JHEP05(2011)112 µ µ B B and Springer µ May 4, 2011 May 7, 2011 May 25, 2011 : : : , b Received Accepted Published 10.1007/JHEP05(2011)112 doi: , problem of gauge mediation. In µ B - µ Published for SISSA by Gian Francesco Giudice, a a [email protected] , [email protected] , and Riccardo Rattazzi Roberto Franceschini, a a Supersymmetry Phenomenology It has been recently pointed out that the unavoidable tuning among supersym- [email protected] . We build an explicit model and we study the constraints coming from LEP and β CH-1211 Geneva 23, Switzerland E-mail: gian.giudice@cern.ch [email protected] Institut de EPFL, Th´eoriedes Ph´enom`enesPhysiques, CH-1015 Lausanne, Switzerland CERN, Theory Division, b a Open Access Keywords: of tan Tevatron. We show that insuperfields, spite the of theory new can interactions remain between perturbativecoupling up the unification. to Higgs very and large the scales, thus messenger retaining gauge are generated at one-loopoffers order. an This interesting class alternativea of to strikingly models, conventional different called gauge phenomenology, Lopsided mediationmass, with Gauge and and light Mediation, moderately is light higgsinos, characterized sleptons. veryfine-tuning by We of large discuss the general Higgs parametric model pseudoscalar relations and involving various the observables such as the chargino mass and the value Abstract: metric parameters required toopens raise up the new avenues Higgs forfact, boson dealing it mass with allows the beyond so for itsand called accommodating, experimental of with limit the no other further Higgs-sector parameter soft tuning, masses, large as values predicted of in models where both Andrea De Simone, Duccio Pappadopulo Lopsided gauge mediation JHEP05(2011)112 7 8 (1.1) originate is a gauge µ g B . Many authors 2 and µ µ and µ B ] is the calculability of the soft 9 – 1 24 14 ] 4 10 3 , Λ µ = – 1 – µ B ) follows from the fact that both 1.1 20 13 17 ]. is the characteristic size of soft masses and 18 16 µ – soft B m 10 , ) leads to a one-loop mismatch between 8 , terms. This mechanism is problematic in gauge mediation because, 7 1.1 23 µ B , where µ 1 22 soft and m ) µ 2 /g 2 π ); see e.g. refs. [ (16 1.1 The problematic relation in eq. ( 7.2 Sneutrino NLSP spectrum 7.1 Higgsino NLSP spectrum 3.1 Naturally vanishing ∼ have addressed this problemeq. of ( gauge mediation by proposing solutions thatat the circumvent same order in perturbationin theory upon the integrating out simplest the case messengers. where For instance, the messenger threshold is controlled by a spurion superfield where Λ is theΛ effective scale of supersymmetrycoupling breaking. constant, eq. Since ( gauge mediation predicts 1 Introduction One of the most appealingterms. features However, of this gauge mediation predictivegenerates [ power the is deficient unlesson one fairly defines general grounds, the one mechanism obtains that the relation [ B Tree-level threshold corrections to unification 8 Conclusions A Effective K¨ahler 5 Higgs mass and fine-tuning 6 Other consequences of a7 heavy pseudoscalar Collider phenomenology 3 A Higgs sector coupled to the messengers 4 The maximal Contents 1 Introduction 2 Characterizing lopsided gauge mediation JHEP05(2011)112 The (1.2) either 1 ) stems ]. µ ] is that problem, B 1.1 17 µ 22 , B - ). Our study 21 µ , but the conclusion 1.1 and the sparticle † Z X m 100 GeV, the relation and is proportional to the inverse X c µ > . This large value of µ , B ) † . This class of models will be called ] or gaugino mediation [ µ B X/X 20 ( , ], on which we shall elaborate. The main F . This indeed shows that the 19 d 17 c H – 2 – u θ H is typically a one-loop factor, but eq. ( 4 c d Z ) is replaced by a generic function of c problem of gauge mediation cannot be ignored. 1.2 µ . . Here 0 B - F µ ) in eq. ( ) does not necessarily imply any further tunings in the theory. In ]). Thus, on rather general grounds, it seems that supersymmetric ∝ † 23 µ 1.1 B X/X ), holds in a more general context than just gauge mediation and it ( ]. It is interesting that the assumption of calculability renders lopsided F implies an anomalously large value of ∝ 1.1 10 ) becomes satisfactory only when Λ is of the order of the soft masses (as ), by dimensional analysis the generic finite 1-loop correction to the K¨ahler 2 2 µ θ 1.1 /g ), independently of the value of 2 ) 1.2 πµ (1 +Λ (4 problem. M lopsided gauge mediation . Such a separation is often needed in order to address the flavor problem and it is An important difference of our study with respect to the analysis of ref. [ The latter consideration prompts us to revisit the viability of eq. ( The situation about fine tuning in supersymmetry has changed significantly in the last Since the experimental searches for charginos constrain µ ∼ In those models = 1 B soft µ - gauge mediation viable only inmasses. the In presence of particular a it separation becomes between viable in theare similar. presence In of the theof simplest the well gaugino volume known mediated of models tuning the compact the that extra small is dimension parameter through which supersymmetry breaking is mediated. both the Higgs masshere and the anomalously large we shall insist on calculabilitywe (weak shall coupling) work and perturbative outexample unification. an of In ref. explicit particular [ model overcoming a quantitative problem of the minimal is directly motivated byobservation the is analysis that, in once ref.limit [ one from LEP, implements eq. the ( certain tuning cases necessary that will to be evade studied the in Higgs-mass this paper, a single tuning is sufficient to take care of Higgs boson have constrained thepercent models seem to almost such inescapable. a The degree problemmediation, that is tunings where particularly at acute the the in soft level modelssome of such terms as special, a gauge are few but tightly propitious, determined relationsgauge capable by mediation of the is partly realized theory alleviating in and the nature, cannot tuning it satisfy problem. must endure If some accidental tuning. rameter tuning in order tosource adjust of the tuning Higgs in vacuum the expectationoriginally Higgs value. introduced, sector As the is eliminating the any very reason why low-energy supersymmetrydecade. was The unsuccessful searches for supersymmetric particles and, especially, for a light model building is facing aµ clash between natural solutions to the flavor problem andB to the destabilizes the Higgs potential along the D-flat direction or requires a considerable pa- potentially exists in anym theory in whichpresent there in is theories with arelation anomaly separation eq. mediation of ( [ scalesin between gravity mediation Λ [ and potential has the form giving rise to from eq. ( defined by eq. ( X JHEP05(2011)112 u ]. 2 H 24 (2.1) (2.2) (2.3) or by δm 2 µ U(1) [ × in the range of A m . ! 2 µ 2 because the stop mass has | B µ | 2 Z ' µ + M
B of the order of the weak scale are 2 d | . µ 2 H µ ˜ t | B m M m + 2 | d . The phenomenology of lopsided gauge µ and log 2 H β | ˜ 2 t µ m + m , can be expressed as a requirement of near u
t 2 2 H α | π µ soft – 3 – 3 µ B | m δm ' + − u u u is large compared to 2 H and H 2 0. For sufficiently low messenger scale one has 2 H u Z ' δm m δm 2 H m

2 H − δm u = M H 2 2 H m M the Higgs mass matrix at vanishing background field value, the condition 2 H , leading to light higgsinos, by a pseudoscalar-Higgs mass M are the usual Higgs soft masses evaluated at the messenger scale and ), there are two extreme ways of implementing the tuning required by near µ . The first option corresponds to ordinary gauge mediation; the second one to u 2.3 u,d 2 H 2 H m m In spite of the necessary tuning, one cannot regard lopsided gauge mediation inferior to this term is atfrom the eq. origin ( of thecriticality: usual the fine negative tuning radiative correction problema can large of be supersymmetry. compensated either Aslopsided by evident gauge a large mediation. Let us consider the two cases. The condition of criticality then becomes The stop induced correction to be close to 1 TeV to lift the Higgs boson mass above the LEP limit. The largeness of where is the radiative correction proportional to the stop square mass lie very close to theIndicating by line separating the brokenof and near unbroken criticality phases is of just SU(2) det 2 Characterizing lopsided gaugeThe mediation condition for electroweak symmetryunavoidable breaking hierarchy in between supersymmetry, in thecriticality: presence in of the an phase diagram spanned by the model parameters, the soft terms have to several TeV and a moderatemediation, to large discussed value of in tan collaborations, this since paper, it represents is justthe worth another theoretical side serious point of of scrutiny gauge view, mediation, by but equally the much likely less from LHC explored. experimental the ordinary gauge mediation scheme,put in in which by hand or generatedmediation by new is interactions equally at good some (or intermediate equally scale.is bad) however Lopsided an since gauge important the degree difference of withsince fine respect the tuning to is phenomenology the identical. is ordinary scheme There totallya of distinct. very gauge small mediation, Lopsided gauge mediation is characterized by that lopsided gauge mediation isthis not tuning “natural”: simply a amounts fine topresent tuning is what experimental actually is limit. present. required It However, of in corresponds gauge order mediation. to to lift the the inevitable Higgs tuning mass present beyond in its any model necessary to lift the Higgs mass above the LEP bound. In this respect, we must stress JHEP05(2011)112 , 2 ∝ π d (3.1) (3.2) (3.3) (2.4) (2.5) 2 H . Or- 2 m π u 2 H 16 effectively m k in the mass ∼ . In this case, 2 d . Electroweak µ 2 . The condition 2 H µ /µ m u,d µ 2 H B  ], while , m 2 25  u,d can be as large as 16 a  2 H π of the SM gauge group. 2 α 4 2 . m I µ /µ  d ) µ . I H ( Φ, Φ) are all controlled by a a B 1 and 2 m and C u µ < u u H 3 =1 2 H 2 2 H a , though in several realistic models X / m 1 ) which requires π i 2 G 2 δm  δm 4 / Λ u 1 1.1 . NL ' ' < ∼ G problems and requires a special solution.  2 H 2 ), because X n u u µ 2 µ k u h 2 H . 2 H ΦΦ µ δm k 2 H 1.1 B - 5 = 2 m X m + µ − δm 2 I – 4 – 1 = ˜ − m M  1 W = '  ) via the superpotential µ 2 X , ' 2 µ B θ 6= 0 arises in an unspecified “secluded sector”, which con- a π µ 2 G α 4 the soft masses are given by µ B X is the canonical Goldstino superfield [ G Λ 2 M ) G F (1 + Λ n 2 = M √ is sufficiently large. Note that the largeness of a = χ/ d M 2 H X + m θ ) does not necessarily imply a further tuning for the theory, if ( F 2.1 1. At the scale are the quadratic Casimirs of the representation = )  I is defined as an effective messenger number which is, for instance, equal to 1 if ( . We will show in the next section that the simplest model of lopsided gauge NL k 2 µ G C X matrix ( B mediation automatically predicts thisfine tuning relation will of be proportionality. addressed in The section issue about This result is not incompatibleas with eq. long ( as This result is clearlydinary incompatible gauge with mediation eq. suffers ( from a In lopsided gauge mediation, we choose of criticality now becomes breaking is achieved at thefrom price of the a condition tuning of involving criticality a we large find value of In ordinary gauge mediation, one takes n as the expectation value (i) (ii) where Here where describes the coupling ofnaive dimensional the analysis messengers (NDA) toone one the has expects supersymmetry breaking dynamics. By The dynamics that gives rise to tains also the supersymmetry breakingX dynamics. On general grounds one should interpret 3 A Higgs sectorIn coupled order to to the set messengers mediation. the The stage masses andspurion fix and superfield the splittings notation of let the us messengers recall ( the basics of minimal gauge JHEP05(2011)112 S X (3.9) (3.4) (3.5) (3.12) (3.10) (3.13) (3.11) (these . The and 10

