Journal articles on the topic 'S-Matrix Bootstrap'
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1
Sivers, Dennis, and Jack L. Uretsky. "S-matrix bootstrap of a scalar Higgs boson." Physical Review Letters 68, no. 11 (March 16, 1992): 1649–52. http://dx.doi.org/10.1103/physrevlett.68.1649.
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KOUBEK, A., G. MUSSARDO, and R. TATEO. "BOOTSTRAP TREES AND CONSISTENT S MATRICES." International Journal of Modern Physics A 07, no. 15 (June 20, 1992): 3435–45. http://dx.doi.org/10.1142/s0217751x92001526.
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We analyze the tree structure arising from the recursive bootstrap equations, given the S matrix of the lightest particle. When S11 contains only one singularity, among all possible bootstrap systems, the only ones which give rise to a consistent set of S matrices coincide with those of sine–Gordon breathers at the reduction point ξ = 2π/(2n + 1). We also present our investigation of bootstrap systems defined by an S11 with a higher number of singularities. The only consistent examples we found belong to the set of minimal S matrices corresponding to Dynkin diagrams.
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KOUBEK, A. "S MATRICES OF ϕ1,2-PERTURBED MINIMAL MODELS: IRF FORMULATION AND BOOTSTRAP PROGRAM." International Journal of Modern Physics A 09, no. 11 (April 30, 1994): 1909–27. http://dx.doi.org/10.1142/s0217751x94000820.
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We analyze the algebraic structure of ϕ1,2-perturbed minimal models, relating them to graph state models with an underlying Birman-Wenzl-Murakami algebra. Using this approach many physical properties of the IRF amplitudes can be derived, as the phase shifts in the ultraviolet limit, crossing relations and a simple formulation of the bootstrap equations. These are used to calculate explicitly the S matrix elements of higher kinks, and to determine the breather spectrum of the ϕ1,2 perturbations of the unitary minimal models ℳr,r+1. Finally it is checked that the bootstrap closes on those particles.
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Cadamuro, Daniela, and Yoh Tanimoto. "Wedge-local observables for factorizing S-matrix with gap in the coupling constant." Reviews in Mathematical Physics 30, no. 04 (April 19, 2018): 1850010. http://dx.doi.org/10.1142/s0129055x18500101.
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In the bootstrap approach to integrable quantum field theories in the [Formula: see text]-dimensional Minkowski space, one conjectures the two-particle S-matrix and tries to study local observables. We find a family of two-particle S-matrices parametrized by two positive numbers, which are separated from the free field or any other known S-matrix. We propose candidates for observables in wedge-shaped regions and prove their commutativity in the weak sense. The sine-Gordon model is conjectured to be equivalent to the Thirring model, and its breather–breather S-matrix components (where the first breather corresponds to the scalar field of the sine-Gordon model) are closed under fusion. Yet, the residues of the poles in this breather–breather S-matrix have wrong signs and cannot be considered as a separate model. Our S-matrices differ from the breather–breather S-matrix in the sine-Gordon model by CDD factors which adjust the signs, so that this sector alone satisfies reasonable assumptions.
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SASAKI, RYU, and FREDDY PERMANA ZEN. "THE AFFINE TODA S-MATRICES vs PERTURBATION THEORY." International Journal of Modern Physics A 08, no. 01 (January 10, 1993): 115–34. http://dx.doi.org/10.1142/s0217751x93000059.
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We present perturbative calculations for the Affine Toda Field Theory (ATFT) S-matrices to the second order in the coupling constants for [Formula: see text] and [Formula: see text] in general, to the fourth order for [Formula: see text] theory as well as to the sixth order for [Formula: see text] theory. Conventional Feynman–Dyson calculation method and the dispersion approach are used to calculate the complete form of the perturbation amplitudes in contrast to the pole residues in previous papers. The results agree with those S-matrices obtained in the S-matrix approach, namely those based on analyticity, unitarity, crossing and bootstrap equation.
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Sauer, Lauren, Yuhlong Lio, and Tzong-Ru Tsai. "Reliability Inference for the Multicomponent System Based on Progressively Type II Censored Samples from Generalized Pareto Distributions." Mathematics 8, no. 7 (July 17, 2020): 1176. http://dx.doi.org/10.3390/math8071176.
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In this paper, the reliability of a k-component system, in which all components are subject to common stress, is considered. The multicomponent system will continue to survive if at least s out of k components’ strength exceed the common stress. The system reliability is investigated by utilizing the maximum likelihood estimator based on progressively type II censored samples from generalized Pareto distributions. The confidence interval of the system reliability can be obtained by using asymptotic normality with Fisher information matrix or bootstrap method approximation. An intensive simulation study is conducted to evaluate the performance of maximum likelihood estimators of the model parameters and system reliability for a variety of cases. For the confidence interval of the system reliability, simulation results indicate the bootstrap method approximation outperforms over the asymptotic normality approximation in terms of coverage probability.
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YOUNG, CATHERINE J. "Characterisation of the Australian Nacophorini using adult morphology, and phylogeny of the Geometridae based on morphological characters." Zootaxa 1736, no. 1 (March 31, 2008): 1. http://dx.doi.org/10.11646/zootaxa.1736.1.1.
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The Australian Nacophorini and related taxa are described using a matrix of 116 adult morphological characters. Adults of 72 species are illustrated using photographs and electron micrographs. Subsets of the characters are used to conduct a phylogenetic analysis based on cladistic principles. The adult morphological character set was augmented with 17 characters from eggs and 27 from larvae. The resulting phylogeny is poorly resolved but provides support for many of the relationships recovered by previous molecular analyses of the group, including basal derivations for characters of Larentiinae and Sterrhinae relative to those of the rest of Geometridae, and the monophyly of the Geometrinae + Oenochrominae s. str. Combining 28S D2 datawith morphological data produced a matrix of 60 taxa and 590 characters. The majority rule consensus tree produced by the combined morphological and 28S D2 data is almost identical to the majority rule consensus tree produced by the 28S D2 data alone, except that bootstrap support is lower for most nodes. Common clades obtained from the molecular and morphological trees are described in terms of morphological data. On this basis a concept of the Australian Nacophorini includes Lithinini and Australian Archiearinae. Two robust groups within the tribe also are delimited using characters from all data sources. Comparsions are made between the Nearctic and Neotropical Nacophorini on the basis of shared morphological characters. Australian Boarmiini are defined by synapomorphies.
