Stringy Naturalness: SUSY Landscape Insights

• Stringy naturalness is a framework that defines naturalness through the statistical abundance of string vacua satisfying anthropic and phenomenological criteria.
• It predicts specific SUSY mass parameters, including light higgsinos and heavy gluinos, by balancing large soft terms with an atomic bound on electroweak fine-tuning.
• It differs from conventional naturalness measures by using a landscape-based approach that avoids overestimating fine-tuning and aligns with observed electroweak values.

Stringy naturalness is a measure of naturalness rooted in the statistical properties of the string theory landscape, where the physical observables of low-energy effective theories, such as supersymmetric extensions of the Standard Model, are judged by the relative abundance of phenomenologically viable vacua that realize those observables. This statistical framework formalizes and extends conventional naturalness by combining a landscape prior, derived from the distribution of supersymmetry (SUSY) breaking parameters across string vacua, with an anthropic (“atomic principle”) constraint that ensures the resulting universes can support complex chemistry. Stringy naturalness provides a predictive statistical rationale for the observed electroweak scale and superpartner spectra, yielding sharply distinct expectations for collider signatures and dark matter compared to conventional bottom-up or high-scale fine-tuning criteria (Baer et al., 2023, Baer et al., 2019, Baer et al., 2020).

1. Definition and Formal Criterion

In the paradigm of flux compactifications and moduli stabilization in string theory, each vacuum corresponds to a specific assignment of background fluxes, moduli vacuum expectation values (vevs), and hidden-sector SUSY-breaking fields, collectively determining the low-energy soft SUSY-breaking terms $m_{\rm soft}$. The core principle is: “An observable $\mathcal{O}$ is more (stringy) natural than $\mathcal{O}'$ if more phenomenologically viable string vacua lead to $\mathcal{O}$ than to $\mathcal{O}'$.” This criterion is quantified by counting the number $N(\mathcal{O})$ of landscape vacua with low-energy parameters compatible with $\mathcal{O}$ after applying anthropic and phenomenological cuts (Baer et al., 2023, Baer et al., 2019).

The landscape measure is constructed as

$$dN_{\text{vac}} \sim f_{SUSY}(m_{\text{soft}})\; f_{EWSB}(m_{\text{soft}})\; dm_{\text{soft}},$$

where:

• $$f_{SUSY}(m_{\text{soft}}) \propto m_{\text{soft}}^n,\; n=2n_F+n_D-1$$ encodes the statistical draw to large soft terms, given $n_F$ independent $F$-term and $n_D$ $D$-term SUSY-breaking fields.
• $f_{EWSB}(m_{\text{soft}})$ is the anthropic/atomic window, typically implemented as a Heaviside function $\Theta(\Delta_{\rm max} - \Delta_{EW})$ restricting attention to vacua with acceptable electroweak symmetry breaking (EWSB) and a weak scale compatible with complex atoms.

2. The Role of the Atomic Principle and Electroweak Fine-Tuning

The atomic principle, introduced by Agrawal et al., posts that atomic structure and nuclei become unstable if the weak scale $m_{\rm weak}$ deviates significantly from its observed value. Operationally, one imposes a bound on the electroweak fine-tuning parameter, defined as

$$\Delta_{EW} \equiv \max_i\left|\text{each term on RHS of } m_Z^2/2\right| / (m_Z^2/2)$$

where $$m_Z^2/2 = - m_{H_u}^2 - \mu^2 - \Sigma_u^u(\tilde{t}_{1,2}) + \ldots$$ incorporates both tree-level and one-loop corrections. A typical anthropic cut is $\Delta_{EW} \lesssim 30$, corresponding to weak scale values not deviating by more than a factor of $4-5$ from $100$ GeV, as required for viable nuclei and chemistry (Baer et al., 2023, Baer et al., 2019, Baer et al., 2020).

The nucleation of vacua thus proceeds as a competition between the statistical prior pulling toward large $m_{\rm soft}$ and the atomic constraint vetoing vacua with $\Delta_{EW}$ above the threshold. The multiverse is thus populated predominantly at the “edge” of this anthropic boundary, leading to critical spectra with large but not excessive soft parameters.

3. Distinction from Conventional Naturalness Measures

Traditional measures, such as Barbieri-Giudice (BG) or EENZ sensitivity $$\Delta_{BG} = \max_i \left| \frac{\partial \ln m_Z^2}{\partial \ln p_i} \right|$$, and high-scale (HS) fine-tuning $\Delta_{HS} = \delta m_{H_u}^2 / m_h^2$, assess naturalness with respect to (generally arbitrary) high-scale parameters or radiative corrections. These measures can greatly overestimate fine-tuning—often by factors of $10^2$–$10^3$—due to treating dependent quantities as independent and the ambiguity in parameter choices (Baer et al., 2023).

