UUCI-TR-2013-01
Naturalness and the Status of Supersymmetry
Jonathan L. Feng
Department of Physics and AstronomyUniversity of California, Irvine, CA 92697, USA
Abstract
For decades, the unnaturalness of the weak scale has been the dominant problem motivatingnew particle physics, and weak-scale supersymmetry has been the dominant proposed solution.This paradigm is now being challenged by a wealth of experimental data. In this review, we beginby recalling the theoretical motivations for weak-scale supersymmetry, including the gauge hierar-chy problem, grand unification, and WIMP dark matter, and their implications for superpartnermasses. These are set against the leading constraints on supersymmetry from collider searches,the Higgs boson mass, and low-energy constraints on flavor and CP violation. We then criticallyexamine attempts to quantify naturalness in supersymmetry, stressing the many subjective choicesthat impact the results both quantitatively and qualitatively. Finally, we survey various proposalsfor natural supersymmetric models, including effective supersymmetry, focus point supersymme-try, compressed supersymmetry, and R -parity-violating supersymmetry, and summarize their keyfeatures, current status, and implications for future experiments. Keywords: gauge hierarchy problem, grand unification, dark matter, Higgs boson, particle colliders a r X i v : . [ h e p - ph ] M a y ontents I. INTRODUCTION II. THEORETICAL MOTIVATIONS
III. EXPERIMENTAL CONSTRAINTS R -Parity Violation 94. Sleptons, Charginos, and Neutralinos 10B. The Higgs Boson 11C. Flavor and CP Violation 131. Flavor-Violating Constraints 132. Electric Dipole Moments 14D. Hints of New Physics 15 IV. QUANTIFYING NATURALNESS
V. MODEL FRAMEWORKS R -Parity-Violating Supersymmetry 27 VI. CONCLUSIONS ACKNOWLEDGMENTS References . INTRODUCTION
Good physical theories are expected to provide natural explanations of experimental dataand observations. Although physicists disagree about the definition of “natural,” the ideathat the criterion of naturalness exists and is a useful pointer to deeper levels of understand-ing has a long and storied history. In 1693, for example, when asked by clergyman RobertBentley to explain how the law of universal gravitation was consistent with a static universe,Isaac Newton wrote [1]:That there should be a central particle, so accurately placed in the middle,as to be always equally attracted on all sides, and thereby continue withoutmotion, seems to me a supposition fully as hard as to make the sharpest needlestand upright on its point upon a looking-glass.Newton went on to conclude that this unnatural state of affairs could be taken as evidencefor an infinite universe with initial conditions set by a divine power. Three hundred yearslater, the assumption of a static universe appears quaint, but we are no closer to a naturalexplanation of our accelerating universe than Newton was to his static one. More generally,the image of a needle balanced upright on a mirror remains the classic illustration of apossible, but unnatural, scenario that cries out for a more satisfactory explanation, and thenotion of naturalness continues to play an important role in many areas of physics.In particle physics today, the role of naturalness is nowhere more central than in thestatement of the gauge hierarchy problem, the question of why the weak scale m weak ∼ . − M Pl = (cid:113) ¯ hc/ (8 πG N ) (cid:39) . × GeV.For many years, this has been the dominant problem motivating proposals for new particlesand interactions. Chief among these is supersymmetry, which solves the gauge hierarchyproblem if there are supersymmetric partners of the known particles with masses not farabove the weak scale. This has motivated searches for superpartners at colliders, in low-energy experiments, and through astrophysical observations. So far, however, no compellingevidence for weak-scale supersymmetry has been found, and recent null results from searchesat the Large Hadron Collider (LHC) have disappointed those who find supersymmetry toobeautiful to be wrong and led its critics to declare supersymmetry dead.In this article, we review the status of weak-scale supersymmetry at this brief momentin time when a Higgs-like particle has been discovered at the 8 TeV LHC, and the LHC hasentered a two-year shutdown period before turning on again at its full center-of-mass energy.The field of supersymmetry phenomenology is vast, and we will necessarily review only asmall subset of its many interesting aspects. As we will see, however, weak-scale supersym-metry is neither ravishingly beautiful (and hasn’t been for decades), not is it excluded byany means; the truth lies somewhere in between. The goal of this review is to understandthe extent to which naturalness and experimental data are currently in tension and exploremodels that resolve this tension and their implications for future searches.We begin in Sec. II with a brief discussion of some of the longstanding theoretical moti-vations for weak-scale supersymmetry and their implications for superpartner masses. Wethen discuss some of the leading experimental constraints on weak-scale supersymmetry inSec. III. In Sec. IV, we critically review attempts to quantify naturalness. Naturalness isa highly contentious subject with many different approaches leading to disparate conclu-sions. We will highlight some of the assumptions that are often implicit in discussions ofnaturalness and discuss the various subjective choices that impact the conclusions, bothqualitatively and quantitatively. 3 ffectiveSUSY Focus PointSUSY CompressedSUSY R p -ViolatingSUSYNaturalness √ √ √ √ GrandUnification √ √ √ √
WIMPDark Matter √ √ √
LHC NullResults √ √ √ √
Higgs Mass √ Flavor/CPConstraints √ √
TABLE I: Some of the supersymmetric models discussed in this review, the virtues they are in-tended to preserve, and the constraints they are designed to satisfy, with varying degrees of success.For the rationale behind the check marks, see Secs. II, III, and V for discussions of the virtues,constraints, and models, respectively.
With all of these considerations in hand, we then turn in Sec. V to an overview of modelframeworks that have been proposed to reconcile naturalness with current experimental con-straints, summarizing their key features, current status, and implications for future searches.As a rough guide to the discussion, these models and the problems they attempt to addressare shown in Table I. We conclude in Sec. VI.
II. THEORETICAL MOTIVATIONS
To review the status of supersymmetry, we should begin by recalling the problems itwas meant to address. Supersymmetry [2–4] has beautiful mathematical features that areindependent of the scale of supersymmetry breaking. In addition, however, there are threephenomenological considerations that have traditionally been taken as motivations for weak-scale supersymmetry: the gauge hierarchy problem, grand unification, and WIMP darkmatter. In this section, we review these and their implications for superpartner masses.
A. The Gauge Hierarchy Problem
1. The Basic Idea
The gauge hierarchy problem of the standard model [5–7] and its possible resolutionthrough weak-scale supersymmetry [8–11] are well-known. (For reviews and discussion, see, e.g. , Refs. [12–16].) The standard model includes a fundamental, weakly-coupled, spin-0particle, the Higgs boson. Its bare mass receives large quantum corrections. For example,4iven a Dirac fermion f that receives its mass from the Higgs boson, the Higgs mass is m h ≈ m h − λ f π N fc (cid:90) Λ d pp ≈ m h + λ f π N fc Λ , (1)where m h ≈
125 GeV is the physical Higgs boson mass [17, 18], m h is the bare Higgs mass,and the remaining term is m h , the 1-loop correction. The parameters λ f and N fc arethe Yukawa coupling and number of colors of fermion f , Λ is the largest energy scale forwhich the standard model is valid, and subleading terms have been neglected. For large Λ,the bare mass and the 1-loop correction must cancel to a large degree to yield the physicalHiggs mass. Attempts to define naturalness quantitatively will be discussed in detail inSec. IV, but at this stage, a simple measure of naturalness may be taken to be N ≡ m h m h . (2)For Λ ∼ M Pl and the top quark with λ t (cid:39)
1, Eq. (1) implies N ∼ , i.e. , a fine-tuningof 1 part in 10 .Supersymmetry moderates this fine-tuning. If supersymmetry is exact, the Higgs massreceives no perturbative corrections. With supersymmetry breaking, the Higgs mass becomes m h ≈ m h + λ f π N fc (cid:16) m f − m f (cid:17) ln (cid:16) Λ /m f (cid:17) , (3)where ˜ f is the superpartner of fermion f . The quadratic dependence on Λ is reduced toa logarithmic one, and even for Λ ∼ M Pl , the large logarithm is canceled by the loopsuppression factor 1 / (8 π ), and the Higgs mass is natural, provided m ˜ f is not too far above m h . Requiring a maximal fine-tuning N , the upper bound on sfermion masses is m ˜ f < ∼
800 GeV 1 . λ f (cid:34) N fc (cid:35) (cid:34) /m f ) (cid:35) (cid:34) N (cid:35) , (4)where λ f and N fc have been normalized to their top quark values, the logarithm has beennormalized to its value for Λ ∼ M Pl , and N has been normalized to 100, or 1% fine-tuning.