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Kim, Byungsuk, Youngjoo Byun, and Eun Hee Lee. "DoE-Based Design of a Simple but Efficient Preparation Method for a Non-Effervescent Gastro-Retentive Floating Tablet Containing Metformin HCl." Pharmaceutics 13, no. 8 (August 8, 2021): 1225. http://dx.doi.org/10.3390/pharmaceutics13081225.
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A sustained-release non-effervescent floating matrix tablet was prepared using a simple and efficient direct compression of spray-dried granules containing metformin hydrochloride and cetyl alcohol with hydroxypropyl methylcellulose K15M (HPMC K15M). The design of experiments was employed to explore the optimal composition of the tablet. The similarity factor was employed to evaluate the equivalence in dissolution profiles between the test tablets and Glucophage XR as a reference. Bootstrap analysis was used to eliminate the formulations for which the dissolution profile was potentially inequivalent to that of the reference. The optimized tablet consisting of 150 mg of cetyl alcohol and 17% HPMC K15M showed a dissolution profile comparable with that of the reference with a similarity factor of 52.41, exhibited a floating lag time of less than 3 s in buffer media, remained floating for 24 h, and reduced the tablet weight by about 20% compared to that of the reference. The current study sheds light on the potential use of non-effervescent gastro-retentive extended-release tablets for high-dose drugs using a simple and efficient direct compression method, and as a potential alternative treatment for Glucophage XR. This study also highlights the importance of a systematic approach to formulation optimization and the evaluation of the dissolution profile.
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Freudenberg, J., H. Boriss, and D. Hasenclever. "Comparison of Preprocessing Procedures for Oligo-nucleotide Micro-arrays by Parametric Bootstrap Simulation of Spike-in Experiments." Methods of Information in Medicine 43, no. 05 (2004): 434–38. http://dx.doi.org/10.1055/s-0038-1633893.
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Summary Objective: Due to scarcity of calibration data for micro-array experiments, simulation methods are employed to assess preprocessing procedures. Here we analyze several procedures’ robustness against increasing numbers of differentially expressed genes and varying proportions of up-regulation. Methods: Raw probe data from oligo-nucleotide micro-arrays are assumed to be approximately multivariate normally distributed on the log scale. Chips can be simulated from a multivariate normal distribution with mean and variance-covariance matrix estimated from a real raw data set.A chip effect induces strong positive correlations. In reverse, sampling from a normal distribution with strong correlation variance-covariance matrix generates data exhibiting a chip effect. No explicit model of chip-effect is needed. Differences can be artificially spiked-in according to a given distribution of effect sizes.Thirty preprocessing procedures combining background correction, normalization, perfect match correction and summarization methods available from the BioConductor project were compared. Results: In the symmetrical setting “50% differentially expressed genes, 50% of which up-regulated” background correction reduces bias, but inflates low intensity probe variance as well as the mean squared error of the estimates. Any normalization reduces variance and increases sensitivity with no clear winner. Asymmetry between up and down regulation causes bias in the effect-size estimate of non-differentially expressed genes. This markedly inflates the false positive discovery rates. Variance stabilizing normalization (VSN) behaved best. Conclusion: A simple parametric bootstrap was used to simulate oligo-nucleotide micro-array raw data. Current normalization methods inflate the false positive rate when many genes show an effect in the same direction.
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Miró, Joan Elias, Andrea L. Guerrieri, Aditya Hebbar, João Penedones, and Pedro Vieira. "Flux Tube S -Matrix Bootstrap." Physical Review Letters 123, no. 22 (November 27, 2019). http://dx.doi.org/10.1103/physrevlett.123.221602.
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Doroud, N., and J. Elias Miró. "S-matrix bootstrap for resonances." Journal of High Energy Physics 2018, no. 9 (September 2018). http://dx.doi.org/10.1007/jhep09(2018)052.
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Hebbar, Aditya, Denis Karateev, and João Penedones. "Spinning S-matrix bootstrap in 4d." Journal of High Energy Physics 2022, no. 1 (January 2022). http://dx.doi.org/10.1007/jhep01(2022)060.
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Abstract We review unitarity and crossing constraints on scattering amplitudes for particles with spin in four dimensional quantum field theories. As an application we study two to two scattering of neutral spin 1/2 fermions in detail. Assuming Mandelstam analyticity of its scattering amplitude, we use the numerical S-matrix bootstrap method to estimate various non-perturbative bounds on quartic and cubic (Yukawa) couplings.
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Ghosh, Sudip, Prashanth Raman, and Aninda Sinha. "Celestial insights into the S-matrix bootstrap." Journal of High Energy Physics 2022, no. 8 (August 22, 2022). http://dx.doi.org/10.1007/jhep08(2022)216.
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Abstract 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 that seems to be present in a wide class of theories. We prove that this positivity arises only in theories with a spin-0 dominance. The crossing symmetric partial waves with spurious singularities removed, dubbed as Feynman blocks, have remarkable properties in the Celestial variable, namely typically realness, in the sense of Geometric Function Theory (GFT). Using GFT techniques we derive non-projective bounds on Wilson coefficients in terms of partial wave moments.
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Bose, Anjishnu, Aninda Sinha, and Shaswat Tiwari. "Selection rules for the S-Matrix bootstrap." SciPost Physics 10, no. 5 (May 31, 2021). http://dx.doi.org/10.21468/scipostphys.10.5.122.
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We examine the space of allowed S-matrices on the Adler zeros' plane using the recently resurrected (numerical) S-matrix bootstrap program for pion scattering. Two physical quantities, an averaged total scattering cross-section, and an averaged entanglement power for the boundary S-matrices, are studied. Emerging linearity in the leading Regge trajectory is correlated with a reduction in both these quantities. We identify two potentially viable regions where the S-matrices give decent agreement with low energy S- and P-wave scattering lengths and have leading Regge trajectory compatible with experiments. We also study the line of minimum averaged total cross section in the Adler zeros' plane. The Lovelace-Shapiro model, which was a precursor to modern string theory, is given by a straight line in the Adler zeros' plane and, quite remarkably, we find that this line intersects the space of allowed S-matrices near both these regions.
