Spin-dependent constraints on blind spots for thermal singlino-higgsino dark matter with(out) light singlets

Journal of High Energy Physics, Jul 2017

The LUX experiment has recently set very strong constraints on spin-independent interactions of WIMP with nuclei. These null results can be accommodated in NMSSM provided that the effective spin-independent coupling of the LSP to nucleons is suppressed. We investigate thermal relic abundance of singlino-higgsino LSP in these so-called spin-independent blind spots and derive current constraints and prospects for direct detection of spin-dependent interactions of the LSP with nuclei providing strong constraints on parameter space. We show that if the Higgs boson is the only light scalar the new LUX constraints set a lower bound on the LSP mass of about 300 GeV except for a small range around the half of Z 0 boson masses where resonant annihilation via Z 0 exchange dominates. XENON1T will probe entire range of LSP masses except for a tiny Z 0-resonant region that may be tested by the LZ experiment. These conclusions apply to general singlet-doublet dark matter annihilating dominantly to \( t\overline{t} \). Presence of light singlet (pseudo)scalars generically relaxes the constraints because new LSP (resonant and non-resonant) annihilation channels become important. Even away from resonant regions, the lower limit on the LSP mass from LUX is relaxed to about 250 GeV while XENON1T may not be sensitive to the LSP masses above about 400 GeV.

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Spin-dependent constraints on blind spots for thermal singlino-higgsino dark matter with(out) light singlets

Received: May Spin-dependent constraints on blind spots for thermal singlino-higgsino dark matter with(out) light singlets Marcin Badziak 0 1 2 3 4 Marek Olechowski 0 1 2 4 Pawel Szczerbiak 0 1 2 4 0 University of California , Berkeley, CA 94720 , U.S.A 1 and Theoretical Physics Group, Lawrence Berkeley National Laboratory 2 ul. Pasteura 5, PL-02-093 Warsaw , Poland 3 Berkeley Center for Theoretical Physics, Department of Physics 4 Institute of Theoretical Physics, Faculty of Physics, University of Warsaw The LUX experiment has recently set very strong constraints on spinindependent interactions of WIMP with nuclei. These null results can be accommodated in NMSSM provided that the e ective spin-independent coupling of the LSP to nucleons is suppressed. We investigate thermal relic abundance of singlino-higgsino LSP in these so-called spin-independent blind spots and derive current constraints and prospects for direct detection of spin-dependent interactions of the LSP with nuclei providing strong constraints on parameter space. We show that if the Higgs boson is the only light scalar the new LUX constraints set a lower bound on the LSP mass of about 300 GeV except for a small range around the half of Z0 boson masses where resonant annihilation via Z0 exchange dominates. XENON1T will probe entire range of LSP masses except for a tiny Z0-resonant region that may be tested by the LZ experiment. These conclusions apply to general singlet-doublet dark matter annihilating dominantly to tt. Presence of light singlet (pseudo)scalars generically relaxes the constraints because new LSP (resonant and non-resonant) annihilation channels become important. Even away from resonant regions, the lower limit on the LSP mass from LUX is relaxed to about 250 GeV while XENON1T may not be sensitive to the LSP masses above about 400 GeV. Supersymmetry Phenomenology - HJEP07(21)5 1 Introduction 2 3 4 5 6 Blind spots without interference e ects and relic density Blind spots and relic density with light singlets Z3-symmetric NMSSM 5.1 5.2 Heavy singlet scalar Light singlet scalar Conclusions A LSP-nucleon cross sections B The LSP (co)annihilation channels B.1 Resonance with the Z0 boson (unitary gauge) B.2 Annihilation into tt via Z0 C Improved formula for h2 near a resonance eter space still exist but they reside very close to blind spots for SI scattering cross-section. In the present article we study implications of the assumption that the SI scattering of the LSP is so small (below the neutrino background) that probably it will never be detected in direct detection of its SI interactions with nuclei. We also demand that the LSP has W = W MSSM + SHuHd + f (S) ; Lsoft = Lsoft MSSM + m2Hu jHuj2 + m2Hd jHdj2 + m2S jSj2 1 3 A 1 2 + A HuHdS + S3 + m32HuHd + m0S2S2 + SS + h:c: ; (2.1) (2.2) general models f (S) F S + 0S2=2 + S3=3. where S is a chiral SM-singlet super eld. In the simplest version, known as the scaleinvariant or Z3-symmetric NMSSM, m23 = m0S2 = S = 0 while f (S) S3=3. In more { 2 { The mass squared matrix for the neutral CP-even scalar elds, in the basis h^; H^ ; s^ related to the interaction basis by a rotation by angle (see [29] for details), reads: + H^ H^ ; S + vs vs s^s^ ; Ms^2s^ = vh2 sin 2 + A vs 1 2 1 2 Mh^2H^ = (MZ2 Mh^2s^ = vh(2 2vh2) sin 4 + h^H^ ; 2 A + h@Sf i sin 2 ) + h^s^; H^ s^; vs h^ih^j are radiative corrections, vs, vh sin and vh cos are VEVs of the singlet and the two doublets, respectively. The mass eigenstates of Ms2 are denoted by hi with hi = h; H; s (h is the 125 GeV scalar discovered by the LHC experiments). These mass eigenstates are expressed in terms of the hatted elds with the help of the diagonalization matrix S~: The mass squared matrix for the neutral pseudoscalars, after rotating away the Goldstone boson, has the form where MA2^A^ = 2 sin (2 ) A + vs MA2^a^ = vh A A^a^ ; Ma^2a^ = 1 2 vh2 sin 2 3A vs vs 2m0S2 S + vs a^a^; : hi = S~hih^h^ + S~ hiH^ H^ + S~his^s^ : Mp2 = MA2^A^ MA2^a^ ! MA2^a^ Ma^2a^ ; + A^A^ ; { 3 { vs (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) Diagonalizing Mp2 with the matrix P~ one gets the mass eigenvalues aj = a; A: aj = P~aja^a^ + P~ ajA^A^ : After decoupling of the gauginos (assumed in this work) the neutralino mass sub-matrix describing the three lightest states takes the form: M 0 = BB 0 vh sin 0 vh cos Trading the model dependent term h@S2 f i for one of the eigenvalues, m j , of the above neutralino mass matrix we nd the following (exact at the tree level) relations for the neutralino diagonalization matrix elements: cos 2 sin 2 ; ; 1 CC : A where Nj3, Nj4 and Nj5 denote, respectively, the two higgsino and the singlino components of the j-th neutralino mass eigenstate while j = 1; 2; 3 and jm 1 j jm 2 j jm 3 j. Using the last two equations and neglecting contributions from decoupled gauginos one can express the composition of the three lighter neutralinos in terms of: vh, m = and tan . Later we will be interested mainly in the LSP corresponding to j = 1, so to simplify the notation we will use m m 1 . The physical (positive) LSP mass is