Selectron NLSP in Gauge Mediation
We discuss gauge mediation models in which the smuon and the selectron are mass-degenerate co-NLSP, which we, for brevity, refer to as selectron NLSP. In these models, the stau, as well as the other superpartners, are parametrically heavier than the NLSP. We start by taking a bottom-up perspective and investigate the conditions under which selectron NLSP spectra can be realized in the MSSM. We then give a complete characterization of gauge mediation models realizing such spectra at low energies. The splitting between the slepton families is induced radiatively by the usual hierarchies in the Standard Model Yukawa couplings and hence, no new sources of flavour misalignment are introduced. We construct explicit weakly coupled messenger models which give rise to selectron NLSP, while accommodating a 126 GeV MSSM Higgs mass, both within the framework of General Gauge Mediation and in extensions where direct couplings between the messengers and the Higgs fields are present. In the latter class of models, large A-terms and relatively light stops can be achieved. The collider signatures of these models typically involve multilepton final states. We discuss the relevant LHC bounds and provide examples of models where the decay of the NLSP selectron is prompt, displaced or long-lived. The prompt case can be viewed as an ultraviolet completion of a simplified model recently considered by the CMS collaboration.
The Large Hadron Collider (LHC) experiments, employing data from the =7 and 8 TeV runs, have placed considerable constraints on the strongly-interacting sector of supersymmetric (SUSY) models, with lower limits being set at about 11.5 TeV on the masses of the gluino and the first two generations of squarks Chatrchyan:2014lfa; Aad:2014pda, and up to around GeV for stops CMS:2013cfa; ATLAS:2013cma. Together with the requirements on the stop sector coming from the Higgs mass measurements Draper:2011aa, this forces the colored states to be heavy, putting under stress the paradigm of naturalness, at least for minimal realizations of low-energy SUSY.
On the other hand, the electroweak (EW) sector of SUSY models is less constrained by direct SUSY searches and the purely EW states such as sleptons, neutralinos and charginos might in principle be significantly lighter than the rest of the spectrum. However, the LHC has recently started to set impressive bounds also on the EW sector, often far beyond the LEP limits. The present bounds for sleptons are around GeV and 500600 GeV for charginos CMS:2013dea; Chatrchyan:2014aea; Aad:2014vma; Aad:2014nua, and further improvements are expected from the upcoming =13/14 TeV run. We find it important to survey non-standard spectra and signatures, in order to fully exploit the LHC discovery potential in terms of EW SUSY particles.
In this paper we discuss models of gauge mediation (GM) in the Minimal Supersymmetric Standard Model (MSSM) which have the non-standard property that the right-handed (RH) selectron and smuon are the (mass-degenerate) next-to-lightest superpartners (NLSP). The lightest SUSY particle (LSP) is the approximately massless gravitino, while the remaining superpartners, including the RH stau, are all parametrically heavier. We will refer to this exotic GM spectrum as the selectron NLSP scenario, in order to distinguish it from the so-called slepton co-NLSP scenario which refers to the case where the RH selectron, smuon and stau are all nearly mass-degenerate co-NLSP Ruderman:2010kj. Throughout this paper, “slepton” refers only to either a selectron or a smuon, in accordance with the usual experimental separation between leptons and taus.
We take a bottom-up perspective and explore how the selectron NLSP scenario can be realized in the framework of General Gauge Mediation (GGM) Meade:2008wd. We will also consider extensions of GGM in which the GM messenger sector is directly coupled to the Higgs sector of the MSSM Chacko:2001km; Komargodski:2008ax; Evans:2010kd; Evans:2011bea; Evans:2012hg; Kang:2012ra; Craig:2012xp; Evans:2013kxa. One virtue of the UV completions we present is that the desired hierarchies in the slepton sector are induced by the already existing flavor texture of the Yukawa couplings in the Standard Model (SM). Thus, in these setups, the selectron NLSP scenario is realized without introducing any new source of flavor misalignment in the slepton sector.111In models where an intrinsic flavor violation in the SUSY breaking mechanism is present, it is natural to get small soft masses for the light generations as a result of a mechanism of flavour violation suppression Nomura:2007ap. Spectra with selectron NLSP were obtained in this context in Shadmi:2011hs.
From the collider point of view, the stable LSP gravitino generically gives rise to missing transverse energy (). Therefore, the key role in characterizing the collider phenomenology of GM models is played by the NLSP, which decays to its SM partner and the gravitino. In this paper we will assume R-parity conservation. Depending on the SUSY breaking scale, the decay of the NLSP can be either prompt, displaced or long-lived on collider scales. We discuss the multilepton final states arising in the case where the NLSP decay is prompt, the charged tracks arising in the long-lived case, as well as the charged tracks ending with displaced lepton-vertices arising in the intermediate case. In the prompt case, since the dominant decay channel of the stau typically is the 3-body decay, via an off-shell Bino, to a tau, a lepton and an NLSP slepton (which subsequently decays into a lepton and a gravitino), stau pair production gives rise to the final state Ambrosanio:1997bq. A simplified model with stau NNLSP and selectron NLSP was recently employed by the CMS collaboration in order to interpret the results of a multileptons search Chatrchyan:2014aea, see also D'Hondt:2013ula for further discussions concerning this simplified model. The messenger models we present, which realize this spectrum at low energies, can be viewed as possible UV completions of such a simplified model.
