SUSY's Ladder: Reframing Sequestering at Large Volume

SUSY’s Ladder: reframing sequestering at Large Volume

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Citation Reece, Matthew, and Wei Xue. “SUSY’s Ladder: Reframing Sequestering at Large Volume.” Journal of High Energy Physics 2016.4 (2016): n. pag.

As Published http://dx.doi.org/10.1007/JHEP04(2016)045

Publisher Springer Berlin Heidelberg

Version Final published version

Citable link http://hdl.handle.net/1721.1/103156

Terms of Use Creative Commons Attribution

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP04(2016)045 Springer April 7, 2016 : March 23, 2016 : January 24, 2016 : Published Accepted 10.1007/JHEP04(2016)045 Received doi: Published for SISSA by [email protected] , b . 3 and Wei Xue 1512.04941 a The Authors. Effective field theories, Supergravity Models, Supersymmetric Effective c

Theories with approximate no-scale structure, such as the Large Volume Sce- , [email protected] E-mail: Department of Physics, HarvardCambridge, University, MA 02138, U.S.A. Center for Theoretical Physics, MassachusettsCambridge, Institute MA of 02139, Technology, U.S.A. b a Open Access Article funded by SCOAP Keywords: Theories, Field Theories in Higher Dimensions ArXiv ePrint: two special higher-dimensional theories: five-dimensionaltype supergravity and IIB ten-dimensional supergravity. This givesultraviolet a physics phenomenological which argument is in favor differentsuperstring of from ten theory. standard dimensional arguments based on the consistency of study SUSY’s Ladder using a superspacein formalism previous that computations makes manifest. the mysterious Thistheory cancelations opens understanding the of possibility the ofin a phenomenology the consistent of effective small field these ratio scenarios,with of approximate based string no-scale to on structure Planck power-counting enforced scales. by We a also single show volume that modulus four-dimensional arise theories only from and the Planck scale, which weSupersymmetry call in SUSY’s which Ladder. the Thisscalar same is small a soft parameter particular masses, realization suppresses of scalaror gaugino Split masses soft string relative scale masses to relative relativeinteresting to properties, to and the the can Planck gravitino avoidlems, scale. dangers mass, and including the and the This moduli-induced gravitino LSP the scenario problem problem, UV that has flavor plague prob- cutoff many other supersymmetric phenomenologically theories. We Abstract: nario, have a distinctive hierarchy of multiple mass scales in between TeV gaugino masses Matthew Reece SUSY’s Ladder: reframing sequesteringVolume at Large JHEP04(2016)045 18 6 9 32 15 11 6 13 26 21 11 23 27 15 28 – 1 – 29 22 6 10 8 27 18 31 24 25 17 26 21 2 26 28 and b terms 6.4.3 Superpotential contributions µ 6.4.1 Phenomenological6.4.2 requirements K¨ahlercontributions 5.2.1 KKLT 5.2.2 Large Volume Scenario 7.3 Further explorations 7.1 Phenomenology of7.2 SUSY’s Ladder Building an effective field theory 6.2 Gaugino masses 6.3 A terms 6.4 5.2 Moduli stabilization in keinstein frame 6.1 Soft scalar masses 4.1 Limiting hierarchies4.2 with quadratic divergences SUSY’s ladder 5.1 Effective field theory 3.1 Three frames3.2 for Kaluza-Klein reductions Chiral superfields and no-scale structure 2.1 Single-field no-scale2.2 structure Kinetic unmixing2.3 and the Cheung/D’Eramo/Thaler Sequestered gauge chiral2.4 superfields Implications for2.5 the moduli-induced dark Integrating out matter heavy problem moduli A The heterotic string, M-theory, and no-scale structure 7 Discussion 6 Soft SUSY breaking from superspace 5 Deviations from no-scale structure and moduli stabilization 3 Dimensional reduction: why no-scale? 4 Hierarchies in sequestered theories Contents 1 Introduction 2 Pure no-scale: phenomenological results JHEP04(2016)045 . ] ], 0 1 χ 32 19 , (1.1) m 18 ]. Light 24 . 2 / 3
]. Again, adding moduli to the sce- 33 ] are readily understood by working in 100 TeV / ]. This motivates the question: how much 8 ] problems. We will focus our discussion on 2 , 3 ]. In Regime III, decays of the gravitino are / 7 – 3 11 1 31 – m – ]. This implies that gravitinos in a radiation- 9 – 2 – 28 ], decays of superpartners after freeze-out [ 13 , 17 ], so there is a potentially problematic population of 12 [ 23 10 MeV – 2 Pl ≈ 20 2 . Gravitinos can be produced in the early universe through /M / ] and moduli [ 2 3 1 / 3 T 3 3 – 1 ], freeze-in [ m π 16 ] enforced by the volume modulus of either a single extra dimension – 193 384 6 ]. We will not discuss this low-scale SUSY breaking scenario further – 14 4 = 27 – 2 / 3 25 We are motivated to study sequestered theories due to the prospect for a clean solution in this paper.already Regime begun, II, is particularly in dangerousunconstrained which [ by gravitinos BBN decay but invalidate aftergenerally the Big leading standard to thermal Bang a WIMP larger Nucleosynthesis freeze-out darkespecially has calculations, matter in abundance that light risks of overclosing indirect the universe detection [ constraints [ thermal scattering [ and inflaton or modulithese decays particles [ in the early universe.In The this first case, case Standard is Model whengravitinos the superpartner pose gravitino masses is a must lighter lie than varietyare below of about also cosmological 10 light TeV problems [ [ that are compounded if moduli fields This separates the cosmology ofpending supersymmetric on theories the gravitino into mass several andmass, different its as regimes relation de- illustrated to in the lightest figure Standard Model superpartner the gravitino, since it necessarilyModel. exists in (We expect any supersymmetric that moduli completionis of fields less the also obvious.) Standard exist In inall the any theory MSSM, decays of the is quantum gravitino Γ gravity, decaydominated but width this universe decay assuming when ample the phase temperature space is for about frame. We view our resultsory as and a step top-down toward string bridgingversion the constructions. of gap split between Phenomenologically, bottom-up SUSY we field with are the- scalar led superpartners to at an theto interesting cosmological PeV gravitino scale. [ argument that singles outrelated 10d to string Type theory. IIB We willbreaking is explore effects, how completely no-scale showing structure independent that can a suppress ofthe various variety any Large SUSY- of consideration Volume initially surprising Scenariosuperspace results of in previously string the derived conformal theory in compensator [ formalism and choosing not to work in Einstein can we reasonably expectwill the gravitino explore to this be question,arguments decoupled largely suggest from from that gauginos? a the simplest, In bottom-upno-scale most this effective structure robust paper, field [ way we theory to decouple viewpoint.or the gravitino Our of is six through extra dimensions arising from Type IIB supergravity in ten dimensions. The An appealing feature ofcandidates supersymmetry in is the the form automaticscenario of presence assuming of gauginos the WIMP or universe darkitino, higgsinos. is as matter populated an Often by extremely these weakly acomplicate coupled are hot or particle even discussed plasma that ruin in of will this not MSSM a appealing be story fields. thermal in [ thermal The equilibrium, grav- can 1 Introduction JHEP04(2016)045 ). ψ Q TeV. F [TeV] (1.3) (1.2) 4 2 2 / / 3 3 10 m > ∼ < T 2 / 3 FO T m Safe region IV: Conventional thermal history. Calls for strong sequestering. TeV 4 10 FO ]. This suggests that what ]. Throughout the paper we < T 45 44 . , [ and so this term immediately ] and indirect detection makes 2 for the auxiliary field (e.g. / Φ 3 42 ]. It is only in Regime IV that 2 Φ 37 [ θ F – 2 2 41 ]. This is an interesting scenario, < T / / – 34 3 3 . ) in boldface, using the same notation . 38 m m 43 Q Q , ) BBN † T TeV 42 4 Dark matter concerns α/π Nonthermal histories. Moduli dilution. Data increases tension. III: ΦQ ( † 10 = 1 + component of Φ ∼ – 3 – > ∼ θ 2 100 TeV λ Φ 4 2 θ . Similarly, one-loop corrections dependent on / d m 3 Q TeV, gravitino production must be suppressed to have a † Z m 4 BBN Q ), and a subscripted 2 10 / Q < T 2 3 < ∼ ψ 2 / m 2 3 / 3 T BBN concerns II: < m ) for the scalar component of a chiral multiplet, a subscripted 0 1 Must avoid producing gravitinos, thermally and during reheating. 0 1 Q χ χ m m a- TeV for weak-to-TeV scale WIMPs. We need a more powerful version fl 3 0 1 χ ) produce gaugino masses to 10 < m 2 2 RG / . Regimes of gravitino mass with distinct cosmology. Conventional thermal relic SUSY 3 10 /µ Low-scale m ∼ Φ I: Scenarios in which the gravitino mass is parametrically larger than the Standard Model 2 / 3 tion? Gravitino overclosure. Late-time thermal in In the simplestproduces theories one a finds scalarlog(Λ that mass term we need is a mechanism to suppress the will follow a convention of writingwithout superfields boldface (like (e.g. for the fermion component (e.g. In superspace, kinetic terms for chiral multiplets take the form lowing anomaly-mediated termsbut to it dominate does [ notm immediately solve the gravitinoof problem, sequestering. because One it waywork leads to in superspace to see with Regime the the III: generic conformal compensator difficulty field of sequestering is the following: we partners have masses at around thewhile 100 GeV the to gravitino TeV mass scale and is constitute at WIMP least candidates four orderssuperpartner of masses magnitude are larger. knownthey as led “sequestered” theories. to gravitino In masses their original one incarnations, loop factor larger than SM superpartner masses, al- Cosmological moduli (or modulinos,present saxions, similar or other issues long-lived and weakly-coupledproblem generally particles) already make the gives gravitino us problem a worse. clear justification But for the seeking gravitino a scenario in which some super- nario tends to increasethis the “moduli-induced dark LSP matter problem”gravitinos abundance are, [ more from severe a [ WIMP cosmological viewpoint, freeze-out a temperature, non-issue: andare they directly so decay applicable. conventional above The WIMP the threshold relic thermal gravitino abundance mass for calculations which this applies is Figure 1 WIMPs are most straightforwardlyIn achieved in Regimes the II rightmost andconventional region, III, thermal Regime relic IV: WIMP. JHEP04(2016)045 can (1.4) T . Scalar mass Pl M . (Such a spectrum was  = 0. This eliminates the Φ : a stratified spectrum with . Of course, the real challenge F ]. This is readily understood 2 2 / 52 3 ].) Following the line of effective – m 47 , we study the one-loop corrections , 50 ) 4 † T + T ( controlling no-scale structure can have highly = 0, so we refer to the case with a linear term Φ , which in realistic examples is a physical field † T – 4 – , we ask when single-field no-scale structure can Φ ) in the action, and proceeds entirely at the level T Φ F 3 θ 1.4 4 d Z -term, but does not decay to the superpartner, higgsinos. µ we show possible ways to deviate from pure no-scale structure, ]: the simplest realization of no-scale structure is 5 54 , 53 , 50 ] and even anomaly-mediation effects [ , we review the basic mechanism by which no-scale structure suppresses soft 49 , 2 48 Our goal is to explore sequestered scenarios driven by no-scale structure, largely taking It has been appreciated for many years that no-scale structure can suppress soft . Meanwhile, the gravitino mass is related to the superpotential and no-scale structure. 2 2 Pl Λ M several large gaps all setfirst by obtained a in single the parametricallyfield string small number phenomenology theory, context in in section [ such as by introducingthe other moduli moduli; stabilization. and we compute Having the the F-terms F-terms of for moduli the and moduli study fields, we study the soft from a heavy gravitino, andscalar find that and it gaugino naturally when introduces theresquares hierarchies is between and gravitino, a gaugino cutoff masses Λ are well below suppressed the by PlanckAll the scale these quadratic things divergences lead from to one-loop, SUSY’s Ladder as shown in figure of ten-dimensional Type IIBon supergravity. the Importantly, desire this to is findof a an linear effective argument term field based like theory. purely ( consistency As arguments such, that this singlecompelling argument out that is ten bottom-up, completely dimensions phenomenologically independentIIB in motivated of as reasoning the the a highlights string special sort 10d theory theory, of Type without context. UV input. It In is section Then the production of dark matterthe and standard the WIMP process scenario. of thermalizationarise In will section be from different compactifying from extracorresponds to dimensions. an isotropic rescaling We ofstructure focus the arises compactified on in directions, the and two argue special case that cases: no-scale when compactification the of single five-dimensional field supergravity or a bottom-up viewpoint that complementsIn work done section in the stringterms. phenomenology We context. also look atSUSY-violating how the decays. scalar field Indecay the to pure higgs fields no-scale through limit, the it turns out that the modulus as “single-field no-scale structure.” Becausewe expect it is that the it simplest isstraightforward version the way). of most no-scale robust In structure, (i.e. this thatof paper the no-scale corrections we structure. can will be limit controlled our in the attention most to the single-field version most dangerous effects that areis naively to proportional to ask whatand subleading not terms just involve aa Lagrange dominant influence multiplier. oninvolving One multiple the fields must physics. can ensure also Theories lead that to with the more leading complicated linear kinetic term functions has masses [ in superspace [ essentially just introducing a Lagrange multiplier to force JHEP04(2016)045 in 3 10 GeV GeV GeV GeV GeV ∼ 3 18 15 9 6  10 10 10 10 10 2 and PeV ≈ β 2 in 10d and     4 5 D 10 ∼  2 Pl / 3 M scalar string m gaugino m M . We find two important results. The m 6 ]). The logic leading to this picture is explained 47 to lead to small corrections to our tree-level  – 5 – GeV GeV GeV GeV 6 18 14 10 10 100 GeV 10 10 10     10 D ] (following earlier work in [ 46 with a brief sketch of two directions that will require future work. The first 2 . Pl / 7 3 . “SUSY’s Ladder”: predicted spectrum in 10d (left) and 5d (right). The Higgs is tuned 4 M scalar string m gaugino m M (Higgs) m (most moduli) (volume modulus) scenario. Ideally, we would havetree-level a supersymmetric use formalism of on superspace thewe same and compute footing the the with effective conformal our K¨ahlerpotential andhave compensator. show the that As the right quadratically a parametric divergentresults. step terms dependence This in on is this an direction, encouraging indication of the consistency of the whole picture. scenario works very wellapproximately for universal scalar achieving masses ascalars. at realistic a The Higgs high scenario massnovel scale also of cosmological which is possibilities 125 lead GeV, appealing for toin starting obtaining from tan accord from the the with correct our standpointdirection WIMP original of motivation dark is flavor of matter a physics avoiding abundance, more gravitino and problems. thorough offers The study second of future effective field theory and loop corrections to the conformal compensator, rather thanas going is to typically Einstein done frameof in and all the working the literature, in SUSY-breaking makesin components effects it section without easy any to apparent simplyfuture cancelations. read direction off Finally, is we the the conclude parametric phenomenology size resulting from SUSY’s Ladder. We argue that this SUSY breaking terms infirst more is detail that in itthat section is we estimated actually on possible generic to grounds. realize The second the is SUSY’s that Ladder working spectrum in superspace of with hierarchies the Figure 2 to be light, but5d. other We fields will are focus at“local most their scenario” of of natural our [ values attentionin for on section the the 10d case scenario, which was previously obtained as the JHEP04(2016)045 . = . 2 Pl has
. M (2.1) (2.2) (2.3) (2.4) . In Ω c -term . T T F = + h 2 ∗ 0 M ]). In later W