given by mLSP In the present work we are interested mainly in two properties of the LSP particles: their cross-sections on nucleons and their relic abundance. The spin-dependent LSPnucleon scattering cross section is dominated by the Z0 boson exchange and equals S(ND) = C(N) 10 38 cm2 (N123 N124)2 ; where C(p) 3:1 [42, 43]. The spin-independent cross-section for the LSP interacting with the nucleus with the atomic number Z0 and the mass number A is given by where red is the reduced mass of the nucleus and the LSP. When the squarks are heavy, as assumed in the present work, the e ective couplings f (N) (N = p; n) are dominated by the t-channel exchange of the CP-even scalars [44]: SI = f (N) 4 r2ed Zf (p) + (A A2 Z)f (n) 2 ; 3 X f (N) hi i=1 3 X i=1 hi 2m2hi hiNN : Further details may be found in appendix A. Formulae for the LSP annihilation cross section and its relic density are much more complicated (some of them are collected in appendices B and C). { 4 { (2.15) (2.16) (2.17) (2.18) jm j. (2.19) (2.20) (2.21) The main goal of the present work is to identify regions of the NMSSM parameter space for which the singlino-higgsino LSP particles ful ll three conditions: 1) have very small, below the neutrino background, SI cross-section on nuclei (SI blind spots); 2) have small SD cross-sections to be consistent with present experimental bounds; 3) have relic density close to the experimentally favored value 0:12 so the LSP can play the role of a dominant component of DM.1 Of course, such points in the parameter space must be consistent with other experimental constrains e.g. those derived from the LHC [48] and LEP h 2 data [49, 50]. In the next sections we discuss solutions ful lling all the above mentioned conditions, starting with the simplest case of blind spots without interference e ects and then for blind spots for which such e ects are crucial. We investigate also modi cations Blind spots without interference e ects and relic density In this section we consider situation when the SI blind spot (BS) takes place without interference e ects i.e. when all three contributions to the e ective coupling f (N) in eq. (2.21) are very small. Two of them are small because the corresponding scalars, s and H, are very heavy while fh is suppressed due to smallness of h . In the next subsection we discuss the simplest case in which the mixing among the scalars may be neglected. Then the e ects of such mixing will be taken into account. 3.1 Without scalar mixing When the interference e ects and scalar mixing may be neglected the SI blind spot condition has the following simple form: m sin 2 = 0 ; (3.1) which corresponds to vanishing Higgs-LSP-LSP coupling for ms; mH ! 1. In gure 1 the dependence of h2 on mLSP and tan for such blind spots is shown for some speci c values of and (the latter parameter does not in uence the situation as long as the resonance with the lightest pseudoscalar is not considered). Some parts of the (mLSP; tan ) plane are excluded by the upper bounds on SD from LUX [41] and IceCube [51] experiments (and also by the LEP data). Particularly important are the new LUX constraints which exclude large part of the parameter space with h2 = 0:12 0:02. We note that in the allowed part of the parameter space presented in gure 1 correct thermal relic abundance is obtained for singlino-dominated LSP. It is not di cult to understand the results shown in gure 1 using (approximate) analytic formulae. For given values of m and tan , the blind spot condition (3.1) together with eqs. (2.17) and (2.18) may be used to obtain the LSP composition. For example, the combination 1In order to take into account theoretical uncertainties in the relic abundance calculations we consider parameters leading to h 0:02 when calculated with the help of MicrOMEGAs 4.3.1 [45] and NMSSMTools 5.0.2 [46, 47] which we use to calculate the NMSSM spectrum. { 5 { .80 ¡ iZ n 0v L U X X E N O N 1 T 04 .01 .012 1 .014 05 03 47]) as functions of mLSP and tan for blind spots with decoupled all Higgs particles except the SM-like one. Dashed and dotted blue lines correspond to eq. (3.6) and (3.7), respectively, after substituting (3.2), for h 2 = 0:12. Red region depicts the points with m < 103 GeV which are ruled out by LEP [49]. Yellow area is forbidden because of eq. (3.9), whereas green/cyan one due to LUX/IceCube (IC) limits on spin-dependent LSP interaction with nucleons. Vertical red lines 1 correspond to (from left to right) mLSP = mZ0 =2; mh=2; mW ; mt. which determines the LSP coupling to the Z0 boson is given by (see appendix A)2 N123 The above expressions are valid as long as blind spot condition (3.1) is satis ed. For the LSP masses for which the annihilation cross section is dominated by the s-channel Z0 exchange this is enough to calculate the LSP relic density to a good accuracy. The approximate formulae are (see the appendix B for the details): h 2 0:1 0:3 N123 N124 2There are some corrections caused by not totally decoupled particles. In our numerical scan we took I c e C u b e ( W + W ¡ ) j j N N ¡ ¡ 2 14 2 13 jN 0 : 1 4 0 : 1 I c e C u b e ( t „t ) 05 06 LSP mass regimes (blue and red dashed lines in the left plot were obtained from eq. (3.6) whereas dotted lines in the right plot from eq. (3.7)). Yellow area is forbidden because of eq. (3.9). Grey color in the left plot denotes the regions in which h resonance and W +W =Z0Z0 channels may be important and a ect the results. Green/cyan areas correspond to LUX/IceCube limits on spindependent LSP interaction with nucleons [41], [51]. Dashed (continuous) green lines in both plots correspond to the precision of the future XENON1T (LZ) experiment of SD direct interaction of the LSP with neutrons [52]. for mLSP of order mZ0 =2 (and below the W +W threshold) and h 2 0:1 0:05 N123 N124 2 s 2 6 6 4 1 above the tt threshold. They reproduce very well the relic density calculated numerically using MicrOMEGAs, as may be seen in gure 1. The composition of the LSP is crucial not only for its relic density but also for some of the experimental constraints. One gets the following upper bounds on the combination (3.2): from the LEP bound for invisible decays of Z0 [18, 50] and N123 . (N) exp SD C(N) 10 38cm2 (3.8) (3.9) from experiments sensitive to the SD interactions of the LSP with protons or neutrons for which C(p) 3:1, respectively [42, 43]. These upper bounds on N123 functions of the LSP mass are shown in gure 2. One can nd the allowed range of mLSP in the vicinity of mZ0 =2. The points in the left plot in gure 2 at which the blue and red dashed lines enter the green region (excluded by LUX constraints on the SD cross-section) determine the limiting values of mLSP for which { 7 { n a t 02 01 51 5 04 0 : 1 .40 .45 .50 .25 .35 .20 .15 (dashed lines in the left plot was obtained from eq. (3.6) after substituting (3.2) whereas dotted lines in the right plot from eq. (3.7) after substituting (3.2)) for blind spots with decoupled all Higgs particles except the SM-like one. Left plot: green areas and thick red lines denote the points excluded by LUX and the LEP chargino searches, respectively; thick blue points correspond to eq. (3.10) and (3.11). Right plot: green region and thick orange line depict the points excluded by LUX and points for which the stop masses above 5 TeV are necessary to obtain the correct Higgs mass (even when the contribution from the stop mixing is maximized), respectively. The red lines denoted by LP depict regions with a Landau pole below the GUT scale. the Z0 resonance may give the correct relic density of the LSP. To be more accurate we used the results for h2 obtained from MicrOMEGAs and found the limiting LSP masses to be approximately 41 and 46.5 GeV.3 Substituting the corresponding values of jN123 into eq. (3.2) we nd the following linear dependence: tan tan 42 44 for m for m 41 GeV ; 46:5 GeV : (3.10) (3.11) The values of tan necessary to obtain good relic abundance of the LSP become larger when moving to values of mLSP closer to the peak of the resonance (which is slightly below mZ0 =2). The situation is illustrated in the left plot of gure 3. One can see that in the region allowed by LUX large tan is required unless is small ( 0:1 in this case). Of course the above limits may become stronger (i.e. for a given larger tan might be required) when SD direct detection experiments (especially based on the interactions with neutrons) gain better precision. For instance, the LZ experiment will be able to explore the entire region considered here | see the left plot in gure 2. In the LSP mass range between the W +W and tt thresholds the annihilation cross section is dominated by gauge boson (W +W =Z0Z0) nal states with the chargino/neutralino exchanged in the t channel. The related couplings are proportional to the higgsino components of the LSP (the gauginos are decoupled) which for the blind spot (3.1) are related to the LSP-Z0 coupling by eqs. (3.3) and (3.4). The values of N14 (N13 is smaller by factor 3These results were obtained for the case of h2 = 0:12 but they do not change much when the uncertainty in the calculation of the relic abundance is taken into account. { 8 { 1= tan ) necessary to get h 0:12 lead to too large SD and are excluded by both LUX and IceCube data (see gure 1). Thus, the LSP masses in the range mW + . mLSP . mt are excluded. The only way to have correct relic abundance consistent with all experimental constraints is to go to very small values of in order to suppress SI cross-section below the LUX constraint also away from the blind spot (3.1) and increase tan such that the Higgs mass constraint is ful lled.4 More exibility in the parameter space may appear if some additional particles are exchanged and/or appear in the nal state of the LSP annihilation (such situations will be discussed in the next sections). For mLSP & 160 GeV annihilation into tt (via s-channel Z0 exchange) starts to be kinematically accessible so smaller higgsino component su ces to have large enough ansection is predicted. As a result, IceCube [51] constraints are satis ed for mLSP & 175 GeV. depending on tan , translates to N125 However, the new LUX constraints exclude mLSP up to about 300 GeV for This lower bound on mLSP may change by about 50 GeV when the uncertainties in the calculation of the relic abundance are taken into account. It becomes stronger (weaker) for smaller (bigger) values of h2. We should also emphasize that the lower bound on the LSP mass from SD constraints are the same for the whole class of singlet-doublet fermion DM as long as it annihilates dominantly to tt. In particular, similar lower bound of 300 GeV on the LSP mass was recently set by LUX on the well-tempered neutralino in MSSM [31]. One can also see in gure 2 that the correct relic abundance requires jN123 0:95 (see eq. (3.5)) and such values may be explored by XENON1T. The right panel of gure 3 shows values of tan necessary to get h 2 = 0:12 as a function of mLSP and . Contrary to the Z0 resonance case small values of tan are preferred and hence moderate or large (in order to have big enough Higgs mass at the tree level). However, too small values of tan lead, for a given big value of , to a Landau pole below the GUT scale. Thus, the assumption of perturbativity up to the GUT scale and the requirement h2 = 0:12 give constraints which result in a -dependent upper bound on the mass of the LSP. For example, mLSP . 700 GeV for = 0:7 (see gure 3) and mLSP . 800 GeV for = 0:6. Let us also note that for large LSP masses coannihilation becomes non-negligible. This e ect relaxes the upper bound on tan and is increasingly important as decreases, as can be seen in gure 3 from comparison of full result by MicrOMEGAs and the approximated one with only tt included. h 2 Let us comment on two features of the 0:12 curves in gure 1. First: there are no signs of a resonant annihilation with the h boson exchange in the s-channel. This is simply the consequence of the blind spot condition leading to vanishing (or at least very small) LSP-Higgs coupling. This is characteristic of all blind spots without interference e ects. Second: tan decreases with m for all m > mZ0 =2 with the exception of the vicinity of the tt threshold. This is related to the fact that the annihilation cross section is directly (Z0 in the s channel) or indirectly (the V V nal states) connected to 4Note that a well-tempered bino-higgsino LSP in MSSM with the mass between the W and t masses cannot accommodate all constraints since in that case the mixing in the neutralino sector is controlled by the gauge coupling constant which is xed by experiment, see e.g. refs. [9, 31]. { 9 { the value of (N123 N124) given by eq. (3.2). The r.h.s. of (3.2) is a decreasing function of m and decreasing function of tan (for tan > 1 + vh=m ). Thus, in order to keep it approximately unchanged the increase of m must be compensated by the decrease of tan (other parameters determining the annihilation cross section may change this simple relation only close to the tt threshold and below the Z0 resonance). Another comment refers to constraints obtained from the indirect detection experiments. The IceCube upper bounds on SD change by orders of magnitude depending on what channels dominate the LSP annihilation. This can be already seen in the simple case discussed in this subsection. The lower bound on tan obtained from the IC data (as a function of mLSP) visible in gure 1 drops substantially above the tt threshold because the tt