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Homrich, Alexandre, João Penedones, Jonathan Toledo, Balt C. van Rees, and Pedro Vieira. "The S-matrix bootstrap IV: multiple amplitudes." Journal of High Energy Physics 2019, no. 11 (November 2019). http://dx.doi.org/10.1007/jhep11(2019)076.
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Córdova, Lucía, and Pedro Vieira. "Adding flavour to the S-matrix bootstrap." Journal of High Energy Physics 2018, no. 12 (December 2018). http://dx.doi.org/10.1007/jhep12(2018)063.
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Miró, Joan Elias, Andrea Guerrieri, and Mehmet Asım Gümüş. "Bridging positivity and S-matrix bootstrap bounds." Journal of High Energy Physics 2023, no. 5 (May 2, 2023). http://dx.doi.org/10.1007/jhep05(2023)001.
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Abstract 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 boundary of the allowed amplitude space by introducing suitable observables. Finally, we show that this construction also leads to novel bounds on operators of dimension less than or equal to six.
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Córdova, Lucía, Yifei He, and Miguel F. Paulos. "From conformal correlators to analytic S-matrices: CFT1/QFT2." Journal of High Energy Physics 2022, no. 8 (August 19, 2022). http://dx.doi.org/10.1007/jhep08(2022)186.
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Abstract We study families of one-dimensional CFTs relevant for describing gapped QFTs in AdS2. 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 satisfy extended unitarity conditions. We also carefully prove that for physical kinematics the S-matrix may be more directly described by a phase shift formula. Our results crucially depend on the assumption of a certain gap in the spectrum of operators. We bootstrap perturbative AdS bubble, triangle and box diagrams and find that the presence of anomalous thresholds in S-matrices are precisely signaled by an unbounded OPE arising from violating this assumption. Finally we clarify the relation between unitarity saturating S-matrices and extremal CFTs, establish a mapping between the dual S-matrix and CFT bootstraps, and discuss how our results help understand UV completeness or lack thereof for specific S-matrices.
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Guerrieri, Andrea L., João Penedones, and Pedro Vieira. "S-matrix bootstrap for effective field theories: massless pions." Journal of High Energy Physics 2021, no. 6 (June 2021). http://dx.doi.org/10.1007/jhep06(2021)088.
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Abstract We use the numerical S-matrix bootstrap method to obtain bounds on the two leading Wilson coefficients (or low energy constants) of the chiral lagrangian controlling the low-energy dynamics of massless pions. This provides a proof of concept that the numerical S-matrix bootstrap can be used to derive non-perturbative bounds on massless EFTs in more than two spacetime dimensions.
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Dutta Chowdhury, Subham, Kausik GHOSH, Parthiv Haldar, Prashanth Raman, and Aninda Sinha. "Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds." SciPost Physics 13, no. 3 (September 5, 2022). http://dx.doi.org/10.21468/scipostphys.13.3.051.
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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 amplitudes and locality constraints for the massive neutral Majorana fermions and parity violating photon and graviton theories. The techniques developed in this paper will be useful for considering numerical S-matrix bootstrap in the future.
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Correia, Miguel, Amit Sever, and Alexander Zhiboedov. "An analytical toolkit for the S-matrix bootstrap." Journal of High Energy Physics 2021, no. 3 (March 2021). http://dx.doi.org/10.1007/jhep03(2021)013.
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Abstract We revisit analytical methods for constraining the nonperturbative S-matrix of unitary, relativistic, gapped theories in d≥ 3 spacetime dimensions. We assume extended analyticity of the two-to-two scattering amplitude and use it together with elastic unitarity to develop two natural expansions of the amplitude. One is the threshold (non-relativistic) expansion and the other is the large spin expansion. The two are related by the Froissart-Gribov inversion formula. When combined with crossing and a local bound on the discontinuity of the amplitude, this allows us to constrain scattering at finite energy and spin in terms of the low-energy parameters measured in the experiment. Finally, we discuss the modern numerical approach to the S-matrix bootstrap and how it can be improved based on the results of our analysis.
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Bercini, Carlos, Matheus Fabri, Alexandre Homrich, and Pedro Vieira. "S-matrix bootstrap: Supersymmetry, Z2 , and Z4 symmetry." Physical Review D 101, no. 4 (February 28, 2020). http://dx.doi.org/10.1103/physrevd.101.045022.
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Paulos, Miguel F., Joao Penedones, Jonathan Toledo, Balt C. van Rees, and Pedro Vieira. "The S-matrix bootstrap II: two dimensional amplitudes." Journal of High Energy Physics 2017, no. 11 (November 2017). http://dx.doi.org/10.1007/jhep11(2017)143.
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Guerrieri, Andrea L., Alexandre Homrich, and Pedro Vieira. "Dual S-matrix bootstrap. Part I. 2D theory." Journal of High Energy Physics 2020, no. 11 (November 2020). http://dx.doi.org/10.1007/jhep11(2020)084.
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Abstract Using duality in optimization theory we formulate a dual approach to the S-matrix bootstrap that provides rigorous bounds to 2D QFT observables as a consequence of unitarity, crossing symmetry and analyticity of the scattering matrix. We then explain how to optimize such bounds numerically, and prove that they provide the same bounds obtained from the usual primal formulation of the S-matrix Bootstrap, at least once convergence is attained from both perspectives. These techniques are then applied to the study of a gapped system with two stable particles of different masses, which serves as a toy model for bootstrapping popular physical systems.
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Chen, Hongbin, A. Liam Fitzpatrick, and Denis Karateev. "Bootstrapping 2d ϕ4 theory with Hamiltonian truncation data." Journal of High Energy Physics 2022, no. 2 (February 2022). http://dx.doi.org/10.1007/jhep02(2022)146.
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Abstract We combine the methods of Hamiltonian Truncation and the recently proposed generalisation of the S-matrix bootstrap that includes local operators to determine the two-particle scattering amplitude and the two-particle form factor of the stress tensor at s > 0 in the 2d ϕ4 theory. We use the form factor of the stress tensor at s ≤ 0 and its spectral density computed using Lightcone Conformal Truncation (LCT), and inject them into the generalized S-matrix bootstrap set-up. The obtained results for the scattering amplitude and the form factor are fully reliable only in the elastic regime. We independently construct the “pure” S-matrix bootstrap bounds (bootstrap without including matrix elements of local operators), and find that the sinh-Gordon model and its analytic continuation the “staircase model” saturate these bounds. Surprisingly, the ϕ4 two-particle scattering amplitude also very nearly saturates these bounds, and moreover is extremely close to that of the sinh-Gordon/staircase model.
