1 Introduction

CERN-PH-TH/2008-106, UMN–TH–2645/08, FTPI–MINN–08/14


Varying the Universality of Supersymmetry-Breaking Contributions to MSSM Higgs Boson Masses


John Ellis, Keith A. Olive and Pearl Sandick

TH Division, PH Department, CERN, CH-1211 Geneva 23, Switzerland

William I. Fine Theoretical Physics Institute,

University of Minnesota, Minneapolis, MN 55455, USA

Abstract

We consider the minimal supersymmetric extension of the Standard Model (MSSM) with varying amounts of non-universality in the soft supersymmetry-breaking contributions to the Higgs scalar masses. In addition to the constrained MSSM (CMSSM) in which these are universal with the soft supersymmetry-breaking contributions to the squark and slepton masses at the input GUT scale, we consider scenarios in which both the Higgs scalar masses are non-universal by the same amount (NUHM1), and scenarios in which they are independently non-universal (NUHM2). We show how the NUHM1 scenarios generalize the planes of the CMSSM by allowing either or to take different (fixed) values and we also show how the NUHM1 scenarios are embedded as special cases of the more general NUHM2 scenarios. Generalizing from the CMSSM, we find regions of the NUHM1 parameter space that are excluded because the LSP is a selectron. We also find new regions where the neutralino relic density falls within the range preferred by astrophysical and cosmological measurements, thanks to rapid annihilation through direct-channel Higgs poles, or coannihilation with selectrons, or because the LSP composition crosses over from being mainly bino to mainly Higgsino. Generalizing further to the NUHM2, we find regions of its parameter space where a sneutrino is the LSP, and others where neutralino coannihilation with sneutrinos is important for the relic density. In both the NUHM1 and the NUHM2, there are slivers of parameter space where the LHC has fewer prospects for discovering sparticles than in the CMSSM, because either and/or may be considerably larger than in the CMSSM.

CERN-PH-TH/2008-106

May 2008

1 Introduction

The simplest supersymmetric model is the minimal supersymmetric extension of the Standard Model (MSSM), and it is commonly assumed that the soft supersymmetry-breaking contributions to the squark, slepton and Higgs scalar masses are universal at some GUT input scale (CMSSM) [1, 2]. This is certainly the simplest assumption, but it is neither the only nor necessarily the most plausible version of the MSSM. For example, universality might hold at some lower renormalization scale [3], as in some mirage unification scenarios [4]. Alternatively, the soft supersymmetry-breaking masses may not be universal at any renormalization scale, as occurs in some string scenarios for supersymmetry breaking [5]. The suppression of flavour-changing supersymmetric interactions suggests that the soft supersymmetry-breaking masses of all generations of squarks and sleptons with the same electroweak quantum numbers may be the same, i.e., , , and similarly for the of charges and [6]. However, this argument does not motivate universality between sleptons and squarks, or even between left- and right-handed sleptons or squarks. Some degree of universality would be expected in supersymmetric GUTs. For example, in supersymmetric SU(5) one would expect and . Supersymmetric SO(10) would further predict universality between all the soft supersymmetry-breaking squark and slepton masses. However, supersymmetric GUTs do not give any reason to think that the soft supersymmetry-breaking contributions to the Higgs scalar masses should be universal with the squark and slepton masses. This full universality, postulated in the CMSSM, would occur in minimal supergravity (mSUGRA) scenarios [7], but not in more general effective no-scale supergravity theories such as those derived from string models [8].

On the basis of the above discussion, it is natural to consider models with non-universal soft supersymmetry-breaking contributions to the Higgs scalar masses [9]. In general, one may introduce two independent non-universality parameters, scenarios which can be termed NUHM2 [10], but one could also consider scenarios with equal amounts of non-universality for the two Higgs doublets, scenarios which can be termed NUHM1 [11]. Such scenarios would be natural in a supersymmetric SO(10) GUT framework, since the two Higgs multiplets occupy a common vectorial 10-dimensional representation, while each matter generation occupies a common spinorial 16-dimensional representation of SO(10).

CMSSM scenarios have four continuous parameters, which may be taken as , with the values of and then being fixed by the electroweak vacuum conditions. Correspondingly, NUHM1 scenarios have one additional parameter, that may be taken as either or , whereas both and are free parameters in NUHM2 scenarios. The full six-dimensional NUHM2 parameter space has been explored in a number of studies [10], but its higher dimensionality renders its complete characterization quite complicated, and it is less amenable to a Markov Chain Monte Carlo analysis than the NUHM1 and particularly CMSSM scenarios [12]. The main purpose of this paper is to discuss how the CMSSM, NUHM1 and NUHM2 scenarios may be related by processes of dimensional enhancement: CMSSM NUHM1 NUHM2 and reduction: NUHM2 NUHM1 CMSSM, laying the basis for more complete understanding of the NUHM1 and NUHM2 parameter spaces. Accordingly, in the following sections we focus first on the relationship between the CMSSM and NUHM1 scenarios, and subsequently on the relationship between the NUHM1 and NUHM2 scenarios.

The most important contributions to most sparticle masses are those due to and , so studies of the phenomenological constraints on the CMSSM parameter space [13, 14] and the prospects for experimental searches at the LHC and elsewhere are frequently displayed in planes for different values of , and the sign of . The values of and then vary across these planes according to the electroweak vacuum conditions. In our first exploration of the NUHM1 parameter space, we display and discuss planes for different choices of fixed values of and positive , seeking to understand, in particular, the dependences on and of the strips of parameter space compatible with the cold dark matter density inferred from WMAP and other observations [15]. A key question here is whether the good (but not complete) LHC coverage of the CMSSM WMAP strips [13] is repeated also in NUHM1 scenarios. We find that there are extensions of the preferred regions of the planes to larger values of these parameters that are affected by the choices of or , whereas the preferred regions of these latter parameters are more sensitive to the choices of the other NUHM1 parameters. In some of the extensions, the LHC would either have difficulty in detecting supersymmetry at all, or would only provide access to a limited range of sparticles. Since the interest of NUHM1 scenarios lies largely with the new possibilities for varying and , which have in turn important implications for the spectrum of heavy MSSM Higgs bosons and gauginos, we also display explicitly the variations of the various phenomenological constraints in planes correlating or with or .

