Supersymmetric Naturalness Beyond MSSM

The experiments at the Large Hadron Collider (LHC) have pushed the limits on masses of supersymmetric particles beyond the $\sim$TeV scale. This compromises naturalness of the simplest supersymmetric extension of the Standard Model, the minimal supersymmetric Standard Model (MSSM). In this paper we advocate that perhaps the current experimental data are actually hinting towards the physics beyond MSSM. To illustrate this, we treat MSSM as a low energy limit of a more fundamental yet unspecified theory at a scale $\Lambda$, and compute the fine-tuning measure $\Delta$ for generic boundary conditions on soft SUSY breaking parameters and various cut-off scales. As a general trend we observe reduction in fine-tuning together with lowering $\Lambda$. In particular, perfectly natural [$\Delta \lesssim \mathcal{O}(10)$] theories with a multi-TeV spectrum of supersymmetric particles and consistent with all current observations can be obtained for $\Lambda \sim \mathcal{O}(100)$TeV. The lowering of the fine-tuning for large cut-off scales can also be observed in theories exhibiting special quasi-fixed point behaviours of parameters. Our observations call for a more throughout exploration of possible alternative ultraviolet completions of MSSM.


Introduction
Supersymmetry (SUSY) is a very attractive theoretical framework for physics beyond the Standard Model as it represents the unique non-trivial extension of relativistic invariance and provides with an unified description of particles with different statistics. As an important byproduct, SUSY quantum field theories exhibit improved short scale behaviour. Namely, the notorious quadratic divergences are completely absent even in theories with softly broken SUSY. This later feature motivates studies of the SUSY extension of the Standard Model (MSSM) with additional supersymmetric particle in the mass range ∼ 100 − 1000 GeV, since such theory would naturally incorporate the electroweak scale without need to fine tune quantum corrected parameters.
Contrary to these theoretical expectations, there has not been any evidence of SUSY particles at LHC. With some simplified assumptions, the experiments exclude gluinos and first and second generation squarks with masses up to ∼ 2 TeV [1]. Taking at face value, this compromises naturalness of the electroweak scale -a fine-tuning ∆ 300 is required to accommodate the correct electroweak scale.
It should be clear however, that the above conclusion is no way falsifies SUSY as the theoretical framework, but rather MSSM, as its particular realisation (see also the recent related discussion in [2]). In fact, we would like to advocate here that the failure of the natural MSSM may indicate towards the physics beyond MSSM. There are two major ways how the parameters involved in fine-tuning can be modified due to the new physics. First is through the modification of the renormalisation group (RG) running and second is through the modification of boundary conditions, e.g., due to the enhanced symmetries in the fundamental theory which are traced down to the low-energy theory. In this paper we take the attitude that MSSM is a low-energy approximation of a more fundamental yet unspecified theory at a scale Λ. We than compute the standard Barbieri-Giudice [3] fine-tuning measure ∆ for various scales Λ and assuming arbitrary boundary conditions on MSSM parameters defined at Λ. This way we parametrise our ignorance of the fundamental theory. As a general trend we observe the reduction of fine-tuning with lowering Λ from ∆ ∼ O(100) to ∆ ∼ O(10) , which may hint towards beyond MSSM physics at a scale ∼ 100 TeV 1 . We also discuss an example of quasi-fixed point running of parameters, which results in a low fine-tuning for large cut-off scales. These examples motivate further search of specific extensions of MSSM which result in natural electroweak scale.
The paper is organised as follows. In Section 2, we describe the measure of fine-tuning in the context of the MSSM and subsequently the relationship to the parameter RGEs. In Section 3 we perform a general scan over the MSSM, computing the fine-tuning measure along with relevant experimental constraints. Similarly in Section 4 we scan over a narrower region of parameter space corresponding to an MSSM quasi-infrared fixed point resulting in low fine-tuning. In Section 5 we present our conclusions.