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Paulos, Miguel F., Joao Penedones, Jonathan Toledo, Balt C. van Rees, and Pedro Vieira. "The S-matrix bootstrap. Part III: higher dimensional amplitudes." Journal of High Energy Physics 2019, no. 12 (December 2019). http://dx.doi.org/10.1007/jhep12(2019)040.
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Bose, Anjishnu, Parthiv Haldar, Aninda Sinha, Pritish Sinha, and Shaswat Tiwari. "Relative entropy in scattering and the S-matrix bootstrap." SciPost Physics 9, no. 5 (November 24, 2020). http://dx.doi.org/10.21468/scipostphys.9.5.081.
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We consider entanglement measures in 2-2 scattering in quantum field theories, focusing on relative entropy which distinguishes two different density matrices. Relative entropy is investigated in several cases which include \phi^4ϕ4 theory, chiral perturbation theory (\chi PTχPT) describing pion scattering and dilaton scattering in type II superstring theory. We derive a high energy bound on the relative entropy using known bounds on the elastic differential cross-sections in massive QFTs. In \chi PTχPT, relative entropy close to threshold has simple expressions in terms of ratios of scattering lengths. Definite sign properties are found for the relative entropy which are over and above the usual positivity of relative entropy in certain cases. We then turn to the recent numerical investigations of the S-matrix bootstrap in the context of pion scattering. By imposing these sign constraints and the \rhoρ resonance, we find restrictions on the allowed S-matrices. By performing hypothesis testing using relative entropy, we isolate two sets of S-matrices living on the boundary which give scattering lengths comparable to experiments but one of which is far from the 1-loop \chi PTχPT Adler zeros. We perform a preliminary analysis to constrain the allowed space further, using ideas involving positivity inside the extended Mandelstam region, and other quantum information theoretic measures based on entanglement in isospin.
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Paulos, Miguel F., Joao Penedones, Jonathan Toledo, Balt C. van Rees, and Pedro Vieira. "The S-matrix bootstrap. Part I: QFT in AdS." Journal of High Energy Physics 2017, no. 11 (November 2017). http://dx.doi.org/10.1007/jhep11(2017)133.
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Kruczenski, Martin, and Harish Murali. "The R-matrix bootstrap for the 2d O(N) bosonic model with a boundary." Journal of High Energy Physics 2021, no. 4 (April 2021). http://dx.doi.org/10.1007/jhep04(2021)097.
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Abstract The S-matrix bootstrap is extended to a 1+1d theory with O(N) symmetry and a boundary in what we call the R-matrix bootstrap since the quantity of interest is the reflection matrix (R-matrix). Given a bulk S-matrix, the space of allowed R-matrices is an infinite dimensional convex space from which we plot a two dimensional section given by a convex domain on a 2d plane. In certain cases, at the boundary of the domain, we find vertices corresponding to integrable R-matrices with no free parameters. In other cases, when there is a one-parameter family of integrable R-matrices, the whole boundary represents integrable theories. We also consider R-matrices which are analytic in an extended region beyond the physical cuts, thus forbidding poles (resonances) in that region. In certain models, this drastically reduces the allowed space of R-matrices leading to new vertices that again correspond to integrable theories. We also work out the dual problem, in particular in the case of extended analyticity, the dual function has cuts on the physical line whenever unitarity is saturated. For the periodic Yang-Baxter solution that has zero transmission, we computed the R-matrix initially using the bootstrap and then derived its previously unknown analytic form.
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He, Yifei, and Martin Kruczenski. "S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem." Journal of High Energy Physics 2021, no. 8 (August 2021). http://dx.doi.org/10.1007/jhep08(2021)125.
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Abstract The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the 2 → 2 scattering matrix S2→2 such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider 3 + 1 dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves kℓ(s) that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves fℓ(s), for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.
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Bielli, Daniele, Vaibhav Gautam, and Alessandro Torrielli. "A study of integrable form factors in massless relativistic AdS2." Journal of High Energy Physics 2023, no. 6 (June 1, 2023). http://dx.doi.org/10.1007/jhep06(2023)005.
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Abstract In this paper we initiate the study of form factors for the massless scattering of integrable AdS2 superstrings, where the difference-form of the S-matrix can be exploited to implement the relativistic form factor bootstrap. The non-standard nature of the S-matrix implies that traditional methods do not apply. We use the fact that the massless AdS2S-matrix is a limit of a better-behaved S-matrix found by Fendley. We show that the previously conjectured massless AdS2 dressing factor coincides with the limit of the De Martino-Moriconi improved dressing factor for the Fendley S-matrix. We then solve the form factor constraints in the two-particle case. Along the way we find a method to construct integral representations of relativistic dressing factors satisfying specific assumptions, and use it to obtain analytic proofs of crossing and unitarity relations.
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Correia, Miguel, João Penedones, and Antoine Vuignier. "Injecting the UV into the bootstrap: Ising Field Theory." Journal of High Energy Physics 2023, no. 8 (August 18, 2023). http://dx.doi.org/10.1007/jhep08(2023)108.
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Abstract We merge together recent developments in the S-matrix bootstrap program to develop a dual setup in 2 space-time dimensions incorporating scattering amplitudes of massive particles and matrix elements of local operators. In particular, the stress energy tensor allows us to input UV constraints on IR observables in terms of the central charge cUV of the UV Conformal Field Theory. We consider two applications: (1) We establish a rigorous lower bound on cUV of a class of ℤ2 symmetric scalar theories in the IR (including ϕ4); (2) We target Ising Field Theory by, first, minimizing cUV for different values of the magnetic field and, secondly, by determining the allowed range of cubic coupling and one-particle form-factor for fixed cUV = 1/2 and magnetic field.
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He, Yifei, Andrew Irrgang, and Martin Kruczenski. "A note on the S-matrix bootstrap for the 2d O(N) bosonic model." Journal of High Energy Physics 2018, no. 11 (November 2018). http://dx.doi.org/10.1007/jhep11(2018)093.
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Gabai, Barak, and Xi Yin. "On the S-matrix of Ising field theory in two dimensions." Journal of High Energy Physics 2022, no. 10 (October 26, 2022). http://dx.doi.org/10.1007/jhep10(2022)168.
