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UC Berkeley UC Berkeley Previously Published Works UC Berkeley UC Berkeley Previously Published Works Title Operator bases, S-matrices, and their partition functions Permalink https://escholarship.org/uc/item/31n0p4j4 Journal Journal of High Energy Physics, 2017(10) ISSN 1126-6708 Authors Henning, B Lu, X Melia, T et al. Publication Date 2017-10-01 DOI 10.1007/JHEP10(2017)199 Peer reviewed eScholarship.org Powered by the California Digital Library University of California Published for SISSA by Springer Received: July 7, 2017 Accepted: October 6, 2017 Published: October 27, 2017 Operator bases, S-matrices, and their partition functions JHEP10(2017)199 Brian Henning,a Xiaochuan Lu,b Tom Meliac;d;e and Hitoshi Murayamac;d;e aDepartment of Physics, Yale University, New Haven, Connecticut 06511, U.S.A. bDepartment of Physics, University of California, Davis, California 95616, U.S.A. cDepartment of Physics, University of California, Berkeley, California 94720, U.S.A. dTheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, U.S.A. eKavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, Japan E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: Relativistic quantum systems that admit scattering experiments are quan- titatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary de- scriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) mo- mentum 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 op- erator 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. Open Access, c The Authors. https://doi.org/10.1007/JHEP10(2017)199 Article funded by SCOAP3. 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 na¨ıve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations. 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. JHEP10(2017)199 Keywords: Effective Field Theories, Differential and Algebraic Geometry, Scattering Amplitudes ArXiv ePrint: 1706.08520 Contents 1 Introduction1 2 The operator basis6 2.1 Single particle modules6 2.2 The operator basis via cohomology and conformal representation theory9 2.3 The operator basis as Feynman rules 10 JHEP10(2017)199 2.4 Non-linear realizations 12 2.5 Gauge symmetries, topological terms and discrete spacetime symmetries 13 2.6 A partition function for the operator basis 14 3 Conformal representations and characters 14 3.1 Unitary conformal representations 15 3.2 Character formulae 17 3.3 Character orthogonality 21 3.4 Summary 22 4 Counting operators: Hilbert series 23 4.1 Deriving a matrix integral formula for the Hilbert series 24 4.2 Multiple fields and internal symmetries 29 4.3 Parity invariance: from SO(d) to O(d) 30 4.4 Reweighting and summary of main formulas 33 5 Constructing operators: kinematic polynomial rings 35 5.1 Polynomial ring in particle momenta 35 5.2 Tour of the four point ring 39 5.3 Interpreting the ring in terms of operators and amplitudes 42 5.4 Constructing the basis at higher n: generators and algebraic properties 43 5.4.1 The Cohen-Macaulay property 43 5.4.2 A set of primary invariants in the absence of Gram conditions 47 5.4.3 Relationship between and 47 J K 5.4.4 Generalizations with spin 47 G 5.5 Conformal primaries and elements of Mn;K 48 5.6 Algorithms for operator construction and the n = 5 basis 49 5.6.1 The n = 5 operator basis 51 5.7 Summary 54 6 Applications and examples of Hilbert series 54 6.1 Hilbert series for a real scalar field in d = 4 55 6.2 Closed form Hilbert series for n distinguishable scalars in d = 2; 3 57 6.3 Hilbert series for spinning particles 61 6.4 Quantifying the effects of EOM, IBP, and Gram redundancy 64 { i { 7 Non-linear realizations 65 7.1 Linearly transforming building blocks 66 7.2 Computing the Hilbert series 68 7.3 Summary 72 8 Discussion 73 A Character formulae for classical Lie groups 75 B Weyl integration formula 79 JHEP10(2017)199 B.1 A simplification when f(g) is Weyl invariant 82 B.2 The measures for the classical Lie groups 83 C Parity 84 C.1 Representations and characters of O(2r) 84 C.2 Understanding the parity odd character: folding so2r 86 C.3 Hilbert series with parity 89 C.4 Examples 92 D Computation of ∆H 94 S E Hilbert series for C[s12; s13; s14; s23; s24; s34] 4 96 F Counting helicity amplitudes 97 G EOM for non-linear realizations 102 1 Introduction The basic tenets of S-matrix theory and effective field theory are equivalent: the starting point is to take assumed particle content and parameterize all possible scattering experi- ments. This is dictated by kinematics and symmetry principles. In the context of effective field theory (EFT), this parameterization is embodied in the operator basis of the EFT, K which is defined to be the set of all operators that lead to physically distinct phenomena. The purpose of this work is to formalize the rules governing the operator basis and investigate the structure they induce on . By considering the set in its own right, we K K are aiming to get as much mileage from kinematics and selection rules as possible before addressing specific dynamics. At a practical level, as well as imposing Lorentz invariance (and other internal symmetries), it amounts to dealing with redundancies associated with equations of motion (EOM) and total derivatives (also called integration by parts (IBP) redundancies). At a more fundamental level, we are accounting for Poincar´ecovariance of single particle states together with Poincar´einvariance of the S-matrix. { 1 { We focus on the exploration of operator bases in a wide class of relativistic EFTs in d 2 spacetime dimensions. Our main purpose is to introduce and develop a number of ≥ systematic methods to deal with EOM and IBP redundancies at all orders in the EFT expansion. This treatment of course also provides a standard procedure to determine low-order terms of an operator basis, a continuing topic of interest for various relativistic phenomenological theories as experimental energy and precision thresholds are crossed. One basic and very useful means of studying the operator basis is to consider the partition function on K H = TrKw = w ; (1.1) O JHEP10(2017)199 O2K X where w is some weighting function. We callb H thebHilbert series of the operator basis, or simply Hilbert series for short. The Hilbert series is a counting function, just as the usual physicalb partition function (path integral) counts states in the Hilbert space weighted by e−βE. In many ways, the Hilbert series can be thought of as a partition function of the S- matrix. Loosely speaking, it captures the rank of the S-matrix by enumerating independent observables. Hilbert series will play a central role in our study of operator bases. Effective field theory is a description of the S-matrix, and as such can address questions concerning the use of general physical principles to derive and implement | bootstrap | consistency conditions with the goal of probing physical observables. A revival of older S- matrix theory ideas, e.g. [1], in the past 25 years has seen remarkable, useful, and beautiful results based on unitarity and causality (for recent reviews see e.g. [2,3 ]). Similar types of general considerations, also discussed last century [4{6], found concrete implementation in d > 2 CFTs a decade ago [7], spawning the modern CFT bootstrap program (see e.g. [8] for an overview). The AdS/CFT correspondence [9] has demonstrated underlying connections between these two areas [10{25]. More concretely, the AdS/CFT correspondence in the flat space limit of AdS indicates that there is a (not entirely understood) correspondence between d-dimensional scattering amplitudes and (d 1)-dimensional conformal correlators [10, 26{28]. In [10] it was shown − that solutions to the crossing equations for scalar four point functions in CFTd−1 are in one-to-one correspondence with operators in d-dimensions consisting of four scalar fields and derivatives. In our analysis, this set is described by M4 in eq. (2.12), whose solution is (see section5 ) a freely generated ring in st + su + tu and stu where (s; t; u) are the usual Mandelstam variables. The initial clue in [10] was a counting argument, that showed the number of objects in M4 is the same as the number of solutions to the crossing equations.
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