{'fraction_non_alphanumeric': 0.04206918043004051, 'fraction_numerical': 0.019917939129531527, 'mean_word_length': 4.062311029315104, 'pattern_counts': {'":': 0, '<': 26, '<?xml version=': 0, '>': 16, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Weak-scale phenomenology of models with gauge-mediated supersymmetry breaking

We study in some detail the spectral phenomenology of models in which supersymmetry is dynamically broken and transmitted to the supersymmetric partners of the quarks, leptons and gauge bosons, and the Higgs bosons themselves, via the usual gauge interactions. We elucidate the parameter space of what we consider to be the minimal model, and explore the regions which give rise to consistent radiative electroweak symmetry breaking. We include the weak-scale threshold corrections, and show how they considerably reduce the scale dependence of the results. We examine the sensitivity of our results to unknown higher-order messenger-sector corrections. We compute the superpartner spectrum across the entire parameter space, and compare it to that of the minimal supergravity-inspired model. We delineate the regions where the lightest neutralino or tau slepton is the next-to-lightest supersymmetric particle, and compute the lifetime and branching ratios of the NLSP. In contrast to the minimal supergravity-inspired model, we find that the lightest neutralino can have a large Higgsino component, of order 50%. Nevertheless, the neutralino branching fraction to the gravitino and the light Higgs boson remains small,<10^{-4}, so the observation of such a decay would point to a non-minimal Higgs sector.


Introduction
Most studies of supersymmetric phenomenology have focused on models in which supersymmetry is broken in a hidden sector at a scale of order M SUSY ≃ 10 10 GeV. In these models, supersymmetry breaking is communicated to the visible sector by gravitational interactions. Such models give rise to superpartner masses in the 100 -1000 GeV range, and to a gravitino with approximately the same mass. The couplings of the gravitino to matter are negligible, suppressed by (E/M SUSY ) 2 ≃ 10 −16 . The now-standard "minimal supergravity" model is of this type, with the familiar unification-scale boundary conditions on M 0 , M 1/2 , and A 0 .
Recently, there has been a resurgence of interest in models where supersymmetry is broken at a scale of order M SUSY > ∼ 10 5 GeV [1] - [12]. In these models, supersymmetry breaking is transmitted to the superpartners of the quarks, leptons, and gauge bosons (and to the Higgs bosons themselves) via the usual SU(3) × SU(2) × U(1) gauge interactions. Because gauge interactions are flavor diagonal, these models naturally suppress the flavor-changing neutral currents associated with the soft squark and slepton masses.
Models with gauge-mediated symmetry breaking also have superpartner masses in the 100 -1000 GeV range. However, because of the low scale of supersymmetry breaking, the gravitino is essentially massless; its couplings to the superpartners are suppressed by (E/M SUSY ) 2 < ∼ 10 −6 . Typically, superparticles decay by cascading down to the next-to-lightest supersymmetric particle (NLSP), which in turn decays to its partner and the gravitino. Such decays give rise to characteristic experimental signatures, e.g. final states containing two photons and missing energy [2,3].
In this paper we take a detailed look at the low-energy spectrum of the simplest models with gauge-mediated supersymmetry breaking. We begin by describing the models, elaborating on the issues associated with electroweak symmetry breaking. We use two-loop renormalization group equations and the full one-loop threshold corrections to determine the parameter space which gives rise to consistent radiative electroweak symmetry breaking. We find the Higgs boson and superpartner masses to the same level of accuracy. We comment on how our results are affected by higher-order messenger-sector corrections. We compare the spectra of the simplest gauge-mediated models with those from the minimal supergravity-inspired model. Finally, we identify the next-to-lightest supersymmetric particle and compute its lifetime and branching ratios across the allowed parameter space.

Electroweak symmetry breaking
In models of gauge-mediated supersymmetry breaking, the SU(3) × SU(2) × U(1) gauge interactions of "messenger" fields communicate supersymmetry breaking from a hidden sector to the fields of the visible world. In the simplest models [1], the messenger sector contains a set of vector-like fields, M i and M i , coupled to a standard-model singlet, S, through the superpotential interaction The lowest and F -components of the singlet superfield S acquire vevs through their interactions with the hidden fields. This breaks supersymmetry and R-symmetry.
To maintain the near unification of the standard-model gauge couplings, we will take the fields M i and M i to transform in complete SU (5) representations. In the same spirit, we will also require that the gauge couplings remain perturbative up to the unification scale. This implies that we can consider at most four 5 + 5 pairs, or one 10 + 10 and one 5 + 5 pair.