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Abstract We explore the analytic structure of the non-perturbative S-matrix in arguably the simplest family of massive non-integrable quantum field theories: the Ising field theory (IFT) in two dimensions, which may be viewed as the Ising CFT deformed by its two relevant operators, or equivalently, the scaling limit of the Ising model in a magnetic field. Our strategy is that of collider physics: we employ Hamiltonian truncation method (TFFSA) to extract the scattering phase of the lightest particles in the elastic regime, and combine it with S-matrix bootstrap methods based on unitarity and analyticity assumptions to determine the analytic continuation of the 2 → 2 S-matrix element to the complex s-plane. Focusing primarily on the “high temperature” regime in which the IFT interpolates between that of a weakly coupled massive fermion and the E8 affine Toda theory, we will numerically determine 3-particle amplitudes, follow the evolution of poles and certain resonances of the S-matrix, and exclude the possibility of unknown wide resonances up to reasonably high energies.
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Tourkine, Piotr, and Alexander Zhiboedov. "Scattering from production in 2d." Journal of High Energy Physics 2021, no. 7 (July 2021). http://dx.doi.org/10.1007/jhep07(2021)228.
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Abstract In 1968, Atkinson proved the existence of functions that satisfy all S-matrix axioms in four spacetime dimensions. His proof is constructive and to our knowledge it is the only result of this type. Remarkably, the methods to construct such functions used in the proof were never implemented in practice. In the present paper, we test the applicability of those methods in the simpler setting of two-dimensional S-matrices. We solve the problem of reconstructing the scattering amplitude starting from a given particle production probability. We do this by implementing two numerical iterative schemes (fixed-point iteration and Newton’s method), which, by iterating unitarity and dispersion relations, converge to solutions to the S-matrix axioms. We characterize the region in the amplitude-space in which our algorithms converge, and discover a fractal structure connected to the so-called CDD ambiguities which we call “CDD fractal”. To our surprise, the question of convergence naturally connects to the recent study of the coupling maximization in the two-dimensional S-matrix bootstrap. The methods exposed here pave the way for applications to higher dimensions, and expose some of the potential challenges that will have to be overcome.
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Cortes Cubero, Axel. "Integrable Floquet QFT: Elasticity and factorization under periodic driving." SciPost Physics 5, no. 3 (September 21, 2018). http://dx.doi.org/10.21468/scipostphys.5.3.025.
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In (1+1)-dimensional quantum field theory, integrability is typically defined as the existence of an infinite number of local charges of different Lorentz spin, which commute with the Hamiltonian. A well known consequence of integrability is that scattering of particles is elastic and factorizable. These properties are the basis for the bootstrap program, which leads to the exact computation of S-matrices and form factors. We consider periodically-driven field theories, whose stroboscopic time-evolution is described by a Floquet Hamiltonian. It was recently proposed by Gritsev and Polkovnikov that it is possible for some form of integrability to be preserved even in driven systems. If a driving protocol exists such that the Floquet Hamiltonian is integrable (such that there is an infinite number of local and independent charges, a subset of which are parity-even, that commute with it), we show that there are strong conditions on the stroboscopic time evolution of particle trajectories, analogous to S-matrix elasticity and factorization. We propose a new set of axioms for the time evolution of particles which outline a new bootstrap program, which can be used to identify and classify integrable Floquet protocols. We present some simple examples of driving protocols where Floquet integrability is manifest; in particular, we also show that under certain conditions, some integrable protocols proposed by Gritsev and Polkovnikov are solutions of our new bootstrap equations.
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Antunes, António, Miguel S. Costa, João Penedones, Aaditya Salgarkar, and Balt C. van Rees. "Towards bootstrapping RG flows: sine-Gordon in AdS." Journal of High Energy Physics 2021, no. 12 (December 2021). http://dx.doi.org/10.1007/jhep12(2021)094.
Full textAbstract:
Abstract The boundary correlation functions for a Quantum Field Theory (QFT) in an Anti-de Sitter (AdS) background can stay conformally covariant even if the bulk theory undergoes a renormalization group (RG) flow. Studying such correlation functions with the numerical conformal bootstrap leads to non-perturbative constraints that must hold along the entire flow. In this paper we carry out this analysis for the sine-Gordon RG flows in AdS2, which start with a free (compact) scalar in the UV and end with well-known massive integrable theories that saturate many S-matrix bootstrap bounds. We numerically analyze the correlation functions of both breathers and kinks and provide a detailed comparison with perturbation theory near the UV fixed point. Our bounds are often saturated to one or two orders in perturbation theory, as well as in the flat-space limit, but not necessarily in between.
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Caron-Huot, Simon, Dalimil Mazáč, Leonardo Rastelli, and David Simmons-Duffin. "AdS bulk locality from sharp CFT bounds." Journal of High Energy Physics 2021, no. 11 (November 2021). http://dx.doi.org/10.1007/jhep11(2021)164.
Full textAbstract:
Abstract It is a long-standing conjecture that any CFT with a large central charge and a large gap ∆gap in the spectrum of higher-spin single-trace operators must be dual to a local effective field theory in AdS. We prove a sharp form of this conjecture by deriving numerical bounds on bulk Wilson coefficients in terms of ∆gap using the conformal bootstrap. Our bounds exhibit the scaling in ∆gap expected from dimensional analysis in the bulk. Our main tools are dispersive sum rules that provide a dictionary between CFT dispersion relations and S-matrix dispersion relations in appropriate limits. This dictionary allows us to apply recently-developed flat-space methods to construct positive CFT functionals. We show how AdS4 naturally resolves the infrared divergences present in 4D flat-space bounds. Our results imply the validity of twice-subtracted dispersion relations for any S-matrix arising from the flat-space limit of AdS/CFT.
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Duary, Sarthak. "Celestial amplitude for 2d theory." Journal of High Energy Physics 2022, no. 12 (December 12, 2022). http://dx.doi.org/10.1007/jhep12(2022)060.
