={\eta^{\prime}-1\over 2}(s-4m^{2})\ ,$ (4.1)

where $t^{\prime}$ is not to be confused with the external $t$ in (3.9).
Indeed, unitarity implies that ${\rm Im}f_{J}(t^{\prime})\geq 0$ and
$P_{J}^{(d)}\left(1+{2s\over t^{\prime}-4m^{2}}\right)$ is positive for
$s,t^{\prime}>4m^{2}$. Consider the function $g\left(1+{2s\over
t^{\prime}-4m^{2}}\right)\equiv T_{t}(s,t^{\prime})$ for fixed $t^{\prime}$.
Provided that $g(z)$ is analytic inside an ellipse in the complex $z$-plane
with foci at $z=\pm 1$, it then follows from Neumann’s argument, see footnote
10, that the partial wave expansion (4.1) converges inside that ellipse. Under
the assumption of extended analyticity and for fixed $t^{\prime}$, the first
singularity is where $T_{t}(s,t^{\prime})$ develops a discontinuity with
respect to $s$. Namely, the partial wave expansion (4.1) converges below the
first Landau curve, red in figure 4,161616As a consistency check, we can also
study the convergence region of (4.1) using the large $J$ asymptotics of the
partial wave coefficients. Using (3.19) and the shape of the first Landau
curve we have arrived at the same conclusion.

$t^{\prime}<t_{1}(s)={16m^{2}s\over s-4m^{2}}\qquad\text{or
equivalently}\qquad\eta^{\prime}<1+{36m^{2}\over(s-4m^{2})^{2}}\ ,\qquad
4m^{2}<s<16m^{2}\ .$ (4.2)

This region of convergence is called the large $s$-channel Martin-Lehmann
ellipse [28]. Note that the standard argument of [28] refers to the
convergence of (3.19) in the crossed region of $s<s_{1}(t^{\prime})$,
$4m^{2}<t^{\prime}<16m^{2}$ and does not require the extended analyticity
assumption. Here, we are saying that under this assumption, the partial wave
expansion (4.1) converges in the union of the $t$\- and $s$-channel large
Martin-Lehmann ellipses.

The maximal value of the $\eta^{\prime}$ integration is given in (3.13).
Hence, we conclude that the double spectral density is non-negative for

$zz_{1}-\sqrt{(z^{2}-1)(z_{1}^{2}-1)}\leq 1+{36m^{2}\over(s-4m^{2})^{2}}\ .$
(4.3)

By solving this condition for $t$ we conclude that

$\displaystyle\boxed{\rho(s,t)>0\ ,\qquad 4m^{2}<s<16m^{2}\
,\qquad{16m^{2}s\over s-4m^{2}}<t\leq
4m^{2}{(3s+4m^{2})^{2}\over(s-4m^{2})^{2}}}\ .$ (4.4)

where the lower bound comes from (3.12), and the upper bound from (4.3). This
region of positivity was first derived in [50] and is plotted in figure
6.171717Note that this is the regime netween the curves $t_{\\{2,0,\dots\\}}$
and $t_{\\{3,0,\dots\\}}$ in (3.36).

Note that at the upper bound of that region $\rho(s,t)$ is strictly positive.
Hence, the region of positivity can always be extended from above. A question
to which we do not know the answer is whether the region of positivity of the
double spectral density can be extended to arbitrary large $s$ and $t$. Here
we only list some known implications of such a scenario. In [51] A. Martin
argued that positivity of the double spectral density for all $s$ and $t$
implies that the total cross-section at high energies satisfies the following
bound

$\displaystyle\sigma_{tot}(s)\leq{C\over\log s}\ ,\qquad s\to\infty\ .$ (4.5)

This immediately implies that for theories that saturate the Martin-Froissart
bound $\sigma_{tot}(s)\sim\log^{2}s$ [52, 28], the double spectral density is
not positive-definite.

Similarly, in [53] A. W. Martin explored implications of the positive-definite
double spectral density for scattering at finite angles. He showed that
amplitudes with positive double spectral density do not exhibit diffraction
peak in the near-forward scattering, which is

$\displaystyle T_{t}(s,t)\sim e^{sb(t)+\dots}\ ,$ (4.6)

where $b(t)$ is a slowly-varying function. Such a diffraction peak is
observed, for example, in the scattering of pions. Therefore, the double
spectral density for pion scattering cannot be positive for arbitrary $s$ and
$t$.

It would be interesting to understand positivity properties of $\rho(s,t)$ in
physical theories more systematically.

###  4.2 The Aks Theorem and Necessity of Particle Production

We now review an elegant argument for scalar particles by Aks [14].181818For
the generalization to the case of spinning particles or particles in
nontrivial representations of some global symmetry, see [54]. It states that
scattering implies particle production. Namely, provided that $T_{2\to 2}\neq
0$, also $T_{2\to n}\neq 0$ with $n>2$. The theorem applies to any crossing
symmetric scalar scattering amplitude in $d\geq 3$ that satisfies extended
analyticity in a finite region above the leading Landau curve.

To derive Aks’s result, let us therefore assume that we have a nontrivial
scattering amplitude $T(s,t)$, but $T_{2\to n}$ are identically zero for
$n>2$. This implies that elastic unitarity in the form of Mandelstam (3.9)
holds for any $s\geq 4m^{2}$, whereas in the the theories with particle
production elastic unitarity only holds below the first threshold $s_{0}>s\geq
4m^{2}$. It then follow that $\rho(s,t)=0$ below the first $s$-channel Landau
curve $4m^{2}<t<t_{1}(s)$ (3.12) for any $s$, see red curve in figure 4. This
region however includes the crossed Mahoux-Martin region of positive
$\rho(s,t)>0$ (4.4), $s<4m^{2}{(3t+4m^{2})^{2}\over(t-4m^{2})^{2}}$, see gray
region in figure 6.191919Recall that positivity of $\rho(s,t)>0$ is a direct
consequence of our assumption about nontriviality of scattering. Therefore, we
conclude that our assumptions of crossing symmetry and scattering without
production are not consistent.

Note that in the case of an infinite $J=0$ scattering length, the derivation
of the positivity of $\rho(s,t)$ in the Mahoux-Martin region does not always
apply. That is because in that case, the integral in (3.9) fails to converge
for $d\geq 5$. However, when the $J=0$ scattering length is infinite,
$\rho(s,t)$ is already non-zero at the leading Landau curve and we reach the
same conclusion that there must be particle production.

A way to relax the assumption of extended analyticity was explained in [55] by
exploiting polynomial boundedness and continuity assumptions. The idea is to
use elastic unitarity to extend the region of analyticity of the amplitude.
One starts with fixed $t<4m^{2}$ dispersion relations and then use elastic
unitarity to first extend the analyticity domain of $T_{s}$ and then use
dispersion relations to continue the scattering amplitude itself. The key
point being that assuming no production we can use elastic unitarity to
continue the discontinuity at arbitrary energy, which is necessary if we want
to use this inside the dispersion relations.

Note that the argument above also implies that we must have four-particle
production. That is because the crossed region of positivity starts at
$s_{1}(16m^{2})={64\over 3}m^{2}<36m^{2}$, it is enough to assume that
$T_{2\to 4}=0$ to reach a contradiction. One can wonder if having $T_{2\to 4}$
is enough to fix the problem, or $T_{2\to 2n}$ with $n>2$ are also necessary?
To address this question, we can then proceed via crossing.202020Note that
crossing has been only rigorously proven within the standard QFT framework for
$2\to 2$ amplitudes [23]. For some further progress in the multi-particle case
$2\to n$ see [32]. By unitarity of the $4\to 4$ amplitude, ${\rm Im}T_{4\to
4}\sim|T_{2\to 4}|^{2}$, non-vanishing $T_{2\to 4}$ implies that we have a
non-vanishing $T_{4\to 4}$ amplitude. Applying crossing this becomes $T_{2\to
6}$. Continuing this recursion we conclude that all $T_{2\to 2n}$ should be
non-zero. Therefore, not only scattering implies production but it requires
all possible production (here we assumed $Z_{2}$ symmetry so that only an even
number of particles is present in the final state).