In what follows we will take the fields M i and M i to lie in (n 5 , n 10 ) 5 + 5 and 10 + 10 SU(5) representations. We will assume that these fields couple to the singlet S through a single Yukawa coupling λ at the unification scale. (See ref. [4,5] for a discussion of variations.) Below that scale, the SU(5) representations split apart, and the Yukawa couplings evolve according to their own renormalization group equations [6]. We will ignore this splitting in most of what follows; we will remark on it briefly in sect. 3.
The lowest and F components of the superfield S acquire vevs through interactions with the sector which dynamically breaks supersymmetry. These vevs induce masses and mixings for the messenger fields. The messenger fermions gain mass M = λ S , while the messenger scalars obtain the mass matrix From this point on, we omit the brackets which denote the vevs. Following the philosophy of dynamical models, we will assume that the standardmodel µ-term and all soft supersymmetry-breaking terms arise dynamically. For the case at hand, the messenger fields transmit the supersymmetry breaking to the visible sector through loop diagrams which contain insertions of the S superfield. Such diagrams induce weak-scale masses for the gauginos and scalars of the minimal supersymmetric standard model. However, they cannot give sizable values for the soft supersymmetry-breaking A parameters. In what follows, we shall set all A-terms to zero at the messenger scale.
The messenger fields induce the following gauginõ and scalarm masses at the scale M, where Λ = F S /S and k i = 1, 1, 3/5 for SU(3), SU(2), and U(1), respectively. The C i are zero for gauge singlets, and 4/3, 3/4, and Y 2 for the fundamental representations of SU(3), SU(2), and U(1). Here Y = Q − I 3 denotes the usual hypercharge and we use the grand unification normalization for α 1 . Because a pair of 10 + 10 fields contributes to the soft masses as if n 5 = 3, we will set n 10 = 0 and only consider changes in n 5 . The messenger-scale threshold functions [5,7] g(x) = 1 + x have the property that g(x), f (x) → 1 as x → 0, or Λ ≪ M. In this limit, the expressions (3) and (4) take the characteristic simple forms [1] that are often associated with gauge-mediated models. The region Λ → M corresponds to x → 1, where g(1) ≃ 1.4 and f (1) ≃ 0.7. We must exclude the limit M = Λ because it gives rise to a massless messenger scalar. For the purposes of this paper, we restrict ourselves to the region x < 0.97, or M/Λ > 1.03. This corresponds to an upper limit on the fine tuning of the messenger masses; it is obtained by requiring that the average scalar mass-squared be less than 30 times the light scalar mass-squared.
Equations (3) and (4) serve as boundary conditions for the renormalization group equations at the messenger scale, M. They give rise to rather generic predictions for the soft supersymmetry-breaking gaugino and scalar masses. In contrast, the boundary conditions for B(M) and µ(M) are more model-dependent. 1 New interactions are necessary to induce these terms because they violate a Peccei-Quinn symmetry in the effective action [8,9]. The new interactions can give rise to additional contributions to the scalar masses beyond those in (4). In particular, they can give additional contributions to the Higgs masses, m 2 H 1 and m 2 H 2 . In this paper, we will not commit ourselves to a particular model for B(M) and µ(M). Instead, we will take a more phenomenological approach and treat them as free parameters. We will, however, assume that the soft Higgs masses m 2 H 1 and m 2 H 2 are given by eq. (4). We will also require that electroweak symmetry be radiatively broken.
Our approach is as follows. We start at the messenger scale, M, and fix the boundary conditions (3)   (We take the top-and bottom-quark pole masses to be m t = 175 GeV, m b = 4.9 GeV, and the MS value of the strong coupling to be α s (M Z ) = 0.118.) We note that there are messenger-scale threshold corrections which we do not take into account, even though they are formally of the same order as the weak-scale threshold corrections. We choose to ignore these corrections because they are model dependent. We will, however, estimate their importance by determining the sensitivity of our results to small changes in the messenger-scale boundary conditions.
With these assumptions, the parameter space of this minimal model is described by tan β, the scale Λ, the messenger scale M, along with n 5 , the effective number of 5 + 5 messenger fields, and the sign of µ. We will first set n 5 = 1 and M = 2Λ. This leaves a two-dimensional parameter space, (Λ, tan β), for each sign of µ.