pairs give softer neutrinos as compared to the W +W pairs [51]. Moreover, the latest LUX results on SD lead to stronger bounds in almost all cases. Only for quite heavy LSP the IC limits are marginally stronger, as may be seen in the right panel of gure 2. To sum up, in this section we identi ed two crucial mechanisms (Z0 resonance and annihilation into tt) which may give correct relic density and are allowed by the experiments. However, both of them rely on the Z0 boson exchange in the s channel and therefore are proportional to the LSP-Z0 coupling, which controls also the SD cross section of the LSP scattering on nucleons. Therefore, the future bounds on such interaction will be crucial in order to constrain the parameter space. In fact, XENON1T is expected to entirely probe regions of the parameter space in which annihilation into tt dominates while LZ will be able to explore the entire region of Z0 resonance. It is also worth noting that the situation presented in gure 1 may change if we consider light pseudoscalar a with mass ma 2m . Such resonance for singlino-dominated LSP (we require 6 = 0) is controlled mainly by the mixing in (pseudo)scalar sector and hence may not be so strongly limited by the SD direct detection experiments. For instance, we checked with MicrOMEGAs that for ma in a few hundred GeV range we can easily obtain points in parameter space with correct relic density and SD below the future precision of the LZ experiment. In principle, the e ect of light pseudoscalar may be also important for 2m & mh + ma when the LSP starts to annihilate into ha state which, depending on , may suppress the tt channel and may weaken the IceCube limits. However, in the case considered in this subsection, the contribution from the ha channel may be important only for large mixing in the pseudoscalar sector. This requires quite large values of A which leads to unacceptably small values of the Higgs mass. We will come to these points in the next sections where the annihilation channels involving the singlet-dominated pseudoscalar may play a more important role. 3.2 With scalar mixing Next we consider the case when the mixing among scalars is not negligible and a ects the blind spot condition (3.1), which is now of the form: where , de ned by S~hs^ S~hh^ N13N14 N125 N15(N13 sin + N14 cos ) ; (3.12) (3.13) depends on the LSP composition (some formulae expressing in terms of the model parameters may be found in appendix A). In equation (3.12) we introduced also parameter describing the mixing of the SM-like Higgs with the singlet scalar. This mixing can be expressed (for ms mh assumed in this section) in terms of the NMSSM parameters as: S~hs^ S~hh^ v 2 ms2 sgn 2 A + h@Sf i sin 2 2 p2 j mixjmh ms ; (3.14) where mix mh M^ hh is the shift of the SM-like Higgs mass due to the mixing [53]. For ms > mh this shift is negative so we prefer it has rather small absolute value. The scenario of higgsino-dominated LSP with 0:12 is very similar to the analh 2 ogous case in the MSSM model and requires j j 1:2 TeV. Even the present results from the direct and indirect detection experiments constrain possible singlino admixture in the higgsino-dominated LSP to be at most of order 0.1. So small singlino component leads to negligible changes of necessary to get the observed relic density of DM particles.5 Thus, similarly as before, we focus on SI blind spots with h 2 0:12 for singlinodominated LSP. In this case and for non-negligible j j N125) the blind spot condition may be approximated by: (more precisely, when jN13N14j m sin 2 1 m 2! : This condition is a quadratic equation for and has solutions only when (3.15) (3.16) # 2 : (3.17) One can see from (3.15) that for > 0 we have always m > 0 and the LSP has more higgsino fraction than when condition (3.1) holds. In the opposite case i.e. < 0 we can have either m dominated LSP with m > 0 with slightly smaller higgsino fraction or strongly singlino < 0. However, for values of j j small enough not to induce large negative mix the higgsino component of the LSP with m < 0 is too small to obtain h 2 0:12. Therefore, from now on we focus on the case m > 0. Solving eq. (3.15) for the ratio m = and substituting the solution into eq. (3.2) one can nd the di erence of two higgsino components of the LSP. For small j j it can be approximated as (3.2) with a small correction: 5In fact, bigger changes of come from not totally decoupled gauginos, e.g. for M1, M2 of order 5 TeV. m Z 0 = 2 54 ‚ = 0 : 6 ; • = 0 : 3 ; j j • 0 : 0 2 5 ; fi h ´ ´ = 0 04 06 07 08 + As = < 0 (> 0) refers to the lower (upper) limit of a given area. The shift of the orange region with respect to the yellow one for LSP masses above about ms is due to the annihilation channels containing s in the nal state and co-annihilation e ects which become more important for larger LSP masses. As we already mentioned discussing eq. (3.2), the rst term in the r.h.s. of the last equation is a decreasing function of tan (unless tan is very close to 1). So, in order to keep the value of N123 N124 necessary to get h 2 0:12, the contribution from the second line of (3.17) may be compensated by increasing (decreasing) the value of tan when > 0 (< 0).6 This e ect from the s-h mixing is illustrated in gure 4. One can see that indeed, depending on the sign of , we can have smaller or larger (in comparison to eq. (3.1)) values of tan for a given LSP mass, while keeping h 2 0:12. In particular, nonnegligible Higgs-singlet mixing may relax the upper bound on the LSP mass arising from the perturbativity up to the GUT scale. It is also important to emphasize that the resonant annihilation with s scalar exchanged in the s channel is quite generic for this kind of blind spot (see the right plot in gure 4, where we have chosen ms = 600 GeV) because it is easier to have substantial s-h mixing when the singlet-dominated scalar is not very heavy. Moreover, the presence of resonant annihilation via s exchange can relax the lower limit on the LSP mass from LUX constraints on the SD cross-section, as seen in the right panel of gure 4, because in such a case smaller higgsino component is required to obtain correct relic density. 