Focus Points and Naturalness in Supersymmetry

We analyze focus points in supersymmetric theories, where a parameter's renormalization group trajectories meet for a family of ultraviolet boundary conditions. We show that in a class of models including minimal supergravity, the up-type Higgs mass has a focus point at the weak scale, where its value is highly insensitive to the universal scalar mass. As a result, scalar masses as large as 2 to 3 TeV are consistent with naturalness, and {\em all} squarks, sleptons and heavy Higgs scalars may be beyond the discovery reaches of the Large Hadron Collider and proposed linear colliders. Gaugino and Higgsino masses are, however, still constrained to be near the weak scale. The focus point behavior is remarkably robust, holding for both moderate and large \tan\beta, any weak scale gaugino masses and A parameters, variations in the top quark mass within experimental bounds, and for large variations in the boundary condition scale.


I. INTRODUCTION
An understanding of electroweak symmetry breaking is currently one of the most important objectives in high energy particle physics. Renormalizability requires that electroweak symmetry be spontaneously broken; in the minimal standard model, this is realized by the condensation of the elementary Higgs field. In such a theory, however, the squared Higgs mass receives quadratically divergent radiative corrections. The Higgs mass, and with it the weak scale, is therefore expected to be of order the cut-off scale, which is typically identified with the grand unified theory (GUT) or Planck scale. The fact that the weak scale is much smaller than the cut-off scale requires a large fine-tuning and is therefore considered unnatural in the minimal standard model [1].
Supersymmetry removes quadratic divergences and therefore provides a framework for naturally explaining the stability of the weak scale with respect to radiative corrections [2]. However, the requirement of naturalness constrains supersymmetric models, as, in these models, the weak scale is generated when electroweak symmetry breaking is induced by (negative) squared mass parameters for the Higgs scalars. The weak scale is therefore related to these supersymmetry breaking parameters. It is hoped that an understanding of the mechanism of supersymmetry breaking will shed light on the origin of the weak scale. Even without this knowledge, though, it is clear that naturalness in supersymmetric theories requires that the supersymmetry breaking parameters in the Higgs potential be not too far above the weak scale.
As naturalness implies upper bounds on supersymmetry breaking parameters and superpartner masses, its implications are obviously of great importance for supersymmetry searches. These implications depend on the assumed structure of supersymmetry breaking. With respect to the scalar masses, broadly speaking, three possibilities exist. In the first, all supersymmetry breaking scalar mass parameters are roughly of the same order; for example, they may be of the same order when generated at some high scale and then remain of the same order when evolved through renormalization group (RG) equations to the weak scale. Naturalness then demands that scalar Higgs, squark, and slepton masses all be near the weak scale. Such light particles are within the discovery reach of the Large Hadron Collider (LHC) and future lepton colliders.
Another possibility is that a hierarchy exists between the various scalar masses. This hierarchy may be present at the scale at which supersymmetry breaking parameters are generated [3] or may be generated dynamically through RG evolution [4]. In either case, one finds that naturalness bounds on the first and second generation squarks and sleptons are much weaker than those for the third generation [5]. First and second generation sfermions may then be much heavier than 1 TeV and far beyond the reach of near future colliders. However, top and bottom squarks, for example, are still constrained to have masses of order the weak scale, and should be discovered by the LHC.
A third possibility, however, is that the RG trajectories of the Higgs mass parameters may meet at a "focus point" [6,7], where their values are independent of their ultraviolet boundary values. 1 If this focus point is near the weak scale, the Higgs potential at the weak scale may be insensitive to the ultraviolet values of certain supersymmetry breaking parameters, including the scalar masses. In this case, naturalness, while constraining (unphysical) Higgs mass parameters, may lead to very weak upper bounds on the squark, slepton, and heavy Higgs boson masses, and these scalars may all be beyond the reach of near future colliders.
The last possibility is the subject of this study. We will show that it is realized in a class of models that includes minimal supergravity. Minimal supergravity is at present probably the single most widely-studied framework for evaluating the potential of new experiments to probe physics beyond the standard model. We therefore concentrate on this model and explore the implications of focus points in minimal supergravity for naturalness and the superpartner spectrum.
The organization of this paper is as follows. In Sec. II, we analyze the focus point behavior of the RG evolution of supersymmetry breaking parameters. In Sec. III, we discuss the implications of the up-type Higgs focus point for naturalness in minimal supergravity. In particular, we will see that multi-TeV scalar masses are consistent with naturalness. The implications of these results for superpartner searches are considered in Sec. IV. Finally, we conclude in Sec. V with a summary of our results and some philosophical comments concerning the concept of naturalness.