In Fig. 1 we show our results for B(M)/µ(M) and µ(M) in the (Λ, tan β) plane. We see that consistent electroweak symmetry breaking requires |µ(M)| to be in the range from 150 GeV for small Λ to over 1 TeV for large Λ. We also find that B(M)/µ(M) ranges from near zero, at the largest values of tan β, to > ∼ 500 GeV at small tan β. We do not consider Λ > 300 TeV because of fine tuning considerations.
Various constraints exclude the border regions in Fig. 1 (and the remaining contour plots). The region of small tan β is excluded because the top Yukawa coupling diverges below the unification scale. This is not necessarily fatal, but since we wish to preserve perturbative grand unification, we require λ t (M GUT ) < 3.5. The region of very large tan β is excluded because electroweak symmetry is not broken. Typically, we find m 2 A < 0 in the large tan β region.
The region at small Λ is excluded because of the usual experimental bounds on the masses of supersymmetric particles, most notably mχ+ > 65 GeV. (We use the bounds listed in Ref. [13].) These bounds hold when the LSP is the lightest neutralino and is stable. They also hold in gauge-mediated models when the NLSP isχ 0 1 and it decays outside the detector. In sect. 4 we delineate the regions of the parameter space where this applies. If the NLSP decays inside the detector, the collider bounds are stronger because the decay modes (e.g.χ 0 1χ 0 1 → γγ+ missing energy) have much smaller backgrounds [2].
In Fig. 2 we plot the scale dependence of B(M)/µ(M) and µ(M) to illustrate the importance of the weak-scale threshold corrections. We see that our results, which consistently incorporate the weak-scale thresholds, have significantly less scale dependence than they would if the corrections were ignored. Furthermore, we see that the threshold corrections are especially important at large tan β, where B(M) is small. The largest threshold corrections arise from squark loops, so the threshold corrections are smallest in the vicinity of the squark scale, Mq = √ n 5 Λ/90. Indeed, in Fig. 2 we see that the tree-level result is nearly equal to the full result at the scale Mq/2. If we vary the scale over a reasonable range, e.g. Mq/2 to 2Mq, we find that the weak-scale threshold corrections (between 5% and 15%) are generally larger than the corrections from the two-loop evolution (about 2%), as might be expected because of the relatively small amount of running.
The region B(M) ≃ 0 is of considerable phenomenological interest [9,10]. For instance, models where B(M) ≃ 0 give rise to large tan β without fine tuning [9]. Furthermore, since the A-terms are also small at the scale M, such models have a naturally small neutron electric dipole moment [9]. Indeed, only the small O(αM 2 /π) starting values of A(M) and B(M)/µ(M) can contribute to the neutron EDM. The pieces of A and B which are generated by the renormalization group, A RG and B RG , do not contribute because they are proportional to M 2 and µM 2 , so the invariant CP-violating phases [14] arg(A * RG M 2 ) and arg(B * RG µM 2 ) vanish. The region of B(M) ≃ 0 is associated with the region of large tan β. At tree level, we have which implies that B decreases as tan β increases. From Fig. 1 we find, for n 5 = 1 and M = 2Λ, that B(M) is small (but non-zero) along the excluded region at large tan β.
(Our results should be contrasted with those of ref. [10], which do not include one-loop thresholds. The authors of ref. [10] find B(M) = 0 for tan β ≃ 20 and µ < 0.) Until now we have restricted our attention to the simple case where M/Λ = 2. In principle, M/Λ can be much larger. A large hierarchy M ≫ Λ can arise from a small Yukawa coupling or from loop factors [11]. An upper bound on this splitting can be obtained from the cosmological constraint that the gravitino relic density not overclose the universe. In the usual cosmological scenario, this translates into an upper bound on the gravitino mass of about 1 keV [15]. (Note, however, that larger masses can be accommodated if the gravitino relic density is suppressed by a brief inflationary epoch.) If we assume that the singlet F -term is of the same order of magnitude as the largest F -term in the theory, we have Setting Λ = 30 TeV and λ = 1, we find that mG ≃ 1 keV if M/Λ ≃ 10 4 . Hence we will not consider values of M/Λ larger than 10 4 . Note that for λ < 1 the upper bound on M/Λ is correspondingly reduced. As we increase M for a fixed Λ, there is more renormalization group evolution, so there are larger splittings between the soft Higgs masses. These, in turn, give somewhat larger values of µ(M). We show this in Figs In Fig. 4 we plot the pole mass of the lightest Higgs boson, m h , and of the CP-odd Higgs boson, m A , in the (Λ, tan β) plane, for M/Λ = 2 and 10 4 . We see that m h < ∼ 130 GeV (see, however, Ref. [12]), and that m A is nearly always larger than 200 GeV. For such large values of m A , all three heavy Higgs bosons are nearly degenerate in mass.