) 55 Φ † F . and absent from T 2 are  2 + . Ω 3Φ θ c | . T T + Φ( + h † =  . 0 Φ c T .

W F 3 , it would appear that : the scalar modulus can have Φ + h Φ ψ T Φ θ † T Φ and T 2 ψ 2 d ψ θ † Φ | ψ − Z Φ Φ †  2 ∗ 0 F . We show that this structure can lead = W M Ω + Φ ) + 2 † † ) T F † T Φ F . Φ (Φ – 6 – + F † = 0 = T ( Φ 2 T Φ ∗  F F 2 ∗ M M Φ : : components † 3 ) determine the kinetic term of the scalar component of † † † is not the K¨ahlerpotential but is related to it via T Φ Φ F − δ δ T θ has the unusual feature of being linear in the field ) δF δF Ω 4 appearing linearly in the kinetic function † + d † T T T T Z + 3 + T log( − ( T 2 ∗ 2 Pl = M = 0 involving the = 0 removes many effects of SUSY-breaking, including anomaly mediated M 2 linearly in the kinetic function | 3 L )). The familiar component formalism for supergravity tells us that deriva- 0 φ Φ Ω − 2 F L Pl F | 3 M = only − (3 K / = K F | − 0 appears L . One way to see this from the superspace viewpoint is to recall that, although we have particular, if we ignoreno the kinetic conformal term. compensatorexp( field However, tives of T The fact that effects. 2.2 Kinetic unmixingThe and the function Cheung/D’Eramo/Thaler gauge The normalization of theThe graviton terms kinetic in term requires that From this we can read off immediately two equations of motion: It has a chiralthe superfield superpotential: sections we will lookwhile at still how preserving additional the terms dominant can phenomenological features lead at2.1 to leading violations order. of Single-field no-scale no-scale structure We structure begin with a toy Lagrangian capturing the idea of pure, single-field, no-scale structure. allows highly supersymmetry-violating decays ofa the nonzero decay field width towidth the to scalar the component fermionic of component. astreamlined These chiral presentation results supermultiplet of are but them known a inthat in zero the superspace decay have literature, that but appeared we eliminates give in the a apparent some cancelations of the previous derivations (though not all [ In this section, wepossible will context review of some pure,T of single-field, the no-scale structure: features ofto that suppression no-scale in of structure which soft a in SUSY-breaking(which chiral effects in the including turn superfield simplest suppresses the anomaly conformal mediated compensator effects). We also point out that this structure 2 Pure no-scale: phenomenological results JHEP04(2016)045 the (2.8) (2.9) (2.5) (2.6) (2.11) (2.10) R , because Φ 6= 0. Φ f , . )
i , 2 i T θ ) Φ ) is any function with

i f − h T . with the graviton, which . We have altered their ( . X 2 ) ξ T 2

/ 0 T ( i 3 1 + . † i − ) in components, with 2 † 0 m T T i 3 g † / ( T ) /T δT/T X + = 0 3 T ) ( R + ]). We will refer to it as the CDT g i i W T i ) where h T δT Φ − 58 h ( T f (1 + K 2 Pl ( h µ (arg h √ i ξ − to normalize the graviton kinetic term ∂ R M − ) + ∝ g 0 i 3 µν † Pl

g − . This produces an independent healthy W δT T 2 = 0, despite the fact that ⊃ ( M ) √ + W = µ ) 2 T x † θ ∂ component of the chiral superfield arg h | = 4 δT / arg i 2 T d – 7 – 2 Pl ( µν ) Φ ∗ i θ g 2 Pl O − + T M Z M ≡ ) 3 −h + T 2 E iM g Φ Pl the ( ) (2.7) T † 2 0 , we would have the problem of removing the quadratic ( 2 F − − Φ M T θ − † √ e 2 Φ δT not − + x f Φ 2 / 4 + δT/T is 2 = T 1 Pl K/ 0 d i

D T † Φ M h f Z (1 + equation of motion sets 3 T 1 = 3 2 Pl / † log( mixed φ 1 − + . The procedure is to fix the gauge so that Z T F L M 2 3 e T ) = 1 + h component. By our previous observation, for pure no-scale structure − = = ΦΩ T 2 † ( = = θ Z Φ ξ Φ Z Φ in superspace there will be a term ] (improving on an earlier related idea [ Φ † 57 , Φ were a real field 56 θ equation of motion enforces 4 T † d T In the case of single-field no-scale structure, we have It would be more convenient, however, to maintain manifest supersymmetry through- If F R itself can have a With this gauge choiceappropriately, we and should expanding take out the exponential we find a term and as CDT shownotation to the emphasize that Z the from the product out this process ratherelegant than procedure for writing achieving down thisThaler kinetic has been [ terms recently in explained by components.gauge. Cheung, D’Eramo, Fortunately, This and an gauge choicethe essentially choice builds of the conformal supersymmetric compensator Weyl transformation field, into removing all linear terms in chiral superfields (positive) kinetic term for the scalar, In principle, we could proceed in this manner to disentangle kinetic mixing effects. This is accomplished withTaylor the expansion rescaling ˜ see Ricci scalar. As awe result, can there remove is by a going kinetic to mixing Einstein of frame. the scalar mixing from the action not explicitly written the vielbein multiplet in our action, it is present and wherever we JHEP04(2016)045 + T (2.17) (2.18) (2.19) (2.12) (2.13) (2.14) (2.15) (2.16) in order log( 2 Pl T M is zero while 3 − Φ = . -component, which i term. However, this F K
. . One way to say this . b c ,  . Ω : , 2 2 Pl θ ¯ Q  . From now on, we will set + h M 2 Pl 2 θ Pl 2 . , ... | component of / | M c that are induced by the Weyl and 3 Φ 0 2 3 Pl ¯ + term, or i 2 QQ T /M † θ ¯ z c √ W Q i Q M | µ T . ˜ T 2 i i / W | + † + 1 h Pl 0 1 + T i and T i ¯ † T Q )  M W 1 h † | T 2 ) 3 + Pl − h Q ¯ Q Pl M √ = + T T h M (2 + i 3 T − † components of the chiral multiplet Pl h √ dependence of K/ i 1 Q ( T h . † / T – 8 – M 2  e = c T 3 + F θ Q 2 T h † h 2 = / 0 2 √ Pl T − other θ 1 h , | e 2 i Φ W c / M = † 2 † to the fields 3 2 Pl / 2 c = T T θ 1 1 Φ | i

m i i , but only to the conformal compensator. In order to M c T † T † θ + − Φ 3 Φ 4 T h Φ T T c f 1 d -terms, we add two fields √ T h h T µ + ) have the consequences that Z = = = = ≡ T i 2 h Φ contains a modified modulus field with no 2.4 + c / T f 3 Φ 0 ˜ T F = Φ L ), ( h m Φ = 2.3 ). Our first observation is that in the pure no-scale limit, the fact that L ... has linear pieces in all − Φ : Q c † ˜ T Q 2 Pl = 0 for simplicity. In terms of the canonically normalized field, we have 1 M 0 3 = 0 implies that these terms lead to no soft mass, W − † Φ F does not mean thatin SUSY-breaking the effects couplings are of completelytransformation the that absent removes here. modulus the kinetic They mixingthese of are couplings the present modulus are and manifest gravity. through In the CDT gauge, This corresponds to aT K¨ahlerpotential with the sequestered form 2.3 Sequestered chiral superfields Suppose now that wethat add do to the not pure directlydiscuss no-scale both couple Lagrangian soft to one masses or and more chiral superfields is that at linearwe order, denote Notice that CDT gaugeensuring maintains that the fact thatto the disentangle net the kinetic mixing arising from the linear terms in in accord with thearg familiar result Our earlier results ( Thus we define a canonically normalized modulus superfield JHEP04(2016)045 . . c i
˜ T . c , the . (2.23) (2.24) (2.20) (2.22) (2.21) ¯ +h QQ ) -term like † µ ¯ X QQ ]. The utility z + 59 + X ( ¯ Q θ † 4 ¯ d Q R + . In terms of this field, Q Q , † 2 i / , Q 1 † c i
h , . † 1 c i Q ! T component that is absent in . . decays to squarks is proportional 2 c . This result follows from a su- . +  † 2 . c c ) T θ 3 X θ c h T + h Q 2 m . This ensures that the two decay † Q † c + h c θ c =