In our discussion of the relationship between the NUHM1 and NUHM2 scenarios, we display the allowed regions of parameter space as explicit functions of the degrees of non-universality of the soft supersymmetry-breaking scalar mass parameters of the two MSSM Higgs multiplets. We find that the WMAP relic density constraint, in particular, generally favours models with a relatively high degree of non-universality, close to the boundaries of the NUHM2 parameter space imposed by other theoretical and phenomenological constraints such as the breakdown of electroweak symmetry breaking or the absence of charged dark matter. This reflects the fact, known already from studies of the CMSSM with GUT-scale universality, that the supersymmetric relic density is too large in generic domains of parameter space, being brought down into the WMAP range in particular cases such as the coannihilation [16] and focus-point regions (close to the charged dark matter and electroweak symmetry breaking boundaries, respectively) [17], or in rapid-annihilation funnel regions [1].

2 From the CMSSM to the NUHM1

In the CMSSM, the weak-scale observables are determined by four continuous parameters and a sign; the universal scalar mass , the universal gaugino mass , the universal trilinear coupling , the ratio of the Higgs vacuum expectation values , and the sign of the Higgs mass parameter . We consider the values of the parameters , and to be specified at the SUSY GUT scale. The effective Higgs masses-squared, and are responsible for generating electroweak symmetry breaking through their running from the input scale down to low energies. In the CMSSM, , and and are calculated from the electroweak vacuum conditions,

(1)

and

(2)

where and are loop corrections [18, 19, 20], , and all quantities in (2) are defined at the electroweak scale, . Unless otherwise noted, and . The values of the parameters in (1) and (2) are related through well-known radiative corrections [18, 21, 22] , and such that

(3)

In the NUHM1 one still has , but these are no longer identified with the universal scalar mass, , so an additional parameter is necessary to fix the common GUT-scale value of the Higgs masses-squared. This additional parameter may be taken to be either or , and the relationship between and at the weak scale can be calculated from (1) - (3) so as to respect the electroweak boundary conditions at and the weakened universality condition at .

If is taken to be the free parameter (input), then at we have

(4)

Alternatively, if is taken as the free parameter, then at we have

(5)

In each case, the boundary condition at is . Clearly, for some specific input values of and , one finds , thereby recovering the CMSSM. The characteristics of the parameter space as one deviates from this scenario are the subjects of the following subsections.

2.1 The NUHM1 with as a Free Parameter

We begin our characterization of the relationship between the CMSSM and NUHM1 scenarios by taking as the additional free parameter, and assume positive , as suggested by and , at least within the CMSSM.

As a basis for the comparison, in Fig. 1 we show in panel (a) a CMSSM plane with and . We have plotted (pink) contours of constant and of 300, 500, 1000, and 1500 GeV, with contours appearing roughly vertical and contours appearing as quarter-ellipses centered at the origin. There are also several phenomenological constraints shown in panel (a) 1. In the region at low and large there is a (dark pink) shaded region where there are no consistent solutions to the electroweak vacuum conditions, since they would require . An additional unphysical region is found along the bottom of the plane at larger and low , where the lightest supersymmetric particle (LSP) is a charged stau (brown shading). Contours of  GeV (red dot-dashed) and  GeV (black dashed) mark, approximately, the edges of the regions excluded by unsuccessful searches at LEP [23]. Both and increase with , so portions to the right of these contours are allowed. The region favored by the measurement of the muon anomalous magnetic moment [24], , at the two- level (light pink shading bounded by solid black lines) is also visible at very low , and the region disfavoured by [25] is shaded green.

Finally, the regions of the plane where the relic density of neutralino LSPs falls in the range favoured by WMAP and other observations for the dark matter abundance appear as thin turquoise strips. For the chosen value of , the relic density of neutralinos is too large over the bulk of the plane, and falls within the WMAP range in two distinct regions. In the upper left corner, tracking the region excluded by the electroweak vacuum conditions, lies the focus-point region [17], where the lightest neutralino is Higgsino-like and annihilations to gauge bosons bring the relic density down into the WMAP range. Alongside the forbidden -LSP region lies the coannihilation strip [16], where - coannihilations reduce the relic density of neutralinos. At larger , a rapid-annihilation funnel [1] may exist where and -channel annihilations mediated by the pseudoscalar Higgs decrease drastically the relic density of neutralino LSPs, though not for . We see that the CMSSM predicts values of between  GeV and  GeV and between  GeV and  GeV in the parts of the coannihilation strips compatible with the LEP constraints, while values of  GeV and  GeV are favoured in the focus-point region for TeV.

Figure 1: Panel (a) shows the plane for the CMSSM for , with contours of and of 300, 500, 1000, and 1500 GeV as described in the text. Panels (b), (c), and (d) show the NUHM1 planes for with , 1000, and 1500 GeV, respectively. Constraints and contours are as described in the text.

Panels (b), (c), and (d) of Fig. 1 show NUHM1 planes for and with , 1000, and 1500 GeV, respectively, and calculated using (2). In addition to the constraints discussed above, we also plot contours of , 500, 1000, and 1500 GeV (light pink). The most prominent departure from the CMSSM is that the requirement of electroweak symmetry breaking constrains the plane at low rather than at large . In this region (below the CMSSM contour), is fixed to be larger than its CMSSM value, resulting in correspondingly larger and . We see from (2) that, with and weighted by , the effect is to drive smaller, and eventually negative. The excluded region grows with as and are pushed farther from their CMSSM values, and is flanked by concentric contours of constant . The stau LSP exclusion regions are qualitatively similar to those in the CMSSM, shown in panel (a), however for moderate values of there is a (black shaded) region of the plane where the lighter selectron is the LSP. Also apparent in panel (b) for  GeV is a small region at low and that is favored by , which disappears for larger beneath the expanding region where electroweak symmetry breaking is not possible. There is no region of this or the following panels that is excluded by .