2. First Implications
Even given this quick and simple analysis, Eq. (4) already has interesting implications: • Naturalness constraints vary greatly for different superpartners. As noted as early as1985 [19], the 1-loop contributions of first and second generation particles to the Higgsmass are suppressed by small Yukawa couplings. For the first generation sfermions,naturalness requires only that they be below 10 TeV! In fact, this upper bound isstrengthened to ∼ −
10 TeV by considerations of D -term and 2-loop effects, asdiscussed in Sec. IV C. Nevertheless, it remains true that without additional theoreti-cal assumptions, there is no naturalness reason to expect first and second generationsquarks and sleptons to be within reach of the LHC .5 Naturalness bounds on superpartner masses are only challenged by LHC constraintsfor large Λ, such as Λ ∼ M Pl or Λ ∼ m GUT (cid:39) × GeV, the grand unifiedtheory (GUT) scale. For low Λ, the loop suppression factor is not compensated by alarge logarithm, and naturalness constraints are weakened by as much as an order ofmagnitude. For example, even for top squarks, for low Λ such that ln(Λ /m f ) ∼ N = 1 becomes m ˜ t < ∼
700 GeV, beyond current LHCbounds, and for N = 100, the bound is m ˜ t < ∼ O (1) fine-tuning requires m ˜ t ∼ m h assumes implicitly that the 1-loop suppressionfactor is compensated by a large logarithm.Of course, supersymmetry makes it possible to contemplate a perturbative theory allthe way up to m GUT or M Pl , and grand unification, radiative electroweak symmetrybreaking, and other key virtues of supersymmetry make this highly motivated. Thereare therefore strong reasons to consider Λ ∼ m GUT , M Pl . This observation, however,suggests that if the enhancement from large logarithms may somehow be removed,supersymmetric theories with multi-TeV superpartners may nevertheless be considerednatural; this is the strategy of models that will be discussed in Sec. V B. • Last, all naturalness bounds depend on what level of fine-tuning is deemed acceptable,with mass bounds scaling as (cid:113) N . This is an irreducible subjectivity that must beacknowledged in all discussions of naturalness.There is a sociological observation perhaps worth making here, however. In the past,in the absence of data, it was customary for some theorists to ask “What regions ofparameter space are most natural?” and demand fine-tuning of, say, less than 10%( N = 10). This requirement led to the promotion of models with very light su-perpartners, and heightened expectations that supersymmetry would be discovered assoon as the LHC began operation.In retrospect, however, this history has over-emphasized light supersymmetry modelsand has little bearing on the question of whether weak-scale supersymmetry is stilltenable or not. As with all questions of this sort, physicists vote with their feet.Rather than asking “What regions of parameter space are most natural?”, a moretelling question is, “If superpartners were discovered, what level of fine-tuning would besufficient to convince you that the gauge hierarchy problem is solved by supersymmetryand you should move on to researching other problems?” An informal survey ofresponses to this question suggests that values of N = 100 , N in Eq. (4) isreasonable, and current bounds from the LHC do not yet preclude a supersymmetricsolution to the hierarchy problem, especially given the many caveats associated withattempts to quantify naturalness, which will be discussed in Sec. IV. B. Grand Unification
The fact that the standard model particle content fits neatly into multiplets of largergauge groups, such as SU(5), SO(10), or E , is striking evidence for GUTs [20–23]. In thestandard model, the strong, weak, and electromagnetic gauge couplings do not unify at anyscale. However, in the minimal supersymmetric standard model (MSSM), the supersymmet-ric extension of the standard model with minimal field content, the gauge coupling renor-6alization group equations (RGEs) are modified above the superpartner mass scale. Withthis modification, if the superpartners are roughly at the weak scale, the gauge couplingsmeet at m GUT (cid:39) × GeV, further motivating both grand unification and weak-scalesupersymmetry [24–28].Gauge coupling unification is sensitive to the superpartner mass scale, since this governswhen the RGEs switch from non-supersymmetric to supersymmetric. However, the sensitiv-ity to the superpartner mass scale is only logarithmic. Furthermore, full SU(5) multiplets ofsuperpartners, such as complete generations of squarks and sleptons, may be heavy withoutruining gauge coupling unification. Note, however, that the MSSM particles do not com-pletely fill SU(5) multiplets; in particular, the Higgs bosons must be supplemented withHiggs triplets. One might therefore hope that the masses of SU(2) doublet Higgsinos mightbe stringently constrained by gauge coupling unification, but even this is not the case. A fulljustification requires a complete discussion of GUTs and proton decay [29, 30], but roughlyspeaking, heavy sfermions suppress the leading contributions to proton decay, and there issufficient freedom in threshold corrections from the GUT-scale spectrum to allow unificationeven for relatively heavy Higgsinos; see, e.g. , Refs. [31, 32].In summary, grand unification is a significant motivation for supersymmetry, but gaugecoupling unification is a blunt tool when it comes to constraining the superpartner massscale. Note, however, that the relations imposed by grand unification may have a strongimpact on naturalness bounds, either weakening or strengthening them significantly; seeSec. IV C.
C. Dark Matter
Supersymmetry provides an excellent WIMP dark matter candidate when the neutralinois the lightest supersymmetric particle (LSP) [33, 34]. Neutralinos naturally freeze out withapproximately the correct thermal relic density. This density is inversely proportional to thethermally-averaged annihilation cross section, which, on dimensional grounds, is inverselyproportional to the superpartner mass scale squared:Ω χ ∝ (cid:104) σv (cid:105) ∝ ˜ m . (5)The requirement Ω χ ≤ .
23 therefore places an upper bound on the superpartner mass scale˜ m . Unfortunately, when the constants of proportionality are included, the upper bounds forsome types of neutralinos are far above current LHC sensitivities. For example, for mixedBino-Higgsino [35–37] and pure Wino-like [38] neutralino dark matter, the upper bounds are m ˜ B − ˜ H < . m ˜ W < . − . . (6)Such neutralinos may be produced in the cascade decays of squarks and gluinos, but thisis model-dependent. The model-independent search strategy is to consider Drell-Yan pro-duction of neutralino pairs with a radiated jet or photon, which contributes to mono-jetand mono-photon searches [39–43]. The limits in Eq. (6) are far above current LHC sensi-tivities [44, 45]. The spin-independent and spin-dependent scattering cross sections of suchneutralinos are also consistent with current bounds from direct search experiments [46].7f course, dark matter may be composed of other particles, such as axions, sterile neu-trinos, hidden sector dark matter, or gravitinos [47]. There is no requirement that super-partners be light in these dark matter scenarios. In fact, some scenarios in which gravitinosare the dark matter provide motivation for extremely heavy superpartners, which freeze outwith Ω (cid:29) .
23, but then decay to gravitinos with Ω ˜ G (cid:39) .
23 [48].In summary, the requirement of WIMP dark matter provides upper bounds on superpart-ner masses, but these upper bounds are high and far beyond the reach of current colliders.In addition, the dark matter doesn’t have to be made of supersymmetric WIMPs. As withthe case of grand unification, the possibility of WIMP dark matter is a significant virtue ofweak-scale supersymmetry, but it does not provide stringent upper bounds on superpartnermasses.
III. EXPERIMENTAL CONSTRAINTSA. Superpartner Searches at Colliders
The search for weak-scale supersymmetry has been ongoing for decades at many collid-ers. Before the 2013-14 shutdown, however, the LHC experiments ATLAS and CMS eachcollected luminosities of more than 5 fb − at √ s = 7 TeV and 20 fb − at √ s = 8 TeV,and the resulting LHC limits supersede previous collider constraints in almost all scenarios.We will therefore confine the discussion to LHC results and focus on a small subset that isparticularly relevant for the following discussion. For a summary of pre-LHC constraints,see Ref. [49], and for the full list of LHC analyses, see Refs. [50, 51].
1. Gluinos and Squarks
The greatest mass reach at the LHC is for strongly-interacting particles, such as gluinosand squarks, which are produced through pp → ˜ g ˜ g, ˜ g ˜ q, ˜ q ˜ q . The limits depend, of course,on the decays. Limits in the ( m ˜ g , m ˜ q ) plane, assuming the decays ˜ g → q ¯ qχ and ˜ q → qχ ,leading to jets + /E T , are shown in Fig. 1. The results imply m ˜ q > ∼ . m ˜ g > ∼ . m ˜ g = m ˜ q > ∼ . m , a unified gaugino mass M / ,a unified tri-linear scalar coupling A , the ratio of Higgs boson vacuum expectation val-ues tan β ≡ (cid:104) H u (cid:105) / (cid:104) H d (cid:105) ), and one discrete choice, the sign of the Higgsino mass parameter µ . Constraints in this model parameter space are shown in Fig. 1. In the limit of heavysfermions (large m ), the jets + /E T search implies m ˜ g > ∼ . luino mass [GeV]800 1000 1200 1400 1600 1800 2000 2200 2400 s qua r k m a ss [ G e V ] ) = 0 GeV χ∼ Squark-gluino-neutralino model, m( =8 TeVs, -1 L dt = 5.8 fb ∫ ATLAS ) theorySUSY σ ± Observed limit ( ) exp σ ± Expected limit ( , 7 TeV) -1 Observed limit (4.7 fb
Preliminary
