Search

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 $\phi^4$ theory, chiral perturbation theory ($\chi PT$) 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 $\chi PT$, 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 $\rho$ 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 $\chi PT$ Adler zeros. We perform a preliminary analysis to constrain the allowed space further, using ideas involving positivity inside the extended Mandelstam region, and elastic unitarity., Comment: v5: 83 pages, 21 figures. Typo fixed, reference updates

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 $\phi^4$ theory, chiral perturbation theory ($\chi PT$) 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 $\chi PT$, 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 $\rho$ 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 $\chi PT$ Adler zeros. We perform a preliminary analysis to constrain the allowed space further, using ideas involving positivity inside the extended Mandelstam region, and elastic unitarity., Comment: v5: 83 pages, 21 figures. Typo fixed, reference updates

Block copolymers are macromolecules consisting of two or more chemically different polymer segments of a single type of monomer unit, covalently bound together. They represent a versatile class of functional materials for a multitude of applications because they combine the properties of incompatible but well known polymers. Among other properties, the ability of these polymers to modify interfacial properties and to enhance the compatibility of polymer blends makes this polymer type attractive for applications ranging from thermoplastic elastomers, information storage, drug delivery and photonic materials. With the development of living anionic polymerisation1 the synthesis of block copolymers, especially those with complex architectures, has recently received increased attention due to interests in both academia and industry. Diblock copolymers of polystyrene (PS) and poly (methyl methacrylate) (PMMA) [PS-b-PMMA] have been extensively used to make templates for fabrication of nanostructured materials.2 Block copolymers of polyisoprene (PI) and PMMA [PI-b-PMMA] have been used as emulsifiers for the fabrication of polyester nanoparticles.3 Copolymers of PI-b-PMMA are interesting because they can be used for rubber production, as effective compatibilisers for natural rubber/acrylic polymer blends4, and as potential materials for medical applications5. Diblock copolymers of PS and PI [PS-b-PI] are thermoplastic elastomers. Chemical modification of these polymers, for example sulphonation, can give access to functional materials. These block copolymers can be used as templates for nanolithographic processes6. Block copolymers are complex materials. The physical properties of block copolymers are determined by their molecular characteristics, such as molar mass, chemical composition and chain architecture. In order to establish a detailed relationship between the molecular characteristics and macroscopic properties of a block copolymer, it is essential to perform a comp

Block copolymers are macromolecules consisting of two or more chemically different polymer segments of a single type of monomer unit, covalently bound together. They represent a versatile class of functional materials for a multitude of applications because they combine the properties of incompatible but well known polymers. Among other properties, the ability of these polymers to modify interfacial properties and to enhance the compatibility of polymer blends makes this polymer type attractive for applications ranging from thermoplastic elastomers, information storage, drug delivery and photonic materials. With the development of living anionic polymerisation1 the synthesis of block copolymers, especially those with complex architectures, has recently received increased attention due to interests in both academia and industry. Diblock copolymers of polystyrene (PS) and poly (methyl methacrylate) (PMMA) [PS-b-PMMA] have been extensively used to make templates for fabrication of nanostructured materials.2 Block copolymers of polyisoprene (PI) and PMMA [PI-b-PMMA] have been used as emulsifiers for the fabrication of polyester nanoparticles.3 Copolymers of PI-b-PMMA are interesting because they can be used for rubber production, as effective compatibilisers for natural rubber/acrylic polymer blends4, and as potential materials for medical applications5. Diblock copolymers of PS and PI [PS-b-PI] are thermoplastic elastomers. Chemical modification of these polymers, for example sulphonation, can give access to functional materials. These block copolymers can be used as templates for nanolithographic processes6. Block copolymers are complex materials. The physical properties of block copolymers are determined by their molecular characteristics, such as molar mass, chemical composition and chain architecture. In order to establish a detailed relationship between the molecular characteristics and macroscopic properties of a block copolymer, it is essential to perform a comp

Block copolymers are macromolecules consisting of two or more chemically different polymer segments of a single type of monomer unit, covalently bound together. They represent a versatile class of functional materials for a multitude of applications because they combine the properties of incompatible but well known polymers. Among other properties, the ability of these polymers to modify interfacial properties and to enhance the compatibility of polymer blends makes this polymer type attractive for applications ranging from thermoplastic elastomers, information storage, drug delivery and photonic materials. With the development of living anionic polymerisation1 the synthesis of block copolymers, especially those with complex architectures, has recently received increased attention due to interests in both academia and industry. Diblock copolymers of polystyrene (PS) and poly (methyl methacrylate) (PMMA) [PS-b-PMMA] have been extensively used to make templates for fabrication of nanostructured materials.2 Block copolymers of polyisoprene (PI) and PMMA [PI-b-PMMA] have been used as emulsifiers for the fabrication of polyester nanoparticles.3 Copolymers of PI-b-PMMA are interesting because they can be used for rubber production, as effective compatibilisers for natural rubber/acrylic polymer blends4, and as potential materials for medical applications5. Diblock copolymers of PS and PI [PS-b-PI] are thermoplastic elastomers. Chemical modification of these polymers, for example sulphonation, can give access to functional materials. These block copolymers can be used as templates for nanolithographic processes6. Block copolymers are complex materials. The physical properties of block copolymers are determined by their molecular characteristics, such as molar mass, chemical composition and chain architecture. In order to establish a detailed relationship between the molecular characteristics and macroscopic properties of a block copolymer, it is essential to perform a comp

