Impact of the Bounds on the Direct Search for Neutralino Dark Matter on Naturalness

In the Minimal Supersymmetric Extension of the Standard Model (MSSM) the higgsino mass parameter $\mu$ appears both in the masses of the Higgs bosons and in the neutralino mass matrix. Electroweak finetuning therefore prefers small values of $|\mu|$. On the other hand, bino--like neutralinos make a good dark matter candidate. We show that current direct search limits then impose a strong lower bound on $|\mu|$, in particular for $\mu>0$ or if the masses of the heavy Higgs bosons of the MSSM are near their current limit from LHC searches. There is therefore some tension between finetuning and neutralino dark matter in the MSSM. We also provide simple analytical expressions which in most cases closely reproduce the numerical results.


Introduction
Softly broken supersymmetry alleviates the electroweak hierarchy problem [1,2]. In the limit of exact supersymmetry there are no quadratically divergent radiative corrections to the masses of Higgs bosons. Increasing lower bounds of the masses of superparticles from searches at the LHC [3] imply sizable loop corrections to the masses of Higgs bosons in realistic supersymmetric models. This leads to finetuning if soft breaking masses are treated as uncorrelated [4,5]. However, this source of finetuning might be greatly reduced in models with a small number of independent soft breaking parameters, typically defined at a very large renormalization scale, which introduce correlations between weak-scale parameters [6].
On the other hand, the minimal supersymmetric extension of the standard model (MSSM) [7,8] also requires a supersymmetric contributions to Higgs and higgsino masses; this mass parameter is usually called µ. Searches for charginos imply [3] |µ| > ∼ 100 GeV. Since µ enters the Higgs potential already at tree-level, a value of |µ| much above M Z inevitably leads to finetuning. There is thus general agreement in the discussion of finetuning issues in the MSSM that -given the experimental lower bound -smaller values of |µ| are preferred, with the necessary degree of finetuning increasing like µ 2 .
The same parameter µ also sets the mass scale for the higgsinos in the MSSM; 1 as already noted, this is the origin of the lower bound on |µ|. This establishes a connection between finetuning and the phenomenology of the neutralinos and charginos in the MSSM.
One of the attractive features of the MSSM with exact R−parity is that it automatically contains a candidate particle to form the cosmological dark matter whose existence can be inferred from a host of observations, assuming only that Einsteinian (or indeed Newtonian) gravity is applicable at the length scales of (clusters of) galaxies [10]. This candidate is the lightest neutralinoχ 0 1 [11], whose mass is bounded from above by |µ|. Given that naturalness arguments prefer a small value of |µ|, one might assume that the most natural dark matter candidate in the MSSM is a light higgsino-like neutralino. However, in minimal cosmology a higgsino-like neutralino has the correct relic density only for |µ| 1.2 TeV [12], which would lead to permille-level electroweak finetuning. A wino-like LSP would have to be even heavier.
One can appeal to non-standard cosmologies, e.g. including non-thermal production mechanisms, in order to give a lighter higgsino-light neutralino the correct relic density; however, such scenarios are already quite strongly constrained by indirect dark matter searches [13].
In this article we therefore assume that the bino mass parameter M 1 < |µ|, so that the LSP eigenstate is dominated by the bino component; M 1 can be taken positive without loss of generality. Also in this case finetuning would prefer |µ| to be not far above M 1 . On the other hand, if |µ| M 1 the LSP has sizable higgsino and bino components, and hence generically sizable couplings to the neutral Higgs bosons of the MSSM. Such mixed neutralinos tend to have rather large scattering cross sections on matter, in potential conflict with strong lower bounds on this quantity from direct dark matter searches [3]. This is true in particular for the so-called "well tempered neutralino" [14], a bino-higgsino mixture with the correct relic density in minimal cosmology.
In this article we explore this connection quantitatively, by deriving a lower bound on the difference |µ| − M 1 from the upper bound on the neutralino-nucleon scattering cross section found by the Xenon collaboration [15]. We do this both numerically, and using a simple approximation for the bino-like neutralino eigenstate which is very accurate in the relevant region of parameter space. The resulting lower bound on |µ| is much stronger than the trivial constraint |µ| > M 1 which follows from the requirement of a bino-like LSP; this is true in particular if M 1 and µ have the same sign. An upper bound on |µ| from finetuning considerations therefore leads to a considerably stronger upper bound on M 1 from direct dark matter searches.
