Abstract
We introduce a systematic framework for counting and finding independent operators in effective field theories, taking into account the redundancies associated with use of the classical equations of motion and integration by parts. By working in momentum space, we show that the enumeration problem can be mapped onto that of understanding a polynomial ring in the field momenta. All-order information about the number of independent operators in an effective field theory is encoded in a geometrical object of the ring known as the Hilbert series. We obtain the Hilbert series for the theory of N real scalar fields in (0+1) dimensions—an example, free of space-time and internal symmetries, where aspects of our framework are most transparent. Although this is as simple a theory involving derivatives as one could imagine, it provides fruitful lessons to be carried into studies of more complicated theories: we find surprising and rich structure from an interplay between integration by parts and equations of motion and a connection with SL(2,\({{\mathbb{C}}}\)) representation theory, which controls the structure of the operator basis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Jenkins, E.E., Manohar, A.V.: Algebraic structure of lepton and quark flavor invariants and CP violation. JHEP 0910, 094 (2009). arXiv:0907.4763
Hanany, A., Jenkins, E.E., Manohar, A.V., Torri, G.: Hilbert series for flavor invariants of the standard model. JHEP 1103, 096 (2011). arXiv:1010.3161
Benvenuti, S., Feng, B., Hanany, A., He, Y.-H.: Counting BPS operators in gauge theories: quivers, syzygies and plethystics. JHEP 0711, 050 (2007). arXiv:hep-th/0608050
Feng, B., Hanany, A., He, Y.-H.: Counting gauge invariants: the plethystic program. JHEP 0703, 090 (2007). arXiv:hep-th/0701063
Gray, J., Hanany, A., He, Y.-H., Jejjala, V., Mekareeya, N.: SQCD: A geometric apercu. JHEP 0805, 099 (2008). arXiv:0803.4257
Lehman, L., Martin, A.: Hilbert series for constructing Lagrangians: expanding the phenomenologist’s toolbox. Phys. Rev. D 91, 105014 (2015). arXiv:1503.07537
Politzer H.D.: Power corrections at short distances. Nucl. Phys. B 172, 349 (1980)
Georgi H.: On-shell effective field theory. Nucl. Phys. B 361, 339–350 (1991)
Sturmfels B.: Algorithms in Invariant Theory. Texts and Monographs in Symbolic Computation. Springer Vienna, Berlin (1993)
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
Cox D., Little J., O’Shea D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, Berlin (1992)
Schenck H.: Computational Algebraic Geometry, vol. 58. Cambridge University Press, London (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
About this article
Cite this article
Henning, B., Lu, X., Melia, T. et al. Hilbert series and operator bases with derivatives in effective field theories. Commun. Math. Phys. 347, 363–388 (2016). https://doi.org/10.1007/s00220-015-2518-2
Received:
Accepted:
Published:
Issue date:
DOI: https://doi.org/10.1007/s00220-015-2518-2