Moduli Stabilization and Supersymmetry Breaking In

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This content was downloaded from IP address 131.215.225.165 on 11/06/2019 at 19:59 JHEP08(2008)102 tributions August 1, 2008 August 18, 2008 August 28, 2008 try , ek Received: Accepted: Published: ectra. The approach sets e gauge singlet field which comparable. We term this demonstrate explicitly that alrmodulus.ähler Turning to oup evolution of the super- rn of soft parameters at low ry breaking at the LHC. ble stabilization mechanisms, butions. These contributions hin phenomenological models identical in most cases to that t stops; in other regions of pa- ation of the KKLT-motivated [email protected] , [email protected] Published by Institute of Physics Publishing for SISSA , Supersymmetry Breaking, Supergravity Models, Supersymme We present a model of supersymmetry breaking in which the con [email protected] Department of Physics, UniversityMadison, of WI Wisconsin, 53706, U.S.A. E-mail: [email protected] Phenomenology. Lisa L. Everett, Ian-Woo Kim, Peter Ouyang and Kathryn M. Zur Abstract: from gravity/modulus, anomaly, andscenario gauge “deflected mediation mirage are mediation,” all mirage which mediation is scenario a to generaliz deflect include the gauge gaugino mediated mass contri energies. unification In scale some and cases, alter thisrameter the results space, patte in the LSP a can gluinocompetitive be LSP a gauge-mediated and well-tempered ligh terms neutralino. canbased We naturally on appear the wit KKLT setupis by responsible addressing for the thethe stabilization masses relation of of between th the the gauge messenger andof fields. anomaly deflected contributions For anomaly is via mediation,TeV despite scale the phenomenology, presence we ofsymmetry analyze the breaking the K terms renormalization and gr the the stage resulting for low studies energy of mass such mixed sp scenarios ofKeywords: supersymmet Moduli stabilization and supersymmetrydeflected breaking mirage in mediation JHEP08(2008)102 5 7 9 1 19 20 27 28 ra 18 is broken in a hidden or N. As the phenomenology hortly face unprecedented lues (vevs). The mediators n of the hierarchy problem, nergy supersymmetry, such mmetry (SUSY) (for recent gest auxiliary field vevs, and at the LHC. d predicted dark matter can- en the mediators and the SM otivated candidate for physics ccurs at a high scale, the low onrenormalizable interactions. um, which in turn is governed ow SUSY is broken is the most set by: (i) the specific mediation by mediator fields, which develop X – 1 – In viable models of supersymmetry breaking, supersymmetry 2.2 Review of2.3 KKLT modulus stabilization Stabililization of the matter modulus 2.1 Theoretical background 4.1 (Deflected) Mirage4.2 unification Examples reviews, see [1, 2]) hasbeyond long the been considered Standard to Model beradiative (SM), the mechanism due best-m for to electroweak its symmetrydidate elegant breaking, (assuming resolutio an a conservedas R-parity). the minimal Theories supersymmetric withexperimental standard low tests model e at (MSSM), the will Largeof s Hadron such Collider theories (LHC) is at dictatedby the CER by soft the supersymmetry superpartner breaking massimportant sector, spectr understanding question h for studies of low energy supersymmetry 1. Introduction and motivation Low energy (electroweak to TeV scale) softly broken supersy 2. Moduli stabilization and supersymmetry breaking 3 secluded sector and is transmitted toauxiliary the observable component sector (F and/or Dgenerically term) couple vacuum to expectation SM va fieldsIn via each loop-suppressed model, the and/or characteristic n spectrummechanism of of soft SUSY masses is breaking, (ii)(iii) which the mediators get dominant the effects lar whichfields. produce the Hence, couplings as betwe supersymmetry breaking itself typically o Contents 1. Introduction and motivation 3. MSSM soft supersymmetry breaking4. terms Renormalization group analysis and low energy mass spect 5. Conclusions A. Anomalous dimensions 13 JHEP08(2008)102 , include problem, µ the anomaly → ∞ P M than one mediation . The gauge-mediated mediation [11]. Since an F term vev purely ivated within the top- ely focus on models in soft masses depends on ′ ere we provide a detailed KKLT-motivated mirage s and anomaly mediated involving new messenger Z ifies the values of the soft sses are generated. Mirage king terms are suppressed M (such as the re recent analysis see [25]). [20]. tivated setup. In particular, ck mass mechanisms. These ratios, e IIB string theory [14]. In se from bulk mediator fields n, and gauge mediation [21] ssified into gravity mediated, cation, in which the gaugino g; the prototype example of igh scale physics, not directly ch include the pure anomaly most-studied supersymmetry the bottom-up approach, this ed models. Gravity mediated ete values in string-motivated rs. rmalization group trajectories, om the Kachru-Kallosh-Linde- 18], including a non-standard necessary ingredients for gauge s to demonstrate explicitly that KKLT setup and demonstrated ) anomaly mediation terms [6], s addressed. o bulk mediation models. – 2 – deflected mirage mediation , which is numerically of the order of a loop factor, such that 2 / 3 /m pl M Phenomenological studies of low energy supersymmetry larg In this spirit, we recently outlined a framework in which the This paper is a companion to our previous paper [21], in that h A complementary approach is to consider models in which more scale pattern of soft masses may tell us something about the h reachable by experiments, which determines these paramete which one mediation mechanism is dominantis (see often e.g. to [3]); solve in a given phenomenological problem of the MSS mediated terms are competitive.and This scalar masses results unify in at miragemediation a scale unifi has much below distinctive wheregaugino the phenomenological soft mass features ma pattern [19] [15 – and reduced low energy fine-tuning mediation scenario was extended tobreaking incorporate mechanisms: three modulus of mediation, the (for anomaly a similar mediatio scenario, seemediated [22]). contributions We can argued naturally in be [21] presentthe that within gauge the the mediated termsterms, are leading indeed to a comparable scenario to we denoted the as modulu terms deflect the soft terms fromin their mirage analogy mediation with reno deflected anomalyThis mediation effect [23, deflects 24] the (forterms gaugino a at mass mo unification low scale energies.the and ratios mod As of inwhich the mirage are contributions parameters mediation, from of the the thescenarios model, pattern in different generically which of mediation take the on stabilization discr of the mediator fieldsdiscussion of i deflected mirage mediation. Ourthe primary approach is goal on i firm theoreticalwe ground within will the KKLT-mo show that the mediator of gauge mediation can acquire gravity is a bulkmediation field, scenario certain (which gravity requires mediated sequestering), models, are als whi mechanism plays andown, important string-motivated role. approach to Suchthis supersymmetry type scenarios is breakin can mirage mediationTrivedi be [12, (KKLT) 13], mot approach which to is modulimirage motivated fr stabilization mediation, within the Typ modulusby ln mediated supersymmetry brea the flavor/CP problems, etc.). Suchgauge models can mediated, be and roughly (braneworld-motivated) cla “bulk”terms mediat [4], which arisemodulus from mediation couplings contributions that [5]among vanish and as others. (loop-suppressed the Gauge Plan mediatedfields with terms SM arise charges from [7 –in loop 9], whereas diagrams braneworld bulk scenarios, mediated terms such ari as gaugino mediation [10] and JHEP08(2008)102 (2.1) (2.2) = 1 supersymmetry N diation results. Since of this string-motivated aking parameters of the as found [12, 13] that the ss the theoretical motiva- at geometry produced by e compute the MSSM soft ly mediation [23]. We find two mediation mechanisms: r conclusions and outlook. ization, we consider several or theory-motivated choices entical in most cases to what is of the order o the observable sector soft ntext of the KKLT approach on is of the order ther phenomenological studies retical background and model- d by various results in Type IIB n for mirage mediation models. ler modulus, the ratio between on, focusing on the stabilization n investigation of the renormal- of the gravity multiplet. As the tacks of D branes located in the arate dynamical supersymmetry the parameters and the resulting esting case in which all contribu- tential self-couplings, which have imensional , C , C , and (ii) anomaly mediation, which ion through radiative supersymmetry C T F T T 2 + F π 1 T 16 ∼ ∼ – 3 – (modulus) soft (anomaly) soft m m In phenomenological models based on this construction, it w The outline of this paper is as follows. In section 2, we discu while that of the (loop-suppressed) anomaly mediated terms observable sector soft SUSY breaking(i) terms modulus are mediation, dominated by due to the modulus Kähler is broken by anti-branesthree-form located fluxes, while at the the observablebulk sector tip Calabi-Yau arises space, of from as the s shown warped schematically thro in figure 1. can be represented by the conformal compensator field through supergravity effects, eliminating the need for a sep breaking sector.terms This from leads gauge tothis mediation competitive mediator to contributions field the t ispossible previously a stabilization mechanisms, known matter including stabilizat modulus miragebreaking which me effects requires and stabilization stabil bypreviously higher-order superpo been considered inthe the interesting result context that ofthe despite deflected gauge the anoma mediated presence and anomaly ofwas mediated the found contributions Käh in is deflected id anomalyterms, mediation. and study With the this renormalization in grouplow hand, trajectories energy w of mass patterns. Theof analysis low sets energy the supersymmetry stage within for fur this general framework. tion and model-building aspects of deflectedof mirage the mediati mediator fields.MSSM fields We compute in thescenario section soft are 3. presented supersymmetry in bre sectionization The 4. group phenomenological Our running analysis implications of includesof a the the parameters parameters, and including sample thetions spectra phenomenologically are inter f roughly of the same size. In section2. 5, Moduli we provide stabilization ou and supersymmetry breaking In this section, we providebuilding a aspects detailed of discussion deflected ofto mirage the moduli mediation stabilization, theo within which has theRecall been co that a in primary the motivatio KKLT construction [14],string which theory was inspire flux compactifications (see e.g. [26]), four-d scale of observable sector soft terms due to modulus mediati JHEP08(2008)102 can (2.3) (2.4) X nerically . Our goal in diation. field can thus be 3 X (grav) soft D m can have the same order is generalized to include vectorlike pairs of “mes- , the within the context of the rgravity effects, such that represents an open string l demonstrate that some mechanism. and n open string mode which their inclusion is not with- N T /X [12, 13]. mechanisms can in principle X e to supersymmetry breaking X ainly be interesting. To this n of the modulus F term; the ution to the observable sector 2 , cally needed in gauge mediation and π F and 4 (anom) soft X X m ∼ . ) warped throat X 2 X / F 3 , is thus sequestered from the hidden sec- 2 C π /m 1 C X F P 16 M ∼ ∼ – 4 – X X F (gauge) soft T m Calabi-Yau 7 Ψ D , can generically be comparable to Ψ 3 or MSSM X, The KKLT-motivated braneworld setup for deflected mirage me Ψ with SM gauge charges, as shown in figure 1. Such states are ge D (gauge) soft m Figure 1: In deflected mirage mediation, this KKLT-motivated picture Interestingly, if the stabilization mechanism is mainly du additional observable sector superfields, a gauge singlet the comparable size ofsuppression the factor two was scales determined indicates to a be suppressio ln( senger” fields Ψ, this paper is simplycoexist, to demonstrate although that more allend, detailed three in mediation model-building what follows would we cert will address the stabilization of acquire an F term vacuumthe separate expectation dynamical value supersymmetry purely breaking sector duemodels typi to is supe not necessary.soft This terms leads through to gauge-mediated an messenger additional loops: contrib We will also show that in a broad class of models, and hence mode which starts andconnects ends observable on and observable hidden sector sector branes, branes. not a tor in the same way as the MSSM matter content. More generally regarded as a matter modulus which must be also stabilized by effects, then the supersymmetry breaking order parameter present in string theory modelsout along top-down with motivation. the MSSM In fields, so this string-motivated context of magnitude as other SUSY breaking order parameters. We wil JHEP08(2008)102 , i T C , (2.7) (2.5) (2.9) (2.6) ). In (2.8) i T (2.11) (2.10) and Φ (Φ X 1 messenger (Φ) is the X , W N are given by , ed group which i ) = 3. The Kähler Z 4 | rfield. In eq. (2.5), p , the , and X GeV is set to unity, but } X | and , Y (Φ) + h.c. 18 4 | X 10 W (1) Φ Z 3 | es the modulus Kähler setting × nglet U ( , 4 , i ation), and the MSSM fields, . icity. Higher order terms are neralization: θC L n 2 O according to , ) me. d al compensator formalism, T-motivated scenario. = 2 i T . , 3) + 1 (which can be field-dependent) is ) ) P / i Z + T T a M Φ SU(2) i + Φ) f ) is the metric Kähler of T , i , + + ( . a α C Z )Φ 2 T ( T a (Φ W = T π θ 8 i K X i ing orders of magnitude. a α T, log( Z − ( SU(3) i + 3 log( p W { Z , Z 2 − − a 1 – 5 – g X + = (Φ) n , and 3 exp( a ) parameter via = )= )= a f X T − = 1 effective supergravity within this class of KKLT T a i 1 4 1 T T G θ a X are the (generically Φ-dependent) gauge field strength θ + ) f N 2 are assumed to be negligible compared with the Planck T, T, h a α T ( ( d T i 0 0 ( Φ) = and W , T, Z K K ( a = g (Φ ) are dictated by the geometric scaling behavior of X X and Φ T Z G generically is assumed to have the “no-scale” form . Furthermore, we consider the situation in which only i Z 0 T, X Φ)+ ( )+ , K i T Z ( (Φ 0 K CG = ) and θC T 4 K is the potential Kähler of d T, is the chiral compensator and Φ denotes a generic chiral supe ( 0 Z X K Ψ (usually taken to be complete multiplets under a grand unifi is the potential, Kähler C Z = develop sizable F term vacuum expectation values. K L X The potential Kähler can be expanded in powers of Φ Mostly, we will use units in which the reduced Planck mass 1 pairs Ψ, metrics As stated previously, we study the scenario in which Φ includ associated with the volume of the compact space, the gauge si constructions, which can be expressed concisely in the chir in which superpotential. The vev of the gauge kinetic function related to the gauge coupling four-dimensional effective supergravity theory of this KKL 2.1 Theoretical background We begin with the four-dimensional contains the SM gauge group towhich preserve we gauge now coupling unific denoteand as Φ superfields of the SM gauge groups such that we can examine the results with the no-scale form by To see deviations from this form, we will use the following ge where with respect to changes in the overall compactification volu eq. (2.8), we have assumedignored, a since diagonal the matter vevs metric of for simpl will restore it occasionally when discussing issues regard scale. At leading order, in which JHEP08(2008)102 0 W can 0 (2.13) (2.12) (2.14) (2.15) (2.16) w is given by ections, and 0 W actification, and -dependent super- and the messenger perturbatively gen- ). In the standard ilizing potential X 3 P X cal constant. To break M competitive with it for ( lization, such as through ely generated terms, Λ is er order, or can be gener- batively generated , ) denotes the singlet self- mong ∼ O X ( , ses from gaugino condensation ): D3 branes are put at the tip of 1 2 MSSM A We will study three possibilities ) X s straightforward. 0 W , respectively. The superpotential W ( i , 1 X , 3 . T n W , − − . This corresponds to supersymmetric ) and Ψ+ n 2 X aT X Ψ Λ X and Φ π ( − ) = 0 (8 m X λX Ae is the standard MSSM superpotential. Here X )= 1 2 ( 1 − X )+ ( ∼ O – 6 – 0 W 1 )= X w a MSSM ( W X 1 ( = W 1 W 0 W W + 0 W = W (1), as discussed in [17]. is stabilized due to radiative corrections to the may have a mass term, such that at leading order U are the modular weights of . This assumption justifies the neglect of higher string corr X i X T n is the vacuum expectation value of is the stabilizing superpotential for 0 and 0 is a constant superpotential term determined by the flux comp X 0 X W n w The singlet stabilization via F terms; one can also consider D term stabi an anomalous The singlet may have no self-interactions, such that where In this case, in which Λ iserated the terms, scale Λ at is which thethe these cutoff scale terms scale, associated are while with generated. for dynamical nonperturbativ For symmetry breaking. symmetry breaking potential. The self-interactions may appear atated renormalizable by or a high nonperturbative mechanism: leads to a supersymmetric minimum with a negative cosmologi 3-instanton corrections, such that In the KKLT setup, the combination of the two terms of the stab • • • T D pairs; a generalization to higher order interaction terms i be chosen to bemoderately large small enough that the nonperturbative term is we have assumed a renormalizable superpotential coupling a allows us to use thefor no-scale potential the Kähler of functional eq. form (2.9). of the singlet self-interaction in which the second term is aor nonperturbative contribution which ari KKLT model, one makes the further assumption that the pertur in which interaction superpotential terms, and in which takes the form for supersymmetry and cancel the cosmological constant, anti- JHEP08(2008)102 thus (2.20) (2.18) (2.21) (2.23) (2.17) (2.19) (2.22) 0 . This t | form W | 2 construction, K/ e = 2 / 3 m plays an important role ay of the flux-generated ll generalize this form to T ow energy supersymmetry ture [12, 13]). Heuristically ngian terms with nonlinearly mass lifting potential of eq. (2.19), , , r potential: = 0). The modulus vev tered from the observable sector 2 ¯ θ 0 2 0 = 0 t . w θ . , = 0 P ! T P . 2

