IFT-UAM/CSIC-13-008

FTUAM-13-126

The Higgs Mass as a Signature of Heavy SUSY

[3mm]

Luis E. Ibáñez and Irene Valenzuela

[3mm] Departamento de Física Teórica and Instituto de Física Teórica UAM-CSIC,

[-0.3em] Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

[12mm] Abstract

[7mm]

We compute the mass of the Higgs particle in a scheme in which SUSY is broken at a large scale well above the electroweak scale . Below one assumes one is just left with the SM with a fine-tuned Higgs potential. Under standard unification assumptions one can compute the mass of the Higgs particle as a function of the SUSY breaking scale . For GeV one obtains GeV, consistent with CMS and ATLAS results. For lower values of the values of the Higgs mass tend to those of a fine-tuned MSSM with GeV. These results support the idea that the measured value of the Higgs mass at LHC may be considered as indirect evidence for the existence of SUSY at some (not necessarily low) mass scale.## 1 Introduction

The evidence [1],[2] obtained by the CMS and ATLAS experiments at CERN of a scalar particle with the properties of a Standard Model (SM) Higgs particle with mass GeV is a crucial piece of information to unravel the origin and characteristics of the Electroweak (EW) symmetry breaking. This mass value is compatible with the region allowed by the MSSM which is GeV. Still getting a value of the Higgs mass of order 125 GeV in the MSSM requires a certain amount of fine-tuning. On the other hand within the SM any value from the LEP bound up to almost 1 TeV could have been possible. Thus one might interpret the experimental results as pointing in the direction of some sort of (fine-tuned) SUSY.

Building on ideas discussed in [3], in the present paper we try to answer the following question. Imagine the SM is extended to the MSSM above a certain scale not necessarily tied to the EW scale, but possibly much higher. If that is the case, a fine-tuning of the underlying theory would be required in order for a Higgs doublet to remain massles. Under those circumstances, what would be the mass of the fine-tuned Higgs?

Although the question sounds too generic to have a sharp answer it turns out that under standard unification assumptions a concrete answer may be given. In particular, assuming the unification of Higgs mass parameters at the GUT/String scale and a minimally fine-tuned Higgs below the SUSY breaking scale , then one obtains a definite prediction for the Higgs mass as a function of the SUSY breaking scale. Although the experimental error from the top quark mass as well as the SUSY spectra introduce some degree of uncertainty, the results, exemplified in fig.LABEL:plotguay, show that for GeV the value of the Higgs mass is centered around GeV. Below that scale this mass depends more on the details of the SUSY breaking mass parameters but the maximum value is bound by 130 GeV, corresponding to a standard fine-tuned MSSM with TeV.

This predictivity is remarkable, since the SM by itself would allow for a large range of consistent values with e.g. much higher values for the Higgs mass of order e.g. 150-300 GeV or higher. The fact that experimentally GeV then renders strong support to the idea of SUSY being realized at some, possibly large, mass scale. Even if SUSY particles are not found at LHC energies the particular value of the Higgs mass points to an underlying SUSY at some higher scale. This is of course due to the fact that SUSY, even spontaneously broken at an arbitrarily high energy scale, relates de quartic Higgs selfcoupling to the EW gauge couplings.

## 2 Traces from high energy SUSY and a minimally fine-tuned Higgs

There is at present no experimental evidence at LHC for the existence of SUSY particles. This, combined with earlier experimental limits, severely constraint the idea of low energy SUSY and indicates the necessity of some degree of fine-tuning of parameters of the order of at least 1-0.1 percent [4, 5]. If no evidence of SUSY particles is found at the 14 TeV LHC the general idea of low-energy SUSY as a solution to the hierarchy/fine-tuning problem will be strongly questionable. On the other hand, as recently emphasized in [3], even if SUSY is not present at the EW scale to solve the hierarchy problem, there are at least three reasons which suggest that supersymmetry could be present at some scale above the EW scale and below the unification/string scale. The first is the fact that SUSY is a substantial ingredient of string theory which is, as of today, the only serious contender for an ultraviolet completion of the SM. The second reason is that SUSY guarantees the absence of scalar tachyons which are generic in non-SUSY string vacua. Thirdly, and independently from any string theory consideration, a detailed study of the non-SUSY SM Higgs potential consistent with the measured Higgs mass indicates that there is an instability at scales above GeV [6, 7]. Although in principle one can live in a metastable vacuum, supersymmetry would stabilize the vacuum in a natural way if present at an energy scale GeV.