A bino-like neutralino will often have too large a relic density in minimal cosmology. This can be cured either by assuming non-standard cosmology (e.g. a period of late entropy production) [16,17], or -for not too small M 1 -by arranging for co-annihilation with a charged superparticle, e.g. aτ slepton [18], which can still have escaped detection by collider experiments if it is close in mass to the lightest neutralino. Neither of these modifications changes the cross section for neutralino-proton scattering significantly. By not imposing any relic density constraint our result thus becomes less model-dependent. This, as well as the use of the more recent, and considerably stronger, Xenon-1T constraint and the approximate analytical derivation of the constraint on |µ|, distinguishes our analysis from that of ref. [19].
The rest of this article is organized as follows. In the following section we review neutralino mixing, both exact and using a simple approximation. We also give the relevant expressions for the neutralino-nucleon scattering cross section. In section 3 we present the resulting lower bound on the difference |µ| − M 1 as a function of M 1 , before concluding in section 4.

Formalism
In this section we briefly review the neutralino masses and mixings in the MSSM, as well as the spin-independent contribution to neutralino-nucleon scattering.

The Neutralinos in the MSSM
The neutralinos are mixtures of the two neutral gauginos (the binoB and the neutral wino W 3 ) and two neutral higgsinosh 0 d ,h 0 u associated with the two SU (2) Higgs doublets required in the MSSM. The resulting mass matrix in theB, W 3 ,h 0 d ,h 0 u basis is given by [8]: Here M 1 and M 2 are soft breaking masses for the bino and wino, respectively, µ is the higgsino mass parameter, θ W is the weak mixing angle, M Z 91 GeV is the mass of the Z boson, and tan β = H 0 u / H 0 d is the ratio of the vacuum expectation values (VEVs) of the two neutral Higgs fields. The mass matrix is diagonalized by the 4 × 4 matrix N , such that the i−th neutralino eigenstate is given bỹ Here we are interested in scenarios where the lightest neutralinoχ 0 1 , which is our dark matter candidate, is dominated by the bino component. This requires |M 1 | < |M 2 |, |µ|. We will assume that these mass parameters are real; nontrivial complex phases would contribute to CP violation, which is strongly constrained by upper bounds on the electric dipole moments of the electron and neutron [20]. Without loss of generality we take M 1 to be positive, but allow both signs for µ. |M 2 | is constrained significantly by searches for charginos and neutralinos at the LHC [3]; as long as |M 2 | > M 1 the sign of M 2 is essentially irrelevant for our analysis.
Evidently the mixing between the gaugino and higgsino states is controlled by the mass of the Z boson. If the difference between the gaugino masses and |µ| is larger than M Z , all mixing angles will therefore be quite small, allowing for an approximate perturbative diagonalization of the mass matrix (1). In particular, the components of a bino-likeχ 0 1 can then be approximated by [21,22] In Fig. 1 we compare these approximate expression with exact results. Evidently the approximation works very well for µ − M 1 ≥ 2M Z or so. Since we took a very large value of M 2 the wino component essentially vanishes in this example; however, the first eq.(3) shows that it only appears at second order in M Z , and is therefore always much smaller than the higgsino components in the region of interest. In this figure we have chosen µ to be positive. The second and third eq.(3) shows that this increases the higgsino components. As a result, for µ < 0 the approximation (3) becomes very accurate already for |µ| − M 1 ≥ 1.5M Z .

