Electroweak and supersymmetry breaking from the Higgs discovery

We will explore the consequences on the electroweak breaking condition, the mass of supersymmetric partners and the scale at which supersymmetry is broken, for arbitrary values of the supersymmetric parameters tan(beta) and the stop mixing X_t, which follow from the Higgs discovery with a mass m_H\simeq 126 GeV at the LHC. Within the present uncertainty on the top quark mass we deduce that radiative breaking requires tan(beta) \gtrsim 7 for maximal mixing X_t\simeq \sqrt{6}, and tan(beta) \gtrsim 20 for small mixing X_t\lesssim 1. The scale at which supersymmetry is broken \mathcal M can be of order the unification or Planck scale only for large values of tan(beta) and negligible mixing X_t\simeq 0. On the other hand for maximal mixing and large values of tan(beta) supersymmetry should break at scales as low as \mathcal M\simeq 10^5 GeV. The uncertainty in those predictions stemming from the uncertainty in the top quark mass, i.e. the top Yukawa coupling, is small (large) for large (small) values of tan(beta). In fact for tan(beta)=1 the uncertainty on the value of \mathcal M is of several orders of magnitude.


Introduction and summary
After the 7 and 8 TeV runs the Large Hadron Collider (LHC) has firmly established the existence of a scalar boson with a mass m H ≃ 126 GeV. In particular the strengths measured in the W W , ZZ, γγ, bb and τ τ decay channels by the ATLAS and CMS collaborations are consistent with the Standard Model (SM) Higgs with a mass m H = 126 ± 0.4 (stat) ± 0.4 (syst) GeV [1] and m H = 125.3 ± 0.4 (stat) ± 0.5 (syst) GeV [2], respectively. The Higgs discovery is of the upmost importance as it is the first direct experimental confirmation of the mechanism of electroweak (EW) symmetry breaking (EWSB). In the SM it points toward a quartic coupling λ = m 2 H /v 2 , where v = 246 GeV, provided by the Higgs potential where m 2 = λv 2 /2, valid at the EW scale Q EW = m H . From the theoretical point of view we know that the the EW minimum is unstable against quantum corrections (a problem known in the literature as the hierarchy problem) and has to be stabilized by some beyond the SM (BSM) physics, the paradigm of which being supersymmetry and in particular the minimal SM supersymmetric extension (MSSM). Nonetheless another feature of the past LHC runs is that no experimental hints have been found of BSM particles which could stabilize the EW vacuum, but it is putting bounds on the mass of supersymmetric particles [3]. Still, and in view of the forthcoming LHC run at 13-14 TeV, it is interesting to explore the consequences of the present Higgs mass data on a possible underlying supersymmetric theory, in particular on the way supersymmetry triggers EWSB at low energy and on the value of the scale at which supersymmetry is broken.
In this paper we will then consider at face value the present data on the Higgs sector. We will assume that the SM emerges at some scale Q 0 from an underlying MSSM, and will extract the relevant information on the mechanism by which the MSSM triggers EWSB and on the scale M at which supersymmetry is broken in the hidden sector. Consistently with present experimental data we will assume that below the scale Q 0 we just have the SM spectrum and the matching conditions are the ones to enforce EWSB at the EW scale Q EW = m H .
The contents of this paper are as follows. In Sec. 2 we impose the condition that the SM and the MSSM merge at the scale Q 0 and compute it by fixing the Higgs mass in the SM and with fixed values of tan β and the stop mixing X t in the MSSM. We see, not unexpectedly, that for low values of tan β, Q 0 is large and insensitive to the mixing X t while for large values of tan β it can be small and sensitive to X t . In particular, values of Q 0 in the TeV range require both large values of tan β (tan β 5) and of the mixing (X t 1.8). Moreover we can translate the condition of EWSB in the SM to a condition on m 2 2 (Q 0 ) (the squared mass of the Higgs doublet that gives a mass to the top quark) and so we can scrutinize on the nature of EWSB, i.e. radiative versus non radiative breaking 1 . We have found that the nature of EWSB strongly depends on both parameters, tan β and X t . In particular we have found that radiative breaking requires tan β 8 for maximal mixing and X t 1.8 for tan β 20. Another interesting feature we have found, both in the calculation of the matching scale, i.e. Q 0 , and the type of EWSB, i.e. the value of m 2 2 (Q 0 ), is that the main uncertainty in the calculation comes from the uncertainty in the top quark mass and that it affects mainly low values of tan β (it is mostly insensitive to the actual value of the mixing X t ). In particular for tan β = 1 and X t = 0, while for the central value of the top quark mass Q 0 ∼ 10 11 GeV, after inserting the 2σ uncertainty it can vary in the range Q 0 ∈ [10 9 , 10 16 ] GeV and no sharp prediction can be made. Notice that the results in this section imply precise values of m 1 (Q 0 ) and m 2 (Q 0 ) determined by the EWSB condition and by the condition of decoupling of the heavy Higgs at the matching scale Q 0 .
