Renormalisation of the unified theory at eleven parameters
Abstract
The unified theory of gravity and the Standard Model at eleven parameters is, in its Einstein gauge, Einstein gravity coupled to the Standard Model with three Planck-scale operators, and its graviton diagrams are not renormalisable. This paper asks what renormalising it can mean. The word has had several meanings, the subtraction of infinities of Dyson and of Bogoliubov, Parasiuk, Hepp and Zimmermann, the group of Stückelberg, Petermann, Gell-Mann and Low, the flow of Wilson with the effective theories of Weinberg, the non-Gaussian fixed points of asymptotic safety, the finite formulations of Epstein and Glaser and of Connes and Kreimer, the finiteness that supersymmetry or strings supply, and the scale-invariant schemes in which the renormalisation scale is a field; the paper sorts them and states what each promises for a theory that contains gravity. Four of them apply to the theory of the eleven and are written out as complete schemes: the scale-invariant effective renormalisation, in which every counterterm is a weight-zero monomial with the compensator supplying the inverse Planck masses and the one-loop running of the theory's pure numbers is computed, the one-loop correction to the cosmological number being fifty-four orders of magnitude larger than its value; the quadratic completion, Stelle's renormalisable theory in its dimensionless form, with its ghost within a decade of the edge of the theory's domain; asymptotic safety, whose fixed point freezes the dimensionless momentum of the eleven at √ g * , 0.84 in Litim's scheme; and the non-local completion with a form factor in the eleven's own variable. Each is tested on the two diagrams of the companion paper, the Higgs loop and the graviton loop in the graviton self-energy, and on the two-loop R 3 term. Finally the structure of the theory is examined for a method of its own: its reverse gauging is a compact momentum space, the 4-sphere of the Planck radius, on which every loop integral is ultraviolet finite, and the reciprocity of its reverse reading, Born's, is the reflection of that sphere through its equator, an isometry that makes the region beyond its equator, at twice the Planck momentum, the image of the region inside it and relates the two halves of every loop integral. Tested on the two diagrams in three models of the propagator, this reciprocal renormalisation turns every logarithmically divergent coefficient into a finite number, predicted up to the constant of the model, the logarithm running from the mass to its image, and every power-law one into a Planckian number, excluded for the Higgs mass and the vacuum energy by thirty-three and eighty-eight orders of magnitude, which the symmetry pairs with its image and doubles rather than cancels; it is a method for the local logarithmic parts and not for the hierarchy, unless the reverse theory's loops enter with the sign opposite to that of their images, a condition the paper states and the eleven do not meet by themselves, the reciprocity acting on the representations of their algebra as the parity does. The method mixes with the scale-invariant, the subtraction, the quadratic and the asymptotically safe schemes, in each case with a definite result, and not with the non-local one; the first mixing, computed at one loop, gives every pure number of the theory a finite correction and shows that the Higgs loop induces an Einstein term a hundred million times the tree one for the curvature coupling of Palatini Higgs inflation, so that a physical compact momentum space and that inflation are not perturbatively compatible.
Related
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- The unified theory at eleven parameters and the experiments: instrument by instrument, what is tested and what would refute
- The new particle allowed by the unified theory at eleven parameters: the second doublet of the empty slot and where to look for it
- The unified theory at eleven parameters: general relativity and the Standard Model on one algebra
- The inverse of the algebraic element of the unified theory at eleven parameters, M 4 (C) ⊕ M 4 (H) ⊕ M 12 (C)
BibTeX
@misc{reynaud2026renormalisation,
title = {{Renormalisation of the unified theory at eleven parameters}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749777v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Renormalisation of the unified theory at eleven parameters
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749777v2
AB - The unified theory of gravity and the Standard Model at eleven parameters is, in its Einstein gauge, Einstein gravity coupled to the Standard Model with three Planck-scale operators, and its graviton diagrams are not renormalisable. This paper asks what renormalising it can mean. The word has had several meanings, the subtraction of infinities of Dyson and of Bogoliubov, Parasiuk, Hepp and Zimmermann, the group of Stückelberg, Petermann, Gell-Mann and Low, the flow of Wilson with the effective theories of Weinberg, the non-Gaussian fixed points of asymptotic safety, the finite formulations of Epstein and Glaser and of Connes and Kreimer, the finiteness that supersymmetry or strings supply, and the scale-invariant schemes in which the renormalisation scale is a field; the paper sorts them and states what each promises for a theory that contains gravity. Four of them apply to the theory of the eleven and are written out as complete schemes: the scale-invariant effective renormalisation, in which every counterterm is a weight-zero monomial with the compensator supplying the inverse Planck masses and the one-loop running of the theory's pure numbers is computed, the one-loop correction to the cosmological number being fifty-four orders of magnitude larger than its value; the quadratic completion, Stelle's renormalisable theory in its dimensionless form, with its ghost within a decade of the edge of the theory's domain; asymptotic safety, whose fixed point freezes the dimensionless momentum of the eleven at √ g * , 0.84 in Litim's scheme; and the non-local completion with a form factor in the eleven's own variable. Each is tested on the two diagrams of the companion paper, the Higgs loop and the graviton loop in the graviton self-energy, and on the two-loop R 3 term. Finally the structure of the theory is examined for a method of its own: its reverse gauging is a compact momentum space, the 4-sphere of the Planck radius, on which every loop integral is ultraviolet finite, and the reciprocity of its reverse reading, Born's, is the reflection of that sphere through its equator, an isometry that makes the region beyond its equator, at twice the Planck momentum, the image of the region inside it and relates the two halves of every loop integral. Tested on the two diagrams in three models of the propagator, this reciprocal renormalisation turns every logarithmically divergent coefficient into a finite number, predicted up to the constant of the model, the logarithm running from the mass to its image, and every power-law one into a Planckian number, excluded for the Higgs mass and the vacuum energy by thirty-three and eighty-eight orders of magnitude, which the symmetry pairs with its image and doubles rather than cancels; it is a method for the local logarithmic parts and not for the hierarchy, unless the reverse theory's loops enter with the sign opposite to that of their images, a condition the paper states and the eleven do not meet by themselves, the reciprocity acting on the representations of their algebra as the parity does. The method mixes with the scale-invariant, the subtraction, the quadratic and the asymptotically safe schemes, in each case with a definite result, and not with the non-local one; the first mixing, computed at one loop, gives every pure number of the theory a finite correction and shows that the Higgs loop induces an Einstein term a hundred million times the tree one for the curvature coupling of Palatini Higgs inflation, so that a physical compact momentum space and that inflation are not perturbatively compatible.
ER -