The above results depend sensitively on n 5 , the number of 5 + 5 pairs. In Fig. 5 we show, for M = 2Λ, contour plots with n 5 = 2, 3 and 4. The change in the parameter The results presented here have varying degrees of sensitivity to the input parameters. We illustrate this sensitivity by computing the changes in B(M)/µ(M) and µ(M). We first vary the quark masses, m b = 4.9 ± 0.5 GeV and m t = 175 ± 5 GeV, and find a shift ∆|B/µ| < ∼ 10 GeV and |∆µ/µ| < ∼ 7% in the region B(M) ≃ 0. To estimate the sensitivity to messenger-scale threshold corrections, we randomly vary the soft masses at the messenger scale by 5%. In the region B(M) ≃ 0, we find shifts in B(M)/µ(M) of up to 10 GeV, and changes in µ of order 5%.

The spectrum
In this section we find the masses of the supersymmetric particles that arise in the gauge-mediated scheme. We examine how the spectrum depends on the parameter space, and comment on the sensitivity to low-and high-energy threshold corrections. As above, we follow a self-consistent procedure. We start with the masses (3) and (4) at the messenger scale, M. We then use the two-loop renormalization group equations to run these masses to the squark scale, Mq = √ n 5 Λ/90. At that scale we apply the one-loop threshold corrections and impose electroweak symmetry breaking. We then calculate the superpartner masses, and run the gauge couplings back to the messenger scale. We iterate this procedure to determine the consistent one-loop superpartner pole masses. The weak-scale threshold corrections for all the superpartner masses are contained in ref. [13]. Figure 6 shows the spectrum in the canonical case where the messenger scale M is equal to 2Λ. We plot the various masses against the mass of the lightest neutralino, χ 0 1 . The first point to note is that, for mχ0 1 > ∼ 100 GeV, most of the curves are rather flat. This simply reflects the fact that most of the masses, including mχ0 1 , scale with Λ. The only exceptions are the light Higgs, h, whose mass is determined by radiative corrections (the correction to its mass-squared grows like ln Λ), and the light tau slepton, whose mass is significantly affected by left-right mixing.
Because of the large mixing in the tau slepton mass matrix, the light tau slepton mass is especially sensitive to radiative corrections. The one-loop self-energy correction is typically less than O(3%) (see Ref. [13]). However, for large tan β, the large threshold corrections to µ (see Fig. 2) induce a significant shift in the light tau slepton mass. In Fig. 7(a) we plot the light tau slepton mass versus tan β. We show the one-loop and tree-level results, for various choices of the renormalization scale, Q. At large tan β, the corrections can be of order 30%. In Fig. 7(b) we show the scale dependence of the one-loop and tree-level tau slepton masses, as well as the effect of the two-loop renormalization group evolution. The weak-scale threshold corrections significantly reduce the scale dependence of the pole mass. As expected, the two-loop renormalization group effects are small because of the small amount of running.
The tau slepton mass matrix should be contrasted with those of the top and bottom squarks. These matrices can also have substantial mixing, but the mixing does not significantly change the mass eigenvalues. This implies that the masses of the light top and bottom squarks cannot be much less than the masses of the other squarks. Indeed, we find that the mass of the light top squark is typically 10 to 20% less than that of the other squarks, although it can be as much as 35% less at the smallest values of tan β. Similarly, the light bottom squark mass is usually 0 to 10% less than that of the other squarks, but it can be as much as 25% less at the largest values of tan β.
In Fig. 6 we see that for mχ0 1 < ∼ 100 GeV, the curves leave their linear trajectories, especially for tan β = 2 with µ negative. This is because for small mχ0 1 , the mixing in the neutralino mass matrix become increasingly important, so theχ 0 1 mass no longer scales with Λ. This is illustrated in Fig. 8, where we show theχ 0 1 mass as a function of Λ.
The previous spectra were all computed with M = 2Λ. If M is increased with respect to Λ, the masses change in two ways. First, the initial conditions are different because the masses at the scale M depend on α i (M). Second, more running is needed to reach the squark scale, Mq.