† X Q and a coupling † T c ∝ Q c c c Pl X † 2 Q ¯ ) c Q T Q † M + ( m Pl c X z Q ˜ T 3 c + ¯ ψ X Q M 2 † √ = ¯ ) + Q 3 Q c θ will proceed exactly as we have derived above. c M c Pl ψ  – 9 – 2 ˜ √ 1 2 T T † † X ˜ θ T can decay to scalar fields through a | ) M  F c c θ ¯ + 3 → ( QQ 4 c T T Q z d √ c h ˜ T → X T θ Z ( Pl 1+ 4 -term like) couplings, we get a similar structure X h

d − µ M 1 ) ] and ensures that for “typical” moduli, even after taking 3 Pl Pl ) for the conformal compensator, we see that the new terms Z 61 √ ) = Γ( M M 1 3 − 3 √ 2.16 ¯ ( √ QQ ]. / ) † c mass. Importantly, however, there is no corresponding coupling to → 60 , T + with a superpotential c X 55 T having a comparable decay rate to the fermionic superpartners of those ( X − e i modulus † T and the equation of motion 1 without + Q T h ψ ¯ θ Q In the case of holomorphic ( Using the expression ( 4 ψ d † X Z derivation of the decay toThe scalars decay to fermionsF will berates dramatically are different, equal, however, Γ( becausepersymmetric Ward of identity the [ coupling Let us pause herethe pure to no-scale emphasize limit, thecoupling the importance modulus ofscalars. the result This we is havechiral a superfield just dramatic derived. violation In of supersymmetry. For instance, in the case of a so that the decay amplitudetional of to the the modulusfermions, to because scalars such through a such coupling a would coupling arise is from the propor- 2.4 Implications for the moduli-induced dark matter problem giving rise to to the squark mass-squared.from A the similar fermion kinetic statement term. holdsof for Similar the observations the were decays conformal already toremarked made compensator fermions upon in formalism arising in [ [ for studying questions like this one has been This gives rise to couplings so that for on-shell decays we see that the amplitude for one of the terms in our Lagrangian is in our Lagrangian correspond to We define canonically normalized quark fields, JHEP04(2016)045 - . is  F . S and c . W (2.27) (2.25) (2.26) T -parity ! + h R 2 we relate ) 2  † 0 S ) 2 † S ) F s 0 + S -even particles. + 0 R s − ]. S ( S 9( . ( 66 ,  S . − c -odd particles are pro- 65 . W ) † : 0 R 1 2 s controls the strength of a † S + h s + ]. It is potentially of great + S 0  + 0 . α s W c 63 S . ,  W 3( 3 55 α + h , and that to the extent that Φ 0 α W θ 1 + S ) 2 s

d W = 3 α + / , which is nearly zero to the extent that Z 1 S 0 ) S  W † S 0 F S S + θ 3 2 + ( – 10 – + / d 1 -even particles, so that 0  ) 2 † R S Z s S )( S † = ], the modulus has a significant decay rate to + , so that rather than explicitly solving for W T ]. The unique possibilities of no-scale structure for the f 34 1 2 S and expand the Lagrangian in + 64 )( s gauge + † . But a nicer way is to supersymmetrically integrate out the T L 0 + ( T S 0 2 W Pl + S  M ]. T 3 = . ( Φ T Φ † S 2 Pl 62 F ,  Φ M θ θ 13 , so that in the low-energy effective theory we can make the replacement 4 2 Φ S d † d Φ ) for some function Z Z should then decouple and leave the no-scale structure intact. Let us be more θ  T 3 4 ( S d − + f Z = + 3 0 L S − is very large, but in general will be nonzero due to the interaction between To do this, write On the contrary, decays of a no-scale modulus — whose couplings inherently violate = S → . One common way to approach this problem is simply to solve for the VEVs and L terms of all fields, including chiral superfield S the gaugino mass to In this case,W the gaugino mass dependsS on It is clear thatvery the large, superpotential wants toexplicit set about this.gauge In interaction: particular, it could be the case that As a first stepanother toward modulus investigating which the is robustness stabilized of with a the large above mass, results, e.g. suppose there is If the branching fraction isin sufficiently of small, such the scatteringnonthermal final processes dark abundance [ matter could scenario arise have from only freeze- begun to be2.5 explored [ Integrating out heavy moduli importance for the phenomenologythermal of modulus supersymmetric decay dark scenario matter. [ odd In particles the which standard can non- to then a annihilate sufficiently significantly smallpotential (e.g. final to for abundance. decay wino overwhelmingly A darkduced to no-scale matter) either modulus, to with on lead a the small other branching hand, ratio has or the through scattering of the comparable rates [ supersymmetry — cana be particle maximally while far having,perpartner. from to democratic. leading This approximation, has The zero been modulus decay previously can rate pointed to decay out that to in particle’s [ su- supersymmetry breaking into account, decays to particles and their superpartners will have JHEP04(2016)045 and (3.1) µ (2.28) (2.30) (2.29) x = 0. spacetime i 6 Φ d -dimensional . F d h ) (the “volume ! x ) ( . But our imme- † ) 0 2 † L ) S s : α s + . + 0 W s  . α S . c 2( . m 9( , in this context correctly W ( 2 dy Pl − S + h l . 1 W M α 1 6 dy 3