The LSP mass and composition are roughly the same as they are in the CMSSM at large : at all but the smallest values of , the LSP is bino-like in the CMSSM. At moderate and large , the masses of the sparticles are only minimally affected by the fact that is fixed, causing several of the constraints to appear similar to the CMSSM case. In particular, the LEP chargino and Higgs constraints again exclude smaller values of , though the shape of both the Higgs and the chargino exclusions change with increasing .

The strip where the relic LSP density falls within the range preferred by WMAP and other data stays, in general, close to the regions excluded by the requirement that the LSP be neutral and by the electroweak vacuum conditions. However, one difference from the CMSSM for that is very prominent in panel (b) is a rapid-annihilation funnel, straddling the dark blue contour where , that rises out of the coannihilation strip at  GeV, reaching GeV. Branches of good relic density form the inner and outer funnel walls, between which the relic density falls below the WMAP range. At larger , the dark matter strip changes somewhat. For GeV, shown in panel (c) of Fig. 1, at GeV. However the coannihilation strip has essentially terminated at lower , so there is no prominent rapid-annihilation funnel. Finally, at GeV, shown in panel (d), at GeV, well beyond the end of the coannihilation strip. The relic density still decreases in these regions, but it remains above the WMAP range, so there is no visible funnel.

We have already emphasized that the parameter space expands by one dimension between the CMSSM and the NUHM1. In each plane (b)-(d) of Fig. 1, there is a green dot-dashed contour tracking the CMSSM parameters in the NUHM1 plane. The change in position of this contour as is increased can be understood by comparison with the contours of constant in the CMSSM panel (a). As an example, we consider the variation in on the CMSSM contour and how its position changes in the NUHM1 plane. Examining the contour of  GeV in the CMSSM plane, we find that in the -LSP region, the value of along the contour reaches a maximum of about 860 GeV. Following the curve to larger , we see that it terminates at the boundary of the region where . So we expect that the CMSSM contour in the NUHM1 plane with GeV runs smoothly through the contours of constant from  GeV in the -LSP region to the boundary of the electroweak symmetry breaking region. As increases, the CMSSM contour begins near the coannihilation strip at correspondingly larger values of , but it always terminates at . The points of intersection of the CMSSM line with the electroweak vacuum boundary move to larger values of and as increases in panels (b), (c) and (d), tracking the focus-point region in panel (a).

It is clear from panels (b) to (d) that the NUHM1 shares some small pieces of the cosmologically preferred regions of the parameter space of the CMSSM for moderate and large values of . Only for 500 GeV GeV does the CMSSM contour intersect a phenomenologically viable portion of the coannihilation strip, and only for  GeV does it intersect the focus-point region. Moving away from the CMSSM contours in the NUHM1 planes, we find that cosmologically preferred areas in the focus-point regions are now available at lower . For example, at GeV in the CMSSM, the focus-point is found at low values of where both the Higgs and chargino mass constraints are violated. In the NUHM1, as seen in panel (c), we find a viable focus-point strip at GeV at values of lower than in the CMSSM. Furthermore, we find additional coannihilation strip at both larger and smaller than what would be expected in the CMSSM, and for a range of there is even a rapid-annihilation funnel.

The funnel region is interesting in that it passes all constraints and may have fairly heavy scalars, as does the focus-point region in the CMSSM, but with a bino-like neutralino LSP. A key difference between the two cases is illustrated by the following simple example. If the LHC discovers a gluino weighing 1.5 TeV, which is estimated to be possible with less than 1 fb of integrated luminosity [28, 27, 28], then, in the CMSSM the lightest charged sparticles are encouragingly light with GeV in the focus-point region and GeV in the coannihilation strip. However, in the NUHM1, although we will discover charged staus easily if Nature has chosen the coannihilation strip, at the peak of the funnel in panel (b) the lighter chargino could be heavier than 900 GeV, and staus would be as heavy as  GeV. In this case, the rapid-annihilation funnel represents a continuum of viable sparticle masses between the two extremes. Both the CMSSM points and the NUHM1 points have a light LSP with 250 GeV 280 GeV, but the pseudoscalar Higgs mass is quite large in the CMSSM and highly dependent on the value of , whereas in the NUHM1  GeV in this case. According to previous studies in the CMSSM, detecting supersymmetry at the LHC should be possible along the rapid-annihilation strip in panel (b) for  GeV with roughly 10 fb of integrated luminosity, though the number of sparticles accessible with dedicated follow-up searches would decrease as increases.

Figure 2: Examples of NUHM1 planes with , , , and , 500, 1000, and 1500 GeV in Panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Figure 1.

Fixed

Alternative ways to view the NUHM1 parameter space include fixing either or and scanning over . We first examine the former option.

We show in Fig. 2 examples of the planes for , 500, 1000, and 1500 GeV. The unfamiliar appearances of the constraints can once again be understood by comparison with panel (a) of Fig. 1. For example, for  GeV, as seen in panel (a), we note that the upper third of the plane is excluded due to a charged LSP. This reflects the fact that in the CMSSM plane, for fixed , increases more slowly than as increases, so that at large the becomes the LSP. Increasing postpones the -LSP region to larger , so that this constraint almost disappears in panel (b) where  GeV, and does not appear at all in panels (c) and (d), where and 1500 GeV, respectively. While there is no -LSP region in the CMSSM plane, as seen in panel (a) of Fig. 1, the selectron mass renormalization is similar to that of the stau, so the selectron-LSP regions in the NUHM1 planes shift similarly to larger .

The other unphysical regions in CMSSM planes occur in their upper left corners, where there is no consistent electroweak vacuum. As seen in panel (a) of Fig. 1, this issue arises at low and large . As is increased, the boundary of this region moves to larger and . The positive correlation between and along this boundary is seen clearly in all the panels of Fig. 2. We also see that, particularly at small , this boundary also retreats to larger as increases. Following the boundary of this excluded region are the contours of constant , which converge slightly as and increase. Also apparent in panel (a) for  GeV is a small region at low and that is favored by , which disappears for larger . We also see at very low the LEP chargino bound. The dominant experimental constraints in these planes are the LEP limits on the Higgs mass and the branching ratio of , which exclude the areas below the dot-dashed red contour and in the green shaded region, respectively.

