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PHYSICAL REVIEW D 99, 035046 (2019)

Partially composite

† ‡ Yusuf Buyukdag,* Tony Gherghetta, and Andrew S. Miller School of and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA

(Received 9 December 2018; published 28 February 2019)

We consider a supersymmetric model that uses partial compositeness to explain the mass hierarchy and predict the mass spectrum. The Higgs and third-generation superfields are elementary, while the first two matter generations are composite. Linear mixing between elementary superfields and supersymmetric operators with large anomalous dimensions is responsible for simulta- neously generating the fermion and sfermion mass hierarchies. After supersymmetry is broken by the strong dynamics, partial compositeness causes the first- and second-generation to be split from the much lighter and third-generation sfermions. This occurs even though the tree-level soft masses of the elementary fields are subject to large radiative corrections from the composite sector, which we calculate in the gravitational dual theory using the AdS=CFT correspondence. The sfermion mass scale is constrained by the observed 125 GeV Higgs , leading to stop masses and gauginos around 10–100 TeV and the first two generation sfermion masses around 100–1000 TeV. This gives rise to a splitlike supersymmetric model that explains the fermion mass hierarchy while simultaneously predicting an inverted sfermion mass spectrum consistent with LHC and flavor constraints. Finally, the lightest supersymmetric is a in the keV to TeV range, which can play the role of dark matter.

DOI: 10.1103/PhysRevD.99.035046

I. INTRODUCTION A knowledge of the sfermion mass spectrum has The discovery at the LHC [1–3] has led to important implications for collider and flavor experiments new constraints on the parameter space of supersymmetric seeking to discover supersymmetry. In split supersymmet- models. In particular, the explanation of the 125 GeV Higgs ric models, the supersymmetry-breaking scale occurs near – boson requires stop masses (or A-terms) which are large, the PeV scale, with sfermion masses in the range 10 ≳1 TeV, causing a significant increase in the tuning of 1000 TeV. Even though this hierarchy of sfermion masses supersymmetric models. In addition, in order to ameliorate seems unrelated to the fermion mass hierarchy, it begs the the supersymmetric flavor problem without any additional question as to whether these two hierarchies could in fact be structure, the first- and second-generation sfermions are explained by the same mechanism. For example, a novel required to have masses in the 100–1000 TeV range. way to account for the fermion mass hierarchy is the idea of Satisfying these two requirements leads to a version of partial compositeness [10]. New strong dynamics is respon- [4,5] dubbed minisplit [6,7] which sible for operators with large anomalous dimensions that explains the 125 GeV Higgs boson while simultaneously linearly mix with elementary . Assuming the maintaining the successful features of supersymmetric Higgs boson is elementary, a large Yukawa coupling then models such as gauge coupling unification and a dark arises for mostly elementary fermion mass eigenstates, matter candidate. Of course, this comes at the price of a while mostly composite fermion mass eigenstates have a meso-tuning, which may be a sign that we live in a correspondingly smaller Yukawa coupling. The hierarchy of Yukawa couplings is therefore explained by a set of multiverse [5], or instead could possibly be explained by operators with large, order-one anomalous dimensions. a relaxion mechanism [8,9]. If one now further assumes that the strong dynamics is responsible for breaking supersymmetry, then an interesting *[email protected] correlation between fermion and sfermion masses results † [email protected] from partial compositeness. Supersymmetric operators that ‡ [email protected] linearly mix with elementary fermions can now communi- cate supersymmetry breaking to the elementary sector. In this Published by the American Physical Society under the terms of way, composite sfermions obtain large supersymmetry the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to breaking masses, while elementary sfermions obtain hier- the author(s) and the published article’s title, journal citation, archically smaller soft masses. The fermion mass hierarchy is and DOI. Funded by SCOAP3. therefore inversely related to the sfermion mass hierarchy:

2470-0010=2019=99(3)=035046(43) 035046-1 Published by the American Physical Society BUYUKDAG, GHERGHETTA, and MILLER PHYS. REV. D 99, 035046 (2019) light (elementary) stops correspond to heavy (elementary) masses arise from a singlet spurion and the other in a top , while heavy (composite) selectrons are related to nonsinglet spurion case. We find that the observed Higgs the light (composite) . Together with the fact that mass naturally accommodates sparticle spectra that hier- gauginos and are predominantly elementary—and archically suppress the masses of the stops and the other therefore lighter than the composite sfermions—a “split” third-generation sfermions below the mass scale of the supersymmetric spectrum arises where the fermion mass first- and second-generation sfermions. In particular, if the hierarchy is naturally explained. It is the anomalous dimen- masses of the first- and second-generation sfermions are sions of the corresponding supersymmetric operators that restricted to be above 100 TeV to additionally suppress simultaneously controls the fermion and sfermion masses. flavor-changing neutral currents that arise in supersym- This contrasts with an alternative approach that radiatively metric models, the stop masses lie in the range 20–100 TeV, generates fermion masses from a sfermion anarchy [11]. while the masses of the lightest stau and may be A four-dimensional (4D), holographic description of as low as 10 TeV. Previous attempts to explain the sfermion partial compositeness for the fermion mass hierarchy mass hierarchy in a slice of AdS5 before the Higgs mass was previously considered in Refs. [12,13]. In this paper, was known were considered in Refs. [17,18]. The results we generalize this description to the supersymmetric case, obtained in this paper are the first predictions for the using results from Ref. [14]. Assuming that electroweak sfermion mass spectrum from partial compositeness that symmetry is broken in the elementary sector (with an are compatible with a 125 GeV Higgs boson. Additionally, elementary Higgs boson), we consider the linear mixing of we include for the first time the full one-loop radiative elementary superfields with supersymmetric operators in corrections to the bulk scalar soft masses squared, the Higgs- the 4D dual . This mixing is responsible not sector soft terms, and the soft trilinear scalar couplings. For only for the fermion mass hierarchy but also for the stops and other UV-localized sfermions, these corrections transmission of the (dynamical) supersymmetry breaking provide the dominant soft mass contributions, and accord- in the composite sector to the elementary sector. This leads ingly have important phenomenological consequences. For to relations that determine the fermion and sfermion mass the Higgs sector, they control the breaking of electroweak hierarchy in terms of the anomalous dimensions of super- symmetry. symmetric operators. These type of theories are similar to A schematic diagram of a possible mass spectrum in the single-sector models of supersymmetry breaking which partially composite supersymmetric model is depicted in were originally proposed in Refs. [15,16] and further Fig. 1. It assumes an elementary Higgs sector interacting studied in Refs. [17–21]. A shortened version of this work with an elementary top sector (UV localized). The summarizing the main results can be found in Ref. [22]. first- and second-generation fermions are composites (IR The nonperturbative nature of the strong dynamics only localized) of a strongly coupled sector that is also respon- allows the sfermion mass spectrum to be qualitatively sible for dynamically breaking supersymmetry. The strong determined. To obtain a more detailed sfermion spectrum dynamics therefore leads to a sfermion mass spectrum that is consistent with a 125 GeV Higgs boson, we use the that is inversely related to the fermion mass spectrum. anti-de Sitter/conformal field theory (AdS=CFT) corre- Furthermore, the supermultiplet is elementary, spondence to model the strong dynamics associated with and, since it couples gravitationally, the gravitino receives partial compositeness in a slice of five-dimensional anti-de Sitter space (AdS5) [23]. The supersymmetric Higgs sector is localized on the UV brane, while supersymmetry break- ing is confined to the IR brane. The supersymmetric matter fields are bulk fields with the (light fermions) localized near the UV (IR) brane. In the dual, five-dimen- sional (5D) gravity description, the overlap of fermion profiles [24] with the UV-localized Higgs fields mimics the partial compositeness and explains the fermion mass hierarchy [25]. The Yukawa couplings are used to constrain the bulk fermion profiles, which then determine the soft masses at the IR scale. Both the Higgs- sector soft masses and the soft trilinear scalar couplings arise at loop order, due to radiative corrections from the bulk that transmit the breaking of supersymmetry. FIG. 1. Schematic diagram depicting a possible particle spec- Renormalization group evolution is then used to run the trum of the partially composite supersymmetric model. The left soft masses down to the electroweak scale and obtain the (right) column depicts the fermions (). The sfermion mass 125 GeV Higgs mass. Using this procedure, we analyze hierarchy is inversely related to the fermion mass hierarchy and two benchmark scenarios: one for the case that the the LSP is the gravitino.

035046-2 PARTIALLY COMPOSITE SUPERSYMMETRY PHYS. REV. D 99, 035046 (2019)

3 a small supersymmetry-breaking contribution, becoming is written as 2 þ δ, where δ denotes the deviation from the the lightest supersymmetric particle (LSP) with a mass canonical scaling dimension. The Lagrangian at the UV ≳1 Λ keV. This differs from other split-supersymmetry scale, UV, is taken to have the form models where the gravitino is usually the heaviest super- 1 partner. Finally, even though the first two generations of † μ c Lψ ¼ iψ σ¯ ∂μψ − ðψOψ þ : :Þ; ð Þ matter are composite, gauge coupling unification still Λδ−1 H c 1 UV occurs at approximately 1016 GeV, as in the usual super- symmetric standard model [assuming any underlying where the Minkowski metric ημν ¼ diagð−; þ; þ; þÞ and strong dynamics is SU(5) symmetric]. It should also be an order-one UV coefficient has been assumed in the noted that the elementary matter is present at the grand second term. The mixing term in (1) means that after unified theory (GUT) scale with order-one Yukawa cou- Λ confinement at an IR scale, IR, the mass eigenstates are an plings. This helps one to avoid the tension that occurs from admixture of elementary and composite states. This is Yukawa coupling unification in usual grand unification analogous to γ-ρ mixing in QCD. To obtain analytic scenarios, where the lighter first two fermion genera- estimates of this mixing, the strong dynamics is assumed tions are elementary (do not mix with a composite sector), to be described by a large-N gauge theory, where N is the and consequently have tiny Yukawa couplings at the number of colors. In the large-N limit, the two-point GUT scale. function for a compositeP operator, O can be written as The rest of this paper is summarized as follows: In hOðpÞOð−pÞi ¼ a2=ðp2 þ m2Þ to leading order in 1, Sec. II, we review the idea of partial compositeness in the n npffiffiffi n N ¼h0jOj i ∝ N O context of the fermion mass hierarchy. This is then where an n 4π is the matrix element for to th generalized to a supersymmetric model when supersym- create the n state from the vacuum and mn is the mass of metry is assumed to be broken by the strong dynamics. This that state [26]. gives rise to a sfermion mass hierarchy that inverts the Applying these results to fermions, we consider a simple fermion mass ordering. We then construct the gravitational three-state system containing an elementary Weyl fermion ψ dual of this model in a slice of AdS5 in Sec. III. The soft , together with a lowest-lying composite Dirac fermion ðψ ð1Þ ψ cð1ÞÞ ð1Þ ¼ ð1ÞΛ ð1Þ masses resulting from supersymmetry breaking on the IR ; , with mass, mψ gψ IR, where gψ is an brane are calculated, including one-loop radiative correc- order-one coupling. Note that having a composite Dirac tions from the bulk theory to the sfermions and Higgs fermion follows from assuming that the strong dynamics fields. In Sec. IV, we develop the parameter space available does not break any Standard Model gauge symmetries. The Λ in the 5D model and estimate the constraints arising Lagrangian at the scale IR is given by on this space from various phenomenological and † μ †ð1Þ μ ð1Þ †cð1Þ μ cð1Þ theoretical requirements. We select two representative Lψ ¼ iψ σ¯ ∂μψ þiψ σ¯ ∂μψ þiψ σ¯ ∂μψ benchmark scenarios and perform a full numerical analysis. −ε Λ ðψψcð1Þ þ Þ− ð1Þðψ ð1Þψ cð1Þ þ Þ ð Þ We conclude by presenting the resulting spectra. In ψ IR H:c: mψ H:c: : 2 Appendix A, we write down the partially composite theory below the confinement scale in terms of component fields, The dimensionless coupling εψ is defined at the IR scale and then determine the mass eigenstates resulting from the to be mixing of the elementary and composite sectors. Next, in pffiffiffiffi   pffiffiffiffi 1 Λ δ−1 Appendix B, we summarize the zero-mode profiles for N IR N εψ ≡ ε˜ψ ðΛ Þ ¼ pffiffiffiffiffiffi fields in the bulk of a slice of AdS5, including their IR 4π Λ 4π Zψ UV deformations due to the presence of supersymmetry break- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 δ − 1 ing on the IR brane. Finally, in Appendix C, we derive the ≃ pffiffiffiffiffi ð Þ Λ ; 3 ζ IR 2ð1−δÞ one-loop radiative corrections to scalar soft masses ψ ðΛ Þ − 1 squared, the Higgs soft b-term, and the soft trilinear scalar UV couplings arising in the 5D theory after supersymmetry is ε˜ ε˜ ðμÞ μ d ψ ¼ broken on the IR brane. where the running parameter ψ satisfies dμ ðδ − 1Þε˜ þ ζ N ε˜3 ζ ψ ψ 16π2 ψ , ψ is an order-one constant due to II. PARTIAL COMPOSITENESS the (unknown) strong dynamics [12], and the coefficient Zψ is the wavefunction renormalization of the elementary A. The fermion mass hierarchy fermion. In the final expression we have taken the large-N We begin by briefly reviewing how partial composite- approximation of Zψ . ness explains the fermion mass hierarchy [12,13]. Consider The diagonalization of the Lagrangian (2) generates two sectors, an elementary sector with a Weyl fermion ψ fermionic admixtures of the elementary and composite and a composite sector with a (charge-conjugate) fermion states [27]. In particular, the massless fermion eigenstate is c operator Oψ. The scaling dimension of the fermion operator given by

035046-3 BUYUKDAG, GHERGHETTA, and MILLER PHYS. REV. D 99, 035046 (2019)   pffiffiffi ε Φ ¼ ϕ þ 2θψ þ θθ ϕ jψ i ≃ N jψi − ψ jψ ð1Þi superfield F, where is a complex 0 ψ ð1Þ scalar field, ψ is a Weyl fermion, and F is an auxiliary field. gψ ( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) In addition,p weffiffiffi introduce a supersymmetric chiral operator 1 1 δ − 1 O ¼ O þ 2θO þ θθO ≃ N jψi − pffiffiffiffiffi jψ ð1Þi ð Þ ϕ ψ F. The scaling dimension of the ψ ð1Þ Λ 4 ζ IR 2ð1−δÞ O ¼ 1 þ δ gψ ψ ðΛ Þ − 1 scalar operator is dim ϕ O, the scaling dimension UV 3 of the fermion operator is dim Oψ ¼ 2 þ δO, and the scaling where N ψ is a normalization constant. This expression dimension of the auxiliary operator is dim O ¼ 2 þ δO, shows that for δ ≥ 1 the mass eigenstate is mostly elemen- F where δO ≥ 0 [14]. 0 ≤ δ 1 tary, while for < the state has a sizeable composite The supersymmetric Lagrangian contains separate admixture. These admixtures play a crucial role in deter- elementary and composite sectors, together with linear mining mass hierarchies. mixing terms of the form ½ΦOc for each chiral superfield. ψ F At the UV scale, the chiral elementary fermions L;R are The superfield Lagrangian at the scale Λ is given by coupled to the elementary Higgs field H via the Yukawa UV interaction λψ ψ H, where λ is an order-one proto-Yukawa 1 L R † c LΦ ¼½Φ Φ þ ð½ΦO þ H:c:Þð6Þ coupling and the Higgs field is assumedpffiffiffi to develop a vacuum D Λδ−1 F UV expectation value (VEV) hHi¼v= 2. Diagonalizing the fermion Lagrangian at the IR scale with the Higgs contri- where Oc is the charge-conjugate composite operator with bution gives the fermion mass expression anomalous dimension δ and we have assumed an order-one UV coefficient.1 The composite sector is assumed to λ v 2 mψ ≃ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi N ψ confine at the infrared scale Λ , and thus, for the Z Z 2 IR 8 L R large-N strong dynamics, the two-point function hOOi is 2 > λ ðδ − 1Þ 16π pvffiffi δ ≥ 1; again assumed to be a sum over one-particle states. The < ζψ N 2   Lagrangian at the IR scale can be written as ≃ 2ð1−δÞ ð5Þ 2 Λ > λ 16π pvffiffi IR : ζ ð1 − δÞ Λ 0 ≤ δ < 1; ψ N 2 UV L ¼½Φ†Φ þ½Φcð1Þ†Φcð1Þ þ½Φð1Þ†Φð1Þ Φ D D D δ ≡ δ ¼ δ þε Λ ð½ΦΦcð1Þ þ Þþ ð1Þð½Φð1ÞΦcð1Þ þ Þ where for simplicity we have assumed that L R, Φ IR H:c: mΦ H:c: ; ð1Þ ð1Þ ð1Þ F F gψ ≡ gψ ¼ gψ , εψ ≡ εψ ¼ εψ , and explicitly included L R L R ð7Þ the normalization factor from (4). The wavefunction coef- 0 ≤ δ 1 ficients ZL;R are approximated as in (3), and for < we Φð1Þ Φcð1Þ ð1Þ where ( ) is the lowest-lying composite chiral have assumed εψ ≲ gψ . Notice that when δ ≥ 1 there is no O Oc ð1Þ ¼ ð1ÞΛ superfield corresponding to ( ) and mΦ gΦ IR is power-law suppression in the Yukawa coupling since the ð1Þ the lowest-lying resonance mass, with g an order-one mass eigenstates are mostly elementary and have order-one Φ coupling. Note that we have neglected heavier resonances couplings to the elementary Higgs field. This, for instance, and higher-order terms in (7). The Lagrangian contains would explain the Yukawa coupling of the top quark. This mixing terms between the elementary superfield Φ and the contrasts with the case 0 ≤ δ < 1, where the mass eigenstates Φcð1Þ have a sizeable composite admixture with a power-law lowest-lying composite superfield . The dimension- ε suppressed Yukawa interaction (due to the fact that the less constant Φ is defined at the IR scale to be proto-Yukawa coupling λ in the elementary sector is divided pffiffiffiffi   pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 1 Λ δ−1 by the large wavefunction renormalization factor, Z Z ,at ε ≡ ε˜ ðΛ Þ N ¼ pffiffiffiffiffiffi IR N L R Φ Φ IR 4π ZΦ Λ 4π the IR scale). These states describe the remaining light 8   UV pffiffiffiffiffiffi δ−1 fermions in the Standard Model, where for each flavor i there > δ−1 Λ <> pffiffiffiffi IR δ > 1 is a corresponding operator with anomalous dimension δ . ζ Λ i ≃ Φ UV ð Þ Therefore, the hierarchical Yukawa couplings result from > pffiffiffiffiffiffi 8 > p1ffiffiffiffi−δ order-one anomalous dimensions of operators. In the super- : 0 ≤ δ < 1; ζΦ symmetric generalization, these anomalous dimensions also determine the magnitude of the corresponding sfermion dε˜Φ N 3 where ε˜Φ satisfies μ ¼ðδ − 1Þε˜Φ þ ζΦ 2 ε˜Φ, with ζΦ masses. dμ 16π an order-one constant. Note that the supersymmetric nonrenormalization theorem guarantees that εΦ only B. Supersymmetric partial compositeness depends on the wavefunction renormalization, unlike the 1. Chiral supermultiplet 1A kinetic mixing between the elementary and composite Next, we consider the supersymmetric generalization of Λ−δO ½Φ†O þ sectors of the form UV H:c: D has been omitted in our partial compositeness. Consider the elementary chiral minimal setup.