2 ! W n ) / P T 2 T 2 3 − n / C 2 T W 2 ) 2 3 m K K ) m 2 T C m T T ¯ ( m D ∂ T +

∂ θ 2 +

4 0 T + d ∂ K/ T , assuming that only – 7 – W e T ( by an amount of order ∼ O ( T T Z ∼ O ∂ T T of the order = would be a flat direction in the absence of the non- T T ≡ 0 P ∼ ≃− L t + is the warped string scale at the tip of the throat. ≈− T F 0 . As we will see explicitly below, the uplifting potential ∆ T V T ( 2 T P D W m 0 T M (NL) 2 6= 0. This results in a supersymmetric AdS vacuum, with W D / L 2 3 0 m W 3 = 0 and − ). Eq. (2.18) leads to a so-called uplifting potential of the i P and n T ,Φ K X T ∂ = generically can be a function of the superfields Φ. In the KKLT T K P is stabilized at a supersymmetric minimum due to the interpl is effectively constant due to the warped geometry. Here we wi satisfies the F-flatness condition: perturbative term; a runaway behavior would result if vacuum energy given by in which and nonperturbative terms in shifts the vacuum expectation value of (further details can bespeaking, the found argument in is as theT follows. mirage In mediation the absence litera of the up The nonlinearly realized sector is assumedmatter to fields be (e.g. well seques realized supersymmetry. Thebreaking set is given of by such the terms following contribution relevant to for the l scala the warped throat. Such breaking effects are encoded in Lagra in which the mass of in which 2.2 Review of KKLT modulusLet stabilization us briefly review the stabilization of P In the KKLT model, will turn out to be parametrically larger than the gravitino vacuum shift induces an F term for JHEP08(2008)102 . z ze (2.28) (2.29) (2.26) (2.27) (2.31) (2.25) (2.24) (2.30) = x rvature of about the ) as follows: reaking, the 0 ibutes to the T t , 0 w 2 0 . to cancel the negative t p/ xpanding t keep in mind that the of the equation , ) = 1 2 0 P t T z t 0

he string scale, such as the t ) M = 0 ∆ (2 2 2 / aT tric minimum; this procedure assuming a real / 2 3 ≈ V scale phenomenology. 2 2 3 − . The unperturbed minimum is / , ) yields m 2 3 2 m 0 2 p / Ae ) 3 2. Since this expression should not m at , P ) − ) − − m p p/ n 2 0 , n / . 0 2 3 t Ae 2 − w ≫ / ≫ ( − 3 0 p/ 0 ) − 0 T /m A m ¯ w 0 3(2 T pw at P 0 m Ae 3(2 w 2 p + leads to − at M − 2 t ln − 2 . From eq. (2.22) (still neglecting the uplifting T – 8 – 2 δt ( 2 ) ∆ P p/ ≈ ≃ t log( , we take the standard approach of treating the ) 2 T 1 2 0 0 0 − M T T t t 2 ∼ (∆ 2 −W / at m m pm 0 2 t (2 aT 3 − = m at m must be of the order 3 = = function, which is the solution 0 2 0 t p 0 ∼ L δt 4 at aAe W ) and V + 0 W is indeed parametrically heavier than the gravitino mass. 0 2 t T = w = /

