Positivity bounds are powerful tools to constrain effective field theories. Utilizing the partial wave expansion in the dispersion relation and the full crossing symmetry of the scattering amplitude, we derive several sets of generically nonlinear positivity bounds for a generic scalar effective field theory: We refer to these as the $PQ$, $D^{\rm su}$, $D^{\rm stu}$ and $\bar{D}^{\rm stu}$ bounds. While the $PQ$ bounds and $D^{\rm su}$ bounds only make use of the $s\leftrightarrow u$ dispersion relation, the $D^{\rm stu}$ and $\bar{D}^{\rm stu}$ bounds are obtained by further imposing the $s\leftrightarrow t$ crossing symmetry. In contradistinction to the linear positivity for scalars, these inequalities can be applied to put upper and lower bounds on Wilson coefficients, and are much more constraining as shown in the lowest orders. In particular we are able to exclude theories with soft amplitude behaviour such as weakly broken Galileon theories from admitting a standard UV completion. We also apply these bounds to chiral perturbation theory and we find these bounds are stronger than the previous bounds in constraining its Wilson coefficients.
Triple crossing positivity bounds for multi-field theories
University of Science and Technology of China（中国科学技术大学）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Center for High Energy Physics, Peking University（北京大学高能物理中心）；Peng Huanwu Center for Fundamental Theory（彭桓武理论物理中心）
This paper presents the fascinating correspondence between the geometric function theory and the scattering amplitudes with $O(N)$ global symmetry. A crucial ingredient to show such correspondence is a fully crossing symmetric dispersion relation in the $z$-variable, rather than the fixed channel dispersion relation. We have written down fully crossing symmetric dispersion relation for $O(N)$ model in $z$-variable for three independent combinations of isospin amplitudes. We have presented three independent sum rules or locality constraints for the $O(N)$ model arising from the fully crossing s…
Crossing symmetry (CS) is the main tool in the bootstrap program applied to CFT models. This consists in an equality which imposes restrictions on the CFT data of a model, i.e, the OPE coefficients and the conformal dimensions. Reflection positivity (RP) has also played a role, since this condition lead to the unitary bound and reality of the OPE coefficients. In this paper we show that RP can still reveal more information, showing how RP itself can capture an important part of the restrictions imposed by the full CS equality. In order to do that, we use a connection used by us in a previous w…
A program of wide interest in modern conformal bootstrap studies is to numerically solve general conformal field theories, based on a critical assumption that the dynamics is encoded in the conformal four-point crossing equations and positivity condition. In this letter we propose and verify a novel algebraic property of the crossing equations which provides strong restriction for this program. We show for various types of symmetries $\cal G$, the crossing equations can be linearly converted into the $SO(N)$ vector crossing equations associated with the $SO(N)\rightarrow \cal G$ branching rule…
We derive positivity bounds for scattering amplitudes of particles with arbitrary spin using unitarity, analyticity and crossing symmetry. The bounds imply the positivity of certain low-energy coefficients of the effective action that controls the dynamics of the light degrees of freedom. We show that low-energy amplitudes strictly softer than $O(p^4)$ do not admit unitary ultraviolet completions unless the theory is free. This enforces a bound on the energy growth of scattering amplitudes in the region of validity of the effective theory. We discuss explicit examples including the Goldstino f…
For a low energy effective theory to admit a standard local, unitary, analytic and Lorentz-invariant UV completion, its scattering amplitudes must satisfy certain inequalities. While these bounds are known in the forward limit for real polarizations, any extension beyond this for particles with nonzero spin is subtle due to their non-trivial crossing relations. Using the transversity formalism (i.e. spin projections orthogonal to the scattering plane), in which the crossing relations become diagonal, these inequalities can be derived for 2-to-2 scattering between any pair of massive particles,…
Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique（巴黎萨克雷大学）；University of Michigan（密歇根大学）；University of California Los Angeles（加州大学洛杉矶分校）；Université de Genève（日内瓦大学）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）
We discuss the general method for obtaining full positivity bounds on multi-field effective field theories (EFTs). While the leading order forward positivity bounds are commonly derived from the elastic scattering of two (superposed) external states, we show that for a generic EFT containing 3 or more low-energy modes, this approach only gives incomplete bounds. We then identify the allowed parameter space as the dual to a spectrahedron, constructed from crossing symmetries of the amplitude, and show that finding the optimal bounds for a given number of modes is equivalent to a geometric probl…
Full positivity bounds for anomalous quartic gauge couplings in SMEFT
University of Science and Technology of China（中国科学技术大学）；Interdisciplinary Center for Theoretical Study, University of Science and Technology of China（中国科学技术大学交叉理论研究中心）；Peng Huanwu Center for Fundamental Theory（彭桓武基础研究中心）
For 2-2 scattering in quantum field theories, the usual fixed $t$ dispersion relation exhibits only two-channel symmetry. This paper considers a crossing symmetric dispersion relation, reviving certain old ideas in the 1970s. Rather than the fixed $t$ dispersion relation, this needs a dispersion relation in a different variable $z$, which is related to the Mandelstam invariants $s,t,u$ via a parametric cubic relation making the crossing symmetry in the complex $z$ plane a geometric rotation. The resulting dispersion is manifestly three-channel crossing symmetric. We give simple derivations of …