4 Blind spots and relic density with light singlets Now we move to the case when the singlet-dominated scalar is lighter than the SM Higgs. Neglecting the e ect from the heavy doublet H exchange for the SI cross section (i.e. setting fH to zero) the blind spot condition may be written in the following form [29]: 6The sign of the second term in the r.h.s. of (3.17) is determined by the sign of product m but for considered here case of m > 0 it is the same as that of (4.1) measure (in the large tan limit) the ratio of the couplings, normalized to the SM values, of the hi (= s; h) scalar to the b quarks and to the Z0 bosons. In the rest of this section we will consider singlino-like LSP, because the case of higgsinodominated LSP does not di er much from the one described in section 3.2. The blind spot condition (4.1), analogously to a simpler case (3.12), may be approximated by a quadratic equation for m sin 2 1 + As As which has solutions only if cos2 > 1 2 1 2 + (4.2) (4.3) (4.4) (4.5) As 1 + cs 1 + ch mh ms chi 1 tan 1 1+cs 1+ch m 2! mh 2 ms mh 2 : ms vh 1 m vh 1 + 2 1+cs where and parameters The last condition may be interpreted as the upper bound on tan (lower bound on cos ). It is nontrivial when its r.h.s. is positive i.e. when the second term under the absolute value is negative but bigger than 1. The bound is strongest when that term equals However, usually the absolute value of that term is smaller then 12 because the h-s mixing measured by is rather small. So, typically the bound on tan becomes stronger with increasing LSP mass or increasing j j (with other parameters xed). We focus again on the more interesting case m > 0 because for m < 0 the blind spot condition may be satis ed only for the LSP strongly dominated by singlino which typically leads to too large relic density.7 Since in this section we consider ms < mh, jAsj is typically larger than j j (unless cs and/or ch deviate much from 1, which under some conditions may happen which we discuss in more detail later in this section) { in such a case the condition (4.5) is always ful lled for < 0 (see gure 5), whereas for > 0 (see comment in footnote 6 ) there 1 2 . is an upper bound on tan which gets stronger for larger values of j j . In the following discussion we focus on big values of j j because they lead to a relatively big positive contribution to the Higgs mass from the Higgs-singlet mixing [53]. In our numerical analysis we take j j = 0:4 which corresponds to mix 4 GeV. In order to emphasize new features related to the modi cation of the BS condition we also consider rather large values of j j O(0:1). For such choices of the parameters there are no viable blind spots for > 0 in accord with the discussion above so we focus on < 0. 7For m < 0, h 2 0:12 may be obtained only when resonant annihilation via singlet (pseudo)scalar exchange is dominant (we describe this phenomenon for m > 0) or which is less interesting from the point of view of the Higgs mass. HJEP07(21)5 at 4 fl n fl n 01 8 8 3 5 5 7 7 2 2 05 05 at 4 3 L H C L E P 01 01 I C .01 .012 .014 510 510 .01 .012 .014 I C L H C 520 520 .01 .014 I C L U X 510 510 520 520 L H C 02 02 mix 4 GeV. In brown areas NMSSMTools reports unphysical global minimum (UM) while in grey ones the LHC constraints on the Higgs production and decay are violated (LHC). Black regions in the upper right corners denote the points where the condition (4.5) does not hold. The fact that our blind spot condition now comes from destructive interference between fh and fs amplitudes (rather than vanishing Higgs-LSP coupling) in uences strongly the relation between the DM relic density, especially in small LSP mass regime, and other experimental constraints. Now the Higgs-LSP coupling is not negligible so the LSP mass below mh=2 is forbidden, or at least very strongly disfavored, by the existing bounds on the invisible Higgs decays [54]. Thus, resonant annihilation with Z0 or s (in this section we consider s lighter than h) exchanged in the s channel can not be used to obtain small enough singlino-like LSP relic abundance. As concerning the h resonance: it may be used but only the \half" of it with mLSP & mh=2 (this e ect is visible in all panels of gure 5). However, we found that other experimental constraints, such as the ones from the LHC and/or LUX exclude even this \half" of the h resonance when the mixing parameter j j is large. In general NMSSM the masses of singlet-like scalar s and pseudoscalar a are independent from each other so let us rst consider the situation when a is heavy. The case with ma = 1 TeV is presented in the upper left panel of gure 5. The contours of h 2 = 0:10; 0:12; 0:14 above the tt threshold are quite similar to the case with heavy singlet. The only di erence is that now somewhat larger values of tan are preferred but even in this case they cannot exceed about 5. Let us now check what happens when the lighter pseudoscalar is also singlet-dominated (i.e. a1 = a) and relatively light. The existence of such light pseudoscalar is very important for both the relic abundance of the LSP and the constraints from the IceCube experiment. Let us now discuss these e ects in turn. The DM relic density is in uenced by a pseudoscalar in two ways. First is given by possible resonant annihilation with a exchanged in the s channel. This possibility is interesting only for ma & mh because for lighter a one still has problems with non-standard Higgs decays constrained by the LHC data (see above).8 However, as for any narrow resonance, the DM relic density may be in agreement with observations only for a quite small range of the DM mass (for a given a mass). One can see this in three panels (except the upper left one) of gure 5. Second e ect is related to new annihilation nal states including the singlet-dominated pseudoscalar, namely sa, ha, aa (in addition to similar HJEP07(21)5 channels involving only scalars: ss, sh, hh). It is best illustrated in the upper right panel of gure 5 for ma = 260 GeV. In this case the sa threshold roughly coincides with the tt threshold. Near this threshold the curves of constant h 2 0:12 go up towards bigger and leave the region excluded by the LUX data for smaller LSP mass than in the case with heavy singlets. The reason is quite simple. With increasing contribution from the annihilation channels mediated by (non-resonant) pseudoscalar exchange smaller contribution from the channels mediated by Z0 exchange is enough to get the desired value h 2 0:12. Moreover, smaller LSP-Z0 coupling is obtained for bigger values of tan so larger values of tan are preferred than in the case with heavy a. As a result the lower possible LSP mass consistent with the LUX SD limits is almost 100 GeV smaller when a is relatively light. The precise values depend on the relic density and for around 200 GeV instead of around 300 GeV as in the case with heavy singlets. The behavior of the h2 = 0:12 0:02 curves close to and slightly above the tt threshold depends on the parameters. Particularly important is the sign of . We see from the right panels of gure 5 that even for the same mass of the pseudoscalar, ma = 260 GeV in this case, the plots are very di erent for di erent signs of . Most di erences originate from the fact that > 0 implies ch > 1 and cs < 1 while for < 0 the inequalities