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REVIEW 3 major objections 5 minor 48 references

The Higgs mass squared power-runs as the square of the energy scale, so an order-one high-scale boundary value produces the electroweak scale without fine-tuning or protective symmetries.

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2026-07-14 20:41 UTC pith:AD2PRHVM

load-bearing objection Clean one-loop SM evaluation of the mass function at GUT scale, but the no-symmetry claim leaves residual GUT cancellations unaddressed. the 3 major comments →

arxiv 2603.15081 v4 pith:AD2PRHVM submitted 2026-03-16 hep-ph hep-th

Natural Higgs Mass from Power-Law Running

Kang-Sin Choi This is my paper
classification hep-ph hep-th
keywords hierarchy problemHiggs masspower-law runningmass functionnaturalnessdecouplingStandard ModelGUT matching
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that the fully renormalized Higgs mass is a physical function of the external momentum that grows as the square of the energy at which it is probed. Because of that power-law running, the ratio of mass-squared to energy-squared stays order one from the electroweak scale all the way to the unification scale. An order-one boundary condition supplied by whatever completes the Standard Model is then mapped, through the small loop coefficient dominated by the top Yukawa, down to the observed Higgs pole mass—exactly as dimensional transmutation produces the QCD scale from a dimensionless coupling. The familiar 28 orders of magnitude therefore measure the smallness of that anomalous dimension rather than a cancellation that must be protected by new TeV-scale symmetries. A sympathetic reader cares because this recasts naturalness as an ordinary question about an order-one ratio and removes the theoretical pressure for supersymmetry, compositeness, or extra dimensions at the LHC.

Core claim

Once the scalar self-energy is completely renormalized, the physical mass function m^{2}(q^{2}) itself runs as q^{2}. Evaluated in units of the scale at which it is probed, it remains O(1) at every energy; at the GUT scale the Standard Model content alone yields m^{2}(M_GUT^{2})/M_GUT^{2} o +0.86. Matching an order-one GUT boundary condition therefore produces an electroweak-scale pole mass through ordinary SM running, with no residual cancellation among huge numbers and no protective symmetry required.

What carries the argument

The renormalized mass function m^{2}(q^{2}) = m_h^{2} + Σ_ren(q^{2}), defined by on-shell subtraction of the self-energy and its first derivative so that the pole mass is fixed while the function retains its full momentum dependence; this object power-runs as q^{2} log q^{2} and decouples heavy states as (q^{2} – m_h^{2})^{2}/M^{2}.

Load-bearing premise

That the fully renormalized mass function truly erases all residual sensitivity to heavy ultraviolet physics, so an order-one matching condition at the unification scale is automatically natural and sufficient.

What would settle it

A precision measurement of the off-shell Higgs propagator (or an explicit two-loop or non-perturbative calculation of the same mass function) that shows m^{2}(q^{2})/q^{2} is not order one, or that residual quadratic sensitivity to heavy states survives complete renormalization.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper argues that the fully renormalized Higgs mass function m²(q²), defined by on-shell subtraction in Eq. (2), power-runs as q² rather than logarithmically. Explicit one-loop SM self-energies (Appendix) yield m²(M_GUT²)/M_GUT² ≈ 0.858 at 10^16 GeV, an O(1) ratio. Decoupling of heavy states after complete renormalization (Eq. (3)) is used to claim that an order-one GUT-scale boundary condition r on this ratio is natural, and that SM running then maps it to an electroweak-scale pole mass via an exponential relation (Eq. (11)) controlled by the small loop coefficient c ≈ 0.0142 (dominated by the top Yukawa). Naturalness is thereby recast as the value of an O(1) ratio, with no protective symmetry required; the 28 orders of magnitude measure the smallness of that anomalous dimension, analogous to dimensional transmutation of Λ_QCD.

Significance. If the reinterpretation is correct, it would substantially change how the hierarchy problem is framed: the absence of symmetry protection would explain rather than threaten the hierarchy, and TeV-scale stabilizers would be unnecessary. The manuscript supplies concrete, checkable one-loop integrals and a sector-by-sector numerical evaluation (Eq. (5), Appendix), a two-loop order-of-magnitude estimate, and points to an off-shell Higgs test developed in prior work. The dimensional-transmutation analogy (Eq. (11)) is a sharp, falsifiable way of stating the claim. These are real strengths even if the conceptual step from the mass function to GUT naturalness needs more work.