035046-4 PARTIALLY COMPOSITE SUPERSYMMETRY PHYS. REV. D 99, 035046 (2019)   1 1 nonsupersymmetric case, where the vertex renormalization L ¼ ½ α þ ½ ð1Þα ð1Þ þ piece in (3) was neglected. V 4 W Wα F 4 W Wα F H:c: Just as for the fermions in Sec. II A, the mixing terms in    ð1Þ ð1Þ† 2 ð1Þ Φ þ Φ the Lagrangian (7) cause the scalar mass eigenstates to be þ Λ2 ε V þ g Vð1Þ þ Vpffiffiffi V ; ð11Þ IR V V 2Λ admixtures of elementary and composite states. The details IR D of this mixing are given in Appendix A. Using the result ð1Þ (A4) and (8), the massless scalar eigenstate ϕ0 is given by where gV is the composite vector coupling,pffiffiffi and the ε ≡ ε˜ ðΛ Þ N ( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) dimensionless constant, V V IR 4π parametrizes 1 δ − 1 ε˜ ðμÞ jϕ i ≃ N jϕi − jϕð1Þi ð Þ the mixing at the IR scale. The running parameter V 0 Φ pffiffiffiffiffiffi Λ ; 9 dε˜ 3 ð1Þ IR 2ð1−δÞ μ V ¼ ζ N ε˜ ζ g ζΦ ð Þ − 1 satisfies μ V 16π2 V, with V an order-one constant Φ ΛUV d that comes from the (unknown) strong dynamics. This where ϕ is an elementary scalar, ϕð1Þ is the lowest-lying immediately leads to the solution composite scalar, and N Φ is a normalization constant. 1 1 4π Similarly to the fermion case (4), the massless scalar ε˜ ðΛ Þ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≃qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffi; ð12Þ V IR 1 ζ Λ Λ þ V N ð UVÞ UV N eigenstates are mostly elementary for δ > 1, whereas for ε˜2 8π2 log Λ 2ζV logð Λ Þ V IR IR 0 ≤ δ < 1 they are an admixture of elementary and composite states. In fact, supersymmetry guarantees that where we have taken the large-N limit in the last term. the admixtures for both the fermion and scalar field in the The Lagrangian (11) can be diagonalized to obtain the chiral multiplet are the same. mass eigenstates (the details are given in Appendix A). Using (A15), the massless eigenstate is found 2. Vector supermultiplet to be 8 9 Next, we consider the supersymmetric generalization of < = partial compositeness for gauge fields. For simplicity, we 1 ð1Þ jAμ0i ≃ N jAμi − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jAμ i ; ð13Þ V: ð1Þ Λ ; consider the case of an Abelian U(1) gauge field, and 2ζ ð UVÞ gV V log Λ assume our discussion can be generalized to non-Abelian IR gauge fields. In the nonsupersymmetric case, the source ð1Þ where N V is the normalization constant and Aμ is the field Aμ mixes with the conserved U(1) current Jμ and μν ð1Þ lowest-lying composite gauge boson. Assuming a large induces a kinetic mixing of the form F Fμν [27,28], where hierarchy (Λ ≫ Λ ) and order-one values for the ð1Þ UV IR Fμν is the field strength of the lowest-lying composite field, couplings, this expression shows that the massless eigen- ð1Þ Aμ . To supersymmetrize this coupling we introduce state is mostly elementary, but with a sizeable composite † μ † † † † admixture. The corresponding gauge coupling for this the vector superfield, V ¼ θ σ¯ θAμ þ θ θ θλ þ θθθ λ þ 1 † † massless state is given in (A20). θθθ θ D, with field strength superfield Wα ¼ λαþ 2 The gaugino part of the Lagrangian (11) is similarly θ þ i ðσμσ¯ νθÞ þ θθðσμ∂ λ†Þ αD 2 αFμν i μ α. The conserved cur- given in Appendix A, where it is shown to be a special case rent Jμ can be embedded into a linear supermultiplet J , (δ ¼ 1) of the fermion Lagrangian (2). After diagonalizing 2 †2 which satisfies the condition D J ¼ D J ¼ 0 and guar- the mass terms (see Appendix A), the massless gaugino antees current conservation [29]. eigenstate becomes The supersymmetric Lagrangian at the UV scale is 8 9 given by < = 1 ð1Þ jλ0i ≃ N jλi − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jλ i ; ð14Þ   V: ð1Þ Λ ; 1 UV α g 2ζV logð Λ Þ L ¼ ½ þ þ 2ε˜ ½ J ð Þ V IR V 4 W Wα F H:c: V V D; 10 where we have used (A24) and λð1Þ represents the lowest- ε˜ ½ J where V is the mixing parameter. Since V D can be lying composite gaugino. By supersymmetry, the overall ½ αW þ N transformed into a kinetic term such as W α F H:c:, normalization constant, V, and composite admixture are where W is the field-strength superfield operator associated the same as in (13). with a composite vector superfield operator V,weomit such a term in the Lagrangian. 3. Gravity supermultiplet After confinement, the IR Lagrangian for the source V, ð1Þ Next, we consider the supersymmetric generalization of together with the lowest-lying composite vector V , field- partial compositeness for gravity fields. In the nonsuper- ð1Þ Φð1Þ strength superfield Wα , and chiral adjoint superfield V , symmetric case the graviton hμν mixes with the conserved can be written as energy-momentum tensor Tμν, inducing a kinetic mixing

035046-5 BUYUKDAG, GHERGHETTA, and MILLER PHYS. REV. D 99, 035046 (2019)   1 Λ between the graviton and massive -2 resonances [27] IR ð1Þ jhμν0i ≃ N jhμνi − pffiffiffiffiffiffi jhμν i : ð20Þ (which is completely analogous to graviton-f2 mixing in H ð1Þ ζ Λ gH H UV QCD). In the supersymmetric extension, the graviton hμν and gravitino ψ μ can be embedded into the real super- Λ ≪ Λ Assuming IR UV, this shows that the massless eigen- gravity field, state is mostly elementary with a tiny composite admixture. The gravitino part of the Lagrangian (17) is given in 1 † ν † † † † Appendix A, where it is shown to be a special case (δ ¼ 2) Hμ ¼ −pffiffiffiðθ σ¯ θÞhμν − iθθθ λμ þ iθ θ θλμ þ; ð15Þ 2 of the fermion Lagrangian (2). After diagonalizing the mass terms (see Appendix A), the massless gravitino eigenstate 1 1 ρ where we have assumed the gauge 2 ψ μ ≡ λμ þ 3 σμσ¯ λρ becomes and only written the graviton and gravitino parts. The   1 Λ conserved energy-momentum tensor Tμν can be embedded jψ i ≃ N jψ i − IR jψ ð1Þi ð Þ μ0 H μ ð1Þpffiffiffiffiffiffi μ ; 21 into the supercurrent Θμ, which satisfies the condition Λ g ζH UV μ † H −iσ¯ DΘμ ¼ D X, where X is an antichiral superfield, and μν ð1Þ guarantees current conservation: ∂μT ¼ 0 [29]. The where we have used (A38) and ψ μ represents the lowest- supersymmetric Lagrangian at the UV scale is given by lying composite gravitino. By supersymmetry, the composite admixture is the same as in (20), and thus the 4 2ε˜ μ H μ gravitino is again mostly elementary. L ¼ ½HμE þ ½HμΘ ; ð16Þ H 3 D Λ D UV

μ C. Supersymmetry breaking where ε˜H is the mixing parameter and E is the Einstein superfield [30]. In an analogous fashion to the gauge boson Supersymmetry is assumed to be broken in the composite case, the IR Lagrangian after confinement can be written as sector by the strong dynamics and will be parametrized by a composite spurion, chiral superfield, X ¼ θθF, where the vacuum expectation value hFi ≡ FX ≠ 0.Inthelarge-N 4 μ 4 ð1Þ ð1Þμ pffiffiffi L ¼ ½ þ ½ μ 2 H HμE D H E D ∝ N Λ 3 3 limit, we have FX 4π IR [31]. 2 μ ð1Þ ð1Þ 2 þ 2Λ ½ðεHH þ g Hμ Þ ; ð17Þ IR H D 1. Sfermion masses pffiffiffi ε ¼ ε˜ ðΛ Þ N ð1Þ Supersymmetry-breaking scalar masses are only gener- where H H IR 4π and Hμ is a composite real super- field with corresponding Einstein superfield Eð1Þμ.The ated for the composite sector fields, since there is no direct ε˜ μ ε˜ coupling of the supersymmetry-breaking spurion to the pffiffiffiffiffiH d H running parameter ε˜HðμÞ ≡ Λ satisfies μ μ ¼ ε˜H þ ZH UV d elementary fields. For instance, the massive chiral super- ζ N ε˜3 ζ ð1Þ H 16π2 H,with H an order-one constant that arises from the field Φ directly couples to the supersymmetry breaking (unknown) strong dynamics. The solution is then given by in the composite sector via the following D-term:

Λ 1 1 Λ 4π ð1Þ2 2 IR rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi IR pffiffiffiffi gΦ † ð1Þ† ð1Þ ð1Þ2 jFX j ð1Þ† ð1Þ ε˜HðΛ Þ¼   ≃ ; ξ ½X XΦ Φ ¼ ξ ϕ ϕ ð Þ IR Λ ζ Λ 4 2 D 4gΦ 2 ; 22 UV 1 ζ Λ 2 H UV N Λ Λ þ HN 1 − ð IR Þ IR IR ε˜2 16π2 Λ H UV ð Þ where ξ4 is a dimensionless parameter, which for a large-N 18 16π2 gauge theory is proportional to N [26]. This D-term where, in the last term, we have taken the large-N limit. Since interaction gives a supersymmetry-breaking mass to the the graviton couples to the energy-momentum tensor with composite scalar field ϕð1Þ. Given the scalar admixture (9), 18 strength 1=MP (where MP ¼ 2.4 × 10 GeV is the reduced the corresponding sfermion mass squared becomes 1 εH Planck mass), we obtain the matching condition ≡ Λ at 8 MP IR ðδ−1Þ Λ 2ðδ−1Þ j j2 < IR FX 2 ζ ðΛ Þ ξ4 2 δ ≥ 1; the IR scale, which leads to the relation jFX j Φ UV Λ ˜ 2 ≃ N 2 ε2 ξ ≃ IR m Φ Φ 4 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Λ : ð1−δÞ jFX j   IR ξ4 2 0 ≤ δ < 1; 2 ζΦ Λ pffiffiffiffiffiffi Λ pffiffiffiffiffiffi IR ≃ ζ Λ 1 − IR ≃ ζ Λ ð Þ MP H UV Λ H UV: 19 ð Þ UV 23

ð1Þ The Lagrangian (17) contains mass mixing terms, and the where we have assumed εΦ ≲ gΦ . Note that the sfermion diagonalization details are given in Appendix A. Using mass is power-law suppressed for δ > 1. This is because the (A32), the massless graviton eigenstate is found to be mass eigenstate is mostly elementary and therefore can

035046-6 PARTIALLY COMPOSITE SUPERSYMMETRY PHYS. REV. D 99, 035046 (2019) only obtain a supersymmetry-breaking mass via mixing generically lighter than the heavy composite sfermions with the composite sector. This contrasts with the case with 0 ≤ δ < 1, leading to a “split” spectrum. 0 ≤ δ < 1, where the mass eigenstate has a sizeable However when δ is greater than the critical value δ, the composite admixture and therefore directly feels the gaugino mass radiative correction (24) gives the dominant supersymmetry breaking from the composite sector without contribution to the scalar masses. The critical value δ any power-law suppression. In this way, a sfermion mass occurs when δm˜ 2 ≃ m˜ 2, and using (26) it takes the value hierarchy can be explained by anomalous dimensions of h i 4π Λ ð UVÞ supersymmetric operators. log g log Λ δ δ ≃ 1 þ i IR ; ð27Þ Note, however, that as is increased beyond one, the ΛUV logð Λ Þ scalar mass squared becomes increasingly small, since the IR scalar is becoming more elementary [using (9)]. Eventually, where we have included only the dominant gauge-coupling radiative corrections are sufficiently large that they provide contribution. Thus, for δ > δ , the gaugino is heavierpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi than the dominant contribution to the sfermion mass squared. α ð4πÞ the corresponding sfermionpffiffiffiffiffiffiffi mass by a factor of i= . For instance, the sfermion-fermion-gaugino interaction ≫ Λ Note that when FX IR, the lightest gaugino with a massive gaugino leads to the one-loop contribution becomes a Dirac fermion with a mass 2 Λ 2 g 2 IR δ ˜ ≃ i ð Þ Mλ ≃ ε Λ ≃ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð28Þ m 2 Mλ ; 24 V IR Λ 16π i 2ζ logð UVÞ V ΛIR ¼ 1 ð1Þ where gi (i , 2, 3) are the Standard Model U Y × This value parametrically matches the gaugino mass that ð2Þ ð3Þ SU L ×SU gauge couplings, respectively. Note that is found in a five-dimensional (5D) supersymmetric stan- this radiative correction is assumed to be finite. Therefore, dard model where supersymmetry breaking arises from δ there is a maximum value of the anomalous dimension , “twisted” boundary conditions [17]. The Dirac limit can beyond which the gaugino-mediated contribution (24) also be seen in Fig. 2, which shows the elementaryp stateffiffiffiffiffiffiffi dominates. ð1Þ λ marrying the composite Weyl fermion χ as FX ∼ ðΛ ÞΛ becomes larger, giving rise to a Dirac mass gs IR IR 2. Gaugino masses pthatffiffiffiffiffiffiffi is smaller than the mass (A26) that applies when ≪ Λ The spurion superfield X can also be used to generate FX IR. gaugino masses. Since the massless gaugino (14) contains a λð1Þ admixture, supersymmetry breaking in the composite 3. Gravitino masses sector is transmitted by the following interactions: The supersymmetry breaking from the composite sector gives rise to a positive vacuum energy. This contribution ξ ð1Þ2 can be cancelled by introducing a constant superpotential 3 gV ð½X αð1Þ ð1Þ þ Þ ð Þ 2 Λ W Wα F H:c: ; 25 W, which induces a mass term for the elementary gravitino IR 1 4π W μ ρ where ξ3 is proportional to pffiffiffi for a large-N gauge theory. − ψ ρ½σ ; σ¯ ψ μ þ H:c: ð29Þ N 4 M2 The supersymmetry-breaking interaction (25), adds the P ð1Þ2 FX ð1Þ ð1Þ 10 Majorana mass term ξ3g λ λ þ H:c: to the gau- V 2ΛIR gino mass terms in (A21). Note that there is no χð1Þ mass ð1Þ2 ½X 1 term because the interaction V D is not gauge invariant. Diagonalization of the mass matrix then leads to a gaugino mass

/ 0.100 M ¼ N 2 ε2 ξ FX ≃ 2ξ FX ð Þ Mλ V V 3 Λ g 3 Λ ; 26 IR IR 0.010 where we have used the gauge coupling relation (A20) and ε ≃ ðΛ Þ 0.001 V gs IR . The result (26) is consistent with simply 0.01 0.10 1 10 100 using (14) and the fact that the massless mode only has a F / λð1Þ admixture. Notice that the gaugino mass is suppressed by a logðΛ =Λ Þ factor due to the fact that the elemen- ð1Þ ¼ 2 ε ¼ 0 3 UV IR FIG. 2. Gaugino mass eigenvalues for gV , V . , and tary gaugino mixes with the composite sector via a ξ3 ¼ 1 as a function of the supersymmetry-breaking order marginal coupling. This causes the gauginos to be parameter FX.

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At the IR scale, this term becomes the gravitino is generically much lighter than the heavy composite sfermions with 0 ≤ δ < 1. There is also a heavy 1 ð1Þ ð1Þ − W ψ ½σμ σ¯ ρψ þ Dirac gravitino mostly comprised of ψ μ and χμ with mass 2 ρ ; μ H:c: ð1Þ2 4ZH M ∼ð þ ε2 Þ1=2Λ P gH H IR. 1 4π FX ≫ Λ ≃− pffiffiffiffi pffiffiffi ψ ½σμ σ¯ ρψ þ ð Þ In the opposite limit FX IRMP, one gravitino Weyl 2 ρ ; μ H:c:; 30 4ζ ε˜ state decouples, and the elementary state ψ μ marries the H H N 3MP ð1Þ composite state ψ μ [17] such that the lightest gravitino ε˜ ðΛ Þ ð1Þ where ZH is defined from H IR in Sec. II B 3 and in ≃ Λ 2 then becomes a Dirac fermion with a mass m3=2 gH IR. j j2 N ≃ 3 jWj the second equation, we have tuned FX 16π2 2 so that This is similar to what occurs for the gaugino, except that MP the cosmological constant vanishes. Note that this means the since the mixing is very small, the Dirac limit is reached elementary sector is supersymmetric AdS4, with Minkowski much more slowly for the gravitino. space obtained after the “uplift” from the supersymmetry- breaking composite sector. Diagonalization of the mass 4. Higgs soft mass matrix [given in (A36)] with the inclusion of (30) then leads There is no direct coupling of the supersymmetry- to a gravitino mass breaking spurion to the elementary fields, and therefore the Higgs soft mass can only be generated radiatively, via FX m3 2 ≃ ξ3 pffiffiffi ; ð31Þ loops of fields with a composite admixture. In particular, = 3 MP gaugino and sfermion loops can transmit the supersym- metry-breaking effects to the elementary sector. ≪ Λ N ≃ 1 ξ ¼ 1 p4πffiffiffi where FX IRMP, H , and 3 ζ ε˜2 . The result H H N The first type of sfermion contributions are due to (31) is consistent with simply using (21) and the fact that Yukawa interactions. These lead to the Higgs soft mass Λ ≪ the massless mode is mostly elementary. Since IR MP, squared

  y2ðΛ Þ Λ ðΔm2 Þ ≃ δ IR m˜ 2 log IR H y 16π2 TeV   λ2 N 2 Λ ≃ Φ m˜ 2 log IR 16π2 Z2 TeV Φ       16π2λ2 N 4 δ − 1 3 Λ 2ðδ−1Þ j j2 Λ ≃ Φ IR ξ FX IR ð Þ 2 3 Λ 4 2 log ; 32 ζ IR 2ðδ−1Þ Λ Λ N Φ 1 − ð Þ UV IR TeV ΛUV ðΛ Þ ≡ h i δ ¼ δ ¼ δ λ where yδ IR mψ = H is the Yukawa coupling from Sec. II A with L R chosen for simplicity, is the proto- Yukawa coupling, N Φ is a normalization constant given in (A4), and ZΦ is defined in (8). − Φð1Þ The second type of sfermion corrections are due to D-term gauge interactions and the mixing term V V . Since the Φð1Þ scalar and the auxiliary field component of V mediates the supersymmetry breaking, the easiest way to estimate the Φð1Þ D-term contribution is to approximately solve for V in order to determine how the elementary D-term is coupled to the composite sfermions. This procedure generates the following correction: 0 1   2 g1ðΛ Þ B Λ N C ðΔm2 Þ ≃ YðHÞYðϕÞ IR ε @log IR þ  Φ Am˜ 2 H D 8π2 V m2 TeV 2ε2 1 þ Φ1 Φ Λ2 0 IR 1

ΛIR 2 2 g1ðΛ Þ Blogð Þ δ − 1 N C jFX j ≃ ð Þ ðϕÞ IR ε N 2 @ TeV þ  Φ Aξ ð Þ Y H Y 2 V Φ Λ 2ð1−δÞ 2 4 2 ; 33 8π ζ ð IR Þ − 1 mΦ Λ Φ Λ 2 1 þ 1 IR UV Λ2 IR ð1Þ 2 ¼ðε2 þ ð1Þ2ÞΛ2 where g1 is the U Y gauge coupling, Y denotes the , and mΦ1 Φ gΦ IR. Finally, the corrections that involve gauginos arise from gauge interactions. The dominant contribution is given by   g2ðΛ Þ Λ ðΔm2 Þ ≃ 4CðR ÞN 2 2 IR M2 log IR ; ð34Þ H g H V 8π2 λ TeV

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100 κ ( m ) where μ is a dimensionless coupling. Assuming that the 50 H g2 Peccei-Quinn symmetry is spontaneously broken by a ( m ) H g1 nonzero vacuum expectation value hSi ∼ f, an effective 10 μ-term of size 5

[TeV] 2 κμf H μ ≃ ð36Þ m ( m ) H y 2MP 1 0.5 ( mH)D is then generated. Since the global symmetry is anomalous (and assuming all other sources of breaking are small), the

0.1 pseudo Nambu- associated with the spon- 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 taneous symmetry breaking can be identified with the . This axion is of the invisible DFSZ type, and is FIG. 3. One-loop corrections to the Higgs mass parameter that consistent with the present astrophysical constraints pro- 109 ≲ ≲ 1012 arise from gauge, Yukawa, and D-term interactions due to a vided that GeV f GeV. For this range of f, single sfermion or gaugino as a function of the anomalous the μ-term (36) can easily accommodate values in the dimension δ. range 0.1 TeV ≲ μ ≲ 100 TeV. Thus, using the Kim-Nilles mechanism, we can solve the strong-CP problem and generate the required values of the μ term. where g2 is the SU(2) gauge coupling and CðRÞ is the quadratic Casimir of the representation R. In Fig. 3,we E. The sfermion mass hierarchy plot the approximate expressions (32), (33), and (34) In a supersymmetric model, partial compositeness relates for a singlepffiffiffiffiffiffiffiffiffiffiffiffi sfermion [with YðϕÞ¼1] and gaugino, ξ ζ ¼ξ ¼ð4 75 1010 Þ2 Λ ¼ the fermion and sfermion mass hierarchies. To explicitly assuming 4= ΦFX 3FX . × GeV , IR ð1Þ ð1Þ 2 ð1Þ2 see this relation we consider a numerical example involving 16 ΛIR 16π 2 2 × 10 GeV, ≃ 0.026, gΦ gΦ λ ≃ 1, g ≃ g5k, ΛUV L R N V the electron and the top quark. In order to explain the ð1Þ2 and ζΦ ≃ 1=g , where i ¼ L, R. Furthermore, for later fermion mass hierarchy, the top quark must be mostly i;V Φi;V 2 elementary with δ > 1 (i.e., irrelevant mixing), while the comparison with the 5D calculation, we use values for m t Φ1 electron must have a sizeable composite admixture with and m2 corresponding to the 5D Kaluza-Klein masses. V1 0 < δe < 1 (i.e., relevant mixing). Using the fermion mass Only the maximal contribution arising from the Yukawa expressions (5), the ratio of the Yukawa couplings at the IR δ ¼ δ δ ¼ 1 interaction (corresponding to L and R ) is shown scale is then given by in the figure. All contributions are qualitatively similar to   2ð1−δ Þ the precise 5D results that are given in Sec. III C and 1 − δ Λ e ye ≃ e IR ð Þ calculated in Appendix C. δ − 1 Λ : 37 yt t UV

D. mass The ratio of the electron to the top-quark Yukawa coupling is determined in terms of δ , and for a sufficiently large Since the Higgs supermultiplet is elementary, there is no e;t hierarchy between Λ and Λ , depends sensitively on the direct coupling to the supersymmetry-breaking spurion in IR UV anomalous dimension δ . It is plotted in Fig. 4 for various the composite sector, and therefore the generation of a μ- e values of Λ =Λ , assuming that the Yukawa coupling term via a Giudice-Masiero mechanism [32] is forbidden. IR UV ratio is approximately y =y ≃ 10−5 over most IR scales. Instead, to generate a sizeable μ-term, we consider the e t Note that (37) is a simplified expression for δ values not Kim-Nilles mechanism [33] and introduce an elementary very close to one, but exact expressions are used in the Standard Model singlet S, together with a global U(1) figure. Peccei-Quinn symmetry, where H , H , S have the charges u d Using the sfermion mass expressions (23) and (24),we þ1, þ1, −1, respectively.2 This global symmetry then similarly obtain the ratio of the selectron mass squared to allows the following nonrenormalizable superpotential the stop mass squared at the IR scale: term: 8   2ð1−δ Þ > 1−δ Λ t κ 2 < e IR δ δ μ 2 δ −1 Λ t < t ; W ¼ S H H ; ð35Þ me˜ ¼ t UV ð Þ KN 2 u d 2   38 MP m > 4π 2 Λ t˜ :> ð1 − δ Þlog UV δ ≥ δ : α3 e ΛIR t t 2To allow for Yukawa interactions with the Higgs fields, the fermions Q, L, u¯, d¯, e¯ must have charges −1, −1,0,0,0, Note that this expression is separately valid for left- and respectively. right-handed scalar masses and only the contribution

