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Physics Letters B

www.elsevier.com/locate/physletb

The Higgs mass and natural supersymmetric spectrum from the landscape

Howard Baera *, Vernon Bargerb, Michael Savoya, Hasan Sercea

CrossMark

a Dept. of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA b Dept. of Physics, University of Wisconsin, Madison, WI 53706, USA

A R T I C L E I N F 0

Article history: Received 15 April 2016 Accepted 3 May 2016 Available online 6 May 2016 Editor: M. Cvetic

A B S T R A C T

In supersymmetric models where the superpotential // term is generated with / ^ msoft (e.g. from radiative Peccei-Quinn symmetry breaking or compactified string models with sequestration and stabilized moduli), and where the string landscape 1. favors soft supersymmetry (SUSY) breaking terms as large as possible and 2. where the anthropic condition that electroweak symmetry is properly broken with a weak scale mW Z h ~ 100 GeV (i.e. not too weak of weak interactions), then these combined land-scape/anthropic requirements act as an attractor pulling the soft SUSY breaking terms towards values required by models with radiatively-driven naturalness: near the line of criticality where electroweak symmetry is barely broken and the Higgs mass is ~ 125 GeV. The pull on the soft terms serves to ameliorate the SUSY flavor and CP problems. The resulting sparticle mass spectrum may barely be accessible at high-luminosity LHC while the required light higgsinos should be visible at a linear e+e- collider with .s/s > 2m(higgsino).

© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

The Standard Model is afflicted with several naturalness problems:

1. in the electroweak sector, why is the Higgs mass mh — 125 GeV so light when quadratic divergences seemingly destabilize its mass [1] and

2. why is the QCD Lagrangian term Ga/v Gxv so tiny (0 < 10-10 from measurements of the neutron electric dipole moment) when its existence seems a necessary consequence of the 0 vacuum solution to the U(1)A problem (the strong CP problem) [2]?

3. A third naturalness problem emerges when gravity is included into the picture: why is the cosmological constant A — 10-47 GeV4 ^ M4 so small when there is no known mechanism for its suppression [3]?

Each of these problems requires an exquisite fine-tuning of parameters to maintain accord with experimental data. Such fine-tuning is thought to represent some pathology with or missing element

* Corresponding author.

E-mail addresses: baer@nhn.ou.edu (H. Baer), barger@pheno.wisc.edu (V. Barger), savoy@nhn.ou.edu (M. Savoy), serce@ou.edu (H. Serce).

within the underlying theory and cries out for a "natural" solution in each case.

The most compelling solution to problem #1 is to extend the spacetime symmetry structure which underlies quantum field theory to include its most general structure: the super-Poincare group which includes supersymmetry (SUSY) transformations [4,5]. The extended symmetry implies a Fermi-Bose correspondence which guarantees cancellation of quadratic divergences to all orders in perturbation theory. Supersymmetrization of the SM implies the existence of superpartner matter states with masses of order MS ~ 1 TeV [5,6]. Searches are underway at the CERN LHC for evidence of the superpartner matter states.

The most compelling solution to problem #2 is to postulate an additional spontaneously broken global Peccei-Quinn (PQ) symmetry and its concomitant axion field a which induces additional potential contributions that allow the offending CP violating term to dynamically settle to a tiny value [7-9]. Searches for the physical axion field are proceeding at experiments like ADMX [10] but so far sensitivity has barely reached parameter values needed to solve the strong CP problem.

At present the leading solution to problem #3 is the hypothesis of the landscape: a vast number of string theory vacua states each with different physical constants [11]. In this case, the cosmolog-ical constant ought to be present, but if it is too large, then the universe would expand too quickly to allow for galaxy condensa-

http://dx.doi.org/10.1016/j.physletb.2016.05.010

0370-2693/© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

tion and there would be no observers present to measure A. This "anthropic" explanation for the magnitude of A met with great success by Weinberg [12] who was able to predict its value to within a factor of a few even well before it was measured [13].

While the SUSY solution to the scalar mass problem seems convincing at the level of quadratic divergences, there is a high level of concern that the fine-tuning problem has re-arisen in light of 1. the apparently severe LHC bounds on sparticle masses and 2. the rather high measured value of mh. This perception arises from two viewpoints on measuring naturalness.

• Log-divergent contributions to the Higgs mass Smh ~

m2 log (a2/ m?) become large for TeV-scale top squark

masses m~t and A as high as mGUT ~ 2 x 1016 GeV [14]. This argument has been challenged in that a variety of inter-dependent log terms, some positive and some negative, contribute to the Higgs mass. Evaluation of the combined log terms via renormalization group equations reveals the possibility of large cancellations in evaluation of the Higgs mass [15,16].

