Higher-Point Positivity

Venkatesa Chandrasekaran, Grant N. Remmen,

and Arvin Shahbazi-Moghaddam

[7mm] Center for Theoretical Physics and Department of Physics

University of California, Berkeley, CA 94720, USA and

Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA e-mail:,,


We consider the extension of techniques for bounding higher-dimension operators in quantum effective field theories to higher-point operators. Working in the context of theories polynomial in , 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 . Under weak-coupling assumptions that we clarify, we show using all three methods that in theories in which the coefficient of the term for some is larger than the other terms in units of the cutoff, must be positive (respectively, negative) for even (odd), in mostly-plus metric signature. Along the way, we present a first-principles derivation of the propagator numerator for all massive higher-spin bosons in arbitrary dimension. We remark on subtleties and challenges of bounding 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.

1 Introduction

A dramatic development in our knowledge of quantum field theory has been the discovery that not all effective field theories are consistent with ultraviolet completion in quantum gravity. Certain Lagrangians that one can write down possess pathologies that are a priori hidden, but that can be elucidated though careful consideration of consistency conditions that can be formulated in the infrared and that are thought to be obeyed by any reasonable ultraviolet completion. Such infrared conditions include analyticity of scattering amplitudes, quantum mechanical unitarity, and causality of particle propagation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], as well as self-consistency of black hole entropy in the context of the recent proof of the weak gravity conjecture [14]. Delineating the space of consistent low-energy effective field theories is of great current interest in the context of the swampland program [15, 16, 17], which seeks to characterize and bound in theory space the possible effective field theories amenable to ultraviolet completion in quantum gravity. Infrared requirements form a powerful set of tools, giving us rigorous positivity bounds that complement intuition from ultraviolet examples. Such self-consistency constraints have been used to bound the couplings of many different higher-dimension operators in scalar field theory [1], gauge theory [1], Einstein-Maxwell theory [5, 14], higher-curvature corrections to gravity [3, 7, 9], and massive gravity [8].

The simplest positivity bound on effective theories applies to the coupling of the operator. In a massless theory of a real scalar with a shift symmetry, the first higher-dimension operator that one can add to the kinetic term is the operator


In a theory given by , the forward amplitude for two-to-two scattering is . A standard dispersion relation argument [1] then relates the coefficient of 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 is positive. Similarly, one can compute the speed of propagation of perturbations in a nonzero background: one finds that subluminality requires and that if it is straightforward to build causal paradoxes involving superluminal signaling between two bubbles of 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 theory, whose Lagrangian is simply a polynomial in


which in the effective field theory we can write as111We will use mostly-plus metric signature throughout.


A case of particular tractability is an th-order theory, in which the are very small or zero for for some , where is the first nonnegligible higher-order term in the polynomial:


We use a weak-coupling assumption from the ultraviolet to the infrared to guarantee a well-defined counting at all energy scales, as in LABEL:Cheung:2016wjt, so that the vanishing of the tree-level for is well defined.

We will show that analyticity of scattering amplitudes and causality of signal propagation imply the same positivity bound on the theory in Eq. (4):


We will also find that Eq. (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 space of self-consistent low-energy effective theories, constituting a generalization of the well known 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 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 . If there is no coupling of massive states to for , then the tree-level value of vanishes for . We can then place the positivity bound in Eq. (5) on using the tree-level amplitude. Note that this logic does not contradict the positivity bound on in LABEL:Adams:2006sv, since could still be generated at loop level, though from the tree-level completion would be parametrically larger in units of the cutoff.222We 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]. Moreover, from the perspective of the effective field theory, the higher-dimension operators in the th-order theory in Eq. (4) can be viewed as a sector of a larger theory. For example, taking a complex scalar with a symmetry for integer , the allowed higher-dimension operators are of the form , , and for integer , where and . In particular, all operators for would be forbidden and the scattering of particles at tree level would occur only through the contact operator, just as in the th-order theory in Eq. (4).

This paper is organized as follows. In Sec. 2, we consider the application of analyticity bounds for higher-point amplitudes and derive our bound (5) on the th-order theory. Next, in Sec. 3 we find that the bound (5) also follows from demanding the absence of causal paradoxes. In Sec. 4 we consider a particular class of tree-level completions and find that the couplings obey Eq. (5) as a consequence of unitarity of the spectral representation. Along the way, we present an elegant derivation of the propagator for higher-spin massive bosons in arbitrary spacetime dimension. We discuss the obstacles, in the form of kinematic singularities, that preclude straightforward generalization of some of these bounds to arbitrary (i.e., not strictly th-order) theories in Sec. 5. In Sec. 6 we show that there is a deep relationship between positivity bounds and the involution property of the Legendre transform relating the Lagrangian and Hamiltonian formulations of the mechanics of the theory. We conclude and discuss future directions in Sec. 7.

