Cosmologically Viable Low-Energy Supersymmetry Breaking

PHYSICAL REVIEW D 98, 115036 (2018)

Cosmologically viable low-energy supersymmetry breaking

Anson Hook* Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, Maryland 20742, USA † Robert McGehee Department of Physics, University of California, Berkeley, California 94720, USA and Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA ‡ Hitoshi Murayama Department of Physics, University of California, Berkeley, California 94720, USA Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA and Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, Japan

(Received 14 September 2018; published 28 December 2018)

A recent cosmological bound on the gravitino mass, m3=2 < 4.7 eV, together with LHC results on the Higgs mass and direct searches, excludes minimal gauge mediation with high reheating temperatures. We discuss a minimal, vector-mediated model which incorporates the seesaw mechanism for neutrino masses, allows for thermal leptogenesis, ameliorates the μ problem, and achieves the observed Higgs mass and a gravitino as light as 1–2eV.

DOI: 10.1103/PhysRevD.98.115036

I. INTRODUCTION For most of the gravitino-mass range, if it is the stable Supersymmetry (SUSY) is a well motivated theory lightest supersymmetric particle (LSP), it overcloses the beyond the standard model. It offers a solution to the Universe without an unnaturally low reheating temperature hierarchy problem and allows for gauge unification (see, [6,7]. Such a low reheating temperature makes baryo- e.g., [1]). Models with SUSY breaking at low energies are genesis difficult as well. In particular, thermal leptogenesis 109 especially interesting because they imply light gravitinos requires TR > GeV [8]. A very light gravitino, 0 24 which can be produced at colliders, enabling experimental m3=2 < . keV, does not overclose the Universe even tests of the SUSY-breaking mechanism [2]. Gauge media- when it is thermalized. But it constitutes a hot (or warm) tion is the most studied way to do this [3–5]. dark matter component and suppresses the structure of the However, the window on the usual gauge mediation is Universe at small scales. According to a very recent study, quickly closing as cosmological and LHC bounds eliminate cosmic microwave background (CMB) lensing and cosmic parameter space from both ends. Decreasing upper bounds shear constrain its mass to be m3=2 < 4.7 eV [9]. Figure 1 on the gravitino mass from cosmological data arepffiffiffiffi decreasing summarizes these cosmological bounds on m3=2, as well as the upper boundpffiffiffi on the SUSY-breaking scale, F, because others, and demonstrates the shrinking parameter space for ¼ 3 5 m3=2 F= MPl. Increasing bounds on the gaugino any models with gravitinos lighter than roughly 10 GeV. masses from the LHC and the 125 GeV Higgspffiffiffiffi mass are In minimal, gauge-mediated models, on the other hand, simultaneously increasing the lower bound on F. the 125 GeVpffiffiffiffi Higgs mass requires a large stop mass. This implies F ≳ 103 TeV. Hence, the gravitino is heavier * than 360 eV for one messenger or 60 eV for five messengers, [email protected] † [email protected] which is the maximum number allowed by perturbative ‡ [email protected]; [email protected] gauge unification [13]. Therefore, minimal, gauge-mediated models are excluded if the reheating temperature is high. Published by the American Physical Society under the terms of This gravitino problem is often ignored in literature which the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to attempts to achieve the 125 GeV Higgs mass via gauge the author(s) and the published article’s title, journal citation, mediation, e.g., using A terms from renormalization group and DOI. Funded by SCOAP3. equation flow [14] or nondecoupling D terms [15,16].

