We establish that unitarity of scattering amplitudes imposes universal entropy bounds. The maximal entropy of a self-sustained quantum field object of radius R is equal to its surface area and at the same time to the inverse running coupling evaluated at the scale R. The saturation of these entropy bounds is in one-to-one correspondence with the non-perturbative saturation of unitarity by 2-to-N particle scattering amplitudes at the point of optimal truncation. These bounds are more stringent than Bekenstein's bound and in a consistent theory all three get saturated simultaneously. This is true for all known entropy-saturating objects such as solitons, instantons, baryons, oscillons, black holes or simply lumps of classical fields. We refer to these collectively as "saturons" and show that in renormalizable theories they behave in all other respects like black holes. Finally, it is argued that the confinement in SU(N) gauge theory can be understood as a direct consequence of the entropy bounds and unitarity.
We show that non-perturbative entities such as solitons and instantons saturate bounds on entropy when the theory saturates unitarity. Simultaneously, the entropy becomes equal to the area of the soliton/instanton. This is strikingly similar to black hole entropy despite absence of gravity. We explain why this similarity is not an accident. We present a formulation that allows to apply the entropy bound to instantons. The new formulation also eliminates apparent violations of the Bekenstein entropy bound by some otherwise-consistent unitary systems. We observe that in QCD, an isolated instanto…
The unitarity condition for scattering amplitudes in a non-anticommutative quantum field theory is investigated. The Cutkosky rules are shown to hold for Feynman diagrams in Euclidean momentum space and unitarity of amplitudes can be satisfied. An analytic continuation of the diagrams to physical Minkowski spacetime can be performed without invoking unphysical singularities in amplitudes. The high energy behavior of amplitudes is found to be regular at infinity provided that only space-space non-anticommutativity is allowed.
In this letter, we exploit generalised unitarity in order to calculate the cut-constructible part of one-loop amplitudes in non-supersymmetric Yang-Mills theory. In particular, we rederive the n-gluon MHV amplitudes for both the adjacent and non-adjacent gluon helicity configurations from 3- and 4-particle cuts alone.
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…
We present advances in the development of the numerical unitarity method for the computation of multi-loop amplitudes in QCD. As an application, we show results for all the leading-color two-loop five-gluon helicity amplitudes. The amplitudes are reduced to a linear combination of master integrals by employing unitarity-compatible integration-by-parts identities, and the corresponding integral coefficients are computed in an exact manner on rational phase-space points through finite fields arithmetics.
We show how generalised unitarity cuts in D = 4 - 2 epsilon dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions.
We present a calculation of the planar two-loop five-gluon amplitudes. The amplitudes are obtained in a variant of the generalized unitarity approach suitable for numerical computations, which we extend for use with finite field arithmetics. Employing a new method for the generation of unitarity-compatible integration-by-parts identities, all helicity amplitudes are reduced to a linear combination of master integrals for the first time. The approach allows us to compute exact values for the integral coefficients at rational phase-space points. All required master integrals are known analytical…
Here we discuss color-kinematics duality for higher-derivative QCD-like amplitudes. We explicitly show that the duality still holds in this case and it can be instrumental in constructing the associated quadratic-gravity amplitudes by using the double-copy prescription. This allows one to drastically simplify calculations. We also evaluate some tree-level Compton scattering amplitudes in higher-derivative Yang-Mills and quadratic gravity coupled with matter. Furthermore, we illustrate the application of generalized unitarity method for both cases by studying a specific one-loop amplitude.
We compute the $2\rightarrow 2$ gravitino scattering amplitudes at tree level in supergravity theories where supersymmetry is spontaneously broken. In the unitary gauge, the gravitino becomes massive (of mass $m_{3/2}$) by absorbing the Goldstino, and the scattering amplitudes of its longitudinal polarisations grow with energy as $κ^2 E^4/m_{3/2}^2$, signaling a potential breakdown of unitarity at a scale $Λ^2\sim m_{3/2}/κ\sim M_{\mathrm{SUSY}}^2$. As we show explicitly in the Polonyi model, this leading term is cancelled by the contributions coming from the scalar partner of the Goldstino (s…
We show, for previously uncalculated examples containing a uniform mass in the loop, that it is possible to obtain complete massive one-loop gauge theory amplitudes solely from unitarity and known ultraviolet or infrared mass singularities. In particular, we calculate four-gluon scattering via massive quark loops in QCD. The contribution of a heavy quark to five-gluon scattering with identical helicities is also presented.