are reversed. There are two important implications of these correlations which we describe in h 2 = 0:12 is the following. Firstly, the LHC constraints from the Higgs coupling measurements are stronger for > 0 because in such a case the Higgs coupling (normalized to the SM) to bottom quarks is larger than the one to gauge bosons. In consequence, the Higgs branching ratios to gauge bosons is suppressed as compared to the SM. Moreover, non-zero results in suppressed Higgs production cross-section so if j j is large enough the Higgs signal strengths in gauge boson decay channels is too small to accommodate the LHC Higgs data which agree quite well with the SM prediction. Moreover, a global t to the current Higgs data shows some suppression of the Higgs coupling to bottom quarks [48]9 which disfavors ch > 1, hence also large > 0. It can be seen from the upper right panel of gure 5 that for = 0:4 the 8The situation may be di erent if the blind spot condition is of the standard form (3.1) | we will come to this point at the end of this section. 9The current t also indicates an enhancement of the top Yukawa coupling. In NMSSM suppression of the bottom Yukawa coupling is correlated with enhancement of the top Yukawa coupling which has been recently studied in refs. [55, 56]. LHC excludes some of the interesting part of the parameter space which is allowed by LUX due to the LSP annihilations into sa nal state. The LHC constraints can be satis ed for small values of but this comes at a price of smaller mix, hence somewhat heavier stops. Secondly, jAsj is larger for < 0 than in the opposite case (see eq. (4.2)). Moreover, since deviations of cs and ch from 1 grow with tan , jAsj increases (decreases) with tan for negative (positive) . For > 0, this implies that for large enough tan the r.h.s. of the blind spot condition (4.4) changes sign and < 0 is no longer preferred. Equivalently, the upper bound on tan in eq. (4.5) gets stronger as tan grows so it is clear that at some point condition (4.5) is violated. The appearance of the violation of the blind spot condition is clearly visible at large tan in the upper panels of gure 5 (the black < 0 instead the blind spot condition may be always ful lled for (by taking e.g. appropriate value of ).10 Moreover, there is interesting phenomenon that may happen for < 0 above the tt threshold which is well visible in the lower right panel of gure 5. The values of tan corresponding to h2 = 0:12 grow rapidly just above the tt threshold and there is a gap in the LSP masses for which there are no solutions with SI BS and observed value of . Such solutions appear again for substantially bigger mLSP (above 300 GeV in this case). The reason why h 2 = 0:12 curve is almost vertical near mLSP mt and the gap appears is related to the fact that m = varies very slowly with tan , which results in fairly constant jN123 N124j which determines the tt annihilation cross-section, hence also h2. The weak dependence of m = on tan originates from the fact that for increasing tan both sides of the blind spot condition (4.4) grow. The l.h.s. grows because of decreasing sin(2 ) while the r.h.s. due to increasing cs. Of course, the fact that these two e ects approximately compensate each other relies on speci c choice of parameters and does not necessarily hold e.g. for di erent values of . The presence of light a in uences also the IceCube constraints in a way depending on the LSP mass. For m & (ma + ms)=2 (assuming ma > ms) the IceCube constraints are very much relaxed and become practically unimportant for the cases discussed in this section. This is so because the additional annihilation channels (into sa, ha, ss, sh) at v = 0 lead to softer neutrinos as compared to otherwise dominant V V channels (or tt channel for even heavier pseudoscalar). The situation is di erent (and more complicated) for LSP masses between the W +W and as thresholds. In this region one can have destructive/constructive interference between Z0 and a-mediated amplitudes11 for bb annihilation at v = 0 which strengthens/reduces the constraints (IceCube limit on SD cross section are two orders of magnitude stronger for V V than for bb). In our case the e ect depends on the sign of : for and vice versa for < 0 the IceCube limits are strengthen (relaxed) for m . ma (& ma) > 0 | see the lower panels of gure 5. In the examples considered in this section and shown in gure 5 we have chosen the sign of to be positive (then m is also positive because, as discussed earlier, there are no interesting solutions with m < 0). The corresponding solutions with negative , and also changed signs of other parameters like , and A , are qualitatively quite similar. 10The only exception is when ch < 1 but this may happen only for very large tan and/or very light H. 11Note that these are the only non-negligible amplitudes which have non-zero a term in v expansion { see (B.1). Of course there are some quantitative changes. Contours in the corresponding plots are slightly shifted towards smaller or bigger (depending on the signs of other parameters) values of tan . As explained in detail in ref. [29], one can also get vanishing spin-independent cross section when the standard BS condition (3.1) is ful lled. Then, the scalar sector has no e ect on the blind spot condition. In such a case, we can have as large mix as is allowed solely by the LEP and LHC constraints, irrespective of the DM sector. Interestingly, the standard BS condition appears also, in a non-trivial way, when j j is relatively small (we still consider singlino-dominated LSP) and both terms in the denominator of (3.13) are comparable and approximately cancel each other. The blind spot condition (3.12) xed and not very large. From eq. (A.4) we see that in such a case a small denominator, for a singlino-dominated LSP, may be compensated by small factor (m = sin 2 ) which means that the simplest BS condition (3.1) is approximately ful lled. In both cases, the analysis performed in subsection 3.1 holds. However, it should be noted S3 and hence may decrease the abovethat small j j weakens the singlet self-interaction mentioned e ects from the a exchange. 