An alternative argument that does not use crossing symmetry of higher-point
amplitudes was put forward in [56]. This argument relies on an unproven
assumption that Landau curves can only asymptote to normal thresholds for the
scattering of lightest particles, see section 3.5. This assumption is
consistent with the perturbative analysis of [29, 30, 31] and it would be
interesting to establish it rigorously.

###  4.3 Bounding Inelasticity

Figure 7: a. We consider a scattering experiment at fixed impact parameters
$b$. This process is controlled by the exchanged momentum $t\sim{1\over
b^{2}}>0$. b. At large impact parameters the amplitude can be organized as a
sum over on-shell particles exchange. The dominant contribution comes from the
exchanged of the lightest on-shell particle.

There is another, more intuitive way to think about the result of Aks and
necessity of particle production in higher dimensions. Ideally, one would like
to take a discontinuity of the $2\to 4$ amplitude that is given by a product
of two $2\to 2$ amplitudes. For physical kinematics however, such a
discontinuity only exist for the $3\to 3$ setup. Instead, let us discuss
impact parameter scattering, which is only possible in $d\geq 3$.

As reviewed for example in appendix E of [57], the effect of going to the
impact parameter space is the same as continuing the conjugate momentum
invariant to the unphysical kinematics. It follows that inelasticity in a
gapped theory cannot be exactly zero at very large impact parameters. To see
this, decompose the four particles in the final state into a pair of dipoles.
Then consider a scattering in which the two dipoles in the final state, as
well as the pair of incoming particles in the initial state, are separated by
a finite distance $b$ in the transverse space, see figure 7.a. Unitarity
becomes, in the impact parameter space, the expansion in Yukawa potential
suppressed terms $T_{2\to 4}(s,b)\sim T_{2\to 2}^{2}\times
e^{-bm}+[\text{multi-particle}\sim e^{-nbm}]$. At large separation, this
expansion is dominated by the one-particle exchange $e^{-bm}$, while the
multi-particle corrections are further exponentially suppressed.212121In terms
of partial waves, large impact parameter scattering corresponds to the large
spin limit and therefore we expect to have inelasticity at large spin which
will be analyzed in detail in the sections below. In that way, a non-trivial
(analytically continued) four-point amplitude imply a non-trivial $2\to 4$
amplitude. We would then like to bound the $2\to 4$ amplitude from below.

There is a convenient way of bounding the integrated discontinuity of $T_{2\to
4}$ in the kinematical regime of figure 7.b from below. It is based on the
discussion in the previous sections, see in particular sections 3.2 and 4.1,
4.2. It also highlights how the crossing of $\rho(s,t)$ discussed in the
section above works microscopically. Let us start with the the square of
$T_{2\to 4}$ that appears in the discontinuity of the $2\to 2$ amplitude,
$T_{s}(s,t)$

$\displaystyle T^{\text{inel},2\to 4}_{s}(s,t)\equiv{1\over 2}{1\over
4!}\int\prod_{i=1}^{4}{d^{d-1}\vec{q_{i}}\
\over(2\pi)^{d-1}(2E_{\vec{q_{i}}})}(2\pi)^{d}\delta^{d}(p_{1}+p_{2}-\sum_{i=1}^{4}q_{i})T_{2\to
4}^{(+)}(p_{1},p_{2}|q_{i})T_{2\to 4}^{(-)}(q_{i}|p_{3},p_{4})\ .$ (4.7)

By construction, the unitarity integral in the right-hand side of (4.7)
depends only on $s$ and $t$. For physical scattering we consider $s>16m^{2}$
and $t<0$.

We would like next to analytically continue (4.7) to the unphysical Martin-
Mahoux region discussed above

$\displaystyle 4m^{2}<t<16m^{2}\ ,\qquad{16m^{2}t\over
t-4m^{2}}<s<4m^{2}{(3t+4m^{2})^{2}\over(t-4m^{2})^{2}}\ .$ (4.8)

We also would like to consider $16m^{2}<s<36m^{2}$ to focus on the $T_{2\to
4}$ amplitude. This condition together with (4.8) imply ${36m^{2}\over 5}<t$.
For the $2\to 2$ case we reviewed the procedure in the sections above. For the
multi-particle case it was discussed in [46, 47, 48, 49].

After taking a discontinuity in $t$ and using crossing symmetry of the double
spectral density, we arrive at the following schematic form

$\displaystyle\rho(s,t)={(t-4m^{2})^{d-3\over 2}\over
4\pi^{2}(4\pi)^{d-2}\sqrt{t}}$
$\displaystyle\int\limits_{\bar{z}_{1}}^{\infty}d\eta^{\prime}\int\limits_{\bar{z}_{1}}^{\infty}d\eta^{\prime\prime}\,{\rm
Disc}_{s}{\cal T}_{2\to 2}^{(+)}(t,\eta^{\prime})\,{\rm Disc}_{s}{\cal
T}_{2\to 2}^{(-)}(t,\eta^{\prime\prime})\times{\rm
Disc}_{\bar{z}}K_{d}(\bar{z},\eta^{\prime},\eta^{\prime\prime})$
$\displaystyle=$ $\displaystyle\int d{\rm LIPS}_{4}\times\,{\rm
Disc}_{t}T_{2\to 4}^{(+)}\,{\rm Disc}_{t}T_{2\to 4}^{(-)}\times
K_{\text{Mandelstam}}^{2\to 4}\ ,$ (4.9)

where in the formula above we switched to $\bar{z}=1+{2s\over t-4m^{2}}$ and
$\bar{z}_{1}=1+{8m^{2}\over t-4m^{2}}$. Here in the right-hand side each ${\rm
Disc}T_{2\to 4}$ contains a delta-function that puts the exchanged particle in
figure 7 on-shell. The phase space integral $d{\rm LIPS}_{4}$ should be
understood in terms of the analytic continuation a-la Mandelstam, and we will
discuss it in more detail elsewhere [58].

Note that since we are in the Mahoux-Martin region, the partial wave expansion
of ${\rm Disc}_{s}{\cal T}^{(\pm)}_{2\to 2}(t,\eta)$ converges and both the
Mandelstam kernel and the Legendre polynomials that enter into partial waves
are positive. Therefore we can write

$\displaystyle T_{s}(t,s)=\sum_{J=0}^{\infty}n_{J}^{(d)}{\rm
Im}f_{J}(s)P_{J}(1+{2t\over s-4m^{2}})\geq\sum_{J=0}^{J_{0}}n_{J}^{(d)}{\rm
Im}f_{J}(s)P_{J}(1+{2t\over s-4m^{2}})\ .$ (4.10)

By plugging (4.10) into (4.3) and using positivity of the Mandelstam kernel
${\rm Disc}_{\bar{z}}K_{d}(\bar{z},\eta^{\prime},\eta^{\prime\prime})$ we
arrive at the lower bound on the integrated discontinuity ${\rm
Disc}_{t}T_{2\to 4}$.