These effects almost cancel for the gaugino masses because the gaugino masses obey the same one-loop renormalization group equations as the gauge couplings. They do, however, change the scalar masses. For squarks, the two effects work in opposite directions. It turns out that for n 5 = 1 the change in the boundary conditions is more important, so the squark masses are reduced. For n 5 = 3 the two effects largely cancel. For sleptons, the effects work in the same direction, and give a slight increase in the slepton masses. The shifts in the scalar masses are illustrated in Fig. 9. The small changes indicate that it is safe to ignore any factor of two or three change in the messenger quarks scale with respect to the messenger lepton scale, such as might be induced by renormalization group evolution of the messenger Yukawa coupling λ. Figure 9 also illustrates the importance of the threshold functions f (x) and g(x). In the region 1.03 < ∼ M/Λ < ∼ 2 the threshold functions lead to significant increases in the gaugino masses, and to small decreases in the scalar masses, relative to the case when M/Λ > ∼ 2. The threshold corrections are so important that the light tau slepton becomes the NLSP for M near Λ.
In Fig. 9 we also show the effect of increasing n 5 from 1 to 3. The gaugino masses scale like n 5 , while the scalar masses go like √ n 5 . Because of this fact, for the case of a 10 + 10 pair of messenger fields (n 5 = 3), the light tau slepton is the NLSP over most of the parameter space. We discuss this in more detail in the next section. Note that the squark masses are light enough that they will be produced in the next generation of colliders over most of the parameter space.
As above, we can estimate the sensitivity of the supersymmetric spectrum to messenger-scale thresholds by varying the soft masses at the messenger scale by 5%. We find that most of the pole masses vary by about 5%, except for the light Higgs, which varies by < ∼ 1%, and the top and bottom squarks, which change by up to 10%. The light tau slepton mass varies by up to 15%, except at a few exceptional points, where the variation can be as large as 70%. This large sensitivity in the tau slepton mass occurs at large tan β, in the region where it is potentially the NLSP.
We will conclude this section by contrasting the predictions of the gauge-mediated models with the predictions of the minimal supergravity model. At the unification scale, the inputs to the supergravity model are a universal scalar mass, M 0 , a common gaugino mass, M 1/2 , and a trilinear scalar coupling A 0 . As in the gauge-mediated model, radiative breaking of electroweak symmetry breaking is imposed. In general, the two models predict qualitatively different spectra. The supergravity model includes separate scales for the gauginos and the scalars, so the scalars can be much heavier than the gauginos. The minimal gauge-mediated model has only one scale, so the ratios of the gaugino to the scalar masses are more or less fixed. We illustrate this in Fig. 10(a), where we show the ratio of the (first or second generation) squark mass to the gluino mass in the two models. 2 In the supergravity case, this ratio varies from about 0.9 to 5. For gauge-mediated models, the ratio depends on n 5 . For n 5 = 1, the squarks are 20 to 40% heavier than the gluinos, while for n 5 = 4 the squarks are about 5 to 10% lighter. (These numbers and the bands on the figure ignore the region M < ∼ 2Λ, where the ratio mq/mg falls by 20%.) Note that gauge-mediated models with n 5 = 4 predict the same ratio of squark to gluino mass as the supergravity models with M 0 < ∼ M 1/2 .
In the region where the two models predict the same gluino and squark masses, they do not typically predict the same slepton masses. We illustrate this in Fig. 10(b), where we show a supergravity scatter plot of the ratio of the left-handed to the right-handed selectron mass, versus M 0 /M 1/2 . On the scatter plot we superimpose the prediction of the gauge-mediated model, for n 5 = 4, divided up into regions of small and large right-handed selectron mass (mẽ R < M Z and mẽ R > M Z ). In the small mẽ R region, the D-term contributions to the slepton masses are important. The D-term contributions can make the selectron ratios coincide with those from the supergravity model in the region M 0 ≪ M 1/2 if tan β is less than 3 in supergravity model and larger than 5 in the gauge-mediated model. If we go even further, we find that the two models can predict the same gluino, squark, charged slepton and light chargino masses. We give an example in Table 1. This table lists the spectra for the supergravity model parameters tan β = 2.1, M 0 = 5 GeV, M 1/2 = 145 GeV, A 0 = 686 GeV, and µ < 0, and for the gauge-mediated model parameters tan β = 14, Λ = 11 TeV, M/Λ = 3716, n 5 = 4 and µ < 0. The overlap can occur only for small selectron andχ 0 1 masses, where the slepton D-terms and the gaugino/Higgsino mixing come into play. Table 1 also illustrates that the models differ in their predictions for other observables. If we scan over the parameter spaces of the two models and match the gluino masses to 8%, and the light chargino and (first and second generation) squark and selectron masses to 2%, we find that the two models predict different values of tan β, as well as significantly different masses for the neutralinos, the sneutrinos, the heavy chargino and the light Higgs boson. They also predict different values for the Higgs parameters µ and m A , and, typically, different masses for the third-generation squarks and sleptons.
