A Cosmological Upper Bound on Superpartner Masses

If some superpartners were in thermal equilibrium in the early universe, and if the lightest superpartner is a cosmologically stable gravitino, then there is a powerful upper bound on the scale of the superpartner masses. Typically the bound is below tens of TeV, often much lower, and has similar parametrics to the WIMP miracle.

where T eq 1.5 eV is the temperature of matterradiation equality, M Pl 2.4 × 10 18 GeV is the re-duced Planck mass, and x f ism divided by the freezeout temperature. The coupling strength α eff , appearing in the thermally averaged LSP annihilation cross section at freeze-out, is defined by σv = 4πα 2 eff /m 2 . It is 0.03 (0.01) for wino (Higgsino) LSP, but can have a much larger variation.
The TeV scale from freeze-out thus results parametrically as the geometric mean of T eq and M P l and is independent of the weak scale. The equality holds for LSP dark matter. Eq. (1) is a very important result: in models where superpartner masses are characterized by a single scale,m is likely in the 1-10 TeV window, and in Split Supersymmetry [7] the fermonic superpartners lie in the TeV region.
From the above list of assumptions it is clear how to evade the bound in Eq. (1), allowingm many orders of magnitude above the TeV scale. In particular, violating assumption (i) through, e.g., R parity breaking, may void the bound entirely. In that case, DM could arise, for instance, from a hidden sector or from axions. Violating assumption (ii), havingm well above T R , allows the superpartners to have no cosmological role, hence evading the bound. Finally, assumption (iii) may not hold if additional late-decaying states reheated the universe.
In this paper we study the intriguing possibility of violating assumption (iv-A). Indeed, there are numerous scenarios where DM is only very weakly coupled so that its abundance does not follow from thermal freeze-out, invalidating Eq. (1).
The most common scenario of this kind has (iv-A) replaced by (iv-B). The gravitino is the LSP (and the Lightest Observable-sector SuperPartner (LOSP) decays predominantly to gravitinos).
The gravitino, present in all supersymmetric theories, has interactions that are highly constrained and very weak. The gravitino has a cosmological abundance determined by thermal scattering, freeze-in, and freeze-out and decay, and reaches thermal equilibrium, in accordance with (iv-A), only when it is very light. The gravitino abundance has been studied in detail for the case of weakscale superpartners, for example leading to bounds on T R as a function of the gravitino mass [8]. In this let- . Even when TR is as low asm, gravitinos provide too much dark matter in the red region, which has borders labelled by the relevant process Th, FI or FO. As TR is increased the overclosed region becomes larger, as illustrated by the dashed blue lines, because UV scattering at TR produces more gravitinos than freeze-in. At the edge of the red region (suitably enlarged for TR >m) gravitinos provide the observed dark matter. In the region to the right of the slanted black dashed line the gravitino is not the LSP; this is the conventional WIMP LSP freeze-out region, with a limit of 2.3 TeV for a wino LSP. The green region is excluded by the effects of late decays of LOSPs to gravitinos during big bang nucleosynthesis (BBN) [18]; light green shading corresponds to a neutral LOSP with 100% hadronic BR, and dark green shading to a neutral LOSP with 1% hadronic BR and 99% electromagnetic BR. The BBN limits when the gravitino is not the LSP are model dependent and are not shown [19]. The purple region next to the "Th" contour is excluded as the gravitino component of dark matter is too warm [20]. The gray shading (and corresponding gray dashed and dotted lines) shows the regions with g 2 susy > 10 (3, 1), which are excluded as described in the text.
ter, however, we take a different approach and derive the cosmological bound on the superpartner mass scale for a gravitino LSP. We find this bound to be strong, so that under the quite mild assumptions of (i), (ii), (iii) and (iv-A) or (iv-B), supersymmetry, if it exists, must be in the (multi-) TeV domain. We also derive bounds for the split spectrum case and scenarios where the LOSP does not predominantly decay to the gravitino. SINGLE SCALE SUSY. In this section we take all superpartners of the observable sector to be characterized by a single mass scale,m, and leave the case of a nondegenerate spectrum to the next sections. Our aim is to derive a general bound on the scalem from overproduction of gravitinos. We ignore other possible components to DM since they would only strengthen the bound. A key superpartner is the LOSP, since it undergoes freezeout. We allow a very wide variation in the (m 3/2 ,m, T R ) space.