We show how calculable IR loop effects impact positivity bounds in Effective Field Theories with causal and unitary UV completions. We identify infrared singularities which appear in dispersion relations at $|t|\lesssim m^2$. In the massless limit, they weaken two-sided bounds based on crossing symmetry, such as the lower bound on the amplitude for Galileon scattering. For amplitudes that are analytic in $s$ even for large negative $t$, i.e. $|t|\gg m^2$, we propose a new simple analytic approach to dispersive bounds, which are instead insensitive to the singularities, and explicitly compute t…
We consider 2-2 scattering in four spacetime dimensions in Celestial variables. Using the crossing symmetric dispersion relation (CSDR), we recast the Celestial amplitudes in terms of crossing symmetric partial waves. These partial waves have spurious singularities in the complex Celestial variable, which need to be removed in local theories. The locality constraints (null constraints) admit closed form expressions, which lead to novel bounds on partial wave moments. These bounds allow us to quantify the degree of low spin dominance(LSD) for scalar theories. We study a new kind of positivity t…
In this note, we study the connections between infrared (IR) and ultraviolet (UV) behaviors of scattering amplitudes of massless channels by exploiting dispersion relations and positivity bounds. Given forward scattering amplitudes, which scale as $\mathcal{A}(s)\sim s^M$ in the IR ($s\to0$) and could be embedded into UV completions satisfying unitarity, analyticity, crossing symmetry and polynomial boundedness $|\mathcal{A}(s)|< c\, |s|^N$ ($|s|\to\infty$), with $M$ and $N$ integers, we show that the inequality $2\ceil*{\frac{N}{2}}\ge M \ge \ceil*{\frac{N}{2}}$ must hold, where $\ceil*{x}$ i…
We develop crossing symmetric dispersion relations for describing 2-2 scattering of identical external particles carrying spin. This enables us to import techniques from Geometric Function Theory and study two sided bounds on low energy Wilson coefficients. We consider scattering of photons, gravitons in weakly coupled effective field theories. We provide general expressions for the locality/null constraints. Consideration of the positivity of the absorptive part leads to an interesting connection with the recently conjectured weak low spin dominance. We also construct the crossing symmetric a…
The presence of a massless spin-2 field in an effective field theory results in a $t$-channel pole in the scattering amplitudes that precludes the application of standard positivity bounds. Despite this, recent arguments based on compactification to three dimensions have suggested that positivity bounds may be applied to the $t$-channel pole subtracted amplitude. If correct this would have deep implications for UV physics and the Weak Gravity Conjecture. Within the context of a simple renormalizable field theory coupled to gravity we find that applying these arguments would constrain the low-e…
We constrain effective field theories by going beyond the familiar positivity bounds that follow from unitarity, analyticity, and crossing symmetry of the scattering amplitudes. As interesting examples, we discuss the implications of the bounds for the Galileon and ghost-free massive gravity. The combination of our theoretical bounds with the experimental constraints on the graviton mass implies that the latter is either ruled out or unable to describe gravitational phenomena, let alone to consistently implement the Vainshtein mechanism, down to the relevant scales of fifth-force experiments, …
We consider the four-point correlator of the stress tensor multiplet in N=4 SYM. We construct all solutions consistent with crossing symmetry in the limit of large central charge c ~ N^2 and large g^2 N. While we find an infinite tower of solutions, we argue most of them are suppressed by an extra scale Δ_{gap} and are consistent with the upper bounds for the scaling dimension of unprotected operators observed in the numerical superconformal bootstrap at large central charge. These solutions organize as a double expansion in 1/c and 1/Δ_{gap}. Our solutions are valid to leading order in 1/c an…
We consider positivity constraints applicable to the Effective Field Theory (EFT) of gravity in arbitrary dimensions. By considering scattering of indefinite initial and final states, we highlight the existence of a gravitational scattering amplitude for which full crossing symmetry is manifest and utilize the recently developed crossing symmetric dispersion relations to derive compact non-linear bounds. We show that the null constraints built into these dispersion relations lead to a finite energy sum rule for gravity which may be extended to a one-parameter family of continuous moment sum ru…
The main objective of this work is to isolate Effective Field Theory scattering amplitudes in the space of non-perturbative two-to-two amplitudes, using the S-matrix Bootstrap. We do so by introducing the notion of Effective Field Theory cutoff in the S-matrix Bootstrap approach. We introduce a number of novel numerical techniques and improvements both for the primal and the linearized dual approach. We perform a detailed comparison of the full unitarity bounds with those obtained using positivity and linearized unitarity. Moreover, we discuss the notion of Spin-Zero and UV dominance along the…
We derive new crossing-symmetric dispersion formulae for CFT correlators restricted to the line. The formulae are equivalent to the sum rules implied by what we call master functionals, which are analytic extremal functionals which act on the crossing equation. The dispersion relations provide an equivalent formulation of the constraints of the Polyakov bootstrap and hence of crossing symmetry on the line. The built in positivity properties imply simple and exact lower and upper bounds on the values of general CFT correlators on the Euclidean section, which are saturated by generalized free fi…
