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Cornering Natural SUSY at a Tera-$Z$ Factory

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read FCC-ee's trillion-Z run, through the single observable Rb, could indirectly probe natural supersymmetry at multi-TeV scales, outperforming its Higgs program and closing LHC blind spots.

desk verdict Solid, useful projection study with a real unresolved Rb discrepancy that needs fixing before the headline number is quoted. read the letter →

arxiv 2507.03073 v2 pith:GLJ5MGSP submitted 2025-07-03 hep-ph hep-exhep-th

Admir Greljo , Ben A. Stefanek , Alessandro Valenti This is my paper · ORCID
classification hep-phhep-exhep-th
keywords supersymmetryMSSMnaturalnessFCC-eeTera-ZfactoryelectroweakprecisionobservablesZtobottom-quarkcoupling(Rb)SMEFT
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

FCC-ee's Tera-Z run—over a trillion Z bosons at the Z pole—could indirectly probe natural supersymmetry more powerfully than the collider's own Higgs program. The paper computes one-loop effects of three MSSM sectors (heavy Higgs doublet, stops, gauginos/higgsinos) on electroweak precision and Higgs observables, using updated FCC-ee projections and three scenarios for Standard Model theory errors. In the experimental-only S3 scenario, the single observable $R_b$ (the $Z\to b\bar b$ rate) dominates: it reaches heavy-Higgs masses $M_A\gtrsim 9$ TeV at $\tan\beta\approx 50$, stops up to about 5 TeV, and Higgsino/gaugino masses $\mu\gtrsim 500$ GeV, $M_2\gtrsim 850$ GeV—while also covering compressed spectra that hide from LHC searches. If correct, this redirects attention to the Z run as the discovery mode, and to $R_b$ as the top target for Standard Model theory improvements.

What carries the argument

The machinery is one-loop matching of the three MSSM sectors onto the dimension-6 Standard Model Effective Field Theory (SMEFT), fed into a single all-FCC-ee likelihood. For the heavy Higgs doublet, the non-oblique operator $Q_{Hd}$ is generated with Wilson coefficient $[C_{Hd}]_{33}(M_Z) \supset -\frac{y_t^2 y_b^2 \tan^2\beta}{32\pi^2 M_A^2}\left(1+\log\frac{M_Z^2}{M_A^2}\right)$, which shifts the $Z b_R \bar b_R$ vertex and makes $R_b$ the dominant Tera-Z observable at large $\tan\beta$. Stops enter through oblique $S,T$ and through mixed stop-higgsino loops that shift $[C_{Hq}^{(1)}+C_{Hq}^{(3)}]_{33}$, affecting $Z b_L \bar b_L$; gaugino and higgsino contributions are captured mainly by the oblique parameter $\hat W \propto 1/\mu^2 + 2/M_2^2$. The likelihood combines Z- and W-pole observables at one loop in SMEFT, Higgs-coupling measurements, and above-Z fermion pair production; the three theory scenarios S1-S3 set the error budget that decides which observable leads.

What would settle it

Compute the Standard Model prediction for $R_b$ at the projected relative precision of $0.18\times 10^{-5}$; if the irreducible theory error is larger than roughly that size, the S3 scenario cannot be realized and the $M_A\simeq 9$ TeV and stop $\simeq 5$ TeV claims collapse. Independently, a future $b\to s\gamma$ measurement that disagrees with minimal flavor violation at the 10 percent level would invalidate the flavor-suppression premise.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Tera-Z electroweak precision run at FCC-ee offers systematically greater indirect sensitivity to natural MSSM parameter space than the Mega-Higgs run, and that in the experimental-error-only limit $R_b$ is the single most powerful observable. At large $\tan\beta$ the heavy-Higgs doublet generates a $\tan^2\beta$- and log-enhanced correction to the $Z b_R\bar b_R$ vertex, so $R_b$ alone bounds $M_A\gtrsim 9$ TeV at $\tan\beta\approx 50$, with an overall floor $M_A\gtrsim 2$ TeV. For stops, oblique $T$ and $R_b$ together reach $\sim 5$ TeV for order-one mass splittings, testing fine-tuning at the $10^{-4}$ level; for gauginos and higgsinos, the $\hat W$ parameter gives decay-channel-independent limits $\mu\gtrsim 500$ GeV and $M_2\gtrsim 850$ GeV that remain valid in compressed regimes. Under the conservative S1 theory scenario the reach degrades to $M_A\gtrsim 1.7$ TeV and stops near $1.5$–$2.5$ TeV, making FCC-ee complementary rather than superior to the LHC.