Full textAbstract:
Abstract We explore celestial amplitude corresponding to 2d bulk $$ \mathcal{S} $$ S -matrix. We consider scalar particles with identical mass and show that the celestial amplitude becomes the fourier transform of the 2d$$ \mathcal{S} $$ S -matrix written in the rapidity variable. We translate the crossing and unitarity conditions into the conditions on the celestial amplitude. For the 2d Sinh-Gordon model, we calculate the celestial amplitude perturbatively in coupling constant and check that the crossing and unitarity conditions are satisfied for the celestial amplitude. Imposing the crossing and unitarity conditions to the celestial amplitude, we want to find amplitudes to the higher order in perturbation theory from the lower order i.e., to provide a “proof of principle” to show we can apply the bootstrap idea to the celestial amplitude. We find that imposing the crossing and unitarity conditions is not enough for bootstrapping celestial amplitude, there is an extra term which can’t be fixed by the crossing and unitarity conditions. We also study the gravitational dressing condition in 2d QFT for massless particles in celestial space and see that for the gravitationally dressed celestial amplitude, the poles on the right half-plane get erased for several ansatzes.
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Haldar, Parthiv, Aninda Sinha, and Ahmadullah Zahed. "Quantum field theory and the Bieberbach conjecture." SciPost Physics 11, no. 1 (July 8, 2021). http://dx.doi.org/10.21468/scipostphys.11.1.002.
Full textAbstract:
An intriguing correspondence between ingredients in geometric function theory related to the famous Bieberbach conjecture (de Branges’ theorem) and the non-perturbative crossing symmetric representation of 2-2 scattering amplitudes of identical scalars is pointed out. Using the dispersion relation and unitarity, we are able to derive several inequalities, analogous to those which arise in the discussions of the Bieberbach conjecture. We derive new and strong bounds on the ratio of certain Wilson coefficients and demonstrate that these are obeyed in one-loop \phi^4ϕ4 theory, tree level string theory as well as in the S-matrix bootstrap. Further, we find two sided bounds on the magnitude of the scattering amplitude, which are shown to be respected in all the contexts mentioned above. Translated to the usual Mandelstam variables, for large |s||s|, fixed tt, the upper bound reads |\mathcal{M}(s,t)|\lesssim |s^2||ℳ(s,t)|≲|s2|. We discuss how Szeg"{o}’s theorem corresponds to a check of univalence in an EFT expansion, while how the Grunsky inequalities translate into nontrivial, nonlinear inequalities on the Wilson coefficients.
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Alday, Luis F., Shai M. Chester, and Tobias Hansen. "Modular invariant holographic correlators for $$ \mathcal{N} $$ = 4 SYM with general gauge group." Journal of High Energy Physics 2021, no. 12 (December 2021). http://dx.doi.org/10.1007/jhep12(2021)159.
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Abstract We study the stress tensor four-point function for $$ \mathcal{N} $$ N = 4 SYM with gauge group G = SU(N), SO(2N + 1), SO(2N) or USp(2N) at large N . When G = SU(N), the theory is dual to type IIB string theory on AdS5× S5 with complexified string coupling τs, while for the other cases it is dual to the orbifold theory on AdS5× S5/ℤ2. In all cases we use the analytic bootstrap and constraints from localization to compute 1-loop and higher derivative tree level corrections to the leading supergravity approximation of the correlator. We give perturbative evidence that the localization constraint in the large N and finite complexified coupling τ limit can be written for each G in terms of Eisenstein series that are modular invariant in terms of τs ∝ τ, which allows us to fix protected terms in the correlator in that limit. In all cases, we find that the flat space limit of the correlator precisely matches the type IIB S-matrix. We also find a closed form expression for the SU(N) 1-loop Mellin amplitude with supergravity vertices. Finally, we compare our analytic predictions at large N and finite τ to bounds from the numerical bootstrap in the large N regime, and find that they are not saturated for any G and any τ , which suggests that no physical theory saturates these bootstrap bounds.
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Maity, Pronobesh. "Positivity of the Veneziano amplitude in D = 4." Journal of High Energy Physics 2022, no. 4 (April 2022). http://dx.doi.org/10.1007/jhep04(2022)064.
Full textAbstract:
Abstract The Veneziano amplitude was put forward as a solution to the axioms of the S-matrix bootstrap. However, unitarity, reflected in the positivity of the coefficients in the Gegenbauer expansion of the amplitude is not obvious. In this note we compute the generating function of these coefficients in terms of the Appell hypergeometric function. We use this to read off an exact form of this coefficient on the leading Regge trajectory in D = 4. We find that it decays with the spin but always remains positive. Since for large spin these coefficients are expected to be smaller than those on the subleading trajectories, our result indicates the positivity of the full Veneziano amplitude in D = 4.
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Lencsés, Máté, Giuseppe Mussardo, and Gabor Takacs. "Quantum integrability vs experiments: correlation functions and dynamical structure factors." Journal of Physics A: Mathematical and Theoretical, August 21, 2023. http://dx.doi.org/10.1088/1751-8121/acf255.
Full textAbstract:
Abstract Integrable Quantum Field Theories can be solved exactly using bootstrap techniques based on
their elastic and factorisable S-matrix. While knowledge of the scattering amplitudes reveals the
exact spectrum of particles and their on-shell dynamics, the expression of the matrix elements of
the various operators allows the reconstruction of off-shell quantities such as two-point correlation
functions with a high level of precision. In this review, we summarise results relevant to the contact
point between theory and experiment providing a number of quantities that can be computed
theoretically with great accuracy. We concentrate on universal amplitude ratios which can be
determined from the measurement of generalised susceptibilities, and dynamical structure factors,
which can be accessed experimentally e.g. via inelastic neutron scattering or nuclear magnetic
resonance. Besides an overview of the subject and a summary of recent advances, we also present
new results regarding generalised susceptibilities in the tricritical Ising universality class.
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Alday, Luis F., Shai M. Chester, and Himanshu Raj. "6d (2, 0) and M-theory at 1-loop." Journal of High Energy Physics 2021, no. 1 (January 2021). http://dx.doi.org/10.1007/jhep01(2021)133.
Full textAbstract:
Abstract We study the stress tensor multiplet four-point function in the 6d maximally supersymmetric (2, 0) AN−1 and DN theories, which have no Lagrangian description, but in the large N limit are holographically dual to weakly coupled M-theory on AdS7× S4 and AdS7× S4/ℤ2, respectively. We use the analytic bootstrap to compute the 1-loop correction to this holographic correlator coming from Witten diagrams with supergravity R and the first higher derivative correction R4 vertices, which is the first 1-loop correction computed for a non-Lagrangian theory. We then take the flat space limit and find precise agreement with the corresponding terms in the 11d M-theory S-matrix, some of which we compute for the first time using two-particle unitarity cuts.