/ -dimensional coordinates ) Φ 1 d W y 3 i ( / α θ † 2 2 ) S † lm d † 0 W h T + S R 2 S ) † + S + x W h ( 2 Φ 1 0 2 T spacetime dimensions down to L / S Φ 1 i n + 3(  † ν 3 2 Pl – 11 – T + / 2 d dx † M ) + µ † 0 T Φ = T S † h dx + ) Φ + D x ¯ θ = T 0 ( 2 i S d ( µν Φ and consider a scalar metric perturbation g h 2 Z Pl is a good approximation only for two very special classes of m = -dimensional coordinates into y 2 2 θM = 3 D 4 ds d α gaugino mass associated with this term! The gaugino mass comes in Z W α no ) corresponds to an isotropic change of the length scale of the internal W x ]. In general, the lower-dimensional theory contains a variety of fields arising ( + L s 67 3 Φ S W Notice that, due to the additional factor in front of -dimensional coordinates modulus”) of the form: Notice that dimensions (and that it is dimensionless). The Einstein-Hilbert term in the from the higher-dimensionalvector metric, modes, including and their the heavierwe lower-dimensional Kaluza-Klein will metric, cousins. only consider For scalar the fluctuationswords, and purposes of we of will the this break overall the discussion, volumen of the higher dimensions. In other to the standard arguments for the critical dimension3.1 of superstrings. Three framesConsider for a Kaluza-Klein compactification reductions from dimensions [ that we studied infour-dimensional section theories: those arisinggravity from theory compactification of and a those five-dimensionaltheory arising super- that from contains compactification four-form of gaugeterestingly, this fields a argument (which ten-dimensional for singles considering supergravity out an underlying Type ten IIB dimensional supergravity). theory is In- unrelated 3 Dimensional reduction: whyIn no-scale? this section,choices we of will frame in review which some to study aspects them. of We argue compactified that gravitational the theories single-field no-scale and structure only at higher order, after we add no-scale breakingnormalizing terms the that Einstein-Hilbert will term leadcompensator will to field, lead to an additional factor in the VEV of the diate interest is mostly in the gaugino mass, which will arise from the term: However, there is Substituting this solution backincluding into higher-dimension the gauge interactions original like Lagrangian, we find a variety of terms, From this we find the equation of motion by varying with respect to JHEP04(2016)045 = . 2 = 1, L (3.4) (3.5) (3.2) (3.3) − d is the Pl n M D . G . , is an integer , )) α  L ) α µν L g d L ]. We will return . Dropping total R (log g 71 µν ν (log g – ∂ − µ ) 2) -dimensional Newton’s ∂ √ L − 68 ) d x , d ( d L d for the lower-dimensional For the special case n/ 54 (log , 2 that leads to kinetic mixing µ 1 Z 2) − (log ∂ may have an interpretation as 53 ) n d µ − i ) d d ∂ µν . So cases where , is Ricci-flat (as it is e.g. for a ( L x g n / 1 R h ( πG 1 , and the 2) lm L | -dimensional internal space. There / 1) L − , taking us to Einstein frame. h ) h 16 ) n − | 2 L -dimensional metric x d − d − ( d . p − n L n y ( + πG = ( i d n has no kinetic term aside from its kinetic n n D L d 2) ( h L → + − G R 1 n n d d − = (8 + V d sets a characteristic length scale in the internal µν = n − R ( g – 12 – ( for the higher-dimensional one, so that d n L = n Pl ), so that it propagates only by mixing with the n V d R ) q 2) x M ( x − L  G ( d L D g ( L R / − n 1 g − √ = 1, the field gV − 1). Now, ) x d n − D √ − d x √ d d and the graviton. A Weyl rescaling of the metric disentangles ( x πG d Z d / d d L Z 2) = (8 d Z 1 πG − , one can show that an appropriate Weyl rescaling leads to Pl D d n 1 -dimensional structure within the 16 πG ˜ α M + 1 − -dimensional volume proportional to 16 πG the kinetic term for n n = ( − 16 . Notice that if n 2 − = − p = 4 dimensions: D Pl is the Ricci scalar associated with the = d ˜ ≡ kein M d Einstein L removes EH α n L R i L For our purposes it is interesting to consider a different Weyl rescaling, namely one We can eliminate the kinetic mixing between the volume modulus and the graviton Assuming that the metric on the internal space, L Leaving no stone unturned in the pursuit of bad physics puns. h 1 n can be particularly interesting, asthe the volume factor of multiplying an are precisely two suchdown integer to cases, for compactifications of higher dimensional theories For other values of where dimensions: the internal volume, in particular, goes as which graviton. In some sense thisis is a the very frame useful that one;we is we already maximally can saw far refer that from to the Einstein simple it frame, compactification as but ansatz the it led keinstein frame. directly to such a Lagrangian. derivatives, this results in the action This is the standard Einsteintions; frame a Lagrangian simple which rescaling is by most a commonly constant used now for suffices calcula- to canonically normalize the field mixing with the graviton.compactifying This 5d results supergravity in theories an toto approximate this four no-scale point dimensions structure in [ arising more when detail below. by doing a Weyl rescaling of the metric, We will alsoPlanck find scale and it usefulV to write reduces to Here higher-dimensional Newton’s constant, constant is a factor of the between the scalar field the kinetic term of the metric and that oftorus the or a scalar Calabi-Yau manifold), we find that the higher-dimensional Einstein-Hilbert term action resulting from this ansatz will not be canonically normalized, as it will contain JHEP04(2016)045 α (3.6) (3.7) -form -cycle p p = 3; and , the imag- = 1. If we , beginning † , α D T α for any + = 5 B T Σ R , n = in the original theory. b = 6 , lives in the supergravity B ? We could approach this d R = 6; = 1 always yields n , α + derivatives , T = 8 † is compact, i.e. it has a gauged discrete F T 2 b θ , n + + T = 3 T – 13 – = d θψ α 2 L . Such a symmetry in the compactified theory should √ (mostly) through the combination b has an (approximate) shift symmetry. In other words, + T real part of a chiral superfield T ib theory lead to ). Because the Ricci scalar † + T τ = 1, and we reproduce our earlier result for compactifications = 4, and we find that the action scales as the fourth power of + = α α 11, to have the potential of finding supergravity theories, there T T ( ≤ Φ n † supergravity . After we compactify, we obtain a scalar field ]. p Φ + 75 = 2. Intriguingly, all of the special cases originate in theories in 10 or d ...µ – 1 µ = 72 , α B D = 3 = 1: in this case, = 6: in this case, is a more useful parametrization of the underlying field for any particular value of , n = 16, but it seems unlikely that there is interesting physics associated with them. from five dimensions down to four. n the internal length scalecorresponds or, to equivalently, compactifications the from volume tentheory to dimensions [ down the to two-thirds four, power. as in This superstring n α If the Lagrangian depends on ) • • D x = 7 ( originate with some sort of symmetryshift in symmetries the are parent higher-dimensional very theory. familiargauge In consequences fields fact, of such supergravity theoriesΣ containing in the compactificationshift manifold. symmetry originating The from field large gauge transformations of inary part of thegiven complex the scalar parametrization our action is (nearly) independent of so that the actionquestion is by linear systematically in studying the theand different repeating higher-dimensional the exercise supergravity above theories inthat the we SUSY expect setting. will Instead, let identify us all give of a faster the argument relevant cases. approximated by multiplet, this means thathigher-dimensional we would like to ask: when does Kaluza-Klein reduction of a The above discussion at firsttranslate glance between seems frames to as be weL a like, purely and academic there exercise: isor we no that are deep keinstein frame free physical is to principle preferred toWe telling any would us other. like that This to changes in find the approximate case of pure supersymmetry. no-scale structure, i.e. a Lagrangian that is well- at This is as farsupersymmetry as to it see is why this useful frame to is go of3.2 in particular the interest. nonsupersymmetric Chiral setting. superfields and Now let no-scale us structure turn to More generally, away fromrestrict four to dimensions, theare case precisely three otherd special cases: 11 dimensions. There are many other integer solutions at larger values of JHEP04(2016)045 ]. .) 76 A (3.8) -form p ] make = 6 has ) that is 78 , p x, y ( 5 is obtained by = 8 to be related to b , n τ dyA H = 3 d ) = x ( a . 2) -brane worldvolume theory [ − 1 = 2. D d − , p , so it remains to check whether a + d α n ( = 3 n , n – 14 – r = = 7 the supergravity multiplet. Because , and the six-dimensional internal volume scales as α d T -dimensional submanifold, we expect = p p part of is a mode of the higher-dimensional graviton, the gauge field = 3 and that gives rise to a four-dimensional axion when integrated over L , p ) gives rise to a four-dimensional axion = 5 and . x, y MNPQ α 2 ( , n / L C 3 M ) A ∼ = 6 = 1: this is the case of a five-dimensional supergravity theory, in which a T = 4: this is the case of 10-dimensional Type IIB supergravity, which contains d τ ]. It is unclear whether any physical significance can be attached to the other , p , p 77 (Re = 1 = 6 a four-dimensional cycle inpart the of geometry. the TheV K¨ahlermodulus volume ∝ of this four-cycle is the real one-form paired with the radion in a 4dn chiral superfield. a four-form n It is noteworthy that no-scale structure, which has interesting phenomenological Because • • -form gauge fields with the way) to a bottom-up argument for Type IIB superstrings as a UV completion. other hand, we expecttence that of the the existence fundamental of stringit as branes essentially a inevitable as soliton that supergravity of the solitonsType only the and IIB UV the superstring. complete exis- version Attemptingtaining of to Type gauginos decouple IIB near supergravity the the is gravitino weak the from scale cosmology has while led main- us (with a few plausible assumptions along special cases properties, provides an argumentdimensional supergravity for theory. considering Notice the thatconsistency superstring arguments compactification theory selecting of and ten Type the dimensions usual IIB played worldsheet ten- no role in our discussion. On the dent of one’s choicethe of relationship frame, of is pastply not claims acquired only of by after no-scale mixing certainAway structure with from in fields the four heterotic are dimensional graviton. string integrated compactifications,been theory, (We observed out, the which to explain special to lead ap- case to ourfour-folds no-scale of structure [ statements in here compactifications of in M-theory appendix on Calabi-Yau ten-dimensional Type IIB superstringThe theory absence compactified of on no-scaleerotic Calabi-Yau structure strings) manifolds in and [ ten-dimensional Type IIA Typethe supergravity, I supergravity for multiplet, supergravity instance, so (including is thatthemselves het- due the into to chiral the multiplets superfields lack which corresponding of always to a have moduli four-form an organize in independent kinetic term that, indepen- The arguments givenstudied here in clarify the why two four-dimensional contexts of no-scale five-dimensional supergravity supergravity has compactified on been a circle and We have already identified thegauge cases field of of integer appropriate rankdimensional exists theories in we the corresponding findinternal supergravity dimensions precisely theories. can two be For four examples associated with in a which chiral superfield isotropic with rescaling no-scale of structure: the we are looking forintegrating should the be gauge field overthe a volume of that submanifold.p Hence, we are looking for supergravity theories containing JHEP04(2016)045 , we 1 1 TeV. < ∼ = 0. This 0 1 χ Φ F m that no-scale 3 no-scale structure: the ] and are closely related One interesting possible 91 2 – . 88 5 ] gives an argument that even in IIB = 0 is enforced by multiple fields, 79 single-field Φ F ]. These divergences have the potential to 87 – that no-scale structure can suppress tree-level 81 – 15 – 2 ]. 92 , 54 approximately acts as a Lagrange multiplier setting T F TeV, while a neutralino dark matter candidate will have 4 10 > ∼ 2 / . We have nothing to say about cases where ], that realize similar physics but where the chiral superfield enforcing no-scale 3 A m 80 The potential problem with consistent no-scale structure arises from divergent radiative We emphasize that our arguments are restricted to An anonymous referee has pointed out to us that section 4.1 of [ 2 to spurion arguments in refs. [ theories with many K¨ahlermoduli, preciselybreaks the SUSY isotropic in rescaling the field no-scale weapproach. limit have (i.e. considered its is superpartner the is one the that goldstino). This clarifies the generality of our ric theories do contain powerdestabilize divergences large [ hierarchies anddimensional invalidate analysis extreme argument realizations sheds of lightexpect sequestering. on the to A largest be simple gaps consistent inthe the with expected spectrum the size that existence of weof can loop of gravitons. corrections such Similar to divergences. estimates scalar have We and been will performed gaugino in simply soft refs. masses compute [ arising from loops argument that it can,— precisely a in situation a that theory obtains with automatically a when UV we compactify cutoffcorrections. large well Although extra below soft dimensions! the supersymmetry Planckfrom breaking renormalizable scale removes interactions quadratic in divergences the arising Standard Model, nonrenormalizable supersymmet- and gaugino masses? Wegaugino saw masses in (as section wellstructure can as arise other at SUSY-breaking tree-leveltheories from effects) in dimensional higher and reduction dimensions. of inWill certain But section such special we a supergravity are theory seeking be a consistent quantum beyond theory, tree-level? not In a this classical subsection one. we will present an 4.1 Limiting hierarchies withLet quadratic us divergences step back andgravitino review is our relatively logic heavy so comparedrequired far. to other We superpartners; would in like to RegimeHow IV consider can of a figure theory we where achieve the a hierarchy of at least four orders of magnitude between gravitino structure no longer has such a simple geometric origin. 4 Hierarchies in sequestered theories the modulus controlling theappendix radius of the M-theoryalthough interval. these cases We are comment also on referredwe to this as have case no-scale assumed structure in in a thestring literature. specific duality Furthermore, geometric web suggests origin that in therecontext may the [ be rescaling related of theories, for internal instance dimensions. in the The Type IIA simple case in which includes cases where additionalor moduli play are a subleading presentexample role, but is obtaining which are the 5d we either no-scalethe scenario will massive from moduli consider 11d and heterotic controlling in M-theory decouple a in section the Calabi-Yau limit factor where in the compactification are much heavier than JHEP04(2016)045 . At (4.2) (4.1) 3 strongly ) for some . These pic- 2 φ π µ (16 ψ / 2 g φ . may be somewhat less b . ,   and . Again, dimensional analysis a 2 log Λ log Λ / 4 3 µ 2 2 ψ × / / ψ m 2 3 2 3 may be of order bm dm b µ ψ + + 2 2 and λ φ Λ Λ a c a   2 2 Pl Pl – 16 – 2 2 / / M M 2 3 3 2 2 π π m m 16 16 2 / µν 3 h ≈ ≈ × m λ 2 φ m m µ ψ grav grav δ δ φ λ symmetries. . Because this is a power divergence, a detailed number cannot be g R manifest. (like the Higgs boson or the scalar superpartners of the Standard Model) and φ φ . Gravitational loop corrections to scalar masses. Similar diagrams exist at two loops . Gravitational loop corrections to gaugino masses. A gravitino mass insertion is required µν For gaugino masses, the estimate is even simpler, since we require an insertion of In the case of scalar masses, we consider diagrams such as those shown in figure h φ the gravitino mass toprovides break the chiral estimate symmetry, of as the in divergent radiative figure correction to the gaugino mass computed without a UVloops, completion such that a UV regulates completion the isthe not loops. effective easy theory to Because in come these a by, andshould detailed are will string be presumably graviton vacuum. require trustworthy. embedding Nonetheless, From thesuppressing this general scalar we scaling masses argument immediately requires a extract cutoff one well important below lesson: the Planck scale result, some aspect of thescalar calculation field must be ablea to field tell with the a differenceprovide shift between the symmetry. a necessary generic Shift-symmetry spurion to breakinggo make effects to the like diagrams two Yukawa nonzero, couplings loops. butmarginal can we As coupling will a generally result, have the to coefficients This estimate is genericallythan correct, although order the one. numbers shift The symmetries reason (e.g. is for that an we axion do field not which expect can perturbative get corrections a to mass violate only via instantons). As a first glance, dimensional analysis tellscally us divergent that in the a graviton nonsupersymmetricthe loop divergence. diagrams theory, but would The be adding divergent quarti- the part gravitino of loop the ameliorates result scales as Figure 4 to break chiral and tures are an oversimplification: thesymmetry blobs on must include enough structure to make the lack of a shift Figure 3 attaching graviton lines to one-loop diagrams involving renormalizable couplings of JHEP04(2016)045 ). 2 π (4.6) (4.3) (4.4) (4.5) (16 / 2 , where Λ g Pl Λ as nearly-free πM 4 λ , we see that the ≡  4.1 and φ . 2 / ], which cuts off the loops in 3 . m 94 , ∼ ? One approach is to UV complete . 93  , 2 / 2 3 / 43 . 3 scalar m Pl 2 m  Λ )). The small parameter 2 > > ∼ ∼ πM , m π 4 2 / – 17 – 3 ≡ (16 scalar  m / 2 gaugino m  m ∼ gaugino m are again constant factors that we expect to be order-one or at least : dimensional reduction of 5d supergravity or 10d Type IIB supergravity with sequestering gravitino masses requires a low cutoff 3 c, d The arguments we have just given strengthen the case for the scenarios discussed in These simple dimensional estimates provide the first indication that a consistent power- should take Λ to4d be theory the to KK a scaletheory perturbative (at breaks 5d which down point theory) entirely). we oroverall have volume the In to is 5d 10d large switch Planck compactifications, there fromapproaches scale can we a the (at be will perturbative substructures which 10d find within point Planck that the local scale, even manifold field where so when the the we KK expect scale that the 10d Planck scale is (for generic Here we assumesparticles, that or loop at corrections leastnon-parametric are are numerical parametrically factors the of like dominant the 1 is same contribution the order Planck to as scale the the in other masses 10d ones (or of (discounting the string scale). In 5d compactifications, it is unclear if we Given our finding ofand no-scale the dimensional structure analysis fromdifferent of scales a gravitino may large be loop separated extra correctionshierarchy naturally. dimensional among in The gauginos, section compactification gravitino scalars loops and tell gravitinos us are the largest possible section large internal volumes will producethe precisely 4d the Planck sort scale of that hierarchies are between the needed cutoff for4.2 and consistently small radiative SUSY’s corrections. ladder a different way:propagating fields it is postulates incorrect due thatdimensions. to the the This presence approach treatment of to strongeffective of sequestering dynamics field and has fields theory large received like anomalous viewpoint. ascales great (large We deal extra will of dimensions focus attention and on from light the an strings). other case of low quantum gravity four-dimensional gravity at a scalepresence well of below extra the dimensions 4d bigger Plancktheory than scale. has Planck This size, a can in smaller which happentwo Planck case in scale, effects the the higher-dimensional or happen if atinteresting string alternative similar states is enter scales the conformal in calculation. sequestering models [ (In with fact, these an order-one string coupling.) An How can we achieve a low cutoff and thus a small Again, counting may exist in If we are to avoid fine-tuning, we have the two inequalities: where JHEP04(2016)045 ) † T (4.7) (4.8) (4.9) + (4.10) (4.11) T log( ].) In the 5d 2 Pl 95 M 3 . − Pl = M /n K away from zero. Having , Φ 2 1+2 / F  3 i † ∼ Pl T -dimensions (which we will con- . Pl M D + Pl M T h /n Pl Pl M M M 1 ' 2+1 2 3 i ∼ /   1 Pl W V ∼ ∼ ) M 2 2 takes values near the Planck scale. Therefore, 2 Pl ∼ / / V 1 3 3 M – 18 – , √ W 2 m m (2 / /n : : 3 ∼ 1 K/ string d d e V m 5 h M 10 2 Pl string 1 ∼ and M M Pl = L string 2 M / 3 M m ∼ KK M is the volume of the internal geometry in string units. The next lower scale V . 2 The dimensional reduction links the Planck scale in terms in the following section. 5.1 Effective field theory There are several possiblewhich include: sources of corrections to the leading-order no-scale structure, In this section,effective we field will theory point add ofthe some view, corrections which sources lead in of to effectivelarge deviations corrections of volume field to modulus theory, and no-scale weand other structure study moduli. their the from masses. The moduli the stabilization Based stabilization gives on the including these moduli the F-terms, field we F-terms are able to study the soft SUSY breaking In both cases, the gravitino mass isus less to than or complete equal to the thein parametric KK figure mass. estimates These of estimates allow the spectrum of SUSY’s Ladder5 as presented Deviations from no-scale structure and moduli stabilization we have a relation between where the second equality usesand the fact we that take the no-scale structure natural has assumption that In order to havemoduli a stabilization to valid fill 4d in effective the supergravity details, theory, we can Although estimate we the need mass a of the model gravitino of the string scale is much smaller than the Planck scale: Recall that is the mass of Kaluza-Klein modes, case, for the moment wea will detailed take explicit Λ example to and the be KK the scale 5d may Planck proveflate scale, more with although appropriate the in we string such will scale, a not assumingin model. order-one study 4d, coupling and for the meanwhile moment) introduces to the the KK Planck scale scale. Due to the large volume compactification, questions) a reasonable choice of cutoff. (Similar reasoning was used in [ JHEP04(2016)045 ] 46 are 2 / 3 ) T -form in the p (Re , we will take a consistent. That is, These arise, for ex- V ∝ 7.2 The supergravity ac- ) in general can give rise to internally x, y ( lm h Fluxes, warping, and other effects can Assumptions like superpotential terms. T b − – 19 – θ e 2 d R . Considering only one volume modulus will lead us to the KKLT T b − In the end, we would like to obtain a 4d vacuum with nearly zero cos- θ e 2 d R for promising models). for phenomenology but we will have little toUplifting. say about it here.) mological constant. This requiresAdS adding vacuum extra we sources will of find.paper, SUSY We but will breaking it not to is discuss lift important details the to of check the that uplifting it scenario does in this not spoil no-scale structure (see [ give rise to modifications of the K¨ahlerpotential afterDeviations reducing from to Ricci-flat four internal geometries. dimensions. modify the internal geometry away fromifying our the assumption four-dimensional of Ricci-flatness, effective again theory. mod- (Warping is potentially very interesting effective K¨ahlerpotential in four dimensions. Higher-dimension operators in thetion higher-dimensional we theory. started withinclude includes corrections involving the higher Einstein-Hilbert powers of action the at Riemann lowest curvature order, tensor. but These can ample, from wrapping thehigher-dimensional theory worldvolumes around of the fields compactified charged dimensions. under the Additional moduli of thegenerally internal not geometry. exact, asmultiple light the scalar internal fields geometry in the lower-dimensional theory. These also modify the Instanton effects contributing First of all, the instanton effects are non-perturbative effects, where the instanton • • • • • action is setup. For twovolume or moduli more is moduli dominantscale over fields, structure; the it thus action we is of will possible the neglect that the large the large volume volume modulus instanton modulus controlling action instantons. no- of Starting small from is there a consistent power-countingpresence for of the suppressed higher-dimension effects operatorsIn such and that the corrections loops remainder in doby the of not compactifying this change higher-dimensional the section, supergravity.preliminary qualitative we look results? Later, at will in loop takeabout corrections section quantum a to consistency the largely from effective the top-down K¨ahlerpotential bottom-up. that approach allow motivated us to ask literature, is to attemptaspects to of derive the the literature structureapproach on is of moduli to corrections work stabilization fromconstruction from remain the gives the rise controversial. top bottom to down. a A up: controlledeffective complementary theory Some rather theory with than with stabilized approximate asking moduli, no-scale we if structure can a ask can particular whether be an string theory All of these considerationsfour-dimensional lead us theory to only expect approximately.us that Deviations in no-scale two from structure different no-scale directions. will structure be The can first, present which in lead has the been widely pursued in the string theory JHEP04(2016)045 ) . 4 2 D D L  . R R c after (5.4) (5.3) (5.1) (5.2) . 2 (log , where and the − µ = 4, the + h b ∂ L . Writing +1 )
. Without α 2 T s ) m D L L = 0, the 2 L T R ].) And after L a µ ˜ ∂ R − ( 96 (log ˜ for the curvature = µ R Ae 2 ∂ , are the same order, D − + mn 2 α case to R 0 Pl ˜ terms can modify the R )) ˜ 3 D . By only checking the xL W p M L 2 d D ) R † b ) gives the corrections to d 3 R L 3.1 T R Φ (log , and θ d µ + 2 2 ˜ ∂ R / b d 1 (log ( πG 3 µ T  16 m terms are incompatible with the ( ∂ Z † s ) ξ  ! T 2 ∗ L 3 D + 2 Pl + ∝ − ˜ R M R # ˜ s M 2 ). When 2 Φ (log / / T s † in the 4d Lagrangian. Accordingly, it does 3 1