In Sec. 3 we computed the scale at which supersymmetry is transmitted M by imposing the condition that the MSSM Higgs mass parameters are equal at that scale: m 1 (M) = m 2 (M). All of our results in this section are based on this assumption. This one is a natural assumption in most existing models of supersymmetry breaking, including those coming from string theories. Of course, should the condition on m 1 (M) and m 2 (M) be changed our results would correspondingly be modified. To compute the value of the scale at which supersymmetry is broken M we have first followed a bottom-up approach where we assume the ideal conditions that all supersymmetric particles decouple exactly at the scale Q 0 (with no thresholds). In the second, top-down, approach we have instead assumed that the supersymmetric parameters are the ones obtained in various models of supersymmetry breaking transmission to the observable sector. In these cases there are different thresholds around the matching scale Q 0 but the results are in all cases consistent with the first approach. The main result in the bottom-up approach is (not unexpectedly) that the value of M depends to a large extent on the supersymmetric parameters (tan β(Q 0 ), X t (Q 0 )). In particular for X t ≥ 0 large values of M close to the unification or Planck scale can only be obtained for large values of tan β and small mixing. Of course for those small values of X t the constraint on the Higgs mass impose large values of Q 0 , say in the 10 -100 TeV region for which the supersymmetric spectrum would be outside the reach of LHC. On the contrary for large values of X t , for instance for maximal mixing, for which Q 0 is in the TeV range and the supersymmetric spectrum is inside the reach of LHC, the scale at which supersymmetry breaking is transmitted can go down to the low scales such that gravity mediation mechanisms are precluded. For negative values of X t there is room for large GUT or Planckian values of M provided that X t is in some intermediate region, e.g. X t ≃ −1.5 which can accommodate lower values of Q 0 for large values of tan β, e.g. Q 0 = O(few) TeV, inside the LHC reach. Again our predictions are affected by the top quark mass uncertainty ∆m t (m t ). As it was the case for the Q 0 prediction, the uncertainty affects mainly small values of tan β and it is rather insensitive to the value of X t . In the second, top-down, approach we have considered two different cases where supersymmetric parameters unify at the scale M. First we have considered the case of universal soft parameters, by which all squark masses (m 0 ), all gaugino masses (m 1/2 ) and all Higgs mass parameters (m H ) unify at the scale M. This is a general CMSSM where we have separated the Higgs from the sfermion masses and which can appear in gravity mediated supersymmetry breaking theories. We have considered two examples with tan β = 10 and X t = 0, 2. In agreement with the results of the previous section the case X t = 0 is consistent with EWSB and the Higgs mass m H = 126 GeV for M ≃ 10 18 GeV, while the case X t = 2 requires supersymmetry breaking at low scale M ≃ 10 6 GeV, hard to reconcile with gravity mediation. The second case we have considered is the minimal gauge mediated supersymmetry breaking (GMSB) where the mass of scalars transforming under a gauge group G a , with gauge coupling α a , and the corresponding gaugino is proportional to α a (M)/4π and the trilinear coupling is A t (M) = 0. Below M, A t is generated by the MSSM RGE and therefore it gets negative values at the scale Q 0 , giving then X t < 0. We have presented two cases with N = 4 messengers, tan β = [15,8]

The matching and electroweak breaking
The quadratic terms in the MSSM potential can be written as where m H i is the soft breaking mass for H i and µ is the supersymmetric Higgsino mass. They can also be written as consequently the mass squared of its (heavy) orthogonal combination H with m 2 + ≡ m 2 H = m 2 1 + m 2 2 + m 2 . This can be done by the fixing leading to the mixing angle β given by where all quantities are evaluated at the matching scale Q = Q 0 , which rotates the Higgs basis (H 1 , H 2 ) into the mass eigenstates (H, H) as The potential for the SM Higgs then reads as In order to make a precise calculation of the Higgs mass we have to first match the SM quartic coupling λ and the supersymmetric parameters at the scale Q 0 . We will improve over the tree-level (ℓ = 0) matching by considering the one-loop (ℓ = 1) and leading two-loop (ℓ = 2) threshold effects as given by [4] and we are using the notation X t = (A t (Q 0 ) − µ(Q 0 )/ tan β)/Q 0 , and s β ≡ sin β and so on. For the numerical calculation we are also taking into account the O(y 6 t s 4 β , . . . ) two-loop threshold corrections whose explicit expression can be found in Ref. [4]. We are neglecting the corrections proportional to y 4 τ as we are not envisaging values of the parameter tan β such that y τ is relevant.