In GM models, the MSSM soft masses are determined by the gauge quantum numbers of the corresponding superpartner. Even though the right-handed (RH) sleptons are mass-degenerate with the RH stau at the messenger scale, the stau mass is usually driven lighter than the first two slepton generations at low energies due to the contributions from the Yukawa interactions to renormalisation group (RG) evolution. Moreover, the lightest stau mass eigenstate can be further separated from the RH sleptons due to stau mass-mixing. However, a closer inspection of the MSSM RG equations (RGEs) reveals that the sign of is crucial for determining whether the stau is driven lighter or heavier than the mass-degenerate sleptons at low energies. In standard GM models, is positive along the RG flow and, as a consequence, the stau is driven lighter than the selectron/smuon. Instead, if remains negative during most of the flow, the stau is driven heavier than the selectron, thereby making possible the realization of the selectron NLSP scenario at low energies.
A negative implies tachyonic scalar masses at high energies and along parts of the RG flow. The presence of tachyons along the flow reverses the usual effect of the Yukawa interactions on the scalar masses, making heavier the scalar particles which have the largest Yukawa coupling. In this way, the appearance of a selectron NLSP at low energy is linked to the usual hierarchy among the SM Yukawa couplings .
In GGM, the squared soft masses for the two Higgs doublets are equal to the left-handed (LH) slepton soft mass in the UV, since they carry the same gauge quantum numbers: . Hence, given the definition of , in order to realize a selectron NLSP in GGM, one needs tachyonic boundary conditions for the sleptons at the messenger scale. Clearly, this possibility goes beyond the minimal GM paradigm in which the squared soft masses for the sleptons are always positive at the messenger scale. Furthermore, it requires a considerable amount of gaugino mediation to push up the scalar masses at low energies such that a tachyon free spectrum is obtained.
Alternatively, if one allows the Higgs doublets to couple directly to the messengers through a superpotential interaction, i.e. not only radiatively via gauge interactions, then the Higgs soft masses can get new contributions in addition to the gauge mediated ones (we refer to this class of models as “deflected”). In this case, a large and negative additional soft mass contribution to the down-type Higgs doublet can induce a selectron NLSP at low energies.222The possibility of having slepton NLSP realized within the MSSM, by allowing the soft masses of the two Higgs doublets to be independent from the soft masses of all the other sfermions, was considered in Evans:2006sj; Grajek:2013ola.
We show that the selectron NLSP scenario indeed is a possible low energy spectrum for GGM models with a Higgs mass at 126 GeV and with the colored sector being significantly heavier. We also show that the selectron NLSP scenario can be realized in models with extra Higgs-messenger interactions, in which the stops (and possibly also other colored states) are kinematically accessible at the LHC. In particular, we study in some detail a simple model originally proposed in Evans:2011bea as a possible way of getting the correct Higgs mass in the MSSM through the generation of a large . Here the down-type Higgs mixes with the messengers and it acquires a tachyonic mass at tree level. This tree-level effect is enhanced for low messenger scales and therefore, this model becomes a natural UV completion of the case where the selectron NLSP is decaying promptly.
The paper is organized in the following way: in Section 2 we study the MSSM RG equations, together with the low energy constraints, and we characterize how the messenger scale soft spectrum should look like in order to realize the selectron NLSP scenario at low energies. We first outline the qualitative features of the different possibilities in GGM and in models with Higgs-messenger couplings. We then study numerically the parameter space of different models which give rise to selectron NLSP. In Section 3, as a proof of principle, we construct some explicit and minimal messenger models which realize this scenario. Finally in Section 4 we discuss the collider signatures of spectra with selectron NLSP treating separately the cases in which the NLSP decay is prompt, displaced or long-lived on collider scales.
2 Selectron NLSP scenario from the RG-flow
In this section we discuss the RGEs for the MSSM soft masses and the low-energy constraints coming from the requirements of EW symmetry breaking (EWSB) and the absence of tachyons. The purpose is to characterize the parameter space that gives rise to the selectron NLSP scenario at the EW scale. In particular, our aim is to determine the minimal requirements on the soft masses, at the messenger scale, which are necessary in order to get the stau heavier than the selectron/smuon. Note that we distinguish between sleptons and stau , as well as between leptons and tau .
A necessary ingredient - though not sufficient - to realise the selectron NLSP scenario is the following condition on the low energy soft SUSY-breaking mass terms:
which can be mapped into specific conditions on the soft masses at the messenger scale by solving the MSSM RGEs. The splitting between the stau and the selectron gauge eigenstates can easily be derived from in the case where it is small compared to the slepton masses: .