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1 Z –16 compactified on a S = 2 orbifold of radius R. The 10 spacetime metric is anti-de Sitter, given by 0.8 10–10 2 −2kjyj μ ν 2 M N ds ¼ e ημνdx dx þ dy ≡ gMNdx dx ; ð39Þ 10–7 0.6 where k is the AdS curvature scale and capital Latin indices M ¼ðμ; 5Þ label the 5D coordinates. The 5D spacetime is a –5 10 slice of AdS5 geometry, bounded by two 3-branes located e at the orbifold fixed points: a UV brane at y ¼ 0 and an IR 3 0.4 10–4 brane at y ¼ πR. Λ ¼ The cutoff scale of the UV brane is UV M5, where 350 100 200 40 M5 is the 5D Planck scale, while the scale of the IR brane is Λ ¼ Λ e−πkR. The 4D reduced Planck mass, M ,is 0.2 10 IR UV P 10–3 given by [23]

M3 M3 M2 ¼ 5 ð1 − e−2πkRÞ ≃ 5 ; ð40Þ 0.0 P k k 1.0 1.2 1.4 1.6 1.8 where we are assuming πkR ≫ 1. Note that this expression t is consistent with the result (19) derived from partial compositeness. In order for the classical metric solution FIG. 4. Values of the anomalous dimensions δe and δt that give rise to the electron to top-quark Yukawa coupling ratio. The to be valid, the AdS curvature must be small enough approximately horizontal lines are contours of the ratio ΛIR=ΛUV, compared to the 5D Planck scale so that higher-order while the approximately vertical lines are contours of the tree- curvature terms in the 5D gravitational action can be level sfermion mass ratio me˜ =mt˜. The running of the Yukawa neglected. This requires k=M5 ≲ 2 [37], but, in the follow- Λ ¼ 1018 couplings has been included, assuming UV GeV, ing, we have taken k to be generically smaller, choos- β ¼ 3 tan , and a supersymmetric mass threshold at 50 TeV. ing k=M5 ¼ 0.1. Besides gravity, we introduce the matter and gauge field is included in (24) for the gaugino-mediated contribution to content of the minimal supersymmetric standard model the stop mass squared. Interestingly, the two expressions (MSSM) in the bulk. Since only Dirac fermions are allowed by the 5D Lorentz algebra, the bulk supersymmetry has (37) and (38) differ only in the exponents for δt < δt , while N ¼ 2 for δt ≥ δt the ratio no longer depends on δt. As shown in eight supercharges, corresponding to supersym- Fig. 4, the allowed region is 0 ≲ δe ≲ 0.9 and 1 ≲ δt ≲ 1.8, metry (SUSY) from the 4D perspective. All fields that Λ Λ N ¼ 2 depending on the value of IR= UV. The largest value of propagate in the AdS bulk are thus in representa- Λ the ratio me˜ =mt˜ is approximately 140 (390) for IR= tions of supersymmetry, but the orbifold compactification Λ ≃ 10−3ð10−16Þ breaks this to an N ¼ 1 supersymmetry at the massless UV . Note that the Yukawa coupling ratio level, preserving four supercharges [17,25]. The massless contours in Fig. 4, end at δt because the sfermion mass ratio becomes approximately constant as seen in (38). modes which form this 4D MSSM are the zero-mode Although these are naïve tree-level results obtained at the solutions in the Kaluza-Klein (KK) decompositions of the IR scale, they clearly reveal an inverted mass hierarchy for 5D fields. the sparticle spectrum where me˜ =mt˜ ∼ 10–350. To obtain a The zero-mode profiles for the bulk fields are summa- physical mass spectrum, this range is further restricted due rized in Appendix B. In a bulk hypermultiplet, the fermion to renormalization group (RG) effects. We next include and scalar zero-mode profiles depend on the bulk fermion these effects, as well as a number of phenomenological mass parameter c. By the AdS=CFT correspondence, the constraints, such as the 125 GeV Higgs boson. Since the scaling dimension of fermionic operators in the 4D dual 3 1 strong dynamics is nonperturbative, the AdS=CFT corre- theory is dim Oψ ¼ 2 þjc 2 j, and for scalar operators it 1 spondence is used to calculate the mass spectrum in a slice is dim Oϕ ¼ 1 þjc 2 j, where the upper (lower) sign is of five-dimensional AdS. used for left-handed (right-handed) fields. Thus, there is 1 direct relation δi ¼jci 2 j between the anomalous dimen- III. THE FIVE-DIMENSIONAL PICTURE sions δi in the 4D dual theory (introduced in Sec. II B) and A. Supersymmetry in a slice of AdS 3We do not specify a particular mechanism to stabilize We consider a warped five-dimensional spacetime μ the extra dimension. In a supersymmetric theory, one possibility ðx ;yÞ, where μ ¼ 0, 1, 2, 3 are the 4D coordinates and is supersymmetrization of the Goldberger-Wise mechanism −πR ≤ y ≤ πR is the coordinate of an extra dimension [34–36].

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second-generation sfermions are IR localized. Due to the properties of localization, the effective coupling strength of each superfield on the IR brane is inversely related to its coupling strength on the UV brane. Therefore, when supersymmetry is broken on the IR brane, as discussed in Sec. III C, the localization of the sfermions induced by the fermion mass spectrum results in an inverted scalar soft mass spectrum: light fermions have heavy , while heavy fermions have light superpartners. We next construct the details of this distinctive supersymmetric particle spectrum.

FIG. 5. Schematic diagram of the 5D model with the Higgs B. The fermion mass hierarchy fields, Hu;d, localized on the UV brane and supersymmetry broken on the IR brane. The Yukawa coupling hierarchy requires Consider first the generation of the fermion mass ψ ð0Þ hierarchy [25,38]. In our 5D spacetime, the SM fermion that the lighter first- and second-generation fermions 1;2 (and ð0Þ mass hierarchy is determined from the overlap of the bulk their superpartners ϕ˜ ) are IR localized (dark gray line) and the 1;2 SM fermion zero modes with the UV-localized Higgs ψ ð0Þ heavier, third-generation fermions 3 (and their superpartners fields. The Yukawa interactions take the form ϕ˜ ð0Þ 3 ) are UV localized (light gray line). The vector super- Z ð0Þ ð0Þ pffiffiffiffiffiffi multiplet, Aμ and λ , (dashed line) is not localized, while ¼ 5 − ð5Þ½Ψ¯ ð μ ÞΨ ð μ Þþ ð μÞδð Þ ð0Þ ð0Þ S5 d x gYij iL x ;y jR x ;y H:c: H x y the gravity multiplet, hμν and ψ˜ μ , (dotted line) is UV localized. Z ≡ 4 ½ ψ¯ ð0Þð μÞψ ð0Þð μÞ ð μÞþ þ ð Þ d x yij iL x jR x H x H:c ; 41 the bulk fermion mass parameters ci. Furthermore, for a bulk vector supermultiplet, the zero-mode profiles of the ð5Þ ¼ 1 where the Yij (with flavor indices i, j , 2, 3) are gauge field and gaugino field are flat, such that the bulk dimensionful (inverse mass) 5D Yukawa couplings, Ψ ð Þ 1 L R mass parameter of the gaugino is c ¼ . ð2Þ 2 is a Dirac spinors that contains an SU L doublet (singlet) Similarly, the gravity supermultiplet contains a graviton of the MSSM as its zero mode, and H is the appropriate −ky 3 4 with the UV-localized zero mode (∝ e ) and a spin-2 Higgs field. Using the 5D fermion zero-mode profiles Rarita-Schwinger fermion (the gravitino), whose bulk mass (B3), the effective 4D SM Yukawa couplings yij are then 3 parameter is fixed to be c ¼ 2. given by [25] The warped geometry naturally generates a separation of ¼ ð5Þ ˜ ð0Þð0Þ˜ ð0Þð0Þ scales that can be used to explain the hierarchy between the yij Yij fiL fjR scale of supersymmetry breaking and the Planck scale. The sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − c 1 þ c IR brane is therefore identified with the scale where ¼ ð5Þ 2 iL 2 jR ð Þ Yij k 2ð1− Þπ 2ð1þ Þπ : 42 supersymmetry is broken; the bulk and UV brane remain e 2 ciL kR − 1 e 2 cjR kR − 1 supersymmetric. The Higgs fields are localized on the UV ð5Þ brane, while the rest of the MSSM fields propagate in the By assuming that the dimensionless 5D couplings Y k are bulk and couple to the Higgs fields with brane-localized ij of order one, and since πkR ≫ 1, the hierarchy in the 4D Yukawa couplings. As discussed in Sec. III B, the degree of Yukawa couplings y is generated by the order one bulk overlap between a particular bulk fermion zero-mode ij profile and a UV-localized Higgs field determines its mass parameters ci of the fermions. Recall that in the 4D effective 4D Yukawa coupling and thus the size of the dual theory, this is equivalent to choosing the anomalous δ corresponding fermion mass. In this setup, third-generation dimensions i. fermions are therefore UV localized, whereas the lighter After electroweak symmetry breaking, the neutral com- ¼ first- and second-generation fermions are more IR local- ponents of each MSSM Higgs doublet acquire VEVs, vu h 0i ¼h 0i β ≡ ized. Including the SM gauge fields, the gauginos, and Hu and vd Hd , which are related by tan vu=vd, gravity, a full schematic diagram of the 5D model is and the fermions obtain masses depicted in Fig. 5. ð Þ ¼ð Þ β ð Þ Since the localization of the fermion in each chiral me ij ye ijv cos ; 43a supermultiplet determines the corresponding scalar localization, this 5D fermion geography has distinctive 4Use of the zero-mode approximation for the profiles, where consequences in the SUSY sector: the third-generation the backreaction of the boundary Higgs-generated fermion mass 2 Λ2 ≪ 1 sfermions are generally UV localized and the first- and is neglected, is a valid approximation provided v = UV .

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ð Þ ¼ð Þ β ð Þ md ij yd ijv cos ; 43b

ð Þ ¼ð Þ β ð Þ mu ij yu ijv sin ; 43c ð Þ ð Þ where mu;d;e ij and yu;d;e ij are the SM fermion mass and 2 2 Yukawa coupling matrices, respectively, and v ≡ vu þ 2 ≃ ð174 Þ2 vd GeV is the SM Higgs VEV. In the mass basis, these matrices are diagonal. Neglecting quark mixing (see Refs. [39,40] for a fuller treatment) the interaction basis coincides with the mass basis, resulting (given values for β Λ ð ð5Þ Þ tan and IR) in a system with 24 free parameters Ye;d;u ii and cLi;ei;Qi;di;ui and nine constraint equations following from (43). If we take a universal value for the 5D Yukawa ð ð5Þ Þ ¼ ð5Þ couplings such that Ye;d;u ii Y k, there remains one parameter degree of freedom within each generation of and within each generation of quarks, which we choose as the doublet c parameters cLi;Qi , without loss of generality. The relations (43) hold for the running masses. Ultimately, we are interested in the bulk mass parameters at the IR-brane FIG. 6. Contours of the effective 4D Yukawa coupling (42) at scale, where they determine the soft masses received by the the IR-brane scale as a function of the localization parameters cL sfermions when supersymmetry is broken. Therefore, before and cR of the bulk fermion fields for tan β ¼ 3, ΛIR ¼ we perform the 4D-5D Yukawa matching, we first evolve the 2 × 1016 GeV, and Yð5Þk ¼ 1. The dashed gray lines give 4D Yukawa couplings to the IR-brane scale. We discuss the contours of the Yukawa coupling strength. In color are contour procedure we use to consistently perform the renormaliza- lines corresponding to the coupling strengths of the SM Yukawa tion in Sec. IV B 1.InFig.6, we give an example of the couplings at the IR-brane scale. The region in which each field is resulting matching, showing the allowed range of local- IR localized is shaded light gray and the region where both fields izations (c parameters) of the SM leptons and quarks when are IR-localized is darker gray. the 5D Yukawa coupling takes the universal value Yð5Þk ¼ 1 Λ ¼2 1016 and IR × GeV. Generally, we see that the largeness include Yukawa couplings for the . Neutrinos can of the third-generation Yukawa couplings requires the be naturally incorporated in a warped extra dimension to third-generation fermions for both leptons and quarks to generate the required masses [24,41–44]. be UV-localized (white region), while the smaller Yukawa couplings of the first and second generations lead to IR C. Supersymmetry breaking localization (darker gray region). Although it is always Supersymmetry is assumed to be broken on the IR brane possible to make one of the chiral fermions in each and is parametrized by the introduction of a spurion generation UV-localized, only for the third generation can superfield X ¼ θθF that couples to the sfermions and this be done without making at least one of the other chiral X the gauginos. The sfermions and gauginos acquire tree- fermions IR localized. level soft masses with a characteristic scale F=Λ , where We note in passing that in the quark sector, since both IR F ¼ F e−2πkR, modulated by their overlap with the IR singlet fields in a given generation must be separately X brane. A gravitino mass of order F=M is generated by the matched to the same doublet field according to (42), the P super-Higgs effect. The Higgs fields receive no tree-level asymmetry between the 4D couplings yui and ydi precludes ð5Þ ð5Þ soft masses, as they are confined to the UV brane and do the solution c ¼ −c ¼ −c unless y =y ¼ Yu =Y . Qi di ui ui di i di not couple directly to the supersymmetry-breaking spurion. In this case, the 4D Yukawa coupling hierarchies are simply We do not include any mechanism to generate tree-level moved into the 5D couplings. This behavior is an indication trilinear soft scalar couplings, although, like the Higgs- of the universality of the warped extra dimension: while it sector soft terms, they arise radiatively. can explain the magnitude of the Yukawa coupling hier- Contributions to the soft masses also generically arise archies, since it is flavor-blind, the underlying flavor from anomaly mediation. Since the gravitino mass is structure remains as an order-one feature. In practice, we Planck-scale suppressed (as opposed to the other soft take Yð5Þk to have a universal value and absorb the flavor masses, which are suppressed by the IR-brane scale), the structure completely into the quark c parameters. A similar anomaly-mediated contribution typically is subdominant situation does not arise in the sector as we do not to the effects (both at tree level and loop level) of the

035046-12 PARTIALLY COMPOSITE SUPERSYMMETRY PHYS. REV. D 99, 035046 (2019) supersymmetry-breaking sector on the IR brane. An addi- 2. Gaugino masses tional source of supersymmetry breaking arises due to the For a field strength superfield Wa of a vector super- stabilization of the radion of the extra dimension, which a a multiplet V containing a standard model gauge field Aμ generically requires a nonzero F-term for the radion and its Majorana fermion gaugino λa (where a superfield (equivalently, the introduction of a constant is the gauge index), we introduce the interaction superpotential on the IR brane). The scale of the radion- Z Z   mediated contribution to the soft masses depends on the pffiffiffiffiffiffi 1 5 2 X αa a S5 ¼ d x −g d θ W Wα þ H:c: δðy −πRÞ: details of the stabilization model. We are interested in the 2Λ k regime where such effects are subdominant to the effects UV ð Þ of the supersymmetry-breaking sector on the IR brane. In 47 the model of Ref. [36], this can be accomplished if the Goldberger-Wise bulk hypermultiplet is sufficiently UV This term gives rise to a boundary Majorana mass for the localized. gaugino field and breaks supersymmetry, shifting the masses of the Kaluza-Klein modes up such that there is no longer a massless gaugino zero-mode solution. At tree 1. Gravitino mass level, the lightest KK mass is

When supersymmetry is spontaneously broken on the IR 2 g5k F 2 F boundary, the effective 4D cosmological constant receives a Mλ ≃ ¼ g ; ð48Þ 2πkR Λ Λ positive contribution from the VEV of FX. In the 5D IR IR warped geometry, this contribution can be canceled by the where g2k ¼ð2πkRÞg2. This mass expression (48) assumes addition of a constant superpotential W on the UV brane 5 the zero-mode approximation for the profiles, where the [36,45–50], which introduces a boundary mass term for the back reaction of the boundary Majorana mass is neglected, gravitino: pffiffiffiffi an approximation that is valid provided F=Λ ≲ 1.In Z   pffiffiffiffi IR practice, we include terms higher order in F=Λ . The 5 pffiffiffiffiffiffi 1 W μ ν IR S5 ¼ d x −g ψ μ½σ ; σ¯ ψ ν þ H:c: δðyÞ: ð44Þ mass for arbitrary F can be determined by solving the full 4 M3 5 KK mass quantization condition (see Refs. [17,25]). Note Λ that the gaugino masses are suppressed relative to F= IR by The cosmological constant vanishes when 2 2 6 g ∼ g5=πkR, the square of the 4D gauge coupling [49], and hence the gauginos in general obtain masses suppressed jWj2 jFj2 ≃ 3 ; ð45Þ below those of sfermions with flat profiles (c ¼ 1). This M2 2 P suppression matches that found in (26), as expected from the AdS=CFT dictionary. such that the lightest gravitino obtains a Majorana mass: If the supersymmetry-breaking sector does not contain any singlets with large F-terms, the interaction (47) is F m3 2 ≃ pffiffiffi : ð46Þ forbidden. In this case, with a nonsinglet spurion X, the = 3 MP leading contribution to the gaugino masses is [51] Z Z   This is the super-Higgs effect.5 Again, higher-order terms pffiffiffiffiffiffi 1 † 5 4 X X αa a S5 ¼ d x −g d θ W Wα þ H:c: δðy −πRÞ; can be included to account for the back reaction of the 2Λ3 k gravitino boundary mass, although in practice this is not UV ð Þ necessary in the relevant regions of our parameter space. As 49 expected, due to the universality of gravity, this matches the such that usual 4D result. Since the gravitational coupling is Planck- scale suppressed, the gravitino mass is lower than the 2 2 2 Λ g5k F 2 F characteristic soft mass scale F= IR by a warp factor, and Mλ ≃ ¼ g : ð Þ 2π Λ3 Λ3 50 the gravitino is therefore always the LSP in the relevant kR IR IR regions of parameter space. This is consistent with the pffiffiffiffi ∼ Λ partial compositeness result (31) in the 4D dual theory, Except in the regime F IR, this mass is highly sup- where the gravitino is mostly an elementary state. pressed, and other supersymmetry-breaking contributions such as radion mediation may dominate. 5Note that a constant superpotential can also be introduced on the IR brane, as is generically expected in the context of radion 6The gauge-coupling dependence arises since we assume a stabilization. However, such a superpotential provides a positive generic GUT symmetry that is broken by the Higgs mechanism contribution to the cosmological constant, and so it cannot be the on the UV brane, separated from the supersymmetry-breaking sole source for the gravitino mass. sector on the IR boundary.

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3. Sfermion masses For a chiral supermultiplet Φ containing a Weyl fermion ψ and its complex scalar superpartner ϕ, we introduce the interaction

Z Z † 5 pffiffiffiffiffiffi 4 X X † S5 ⊃ d x −g d θ Φ Φδðy − πRÞ: ð Þ Λ2 51 UVk

As with the gauginos, adding this boundary mass breaks supersymmetry. At tree level, the lightest KK mass is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ∓ F 2 c ð1∓ Þπ tree ≃ 2 c kR mϕ 2ð1∓ Þπ e L;R Λ 2 c kR − 1 IR e FIG. 7. Plot of the magnitude of the tree-level scalar soft 8   > 1 1=2 ð1∓ Þπ 1 mass and the gauge and Yukawa one-loop radiative corrections as <> − F 2 c kR c 2 Λ e c>2 ; a function of hypermultiplet localization when the gauginopffiffiffiffi ∼   IR ð Þ Λ ¼ 2 1016 ¼ 1 2 52 masses are given by (48). We take IR × GeV, F > 1 = F 1 10 : ∓ c c< ; 4.75 × 10 GeV, and tan β ¼ 3. 2 ΛIR 2 where the upper (lower) signs refer to the L (R) states. As pwithffiffiffiffi the gauginos, the scalar mass is valid in the limit Λ ≲ 1 F= IR , and in our numerical calculations we include Λ2 terms higher order in F= IR to account for the back reaction of the sfermion boundary mass. For simplicitly, we have taken these interactions to be flavor-diagonal, although this assumption can be relaxed. Note that the 1 UV-localized ðc>2Þ sfermion masses are suppressed by a warp factor relative to the IR-localized sfermion 1 masses ðc<2Þ because supersymmetry is broken on the IR brane. This behavior is illustrated in Figs. 7 and 8. Λ Λ ¼ −πkR δ ¼j 1 j Using the relations IR= UV e and i ci 2 , the expressions (52) are seen to be consistent with the FIG. 8. Plot of the magnitude of the tree-level scalar soft mass masses (23) obtained in the 4D dual theory. and the gauge and Yukawa one-loop radiative corrections as a Since the soft masses generated at tree level by (51) can function of hypermultiplet localization when thepffiffiffiffi gaugino masses 6 6 be exponentially small for UV-localized bulk scalar fields, are given by (50). We take ΛIR ¼6.5×10 GeV, F¼2×10 GeV, quantum corrections become significant when c is and tan β ¼ 5. sufficiently large. At the one-loop level, supersymmetry breaking is transmitted to the bulk scalars via interactions with other bulk scalars and gauginos. In Appendix C1,we masses squared at the IR-brane scale, arising when the KK derive the resulting contributions to the bulk scalar masses modes of the theory are integrated out. Parametrized in squared in the bulk theory. From the 4D perspective, these terms of the gaugino and sfermion tree-level soft masses, appear as one-loop threshold corrections to the scalar soft the corrections take the forms

32 ˜ ˜ 2 ˜ ˜ ˜ 1 16π2ðΔ 2 Þ ¼ Qi 2 2 þ 6 Qi 2 2 þ Qi 2 2 − 2 Qi 2 2 − 2 Qi 2 2 − 2Δ ð Þ m ˜ 1-loop rg3 g3M3 rg2 g2M2 rg1 g1M1 ryu yu mu˜ ryd y m˜ g1 S; 53a Qi 3 15 i i i i di di 5

32 ˜ 32 ˜ ˜ 4 16π2ðΔ 2 Þ ¼ ui 2 2 þ ui 2 2 − 4 ui 2 2 þ 2Δ ð Þ mu˜ 1-loop rg3 g3M3 rg1 g1M1 ryu yu m ˜ g1 S; 53b i 3 15 i i Qi 5

