Bayesian analysis and naturalness of (Next-to-)Minimal Supersymmetric Models

The Higgs boson discovery stirred interest in next-to-minimal supersymmetric models, due to the apparent fine-tuning required to accommodate it in minimal theories. To assess their naturalness, we compare fine-tuning in a $\mathbb{Z}_3$ conserving semi-constrained Next-to-Minimal Supersymmetric Standard Model (NMSSM) to the constrained MSSM (CMSSM). We contrast popular fine-tuning measures with naturalness priors, which automatically appear in statistical measures of the plausibility that a given model reproduces the weak scale. Our comparison shows that naturalness priors provide valuable insight into the hierarchy problem and rigorously ground naturalness in Bayesian statistics. For the CMSSM and semi-constrained NMSSM we demonstrate qualitative agreement between naturalness priors and popular fine tuning measures. Thus, we give a clear plausibility argument that favours relatively light superpartners.

Checking such claims by calculating fine-tuning in various supersymmetric models, however, is somewhat futile, as the results would completely depend upon the definition of fine-tuning itself. This subjectivity is a common criticism of naturalness arguments.
Rather than abandoning naturalness or relying on heuristic judgments, we instead advocate for an approach that is based on Bayesian statistics. In this approach, one has a welldefined means of quantifying both how plausible a particular parameter space point is in the context of a given model and which model in a given set is the most plausible in light of experimental data. Apart from these being the most germane questions to pose, we argue that they also capture the essence of ordinary naturalness arguments whilst evading arbitrary aspects of naturalness by utilizing a unique logical framework in Bayesian inference (see e.g., Ref. [74][75][76]).
Such calculations automatically incorporate so-called naturalness priors that contain factors strongly resembling some traditional measures of fine-tuning, but which have a rigorous probabilistic interpretation. In addition to being a well-founded fine-tuning measure, the appearance of these naturalness priors also leads to posterior probability densities that tend to favor regions of parameter space that would be considered as having low fine-tuning according to the naïve tuning measures, as we show below. Thus Bayesian plausibility analyses automatically take into account fine-tuning in a model and the effects of new experimental data on this tuning. Moreover, through comparing the Bayesian evidence for different models it is possible to make statistically meaningful comparisons between models. The role of the naturalness priors in these comparisons is to ensure that the outcome of such a comparison is reflective of whether one model is more natural than another for a given set of experimental data.
The Bayesian interpretation of naturalness was advocated numerous times over the last decade [77][78][79][80][81][82][83][84][85]. However, since it remains much less common than traditional fine-tuning measures, we recapitulate the essential points in Sec. 2. We illustrate this methodology with a warm-up example of the hierarchy problem in the SM in Sec. 3, define our semiconstrained NMSSM and the CMSSM models in Sec. 4, and describe results from our fully-fledged Bayesian analysis in Sec. 5. This completes our previous study [86] and complements previous Bayesian analyses of the semi-constrained NMSSM [82,87,88] and CMSSM [13,. We close by summarizing our findings in Sec. 6.

Bayesian inference
Bayesian statistics is a framework for quantifying the plausibility of a hypothesis, such as a scientific theory (see e.g., Ref. [74]). The central equation for our analysis is Bayes' theorem for continuous variables, As simple as it seems, marginalization captures the traditional idea in physics that finetuned parameters are relatively implausible. We may rewrite Eq.
which states that the posterior density for x is the average conditional density p(x | y, . . . , M, D).
For a given x, it may be possible to fine-tune the value of y, . . . such that the conditional density is substantial. The average conditional density and thus the posterior, though, may be negligible. As we shall see in Sec. 3, in this way marginalization automatically penalizes fine-tuning related to the hierarchy problem.
The second equation for our statistical analysis is Bayes' theorem for a discrete hypothesis, We see that the plausibility of a model is updated by a factor known as the evidence, which may be expressed as The evidence is a functional of the priors for the model's parameters. Model selection by evidences is somewhat controversial, partly since evidences may be sensitive to the diffuseness of prior densities and this sensitivity cannot be compensated by sufficient data.
For that reason, we focus upon posterior distributions, though briefly compare models with evidences, which are a byproduct of our analysis.
Computationally, the evidence is the average likelihood. As such, it penalizes finetuning automatically, since if, for a particular model, agreement with data is found in only a small region of the prior volume, the average likelihood will be small relative to a model in which agreement is found everywhere or more readily.