II. FOCUS POINTS
In this section, we explore the phenomenon of focus points in the RG evolution of supersymmetry breaking parameters. We will see that, in certain circumstances, the supersymmetry breaking up-type Higgs mass has such a focus point at the weak scale, where it becomes insensitive to its boundary value at the high scale (for example, the GUT scale). Since the supersymmetry breaking masses for the Higgses are related to the weak scale, this fact has implications for naturalness, as we will discuss in Sec. III.
We start by considering the RG behavior of supersymmetry breaking scalar masses. Denoting the mass of the i-th scalar field by m i , the one-loop RG evolution of scalar masses is given schematically 2 by the inhomogeneous equations where g a and y j are gauge and Yukawa coupling constants, respectively, Q is the renormalization scale, and the summation is over all chiral superfields coupled to the i-th chiral superfield through Yukawa interactions. The gaugino masses M a and supersymmetry breaking trilinear scalar couplings A j have RG equations As one can see, the evolution of the m 2 i parameters depends on the gaugino masses and A parameters, as well as on the scalar masses themselves. On the other hand, the gaugino masses and A parameters evolve independently of the scalar masses. 3 This structure implies that, if m 2 i | p is a particular solution to Eq. (1) with fixed values of the gaugino masses and A parameters, then for arbitrary constant ξ, is also a solution if the ∆ 2 i obey the following linear and homogeneous equation: The evolution of the ∆ 2 i depends only on the ∆ 2 i , and the ∆ 2 i are themselves solutions to the RG equations in the limit M a , A j → 0.
With a given boundary condition, ∆ 2 i may vanish for some i at some renormalization scale Q (i) F . At this scale, m 2 i is given by m 2 i | p irrespective of ξ, and the family of boundary conditions parameterized by m 2 This concludes our general discussion of focus points. We now consider the existence of focus points in the minimal supersymmetric standard model. Before showing the results of detailed numerical calculations, we first analyze the focus point behavior by using oneloop RG equations. In the minimal supersymmetric standard model, the possibly large Yukawa couplings are those for the third generation quarks, y t and y b . 4 The system of RG equations for the ∆ 2 i is then 3 Note that this implies that if a hierarchy M a , A j ≪ m i is generated at some high scale, for example, by an approximate R-symmetry, it will not be destabilized by RG evolution.
where Q 3 , U 3 , and D 3 represent the third generation SU(2) L doublet, singlet up-type, and singlet down-type squarks, and H u and H d are the up-and down-type Higgses, respectively. All other ∆ 2 i 's are not coupled to large Yukawa coupling constants and hence are (almost) scale independent.
To find the focus point, it is simplest to begin by considering small or moderate values of tan β, for which y b is negligible. In this case, ∆ 2 D 3 and ∆ 2 H d remain constant, but the RG equations for ∆ 2 Hu , ∆ 2 U 3 , and ∆ 2 where The κ's are constants determined by the boundary conditions at the scale Q 0 and are independent of the renormalization scale Q. For given κ's, the focus point for m 2 Hu is given by For large tan β, we cannot neglect y b , and the above arguments do not apply. For a general large tan β, the evolution of the parameters is complicated and will be studied numerically below. However, for the specific case tan β ≃ m t /m b , we can assume y b = y t and follow an analysis similar to the one above. 5 In this case, the ∆ 2 i 's evolve according to 5 In fact, y t and y b , even if initially identical, will be slightly split in their RG evolution by y τ and the U(1) Y gauge interaction. In this discussion, we neglect this difference. This approximation is justified by the numerical calculations to follow.
The focus point for m 2 Hu is then given by The actual focus point depends on the boundary condition. Here, we first consider the well-studied case of minimal supergravity. In this framework, the supersymmetric Lagrangian is specified by five new fundamental parameters: the universal scalar mass m 0 , the unified gaugino mass M 1/2 , the supersymmetric Higgs mass µ 0 , the universal trilinear coupling A 0 , and the bilinear Higgs scalar coupling B 0 . These parameters are given at the GUT scale M GUT , which, in our analysis, is defined as the scale where the SU(2) L and U(1) Y gauge couplings meet. (Numerically, M GUT ≃ 2 × 10 16 GeV.) All the supersymmetry breaking scalar masses are universal at M GUT , and we may take 6 With these boundary conditions, the coefficients κ are and the focus point scale is determined by e 7I(Q) = 2/9 , for y b = y t .
