We show that in the quadratic curvature theory of gravity, or simply $R_{μν} ^2$ gravity, the tree-level unitariy bound (tree unitarity) is violated in the UV region but an analog for $S$-matrix unitarity ($SS^{\dagger} = 1$) is satisfied. This theory is renormalizable, and hence the failure of tree unitarity is a counter example of Llewellyn Smith's conjecture on the relation between them. We have recently proposed a new conjecture that $S$-matrix unitarity gives the same conditions as renormalizability. We verify that $S$-matrix unitarity holds in the matter-graviton scattering at tree level in the $R_{μν} ^2$ gravity, demonstrating our new conjecture.
In this paper we discuss massive gravity in Minkowski space via gravitational Higgs mechanism, which provides a non-perturbative definition thereof. Using this non-perturbative definition, we address the issue of unitarity by studying the full nonlinear Hamiltonian for the relevant metric degrees of freedom. While perturbatively unitarity is not evident, we argue that no negative norm state is present in the full nonlinear theory.
We discuss aspects of non-perturbative unitarity in quantum field theory. The additional ghost degrees of freedom arising in "truncations" of an effective action at a finite order in derivatives could be fictitious degrees of freedom. Their contributions to the fully-dressed propagator -- the residues of the corresponding ghost-like poles -- vanish once all operators compatible with the symmetry of the theory are included in the effective action. These "fake ghosts" do not indicate a violation of unitarity.
We prove the unitarity of the Euclidean nonlocal scalar field theory to all perturbative orders in the loop expansion. The amplitudes in the Euclidean space are calculated assuming that all the particles have purely imaginary energies, and afterwards they are analytically continued to real energies. We show that such amplitudes satisfy the Cutkowsky rules and that only the cut diagrams corresponding to normal thresholds contribute to their imaginary part. This implies that the theory is unitary. This analysis is then exported to nonlocal gauge and gravity theories by means of Becchi-Rouet-Stor…
Resumming quantum fluctuations at the level of the gravitational path integral is expected to result in non-local effective actions and thus in a non-trivial momentum dependence of the propagator. Which properties the (dressed) graviton propagator has to satisfy and whether they can all be met are key open questions. In this work we present criteria and conditions for the momentum dependence of a graviton propagator which is consistent with unitarity, causality, and stability in a non-perturbative setting. To this end, we revisit several aspects of these conditions, highlighting some caveats a…
We argue that the tree-level graviton-scalar scattering in the Regge limit is unitarized by non-perturbative effects within General Relativity. At Planckian energy the back reaction of the incoming graviton on the background geometry produces a non-perturbative plane wave which, in turn, softens the UV-behavior. Our amplitude then interpolates between the perturbative graviton-scalar scattering at low energy and scattering on a classical plane wave in the Regge limit.
We investigate the relation between the $S$-matrix unitarity ($SS^{\dagger}=1$) and the renormalizability, in theories with negative norm states. The relation has been confirmed in many theories, such as gauge theories, Einstein gravity and Lifshitz-type non-relativistic theories by analyzing the unitarity bound, which follows from the $S$-matrix unitarity and the norm positivity. On the other hand, renormalizable theories with a higher derivative kinetic term do not necessarily satisfy the unitarity bound essentially because the unitarity bound does not hold due to the negative norm states. I…
We consider the relationship between renormalizability and unitarity at a Lifshitz point in d dimensions. We test tree unitarity for theories containing only scalars and fermions, and for pure gauge theory. In both cases, we find the requirement of weighted power-counting renormalizability is equivalent to that of tree unitarity.
We investigate unitarity within the Complex-Mass Scheme, a convenient universal scheme for perturbative calculations involving unstable particles in Quantum Field Theory which guarantees exact gauge invariance. Since this scheme requires to introduce complex masses and complex couplings, the Cutkosky cutting rules, which express perturbative unitarity in theories of stable particles, are no longer valid. We derive corresponding rules for scalar theories with unstable particles based on Veltman's Largest-Time Equation and prove unitarity in this framework.
We derive cutting rules for loop integrals containing propagators with complex masses. Using a field-theoretical model of a heavy vector boson interacting with a light fermion, we demonstrate that the complex-mass scheme respects unitarity order by order in a perturbative expansion provided that the renormalized coupling constant remains real.