There are two viable WMAP-compatible regions in these planes. One is the upper portion of the rapid-annihilation funnel, which is oriented diagonally in the planes, close to the diagonal blue line where . Since the position of the funnel is defined by the LSP mass, which in this case depends primarily on due to its bino-like character, and the pseudoscalar Higgs mass, which forms the -axis, the rapid-annihilation funnel is fixed in the plane as is varied. The other viable WMAP-compatible region (less immediately apparent in these plots) is the focus-point region which tracks the boundary of the region where electroweak symmetry breaking is not possible.

In each plane of Fig. 2, the CMSSM contour runs diagonally through the contours of constant . For GeV, the CMSSM contour starts in the bulk region at low . Many of these points lie in the region favored by , but this portion of the plane is excluded by the LEP bound on the Higgs mass. As we follow the CMSSM contour to larger (larger ), we see that is increasing along the contour. This corresponds to following a contour of constant horizontally across the CMSSM plane. Eventually, at large and any fixed value of , the CMSSM contour intersects the region where the is the LSP, but not the -LSP region. As we increase , the -LSP region is postponed to larger . The CMSSM contours at large lie above the bulk region, but the LEP constraint on the Higgs mass is still important, as it is only very weakly dependent on . The rapid-annihilation funnel region of WMAP-compatible neutralino relic density is bounded at low by the LEP Higgs constraint and, for low , at large by the -LSP region. The funnel occurs at larger than we expect in the CMSSM.

According to previous studies [28, 27], the LHC should find a signal of supersymmetry in the CMSSM scenario with 10 fb of integrated luminosity if  GeV for  GeV. In the NUHM1, for fixed and , the spectrum of charged scalars and gauginos is only affected through loop corrections to the RGEs, so we expect a similar LHC reach for these values of , shown in panels (a, b, c) and (d) of Fig. 2. This means that progressively shorter sections of the rapid-annihilation funnels and focus-point strips are likely to be accessible to the LHC.

Figure 3: Examples of NUHM1 planes with , , , and , 500, 1000, and 1500 GeV in Panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Figure 1.

Fixed

We now discuss NUHM1 parameter space for various fixed values of , as shown in the planes in Fig. 3. We note first that the forbidden stau LSP region is absent for low  GeV, as seen in panel (a), puts in an appearance at low when  GeV, as seen in panel (b), and reaches progressively to larger at larger , as seen in panels (c) and (d). This behaviour was to be expected from the analogous feature in the CMSSM, shown in panel (a) of Fig. 1, and reflects the fact that increases more rapidly with than does . At larger we see the emergence of the selectron LSP region at low . We also note that the electroweak vacuum exclusion retreats to smaller and larger as increases, disappearing altogether for and 1500 GeV, again reflecting the CMSSM feature seen in panel (a) of Fig. 1.

One of the dominant experimental constraints on the parameter space is that due to the LEP Higgs mass bound, which excludes most of the plane for  GeV and low for  GeV, as seen in panels (a) and (b), respectively. The Higgs mass is more sensitive to variations in at lower , whereas at large the Higgs mass is primarily sensitive to and less dependent on (as in the CMSSM). We also note that the branching ratio of excludes a strip of parameter space that expands slowly with .

There are three distinct regions of WMAP-compatible relic density in these planes. The first is the vertical rapid-annihilation funnel, where the relic density decreases drastically. This moves to larger as increases, reflecting the movement of the blue line where . The second region of good relic density is the coannihilation strip, which is present when  GeV. In fact, we see that the rapid-annihilation funnel rises directly out of the coannihilation strip where the two coincide, as also seen in Fig. 1. Finally, the third is the focus-point strip, which tracks the region excluded by the requirement of electroweak symmetry breaking. As continues to increase, this strip is pushed to values of beyond those plotted.

The CMSSM contours in the planes correspond to following a strip of constant in the plane shown in panel (a) of Fig. 1 upwards from the coannihilation strip. Since depends strongly on , but has little sensitivity to the value of , these contours appear to be roughly contours of constant in each case. For low values of , the CMSSM contour begins in the bulk region at low . This is a region favoured by but strongly excluded by the LEP Higgs bound. Eventually, we find the focus-point region at very large . In panel (b), the CMSSM line arches up from the -LSP region towards the region where there is no electroweak symmetry breaking. In Panels (c) and (d), the CMSSM contour begins at low and large in the -LSP region, but there are no further visible features of interest. As already noted, both the CMSSM contour and the rapid annihilation funnel move to larger as increases. However, since the CMSSM contour moves more quickly than the funnel, there is no rapid annihilation funnel in the CMSSM for , unlike the NUHM1 case.

According to previous studies [28, 27], the LHC should find a signal of supersymmetry in the CMSSM scenario with 10 fb of integrated luminosity if  GeV for  GeV. As discussed in section 2.1.1, we expect a similar reach in the NUHM1 for comparable values of , as shown in panels (a) and (b) of Fig. 1. This means that all of the visible parts of these planes should be accessible to the LHC. On the other hand, previous analyses [28, 27] suggest that in the CMSSM, the parameter space with  GeV would be inaccessible without an increase in the integrated luminosity. In the NUHM1 planes, due to the appearance of the rapid-annihilation funnel, one may find fairly light charged scalars even if  GeV, as shown in panels (c) and (d).

Varying

Finally, we discuss the characteristics of the NUHM1 parameter space as we vary . We recall that in the CMSSM at large a rapid-annihilation funnel appears in the plane when , extending from the coannihilation strip to larger . In addition, at large the excluded -LSP region becomes more prominent in the plane at low , and the branching ratio of excludes more of the plane at low  2. The effects of variations in on these constraints alter the appearance of the NUHM1 planes, as well.

Figure 4: Examples of NUHM1 planes with GeV, , , and , 20, 35, and 50 in Panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Figure 1.

In Fig. 4, we show NUHM1 planes with  GeV and , 20, 35, and 50 in panels (a), (b), (c), and (d), respectively. Note that panel (a) of Fig. 4 is the same as panel (b) of Fig. 2. As is increased, decreases, as is evidenced by the movement of the contours of constant out into the plane and the expansion of the region where there are no consistent solutions to the electroweak vacuum conditions. As a result, the CMSSM contour is pushed to lower for fixed , moving closer to the rapid-annihilation funnel. In the CMSSM, however, the rapid-annihilation funnel begins at roughly  GeV, so the CMSSM contour does not cross the rapid-annihilation funnel even at in these planes with  GeV. In these NUHM1 planes, the location of the rapid-annihilation funnel is almost independent of .