• The EENZ/BG fine-tuning measure [17] ABG = maxj | d'oggmo^ I (where pj are fundamental parameters of the theory) is traditionally evaluated using the various soft SUSY breaking terms as fundamental parameters. In this case, low ABG favors sparticle masses in the 100 GeV range. These evaluations have been challenged in that in more fundamental theories, the soft terms are not independent, but are derived in terms of more fundamental quantities, for instance the gravitino mass m3/2 in supergravity theories. Evaluation of ABG instead in terms of ¡2 and m2/2 allows for just i and mHu to be ~ 100 GeV while the other sparticles can safely lie at or beyond the TeV scale [15,16].

A more conservative measure which is in accord with the above (corrected) measures is to evaluate just the weak scale contributions to the Z mass. The minimization condition for the Higgs potential Vtree + AV in the minimal supersymmetric Standard Model (MSSM) reads

+ & — (ml + ^U) tan2 p

tan2 p — 1

— ¡

The radiative corrections and include contributions from various particles and sparticles with sizeable Yukawa and/or gauge couplings to the Higgs sector. Expressions for the and are given in the Appendix of Ref. [18].

A naturalness measure AEW has been introduced [18,19] which compares the largest contribution on the right-hand-side of Eq. (1) to the value of m2Z/2. If they are comparable (AEW < 10-30), then no unnatural fine-tunings are required to generate mZ = 91.2 GeV. The main requirement for low fine-tuning is then that

• \¡\ ~ mZ [20-22] (with ¡ > 100 GeV to accommodate LEP2 limits from chargino pair production searches) and also that

• m2HU is driven radiatively to small, and not large, negative values" [18,19]. Also,

• the top squark contributions to the radiative corrections S¡¡ (t1>2) are minimized for TeV-scale highly mixed top squarks [19]. This latter condition also lifts the Higgs mass to mh ~ 125 GeV.

• First and second generation squark and slepton masses may range as high as 10-20 TeV with little cost to naturalness [18,23].

The typical low AEW SUSY mass spectra is characterized by 1. a set of light higgsinos W± and Z12 with masses ~ 100-200 GeV, 2. gluinos with mass mg ~ 1.5-4 TeV, 3. highly mixed stops with mass m~t1 < 3 TeV and m~t2 < 8 TeV. Several versions of super-gravity GUT models have been found to generate such "natural" spectra [24]. For instance, the two-extra-parameter nonuniversal Higgs mass model [25] (NUHM2) with matter scalars m0 ~ 3-10 TeV, m1/2 ~ 0.5-2 TeV, A0 ~ ±(1-2)m0 and tanp ~ 10-30 with mHu ~ (1.3-2)m0 and mHi ~ mA ~ 1-8 TeV produces spectra with AEW < 30. In particular, the up-Higgs soft mass is as large as possible such that the RG running of m2HU nearly cancels out its GUT-scale boundary value m2H (A), i.e. mHu runs to small weak scale values ~ —(100-200)2 GeV2 so that electroweak symmetry is barely broken. The soft terms, especially mHu, lie on the edge of criticality: if mHu is much bigger, then EW symmetry does not get broken while if m2H (A) is much smaller, then it would likely generate a value of mZ far beyond its measured value of 91.2 GeV.

While such effective theory parameters can successfully generate natural SUSY mass spectra, the question arises: is there some mechanism which favors parameters which barely break EW symmetry, and which generate a weak scale mW,Z h ~ 100 GeV rather than say in the TeV range? In this letter, we argue that the string landscape - which provides some understanding for the small but non-zero cosmological constant - also favors soft SUSY breaking terms as large as possible such that they generate a universe which is habitable for observers: if the soft parameters were much larger, then they would lead to a vacuum state with color breaking minima, or unbroken EW symmetry or if they were much smaller they would generate a weak scale characterized by the TeV regime. In the latter case, with mW,Z h ~ 1-10 TeV, then weak interactions would be far weaker than in our universe: then for instance nuclear fusion reactions would be sufficiently suppressed so that heavy element production in stars and in the early universe would be far different from that of our universe, likely leading to a universe with chemistry unsuitable for life forms as we known them.