In contrast, stringy naturalness is governed by the landscape distribution and anthropic veto:

• It does not exceed the anthropic limit ($\Delta_{EW} \lesssim 30$).
• It selects for the largest soft terms $m_{\text{soft}}$ still allowed by $\Delta_{EW}$, so that most vacua exist on the critical boundary.
• It endows bottom-up electroweak naturalness with a probabilistic landscape interpretation, and it never overestimates fine-tuning beyond the anthropic cutoff (Baer et al., 2023, Baer et al., 2019).

4. Landscape Predictions for SUSY Spectra and Phenomenology

Stringy naturalness statistically favors spectra where:

• $\mu \sim 100$–$350$ GeV, leading to light higgsinos ($$\tilde{\chi}_1^0, \tilde{\chi}_2^0, \tilde{\chi}_1^\pm$$).
• Gluinos ($\tilde{g}$) and stops ($\tilde{t}_{1,2}$) are heavy, with $m_{\tilde{g}} \sim 2$–$6$ TeV, $m_{\tilde{t}_1} \sim 1$–$3$ TeV, and first/second generation sfermions even heavier (up to 10–30 TeV).
• The Higgs boson mass is naturally $m_h \sim 125$ GeV, achieved via large $A_t$ (maximal mixing) at the boundary of radiative EWSB.
• Mass gaps such as $$\Delta m^0 = m_{\tilde{\chi}_2^0} - m_{\tilde{\chi}_1^0}$$ are most naturally in the $4$–$10$ GeV range; gaugino masses are driven as large as allowed by the $\Delta_{EW}$ cut, further compressing this gap (Baer et al., 2020).

Collider phenomenology is characterized by:

• The most promising LHC discovery channel is higgsino pair production with compressed spectra: events with very soft dileptons plus MET, arising from $$\tilde{\chi}_2^0 \to \tilde{\chi}_1^0 \ell^+ \ell^-$$.
• Gluino and stop pair production rates are suppressed due to their multi-TeV masses, likely beyond near-term LHC reach.
• The soft-dilepton plus jet plus MET signature probes the bulk of the stringy-natural region, which is challenging but targeted by dedicated low-mass-gap searches at both HL-LHC and proposed HE-LHC (Baer et al., 2020).

In dark matter, stringy-natural vacua yield under-abundant higgsino WIMPs (10% of $\Omega_{DM}$), with axions expected to supplement the remainder (Baer et al., 2019).

5. Application to Specific String-Inspired and D-brane Models

The framework applies to D-brane inspired constructions as well, such as flipped $SU(5)\times U(1)_X$ with TeV-scale vectorlike “flippon” fields that enable string-scale gauge coupling unification. These models generate:

• Predominantly Higgsino LSPs with compressed light stop spectra ($\Delta M(\tilde{t}_1, \tilde{\chi}_1^0) < 5$ GeV) and large negative $A_t\tiny (EW) = -3$ to $-4$ TeV at the weak scale.
• $\Delta_{EW}$ below 30, reflecting extremely low fine-tuning, due almost entirely to the $\mu$ and $M_{H_u}^2$ sectors; contributions from stops are $\mathcal{O}(1)$ and not significant (Benedetti et al., 2019).
• Higgs mass uplifted to $\sim125$ GeV by 1-loop flippon effects even with light stops, preserving both naturalness and the Higgs mass prediction.

Parameter space scans of these constructions confirm that the stringy naturalness conditions—compressed stop-LSP system, low $\mu$, and appropriate $A_t$ and flippon Yukawa—uniquely target regions that evade current collider bounds while minimizing fine-tuning (Benedetti et al., 2019).

6. Broader Implications and Theoretical Interpretation

Stringy naturalness recasts the notion of naturalness as a problem of maximizing the number of viable vacua, rather than minimizing cancellations among Lagrangian parameters. Once the landscape’s statistical preference for large soft terms is combined with the atomic principle, the most probable vacua “pile up” at the boundary of electroweak symmetry breaking:

• The result naturally explains the observed alignment between low $\Delta_{EW}$ and phenomenological targets such as $m_h \sim125$ GeV and heavy, but not inaccessible, superpartners.
• Models with highly fine-tuned EWSB, or with low-scale soft terms ($\Delta_{EW}\ll1$), are exponentially rare, as are those with too large a weak scale (which fail the anthropic cut).
• The Standard Model (without new physics below high scales) is disfavored in the string landscape, as attaining $m_H\sim125$ GeV in the presence of a large cutoff requires extreme tuning, resulting in a negligible number of viable vacua (Baer et al., 2019).

In summary, stringy naturalness is not an alternative but rather the statistical realization of conventional electroweak naturalness—explaining why the parameter region $\Delta_{EW} \sim10$–$30$ is both theoretically favored in the landscape and in concordance with experimental non-observation of superpartners up to current limits (Baer et al., 2023, Baer et al., 2019).