Note that the I(Q) in Eqs. (16) and (17) are not identical, as y t runs differently for small and large tan β. Eqs. (16) and (17) determine the focus point scale in terms of the gauge couplings and the top quark Yukawa coupling, or equivalently, the top quark mass. Remarkably, for the physical gauge couplings and top quark mass m t ≈ 174 GeV, both conditions yield focus points that are very close to the weak scale! Thus, in minimal supergravity, the weak scale value of m 2 Hu is highly insensitive to the universal scalar mass m 0 . We now show numerically that the focus point is near the weak scale for m t ≈ 174 GeV. (For an analytical discussion, see the Appendix.) To study the focus point more carefully, we have evolved the supersymmetry breaking parameters with the full two-loop RG equations [9]. The one-loop threshold corrections from supersymmetric particles 6 We choose m 2 i | p (M GUT ) = 0 so that m 2 i | p is independent of m 0 and is always O(M 2 1/2 ) (or O(A 2 0 )) or smaller. to the gauge and Yukawa coupling constants are also included [10,11]. We take as in- Hu is determined by the other fundamental parameters M 1/2 and A 0 , and hence at least one of these parameters is required to be O(100 GeV).
In Fig. 1, two values of tan β were presented. In Fig. 2, we show the focus point scale of m 2 Hu as a function of tan β. The focus point is defined here as the scale where ∂m 2 Hu /∂m 0 = 0. As noted above, we have included the low-energy threshold corrections to the gauge and Yukawa coupling constants, which depend on the soft supersymmetry breaking parameters. As a result, the RG trajectories do not all meet at one scale, and the focus point given in Fig. 2 has a slight dependence on m 0 . For small values of tan β, say, tan β ∼ 2 − 3, the focus point is at very large scales. However, the important point is that, for all values of tan β > ∼ 5, including both moderate values of tan β and large values where y b and y τ are not negligible, Q So far, we have considered only the case of a universal scalar mass. However, the m 2

Hu
focus point remains at the weak scale for a much wider class of boundary conditions. For example, for small tan β, Eq. (7) shows that the parameter κ ′ 0 does not affect the evolution of m 2 Hu . As a result, the focus point of m 2 Hu does not change even if we vary κ ′ 0 , and the weak scale focus point is realized with any boundary condition of the form with x an arbitrary constant. Similarly, for the case of y b = y t , the possible variation is with both x and x ′ arbitrary. Another possible modification of the boundary conditions may be seen by viewing the m 2 i | p (Q 0 ) as perturbations. Since it was never necessary to specify the particular solution in the general focus point analysis, we may consider arbitrary m 2 i | p (Q 0 ) without changing the focus point scale. The only constraint on m 2 i | p is from naturalness. As will be discussed in the next section, m 2 Hu is required to be O((100 GeV) 2 ) at the weak scale for natural electroweak symmetry breaking. As a result, m 2 Hu | p , m 2 U 3 | p , and m 2 Q 3 | p (and also m 2 H d | p and m 2 D 3 | p for large tan β) are required to be of the order of the weak scale. Therefore, deviations from the boundary conditions of Eqs. (18) and (19) of order δm 2 ∼ O((100 GeV) 2 ) are acceptable and do not lead to fine-tuning problems. Similar arguments show that deviations M a , A j ∼ O(100 GeV) are allowed. In particular, the focus point is independent of gaugino mass or A parameter universality, and may therefore be found in many other frameworks. For example, focus points also exist in anomaly-mediated supersymmetry breaking models with additional universal scalar masses [6].
Before closing this section, we discuss another way of formulating the focus point, which was originally used in Ref. [12] for the specific case of minimal supergravity. By dimensional analysis, the evolution of m 2 Hu may be parameterized as where the coefficients η are determined by the (dimensionless) gauge and Yukawa coupling constants, and are independent of the (dimensionful) supersymmetry breaking parameters. In this formulation, the focus point is given by the scale where η m 2 0 = 0, since at that scale, the value of m 2 Hu is insensitive to m 0 . 7 Based on this observation, it was noted in Ref. [12] that, for minimal supergravity, and neglecting y b , η m 2 0 = 0 at the weak scale, and the weak scale becomes very insensitive to the universal scalar mass m 0 , for m t ≃ 160 − 170 GeV. As may be seen from the general analysis of focus points above, however, this conclusion holds much more generally: it is valid even when y b is not negligible, holds for the more general boundary conditions given in Eqs. (18) and (19), and is independent of all other scalar masses. In addition, as noted above and as is evident from Eq. (20), the conclusion is valid also for non-universal gaugino masses and A parameters, as long as they are not too much larger than the weak scale.