2 Bounds from Analyticity

In this section, we derive the bound in Eq. (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 -to- 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 positivity bounds come from the optical theorem, where is the forward amplitude, is the center-of-mass energy of the incoming particles, and is the cross section, which is mandated physically to be positive. Taking a four-point operator, kinematics allows only one forward limit (module polarization or other, internal degrees of freedom):


working in the convention of all momenta incoming.

However, at higher-point, there are multiple forward kinematic configurations, given by the angles that the various momenta make with respect to each other. In particular, considering -to- particle scattering, going to forward kinematics so that for , there is a family of forward limits parameterized by independent angles and independent energies. The reason for this counting is as follows. A priori, we choose an angle on the celestial sphere for the direction associated for each of the , . Momentum is conserved automatically by the forward condition. Moreover, we can use Lorentz invariance to fix two of the directions: one angle is fixed by rotational invariance and another is fixed by boost symmetry, which allows us to take two of the pairs to be back-to-back with equal energy. Hence, we can fix points on the celestial sphere, each of which requires angular coordinates in spacetime dimensions.

This large number of possible forward limits means that higher-point amplitudes have significant power to constrain the couplings of higher-point operators, despite the larger number of operators one can write down.

2.2 Higher-Point Dispersion Relations and Bounds for

Placing positivity bounds using higher-point amplitudes follows a generalization of the argument bounding four-point operators. First, let us define the Mandelstam invariants


There are independent Mandelstam invariants for the -point amplitude (i.e., -to- scattering), taking into account momentum conservation and the on-shell conditions. Choosing a particular forward limit, by fixing all of the angular parameters, becomes a function of the remaining nonzero . In particular, we will choose as our variable for analytic continuation the center-of-mass energy squared,


We wish to place a bound on the couplings of the th-order theory (4) for even or odd , where the first nonnegligible coefficient of the operator occurs at . Making particular concrete choices for the kinematics will allow us to bound the coefficient . We will find that different choices of kinematics and dispersion relations are needed for even or odd.

At general kinematics, the -point tree-level amplitude for the th-order theory is


where runs over the the different possible groupings of into an ordered list of unordered pairs. Throughout this section, we will work with a weak-coupling assumption from the infrared to the deep ultraviolet, above the cutoff, implying a well defined expansion at all scales [9]. For our th-order theory, such an assumption will allow us to ignore disconnected components of the amplitude in the generalized optical theorem, since the loop contributions to the disconnected amplitude will be negligible and the tree-level components will vanish except for the contact diagram.

2.2.1 Even

If is even, we choose the following forward kinematics:

  for (10)

for all , . Then the center-of-mass energy is


and the forward amplitude, within the regime of our weakly-coupled effective field theory, is


In the complex plane, we consider the contour integral


where is a small contour around the origin. Similarly, we can define


where is a contour running just above and below the real axis, plus a boundary contour at infinity.

The standard analyticity assumptions of the S-matrix imply that is analytic everywhere except for poles in the where massive states in the ultraviolet completion go on-shell and, at loop level, branch cuts associated with massive states in loops. See Refs. [19, 20] for a discussion of analyticity for 3-to-3 scattering and LABEL:Eden:1966dnq for a more general treatment of the analytic S-matrix. (If we made the more restrictive assumption of a tree-level ultraviolet completion, then the nonanalyticities would only occur at the poles of the massive states, as one could see by explicit construction of the Feynman diagrams.) Given the choice of kinematics in Eq. (10), the only independent nonzero is , which is equivalent to a rescaled version of by Eq. (11). Hence, all nonanalyticities in the complex plane occur at a set of poles (and branch cuts) on the real axis. That is, Cauchy’s theorem implies that . We assume that the boundary integral at infinity vanishes. For a massive theory, this would follow from the Froissart bound at large [22, 23]. Even though we are considering a massless theory, it is reasonable to assume some form of polynomial boundedness that forbids the amplitude from diverging too quickly with at large ; in essence, discarding the boundary integral is equivalent to demanding that the term in the action is in fact ultraviolet completed, i.e., forbidding primordial terms by demanding that the higher-dimension operator originate from the exchange of states at some scale.

Equating , we thus have


where is some regulator below which we take the amplitude to be analytic and . For example, if we use counting to restrict to the tree-level scattering amplitude, we can take to be of order the scale of the ultraviolet completion.