The RGEs for the soft masses of the LH and RH sleptons and stau are, at one loop, given by
where the contributions induced by the Yukawa couplings enter via the combination
where the trace Tr is taken over the flavour indices of the MSSM soft mass matrices, , , .
If we neglect the contributions from the (small) lepton Yukawa couplings, we see that the RG evolution for the difference between the stau and slepton soft masses is determined by :
where . In leading-log approximation, the solution to Eq. (6) reads:
where and are the messenger scale and a typical low-energy soft mass, respectively, and where we have assumed flavour blind soft terms at the scale by imposing and . Here, the soft masses on the RHS of (7) take the values they have at the messenger scale and we neglect their running. This approximation allows us to make some useful rough estimates, but it is clearly not accurate in the case where the running of the slepton masses and/or of the down-type Higgs mass is non-negligible.
In minimal GM models, where all the three (squared) soft masses on the RHS of Eqs. (6, 7) are positive at the messenger scale, , and the two stau soft masses are driven smaller than the slepton masses at low energies. Instead, from Eqs. (6, 7) we can also conceive the possibility of realizing situations where such that the stau masses are driven heavier than the slepton masses at low energies. This is the effect we need in order to realize the selectron NLSP scenario. As is manifest in Eqs. (6, 7), the key ingredient is the presence of tachyonic masses for and/or along the RG flow, sufficiently large to render negative.
In Figure 1 we show both the leading-log estimates and the exact 1-loop solution for the mass splitting as a function of the high-energy values of (left) and (right). The two plots correspond to the two prototypical spectra that we will study. Figure 1 (left) corresponds to deflected spectra where the splitting is triggered by alone and where there is no need for large values of . Instead, Figure 1 (right), where and where is always sizeable, corresponds to GGM spectra.
Comparing the two plots in Figure 1, we see that the leading-log estimate predicts the splitting effect in the right plot to be a factor of larger than in the left plot. However, it turns out that this approximation is accurate only for small values of . In fact, by increasing , the gaugino mediation effect from the Wino on the LH sleptons and on the down-type Higgs becomes relevant and, as a consequence, greater tachyonic values for are needed in order to get the same splitting effect. Note that the value of the squark masses controls the running of and determines the splitting of the turning points for the different curves, which is not captured by the leading-log approximation.
We are interested in spectra in which the RH sleptons are co-NLSP.333Another interesting option could be to have the LH sleptons and their corresponding sneutrinos as co-NLSP. Many of the considerations of this paper can be easily adapted to this case, which we leave for future studies. In order for the RH sleptons to be co-NLSP they should be lighter than all the other sparticles and, in particular, lighter than the physical mass of the lightest stau. In order to take into account the left-right mixing we should consider the mass matrices for the sleptons/staus, which are given by,
where we have neglected the contributions from the corresponding -terms, as well as the contributions that arise upon the EW symmetry breaking. For the sleptons, since the off-diagonal entries are negligible, we will denote the two mass eigenvalues also by and . For the staus, since the off-diagonal entries can be relevant - especially for large values of - we are interested in the smallest stau mass eigenvalue, given by
where all the soft masses are evaluated at the EW scale. The selectron NLSP scenario requires at the EW scale.
Since the tau Yukawa coupling in (6) is enhanced at large values of , it is possible to increase the separation between the RH stau and slepton by increasing . However, from (8) we see that can not be too large since the mixing in the stau mass matrix then increases as well – precisely how much also depends on the values of the mass parameters in (8) – and consequently, the lightest stau, whose mass is given in (9), is pushed lighter. In other words, for a given configuration of the parameters, we will have an upper bound on from the requirement that the LR stau mixing can not be too large in order to realize . The dependence of such a bound on the slepton masses and the splitting parameter is shown in Figure 2. As we can see, this bound can be quite stringent, especially for a light LH stau. Interestingly, it can be translated into an upper bound on , i.e. on the Higgsino masses, taking into account that at least moderate values of are typically needed in order to radiatively generate a sizeable mass splitting , as is shown in Figure 1. As we can see from Figure 2, the bound gets significantly relaxed if we allow for a heavy LH stau. In this figure, we also show the exact dependence of on the values of and , and we see how the splitting decreases as we increase the mass of the RH sleptons, at fixed .
Let us now discuss how these two features of the selectron NLSP scenario, i.e. the necessity of having tachyonic masses for and/or and to minimize the left-right mixing in the stau mass matrix, can be realized in different classes of models. In particular we are going to consider GGM models in Section 2.1 and models with deflections for the soft masses in the Higgs sector in Section 2.2.
Each section will be organized as follows: we first give a brief summary of the structure of the parameter space. We then give some qualitative understanding of the RG-flow effects for points with selectron NLSP. In order to do that we solve the 1-loop RG-equations semi-analytically imposing the EWSB conditions at tree level.444We thank Simon Knapen and David Shih for sharing with us a private Mathematica code in which this is computed. We then complement our analysis with a full numerical scan of the parameter space with SOFTSUSY 3.3.9 Allanach:2001kg, taking into account low-energy threshold corrections and 2-loop effects. The full numerical approach is going to confirm our qualitative understanding and to realize selectron NLSP scenarios with .
