JHEP08(2019)105 Springer June 4, 2019 : August 5, 2019 August 21, 2019 : : (100)Tev

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JHEP08(2019)105 Springer June 4, 2019 : August 5, 2019 August 21, 2019 : : (100)TeV. The low- Received Accepted ∼ O Published TeV scale. This compromises nat- ∼ Published for SISSA by https://doi.org/10.1007/JHEP08(2019)105 [email protected] , . 3 1806.08502 The Authors. (10)] theories with a multi-TeV spectrum of supersymmetric particles that c O Supersymmetry Phenomenology

The experiments at the Large Hadron Collider (LHC) have pushed the limits , . [email protected] E-mail: ARC Centre of Excellence forSchool of Particle Physics, Physics The atNSW University the 2006, of Terascale, Australia Sydney, Open Access Article funded by SCOAP thorough exploration of possible alternative ultraviolet completions of MSSM. Keywords: ArXiv ePrint: trend we observe reduction in fine-tuningnatural together [∆ with lowering Λ.are In consistent particular, with perfectly all currentering observations of can be the obtained fine-tuning forspecial for Λ quasi-fixed large point cut-off behaviour for scales certain can parameters. also Our be observations call observed for in a more theories exhibiting supersymmetric Standard Model (MSSM).current experimental In data this areillustrate paper actually this, we we hinting advocate treat towardsspecified that the the theory perhaps at MSSM physics a the as beyond scaleconditions a Λ, MSSM. with and low-energy compute soft limit the To SUSY of fine-tuning measure breaking a ∆ parameters more for and generic fundamental boundary various yet cut-off un- scales. As a general Abstract: on masses of supersymmetricuralness particles of beyond the the simplest supersymmetric extension of the Standard Model, the minimal Archil Kobakhidze and Matthew Talia Supersymmetric naturalness beyond MSSM JHEP08(2019)105 300 ] fine- & 3 ]. Taking this at face 1 2 TeV [ 6 ∼ – 1 – 2 ]). In fact, we would like to advocate here that the failure 2 4 1 8 1000 GeV, since such a theory would naturally incorporate the electroweak scale − It should be clear however, that the above conclusion in no way falsifies SUSY as a Contrary to these theoretical expectations, there has not been any evidence of SUSY 100 the fundamental theory which aretake traced the down attitude to that the theunspecified MSSM low-energy is theory theory. a at In low-energy this a approximationtuning of paper scale a measure we Λ. more ∆ fundamental for We yet then variousparameters scales compute defined Λ, at the Λ. assuming standard arbitrary This Barbieri-Giudice boundary way we [ conditions parametrize on our ignorance MSSM of the fundamental theory theoretical framework, butrecent rather related the discussion MSSM, inof in [ the its particular naturalways realization MSSM that (see may the parameters also indicate involvedFirst the physics in is fine-tuning through beyond can the the modification bethrough of modified MSSM. the the due modification renormalisation to of group There the boundary (RG) new running are conditions, physics. and e.g., two second due is major to the enhanced symmetries in particles at the LHC.and With first some and simplified second assumptions, generationvalue, the squarks this experiments with compromises exclude masses up gluinos naturalness to is of required the to electroweak accommodate scale the — correct a electroweak fine-tuning scale. of ∆ theories with softly broken SUSY.the This Standard feature Model motivates studies (MSSM) of∼ with the additional SUSY extension supersymmetric of particleswithout in the the need mass to range fine tune quantum corrected parameters. Supersymmetry (SUSY) is aStandard very Model attractive as theoretical it framework represents for aprovides unique physics a non-trivial beyond extension unified the of description relativisticby-product, invariance of and supersymmetric particles quantum with field differentbehaviour. theories spin-statistics. exhibit Namely, improved As short-distance the an scale notorious important quadratic divergences are completely absent even in 1 Introduction 3 Parameter scan 4 MSSM quasi infrared fixed-point5 and fine-tuning Conclusions Contents 1 Introduction 2 Supersymmetric naturalness JHEP08(2019)105 × (2.1) (2.2) (2.3) , . d ) holds to a very good 2 we scan over a narrower and the supersymmetric µ H 4 d u 2.2 − ,H u u µH H 2 H ]: + m 3 m , d