32 ˜ 8 ˜ ˜ 2 16π2ðΔ 2 Þ ¼ di 2 2 þ di 2 2 − 4 di 2 2 − 2Δ ð Þ m˜ 1-loop rg3 g3M3 rg1 g1M1 ryd yd m ˜ g1 S; 53c di 3 15 i i Qi 5

˜ 6 ˜ ˜ 3 16π2ðΔ 2 Þ ¼ 6 Li 2 2 þ Li 2 2 − Li 2 2 þ 2Δ ð Þ m ˜ 1-loop rg2 g2M2 rg1 g1M1 rye ye me˜ g1 S; 53d Li 5 i i i 5

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24 ˜ ˜ 6 16π2ðΔ 2 Þ ¼ ei 2 2 − 4 ei 2 2 − 2Δ ð Þ me˜ 1-loop rg1 g1M1 rye ye m ˜ g1 S; 53e i 5 i i Li 5 where ΔS, defined in (C22),is X h i Δ ¼ ðϕ Þ D 2 ¼ D 2 − 2 D 2 þ D 2 − D 2 þ D 2 ð Þ S Y i rϕ mϕ Tr r ˜ m ˜ ru˜ mu˜ r˜ m˜ r ˜ m ˜ re˜ me˜ ; 54 i i Qi Qi i i di di Li Li i i i

ϕi ϕi and the sum is over the sfermions. The coefficients rg , ry , with the inclusion of D-term and Yukawa radiative con- and rD , defined in (C7), (C14), and (C23), respectively, are tributions. The moderate size of this hierarchy is an ϕi numerical parameters which encode the effect of the extra important result of the inclusion of radiative corrections, dimension. which wash out the exponential localization dependence of The radiative corrections to the bulk scalar soft masses the sfermion soft masses that is a tree-level feature of the can be divided into three types of contributions: gauge extra dimension. corrections arising from loops involving bulk vector super- A larger hierarchy can be achieved if the supersymmetry- multiplets, Yukawa corrections arising from loops of bulk breaking sector does not contain any singlets with large hypermultiplets and boundary Higgs fields, and D-term F-terms as in Fig. 8. In this case, the gaugino masses take corrections arising from the Fayet-Iliopoulos D-term for the form (50), and the characteristic sfermion hierarchy is UV IR ∼ 2 Λ weak hypercharge. In Fig. 7 we plot the magnitudes of the mϕ =mϕ gaF= IR. The maximum splitting that can be various contributions as functions of the hypermultiplet accommodated between the two sfermion mass scales is localization when the gaugino mass is given by (48) limited by the requirement that no sfermions receive (singlet spurion), and the magnitudes when the supersym- negative soft masses at the IR-brane scale or in the metry-breaking sector does not contain any singlets with subsequent RG evolution of the theory to lower scales. large F-terms and the gaugino mass takes the form (50) in The constraints this condition imposes on the pattern of Fig. 8. In each case, the tree-level sfermion mass is plotted sfermion mass spectrum at the IR-brane scale are discussed in grey, the one-loop U(1), SU(2), and SU(3) gauge in Sec. IVA 2. radiative corrections are plotted in green, and the magni- In the sfermion sector, localization in the extra dimen- tude of the maximal contribution from a single Yukawa sion thus distinguishes between two scales: a tree-level 1 1 coupling (this corresponds to cL ¼ 2 or −cR ¼ 2 for the mass scale associated with IR-localized sfermions and a corresponding doublet and singlet hypermultiplets), lower mass scale arising from radiative corrections for UV- neglecting all color multiplicity factors and modulo the localized sfermions. When the localizations of the matter 5D Yukawa coupling, is plotted in orange. Also shown is hypermultiplets are chosen to explain the SM fermion mass the magnitude of the one-loop D-term radiative contribu- hierarchy (i.e., the third-generation fermions must be tion due to a single scalar mode (yellow), modulo hyper- predominantly UV-localized, while the lighter generations charge factors.7 are mostly IR-localized), the result is a split sfermion Due to the conformal flatness of the vector supermul- spectrum, with the third-generation sfermions hierarchi- tiplet, the gauge corrections take a universal value for cally lighter than the first two generations. We note that the UV-localized sfermions that is of order of the gaugino sfermion spectrum inverts the ordering of the fermion masses. These contributions are positive and set the spectrum, a consequence of the separation of the super- characteristic scale of the radiative corrections. Since the symmetry-breaking sector and the Higgs sector on opposite tree-level contribution is dominant for IR-localized sfer- orbifold fixed points. Additionally, although both are mions, which accordingly receivepffiffiffiffi a mass of order of the explained by the same localization mechanism, the sfer- Λ characteristic soft mass scale F= IR, the sfermion sector mion hierarchy is necessarily less split than the fermion UV IR ∼ Λ thus accommodates a hierarchy mϕ =mϕ Ma IR=F. hierarchy. This is because the Yukawa couplings only When the gaugino mass is given by (48) as in Fig. 7, receive wave function renormalization, while the scalar UV IR 2 masses are soft parameters and can receive large radiatve the sfermion hierarchy is mϕ =mϕ ∼ ga ∼ Oð0.4 − 1Þ, which may be increased modestly in individual families corrections from the extra dimension and MSSM running.

7As seen in Appendix C1c, one-loop D-term corrections are 4. Higgs sector independent of the localization of the external scalar, and so here The Higgs sector, confined to the UV brane, does not we plot the contribution as a function of the localization of the scalar contributing in the loop (unlike the other radiative con- couple directly to the supersymmetry-breaking sector, and tributions, which depend on the localization of the external thus the Higgs soft terms at the IR-brane scale are zero at scalar). tree level. Nevertheless, as with the sfermions, the breaking

035046-15 BUYUKDAG, GHERGHETTA, and MILLER PHYS. REV. D 99, 035046 (2019) is transmitted to the Higgs fields at the quantum level. We derive the one-loop corrections to m2 , m2 , and b from the Hu Hd bulk theory in Appendix C2. Parametrized in terms of the tree-level gaugino and sfermion soft masses, the one-loop Higgs masses take the form

6 16π2 2 ¼ 6 H 2 2 þ H 2 2 m rg2 g2M2 rg1 g1M1 Hu 5 3 − 6 ½ H 2 ð 2 þ 2 Þ − 2Δ ð Þ Tr ry yu m ˜ mu˜ g1 S; 55a ui i Qi i 5

6 16π2 2 ¼ 6 H 2 2 þ H 2 2 −6 ½ H 2 ð 2 þ 2 Þ mH rg2 g2M2 rg1 g1M1 Tr ry y m ˜ m˜ d 5 di di Qi di 3 FIG. 9. Plot of the magnitude of the one-loop radiative −2 ½ H 2 ð 2 þ 2 Þþ 2Δ ð Þ Tr ry ye m ˜ me˜ g1 S; 55b corrections that generate the Higgs soft masses as a function ei i Li i 5 of hypermultiplet localizationpffiffiffiffi in the singlet spurion case. We take   Λ ¼ 2 × 1016 GeV, F ¼ 4.75 × 1010 GeV, and tan β ¼ 3. 6 IR 2 b 2 b 2 16π b ¼ −μ 6r g2M2 þ r g1M1 ; ð55cÞ λ1 5 λ2

H H b where rg , ry , and rλ are defined in (C29), (C34), and (C41), respectively. The origin of the μ-term on the UV brane is assumed to arise from the Kim-Nilles mechanism, as discussed in Sec. II D; its magnitude is determined as necessary to ensure that electroweak symmetry is broken, along with the value of tan β, as described in Sec. III D. In Figs. 9 and 10, we plot the magnitudes of the various one-loop contributions to the Higgs soft masses as func- tions of hypermultiplet localization in the singlet spurion and nonsinglet spurion cases, respectively. The U(1) (lighter green) and SU(2) (darker green) gauge-sector contributions are independent of localization. In orange, we give the maximal contribution from a single Yukawa ¼ 1 − ¼ 1 FIG. 10. Plot of the magnitude of the one-loop radiative coupling (this corresponds to cL 2 or cR 2 for the corrections that generate the Higgs soft masses as a function corresponding doublet and singlet hypermultiplets), of hypermultiplet localizationp inffiffiffiffi the nonsinglet spurion case. We Λ ¼ 6 5 106 ¼ 2 106 β ¼ 5 neglecting all color multiplicity factors and modulo the take IR . × GeV, F × GeV, and tan . 5D Yukawa coupling. These contributions are negative. In yellow is the D-term contribution from a single bulk scalar, modulo the scalar hypercharge. These individual contribu- 5. Trilinear soft scalar couplings (a-terms) tions can be either positive or negative depending on the Soft a-terms are also generated radiatively. We derive the relative sign between the hypercharge of the Higgs field one-loop corrections in Appendix C4. Parametrized in terms and the hypercharge of the scalar. of the tree-level gaugino masses, these take the forms:

    32 2 8 8 2 a 2 a 2 a a a 2 16π a ¼ −y ðrλ Þ g3M3 þ 6ðrλ Þ g2M2 þ − ðrλ Þ þ ðrλ Þ þ ðrλ Þ g1M1 ; ð56aÞ ui ui 3 3 Qiui 2 Qi 5 1 Qi 5 1 ui 15 1 Qiui     32 2 4 4 2 a 2 a 2 a a a 2 16π a ¼ −y ðrλ Þ g3M3 þ 3ðrλ Þ g2M2 þ ðrλ Þ þ ðrλ Þ − ðrλ Þ g1M1 ; ð56bÞ di di 3 3 Qidi 2 Qi 5 1 Qi 5 1 di 15 1 Qidi     6 12 12 2 a 2 a a a 2 16π a ¼ −y 6ðrλ Þ g2M2 þ − ðrλ Þ þ ðrλ Þ þ ðrλ Þ g1M1 ; ð56cÞ ei ei 2 Li 5 1 Li 5 1 ei 5 1 Liei

a where the rλ are defined in (C46a) and (C46b).

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D. Electroweak symmetry breaking model is in general quite large. The overall mass scale of Λ In the MSSM, the tree-level scalar potential has a our sparticle spectrum isp jointlyffiffiffiffi determined by IR, the minimum breaking electroweak symmetry if the following scale of the IR brane, and F, the scale of supersymmetry two equations are satisfied: breaking. Together these two parameters fill the roles associated with MGUT, m1=2, and m0 in classic universal 2 2 1 2 2 2 supergravity (SUGRA) models. As we discuss in Sec. III D, m þjμj − b cot β − ðg1 þ g2Þv cos 2β ¼ 0; ð57aÞ Hu 8 we do not have the usual freedom in tan β and the sign of μ, which are in this case determined by electroweak symmetry 1 2 þjμj2 − β þ ð 2 þ 2Þ 2 2β ¼ 0 ð Þ breaking. In addition to these universal parameters, our mH b tan g1 g2 v cos : 57b d 8 model features nonuniversal IR-scale boundary conditions 2 2 for the sfermion soft masses, which we specify in a flavor- In our model, m , m , and b,givenby(55a), (55b), Hu Hd dependent way by choosing field localizations to explain and (55c), respectivelt, are radiatively generated at the IR- the SM fermion mass spectrum, as described in Sec. III B. brane scale when the extra dimension is integrated out. ð5Þ Solving (57) determines two parameters: the magnitude of We choose a universal value Y k for all 5D Yukawa the Higgsino mass parameter jμj and the ratio of Higgs couplings, such that this specification requires fixing six VEVs tan β. In the limit that the scale of supersymmetry additional free parameters, which we take as the doublet c parameters c and c . In this section we discuss various breaking is much larger than the scale of electroweak Li Qi symmetry breaking the physical solutions are phenomenological and theoretical constraints that impose qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi limits on the set of model parameters. 2 2 2 2 2 2   ðm − m Þþ ðm − m Þ þ 4b 2 Hd Hu Hd Hu v tan β ≃ þ O ; 1. Phenomenological considerations 2b b ð Þ We first consider five phenomenological desiderata that 58a constrain our model. (1) Gravitino dark matter.—Since the gravitino mass is m2 − m2 tan2 β jμj2 ≃ Hd Hu þ Oð 2Þ ð Þ Planck-scale suppressed, it is the LSP throughout 2 v ; 58b tan β − 1 our parameter space. In the absence of R-parity violation, the LSP is absolutely stable, and as such, where we require signðμÞ¼−1 and m2 < minðm2 ;m2 = Hu Hd Hd the gravitino makes an attractive dark matter can- tan2 βÞ. Solutions with signðμÞ¼þ1 are excluded at tree didate. However, the stability of a gravitino LSP can level since b is constrained to have the opposite sign from μ lead to cosmological problems, as the thermal at the IR-brane scale according to (55c) (and this typically gravitino mass density arising from freeze-out is remains true under RG evolution). Although these equa- sufficient to overclose the universe unless the grav- tions are modified when loop corrections to the Higgs itino is very light ½Oð100Þ eV [52,53]. In this case, scalar potential are included, they remain practical con- observations of the matter power spectrum at small straints since the iterative method employed to determine cosmological scales limit the free-streaming length electroweak symmetry breaking (EWSB) in the numerical of the gravitino, further requiring m3=2 < 4.7 eV in renormalization procedure (see Sec. IV B 1) requires an order for the gravitino to be adequately cold [54], initial tree-level solution. It is not guaranteed that the Higgs and in this scenario cannot therefore soft terms (55a), (55b), and (55c) permit a tree-level account for all of the observed dark matter density. solution at the IR-brane scale, in which case electroweak Throughout the relevant parameter space of our symmetry must be broken radiatively. model, the gravitino is sufficiently heavy that we The conditions (58) for EWSB strongly favor m2 < 0. require inflation to dilute the initial thermal pop- Hu In order for this to occur, the combined radiative correc- ulation [55] and must restrict the reheating temper- tions to m2 both from the KK modes at the IR-brane scale ature so that the gravitino does not subsquently Hu and from the MSSM running to lower scales must be come back into thermal equilibrium. In this case, overall negative. We discuss the constraints this imposes on structure-formation constraints are weaker, requir- ≳ Oð1Þ the parameter space of our theory in Sec. IVA 2. ing m3=2 keV for generic warm dark matter candidates [56–58], such the gravitino may be the IV. SUPERPARTNER MASS SPECTRUM dominant component of dark matter. In this sce- nario, gravitinos may still be produced from the A. Model parameter space scattering of in thermal equilibrium with As we have seen in generating the fermion mass the plasma. The largest contribution arises from hierarchy and breaking supersymmetry, the parameter , such that the thermal gravitino density space available for our partially composite supersymmetric takes the form

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pffiffiffiffi Λ FIG. 11. Plot of the gravitino dark matter constraints on the parameter space of our model in the ( IR; F) plane in the singlet spurion case for Yð5Þk ¼ 1.      2 100 GeV m˜ T component of dark matter. If nonthermal production Ωthermal 2 ∼0 3 g R 3=2 h . 10 : is significant, the reheating temperature must be m3=2 1 TeV 10 GeV lowered to suppress the contribution from thermal ð59Þ production. In the hatched yellow regions, the thermal relic density is insufficient to provide all For m3=2 ≲ 1 keV, thermal scattering production of of the dark matter, and some level of nonthermal gravitinos cannot supply all of the observed dark production is required to obtain the observed dark matter density unless TR is high enough to bring matter density. the gravitino into thermal equilibrium. Further parameter space constraints arise from big Gravitinos are also produced nonthermally, bang nucleosynthesis (BBN), which strongly limits via decays of the next-to-lightest supersymmetric the NLSP energy density if the NLSP is long-lived particle (NLSP), which in our model is typically enough to decay to the gravitino during or after the either a (Bino-like or Higgsino-like) neutralino or a (mainly right-handed) stau. In both cases, the NLSP is sufficiently long-lived throughout our parameter space to decay after freeze-out, such that the resulting nonthermal gravitino population takes the form (superWIMP scenario [59]):

m3 2 Ωnonthermal 2 ¼ = Ω 2 ð Þ 3=2 h NLSPh : 60 mNLSP The initial NLSP population that contributes to Ω 2 NLSPh is moderated by TR. In particular, if the reheating temperature is low enough that the NLSP never comes into thermal equilibrium after infla- ≲ 20 tion (TR mNLSP= ), the initial NLSP population is Boltzmann-suppressed. Ω 2 ¼ The observed dark matter abundance ( dmh 0.1186 0.0020 [60]) thus places an upper limit on the reheating temperature. We showpffiffiffiffi an estimate ðΛ Þ of these limits in the space of IR; F in Figs. 11 and 12. In each case, the yellow contours give the FIG. 12. Plot of the gravitino dark matterp constraintsffiffiffiffi on the Λ reheating temperature necessary for the thermal parameter space of our model in the ( IR; F) plane in the gravitino relic density (59) to provide the dominant nonsinglet spurion case for Yð5Þk ¼ 1.

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formation of the light elements. To avoid altering the flavor-changing neutral currents (FCNCs) and successful predictions of the standard BBN scenario, CP-violation at levels above current experimental decays to the gravitino must be prompt, limiting the limits. This problem can be alleviated if the sfermions τ Oð0 1 − 100Þ lifetime of the NLSP to NLSP < . s, or, of the first and second generations are heavy, ≪ for m3=2 mNLSP, Oð100Þ TeV [63–66], or even heavier [67–69],a   2 5 solution that has gained popularity in the absence of m3 2 = Oð1 − 4Þ = ð Þ any low-energy experimental signatures of supersym- mNLSP > TeV ; 61 10 GeV metry. Such a solution is a natural and elegant choice in our model, as the inverted hierarchy in the sfermion as a conservative estimate. This condition places pffiffiffiffi soft mass spectrum naturally separates the scale of the constraints in the space of Λ and F. We show the IR first two generations from that of the third, which can regions where τ < 0.1 s in the neutralino NLSP remain light enough to explain the Higgs mass and case (dark orange) and in stau case (magenta) in offer the possibility of experimental detection. Figs. 11 and 12. Evading these limits restricts the Thus, to ameliorate the flavor problem, we restrict reheating temperature for our model to the range the masses of the first- and second-generation sfer- T ∼ Oð102–106Þ GeV, necessitating an alterna- R mions to be at least 100 TeV. As a constraint, this tive to thermal leptogenesis to generate the condition places an upper (lower) limit on the asymmetry. c parameters of the first- and second-generation If the NLSP is the stau and is sufficiently stable to left-handed (right-handed) bulk hypermultiplet fields. survive into BBN, the formation of bound states with nuclei can catalyze the production of light elements This is a further limit on the hypermultiplet local- 3 3 izations beyond the structure necessary to explain the [61]. In particular, if 10 s ≲ ττ˜ ≲ 5 × 10 s, the 1 SM fermion mass spectrum. In Figs. 13 and 14,we catalytic enhancement for 6Li can solve the lithium showp anffiffiffiffi estimate of the region in the space of problem in the standard BBN scenario [62]. We show ðΛ Þ an estimate of the region in which the stau lifetime IR; F where this limit is incompatible with the falls within these limits in teal in Fig. 11. We note SM fermion mass for at least one field (the strictest however, that this solution to the lithium problem is constraint typically comes from the , the heaviest ruled out for our model, as the entirety of the catalytic of the first- and second-generation fermions). — region is excludedby currentLHC limits (see Fig. 13). (3) 125 GeV Higgs mass and collider exclusion limits. In the nonsinglet spurion case, the stau is not In the MSSM, the tree-level mass of the neutral scalar Higgs boson is bounded from above by sufficiently long-lived to survive into BBN anywhere tree in the relevant parameter space. mh

pffiffiffiffi Λ FIG. 13. Plot of the constraints on the parameter space of our model in the ( IR; F) plane in the singlet spurion case for Yð5Þk ¼ 1.

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freedom present in the theory at a given energy scale; in the MSSM, unification is most sensitive to the Higgsino mass μ as well as the ratio of the Wino mass to the gluino mass [76], and it can be spoiled if the magnitude of μ is larger than a few hundred TeV [6]. As discussed in Sec. III D, μ is determined as necessary to achieve electroweak symmetry break- ing. Generically, this implies that the scale of μ is of the same order of magnitude as the soft masses in the Higgs sector, i.e., jμj2 ∼ jbj ∼ jm2 j ∼ jm2 j. Since Hu Hd the Higgs soft masses are generated radiatively (and therefore characteristically of the scale of the gau- gino masses) a first-order estimate of this constraint is jμj ∼ M2 ≲ 100 TeV. A more precise constraint can be obtained by solving the tree-level EWSB equations (57). We show an estimatepffiffiffiffi of the excluded ðΛ Þ jμtreej ≳ region in the space of IR; F where 100 TeV in Figs. 13 and 14. Λ FIG. 14. Plot of thepffiffiffiffi constraints on the parameter space of our Note that for the case where IR is much below ∼1016 GeV, we are implicitly assuming that the model in the (ΛIR; F) plane in the nonsinglet spurion case for Yð5Þk ¼ 1. gauge boson Kaluza-Klein states form complete SU(5) multiplets so that there is a universal shift in the running of the gauge couplings. In the warped on the sparticle spectrum, particularly on the masses extra dimension this can be modeled by considering of the stops. In the MSSM, the observed Higgspffiffiffiffiffiffiffiffiffiffiffiffiffi mass the full SU(5) gauge symmetry in the bulk, although ˜ ˜ constrains the general stop mass scale mt1 mt2 there are no Kaluza-Klein states for the UV- and is broadly compatible with stop masses ranging localizeqd Higgsino. For simplicity, we will not from Oð1Þ TeV for tan β ∼ 50 to Oð100Þ TeV for consider the full SU(5) extension here, since it does tan β ∼ 3 [71]. In our model, the stop masses depend not affect the details of our low-energy spectrum.9 critically [at least Oð1Þ] on the localizations of the (5) Minimal supersymmetric particle content.—In the bulk hypermultiplets, and hence the Higgs mass construction of our model, we are motivated to principally induces constraints on the c parameters explain the observed Higgs mass using only the of the theory. The precise calculation of the Higgs minimal supersymmetric particle content at low mass requires a complete numerical analysis (see energy. While the orbifold compactification allows Sec. IV B). To obtain a conservative evaluation of the us to recover this particle content as the zero modes constraint induced by the Higgs mass on our model of the 5D N ¼ 1 supersymmetric theory, the essen- we can exclude the region where the stop masses are tially Dirac nature of fermions in five dimensions is a always greater than 100 TeV. We showpffiffiffiffi an estimate nontrivial feature of the model and can have phe- ðΛ Þ of this region in the space of IR; F in Figs. 13 nomenological implications when the scale of and 14. pNffiffiffiffi¼ 1 supersymmetry breaking on the IR brane, Further limits on sparticle masses are set by F, approaches the local compactification scale, experiments searching for direct sparticle production Λ IR. In this case, the backreaction of the supersym- at colliders. In the context of our model, the strictest metry-breaking boundary mass on the wavefunction of constraints arise from the exclusion limits on the profiles of the gaugino and sfermion fields cannot be masses of the gluino and the squarks, which must be neglected.pffiffiffiffi In particular for the gauginos, the effect of heavier than Oð1Þ TeV when the LSP is the gravitino Λ – 8 larger F= IR is to increase the zero-mode mass, [72 75]. Qualitatively, these limits place an effective but at the same time to decrease the magnitude of the lower bound on thepffiffiffiffi soft mass scale of our model, ð1Þ Λ mass of the next-to-lightest KK mode, mλ . This restricting the ratio F= IR. We show an estimate of the excluded region in Figs. 13 and 14. — (4) Gauge coupling unification. Gauge coupling uni- 9We note that in a theory with IR-localized bulk hyper- fication is a generic feature of the minimal super- multiplets, higher-dimension operators may only be suppressed symmetric model. The renormalization of gauge by the IR-brane scale, which, in the context of grand unification, couplings depends on the number of degrees of can lead to decay constraints [25]. These may be addressed by the introduction of an additional global symmetry ð1Þ in the bulk, such as a U B baryon number symmetry as in 8These limits are weakly model-dependent: see the review [60]. Refs. [77,78], or by an orbifold GUT scenario [41].

