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.


Introduction
A natural weak scale, precision gauge coupling unification and dark matter provide powerful arguments for weak-scale supersymmetry. However, to date, direct evidence for supersymmetry is still missing and thus whether or not low-scale supersymmetry is realized in nature remains unknown. In fact, the recent discovery of a 125 GeV Higgs boson [1,2] implies a fine-tuning in the MSSM worse than 1% [3], and searches for supersymmetry at the LHC are placing limits on colored superpartners in the region of 1 TeV [4,5]. Therefore, the superpartner mass scale,m, may be decoupled from the weak scale and could, in principle, be anywhere between the present experimental bound near 1 TeV up to the Planck scale. With the naturalness reasoning aside, the question arises: Are there arguments for superpartners at the TeV scale that are unrelated to the stabilization of the weak scale?
The argument for TeV superpartners from gauge coupling unification alone is weak, as logarithmic running implies that the precision changes only mildly asm increases well above 1 TeV. On the other hand, there is a powerful and well-known argument for TeV-scale superpartners from the cosmological abundance of the lightest supersymmetric particle (LSP) [6]. This results from LSP freeze-out and follows from three assumptions: (i). The LSP is cosmologically stable.
(ii). The reheat temperature of visible particles after inflation, T R , was sufficiently high, T R m.
(iii). There is no substantial late-time dilution of the LSP abundance.
The second assumption implies that the standard model superpartners were in thermal equilibrium. If we further assume JHEP02(2015)094 (iv-A). The LSP reached thermal equilibrium, then the thermal freeze-out relic abundance leads to the overclosure bound on the LSP mass, where T eq 1.5 eV is the temperature of matter-radiation equality, M Pl 2.4 × 10 18 GeV is the reduced Planck mass, and x f ism divided by the freeze-out 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. (1.1) 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. (1.1), 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. (1.1).
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. Additional sources for the gravitino production occurs in non-standard cosmological scenarios [8][9][10][11][12][13][14][15]. The gravitino abundance has been studied in detail for the case of weak-scale superpartners, for example leading to bounds on T R as a function of the gravitino mass [16]. In this letter, 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.  Even when T R 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 T R is increased the overclosed region becomes larger, as illustrated by the dashed blue lines, because UV scattering at T R produces more gravitinos than freeze-in. At the edge of the red region (suitably enlarged for T R >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) [24]; 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 [25]. The purple region next to the "Th" contour is excluded as the gravitino component of dark matter is too warm [26]. 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.

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 non-degenerate 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 freeze-out. We allow a very wide variation in the (m 3/2 ,m, T R ) space.

JHEP02(2015)094
Left: 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 T R =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 T R is raised, the bounds become more stringent as indicated by the blue dashed lines of figure 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 T R is increased. We do not analyze the region withm < m 3/2 as the results are model-dependent.
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 [17][18][19], Y U V 3/2 , gravitino "freeze-in" from decays of visible sector superpartners at T ∼m [20,21], Y F I 3/2 , and LOSP freeze-out and decay [22], 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 [8,9]. 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 freeze-out and decay and occur with rate constants

JHEP02(2015)094
0.13 is related to the thermal corrections of the scattering process [19], 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. (2.1) corresponds to the case that these processes yield the observed DM abundance. If gravitinos do thermalize, the overabundance constraint becomes [23] Here g * s g * = 228.75. The resulting bound onm as a function of m 3/2 is shown in figure 1 for α eff = 0.03, relevant for a (perturbative) wino LOSP. We do not include the non-perturbative Sommerfeld effect [27], 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. (2.1) onm 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 figure 1 with slopes of opposite signs. Hence there is an upper bound, A second allowed region occurs at very low m 3/2 in figure 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 figure 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 [28].
We recall that it is also possible to derive an upper bound on the superpartner mass scale when the gravitino is not the LSP,m < m 3/2 , as shown to the right of the black dashed line of figure 1. The condition that the LSP not yield too large of an abundance is where the first term corresponds to the UV production of gravitinos which subsequently decay to the LSP and the second term is the usual LSP abundance from freeze-out. The two terms in parentheses capture production of transverse and longitudinal gravitinos, respectively. In the limit m 3/2 m, transverse production of gravitinos dominates and the JHEP02(2015)094 gravitino mass drops out of the LSP abundance. For sufficiently low reheat temperatures, the usual overclosure constraint from freeze-out applies, as in eq. (1.1). For T R 10 5m , the LSP abundance is dominated by gravitino decays, strengthening the bound, as shown by the dashed blue lines in figure 1.

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 + , 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. (2.1) becomes

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 [29]. The freeze-in process dominates over the scattering process as long as T R >m s [29,30]. Using eq. (3.1), withm s =m + andm f =m − , yields the bound oñ m f shown in the center panel of figure 3 for various values ofm s /m f . To compute the bound, the split-SUSY 1-loop RGEs were used [31,32]. The bound onm s is in the region of 100 TeV, as shown in the right panel of figure 3, and hence arbitrary flavor and CP violation in the squark mass matrix 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 figure 1 up to O(1) corrections stemming from the absence of some diagrams in the finite-temperature thermal production of the gravitinos [17].
The non-degeneracies explored in the left and center panels of figure 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.

JHEP02(2015)094 5 Natural SUSY
Another motivated possibility for a non-degenerate spectrum is the "Natural SUSY" scenario [33][34][35][36]. The superpartners whose masses are most constrained by naturalness are assumed to be light (the Higgsino, stops, left-handed sbottom, and gluino), with masses below ∼ 1 TeV, while the superpartners whose masses are less constrained by naturalness are allowed to be heavy, alleviating LHC constraints. The superpartner masses that are least constrained by naturalness are the sfermions of the first two generations, right handed sbottom, and staus. In order to avoid a fine-tuning worse than ∼ 10%, these sfermions should be lighter than ∼ 3−5 TeV in models where SUSY breaking generates a hypercharge D-term, and ∼ 10 − 20 TeV in models where the hypercharge D-term vanishes (in which case the sfermion masses dominantly correct the Higgs potential at 2-loop order). When the sfermion masses get heavy, freeze-in production of gravitinos is enhanced, analogous to the split-SUSY scenario discussed above.
In figure 4 we show the bound on a simplified spectrum, motivated by Natural SUSY, where the sfermions of the first two generations, right handed sbottom, and staus have a common mass,m H , which is allowed to be heavier than the common mass of the remaining superpartners,m L . Note that for this figure, we have assumed the field content of the MSSM. In order to explain the observed Higgs mass, m h ≈ 125 GeV, the MSSM requires a fine tuning of 1% or worse, while extensions with extra contributions to the Higgs quartic, such as the NMSSM, remain more natural [3]. Extra fields, beyond the MSSM, enhance gravitino production from freeze-in and scattering, and therefore figure 4 can be conservatively applied to theories where extra states couple to the Higgs. 6 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 [37,38]. 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 [39,40] 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 figure 2. The maximalm occurs at m 3/2 =m, when eq. (2.1) 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 JHEP02(2015)094 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 freeze-out 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 figure 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 figure 5 we show a particular example: a sneutrino LOSP withm c /m nc = 6. BBN is affected dominantly by rare sneutrino decays with a radiated Z or W , so the excluded green region is quite small [25,41], 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 [42]. In this case the colored superpartners JHEP02(2015)094 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.