We present and numerically implement a computational method to construct relativistic scattering amplitudes that obey analyticity, crossing, elastic and inelastic unitarity in three and four spacetime dimensions. The algorithm is based on the Mandelstam representation of the amplitude and iterations of unitarity. The input for the iterative procedure is given by the multi-particle double spectral density, the S-wave inelasticity, and the value of the amplitude at the crossing-symmetric point. The output, obtained at the fixed point of the iteration of unitarity, is a nonperturbative scattering…
The numerical unitarity approach has been important for obtaining reliable QCD predictions for the LHC. Here I discuss the extension of the approach beyond the leading quantum corrections for computing multi-loop amplitudes. The numerical unitarity approach requires a suitable parametrisation of the loop integrands as a sum of terms that integrate to zero (surface terms) and master integrands. The construction and classification of suitable surface terms which match the propagators structures of Feynman amplitudes is the main technical advance discussed. A number of spin-offs for integral redu…
We present a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory. As an example, we obtain the one-loop contribution to amplitudes for $n$ gluon scattering in $N=4$ supersymmetric Yang-Mills theory with the helicity configuration of the Parke-Taylor tree amplitudes. We prove that our $N=4$ ansatz is correct using general properties of the relevant one-loop $n$-point integrals. We also give the ``splitting amplitudes'' which govern the collinear behavior of one-loop helicity amplitudes in gauge theories.
We present the first numerical computation of two-loop amplitudes based on the unitarity method. As a proof of principle, we compute the four-gluon process. We discuss the new method, analyze its numerical properties and apply it to reconstruct the analytic form of the amplitudes. The numerical method is universal, and can be automated to provide multi-scale two-loop computations for phenomenologically relevant signatures at hadron colliders.
We present an alternative reduction to master integrals for one-loop amplitudes using a unitarity cut method in arbitrary dimensions. We carry out the reduction in two steps. The first step is a pure four-dimensional cut-integration of tree amplitudes with a mass parameter, and the second step is applying dimensional shift identities to master integrals. This reduction is performed at the integrand level, so that coefficients can be read out algebraically.
We present the first public version of Caravel, a C++17 framework for the computation of multi-loop scattering amplitudes in quantum field theory, based on the numerical unitarity method. Caravel is composed of modules for the $D$-dimensional decomposition of integrands of scattering amplitudes into master and surface terms, the computation of tree-level amplitudes in floating point or finite-field arithmetic, the numerical computation of one- and two-loop amplitudes in QCD and Einstein gravity, and functional reconstruction tools. We provide programs that showcase Caravel's main functionaliti…
It is well known that under the color-decomposition, one-loop amplitude of gluons contains partial amplitudes of single and double trace structures, and particularly all partial amplitudes of double trace structure can be expressed as a linear combination of partial amplitudes of single trace structure. Using unitarity cut method, we prove that this result is the natural consequence of tree-level Kleiss-Kuijf relation. Generalizing the unitarity cut method to two-loop (triple cut in this case), we show that, unlike the one-loop case, partial amplitudes of double and triple trace structures can…
We discuss the unitarity relation of the Aharonov-Bohm scattering amplitude with the hope that it distinguishes between the differing treatments which employ different incident waves. We find that the original Aharonov-Bohm scattering amplitude satisfies the unitarity relation under the regularization prescription whose theoretical foundation does not appear to be understood. On the other hand, the amplitude obtained by Ruijsenaars who uses plane wave as incident wave also satisfies the unitarity relation but in an unusual way.
We explain how one-loop amplitudes with massive fermions can be computed using only on-shell information. We first use the spinor-helicity formalism in six dimensions to perform generalised unitarity cuts in $d$ dimensions. We then show that divergent wavefunction cuts can be avoided, and the remaining ambiguities in the renormalised amplitudes can be fixed, by matching to universal infrared poles in $4-2ε$ dimensions and ultraviolet poles in $6-2ε$ dimensions. In the latter case we construct an effective Lagrangian in six dimensions and reduce the additional constraint to an on-shell tree-lev…
The unitarity equations for the boson interaction amplitudes in the superstring theory are used to calculate the interaction amplitudes including the Ramond states, which are 10-fermion and Ramond bosons. The n-loop, 4-point amplitude with two massless Neveu-Schwarz bosons and two massless Ramond states is obtained explicitly. It is shown that, in addition, the unitarity equations require some integral relations for local functions determining the amplitude. For the tree amplitude the validness of the above integral relations is verified.
