Testing SUSY at the LHC: Electroweak and Dark matter fine tuning at two-loop order

In the framework of the Constrained Minimal Supersymmetric Standard Model (CMSSM) we evaluate the electroweak fine tuning measure that provides a quantitative test of supersymmetry as a solution to the hierarchy problem. Taking account of current experimental constraints we compute the fine tuning at two-loop order and determine the limits on the CMSSM parameter space and the measurements at the LHC most relevant in covering it. Without imposing the LEPII bound on the Higgs mass, it is shown that the fine tuning computed at two-loop has a minimum $\Delta=8.8$ corresponding to a Higgs mass $m_h=114\pm 2$ GeV. Adding the constraint that the SUSY dark matter relic density should be within present bounds we find $\Delta=15$ corresponding to $m_h=114.7\pm 2$ GeV and this rises to $\Delta=17.8$ ($m_h=115.9\pm 2$ GeV) for SUSY dark matter abundance within 3$\sigma$ of the WMAP constraint. We extend the analysis to include the contribution of dark matter fine tuning. In this case the overall fine tuning and Higgs mass are only marginally larger for the case SUSY dark matter is subdominant and rises to $\Delta=28.7$ ($m_h=116.98\pm 2$ GeV) for the case of SUSY dark matter saturates the WMAP bound. For a Higgs mass above these values, fine tuning rises exponentially fast. The CMSSM spectrum that corresponds to minimal fine tuning is computed and provides a benchmark for future searches. It is characterised by heavy squarks and sleptons and light neutralinos, charginos and gluinos.