T 0 t t 0 + K m ∆ is of the order e W +∆ ∂V T and ∂ 0 T ( t T | = / 3 P D m T V 2 is given by the expression M / F 3 0 ) dictates the size of the observable sector supersymmetry b t generically has a large vacuum expectation value, and the cu ∼ m is exponentially small, eq. (2.25) reduces to T modulus governs the extent to which modulus mediation contr A 0 T ) is the Lambert + given in eq. (2.25), the scalar potential takes the form w x T ( T 0 ( t / W T F Turning now to the effects of the uplifting potential, one mus To determine determined from eq. (2.20) (using eqs. (2.10) and (2.13) and potential), the mass recalling that vacuum expectation value of the potential at this minimum is governed by which demonstrates that The gravitino mass is change in the presence of the uplifting potential, eq. (2.26 in which the last approximate equality holds if soft terms. Hence, the Fdilaton terms and of complex moduli structure with moduli, masses are of irrelevant order for t Te Since mass of the cosmological constant of the supersymmetric AdS minimum. E in which The solution for uplifting potential as a perturbation about the supersymme is valid since Minimizing this potential with respect to Note that if minimum JHEP08(2008)102 2 o − ) as 0 0 X fting term X ( X (2.34) (2.33) (2.35) (2.36) (2.32) X − B = m X 2 ( / , the mass X he vacuum 3 X X m m ( 2 1 O = ) is suppressed with W ¯ ). However, here such T . T 0 One example is F term + t y is broken, ilization of 2 a Taylor expansion of the T / T ( 2 3 / terms are comparable, which m . m T 2 ravitino mass is approximately ) F n has the form p/ P ) . n 0 T . C m of the form p We first consider the case in which w C − 2 F + / 2 T 2 3 . 0 T m 1 6(2 C m ( at C ) F ) C induces an F term of the form = X p P 0 ∗ n n 1 . Since this factor is numerically of the order ≃ T at 2 p p − within each of the three previously discussed − W 2 π aT = 4 – 9 – T − X D 3(2 6(2 ∼ T ; higher-order terms will also maintain and the messenger pairs are present, we will now T T ) 0 T + = = 0, = aAe 2 F ) / X 2 X 0 p t 3 T 3 0 T ¯ n C t K ∆ T at . For example, the soft supersymmetry breaking ( + /m would be induced (as in the case of 2 F = X / P X T 0 clearly vanishes. If supersymmetry breaking effects acted t X C M K F F e X ln( − = 3 and p ≃ = . Hence, 0 should be less than the Planck mass. In the absence of the upli T /C 0 at F C X , a nonzero by F must be hierarchically larger than the gravitino mass, and t X ≈ /C X 2 , the F term of C / m 3 L F V m ), which keeps the vev of 2 is stabilized by a supersymmetric mechanism at a high scale. ) 0 associated with eq. (2.14) generated from anomaly mediatio X as in eq. (2.14), whichpotential. may be For thought of this as term the to leading term play in an important role in the stab Comparing eq. (2.34) with eq. (2.27)given demonstrates by that the g effects do not lead to a shift in shift the vev of parameter potential such that expectation value The compensator also gets a nonzero F term when supersymmetr In the KKLT case with Eq. (2.36) shows that the modulus mediation contribution The shift in the vacuum expectation value of stabilization through an explicit superpotential mass ter respect to of a loop suppression, the modulus and anomaly mediated soft To generalize to the case in which is the standard mirage mediation result. 2.3 Stabililization of the matter modulus address the issue of the stabilization of stabilization mechanisms. Supersymmetric stabilization at aX high scale. JHEP08(2008)102 . is is X sym- mply /X (2.39) (2.37) (2.38) should R couples X a purely F X (1) gauge (1) X U U (1) was studied 17]. U , 2 oup running, X, | lization via Fayet- 2 / X | 3 m ) is massless before super- ts the mass also sets the X ( irection. If this soft mass X . uch anomalous 2 ≃− a canceled at one-loop byenergy the effective field theory (see eq. (2.12)), re precisely, since g ng out the messengers Ψ at 0 will be zero, resulting in no a (1) gauge symmetry at a scale o a moduli-dependent FI term. λ C W U her from (i) gauged X ersymmetry is broken, ¯ X − F K . ) , 2 ¯ X 2 / X / 3 ( X 3 2 m λ K m 0 2 / A (1), it can be triggered to acquire a vev by 0 ≫ ≈− K U 2 e ) C X C – 10 – F X m ( ≃− − can be stabilized at a scale anywhere between the = Nλ W X 2 ¯ X

degree of freedom is absorbed in the vector multiplet X , (1) symmetry, under which a subset of the moduli (the X C a can be induced if the Green-Schwarz modulus has an F Ψ with coupling strength D F U C g ¯ X F X at a high scale either by F terms or D terms does not yield X

X F 2 . One can think of this potential as being generated by the We now consider the case in which X results in comparable soft terms to that of mirage mediation can be a Green-Schwarz modulus, but the induced a K and π (1) gauge groups. The case of the anomalous 1 2 C / has no superpotential self-interactions, as in eq. (2.15), U T λ 16 0 /X K induced by this radiative stabilization mechanism can be si X ∼ e X and ) F X (1), and 0 X ≃− U ( A V X F )), which gives a subleading contribution to the soft terms [ is charged under the anomalous T changes sign at a particular scale due to renormalization gr + X X T ( / T In summary, stabilizing The situation is more involved when considering D term stabi The F term of F ( nonsupersymmetric stabilization results because once sup Hence, we now turn to the other twoRadiative stabilization stabilization. mechanisms a situation in which symmetry breaking. If acquire a soft massterm for from anomaly mediation,stabilized lifting at a its value flatto of d the the vectorlike order matter of pairs this Ψ, crossover point. Mo a (meta-)stable vacuum. Therefore, if gauge-mediated contributions to the MSSM soft terms. Iliopoulos (FI) terms, which can result in supergravity eit metries or (ii) anomalous O the FI term. In this case, the given by the string scale dividedHence, by if a loop factor, and leads t in [17], we summarize their results here for completeness. S for some positive Green-Schwarz moduli) transform. This effect breaks the electroweak scale and thescale UV of cutoff the scale messengers [23]. Ψ, as The seen vev in eq. that (2.12). se Coleman-Weinberg mechanism [27]; it arises from integrati one loop. Depending on of the anomalous term. In KKLT models, groups are ubiquitous inGreen-Schwarz string mechanism, models. which manifests Theiras itself anomalies a in are nonlinear the realization low of the obtained by such that JHEP08(2008)102 2 are C (2.43) (2.42) (2.41) (2.40) and T in eq. (2.12). is stabilized by 1 are comparable, W X ¯ C F X X ˜ 3 X, and of the form of eq. (2.16) ˜ p/ ¯ X ive; as a result, deflected uge-mediated interactions 1 X − . The F terms are given by 3 constraints for F X 3]. As we will see shortly, W G nger vevs must vanish; these p/ ence, the terms Kähler of the n ¯ that this term originates from ) B − 1 -mediated terms. This result is symmetry breaking effects and o of Pomarol and Rattazzi [23], ected anomaly mediation. Here ch are relevant for the stabiliza- as given in eq. (2.8), eq. (2.10), ∂ X T hat n A ) 1 + ∂ X to simplify the potential. Kähler , T ) T ≡ ( + , W and B . The feature that the F terms for this C CX 3 ..., B T A can have an arbitrary value which may ( T C T C F 3 G = + ( n A + + B 3 K/ ˜ 3 3 F n ∂ X − and − n B B a more complicated superpotential. It is also p/ p/ X ) ) , and A A Λ X – 11 – Ce ˜ T T X G G X C = − + + 3 = 1 − . We pause to comment here that the F term equa- T T = W V ( ( B C, T, A θ C C A 4 Let us finally consider the case in which F C C G d 3 3 Z is opposite to that of [23]. = ≃ − ≃ − run over ˜ X θG 4 and d A, B X Z is stabilized below the Planck scale. self-interaction superpotential terms, as represented by is the inverse of X X is holomorphically redefined to B A are the leading order contributions for X G X Our definition of 2 This situation was also considered inwe [23] limit for ourselves the to case of the defl case in which only a single term in dominates the stabilization: the interplay between the scalarhigher-order potential induced by super and eq. (2.11). Revealing thesuperspace comformal action compensator are depend and also that Higher-order stabilization. model are equalanomaly in magnitude mediation but isthis opposite also relation in called can sign “anti-gauge beworthwhile is mediation” to modified distinct note [2 by that the giving standardonly modified mirage by mediation Planck-suppressed F corrections, term provided t in which in which the indices The scalar potential is of the form be positive or negative; anonperturbative negative dynamics. exponent The potential would Kähler terms indicate whi tion of involving messenger loops is thus comparableidentical to the to anomaly that of thedespite deflected anomaly the mediation nontrivial potential scenari Kähler of The contribution to the observable sector soft terms from ga in which tions for the messengers impose the condition that the messe in which Λ is the cutoff scale. The exponent JHEP08(2008)102 for , we ˜ ¯ , and B X (2.44) (2.49) (2.50) (2.48) (2.45) (2.47) (2.46) 0 / ¯ F n A ˜ X ˜ 2 X δF / 3 ˜ ¯ B X A m Λ (or Λ X n G # ˜ can be treated as ˜ + X/ X, , such that higher- O 3 p = 0 , ˜ P ) X , : X P 1 T F W M X 2 + h.c. δV − − + C X 2 n is n ˜ 2 − T . and X/M n n ( ˜ the following analysis. 2 A , 3+ 2 X ˜ y 0 X − C C 2 3 − X . p/ n / 3 n C δF 1 2 ) 0 3 p/ X 0 W − O ) − F try breaking as well. Hence, the T n n m n T C 3 W ) ! Λ 9 + 2 ∼ ˜ " are + − X C T ( 2 , , p ( is smaller than T ˜ ) 3 − 2 X C ( T n X + h.c. p/ . in the absence of − 2 . C X 3 1 − ), we find = 3 δF ˜ δF 3 C 1 X − 0 2 − nn T X X − 0 C / n n − p/ F n + + , n n 3 ) ) n ∼ ) − 1 p 3 C ) W T C ˜ ˜ 0 0 3 3 T − T p n 3 X X T and /m ( n / − − F F CC , ˜ ˜ ˜ 0 – 12 – + X + 3 X n P X ˜ 2 p + 2 X ( C 2 2 − K − 1 − Λ = = C / / 0 T 3 T e ˜ n M n n 3 T 3 3 X ( + ( 2 F ( 3) ˜ T − p/ C 2 Λ 3 X n n m a m n C ln( − − can be safely neglected, F F − ) p/