2470-0010=2018=98(11)=115036(9) 115036-1 Published by the American Physical Society HOOK, MCGEHEE, and MURAYAMA PHYS. REV. D 98, 115036 (2018)

FIG. 1. A compilation of bounds in the m3=2-Tmax plane, where Tmax is the maximum temperature from which the usual, radiation- dominated universe starts. In inflationary models, Tmax corresponds to the reheat temperature, TR. The bound from CMB lensing and cosmic shear is in light blue [9]. The Lyman-α bound from Ref. [10] is in dark green. The overclosure bound for gravitinos heavier than ≳0.2 keV is in light purple [6,10]. We recalculated this limit with current measurements of the Hubble scale factor and dark matter abundance assuming M1 ¼ 417 GeV and the ratio M1∶M2∶M3 ¼ 1∶2∶7. The bound from light-element photodestruction is in dark purple [11]. The bounds from a long-lived gravitino affecting light-element abundances after Big Bang nucleosynthesis are taken from ¼ ¼ 417 Fig. 15 of Ref. [12] for mLSP M1 GeV. The lower bound on the reheat temperature from thermal leptogenesis is the dark- purple dashed line [8].

A nonminimal Higgs sector or strongly coupled messengers masses at loop level. Then the low, SUSY-breaking scale may achieve m3=2 ≲ 16 eV [17], but it is unclear if the results in spartners too light for the current LHC bounds. gravitino mass can be pushed below 4.7 eV.These problems Thus, we want a low-energy, gauge-mediated model with can be ameliorated by using the next-to-minimal super- scalar masses at tree level. symmetric Standard Model (NMSSM) or Dirac-NMSSM However, there is a “no-go theorem” against models [18], coupling the Higgs to messengers [19], or having a which break SUSYat tree level [30–32]. The problem is the 2 strongly coupled messenger sector [20–23]. However, these tree-level identity STrðM Þ¼0 which usually implies models are quite intricate. SUSY breaking cannot occur at the tree level. This is In this paper, we elaborate on a simple, vector-mediated because anomaly cancellation requires both positive and model proposed by Hook and Murayama which breaks negative Uð1Þ charges which gives some scalars negative SUSY at tree level with a Uð1Þ D term and evades all soft masses at tree level from a Uð1Þ D term. gravitino bounds [24]. We discuss electroweak symmetry In vector mediation, there are vectorlike messenger 3 breaking (EWSB) in detail to obtain the correct Higgs mass fields. We assign positive Uð1Þ charges to all sfermions while avoiding all LHC bounds on minimal supersymmetric and negative charges to all vectorlike messengers. Their Standard Model (MSSM) particles. We also introduce a new vectorlike masses overcome their negative soft masses so Uð1Þ charge assignment where the right-handed neutrino is that no scalars are tachyonic. Thus, vector mediation allows neutral to allow the seesaw mechanism for neutrino masses tree-level scalar masses and a lower SUSY-breaking scale. and thermal leptogenesis [25]. Additionally, this model This gives a lighter gravitino while getting the Higgs mass produces a split spectrum usually found in gravity-mediated correct and avoiding current bounds on MSSM particles. models [26]. The lower scale also ameliorates the so-called μ problem. We first review the basics of SUSY breaking via vector mediation.1 If we want a cosmologically viable, light II. THE MODELS gravitino and a naturally high reheating temperature, 2 The vector-mediated models [24] employ an E6-inspired Fig. 1 clearly shows it must be very light. A first attempt particle content that consists of three families of the is to use a low-energy, gauge-mediated model. However, fundamental representation decomposed into SOð10Þ × normal, gauge-mediated models have gaugino and scalar Uð1Þψ as

1Not to be confused with the identically named theory in 3References [33,34] also used a Uð1Þ D term to mediate SUSY Ref. [27] which has a different mechanism. breaking at tree level, but at the grand unified theory scale. This 2An alternative way to achieve a high reheating temperature is resulted in a very weakly coupled gravitino and thus, very anomaly mediation [28,29]. different collider and cosmology signatures [35].