Let us then consider a situation in which SUSY is broken at some high scale with , where is the unification/string scale. For previous work on a fine-tuned Higgs in a setting with broken SUSY at a high scale see e.g.[8, 9, 10, 11, 3, 12]. With generic SUSY breaking soft terms one is just left at low-energies with the SM spectrum. In addition the scalar potential should be fine-tuned so that a Higgs doublet remains light so as to trigger EW symmetry breaking. One would say that no trace would be left from the underlying supersymmetry. However this is not the case [9]. Since dimension four operators are not affected by spontaneous SUSY breaking, the value of the Higgs self-coupling at the scale will be given in the MSSM by the (tree level) boundary condition

(2.1) |

which is inherited from the D-term scalar potential of the MSSM. Here are the EW gauge couplings and is the mixing angle which defines the linear combination of the two doublets of the MSSM which remains massless after SUSY breaking, i.e.,

(2.2) |

Thanks to this boundary condition, for any given value of tan one can compute the Higgs mass as a function of the SUSY breaking scale .

Schematically the idea is to run in energies the values of up to the given scale. For any value of tan one then computes from eq.(2.1). Starting with this value we then run down in energies and obtain the value for the Higgs mass from . Threshold corrections at both the EW and SUSY scales have to be included. This type of computation for different values of tan was done e.g in ref.[13], [14], [15], [16]. We show results for a similar computation in fig.LABEL:plotguay (grey bands). The Higgs mass may have any value in a broad band below a maximum around 140 GeV. One may easily understand the general structure of these curves. The mass is higher for higher tan since the tree level contribution to the Higgs mass through eq.(2.1) is higher. On the other hand the Higgs mass slowly grows with larger as expected.

What we want to emphasize here is that the natural assumption of Higgs soft mass unification at the unification scale , i.e.

(2.3) |

leads to a much more restricted situation with trajectories in the plane rather than a wide band. Note that this equality is quite generic in most SUSY, unification or string models. In particular it appears in gravity mediation as well as in almost all SUSY breaking schemes, including those arising from compactified string theory, see e.g.[17].

Indeed, to see this let us recall what is the general form for Higgs masses in the MSSM at the scale ,

(2.4) |

where we will take real.
If all these mass terms were zero we would get two Higgs doublets
in the massless spectrum. However this would require extra unnecessary fine-tuning.
The minimal Higgs fine-tuning would only require a single Higgs doublet
to remain at low-energies ^{1}^{1}1This is a particular realization of the Extended Survival Hypothesis of ref.[18](see also [19]).
. This is achieved for a single fine-tuning
. The massless eigenvector is then

(2.5) |

with

(2.6) |

If the origin of this fine-tuning is understood in terms of the fundamental SUSY breaking parameters scanning in a landscape of possibilities, the diagonal Higgs masses are supposed to scan in a way consistent with the boundary condition (2.3). One can then compute the value of tan by running the ratio in (2.6) from the unification scale down to the SUSY breaking scale . One computes the value of the Higgs self-coupling from eq.(2.1) and then runs down in energies to compute the Higgs mass for any given value of . In a general MSSM model we can compute this in terms of the underlying structure of soft terms at . In particular one expects generic SUSY-breaking soft terms of order . For definiteness we will assume here a universal structure of soft terms with the standard parameters (3-d generation scalars masses), (gaugino masses), (3-d generation trilinear parameter) and (mu-term). As we will see, the results will have very little dependence on this universality assumption which simplifies substantially the computations. This universality assumption is also consistent with the (weaker) assumption of Higgs mass unification, eq.(2.3).

Let us remark that in this approach the only relevant condition is at the unification scale . There is no need for a shift symmetry which imposses at the unification scale as in ref.[11], since then the fine-tuning would be destroyed by the running from to . The idea is that enviromental selection should ensure that at the scale (not ) the fine-tuning condition is impossed with high accuracy.

