Sparticle and Higgs boson masses from the landscape: dynamical versus spontaneous supersymmetry breaking

Perturbative supersymmetry breaking on the landscape of string vacua is expected to favor large soft terms as a power-law or log distribution, but tempered by an anthropic veto of inappropriate vacua or vacua leading to too large a value for the derived weak scale -- a violation of the atomic principle. Indeed, scans of such vacua yield a statistical prediction for light Higgs boson mass m_h~ 125 GeV with sparticles (save possibly light higgsinos) typically beyond LHC reach. In contrast, models of dynamical SUSY breaking (DSB) -- with a hidden sector gauge coupling g^2 scanned uniformly -- lead to gaugino condensation and a uniform distribution of soft parameters on a log scale. Then soft terms are expected to be distributed as $m_{\rm soft}^{-1}$ favoring small values. A scan of DSB soft terms generally leads to $m_h\ll 125$ GeV and sparticle masses usually below LHC limits. Thus, the DSB landscape scenario seems excluded from LHC search results. An alternative is that the exponential suppression of the weak scale is set anthropically on the landscape via the atomic principle.


Introduction
One of the mysteries of nature is the origin of mass scales. At least in QCD, we have an answer: the hadronic mass scale can arise when the gauge coupling evolves to large values such that the fundamental constituents, the quarks, condense to bound states. From dimensional transmutation, the proton mass can be found even in terms of the Planck mass m P l via m proton m P l exp(−8π 2 /g 2 ) which gives the right answer for g 2 ∼ 1.8.
Another mass scale begging for explanation is that associated with weak interactions: m weak m W,Z,h ∼ 100 GeV. In the Standard Model (SM), the Higgs mass is quadratically divergent so one expects m h to blow up to the highest mass scale Λ for which the SM is the viable low energy effective field theory (EFT). Supersymmetrization of the SM eliminates the Higgs mass quadratic divergences so any remaining divergences are merely logarithmic [1,2]: the minimal supersymmetric Standard Model, or MSSM [3], can be viable up to the GUT or even Planck scales. In addition, the weak scale emerges as a derived consequence of the visible sector SUSY breaking scale m sof t . So the concern for the magnitude of the weak scale is transferred to a concern for the origin of the soft breaking scale. In gravity mediated SUSY breaking models 1 , it is popular to impose spontaneous SUSY breaking (SSB) at tree level in the hidden sector, for instance via the SUSY breaking Polonyi superpotential [9]: W = m 2 hidden (ĥ + β) whereĥ is the lone hidden sector field. For β = (2 − √ 3)m P (with m P the reduced Planck mass m P ≡ m P l / √ 8π and m hidden ∼ 10 11 GeV) then one determines m sof t ∼ m 3/2 ∼ m weak . Thus, the exponentially-suppressed hidden sector mass scale must be put in by hand, so SSB can apparently only accommodate, but not explain, the magnitude of the weak scale. 2 A more attractive mechanism follows the wisdom of QCD and seeks to generate the SUSY breaking scale from dimensional transmutation, which automatically yields an exponential suppression. This is especially attractive in string models where the Planck scale is the only mass scale available. Then one could arrange for dynamical SUSY breaking (DSB) [1,13,14] (for reviews, see [15][16][17]) wherein SUSY breaking arises non-perturbatively. 3 Some possibilities include hidden sector gaugino condensation [19], where a hidden sector gauge group such as SU (N ) becomes confining at the scale Λ GC and a gaugino condensate occurs with λλ ∼ Λ 3 GC leading to SUSY breaking with soft terms m sof t ∼ Λ 3 GC /m 2 P . The associated hidden mass scale [20] is given by where then m 2 hidden ∼ Λ 3 GC /m P . Another possibility is non-perturbative SUSY breaking via instanton effects which similarly leads to an exponential suppression of mass scales [21]. Of 1 In days of yore, gauge mediated SUSY breaking (GMSB) models [4] were associated with dynamical SUSY breaking in that they allowed much lighter gravitinos. In GMSB models, the trilinear soft term A 0 is expected to be tiny, leading to too light a Higgs boson mass unless soft terms are in the 10-100 TeV regime [5][6][7]. Such large soft terms then lead to highly unnatural third generation scalars. For this reason, we focus on DSB in a gravity-mediation context [8].