We illustrate these differences in Fig. 11, where we show scatter plots of the values of tan β, m h , m A , and mχ+ 2 in the two models. We see that the models are clearly distinguished. Note that since both models have essentially one scale, there is a strong correlation between the heavy Higgs and Higgsino masses. (The two scales are Λ and M 1/2 , since M 0 ≃ 0.) We conclude that it is possible, but not probable, that measurements of the gluino, squark, slepton and light chargino masses may not be enough to discriminate the gauge-mediated from the supergravity model. However, additional measurements would easily rule out one (or both) of the models.

Properties of the NLSP
In the preceding section, we saw that the NLSP is either the light tau slepton,τ 1 , or the lightest neutralino,χ 0 1 . Over most of the parameter space, the number of 5 + 5 pairs determines the NLSP. For n 5 = 1, theχ 0 1 tends to be the NLSP. For larger n 5 , the balance tips towards theτ 1 because as n 5 increases, the scalar masses increase less than their gaugino counterparts.
In Fig. 12 we plot contours of mχ0 1 = mτ 1 in the (mχ0 1 , tan β) plane, for each value of n 5 between 1 and 4. Each subfigure contains four contours, two for each sign of µ, and two for M/Λ = 2 and 10 4 . We see that for n 5 = 1, the neutralino is the NLSP, except for small regions at large tan β > ∼ 25 and small mχ0 1 . For n 5 = 2 and M/Λ = 10 4 , χ 0 1 is the NLSP for tan β < ∼ 20 − 30. For n 5 = 2 and M/Λ = 2, the light tau slepton is the NLSP over most of the parameter space, as it is for n 5 = 3 or 4 and arbitrary M/Λ. For n 5 > 2, the neutralino is the NLSP at very small mχ0 1 < ∼ 100 GeV. These results are subject to significant uncertainties from the unknown messengerscale thresholds. In particular, the line where mχ0 1 = mτ 1 is sensitive to potentially large one-loop corrections to the mass of the light tau slepton. We find that 5% variations in the messenger-scale boundary conditions give rise to a mχ0 1 − mτ 1 mass difference of up to 30 GeV in the region where the difference is less than 100 GeV.
The collider phenomenology in gauge-mediated models depends crucially on the nature of the NLSP, especially when it decays inside the detector. The lifetime of the NLSP depends on its mass, m, and on the ratio M/Λ, as follows, For the case at hand, where the NLSP is either the B-ino or theτ 1 , we can put in the For βγ = E 2 /m 2 − 1 ≃ 1, this is the approximate decay length. We see that for βγ ≃ 1, λ ≃ 1 and M/Λ < ∼ 100, the NLSP will decay inside the detector. However, for sufficiently small λ it will decay outside the detector. In Fig. 13 we illustrate the neutralino lifetime in different regions of parameter space. In Fig. 13(a) we plot the lifetime versus the neutralino mass, for M/Λ = 2 and 100, with n 5 = 1, λ = 1 and tan β = 2. In Fig. 13(b) we show a scatter plot of the neutralino lifetime versus M/Λ. The lifetime increases by eight orders of magnitude as we increase M/Λ from 1 to 10 4 . In this and the next scatter plot, we vary the parameters n 5 from 1 to 4, Λ from 16 TeV/n 5 to 300 TeV/ √ n 5 , M/Λ from 1.03 to 10 4 , and tan β from 1.2 to 70. The latter three variables are sampled on logarithmic measures, subject to the phenomenological constraints discussed in sect. 2.