## 3 The Higgs mass from minimal fine-tuning

We now turn to a description of the different steps required to compute the Higgs mass as a function of the SUSY-breaking scale .

### 3.1 Computing the couplings at

We start by computing the electroweak couplings at the scale. We take the central values for the masses (in GeV) and couplings at the weak scale

(3.1) |

(3.2) |

We will allow to vary the top mass with an error GeV obtained from the average from Tevatron [20] and CMS and ATLAS results as in ref.[21]. We will neglect the error from which is much smaller than that from the top quark mass. To extract the value of the top Yukawa coupling we take into account the relationship between the pole top-quark mass and the corresponding Yukawa coupling in the scheme [22]

(3.3) |

where the dominant one-loop QCD corrections may be estimated ([22], [14, 16])

(3.4) |

One then obtains . We run now the couplings , and up to the given scale . We do this by solving the RGE at two loops for the SM couplings. Those equations are shown for completeness in appendix A.

### 3.2 Computing tan and

With at hand we want now to compute the value of from eq.(2.1). To do that we need to compute tan from eq.(2.6), which in turn requires the computation of the running of the masses , from the unification scale at which down to .

The value of the unification scale is usually obtained from the unification of gauge coupling constants. In our case, with two regions respectively with the SM (below ) and the MSSM (in between and ) the value of is not uniquely determined. In fact it is well known that precise unification is only obtained for TeV, as in standard MSSM phenomenology [23]. However, approximate unification around a scale GeV is anyway obtained for much higher values of , even in the limiting case with in which case SUSY is broken at the unification scale, so a simple approach would be to take GeV to compute the runing of tan. We find more interesting instead to achieve consistent gauge coupling unification from appropriate threshold corrections. In particular, in a large class of string compactifications like F-theory GUT’s there are small threshold corrections respecting the boundary condition at the GUT scale [24, 25, 3]

(3.5) |

This boundary condition is consistent (but more general) than the usual GUT boundary conditions . It arises for example from F-theory GUT’s [26] once fluxes along the hypercharge direction are added to break the symmetry down to the SM [25, 27]. Using the RGE for gauge couplings in both SM and MSSM regions (at two loops for the gauge couplings and one loop for the top Yukawa) one finds that unification of couplings is neatly obtained at a scale related with by the approximate relationship

(3.6) |

As one varies in the range TeV- one obtains GeV. This relation changes very little compared to the one obtained just using the RGE at one loop in ref.[3]. To compute tan we will use as unification scale the obtained from eq.(3.6) consistent with gauge coupling unification. It is important to remark though that this has very little impact in the numerical results obtained, there is no detail dependence on the value of as long as it remains in the expected GeV region.

To compute tan at one solves the RGE for the Higgs mass parameters . At this point one needs to make some assumptions about the structure of the SUSY-breaking soft terms of the underlying MSSM theory. We will thus assume a standard universal SUSY breaking structure parametrized by universal scalar masses , gaugino masses and trilinear parameter . The results are independent from the value of the B parameter which is fixed by the fine-tuning condition at . Given these uncertainties it is enough to use the one-loop RGE for the soft parameters, which were analytically solved in ref.[28]. As described in [3] one has tan with

(3.7) |

where are the standard universal CMSSM parameters at the unification scale , and are known functions of the top Yukawa coupling and the three SM gauge coupling constants. Except for regions with large tan, appearing only for low , one can safely neglect the bottom and tau Yukawa couplings, . For completeness these functions are provided in Appendix B. The value taken for to perform the running of soft terms is a bit subtle since at one has to match the value obtained from the SM running up to with the SUSY value which are related by

(3.8) |

Since the value of depends on through eq.(3.8), the computation of tan is done in a self-consistent way: a value is given to sin, is run up in energies and one has a tentative . One then runs down in energies and computes tan at . When both values for at agree the computation of tan is consistent.