2 A related problem is how the SUSY conserving µ parameter is also generated at or around the weak scale. A recent explanation augments the MSSM by a Peccei-Quinn (PQ) sector plus a Z R 24 discrete R-symmetry [10] which generates a gravity-safe accidental approximate U (1) P Q which solves the strong CP and SUSY µ problems, and leads to an axion decay constant f a ∼ m hidden whilst µ ∼ m weak [11]. A recent review of 20 solutions to the SUSY µ problem is given in Ref. [12]. 3 The DSB scenario has been made more plausible in recent years with the advent of metastable DSB [17,18].
course, now the mass scale selection problem has been transferred to the selection of an appropriate value of g 2 hidden . A solution to the origin of mass scales also arises within the string landscape picture [22,23]. This picture makes use of the vast array of string vacua found in IIB flux compactifications [24]. Some common estimates from vacuum counting [25] are N vac ∼ 10 500 − 10 272,000 [26,27]. The landscape then provides a setting for Weinberg's anthropic solution to the cosmological constant problem [28]: the value of Λ cc is expected to be as large as possible such that the expansion rate of the early universe allows for galaxy condensation, and hence the structure formation that seems essential for the emergence of life.
Can similar reasoning be applied to the origin of the weak scale, or better yet, the origin of the SUSY breaking scale? This issue has been explored initially in Ref's [29], [30] and [31]. Here, one assumes a fertile patch of the landscape of vacua where the MSSM is the visible sector low energy EFT. The differential distribution of vacua is expected to be of the form where f SUSY (m 2 hidden ) contains the distribution of SUSY breaking mass scales expected on the fertile patch and f EW SB contains the anthropic weak scale selection criteria. Denef and Douglas have argued that the cosmological constant selection acts independently and hence does not affect landscape selection of the SUSY breaking scale [26].
For SSB, then SUSY breaking F i -and D α -terms are expected to be uniformly distributed across the landscape, the first as complex numbers and the latter as real numbers [30]. This would lead, in the case of spontaneous SUSY breaking, to a power law distribution of soft terms where n = 2n F + n D − 1 and n F are the number of hidden sector SUSY breaking F -fields and n D is the number of hidden sector D-breaking fields contributing to the overall SUSY breaking scale. Such a distribution would tend to favor SUSY breaking at the highest possible mass scales for n ≥ 1. Also, Broeckel et al. [32] analyzed the distributions of SUSY breaking scales from vacua for KKLT [33] and LVS [34] flux compactifications and found for the KKLT model that f SUSY ∼ m 2 sof t while the LVS model gives f SUSY ∼ log(m sof t ) [35]. For the anthropic selection, an initial guess was to take f EWSB = (m weak /m sof t ) 2 corresponding to a simple fine-tuning factor which invokes a penalty for soft terms which stray too far beyond the measured value of the weak scale. As emphasized in Ref. [36] and [37], this breaks down in a number of circumstances: 1. soft terms leading to charge-or-color-breaking (CCB) vacua must be vetoed, not just penalized, 2. soft terms for which EW symmetry doesn't even break also ought to be vetoed (we label these as noEWSB vacua), 3. for some soft terms, the larger they get, then the smaller becomes the derived value of the weak scale. To illustrate this latter point, we write the pocket universe (PU) [38] value of the weak scale in terms of the pocket-universe Z-boson mass m PU Z and use the MSSM Higgs potential minimization conditions to find: where m 2 H u,d are Higgs soft breaking masses, µ is the superpotential Higgsino mass arising from whatever solution to the SUSY µ problem is invoked, and tan β ≡ v u /v d is the ratio of Higgs field vevs. The Σ u u and Σ d d contain over 40 1-loop radiative corrections, listed in the Appendix of Ref. [39]. The soft term m 2 Hu must be driven to negative values at the weak scale in order to break EW symmetry. If its high scale value is small, then it is typically driven deep negative so that compensatory fine-tuning is needed in the µ term. If m 2 Hu is too big, then it doesn't even run negative and EW symmetry is unbroken. The landscape draw to large soft terms pulls m 2 Hu big enough so EW symmetry barely breaks, corresponding to a natural value of m 2 Hu at the weak scale. (this can be considered as a landscape selection mechanism for tuning the high scale value of m 2 Hu to such large values that its weak scale value becomes natural.) Also, for large negative values of trilinear soft term A t , then large cancellations occur in Σ u u (t 1,2 ) leading to more natural Σ u u values and a large m h ∼ 125 GeV due to large stop mixing in its radiative corrections. Also, large values of first/second generation soft scalar masses m 0 (1, 2) cause stop mass soft term running to small values, thus also making the spectra more natural [40].
