We investigate the problem of deriving bounds on strong interaction scattering amplitudes from the results of axiomatic field theory. The bounds on the π-π scattering amplitude at points within its analyticity domain which have been obtained by Łukaszuk and Martin are especially interesting because they contain no free parameters, and they impose absolute limits on the size of the renormalized coupling constant for π-π scattering. We improve the rigorous upper bound derived by Łukaszuk and Martin for the π⁰-π⁰ scattering amplitude at the symmetric point. Our principle new tool is the \"parametric dispersion relation\" of Auberson and Khuri. Also, for a ϕ⁴ type field theory with a scalar bound state which is not too tightly bound, we generalize the methods developed by Martin for π-π scattering to establish upper and lower bounds on the renormalized coupling constant and an upper bound on the physical coupling constant to the bound state. These new bounds are functions of the mass of the bound state. Numerical examples of the bound on the physical coupling constant are given for several bound state masses. Finally, we discuss the relevance of our results to constructive field theory. We point out that, while our bounds do not apply to a ϕ⁴ field theory in 1 space +1 time dimensions, they do limit the values of the renormalized coupling constant for which one could construct a ϕ⁴ field theory in 2 space +1 time (and, of course, 3 space +1 time) dimensions.
","abstract_html":"<p>We investigate the problem of deriving bounds on strong interaction scattering amplitudes from the results of axiomatic field theory. The bounds on the π-π scattering amplitude at points within its analyticity domain which have been obtained by Łukaszuk and Martin are especially interesting because they contain no free parameters, and they impose absolute limits on the size of the renormalized coupling constant for π-π scattering. We improve the rigorous upper bound derived by Łukaszuk and Martin for the π⁰-π⁰ scattering amplitude at the symmetric point. Our principle new tool is the "parametric dispersion relation" of Auberson and Khuri. Also, for a ϕ⁴ type field theory with a scalar bound state which is not too tightly bound, we generalize the methods developed by Martin for π-π scattering to establish upper and lower bounds on the renormalized coupling constant and an upper bound on the physical coupling constant to the bound state. These new bounds are functions of the mass of the bound state. Numerical examples of the bound on the physical coupling constant are given for several bound state masses. Finally, we discuss the relevance of our results to constructive field theory. We point out that, while our bounds do not apply to a ϕ⁴ field theory in 1 space +1 time dimensions, they do limit the values of the renormalized coupling constant for which one could construct a ϕ⁴ field theory in 2 space +1 time (and, of course, 3 space +1 time) dimensions.</p>","abstract_has_math":false,"creators":["Healy, James Bruce"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Thesis","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Nicola Khuri"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1973,"date_issued":"1973-01-01T08:00:00Z","date_published":"1973-01-01T08:00:00Z","updated_at":"2026-07-24T04:11:11Z","subjects":["scattering amplitudes","axiomatic field theory","pion scattering","renormalized coupling constant","dispersion relations","constructive field theory","Life Sciences"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.rockefeller.edu/student_theses_and_dissertations/539","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Nicola Khuri"]},{"key":"dc:creator","label":"Author","values":["Healy, James Bruce"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["scattering amplitudes","axiomatic field theory","pion scattering","renormalized coupling constant","dispersion relations","constructive field theory","Life Sciences"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.rockefeller.edu/student_theses_and_dissertations/539"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We investigate the problem of deriving bounds on strong interaction scattering amplitudes from the results of axiomatic field theory. The bounds on the π-π scattering amplitude at points within its analyticity domain which have been obtained by Łukaszuk and Martin are especially interesting because they contain no free parameters, and they impose absolute limits on the size of the renormalized coupling constant for π-π scattering. We improve the rigorous upper bound derived by Łukaszuk and Martin for the π⁰-π⁰ scattering amplitude at the symmetric point. Our principle new tool is the \"parametric dispersion relation\" of Auberson and Khuri. Also, for a ϕ⁴ type field theory with a scalar bound state which is not too tightly bound, we generalize the methods developed by Martin for π-π scattering to establish upper and lower bounds on the renormalized coupling constant and an upper bound on the physical coupling constant to the bound state. These new bounds are functions of the mass of the bound state. Numerical examples of the bound on the physical coupling constant are given for several bound state masses. Finally, we discuss the relevance of our results to constructive field theory. We point out that, while our bounds do not apply to a ϕ⁴ field theory in 1 space +1 time dimensions, they do limit the values of the renormalized coupling constant for which one could construct a ϕ⁴ field theory in 2 space +1 time (and, of course, 3 space +1 time) dimensions.
"]},{"key":"dc:title","label":"Title","values":["Rigorous Bounds on Strong Interaction Coupling Constants"]}]}],"canonical_facts":{"dc:contributor":["Nicola Khuri"],"dc:creator":["Healy, James Bruce"],"dc:description.abstract":["We investigate the problem of deriving bounds on strong interaction scattering amplitudes from the results of axiomatic field theory. The bounds on the π-π scattering amplitude at points within its analyticity domain which have been obtained by Łukaszuk and Martin are especially interesting because they contain no free parameters, and they impose absolute limits on the size of the renormalized coupling constant for π-π scattering. We improve the rigorous upper bound derived by Łukaszuk and Martin for the π⁰-π⁰ scattering amplitude at the symmetric point. Our principle new tool is the \"parametric dispersion relation\" of Auberson and Khuri. Also, for a ϕ⁴ type field theory with a scalar bound state which is not too tightly bound, we generalize the methods developed by Martin for π-π scattering to establish upper and lower bounds on the renormalized coupling constant and an upper bound on the physical coupling constant to the bound state. These new bounds are functions of the mass of the bound state. Numerical examples of the bound on the physical coupling constant are given for several bound state masses. Finally, we discuss the relevance of our results to constructive field theory. We point out that, while our bounds do not apply to a ϕ⁴ field theory in 1 space +1 time dimensions, they do limit the values of the renormalized coupling constant for which one could construct a ϕ⁴ field theory in 2 space +1 time (and, of course, 3 space +1 time) dimensions.
"],"dc:identifier":["https://digitalcommons.rockefeller.edu/student_theses_and_dissertations/539"],"dc:subject":["scattering amplitudes","axiomatic field theory","pion scattering","renormalized coupling constant","dispersion relations","constructive field theory","Life Sciences"],"dc:title":["Rigorous Bounds on Strong Interaction Coupling Constants"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T04:11:11Z"}