Operator bases, Smatrices, and their partition functions
Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where Smatrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. Here in this paper we use the Smatrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (onshell) and integration by parts (momentum conservation). We introduce a partition function, termed the Hilbert series, to enumerate the operator basis — correspondingly, the Smatrix — and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries. In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to n = 5 scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of npointmore »
 Authors:

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^{[2]}
;
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^{[3]}
 Yale Univ., New Haven, CT (United States). Dept. of Physics
 Univ. of California, Davis, CA (United States). Dept. of Physics
 Univ. of California, Berkeley, CA (United States). Dept. of Physics; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of Tokyo (Japan). Kavli Inst. for the Physics and Mathematics of the Universe (WPI), Todai Inst. for Advanced Study
 Publication Date:
 Grant/Contract Number:
 AC0205CH11231; SC0009999; 291377; PHY1316783; PHY1638509; 26400241; 17K05409; 15H05887; 15K21733
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of High Energy Physics (Online)
 Additional Journal Information:
 Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 10; Journal ID: ISSN 10298479
 Publisher:
 Springer Berlin
 Research Org:
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC); Japan Society for the Promotion of Science (JSPS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Effective Field Theories; Differential and Algebraic Geometry; Scattering Amplitudes
 OSTI Identifier:
 1421808
Henning, Brian, Lu, Xiaochuan, Melia, Tom, and Murayama, Hitoshi. Operator bases, Smatrices, and their partition functions. United States: N. p.,
Web. doi:10.1007/JHEP10(2017)199.
Henning, Brian, Lu, Xiaochuan, Melia, Tom, & Murayama, Hitoshi. Operator bases, Smatrices, and their partition functions. United States. doi:10.1007/JHEP10(2017)199.
Henning, Brian, Lu, Xiaochuan, Melia, Tom, and Murayama, Hitoshi. 2017.
"Operator bases, Smatrices, and their partition functions". United States.
doi:10.1007/JHEP10(2017)199. https://www.osti.gov/servlets/purl/1421808.
@article{osti_1421808,
title = {Operator bases, Smatrices, and their partition functions},
author = {Henning, Brian and Lu, Xiaochuan and Melia, Tom and Murayama, Hitoshi},
abstractNote = {Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where Smatrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. Here in this paper we use the Smatrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (onshell) and integration by parts (momentum conservation). We introduce a partition function, termed the Hilbert series, to enumerate the operator basis — correspondingly, the Smatrix — and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries. In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to n = 5 scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of npoint amplitudes. We discuss in detail the operator basis for nonlinearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the S matrix in the form of soft limits. The most naÏve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for nonlinear realizations. Finally, although primarily discussed in the language of EFT, some of our results — conceptual and quantitative — may be of broader use in studying conformal field theories as well as the AdS/CFT correspondence.},
doi = {10.1007/JHEP10(2017)199},
journal = {Journal of High Energy Physics (Online)},
number = 10,
volume = 2017,
place = {United States},
year = {2017},
month = {10}
}