Operator bases, S-matrices, and their partition functions

Journal Article · Fri Oct 27 00:00:00 EDT 2017 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/JHEP10(2017)199· OSTI ID:1421808

Henning, Brian ^[1]; Lu, Xiaochuan  ^[2]; Melia, Tom ^[3]; Murayama, Hitoshi ^[3]

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1. Yale Univ., New Haven, CT (United States). Dept. of Physics
2. Univ. of California, Davis, CA (United States). Dept. of Physics
3. 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

Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. Here in this paper we use the S-matrix 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 (on-shell) and integration by parts (momentum conservation). We introduce a partition function, termed the Hilbert series, to enumerate the operator basis — correspondingly, the S-matrix — 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 n-point amplitudes. We discuss in detail the operator basis for non-linearly 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 non-linear 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.

• Research Organization: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
• Sponsoring Organization: USDOE Office of Science (SC); Japan Society for the Promotion of Science (JSPS)
• Grant/Contract Number: AC02-05CH11231; SC0009999; 291377; PHY-1316783; PHY-1638509; 26400241; 17K05409; 15H05887; 15K21733
• OSTI ID: 1421808
• Journal Information: Journal of High Energy Physics (Online), Vol. 2017, Issue 10; ISSN 1029-8479
• Publisher: Springer Berlin
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
Effective Field Theories
Differential and Algebraic Geometry
Scattering Amplitudes