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PHYSICAL REVIEW D 99, 075015 (2019)

Higgsino dark in an economical Scherk-Schwarz setup

Antonio Delgado,1 Adam Martin,1 and Mariano Quirós1,2 1Department of Physics, University of Notre Dame, 225 Nieuwland Hall Notre Dame, Indiana 46556, USA 2Institut de Física d’Altes Energies (IFAE) and BIST, Campus UAB 08193, Bellaterra, Barcelona, Spain

(Received 14 January 2019; published 15 April 2019)

We consider a minimal natural supersymmetric model based on an extra dimension with breaking provided by the Scherk-Schwarz mechanism. The lightest supersymmetric is a neutral, quasi-Dirac Higgsino and, unlike in previous studies, we assume that all fields are propagating in the bulk. The resulting setup is minimal, as neither extra matter, effective operators, nor extra Uð1Þ groups are needed in order to be viable. The model has three free parameters which are fixed by the Higgsino mass—set to the range 1.1–1.2 TeV so it can play the role of , and by the requirements of correct electroweak breaking and the mass of the Higgs. After imposing the previous conditions we find a benchmark scenario that passes all experimental constraints with an allowed range for the supersymmetric parameters. In particular we have found in the range 2.0–2.1 TeV mass, electroweakinos and sleptons almost degenerate in the range 1.7–1.9 TeV and squarks degenerate in the range 1.9–2.0 TeV. The best discovery prospects are: (i) detection at the high luminosity LHC (≳3 ab−1), and (ii) Higgsino detection at next-generation dark matter direct detection experiments. The model is natural, as the fine-tuning for the fixed values of the parameters is moderate mainly because supersymmetry breaking parameters contribute linearly to the Higgs mass parameter, rather than quadratically as in most models.

DOI: 10.1103/PhysRevD.99.075015

I. INTRODUCTION scenario relies on the fact that its capability to reproduce Ωh2 0 1186 0 0020 In spite of its experimental elusiveness, low-scale super- the observed value of ¼ . . [7] comes symmetry still (arguably) remains as the most complete and from the gauge interactions of the Higgsino multiplet alone 1 and does not require a delicate mixture of different neu- best motivated beyond the Standard Model (BSM) theory. “ ” On top of solving the problem, supersymmetric tralino states (so-called well-tempering ) [8]. theories with R parity conservation have naturally candi- In this paper we will consider a very predictive low-scale dates for thermal dark matter (DM), the . A supersymmetry breaking model with an LSP Higgsino. The number of recent works [1–6] have pointed out that, out of paradigmatic mechanism of natural supersymmetry break- the different spectra, a nearly pure Higgsino ing is the Scherk-Schwarz (SS) twist of boundary con- lightest supersymmetric particle (LSP) with a 1.1–1.2 TeV ditions in a supersymmetric five-dimensional (5D) theory – mass range remains as the most phenomenologically [9 11]. Due to the geometric nature of the SS mechanism, appealing candidate to DM.2 The attractiveness of this the supersymmetry breaking contributions to the Higgs mass are linear, instead of quadratic (as in gravity or gauge

1 mediation). As a result, the fine tuning is proportional to a Although justice should be done to other BSM theories which mass ratio (δm=m), instead of the square mass ratio also show themselves elusive with respect to experimental searches. δ 2 2 2We have encoded, in the given Higgsino mass interval, the ( m =m ), and is thus significantly smaller. Moreover, theoretical uncertainty in the calculation of the thermal relic because of the low-scale character of supersymmetry abundance Ωh2: (i) There is a small mixing effect of the Higgsino breaking, radiative corrections below the compactification and the wino which tends to increase the annihilation cross section; scale are moderate, along with their corresponding con- (ii) There is a small effect of the running of the Higgsino mass between the scale 1=R, where the boundary conditions are set, and tributions to the tuning. Unlike previous studies [12–15], the considered model is the tree level Higgsino mass scale qH=R, by which the Higgsino mass tends to decrease as its beta function is positive. a 5D supersymmetric theory with all matter and gauge fields in the bulk. It has three free parameters: the Published by the American Physical Society under the terms of compactification scale 1=R, the supersymmetry breaking the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to masses proportional to a real parameter qR and a super- the author(s) and the published article’s title, journal citation, symmetric mass in the Higgs sector proportional to another 3 and DOI. Funded by SCOAP . real parameter qH [12–14]. The mass of the lightest KK