˜ n − can be perturbed about the values which were obtained / X X ) ∼ ∼ = ≈ C ) n n 3 C X 3 T C ) = 3 2 X C ˜ p/ − C X 0 T ( ) T F − ˜ 3)( ˜ n + n − X X T n F 3 0 X 1 3) 2 Λ + + − T ) F 3 / Λ F n − 2 ( 2 3 − n 3 p T and 3 T T − ( 3) ( p n m 2( − 2 ( 1 T 3 − + n Λ C − n n p F Λ − T ( C n 3 , are the F terms of ( + + n − T T X n 0 3)( = F C − nn p C δV p 9 − ( and . Hence, we expect 9 1 p − C 0 2 = = F ˜ X T C 2 / 2 3 δF δF In the calculable scenario in which the vev of m nonperturbative corrections) and 1 small numbers, and in which order potential Kähler terms for in the previous subsection: Ignoring the subleading order terms, which are suppressed b constraints can be consistently satisfied after supersymme messenger fields and their F-terms can be safely set to zero in Given that the perturbation to the scalar potential due to can see that the leading order terms in powers of Therefore, we obtain O and JHEP08(2008)102 (3.1) (2.52) (2.53) (2.54) (2.51) . ˜ X takes the form and i . , which result in 1 X − /C n soft mass spectrum superpotential mass C r-order superpotential ˜ F X t recall the form of the . ˜ vevs of the modulus and X 2) e nature of the soft terms down model are no longer − itions of ral mixtures of the moduli, ssengers, in a manner which parable (see eq. (2.36)), this 3 n ( , 5), we obtain We now turn to the issue of / 3) p/ to simply take the F terms as ent in the superpotential, and k MSSM fields Φ 1 are the (unnormalized) Yukawa − − Φ n X j ∗ ghly equal for a very general class n 2( 0 same as that obtained in the case ijk Φ C ) C i y C F T Φ . 3) 3 . + C − 0 ijk p/ C C n 3 y F C T − − F 2( ( 1 n X 2 + Λ 1 3 n n j C 2 ) − 3 − − Φ T n i − + n n n – 13 – Φ + − n which are of the order of Λ 0 ij = ˜ T X = , in which case we obtain µ 2 ( ˜ C 0 X − /X C ˜ X = X n F 1) F X X 3 F = C F − + 3 − C − n n n n ( ˜ MSSM X n Λ C W 0 and − F ˜ X 3) = = − ˜ thus leads to the result X ˜ n X X = ( shows with δV are supersymmetric mass parameters, and δV ˜ , both radiative stabilization and stabilization via highe X 0 ij X µ In summary, we have demonstrated that in the absence of a bare Before we present the derivation of the soft terms, let us firs Using this result in eq. (2.48) and comparing it with eq. (2.4 Let us assume that Replacing Since the modulus and anomalyresult contributions indicates are that already all com threeof contributions should superpotentials. be rou Interestingly, thisof result deflected is anomaly also mediation [23], the keeping in mind our defin term for self-interactions lead to values of Minimizing comparable contributions to observablecomputing sector the soft soft masses terms. generated by these F terms. 3. MSSM soft supersymmetry breaking terms The F term vevsinduced studied by in supersymmetry the breaking.essential previous for The subsection our specifics control phenomenological th ofparameters. purposes, the and By top- it varying suffices thesegauge, parameters, and we can anomaly obtaincan mediation. gene be computed With usingcompensator a the fields, given spurion the technique; messenger set scale, itwe of and depend will the F on now number review. of terms, the me the MSSM superpotential. In general, the superpotential of the couplings. We assume that only trilinear couplings are pres in which JHEP08(2008)102 rm and (3.7) (3.2) (3.6) (3.9) (3.3) (3.4) (3.8) (3.5) (3.10) plings T F , C , B F are the coeffi- se F i A Y F , , fields have Kähler 0

i i X Y re defined in the field B F Φ 0 are the gaugino masses, + h.c. | ∂ . the Φ erms of the form a k a A f Φ problem of the MSSM) until Φ erms, the quantities the presence of 3)) M X e superspace action: j ian are regarded as functions ∂ to use the spurion technique, ∂ , µ Φ . 2 ; the physical Yukawa couplings i ¯ K/ llows: θ A ! ion effects. The + 2 Φ . − k i F θ T . The superpotential Yukawa cou- Y Y 0 ijk j .

F + , 0 ijk X y Y ) p 3 0 y = 0. The metrics K¨ahler of the matter i | B a 0 ijk log − ijk i Y a ¯ 3 exp( f θ y i F f B Z n A A and −