Fine-tuning in the Standard Model
We now consider fine-tuning of the weak scale in the Standard Model (SM) interpreted as an effective field theory with quadratic corrections from new physics. Our toy-model of the effective SM is defined by a cut-off Λ 2 and parameters µ 2 and λ in the Higgs potential, This toy model predicts that whereḡ 2 = g 2 + g 2 , g and g being the SU (2) L and (non-GUT normalized) U (1) Y gauge couplings, respectively. We assume, as happens in many specific cases, that new physics at the cut-off scale results in quadratic corrections to µ 2 . To keep the toy model as simple as possible, we do not consider any coefficient from a loop-factor in front of the quadratic correction and neglect the new physics corrections to λ.
The most common measure of fine-tuning in particle physics is the Barbieri-Giudice-Ellis-Nanopoulos (BGEN) measure [4,5], which is based upon measuring the sensitivity of some observable quantity to variations in the underlying, assumed to be fundamental, model parameters. In discussions of the hierarchy problem, the measure is conventionally formulated in terms of the predicted Z-boson mass, leading to the tuning sensitivities defined by for each model parameter p. This traditional measure leaves many questions. Are finetuned theories implausible? And if so, why? How much fine-tuning is too much and why?
How should we adjust our conclusion in light of new experimental evidence? There are no answers to these questions because the measure is only intuitively connected to physics and lacks rigorous mathematical roots. In contrast, it is well known that in Bayesian statistics fine-tuning is intimately connected to model plausibility by a fine-tuning penalty automatically incorporated in the evidence [78,81,126].
Applying the traditional BGEN measure to the cut-off Λ 2 in our toy model of the SM we find 1 which indicates that fine-tuning mounts as the cut-off exceeds the weak scale, that is if Λ M Z . This is the SM hierarchy problem.
To illustrate that Bayesian statistics captures essential aspects of the hierarchy problem and fine-tuning, we consider the posterior for the SM cut-off, conditioned upon the measured Z-boson and Higgs mass, M exp Z and m exp h . If Bayesian statistics quantifies the hierarchy problem, the posterior should favor an SM cut-off close to the weak scale. We begin by applying Bayes' theorem to calculate the posterior for Λ 2 given our toy version of the SM and the experimental measurement of the mass of the Z boson, 1 The sensitivities ∆ p are more commonly calculated with respect to the Lagrangian parameters in a model. In a realistic model, one might consider applying the measure to a heavy mass parameter characterizing the scale of new physics; we use the generic cut-off here simply to illustrate the effects of these (unspecified) parameters. M exp Z = 91.1876 ± 0.0021 GeV [131], where Z ≡ p(M exp Z | SM) is the evidence. We continue by replacing the likelihood function p(M exp Z | Λ 2 , µ 2 , λ, SM) with a Dirac δ-function, because M Z is measured with such high precision, 2 and change the variable in the Dirac δ-function from M Z to µ 2 , where µ 2 Z reproduces the measured M Z , We identify the integral over µ 2 as an effective prior for the SM quartic and cut-off scale, In Sec. 4.2.1 we identify similar effective priors in supersymmetric models. The effective prior is conditioned upon measurement of M Z . By using an effective prior with one Lagrangian parameter fixed such that the measured M Z is obtained, one obtains a prior which is logically identical to the case in which no parameters are fixed and M Z is simply input as a constraint in the likelihood. However, the effective prior allows for vastly more efficient scanning, since one can scan only the hypersurface of parameter space in which the correct M Z is predicted. In the SM, the fixed parameter was the Higgs Lagrangian mass, µ 2 , and the specific form of the effective prior obtained contains the same derivative that would appear when the traditional fine-tuning measure, Eq. (8), is applied to the parameter µ 2 . 