OU-HEP-211130

Comparison of SUSY spectra generators for
natural SUSY and string landscape predictions
Howard Baer1,2111Email<EMAIL_ADDRESS>, Vernon Barger2222Email:
<EMAIL_ADDRESS>and Dakotah Martinez1444Email:
<EMAIL_ADDRESS>
1Homer L. Dodge Department of Physics and Astronomy,
University of Oklahoma, Norman, OK 73019, USA
2Department of Physics, University of Wisconsin, Madison, WI 53706 USA

Models of natural supersymmetry give rise to a weak scale $m_{weak}\sim
m_{W,Z,h}\sim 100$ GeV without any (implausible) finetuning of independent
contributions to the weak scale. These models, which exhibit radiatively
driven naturalness (RNS), are expected to arise from statistical analysis of
the string landscape wherein large soft terms are favored, but subject to a
not-too-large value of the derived weak scale in each pocket universe of the
greater multiverse. The string landscape picture then predicts, using the
Isajet SUSY spectra generator Isasugra, a statistical peak at $m_{h}\sim 125$
GeV with sparticles generally beyond current LHC search limits. In this paper,
we investigate how well these conclusions hold up using other popular spectra
generators: SOFTSUSY, SPHENO and SUSPECT (SSS). We built a computer code
DEW4SLHA which operates on SUSY Les Houches Accord files to calculate the
associated electroweak naturalness measure $\Delta_{EW}$. The SSS generators
tend to yield a Higgs mass peak $\sim 125-127$ GeV with a superparticle mass
spectra rather similar to that generated by Isasugra. In an Appendix, we
include loop corrections to $\Delta_{EW}$ in a more standard notation.

##  1 Introduction

Supersymmetrization of the Standard Model (SM) elegantly solves the gauge
hierarchy problem (stabilizing the newly discovered Higgs boson mass under
quantum corrections) but at the expense of including a host of new matter
states, the so-called superpartners. Early expectations from naturalness
predicted superpartners at or around the weak scale[1, 2, 3, 4]. For instance,
the naturalness upper bound for the gluino was predicted (under the
naturalness measure $\Delta_{BG}\lesssim 30$) to be $m_{\tilde{g}}\lesssim
400$ GeV. In contrast, the current mass limits from LHC Run 2 searches with
139 fb-1 claim $m_{\tilde{g}}\gtrsim 2.25$ TeV[5, 6]. The yawning gap between
the weak scale and the superpartner mass scale– the Little Hierarchy Problem
(LHP)[7]– has lead many authors to conclude[8, 9, 10] that the weak scale
supersymmetry[11] hypothesis is under intense pressure, and possibly even
excluded.

However, it has been pointed out that the resolution to the LHP lies instead
in that conventional early measures of naturalness over-estimated the
finetuning[12, 13, 14]. The BG log derivative measure[2], $\Delta_{BG}\equiv
max_{i}|\frac{\partial\log m_{Z}^{2}}{\partial\log p_{i}}|$ where the $p_{i}$
are fundamental parameters of the low energy effective field theory (EFT),
depends strongly on what one assumes are independent parameters. To derive the
bounds in Ref’s [1, 2, 3, 4], the authors adopted common scalar masses
$m_{0}$, gauginos masses $m_{1/2}$ and $A$-terms as independent parameters.
However, in more ultraviolet complete theories, such as string theory, these
parameters are all correlated. Adopting correlated soft terms then greatly
reduces the amount of finetuning which is calculated, often by 1-2 orders of
magnitude. An alternative measure $\Delta_{HS}\equiv\delta
m_{H_{u}}^{2}/m_{h}^{2}$, (which is inconsistent with $\Delta_{BG}$ in that it
splits $m_{H_{u}}^{2}(weak)$ into $m_{H_{u}}^{2}(HS)+\delta m_{H_{u}}^{2}$
which destroys the focus point behavior inherent in $\Delta_{BG}$) discards RG
contributions which show the interdependence of $m_{H_{u}}^{2}$ and $\delta
m_{H_{u}}^{2}$.

An alternative measure for naturalness $\Delta_{EW}$ was proposed in [15, 16]
based on the notion of practical naturalness[17]: that all independent
contributions to an observable ${\cal O}$ should be comparable to or less than
${\cal O}$. For instance, if ${\cal O}=o_{1}+\cdots+o_{n}$ where the $o_{i}$
are independent contributions to ${\cal O}$, and if $o_{1}\gg{\cal O}$, then
some other unrelated contribution would have been a huge opposite sign
contribution of precisely the right value such as to maintain ${\cal O}$ at
its measured value. Such finetunings, while logically possible, are thought to
be highly implausible unless the contributions $o_{i}$ are related by some
symmetry, in which case they would not actually be independent. Practical
naturalness has been successfully applied for instance by Gaillard and Lee in
the case of the $K_{L}-K_{S}$ mass difference to correctly predict the value
of the charm quark mass[18]. It is also closely related to predictivity in
physical theories in that missing contributions to an observable, such as
higher order corrections in perturbation theory, should be (hopefully) small
so that leading order terms provide a reliable estimate to any perturbatively
calculated observable.

The minimization conditions for the MSSM Higgs potential allows one to relate
the observed value of the weak scale to terms in the minimal supersymmetric
standard model (MSSM) Lagrangian:

$m_{Z}^{2}/2=\frac{(m_{H_{d}}^{2}+\Sigma_{d}^{d})-(m_{H_{u}}^{2}+\Sigma_{u}^{u})\tan^{2}\beta}{\tan^{2}\beta-1}-\mu^{2}$
(1)

where $m_{H_{u}}^{2}$ and $m_{H_{d}}^{2}$ are Higgs sector soft breaking
masses, $\mu$ is the (SUSY-conserving) $\mu$ parameter and
$\tan\beta=v_{u}/v_{d}$ is the ratio of Higgs field vacuum expectation values.
The $\Sigma_{d}^{d}$ and $\Sigma_{u}^{u}$ terms contain a variety of loop
corrections to the Higgs potential and are detailed in Ref. [16] in the
notation of Weak Scale Supersymmetry (WSS)[11] and given in the Appendix of
this paper in the more standard notation from S. P. Martin[19]. The most
important of the loop corrections typically comes from the top-squark sector,
$\Sigma_{u}^{u}(\tilde{t}_{1,2})$. Note that all contributions in Eq. 1 are
evaluated at the weak scale typically taken as
$Q^{2}=m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}$ such as to minimize the logs which
are present in the $\Sigma_{u}^{u}(\tilde{t}_{1,2})$ contributions.

The $\Delta_{EW}$ measure is defined as

$\Delta_{EW}\equiv|{\rm largest\ contribution\ to\ RHS\ of\
Eq.~{}\ref{eq:mzs}}|/(m_{Z}^{2}/2).$ (2)

One can quickly read off the consequences for a low value of $\Delta_{EW}$:

  * •

$m_{H_{u}}^{2}$, which in the decoupling limit functions like the SM Higgs
doublet and gives mass to the $W$, $Z$ and $h$ bosons, must be driven under
radiative EWSB to small negative values, a condition known as radiatively-
driven naturalness (RNS). Thus, electroweak symmetry is barely broken.

  * •

The $\mu$ parameter, which feeds mass to the $W$, $Z$ and $h$ bosons as well
as to the higgsinos, must be within a factor of several of $m_{W,Z,h}\sim 100$
GeV.

  * •

$m_{A}\sim m_{H_{d}}$ in the decoupling limit can live in the TeV regime since
the contribution of $m_{H_{d}}^{2}$ is suppressed by a factor $\tan^{2}\beta$.

  * •

Top squark contributions to the weak scale are loop suppressed and so can live
in the TeV range while maintaining naturalness.

  * •

The gluino contributes at two-loops[20] and via RG running contributions to
the stop soft masses[21, 22] and so also can live in the TeV range,

  * •

First and second generation sfermion contributions to the weak scale are via
Yukawa-suppressed 1-loop terms and via 2-loop RG contributions (which are
dominant)[23]. Thus, they can live in the 10-50 TeV regime which helps solve
the SUSY flavor and CP problems[24].

An advantage of $\Delta_{EW}$ is that it is model independent insofar as it
only depends on the weak scale sparticle and Higgs mass spectrum and not on
how they are arrived at. Thus, a given spectrum will generate the same value
of $\Delta_{EW}$ whether it was computed from the pMSSM or some high scale
model. Also, requiring the contributions to $m_{Z}^{2}/2$ to be comparable to
or less than its measured value typically corresponds to an upper limit of
$\Delta_{EW}\lesssim 30$. The turn-on of finetuning for $\Delta_{EW}\gtrsim
30$ is visually displayed in Fig. 1 of Ref. [17].

While WSS seems ruled out under the older naturalness measures[1, 2, 3, 4],
there is still plenty of natural parameter space left unexplored by LHC under
the $\Delta_{EW}$ measure[25]. However, the $\Delta_{EW}$ measure does predict
the existence of light higgsino-like EWinos $\tilde{\chi}_{1}^{\pm}$ and
$\tilde{\chi}_{1,2}^{0}$ with mass $\sim 100-350$ GeV. The light higgsinos can
be produced at decent rates at LHC, but owing to their small mass gaps
$m_{\tilde{\chi}_{2}^{0}}-m_{\tilde{\chi}_{1}^{0}}\sim 5-10$ GeV, there is
only small visible energy released in their decays, making detection a
difficult[26] (but not impossible[27, 28]) prospect. The higgsino-like LSP
$\tilde{\chi}_{1}^{0}$ is thermally underproduced as dark matter, leaving room
for axionic dark matter as well[29].

The $\Delta_{EW}$ naturalness measure is built in to the Isajet/Isasugra[30,
31] event/spectrum generator. Also, the crucial 1-loop corrections to the
Higgs potential have been calculated within the (non-standard) notation of
WSS[16]. As a result, of the spectrum generators available, Isasugra has been
used the most for such studies. These include sparticle mass bounds from
naturalness, and parameter space limits and lucrative collider signatures from
natural SUSY. However, a variety of other SUSY/Higgs spectra generators are
available, including SUSPECT[32], SOFTSUSY[33] and SPHENO[34]. Some special
Higgs spectrum calculators include FeynHIGGS[35] and SUSYHD[36] and
others[37]. Thus, it would be useful to know how other spectrum generators
compare to Isasugra in their natural SUSY spectra. For this reason, we have
built a computer code DEW4SLHA which operates on a SUSY Les Houches Accord
file (SLHA)[38] which is the standard output of spectrum generators. The
program computes the associated value of $\Delta_{EW}$ and all the various
contributions. In Sec. 2 of this paper, we introduce the code DEW4SLHA along
with pointers on its accessibility.