0 ) ∂ T 3 = (

Φ / ¯ i ∂ 1 A + T 0 ∂ T ¯ θ C Y , ∂ log a K 2 + + B B λ C θ log(Re − A Φ a θC e C ∂ F – 14 – T 4 ∂ A λ + ( A A F d ∂ a 2 = 0 0 | ¯ F A F θ i | = M Z a Y − F − i 1 2 0 ijk + f denote Y y = C can be analytically continued to a superspace function A = = = ∂ = a − F 2 i a f ¯ 2 0 ijk CG 2 | | A, B m M i i 0 ijk θ A y Y Φ θC | + A 4 2 i Φ 0 d | -dependent parts of these couplings are generated by the F te ∂ m are given by a θ are in the definition of trilinear terms. From the spurion cou Z 2 f − i 2 θ Y / 1 = = + ) k a 0 | f Z soft i j Y L Z are the soft scalar mass-squared parameters, i can in principle be a function of the moduli fields, in which ca = Z 2 i ( indicates that the value is taken at i 0 ijk / m Y are trilinear scalar interaction parameters. These terms a 0 y | 0 ijk y ijk A = , the gauge kinetic function X ijk vevs of the theory (for a review, see [9]). More precisely, in F The MSSM soft supersymmetry breaking Lagrangian includes t later in the paper. In addition to the couplings of eq. (3.1), metrics given by eq. (2.11). For the computation of the soft t are useful for discussing the renormalization group evolut where once again, the indices plings in defer the discussion of supersymmetric mass terms (i.e. the and fields are also analytically continued into superspace as fo For our scenario, the cients of the bilinear terms in the nonholomorphic part of th To derive the observablein sector which soft the terms, couplings itin of is superspace, the convenient with effective the supergravity Lagrang in which in which basis in which the kineticy terms are canonically normalized of eqs. (3.5)–(3.7), we obtain JHEP08(2008)102 5 T (3.17) (3.12) (3.16) (3.11) (3.13) (3.15) (3.14) e gauge Ψ are GeV), . It can also 16 , 10 2 UV is normalized to Λ × C 5 C = 2 Ψ are complete GUT G d to tation M ge mediation. For general can be extracted from ion at the GUT scale, we explicitly breaks conformal s Ψ, i . Therefore, the dependence ion in this work. Y dependence due to the super- tions on the compensator field with mass terms of the form UV UV C , φ Λ , . First, note that the renormaliza- a and , C 3 p f φ . a , 0 ijk i − f k i y Re Y . n i A µ µ is also assumed to be independent of ) φ. Y 1 T must be changed to + φ µ∂ µ∂ (SU(5) Dynkin index of Ψ) and . 0 ijk j 1 2 2 1 + i , log y as UV × a A Y 2 UV l − − A 2 UV T , there is no Λ , ( 2 ∂ T N 3 + Λ a µ = = A i 0 f ijk – 15 – C y i a θC A F C Y 2 f ) = ) = d = = C G G i Re Z M M A C∂ ijk ( ( C i a has an effective mass A Y f C∂ ˜ φ is always accompanied by , µ Cφ = ˜ φ 1 depending on the type of D branes from which the gauge groups , which we will take to be the GUT scale ( , is the number of messenger pairs for the case in which Ψ and G M N = 0 are the same for each of the SM gauge group factors. To preserv a = (number of messenger pairs) l a l N representations; we will restrict ourselves to this situat , the expression for trilinear terms eq. (3.10) can be reduce 5 X . This effect arises because introducing a cut-off scale Λ and unity. Note that in which the SU(5) Dynkin index for the fundamental represen tion procedure gives an additionalC dependence of these func Therefore, the conformal conpensator dependence of assume that originate. Since we wish to maintain gauge coupling unificat Let us now consider the functional form of invariance, such that to formally restore it, Λ coupling unification, we also assume that the messenger pair In the above, multiplets e.g. of SU(5),sets as of is messengers, standard it in is useful many to models (re)define of gau With the redefinition symmetric nonrenormalization theorem. Since be understood by introducing Pauli-Villars regulators For the unnormalized Yukawa couplings and on the renormalization scale At the high scale in which JHEP08(2008)102 ). G op: s the M by (3.18) (3.22) (3.21) (3.19) (3.23) (3.20) a to b , ). Below µ . Y , such that b π 5 ) mess 4 i / C / scale M 2 a (Φ µ C g a = 3 ion of scale. √ C = 1 . 2 GUT a b a g C 1 M α − a 2 ion to denote the group α µ µ dµ XC # ln C k X s light degrees of freedom of 0 y integrable: ijk Ψ C Y − tly integrated, but this is not the gauge-loop-induced term ¯ j y ln ) √ Y 2 i G 0 ijk µ/ X a π Y y , ( M b 1 Z " 16 − a 5 α a µ 33 + dµ X . , ln a 1 2 2 C G X MSSM + Ψ, , which is determined by the stabilization f , π C 1 3 M N, X a i a √ 16 c Re − b mess G X + 2 ln µ/ h + π M a M λ 2 b Z a ′ a π – 16 – X = b = 4 = 16 )= j,k − 1 X ′ a 1 b − a 2 mess , b ) α π )+ 2 1 functions) specific to this scale range ( M X G , b ( 16 β 3 1 (with GUT normalization, such that M b − a ( ( are related to the MSSM beta function coefficients a Y α )+ f ′ a G b ln , the gauge kinetic function can be easily obtained at one-lo (1) M − U ( i ) MSSM = Re mess Y G a , the matter content charged under the SM gauge group include M and i f M ( X L 1 ≪ Re − a The particle content of deflected mirage mediation as a funct ) = ln µ α ≡ h µ . We rewrite the last term of eq. (3.21) by ( i ln is the conventionally defined gauge coupling parameter X Y EW , SU(2) mess a a ln ′ a C M c α b M 2 -function coefficients Figure 2: by a β Above the mass scale of the messengers X X − MSSM fields and the messengertheoretical pairs. parameters We (e.g. will the use a primed notat The of in which in which The last term due topossible the gauge for loop the interaction Yukawa-dependent candepends term. be on explici Fortunately, only for SU(3) For the metric, Kähler the one-loop RG equation is not simpl the messenger scale, only thethe MSSM matter theory. fields For are present a JHEP08(2008)102 (3.27) (3.26) (3.24) (3.25) (3.28) (3.29) (3.30) , ssenger 2

, C C ! F

2 ) 2 ′ i + h.c. π ˙ γ C s for the soft mass , C (16 : F C are given by eq. (3.18) ribute to the threshold C ion, including the values F − ′ a X . b X identify the contributions 2 mess , F C i a π C ! γ F M ugh eq. (3.21) is an integral X + M 16 ′ a X 2 F b

, and 2 n of the soft parameters. This − G C , 2 + 0 π + h.c. . Note that there are no threshold C k g T i F )+∆ M C 16 T A

C C + , C F F F 2 i + + + + mess T j 2 m T

T ) M T A i T ( 2 + X

+ F a n , it is necessary to take into account the 2 X F + F T mess T π i T

M − )+∆ T

A M mess +

16 p 3 ( F 2 – 17 – ) 2 a = + M T ′ mess i π ) = ) = g 2

θ ) G M 32 2 N ijk mess i − π ) = M mess A − − = n ( M G a ( M 0 for asymptotically free theories). µ − = (16 4 ( a and their primed counterparts are given in appendix A. ( M M a g 2 i a < p 3 i derived from eqs. (3.20) and (3.21) using eqs. (3.9- 3.10) θ = ′ a M m N M , b The scalar mass-squared parameters are given by a µ µ i c ∆ ( γ i 2 , ˙ The gaugino mass parameters are given by ) = ) = A i Recalling that G a γ X and the messenger threshold effects at is the unified gauge coupling at − M mess 0 = G by comparing its value infinitesimally above and below the me M = g 2 i M µ = denotes the anomalous dimension of Φ ( m 2 i i mess µ ∆ γ ( m 2 i M m Gaugino Masses. we have Trilinear terms. contributions to the trilinear terms due toSoft the scalar messengers. masses. (our convention is that In the above, in which the threshold corrections are in which where the threshold corrections are For completeness, • • • scale. We now present the MSSMat soft the terms for GUT deflected scale mirage mediat equation, we can easilycorrection at see that the Yukawa term does not cont presence of the messenger fields in the beta functions. Altho automatically include the renormalizationimplies group that we evolutio obtainfrom the correct the soft threshold massparameters. formulae scale once One and we caveat then is apply that the above usual RG equation The soft terms at the scale JHEP08(2008)102 , , e m ng α /X (4.1) (4.2) X F , and (see eq. (2.8)) ra , we define i /C term of the order C F B nd gauge mediated his scenario. As pre- ), (1), the electron and T O . 5)–(3.30) clearly demon- + o be absent for the light 0 n, which is governed by which the potential Kähler T m generation. Here we do not me that the F terms are all running and the low energy ( qs. (3.25)–(3.30). he best option for addressing 2 . / s are ontext may be worthwhile. dent modular weights. Here simplicity, although the most in [18]. ng if these diagonal terms are th this form of the leading or- ll known that flavor-violating / P 0 MSSM fields Φ r a recent discussion see [22]. P-violating phases in the soft T 3 bsent, such that the only terms bounds unless the superpartner owever, gauge mediated models m M F m 2 / P n denote the relative importance of plings may exhibit nontrivial flavor 3 ln M g m m α α ln = m and α T g problem [28] results in a T α m + µ F α = T – 18 – 2 C / P 3 C F M m g α ln = 3). For the MSSM superpotential, we assume that R- = m p α X problem of the MSSM, except to say that in a given string X F = of mirage mediation. To account for the gauge mediated terms sets the overall mass scale of the soft supersymmetry breaki C problem. Furthermore, due to the anomaly mediated terms, th 0 α C F m µ/Bµ by g is the µ/Bµ α (see eq. (2.1)). Following the mirage mediation literature 0 m α m ≡ ) T + T is of the “no-scale” form ( ( In what follows, we will now restrict ourselves to the case in With these assumptions, it is useful to express the anomaly a Turning to effects of CP violation, an inspection of eqs. (3.2 We also comment here on the issue of flavor and CP violation in t / T T we will also define are the standard Yukawa couplingsexplicitly required address for the fermion mass parity is conserved and bare superpotential mass terms are a Our parameter 4. Renormalization group analysis and low energy mass spect In this section, wemass will spectra show the for the renormalization supersymmetry group breaking (RG) parameters of e of theory model, supersymmetric(i.e., bare below mass string terms scale) arehave fields a expected of well-known the t observable sector. H With these definitions, terms, and the dimensionless parameters as there will generically besupersymmetry breaking irremovable flavor-independent terms. C Generically, if these phase contributions in termsF of the modulus mediated contributio neutrino electric dipole moments willmasses exceed are experimental in the multi-TeVreal, range. though In a this thorough paper, exploration of we will CP assu violation in this c Giudice-Masiero mechanism for solving the to avoid flavor-changing neutral currents.effects can However, also it be is inducednot from we universal, renormalization group which runni wouldwe will correspond assume to generation-independent generation-depen modularstringent bounds weights occur for for theder first terms, two higher-order generations. corrections potential Even Kähler cou wi violation; a more comprehensive study of this issue is found strates that if there are nontrivial relative phases betwee of the gravitino mass, which leadsthis to issue a may fine-tuning then problem. be T the addition of extraviously singlets stated, we [29]; assumed fo a diagonal metric Kähler for the JHEP08(2008)102 (4.3) (4.4) (4.8) (4.5) (4.7) (4.6) s and pecific riguing , te that in # 2 ) = 1 for fields terms) and the 2 l / P n 3 A . M − m 2 (1 ln 2 / P m 3 i,j,k α M , and defer an analysis of scenario, one of the most m = g l , α 2 2 ter sets, we will first address P ew features which arise from ) this parametrization, the soft / ) ln P generations; further details on ing. However, it is well known 2 3 f g ′ i π ˙ M . γ te values. In this paper, we will α . m and he gaugino masses at the mirage r couplings (the m 2 (16 m α ) ln , α (1 + g − m 2 α 2 2 . Whether mirage unification happens / P / α / P P 3 3 3 , = 1, which is the value predicted in the M M m M ) m m (1+ 2 m 2 values which result from the stabilization of / mirage P ) ln α ln 3 m 2 G mess g M M α π m m m α ) – 19 – M M α α ), or on whether the effects of the Yukawas on the ( ln 2 2 (16 4 a = 2 ′ i i π π 1 (see eq. (2.39)). For stabilization via higher-order g mess ijk m π θ γ y − α 16 16 N M 16 ′ a ( a b = 2 a c − − mirage 2 g 2 g ) ) 2 0 π i i M α g N a n n 0 16 X − − 2 0 m = 2 is needed to have mirage unification at a TeV scale, which is − m (1 1+ (1 can take on different discrete values depending on the detail m " can be considered to be continuous parameters, although in s g α = = 2 0 0 0 g α : 2 i a via the stabilization mechanisms discussed in section 2. No is not easily obtained within the KKLT approach. Another int m m m α m M mir m X ∆ α ∆ M ) = ) = ) = and G G G m take the form M M M α ( ( ( G i 2 i a A M m M soft scalar mass-squares can also unify at Eq. (4.8) indicates that terms at anomaly mediation and gauge mediation, respectively. With and the threshold terms are given by desirable from the point ofthat view this of value electroweak of scale fine tun In principle, string-motivated models they are typicallyconsider given specific by (discrete) discre string-motivated values of more general possibilities forthe future gauge work. mediated To focus terms, on here the we n will fix feature of mirage mediation is that the soft trilinear scala for these parameters depends on the modular weights (i.e., i running are negligible, as is the case for the first and second this issue can be found in [15]. standard KKLT model, and focus on the the mediator field with nonvanishing Yukawa couplings the radiative stabilization case dynamical origin of the higher-order terms (see4.1 eq. (2.54)) (Deflected) Mirage unification Before turning to athe more issue detailed of analysis of miragedistinctive specific unification. features parame In of the the soft KKLTunification terms mirage scale is mediation the unification of t superpotential terms, JHEP08(2008)102 in . del. a = 0. eory l (4.9) ctions (4.11) (4.10) = 1. N ). Here β m )=+1 α µ ; the discrete on behavior no l be shown later sign( and tan mess Z M s typically no longer , , the exponents m tion phenomenon for i n , which govern the size = 1 ediated terms. To show for ) and eq. (4.6), the new cale, even for zation group evolution in 3 l mess = 0, which is the standard The exception is when the B from the mirage mediation cific examples, the deflected ched when gauge mediation 8) is obtained only if = M essions for the soft terms), in , N 2 l h would occur in the absence of and , ly the same as mirage mediation ρ = = 3 . he masses do not vary with scale) µ 2 m , and 2 1 / P α 3 N M m , mess GUT 2 g / P M M ln α 3 , l 2 0 M m ln (we trade 2 2 Ng 0 = 1 2 π g , the modular weights π d µ α G , N Ng 16 , and the messenger scale 16 H N 2 m – 20 – = 0, mirage unification occurs at the usual scale M β n α and fix the other parameters as follows: = 3 the mirage unification scale is deflected to a = N = = 10 − = 1 TeV and (a)–(b) 1+ N u 1 0 β mess , tan H 0 m M = tan mirage m ρ M = 0, = 3. For , n g 2 1 N α , there are both continuous and discrete parameters of the mo = g GeV). α = 1), while for E 0); it only occurs when the messengers are removed from the th 8 = 1, n m m → α = and α g L = 0). The reason is that the messengers affect the MSSM beta fun m α n terms and the soft scalar mass-squares, the mirage unificati , α in certain cases. Examples which display these features wil N = A . While the soft scalar mass-squares of the light generation is given by D ρ n GeV (for mess mess 9 M = 1 TeV = M 10 0 U In deflected mirage mediation, we find a similar mirage unifica For the In eqs. (4.9)–(4.10), the mirage mediation result of eq. (4. n m ∼ the gauge kinetic functions, and the sign of parameters are the number of messengers in which of the messenger thresholds.mirage As gaugino we mass will unification demonstrate scale with can spe be as low as the TeV s The mirage unification scaleresult. of The the size gauginos of is the thus deflection deflected is dependent on of the gaugino masses.mirage From unification the scale form for of the the gauginos soft (see terms also [21]) of is eq. (4.3 at all scales ( The continuous parameters are above the messenger scale,this which feature in explicitly turn (whichfigure can affects 3 also the we be anomaly show seen the m two from gauge models the coupling with expr and gaugino mass renormali we will consider several values for KKLT model, and (c)–(d) lower value (below 10 4.2 Examples In addition to This demonstrates that theis mirage switched mediation off limit ( is not rea longer happens in generalmessenger in scale is the below presence the scalethe of of messenger the mirage thresholds, unification messengers. since whic thebelow theory is then effective unify, they can display abelow quasi-conformal behavior (i.e., t in the paper (additional examples can be found in [21]). JHEP08(2008)102 (for = 0. = 3, in N has no mess N X M 3 3 2 2 1 M 1 M M (GeV)) (GeV)) M M µ µ M and ( ( dependence and 10 10 g log log g α α ls are not specifically and gaugino masses, in on the gaugino mirage g = 0, and (c)–(d) iation literature, though for future exploration. α d the resulting low energy matter allowed parameter (d) (b) i N nction of n hibit the es. l s paper, we focus on the effects s a value of the neutralino relic ues of htforward to scan the parameter rd KKLT model is obtained for bilized by radiative supersymmetry 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16