2 We approximate the likelihood function by a Dirac δ-function under integration, i.e., (12) Performing the µ 2 integration to obtain the marginal density for Λ 2 in Eq. (13) we find We pick logarithmic priors for the SM parameters, such that The prior for e.g., the SM cut-off favors no particular scale -logarithmic priors equally weight every order of magnitude, i.e. p(ln Λ | SM) = const. The normalization factor N is defined such that the integral of the prior over the chosen prior ranges is unity. We take the prior ranges to be 10 −4 < λ < 10 and 10 −10 GeV 2 < |µ 2 | < 10 40 GeV 2 . The prior range for the cut-off affects only the overall normalization of the posterior and the ranges outside of which it is zero.
Thus with our priors the posterior is, The prior distribution is substantially updated by the data because we have taken µ to have a logarithmic prior instead of fixing it at the outset to reproduce the measured M Z and then treating the latter as a nuisance parameter. As a result, after the µ integration a factor of 1 |µ 2 Z | appears in the remaining integrand which is approximately (ḡ 2 Λ 2 ) −1 when Λ M exp Z . The impact of this is to update the prior distribution such that large values of Λ are strongly disfavored. This is illustrated in Fig. 1 where this posterior distribution and a similar one from a calculation that includes the Higgs boson mass m exp h 125 GeV in the likelihood are plotted as functions of log Λ. We find, as expected, that the application of Bayes' theorem captures the gist of the hierarchy problem: quadratic corrections in the SM Higgs mass mean that we ought to expect new physics close to the measured weak scale. The prior distribution for the magnitude of the cut-off was flat, but once conditioned upon the weak scale (i.e., measurements of the Z-boson mass and the Higgs mass), a sub-TeV SM cut-off was favored.
Finally, we calculate Z, the evidence of the SM in light of the measurement of M Z , by integrating Eq. (19) with respect to the SM cut-off and rearranging to find the evidence,  Z. We find that if the lower limit on the prior for Λ is Λ SM and Λ SM M exp Z , then where c is a coefficient determined by factors from integration and priors, which we calculated to be O(1) for our choices. For comparison with dimensionless fine-tuning ratios, we expressed the evidences as e.g., p(log M exp . This tells us that if the cut-off is of the order of the Planck scale, M Pl , then the evidence is very small, O(10 −34 ). But if the model allows the cut-off to be of order M Z then the evidence is O(1). Therefore the evidence strongly prefers an SM effective theory that is valid only up to the electroweak or TeV scale (with new physics such as supersymmetry appearing at that scale) to an SM effective theory with no new physics below M Pl . This is the essence of the well-known hierarchy problem, but expressed in a statistically rigorous manner.
Besides its coherency and connection to statistics, an advantage of this formulation over ad-hoc fine-tuning measures is that the evidence calculation can be repeated in any new extension of the SM, and consistently compared to the evidence computed in other models. If there is no cancellation of the quadratic divergences within that model then one should obtain a similar result as obtained in the toy example. In supersymmetry the quadratic divergences are cancelled; however, soft-breaking introduces corrections of order In supersymmetry the quadratic divergences are cancelled; however, soft-breaking introduces corrections of the order of the squared soft masses, which may result in fine-tuning if these soft masses are required to be substantially larger than M Z . For this reason one should expect that in supersymmetric models a similar result approximately holds, i.e., where in this expression m SUSY characterizes the minimal size of the soft masses consistent with the likelihood and chosen priors. We now explicitly repeat our calculations in two supersymmetric models to see whether this is the case.

