Effective field theory of gravity to all orders
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Springer
Received: November Revised: March Accepted: April Published: May
18, 29, 25, 19,
2019 2020 2020 2020
Maximilian Ruhdorfer, Javi Serra and Andreas Weiler Physik-Department, Technische Universit¨ at M¨ unchen, James-Franck-Strasse 1, 85748 Garching, Germany
E-mail: [email protected], [email protected], [email protected] Abstract: We construct the general effective field theory of gravity coupled to the Standard Model of particle physics, which we name GRSMEFT. Our method allows the systematic derivation of a non-redundant set of operators of arbitrary dimension with generic field content and gravity. We explicitly determine the pure gravity EFT up to dimension ten, the EFT of a shift-symmetric scalar coupled to gravity up to dimension eight, and the operator basis for the GRSMEFT up to dimension eight. Extensions to all orders are straightforward. Keywords: Beyond Standard Model, Effective Field Theories, Models of Quantum Gravity ArXiv ePrint: 1908.08050
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)083
JHEP05(2020)083
Effective field theory of gravity to all orders
Contents 2
2 Method 2.1 Hilbert series 2.2 Hilbert series for EFTs 2.3 Example: generalized Euler-Heisenberg Lagrangian
3 3 5 8
3 Gravity 3.1 General relativity as an EFT 3.2 Building blocks for the gravity EFT
10 10 12
4 Applications 4.1 Gravity in vacuum 4.2 Shift-symmetric scalar coupled to gravity
14 14 17
5 Standard Model coupled to gravity 5.1 Comments on the GRSMEFT operator basis
18 21
6 Summary
23
A Group characters A.1 Integration measures A.2 Characters for SM gauge representations A.3 Conformal characters
24 24 24 25
B Operator redundancies
25
C Plethystic exponential C.1 Bosonic plethystic exponential C.2 Fermionic plethystic exponential
26 26 27
D Dimension 8 GRSMEFT basis
28
E Gravity EFT in d > 4 spacetime dimensions E.1 Character for the single particle module E.2 Gravity in vacuum in d = 5
30 31 32
–1–
JHEP05(2020)083
1 Introduction
1
Introduction
–2–
JHEP05(2020)083
Effective field theory (EFT) lies at the core of our modern understanding of the fundamental interactions in nature. EFTs encode the dynamics of the relevant degrees of freedom at the scales of interest, and enable the systematic exploration of the effects of heavy states via an infinite set of local operators built out of the light fields. Crucially, the higher the operator’s dimension, the smaller the departure it introduces from the leading order dynamics, the latter understood, from this point of view, as a standard quantum field theory. One of the most elegant quantum field theories is Einstein’s theory of general relativity (GR). Formulated in 1915, it has survived all experimental tests, both in measurements on Earth and using precision astrophysical observations at various scales. An important question is how one can systematically test departures from GR, or even at a more basic level, what is the set of independent IR departures one could possibly test. Since most
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