Charting generalized supersoft supersymmetry

Journal of High Energy Physics, May 2018

Abstract Without any shred of evidence for new physics from LHC, the last hiding spots of natural electroweak supersymmetry seem to lie either in compressed spectra or in spectra where scalars are suppressed with respect to the gauginos. While in the MSSM (or in any theory where supersymmetry is broken by the F-vev of a chiral spurion), a hierarchy between scalar and gaugino masses requires special constructions, it is automatic in scenarios where supersymmetry is broken by D-vev of a real spurion. In the latter framework, gaugino mediated contributions to scalar soft masses are finite (loop suppressed but not log-enhanced), a feature often referred to as “supersoftness”. Though phenomenologically attractive, pure supersoft models suffer from the μ-problem, potential color-breaking minima, large T-parameter, etc. These problems can be overcome without sacrificing the model’s virtues by departing from pure supersoftness and including μ-type effective operators at the messenger scale, that use the same D-vev, a framework known as generalized supersoft supersymmetry. The main purpose of this paper is to point out that the new operators also solve the last remaining issue associated with supersoft spectra, namely that a right handed (RH) slepton is predicted to be the lightest superpartner, rendering the setup cosmologically unfeasible. In particular, we show that the μ-operators in generalized supersoft generate a new source for scalar masses, which can raise the RH-slepton mass above bino due to corrections from renormalisation group evolutions (RGEs). In fact, a mild tuning can open up the bino-RH slepton coannihilation regime for a thermal dark matter. We derive the full set of RGEs required to determine the spectrum at low energies. Beginning with input conditions at a high scale, we show that completely viable spectra can be achieved.

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Charting generalized supersoft supersymmetry

Revised: May Charting generalized supersoft supersymmetry Sabyasachi Chakraborty 0 1 4 Adam Martin 0 1 2 Tuhin S. Roy 0 1 3 4 0 Los Alamos , NM 87545 , U.S.A 1 Mumbai 400005 , India 2 Department of Physics, University of Notre Dame 3 Theory Division T-2, Los Alamos National laboratory 4 Department of Theoretical Physics, Tata Institute of Fundamental Research Without any shred of evidence for new physics from LHC, the last hiding spots of natural electroweak supersymmetry seem to lie either in compressed spectra or in spectra where scalars are suppressed with respect to the gauginos. While in the MSSM (or in any theory where supersymmetry is broken by the F -vev of a chiral spurion), a hierarchy between scalar and gaugino masses requires special constructions, it is automatic in scenarios where supersymmetry is broken by D-vev of a real spurion. In the latter framework, gaugino mediated contributions to scalar soft masses are nite (loop suppressed but not log-enhanced), a feature often referred to as supersoftness". Though phenomenologically attractive, pure supersoft models su er from the -problem, potential color-breaking minima, large T -parameter, etc. These problems can be overcome without sacri cing the model's virtues by departing from pure supersoftness and including -type e ective operators at the messenger scale, that use the same D-vev, a framework known as generalized supersoft supersymmetry. The main purpose of this paper is to point out that the new operators also solve the last remaining issue associated with supersoft spectra, namely that a right handed (RH) slepton is predicted to be the lightest superpartner, rendering the setup cosmologically unfeasible. In particular, we show that the -operators in generalized supersoft generate a new source for scalar masses, which can raise the RH-slepton mass above bino due to corrections from renormalisation group evolutions (RGEs). In fact, a mild tuning can open up the bino-RH slepton coannihilation regime for a thermal dark matter. We derive the full set of RGEs required to determine the spectrum at low energies. Beginning with input conditions at a high scale, we show that completely viable spectra can be achieved - 2 3 1 Introduction The framework 4 Solutions 4.1 Analytical solutions Renormalization group equations 3.1 A toy model 3.1.1 3.1.2 Detour: consistency check Towards the full Lagrangian 3.2 Renormalization group equations in the full model 4.1.1 4.1.2 4.1.3 4.1.4 First and second generation sfermions Colored sector Electroweak sector Electroweak symmetry breaking 5 Numerical results 5.1 5.2 5.3 5.4 5.5 5.6 Right-handed slepton masses Dark matter constraints Higgsino masses Dark matter direct detection Higgs mass and new superpotential terms Benchmark points 6 Summary and conclusion A Neutralino and chargino mass matrices B Switching on the Yukawa coupling C Higgs mass matrix D RGEs with S, T E Tadpole issue { 1 { Introduction Supersymmetry at the electroweak scale remains one of the most celebrated solutions to the hierarchy problem of the Standard Model (SM) to date. However, the LHC experiments have not yet found any signi cant excesses in their pursuit of superpartners. After the most recent run, containing an integrated luminosity of nearly 36 fb 1 at a center of mass energy of 13 TeV, the constraints on the superpartner masses are becoming quite stringent. For example, the lack of events in the jets plus missing energy search has excluded degenerate squark/gluino scenarios up to masses of 2 TeV, and squarks (including the stop) are now ruled out up to a TeV or so provided they can decay to comparatively lighter neutralinos [1{3]. However, it should be emphasized that these experimental exclusions are drawn assuming simpli ed scenarios within the paradigm of the Minimal Supersymmetric Standard Model (MSSM), and are subject to change with more involved topologies. Within the MSSM, the increased top squark and gluino masses have profound implications for naturalness, as both scales feed into the soft mass of the Higgs via renormalization, dragging it upwards and requiring ne-tuned cancellations among parameters.1 In short, the promise of a natural MSSM explanation for the weak scale is fading, even if we neglect other generic MSSM problems such as rapid proton decay from dimension ve operators, excessive avor changing neutral currents, and additional CP violating phases. While the MSSM is the most well studied framework of weak scale supersymmetry, it does not provide all aspects of a general model of electroweak scale supersymmetry. Speci cally, the MSSM is an infrared e ective supersymmetric framework with superpartner masses sourced by the supersymmetry-breaking vacuum expectation value (vev) of the F -component of a chiral spurion. Alternative theories, where supersymmetry breaking is sourced from D-vev of a real spurion, are qualitatively distinct and bring several new niceties: The primary operators sourced by the D-vev generate Dirac masses for gauginos. This requires one to extend the theory to include new chiral super elds in the adjoint representations, as the D-vev mass terms pair up the gauginos with the fermion components of the appropriate adjoint super eld, e.g. gluino with color octet fermion, wino with SU(2) triplet fermion, etc. In the absence of any other mass terms, gauginos are purely Dirac. This can be contrasted to the MSSM, where gauginos are purely Majorana. It turns out that one can not write similar D-vev sourced operators that can generate scalar masses at the messenger scale. Further, because of the speci c structure of gaugino masses, the gaugino-mediated contribution to scaler masses do not get any log-enhancement and remain nite. This property that the gaugino generated sfermion masses are insensitive to even logarithmic dependence of the ultraviolet (UV) scale, is often referred to as `supersoft' [15] | as opposed to the MSSM, where 1Exceptions exist within MSSM where scalar masses can somewhat decouple from gaugino masses. Examples include models with double protection [4{10], scalar sequestering [11{13], twin SUSY [14] etc. { 2 { a logarithmic dependence remains (`soft'). As a result, the scalar masses at the weak scale are loop suppressed with respect to the gaugino masses. The resultant mild split in the squark-gluinos masses are of utmost importance, especially when one calculates the bound in the jets + missing energy (MET) channel. Heavier gluinos imply suppressed gluino pair and squark-gluino pair production. Additionally, pair production of same-chirality squarks (pp ! q~Lq~L) is not possible if gluinos are pure Dirac, as it requires a chirality ipping Majorana mass insertion. The reduced cross-sections relax the bound on the rst two generation squark masses [16, 17]. As the gaugino are Dirac particles, the theory (minus the Higgs sector) respects a continuous U(1)R symmetry. This symmetry suppresses supersymmetric contributions to avor changing neutral currents and electric dipole moments [15, 18]. More model building can elevate the U(1)R symmetry to be a symmetry of the full Lagrangian, which ameliorates many of the avor and CP di culties of the prototypical MSSM [19, 20]. While the above features are attractive, the supersoft framework does have its share of issues: The ` ' problem is severe for supersoft models. A non-zero term is essential in a supersymmetry framework as it controls the mass of the charginos. The LEP experiments have placed model independent bounds on light charged states, e.g., m ~+ > 97 GeV at 95% C.L [21], therefore a small 1 elegant solution | at least in the framework of the MSSM | is given by the GuidiceMasiero mechanism [22], however this mechanism requires an F -type spurion and thus will not work for supersoft models. Another solution was proposed in ref. [15], where the conformal