Quasi-fixed points from scalar sequestering and the little hierarchy problem in supersymmetry

In supersymmetric models with scalar sequestering, superconformal strong dynamics in the hidden sector suppresses the low-energy couplings of mass dimension two, compared to the squares of the dimension one parameters. Taking into account restrictions on the anomalous dimensions in superconformal theories, I point out that the interplay between the hidden and visible sector renormalizations gives rise to quasi-fixed point running for the supersymmetric Standard Model squared mass parameters, rather than driving them to 0. The extent to which this dynamics can ameliorate the little hierarchy problem in supersymmetry is studied. Models of this type in which the gaugino masses do not unify are arguably more natural, and are certainly more likely to be accessible, eventually, to the Large Hadron Collider.

The setup: • SUSY is broken in a hidden sector, parameterized by F S , • The chiral superfield S that contains F S is part of a strongly coupled theory , • SUSY breaking is communicated to the MSSM (visible) sector by non-renormalizable Lagrangian terms suppressed by a scale M * , • Above a scale Λ ∼ √ F S , which is supposed to be much less than M * , the strongly coupled theory is approximately conformal, so there is power-law renormalization group running , • Scalar squared masses are driven towards 0 by renormalization group running.
This is scalar sequestering.
We now know Γ can't be too large:
Unfortunately, the classic prediction is somewhat too naive. Some issues that limit the power-law suppression: • Γ cannot be very large (now know < ∼ 0.3), • The range of scales over which the superconformal scaling takes place is limited to Q > Λ ∼ √ F S > ∼ 10 10 GeV.
• Need to include visible sector running as well.
Instead of power-law running to 0 in the infrared, dimension-2 terms will run towards quasi-fixed trajectories where the beta functions vanish: For squarks, including only gluino contribution for simplicity:

Mg.
This quasi-fixed point is often reached, but running below the scale Λ increases the squark masses substantially.
Still, M squark < M gluino is a fairly robust prediction.

(See numerical examples below.)
More importantly, what about quasi-fixed point for Higgs squared mass?
For two reasons, I don't view this as a complete solution to the SUSY little hierarchy problem: • Prefactor 3 8π 2 Γ is no smaller than about 0.12 • Running below scale Λ is also significant However, it has some helpful features: • Terms of both signs, so cancellation can occur Higgsino still very heavy, Winos could be the heaviest superpartners.
Model not excluded by the LHC, but not hopeless for eventual LHC discovery.

Conclusion:
• Interplay between visible sector renormalization and hidden sector superconformal scaling: quasi-fixed point behavior with predictive power • According to my subjective standards, some improvement in the SUSY little hierarchy problem, but not a completely satisfying "solution".
• Results are more optimistic with non-unified gaugino masses, in particular M 2 > M 3 .
Running below the scale Λ increases the selectron mass, but naively the LSP (Lightest SUSY Particle) is a charged slepton. To avoid disaster in cosmology from charged stable particle: • R-parity violation allows slepton LSP to decay How small can the scale Λ be? (Knapen and Shih, 1311.7107) Gaugino mass estimate at the scale Λ is So, using Λ > ∼ √ F S , and taking c a of order unity, and requiring M gaugino > ∼ 1000 GeV, we need: Using the indications from the conformal bootstrap for γ S = 3/7, and taking M * = M GUT = 2.5 × 10 16 GeV, we need: In the following, for numerical examples I will optimistically take: Communication of supersymmetry breaking to the MSSM sector: Key feature: the last two terms are non-holomorphic in S, so they have an additional scaling factor Z S * S ∼ (Q/Q 0 ) Γ .

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To realize this, need a positive exponent from scaling dimensions: • ∆ S * S is the scaling dimension for the operator S * S, and • ∆ S = 1 + γ S is the scaling dimension for S.
Does such a superconformal theory exist?
If so, what can one say about Γ and ∆ S ?
No actual models with positive Γ are known, but. . .
• For ∆ S = 10/7, find that Γ < ∼ 0.3 • For smaller ∆ S , Γ is constrainted to be (much) smaller Example Model Line 1: unified gaugino masses • Free parameters: m 1/2 , m 0 , tan β • Solved for using electroweak symmetry breaking: µ, A 0 It turns out that one can only get the correct M Z = 91.2 GeV with small positive A 0 , so that top-squark mixing is moderate.
This in turn requires that m 1/2 is large, to give heavy top squarks, to allow M h = 125 GeV.
Small B is easy to achieve; one of the classic motivations for scalar sequestering.
Renormalization group running of squark, slepton masses, for m 1/2 = 4.5 TeV: The squarks are lighter than gluino; quasi-fixed point not so important for squarks, because SUSYQCD running below Λ dominates.
Slepton masses less strongly attracted to quasi-fixed point, running below Λ is weak.
New particles safely out of reach of present LHC and future upgrades.