Models
We consider two models: a semi-constrained NMSSM and the constrained MSSM (CMSSM).
The models are tractable examples of a minimal supersymmetric model and a singlet extension that we investigate with Bayesian statistics.

Semi-constrained NMSSM
The NMSSM solves the µ-problem [132] of the MSSM by replacing the MSSM superpotential term µĤ d ·Ĥ u by one of the form λŜĤ d ·Ĥ u , whereŜ is a new gauge singlet superfield. 3 An effective µ-term, given by is then generated when the scalar component S of this singlet superfield develops a vacuum expectation value (VEV), S . The most general renormalizable superpotential of the NMSSM should also contain additional terms beyond those found in the MSSM involving the singletŜ. Here we restrict our attention to the Z 3 -conserving NMSSM (see e.g., 3 We use the notationÂ ·B ≡ αβÂ αBβ =Â 2B1 −Â 1B2 to denote a contraction between SU (2) L doublets.
Ref. [33,34]), for which the full superpotential iŝ Here the notationŴ MSSM | µ=0 refers to the usual MSSM superpotential, i.e., evaluated with µ = 0. The cubic singlet coupling κ is required to explicitly break a global U (1) Peccei-Quinn symmetry, which would otherwise give rise to a massless axion when it is spontaneously broken by the scalar field S acquiring a VEV.
As usual in phenomenological SUSY models, in the NMSSM SUSY is softly broken by a set of explicit soft terms, where the soft scalar masses, gaugino masses and soft trilinear terms are taken to be respectively. To construct the semi-constrained NMSSM that we consider here, the above soft parameters are assumed to satisfy a set of relationships at the grand unification (GUT) scale M GUT motivated by those found in minimal supergravity (mSUGRA) [133,134].
These GUT scale boundary conditions are as follows: • The soft-breaking trilinears are parameterized by where the reduced trilinear couplings are partially unified at the GUT scale, that is, while A λ and A κ are allowed to vary separately.
• The soft-breaking scalar masses are partially unified at M GUT , The exception is the soft-breaking scalar mass for the singlet, m 2 S (M GUT ) ≡ m 2 S 0 , which is taken to be free.
• The soft-breaking gaugino masses are unified at the GUT scale, In addition to the GUT scale values of the soft parameters, the values of the Yukawa couplings λ and κ at the GUT scale, λ(M GUT ) ≡ λ 0 and κ(M GUT ) ≡ κ 0 , must also be specified. This semi-constrained model is therefore described by the nine GUT scale In the MSSM, the effects of A terms are absorbed into the RG evolution of the soft terms such that the VEVs have no explicit dependence on them. On the other hand, the VEVs depend on A terms directly in the NMSSM. It is, therefore, important to have flexible constraints on A terms in the semi-constrained NMSSM.
Note that, depending on the literature, the semi-constrained NMSSM is defined by slightly different assumptions. A more strict convention allows only the singlet specific parameters to be unconstrained such that A λ = A 0 is implied at the GUT scale [37], while the more flexible version lets non-universal Higgs masses be free parameters as well as Hereafter, NMSSM is used to simply denote the semi-constrained NMSSM, if there is no special reason to distinguish it from the general NMSSM.

CMSSM
For comparison purposes we use the CMSSM [133,134,136], one of the most-studied supersymmetric models. In the parameterization that we consider, the model can be

Likelihood and priors
We include Particle Data Group (PDG) world-averages [1] of measurements of the Higgs and Z-boson masses in Table 1  Our chosen priors for the parameters of the CMSSM and the semi-constrained NMSSM are shown in Table 2. Because we are ignorant of the soft-breaking mass scale, we pick logarithmic priors, where possible, that equally weight every order of magnitude. Note that the trilinear couplings in both models, A 0 , A λ , and A κ , are allowed to take both signs, and we use the piecewise prior This choice corresponds to a logarithmic prior with special treatment at |A| 0 such that the prior remains proper.
In addition to the relevant GUT scale parameters, the models share nuisance parameters that are not of particular interest in this analysis, but which could impact our results.
The most important nuisance parameters are the top quark mass, m pole t , and the strong coupling, α s (M Z ) MS . We pick Gaussian priors for them, with means and standard deviations determined by PDG world-averages of experimental measurements [1], as shown in