major comments (3)
  1. [Constraint on GUT / Eqs. (7)–(10)] Eqs. (7)–(10): The central claim that 0 < r ≤ 1 is automatically natural and requires no protective symmetry is not demonstrated on the GUT side. The paper evaluates only the SM-side ratio (≈0.86) and asserts that any order-one GUT boundary condition is free of tuning. In a concrete completion (e.g. minimal SU(5)), the combination μ² − (3/2)λ₅ V + Σ_heavy is a sum of individually O(M_GUT²) terms; showing that their dimensionless ratio lands in (0,1] without a residual cancellation that reintroduces the hierarchy problem is load-bearing. An explicit matching calculation in at least one prototype GUT, or a clear argument why no such cancellation appears, is needed.
  2. [The mass function / Decoupling / Explaining the electroweak scale] The manuscript must distinguish the momentum-dependent mass function m²(q²) from the matching of the local |H|² Wilson coefficient in the EFT below M_GUT. Standard EFT reasoning identifies the hierarchy problem with O(M_GUT²) corrections to that local relevant operator; after on-shell subtraction (Eq. (2)), those corrections are absorbed into the input pole mass, so the high-q² growth of Σ_ren is largely a kinematic feature of light-field loops. Without a precise account of how power-law running of m²(q²) replaces or modifies local-operator matching (including why MS-bar logarithmic running of m²(μ) is misleading for naturalness), the claim that protective symmetries are unnecessary remains incomplete.
  3. [Explaining the electroweak scale / Eqs. (9), (11)] Eqs. (9) and (11) are not obviously equivalent. Equation (9) is a linear subtraction, m_h² = r M_GUT² − Σ_SM_ren(M_GUT²), which for independent O(1) values of r and Σ/M² appears to require a 10^{-28}-level cancellation to produce the observed pole mass. Equation (11) instead gives the exponential map m_h² ∼ e^{-r/c} M_GUT² characteristic of dimensional transmutation. The text should derive (11) explicitly from the implicit relation (6) (including the dependence of thresholds and logs on the IR scale) and state under what conditions the linear form does not reintroduce fine-tuning.
minor comments (5)
  1. [Eq. (6) and Appendix] Clarify whether “log” in Eq. (6) and the Appendix is the natural logarithm (as the numerical value ≈63 indicates) and use a consistent notation (ln vs log) throughout.
  2. [Fig. 1] Figure 1: state explicitly that the curve is the one-loop SM mass function with fixed measured inputs; the caption’s bound ∼10^18 GeV is useful but should note the sensitivity to the two-loop shift discussed in the text.
  3. [Fig. 2] Figure 2: specify how gauge and Yukawa couplings are held fixed while v is varied, and whether thresholds are rescaled self-consistently; this affects the interpretation that “any scale is natural.”
  4. [Explaining the electroweak scale] The doublet-triplet splitting remark is intriguing but purely qualitative; either move it to a footnote or flag more clearly that the explicit running coefficients are left for future work.
  5. [The mass function] References [13] (“to appear”) and the chain of prior works carry a large fraction of the technical construction of the mass function; a short self-contained summary of the gauge-invariant definition would help the reader.

Circularity Check

2 steps flagged

Moderate circularity: load-bearing self-citations define the physical mass function and its decoupling, while the O(1) high-scale ratio is computed bottom-up from the measured pole mass fixed by construction of Eq. (2).

specific steps
  1. self citation load bearing [p.1–2, paragraphs introducing the mass function and decoupling; Refs. [13–15]]
    "What is new in [14, 15] is the extraction of a finite, physical mass function in which the q^{2} dependence is not a cutoff artifact or a source of fine-tuning but the entire observable content of the scalar mass. The mass function constructed in Ref. [13] defines the scalar mass at every external momentum q. … The complete renormalization (2) removes the leading sensitivity to heavy fields. A field of mass M^{2}≫ q^{2} contributes Σ_heavy_ren(q^{2})=O((q^{2}-m_h^{2})^{2}/M^{2})"

    The central technical premises—that Eq. (2) yields a finite, scheme-independent, physical mass function free of residual UV sensitivity after decoupling—are justified exclusively by citations to the author’s own prior works [13–15] (two arXiv preprints and one “to appear”). No independent derivation or external verification of those properties is supplied; the hierarchy-resolution claim therefore reduces to an unverified self-citation chain.

  2. self definitional [Eq. (2) and the evaluation leading to Eq. (5); also Fig. 2 and the inversion (9)–(11)]
    "On-shell renormalization yields m^{2}(q^{2})=m_h^{2}+Σ(q^{2})-Σ(m_h^{2})-(q^{2}-m_h^{2})dΣ/dq^{2}(m_h^{2})≡m_h^{2}+Σ_ren(q^{2}), … with the dressed propagator i/(q^{2}-m^{2}(q^{2})) and pole mass m_h fixed by m^{2}(m_h^{2})=m_h^{2}. … Using m_h=125.08 GeV, the complete one-loop correction … gives … =+0.858, which is O(1)."