III. NATURALNESS
In the previous section, we saw that the weak scale value of m 2 Hu is highly insensitive to the high scale scalar mass boundary conditions in a class of models that includes minimal supergravity. This fact has important implications for the naturalness of the gauge hierarchy, since m 2 Hu determines, to a large extent, the shape of the Higgs potential. In minimal supergravity, m 2 Hu and other sfermion masses have the same origin, the universal scalar mass m 0 , and it has typically been believed that naturalness constraints on m 2 Hu also give similar bounds on the sfermion masses. Such beliefs have led to great optimism in the search for scalar superpartners at future colliders in the framework of minimal supergravity [13]. However, as we have seen, the relation between m 2 Hu and other sfermion masses is not as trivial as typically assumed. In the following, we therefore reconsider the naturalness bounds on the sfermion masses in the minimal supergravity model.
To begin, it is instructive to start with the tree-level expression for the weak scale. The Z boson mass is determined by minimizing the tree-level Higgs potential to be For all tan β, m 2 Hu ≫ m 2 Z is disfavored by the naturalness criterion, as in that case, a large cancellation between m 2 Hu and µ 2 is needed to arrive at the physical value of the weak scale. For moderate and large values of tan β, however, m 2 H d ≫ m 2 Z does not necessarily lead to fine-tuning.
For more detailed discussions of naturalness, it is convenient to define a quantitative measure of fine-tuning [12,[14][15][16][17]. Following previous work, we use the sensitivity of the weak scale (i.e., m Z ) to fractional variations in the fundamental parameters as such a measure.
In any discussion of naturalness, several subjective choices must be made. The choice of supersymmetry breaking framework is crucial. For example, in GUT models, the gaugino masses are all governed by one parameter, whereas in general, all three gaugino masses may be varied independently, and the sensitivity of the weak scale to each of them must be considered. In the following, we will specialize to minimal supergravity. As noted previously, minimal supergravity introduces five new fundamental parameters: m 0 , M 1/2 , µ 0 , A 0 , and B 0 . All quantities at the weak scale are fixed by these parameters. In particular, the vacuum expectation values of the Higgs bosons depend on these quantities. Therefore, some combination of them is constrained to yield the correct Z boson mass. (At tree level, this constraint is that of Eq. (21). At one-loop, the Higgs masses squared are shifted by the corresponding tadpole contributions [18], as will be discussed below.) From the low energy point of view, it is therefore more convenient to consider tan β and sign(µ) as free parameters, instead of µ 0 and B 0 .
We adopt the following procedure to calculate the magnitude of fine-tuning at all physically viable parameter points: (i) We consider the minimal supergravity framework with its 4 + 1 input parameters Any point in the parameter space of minimal supergravity is specified by these parameters.
(ii) The naturalness of each point is then calculated by first determining all the parameters of the theory (Yukawa couplings, soft supersymmetry breaking masses, etc.), consistent with low energy constraints. RG equations are used to relate high and low energy boundary conditions. In particular, using the relevant radiative breaking condition, |µ 0 | and B 0 are determined consistent with the low energy constraints.
(iii) We choose to consider the following set of (GUT scale) parameters to be free, continuously valued, independent, and fundamental: (iv) All observables, including the Z boson mass, are then reinterpreted as functions of the fundamental parameters a i , and the sensitivity of the weak scale to small fractional variations in these parameters is measured by the sensitivity coefficients where all other fundamental (not input) parameters are held fixed in the partial derivative.
which is taken as a measure of the naturalness of point {P input }, with large c corresponding to large fine-tuning. Among the choices made in the prescription above, the choice of fundamental parameters a i is of particular importance. This choice varies throughout the literature [12,[14][15][16][17]. Since we are interested in the naturalness of the supersymmetric solution to the gauge hierarchy problem, we find it most reasonable to include only supersymmetry breaking parameters (and µ, as its origin is likely to be tied to supersymmetry breaking) and not standard model parameters, such as y t or the strong coupling. We will return to this issue in Sec. V.
Given the prescription for defining fine-tuning described above, we may now present the numerical results. In minimizing the Higgs potential, we use the one-loop corrected Higgs potential, calculated with parameters evaluated with two-loop RG equations. Denoting the physical stop masses by mt 1 and mt 2 , we choose to minimize the potential at the scale where the one-loop corrections to the Higgs potential tend to be smallest [15]. In terms of the fundamental parameters, Qt ≃ 0.5(m 2 0 + 4M 2 1/2 ) 1/2 . In Figs. 3-5, we give contours of constant c m 0 , c M 1/2 , and c µ 0 in the (m 0 , M 1/2 ) plane for tan β = 10 and 50. (The parameters c A 0 and c B 0 are typically negligible relative to these, and are so for the parameter ranges displayed.) Several features of these figures are noteworthy. First, as is evident from Fig. 3, the fine-tuning coefficient c m 0 is small even for scalar masses as large as m 0 ∼ 2 TeV. This is a consequence of the focus point behavior of m 2 Hu : for moderate and large tan β, m 2 Hu is insensitive to m 0 , and, therefore, so is m 2 Z . The deviation of c m 0 from zero for very large m 0 is a consequence of the fact that the focus point does not coincide exactly with the electroweak scale, or, more precisely, with Qt. We will explain this statement in more detail below.