This topic of anthropic selection of soft SUSY breaking terms has been addressed previously by Giudice and Rattazzi [26] with some follow-up work in Refs. [27] (for mixed moduli-anomaly mediated SUSY breaking models) and [28] (for mSUGRA/CMSSM model). One of the main differences of our work here is in the treatment of the superpotential ¡ parameter and the so-called SUSY i problem. Under the Giudice-Masiero mechanism [29], where ¡ arises from Higgs doublet couplings to the hidden sector via the Kahler potential, then ¡ is expected to have magnitude of order the other soft terms: \¡\ ~ m3/2. Alternatively, in the Kim-Nilles mechanism [30] - which is assumed here as an axionic solution to the strong CP problem - ¡ is initially forbidden by the requirement of Peccei-Quinn symmetry, but is then re-generated upon spontaneous PQ symmetry breaking at a scale fa ~ 1011 GeV with a value ¡i ~ f2/MP ^ m3/2. In models where PQ symmetry breaking is induced radiatively, then values of m3/2 ~ 10 TeV easily produce ¡ values around 100-200 GeV [31,32]. In classes of compactified string models with sequestration between the visible sector and the SUSY breaking sector and with stabilized moduli fields [33], it is also found that ¡i ^ MS where MS stands for the approximate scale of the collective soft SUSY breaking terms. In this letter we will assume

• the superpotential ¡ term has been generated by some mechanism such as Ref. [32] or Ref. [33] to be small, comparable to mh = 125 GeV.

Then, instead of fixing mZ at its measured value, we will invert the usual usage of Eq. (1) to calculate mW,z,h ~ mweak as an output depending on high scale values of the soft terms and a small value of f.1

In the following, we will assume gravity-mediated supersym-metry breaking [34]. Gravity-mediation is supported by the large value of mh ~ 125 GeV which requires a large trilinear A0 term (generic in gravity-mediation) to provide substantial mixing in the stop sector and consequently a boost in the radiative corrections to the light Higgs mass [35,36]. Gravity-mediated SUSY breaking can be parametrized by the presence of a spurion superfield X = 1 + 62 FX where the auxiliary field FX obtains a vev which we also denote by FX (here 6 are anti-commuting superspace coordinates). Under SUSY breaking via the superHiggs mechanism, then the gravitino gains a mass m3/2 ~ FX/MP where MP = 2.4 x 1018 GeV is the reduced Planck mass. The soft SUSY breaking terms are then all calculable as multiples of m3/2 [37]. Motivated by su-pergravity grand unified theories (SUSY GUTs), here we assume the soft breaking terms valid at Q = mGUT ~ 2 x 1016 GeV include m0 (a common matter scalar mass term), m1/2 (a common gaugino mass), A0 (a common trilinear soft term) and B. The latter soft term can be traded for the more common ratio of Higgs vevs tan p = vu /vd via the electroweak minimization conditions. We also assume separate Higgs scalar soft terms m2Hu and m2Hi since the Higgs superfields live in different GUT representations than matter superfields [25]. It is convenient to denote collectively the superpartner mass scale MS as the generic scale of soft terms.

It is reasonable to assume in the landscape that any value of the complex-valued field FX is equally likely. In this case, one expects the magnitude of soft breaking terms to statistically scale linearly in MS (the likelihood of a given value of MS is proportional to the area of an annulus 2n FX8 FX in the complex FX plane). This is important because then we see a statistical draw of soft terms towards their largest values possible (while f remains far smaller). In Ref. [26], additional arguments are presented that the likelihood of soft terms MS scale as a power of MS; for our purposes here, we merely rely on a likely statistical draw by the landscape of vacua towards higher values of soft terms. This draw is to be balanced by the anthropic requirements that 1. electroweak symmetry is appropriately broken (no charge or color breaking minima of the Higgs potential) and 2. that the weak scale is typified by the values of mweak ~ mW,z,h ~ 100 GeV. Rates for nuclear fusion reactions and beta decays all scale as 1/mWeak so that heavy element production in BBN and in stars would be severely altered for too large a value of mweak; see Ref. [38] for discussion.

Armed with a notion of both the statistical and anthropic pull from the landscape, we may examine the soft SUSY breaking terms. First, we expect the matter scalar mass m0 as large as possible while maintaining mweak ~ 100 GeV. If m0 gets much beyond the 10 TeV scale, then the weak scale top squark masses m^ 2 become too large, increasing the radiative corrections SU (t1t2) in Eq. (1). For fixed f ~ 100-200 GeV, then this increases the resultant weak scale well beyond the anthropic target 100-200 GeV. Re-interpreting the limits on m0 from Refs. [18,23] requires m0 < 10 TeV for mweak ~ 100 GeV. Such large values of m0 go a long ways towards solving the SUSY flavor and CP problems via a decoupling solution [39].

Likewise, we expect the gaugino mass m1/2 as large as possible whilst maintaining mweak ~ 100-200 GeV. If the gaugino masses are too large, then they feed into the stop masses via RG running

Fig. 1. Contours of mweak in the A0 vs. m0 plane for m1/2 = 1 TeV, mHu = 1.3mo, tan p = 10 and mHd = 1 TeV. The arrows show the direction of statistical/anthropic pull on soft SUSY breaking terms. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

1 In this case, low values of AEW can be re-interpreted as the likelihood to gen-

erate the weak scale mweak ~ 100 GeV: i.e. mweak = ^AEWm2Z/2.