compensator eld generates the mass for the higgsinos. While viable, this mechanism relies on a conspiracy among the supersymmetry breaking scale and the Planck scale. A third solution is to include a gauge singlet S in the theory and the superpotential interaction SHuHd (e.g., NMSSM [23]) which becomes an e ective -term once the scalar component of S acquires a vev. Supersoft models automatically include the gauge singlet required for this approach as the Dirac partner of the bino. Gauge singlets do require care, however, and may lead to power law UV term is catastrophic. The most sensitivity [24]. The infamous `lemon-twist' operators [15, 25, 26] can break color or generate a too large a T -parameter. Upon replacing the real spurion by its D-vev, one nds mass squared of the adjoint scalars to be linear in the coupling constants of these operator. In a large part of the parameter space, the imaginary components of these adjoints may acquire vevs leading to dangerous charge and color breaking vacuum. Previous solutions to this problem require deviation from the supersoftness, resulting in a low energy theory with Majorana gauginos or extended messenger sectors [26, 27]. A viable solution of this problem was put forward in terms of Goldstone gauginos [28], { 3 { HJEP05(218)76 where the right handed gaugino is a pseudo-Goldstone eld originating from the spontaneous breaking of an anomalous avor symmetry. In the framework of pure supersoft supersymmetry breaking, D- atness is a natural direction, leading to vanishing D terms in the potential [15]. In the generic MSSM, the D-term contribution leads to a tree level mass of the Higgs boson as MZ cos 2 . Given the discovery of a Higgs with mass close to 125 GeV, a vanishing D-term is not a good starting point, as one needs to depend on large quartic corrections at the tree level (NMSSM like) or at the one-loop level to achieve the correct value. Most R-symmetric models with Dirac gauginos therefore include both F - and Dtype breaking to enhance the Higgs quartic term, which consequently increases the Higgs mass. A partial list of such frameworks can be found in the literature [29{79]. Another problematic issue of the supersoft scenario is dark matter (DM). If the D-vev spurion remains the only source of superpartner masses and the mediation scale is high, then the lightest supersymmetric particle (LSP) is a right handed (RH) slepton, leading to contradictions with cosmological observations. Lowering the mediation scale, the gravitino becomes the LSP, but this brings its own issues: i.) a lower mediation scale means the conformal compensator mechanism for generating a fails, and ii.) the gravitino LSP scenario is highly constrained because of the long lived charged particle searches [ 80 ] or | if the gravitino mass is less than a few keV | constraints arise from prompt multi-jet or multi-lepton nal states [81, 82]. A recently proposed framework, `generalized supersoft supersymmetry' (GSS) [83] is free of many of these issues yet maintains key phenomenological features of pure supersoft models, such as nite gaugino mediated contribution to scalar soft masses. Crucially, GSS ameliorates the above issues without introducing additional matter or additional sources of supersymmetry breaking. Instead, in GSS one adds a set of new operators formed from matter super elds and the same D-type spurion used to give gauginos mass. When written with chiral adjoints, the new operators give supersymmetric masses to the adjoints and allow one to avoid color-breaking and T -parameter issues. When written using Higgses, one nds two -parameters (namely, u and d) are generated, both of which scale with the supersymmetry breaking D-vev. If u = d, these operators can be replaced by the supersymmetric -term with = u = d, which solves the problem in the same spirit as that of Guidice-Masiero mechanism. However when d, supersoftness is lost as the hyperchage D-term gets a vev and log sensitivity (i.e., soft) creeps back into the scalar masses. The goal of this paper is to esh out the spectrum of generalized supersoft theories and to determine the parameter regions where they can be viable. While at rst glance the loss of supersoftness when u 6= d appears to be a aw in the idea, we will show that it turns out to be a feature that can be utilized to resolve the DM issue that plagues pure supersoft setups. It turns out that this theory with usual supersoft picture with a supersymmetric ` ' term term with DY / , and non-holomorphic trilinears (such as 2 u d 6= 0 is equivalent to the = u+ d , a hypercharge D hyuq~u~, etc.). In particular { 4 { the non-zero D-terms provide a boost to all scalar soft masses, while D-term operators involving the adjoint super elds allow more exibility in the gaugino masses. These e ects combine to open up a swath of parameter space where the theory satis es the observed Higgs mass and achieves the correct thermal relic dark matter abundance through binoslepton coannihilation, all in addition to the usual supersoft phenomenological bene ts. Additionally, whenever d, non-standard supersymmetry breaking operators arise which give subtle e ects in the running of soft parameters. To map out the e ects of the GSS UV inputs, we derive the full set of renormalization group equations (RGEs) and provide full numerical results for important outputs such as the Higgs mass and relic abundance. It turns out that RGEs with these unconventional operators are quite subtle and previous attempts [84, 85] miss several important e ects. To give a more intuitive understandings of electroweak parameters in terms of high scale inputs, we also derive analytical solutions to the RGE under simplifying assumptions. The rest of this paper is organized as follows: in section 2 we begin by laying out the elds, scales, and operators we will consider. Next, in section 3 we examine how the radiative generation of soft masses in our theory di ers from the conventional MSSM and pure supersoft setups, using a toy model to illustrate the key features. In section 4 and 5, we turn to the IR spectra. Many of the gross features of the spectrum can be understood using simpli ed renormalization group equations that admit analytic solutions, however we will resort to numerics when calculating quantities like the Higgs mass and dark matter constraints. This section concludes with a few benchmark points that pass all tests. Finally, a summary and discussion of our results is given in section 6 . 2 The framework The skeleton of any weak scale model of supersymmetry typically consists of two sectors | one containing the MSSM elds, and the other, known as the hidden sector, responsible for supersymmetry breaking, linked by a \messenger" sector.2 The messenger sector provides a scale (namely, mess) that characterizes all contact operators among the hidden and visible elds. The supersymmetry breaking scale (or rather the supersymmetry breaking vev, namely int) is generated in the hidden sector. Note that the observables at the electroweak scale are various superpartner masses, which are functions of both int and mess, as well as dimensionless numbers, that represent details of messenger mechanisms, and renormalization e ects. In fact, renormalization can be quite tricky especially if the mess is taken as the input scale. As shown in refs. [11, 87], the superpartner masses are renormalized due to interactions of hidden sectors from the scale mess to int, in addition to the usual e ects from visible sector interactions. The lack of knowledge of the exact nature of hidden sector dynamics can actually lead to order one uncertainty in the superpartner masses as calculated using mass-market softwares such as SOFTSUSY [88], SPheno [89, 90], SuSpect [91] etc, which completely neglect these e ects. Therefore, out the two natural choices for setting the UV input scale of the theory ( mess or int) we choose int; below this scale the hidden sector decouples and the superpartner masses renormalize only due to 2Examples of single site model do exist, see ref. [86]. { 5 { visible sector interactions. In other words, we use the superpartner masses and couplings at the scale of renormalization r = int to be the input conditions in order to evaluate the spectrum at the electroweak scale. Directly using int as the input scale naturally raises questions regarding the nature of hidden sector interactions and the details of the messengers that result in the used boundary conditions. We leave this discussion (and further UV completion) for future work. We reiterate that int is the scale at which contact operators are turned into masses for visible sector elds. In order to simplify, we begin with a hidden sector where the supersymmetry breaking is captured in the vev of a single real eld R, with the following conditions: (2.1) HJEP05(218)76 (2.2) (2.3) R y = R and D 8 Here, D's are the usual chiral covariant derivative and D is the gauge auxiliary Further, we rede ne R ! R= dmess, where d is the engineering dimension of the operator elds. at mess. This rede nition sets the engineering dimension of R to be 0. The visible sector in generalized supersoft supersymmetry extended beyond the MSSM content to include three extra chiral super elds in the adjoint representation of the three SM gauge groups, i.e. we include a color octet 3, a weak triplet 2, and a charge-neutral 1 . Given this eld content, we will work with the minimal set of contact operators (between the hidden and visible sector elds) capable of generating a viable IR spectrum. First, consider the conventional supersoft operator [15] that gives rise to Dirac type gaugino masses. 