' − LH e 2 Z i 2 y , we describe the measure of ﬁne-tuning µ e ∂a ∂m 2 − + ¯ 2 Z i β d a m 2 at low-energies must be adjusted in a way to

(10), which may hint towards physics beyond – 2 – µ QH 1 tan d − u y ¯ d 2 H ∼ O β and 2 m + u This is roughly consistent with the observations made u H − 91 GeV. This adjustment is natural if not very sensitive 1 tan ∆ = max d m ' we present our conclusions. 2 H QH u Z 5 m y m u , the last approximate equation in ( = | (100) to ∆ 100 TeV. u = ¯ 2 Z 2 H ∼ m 3. Hence, W m ∼ O | & . β , and the low energy soft breaking masses | ]. We also discuss an example of quasi-fixed point running of parameters, d Z 5 are the 3x3 Yukawa matrices in flavor space, the supersymmetric SU(3) H run over the “fundamental” parameters of the effective low-energy MSSM. m i e m , | a ], the reduction of fine-tuning was also observed within the effective MSSM with high dimension d 4 , we perform a general scan over the MSSM, computing the fine-tuning measure U(1) gauge interactions, and the standard MSSM soft SUSY breaking terms. The u 3 y × The relation between low energy and “fundamental” parameters are defined by the Minimization of the tree-level potential gives the following relation between the Z- The paper is organized as follows. In section In ref. [ 1 parameter: where the solution of the respective RG equations and matching conditions at the cut-offoperators scale in Λ. the The Higgs sector. reproduce the Z-pole mass to the variationsensitivity of is “fundamental” the parameters Barbieri-Giudice fine-tuning at parameter Λ. [ The quantitative measure of this µ Assuming accuracy for tan SU(2) effective running parameters evaluated at thetal” cut-off parameters scale of Λ the we effective identify as MSSM. the “fundamen- boson mass, violet theory with atheory cut-off are scale those Λ. of the The MSSM relevant superpotential, and marginal operators in the effective where in low fine-tuning. In section 2 Supersymmetric naturalness We consider the MSSM as an effective low-energy approximation of an unspecified ultra- search of specific extensions of the MSSM which resultin in the natural context electroweak of scale. thesection MSSM and subsequently thealong relationship with to relevant the experimental constraints. parameter RGEs. Similarlyregion in In of section parameter space corresponding to an MSSM quasi-infrared fixed point resulting important at energies Λ.lowering As Λ a from general trend ∆ the we MSSM observe at the a reductionpreviously scale of in fine-tuning [ with which results in a low fine-tuning for large cut-off scales. These examples motivate further as well as the effects of higher-dimensional irrelevant operators which are expected to be JHEP08(2019)105 is . In µ (2.8) (2.6) (2.7) (2.4) (2.5) 4 = 0 and µ ) is generi- 2.3 , = 0. This obser- . , such a behaviour u u S 2 ¯e H 2 H 2 1 g m m m 3 5 + + ), the RG running cannot ¯ 2 d 2 | m 2.2 1 , + M 2 | | ¯u t 2 2 1 . g A 2 3 | 6 5 , exhibits a fixed-point at 2m u M + , − 2 µ 3 2 H − 3 g 2 ¯ 2 u | 2 m 2 L 2 | 2 2 2 t m π 2 t m π M A y is sensitive to variations of different mass | | + − 6 − 2 2 16 3 u g + dominates over other mass parameters at high Σ 2 Q 2 Q 6 2 H – 3 – 3 2 2 = t u ¯ 2 u m y m π m − u 2 H 3 4 t m 2 H + m X u + m T r 2 | 2 H 3 t d Σ = ) at Λ, it will stay small at low-energies. Small dt + 2 Q y m | Z d d ), we can determine another infrared fixed-point by defin- dt 3 m 2 H m ) takes the approximate form: + an arbitrary renormalization scale) and 2.4 m 2 Σ = ∼ u π ). We confirm this by numerical analysis — the required fine 2.4 µ 1 − 2 H µ 16 u m 2.2 2 H = m (here ˜ u parameter) exhibits an infrared fixed-point at 2 H = 2 ≡ 2 t µ m ˜ S µ X / d dt 2 Λ = ln t From the observation in ( Alternatively, if one assumes that One can think of two ways to reduce the required fine-tuning in models with large spar- The running of the supersymmetric parameter from which weparameters can are compute subdominant the beta function for Σ in the limit that all other mass vation motivates us to scanaccord a with