We study the three-particle and four-particle scattering amplitudes for an arbitrary, finite number of massive scalars, spinors and vectors by employing the on-shell massive spinor formalism. We consider the most general three-particle amplitudes with energy growing behavior at most of $\mathcal{O} (E)$. This is the special case of the requirement of tree unitarity, which states that the $N$-particle scattering amplitudes at tree-level should grow at most as $\mathcal{O} (E^{4-N})$ in the high energy hard scattering limit, i.e. at fixed non-zero angles. Then the factorizable parts of the four-…
We study the impact of full unitarity on the moment structure of forward scattering amplitudes. We introduce the semiarcs, calculable quantities in the EFT dispersively related to both real and imaginary parts of the UV amplitude for a fixed number of subtractions. It is observed that large hierarchies between consecutive moments are forbidden by unitarity. Bounds from full unitarity compete with the ones stemming from convexity, and become more important in EFTs where the loop expansion is more important than the derivative expansion.
Any loop QCD amplitude at full colour is constructed from kinematic and gauge-group building blocks. In a unitarity-based on-shell framework, both objects can be reconstructed from their respective counterparts in tree-level amplitudes. This procedure is at its most powerful when aligned with flexible colour decompositions of tree-level QCD amplitudes. In this note we derive such decompositions for amplitudes with an arbitrary number of quarks and gluons from the same principle that is used to bootstrap kinematics - unitarity factorisation. In the process we formulate new multi-quark bases and…
We compute a complete set of independent leading-color two-loop five-parton amplitudes in QCD. These constitute a fundamental ingredient for the next-to-next-to-leading order QCD corrections to three-jet production at hadron colliders. We show how to consistently consider helicity amplitudes with external fermions in dimensional regularization, allowing the application of a numerical variant of the unitarity approach. Amplitudes are computed by exploiting a decomposition of the integrand into master and surface terms that is independent of the parton type. Master integral coefficients are nume…
In theories with unstable particles, unitarity is satisfied by the inclusion of only stable states in unitarity sums. Hence unitarity cuts are not to be taken through unstable particles. This raises a challenge to the generalized unitarity method, whose aim is to reconstruct amplitudes by analyzing sets of unitarity cuts. Nevertheless, under some general physical conditions, and perhaps some methodological modifications, we prove that the method is still reliable for one-loop amplitudes containing resonances. We discuss some simple examples which illustrate these features.
We show how the holomorphic anomaly found in hep-th/0409245 can be used to efficiently compute certain classes of unitarity cuts of one-loop N=4 amplitudes of gluons. These classes include all cuts of n-gluon one-loop MHV amplitudes and of n-gluon next-to-MHV amplitudes with helicities (1+,2+,3+,4-,..., n-). As an application of this method, we present the explicit computation of the (1,2,3)-cut of the n-gluon one-loop N=4 leading-color amplitude A_{n;1}(1+,2+,3+,4-,..., n-). The answer is given in terms of scalar box functions and provides information about the corresponding amplitudes. A pos…
We develop a manifestly supersymmetric version of the generalized unitarity cut method for calculating scattering amplitudes in N=4 SYM theory. We illustrate the power of this method by computing the one-loop n-point NMHV super-amplitudes. The result confirms two conjectures which we made in arXiv:0807.1095 [hep-th]. Firstly, we derive the compact, manifestly dual superconformally covariant form of the NMHV tree amplitudes for arbitrary number and types of external particles. Secondly, we show that the ratio of the one-loop NMHV to the MHV amplitude is dual conformal invariant.
We show that one-loop amplitudes in massless gauge theories can be determined from single cuts. By cutting a single propagator and putting it on-shell, the integrand of an n-point one-loop integral is transformed into an (n+2)-particle tree level amplitude. The single-cut approach described here is complementary to the double or multiple unitarity cut approaches commonly used in the literature. In common with these approaches, if the cut is taken in four dimensions, one finds only the cut-constructible parts of the amplitude, while if the cut is in D=4-2 epsilon dimensions, both rational and c…
Recently, it has been shown that the holomorphic anomaly of unitarity cuts can be used as a tool in determining the one-loop amplitudes in N=4 super Yang-Mills theory. It is interesting to examine whether this method can be applied to more general cases. We present results for a non-MHV N=1 supersymmetric one-loop amplitude. We show that the holomorphic anomaly of each unitarity cut correctly reproduces the action on the amplitude's imaginary part of the differential operators corresponding to collinearity in twistor space. We find that the use of the holomorphic anomaly to evaluate the amplit…
Recently, the spinor helicity formalism and on-shell superspace were developed for six-dimensional gauge theories with (1,1) supersymmetry. We combine these two techniques with (generalised) unitarity, which is a powerful technique to calculate scattering amplitudes in any massless theory. As an application we calculate one-loop superamplitudes with four and five external particles in the (1,1) theory and perform several consistency checks on our results.