In contrast to the CMSSM, in these particular NUHM1 planes the constraint due to the branching ratio of becomes insignificant at large . On the other hand, the region favored by expands such that a significant portion of the rapid-annihilation funnel falls within it, as well as the LEP constraint on the Higgs mass. In addition to the fixed rapid-annihilation funnel, in each panel of Fig. 4 there is a narrow WMAP strip close to the electroweak symmetry-breaking boundary. For , portions of the funnel and this boundary strip are compatible with all these constraints, except , for  GeV. When ,  GeV is allowed by the Higgs constraint, and part of this boundary strip is also compatible with . When , the region allowed by extends to larger , and parts of both the rapid-annihilation funnel and the boundary strip are compatible with it and with .

In Fig. 5, we show NUHM1 planes with  GeV and , 20, 35, and 50 in panels (a), (b), (c), and (d), respectively. Note that panel (a) of Fig. 5 is the same as panel (b) of Fig. 2. As increases, we see that the boundary of the electroweak symmetry-breaking region moves to lower values of , while the -LSP region changes its shape, becoming less important at small but more important at larger . In contrast, the -LSP region is fixed at very low as is increased, becoming visible as the -LSP region shifts, and it is bordered by a coannihilation strip. The LEP Higgs constraint excludes only a narrow strip at small , almost independent of , that narrows as increases. The constraint is visible only for , at small . There is no region favoured by when , but this appears and expands as increases. The CMSSM line arches up and outwards in each panel, following and gradually approaching the boundary of electroweak symmetry breaking.

Figure 5: Examples of NUHM1 planes with GeV, , , and , 20, 35, and 50 in Panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Figure 1.

The strip where the dark matter density falls within the WMAP range exhibits the familiar features of a rapid-annihilation funnel, which is near-vertical and straddles the blue line where , a coannihilation strip near the boundary of the charged LSP regions, and a strip near the boundary of the region where there is no electroweak symmetry breaking. This region is compatible with all the phenomenological constraints, including also when or more. There are in general two intersections with the CMSSM line, corresponding to the coannihilation and fixed-point strips in the planes for different values of . The rapid-annihilation funnel is in general at lower than the CMSSM line, except for . The analogous planes for larger would exhibit more intersections between the CMSSM line and the rapid-annihilation funnel.

According to previous studies [28, 27], the LHC should find a signal of supersymmetry in the CMSSM scenario for with 10 fb of integrated luminosity if  GeV for  GeV. Given the sensitivity of the sparticle spectrum to the value of , we estimate that the visible parts of the planes in Fig. 5 should be accessible to the LHC.

2.2 The NUHM1 with as a Free Parameter

Figure 6: Panel (a) shows the plane for the CMSSM, with contours of and of 300, 500, 1000, and 1500 GeV as described in the text. Panels (b), (c), and (d) show the NUHM1 plane with , 1000, and 1500 GeV, respectively. Constraints are displayed as in Figure 1.

As discussed above, in the NUHM1, one may choose either or as the additional input to those of the CMSSM. In this subsection, we re-examine the parameter space, this time choosing as a free parameter. We begin, as in Section 2.1, with a comparison of the CMSSM planes with NUHM1 planes, now at fixed . In Fig. 6, we show in panel (a) the CMSSM plane (identical to panel (a) of Fig. 1), including the contours of constant and of 300, 500 1000, and 1500 GeV. Panels (b), (c), and (d) show the NUHM1 planes with , 1000, and 1500 GeV, respectively.

At first glance, the planes with fixed have some similarities with those with fixed . There are excluded regions at very low where the pseudoscalar Higgs mass squared is negative, corresponding to the absence of electroweak symmetry breaking, surrounded by four contours of fixed , 500, 1000, and 1500 GeV. At small values of , extending out to large , there are excluded -LSP regions resembling those in the CMSSM. As usual, the LEP chargino and Higgs constraints exclude regions at small , and excludes strips near the electroweak symmetry-breaking boundaries for  GeV, shown in panels (b) and (c), respectively. We also see in these planes regions at low and that are favoured by .

There are three generic parts of the WMAP relic density strips in panels (b, c) and (d) of Fig. 6. There are coannihilation strips close to the -and -LSP boundaries, and other strips close to the electroweak symmetry-breaking boundaries. Arching between these are curved rapid-annihilation funnels that appear at low , with strips of good relic density forming the funnel walls. For GeV, the rapid-annihilation funnel is partially excluded by the branching ratio of and even more so by . Additionally, in panel (b) for  GeV, there is a fourth, near-vertical strip, where the relic density is brought down into the WMAP range because of the large mixing between the bino and Higgsino components in the LSP. For smaller  GeV, the LSP is almost pure bino, and the relic density is too large except in the narrow strips mentioned previously. This is the opposite of what happens in the CMSSM, where the Higgsino fraction increases at smaller at large . On the other hand, for larger  GeV, the LSP is almost pure Higgsino, and the relic density falls below the WMAP range 3. At large in panel (b) of Fig. 6, it is only in the ‘crossover’ strip that the relic density falls within the WMAP range. Analogous near-vertical crossover strips are not visible in panels (c) and (d) of Fig. 6, but would in principle appear at larger  GeV, respectively.

The CMSSM contour in each of panels (b, c) and (d) of Fig. 6 is a roughly vertical line, the position of which is determined by the value of that one would find from the electroweak vacuum conditions in the standard CMSSM. Since the contours of constant in these NUHM1 planes look very similar to the corresponding contours in the CMSSM plane shown above, the CMSSM contours here in turn look qualitatively similar to contours of constant in the CMSSM plane. The CMSSM lines are compatible with WMAP only in infinitesimal cuts across the coannihilation strips, missing all the excitement occurring elsewhere in the planes, namely the focus-point, rapid-annihilation and crossover strips.