specific the region expectation, oflarge we “fundamental” observe Λ parameters significant and in reduction heavy section in sparticles. fine-tuning measure for a ing the sum: energies, the RG equation ( which (similar to scale Λ such that thedown “fundamental” to parameters low do energies. notfine evolve tuning In significantly is this when required running case, todestabilize if satisfy the the the relation minimization fundamental ( conditiontuning theory ( is is significantly such reduced that for no low Λ, significant even for a rather heavy spectrum of sparticles. As a result,parameters, the and low-energy if the parameter sparticlecally spectrum large. is heavy, the fine-tuning measure ( ticle masses. First, one assumes that the physics beyond the MSSM enters at a low enough where therefore if taken smalltherefore natural. ( However, for pure scalaris mass atypical parameters, such due as to thebeta-function. additive More contribution specifically, of heavy particle masses to the corresponding we are working within theof effective the field theory ultraviolet framework, physicsrange) we by parameterize values considering of our an the ignorance 20 unconventionalmost “fundamental” and relevant parameters arbitrary for of computing (within the the certain MSSM, low-energy which parameters are for potentially the different values of Λ. latter can only be computed if the ultraviolet completion of the MSSM is known. Since JHEP08(2019)105 3 ], M 12 , all [ (3.1) (3.2) µ B and 3 g SARAH and µ ]. Here we allow 10 ) are the gaugino and fine tuning. – 8 ˜ g 2.3 ]. 2 2 2 M is kept large at the grand 13 [ t y GeV GeV GeV ], combined with 2 2 2 , 3rd generation scalar masses , and the terms 11 d τ [ , . 2 H 1 ,A b 2000 GeV (3000) (3000) (3000) 3000 GeV 50 3000 GeV ,M u GeV ,A < < < < < < 2 H t d τ A 16 M ) = 2 2 H µ 2 β < ,A , 3 10 ( 3 b 2 i 1 , 2 i , m ,M 1 u M m micrOmegas-4.3.2 10 m ,A tan 2 H SPHENO-3.3.8 t – 4 – sign 10 , and spin-independent WIMP-nucleon cross-section 5 < m < A < < M 2 10 2 h 1 ∈ ]. More specifically, upon computing the beta-functions GeV 7 Λ , 2 6 3000 GeV 3000 GeV 18 of the corresponding RG equations. This procedure has also − − / , the trilinear couplings (3000) 3 ). The first and second generation scalar soft masses are taken to ¯ 2 e − ¯ e = 7 2 3 , m , Higgs soft-breaking masses 3 3 /g ¯ 2 L d, L, 2 t y ,M ¯ u, , m 2 up to one-loop and without electroweak contribution, one finds an infrared 3 ¯ 2 d 3 Q, 0 this expresses an infrared fixed-point at Σ = 0. However, since g ,M 1 , m 3 = ( → M ¯ 2 u i 3 and t , m We employ full two-loop RGEs using We choose the input scale Λ in which the parameters are defined for the following Before we proceed with our numerical analysis, we note that the infrared quasi-fixed M y 3 2 Q masses m computed at thealso corresponding compute the scale DM Λ. relicassuming density a Ω The neutralino DM top candidate (pole) using mass is set to 173 GeV. We three cases: in order toincluded compute in the the MSSM calculation spectrum and of fine-tuning the measure. fine-tuning measure The in parameters eq. ( where be degenerate and we assume no flavour mixing at the input scale. scan over the following space: 3 Parameter scan In this section we presentof our Λ. results for We a retain generic a scan full of 20 parameters and parameter different version values of the MSSM and perform a broad random for stable point at been carried out formore other generic couplings variation and of soft-massesfocusing fundamental in parameters on the model-dependent at MSSM correlations the (such [ high as energy in scale grand Λ unified rather theories) among than them. increase in the infrared, once canin expect a the significant numerical positive contribution analysis to the Σ. correlation We confirm between the gluinopoint mass solution in the MSSMunified in which scale, the are top-Yukawa coupling well known [ For JHEP08(2019)105 (3.3) (3.4) ], 16 , 15 006. For points with . 