On the other hand, large gaugino masses lead to large m 2 Hu through RG evolution, and hence c M 1/2 increases as M 1/2 increases, as shown in Fig. 4. Multi-TeV values of M 1/2 are therefore inconsistent with naturalness.
The behavior of c µ 0 , presented in Fig. 5, is also interesting. Since µ ∝ µ 0 , Eq. (21) implies c µ 0 ≈ 4µ 2 /m 2 Z . In particular, c µ 0 is small in the region bordering the excluded and down-type Higgs scalars, respectively, may be a more accurate choice, especially if variations of the gauge coupling constants are considered. In this paper, however, we follow the literature and define sensitivity coefficients as in Eq. (24). right-hand region, as there µ is suppressed by a cancellation between the η m 2 0 and η M 2 1/2 terms in Eq. (20). However, in this region, c m 0 and c M 1/2 are large and this region is fine-tuned; the simple criterion of requiring low µ for naturalness [19] fails here. In Fig. 6, we show the overall fine-tuning parameter c, the maximum of c a , in the (m 0 , M 1/2 ) plane. The fine-tuning c is determined by c µ 0 , c M 1/2 , and c m 0 . For small m 0 , c µ 0 is the largest. As m 0 increases, however, c M 1/2 becomes dominant, and c is therefore almost independent of m 0 in this region. Finally, for extremely large m 0 , c m 0 becomes important. (For large tan β, large m 0 is excluded by the chargino mass limit before c m 0 becomes dominant. As a result, we do not see the c m 0 segment in Fig. 6b.) Note that, for fixed M 1/2 in Fig. 6, values of m 0 > ∼ 1 TeV are actually more natural than small m 0 : for large m 0 , the parameter µ, and therefore c µ 0 , is reduced. Of course, eventually as m 0 increases to extremely large values, either µ becomes so small that the chargino mass bound is violated or c m 0 becomes large, and so very large m 0 is either excluded or disfavored.
As one can see, regions of parameter space with m 0 ∼ 2 − 3 TeV are as natural as regions with m 0 < ∼ 1 TeV. As will be discussed more fully in Sec. IV, in the region of parameter space with m 0 ∼ 2 − 3 TeV, all squarks and sleptons have multi-TeV masses, and discovery of these scalars will be extremely challenging at near future colliders. On the other hand, the gaugino mass M 1/2 cannot be multi-TeV since it generates unnaturally large m 2 Hu . Although the focus point mechanism allows multi-TeV scalars consistent with naturalness, the same conclusion does not apply to gauginos and Higgsinos.
As discussed in Sec. III, we expect also that the A parameters should be bounded by naturalness to be near the weak scale. In Fig. 7, we present contours of constant c in the (m 0 , A 0 ) plane. As expected, large A terms lead to large m 2 Hu , and A 0 is also required to FIG. 6. As in Fig. 3, but for the fine-tuning parameter c.
be O(100 GeV). In Fig. 7, for increasing m 0 , c is determined successively by c µ 0 , c A 0 and c m 0 . (The c m 0 segments are missing for the A 0 > 0 contours in Fig. 7b.) The dependence of the fine-tuning parameter c on tan β is illustrated in Fig. 8. From this figure, for a given tan β and maximal allowed c, we can determine the upper bound on m 0 . The exact range of c required for a natural model is, of course, subjective. However, taking as an example the requirement c ≤ 50, corresponding to µ < ∼ 300 GeV at parameter points where c = c µ 0 , we find m 0 < ∼ 2 TeV for tan β = 10. So far, we have assumed m t = 174 GeV in our calculations. However, given the experimentally allowed range m t = 173.8 ± 5.2 GeV [20], we now consider the top quark mass dependence of the naturalness constraint on m 0 . This can be understood only after accounting for the one-loop corrections to the Higgs effective potential. 9 At one-loop, the relation between m Z and the Higgs mass parameters, Eq. (21), is modified to [11,18] where T u and T d are the tadpole contributions to the effective potential and Π T ZZ (p), the transverse part of the Z boson self-energy, is negligible. In minimal supergravity, the dominant terms in T u and T d are from third generation squark loops and have the generic form Using Eqs. (27), (20) and (28), we find where η ′ m 2 0 (Q) encodes the dependence on m 0 arising at one-loop through the tadpole.