800 600 800

Mass (GeV) Mass Mass (GeV) Mass 2200 2000 1800 1600 1400 1200 1000 1600 1400 1200 1000 = 1 TeV, and (a)–(b) – 21 – First, we will consider the case in which 0 m (GeV)) (GeV)) µ µ GeV, ( ( 10 10 12 log log = 10 mess (c) (a) M = 0, g α 3 3 2 g g 2 1 1 g g g g The renormalization group evolution of the gauge couplings 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 1 1 = 1,

1.1 1.1 0.9 0.8 0.7 0.6 0.5 0.9 0.8 0.7 0.6 0.5 coupling coupling m α We will now describe the renormalization group evolution an = 1), with the other parameters fixed by eq. (4.11). These mode m superpartner mass spectrum for sample parameter sets as a fu This choice of modularnonuniversal modular weights weights is may also often befrom used of the in interest. messenger In the thresholds, thi mirage and leave med the issue of genera α chosen to obtain certain desireddensity low within energy the features, allowed such experimental a space range. and It find is straig suchsets allowed were points; previously two presented examples in of [21]. such dark- Here the goal is to ex to examine the implications of theoretically motivated val which which case there are messenger threshold effects. The standa unification scale and the pattern of soft masses atExample low 1: energi radiative stabilization. Figure 3: superpotential self-interaction terms, and instead is sta JHEP08(2008)102 r A 1. It − terms and = GeV. E1 L1 D1 U1 Q1 M M M M M A 12 g α (GeV) ) (GeV) ) (GeV) ) µµ µ ( ( ( 1010 10 = 10 vely light stop due LogLog Log mess M t masses, in figure 4 we es, (b) the first family soft ions do not depend on the , and hence renormalization group flow h scale, and the ks are driven to be relatively n behavior (due to ourzation choice group evolution drives 1, and masses, (b)–(c) the soft scalar /C neration trilinear scalar terms, ently eqs. (4.6)–(4.7)) that the − C ct of the messengers is to change = F g − t α b τ squares, and (d) the third generation A A A 22 4 4 6 6 8 8 10 10 12 12 14 14 16 16 2 4 6 8 10 12 14 16 = 00 0 (b) First family soft scalar masses.

(d) Third family soft trilinear terms. = 1,

500500 500 Mass ( GeV ) ) GeV GeV ( ( Mass Mass Mass ( GeV ) GeV ( Mass -500 25002500 20002000 15001500 10001000 1500 1000 /X m α X F – 22 – E3 L3 D3 U3 Q3 3 2 M M M M M 1 M M (GeV)) (GeV) ) M µ µ ( ( 10 10 log Log GeV. Figure 4 shows that the resulting physics is then simila 12 = 10 mess (a) Gaugino masses. M terms can also be relatively large, which can lead to a relati A 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 The renormalization group evolution of (a) the gaugino mass 0 (c) Third family soft scalar masses.

800 500 Mass ( GeV ) GeV ( Mass Mass (GeV) Mass -500 2200 2000 1800 1600 1400 1200 1000 1500 1000 To see the resulting effects on the low energy values of the sof Figure 4: scalar mass-squares, (c) the third family soft scalar mass- terms, for the case of radiative stabilization, with to that of mirage mediation: the gaugino masses unify at a hig the soft scalar masses showof an modular approximate mirage weights in unificatio eq.the (4.11)). gluino to The subsequent befor renormali rather many of heavy; the this othergluino heavy soft mass gluino terms. at controls The one the sleptons, loop,heavy. whose tend The RG to equat remain light, while the squar breaking effects. As shown in eq. (2.39), show the renormalization group evolutionmasses of of (a) the the first gaugino and thirdin generations, a and model (d) with the third ge can immediately be seen fromthreshold effects eqs. identically (3.26)–(3.30) vanish, (or and equival hencethe the beta only functions effe above the messenger scale. JHEP08(2008)102 . , 2 A / 1 ) mess GeV = 1, = 3), 2 / M 10 3 m N α m . E3 L3 D3 U3 Q3 (Λ = 10 M M M M M (GeV) ) µ ( = 0 for i ∼ 10 mess ′ 3 X GeV. Log M b h For our second ex- 12 GeV, the mirage uni- GeV in panels (a) and . es and the (b) the third 8 eq. (2.54), this leads to nonrenormalizable oper- 10 GeV. f the gaugino masses and = 10 8 ar mass-squares. Since the ter superpartners, the relic (in fact, = 10 n the case of a higher ing of the messengers; the = 10 with dark matter constraints. ters, the relic density remains mess . In figure 5, we show the RG = 10 Bino annihilation tends to give ierarchy to mix strongly, so the owever, it is possible to adjust 1). We will consider the case M mess − mess USY mass spectrum. The lightest mess M n M ( M / 2 4 6 8 10 12 14 16 stabilization occurs through superpo- 2 0 (b) Third family soft scalar masses.

− 500 Mass ( GeV ) GeV ( Mass -500 X 1000 -1000 1, and = − GeV. In this example, we will set g = α 10 – 23 – g α 10 is light enough coannhilation eﬀects can reduce the ˜ t 3 ∼ 2 1 M or = 1, M (GeV)) M µ τ ( m 10 mess α log 3. With this stabilization mechanism, , such that M / 2 /C − C F = , then g P 1)) α M − GeV, which corresponds to a smaller value of the cutoﬀ scale Λ n ∼ (a) Gaugino masses. 6 ( / 2 − = 10 2 4 6 8 10 12 14 16 The renormalization group evolution of (a) the gaugino mass 3, and consider two possible values of the messenger scale: /

800 600

= ( 2 2600 (GeV) 2400 2200 Mass 2000 1800 1600 1400 1200 1000 mess − = 4, corresponding to stabilization through a perturbative M If the messenger scale is lowered, for example to In figure 6, we present the renormalization group evolution o /X = n X g fication behavior for theterms gauginos and occurs scalar mass-squares before alsorunning the unify of decoupl at (a) the the miragepresence scale gauginos of and the (b) messengers the flattens third the family running of soft the scal gluino family soft scalar mass-squares for Figure 5: to left-right splitting. The gauginoslightest have supersymmetric too partner large (LSP) a is masstoo nearly h large pure a bino. relic density, but when ˜ too large since coannihilationother effects parameters to are obtain not a significant; low energy h spectrum consistent relic density to an acceptable value. With this set of parame the mirage unification behavior occurs at a lower scale than i The resulting slightly lighter gluinosuperpartner results is in a again lighter almostabundance S purely is bino, larger than and it due was to in the the ligh previous case with tential couplings of theF form given in eq. (2.16). As shown in of ator, which implies α Example 2: stabilizationample, via we nonrenormalizable will operators consider a model in which the and hence if Λ and the soft scalar mass-squares of the first generation for JHEP08(2008)102 hat and by a 1 X M E1 L1 D1 U1 Q1 E1 L1 D1 U1 Q1 M M M M M M M M M M (GeV) ) (GeV) ) GeV (the scale µ µ ( ( 9 10 10 Log Log es and (b) the first gen- ss splitting of e in the wino component ark matter constraints, it is once again almost pure bino, 3. / f the lighter spectrum, which he messenger scale is smaller, 2 ameter set, we find that while ressed superpartner spectrum. ale of about 10 − 0 again lead to mirage unification large masses for the superpartners = g < α g 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 α 0 0 (d) First family soft scalar masses. (b) First family soft scalar masses. 1 example of figure 4 and figure 5). The

500 500 0. For the case with a higher messenger Mass ( GeV ) GeV ( Mass Mass ( GeV ) GeV ( Mass -500 -500 1000 1500 1000 − -1000 GeV, and (c) the gaugino masses and (d) the < = GeV, for the case of the stabilization of = 1 and 10 g g 6 α m α terms of the third generation are not displayed – 24 – α = 10 A = 10 3 2 1 3 M M M 2 1 mess M mess (GeV)) (GeV)) M M µ µ M ( ( M 10 10 log log GeV in panels (c) and (d). The physics of this case resembles t 6 = 10 (c) Gaugino masses. (a) Gaugino masses. mess M 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 The renormalization group evolution of (a) the gaugino mass

500 800 600

Mass (GeV) Mass Mass (GeV) Mass 2500 2000 1500 1000 2400 2200 2000 1800 1600 1400 1200 1000 . Though the LSP remains almost pure bino, the slight increas 2 nonrenormalizable superpotential term, with scale, the small threshold effects which result when first family scalar mass-squares for (b), and for eration soft scalar mass-squares for Figure 6: of the radiative stabilization model, since of the gauginos and soft scalar masses at a relatively high sc explicitly, as they too resemble that of the M dependence the soft mass-squares and acts to lessen theact relic to density; increase this the competesthe relic neutralino with abundance. relic effects For abundance o this is particular too par large to be allowed by d with SU(3) charges and a largewith stop mass a splitting. characteristically The too LSPthe is large gluino relic is abundance. significantlyHowever, When lighter, the t resulting in gauginos a mix more more comp strongly due to the smaller ma gluino is relatively heavy, which again leads to relatively JHEP08(2008)102 1, 0. by − > = 5), GeV. X g = n α 12 n 1, just as − , while still 30 TeV and resulting in g = 10 = α n ∼ 2 (i.e., field originates g / 2 α 1 mess / or that results in X 3 − M m = g GeV. , if ino masses, (b) the first α 3 he messenger thresholds e to stop coannihilation. 8 ly participate in the RG / GeV. For our last example, we uino is light, the squarks terms no longer unify at a 1 o masses, and particularly scalar masses, which is the e case in which the messen- ) s become nearly degenerate respond in most cases to a elic density falls within the = 1 and 10 2 -conformal behavior for the A ll-tempered neutralino” [30]. n our previous paper [21], we erators, clearly / g = 10 senger scale. This feature can emerge for the case of 3 wed by experimental bounds.) till not suitable for benchmark come important and can drive to a lesser extent for the third on of the messenger threshold α neutralino is a general mixture raction of the than the gluino. The very light = 10 = 1 and e mechanism of stabilizing /m = ion soft scalar masses, and (d) the mess 4 e, the m m = 3 case results in M α (Λ α mess n ∼ M 0, with higher powers of 0 in eq. (2.16). From eq. (2.54), we see < g GeV and n < < α – 25 – 12 1 − = 10 GeV. As in previous examples, we will consider two GeV, as shown in figure 8. In this case, we see that the is now given by 8 12 mess i GeV than it is for 10 is larger than the original mirage unification scale. 6 M X 0. To obtain a concrete example, we will choose h = 10 ∼ > mess = 10 g mess M α mess M (note that the renormalizable mess M X M = 1. Since g α = 4 case corresponds to the lowest nonrenormalizable operat GeV, then n 10 10 The Since the messenger threshold corrections drive the gaugin In figure 7, the renormalization group running of (a) the gaug Several of the qualitative features shown above change in th approaching its limiting value of zero. For higher-order op ∼ g slightly smaller for in the case ofnonrenormalizable radiative operators stabilization). leads More to generally, th the stabilization of α Λ negative, has a smallereffects magnitude, is such less that efficient and thepresented more cancellati a dramatic effects sample can model appear. with similar I features, with values of the messenger scale: in which case In particular, the threshold effectsthe from gauginos to gauge unify mediation at be and a low mix scale strongly of with order each 1between TeV. other; The the accordingly, gaugino bino, the lightest wino,experimental and limits, higgsinos. indicating In that(We this the note example, in LSP passing, the is however, r that indeedstudies, this a because parameter “we the choice is lightest s Higgs mass is too low tothe be gluino allo mass, to“compressed” be SUSY lighter, spectrum the (seealso low energy remain e.g. soft relatively [31, masses light, 32]).gluino cor though leads significantly to Since heavier an the intriguingquasi-conformal gl feature fixed of point behavior the at RGbe scales flow seen below of clearly the the for mes soft thegeneration lighter in generations panel in (c) panellower (b), due the and to gluino Yukawa mass couplings,evolution to of of such figure the an 7. other extentscalar sparticles that soft T (particularly it parameters, the does squarks). leading not Furthermor to strong the quasi in which the dark matter abundance was inExample the allowed 3: range du stabilizationconsider via models in nonperturbative which effects. from that the nonperturbative superpotential dynamics, self-inte suchthat that this will result in high mirage scale, since generation soft scalar mass-squares, (c)third the generation third generat trilinear scalar terms, are shown for ger scale is lowered to Here we can immediately see dramatically different features JHEP08(2008)102 = 1, pton m α 800 GeV) E1 L1 D1 U1 Q1 ∼ M M M M M (GeV) ) (GeV) ) (GeV) ) µµ µ ( ( ( 1010 10 LogLog Log GeV. The threshold es, (b) the first family 9 rong enough to drive to (i) very light states, which he gluino is the LSP. This -squares. The superpartner nce the gauginos, soft scalar o-dominated next lightest su- ely heavy states ( d (iii) heavier squarks, of order ch the gluino becomes very light superpotential terms, with b τ t A ass-squares, and (d) the third generation A A 22 4 4 6 6 8 8 10 10 12 12 14 14 16 16 2 4 6 8 10 12 14 16 00 0 (b) First family soft scalar masses.

(d) Third family soft trilinear terms.

800800 600600 400400 200200 800 600 400 200 Mass ( GeV ) ) GeV GeV ( ( Mass Mass Mass ( GeV ) GeV ( Mass -200 -400 -600 -800 12001200 10001000 1000 – 26 – E3 L3 D3 U3 Q3 3 2 M M M M M 1 M M (GeV)) (GeV) ) M µ µ ( ( 10 10 log Log terms all unify at a high scale of order 10 GeV. 0, the threshold eﬀects can either lead to a gluino LSP or a sle A 12 > g α = 10 (a) Gaugino masses. mess M The renormalization group evolution of (a) the gaugino mass 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 0 (c) Third family soft scalar masses.