2 1 Z d2 !i( int) 1 mess D2D RW ;i i ! MDi ( int) i i ; where MDi ( int) = !i( int) D mess ; where Wi is the eld-strength chiral super elds for the i-th SM gauge group (i is not summed over, but spinor Lorentz indices are summed) and !i are dimensionless coupling constants. In eq. (2.2) we have suppressed any gauge group indices. By design, the supersymmetry breaking vev of R picks out gauginos ( i) from Wi and the fermionic component of i elds (denoted by i). Another operator often used in this context is known as the `lemon-twist' operators [15]: 1 Z d2 !qq0 2mess 1 2 where D2DR 2 QQ0 ! bqq0 ( int) q 0q ; bqq0 ( int) = !qq0 ( int) D2mess ; 2 where Q and Q0 represent visible sector chiral super elds with scalar components q; 0q, and the coupling constant !qq0 is nonzero only if QQ0 is a gauge singlet. Examples of such operators include i2, and HuHd. These operators are problematic because they give opposite sign mass to the real and imaginary parts of the scalars and thus can drive elds tachyonic [15]. { 6 { In addition to eq. (2.2) and (2.3), we use the operators proposed in ref. [83] to generate the term. 4 1 Z d2 !Dqq0 1 mess D2 (D RD Q) Q0 ! Dqq0 ( int) 2 q q0 + FQ Q0 ; where Dqq0 ( int) = !Dqq0 ( int) D mess 1 ; where FQ represents the auxiliary component of the eld Q. As in eq. (2.3), the coupling constant !Dqq0 are nonzero only if the chiral operator QQ0 is a gauge singlet. The potential generated after eliminating the auxiliary elds is given by crucially, if Q and Q0 represent di erent elds, the equation above only gives mass for Q0-scalars. For non-zero @@WQ , eq. (2.4) also gives rise to trilinear scalar mass terms. As an explicit example, substituting Q ! Hu and Q0 ! Hd, eq. (2.4) generates masses for higgsinos, the scalar hd, and a non-holomorphic scalar trilinear operator hy q~u~. If we ip d Hu and Hd (namely, Q ! Hd and Q0 ! Hu), eq. (2.4) instead gives rise to masses for the higgsinos and hu scalar, and operator hyuq~d~. Including both possibilities and denoting the mass scales of the two operators by d and u, we nd the following terms in the Lagrangian: 1 2 L ( d + d) H~uH~d a result identical to what we would get from superpotential HuHd. To make this supersymmetric limit more apparent we can rewrite eq. (2.6) in terms of a new superpotential term (namely, the -term) and soft supersymmetry breaking mass terms: W Lsoft HuHd way, the supersymmetric limit corresponds to Lsoft ! 0. In addition, supersoft models are well motivated as the avor and CP violating e ects are suppressed. In our scenario, we are generating non-standard trilinear scalar terms proportional to Yu and Yd. As a result, minimal avor violation (MFV) would ensure that avor issues are under control. As pointed out in [83], the operators in eq. (2.4) can also be used to solve potential color breaking or a large T -parameter issues in supersoft models by preventing scalars from elds from becoming tachyonic. In detail, one needs to include the operators in eq. (2.4), with QQ0 replaced by Tr( 2a). Repeating the same steps as shown after eq. (2.4), this results in a mass for both the fermion and scalar components of a. The Majorana mass for a upsets the cancellation of the SU(2) and U(1) D-terms that occurs in pure supersoft { 7 { (2.4) (2.5) scenarios, regenerating the tree-level Higgs quartic. Additionally, the scalar mass squared from eq. (2.4) is positive for both the real and imaginary parts of the adjoint scalar, thus the issue of tachyonic masses can be avoided provided contributions from eq. (2.4) are larger than any `lemon twist' adjoint masses. At this point, let us reiterate that we are still describing the theory at the scale of renormalization r = IR. In order to calculate the spectrum at the electroweak scale, these operators must be renormalized below int. Renormalization gives rise to new counterterms and changes the couplings of various operators. In particular, one nds that the nice relationship between ; ~u;d; m~ hu;d , are modi ed and need to be treated independently. We therefore, collect all the terms described above, include new operators to account for generated counter-terms as follows: W = Yu QHuU + Yd QHdD + Y LHdE + HuHd + stands for q~i; u~i; d~i; ~li; e~i, and i denotes the family. Coe cients of many of these operators are either zero or related to each other at the input scale int, as discussed before. Before we proceed any further, let us completely specify the coe cients at the input scale as well as establish our notation. Our UV inputs are: The Dirac gaugino masses and supersymmetric masses for the adjoints ( elds), as well as the supersymmetric parameter: MDa ( int) = M D0a ; M a ( int) = M 0a ; ( int) = 0 ; (2.9) where a runs over the three gauge groups in the SM. Soft masses for scalars m~ 2 ( int) = 0; m~ 2hu ( int) = Another important quantity is the Hypercharge D-term de ned as S where runs over all particles, and q represent corresponding hyperchages. Therefore, we nd the boundary value of S to be P : (2.10) q m~ 2 , (2.11) (2.12) The initial conditions for the soft trilinear ~ operators are completely speci ed in terms of the Higgs sector parameters: S ( int) = ~u ( int) = ~ d= ( int) = We devote the next section for deriving the renormalization group equations for the coupling constants given in eq. (2.8). { 8 { help us to disregard complexities due to the gaugino masses. 3 Renormalization group equations With the exception of the non-holomorphic trilinear terms (namely the -operators) in eq. (2.8), RGEs of all other operators are well known and widely used. The e ect of the operators, on the other hand, are extremely non-trivial and subtle. Early e orts in deriving these missed several e ects [84, 85], and the RGEs of these operators and their e ects in RGEs of other operators get more complicated in the presence of various Yukawa terms, gauge couplings, and, in particular, the hypercharge D-term. In order to get things correct, we begin with a simple toy model consisting of: HJEP05(218)76 1. A minimal set of degrees of freedoms; 2. Only a single Yukawa coupling; 3. Only global symmetries: this assumption allows us to disregard complexities due to gaugino masses. Towards the end we will gauge one of the global U(1) in the model, and study the e ects of -operators, in the presence of D-term. RGEs of the full model of eq. (2.8) is given in section 3.1.2. Readers interested in seeing the nal form can simply jump to section 3.1.2. 3.1 A toy model 1 2 In order to understand the non-trivial e ects of the operator in eq. (2.4) we construct a toy model of the visible sector described by ve multiplets (namely, Q; U; D; Hu and Hd) charged under the global groups G1 G2 U(1)X U(1)R as described in table 1, along with a single superpotential term (namely, Y QHuU ). The non-generic nature of the holomorphic superpotential allows us to get away with not writing any other marginal interaction. With this particle content, the only invariant, holomorphic bilinear we can form is HuHd. Consequently, there can be two operators of the form eq. (2.4) (namely, one term with derivative acting on Hu, and the second with derivative on Hd). At the input scale int, this give rise to masses for higgsinos, scalar Higgses as well as non-holomorphic scalar trilinear. r = int : L ( u + d) H~uH~d u 2 jhuj2 d 2 jhdj2 Y dhydq~u~ + h.c.: (3.1) { 9 { soft terms: with In eq. (3.1) we have explicitly written down r = int in order to indicate the fact that the relationships exhibited amongst masses of Higgs scalars, higgsinos, as well as the couplings is only valid at the scale int. In order to renormalize the theory below int, additional counter-terms are needed, and this is where the full symmetry structure in table 1 is helpful since it restricts the number of operators we need to consider considerably. Further, the D multiplet does not have any interaction at all, and as a result no new counter-term involving D needs to be written down (another way to see it is the fact that with the current interactions, there is an additional U(1) symmetry under which only D is charged, and this restricts more counter-terms). Including all of the counter-terms we need to take into account (and that are invariant under all global symmetries mentioned above), the Lagrangian in this toy model at any scale r int, is given by: W = Y QHuU ; L H~uH~d Terms corresponding to traditional a-terms, such as huq~u~ (hyuq~u~), or the b -term break U(1)R (U(1)X ) and therefore will not be generated via loops. Note that the masses-squared parameters in eq. (3.2), such as m2hu , refer to the full masses of the scalar elds. Following the logic of eq. (2.7), this Lagrangian can be re-expressed as a superpotential piece plus W = Y QHuU + HuHd ; Lsoft m~q2~;u~;d~ = mq2~;u~;d~ : (3.4) supersymmetric limit corresponds to: m2hu;d ! j j2 ; ! ; m2 q~;u~;d~ ! 0. The supersymmetric limit of eq. (3.3) is obvious, namely Lsoft ! 0, while for (3.2), the It is instructive to derive the RGEs both in term of mass-parameters from eq. (3.2) and in terms of soft-mass parameters in eq. (3.3), then match the two apprroaches using the substitutions stated above for a consistency check. However, to save space in this write-up we show only one derivation, the -functions of the operators in eq. (3.2); the -functions for the soft parameters can be derived using the substitutions in eq. (3.4) At one loop order, the -functions can be evaluated diagrammatically. In addition to the standard diagrams one encounters in MSSM calculations, we need to take into account new diagrams due to the operators (e.g., gure 1). Starting from eq. (3.2), the full functions for the Yukawa coupling Y , the higgsino mass parameter , the non-holomorphic q˜, u˜, hd q˜, u˜, hd ⇠ u⇤Y ⇠ uY ⇤ q˜, u˜, hd qe, ue ˜ Hu q, u Y Y ⇤ qe, ue scalar trilinear parameter u, and various scalar mass-squared parameters are given below. mq2~ + m2u~ + m2hu + j uj 2 2 2 j j 2 j j 2 2 ; ; 3.1.1 Detour: consistency check (namely, j j2) at all scales, As a consistency check for the RGEs derived above due to the unconventional operator, consider taking the supersymmetric limit. In particular, we look at the mass parameter m2hd | in the supersymmetric limit, m2hd should be equal to the higgsino mass parameter d lim ! dt m2hd j j 2 = 0 : Staring at eq. (3.6), (3.12), we see our RGE pass this check. 