D 2 ⊕ X x G,D,S ) ]. 10 log are equal at the 3.4 29  – 2 d ) can also be chosen H 2 26 x S m ¯ . S. (1 +  and S to be SU(2) doublets with 2 y and Λ S u x ¯ to matrices under the unique − D H  X D 2 ) k m 2 x + . ) log y (3.7) . (3.8) (3.6) ) ¯ and ) )  ) y D ) 2 2 and θ D ) log x − D 2 x, y x, y x, y (1 + D x, y ( ( ( M x 2 ( (1 D,S X 2 x Q R is adding direct couplings of the Higgs P S − ) log 2 x 2 + D 2 D  2 D D µ y x − ¯ Λ Λ Λ Λ S B + ) d d 2 2 2 2 ¯ ) can be eliminated by field redefinitions up (1 + Λ 2 D y λ λ π π π π and defined the functions ) d 2 u,d 2 u,d 2 ) + ( u u – 5 – y 3.5 λ λ x y 16 16 and 16 λ λ 16 D,S H D d ] and the soft parameters of the Higgs sector are − − − Λ M µ λ = = = = ) + (1 + / 30 2 S components which are conveniently parametrized as = + µ )(1 µ with a single chiral superfield. In eq. ( x 4 u,d u,d 1)(1 1)(1 B ¯ ) and ( F S x 2 H − A for details) = Λ D,S DS − − − m 3.4 u 2 2 X A , y 2(1 and x x H (1  U(1) gauge invariance. Indeed in that way one covers a chunk ( ( D u    S of SU(5) or to 3 in the case they belong to a 2 are vanishing while the masses λ × 3 ) 2 3 2 y µ 5 /M 1) = 1) 1) 1) S B − ⊕ − 1 x x are weak and hypercharge (as well as GUT) singlets. In this case it is − − − M W 2 5 SU(2) (1 2 2 2 ¯ S x 2 = and x x x × ( x ( ( ( x µ and 1 (being part of a complete GUT representation, such as the fundamental of ) = ) = ) = ) = S ± x, y x, y x, y x, y ( ( ( ( . A possible way to generate S R P Q M At this level Here we have set real. After integrating outrenormalized the in messengers a at calculable one-loop,easily the fashion calculated low-energy [ (see K¨ahlerpotential appendix is Notice that all phasesto in the eqs. relative phases ( parameters between have the indeed the various same supersymmetry-breaking quantumThe numbers masses under remaining Λ all phases spurionic explicitly global symmetries). breakthus CP assume and that dangerously the contributes secluded to sector edms. respects We CP, shall so that Λ SU(5)), while more economical to identify are spurions with both scalar and doublets to two pairs of messenger fields The simplest choice for thehypercharge messenger fields is taking minimal scenario is easily generalizedconstraint by of promoting SU(3) of the full 6-dimensional parameter space of general gaugescale mediation [ the messenger are a JHEP05(2011)112 ¯ S is P S y a and µ (3.23) (3.15) (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) (3.22) (3.14) satisfy P ,
2 forbids contributions ¯ . and S ξ ¯ S
S ¯ R S + S D 2 D 2 ¯ 2 D S 2 D and → − ]. Notice also that, , the superpotential Λ Λ S S Λ Λ S is related to the . d 2 d 2 a S S 31 2 2 2 2 ) λ [ | π λ π | X S µ π π u u u . d 2 S 16 B λ 16 X charge, so that one could λ λ + 2 λ θ θ 16 | | x, y 16 and 2 3 2 X ( 2 2 u 1 d 1 are also positive. However, R a a + u 2 ¯ S . Also worth noticing is that a a ¯ ∝ S ¯ y R and H ≡ ≡ D tan tan u,d a ≡ ≡ . ¯ − 2 H D µ S S D )] )]     + a a ) ) ) D ) ) X m and → − 0 so that electroweak symmetry is 2 − − X 2 S u ξx, y x, y ξx, y < x ¯ S S S , but with a superpotential featuring ( ( ( ξx, y ξx, y H ξx, y ξx, y ¯ a a a ) and rescaling 2 ( ( S ) + ( 2 H ( Q R ¯ P P P S S S , = S possesses U(1) θ 2 θ 2 θ M θ 2 2 θ µ 2 3.15 c ∝ c s ) + ) + c s X and ) R 2 µ + S + B x, y x, y ) + ∝ ) + 3.4 ) + ) + S – 6 – ( ( ¯ θ D µ , ξ ) are smaller than one in absolute value, with − s Q R x, y x, y d B D S x, y x, y − − − ( ( ( ( ¯ a 2 H [ [ ( S D is a function of 3.13 S P S P S θ θ ¯ m D X − c c θ 2 θ 2 2 θ 2 θ a . This relation is problematic because at the messenger d u θ θ c c s s ¯ 2 , one has det )–( S y s s +     2 H 2 H a ). This is an instance of the known result that one-loop d ¯ D 2 D S D 2 D P D D 2 m QRS S λ = ¯ 3.10 Λ Λ Λ Λ Λ Λ and D is not. Consistently with that, one has that the rephasing ' ∝ d d 2 2 2 2 2 2 2 2 symmetry under which 2 2 d θ | | | | x and , leading to a spectrum with far too light sparticles. λ λ ∗ π π π π π π µ 2 ) + x = Λ d d u u 2 H H u u ¯ θ d λ λ λ λ Z S M 16 16 16 16 | | 16 λ λ | 16 | θ D λ M tan 2 s AµB cos ======+ + det d d u u µ ≡ µ = 0) we find that, for small S H 2 2 H ) for any A A B θ θ DS = 0 the u = 1 (Λ c c u m m is positive and thus the square masses ( ¯ 2 H , S y x, y H θ S D ( u P a u δm λ P sin 2 H = u is always bigger than ≡ λ 2 | ≥ θ W ) s = R It is clear that it must be possible to overcome the above difficulty by a slight com- W x, y . After diagonalizing the mass term in eq. ( vanishes for ( µ R Again we have assumed CPditions invariance for and the chosen soft all parameters parameters in real. the Higgs The sector boundary are con- the sum of the coefficient. If B becomes where it is enough toadditional consider mass two terms gauge with singlets respect to eq. ( The generation of a non-vanishing contribution to both Since broken at the large scale plication of the model.conceive This of is contributions with because opposite signs due to the presence of several spurions. Indeed all the four functionsbeing in eqs. somewhat ( smaller than| the others. Anscale (where important feature is that P soft masses vanish inby the the presence argument of onphysical just the while one elimination the spurion of sign unphysicalinvariant combination phases, of it follows that the sign of The function JHEP05(2011)112 2 1 R D of Z H 0 ,M D S , 0 D X,M ). and preserves ) (3.24) 3.24 ¯ D 0 PQ D breaks PQ. On the + 0 X X ¯ D D 0 ( D D = 0. The model contains an M breaks PQ, and one can easily 0 µ + X B 2 S . Indeed in order to preserve U(1) -symmetry is thus exact in the above SD S d R 2 H M will be suppressed. is broken in the sector that couples m , while µ + , unlike diagonal masses, transforms under d µ ¯ 2 H PQ D 1 0 0 1 1 2 and B m DS 1, such that the electroweak critical condition , and with superpotential − u ¯ – 7 – -neutral. S XD is not. Of course in order to generate gaugino 2 H → R 1 + and m R . The expectation value of ξ D is − and u 1 more plausible. On the other hand, in our case one ¯ F 2 H DS S ) must be controlled by multiple spurions d d m H H , soft d µ is here tied to supersymmetry breaking. This seems more 3.24 λ m charge assignments for the superpotential in eq. ( B u µ 1 1 1 1 0 0 1 1 2 2 2 + ∼ F/M H in the superpotential, which also breaks U(1) PQ term can be made arbitrarily small either by approaching the . The above superpotential is the most general one compatible 2 µ ∼ 2 DS µ symmetry. H u ) that at one-loop the renormalization of the Kahler potential gives 0), or by taking 1 µ R B and F θ PQ H R -charges in table has R-charge 2, A u → R µH = λ . θ X i PQ = ) it is clear that we can now avoid the relation that lead to a negative must be broken by other messengers which for some reason do not couple to NL W and R X 3.21 ) can be satisfied. In these limits also Table 1 R to the messengers, while U(1) ≡ h . Indeed, the 2.3 phase rotations. Then its overall relative size must basically be a free parameter, emerges as an accidental symmetry of the Higgs-messenger sector, while it is bro- 2 2 H , in that the origin of X H R R R It remains an issue of model building to come up with a fundamental theory where M simple addition of U(1) satisfactory as it makes necessarily has 1-loop contributionsthe to messenger masses in eq. ( low-energy spectrum if PQ is onlyas an in approximate symmetry the of case the Higgs-messenger of sector, the U(1) ken elsewhere. It doesinvestigated not specific seem implementations. implausible that Notice it also could that happen, our although mechanism, we differs have from not the lagrangian. On the otherverify hand (see the appendix expectation valuerise of to non-vanishing axion, which could be interpreted asbreaking the benign is invisible sufficiently axion if high. the scale Alternatively, of the supersymmetry would-be axion could eliminated from the with with the other hand, since masses U(1) the Higgses. An exampleby of a a variation Higgs-messenger ofdoublets, sector the with with the models the identification above desired of with features is the offered simple addition of an extra pair The remark leading to theU(1) last model is that given sufficient freedom tocould pick indeed a conceive model. aand model Taking in the which same U(1) argument to its extreme we det symmetric limit ( in eq. ( 3.1 Naturally vanishing From eq. ( JHEP05(2011)112 µ ) is η (4.1) (4.2) ) cor- ) since 4.2 , one has 4.1 D was already Λ , that is two 4 µ ∼ , ) vanishes at the B π . The smaller is # G µ d 4 ), and the hyper- / (1)), eq. ( λ B . The limit can be u 3 3.3 O Z g 2 D H 2(FI) Λ m m = ) turns out to be quite is generated dominantly ˜ ∆ t & M µ 4.1 m ) we obtain whose smallness quantifies µ are weak-size contributions − B ˜ 2 t in protecting log over a surprisingly large span ], u 3.20 t u /m 2(FI) H R β 32 α 2 D . Eq. ( 2 Z H 2(GM) η . m Λ u # m m 2 H to be roughly ( ˜ t can vary extensively throughout the π ∆ ) and ( m = 2 ) M m ) η µ − η π 2 B 4.1 ˜ t 4 , and ∆ a at 1-loop makes the above model a special log / M m t 6= 1. For the semianalytic discussion of this d u − α λ 2 H µ u , which is legitimate for perturbative values of 2 2 B (1 log 2(GM) H m is obtained from the experimental lower bound is not too close to 1, lopsided gauge mediation 2 a – 8 – u 1 a ˜ t 2 µ 3 d m a µ 2 D λ m B " t and Λ a α π & u 3 2 H 2 u " 2 d π λ m ) λ . Here ∆ µ 16 d 1 2 B a a u 1 1 − /a (20%) range, if 2 3 O (1 a . Thus, the very existence of the tuning (i.e. the smallness of u 1 2 ]. However, following that remark, in the models of that paper / a = µ 1 ) hold true, but with the additional property that µ 18 d ' . Thus, as long as 2 B ) to get an approximate analytical understanding of how the electroweak πη 2 H 4 2 4 a ) 2 u π π 2.3 & λ 4 16 < m / d is fixed, a lower bound on 3 λ u g u 2 H ( λ m ∼ couplings, we can solve for  2 D Once The necessary appearance of 2 Λ λ / µ ˜ 2 t In our model theparameter numerical space. pre-factor However, in parametrically frontresponds (and of to treating the key element that allows for the existence of perturbative values of on the chargino mass,expressed which in can be terms approximately ofthe written a tuning as dimensionless of the parameter supersymmetric model. Using ( they are subdominant. Note thatm under the reasonable assumptionrequires that Λ the one-loop expansionloops parameter in ( QCD. where we defined coming respectively from thecharge standard Fayet-Iliopoulos gauge contribution mediated (see part, below)reliable, see typically to eq. in ( the session we shall however drop the second and third term within brackets in eq. ( the 4 The maximal We can use eq. ( breaking condition is satisfied. Dropping case of lopsided gauge-mediation:as all the relations discussedmessenger in scale. the Similarly previous to sectionvia the (such RG model running, discussed leading inof to ref. messenger a [ masses. naturally large value of tan ing the vanishing of 1-loopwhole masses spectrum in and the itsemphasized presence mass of in splittings. a ref. single The [ originates spurion role from field of the controlling U(1) superpotential the directly rather related than to from supersymmetry the breaking. K¨ahlerpotential, and is thus not with different global quantum numbers. Then one cannot rely on the usual theorem insur- JHEP05(2011)112 2 (4.5) (4.6) (4.7) (4.3) (4.4) , which we . and with a d
µ ) 2 H 2 E m m is independent of 3.19 . The dependence + . For instance the 1 ˜ which is explicit in 2 t M to stay perturbative 2 D . These will add to the m d d G /M m H λ n . We show these effects + 1 m max 2 U µ from eq. ( and log /M m d 2 , G , λ µ . n max d d , a feature of the spectrum that − , µ d H R H 2 L ˜ 2 t 2 d M H M M 2 D m m m m m Λ 2 G 1 2 G − < m log log log α Λ Λ d 2 Q ) R G G S µ 2 u 2 H 2 D m n 1 n 2 B π Λ m m 2 2 2 2 α a is fully fixed only when the content of the obtained from the criticality of electroweak ˜ d 1 1 f α π α 10 is large enough. a t − Y d α 5 u 1 3 3 – 9 – 20 λ α λ + Tr M (1 2 µ u − 2 ) translates into a large value for πα a 2 H π 8 ' − = 8 15 m 4.2 spoils the naive scaling with . ) we observe that the ratio + . ˜ 2 2 t ˜ 2(FI) f 2(FI) L d d 2 2 π 4.7 m 2 H λ µ compatible with the experimental bound on m m 16 results in bigger values of m d ∆ ∆ λ G − n =
˜ 2 f ), and only mildly dependent on both m . From eq. ( can be further bounded from above requiring ˜ required by eq. ( f D d 1 Y d a µ Λ ), to get an upper bound for the value of u 1 λ (1), which is negligible when / O G 4.1 /a 2 2 = Tr a S , for specific choices of the parameters. = 1 2 µ ): bigger values of a 4.7 The value of A much more easily observable effect occurs among sleptons. Indeed the left-handed The sizable (for fixed Λ In the formula we neglect the finite threshold corrections at the scale 2 G , the wider is the range of in figure up to the GUT scale. Thelogarithm evolution a of term where Λ on these latter parameterspositive comes gluino mainly contribution from to eq. extra ( logarithms in This can be combinedbreaking, eq. with ( the value of as opposed toslepton the square positive masses sets shift an to upper the bound on right-handed the sleptons. value of The request of positive sleptons get a large and negative contribution to their square masses Such effects are largesparticles in masses lopsided predicted gauge by mediationthe ordinary first and gauge and can mediation. second easily generationis reverse For squark nevertheless instance, the masses, hard we ordering to find study of at that, the for LHC. where perturbative extrapolation of the theory up to high energies. recall arises already atlopsided 1-loop gauge mediation order and unlike can have thethrough significant the sfermions impact induced on masses. hypercharge the Fayet-Iliopoulos term running This (FI). of This is the contribution soft a is masses given trademark by of η JHEP05(2011)112 . ). )– 10 3 GeV 4.7 . As (4.8) GUT 10 3 could 3.10 M 10 0.9 : on the ≤ GUT = Μ d < ∼ 1 B M and a , 8 (left panel) . 4 = 1 chiral fields. M = together with G d and n = 0 2 G M , θ L spurions originate from . This is confirmed  5 1 . D 3 X 7 L

