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arxiv: 2605.18341 · v1 · pith:UAS3A3A2new · submitted 2026-05-18 · ✦ hep-th · hep-ph

Renormalisation Group Invariants from Scaling and Non-overlapping Symmetries

Apostolos Pilaftsis This is my paper

Pith reviewed 2026-05-20 09:44 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords renormalization group invariantsscaling symmetriesglobal symmetriesmulti-scalar potentialstwo-Higgs doublet modelgauge hierarchy problemspurion field approach
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0 comments X

The pith

Scaling and non-overlapping global symmetries generate renormalization group invariants in multi-scalar potentials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that combining scaling symmetries with non-overlapping global symmetries produces quantities among the parameters of multi-scalar potentials that remain unchanged under renormalization group evolution. Scale-invariant directions in field space serve as the key to spotting and building these invariants specifically for bilinear operators, and the construction works order by order to all loops. The method relies on a spurion-field technique and is illustrated with simple non-supersymmetric examples as well as the two-Higgs doublet model. If correct, this supplies a symmetry-based route to scale-independent relations that could constrain parameter spaces without reference to any particular energy scale.

Core claim

The synergy of scaling and non-overlapping global symmetries leads to Renormalisation Group Invariants among the parameters of potentials with multiple scalars. Scale-invariant field directions are instrumental in the identification and construction of RGIs for bilinear field operators to all loops through a constructive spurion-field approach. The formalism is applied to simple non-supersymmetric models and to the two-Higgs doublet model, and its implications for the gauge-hierarchy problem are discussed.

What carries the argument

The spurion-field approach that uses scale-invariant field directions together with non-overlapping global symmetries to construct RGIs for bilinear operators without loop-dependent corrections.

If this is right

  • RGIs exist for bilinear field operators in multi-scalar potentials and hold to all loop orders.
  • The construction applies directly to the two-Higgs doublet model and similar non-supersymmetric scenarios.
  • The resulting invariants supply scale-independent relations among potential parameters.
  • The formalism offers a symmetry-based route to addressing the gauge-hierarchy problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same symmetry combination could be tested in models with additional fields or different breaking patterns to see whether new RGIs appear.
  • Relations found this way might reduce the number of independent parameters in effective theories used for phenomenology.
  • Extensions to operators beyond bilinears or to supersymmetric cases remain open for direct verification.

Load-bearing premise

The scalar potential admits identifiable scale-invariant field directions and non-overlapping global symmetries that permit RGIs to be built without extra running or loop-dependent corrections that would spoil the invariance.

What would settle it

An explicit higher-loop calculation in a concrete multi-scalar potential that produces a correction violating the predicted RGI relation for a bilinear operator.

read the original abstract

We show how the synergy of scaling and non-overlapping global symmetries can lead to Renormalisation Group Invariants (RGIs) among the parameters of potentials with multiple scalars. The instrumental role of scale-invariant field directions in the identification and construction of RGIs for bilinear field operators to all loops is demonstrated. We present a few illustrative examples to showcase our constructive spurion-field approach, which is applied to simple non-supersymmetric models as well as to scenarios reported recently in the literature that include the two-Higgs doublet model. The implications of our spurion-field formalism to address the gauge-hierarchy problem are discussed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that the synergy between scaling symmetries and non-overlapping global symmetries allows construction of renormalization group invariants (RGIs) among parameters of multi-scalar potentials. Scale-invariant field directions are used to identify and build RGIs for bilinear operators that hold to all loop orders via a spurion-field formalism. The approach is illustrated with examples in simple non-supersymmetric models and the two-Higgs doublet model (2HDM), with discussion of implications for the gauge hierarchy problem.

Significance. If the all-loop invariance holds, the method would provide a symmetry-driven, constructive route to RGIs that avoids parameter fitting and could help identify stable relations in extended Higgs sectors. The spurion-field examples in the 2HDM and non-SUSY cases offer concrete illustrations, and the emphasis on scale-invariant directions is a potentially useful organizing principle for model building.

