Precision Corrections to Fine Tuning in SUSY

JHEP06(2017)103 Springer , June 6, 2017 May 22, 2017 June 20, 2017 : : : January 16, 2017 : Revised Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP06(2017)103 [email protected] , . 3 1611.05873 The Authors. c Supersymmetry Phenomenology

Requiring that the contributions of supersymmetric particles to the Higgs mass , [email protected] 136 Frelinghuysen Rd, Piscataway, NJ 08854,E-mail: U.S.A. [email protected] NHETC, Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Open Access Article funded by SCOAP Keywords: ArXiv ePrint: sensitive to the massescan of be the decoupled 1st in and Effectivebounds SUSY 2nd scenarios. can generation allow We squarks, find for which that,the heavier limits for natural gluinos a how fixed far and region level these stops of ofhigh than tuning, messenger supersymmetry our previously scales is particularly considered. under disfavored. Despite pressure this, from the LHC constraints, with the higgsino, stop, andduce gluino. a We number revisit of thetreatment improvements, of details including UV of RGE vs. the resummation, IRaccurately tuning masses, two-loop and connects calculation effects, threshold the and a corrections. intro- tuning proper ThisLHC-accessible measure improved energies. with calculation After more the these physical refinements, the masses tuning of bound the on superpartners the stop at is now also Abstract: are not highly tuned places upper limits on the masses of superpartners — in particular Matthew R. Buckley, Angelo Monteux and David Shih Precision corrections to fine tuning in SUSY JHEP06(2017)103 5 ]. 2 (1.1) ]). Given the 1 ], reformulated 2 6

2 2 h 15 12 ]. We will apply a number of precision m M 4 , ], in order to better take into account the 10 3 5 log log [ 8 16 ∂ ∂ 5

2 Z – 1 – m = 2 M u ∆ 2 H 1 m instead of 2 12 (125 GeV) 17 ≈ 2 h m For the tuning calculation, we will use the Barbieri-Giudice measure [ In this paper, we will revisit the naturalness bounds on the gluino and stop masses, 3.1 Natural regions3.2 in stop/gluino mass Absolute plane upper3.3 limits on gluinos Comparison and with stops LL approximation 2.2 Transfer matrix2.3 RGEs: stop and Higgs gluino potential2.4 masses threshold corrections Gluino and stop pole mass corrections 2.1 Transfer matrix RGEs: in terms of radiative corrections to the Higgs quartic: masses. The essential point here isparameters, that defined the at tuning the measure messenger is scaleStandard calculated Λ with Model where respect superpartners, SUSY to breaking UV while isin the communicated LHC the to the is IR sensitive at tothreshold corrections, the the and physical weak together these masses, scale. can defined have sizable The effects on two the are naturalness bounds. related through the RGEs and through finite less than some fixed amount, then we can derive upper limits on thewhich superpartner (together masses [ with theboth Higgsino fine-tuning mass) and are collidercorrections among phenomenology to [ the the standard most SUSYboth important tuning calculation, a parameters and quantitative for we and will show a that qualitative they difference can make to the tuning limits on the superpartner the-Standard-Model physics at thethe TeV most scale, promising with candidates supersymmetry (forcontinued a (SUSY) review non-observation being and of among original references, superpartners,the see weak the e.g. [ scale supersymmetric cannot cancellationof be the protecting perfect, weak with scale. heavier If SUSY we particles define implying a measure more of fine-tuning fine-tuning ∆, and require that this tuning be 1 Introduction and summary The naturalness of the weak scale has long been one of the best motivations for beyond- 4 Conclusions 3 Putting it all together Contents 1 Introduction and summary 2 Precision corrections to fine-tuning JHEP06(2017)103 . 2 i M 10), (1.4) (1.5) (1.6) (1.7) (1.2) (1.3) ∆ ≤ } 2 i M ], we will { 7 , 6 ). When multiple sources 3 Λ Q 2 Q 2 m is the IR scale, conventionally ) log Λ Q 3 4 Q 2 U , or | 2 3 log m

] for notable exceptions). In this ap- H ]. This correction alone relaxes their gluino | 2 M 2 2 H + | λ 10 11 2 , . | – µ 3 2 3 | + 8 2 Q ∂M µ ∂m ]). For deﬁniteness, we are assuming here 2 M = | 2 | m 2 h ( 2 t 11 4 H 2 H 2 , t M | y m π y 2 2 3 2 H 4 10