We study the unitarity bounds of the scattering amplitudes in the extra dimensional gauge theory where the gauge symmetry is broken by the boundary condition. The estimation of the amplitude of the diagram including four massive gauge bosons in the external lines shows that the asymptotic power behavior of the amplitude is canceled. The calculation will be done in the 5 dimensional standard model and the SU(5) grand unified theory, whose 5th dimensional coordinate is compactified on $S^1/Z_2$. The broken gauge theories through the orbifolding preserve the unitarity at high energies similarly t…
We review the Four-Dimensional-Formulation variant of the Four-Dimensional-Helicity scheme, by showing two applications of this regularisation scheme. The first one is the computation of one-loop helicity amplitudes, for which we present preliminary results for the analytic expressions of the one-loop Higgs plus five- gluon amplitudes. In the second part, we study the Colour-Kinematics duality for off-shell diagrams in gauge theories coupled to matter, showing in a diagrammatic way that the Jacobi relations for the kinematic numerators of off-shell diagrams, built with Feynman rules in axial g…
We compute the polylogarithmic parts of the two-loop four and five-graviton amplitudes where the external helicities are positive and express these in a simple compact analytic form. We use these to extract the $\ln(μ^2)$ terms from both the four and five-point amplitudes and show that these match the same $R^3$ counterterm.
We combine the unitarity method with the six-dimensional helicity formalism of Cheung and O'Connell to construct loop-level scattering amplitudes. As a first example, we construct dimensionally regularized QCD one-loop four-point amplitudes. As a nontrivial multiloop example, we confirm that the recently constructed four-loop four-point amplitude of N=4 super-Yang-Mills theory, including nonplanar contributions, is valid for dimensions less than or equal to six. We comment on the connection of our approach to the recently discussed Higgs infrared regulator and on dual conformal properties in s…
We describe the unitarity approach for the numerical computation of two-loop integral coefficients of scattering amplitudes. It is well known that the leading propagator singularities of an amplitude's integrand are related to products of tree amplitudes. At two loops, Feynman diagrams with doubled propagators appear naturally, which lead to subleading pole contributions. In general, it is not known how these contributions can be directly expressed in terms of a product of on-shell tree amplitudes. We present a universal algorithm to extract these subleading pole terms by releasing some of the…
This thesis describes some of the recent (and some less recent) developments in calculational techniques for scattering amplitudes in quantum field theory. The focus is on on-shell recursion relations in complex momenta and on the use of unitarity methods for loop calculations. In particular, on-shell recursion is related to the MHV rules for computing tree-level gauge amplitudes and used to extend the MHV rules to graviton scattering. Combinations of unitarity cut techniques and recursion are used to argue for the "No-Triangle Hypothesis" in N=8 supergravity which is related to its UV behavio…
A spontaneously broken SU(2) theory is the simplest generalization of the Abelian Higgs model, containing three equally massive vector bosons and a single Higgs scalar. A strictly diagrammatic proof is presented of the tree-level unitarity of this model in the unitary gauge, i.e. employing only physical fields. A new Ward-like identity is used to show that the high-energy behaviour of all amplitudes cannot be more than quadratic; the use of generating functions for all off-shell amplitudes then leads to the unitarity proof.
Gravitational amplitudes in the Regge limit: waveforms, shock waves and unitarity cuts
INFN, Laboratori Nazionali di Frascati（意大利国家核物理研究所，弗拉斯cati国家实验室）；Queen Mary University of London（伦敦大学玛丽女王学院）；The University of Edinburgh（爱丁堡大学）；Sapienza Università di Roma（罗马第一大学）
We analyse the high-energy behavior of tree-level graviton Compton amplitudes for particles of mass m and arbitrary spin, concentrating on a combination of forward amplitudes that will be unaffected by eventual cross- couplings to other, higher spins. We first show that for any spin larger than 2, tree-level unitarity is already violated at energies well below the Planck scale M, if m << M. We then restore unitarity to this amplitude up to M by adding non-minimal couplings that depend on the curvature and its derivatives, and modify the minimal description - including particle gravitational qu…
We propose a systematic way to carry out the method introduced in hep-th/0410077 for computing certain unitarity cuts of one-loop N=4 amplitudes of gluons. We observe that the class of cuts for which the method works involves all next-to-MHV n-gluon one-loop amplitudes of any helicity configurations. As an application of our systematic procedure, we obtain the complete seven-gluon one-loop leading-color amplitude A_{7;1}(1-,2-,3-,4+,5+,6+,7+).
We discuss the black hole information problem, including the recent claim that unitarity requires a horizon firewall, emphasizing the role of decoherence and macroscopic superpositions. We consider the formation and evaporation of a large black hole as a quantum amplitude, and note that during intermediate stages (e.g., after the Page time), the amplitude is a superposition of macroscopically distinct (and decohered) spacetimes, with the black hole itself in different positions on different branches. Small but semiclassical observers (who are themselves part of the quantum amplitude) that fall…
From the low-energy effective theory of dilatons, consistent with the scale anomaly, we calculate the $2\to2$ scattering amplitudes of dilatons. We find that the one-loop amplitude violates the unitarity bound as the scattering energy approaches the cutoff of the effective theory, $\sqrt{s}=\sqrt{4π} f_D$. We then show that the inclusion of the next-to-lightest state, namely the spin-2 state, of mass around the cutoff improves the unitarity. The unitarity argument suggests that the mass ratio of the dilaton and the spin-2 state is proportional to the square of the Miransky-BKT scaling of the n…
In this paper, we generalize the unitarity method to two-loop diagrams and use it to discuss the integral bases of reduction. To test out method, we focus on the four-point double-box diagram as well as its related daughter diagrams, i.e., the double-triangle diagram and the triangle-box diagram. For later two kinds of diagrams, we have given complete analytical results in general (4-2\eps)-dimension.
We find a permutation relation among Yangian Invariants -- two Yangian Invariants with adjacent external lines exchanged are related by a simple kinematic factor. This relation is shown to be equivalent to U(1) decoupling and Bern-Carrasco-Johansson (BCJ) relation at the level of maximal helicity violating (MHV) amplitudes. Together with the newly found permutation relation and using unitarity cuts to remove ambiguity in the definition of loop momenta of cut amplitudes due to the nonplanar legs, we propose a systematic way of reconstructing the the integrands of nonplanar MHV amplitudes up to …
Four-particle tree-level scattering amplitudes in string theory are magically consistent with unitarity, reflected in the non-trivial fact that beneath the critical dimension, the residues of the amplitudes on massive poles can be expanded in partial waves with all positive coefficients. While this follows (rather indirectly) from the no-ghost theorem, the simplicity of the statement and its fundamental importance for the physical consistency of string theory begs for a more direct and elementary understanding. In this note we take a step in this direction by presenting a new expression for th…
In two recent papers, a new method was developed for calculating ten-dimensional superstring amplitudes with an arbitrary number of loops and external massless particles, and for expressing them in manifestly Lorentz-invariant form. By explicitly checking for divergences when the Riemann surface degenerates, these amplitudes are proven to be finite. By choosing light-cone moduli for the surface and comparing with the light-cone Green-Schwarz formalism, these amplitudes are proven to be unitary.
We consider the S-matrix of c=1 Liouville theory with vanishing cosmological constant. We examine some of the constraints imposed by unitarity. These completely determine (N,2) amplitudes at tree level in terms of the (N,1) amplitudes when the plus tachyon momenta take generic values. A surprising feature of the matrix model results is the lack of particle creation branch cuts in the higher genus amplitudes. In fact, we show the naive field theory limit of Liouville theory would predict such branch cuts. However, unitarity in the full string theory ensures that such cuts do not appear in genus…
The interplay of unitarity and analyticity has long been known to impose strong constraints on scattering amplitudes in quantum field theory and string theory. This has been highlighted in recent times in a number of papers and lecture notes. Here we examine such conditions in the context of superstring tree-level scattering amplitudes, leading to positivity constraints on determinants of Hankel matrices involving polynomials of multiple zeta values. These generalise certain constraints on polynomials of single zeta values in the mathematics literature.