While natural SUSY is highly interesting in its own right, some authors
maintain that naturalness should cede ground to the emergent
landscape/multiverse picture of string theory: if the cosmological constant
$\Lambda_{cc}$ is finetuned to tiny values via anthropic selection in the
multiverse, then why not also the weak scale? There is expected to be a
statistical pull to large soft SUSY breaking terms via a power law[39, 40, 41]
or log distribution[42] in the landscape of string theory vacua. However, one
of the most important predictions of SUSY theories is the magnitude of the
weak scale $m_{weak}$. Agrawal et al.[43, 44] have shown that if the pocket
universe value of the weak scale is greater than a factor of 2-5 times our
universe’s measured value, then complex nuclei, and hence atoms as we know
them, would not arise. Now in a subset of vacua with the MSSM as low energy
EFT but with variable soft terms, then absent finetuning, the pocket universe
value of the weak scale $m_{weak}^{PU}$ will nearly be the maximal
contribution to the RHS of Eq. 1. Thus, a value $m_{Z}^{PU}\sim 4m_{Z}^{OU}$
corresponds to a value $\Delta_{EW}\lesssim 30$. This anthropic veto has been
used along with a landscape pull to large soft terms to make statistical
predictions from the string landscape for the SUSY and Higgs boson masses. It
is found using Isasugra that the Higgs mass $m_{h}$ rises to a peak at
$m_{h}\sim 125$ GeV while sparticles such as the lightest stop and gluino are
pulled to values beyond LHC13 search limits. It would also be of interest to
confirm or refute these results using other spectra/Higgs mass calculators.

Thus, in this paper we first introduce the public code DEW4SLHA in Sec. 2. In
Sec. 3, we apply this code to a natural SUSY benchmark point to compare
spectra from Isasugra against results from SOFTSUSY, SUSPECT and SPHENO. As
such, our paper follows previous comparison work but within the context of
natural SUSY and string landscape phenomenology[45, 46]. In Sec. 3, we also
move beyond benchmark points to compare Higgs mass and naturalness contours in
the scalar mass vs. gaugino mass parameter planes for just the SOFTSUSY
spectrum generator. In Sec. 4, we use SOFTSUSY to generate statistical
landscape predictions to compare against earlier work from Isasugra. A summary
and conclusions are given in Sec. 5. In an Appendix A, we present expressions
for the $\Sigma_{u}^{u}$ and $\Sigma_{d}^{d}$ contributions in the standard
notation of S. P. Martin’s SUSY primer [19].

##  2 The DEW4SLHA code

A new code, DEW4SLHA, has been developed in Python 3 by D. Martinez to
evaluate $\Delta_{EW}$ from any user-supplied SLHA-format output from a
spectrum generator such as Isajet, SOFTSUSY, SUSPECT, or SPHENO. The
standalone executable can be found at https://dew4slha.com, along with
instructions on how to run the program from a Linux terminal. The
computational routine of the program follows the equations presented in the
Appendix A and then orders the 44 1-loop contributions to the Higgs
minimization condition by magnitude. Two corrections at the 2-loop level are
included in the routine to include the effects of the gluino mass on the DEW
measure[20]. Similar codes have been developed but are not to our knowledge
publicly available[47, 48].

##  3 Natural SUSY benchmark points

Using the code DEW4SLHA, we can now compare spectra generated from the various
spectra calculators for a particular natural SUSY benchmark point. For the BM
point, we adopt the two-extra-parameter non-universal Higgs model (NUHM2)[49,
50] with input parameters

$m_{0},\ m_{1/2},\ A_{0},\ \tan\beta,\ \mu,\ m_{A}$ (3)

where we have traded the high scale Higgs soft masses $m_{H_{u}}^{2}$ and
$m_{H_{d}}^{2}$ for the more convenient weak scale parameters $\mu$ and
$m_{A}$. Then we adopt the benchmark parameter values $m_{0}=5$ TeV,
$m_{1/2}=1.2$ TeV, $A_{0}=-8$ TeV, $\tan\beta=10$, $\mu=200$ GeV and $m_{A}=2$
TeV. A pictorial representation of the spectra using SOFTSUSY is shown in Fig.
1 where we see that indeed the higgsinos and Higgs boson $h$ lie in the
$100-200$ GeV range whilst the top-squarks and gluino live in the several TeV
regime.

Figure 1: Sparticle and Higgs mass spectra for a natural SUSY benchmark point
from SOFTSUSY.

In Table 1, we list the mass spectra and $\Delta_{EW}$ values from each of
four spectra generators. The gluino masses are all within 1.5% of each other.
The naturalness parameters for three codes are all less than thirty; the
outlier here is SPHENO where also the light top squark mass
$m_{\tilde{t}_{1}}$ is somewhat higher than the other codes. Here, the top
squark masses are highly sensitive to mixing which comes from the weak scale
value of $A_{t}$ and indeed the values of $A_{t}(Q)$ for
Isasugra/SOFTSUSY/SUSPECT/SPHENO are -4898/-4830/-4894/-5090 GeV,
respectively. Thus, SPHENO has slightly more stop mixing than the other codes
which increases $\Delta_{EW}$ somewhat. Another difference comes from the
value of $m_{h}$ generated: both SOFTSUSY and SUSPECT generate $m_{h}\sim
127.4$ GeV whilst SPHENO generates $m_{h}=125.2$ GeV and Isasugra generates
$m_{h}=124.7$ GeV. It can be remarked that Isasugra has the least
sophisticated light Higgs mass calculation, and includes only third generation
sparticle 1-loop contributions to $m_{h}$. Another feature is that the
Isasugra value of $m_{\tilde{\chi}_{1}^{\pm}}$ is about six GeV higher than
SOFTSUSY and SUSPECT while the SPHENO is six GeV lower. These values depend
sensitively on the scale choice at which each EWino mass is calculated. For
instance, Isasugra uses the Pierce et al. (PBMZ)[51] recipe to calculate each
mass separately at each mass scale.

parameter | Isasugra | SOFTSUSY | SUSPECT | SPHENO
---|---|---|---|---
$m_{\tilde{g}}$ | 2830.7 | 2794.3 | 2838.6 | 2827.6
$m_{\tilde{u}_{L}}$ | 5440.3 | 5403.2 | 5406.0 | 5412.8
$m_{\tilde{u}_{R}}$ | 5561.7 | 5521.3 | 5523.0 | 5521.8
$m_{\tilde{e}_{R}}$ | 4823.0 | 4817.3 | 4818.1 | 4825.8
$m_{\tilde{t}_{1}}$ | 1714.3 | 1682.8 | 1746.9 | 1942.1
$m_{\tilde{t}_{2}}$ | 3915.1 | 3879.0 | 3899.2 | 3947.0
$m_{\tilde{b}_{1}}$ | 3949.1 | 3871.6 | 3891.7 | 3939.1
$m_{\tilde{b}_{2}}$ | 5287.5 | 5266.4 | 5277.2 | 5281.7
$m_{\tilde{\tau}_{1}}$ | 4745.7 | 4746.1 | 4749.1 | 4757.4
$m_{\tilde{\tau}_{2}}$ | 5110.2 | 5109.7 | 5110.8 | 5107.2
$m_{\tilde{\nu}_{\tau}}$ | 5116.8 | 5108.7 | 5113.8 | 5106.2
$m_{\widetilde{\chi}^{\pm}_{2}}$ | 1020.2 | 1027.5 | 1030.6 | 1031.9
$m_{\widetilde{\chi}^{\pm}_{1}}$ | 209.7 | 203.1 | 203.0 | 197.3
$m_{\widetilde{\chi}^{0}_{4}}$ | 1033.5 | 1027.3 | 1031.1 | 1032.0
$m_{\widetilde{\chi}^{0}_{3}}$ | 540.1 | 536.4 | 537.2 | 538.1
$m_{\widetilde{\chi}^{0}_{2}}$ | -208.3 | -208.6 | -208.7 | -203.0
$m_{\widetilde{\chi}^{0}_{1}}$ | 197.9 | 197.2 | 197.1 | 191.9
$m_{h}$ | 124.7 | 127.3 | 127.5 | 125.2
$\Delta_{\rm EW}$ | 24.8 | 23.0 | 28.2 | 44.1
Table 1: Sparticle and Higgs mass spectra from four spectra generators for a
natural SUSY benchmark point with $m_{0}=5$ TeV, $m_{1/2}=1.2$ TeV, $A_{0}=-8$
TeV, $\tan\beta=10$ with $\mu=200$ GeV and $m_{A}=2$ TeV.

In Table 2, we list the top 46 contributions to $\Delta_{EW}$ from each of the
spectra codes. We see from line 1 that the largest contribution comes for each
code from $\Sigma_{u}^{u}(\tilde{t}_{2})$ which sets the value of
$\Delta_{EW}$, and where we see that SPHENO gives the largest value. The
second largest contribution comes from $\Sigma_{u}^{u}(\tilde{t}_{1})$ as
might be expected. The next several largest contributions come from $H_{d}$,
$\mu$ and $H_{u}$ and $\Sigma_{u}^{u}(\tilde{b}_{1,2})$ although the ordering
of these differs among the codes. In general, the agreement for the remaining
contributions is typically within expectations.