Supersymmetric naturalness
We consider MSSM as an effective low energy approximation of an unspecified ultraviolet theory with a cut-off Λ. The relevant and marginal operators in the effective theory are those of the MSSM superpotential, where y u,d,e are the 3x3 Yukawa matrices in flavor space, supersymmetric SU (3)×SU (2)×U (1) gauge interactions, and the standard MSSM soft SUSY breaking terms. The effective running parameters evaluated at the cut-off scale Λ we identify as "fundamental" parameters of the effective MSSM. Minimization of the tree-level potential gives the following relation between Z-boson mass, m Z , low energy soft breaking masses m Hu,H d and the supersymmetric µ parameter: Assuming |m H d | |m Hu |, the last approximate equation in (2) holds to a very good accuracy for tan β 3. Hence, m Hu and µ at low energies must be adjusted in a way to reproduce Z-pole mass m Z 91 GeV. This adjustment is natural if not very sensitive to the variation of "fundamental" parameters at Λ. The quantitative measure of this sensitivity is the Barbieri-Giudice fine-tuning parameter: [3]: where the a i run over the "fundamental" parameters of the effective low-energy MSSM. The relation between low energy and "fundamental" parameters defined by the solution of the respective RG equations and matching conditions at the cut-off scale Λ. The later can only be computed if the ultraviolet completion of the MSSM is known. Since we are working within the effective field theory framework, we parameterise our ignorance of the ultraviolet physics by considering an unconventional and arbitrary (within certain range) values of the 20 "fundamental" parameters of the MSSM, which are potentially the most relevant for computing the low energy parameters for different values Λ.
The running of the supersymmetric parameter µ exhibits a fixed point µ = 0 and therefore if taken small (µ ∼ m Z ) at Λ, it will stay small at low energies. The small µ therefore is natural. However, for pure scalar mass parameters, such as m Hu , such a behaviour is atypical due to the additive contribution of heavy particle masses to the corresponding beta-function. More specifically, d dt where t = ln (Λ 2 /μ 2 ) [hereμ an arbitrary renormalisation scale] and As a result, the low-energy parameter m 2 Hu is sensitive variations of different mass parameters, and if the sparticle spectrum is heavy the fine-tining measure (3) is generically large.
One can think of two ways to reduce the required fine-tuning in models with large sparticle masses. First, one assumes that the physics beyond the MSSM enters at low enough scale Λ, such that the "fundamental" parameters do not evolve significantly when running down to low energies. In this case, if the fundamental theory is such that no significant fine tuning is required to satisfy the minimization condition (2), the RG running cannot destabilize the relation (2). We confirm this by numerical analysis -the required fine tuning is significantly reduced for low Λ, even for a rather heavy spectrum of sparticles.
Alternatively, if one assumes that m 2 Hu dominates over other mass parameters at high energies, the RG equation (6) takes the approximate form: which (similar to µ parameter) exhibits an infrared fixed point at m 2 Hu = 0. This observation motivates us to scan a specific region of "fundamental" parameters in section 4. In accord with the expectation, we observe significant reduction in fine-tuning measure for a large Λ and heavy sparticles.
From the observation in (6), we can determine another infrared fixed-point by defining the sum from which we can compute the beta function for Σ in the limit that all other mass parameters are subdominant For M 3 → 0 this expresses an infrared fixed-point at Σ = 0. However, since g 3 and M 3 increase in the infrared, once can expect a significant positive contribution to Σ. We confirm in the numerical analysis the correlation between the gluino mass Mg and fine tuning. Before we proceed with our numerical analysis, we note that infrared quasi-fixed point solution in the MSSM , in which the top-Yukawa coupling y t is kept large at the grand unified scale, are well known [5,6]. More specifically, upon computing the beta-functions for y t and g 3 up to one-loop and without electroweak contribution, one finds an infrared stable point at y 2 t /g 2 3 = 7/18 of the corresponding RG equations. This procedure has also been carried out for other couplings and soft-masses in the MSSM [7,8,9]. Here we allow more generic variation of fundamental parameters at the high energy scale Λ rather than focusing on model-dependent correlations (such as in grand unified theories) among them.