Neutralino-Nucleon Scattering in the MSSM
In the limit of vanishing neutralino velocity only two kinds of interactions contribute to neutralinonucleon scattering. One of them depends on the spin of the target nucleus; this contribution is usually sub-dominant for heavy target nuclei like Xenon or Germanium, which currently yield the tightest constraints for neutralino masses above 10 GeV or so. The spin independent contributions dominate because their contribution to the scattering cross section of heavy nuclei scales like the square of the nucleon number. They originate from the effective Lagrangian [11] L eff which describes the interactions of neutralinos with quarks. Here we have limited ourselves to the leading, dimension-6, operator; strong lower bounds on squark masses [3] imply that dimension-8 operators due to squark exchange [23] can safely be ignored. Squark exchange also contributes at dimension 6. However, this contribution is proportional to the mass of the quark [11]. That is of course also true for Higgs exchange contributions. However, at least the lighter neutral Higgs boson, whose mass we now know to be 125 GeV, lies about an order of magnitude below the current lower bound on first generation squark masses. Squark exchange diagrams therefore contribute only at the 1% level at best, which is well below the uncertainty of the Higgs exchange contribution. We therefore ignore them in our analysis.
However, we do allow for the contribution of the heavier neutral Higgs boson. The effective neutralino-quark coupling f q is thus given by Note that we factored the quark mass out of the Higgs couplings to quarks, making the latter independent of the quark mass; the couplings to up-and down-type quarks still differ, however. They are given by [24]: Here α is the mixing angle between the two neutral Higgs bosons, g is the SU (2) gauge coupling and M W 80 GeV is the mass of the W boson. The couplings of the 125 GeV Higgs boson are known to be quite close to those of the SM Higgs boson [3]. Moreover, none of the heavier Higgs bosons of the MSSM have yet been found. Both observations can easily be satisfied in the so-called decoupling limit, where the mass of the neutral CP-odd Higgs boson satisfies m 2 A M 2 Z . In that limit the other heavy MSSM Higgs bosons also have masses very close to m A , and the mixing angle α satisfies cos α sin β, sin α − cos β. This leads to the simplifications in the Higgs couplings to quarks given after the signs in eqs. (6). In particular, the couplings of the lighter Higgs boson h then approach those of the SM Higgs, in agreement with observation. The couplings of the heavier neutral Higgs boson H to up-type quarks is suppressed by 1/ tan β, while its couplings to down-type quarks are enhanced by tan β.
The Higgs bosons couple to one gaugino and one higgsino current state. As a result, their couplings to neutralino current eigenstates are proportional to the product of gaugino and higgsino components [24]: where g is the U (1) Y coupling. The total spin-independent neutralino-proton scattering cross section can be written as [11] σ χp SI = where µ χp = m p mχ0 1 /(m p + mχ0 1 ) is the reduced mass of the neutralino-proton system, and the effective neutralino-proton coupling is given by For the light u, d, s quarks, the hadronic matrix elements have to be computed using nonperturbative methods. Once these are known, the contribution from heavy c, b, t quarks can be computed perturbatively through a triangle diagram coupling to two gluons [25]. One usually parameterizes N |m qq q|N = f Tq m p . We use the numerical values from DarkSUSY [26]: We note that these numbers are somewhat uncertain, but our values are rather conservative [27].
Putting everything together, using the approximate expressions (3) for the lightest neutralino eigenstate and assuming the decoupling limit, we find: .
Here we have introduced the constant It should be noted that tan β > 1 implies cos(2β) < 0. Eqs. (10) imply that the term ∝ f T D always dominates H exchange. Hence h and H exchange contribute with the same sign if µ > 0 or µ sin 2β < −M 1 ; they interfere destructively for 0 > µ > −M 1 / sin 2β.