Order | Isajet | SoftSUSY | Suspect | Spheno
---|---|---|---|---
1 | 24.819, $\Sigma_{u}^{u}(\widetilde{t}_{2})$ | 23.015, $\Sigma_{u}^{u}(\widetilde{t}_{2})$ | 28.227, $\Sigma_{u}^{u}(\widetilde{t}_{2})$ | 44.062, $\Sigma_{u}^{u}(\widetilde{t}_{2})$
2 | 19.367, $\Sigma_{u}^{u}(\widetilde{t}_{1})$ | 18.318, $\Sigma_{u}^{u}(\widetilde{t}_{1})$ | 20.372, $\Sigma_{u}^{u}(\widetilde{t}_{1})$ | 27.465, $\Sigma_{u}^{u}(\widetilde{t}_{1})$
3 | 10.449, $\Sigma_{u}^{u}(\mathcal{O}(\alpha_{s}\alpha_{t}))$ | 10.074, $H_{d}$ | 10.294, $H_{d}$ | 11.205, $H_{u}$
4 | 10.424, $H_{d}$ | 9.618, $\mu$ | 9.621, $\mu$ | 10.298, $H_{d}$
5 | 9.625, $\mu$ | 6.985, $\Sigma_{u}^{u}(\mathcal{O}(\alpha_{s}\alpha_{t}))$ | 7.405, $\Sigma_{u}^{u}(\mathcal{O}(\alpha_{s}\alpha_{t}))$ | 9.621, $\mu$
6 | 5.861, $H_{u}$ | 4.557, $\Sigma_{u}^{u}(\widetilde{b}_{2})$ | 4.044, $\Sigma_{u}^{u}(\widetilde{b}_{2})$ | 8.321, $\Sigma_{u}^{u}(\mathcal{O}(\alpha_{s}\alpha_{t}))$
7 | 4.164, $\Sigma_{u}^{u}(\widetilde{\tau}_{2})$ | 4.316, $\Sigma_{u}^{u}(\widetilde{\tau}_{2})$ | 3.761, $\Sigma_{u}^{u}(\widetilde{\tau}_{2})$ | 3.604, $\Sigma_{u}^{u}(\widetilde{b}_{1})$
8 | 3.933, $\Sigma_{u}^{u}(\widetilde{b}_{2})$ | 3.252, $\Sigma_{u}^{u}(\widetilde{\tau}_{1})$ | 2.801, $\Sigma_{u}^{u}(\Sigma$ 2nd gen. $\widetilde{q})$ | 2.505, $\Sigma_{u}^{u}(\widetilde{\tau}_{2})$
9 | 2.970, $\Sigma_{u}^{u}(\widetilde{\tau}_{1})$ | 2.909, $\Sigma_{u}^{u}(\Sigma$ 2nd gen. $\widetilde{q})$ | 2.801, $\Sigma_{u}^{u}(\Sigma$ 1st gen. $\widetilde{q})$ | 2.486, $\Sigma_{u}^{u}(\widetilde{b}_{2})$
10 | 2.912, $\Sigma_{u}^{u}(\Sigma$ 2nd gen. $\widetilde{q})$ | 2.909, $\Sigma_{u}^{u}(\Sigma$ 1st gen. $\widetilde{q})$ | 2.653, $\Sigma_{u}^{u}(\widetilde{\tau}_{1})$ | 2.468, $\Sigma_{u}^{u}(\Sigma$ 2nd gen. $\widetilde{q})$
11 | 2.912, $\Sigma_{u}^{u}(\Sigma$ 1st gen. $\widetilde{q})$ | 2.761, $H_{u}$ | 2.507, $\Sigma_{u}^{u}(\widetilde{b}_{1})$ | 2.468, $\Sigma_{u}^{u}(\Sigma$ 1st gen. $\widetilde{q})$
12 | 2.003, $\Sigma_{u}^{u}(\widetilde{b}_{1})$ | 2.101, $\Sigma_{u}^{u}(\widetilde{b}_{1})$ | 1.212, $\Sigma_{u}^{u}(\widetilde{\chi}_{2}^{\pm})$ | 1.263, $\Sigma_{u}^{u}(\widetilde{\chi}_{2}^{\pm})$
13 | 1.169, $\Sigma_{u}^{u}(\widetilde{\chi}_{2}^{\pm})$ | 1.191, $\Sigma_{u}^{u}(\widetilde{\chi}_{2}^{\pm})$ | 9.235e-1, $\Sigma_{u}^{u}(\widetilde{Z}_{3}^{0})$ | 1.133, $\Sigma_{u}^{u}(\widetilde{\tau}_{1})$
14 | 9.765e-1, $\Sigma_{u}^{u}(\widetilde{Z}_{3}^{0})$ | 9.114e-1, $\Sigma_{u}^{u}(\widetilde{Z}_{3}^{0})$ | 7.312e-1, $H_{u}$ | 9.538e-1, $\Sigma_{u}^{u}(\widetilde{Z}_{3}^{0})$
15 | 6.987e-1, $\Sigma_{u}^{u}(\widetilde{Z}_{4}^{0})$ | 6.924e-1, $\Sigma_{u}^{u}(\widetilde{Z}_{4}^{0})$ | 7.076e-1, $\Sigma_{u}^{u}(\widetilde{Z}_{4}^{0})$ | 7.381e-1, $\Sigma_{u}^{u}(\widetilde{Z}_{4}^{0})$
16 | 5.98e-1, $\Sigma_{u}^{u}(H^{\pm})$ | 6.083e-1, $\Sigma_{u}^{u}(H^{\pm})$ | 6.264e-1, $\Sigma_{u}^{u}(H^{\pm})$ | 6.755e-1, $\Sigma_{u}^{u}(H^{\pm})$
17 | 1.532e-1, $\Sigma_{u}^{u}(t)$ | 1.438e-1, $\Sigma_{u}^{u}(t)$ | 1.440e-1, $\Sigma_{u}^{u}(t)$ | 2.064e-1, $\Sigma_{u}^{u}(\widetilde{Z}_{1}^{0})$
18 | 5.924e-2, $\Sigma_{u}^{u}(\widetilde{Z}_{1}^{0})$ | 7.522e-2, $\Sigma_{u}^{u}(\widetilde{Z}_{1}^{0})$ | 7.687e-2, $\Sigma_{u}^{u}(\widetilde{Z}_{1}^{0})$ | 1.361e-1, $\Sigma_{u}^{u}(t)$
19 | 5.543e-2, $\Sigma_{d}^{d}(H^{0})$ | 5.305e-2, $\Sigma_{d}^{d}(H^{0})$ | 5.564e-2, $\Sigma_{d}^{d}(H^{0})$ | 5.831e-2, $\Sigma_{d}^{d}(H^{0})$
20 | 4.758e-2, $\Sigma_{d}^{d}(\widetilde{Z}_{3}^{0})$ | 4.397e-2, $\Sigma_{d}^{d}(\widetilde{Z}_{3}^{0})$ | 4.507, $\Sigma_{d}^{d}(\widetilde{Z}_{3}^{0})$ | 4.649e-2, $\Sigma_{d}^{d}(\widetilde{Z}_{3}^{0})$
21 | 4.3e-2, $\Sigma_{u}^{u}(Z^{0})$ | 4.175e-2, $\Sigma_{d}^{d}(\widetilde{b}_{2})$ | 3.909e-2, $\Sigma_{d}^{d}(\widetilde{b}_{2})$ | 4.341e-2, $\Sigma_{d}^{d}(\widetilde{t}_{1})$
22 | 4.3e-2, $\Sigma_{d}^{d}(\widetilde{b}_{2})$ | 3.783, $\Sigma_{u}^{u}(Z^{0})$ | 3.825e-2, $\Sigma_{u}^{u}(Z^{0})$ | 3.889e-2, $\Sigma_{u}^{u}(Z^{0})$
23 | 3.748e-2, $\Sigma_{d}^{d}(\widetilde{t}_{1})$ | 3.438, $\Sigma_{d}^{d}(\widetilde{t}_{1})$ | 3.713e-2, $\Sigma_{d}^{d}(\widetilde{t}_{1})$ | 2.793e-2, $\Sigma_{d}^{d}(\widetilde{b}_{2})$
24 | 3.198e-2, $\Sigma_{d}^{d}(\Sigma$ 2nd gen. $\widetilde{q})$ | 3.128e-2, $\Sigma_{d}^{d}(\Sigma$ 2nd gen. $\widetilde{q})$ | 3.075e-2, $\Sigma_{d}^{d}(\Sigma$ 2nd gen. $\widetilde{q})$ | 2.706e-2, $\Sigma_{d}^{d}(\Sigma$ 2nd gen. $\widetilde{q})$
25 | 3.198e-2, $\Sigma_{d}^{d}(\Sigma$ 1st gen. $\widetilde{q})$ | 3.128e-2, $\Sigma_{d}^{d}(\Sigma$ 1st gen. $\widetilde{q})$ | 3.075e-2, $\Sigma_{d}^{d}(\Sigma$ 1st gen. $\widetilde{q})$ | 2.706e-2, $\Sigma_{d}^{d}(\Sigma$ 1st gen. $\widetilde{q})$
26 | 2.329e-2, $\Sigma_{u}^{u}(h^{0})$ | 2.377e-2, $\Sigma_{u}^{u}(h^{0})$ | 2.395e-2, $\Sigma_{u}^{u}(h^{0})$ | 2.323e-2, $\Sigma_{u}^{u}(h^{0})$
27 | 1.875e-2, $\Sigma_{d}^{d}(\widetilde{Z}^{0}_{4})$ | 1.841e-2, $\Sigma_{d}^{d}(\widetilde{Z}^{0}_{4})$ | 1.895e-2, $\Sigma_{d}^{d}(\widetilde{Z}^{0}_{4})$ | 2.152e-2, $\Sigma_{u}^{u}(\widetilde{Z}^{0}_{2})$
28 | 1.669e-2, $\Sigma_{d}^{d}(\widetilde{\tau}_{1})$ | 1.787e-2, $\Sigma_{d}^{d}(\widetilde{\tau}_{1})$ | 1.504e-2, $\Sigma_{d}^{d}(\widetilde{\tau}_{1})$ | 1.974e-2, $\Sigma_{d}^{d}(\widetilde{Z}^{0}_{4})$
29 | 1.279e-2, $\Sigma_{d}^{d}(\widetilde{\chi}_{2}^{\pm})$ | 1.276e-2, $\Sigma_{d}^{d}(\widetilde{\chi}_{2}^{\pm})$ | 1.326e-2, $\Sigma_{d}^{d}(\widetilde{\chi}_{2}^{\pm})$ | 1.719e-2, $\Sigma_{d}^{d}(\widetilde{Z}^{0}_{1})$
30 | 1.102e-2, $\Sigma_{d}^{d}(\widetilde{\tau}_{2})$ | 1.107e-2, $\Sigma_{u}^{u}(H^{0})$ | 1.079e-2, $\Sigma_{u}^{u}(H^{0})$ | 1.553e-2, $\Sigma_{d}^{d}(\widetilde{b}_{1})$
31 | 1.095e-2, $\Sigma_{d}^{d}(\mathcal{O}(\alpha_{s}\alpha_{t}))$ | 1.101e-2, $\Sigma_{d}^{d}(\widetilde{\tau}_{2})$ | 1.034e-2, $\Sigma_{d}^{d}(\widetilde{b}_{1})$ | 1.380e-2, $\Sigma_{d}^{d}(\widetilde{\chi}_{2}^{\pm})$
32 | 9.869e-3, $\Sigma_{u}^{u}(\widetilde{Z}_{2}^{0})$ | 8.412e-3, $\Sigma_{d}^{d}(\widetilde{b}_{1})$ | 9.897e-3, $\Sigma_{d}^{d}(\widetilde{\tau}_{2})$ | 8.754e-3, $\Sigma_{u}^{u}(H^{0})$
33 | 8.366e-3, $\Sigma_{d}^{d}(\widetilde{b}_{1})$ | 7.381e-3, $\Sigma_{d}^{d}(\mathcal{O}(\alpha_{s}\alpha_{t}))$ | 7.391e-3, $\Sigma_{d}^{d}(\mathcal{O}(\alpha_{s}\alpha_{t}))$ | 8.132e-3, $\Sigma_{d}^{d}(\mathcal{O}(\alpha_{s}\alpha_{t}))$
34 | 8.083e-3, $\Sigma_{u}^{u}(H^{0})$ | 7.315e-3, $\Sigma_{u}^{u}(\widetilde{Z}_{2}^{0})$ | 7.180e-3, $\Sigma_{u}^{u}(\widetilde{Z}_{2}^{0})$ | 7.408e-3, $\Sigma_{d,u}^{d,u}(H^{\pm})$
35 | 6.658e-3, $\Sigma_{d,u}^{d,u}(H^{\pm})$ | 6.542e-3, $\Sigma_{d,u}^{d,u}(H^{\pm})$ | 6.877e-3, $\Sigma_{d,u}^{d,u}(H^{\pm})$ | 6.470e-3, $\Sigma_{d}^{d}(\widetilde{\tau}_{2})$
36 | 5.469e-3, $\Sigma_{u}^{u}(W^{\pm})$ | 5.400e-3, $\Sigma_{u}^{u}(W^{\pm})$ | 5.467e-3, $\Sigma_{u}^{u}(W^{\pm})$ | 6.324e-3, $\Sigma_{d}^{d}(\widetilde{\tau}_{1})$
37 | 2.611e-3, $\Sigma_{d}^{d}(\widetilde{Z}_{2}^{0})$ | 2.660e-3, $\Sigma_{d}^{d}(\widetilde{Z}_{2}^{0})$ | 2.717e-3, $\Sigma_{d}^{d}(\widetilde{Z}_{2}^{0})$ | 5.561e-3, $\Sigma_{u}^{u}(W^{\pm})$
38 | 1.081e-3, $\Sigma_{d}^{d}(\widetilde{Z}_{1}^{0})$ | 2.305e-3, $\Sigma_{d}^{d}(\widetilde{Z}_{1}^{0})$ | 2.428e-3, $\Sigma_{d}^{d}(\widetilde{Z}_{1}^{0})$ | 2.441e-3, $\Sigma_{u}^{u}(\widetilde{\chi}_{1}^{\pm})$
39 | 7.420e-4, $\Sigma_{d}^{d}(h^{0})$ | 2.044e-3, $\Sigma_{u}^{u}(\widetilde{\chi}_{1}^{\pm})$ | 2.336, $\Sigma_{u}^{u}(\widetilde{\chi}_{1}^{\pm})$ | 1.630e-3, $\Sigma_{d}^{d}(\widetilde{Z}_{2}^{0})$
40 | 4.723e-4, $\Sigma_{d,u}^{d,u}(Z^{0})$ | 7.568e-4, $\Sigma_{d}^{d}(h^{0})$ | 7.776e-4, $\Sigma_{d}^{d}(h^{0})$ | 7.394e-4, $\Sigma_{d}^{d}(h^{0})$
41 | 4.205e-4, $\Sigma_{u}^{u}(\widetilde{\chi}_{1}^{\pm})$ | 4.069e-4, $\Sigma_{d,u}^{d,u}(Z^{0})$ | 4.199e-4, $\Sigma_{d,u}^{d,u}(Z^{0})$ | 4.265e-4, $\Sigma_{d,u}^{d,u}(Z^{0})$
42 | 1.000e-4, $\Sigma_{d}^{d}(\widetilde{t}_{2})$ | 2.013e-4, $\Sigma_{d}^{d}(\widetilde{t}_{2})$ | 2.673e-4, $\Sigma_{d}^{d}(\widetilde{t}_{2})$ | 4.215e-4, $\Sigma_{d}^{d}(\widetilde{t}_{2})$
43 | 6.007e-5, $\Sigma_{d,u}^{d,u}(W^{\pm})$ | 5.808e-5, $\Sigma_{d,u}^{d,u}(W^{\pm})$ | 6.002e-5, $\Sigma_{d,u}^{d,u}(W^{\pm})$ | 6.098e-5, $\Sigma_{d,u}^{d,u}(W^{\pm})$
44 | 9.197e-6, $\Sigma_{d}^{d}(\widetilde{\chi}_{1}^{\pm})$ | 2.608e-5, $\Sigma_{d}^{d}(\widetilde{\chi}_{1}^{\pm})$ | 2.986e-5, $\Sigma_{d}^{d}(\widetilde{\chi}_{1}^{\pm})$ | 3.085e-5, $\Sigma_{d}^{d}(\widetilde{\chi}_{1}^{\pm})$
45 | 2.315e-8, $\Sigma_{d}^{d}(b)$ | 2.302e-8, $\Sigma_{d}^{d}(b)$ | 2.282e-8, $\Sigma_{d}^{d}(b)$ | 1.895e-8, $\Sigma_{d}^{d}(b)$
46 | 9.579e-9, $\Sigma_{d}^{d}(\tau)$ | 7.904e-9, $\Sigma_{d}^{d}(\tau)$ | 7.812e-9, $\Sigma_{d}^{d}(\tau)$ | 7.783e-9, $\Sigma_{d}^{d}(\tau)$