While this argument bounds the discontinuity of $T_{2\to 4}$ in the unphysical
kinematics of figure 7.b, for ${36m^{2}\over 5}\leq t<16m^{2}$ from below, it
does not tell us anything about $T_{2\to 4}$ in the physical kinematics $t<0$.
Let us however comment why one expects to have $T_{2\to 4}$ of the same order
also in the physical kinematics. Schematically, we can write the following
representation of the $2\to 4$ impact parameter amplitude

$\displaystyle T_{2\to 4}(s,b)=(T_{2\to
2})^{2}e^{-bm}+\int\limits_{3m}^{\infty}dM\rho(M)e^{-Mb}.$ (4.11)

We can rewrite this as follows

$\displaystyle|T_{2\to 4}(s,b)-(T_{2\to 2})^{2}e^{-bm}|$ $\displaystyle\leq
c_{\text{MP}}\,e^{-3mb}\ ,\qquad
c_{\text{MP}}=\int\limits_{3m}^{\infty}dM\,|\rho(M)|e^{-(M-3m)b}\ ,$ (4.12)

where $c_{\text{MP}}$ encodes the contribution of the multi-particle
exchanges.

Without extra fine-tuning we expect that in a strongly coupled theory $T_{2\to
2},c_{\text{MP}}\sim{\cal O}(1)$ and therefore the one-particle exchange to
dominate for $b\gtrsim{1\over m}$. This would make $T_{2\to 4}\sim O(1)$. On
the other hand, making $T_{2\to 4}\ll 1$ given $T_{2\to 2}\sim O(1)$ would
require a very fine-tuned cancellation between the exchange of one-particle
state and the multi-particle state as well as $c_{\text{MP}}\gg 1$. It seems
quite possible that such a scenario is not consistent with multi-particle
unitarity, but it is very hard to show it explicitly due to the complexity of
the latter.

One scenario in which the regime of single particle dominance can be delayed
to arbitrary large impact parameters is if the theory contains extended
objects, such as strings. In such case we can choose
$l_{\text{string}}\gg{1\over m}$, and the one-particle exchange is expected to
dominate only for $b\gtrsim l_{\text{string}}$. In this case however, the
spectrum is expected to contain particles of mass ${1\over
l_{\text{string}}}\ll m$ which contradicts our assumption about $m$ being the
lightest particle.

In some numerical applications that we discuss in more detail below, see e.g.
[15], it was observed that small $J$ partial wave converge very well.
Therefore we can set in the formula (4.10) $J_{0}=2$ and then use it in (4.3).
We can then evaluate the integral above to rigorously bound from below ${\rm
Disc}T_{2\to 4}$ in the Mahoux-Martin kinematical region. In the numerical
analysis when maximizing some couplings, one usually finds ${\rm
Im}f_{0}(s)\sim 1$ therefore the formulas (4.3), (4.10) will produce ${\rm
Disc}T_{2\to 4}\sim 1$. It is however still an open problem to translate this
fact into a rigorous statement about $T_{2\to 4}$ for physical kinematics.

Independently of the discussion above, in section 7 we bound inelasticity
using an additional input about the discontinuity of the $2\to 2$ amplitude.

##  5 Threshold Expansion

5.15.25.36.16.2765 Figure 8: The logical structure of sections 5, 6 and 7. We
start in section 5 by developing the idea of the threshold expansion. We solve
elastic unitarity for $f_{J}(s)$ close to $s=4m^{2}$. This naturally leads to
the threshold expansion both for the discontinuity of the amplitude
$T_{s}(s,t)$ and its double discontinuity $\rho(s,t)$. In section 6 we use the
Froissart-Gribov formula and crossing symmetry of $\rho(s,t)$ to translate the
threshold expansion of discontinuity $T_{s}(s,t)$ into the large $J$ expansion
of $f_{J}(s)$ and the threshold expansion of $\rho(s,t)$ into the large $J$
expansion of ${\rm Im}f_{J}(s)$ correspondingly. In section 7 we give an error
bound on these expansion in terms of an extra phenomenological bound on the
discontinuity of the amplitude.

To further analytically constrain a nonperturbative amplitude a small
parameter is needed. In this section, as well as the next one, we will study
the expansions of the amplitude in two kinematical small parameters, as well
as the relation between the two expansions.

One kinematical small parameter that always exists in a gapped theory is the
energy distance from the two-particle thereshold in units of the mass gap,

$\sigma_{s}={s\over 4m^{2}}-1={\vec{p}^{2}\over m^{2}}\ ,$ (5.1)

where in the last equality we evaluated $\sigma_{s}$ in the center-of-mass
frame (2).

The threshold expansion is, thus, an expansion in powers of $\sigma_{s}$. The
small $\sigma_{s}$ expansion is known in nuclear physics as the effective
range expansion [59]. As we take $\sigma_{s}\ll 1$ or, equivalently,
$|\vec{p}|\ll m$ scattering becomes non-relativistic and we can characterize
it by some effective potential whose properties are captured by the threshold
expansion parameters [60]. In this section we use elastic unitarity, extended
analyticity and crossing to argue that both the discontinuity and the double
discontinuity of the scattering amplitude admit a natural threshold expansion.
Importantly, the parameters that enter the threshold expansion are controlled
by the low energy physics which is well-known from the experiments or lattice
simulations, see e.g. [61].

Whenever a theory has a conserved charge, there is a natural small parameter –
one over the charge of states that are being exchanged. In the case at hand,
the only symmetry we assume is Lorentz symmetry. Correspondingly, the only
available conserved charge is the spin. Indeed, in the next section we will
show that the partial wave coefficients admit a systematic large $J$
expansion. Moreover, by combining the Froissart-Gribov formula with crossing
we will relate the threshold expansion to the large spin expansion.

The logical structure and steps of this and following sections is summarized
in figure 8. We start by constructing a threshold expansion of the partial
wave coefficients that is consistent with elastic unitarity.

###  5.1 Threshold Expansion of $f_{J}(s)$

To expand the the partial wave coefficients close to the threshold in a way
that is consistent with elastic unitarity, it is first convenient to solve the
latter in a different fashion than (2.39). After dividing by $|f_{J}(s)|^{2}$,
the elastic unitarity condition (2.37) takes the form

$2{\rm Im}{1\over f_{J}(s)}={1\over i}\left({1\over
f_{J}(s+i\epsilon)}-{1\over f_{J}(s-i\epsilon)}\right)=-{(s-4m^{2})^{d-3\over
2}\over\sqrt{s}}\times\left\\{\begin{array}[]{lc}1&4m^{2}<s<16m^{2}\\\
0&0<s<4m^{2}\end{array}\right.\ .$ (5.2)

Hence, $1/f_{J}$ has a branch cut at threshold for even $d$ and a logarithmic
cut for odd $d$. The general solution to (5.2) combined with real analaticity
(2.35) takes the form

${1\over f_{J}(s+i\epsilon)}=b_{J}(\sigma_{s})-{i\over
2}{(4m^{2}\sigma_{s})^{d-3\over
2}\over\sqrt{s}}\times\left\\{\begin{array}[]{ll}1&d\text{ even}\\\
{i\over\pi}[\log\sigma_{s}-i\pi]&d\text{ odd}\end{array}\right.\ ,$ (5.3)

where $b_{J}(\sigma_{s})$ is a real analytic and single-valued function in
some finite neighborhood around the origin, except for potentially isolated
singularity at $\sigma_{s}=0$. Note that in writing (5.3) we used the
continuity assumption for $f_{J}(s)$, see assumption 5 at the introduction and
section 2.5.

As we have learned from the Froissart-Gribov representation (see (2.54)), as
$\sigma_{s}\to 0^{+}$ from above, $f_{J}(s)\to a_{J}\sigma_{s}^{J}$ (2.55). In
terms of the function $b_{J}(\sigma)$, this behavior is

$\displaystyle b_{J}(\sigma)={m^{d-4}\over a_{J}\sigma_{s}^{J}}+{\cal
O}(\sigma_{s}^{1-J})\ ,\qquad J\geq 2\ ,$ (5.4)

where we have assumed that $J_{0}(4m^{2})<2$. For $J=0$ we do not expect the
Froissart-Gribov representation to hold close to $\sigma_{s}=0$. Therefore we
do not have a similar prediction. After factoring out the leading threshold
singularity of $b_{J\geq 2}(\sigma)$, it has a regular expansion

$b_{J\geq 2}(\sigma_{s})={m^{d-4}\over\sigma_{s}^{J}}\left({1\over
a_{J}}+\sum_{i=1}^{\infty}b_{J,i}\sigma_{s}^{i}\right)\qquad\text{and}\qquad
b_{0}(\sigma_{s})={m^{d-4}\over\sigma_{s}^{\tilde{J}_{0}}}\sum_{i=0}^{\infty}b_{0,i}\,\sigma_{s}^{i}\
,$ (5.5)

where $\tilde{J}_{0}$ is an unconstrained integer at this point and $a_{J}$,
$b_{J,i}$ are real coefficients. Reality of scattering lengths $a_{J}$ and
effective ranges $b_{i,J}$ follows from real analyticity property of the
partial waves coefficients (2.35). For actual values of these parameters in
various QCD processes see e.g. [62].

Similarly, using real analaticity of $b_{J}(s)$, we have the following
equations for the imaginary part of the partial wave in the elastic unitarity
region

$\displaystyle{\rm Im}f_{J}(s)$ $\displaystyle={(4m^{2}\sigma_{s})^{d-3\over
2}\over 2\sqrt{s}}\left[b_{J}^{2}(\sigma_{s})+{(4m^{2}\sigma_{s})^{d-3}\over
4s}\right]^{-1}\qquad\qquad\qquad\qquad\qquad\qquad\quad\ d\text{ even}\ ,$
$\displaystyle{\rm Im}f_{J}(s)$ $\displaystyle={(4m^{2}\sigma_{s})^{d-3\over
2}\over 2\sqrt{s}}\left[\left(b_{J}(\sigma_{s})+{(4m^{2}\sigma_{s})^{d-3\over
2}\over 2\pi\sqrt{s}}\log\sigma_{s}\right)^{2}+{(4m^{2}\sigma_{s})^{d-3}\over
4s}\right]^{-1}\qquad d\text{ odd}\ .$ (5.6)

To summarize, by plugging (5.5) into (5.3) and (5.1) we have constructed a
threshold expansion of $f_{J}(s)$ and ${\rm Im}f_{J}(s)$ that automatically
solves elastic unitarity for any real coefficients.

Note that the most singular possible threshold behavior is completely fixed by
elastic unitarity, with no free coefficient. It comes from $J=0$ and
corresponds to setting $b_{0}(\sigma_{s})=0$ in (5.3), so that the partial
wave is dominated by the universal term

${\rm Im}f_{0}(s)={2^{5-d}m^{4-d}\over\sigma_{s}^{{d-3\over
2}}}\times\left\\{\begin{array}[]{ll}1+\dots&\qquad d\text{ even}\\\
{\pi^{2}\over\log^{2}\sigma_{s}+\pi^{2}}+\dots&\qquad d\text{
odd}\end{array}\right.\ .$ (5.7)

In particular, this implies that the spin zero scattering length is infinite.
In $d=3$ the threshold behavior (5.7) is realized in massive $\phi^{4}$ theory
[63]. In $d=4$ the asymptotic behavior (5.7) appeared in the study of coupling
maximization in [15] and corresponds to having a bound state at
$s=4m^{2}$.222222By tuning the parameters of the theory it could be possible
to reach this situation in QCD as well. We thank Mattia Bruno and Maxwell
Hansen for the discussion. For $J\geq 2\geq J_{0}(s)$ the singular behavior
(5.7) does not occur due to (2.54).

Below we will also be interested in the case of finite ${\rm Im}f_{0}(s)$. In
this case, the spin zero scattering length is finite, namely
$b_{0}(s)={m^{d-4}\over a_{0}}$, and we get the following leading behavior of
${\rm Im}f_{0}(s)$

$\displaystyle{\rm Im}f_{0}(s)=2^{d-5}m^{4-d}a_{0}^{2}\,\sigma_{s}^{{d-3\over
2}}+\dots\ ,\qquad d>3\ .$ (5.8)

As a final remark, note that the analysis of this section can also be
generalized to non-integer spin, see appendix F.

###  5.2 Threshold Expansion of Discontinuity

For our purposes we will be interested in a closely related expansion, that of
the discontinuity of the amplitude $T_{s}(s,t)$ for $t>4m^{2}$ and
$\sigma_{s}\to 0$. As we will see, the Froissart-Gribov formula (2.53) relates
this expansion to the large $J$ behavior of the partial wave coefficients.

We proceed by considering the $s$-channel partial wave expansion for the
discontinuity of the amplitude

$\displaystyle T_{s}(s,t)=n_{J=0}^{(d)}{\rm
Im}f_{0}(s)+\sum_{J=2}^{\infty}n_{J}^{(d)}{\rm
Im}f_{J}(s)P_{J}^{(d)}\left(1+{2t\over s-4m^{2}}\right).$ (5.9)

For $s$ close to $4m^{2}$, the sum over $J$ above converges inside the large
$s$-channel Lehmann ellipse which is for $t<{16m^{2}s\over s-4m^{2}}$. Given
some fixed $t$, and considering the limit $s\to 4m^{2}$ we stay within the
convergence region.

We can now plug (5.1) into (5.9) and perform the threshold expansion under the
sum. We get

$\displaystyle T_{s}(s,t)=n_{0}^{(d)}{\rm
Im}f_{0}(s)+n_{2}^{(d)}a_{2}^{2}m^{4-d}\left({t\over
m^{2}}\right)^{2}{2^{d-7}(d-1)\over(d-2)}\sigma_{s}^{2+{d-3\over 2}}+\dots\ ,$
(5.10)

where we can systematically expand $T_{s}(s,t)$ using the threshold expansion
of partial waves (5.5) which is based on elastic unitarity. Note that the
expansion parameter in even dimensions is simply $\sigma_{s}$, whereas in odd
dimensions we have in addition powers of $\log\sigma_{s}$, as well as inverse
powers ${1\over\log\sigma_{s}}$, which we do not write here explicitly.

Let us emphasize that in the argument above it was absolutely crucial to
consider the discnontinuity of the amplitude $T_{s}(s,t)$ and not the
amplitude $T(s,t)$ itself. Indeed, if we were to try repeating the argument
above for $T(s,t)$ itself, we would run into the following problem. In (5.9)
we have partial waves $f_{J}(s)\sim(s-4m^{2})^{J}$ and Legendre polynomials
behaving as $P_{J}^{(d)}\left(1+{2t\over s-4m^{2}}\right)\sim(s-4m^{2})^{-J}$.
Therefore the expansion for $s\to 4m^{2}$ requires re-summation of partial
waves of all spins which is beyond our control.

The conclusion is that $T_{s}(s,t)$ admits a systematic threshold expansion in
terms of the solution to elastic unitarity in the $s$-channel. Moreover, the
contribution of higher spin partial waves are suppressed by an additional
factor of $\sigma_{s}^{J}$. We expect the threshold expansion converges as
long as we stay below the leading Landau curve, namely for $t<{16m^{2}s\over
s-4m^{2}}$. Luckily for us, it is the discontinuity of the amplitude and not
the amplitude itself that enters the Froissart-Gribov inversion formula.
Therefore the results of this section can be readily put to use.

###  5.3 Threshold Expansion for Double Spectral Density

We now turn to the study the double spectral density (3.7) close to the
leading Landau curve. We consider the Mandelstam equation (3.9) that is
repeated here for convenience

$\displaystyle\rho(s,t)$ $\displaystyle={(s-4m^{2})^{d-3\over 2}\over
4\pi^{2}(4\pi)^{d-2}\sqrt{s}}\int\limits_{z_{1}}^{\infty}d\eta^{\prime}\int\limits_{z_{1}}^{\infty}d\eta^{\prime\prime}\,{\cal
T}_{t}^{(+)}(s,\eta^{\prime})\,{\cal T}_{t}^{(-)}(s,\eta^{\prime\prime})\,{\rm
Disc}_{z}K(z,\eta^{\prime},\eta^{\prime\prime})\ ,$ (5.11)

where $4m^{2}<s<16m^{2}$. The crucial observation is that when we are very
close to the leading Landau curve $t={16m^{2}s\over s-4m^{2}}$, the integral
in (5.11) is again controlled by the threshold expansion of ${\cal
T}^{(+)}_{t}(s,\eta^{\prime})$ and ${\cal
T}_{t}^{(-)}(s,\eta^{\prime\prime})$. The reason for this is that the region
of integration starts at the threshold, $\eta^{\prime},\eta^{\prime\prime}\geq
z_{1}$, and ends at the boundary of support of the kernel,
$\eta_{+}(\eta^{\prime},\eta^{\prime\prime})\leq z$. As $t$ approach the
leading Landau curve, this region shrinks close to the threshold for both
$\eta^{\prime}$ and $\eta^{\prime\prime}$ integrals, see figure 3.

A convenient small parameter in the problem is the dimensionless distance from
the Landau curve, see (3.11)

$\delta z\equiv z-(2z_{1}^{2}-1)\propto t-{16m^{2}s\over s-4m^{2}}\ .$ (5.12)

We can now plug the threshold expansion of the discontinuity (5.5) into (5.11)
and expand the result in powers of $\delta z$. At any given order in that
expansion only finite number of terms in the threshold expansion of the
discontinuity contribute. For even spacetime dimension, the relevant integrals
that appears in the expansion are

$\displaystyle\tilde{I}_{n_{1},n_{2}}^{(d)}(z)\equiv\int\limits_{z_{1}}^{\infty}d\eta^{\prime}\int\limits_{z_{1}}^{\infty}d\eta^{\prime\prime}\sigma_{t(\eta^{\prime})}^{n_{1}-{d-3\over
2}}\sigma_{t(\eta^{\prime\prime})}^{n_{2}-{d-3\over 2}}{\rm
Disc}_{z}K(z,\eta^{\prime},\eta^{\prime\prime})\ .$ (5.13)

To leading order in $\delta z$ this integral is given by232323To derive (5.15)
it is useful to do the following change of integration variables close to
$\delta z=0$ $\displaystyle\eta^{\prime}$ $\displaystyle=z_{1}+{\delta z\over
2z_{1}}\alpha x\ ,\qquad\eta^{\prime\prime}=z_{1}+{\delta z\over
2z_{1}}\alpha(1-x)\ ,$ (5.14) where the integration in (5.11) for $\delta z\ll
1$ is restricted to $0\leq x$ and $\alpha\leq 1$.

$\displaystyle\tilde{I}_{n_{1},n_{2}}^{(d)}(z)=2{\pi^{{d+1\over
2}}(z_{1}-1)^{d-3-n_{1}-n_{2}}\over
z_{1}^{n_{1}+n_{2}+1}(z_{1}^{2}-1)^{1/2}}\left({\delta z\over
2}\right)^{n_{1}+n_{2}+{5-d\over 2}}{\Gamma(n_{1}+{5-d\over
2})\Gamma(n_{2}+{5-d\over 2})\over\Gamma(n_{1}+n_{2}+{7-d\over
2})}\left(1+{\cal O}(\delta z)\right)\ .$ (5.15)

For $n_{1},n_{2}\leq{d-5\over 2}$ the integral (5.13) diverges, but this
apparent divergence is not physical. Namely, the integral in (5.15) originates
from the contour integral that wraps around
$\eta^{\prime},\eta^{\prime\prime}=z_{1}$. Therefore, the integral can be
safely deformed to a keyhole contour around the dangerous region, see figure
9. The result of the keyhole integration is equivalent to analytically
continuing (5.15) in $m,n$ (as long as the final result is finite).

Figure 9: The discontinuity of the Mandelstam equation (3.9) has been obtained
by deforming the contour of integration in the finite integral (3.8). Hence,
the result cannot diverge. This means that an apparent divergence in (5.13)
that comes from the region of integration close to the threshold should be
understood as a finite keyhole contour integral plotted here.

We see that indeed, higher powers in the threshold expansion result in higher
powers of $\delta z$. For odd spacetime dimension, due to the presence of
${1\over\log^{2}\sigma_{s}+\pi^{2}}$ in (5.1), the relevant integral is more
complicated, but the power suppression in $\delta z$ is of course the same. We
discuss it more in appendix H.2.

The most singular possible behavior close to the Landau curve comes from the
corresponding universal threshold behavior of ${\rm Im}\,f_{0}(t)$ in (5.7)
and corresponds to $n_{1}=n_{2}=0$ in (5.13). In that case we find

$\displaystyle\rho(s,t)=(2^{5-d}n_{0}^{(d)})^{2}\frac{(d-3)\pi^{\frac{3-d}{2}}m^{4-d}(z_{1}-1)^{\frac{d-3}{2}}}{32(5-d)\cos\left(\frac{\pi
d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)z_{1}(z_{1}+1)}\delta
z^{\frac{5-d}{2}}+\dots\ ,\qquad d\text{ even}\ .$ (5.16)

On the other hand, for the case where the spin zero scattering length is
finite (5.8), we have $n_{1}=n_{2}=d-3$ in (5.13) and correspondingly the
approach to the Landau curve is much softer

$\displaystyle\rho(s,t)=\frac{a_{0}^{4}\pi^{\frac{1-d}{2}}m^{4-d}(z_{1}-1)^{\frac{9-3d}{2}}z_{1}^{5-2d}\Gamma\left(\frac{d-1}{2}\right)^{2}(n_{0}^{(d)})^{2}}{512(z_{1}+1)\Gamma\left(\frac{3d-5}{2}\right)}\delta
z^{\frac{3d-7}{2}}+\dots\ .$ (5.17)

###  5.4 Radius of Convergence

Let us briefly discuss the radius of convergence of the threshold expansion
introduced above. As usual the radius of convergence is controlled by the
analytic properties of the function at hand.

Consider first $f_{J}(s)$. It has a normal threshold cut starting at
$s=4m^{2}$ and multi-particle cuts at $s\geq 16m^{2}$, as well as the
$u$-channel cut for $s\leq 0$. In addition to that we can have bound states
and resonances (which correspond to poles on the second sheet). In a given
theory, we expect the singularity that is the closest to $s=4m^{2}$ to control
the convergence radius of the threshold expansion of $f_{J}(s)$. In this paper
we assume for simplicity that there are no bound states, however it should be
easy to include them in the analysis.

Consider next $T_{s}(s,t)$. In this case, the partial wave expansion (5.9)
converges below the first Landau curve. For $4m^{2}<s<16m^{2}$ and
$t>16m^{2}$, that is the regime of $\sigma_{s}<{16m^{2}\over t-16m^{2}}$.
Moreover, if we are to first expand each ${\rm Im}f_{J}(s)$ close to the
threshold under the sum, we will have to argue that the sum over $J$ and the
threshold expansion commute. It is not clear to us how to do it. Instead below
we adopt a different approach. We separate a few low spin partial waves in
(5.9) and apply the threshold expansion to them. We then bound the sum over
spins without doing the threshold expansion under the sum over spins.

##  6 Large $J$ Expansion

In this section we map the threshold expansion developed in the previous
section to the large $J$ expansion of the partial waves coefficients
$f_{J}(s)$. 242424In $d=4$ this was pioneered by Dragt [64], see also [65],
and further developed by A. W. Martin in [66]. Here we consider a slightly
more general form of the amplitude, as well as generalize the analysis to any
spacetime dimension.