Results
We are now ready to present our numerical results. We wish to determine the lower bound on |µ| that follows from the non-observation of neutralinos, which we assume to form all of (galactic) dark matter. We will not be concerned with very light neutralinos, where current bounds from direct dark matter search are still quite poor [3]. For masses above 20 GeV the most stringent current bound comes from the Xenon-1T experiment [15]. In this range the bound is well parameterized by  Fig. 3 shows that this bound constrains the MSSM parameter space quite severely, if we assume that the lightest neutralinoχ 0 1 forms all of dark matter. Here we have chosen M 1 = 150 GeV, m A = 1.8 TeV, M 2 = 3 TeV and tan β = 10; the exact value of M 2 is basically irrelevant as long as it is significantly larger than M 1 . As expected the predicted cross section is largest for |µ| M 1 , which leads to strong bino-higgsino mixing. However, the Xenon-1T bound requires |µ| well above M 1 , i.e. the lightest neutralino has to be bino-like; note that values of |µ| below 100 GeV have not been considered here since they are excluded by chargino searches at LEP for the given (large) value of M 2 . In the allowed range of |µ| the approximate diagonalization of eqs.(3) works quite well. Eq.(11) then explains why the lower bound on |µ| is considerably weaker for µ < 0: evidently the two terms in the numerator of G p S | h tend to cancel (add up) for positive (negative) µ. As a result, for µ < 0 the contribution from H exchange, which contributes with opposite sign than the dominant h exchange term, is relatively more important and further reduces the cross section. In fact, the h exchange contribution vanishes at µ = −M 1 / sin(2β) 760 GeV. Due to H exchange the actual zero of the cross section [23] -the so-called "blind spot" [28] -already occurs at µ −660 GeV. In contrast, for µ > 0 the (small) H exchange contribution slightly strengthens the lower bound on µ, which saturates at ∼ 590 GeV for m H → ∞. This value is already uncomfortably large in view of finetuning considerations. This conclusion is reinforced by Fig. 3, which shows the lower bound on µ − M 1 in units of M Z as a function of M 1 for four different values of tan β; the values of M 2 and m A are as in Fig. 2. The solid colored lines have been derived numerically using DarkSUSY, whereas the black dashed lines are based on the approximate diagonalization of the neutralino mass matrix. Recall that we ignore squark exchange, so that only h and H exchange contribute to the spin-independent scattering cross section. Using eqs. (11) the extremal values of µ that saturate the experimental upper bound on the cross section can be computed analytically. To The quantity A has been defined in eq. (12), and the normalized hadronic matrix elements f T D and f T U in eqs. (10). As noted above, the contribution ∝ f T U to the H exchange contribution is essentially negligible. c µ collects terms in the Higgs exchange amplitude that are proportional to µ; only h exchange contributes to c 1 , which gets multiplied with M 1 in this amplitude. The extremal values of µ are then given by The dashed lines in Fig. 3 correspond to the positive solution µ + . We see that this approximation describes the numerical results very well even for the smallest value of M 1 we consider. In particular, the rather strong dependence on tan β originates from the sin 2β factor in the h exchange contribution to c µ , see the second eq. (14); H exchange is always subdominant for our choice m H = 1.8 TeV. The tan β dependence becomes somewhat weaker for larger values of M 1 , where the c 1 term becomes more important which is independent of tan β.
Evidently the Xenon-1T bound is quite constraining for µ > 0. For example, if we interpret electroweak finetuning considerations as requiring |µ| < 500 GeV, one finds tan β > 10 for M 1 ≥ 20 GeV, and M 1 < 115 (165) GeV for tan β = 20 (50). In eq.(15) we have assumed that c µ µ + c 1 M 1 > 0, which is always true for µ > 0. It remains true for values of |µ| below the "blind spot"; if H exchange is negligible this corresponds to |µ| sin 2β < M 1 . The negative solution in eq.(15) then gives the value of µ where the cross section decreases below the lower bound when coming from µ = 0. Fig. 4 shows that this again describes the exact numerically derived bound quite well: although the bound on |µ| − M 1 is considerably weaker than for positive µ, we saw in eqs.(3) that there are cancellations in the small entries of theχ 0 1 eigenstate if µ < 0; since corrections to this approximation are of order of the squares of these small entries, for given |µ| the approximation works better for µ < 0. In this figure we only show results for M 1 ≥ 40 GeV, since for smaller values of M 1 the bound on |µ| often is below chargino search limit of about 100 GeV. Note that now the lower bound on |µ| increases with increasing tan β. This is because the "blind spot" µ = −M 1 / sin 2β moves to larger values of |µ| for larger tan β. In the limit tan β → ∞ the bound on |µ| becomes independent of the sign of µ.