µ g  ) ) ∂ and Φ † s † b L m 4 − 2 µ T T keinstein 2 D terms, the leading order of 2 − dθ / + + L α R L∂ 3 b – 20 – s 3 D µ  Z † ∂ does not include the volume modulus T R T xL b for the extra dimensions and 3 . In Type IIB supergravity the leading correction ( ( d , we will know the terms relevant to the corrections T ˜ d m R 2 3 L lm 2 + h − can be the Ricci or scalar curvatures or the Riemann ' − Z − b α d T  D L † b  R 1 order T πG term coupled to = is the Planck scale in 10d (or is the string scale — at the − + 16 3 b V pl ˜ and R , T ˜ s M . It is straightforward to generalize the  ˜ higher R ∝ − T L " L 2 ∗ operators, since the , and the correction is kein M λ 4 D L Φ R † for the curvature of Φ ˜ θ R 4 d 0) supersymmetry of the 10d Lagrangian. (The specific coefficient of the Z , 3 2. Each additional curvature tensor necessitates an additional inverse metric to raise Secondly, higher order terms of the curvatures in the action will modify the no-scale = (2 ≥ L ' − kinematic term of these corrections in superfields andhigher recalling order that terms in of the the 10d curvatures context modify we the require K¨ahlerpotential as follows: this term is onlyoriginates suppressed from by N terms is calculable whenthe supergravity compactification, is the UV completedK¨ahlerpotential. to By string removing theory the [ Ricci-flat assumption, the where Lorentz contractions aredifferent implied. contractions.) (Inmoment we general are there not being may careful be with factors multiple of terms with m an index when we formdimensional a reduction. 10d Lorentz In invariant, and theschematically this keinstein as brings frame along the a action factor after of compactification is written to K¨ahlerpotential. Considering the Ricci-flatterms extra will dimension, not introduce i.e. the kinetic term not contribute to K¨ahlerpotential. For is given by contractingby three the kinetic Riemann term curvature of tensors, one of which can be replaced to explore in morethe detail). notation Since there arein several the different original curvatures 10d to theory.introducing consider, Note we ambiguity, that use tensor. We start withkinetic the terms of dimensional the reduction volume as modulus in section structure as well. We onlymensional consider curvature the is trivial higher (unless order we terms have in warped 10d, geometry, which since would in be 5d interesting the extra di- small volume modulus the Lagrangian having a small and large volume modulus is written as, the volume in string units of “swiss-cheese” type with the big volume modulus JHEP04(2016)045 . . # 2.  2 . / / c 1 . 1 (5.9) (5.6) (5.7) (5.8) (5.5) − ]. We − ) † b +h 76 = (Einstein) MN T T g p F φ . + 1 2 results from b  aT e . . 2 T − c / ( . 3 2 → ) / = 0 † 3 aAe + h )  . Adding the above S 3 terms are the same † † 0
Φ S + T 4 D aT iC a − + (string) MN − S R terms lead to
− + g S φ ( 4 D Ae aT ξ − Ae R − e and + + + 2 2 0 = D Ae 0 / / 3 1 R W S + ) ) W † s † b 0  is the leading nonzero correction .. 2 3 T T c W † . 4 D Φ + + 2. b s θ R / Φ 2 1 + h F T T , d + 3Φ ( ( 2 −  2 3  2 Z 3Φ ) = = 0 T  = 0 (logarithm);  − 0 † F ], but we will defer preliminary comments on p + S  p + † b  aT – 21 – †  − T 97 T − , † + Φ F serves as a stand-in for the physics that stabilizes S + correction). The factor of ( T (  Φ 95 2 b s † 0 ) F aAe ], + 0 T † T α 3 W †  S 1 2 T + 98 " Φ ,  +Φ − 3 / + T 2 − ∗ T 1 , the K¨ahlerpotential and superpotential are written as, S  terms lead to s 70  ( F , Φ S M † s T Φ Φ 3 D a F S Φ 50 F † † Φ W − R in superspace. Because †  1 2 + Φ F term is an 2 2 Φ 2 ∗ ∗ S 2; Ae + / θ /  4 D 1 4 M M  ) + 2 d ∗ † R 3 3 † b and leave a more detailed study for future work. 0 factor coming from each additional Riemann tensor in 10d becomes a T = 1 − − T M Z W : : 2 p + Φ 3 7.2 +  † † † T T Φ − b /L 3 δ δ  Φ 4 T δF δF Φ = Φ ( 2 F / dθ L † Φ dθ F Z Z  3 2 ∗ + Lastly, in Type IIB supergravity we must also include an additional modulus arising terms lead to M 3 2 D L ' − − Varying with respect to the auxiliary fields leads to two equations of motion: Reading off terms involving only homogeneous background values of the fields, this is: 5.2.1 KKLT We begin by consideringKKLT-like superpotential the [ theory of a single modulus with no-scale K¨ahlerterm and 5.2 Moduli stabilizationAfter in having keinstein the frame Lagrangianterms derived and from the effective potentialwe field of will moduli theory, derive we in F-terms order can inthe to moduli. KKLT compute and study The the in moduli computation F- Large becomes stabilization. Volume much scenario, In more and clear this estimate by section the using mass the of CDT gauge. The superpotential term the dilaton in awrite supersymmetric a manner, simple which mass canconsider term be string loop accomplished because corrections via itthis as fluxes leads until in [ section to [ the right qualitative results. We can also The way that thesupergravity. dilaton The couples leading order to(in of gravity the string is string dictated theory, couplingi.e. by this to that the is the symmetries the leading ofthe statement Type transformation that IIB from 10d both string terms frame exist to 10d already Einstein at frame, tree level — corrections and the modulus In short, the 1 factor of 1 allowed by the 10d supersymmetry, we take from the 10d graviton multiplet, namely the dilaton-axion, R JHEP04(2016)045 , , #  : † s ) 2 T † / aT 1    F T 2 − 2 − (5.15) (5.13) (5.14) (5.10) (5.11) (5.12) / / ) + 3 1 † b   T T aAe † † ( s b is strongly 3 a + T T † b T Φ + + s T 5+ b s † (  T T † T 2  . F /  †  0 3 aT † S − ) s i − † † 2 2 T s / / S T T 3 1 + Ae i − F + s   † 3 1 + † ), which we repeat here † s b T S S − ( h T T T 1 2 ξ  5.5 =  + + ]. In contrast to KKLT, it s + b + s † 2 2 T T W 8 2 M , / / aT /   3 W 3 1 7 , which will be verified to be self- 3 − † 3 ) 3 ) = 0 from pure no-scale structure i
† † s / i † b Φ † Φ 1 .. Φ Φ † Φ ) F T S T h c S h † . † aAe F † Φ F + as in eq. ( + , + S are so tiny that they can be neglected, Φ F i b s + † 4 S b i 2 + + h T T S / S ) +  R ( ( T 1 S † † + F ξ  ( h  2 3 S gives us that 2 T † ) + 3 † h + aT b ) † 2 − 0 − T T + / T and then use this to solve the second for S  ΦΦ 3  + † b T Ae  b + i – 22 – Φ ( − ! † † T s b T 3 i † F s + T / b T T S  + ( 1 T T ( s 0 + b ) 3 F s + b T , † h / + s T T W † 1 F h s S W T )   T † 2 1 aT  † " + S 1 2 3 − 2 3 / + e S 1 3Φ − + − ( 2 s ∗  h † † b † † T b S † M a T b ( is via the K¨ahlerpotential. We start from the Lagrangian aA S T b Φ † 3 T Φ − T ΦΦ + b + 2 + F 2 † . It deviates from the result ∗ F b b † b i S T Ae T ΦΦ T † 1 T Φ b i M  † T b T 1 2 T 3 3 4 + − 2 + ∗ T + F F b 0 h