The basic idea is very simple. The Froissart-Gribov formula (2.53) directly
maps the threshold expansion of the discontinuity of the amplitude (5.10) to
the large $J$ expansion of the partial wave coefficients in the crossed
channel.252525This is analogous to the analytic bootstrap in CFTs [67, 68,
69]. Interestingly, once combined with crossing, the Froissart-Gribov formula
allows us to map the low energy threshold expansion, that is dominated by the
low spins, to the large spin behavior of the partial wave coefficients for any
energy and in the same channel. In particular, it automatically predicts the
amount of inelasticity at large $J$.

In this section we will use the results for the threshold expansion of the
double spectral density and the discontinuity of the amplitude in the crossed
$t$-channel as opposed to the $s$-channel threshold expansion of the previous
section. We hope this will not cause any confusion. In sections 6.1 and 6.2 we
work out the leading large $J$ expressions for $f_{J}(s)$ and ${\rm
Im}f_{J}(s)$ correspondingly. In section 6.3 we compute the exact contribution
of a given term in the threshold expansion of the amplitude to the partial
waves. This includes infinitely many ${1\over J}$ corrections to the results
of 6.1 and 6.2.

Note that in this section we only concern ourselves with the threshold
expansion of the amplitude close to the two-particle threshold. Of course, in
addition there are contributions from multi-particle thresholds. These lead to
further nonperturbative in ${1\over J}$ corrections to the results of this
section and are potentially important if we would like to discuss partial
waves at finite $J$ which is the subject of the next section.

###  6.1 Large Spin Expansion of $f_{J}(s)$

We start by analyzing the Froissart-Gribov integral (2.53) in the large spin
limit $J\gg 1$. For convinience, we quoting the integral here

$f_{J}(s)={2\,{\cal
N}_{d}\over\pi}\int\limits_{z_{1}}^{\infty}dz\,(z^{2}-1)^{{d-4\over
2}}Q^{(d)}_{J}(z)T_{t}(s,t(z))\ ,\qquad{\rm Re}[J]>J_{0}(s)\ .$ (6.1)

The large $J$ behavior of $Q_{J}^{(d)}(z)$ is given by262626An efficient way
to systematically expand $Q_{J}^{(d)}(z)$ to an arbitrary order in the large
$J$, fixed $z$ expansion is to start from its representation in terms of the
hypergeometric function [70] $\displaystyle
Q^{(d)}_{J}(z)=2^{d-4}\sqrt{\pi}{\Gamma\left(\tfrac{d-2}{2}\right)\Gamma(J+1)\over\Gamma\left(J+{d-1\over
2}\right)}{\lambda(z)^{-J}\over(\lambda(z)^{2}-1)^{d-3\over
2}}\,_{2}F_{1}\left(1-\tfrac{d-3}{2},\tfrac{d-3}{2};J+\tfrac{d-1}{2};\tfrac{1}{1-\lambda(z)^{2}}\right)\
,$ (6.2) and then use the series representation for the hypergeometric
function as
$\,{}_{2}F_{1}(a,b;c;x)=\sum_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}}{k!(c)_{k}}x^{k}$.
The spin, $J$, only enters in the Pochhammer $(c)_{k}=(J+{d-1\over 2})_{k}$
and therefore the $k$’th term only contributes at order $1/J^{k}$ and higher.
Note that for $d=3$ there are no ${1\over J}$ corrections to (6.3). More
generally, in odd $d$ the large $J$ properties of $Q^{(d)}_{J}(z)$ can be made
manifest, see appendix C.

$\displaystyle
Q^{(d)}_{J}(z)=2^{d-4}\sqrt{\pi}\,{\Gamma\left(\tfrac{d-2}{2}\right)\over
J^{{d-3\over 2}}}{\lambda(z)^{-J}\over(\lambda(z)^{2}-1)^{d-3\over
2}}\left(1+{\cal O}(1/J)\right)\ ,$ (6.3)

where

$\lambda(z)\equiv z+\sqrt{z^{2}-1}=e^{\theta}\qquad\text{for}\qquad|z|>1\
,\quad{\rm Re}\,\theta>0\ .$ (6.4)

Crucially, $Q^{(d)}_{J}(z)$ decays very fast for fixed $z>1$ at large $J$. The
integral is therefore dominated by the region close to the threshold $z_{1}$.
Explicitly, for $J\gg 1$ we have

$\displaystyle Q^{(d)}_{J}\left(z_{*}+{\sqrt{z_{*}^{2}-1}\delta z\over
J}\right)=2^{d-4}\sqrt{\pi}\,{\Gamma\left(\tfrac{d-2}{2}\right)\over
J^{{d-3\over 2}}}{\lambda(z_{*})^{-J}\over(\lambda(z_{*})^{2}-1)^{d-3\over
2}}e^{-\delta z}\left(1+{\cal O}(1/J)\right)\ ,$ (6.5)

and therefore the integral in (6.1) is controlled by the region of the size
$\sim{1\over J}$ close to $z_{*}=z_{1}$. In that way, the large $J$ behavior
is controlled by the threshold expansion of $T_{t}(s,t(z))$ established in the
previous section in a manifest way.

Next, we explicitly plug the threshold expansion into (6.1) and compute the
leading large $J$ behavior of the partial wave coefficients. For that aim, it
is convenient to express the integral as follows

$\displaystyle f_{J}(s)\equiv{{\cal
N}_{d}\over\sqrt{\pi}}{\Gamma\left(\tfrac{d-2}{2}\right)\over J^{{d-1\over
2}}}{(\lambda(z_{1})^{2}-1)^{{d-3\over
2}}\over\lambda(z_{1})^{J+d-3}}\hat{f}_{J}(s)\
,\qquad\hat{f}_{J}(s)=\int\limits_{0}^{\infty}d\delta z\,T_{t}(s,t(\delta
z,J))\,e^{-\delta z}\left(1+{\cal O}(1/J)\right)\ ,$ (6.6)

where we have272727Note that $\delta z$ in (6.7) is different from the one
used in the previous section.

$\displaystyle\sigma_{t}={1\over J}\left({z_{1}+1\over
z_{1}-1}\right)^{1/2}\delta z\ .$ (6.7)

Note that in deriving (6.6) we took into account the integration measure
$(z^{2}-1)^{{d-4\over 2}}$ and the Jacobian from switching the integration
variable to $\delta z$.

After performing the crossing transformation $s\leftrightarrow t$, we can now
plug the leading threshold expansion expressions from the previous section
(5.10) to get the $J\gg 1$ limit of $f_{J}(s)$. Let us start with the
universal and most singular case (5.7). For this case we get

$\hat{f}_{J}(s)=2^{5-d}n_{0}^{(d)}m^{4-d}J^{{d-3\over 2}}\left({z_{1}-1\over
z_{1}+1}\right)^{{d-3\over 4}}\left(1+{\cal
O}(1/J)\right)\times\left\\{\begin{array}[]{ll}\pi^{2}g_{{d-3\over
2}}\left(\log J\sqrt{{z_{1}-1\over z_{1}+1}}\right)&\qquad d\text{ odd}\\\
\quad\Gamma\left(\tfrac{5-d}{2}\right)&\qquad d\text{ even}\end{array}\right.\
,$ (6.8)

where the explicit form of the slowly-varying function $g_{{d-3\over 2}}(\log
x)$ is computed in appendix H.