Once computed the value of tan as described above, one then obtains from eq.(2.1). In addition there are threshold corrections at induced by loop diagrams involving the SUSY particles. The leading one-loop correction is given by

(3.9) |

where is the SUSY top Yukawa coupling at and the stop mixing parameter is given by

(3.10) |

with the left(right)-handed stop mass. This term comes from finite corrections involving one-loop exchange of top squarks. There are further correction terms which are numerically negligible compared to this at least for not too low , in which case the SUSY spectrum becomes more spread and further threshold corrections become relevant, see e.g. [14]. We have computed this parameter using the one loop RGE for the soft parameters that are provided in Appendix B and the value of tan obtained above.

### 3.3 Computing the Higgs mass

Starting from one runs back the self-coupling down to the EW scale (using the SM RGE at two loops) and computes the Higgs mass at a scale (taken as ) through

(3.11) |

where GeV is the Higgs vev and are additional EW scale threshold corrections. At one-loop these corrections are given by [29]

(3.12) |

where and the functions y depend only on EW parameters and are shown in appendix C for completeness. This completes the computation procedure for the Higgs mass as a function of .

Figure LABEL:plotguay plots the value of as a function of . For definiteness we plot the results for
universal soft terms with , . This choice of values is motivated by modulus dominance
SUSY breaking in string scenarios, see e.g. [30],[17].
However, as we will explain later, other different choices for soft parameters lead to analogous results.
The grey bands correspond to the computation of the mass for
tan and . The results are similar to those obtained in
ref.[13], [14], [15], [16].
The other colored bands correspond to the
Higgs mass values obtained under the assumption of Higgs parameter unification as in eq.(2.3).
Results are displayed for a mu-term with the value for computed from the
obtained running soft terms
^{2}^{2}2The results are very weakly dependent on the sign of through
the appearing in the threshold corrections.. The width of the grey and colored bands corresponds to the
error from the top quark mass. Finally the horizontal band corresponds to
the average CMS and ATLAS results for the Higgs mass (we take , see
[16]).

The figure shows that above a scale GeV the value of the Higgs mass is contained in the range

(3.13) |

This is remarkably close to the measured value at LHC and supports the idea that SUSY and unification underly the observed Higgs mass. This result is quite independent of the details of the soft terms. Below GeV the Higgs mass becomes more model dependent. In particular the Higgs mass is reduced as increases. This is easy to understand from eq.(3.7) since for larger the ratio approaches one, yielding tan. One still gets a Higgs mass consistent with LHC results for not too large . As one approaches TeV one reaches the region of standard fine-tuned MSSM with a Higgs mass which may be as large as 130 GeV. As we approach that region our treatment becomes incomplete since some neglected SUSY threshold corrections beyond those in (3.12) become important, and the SUSY spectrum spreads out. However, that region corresponds to the well understood situation of the MSSM with a heavy SUSY spectrum with masses in the 10-100 TeV region.

Let us finally note that, within uncertainties, the figure also favours values for the SUSY breaking scale GeV.

One may also interpret graphically the above results in terms of the unification of the SM Higgs self-coupling and the SUSY predicted self-coupling . This is depicted in fig.LABEL:plotmass2, in which we have not included the uncertainty from the error to avoid clutter. Note that the dependence of on is qualitatively similar to the running of . This may be understood as follows. In the definition of , runs very little and remains practically constant. On the other hand one has . The difference on the numerator goes like , which is also the order of the leading correction to the coupling.

## 4 Model dependence

In this section we discuss different model dependent possibilities which arise depending on the structure of the underlying soft terms. With sufficiently precise information about the top quark and Higgs masses one may obtain interesting constraints on the possible structure of the SUSY-breaking terms.

Let us concentrate first in the case with universal soft terms and but still keeping the
relationships , . As we said these values are interesting since, as discussed in ref.[3], they
may be understood as arising from a Giudice-Masiero mechanism in a modulus dominance SUSY breaking scheme.
The dependence of the Higgs mass as a function of in this particular case
is shown in fig.LABEL:plotguay with the red band, a zoom is provided in fig.LABEL:plotmas2.
Given the uncertainties, in this particular case () essentially any value for in the
GeV region is consistent with the observed Higgs mass, although regions around and
GeV are slightly favoured.
This second possibility with GeV was explored in [3] (see also[11])
in which it was argued
that such intermediate SUSY breaking may be interesting for two additional reasons
^{3}^{3}3An additional interesting property is that for GeV such models do not require the implementation of doublet-triplet splitting nor
R-parity preservation. No Polony problem is present either [3]..
On one hand this scale naturally appears in string compactifications in which SUSY breaking is induced by closed string fluxes.
Indeed in such a case one has [3]