Table 2: Top 46 contributions to $\Delta_{EW}$ for our natural SUSY benchmark
point for four different spectra calculator codes.

In Fig. 2, we show the values of a) $A_{t}$, b) $\Delta_{EW}$, c) $m_{h}$ and
d) $m_{\tilde{t}_{1,2}}$ versus $A_{0}/m_{0}$ for the NUHM3 model with
parameters as in the caption but with varying $A_{0}$. (NUHM3 splits
first/second generation sfermion soft terms from third generation ones so that
$m_{0}(1,2)\neq m_{0}(3)$.) These plots are obtained using SOFTSUSY and can be
compared to similar plots in Ref. [15] using Isasugra. We see from frame a)
that the value of $m_{h}$ is actually maximal at large negative $A_{t}$ values
(which are shown in frame d)). The large mixing in the stop sector lifts the
value of $m_{h}$ to the 125 GeV regime, but in this case only for negative
$A_{t}$ values. The stop mass eigenstates are shown in frame b) where again,
when there is large mixing, the eigenstates have the largest splittings and
$m_{\tilde{t}_{1}}$ becomes lowest in value. In frame c), we show the
corresponding value of $\Delta_{EW}$. Here we see that for large trilinear
$A_{t}$, then there can be large cancellations in
$\Sigma_{u}^{u}(\tilde{t}_{1,2})$ which lead to decreased finetuning. The low
value of $\Delta_{EW}$ coincides with the uplift in $m_{h}$ to $\sim 125$ GeV
for large negative values of $A_{t}$.

Figure 2: Values of a) $m_{h}$, b) $m_{\tilde{t}_{1,2}}$, c) $\Delta_{EW}$ and
d) $A_{t}(Q)$ vs. $A_{0}/m_{0}(3)$ for the NUHM3 model with $m_{0}(1,2)=10$
TeV, $m_{0}(3)=5$ TeV, $m_{1/2}=0.7$ TeV, $\tan\beta=10$ with $\mu=150$ GeV
and $m_{A}=1$ TeV.

In Fig. 3, we show the third generation contributions to $\Delta_{EW}$ vs.
$A_{0}/m_{0}(3)$ for the same parameters as in Fig. 2, but using SOFTSUSY.
These can be compared with the same plot using Isasugra in Fig. 2 of Ref.
[15]. Here, we see that the contributions from staus and sbottoms are
generally rather small, and the top-squark contributions typically dominate.
But for large $|A_{0}/m_{0}(3)|$, then cancellations in both
$\Sigma_{u}^{u}(\tilde{t}_{1})$ and $\Sigma_{u}^{u}(\tilde{t}_{2})$ occur, and
the stop contributions become comparable to those of the other third
generation sparticles, giving reduced finetuning and greater naturalness.

Figure 3: Third generation contributions to $\Delta_{EW}$ for the same model
parameters as Fig. 2 vs. $A_{0}/m_{0}(3)$ in the $m_{0}$ vs. $m_{1/2}$ plane
of the NUHM2 model with $\mu=200$, $\tan\beta=10$, $A_{0}=-1.6m_{0}$ and
$m_{A}=2$ TeV.