Introduction
With the Large Hadron Collider up and running, the search for TeV-scale SUSY is now significantly closer. Many models of physics beyond the Standard Model (SM), in particular its minimal supersymmetric version (MSSM), will be directly tested. To do so, one has to quantify the viable range of parameters entering in these models, which impact in particular on the scale of its low-energy supersymmetric spectrum. Previous theoretical and experimental constraints such as those from LEP have already tested a considerable amount of the MSSM parameter space and identified bounds on it. To study these bounds further it is instructive to investigate from a quantitative perspective the impact of the, so far, negative searches for low-energy supersymmetry. A quantitative measure of this impact is the fine-tuning measure ∆ [1,2], that quantifies the degree of cancellation between unrelated parameters that is needed to fix the electroweak scale and can be extended to include the fine tuning needed to obtain an acceptable dark matter abundance. In this paper we shall perform such an investigation, computing ∆ at two-loop leading log order in the Constrained MSSM (CMSSM). For previous studies of the fine-tuning problem in a similar context see [1]- [18].
The electroweak fine-tuning measure, ∆, provides a measure of the probability of unnatural cancellations of soft masses in the expression of the electroweak scale v 2 ∼ − i m 2 sof t,i /λ, (λ is Higgs quartic coupling) after including quantum corrections. So ∆ measures the stability of the MSSM electroweak scale at the quantum level, with all available experimental and theoretical constraints imposed. These include the LEP mass bounds on supersymmetry masses, charge/colour breaking constraints, the dark matter relic density constraint, and the measurement of b → sγ and the anomalous magnetic moment of the muon. In what follows we identify the constraints with the largest impact on ∆. We also extend the analysis to include the fine tuning, ∆ Ω , needed to satisfy the constraints on the SUSY dark matter abundance.
The method can be readily extended to other models that claim to solve the hierarchy problem.
For the case that the fine tuning is reduced by new states with mass well above the TeV range, one may extend the analysis using the effective Lagrangian in which the very heavy states have been integrated out. For fine tuning and related issues in such scenarios see for example [17,20,21,22,23], where the MSSM Higgs mass can be increased nearer the LEP bounds by classical effects due to new physics beyond few TeV, which ultimately reduces the fine tuning.
The fine-tuning problem in the MSSM is important not only for supersymmetry searches, but also for the Higgs physics since it is intrinsically related to the value of the lightest Higgs mass m h , currently restricted by the LEPII lower bound of 114.4 GeV [19]. As a result searches for m h are relevant to supersymmetry phenomenology. In particular, the need to increase the SUSY prediction for the Higgs mass by radiative corrections above the LEPII bound means that the electroweak fine tuning measure rises exponentially with the Higgs mass. If ∆ becomes too large one can conclude that SUSY fails to provide a solution to the hierarchy problem. The parameter configuration (consistent with the non-observation of SUSY states) that minimises ∆ gives an indication of its most likely values. We identify this configuration and investigate its phenomenological implications. Also for a given upper value of ∆ one can extract the corresponding range of parameter space of the CMSSM and of the superpartners masses.
In Section 2 we present the calculation of the electroweak fine tuning measure to two loop order. We are not aware of a similar analysis of the fine tuning problem at this level of precision (two-loop leading log), which is responsible for a ∆ smaller than that usually found in the literature. In Section 3 we discuss the dependence of the fine tuning on tanβ and the impact on the fine tuning coming from imposing the bounds on the SUSY spectrum and the limits on b → sγ. Using the dependence of the fine tuning measure on the parameters of the CMSSM we then determine their allowed range consistent with a given value of ∆. This provides a quantitative measure of the remaining parameter space range that remains to be tested. Next we determine the dark matter abundance as a function of the fine tuning measure showing that low fine tuning is consistent with acceptable SUSY dark matter abundence. We conclude this Section by considering the implications for the Higgs mass following from requiring low fine tuning. We show that, without imposing the LEPII bound on the Higgs mass, ∆ has a minimum for a region of m h near the LEPII bound. In Section 4 we extend the fine tuning analysis to include the fine tuning needed either to satisfy the dark matter bound or to saturate the bound with SUSY dark matter. Finally in Section 5 we discuss the predictions for the superpartner mass spectrum from the fine tuning bound. We also determine the most likely spectrum that minimises fine tuning and discuss the relative importance of various LHC measurements in the test of the CMSSM. Section 6 presents a summary and our conclusions.
