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JHEP11(2018)015

Published for SISSA by Springer

Received: April 26, 2018

Revised: September 7, 2018

Accepted: October 24, 2018

Published: November 6, 2018

Higher-point positivity

Venkatesa Chandrasekaran, Grant N. Remmen and Arvin Shahbazi-Moghaddam

Center for Theoretical Physics and Department of Physics, University of California,

Berkeley, CA 94720, U.S.A.

Lawrence Berkeley National Laboratory,

Berkeley, CA 94720, U.S.A.

E-mail: ven chandrasekaran@berkeley.edu, grant.remmen@berkeley.edu,

arvinshm@berkeley.edu

Abstract: We consider the extension of techniques for bounding higher-dimension oper-

ators in quantum eﬀective ﬁeld theories to higher-point operators. Working in the context

of theories polynomial in X = (∂φ)

, we examine how the techniques of bounding such

operators based on causality, analyticity of scattering amplitudes, and unitarity of the

spectral representation are all modiﬁed for operators beyond (∂φ)

. Under weak-coupling

assumptions that we clarify, we show using all three methods that in theories in which the

coeﬃcient λ

of the X

term for some n is larger than the other terms in units of the

cutoﬀ, λ

must be positive (respectively, negative) for n even (odd), in mostly-plus metric

signature. Along the way, we present a ﬁrst-principles derivation of the propagator numer-

ator for all massive higher-spin bosons in arbitrary dimension. We remark on subtleties

and challenges of bounding P (X) theories in greater generality. Finally, we examine the

connections among energy conditions, causality, stability, and the involution condition on

the Legendre transform relating the Lagrangian and Hamiltonian.

Keywords: Eﬀective Field Theories, Scattering Amplitudes

ArXiv ePrint: 1804.03153

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP11(2018)015

JHEP11(2018)015

The simplest positivity bound on eﬀective theories applies to the coupling of the (∂φ)

operator. In a massless theory of a real scalar φ with a shift symmetry, the ﬁrst higher-

dimension operator that one can add to the kinetic term −∂

φ∂

φ/2 is the operator

(∂φ)

= ∂

φ∂

φ. (1.1)

In a theory given by −

(∂φ)

+ λ(∂φ)

, the forward amplitude for two-to-two φ scattering

is A(s) = 16λs

. A standard dispersion relation argument [1] then relates the coeﬃcient

of s

in this forward amplitude at low energies to an integral over the cross section at high

energies, which physically must be positive. That is, analyticity of scattering amplitudes

guarantees that λ is positive. Similarly, one can compute the speed of propagation of φ

perturbations in a nonzero φ background: one ﬁnds that subluminality requires λ > 0

and that if λ < 0 it is straightforward to build causal paradoxes involving superluminal

signaling between two bubbles of φ background with a relative boost. A litany of other

examples of analyticity and causality bounds focuses on similar four-point interactions,

though for more complicated theories and ﬁelds involving gauge bosons and gravitons.

In this paper, we explore a new direction in the space of positivity bounds: higher-

point operators. In particular, we will bound the P (X) theory, whose Lagrangian is simply

a polynomial in

X = ∂

φ∂

φ, (1.2)

which in the eﬀective ﬁeld theory we can write as

L = −

X +

∞

i=2

. (1.3)

A case of particular tractability is an nth-order P(X) theory, in which the λ

are very

small or zero for i < n for some n > 1, where n is the ﬁrst nonnegligible higher-order term

in the P(X) polynomial:

L = −

X +

∞

i=n

. (1.4)

We use a weak-coupling assumption from the ultraviolet to the infrared to guarantee a

well-deﬁned ~ counting at all energy scales, as in ref. [9], so that the vanishing of the

tree-level λ

for i < n is well deﬁned.

We will show that analyticity of scattering amplitudes and causality of signal propa-

gation imply the same positivity bound on the theory in eq. (1.4):

> 0 if n is even,

< 0 if n is odd.

(1.5)

We will also ﬁnd that eq. (1.5) comes about as a consequence of unitarity of quantum

mechanics in the context of spectral representations for a particular class of ultraviolet

completions. This bound represents progress for the program of constraining the allowed

We will use mostly-plus metric signature throughout.

– 2 –

JHEP11(2018)015

space of self-consistent low-energy eﬀective theories, constituting a generalization of the

well known (∂φ)

bound. Further, the formalism we develop along the way for applying

infrared consistency bounds to higher-point operators is useful in its own right.

Considering X

as the ﬁrst nonnegligible operator in the eﬀective ﬁeld theory can be

motivated physically in several diﬀerent ways. We can consider tree-level completions of the

operators through massive states coupling to (∂φ)

. If there is no coupling of massive

states to (∂φ)

for i < n, then the tree-level value of λ

vanishes for i < n. We can then

place the positivity bound in eq. (1.5) on λ

using the tree-level amplitude. Note that this

logic does not contradict the positivity bound on (∂φ)

in ref. [1], since λ

could still be

generated at loop level, though λ

from the tree-level completion would be parametrically

larger in units of the cutoﬀ.

Moreover, from the perspective of the eﬀective ﬁeld theory,

the higher-dimension operators in the nth-order P (X) theory in eq. (1.4) can be viewed as

a sector of a larger theory. For example, taking a complex scalar φ with a Z

symmetry

φ → e

2πim/n

φ for integer m, the allowed higher-dimension operators are of the form X

, and

for integer p, where

X = ∂

∗

∂

∗

and

X = ∂

φ∂

∗

. In particular, all

operators X

for i < n would be forbidden and the scattering of 2n φ particles at tree level

would occur only through the X

contact operator, just as in the nth-order P (X) theory

in eq. (1.4).

This paper is organized as follows. In section 2, we consider the application of ana-

lyticity bounds for higher-point amplitudes and derive our bound (1.5) on the nth-order

P (X) theory. Next, in section 3 we ﬁnd that the bound (1.5) also follows from demanding

the absence of causal paradoxes. In section 4 we consider a particular class of tree-level

completions and ﬁnd that the couplings obey eq. (1.5) as a consequence of unitarity of

the spectral representation. Along the way, we present an elegant derivation of the prop-

agator for higher-spin massive bosons in arbitrary spacetime dimension. We discuss the

obstacles, in the form of kinematic singularities, that preclude straightforward generaliza-

tion of some of these bounds to arbitrary (i.e., not strictly nth-order) P (X) theories in

section 5. In section 6 we show that there is a deep relationship between positivity bounds

and the involution property of the Legendre transform relating the Lagrangian and Hamil-

tonian formulations of the mechanics of the P (X) theory. We conclude and discuss future

directions in section 7.

2 Bounds from analyticity

In this section, we derive the bound in eq. (1.5) through a generalization of the dispersion

relation argument that has been previously applied to two-to-two scattering amplitudes [1].

We ﬁrst discuss formalism for general n-to-n particle scattering, before considering our

speciﬁc theory of interest and deriving the bounds.

2.1 The forward limit

Consider a general eﬀective ﬁeld theory for which one wishes to bound the couplings of

higher-dimension operators using analyticity of scattering amplitudes. Fundamentally, such

We assume a suﬃciently weak coupling that it is consistent to drop the lower-point operators that are

suppressed by loop factors, despite additional bounds coming from inelastic scattering [18].

– 3 –

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