In the two-to-two scattering case, the integrals over the positive and negative real axis are related by the crossing symmetry associated with swapping and , i.e., by swapping the and channels for forward kinematics. For our present calculation involving -to- scattering, crossing symmetry implies that the amplitude is invariant under swapping legs and . With the choice of kinematics in Eq. (10), this is equivalent to swapping legs for for all even between and , which has the effect of swapping while leaving unchanged, so and . Hence, as in the two-to-two case, crossing symmetry implies that with our choice of kinematics is an even function of , even in the ultraviolet. We thus have and


Using the Schwarz reflection principle , we have . In two-to-two scattering, the optical theorem relates the cross-section to the imaginary part of the forward amplitude. Generalized to an initial multiparticle state with center-of-mass energy , the optical theorem implies


where the sum is over all intermediate states , is the Lorentz-invariant phase space measure for the intermediate state [24], and is the amplitude for the -particle initial state with center-of-mass energy going to the final state . Note that, in general, the appropriate amplitude appearing in the generalized optical theorem (17) is the full -to- amplitude, including disconnected diagrams [25]. However, as noted above, for the th-order theory we consider, the only contribution to the -point amplitude comes from the tree-level diagrams. Hence, for the theory at hand, Eq. (17) applies to the connected component of the amplitude alone.

The right-hand side of Eq. (17) is manifestly positive. Thus, we have a bound on in the th-order theory for even :


2.2.2 Odd

For the th-order theory where is odd, we choose the kinematics

  for (19)

With these choices of kinematics, we have the center-of-mass energy


and the forward amplitude, in our weakly-coupled low-energy effective field theory, is


We can make a further choice of kinematics to set , which we will for brevity call , and analytically continue in while holding constant. That is, the center-of-mass energy is , so analytic continuation in is equivalent to analytic continuation in .333In LABEL:Elvang:2012st, a related choice of kinematics was made for six-point scattering, in a dilaton effective action relevant for the -theorem in . Note that for physical kinematics, . The forward amplitude is


In contrast with Sec. 2.2.1, we define the contour integrals for odd as


for a small contour around the origin and


for a contour running just above and below the real axis, plus a boundary contour at infinity that we drop as before.

Crossing symmetry under swapping legs and is equivalent under our choice of kinematics to swapping , i.e., swapping while holding (and thus ) fixed. That is, the forward amplitude, even in the ultraviolet, must be an even function of . Equivalently, the full forward amplitude satisfies


We therefore have

Using analyticity to equate and in Eqs. (23) and (24) and using the Schwarz reflection principle and the optical theorem as before, we obtain a bound on in the th-order theory for odd :


3 Bounds from Causality

Next, let us consider how bounds on the theory can be derived from causality. For now, we will consider an arbitrary theory, with no assumptions about the relative sizes of the various higher-dimension operators. The equation of motion for this theory is:


which is solved by a constant background condensate, . We will use bars to denote background vaues of fields, so and . The leading-order action for the fluctuation can be written as




The term in the action zeroth-order in is a cosmological constant , which can be dropped, while the term first-order in is a tadpole, which vanishes because satisfies the background equations of motion (27).

Let us compute the speed of propagation for fluctuations about this background. The equation of motion for in the background is


Taking a plane-wave ansatz for , we have the dispersion relation , that is,


Writing , the speed of propagation is , which satisfies


where and . We note that is always nonnegative and can be chosen to be strictly positive for nonzero by choosing the direction of . Moreover, we choose so that is nonzero. It follows that if and only if


If we are to impose a causality condition on the fluctuations , to be conservative we should for consistency impose a similar condition on the background itself. That is, we should require that the background energy-momentum not propagate faster than light. The relevant energy condition mandating this causal flow of energy-momentum is the null dominant energy condition (NDEC) [26], which is the statement of the null energy condition (NEC), for all null , along with the requirement that be timelike or null. The background energy-momentum tensor is


so the NEC implies . Hence, we conclude that


in order to guarantee (see also LABEL:Adams:2006sv). As shown in Refs. [1, 5], if one can immediately form a causal paradox by highly boosting two bubbles of background condensate relative to each other in an otherwise empty region of space; sending superluminal signals back and forth between the two forms a closed signal trajectory in spacetime.

In addition to the NEC, the NDEC implies that is causal (i.e., timelike or null). The reason for this is as follows. Suppose that is spacelike, . Then, defining for some null , we have


Now, taking to contain no cosmological constant, we have , so since , it follows that for . Moreover, since we are interested in an interacting theory, for , so , , and . As a result, for and, since we can choose the orientation of so that , we have . That is, is spacelike, contradicting the NDEC. We conclude that cannot be positive, so is causal.