2 Computing electroweak scale fine-tuning ∆ at two-loop level In this section we present the strategy for evaluating the MSSM fine-tuning at tree, one-loop and two-loop (leading log) level; particular attention is paid to clarifying the impact on fine tuning of the quantum corrections to couplings and masses. With the standard two-Higgs doublet notation, the scalar potential is The couplings λ j and the soft masses receive one-and two-loop corrections that for the MSSM are found in [24,25]. We shall use these results to evaluate the overall amount of fine-tuning of the electroweak scale. Technical details of the procedure can be found in the Appendix.
To evaluate the fine-tuning, it is convenient to introduce the notation with the assumption that, in the MSSM, at the UV scale m 2 1 = m 2 2 = m 2 0 + µ 2 0 while m 2 3 = B 0 µ 0 . The couplings λ j are assumed to be real and λ 345 = λ 3 + λ 4 + λ 5 . In 1 the MSSM, at the tree level they are where g 1,2 are the U (1) and SU (2) gauge couplings respectively.
The fine-tuning amount wrt to a set of parameters {p} of the MSSM is then [1,2] ∆ ≡ max ∆ p p={µ 2 where all p are input parameters at the UV scale of CMSSM, in the standard notation 2 The which fix v and β as functions of the above MSSM bare parameters. Taking into account that m 2 = m 2 (p, β), λ = λ(p, β) we can find ∂β/∂p from the second minimum condition for V .
Using this, one finds 3 [11], see also [17]: 1 When using the Yukawa couplings at the low energy scale as an input, expressed in terms of the Higgs vev and quark masses, then m and λ pick up additional, implicit dependence on v = h and tan β. Neglecting this dependence when evaluating ∆p does not bring significant changes to the results for final ∆.
2 One could also include in the set of parameters p, the top Yukawa coupling or the strong coupling α3. For such parameters which are measured, one can use the modified fine tuning definition [26] and with this, it turns out that in the cases we consider their associated fine tuning never dominates. 3 Later the min condition (fixing β) is used to replace B0 by tan β as an independent parameter.
This result takes into account the dependence of β on the MSSM set of parameters (p). This formula also takes into account the loop-effects to the quartic couplings as well as the tan β dependence of the radiative corrections on the parameter "p". As we shall see later, such effects tend to reduce fine-tuning, in many cases rather significantly 4 .
For comparison with similar studies of fine-tuning, a comment is in place here. After some algebra, one can show that the general formula of ∆ (eqs.(4), (6)) reduces, in the limit of removing the loop corrections to quartic couplings λ i , i = 1, 2..., to the more familiar "master formula" [28] (see also [4,5]) This formula is sometimes used as the starting point in works that evaluate electroweak scale fine-tuning, by using in it the loop corrected soft masses. However, for accurate estimates, it is necessary to take full account of radiative corrections, as done by eqs.(4), (6). Indeed, the loop corrections to the quartic couplings significantly reduce the amount of fine-tuning, in some cases by a factor as large as 2, and these corrections are not accounted for by eq.(7).
This can be seen by considering the one-loop correction δ to λ 2 → λ 2 (1 + δ), due to stop/top Yukawa couplings. Usually δ = O(1). Including it one finds (for details see eq. (26) in [17]): This is showing that one-loop corrections to the quartic coupling reduce the amount of finetuning significantly. In fact it is the smallness of the quartic Higgs coupling (fixed in the MSSM by gauge interactions) that is at the origin of substantial tree-level fine tuning. That this is so can be seen from the relation v 2 ∼ −m 2 i /λ where v is of O(100) GeV, m i ∼ O(T eV ) while at the same time λ < 1, which makes it difficult to separate the EW and SUSY breaking scales (for a discussion see [7]). Loop corrections increase the quartic couplings and in most cases reduce the overall amount of fine tuning.
For a general two-Higgs doublet model formulae (4), (6) can be expressed in terms of only derivatives of couplings and of masses wrt to the corresponding parameter, see Appendix in [17] (also Appendix A.1). For the CMSSM we use this result, in which we consider the full two-loop (leading log) corrections to the quartic couplings and masses. This defines unambiguously our procedure for evaluating the EW fine-tuning in CMSSM.