2.1 General Gauge Mediation
The General Gauge Mediation (GGM) framework consists of a hidden sector that breaks SUSY spontaneously and a visible sector that we choose to be the MSSM. The decoupling limit between the two sectors is achieved when all the SM gauge interactions are switched off: , for . Consequently, the parameter space is defined at the messenger scale by two independent sum-rules and , which follows from the two non-anomalous symmetries of the MSSM. Using these two relations we can write two of the five MSSM soft terms in terms of the others:
The independent GGM soft parameters at the messenger scale are then reduced to three complex gaugino masses , which are here taken to be real, and three real sfermion masses . These soft masses can be written as
where is the quadratic Casimir for the representation under the gauge group of the SM, with the GUT normalization for . and are model-dependent functions of the SUSY-breaking scales of the hidden sector and of the characteristic UV scale , which we take to be unique.
In the Higgs sector, the soft masses for the two doublets are fixed to be equal to the soft mass for the left-handed sleptons:
If we take to be a free parameter in the superpotential, independent of the SUSY breaking mechanism, then GGM sets
at the messenger scale. Moreover, the A-terms are always suppressed in gauge mediation and can be set to zero at the messenger scale. The GGM parameter space is then determined by parameters, where 6 parameters describe the soft masses for gauginos and sfermions, 2 parameters characterise the EWSB and and 1 is simply the messenger scale , which sets the length of the RG-flow.
Because of the condition (14), the easiest way to realize the selectron NLSP scenario is to have a tachyonic mass for the left-handed sleptons at high energy, resulting in a negative . A sufficient amount of gaugino mediation, in particular a heavy Wino, can then drive the left-handed slepton mass positive at low energies, and even heavier than the right-handed ones. This effect is displayed in both of the RG flow examples shown in Figure 3, where we have chosen and at low energy, which correspond to a splitting of , as can be derived from Figure 2.555In order to get a light RH selectron NLSP at low energy we may need the RH slepton mass to start tachyonic at the messenger scale in order to counteract the effect of gaugino mediation controlled by the Bino mass. Such a tachyonic start also helps getting the desired splitting effect between the stau and the sleptons, as can be seen from Eq. (6).
The two examples in Figure 3 are distinguished by the behaviour of the squarks along the flow. In the left panel, the squark masses are positive at the messenger scale and, as a consequence, and are driven negative along the flow. Since is already forced to start tachyonic because of the GGM condition (14), which set it equal to , this scenario is characterized by a large , which is fixed to be by the EWSB condition. In the left panel of Figure 3 we get but the selectron NLSP scenario is still possible since is large enough to make the LH sleptons heavy () and thereby counteract the mixing effects in the stau mass matrix, as can be seen in Figure 2.
A possibility of getting a small and selectron NLSP in GGM is depicted in the right panel of Figure 3, where . The idea is to start with tachyonic masses for the squarks at the messenger scale, which can then be driven positive along the flow by gluino mediation. In this way, the usual effect of the stops on is partially reversed since is pulled up until the scale for which the stop masses become non-tachyonic again. Analogous spectra have been proposed as possible ways to minimize the tuning in GGM and getting large A-terms to enhance the MSSM Higgs mass Dermisek:2006ey; Draper:2011aa. In order to get selectron NLSP, this spectrum is quite natural since it is the only way of starting with a tachyonic without getting a large at the EW scale, hence automatically minimizing the mixing effects in the stau mass matrix (we get in our benchmark). Therefore, we expect the selectron NLSP scenario to be a possible spectrum in the GGM scenarios proposed in Dermisek:2006ey; Draper:2011aa.
In principle one can envisage a third possibility of getting the selectron NLSP scenario in GGM by starting with a fully non-tachyonic spectrum and triggering a negative squared mass for (large enough to make negative) via the RG-flow. The effect can be understood from the following RG equation:
From this equation, we see that when the squark masses are heavy, , and the RGEs drive . This effect becomes relevant for large values of because it is controlled by . However, heavy squark masses would also induce a very large , which would enhance the stau left-right mixing term. Getting a selectron NLSP is then a matter of balancing these two effects. We checked numerically that this indeed is feasible, but it requires very fine-tuned spectra. For this reason we do not consider this possibility in what follows.
In Figure 4, we show the result of a full numerical scan over the GGM parameter space in the plane defined by the high-energy values of and . The GGM parameters were varied independently in the following ranges:
Notice that both signs of were considered. In Figure 4, the light-green points correspond to a selectron NLSP with GeV and which fulfill the basic phenomenological requirements: no tachyons at the EW scale, successful EWSB etc. Moreover we discard all points which have superpartners heavier than , thus imposing an indirect mild upper bound on the soft parameters at the messenger scale. The dark-green points, in addition, account for the observed Higgs mass, up to theoretically uncertainties: GeV.