    By construction of the on-shell subtraction, m^{2} equals the measured pole mass at the pole. Computing the high-scale ratio r=m^{2}(M_GUT^{2})/M_GUT^{2} bottom-up from that same measured m_h therefore produces an O(1) number whose hierarchy relative to M_GUT^{2} has already been absorbed into the definition of the running function. The subsequent claim that any O(1) GUT boundary condition is natural (and that Fig. 2 shows every electroweak scale is natural) is consequently partly definitional rather than an independent dynamical prediction.

full rationale

The paper’s explicit one-loop SM calculation (Appendix, Eq. (5)) is self-contained and non-circular: it evaluates standard integrals with measured inputs and obtains r≈0.86. The exponential mapping formula (11) likewise follows algebraically from the asymptotic form of Σ_ren. However, the interpretive claim that this fully solves the hierarchy problem (no residual UV sensitivity, any O(1) GUT-side r is automatically natural, no protective symmetry needed) rests on two load-bearing elements that reduce to the paper’s own prior framework. First, the very definition of the finite, physical, regularization-independent mass function m^{2}(q^{2}) that “power-runs as its square” and obeys the decoupling (3) is imported from the author’s Refs. [13–15]; those properties are not re-derived here. Second, because m^{2} is defined by on-shell subtraction so that m^{2}(m_h^{2})=m_h^{2} by construction, feeding the measured pole mass into the bottom-up evaluation of r at M_GUT necessarily yields an O(1) number whose smallness relative to M_GUT^{2} has already been absorbed into the running; Fig. 2’s demonstration that “any v gives O(1) r” inherits the same SM content. These steps do not render the whole argument tautological—the power-law growth itself is a non-trivial dynamical statement—but they make the naturalness conclusion partially definitional and self-referential, justifying a moderate score of 4 rather than 0 or 8+.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The central claim rests on the physicality and power-law behavior of a mass function constructed in the author's previous works, on an extension of decoupling to super-renormalizable operators, and on the assertion that an order-one GUT matching condition requires no further explanation. No free parameters are fitted; measured SM masses are used as inputs. The only new entity is the interpretation of that mass function as the complete solution to the hierarchy problem.

axioms (4)
  • domain assumption The renormalized mass function defined by on-shell subtraction (Eq. (2)) is finite, scheme-independent and exhausts the physical content of the scalar mass at every scale.
    Imported from the author's prior papers [13–15]; used throughout as the starting point for all running and matching statements.
  • domain assumption After complete renormalization, a heavy field of mass M ≫ q contributes only O((q^{2}–m_h^{2})^{2}/M^{2}) to the mass function (Eq. (3)), extending Appelquist–Carazzone to the scalar mass.
    Stated as a theorem in the author's earlier work; load-bearing for the claim that no residual hierarchy remains.
  • ad hoc to paper An order-one value of the dimensionless ratio r at the GUT scale is natural and requires no protective symmetry or cancellation among GUT parameters.
    Asserted after Eq. (10); converts the traditional fine-tuning problem into a flavor-like question about O(1) couplings.
  • standard math Standard Model one-loop self-energies (top, W, Z, Higgs) dominate the running up to the GUT scale and remain perturbative (r < 1).
    Explicit integrals given in the Appendix; numerical evaluation uses PDG masses.
invented entities (1)
  • Physical mass function m^{2}(q^{2}) no independent evidence
    purpose: To provide a finite, momentum-dependent, gauge-invariant object that power-runs as q^{2} and can be matched at the GUT scale.
    Constructed by the author in prior papers; claimed to be measurable via off-shell Higgs production, but no independent experimental confirmation is presented here.

pith-pipeline@v1.1.0-grok45 · 13432 in / 3089 out tokens · 49442 ms · 2026-07-14T20:41:44.643534+00:00 · methodology

0 comments
read the original abstract

The renormalized scalar mass squared is a function of the energy scale and power-runs as its square. Well above the electroweak scale, its dimensionless couplings evolve only slowly, so the Standard Model is approximately scale invariant, and an order-one boundary value supplied at the unification scale is mapped exponentially down to the electroweak scale, much as the QCD scale arises from a dimensionless coupling alone. The 28 orders of magnitude separating the two then measure the smallness of an anomalous dimension, dominated by the top-quark Yukawa coupling, rather than a tuning. Naturalness is thereby recast as the value of an order-one ratio, with no protective symmetry required.

Figures

Figures reproduced from arXiv: 2603.15081 by Kang-Sin Choi.

Figure 1
Figure 1. Figure 1: FIG. 1. The Higgs mass function in units of the external mo [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

discussion (0)

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