Electroweak fine tuning and its effects on the Higgs mass
In the following, the numerical results we present for ∆ include two-loop corrections with: • radiative electroweak breaking (EWSB), • non-tachyonic SUSY particle masses (avoiding colour and charge breaking (CCB) vacua).
• consistency of m h with the LEPII bound (114.4 GeV) and/or consistency with thermal relic density constraint, only if stated explicitly.

Constraint
Reference SUSY particle masses Routine in MicrOmegas 2.2, "MSSM/masslim.c" δa µ < 366 × 10 −11 PDG (sys. and stat. 1σ errors added linearly) 3.20 < 10 4 Br(b → sγ) < 3.84 PDG (sys. and stat. 1σ errors added linearly) Br(b → µµ) < 1.8 × 10 −8 Particle Data Group −0.0007 < δρ < 0.0012 Particle Data Group Using these constraints we evaluated ∆ numerically. The LEPII bound on the mass of the   [34]. This is the "worst" case scenario, when such deviations combine such as to give the largest variation of ∆. An increase of α 3 (m Z ) or reduction of m t (m Z ) by 1σ have similar effects, which can be also understood from the relation between the mass of top evaluated at m Z and at m t . Keeping either α 3 or m t fixed to its central value and varying the other within 1σ brings a curve situated half-way between the continuous line and the corresponding dashed or dotted line. The LEPII bound of 114.4 GeV is indicated by a vertical line. Note the steep (≈ exponential) increase of ∆ on both sides of its minimum value situated near the LEPII bound. stated otherwise, the LEPII bound on m h is not imposed. The relic density constraint is imposed only after all the constraints other than the LEPII bound on m h are satisfied and, when done, this is stated explicitly.
Before proceeding to present our numerical results obtained with the above constraints, let us mention the details of the procedure followed. First the fine-tuning is evaluated at twoloop order, including the dominant third generation supersymmetric threshold effects to the scalar potential. The scan is done over all parameter space using a slightly simplified two-loop calculation performed by a Mathematica code, based on the formulae in the Appendix, and optimised to run quickly. For the points in phase space that have the smallest fine-tuning (say with ∆ < 1000), the analysis is re-done using (the slower) SOFTSUSY 3.0.10 [29] that includes all the two-loop radiative effects mentioned. This two-step procedure is extremely important, since otherwise the CPU run time using SOFTSUSY alone would be about 6 years (when run on 30 parallel processors at 3GHz each), which prevented previous investigations at this precision level. Our two-loop analysis is also important because there is a significant reaching similar values near 120 GeV). The transition between the two regions is happening at about 114.5 GeV. Note that in this plot the LEPII bound is not imposed at any time. Although ∆ h 2 t (purple) is presented above for illustration, this contribution is always sub-dominant when assuming the modified definition of fine-tuning [26], appropriate for measured parameters (as we do in the text). difference between one-loop and two-loop values for overall ∆, and was not performed in the past. QCD effects can compete at two-loop with Yukawa couplings effects, can dominate them and also displace the minimum of the fine tuning from its one-loop value. Regarding the Higgs mass, its value computed with SOFTSUSY agrees with that found using SuSpect [30] within 0.1 GeV, but can differ by ±2 GeV [31] from the value found using FeynHiggs [32].
We use the SOFTSUSY Higgs mass for all figures in the paper. Given the small discrepancy with FeynHiggs, coming from higher order terms in the perturbative expansion, the LEPII bound should be interpreted as m h > 114.4 ± 2 GeV. In the following analysis the results are always quoted with respect to the central value of m h .
Turning now to the numerical results, Figure 1  corresponds to a larger ∆ for a same, fixed value of m h . In this sense one could even say that the minimal value of ∆ is situated at the transition region between dominant effects, Yukawa versus QCD interactions. Away from the minimum of ∆, fine tuning increases dramatically, roughly exponentially. This is because, as discussed above, ∆ depends exponentially on m h .
In conclusion, the fine tuning at two-loop with all the latest constraints is minimised for with the theoretical uncertainty of ±2 GeV, explained earlier.
Note that our analysis also investigated the contribution to ∆ coming from the uncertainty in measured "parameters" such as top Yukawa and strong coupling. Using the modified 6 definition [26], appropriate for measured parameters, we find their fine-tuning is sub-dominant.
Finally, let us also mention that the reduction of fine-tuning that we have seen is mostly due to (two-) loop corrections considered, particularly to quartic couplings and is actually very significant, given the conservative scenario considered here assuming universal gaugino mass structure of the CMSSM; relaxing this condition could reduce [6] ∆ further.

Constraints on ∆ from fixing tan β
It is interesting to examine the fine-tuning for fixed values of some of the parameters present, to see the individual impact of such constraints on ∆. Here we do this for a fixed value of tan β. This is shown 7 in Figure 3 for increasing vales of tan β, with tan β = 2 (blue), tan β = 3 (red), tan β = 10 (green). Increasing tan β shifts the curves of ∆ towards larger m h and, to a limited extent, to lower fine-tuning, reached for medium tan β ∼ 10, also for larger tan β ∼ 40 (see later, Fig.7 (a)). As may be seen from the figure, the two-loop expressions for soft masses 5 Larger m h requires larger m 2 t ∼ m 2 0 , and a larger m0, above the focus point region, increases ∆ m 2 0 .   Table 1 are included. The upper and lower lines associated with the coloured regions are the 1-loop without thresholds for λ and soft masses and "full" 1-loop results respectively (similarly for the grey region, for all tan β). The minimum 2-loop fine tuning is found between these two cases. The solid lines refer to the scan 2 ≤ tan β ≤ 55. The black points give the positions of minimum ∆ for fixed tan β from 2 to 4 inclusive in steps of 0.5. and couplings bring values of ∆ which are situated between the higher "tree-level" curve (i.e. tree-level for λ i , one-loop for soft masses without field dependent threshold effects) and the lower one-loop case (one-loop for both λ i and soft masses). This is consistent with what one would expect from a convergent, perturbative, loop expansion. The roughly exponential behaviour of ∆ at large m h for fixed tan β can bring a significant variation of ∆.