Beyond the blind spot the sign of c µ µ + c 1 M 1 flips. This region of parameter space can still be described by eq.(15), by simply setting κ → −κ everywhere. If H exchange is negligible, this can easily make the argument of the root in eq.(15) negative, signaling that no solution exists. In this case the cross section remains below the experimental bound for all values of µ below the µ − solution in the original eq. (15). For the parameters of Fig. 4 we find that this is true for tan β > 8. For smaller values of tan β a sizable region of parameter space (to the left and below the red and green solid lines) beyond the blind spot is again excluded. 1 So far we have assumed that the heavy Higgs boson is very heavy, so that its contribution to neutralino-proton scattering is subdominant. In fact, there are several constraints on the masses of the heavy Higgs bosons in the MSSM, which can be characterized by the mass of the CP-odd Higgs boson, m A . The most robust bounds come from direct searches for the heavy neutral Higgs bosons; the most sensitive ones exploit their decay into τ + τ − pairs. In particular, a recent ATLAS analysis [29] is sensitive to m A up to about 2 TeV, for very large tan β. For τ + τ − invariant masses around 400 GeV there seems to be some excess of events. While not statistically significant, it leads to a bound which is somewhat worse than the expected sensitivity. 2 We therefore chose a parameter set just on the exclusion line, m A = 400 GeV and tan β = 8, in order to illustrate the maximal possible effect from heavy Higgs exchange. It should be noted that this choice leads to a sizable contribution from charged Higgs boson loops to radiative b → sγ decays [30,31]; however, these can be compensated by postulating some amount of squark flavor mixing [32]. The bound on |µ| that results from the Xenon-1T constraint for this choice of parameters is shown by the black lines in Figs. 5; for comparison we also show results for negligible H exchange (green curves). As noted earlier for µ > 0 both Higgs bosons always contribute with equal sign, so maximizing the H exchange contribution greatly strengthens the lower bound on µ. The effect is especially strong at smaller M 1 since H exchange only contributes to c µ , not to c 1 . The resulting lower bound on µ is always above 950 GeV, i.e. in this region of parameter space a bino-like lightest neutralino would yield little benefit regarding electroweak finetuning compared with the canonical thermal higgsino-like neutralino with µ 1.2 TeV. Of course, the black line shows the maximal effect from H exchange. For somewhat larger m A , away from the ATLAS lower bound, the bound on µ will fall somewhere between the black and green lines.
In sharp contrast, for µ < 0 the H exchange contribution reduces the lower bound on |µ| even further, by moving the blind spot to smaller values of |µ|. However, for M 1 ≤ 170 GeV the cross section beyond the blind spot again increases above the Xenon-1T bound, leading to a second excluded region. In fact, the allowed region around the blind spot is very narrow for M 1 ≤ 150 GeV. The right frame of Fig. 5 therefore shows that for m A = 400 GeV values of M 1 below about 150 GeV require significant finetuning, either to hone in on the blind spot, or in the electroweak sector due to the large values of |µ| required by the Xenon-1T constraint away from the blind spot.

Summary and Conclusions
In this paper we have shown that the upper bound on the neutralino-proton cross section from the Xenon-1T experiment leads to strong lower bounds on |µ| if the bino mass parameter M 1 exceeds 20 GeV. We have assumed that the lightest neutralino forms all of dark matter, but did not require the correct thermal relic density in minimal cosmology. This constraint causes tension with electroweak finetuning arguments, since in the MSSM with holomorphic soft breaking terms the higgsino mass parameter µ also contributes to the masses of the Higgs bosons. The bound is usually significantly stronger for µ > 0; however, also for µ < 0 we found very strong constraints if the heavy Higgs bosons are close in mass to current experimental constraints and M 1 ≤ 150 GeV. This argument can be turned around to derive an upper bound on M 1 from an upper bound on |µ| from electroweak finetuning; the latter is, however, not easy to quantify unambiguously [33]. Our analytical expressions will help to easily update these constraints when future direct dark matter searches are published.
The Xenon-1T bound becomes considerably weaker for neutralino masses below 20 GeV, which we did not consider in this paper. While even very small values of M 1 remain experimentally allowed as long as all sfermions are sufficiently heavy [34], they do not appear particularly plausible given the ever strengthening lower bounds on the masses of the other gauginos (electroweak winos as well as gluinos), chiefly from searches at the LHC [3]. In fact, many models of supersymmetry breaking predict fixed ratios between these masses [35], leading to strong lower bounds on M 1 . Our analysis would then lead to even stronger lower bounds on |µ|.