In the NUHM1 planes with fixed , the crossover strip and the rapid-annihilat- ion funnel comprise regions of interest in addition to those commonly found in the CMSSM. Whereas the standard CMSSM regions will be fairly well-covered by the LHC, there are significant regions of the NUHM1 plane which may not be so easily accessed. For example, for  GeV, as shown in panel (b) of Fig. 6, the crossover strip runs at  GeV from  GeV, where it is terminated by the -LSP region, to well beyond 10 TeV, crossing the CMSSM contour at  GeV. Since the strip is roughly constant in , at any point along it one finds  GeV and  GeV. The gluino mass is 2.2 to 2.3 TeV along this strip, which is expected to be within the LHC’s reach with just over 10 fb of integrated luminosity [28, 27]. If is low, then charged scalar particles may be accessible, with masses as low as 450 GeV. Above the CMSSM contour, however, all scalar particles have masses well above 3 TeV.

Turning to panel (d), when  GeV, we find a different situation. The rapid-annihilation funnel represents a cosmologically preferred region that occurs at moderate values of both and , in contrast to the CMSSM, where cosmologically-preferred regions generally occur at either small or small . Taking as an example the point  GeV, we find a rather light neutralino with  GeV. The chargino and psuedoscalar Higgs are somewhat heavier at 545 and 570 GeV, respectively, and charged scalars have masses of 735 GeV. This point is particularly interesting in that  GeV, which should be accessible at the LHC with only 1 fb of integrated luminosity. In the CMSSM, a gluino of 1480 GeV would imply either the coannihilation strip, where  GeV and  GeV, or the focus-point region, where charged scalars are much heavier. In the NUHM1, several sparticles may have masses below 1 TeV, and points on the rapid-annihilation strip should be distinguishable from points on the CMSSM coannihilation strip.

Fixed

Figure 7: Examples of planes with , , and , 500, 1000, and 1500 GeV in Panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Figure 1.

Analogously to the discussion in Section 2.1, alternative ways to view the parameter space are to fix either or and scan over . In Fig. 7, we show examples of planes for fixed , 500, 1000, and 1500 GeV in panels (a), (b), (c), and (d), respectively. For the first time, we display here both positive and negative values of . The unphysical regions excluded by not having electroweak symmetry breaking or by having a charged LSP cover a large part of the plane for  GeV and recede out of the visible part of the plane as increases. Triangular regions in the lower right and left corners are forbidden because the pseudoscalar Higgs mass-squared is negative. For fixed and , as increases, increases slightly. As a result, the regions at small that had been excluded due to unphysical negative recede to larger , dragging along the contours of constant , 500, 1000, and 1500 GeV. For GeV, the upper right and left portions of the plane are forbidden because the stau is the LSP, though these regions move quickly to large as is increased, almost disappearing for  GeV and becoming invisible for larger . Bordering these regions of the plane (but away from the CMSSM contours) the selectron is lighter than the stau, forming a second region forbidden by the presence of a charged LSP.

In each panel, there is a strip at low that is excluded by the LEP chargino constraint. Additionally, at low (slightly dependent on ), there is a region where the light Higgs mass falls below the LEP limit. Since increases with , the region below the Higgs mass contour is excluded, a constraint that is slightly stronger for . The branching ratio of constrains significantly more strongly the half of the plane, with the green area being excluded. However, the half-planes with are not all excluded in the NUHM1. The region favored by is found at small positive and low . However, it lies below the Higgs mass contour even at GeV, and shrinks and then evaporates as is increased.

There are two cosmologically preferred regions in each plane 4. Crossover regions form a long, narrow ‘Vee’ at relatively small , roughly proportional to . The relic density of neutralinos is below the WMAP range inside the crossover ‘Vee’, and above the WMAP range at larger . In addition, rapid-annihilation funnels occur along diagonals that form a broader ‘Vee’ with slightly curved walls. These are very thin cosmologically preferred strips on either side of the blue lines where , and the relic density is again below the WMAP range between the two strips of each rapid-annihilation funnel. We see that there are allowed regions of both the crossover strips and the rapid-annihilation funnels when , as well as in the conventionally favoured case . However, the latter also include lower values of where (in panel (a) for  GeV and panel (b) for  GeV) the preferred range for may also be obtained.

Comparison with the CMSSM case shown in panel (a) of Fig. 6 yields insight into the appearance of the CMSSM contours in the NUHM1 planes of Fig. 7. Following a contour of constant , at low we begin in either the bulk region excluded by the LEP Higgs and/or chargino bounds and in the unphysical region. As we move to larger , the sparticle masses and relic density generally increase, until one reaches the forbidden -LSP region at very large . Thus, the CMSSM contours in Fig. 7 begin at in a portion of the plane excluded by LEP, rising up to larger and . In the CMSSM, for  GeV,  GeV and is sensitive to only at the level of for 300 GeV 1500 GeV. It is well-known that in the CMSSM there is no rapid-annihilation funnel for , so we do not expect the funnel regions in the NUHM1 to cross the CMSSM contours, as seen in all the panels of Fig. 7. At large , however, the CMSSM crossover WMAP strip appears at very low , so there is a crossing between each crossover WMAP strip and the CMSSM contour for  GeV.

According to previous studies [28, 27], the range of accessible to the LHC depends on the value of chosen, being roughly  GeV for the choices  GeV shown in Fig. 7. This implies that there are increasing portions of the crossover and rapid-annihilation strips that are likely to be inaccessible as increases from panels (a) and (b) to panels (c) and (d).

Fixed

Figure 8: Examples of planes with , , and , 500, 1000, and 1500 GeV in Panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Figure 1.

Fig. 8 shows NUHM1 planes with fixed to be 300, 500, 1000, and 1500 GeV in panels (a), (b), (c), and (d), respectively. Again, regions at large excluded because there is no electroweak symmetry breaking (since ) are bordered by contours of constant and parallel rapid-annihilation funnels. These regions recede and disappear for  GeV. There are also excluded charged LSP regions at small , which expand as increases.