Load-bearing premise

The headline reach assumes the Standard Model theory uncertainty on $R_b$ can be reduced to the experimental-error-only S3 level, and that minimal flavor violation keeps flavor and CP observables from dominating over the electroweak constraints.

Editorial extensions

If this is right

  • In the S3 scenario, the Tera-Z run alone can probe natural SUSY into the multi-TeV range: the heavy-Higgs mass reach is about 9 TeV at $\tan\beta\simeq 50$, and even the conservative S1 scenario reaches about 1.7 TeV, complementing the HL-LHC direct reach.
  • The single observable $R_b$ becomes the dominant indirect probe for both the heavy Higgs and, in parts of parameter space, stops; reducing its Standard Model theory uncertainty is the highest-leverage target for future calculations.
  • Stop masses up to about 5 TeV with order-one mass splittings are within reach, improving the implied fine-tuning constraint by more than an order of magnitude over current LHC bounds.
  • Higgsino and gaugino lower bounds ($\mu\gtrsim 500$ GeV, $M_2\gtrsim 850$ GeV) are decay-channel independent, so they remain valid for compressed spectra where LHC chargino and neutralino searches lose sensitivity.
  • FCC-ee's Higgs-coupling measurements become the second-most-powerful probe of these sectors; the Z-pole program carries the main discovery potential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The $\tan^2\beta$-enhanced $Z\to b\bar b$ effect is not specific to the MSSM; any two-Higgs-doublet or extended-scalar sector with a large hierarchy in bottom versus top couplings would generate the same $Q_{Hd}$ operator, so the Tera-Z $R_b$ reach generalizes beyond supersymmetry.
  • The ranking of observables is itself a testable prediction of the error budgets: if the Standard Model theory error on $R_b$ cannot be pushed below the S3 level, the lead role shifts to the $T$ parameter and Higgs couplings, flipping the paper's headline hierarchy.
  • A future high-precision $b\to s\gamma$ measurement would directly test the minimal-flavor-violation premise; a deviation there could make flavor observables dominate over $R_b$ for the same parameter space.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents FCC-ee Tera-Z projections for indirect MSSM searches in three sectors: the heavy Higgs doublet, scalar top partners, and gauginos/higgsinos. Using one-loop SMEFT matching in the decoupling limit, the authors compute the relevant Wilson coefficients, including non-oblique Z->bb vertex corrections, and combine them in a global 'all-FCC-ee' likelihood with three theory-error scenarios (S1, S2, S3) built from EW PPG inputs and FCC-ee FSR projections. The central finding is that in the experimental-only S3 scenario Rb dominates the Z-pole fit, excluding mA >~9 TeV at tan(beta)~50, probing stops up to about 5 TeV through the T parameter, and setting mu and M2 bounds around 500 and 850 GeV through the W parameter. The paper argues that Tera-Z therefore provides superior or complementary sensitivity relative to the Mega-h Higgs program and closes compressed-spectrum blind spots left by the LHC.