127 GeV [ 0 30 GeV to be consistent ], < > , h 0 1 17 ) e χ 112 ]: . m e, µ < m 14 = = 0 l 2 h . P lanck 100 GeV ( 105 GeV as the LSP and – 5 – 0 1 > > e χ R 1 l e e χ m , m L l e ] given by Ω m 18 -Physics and Higgs precision constraints. The green triangles are a subset of these B . Fine-tuning measure as a function of the gluino mass (top) and lighter stop mass with the bound on light MSSM neutralinoWe dark satisfy matter the [ 3PLANCK sigma collaboration upper-bound [ onunderabundant dark dark matter matter, relic wefrom assume density non-thermal there observed candiates, may by such be the as some the additional axion. contribution We require the lightest Higgs mass inWe the require range the 122 lightest neutralino Direct searches for the sleptonfirst and two generation chargino sleptons at and LEP lightest produce chargino the [ mass limits on the 1000 are subsequently passed through the following constraints: Points which have a vacuum in the electroweak broken phase are chosen which satisfy • • • • ≤ as well as containing DM relic density and direct detection constraints. ∆ Figure 1 (bottom) for three representative NP scales. Yellow squares contain LEP and Higgs mass constraints JHEP08(2019)105 with 55 (95% . ) and 8 SI γ s − σ ] effective = (3 X 10 can actually 26 × , exp 2 → SPheno/SARAH ) 25 γ 08 . s 1 B X ( < → at the scale Λ we allow BR u exp B ) 2 H ( − m µ GeV, there is less constraint 1 at the high-scale to enhance BR + 5 ] as part of the µ > t 24 [ y → S ] to constrain the points from direct B ), ( 2 19 h BR FlavorKit P lanck – 6 – Ω / 2 h -physics constraints, namely ], B 21 [ . As expected, one finds that when the “fundamental” 1 ˜ t M , since the contributions from the electroweakino masses to the RG ). The measured values we use are ] and the upper bound 2 − µ 22 [ . In order to enhance the contribution from + u µ 4 2 H of the quoted value. − 100 TeV which motivates further searches for physics beyond the MSSM we show the dependence on the fine-tuning measure on the gluino mass m → σ 10 ]. These are calculated using ∼ 1 S HiggsBounds-4.3.1 × 23 10 in this case. Remarkably, this is true for gluinos and stops with multi-TeV B ( ∼ 26) . within 3 We also checkBR important 0 CL) [ package. Where an upper and lower bound are shown, we constrain our points to detection experiments, where we rescalethe the observed spin-independent relic cross-section density by (Ω We check the bounds fromusing Higgs searches at LEP, Tevatron and LHC implemented We use the recent data from XENON1T [ min and lighter stop mass We conclude this section by stressing that perfectly natural theories are possible even The constraints on relic density and direct-detection of DM can be satisfied relatively In figure • • • ˜ g 4 MSSM quasi infraredIn fixed-point the and following, we fine-tuning choose athe large running top of Yukawa coupling, in the early universe. for multi-TeV spectra ofin sparticles, some which cases not arecut-off only quite at beyond satisfy Λ the the reach currentwith of a LHC the natural bounds LHC. electroweak but scale. This is especially true for the low running of the up-type Higgsthat mass the soft-breaking dark term is matterthe mild. abundance, cosmological We besides would evolution the also of like microscopicwith to the over-abundant properties, dark stress universe. depend matter (ie. crucially Inbe a on mostly particular, consistent bino-like with LSP) the shown observation region in with figure of depopulation parameter mechanisms space shown in [ in the parameters from renormalizationas group ∆ evolution even allowsmasses, for well fine-tuning beyond as the low reach of current experiments. easily, shown in figure gluino masses can be easily satisfied in many cases. M parameters are entered aton the a low heavier scale, spectrum chosen whilst at the Λ electroweak = scale 10 still remains natural. Little variation We do not impose constraintslimits from are gluino/squark model-dependent searches and fromBesides, would ATLAS require there and a are CMS dedicated many asthese recasting cases the LHC of in search the constraints. which collider the However, limits. as spectrum we may will be see the compressed LHC to constraints easily on squark avoid and JHEP08(2019)105 tend (4.1) ]. We 3 . Most ¯ 2 u 28 3 m . The solid and 3 ] results. Similar P lanck 2 Q 20 m = Ω c year [ ] and the MSSM [ × 2 GeV 27 2 where Ω c , run to more negative values for Ω / ]. u 2 H 3000 GeV 3000 GeV 3000 GeV 0 (10000) 29 GeV m < < < < < 19 