For  GeV, shown in panel (a) of Fig. 8, the LEP constraint on the Higgs mass excludes all of the plane below the contour at  GeV. The branching ratio of also excludes a region with at lower . The chargino mass bound from LEP appears as vertical black dot-dashed lines at small , and a region favored by is visible at small positive . For  GeV, shown in panel (b), the Higgs constraint is weakened for and disappears for , and the region favoured by for contracts. The Higgs and constraints disappear completely when  GeV.

The relic density of neutralinos may fall in the range favoured by WMAP in three regions of each plane: along the rapid-annihilation funnels that straddle the blue lines where , in the thin crossover strips that run outside and roughly parallel to the LEP chargino limits, and, at small , along coannihilation strips close to the excluded - and -LSP regions.

The CMSSM contours appear in these planes as parabolas, symmetric about , with a peak height that increases dramatically with . Since is constant in each of the planes, each half of each parabola may be regarded as tracing a line of constant in the standard CMSSM plane. When  GeV, at low one encounters the bulk region that is excluded by the Higgs constraint and (for ) the constraint. The only points compatible the dark matter and all other constraints are at  GeV and  GeV, barely satisfying the Higgs constraint. As increases, these CMSSM WMAP-compatible points move up to very large  GeV, a relic of the focus-point region in the familiar CMSSM plane. However, for  GeV  GeV we also encounter WMAP-compatible -coannihilation points at the bottoms of the parabolae, which are compatible with all the other constraints (except the Higgs when  GeV and ). The CMSSM contours never cross the rapid-annihilation funnels for this value of .

According to previous studies [28, 27], the range of accessible to the LHC depends on the value of chosen, being above  GeV for the choices  GeV shown in panels (a) and (b) of Fig. 8. In the CMSSM, we do not expect to be able to probe supersymmetry with  GeV, however in the NUHM1, there are regions of parameter space with heavy gauginos and much lighter scalars that may be accessible, specifically the lower portions of the crossover strips shown in panels (c) and (d).

Varying

Figure 9: Examples of planes with GeV, , and , 20, 35, and 50 in Panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Figure 1.

We now consider the effect of varying , initially at fixed  GeV. Panels (a), (b), (c), and (d) of Figure 9 show NUHM1 planes for , 20, 35, and 50, respectively. In all panels, the requirement of electroweak symmetry breaking appears identically as a triangular excluded region at large and low . The -LSP regions, while remaining similar in shape, become more prominent at large , as in the CMSSM. Focusing on , we see the constraint due to the branching ratio of grows with , while the LEP Higgs constraint, for fixed , has little dependence on . In panels (c) and (d), we display only the half of the plane for and 50, since solutions are not reliably found for large with .

In all panels, the crossover strips and the rapid-annihilation funnels are viable cosmologically preferred regions, both appearing as diagonals forming ‘Vee’ shapes in the planes. The CMSSM contours lie between the two ‘Vees,’ intersecting WMAP strips only in regions excluded by collider constraints in panels (a) through (c). As increases, the CMSSM contours shift to smaller , while the rapid-annihilation funnel becomes more prominent and is deformed to lower . At , where a rapid-annihilation funnel is natural in the CMSSM, the coannihilation strip connects the crossover strip with the enlarged funnel region. For this fixed value of  GeV, the CMSSM contour does not intersect the rapid-annihilation funnel, however an intersection would occur for larger . At , the region favoured by has expanded to encompass large regions of the plane where collider constraints are evaded and the dark matter density is in agreement with astrophysical measurements.

We recall that in the CMSSM, none of the regions of parameter space with  GeV may be within the 10 fb reach of the LHC [28, 27] regardless of the value of . Extrapolating to the NUHM1, it is clear from Fig. 9 that portions of the crossover and rapid-annihilation strips, and possibly part of the coannihilation strip at , will be beyond the reach of the LHC. For comparison, in the CMSSM the corresponding coannihilation strips would be accessible, but not portions of the focus-point and rapid-annihilation funnels.

Figure 10 shows examples of the plane at fixed  GeV for four choices of . Progressing from shown in panel (a), which is the same as panel (b) of Fig. 8, we see that, as increases to 20 in panel (b), the regions excluded by the absence of electroweak symmetry breaking and the presence of a or LSP are little changed 5. However, the Higgs constraint essentially disappears, whereas the constraint is much more aggressive at and a larger region is favoured by at . Again, in panels (c) and (d), we display only the half of the plane.

The regions favoured by the dark matter density are crossover strips at  GeV, rapid-annihilation funnels arching up close to the region excluded by the absence of electroweak symmetry breaking, and coannihilation strips close to the charged LSP regions. For , separate and coannihilation strips are easily discerned, separated by the rapid-annihilation funnel.

The CMSSM lines in the planes remain essentially unchanged as increases. They always have intersections with the crossover strips at large  GeV, for both signs of , and also intersect the coannihilation strip for . This intersection is in the region favoured by , whereas the corresponding intersection for is excluded either by the LEP Higgs limit (for ) or (for ). There are no intersections with the rapid-annihilation funnels or the -coannihilation regions.

For the choice of  GeV made in Fig. 10, all the range of  GeV should be accessible to the LHC [28, 27]. However, fewer of the heavier neutralinos, charginos and Higgs bosons would be detectable at larger values of (horizontal axis) and (pink contours).

Figure 10: Examples of planes with GeV, , and , 20, 35, and 50 in panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Fig. 1.

3 From the NUHM1 to the NUHM2

Having situated the NUHM1 relative to the CMSSM, we now discuss the extension to the NUHM2, in which the soft supersymmetry-breaking contributions to both the Higgs scalar masses are regarded as free parameters. These two extra parameters imply that each point in a CMSSM plane can be ‘blown up’ into a plane, as displayed in Figs. 11, 12 and 13. Alternatively, one may display the NUHM2 parameter space directly in planes, as we do in Figs. 14, 15 and 16. In the following, we use these ‘blow-ups’ to relate the NUHM2 to the NUHM1 and the CMSSM, noting that, in each plane, the NUHM1 subspace may be represented as a line, and the CMSSM as one or two points on this line.