###  3.1 Natural regions of $m_{0}$ vs. $m_{1/2}$ plane

In Fig. 4, we show the $m_{0}$ vs. $m_{1/2}$ parameter plane for the NUHM2
model with $A_{0}=-1.6m_{0}$, $\mu=200$ GeV and $m_{A}=2$ TeV. The plot is
generated using SOFTSUSY but can be compared with similar results from
Isasugra in Fig. 8b of Ref. [52]. From the plot, we see the lower-left corner
is actually excluded due to charge-or-color-breaking (CCB) vacua which occur
for too large $A_{0}$ values. Both SOFTSUSY and Isasugra generate CCB regions
there. We also show contours of Higgs mass $m_{h}=123$ and $127$ GeV. These
are qualitatively similar to the Isasugra results but shifted to the right by
a couple hundred GeV in $m_{0}$. Thus, much of the parameter space allows for
the measured Higgs mass $m_{h}\sim 125$ GeV. We also show naturalness contours
for $\Delta_{EW}=15$ and 30. These can also be compared against the LHC Run 2
gluino mass limit $m_{\tilde{g}}\gtrsim 2.25$ TeV as shown by the light blue
contour. The important point is that both SOFTSUSY and Isasugra agree that the
bulk of this parameter space plane is EW natural, in accord with LHC gluino
mass limits, and in accord with the measured Higgs mass. This is in contrast
to older naturalness measures which required much lower gluino masses[1, 2, 3,
4] and also Higgs boson masses[53].

Figure 4: Contours of naturalness measure $\Delta_{EW}$ and $m_{h}$ in the
$m_{0}$ vs. $m_{1/2}$ plane of the NUHM2 model with $\mu=200$, $\tan\beta=10$,
$A_{0}=-1.6m_{0}$ and $m_{A}=2$ TeV.

##  4 String landscape distributions from SOFTSUSY

In this section, we wish to compare SUSY landscape predictions using a
spectrum calculator other than Isasugra. Here, we choose SOFTSUSY. The
assumption is that the MSSM is the low energy EFT in a fertile patch of
landscape vacua, but with different sets of soft SUSY breaking terms in each
pocket universe, and hence a different value for the weak scale
$m_{weak}^{PU}\neq m_{weak}^{OU}$ in each pocket universe (here, $OU=$ our
universe). Following Douglas[39], Susskind[40] and Arkani-Hamed, Dimopoulos
and Kachru[41], we will assume the soft terms scan in the landscape as a
power-law: $m_{soft}^{n}$ where $n=2n_{F}+n_{D}-1$ with $n_{F}$ the number of
$F$ breaking fields and $n_{D}$ the number of $D$ breaking fields. Here, we
assume $n=1$ corresponding to SUSY breaking by a single $F$ term, where $F$ is
distributed as a random complex number. As in Ref. [54], we expect each soft
term in the NUHM3 model to scan independently.

We perform the linear soft term scan over NUHM3 space as follows:

  * •

$m_{0}(1,2):\ 0.1-60$ TeV,

  * •

$m_{0}(3):\ 0.1-20$ TeV,

  * •

$m_{1/2}:\ 0.5-10$ TeV,

  * •

$A_{0}:\ -50\ -\ 0$ TeV,

  * •

$m_{A}:\ 0.3-10$ TeV

with $\mu=200$ GeV and $\tan\beta$ scanned uniformly between $3-60$. The goal
is to set upper limits on scan parameters that are beyond the upper limits
that will result from imposing the anthropic conditions. We also require
appropriate EWSB and so veto vacua with CCB minima or with no EWSB. We also
require the Agrawal condition on the magnitude of the weak scale[43]:

  * •

$m_{Z}^{PU}<4m_{Z}^{OU}$

which corresponds to $\Delta_{EW}<30$. Thus, the following results generated
using SOFTSUSY can be compared to comparable results in Ref. [55] using
Isasugra.

The resultant distribution in input parameters from the SOFTSUSY scan is shown
in Fig. 5. In frame a), we see the distribution in first/second generation GUT
scale soft masses peaks around 20 TeV and spans $\sim 5-40$ TeV. While
first/second generation scalars contribute to the weak scale via Yukawa
suppressed terms, they also contribute via EW $D$-term contributions (which
largely cancel due to cancellation of EW quantum numbers) and via two-loop RG
terms which, when large, drive third generation scalars to tachyonic
values[23]. The last of these effectively sets the upper bound, allowing for
$m_{0}(1,2)$ as high as $40-50$ TeV. This provides a mixed decoupling/quasi-
degeneracy solution to the SUSY flavor and CP problems[24] since the upper
bound is flavor independent. In frame b), the third generation soft masses are
bounded by much lower values: $m_{0}(3)\sim 1-10$ TeV with a peak around 5
TeV. Here, the upper bound comes from requiring not-too-large values of
$\Sigma_{u}^{u}(\tilde{t}_{1,2})$ values. In frame c), we see the distribution
in $m_{1/2}$, which ranges form $0.5-3$ TeV. The upper bound is set because if
$m_{1/2}$ is too large, it drives the stop soft terms to large values and
again $\Sigma_{u}^{u}(\tilde{t}_{1,2})$ gets too big. In frame d), we plot the
distribution in $-A_{0}$. There is hardly any probability around $A_{0}\sim 0$
so we expect large mixing in the stop sector, which ends up driving $m_{h}$ to
large values. But $A_{0}$ cannot become too large (negative) lest it pushes
the top squark soft terms to tachyonic values via RG running.

Figure 5: Distribution of soft SUSY breaking terms in subset of landscape
vacua with the NUHM3 model as low energy EFT and an $n=1$ statistical draw. We
show distributions in a) $m_{0}(1,2)$, b) $m_{0}(3)$, c) $m_{1/2}$ and d)
$A_{0}$.

In Fig. 6, we show $n=1$ landscape scan probability distributions from the EW
sector. In frame a), we show the distribution in light Higgs mass $m_{h}$.
Using SOFTSUSY, the distribution rises to a peak $m_{h}\sim 128$ GeV, which is
several GeV higher than the result from Isasugra. This is consistent with
SOFTSUSY generating $m_{h}$ typically a couple GeV higher than Isasugra. In
frame b), we see the distribution in $m_{A}$ which runs from $1-9$ TeV with a
peak around 4 TeV. Thus, we expect a decoupled SUSY Higgs sector with the
couplings of $h$ being very close to their SM values. In frame c), the
distribution in $\tan\beta$ peaks around $10-20$. The upper bound is set
because if $\tan\beta$ gets too big, then the
$\Sigma_{u}^{u}(\tilde{b}_{1,2})$ terms become large (large $b$ and $\tau$
Yukawa couplings) and the model is more likely to generate a large
$m_{weak}^{PU}$. In frame d), we show the
$\tilde{\chi}_{2}^{0}-\tilde{\chi}_{1}^{0}$ mass difference which is important
for LHC higgsino-pair searches[27]. In this case, the landscape predicts
$m_{\tilde{\chi}_{2}^{0}}-m_{\tilde{\chi}_{1}^{0}}\sim 5-15$ GeV.

Figure 6: Distribution of Higgs and EWino sector from a subset of landscape
vacua with NUHM3 model as low energy EFT and an $n=1$ statistical draw. We
show distributions in a) $m_{h}$, b) $m_{A}$, c) $\tan\beta$ and d)
$m_{\tilde{\chi}_{2}^{0}}-m_{\tilde{\chi}_{1}^{0}}$.

In Fig. 7, we show $n=1$ landscape distributions for strongly interacting SUSY
particles using SOFTSUSY. In frame a), we see the gluino mass
$m_{\tilde{g}}\sim 2-7$ TeV. The LHC13 limit of $m_{\tilde{g}}\gtrsim 2.25$
TeV just excludes the lower edge of the expected values. Thus, from the
landscape point of view, it is no surprise that LHC has so far failed to
detect gluinos. In frame b), we show the distribution in left up-squark mass
(which is indicative of both first and second generation sfermion masses). The
distribution ranges from $m_{\tilde{u}_{L}}\sim 10-40$ TeV, so these
sparticles are likely far beyond LHC reach. In frame c), we show the
distribution in light top-squark mass. Here we find $m_{\tilde{t}_{1}}\sim
1-2$ TeV, so again it may come as no surprise that LHC has so far not
discovered evidence of top-squark pair production. The heavier top squark
distribution $m_{\tilde{t}_{2}}$ is shown in frame d). Here we see it ranges
between $2-5$ TeV. All these results are in qualitative agreement with
previous results generated using Isasugra[55].

Figure 7: Distribution of strongly interacting sector masses from a subset of
landscape vacua with NUHM3 model as low energy EFT and an $n=1$ statistical
draw. We show distributions in a) $m_{\tilde{g}}$, b) $m_{\tilde{u}_{L}}$, c)
$m_{\tilde{t}_{1}}$ and d) $m_{\tilde{t}_{2}}$.

##  5 Conclusions

We have created a publicly available computer code DEW4SLHA which computes the
electroweak finetuning measure $\Delta_{EW}$ from any SUSY/Higgs spectrum
generator which produces the standard SUSY Les Houches Accord output file. The
code then allows us to compare naturalness and landscape predictions from
various spectra codes against each other.

We have used the DEW4SLHA code in Sec. 3 to compare natural SUSY spectra from
Isasugra, SOFTSUSY, SUSPECT and SPHENO. The outputs for a natural SUSY
benchmark point are generally in good agreement although SOFTSUSY and SUSPECT
gives values of $m_{h}$ a couple GeV higher than Isasugra or SPHENO. We also
computed the top 44 contributions to $\Delta_{EW}$ which are typically in
agreement although SPHENO generates slightly more stop mixing than the other
codes. The SOFTSUSY code was used to display cancellations in
$\Sigma_{u}^{u}(\tilde{t}_{1,2})$ that lead to increased naturalness for large
stop mixing (which also lifts the value of $m_{h}\sim 125$ GeV).

We also used SOFTSUSY to corroborate predictions for sparticle and Higgs boson
masses from the string landscape where a draw to large soft terms along with
an anthropic requirement on the weak scale $m_{weak}^{PU}<4m_{weak}^{OU}$
leads to statistical predictions from compactified string models with the MSSM
as the low energy EFT. SOFTSUSY generates a Higgs mass peak $m_{h}\sim
127-128$ GeV, slightly higher than Isasugra. SOFTSUSY also generates sparticle
mass spectra typically beyond LHC13 reach, confirming earlier Isasugra
results.

## Acknowledgements

This work has been performed as part of a contribution to the Snowmass 2022
workshop. This material is based upon work supported by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences Energy Frontier
Research Centers program under Award Number DE-SC-0009956 and U.S. Department
of Energy Grant DE-SC-0017647.