Figure 7: A simplified MSSM scenario with only strong production of gluinos and first- and second-generation squarks, with direct decays to jets and neutralinos. Exclusion limits are obtained by using thesignal region with the best expected sensitivity at each point. The blue dashed lines show the expectedlimits at 95% CL, with the light (yellow) bands indicating the 1 σ experimental uncertainties. Observedlimits are indicated by medium (maroon) curves, where the solid contour represents the nominal limit,and the dotted lines are obtained by varying the cross section by the theoretical scale and PDF uncertain-ties. Previous results from ATLAS [17] are represented by the shaded (light blue) area. Results at 7 TeVare valid for squark or gluino masses below 2000 GeV, the mass range studied for that analysis.set to 0.96 times the mass of the gluino.In the CMSSM / MSUGRA case, the limit on m / is above 340 GeV at high m and reaches 710 GeVfor low values of m . Equal mass light-flavor squarks and gluinos are excluded below 1500 GeV inthis scenario. The same limit of 1500 GeV for equal mass of light-flavor squarks and gluinos is foundfor the simplified MSSM scenario shown in Fig. 7. In the simplified model cases of Fig. 8 (a) and (c),when the lightest neutralino is massless the limit on the gluino mass (case (a)) is 1100 GeV, and thaton the light-flavor squark mass (case (c)) is 630 GeV. Mass limits for the direct production of light-flavor squarks (case (c)) hardly improve with respect to the 7 TeV data analysis because of increasedbackground predictions and uncertainties at 8 TeV in the low m e ff and low jet multiplicity channels usedto provide exclusions for these models. This note reports a search for new physics in final states containing high- p T jets, missing transversemomentum and no electrons or muons, based on a 5.8 fb − dataset recorded by the ATLAS experiment atthe LHC in 2012. Good agreement is seen between the numbers of events observed in the data and thenumbers of events expected from SM processes.The results are interpreted both in terms of MSUGRA / CMSSM models with tan β = A = µ >
0, and in terms of simplified models with only light-flavor squarks, or gluinos, or both, togetherwith a neutralino LSP, with the other SUSY particles decoupled. In the MSUGRA / CMSSM models,values of m / <
350 GeV are excluded at the 95% confidence level for all values of m , and m / < m . Equal mass squarks and gluinos are excluded below 1500 GeV in this scenario. Whenthe neutralino is massless, gluino masses below 1100 GeV are excluded at the 95% confidence level ina simplified model with only gluinos and the lightest neutralino. For a simplified model involving thestrong production of squarks of the first two generations, with decays to a massless neutralino, squarkmasses below 630 GeV are excluded. 12 [GeV] m500 1000 1500 2000 2500 3000 3500 [ G e V ] / m ( G e V ) q ~ ( G e V ) q ~ ( G e V ) q ~ ( G e V ) g ~ ( G e V ) g ~ ( G e V ) g ~ ( G e V ) g ~ >0 µ = 0, = 10, A β MSUGRA/CMSSM: tan =8 TeVs, -1 L dt = 5.8 fb ∫ ATLAS ) theorySUSY σ ± Observed limit ( ) exp σ ± Expected limit ( , 7 TeV) -1 Observed limit (4.7 fbNon-convergent RGENo EW-SB
Preliminary gluino mass [GeV]200 400 600 800 1000 1200 1400 1600 1800 s qua r k m a ss [ G e V ] < µ = , β CD F , R un II, t an < µ = , β D , R un II, t an >0 µ = 0, = 10, A β MSUGRA/CMSSM: tan =8 TeVs, -1 L dt = 5.8 fb ∫ ATLAS ) theorySUSY σ ± Observed limit ( ) exp σ ± Expected limit (Theoretically excludedStau LSP
Preliminary
Figure 6: 95% CL exclusion limits for MSUGRA / CMSSM models with tan β = A = µ > m – m / plane and (right) in the m ˜ g – m ˜ q plane. Exclusion limits are obtained byusing the signal region with the best expected sensitivity at each point. The blue dashed lines show theexpected limits at 95% CL, with the light (yellow) bands indicating the 1 σ excursions due to experimen-tal uncertainties. Observed limits are indicated by medium (maroon) curves, where the solid contourrepresents the nominal limit, and the dotted lines are obtained by varying the cross section by the the-oretical scale and PDF uncertainties. Previous results from ATLAS [17] are represented by the shaded(light blue) area. The theoretically excluded regions (green and blue) are described in Ref. [63].Data from all the channels are used to set limits on SUSY models, taking the SR with the bestexpected sensitivity at each point in several parameter spaces. A profile log-likelihood ratio test in com-bination with the CL s prescription [59] is used to derive 95% CL exclusion regions. Exclusion limits areobtained by using the signal region with the best expected sensitivity at each point. The nominal signalcross section and the uncertainty are taken from an ensemble of cross section predictions using di ff erentPDF sets and factorisation and renormalisation scales, as described in Ref. [52]. Observed limits arecalculated for both the nominal cross section, and ± σ uncertainties. For each of these three individuallimits, the best signal region at each point is taken. Numbers quoted in the text are evaluated from theobserved exclusion limit based on the nominal cross section less one sigma on the theoretical uncertainty.In Fig. 6 the results are interpreted in the tan β = A = µ > / CMSSM models . For the nominal cross sections, the best signal region is E-tight for high m values, C-tight for low m values and D-tight between the two. Results are presented in both the m – m / and m ˜ g – m ˜ q planes. Thesparticle mass spectra and decay tables are calculated with SUSY-HIT [60] interfaced to
SOFTSUSY [61]and
SDECAY [62].An interpretation of the results is presented in Figure 7 as a 95% CL exclusion region in the ( m ˜ g , m ˜ q )-plane for a simplified set of SUSY models with m ˜ χ =
0. In these models the gluino mass and the massesof the squarks of the first two generations are set to the values shown on the axes of the figure. All othersupersymmetric particles, including the squarks of the third generation, are decoupled.In Fig. 8 limits are shown for three classes of simplified model in which only direct production of (a)gluino pairs, (b) ‘light’-flavor squarks (of the first two generations) and gluinos or (c) light-flavor squarkpairs is kinematically possible, with all other superpartners, except for the neutralino LSP, decoupled.This forces each light-flavor squark or gluino to decay directly to jets and an LSP. Cross sections areevaluated assuming decoupled light-flavor squarks or gluinos in cases (a) and (c), respectively. In allcases squarks of the third generation are decoupled. In case (b) the masses of the light-flavor squarks are Five parameters are needed to specify a particular MSUGRA / CMSSM model: the universal scalar mass, m , the universalgaugino mass m / , the universal trilinear scalar coupling, A , the ratio of the vacuum expectation values of the two Higgs fields,tan β , and the sign of the higgsino mass parameter, µ = ± . FIG. 1: Constraints on gluinos and first and second generation squarks from ATLAS at the LHCwith L = 5 . − and √ s = 8 TeV [52]. Left: Limits in the ( m ˜ g , m ˜ q ) plane from pp → ˜ g ˜ g, ˜ g ˜ q, ˜ q ˜ q followed by ˜ g → q ¯ qχ and ˜ q → qχ , leading to jets+ /E T . The analysis assumes m ˜ q ≡ m ˜ u L,R = m ˜ d L,R = m ˜ s L,R = m ˜ c L,R , m χ = 0, and that all other superpartners, including the top and bottom squarks,are very heavy. The shaded region boundaries at m ˜ g , m ˜ q = 2 TeV are artifacts of the previous 7TeV analysis. Right: Limits from the jets + /E T search in the ( m , M / ) plane of mSUGRA, withtan β = 10, A = 0, and µ >
2. Top and Bottom Squarks
The constraints of Fig. 1 might appear to require all superpartners to be above the TeVscale. As noted in Sec. II A, however, naturalness most stringently constrains the top andbottom squarks, but allows effectively decoupled first and second generation squarks, exactlythe opposite of the assumptions made in deriving these bounds. It is therefore importantto consider other analyses, including searches for light top and bottom squarks. Resultsfrom such searches are shown in Fig. 2. Limits from direct stop pair production followedby ˜ t → tχ are shown, as are limits from gluino pair production followed by ˜ g → ˜ t ∗ ¯ t → t ¯ tχ ,which is the dominant decay mode if stops are significantly lighter than all other squarks.In the case of stop pair production, we see that stops as light as 500 GeV are allowed formassless neutralinos, and much lighter stops are allowed if one approaches the kinematicboundary m ˜ t − m χ = m t . In the case of gluino pair production, the bound is m ˜ g > ∼ . m χ = 0, but again, there are allowed regions with much lighter gluinos near the kinematicboundary m ˜ g − m χ = 2 m t . Searches for light stops in other channels, as well as searches forbottom squarks, yield roughly similar constraints [50, 51]. R -Parity Violation The search results presented so far require missing transverse energy. Although WIMPdark matter motivates this possibility, large /E T is far from a requirement of supersymme-try, and /E T signals may be degraded in a number of ways, for example, by compressedsuperpartner spectra, a possibility discussed in Sec. V C.Perhaps the most dramatic way is with R -parity ( R p ) violation. When the standard modelis extended to include supersymmetry, there are many new gauge-invariant, renormalizableinteractions. If any one of these is present, all superpartners decay, effectively eliminating9 [GeV] t~ m
200 250 300 350 400 450 500 550 600 [ G e V ] c~ m -2 -1 mixture R / t L
50 / 50 t c~ t fi t~*, t~ t~ fi pp NLO-NLL exclusions theory s – Observed s – Expected B R [ pb ] ·s % C . L . U L on -1 Ldt = 9.7 fb (cid:242) = 8 TeV, sCMS Preliminary ( pb ) σ % C . L . uppe r li m i t on -3 -2 -1 gluino m400 600 800 1000 1200 1400 ( G e V ) L SP m exp. σ ± Expected Limit theory σ ± NLO+NLL σ = 8 TeVs, -1 CMS Preliminary, 11.7 fb )g~)>>m(t~; m( χ∼ t t → g~, g~ g~ → pp FIG. 2: Constraints on gluinos and top squarks from CMS at the LHC. Left: Limits from L =9 . − at √ s = 8 TeV in the ( m ˜ t , m χ ) plane from pp → ˜ t ˜ t ∗ , followed by ˜ t → tχ , leading to thesignature of l + b -jet + /E T [53]. Right: Limits from L = 11 . − at √ s = 8 TeV in the ( m ˜ g , m χ )plane from pp → ˜ g ˜ g , followed by ˜ g → t ¯ tχ , leading to signatures of N j jets + N b b -jets + /E T , where2 ≤ N j ≤ N j ≥
4, and N b = 0 , , ,
3, or ≥ the /E T signature. These R p -violating (RPV) terms arise from superpotentials of the form W R p / = λ ijk L i L j E k + λ (cid:48) ijk L i Q j D k + λ (cid:48)(cid:48) ijk U i D j D k + µ i L i H u , (7)where the first three types of terms are categorized as leptonic, semi-leptonic, and hadronic,and i, j, k = 1 , , pp → ˜ g ˜ g followed by the RPV decay ˜ g → qqq through a λ (cid:48)(cid:48) operator [55]. The resultingbound on the gluino mass is 670 GeV, far weaker than in cases where the gluino cascadedecay includes significant /E T .
4. Sleptons, Charginos, and Neutralinos
Finally, the mass reach for superpartner searches is, of course, greatly reduced for un-colored superpartners. In Fig. 4, we show constraints from CMS on Drell-Yan slepton pairproduction and associated chargino-neutralino pair production [56]. The limits are im-pressive, as they extend LEP bounds of ˜ m > ∼
100 GeV to much higher masses, requiring m ˜ e , m ˜ µ > ∼
275 GeV and m χ ± = m χ > ∼
330 GeV for m χ = 0. Note, however, that these limitsdo not apply to staus, they degrade significantly for larger m χ and more degenerate spectra,and they bound superpartner masses that are not highly constrained by naturalness in theabsence of additional theoretical assumptions.10 [GeV] g~ m100 200 300 400 500 600 700 800 ) [ pb ] fi g ~ g ~ fi ( pp s -1 Exp Limit (Resolved) s – Exp Limit (Resolved) s – Cross-Section (NLO+NLL)g~g~
All limits at 95% CL=7 TeVs, -1 L dt = 4.6 fb (cid:242)
Exp Limit (Resolved)Obs Limit (Resolved)Obs Limit (CMS 2010)Obs Limit (CMS 2011)
ATLAS
Figure 5 . The expected and observed 95% confidence limits are shown for the resolved analyseschannel. The published CMS results using 35 pb − of 2010 data and using 5 fb − of 2011 data areshown for comparison. – 19 – FIG. 3: Constraint on gluinos in supersymmetry with R p violation from ATLAS at the LHC with L = 4 . − and √ s = 7 TeV [55]. The constraint is on gluino pair production pp → ˜ g ˜ g followedby the hadronic RPV decay ˜ g → qqq , leading to 6 jets with no /E T , and implies m ˜ g >
670 GeV. [GeV] l~ m
100 150 200 250 300 [ G e V ] c~ m [f b ] s % C L uppe r li m i t on -1 = 9.2 fb int = 8 TeV, LsCMS Preliminary
95% C.L. CLs NLO Exclusions theory s – Observed s – Expecteded L m~ L m~ , L e~ L e~ fi pp ) = 1 c~ l fi L l~ ( Br [GeV] c~ =m – c~ m
100 150 200 250 300 350 400 [ G e V ] c~ m [f b ] s % C L uppe r li m i t on Z < m c~ - m c~ m -1 = 9.2 fb int = 8 TeV, LsCMS Preliminary
95% C.L. CLs NLO Exclusions theory s – l +3 j l Observed 2 s – l l Expecteded 2 only l Observed 3 only j l Observed 2 – c~ c~ fi pp c~ Z fi c~ c~ W fi – c~ FIG. 4: Constraints on sleptons and electroweak gauginos from Drell-Yan production at CMSat the LHC with L = 9 . − and √ s = 8 TeV [56]. Left: Limits in the ( m ˜ l , m χ ) plane from pp → ˜ l L ˜ l ∗ L , where l = e, µ , followed by ˜ l L → lχ , leading to 2 l + /E T events. Right: Limits inthe ( m χ ± = m χ , m χ ) plane from pp → χ ± χ , followed by χ ± → W χ and χ → Zχ , leading to2 j l + /E T and 3 l + /E T events. B. The Higgs Boson
The Higgs boson, or at least an eerily similar particle, has been discovered at the LHC [17,18]. Constraints on the Higgs boson mass and the signal strength in the h → γγ and h → ZZ ∗ → l channels are shown in Fig. 5. Early hints of slight inconsistencies between themass measurements and signal strengths in various channels have now largely disappeared.At ATLAS, the γγ mass is slightly larger than the ZZ ∗ mass, and both signal strengths areslightly above SM expectations. None of these discrepancies is significant, however, and the11 [GeV] H m122 123 124 125 126 127 128 129 ) μ S i gna l s t r eng t h ( Preliminary
ATLAS -1 Ldt = 4.6-4.8 fb ∫ = 7 TeV: s -1 Ldt = 20.7 fb ∫ = 8 TeV: s Combined γγ → H l → (*) ZZ → H Best fit68% CL95% CL (GeV) X m
124 125 126 127 S M s / s Combined gg fi H ZZ fi H CMS Preliminary -1 £ = 8 TeV, L s -1 £ = 7 TeV, L s ZZ fi + H gg fi H FIG. 5: Constraints in the ( m h , σ/σ SM ) plane for the h → γγ and h → ZZ ∗ → l channels fromATLAS [57] (left) and CMS [58] (68% CL) (right). results of the two experiments are also quite consistent, as evident in Fig. 5.At present, the most pressing concern for supersymmetry is simply the Higgs boson mass.In the MSSM the Higgs boson is generically light, since the quartic coupling in the scalarpotential is determined by the electroweak gauge couplings. Indeed, the tree level value m h (tree) = m Z | cos 2 β | cannot exceed the Z boson mass. However, the Higgs mass may beraised significantly by radiative corrections [59–61]. For moderate to large tan β , a 2-loopexpression for the Higgs mass is [62, 63] m h ≈ m Z cos β + 3 m t π v (cid:40) ln M S m t + X t M S (cid:32) − X t M S (cid:33) + 116 π (cid:32) m t v − πα s (cid:33)(cid:34) X t M S (cid:32) − X t M S (cid:33) ln M S m t + (cid:32) ln M S m t (cid:33) (cid:35)(cid:41) , (8)where v (cid:39)
246 GeV, M S ≡ √ m ˜ t m ˜ t , X t ≡ A t − µ cot β parameterizes the stop left-rightmixing, and α s ≈ .
12. Several codes incorporate 2-loop [64–66], or even 3-loop [67, 68],corrections.Equation (8) has several interesting features. First, increasing tan β increases the tree-level Higgs mass; this effect saturates for tan β > ∼
13, where the tree-level mass is within1 GeV of its maximum. Second, the Higgs boson mass may be greatly increased either bylarge stop mixing ( X t ≈ ±√ M S ) or by heavy stops ( M S (cid:29) m t ). Numerical results areshown in Fig. 6. For negligible stop mixing, stop masses M S > ∼ M S are possible,but such large mixing is highly fine-tuned with respect to the A t parameter [70]. The genericlesson to draw is that the measured Higgs mass favors stop masses well above a TeV.At present, the Higgs mass measurement is at least as significant a challenge to natural-ness as the absence of superpartners at the LHC. First, the Higgs mass can only be raisedto ∼
125 GeV by raising the masses of superpartners that couple strongly to the Higgs. Butit is exactly these particles that, at least at first sight, must be light to preserve naturalness.Second, because the Higgs mass is only logarithmically sensitive to the top squark mass,12 X t /M S MSSM m ( G e V ) t˜ -4 -3 -2 -1 0 1 2 3 4 X t /M S NMSSM ( l < -4 -3 -2 -1 0 1 2 3 4 X t /M S NMSSM ( l > FIG. 1: The scatter plots of the samples in the MSSM and NMSSM satisfying all the requirements(1-7) listed in the text (including 123GeV ≤ m h ≤ m ˜ t versus X t /M S with M S ≡ √ m ˜ t m ˜ t and X t ≡ A t − µ cot β . SUSY is disfavored by naturalness, we in the following concentrate on the implication of m h ≃ B. Implication of m h ≃ in sub-TeV SUSY In this section, we study the implication of m h ≃ λ > .
53. Our scans over the parameter spaces are quite similar to those in Eq.(15)and Eq.(16) except that we narrow the ranges of M Q , M U and A t as follows:100 GeV ≤ ( M Q , M U ) ≤ , | A t | ≤ . (17)In Fig.2 we project the surviving samples of the models in the plane of m ˜ t versus A t ,showing the results with R γγ < R γγ > m h ≃ m ˜ t and | A t | must be largerthan about 300GeV and 1 . T eV respectively, and the bounds are pushed up to 600GeV and1 . T eV respectively for R γγ >
1. While in the NMSSM, a ˜ t as light as about 100GeV (in9 FIG. 6: Values of top squark parameters that give 123 GeV < m h <
127 GeV in viable MSSMmodels [69]. The parameters are m ˜ t , the mass of the lighter stop, and X t /M S , where X t ≡ A t − µ cot β parameterizes left-right stop mixing and M S ≡ √ m ˜ t m ˜ t is the geometric mean of thephysical stop masses. it has tremendous reach, favoring, in the no-mixing case, stop masses that are far abovecurrent LHC bounds and even challenging for all proposed future colliders. C. Flavor and CP Violation
Bounds on low-energy flavor and CP violation stringently constrain all proposals fornew physics at the weak scale. For supersymmetry, these longstanding constraints areextremely stringent and are a priori a strong challenge to naturalness. The constraints onsupersymmetry may be divided into two qualitatively different classes.
1. Flavor-Violating Constraints
The first are those that require flavor violation. Supersymmetry breaking generatessfermion masses that generically violate both flavor and CP. For example, for the left-handed down-type squarks, the mass matrix m ij , where i, j = ˜ d L , ˜ s L , ˜ b L , generically hasoff-diagonal entries that mediate flavor violation and complex entries that violate CP. Flavorand CP violation may also arise from all of the other mass matrices, as well as from thesupersymmetry-breaking A -terms.The constraints from low-energy flavor violation have been analyzed in many works. InRef. [71], for example, constraints are derived by requiring that the supersymmetric boxdiagram contributions to meson mass splittings not exceed their observed values, and thesupersymmetric penguin diagram contributions to radiative decays l i → l j γ not exceedcurrent bounds. A small sample of these results include (cid:34)
12 TeV m ˜ q (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re (cid:32) m d ˜ s m q (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ ∆ m K . × − MeV (9)13
16 TeV m ˜ q (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re (cid:32) m u ˜ c m q (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ ∆ m D . × − MeV (10) (cid:34) . m ˜ q (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re (cid:32) m d ˜ b m q (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ ∆ m B . × − MeV (11) (cid:34)
160 TeV m ˜ q (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im (cid:32) m d ˜ s m q (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ (cid:15) K . × − (12) (cid:34) . m ˜ l (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m e ˜ µ m l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ B ( µ → eγ )2 . × − (13) (cid:34)
150 GeV m ˜ l (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m e ˜ τ m l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ B ( τ → eγ )3 . × − (14) (cid:34)
140 GeV m ˜ l (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m µ ˜ τ m l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∼ B ( τ → µγ )4 . × − , (15)where the constraints apply to both left- and right-handed fermions and arise from theindicated observables, which have been normalized to current values [72]. Here m ˜ q and m ˜ l are average masses of the relevant squark and slepton generations, and we have set m ˜ g = m ˜ q and m ˜ γ = m ˜ l . For O (1) flavor violation, low-energy constraints require that the first andsecond generation sfermions have masses > ∼
10 TeV, and if there are additionally O (1)phases, the down-type squarks must have masses > ∼
100 TeV. In contrast, constraints fromprocesses involving third generation squarks and sleptons are generally much less severe,and are typically satisfied for sub-TeV masses.
2. Electric Dipole Moments
The second class of constraints arises from flavor-conserving, but CP-violating, observ-ables, namely the electric dipole moments (EDMs) of the electron and neutron. There arewell-known frameworks, e.g. , gauge-mediated supersymmetry breaking [73–78] and anomaly-mediated supersymmetry breaking [79, 80], in which the sfermion mass matrices are essen-tially diagonal, and all of the flavor-violating observables discussed above may be suppressed.Even in these frameworks, however, the gaugino masses M i , A -terms, and the µ and B parameters may have CP-violating phases, and these generate potentially dangerous contri-butions to the EDMs.The EDMs of the electron and neutron are generated by penguin diagrams with gaug-inos, Higgsinos and sfermions in the loop. The dominant diagram involves Wino-Higgsinomixing. The electron EDM is d e and the neutron EDM is, assuming the naive quark model, d n = (4 d d − d u ) /
3. The electron and down quark EDMs are particularly dangerous insupersymmetry, as they are enhanced for large tan β , and have the form [81] d f ∼ e g π m f | µM | m f tan β sin θ CP , (16)where f = e, d , m ˜ f is the mass scale of the heaviest superpartners in the loop, which wetake to be ˜ f , and θ CP ≡ Arg( µM a B ∗ ) is the CP-violating phase. Given the tan β -enhanced14DMs, and setting m d = 5 MeV, the EDM constraints are (cid:32) . m ˜ l (cid:33) | µM | m l tan β
10 sin θ CP . < ∼ d e . × − e cm (17) (cid:32) . m ˜ q (cid:33) | µM | m q tan β
10 sin θ CP . < ∼ d n . × − e cm , (18)where the electron and neutron EDMs are normalized to their current bounds [72].The EDM constraints are extremely robust. The CP-violating phase can be suppressedonly by a mechanism that correlates the phases of the supersymmetry-breaking gauginomasses, B , and the supersymmetry-preserving µ parameter. In many frameworks, suchas gauge-mediated supersymmetry breaking, it is already challenging to generate µ and B parameters of the correct magnitude, much less to correlate their phases with the gauginomasses, and, of course, some CP-violation is desirable to generate the matter–anti-matterasymmetry of the universe. Although CP-conserving mechanisms have been proposed [82–84], they are typically far from the simple and elegant ideas proposed to eliminate flavorviolation. In the absence of such mechanisms, the EDM constraints require multi-TeV firstgeneration superpartners to be consistent with O (0 .
1) phases.
D. Hints of New Physics
In addition to constraints excluding large effects from new physics beyond the standardmodel, there are also experimental signals that may be taken as indications for new physics.Chief among these is the anomalous magnetic moment of the muon, a µ ≡ ( g µ − / σ to 3 . σ [86, 87]. If supersymmetry is to resolve thisdiscrepancy, the mass of the lightest observable superpartner, either a chargino or a smuon,must satisfy [88] m LOSP <
480 GeV (cid:34) tan β (cid:35) (cid:34) × − ∆ a µ (cid:35) , (19)where ∆ a µ has been normalized to the current discrepancy.The anomalous magnetic moment of the muon is not the only potential signal for newphysics, however. For example, A b FB , the forward-backward asymmetry in Z → b ¯ b deviatesfrom the standard model prediction by 2.8 σ [89], the Higgs signal strength in γγ is 1 σ to2 σ too large, and the various Higgs mass measurements discussed above disagree with eachother at the 1 σ to 3 σ level.In this review, as tempting as it is be optimistic, we do not consider these results to becompelling evidence for new physics. Of course, if well-motivated supersymmetric modelselegantly explain a tantalizing anomaly, that should be noted, but here we will take a moreconservative view and will not require supersymmetry to resolve these tentative disagree-ments between experiment and the standard model. IV. QUANTIFYING NATURALNESS
We now return to naturalness and discuss attempts to quantify it in more detail. Allsuch attempts are subject to quantitative ambiguities. However, this fact should not ob-15cure the many qualitative differences that exist in naturalness prescriptions proposed inthe literature. In this section, we begin by describing a standard prescription for quantify-ing naturalness. We then critically review some of the many alternative prescriptions thathave been proposed, stressing the qualitative differences and their implications. After thislengthy discussion has highlighted the many caveats in any attempt to quantify naturalness,we present some naturalness bounds on superpartner masses that may serve as a rough guideas we turn to models in Sec. V.
A. A Naturalness Prescription
We begin by describing a general five-step prescription for assigning a numerical measureof naturalness to a given supersymmetric model. So that clarity is not lost in abstraction,we also show how it is typically applied to mSUGRA, as implemented in software programs,such as SoftSUSY [65]. • Step 1: Choose a framework with input parameters P i . In mSUGRA, the input pa-rameters are { P i } = { m , M / , A , tan β, sign( µ ) } . • Step 2: Specify a model.
A model is specified by choosing values for the input parame-ters and using experimental data and RGEs to determine all the remaining parameters.One key constraint on the weak-scale parameters is the relation m Z = 2 m H d − m H u tan β tan β − − µ , (20)suitably improved to include subleading corrections. • Step 3: Choose a set of fundamental parameters a i . These parameters are independentand continuously variable; they are not necessarily the input parameters. In mSUGRA,a common choice is the GUT-scale parameters { a i } = { m , M / , A , B , µ } . • Step 4: Calculate the sensitivity parameters N i . These parameters are [90, 91] N i ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ln m Z ∂ ln a i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i m Z ∂m Z ∂a i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (21)They measure the sensitivity of the weak scale, represented by the Z mass, to variationsin the fundamental parameters. • Step 5: Determine the overall measure of naturalness
N ≡ max {N i } . In mSUGRA,the overall measure of naturalness is, then,
N ≡ max {N m , N M / , N A , N B , N µ } . B. Subjective Choices
There are many subjective choices and caveats associated with each of the steps outlinedin Sec. IV A. Here we highlight some of these for each step in turn.
1. Choosing a Framework
This initial step is absolutely crucial, as all naturalness studies are inescapably model-dependent. In any supersymmetry study, some fundamental framework must be adopted. In16tudies of other topics, however, there exists, at least in principle, the possibility of a model-independent study, where no correlations among parameters are assumed. This model-independent study is the most general possible, in that all possible results from any other(model-dependent) study are a subset of the model-independent study’s results. In studiesof naturalness, however, the correlations determine the results, and there is no possibility,even in principle, of a model-independent study in the sense described above.Given this caveat, there are two general approaches, each with their advantages and dis-advantages. The first is a bottom-up approach, in which one relaxes as many theoreticalassumptions as is reasonable in the hope that one might derive some generic insights. Thedrawback to this approach is that, since generic weak-scale supersymmetry is excluded byexperimental constraints, we expect there to be structure in the supersymmetry-breakingparameters, which implies correlations, which impact naturalness. Ignoring these correla-tions is analogous to ignoring constraints from, say, the CPT theorem, allowing the electriccharges of the electron and positron to be independent parameters, and concluding thatthe neutrality of positronium is incredibly fine-tuned. Of course, for supersymmetry, we donot know what the underlying correlations are, but we know there are some, and the onlyassumption that is guaranteed to be wrong is that the supersymmetry-breaking parametersare completely uncorrelated.The second approach is a top-down approach, in which one takes various theoreticalframeworks seriously and analyzes their naturalness properties, incorporating all the as-sumed correlations of the framework. The hope is that by examining various frameworks insufficient detail and sampling enough of them, one can derive new insights to resolve knownproblems. The disadvantage here, of course, is that it is unlikely that any of the knownframeworks correctly captures all the correlations realized in nature.
2. Specifying a Model
As noted above, it is important to include subleading corrections to the tree-level ex-pression for m Z . For example, it is important to use 2-loop RGEs and 1-loop thresholdcorrections, decouple superpartners at their masses, and minimize the electroweak potentialat an appropriate scale (typically the geometric mean of the stop masses). The tree-levelexpression of Eq. (20) is very useful to obtain an intuitive understanding of many natu-ralness results, but it does not capture many dependencies, especially in the case of heavysuperpartners.
3. Choosing a Set of Fundamental Parameters
Many naturalness studies differ at this step. As an example, let’s consider mSUGRA. Thechoice given above follows from the view that GUT-scale parameters are more fundamentalthan weak-scale parameters and that sensitivity of the weak scale to variations in any of theparameters m , M / , A , B , and µ is an indication of fine-tuning.Another choice is simply { a i } = { µ } . The advantage of this choice is that it is ex-tremely simple to implement. The µ parameter is (barely) multiplicatively renormalized,and so c µ ≡ ∂ ln m Z /∂ ln µ = ∂ ln m Z /∂ ln µ (cid:39) µ /m Z . With this choice, naturalnessis, therefore, deemed equivalent to low µ . Some string-inspired models in which all squarksmasses are ∼
10 TeV are claimed to be natural based on this prescription [92].17uch claims are subject to caveats, however. Given our current understanding, the µ pa-rameter is typically assumed to have an origin separate from the supersymmetry-breakingparameters. It is therefore reasonable to assume that it is not correlated with other pa-rameters, and so low µ is a necessary condition for naturalness. (Note, however, that thediscussion of EDMs in Sec. III C 2 provides a counterargument.) Much more problematic,however, is that low µ is certainly not a sufficient condition for naturalness. In the mod-erate to large tan β limit, Eq. (20) becomes m Z ≈ − m H u − µ . It is certainly possiblefor m H u to be small as the result of large cancellations. In this case, µ will be small. Butthis does not imply there is no fine-tuning: the relation a − b − c = 1 with a = 1 , , b = 999 , c = 1 is fine-tuned, despite the fact that c is small. Claims that suchtheories are natural are implicitly assuming that some unspecified correlation explains thelarge cancellation that yields low m H u .A third possible choice for the fundamental parameters is to include not only the di-mensionful supersymmetry-breaking parameters, but all of the parameters of the standardmodel. Some naturalness studies include these [91, 93–96], while others do not [90, 97–101].From a low-energy point of view, one should include all the parameters of the Lagrangian.However, by assuming some underlying high energy framework and defining our parametersat m GUT , we have already abandoned a purely low-energy perspective. Once we considerthe high-energy perspective, the case is not so clear. For example, the top Yukawa coupling y t may be fixed to a specific value in a sector of the theory unrelated to supersymmetrybreaking. An example of this is weakly-coupled string theory, where y t may be determinedby the correlator of three string vertex operators and would therefore be fixed to some dis-crete value determined by the compactification geometry. The fact that all of the Yukawacouplings are roughly 1 or 0 helps fuel such speculation. In such a scenario, it is clearlyinappropriate to vary y t continuously to determine the sensitivity of the weak scale to varia-tions in y t . Dimensionless couplings may also be effectively fixed if they run to fixed points.Other such scenarios are discussed in Ref. [31]. In the end, it is probably reasonable toconsider the fundamental parameters both with and without the dimensionless parametersand see if any interesting models emerge. Note that the question of which parameters toinclude in the { a i } is independent of which parameters have been measured; see Sec. IV B 4.A final alternative approach is to choose the fundamental parameters to be weak-scaleparameters. This is perhaps the ultimate bottom-up approach, and it has the advantageof being operationally simpler than having to extrapolate to the GUT or Planck scales.However, as noted above, many of the motivating virtues of supersymmetry are tied to highscales, and some structure must exist if weak-scale supersymmetry is to pass experimentalconstraints. Working at the weak-scale ignores such structure. It is possible, however, toview sensitivity to variations in electroweak parameters as a lower bound on sensitivitiesto variations in high-scale parameters, as they neglect large logarithm-enhanced terms; see, e.g. , Ref. [102]. Of course, one might argue that some more fundamental theory will fix all parameters, including thosethat break supersymmetry. There are no known examples, however. . Calculating the Sensitivity Parameters Alternative choices, sometimes found in the literature, are N i ≡ | ∂ ln m Z /∂ ln a i | or N i ≡| ∂ ln m Z /∂ ln a i | . There is little reason to choose one over the others, except in the case ofscalar masses, where m is the fundamental parameter, not m ( m may be negative, forexample). In any case, these definitions differ by factors of only 2 or 4, which should beignored. This is easier said than done: for example, one definition may yield N i = 20, or O (10)% fine-tuning, while the other definition yields N i = 80 or O (1)% fine-tuning, leadingto a rather different impression. Such examples serve as useful reminders to avoid grandconclusions based on hard cutoffs in naturalness measures.There are other caveats in defining the sensitivity parameters. The role of the sensitivitycoefficients is to capture the possibility of large, canceling contributions to m Z . In principle,it is possible to have a contribution to m Z that is small, but rapidly varying, or large,but slowly varying. It is also possible that m Z is insensitive to variations of any singleparameter, but highly sensitive to variations in a linear combination of parameters. In allof these cases, the sensitivity coefficients are highly misleading, and these possibilities againserve as reminders of how crude naturalness analyses typically are.Finally, some studies have advocated alternative definitions of sensitivity parameters thatincorporate experimental uncertainties. For example, some authors have proposed that thedefinition of Eq. (21), be replaced by [97, 100] N exp i ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ a i m Z ∂m Z ∂a i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (22)where ∆ a i is the experimentally allowed range of a i . The intent of this definition is toencode the idea that naturalness is our attempt to determine which values of parametersare most likely to be realized in nature.To contrast this definition with the conventional definition, consider, for example, thehypothetical scenario in which our theoretical understanding of supersymmetry has notimproved, but the µ parameter is measured to be 10 GeV with very high accuracy. Withthe standard definition of Eq. (21), this model is fine-tuned, but with Eq. (22), it is not.In our view, Eq. (22) encodes an unconventional view of naturalness. Naturalness is not ameasure of our experimental knowledge of the parameters of nature. Rather it is a measureof how well a given theoretical framework explains the parameters realized in nature. It isperfectly possible for values of parameters realized in nature to be unnatural — this is whatthe gauge hierarchy problem is! — and once a parameter’s value is reasonably well-known,naturalness cannot be increased (or decreased) by more precise measurements.
5. Determining the Overall Measure of Naturalness
There are many possible ways to combine the N i to form a single measure of naturalness.A simple variation, advocated by some authors, is to add the N i in quadrature.There are also reasons to consider normalizing the N i either before or after combiningthem. The rationale for this is that in certain cases, all possible choices of a fundamentalparameter may yield large sensitivities. A well-known example of this is the hierarchybetween M Pl and Λ QCD ∼ M Pl e − c/g , which is often considered the textbook example of how19o generate a hierarchy naturally. The related sensitivity parameter, c g ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ln Λ QCD ∂ ln g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ln( M / Λ ) ∼ , (23)however, is large. The authors of Refs. [94, 95, 103] have argued that in such cases, only relatively large sensitivities should be considered fine-tuned, and conclusions based on sensi-tivity parameters consistently overestimate the degree of fine-tuning required. These authorspropose replacing the sensitivity parameters N i defined above, with fine-tuning parameters,defined as γ i ≡ N i / ¯ N i , where ¯ N i is an average sensitivity. These γ i are then combined toform an overall measure of naturalness.Unfortunately, the averaging procedure brings additional complications. If it is done onlyover a subspace of parameter space, it may mask important features [31], and so it should becarried out over the entire parameter space, which is computationally intensive. In addition,it requires defining a measure on the parameter space and defining its boundaries. These ad-ditional complications have dissuaded most authors from including an averaging procedure.Nevertheless, many would agree that the sensitivity parameters should, in principle, be nor-malized in some way, and the naturalness parameter derived from un-normalized sensitivityparameters exaggerates the fine-tuning required for a given model. C. Naturalness Bounds
We now derive upper bounds on superpartner masses from naturalness considerations.Given all the caveats of Sec. IV B, it should go without saying that these should be consideredat most as rough guidelines. The goal here is to give a concrete example of how natural-ness bounds may be derived, compare these with the other theoretical and experimentalconstraints discussed in Secs. II and III, and provide a starting point for the discussion ofmodels in Sec. V.We will consider a bottom-up approach, following the general prescription ofSec. IV A. We consider a model defined at the GUT-scale with input parameters P i = M , M , M , m H u , m H d , m Q , m U , m D , A t , . . . , sign( µ ). These include the gaugino masses M i , the soft SUSY-breaking scalar masses, and the A -terms, all treated as independent.The weak-scale value of | µ | is determined by m Z . The fundamental parameters are taken tobe the GUT-scale values of the input parameters, with sign( µ ) replaced by the GUT-scalevalue of µ . Sensitivity parameters are defined as in Eq. (21), and the overall naturalnessparameter is defined as the largest one.The weak-scale values of supersymmetry-breaking parameters may be determined ana-lytically or numerically in terms of their GUT-scale values [104–108]. Recent analyses fortan β = 10 and using 1- and 2-loop RGEs find [106, 107] M ( m weak ) = 0 . M (24) M ( m weak ) = 0 . M (25) M ( m weak ) = 2 . M (26) − µ ( m weak ) = − . µ (27) − m Hu ( m weak ) = 3 . M + 0 . M M + 0 . M M − . M +0 . M M − . M − . M A t − . M A t . M A t + 0 . A t + 0 . M A b − . m H u − . m H d +0 . m Q + 0 . m U + 0 . m D − . m L + 0 . m E +0 . m Q − . m U + 0 . m D − . m L + 0 . m E +0 . m Q − . m U + 0 . m D − . m L + 0 . m E , (28)where all the parameters on the right-hand sides of these equations are GUT-scale param-eters. The RGEs mix the parameters. Although H u does not couple to gluinos directly,the gluino mass enters the squark mass RGEs and the squark masses enter the H u RGE,and so m H u ( m weak ) depends on the gluino mass M . For the first and second generationsfermions, their Yukawa couplings are so small that their main impact on the Higgs potentialis through hypercharge D -term contributions or, if GUT or other boundary conditions causethese terms to vanish, through 2-loop effects in the H u RGE [109, 110].The naturalness prescription of Sec. IV A is applicable to complete models, but we mayderive rough bounds on individual superpartner masses by neglecting other parameters whenderiving the bound on a given superpartner mass. As an example, keeping only the M termin Eq. (28), we find N M ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ln m Z ∂ ln M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ M m Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ [ − m H u ( m weak ) − µ ( m weak )] ∂M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 3 . M m Z . (29)Requiring N M < N max and using Eq. (26), we can derive a naturalness bound on the physicalgluino mass m ˜ g ≈ M ( m weak ). Proceeding in a similar way for all the parameters, and using m Q ( m weak ) = 0 . m Q + . . . , m U ( m weak ) = 0 . m U + . . . , and m f ( m weak ) ≈ m f + . . . forall other sfermions [106], we find m ˜ H < ∼
640 GeV ( N max / / (30) m ˜ B < ∼ . N max / / (31) m ˜ W < ∼ . N max / / (32) m ˜ g < ∼ . N max / / (33) m ˜ t L , ˜ b L < ∼ . N max / / (34) m ˜ t R < ∼ . N max / / (35) m ˜ b R < ∼ . N max / / (36) m ˜ τ L , ˜ ν τ < ∼ . N max / / (37) m ˜ τ R < ∼ . N max / / (38) m ˜ c L , ˜ s L , ˜ u L , ˜ d L < ∼ . N max / / (39) m ˜ c R , ˜ u R < ∼ . N max / / (40) m ˜ s R , ˜ d R < ∼ . N max / / (41) m ˜ µ L , ˜ ν µ , ˜ e L , ˜ ν e < ∼ . N max / / (42) m ˜ µ R , ˜ e R < ∼ . N max / / . (43)If the standard model particles are unified into GUT multiplets at the GUT scale, thecorrelations lead to significantly different conclusions. For example, assuming M / = M =21 = M , m i = m Q i = m U i = m E i and m i = m D i = m L i , where i = 1 ,
2, we find, withinthe accuracy of these numerical results, − m H u ( m weak ) = 3 . M / + 0 . m + 0 . m + 0 . m + 0 . m + . . . , (44)implying m ˜ B < ∼
190 GeV ( N max / / (45) m ˜ W < ∼
380 GeV ( N max / / (46) m ˜ g < ∼ . N max / / (47) m ˜ u L , ˜ d L , ˜ c L , ˜ s L , ˜ u R , ˜ c R , ˜ e R , ˜ µ R < ∼
11 TeV ( N max / / (48) m ˜ d R , ˜ s R , ˜ ν e , ˜ e L , ˜ ν µ , ˜ µ L < ∼
25 TeV ( N max / / . (49)These results may be understood as follows: The ˜ t L , ˜ b L , and ˜ t R masses enter the H u RGE through top Yukawa couplings, and their bounds in Eqs. (34) and (35) are consistentwith those of Eq. (4). For the other sfermions, the naturalness constraints are weaker.Generically, these masses enter the H u RGE dominantly through hypercharge D -terms, andso one expects constraints on them to be weaker by a factor of (cid:113) α y t /α = y t /g ∼ M ˜ B : M ˜ W : M ˜ g ≈ (cid:113) π/α ∼
10, as seen in Eqs. (48) and (49) [109, 110]. Note thatthe GUT correlations greatly strengthen the naturalness bounds on Binos and Winos, butgreatly weaken the bounds on first and second generation scalars: the choice of underlyingframework can have enormous qualitative implications for naturalness bounds.
V. MODEL FRAMEWORKS
We now discuss a few classes of models that have been proposed to relieve the tensionbetween the various constraints discussed so far. To set the stage, we present all of thetheoretical and experimental constraints discussed in this review in Fig. 7.
A. Effective Supersymmetry
As evident from Fig. 7, naturalness most stringently restricts the masses of scalars withlarge Yukawa couplings, since these are most strongly coupled to the Higgs sector. Atthe same time, experimental constraints are most stringent for scalars with small Yukawacouplings, since light fermions are most easily produced and studied. This suggests thatlight fermions have heavy superpartners and vice versa, which provides a promising wayto reconcile naturalness with flavor and CP constraints [19, 109–111]. A representativespectrum for such models, known as “effective supersymmetry” [112] or, alternatively, “moreminimal supersymmetry” or “inverted hierarchy models,” is shown in Fig. 8.The effective supersymmetry spectrum may be realized in many ways [112–126]. Forexample, if there is an extra anomalous or non-anomalous U(1) gauge group under which22
HC LHC LHC LHCLHC LHC LHCLHC LHCNATNAT NATEDMFLAVNAT NAT NAT NAT NAT NAT NAT NAT NATDM DM EDM FLAV HIGGS
FIG. 7: A sample of constraints on the superpartner spectrum from naturalness (NAT), darkmatter (DM), collider searches (LHC), the Higgs boson mass (HIGGS), flavor violation (FLAV),and EDM constraints (EDM). The constraints assume a moderate value of tan β = 10. Thenaturalness constraints derive from a bottom-up analysis and scale as ( N max / / , where N max is the maximally allowed naturalness parameter; see Sec. IV. All of the constraints shown aremerely indicative and subject to significant loopholes and caveats; see the text for details. LHC LHC LHC LHCLHC LHC LHCLHC LHCNATNAT NATEDMFLAVNAT NAT NAT NAT NAT NAT NAT NAT NATDM DM EDM FLAV HIGGS
FIG. 8: Example superpartner mass ranges for effective supersymmetry (shaded) with constraintsas given in Fig. 7. Heavy and degenerate first and second generation sfermions satisfy flavorand EDM constraints, and light third generation sfermions satisfy naturalness. The Higgs massconstraint requires either near-maximal stop mixing, or physics beyond the MSSM. ∼
10 TeV at the GUT scale, but for particular GUT-scale boundary conditions, those withlarge Yukawa couplings may be driven to low values at the weak scale. In these scenarios, thelarge Yukawa coupling produces both heavy fermions and light sfermions, again naturallyexplaining the inverted hierarchy structure.Effective supersymmetry predates not only the Higgs discovery, but even the most strin-gent LEP limits on the Higgs mass, and was not originally intended to explain the largeHiggs boson mass. As discussed in Sec. III B, the Higgs boson mass may be consistent withsub-TeV stops, but only in the highly fine-tuned case when there is large left-right mixing;for a recent discussion of this in the context of effective supersymmetry, see Ref. [126]. Alter-natively, more minimal supersymmetry may be made less minimal by adding extra fields toraise the Higgs mass; see, e.g. , Refs. [127, 128]. Effective supersymmetry with this extensionhas attracted renewed attention, sometimes under the confusingly generic moniker “naturalsupersymmetry,” as a strategy to reconcile naturalness with LHC constraints [129].In effective supersymmetry, the first and second generation sfermions are beyond the reachof the LHC, but gluinos, stops, and sbottoms may be within reach. The most promisingcollider signals are therefore direct stop and sbottom squark production, or gluinos withtop- and bottom-rich cascade decays [130]. Effects in low-energy B physics may also beaccessible [131]. Finally, new particles added to raise the Higgs mass may have associatedsignals. B. Focus Point Supersymmetry
In focus point supersymmetry [31, 132, 133], correlations between parameters allow spar-ticle masses to be larger than their naive naturalness bounds. A representative spectrumwith heavy scalars is given in Fig. 9. Heavy first and second generation scalars suppressflavor and CP violation, as in effective supersymmetry. In contrast to effective supersymme-try, however, the third generation is also heavy, naturally raising the Higgs mass to withincurrent bounds. There are many theoretical reasons for expecting scalar superpartners tobe heavier than the gauginos. For example, such a hierarchy follows from an approximateU(1) R symmetry, which suppresses the gaugino masses (and A -terms) but not the scalarmasses. It also results if there are no singlet supersymmetry-breaking fields [79, 80]. Notethat gaugino masses enter the scalar mass RGEs, but scalar masses do not enter the gaug-ino mass RGEs; the hierarchy m (cid:29) M / is therefore stable under RGE evolution, whereas M / (cid:29) m is not.The obvious difficulty is that heavy top squarks naively contradict naturalness. In focuspoint supersymmetry, correlations in GUT-scale parameters are invoked to alleviate this fine-tuning. A simple example is evident from Eq. (28). The weak-scale value of m H u ( m weak ) ishighly sensitive to the GUT-scale values of m H u , m Q , and m U , but if these have a unifiedvalue m at the GUT scale, − m Hu ( m weak ) = − . m H u + 0 . m Q + 0 . m U + . . . =0 . m + . . . , and the weak scale becomes highly insensitive to variations in these GUT-scaleparameters, even if they are large. The reasoning here is similar to that leading to natural ∼
10 TeV first and second generation sfermions with N max ∼
100 in the GUT case analyzed24
HC LHC LHC LHCLHC LHC LHCLHC LHCNATNAT NATEDMFLAVNAT NAT NAT NAT NAT NAT NAT NAT NATDM DM EDM FLAV HIGGS
FIG. 9: Example superpartner mass ranges for focus point supersymmetry (shaded) with con-straints as given in Fig. 7. Heavy and degenerate first and second generation sfermions satisfyflavor and EDM constraints and heavy third generation sfermions raise the Higgs mass, whilenaturalness is preserved despite heavy stops by correlations between GUT-scale parameters. in Sec. IV C.This behavior may be understood as a property of the RGEs. The m H u RGEs in afocus point model are shown in Fig. 10. The RG trajectories have a focus point at theweak scale, and so the weak-scale value of m H u is insensitive to variations in the GUT-scale parameters. The weak scale still receives quadratic contributions from heavy stops,but the large logarithm enhancement from RG evolution in Eq. (4) is absent, reducing thefine-tuning associated with multi-TeV stops by a factor of ∼ ln( m /m ) ∼
60. Suchfocusing occurs if the GUT-scale parameters satisfy [132]( m H u , m t R , m t L ) ∝ (1 , x, − x ) (50)for moderate values of tan β , and( m H u , m t R , m t L , m b R , m H d ) ∝ (1 , x, − x, x − x (cid:48) , x (cid:48) ) (51)for large values of tan β , where x and x (cid:48) are arbitrary constants. Note that the scale atwhich focusing occurs is sensitive to dimensionless couplings, particularly the top Yukawa y t . As discussed in Sec. IV B 3, one may include y t as a fundamental parameter or not. If itis included, N y t is large for large superpartner masses, but it is large throughout parameterspace [133]. If one adopts the averaging procedure described in Sec. IV B 5 to identify onlyrelatively large sensitivities, the effect of including y t as a fundamental parameter is greatlymoderated.A universal scalar mass obviously satisfies both Eqs. (50) and (51), and the large m region of mSUGRA has become the canonical example of focus point supersymmetry [99,132, 133]. Focus point supersymmetry is, however, a far more general phenomenon, as onemay postulate many relations between the GUT-scale parameters to reduce the fine-tuning25 IG. 1. The RG evolution of m H u for (a) tan β = 10 and (b) tan β = 50, several values of m (shown, in GeV), M / = 300 GeV, A = 0, and m t = 174 GeV. For both values of tan β , m H u exhibits an RG focus point near the weak scale, where Q ( H u )F ∼ O (100 GeV), irrespective of m . to the gauge and Yukawa coupling constants are also included [10,11]. We take as in-puts α − = 137 . G F = 1 . × − , α s ( m Z ) = 0 . m Z = 91 .
