We show that in most conformal field theories the condition of the energy flux positivity, proposed by Hofman and Maldacena, is equivalent to the absence of ghosts. At finite temperature and large energy and momenta, the two-point functions of the stress energy tensor develop lightlike poles. The residues of the poles can be computed, as long as the only spin two conserved current, which appears in the stress energy tensor OPE and acquires nonvanishing expectation value at finite temperature, is the stress energy tensor. The condition for the residues to stay positive and the theory to remain ghost free is equivalent to the condition of positivity of energy flux.
Recent programs on conformal bootstrap suggest an empirical relationship between the existence of non-trivial conformal field theories and non-trivial features such as a kink in the unitarity bound of conformal dimensions in the conformal bootstrap equations. We report the existene of non-trivial kinks in the unitarity bound of scalar operators in the adjoint representation of the $SU(N)$ symmetric conformal field theories. They have interesting properties (1) the kinks exist in $d<6$ dimensions (2) the location of kinks are when the unitarity bound hits the space-time dimension $d$ (3) there …
Unitary conformal field theories (CFTs) are believed to have positive (non-negative) energy correlators. Energy correlators are universal observables in higher-dimensional CFTs built out of integrated Wightman functions of the stress-energy tensor. We analyze energy correlators in parity invariant four-dimensional CFTs. The goal is to use the positivity of energy correlators to further constrain unitary CFTs. It is known that the positivity of the simplest one-point energy correlator implies that 1/3 <= a/c <= 31/18 where a and c are the Weyl anomaly coefficients. We use the positivity of high…
We consider the four-point correlator of the stress tensor multiplet in ${\cal N}=4$ SYM in the limit of large central charge $c \sim N^2$. For finite values of $g^2N$ single-trace intermediate operators arise at order $1/c$ and this leads to specific poles in the Mellin representation of the correlator. The sign of the residue at these poles is fixed by unitarity. We consider solutions consistent with crossing symmetry and this pole structure. We show that in a certain regime all solutions result in a negative contribution to the anomalous dimension of twist four operators. The reason behind …
In this work we expand on the holographic description of CFTs on de Sitter (dS) and anti-de Sitter (AdS) spacetimes and examine how violations of the unitarity bound in the boundary theory are recovered in the bulk physics. To this end we consider a Klein-Gordon field on AdS_(d+1) conformally compactified such that the boundary is (A)dS_d, and choose masses and boundary conditions such that the corresponding boundary operator violates the CFT unitarity bound. The setup in which the boundary is AdS_d exhibits a particularly interesting structure, since in this case the boundary itself has a bou…
We explore the emergence of gravitation from entanglement in holographic CFTs with gravitational anomalies. More specifically, the holographic correspondence between topologically massive gravity (TMG) with gravitational Chern-Simons term in the 3D bulk and its dual CFT with unbalanced left and right moving central charges on the 2D boundary, is studied from the quantum entanglement perspective. Using the first law of entanglement, we derive the holographic dictionary of the energy-momentum tensor in TMG, including the chiral case with logarithmic mode. Furthermore, we show that the linearized…
Various observables in compact CFTs are required to obey positivity, discreteness, and integrality. Positivity forms the crux of the conformal bootstrap, but understanding of the abstract implications of discreteness and integrality for the space of CFTs is lacking. We systematically study these constraints in two-dimensional, non-holomorphic CFTs, making use of two main mathematical results. First, we prove a theorem constraining the behavior near the cusp of integral, vector-valued modular functions. Second, we explicitly construct non-factorizable, non-holomorphic cuspidal functions satisfy…
We derive the Cutkosky rules for conformal field theories (CFTs) at weak and strong coupling. These rules give a simple, diagrammatic method to compute the double-commutator that appears in the Lorentzian inversion formula. We first revisit weakly-coupled CFTs in flat space, where the cuts are performed on Feynman diagrams. We then generalize these rules to strongly-coupled holographic CFTs, where the cuts are performed on the Witten diagrams of the dual theory. In both cases, Cutkosky rules factorize loop diagrams into on-shell sub-diagrams and generalize the standard S-matrix cutting rules. …
We study families of one-dimensional CFTs relevant for describing gapped QFTs in AdS$_2$. Using the Polyakov bootstrap as our main tool, we explain how S-matrices emerge from the flat space limit of CFT correlators. In this limit we prove that the CFT OPE density matches that of a generalized free field, and that this implies unitarity of the S-matrix. We establish a CFT dispersion formula for the S-matrix, proving its analyticity except for singularities on the real axis which we characterize in terms of the CFT data. In particular positivity of the OPE establishes that any such S-matrix must…