The couplings y t and y b are the top and bottom Yukawa couplings in the MSSM. They are related to the corresponding SM couplings h t and h b by [4] we are assuming nearly degenerate spectrum at Q 0 , and only the leading one-loop QCD and top Yukawa coupling corrections are kept. The function I(x, y, z) can be found in Ref. [4]. The parameters of the potential (2.8) have to be run with the SM RGE down to the scale Q EW = m H , where minimizing the SM potential should lead to For a similar analysis see Ref. [5]. Here in agreement with the used threshold corrections we are using the two-loop RGE as given in [6].
Finally going from the running Higgs mass m H to the pole Higgs mass M H requires the calculation of the Higgs boson self energy Π(p 2 ) as M 2 Here we keep only the leading correction to ∆Π coming from the top quark loop exchange given by [7] For the actual values of M t ≃ m(m t ) + 10 GeV [8] (the pole top quark mass) and M H , the correction in (2.12) is of the order of the experimental error in the Higgs mass. Notice that, for fixed values of the supersymmetric parameters tan β and X t , Q 0 is a function of the Higgs mass m H . This prediction comes from the intersection of the function λ(Q), which is determined mainly by the value of the Higgs mass [with some dependence on the actual values of h t (m H ) and α 3 (m H )], with the value λ(Q 0 ) given by Eq. (2.10). So given that the Higgs mass is fixed to m H = 126 GeV, we can predict Q 0 = Q 0 (tan β, X t ) as it is shown in the left panel plot of Fig. 1. We have used as an input the running top mass in the MS scheme evaluated at the top mass m t (m t ) = 163.5 GeV. We can see that for small values of tan β the values of Q 0 are large and insensitive to the values of the mixing X t . This is due to the fact that the threshold effect is proportional to h 2 t (Q 0 ) and the Standard Model RGE leads to small values of h t (Q 0 ) for large values of the scale Q 0 . On the other hand for large values of tan β the values of Q 0 are smaller and consequently the RGE running is small and Q 0 becomes sensitive to the mixing X t . In particular values of Q 0 in the TeV region require large values of tan β (tan β 5) and large values of X t (X t 1.8).
As for the error in m t (m t ) it is safe to consider the experimental range of the running top mass to be given by ∆m t = ±2 GeV at 2σ [8,9]. In order to see the relevance of the error in m t (m t ) we plot, in the left panel of Fig. 2, Q 0 as a function of tan β for various values of X t , and in the right panel of Fig. 2, Q 0 as a function of X t for different values of tan β. In fact the upper border of each band corresponds to ∆m t = −2 GeV and the lower border to ∆m t = +2 GeV. We can see from both panels of Fig. 2 that the error in the determination of Q 0 , ∆Q 0 arising from the error in m t (m t ) is large (small) for small (large) values of tan β. The reason for this behavior is that the error m t (m t ) is amplified by the RGE running and it is consequently large (small) for large (small) running, which means small (large) values of tan β. In the same way, as we can see from the right panel of Fig. 2, the error ∆Q 0 is uncorrelated with X t as it has little influence on the RGE running. This translates into a big overlapping in the left panel of Fig. 2 for small values of tan β and different values of X t . In fact notice that for the limiting case tan β = 1 and X t = 0 we have that λ(Q 0 ) 0 and the Standard Model potential is unstable. This corresponds, for the central value of the quark top mass, to Q 0 ∼ 10 11 GeV. However for the lowest allowed value of the top quark mass the instability scale can go to Planckian values in agreement with various calculations in the literature [10,11]. In this case it has been shown that the Veltman condition [12] (or absence of quadratic divergences) can also be satisfied [13]. In Fig. 3 we plot the absolute value of m 2 , |m 2 (Q 0 )|, as a function of tan β for different values of X t (left panel) and as a function of X t for different values of tan β (right panel). Notice that points where electroweak breaking becomes radiative are characterized by the fact that |m 2 | = 0 and for larger values of tan β (left panel of Fig. 3) or larger values of X t (right panel of Fig. 3), m 2 2 becomes negative and thus |m 2 | takes on positive values. Again we can see that, as for the results in Fig. 2, the effects of the error ∆m t (m t ) are amplified for small values of tan β while they stay small for large values of tan β. We can also see that radiative breaking only occurs for large values of tan β, tan β 8, and/or large values of the mixing X t 1.8 in the range tan β 20.