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Guerrieri, Andrea, Harish Murali, Joao Penedones, and Pedro Vieira. "Where is M-theory in the space of scattering amplitudes?" Journal of High Energy Physics 2023, no. 6 (June 13, 2023). http://dx.doi.org/10.1007/jhep06(2023)064.
Full textAbstract:
Abstract We use the S-matrix bootstrap to carve out the space of unitary, analytic, crossing symmetric and supersymmetric graviton scattering amplitudes in nine, ten and eleven dimensions. We extend and improve the numerical methods of our previous work in ten dimensions. A key new tool employed here is unitarity in the celestial sphere. In all dimensions, we find that the minimal allowed value of the Wilson coefficient α, controlling the leading correction to maximal supergravity, is very close but not equal to the minimal value realized in Superstring theory or M-theory. This small difference may be related to inelastic effects that are not well described by our numerical extremal amplitudes. Although α has a unique value in M-theory, we found no evidence of an upper bound on α in 11D.
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Alday, Luis F., Shai M. Chester, and Himanshu Raj. "M-theory on AdS4 × S7 at 1-loop and beyond." Journal of High Energy Physics 2022, no. 11 (November 15, 2022). http://dx.doi.org/10.1007/jhep11(2022)091.
Full textAbstract:
Abstract We study graviton scattering on AdS4× S7, which is dual to the stress tensor multiplet four-point function in the maximally supersymmetric 3d U(N)1× U(N)−1 ABJM theory. We compute 1-loop corrections to this holographic correlator coming from Witten diagrams with supergravity R and higher derivative R4 vertices, up to contact term ambiguities, and find that the flat space limit matches the corresponding terms in the 11d M-theory S-matrix. We then use supersymmetric localization to show that all the 1-loop contact terms vanish, as was previously observed for the AdS4× S7/ℤ2 theory dual to U(N)2× U(N)−2 ABJM. Finally, we use the recent localization results of Gaiotto and Abajian, as inspired by twisted M-theory, to compute all the short OPE coefficients in correlators of the stress tensor multiplet and the next lowest half-BPS operator, which we find saturate the bootstrap bounds on these mixed correlators for all N.
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Mojahed, Martin A., and Tomáš Brauner. "Nonrelativistic effective field theories with enhanced symmetries and soft behavior." Journal of High Energy Physics 2022, no. 3 (March 2022). http://dx.doi.org/10.1007/jhep03(2022)086.
Full textAbstract:
Abstract We systematically explore the landscape of nonrelativistic effective field theories with a local S-matrix and enhanced symmetries and soft behavior. The exploration is carried out using both conventional quantum field theory methods based on symmetry arguments, and recently developed on-shell recursion relations. We show that, in contrary to relativistic theories, enhancement of the soft limit of scattering amplitudes in nonrelativistic theories is generally not a byproduct of symmetry alone, but requires additional low-energy data. Sufficient conditions for enhanced scattering amplitudes can be derived by combining symmetries and dispersion relations of the scattered particles. This has direct consequences for the infrared dynamics that different types of nonrelativistic Nambu-Goldstone bosons can exhibit. We then use a bottom-up soft bootstrap approach to narrow down the landscape of nonrelativistic effective field theories that possess a consistent low-energy S-matrix. We recover two exceptional theories of a complex Schrödinger-type scalar, namely the ℂP1 nonlinear sigma model and the Schrödinger-Dirac-Born-Infeld theory. Moreover, we use soft recursion to prove a no-go theorem ruling out the existence of other exceptional Schrödinger-type theories. We also prove that all exceptional theories of a single real scalar with a linear dispersion relation are necessarily Lorentz-invariant. Soft recursion allows us to obtain some further general bounds on the landscape of nonrelativistic effective theories with enhanced soft limits. Finally, we present a novel theory of a complex scalar with a technically natural quartic dispersion relation. Altogether, our work represents the first step of a program to extend the developments in the study of scattering amplitudes to theories without Lorentz invariance.
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Mohd. Azmi, Nurulhuda Firdaus, Habshah Midi, and Noranita Fairus Ismail. "The Performance of Clustering Approach with Robust MM–Estimator for Multiple Outlier Detection in Linear Regression." Jurnal Teknologi, January 20, 2012. http://dx.doi.org/10.11113/jt.v45.320.