3.1.2 Towards the full Lagrangian To formulate the RGEs of the full theory as given in eq. (2.8), one needs to incorporate important complexicites: If the superpotential in eq. (3.2) is expanded to include new marginal interactions involving the D super eld, such as Y QHdD, the accidental global U(1) symmetry with D is lost and new counter-terms are needed. For example, in the presence of both Y and Y , the operator hydq~u~ shown in gure 2 is permitted. Y ⇠ u⇤ q˜ Y¯ 2 hd q˜ If one gauges U(1)X , the e ect of its D-terms need to be taken into account, even if we refrain from adding mass term for the gaugino (so that the U(1)R remains unbroken, and we can keep using the symmetry arguments in order to restrict counter-terms). The impact of gauging shows up in two places. First, the anomalous dimensions of all super elds charged under U(1)X changes because of the gauge elds. Second, new additive contributions to the RGEs arise because of the U(1)X D-term. The contribution can be summarized in terms of the parameter SX . HJEP05(218)76 SX m~2 X q m~ 2 ; ! m~ 2 gX2 q SX ; (SX ) = gX2 SX 12 jY j2 u + ~u + ~u + 12 Y 2 d + ~d + ~d (3.14) (3.15) ; (3.16) where runs over all scalars of the model, q represents 's charge under U(1)X , and gX represents the gauge coupling constant. Importantly, these RGE hold as long as U(1)R remains unbroken. Therefore, even in the presence of U(1)R preserving gaugino mass terms (i.e., Dirac gaugino masses) the above one-loop results prevail. if one adds a b term to the toy model, it is multiplicatively renormalized and does not enter the RGE for any other parameters. Both e ects stem from the fact that b is the only U(1)R breaking term and has the wrong mass dimension to radiatively generate a-terms. The experience with the toy model (with or without added complications) paves way for us to write down the RGEs of the full model, as shown in the next section. 3.2 Renormalization group equations in the full model We will take the approximation that only the third generation of the Yukawa couplings are non-zero. This reduces Yukawa coupling matrices to 0 0 y (3.17) + ~u + ~d m~ 2u~3 m~ 2~ d3 m~ 2~ `3 m~e2~3 = 4 jytj2 m~ q2~ + m~2u~ + m~2hu +4 jytj2 = 4 jybj2 = 2 jy j2 = 4 jy j2 m~ q2~ + m~2~ + m~2hd d +4 jybj2 m~ 2L~ + m~e2~ + m~2hd m~ 2L~ + m~e2~ + m~2hd +2 jy j2 +4 jy j2 = 6 jytj2 m~ q2~ + m~2u~ + m~2hu + 6 jybj2 + 2 jy j2 + 6 jytj2 + ~u + ~ + ~u + 51 g12S ; u d + ~u + ~d + ~d + 53 g12S ; 53 g12S : When expressing the RGE in the full model, we will work in the more familiar language of running supersymmetric and supersymmetry breaking parameters, in the spirit of eq. (3.3). To split RGE for full masses into supersymmetric and soft pieces, one needs to express the soft mass parameters in terms of the mass parameters in eq. (3.2) using eq. (3.4), then apply equations eqs. (3.5){(3.12). Written in this form and using the reduced Yukawa matrices, the -functions of the soft masses are: 16 2 = 2 jytj2 m~ q2~ + m~2u~ + m~2hu + 2 jybj2 + ~u + ~d + ~ + ~ + ~d (3.18) + 54 g12S ; (3.19) 52 g12S ; (3.20) 53 g12S ; (3.21) 56 g12S ; (3.22) + The soft mass RGE must be complemented by the RGE for the Higgs sector parameters, ~u; ~d; ~ ; and b : = 3 jytj2 + 3 jybj2 + jy j2 ~ u 2 jybj2 ~u + ~d + ~u 3g22 + = 3 jytj2 + 3 jybj2 + jy j2 ~ d 2 jytj2 ~u + ~d + 2 jy j2 ~ ~ d + ~d 3g22 + = 3 jytj2 + 3 jybj2 + jy j2 ~ + 6 jybj2 ~ d ~ + ~ 3g22 + The notation for left and right handed sleptons are `~ and e~ respectively. The RGEs of Dirac gauginos and Majorana adjoint fermions can be found in [56]. Examining these RGE, there are several features worth mentioning. First, the rst and second generation squarks and sleptons can be found by setting the Yukawa couplings in eq. (3.18){(3.22) to zero. As the traditional gaugino mediated contribution to the soft mass RGEs is absent because of the Dirac nature of our gauginos, the rst and second generation squarks/sleptons only renormalize due to hypercharge D-term. Written compactly, is a rst or second generations sfermion with hypercharge Y . A second feature of these RGE is the unusual piece proportional to u seen in e.g. eq. (3.18). As shown in gure 1, this piece can be traced to insertions of trilinear scalar term from -operator and The nal ingredient needed to complete the RGE for this theory is the running of S: u d + ~d + ~u + ~d + ~u + 4y2 d + ~d + ~d : (3.31) In the limit the ~ parameters are taken to zero, eq. (3.31) reduces to its conventional Our objective in this section is to determine the spectra at IR and other parameters after solving equations listed in eqs. (3.18){(3.31) using all the initial conditions speci ed at the boundary int (eq. (3.1)). Before we proceed any further, however, we give the schematics of the scales we target. We think that having a prior understanding of the scales (i.e., sizes of various terms) allows one to visualize and consequently appreciate the solutions, especially the analytical part, better. We have plotted a schematic representation of superpartner mass spectrum in gure 3. Here is a simpli ed summary of the mass scales. The key spectral features are: As expected, the lowest lying scale represents the mass of the LSP. For the relic abundance to work out we rely on co-annihilation of the LSP (mostly bino) with RH sleptons. This scale, therefore, also represents masses of the RH sleptons. We take this opportunity to reiterate that gaugino mediated contribution (loop suppressed and nite) from bino can not generate this mass-scale. In this work, we generate this scale by the hypercharge D-term S gives the size of the S-term. . The mass of dark matter therefore also directly Decays of left handed sleptons to LSP give rise to hard leptons and consequently these need to be signi cantly heavier than the LSP mass scale (or the scale of RH ⇠ 1 TeV ⇠ 500 GeV ⇠ 100 GeV Gluinos Weak adjoint fermions Weak gauginos Squarks LH slepton masses RH slepton masses bino mass Directly set by initial conditions Dominant contribution comes from threshold corrections from gluinos Dominant contribution comes from threshold corrections from weak gauginos gives the size of S Dominant contribution comes from S See-saw in neutralino mass matrix resulting in a mostly Majorana bino LSP HJEP05(218)76 sleptons). Setting the LH sleptons above the LHC bound, therefore, gives rise to a second scale in the spectrum. Masses for the LH sleptons are generated by nite corrections from Wino masses. This allows us, in-turn, to set the scale for wino mass. Finally, LHC bounds on colored particles imply that squarks need to be signi cantly heavier. The primary source for squark masses are nite corrections from gluinos. Setting the squark masses to be around the TeV scale, one then nds masses for gluinos. The nal piece is the mass scale of the higgsinos. This can be determined either by dark matter direct detection constraints which limits the higgsino fraction in the lightest bino-like neutralino or by direct LHC searches. At LHC, heavy higgsinos can decay to the LSP associated with Higgs or Z-boson. The non-observation of such events puts an upper bound on the higgsino masses. 4.1 Analytical solutions It is clear that, even at one loop order, we need to solve the RGE eq. (3.18){(3.31) numerically. However, in order to develop some intuition for the gross features of the spectrum, we start by making some simplifying assumptions which allow us to solve the RGE analytically. As we show later in this subsection, most of the phenomenological aspects of this work, such as nding a viable candidate for dark matter or the spectrum of colored particles, can be understood within this simpli ed picture. Calculating the Higgs mass, however, requires more careful considerations and will only be discussed in the context of full numerical solutions. To simplify the RGE, in the following subsections we will ignore all Yukawa couplings except for the top Yukawa yt. Even though we use non-zero S0, we will use ~0 = 0, which ensures that none of the operators will play a role. While this approximation may seem unjusti ed given our initial conditions, we nd that the full, numerical solution derived later is well approximated by the results we derive with ~0 = 0. First and second generation sfermions As mentioned earlier, the rst and second generation sfermions only run because of the hypercharge D-term S . With the Yukawa couplings zeroed, the running of S is easy to work out: S( IR) = S0 g1( int) ; where b1 is the beta function for hypercharge and we have imposed the boundary conditions from eq. (2.10). Plugging eq. (4.1) into the sfermion RGE (eq. (3.30)), we nd m~ 2 ( IR) = 6 5 qY S0 4 1( IR) log int IR ; where qY is the hypercharge of the sfermion. No matter what sign we choose for S0, this (radiative) mass squared will be negative for some matter elds (q~i; u~i; d~i; ~li; e~i), simply because they don't all have the same sign hypercharge. If eq. (4.2) were the only contribution to the sfermion masses, this result would be fatal as charge/color breaking minima would occur. Fortunately, as illustrated in gure 3, the LH sfermion masses receive positive de nite and nite contributions from loops of gauginos [15]. As eq. (4.2) is proportional handed sleptons. For example, if int = 1011 GeV we need jS0j to the hypercharge coupling and only logarithmically sensitive to int, it is entirely possible for the nite contribution proportional to 2( IR); 3( IR) to dominate over eq. (4.2). This logic suggests that we should choose S0 < 0, so that eq. (4.1) is positive for the right (700 GeV)2 in order to achieve 100 GeV right handed slepton masses. Such a value of S can be obtained by properly choosing UV inputs . This choice for S0 renders the contribution of eq. (4.2) to m~q2~ and m~2~ also positive, while m~2~ and m~2u~ receive a negative contribution. As d ` we will show, negative m~2~ ` ; m~ 2u~ must be o set by the requirement that m~2~( IR) > 0; m~ 2u~( IR) > 0 can be used to restrict the input wino/gluino ` mass parameters. In order to impose this restriction, we rst need to know how the IR nite wino and gluino loops, and the gaugino masses depend on the int inputs. 