2 2 relative to what will be quoted in this work. g 300 GeV – 2 – δm 3 π m √ ≈ 8 . -terms (i.e. we are assuming they are small). We will also 2 A ∼ − µ ) = M 2 H ∼ − ) becomes H ∆ ( 2 H δm V 1.1 δm , it has been conventional in much of the literature to work in H 2 and ( 2, which is numerically quite significant. m √ 2 H m 2 − as separate UV parameters for the purposes of the tuning computation. In some UV = 3 2 h 2 U 2 m m 1 is a UV mass-squared parameter (e.g. and 2 3 M 2 Q Gluinos: Higgsinos: Stops: m The higgsino formula is fairly accurate as is. For better-than-10% tuning (∆ For the calculation of Motivated by the increasingly SM-like Higgs coupling measurements [ In this paper, we will be neglecting the Notice that our formula corrects a factor of 2 mistake in [ • • • 1 2 treat completions, such as gauge mediation,the they tuning may bounds in on fact stops be correlated by or up even totuning equal. a bounds factor This by would of a strengthen factor of we need Meanwhile, the stop andpaper is gluino to formulas include are ain number this rather of paper: imprecise, higher order and corrections. the We purpose identify five of such corrections this Here Λ is thetaken to messenger be scale 1 TeV of in(and many SUSY throughout works this breaking, (see work) e.g. and that [ the stops and gluinos contribute as in the MSSM. stop and gluino soft masses arises at tree level, one-loop and two-loops respectively: the leading-log (LL) approximation (seeproximation, however [ the quadratic sensitivity of the Higgs mass-squared parameter to the higgsino, work in the decoupling limit of the Higgs sector: In that case, of tuning are present,Note we that take there the maximum isit tuning an is as inherent not our ambiguity measure, in reparametrizationarbitrariness ∆ the invariant. to = definition it, But max once of it while the is the measure. decided choice upon, In of one particular, tuning should attempt measure to has compute some it precisely. where JHEP06(2017)103 , 3 2 Q ] for (1.8) m 13 2 heavier and give √ ] respectively for . ∼ 15 Λ Q log . 2 | ] and [ 3 dependence, effectively re- 14 gluino M both decrease considerably in | Q t 2 t M 4 y y π 2 3 4 or g and − s stop 2 α m Λ Q log – 3 – 2 | 3 , but these run considerably with the RG. In partic- s M in the MSSM, | α that would remove the 2 t defined at the weak scale, one can considerably overes- β 4 -parity conservation and flavor-degenerate sfermions), ]) for items 1–3, the two-loop effective potential [ y u π s 2 3 R 4 and 2 H α g 12 t m y ], we have reinterpreted the latest LHC searches post-ICHEP and ∼ − 16 t However, as will be shown here, this does not come without a H 2 y 3 δm ]. is an IR renormalization group scale. A proper treatment includes ]. In this case, alignment between 1st and 2nd generation squarks would be needed 4 , 17 and the stop and gluino masses. 3 Q [ , we will address these issues in turn, using the fully integrated RGEs 2 H R evaluated in the UV and in the IR. In reality, these masses evolve quite a ˜ t m 2 3 , L M b ˜ , , L 3 ˜ t 2 U ]. We will expand on these results, in particular adding in important finite threshold placing it with a physical scale, for example leading to a difference between the running IR masses and the pole masses. bit with the renormalization group.that In the fact, IR as massesrelax we are the will tuning see considerably bounds. below, larger it than is the often the UV case masses. Thisthreshold can corrections further to ular, for moderate to large tan the UV. If one uses timate the tuning. This effect tends to be more important at higher messengerm scales. to-leading-log (NLL) correctionscales: can be numerically important at lower messenger 10 Finally, the gluino and stop masses are subject to their own threshold corrections, The energy Even more importantly, in the LL approximation, there is no difference between The LL formulas refer to While it is common in the literature to use the LL approximation for gluinos, the next- In a companion paper [ In section , We also note that in order to reduce squark production rates, one could also decouple only the squarks 8 5. 4. 3. 2. 1. 3 cost. Through the threshold corrections and theof RG the first running, generation heavy [ to 1st/2nd avoid large generation contributions tostop kaon mass mixing, would but be on reducedthe the by same half. other tuning hand Up bounds the and on RGE down the contributions squarks lowering stop could the then mass IR be discussed a here. factor of and then proceeding toLHC. more Because complicated of scenariosto the which relax large can LHC valence better squark limitsSUSY”) hide cross is and sections, SUSY to only we decouple at found keep thenamely the that first light a and the good second squarks generation strategy which of are squarks most (“Effective important for the fine-tuning, and used them toparameter understand space, as the determined current by experimental towork. the constraints improved We fine-tuning on considered calculation the a described“vanilla” natural in set this case SUSY of simplified (the models MSSM for with natural SUSY, starting from the most (called the “transfer matrix” initem [ 4, and theitem one- 5. and two-loop Some pole ofin [ mass these corrections, formulas from having tocorrections [ to do with RG effects, were previously studied JHEP06(2017)103 6 (1.9) ], we find that a to 125 GeV. This 16 h m 2–5 TeV range. ∼ 2 H m ], the 125 GeV Higgs in the MSSM 26 – + ∆ ]). Our implicit assumption here is that 10) in the gluino-stop mass plane, as a 2 21 | 10 µ ≤ | is the additional contribution that depends + 2 H – 4 – u 10 TeV stops; at best the resulting fine-tuning is 2 H m 10 to be meaningful, there should be an additional m & ∼ = ] for a review and original references) or non-decoupling 2 H 2) TeV for Λ = 100 TeV. This is close to the current LHC m 18 . 1 , -terms or 5 . A (1 , we conclude with a brief summary of our results, together with < 4 ) ˜ t , m ˜ g , we combine all of the precision corrections to tuning and present the ]. Otherwise, as is well known [ ]. In the light of this, we conclude that fully-natural supersymmetry requires m 3 16 20 is as in the MSSM, and ∆ , u 19 2 H m for examples of this behavior). For very low messenger scales, Λ = 20 TeV, we find Finally, in section We should note here that for ∆ In section 7 -terms [ a discussion of someways well-motivated to directions achieve for anas future effective loopholes model-building. SUSY to spectrum the Thesegluinos) conventional with include tuning with an bounds reduced ultra-low —renormalization contributions messenger models of scale, of beyond the weak as the the scale. MSSM well higgsino, (e.g. stop Dirac and/or gluino masses to the at most weakly onthe the tuning stop calculation and shouldconsidered gluino in be this masses. paper revisited. would need Ifmore to Nevertheless, complete these be or we taken assumptions otherwise. into expect account dotemplate So in the not for any at supersymmetric effects future the hold, model, works. very we then least, have the treatment here should be taken as a respect to the stops and gluinos. In other words, we assume that where could be e.g. the NMSSMD (see [ requires either multi-TeV a few percent (forthis a recent additional discussion, sector see is e.g. such [ that it does not modify the calculation of the tuning with messengers at or belowheavier 100 than TeV, the likely stops. with (Onconstrained, 1st the with and other much 2nd hand, higher generation percent-level-tuned messenger SUSY squarks scales is significantly still considerably allowed.) less contribution coming from a sector beyond the MSSM in order to raise that fully-natural gluinos (stops) shouldthe be 1st/2nd below generation 2.2 squark (1.5) masses.shrinks, TeV, For with with larger some messenger ( dependence scales, on thelimits fully-natural even region in the bestthe higgsino possible [ scenarios, e.g. with decoupled squarks and RPV decays of fully-natural regions (taken herefunction of to the mean mass ∆ of of the the 1st/2nd effects generation discussed,large squarks namely gluino and contribution the the to messenger thresholdshaped the scale. corrections like RGE wedges Because to for with gluinos heavy theand gluinos from stop favoring mass, stops heavy squarks we and and find the vice that versa natural (see regions figures are That is to say, astuning the allows 1st/2nd for generation heaviergluino squark gluinos and masses but stop are tuning, increased, requires“sweet and a lighter spot” given taking stops. for amount into 1st/2nd of account Given generation the squark this LHC masses tension constraints is [ between in the squarks improve the tuning on gluinos, but worsen the tuning for stops by a larger degree. JHEP06(2017)103 ]. ). Q )– (Λ) 10 2 1.6 – (2.3) (2.4) 1.5 8 m -terms )–( A ; Λ), etc. Q 1.5 and let the ( u = 1 TeV and 3 2 Q 2 H M .... Q (Λ) (2.2) m m ) have a number 00 + A 2 M | 1.6 )–( (Λ) (Λ) 0 3 1.5 M M , being suppressed either | u 3 2 H ; Λ) M u Q m 3 2 H ( 00 M m | (Λ) (2.1) A 0 M u 0 2 2 H M m M m m m ∆ A A | 00 ; Λ) (Λ) + and Λ dependence of the transfer matrix . For each dimension-one soft parameter 2 3 2 , the matrix takes the form Q ,M 2 U 0 ( 2 Q X × 0 m m M m 2 h M M u 3 A – 5 – 2 U 2 H 0 m m < m (note also that we have assumed small M | (Λ) + X A 2 2 , 0 u 1 2 H (Λ) m α ) = 2 m Q m (Λ) + ( | ; Λ) , as a function of the messenger scale Λ, for 3 Q 2 Q 1 M ( 2 m 0 2 m m u 3 Λ reproduces the leading order behavior shown in ( 2 Q 2 H A m m 2 0 Q/ A m X ), one can compute the upper bound on a UV mass parameter ) = 2.3 ) = Q = log Q ( ( u t 2 , the dominant terms in the transfer matrix relation are given by: 2 H u m m 2 H ) and ( ]) instead of their LL approximation, and then translating the tuning bounds into m . In this subsection and the next, we will remedy the first two deficiencies (items 1 1.3 27 [ Q A plot of the transfer matrix coefficients, together with a comparison to the LL ap- For As is well-known, integrating the MSSM RGEs between a UV scale Λ and an IR scale ). Indeed, one can check analytically that expanding the coefficients , the transfer matrix takes the form 1.6 In the following subsections, we willthis apply into a number a of bound corrections and on successively the improve physical masses. proximation, is shown in figure in powers of Using ( for a given fine-tuning level ∆: The masses of theby other small superparticles Yukawa couplings contributein or less our by analysis). to This( is the transfer matrix upgraded version of the LL formulas ( In what follows, we willcoefficients generally to suppress avoid the cluttering the equations. while for each dimension-two soft parameter results in a (bi)linear mapdefined — at a these sort scales. of “transfer Letdimension-two matrix” the soft — dimension-one that parameters soft relates parameters be the be denotedM soft denoted by parameters by group and 2 in the listSARAH above) by employing the fully-integratedones two-loop on RGEs running (as IR masses deriveding (item from 3). the tuning The bounds importance in of integrating terms the of RGEs the and IR rephras- parameters was previously emphasized in [ 2.1 Transfer matrix RGEs: As discussed in the Introduction, theof leading-log practical tuning formulas drawbacks. ( Theyparameters neglect at higher-order an terms, indeterminate they scale, refer to and couplings they and refer soft to an arbitrary IR renormalization 2 Precision corrections to fine-tuning JHEP06(2017)103 s α ; it 2 2 GeV to Λ, , with M and 16 R Q ˜ t t y �� throughout and β �� L ˜ t ) with are similar, with u 1.6 2 H 3 u 2 U 2 H �� m )–( m m A �� 1.5 10, this gives an upper bound �� . Shaded bands show the intrinsic < �� 2 2 2 2 as function of the messenger scale Λ, M 2 M u M u , while the short-dashed and dotted lines ] contributions to 2 2 H 3 2 H M m 2 U 3 and m A �� U m - the transfer matrix coefficient for 2 3 �� [ �� 1 ], a natural wino will typically participate in M Λ and and 16 3 3 – 6 – 2 Q Q m � �� that the 1 � �� 3 u 3 = 20. The solid and long-dashed lines show coefficients for the terms 2 Q 2 H M u m m 3 2 H when showing bounds on the stop. β A M m A L � ˜ t �� itself does not run more than 20% between the messenger scale Λ and the - �� -