Constraints on ∆ from the SUSY spectrum
Here we examine the impact on ∆ due to constraints related to the supersymmetric spectrum.
The key features of the impact of this spectrum on ∆ can be seen from the limits on the chargino mass considered in Figure 4. Currently, the (lightest) chargino mass bound is the most important, followed by that of the neutralino. It also turns out that the gluino mass limit is not very constraining. These results follow recent experimental data, since using the 1998 data it was the neutralino mass bound that was more constraining for ∆.
In Figure 4 Table 1 and to the SUSY spectrum limits respectively shows that the latter are the main constraints at the moment for ∆ at low tan β. The graph presents ∆ and m h computed using quartic couplings and mass expressions evaluated at two-loop. Figure 5 gives the impact of the b → sγ constraint on ∆. The lower limit of the b → sγ constraint for a given coloured area (fixed tan β) restricts the right hand edges of the plot, while the upper limit restricts its left hand side. These curves also depend on the mass limitsthese constraints are not fully independent. For the experimentally allowed area of m h > 114.4

Constraints on ∆ from b → sγ
GeV, the impact of the constraint b → sγ is rather small; in this case its effect is overlapping that of the SUSY mass limits, as can be seen from the rhs of the plots for individual tan β plots. The combination of the SUSY mass limits and b → sγ constraint currently dominate the restriction on how small the fine tuning could be, see also Figure 6. In this last figure one can easily see, at two-loop, the impact on ∆ of removing the b → sγ constraint. For a related analysis of b → sγ see recent [27]. The results are at 2-loop with the lighter shading connecting the case of only applying the SUSY spectrum constraints (lower line) to that with also the b → sγ constraint listed in Table 1 (upper  line). This upper edge of this shading is indistinguishable from the solid line which includes all constraints in Table 1. The darker shading extends up to the minimum fine tuning limits for the case, 3.52 < 10 4 Br(b → sγ) < 3.77, with the other constraints as given in Table 1. The results for the scan 2 ≤ tan β ≤ 55 are also shown, between the two continuous and almost parallel lower curves.
The other constraints listed in Table 1 do eliminate further mSUGRA points, but have a negligible effect on the fine tuning limits. With the current mass limits, a change in the δa µ constraint by factors of 2 or more does not affect these results significantly.

Constraints on ∆ and the CMSSM parameters
The fine tuning measure can be easily applied to establish the remaining allowed range for the MSSM SUSY parameters. In Figure 7 we plotted the dependence of the total fine tuning wrt various parameters. To understand some aspects of the dependence of the electroweak scale fine tuning on the MSSM parameters, let us use, for the sake of discussion, one of the two minimum conditions which, when ignoring quantum corrections to quartic couplings, simplifies to: Using 2-loop RGE solutions at tan β = 10 (for details see the Appendix), one has It is the large cancellation between the µ 2 0 and m 2 1/2 terms that is often responsible for the large fine tuning (note however that this argument ignores the impact of quantum corrections to quartic couplings, known to reduce the fine-tuning). This leads to the approximate relation   |A 0 | requires larger cancellations with µ to set the electroweak scale. However, once the Higgs mass constraint is applied, A 0 is driven negative for small tan β in order to maximise the stop mixing. The related increase in the minimum fine tuning from being in this region of parameter space then follows. This is important for small tan β where the tree level Higgs mass is smallest. The sign structure of the UV parameter coefficients in m 2 leads to a preference in a small, positive A 0 .
As mentioned earlier, the fine tuning measure ∆ can be used to establish the remaining parameter space of the CMSSM compatible with a solution to the hierarchy problem. Assuming that ∆ = 100 is the upper limit beyond which we consider that SUSY failed to solve the hierarchy problem, we obtain the following bounds: m h < 121 GeV 5.5 < tan β < 55 µ < 680 GeV 120 GeV < m 1/2 < 720 GeV These values can be easily re-calculated for a different value of ∆.