4.1.2 Colored sector The gluino sector (gluino + adjoint partner) masses at IR are straightforward to calculate. Using the boundary conditions set in section 4, the renormalized masses are. s ( int) = s ( IR) ; ( IR) = s s ( IR) s ( int) s Z 3 Z 3 ( int) M 3 ( IR) 3 s Z 3 Z 3 ; ( IR) ( int) MD3 ( int) = MD3 ( int) 3 s 2 ; ( int) = M 3 ; (4.1) (4.2) HJEP05(218)76 (4.3) (4.4) (4.5) (4.6) ˜ R q˜ g˜ elds. The purely scalar loop cancels the logarithmic divergence which appears in the prototypical gaugino mediated correction to squark masses. where Z 3 is the eld strength renormalization of the color adjoint and eq. (4.3) contains the well-known result that the QCD gauge coupling does not run with the supersoft eld content. Using 3 to denote the fermion within 3, the color adjoint fermion masses can be written in matrix form as3 Depending on the relative strength of the Dirac mass of gluino (MD3 ) and the Majorana mass of 3 (M 3 ), three qualitatively distinct spectrum at the scale IR emerge. i.) primarily Majorana gluinos, ii.) primarily Dirac gluinos, and iii.) mixed Majorana-Dirac gluinos. For the coupling structure and some collider implications of the di erent possibilities, see ref. [92]. The gluino mass matrix can acquire all the three types of textures even if M 3 and MD3 start out being equal at the messenger scale (depending on whether 3 self interactions are present or not). It is instructive to properly diagonalize the mass matrix and write the e ective theory in terms of mass eigenstates g~ ! l g~h = cos2 g = 0 1 cos g sin g sin g cos g ! A ; In the above, g~l and g~h represent the light and heavy gluino spinors with masses Mgl and Mgh respectively. L Mgl = 1 2 1 2 Mgl g~lg~l + Mgh g~hg~h ; 1 2 q and Mgh = 1 2 M 3 + q M 23 + 4M D23 : and sin g = cos g = 1=p2. This decomposition is valid even if gluinos are purely Dirac; there, M 3 = 0, Mgl = Mgh , One of the important features of the generalized supersoft spectrum is that | regardless of the Majorana/Dirac composition of the gluino in all these three cases | squark 3The gluino g~ does not acquire a Majorana mass. (4.7) (4.8) (4.9) (4.10) masses remain supersoft, i.e. the gluino mediated squarks masses do not pick up any log int sensitivity. Below, we verify this statement using the diagrams in gure 4: m~q2 = s C2 (r) Z 4 d k cos2 g k2 Mg2l + k2 sin2 g Mg2h + s C2 (r) Z 4 d k M D23 k2(k2 m2R ) s C2 (r) cos2 gMg2l + sin2 gMg2h log i2nt + s C2 (r) M D23 log i2nt : (4.11) Here, m~q2 is the nite correction squark mass generated at IR and C2(r) is the quadratic casimir. Examining eq. (4.11), the rst integral is due to gl and gh running in the loop, and the second integral is due to the real part of the scalar octet of mass m R . The log i2nt term is cancelled between the two terms, as Cleaning up eq. (4.11), the gluino-induced contributions to the soft masses (squared) of , m~2~) can be written in terms of the mass eigenvalues and the gluino d m~ q2~ = s C2 (r) M D23 log M D23 m23 + M D23 cos 2 g log tan2 g : (4.13) where m 3 is the mass of the scalar adjoint component in 3 . At tree level, m 3 = 2MD3 , though running and the existence of Majorana masses will change the relationship somewhat. If we assume that nite gluino contribution represents the squark masses, we can compare eq. (4.13) to the LHC bounds on colored sparticles to get an idea for the allowed ranges of UV inputs M D03 ; M 03 . The IR masses for the squarks and lightest gluino sector are shown below in gure 5 as a function of the input masses. As the squark masses are radiatively generated and nite, the squarks are signi cantly lighter than the gluinos for a given set of inputs. The shape of the squark mass curves can be understood from the fact that the Majorana gluino sector mass runs signi cantly faster than the Dirac mass. For M 03 M D03 , M 3 ( IR) MD3 ( IR), e ectively, the gluino is Majorana like with eigenvalue M D23 =M 3 . In such a scenario, the log emerges in the gluino mediated correction to the scalar masses but with a cut o M 3 , i.e., Log (M 3 =MD3 ). In assuming the squark masses are set by eq. (4.13), we have ignored: i.) nite (positive) contributions from loops of SU(2) or U(1) gauginos, ii.) the log-enhanced hypercharge D-term contribution proportional to S0. While necessary for getting the exact spectrum of the theory, these contributions are both subdominant to eq. (4.13); the SU(2) and U(1) gaugino loops are suppressed by the smaller EW couplings, and the S0 contribution is small because, as explained in section 4.1.1, it sets the mass of the lightest sfermions. Therefore, our assumption that eq. (4.13) sets the squark mass is justi ed, and we can use gure 5 to rule out M D03 . 2 TeV. These bounds are rough, as the detailed phenomenology will depend on how `Dirac-like' vs. `Majorana-like' the lightest gluino eigenstate is; see refs. [16, 92]. 0 ⌃ (G1000 1500 1000 500 0 1000 Dirac mass M D03 and adjoint Majorana mass M 03 . Right: mass of the lightest eigenvalue of the gluino sector as a function of the same UV inputs. For both plots, we used the tree level relation m 3 = 2MD3 for simplicity and took IR = 103 GeV, int = 1011 GeV and 3( IR) = 3( int) = 0:118 as inputs. The starred point (M D03 ; M 03 ) = (2:48 TeV; 0 TeV) for BP1 and (M D03 ; M 03 ) = (2:65 TeV; 940 GeV) for BP2, yields m~q~ ' 1:8 TeV which satisfy the present LHC bounds [1, 2], and a gluino mass of 7 TeV and 4:6 TeV respectively. 4.1.3 Electroweak sector Following the same procedure as above, we can calculate the electroweak gaugino masses (both Dirac and Majorana pieces). One di erence in the electroweak case is that the gauge couplings do run in a supersoft theory, with beta function coe cients b1 = 33=5 and b2 = 3. Working in the yt ! 0 limit, the masses at the IR scale are: 1 i ( int) = 1 i ( IR) 2 ba log Z a ( IR) = Z a ( int) MDa ( IR) = M D0a M a ( IR) = M 0i i ( R) i ( int) a ( IR) 2 i ( int) i ( IR) i ( int) 1 C2(Adj) ba 2C2(Adj) ba int IR 2C2(Adj) ba ; ; ; ; (4.14) (4.15) (4.16) (4.17) where a = 1; 2 and C2(Adj) is the quadratic Casimir of the adjoint representation. Focusing rst on the SU(2) sector, loops of winos and SU(2) adjoints generate a nite correction to the mass of all SU(2) charged sfermions. The contribution has the same form as eqs. (4.7){(4.13), but with MD2 replacing MD3 , 2 replacing 3, and 2 replacing g: m~ `2~ = 2 C2 (r) M D22 log M D22 m22 + M D22 cos 2 2 log tan2 2 : (4.18) This e ect is particularly important for the left-handed sleptons, as the contribution to their masses from the S0 piece is negative (see discussion following eq. (4.2)) and they 0.025 ˜ m 1 2 e˜ ˜ m 3 2 e˜ ˜ m (5.4) (5.5) 0103 104 105 106 107 108 109 1010 1011 by S0 as a function of the intermediate scale for di erent tan values of 2.5 (red-solid), 20 (bluedashed) and 40 (black-dotted). In the left panel we present the mass splitting between the rst and third generation of right handed sleptons as a function of tan . The tan dependence manifests itself in the RGEs. where | as discussed earlier | the largest regions parameter space will have the bino (LSP) and lightest right handed selectron nearly degenerate. As such, a mass splitting between slepton generations means the heavier sleptons are slightly heavier than the LSP and can decay `~ ! ` ~01. For sleptons in the 100 GeV range, bounds from lepton plus missing energy searches are quite stringent [100, 101]. To avoid these bounds without raising the overall mass scale, we need to quench the splitting by restricting parameters to low to moderate tan . As we shall see in the next section, the small tan region also well motivated from the perspective of Higgs mass. To obtain a rough estimate on the size of S0, we present two cases with di erent values of tan : 2.5 and 40. To obtain a right slepton IR mass of around 100 GeV for these tan values, one would require p jS0j = m~ e~ = 2:5, the e ect of the tau Yukawa coupling in the RGEs can be neglected to a very good approximation. Hence, typical values such as 0d = 800 GeV and 0u = 500 GeV satis es the left hand side of eq. (5.4). Furthermore, requiring the right-sleptons to be nearly degenerate results in m~ e2~ j m~e~j2 < 1% =) tan . 10: From eq. (5.5) and for m~e~ 100 GeV, the degeneracy between the slepton generations turn out to be around 10 GeV. In the next section we look at the DM constraints which gives a range of the LSP-right slepton masses where relic density can be satis ed. 5.2 In the absence of coannihilation, the relic density of a bino-like neutralino can be approximated as [96] other states, say r < 0:8, the present value of the observed relic density 0.1199 [102] readily limits m~e~ . 100 GeV. Such slepton masses are very tightly constrained from the LEP data [21]. As a result, in most of the allowed parameter space of slepton masses, the bino su ers from overabundance. In our setup, the bino is nearly mass-degenerate with right handed sleptons, and consequently coannihilation becomes important. Roughly, whenever m then these slightly heavier degrees of freedom are thermally accessible and are therefore m~ e~ m~01 Tf , nearly abundant as the relic species. Since Tf degeneracy needed for coannihilation is m~01=25 [96], we nd the degree of The higgsino mass is essentially tied with the right handed slepton masses through their initial conditions. Assuming again 0u = 500 GeV and 0d = 800 GeV, the right panel of the gure 8 shows that the higgsino mass at IR is driven by ( IR) tan =2:5 1:025 0d + 0u 2 660 GeV: (5.9) Another way to make higgsinos heavy would rely on the modi cation of the messenger scale. diately sets d( int) again u( int) Higgsino masses m~ e~ m~01 = m~01 m m~01 5%: Once coannihilation becomes important, the slepton self interactions and interactions with the bino [103] set the relic abundance. These additional interactions relax the bound on the slepton masses and overabundance issue can be avoided if m~ e~ me01 . 