We derive rigorous bounds on corrections to Einstein gravity using unitarity and analyticity of graviton scattering amplitudes. In $D\geq 4$ spacetime dimensions, these consistency conditions mandate positive coefficients for certain quartic curvature operators. We systematically enumerate all such positivity bounds in $D=4$ and $D=5$ before extending to $D\geq 6$. Afterwards, we derive positivity bounds for supersymmetric operators and verify that all of our constraints are satisfied by weakly-coupled string theories. Among quadratic curvature operators, we find that the Gauss-Bonnet term in …
We derive new effective field theory (EFT) positivity bounds on the elastic $2\to2$ scattering amplitudes of massive spinning particles from the standard UV properties of unitarity, causality, locality and Lorentz invariance. By bounding the $t$ derivatives of the amplitude (which can be represented as angular momentum matrix elements) in terms of the total ingoing helicity, we derive stronger unitarity bounds on the $s$- and $u$-channel branch cuts which determine the dispersion relation. In contrast to previous positivity bounds, which relate the $t$-derivative to the forward-limit EFT ampli…
Scalars in AdS${}_{d+1}$ with squared masses in the Breitenlohner-Freedman window $-d^2/4 \le m^2 < -d^2/4 +1$ (in units with the AdS scale $\ell$ set to 1) are known to enjoy a variety of boundary conditions. For larger masses $m^2 > -d^2/4 +1$, unitarity bounds in possible dual CFTs suggest that such general boundary conditions should lead to ghosts. We show that this is not always the case as, for conformally-invariant boundary conditions in Poincaré AdS that would naively violate unitarity bounds, the system is generically ghost-free. Conflicts with unitarity bounds are avoided due to the …
Sum rules in effective field theories, predicated upon causality, place restrictions on scattering amplitudes mediated by effective contact interactions. Through unitarity of the $S$-matrix, these imply that the size of higher dimensional corrections to transition amplitudes between different states is bounded by the strength of their contributions to elastic forward scattering processes. This places fundamental limits on the extent to which hypothetical symmetries can be broken by effective interactions. All analysis is for dimension $8$ operators in the forward limit. Included is a thorough …
The Coon amplitude is a deformation of the Veneziano amplitude with logarithmic Regge trajectories and an accumulation point in the spectrum, which interpolates between string theory and field theory. With string theory, it is the only other solution to duality constraints explicitly known and it constitutes an important data point in the modern S-matrix bootstrap. Yet, its basics properties are essentially unknown. In this paper we fill this gap and derive the conditions of positivity and the low energy expansion of the amplitude. On the positivity side, we discover that the amplitude switche…
The study of conformal boundary conditions for two-dimensional conformal field theories (CFTs) has a long history, ranging from the description of impurities in one-dimensional quantum chains to the formulation of D-branes in string theory. Nevertheless, the landscape of conformal boundaries is largely unknown, including in rational CFTs, where the local operator data is completely determined. We initiate a systematic bootstrap study of conformal boundaries in 2d CFTs by investigating the bootstrap equation that arises from the open-closed consistency condition of the annulus partition functio…
Gauging extra matter is a common way to couple two CFTs discontinuously. We may consider gauging matter by strongly coupled gauge theories at criticality rather than by weakly coupled (asymptotic free) gauge theories. It often triggers relevant deformations and possibly leads to a non-trivial fixed point. In many examples such as the IR limit of SQCDs (and their variants), the relevant RG flow induced by this strong gauging makes the total central charge $a$ increase rather than decrease compared with the sum of the original decoupled CFTs. The dilaton effective field theory argument given by …
What is the space of weakly-coupled, gravitational theories which contain massive, higher-spin particles? This class of theories is highly constrained and it is conjectured their ultraviolet completion must be string theory. We provide more evidence for this conjecture by studying the Regge limit in large $N$, $4d$ CFTs with single-trace operators of unbounded spin. We show that in the Regge limit, these theories have bulk scattering amplitudes which are consistent with the string theory prediction to all orders in $1/N$ for large, but finite, coupling. In the language of Regge theory, we show…
The EEC-Hedron: Positivity Bounds on Energy Correlators
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Yale University（耶鲁大学）；Heriot-Watt University（赫瑞-瓦特大学）；Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Carnegie Mellon University（卡内基梅隆大学）
Positivity constraints are derived on pion-nucleon scattering amplitudes and their even-order derivatives inside the Mandelstam Triangle with the help of dispersion relations. Fairly interesting constraints are obtained on some of the low energy constants, by a combination of the chiral perturbation theory for heavy baryons and existing fitting results from available pion-nucleon phase shifts at intermediate energies.