=10 = 4

M . The bound from pertur- 1 because of the larger energy µ < M

6 , ξx , 2 µ .

10 1 is assumed. On the red dashed = , there is clearly no effect on the RG µ/M M  2 2 

5 = 1 g GUT 3 10 y spurions originate from x

= M

M − X 1 2 1 somewhere between g

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

3 5 max 1 M  Μ π is not much bigger than one, see eq. ( − G 2 d Λ λ / 4 – 10 – 10 , are part of a fundamental and a singlet of SU(5) D  shown in the left panel of figure 0 ) 2 1 2 d 1 π is λ , . Somewhere below the GUT scale this effective description will 8 0.6 1 d S = gets non perturbative at some point between Μ λ X µ/M = B we discuss here requires qualifications. The point is that, in our , d a hits the value 4 = ( 2 d , λ D d 4 d dt in which the running coupling must remain perturbative. λ = X dλ λ G ,S n 1 7 (right panel). In both cases . − G ) GUT L  2 and in principle relaxes our bound. On the other hand we do not expect that = 0 D 3 M , d L 1 in particular. However for the simple case in which the λ , θ for a specific choices of parameters: d 7 . λ and 1 = ( has a broad validity. spurions. The other possibility is that the D d /M = 6 M . The significance of this perturbativity bound is shown in figure λ 9 . In this set up, assuming that the original scale of supersymmetry breaking is X

5 Q 10 max u,d

= , ξx 10 µ λ

M = 7

13 . M . =10 M = log = 0 t 1 ). In the model of section 3 we find, quite generically, These general considerations imply an upper limit on x

The perturbativity bound on

0.2 0.1 0.0 0.3 0.4

3

1 max M Μ  3.13 evolution of to suppress gravity mediated terms,give one rise finds to that the hiddenIn sector that chiral case operators the ofthe presence dimension of upward evolution additional of Yukawa interactionsthis involving effect the can messengers become generally dramaticupper slows without bound down those on other couplings also becoming large. We thus think that our description of the models, the dynamicseffectively that in breaks terms supersymmetry of is the nothave specified spurions to but simply be parametrized replacedour by couplings, some and of physicalnon-renormalizable interacting operators fields. suppressed by This a will scale in larger principle than affect the RG evolution of The precise limit is ultimately( related to the structureby of the the scatter loop plot functionsthe in for region eq. the excluded ( ratio viable by spectrum, the free perturbative of requirement. tachyons. All We restricted the the points scanning showed of provide the a parameter space to the messenger scale is lowered,range we between expect a smaller value of bativity is generally strongermasses, than and the this one is imposed particularly by true the if experimental Λ limit on slepton with red dashed part of the lines and branch of the curves the coupling messenger sector coupled to thethe Higgs messengers fields is specified.respectively, the In RG the equation the for simplest case in which Figure 1 JHEP05(2011)112 1 ). or  (4.9) 3.16 y (4.10) GUT 1. This , for the M ≤ d 3 θ H ), while the M are obtained m ≤ 1 4.1 5 . log : ˜ t β 2 2 in figure µ/M ), we obtain M m α 10, 0 d 2 β 2 H π ≤ log m 2 3 ξx 4 is favored to avoid extra + α 1 u ≤ π/ α 2 H t ). Under the assumption of a ∼ α m 4, 4 . The lower bound, on the other 2 θ +  G 3.21 ≤ ). Due to the cancellation in the 2 µ Λ | B x / µ 2 )–( µ a | B D . ), bigger values for ≤ a − (2 d u 2 1 3.18 / . λ λ as discussed in the text. Furthermore µ d 1 3  3.21 , while the other superpartners are much attainable in the model described by eq. ( a a 1 B 5 4 Z 1 )–( ' m . = 2 µ/M – 11 – β µ/M β β 3.20 2 tan tan . . The upper bound, obtained by requiring positive square for a given choice of the cutoff scale of the theory: ˜ t 1 d M m λ , see eqs. ( 2 , using sin 2 µ d B λ η log 2 D , together with the mass relations imposed by gauge mediation, 2 G is is determined by eqs. ( varies between 0.1 and 10. The red shaded regions are those excluded by Λ µ Λ and and sufficiently far from 1. Furthermore β 4 ). These constraints translate into a range for tan G G u µ π ξ µ n Λ λ / 2 3 is fixed by the electroweak breaking condition as in eq. ( D α . 2 t u µ λ α = 1% we plot the bands for the allowed values of tan 2 is bounded both from above (slepton masses and perturbativity) and below  ˜ 2 t d µ µ λ B . Scatter plot showing the values of B /m a a 2 Z µ − m a 1 The value of tan The smallness of GeV.  = 9 3 2 where we have usedη the uppersame bound parameter choices coming of figure frommasses the for the FI slepton term. doublets, is independent For of a Λ fixed value of The coupling coupling (minimum value of hierarchy between requiring a value of suppression in give significant constraints toleft-handed the sleptons spectrum. have massesheavier. Typically close the to higgsinos, the bino and the the perturbative requirement on 10 the most favorable regionboundary (see values of the caption in figure Figure 2 The points are chosenpicks randomly the among most those favorable withis region 0 assumed, for and the Λ ratio JHEP05(2011)112 . . ‰ 1 1.0 M 1 4 = = µ/M ‰ G G . The , turns n n for same θ gets non ‰ : : , that is ‰ 4 o ‰ d d G 0.8 and ‰ λ ‰ λ ‰ Λ ‰ ‰ / β ‰ ‰ ‰ , and D o ξ o ‰ , o ‰ ‰ ‰ y 0.6 ‰ , ‰ 1 o ‰ ‰ o x while on the right the M  ‰ ‰ o ‰ ‰ ‰ ‰ Μ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ GUT o ‰ o 0.4 ‰ o o ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o o o o ‰ ‰ o ‰ o ‰ M ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o o o ‰ ‰ ‰ o ‰ ‰ o o ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ o o ‰ ‰ o ‰ o ‰ o o ‰ ‰ o ‰ o o ‰ ‰ ‰ ‰ o ‰ o ‰ o ‰ o o o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ o o ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o o o o o o ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o o o ‰ o o ‰ o ‰ o ‰ o ‰ o o ‰ o ‰ o ‰ o o ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ o as a function of Λ o ‰ ‰ ‰ o o o ‰ o ‰ ‰ o o o o ‰ o o o ‰ o o ‰ o o ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ o ‰ o o ‰ o o ‰ ‰ ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ o o o ‰ o ‰ ‰ ‰ ‰ ‰ o o ‰ o ‰ ‰ o o o o o ‰ o o o ‰ o ‰ o ‰ ‰ ‰ o ‰ o ‰ o o ‰ ‰ ‰ o o o ‰ o ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ o o o o ‰ ‰ ‰ o o o ‰ ‰ ‰ ‰ o o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o o o ‰ ‰ o o o ‰ o o ‰ o o ‰ o ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ o o ‰ o ‰ o ‰ o o ‰ o o o ‰ o o o o ‰ ‰ o o ‰ o o o o o o ‰ o ‰ o o ‰ ‰ o o ‰ o o o ‰ o ‰ 0.2 ‰ ‰ ‰ ‰ ‰ o ‰ o o o o o o o o o o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o o o ‰ o o o o ‰ ‰ o o ‰ o ‰ ‰ o ‰ o o o ‰ ‰ o o ‰ ‰ o o o o ‰ o o o o ‰ o o o o o o o ‰ o ‰ o o o o o o ‰ ‰ ‰ o ‰ ‰ o o o o o ‰ o o ‰ ‰ o o ‰ o ‰ ‰ o o o ‰ ‰ ‰ o o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ h ‰ o o o ‰ ‰ ‰ ‰ o o o ‰ ‰ ‰ o o o o ‰ o ‰ o o o o ‰ ‰ ‰ ‰ o o ‰ o ‰ o ‰ o ‰ o ‰ o ‰ ‰ o o o ‰ ‰ o o o o ‰ ‰ o ‰ o o ‰ o ‰ o o o ‰ o ‰ o ‰ o ‰ ‰ o ‰ o ‰ o o o o o o o o ‰ o o o ‰ ‰ ‰ ‰ ‰ ‰ o o o ‰ ‰ o o o o ‰ ‰ o ‰ ‰ ‰ ‰ o o o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ o o o ‰ o ‰ o o ‰ ‰ o o o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o o o ‰ o o o ‰ ‰ ‰ ‰ o o ‰ ‰ o o ‰ o ‰ o o o o ‰ o ‰ o ‰ o o ‰ ‰ ‰ o o o ‰ o o o ‰ ‰ ‰ o ‰ o ‰ o ‰ o o o ‰ ‰ ‰ o o o o ‰ o o o o o ‰ o ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ o o o o o o ‰ ‰ o ‰ o ‰ o ‰ ‰ o o o o o o ‰ o o o o o ‰ ‰ o ‰ ‰ o o ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o o o o o ‰ o o o ‰ o o o ‰ o o ‰ o o ‰ o o ‰ o o o ‰ o o o o ‰ ‰ o ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ o o o ‰ o o o o o o o o o o o ‰ o o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o o o ‰ o ‰ ‰ ‰ o ‰ o ‰ o o o ‰ o ‰ o ‰ ‰ o o o ‰ o o o o o o ‰ ‰ ‰ ‰ o o ‰ o o ‰ o o o ‰ o o o o o o o ‰ o ‰ ‰ o o ‰ o ‰ o o o o ‰ o o o is simply controlled by the ratio of o o o ‰ o ‰ o o o ‰ o ‰ ‰ ‰ o o o ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ o o o o ‰ ‰ o ‰ ‰ o ‰ o o o o ‰ o o o ‰ o ‰ o o o o o ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o o o o o ‰ o ‰ o o o o ‰ o o ‰ o o o ‰ o ‰ ‰ o o o ‰ o o o ‰ o ‰ o o ‰ o ‰ o o o o o o o o ‰ o o ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ o o o o o o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ o ‰ o ‰ o o o ‰ o ‰ ‰ o ‰ o ‰ ‰ o ‰ o o o o o ‰ ‰ o o o ‰ o o ‰ ‰ o o o ‰ ‰ o o ‰ o o o o o o o ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ o o o o o o ‰ ‰ o ‰ ‰ ‰ ‰ o o ‰ o ‰ o o o ‰ o o o ‰ ‰ o o ‰ o ‰ ‰ o ‰ ‰ o o ‰ o o o ‰ o ‰ o ‰ ‰ o ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o o o ‰ o o o o ‰ o ‰ ‰ ‰ ‰ o o ‰ ‰ o o o o o o ‰ o ‰ o ‰ o ‰ o o ‰ o o o o o o ‰ o o o o ‰ o ‰ o ‰ ‰ ‰ o ‰ o o ‰ o o o o o o o o o ‰ o o o o o ‰ ‰ ‰ o o o o ‰ o o o o o o o ‰ o ‰ ‰ o o ‰ o ‰ ‰ o ‰ ‰ o o o o o o ‰ o o o o ‰ ‰ o ‰ ‰ o o o o o o o o o o o o ‰ o o o o ‰ ‰ o o ‰ o ‰ ‰ o o o ‰ ‰ o ‰ ‰ o ‰ o o o ‰ ‰ ‰ ‰ o ‰ o o ‰ o ‰ ‰ ‰ o ‰ ‰ o o ‰ ‰ o ‰ ‰ ‰ o o ‰ o o ‰ o o ‰ ‰ ‰ o o o o o ‰ o o o o ‰ o ‰ ‰ o o o ‰ ‰ o o ‰ o ‰ o o o o o ‰ ‰ o ‰ o o o o ‰ o o o ‰ o o o o o o ‰ o ‰ o ‰ o o o o o o o o ‰ o ‰ ‰ o o o ‰ o o ‰ ‰ o ‰ ‰ ‰ o ‰ o ‰ o ‰ ‰ ‰ o o o ‰ o o o o o o o o o ‰ o o o o o o ‰ ‰ o ‰ o o o ‰ ‰ ‰ o o ‰ o o o ‰ o o ‰ o ‰ o ‰ o o ‰ o ‰ ‰ o o o ‰ o o o o o ‰ o o o ‰ ‰ o o ‰ o o o o ‰ o ‰ o ‰ o ‰ o ‰ o o o ‰ ‰ ‰ o ‰ ‰ ‰ o o o ‰ ‰ o ‰ o o o o ‰ o o ‰ o o o ‰ ‰ o o ‰ ‰ = 1 o ‰ o o o ‰ o ‰ o o o ‰ o o ‰ o o o o ‰ ‰ ‰ o ‰ o ‰ o o o o o ‰ ‰ ‰ o ‰ ‰ o o o o ‰ o ‰ ‰ o ‰ o o o ‰ ‰ o o o ‰ o o o o o o ‰ o o o o o o o ‰ o ‰ o o ‰ ‰ ‰ o o o ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ o o o ‰ o o ‰ ‰ o o ‰ o ‰ o ‰ o ‰ ‰ o ‰ o o o o ‰ o ‰ ‰ o o o ‰ o o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o o o o ‰ o o ‰ o o ‰ ‰ o ‰ ‰ o ‰ o o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o o o o o ‰ ‰ o ‰ o ‰ ‰ o o o o o o ‰ ‰ o o ‰ ‰ o ‰ o o o o ‰ o ‰ o o ‰ ‰ ‰ ‰ o o o ‰ o ‰ ‰ ‰ o o o o o ‰ ‰ o o o o ‰ o o o o ‰ ‰ ‰ o ‰ o o o ‰ o o o o o o o o ‰ o o o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o o o ‰ o ‰ ‰ ‰ o o o o ‰ o ‰ o o o o ‰ o ‰ ‰ o ‰ ‰ o o ‰ ‰ o ‰ ‰ o o ‰ β o o ‰ o ‰ ‰ ‰ ‰ o o ‰ o o o ‰ o ‰ o ‰ ‰ o o ‰ ‰ ‰ o o ‰ ‰ ‰ o o o o o o ‰ o ‰ o o ‰ ‰ ‰ ‰ o ‰ ‰ o o o ‰ o ‰ ‰ ‰ o o o o ‰ o o ‰ o ‰ o o o o ‰ ‰ o o ‰ o o ‰ ‰ o o ‰ ‰ o ‰ o ‰ ‰ o o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o o o o ‰ o o o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ o o ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ o ‰ ‰ o o o o o ‰ ‰ o o ‰ ‰ o o ‰ ‰ ‰ o o ‰ ‰ o ‰ o ‰ o ‰ o ‰ o o o o o ‰ ‰ ‰ o o ‰ o ‰ o ‰ o o o o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o o o ‰ o ‰ o o o o ‰ ‰ ‰ ‰ ‰ o o o ‰ o ‰ ‰ o o ‰ o ‰ o o ‰ ‰ o ‰ o o ‰ o o o o o ‰ o o ‰ ‰ ‰ ‰ o ‰ o o o ‰ ‰ ‰ ‰ o o o o o o ‰ o ‰ o ‰ o ‰ o o ‰ o o o ‰ o ‰ ‰ o ‰ o ‰ ‰ o ‰ o ‰ o o ‰ ‰ o o ‰ o ‰ ‰ ‰ o ‰ o ‰ ‰ o o o o o ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ o o o ‰ ‰ o ‰ o o ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ o o o ‰ o o ‰ ‰ ‰ o o o ‰ ‰ ‰ o o o ‰ o o ‰ o o o o o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ o o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ o o o ‰ ‰ o o ‰ o o ‰ o ‰ o ‰ ‰ ‰ ‰ o o o ‰ o o ‰ ‰ o ‰ ‰ ‰ ‰ o o ‰ o o ‰ o o ‰ ‰ o o o ‰ o o o o o ‰ ‰ ‰ o o ‰ o ‰ o ‰ ‰ ‰ o ‰ o ‰ o ‰ o o o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ o ‰ o o o o ‰ o o ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o o o o o ‰ o o ‰ ‰ o o o ‰ ‰ ‰ o ‰ o ‰ o o o ‰ ‰ ‰ ‰ o o o o o ‰ o ‰ o o ‰ ‰ o ‰ o o o ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o o o o o o o ‰ ‰ ‰ o o ‰ ‰ ˜ ‰ ‰ o o o o o ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o o o o 2 t ‰ o o o ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ o o o o o o ‰ ‰ o ‰ ‰ ‰ ‰ o o o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ o ‰ o o o o o o o o o o ‰ ‰ o ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o o o ‰ o ‰ o ‰ ‰ o o ‰ ‰ ‰ o o o o ‰ o ‰ o o o ‰ ‰ ‰ ‰ o o o ‰ o o ‰ ‰ ‰ ‰ o o o ‰ ‰ o o ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ o o o o ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ o o ‰ ‰ ‰ o o o o o ‰ o o ‰ o ‰ o o ‰ ‰ ‰ o o o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ o ‰ o ‰ o o ‰ o o ‰ ‰ o o ‰ o ‰ ‰ o o o o o o ‰ o ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o o ‰ o ‰ o o o ‰ o o ‰ o ‰ ‰ o ‰ o ‰ o ‰ ‰ o o o o ‰ o o ‰ o o o ‰ o ‰ o ‰ ‰ o ‰ o o ‰ o o ‰ o ‰ ‰ o ‰ o ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o ‰ ‰ o o o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ o o ‰ ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ o o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ ‰ ‰ o o ‰ o o ‰ ‰ o ‰ o o ‰ ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ o o ‰ o o o o ‰ o o ‰ ‰ ‰ ‰ o o o o o ‰ ‰ o ‰ ‰ o ‰ ‰ o o ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ o o o o ‰ ‰ ‰ o o ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ o o ‰ o ‰ o ‰ ‰ o o ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ o o o ‰ ‰ o o o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o o ‰ o ‰ o o ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ o ‰ o o ‰ ‰ ‰ o ‰ ‰ ‰ o o o o ‰ ‰ o ‰ o o ‰ ‰ o o o o o ‰ ‰ ‰ ‰ ‰ o ‰ o o ‰ ‰ o o ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ o o ‰ o o ‰ ‰ o ‰ ‰ ‰ o o o o o o o o ‰ o o ‰ o o ‰ o o ‰ o o ‰ o ‰ ‰ ‰ o o o o ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o o ‰ o ‰ o ‰ o ‰ o o ‰ ‰ ‰ ‰ o o o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ o o o ‰ ‰ o ‰ o ‰ o o ‰ ‰ o o o o ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ ‰ o ‰ o o o o ‰ o o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ o o o ‰ ‰ o ‰ o ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ o o o ‰ o o ‰ o o o o o ‰ o ‰ ‰ o o ‰ o o ‰ o o o ‰ o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o o o ‰ o o ‰ o o ‰ o ‰ o ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ o o ‰ o ‰ o ‰ o o ‰ o ‰ o o ‰ o ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ o o o o ‰ o o o ‰ ‰ o o o ‰ ‰ ‰ o ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o o ‰ ‰ ‰ o o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ o o o ‰ ‰ o ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ o o o o o ‰ ‰ o o ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ o o ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ o ‰ ‰ o o o o ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ o ‰ o ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ o ‰ ‰ o ‰ ‰ o ‰ o o o ‰ ‰ o o o o ‰ ‰ o o ‰ o ‰ ‰ o ‰ ‰ ‰ o o ‰ o o ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ o o o ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ o o ‰ o ‰ o o o o ‰ ‰ ‰ ‰ o o o o o o ‰ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

0.0 0 5

/m 30 25 20 15 10

tan 2 Z Β on the UV parameters . . The results are shown in figure m β correlation. The domain of the scatter-plot is the β = β GUT η – 12 – 1.0 M 1 4 , relaxing the perturbativity bound on -tan = = 1 G G 4 and tan n n and and tan follows simply from the fact that their product depends : : o ‰ , populates the region at larger values of tan µ 1 0.8 µ µ/M M ‰ . On the red (dark) shaded regions the coupling for a ratio ‰ 1 GUT β µ/M and M β ‰ 0.6 1 , its moderately large value is natural, as it does not require any and both limits are approximately inversely proportional to log M u  Μ ‰ λ ‰ ‰ ‰ G ‰ Λ ‰ ‰ ‰ ‰ 0.4 o / ‰ GeV. ‰ o ‰ ‰ . On the left we require perturbativity up to Λ = ‰ ‰ ‰ and ‰ D ‰ ‰ 9 ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ 2 o d ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ o ‰ ‰ ‰ λ o o o o ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o o ‰ o ‰ o ‰ o ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ o o o ‰ ‰ o o ‰ o o ‰ ‰ o o o ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ o o ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ o o ‰ o ‰ o ‰ ‰ ‰ o ‰ o o ‰ o ‰ o ‰ o ‰ o o ‰ ‰ ‰ 0.2 ‰ o o o o o ‰ o ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ o o ‰ o o o o ‰ o ‰ o o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ o o o ‰ ‰ o ‰ ‰ o o ‰ ‰ o o o ‰ ‰ o o ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o o ‰ o ‰ o o ‰ o ‰ ‰ ‰ ‰ o o o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o o ‰ ‰ o o ‰ o o ‰ o ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o o ‰ ‰ ‰ o ‰ o ‰ o o ‰ o ‰ o o o ‰ o o ‰ o ‰ ‰ o o o ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ o o o o o o o o o ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ o ‰ o o o o o ‰ o o ‰ ‰ o o ‰ ‰ ‰ o ‰ ‰ o ‰ o o o o ‰ o ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ o o o ‰ o o o o o o o ‰ o o o o o o o ‰ o ‰ o ‰ o ‰ o ‰ o ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ o o o o ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o o ‰ o o o ‰ o . Scatter plot showing the o o . Allowed values of tan ‰ o ‰ o o ‰ o o ‰ o o o ‰ ‰ o o ‰ ‰ o o o o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o o o ‰ ‰ ‰ ‰ o o o o o o o ‰ o ‰ ‰ ‰ ‰ o o o ‰ o o o ‰ o o ‰ ‰ o o o ‰ ‰ ‰ o ‰ ‰ o ‰ o ‰ ‰ o o o o ‰ o ‰ o o o ‰ ‰ ‰ o o o o o ‰ o ‰ o ‰ o o o ‰ o o o o o ‰ ‰ ‰ ‰ o ‰ o ‰ o o o o ‰ ‰ o o ‰ o o ‰ ‰ o o ‰ ‰ o ‰ ‰ o o o o ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ o o ‰ o ‰ o o o o o o o o ‰ o o ‰ o ‰ o ‰ ‰ o ‰ ‰ o o ‰ o ‰ o o ‰ o o o o ‰ ‰ ‰ ‰ o o o ‰ o o o o o o ‰ ‰ o ‰ ‰ ‰ o o ‰ o o ‰ o ‰ o ‰ ‰ o ‰ o ‰ o o o o ‰ ‰ o ‰ o o ‰ o o o ‰ ‰ o o ‰ o ‰ ‰ ‰ o o o ‰ o o ‰ ‰ o ‰ ‰ o ‰ o o o o ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ o o o o ‰ o o o o o ‰ ‰ o ‰ ‰ o o o ‰ o o o o o o o o o ‰ ‰ ‰ o o ‰ o o o o o ‰ o o o o o o o ‰ o o o o o ‰ o o o o o o ‰ o ‰ o o ‰ o ‰ ‰ ‰ o o o ‰ o o ‰ o o o o o ‰ ‰ ‰ ‰ o o ‰ ‰ o o o ‰ o o o ‰ o ‰ o o o o o o o ‰ o o o o ‰ o ‰ ‰ o o ‰ ‰ o o ‰ o o o o o o ‰ o ‰ o ‰ ‰ o o o ‰ o o o o ‰ o o o ‰ o o ‰ ‰ o o ‰ o o ‰ o o ‰ o ‰ ‰ ‰ o o ‰ o ‰ o o o o ‰ ‰ ‰ ‰ o ‰ o ‰ o o o ‰ o ‰ o ‰ o o o o o o o ‰ o o o o o o o ‰ ‰ ‰ o ‰ o ‰ ‰ o o ‰ o o o ‰ o ‰ o o o o o ‰ ‰ o ‰ ‰ o o o o ‰ ‰ o ‰ ‰ o o o o o o o o o ‰ ‰ o o o ‰ o o ‰ ‰ o ‰ ‰ o o ‰ ‰ o o o ‰ o o o o ‰ ‰ ‰ o o o o o o o o o o o o o o ‰ o o ‰ ‰ o o o ‰ o ‰ ‰ ‰ ‰ o o o o ‰ ‰ o o o ‰ o o o o ‰ o o o o ‰ o ‰ o o ‰ ‰ o ‰ ‰ ‰ o o o o o ‰ ‰ o ‰ o ‰ o ‰ ‰ o ‰ ‰ ‰ o o o ‰ ‰ o ‰ ‰ ‰ o o ‰ o ‰ ‰ ‰ o o o o ‰ o ‰ ‰ o ‰ ‰ ‰ o ‰ o o o o ‰ o o o o ‰ o o o o ‰ o ‰ o o o ‰ ‰ o o ‰ ‰ o o ‰ ‰ o ‰ o ‰ o ‰ o o o o ‰ ‰ o o o o o o ‰ ‰ o o ‰ o o ‰ o o o ‰ o o o o o o o o o o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ o o ‰ ‰ o ‰ o ‰ o o o o ‰ ‰ o o o ‰ o o ‰ o ‰ ‰ ‰ o ‰ o ‰ o o o ‰ o o o o o o ‰ o o ‰ o o o ‰ o ‰ o ‰ ‰ o o ‰ o ‰ ‰ o o ‰ o o ‰ ‰ ‰ o ‰ ‰ ‰ o o o o o o o o ‰ ‰ o o ‰ o o ‰ ‰ o ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ o o o o ‰ o ‰ ‰ o o o ‰ ‰ o ‰ ‰ o o o o ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ o o ‰ o o ‰ o o ‰ o o ‰ ‰ o o ‰ ‰ ‰ o o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o o o o o o ‰ ‰ o o ‰ o ‰ o o o o o o o ‰ ‰ o o o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o o ‰ o o ‰ o o o ‰ o ‰ o ‰ ‰ o ‰ o o ‰ o o o o ‰ o ‰ ‰ o o ‰ ‰ o o o ‰ ‰ ‰ o o ‰ ‰ o ‰ ‰ ‰ o ‰ o ‰ o ‰ ‰ o o ‰ o o o o ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ o o ‰ o ‰ o o ‰ ‰ ‰ o o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ o o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ o ‰ ‰ o ‰ o o ‰ o ‰ o o o o o o ‰ ‰ o ‰ o o o o o ‰ o ‰ o ‰ o o o o o o ‰ o ‰ ‰ ‰ ‰ o o o ‰ ‰ o o o o o ‰ o o o ‰ o o ‰ o o ‰ o o o ‰ o ‰ o o o ‰ ‰ ‰ o o o o o o ‰ ‰ o o o o o o o o o o ‰ ‰ o ‰ o ‰ o ‰ o o o o ‰ o o ‰ o o ‰ ‰ ‰ ‰ o o ‰ o o ‰ o ‰ ‰ o o ‰ ‰ o o ‰ o o o o o o o o o o o ‰ ‰ o ‰ o ‰ ‰ ‰ o o o o o o ‰ ‰ o ‰ ‰ ‰ ‰ o o o ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ o o ‰ ‰ o ‰ o o ‰ o o o o ‰ ‰ o o ‰ ‰ ‰ o o o o o o o ‰ ‰ o o o o ‰ o o o ‰ o o o o o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ ‰ o ‰ ‰ ‰ o o o ‰ o ‰ ‰ o ‰ o o ‰ ‰ o ‰ o o o ‰ ‰ o ‰ ‰ o o o ‰ ‰ ‰ o ‰ o ‰ o o ‰ o ‰ ‰ ‰ ‰ o o ‰ ‰ o o ‰ ‰ ‰ o o ‰ ‰ ‰ o o o o o o ‰ ‰ o o ‰ o o ‰ o o ‰ ‰ ‰ o o ‰ o o ‰ o ‰ o o o o o o ‰ ‰ o ‰ ‰ o o ‰ o o ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ o o o o o o ‰ ‰ o ‰ o o ‰ o o o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ o ‰ o o o o o o o o ‰ o o o o ‰ o o ‰ ‰ o ‰ o ‰ o o o o o o o o ‰ ‰ ‰ o o o ‰ ‰ ‰ o o ‰ o o ‰ ‰ o o o o o o ‰ ‰ ‰ ‰ ‰ o o o ‰ o ‰ o ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ o o ‰ o ‰ o o o o o ‰ ‰ ‰ ‰ ‰ o ‰ o o ‰ o ‰ ‰ ‰ o o ‰ ‰ ‰ o ‰ o o ‰ o o ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ o o o o o o ‰ ‰ o o ‰ o ‰ ‰ ‰ ‰ ‰ o o o o ‰ ‰ o ‰ ‰ o ‰ o o ‰ ‰ ‰ ‰ o ‰ o ‰ o o o ‰ o o o ‰ o o o o o o ‰ ‰ o ‰ ‰ o ‰ ‰ o o o o o o o ‰ ‰ o o ‰ ‰ ‰ ‰ o o o o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o o o o ‰ ‰ ‰ o ‰ o o ‰ o ‰ ‰ o o o o ‰ o o o ‰ ‰ o ‰ o ‰ o ‰ ‰ o o ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o o o o ‰ o o ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ o o o o ‰ o ‰ ‰ ‰ o ‰ o o ‰ o o ‰ ‰ o ‰ ‰ o o o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o o ‰ o o ‰ o o ‰ o o o o ‰ ‰ ‰ ‰ o ‰ o o ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ o ‰ ‰ o o o o ‰ o o ‰ ‰ ‰ ‰ o o ‰ ‰ o o ‰ ‰ o ‰ o o ‰ o ‰ o o ‰ ‰ o ‰ o o ‰ ‰ ‰ o o o o ‰ ‰ ‰ ‰ o o ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o o o o ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o o o ‰ o o ‰ ‰ ‰ o o o ‰ ‰ o o ‰ o o o ‰ o o ‰ o o o o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ ‰ o o ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ o ‰ o ‰ o o ‰ o ‰ ‰ o o o ‰ o o ‰ ‰ o o o ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o o ‰ ‰ o ‰ o ‰ o o o o o o ‰ ‰ o ‰ o o o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o o ‰ o o o ‰ ‰ o o ‰ o ‰ ‰ o o ‰ o ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ o ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ o o ‰ ‰ o ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o o o ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ o o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ o o o ‰ ‰ o ‰ ‰ o ‰ ‰ o ‰ o o ‰ ‰ ‰ o o ‰ o o o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ o ‰ ‰ o o o ‰ o ‰ ‰ o ‰ ‰ o o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o o o o ‰ o o ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ o ‰ ‰ o o ‰ o ‰ o ‰ o ‰ ‰ o o o ‰ o o ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ o o ‰ ‰ o o o o o ‰ o o o ‰ o ‰ o o o ‰ ‰ ‰ ‰ o ‰ o o o o o ‰ ‰ o ‰ ‰ ‰ o o o o o ‰ ‰ o o o ‰ o ‰ ‰ ‰ o ‰ ‰ o o ‰ ‰ ‰ ‰ o ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o o o o ‰ o ‰ ‰ o ‰ ‰ o ‰ o o ‰ o o ‰ ‰ ‰ o o ‰ o ‰ ‰ ‰ ‰ o o ‰ ‰ o o o ‰ o ‰ o ‰ ‰ o ‰ ‰ o ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ o o ‰ o ‰ o o o o ‰ ‰ o o ‰ o o ‰ ‰ ‰ o o ‰ ‰ ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o o ‰ ‰ o ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ o ‰ ‰ o ‰ ‰ o o o o ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ o o ‰ o o o o ‰ ‰ o o o o ‰ ‰ ‰ o o o o o o o o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ o ‰ ‰ o o o ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ o o ‰ o o ‰ ‰ o o ‰ ‰ ‰ ‰ o ‰ o o o ‰ o o o o ‰ o o ‰ o ‰ o ‰ ‰ o o o o o ‰ o o ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ o o ‰ o o o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o o ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o o o ‰ ‰ ‰ o o o o ‰ o o ‰ ‰ ‰ ‰ o o o o ‰ o o ‰ ‰ o o o ‰ ‰ ‰ ‰ o ‰ o o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ o o o o o o ‰ o ‰ o o ‰ ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ o o ‰ ‰ o o ‰ o o o o ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ o o o ‰ ‰ ‰ ‰ ‰ o o o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o ‰ o ‰ o o ‰ o o o o o o ‰ o ‰ ‰ ‰ o o o o ‰ o o ‰ o ‰ o o ‰ o o ‰ o o o ‰ ‰ ‰ o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ ‰ o o o ‰ ‰ o o ‰ o o ‰ o ‰ o ‰ ‰ ‰ ‰ o o ‰ o ‰ ‰ ‰ o ‰ o o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o o ‰ o ‰ ‰ o ‰ o ‰ o ‰ o ‰ o ‰ ‰ o o ‰ o o o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o o o o o o o o ‰ o ‰ ‰ o ‰ o o o ‰ ‰ o o ‰ ‰ ‰ o o o ‰ ‰ ‰ o ‰ o ‰ ‰ o o o ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ o ‰ o o ‰ ‰ ‰ o o ‰ o ‰ ‰ ‰ o ‰ o o o o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ o o ‰ o ‰ ‰ ‰ o o o o ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o o ‰ o o ‰ ‰ ‰ o o ‰ ‰ ‰ o o o o ‰ o o o o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o o o ‰ o ‰ ‰ o ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ ‰ o o o ‰ ‰ o o o o ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o o o ‰ o ‰ ‰ ‰ o o ‰ ‰ ‰ o ‰ ‰ ‰ o ‰ ‰ o ‰ o o ‰ ‰ o ‰ ‰ o ‰ o o ‰ ‰ o ‰ ‰ ‰ ‰ o o o o o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ o o ‰ ‰ o o ‰ o ‰ ‰ o ‰ ‰ o ‰ ‰ ‰ ‰ o o o ‰ o ‰ ‰ ‰ ‰ ‰ o ‰ ‰ o ‰ o ‰ o ‰ ‰ ‰ ‰ o ‰ ‰ o o ‰ o o o o ‰ ‰ ‰ o o o o ‰ ‰ o o o ‰ ‰ ‰ ‰ ‰ o o o ‰ ‰ o ‰ o o o ‰ ‰ ‰ ‰ o o o ‰ o ‰ o ‰ ‰ ‰ o ‰ ‰ ‰ ‰ o o o o o ‰ ‰ o o ‰ o ‰ ‰ ‰ ‰ ‰ ‰ ‰ o ‰ o o ‰ ‰ o ‰ o ‰ o ‰ o ‰ ‰ o o o o ‰ ‰ o ‰ ‰ o o o o o o o o ‰ o ‰ o ‰ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o Since the dependence of both

0.0 0 5

10 30 25 20 15

tan Β for each point,anticorrelation the between tan value of on a combination of theobserved variables in which the is bounded rightimposing on panel a cutoff the of lower domain than figure of the scatter plot. As We stress that since intwo lopsided Yukawas gauge mediation tan additional tuning or cancellation among different contributions. out to be quite involved, we perform a scanning of such parameter space and calculate, Figure 4 same as in figure cutoff is at Λ = 10 hand, scales as Λ Figure 3 choices of parameters asperturbative in at figure some point between JHEP05(2011)112 : µ . µ µ G for B n 1 , ). As β 4.2 µ/M determines µ such that the ˜ 2 t 1.0 /m 2 Z m = as a function of η becomes independent of 0.8 1 η η = G √ n , , as in ordinary gauge mediation. 3 = Β M

GeV and various choices of tan 1 ,

9

tan =

G G

0.6 n

n , 1 ,

= 10 10 M

G

 =

Μ Β approaches 1, M

is fixed once and for all when the parameters tan 1 1 the mass of the lightest Higgs boson depends is then fixing the overall normalization of the

is fixed by the bound on the Higgs mass. Moving 4

– 13 – 4 0.4 and light higgsinos and sleptons.

η = µ/M

G

µ/M ). G

d

n 1

, H 10

m are simultaneously satisfied. = Β µ/M h

we reach a point where the LEP bound on tan m . When 1 0.2 β and µ/M µ ) are chosen. Λ as defined in the text for 1 2 is fixed there is thus a maximal value of - - and tan η , the latter being fixed by Λ 1 ˜

t 10 10

√ G 3.15 Η m n µ/M we display the behaviour of this maximal and 115 GeV. This is a well-known source of tuning for supersymmetric theories 5 β > . Value of h m The goal of lopsided gauge mediation is fully reached only if the lower bound on In figure LEP constrains the higgsino mass term to be bigger than about 100 GeV. Although A-term contributions to the stop masses, coming from the new couplings of the higgses to the messen- 4 , and the curve starts to fall as ( Figure 5 η does not introduce additional fine-tuning with respect to the onegers, required are to always lift irrelevant. the Higgs experimental bounds on definite values of we are in the ordinary situationtowards in smaller which values of shown in the previous section,appearing the ratio in eq.spectrum. ( To a veryonly on good tan approximation Once the ratio typically found in models wherethe all possibly the simplest soft implementation terms of lopsided arephenomenological gauge calculable. features: mediation In and a we section large observed 3 its we main presented this is never an issue inof ordinary gauge tuning mediation, in it can the become setup the most we important are source describing, as we already pointed out below eq. ( Consistency with experimental databound requires the Higgs bosonin mass general to and lie forintroduction, above lopsided those the gauge LEP mediation employing makes gauge nois exception mediation to to in this. hide particular. What behind it this does, As unavoidable however, explained tuning the in one the needed to accommodate the large 5 Higgs mass and fine-tuning JHEP05(2011)112 = 10 β , where ) is then , besides A η proportional GUT < m h M m 3 and tan . is not particularly 0 SUSY and 1 m . Moreover it should ≈ Z 1 automatically imply a 1 m µ/M 2 A is smaller than the upper µ/M m µ/M (several TeV) together with d λ d at H η shows the contours for the rela- 6 generically requires an extra tuning of β , lopsided gauge mediation requires no fur- . Figure , the overall tuning decreases, although lop- β Z β to the value required by the Higgs mass is . m is on the value of the light Higgs boson mass. η 5 – 14 – ) with respect to the experimental measurement A Z doublet is integrated out, and m m d ( 3 , the sfermion masses, and H µ α ]. Note that a heavy pseudoscalar gives a negative con- 33 . The request that the tuning is dominated by the Higgs ). Fixing β 4.2 006) [ . 0 , where the A ± m (1 ) which tends to spoil unification. On the other hand an improvement Z × m ( without additional tuning. This shows that lopsided gauge mediation can in the case where all soft terms are comparable, in lopsided gauge mediation 3 ]. Notice that it is consistent to single out this effect among others which are 1184 α shows that, at small values of tan shows that, for a large fraction of the points, the ratio β β . 5 4 34 tan / ) = 0 Z Another possible effect of a large Let us consider, for concreteness, only two thresholds between Figure Figure m ( 3 The effect is analogous,and but smaller, can to be the treatedpotential one in [ obtained the by integrating same outnumerically way, of the using the two the stops same renormalization order, group as it improved represent effective the leading contribution to These latter particles are assumedtive variation to of sit the at predictionα for tribution to is obtained by splitting theof higgsinos the and supersymmetric the spectrum. left-handed sleptons with respect to the rest typical supersymmetric spectrum.threshold effects The are relevant given in equations appendix needed B. to computethe messenger the scale: new all other superpartners except the bino, the higgsinos and the left-handed sleptons live. 6 Other consequences ofThe a special heavy spectrum pseudoscalar we arelight dealing higgsinos with — and apeculiar left-handed very pattern heavy sleptons of (around deviations 100 GeV) in — the running is of expected gauge to couplings imprint with a respect to a more the structural hierarchies between sizeable tan give spectra with tunings comparablefeatures to are the quite ordinary distinct. case, although its phenomenological sided gauge mediation can makethat the the situation effect is slightly rather worse.does modest. However, not it For significantly instance, should differ the be frombe value noted the of remarked plateau that, at while largerorder a values 1 moderate of value of tan not always sufficient tobound guarantee from that perturbativity. the A further lowerrequired. increase bound in on tuning (obtained by decreasing ther tuning than ordinary gaugetuning mediation. is In rather this severe. case, For however, large the values of overall degree tan of fine (or nearly flat) part of the curves in figure large, especially for largermass tan constraint severely restricts the allowedcan points. be Another way extracted to describe from the situation eq. ( boson mass above the LEP experimental limit. This happens for models living on the flat JHEP05(2011)112 . is A (6.1) (6.2) (6.3) which < m SUSY ) at the A m m 6.3 SUSY m and is identified with β . β . H 2 2 SUSY cos m 006. . 2 0 . Here supersymmetry fixes g A = 0 8 + m 2 in terms of two fields exp g | 3 δλ u,d /α turns out to be always smaller than H ) = 3 βH , βH , ) + A h δα A | m sin m threshold as explained in the text. In green . We can thus calculate the RG improved ( m . For all the values of tan ( − + cos A λ λ m 2 βh SUSY v βh , where – 15 – ) = , λ 3 /m 4 = 4 | A /α we show the value of the Higgs mass at the weak h 2 h 3 = cos = sin | m SUSY 7 gets a vacuum expectation value. λ m using the new boundary condition in eq. ( ∗ u δα d m h ( + H Z log H λ 2 2 | M ) h ) including the π | 4 Z 2 / m m 2 ( g 3 − α ) = , so that only h d ( ) allowed range for /v V σ u v (2 σ = , running it down to the weak scale using the SM RGE equations. After this β is of the order ( is not present in the effective theory the quartic coupling at the scale . Prediction of SUSY δλ . It is convenient to redefine the two doublets H m A m scale we obtain the Higgs bosonare mass relevant as for our1 GeV model and this thus contribution negligible.scale to assuming In all figure the superpartners to be degenerate at a scale not fixed to its supersymmetric form but has an extra contribution where effective potential at the scale We then assume allSince the other superpartners to be degenerate at the scale where tan the heavy Higgs doublet andthe is boundary integrated condition out for at the the Higgs scale boson quartic to Figure 6 (yellow) the 1 JHEP05(2011)112 that does d λ whose lightness 5 of lopsided gauge 4 = 10 (full), assuming µ β 10 terms are never relevant enough τ A , or a sneutrino 1 value. L d = 1 (dashed), tan µ/M H β . 3 GeV m H 10 SUSY m SUSY -terms of 10–20 GeV. – 16 – m D , so that the NLSP is always the sneutrino. The three sneutrino τ ν and ˜ τ 2 10 very close to the extreme values allowed by the experimental limits 40 20 80 60

120 100

h D m GeV L H Λ , G , n d λ . Higgs boson mass at the weak scale for tan After its production any supersymmetric state decays promptly until the NLSP state The generic spectrum of lopsided gauge mediation have squarks heavier than 1.5 TeV The identity of the NLSP depends on the parameters choice. In the majority of the The slepton doublet is splitted by the SU(2) 5 of the mass of the colored states. to invert the mass hierarchy between ˜ flavors can be considered degenerate for all practical purposes. the production of colored statesand at the negligible LHC with cross-section 14electroweak TeV at sparticles of in center the the of cascade LHC mass of energyhand, the with (LHC14) the colored 7 generic ones TeV is lightness (LHC7). not ofdirect very the abundant. production higgsinos Thus, On in and the the the Drell-Yan other processes sleptons, production can mediated give of by sizable electroweak rates bosons, for the independently on the sleptons. Wenot also remain notice perturbative that up typically thesome this GUT other corresponds scale, new to thus physics a a happening sneutrino value well NLSP of below might the be GUT suggestiveand scale. of gluinos typically heavier than 2 TeV, which results in only a few fb of cross-section for is the result of the FI term induced byparameter the large space we findmediation. a Indeed to higgsino have NLSP, aparameters sneutrino as lighter a than the result Higgsino of one has the to small pick a choice of In gauge mediationHowever, the in lightest contrast supersymmetric with thepossibilities particle for ordinary (LSP) the case, next-to-lightest is supersymmetric in particleor always lopsided (NLSP) less the gauge are mixed the mediation gravitino. with higgsino, the more the two bino typical depending on the ratio Figure 7 all the superpartners to degenerate at the scale 7 Collider phenomenology JHEP05(2011)112 6 or (1). (7.5) (7.2) (7.3) (7.1) O . = k kF < M if √ 2 TeV TeV the NLSP F & 3 ˜ g √ . 0 1 ; (7.4) νχ , m ± ∓ ` ` 0 3 (which can be larger than ± , χ → 4 5 TeV  m . ± NL 2 1 ν , χ smaller than 100 TeV give rise 2 X , ' & 0 1 F F ˜ q 0 4 χ √ χ √ , χ 100 TeV → 0 1  ˜ ν 5 , m ¯ νχ  0 3 ν χ , m → ν , 0 2 is larger than a few times 10 < m NLSP ± – 17 – L m F l ˜ 100 GeV , χ √ m 400 GeV  2 , , χ 0 1 . > cm 0 1 χ 0 3 ˜ 100 GeV, values of ν 2 χ ± χ ` − ∓ i ≈ ` 10 10 TeV. The absence of tachyonic messengers, ± i → ` , m ≈ ± 0 3 : ˜ ` NLSP χ → 0 1 L χ 0 2 m χ , m < m , thus implies a lower bound of roughly 10 TeV on kF/M > 0 2 χ = m < M G . G 150 GeV ± . χ m µ is the scale appearing in the goldstino superfield . ' F final states with a light chargino and neutralinodecay of states, the sleptons, either decaying, through directly off-shell vector produced bosons or or sleptons, in coming leptonic from the charged and neutral sleptonsone promptly chargino decaying or in neutralino: final states with one lepton and 0 1 0 1 = 1 will be assumed in the following. χ χ k • • 6 m m ored objects, the characteristicneutralinos signals and arise sleptons and from their the decay Drell-Yan to production final of states charginos, with many leptons: Disregarding the signals from the production of electroweak states in the cascades of col- bottom of the spectrum,state parametrically the heavier two than neutralslepton the and doublets higgsinos the are and charged lighterstate. a higgsinos, than The yet the a typical heavier singlets neutral spectrum wino-like and in bino-like triplet. are the less typically The tuned lighter region than of the the bino-like model looks as follows: supersymmetry breaking scale the decay occurs via a displaced vertex. 7.1 Higgsino NLSP spectrum The typical spectrum with a higgsino NLSP is characterized by a triplet of states at the the mass splittingmass among requires the Λ messengerequivalently fields). Λ TheIn lower bound this range on andto the with prompt Higgs NLSP boson decays,decay while takes when place most of the time outside the detector. For intermediate values of the where is reached. The NLSP decay length is then determined by JHEP05(2011)112 ) (7.6) (7.7) (7.8) (7.9) 7.13 (7.12) (7.13) (7.10) (7.11) s as in the ) and ( W 10 GeV which ). Moreover, a 7.12 ∼ 0 1 7.8 χ m − ) or ( 0 2 ν , χ 0 1 , 7.6 ` χ m ν 2 0 1 ` ν , ∼ χ 5 2 T 0 1 ±∗ p χ 2 W ν , pair in case they are higgsino-like or ` ∓ i 4 0 1 ` , ν , χ 2 or 2 0 2 ν , , . ∓∗ → 0 1 χ 0 1 ν ν , 0 1 0 1 l χ 2 χ 0 1 ν W χ χ 2 2 0 1 0 1 2 χ 2 ± ` ` 2 χ χ ` ` 2 χ 3 2 6 ` 2 ± ∓ i from the decay chains of the colored sparticles, – 18 – ` 3 ` ` → 4 → 0 2 → → χ → → 0 2 ∓ → 0 1 ∓ → χ χ ], lopsided gauge mediation does not benefit from a ˜ ˜ ν ` χ ∓ 1 ± i ± ± 0 2 ˜ ν ˜ ± ± χ ` 43 ˜ ˜ ˜ ν χ χ χ ` ` – ± ∓ i ` ` 39 can be used to determine the higgsino-like nature of both → → → → → → → → ¯ ¯ ¯ ¯ ¯ ¯ 0 2 p p p p p p p p p p p p χ 4 ˜ ν , 0 3 channel which has a negligible production cross section for a or or or or or or χ 0 2 pp pp pp pp pp pp χ 0 2 χ ]. This allows, in principle, for testing the prediction of lopsided gauge 43 . Focusing on events with at least three leptons, at the LHC7 we ex- (10 fb) of cross section, for sleptons and higgsino masses not excluded by – 0 2 O χ 39 , as the invariant mass distribution of the di-lepton system from the decay of 0 1 χ is sensitive to the composition of the two neutralinos. This is due to CP invariance 2 0 A marked difference between lopsided gauge mediation and ordinary gauge mediation is The decay products of χ and 0 2 lead to both opposite-sign same-flavourfinal (OSSF) states. and The opposite-sign invariant opposite-flavor massare (OSOF) distribution characteristic of these of OS their di-leptonsSM displays production processes features process. which butfeatureless also Backgrounds OS from are and SS other expected di-lepton invariant supersymmetric not mass distribution only production with from similar mechanisms, shapes. which Therefore, yield following processes: which are flavor universal sources of charged leptons. The processes eq. ( the presence of doublet sleptons whichcan are in lighter than principle the singlet becrucial ones. discovered feature The through light for sneutrino the distinguishingthe processes the relevance of left-handed in charged sleptons eqs. currents from ( in the the interactions. right-handed These ones, leads is to off-shell and therefore the leptons relevantmakes for them this detectable analysis but have LHC rather to soft. measure A the quantitativebe details assessment needed of of before the the concluding invariant potentialgauge mass that of mediation distribution the can the of higgsino be such nature tested soft at of leptons the the would LHC. light neutralinos of lopsided the which requires different intrinsic paritiesgaugino-like for [ the mediation of a light higgsino-likefrom neutralino. the However, case we have studiedcopious to in remark source refs. that, of differently [ moderately boosted pect at most LEP. Therefore the observability of suchthe signals LHC. seems challenging in the 2011–2012 run of χ We neglected the higgsino-like Altogether the resulting lepton-rich final states are: JHEP05(2011)112 ] , is 7 T ). / 0 1 51 E , χ (7.14) (7.15) (7.16) using a + 7.9 50 is several − . e γγ 0 1 ∗ µ − χ µ B , + ˜ λ µ  ). The searches 2 µM 5 χ and 7.9 ] and CDF [ πF m − 49 µ 16 −  µ . In our case we expect , 2 , ]. The production cross- + µ  e   , 1 Z . 2 2 51 − , , the NLSP. In the following 1 M m 5 χ µ 0 1 5 χ ) result in this case in multi- 50 πF − χ πF  M m m µ . 16 + 16 W of data with 0.4 expected. Also in this 1 θ µ 7.11 1  −  2 4 − 4 100 GeV. We believe, however, that   )–( sin 2 χ 2 h 2 Z 2 χ ' 7.6 m m W m m µ θ ] and CDF [ 2 − − 49 1 1 cos 15 GeV. While this requirement is typically   2 2 2 ) ) ) – 19 – & β β β T . As such their results cannot be translated into a limit for p cos cos cos ± 1    χ + + − β β β ) one can subtract the OSSF and SSSF di-lepton invariant and the (sin (sin (sin ) is in principle a source of tri-lepton events likely to pass the 0 2 1 2 1 4 1 4 7.13 χ limits the hardness of the leptons from the process in eq. ( 7.6 ± ) = ) = ) = e e e , χ G G G 0 1 ) and ( γ h Z , χ 0 2 ] performed a similar analysis on the final states → → ) in lopsided gauge mediation for → as low as 5 GeV and observed 1 event in 0.96 fb 7.12 χ 52 χ χ T χ 7.9 parameter is the sign of the rephasing invariant combination p Γ( Γ( Γ(  The minimal gauge mediation scenario motivated a lot of effort on the signal A meta-stable neutralino can leave displaced vertex signatures, leading to striking The process in eq. ( In the first case, typical of high-scale models of supersymmetry breaking, the All signatures discussed so far do not rely on the fate of The search [ 7 which is the dominant channel for a bino-like NLSP, i.e. for thresholds for case the limits aretens given of only for GeV a lighterour spectrum than case. that the comes from the mSUGRA model in which where the case of a higgsino NLSP, the relevant decay widths are section for such process isthe however final significantly lower TeVatron’s integrated than luminosity, the around bound 10 attainable fb even with signals of low-scale supersymmetryare breaking. still usable, While the the displaced leptonic vertex signature signals is described an above additional characteristic signal. In the must have at leastfulfilled by one the lepton mSUGRA with degeneracy signal, of this is not the case for lopsidedselection cuts gauge employed mediation in the as searches the at D0 [ searched for new physics in the tri-leptonoptimize channel the from the cuts process for in eq.the a ( typical source mSUGRA of spectrum tri-leptonprocess and events. eq. put ( a This boundthis bound at kind around is of 100 search comparable fb is for with not the effective in cross-section our of case because the the events selected in these analyses we shall examine theand three a different promptly cases decaying NLSP. of Along withthe a the current collider-stable discussion limits of NLSP, the from a signatures searches meta-stable we at shall NLSP, present the TeVatron. a massive invisible particle.leptons and The missing transverse signals energy. in The TeVatron experiments eqs. D0 ( [ mass distribution (or equivalentlybackgrounds the is OSOF and expected SSOF), topoints, so are cancel. connected that to the the This contributionto mass leaves differences of discover in a a the the sneutrino distribution chain. NLSP, Similar as whose reasoning discussed features, can in be e.g. the applied following end- section. to isolate eq. ( JHEP05(2011)112 ] 48 ] used . Neu- ]. The T 35 47 / E . boson plus h of integrated Z m ) + 1 T or jj − p Z )( − m µ + µ using the full integrated )( µ − e ] for a high- production at most of about +  0 1 e 53 from a few hundreds to a few pairs. Unfortunately the search [ 2 χ ( , ) but with the inversion of the 1 b ¯ F b → M √ 7.2 is excluded by this search. . Their result is not applicable to our case e G b ¯ m Z bb ¯ 2 b does not put in our case any significant . → 0 → ˜ G 0 1 ∼ ] claims that for decay lengths going from χ ηη + 36 there are further final state tracks originating cτ γ 150 GeV for → – 20 – 0 1 s χ pairs with invariant mass of a few tens of GeV originating → and µ < b ¯ 0 1 b χ µ have been searched at the D0 experiment [ 1000 TeV, we expect that a few signal events will be + e . − , where the gravitino escapes the detector. Ref. [ ] finds that the CDF search [ In lopsided gauge mediation only electroweak particles are e ˜ F G 35 8 √ . + → m . Z 2 . 100 TeV would be excluded. 0 Z/H mm (corresponding to a range in . ' and Higgs bosons to dominate over the photon channel. Displaced 5 → ] considers the displaced decay of a neutralino taking place inside the F 0 1 cτ Z √ χ 36 decay. For LHC7, considering the electroweak production of a meta-stable to 10 100 GeV. Therefore only a very limited portion of the parameter space close 1 Z & − µ invariant mass of interest for the lopsided gauge mediation case is around b ¯ b All supersymmetric particle production will eventually contribute to a pair of sneutri- In the case of a promptly decaying Reference [ D0 searched, in addition, for displaced production of resonant 8 acting always as sources of missing transverse energy. Once produced,considered the only higgsino the states displacedfrom production the of chain decay ofas a the heavier resonance, e.g. higgsino and the sneutrino atthe the bottom. higgsino Although at the a sneutrinosneutrino generic is NLSP typically spectrum point heavier deserves than of special the study parameter due space to its ofnos distinctive in lopsided features. the gauge final state. mediation, These the will decay, either promptly or not, in neutrinos and gravitinos, 7.2 Sneutrino NLSP spectrum For particular choices ofbe the lighter parameters than ofthe the lopsided spectrum higgsino, gauge will resulting mediation be the in very sneutrino a similar can sneutrino to NLSP the spectrum. one in In eq. these ( cases bound. On the contrary ref.missing [ transverse energy can exclude luminosity of TeVatron data. Ifa this result significant is portion confirmed by ofmediation the with the experimental most collaborations natural part of the parameter space of lopsided gauge from the primary vertex.prompt decays The interesting channelsthe for results lopsided of gauge non-dedicatedhiggsino mediation searches NLSP. are from The the photonic TeVatron and decay estimated the limits on a pure roughly 10 thousands TeV) at least aluminosity. few signal Thus, events in should the bemediation observable most with with natural 100 1 TeV fb portionproduced of in the the 2011–2012 parameter run space of of the lopsided LHC. gauge to the LEP experimental bounds on ATLAS detector. The studiedtralino signatures decays are to finala states hadronic containing ahiggsino-like Higgs NLSP boson of can mass be 250 assimilated GeV, to ref. those [ with vertices from the decay bounds strongly depend onaround the 1 lifetime pb of for accessible the at NLSP the TeVatron, and yielding a the1 total tightest cross-section pb for for bound from D0 is the decays into JHEP05(2011)112 (7.24) (7.25) (7.17) (7.18) (7.19) (7.20) (7.21) (7.22) (7.23) ). Another 7.11 ). Differently from . )–( 1 − . This cross-section 7.6 ± 1 7.21 χ . 2 , ∗ 0 1 , , χ ± ± ˜ ` , . W T , 0 1 T / χ E T ˜ ν , ν / E / ˜ E ν . ± T + → ` + / E + ∓∗ + (20) fb, which is significantly less than the ± ` + ` → + O ` − W 3 ` ` − ± 1 . ± ` 4 ∗ ` → ˜ ν . → ˜ ν → → ± 1 , χ → ˜ − 1 νW 9 0 1 χ ∗ − , → χ ∓ 2 ˜ ` χ , ˜ ` ∓ – 21 – 0 2 0 1 + 1 → + ˜ ν , χ ± ˜ pp χ χ χ ` ˜ ` ` W ± , ν ]. The interesting phenomenon resides in the decay → → → → → ± ˜ ¯ ¯ ¯ ¯ , χ ` 2 p p p p , ] using the final luminosity of about 10 fb 45 p p p p ∗ ] are sensitive to the signal in eq. ( , 0 1 ∓ ` 54 χ Z 51 44 0 1 or or or or – → χ 49 2 pp pp pp pp , → 0 1 χ 2 χ and a sneutrino W ]. 51 – 49 The presence of a sneutrino NLSP leads to peculiar flavor and charge correlations in The TeVatron searches [ The production of sneutrino at TeVatron is less than 9 been studied in detailchain in refs. [ sensitivity attainable with the search [ corresponds however to aon higgsino the whose sleptons. massrefs. This lies [ leads very to close soft to leptons the and experimental thusthe limit invalidates multi-lepton the final states. potential These limit originates from from the SU(2) charge of the NLSP and have the case of higgsino NLSPdifference spectrum, between the the higgsino hardness and of thecan the sleptons be leptons which, being sufficiently is in controlled large principlea by to a the bound yield free mass parameter, hard of leptons about that 100 pass fb the for selection. the Therefore production we cross-section have of through the reaction This results in anemission invisible of final QCD staterequires initial which large state can luminosity for radiation. in an principle observation. The be observed resulting thanks cross to section the (few fb) certainly Comparing with thesignals higgsino with NLSP case fewerdifference charged we is leptons notice the that with possibility these respect to processes to pair typically produce those yield the in NLSPs eqs. with ( a non-vanishing cross sectio nal as: The charged sleptons, produced eitheran directly off-shell or from the decay of a higgsino, decay into Altogether the production and decay of higgsinos and sleptons lead to multi-leptons sig- and sub-dominantly via three-body decays decay dominantly to sleptons via two-body decays JHEP05(2011)112 µ B and µ , this may or ] and consid- . We find, in β β 17 ). Unfortunately 7.25 ) looking at the differences ). In this way, a viable Higgs ¯ 7.25 S ) results in a final state with (at S found in models where ¯ S µ S ), while the supersymmetry-breaking 7.25 c . The superpotential couplings of the B ¯ S 3.15 + 2 S, 2 ¯ S ¯ S c . Depending on the value of tan – 22 – 1 + 2 S µ/M S c )( /ϕ † ϕ boson decays leptonically ( ] are not immediately applicable to our spectrum, as the colored soft mass can be achieved. W 45 , d is automatically large compared to the experimental limit, being fixed H , light left-handed sleptons, and moderate to large tan 44 µ d H of the kind ( m ϕ ] consider the relevant backgrounds to multi-lepton events and show how it ). 45 , ) are absent. In that case opposite sign lepton pairs are always of the same flavor, 2.4 , large 44 ] with the addition of two chiral singlets µ 46 7.25 The phenomenology of lopsided gauge mediation presents crucial differences with re- Although we have restricted ourselves to gauge-mediated supersymmetry breaking, the We used an explicit model to analyze the distinctive features of our generic setup: spect to ordinary gaugeexperimental bounds), mediation. large pseudo-scalar It Higgsleft-handed is mass sleptons. (in characterized the We by several have TeV lightof briefly range) models higgsinos discussed and deserving some light (close experimental possible to search. signatures their of this new class ref. [ higgses to the messengers can bemass chosen terms as for in eq. the ( compensator singlets get generated bysector K¨ahlerpotential with couplings a to large the conformal may not require anwhere increase the of value of the fine-tuningby with eq. respect ( to ordinary gauge mediation, same mechanism can be implemented into the the case soft where masses the anomaly-mediated is contribution dominant. In particular, one possibility is to extend the realization of originate at the same loop order. small particular, that the perturbativity constraintstringent on upper the higgs-messenger bounds couplings on puts the rather ratio In this paper weered have a expanded new on class thegauge of mechanism mediation), models first where of proposed a gauge-mediated inexperimental single supersymmetry ref. fine-tuning limit breaking [ is and (dubbed sufficient for both lopsided accomodating for the evading large the Higgs mass to conclude that theNLSP discovery are of within the the flavor reach and of charge the correlations LHC. typical of a sneutrino 8 Conclusions of ref. [ is possible tobetween reveal the the same-sign presence and the ofsubtraction opposite-sign the reveal di-lepton edges process invariant mass which inthe distributions. are studies eq. Their of characteristic ( ref. ofparticles [ the in chain our eq. setup ( are well above the TeV. A dedicated study would thus be needed least) two leptons ofwith opposite the sign ordinary whose situation flavor whereeq. is the ( lightest uncorrelated. slepton This is an isas SU(2) they in singlet arise sharp from and contrast the the decayslepton. chains of of a For heavier a neutralino into sneutrino the NLSP NLSP through spectrum an with intermediate colored particles at the TeV, the authors When the the virtual JHEP05(2011)112 )– 3.6 (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.1) (A.2) .  d 2 ). The mes- | H 2 D u 3.5 Λ X | µH )–( 2 log , 3.4 dθ 2 | Z D 2 X , + | + h.c.)  d D − | d =0 H X 2 2 . | M ¯ u 2 θ | H † u , S = Λ H 2 =0 µ θ H X 2 M |