500GeV; which is well above the current LHC bound on sleptons nearly degenerate with the LSP [94]. For a dedicated and more robust analysis, we implemented the e ective model in SARAH-4.11.0 [104, 105] and generated the spectrum using SPheno-3.3.3 [89, 90]. We have varied the masses of the lightest neutralino state and the slepton masses (assumed to be degenerate). If the LSP is neutralino then only the spectrum le is fed to micrOMEGAs-v3 [106] for computing relic abundance and dark matter direct detection rates. We nd that coannihilation works e ciently for m~01 me~ 400 GeV. This imme2 TeV and ( IR) 600 GeV. This limit is obtained by considering (5.6) (5.7) (5.8) IR μ (μIR) 1.03 10 5 μ (μIR) m˜e˜ (μIR) 1103 104 105 106 107 108 109 1010 1011 0104 105 106 107 108 109 1010 1011 gure we show the running of the higgsino mass scaled with its initial condition for a xed value of tan = 2:5. On the right panel we elucidate how the low energy constraints can be translated to a bound on the intermediate scale. These constraints can come from either DM direct detection results or collider experiments. The yellow shaded region is excluded from the direct collider searches for right slepton/LSP masses close to 100 GeV for the same initial conditions as discussed before. In the right panel of gure 8 we show how low energy constraints can signi cantly constrain the messenger scale in our scenario. The constraints are two folds, rst from direct detection experiments. The limit from direct detection experiment constrains the higgsino fraction in the lightest neutralino. As a result, higgsino should be heavier compared to the LSP or the lightest slepton. In our case, the lightest neutralino is a predominant mixture of the bino and singlino gauge eigenstates and the higgsino admixture is negligible. Therefore, the stringent constraint from the DM direct detection can be avoided easily. However, to push the Higgs mass to the observed value, as elaborated in the next section, we require introducing new superpotential terms. For larger couplings, this increases the higgsino component in the lightest neutralino state. Secondly, collider experiment can also provide stringent constraint on the ratio of the higgsino and slepton masses. For example, our spectrum has the following hierarchical structure where NLSPs are the right handed slepton and LSP is the bino-singlino admixture neutralino. Higgsinos are heavier than the sleptons. Such a higgsino after electroweak production can decay to a Z ~01 or h ~01 [93, 107]. However, all these modes are phase space suppressed for ~02 the dominant decay mode would be `~`. The limits on this particular ~ 0 3 robust [101]. In our case, we took a conservative approach and used 200 GeV. In such cases nal state is very ( IR)=m~ `~R ( IR) For our choice of parameters, we observe from gure 8 that the UV scale should be less than 1011 GeV or so. Changing the initial values would modify the results as the dependence of these two parameters are di erent for higgsino and slepton mass runnings. 5.4 Dark matter direct detection Our framework also needs to be consistent with the null results in DM direct detection. When the squarks are heavy, the spin-independent interaction between DM and nuclei comes from Higgs exchange and thus it depends crucially on the Higgs coupling to the lightest neutralino. The Higgs-neutralino coupling, in turn, depends on the higgsino mass parameter. For a given LSP bino mass we can translate limits from direct detection directly into limits on the higgsino mass. The spin-independent cross-section can be well approximated by the following relation [108] SI ' where GF ; MZ ; mred are the Fermi coupling constant, Z boson mass, and DM-nucleon reduced mass, and Fh, Ih are coupling and kinematic factors. In the limit when the wino and heavy Higgs are heavy and e ectively decoupled (in addition to the squarks), HJEP05(218)76 Fh = N11N14 sin W ; Ih = X kqhmqhN jqqjN i: q h Here, N11; N14 are the bino and higgsino fraction in the lightest neutralino, kup-type = h cos = sin , kdown-type = sin = cos , and we have already assumed cos ! 1 and sin ! 0 by decoupling the heavy Higgs. Because of the Dirac nature of the gaugino masses and the extra interactions involving the adjoint (e.g., SHuHd coupling in the superpotential), the neutralino mass matrix no longer has the MSSM form. Taking the ratio of the DM direct detection in GSS to the prototypical MSSM, the only factors that don't drop out are the N11; N14, (5.10) (5.11) (5.12) GSS SI SI MSSM ' N1G1SS N1G4SS 2 jN11N14j2 : For typical values = 300 GeV, M ~01 section in GSS turns out to be an order of magnitude less than in the MSSM, as the binosinglino mixing in GSS dominates over bino-higgsino mixing. This implies lower higgsino 150 GeV, tan = 4, the direct detection crossmasses are possible in our framework. The constraints on the right handed slepton masses and the higgsino mass can be the contours of right handed slepton mass (red) and higgsino mass (blue) in the 0u e ectively translated to constrain the two input parameters 0u and 0d. In gure 9 we show - 0d plane. As already stated, one needs to have 0d > 0u in order to get positive de nite right handed slepton mass. As a result, the grey shaded region is not viable. The masses of right handed sleptons and the LSP should be nearly degenerate in order to satisfy the relic abundance. Moreover, DM direct detection experiments sets a limit on the bino-higgsino mixing and e ectively on the higgsino mass parameter, . Therefore, given the values of m`~R and one can readily constrain the input parameters gure in the left panel. In such cases, the lightest neutralino is a predominant admixture of the bino and the singlino instead of bino-higgsino. As a result, the dark matter direct detection constraints are irrelevant. However, collider constraints are certainly applicable and we discuss those issues in detail later. The gure in the right panel is obtained by xing S = 0:8. A non-zero S is important to satisfy the Higgs mass constraints as discussed in section 5.5. It is obvious, that order one value of S increases the higgsino fraction in the lightest neutralino and hence direct detection constraints become important. The yellow T 0μu 0.8 0 u μ 5 0 0 Direct detection constraint μ0d (TeV) u- 0d plane. To obtain positive de nite mass for the right handed sleptons one needs to have 0 0 u, therefore, ruling out the shaded region. For de nite values of right handed slepton and higgsino masses, designated by the crossing of the contours, the two input parameters can be readily obtained. We have xed int = 1011 GeV and tan = 2:5. We have xed 0u and 0d S = 0, for the gure in the left panel. In such a scenario, the direct detection constraints are not relevant. However, for the gure in the right panel, we have xed S = 0:8 and show the excluded region after taking into account constraints from direct detection experiments shaded region in the right panel plot of gure 9 shows the excluded parameter space when direct detection results (LUX WS 2014-16) [97] are taken into consideration. We reiterate that the LEP results exclude parameter space where slepton masses are below 100 GeV. mass and degree of etL 5.5 Higgs mass and new superpotential terms In previous sections we have shown how the inputs 0u ; 0d, the Dirac and Majorana gaugino masses and, to some extent, tan are constrained by collider physics and dark matter. What remains to be done is to see what Higgs masses are possible in the `surviving' regions. As already mentioned in section 4.1.4, the traditional MSSM Higgs quartic terms get depleted due to the presence of Dirac gauginos. However, new superpotential couplings such as S generates additional quartic terms which are NMSSM like. Under some simpli ed assumptions such as i.) integrating out the adjoint scalar elds, ii.) assuming T = 0, for simplicity, the Higgs mass can be obtained by diagonalizing the scalar mass matrix in the basis (hu; hd) given in appendix C. To fully answer the question of the Higgs mass, we need to go beyond tree-level. The largest loop-level contribution comes from the top squarks and is governed by their overall etR mixing. In GSS, the top squark mass matrix, neglecting D-term mt2~ = m2Q3 + mt2 ~uytvd ~uytvd m2u3 + mt2 ! ; (5.13) where the terms ~u is the supersymmetry breaking trilinear interaction originating from eq. (2.8). It is well known that the top squark contribution to the Higgs mass is largest when there is substantial etL etR mixing [ 98 ]. In GSS, the mixing angle in the stop-sector, tan 2 = m2Q3 2 ~uymtvd2u3 ; is proportional to vd cos , while the mixing angle in the MSSM sin . The di erence can be traced to the unusual structure of the scalar trilinears in GSS and, since one usually wants to take tan large to maximize the tree level Higgs quartic, leads to suppressed stop sector mixing. Suppressed stop sector mixing can be overcome by taking large ~u, though this is an unattractive option as it will increase the traditional ne tuning measure of the setup. Moreover, even if we set aside tuning concerns for the moment, we nd that even very large values of ~u are unable to push the Higgs mass to more than 100 GeV (recent works, studying the phenomenological implications of such non-standard soft supersymmetry breaking terms in MSSM can be found in [109, 110]). In addition to the tree level terms, we have also included two loop corrections from the stop sector [111] (for three loop corrections in MSSM see [112]) and considered mt~1 are possible if we consider heavier stops, though at the expense of increased tuning. 