b    T T = = M h T + † † s b W F † b Φ T Φ T T s b  † † † T S  T T F F † + 3 S + † † Φ F F i Φ s b i + F 4 Φ Φ Φ F T T Φ 2 F h S 2 2 2 h ∗ ∗ ∗ dθ 2 ∗ dθ M M M Z 3 M 3 3 Z 3 − − − − + : : : : b s † † S Φ † † T T F F F F L ' − consistent afterwards. We read off the leading auxiliary terms (neglecting fermion fields), We work under the assumption that so the stabilizationderived of from effective fieldmodulus, theory having and large a and higherfor small dimension convenience: volume operator moduli, from a dilaton-axion 5.2.2 Large VolumeNow, Scenario let us considerhas the small Large deviation Volumesuperpotential from Scenario terms the involving (LVS) the [ no-scale large modulus structure. Essentially, the key point is that the The minimization of potential with respect to leading to too much, andexample loses illustrates that many no-scale structure of maystabilized be the easily by broken desired the if the superpotential. phenomenological modulus consequences. This simple We solve the first of these equations for JHEP04(2016)045 (5.16) (5.18) (5.19) (5.20) (5.21) (5.22) (5.17) ) gives 5.5 by varying , i b † T i † S b F T + + S b h 2 ], following the lines / T  3 h V † 0 3 2 m / S / ) in CDT gauge gives us 1 3 100 . This can be derived by V i , . Its mass can be derived i ∼ † 2 m 5.5 † i − / S 2 † 3 S 99 / ∼ , S 5 2.5 + m h i + i i i † b  S † † s b ∼ ), we can see that 79 S T , W S T ih T h i h † Φ + 2  s M + + h / † i , 0 63 5.12 b 2 T 3 , 2 b s s ∗ i S i T T . † T . T Pl † h + h h M F 3 2 2 Φ 2 55 S s h / / / h 3 , M / ) respectively, i i − 1 3 3 T / 1 † S † s V i + h 1 † T V ' i m m S h 47 W † S h , S a 5.14 × i h S ∼  † √ ∼ − b ξ . We have two ways to compute + – 23 – 2 s S T b 2 46 ), ( + i . In all of these cases, the scaling of the mass in (1) T , T ∼ ) S † 8 0 M + S Pl aAe m S m ih Φ S 2 b h ih i 5.13 direction, or the leading order can be given by the M Φ T m ∼ O 3 † h − h Φ i i ' b 2 h i 3 ∗ Φ − ∼ 0 † / 2 T S ∗ b = 0. From the eq. ( , the second term of the K¨ahlerpotential in eq. ( 1 ( i W M S T s i −h W s h b M 4 † h 2 T T W W in eq. ( + i W S F † S canonically leads to ' ' 1 2 b h † if Φ Φ + b T D i i h h F i 2 + † T Φ S / , i Φ s S 3 F supersymmetrically as in in section † h S S ih h aT m + F F Φ S h − h S 2 has larger supersymmetric mass than its soft mass, which is consistent ∗ h Ae S M + ' 0 ) as well: i W † b i T 5.5 b = T + F W b h T h Having calculated all the F-terms, we can compute the mass of moduli. For the large . Normalizing the field b without apparent cancelations. 6 Soft SUSY breakingComputations from of superspace the softthe SUSY LVS breaking have effective been Lagrangian discussed in in various refs. incarnations [ of from eq. ( on the same orderterms as of powers of volume is straightforwardly read off from the superspace Lagrangian, both the kinetic term and mass, hence It turns out that with integrating out its leading kinetic term, andT the second term is the leading mass term of (the real part of) For the small volume modulus volume modulus, the first term of the K¨ahlerpotential in eq. ( The other F-terms may bevolume found expansion by of considering the the equation next-to-leading of order motion: term in a large- minimizing the potentialsupersymmetric in minimum the the auxiliary fields of which leads to the relation of where JHEP04(2016)045 , b – 1 T − ) (6.2) (6.3) (6.4) (6.1) † b 101 , T (which + Q 45 , c b T 42 with other terms 2 / 3 . . There are additional m Q 4 † , we saw that at leading Q 2 Q. ! † 2 Q / 6= 0 and we may also have new 3 2 |
φ . b † c F T b . Suppose the kinetic term for the Q F Q b | † T c c 2 2 T / + / Q 3 7 2 b /  2 3 † m T b -term in the theory, let us first consider the Q T m c ∼ F Q + , so the role of these new factors is to translate c V 15 – 24 – b E 1 + 4 ) T 15 †

b scalar  T ⊃ 4 Φ m . Finally, we observe that the remaining powers of † 2 + Φ / † Φ b kin 3 T θ Φ L m ( 4 = 0 in the pure no-scale limit. In this section, we will look , but they are subdominant. / d ⊃ 2 b Φ Φ T . These arise from taking two derivatives: F Z b F is the dominant F kin T D b ∼ L = F T b , we see that this is consistent with the scaling † ≈ . T F V kin F 2 4 and / that in the pure no-scale limit soft masses are absent for sequestered √ Φ appears in the leading kinetic term and hence serves to canonically L 3 † . / c . m into factors of components of the moduli fields. In section 2.3 = 1 b T + h F  F . Because T F † Φ 4.1 F ∼ Here we have added a volume-suppressed term proportional to a coefficient ]. Many of the computations presented in the literature involve nontrivial cancelations Recalling that that we estimated fromterms loops and dimensional analysis in section in the denominator areunits. approximately As just a the result, volume we of have the that internal geometry in string Now, the factor of Φ normalize the scalar field.but Taking we derivatives recall brought that inthe two factors new of factors of ( consistent with the effective fieldin theory section perspective, which is estimatedterms by proportional gravitino to loops may in general dependa on constant). complex structure The moduli, reason but for to the add moment the we volume-suppressed take to term be is that this correction is chiral superfields. Away fromcouplings the of pure the no-scale chiral limit, chiral superfields superfield is to the modulus at corrections beyond theis pure easily no-scale read off limit fromcalculations and the in see Lagrangian section that and is the compatible suppression with of6.1 the soft estimates terms based on Soft loop scalarWe masses saw in section 104 between different terms, often compensatingrelated terms to proportional the to order these somewhat mysterious cancelationsoriginate are from absent when the working fact in that superspace, and of general earlier work on supersymmetry breaking in supergravity theories [ JHEP04(2016)045 ∼ = i (6.7) (6.8) (6.9) (6.5) (6.6) a a λ f h m by a constant )). Expanding . 2 Pl , we will obtain , Q s a in the future. M f T ··· is controlled by the (3 Q + c / or a ] and further studied f = 0 when the volume- b K Q 46 † T − Q Q c scalar and gaugino masses ! exp( ˆ ξ V s 2 Pl both . c ., 2 M / c − . 3 3 V . 1 − m of the smallest possible gaugino-to- 