Recall that the focus point Q (Hu) F is defined by while empirically we find which may also be understood to a good approximation from Eq. (29). We have already seen from Fig. 2 that for m t = 174 GeV, Q Hu to m 0 may then be understood in two ways: either we may minimize the potential at Qt where the tadpole contributions are negligible, but η m 2 0 is non-vanishing, or we may choose to minimize the potential at Q (Hu) F , in which case η m 2 0 = 0, but m 0 dependence arises from non-vanishing tadpole contributions. Either way, there is some residual dependence on m 0 , as can be seen in Fig. 3.
If Q Hu ). In Fig. 9, we present contours of fine-tuning c in the (m 0 , m t ) plane. On the dotted contour, the focus point and Qt coincide, and c m 0 = 0. This occurs for m t above 174 GeV, in accord with the discussion above. More generally, the sensitivity c m 0 is indeed reduced for larger m t . As a result, the upper bound on m 0 is increased for larger m t , and for m t ≃ 180 GeV, near the 1σ upper bound, m 0 ∼ 3.4 TeV is allowed for c ≤ 50. For smaller top quark mass, the naturalness bound on m 0 becomes more stringent. Thus, while the focus point behavior persists for all m t within current experimental bounds, future improvements in top mass measurements [21] may provide important information about the extent to which multi-TeV scalars are allowed in minimal supergravity.
We may also consider variations in the high scale Q 0 where the supersymmetry breaking parameters are assumed to be generated. So far we have assumed Q 0 = M GUT . It is interesting to consider the effects of assuming that the boundary conditions are specified at a different scale, e.g., the string or Planck scale. To investigate this, we have taken a simple approach, and evolved the gauge couplings up to some fixed Q 0 , set the supersymmetry breaking parameters at that scale, and then evolved them down to the weak scale. The minimal field content is assumed throughout the RG evolution; in particular, no additional GUT particle content is assumed for Q 0 > M GUT , and the unification of gauge couplings at M GUT is unexplained. In Fig. 10 we show contours of constant c in the (m 0 , Q 0 ) plane. Just as in Fig. 6, the fine-tuning parameter c is determined successively, for increasing m 0 , by c µ 0 , c M 1/2 and c m 0 . We see that increasing the scale Q 0 also allows even larger scalar masses. For example the requirements c ≤ 50 and Q 0 < M Pl allow m 0 as large as 2.9 TeV.
In Fig. 11 we show contours of constant c in the (m t , Q 0 ) plane. As expected, smaller values of m t can be compensated for by larger evolution intervals, and vice versa. Notice, however, that varying m t within its current experimental uncertainty leads to changes in c that are as large as those caused by varying Q 0 by several orders of magnitude.

IV. IMPLICATIONS FOR SUPERSYMMETRY SEARCHES
We have seen that the naturalness bound on m 0 (i.e., the typical sfermion mass) may be as large as a few TeV. In this section, we discuss the implication of these results for the superpartner spectrum and, in particular, the discovery prospects for scalar superpartners at future colliders. (Of course, it is clear that such heavy scalars also drastically reduce the size of supersymmetric effects in low energy experiments, but we will not address this

further.)
The implications for sleptons are fairly straightforward. Sleptons have small Yukawa couplings (with the possible exception of staus for large tan β), and so their masses are virtually RG-invariant, with ml ≈ m 0 in this scenario. Multi-TeV sleptons are beyond the kinematic limit ml < √ s/2 of all proposed linear colliders. They will also escape detection at hadron colliders, as they are not strongly produced, and will not be produced in large numbers in the cascade decays of strongly interacting superparticles. We now turn to squarks. Multi-TeV squarks will, of course, also evade proposed linear colliders. Traditionally, however, it has been expected that future hadron colliders, particularly the LHC, will discover all squarks in the natural region of minimal supergravity parameter space [13]. This conclusion is based on the expectation that all squarks have masses < ∼ 1 − 2 TeV. In Fig. 12, we present contours for mũ L . 10 (All first and second generation squarks are nearly degenerate.) In the same figure, we have also included contours of the fine-tuning parameter c. For c ≤ 50, we see that squark masses of mũ L = 2.2 TeV are allowed, and more generally, the parameter space with multi-TeV squarks is as natural as parameter space with mũ L < ∼ 1 TeV. Recall also that these mass limits may be extended to as large as ∼ 3 TeV for variations of m t within its current bounds. Squarks of mass ∼ 2 TeV may be detected at the LHC with large integrated luminosities of several 100 fb −1 . However, squarks with masses significantly beyond 2 TeV are likely to escape detection altogether [22].