1:8 TeV.5 Higher Higgs masses Hence, the most natural way to increase the Higgs mass is to extend the theory with additional F -terms, as in the NMSSM. In GSS, this extension requires no new matter as the theory already contains a SU(2) singlet and triplet super eld. Including interactions between these super eld and the Higgses, the GSS superpotential is modi ed to WGSS = W + SS Hu Hd + T Hu T Hd; (5.15) with W given in eq. (2.8). The modi ed superpotential generates a new tree level quartic for the Higgs. Speci cally, taking S 6= 0; T = 0 for simplicity, two new Higgs potential terms are generated, V2 = j Sj V3 = 2 " 4 p2g0 S 4 In the limit MD 4M D21 MD1 M 21 + 4M D21 (5.14) HJEP05(218)76 M 21 +4M D21 h2uh2d + M 21 + 4M D21 huhd h2u +h2d M 1 2 M 21 +4M D21 h2u + h2d M 1 huhd h 2 d h 2u : h2u + h2d 2 ; (5.16) M ; , the quartic no longer vanishes but instead takes on a NMSSM form, e.g., V = j sj2 h2uh2d and can generate a large tree level Higgs mass.6 4 In the left panel of gure 10 we show the Higgs mass with new tree level quartic contributions after taking into account two loop corrections from the stop sector ( m~t~ 5In the presence of additional superpotential interactions (eq. (5.15)), the electroweak adjoints also contribute to the Higgs mass and can in principle be included. At one loop, these contributions are / and are logarithmically sensitive to the di erence between the adjoint fermion and scalar masses. However, unlike in the top-stop sector, the ratio of adjoint fermion to scalar masses is O(1), thereby suppressing the adjoint contribution to the Higgs mass. We have therefore neglected this (positive de nite) piece for 4 S ; 4T 6The -dependent pieces do not vanish in the D- at limit so they will alter the stabilization conditions, as well as the relation between the b and the other inputs. 4.5 3.5 2.5 b 24G eV 1 26 G eV BP1 2 1 HJEP05(218)76 0.6 loop corrections from the stop sector ( m~t~ 1:8 TeV). The MSSM tree level contribution is depleted due to our choice of M and MD's (same as BP1). However, additional F -term contributions occur from the superpotential coupling S. Right: running of superpotential coupling S (shown by the red-solid curve on the left panel) and top Yukawa coupling (blue-solid curve). We have xed ( IR) = 0:8 and tan = 3:6 (same as BP1) to satisfy Higgs mass constraints. 1:8 TeV). The MSSM tree level contribution gets reduced due to the presence of M i and MDi , however additional F -term contributions from the superpotential coupling S helps. The values shown in gure 10 are the IR values, so one might worry that the relatively large values needed for mh = 125 GeV grow even larger in the UV and lead to a violation of perturbativity. In addition to their own running, the couplings modify the anomalous dimensions of the Higgs elds and thus contribute to the running of the top and bottom Yukawa couplings: We reiterate, the RGEs of the scalar masses are shown to run from therefore integrating out any hidden sector e ects. However, 's are superpotential coupling and should remain perturbative upto the GUT scale. The running of the singlet coupling to the Higgs elds is shown in eq. (5.17) and depicted in the right panel of gure 10. We have chosen ( IR) = 0:8 and tan = 3:6 (same as BP1) to satisfy Higgs mass constraints. The coupling remains perturbative up to the GUT scale. However, due to larger group theory factors in the anomalous dimension the triplet coupling diverges more rapidly, becoming nonperturbative at an energy close to 1010 GeV unless additional structure is added to the theory. The new superpotential couplings generate new trilinear interactions in the scalar potential and modify the running of the Higgs soft masses m~hu , m~hd and couplings ~u; ~d. : int (5.17) 1011 GeV, The rest remains una ected. The additional couplings in the Lagrangian L ! L S d jhuj2 S S u jhdj2 S + triplet trilinear terms (5.18) The modi ed RGE are presented in appendix D. One important thing to note that the presence of the singlet eld and non-standard soft terms can give rise to dangerous tadpole diagrams which might destabilize the hierarchy [113, 114]. We discuss such issues in appendix E. Finally, we provide two benchmark points to show that the framework is consistent with all the phenomenological observations including the Higgs mass, DM relic density, DM direct detection and collider results. The UV and IR parameters for these benchmarks, along with mh; h2 and the DM-nucleon spin-independent cross section are shown in table 2. As discussed in section 5.5, a viable Higgs mass is most naturally reached in GSS when one admits superpotential interactions between the SU(2); U(1) adjoint super eld and the Higgses. While either S, T or both can be used, however, T = 0 is the safer choice, in the sense that it requires fewer assumptions about the UV. Therefore, both of the benchmark setups have S 6= 0; T = 0. Inspecting the benchmarks, both points share the feature that q~; u~; d~; `~; e~ are massless at int and have 0d;u split in a way that yields positive RH slepton masses, 0d 0 100 GeV. To augment the Higgs mass, both benchmarks have S 6= 0; O(1). Order one values of S enhance the higgsino fraction of the lightest bino sector eld [115]. Hence, to overcome stringent constraints from the DM direct detection,7 the higgsino mass | set by needs to be around a TeV. We note in passing that such a problem does not arise when one makes the triplet coupling T large to x the Higgs mass. As this coupling only increases the higgsino fraction in the wino-tripletino like neutralino, considered to be heavy in our case. For both benchmarks, the b ( int) values have been determined numerically following the logic established in section 4.1.4 with re nements due to S 6= 0 and are similar in size 0 u;d | to the other UV inputs. The largest di erence between the points is the mass of the triplet and octet fermions. In BP1, M 2 = M 3 = 0 in the UV, which, since the running of these masses is multiplicative, implies M 2 = M 3 = 0 in the IR as well.8 We cannot further simplify things by choosing M 1 = 0 without destroying the see-saw mechanism in the bino sector. A second di erence between the benchmarks is the LSP mass, set to be 100 GeV in BP1 and 150 GeV in BP2. The fact that the LSP mass is higher in BP2 while the higgsino mass remains the same as in BP1 results in slightly enhanced direct detection limit. Finally, one set of bounds we have yet to address are the LHC limits on electroweakinos. The bounds on chargino/heavier neutralinos ( ~02; ~0) production can be quite stringent, 3 1 TeV if the chargino/neutralinos decay primarily to sleptons [116, 117]. Fortunately, 7We have used micrOMEGAs-v3 for DM direct detection cross-section. For 100 GeV neutralino, the cross-section should be less than 9 8There is no phenomenological disaster associated with M 2 = M 3 = 0 as the 2;3 elds will receive a nite, loop-level contribution from gaugino loops, see section 4.1.2 and 4.1.3 m~ q~;u~;d~;`~;e~ Parameters ( IR) int MD3 MD2 MD1 b 0 d 0 MD3 MD2 MD1 M 3 M 2 M 1 b S tan Outputs m~ g~1;g~2 m~ q~;u~;d~ m~ t~1;2 m~ ~2;3 m~ ~ ` m~ e~ m~ ~01 mh;H h 2 SI 02 ! h ~01 03 ! Z ~01 03 ! h ~01 1011 GeV 2:48 TeV 4:72 TeV 556 GeV 0 0 0 0 0 1.97 TeV 1.39 TeV 1.15 GeV 0:7392 TeV2 BP1 7:00 TeV 5:00 TeV 515 GeV 2.50 TeV 1.026 TeV 0:7492 TeV2 1011 GeV 2.65 TeV 567 GeV 940 GeV 3.45 TeV 1.30 TeV 0 1.475 TeV 1.0 TeV 0:8582 TeV2 BP2 7.50 TeV 5.50 TeV 525 GeV 7.50 TeV 5.50 TeV 1.65 TeV 1.076 TeV 0:8672 TeV2 the Higgs mass and DM relic density and direct detection constraints. The DM direct detection limit for a 100 GeV WIMP is around 9 on left handed sleptons push the mass of the winos into the multi-TeV range, while DM and Higgs constraints prefer higgsinos at 1 TeV. As a result, ~ ; ~02; ~03 are predominantly higgsino and thus have Yukawa-suppressed couplings to SM fermions (this is exacerbated by the small tan in BP1 and BP2). With sfermion-fermion decays suppressed, the charginos and neutralinos decay preferentially to gauge bosons and Higgses, modes with much looser constrains. The dominance of the gauge/Higgs boson branching ratios of ~ ; ~02; ~03 in BP1 and BP2 can be seen in the bottom rows of table 2. These two benchmarks have been chosen with particles sitting just outside the existing bounds. In the near future, both points would be exposed through jets plus missing energy searches (sensitive to q~; u~; d~) or leptons plus missing energy (sensitive to `~). However, the squarks and left handed sleptons can easily been taken heavier without ruining the main features of GSS, namely the near degeneracy of the right handed sleptons with the LSP. Compressed spectra, especially among weakly interacting states, are hard to probe at the LHC, though studies with displaced vertices [118, 119], soft-tracks [120, 121], or initial state radiation [122{126] can be a useful tool to look for such regions. Should a compressed slepton-LSP sector be discovered at the LHC, the next step towards singling out GSS as the underlying framework would be to verify the hierarchical structure of the spectrum, for example, a right handed slepton signal without any sign of squarks, left handed sleptons, or gauginos. As the absence of other states is a rather unsatisfactory discriminator among models, a more concrete signal is the presence of SU(2), SU(3) adjoint scalars. The masses of these states is more model dependent (e.g. compare BP1 and BP2) and it is possible that they are light enough to yield a signal at the LHC [127], however it is likely that a future, higher energy machine is required to explore the spectrum fully. 