Given causal, let us consider the question of stability of the condensate background and write . First, suppose that is timelike, so . We can go to the condensate rest frame, so . Then we have


If , there are ghosts in theory, resulting in a quantum mechanical pair-production instability [27]. We thus conclude that if . If is null, then we simply have . Hence, stability guarantees that is always nonpositive. Since in the case is trivial, we hereafter take to be timelike.

Let us now apply the causality bound (35) to the th-order theory, where all the are negligible at leading order for . By taking sufficiently small, we guarantee that is dominated by the term, which we take to be nonzero. We have


so since ,


4 Bounds from Unitarity

Let us again consider the th-order theory in which the first higher-dimension operator with nontrivial coefficient is . For such a theory, we can consider a family of tree-level completions of the operator that takes the form of some combination of operators , where


We generate whenever there is some part of and that are the same field (up to some extraneous metrics) for . The coupling of will thus receive contributions that go as for . Of course, in that case the operator is also generated via the exchange of a between two of the operators and similarly for . Thus, in a theory in which the tree-level coefficients for are negligible, in units of the cutoff, compared to for , we must consider a completion in which the coefficients vanish for . In such an th-order theory, the operator is generated by integrating out , joining two copies of .444We will not consider theories in which the -point operators in the completion vanish on-shell, e.g., for a traceless, spin-two massive state , a coupling of the form . Completions comprised purely of such operators do not have poles in their forward amplitudes associated with the massive state going on-shell.

Let us consider the structure of our massive states . Without loss of generality, we can take to be symmetric on its indices, since the interaction with effectively projects out any nonsymmetric component. We can split up into its traces and traceless components by defining


where parentheses around subscripts denotes normalized symmetrization, i.e., .

We will bound via an argument involving the Källén-Lehmann form of the exact propagator for the states.

4.1 All Massive Bosonic Higher-Spin Propagators in Arbitrary

We now build the propagator numerator for . This is a canonical higher-spin state, that is, a symmetric tensorial rank- representation of the little group for a massive state in dimensions.555We will derive the unitary-gauge propagator numerator in the form of a Lorentz-covariant tensor; for a spin representation in , see LABEL:Weinberg:1964cn. We require that satisfy the Fierz-Pauli conditions [29], so that at leading order in in the equations of motion we have


Equivalently, the propagator numerator must be transverse and traceless on shell, when , where is the mass of . We will write the propagator numerator for as . Considering , the index with which contracts can either be on a metric, which by index symmetry on the indices we can write as , or another momentum . We can therefore write , up to symmetrization, for some tensors and that are themselves built out of metrics and momenta. In order for this object to vanish on shell while leaving a nontrivial propagator, we must have and . That is, on-shell transversality requires that the propagator numerator be built out of the projector [30]


Without loss of generality, we can use symmetry of the propagator on the and indices separately, along with symmetry on the interchange of the sets of and indices, to write the general form the propagator numerator must take:


That is, if is even, the final term is , while if is odd, the final term is . Next, we enforce the tracelessness condition, which requires that on shell. We note that, on shell, and . We find


where, taking careful account of the combinatorics,


To enforce tracelessness, we thus require that each , so


That is,


Now, we need to determine , equivalently, the overall normalization of the propagator. Let us first count the number of degrees of freedom in . A tensor that is symmetric on indices in dimensions will have


independent components. The transverse condition restricts us to setting (i.e., going to the rest frame, we must have only spatial components). Furthermore, the tracelessness condition removes components, so the number of independent components is


which matches the counting of LABEL:Cortese:2013lda. In , this expression reduces to the expected .

Unitarity implies that, on shell, the propagator numerator can be written as a sum over a tensor product of the physical polarization states,


where are the unit-normalized spin- polarization states and is a label for the different states, with [24]. Hence, the full trace of the massive propagator numerator counts the number of physical degrees of freedom. As one can verify by computation, for the propagator numerator given in Eq. (44), with the coefficients given in Eq. (48), we have


Thus, in arbitrary dimension, for arbitrary integer spin. That is, the propagator numerator for all massive higher-spin bosons in arbitrary dimension is


In the special case of , Eq. (53) matches the result of LABEL:Miyamoto.666See also Refs. [32, 33] for . While, Eq. (A.3) of LABEL:Nayak gives an expression for the arbitrary- propagator, their coefficient contains an error inherited from Eq. (32) of LABEL:Singh.

For example, the propagator numerator for a massive vector is just , while the propagator numerators for massive states of spin , , , and are:

spin 2: (54)
spin 3:
spin 4: