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Naturalness Versus Stringy Naturalness with Implications for Collider and Dark Matter Searches

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Naturalness Versus Stringy Naturalness with Implications for Collider and Dark Matter Searches PHYSICAL REVIEW RESEARCH 1, 023001 (2019) Naturalness versus stringy naturalness with implications for collider and dark matter searches Howard Baer ,1,* Vernon Barger,2,† and Shadman Salam1,‡ 1Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019, USA 2Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA (Received 21 June 2019; published 3 September 2019) The notion of stringy naturalness—that an observable O2 is more natural than O1 if more (phenomenologically acceptable) vacua solutions lead to O2 rather than O1—is examined within the context of the Standard model (SM) and various supersymmetric models (SUSY) extensions: Constrained minimal supersymmetric standard model (CMSSM)/minimal supergravity model (mSUGRA), high-scale SUSY, and radiatively driven natural SUSY (RNS). Rather general arguments from string theory suggest a (possibly mild) statistical draw towards vacua with large soft SUSY breaking terms. These vacua must be tempered by an anthropic veto of nonstandard vacua or vacua with too large a value of the weak scale mweak. We argue that the SM, the CMSSM and various high-scale SUSY models are all expected to be relatively rare occurrences within the string theory landscape of vacua. In contrast, models with TeV-scale soft terms but with mweak ∼ 100 GeV and consequent light higgsinos (SUSY with radiatively driven naturalness) should be much more common on the landscape. These latter models have a statistical preference for mh 125 GeV and strongly interacting sparticles beyond current LHC reach. Thus, while conventional naturalness favors sparticles close to the weak scale, stringy naturalness favors sparticles so heavy that electroweak symmetry is barely broken and one is living dangerously close to vacua with charge-or-color breaking minima, no electroweak breaking or pocket universe weak-scale values too far from our measured value. Expectations for how landscape SUSY would manifest itself at collider and dark matter search experiments are then modified compared to usual notions. DOI: 10.1103/PhysRevResearch.1.023001 I. INTRODUCTION up a Little hierarchy problem (LHP) [12]: why does the weak scale not blow up to the energy scale associated with soft Naturalness. The gauge hierarchy problem (GHP) [1]— SUSY breaking, i.e., why is m m ? what stabilizes the weak scale so that it does not blow up to weak soft Early upper bounds on sparticle masses derived from the GUT/Planck scale—is one of the central conundrums of naturalness were usually computed using the EENZ/BG particle physics. Indeed, it provides crucial motivation for the logarithmic-derivative measure [7,8]: for an observable O, premise that new physics should be lurking in or around the then weak scale. Supersymmetric models (SUSY) with weak-scale soft SUSY breaking terms provide an elegant solution to ∂ O ∂O O ≡ ln = pi , the GHP [2,3] but so far weak-scale sparticles have failed BG( ) maxi maxi (1) ∂ ln pi O ∂ pi to appear at the CERN Large Hadron Collider (LHC) and WIMPs have failed to appear in direct detection experiments where the pi are fundamental parameters of the underly- [4]. The latest search limits from LHC Run 2 require gluinos ing theory. For an observable depending linearly on model . . O = +···+ ∂O/∂ = with mg˜ 2 25 TeV [5] and top squarks mt˜1 1 1TeV parameters, a1 p1 an pn, then pi ai and [6]. These lower bound search limits stand in sharp contrast BG(O) just picks off the maximal right-hand-side contri- to early sparticle mass upper bounds from naturalness that bution to O and compares it to O. In the case where one . 1 O seemingly required mg˜,t˜1 0 4TeV[7–10]. Thus LHC limits contribution ai pi , then some other contribution(s) will seem to imply the soft SUSY breaking scale msoft lies in the have to be finetuned to large opposite-sign values such as multi-TeV rather than the weak-scale range. This then opens to maintain O at its measured value. Such finetuning of fundamental parameters seems highly implausible in nature absent some symmetry or parameter selection mechanism. * Thus the logarithmic derivative is a measure of [13] practical [email protected] O †[email protected] naturalness. An observable is natural if all independent O O ‡[email protected] contributions to are comparable to or less than . For the case of the LHP, the observable O is traditionally 2 μ Published by the American Physical Society under the terms of the take to be mZ and the pi are taken to be the MSSM term and Creative Commons Attribution 4.0 International license. Further soft SUSY breaking terms so that naturalness then requires 2 2 distribution of this work must maintain attribution to the author(s) all contributions to mZ to be comparable to mZ : this is the and the published article’s title, journal citation, and DOI. basis for expecting sparticles to occur around the 100-GeV 1 For a recent analysis in the context of EENZ/BG, see, e.g., scale. A fundamental issue in computing BG is [14–17]: Ref. [11]. what is the correct choice to be made for the pi? If soft terms 2643-1564/2019/1(2)/023001(13) 023001-1 Published by the American Physical Society HOWARD BAER, VERNON BARGER, AND SHADMAN SALAM PHYSICAL REVIEW RESEARCH 1, 023001 (2019) 2 / are correlated with one another, as expected in fundamental is then low provided all weak-scale contributions to mZ 2are supergravity/superstring theories, then one gets a very 2 / u d comparable to or less than mZ 2. The u and d contain different answer than if one assumes some effective 4 − d over 40 radiative corrections which are listed in Appendix of SUSY theory with many independent soft parameters which Ref. [23]. The conditions for natural SUSY (for, e.g., EW < are introduced to parametrize one’s ignorance of their origin.2 30)4 can then be read off from Eq. (6). Alternatively, even if the parameters pi are independent, it is (1) The superpotential μ parameter has magnitude not too possible that some selection mechanism is responsible for the far from the weak scale, |μ| 300 GeV. This implies the values towards which they tend. χ0 χ± χ0 , χ± ∼ existence of light higgsinos 1,2 and 1 with m( 1,2 1 ) It is also common in the literature to apply practical natu- 100–300 GeV. ralness to the Higgs mass: (2) m2 is radiatively driven from large high-scale values Hu to small negative values at the weak scale (SUSY with radia- m2 m2 (weak) + μ2(weak) + mixing + rad. corr., (2) h Hu tively driven naturalness or RNS [22]). u where the mixing and radiative corrections are both compa- (3) Large cancellations occur in the u (t˜1,2 )termsfor 2 2 = 2 + δ 2 ∼ rable to m .Also,m (weak) m ( ) m where it is large At parameters which then allow for mt˜1 1–3 TeV for h Hu Hu Hu < common to estimate δm2 using its renormalization group EW 30. The large At term also lifts the Higgs mass mh into Hu equation (RGE) by setting several terms in dm2 /dt (with the vicinity of 125 GeV. The gluino contributes to the weak Hu scale at two-loop order so its mass can range up to m 6TeV t = ln Q2) to zero so as to integrate in a single step: g˜ with little cost to naturalness [23–25]. 3 f 2 (4) Since first/second generation squarks and sleptons δm2 ∼− t m2 + m2 + A2 ln 2/m2 . (3) Hu 8π 2 Q3 U3 t soft contribute to the weak scale through (mainly canceling) D terms, they can range up to 10–30 TeV at little cost to ∼ Taking mGUT and requiring the high-scale measure naturalness (thus helping to alleviate the SUSY flavor and CP ≡ δm2 /m2. (4) problems) [26]. HS Hu h By combining dependent soft terms in BG or by combin- 2 2 HS 1 then requires three third generation squarks lighter ing the dependent terms m () and δm in , then these Hu Hu HS than 500 GeV [18,19] (now highly excluded by LHC top- measures roughly reduce to EW (aside from the radiative squark searches) and small A terms (whereas m 125 GeV u,d t h contributions u,d ). Since EW is determined by the weak- typically requires large mixing and thus multi-TeV values scale SUSY parameters, then different models which give rise of A0 [20,21]). The simplifications made in this calculation to exactly the same sparticle mass spectrum will have the same ignore the fact that δm2 is highly dependent on m2 () Hu Hu finetuning value (model independence). Using the naturalness (which is set to zero in the simplification) [14,16,17]. In fact, measure EW, then it has been shown that plenty of SUSY the larger one makes m2 (), then the larger becomes the can- Hu parameter space remains natural even in the face of LHC Run celing correction δm2 . Thus these terms are not independent: 2 Higgs mass measurements and sparticle mass limits [23]. Hu one cannot tune m2 () against a large contribution δm2 . String theory landscape. Weinberg’s anthropic solution to Hu Hu Thus weak-scale top squarks and small At are not required the cosmological constant (CC) [27], along with the emer- by naturalness. gence of the string theory landscape of vacua [28,29], have To ameliorate the above naturalness calculational quan- presented a challenge to the usual notion of naturalness. daries, a more model independent measure EW was intro- In Weinberg’s view, in the presence of a vast assortment duced [22,23].3 By minimizing the weak-scale SUSY Higgs (10120) of pocket universes which are part of an eternally potential, including radiative corrections, one may relate the inflating multiverse, it may not be so surprising to find our- ∼ −120 4 measured value of the Z-boson mass to the various SUSY selves in one with cc 10 mP since if it were much contributions: larger, then the expansion rate would be so large that galaxies would not be able to condense, and life as we know it would be m2 + d − m2 + u tan2 β m2 /2 = Hd d Hu u − μ2 unable to emerge.
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