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behaviorpffiffiffiffi smoothly approaches the twisted limit left-handed (right-handed) sfermions with hypercharge of Λ ≫ 1 Δ ( F= IR ), where the magnitudes of the masses the same sign as the sign of S, which is determined by the of the lowest two gaugino KK modes meet at a heaviest (most IR-localized) sfermions. common valuepffiffiffiffi and the two states form a Dirac Scalars in our theory also receive negative Yukawa Λ ≲ 1 spinor. For F= IR , the gaugino mass is only corrections of the form (C8) from the bulk. As a result of approximately Dirac, but the first KK mode is light the bulk hypermultiplet c-parameter structure necessary to enough that it must be included in the spectrum. explain the SM fermion mass hierarchy, the magnitude of the While the presence of such modes in the theory at Yukawa contribution to a left-handed (right-handed) field low energy can be helpful to achieve a more natural grows as that field becomes more UV-localized (see Figs. 7 model, we leave the exploration of this region of and 8). These corrections can become large, particularly parameter space for the future. Under this criterion, for the third-generation fields, such that upper (lower) ð1Þ −πkR c-parameter limits for each left-handed (right-handed) field we exclude the region in which mλ1

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Ref. [68] that the two-loop contributions from heavy scalars contributions to destabilize the Higgs VEV, setting a lower provide considerable tachyonic constraints on the allow- limit on the net correction (ΔS and S together). The able hierarchy among the scalar soft mass parameters predicted Higgs boson mass in this case is also correlated unless the effect can be balanced by positive contributions with the D-term corrections, and consistency with the from the gauginos. This analysis assumed a common mass observation may require additional limits on ΔS and S. for the heavy scalars. When this assumption is lifted, we Together, these requirements introduce a global constraint note that the presence of nonuniversality among the soft on the c parameters of the heavy (typically, first- and scalar masses generically induces a nonzero value for the second-generation) sfermions which give the dominant trace S. The MSSM D-term corrections have the same contributions to ΔS and S and can also induce additional hypercharge dependence as the D-term corrections from tachyonic limits on the matter bulk hypermultiplet c the bulk. Thus, the resulting contributions to scalars with parameters, primarily constraining the families Q, d, and e. hypercharge of sign opposite to that of the trace S are In some regions of the parameter space of our model, the positive, and consequently may ameliorate the effect of the union of all limits imposed on the bulk hypermultiplet c negative two-loop contributions. This comes, however, at parameters by these effects excludes all solutions compat- the cost of worsening the constraints for scalars with ible with the SM fermion mass spectrum.When the gaugino hypercharge of the same sign as that of S, which receive mass is given by (48) (singlet spurion), the splitting negative corrections in the running. between the masses of the third-generation sfermions For our model, the MSSM corrections further restrict the and the heavier mass scale of the first and second accessible c-parameter ranges of the matter bulk hyper- generations that results from the explanation of the SM multiplets. Since the sign of S is typically the same as that fermion masses is small enough throughout the parameter of ΔS, the MSSM limits generally reinforce the bulk limits, space that tachyonic constraints are not significant, but further restricting the viable IR-brane hierarchies for when the gaugino mass is given by (50) (nonsinglet sfermions (particularly squarks) with hypercharge of the spurion), larger hierarchies arise, excluding some areas same sign as S. Large hierarchies in the high-scale of the parameter space. We show anpffiffiffiffi estimate of the ðΛ Þ spectrum are only allowed for sfermions with hypercharge excluded region in the space of IR; F in Fig. 14. sign opposite to the sign of S. For example, if S > 0, then Hierarchies in the sfermion mass spectrum (and separa- 2 u˜ 3 (hypercharge − 3) may receive large negative correc- tions of scale in general) also complicate the numerical tions, and therefore cannot have a mass at the IR-brane renormalization procedure, since they necessitate a careful scale significantly lower than the scale of the other account of particle decoupling if precision in mass spec- sfermions. Such high-scale sfermion structure is a generic trum calculations is to be obtained. In mass-independent signature of a nonuniversal split sfermion spectrum. renormalization schemes such as DR, the effects of heavy A similar analysis can be performed in the Higgs sector, particles do not decouple, and hence, at renormalization where, conversely, large negative corrections are typically scales small compared to the particle masses, finite quan- favorable for electroweak symmetry breaking. Negative tum corrections may involve terms with large logarithms of corrections to the Higgs soft masses squared m2 and m2 the masses of these particles (see [79] for MSSM expres- Hu Hd may arise from both the bulk, in the form of Yukawa sions). In order for mass calculations to be precise when contributions (C30) and D-term contributions (C35) at large hierarchies in the soft mass parameters are present, one loop, or from the MSSM, in the form of Yukawa such large logarithmic corrections need to be resummed, a corrections process which is most naturally accomplished by the use of an effective theory, or of a tower of effective theories [80]. 2 2 2 2 2 16π ðβ 2 Þ ⊃ 6j j ð þ þ Þ ð Þ Precision in the case of scalar hierarchies is especially m 1-loop yt mH m ˜ mu˜ 3 ; 66a Hu u Q3 critical, since the light scalar masses depend crucially on 2 2 2 2 2 the heavy scalar masses through the factors such as (62) 16π ðβ 2 Þ ⊃ 6j j ð þ þ Þ m 1-loop yd mH m ˜ m˜ Hd d Q3 d3 and (64). It is important to note as well that the scale of þ 2j j2ð 2 þ 2 þ 2 Þ ð Þ supersymmetry breaking in our model, the IR brane, which yτ mH m ˜ me˜3 ; 66b d L3 can be significantly lower than the Planck scale or the GUT and D-term corrections (62) at one loop and the corrections scale due to the warped 5D geometry, is the natural cutoff (64) at two loops. As discussed in Sec. III D, EWSB in our for the IR-localized (or composite) part of the 4D MSSM. 2 2 2 2 Thus, the effects of the heavy scalars, which may receive model requires m < minðm ;m = tan βÞ. If the sfer- Hu Hd Hd masses very near (or even above, depending on the choice of mion hierarchy is relatively modest, such that at least one of 2 2 F and k) the IR-brane scale, may decouple after a little m ˜ or m˜ is relatively heavy, this may be achieved in the Q3 u3 running, minimizing the effect of the heavy scalar contri- familiar way in the MSSM through Yukawa radiative butions. In an effective field theory (EFT) approach, these corrections. If, however, the sfermion splitting is large, heavy scalars are integrated out, introducing threshold the MSSM Yukawa contributions may be suppressed, and corrections to the lighter scalar masses. At the one- and successful EWSB may rely on the presence of D-term two-loop level, such corrections can be large and negative as

035046-22 PARTIALLY COMPOSITE SUPERSYMMETRY PHYS. REV. D 99, 035046 (2019) a result of the effects mentioned above, but overall, the TABLE I. Selected parameter space sampling regions. decoupling procedure may substantially relax the tachyonic bounds indicated in purely MSSM DR renormalization AB Λ 2 1016 6 5 106 [81,82]. Our renormalization procedure, discussed in pIRffiffiffiffi × GeV . × GeV Sec. IV B 1 does not implement decoupling; hence, we F 4.75 × 1010 GeV 2 × 106 GeV expect that some regions of our parameter space with light tan βa ∼3 ∼5 scalars may be unnecessarily excluded on tachyonic sign μ −1 −1 grounds, solely as an artifact of the numerical renormaliza- Yð5Þk 11 tion method. spurion singlet nonsinglet a M1 52.9 TeV 14.60 TeV a 3. Parameter space constraints M2 50.7 TeV 22.9 TeV a M3 49.85 TeV 38.94 TeV Together, the constraints discussed in Secs. IVA 1 and pffiffiffiffi m3=2 535 GeV 1 keV IVA 2 lead to restrictions on the ðΛ ; FÞ parameter space, IR a Λ which is shown in Figs. 13 and 14 for the singlet spurion and At scale IR. nonsinglet spurion cases, respectively. pffiffiffiffi In the light gray region of each plot F > Λ , which is predicted by the partially composite supersymmetric IR Λ excluded as the dynamics of the spurion are restricted by model. The IR branepffiffiffiffi scale, IR, and the scale of super- Λ symmetry breaking, F, set the overall soft mass scale, and IR as a cutoff scale. Along the edges of these regions, the shaded strip gives the area in which the next-to-lightest are chosen to comply with all phenomenological con- gaugino KK mode must be included in the low-energy straints in Sec. IVA 1. tan β is determined by the measured spectrum. The dark gray/blue regions give an estimate of Higgs boson mass and the sign of μ is set in order to achieve the exclusions due to collider direct-detection limits. The the correct pattern of EWSB. yellow region of Fig. 13 shows our estimate of the BBN exclusions when the NLSP is the lightest neutralino or the 1. Renormalization procedure lightest stau (also see Fig. 11). The corresponding limits in To obtain pole mass predictions for the superpartners, we the nonsinglet spurion case (see Fig. 12) are less strict than use the spectrum calculator FLEXIBLESUSY [83,84], which the collider constraint, and therefore not visible in Fig. 14. incorporates elements of SARAH [85–88] and SOFTSUSY jμj 100 In blue are the regions in which > TeV, which is [89,90], to run selected points down from the input scale excluded in order to preserve gauge coupling unification. In (IR brane) to lower energy. To solve the renormalization the red regions, the bulk hypermultiplet localization cannot boundary value problem, FLEXIBLESUSYemploys a nested be chosen such that all of the first- and second-generation iterative algorithm, using the three-loop MSSM β functions sfermions have masses that are at least 100 TeV. In green (the renormalization procedure includes components and are the regions where we estimate that the stop masses are corrections from [91–97]) between boundary conditions heavier than 100 TeV, and hence we expect the resulting Λ imposed at the high scale, IR, and the SM at the electro- Higgs boson mass to be too heavy to match the observed weak scale. Electroweak symmetry breaking is determined value. In the purple region of Fig. 14, one or more of the by numerical minimization of the loop-corrected Higgs sfermions receives a tachyonic mass, either on the IR brane, potential, with the value of tan β and the Higgsino mass or in the subsequent MSSM running. The remaining white parameter μ determined iteratively. Loop-corrected pole areas are the regions of interest, simultaneously satisfying masses are calculated from the full self-energies for each all constraints. Within these regions, the flavor constraint, particle. the observed Higgs boson mass, and radiative corrections The renormalization procedure is made more complicated impose additional restrictions on the hypermultiplet c by the fact that the soft mass spectrum at the IR-brane scale parameters. depends on the values of the supersymmetric parameters (the Λ ∼ 107 The constraints favor two regions: either IR GeV, gauge and Yukawa couplings and the Higgsino mass with a keV-scale gravitino and a singlet spurion, or a GUT- parameter) at that scale. As discussed in Sec. III B, the value Λ ∼500 scale value for IR, with an GeV gravitino and a of the 4D Yukawa couplings at the IR-brane scale must be nonsinglet spurion. In Sec. IV B, we calculate detailed known in order to choose the localizations of the bulk sparticle spectra for two benchmark scenarios (marked as hypermultiplets to explain the fermion mass hierarchy; A and B in Figs. 13 and 14, respectively). additionally, the radiative corrections included for the soft masses at the IR-brane scale explicitly incorporate gauge and B. Numerical results Yukawa couplings as well as the Higgsino mass parameter. Based on the constraints considered in Sec. IVA,we The IR-brane values of these parameters in turn depend select the regions of parameter space given in Table I as our (weakly) on the resulting sparticle pole masses. In order to benchmark scenarios. With these parameters we determine consistently determine the sparticle mass spectrum, we must the sparticle mass spectrum and Higgs boson mass therefore apply the renormalization procedure iteratively.

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To do this, we first obtain an initial estimate of the high- procedure includes threshold effects from the soft masses, scale theory using FLEXIBLESUSY to extract the Yukawa and the running values of the resulting supersymmetric and gauge couplings from low-energy experimental data and parameters are spectrum-dependent. The values of the run them in the SM (including components and corrections parameters extracted at the IR-brane scale are then used to from Refs. [98–108]) up to a common supersymmetry scale, construct an updated input spectrum and the procedure is mSUSY, where they are matched at tree level to the MSSM repeated until the values of the input parameters converge. DR couplings. The evolution is then continued under the Λ two-loop MSSM RGEs (extracted from SARAH)upto IR. 2. Higgs mass calculation The IR-scale couplings calculated using this procedure are Due to the high scale of supersymmetry in our model, to insensitive to the details of the sparticle spectrum, since the calculate the neutral scalar Higgs pole mass we use HSSUSY tree-level matching procedure neglects threshold corrections [83–90,98–109] and FLEXIBLEEFTHIGGS [83–95,98–108, at the scale mSUSY, where the MSSM is matched to the SM. 110]. Both numerical methods are based on an effective We accordingly use this procedure to provide coupling field theory approach in which all non-SM particles are ¼ estimates in cases where we wish to make a general integratedpffiffiffiffiffiffiffiffiffiffiffiffiffi out at a common threshold (namely, mSUSY calculation without reference to a particular spectrum. The mt˜1 mt˜2 ), at which point the theory is matched to the uncertainty resulting from the tree-level matching approxi- SM and run down to the electroweak scale, where the SM mation can be quantified by varying mSUSY and observing the couplings are matched to experimental data and the Higgs variation in the renormalized couplings. This is illustrated in pole mass is extracted. Theoretical uncertainty in this type Fig. 15, where we plot the percent deviation of the Yukawa of procedure arises in three areas: couplings under this renormalization procedure, relative to (1) SM uncertainty: uncertainty due to neglected higher- ¼ 10 — mSUSY TeV. By this measure, the uncertainty in this order corrections at the electroweak scale. We Yukawa coupling estimate is less than 6% throughout our consider two sources. The first source is missing parameter space. corrections in the extraction of the SM running To obtain higher precision coupling values, a particular parameters from experimental data at the electroweak spectrum must be specified by choosing localizations for the scale, which induce uncertainty that can be estimated doublet bulk hypermultiplets. Given these, the IR-brane as the effect of higher loop corrections on the Higgs couplings calculated as above can be combined with pole mass. Here, we estimate the uncertainty by educated guesses for tan β and μ to obtain an initial estimate comparing the Higgs pole mass when 3-loop QCD of the IR-brane scale soft mass spectrum. This spectrum is corrections to the top Yukawa coupling are included then renormalized and EWSB computed using the full the to the result for 2-loop top Yukawa coupling corrections. The second source is missing corrections FLEXIBLESUSY MSSM routine. Unlike our spectrum- in the calculation of the Higgs pole mass itself. agnostic procedure above, this renormalization and matching Since such missing corrections lead to residual renormalization-scale dependence in the pole mass, we estimate this uncertainty by varying the scale at which the pole mass is calculated over the range 1 ð2 mt; 2mtÞ and taking the difference between the maximum and minimum. The total SM uncertainty is the linear sum of these two contributions. (2) SUSY uncertainty: uncertainty due to neglected threshold corrections in the matching of the SM to the MSSM.—In the EFT approach, the Higgs pole mass is primarily sensitive to new physics via thresh- old corrections from supersymmetric particles to the Higgs quartic coupling, at the scale where the Stan- dard Model is matched to the MSSM. The matching in HSSUSY and FLEXIBLEEFTHIGGS is complete up through the 2-loop level, and includes some 3-loop corrections. The uncertainty due to the neglected higher-order corrections to the Higgs quartic coupling can be estimated by the residual matching-scale dependence in the Higgs pole mass. To calculate this, FIG. 15. Plot of the percent deviation of the Yukawa couplings we vary the scale at which the SM is matched to the ð1 2 Þ in the DR scheme for mSUSY ¼ 5 TeV (dotted) and mSUSY ¼ MSSM over the interval 2 mSUSY; mSUSY and take 100 ¼ 10 TeV (dashed) from the value for mSUSY TeV, the difference between the maximum and minimum with tan β ¼ 5. value of the Higgs pole mass.

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(3) EFT uncertainty: uncertainty due to neglected Sec. IVA 2 impose further limits on the c parameters that higher-dimensional operators in the SM EFT can only be determined a posteriori in the numerical below the matching scale.—Both HSSUSY and renormalization. These constraints can further limit the FLEXIBLEEFTHIGGS only include terms up to order c-parameter ranges and introduce correlations among Oð Þ v=mSUSY in the SM EFT. In a pure EFT the c parameters of successful spectra. Once the doublet approach, the uncertainty due to the missing terms c parameters are specified, the singlet c parameters are fixed Oð 2 2 Þ of order v =mSUSY and higher can be estimated according to (42) to generate the SM fermion mass spec- as a shift in the SUSY threshold corrections to the trum. In order to avoid introducing any new hierarchies Higgs quartic coupling. Accordingly, in HSSUSY,we with this mechanism, we additionally require that all the calculate this uncertainty as the shift in the Higgs pole c parameters are order-one numbers. In practice, we (gen- mass induced by multiplying all 1-loop threshold erously) require c ≲ 10, providing effective upper and corrections to the Higgs quartic coupling at the scale lower limits on all sfermion masses that hold in the absence ð1 þ 2 2 Þ mSUSY by v =mSUSY .InFLEXIBLEEFTHIGGS, of stronger constraints. conversely, this uncertainty is not present, as the In Fig. 16 we present the resulting superpartner pole mass calculation departs from the pure EFT approach at spectra obeying all phenomenological constraints and con- low energy by switching to a diagrammatic calcu- sistent with the measured value of the Higgs boson mass lation, which correctly resums leading and sublead- (see Sec. IV B 2 for details of the Higgs mass calculation). ing logarithms to all orders. The corresponding ranges for the sfermion, gaugino, and The total uncertainty is taken to be the linear sum of the SM, Higgsino masses in the gauge-eigenstate basis are shown in SUSY, and EFT uncertainties. As our Higgs mass estimate, Fig. 17. In general, the spread in the masses is a result of the we take the average of the HSSUSY and FLEXIBLEEFTHIGGS freedom in the bulk hypermultiplet localizations (c param- results, with uncertainty given by the union of the two eters) remaining after the application of all constraints, calculated ranges. combined with the uncertainty in the numerical calculations. In both cases, the allowed mass ranges for the third- 3. Superpartner mass spectrum generation sfermions are relatively unconstrained on phenom- enological grounds, and their limits are principally determined To explore the parameter space of the benchmark scenar- by the restriction of c-parameters to order-one numbers. In ios given in Table I, we randomly sample over the estimated particular, we note that for the stops (below 100 TeV,these can ranges of doublet c parameters (c ) that are consistent Li;Qi be identified unequivocally with u˜ 1 2), the general mass scale with all phenomenological constraints. The allowed ranges pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ( m˜ m˜ ) is broadly consistent with the observed Higgs mass are principally determined by the FCNC constraints on the t1 t2 c first- and second-generation sfermions and by the Higgs throughout the allowed -parameter ranges. The observed Higgs boson mass provides stronger con- mass constraints on the third-generation squarks. EWSB straints indirectly, since the particular structure of EWSB in and the large D-term radiative corrections discussed in

FIG. 16. Predicted superpartner pole mass spectra for benchmark scenarios A (hatched) and B (solid) given in Table I.

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FIG. 17. Predicted sfermion mass spectra in the gauge-eigenstate basis for benchmark scenarios A (hatched) and B (solid) given in Table I.

Δ S∼ 2 −2 2 þ 2 − 2 þ 2 our model makes it sensitive to the scalar D-term corrections that S; m ˜ mu˜ m˜ me˜ me˜ are of the cor- u˜ L;dL R dR L R arising either from the extra dimension or in the MSSM rect scale. running (see Sec. IVA 2). We show the Higgs boson mass We show this correlation for scenario A in Fig. 19.For estimates for both scenarios in Fig. 18 as smoothed functions scenario A, the explanation of the SM fermion mass of tan β. For both benchmark scenarios, consistency with hierarchy typically requires that either u˜ L or u˜ R is the the observed Higgs boson mass restricts the allowed range of heaviest sfermion. Thus, in order to obtain ΔS; S > 0,as D-term corrections, such that the necessary value of tan β 2 preferred by the Higgs boson mass, m˜ must be heavy results from EWSB. This, first of all, introduces correlations eR enough to compensate for the negative contribution of among the heavy sfermion masses, such that the necessary 2 m˜ ,ifmu˜ >mu˜ . When mu˜ >mu˜ , then me˜ >me˜ is corrections are obtained. The primary constraint arises on the uR R L L R L R c parameters of the first-generation sfermions (the heaviest allowed. Note that the Higgs mass measurement also sfermions) cL1 and cQ1 , which must be correlated such

(a) (b)

FIG. 18. Predicted Higgs boson mass and uncertainty for FIG. 19. Correlation between first-generation slepton and benchmark scenarios A (left)pffiffiffiffiffiffiffiffiffiffiffiffiffi and B (right) as functions of up-squark masses for benchmark scenario A (given in Table I). β ¼ tan at the scale mSUSY mt˜1 mt˜2 . The horizontal blue line The green region gives a smoothed estimate of the region and surrounding region shows the observed Higgs mass value and preferred by the experimentally measured Higgs boson. The its uncertainty. horizontal (vertical) gray lines give the range of cL1 (cQ1 ).