(4.1) |

where is the string coupling, is the unified fine structure constant and is the Planck mass. For and GeV one indeed gets GeV. The second reason is that in those constructions an axion with a scale GeV appears, which is consistent with axions providing for the dark matter in the universe. In this case, using eqs.(3.6),(4.1) one obtains GeV and GeV. Values this low for the unification scale can still be compatible with proton decay constraints [3]. Computing the Higgs mass following the procedure described in the previous section one obtains in this case

(4.2) |

where the error includes only that coming from the top mass uncertainty. This is clearly consistent with the findings at ATLAS and CMS. In this scheme with an intermediate scale the Higgs self-coupling unifies with its SUSY extension as depicted in fig.LABEL:plotmass3 (left) . The soft masses evolve logarithmically from down to as depicted in fig.LABEL:plotmass3 (right). The value of tan increases as the value of decreases and remains almost constant, so that tan increases as decreases.

It is interesting to explore how relaxing the above mentioned relationships , modify the results for the Higgs mass. In fig.LABEL:plotmass6 (up) we show how the prediction for the Higgs mass is changed as one varies the value of away from . The figure remains qualitatively the same but one observes that as increases the Higgs mass tends to be lighter. Above GeV the Higgs mass remains in the region GeV. The effect of varying away from is also shown in fig.LABEL:plotmass6 (down). Although we have not included the error coming from the top quark mass to avoid clutter, one concludes that the overall structure remains the same and the Higgs mass stays around GeV for GeV. However now values of in between 100 TeV and GeV are consistent with the observed Higgs mass for particular choices of soft terms.

Let us finally comment that our results do not directly apply to the case of Split SUSY [8, 12] in which one has , since then the effect of light gauginos and Higgssinos should be included in the running below . In that case however it has been shown (see e.g.[13, 14, 15]) that split SUSY is only consistent with a 126 GeV Higgs for TeV and no intermediate scale scenario is possible. Essentialy Split SUSY becomes a fine-tuned version of the standard MSSM. One relevant issue is also that in Split SUSY, due to the smallness of gaugino masses, in running down from the unification scale the scalar quarks of the third generation may easily become tachyonic, which restricts a lot the structure of the possible underlying SUSY breaking terms [12].

## 5 Discussion

In this paper we have argued that the evidence found at LHC for a Higgs-like particle around GeV supports the idea of an underlying Supersymmetry being present at some (not necesarily low) mass scale. Even if the SUSY breaking scale is high, SUSY identities for dimension four operators remain true to leading order. In particular, the quartic Higgs self-coupling is related to the EW gauge couplings at the SUSY breaking scale, yielding constraints on the Higgs mass. The presence of SUSY restores the stability of the SM Higgs potential which otherwise becomes unstable at high scales.

It is remarkable that the simple assumption of Higgs mass parameter unification at the unification scale and minimal fine-tuning directly predict a Higgs mass in the range GeV, consistent with LHC results, for a SUSY scale GeV. For smaller values of the Higgs mass tends to the value of a standard fine-tuned MSSM scenario with GeV. Both situations with (relatively) low and High scale SUSY are consistent with the Higgs data (see e.g. fig.LABEL:plotmas2). Since in the context of the SM any value from e.g 100 GeV to 1 TeV would have been possible, one may interpret this result as indirect evidence for an underlying SUSY.

It has been argued that the fact that near the Planck scale and that the SM Higgs potential seems to be close to metastability could have some deep meening [31] . In our setting with SUSY at a large scale the quartic coupling , is always positive definite and no such instability arises. The smallness of is due to the fact that the EW couplings are small and that the boundary condition at the unification scale keeps tan close to one for GeV. As discussed in the previous section, such a situation with an intermediate scale SUSY breaking and gauge coupling unification may be naturally embedded in string theory compactifications like those resulting from F-theory unification. The embedding into string theory is also suggested in order to understand the required fine-tuning in terms of the string landscape of compactifications.