The correct anthropic condition we believe was set down by Agrawal, Barr, Donoghue and Seckel (ABDS) in Ref. [41]. In that work, they show that for variable values of the weak scale, then nuclear physics is disrupted if the pocket-universe value of the weak scale m PU weak deviates from our measured value m OU weak by a factor 2 − 5. For values of m PU weak outside this range, then nuclei and hence atoms as we know them wouldn't form. In order to be in accord with this atomic principle, then to be specific, we require m PU weak < 4m OU weak . In the absence of fine-tuning of µ, this requirement is then the same as requiring the electroweak fine-tuning measure [39,42] ∆ EW < 30. Thus, we require as the anthropic condition while also vetoing CCB and noEWSB vacua. For the case of dynamical SUSY breaking, the SUSY breaking scale is expected to be of the form m 2 hidden ∼ m 2 P exp(−8π 2 /g 2 hidden ) where in the case of gaugino condensation, g hidden is the coupling constant of the confining hidden sector gauge group. It is emphasized by Dine et al. [43][44][45] and by Denef and Douglas [46] that the coupling g 2 hidden is expected to scan uniformly on the landscape. According to Fig. 1, for g 2 hidden values in the confining regime ∼ 1 − 2, we expect a uniform distribution of soft breaking terms on a log scale: i.e. each possible decade of values for m sof t is as likely as any other decade. Thus, with m sof t ∼ m 2 hidden /m P ∼ Λ 3 GC /m 2 P , we would expect which provides a uniform distribution of m sof t across the decades of possible values 4 . Such a distribution of course favors the lower range of soft term values.
We adopt the Isajet [54] code for calculation of the Higgs and superparticle mass spectrum [55] based on 2-loop RGE running [56] along with sparticle and Higgs masses calculated at the RG-improved 1-loop level [57]. To compare our results against similar calculations which were presented in Ref. [37]-but using f SUSY = m n sof t -we will scan over the same parameter space using the f DSB SUSY distribution for soft terms with µ = 150 GeV while tan β : 3 − 60 is scanned uniformly. The goal here was to choose upper limits to our scan parameters which will lie beyond the upper limits imposed by the anthropic selection from f EWFT . Lower limits are motivated by current LHC search limits, but also must stay away from the singularity in the f DSB SUSY distribution. Our final results will hardly depend on the chosen value of µ so long as µ is within an factor of a few of m W,Z,h ∼ 100 GeV. We expect the different classes of soft terms to scan independently as discussed in Ref. [58]. We will compare the f DSB SUSY results against the f SSB SUSY results from Ref. [37] using an n = 2 power-law draw. In Fig. 2, we first show probability distributions for various soft SUSY breaking terms for f DSB SUSY and also for f SSB SUSY = m 2 sof t . In frame a), we show the distributions versus first/second generation soft breaking scalar masses m 0 (1, 2). We see the old SSB n = 2 result gives a peak distribution at m 0 (1, 2) ∼ 25 TeV with a tail extending to over 40 TeV. This distribution reflects the mixed decoupling/quasi-degeneracy landscape solution to the SUSY flavor and CP problems [59]. In contrast, the distribution from f DSB SUSY peaks at the lowest allowed m 0 (1, 2) values albeit with a tail extending out beyond 10 TeV. Thus, we would expect relatively light, LHC accessible, squarks and sleptons from gravity-mediation with DSB in a hidden sector. In frame b), we show the distribution in third generation soft mass inputs: m 0 (3). Here also the soft terms peak at the lowest values, but this time the tail extends only to ∼ 4 TeV (lest Σ u u (t 1,2 ) becomes too large). In contrast, the SSB n = 2 distribution peaks around 7 TeV. In frame c), the distribution in unified gaugino soft term m 1/2 is shown. Here again, gaugino masses peak at the lowest allowed scales for DSB while the n = 2 distribution peaks just below 2 TeV. Finally, in frame d), we show the distribution in trilinear soft term −A 0 . Here, the DSB distribution peaks at −A 0 ∼ 0 leading to little mixing in the stop sector and consequently lower values of m h [60,61]. In contrast, the n = 2 distribution has a double peak structure with peaks at ∼ −4 and −7 TeV with a tail extending to ∼ −15 TeV: thus, we expect large stop mixing