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27 ¼ Ψð16; þ1Þ ⊕ Φð10; −2Þ ⊕ Sð1; þ4Þ; ð1Þ charge assignment generally. We explore the viability of models with each of these charge assignments separately. The right-handed neutrino has charge þ1 under Uð1Þ , where Ψ contains the SM fermions and Φ contains two ψ as do all SM fermions by virtue of Eq. (1). Thus, gauging families of messengers and the MSSM Higgs doublets with Uð1Þ does not allow the seesaw mechanism to generate their color triplets. Because the messengers and Higgs color ψ triplets have different couplings than the Higgs doublets, neutrino masses. However, the right-handed neutrino can we distinguish between generations using subscripts with be made neutral by instead gauging a linear combination of Uð1Þψ and Uð1Þχ, where a given SOð10Þ representation is Φ1 ≡ ðT ;H ÞþðT ;H Þ where T are the color triplets. u u d d ð5Þ ð1Þ Since these Higgs color triplets’ masses are not at the decomposed into SU × U χ. The linear combination grand-unified-theory scale, they prevent gauge unification. we gauge is However, we can add two new electroweak doublets, 1 uncharged under our Uð1Þ, with the same mass as the ¼ ð5 þ Þ ð Þ QSS 4 Qψ Qχ : 3 Higgs color triplets in order to form complete SUð5Þ multiplets and restore gauge unification. We also include Using this charge, we find the decomposition of Ψð16; þ1Þ a neutral particle Zð0Þ and a vectorlike multiplet charged ð5Þ ð1Þ into SU × U SS is under Uð1Þψ, Xð−4Þ, and Yðþ4Þ. These particles are responsible for SUSY breaking. 16 ¼ 10ðþ1Þ ⊕ 5¯ðþ2Þ ⊕ 1ð0Þ: ð4Þ The superpotential for our model is The right-handed neutrino of each family of Ψ is neutral ð1Þ W ¼ MSX þ λZðXY − v2Þ under U SS by virtue of Eq. (4), allowing the seesaw S mechanism for neutrino masses. The decomposition of g k g k Φð10 −2Þ ð5Þ ð1Þ − 2 þ − 2 H þ H ; into SU × U SS is TuTd 2 Y 2 S HuHd 2 Y 2 S ¯ X3 10 ¼ 5ð−2Þ ⊕ 5ð−3Þ: ð5Þ − Φ Φ g þ k a · a 2 Y 2 S ¼2 a Additionally, each S has charge QSS ¼ 5, XðQSS ¼ −5Þ, 3 2 2 ð ¼þ5Þ λS X X X Y QSS , and Z remains neutral. þ Φ Φ þ κ ð Þ μ 2 a · a Sb X NaSa: 2 We do not have the usual -term as it is not gauge a¼2 ¼1 a¼1 ð1Þ ð1Þ b invariant under U ψ or U SS. Similar to the NMSSM, it is generated by the expectation values of Y and S. For the The three generations of S are denoted S1, S2, and S. The two charge assignments, we have to introduce different first line in Eq. (2) breaks SUSY with F terms and interactions and discrete symmetries to allow the color a positive D term. It can be dynamically realized as the triplets to decay while preventing proton decay. low-energy effective theory of an Izawa-Yanagida- For the seesaw-charge assignment, we introduce the Intriligator- Thomas (IYIT) model with 4 SUð2Þ doublets interaction W ⊃ 1 SQT H to allow for the color triplets MPl d d with appropriate Uð1Þ charges [36,37]. The second and in Φ to decay. We assign negative matter parity to all SM third lines are standard messenger interactions and generate multiplets in Ψ and to the color triplets in Φ, but positive the usual, gauge-mediated gaugino masses at one loop. We matter parity to the electroweak doublets in Φ to prevent 4 ¼þ1 3 have allowed the MSSM Higgs doublets, Hu and Hd,to proton decay. This is consistent with assigning B = have different couplings to Y and S than their color triplets, to Tu and B ¼ −1=3 to Td. T and T , in order to satisfy EWSB conditions. Each For the original-charge assignment, we introduce the u d ¯ generation of Φ could have different couplings to Y and S, interaction W ⊃ Tdu¯ d to allow for the color triplets in Φ to but we set them equal to maximize the gaugino masses decay. We enforce lepton-number conservation to prevent (see Sec. III). The fourth line gives the S1;2 fermion masses proton decay. Since the right-handed neutrinos are charged ð1Þ by introducing two neutral fields N1;2, while the S1;2 scalars under U ψ, this requirement is consistent with the also acquire a mass from the D term. We have the neutrinos having Dirac masses. interaction W ⊃ S1;2ΦΦ in line four to allow S1;2 and For the seesaw-charge assignment in Eq. (5), the MSSM 2 2 N1 2 to decay so that they do not overclose the universe. Higgs soft masses satisfy m >m at tree level since the ; Hu Hd In this paper, we consider two different charge assign- D term is positive and −2 > −3. This is problematic for ments under a new Uð1Þ: the original-charge assignment EWSB. To help, we introduce kinetic mixing between our corresponding to Uð1Þψ and a new, seesaw-charge assign- ð1Þ ment corresponding to a different U SS. For clarity, we 4R. M. thanks T. Yanagida for pointing out this issue. Matter- distinguish between these charge assignments using these parity assignment is related to R-parity assignment via PM ¼ 2s subscripts and omit a subscript when talking about either PR × ð−1Þ , where s is the spin of the particle in question.

115036-3 HOOK, MCGEHEE, and MURAYAMA PHYS. REV. D 98, 115036 (2018) Z Uð1Þ [or Uð1Þ for the original-charge assignment] and ∞ X1 dϕ 2 SS ψ S ðϕðrÞÞ¼2π2 drr3 i þVðϕðrÞÞ ; ð11Þ Uð1Þ . Following Ref. [38], this kinetic mixing can be E 2 Y 0 i dr written as a cross term between the D terms associated with the two Uð1Þ’sas where the sum is over all field dimensions ðX; Y; Z; S; ΦÞ. X The tunneling rate per unit volume is exponentially ⊃ 0χ i jjϕ j2jϕ j2 ð Þ ¯ V g e QYQ i j ; 6 sensitive to the bounce action B ¼ SEðϕÞ − SEðϕ0Þ as Γ ∝ i;j exp ð−BÞ [39]. ¯ The thermal bounce action, B ¼ S ðϕÞ − S ðϕ0Þ,is i ð1Þ ϕ j th th th where QY is the U Y charge of the scalar i, Q is the similar: the action is given by the three-dimensional (3D) ð1Þ ð1Þ U charge, and e is the U electric charge. The version of (11) and the bounce profile solves the 3D dimensionless coupling χ is a measure of the mixing, 2 versions of (9) and (10). The thermal tunneling rate is and we have dropped terms proportional to χ . χ is naturally Γ ∝ 4 ð− Þ th T exp Bth=TC , where TC is the critical temper- small and is generated by one-loop diagrams with particles ature at which the SUSY-invariant vacuum becomes the ð1Þ’ charged under both U s [38]. Note that the addition of true minimum. this mixing does not change the locations of the false and When approximating both the vacuum and thermal ϕ true vacua. The D term gives every scalar i a mass bounce actions, we calculate the tunneling along the contribution from Eq. (6), straight line between the true and false vacua. The 4D vacuum and 3D thermal bounce actions are calculated δ 2 ¼ i 0χh i ð Þ mi QYg D : 7 semianalytically using results from Ref. [40]. In order for these tunneling rates to be sufficiently small, we conserva- H H Uð1Þ Since u and d have opposite Y charges, Eq. (7) aids tively require that B>450 and B =T > 250. We use the EWSB in models with the seesaw-charge assignment. th c high-temperature approximation for the thermal effective potential to calculate TC. We also assume Bth is approx- III. CONSTRAINTS PRIOR TO EWSB imately constant and calculate it using the zero-temperature ≲ We explore the parameter space ðe; M; λ;vS; χ;k; potential. For much of our parameter space, TC vS where g; kH;gH; tan βÞ for both charge assignments to find viable this approximation breaks down. However, we find that Bth models with light gravitinos which avoid the cosmological is so large in our region of interest that this does not matter. 2 bound m3=2 < 4.7 eV. Before we verify EWSB and the We verify that STrðM Þ¼0 along the straight line Higgs mass, we must impose some constraints on our between the true and false vacua. We set κ ¼ 0.1 for parameter space. The first suppresses tunneling to the true concreteness. vacuum. To illustrate that the vacuum and thermal tunneling rates The interactions in lines two and three of Eq. (2) which are highly suppressed, we have chosen the representative give rise to the gaugino masses at one loop also generate a values (M ¼ 2, λ ¼ 4π, χ ¼ 0, k ¼ 4π) and set g as small supersymmetric vacuum. This true vacuum appears at as possible before the lightest messengers go tachyonic [see Eq. (12) below]. See Figs. 2 and 3. 2 Φ2 2 2 ¼ vs ¼ k 0 ¼ gM ¼ − gMvS ð Þ Both vacuum and thermal bounce actions are approxi- X0 ;Z0 ;S0 2 2 ; 8 Y0 2M kλ k Φ0 mately independent of χ over the range of values we consider later. While χ ¼ 0 is not viable for the seesaw-charge where Φ0 is set by requiring the D term to vanish. We require the parameter space we consider to disallow substantial vacuum or thermal tunneling between the false and true vacua. To calculate the vacuum tunneling rate, we calculate the bounce profile ϕ¯ ðrÞ which solves 2ϕ¯ 3 ϕ¯ d i þ d i ¼ dV ð Þ 2 ¯ 9 dr r dr dϕi with the boundary conditions ϕ¯ d i ¯ ðr ¼ 0Þ¼0; ϕ ðr → ∞Þ¼ϕ0; ð10Þ dr i FIG. 2. Vacuum bounce action for both charge assignments. We take the representative values (M ¼ 2, λ ¼ 4π, χ ¼ 0, k ¼ 4π) where ϕ0 ¼ðX0;Y0; 0; 0; 0Þ is the false vacuum. The and set g as small as possible before the lightest messengers go spherically symmetric, four-dimensional (4D) Euclidean tachyonic. We easily satisfy B>450 for the Universe to be action of a profile ϕðrÞ is stable.

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2 2 D ¼ eQXðX0 − Y0Þ > 0. QX ¼ −QY since X and Y form a vectorlike multiplet. For the bosons in the color triplets, this is the correct mass matrix under the appropriate replacement QY ¼þ1 → QY ¼ − 1. We leave the Uð1Þ Hu 2 d 3 charges unspecified since we find viable models for both charge assignments. The MSSM Higgses have the same mass matrix under the replacement k → kH, g → gH [see Eq. (2)]. We calculate the one-loop gaugino masses using Eq. (2.3) in Ref. [43]. All three families’ color triplets FIG. 3. Bth=TC for both charge assignments. We take the same 250 and electroweak doublets contribute to the gaugino masses. values as in Fig. 2. We easily satisfy Bth=TC > for the The on-shell gluino mass enhancement from (s)top loops is Universe to be stable. calculated using SOFTSUSY (discussed below) [44–47]. The gaugino masses are maximized when there is a large mass hierarchy in Eq. (12). We ensure a large hierarchy by assignment due to EWSB failure, these figures do not choosing parameters where the lighter scalar is light change for realistic values of χ. We allow different values (≈1 TeV). The gaugino masses are also proportional to ð λ χÞ of M; ; when conducting a full parameter-space search. the fermion masses so we prefer larger g and larger Y0.In ¼ 4π However, we always take k and minimize g in order to order to increase the gaugino masses in units of vS, we thus maximize the gaugino masses (see Sec. III). set k ¼ 4π and choose g such that the lighter scalar in Both actions are dramatically larger than in Ref. [24] Eq. (12) is near 1 TeV. These are the prescriptions we used because we are considering much smaller values of e. in Figs. 2 and 3 and allow smaller vS. One can guess that a smaller value of e makes the bounce We set vS by satisfying all current, gaugino-mass actions much larger because when e → ∞, Φ is tachyonic bounds. The ATLAS bound on the gluino mass is roughly and there is no barrier between the false and true vacua. 2.03 TeV [48], while the CMS bound is 1.95 TeV [49]. The As there is a barrier for small e, by continuity, one ATLAS bound on the Wino mass is roughly 620 GeV, expects that smaller e gives larger barriers. In the limit unless the Bino is heavier than 350 GeV, in which case it is e → 0, we numerically find that the barrier height 720 GeV [50]. The Wino bound is the most stringent between the two vacua increases. This is the thin-wall from direct searches. So, we use it to estimate vS before limit and the bounce actions increase dramatically EWSB. But we find getting the correct Higgs mass is [39,41,42]. generally harder and therefore, determines vS after EWSB The scalar potential along the straight-line, tunneling (see below). path is quartic in one scalar field. This potential is The gravitino mass is given by characterized by a single, dimensionless parameter δ which can take values between 0 and 2 [40]. The thin-wall limit 2 V m ¼ ; ð13Þ δ → 2 3=2 3 2 corresponds to the limit . In this limit, the analytic, MPl vacuum bounce action is proportional to ð2 − δÞ−3 and the −2 ¼ 2 4 1018 analytic, thermal bounce action is proportional to ð2 − δÞ . where MPl . × GeV. For a viable model, we 4 7 For the parameters taken in Ref. [24], δ ¼ 1.6. We consider require m3=2 < . eV [9]. χ 2 points in parameter space which are much closer to the thin- We always set negative to make mH lighter and help ¯˜ u ð1Þ wall limit with δ ≳ 1.996. EWSB [see Eq. (12)]. Since e has the greatest U Y The gaugino masses are generated from one-loop dia- charge, Qe¯ ¼þ1, we determine how negative χ can be by grams with the messengers in the three families of Φ.All requiring that e¯˜ is not tachyonic. fermions in Φ have the same mass MΦ ¼ gY0. Taking into We also require that the massive Uð1Þ boson does not account the kinetic mixing in Eq. (7), the boson compo- mix too much with W and Z to affect the ρ parameter.5 We nents all have the mass matrix conservatively require that δρ < 10−3 which imposes that ð ev Þ2 < 10−3, where v is the Higgs vacuum expectation ! MV 2 þ þ Y 0χ value and M is the mass of our Uð1Þ boson [51]. MΦ QH eD QH g DkFS V u u ; 2 Y 0 kF MΦ þ Q eD − Q g χD S Hd Hu IV. CONSTRAINTS AFTER EWSB ð Þ 12 We use SOFTSUSY to calculate radiative EWSB. Our inputs into SOFTSUSY are the gaugino masses, trilinear ð1Þ 5 Φ where QHu is the U charge of fields in the of , QHd is ¯ 5 the charge of fields in the 5 of Φ, FS ¼ MX0, and R. M. thanks Simon Knapen on this point.

115036-5 HOOK, MCGEHEE, and MURAYAMA PHYS. REV. D 98, 115036 (2018) couplings, sfermion masses, m2 , m2 , and tan β. We can models only occur for smaller values of e. This is a nontrivial Hu Hd read off the MSSM Higgs parameters from Eq. (12): check because ðkH;gHÞ are set by the output boundary conditions from SOFTSUSY. 1 2 0 Requiring the Higgs mass to be 125 GeV is the strongest μ ¼ g Y0;m¼ Q eD þ g χD; H Hu Hu 2 constraint and sets the overall scale vS in our models. If vS 2 1 0 is set to satisfy the gaugino constraints, the stop masses are Bμ ¼ kHFS;m¼ QH eD − g χD: ð14Þ Hd d 2 too light to lift the Higgs mass. When searching for viable models, we estimate vS by satisfying the Wino constraint The soft, SUSY-breaking A terms are all 0 in our models at and then explore parameter space by increasing this vS by tree level. SOFTSUSY effectively sets μ and Bμ as output some factor between ∼2 and 3. boundary conditions and therefore, kH and gH in our We require that the output top Yukawa is still perturba- models. tive. We calculate yt using Eq. (62) in Ref. [52] using the We only input the above boundary conditions at tree top mass in Ref. [53]. We require that yt < 0.94. We also level. We input the gaugino masses at one-loop level, but do require that the other neutral scalar Higgses are not too not include QCD enhancements to M3 or other, higher- light. As a conservative constraint, we require the mass of order corrections. This is slightly inconsistent as we run the heavier CP-even neutral scalar, H0, to be larger than SOFTSUSY with the 3-loop renormalization group equa- 580 GeV [54]. We require the mass of the CP-odd neutral tions and calculate the lightest MSSM Higgs both at 2-loop scalar, A0, to avoid the tan β-dependent, ATLAS bounds in and 3-loop levels. When calculating the Higgs at 3-loop, Fig. 10(b) of Ref. [55]. SOFTSUSY also defaults to adding 2-loop Yukawa and g3 Since SOFTSUSY includes loop corrections to the threshold corrections to mt and mb. gluino, we make sure the output gluino is still heavier EWSB itself is a strong constraint. If a point in parameter than 2.03 TeV. SOFTSUSY also gives the spectrum of the space has successful radiative EWSB, there are still many neutralinos and charginos. We make sure to avoid the 0 0 additional constraints we check before considering it bound on χ˜1 and χ˜2 as a function of χ˜1 given in Ref. [50]. viable. In order to approximate EWSB using SOFTSUSY, the correction to the MSSM, Higgs-quartic coupling, V. SUSY SPECTRA AND BEST MODELS λ ¼ 2 2 h mh=v , must be sufficiently small. We require our Higgs mass to be accurate to 0.5 GeV. This places an upper Figure 4 shows our search through parameter space for bound at roughly viable models with the seesaw-charge assignment, QSS, and the original-charge assignment, Qψ . The points correspond 1 1 ð λ χ δλ ¼ 2 þ 2 2 þ 0χ 0 002 ð Þ to viable points in the parameter space e; M; ;vS; ;k; h gH QH e eg < . : 15 2 d 2 g; kH;gH; tan βÞ for which all of the constraints are sat- isfied. We show viable points when we calculate the Higgs This is a conservative limit because the quartic couplings due mass at both the 2-loop and 3-loop levels in SOFTSUSY. to the F term and the D term of heavy fields decouple and the The 3-loop calculation generally yields a lighter Higgs effect is in general smaller. We do not bother including a term which requires a greater vS to obtain 125 GeV. This yields a 2 2 ≪ 2 proportional to kH since kH gH generically. Equation (15) heavier gravitino and explains the separation of viable 2 is also conservative because the gH piece vanishes in the limit points between the 2-loop and 3-loop calculations. We find of large or small tan β. Due to the upper bound on δλh, vS ∈ ½55; 240 TeV for the seesaw-charge assignment and Eq. (15) lets us only consider e<0.021. We find viable vS ∈ ½44; 240 TeV for the original-charge assignment.

FIG. 4. m3=2 vs e for viable models with the seesaw-charge and original-charge assignments which satisfy all aforementioned constraints. Viable points are shown when the Higgs is calculated at both the 2-loop and 3-loop levels in SOFTSUSY.

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FIG. 5. m3=2 vs e for models with the seesaw-charge and original assignments which are excluded by the aforementioned constraints. Excluded points are shown along with their reason for exclusion when the Higgs is calculated at the 3-loop level in SOFTSUSY.

χ 2 2 2 For the seesaw-charge assignment, is always close to N y m˜ m˜ 2 2 Δ 2 ¼ C t 2 tR þ 2 tL m 2 m˜ ln 2 m˜ ln 2 its maximum value to help alleviate mH >mH , namely Hu ð4πÞ tR μ tL μ u d R R χ ∈ ½−0.033; −0.014. This is naturally small and generated by one-loop diagrams of particles charged under both − 2 − 2 ð Þ mt˜ mt˜ ; 16 Uð1Þ’s [38]. However, it is not possible to take χ large R L enough to make m2 ≤ m2 without making e¯˜ tachyonic. Hu Hd ¼ 3 2 where NC is the number of colors. At the renormal- EWSB works due to negative, one-loop corrections to m 2 Hu μ ¼ ˜ ˜ ization scale, R mtR mtL , this correction is negative as from (s)tops at the geometric mean of the stop masses. long as the stop masses are not too separated. This negative In particular, the one-loop corrections to m2 from the Hu correction, in addition to the kinetic mixing χ, enables (s)tops are EWSB in models with the seesaw-charge assignment. Figure 5 shows why representative points in parameter space are excluded by the aforementioned constraints for the seesaw-charge and original-charge assignments when we calculate the Higgs mass at the 3-loop level. The most exclusive constraint is getting the Higgs mass correct. Points toward the bottom and left of these figures correspond to a Higgs which is too light. For the seesaw-charge assignment, the other MSSM Higgs scalars are often too light when the Higgs is too light. Points to the far right are excluded because δλh from Eq. (15) is too large and the MSSM approximation breaks down. The viable models with the lightest gravitinos are illus- trated in Fig. 6. The equivalent figures with mh calculated at the 3-loop level are similar, with their spectra shifted slightly up. SOFTSUSY requires μ to be 9.74 and 11.4 TeV for the original- and seesaw-charge assignments, respectively, reducing the μ problem from the Planck scale to the 10-TeV scale. One might be confused that the gauginos are near the sfermions. This is simply an artifact of maximizing the gaugino masses in units of vS before EWSB. The lightest gravitino masses for these 3-loop calculations are 1.9 eV and 2.3 eV for the original- and seesaw-charge assignments, respectively. Optimistically, we see that we can obtain gravitino masses as light as 1.0 and 1.4 eV for the original- and seesaw-charge assignments when we calculate the Higgs at the 2-loop level. FIG. 6. Low-lying spectra for the seesaw-charge and original- Even being conservative and calculating the Higgs mass at charge models with the lightest gravitino when mh is calculated at the 3-loop level still yields a sufficiently light gravitino to the 2-loop level. evade the current cosmological bound m3=2 < 4.7 eV.