The upper bound onm follows from the three assumptions (i), (ii) and (iii). Assumption (ii) implies that the observable sector produces gravitinos from three sources: gaugino scattering at T R [9][10][11], Y U V 3/2 , gravitino "freeze-in" from decays of visible sector superpartners at T ∼m [12,13], Y F I 3/2 , and LOSP freeze-out and decay [14], Y F O 3/2 . For sufficiently small m 3/2 , the gravitinos are in thermal equilibrium when T =m; in this case Y U V 3/2 + Y F I 3/2 are replaced by a thermal abundance, and Y F O 3/2 may be neglected. Below, in accordance with assumption (iv-B), we assume the LOSP branching ratio to the gravitino is O(1). In the final section we discuss how our bound is weakened when this assumption is relaxed. Gravitinos may also be produced from other sectors or they may arise from an initial condition [15,16]. However, these additional sources of gravitinos only strengthen our bound, and to be conservative we ignore them.
If gravitinos do not thermalize, the condition that they not yield too large a DM abundance is where a = 0.27 and α eff is now the coupling relevant for LOSP annihilation. The three terms labelled UV, FI and FO correspond to scattering at T R , freeze-in and freezeout and decay and occur with rate constants C U V = The bound onm in the single-scale SUSY case, for α eff = 0.03, 10 −2 and 10 −3 in blue, green and purple respectively, assuming TR =m. As α eff decreases freeze-out yields a larger abundance, so the FO boundary and the BBN constraints (shown shaded in the corresponding colors) both become more stringent. As TR is raised, the bounds become more stringent as indicated by the blue dashed lines of Fig. 1. Right: The bound onm when the contribution to the gravitino abundance from freeze-out and decay is negligible. This may be the case in several scenarios, as discussed in the last section.
The dashed blue lines demonstrate the strengthening of the bound as TR is increased. We do not analyze the region with m < m 3/2 as the results are model-dependent. 0.13 x f 23 . Here γ 3 0.36 is related to the thermal corrections of the scattering process [11], g * = 228.75, and n F I counts the number of fermions and complex scalars participating in the freeze-in with massm; with degenerate MSSM sparticles, n F I = 36+9+12+4 = 61. The equality in Eq. (2) corresponds to the case that these processes yield the observed DM abundance. If gravitinos do thermalize, the overabundance constraint becomes [17] with Here g * s g * = 228.75. The resulting bound onm as a function of m 3/2 is shown in Fig. 1 for α eff = 0.03, relevant for a (perturbative) wino LOSP. We do not include the nonperturbative Sommerfeld effect [21], which results in an O(1) shift in α eff . When gravitinos are not thermalized, the key point is the differing dependences of the three terms in Eq. (2) oñ m and m 3/2 . While all three terms have a positive power ofm, the UV and FI terms are proportional to 1/m 3/2 while the FO term is proportional to m 3/2 , leading to contours in Fig. 1 with slopes of opposite signs. Hence there is an upper bound, where 38 TeV for T R =m. Decreasing α eff makes the FO term larger, as shown in the left panel of Fig. 2 for T R =m. The parametrics of Eq. (4) is similar, but not identical, to that in the so-called "WIMP Miracle", Eq. (1).
A second allowed region occurs at very low m 3/2 in Fig. 1, where the gravitinos are thermalized for any T R ≥m. Here the bound onm arises from theory rather than cosmology:m ≤ (g susy /4π) 2 √ F , where g susy is the strength of the coupling between obervable and supersymmetry breaking sectors, and F = √ 3m 3/2 M Pl is the supersymmetry breaking scale. The bound results when the messenger scale takes its minimal value of √ F , and is shown in Fig. 1 for g 2 susy = 1, 3 and 10. We note that it may be possible to construct realistic models of composite quarks and leptons having non-perturbative couplings, g susy ∼ 4π [22]. NON-DEGENERATE SPECTRUM. The completely degenerate spectrum discussed above is special because non-degeneracies typically arise from renormalization group effects or the dynamics of the mediation of supersymmetry breaking. How do non-degeneracies affect the above bounds?
Non-degeneracies induce independent changes in the three gravitino production mechanisms. The freeze-in process is dominated by the heaviest superpartners,m + , 10 2 and is suppressed compared to the degenerate case by n + F I /n F I , where n + F I is the number of these heavy superpartners. The scattering process, dominated by gluino scattering, is proportional to the square of the gluino mass, M 2 3 . Finally, the freeze-out abundance is proportional to the LOSP mass,m − , with σv = 4πα 2 eff /m 2 − , so that Eq. (2) becomes (5) While pure FO of Eq. (1) bounds m LSP , with a gravitino LSP the bound depends on the m LOSP , M 3 , and the mass dominating FI.