Supersymmetry breaking scale
In the previous section we have computed, using the measured value of the Higgs mass, the value of the scale Q 0 at which the MSSM matches with the Standard Model and the value of the parameter m 2 2 (Q 0 ) which guarantees a correct electroweak Standard Model breaking at the scale Q EW = m H . We are here making the conservative assumption (alas, consistent with present experimental data!) that only the SM states survive below the matching scale Q 0 . For large values of Q 0 this amounts to assume a high-scale MSSM beyond Q 0 , in contradistinction with other possibilities, as those dubbed as split (or minisplit) supersymmetry. Using these tools we will now get information on the scale at which supersymmetry breaking is transmitted M.
As we have seen both Q 0 and m 2 2 (Q 0 ) are (for fixed values of the Standard Model parameters) functions of the MSSM parameters tan β and X t defined at the scale Q 0 : Q 0 ≡ f 0 (tan β, X t ) and m 2 2 (Q 0 ) ≡ f 2 (tan β, X t ). Now from the EWSB condition (2.6) one can also compute m 2 1 (Q 0 ) ≡ f 1 (tan β, X t ) as so that both squared mass parameters m 2 1 and m 2 2 are fixed at the scale Q 0 for fixed values of tan β and X t . We will now define the scale at which supersymmetry is transmitted M as the scale at which  where we are running the MSSM parameters from the scale Q = Q 0 to the scale Q = M by using the two-loop RGE [14]. Notice that this condition is rather generic in most models of supersymmetry breaking, as models based on gravity mediation or minimal gauge mediation, as well as in string constructions [15][16][17].
As we are assuming that the effective theory below Q 0 is just the Standard Model we are implicitly assuming that, at the matching scale the heavy Higgs H decouples, so that m H (Q 0 ) = Q 0 . On the other hand the scale at which supersymmetry breaking is transmitted, given by (3.2), does have little dependence on the boundary conditions imposed for the rest of the supersymmetric spectrum. Thus we will next consider two generic situations.

Bottom-up approach
The most precise (and ideal) way by which the Standard Model will emerge as the low energy effective theory below the matching scale Q 0 is when all supersymmetric particles are (approximately) degenerate at the decoupling scale 4 . So we will here assume for all sfermions ( f ), Higgsinos (with mass µ) and gauginos a degenerate mass at the matching We will leave X t (Q 0 ) [and consequently the mixing A t (Q 0 )] and tan β(Q 0 ) as free parameters in the plots. Note that by imposing the matching scheme in Eq.

Top-down approach
In the previous section we have assumed that all supersymmetric particles exactly decouple at the matching scale Q 0 , by which we were assuming a degenerate spectrum at this scale. Of course this is not the generic case in (realistic) models of supersymmetry breaking which provide some pattern of masses at the scale M. These masses run, with the MSSM RGE, from the scale M to Q 0 and thus they decouple at the scale ∼ Q 0 with different thresholds.