##  Appendix A Appendix of corrections

Starting with the effective potential, we have to one-loop order

$\displaystyle V_{\text{Higgs}}$ $\displaystyle=V_{\text{tree}}+\Delta V$
(A.1)

$\displaystyle V_{\text{tree}}$
$\displaystyle=\left(\left\lvert\mu\right\rvert^{2}+m_{H_{u}}^{2}\right)\left\lvert
H_{u}^{0}\right\rvert^{2}+\left(\left\lvert\mu\right\rvert^{2}+m_{H_{d}}^{2}\right)\left\lvert
H_{d}^{0}\right\rvert^{2}-\left(bH_{u}^{0}H_{d}^{0}+\text{c.c.}\right)+\frac{\left(g^{2}+g^{\prime
2}\right)}{8}\left(\left\lvert H_{u}^{0}\right\rvert^{2}-\left\lvert
H_{d}^{0}\right\rvert^{2}\right)^{2}$ (A.2)

where $b\equiv B\mu$.

In the $\overline{\text{DR}}^{\prime}$ scheme, to remove the dependence of the
1-loop effective potential on $m_{\epsilon}^{2}$, we can write[56]

$\displaystyle\Delta V$
$\displaystyle=\frac{1}{16\pi^{2}}V^{(1)}_{\overline{\text{DR}}^{\prime}}$
(A.3)

where

$V^{(1)}_{\overline{\text{DR}}^{\prime}}=\sum\limits_{n}(-1)^{2s_{n}}\left(2s_{n}+1\right)h\left(m_{n}^{2}\right)\equiv\text{STr}(h(m_{n}^{2})),$
$h(x)=\frac{x^{2}}{4}\left[\overline{\ln}(x)-\frac{3}{2}\right]$

and

$\overline{\ln}(x)\equiv\ln\left(\frac{x}{Q^{2}}\right)$

with $Q$ being the renormalization scale. One must also be careful to account
for color multiplicity and charge multiplicity factors, where

$n_{\widetilde{f}}=\begin{cases}1,&\widetilde{f}=\text{color neutral
sparticle}\\\ 3,&\widetilde{f}=\text{squark}\\\ \end{cases}$

represents the number of colors of $\widetilde{f}$ and

$n_{\text{charge}}=\begin{cases}1,&\widetilde{f}\text{ is uncharged}\\\
2,&\widetilde{f}\text{ is charged}\\\ \end{cases}.$

For notation purposes, we may write

$v_{u}=\left\langle H_{u}^{0}\right\rangle\ \ \ {\rm and}\ \ \
v_{d}=\left\langle H_{d}^{0}\right\rangle$

such that

$v_{u}^{2}+v_{d}^{2}=v^{2}=\frac{2m_{Z}^{2}}{g^{2}+g^{\prime
2}}\approx(174\text{ GeV})^{2}.$

Then, defining the parameter $\beta$ through

$\tan(\beta)\equiv\frac{v_{u}}{v_{d}}$

one may write

$v_{u}=v\sin(\beta)\ \ \ {\rm and}\ \ \ v_{d}=v\cos(\beta).$

Therefore, the minimization conditions are:

$\displaystyle
m_{H_{u}}^{2}+\Sigma_{u}^{u}+\left\lvert\mu\right\rvert^{2}-b\cot(\beta)-\frac{m_{Z}^{2}}{2}\cos(2\beta)$
$\displaystyle=0$ (A.4)

and

$\displaystyle
m_{H_{d}}^{2}+\Sigma_{d}^{d}+\left\lvert\mu\right\rvert^{2}-b\tan(\beta)+\frac{m_{Z}^{2}}{2}\cos(2\beta)$
$\displaystyle=0$ (A.5)

where we treat $v_{u,d}$ as real variables in the differentiation such that
the derivatives of $\Delta V(v_{u},v_{d})$ can be written as

$\displaystyle\Sigma_{u}^{u}$ $\displaystyle=\frac{\partial\Delta
V\left(v_{u},v_{d}\right)}{\partial v_{u}^{2}}$ (A.6)
$\displaystyle=\frac{\partial\Delta V}{\partial
v_{u}}\underbrace{\frac{\partial v_{u}}{\partial
v_{u}^{2}}}_{\frac{1}{2v_{u}}}+\frac{\partial\Delta V}{\partial
v_{d}}\underbrace{\frac{\partial v_{d}}{\partial v_{u}^{2}}}_{0}$
$\displaystyle=\frac{1}{2v_{u}}\frac{\partial\Delta V}{\partial v_{u}},$
$\displaystyle\Sigma_{d}^{d}$ $\displaystyle=\frac{\partial\Delta
V\left(v_{u},v_{d}\right)}{\partial v_{d}^{2}}$ (A.7)
$\displaystyle=\frac{\partial\Delta V}{\partial
v_{u}}\underbrace{\frac{\partial v_{u}}{\partial
v_{d}^{2}}}_{0}+\frac{\partial\Delta V}{\partial
v_{d}}\underbrace{\frac{\partial v_{d}}{\partial
v_{u}^{2}}}_{\frac{1}{2v_{d}}}$
$\displaystyle=\frac{1}{2v_{d}}\frac{\partial\Delta V}{\partial v_{d}},$
$\displaystyle\Sigma_{u}^{d}$ $\displaystyle=\frac{\partial\left(\Delta
V\right)}{\partial(v_{u}v_{d}+c.c.)},$ (A.8)

with

$\displaystyle\Sigma_{u}^{d}$ $\displaystyle=\Sigma_{d}^{u}.$ (A.9)

With these notes in mind, we can write the minimization conditions as

$\displaystyle m_{Z}^{2}$ $\displaystyle=\frac{\left\lvert
m_{H_{d}}^{2}+\Sigma_{d}^{d}-m_{H_{u}}^{2}-\Sigma_{u}^{u}\right\rvert}{\underbrace{\sqrt{1-\sin^{2}(2\beta)}}_{\cos(2\beta)}}-m_{H_{u}}^{2}-\Sigma_{u}^{u}-m_{H_{d}}^{2}-\Sigma_{d}^{d}-2\left\lvert\mu\right\rvert^{2}$
(A.10)

and

$\displaystyle\sin(2\beta)$
$\displaystyle=\frac{2b}{m^{2}_{H_{u}}+\Sigma_{u}^{u}+m_{H_{d}}^{2}+\Sigma_{d}^{d}+2\left\lvert\mu\right\rvert^{2}}$
(A.11)

or equivalently

$b=\left[\left(m_{H_{u}}^{2}+\Sigma_{u}^{u}+\left\lvert\mu\right\rvert^{2}\right)+\left(m_{H_{d}}^{2}+\Sigma_{d}^{d}+\left\lvert\mu\right\rvert^{2}\right)\right]\sin(\beta)\cos(\beta).$

The $\Sigma$ contributions are, explicitly,

$\displaystyle\Sigma_{u}^{u}$
$\displaystyle=\sum\limits_{n}\frac{(-1)^{2s_{n}}}{32\pi^{2}}\left(2s_{n}+1\right)m_{n}^{2}\left(\frac{\partial
m_{n}^{2}}{\partial
v_{u}^{2}}\right)\left(\overline{\ln}\left(m_{n}^{2}\right)-1\right),$ (A.12)
$\displaystyle\Sigma_{d}^{d}$
$\displaystyle=\sum\limits_{n}\frac{(-1)^{2s_{n}}}{32\pi^{2}}\left(2s_{n}+1\right)m_{n}^{2}\left(\frac{\partial
m_{n}^{2}}{\partial
v_{d}^{2}}\right)\left(\overline{\ln}\left(m_{n}^{2}\right)-1\right),$
$\displaystyle\Sigma_{u}^{d}$ $\displaystyle=\Sigma_{d}^{u}$
$\displaystyle=\sum\limits_{n}\frac{(-1)^{2s_{n}}}{64\pi^{2}}\left(2s_{n}+1\right)m_{n}^{2}\left(\frac{\partial
m_{n}^{2}}{\partial\left(v_{u}v_{d}\right)}\right)\left(\overline{\ln}\left(m_{n}^{2}\right)-1\right),$

with

$\displaystyle F(m^{2})$
$\displaystyle=m^{2}\left[\overline{\ln}\left(m^{2}\right)-1\right].$ (A.13)

The color and charge factors are accounted for as mentioned above, and each
individual contribution is given below.

### Squarks and sleptons

The stop squark squared mass matrix is given by

$\displaystyle\textbf{m}_{\widetilde{t}}^{2}$
$\displaystyle=\begin{bmatrix}m^{2}_{\widetilde{t}_{L}}+m_{t}^{2}+\Delta_{\widetilde{u}_{L}}&a^{*}_{t}v_{u}-\mu
y_{t}v_{d}\\\
a_{t}v_{u}-\mu^{*}y_{t}v_{d}&m^{2}_{\widetilde{t}_{R}}+m_{t}^{2}+\Delta_{\widetilde{u}_{R}}\\\
\end{bmatrix}$ (A.14)

where

$\displaystyle\Delta_{\phi}$
$\displaystyle=\left(T_{3\phi}-Q_{\phi}\sin^{2}(\theta_{W})\right)\left(\frac{g^{2}+g^{\prime
2}}{2}\right)(v_{d}^{2}-v_{u}^{2})$ (A.15)

where $T_{3\phi}$ is the third component of the weak isospin of $\phi$, and
$Q_{\phi}$ is the electrical charge of $\phi$. Hence,

$\frac{\partial\Delta_{\phi}}{\partial
v_{u}^{2}}=-(T_{3\phi}-Q_{\phi}\underbrace{\sin^{2}(\theta_{W})}_{x_{W}})\left(\frac{g^{2}+g^{\prime
2}}{2}\right)$

and

$\frac{\partial\Delta_{\phi}}{\partial
v_{d}^{2}}=\left(T_{3\phi}-Q_{\phi}x_{W}\right)\left(\frac{g^{2}+g^{\prime
2}}{2}\right).$

At tree-level, the following mass relations hold for the running masses:

$\displaystyle m_{t}$ $\displaystyle=y_{t}v_{u},$ (A.16) $\displaystyle m_{b}$
$\displaystyle=y_{b}v_{d},$ (A.17) $\displaystyle m_{\tau}$
$\displaystyle=y_{\tau}v_{d}.$ (A.18)

The eigenvalues of Eq. (A.14) are

$\displaystyle m^{2}_{\widetilde{t}_{1,2}}$
$\displaystyle=\frac{1}{2}\Bigg{(}m^{2}_{\widetilde{t}_{L}}+2m_{t}^{2}+m^{2}_{\widetilde{t}_{R}}+\Delta_{\widetilde{u}_{L}}+\Delta_{\widetilde{u}_{R}}$
(A.19)
$\displaystyle\mp\sqrt{(m^{2}_{\widetilde{t}_{L}}-m^{2}_{\widetilde{t}_{R}}+\Delta_{\widetilde{u}_{L}}-\Delta_{\widetilde{u}_{R}})^{2}+4\left[|a_{t}|^{2}v_{u}^{2}-v_{d}v_{u}y_{t}\mu^{*}a_{t}^{*}+v_{d}v_{u}y_{t}\mu
a_{t}+v_{d}v_{u}+|\mu|^{2}v_{d}^{2}y_{t}^{2}\right]}\Bigg{)}$

where $m^{2}_{\widetilde{t}_{1}}<m^{2}_{\widetilde{t}_{2}}$.