If, on the other hand, we consider the situation with a finite spin zero
scattering length (5.8), we get

$\displaystyle\hat{f}_{J}(s)$
$\displaystyle=2^{d-5}n_{0}^{(d)}a_{0}^{2}\,\Gamma\left(\tfrac{d-1}{2}\right)m^{4-d}{1\over
J^{{d-3\over 2}}}\left({z_{1}+1\over z_{1}-1}\right)^{{d-3\over 4}}\
\left(1+{\cal O}(1/J)\right).$ (6.9)

It is clear that we can systematically include ${1\over J}$ correction in this
expansion. Note that in all cases, the leading large $J$ result for $f_{J}(s)$
is exponentially small and purely real. In the next section we compute the
leading $J$ contribution to ${\rm Im}f_{J}(s)$.

###  6.2 Large Spin Expansion of ${\rm Im}f_{J}(s)$

We now repeat the analogues large $J$ expansion for ${\rm Im}f_{J}(s)$ in
terms of the threshold expansion of the double spectral density. In this case,
the Froissart-Gribov formula can be written as

$\displaystyle{\rm Im}f_{J}(s)$ $\displaystyle={2{\cal
N}_{d}\over\pi}\int\limits_{z_{1}}^{\infty}dz(z^{2}-1)^{{d-4\over
2}}Q^{(d)}_{J}(z)\rho(s,t(z))\ ,\qquad{\rm Re}[J]>J_{0}(s)\ .$ (6.10)

It is instructive to separate $s$ into three regions, $4m^{2}<s<16m^{2}$,
${64\over 3}m^{2}>s>16m^{2}$ and $s>{64\over 3}m^{2}$.

#### The Elastic Region $4m^{2}<s<16m^{2}$

In this elastic strip the double spectral density is zero for
$t<{16m^{2}\,s\over s-4m^{2}}$ or, equivalently, for $z<2z_{1}^{2}-1$, see
figure 4. The large $J$ limit of ${\rm Im}f_{J}(s)$ is thus dominated by the
expansion of double spectral density close to the leading Landau curve which
we worked out in section 5.3. Hence, we can simply use (6.5) with
$z_{*}=2z_{1}^{2}-1$, otherwise the consideration is identical to the one in
the previous subsection. Explicitly, we have

$\displaystyle{\rm Im}f_{J}(s)$ $\displaystyle={{\cal
N}_{d}\over\sqrt{\pi}}{\Gamma\left(\tfrac{d-2}{2}\right)\over J^{{d-1\over
2}}}{([\lambda(2z_{1}^{2}-1)]^{2}-1)^{{d-3\over
2}}\over[\lambda(2z_{1}^{2}-1)]^{J+d-3}}{\rm Im}\hat{f}_{J}(s)\ ,\qquad
4m^{2}<s<16m^{2}\ ,$ $\displaystyle{\rm Im}\hat{f}_{J}(s)$
$\displaystyle=\int\limits_{0}^{\infty}d\delta ze^{-\delta z}\rho(s,t(\delta
z,J))\left(1+{\cal O}(1/J)\right)\ ,$ (6.11)

where

$\displaystyle z=(2z_{1}^{2}-1)+{2z_{1}\sqrt{z_{1}^{2}-1}\over J}\delta z\
,\qquad t(\delta z,J)=8m^{2}\left(z_{1}+1+{z_{1}\over J}\left({z_{1}+1\over
z_{1}-1}\right)^{1/2}\delta z\right)\ .$ (6.12)

Therefore, upon making a substitute $\delta z\to{2z_{1}\sqrt{z_{1}^{2}-1}\over
J}\delta z$ in the formulas of section 5.3, we can directly plug them into
(6.2). The resulting large $J$ behavior with the universal threshold expansion
(5.7) takes the following form

$\displaystyle{\rm Im}\hat{f}_{J}(s)=2^{-5{(d-3)\over
2}}\left(n_{0}^{(d)}\Gamma\left(\tfrac{5-d}{2}\right)\right)^{2}m^{4-d}\pi^{{1-d\over
2}}J^{{d-5\over 2}}z_{1}^{{3-d\over 2}}\left({z_{1}-1\over
z_{1}+1}\right)^{d-1\over 4}\left(1+{\cal O}(1/J)\right)\ ,\qquad d\text{
even}\ ,$ (6.13)

where again the case of $d$ odd should be considered separately using the
results of appendix H.

Turning to the case with a finite spin zero scattering length (5.8), we get

$\displaystyle{\rm Im}\hat{f}_{J}(s)$ $\displaystyle={2^{{3d-25\over
2}}\left(n_{0}^{(d)}a_{0}^{2}\Gamma\left(\tfrac{d-1}{2}\right)\right)^{2}\over\pi^{d-1\over
2}m^{d-4}J^{{3d-7\over 2}}}{1\over z_{1}^{d-3\over 2}}\left({z_{1}+1\over
z_{1}-1}\right)^{{3d-11\over 4}}\left(1+{\cal O}(1/J)\right)\ .$ (6.14)

An alternative way to arrive at (6.13) and (6.14) is to start from the large
$J$ expansion of $f_{J}(s)$ discussed in the previous section (6.8) and use
elastic unitarity. Indeed, one can check that (6.8) and (6.13), (6.14) are
consistent with elastic unitarity

$\displaystyle\log{2{\rm Im}f_{J}(s)\over{(s-4m^{2})^{d-3\over
2}\over\sqrt{s}}|f_{J}(s)|^{2}}=0\ ,\qquad 4m^{2}<s<16m^{2}\ .$ (6.15)

#### The Inelastic Region $16m^{2}<s<{64\over 3}m^{2}$

This regime is in the multi-particle part of figure 4. To analyze it one
should first identify the leading Landau curve in that segment, which is
beyond the scope of this paper.

#### The Inelastic Region $s>{64\over 3}m^{2}$

As $s$ increases passed ${64m^{2}\over 3}$, we know from crossing that the
leading Landau curve is the one of the dual channel (the blue curve in figure
4). A remarkable consequence of this fact is that crossing allows us to
measure the non-zero inelasticity at large spin $J$.

As before the leading large $J$ behavior of ${\rm Im}f_{J}(s)$ is controlled
by the leading Landau curve in the relevant kinematics. For $s>{64\over
3}m^{2}$ the leading Landau curve is at $t=t_{1}(s)={4m^{2}s\over s-16m^{2}}$,
or equivalently at

$\displaystyle z=\tilde{z}_{1}\equiv{4-3z_{1}+z_{1}^{2}\over 5-3z_{1}}\ .$
(6.16)

See blue curve on figure 4.

In this case we get that the Froissart-Gribov integral takes the same form as
in (6.10) with

$z_{1}\to\tilde{z}_{1}\ ,\qquad t(\delta
z,J)=4m^{2}\left({\tilde{z}_{1}-1\over z_{1}-1}+{1\over
J}{\sqrt{\tilde{z}_{1}^{2}-1}\over z_{1}+1}\delta
z\right)\qquad\text{and}\qquad s>{64\over 3}m^{2}\ ,$ (6.17)

where $t(\delta z,J)$ is a definition of $\delta z$ that we will use below.

The double spectral density close to $\tilde{z}_{1}$ is given by the results
of section 5.3 upon application of crossing $s\leftrightarrow t$.282828By
taking $s\to\infty$ with $x={m^{3}\over s^{3/2}}J$ kept fixed in (6.2), (6.17)
one gets the $d$-dimensional version of the Haan-Mütter scaling law [71, 72].
Considering the amplitudes with the universal threshold expansion (5.7) we get

$\displaystyle{\rm Im}\hat{f}_{J}(s)$ $\displaystyle={J^{{d-5\over
2}}\left(n_{0}^{(d)}\Gamma\left(\frac{5-d}{2}\right)\right)^{2}\over
8(2m)^{d-4}(\tilde{z}_{1}^{2}-1)^{{d-5\over 4}}\pi^{{d-1\over
2}}}{(z_{1}-1)^{{3d-13\over
2}}\over(1+z_{1})(3-z_{1})(5-3z_{1})^{d-6}}\left(1+{\cal O}(1/J)\right)\
,\qquad d\text{ even}\ .$ (6.18)

In odd $d$ the situation is more complicated due to the logarithmic nature of
the threshold singularity. While the leading power dependence on $J$ can be
easily computed, we do not give an explicit form for the dependence on $\log
J$ that multiplies the leading power.292929It should be possible to use the
integral form for $\rho(s,t)$ given in appendix H.2 to efficiently evaluate
the integral numerically for given $J$ but we do not pursue it here.