Constraints on ∆ from the relic density
It is interesting to see the impact on ∆ and on the CMSSM parameters from the presence of the dark matter relic density constraint (examined using micrOMEGAs2.2 [35]) and the LEPII constraint on m h . These are rather strong constraints, particularly in the "restrictive" context of CMSSM that we study, with universal gaugino mass. The results are presented in and m 0 , but has a rather small impact on A 0 .

Prediction for m h from minimising ∆ and saturating the relic density
The relic density constraint can be combined with that of minimal electroweak fine-tuning ∆ to make an interesting prediction for m h . Figure 9 shows the impact of non-baryonic relic density constraint on ∆ presented in Figure 1. Obviously, not all initial points in ∆ satisfy this constraint, and this is shown in Figure 9 by the red and blue points which do not fill the whole area above the continuous line of minimal ∆. As expected, the additional dark    Table 1 are sufficient to also satisfy the thermal relic density; note that the red points in the left (right) plots in Figure 9 satisfy the relic density within 1σ and 3σ WMAP bounds [33], respectively. The results are obtained as usual, using two-loop values for the quartic couplings and soft masses and corresponding threshold effects. (with the SOFTSUSY and micrOMEGAs codes).
It is important to notice that, without imposing the LEPII bound, at the two-loop level, the smallest fine tuning ∆ consistent with the relic density WMAP bounds [33] predicts a mass for the lightest Higgs as follows:   To conclude, minimising the fine-tuning together with the constraints from precision electroweak data, the bounds on SUSY masses and the requirement of the observed dark matter abundance lead to a prediction for m h , without imposing the LEPII bound. This is an interesting result, and represents our prediction for the CMSSM lightest Higgs mass based on assuming ∆ as a quantitative test of SUSY as a solution to the hierarchy problem.

Dark matter fine tuning and its effect on the Higgs mass
The dark matter abundance can be very sensitive to the choice of parameters and can introduce a new fine tuning to the model. To quantify this it is interesting to consider the dark matter fine tuning ∆ Ω wrt the CMSSM parameters, and to determine its impact on the overall fine tuning (for earlier studies see [15,16] and references therein). Its definition is similar to that of ∆: In Figure 10 we evaluated ∆ Ω at two-loop level and presented as a function of the Higgs mass, without imposing any restriction on the latter. It turns out that ∆ Ω can have acceptable values even for m h close to 120 GeV. In Figure 11 the total fine-tuning, defined as max{∆, ∆ Ω } is presented as a function of m h . Its value is only slightly larger than that found earlier for ∆ alone with WMAP saturated dark matter abundance (in red in the plots). From Figure 11 we predict, from minimising max{∆, ∆ Ω } and from consistency with the 3σ WMAP bound: The last predicted value of m h is only marginally above that predicted in (14), based on minimised electroweak fine-tuning and right dark matter abundance.

Predictions for the superpartners from fine tuning limits
The results so far demonstrate that electroweak fine-tuning has a strong sensitivity to parameters such as µ, m 1/2 , with a preference for lower values. Regarding the m 0 dependence, ∆ has a rather flat dependence when we are in the focus point region. The states that are dominantly controlled by the µ, m 1/2 parameters are then the most important in determining the naturalness of the proposed theory. These include the neutralinos, charginos and the gluino states. Further, setting an upper bound on electroweak ∆ gives a bound on the spectrum. If any of these states have masses in excess of those given in Table 2, it will require less than 1% tuning (∆ > 100) for the MSSM.  These upper mass limits scale approximately as ∆ min , so they may be adapted depending on how much fine tuning the reader is willing to accept. Overall low fine tuning prefers a Finally we return to the intriguing fact that minimum electroweak fine tuning plus correct dark matter abundance corresponds to a Higgs mass just above the LEPII bound 8 . As we noted above this point is fixed by the current bounds on the SUSY spectrum and not by the current Higgs mass bound which is not included when doing the scans giving Figs 1, 2, 9, 11.
One may interpret the SUSY parameters corresponding to this point as being the most likely given our present knowledge and so it is of interest to compute the SUSY spectrum for this parameter choice as a benchmark for future searches. This is presented in Table 3 where it may be seen that it is somewhat non-standard with very heavy squarks and sleptons and lighter neutralinos, charginos and gluinos. This has similarities to the SPS2 scenario [36].