If the tau slepton is the NLSP, it will decay to a τ lepton and a gravitino with a branching fraction that is essentially 100%. The lightest neutralino has many more decay modes because it has Higgsino and gaugino components. Theχ 0 1 can decay to either a Higgs or a gauge boson, plus a gravitino,G. In Fig. 14 we show the Higgsino and photino components of theχ 0 1 . In Figs. 14(a) and 14(b) we set n 5 = 1 and see that the NLSP is approximately 60 to 90% (40% to 70%) photino for µ > 0 (µ < 0  is larger, because |µ| is reduced relative to the gaugino masses. In Fig. 14(c) we see that for n 5 = 4 and µ > 0, the Higgsino component of the NLSP can be large. In fact, it can be as large as 50%. Theχ 0 i partial widths can be readily computed. We express them in units of We first write the well-known decay rate to a photon and the gravitino, For the remaining decay modes, we give the three body decay formula to account for the decay to standard model fermions via real or virtual boson exchange. If we sum over final-state standard-model fermions, we find where the sum f is over all fermions with mass less than mχ0 i /2, and N f c is the number of colors: 3 for quarks and 1 for leptons. In the virtual photon contribution, s f min is a detector dependent cut-off on the fermion pair invariant mass. Also, we define In writing the H-h interference contribution, we have neglected all Yukawa couplings except those of the the bottom and tau. The γ-Z (H-h) interference terms go to zero in the limit mχ0 , H, A), the decay rate A −1 Γ(χ 0 i →Gϕ) = κ iϕ I 1 (ϕ). In these expressions, the integrals I n are given by small ǫ ϕ (i.e. in the narrow width approximation). The integral I 10 (h, H) is Our results complete the formulae for the two-body decay rates of the χ 0 i given in ref. [3]. Our results include the contribution of the virtual photon and the contributions from Z-γ and H-h interference. Note that the formula for Γ(χ 0 i →G+ Higgs boson) contains the function I 1 , while the formula for Γ(χ 0 i →G+ longitudinal Z boson) contains the function I 0 . At first sight one might think that this violates the electroweak equivalence theorem, since the decay rate to the longitudinal Z is not equal to the decay rate to the Goldstone boson. However, the equivalence principle does indeed hold in the applicable regime, since the two functions approach each other for mχ0 i ≫ M Z . In Fig. 15(a) we show the branching fractions of theχ 0 1 versus mχ0 1 , for tan β = 2, n 5 = 1, and M = 2Λ. (We plot the magnitude of the Z-γ interference contribution, |Γ Zγ |/Γ tot , and we take into account 5 quark flavors.) The decay to the photon dominates, as expected, since theχ 0 1 has a large photino component. The branching fraction to fermions via an off-shell photon varies from about 3 to 4% for s f min = 1 GeV. In the parameter space associated with the minimal models, the branching ratio to fermion pairs via heavy Higgs boson exchange is negligible. For the parameters corresponding to Fig. 15, the heavy Higgs exchange branching ratio is of order 10 −12 . The branching ratio associated with H-h interference and the branching fraction associated Because of the kinematical suppression, the branching fraction to the Z-boson rises slowly above threshold, to about 15% (20%) for mχ0 1 = 2M Z (3M Z ). We illustrate this in Fig. 15(b), where we show a scatter plot of the branching fraction of the lightest neutralino to a gravitino and a fermion pair. We see that the branching ratio depends only on the neutralino mass. This would not be true if the part of parameter space in which theχ 0 1 has a large Higgsino component (n 5 = 4, µ > 0,small Λ, see Fig. 14(c)) led to a sizable branching ratio to the Higgs boson. However, in this region of parameter space either theτ 1 is the NLSP or theχ 0 1 mass is below the h-boson threshold. Hence, the associated partial width is negligible.
As discussed above, the mechanism that generates B(M) and µ(M) might result in extra terms to the scalar masses beyond those in eq. (4). In this case, the NLSP can decay predominately into a Higgs boson. To study this possibility, we take µ and B to be independent low-energy parameters, and focus our attention on the case of small µ, so the lightest neutralino is predominantly Higgsino. In Fig. 16 we plot the lifetime and branching fractions of theχ 0 1 as a function of mχ0 1 , for various choices of parameters. Note that in the region mχ0 1 < ∼ 250 GeV the lightest neutralino is largely Higgsino, and mχ0 1 ≃ |µ|. The lifetime and branching fractions of the NLSP in the Higgsino region are shown in Fig. 16. In Fig. 16(a) we see that the lifetime varies by 6 or 7 orders of magnitude as mχ0 1 varies over less than one order of magnitude. This can be contrasted with the case where the neutralino is dominantly B-ino, where the lifetime varies by less than one order of magnitude (see eq. (9) and Fig. 13). The branching fractions of the lightest neutralino are shown in Figs. 16(b-d).
These results can be understood as follows. In the small µ region the NLSP is primarily Higgsino, so it prefers to decay to a Higgs boson. Below h and Z threshold, however, it is forced to decay through its suppressed photino component. This accounts for the long lifetime and the large photino branching fraction for small mχ0 1 . As the mass increases, the h and Z channels open, so the lifetime decreases and the branching fractions are determined by κ 1Z L and κ 1h . For small tan β, the decay to h (Z) dominates for µ positive (negative). For large tan β, κ 1Z L ≃ κ 1h for each sign of µ. In this case the branching fraction to Z is larger because the Higgs is heavier, so I 0 (Z) > I 1 (h). Finally, as mχ0 1 increases still further, the decay to photons again dominates, because in this region, the NLSP is predominately photino. For mχ0 1 ≃ 300 GeV, the decay length approaches 10 −5 meters.