Note that

$g_{Z}^{2}\equiv\frac{g^{2}+\left(g^{\prime}\right)^{2}}{8}\ \ {\rm so\ that}\
\ \ m_{Z}^{2}\equiv 4v^{2}\cdot g_{Z}^{2}.$

The trilinear couplings are written in “reduced” form, i.e.,

$a_{i}\equiv A_{i}\cdot y_{i}$

with $y_{i}$ the corresponding Yukawa coupling.

The radiative correction terms are then

$\displaystyle\Sigma_{u}^{u}(\widetilde{t}_{1,2})$
$\displaystyle=\frac{3}{16\pi^{2}}F(m_{\widetilde{t}_{1,2}}^{2})\cdot\left[y_{t}^{2}-g_{Z}^{2}\mp\frac{a_{t}^{2}-2g_{Z}^{2}\cdot\Delta_{\widetilde{t}}}{m_{\widetilde{t}_{2}}^{2}-m_{\widetilde{t}_{1}}^{2}}\right]$
(A.20)

with

$\Delta_{\widetilde{t}}\equiv
2\left(\frac{1}{2}-\frac{4}{3}\sin^{2}(\theta_{W})\right)\cdot\left[\frac{m_{\widetilde{t}_{L}}^{2}-m_{\widetilde{t}_{R}}^{2}}{2}+\left(m_{Z}^{2}\cos(2\beta)\cdot\left[\frac{1}{4}-\frac{2}{3}\sin^{2}(\theta_{W})\right]\right)\right]$

and

$\displaystyle\Sigma_{d}^{d}(\widetilde{t}_{1,2})$
$\displaystyle=\frac{3}{16\pi^{2}}F(m_{\widetilde{t}_{1,2}^{2}})\left[g_{Z}^{2}\mp\frac{y_{t}^{2}\cdot\mu^{2}+2g_{Z}^{2}\cdot\Delta_{\widetilde{t}}}{m_{\widetilde{t}_{2}}^{2}-m_{\widetilde{t}_{1}}^{2}}\right]$
(A.21)

where the minus (plus) sign corresponds to $\widetilde{t}_{1(2)}$.

Next, with the bottom squark mass matrix

$\displaystyle\textbf{m}_{\widetilde{b}}^{2}$
$\displaystyle=\begin{bmatrix}m^{2}_{\widetilde{b}_{L}}+\Delta_{\widetilde{d}_{L}}&a^{*}_{b}v_{d}-\mu
y_{b}v_{u}\\\
a_{b}v_{d}-\mu^{*}y_{b}v_{d}&m^{2}_{\widetilde{b}_{R}}+\Delta_{\widetilde{d}_{R}}\\\
\end{bmatrix},$ (A.22)

the eigenvalues are computed to be

$\displaystyle\leavevmode\resizebox{422.77661pt}{}{$m^{2}_{\widetilde{b}_{1,2}}=\frac{1}{2}\left(m^{2}_{\widetilde{b}_{L}}+\Delta_{\widetilde{d}_{L}}+m^{2}_{\widetilde{b}_{R}}+\Delta_{\widetilde{d}_{R}}\mp\sqrt{(-m^{2}_{\widetilde{b}_{L}}-\Delta_{\widetilde{d}_{L}}+m^{2}_{\widetilde{b}_{R}}+\Delta_{\widetilde{d}_{R}})^{2}+4(a_{b}^{*}v_{d}-\mu
v_{u}y_{t})(a_{b}v_{d}-\mu^{*}v_{u}y_{t})}\right).$}$ (A.23)

Thus, the radiative correction terms are

$\displaystyle\Sigma_{u}^{u}(\widetilde{b}_{1,2})$
$\displaystyle=\frac{3}{16\pi^{2}}F(m_{\widetilde{b}_{1,2}}^{2})\cdot\left[y_{b}^{2}-g_{Z}^{2}\mp\frac{a_{b}^{2}-2g_{Z}^{2}\cdot\Delta_{\widetilde{b}}}{m_{\widetilde{b}_{2}}^{2}-m_{\widetilde{b}_{1}}^{2}}\right]$
(A.24)

with

$\Delta_{\widetilde{b}}=2\left(\frac{1}{2}-\frac{2}{3}\sin^{2}(\theta_{W})\right)\cdot\left(\frac{m_{b_{L}}^{2}-m_{b_{R}}^{2}}{2}-m_{Z}^{2}\cos(2\beta)\cdot\left[\frac{1}{4}-\frac{1}{3}\sin^{2}(\theta_{W})\right]\right)$

and

$\displaystyle\Sigma_{d}^{d}(\widetilde{b}_{1,2})$
$\displaystyle=\frac{3}{16\pi^{2}}F(m_{\widetilde{b}_{1,2}}^{2})\cdot\left[g_{Z}^{2}\mp\frac{y_{b}^{2}\cdot\mu^{2}+2g_{Z}^{2}\cdot\Delta_{\widetilde{b}}}{m_{\widetilde{b}_{2}}^{2}-m_{\widetilde{b}_{1}}^{2}}\right].$
(A.25)

Similarly, one can obtain the result for the staus by exchanging
$b\rightarrow\tau$, $n_{\widetilde{\tau},\text{ col.}}\rightarrow 1$, and

$\Delta_{\widetilde{\tau}}\equiv
2\left(\frac{1}{2}-2\sin^{2}(\theta_{W})\right)\cdot\left[\frac{m_{\widetilde{\tau}_{L}}^{2}-m_{\widetilde{\tau}_{R}}^{2}}{2}-\left(\frac{1}{4}-\sin^{2}(\theta_{W})\right)\right].$

### Sfermions

For a general sfermion $\widetilde{f}_{L,R}$ in the first or second
generation, the masses can be parameterized based on the boundary conditions
of the model. It can be shown that for these sfermions,

$\displaystyle\frac{\partial m_{\widetilde{f}}^{2}}{\partial
v_{u,d}^{2}}=\overbrace{\underbrace{\pm}_{v_{u}^{2}}}^{v_{d}^{2}}(T_{3\widetilde{f}}-Q_{\widetilde{f}}x_{W})\left(\frac{g^{2}+g^{\prime
2}}{2}\right)$ (A.26)

which leads to the cancellation between first and second generation sfermion
corrections, where the cancellation arises between
$\Sigma_{u}^{u}(\widetilde{f})$ and $\Sigma_{d}^{d}(\widetilde{f})$. Indeed,
one finds that for the squarks,

$\displaystyle\Sigma^{u}_{u}(\widetilde{f}_{L,R})$
$\displaystyle=\frac{-3}{4\pi^{2}}(T_{3\widetilde{f}_{L,R}}-Q_{\widetilde{f}_{L,R}}x_{W})g_{Z}^{2}F(m_{\widetilde{f}_{L,R}}^{2})$
(A.27)

and

$\displaystyle\Sigma^{d}_{d}(\widetilde{f}_{L,R})$
$\displaystyle=\frac{+3}{4\pi^{2}}(T_{3\widetilde{f}_{L,R}}-Q_{\widetilde{f}_{L,R}}x_{W})g_{Z}^{2}F(m_{\widetilde{f}_{L,R}}^{2}).$
(A.28)

For the sleptons in the 1st and 2nd generations, replace the color factor of
$3$ in the numerator with $1$, and for the slepton sneutrinos, replace the
color factor of $3\rightarrow 1$ and the charge factor from $2\rightarrow 1$.

### Neutralinos

For the neutralinos, the _unsquared_ mass matrix can be given by the
following, where $c_{\beta}=\cos(\beta),s_{\beta}=\sin(\beta)$, and likewise
for other angles:

$\displaystyle\textbf{m}_{\widetilde{N}}$
$\displaystyle=\begin{bmatrix}M_{1}&0&-c_{\beta}s_{W}m_{Z}&s_{\beta}s_{W}m_{Z}\\\
0&M_{2}&c_{\beta}c_{W}m_{Z}&-s_{\beta}c_{W}m_{Z}\\\
-c_{\beta}s_{W}m_{Z}&c_{\beta}c_{W}m_{Z}&0&-\mu\\\
s_{\beta}s_{W}m_{Z}&-s_{\beta}c_{W}m_{Z}&-\mu&0\\\ \end{bmatrix}.$ (A.29)

One could square this matrix and solve for the eigenvalues by brute force
using the Ferrari method, and then differentiating the resultant squared mass
eigenvalues. Instead, we use the method proposed by Ibrahim and Nath for
taking the derivatives of eigenvalues[57]. Consider the characteristic
polynomial of the squared neutralino mass matrix. The coefficients will each
be functions of $v_{u},v_{d}$. As such, one may write in general form

$\displaystyle F(\lambda)$
$\displaystyle=\lambda^{4}+b_{\lambda}(v_{u},v_{d})\lambda^{3}+c_{\lambda}(v_{u},v_{d})\lambda^{2}+d_{\lambda}(v_{u},v_{d})\lambda+e_{\lambda}(v_{u},v_{d})$
(A.30) $\displaystyle=0.$