6 Summary and conclusion The recent run of the LHC has put stringent constraints on the superpartner masses. In fact, the strongest limit is drawn when the gluinos/squark masses are well separated from the LSP mass, reaching almost 2 TeV. A heavy gluino raises the soft mass of Higgs elds through renormalization group evolutions. As these parameters are a measure of the ne tuning, the non-observation of the superpartners has resulted into nely tuned regions of parameter space for most supersymmetry models. Frameworks with supersoft supersymmetry with Dirac gauginos are well motivated in this light. The supersoft nature of the gluinos ensure that the gaugino mediated correction to the squark masses are nite and not log enhanced. Therefore, Dirac gluinos can be naturally heavy. Consequently, the pair production of the gluinos goes down signi cantly due to kinematic suppression. Moreover, the production of same chirality squarks are also less as this requires chirality ipping Majorana gaugino masses in the propagator. The reduction in the production cross-section of the squarks weakens the constraints on the squark masses signi cantly. Hence, supersoft models are often coined as `supersafe' in the literature. An additional virtue of this framework is avor and CP violating e ects are well under control. However, supersoft frameworks also su er from a few drawbacks. First, the GuidiceMasiero mechanism is unavailable so viable values require a conspiracy between the supersymmetry breaking scale and the Planck scale. Further, the natural D- at direction of the Higgs potential sets the tree level quartic term to zero, making it very di cult to t the observed Higgs mass of 125 GeV. Finally, supersoft models contain additional scalars in the adjoint representation of the SM gauge group that often acquire negative squared masses, resulting in a color breaking vacuum. An interesting way to resolve these three issues is to supplement the theory with additional, potentially non-supersoft operators involving the same D-term vev used to generate gaugino masses, the so-called generalized supersoft framework. The additional operators in GSS generate -terms proportional to the supersymmetry breaking vev and positive de nite masses (squared) for the scalar components of the adjoint chiral super elds. While economically solving the and adjoint masses issues is a step forward, dark matter remains an issue in GSS; right handed sleptons only receive mass through the nite correction from the bino and therefore seem to be destined to be the LSP unless the mediation scale is low. In this work we mapped out the parameter space of GSS, paying particular attention to the DM problem. We have shown that bino LSP can be avoided while maintaining a high mediation scale and without new elds in parameter regions where the two GSS -terms are unequal. If u 6= d, supersoftness is lost and loop suppressed, log divergent pieces from the hypercharge D-term contribute to the running of the scalar masses. For the right choice of inputs, these hypercharge contributions can lift up the RH slepton mass above the bino. Focusing on the region where the bino is the LSP, we explored how well the setup can satisfy additional constraints such as the 125 GeV Higgs mass, correct relic abundance | achieved whenever the bino and RH slepton masses are similar and coannihilation becomes important | and compatibility with the latest LHC results. The RH slepton masses are controlled by the di erence of the UV parameters while the higgsino mass is set by the sum. If we insist that the LSP bino is a valid thermal relic DM that escapes all direct detection bounds, the two constraints become tightly correlated, since the slepton mass controls the degree of coannhiliation while the higgsino mass controls the strength of the Higgs-exchange DM-nucleon interaction. Within this parameter region, we nd squark and slepton collider bounds can be satis ed, but the Higgs mass is generically too low unless one resorts to large loop corrections. Rather than resort to heavy stops, we showed how additional, NMSSM-type interactions can be used to lift the Higgs mass. These NMSSM-type interactions require no additional eld content, as the bino's Dirac partner is a gauge singlet. The interpolation between IR constraints and UV inputs requires some care, as non-standard supersymmetry breaking interactions are generated whenever u 6= d and enter non-trivially into the RGE. Even though we supplement this work with full numerical solutions, we focus rather on calculating general features of the spectrum, which we derive using analytical solutions whenever we can after making various simplifying assumptions. In fact, most of the features of the electroweak spectra get captured even after these simpli cations. Instead of scanning the full parameter space numerically for allowed regions, this approach allows us to generate intuitions about how to convert various experimental bounds into bounds in results of this paper came from taking the masses and parameters of the theory at the supersymmetry breaking scale ( D) to be inputs. The lack of knowledge in the exact nature of supersymmetry breaking dynamics does not allow us directly to consider the parameters at the messenger scale as inputs. A derivation of renormalization group equations of these new operators | including the operators proposed in eq. (2.4) | in the presence of arbitrary hidden sector dynamics (in the fashion of [87]) would allow one to relate these masses and parameters of the messenger scale and the supersymmetry breaking scale. To further UV complete this framework, more detailed messenger sector model building is required. Acknowledgments No. PHY-1520966. The work of AM was partially supported by the National Science Foundation under Grant A Neutralino and chargino mass matrices The fermion sector in our case is di erent from the usual MSSM structure. The neutralino mass matrix in the basis (B~; S~; W~ 0; T~0; H~u0; H~d0) looks like HJEP05(218)76 16 2 dS = dt 16 2 d ~u 16 2 dyt dt ' 3 jytj2 ~u; dt ' yt 6 jytj2 ~ 2 u + ~u + ~u 136 g32 : M ~0 = B B B BB MD1 BB g0vu MD1 M 1 0 0 MD2 gvu 2 2 2 pSvu gvd 2 MD2 M 2 T vd T vu 2 2 g0vu 2 pSvd 2 gvu T vd 2 2 0 g0vd 1 2 pSvu C gvd2 CC C 2T vu CC : M ~+ = BMD2 0 0 p 2 MD2 M 2 2 gvu 1 p 2 pTv2d CA : Also the chargino mass matrix written in the basis (W~ ; T~ ; H~d ) and (W~ +; T~+; H~u+) takes the following shape B Switching on the Yukawa coupling We now switch on the Yukawa couplings and show how the right handed slepton masses get generated through RGEs. For this the following equations are required to be solved through some approximate means. Such as (A.1) (A.2) (B.1) (B.2) (B.3) Eq. (B.3) can be simpli ed to obtain the following form Z d log jytj2 = 4 2 dt jytj2 2g32 Z 3 2 dt: This can be further simpli ed to and used for the solution of ~u, which we nd 4 2 dt jytj2 = log jyt( )j jyt( )j 2 ! 2 8 S=3 # ; (B.5) ; 16 2 dS dt 66 2 5 g1S ' 12 jytj2 ~ 2 u : The nal part is the computation of S which directly goes into the RGEs of the scalar masses. We simplify by considering (B.4) (B.6) (B.7) (B.9) (B.10) S(t) = 2 g1(t) 2 2 ~u(t3) 2 6 4 jg1(t3)j2 + b1 Z 8 2 dt3 ~u(t3) 7 : 2 3t 5 t0 Although, the nal result is not really transparent from eq. (B.10), however it is conspicuous that a non-zero S requires a non-zero ~u=d at int. One can treat the above equation as the most general linear rst order ordinary di erential equations. The term in the right hand can be regarded as a source or a driving term for the inhomogeneous ordinary di erential equation. The solution is straightforward which involves a integrating factor. The nal solution can be written in the closed form as Z t t0 g1(t) 2 Z t 2 d ~u(t3) dt3 3 27 : 5 Z t 2 Using integration by parts one can further simply this to obtain 2 Z t3 d log g1(t2) ; (B.8) The scalar mass matrix elements written in the basis (hu; hd) after integrating out the adjoint scalars and assuming T = 0, turns out to be M121 = b cot 1 g2M 22 4 4M D22 + M 22 4M D21 + M 21 v2 sin2 2 2 M222 = b tan g0 s MD1 M 1 v + p 2 2 4M D21 + M 21 ( 2 + cos 2 ) cot p +2 2 sg0 2 MD1 v 4M D21 + M 21 sin2 1 g2M 22 4 4M D22 + M 22 4M D21 + M 21 2 2 v 2 s2 4M D21 + M 21 sin2 ; v2 cos2 M 1 v 2 g0 s MD1 M 1 v + p 2 2 4M D21 + M 21 (2 + cos 2 ) tan 2 4M D21 + M 21 sin 3 sec p +2 2 sg0 2 MD1 v 4M D21 + M 21 cos2 2 s2 4M D21 + M 21 cos2 ; M122 = b 1 g2M 22 4 4M D22 + M 22 v2 sin cos 2 s 2 s 2 2 v M 1 v 2 2 4M D21 + M 21 cos 3 csc 2p2 4M D21 + M 21 cos 2 + 2 2v2 3M 1 s 2 + 4M D21 4M D21 + M 21 2 2 sin 2 : (C.1) Furthermore, integrating out the Dirac adjoint scalars and adding the two minimization equations for hu and hd we nd 2b sin 2 = j uj2 + j dj2 + 2 s2v2 M D21 2 4M D21 + M 21 g0 sv2 p MD1 M 1 2 4M D21 + M 21 cot 2 (C.2) p2g0 sv 2 MD1 4M D21 + M 21 cos 2 M 1 4 4M D21 + M 21 ( 7 + 5 cos 2 ) cot : v For a xed right-slepton and higgsino masses, u and d is completely xed. Therefore, given the values of gaugino masses and tan , one completely xes b . The size of b also controls the heavy and charged Higgs masses. Hence, the whole spectrum gets determined. D RGEs with S, T The inclusion of the superpotential and non-standard soft supersymmetry breaking terms proportional to S and T modi es the anomalous dimensions of the Higgs elds. This in turn modi es the RGEs of the soft supersymmetry breaking Higgs mass parameters and obviously the -term and non-standard supersymmetry breaking terms proportional S S ˜ Hu ˜ Hd μ gauge singlet chiral super eld. Tadpole issue h ~ i d + 2 j Sj2 + 6 j T j 2 ~ ; d u + 2 j Sj2 + 6 j T j 2 ~u: + 2 j Sj2 + 6 j T j + 2 j Sj2 + 6 j T j 2 hm~ 2hu + m~2hd + 2 nj ~dj2 + ~d 2 hm~ 2hu + m~2hd + 2 nj ~uj2 + ~u + ~d + ~u oi ; oi ; Another important aspect of our case is the e ect of non-standard soft terms in the presence of the singlet. 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Sabyasachi Chakraborty, Adam Martin, Tuhin S. Roy. Charting generalized supersoft supersymmetry, Journal of High Energy Physics, 2018, 176, DOI: 10.1007/JHEP05(2018)176