Significance. The study is a careful, largely reproducible phenomenological projection. Its strengths include the use of updated FCC-ee FSR inputs, the explicit one-loop Wilson coefficient formulas in Appendix C, the cross-check of the Matchete matching against Ref. [43], and the transparent S1-S3 framework that separates experimental from theoretical uncertainties. The projected observables are taken from external FCC-ee studies rather than fitted, so there is no circularity in the analysis. The core physics message—that non-oblique Z->bb vertex corrections, especially Rb, deserve targeted Standard Model theory work—is timely and potentially important for the FCC-ee program. However, the headline S3 reach is conditional on an unresolved discrepancy with Ref. [31] in the Rb calculation, so the quantitative claims are not yet fully secured.

major comments (3)
  1. [Section 3, footnote 3 and Appendix C] The paper states that its Rb result agrees with Ref. [43] but disagrees with Delta Rb in Appendix A.3 of Ref. [31] when the small-angle EFT limit is taken, without identifying the source of the discrepancy. This is not a peripheral issue: Rb is the observable that drives the S3 heavy-Higgs bound in Fig. 2 and the stop Rb regions in Figs. 3, 5, and 6, and hence the paper's headline claim of mA > 9 TeV. The authors should identify the origin of the difference (for example, the y_b renormalization scheme, tan(beta) resummation, a sign error, or the validity of the small-mixing EFT limit) and provide a quantitative comparison against Ref. [31] in a common limit. Until this is resolved, an independent group cannot reproduce the central result.
  2. [Table I and Figs. 2-3] The S3 scenario assigns Rb a relative uncertainty of 0.18 x 10^-5, which is about a factor of nine smaller than the S2 value. The paper presents S3 as the ultimate reach and uses it for the strongest conclusions, but it does not identify which missing higher-order calculations would be needed to reduce the Standard Model theory uncertainty to this level, nor does it show how the limits in Figs. 2 and 3 degrade between S2 and S3. The authors should quantify this interpolation and state more precisely what theoretical progress is required to exploit the Rb sensitivity; without this, the practical significance of the 'Tera-Z beats Mega-h' conclusion is somewhat overstated.
  3. [Appendix B and Section 4] The W-parameter limits in Section 4 are driven by fermion pair-production above the Z-pole, but Appendix B states that RGE is neglected for the datasets at 163, 240, and 365 GeV. Since the projected precision reaches 10^-5 and the relevant SMEFT operators run between MZ and these scales, the authors should quantify the numerical impact of this approximation on the mu and M2 bounds. If the impact is non-negligible, the fit should include the running; if it is negligible, a quantitative statement to that effect would strengthen the presentation.
minor comments (5)
  1. [Introduction] The sentence 'For the less precise Higgs observables, we assume theoretical errors will be negligible' should be quantified or referenced, since the Higgs observables enter some of the main plots and the S1-S3 framework is otherwise defined only for pole observables.
  2. [Appendix C] The Wilson coefficient formulas in Eqs. (C1) and (C2) do not explicitly state the renormalization scale at which they are evaluated or the log terms that implement the matching at MZ; adding this information would make the calculation easier to reproduce.
  3. [Section 2] The phrase 'Rb alone will dominate' should be understood as 'the Rb constraint alone will dominate'; a short clarification in the text would avoid the impression that a single-observable fit is being used.
  4. [References] Reference [29] is incomplete: 'FCC Collaboration, Prospects in BSM physics at FCC' has no arXiv number or publication identifier; please complete the entry.
  5. [Fig. 2 caption] In the right panel, the red-shaded region 'requires heavy stops to fit the Higgs mass' is mentioned in the text but not described in the caption; please clarify how this region is obtained and why it adds fine-tuning.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's projected FCC-ee sensitivities follow from external FSR/EW PPG inputs and one-loop MSSM–SMEFT matching, with no fitted input renamed as prediction.