3.1 Nuhm2 Planes

We start by considering the ‘blow-ups’ of points with the relatively small values  GeV, shown in Fig. 11. Panel (a) is for . We see (brown) regions excluded because of a LSP at small values of and , and other regions at large values of and excluded because either the (brown) or sneutrino (dark blue) is the LSP. Most of the half-plane with is excluded by , and also a small region with small and . The Higgs mass is slightly below the LEP constraint over the entire plane in all four panels of Fig. 11. In panel (a) compatibility with is found for . The dark matter density favoured by WMAP et al. is attained in narrow strips that stretch around the non-excluded regions. They feature a gaugino-Higgsino crossover at small and large , sneutrino coannihilation at large and , rapid-annihilation funnels at  GeV, and coannihilation at small and .

Figure 11: Examples of NUHM2 planes with  GeV,  GeV, , and , 20, 35, and 50 in panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Fig. 1.

The NUHM1 line is a symmetric parabola passing through  GeV and  GeV. For , this passes through the WMAP strip in three locations, once in the crossover strip at  GeV, and once on either side of the rapid-annihilation funnel at  GeV. These NUHM1 WMAP-preferred region crossings are visible in the NUHM1 planes, as well. For example, in panel (a) of Figure 3, where  GeV, by following  GeV, one encounters precisely these three WMAP preferred strips, one at  GeV near the boundary of the region where electroweak symmetry breaking is not obtained, plus both walls of the rapid-annihilation funnel at lower . The same crossings can be observed in the plane when is fixed to be 300 GeV, by examining the  GeV contour in a similar manner. On the other hand, the NUHM1 line in the NUHM2 plane completely misses the sneutrino coannihilation region at large and , which is a new feature for the NUHM2. In this case, the CMSSM point (marked by a + sign) is in a region interior to the WMAP strip, where the relic LSPs are overdense.

Turning now to the corresponding plane for  GeV,  GeV and , shown in panel (b) of Fig. 11, we see that the -LSP regions at low and have expanded somewhat, and the -LSP regions at large have changed little, whereas the -LSP region has concentrated at large . The constraint is of reduced importance compared to panel (a), and now favours a region of small . The WMAP strip is qualitatively similar to that in panel (a), except that there are now separate and coannihilation regions at large .

The NUHM1 line follows closely the coannihilation strip at low and missing, in this case, both the crossover strip and the coannihilation strip. In particular, the CMSSM points for both positive and negative would, with only minor adjustment, satisfy the WMAP constraint as well as the phenomenological constraints including . The CMSSM point with also lies in the region favoured by , as does a portion of the NUHM1 strip extending from to 500 GeV.

For larger values of , as seen in panels (c) and (d) of Fig. 11, the half-plane with and a large part of the half-plane with are excluded because the is the LSP. The -LSP region at large has also expanded, leaving only a (curved) triangle of allowed parameters at . The WMAP strip now consists of a coannihilation strip and a coannihilation strip, linked by a rapid-annihilation funnel. Since the values of and chosen for Fig. 11 are not large, all the WMAP-compatible points are accessible to the LHC [28, 27], and several types of sfermions should be detectable. Some neutralinos, charginos and heavy Higgs bosons should also be detectable in the coannihilation strip and the rapid-annihilation funnel, but this would be more difficult in the coannihilation strip.

In both panels (c) and (d), only a small portion of the NUHM1 line is allowed. It intersects the WMAP strip close to a junction between the coannihilation strip and the rapid-annihilation funnel. The CMSSM points in both panels are well within the excluded -LSP region, as could have been anticipated from the well-known fact that this region extends to higher (at fixed ) as increases.

The configurations of the planes change significantly for GeV, as seen in Fig. 12. The - and -LSP regions disappear completely in panels (a) and (b) for and 20, respectively. There is only a small excluded region in panel (c) for , which grows finally in panel (d) for . Much of the half-plane is excluded by for and 20, but this constraint disappears for larger . The LEP Higgs constraint is not important in the regions allowed by . The half-planes are favoured by for . The regions favoured by WMAP are rapid-annihilation funnels for all values of , crossover strips for and , and coannihilation strips for and (fleetingly) for .

Figure 12: Examples of NUHM2 planes with  GeV,  GeV, , and , 20, 35, and 50 in panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Fig. 1.

The NUHM1 lines are again (approximate) parabolae in all four panels. They intersect the WMAP strips in crossover and rapid-annihilation regions in panels (a, b) and (c), for , and in the rapid-annihilation and coannihilation regions for in panel (d). We note that in panel (d) the approximate NUHM1 parabola has shifted such that for some values of there is no unique solution for  6 The CMSSM points are in strongly overdense regions in panels (a, b) and (c), but in the forbidden -LSP region of panel (d). However, this point is close to an allowed region where the relic density would be within the favoured range. Therefore, there are nearby CMSSM points with similar values of and that are consistent with all the constraints. All these planes in Fig. 12 should be accessible to the LHC [28, 27], because of the moderate values chosen for and , but some heavier neutralinos, charginos and Higgs bosons would only be accessible for relatively small values of and .

Finally, we present in Fig. 13 some planes for the choices GeV. Unlike the previous cases, these choices are in a region of the plane that is far from the coannihilation strip in the CMSSM. No parts of any of the planes are excluded by the absence of electroweak symmetry breaking or the presence of a charged LSP. We see explicitly in the panels (a, b) and (c) for that again excludes most of the half-plane with . For , shown in panel (d), reliable solutions are not found with . The LEP Higgs limit does not exclude a significant extra region of the plane in any of the panels. In panel (d) for there is a region at  GeV that is favoured by , but not for the lower values of . In each of the panels, the region favoured by WMAP consists of a crossover strip at  GeV and a rapid-annihilation funnel with 400 GeV 450 GeV. These planes should also be accessible to the LHC [28, 27], though more luminosity would be required than in the previous cases because of the larger value of , in particular. This would also render more difficult the searches for some heavier neutralinos, charginos and Higgs bosons.

Figure 13: Examples of NUHM2 planes with  GeV,  GeV, , and , 20, 35, and 50 in panels (a), (b), (c), and (d), respectively. Constraints are displayed as in Fig. 1.

The NUHM1 lines are again parabolae, reaching values of that decrease from  GeV to  GeV as increases, and becoming increasingly asymmetric in  7. They intersect the WMAP regions in both the rapid-annihilation strips and the crossover strips (the latter at