5 All the analysis presented till this point apply to a general NMSSM. In this section we focus on the most widely studied version of NMSSM with Z3 symmetry. In this model there is no dimensionful parameters in the superpotential: WNMSSM = SHuHd + S3=3 ; while the soft SUSY breaking Lagrangian is given by (2.2) with m23 = m0S2 = S = 0. This model has ve free parameters less than general NMSSM which implies that some physical parameters important for dark matter sector are correlated. The main features of Z3-symmetric NMSSM relevant for phenomenology of neutralino dark matter are summarized below: (5.1) (5.2) (5.3) sgn(m ) = sgn( ) LSP dominated by singlino implies j j < 1 2 : Neither singlet-like scalar nor singlet-like pseudoscalar can be decoupled due to the following tree-level relation (for singlino LSP after taking into account the leading contributions from the mixing with both scalars coming from the weak doublets, h^ and H^ ): 1 3 ms2 + m2a m2LSP + 2(ms2 m2h) : Masses of both singlet-dominated scalar and pseudoscalar are at most of order mLSP. 0 : 7 0 : 6 0 : 6 03 M S U S Y > 5 T e V 04 > 0 in the Z3-invariant NMSSM for several values of with ms = 200 (left panel) and 500 GeV (right panel). The color code is the same as in the right panel of gure 3. The lines denoted by UM depict regions in which NMSSMTools reports unphysical global minimum. Phenomenologically viable (small) Higgs-singlet mixing leads to the following treelevel relation: MA jMA^A^j 2j j sin(2 ) r 1 sin(2 ) 2 2j j ; sin(2 ) (5.4) where the last approximation is applicable for large tan and/or singlino-like LSP and forbids resonant LSP annihilation via heavy Higgs exchange. Such resonance may be present only for 1 since only in such a case signi cant deviation from relation (5.4) is possible. Important constraints on dark matter sector of Z3-symmetric NMSSM follow from relation (5.3) which we discuss in more detail in the following subsections. 5.1 Heavy singlet scalar Let us rst discuss the case of heavy singlet scalar in which only the Higgs exchange is relevant in the SI scattering amplitude and the SI blind spot has the standard form (3.1). In this case j j must be close to zero to avoid large negative correction to the Higgs mass and eq. (5.3) implies mLSP > ms. This is demonstrated in gure 6 where it is clearly seen that for mLSP . ms there are no solutions (due to a tachyonic pseudoscalar).12 We also note that eq. (5.3) implies that resonant LSP annihilation via singlet-like scalar or pseudoscalar is typically not possible in this case.13 On the other hand, eq. (5.3) implies that the LSP annihilation channel into sa via a exchange is almost always open (for small and ms > mh this channel is kinematically forbidden only in a small region of the parameter space for which ma 3ms). This allows for smaller annihilation rate into 12In gure 6 vanishing h^-s^ mixing is assumed but for small non-zero mixing (preferred by the Higgs mass) the results are similar. 13Due to loop corrections to eq. (5.3) one may nd some small regions of resonant annihilation mediated by a singlet for ms not far above mh and large close to the perturbativity bound. We discuss this e ect in more detail in subsection 5.2 because it is more generic for ms < mh. tt, hence also for smaller higgsino component of the LSP and larger tan . In consequence, larger LSP masses consistent with h 2 = 0:12 and perturbativity up to the GUT scale are possible than in the case with both singlets decoupled (compare gure 6 to gure 3). For the same reason large enough LSP masses are beyond the reach of XENON1T, as seen from gure 6. The situation signi cantly changes when singlet-like scalar is light, especially if the Higgssinglet mixing is not small (which enhances the Higgs mass if ms < mh). This is because the blind spot condition changes to eq. (4.4). Moreover, for light singlet the loop corrections to condition (5.3) can no longer be neglected which under some circumstances allows for resonant LSP annihilation via the s-channel exchange of a. In the Z3-symmetric NMSSM the singlet-dominated pseudoscalar a plays quite important role for the relic density of the singlino-dominated LSP. First we check if and when the s-channel exchange of a may dominate the LSP annihilation cross section and lead to the observed relic density. Of course, this may happen if we are quite close to the resonance, i.e. when ma 2mLSP. It occurs that it is not so easy to ful ll this requirement in the Z3-symmetric model. This is related to the condition (5.3) which, for ma 2mLSP and after taking into account the loop corrections in eqs. (2.6) and (2.14), may be rewritten in the form 1 3 ms2 + m2LSP + 2 m2h ms2 1 3 s^s^ + a^a^ : The l.h.s. of the above expression is positive so this condition can not be ful lled without the loop contributions. The last equation may be treated as a condition for the size of the loop corrections necessary to have resonant annihilation of the LSP mediated by the pseudoscalar a. In order to understand qualitatively the impact of condition (5.5) on our analysis it is enough to consider the following simple situation: we assume that the scalar is negligible and the BS is approximated by (3.1). On the r.h.s. of eq. (5.5) we take into account only the rst term of the loop correction s^s^ [57] (5.5) (5.6) (5.7) s^s^ 1 ms2 m2LSP " tan 2 2 ln 2MSUSY mLSP tan MS2USY m2LSP 1 3 # : (the second term is subdominant because m2LSP 2 < 14 2 due to condition (5.2)). In such approximation and for tan 2 for a singlino-dominated LSP and 1 condition (5.5) simpli es to For given values of and ms any change of mLSP must be compensated by appropriate change of tan . The expression in the square bracket has a maximum as a function of tan approximately at 1:2MSUSY=mLSP. Thus, to keep the r.h.s. constant in order to stay close to the resonance one has to decrease tan for small mLSP and increase for large mLSP. In our numerical examples presented in gure 7 we x MSUSY = 4 TeV so the maximum of the square bracket corresponds to tan about 30 (10) for the LSP masses of 150 (500) 01 21 nta 5 9 6 8 4 3 7 2 ‚ = 0 : 5 ; = 0 ; m s = 9 5 G e V ; „ > 0 ; 3 ¡ N M S S M ; f s + f h = 0 ‚ = 0 : 5 ; = 0 : 1 ; m s = 9 5 G e V ; „ > 0 ; 3 ¡ N M S S M ; f s + f h = 0 01 150 20 250 L U X 41 61 31 51 21 at nfl 7 9 8 6 4 3 5 I C I C 01 4 9 6 8 3 7 2 GeV. For small j j the a resonance occurs at the blind spot for tan of order 10. That is why tan typically decreases with mLSP, as can be seen from gure 7. Local minimum for tan is present only in the lower panel of gure 7 because more negative values of lead in general to larger tan (see gure 8 and discussion at the end of this section). Nevertheless, in every case the h2 = 0:12 curves corresponding to the a resonance have horizontal-like behavior: do not change very much with the LSP mass (and have values of tan of order 10 for MSUSY = 4 TeV that we use in our numerical examples). This should be compared to the general case when such curves are almost vertical (narrow ranges of the LSP mass but wide ranges of tan ) | see gure 5. This di erence comes from the fact that in the general model there are more parameters and eq. (5.3) is not ful lled. Figure 7 shows that there are two situations for which BS and correct value of DM relic density are still compatible with the latest bound