- �� M , so the bounds on UV and IR parameters are similar). While natural SUSY . Relevant transfer matrix coefficients for Q -term assumption, this will correspond to similar tuning bounds on = 1 TeV and tan A = 20. (Unless otherwise stated, this will be our benchmark value of tan We also show with a dotted line in figure It can be seen from figure β Q 2.2 Transfer matrix RGEs:Next stop we and will gluino address∆ masses is the calculated difference with betweenespecially respect the for to) UV the and soft the collider IR phenomenology). mass soft (which This mass the (which is BG is item more measure #3 physically on relevant, the list presented in the (note that IR scale spectra usually focus on stops,collider gluinos constraints and on higgsinos, gluinos it should andthe be squarks cascade noted [ that, decays given of present superpartners. small the bound on the latterwill slightly be weaker referring than to on the former. Unless explicitlyis specified, actually we not muchon smaller the than wino the mass others. ranging For from ∆ 1.5 TeV to 500 GeV for Λ between 10 TeV and 10 tan the paper.) For thedemonstrating latter, one of the the RG inherent scale ambiguities of in the the running LLdifferences couplings approximation amounting formulas. is to only varied about from 10% (20%) at low (high) messenger scales. Given our proportional to the squaredare squark masses the coefficient foruncertainties the in terms the porportionalevaluated LL at to approximations, the weak by scale taking (lower boundary) the or LL at the formulas messenger ( scale (upper boundary). Figure 1 for JHEP06(2017)103 So vs. , so u 4 3 (2.7) (2.8) (2.5) (2.6) 2 H 3 M m 3 2 Q M m A , 2 ... , ]. 3 1 ˜ 2 q 2 Q + m m 28 2 , | A 15 , 3 3 (Λ) is asymptotically free 3 2 Q 2 Q 3 m m g M | A 3 , 3 3 2 M 3 , 3 1 3 M M 2 Q ) contribute at one-loop to 2 2 D 2 Q , R M m A 1 ˜ t . Since m m 2 U A 3 . A m M and + L (Λ) = 6= 2 b ˜ 3 , 2 3 , (Λ) + 0 1 , 2 1 2 U 2 Q L M , ˜ t , we show ratios of the IR to UV stop and 1 ) 2 D M m m ˜ 2 q 3 Q A m m (Λ) ( grows in magnitude in the IR. So the tuning 2 3 2 3 + 2 = g g , 3 3 0 for 1 – 7 – 2 2 ˜ , 2 q 2 Q , 1 3 1 M ≈ m m ≈ 2 Q 2 Q 2 Q ) 0 A 0, that is, the first and second generations reduce 3 m m m Q M M A ( < 3 ≡ A ≡ 2 2 M , , 3 2 1 (Λ) + 1 , ˜ , and can become important if they are much heavier. 3 2 q 2 Q ˜ 2 q 3 1 4 3 ˜ q 2 Q 2 5 m m (setting the masses of 2 Q with g m m m = 1 TeV. In figure 3 A m ) U A Q Q 3 3 (Λ) ( m ], but these are absent if the 1st/2nd generation squark masses are decoupled 2 3 2 2 Q 2 Q 3 0, i.e. the gluino always pulls up the squark masses. This is a g g 4 m m , . Here and below, we have taken the first and second generation 3 and are the transfer matrix coefficients > A ≈ 3 3 2 U 2 3 3 3 Q M 3 m M M ) = m without directly affecting tuning. However, as with the other squarks, the right-handed 3 2 Q A Q 3 -terms [ M m ( D 3 D A m 2 Q m to be at the same scale as other third generation squarks to minimize this effect. Having all 3rd R b Shown in figure For stops, the most important terms are the stop mass-squared itself and the gluino For the gluinos the translation from UV to IR is straightforward. As is well-known, at ˜ Technically only There is also a potentially dangerous 1-loop contribution to the stop and Higgs soft mass-squareds from 5 4 the hypercharge in degenerate SU(5) multiplets, which we assume throughout thisone could work. raise sbottom enters the stoptake RGEs and pullsgeneration down squarks the at IR one stop scale mass, should also worsening be the simpler tuning. from the In model-building this prospective. work we scales, the stop squarks can even become tachyonic due to thisthe effect messenger [ scale Λ, for Importantly, significant effect in the contextstops. of On natural the SUSY, otherthe as hand, IR it stop can mass,third allow generation worsening heavier-than-expected squarks. the For very fine-tuning heavy for 1st/2nd generation large squarks or hierarchies too-high between messenger them and the so that the transfer matrix coefficient is and similarly for squark masses to be the same, mass. The 1st/2ndin generation the squark RGEs, squared proportional masses to we contribute have irreducibly at two-loop In other words, in the MSSM, itbound is on always the the IR case gluinogluino that mass mass. will always be relaxed as compared to the bound on the UV the messenger scale is generally negligible. one-loop, the running gaugino masses simply scale with the gauge-couplings squared: Introduction. We will focus on the stop and gluino masses; for higgsinos, the running from JHEP06(2017)103 ). 2.4 (2.9) �� ) �� i ˜ 2 t , m . The function (red), as function of ˜ 2 g i -dependence in the ˜ t 3 3 3 i is negative, unlike the Q 2 Q 2 Q �� ˜ t 2 Q , m m m 2 2 t , X A 3 m �� 1 i ˜ 2 q 2 Q m ( m m P 3 A 1,2 ¯ ∼ 2 q 2 Q FS m m ¯ F = A �� F - ] i ˜ ∗ t �� X R ˜ t ) one has to also specify the gluino i ˜ t L ] 2.6 [ (solid blue) and 3 �� )) Re 2 t M 2 �� ] and describes the one-loop stop/top cor- [ �� = 20. Note that m − ( 13 ]. Up to two-loops, the most important terms Λ β ) h i – 8 – ˜ 2 t 2 13 − , m � ) ˜ 2 g 2 ˜ 2 t �� 3 3 ) , m m M M 2 t u ( A 2 H h m ( threshold corrections that remove the m u ) + � + was defined in [ 1 = 1 TeV and tan 2 H are ˜ 2 t FFS 2 ), for the IR stop mass ( �� | 3 F m 3 3 m 3 2 Q V µ M 3 ( | 3 2 Q M M 2 ( 3 h M m − 2 g u A A ) v 6( 8 2 are the stop mixing matrices, � ⊃ ⊃ ⊃ . For the stops, the gluino lifts the IR mass, while the 1st/2nd generation � �� x/Q 3 (0) (1) (2) g �� L, R V V V

