From Nonmetricity Operators to the Physical Spectrum: A Branch-Complete Analysis of Local Four-Derivative Symmetric Teleparallel Gravity
Abstract
We develop a branch-complete classical free-spectrum analysis of local parity-even symmetric teleparallel gravity containing all pure-gravity nonmetricity operators through four derivatives. After canonicalisation and integration by parts, the nonlinear action contains 116 independent operators distributed as 5+13+29+69 among Q2, (∇ Q)2, Q2∇ Q and Q4. Around Minkowski spacetime in the coincident gauge only the first two sectors contribute to the quadratic action, which reduces to four two-derivative and five four-derivative bilinears. We map the original coefficients to this nine-dimensional basis, construct the Hessian and momentum kernel, and classify local linear gauge symmetries. Unrestricted linearised diffeomorphism invariance imposes six independent relations and leaves a three-parameter family (A,B,C). We also recover TDiff, Weyl, WTDiff and additional accidental symmetry branches. Barnes--Rivers decomposition, gauge fixing, exact inversion and conserved-source saturation reduce the physical exchange to spin-two and scalar factors A+2Cq and 2A+(3B−2C)q. The generic regular spectrum contains a massless graviton, a massive spin-two state and a massive scalar, with eight degrees of freedom. The massive spin-two residue is necessarily opposite to that of the healthy massless graviton. The regular classification leaves only GR/STEGR, A>0 with B=C=0, and the scalar extension, A>0, B>0, C=0, as fully healthy Minkowski loci. The latter has three degrees of freedom and mixed ultraviolet behaviour, with tensor exchange scaling as k−2 and scalar exchange as k−4. Thus, within this finite local metric theory, full four-derivative tensor suppression requires the finite massive spin-two pole with opposite residue.
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BibTeX
@misc{pala2026from,
title = {{From Nonmetricity Operators to the Physical Spectrum: A Branch-Complete Analysis of Local Four-Derivative Symmetric Teleparallel Gravity}},
author = {Caglar Pala},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04448},
eprint = {2609.04448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04448},
}
RIS
TY - UNPB
TI - From Nonmetricity Operators to the Physical Spectrum: A Branch-Complete Analysis of Local Four-Derivative Symmetric Teleparallel Gravity
AU - Caglar Pala
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04448
UR - https://arxiv.org/abs/2609.04448
AB - We develop a branch-complete classical free-spectrum analysis of local parity-even symmetric teleparallel gravity containing all pure-gravity nonmetricity operators through four derivatives. After canonicalisation and integration by parts, the nonlinear action contains 116 independent operators distributed as 5+13+29+69 among Q^2, (∇ Q)^2, Q^2∇ Q and Q^4. Around Minkowski spacetime in the coincident gauge only the first two sectors contribute to the quadratic action, which reduces to four two-derivative and five four-derivative bilinears. We map the original coefficients to this nine-dimensional basis, construct the Hessian and momentum kernel, and classify local linear gauge symmetries. Unrestricted linearised diffeomorphism invariance imposes six independent relations and leaves a three-parameter family (A,B,C). We also recover TDiff, Weyl, WTDiff and additional accidental symmetry branches. Barnes--Rivers decomposition, gauge fixing, exact inversion and conserved-source saturation reduce the physical exchange to spin-two and scalar factors A+2Cq and 2A+(3B−2C)q. The generic regular spectrum contains a massless graviton, a massive spin-two state and a massive scalar, with eight degrees of freedom. The massive spin-two residue is necessarily opposite to that of the healthy massless graviton. The regular classification leaves only GR/STEGR, A>0 with B=C=0, and the scalar extension, A>0, B>0, C=0, as fully healthy Minkowski loci. The latter has three degrees of freedom and mixed ultraviolet behaviour, with tensor exchange scaling as k^−2 and scalar exchange as k^−4. Thus, within this finite local metric theory, full four-derivative tensor suppression requires the finite massive spin-two pole with opposite residue.
ER -