Our scan confirms that a selectron NLSP can be obtained as a consequence of large negative values of , while also has to be large in order to avoid tachyonic LH sleptons in the IR. In particular, we see from Figure 4 that the lowest possible value of that is compatible with a non-tachyonic spectrum, rapidly increases as increases. The observed Higgs mass needs rather heavy stops in GGM, and thereby a large . Therefore we can obtain and GeV (dark points) only for a sizeable (i.e. large ), as we expected from the results shown in Figure 2.
From this scan, we observed that the low-energy value of the selectron mass can be as light as GeV and a selectron NLSP in GGM models is only possible for , i.e. for a sufficiently long RG running. This implies that the decay to lepton and gravitino is never prompt, as will be discussed in Section 4.
2.2 Deflected Models
We define “deflected” models as those models of GM that feature additional contributions to the Higgs masses at the messenger scale, besides the GGM one of Eq. (14). These additional contributions are due to the presence of extra superpotential interactions between the hidden sector and the Higgs sector. In particular, the new interactions can generate and at the messenger scale, thus being good candidate to solve the problem in GM Dvali:1996cu; Csaki:2008sr; Komargodski:2008ax; DeSimone:2011va. Another nice feature of these kind of models is the possibility of generating non-zero A-terms at the messenger scale Chacko:2001km; Evans:2010kd; Evans:2011bea; Evans:2012hg; Kang:2012ra; Craig:2012xp. A complete study of the threshold corrections at the messenger scale that one can get in this class of models have been performed in Evans:2013kxa. Without entering the details of any specific model, we discuss here the general features of this setup, which are relevant for the selectron NLSP scenario. In Section 3, we will discuss simple messenger models that explicitly realize this spectrum.
Equation (6) shows that the selectron NLSP scenario can be achieved by means of tachyonic boundary conditions for the down-type Higgs, . For this reason we focus our attention on modifications of the high-energy thresholds of GGM only for the Higgs soft masses, assuming for the moment that all the other soft masses are not deflected. This simplifying assumption is not necessarily realized in concrete models, as we will show in Section 3. We can account for the Higgs deflections by adding two independent parameters at the messenger scale:
Clearly, these new contributions invalidate the GGM relation (14). Moreover, they deform the hypercharge sum rule in the sfermion sector by introducing a non-zero Fayet-Ilioupoulos term at the messenger scale, .
If is negative and sufficiently large, then the RHS of (6) can be negative even if both the RH and the LH slepton squared masses are positive, leading to selectron NLSP. This is depicted in the left panel of Figure 1, where we plot both the leading-log estimate of Eq. (7) and the exact RG-solution for the splitting of the RH stau and selectron soft masses.
However, having tachyonic can affect the EWSB. In the MSSM, the two EWSB conditions can be written as
where every parameter is evaluated at the EW scale. For large to moderate values of , the term in (19) containing the down-type Higgs soft mass can be neglected and one obtains the approximate expression
Inserting this relation into the mass formula for the CP-odd Higgs, one gets that, at the EW scale, in the large limit
indicating that can potentially lead to a tachyionic CP-odd Higgs.
As a first “tree level” solution to this problem, equation (22) suggests that, in order to obtain ,666The current bound on from direct searches CMS-PAS-HIG-12-050 – setting a more stringent constraint – will be taken into account in the numerical analysis. should also be negative at the messenger scale and, in absolute value, larger than . This is indeed a viable case and it is displayed in the left panel of Figure 5. Note that, even if starting with a large and negative , which induces a large at the EW scale, the left-right mixing in the stau mass matrix can always be suppressed by a large , which is not forced to start tachyonic, in contrast to the GGM case.
Another possibility to circumvent the obstruction given by the requirement of a non-tachyonic is to enhance the negative contributions to (driven by terms ) from the RG running, which are actually responsible for the radiative EWSB. These radiative effects are summarized by the following RGEs:
where the Fayet-Iliopoulos term is given in (5) and where
Here we see that, beside the terms proportional to , the contributions from the gauge interactions to the difference vanish. The RG equation for the difference in (22) is then given by
Hence, in models with negative, the problem of having at the EW scale can be alleviated for instance by heavy stops, a large or a large gluino mass, which drive up the stop masses at low energies. Interestingly, within the MSSM, the very same conditions are required in order to accommodate a SM-like Higgs scalar with mass around 126 GeV. In the right panel of Figure 5 we display a possible solution with heavy stops in which the entire soft spectrum at the messenger scale is non-tachyonic, except for , which is responsible for triggering the desired effect. This benchmark realizes, in a concrete setup, the spectrum of Figure 1 (left).