2470-0010=2019=99(7)=075015(10) 075015-1 Published by the American Physical Society DELGADO, MARTIN, and QUIRÓS PHYS. REV. D 99, 075015 (2019) states in the Higgsino sector is given by qH=R, and the mass The spectrum, and some experimental prospects to detect of the lightest KK propagating in the bulk is it, is presented in Sec. V. Finally some concluding remarks qR=R. The mass of the Higgs sector depends on both are postponed to Sec. VI. parameters qR and qH. We will fix q so that the Higgsino is the DM particle, i.e., H II. THE MODEL the LSP with a mass of 1.1 to 1.2 TeV. The two other 1 parameters, qR and =R can be set by demanding correct Our starting point is a flat, five-dimensional space where 1 electroweak symmetry breaking and that the mass of the the fifth dimension y is compactified on the orbifold S =Z2, physical Higgs be 125 GeV. We find that for certain with branes at the two fixed points y ¼ 0, πR. We are going 1 values of the parameters in the (qR, =R) plane, the (bulk to embed the SM into a supersymmetric model in 5D propagating) stop sector is capable of radiatively triggering [9–11]. Since the minimal (N ¼ 1) supersymmetry in 5D is electroweak breaking—much as it happens in the minimal the equivalent of N ¼ 2 in four-dimensions (4D), we have supersymmetric standard model (MSSM) for high-scale to incorporate new fields into every multiplet to satisfy this supersymmetry breaking—despite the fact our model has extended algebra. As a result of the orbifold compactifi- a low supersymmetry breaking scale. This is an advantage cation, one can decompose every N ¼ 2 multiplet into two over other similar 5D SS constructions with stops localized N ¼ 1 4D multiplets, each with a definite transformation 0 at the y ¼ brane, where higher dimensional operators with respect to the Z2 symmetry. In particular, (on-shell) localized at the y 0 brane [16,17] or extra triplets in the ¼ vector multiplets in the bulk are V ¼ðV ; Σ; λiÞ ≡ bulk [15] are required in order to drive electroweak sym- M λ1 þ ⊕ Σ λ2 − 1 metry breaking (EWSB). Moreover, integrating out the ðVμ; LÞ ð þ iV5; LÞ where i ¼ , 2 transforms as 2 Z top/stop sector, including all KK-modes, provides a thresh- a doublet of SUð ÞR and the parities under 2 for the two old effect for the Higgs quartic coupling λ. After evolving λ N ¼ 1 multiplets are specified by the superscripts. to the electroweak scale (by the SM radiative corrections) it Similarly there are two bulk Higgs hypermultiplets Ha a Ψa 1 is sufficiently large that the model accommodates a 125 GeV ¼ðHi ; Þ, where the index a ¼ , 2 transforms as 2 Ψa mass for the Higgs. This is again an advantage over similar a doublet of a global group SUð ÞH, and are Dirac 0 Z ≡ σ ⊗ γ constructions with stops localized in the y ¼ brane, where spinors. The parity, 2 3jSUð2Þ 5, decomposition 0 H an extra Uð1Þ [16,17] or singlets and/or triplets [15] had to H2 ≡ 2 Ψ2 þ ⊕ 2 Ψ2 − H1 ≡ 1 Ψ1 þ ⊕ is ðH2; LÞ ðH1; RÞ and ðH1; RÞ be introduced to accommodate the physical value of the 1 Ψ1 − H 2 Ψ2 ðH2; LÞ . As such, the chiral multiplets 2 ¼ðH2; LÞ Higgs mass. The Higgs mass condition will also carve out 1† ¯ 1 and H1 ¼ðH1 ; Ψ Þ have zero modes and play the role of contours in the (qR, 1=R) plane, so that the phenomeno- R logically interesting values of qR and 1=R are given by the the Higgs sector of the MSSM. 1 – – intersection of the Higgs mass (qR, =R) curves with the In Refs. [12 14], the Scherk-Schwarz mechanism [9 11] 1 ⊗ (qR, 1=R) contours from correct electroweak breaking. was used to break supersymmetry by means of a Uð ÞR 1 In short, the simultaneous conditions of a 1.1 to Uð ÞH symmetry. The mass spectrum one gets from this 1.2 TeV Higgsino, a 125 GeV Higgs, and correct procedure depends on the charges ðqR;qHÞ. In fact, only qR electroweak breaking provide a discrete set of values breaks supersymmetry; qH ≠ 0 generates a Higgsino mass for the three parameters (q , q , 1=R)—this is a non- μ R H qH=R, thus providing a solution to the problem of the trivial statement since it was not guaranteed a priori that a MSSM. More specifically, after SS supersymmetry break- viable solution would exist. Moreover, for the q , q ,and H R ing the mass eigenstates are ffiffiffi 1 1 2 p =R values consistent with these conditions, we find (i) Two Majorana λðnÞ ¼ðλ ðnÞ λ ðnÞÞ= 2, solutions that have spectra that are completely compatible L L with masses jqR nj=R. pffiffiffi with current LHC and direct dark matter (ii) Two Dirac Higgsinos H˜ ðnÞ ¼ðΨ1ðnÞ Ψ2ðnÞÞ= 2, searches [18,19]. The net result is a model with three with masses jqH nj=R. free parameters (q , q , 1=R) that is able to reproduce ðnÞ 1ðnÞ 2ðnÞ 1ðnÞ R H (iii) Two Higgses h ¼½H1 þ H2 ∓ ðH2 − the correct electroweak breaking, the correct Higgs mass, 2 H ðnÞ=2,withmassesjq − q nj=R. provide a viable DM candidate, and passes all exper- 1 R H ðnÞ 1ðnÞ− 2ðnÞ∓ 1ðnÞ 2ðnÞ imental bounds. (iv) Two Higgses H ¼½H1 H2 ðH2 þH1 = 2 The plan of this paper goes as follows. In Sec. II we will , with masses jqR þ qH nj=R. introduce in some detail the 5D model and its mass where positive (þn) and negative (−n) modes combine into spectrum. In Sec. III we will describe the conditions on whole towers with n ∈ Z. electroweak symmetry breaking. In Sec. IV we will The main difference with respect to the scenario pro- compute the threshold corrections to the Higgs quartic posed in Refs. [12–14]3 is that we will consider all matter coupling and the physical value of the Higgs mass. In particular we will impose on the light Higgs a mass of 3In Ref. [14] the third generation of and was 125 GeV, according to experimental measurements. localized in the y ¼ 0 brane.