Parameter Scan
In this section we present our results for a generic scan of parameters and different values of Λ. We retain a full 20 parameter version of the MSSM and perform a broad random scan over the following space: where i = (Q,ū,d, L,ē). The first and second generation scalar soft masses are taken to be degenerate and we assume no flavour mixing at the input scale. We choose the input scale Λ in which the parameters are defined for the following three cases: Λ ∈ 10 5 , 10 10 , 10 16 GeV We employ full two-loop RGEs using SPHENO-3.3.8 [10], combined with SARAH [11], in order to compute the MSSM spectrum and fine-tuning measure. The parameters included in the calculation of the fine-tuning measure in Eq. 3 are the gaugino masses M 1 , M 2 , M 3 , higgs soft-breaking masses M 2 Hu , M 2 H d , 3rd generation scalar masses m 2 Q 3 , m 2 u 3 , m 2 d 3 , m 2 L 3 , m 2 e 3 , the trilinear couplings A t , A b , A τ , and the terms µ and B µ , all computed at the corresponding scale Λ. The top (pole) mass is set to 173 GeV. We also compute the DM relic density Ωh 2 and spin-independent WIMP-nucleon cross-section assuming a neutralino DM candidate using micrOmegas-4.3.2 [12].
Points which have a vacuum in the electroweak broken phase are chosen which satisfy ∆ ≤ 1000 are subsequently passed through the following constraints: • Direct searches for the slepton and chargino at LEP produce the mass limits on the first two generation sleptons and lightest chargino [13]: • We require the lightest Higgs mass in the range 122 < m h < 127 GeV [14,15], • We require the lightest neutralino χ 0 1 as the LSP and m χ 0 1 > 30 GeV to be consistent with the bound on light MSSM neutralino dark matter [16], • We satisfy the 3 sigma upper bound on dark matter relic density observed by the PLANCK collaboration [17] given by Ω P lanck h 2 = 0.112 ± 0.006. For points with underabundant dark matter, we assume there may be some additional contribution from non-thermal candiates, such as the axion. • We use the recent data from XENON1T [18] to constrain the points from direct detection experiments, where we rescale the spin-independent cross-section σ SI to the observed relic density by (Ωh 2 /Ω P lanck h 2 ), • We check the bounds from higgs searches at LEP, Tevatron and LHC implemented using HiggsBounds-4.3.1 [20], • We also check important B-physics constraints, namely BR(B → X s γ) and BR(B S → µ + µ − ). The measured values we use are BR(B → X s γ) exp = (3.55 ± 0.26) × 10 −4 [21] and the upper bound BR(B S → µ + µ − ) exp < 1.08 × 10 −8 (95% CL) [22]. These are calculated using FlavorKit [23] as part of the SPheno/SARAH package. Where an upper and lower bound are shown, we constrain our points to within 3σ of the quoted value.
We do not impose constraints from gluino/squark searches from ATLAS and CMS as the limits are model-dependent and would require a dedicated recasting of the collider limits. Besides, there are many cases in which the spectrum may be compressed to easily avoid these LHC search constraints.
In Figure 1 we show the dependence on the fine-tuning measure on the gluino mass Mg and lighter stop mass Mt 1 . As expected, one finds that when the "fundamental" parameters are entered at the low scale, chosen at Λ = 10 5 GeV, there is less constraint on a heavier spectrum whilst the electroweak scale still remains natural. Little variation in the parameters from renormalization group evolution even allows for fine-tuning as low as ∆ ∼ 20 in this case.  Figure 1. The dotted line corresponds to the PLANCK measurement of Ω P lanck h 2 = 0.112 ± 0.006 [17]. Right: WIMP-nucleon spin-independent crosssection as a function of the LSP mass for the points shown in the left panel. Since we allow the LSP to be underabundant after freeze-out, we rescale the cross-section by the factor Ω/Ω c . The solid lines corresponds to the XENON1T 2017 [18] and the recent 1 tonne × year [19] results. Similar plots exist for Λ = 10 5 and 10 10 GeV.
The constraints on relic density and direct-detection of DM can be satisfied relatively easily, shown in Figure 2, since the contributions from the electroweakino masses to the RG running of the up-type Higgs mass soft-breaking term is mild. We would also like to stress that the dark matter abundance, besides the microscopic properties, depend crucially on the cosmological evolution of the universe. In particular, the region of parameter space with over-abundant dark matter (ie. a mostly bino-like LSP) shown in Figure 2 can actually be consistent with observation with depopulation mechanisms shown in [24,25] effective in the early universe.

MSSM Quasi infrared fixed-point and fine-tuning
In the following, we choose a large top Yukawa coupling, y t > 1 at the high-scale to enhance the running of m 2 Hu . In order to enhance the contribution from m 2 Hu at the scale Λ we allow it to be dominant over m 2 i where i runs over the scalar mass squared values, excluding the 3rd generation squark soft-masses. The 3rd generation squark masses m 2 Q 3 and m 2 u 3 tend to de-stabilize m 2 Hu as they largely contribute a positive value toward the infrared, leading to a large value of m 2 Hu at M SU SY . One can avoid this with negative scalar mass squared parameters at the input scale. Negative stop mass-squared at the GUT scale have been previously studied in some gauge messenger models [26] and MSSM [27]. We present an example of the RGE evolution of these soft mass parameters in Figure 3. Most notably, the gluino mass parameter significantly raises the fixed-point value of Σ in the infrared, whilst simultaneously enhancing the running of the stop mass parameters into positive values. The fixed-point behavior requires m 2 Hu run to more negative values for large M 3 , increasing the fine-tuning.   Figure 1 with higher NP scales and in a narrower scan range supporting the infrared fixed-point behavior for Σ. Small fine tuning of ∆ < O(100) can be achieved even for heavier sparticle masses > 1 TeV With this as our motivation, we more precisely scan over the following modified space: Λ ∈ 10 10 , 10 16 , 10 19 GeV −3000 GeV < M 1 , M 2 < 3000 GeV M 3 < 3000 GeV −(3000) 2 GeV 2 < m 2 The plots in Figure 4 confirm our expectation with significant reduction in fine-tuning observed in models with quasi-fixed point running of m 2 Hu and µ. In particular, low-sensitivity towards Λ (∆ O(100)) can even be maintained in models where Λ is as high as 10 19 GeV.

Conclusions
The major conclusion we would like to draw in this paper is that the naturalness considerations within the supersymmetric theories in light of current experimental data may be indicating towards physics beyond MSSM that enters at scales as low as Λ 100 TeV. To demonstrate this point we have treated MSSM as an effective theory below Λ without any a priory assumption on soft-breaking parameters at Λ. The general scan of 20 MSSM parameters shows a reduction of the fine tuning measure from ∆ ∼ O(100) for Λ 10 16 GeV down to ∆ ∼ O(10) for Λ 10 5 GeV, even for sparticle spectrum lying in the multi-TeV region. We have also argued, that theories with a special quasi-fixed point behaviour of running parameters may also have reduced (∆ < O(100)) fine tuning. Our results call for further exploration of non-standard theories beyond MSSM.