Block copolymers are macromolecules consisting of two or more chemically different polymer segments of a single type of monomer unit, covalently bound together. They represent a versatile class of functional materials for a multitude of applications because they combine the properties of incompatible but well known polymers. Among other properties, the ability of these polymers to modify interfacial properties and to enhance the compatibility of polymer blends makes this polymer type attractive for applications ranging from thermoplastic elastomers, information storage, drug delivery and photonic materials. With the development of living anionic polymerisation1 the synthesis of block copolymers, especially those with complex architectures, has recently received increased attention due to interests in both academia and industry. Diblock copolymers of polystyrene (PS) and poly (methyl methacrylate) (PMMA) [PS-b-PMMA] have been extensively used to make templates for fabrication of nanostructured materials.2 Block copolymers of polyisoprene (PI) and PMMA [PI-b-PMMA] have been used as emulsifiers for the fabrication of polyester nanoparticles.3 Copolymers of PI-b-PMMA are interesting because they can be used for rubber production, as effective compatibilisers for natural rubber/acrylic polymer blends4, and as potential materials for medical applications5. Diblock copolymers of PS and PI [PS-b-PI] are thermoplastic elastomers. Chemical modification of these polymers, for example sulphonation, can give access to functional materials. These block copolymers can be used as templates for nanolithographic processes6. Block copolymers are complex materials. The physical properties of block copolymers are determined by their molecular characteristics, such as molar mass, chemical composition and chain architecture. In order to establish a detailed relationship between the molecular characteristics and macroscopic properties of a block copolymer, it is essential to perform a comp

Block copolymers are macromolecules consisting of two or more chemically different polymer segments of a single type of monomer unit, covalently bound together. They represent a versatile class of functional materials for a multitude of applications because they combine the properties of incompatible but well known polymers. Among other properties, the ability of these polymers to modify interfacial properties and to enhance the compatibility of polymer blends makes this polymer type attractive for applications ranging from thermoplastic elastomers, information storage, drug delivery and photonic materials. With the development of living anionic polymerisation1 the synthesis of block copolymers, especially those with complex architectures, has recently received increased attention due to interests in both academia and industry. Diblock copolymers of polystyrene (PS) and poly (methyl methacrylate) (PMMA) [PS-b-PMMA] have been extensively used to make templates for fabrication of nanostructured materials.2 Block copolymers of polyisoprene (PI) and PMMA [PI-b-PMMA] have been used as emulsifiers for the fabrication of polyester nanoparticles.3 Copolymers of PI-b-PMMA are interesting because they can be used for rubber production, as effective compatibilisers for natural rubber/acrylic polymer blends4, and as potential materials for medical applications5. Diblock copolymers of PS and PI [PS-b-PI] are thermoplastic elastomers. Chemical modification of these polymers, for example sulphonation, can give access to functional materials. These block copolymers can be used as templates for nanolithographic processes6. Block copolymers are complex materials. The physical properties of block copolymers are determined by their molecular characteristics, such as molar mass, chemical composition and chain architecture. In order to establish a detailed relationship between the molecular characteristics and macroscopic properties of a block copolymer, it is essential to perform a comp

Block copolymers are macromolecules consisting of two or more chemically different polymer segments of a single type of monomer unit, covalently bound together. They represent a versatile class of functional materials for a multitude of applications because they combine the properties of incompatible but well known polymers. Among other properties, the ability of these polymers to modify interfacial properties and to enhance the compatibility of polymer blends makes this polymer type attractive for applications ranging from thermoplastic elastomers, information storage, drug delivery and photonic materials. With the development of living anionic polymerisation1 the synthesis of block copolymers, especially those with complex architectures, has recently received increased attention due to interests in both academia and industry. Diblock copolymers of polystyrene (PS) and poly (methyl methacrylate) (PMMA) [PS-b-PMMA] have been extensively used to make templates for fabrication of nanostructured materials.2 Block copolymers of polyisoprene (PI) and PMMA [PI-b-PMMA] have been used as emulsifiers for the fabrication of polyester nanoparticles.3 Copolymers of PI-b-PMMA are interesting because they can be used for rubber production, as effective compatibilisers for natural rubber/acrylic polymer blends4, and as potential materials for medical applications5. Diblock copolymers of PS and PI [PS-b-PI] are thermoplastic elastomers. Chemical modification of these polymers, for example sulphonation, can give access to functional materials. These block copolymers can be used as templates for nanolithographic processes6. Block copolymers are complex materials. The physical properties of block copolymers are determined by their molecular characteristics, such as molar mass, chemical composition and chain architecture. In order to establish a detailed relationship between the molecular characteristics and macroscopic properties of a block copolymer, it is essential to perform a comp