Following the above discussion, we have performed a scan of the UV parameters, again requiring a sizeable mass-splitting between the lightest stau and the selectron NLSP, GeV, as well as the other constraints. The points in the scan that exhibit a selectron NLSP are shown in Figure 6. In the left panel we display the plane , defined by:
As is highlighted in the figure, three different viable regions with selectron NLSP can be identified, which correspond to different realizations of the EWSB:
Region (i), with : this corresponds to the simplest (“tree-level”) solution to avoid at the EW scale, in which we allow for tachyonic up-type Higgs masses, at the messenger scale, which are larger in modulus than the down-type one, cf. the left panel of Figure 5.
Region (ii), with : the negative contribution to the up-type Higgs mass at the messenger scale is lower in modulus than the one to the down-type Higgs, so that we access an intermediate region in which the problem of the tachyonic mass for the CP-odd Higgs is solved partially by radiative contributions that can be induced by large stops or a large .
Region (iii), : in this case, a non-tachyonic value of the CP-odd Higgs mass, at the EW scale, is realized purely by radiative contributions coming from large stop masses (cf. right panel of Figure 5) or a large .
In the right panel of Figure 6, the result of the scan is shown in terms of the messenger scale and the low-energy value of the selectron NLSP mass. Important hints on the model building requirements can be extracted by identifying the parameters in Equations (12) and (13) which have to be independent at the messenger scale in order to realize each of the above regions. Points with different colors in Figure 6 correspond to different choices: the red points correspond to the simplest models of GM with , the purple points to a two-scale setup with two separate parameters and controlling the gaugino and sfermion masses, respectively, and the light-blue points correspond to the case where sfermion mass unification is relaxed by taking . The SUSY breaking parameters were varied in the range GeV for all the three cases. For the other parameters we took: , GeV.
As we can see from Figure 6, the selectron NLSP scenario in region (i) can be obtained even with (red points), while region (ii) only marginally and region (iii) is not accessible in this case. In fact, effective radiative corrections are very much constrained by the fact that there is only one scale that controls the whole soft spectrum. A large would be needed in order to obtain a non-tachyonic CP-odd Higgs with , but that would also increase the universal contribution to the slepton masses, washing out the effects of the Yukawa interactions that might give a selectron NLSP. Moreover, this scenario shares the phenomenological problems of minimal GM, in particular the requirement of needed in order to have GeV. As is shown by the right panel of Figure 6, this translates into a lower bound on the scalar masses, in particular on the slepton mass ( TeV), again because the spectrum is essentially controlled by a single parameter. As a consequence, testing such a scenario at the LHC would be very challenging.
In order to access the region (iii), large radiative corrections to are necessary. The simplest possibility is to rely on the gaugino-driven contribution of the running, through a quite heavy gluino. This is possible by splitting and , as is shown in Figure 6 (purple points). Notice that this scenario realizes automatically gaugino and sfermion mass unification at the messenger scale and hence, it can be easily embedded in messenger models with a complete GUT structure. A common mass parameter for the gauginos however implies a rather heavy spectrum, in particular GeV, cf. the right panel.
In order to realize region (iii) with lighter sleptons, we need further contributions from large stop masses and/or large . This latter possibility will be discussed in an explicit model in the next section, since the extra interactions which generate large A-terms will also typically contribute to the sfermion masses, resulting in a sizeable deflection of the spectrum from the usual GM one. Here we consider only the possibility of splitting the colored sector so that heavy squarks can be obtained, while keeping the sleptons light. This can be done in two ways: either by relaxing the hypothesis of sfermion mass unification, i.e. , or by dropping gaugino mass unification, i.e. . For illustrative purposes, in Figure 6, we adopted the first possibility (light-blue points). As we can see, region (iii) can now be easily accessed. Moreover, the light-blue points correspond to slepton masses down to GeV.
As a final remark, Figure 6 shows that, within these models it is rather difficult to obtain our effect for GeV, especially for light sleptons. Therefore, like in the case of GGM, these models typically predict the NLSP decay to be displaced from the interaction point (either inside or outside of the detector), as will be clear from the discussion in Section 4. Note, however, that the lower bound on the messenger scale can in principle be circumvented if we allow the soft masses for the scalars, other than the Higgses, to be tachyonic at the messenger scale.
3 Realizations in terms of messengers models
In this section we investigate possible concrete realizations of the selectron NLSP scenario in terms of weakly coupled messenger models, possibly directly coupled to the two Higgs doublets of the MSSM.
As mentioned in the previous section, it is possible to obtain selectron NLSP with standard (non-tachyonic) UV boundary conditions for the soft masses at the price of accepting a large tuning of the UV parameters. The only requirements consist of long running, i.e. a large messenger scale, and large . From a model building perspective, this case corresponds to usual models of (general) gauge mediation, and we will not discuss it any further here.
The other two cases we have described rely on negative squared masses in the UV, and are in some sense complementary. In the first case, which is within the definition of GGM, the squared masses of the left-handed sleptons are negative in the UV, and equal to the down-type Higgs squared mass; we will discuss this in Subsection 3.1. In the second case, we generate negative squared masses only for the down-type Higgs by coupling the Higgses to some hidden sector fields, thus going beyond the pure GGM paradigm; we will explore this option in Subsection 3.2.
