Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields
We develop the BRST approach to construct the general off-shell local Lorentz covariant cubic interaction vertices for irreducible massless and massive higher spin fields on <i>d</i>-dimensional Minkowski space. We consider two different cases for interacting higher spin fields: with one...
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MDPI AG
2023-11-01
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| author | I. L. Buchbinder A. A. Reshetnyak |
| author_facet | I. L. Buchbinder A. A. Reshetnyak |
| author_sort | I. L. Buchbinder |
| collection | DOAJ |
| description | We develop the BRST approach to construct the general off-shell local Lorentz covariant cubic interaction vertices for irreducible massless and massive higher spin fields on <i>d</i>-dimensional Minkowski space. We consider two different cases for interacting higher spin fields: with one massive and two massless; two massive, both with coinciding and with different masses and one massless field of spins <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><msub><mi>s</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula>. Unlike the previous results on cubic vertices we extend our earlier result in (Buchbinder, I.L.; et al. <i>Phys. Lett. B</i> 2021, 820, 136470) for massless fields and employ the complete BRST operator, including the trace constraints, which is used to formulate an irreducible representation with definite integer spin. We generalize the cubic vertices proposed for reducible higher spin fields in (Metsaev, R.R. <i>Phys. Lett. B</i> 2013, 720, 237) in the form of multiplicative and non-multiplicative BRST-closed constituents and calculate the new contributions to the vertex, which contains the additional terms with a smaller number of space-time derivatives. We prove that without traceless conditions for the cubic vertices in (Metsaev, R.R. <i>Phys. Lett. B</i> 2013, 720, 237) it is impossible to provide the noncontradictory Lagrangian dynamics and find explicit traceless solution for these vertices. As the examples, we explicitly construct the interacting Lagrangians for the massive spin of the <i>s</i> field and the massless scalars, both with and without auxiliary fields. The interacting models with different combinations of triples higher spin fields: massive spin <i>s</i> with massless scalar and vector fields and with two vector fields; massless helicity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> with massless scalar and massive vector fields; two massive fields of spins <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></semantics></math></inline-formula> and massless scalar is also considered. |
| first_indexed | 2024-03-08T20:19:54Z |
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| spelling | doaj.art-407590ab346f4ebca7cd4b4005c54ae72023-12-22T14:45:07ZengMDPI AGSymmetry2073-89942023-11-011512212410.3390/sym15122124Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin FieldsI. L. Buchbinder0A. A. Reshetnyak1Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, RussiaBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, RussiaWe develop the BRST approach to construct the general off-shell local Lorentz covariant cubic interaction vertices for irreducible massless and massive higher spin fields on <i>d</i>-dimensional Minkowski space. We consider two different cases for interacting higher spin fields: with one massive and two massless; two massive, both with coinciding and with different masses and one massless field of spins <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><msub><mi>s</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula>. Unlike the previous results on cubic vertices we extend our earlier result in (Buchbinder, I.L.; et al. <i>Phys. Lett. B</i> 2021, 820, 136470) for massless fields and employ the complete BRST operator, including the trace constraints, which is used to formulate an irreducible representation with definite integer spin. We generalize the cubic vertices proposed for reducible higher spin fields in (Metsaev, R.R. <i>Phys. Lett. B</i> 2013, 720, 237) in the form of multiplicative and non-multiplicative BRST-closed constituents and calculate the new contributions to the vertex, which contains the additional terms with a smaller number of space-time derivatives. We prove that without traceless conditions for the cubic vertices in (Metsaev, R.R. <i>Phys. Lett. B</i> 2013, 720, 237) it is impossible to provide the noncontradictory Lagrangian dynamics and find explicit traceless solution for these vertices. As the examples, we explicitly construct the interacting Lagrangians for the massive spin of the <i>s</i> field and the massless scalars, both with and without auxiliary fields. The interacting models with different combinations of triples higher spin fields: massive spin <i>s</i> with massless scalar and vector fields and with two vector fields; massless helicity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> with massless scalar and massive vector fields; two massive fields of spins <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></semantics></math></inline-formula> and massless scalar is also considered.https://www.mdpi.com/2073-8994/15/12/2124higher spin field theorymultidimensional Minkowski spacesLagrangian formulationBRST operatorgauge invariancecubic interaction vertices |
| spellingShingle | I. L. Buchbinder A. A. Reshetnyak Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields Symmetry higher spin field theory multidimensional Minkowski spaces Lagrangian formulation BRST operator gauge invariance cubic interaction vertices |
| title | Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields |
| title_full | Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields |
| title_fullStr | Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields |
| title_full_unstemmed | Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields |
| title_short | Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields |
| title_sort | covariant cubic interacting vertices for massless and massive integer higher spin fields |
| topic | higher spin field theory multidimensional Minkowski spaces Lagrangian formulation BRST operator gauge invariance cubic interaction vertices |
| url | https://www.mdpi.com/2073-8994/15/12/2124 |
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