New Classes of Solutions for Euclidean Scalar Field Theories
This paper presents new classes of exact radial solutions to the nonlinear ordinary differential equation that arises as a saddle-point condition for a Euclidean scalar field theory in <i>D</i>-dimensional spacetime. These solutions are found by exploiting the dimensional consistency of...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-02-01
|
| Series: | Universe |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2218-1997/10/2/72 |
| _version_ | 1797296848462938112 |
|---|---|
| author | Carl M. Bender Sarben Sarkar |
| author_facet | Carl M. Bender Sarben Sarkar |
| author_sort | Carl M. Bender |
| collection | DOAJ |
| description | This paper presents new classes of exact radial solutions to the nonlinear ordinary differential equation that arises as a saddle-point condition for a Euclidean scalar field theory in <i>D</i>-dimensional spacetime. These solutions are found by exploiting the dimensional consistency of the radial differential equation for a single <i>massless</i> scalar field, which allows it to transform into an autonomous equation. For massive theories, the radial equation is not exactly solvable, but the massless solutions provide useful approximations to the results for the massive case. The solutions presented here depend on the power of the interaction and on the spatial dimension, both of which may be noninteger. Scalar equations arising in the study of conformal invariance fit into this framework, and classes of new solutions are found. These solutions exhibit two distinct behaviors as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>→</mo><mn>2</mn></mrow></semantics></math></inline-formula> from above. |
| first_indexed | 2024-03-07T22:10:45Z |
| format | Article |
| id | doaj.art-16ac5476c2e74d7fad5ace969b6c76b9 |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-07T22:10:45Z |
| publishDate | 2024-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-16ac5476c2e74d7fad5ace969b6c76b92024-02-23T15:36:46ZengMDPI AGUniverse2218-19972024-02-011027210.3390/universe10020072New Classes of Solutions for Euclidean Scalar Field TheoriesCarl M. Bender0Sarben Sarkar1Department of Physics, Washington University, Saint Louis, MO 63130, USATheoretical Particle Physics and Cosmology, King’s College London, Strand, London WC2R 2LS, UKThis paper presents new classes of exact radial solutions to the nonlinear ordinary differential equation that arises as a saddle-point condition for a Euclidean scalar field theory in <i>D</i>-dimensional spacetime. These solutions are found by exploiting the dimensional consistency of the radial differential equation for a single <i>massless</i> scalar field, which allows it to transform into an autonomous equation. For massive theories, the radial equation is not exactly solvable, but the massless solutions provide useful approximations to the results for the massive case. The solutions presented here depend on the power of the interaction and on the spatial dimension, both of which may be noninteger. Scalar equations arising in the study of conformal invariance fit into this framework, and classes of new solutions are found. These solutions exhibit two distinct behaviors as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>→</mo><mn>2</mn></mrow></semantics></math></inline-formula> from above.https://www.mdpi.com/2218-1997/10/2/72quantum field theorypath-integralvacuum decaybounceinstantonsscale invariance |
| spellingShingle | Carl M. Bender Sarben Sarkar New Classes of Solutions for Euclidean Scalar Field Theories Universe quantum field theory path-integral vacuum decay bounce instantons scale invariance |
| title | New Classes of Solutions for Euclidean Scalar Field Theories |
| title_full | New Classes of Solutions for Euclidean Scalar Field Theories |
| title_fullStr | New Classes of Solutions for Euclidean Scalar Field Theories |
| title_full_unstemmed | New Classes of Solutions for Euclidean Scalar Field Theories |
| title_short | New Classes of Solutions for Euclidean Scalar Field Theories |
| title_sort | new classes of solutions for euclidean scalar field theories |
| topic | quantum field theory path-integral vacuum decay bounce instantons scale invariance |
| url | https://www.mdpi.com/2218-1997/10/2/72 |
| work_keys_str_mv | AT carlmbender newclassesofsolutionsforeuclideanscalarfieldtheories AT sarbensarkar newclassesofsolutionsforeuclideanscalarfieldtheories |