035046-26 PARTIALLY COMPOSITE SUPERSYMMETRY PHYS. REV. D 99, 035046 (2019) constrains the allowed spread of the first-generation sfer- V. CONCLUSION mion masses more than the condition imposed to suppress We have presented a minimal supersymmetric model that FCNCs, which merely restricts the masses to be above uses partial compositeness to relate the SM fermion mass 100 TeV. Similarly, for scenario B, the Higgs boson mass hierarchy to the sfermion mass hierarchy. This occurs by prefers ΔS; S < 0, which is accomplished when u˜ is the R assuming that the SM gauge fields, Higgs sector, and the heaviest sfermion in the theory. ΔS; S > 0 is also allowed, third-generation matter are (mostly) elementary, while the but in this case, m˜ must be tuned against m˜ . For this uR eR first two generations of matter are composite due to some point we note that the Higgs boson mass also limits the Λ unknown strong dynamics that confines at a scale IR. mass ranges of the other first- and second-generation Hierarchies are then generated when elementary superfields sfermions more strictly than necessary to suppress FCNCs. linearly mix with supersymmetric operators that have large For the third-generation sfermions, the D-term correc- anomalous dimensions. Since the Higgs fields are elemen- tions necessary to obtain the observed Higgs boson mass tary, the more composite the fermion, the lighter the may necessitate additional constraints in order to avoid corresponding fermion mass. The strong dynamics is also tachyonic masses. For both scenarios, this effect imposes assumed to dynamically break supersymmetry, such that the the lower limit on mτ˜ , and for scenario B it also imposes R composite sparticle states directly feel the supersymmetry the lower limit on mτ˜ . These are the only constraints L breaking. The predominantly elementary states, such as the on the third-generation sfermion masses that are stronger third-generation sfermions, Higgsinos, and gauginos, are than those set by the order-one limit on the c parameters. therefore split from the much heavier first- and second- For both scenarios, the right-handed stau may thus generation composite sfermions. Thus, the partially be the lightest sfermion, and τ˜1 may accordingly be composite supersymmetric model generically predicts that the NLSP. light (heavy) SM fermions, have heavy (light) sfermion ˜0 When τ˜1 is heavy, the NLSP for both scenarios is χ1.For superpartners. Moreover, since the gravity multiplet mixes 0 0 scenario A, χ˜1 is Bino-like and χ˜2 , χ˜2 are Wino-like, while with the stress-energy tensor (via an irrelevant term), the χ˜ χ˜0 the heavy , 2 , and the heavy , 3;4, are gravitino is much lighter than the gauginos. It therefore χ˜0 becomes the LSP that can play the role of dark matter. Higgsino-like. This is reversed for scenario B, where 1;2;3 0 To obtain quantitative predictions and model the unknown and χ˜1 are Bino-like or Higgsino-like and χ˜2 , χ˜4 are Wino- like. The spread in the masses of the Higgsino-like states is strong dynamics responsible for the composite states and due to the spread of the Higgsino mass parameter μ, which large anomalous dimensions, we use the AdS=CFT corre- is fixed by EWSB. In both cases, μ correlates precisely with spondence to study a 5D version of our 4D model [where the 2 strong dynamics is specifically due to a large-N gauge theory the soft mass parameter m , which is predominantly Hu 2 (CFT)]. In a slice of AdS5, the Higgs sector is confined to determined by Yukawa radiative corrections from m ˜ Q3 the UV brane, while the remaining MSSM superfields are 2 μ and mu˜ 3 . Thus, the spread of is ultimately tied to freedom located in the bulk. Supersymmetry breaking occurs on the IR brane. The MSSM fields are identified with the zero in cQ3 , which as we discussed above, is limited only by the constraint that it is an order-one number. The masses of the modes of the corresponding 5D fields. The zero-mode profile heavy Higgs, H0, H, and A0 also scale with μ, but about depends on a bulk mass (dimensionless) parameter c that can 10% of their spread is due to the running of the soft masses be arbitrarily varied to localize the zero-mode superfield m2 and m2 . The spread in the gluino and the Bino-like anywhere in the bulk. The fermion and sfermion mass Hu Hd hierarchy is now dictated by the 5D fermion geography. and Wino-like states is due primarily to the uncertainty in Since the Higgs fields are confined to the UV brane, the third- the gauge and Yukawa couplings. In Figs. 16 and 17 that generation SM fermions are UV-localized, while the first- spread is exaggerated for clarity. and second-generation SM fermions are IR-localized. This In both scenarios, the hierarchical structure of the mass naturally leads to an inverted sfermion mass hierarchy, where spectrum is clear. The largest hierarchy occurs for τ˜1, where the first- and second-generation sfermions are heavy, while we find ratios up to m ˜ =mτ˜ ∼ 13 in the singlet spurion u˜ 6;d6 1 those of the third generation are light. case and up to m˜ =mτ˜ ∼ 35 in the nonsinglet spurion case. u6 1 At tree level, the sfermion hierarchy may be exponentially The hierarchy for the stops is relatively more modest: we large due to the suppressed coupling between the UV- m ˜ =m˜ ∼ 3 find ratios up to u˜ 6;d6 t1 in the singlet spurion case localized fields and the supersymmetry-breaking sector. The ∼ 18 and up to mu˜ 6 =mt˜1 in the nonsinglet spurion case. The mass scale of the third-generation sfermions is therefore set size of these mass splittings, which cannot be generated by by radiative corrections from the heavy states, which trans- MSSM running alone, is a direct consequence of the mit the breaking of supersymmetry at loop order and become hierarchy in the sfermion IR-brane soft mass boundary the dominant soft mass contribution. At one loop in 5D, conditions, and hence is ultimately a signature of the SM these corrections arise from bulk gauginos and scalars. Since fermion mass spectrum, mediated by the radiative correc- the Higgs fields are localized on the UV brane, both the tions of the extra dimension and the MSSM. Higgs-sector soft masses and the soft trilinear scalar

035046-27 BUYUKDAG, GHERGHETTA, and MILLER PHYS. REV. D 99, 035046 (2019) couplings (a-terms) are zero at tree level, but they, too, invoked to dynamically break supersymmetry (e.g., via receive radiative corrections from the bulk. gaugino condensation). The supersymmetry breaking is The overall scales in the 5D model can be fixed by then mediated via gravity (or alternatively, gauge inter- imposing a number of phenomenological constraints: actions) to the visible sector with universal boundary (i) the LSP gravitino is assumed to be the dark matter conditions for the sfermion masses. The difference in with a mass ≳1 keV; (ii) electroweak symmetry is broken, our model is that the first- and second-generations of consistent with a 125 GeV Higgs boson; (iii) the first- and matter are composites of the strong dynamics at some Λ second-generation sfermions are at least as heavy as high scale IR. The composite states also directly feel the 100 TeV to ameliorate the supersymmetric flavor problem; supersymmetry breaking (e.g., perhaps via a nonzero (iv) the gaugino and Higgsino masses are constrained, so as F-term of the underlying constituents), thereby giving rise to preserve gauge coupling unification as in the usual to strongly flavor-dependent sfermion mass boundary con- MSSM [assuming any underlying dynamics preserves ditions. Furthermore, assuming that the strong dynamics is SU(5)]; and (v) only the MSSM fields are present in the SU(5) invariant (similar to what is imposed on the messenger theory below the scale of compactification. The SM sector in gauge-mediated models), gauge coupling unifica- fermion mass spectrum is used to constrain the bulk tion is still preserved at the GUT scale ∼1016 GeV. fermion mass parameters ci. The 5D model then predicts In light of the Higgs boson discovery and its implications Λ the sfermion masses at the IR-brane scale, IR, which are for the supersymmetric spectrum, our model thus provides run down to lower energies using renormalization group a more predictive, split-like supersymmetry scenario by equations. Since the boundary conditions for the sfermion explicitly relating the SM fermion mass hierarchy to the masses are nonuniversal and flavor-dependent, tachyonic sfermion mass spectrum. It would be interesting to con- constraints that avoid charge- and color-breaking minima struct models of the nontrivial dynamics (perhaps going must be imposed to further restrict the parameter space. beyond large-N theories) that may constrain the anomalous The numerical results of our benchmark scenarios, given dimensions even further, and therefore lead to exact in Table I, predict a hierarchical sfermion mass spectrum. predictions for the sparticle spectrum. Nonetheless, the The third-generation sfermions have masses in the approxi- partially composite supersymmetric model provides the mate range 10–100 TeV (20–100 TeV for the stops), while raision d’être for the inverted sfermion hierarchy with a the first- and second-generation sfermions have masses in gravitino LSP. The NLSP is typically a Bino, Higgsino, the range 100–350 TeV. We do not obtain a unique or right-handed stau which decays to the gravitino and prediction because we assume that there is no relation could eventually be probed at a future 100 TeV collider. between the cL;R parameters of the left- and right-handed Alternatively, the heavy first- and second-generation sfer- fermions. Nevertheless, the numerical results reveal some mions could be indirectly probed via rare-decay experiments, interesting features. Most obvious is the hierarchical nature such as the flavor-violating Mu2e experiment [111],or of the spectrum. Typical MSSM running cannot produce a experiments attempting to measure the electric dipole mass spectrum with widely separated sparticle masses, and moment of the electron [112]. Of course, with heavy super- thus, with minimal particle content, the origin of the mass partners, our model is tuned, and the question of why the hierarchy must necessarily reside in the high-scale boun- overall scale of the sparticle spectrum is much heavier than dary conditions. Such conditions are a generic feature of the TeV scale remains a mystery. Perhaps this is just evidence our model and result in a distinctive split spectrum. The of the multiverse, as speculated in split-supersymmetric nonuniversality of the sfermion boundary conditions is also models, or a supersymmetric relaxion mechanism is at play, visible at a finer level, as it is responsible for the presence of or, instead, the tuning could be related to the strong dynamics sizeable D-term radiative corrections to the scalar masses. of the supersymmetry-breaking sector. In any case, we have Although the sign and magnitude of these corrections are attempted to provide further rationale for why low-energy highly constrained on tachyonic grounds, they can be supersymmetry may be lurking at a scale of 10–1000 TeV. favorable for EWSB and can offset negative contributions to the scalars that arise at two loops. Due to the structure of ACKNOWLEDGMENTS EWSB in our model (imposed by radiative corrections from the bulk), the predicted Higgs boson mass is also sensitive We thank Jason Evans, Ben Harling, and Alex Pomarol to D-term corrections, and the experimentally measured for useful discussions. This work was supported by the U.S. mass value can therefore indirectly constrain the heaviest Department of Energy Grant No. DE-SC0011842 at the mass scales in the theory. In fact, since the measured University of Minnesota. Higgs mass is broadly consistent with stop masses in the 10–100 TeV range (as predicted in both benchmark APPENDIX A: PARTIAL COMPOSITENESS scenarios) it is primarily through this effect that it con- strains our benchmark spectra. In this appendix we present details of the partial Our model is not too different from the usual MSSM, compositeness (equivalent to holographic mixing in 5D) where a hidden sector with strong dynamics is typically for the chiral, vector, and gravity supermultiplets.

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1. Chiral supermultiplet mixing basis is shown in Sec. III C to give consistent results a. Complex scalar with the 5D gravity model. At the IR scale, the supersymmetric Lagrangian (7) for the chiral supermultiplet has the component form b. Fermion Similarly, the fermion part of the supersymmetric L ¼−∂ ϕ†∂μϕþ † þε Λ ðϕ cð1Þ þ ϕcð1Þ þ Þ scalar μ F F Φ IR F F H:c: Lagrangian (7) at the IR scale is given by cð1Þ† μ cð1Þ cð1Þ† cð1Þ ð1Þ† μ ð1Þ −∂μϕ ∂ ϕ þF F −∂μϕ ∂ ϕ L ¼ ψ †σ¯ μ∂ ψ þ ψ ð1Þ†σ¯ μ∂ ψ ð1Þ þ ψ cð1Þ†σ¯ μ∂ ψ cð1Þ fermion i μ i μ i μ ð1Þ† ð1Þ ð1Þ ð1Þ cð1Þ ð1Þ cð1Þ þF F þm ðϕ F þF ϕ þH:c:Þ; ð1Þ Φ −ε Λ ðψψcð1Þ þ Þ − ðψ ð1Þψ cð1Þ þ Þ Φ IR H:c: mΦ H:c: : ðA1Þ ðA6Þ ð1Þ ð1Þ where εΦ is a dimensionless constant and m ¼ g Λ . Φ Φ IR ðψ ψ cð1Þ ψ ð1ÞÞ Eliminating the auxiliary fields gives rise to the Lagrangian In the basis ; ; , this leads to the following fermion mass matrix: † μ 2 2 † ð1Þ 2 † ð1Þ 0 1 L ¼ −∂μϕ ∂ ϕ−εΦΛ ϕ ϕ−εΦgΦ Λ ðϕ ϕ þH:c:Þ scalar IR IR 0 εΦ 0 ð1Þ2 B C −∂ ϕð1Þ†∂μϕð1Þ − ϕð1Þ†ϕð1Þ 1 B ð1Þ C μ mΦ ¼ εΦ 0 g Λ ð Þ mψ 2 @ Φ A IR: A7 −∂ ϕcð1Þ†∂μϕcð1Þ −ð ð1Þ2 þε2 ÞΛ2 ϕcð1Þ†ϕcð1Þ ð1Þ μ gΦ Φ IR ; 0 gΦ 0 ðA2Þ The mass eigenstates correspond to a massless ð1Þ ð1Þ Weyl fermion and a massive Dirac state with mass where, in the basis ðϕ; ϕ ; ϕc Þ, the mass matrix is ðε2 þ ð1Þ2Λ Þ1=2 given by Φ gΦ IR . The massless eigenstate is given by   0 1 ε ε2 ε ð1Þ 0 jψ i ≃ N jψi − Φ jψ ð1Þi ð Þ B Φ ΦgΦ C 0 Φ ð1Þ ; A8 B C gΦ 2 ¼ B ε ð1Þ ð1Þ2 0 CΛ2 ð Þ mϕ @ ΦgΦ gΦ A IR: A3 2 ð1Þ2 while the massive eigenstates are 00εΦ þ gΦ ( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) N ε ε2 Note that there is a mass mixing between the elementary pffiffiffiΦ Φ ð1Þ Φ cð1Þ jψ 1;2i ≃ jψiþjψ i 1 þ jψ i : ð1Þ 2 ð1Þ ð1Þ2 state ϕ and the composite state, ϕ . Nevertheless, when gΦ gΦ this matrix is diagonalized, there is a massless eigenstate ðA9Þ which can be written as   ε Thus, partial compositeness leads to a massless complex jϕ i ≃ N jϕi − Φ jϕð1Þi ð Þ scalar and Weyl fermion which combine into a chiral 0 Φ ð1Þ ; A4 gΦ supermultiplet. where N Φ is a normalization constant, while the massive 2. Vector supermultiplet eigenstates are given by The component fields of the vector supermultiplet can be   ε identified by noting that the IR Lagrangian (11) is invariant jϕ i ≃ N Φ jϕiþjϕð1Þi ð Þ 1 Φ ð1Þ ; A5a under the supergauge transformations gΦ V → V þ iðΩ† − ΩÞ; ðA10aÞ cð1Þ jϕ2i ≃ jϕ i: ðA5bÞ Φð1Þ þ Φð1Þ† Φð1Þ þ Φð1Þ† ε Thus, we see that the massless eigenstate is an admix- Vð1Þ þ pVffiffiffi V → Vð1Þ þ pVffiffiffi V − i V ðΩ† − ΩÞ; 2 ð1ÞΛ 2 ð1ÞΛ ð1Þ ture of the elementary and composite states and that the gV IR gV IR gV massive eigenstate is a complex scalar with mass-squared ðA10bÞ ðε2 þ ð1Þ2ÞΛ2 Φ gΦ IR. Note that the eigenstates are expressed in the mass-mixing basis, unlike the kinetic-mixing basis used where Ω is a chiral superfield gauge-transformation param- in Ref. [27]. While both bases are equivalent at the level of eter. Choosing the Wess-Zumino gauge for V, the super- mass eigenstates, supersymmetry breaking in the mass- fields then take the form

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† μ † † † † 1 † † while the massive gauge boson eigenstate is given by V ¼ θ σ¯ θAμ þ θ θ θλ þ θθθ λ þ θθθ θ D; ðA11aÞ 2   εV ð1Þ 1 jAμ1i ≃ N jAμiþjAμ i ; ð Þ ð1Þ † μ ˜ ð1Þ † † ð1Þ † ð1Þ† † † ð1Þ V ð1Þ A16 V ¼ θ σ θAμ þ θ θ θλ þ θθθ λ þ θθθ θ D ; 2 gV ðA11bÞ where N V is a normalization constant. The eigenstates are pffiffiffi expressed in the mass-mixing basis instead of the kinetic- Φð1Þ ¼ ϕð1Þ þ 2θχð1Þ þ θθ ð1Þ þ θ†σ¯ μθ∂ ϕð1Þ V V FV i μ V mixing basis used in Ref. [27], since supersymmetry is assumed to be broken in this basis. The massless eigen- i † μ ð1Þ 1 † † ð1Þ − pffiffiffi θθθ σ¯ ∂μχ þ θθθ θ □ϕ ; ðA11cÞ 2 4 V mode (A15) now transforms under a gauge transformation → þð1 þ ε2 ð1Þ2Þ∂ ζ as Aμ0 Aμ0 V=gV μ , while, as expected, Aμ1 □ ¼ ∂ ∂μ ˜ ð1Þ ϕð1Þ λð1Þ χð1Þ no longer transforms. where μ . Here, Aμ , V , , and are ð1Þ ð1Þ The massive eigenstate Aμ1 obtains a mass dynamical (composite) fields, while FV and D are auxiliary (composite) fields. Next, we diagonalize the mass ð1Þ2 ð1Þ2 m2 ¼ðε2 þ g ÞΛ2 ≃ ½g2ðΛ Þþg Λ2 ; ðA17Þ term for the component Lagrangians. V1 V V IR s IR V IR

where, in thepffiffiffi second expression, we have used a. Gauge field ε˜ ðΛ Þ ≃ pffiffiffiffiffiV N ¼ ε gs IR 4π V, which follows from the large-N Using (11) and (A11), the gauge field component ZV corrections to the elementary-field gauge coupling g . The Lagrangian becomes s diagonal gauge-field Lagrangian for the two-state system is 1 1 then given by L ¼ − μν − ˜ ð1Þμν ˜ ð1Þ gauge 4 F Fμν 4 F Fμν 1 μν 1 μν 1 2 2 1 2 ð1Þ ð1Þ ð1Þ 2 L ¼ − − − ð Þ − Λ ðε þ ˜ þ ∂ φ Þ ð Þ gauge F0 F0μν F1 F1μν mV1 A1μ; A18 2 IR VAμ gV Aμ μ ; A12 4 4 2 which is invariant under the gauge transformations where the gauge coupling of the massless mode is obtained from

Aμ → Aμ þ ∂μζ; ðA13aÞ ðΛ Þψ †σ¯ μ ψ − ð1Þψ ð1Þ†σ¯ μ ð1Þψ ð1Þ gs IR Aμ gV Aμ ð1Þ ∂ φ ð1Þ ð1Þ† μ ð1Þ ð1Þ † μ ð1Þ ð1Þ μ ð1Þ εV þ ψ c σ¯ ψ c ¼ ψ σ¯ ψ þ ð Þ ≡ ˜ þ → − ∂ ζ ð Þ gV Aμ g 0 Aμ0 0 A19 Aμ Aμ ð1Þ Aμ ð1Þ μ ; A13b gV gV Using (A8) and (A15), this leads to the expression ð1Þ i ð1Þ ð1Þ where ζ is a gauge parameter and φ ≡ 2Λ ðϕ − ϕ Þ. IR V V    Note that in the limit of no mixing between the elementary 1 ε2 ε −2 ¼ N 2 N ðΛ Þþ ð1Þ Φ V and composite sectors ðεV ¼ 0Þ, the lowest-lying massive 2 Φ V gs IR gV 2 ð1Þ g gΦ ð1Þ ð1Þ ð1Þ gV Aμ m ¼ g Λ composite state, , has a mass V V IR. The 1 1 appearance of a mass term for Aμ is similar to what ≃ þ : ðA20Þ g2ðΛ Þ ð1Þ2 happens for vector- dominance in QCD [113]. s IR gV The mass eigenstates are obtained by diagonalizing the ð1Þ ð1Þ mass-squared term in (A12), where in the basis ðAμ;Aμ Þ, Finally, note that by eliminating D and D in the scalar the mass matrix is given by field part of the Lagrangian, one can check that the real part ð1Þ ! of the composite scalar field ϕ also obtains a mass, ε2 ε ð1Þ V 2 V V gV 2 Aμ1 m ¼ Λ : ðA14Þ identical to that of the gauge field . A ε ð1Þ ð1Þ2 IR VgV gV b. Gaugino This gives rise to one massless and one massive eigenstate. The gaugino part of the vector supermultiplet Lagrangian For canonical kinetic terms, the massless gauge boson (11) is given by eigenstate becomes   † μ ð1Þ† μ ð1Þ ð1Þ† μ ð1Þ L ¼ iλ σ¯ ∂μλ þ iλ σ¯ ∂μλ þ iχ σ¯ ∂μχ εV ð1Þ gaugino jAμ0i ≃ N jAμi − jAμ i ; ðA15Þ V ð1Þ − Λ ðε λχð1Þ þ ð1Þλð1Þχð1Þ þ Þ ð Þ gV IR V gV H:c: : A21