187 GeV, m DRτ ( m Z ) = 1 . m b = 4 . m t = 174 GeV.The scale dependence of m H u for various values of m in minimal supergravity is shownin Fig. 1. To high accuracy, all of the RG trajectories meet at Q ∼ O (100 GeV). In fact,in this case, the weak value of m H u is determined by the other fundamental parameters M / and A , and hence at least one of these parameters is required to be O (100 GeV).In Fig. 1, two values of tan β were presented. In Fig. 2, we show the focus pointscale of m H u as a function of tan β . The focus point is defined here as the scale where ∂m H u /∂m = 0. As noted above, we have included the low-energy threshold correctionsto the gauge and Yukawa coupling constants, which depend on the soft supersymmetrybreaking parameters. As a result, the RG trajectories do not all meet at one scale, andthe focus point given in Fig. 2 has a slight dependence on m . For small values of tan β ,say, tan β ∼ −
3, the focus point is at very large scales. However, the important point isthat, for all values of tan β > ∼
5, including both moderate values of tan β and large valueswhere y b and y τ are not negligible, Q ( H u )F ∼ O (100 GeV), and the weak scale value of m H u is insensitive to m .So far, we have considered only the case of a universal scalar mass. However, the m H u focus point remains at the weak scale for a much wider class of boundary conditions. Forexample, for small tan β , Eq. (7) shows that the parameter κ ′ does not affect the evolutionof m H u . As a result, the focus point of m H u does not change even if we vary κ ′ , and the7 FIG. 10: The RG evolution of m H u in the focus point region of mSUGRA for tan β = 10 (left)and 50 (right), several values of m (as shown in GeV), M / = 300 GeV, and A = 0. For bothvalues of tan β , m H u exhibits an RG focus point near the weak scale, implying that the weak scaleis insensitive to variations in the GUT-scale supersymmetry-breaking parameters [133]. in Eq. (28). For example, considering the M , M M , and M terms, one finds focusingfor M /M ≈ . , − . A -terms [138], and may emerge from the boundary conditions ofmirage mediation [139–141] or be enforced by a symmetry [136].In the most studied focus point supersymmetry models, all scalars are heavy, but typicallythe stops are slightly lighter. They may be produced in future LHC runs, or may be beyondreach, but light enough to enhance the top content of gluino decays. The most promis-ing LHC signals are therefore again direct stop production with cascade decays throughcharginos and neutralinos, or gluino production, followed by top- and bottom-rich cascadedecays. As the first generation scalars are heavy but not extremely heavy, there may alsobe a signal in EDMs. Last, the prospects for WIMP dark matter detection are extremelypromising in focus point models [37, 142]. In particular, focus point supersymmetry predictsa mixed Bino-Higgsino neutralino with a spin-independent proton cross sections typicallyabove the zeptobarn level, which should be probed in the coming few years.Last, note that the large logarithm enhancement may also be eliminated by adding ad-ditional particles. This is the approach of an entirely different class of models, typicallycalled “supersoft supersymmetry” [143], where the MSSM is extended to include a gaugeadjoint chiral superfield for each gauge group, providing an interesting alternative strategyfor reconciling naturalness with experimental constraints [144, 145]. C. Compressed Supersymmetry
In many supersymmetric models, there is a large mass splitting between the gluino andsquarks at the top of the spectrum and the lighter superparters at the bottom. This reducesthe naturalness of these models in two ways. First, the large mass splittings imply that26he gluino and squark cascade decays produce energetic particles and large /E T , leading todistinct signals and strong bounds on sparticle masses. Second, lower bounds on the massesof the lighter sparticles imply stringent lower bounds on gluino and squark masses, whichdecreases naturalness.The superpartner spectrum may be much more degenerate, however. This has beenexplored in the context of “compressed supersymmetry” [107], in which there are smallsplittings between colored superpartners and an LSP neutralino. For the reasons givenabove, this leads to weaker bounds on sparticle masses and provides an interesting approachto developing viable and natural models [146, 147].There are well-motivated reasons to expect large mass splittings. RG evolution drives upcolored sparticles masses relative to uncolored ones. For example, assuming gaugino massunification at the GUT scale, Eqs. (24)–(26) imply | M | : | M | : | M | ≈ F -terms, but by a multiplet of SU(5),group theoretic factors imply | M | : | M | : | M | ≈ | M | : | M | : | M | ≈ M and M terms enter with opposite signs in Eq. (28), and so when | M | isa little larger than | M | at the GUT scale, these terms partially cancel and naturalness isimproved by essentially the same mechanism discussed in Sec. V B for focus point scenarioswith non-universal gaugino masses.A representative spectrum is shown in Fig. 11. The virtue of compressed supersymmetryis that it decreases the tension between naturalness and LHC superpartner search bounds. Ashortcoming of these models is that the light spectrum exacerbates problems with flavor andCP violation. In particular, ∼
100 GeV superpartners generically require φ CP < ∼ − − − to satisfy EDM constraints, and so these models require some additional mechanism tosuppress CP violation. In addition, the problem of obtaining a 125 GeV Higgs boson massis present in compressed supersymmetry if the stops are light. As in the case of effectivesupersymmetry, physics beyond the MSSM [127, 128] is required to raise the Higgs mass toits measured value, bringing with it additional complications.The collider signals of compressed supersymmetry have been explored in a number ofstudies [150–156]. The relevant signals depend on the degree of compression. For m ˜ t − m χ
500 GeV. Implications for neutralino dark matter have been explored inRefs. [107, 150, 157].Finally, there are many other models in which the /E T signal is reduced. Interestingpossibilities in which cascade decays go through hidden sectors include hidden valley mod-els [158, 159] and stealth supersymmetry [160]. D. R -Parity-Violating Supersymmetry As discussed in Sec. III A 3, the characteristic /E T collider signal of supersymmetry mayalso be degraded in the presence of R -parity violation. If any of the superpotential termsof Eq. (7) is non-zero, all superpartners decay, and, provided the decay length is not toolong, supersymmetric particles do not escape the detector. The phenomenology of RPVsupersymmetry has been studied for a long time [161–163], but it has recently attracted27 HC LHC LHC LHCLHC LHC LHCLHC LHCNATNAT NATEDMFLAVNAT NAT NAT NAT NAT NAT NAT NAT NATDM DM EDM FLAV HIGGS
FIG. 11: Example superpartner mass ranges for compressed supersymmetry and RPV supersym-metry (shaded) with constraints as given in Fig. 7. Light sfermions preserve naturalness andevade LHC bounds because /E T signals are degraded by superpartner degeneracies in compressedsupersymmetry or by LSP decays in RPV supersymmetry. These models require mechanisms toeliminate flavor violation and reduce CP-violating phases to O (10 − ) − O (10 − ), and also requirenear-maximal stop mixing or physics beyond the MSSM to raise the Higgs mass. In addition, inRPV supersymmetry, there is no WIMP dark matter candidate. renewed attention as a way to make light superpartners viable, and thereby reduce fine-tuning.In general, once one allows R p violation, one opens a Pandora’s box of possibilities. Thereare few principled ways to violate R -parity conservation. The RPV couplings cannot all besizable. In fact, there are stringent bounds on individual RPV couplings, and even morestringent bounds on products of pairs of couplings [164, 165]; in particular, if any leptonnumber-violating coupling and any baryon number-violating coupling are both non-zero,proton decay sets extremely stringent constraints.If theory is any guide, one might expect that the RPV couplings follow the pattern of the R p -conserving couplings, with those involving the third generation the biggest, the secondgeneration smaller, and the first smaller still. Realizations of this hypothesis have beenpresented in Refs. [166–170], where models of R p violation based on the principle of minimalflavor violation lead to scenarios in which only the hadronic RPV terms λ (cid:48)(cid:48) ijk U i D j D k aresizable, with λ (cid:48)(cid:48) typically the largest. Such models somewhat moderate the naturalnessmotivation for R p violation, as they imply that, say, the LSP neutralino decays dominantlyto top and bottom quarks, leading to b -jets, leptons, and /E T from neutrinos, all distinctivecharacteristics that one was hoping to avoid. Nevertheless, such RPV signals likely do reduceLHC limits somewhat, and, of course, one may always ignore theoretical bias and consider,say, λ (cid:48)(cid:48) couplings that would lead to decays to light flavor and pure jet signals, such asthose discussed in Sec. III A 3 and Fig. 3.In summary, as with compressed supersymmetry, R p violation provides another possi-bility for reducing the distinctiveness of supersymmetry signals at colliders and potentially28mproving the naturalness of viable models. The shortcomings, however, are also similar: ifall of the superpartners are light, the Higgs boson is generically too light, requiring physicsbeyond the MSSM, and the EDM constraints are generically not satisfied, requiring yet morestructure to remove troubling CP-violating phases. In RPV supersymmetry, one also losesthe motivation of WIMP dark matter, although the gravitino or other candidates may playthis role. VI. CONCLUSIONS
PARABLE. Some children notice that a soap bubble’s width and height are remarkablysimilar. They get excited when they find that this can be explained by surface tension androtational symmetry. Later, with amazing experiments, they find that the width and heightare not identical, but differ by 1 part in . They remember, however, that wind candistort the shape of the bubble and calculate that, given typical winds, one would expectdifferences of 1 part in . Some of the children become despondent and wonder howsuch a beautiful solution could be so wrong; others consider alternative explanations; otherspostulate that bubbles can be any shape, but only nearly spherical ones are compatible withthe presence of children; and others study the wind. Supersymmetry has long been the leading candidate for new physics at the weak scale. Inthis review, we have evaluated its current status in light of many theoretical and experimentalconsiderations.The leading theoretical motivations for weak-scale supersymmetry are naturalness, grandunification, and WIMP dark matter. Each of these prefers supersymmetry breaking at theweak scale, but each argument is subject to caveats outlined in Sec. II. Of course, taken asa whole, these continue to strongly motivate supersymmetry.Current experimental constraints are discussed in Sec. III and summarized in Fig. 7.For some varieties of supersymmetry models, the LHC now requires superpartner masseswell above 1 TeV, but there are also well-motivated examples in which superpartners maybe significantly lighter without violating known bounds. The 125 GeV Higgs boson massprefers heavy top squarks in the MSSM, and longstanding flavor and CP constraints stronglysuggest multi-TeV first and second generation sfermions. We have especially emphasized therobustness of the EDM constraints, which are present even in flavor-conserving theories. Inthe absence of a compelling mechanism for suppressing CP violation, the EDM constraintsrequire first generation sfermions to be well above the TeV scale. Against the backdrop ofthese indirect constraints, LHC bounds on supersymmetry are significant because they aredirect, but they are hardly game-changing. One may like supersymmetry or not, but to havethought it promising in 2008 and to think it much less promising now is surely the leastdefensible viewpoint.In Sec. IV, we have critically examined attempts to quantify naturalness. There aremany studies embodying philosophies that differ greatly from each other. We have expressedreservations about some, but for many, one can only acknowledge the subjective nature ofnaturalness and make explicit the underlying assumptions. Very roughly speaking, however,current bounds are beginning to probe naturalness parameters of
N ∼ ∼
13 TeV in 2015, with initial results available by Summer 2015, and 100 fb − of dataanalyzed by 2018. Such a jump in energy and luminosity will push the reach in gluino andsquark masses from around 1 TeV to around 3-4 TeV, and probe models that are roughlyan order of magnitude less natural. Given these exciting prospects for drastically improvedsensitivity to supersymmetry or other new physics at the weak scale, patience is a virtue.In the grand scheme of things, we will soon know. ACKNOWLEDGMENTS
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