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…
We develop a formalism to study the implications of causality on OPE coefficients in conformal field theories with large central charge and a sparse spectrum of higher spin operators. The formalism has the interpretation of a new conformal collider-type experiment for these class of CFTs and hence it has the advantage of requiring knowledge only about CFT three-point functions. This is accomplished by considering the holographic null energy operator which was introduced in arXiv:1709.03597 as a generalization of the averaged null energy operator. Analyticity properties of correlators in the Re…
The trace anomaly in external gravity is the sum of three terms at criticality: the square of the Weyl tensor, the Euler density and Box R, with coefficients, properly normalized, called c, a and a', the latter being ambiguously defined by an additive constant. Considerations about unitarity and positivity properties of the induced actions allow us to show that the total RG flows of a and a' are equal and therefore the a'-ambiguity can be consistently removed through the identification a'=a. The picture that emerges clarifies several long-standing issues. The interplay between unitarity and re…
We derive constraints on the operator product expansion of two stress tensors in conformal field theories (CFTs), both generic and holographic. We point out that in large $N$ CFTs with a large gap to single-trace higher spin operators, the stress tensor sector is not only universal, but isolated: that is, $\langle TT{\cal O}\rangle=0$, where ${\cal O}\neq T$ is a single-trace primary. We show that this follows from a suppression of $\langle TT{\cal O}\rangle$ by powers of the higher spin gap, $Δ_{\rm gap}$, dual to the bulk mass scale of higher spin particles, and explain why $\langle TT{\cal …
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）
Positivity bounds - constraints on any low-energy effective field theory imposed by the fundamental axioms of unitarity, causality and locality in the UV - have recently been used to constrain scalar-tensor theories of dark energy. However, the coupling to matter fields has so far played a limited role. We show that demanding positivity when including interactions with standard matter fields leads to further constraints on the dark energy parameter space. We demonstrate how implementing these bounds as theoretical priors affects cosmological parameter constraints and explicitly illustrate the …
We derive the first positivity bounds for low-energy Effective Field Theories (EFTs) that are not invariant under Lorentz boosts. "Positivity bounds" are the low-energy manifestation of certain fundamental properties in the UV -- to date they have been used to constrain a wide variety of EFTs, however since all of the existing bounds require Lorentz invariance they are not directly applicable when this symmetry is broken, such as for most cosmological and condensed matter systems. From the UV axioms of unitarity, causality and locality, we derive an infinite family of bounds which (derivatives…
We prove the achronal averaged null energy condition for general quantum field theories in the near horizon geometry of spherical extremal black holes (i.e. ${{\rm AdS}_2\times S^{d-2}}$), de Sitter and anti-de Sitter. The derivation follows from monotonicity of relative entropy after computing the modular hamiltonian of a null deformed region. For incomplete (but maximally extended) achronal null geodesics in ${\rm AdS}_2\times S^{d-2}$, we prove the positivity of a different light-ray operator for arbitrary CFTs. This agrees with a constraint recently derived for the Lorentzian cylinder.