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ε In the limit εV ¼ 0, the massive composite state is a Dirac ð1Þ ð1Þ H Hμ → Hμ − Δμ; ðA27bÞ ð1Þ ¼ ð1ÞΛ ð1Þ fermion with mass mV gV IR. Using the basis gH ðλ; χð1Þ; λð1ÞÞ and the orthogonal matrix Δ 0 1 where μ is a real superfield gauge-transformation param- ð1Þ 0 −ε eter. Choosing an analog of the Wess-Zumino gauge for gV V B qffiffiffiffiffiffiffiffiffiffiffiffiffi C Hμ, the superfields take the form B ð1Þ2 ð1Þ C 1 B ε g þε2 g C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pVffiffi − V V pVffiffi O1=2 ¼ B 2 2 2 C; ðA22Þ 1 ð1Þ2 B C † ν † † 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi Hμ ¼ − pffiffiffi θ σ¯ θhμν − iθθθ λμ g þ ε @ ð1Þ2 ð1Þ A V V ε þε2 2 pVffiffi gV V gpVffiffi 2 2 2 1 þ θ†θ†θλ þ θθθ†θ† þ ð Þ i μ 2 Dμ A28a the mass terms in (A21) can be diagonalized via ð1Þ ð1Þ ð1Þ † ð1Þ† † ν ð1Þ 0 ε 1 0 1 Hμ ¼ Cμ − iθωμ þ iθ ωμ − θ σ¯ θVμν 0 V 0 00 0 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi   B ð1Þ C B 2 ð1Þ2 C † ð1Þ† i ν ð1Þ B ε C B ε þg C − θθθ λ − σ¯ ∂ ω V gV T 0 − V V 0 i μ ν μ O1 2B 0 CO ¼ B 2 C; 2 = @ 2 2 A 1=2 @ pffiffiffiffiffiffiffiffiffiffiffiffiffiA   ð1Þ 2 ð1Þ2 g ε þg ð1Þ i ð1Þ† 0 V 0 00 V V þ θ†θ†θ λ − σν∂ ω 2 2 i μ 2 ν μ ðA23Þ   1 † † ð1Þ 1 ð1Þ þ θθθ θ Dμ þ □Cμ þ ðA28bÞ which gives rise to a massless Weyl fermion and a massive 2 2 Dirac fermion. The massless gaugino eigenstate is given by   where we have neglected terms with auxiliary fields. The ε gravity component fields are then defined to be jλ i ≃ N jλi − V jλð1Þi ð Þ 0 V ð1Þ : A24 g ð1Þ ð1Þ V ð1Þ Vμν þ Vνμ hμν ≡ pffiffiffi ; ðA29aÞ Thus, the massless gaugino is an admixture of the elemen- 2 ð1Þ tary gaugino λ and the composite gaugino λ . ð1Þ 1 1 ð1Þ Cμ ≡ pffiffiffi hμ ; ðA29bÞ The massive Dirac fermion state has the decomposition 3Λ ð1Þ IR gH 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 < = 1 1 N ε ε2 ψ ð1Þ ≡ λð1Þ þ σ σ¯ ρλð1Þ ð Þ jλ i ≃ pffiffiffiV V jλiþjλð1Þi 1 þ V jχð1Þi μ μ μ ρ A29c 1;2 ð1Þ ð1Þ2 ; 2 3 2 :g g ; V V 1 8 9 ð1Þσ ð1Þσ νμκσ ð1Þ b ≡ D þ ϵ ∂κVνμ ; ðA29dÞ N < 1 = 2 ≃ pffiffiffiV qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jλiþjλð1Þijχð1Þi ; : ð1Þ Λ ; 1 1 2 2ζ ð UVÞ ð1Þ ð1Þ gV V log Λ ωμ ≡ χμ : ðA29eÞ IR Λ 2 ð1Þ IR gH ðA25Þ ð1Þ ð1Þ ð1Þ ð1Þ Note that hμν , ψ μ , χμ ,andhμ are dynamical ε ≪ 1 Oðε2 Þ ð1Þ assuming V and dropping terms of V and higher (composite) fields, while bμ is an auxiliary (composite) field. in the second expression, with mass eigenvalues qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ2 a. Graviton m ≃ g2ðΛ Þþg Λ : ðA26Þ V1;2 s IR V IR Using (17) the graviton field component Lagrangian becomes This mass agrees with that obtained for the gauge field Aμ1 ð1Þ and scalar field ϕ , as expected by supersymmetry. 1 μν 1 ð1Þ ð1Þμν V L ¼ pffiffiffi h E þ pffiffiffi hμν E graviton 2 μν 2 3. Gravity supermultiplet 1 2 ð1Þ ð1Þ 2 − Λ ðε hμν þ g hμν Þ ; ðA30Þ To identify the component fields of the gravity super- 2 IR H H multiplet, we note that the IR Lagrangian (17) is invariant ≡ p1ffiffi ð∂ ∂ λ þ □ − ∂ ∂λ − ∂ ∂λ − under the supergauge transformations where Eμν 2 μ νhλ hμν μ hλν ν hλμ λ λ ρ ημν□h þ ημν∂ ∂ hλρÞ. The Lagrangian is invariant under → þ Δ ð Þ λ Hμ Hμ μ; A27a the gauge transformation

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1 ð1Þ ð1Þ where ψ μ and χμ are both contained in the tensor hμν → hμν − ð∂μζν þ ∂νζμÞ; ðA31aÞ 2 ð1Þ supermultiplet Hμ . In the limit εH ¼ 0 the massive ð1Þ ð1Þ ð1Þ 1 ε → þ H ð∂ ζ þ ∂ ζ Þ ð Þ composite state is a Dirac fermion with mass mH . hμν hμν ð1Þ μ ν ν μ ; A31b ð1Þ ð1Þ 2 Using the basis ðψ μ; χμ ; ψ μ Þ, the mass term in (A35) gH can be diagonalized via the transformation where ζμ is a gauge parameter. Note that in the limit of no 0 ε 1 0 1 0 H 0 00 0 mixing between the elementary and composite sectors 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ B ð1Þ C B ε2 þ ð1Þ2 C ðε ¼ 0Þ μν B ε C B H g C H , the lowest-lying massive composite state, h , H gH T 0 − H 0 O3 2B 0 CO ¼ B 2 C; ð1Þ ¼ ð1ÞΛ = @ 2 2 A 3=2 @ pffiffiffiffiffiffiffiffiffiffiffiffiffiA has a mass mH gH IR. ð1Þ ð1Þ2 g ε2 þg 0 H 0 H H The mass eigenstates are obtained by diagonalizing the 2 00 2 mass term in (A30). The massless graviton eigenstate is ð Þ then A36   with the orthogonal matrix εH ð1Þ jhμν0i ≃ N jhμνi − jhμν i ; ðA32Þ H ð1Þ 0 1 gH ð1Þ g 0 −ε B H qffiffiffiffiffiffiffiffiffiffiffiffiffi H C B ð1Þ2 ð1Þ C and the massive graviton eigenstate is given by 1 B ε g þε2 g C ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B pHffiffi − H H pHffiffi C ð Þ   O3=2 B 2 2 2 C; A37 ε ð1Þ2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi H ð1Þ g þ ε @ ð1Þ2 ð1Þ A j i ≃ N j iþj μν i ð Þ H H ε þε2 hμν1 H ð1Þ hμν h ; A33 pHffiffi gH H gpHffiffi 2 gH 2 2 N where H is a normalization constant. The mass- which gives rise to a massless Weyl fermion and a massive less eigenmode hμν0 now transforms under a gauge trans- Dirac fermion. The massless gravitino eigenstate then → −1ð1þε2 ð1Þ2Þð∂ ζ þ∂ ζ Þ becomes formation as hμν0 hμν0 2 H=gH μ ν ν μ , while, as expected, hμν1 no longer transforms. The massive   2 2 ð1Þ2 2 εH ð1Þ ≃ ðε þ ÞΛ jψ i ≃ N jψ i − jψ μ i ð Þ eigenstate hμν1 obtains a mass mH1 H gH IR.The μ0 H μ ð1Þ ; A38 diagonal gravity Lagrangian for the two-state system is then gH given by Thus, the massless gravitino is an admixture of the 1 1 1 ð1Þ μν μν 2 2 ψ ψ L ¼ pffiffiffi h Eμν0 þ pffiffiffi h Eμν1 − m h : ðA34Þ elementary gravitino μ and the composite gravitino μ . graviton 0 1 2 H1 μν1 2 2 The massive Dirac fermion state is given by ð1Þ By eliminating bμ and bμ in the vector-field part of the 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 < 2 = Lagrangian, one can check that the composite vector N ε ð1Þ ε ð1Þ ð1Þ H H H jψ μ1 2i ≃ pffiffiffi jψ μiþjψ μ i 1 þ jχμ i ; field hμ also obtains a mass identical to that of the graviton ; 2 : ð1Þ ð1Þ2 ; gH gH field hμν1. For simplicity, we have not included the 8 9 scalar components, but they obtain a similar mass by N < 1 Λ = supersymmetry. H IR ð1Þ ð1Þ ≃ pffiffiffi pffiffiffiffiffiffi jψ μiþjψ μ ijχμ i ; 2 : ð1Þ ζ Λ ; gH H UV b. Gravitino ðA39Þ The gravitino part of the gravity supermultiplet Lagrangian (17) at the IR scale is given by ε ≪ 1 Oðε2 Þ where H and terms of H and higher have been L gravitino dropped in the second expression. The mass eigenvalues 1 1 are ¼ − ϵμνρκψ σ ∂ ψ † − ϵμνρκψ ð1Þσ ∂ ψ ð1Þ† 2 μ ν ρ κ 2 μ ν ρ κ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1Þ2 1 μνρκ ð1Þ ð1Þ† ¼ ε þ Λ ð Þ − ϵ χ σ ∂ χ mH1;2 H gH IR; A40 2 μ ν ρ κ

1 μ ν ð1Þ ð1Þ ð1Þ μ ν ð1Þ − Λ ðε ψ μ½σ ; σ¯ χν þ g ψ μ ½σ ; σ¯ χν þ H:c:Þ; which agrees with that obtained for the graviton 4 IR H H ð1Þ ð1Þ field hμν and the vector field hμ , as expected by ð Þ A35 supersymmetry.

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APPENDIX B: BULK ZERO-MODE mass terms on the IR brane induce a back reaction on the PROFILES IN A SLICE OF ADS KK profiles of these fields. In this case, the zero-mode profiles are determined in the same manner as the profiles The quadratic part of the 5D bulk action of a hyper- of massive KK states: multiplet containing complex scalar fields ϕ and ϕc and Dirac fermion Ψ living in a slice of AdS5 is given by [25]    ˜ ð0Þ kjyj mϕ kjyj Z f ðyÞ¼Nϕe J2− e pffiffiffiffiffiffi ϕ b 5 2 2 2 c 2 2 c 2 k S5 ¼ d x −g½−j∂ ϕj − m jϕj − j∂ ϕ j − m c jϕ j   M ϕ M ϕ mϕ J1− ð Þ mϕ − b k kjyj ð Þ ¯ M ¯ mϕ Y2−b e ; B6 þ iΨΓ D Ψ − imΨΨΨ; ðB1Þ ð Þ M Y1−b k k g ¼ ðg Þ      where det MN is the determinant of the AdS metric Mλ J0ð Þ (39) and the curved space covariant derivative D ¼ ∂ þ ˜ ð0Þ kjyj Mλ kjyj k Mλ kjyj M M fλ ðyÞ¼Nλe J1 e − Y1 e ; ω ω k ðMλÞ k M includes the spin connection M. The bulk masses are Y0 k given by ðB7Þ 0 mΨ ¼ cσ ; ðB2aÞ where Nϕ;λ are determined by the normalization condi- 2 ¼ 2 þ σ00 ð Þ tions [39] mϕ;ϕc ak b ; B2b Z σ ¼ j j πR ˜ ð0Þ 2 where k y and a, b, c are dimensionless parameters. dyðfϕ ðyÞÞ ¼ 1; ðB8Þ Performing a KK decomposition, the zero-mode profiles −πR (with respect to a flat metric) are [24,25] Z πR 1 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜ ð0Þ 2 2 F ˜ ð0Þ 2 dyðfλ ðyÞÞ ¼ 1 þ g ð2πkRÞ ðfλ ðπRÞÞ : ð1 ∓ Þ −π 2 Λ Mλ k ð0Þ −3 j j ð0Þ 2 c k ð1∓ Þ j j R IR ˜ ð Þ¼ 2k y ð Þ¼ 2 c k y fΨ y e fΨ y 1 e ; L;R L;R 2ð ∓ Þπ ð Þ e 2 c kR − 1 B9 ðB3Þ When supersymmetry is broken, the super-Higgs effect rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gives rise to the gravitino coupling (44) on the UV brane, ð0Þ − j j ð0Þ ðb − 1Þk ð −1Þ j j ˜ ð Þ¼ k y ð Þ¼ b k y ð Þ and the gravitino acquires a mass m3 2 (46). This boundary fϕ y e fϕ y 2ð −1Þπ e ; B4 = e b kR − 1 mass term induces a back reaction on the gravitino KK profiles, such that the zero-mode profile is given where the upper (lower) sign is used for the L (R) component and a ¼ bðb − 4Þ must be satisfied for a    ˜ ð0Þ kjyj m3=2 kjyj massless scalar mode. This is automatic for a hyper- f3 2ðyÞ¼N3=2e J2 e ¼ 3 ∓ = k multiplet, where supersymmetry requires that b 2 c,   ð0Þ ð0Þ ð1∓ Þ j j m3=2 ˜ ˜ 2 c k y J1ð Þ m3 2 such that fΨ ðyÞ¼fϕ ðyÞ ∝ e as expected k = kjyj L;R L;R − Y2 e ; ðB10Þ ðm3=2Þ for the zero-mode fermions and scalar fields in a Y1 k k hypermultiplet. In a vector supermultiplet, the profile for the zero mode where of the gauge boson is Z πR ð0Þ 2 1 F 1 ð0Þ 2 dyðf˜ ðyÞÞ ¼ 1 þ pffiffiffi ðf˜ ð0ÞÞ ; ˜ ð0Þ ð0Þ 1 3=2 2 3=2 f ðyÞ¼f ðyÞ¼pffiffiffiffiffiffiffiffiffi ; ðB5Þ −πR 3MPm3=2 k A A 2πR ðB11Þ ð0Þ while that for the gaugino corresponds to fΨ ðyÞ in (B3) 1 L with c ¼ 2, and therefore matches (B5). Similarly for the determines the normalization N3=2. graviton supermultiplet the zero-mode graviton profile is given by (B4) with b ¼ 0, and for the gravitino the profile ð0Þ 3 APPENDIX C: RADIATIVE CORRECTIONS is fΨ ðyÞ in (B3) with c ¼ 2, which again matches by L IN A SLICE OF ADS supersymmetry. When supersymmetry is broken as in Sec. III C by the In this appendix we calculate the radiative corrections to IR-brane operators (51) and (47), the sfermion and gaugino the scalar masses as well as soft mass parameters in the zero modes acquire soft masses mϕ and Mλ. The boundary Higgs sector arising from the 5D bulk in our model.

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1. Bulk scalar soft masses are combinations of the modified Bessel functions I, K. Although the sfermions receive soft masses at tree level The natural argument of the Bessel functions is the ¼ from their couplings to the supersymmetry-breaking sector dimensionless variable x pEz. In the following, we ð Þ ≡ αð Þ on the IR brane, as discussed in Sec. III C, quantum suppress repeated indices, i.e., Tα x1;x2 Tα x1;x2 . corrections from the bulk can become significant for That the loop contribution to the scalar soft masses UV-localized fields. At one loop, these corrections arise squared is purely a supersymmetry-breaking effect can be from the gauge sector, Yukawa couplings, and D-term seen in the difference interactions. Here, we consider the corrections to a generic ð Þ − −3kjyj ð Þ bulk scalar soft mass squared in a supersymmetric theory. GA pE;y;y e Gλ pE;y;y 1 1 2 0ð Þ ¼ − z S1 xUV;x 2 ð Þ a. Gauge-sector corrections pE zUVzIR T0 xUV;xIR In the bulk, scalars couple to gauge bosons and gauginos, 2π ξ2 0ð Þ ig kR S1 xUV;x ð Þ generating soft mass corrections at one loop that take the × 2 2 0 ; C5 T0ðx ;x Þ − ig πkRξ S1ðx ;x Þ form UV IR UV IR which vanishes (by cancellation) when supersymmetry is 2 2 ϕi ðΔm Þ ¼ 4g CðRϕ ÞΠg ; ðC1Þ ξ ¼ 0 ϕi g i unbroken ( ). When the gaugino acquires a mass, this cancellation is shifted, and the scalar receives a correction where i indexes the scalar field, g is the (4D) gauge that is quadratically divergent. coupling at the IR-brane scale, CðRÞ is the quadratic This divergence arises since the addition of boundary Casimir [in the SU(5) normalization] of the representation masses as in Sec. III C deforms the superpartner KK wave- R, and function profiles, which results in a difference between the

Z 4 effective couplings of the gauge boson and of the gauginos, ϕ 2πkR d p Π i ¼ leading to a parametrically hard breaking of supersymmetry g k ð2πÞ4 Z on the IR brane. The extra dimension protects the UV brane πR and the bulk from the hard breaking, but on the IR brane, ð˜ ð0Þð ÞÞ2ð ð Þ − −3kjyj ð ÞÞ × dy fϕ y GA p; y; y e Gλ p; y; y ; where there is no finite distance separating the scalar mode −π i R from the source of supersymmetry breaking, quantum ð Þ C2 corrections to the scalar masses squared are not finite, and sensitive to the cutoff scale. We note that this divergent in the limit in which we neglect the external momentum. If behavior is a peculiar feature of warped spaces and does not we parametrize the amount of supersymmetry breaking on occur in the flat case, where this type of supersymmetry Λ ¼ ξ2 −πkR the IR brane as F= IR ke , the gauge boson, GA, breaking is globally realized (and hence does not lead to a and gaugino, Gλ, bulk propagators take the forms local distortion of field profiles). Thus, the flat-space break- ing is soft, and quantum corrections are finite and indepen- 1 S0ðx ;xÞS1ðx; x Þ G ðp ;y;yÞ¼ z2k 1 UV 0 IR ; ð Þ dent of the cutoff scale [17,114–116]. A E 2 0ð Þ C3a T0 xUV;xIR In the 4D dual theory, this divergence is a result of the breakdown of the perturbativity of the loop expansion as the 1 ð Þ¼ 5 4 0ð Þ gauge coupling becomes strong. Since we lose control over Gλ pE;y;y z k S1 xUV;x 2 the corrections near the compactification scale, we wish to 1 2 2 S ðx; x Þ − ig πkRξ T1ðx; x Þ extract a well-defined, finite portion of the correction 0 IR IR ; × 0ð Þ − 2π ξ2 0ð Þ T0 xUV;xIR ig kR S1 xUV;xIR associated with the long-range physics, which includes ð Þ the breaking of supersymmetry. The correspondence C3b between the renormalization scale in the 4D dual theory and position in the fifth dimension suggests an appropriate where p ¼ −ip is the Euclidean momentum (the general E regularization procedure in which we scale the effective IR expressions for Euclidean 5D mixed position-momentum brane seen by the propagators in the loop with position in the propagators, normalized to the interval 0 ≤ y ≤ πR are extra dimension [117]. The equivalent procedure in the 5D ¼ kjyj given in Ref. [17]), z e =k is a conformal coordinate perspective is to isolate the portion of the loop correction due ¼ 1 along the fifth dimension such that zUV =k and to the compactification of the theory, absorbing the remain- ¼ πkR zIR e =k, and the auxiliary functions ing infinite part into a counterterm [118–121]. Since the

α1 presence of the IR brane in AdS explicitly breaks 5D Lorentz Sα ðx1;x2Þ¼Iα ðx1ÞKα ðx2ÞþKα ðx1ÞIα ðx2Þ; ðC4aÞ 2 1 2 1 2 symmetry, this finite correction is nonlocal, associated with

α1 a winding around the compact dimension. This purely α ð Þ¼ ð Þ ð Þ − ð Þ ð Þð Þ T 2 x1;x2 Iα1 x1 Kα2 x2 Kα1 x1 Iα2 x2 C4b curvature-dependent contribution to (C2) can be extracted

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ϕ −πkR 2 2 L;R by employing a cutoff Λ ¼ ke on the 4D momentum ðΔm Þ ¼ y Πy ; ðC8Þ ϕL;R y integral. We have checked that both of these renormalization methods are numerically equivalent, and therefore may be where used interchangeably. For calculational convenience, we  ð0Þ  Z ˜ ð0Þ 2 4 employ the simple cutoff scheme. ϕ fϕ d p Π L;R ¼ L;R ð UV ð Þ− UV ð ÞÞ y 4 Gϕ p Gψ p : After regularization, the resulting finite part of the ˜ ð0Þ ˜ ð0Þ ð2πÞ R;L R;L fψ ð0Þfψ ð0Þ correction can be parametrized in terms of the gaugino L R mass as ðC9Þ

ϕi rg UV UV ðΔ 2 Þ ¼ 4 2 ð Þ 2 ð Þ Here, the bulk scalar, Gϕ , and fermion, Gψ , propagators mϕ 2 g C Rϕ Mλ; C6 i g 8π are evaluated on the UV brane and take the forms where ϕ α 4 i 1 1 2 ð Þ − ξ αð Þ ϕ Π pEzIRSβ xUV;xIR T xUV;xIR i 2 Rei g UV ð Þ¼ r ¼ 8π ð Þ Gϕ pE β ; g 2 C7 L;R 2 2 ð Þ − ξ4 ð Þ Mλ pE pEzIRTβ xUV;xIR Sα xUV;xIR is a positive parameter that depends on the amount of ðC10aÞ supersymmetry breaking as well as the localization of the bulk hypermultiplet containing the scalar field (and on the 1 1 Sαðx ;x Þ ϕ UV β UV IR i Gψ ðp Þ¼ ; ðC10bÞ IR-brane scale). We plot rg in Fig. 20 as a function of L;R E pffiffiffiffi 2 p Tβðx ;x Þ Λ Λ ¼ 107 E UV IR F= IR at IR GeV for three cases of the hyper- multiplet localization: the two limits in which the scalar is 1 where α ¼jc 2 j, confined to the UV and IR branes (ci → ∞ and 1 c → −∞) as well as the case (flat) in between. For ðc 1Þðc ∓ 1Þ i 2 β ¼ 2 2 ð Þ each localization, we give the U(1) (light), SU(2) 1 ; C11 jc 2 j (medium), and SU(3) (dark) contributions.p Theffiffiffiffi effect of Λ ≫ 1 the supersymmetry breaking saturates for F= IR and c is the fermion bulk mass parameter that specifies the ϕi and rg approaches a constant value that is independent of localization of fields in the hypermultiplet. the gauge group. The supersymmetry-breaking contribution arises from the difference between the scalar and fermion loops:

b. Yukawa corrections UV UV Gϕ ðpEÞ − Gψ ðpEÞ The bulk scalar soft masses squared also receive con- L;R L;R 1 1 1 1 tributions from their Yukawa couplings on the UV brane. ¼ 2 3 ð Þ At one loop, these corrections involve a boundary Higgs pE zUVzIR Tβ xUV;xIR field and a scalar or fermion field from a different bulk ξ4 hypermultiplet, taking the form × : ðC12Þ 2 ð Þ − ξ4 βð Þ xIRTβ xUV;xIR Sα xUV;xIR When supersymmetry is broken, the correction is finite and negative and can be parametrized in terms of the soft scalar masses as

ϕL;R ðΔ 2 Þ ¼ − ry 2 2 ð Þ mϕ 2 y mϕ ; C13 L;R y 8π R;L

where

ϕ Π L;R ϕL;R 2 Rei y ry ¼ −8π ðC14Þ m2 ϕR;L is positive and depends on the amount of supersymmetry ϕ breaking and the localizations of the bulk hypermultiplets.pffiffiffiffi i ϕ FIG. 20. Plot of the coefficient rg , which parametrizes the L;R In Fig. 21 we plot ry as a function of F=Λ for one-loop gauge corrections to bulk scalar soft masses squared, IR three choices of bulk hypermultiplet localizations: cL ¼ for the U(1), SU(2), and SU(3) gauge groups as a function of the ϕ pffiffiffiffi 1 L;R ϕi Λ −c ¼ 0, , 1. The behavior of ry is similar to that of rg , relative supersymmetry breaking on the IR brane, F= IR, for R 2 pffiffiffiffi Λ ¼ 107 Λ ≫ 1 IR GeV. saturating to a constant for F= IR .