LHC at 13 TeV will be able to test the low SUSY breaking regime for squark and gluino masses of the order of a few TeV. If no direct trace of SUSY or any other alternative new physics is found at LHC, the case for a fine-tuning/landscape approach to the hierarchy problem will become stronger. Still, as we have argued, heavy SUSY may be required for the stability of the Higgs potential and we have shown that the value GeV is generic for GeV (see e.g. figures LABEL:plotguay and LABEL:plotmass6), hinting to a heavy SUSY scale.

One apparent shortcoming of High scale SUSY is that we lose the posibility of using the lightest neutralino as a dark matter candidate. In this context the case of an intermediate scale SUSY breaking GeV is particularly interesting. Indeed, as recently discussed in [3] (see also [32, 33]), such scale may be compatible with an axion with decay constant GeV, appropriate to provide for the required dark matter in the universe. Furthermore, gauge coupling unification may elegantly be accomodated due to the presence of small threshold corrections as discussed in [3].

Although a large scale for SUSY makes it difficult to test this idea directly at accelerators, indirect evidence could be obtained. Improved precission on the measured values of both the top quark and the Higgs masses (e.g. at a linear collider) can make the constraints on specific High SUSY breaking models and the Higgs mass predictions more precise, along the lines discussed in the previous section. Going beyond the next to leading order in the Higgs mass computation would also be required, see [21]. If those measurements were precise enough, specific choices of soft terms and SUSY breaking scenarios could be ruled out or in. Additional evidence in favour of an intermediate scale SUSY secenario could come from dark matter axion detection in microwave cavity search experiments like ADMX [34]. Furthermore, since in models with large SUSY breaking scale the unification scale typically decreases, proton decay rates could also be at the border of detectability [3]. Finally, if any deviation from the SM expectations is found at low energies (like e.g. an enhanced Higgs rate to ) the idea of a large SUSY scale with a fine-tuned Higgs would be immediately ruled out.

Acknowledgments

We thank P.G. Cámara, A. Casas, J.R. Espinosa and F. Marchesano for useful discussions. This work has been partially supported by the grants FPA 2009-09017, FPA 2009-07908, Consolider-CPAN (CSD2007-00042) from the MICINN, HEPHACOS-S2009 /ESP1473 from the C.A. de Madrid and the contract “UNILHC” PITN-GA-2009-237920 of the European Commission. We also thank the spanish MINECO Centro de excelencia Severo Ochoa Program under grant SEV-2012-0249.

## Appendix A Renormalization group equations

Here we first present the renormalization group equations at two loops for the SM couplings (the three gauge couplings, the top Yukawa and the Higgs quartic coupling).

(A.1) | |||||

(A.2) | |||||

(A.3) |

(A.4) | |||||

(A.5) | |||||

And finally the RGE (at 2 loops for gauge couplings, leading order in ) for the SUSY case:

(A.6) | |||||

(A.7) | |||||

(A.8) | |||||

(A.9) |

## Appendix B RGE solutions for the soft terms

Here we display all the functions that appear in the solution of the RGE for the Higgs mass parameters and (see ref.[28]).

First we define the functions

(B.1) |

with and . The beta-functions coefficients for the SUSY case are and we define for , where is the unified coupling at . In our case the couplings do not strictly unify, only up to corrections. In the numerical computations we take the average value of the three couplings at , which is enough for our purposes.

We then define the functions in eqs.(3.7)

(B.2) | |||||

where and . The functions and are independent of the top Yukawa coupling, only depend on the gauge coupling constants and are given by

(B.3) |

where and are defined by

(B.4) |

The low energy of the top mass may be obtained from the solutions of the one-loop renormalization group equations, devided into two pieces, SUSY and non-SUSY, i.e. (here )

(B.5) |

where

(B.6) |

The functions are as defined above, with and , while the functions are analogous to but replacing the and anomalous dimensions by the non-SUSY ones, i.e.

(B.7) |

with , and .
For the anomalous dimensions we have made the change in the definition of
.
And we take the value of computed in eq.(3.3) taking into account the threshold corrections at electroweak scale. For this particular computation we take actually as electroweak scale the top mass, so .

Finally, in order to compute the value of the stop mixing parameter we need the following equations for the running of the soft parameters: