# No-scale SU(5) super-GUTs

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## Abstract

We reconsider the minimal SU(5) grand unified theory (GUT) in the context of no-scale supergravity inspired by string compactification scenarios, assuming that the soft supersymmetry-breaking parameters satisfy universality conditions at some input scale \(M_\mathrm{in}\) above the GUT scale \(M_{\mathrm{GUT}}\). When setting up such a no-scale super-GUT model, special attention must be paid to avoiding the Scylla of rapid proton decay and the Charybdis of an excessive density of cold dark matter, while also having an acceptable mass for the Higgs boson. We do not find consistent solutions if none of the matter and Higgs fields are assigned to twisted chiral supermultiplets, even in the presence of Giudice–Masiero terms. However, consistent solutions may be found if at least one fiveplet of GUT Higgs fields is assigned to a twisted chiral supermultiplet, with a suitable choice of modular weights. Spin-independent dark matter scattering may be detectable in some of these consistent solutions.

## Keywords

Dark Matter Yukawa Coupling Higgs Mass Light Supersymmetric Particle Grand Unify Theory## 1 Introduction

Globally supersymmetric grand unification has long been an attractive framework for unifying the non-gravitational interactions, with the minimal option using the gauge group SU(5) [1, 2, 3]. When incorporating gravity, one must embed such a supersymmetric grand unified theory (GUT) within some supergravity theory, and an attractive option is no-scale supergravity [4, 5, 6, 7]. This has the advantages that it leads to an effective potential without holes of depth \(\mathcal{O}(1)\) in natural units, and emerges in generic string compactifications [8]. No-scale supergravity also allows naturally for the possibility of Planck-compatible cosmological inflation [9, 10]. In general, a no-scale Kähler potential contains several moduli \(T_i\), but here we consider scenarios in which the relevant dynamics is dominated by a single volume modulus field *T*.

The construction of no-scale supergravity GUTs encounters significant hurdles, such as fixing the compactification moduli. Moreover, pure no-scale boundary conditions require that all the quadratic, bilinear and trilinear scalar couplings \(m_0, B_0\) and \(A_0\) vanish, leading to phenomenology that is in contradiction with experimental constraints. However, this issue may be avoided in models with (untwisted or twisted) matter fields with non-vanishing modular weights as we show below.

The simplest possibility for soft supersymmetry breaking is to postulate universal values of \(m_0, B_0\) and \(A_0\), as in the constrained minimal supersymmetric Standard Model (CMSSM) [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. With the inclusion of a universal gaugino mass, \(m_{1/2}\), the CMSSM is a four-parameter theory.^{1} Minimal supergravity places an additional boundary condition, relating \(B_0\) and \(A_0\) (\(B_0 = A_0 - m_0\)) making it a three-parameter theory [39, 40, 41]. No-scale supergravity, however, is effectively a one-parameter theory since we require \(m_0 = A_0 = B_0 = 0\). Another one-parameter theory in this context is pure gravity mediation [42, 43, 44, 45, 46, 47], in which the gaugino masses, *A* and *B* terms^{2} are determined by anomaly mediation [49, 50, 51, 52, 53] leaving only the gravitino mass, \(m_{3/2} = m_0\) as a free parameter.

These boundary conditions may be too restrictive if they are imposed at the GUT scale, \(M_{\mathrm{GUT}}\), defined as the renormalization scale where the two electroweak gauge couplings are unified. There is, however, no intrinsic reason that the boundary conditions for supersymmetry breaking coincide with gauge coupling unification. Separating these two scales opens the door for so-called sub-GUT models [37, 38, 54, 55, 56] where the input universality scale differs from the GUT scale with \(M_\mathrm{in} < M_{\mathrm{GUT}}\) or the possibility that the boundary conditions are imposed at some higher input scale \(M_\mathrm{in} > M_{\mathrm{GUT}}\), a scenario we term super-GUT [57, 58].

However, the regions of parameter space with acceptable relic density and Higgs mass typically require quite special values of the GUT superpotential couplings and rather large values of \(\tan \beta \) [59], and hence a proton lifetime that is unacceptably short. In order to accommodate smaller values of \(\tan \beta \) and hence an acceptably long proton lifetime, we consider non-zero Giudice–Masiero (GM) terms [48] in the Kähler potential. In this way we are able to avoid the Scylla of rapid proton decay and the Charybdis of an excessive density of cold dark matter, while also having an acceptable value of the Higgs mass.^{3} Furthermore, when no-scale boundary conditions are applied at the GUT scale, the lightest sparticle in the spectrum is typically a stau (or the stau is tachyonic). Applying the boundary conditions above the GUT scale as in a super-GUT model can alleviate this problem [62].

The outline of this paper is as follows. In Sect. 2, we review our theoretical framework, with our set-up of the minimal supersymmetric SU(5) model described in Sect. 2.1, our no-scale supergravity framework inspired by string compactification scenarios described in Sect. 2.2 and the vacuum conditions and the relevant renormalization-group equations (RGEs) set out in Sect. 2.3. We describe our key results in Sect. 3. We explore in Sect. 3.1 scenarios in which none of the matter and Higgs supermultiplets are twisted, and we find no way to steer between Scylla and Charybdis with an acceptable Higgs mass in this case. However, as we show in Sect. 3.2, this is quite possible if one or the other (or both) of the GUT fiveplet Higgs supermultiplets is twisted. Spin-independent dark matter scattering may be observable in some of the cases studied. Finally, Sect. 4 discusses our results.

## 2 Super-GUT CMSSM models

### 2.1 Minimal supersymmetric SU(5)

*X*to acquire masses \(M_X = 5 g_5 V\), where \(g_5\) is the SU(5) gauge coupling.

The multiplets *H* and \(\overline{H}\) in Eq. (1) are \(\mathbf{5}\) and \(\overline{\mathbf{5}}\) representations of SU(5), respectively, and contain the MSSM Higgs fields. In order to realize doublet-triplet mass splitting in the *H* and \(\overline{H}\) multiplets, we impose the fine-tuning condition \(\mu _H -3\lambda V \ll V\). In this case, the color-triplet Higgs states have masses \(M_{H_C} = 5\lambda V\), the masses of the color and weak adjoint components of \(\Sigma \) are \(M_\Sigma = 5\lambda ^\prime V/2\), and the singlet component of \(\Sigma \) acquires a mass \(M_{\Sigma _{24}} = \lambda ^\prime V/2\).

The multiplets \({\Phi }_i\) in Eq. (1) are \(\mathbf {\overline{5}}\) representations containing the left-handed SM matter fields \(\overline{D}_i\) and \({L}_i\), and the \({\Psi }_i\) are \(\mathbf {10}\) representations of SU(5) containing the left-handed \({Q}_i\), \(\overline{U}_i\), and \(\overline{E}_i\), where the index \(i = 1,2,3\) denotes the generations.

*A*-terms are proportional to the corresponding Yukawa couplings in the superpotential; we will see in Sect. 2.2 that the

*A*-terms arising in no-scale supergravity actually have this structure.

### 2.2 No-scale framework

^{4}Our starting-point is a no-scale Kähler potential inspired by string compactification scenarios,

*T*, and both untwisted and twisted matter fields, \(\phi _i\) and \(\varphi _a\), respectively, the latter with modular weights \(n_a\). We consider a generic superpotential of the form

*c*is an arbitrary constant, and \(W_{2,3}\) denote bilinear and trilinear terms with modular weights that are in general non-zero and \(\mu _\Lambda \). When \(\langle \phi ,\varphi \rangle =0\), the effective potential for

*T*is completely flat at the tree level, so it has an undetermined vev, and the gravitino mass

^{5}We assume here that some Planck-scale dynamics fixes \(T = \bar{T} = c\), and take \(c = 1/2\) in the following.

*A*-terms proportional to the Yukawa couplings in Eq. (3). We also postulate in what follows generalized Giudice–Masiero terms [48]

*H*, \(\bar{H}\), and \(\Sigma \) are untwisted, these induce corrections to the \(\mu \) and

*B*terms:

*B*terms are quite small, they are crucial for matching the GUT scale

*B*terms onto the MSSM

*B*term at the GUT scale, as we see below.

*B*-terms is included in all calculations in order to satisfy the

*B*-term matching condition.

We conclude this subsection by emphasizing that the parameters introduced in Eqs. (4)–(6) and (9) above are intrinsic to our string-inspired no-scale supergravity framework. The parameters \(n_a\) in (4) characterize the twisted matter fields \(\varphi _a\), the parameters \(\alpha , \beta , \sigma \) and \(\rho \) in (5) are modular weights that are unknown a priori, the value *c* of the volume modulus is assumed to be fixed by some Planck-scale dynamics, and the parameters \(c_{H, \Sigma }\) and \(\gamma _{H, \Sigma }\) are needed to characterize the Giudice–Masiero terms (9) that are a common feature of supergravity models. Many such analogous parameters would appear in any model based on \(N = 1\) supergravity: here they are related to properties of an underlying scenario, namely string compactification, which is necessarily ambiguous at our present level of understanding. Our subsequent phenomenological analysis may serve to give pointers how these ambiguities could be reduced.

### 2.3 Vacuum conditions and renormalization-group equations

Since the *B*-term boundary conditions are specified at \(M_\mathrm{in}\), we cannot use the Higgs minimization equations to determine *B* and the MSSM \(\mu \) term as is commonly done in the MSSM. Instead, as in mSUGRA models, these conditions can be used to determine \(\mu \) and \(\tan \beta \) [40, 41] as was done in the no-scale super-GUT models considered in [59]. In [59], standard no-scale boundary conditions were used to identify regions of parameter space with acceptable relic density and Higgs mass. Typically, rather large values of \(\tan \beta \) were found and, in addition, it was necessary to choose somewhat small values of the coupling \(\lambda = \mathcal{O}(0.01)\) with much larger values of \(\lambda ^\prime = \mathcal{O}(1)\). All of these choices tend to decrease the proton lifetime to unacceptably small values [58]. In order to reconcile the proton lifetime with the relic density and Higgs mass, we need to consider lower values of \(\tan \beta \) [38, 58, 67], which can be accomplished when the GM terms (9) are included [46, 47, 48, 68].

The soft supersymmetry breaking parameters are evolved down from \(M_\mathrm{in}\) to \(M_\mathrm{GUT}\) using the renormalization-group equations (RGEs) of the minimal supersymmetric SU(5) GUT, which can be found in [57, 59, 70, 71, 72, 73], with appropriate changes of notation. During the evolution, the GUT couplings in Eq. (1) affect the running of the soft supersymmetry-breaking parameters, which results in non-universality in the soft parameters at \(M_\mathrm{GUT}\). In particular, the GUT coupling \(\lambda \) contributes to the running of the Yukawa couplings, the corresponding *A*-terms, and the Higgs soft masses. On the other hand, \(\lambda ^\prime \) affects directly only the running of \(\lambda \), \(m_\Sigma \), and \(A_\lambda \) (besides \(\lambda ^\prime \) and \(A_{\lambda ^\prime }\)), and thus can affect the MSSM soft mass parameters only at higher-loop level. Both \(\lambda \) and \(\lambda ^\prime \) contribute to the RGEs of the soft masses of matter multiplets only at higher-loop level, suppressing their effects on these parameters.

*V*, which in turn allows us to fix the gauge and Higgs boson masses as

In the case where \(\lambda '\) is small, the mass splitting of \(M_{\Sigma _{3,8}}\) becomes significant and \(\ln (M_{\Sigma _3}/M_{\Sigma _8})\) is now order one. The threshold correction to the gauge couplings coming from Eq. (17) is now of order \(1/(16\pi ^2)\). However, for small \(\lambda '\), the vev of \(\Sigma \) grows and is now of order \(10^{17}\) GeV.^{6} With a vev this large, \(8V/M_P\) is much larger than a loop factor and we can again safely neglect the contributions coming from the operators in Eq. (17).

*A*-terms of the third generation sfermions, are given by

*B*terms are [85]

*B*that are \(\mathcal{O}(M_\mathrm{SUSY})\) is determined by these last two equations. From Eq. (22), we find that we need to tune \(|\mu _H -3\lambda V|\) to be \(\mathcal{O}(M_\mathrm{SUSY})\). From Eq. (23), \(V\Delta /\mu \) should be \(\mathcal{O}(M_\mathrm{SUSY})\), which requires \(|\Delta | \le \mathcal{O}(M_\mathrm{SUSY}^2/M_\mathrm{GUT})\). In standard no-scale supergravity, \(\Delta = 0\) and this is stable against radiative corrections, as shown in Ref. [86]. As discussed in Ref. [58], in order for Eq. (23) to have a real solution for \(B_\Sigma \), the condition \(A_{\lambda ^\prime }^2 \gtrsim 8m_\Sigma ^2\) should be satisfied for \(\lambda ^\prime \ll \lambda \). We have checked that this condition is always satisfied over the parameter space we consider in Sect. 3.

*B*parameters can be determined by using the electroweak vacuum conditions:

*A*and

*B*-terms set by \(A_0 = B_0 = 0\) (and similarly for \(m_\Sigma ^2\)). Thus, (23) is not satisfied in general. Nevertheless, it is often possible to find a value of \(\tan \beta \) that adjusts \(B\mu \) via (26) to have the correct value at the GUT scale. As noted earlier, this often leads to relatively large values of \(\tan \beta \) and unacceptable low values for the proton lifetime.

*B*is \(\mathcal{O}(M_\mathrm{SUSY})\). Thus the shift in (23) becomes

In the following we assume initially that all fields are untwisted, so that \(m_0 = 0\), and assume vanishing modular weights \(\alpha = \beta = 0\), so that \(A_0 = B_0 = 0\). Later we consider the effects of twisting one or both of the Higgs 5-plets and turning on the trilinear weight \(\alpha \) in order to allow non-zero \(A_0\).

## 3 Results

### 3.1 Standard no-scale supergravity with a GM term

It is well known that the CMSSM with no-scale boundary conditions is not viable. With \(m_0 = A_0 = B_0 = 0\), the particle spectrum almost inevitably contains either a stau lightest supersymmetric particle (LSP) or tachyonic stau. However, this problem can be alleviated if the universal boundary conditions are applied above the GUT scale [62]. In this case, the running from \(M_\mathrm{in}\) to \(M_\mathrm{GUT}\) produces non-zero soft terms that may be sufficiently large to produce a reasonable spectrum.^{7}

The basic no-scale super-GUT model was studied in detail in [59]. There it was found that, for sufficiently large \(M_\mathrm{in}\), not only could a reasonable mass spectrum be obtained, but also regions of parameter space with the correct relic density and Higgs mass were identified. This region was further explored in [92], with the aim of studying possible departures from minimal flavor violation. There, for example, a particular benchmark point was chosen with \(M_5 = 1500\) GeV, \(M_\mathrm{in} = 10^{18}\) GeV, \(\lambda = -0.1, \lambda ^\prime = 2\), which required \(\tan \beta \approx 52\) as no GM term was included. One concern for this benchmark is the proton decay rate, which is enhanced by the combination of large \(\tan \beta \) and small \(\lambda \) (which induced a low value for the Higgs color-triplet mass). Indeed, as we show below, the proton lifetime is far too small in this minimal SU(5) construction.

We show in Fig. 1 two examples of \((m_{1/2}, \tan \beta )\) planes for fixed \(M_\mathrm{in} = 10^{18}\) GeV. In the left panel, we have chosen \(\lambda = -0.1\) and \(\lambda ^\prime = 2\). In the dark blue shaded strip, the neutralino LSP relic density agrees with the value determined by Planck and other experiments. To its left, in the brown shaded region the stau is either the LSP or tachyonic. The red dot-dashed contours show the value of the Higgs mass as computed using the FeynHiggs code [93].^{8} As one can see, there is a region at large \(\tan \beta \in 52\)–55 for \(m_{1/2} \in 1\)–1.5 TeV that corresponds to the preferred region found in [59].^{9} In this region the Higgs mass \(\in 122\)–124 GeV, which is acceptable given the uncertainty in the mass calculated using FeynHiggs. By including a GM term, we are able to probe lower values of \(\tan \beta \) for the same set of input parameters. Unfortunately, the proton lifetime is much too small over the entire left panel, with a value of only \(10^{25}\) years in the upper left corner. We also show (in green) the contours of the GM term. In this case, since \(|\lambda |\lesssim \lambda ^\prime \), we assume \(c_H = 0\) and show the contours of \(c_\Sigma (m_{3/2}/m_{1/2})^2\).^{10} As one can see, the contour for \(c_\Sigma = 0\) runs through the region of good relic density and Higgs mass found in [59].

In the right panel of Fig. 1 we show a similar plane but with different choices of \((\lambda , \lambda ^\prime ) = (1, 10^{-5})\), which are more typical of the values required in [58]. In this case, with \(\lambda \gg \lambda ^\prime \), the value of \(c_\Sigma (m_{3/2}/m_{1/2})^2\) is very near \(-0.25\) all across the plane. As long as \(c_H\) is relatively small, one can see from Eq. (28) that the value of \(c_H\) has little effect on our estimates of \(c_\Sigma (m_{3/2}/m_{1/2})^2\), which are quoted assuming \(c_H = 0\). The large ratio of \(\lambda /\lambda ^\prime \) is beneficial for increasing the proton lifetime, and contours showing the lifetime are seen as solid black curves in the lower right portion of the panel, labeled in units of \(10^{35}\) years;^{11} as the current experimental limit is \(\tau (p\rightarrow K^+ \overline{\nu }) > 6.6\times 10^{33}\) years [97, 98], the region with acceptable proton stability lies below the contour labeled 0.066. Whilst it is encouraging that some region of parameter space exists with a sufficiently long proton lifetime and acceptable Higgs mass, the relic density is far too large in this region: \(\Omega h^2 \sim \mathcal {O}(100)\). Further exploration in the (\(M_\mathrm{in}, \lambda , \lambda ^\prime \)) parameter space does not yield better results. The Higgs mass can be made compatible with either the relic density or the proton lifetime, but not both.

The left panel of Fig. 1 shows that, at fixed \(m_{1/2}\), the value of \(m_h\) decreases rapidly when \(\tan \beta \lesssim 10\). On the other hand, the right panel of Fig. 1 shows that the proton lifetime is unacceptably short for \(\tan \beta \gtrsim 10\). As we discuss below with several examples, these two problems can be avoided simultaneously when \(\tan \beta = 7\), for suitable choices of the other super-GUT model parameters \(M_\mathrm{in}, \lambda \) and \(\lambda ^\prime \). We do not discuss in the following possible variations in the value of \(\tan \beta \), but have checked that values differing from 7 by factors \(\gtrsim 2\) are typically excluded by either \(m_h\) or the proton lifetime.

### 3.2 Twisted *H* and \(\overline{H}\) Higgs fields

In this subsection we consider departures from the minimal model discussed above that allow for more successful phenomenology. We start by considering the consequences of a twisted Higgs sector. As discussed above, \(\tan \beta \) must be relatively low to obtain sufficiently long proton lifetimes. However, in order to obtain a sufficiently large Higgs mass, \(\tan \beta \) should not be too low. Choosing \(\tan \beta = 7\) with \(\lambda ^\prime = 10^{-5}\) optimizes both \(m_h\) and \(\tau _p\), so we fix those values for now. In the following, we take \(\lambda = 0.6\) and 1.

*H*and \(\overline{H}\) are twisted,

*A*and

*B*terms are given at the input renormalization scale by

*A*and

*B*terms at the input scale. We assume \(A_{t,b} = m_{3/2}\), \(A_\lambda = 2 m_{3/2}\), \(A_{\lambda '} = 0\), \(B_H = 2 m_{3/2}\) and \(B_\Sigma = 0\) at the input renormalization scale, \(M_\mathrm{in}\). The (\(m_{1/2}, m_1\)) plane for this case with \(M_\mathrm{in} = M_\mathrm{GUT}\) is shown in the left panel of Fig. 2. This is the limiting case in which the super-GUT scenario reduces to an NUHM1 plane [37, 38, 99, 100, 101] with \(m_0 = 0\) and \(A_0 = m_1\). Note that the values of \(\lambda \) and \(\lambda ^\prime \) are irrelevant when taking \(M_\mathrm{in} = M_\mathrm{GUT}\) as there is no running above the GUT scale in this case. There is narrow band where the LSP is the lightest neutralino and the electroweak symmetry-breaking conditions can be satisfied, through which runs a blue relic density strip.

^{12}At low values of \(m_{1/2}\), the relic density is determined by stau coannihilation [102, 103, 104, 105, 106, 107, 108, 109], and the blue relic density strip lies close to the boundary of the stau LSP region (shaded red). At higher \(m_{1/2}\), the strip moves closer to the region with no electroweak symmetry breaking (shaded pink) and becomes a focus-point strip [110, 111, 112, 113, 114, 115]. The Higgs mass (shown by the red dot-dashed contours between the two excluded regions) has acceptable values along much of the relic density strip. On the other hand, the proton lifetime is too short as the entire strip shown lies at or below the contour corresponding to \(\tau _p = 0.001 \times 10^{35}\) years (which appears as the black curve that enters the allowed region at about 5 TeV at an angle to the relic density strip). The right panel of Fig. 2 shows the corresponding plane with the following choices of modular weights: \(\alpha _{t,b} = 1\), \(\alpha _\lambda = 2\), \(\alpha _{\lambda '} = 0\), \(\beta _H = 2\) and \(\beta _\Sigma = 0\), which correspond to \(A_0 = B_0 = 0\). This exhibits many features similar to the left panel. In particular, the relic density and proton lifetime constraints are incompatible, motivating our exploration of super-GUT scenarios.

Another very obvious difference between the left panel of Figs. 2 and 3 is the value of the proton lifetime. With \(M_\mathrm{in} = M_\mathrm{GUT}\), the entire strip shown has a lifetime \(\tau _p < 10^{33}\) years, as it lies to the left of the contour labeled 0.01. However, the proton lifetime is significantly longer in both panels of Fig. 3, and there are acceptable parts of the relic density strip where \(\tau _p > 0.066 \times 10^{35}\) years. Comparing the two panels allows one to see the effect of increasing \(\lambda \) on the proton lifetime. For \(\lambda = 0.6\), the lifetime is sufficiently long for \(m_{1/2} \gtrsim 5\) TeV, whereas for \(\lambda = 1\) this is relaxed to \(m_{1/2} \gtrsim 2.5\) TeV. Increasing \(\lambda \) much further is not possible due to its effect on the Yukawa couplings, as discussed in [58]. In both cases, the Higgs masses are reasonably consistent with 125 GeV, though due to the increased “bending” of the contours, the Higgs mass along the relic density strip is slightly lower for the larger value of \(\lambda \).^{13} The GM couplings are also acceptably small: in the GUT case shown in the left panel of Fig. 2 they are \(\ll \)1 across the plane, whereas in Fig. 3 \(c_\Sigma (m_{3/2}/m_{1/2})^2\) is of order 0.05 all along the relic density strip.

Since the strips with acceptable relic density in these models resemble the familiar focus-point region [110, 111, 112, 113, 114, 115], one can expect that the spin-independent elastic scattering cross section on protons, \(\sigma ^\mathrm{SI}\), may be relatively large. Concentrating on the right panel of Fig. 3, we have computed \(\sigma ^\mathrm{SI}\) at two points: (\(m_{1/2}, m_1\)) = (3100, 6000) GeV and (4100, 8000) GeV. The resulting cross sections are \(\sigma ^\mathrm{SI} = (1.24 \pm 0.77) \times 10^{-8}\) pb and \((1.90 \pm 1.19) \times 10^{-9}\) pb with \(m_\chi = 930\) GeV and 1400 GeV, respectively, where we have assumed \(\Sigma _{\pi N} = 50 \pm 8\) MeV [119] and \(\sigma _0 = 36\pm 7\) MeV [120]. The central value for the former point is slightly above the recent LUX [121] and PandaX [122] bounds, but remains acceptable when uncertainties in the computed cross sections are taken into account. Furthermore, using nucleon matrix elements computed with lattice simulations as in [123] would reduce the predicted cross section by more than a factor 2 due to the smallness of strange-quark content in a nucleon. However, in both the cases studied one may anticipate a positive signal in upcoming direct detection experiments such as LUX-Zeplin and XENON1T/nT [124, 125].

*A*and

*B*terms. The right panel of Fig. 2 shows the \((m_{1/2}, m_1)\) plane for \(M_\mathrm{in} = M_\mathrm{GUT}\), which is similar to that shown in the left panel when

*A*and

*B*terms are non-zero. The

*A*and

*B*terms are seen to affect somewhat the dependence on \(m_1\) of the Higgs mass and the position of the relic density strip. The same case with \(A_0 = B_0 = 0\) but \(M_\mathrm{in} = 10^{16.5}\) GeV is shown in the left panel of Fig. 4. Comparing this with the right panel of Fig. 3, we see that the proton lifetime shows little dependence on \(A_0\) and is similar in the two cases shown. For larger \(M_\mathrm{in} = 10^{18}\) GeV with \(A_0 = B_0 = 0\), as shown in the right panel of Fig. 4, we see that the relic density strip shifts to larger values of \(m_1\) and the proton lifetime is somewhat longer. Much of the allowed dark matter strip has an acceptably long proton lifetime. The effect of adjusting the modular weights does not have a major effect on the elastic scattering cross section.

*H*twisted,

*A*-terms related via the Yukawa couplings.

*H*remains untwisted, we have \(m_0 = m_2 = 0\) and, since the two Higgs soft masses are unequal, this is an example of a super-GUT NUHM2 model [37, 38, 101, 126, 127].

^{14}The region where one obtains an acceptable relic density could be expected from the upper left panel of Fig. 14 in [101], which shows an example of an \((m_1,m_2)\) plane for relatively low \(m_{1/2}\), \(m_0\) and \(\tan \beta \). For \(m_2 = 0\), we expect that there should be a funnel strip [11, 12, 13, 14, 15] where

*s*-channel annihilation of the LSP through the heavy Higgs scalar and pseudoscalar dominates the total cross section and \(m_\chi \approx m_A/2\). This generally occurs when \(m_1^2 < 0\) at the input scale.

In the left panel of Fig. 5, we have taken \(\alpha _t = 0\), \(\alpha _{b} = 1\), \(\alpha _\lambda = 1\), \(\alpha _{\lambda '} = 0\), \(\beta _H = 1\) and \(\beta _\Sigma = 0\), so that all the *A* and *B* terms vanish at the input scale. In the pink shaded region, the electroweak symmetry-breaking (EWSB) conditions cannot be satisfied as \(m_A^2 < 0\). Indeed, for \(m_1^2 < 0\), we see a blue relic density strip above the shaded region. Whilst the proton lifetime is sufficiently large for \(m_{1/2} \gtrsim 1.8\) TeV, the strip extends (barely visibly) to \(m_h = 123\) GeV (shown by the red dot-dashed contours). In the right panel of this figure, we have set all weights to zero, and therefore \(A_t = 0\), \(A_b = m_1\), \(A_\lambda = m_1\), \(A_{\lambda '} = 0\), \(B_H = m_1\), and \(B_\Sigma = 0\). Qualitatively, the two figures are very similar. The strip extends to slightly larger \(m_h\) but, again, not much past 123 GeV. In both cases, \(A_t = 0\) at the input scale and, although \(A_b \ne 0\) in the right panel, the dominant factor contributing to the Higgs mass is \(A_t\). In both panels \(c_\Sigma (m_{3/2}/m_{1/2})^2 \approx -0.25\) in the allowed regions of the parameter space. We see that the proton lifetime is acceptably long when \(m_{1/2} \gtrsim 1.7\) TeV along the dark matter strip. The elastic cross section near the end point of the relic density strip where \(m_1 \approx -3500\) GeV is quite small: \(\sigma ^\mathrm{SI} \approx 1 \times 10^{-11}\) pb with \(m_\chi \approx 800\) GeV, probably beyond the reach of LUX-Zeplin and XENON1T/nT [124, 125], though still above the neutrino background level.

The Higgs mass can be increased slightly by turning on the weight \(\alpha _t\) controlling \(A_t\). To determine the optimal value for \(\alpha _t\), for all other \(A_i=0\) and \(B_i=0\), we scan over \(\alpha _t\). In the left panel of Fig. 6 the resulting \((A_t, m_{1/2})\) plane for fixed \(m_1 = -3000\) GeV and \(m_0 = m_2 = 0\) is shown. Once again, the pink shaded region is excluded as \(m_A^2 < 0\) and the constraints for electroweak symmetry breaking cannot be satisfied. The blue line (enhanced here for visibility) shows the position of the relic density funnel strip. We see that the largest value of the Higgs mass obtained is slightly larger than 124 GeV, which is reached when \(A_t/|m_1| \sim 1\). The proton lifetime is acceptably long for \(m_{1/2} \gtrsim 1.8\) TeV along the dark matter strip, and the GM coupling shown by the green lines is \(\gtrsim -1.5\) in this region. In the right panel, we show the corresponding \((m_{1/2}, m_1)\) plane with \(A_t = m_1\) and again all other \(A_i = B_i = 0\). Here we see that the funnel strip extends to Higgs masses slightly larger than 124 GeV, where the proton lifetime is about \(10^{34}\) years. Points along the dark matter strip with \(m_{1/2} \gtrsim 1.7\) TeV have an acceptably long proton lifetime. In both cases, displayed, the elastic cross sections are relatively small. Near the end point of the relic density strip where \(m_1 \approx -3000\) GeV, we find \(\sigma ^\mathrm{SI} \approx 2 \times 10^{-11}\) pb with \(m_\chi \approx 950\) GeV. Although this cross section is still above the neutrino background, it may be difficult to detect in the planned LUX-Zeplin and XENON1T/nT experiments.

*H*leaving \(\overline{H}\) untwisted. In this case, \(m_0 = m_1 = 0\), and previous studies lead us to expect the relic density strip to lie at positive values of \(m_2^2\). Once again, we have taken \(\tan \beta = 7\), \(\lambda = 1\), and \(\lambda ^\prime = 10^{-5}\). In the left panel of Fig. 7, we have taken \(\alpha _t = 1\), \(\alpha _{b} = 0\), \(\alpha _\lambda = 1\), \(\alpha _{\lambda '} = 0\), \(\beta _H = 1\) and \(\beta _\Sigma = 0\), so that all

*A*and

*B*terms vanish at the input scale. In the pink shaded region, the EWSB conditions cannot be satisfied, but in this case it is because \(\mu ^2 < 0\). Just to the right of the excluded region, we see the equivalent of the focus-point strip, where the LSP is mostly Higgsino. Still further to the right, we see two closely spaced strips corresponding to the funnel region with a mostly bino-like LSP. For this choice of \(\lambda \) and \(\lambda ^\prime \), the proton lifetime is sufficiently long if \(m_{1/2} \gtrsim 1.8\) TeV, but the Higgs mass is \(\lesssim 123\) GeV unless \(m_{1/2} \gtrsim 2.7\) TeV. In the right panel of Fig. 7, we again take all weights equal to 0, so that \(A_t = m_2\), \(A_b = 0\), \(A_\lambda = m_2\), \(A_{\lambda '} = 0\), \(B_H = m_2\), and \(B_\Sigma = 0\). In this case, the pink shaded region has \(m_A^2 < 0\) and we see the funnel strip running to values of \(m_h > 125\) GeV. Comparing this with the left panel, we see the effect of the non-zero value of \(A_t\) on \(m_h\). In both panels we see that points along the dark matter strips with \(m_{1/2} \gtrsim 1.7\) TeV have an acceptably long proton lifetime.

Since we have both a focus-point strip and a funnel region, there is more variation in the computed elastic cross section. Corresponding to the left panel of Fig. 7, we considered points at \(m_2 = 4000\) GeV with \(m_{1/2} \simeq 2700\) GeV (focus point with \(m_\chi \simeq 900\) GeV) and \(m_{1/2} \simeq 3200\) GeV (funnel with \(m_\chi \simeq 1160\) GeV). We found \(\sigma ^\mathrm{SI} \simeq (2.2 \pm 1.4) \times 10^{-8}\) pb and \((1.2 \pm 0.7) \times 10^{-10}\) pb respectively. At higher \(m_0 = 5000\) GeV, the cross section on the focus point at \(m_{1/2} \simeq 1065\) GeV drops to \((6.4 \pm 4.0) \times 10^{-9}\) pb and on the funnel at \(m_{1/2} \simeq 1530\) GeV drops to \((7.2 \pm 4.5) \times 10^{-11}\) pb. When the weights are set to zero as in the right panel of Fig. 7, we have only a funnel strip and the cross section is quite low. For \((m_{1/2}, m_2)\) = (1920,3000), we find \(\sigma ^\mathrm{SI} \simeq (5.3 \pm 3.3) \times 10^{-11}\) pb and for \((m_{1/2}, m_2)\) = (2965,4400), we find \(\sigma ^\mathrm{SI} = (2.2 \pm 1.4) \times 10^{-11}\) pb.

## 4 Discussion

Working within a no-scale supergravity framework inspired by string compactification scenarios, we have shown in this paper that, *if* the matter and Higgs supermultiplets are all *untwisted*, super-GUT SU(5) models are unable to provide simultaneously a long enough proton lifetime, a small enough relic LSP density and an acceptable Higgs mass in the framework of no-scale supergravity, even in the presence of a Giudice–Masiero term in the Kähler potential. However, all of these phenomenological requirements can be reconciled *if* one or both of the GUT Higgs fiveplets is *twisted*. We have exhibited satisfactory solutions for various values of the input super-GUT scale \(M_\mathrm{in}\), the GUT Yukawa couplings that are important in the RGEs above the GUT scale, and the modular weights of the various matter and Higgs fields. All the examples shown assume \(\tan \beta = 7\): significantly smaller values of \(\tan \beta \) are largely excluded because \(m_h\) is too small, and significantly larger values of \(\tan \beta \) are largely excluded because the proton lifetime is too short. Spin-independent dark matter scattering may be observable in some of the cases studied.

Although, as we have shown, many of the problems of the minimal SU(5) GUT model may be resolved in the no-scale SU(5) super-GUT, including rapid proton decay through dimension-five operators, in a manner compatible with the dark matter density and the Higgs mass, other issues such as neutrino masses/oscillations remain unresolved. Moreover, the resolution of the minimal supersymmetric SU(5) GUT problems within the super-GUT and no-scale supergravity frameworks is quite constrained and somewhat contrived. It also remains unclear how an SU(5) GUT model could be embedded within string theory. However, our analysis may point the way how this could be done successfully.

A natural alternative is the flipped SU(5) \(\times \) U(1) framework proposed in [128, 129, 130, 131, 132, 133], which resolves automatically the problems mentioned above, and can be embedded with string theory. Choosing even the simplest strict no-scale boundary conditions \(m_0=A_0=B_0 = 0\) at \(M_\mathrm{in}\) provides a very interesting flipped SU(5) framework that satisfies all the constraints from present low-energy phenomenology, including the relic dark matter density and the proton lifetime, and makes interesting predictions for Run 2 of the LHC [134, 135, 136, 137]. Moreover, flipped SU(5) also contains a rationale for \(M_\mathrm{in} > M_\mathrm{GUT}\), since the final unification of the SU(5) and U(1) gauge couplings could well occur at the string scale. We therefore plan to consider the possibility of a no-scale flipped SU(5) super-GUT in a forthcoming paper.

## Footnotes

- 1.
In addition, one must choose the sign of \(\mu \), which we take here to positive.

- 2.
In order to get electroweak symmetry breaking to work, the

*B*terms in these models also get a contribution from a Giudice–Masiero term [48]. - 3.
- 4.
- 5.
The parameter \(\mu _\Lambda \) does not play any other role in our construction, and its precise value is unimportant for our analysis.

- 6.
For example, \(V=9\times 10^{16}-3\times 10^{17}\) GeV in Fig. 1 for \(\lambda '=10^{-5}\).

- 7.
- 8.
Note that here we use FeynHiggs version 2.11.3, which gives a slightly lower value of \(m_h\) than the version used in [59]. In addition, since FeynHiggs does not produce stable results in the upper right portion of the plane, the Higgs contours terminate in this region.

- 9.
The slight differences between these and past results arise mostly because here we do not force the strong gauge coupling to be equal to the electroweak couplings at the GUT scale.

- 10.
We make no specific assumption as regards the magnitude of \(m_{3/2}\), except that it is large enough for the LSP to be the lightest neutralino, rather than the gravitino.

- 11.
- 12.
The relic density strip has been enhanced here and in subsequent figures for better visibility by showing regions with \(\Omega _\chi h^2\) lies between 0.06 and 0.2.

- 13.
At higher \(M_\mathrm{in}\), the bending of Higgs mass contours seen in Fig. 3 as they approach the region with no radiative electroweak symmetry breaking (shaded pink) becomes more severe, and the Higgs mass becomes too low all along the relic density strip.

- 14.
The quoted sign of \(m_1\) actually represents the sign of \(m_1^2\) at the input scale.

## Notes

### Acknowledgements

The work of J. E. was supported in part by the UK STFC via the research Grant ST/J002798/1. The work of D. V. N. was supported in part by the DOE Grant DE-FG02-13ER42020 and in part by the Alexander S. Onassis Public Benefit Foundation. The work of N. N. and K. A. O. was supported in part by DOE Grant DE-SC0011842 at the University of Minnesota.

## References

- 1.H. Georgi, S.L. Glashow, Phys. Rev. Lett.
**32**, 438 (1974). doi: 10.1103/PhysRevLett.32.438 ADSCrossRefGoogle Scholar - 2.S. Dimopoulos, H. Georgi, Nucl. Phys. B
**193**, 150 (1981)ADSCrossRefGoogle Scholar - 3.N. Sakai, Z. Phys. C
**11**, 153 (1981)ADSCrossRefGoogle Scholar - 4.E. Cremmer, S. Ferrara, C. Kounnas, D.V. Nanopoulos, Phys. Lett.
**133B**, 61 (1983). doi: 10.1016/0370-2693(83)90106-5 ADSCrossRefGoogle Scholar - 5.J.R. Ellis, A.B. Lahanas, D.V. Nanopoulos, K. Tamvakis, Phys. Lett.
**134B**, 429 (1984). doi: 10.1016/0370-2693(84)91378-9 ADSCrossRefGoogle Scholar - 6.J.R. Ellis, C. Kounnas, D.V. Nanopoulos, Nucl. Phys. B
**247**, 373 (1984). doi: 10.1016/0550-3213(84)90555-8 ADSCrossRefGoogle Scholar - 7.A.B. Lahanas, D.V. Nanopoulos, Phys. Rep.
**145**, 1 (1987). doi: 10.1016/0370-1573(87)90034-2 ADSCrossRefGoogle Scholar - 8.E. Witten, Phys. Lett. B
**155**, 151 (1985)ADSMathSciNetCrossRefGoogle Scholar - 9.J. Ellis, D.V. Nanopoulos, K.A. Olive, Phys. Rev. Lett.
**111**, 111301 (2013). doi: 10.1103/PhysRevLett.111.129902. arXiv:1305.1247 [hep-th] [Erratum: Phys. Rev. Lett.**111**, 129902 (2013) , doi: 10.1103/PhysRevLett.111.111301] - 10.J. Ellis, M.A.G. Garcia, D.V. Nanopoulos, K.A. Olive, Class. Quant. Grav.
**33**, 094001 (2016). doi: 10.1088/0264-9381/33/9/094001. arXiv:1507.02308 [hep-ph]ADSCrossRefGoogle Scholar - 11.
- 12.
- 13.
- 14.H. Baer, M. Brhlik, M.A. Diaz, J. Ferrandis, P. Mercadante, P. Quintana, X. Tata, Phys. Rev. D
**63**, 015007 (2001). arXiv:hep-ph/0005027 ADSCrossRefGoogle Scholar - 15.J.R. Ellis, T. Falk, G. Ganis, K.A. Olive, M. Srednicki, Phys. Lett. B
**510**, 236 (2001). arXiv:hep-ph/0102098 ADSCrossRefGoogle Scholar - 16.G.L. Kane, C.F. Kolda, L. Roszkowski, J.D. Wells, Phys. Rev. D
**49**, 6173 (1994). arXiv:hep-ph/9312272 ADSCrossRefGoogle Scholar - 17.J.R. Ellis, T. Falk, K.A. Olive, M. Schmitt, Phys. Lett. B
**388**, 97 (1996). arXiv:hep-ph/9607292 ADSCrossRefGoogle Scholar - 18.J.R. Ellis, T. Falk, K.A. Olive, M. Schmitt, Phys. Lett. B
**413**, 355 (1997). arXiv:hep-ph/9705444 ADSCrossRefGoogle Scholar - 19.
- 20.L. Roszkowski, R. Ruiz de Austri, T. Nihei, JHEP
**0108**, 024 (2001). arXiv:hep-ph/0106334 ADSCrossRefGoogle Scholar - 21.A. Djouadi, M. Drees, J.L. Kneur, JHEP
**0108**, 055 (2001). arXiv:hep-ph/0107316 ADSCrossRefGoogle Scholar - 22.U. Chattopadhyay, A. Corsetti, P. Nath, Phys. Rev. D
**66**, 035003 (2002). arXiv:hep-ph/0201001 ADSCrossRefGoogle Scholar - 23.J.R. Ellis, K.A. Olive, Y. Santoso, New J. Phys.
**4**, 32 (2002). arXiv:hep-ph/0202110 ADSCrossRefGoogle Scholar - 24.H. Baer, C. Balazs, A. Belyaev, J.K. Mizukoshi, X. Tata, Y. Wang, JHEP
**0207**, 050 (2002). arXiv:hep-ph/0205325 ADSCrossRefGoogle Scholar - 25.R. Arnowitt, B. Dutta, Invited talk at Identification of Dark Matter (IDM 2002) (2002). arXiv:hep-ph/0211417
- 26.J.R. Ellis, T. Falk, G. Ganis, K.A. Olive, M. Schmitt, Phys. Rev. D
**58**, 095002 (1998). arXiv:hep-ph/9801445 ADSCrossRefGoogle Scholar - 27.J.R. Ellis, T. Falk, G. Ganis, K.A. Olive, Phys. Rev. D
**62**, 075010 (2000). arXiv:hep-ph/0004169 ADSCrossRefGoogle Scholar - 28.J.R. Ellis, K.A. Olive, Y. Santoso, V.C. Spanos, Phys. Lett. B
**565**, 176 (2003). arXiv:hep-ph/0303043 ADSCrossRefGoogle Scholar - 29.
- 30.A.B. Lahanas, D.V. Nanopoulos, Phys. Lett. B
**568**, 55 (2003). arXiv:hep-ph/0303130 ADSCrossRefGoogle Scholar - 31.U. Chattopadhyay, A. Corsetti, P. Nath, Phys. Rev. D
**68**, 035005 (2003). arXiv:hep-ph/0303201 ADSCrossRefGoogle Scholar - 32.
- 33.R. Arnowitt, B. Dutta, B. Hu, Plenary talk at Beyond The Desert ’03, Castle Ringberg, Germany, (2003). arXiv:hep-ph/0310103
- 34.J. Ellis, K.A. Olive, in
*Particle Dark Matter: Observations, Models and Searches*(Chap. 8), ed. by G. Bertone, pp. 142–163. Hardback. ISBN 9780521763684. arXiv:1001.3651 [astro-ph.CO] - 35.J. Ellis, K.A. Olive, Eur. Phys. J. C
**72**, 2005 (2012). arXiv:1202.3262 [hep-ph]ADSCrossRefGoogle Scholar - 36.O. Buchmueller et al., Eur. Phys. J. C
**74**(3), 2809 (2014). arXiv:1312.5233 [hep-ph]ADSCrossRefGoogle Scholar - 37.J. Ellis, F. Luo, K.A. Olive, P. Sandick, Eur. Phys. J. C
**73**(4), 2403 (2013). arXiv:1212.4476 [hep-ph]ADSCrossRefGoogle Scholar - 38.J. Ellis, J.L. Evans, F. Luo, N. Nagata, K.A. Olive, P. Sandick, Eur. Phys. J. C
**76**(1), 8 (2016). doi: 10.1140/epjc/s10052-015-3842-6. arXiv:1509.08838 [hep-ph]ADSCrossRefGoogle Scholar - 39.R. Barbieri, S. Ferrara, C.A. Savoy, Phys. Lett. B
**119**, 343 (1982)ADSCrossRefGoogle Scholar - 40.J.R. Ellis, K.A. Olive, Y. Santoso, V.C. Spanos, Phys. Lett. B
**573**, 162 (2003). arXiv:hep-ph/0305212 ADSCrossRefGoogle Scholar - 41.J.R. Ellis, K.A. Olive, Y. Santoso, V.C. Spanos, Phys. Rev. D
**70**, 055005 (2004). arXiv:hep-ph/0405110 ADSCrossRefGoogle Scholar - 42.M. Ibe, T. Moroi, T.T. Yanagida, Phys. Lett. B
**644**, 355 (2007). arXiv:hep-ph/0610277 ADSCrossRefGoogle Scholar - 43.M. Ibe, T.T. Yanagida, Phys. Lett. B
**709**, 374 (2012). arXiv:1112.2462 [hep-ph]ADSCrossRefGoogle Scholar - 44.M. Ibe, S. Matsumoto, T.T. Yanagida, Phys. Rev. D
**85**, 095011 (2012). arXiv:1202.2253 [hep-ph]ADSCrossRefGoogle Scholar - 45.J.L. Evans, M. Ibe, K.A. Olive, T.T. Yanagida, Phys. Rev. D
**91**, 055008 (2015). doi: 10.1103/PhysRevD.91.055008. arXiv:1412.3403 [hep-ph]ADSCrossRefGoogle Scholar - 46.J.L. Evans, M. Ibe, K.A. Olive, T.T. Yanagida, Eur. Phys. J. C
**73**, 2468 (2013). doi: 10.1140/epjc/s10052-013-2468-9. arXiv:1302.5346 [hep-ph]ADSCrossRefGoogle Scholar - 47.J.L. Evans, K.A. Olive, M. Ibe, T.T. Yanagida, Eur. Phys. J. C
**73**(10), 2611 (2013). doi: 10.1140/epjc/s10052-013-2611-7. arXiv:1305.7461 [hep-ph]ADSCrossRefGoogle Scholar - 48.G.F. Giudice, A. Masiero, Phys. Lett. B
**206**, 480 (1988)ADSCrossRefGoogle Scholar - 49.
- 50.L. Randall, R. Sundrum, Nucl. Phys. B
**557**, 79 (1999). arXiv:hep-th/9810155 ADSCrossRefGoogle Scholar - 51.G.F. Giudice, M.A. Luty, H. Murayama, R. Rattazzi, JHEP
**9812**, 027 (1998). arXiv:hep-ph/9810442 ADSCrossRefGoogle Scholar - 52.J.A. Bagger, T. Moroi, E. Poppitz, JHEP
**0004**, 009 (2000). arXiv:hep-th/9911029 ADSCrossRefGoogle Scholar - 53.P. Binetruy, M.K. Gaillard, B.D. Nelson, Nucl. Phys. B
**604**, 32 (2001). arXiv:hep-ph/0011081 ADSCrossRefGoogle Scholar - 54.J.R. Ellis, K.A. Olive, P. Sandick, Phys. Lett. B
**642**, 389 (2006). arXiv:hep-ph/0607002 ADSCrossRefGoogle Scholar - 55.J.R. Ellis, K.A. Olive, P. Sandick, JHEP
**0706**, 079 (2007). arXiv:0704.3446 [hep-ph]ADSCrossRefGoogle Scholar - 56.J.R. Ellis, K.A. Olive, P. Sandick, JHEP
**0808**, 013 (2008). arXiv:0801.1651 [hep-ph]ADSCrossRefGoogle Scholar - 57.J. Ellis, A. Mustafayev, K.A. Olive, Eur. Phys. J. C
**69**, 201 (2010). doi: 10.1140/epjc/s10052-010-1373-8. arXiv:1003.3677 [hep-ph]ADSCrossRefGoogle Scholar - 58.J. Ellis, J.L. Evans, A. Mustafayev, N. Nagata, K.A. Olive, Eur. Phys. J. C
**76**, 592 (2016). doi: 10.1140/epjc/s10052-016-4437-6. arXiv:1608.05370 [hep-ph]ADSCrossRefGoogle Scholar - 59.J. Ellis, A. Mustafayev, K.A. Olive, Eur. Phys. J. C
**69**, 219 (2010). arXiv:1004.5399 [hep-ph]ADSCrossRefGoogle Scholar - 60.J.L. Evans, D.E. Morrissey, J.D. Wells, Phys. Rev. D
**75**, 055017 (2007). doi: 10.1103/PhysRevD.75.055017. arXiv:hep-ph/0611185 ADSCrossRefGoogle Scholar - 61.J.L. Evans, D.E. Morrissey, J.D. Wells, Phys. Rev. D
**80**, 095011 (2009). doi: 10.1103/PhysRevD.80.095011. arXiv:0812.3874 [hep-ph]ADSCrossRefGoogle Scholar - 62.J.R. Ellis, D.V. Nanopoulos, K.A. Olive, Phys. Lett. B
**525**, 308 (2002). arXiv:hep-ph/0109288 ADSCrossRefGoogle Scholar - 63.J. Ellis, M.A.G. Garcia, D.V. Nanopoulos, K.A. Olive, JCAP
**1510**(10), 003 (2015). doi: 10.1088/1475-7516/2015/10/003. arXiv:1503.08867 [hep-ph]ADSCrossRefGoogle Scholar - 64.K. Choi, A. Falkowski, H.P. Nilles, M. Olechowski, Nucl. Phys. B
**718**, 113 (2005). arXiv:hep-th/0503216 ADSCrossRefGoogle Scholar - 65.O. Lebedev, H.P. Nilles, M. Ratz, Phys. Lett. B
**636**, 126 (2006). arXiv:hep-th/0603047 ADSMathSciNetCrossRefGoogle Scholar - 66.O. Lebedev, V. Lowen, Y. Mambrini, H.P. Nilles, M. Ratz, JHEP
**0702**, 063 (2007). arXiv:hep-ph/0612035 ADSCrossRefGoogle Scholar - 67.J.L. Evans, N. Nagata, K.A. Olive, Phys. Rev. D
**91**, 055027 (2015). doi: 10.1103/PhysRevD.91.055027. arXiv:1502.00034 [hep-ph]ADSCrossRefGoogle Scholar - 68.E. Dudas, Y. Mambrini, A. Mustafayev, K.A. Olive, Eur. Phys. J. C
**72**, 2138 (2012); [Erratum: Eur. Phys. J. C**73**, 2430 (2013). arXiv:1205.5988 [hep-ph]] - 69.E. Dudas, A. Linde, Y. Mambrini, A. Mustafayev, K.A. Olive, Eur. Phys. J. C
**73**(1), 2268 (2013). arXiv:1209.0499 [hep-ph]ADSCrossRefGoogle Scholar - 70.N. Polonsky, A. Pomarol, Phys. Rev. Lett.
**73**, 2292 (1994). arXiv:hep-ph/9406224 ADSCrossRefGoogle Scholar - 71.N. Polonsky, A. Pomarol, Phys. Rev. D
**51**, 6532 (1995). arXiv:hep-ph/9410231 ADSCrossRefGoogle Scholar - 72.H. Baer, M.A. Diaz, P. Quintana, X. Tata, JHEP
**0004**, 016 (2000). arXiv:hep-ph/0002245 ADSCrossRefGoogle Scholar - 73.J. Ellis, A. Mustafayev, K.A. Olive, Eur. Phys. J. C
**71**, 1689 (2011). arXiv:1103.5140 [hep-ph]ADSCrossRefGoogle Scholar - 74.J. Hisano, H. Murayama, T. Yanagida, Phys. Rev. Lett.
**69**, 1014 (1992)ADSCrossRefGoogle Scholar - 75.J. Hisano, H. Murayama, T. Yanagida, Nucl. Phys. B
**402**, 46 (1993). arXiv:hep-ph/9207279 ADSCrossRefGoogle Scholar - 76.J. Hisano, T. Kuwahara, N. Nagata, Phys. Lett. B
**723**, 324 (2013). arXiv:1304.0343 [hep-ph]ADSCrossRefGoogle Scholar - 77.J.R. Ellis, K. Enqvist, D.V. Nanopoulos, K. Tamvakis, Phys. Lett. B
**155**, 381 (1985)ADSCrossRefGoogle Scholar - 78.C.T. Hill, Phys. Lett. B
**135**, 47 (1984)ADSCrossRefGoogle Scholar - 79.Q. Shafi, C. Wetterich, Phys. Rev. Lett.
**52**, 875 (1984)ADSCrossRefGoogle Scholar - 80.M. Drees, Phys. Lett. B
**158**, 409 (1985)ADSCrossRefGoogle Scholar - 81.M. Drees, Phys. Rev. D
**33**, 1468 (1986)ADSCrossRefGoogle Scholar - 82.
- 83.B. Bajc, P. Fileviez Perez, G. Senjanovic, arXiv:hep-ph/0210374
- 84.J. Hisano, H. Murayama, T. Goto, Phys. Rev. D
**49**, 1446 (1994)ADSCrossRefGoogle Scholar - 85.F. Borzumati, T. Yamashita, Prog. Theor. Phys.
**124**, 761 (2010). arXiv:0903.2793 [hep-ph]ADSCrossRefGoogle Scholar - 86.Y. Kawamura, H. Murayama, M. Yamaguchi, Phys. Rev. D
**51**, 1337 (1995). arXiv:hep-ph/9406245 ADSCrossRefGoogle Scholar - 87.V.D. Barger, M.S. Berger, P. Ohmann, Phys. Rev. D
**49**, 4908 (1994). arXiv:hep-ph/9311269 ADSCrossRefGoogle Scholar - 88.W. de Boer, R. Ehret, D.I. Kazakov, Z. Phys. C
**67**, 647 (1995). arXiv:hep-ph/9405342 ADSCrossRefGoogle Scholar - 89.M. Carena, J.R. Ellis, A. Pilaftsis, C.E. Wagner, Nucl. Phys. B
**625**, 345 (2002). arXiv:hep-ph/0111245 ADSCrossRefGoogle Scholar - 90.M. Schmaltz, W. Skiba, Phys. Rev. D
**62**, 095005 (2000). arXiv:hep-ph/0001172 ADSCrossRefGoogle Scholar - 91.M. Schmaltz, W. Skiba, Phys. Rev. D
**62**, 095004 (2000). arXiv:hep-ph/0004210 ADSCrossRefGoogle Scholar - 92.J. Ellis, K. Olive, L. Velasco-Sevilla, Eur. Phys. J. C
**76**(10), 562 (2016). doi: 10.1140/epjc/s10052-016-4398-9. arXiv:1605.01398 [hep-ph]ADSCrossRefGoogle Scholar - 93.T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak, G. Weiglein, Phys. Rev. Lett.
**112**(14), 141801 (2014). arXiv:1312.4937 [hep-ph]ADSCrossRefGoogle Scholar - 94.J. Hisano, D. Kobayashi, T. Kuwahara, N. Nagata, JHEP
**1307**, 038 (2013). arXiv:1304.3651 [hep-ph]ADSCrossRefGoogle Scholar - 95.
- 96.J.R. Ellis, M.K. Gaillard, D.V. Nanopoulos, Phys. Lett. B