full rationale

The derivation chain is self-contained and does not reduce to its own inputs. Heavy-Higgs bounds are obtained by full one-loop matching of the heavy doublet onto the dimension-6 SMEFT using Matchete, cross-checked against the independent results of Ref. [43]. Stop constraints use oblique and vertex corrections taken from Refs. [31,53,54], with the benchmark choice μ = 500 GeV justified by the independent limits derived in Section 4; this is a benchmark, not a fitted parameter, and the Rb sensitivity depends only logarithmically on μ. Gaugino/higgsino constraints reproduce the external results of Ref. [56]. The S1–S3 theory-error scenarios are inputs from the EW PPG [58] and FCC FSR [7], not parameters fitted to the MSSM predictions, and the observables themselves are projections taken from those external studies. The footnote disagreement with Ref. [31] over ΔRb is an unresolved external consistency issue, not a circular step: the paper's calculation is matched and cross-checked against Ref. [43] rather than being defined in terms of the claimed result. Self-citations such as Ref. [15] provide auxiliary ingredient for fermion-pair observables but are not the load-bearing derivation of the headline Rb or T bounds, so they do not constitute circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the MSSM particle content, standard SMEFT and EFT tools, and projection inputs from the FCC-ee FSR and the EW PPG. The parameter choices listed are benchmarks fixed by hand, not fits to data, but they shape the displayed exclusion contours. The largest external loads are the FCC-ee FSR experimental projections, the EW PPG theory error numbers, and the assumption of Minimal Flavor Violation. No new particles or entities are introduced.

free parameters (4)
  • Xt (stop trilinear coupling) = Xt = sqrt(6 M_t1 M_t2), maximal mixing
    Fixed in Section 3 to the value maximizing the radiative Higgs mass correction, avoiding regions where the Higgs is too light. This is a benchmark choice, not a fit to projected data.
  • mu (Higgsino mass) = 500 GeV
    Fixed in the stop Rb sensitivity plots (Fig. 3, 5, 6) as a natural benchmark, stated to be consistent with the gaugino mass limits derived in Section 4.
  • M1 (bino mass parameter) = M2/2
    GUT-inspired relation imposed in Section 4 to reduce the parameter space; the paper argues the results depend weakly on this choice.
  • tan beta = 10 (in the gaugino section)
    Fixed to tan beta = 10 in Section 4; S and T depend mildly on it and those contributions are subdominant. In other sections tan beta is scanned as an axis.
assumptions (5)
  • domain assumption The MSSM is a well-established benchmark for natural supersymmetry.
    The entire analysis evaluates MSSM sectors; stated in the Introduction and not derived elsewhere.
  • domain assumption Minimal Flavor Violation suppresses flavor and CP violation, making b to s gamma and other flavor bounds subdominant.
    Stated in the Introduction. This is load-bearing because flavor bounds could otherwise dominate over the electroweak precision constraints.
  • domain assumption The decoupling limit (beta - alpha = pi/2) for the heavy Higgs doublet.
    The Section 2 matching is performed in this limit; corrections from Higgs mixing are neglected.
  • standard math Dimension-6 SMEFT with one-loop leading-log RGE is adequate for the projections.
    Used in the matching to Z and W pole observables; NLO SMEFT corrections from Refs. [44,45] are taken as input.
  • domain assumption The S1-S3 theory uncertainty scenarios from the EW PPG are adopted as inputs.
    These scenarios determine the projected errors; they are inputs from Ref. [58], not derived in this paper.

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Cite this review

Pith. "Pith review of Cornering Natural SUSY at a Tera-$Z$ Factory." pith.science (2026). https://pith.science/paper/GLJ5MGSP

@misc{pith2026250703073,
  author       = {Pith},
  title        = {Pith review of: Cornering Natural SUSY at a Tera-$Z$ Factory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GLJ5MGSP}},
  note         = {Machine review of arXiv:2507.03073}
}
abstract

The future circular $e^+ e^-$ collider (FCC-ee) stands out as the next flagship project in particle physics, dedicated to uncovering the microscopic origin of the Higgs boson. In this context, we assess indirect probes of the Minimal Supersymmetric Standard Model (MSSM), a well-established benchmark hypothesis, exploring the complementarity between Higgs measurements and electroweak precision tests at the $Z$-pole. We study three key sectors: the heavy Higgs doublet, scalar top partners, and light gauginos and higgsinos, focusing on the parameter space favored by naturalness. Remarkably, the Tera-$Z$ program consistently offers significantly greater indirect sensitivity than the Mega-$h$ run. While promising, these prospects hinge on reducing SM uncertainties. Accordingly, we highlight key precision observables for targeted theoretical work.

Figures

Figures reproduced from arXiv: 2507.03073 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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