Effective naturalness priors
As in the SM in Eq. (15), from these initial priors we find effective priors in the CMSSM and NMSSM in which one of the GUT scale parameters is fixed to reproduce the observed value of M 2 Z . This corresponds quite closely to the approach taken in spectrum generators for the MSSM and NMSSM, such as SOFTSUSY, where some of the presumed fundamental parameters are traded for phenomenological parameters at the weak scale. The models can then be parameterized in terms of the remaining GUT scale parameters and a set of precisely known electroweak (EW) parameters. It should be noted, however, that this is equivalent to working directly in terms of the GUT scale parameters and marginalizing with the chosen EW observables using a δ-function likelihood. This provides an economic way to survey the entirety of parameter space, discarding points that lead to hardly justifiable low-energy spectra.
In the MSSM, the effective priors arise from making the conventional trade where as usual tan β ≡ v 2 /v 1 is defined as the ratio of the two Higgs VEVs, In practice, we achieve this trade in two steps. First, the GUT scale parameters are evolved to m SUSY ≡ √ mt 1 mt 2 , the scale of EW symmetry breaking (EWSB), using twoloop renormalization group equations (RGEs). The EWSB conditions, can then be used to exchange the low-energy values of µ and Bµ for M 2 Z and tan β. In these expressions, the one-and two-loop corrections from the Coleman-Weinberg potential [139] have been absorbed into the quantitiesm 2 H d,u , and Re Π T ZZ is the transverse part of the Z-boson self-energy. Since the EWSB conditions cannot fix the phase of the µ-parameter, sign µ is an additional parameter. This trade is convenient, since we may now input the measured Z-boson and fermion masses, the latter being related to their Yukawa couplings via tan β.
The priors for the two choices of parameter sets are related by the Jacobian, J CMSSM , associated with this change of variables, where J CMSSM is given by J CMSSM = | det J CMSSM |, J CMSSM being the appropriate Jacobian matrix. Here we treat the RG evolution from M GUT to m SUSY and the subsequent solution of the EWSB conditions as two consecutive changes of variables, so that J CMSSM may be written as a product of the Jacobian determinant associated with each, i.e., where are the values of the high-scale parameters that result for M Z = M exp Z , for the given value of tan β and all other model parameters. The form of the effective prior is identical to that in the SM in Eq. (15). It is worth noting that we do not develop any nontrivial, or misleading, behavior in the effective prior according to our choice of EW parameters, {M 2 Z , tan β}. This choice is not unique; for instance, one could choose the VEVs {v 1 , v 2 } instead. In this case, the effective prior would differ only by the additional non-singular Jacobian factor 4 where m Z is the tree-level Z-boson mass.
In the NMSSM, the imposed Z 3 symmetry forbids an explicit superpotential bilinear term forĤ d andĤ u , along with the corresponding soft-breaking parameter. We instead make the trade where the effective µ-parameter, µ eff. , is defined in Eq. (23), to express the effective µparameter in terms of M 2 Z and tan β. Since in this approach we retain λ as a free input parameter, this has the effect of determining the singlet VEV, S ≡ s/ √ 2, as a function of M 2 Z and tan β. Second, we trade s for m 2 S via the EWSB condition, where we make the usual definition v 2 = v 2 1 + v 2 2 , and have absorbed the loop-corrections from the Coleman-Weinberg potential intom 2 S . Finally, we make tan β an input parameter by trading κ for tan β via the second MSSM-like EWSB condition, where we define an effective soft-breaking bilinear Thus, ultimately, in our analysis m 2 S plays the role of µ 2 via an effective µ-term and κ plays the role of Bµ via an effective Bµ-term. The final effective prior is defined in the same way as in the CMSSM, i.e., it has the form The effective priors automatically disfavor fine-tuned regions of parameter space. Indeed, from their explicit forms in App. A, we see that the effective priors favor RG evolution that results in weak-scale parameters similar in magnitude to the weak scale. The region of parameter space in which this occurs is known as the "focus point" [140][141][142]. In these regions of parameter space, the RG evolution of the soft masses is such that, at the SUSY scale, m 2 Hu ∼ M 2 Z almost independently of the initial value of m 2 Hu (M GUT ) = m 2 0 . In the CMSSM, the dependence of m 2 Hu (m SUSY ) on the universal soft-breaking masses can be quantified using semi-analytic solutions to the RGEs, which take the form for m 2 = m 2 Hu , m 2 H d and where the coefficients c i j (Q) depend only on the gauge and Yukawa couplings. In the semi-constrained NMSSM the semi-analytic solutions instead take the form for