Then the eigenvalues have derivatives given by

$\displaystyle\frac{\partial m^{2}_{\widetilde{N}_{i}}}{\partial v_{u,d}^{2}}$
$\displaystyle=\frac{-D_{v_{u,d}^{2}}F}{D_{\lambda}F}\Bigg{\rvert}_{\lambda=m^{2}_{\widetilde{N}_{i}}},$
(A.31)

which can be obtained by taking the derivative of Eq. (A.30), with
$i=1,2,3,4$,

$\displaystyle D_{\lambda}F$ $\displaystyle\equiv\frac{dF}{d\lambda},$ (A.32)

and

$\displaystyle D_{v_{u,d}^{2}}F$ $\displaystyle=\frac{\partial
b_{\lambda}}{\partial v_{u,d}^{2}}\lambda^{3}+\frac{\partial
c_{\lambda}}{\partial v_{u,d}^{2}}\lambda^{2}+\frac{\partial
d_{\lambda}}{\partial v_{u,d}^{2}}\lambda+\frac{\partial e_{\lambda}}{\partial
v_{u,d}^{2}}/$ (A.33)

From there, the explicit form of $\Sigma^{u,d}_{u,d}(\widetilde{N}_{i})$ for
$i=1,2,3,4$ can be written, which will take the form

$\displaystyle\Sigma^{u,d}_{u,d}(\widetilde{N}_{i})$
$\displaystyle=\frac{-1}{16\pi^{2}}\left[\frac{-D_{v_{u,d}^{2}}F}{D_{\lambda}F}\Bigg{\rvert}_{\lambda=m^{2}_{\widetilde{N}_{i}}}\right]F(m^{2}_{\widetilde{N}_{i}})/$
(A.34)

### Charginos

The chargino mass matrix is in $2\times 2$ block form on the off-diagonals,
and can be squared to obtain the squared mass matrix,

$\displaystyle\textbf{m}_{\widetilde{C}}^{2}$
$\displaystyle=\begin{bmatrix}M_{2}^{2}+g^{2}v_{d}^{2}&g(M_{2}v_{u}+v_{d}\mu)&0&0\\\
g(M_{2}v_{u}+\mu v_{d})&g^{2}v_{u}^{2}+\mu^{2}&0&0\\\
0&0&M_{2}^{2}+g^{2}v_{u}^{2}&g(M_{2}v_{d}+\mu v_{u})\\\ 0&0&g(M_{2}v_{d}+\mu
v_{u})&g^{2}v_{d}^{2}+\mu^{2}\\\ \end{bmatrix}.$ (A.35)

This has doubly degenerate eigenvalues

$\displaystyle m^{2}_{\widetilde{C}_{1,2}}$
$\displaystyle=\frac{1}{2}\left[|M_{2}|^{2}+|\mu|^{2}+g^{2}(v_{u}^{2}+v_{d}^{2})\mp\sqrt{\left[g^{2}(v_{u}+v_{d})^{2}+(M_{2}-\mu)^{2}\right]\left[g^{2}(v_{d}-v_{u})^{2}+(M_{2}+\mu)^{2}\right]}\right]$
(A.36)

with $m^{2}_{\widetilde{C}_{1}}<m^{2}_{\widetilde{C}_{2}}$, leading to

$\displaystyle\Sigma_{u}^{u}(\widetilde{C}_{1,2})$
$\displaystyle=\frac{-g^{2}}{16\pi^{2}}\left(1\mp\frac{(-2)\cdot
m_{W}^{2}\cdot\cos(2\beta)+M_{2}^{2}+\mu^{2}}{m^{2}_{\widetilde{C}_{2}}-m^{2}_{\widetilde{C}_{1}}}\right)F(m^{2}_{\widetilde{C}_{1,2}})$
(A.37)

and

$\displaystyle\Sigma_{d}^{d}(\widetilde{C}_{1,2})$
$\displaystyle=\frac{-g^{2}}{16\pi^{2}}\left(1\mp\frac{2\cdot
m_{W}^{2}\cdot\cos(2\beta)+M_{2}^{2}+\mu^{2}}{m^{2}_{\widetilde{C}_{2}}-m^{2}_{\widetilde{C}_{1}}}\right)F(m^{2}_{\widetilde{C}_{1,2}}).$
(A.38)

with $m_{W}^{2}$ given below.

### Weak bosons

Using the mass relations

$\displaystyle m_{W}^{2}$ $\displaystyle=\frac{g^{2}}{2}v^{2}\implies\
\frac{\partial m_{W}^{2}}{\partial v_{u,d}^{2}}$
$\displaystyle=\frac{g^{2}}{2}$ (A.39)

and

$\displaystyle m_{Z}^{2}$ $\displaystyle=\frac{(g^{2}+g^{\prime
2})}{2}(v^{2})\implies\ \frac{\partial m_{Z}^{2}}{\partial v_{u,d}^{2}}$
$\displaystyle=\frac{g^{2}+g^{\prime 2}}{2}$ (A.40)

such that the radiative corrections are

$\displaystyle\Sigma^{u}_{u}(W^{\pm})$ $\displaystyle=\Sigma^{d}_{d}(W^{\pm})\
$ $\displaystyle=\frac{3g^{2}}{32\pi^{2}}F(m_{W}^{2})$ (A.41)

and

$\displaystyle\Sigma^{u}_{u}(Z^{0})$ $\displaystyle=\Sigma^{d}_{d}(Z^{0})\ $
$\displaystyle=\frac{3(g^{2}+g^{\prime 2})}{64\pi^{2}}F(m_{Z}^{2}).$ (A.42)

### Higgs bosons

For the Higgs bosons, we have the mass relations (with
$m^{2}_{h_{0}}<m^{2}_{H_{0}}$)

$\displaystyle m^{2}_{h_{0},H_{0}}$
$\displaystyle=\frac{1}{2}\left(m_{A_{0}}^{2}+m_{Z}^{2}\mp\sqrt{(m_{A_{0}}^{2}-m_{Z}^{2})^{2}+4m_{Z}^{2}m_{A_{0}}^{2}\sin^{2}(2\beta)}\right)$
(A.43)

and

$\displaystyle m^{2}_{A_{0}}$ $\displaystyle=\frac{2b}{\sin(2\beta)}\ $
$\displaystyle=2|\mu|^{2}+m^{2}_{H_{u}}+m^{2}_{H_{d}}$ (A.44)

Therefore,

$\displaystyle\Sigma^{u}_{u}(h_{0},H_{0})$
$\displaystyle=\frac{g_{Z}^{2}}{16\pi^{2}}\left[1\mp\frac{m_{Z}^{2}+m_{A_{0}}^{2}\cdot\left(1+4\cos(2\beta)+2\cos^{2}(2\beta)\right)}{m^{2}_{H_{0}}-m^{2}_{h_{0}}}\right]F(m_{h_{0},H_{0}}^{2})$
(A.45)

and

$\displaystyle\Sigma^{d}_{d}(h_{0},H_{0})$
$\displaystyle=\frac{g_{Z}^{2}}{16\pi^{2}}\left[1\mp\frac{m_{Z}^{2}+m_{A_{0}}^{2}\cdot\left(1-4\cos(2\beta)+2\cos^{2}(2\beta)\right)}{m^{2}_{H_{0}}-m^{2}_{h_{0}}}\right]F(m_{h_{0},H_{0}}^{2}).$
(A.46)

Then, for the charged Higgs, with the relation

$\displaystyle m^{2}_{H^{\pm}}$ $\displaystyle=m^{2}_{A_{0}}+m_{W}^{2}$ (A.47)

one simply finds that

$\displaystyle\Sigma^{u,d}_{u,d}(H^{\pm})$
$\displaystyle=\frac{g^{2}}{32\pi^{2}}F(m^{2}_{H^{\pm}}).$ (A.48)

### SM fermions

Finally, we can use the mass relations mentioned in Eqs. (16) - (18) to obtain
the top, bottom, and $\tau$ contributions:

$\displaystyle\Sigma^{u}_{u}(t)$
$\displaystyle=\frac{-3y_{t}^{2}}{16\pi^{2}}F(m_{t}^{2}),$ (A.49)
$\displaystyle\Sigma^{d}_{d}(t)$ $\displaystyle=0,$ (A.50)
$\displaystyle\Sigma^{u}_{u}(b)$ $\displaystyle=0,$ (A.51)
$\displaystyle\Sigma^{d}_{d}(b)$
$\displaystyle=\frac{-3y_{b}^{2}}{16\pi^{2}}F(m_{b}^{2}),$ (A.52)
$\displaystyle\Sigma^{u}_{u}(\tau)$ $\displaystyle=0,$ (A.53)

and

$\displaystyle\Sigma^{d}_{d}(\tau)$
$\displaystyle=\frac{-y_{\tau}^{2}}{16\pi^{2}}F(m_{\tau}^{2}).$ (A.54)

We choose

$\displaystyle Q=\sqrt{m_{\text{stop}_{1}}m_{\text{stop}_{2}}}$ (A.55)

as the renormalization scale used in the function $F$.

From these terms, one can use the minimization conditions to define the
naturalness measure $\Delta_{EW}$:

$\displaystyle\Delta_{EW}$ $\displaystyle\equiv\frac{2\max\left\lvert
C_{i}\right\rvert}{m_{Z}^{2}}$ (A.56)

where the $C_{i}$ terms are the individual contributions from the Higgs
minimization condition ($i=H_{u},H_{d},\mu,\Sigma_{u}^{u},\Sigma_{d}^{d}$).
Specifically, $C_{\mu}=\left\lvert\mu\right\rvert^{2}$,

$C_{H_{u}}=\frac{-m_{H_{u}}^{2}\tan^{2}(\beta)}{\tan^{2}(\beta)-1}\ \ {\rm
and}\ \ C_{H_{d}}=\frac{m_{H_{d}}^{2}}{\tan^{2}(\beta)-1},$

and

$C_{\Sigma_{u}^{u}}=\frac{-\Sigma_{u}^{u}\tan^{2}(\beta)}{\tan^{2}(\beta)-1}\
\ {\rm with}\ \ C_{\Sigma_{d}^{d}}=\frac{\Sigma_{d}^{d}}{\tan^{2}(\beta)-1}.$

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