Explorations of the violation of null energy condition (NEC) in cosmology could enrich our understanding of the very early universe and the related gravity theories. Although a fully stable NEC violation can be realized in the ``beyond Horndeski'' theory, it remains an open question whether a violation of the NEC is allowed by some fundamental properties of UV-complete theories or the consistency requirements of effective field theory (EFT). We investigate the tree-level perturbative unitarity for stable NEC violations in the contexts of both Galileon and ``beyond Horndeski'' genesis cosmology…
We consider the pertubative consistency of inflationary models with features with effective field theory methods. By estimating the size of one-loop contributions to the three-point function, we find the energy scale where their contribution is of the same order of the tree-level amplitude. It is well-known that beyond that scale, perturbative unitarity is lost and the theory is no more under theoretical control. Requiring that all the relevant energy scales of the problem are below this cutoff, we derive a strong upper bound on the sharpness of the feature, or equivalently on its characterist…
We study implications of perturbative unitarity for quasi-single field inflation with the inflaton and one massive scalar. Analyzing high energy scattering, we show that non-Gaussianities with $|f_{\rm NL}|\gtrsim1$ cannot be realized without turning on interactions which violate unitarity at a high energy scale. Then, we provide a relation between $f_{\rm NL}$ and the scale of new physics that is required for UV completion. In particular we find that for the Hubble scale $H\gtrsim 6\times 10^{9}$ GeV, Planck suppressed operators can easily generate too large non-Gaussanities and so it is hard…
We discuss the breakdown of perturbative unitarity of noncommutative quantum field theories in electric-type background in the light of string theory. We consider the analytic structure of string loop two-point functions using a suitable off-shell continuation and then study the zero slope limit of Seiberg and Witten. In this way we pick up how the unphysical tachyonic branch cut appears in the noncommutative field theory. We briefly discuss discontinuities and cutting rules for the full string theory amplitude and relate them to the noncommutative field theoretical results, and also discuss t…
We reconsider perturbative unitarity in quantum field theory and upgrade several arguments and results. The minimum assumptions that lead to the largest time equation, the cutting equations and the unitarity equation are identified. Using this knowledge and a special gauge, we give a new, simpler proof of perturbative unitarity in gauge theories and generalize it to quantum gravity, in four and higher dimensions. The special gauge interpolates between the Feynman gauge and the Coulomb gauge without double poles. When the Coulomb limit is approached, the unphysical particles drop out of the cut…
A perturbative regime based on contortion as a dynamical variable and metric as a (classical) fixed background, is performed in the context of a pure Yang-Mills formulation for gravity in a $2+1$ dimensional space-time. In the massless case we show that the theory contains three degrees of freedom and only one is a non-unitary mode. Next, we introduce quadratical terms dependent on torsion, which preserve parity and general covariance. The linearized version reproduces an analogue Hilbert-Einstein-Fierz-Pauli unitary massive theory plus three massless modes, two of them represents non-unitary …
Because the gravitational Hamiltonian is a pure boundary term on-shell, asymptotic gravitational fields store information in a manner not possible in local field theories. This fact has consequences for both perturbative and non-perturbative quantum gravity. In perturbation theory about an asymptotically flat collapsing black hole, the algebra generated by asymptotic fields on future null infinity within any neighborhood of spacelike infinity contains a complete set of observables. Assuming that the same algebra remains complete at the non-perturbative quantum level, we argue that either 1) th…
In a series of papers, we examine the physical unitarity in a massive Yang-Mills theory without the Higgs field in which the color gauge symmetry is not spontaneously broken and kept intact. For this purpose, we use a new framework proposed in the previous paper Kondo [arXiv:1208.3521] based on a nonperturbative construction of a non-Abelian field describing a massive spin-one vector boson field, which enables us to perform the perturbative and nonperturbative studies on the physical unitarity. In this paper, we present a new perturbative treatment for the physical unitarity after giving the g…
We reconsider perturbative unitarity violation in the standard model Higgs inflation model. We show that the Cutkosky cutting rule implied by perturbative unitarity is fulfilled at one-loop. This is a strong indication that unitarity is restored order by order in perturbation theory. We then resum certain one-loop diagrams and show that the relevant dressed amplitude fulfills the Cutkosky rule exactly. This is an example of the self-healing mechanism. The original Higgs inflation model is thus consistent and does not require any new physics beyond the standard model at least up to the Planck s…
We study the quantum information storage capacity of solitons and baryons in renormalizable quantum field theories that do not include gravity. We observe that a 't Hooft-Polyakov magnetic monopole saturates the Bekenstein bound on information when the theory saturates the bound on perturbative unitarity. In this very limit the monopole entropy assumes the form of an area-law, strikingly similar to a black hole entropy in gravity. The phenomenon appears universal and takes place for other solitons and non-perturbative objects. We observe the same behaviour of entropy of a baryon in QCD with la…
Renormalization of Hamiltonian field theory is usually a rather painful algebraic or numerical exercise. By combining a method based on the coupled cluster method, analysed in detail by Suzuki and Okamoto, with a Wilsonian approach to renormalization, we show that a powerful and elegant method exist to solve such problems. The method is in principle non-perturbative, and is not necessarily unitary.
We study perturbative unitarity constraints on generic Yukawa interactions where the involved fields have arbitrary quantum numbers under an $\prod_i SU(N_i) \otimes U(1)$ group. We derive compact expressions for the bounds on the Yukawa couplings for the cases where the fields transform under the trivial, fundamental or adjoint representation of the various $SU(N)$ factors. We apply our results to specific models formulated to explain the anomalous measurements of $(g-2)_μ$ and of the charged- and neutral-current decays of the $B$ mesons. We show that, while these models can generally still e…
We study perturbative unitarity in the scalar sector of the Myers-Pospelov model. The model introduces a preferred four-vector $n$ which breaks Lorentz symmetry and couples to a five-dimension operator. When the preferred four-vector is chosen in the pure timelike or lightlike direction, the model becomes a higher time derivative theory, leading to a cubic dispersion relation. Two of the poles are shown to be perturbatively connected to the standard ones, while a third pole, which we call the Lee-Wick-like pole, is associated to a negative metric, in Hilbert space, threatening the preservation…
We formulate a generalization of Higgs effective field theory (HEFT) including arbitrary number of extra neutral and charged Higgs bosons (generalized HEFT, GHEFT) to describe non-minimal electroweak symmetry breaking models. Using the geometrical form of the GHEFT Lagrangian, which can be regarded as a nonlinear sigma model on a scalar manifold, it is shown that the scalar boson scattering amplitudes are described in terms of the Riemann curvature tensor (geometry) of the scalar manifold and the covariant derivatives of the potential. The coefficients of the one-loop divergent terms in the ob…
We study the perturbative unitarity of the Lee-Wick models, formulated as nonanalytically Wick rotated Euclidean theories. The complex energy plane is divided into disconnected regions and the values of a loop integral in the various regions are related to one another by a nonanalytic procedure. We show that the one-loop diagrams satisfy the expected, unitary cutting equations in each region: only the physical degrees of freedom propagate through the cuts. The goal can be achieved by working in suitable subsets of each region and proving that the cutting equations can be analytically continued…
Perturbative unitarity of fractional field theories and gravity
Instituto de Estructura de la Materia, CSIC（材料结构研究所，CSIC）；Dipartimento di Architettura, Università Roma Tre（建筑系，罗马第三大学）；Istituto Nazionale di Alta Matematica Francesco Severi, Gruppo Nazionale di Fisica Matematica（弗朗切斯科·塞韦里高等数学研究所，数学物理国家小组）；Istituto Nazionale di Fisica Nucleare, Sezione di Roma 3（意大利国家核物理研究所，罗马三分部）
We revisit the problem of the bulk-boundary unitarity clash in 2 + 1 dimensional gravity theories, which has been an obstacle in providing a viable dual two-dimensional conformal field theory for bulk gravity in anti-de Sitter (AdS) spacetime. Chiral gravity, which is a particular limit of cosmological topologically massive gravity (TMG), suffers from pertur- bative log-modes with negative energies inducing a non-unitary logarithmic boundary field theory. We show here that any f(R) extension of TMG does not improve the situation. We also study the perturbative modes in the metric formulation o…
The coproduct of a Feynman diagram is set up through identifying the perturbative unitarity of the S-matrix with the cutting equation from the cutting rules. On the one hand, it includes all partitions of the vertex set of the Feynman diagram and leads to the circling rules for the largest time equation. Its antipode is the conjugation of the Feynman diagram. On the other hand, it is regarded as the integration of incoming and outgoing particles over the on-shell momentum space. This causes the cutting rules for the cutting equation. Its antipode is an advanced function vanishing in retarded r…
The results of our research on noncommutative perturbative quantum field theory and its relation to string theory are exposed with details. 1) We give an introduction to noncommutative quantum field theory and its derivation from open string theory in an antisymmetric background. 2) We perform a perturbative Wilson loop calculation for 2D NCYM. We compare the LCG results for the WML and the PV prescription. With WML the loop is well-defined and regular in the commutative limit. With PV the result is singular. This is intriguing: in the commutative theory their difference is related to topologi…
Naïve dimensional analysis seems to suggest possible unitarity violations in the framework of the Higgs and new Higgs inflationary scenarios. These violations seem to happen around the value in which the potential energy, per given Higgs boson's vacuum expectation value, crosses the perturbative cut-off scale calculated around the electroweak vacuum. Conversely to these expectations, in this paper we show that, by using an exact analysis of the background dependent cut-off scale, and by including the contribution of the phase-space volume in the perturbative scattering amplitudes, no violation…
In this article we develop an exact(non-adiabatic,non-perturbative) density matrix scattering theory for a two component quantum liquid which interacts or scatters off from a generic spin-dependent quantum potential. We find that the number or charge density in scattered fluid,$\text{Tr}({\varrho}_{sc})$ depends on {non-trivial quantum interference coefficients}, ${\cal{Q}}^{αβ}_{0ijk}$ which arises due to quantum interference between spin-independent and spin-dependent scattering amplitudes and among spin-dependent scattering amplitudes. The effect of quantum interference coefficients can be …
We present a proof of perturbative unitarity for $\mathcal{N}=4$ SYM, following from the geometry of the amplituhedron. This proof is valid for amplitudes of arbitrary multiplicity $n$, loop order $L$ and MHV degree $k$.
Higher-order effects of CPT and Lorentz violation within the SME effective framework including Myers-Pospelov dimension-five operator terms are studied. The model is canonically quantized by giving special attention to the arising of indefinite-metric states or ghosts in an indefinite Fock space. As is well-known, without a perturbative treatment that avoids the propagation of ghost modes or any other approximation, one has to face the question of whether unitarity and microcausality are preserved. In this work, we study both possible issues. We found that microcausality is preserved due to th…
We reconsider, from a novel perspective, how unitarity constrains the corrections to the ratio of shear viscosity η to entropy density s. We start with higher-derivative extensions of Einstein gravity in asymptotically anti-de Sitter spacetimes. It is assumed that these theories are derived from string theory and thus have a unitary UV completion that is dual to a unitary, UV-complete boundary gauge theory. We then propose that the gravitational perturbations about a solution of the UV complete theory are described by an effective theory whose linearized equations of motion have at most two ti…
We perform a model independent analysis of the helicity amplitudes at high energy for all the $2\to2$ scattering processes involving gauge and Higgs bosons in the presence of anomalous $WWV$, $WWVV$, $VVH$, $VVHH$ ($V\equiv Z,γ$ and $W^\pm$), $HHHH$ and $HHH$ interactions. We obtain the perturbative unitarity constraints on anomalous couplings by demanding the vanishing of terms proportional to $s^2$ and $s^{3/2}$ in the helicity amplitudes. Using these constraints, we also compute the upper bound on all the anomalous couplings from terms linear in $s$. Further, assuming all anomalous coupling…
In this review we describe a non-trivial relationship between perturbative gauge theory and gravity scattering amplitudes. At the semi-classical or tree level, the scattering amplitudes of gravity theories in flat space can be expressed as a sum of products of well defined pieces of gauge theory amplitudes. These relationships were first discovered by Kawai, Lewellen and Tye in the context of string theory, but hold more generally. In particular, they hold for standard Einstein gravity. A method based on D-dimensional unitarity can then be used to systematically construct all quantum loop corr…
We study the perturbative unitarity of non-commutative quantum Yang-Mills theories, extending previous investigations on scalar field theories to the gauge case where non-locality mingles with the presence of unphysical states. We concentrate our efforts on two different aspects of the problem. We start by discussing the analytical structure of the vacuum polarization tensor, showing how Cutkoski's rules and positivity of the spectral function are realized when non-commutativity does not affect the temporal coordinate. When instead non-commutativity involves time, we find the presence of extra…
In this paper we lay the foundations of causal quantum gravity (CQG), i.e. of a quantum theory of self-interacting symmetric massless rank-2 tensor gauge fields, the gravitons, on flat space-time, in the framework of causal perturbation theory. The causal inductive construction of the S-matrix for quantum gravity leads to a very satisfactory treatment of the ultraviolet problem. Here we concentrate on some main fundamental issues that concern the quantization of the gravitational interactions: the quantization of the free graviton field, the role of the fermionic ghost vector fields in preserv…
Perturbative spectra and related factorization properties of one-loop open string amplitudes in the presence of a constant external background B are analysed in detail. While the pattern of the closed string spectrum, obtained after a careful study of the properly symmetrized amplitudes, turns out to be unaffected by the presence of B, a series of double open-string poles, which would be absent when B is turned off, can couple owing to a partial symmetry loss. These features are studied first in a bosonic setting and then generalized in the more satisfactory superstring context. When the backg…
The cutting equations are diagrammatic identities that are used to prove perturbative unitarity in quantum field theory. In this paper, we derive algebraic, upgraded versions of them. Differently from the diagrammatic versions, the algebraic identities also holds for propagators with arbitrary, nonvanishing widths. In particular, the cut propagators do not need to vanish off shell. The new approach provides a framework to address unsolved problems of perturbative quantum field theory and a tool to investigate perturbative unitarity in higher-derivative theories that are relevant to the problem…
We comment on recent results of a possible finite theory of Super Gravity from both Feynman graph and global E(7,7) symmetry arguments. The four point amplitude can be written as a series in the gravitational coupling and energy squared, Gs, multiplied by coefficient functions of the scattering angle. Every order of the perturbation expansion violates the unitarity bound for any element of the S matrix by powers for large values of the energy. While this conclusion is not unexpected; the meaning of such a theory is not understood. Either the theory is not finite or a non-perturbative completio…
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 …