¯ =0 θ B 2 ud = ¯ θ 2 − u,d Z − log = θ

2 Z =0 2 | θ 2 . 2

¯ θ W + ( ud S M 2 i

µ † Λ = log d Z X 2 u,d H | S B D 2 θ 2 ∂ H

M Z ¯ ¯ θ i θ † ∂ ∂ d X X  ∂ ∂

ud H log – 23 – H i 2 2 log and Z d 2 Tr and after some manipulations we arrive at ∂ | ∂ A 2 2 log Z µ 2 ∂θ ∂θ ¯ θ D ∂ ∂ 2 π i ∂ ∂θ 1 | + − − M X 2 X † | in the K¨ahlerpotential generate contributions to the D u 32 S = = = = − 2 † X − | H D − | M † X 2 µ u µ 2 | | X | u,d = u,d d − | B u,d H † S S 2 H u A 2 generates H H | X | X eff | Z | m d S d K  2 H X H = are integrated out at one loop giving rise to the effective K¨ahler | 2 u ). The superpotential is described by eqs. ( | m 2 d 2 ¯ eff S π H λ π − u,d 3.9 u K 16 2 λ 16 | | λ u )–( ¯ − − D, S, H | 3.6 = = u D, 2 H ] ud u,d m 30 Z is the field-dependent messenger mass matrix and Λ is some UV cutoff scale. We Z − = M ). L 3.13 These expressions are easily evaluated and( lead to the results given in the text in eqs. ( while the mixed term The terms proportional to Higgs masses and the A-terms To fix the sign conventions we will use the lagrangian where we kept only the quadratic terms in the propagating fields and we have defined where then compute the eigenvalues of In this appendix we providesector some in details eqs. of the ( sengers calculation fields of the parameterspotential of [ the Higgs The work of A.D.S. and R.R.contract is 200021-125237. supported by The theScience Swiss work National Foundation of Science under R.F. Foundation contract under and 200021-116372. D.P. is supported byA the Swiss National Effective K¨ahler Acknowledgments JHEP05(2011)112 . GUT (B.1) (A.10) (A.11) (A.12) M vanishes µ , B  2 S /M is -terms at one-loop. 2 D has no scalar VEV, A 0 G M -symmetry to be exact α X R log . 2 S  and ) (B.2) 2 ) , ) µ M 2 and the S † G ( 2 S 2 i D µ α M ; (B.3) X /M B M 2 i XX +1 2 D 2 i ) − + − † 2 M M 2 M D 4 2 S GUT M 2 2 M M M ) GUT log is thus µ ( log 2 S 2 2 M,M S D + ) M ( i π log M ( 2 u,d S g M M 4 b . Furthermore since Z 2 G π − log 1 M † + 4 n 2 − ) + =1 D 2 , of messenger fields in a unified representation D i 2 † X n D – 24 – + . With the same notation we used above π M M XX M ) + M µ 4 3 ( i G ( MSSM 0 and the K¨ahlerpotential is an analytic function 1 B b G − α − 1 M,M α 2 S ( → 4 + D contains all thresholds that do not depend on = f . The formula can be written as are generically non zero in a model where M i = 0 G M Z ( X 1 0 G d i 2 1 = X α S 1 α 2 H M α  M 2 = u,d m | a possible unified contribution to the running, for instance =  i and all the various thresholds Z † 1 X n 0 α G | 2 X α M and | 2 d 2 π GUT u λ π u,d u 2 H 16 λ M and 16 λ | m − − has to be fixed to 2. This means that the K¨ahlerpotential can only be GUT = = X vanishes accidentaly, the soft masses are generated. M ud g u,d -symmetry can be simply understood. In order for the Z = is the Dynkin index of the representation of the messengers. Z R 1 G n M The running gauge couplings at low scale can be written in terms of the unified coupling The fact that With the previous technique it is also straightforward to derive the effective K¨ahler , the grand scale is an arbitrary scale and the G of the gauge group. If this is the case the relation between where µ Finally we absorbedcoming in from the presence, at some scale where We supply here the necessaryof formulas the to calculate strong the coupling corrections constant to due the to low tree-level energy thresholds. value α and, unless B Tree-level threshold corrections to unification a function of theno invariant mass combination can vanisharound in that the point. limit The most general form of The structure of the potential imply the vanishing of both due to an the charge of potential in the model with vanishing JHEP05(2011)112 , at D ]. B 2 (B.4) (B.6) (B.7) α (B.10) > µ > model A ]. and SPIRES m B 189 [ ].  1 U(1) Phys. Rev. α )] Nucl. Phys. × , µ , ( 2 SPIRES X ][ SPIRES SU(2) [ − (1983) 479 Nucl. Phys. × ) , (B.5) µ ) ( 1 µ ( X 3 B 219 SU(3) [ . X functions in the three relevant as a function of (1981) 353 − 13 12 b b b + ) 3 ) Z α hep-ph/9507378 (B.9) . Z ]. 13 12 [ b b 1 M 3) ( (B.8) 1 M ) 7) B 192 2 ( µ − 2 − 3) α ( , Nucl. Phys. , Low-energy supersymmetry 2 α 6 − , 2 / , − X New tools for low-energy dynamical SPIRES + 5 − 1 ) ) [ , , , j − Z ( MSSM 2 5 5 23 21 ]. b / / / (1996) 2658 b b 1 23 21 M ) ( b b − – 25 – Nucl. Phys. Supersymmetric technicolor 1 ) Z ]. , α µ 1 M ( = (13 = (24 = (33 D 53 (  1 ) (1982) 227 SPIRES 1 i I ( MSSM X π II b α 12 b ][ III b 4 ) where only the SM states together with higgsinos and b − b SPIRES = = [ Supersymmetric extension of the = = Geometric hierarchy Supercolor III B 110 ij ) b Dynamical supersymmetry breaking at low-energies (0) 3 1 A phenomenological model of particle physics based on Z Phys. Rev. α 2 ) where the whole MSSM spectrum is propagating, 2 , ]. GUT µ M I 1 ( M (region ]. (0) 3 (1982) 175 α Z log hep-ph/9303230 Phys. Lett. SPIRES − [ , ) where a combination of the two Higgs doublets has been integrated This article is distributed under the terms of the Creative Commons (region [ ) II SPIRES Z A B 113 [ 1 M ( > µ > m 3 α (1982) 96 µ > m (1993) 1277 (region ´ Alvarez-Gaum´e,M. Claudson and M.B. Wise, SUSY 48 supersymmetry breaking supersymmetry Phys. Lett. 207 (1981) 575 m For the specific setup described in the text we give the With these definitions one obtains a prediction for S. Dimopoulos and S. Raby, M. Dine and A.E. Nelson, M. Dine, A.E. Nelson, Y. Nir and Y. Shirman, C.R. Nappi and B.A. Ovrut, L. M. Dine, W. Fischler and M. Srednicki, S. Dimopoulos and S. Raby, M. Dine and W. Fischler, SUSY [6] [7] [8] [4] [5] [1] [2] [3] Open Access. 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