In this section we will consider different supersymmetric spectra, for which the scale at which supersymmetry breaking is transmitted and the matching scale with the Standard Model satisfy the general values which have been obtained in the previous section: in particular they are consistent with electroweak symmetry breaking with a Higgs mass of 126 GeV. We will not commit ourselves to any particular mechanism of supersymmetry breaking but instead will consider generic pattern of supersymmetric spectra at the scale where supersymmetry breaking is transmitted, which can arise from different mechanisms. In particular we will consider two classes of models, which are simply particular examples while many others can be easily found and studied: • Models with universal soft parameters, typical of gravity mediated-like models, although not necessarily arising from gravity mediation.
• Gauge mediated models, where the values of supersymmetric parameters satisfy, at the scale M, typical ratios provided by gauge mediation.

Universal soft parameters
In this section we are going to consider some universal soft breaking parameters at the scale at which supersymmetry breaking is transmitted M. In particular we will assume the rather general pattern by which all third generation squarks 5 are degenerate at the scale M, as well as the three gauginos and the two MSSM Higgs doublets. We have then considered the common masses m 0 and m 1/2 as free parameters only subject to the constraint of getting a correct electroweak symmetry breaking. We have considered in Fig. 6 two generic models which correspond to tan β = 10, and X t = 0 (left panel) and X t = 2 (right panel). As for the case of X t = 0 a quick glance at the left panel of Fig. 1 shows that the matching scale is Q 0 ∼ 100 TeV while from Fig. 4 the scale where supersymmetry breaks is M ∼ 2 × 10 18 GeV. Also from the right panel of Fig. 1 we see that the breaking is not radiative in the sense that m 2 2 (Q 0 ) > 0 and indeed from Fig. 3 we can see that, according with the correct electroweak symmetry breaking, m 2 (Q 0 ) ≃ 3 TeV. As we can see from the left panel of Fig. 6 the values for the common squark and gaugino masses which fit these conditions are: m 0 ≃ m 1/2 ≃ 30 TeV. Also the value of X t = 0 at the matching scale Q 0 translates into the mixing A t (M) ≃ 1.7 m 0 . Notice that, as the value of M is around the Planck scale, this scenario could arise in models where supersymmetry breaking is transmitted by gravitational interactions.
If we now increase the value of X t , as in the right panel of Fig. 6, in which X t = 2, then looking again at Fig. 1 we see that the matching scale is Q 0 ∼ 1 TeV and the electroweak breaking is (almost) radiative as m 2 (Q 0 ) ∼ 100 GeV. Likewise, from Fig. 4, the scale at which supersymmetry is broken is M ∼ 5 × 10 5 GeV. Here we can see a general phenomenon by which the scale where supersymmetry breaking is transmitted (i.e. the scale of unification of m 1 and m 2 ) strongly goes down when the mixing increases if we fix the correct conditions for electroweak breaking. The reason is the contribution of the mixing to the RGE as To prevent electroweak breaking at high scale (Q ≫ Q 0 ) we then let the scale M go down.
For the same reason we need gauginos heavier than squarks as the former ones contribute with negative sign to β m 2 2 . As we can see in the right panel of Fig. 6 this condition translates into m 0 ≃ 3.3 TeV and m 1/2 ≃ 5.3 TeV while at the matching scale Q 0 all the supersymmetric spectrum is in the interval 3 − 6 TeV.

Gauge mediated models
In this section we will apply the previous results to the particular case in which supersymmetry breaking is transmitted to the observable sector by gauge interactions (GMSB). We will assume in particular the minimal GMSB model whose main features we now summarize. Supersymmetry is broken, in a hidden sector, by a spurion chiral superfield X = F θ 2 which is coupled to a set of pairs, Φ i + Φ i , of messenger fields, in vector like r + r representations of the gauge group with the superpotential Gauginos acquire a Majorana mass, by one loop diagrams, given by [18] where n i is the Dynkin index for the pair Φ i + Φ i 6 , and N = i n i . For the last equality of Eq. (3.6) we are assuming universal messenger masses as M i ≡ λ i M (for ∀i). Likewise supersymmetric scalars (squarks and sleptons) acquire soft breaking squared masses through two loop diagrams as where C f a is the quadratic Casimir of the representation to which f belongs in the group G a 7 , and again for the last equality of Eq. (3.7) we are assuming universal messenger masses. In fact for the case of universal messenger masses the ratio Λ 2 G /Λ 2 S = N is given by the number of messengers, however in more general cases (which can arise e.g. for several X fields overlapping with the Goldstino field) one can treat Λ G and Λ S as free parameters. The soft breaking parameter A t is not generated at one loop so we will fix it as A t (M) = 0 and will let it to develop at the scale Q 0 by the MSSM RGE running, which is equivalent to a two loop effect.