Fig. 2. Contours of mweak (blue) in the mHu vs. m1/2 plane for m0 = 5 TeV, A0 = —8 TeV, tan p = 10 and mHd = 1 TeV. Above the black dashed contour is where mh > 124 GeV. The red region has mweak < 0.5 TeV. The arrows show the direction of the statistical/anthropic pull on soft SUSY breaking terms. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

and again the SU (£1,2) become too large. For mweak ~

100 GeV,

then typically m1/2 < 2 TeV leading to a gluino mass bound mg < 4-5 TeV: well above the reach of LHC14 [40].

What of the trilinear soft term A0? In Fig. 1 we show the A0 vs. m0 plane for the NUHM2 model with m1/2 fixed at 1 TeV, tan p = 10 and mHd = 1 TeV. We take mHu = 1.3m0. The plane is qualitatively similar for different reasonable parameter choices. We expect A0 and m0 statistically to be drawn as large as possible while also being anthropically drawn towards mweak ~ 100-200 GeV, labelled as the red region where mweak < 500 GeV. The blue region has mweak > 1.9 TeV and the green contour labels mweak = 1 TeV. The arrows denote the combined statistical/anthropic pull on the soft terms: towards large soft terms but low mweak. The black contour denotes mh = 123 GeV with the regions to the upper left (or upper right, barely visible) containing larger values of mh. We see that the combined pull on soft terms brings us to the region where mh ~ 125 GeV is generated. This region is characterized by highly mixed TeV-scale top squarks [35,36]. If instead A0 is pulled too large, then the stop soft term mU3 is driven tachyonic resulting in charge and color breaking minima in the scalar potential (labelled CCB). If m0 is pulled too high for fixed A0, then electroweak symmetry isn't even broken.

In Fig. 2, we show contours of mweak in the mHu vs. m1/2 plane for m0 = 5 TeV, A0 = —8 TeV, tanp = 10 and mHd = 1 TeV. The

Fig. 3. Evolution of the soft SUSY breaking mass squared term sign(mHu\m2Hu | vs. Q for the case of no EWSB (upper), criticality (middle) as in radiatively-driven natural SUSY (RNS) and mweak ~ 3 TeV (lower). Most parameters are the same as in Fig. 2.

statistical flow is to large values of soft terms but the anthropic flow is towards the red region where mweak < 0.5 TeV. While m\/2 is statistically drawn to large values, if it is too large then, as before, the ti>2 become too heavy and the SU (ti,2) become too large so that mweak becomes huge. The arrows denote the direction of the combined statistical/anthropic flow. The region above the black dashed contour has mh > 124 GeV. The value of mHu(GUT) would like to be statistically as large as possible but if it is too large then EW symmetry will not break. Likewise, if mHu (GUT) is not large enough, then it is driven to large negative values so that mweak ~ the TeV regime and weak interactions are too weak. The situation is shown in Fig. 3 where we show the running of sign(m2Hu |mHu | versus energy scale Q for several values of mHu(GUT) for mi/2 = 1 TeV and with other parameters the same as Fig. 2. Too small a value of mHu(GUT) leads to too large a weak scale while too large a value results in no EWSB. The combined statistical/anthropic pull is for barely-broken EW symmetry where soft terms teeter on the edge of criticality: between breaking and not breaking EW symmetry. This yields the other naturalness condition that mHu is driven small negative: then the weak interactions are of the necessary strength. These are just the same conditions for supersymmetric models with radiatively-driven natural SUSY (RNS) [18,19].

Summary: The naturalness condition of no large unnatural cancellations in mZ h requires small higgsino mass i ~ 100-200 GeV, mHu driven small rather than large negative and not-too-large radiative corrections SU(i). There are mechanisms where i ^ MS - such as radiative PQ breaking - but is it merely luck that the soft terms are poised to be just large enough to guarantee also that mweak ~ 100 GeV? Here, we argue that the statistical landscape pull towards large soft terms coupled with the anthropic pull towards the Goldilocks condition - small enough to break EW symmetry but not so small as to suppress weak interactions -gives the required conditions for SUSY with radiatively-driven naturalness and barely broken EW symmetry. While sparticles may barely be accessible to LHC, the required light higgsinos should be accessible to an e+e- collider with «fs > 2m(higgsino). We also expect ultimately detection of a higgsino-like WIMP [41] along with the axion.

Acknowledgements

We thank Jake Baer for an inspiring essay on the landscape and Xerxes Tata for comments on the manuscript. This work was supported in part by the US Department of Energy, Office of High Energy Physics.

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