+ h = 1 3 = 0 looks unnatural; the loop estimates ) ∼ a α these are a loop factor smaller than the . Q 4.1 3 Ω −  c S / W s 2 V a S U, S c S δ / ( V aα F 2 α  / = f  ˆ 3 ξ W – 25 – a a 1 3 a m f − δ f ]. They studied a K¨ahlerpotential of the form matches our ansatz if we rescale 1 2 1 4 = 2) 46 θ Ω 2 = Q controls the gauge coupling and theta angle: i ∼ ˆ ξ/ c d a Φ a λ + / f Z Φ m V F . On the other hand, suppose that h ) we must evaluate 2 / kinetic term is simply absent. This is yet another example 3 log( contains a linear term in the moduli 6.1 2 m Pl a Q f M ] found that the scalar masses are suppressed at the special value , we find that ]. It has the potential to produce 2 V 46 − 105 = and make the choice 2 K , but because / = 0 was referred to as the “ultralocal limit” in [ i 1 1 might already be interesting, and could lead to a distorted SUSY’s ladder i Φ Q suggest that it will not hold, barring UV physics that would effectively regulate α / . Clearly, if c , producing CMSSM-like phenomenology in a setting where the gravitino problem f  a Φ 4 V F Θ 3: from the superspace point of view, this is the case Q h / / ∝ h π c i 2 ) 8 / 3 We also expect anomaly mediation to generate contributions of order Let us compare to the results of [ /π − = 1 m a 2 a s α 1 g This is consistentgravitino with mass our ratio estimate that is in not section destabilized by loops. ( tree-level dilaton contributions. Then we obtain gaugino masses where the lowest component of gaugino masses of order dilaton: more detailed understanding of the reasonable size6.2 of the coefficient Gaugino masses Gaugino masses originate from the holomorphic gauge kinetic function is decoupled. From our pointin of section view, however, the loop in a way thathand, seems magical from thewhere low-energy EFT the point of scalar view. rung On is the a other bit lower than expected. It would be interesting to pursue a c suppressed part ofof the how working inThe superspace case can makein more the detail outcome in of∼ [ calculations more transparent. factor This explains why [ To compare to our ansatz ( in the limit of large JHEP04(2016)045 (6.10) (6.11) (6.12) (6.13) (6.16) (6.14) (6.15) and gauginos at to be of the same V µ √ / 2 / 3 , m . ! c . d . 2 A 2 2 H . . µ . , these will have little dynamical 2 m c m to be approximately the geometric + h . 2
µ V / c 3 µ b + 3 b √ ¯u log 2 V + h c / | m , 2 Higgs 2 t 3 2 A 3 µ u / y | : scalars at 3 ¯u Q m M H c 3 u µ m ∼ i u b tr − 2 H Q t 4.2 Φ 2 H y u h m t µ µ y  H  b + = µ t – 26 – 3 can be an order-one number that does not scale Φ y 2 2 c , Φ |  3 t π F u 1 i µ y 2 | Φ α 2 Higgs  H Φ Φ θ h

2 = ∼ , so the canonically normalized fields are rescaled, e.g. M d  t = 2 Φ , θ † 1 A Z 2 det d Φ δM 2 Higgs Z M -terms on order of the gaugino masses times Yukawa couplings. A , this tells us that we would like

d 2 H that we want to obtain. Recall that the mass matrix of the two Higgs m µ

]: b ,

u 107 2 H and , m µ

and b terms 106 . To obtain realistic phenomenology, we must be able to arrange for one light Higgs  µ V / 2 | 2 / µ | 3 -term If mean of theorder soft as masses the squared. scalarmasses soft [ On masses, the there other is hand, a significant if threshold we correction take to the gaugino and the requirement of a light Standard Model-like doublet imposes m doublet to play thevalues role of of thedoublets Standard is Model Higgs. This places requirements on the 6.4 6.4.1 Phenomenological requirements So far we haveSUSY’s found Ladder that that large-volume we SUSY discussed breaking in can section produce the hierarchies of Thus we expect toCompared find to the largerrole scalar to masses play. of order This shows that thewith Yukawa coupling any power ofA the internal volume. From this we also read off that there will be an compensator through a factor and we can write the Yukawa interaction as Consider a superpotential interaction like the top quark Yukawa coupling, Recall that the leading kinetic terms for each of these fields will also involve the conformal 6.3 A terms JHEP04(2016)045 ) to and 1 6.15 0 (6.20) (6.17) (6.18) (6.19) H, c H, c in the end . 2 µ so that they ) b . V c S . √ / 2 + h ., / and c c 3 . b may be problematic. d ], although the Higgs m T ( µ H + h b c there may be additional 108 ∼ u , ]    2 µ H / S b s and 3 V ... T 110 m µ a , . + − -term 2 terms [ and 2 ∼ e / µ / 2 3 V d 3 H V 109  , m  b H √ W † b b † T 1 / 46 2 T ∼ d T 2 3 F i , / H, 2 µ 3 Φ c H Φ  + b h . small because then the requirement (  b 1 m b V θ b µ T T 2 V / H, b T and 2 2  d c  F / 3 3 u µ  + – 27 – Z 1 − m H 0 † = coefficients depend on H, V + H, u . c c 4 † † c Φ . H H from a holomorphic kinetic term 15 Φ    F c µ d * + h = b 0 H d µ u b H, in the H c and u d ΦH = µ H H † c s µ Φ T -terms. a θ 4 − F and and we would predict a too-large Higgs mass given our heavy scalars. e d H u 1000 TeV is in some tension with our desire to have gauginos near the β Z H W c ∼ 3 Φ µ ) for obtaining a light Higgs boson. A shift symmetry in the Higgs kinetic θ 2 d 6.15 Z may depend on moduli but are assumed to be independent of We also read off the leading contribution to From this expression we can immediately read off a 1 H, phenomenological viewpoint. However, therethis are picture. some In dangerous particular,These effects superpotential can that contributions arise could to from spoil a “de-sequestering” term like [ for realistic electroweak symmetry breaking. 6.4.3 Superpotential contributions So far, working into many superspace SUSY-breaking has terms allowed is us precisely to of read the off order that that we the would leading like contributions to see from a This is on thedition same ( order asterms the could even soft guarantee scalarthe a masses, coefficients vanishing as determinant at iscouplings leading explicitly necessary break order to such by a realize relating symmetry the and we con- will still need to fine-tune contributions, but they are still of order which is on theatic same threshold order corrections. as the If gaugino the masses and hence does not lead to problem- where the omitted termsc correspond to further volumedo suppression. not The contain large factors Consider contributions to leads to large tan Thus taking weak scale for interesting phenomenologyAs (including a the result, possibility of we SUSY hope dark to matter). obtain hierarchies 6.4.2 K¨ahlercontributions We cannot suppress this correction by taking JHEP04(2016)045 ) µ ]. ), = b u GeV 6.20 2 H 122 6.20 (6.21) (6.22) , 8 m 121 . 2 3 / / 3 1 ). Thus, our prob- V m are universal and the ∼ b 6.17 in the exponent with a T 3 3. / Pl a 4 -problem of supersymmet- a/ M V µ 5 ∼ > 0 , 1 V a 2 3 / / Pl if 2 3 1 V M m V 2 / / 2 1 / 2 3 i ∼ E s † b ], which is unexpected given the tendency T m T 1 F < ∼ 2. The former regime has received intensive h 114 + b – 28 – W ≈ T µ . It is known that universal scalar masses D β ]. This is encouraging for our scenario if we expect ∼ β . (The scalar masses can be pushed slightly heavier d ∼ 114 2 H W

) s ], if we replace the coefficient ]. The heaviest admissible scalar masses are near 10 m µ ]. It seems fair to say that it has been widely viewed as 2 [ b aT 46 ( ], though this in turn generates a large threshold correction − ≈ 115 120 e – and

– β 116 very close to 1 [ u H 113 2 H β W 116 , m i Φ 45 h at the GUT scale lead to the correct Higgs mass when scalar masses 3 the term becomes safely small. A similar statement holds for the = a/ ]. ... 5 W µ = > , then after canonically normalizing the Higgs fields we read off a contribution 112 0 , 3 Pl a 2 Q M 111 m ∼ or scalars at 1000 TeV with tan = H SUSY’s Ladder has scalars around the 1000 TeV scale, which can explain the Higgs This “de-sequestering” problem implies that the familiar β d W 2 H m are at about auniversal PeV scalar and masses tan at theis GUT whether scale. the leading There couplings aresecond of two is aspects MSSM which scalars to UV to scale this is the question: relevant modulus for the imposing first boundary conditions on the RGEs. These attention in a varietybelow of scalar masses “mini-split” [ scenariosthe most where well-motivated variation gaugino of split massesjust SUSY. barely are It light offers enough a the that tantalizing some loop prospect may be of factor scalars discovered at a futuremass 100 TeV at collider [ relatively small values of tan and require pushing tan of RG running togiven split a large higgsino mass [ to gaugino masses.) More plausibletan parameter regimes include scalars at 20 TeV with large 7 Discussion 7.1 Phenomenology ofThe SUSY’s observation Ladder of themake Higgs sense in boson split SUSY at [ 125 GeV restricts the range of scalar masses that answers whenever there is alem good is inverse merely volume towhich expansion explain as may in the follow ( absencemodels from or [ geometric suppression of properties superpotential of terms the like compactification ( in particular UV again undesirably large, but which becomes ric model-building acquires newmodel-building. aspects What in is encouraging the is context that of the kinetic extra-dimensional, terms sequestered alone lead to unproblematic are absent. As emphasizedcoefficient in [ term which gets a contribution This is larger even thanto the electroweak soft gaugino scalar masses. masses, As and a will result, lead we to must demand large that threshold terms corrections of the form ( If JHEP04(2016)045 (7.1) ] have 125 – 123 ]. The gluino 128 ]. This contains several 86 ] and the 1000 TeV scale could be ˆ K, 127 , which is close to the standard GUT -channel exchange of 1000 TeV squarks, or string t log det M 2 6 2 π / to achieve a light Standard Model-like doublet.) – 29 – Λ 1 16 µ V b = δK ]. Probing 1000 TeV scalars experimentally is an interesting provide some indication that the hierarchical SUSY’s Ladder 126 4 ], which found that an “extended no-scale structure” ensures that is the matrix of second derivatives of the K¨ahlerpotential. Unfortu- j ]. It would be interesting to explore other possibilities, like whether a 129 , ∂Q K † 2 97 i 120 ∂ , , ∂Q 95 116 = j † i ˆ K Ideally, we would like to have a supersymmetric formalism for estimating the size of Our results motivate a closer look at the variation of split SUSY with scalars at where nately, several things are unclear about this expression, including the appropriate choice of potential calculations with an appropriate UV cutoff. higher-dimension operators directly in superspacewe in are the working conformal with. compensator formalism Forloop now effective we K¨ahlerpotential will for settle aterms, for but nonrenormalizable making we theory some will comments [ focus based on on the the one- quadratically divergent contribution directly in the context offields the full and theory their (including couplings bothestimated to Standard in Model each [ fields other). andthe moduli Certain leading-order aspects of results theseogy are loop to not computable effects wildly string have theory altered been loops by (in toroidal loops. orientifolds) and These on Coleman-Weinberg results relied on anal- 7.2 Building anIn effective this field paper theory wefield have taken theory some of steps no-scaleysis in structure. arguments the of direction However, section of therespectrum building is is a more convincing radiatively to effective stable. do. However, it The would dimensional be anal- useful to compute loop corrections contain a sufficient amountcourse, our of starting information point was toogy the can indirectly gravitino also problem, probe probe and the this darkfor scenario. matter direct scalar physics Inflationary access spectrum. and phenomenology to cosmol- perhaps some offers Of aspects the of best the hope moduli physics. lifetime is in thetechnology hundred [ micron range, on100 the TeV edge collider could of measure accessibility theto with gluino measure current pair detector the production interference crosswhether term section arising precision accurately from measurements enough of gaugino and higgsino decay branching fractions could 1000 TeV, which hasmini-split received when less the attention, bottomthermal wino except of dark the when matter [ gaugino viewedchallenge. spectrum as Flavor is and the at CPinteresting upper multiple provide from end TeV indirect the as probes of viewpoint required [ of for explaining the SM flavor structure [ studied threshold corrections inpush string the theory effective and universalityscale concluded scale in that to the large SUSY’s125 threshold Ladder GeV effects scenario. may emerge This in isalways in a an split natural encouraging SUSY, way hint with from that a 10d the tuning no-scale Higgs of compactifications. mass of (Though, as questions are beyond the scope of this paper. However, Conlon and Palti [ JHEP04(2016)045 ] 4 + R (1) T 97 , (7.4) (7.2) (7.3) O . We i 95 log( T an h ... 2 Pl β + M kinetic term ): 3 Q † T − Q Q = , with ) † K , T reasonable cutoff we i 2 string + / γ 3 ) T † . βM ) + , such a term is simply a T ! largest Q 2 2 Ω ) ) † + Q Q . This can only decrease the Q † † T 6 † ( † / Q Q − T 1 Q − − † + + 1 + log( † † . The V − T T T / T ). Thus, taking the largest possible γ T + + 2 + T T 2 . Taking Λ = 6.1 π ( ( 4 β term, like the one we included from T string / 16 2 2 field 3 ) ) 2 ) M / † Q Q + 1 log( † † T h − ) ∼ Q Q ) † ). The leading correction to the Q 2 1 − − † + 2 – 30 – / † † − T / 3 T T T 5.5 1 T ) KK ) + ( † + + † + / 2 T T M Pl T ( ( T T Pl T M +