In Fig. 7 we are presenting two typical cases where GMSB is consistent with the conditions imposed by electroweak breaking for a 126 GeV Higgs mass. The case tan β = 15 is presented in the left panel and tan β = 8 in the right panel. In both cases we have fixed Λ G = 2Λ S which corresponds to four messengers, N = 4, in minimal GMSB models. Both cases are, as we will see, consistent with perturbative unification.
For the case tan β = 15 in the left panel of Fig. 7 we get Λ G ≃ 1.4 × 10 6 GeV, M ≃ 3 × 10 8 GeV, and the scale of supersymmetry breaking √ F ≃ 10 7 GeV while the expansion parameter F/M 2 ≃ 10 −3 is small, and the gravitino mass is m 3/2 ≃ 20 keV. Notice that because of the contribution of µ 2 in m 2 i . This case is perfectly consistent with perturbative unification and the messengers change the value of the gauge couplings at the unification scale by δα −1 GU T ≃ −11. Even if A t (M) = 0 a nonzero (and negative) value is generated at the scale Q 0 such that X t ≃ −1.8.
For the case shown in the right panel of Fig. 7 that corresponds to tan β = 8 we get the following values of the parameters: Λ G ≃ 2 × 10 6 GeV, M ≃ 10 11 GeV, √ F ≃ 3 × 10 8 GeV with the expansion parameter F/M 2 ≃ 4 × 10 −6 and m 3/2 ≃ 20 MeV. This case is also

Conclusions
The Standard Model is consistent with all present experimental data including the recent measurements of the Higgs mass and its couplings to gauge bosons and fermions. By the same token experimental data are putting bounds on possible BSM physics whose aim is to solve the SM hierarchy problem, i.e. to understand the hierarchy Q EW /M P ≃ 10 −16 , or equivalently the stability of the electroweak vacuum. Even if no hint of new physics have been found by the LHC (and even in the case that no positive signal be found by the future LHC runs) still the stability of the big hierarchy between the LHC energy scale and the Planck scale (Q LHC /M P ≃ 10 −14 ) requires a theoretical explanation, although theories aiming to explaining the little hierarchy (Q EW /Q LHC ≃ 10 −2 ) do not receive support from the experimental side. On the other hand the paradigm of theories solving the hierarchy problem is supersymmetry, which has roots in superstring theories for which low-scale supersymmetry is not an essential ingredient. So a possible attitude (that we have adopted in this paper) is to assume that supersymmetry is solving the big hierarchy problem from Q LHC /M P (which amounts to a fine-tuning of one part in 10 28 !) but perhaps not necessarily the little hierarchy problem from Q EW /Q LHC (which amounts to a fine-tuning of around one part in ten thousand) and see what the present data are telling us about the parameters of the supersymmetric theory.
Using then the measured value of the Higgs mass and imposing the conditions for electroweak symmetry breaking we can obtain information on the scale of supersymmetric parameters (Q 0 ) and the conditions on how the supersymmetric theory triggers electroweak breaking. Moreover by making the mild assumption that the mass parameters of both Higgs bosons unify at the scale at which supersymmetry breaking is transmitted M we can obtain rather general information on the latter, as we have described throughout this paper. In models where the former assumption on the Higgs bosons mass at M is not fulfilled the conditions should be accordingly modified.
Our analysis just reflects the present experimental situation concerning the Higgs discovery and the non-observation of any supersymmetric particle in the LHC7 and LHC8 runs. In the future, when the LHC13-14 run will start in 2015, it might happen that supersymmetric signals are found or that they are not. In both cases the present analysis should be correspondingly constrained. In case where supersymmetric signals are found, they would give information about our energy scale Q 0 which in turn will give indirect information about the scale at which supersymmetry breaking is transmitted M. In the other case, in which supersymmetric signals not be found at the LHC13-14, the data will put a lower bound on the scale Q 0 by which also the scale M will be correspondingly constrained, suggesting that perhaps we will need a higher energy collider to uncover BSM physics as the HE-LHC (at 33 TeV) & VHE-LHC (at 100 TeV) [19].