**88**, 320 (1979)ADSCrossRefGoogle Scholar - 97.V. Takhistov, Super-Kamiokande Collaboration, in
*Contribution to the proceedings of the 51st Rencontres de Moriond: Electroweak Interactions and Unified Theories*(La Thuile, Italy, March 12–19, 2016). arXiv:1605.03235 [hep-ex] - 98.K. Abe et al., Super-Kamiokande Collaboration, Phys. Rev. D
**90**(7), 072005 (2014). arXiv:1408.1195 [hep-ex] - 99.H. Baer, A. Mustafayev, S. Profumo, A. Belyaev, X. Tata, Phys. Rev. D
**71**, 095008 (2005). arXiv:hep-ph/0412059 ADSCrossRefGoogle Scholar - 100.H. Baer, A. Mustafayev, S. Profumo, A. Belyaev, X. Tata, JHEP
**0507**, 065 (2005). arXiv:hep-ph/0504001 ADSCrossRefGoogle Scholar - 101.J.R. Ellis, K.A. Olive, P. Sandick, Phys. Rev. D
**78**, 075012 (2008). arXiv:0805.2343 [hep-ph]ADSCrossRefGoogle Scholar - 102.J. Ellis, T. Falk, K.A. Olive, Phys. Lett. B
**444**, 367 (1998). arXiv:hep-ph/9810360 ADSCrossRefGoogle Scholar - 103.J. Ellis, T. Falk, K.A. Olive, M. Srednicki, Astron. Part. Phys.
**13**, 181 (2000). arXiv:hep-ph/9905481 [Erratum-ibid.**15**, 413 (2001)] - 104.R. Arnowitt, B. Dutta, Y. Santoso, Nucl. Phys. B
**606**, 59 (2001). arXiv:hep-ph/0102181 ADSCrossRefGoogle Scholar - 105.M.E. Gómez, G. Lazarides, C. Pallis, Phys. Rev. D
**D61**, 123512 (2000). arXiv:hep-ph/9907261 ADSCrossRefGoogle Scholar - 106.M.E. Gómez, G. Lazarides, C. Pallis, Phys. Lett. B
**487**, 313 (2000). arXiv:hep-ph/0004028 ADSCrossRefGoogle Scholar - 107.M.E. Gómez, G. Lazarides, C. Pallis, Nucl. Phys. B
**B638**, 165 (2002). arXiv:hep-ph/0203131 ADSCrossRefGoogle Scholar - 108.T. Nihei, L. Roszkowski, R. Ruiz de Austri, JHEP
**0207**, 024 (2002). arXiv:hep-ph/0206266 ADSCrossRefGoogle Scholar - 109.M. Citron, J. Ellis, F. Luo, J. Marrouche, K.A. Olive, K.J. de Vries, Phys. Rev. D
**87**, 036012 (2013). arXiv:1212.2886 [hep-ph]ADSCrossRefGoogle Scholar - 110.J.L. Feng, K.T. Matchev, T. Moroi, Phys. Rev. Lett.
**84**, 2322 (2000). arXiv:hep-ph/9908309 ADSCrossRefGoogle Scholar - 111.J.L. Feng, K.T. Matchev, T. Moroi, Phys. Rev. D
**61**, 075005 (2000). arXiv:hep-ph/9909334 ADSCrossRefGoogle Scholar - 112.J.L. Feng, K.T. Matchev, F. Wilczek, Phys. Lett. B
**482**, 388 (2000). arXiv:hep-ph/0004043 ADSCrossRefGoogle Scholar - 113.H. Baer, T. Krupovnickas, S. Profumo, P. Ullio, JHEP
**0510**, 020 (2005). arXiv:hep-ph/0507282 ADSCrossRefGoogle Scholar - 114.J.L. Feng, K.T. Matchev, D. Sanford, Phys. Rev. D
**85**, 075007 (2012). arXiv:1112.3021 [hep-ph]ADSCrossRefGoogle Scholar - 115.P. Draper, J. Feng, P. Kant, S. Profumo, D. Sanford, Phys. Rev. D
**88**, 015025 (2013). arXiv:1304.1159 [hep-ph]ADSCrossRefGoogle Scholar - 116.L. Calibbi, Y. Mambrini, S.K. Vempati, JHEP
**0709**, 081 (2007). arXiv:0704.3518 [hep-ph]ADSCrossRefGoogle Scholar - 117.L. Calibbi, A. Faccia, A. Masiero, S.K. Vempati, Phys. Rev. D
**74**, 116002 (2006). arXiv:hep-ph/0605139 ADSCrossRefGoogle Scholar - 118.E. Carquin, J. Ellis, M.E. Gomez, S. Lola, J. Rodriguez-Quintero, JHEP
**0905**, 026 (2009). arXiv:0812.4243 [hep-ph]ADSCrossRefGoogle Scholar - 119.J.R. Ellis, K.A. Olive, C. Savage, Phys. Rev. D
**77**, 065026 (2008). doi: 10.1103/PhysRevD.77.065026. arXiv:0801.3656 [hep-ph]ADSCrossRefGoogle Scholar - 120.B. Borasoy, U.G. Meissner, Ann. Phys.
**254**, 192 (1997). doi: 10.1006/aphy.1996.5630. arXiv:hep-ph/9607432 ADSCrossRefGoogle Scholar - 121.D.S. Akerib et al., LUX Collaboration, Phys. Rev. Lett.
**118**(2), 021303 (2017). doi: 10.1103/PhysRevLett.118.021303. arXiv:1608.07648 [astro-ph.CO] - 122.A. Tan et al., PandaX-II Collaboration, Phys. Rev. Lett.
**117**(12), 121303 (2016). arXiv:1607.07400 [hep-ex] - 123.A. Abdel-Rehim et al., ETM Collaboration, Phys. Rev. Lett.
**116**(25), 252001 (2016). doi: 10.1103/PhysRevLett.116.252001. arXiv:1601.01624 [hep-lat] - 124.D.C. Malling et al., in
*Contribution to the proceedings of the 2011 APS DPF conference*. arXiv:1110.0103 [astro-ph.IM] - 125.E. Aprile et al., XENON Collaboration, JCAP
**1604**, 027 (2016). arXiv:1512.07501 [physics.ins-det] - 126.J. Ellis, K. Olive, Y. Santoso, Phys. Lett. B
**539**, 107 (2002). arXiv:hep-ph/0204192 ADSCrossRefGoogle Scholar - 127.J.R. Ellis, T. Falk, K.A. Olive, Y. Santoso, Nucl. Phys. B
**652**, 259 (2003). arXiv:hep-ph/0210205 ADSCrossRefGoogle Scholar - 128.S.M. Barr, Phys. Lett.
**112B**, 219 (1982). doi: 10.1016/0370-2693(82)90966-2 - 129.J.P. Derendinger, J.E. Kim, D.V. Nanopoulos, Phys. Lett.
**139B**, 170 (1984). doi: 10.1016/0370-2693(84)91238-3 ADSCrossRefGoogle Scholar - 130.I. Antoniadis, J.R. Ellis, J.S. Hagelin, D.V. Nanopoulos, Phys. Lett. B
**194**, 231 (1987). doi: 10.1016/0370-2693(87)90533-8 ADSCrossRefGoogle Scholar - 131.I. Antoniadis, J.R. Ellis, J.S. Hagelin, D.V. Nanopoulos, Phys. Lett. B
**205**, 459 (1988). doi: 10.1016/0370-2693(88)90978-1 ADSMathSciNetCrossRefGoogle Scholar - 132.I. Antoniadis, J. R. Ellis, J. S. Hagelin and D. V. Nanopoulos, Phys. Lett. B
**208**, 209 (1988). doi: 10.1016/0370-2693(88)90419-4 [Addendum: Phys. Lett. B**213**, 562 (1988)] - 133.I. Antoniadis, J.R. Ellis, J.S. Hagelin, D.V. Nanopoulos, Phys. Lett. B
**231**, 65 (1989). doi: 10.1016/0370-2693(89)90115-9 ADSMathSciNetCrossRefGoogle Scholar - 134.T. Li, J.A. Maxin, D.V. Nanopoulos, J.W. Walker, Phys. Rev. D
**83**, 056015 (2011). doi: 10.1103/PhysRevD.83.056015. arXiv:1007.5100 [hep-ph]ADSCrossRefGoogle Scholar - 135.T. Li, J.A. Maxin, D.V. Nanopoulos, J.W. Walker, Phys. Lett. B
**710**, 207 (2012). doi: 10.1016/j.physletb.2012.02.086. arXiv:1112.3024 [hep-ph]ADSCrossRefGoogle Scholar - 136.T. Li, J.A. Maxin, D.V. Nanopoulos, J.W. Walker, Phys. Rev. D
**84**, 076003 (2011). doi: 10.1103/PhysRevD.84.076003. arXiv:1103.4160 [hep-ph]ADSCrossRefGoogle Scholar - 137.T. Li, J.A. Maxin, D.V. Nanopoulos, Phys. Lett. B
**764**, 167 (2017). doi: 10.1016/j.physletb.2016.11.022. arXiv:1609.06294 [hep-ph]ADSCrossRefGoogle Scholar

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