) in coefficients, we see that the leading correction to 2 + M † 2 T β Pl ( T T 3 = 2 ( + log( M is a ( 2 π , as we assumed in ( + ) term in the effective K¨ahlerpotential, which superficially / γ 2 † , so all of the corrections are suppressed by the volume rather 1 / 16 2 Ω = 3 T 3 T V V 2 π / − − / ˆ β ], we might expect loop effects on small cycles to be sensitive to the ) K + 2 16 Pl † Pl = 95 T T M M − ( / + δK ∼ = Q , for instance, which would completely spoil the phenomenology. The † 2 2 T / Q 1 is volume-suppressed compared to the leading-order K¨ahlerpotential. This ) − string † † T M -dependence of δK T ). In this case the matrix of second derivatives of the K¨ahlerpotential is T + is a constant that depends on the scale of the argument of the logarithm. The + Q † T γ ], which again found sufficient volume suppression to preserve the phenomenology. T ( Q / As emphasized in [ Consider a no-scale with K¨ahlerpotential sequestered chiral matter, = 130 − Q † † Ω importance of loops, sodiscuss our a estimates possible have been 1 appears pessimistic. dangerous On but theconstant, does other so hand, not again [ change superspacepotential the clarifies estimate why potential. has it so far is In turned harmless. up no However, indication our that effective such K¨ahler a term exists. than some other power ofshift the length in scale moduli of masses internalin dimensions. due [ A to related Planck-suppressed estimate couplings of to the gauge fields wasstring-scale performed radius of thosemight cycles be while cut loop effects off from at KK the modes lower of scale the large volume cutoff in the computation ofchoices the effective we K¨ahlerpotential gives have further made justificationQ for above. the Loop correctionsunderlying do reason not for generate this dangerousproportional is operators to easy Λ like to see: the dangerous terms are quadratic divergences Apart from factors ofthe log( pure operators in the 10d theoryis in equation supressed ( by ( where correction can be rephrased as a superspace kinetic function (expanding around large can take is number reflecting our uncertainty about ultraviolet physics, we have The tricky part of the calculationof is the that one-loop Λ effective is K¨ahlerpotential, but viewedstring reasonable as scale choices a in or constant in our KK the context scalewill original include which, discussion the assume in that Planck Λ units, depend is on in the fact expectation a value function of the Let us forge ahead and try to make conservativeT choices. cutoff Λ and the appropriate scale to make the argument of the logarithm dimensionless. JHEP04(2016)045 2 / 3 m  is completely gaugino V m √ / = 1  also suggested the possibility that 5d 3 – 31 – As the LHC continues to test supersymmetry, completely natural models may begin More generally, we hope that our results can help to bridge the gap between activity in Although our first look at loop corrections has been encouraging, there is much more to gravitinos and moduli provide anatural partial explanation universe. for why weSUSY’s If do Ladder not we scenario live allow can inthe a naturally a completely existence avoid single such of cosmological moderate largethe problems hierarchies tuning discovery while in of in explaining nature. a the gluino. The Higgs first We eagerly experimental boson await signal further mass, would data likely from the be the 13 TeV LHC. but K¨ahlercorrections seem to besuperspace, under we control. hope By toto using have phenomenologists the clarified wary formalism the of and underlying delving notation physics into of and the made literature it on more stringto accessible theory. fall by the wayside. It could be that the notorious cosmological problems induced by extra-dimensional theories). Although many ofthe our string phenomenology results literature, have we previously hopeism been that derived where by in presenting the them scaling inmanifest, of a different we various formal- have terms made withautomatic it the more outcomes. small clear parameter No that uncannya apparently “string mysterious gravitational magic” cancelations context. is are at in De-sequestering work, fact terms just in effective the field superpotential theory in are a real concern, string phenomenology. One reason formations this being is used justified will concern standunderstandably up about makes to whether theorists corrections: the worry. approxi- the Ourswered idea results in of suggest an a that effective model these field with a concerns theory cutoff can with that be a is an- controlled parametrically power-counting, below the the key Planck feature scale being (which happens naturally in certain ing sector, which mustcosmological constant. be part of any complete modelLarge that Volume superstring is compactifications capable andmetry. in of The phenomenological canceling phenomenology studies of community the has supersym- to date shown limited interest in the results of compactifications could lead towhether no-scale the structure. cutoffPlanck scale Here scale, in there which would our is lead5d estimates some to scenario ambiguity should rather could different about be phenomenology. naturallyexplore viewed UV exist these completions as in possibilities of the heterotic the in KK M-theory. more scale detail It in or would the the be future. 5d very We interesting have to also omitted the uplift- 7.3 Further explorations We have focused onsupergravity. the Our phenomenology general of argument large in volume section compactifications of Type IIB understand. If different cutoffs Λeffective K¨ahlerpotential appear formalism in will different not parts givethe of correct cutoff, the answers. the calculation, The the uncertain field-dependencea standard argument of gravitational of theory the rather logarithm,powerful than and formalism global the for supersymmetry fact computing all loops. that suggest we are the working need in for a more JHEP04(2016)045 ]. 74 (A.2) ]: ). What lesson . superstrings [ 135 11 2.5 8 dx E coordinates are much 11 × m y dx 8 ) ) (A.1) x E † ( superstring is heterotic M- c S 2 8 e + E + S is the dilaton or string coupling × assumed that we were interested m is the length scale of the six internal φ 8 3 log( dy 2 E l σ/ − dy e ) ) † y )) and ( T x has the desired no-scale kinetic term. If the ( lm + ]. Suppose that we begin with the following h L ) T T , where – 32 – x 4 ( 137 / a 3 – 2 − e a disguised form of one of the examples we derived 3 log( φ 135 + σ − is 3 ν e = dx = 2 µ Pl S that no-scale structure suggests a preferred role for Type IIB dx ) 3 x K/M ( µν g and Re 4 = ¯ / 3 2 φ ], which at low energies is 11-dimensional supergravity compactified on σ ds e 134 were to acquire a large supersymmetric mass, we can consistently set it equal = – S T 131 . We can study the 11-dimensional theory reduced to four dimensions on a Calabi- 2 However, this example in fact Z / 1 ansatz for an 11-dimensional metric, choosing our notation to resemble [ If we work in asmaller regime where than the the six remaining dimensions described five by dimensions, the we can dimensionally reduce to obtain a 5d above. The strongtheory coupling [ limit ofS the heterotic Yau times ana interval, five-dimensional or orbifold alternatively theory we [ can study it reduced on the Calabi-Yau to should we draw fromin this? a single Our field discussionwith describing in two an or section more isotropic fields, lengthpotentially such scale as give the for rise dilaton the to and a internal the wider dimensions. length scale variety Theories of of the ways internal to dimensions, realize no-scale structure. where Re Calabi-Yau dimensions (analogous toconstant. our From this wesuperfield see that theto field its VEV and perhaps obtain no-scale phenomenology (as in section imaginary part of the chiraldiscussion superfield of that no-scale serves structure as into a string K¨ahlermodulus. theory Witten’s However, that the 1985 we first are workWitten aware of on found dates the dimensional all K¨ahlerpotential the reduction way of back heterotic A The heterotic string,We M-theory, have and argued in no-scale section structure superstrings, because of the existence of a four-form gauge field which can give rise to the thanks the organizers and participantsfor of providing that congenial conference settings and to of discusstheory. String the Pheno interplay The between Cosmo work phenomenology 2015 and ofNASA string MR Grant is 14-ATP14-0018. supportedEnergy The in under work part grant Contract of by Numbers the WX DE-SC00012567 NSF is and Grant DE-SC0013999. support PHY-1415548 by and the the U.S. Department of We thank Marcus Berg, Michele Cicoli,Enrico Ben Heidenreich, Pajer, M.C. David Fernando Marsh, Brentespecially Quevedo, Nelson, like to and thank Jesse Michelemodulus Cicoli Thaler couplings for to emphasizing dark for the matter possibility in usefulHidden the of Dark Large suppressed discussions. Matter Volume Scenario volume Conference, in which discussions stimulated at MR MR’s the renewed MCTP would interest in this topic. MR Acknowledgments JHEP04(2016)045 4 T / ] 3 φ (A.3) (A.4) σ ]. e Phys. Rev. Phys. Rev. B 138 , , SPIRE Nucl. Phys. , , i.e. IN [ ,  MN hep-th/0502058 a g [ ) N x ( Phys. Lett. a a∂ where 4 , (1983) 61 − M Systematics of moduli e ∂ . , ) No-Scale Supersymmetric = x 11 Naturally Vanishing (2005) 007 MN ( g a B 133 dx MN 03 11 ]. g + 30˜ ) + 2 dx ) . We can rewrite this in the five- 5 x x ( ) when we repackage all of the fields ˜ ( 11 R JHEP c c SPIRE x 2ˆ ,  ( IN e dx ) Phys. Lett. T [ x No Scale Supersymmetric Guts ) = 11 , ( + a x ν 6 ( dx e ˆ c ) 6 x dx ( µ c – 33 – , ˜ gV 2 e dx − (1984) 429 µν ) ¯ g + x p Is It Easy to Save the Gravitino? a Supergravity ( x 4 ν 5 µν − d ]. g e dx Supersymmetry, Cosmology and New TeV Physics = 1 ]. ), which permits any use, distribution and reproduction in B 134 µ ]. Z = = N dx which becomes Re ) N 11 µν SPIRE ) x g SPIRE x ( 1 dx IN ( SPIRE πG ˆ c IN [ ]. µν M e IN [ g 16 [ dx Phys. Lett. CC-BY 4.0 − = ¯ , SPIRE = Cosmological Constraints on the Scale of Supersymmetry Breaking This article is distributed under the terms of the Creative Commons MN N g d IN 5 [ dx S ]. (1982) 223 (1982) 1303 direction in heterotic M-theory, as measured in the 5d Einstein frame after M describes the remaining dimensions. (1984) 373 2 48 48 Z S dx / 1 SPIRE S MN IN B 247 stabilisation in Calabi-Yau flux[ compactifications (1984) 265 Cosmological Constant in Standard Model Lett. Lett. g J.R. Ellis, C. Kounnas and D.V. Nanopoulos, V. Balasubramanian, P. Berglund, J.P. Conlon and F. Quevedo, E. Cremmer, S. Ferrara, C. Kounnas and D.V. Nanopoulos, J.R. Ellis, A.B. Lahanas, D.V. Nanopoulos and K. Tamvakis, H. Pagels and J.R. Primack, S. Weinberg, M. Yu. Khlopov and A.D. Linde, This shows that the appearance of no-scale structure from a peculiar combination [6] [7] [4] [5] [1] [2] [3] Attribution License ( any medium, provided the original author(s) and source are credited. 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