800 600 400 200 800 600 400 200 Mass ( GeV ) GeV ( Mass Mass (GeV) Mass -200 -400 -600 -800 2200 2000 1800 1600 1400 1200 1000 1000 2 TeV. For . 1 − = 1, and terms, for the case of stabilization due to nonperturbative 1 g . Figure 7: α messenger scale is below the mirage unification scale, and he mass-squares, and the effects drive the gluinoresults mass again to in very aspectrum small is quasi-conformal values, slightly such behavior less compressed, that ofoccurs since t the the at point a soft at later whi mass are point the in gluino the LSP, running. twoperpartner light The (NLSP)), neutralinos spectrum and (including includes the the lighter win chargino, (ii) moderat soft scalar mass-squares, (c) theA third family soft scalar m LSP, if the RGdrive running the above binos the and winos messenger sufficiently threshold light. is not st which include the sleptons,1 charginos, and neutralinos, an JHEP08(2008)102 = 1, m α vectorlike . E1 L1 D1 U1 Q1 T M M M M M N (GeV) ) (GeV) ) (GeV) ) µµ µ ( ( ( 1010 10 and LogLog Log X es, (b) the first family ulus and anomaly medi- alrmodulus ähler ge mediation, the minimal gauge mediated terms arise uge mediation, and anomaly acquires a SUSY breaking F nomenological aspects of de- g-motivated model of super- ontributions, with the propor- n be either nonrenormalizable iator field contributions to the observable X superpotential terms, with b t τ ass-squares, and (d) the third generation A A A 22 4 4 6 6 8 8 10 10 12 12 14 14 16 16 2 4 6 8 10 12 14 16 0 00 (b) First family soft scalar masses.

(d) Third family soft trilinear terms.

500 800800 600600 400400 200200 Mass ( GeV ) ) GeV GeV ( ( Mass Mass Mass ( GeV ) GeV ( Mass -500 -200-200 -400-400 1000 12001200 10001000 -1000 can be stabilized either by radiative correc- X – 27 – E3 L3 D3 U3 Q3 3 2 M M M M M M 1 M (GeV)) (GeV) ) M µ µ ( ( 10 10 log Log GeV. 8 = 10 (a) Gaugino masses. mess M The renormalization group evolution of (a) the gaugino mass 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 0 0 (c) Third family soft scalar masses.

500 500 Mass ( GeV ) GeV ( Mass Mass (GeV) Mass -500 2500 2000 1500 1000 1000 -1000 We have explicitly demonstrated that = 1, and terms, for the case of stabilization due to nonperturbative g Figure 8: pairs of messenger fieldsterm with through SM supergravity charges. effects due The to mediator its mixing with the K tions or higher-order terms in the superpotential, which ca or nonperturbative in origin. Thesector resulting soft gauge terms mediated are proportional to the anomaly mediated c α flected mirage mediation, whichsymmetry is breaking a in which recently modulusmediation proposed (gravity) all strin mediation, ga contribute toKKLT mirage the mediation soft model, terms.ated which terms, predicts In is comparable extended deflected mod tofrom mira include additional gauge visible mediated sector terms. fields: The the gauge singlet med 5. Conclusions In this paper, we have investigated both theoretical and phe soft scalar mass-squares, (c) theA third family soft scalar m JHEP08(2008)102 α (A.1) = 1), m . For the α τ y r perturbative , which is the , and m b α y 8093 and a Cottrell , t y DE-FG-02-95ER40896. ifferent known models of tting in which to investi- ngly depends on the sign these parameters are taken y-generated operators, the d from the value obtained al KKLT model ( a bino-dominated LSP. For oup evolution and resulting three of the standard super- by fixing a small number of ation r values motivated from the ss mirage unification scale to ized framework, in which the supersymmetry at the LHC. ggsino (i.e., “well-tempered”) , ight. In this case, the LSP can is given by stabilization mechanisms. We ork lends itself to more general 2 | low energy supersymmetry. One ects are small and the basic pat- for the soft terms. We computed X ing would be of interest. Deflected ilm y | lm X 2 1 are the normalized Yukawa couplings. Here − ) ). Since the anomaly mediated and modulus i ilm – 28 – y /C (Φ a C c 2 a F g g α a X = = 2 /X i X γ that emerge from each of the F g α (with g 0, which corresponds to stabilization by radiative effects o α < g is the quadratic Casimir, and α a c 0, which corresponds to stabilization by nonperturbativel . For In conclusion, deflected mirage mediation provides a rich se g > α g tionality factor we will consider only the Yukawa couplings of the third gener in which mediated terms in this scenario are comparable (their ratio of mirage mediation), all threethe contributions MSSM are soft relevant termspattern and of investigated the superpartner renormalization masses gr at low energies for the minim threshold effects can be large;TeV this values, and can typically deflect drives the theeven gaugino stops be ma and the gluinos gluino, to though be itneutralino l can which also is be a allowed mixed by bino-wino-hi dark matter constraints. gate the theoretical and phenomenologicalparticular implications direction of of interest is tomirage concrete mediation model also build has thesymmetry advantage breaking that mediation it mechanisms encompasses withinrelative one general contributions from eachparameters. mechanism can Although be here adjusted top-down we approach, we considered can discrete take a paramete to bottom-up approach be in continuous, which whichsupersymmetry allows breaking. us to The resulting dial generalizedstudies between framew many of of the the phenomenologicalWork d along implications these of lines low is energy currently in progress [33]. Acknowledgments This work is supportedP.O. by is the also U.S. supportedScholar Department Award in of from Energy part Research grant Corporation. by the NSF CareerA. Award PHY-034 Anomalous dimensions At one loop, the anomalous dimension is given by focusing on values of find that the gauginoin mass the mirage absence unification ofof scale the is messengers. deflecte The low energy physics stro nonrenormalizable superpotential terms, the thresholdtern eff of soft massesα is similar that of mirage mediation, with JHEP08(2008)102 . i γ the = (A.5) (A.6) (A.4) (A.2) (A.3) ′ i γ 3 i δ ) b , , b 2 1 3 2 b . For the MSSM fields, the i , g y δ ) 7 ilm ilm 16 2 b + , y y y 2 1 t , b b g − 2 τ 2 | t + b 2 2 y 2 τ 13 15 2 t g , ( y ilm y 3 2 τ ( 3 3 y − y − , , i i . | 3 − − i 3 3 δ , δ 2 2 1 2 4 1 i i − t b t b δ − 3 2 g g 3 δ δ g i b b b b τ 2 1 lm g 3 2 2 2 2 1 t b t b δ 9 5 X 4 4 2 2 t b t b b g b 2 τ y y y y 2 τ y y y y 3 − 2 2 y 3 3 16 − 2 2 y 3 3 1 − 1 30 30 2 3 ) , 2 2 − − − − − i g − − − − − − 3 3 g , the anomalous dimensions are i + i + 2 2 2 2 2 1 1 1 1 1 4 4 4 4 4 d 2 3 1 1 1 1 1 τ – 29 – δ 3 δ (Φ 2 g g g g g 2 16 4 g g g g g 2 y 2 τ τ a H g 1 1 1 1 1 g − y b 8 2 3 3 3 c b b b b b 2 2 15 15 10 10 10 3 − + 2 τ 2 a 2 τ b y b 2 2 8 2 3 3 3 y b b 2 15 15 10 10 10 3 4 2 + + + + − + + a y y and g 2 2 2 2 2 2 2 3 3 3 2 1 2 2 + + + + − + + u + 4 + g g g g g g g a + 6 4 4 4 4 4 4 4 3 3 3 2 1 2 2 3 6 8 3 8 3 3 3 2 5 2 2 3 8 2 2 b t H X g g g g g g g 2 y y t , does not change according to eq. (A.1), and hence 2 3 3 3 2 1 2 y c b b b b b b b ======i 6 8 3 3 3 2 3 3 3 2 5 2 8 8 γ E = 2 d u = 3 = 6 = , i , the beta function of the gauge couplings changes because of U,i L,i H E,i Q,i H D,i t b ˙ τ ======γ L γ γ γ γ γ γ b γ b b , d u c mess U,i L,i H H E,i Q,i D,i ˙ ˙ ˙ ˙ ˙ ˙ D ˙ γ γ γ γ γ γ γ M , c U , is the beta function for the Yukawa coupling Q ilm y b ’s are given by the expression i γ ’s are given by i ˙ where MSSM fields respectively. Above messenger fields. However, The ˙ in which γ JHEP08(2008)102 , , , 3 i (A.8) (A.7) δ , in s )) (1995) =1 D Snowmass n plications N 436 (1984) 1. − Q . . n ) 110 ]; , which appears in E − i n θ d . H − ) n k L n n − Erratum ibid. − p red parameters, is given by − ( Phys. Rept. j d 2 b , n y H hep-ph/0202233 n − hep-ph/9508258 i . )+ − n U p ( n (1994) 125 [ − , , Some issues in soft SUSY breaking terms 2 τ 3 3 p y i i − ( 2 δ δ , 2 Locally supersymmetric grand unification ) ) Q | 3 in eq. (A.4). Finally, i − U n D , δ B 422 ijk ) ) n (1997) 157 [ n ) N − y U D , | E ]. − − u 3 n n + n i . H Supergravity as the messenger of supersymmetry – 30 – Towards a theory of soft terms for the (1983) 215; δ Q Soft supersymmetry-breaking terms from supergravity Q a Gauge models with spontaneously broken local − ) − i,j,k − X n b hep-ph/9709356 n C 74 n . E i , Q L Q − − = θ − n − n n n ) (1982) 343; p ′ a u d i ( − b − − − H H B 126 Nucl. Phys. 2 t Q L d u d n n , ( y n a H H H 2( c − − Analysis of the supersymmetry breaking induced by Z. Physik n n n 2 a with − B 119 p p , g (1983) 2359; − hep-ph/0312378 ( ( d are − − − a 2 2 2 t b 1 b H i p a p p hep-ph/9707209 g y y θ ( ( ( n X 4 4 , 2 2 2 ]; t b τ 1 Phys. Lett. 15 The Snowmass points and slopes: benchmarks for SUSY searche y y y The soft supersymmetry-breaking lagrangian: theory and ap D 27 − is the same as − − (1982) 970; , 6 6 2 ′ p i = 4 2 2 1 1 + ( θ g g i − − − (2005) 1 [ 2 2 τ 2 49 , Phys. 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