and higher m h values from the SSB with n = 2 case. In Fig. 3, we show distributions in light and heavy Higgs boson masses. In frame a), we show the m h distribution. For the DSB case, we see a peak at m h ∼ 118 GeV with almost no probability extending to ∼ 125 GeV. This is in obvious contrast to the data and to the n = 2 distribution which we see has a sharp peak at m h ∼ 125 − 126 GeV (as a result of large trilinear soft terms). In frame b), we see the distribution in pseudoscalar Higgs mass m A . In the DSB case, dP/dm A peaks in the ∼ 300 GeV range, leading to significant mixing in the Higgs sector and consequently possibly observable deviations in the Higgs couplings (see Ref. [62]). Alternatively, the SSB n = 2 distribution peaks at m A ∼ 3.5 TeV with a tail extending to ∼ 8 TeV. In the latter case, we would expect a decoupled Higgs sector with a very SM-like lightest Higgs scalar h (as indeed the ATLAS/CMS data seem to suggest). In Fig. 4, we show predictions for various sparticle masses from the DSB and SSB n = 2 cases. In frame a), we show the distribution in gluino mass mg. For the DSB case, the distribution peaks around the ∼ TeV range while LHC search limits typically require mg 2.2 TeV. In fact, almost all parameter space of DSB is then excluded. Had we lowered the lower scan cutoff on m 1/2 , the distribution would shift lower, making matters worse. The SSB n = 2 distribution peaks at mg ∼ 4 − 5 TeV with a tail extending to ∼ 6 TeV; hardly any probability is excluded by the LHC mg 2.2 TeV limit. In frame b), we show the distribution in first generation squark mass mũ L (as a typical example of first/second generation matter scalars). The distribution from DSB peaks in the 0 − 3 TeV range with a tail extending beyond 10 TeV. Coupled with the gluino distribution, most probability space would be excluded by LHC search limits from the mg vs. mq plane. The SSB n = 2 distribution peaks above 20 TeV with a tail extending beyond 40 TeV. In frame c), we show the distribution in lighter top squark mass mt 1 . Here, we see DSB peaks around 1 TeV with a tail to ∼ 2.5 TeV. LHC searches require mt 1 1.1 TeV so that about half of probability space is excluded. For the SSB n = 2 case, the peak shifts to mt 1 ∼ 1.6 TeV so the bulk of p-space is allowed by LHC searches. Finally, in frame d), we show the distribution in heavier stop mass mt 2 . The DSB distribution peaks around ∼ 1.5 TeV whilst the SSB n = 2 distribution peaks around 4 TeV. Thus, substantially heaviert 2 squarks are expected from SSB as compared to DSB.

Conclusions
One of the mysteries of particle physics is the origin of mass scales, especially in the context of string theory where only the Planck scale m P appears. Here, we investigated the origin of the weak scale which is presumed to arise from the scale of SUSY breaking. The general framework of dynamical SUSY breaking presents a beautiful example of the exponentially suppressed SUSY breaking scale (relative to the Planck scale) arising from non-perturbative effects such as gaugino condensation or SUSY breaking via instanton effects. The SUSY breaking scale from DSB is expected to be uniformly distributed on a log scale within a fertile patch of the string landscape with the MSSM as the low energy EFT. In this case, the probability distribution f DSB SUSY ∼ 1/m sof t . Such a distribution, coupled with the ABDS anthropic window, typically leads to Higgs masses m h well below the measured 125 GeV value and many sparticles such as the gluino expected to lie below existing LHC search limits. Thus, the LHC data seem to falsify this approach. That would leave the alternative option of spontaneous SUSY breaking where instead the soft SUSY breaking distribution is expected to occur as a power law or log distribution. These latter cases lead to landscape probability distributions for m h that peak at m h ∼ 125 GeV with sparticles typically well beyond current LHC reach, but within reach of hadron colliders with √ s 30 TeV. For perturbative, or spontaneous, SUSY breaking, then apparently the magnitude of the SUSY breaking scale is set anthropically much like the cosmological constant is: those vacua with too large a SUSY breaking scale lead to either CCB or noEWSB vacua, or vacua with such a large weak scale that it lies outside the ABDS allowed window, in violation of the atomic principle.