Full textAbstract:
Pengenalpastian cerapan data yang terpencil daripada kelompok cerapan merupakan langkah asas dalam membina model regresi. Oleh kerana cerapan data yang terpencil ini memberi kesan kepada model yang dibangunkan, pelbagai ukuran terhadap pengenalpastian cerapan data yang terpencil telah dibina. Sebagai contoh, ukuran residual dan ukuran matrik identiti bagi hat matrix. Walau bagaimanapun, ukuran-ukuran ini hanya dapat mengukur dengan baik jika di dalam set data itu terkandung hanya satu atau sedikit cerapan data yang terpencil, walhal jika data dicerap berdasarkan kepada persekitaran sebenar berkemungkinan terdapat lebih banyak cerapan data yang terpencil. Kertas kerja ini mencadangkan pendekatan alternatif iaitu penggunaan teknik kelompok bersama penganggar statistik tegap di dalam pengenalpastian kumpulan cerapan data terpencil. Penganggar statistik tegap yang dicadangkan ialah penganggar MM. Penilaian terhadap kebolehupayaan pendekatan kelompok bersama penganggar cadangan, diuji melalui perbandingan dengan penganggar klasik Least Square (LS) dan penganggar statistik tegap yang lain iaitu Least Trimmed Square (LTS). Pengujian dilakukan melalui analisis pada kumpulan set data terpencil klasik yang diperolehi daripada kajian literatur dan kumpulan set data yang diperolehi daripada simulasi. Sebagai tambahan, kebolehupayaan bagi ketiga-tiga penganggar ini seterusnya diuji berdasarkan nilai punca kuasa dua ralat (RMSE) dan kebarangkalian liputan bagi selang keyakinan Bootstrap Bias Corrected and Accelerated (BCa) bagi menentukan penganggar yang terbaik. Hasil analisis menunjukkan bahawa penganggar yang dicadangkan memberi prestasi yang baik diikuti dengan penganggar LTS dan LS di dalam pengenalpastian kumpulan cerapan data yang terpencil bagi kumpulan set data terpencil klasik dan data simulasi dengan sebarang nilai peratus cerapan terpinggir, bilangan pembolehubah regreasi dan bilangan saiz data. Selain itu, hasil daripada analisis juga menunjukkan nilai punca kuasa dua ralat (RMSE) bagi penganggar cadangan adalah kecil berbanding dengan kedua jenis penganggar yang lain. Manakala, bagi analisis terhadap kebarangkalian liputan selang keyakinan Bootstrap Bias Corrected and Accelerated (BCa) ia menunjukkan bahawa selang keyakinan penganggar MM adalah yang terbaik kerana ia mempunyai kebarangkalian liputan yang baik, equatailness yang baik dan purata jarak keyakinan yang pendek, diikuti dengan penganggar LTS and LS. Kata kunci: Cerapan terpencil berganda, regresi linear, penganggar teguh, penganggar MM, selang keyakinan Bootstrap Bias Corrected and Accelerated (BCa) Identifying outlier is a fundamental step in the regression model building process. Outlying observations should be identified because of their potential effect on the fitted model. As a result of the need to identify outliers, numerous outlying measures such as residuals and hat matrix diagonal are built. However, these outlying measures works well when a regression data set contains only a single outlying point and it is well established that regression real data sets may have multiple outlying observations that individually are not easy to identify by the same measures. In this paper, an alternative approach is proposed, that is clustering technique incorporated with robust estimator for multiple outlier identification. The robust estimator proposes is MM-Estimator. The performance of clustering approach with proposed estimator is compared with other estimator that is the classical estimator namely Least Square (LS) and other robust estimator that is Least Trimmed Square (LTS). The evaluation of the estimator performance is carried out through analyses on a classical multiple outlier data sets found in the literature and simulated multiple outlier data sets. Additionally, the analysis of Root Mean Square Error (RMSE) value and coverage probabilities of Bootstrap Bias Corrected and Accelerated (BCa) confidence interval are also being conducted to identify the best estimator in identification of multiple outliers. From the analysis, it has been revealed that the MM-Estimator performed excellently on the classical multiple outlier data sets and a wide variety of simulated data sets with any percentage of outliers, any number of regressor variables and any sample sizes followed by LTS and LS. The analysis also showed that the value of RMSE of the proposed estimator is always smaller than the other two estimators. Whereupon, the coverage probabilities of BC a confidence interval also conclude that the MM-Estimator confidence interval have all the criterias to be the best estimator since it has a good coverage probabilities, good equatailness and the shortest average confident length followed by LTS and LS. Key words: Multiple outliers, linear regression, robust estimator, MM-Estimator, Bootstrap Bias Corrected and Accelerated (BCa) confidence interval
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Bajnok, Zoltán, and Ladislav Šamaj. "Introduction to Integrable Many-Body systems III." Acta Physica Slovaca. Reviews and Tutorials 61, no. 2 (April 1, 2011). http://dx.doi.org/10.2478/v10155-011-0002-x.
Full textAbstract:
Introduction to Integrable Many-Body systems IIIThis is the third part of a three-volume introductory course about integrable systems of interacting bodies. The emphasis is put onto the method of Thermodynamic Bethe ansatz. Two kinds of integrable models are studied. Systems of itinerant electrons, forming a part of Condensed Matter Physics, involve the Hubbard lattice model of electrons with short-ranged one-site interactions (Sect. 20) and the s-d exchange Kondo model (Sect. 21), describing the scattering of conduction electrons on a spin-s impurity. Methods and basic concepts used in Quantum Field Theory are explained on the integrable (1 + 1)-dimensional sine-Gordon model. We start with the classical description of the model in Sect. 22, analyze its finite energy field configurations (soliton, anti-soliton and breathers) and show its classical integrability. The model is quantized by using two schemes: the conformal (Sect. 23) and Lagrangian (Sect. 24) quantizations. The scattering matrix of the sine-Gordon theory is derived at the full quantum level in the bootstrap scheme and is compared to its classical limit in Sect. 25. The parameters of the scattering matrix are related to those of the Lagrangian by calculating the ground-state energy in an applied magnetic field in two ways: Conformal perturbation theory and Thermodynamic Bethe ansatz (Sect. 26). The relation of the sine-Gordon theory to the XXZ Heisenberg model, which provides a complete solution of the sine-Gordon model in a finite volume, is pointed out in Sect. 27. The obtained results are applied in Sect. 28. to the derivation of the exact thermodynamics for the (symmetric) two-component Coulomb gas; this is the first classical two-dimensional fluid with exactly solvable thermodynamics.
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Castro-Alvaredo, Olalla A., Cecilia De Fazio, Benjamin Doyon, and Aleksandra A. Ziółkowska. "Generalised hydrodynamics of particle creation and decay." Journal of High Energy Physics 2022, no. 4 (April 2022). http://dx.doi.org/10.1007/jhep04(2022)035.
Full textAbstract:
Abstract Unstable particles rarely feature in conjunction with integrability in 1+1D quantum field theory. However, the family of homogenous sine-Gordon models provides a rare example where both stable and unstable bound states are present in the spectrum whilst the scattering matrix is diagonal and solves the usual bootstrap equations. In the standard scattering picture, unstable particles result from complex poles of the S-matrix located in the unphysical sheet of rapidity space. Since they are not part of the asymptotic spectrum, their presence is only felt through the effect they have on physical quantities associated either to the theory as a whole (i.e. scaling functions, correlation functions) or to the stable particles themselves (i.e. energy/particle density). In two recent publications, the effect of unstable particles in different out-of-equilibrium settings has been studied. It has been shown that their presence is associated with specific signatures in many quantities of physical interest. A good way to select those quantities is to adopt the generalised hydrodynamic approach and to consider the effective velocities and particle densities of the stable particles in the theory. For an initial state given by a spacial gaussian profile of temperatures peaked at the origin, time evolution gives rise to particle and spectral particle densities that exhibit hallmarks of the creation and decay of unstable particles. While these signatures have been observed numerically elsewhere, this paper explores their quantitative and qualitative dependence on the parameters of the problem. We also consider other initial states characterised by “inverted gaussian” and “double gaussian” temperature profiles.