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k G ð0;y;y0Þ¼ ; ðC17cÞ D 2πkR where we have evaluated the auxiliary field propagator in the zero-momentum limit, as there is no momentum flow into the scalar loop. Since the auxiliary field has no y-dependence in this limit, we can use the orthonormality condition for the scalar zero-mode profile (B8) to do the first integral over the fifth dimension, leaving

ϕ ðΠ i Þ → ΠD D ϕ ϕ Z j Z j 4 πR d p −2kjyj ¼ dye ðGϕ ðp; y; yÞ − Gϕc ðp; y; yÞÞ: 4 j j ð2πÞ −πR ϕL;R FIG. 21. Plot of the coefficient ry , which parametrizes the ðC18Þ one-loop Yukawa corrections to bulk scalar soft masses squared as a functionpffiffiffiffi of the relative supersymmetry breaking on the IR This quantity is independent of the localization of the Λ Λ ¼ 107 brane, F= IR,for IR GeV. external scalar, indicating that the sum X D YðϕiÞΠϕ ðC19Þ c. D-term corrections i i In models such as ours, where the pattern of supersym- metry breaking is nonuniversal in flavor-space, Fayet- gives a universal correction for all scalars (bulk and ð1Þ boundary) that are charged under the U(1) gauge symmetry. Iliopoulos U Y D-term corrections to scalar soft masses squared due to supersymmetry breaking arise. At one loop, As with (C1), the loop integral here is divergent. However, the leading divergences in this case arise even the contributing diagrams are of tadpole form, involving a 12 bulk scalar field and a vector supermultiplet auxiliary field: when supersymmetry is unbroken. The D-term contribu- tion to the soft scalar mass squared arising purely as a 3 X supersymmetry-breaking effect can be extracted by con- 2 2 ϕi ðΔmϕ Þ ¼ g1YðϕiÞ YðϕjÞðΠ Þϕ ; ðC15Þ i D 5 D j sidering the difference between loop integrals in broken j and unbroken supersymmetry: schematically,

D D D where ðΠ Þξ ¼ Π − ½Π ξ¼0: ðC20Þ ϕi ϕi ϕi Z Z 4 π ϕ 2πkR d p R ð0Þ 2 This is equivalent to evaluating the loop integral using the ðΠ i Þ ¼ ð˜ ð ÞÞ D ϕ 4 dy fϕ y j k ð2πÞ −π i difference in propagators: Z R πR 0 0 ð0 Þ Gϕ ðp ;y;yÞ − ½Gϕ ðp ;y;yÞ × dy GD ;y;y L;R E L;R E ξ¼0 −πR 4 β 0 1 1 ð Þ −2kjy j 0 0 0 0 z Sα xUV;x × e ðGϕ ðp; y ;yÞ − Gϕc ðp; y ;yÞÞ; ðC16Þ ¼ j j 2 3 ð Þ pE zUVzIR Tβ xUV;xIR ξ4 βð Þ Y is the hypercharge, and the sum is over all bulk scalars. Sα xUV;x c × : ðC21Þ Here ϕ is the N ¼ 2 supersymmetric scalar partner of ψ i, 4 β i 2p z Tβðx ;x Þ − ξ Sαðx ;x Þ a member of the same hypermultiplet, but odd under the E IR UV IR UV IR orbifold symmetry. The bulk scalar, Gϕ and Gϕc , and The supersymmetry-breaking contribution extracted in this auxiliary field, GD, propagators take the forms manner is linearly divergent. After regularizing the inte- grals, the resulting finite part of the correction is negative 1 4 3 β Gϕ ðp ;y;yÞ¼ z k Sαðx ;xÞ and can be parametrized in terms of the scalar mass: L;R E 2 UV 2 αð Þ − ξ4 ð Þ pEzIRSβ x; xIR Tα x; xIR 12 × ; Both quadratic and linear divergences arise in this manner; 2 ð Þ − ξ4 βð Þ pEzIRTβ xUV;xIR Sα xUV;xIR the former on the branes, and the latter in both the bulk and on the branes [122]. The quadratic divergences depend on the hyper- ðC17aÞ charge, and hence vanish in the trace over the scalars, provided that the sum of the scalar hypercharges is zero (as is true for the 1 ð Þ ð Þ MSSM field content). The linear divergences can be absorbed in a 4 3 Tα xUV;x Tα x; xIR Gϕc ðpE;y;yÞ¼ z k ; ðC17bÞ renormalization of the hypermultiplet bulk mass when regular- L;R 2 ð Þ Tα xUV;xIR ized with a position-dependent cutoff.

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bosons and gauginos. At one loop, the corrections involve one bulk field and one boundary field and induce the soft mass correction [115,116,124–126]

ðΔm2 Þ ¼ 4g2CðR ÞΠH; ðC24Þ Hi g Hi g where i ¼ u, d indexes the Higgs field and Z 2πkR d4p ΠH ¼ ðGUVðpÞ − GUVðpÞÞ: ðC25Þ g k ð2πÞ4 A λ

UV UV The gauge boson, GA , and gaugino, Gλ , bulk propa- gators are evaluated on the UV brane where the Higgs fields are localized, taking the forms

D 1 FIG. 22. Plot of the coefficient rϕ , which parametrizes the 1 1 S ðx ;x Þ i GUVð Þ¼ 0 UV IR ð Þ one-loop corrections to a bulk scalar soft masses squared due to a A pE 2 ð Þ ; C26a ϕ pE T0 xUV;xIR D-term coupling with the bulk scalar i as a functionpffiffiffiffi of the Λ relative supersymmetry breaking on the IR brane, F= IR, for 1 1 1ð Þ − 2π ξ2 ð Þ Λ ¼ 107 UV S0 xUV;xIR ig kR T1 xUV;xIR IR GeV. G ðpÞ¼ : λ 2 ð Þ − 2π ξ2 0ð Þ pE T0 xUV;xIR ig kR S1 xUV;xIR 1 3 X ðΔ 2 Þ ¼ − 2 ðϕ Þ ðϕ Þ D 2 ð Þ mϕ 2 g1Y i Y j rϕ mϕ C26b i D 8π 5 j j j We note that this correction (C24) is a special case of the 1 3 2 ≡ − g Yðϕ ÞΔS; ð Þ 8π2 5 1 i C22 general bulk scalar soft mass-squared correction (C1), corresponding to the limit in which the scalar is confined where to the UV brane. D The contribution of the loop integral from supersym- ReiðΠϕ Þξ D ¼ −8π2 i ð Þ rϕ 2 C23 metry breaking is extracted in the propagator difference: i m ϕi GUVðp Þ − GUVðp Þ is positive and depends on the amount of supersymmetry A E λ E 1 1 1 1 breaking as well as the localizations ofpffiffiffiffi the bulk scalars. In ¼ − D 2 3 ð Þ Fig. 22 we plot r as a function of F=Λ , considering p zUVzIR T0 xUV;xIR ϕi IR E four cases for the hypermultiplet localization of the internal 2π ξ2 ig kR ð Þ 1 3 D × 2 2 0 : C27 scalar: c ¼ 0, 2, 4, 1. The largeness of rϕ when the bulk ð Þ − π ξ ð Þ i i T0 xUV;xIR ig kR S1 xUV;xIR hypermultiplet is highly UV-localized is due to the expo- nential smallness of the scalar tree level mass in that case, The loop integral (C25) is finite, unlike the bulk scalar case rather than the absolute magnitude of the correction. (C18), due to the finite separation between the Higgs fields on UV brane and the supersymmetry-breaking sector on the 2. Higgs soft scalar masses IR brane. The resulting contribution to the Higgs soft masses squared can be parametrized in terms of the gaugino When the Higgs fields are confined to the UV brane, they mass as have no direct couplings to the supersymmetry-breaking sector on the IR brane. The Higgs-sector soft mass terms H 2 rg 2 2 are therefore zero at tree level, and are generated instead at ðΔm Þ ¼ 4g CðRH ÞMλ; ðC28Þ Hi g 8π2 i higher loop order by radiative corrections involving bulk fields that transmit the supersymmetry breaking from the where IR brane. Here, we consider the one-loop corrections to the soft mass squared for a generic Higgs field completely ReiΠH H ¼ 8π2 g ð Þ localized on the UV brane. A similar one-loop analysis was rg 2 C29 Mλ performed in the 4D KK formalism for the Higgs sector in unbroken supersymmetry in Ref. [123]. depends on the amount of supersymmetry breaking. We H plot rg in Fig.pffiffiffiffi23 for the U(1) and SU(2) gauge groups as a a. Gauge-sector corrections Λ function of F= IR. This behavior matches the UV-brane The contributions to the Higgs scalar soft masses squared limit of the bulk scalar corrections in Fig. 20 and repro- from the gauge sector arise from loops of bulk gauge duces that found in Ref. [126] up to an order-one shift due

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H H FIG. 23. Plot of the coefficient rg , which parametrizes the one- FIG. 24. Plot of the coefficient ry , which parametrizes the one- loop gauge corrections to the Higgs soft masses squared, for the loop Yukawa corrections to the Higgs soft masses squared, as a U(1) and SU(2) gauge groups as a function ofpffiffiffiffi the relative pfunctionffiffiffiffi of the relative supersymmetry breaking on the IR brane, Λ Λ Λ ¼ 107 supersymmetry breaking on the IR brane, F= IR, for F= IR, for IR GeV. Λ ¼ 107 IR GeV. rH ðΔ 2 Þ ¼ − y 2ð 2 þ 2 Þ ð Þ m 2 y mϕ mϕ ; C33 to a difference in the definition of the supersymmetry- Hi y 8π L R breaking gaugino IR-brane operator and the UV cutoff of where the 4D momentum integration. ReiΠH rH ¼ −8π2 y ðC34Þ b. Yukawa corrections y ðm2 þ m2 Þ ϕL ϕR Similarly, the Higgs soft masses squared receive con- is positive and depends on the amount of supersymmetry tributions via their Yukawa interactions with bulk fermions breaking as well as the localization of thep bulkffiffiffiffi fields. and scalars. At one loop, the corrections each involve two In Fig. 24 we plot rH as a function of F=Λ for bulk fields and take the form y IR three choices of the hypermultiplet localization: cL ¼ −c ¼ 0, 1,1. ðΔm2 Þ ¼ y2ΠH; ðC30Þ R 2 Hi y y where c. D-term corrections   Since, as discussed above, the D-term corrections to the 1 2 ΠH ¼ scalar soft masses squared are independent of the locali- y ˜ ð0Þ ˜ ð0Þ fψ ð0Þfψ ð0Þ zation of the scalar, the Higgs-sector corrections on the Z L R boundary take the same form as the corrections in the bulk: d4p ð UVð Þ UVð Þþ UVð Þ UVð Þ × 4 Gϕ p GF p Gϕ p GF p 3 X ð2πÞ L R R L 2 2 D ðΔm Þ ¼ g1YðHiÞ YðϕjÞΠϕ ; ðC35Þ Hi D 5 j − 2 2 UVð Þ UVð ÞÞ ð Þ j p GψL p Gψ R p : C31 D where Πϕ is defined in (C18). We regulate as discussed in The UV-brane bulk scalar and fermion propagators are given j in (C10) and (C10b). The bulk hypermultiplet auxiliary field Sec. C1c, resulting in a finite negative contribution to the UV Higgs soft masses squared, parametrized in terms of the propagator GF (in the style of Ref. [127]) is likewise evaluated on the UV brane and takes the form scalar mass as 1 3 X UV 2 UV 2 UV ðΔ 2 Þ ¼ − 2 ð Þ ðϕ Þ D 2 ð Þ G ðp Þ¼−p ½G ðp Þξ¼0 ¼ −p Gψ ðp Þ: ðC32Þ mH D 2 g1Y Hi Y j rϕ mϕ ; C36 FL;R E E ϕL;R E E L;R E i 8π 5 j j j

D As with the bulk scalar Yukawa corrections, the con- where rϕ is defined in (C23). tribution to the loop integral from supersymmetry breaking j is extracted in the propagator differences (C12). The resulting contribution to the Higgs soft masses squared 3. Higgs soft b-term is finite and negative and can be parametrized in terms of The Higgs soft b-term, like the Higgs soft masses the scalar masses squared as squared, is zero at tree level, and so is generated at loop

035046-38 PARTIALLY COMPOSITE SUPERSYMMETRY PHYS. REV. D 99, 035046 (2019) order once supersymmetry is broken. In this case, the only is positive and depends on the amount of supersymmetry μ b corrections which contribute at one loop arise in the gauge breakingpffiffiffiffi and . We plot rλ in Fig. 25 as a function of sector, involving one bulk gaugino field and one boundary Λ F= IR for the U(1)pffiffiffiffi (lighter) and SU(2) (darker) gauge Higgsino field Λ ≫ 1 b 1 ξ groups. In the limit F= IR , rλ tends to zero as = . ðΔ Þ ¼ 4 2 að Þ að ÞΠb ð Þ This is a result of the fact that in the twisted limit the b λ g T RHu T RHd λ ; C37 gaugino mass is pure Dirac and the Majoranapffiffiffiffi mixing that að Þ Λ ≪ 1 where T R is the generator of the gauge group in generates the soft coupling disappears. When F= IR , b representation R, and the effect of the supersymmetry breaking saturates, and rλ Z approaches a constant value. 2πkR d4p μ Πb ¼ − GUVðpÞ: ðC38Þ λ k ð2πÞ4 p2 Mλ 4. Trilinear soft scalar couplings (a-terms) The Majorana mass-mixing component of the gaugino The soft a-term interactions, like the Higgs-sector soft UVð Þ propagator, GMλ p , evaluated on the UV brane, is given terms, are zero at tree level but are generated at loop order by [128] once supersymmetry is broken. At one loop, the only nonvanishing corrections arise from loops of bulk gaugi- 1 1 z i GUVðp Þ¼− UV nos, bulk fermions, and Higgsinos: Mλ E 2 2 0ð Þ pE zIR S1 xUV;xIR 1 2 a a a ðΔaÞλ ¼ 4yg ½T ðR ÞT ðRϕ ÞðΠ Þϕ : ð Þ H L λ L × ð Þ − 2π ξ2 0ð Þ C39 T0 xUV;xIR ig kR S1 xUV;xIR a a a þ T ðR ÞT ðRϕ ÞðΠ Þ H R λ ϕR Unlike the previous loop corrections, the b-term one- a a a þ T ðRϕ ÞT ðRϕ ÞðΠλ Þϕ ϕ ; ðC42Þ loop gaugino correction depends explicitly on supersym- L R L R metry breaking through the Majorana gaugino mass, and no complementary loop of superpartners is present. The where resulting contribution is finite and negative, and we para- Z Z ð0Þ ð0Þ μ 4 ˜ ð Þ ˜ ð0Þ metrize it in terms of the Higgsino mass and the gaugino 2πkR d p πR fϕ y fϕ ðΠaÞ ¼ − dy L;R R;L e−3kjyj mass as λ ϕL;R 4 ð0Þ ð0Þ k ð2πÞ −πR ˜ ð0Þ ˜ ð0Þ fψ L;R fψ R;L rb ð 0 Þ ð 0Þ ð Þ λ 2 a a × Gψ p; ;y GMλ p;y; ; C43a ðΔbÞλ ¼ − 4g T ðR ÞT ðR ÞμMλ; ðC40Þ L;R 8π2 Hu Hd Z Z 2πkR d4p πR where ðΠaÞ ¼ − dy λ ϕLϕR 4 k ð2πÞ −πR b Z ð0Þ ð0Þ 2 ReiΠλ π ˜ ð Þ ˜ ð 0Þ b R fϕ y fϕ y 0 rλ ¼ −8π ðC41Þ 0 2 L R −3kjyj −3kjy j μMλ × dy p ð0Þ ð0Þ e e −πR ˜ ð0Þ ˜ ð0Þ fψ L fψ R ð 0 Þ ð 0Þ ð 0 0Þ × GψL p; ;y GMλ p; y; y Gψ R p; y ; ðC43bÞ

are the contributing loop integrals. The bulk gaugino Majorana mass-mixing propagator and the bulk fermion propagator take the forms

1 ðzz0Þ5=2 i G ðp ;y;y0Þ¼− Mλ E 2 4 0ð Þ zUVzIR S1 xUV;xIR iS0ðx ;x ÞS0ðx ;x Þ 1 UV > 1 UV < ; × ð Þ − 2π ξ2 0ð Þ T0 xUV;xIR ig kR S1 xUV;xIR ðC44aÞ b FIG. 25. Plot of the coefficient rλ , which parametrizes the one- loop gaugino corrections to the Higgs soft b-term, as a functionpffiffiffiffi of αð Þ 1 1 Sβ x; xIR the relative supersymmetry breaking on the IR brane, F=Λ , ð 0 Þ¼ ð Þ5=2 ð Þ IR GψL;R pE; ;y zk ; C44b Λ ¼ 107 2 pE Tβðx ;x Þ for IR GeV. UV IR

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a a FIG. 26. Plot of the coefficient ðr Þ , which parametrizes the FIG. 27. Plot of the coefficient ðr Þϕ ϕ , which parametrizes the λ ϕL;R λ L R one-loop gaugino corrections to the trilinear soft scalar couplings, one-loop gaugino corrections to the trilinear soft scalar couplings, as a functionpffiffiffiffi of the relative supersymmetry breaking on the IR as a functionpffiffiffiffi of the relative supersymmetry breaking on the IR Λ Λ ¼ 107 Λ Λ ¼ 107 brane, F= IR,for IR GeV. brane, F= IR,at IR GeV.

a ReiðΠλ Þϕ a 2 L;R where the values z>ð<Þ represent the greater (lesser) of z ðr Þ ¼ −8π ; ðC46aÞ λ ϕL;R and z0. Mλ a As with the soft b-term, this gaugino correction alone ReiðΠλ Þϕ ϕ ð aÞ ¼ −8π2 L R ð Þ depends explicitly on supersymmetry breaking, without the rλ ϕ ϕ C46b L R Mλ presence of the corresponding loop of superpartners. The loop integrals (C43a) and (C43b) are finite, despite their are positive and depend on the amount of supersymmetry extension into the bulk. The resulting correction is negative breaking and the localizations of the bulk hypermultiplets. a a We plot ðrλ Þϕ in Fig. 26 and ðrλ Þϕ ϕ in Fig. 27 as and can be parametrized in terms of the gaugino mass as pL;Rffiffiffiffi L R Λ functions of F= IR for two choices of hypermultiplet 1 1 2 a a a ¼ − ¼ −1 ¼ − ≥ ðΔaÞλ ¼ − 4yg ½ðrλ Þϕ T ðR ÞT ðRϕ Þ localizations: cL cR and cL cR 2 (the cor- 8π2 L H L rection saturates for UV-localized fields). The U(1) (light), a a a þðr Þ T ðR ÞT ðRϕ Þ λ ϕR H R SU(2) (medium), and SU(3) (dark) contributions are given a a a a a in each case. The coefficients ðrλ Þϕ and ðrλ Þϕ ϕ exhibit þðrλ Þϕ ϕ T ðRϕ ÞT ðRϕ ÞMλ; ðC45Þ L;R L R L R L R b 1 ξ pbehaviorffiffiffiffi similar to rλ , vanishing as = in the limit Λ ≫ 1 where F= IR .

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