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1 fields propagating in the bulk. As such, matter fields 2 0 ffiffiffiffiffiffiffiffiffi − Q H2ð Þ¼p ðh HÞ; must be represented by hypermultiplets, e.g., L ¼ 2πR ðQ;˜ Q˜ c;qÞ ≡ ðQ;˜ q Þþ ⊕ ðQ˜ c;q Þ− for the SM left- 1 L R ∂ 2 0 ffiffiffiffiffiffiffiffiffi ˆ − ˆ handed top , where only the even chiral multiplet yH1ð Þ¼p ðh HÞð2:3Þ 2πR Q ˜ ˜ ˜ c T L ¼ðQ; qLÞ admits a zero mode and ðQ; Q Þ transforms 2 as a doublet of SUð ÞR. The SS supersymmetry breaking 1 1 2 2 H1ð0Þ¼pffiffiffiffiffiffiffiffiffi ðh þ HÞ; gives squared masses, equal to ðqR nÞ =Rffiffiffi , to the two 2πR ˜ ˜ p complex scalars QðnÞ ¼ðQðnÞ QcðnÞÞ= 2, which then 1 ∂ 1 0 − ffiffiffiffiffiffiffiffiffi ˆ ˆ become a whole tower of complex scalars with n ∈ Z. yH2ð Þ¼ p ðh þ HÞð2:4Þ 2πR Moreover, the SS breaking does not affect the tower n ∈ Z 2 ðnÞ of (SUð ÞR singlet) Dirac q . Their KK modes 1 1 Q˜ ð0Þ¼pffiffiffiffiffiffi Q; U˜ ð0Þ¼pffiffiffiffiffiffi U; instead have mass jnj=R, so that the zero mode is massless πR πR and can be identified with the left-handed SM 1 1 ð0Þ ∂ ˜ c 0 ffiffiffiffiffiffi ˆ ∂ ˜ c 0 ffiffiffiffiffiffi ˆ q (without a Dirac partner). The same logic applies to yQ ð Þ¼p Q; yU ð Þ¼p U ð2:5Þ L πR πR every other SM , e.g., the SM right-handed quark, U U;˜ U˜ c;u ≡ U;˜ u þ ⊕ U˜ c;u − with R ¼ð Þ ð RÞ ð pffiffiffi LÞ where we have used h, H, Q, etc. to stand for their mass eigenstates UðnÞ ¼ðU˜ ðnÞ U˜ cðnÞÞ= 2. Since we corresponding tower of KK modes (or their derivatives) want to recover a chiral theory at the zero model level evaluated at y ¼ 0. Explicitly, we are going to assume there are no masses in the bulk. X X Interactions among hypermultiplets are forbidden in the h ≡ hðnÞ;H≡ HðnÞ; bulk, but are permitted on the branes where the symmetry n n of the theory is reduced to N ¼ 1 supersymmetry. We X q − q n X q q n ˆ ≡ ð R H þ Þ ðnÞ ˆ ≡ ð R þ H þ Þ ðnÞ include the superpotential at the y ¼ 0 brane: h h ; H H n R n R ˆ ˆ ˆ W ¼ðhtQLH2UR þ hbQLH1DR þ hτLLH1ERÞ δðyÞ; ð2:6Þ ð2:1Þ and ˆ X X where hb;τ;t are the 5D bottom, and top Yukawa Q ≡ QðnÞ;U≡ UðnÞ; couplings. This superpotential will generate mass terms n n for the zero mode fermions once electroweak symmetry is X X ˆ ðqR þ nÞ ˆ ðqR þ nÞ broken. Supersymmetry demands that the Yukawa inter- Q ≡ QðnÞ; U ≡ UðnÞ: ð2:7Þ R R actions present in Eq. (2.1) are accompanied by several n n other interactions among the scalar . To see With these definitions, the potential at y ¼ 0 becomes the full set of scalar interactions, we first integrate out the ˆ auxiliary fields. Neglecting the small effects of h τ, the b; V ¼ h ½ðh − HÞQUˆ þðhˆ − Hˆ ÞQU þðh − HÞQUˆ þ H:c: quartic 4D potential can be written as [20]: t 2 2 2 2 þ ht ½jðh − HÞQj þjQUj þjðh − HÞUj πRδð0Þ; ˆ 2 ˜ 0 2 0 2 ˜ 0 ˜ 0 2 ˜ 0 2 0 2 δ 0 V ¼ ht ðjQð ÞH2ð Þj þjQð ÞUð Þj þjUð ÞH2ð Þj Þ ð Þ ð2:8Þ ˆ ˜ 0 2 0 ∂ ˜ c 0 ˜ 0 ˜ 0 ∂ 2 0 þ htðQð ÞH2ð Þ yU ð ÞþQð ÞUð Þ yH1ð Þ pffiffiffiffiffiffiffiffiffiffiffiffi h hˆ = 2π3R3 ˜ 0 2 0 ∂ ˜ c 0 where t ¼ t is the (4D) SM top-Yukawa þ Uð ÞH2ð Þ yQ ð ÞþH:c:Þ; ð2:2Þ coupling. The other interaction from Eq. (2.1) that we will P need is the Yukawa coupling: π δ 0 ≡ 1 ˜ 0 2 0 where R ð Þ n and Qð Þ, H2ð Þ, etc. are the values of the wave functions for the entire KK tower of Q, h − H, ˆ ffiffiffiffiffiffiffiffiffiffiffiffiht ¯ ¯ etc. on the brane. This potential depends on even fields and LY ¼ p ðh− HÞqLuR þ H:c: ≡ htðh− HÞqLuR þ H:c: 2π3R3 the derivative (∂y) of odd fields. The origin of these ∂y terms resides in the fact that auxiliary components of ð2:9Þ off-shell 5D multiplets localized at the brane are given by the auxiliary field of the corresponding even component where h − H represents the full tower of fields given by ∂ Eq. (2.6), and, similarly, q and u stand for the coherent minus the derivative y of the scalar field of the odd L R P ≡ ðnÞ component [20]. sums of fermionic states, e.g., uL;R nuL;R. Working with mass eigenstates and pulling out normali- With the superpotential set, the model is completely zation factors, the fields in Eq. (2.2) become: determined (apart from the SM couplings) by three different

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1 1 free parameters, qR, qH and =R. We will fix these param- (iv) Finally for scales above =R the theory becomes 5D eters in the following way: first, we will require the lightest and all KK modes start to propagate. neutral Higgsino to be the dark matter. This not only sets the hierarchy qR >qH, as the lightest Higgsino needs to be the LSP,but requiring that the Higgsino achieves the correct relic III. ELECTROWEAK BREAKING abundance also fixes its mass, qH=R ≃ 1.1–1.2 TeV [6,8]. To set the other two parameters, we impose: (i) Correct As explained earlier, in our 5D SS model the SM Higgs H ð0Þ ≤ 1 2 electroweak symmetry breaking, and (ii) A physical Higgs field is identified with h (as we assume qR = ). 0 boson mass of 125 GeV. Both of these conditions can be While the spectrum contains a second “heavy” Higgs H , 1 ð−1Þ expressed as curves in the (qR, =R) plane, so our final identified with the mode H , the large hierarchy between parameter points will be given by the intersection of the (qR, the SM Higgs and the heavy Higgs sector (including the 1=R) curves from the electroweak breaking requirement with KK modes) means we should immediately integrate out the those from the Higgs mass condition. heavy Higgses. The resulting low energy Higgs potential Before detailing the electroweak breaking and Higgs contains only one Higgs doublet H, as in the SM: mass conditions, let us review the general spectrum given the hierarchy qR >qH: V ¼ m2jHj2 þ λjHj4 ð3:1Þ (i) ForscalesbelowqH=R the theory is just the SM. For ≤ 1 2 H ≡ ð0Þ qR = , the SM Higgs h . This mode has a 2 2 − 2 2 This potential yields mH ¼ 2λv , where v ¼ 246 GeV, (tree level) mass squared of ðqR qHÞ =R ,sothat 2 mH ≃ 125 GeV. These inputs fix the numerical value of m tuning qR ¼ qH, as we have done in previous work ffiffiffi 2 p 2 2 [14], makes it massless. However, in this paper wewill for correct EWSB to m ¼ −ðmH= 2Þ ≃−ð88.4 GeVÞ . set the parameters by the condition of electroweak Therefore, the condition of electroweak breaking boils breaking and a 125 GeV Higgs. For parameter sets down to imposing this number on the model and thereby ð0Þ 1 with q ≃ q (q < 1=2), h is light and identified selecting out viable values of the inputs qR and =R. R H R 2 with the SM Higgs. In this case, the mode H0 ≡ Hð−1Þ As we have seen in the previous section, m receives a 2 is identified as the second, “heavy” Higgs, with a mass tree level contribution m0,as 2 2 squared equal to ðqR þ qH − 1Þ =R . If we instead chose q > 1=2, the spectrum stays 2 − 2 2 R m0 ¼ðqR qHÞ =R : ð3:2Þ the same, but the identification of the lightest states shifts.4 Specifically, Hð−1Þ is lighter than hð0Þ and is This mass (squared) is positive definite so that a negative identified with the SM Higgs, and the lightest sfer- 2 mions and gauginos correspond to the n ¼ −1 mode value of m must be induced radiatively. Radiative con- instead of the n ¼ 0 mode. For definiteness, we will tributions coming from gauge interactions, computed in 1 2 Ref. [14], are positive and cannot trigger electroweak assume qR < = from now on. breaking. They are given by Note that the particular case qR ¼ qH ¼ 1=2 would make both hð0Þ and Hð−1Þ massless, in which case the Higgs sector would contain two light doublets, a 3 2 2 Δ 2 g þ gY 9Δ 2 0 3Δ 2 configuration disfavored by present Higgs data except gm ¼ 192π4 ½ m ð Þþ m ðqR qHÞ in the “alignment limit” [21–23].Aswewillsee,wedo − 6Δ 2 − 6Δ 2 1 2 m ðqRÞ m ðqHÞ; ð3:3Þ not have to worry about the qR ¼ qH ¼ = case as it does not fulfill the required conditions of electroweak breaking and correct value of the Higgs mass. where the plus sign corresponds to the hð0Þ square mass, ð−1Þ (ii) For scales between qH=R and qR=R the only extra and the minus sign to that of H , and (on top of the heavy Higgses) are the Dirac Higgsinos with a mass qH=R. 1 2 2πiq (iii) Gauginos, sfermions and the are degenerate Δm ðqÞ¼ ½Li3ðe ÞþH:c:; ð3:4Þ 2R2 at the mass qR=R, although their masses show some splitting due to electroweak breaking contributions P ∞ k n and radiative corrections in the 4D theory below the where LinðxÞ¼ k¼1 x =k are polylogarithm functions. compactification scale. Therefore, between the scales On the other hand, radiative corrections from the Yukawa qR=R and 1=R the theory resembles the MSSM. coupling in Eq. (2.1), are negative and originate from the diagrams in Fig. 1. The result is a finite, negative definite threshold correction that can be thought of as the result of 4The hð0Þ and Hð−1Þ masses are related by the symmetry integrating out the squark and quark KK modes. The → 1 − H H0 qR qR. correction is common for both and , and is given by:

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2.5

2.0

2 2 FIG. 1. Diagrams proportional to ht contributing to ΔtmH. ˆ ˆ Note that in the loops full towers of , Q, U, Q, U, and 1.5 fermions qL and uR, are exchanged. /R (TeV) R q 3 2 μ 2 ht ð Þ 2πiq 2πiq Δ m ¼ ½3Li3ðe R Þ − 3i cotð2πq ÞLi4ðe R Þ t 32π4R2 R 1.0 − 2ζð3ÞþH:c:: ð3:5Þ

To fix qR and 1=R, we first match the high-energy and 0.5 0.15 0.20 0.25 0.30 0.35 0.40 0.45 low-energy theories at the scale μ0 ¼ q =R, the scale where R q we integrate out the stop zero mode. In principle, the net R effect of the matching is that the coupling in Eq. (3.5) FIG. 2. Cyan bands are the electroweak breaking conditions, should be the top-Yukawa, evaluated with SM field content for 1.1 TeV ≤ qH=R ≤ 1.2 TeV. The shadowed gray areas are 2 μ 2 0 only, at qR=R. In practice, the SM running of m ð Þ the corresponding (excluded) regions where mH0 < . The red 2 SM between mt and qR=R has only a small effect on m so thick solid line is the condition that λ ¼ Δλ at the matching μ we neglect it (this will not be the case when we examine the scale 0 ¼ qR=R. λ μ μ Higgs quartic ð Þ). With fixed to qR=R by the matching condition and qH=R fixed to the values between 1.1 and effects which we are not considering in this paper, therefore 1.2 TeV, the net tree-plus-loop Higgs mass is a function of we will ignore it in our numerical calculations. 1 qR and =R alone. 2 1 − 88 4 2 Setting now m ðqR; =RÞ¼ ð . GeVÞ , we can IV. THE HIGGS MASS solve for the regions where EWSB is correctly achieved. q The result, plotted in the (qR, qR=R) plane, is shown in The second condition we impose to nail down R and Fig. 2 below (red solid lines). As we vary the Higgsino 1=R is a physical Higgs mass of 125 GeV. The Higgs mass mass qH=R between 1.1–1.2 TeV, the EWSB curves turn in this theory is determined by the value of the quartic λ λ0 Δλ λ0 into a band (qH=R ¼ 1=2 TeV is the upper boundary, coupling ¼ þ , where is the tree level value and 1.1 TeV is the lower one). For every case, we have to Δλ is the loop contribution. The quartic coupling at tree impose that the heavy Higgs H0 is not tachyonic and heavy level vanishes, λ0 ¼ 0 [12], so that the whole Higgs mass is enough to justify our use of the decoupling limit.5 Regions controlled by the radiative piece. Here we will approximate where this condition, imposed on the full tree plus loop Δλ by the dominant contribution, coming from diagrams level H0 mass, fail are shaded gray in Fig. 2. The red thick involving the top Yukawa. The relevant diagrams are ˆ ˆ line comes from imposing the correct Higgs mass as we shown in Fig. 3, where as before Q, U, Q, U, qL and will see in the next section. Before moving on, there is one subtlety in the Δm2 calculations that we would like to mention. The diagrams in Fig. 1 exist between any two Higgs external states in the KK tower—n ¼ 0 in and n ¼ 0 out, as well as for n ¼ 1 in and n ¼ 0 out, n ¼ 2 in and n ¼ 0 out, etc. As a result, the mass matrix for the KK Higgs states is not diagonal, with 2 2 off diagonal entries all Δtm (in addition to Δtm con- tributions to the diagonal terms). Diagonalizing this mass matrix, the zero mode mass squared eigenvalue shifts by 2 2 2 O½ðΔtm Þ R . However, we have checked that this effect is small and, as it is parametrically comparable to two-loop

5The potential for the heavy Higgs is very steep. Therefore, we 2 0 use the condition mH0 > to approximate where the heavy Higgs 4 is in the decoupling limit. FIG. 3. Diagrams proportional to ht contributing to Δλ.

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2.5 0.065 0.065 2.5 0.1 0.02 0.055

0.11 2.0 2.0 0.055 0.04 ) )

0.12 TeV 1.5 0.05 TeV 1.5 /R ( /R (

R 0.06 R

q 0.06 0.06 q

0.09

1.0 0.045 1.0 0.14 0.08

0.15 0.04 0.5 0.1 0.5 0.13 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.15 0.20 0.25 0.30 0.35 0.40 0.45 qR qR

Δλ μ Δλ μ μ FIG. 4. Left panel: Contour lines of logð 0Þ (blue solid lines) and thð 0Þ (red dashed lines) in the plane (qR, qR=R)for 0 ¼ qR=R. Δλ μ Right panel: Contour lines of ð 0Þ (red thick solid lines) in the plane (qR, qR=R).

uR correspond to whole KK towers of states. The final [24]. In our setup here, the scale of all contributions is fixed expression is given by the Euclidean momentum integral by the compactification scale 1=R, which explains the Z origin of the argument of the logarithm in Eq. (4.4). 3 4 μ ∞ Δλ 1 Δλ ht ð Þ 7 4 0 − 4 The remaining term th has no explicit =R depend- ¼ 2 p ½s ðp; Þ s ðp; qRÞdp; ð4:1Þ 8π 0 ence and is therefore finite (as R → 0). The only 1=R,orμ, dependence is implicit, in the scale where we evaluate the μ where we will fix to the matching scale and the function top Yukawa coupling htðμÞ As such, Δλth is a genuine sðp; qÞ is defined as threshold correction. The form of Δλth is reminiscent of the stop threshold effect in the MSSM, with qR=R playing πR sinh 2pπR ð Þ the role of the left-right mixing parameter At [24–26]. sðp; qÞ¼ 2 π − 2π : ð4:2Þ p½coshð p RÞ cosð qÞ This qR=R → At identification can better understood by expanding out the first line of Eq. (2.8) and focusing As in the previous section, we will set the matching scale to on the zero modes. Among the terms, we find the tri- μ0 q =R where the high energy and low energy theories ð0Þ ð0Þ ¼ R linear ½3ðq =RÞ − q =RHQ ðU Þ þ H:c. coincide. Notice that, for consistency, we have omitted any R H L R The relative weights of Δλ ðμ0Þ and Δλ ðμ0Þ are effect from the mixing among the heavy KK modes as this log th shown in the left panel or Fig. 4. As expected, the value would correspond to a (small) two-loop effect. Δλ of th (red dashed lines) increases when qR increases The integral (4.1) is UV convergent. However, it has a Δλ (keeping fixed qR=R), while log increases when qR logarithmic IR divergence originating from the contribution 1 decreases (for fixed qR=R) as the value of =R increases. of the (massless) top quark zero mode in the loop. This Δλ divergence should be regularized by introducing an IR In this way, the effect is dominated by log for low values of qR, while for large values of qR it is dominated by Δλth. cutoff at the scale mt. A careful inspection reveals that Δλ, can be decomposed as In the right panel of Fig. 4 we show contour lines of the total Δλ (red solid lines). As we can see, the value of Δλ Δλ μ Δλ μ Δλ μ increases with increasing q as the threshold corrections ð 0Þ¼ logð 0Þþ thð 0Þð4:3Þ R become more important in this region. where the IR cutoff and compactification scale are lumped Note that our procedure is conservative. Had we fixed the μ together into one term: matching scale at the top quark mass ( ¼ mt), we would have considered the renormalization group running of the 4 1 3ht ðμ0Þ quartic coupling between the scales =R and mt in the one- Δλ μ0 − m R : : logð Þ¼ 8π2 logð t Þ ð4 4Þ loop approximation. As shown in MSSM Higgs studies [24–26], one-loop running from the cut-off to mt over- From the form of Eq. (4.4), we see that Δλlog is reminiscent shoots the Higgs mass, yielding a larger value than the of a similar contribution which appears in the MSSM, result if all large logarithms are resummed by renormal- where the role of logðmtRÞ is played by logðmt=MSUSYÞ ization group techniques.

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TABLE I. Range from Fig. 2 that satisfies the conditions of correct electroweak breaking for a single Higgs field, the correct value of the Higgs mass at 125 GeV, and the Higgsino with a mass in the range between 1.1 and 1.2 TeV being the LSP. The supersymmetric parameters for the first (second) row is the lower (upper) limit of the range. qR qH 1=R (TeV) qR=R (TeV) qH=R (TeV) Mg˜ (TeV) mH0 (TeV) 0.31 0.2 5.5 1.7 1.1 2.0 2.7 0.31 0.2 5.9 1.9 1.2 2.1 2.9

Finally, the matching condition is then given by:6 (ii) In the scalar Higgs sector we have the SM Higgs H, whose mass has been fixed to the experimental value 2 H0 SM mH 125 GeV, and a heavy inert Higgs doublet .The Δλðμ0Þ¼λ ¼ : ð4:5Þ 2v2 four scalars in the inert doublet—one CP-even, one CP-odd, two charged—are degenerate and with a Using this condition (4.5) with mH and v fixed to their mass in the range 2.7 TeV

m3=2 ¼ qR=R: ð5:1Þ 6As we have already integrated out the top quark in the contribution of the corresponding tower to Eq. (4.2), and in the approximation we are considering where we are neglecting Current searches of gluinos put a bound on their the contribution from the gauge g and gY , and quartic λ, mass around 2 TeV for the case of a 1.1 TeV neutralino SM couplings, λ is given by its value at the scale mt. [29,30]. However, that analysis assumes a 100% branching

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7 ¯ 0 rate g˜ → bb þ χ1, while the degeneracy of the squark strength of the LSP-LSP-Higgs vertex that drives spin- ¯ 0 8 spectrum in our model means BRðg˜ → bb þ χ1Þ ∼ 1=6. independent scattering rate off nuclei. The LSP for our Naively recasting the excluded cross section from [29,30] benchmark points is around 99% pure Higgsino—a result of into a “democratically decaying gluino” scenario, we find the large wino/bino mass—so the spin-independent nuclear −10 the gluino mass bound relaxes to 1.7 TeV for a LSP of cross section for the benchmark range is ∼10 pb [6]; thus, 1.1 TeVand essentially no bound for a 1.2 TeV LSP. Taking the whole range escape the current limit from XENON-1T into account that the pole mass of the gluino in our model is [44]. However, as shown in Ref. [6], Higgsino LSPs of this in the rage between 2.0 TeV and 2.1 TeV we find that our purity will be accessed in the next generation experiments model is safe from current LHC searches. like XENON-nT or LZ. The model may be probed at the HL-LHC and in future colliders. The best place to look for a signal at VI. CONCLUSION the LHC will be in classic gluino pair production channels: → ˜ ˜ → 2 2¯ 2χ In this paper, we have presented an economical, pre- pp g g q q þ . Of these, gluinos that decay dictive supersymmetric model where (5D) supersymmetry g˜ g˜ → into third generation quarks plus missing energy, is broken by the Scherk-Schwarz mechanism. The model 2 2¯ 2 2¯ ¯ ¯ b b þ ET, t t þ ET, bbtt þ ET, etc. provide the most 1 has three free parameters: qR, qH and =R; or, equivalently experimental handles (third generation tags, leptons from t the mass of gauginos and sfermions, q =R, the Higgssino decay, etc) and should be the most effective. Reference [31] R mass qH=R and the KK mass 1=R. The conditions for explored a spectrum similar to ours and showed electroweak breaking, and a physical mass of that the reach of the HL-LHC would be around 2.5 TeV 1 3 −1 125 GeV fix qR=R and =R, while the Higgsino mass is set (after ab ) for our neutralino mass range in the top plus in the range between 1.1 TeV and 1.2 TeV to reproduce the missing energy channel, though as in Ref. [29] this limit observed DM relic abundance (thus fixing q =R). We find came from assuming gluinos decay 100% of the time to H a range in the parameters space that can reproduce only one quark-squark flavor. the aforementioned conditions, which correspond to Of special interest is the LSP Higgsino, in the range sparticle masses in the range between ∼1.7 TeV (for between 1.1 and 1.2 TeV. Higgsino LSPs are best probed at qH=R ¼ 1.1 TeV) and ∼1.9 TeV (for qH=R ¼ 1.2 TeV). colliders through the production of their chargino and heavier By considering that the corresponding range of pole gluino → χχ0 neutralino cousins pp 2, which subsequently decay masses is between 2.0 TeV and 2.1 TeV, the whole range of χ0 back to the LSP 1. However, in our setup the entire Higgsino points seem to pass all experimental bounds. multiplet is highly degenerate, mχ − mχ0 ≲ 4 GeV. A ∼ 1 Moreover there is no chance of detecting the Higgsino 4 GeV splitting is sufficiently large that the decays will be LSP at the LHC as the neutralino/chargino components are prompt and therefore techniques based on displaced vertices highly degenerate. The best chance for discovery is instead [32–36] do not apply. At the same time, a 4 GeV splitting is at next-generation direct detection experiments for dark 0 matter like XENON-nT or LZ. The LHC prospects for the small enough that the particles emitted χ , χ2 decays to the LSP are too soft to pass triggering and identification require- gluino are better, as studies of similar spectra project sensitivity to ∼2.5 TeV after an integrated luminosity ments. If the entire chargino/neutralino system is boosted, the 3 −1 decay products inherit this boost and can be pushed above around ab at the HL-LHC. One of the main features of the SS supersymmetry trigger/identification thresholds, though a large boost breaking mechanism is that masses m ¼fq =R; q =Rg requires a hard object for the chargino/neutralino system i R H contributing to the Higgs mass term m2jHj2, are added to recoil against and significantly decreases the production 0 linearly as in Eq. (3.2), and not quadratically as in other rate [37–40]. The net result is that 1.1 to 1.2 TeV Higgsino mechanisms of supersymmetry breaking such as gravity or discovery at the LHC is essentially impossible due to the gauge mediation. As a consequence of this linear behavior, small cross section of sufficiently boosted Higgsino the fine tuning according to the sensitivity definition [45] pairs. However, at a collider with more energy, the signal cross section is higher and it may be possible to create ∂ 2 ∂ Δ log m0 log m0 mi sufficient amounts of highly boosted /neutralinos i ¼ 2 ¼ ¼ ð6:1Þ ∂ log m ∂ log m m0 for discovery [41–43]. i i A better option for discovering the Higgsino LSP is via is milder. In particular for the benchmark points of Table I dark matter direct detection experiments [41,42]. The the fine-tuning among scales qR=R and qH=R is around detection prospects depend strongly on the bino/wino ðqR − qHÞ=qR ∼ 0.3, while for conventional mechanisms admixture in the LSP, as that admixture controls the where contributions are added quadratically, the tuning

7˜ ¯ 8 g → bb þ ET is the most sensitive gluino decay channel. All Higgsino LSPs can be detected via their spin-dependent other channels were considered in Refs. [29,30], each with 100% scattering off nuclei, although the prospects there are not as branching fraction. good [6].

075015-8 HIGGSINO DARK MATTER IN AN ECONOMICAL SCHERK- … PHYS. REV. D 99, 075015 (2019) would be ≲1%. We can also compare the fine-tuning in our section is too weak and the thermal Higgsino relic density model with the tuning in a Scherk-Schwarz scenario with would overclose the universe. One way to make this heavier the stop localized on the brane. In the latter case the stop scenario viable is to introduce a small source of R-parity squared mass induced by the Scherk-Schwarz mechanism breaking—small enough to be consistent with collider is generated at one-loop level [14] and therefore it con- bounds but enough to make the Higgsino decay in the tributes to the squared H2 Higgs mass at two-loop level. early universe. Of course, in that case, an alternative For the values of the parameters quoted in Table I the stop candidate to dark matter should be provided by the theory. mass would typically be mt˜ ≃ 0.5 TeV, and would provide a similar fine-tuning to the one we have obtained in our model with a much heavier value of the stop mass. ACKNOWLEDGMENTS Finally, while we have focused on the scenario where the The work of A. D. and A. M. is partly supported by the Higgsino mass is set from the outset, it is worth considering National Science Foundation under Grant No. PHY-1820860. what happens if we drop this requirement. It is surprising The work of M. Q. is partly supported by Spanish Ministerio that the present bounds on the gluino mass are set by de Ciencia, Innovación y Universidades (MINEICO) (Grants Higgsino masses ≳Oð1Þ TeV, in the ballpark where they No. CICYT-FEDER-FPA2014-55613-P and No. FPA2017- reproduce the required thermal relic density. For smaller 88915-P), by the Catalan Government under Grant Higgsino masses a second dark matter component would be No. 2017SGR1069, and Severo Ochoa Excellence needed. By the same logic, increasing the Higgsino mass Program of MINEICO (Grant No. SEV-2016-0588). while maintaining its role as the LSP, the annihilation cross

[1] G. B. Gelmini, Light weakly interacting massive particles, [13] I. Antoniadis, S. Dimopoulos, A. Pomarol, and M. Rep. Prog. Phys. 80, 082201 (2017). Quiros, Soft masses in theories with supersymmetry [2] G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, breaking by TeV compactification, Nucl. Phys. B544, M. Pierre, S. Profumo, and F. S. Queiroz, The waning of the 503 (1999). WIMP? A review of models, searches, and constraints, Eur. [14] A. Delgado, A. Pomarol, and M. Quiros, Supersymmetry Phys. J. C 78, 203 (2018). and electroweak breaking from extra dimensions at the TeV [3] T. Plehn, Yet another introduction to dark matter, scale, Phys. Rev. D 60, 095008 (1999). arXiv:1705.01987. [15] A. Delgado, M. Garcia-Pepin, G. Nardini, and M. Quiros, [4] R. Krall and M. Reece, Last electroweak WIMP standing: Natural supersymmetry from extra dimensions, Phys. Rev. Pseudo-Dirac Higgsino status and compact stars as future D 94, 095017 (2016). probes, Chin. Phys. C 42, 043105 (2018). [16] S. Dimopoulos, K. Howe, and J. March-Russell, Maximally [5] L. Roszkowski, E. M. Sessolo, and S. Trojanowski, Natural Supersymmetry, Phys. Rev. Lett. 113, 111802 WIMP dark matter candidates and searchescurrent (2014). status and future prospects, Rep. Prog. Phys. 81, 066201 [17] I. Garcia Garcia, K. Howe, and J. March-Russell, Natural (2018). Scherk-Schwarz theories of the weak scale, J. High Energy [6] K. Kowalska and E. M. Sessolo, The discreet charm of Phys. 12 (2015) 005. higgsino dark matter—a pocket review, Adv. High Energy [18] M. Aaboud et al. (ATLAS Collaboration), Search for Phys. 2018, 1 (2018). squarks and gluinos in final states with hadronically decaying τ-leptons, jets, and missing transverse momentum [7] P. A. R. Ade et al. (Planck Collaboration), Planck 2015 pffiffiffi results. XIII. Cosmological parameters, Astron. Astrophys. using pp collisions at s ¼ 13 TeV with the ATLAS 594, A13 (2016). detector, Phys. Rev. D 99, 012009 (2019). [8] N. Arkani-Hamed, A. Delgado, and G. F. Giudice, The [19] A. M. Sirunyan et al. (CMS Collaboration), Search for new well-tempered neutralino, Nucl. Phys. B741, 108 phenomena with the M 2 variable in the all-hadronic final T pffiffiffi (2006). state produced in -proton collisions at s ¼ 13 TeV, [9] J. Scherk and J. H. Schwarz, Spontaneous breaking of Eur. Phys. J. C 77, 710 (2017). supersymmetry through dimensional reduction, Phys. Lett. [20] E. A. Mirabelli and M. E. Peskin, Transmission of super- 82B, 60 (1979). symmetry breaking from a four-dimensional boundary, [10] J. Scherk and J. H. Schwarz, How to get masses from extra Phys. Rev. D 58, 065002 (1998). dimensions, Nucl. Phys. B153, 61 (1979). [21] N. Craig, J. Galloway, and S. Thomas, Searching for signs [11] M. Quiros, New ideas in symmetry breaking, arXiv:hep-ph/ of the second Higgs doublet, arXiv:1305.2424. 0302189. [22] M. Carena, I. Low, N. R. Shah, and C. E. M. Wagner, [12] A. Pomarol and M. Quiros, The standard model from extra Impersonating the standard model Higgs boson: Alignment dimensions, Phys. Lett. B 438, 255 (1998). without decoupling, J. High Energy Phys. 04 (2014) 015.