Since the top squarks and left-handed bottom squark interact strongly through y t , they are lighter than the other squarks. For small tan β, and small M 1/2 and A 0 parameters, their masses at the focus point of m 2 Hu are given by In general, these squark masses, particularly the stop masses, may also be influenced by left-right mixing. However, because naturalness constrains the A and µ parameters to be of order the weak scale, left-right mixing effects are sub-leading for multi-TeV m 0 . As a result, the lighter (heavier) stop is mostly right-handed (left-handed), and the lighter (heavier) sbottom is mostly left-handed (right-handed). For large tan β, y b also suppresses third generation squark masses. For example, for y b = y t , we obtain In Figs. 13 and 14 we present the masses of the stops and sbottoms, respectively. By comparing with Fig. 12, we see that the stops are always lighter than the first and second generation squarks. The lighter sbottom and heavier stop are nearly degenerate, since they are (approximately) in the same SU(2) L doublet. The heavier sbottom, which is mostly the right-handed sbottom, may also become significantly lighter than the first two generation squarks for large tan β. Therefore, in the multi-TeV m 0 scenario, stop and sbottom production will be the most promising modes for squark discovery at the LHC.
In contrast to the sfermions, gauginos cannot be very heavy in this scenario. This fact is explicit in Fig. 4: for large M 1/2 , the fine-tuning coefficient c M 1/2 becomes unacceptably large, irrespective of tan β. As a result, naturalness requires fairly light gauginos. For example, the constraint c ≤ 50 implies M 1/2 < ∼ 400 GeV, corresponding to M 1 < ∼ 160 GeV, M 2 < ∼ 320 GeV, and M 3 < ∼ 1.2 TeV. Such gauginos will be produced in large numbers at the LHC, and will be discovered in typical scenarios. 11 It is also interesting to consider the implications of the focus point for Higgs masses. In supersymmetric models with minimal field content, the lightest Higgs boson mass m h is bounded by m Z at tree level. However, this upper bound may be significantly violated by radiative corrections [24]. In particular, top-stop loop contributions, approximately proportional to ln(mt/m t ), may be important. Since the focus point allows heavy stops, one may wonder if this affects the upper bound on the lightest Higgs mass. To answer this question, we have calculated the one-loop radiative corrections to the lightest Higgs mass. The result is shown in Fig. 15; we emphasize the A 0 dependence, as the radiative correction is sensitive to left-right mixing through the A parameter. For the region with small fine-tuning parameter, say, c ≤ 50, we find m h < 118 GeV. It is important that a large A parameter is forbidden by naturalness, as this suppresses left-right stop mixing, which usually significantly increases m h . Therefore, even in the focus point scenario with multi-TeV squarks, Run II of the Tevatron will probe much of parameter space in its search for the lightest Higgs boson [25].
Discovery of the heavy Higgs scalars is more challenging. At tree-level, the masses of the heavy Higgs scalars are approximately given by

V. SUMMARY AND DISCUSSION
In this paper, we have explored the existence of focus points in the RG behavior of supersymmetry breaking parameters and their implications for naturalness and experimental searches for supersymmetry.
For the experimentally measured top quark mass, the supersymmetry breaking uptype Higgs mass parameter m 2 Hu has a focus point at the scale Q ∼ O(100 GeV) in a class of models which includes minimal supergravity. The value of m 2 Hu at the weak scale is therefore highly insensitive to the universal scalar mass m 0 at the GUT scale. We have also seen that this focus point behavior exists for all values of tan β > ∼ 5. Since m 2 Hu plays an important role in the determination of the weak scale, this focus point behavior affects the naturalness of electroweak symmetry breaking in minimal supergravity. In particular, because a large m 0 can result in a reasonably small m 2 Hu , naturalness constraints on m 0 are not as severe as typically expected. To discuss this issue quantitatively, we have calculated the fine-tuning parameter c, which is determined by the sensitivity of the weak scale to fractional variations of the fundamental parameters. As we have seen, in regions of parameter space with m 0 ∼ 2 − 3 TeV this fine-tuning parameter may be as small as in regions with m 0 < ∼ 1 TeV. As a result, multi-TeV sfermions are as natural as sfermions lighter than 1 TeV. We note that the region of multi-TeV scalars and light gauginos and Higgsinos is also somewhat preferred by gauge coupling unification in minimal SU (5) [10,27,28], as well as b-τ Yukawa unification at moderate to large tan β [29,30]. The discovery of squarks and sleptons at the LHC and proposed linear colliders may therefore be extremely challenging, and may require some even more energetic machines, such as muon or very large hadron colliders.
In our analysis, we did not include the Yukawa couplings, notably y t , and gauge couplings in the calculation of the the fine-tuning parameter c. In Fig. 17, we present the sensitivity coefficient c yt for the (GUT scale) top Yukawa coupling. For m 0 ≥ 1 TeV, c yt is always larger than 70, and so if c yt is included in the calculation of the fine-tuning parameter c, the naturalness bound on m 0 becomes much more stringent and m 0 > ∼ 1 TeV is disfavored. The inclusion of y t and other standard model parameters in the fine-tuning calculation would thus lead to significantly different conclusions.
A definitive resolution to this question of whether or not to include variations of standard model parameters in fine-tuning calculations cannot, we believe, be achieved without a more complete understanding of the fundamental theories of flavor and supersymmetry breaking. Without such knowledge, any discussion necessarily becomes somewhat philosophical. Nevertheless, several remarks are in order.
First, it is sometimes argued that one should not consider variations with respect to parameters that have been measured or are highly correlated with measured quantities. According to this view, standard model parameters such as the strong coupling constant, and possibly also y t , should not be included among the a i . We do not subscribe to this view. 12 If in the future the Higgsino mass µ is measured to be 10 TeV, given our current notions of naturalness, we believe this should be considered fine-tuned, irrespective of the accuracy with which the Higgsino mass is measured. Of course, if this were the case, the fact that a 10 TeV µ parameter is realized in nature would be a strong motivation to consider alternative, and perhaps more fundamental, theoretical frameworks in which a 10 TeV µ parameter is not unnatural.
There are, however, other considerations which favor the exclusion of standard model parameters from the list of a i . As noted in Sec. III, we are interested in the naturalness of the supersymmetric explanation of the gauge hierarchy. We should not require that supersymmetry also solve the problem of flavor. In fact, in many supergravity frameworks, the supersymmetry breaking parameters and the Yukawa couplings are expected to be determined independently. For example, in hidden sector scenarios, the supersymmetry breaking parameters are determined in one sector, while the Yukawa couplings are fixed in some other sector and by a completely independent mechanism. In this case, it seems reasonable to assume that y t is fixed to its observed value in some sector not connected to supersymmetry breaking, and we therefore should not consider variations with respect to it.
Finally, it is worth noting that there are several possible scenarios in which it is clear that c yt should not be included in c or is at least negligibly small. One possibility is that y t may evolve from some higher scale, such as the Planck scale, to a fixed or focus point at the GUT scale. The weak scale is then highly insensitive to variations in the truly fundamental parameter, i.e., y t at the Planck scale. Alternatively, the top Yukawa coupling may arise as a renormalizable operator with coefficient determined by a correlation function of string vertex operators. The coupling y t would then be fixed to its current value (or possibly one of a discrete set of values), and it is again inappropriate to consider continuous variations with respect to y t . Note that in both of these scenarios, y t may receive additional contributions from non-renormalizable operators of the form δy t ∼ gǫ, where g is a coupling constant, and ǫ is some small expansion parameter, such as v/M P l , where v is some vacuum expectation value. In this case, c g and c ǫ should be included in the definition of fine-tuning, but they will be negligible for small ǫ. In this appendix, we discuss the focus point analytically at one-loop for the two cases y b ≪ y t and y b = y t . Solutions to the RG equations are well-known for these two cases [31]. For both cases, we derive a closed form expression involving the gauge couplings and m t that must be satisfied if the focus point scale Q F is to be at the weak scale. We also show that if the focus point is at the weak scale for y b ≪ y t , it is also at the weak scale for y b = y t .
In the analysis of the y b = y t case, we neglect the effects of y τ and the hypercharge differences in the y t and y b RG equations, so y t and y b remain degenerate throughout their RG evolution. The validity of these approximations is verified only by the numerical results in Sec. II. In addition, the intermediate case, where y b = y t but y b may not be neglected, is not considered. However, the following analysis may be helpful in understanding the numerical results, and in particular, the behavior of Figs. 1 and 2.
For a universal scalar mass, the conditions for the focus point (see Sec. II) are e sI = 1/3 , for y b ≪ y t 2/9 , for y b = y t .
We see that these are simultaneously satisfied at Q = m t if In terms of the top quark mass, this requirement corresponds to where m t (m t ) is the running top quark mass, and v = 174 GeV. Given fixed gauge coupling constants, Eq. (A11) specifies which top quark mass will place the focus point at the weak scale. For Q 0 = M GUT = 2 × 10 16 GeV and α(M GUT ) = 1/24, we obtain F ≃ −130 and E ≃ 13 at Q = 174 GeV. For sin β ≃ 1, the requirement is then m t (m t ) ≃ 160 GeV, which is very close to the running mass corresponding to the physical pole mass m t ≈ 174 GeV. Therefore, for the experimentally measured top quark mass, the focus point of m 2 Hu is close to the weak scale for y b = y t and also for y b ≪ y t .