�� 2 2

�� log( ) �� π 2 = π 2 16 4 . Relevant transfer matrix coefficients for x (16 X ) = 2 x ( Here h rections. The explicit dependence on the gluino mass first enters in at two-loops; these a two-loop effective potential calculationin [ the effective potential considering UV parameters. 2.3 Higgs potential thresholdHere we corrections will consider the LL tuning formulas (item #4 on the list in the Introduction). These can be obtained from section), here we set gluino andWe stop see UV masses that to for theirthe ∆ the running = gluino, 10 of the upper limits IRsquarks given mass by pull ( is it considerably down. higherallow than fully-natural In stops the particular, with UV it high mass can messenger due be scales, to seen an that effect highly which decoupled would be squarks lost do if not only other coefficients. gluino masses vs. Λ.UV and While IR for (given theand by gluino other there squark is masses. a simple For definiteness one-to-one (and mapping in between anticipation of our results in the next Figure 2 the messenger scale Λ, for JHEP06(2017)103 ! ) 3 (2.11) (2.10) Q m x − -terms) but ]. Note that A 16 (1 = 0, and read 13 2 d 10 v = )+2Li u 3 v 14 Q at one-loop, and other m t 10 defined in [ q y ]). The most relevant terms ¯ FS log ¯ 13 F − F 12 (1 dependence through the RG (e.g. 2 2 3 10 3 Q = 1 TeV) vs. UV running mass for the Q m and m M Q x ] = , and we have introduced the following + FFS m Q 2 for a benchmark point (a similar plot can 10 ) F GeV 3 , we expand around 4 [ 10 u Q )+ 2 H 3 m Λ q Q , q – 9 – m m 2 2 3 q log 8 m M − 10 log ≡ (1 − ) and the stop (red lines), for 1st and 2nd generation squarks m − 3 3 2 x (1 ) M M 3 3 6 A 2 Q is shown in figure Q explicitly at two-loop. Here all of the parameters are running ... 2 m 3 m 10 3 2 t x M y M ) + . Using all the terms, it can be verified analytically that the result 2 3 2 π | 3 (log U u 4 v 16

up to two-loop order (equation (5.1) of [ |

→ 2 3

10 − Λ 4 3 Q M u π Q 2

includes other corrections not proportional to 0.0 2.0 1.5 1.0 0.5 2 H

g 32 m Q m ) ( )/ ( 2 t m + ( y ... = . Ratio of the 10% tuning bound on the IR (at eff ) u 2 H The RG stability of ∆ To derive the threshold correction to m ( notation: be made for stops). We see that without including the threshold corrections, the gluino where the corrections not involving couplings and soft masses evaluated at the scale is independent of can be extracted and summarized in a relatively compact formula: to the one-loop term).terms Additional that terms involve will the acquire stopwe and find left-handed sbottom these masses-squared to and be the numerically subleading. off the coefficient of either degenerate with the stop (solid red) or set atall 5 TeV come (dashed from red) the or fermion-fermion-scalarfor 10 TeV terms (dotted each red). sparticle weexample, only there are include two-loop terms its involving leading the stop, contribution but to they are the suppressed effective with respect potential (for Figure 3 gluino (blue line - this is simply JHEP06(2017)103 (2.12) (2.14) ), the scale- (Λ) = 1 TeV, Q 3 ( U u i 5000 2 H m m q m log − (Λ) = 1 3

− i 1 Q . Finally, the solid curve is # − m ˜ q m q (2) x ) (2.13) 4000 V 3 − 3 3 1 M

q M and M ∆ log (1) D 2 3 log ) ) and the threshold corrections, it is very V i − Q − ˜ m g ( g x u GeV (5 H 2 @ 2 3000 − 3 2 3 m π 3 g Q M (1 3 16 M – 10 – ∆ − = i ˜ m g g − x 1 − 3 " = 20. The dashed curve is derived from ]. The one- and two-loop stop thresholds are generally somewhere between the stop and gluino masses in this 3 2 3 β M 14 2000 M Q M ). As can be seen, this is subject to large RG-scale uncertainty. ∆ =1 12 i . = X 2.9 Q 2 2 3 π pole g 3 32 M − = ], even if there is a large splitting between the stop and the ﬁrst two ˜ q 1000 0 q 20 40 60 20 15 , 3 - - -

14 M D M ∆ . The gluino tuning vs RG scale for a benchmark point ]: 14 The finite one-loop corrections to the gluino mass are given by gluino and squark (Λ) = 3 TeV, Λ = 100 TeV, tan 3 and loops [ where Finally, we will consider#5 the on difference the betweenthe list the classic in pole one-loop the results massnegligible Introduction) of and for [ [ running the massgenerations gluinos of (item and squarks, stops. but we include For them gluinos, for we completeness. rely on tuning estimate can varythreshold by corrections, it nearly becomes a stablecorrections to factor better are of than minimized 3 10%. for when Weexample; also this varying see makes that the sense the RG intuitively. threshold scale.2.4 With the Gluino and stop pole mass corrections dependent tree-level term inThe ( dotted curve showsthe the sum threshold of corrections thestable from contributions across to a the wide tuning range from of Figure 4 M JHEP06(2017)103 �� ) = Q ( (2.16) (2.15) 3 = 1 TeV M �� Q ] , and we are 3 2 U , for two different �� m [ | �� 1 or − squark �� 3 , for two different choices m � m = 1 TeV), with 2 Q x | m m Q , left panel, for � in such a way as to cancel q 2 squark , 1 5 ˜ q m Q 1 TeV 3 TeV ) = = 2) log 2 3 3 2 log M M m − − 1 � − m m (Λ). x 3 x + (∆ ) in figure

��

˜ g M � + �

� � � � �

) ( / ) �� 2 m at one-loop order, resulting in a pole mass 2.2 ) log x m stands for either 3 m indexes the 12 squarks (right-handed and left x 2 M + ( i ]: − – 11 – = 20 TeV. This effect would be further amplified m 5 TeV, as a function m + (∆ for fixed �� . x 2 15 , Q m + (2 ) vs. the 1st/2nd generation squark mass (recall our 14 ), and ) = 1 squark 3 + = Q Q m ( ( 2 3 3 3 2 2.11 �� U pole π M ) ] m 2 3 6 /M . The corrections depend on 2 g 1 TeV Q m = �� 20% at ( pole ) = 3 3 = [ Q Q ˜ g m ∼ M ( ) �� 3 2 Q �� 2 TeV m = m � 3 Q (∆ m 5 TeV. We see that the pole mass is generally even higher than the IR . � are defined as in ( i m ) = 1 -dependence of the running mass , q . Left: ratio of gluino pole mass to running IR mass (at Q i Q ( 3 m which is largely independent of = 1 TeV), with x M

Similarly, the stop pole mass (here We show the ratio

��

Q �

� � �

� pole ) ( / 3 �� 5 TeV, as a function of the common 1st/2nd generation squark mass . where the first term isfrom the gluinos, IR running squared mass, the second is the one-loop correction as the stop itself would contribute significantly to theneglecting Higgs stop fine-tuning. mixing) is given by [ and running mass. Atinterestingly, low at squark higher masses, squark masses, theas the high finite finite as threshold threshold an additional corrections correctionsif can are third be negligible, generation much squarks but larger, were also made heavy, but we are not interested in this scenario evaluated at the RGout scale the M assumption about the squark masses in section (at of the gluino mass. Here handed times 6 flavors). Again, all the couplings and masses here are running parameters Figure 5 1 choices of the 3rd generation squark mass. Right: ratio of the stop pole mass to running IR mass JHEP06(2017)103 , ˜ t m given = (2.17) 2 2 , M 1 ˜ q m dependence of Q 5 TeV and show . 2 , and the transfer , 1 ˜ q 2 H m m q ) = 1 ) we have omitted terms log Q , due to a larger coefficient ( ) for − 3 L ˜ t U 2.17 2 m 2.10 − 2 3 π U(1) splittings. The tuning bound ) = + × Q ( E 3 γ Q (1%) of the running mass, for any reason- − (Λ), using ( m O 3 π 2 U ) for the finite threshold corrections. m – 12 – log 4 to translate the running IR parameters to the UV 2.15 4 3 , we set g 2.2 2 4 , 5 (Λ) and 1 π ˜ 2 q 3 10.) We will explore the dependence on the 1st/2nd gen- ), from small SU(2) ) and ( 2 Q / 12 m and m ∆ − 2.3 , = 5 and 10 TeV. 2.12 p 2 2.1 = | 2 , 1 2 , ˜ q are nontrivial functions of both the gluino and stop masses. Heavy 1 ˜ (Λ) q 3 m ) 3 2 Q 3 M 2 Q m . | which are never more than 2 , m are the natural regions in the gluino/stop mass plane. As noted above, and the stop/gluino masses. Similarly, in ( 4 2 1 t = g y 6 (∆ 2 Q and ∆ M dependence of the ratio of stop pole mass vs. running IR mass, again with transfer matrix ( 2 3 u M H 2 ] m ), for 15 squark [ We have provided all the analytic results necessary to reproduce the plots shown in this We will mostly focus on ∆ = 10 as a benchmark value. (Our full calculation indicates On the right panel of figure m 2 1.3 , 1 = 1 TeV. There, we notice the dependence on the gluino mass, with heavier gluinos ˜ q in the 3.1 Natural regionsShown in in stop/gluino figure mass plane for definiteness, we will plotof the the smaller separate of tuning the measures. two stop This masses always according corresponds to to the maximum in ( matrix relations in sections parameters in order toaccessible take masses, the one derivative. should Finally,transfer to convert to matrix, get and the a using gluino bound ( and on stop the pole experimentally masses, again using the eration squark masses, taking asor benchmark decoupled values squarks, either degenerate squarks, section, as well as toshould explore the the reader so parameter desire. space Specifically, for one different should benchmark calculate values the of tuning measure ∆, ∆ etc. stops contribute threshold correctionswhile to gluinos the pull gluino up poleresult, the mass the stops (a natural primarily relatively region through minor becomes the wedge-shaped. effect), RGEs (a muchthat larger the limits effect). on As therescaling masses a all for the other masses values by of ∆ can be very approximately obtained by Having explored several important higher-orderfine-tuning effects which parameter impact ∆, the we calculationbounds can of on the now the combine them physicalwill gluino and see, and derive the more squark combined precise poleRather, natural ∆ naturalness masses region as is a not function a of simple ∆ rectangle and in Λ. the As gluino/stop we plane. (dashed) lifting the stoplarger IR squark mass masses, both the through magnitude the of RGEs the and threshold3 corrections the increases finite as corrections. expected. Putting For it all together proportional to able choice of proportional to the Q m As for the gluino,the the threshold running corrections mass included to here a cancel out high the degree of accuracy. We do not include stop self-corrections and the third term is the two-loop contribution from 1st/2nd generations set at a scale JHEP06(2017)103 GeV. Right: 7 UV stop mass GeV (bottom, right). The and negative 16 – 13 – GeV and 1% fine-tuning, that is, the lines and shaded 16 GeV (bottom, left) and 10 7 (with the upper and lower bounds delimited by blue and strip GeV, where we show 1% tuned regions instead): in green is the 100. 16 ≤ 10 region. Bottom: left: same as the other plots, but for Λ = 10 ≤ . Top: 10% fine tuned regions with Λ = 20 TeV (left) and Λ = 100 TeV (right), with masses is typically less than 5% higher. We take the messenger scales to be 20 TeV Although it does not impact our 10% natural region, it is interesting that the stop R ˜ t and stop regions, i.e. the region in which both gluinosnatural and region stops is are actually natural. a purple lines), with the lowersquared, boundary which have corresponding been to pulled large up by the gluino to be non-tachyonic in the IR. The slope shaded areas mark the regions10% where (except the for contribution Λ to =natural the 10 region Higgs for fine-tuning is the lessdashed gluino, than and while dotted in lines blue indicatesquark we how are see the taken the natural to stop regions 5 natural evolve and as parameter 10 TeV, the space. respectively. 1st/2nd Red The generation lines delimit the intersection of gluino same as the otherregions plots, correspond but to ∆ for Λ = 10 on (top, left), 100 TeV (top, right), 10 Figure 6 respect to the gluinofor mass 1st/2nd generation (green) squarks and degeneratelines) the with and stop the at third mass 10 generation TeVfully (blue, (solid (dotted natural delimited lines), lines). ∆ at by 5 The blue TeV wedge-shaped (dashed and intersection purple (delimited lines), by red lines) is the JHEP06(2017)103 ] at Δ GeV �� 16 16 �� ]; this would be �� 29 �� = = Λ Λ = 10 TeV, there is simply 2 GeV (now the dashed lines , 1 7 ˜ q GeV (bottom, left) and 10 10 m 7 ≥ 100 TeV and ≥ – 14 – �� = = Λ Λ -parity conserving MSSM, it can be seen that 1st and 2nd generation R , we show the level of fine-tuning in the gluino-stop mass plane, with varying 7 10 fully natural region for both gluinos and stops, as the dotted lines do not . Level of Higgs fine-tuning in the gluino-stop mass plane, for different values of the ≤ In figure It can also be seen that raising the 1st/2nd generation squark masses can expand the 1st and 2nd generation masses shown as solid, dashed and dotted lines. We provide these squarks at 2–5 TeV occupy athe “sweet LHC, spot”: but heavy light enoughraising enough to the not to squark be mass not does efficientlycorrections), contribute produced not but too significantly at considerably improve negatively the lowers to gluino the tuning the allowed (via stop stop the mass. tuning. threshold Further scales (longer running): forno example, for ∆ Λ intersect. For squarks atdo 5 TeV, not the intersect). same With happens thearound at 2 experimental TeV Λ for limits the on degenerate squarks presented in [ maximum natural gluino mass throughmaximum natural the stop 1-loop mass, threshold through correction,in the but certain 2-loop extensions it RGE of and reduces the threshold the interesting MSSM, corrections. to this (Perhaps 2-loop explore effect in could future be work.) alleviated [ This trade-off becomes worse at higher messenger dashed and dotted contours corresponding to squarks degenerate with the stops and 5 and 10of TeV. the band increasesRG with running. Λ, due to the increased dependence on the gluino mass from Figure 7 messenger scale Λ = 20(bottom, TeV right). (top, left), Different values 100 of TeVwith (top, fine-tuning shades are right), of color-coded blue 10 according corresponding to toand fine-tuning the yellows/reds levels legend for close on to sub-percent the 10%,and right, fine-tuning whites 2nd in (for the generation definiteness, few squarks percent the range, at color-coding 5 corresponds TeV). to Contours 1st for specific values are also provided, with solid, JHEP06(2017)103 16 10 5 20 10 = 100 = = = Δ Δ Δ Δ 14 10 , the solid, 6 , we also list 12 1 10 ] 10 GeV [ 10 Λ . In table 8 8 10 Stop tuning contours 6 10 4 0 10

3000 2000 1000 4000

GeV Mass Stop ] [ 16 – 15 – 10 5 20 10 = 100 = = = Δ Δ Δ Δ 14 10 12 , but across a wider range of messenger scales.) The result 10 6 ]. ] 16 10 GeV [ 10 Λ 8 10 Gluino tuning contours 6 10 TeV. The plots also demonstrate the danger of increasing the 1st/2nd generation 3 . Tuning contours of IR gluino (left) and stop mass (right) for ∆ = 5 (black), 10 (red), 20 4 10 0 10 ∼

2000 1000 4000 3000 5000

] [ lioMass Gluino GeV the 3rd generation, ordependence at on 5 the TeV messenger and scale 10 — TeVto the Λ respectively. tuning These bounds plots decrease sharply illustratesquark from masses the Λ — strong = the 10 TeV stopswhen go they tachyonic are very quickly not for tachyonic, larger the messenger tuning scales, bounds and become even increasingly restrictive. function of the messenger scale Λ,of for different the values wedge of ∆. regions inof (This amounts figure this to taking projection the tips issome shown reference in values for the the leftdifferent physical gluino and choices and of right stop Λ panel masses anddashed of corresponding the and figure to 1st/2nd dotted ∆ generation = lines squark 10, correspond for masses. to As 1st/2nd in generation figure squark masses degenerate with 3.2 Absolute upper limits on gluinosIt is and also stops informative toaxes project and our two-dimensional obtain tuning the regions absolute onto the upper gluino bounds or on stop the physical gluino and stop masses, as a the fine-tuning of SUSY.as It 5% is will easilysense, hardly noted our that be reference even probed choice relativelycomprehensively of mild exclude at 10% level [ the fine-tuning of LHC also tuning represents such for a messenger target scales that the below LHC 100 TeV; can in this squark masses are set10 equal TeV (dotted to lines). the 3rd generation mass (solid lines), at 5 TeV (dashedfigures lines), as and a reference on which future LHC limits can easily be superimposed to assess Figure 8 (blue) and 100 (green) as a function of messenger scale Λ. In both plots, the 1st and 2nd generation JHEP06(2017)103 ), 16 at 10 3 1.6 g )–( 14 705 1750 ). 10 and 1.5 8 t Λ = 100 TeV y 12 10 = 10 TeV ˜ q ] m requiring ∆ = 10 for 10 GeV ˜ t [ 10 1260 2600 m Λ Λ = 20 TeV 8 = 1 TeV). The successively 10 Q ], with a numerical error resulting 6 10 11 (green), and finally moving converting 1040 1665 u 4 2 H Λ = 100 TeV and stop squark 10 m

1.5 1.0 0.5 0.0 2.5 2.0 and messenger scale Λ (see ﬁgure

= 5 TeV

LL results Full Stops: m / ˜ ˜ q q m m 16 – 16 – 1400 2480 is the ratio of the corrected naturalness bound 10 Λ = 20 TeV 9 14 10 12 1140 1475 10 ˜ t ] m Λ = 100 TeV 10 = GeV [ ˜ 10 q Λ m 8 10 1455 2230 6 Λ = 20 TeV in turn. Shown in ﬁgure 10 2 . The ratio of the ∆ = 10 naturalness bound on the gluino (left) and stop (right) masses, (low messenger scales). Also important are the gluino pole mass corrections from . Upper limits on the mass of the gluino 4 u 2 reduction. 2 H (GeV) 10 (GeV) √

˜ 0.0 1.5 1.0 0.5 3.0 2.5 2.0 3.5 t ˜ g m

= 1 TeV. As in previous plots, we take the UV gluino and stop masses saturating their