Finally, in Subsection 3.3, we study a model which also includes extra contribution to the A-terms. This represents the most economical model that features promptly decaying selectron/smuon co-NLSP, a correct Higgs mass and also relatively light stops.
3.1 Boundary conditions with tachyonic slepton masses
The SUSY breaking parameters and determine the UV pattern of soft masses of GGM, and here we investigate whether it is possible to obtain the desired UV boundary conditions in models with weakly coupled messengers. The purpose of this section is to provide a proof of existence, without the ambition of being complete.
As was explained in Section 2.1, in order to obtain selectron NLSP it is sufficient to consider UV tachyonic boundary conditions for the left-handed sleptons. There are several mechanisms able to generate a negative squared mass for the scalars in GM. One possibility consists of considering gauge messengers, as explained in Intriligator:2010be; Buican:2009vv. This would require to specify the embedding of the SM gauge group into the unification group, as well as the mechanism that breaks it. Another option is to consider models where the Supertrace on the messengers is non-vanishing and positive. This would induce a negative contribution to the scalar soft masses in the MSSM, as for instance in models of direct gauge mediation ArkaniHamed:1997jv. However, as shown in Poppitz:1996xw, in minimal messenger models this contribution is divergent and it introduces logarithmic dependence on the UV cut-off
A possible way to soften the logarithmically divergent contribution is to UV complete the theory, for instance with models of semi-direct gauge mediation, where the messengers couple to the supersymmetry breaking sector only through another extra gauge group Seiberg:2008qj; Argurio:2009ge. However these models are plagued by the gaugino screening problem ArkaniHamed:1998kj; Argurio:2009ge, and hence are not useful in our setup.
Finally, even if the Supertrace on the messenger sector is vanishing, the simultaneous D and F term breaking of at least two pairs of messenger fields can lead to negative squared masses for the sfermions Buican:2008ws. In particular, this requires the pairs of messengers to be oppositely charged under an extra gauge group, with non-vanishing D-term breaking. This is the strategy we adopt in the following.777It is interesting to note that the weakly coupled possibilities discussed here are all characterized by the presence of heavy massive vector bosons in the hidden sector.
We also demand gauge coupling unification to be preserved. This would require the messengers to belong to complete representations of the unification group or to some “magic” set, as discussed in Martin:1995wb; Calibbi:2009cp.
We propose the set of messengers reported in Table 1, which are vectorlike pairs in the fundamental of the SM gauge groups.
|# of pairs|
One can easily show that this set of fields induces the same shift in the beta function coefficients of the , and gauge couplings. As a consequence, unification at the usual MSSM GUT scale is preserved, even though the messengers do not form a complete GUT representation. In the following we will consider slightly different mass scales for some messengers, assuming that the consequent thresholds induced on the running of the gauge couplings are negligible.
As is shown in Table 1, two out of the three charged messengers are also charged (with opposite charge) under an extra gauge group, with a non-vanishing D-term. Moreover, we assume the following superpotential couplings of the messenger fields to some spurions, i.e. the ’s:
where the subscripts refer to the gauge group under which the messenger field is charged and the superscripts of the -charged fields indicate their charge under the gauge group. The spurions take the following form,
with the choice such that the contribution to the soft masses will be negative. This configuration leads to the following SUSY-breaking parameters, which determine the soft terms:
From these expressions it is clear that can be made negative if is sufficiently large. In order to avoid a large tuning, we expect that in such situations, and are of the same order. This is compatible with Figure 4, where in the region of selectron NLSP. Depending on the value of , the left-handed squarks can have positive or negative UV squared masses, and we have seen in Section 2.1 that both cases are possible.
Finally, characterizes the Bino and the right-handed slepton masses. The hierarchy between the Bino and the right-handed sleptons masses is determined by the length of the RG flow, i.e. by the messenger mass . Since the effective number of messengers in the sector is , we can estimate that the Bino will be heavier than the selectron/smuon as long as GeV.888If needed, one can introduce a D-term breaking also for the messengers in order to reduce the contribution to the right-handed sleptons and make the Bino/slepton mass ratio bigger.
In order to verify that the model presented here can realize selectron NLSP, we performed a numerical scan by fixing GeV and varying the other SUSY breaking parameters and the messenger mass. We indeed find selectron NLSP in the expected region, i.e. for large and negative. We do not present the results here since the qualitative features are very similar to the ones discussed after Figure 4 in Section 2.1, once translated in terms of the UV soft masses.
3.2 Tachyonic down-type Higgs mass
Parametrizing the extra contributions to the two Higgs doublets of the MSSM, i.e. in addition to the usual GM contributions, as and , in Figure 6, we already identified the three possible interesting regions, characterized by different values of compared to a negative and typically large . In what follows, we survey the possibilities to induce a negative and large at tree level, or at loop level, in models of weakly coupled messengers coupled to the Higgs sector. For every scenario we comment how these extra couplings in the Higgs sector affect the other dimensionful parameters characterizing the Higgs potential and the sparticle soft masses.