2 2 10 d u , 3 , 1 2 H 2 H 2 i ,M 1 16 M – 7 – 10 ] and the recent 1 tonne , 10 19 < m < M < m < m effect discussed in [ 10 2 0 0 2 runs over the scalar mass squared values, excluding the . One can avoid this with negative scalar mass-squared ∈ i as a function of the LSP mass corresponding to the red squares GeV Λ . The dotted line corresponds to the PLANCK measurement of 2 1 GeV. SUSY DM 3000 GeV 10 M ]. Right: WIMP-nucleon spin-independent cross-section as a function where − 2 i 18 “gluino sucks” (3000) at m − u and 10 H 2 006 [ . as they largely contribute a positive value toward the infrared, leading 5 0 m u 2 H m 112 . = 0 , increasing the fine-tuning. . Left: Relic density Ω 2 3 h M With this as our motivation, we more precisely scan over the following modified space: This is analogous to the 2 P lanck present an example of thenotably, RGE the evolution gluino of mass theseinfrared, parameter soft whilst mass significantly simultaneously parameters raises enhancing inpositive the figure the values. fixed-point running value The of oflarge fixed-point the Σ behavior stop in requires mass the parameters into 3rd generation squark soft-masses.to The de-stabilize 3rd generation squarkto masses a large value of parameters at the inputhave scale. been previously Negative studied stop in some mass-squared gauge parameters messenger at models the [ GUT scale of the LSP mass for theafter points freeze-out, shown in we the rescale leftlines panel. the correspond Since cross-section to we by allow theplots the the XENON1T LSP exist to factor for 2017 be Λ Ω [ underabundant = 10 it to be dominant over Figure 2 in the rightmost panelsΩ of figure JHEP08(2019)105 . 2 = u (4.2) 2 H GeV 5 , m 100 TeV. 10 = 10. The × ∼ 5 β − , 5 = 100 GeV 10 3 − , 4 M . In particular, low- µ 10 2 × 2 5 GeV within all constraints, and GeV − 100 GeV and tan 2 10 u − = 2 H u = m 2 H 1 t 3000 GeV (1000) 3000 GeV 50 A , m , < < < . 2 τ 3 ) = 3] ¯ d 3 , , GeV µ ¯ u β < ,A 3 , ( ¯ b e 5 [1 3 , 3 2 Q 10 ∈ 2 L ,A tan = 29 for Λ = 10 = 2500 GeV t – 8 – − t sign y 3 = M < m < m < A < QF P min 3 GeV. ¯ 2 u 2 0 1 19 m 10 = , GeV 3 2 16 (100)) can even be maintained in models where Λ is as high 2 Q 3000 GeV O m − . . Bottom Row: (1000) 2 − confirm our expectation with significant reduction in fine-tuning 4 GeV 6 GeV with 10 16 − , 5 10 . Evolution of the parameter Σ to the infrared fixed-point and soft mass parameters × GeV. In particular we find ∆ 5 − 19 , 5 10 The major conclusionsiderations we within would supersymmetric like theories toindicating in towards draw physics light in beyond of this theTo current MSSM demonstrate paper experimental that this is point enters data we at that may haveΛ scales treated be the as without the MSSM low any naturalness as as a con- an Λ priori effective assumption theory below on the soft-breaking scale parameters at Λ. The general scan of sensitivity towards Λ (∆ as 10 with similar results for Λ = 10 5 Conclusions The plots inobserved figure in models with quasi-fixed point running of Figure 3 input at Λ =three separate 10 curves are− shown for different initial values of: Top Row: JHEP08(2019)105 D (100) for ∼ O Phys. Rev. , (100)) fine tuning. Our O < (100) can be achieved even for heavier < O ]. TeV with the ATLAS detector GeV, even for a sparticle spectrum lying in 5 10 = 13 SPIRE – 9 – s IN ' √ ][ ), which permits any use, distribution and reproduction in (10) for Λ with higher NP scales and in a narrower scan range supporting the . Small fine tuning of ∆ Search for squarks and gluinos in events with an isolated lepton, jets 1 ∼ O 1 arXiv:1708.08232 CC-BY 4.0 [ This article is distributed under the terms of the Creative Commons 1 TeV. > collaboration, GeV down to ∆ . Same as in figure (2017) 112010 16 ATLAS and missing transverse momentum96 at 10 ' [1] References We would like toAustralian thank Research Lei Council. Wu for usefulOpen discussions. Access. The workAttribution was License ( supported byany the medium, provided the original author(s) and source are credited. behaviour of running parametersresults may call also for have further reduced exploration (∆ of non-standard theories beyondAcknowledgments the MSSM. 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