Predictions for SUSY searches at the LHC
It is clear that there is still a wide range of parameters that needs to be explored when testing the CMSSM. Will the LHC be able to cover the whole range? To answer this note that, for a fine tuning measure ∆ < 100, one must be able to exclude the upper limits of the mass parameters appearing in Table 2. Of course the state that affects fine tuning most is the Higgs scalar and one may see from Figure 1 that establishing the bound m h > 120 GeV will imply that ∆ > 100. However the least fine tuned region corresponds to the lightest Higgs consistent with the LEPII bound and this is the region where the LHC searches rely on the h → γγ channel which has a small cross section and will require some 30 f b −1 at √ s = 14 TeV to explore. Given this it is of interest to consider to what extent the direct SUSY searches will probe the low fine tuned regions. Following the discussion in the previous section, the most significant processes at the LHC will be those looking for gluinos, winos and neutralinos.
Studies of SUSY at the LHC [37] have shown that the LHC experiments have a sensitivity to gluinos of mass 1.9 TeV for √ s = 10 TeV, 2.4 TeV for √ s = 14 TeV and luminosity 10f b −1 .
Of relevance to the first LHC run the limit is 600 GeV for √ s = 10 TeV and luminosity 100 pb −1 . These correspond to probing up to ∆ = 120, 180 and 14 respectively. As we have discussed charginos and neutralinos can be quite light, but their signal events are difficult for LHC to extract from the background, owing in part to a decreasing M W − M Z mass gap as |µ| decreases [38,39]. An Atlas study [40] of the trilepton signal from chargino-neutralino production found that 30f b −1 luminosity at 14 TeV is needed for a 3σ discovery significance for M 2 < 300 GeV and µ < 250 GeV [41].

Summary and Conclusions
Supersymmetry was introduced to solve the hierarchy problem and to avoid the large finetuning in the SM Higgs sector associated with the Planck or GUT scale when quantum corrections are included. While this hierarchy problem is solved by TeV-scale supersymmetry, the non-observation, so far, of SUSY states means that the MSSM has acquired some residual amount of fine-tuning related to unnatural cancellations in the SUSY breaking sector. The goal of this paper was to analyse in detail the level of fine tuning in the CMSSM.
The fine tuning measure ∆ provides a quantitative test of SUSY as a solution to the "little" hierarchy problem and measures the "tension" required to satisfy the scalar potential minimum condition v 2 ∼ −m 2 susy /λ, for a combination of soft masses m 2 susy ∼ TeV, with an effective quartic coupling λ < 1 and v ∼ O(100) GeV. Although the exact upper limit on the fine tuning ∆ beyond which a theory fails to solve the hierarchy problem is debatable, it is preferable, for a given model, to have a parameter space configuration corresponding to the lowest value of ∆. We evaluated ∆ at two-loop order and also paid particular attention to threshold corrections and to the tan β radiative dependence on the parameters. Such effects on fine-tuning were not fully considered in the past and turned out to reduce fine tuning significantly.
Our determination of the fine-tuning measure for the CMSSM included the theoretical constraints (radiative EWSB, avoiding charge and colour breaking vacua), and also the experimental constraints (bounds on superpartner masses, electroweak precision data, b → s γ, b → µ µ and muon anomalous magnetic moment, dark matter abundance). As far as we are aware, our study is the first two-loop analysis of the fine tuning problem in the CMSSM, largely based on SOFTSUSY and micrOMEGAs, SuSpect and our own Mathematica code.
The latter was very important since it reduced to a feasible level the CPU run time necessary to scan the full parameter space.
Not including the dark matter constraint, we found the minimum value is given by ∆ = 8.8.
Remarkably, even without imposing the LEP bound on the Higgs mass, the condition fine tuning should be a minimum predicts m h = 114 ± 2 GeV. Adding the constraint on the dark matter relic density, one finds ∆ = 15 corresponding to m h = 114.7 ± 2 GeV and this rises to ∆ = 17.8 (m h = 115.9 ± 2 GeV) for SUSY dark matter abundance within 3σ of the WMAP constraint. The results are encouraging for the search for SUSY because we considered the "conservative" case of CMSSM, and it is well-known that relaxing gaugino universality can reduce ∆ further [6,9].
The spectrum corresponding to the minimum value of the fine tuning shows similarities to the SPS2 scenario with light neutralinos, charginos and gluinos (corresponding to light µ, m 1/2 ) and heavy squarks and sleptons corresponding to large m 0 , near the focus point limiting value [13,14]. It provides the "best" estimate for the SUSY spectrum given the present experimental bounds.
Increasing m h above the minimum fine tuned value causes ∆ to increases exponentially fast and one leaves the focus point region at the edge of which this minimal value is reached; one obtains ∆ = 100 (1000) for a scalar mass m h = 121 (126) GeV, respectively. Ultimately the question whether the SUSY solution to the hierarchy problem has been experimentally tested relies on what value of fine tuning represents the limit of acceptability. Given a value one can determine the range of parameter space that is still acceptable. For the case that the fine tuning measure should satisfy ∆ < 100, we determined the corresponding superpartners masses and CMSSM parameters values, that can be relevant for SUSY searches.