The neutralino lifetime in the Higgsino region can be readily scaled for other values of n 5 and M/Λ. As before, for fixed mχ0 1 , cτ scales like (M/Λ) 2 . For fixed M/Λ and fixed gaugino masses, it scales like 1/n 4 5 because τ ∝ F 2 S /µ 5 ∝ Λ 4 /µ 5 . These results should be contrasted with those of the minimal models, where we found that, above h threshold, the branching ratio to the h varies from 10 −8 to 10 −4 . Therefore collider events with four b-jets and missing energy would imply a non-minimal Higgs sector in these models.

Conclusions
In this paper we examined the detailed low-energy spectrum of gauge-mediated supersymmetry breaking models. We used two-loop renormalization group equations for the gauge and Yukawa couplings, and for the soft supersymmetry breaking parameters. We imposed consistent one-loop radiative electroweak symmetry breaking under the assumption that the mechanism that generates the µ and B parameters does not induce any extra contributions to the scalar masses.
We examined the phenomenology in the case of an arbitrary number of 5 + 5 messenger fields, and in the case that the messenger scale M is greater than Λ. We began by examining electroweak symmetry breaking. We initially considered the minimal case, n 5 = 1 and M = 2Λ. In the parameter space we considered, with Λ < 300 TeV, we found that |µ(M)| varies from 150 GeV to over 1 TeV. We also found that |B(M)/µ(M)| varies from near (but not equal to) zero to about 500 GeV. We found the phenomenologically interesting region B(M) ≃ 0 to be compatible with electroweak symmetry breaking in the region of large tan β, for either sign of µ. For larger M/Λ, we found B(M) ≃ 0 requires µ < 0.
We examined the spectrum and illustrated how it depends on n 5 and M/Λ. We found that the spectrum is not qualitatively affected by an increase in M/Λ. For n 5 > 1, we found that the light tau slepton is the NLSP over most of the parameter space. It decays to a tau lepton and a gravitino.
By varying the boundary conditions of the soft parameters at the messenger scale by 5%, we determined the sensitivity of the supersymmetric spectrum to higher-order messenger-sector corrections. We found that in the region B(M) ≃ 0, B(M) varies by 10 GeV. Furthermore, the spectrum varies by 5%, except for the light Higgs mass, which is essentially unchanged, and the third-generation squark and slepton masses. In the most extreme case, we found a 70% variation in the light tau slepton mass, which implies a substantial uncertainty in the identification of the NLSP.
We compared the predictions for the spectra in the gauge-mediated and supergravityinspired models. In general, the two models predict qualitatively different spectra. We found that it is possible for the two models to give the same gluino, light chargino, and (first and second generation) squark and slepton masses. Such a match is only possible for very light masses, where D-terms and gaugino/Higgsino mixing is important. In this region, the models can be distinguished by their predictions for other observables, such as tan β, as well as the sneutrino, neutralino, heavy chargino, and Higgs boson masses.
For n 5 = 1, we found that theχ 0 1 is usually the NLSP. Theχ 0 1 decays to the gravitino and either an (on-or off-shell) gauge or Higgs boson. We derived the neutralino decay rate and branching fractions. The lightest neutralino tends to be gaugino-like, and decays to a photon and a gravitino. However, in the region mχ0 1 > ∼ 100 GeV, the branching fraction to the Z can be larger than 20%. Leptons are easier to track than photons, so this mode has the potential to permit a precise neutralino lifetime measurement. We found that for n 5 = 4 the Higgsino component of theχ 0 1 can be as large as 50%. Nevertheless, over the entire parameter space the branching fraction to the light Higgs boson is less than 10 −4 .
We also examined the branching fraction of the neutralino in the Higgsino region. In this case the branching fraction to the on-shell h-boson can reach over 80% well above threshold. Since this Higgsino region can only occur for models with non-minimal Higgs sectors, the observation of 4 b-jets plus missing energy would be an important step towards understanding the origin of the µ term.
Models with gauge-mediated supersymmetry breaking offer the appealing possibility that the origin of supersymmetry breaking is experimentally accessible. Once supersymmetry is discovered, detailed study of the supersymmetric particles, along the lines suggested here, might well prove to be the first step towards uncovering the mechanism of supersymmetry breaking.