Comparison to other fine-tuning measures
As discussed in Sec. 2 where ∆ p is defined as in Eq. (8) and the notation ∆ p Q indicates that the parameters to differentiate with respect to are those defined at the scale Q; here, this is taken to be seek to quantify the size of any cancellations that must take place to reproduce the observed EW scale. An example of this class of measures is the so-called electroweak finetuning [143], which is defined as The The By comparing with Eq. (22) we see that the evidence for a supersymmetric model (written in terms of log M Z ) may be crudely written as The parametric behavior for m 2 SUSY M 2 Z is identical. Thus, in this case, there is reason to expect that fine-tuning measures and Bayes factors may result in similar conclusions.

Numerical methods
We computed statistical quantities -posterior densities and evidences -with MultiNest v3.10 [178][179][180] and plotted them with SuperPlot [181]. For the evidence integration, we modified the convergence criteria by defining the tolerance using the average likelihood of the live points, instead of the maximum. We performed two scans for each model: one with only M Z , and one with M Z and m h in the likelihood. We scanned 10 million and 100 million points for each scan of the CMSSM and NMSSM, respectively. To calculate the likelihoods and effective priors in each model, we computed the mass spectrum and Jacobian factors for each parameter point using a modified version of SOFTSUSY-3.6.2. As described in detail in App. A, the required Jacobian is written as the product of the Jacobian determinants for the change of variables from the GUT scale parameters to the low-energy Lagrangian parameters, and for the transformation from these parameters to the derived parameters  Requiring that m h 125 GeV increases the fine-tuning measures in the CMSSM ( Fig. 9) and NMSSM (Fig. 10), and further structure is revealed. We find diagonal strips of low fine-tuning for Jacobian-based measures at about m 0 ∼ 10 TeV and m 1/2 ∼ 1 TeV.
The GUT scale Jacobian measure furthermore exhibits a vertical strip of low fine-tuning at m 0 ∼ 10 TeV. This indicates that the Jacobian based measure has a much sharper preference for the focus point region than ∆ BG . Note that this is the case even though we have not included the top mass or top Yukawa coupling in the set of parameters for which we take logarthmic derivatives for ∆ BG . The Jacobian based measures in the NMSSM are also visibly smaller than those in the CMSSM.
We summarize the one-dimensional posterior for the dimensionful parameters in Fig. 11.
We see that in the CMSSM and NMSSM with only M Z in the likelihood, the posterior favors m SUSY 1 TeV. Once we consider M Z and m h , however, we require m SUSY 4 TeV and TeV-scale soft-breaking parameters. It is, therefore, not surprising to see no signature of supersymmetric particles until the current data set of the LHC in the regard that our Higgs mass is 125 GeV.
As a byproduct of our investigations, we calculated the Bayes factor between the semiconstrained NMSSM and CMSSM, though with appreciable uncertainty as in Ref. [82].
The Bayes factor measures the change in relative plausibility of two models in light of data. for different data calculated in Ref. [82]. The lower estimates may be more accurate as they were found by importance sampling; however, since there were significant differences between estimates from importance sampling and ordinary summation, we present our results with caution, and do not make a definitive model selection statement. To improve the accuracy of our evidence estimates requires more computational resources, or, possibly, sampling techniques which are more specialised for exploring the very strong degeneracies that can be induced by the naturalness priors in scans constrained only by measurements of M Z and m h .
The minimum fine-tuning measures found in our scan are shown in Table 3

A CMSSM and NMSSM Jacobians
In this appendix we present analytic expressions for the Jacobians that appear in the effective priors as discussed in Sec. 4.2.1.