A.1 Higgs mass and EW fine tuning
We provide technical results used in the text to evaluate ∆. The potential used in (1) Using the notation or, equivalently, One finds for the CP odd Higgs mass: with g 2 = g 2 1 + g 2 2 . In the notation of [42], the CP even Higgs masses are where λ 345j = λ 345 − λ j , λ is that of (2), and, (s β , c β ) = (sin β, cos β). In the limit of large m A , m 2 h reaches an upper limit of 2λv 2 (which tends to λ 2 v 2 for large tan β). At tree level, λ = (g 2 /8) cos 2 2β, and the individual λ j are: The general formula for fine-tuning ∆ p wrt a parameter p in a two-higgs doublet model that we are using can be found in the Appendix of [17]. For large tan β = v 2 /v 1 this reduces to where x ′ = ∂x/∂ ln p is the partial derivative of x wrt p.

A.2 The scalar potential: 1 Loop Leading Log (1LLL) Terms
Here we show how to obtain the one-loop corrected potential, which is "improved" to the two-loop leading log (2LLL) result in the next section. Start with V (0) =m 2 1 |H 1 | 2 +m 2 2 |H 2 | 2 −m 2 3 (H 1 H 2 + h.c.) + This receives (field dependent threshold) corrections, computed using the Coleman-Weinberg potential [43]: where m k is the field dependent mass, the degeneracy factor g k is 6 for squarks, and J k is the particle spin. All parameters in eq.(A-10) are evaluated at the scale Q using the RGEs which ignore the particle masses. The field dependent squark masses are (neglecting O g 4 terms): -12) and where m Q,U,D (M S ) = M S is assumed. Here, M S is the soft SUSY breaking squark mass evaluated at the squark mass scale.
One can expand the non-linear field dependence (log) in V (1) in inverse powers of 1/M S to find the dominant threshold corrections, which come from the third generation squarks: where t = log(M 2 S /Q 2 ) (A-15) When running below the EWSB scale, the inclusion of higher dimensional terms (threshold corrections) lead to a re-summation such that M S is replaced by a mass scale related to the physical particle masses [44]. For the results of this paper, the geometric mean of the stop masses is used in the place of M S .
The above equations are valid down to the top mass scale; below this scale threshold corrections from the top quark should also be included. The dominant effect of running below the top scale can be absorbed by setting Q in the above equations as the "running" top mass evaluated at the scale Q instead of the pole mass.
16π 2 β λ 1 ≈ 6h 2 Note that these results assume that the CP odd Higgs mass is not decoupled. If this is the case, the usual SM β-functions should be used. The effective quartic coupling at the EW scale when m A M S is given by: with the following notation: A similar but distinct result is obtained when m A ∼ M S (notably no δ i terms and a different dependence on tan β and the mixed Yukawa couplings). The threshold corrections are dependent on where the CP odd Higgs decouples. The same procedure has been applied to determine the 2LLL threshold corrections to the mass terms.