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VI. CONCLUSION AND DISCUSSIONS ACKNOWLEDGMENTS We expand on the previous work of Hook and R. M. thanks Simon Knapen for many useful discussions Murayama by finding models with lighter gravitinos, and an introduction to SOFTSUSY. R. M. thanks Jason calculating EWSB, and enabling EWSB in models with Evans and Marcin Badziak for helpful discussions. R. M. the new, seesaw-charge assignment by introducing Uð1Þ also thanks Katelin Schutz for comments on drafts. H. M. kinetic mixing. Unlike typical, gauge-mediated models, we and R. M. thank T. T. Yanagida for discussions. R. M. is find that our vector-mediated models are cosmologically supported by an NSF graduate research fellowship. A. H. is viable with light gravitinos. As can be extrapolated from supported by NSF Grant No. PHYS-1316699 and DOE Fig. 4, going to larger values of e generally allows for Grant No. DE- SC0012012. This material is based upon lighter gravitinos. Since we use SOFTSUSY to approxi- work supported by the National Science Foundation mate EWSB in our models, we only consider e ≲ 0.01 to Graduate Research Fellowship Program under Grant avoid invalidating this approximation. If we calculate No. DGE 1106400. H. M. was supported by the U.S. EWSB without SOFTSUSY, we cannot consider values DOE under Contract No. DE-AC02-05CH11231, and by of e much larger because the vacuum and thermal tunneling the NSF under Grants No. PHY-1316783 and No. PHY- rates begin to matter. For values of e ≳ 0.1, we can no 1638509. H. M. was also supported by the JSPS Grant-in- longer choose k and g to maximize the gaugino masses with Aid for Scientific Research (C) (No. 26400241 and respect to vS. Satisfying the gaugino-mass bounds then No. 17K05409), MEXT Grant-in-Aid for Scientific increases vS which increases the gravitino mass. It is very Research on Innovative Areas (No. 15H05887 and interesting that the best we can do is not too far away from No. 15K21733), and by WPI, MEXT, Japan. Any opinions, current cosmological limits. Near-future improvements in findings, and conclusions or recommendations expressed in cosmological data, such as improvements in cosmic shear this material are those of the author(s) and do not measurements at DES and Hyper Suprime-Cam, could necessarily reflect the views of the National Science completely rule out low-energy SUSY breaking. Foundation.

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