For the case of finite spin zero scattering length (5.8), we get in any $d$

$\displaystyle{\rm Im}\hat{f}_{J}(s)$
$\displaystyle={\left(n_{0}^{(d)}a_{0}^{2}\,\Gamma\left({d-1\over
2}\right)\right)^{2}\over 2^{d+7}m^{d-4}\pi^{d-1\over
2}}{(\tilde{z}_{1}^{2}-1)^{{3d-7\over 4}}\over J^{{3d-7\over
2}}}{(5-3z_{1})^{3(d-2)}(z_{1}-1)^{{11-5d\over
2}}\over(1+z_{1})(3-z_{1})^{2d-5}}\left(1+{\cal O}(1/J)\right)\ .$ (6.19)

We can use the results above for the leading large $J$ behavior of ${\rm
Im}\hat{f}_{J}(s)$ for $s>{64\over 3}m^{2}$ to estimate the amount of
inelasticity that exists at given $s$ and $J$. Recall that if the scattering
were purely elastic partial waves would satisfy $2{\rm
Im}f_{J}(s)={(s-4m^{2})^{d-3\over 2}\over\sqrt{s}}|f_{J}(s)|^{2}$, see (6.15).

Using the explicit results of this section we conclude that elastic unitarity
together with crossing lead to the universal inelasticity ratio at large $J$

$\displaystyle r_{J}(s)\equiv\log{2{\rm Im}f_{J}(s)\over{(s-4m^{2})^{d-3\over
2}\over\sqrt{s}}|f_{J}(s)|^{2}}=J\log{\lambda(z_{1})^{2}\over\lambda(\tilde{z}_{1})}+O(1)\
,\qquad s>{64m^{2}\over 3}\ .$ (6.20)

In figure 10 we plotted $r_{J}(s)$ as a function of $s/m^{2}$ for fixed $J$.
We see that $r_{J}(s)$ approaches $0$ for $s=20m^{2}$ and $s=\infty$. It
acquires its maximal value of about $0.47J$ at $s=40m^{2}$.

Figure 10: The large spin inelastisity ratio $r_{J}(s)$ in (6.20) plotted for
$s>{64\over 3}m^{2}$. The maximum value of the ratio is achieved at
$s=40m^{2}$ for which we get $r_{J}(40m^{2})\simeq 0.473J(1+{\cal O}(1/J))$.

###  6.3 Threshold Expansion in the $J$-space

In this section we evaluate exactly the contribution of a given term in the
threshold expansion to the partial waves. This generalizes the results above
which correspond to the large $J$ limit of the exact formulas that we present
in the current section. For simplicity let us first focus on $d$ even, when
the threshold behavior is of the square-root type.

Consider the threshold expansion of the discontinuity of the amplitude. In the
regime where it converges close to the threshold we can write

$\displaystyle T_{t}(s,t(z))=\sum_{n=0}^{\infty}c_{n}(s)\left({z-z_{1}\over
z_{1}-1}\right)^{n+{3-d\over
2}}=\sum_{n=0}^{\infty}c_{n}(s)\,\sigma_{t}^{n+{3-d\over 2}}\ .$ (6.21)

In practice we truncate this expansion at any desired order and plug it into
the Froissart-Gribov formula. Note that the Froissart-Gribov integral goes all
the way to infinite $z$ where the expansion (6.21) is no longer valid.
However, that regime of integration is exponentially suppressed in $J$ and
therefore only affects the nonperturbative corrections at large $J$. It turns
out that the relevant integral can be done exactly. It is given by

$\displaystyle I_{n,J}^{(d)}(z_{1})$ $\displaystyle\equiv{2{\cal
N}_{d}\over\pi}\int\limits_{z_{1}}^{\infty}\\!dz(z^{2}-1)^{{d-4\over
2}}\sigma_{t(z)}^{n+{3-d\over 2}}Q_{J}^{(d)}(z)={2^{{5\over
2}}\Gamma(n+{5-d\over 2})\over 2^{n}(8\pi)^{d\over 2}\Gamma({3\over
2}+n)}{(z_{1}+1)^{n+{1\over 2}}\over(z_{1}-1)^{{2-d\over 2}}}Q_{J-n+{d-5\over
2}}^{(2n+5)}(z_{1})\ .$ (6.22)

This formula is derived in appendix G and holds for arbitrary $J$, $d$ and
$n$, not necessarily integer. If we expand both sides at large $J$, we
reproduce the formulas of the previous section. In this way, we arrive at the
explicit expression for the large $J$ expansion of $f_{J}(s)$ in terms of the
coefficients of the threshold expansion in the crossed channel

$\displaystyle
f_{J}(s)=\sum_{n=0}^{\infty}c_{n}I_{n,J}^{(d)}(z_{1})+\text{[nonperturbative]}\
.$ (6.23)

Using this expression and the large $J$ expansion of $I_{n,J}^{(d)}(z_{1})$

$I_{n,J}^{(d)}(z_{1})={1\over J^{n+1}}\times{2\,\Gamma(n+{5-d\over
2})\over(8\pi)^{d-1\over 2}\lambda(z_{1})^{J+{d-3\over
2}}}{(z_{1}+1)^{n+{1\over 2}}\over(z_{1}-1)^{2-d\over
2}}\left({2\lambda(z_{1})\over\lambda^{2}(z_{1})-1}\right)^{n+1}\left(1+{\cal
O}(1/J)\right)\ ,$ (6.24)

we see that higher orders in the sum are more suppressed in $J$. Hence, one
can use this representation to explicitly and systematically obtain all the
coefficients in the large $J$ expansion of $f_{J}(s)$, the leading $1/J$
result of course coincides with the previous analysis.

Similarly, we can write down the threshold expansion formula for $\rho(s,t)$.
Based on the previous discussion we have

$\displaystyle\rho(s,t(z))={1\over(z-(2z_{1}^{2}-1))^{{d-5\over
2}}}\sum_{m=0}^{\infty}d_{m}(s)(z-(2z_{1}^{2}-1))^{m}\ .$ (6.25)

The problem of finding $d_{m}(s)$ becomes completely algebraic after we note
that (6.25) implies

$\displaystyle{\rm Im}f_{J}(s)$
$\displaystyle=\sum_{m=0}^{\infty}d_{m}(s)I_{m+1,J}^{(d)}(2z_{1}^{2}-1)+\text{[nonperturbative]}\
,$ (6.26)

where we again performed the Froissart-Gribov integral exactly.

Elastic unitarity (6.15) relates the two expansions, (6.23) and (6.26). It
then allows one to express the $d_{m}(s)$’s in terms of the $c_{n}(s)$’s. This
is possible because the $d_{m}(s)$’s do not depend on $J$. Equivalently, we
can use (5.11) to directly map the threshold expansion of $T_{t}(s,t)$ to that
of $\rho(s,t)$ near the Landau curve. The details of this expansion are
presented in appendix G.

##  7 Finite $J$ and Finite $s$

In practice, one is interested in making statements about partial waves at
some finite (but potentially large) spin $J$ and finite energy $s$. Our
ability to make such statements crucially depends on our ability to estimate