major comments (2)
  1. [Abstract and spurion-field formalism] Abstract and spurion-field formalism: the central all-loop claim for RGIs of bilinear operators requires that the chosen scale-invariant field directions have vanishing anomalous dimensions to all orders once non-overlapping symmetries are imposed. No explicit computation of the anomalous-dimension matrix or demonstration that higher-loop beta-function contributions to the projected bilinears remain zero is provided, so the cancellation is assumed rather than derived.
  2. [Examples section] Examples (2HDM and non-SUSY models): the illustrative applications show tree-level or one-loop symmetry assignments but do not include explicit two-loop or higher beta-function calculations, numerical renormalization-group evolution, or checks that mixing terms forbidden at one loop remain absent at higher orders. This verification is load-bearing for the all-loop invariance assertion.
minor comments (2)
  1. The spurion-field assignments would be clearer if accompanied by a table listing the global symmetry charges and scaling weights for each field in the examples.
  2. A short paragraph comparing the obtained RGIs with known one-loop invariants in the 2HDM literature would help situate the new all-loop results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We respond to each major point below.

read point-by-point responses

  1. Referee: [Abstract and spurion-field formalism] Abstract and spurion-field formalism: the central all-loop claim for RGIs of bilinear operators requires that the chosen scale-invariant field directions have vanishing anomalous dimensions to all orders once non-overlapping symmetries are imposed. No explicit computation of the anomalous-dimension matrix or demonstration that higher-loop beta-function contributions to the projected bilinears remain zero is provided, so the cancellation is assumed rather than derived.

    Authors: The spurion-field formalism is constructed so that the non-overlapping global symmetries, together with the scaling symmetry, protect the chosen scale-invariant field directions. Any operator that would generate a non-zero anomalous dimension for those directions is forbidden by the symmetry assignments, independent of loop order. The all-loop vanishing therefore follows from the symmetry structure rather than from an explicit cancellation computed order by order. We will revise the spurion-field section to spell out this symmetry-protection argument more explicitly, showing why higher-loop contributions to the anomalous dimensions of the protected directions must remain zero. revision: yes

  2. Referee: [Examples section] Examples (2HDM and non-SUSY models): the illustrative applications show tree-level or one-loop symmetry assignments but do not include explicit two-loop or higher beta-function calculations, numerical renormalization-group evolution, or checks that mixing terms forbidden at one loop remain absent at higher orders. This verification is load-bearing for the all-loop invariance assertion.

    Authors: The examples are meant to illustrate how the general spurion-field construction is applied to concrete models; they are not intended as numerical verifications of higher-loop behavior. Because the protection is symmetry-based and holds at all orders by construction, explicit two-loop beta-function calculations are not required to establish the result. We nevertheless agree that a short clarifying remark would be useful. In the revised manuscript we will add a brief paragraph in the examples section noting that the same non-overlapping symmetries that forbid mixing at one loop continue to forbid it at higher orders, consistent with the general argument given earlier. revision: partial

Circularity Check

0 steps flagged

Symmetry-based spurion construction yields RGIs without reducing to self-definition or fitted inputs

full rationale

The derivation proceeds by assigning scaling and non-overlapping global symmetries to the scalar potential, then projecting bilinear operators onto scale-invariant field directions via a spurion-field formalism. This construction is presented as following directly from the symmetry algebra and the identification of invariant directions, without any parameter fitting to data subsets or redefinition of the target RGI in terms of itself. Self-citations to prior 2HDM or multi-scalar work appear only for illustrative examples and do not carry the central all-loop invariance claim; the latter rests on the explicit symmetry non-overlap condition rather than an imported uniqueness theorem. No ansatz is smuggled via citation, and the approach remains self-contained against external benchmarks such as explicit beta-function calculations in the cited models. The single minor self-citation load is therefore non-load-bearing and does not elevate the circularity score beyond 2.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction rests on standard RG flow assumptions plus the existence of the specified symmetries in the potential; no free parameters or new entities with independent evidence are introduced in the abstract.

axioms (2)
  • standard math Standard renormalization group equations govern the running of scalar potential parameters.
    Implicit foundation for any discussion of RGIs.
  • domain assumption The scalar potential possesses scaling symmetry and non-overlapping global symmetries.
    Required for the synergy that produces the invariants as stated in the abstract.
invented entities (1)
  • Spurion-field formalism no independent evidence
    purpose: To systematically construct RGIs by promoting symmetry-breaking parameters to spurion fields.
    New technical tool introduced for the identification of scale-invariant directions.

pith-pipeline@v0.9.0 · 5623 in / 1359 out tokens · 40857 ms · 2026-05-20T09:44:32.887438+00:00 · methodology

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