/ lio:Fl results Full Gluinos: m LL m 10% naturalness bounds. Thethan final the result corresponding is leading a orderat gluino calculation least mass for 50% which the is larger samedifference at value when between least of squarks IR ∆, 10–30% and are and UV larger light. masses ato (high stop For messenger mass the scales) and gluino, the the threshold corrections dominant effects are: the Finally, we exhibit the effect on thein naturalness bounds section of adding each correctionon considered the gluino andas stop each masses higher-order relative effectQ to is the added. leading order For calculation the from LL ( estimate, we have taken between being degenerate withThe 3rd dashed generation horizontal (solid), line orin for a at the 5 gluino TeV is (dashed) the and LL 10 TeV result (dotted). of [ 3.3 Comparison with LL approximation Figure 9 with the higher-order effectsbound outlined in the in LL thisincluded approximation paper effects (with sequentially are: the added,running couplings masses resummed to (red), evaluated two-loop one-loop the at thresholdthe same corrections RGEs to naturalness IR (orange), running resummed mass two-loop to RGEs the (blue), pole IR mass (black). First and second generation squarks are varied Table 1 varying values of 1st and 2nd generation squarks JHEP06(2017)103 ] 16 100 TeV . ], one could put the 36 ], we have explored the , 16 35 2–5 TeV. One very interesting ∼ ]. In this way, the gluino tuning -channel diagram giving large va- t 2 , 35 1 ] for some promising models and [ ˜ q m 34 – 30 – 17 – of what constitutes a fully-natural SUSY spectrum can 7 100 TeV are compatible with 10% fine-tuned SUSY after the first . and figure 6 of 13 TeV LHC data. ]). In the MSSM, we found that heavy 1st and 2nd generation squarks push down 1 − 29 Another possibility is to relax the tuning bound on the stop mass through the addition Given the strong tension between the tuning bounds derived here and the current LHC Motivated by the latest LHC constraints, we have given special attention in this work to 15 fb the mass of a “natural” stop through the RG equations and in the threshold corrections. towards precision corrections tosimplified the models. tuning calculation, combined with recasted limits on of new particles which positively affectas the in stop RGE [ (e.g. the addition of vector-like quarks the dependence on theis ameliorated gluino with mass respect inexperimental to stop the the limits running MSSM, would [ but bechange more the significantly constraining. stop (in tuning As particular,lence is the there squark SUSY actually production), is production worsened, it no rates and would gluino also be very interesting to revisit Dirac gluinos with an eye between the messenger scale andour the tuning weak scale calculation, has andcase. the a potential similarly of For significantly example precise changing gluinos by calculation introducing out should “super-safe” of be reach Dirac carriedHowever, of gluinos out Dirac current gluinos for [ would bounds that also without change incurring the as RGEs much for of the stop a mass-squared, fine-tuning removing price. required by the currentfor LHC further constraints, discussion see of [ this. constraints, another interesting direction would begoing to into challenge the the underlying tuning assumptions calculations. In general, any extension or modification of the MSSM nario. While increasing the 1st/2nd generationtuning squark bound masses through moderately one-loop relaxes the finitestop gluino threshold tuning corrections, bound it through significantly the strengthensthe two-loop the LHC RGEs. constraints) The leads tensionfuture between to direction these a (together for “sweet with model-buildingmoderate spot” Effective will of SUSY be scenario to consistent investigate with viable the low UV messenger completions scales of Λ this messenger scales, Λ ∼ “Effective SUSY” scenarios where thegeneration. 1st/2nd generation We have squarks uncovered are significant new heavier corrections than to the the 3rd tuning bounds in this sce- In this work, we havemass. detailed With several SUSY precision increasingly under corrections pressureestimates to from in the the second figure fine-tuning run of of thebe the LHC, our Higgs used accurate as pointscollider consequences of of reference the as natural more spectra described data here, is and collected. found that In only [ very low difference between IR and UVthe masses drop (especially due the tostop additive 1st/2nd mass boost generation just from squarks), by the with a gluinos the few and other percent. effects4 changing the allowed Conclusions the heavy 1st/2nd generation squarks. Meanwhile, for the stop, the dominant factor is the JHEP06(2017)103 ] 21 ]). 38 , Phys. 37 , ]. Nucl. , (although this SPIRE u IN H ][ ]. ]. in this work, because its µ SPIRE SPIRE IN IN [ hep-ph/9507282 ][ [ ]. ], are a very interesting loophole to 43 – SPIRE would be rounded at the tips and would 39 IN 6 The More minimal supersymmetric standard (1995) 573 ][ – 18 – hep-ph/9709356 hep-ph/9607394 , [ B 357 ]. This tends to be an issue only for high messenger term, e.g. [ 9 Naturalness constraints in supersymmetric theories with ]. µ Upper Bounds on Supersymmetric Particle Masses ), which permits any use, distribution and reproduction in ) and taking the maximum when multiple sources for the Supersymmetry, naturalness and signatures at the LHC hep-ph/0602096 SPIRE (1996) 588 1.3 [ Phys. Lett. IN , [ B 388 CC-BY 4.0 This article is distributed under the terms of the Creative Commons (1988) 63 A supersymmetry primer (2006) 095004 Phys. Lett. ]. B 306 , D 73 16 ], but models of this type may be sufficiently removed from the MSSM that the tuning Rev. nonuniversal soft terms model Phys. Finally, a note on our assumptions about fine-tuning: we have here shown the natural We have not considered the role of the higgsino mass S. Dimopoulos and G.F. Giudice, A.G. Cohen, D.B. Kaplan and A.E. Nelson, R. Kitano and Y. Nomura, S.P. Martin, R. Barbieri and G.F. Giudice, 16 [3] [4] [5] [1] [2] References grant DE-SC0013678. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. bounds [ Acknowledgments We are grateful to J. A.and Casas, J. B. Moreno, Zaldivar K. Rolbiecki, for S. helpful Robles, J. discussions. Ruderman, S. The Thomas work of AM and DS are supported by DOE reduce the maximum allowed masses forour gluinos results qualitatively and unchanged. stops by A separate upcould aspect to also is about be that 200 the susceptible GeV, mass to leaving ofsuffers tuning: the from stop for (a a a scalar) naturalness light stopscales problem and or [ much extremely heavier gluino, light the stops stop — itself neither of which is well-motivated given the latest LHC regions given our measureHiggs ( tuning are present. Ifhope the to UV parameters take arequadrature. assumed into to account In be multiple independent, this one case tunings might the by wedges adding in them figure together somehow, e.g. in precision corrections are rathermass small is in the not MSSM. setthe However, primarily tuning models bounds by where and the a theheavier promising higgsino higgsinos direction have for a futurein potentially work. [ huge Not effect only would oncalculations allowing the for for stops collider and phenomenology gluinos discussed would also be significantly impacted. than considered in this paper.at Similar the conclusions LL hold level, for where the NMSSM, anto as extra fine-tuning described singlet are in lifts reduced [ the due tree-levelshould to Higgs be sizable mass mixing revisited and between in stop the contributions light singlet of and the Higgs couplings being rather SM-like, see e.g. [ New particles could potentially counteract this, allowing ∆ = 10 tuning with heavier stops JHEP06(2017)103 ] 03 , 06 , 04 ] ]. Eur. , JHEP JHEP , JHEP , , SPIRE (2012) 035 IN The Last ][ (1999) 015007 09 ]. arXiv:1112.3028 [ ]. hep-ph/9606211 Light Nondegenerate . [ (2004) 043 D 59 . Implications of a SPIRE JHEP ]. , What is a Natural SUSY IN 02 SPIRE ][ IN Cornering Natural SUSY at (2012) 162 ][ (1997) 3 ]. SPIRE , ]. JHEP IN Precision corrections in the Phys. Rev. arXiv:1212.3328 , ][ , [ ]. 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