Region bounds, error margins and certified positivity in the numerical conformal bootstrap; Or the first mathematically rigorous bounds on the 3d Ising CFT critical exponents
The positivity bounds, derived from the axiomatic principles of quantum field theory (QFT), constrain the signs of Wilson coefficients and their linear combinations in the Standard Model Effective Field Theory (SMEFT). The precise determination of these bounds, however, can become increasingly difficult as more and more SM modes and operators are taken into account. We study two approaches that aim at obtaining the full set of bounds for a given set of SM fields: 1) the traditional elastic positivity approach, which exploits the elastic scattering amplitudes of states with arbitrarily superpos…
Weak vector boson scattering (VBS) is a sensitive probe of new physics effects in the electroweak symmetry breaking. Currently, experimental results at the LHC are interpreted in the effective field theory approach, where possible deviations from the Standard Model in the quartic-gauge-boson couplings are often described by 18 dimension-8 operators. By assuming that a UV completion exists, we derive a new set of theoretical constraints on the coefficients of these operators, i.e. certain combinations of coefficients must be positive. These constraints imply that the current effective approach …
We study the crossing equations in $d=3$ for the four point function of two $U(1)$ currents and two scalars including the presence of a parity violating term for the $s$-channel stress tensor exchange. We show the existence of a new tower of double trace operators in the $t$-channel whose presence is necessary for the crossing equation to be satisfied and determine the corresponding large spin parity violating OPE coefficients. Contrary to the parity even situation, we find that the parity odd $s$-channel light cone stress tensor block do not have logarithmic singularities. This implies that t…
We explore the correspondence between geometric function theory (GFT) and quantum field theory (QFT). The crossing symmetric dispersion relation provides the necessary tool to examine the connection between GFT, QFT, and effective field theories (EFTs), enabling us to connect with the crossing-symmetric EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of Typically Real functions are summarized and shown to be of enormous use in bounding Wilson coefficients in the context of 2-2 scattering. We prove that two-sided bounds on Wilson coefficients are guaranteed to exist …
We examine the plausibility of crossing the cosmological constant ($Ł$) barrier in a two-field quintessence model of dark energy, involving a kinetic interaction between the individual fields. Such a kinetic interaction may have its origin in the four dimensional effective two-field version of the Dirac-Born-Infeld action, that describes the motion of a D3-brane in a higher dimensional space-time. We show that this interaction term could indeed enable the dark energy equation of state parameter $\wx$ to cross the $Ł$-barrier (i.e., $\wx = -1$), keeping the Hamiltonian well behaved (bounded fro…
This article revisits and elaborates the significant role of positive geometry of momentum twistor Grassmannian for planar N=4 SYM scattering amplitudes. First we establish the fundamentals of positive Grassmannian geometry for tree amplitudes. Then we formulate this subject around these four major facets: 1. Deriving the tree and 1-loop BCFW recursion relations solely from positivity, after introducing the simple building blocks called positive components. 2. Applying Grassmannian geometry and Pluecker coordinates to determine the signs in N$^2$MHV homological identities. Most of them in fact…
We consider the number distribution of topological defects resulting from the finite-time crossing of a continuous phase transition and identify signatures of universality beyond the mean value, predicted by the Kibble-Zurek mechanism. Statistics of defects follows a binomial distribution with $\mathcal{N}$ Bernouilli trials associated with the probability of forming a topological defect at the locations where multiple domains merge. All cumulants of the distribution are predicted to exhibit a common universal power-law scaling with the quench time in which the transition is crossed. Knowledge…
This paper discusses the locality and analyticity of the crossing symmetric dispersion relation (CSDR). Imposing locality constraints on the CSDR gives rise to a local and fully crossing symmetric expansion of scattering amplitudes, dubbed as Feynman block expansion. A general formula is provided for the contact terms that emerge from the expansion. The analyticity domain of the expansion is also derived analogously to the Lehmann-Martin ellipse. Our observation of type-II super-string tree amplitude suggests that the Feynman block expansion has a bigger analyticity domain and better convergen…
It is shown that the recently introduced positivity and causality preserving string-local quantum field theory (SLFT) resolves most No-Go situations in higher spin problems. This includes in particular the Velo-Zwanziger causality problem which turns out to be related in an interesting way to the solution of zero mass Weinberg-Witten issue. In contrast to the indefinite metric and ghosts of gauge theory, SLFT uses only positivity-respecting physical degrees of freedom. The result is a fully Lorentz-covariant and causal string field theory in which light- or space-like linear strings transform …
The Hilbert space formulation of interacting $s=1$ vector-potentials stands in an interesting contrast with the point-local Krein space setting of gauge theory. Already in the absence of interactions the Wilson loop in a Hilbert space setting has a "topological property" which is missing in the gauge theoretic description (Haag duality, Aharonov-Bohm effect); the conceptual differences increase in the presence of interactions. The Hilbert space positivity weakens the causal localization properties if interacting fields which results in the replacement of the gauge-variant point-local matter fi…