Tree level contributions to the Higgs soft masses can be obtained by mixing the Higgs fields with messengers coupled to a SUSY-breaking spurion. A generic superpotential realizing this possibility is the one considered e.g. in Komargodski:2008ax; Evans:2011bea:
where is a spurion superfield, and we assume a canonical Kahler potential for the Higgses and the messengers. In the limit of large we can integrate out the messengers, resulting in
where we have assumed real and . This leads to the following soft terms, at leading order in ,
where denote the coefficient of the term in the Lagrangian, giving rise to once we integrate out . The A-terms are nevertheless suppressed by extra powers of with respect to the soft masses. Given the induced negative mass terms for both Higgs doublets, this model covers region (i) and (ii) of Figure 6. However, it also generates non-negligible contributions to and . Hence, the EWSB condition puts some constraint on these parameters, which could possibly be circumvented by adding another sector to provide the appropriate contributions to and only.
A more economical realization, which
suppresses these extra contributions,
can be achieved by
imposing an R symmetry
(broken by the VEV of the spurion )
with charges such that the term of the superpotential is forbidden:
This assignment can realize models of region (ii), with large and negative , vanishing , and with vanishing at the messenger scale.
Notice that the superpotential in (32) also induces soft masses for EW gauginos and sfermions with the usual GM formulas, since and are charged under . In the limit these contributions are the ones of a minimal GMSB model with .
Since the messengers have the same quantum numbers as the Higgs doublets, we could add extra Yukawa couplings between the messengers and the matter superfields. We will explore this option in the explicit example of Section 3.3.
at loop level
In order to realize our selectron NLSP scenario, we have seen that the negative squared mass for the down-type Higgs should generically be sizable, see Figures 1 and 6. This implies that if we aim at obtaining this term from quantum corrections, we typically need a sizeable SUSY breaking scale . Quantum corrections will generate such terms if there is some messenger field, coupled with the Higgs superfields, that acquires split masses when SUSY is broken, or, in other words, which couples directly to the spurion. We would like to realize a scenario with through a modular structure, that can be attached to a given GM model, without affecting the rest of the soft spectrum. This suggests to focus on models with only singlet chiral superfields coupled to a spurion, so that the other soft masses are not modified by the GM effects. The simplest superpotential that can achieve this is:999We could have envisaged more complicated models with other doublets and singlets. Since this simple example already gives rise to negative soft masses for the Higgses, we chose to restrict to the most economical possibility.
The trilinear coupling of the Higgses resembles the one in the NMSSM. However, here the fields and do not acquire any expectation value, so no term is generated, it originates from a separate sector. The SUSY breaking spurion induces one loop soft masses for the Higgses. Note that there is an symmetry, broken by the VEV, such that and , implying that cannot be generated at leading order Komargodski:2008ax. We can compute the one loop Kahler potential for this model, after integrating out the field and ,
where is the SUSY mass matrix. From this we can extract the wave function renormalization for the Higgs superfields, as a function of the spurion by expanding the effective Kahler potential up to the quadratic order in the Higgses:
The relevant part of the Higgs wave function renormalizations is
and . Expanding these expressions in and one can extract the soft terms. Here we show the result in the limit . As expected, and are vanishing, while the Higgs soft masses and the A-terms, at first order in the SUSY breaking scale and in , are
The soft masses are negative, and the -terms are suppressed by . Note that in the opposite regime, i.e. , the soft masses would instead be positive.
The contributions (41) generate negative and equal and at one loop, realizing the scenario at the boundary between region (i) and (ii) in Figure 6. Note that this is the main modification to the sparticle spectrum induced by the extra couplings and superfields, as the latter ones are singlets. In this sense the superpotential (37) can be considered as a module to add to usual GGM scenarios, in order to obtain a selectron NLSP.
3.3 The simplest example: Yukawa-deflected gauge mediation
We discuss here a simple example, based on the previous observations, that generically leads to promptly decaying selectron/smuon co-NLSP. The model also accommodates the observed Higgs mass and features a light stop, as extensively discussed in recent literature Evans:2011bea; Evans:2012hg; Kang:2012ra; Evans:2013kxa; Abdullah:2012tq; Calibbi:2013mka.
We consider the following superpotential for the Higgses and the messengers,
where as usual is a spurion . Beside the mixing terms among the Higgses and the messengers, we consider only one extra top-like matter-messenger coupling, following Evans:2011bea; Evans:2012hg.
Let us remark that, unlike the models considered so far, the presence of matter-messenger couplings generates new contributions to the sfermion mass matrices whose flavour structure can give rise to effects beyond minimal flavour violation, in principle spoiling the usual flavour protection of GM. However, the new flavour effects are completely under control as far as the new couplings feature a hierarchical structure that resembles the ordinary Yukawa matrices Abdullah:2012tq; Calibbi:2013mka, as is the case here where we consider a single