A.1 CMSSM Jacobian
In the CMSSM, the relevant Jacobian arises from making the change of variables By performing this trade in two steps, namely, by first exchanging the high-scale values of the Lagrangian parameters for their values at the EW scale, and subsequently trading these for the parameters M 2 Z and tan β, the full Jacobian factorizes, where the Jacobian determinants on the right-hand side arise from this series of variable changes, i.e., {|µ 0 |, B 0 µ 0 } → {|µ|, Bµ} → {M 2 Z , tan β}. The various prior probability density functions are related by The elements of the two Jacobian matrices that are required read The elements of J CMSSM M GUT can immediately be read from these expressions. The dimensionless coefficients c i j depend only on the running of the gauge and Yukawa couplings; however, in the absence of exact analytic solutions to the two-loop RGEs they must be evaluated by numerical integration of the RGEs.
As noted in Sec. 4.2.1, the subsequent change of variables from {µ, Bµ} to {M 2 Z , tan β} is done by solving the EWSB conditions to write the former pair as functions of M 2 Z and tan β. In the MSSM, the requirement that the neutral scalar Higgs fields acquire VEVs of the form given in Eq. (37) leads to the two EWSB conditions, where for p = µ, Bµ. The coefficients appearing on the left-hand side of Eq. (69) are given by (assuming µ to be real) while the derivatives of the EWSB conditions with respect to the Lagrangian parameters read b (µ) The elements of the Jacobian matrix J CMSSM m SUSY are then related to the solution of Eq. (69) through for each of p = µ, Bµ. In arriving at Eq. (75), we approximate the solution of for M Z by evaluating the Z-boson self-energy at the external momentum p 2 = m 2 Z = g 2 (v 2 1 + v 2 2 )/4. Although it is possible to evaluate the above derivatives completely analytically, the resulting expressions are quite long and unwieldy. As described in Sec. 4.4, for the results presented here we have instead computed these derivatives numerically using

A.2 NMSSM Jacobian
The calculation of the Jacobian in the NMSSM proceeds in a similar fashion to the approach used in the CMSSM. As mentioned in Sec The elements in the last column of this matrix are easily seen to be given by where it should be noted that, since λ remains an input parameter, it is taken to be the case that M 2 Z and tan β are independent of λ. The determinant of this matrix, J NMSSM m SUSY ≡ | det J NMSSM m SUSY |, when combined with J NMSSM M GUT , yields the full Jacobian appearing in the effective priors in the NMSSM, The derivatives of M 2 Z and tan β can once again be expressed in terms of derivatives of the Higgs and singlet VEVs, v 1 , v 2 and s. Eq. (76) continues to hold in the NMSSM, with p = κ, m 2 S , while the dependence on the additional singlet VEV leads to an expression of the form for the required derivatives of M 2 Z . Analytic formulas for the derivatives of the VEVs are most conveniently obtained from the three EWSB conditions, where we take there to be no additional sources of CP-violation, and write the one-and two-loop corrections to the effective potential as ∂s .
The quantities ∂v 1 /∂p, ∂v 2 /∂p and ∂s/∂p are then once again obtained by solving a linear system of the form The elements of the 3 × 3 matrix X are easily found to be given by Similarly, the derivatives of the EWSB conditions with respect to κ and m 2 S appearing on the right-hand side of Eq. (85) are simply 6 y (κ)

B EW Fine-Tuning Contributions
The tuning measure ∆ EW defined in Eq. (54) in Sec. 4.3 quantifies the competition between the terms contributing to the EWSB condition determining m Z . The C i are given by the absolute values of the terms entering into the prediction of m Z in the model, i.e., the terms on the right-hand side of Eq. (38) or Eq. (46), excluding the self-energy correction. In the MSSM we consider the coefficients Hu tan 2 β tan 2 β − 1 , , C t 2 = t 2 tan 2 β v 2 (tan 2 β − 1) .
Here the quantities t 1 and t 2 are the Coleman-Weinberg contributions defined in Eq. (68) and previously absorbed intom 2 H u,d in Sec. 4.3. The coefficients considered in the NMSSM are similar, with the only differences being that µ → µ eff. and t 1 , t 2 are instead given by Eq. (84).
Separating the Coleman-Weinberg pieces allows to see how the loop corrections in the Higgs potential cancel the tree level parameters delicately. The ideal case would be 6 Note that we allow a κ to vary independently of κ.
|C i | ∼ O(m 2 Z ), while reality pushes them to much larger values. In the case of large tan β, the prediction for m Z is well approximated by so that C µ , C Hu and C t 2 play the most important roles in the determining ∆ EW .