075015-9 DELGADO, MARTIN, and QUIRÓS PHYS. REV. D 99, 075015 (2019)

[23] H. E. Haber, S. Heinemeyer, and T. Stefaniak, The impact of [34] V. Khachatryan et al. (CMS Collaboration), Search for long- two-loop effects on the scenario of MSSM Higgs alignment lived neutral particles decayingpffiffiffi to quark-antiquark Pairs in without decoupling, Eur. Phys. J. C 77, 742 (2017). proton-proton collisions at s ¼ 8 TeV, Phys. Rev. D 91, [24] H. E. Haber, R. Hempfling, and A. H. Hoang, Approximat- 012007 (2015). ing the radiatively corrected Higgs mass in the minimal [35] R. Mahbubani, P. Schwaller, and J. Zurita, Closing the supersymmetric model, Z. Phys. C 75, 539 (1997). window for compressed Dark Sectors with disappearing [25] M. Carena, J. R. Espinosa, M. Quiros, and C. E. M. Wagner, charged tracks, J. High Energy Phys. 06 (2017) 119; Analytical expressions for radiatively corrected Higgs Erratum 10 (2017) 61. masses and couplings in the MSSM, Phys. Lett. B 355, [36] J. Liu, Z. Liu, and L. T. Wang, Long-lived Particles at the 209 (1995). LHC: Catching Them in Time, arXiv:1805.05957 [Phys. [26] M. Carena, M. Quiros, and C. E. M. Wagner, Effective Rev. Lett. (to be published)]. potential methods and the Higgs mass spectrum in the [37] G. F. Giudice, T. Han, K. Wang, and L. T. Wang, Nearly MSSM, Nucl. Phys. B461, 407 (1996). degenerate gauginos and dark matter at the LHC, Phys. Rev. [27] M. Cirelli, N. Fornengo, and A. Strumia, Minimal dark D 81, 115011 (2010). matter, Nucl. Phys. B753, 178 (2006). [38] P. Schwaller and J. Zurita, Compressed electroweakino [28] S. P. Martin, Refined gluino and squark pole masses beyond spectra at the LHC, J. High Energy Phys. 03 (2014) 060. leading order, Phys. Rev. D 74, 075009 (2006). [39] Z. Han, G. D. Kribs, A. Martin, and A. Menon, Hunting [29] M. Aaboud et al. (ATLAS Collaboration), Search for quasidegenerate Higgsinos, Phys. Rev. D 89, 075007 supersymmetry in final states with missing transverse (2014). momentum and multiple b-jets in proton-proton collisions [40] C. Han, L. Wu, J. M. Yang, M. Zhang, and Y. Zhang, New pffiffiffi at s ¼ 13 TeV with the ATLAS detector, J. High Energy approach for detecting a compressed bino/wino at the LHC, Phys. 06 (2018) 107. Phys. Rev. D 91, 055030 (2015). [30] A. M. Sirunyan et al. (CMS Collaboration), Search for [41] J. Bramante, P. J. Fox, A. Martin, B. Ostdiek, T. Plehn, T. supersymmetry in multijet events with missing transverse Schell, and M. Takeuchi, Relic neutralino surface at a momentum in proton-proton collisions at 13 TeV, Phys. Rev. 100 TeV collider, Phys. Rev. D 91, 054015 (2015). D 96, 032003 (2017). [42] J. Bramante, N. Desai, P. Fox, A. Martin, B. Ostdiek, and T. [31] L. Shchutska (CMS Collaboration), Prospects for BSM Plehn, Towards the final word on neutralino Dark Matter, searches at the high-luminosity LHC with the CMS detector, Phys. Rev. D 93, 063525 (2016). Nucl. Part. Phys. Proc. 273–275, 656 (2016). [43] M. Low and L. T. Wang, Neutralino dark matter at 14 TeV [32] J. L. Feng, T. Moroi, L. Randall, M. Strassler, and S.-f. Su, and 100 TeV, J. High Energy Phys. 08 (2014) 161. Discovering Supersymmetry at the Tevatron in Wino LSP [44] E. Aprile et al. (XENON Collaboration), First Dark Matter Scenarios, Phys. Rev. Lett. 83, 1731 (1999). Search Results from the XENON1T Experiment, Phys. Rev. [33] J. F. Gunion and S. Mrenna, A study of SUSY signatures at Lett. 119, 181301 (2017). the Tevatron in models with near mass degeneracy of the [45] R. Barbieri and G. F. Giudice, Upper bounds on lightest chargino and neutralino, Phys. Rev. D 62, 015002 supersymmetric particle masses, Nucl. Phys. B306,63 (2000). (1988).

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