As a simple example, on the left of Fig. 3 we show the bound that results by taking all colored states atm c = m + and all non-colored states atm nc =m − , assuming all superpartners are reheated. As can be seen, the bound onm nc becomes much more stringent asm c is raised, being reduced to 7 TeV form c /m nc = 10. Much of the allowed regions in Figs. 1, 2-Left and 3-Left are within the LHC reach. SPLIT SUSY. In the split-SUSY scenario [7], where the scalar superpartner mass,m s , becomes much larger than the fermionic superpartner mass,m f , a bound onm f , with a gravitino LSP, was discussed in [23]. The freezein process dominates over the scattering process as long as T R >m s [23,24]. Using Eq. (5), withm s =m + andm f =m − , yields the bound onm f shown in the center panel of Fig. 3 for various values ofm s /m f . To compute the bound, the split-SUSY 1-loop RGEs were used [25,26]. The bound onm s is in the region of 100 TeV, as shown in the right panel of Fig. 3, and hence arbitrary flavor and CP violation in the squark mass ma-trix requires T R <m s . Finally, we note that if T R is indeed belowm s a bound onm f may still be obtained, and is similar to that shown in Fig. 1 up to O(1) corrections stemming from the absence of some diagrams in the finite-temperature thermal production of the gravitinos [9].
The non-degeneracies explored in the left and center panels of Fig. 3 lead to similar bounds, and forbid large splittings between the light and heavy states (assuming that both are reheated). Indeed, as the splittings increase, the BBN bounds rapidly become very constraining. RELAXING ASSUMPTION (iv-B). We now consider how the bound on superparticle masses is relaxed in theories that violate assumption (iv-B).
LOSP freeze-out and decay may not produce a significant yield of LSP gravitinos, depleting Y F O 3/2 . This occurs, for example, if the LOSP dominantly decays through R-parity violating (RPV) operators, which can still be consistent with gravitino DM for sufficiently small RPV [27,28]. Alternatively, the LOSP may dominantly decay to a light hidden sector, which, if thermalized, may not produce significant gravitinos due to its lighter mass scale. A third possibility is that the LOSP is colored, in which case a late annihilation stage, after the QCD phase transition, can dilute the abundance of R-hadrons [29,30] before the LOSP decays to gravitinos. In these cases, a bound onm results from dropping the FO term and is shown on the right of Fig. 2. The maximalm occurs at m 3/2 =m, when Eq. (2) gives The numerical value above was obtained for T R =m. For larger reheat temperatures the bound is stronger. A more drastic possibility is to consider an LSP that violates both assumptions (iv-A,B) entirely, i.e. a state that is not the gravitino and yet interacts with the observable sector so weakly that it remains out of equilibrium. An example is a light, weakly coupled singlino. In this case, the bound can be completely removed as the singlino couplings can be chosen to be arbitrarily small (removing scattering and freeze-in production) simultaneously with a vanishing mass (thereby removing freezeout and decay), allowing arbitrarily heavy superpartners.
The key characteristic about the gravitino that leads to our bound is that its mass is inversely related to its coupling to observable states, so that the mass and coupling cannot simultaneously be taken too small. GRAVITY MEDIATION. When mediation of supersymmetry breaking occurs at a very high fundamental scale, M * , of order the scale of gauge coupling unification or higher, then m 3/2 /m ∼ M * /M P l ∼ 10 −3 − 1. Thus "gravity mediation" typically has a gravitino LSP and selects a small region of Fig. 1 that is within a few orders of magnitude of the m 3/2 =m dashed line. Part of this region, with M * near M P l , is typically highly constrained by BBN, but smaller values of M * are of interest and include the largest values ofm.
The details of this gravity-mediated region are highly dependent on the LOSP, the superpartner spectrum and T R . In Fig. 4 we show a particular example: a sneutrino LOSP withm c /m nc = 6. BBN is affected dom-inantly by rare sneutrino decays with a radiated Z or W , so the excluded green region is quite small [19,31], allowing various possibilities. One has a light, e.g. 200 GeV, sneutrino, with M * near M P l and a high T R ∼ 10 9 GeV, compatible with Leptogenesis [32]. In this case the colored superpartners may well be in reach of the LHC. Another possibility has M * further from M P l and a much lower T R so that the sneutrino mass can be near its upper bound of 5 TeV.