on DM SD cross-section. One is the above discussed case of resonant annihilation with the light pseudoscalar exchanged in the s channel. The second one occurs for smaller tan but bigger mLSP and corresponds to annihilation via non-resonant exchange of particles in the s channel. Usually the main contribution to the annihilation cross-section in such a case comes from the exchange of Z0 boson decaying into tt nal state. This process allows to avoid the LUX bounds on SD for mLSP above about 300 GeV but is not su cient to push SD below sensitivity of XENON1T, as discussed in section 3. The situation changes when new nal state channels, especially as, open. Then not only the present bounds on SD may be easily ful lled but some parts of the parameter space are beyond the XEXON1T reach. We see from gure 7 that for light singlets the lower limit on the LSP mass from LUX may be relaxed to about 250 GeV. The e ect of annihilation into light singlets is even more important for heavier LSP so XENON1T may not be sensitive to LSP masses above about 400 GeV. 02 12 41 2 61 81 91 01 31 71 51 fln 12 at 1 9 6 8 7 .02 .02 .0 21.0 01. tan 4.0 .90 .80 .60 .70 .20 .50 .30 .15 ‚ = 0 : 5 ; m s = 9 5 G e V ; „ > 0 ; 3 ¡ N M S S M ; f s + f h = 0 ‚ = 0 : 5 ; m s = 9 5 G e V ; „ > 0 ; 3 ¡ N M S S M ; f s + f h = 0 m L S P = 1 5 0 G e V m L S P = 3 0 0 G e V m L S P = 4 5 0 G e V .30 .10 .35 .05 .0 NMSSM. Left panel: resonant annihilation via a exchange for mLSP = 150 GeV. Right panel: nonresonant annihilation for mLSP = 300 and 450 GeV. The green parts of the contours are excluded by LUX. The parabola-like curves show dependence of mix (on the right horizontal axes) on . In both cases discussed above the allowed values of tan are correlated with the LSP mass. The exact form of such correlation depends on the s^-h^ mixing parameter . Quite generally values of tan decrease with . This is illustrated in gure 8 where the bands of allowed tan between tan as functions of are shown for a few values of the LSP mass. This correlation and can be easily understood from eqs. (4.2){(4.4). The rst factor on the r.h.s of (4.4) grows in the rst approximation like . This can not be compensated by decreasing because in the Z3-symmetric NMSSM is xed by the LSP mass. The BS condition (4.4) with increasing r.h.s. may be ful lled by decreasing the absolute value of the negative contribution to its l.h.s. i.e. by increasing tan . 6 Motivated by the recent strong LUX constraints we investigated consequences of the assumption that the spin-independent cross-section of singlino-higgsino LSP scattering o nuclei is below the irreducible neutrino background. We determined constraints on the NMSSM parameter space assuming that the LSP is a thermal relic with the abundance consistent with Planck observations and studied how present and future constraints on spin-dependent scattering cross-section may probe blind spots in spin-independent direct detection. In the case when all scalars except for the 125 GeV Higgs boson are heavy the new LUX constraints exclude the singlino-higgsino masses below about 300 GeV unless the LSP mass is very close to the half of the Z0 boson mass (between about 41 and 46 GeV). In the allowed region LSP dominantly annihilates to tt and tan must be below about 3.5 (assuming perturbative values of up to the GUT scale) with the upper bound being stronger for smaller and heavier LSP. There is also an upper bound of about 700 GeV assuming perturbativity up to the GUT scale. We found that XENON1T has sensitivity to exclude the entire region of dark matter annihilating dominantly to tt. This conclusion apply to general models of singlet-doublet dark matter. On the other hand, the LSP resonantly annihilating via Z0 boson exchange is possible only for large tan unless is very small e.g. for > 0:5, tan & 20. Only small range of LSP masses around the resonance of about 2 GeV is beyond the XENON1T reach while LZ is expected to probe Z0 resonance completely. In all of the above cases the LSP is dominated by singlino. Current and future constraints can be avoided also for very pure higgsino with mass in the vicinity of 1.1 TeV. The situation signi cantly changes when singlet-like (pseudo)scalars are light. Firstly, the presence of light CP-even singlet scalar modi es the condition for spin-independent blind spot when its mixing with other Higgs bosons is non-negligible. Depending on the sign of the mixing angle between the singlet and the 125 GeV Higgs preferred values of tan may be either increased or decreased, as compared to the case with heavy singlet. Interestingly, tan is increased when the Higgs coupling to bottom quarks is smaller than that to gauge bosons which is somewhat favored by the LHC Higgs coupling measurements. Secondly, the presence of light singlets opens new annihilation channels for the LSP. As a result, correct relic abundance requires smaller higgsino component of the LSP which relaxes spin-dependent constraints. We found that resonant annihilation via exchange of singlet pseudoscalar is possible even in the Z3-invariant NMSSM. Interestingly, even far away from the resonant region the lower limit on the mass of LSP annihilating mainly to tt may be relaxed to 250 GeV. For larger LSP masses sa may become dominant annihilation channel and the LSP masses above 400 GeV may be beyond the reach of XENON1T. Acknowledgments This work has been partially supported by National Science Centre, Poland, under research grants DEC-2014/15/B/ST2/02157, DEC-2015/18/M/ST2/00054 and DEC2012/04/A/ST2/00099, by the O ce of High Energy Physics of the U.S. Department of Energy under Contract DE-AC02-05CH11231, and by the National Science Foundation under grant PHY-1316783. MB acknowledges support from the Polish Ministry of Science and Higher Education through its programme Mobility Plus (decision no. 1266/MOB/IV/2015/0). PS acknowledges support from National Science Centre, Poland, grant DEC-2015/19/N/ST2/01697. A LSP-nucleon cross sections In this appendix we collect several expressions useful in discussing the SI and SD crosssections of LSP on nuclei. The couplings of the i-th scalar to the LSP and to the nucleon, appearing in the formula (2.19) for the SI cross-section, after decoupling the gauginos are approximated, respectively, by hi hiNN ai p 2 mN p2v p i 2 S~hih^N15 (N13 sin + N14 cos ) + S~ hiH^ N15 (N14 sin N13 cos ) + S~his^ N13N14 ; S~hih^ F (N) + Fu(N) + S~hiH^ tan F (N) d d 1 tan Fu(N) The LSP couplings to pseudoscalars, important for the relic abundance calculation, are approximated by ref. [57] where P~ij are elements of the matrix diagonalizing the pseudoscalar mass matrix de ned in eq. (2.15). Parameter de ned by eq. (3.13) and convenient for the discussion of SI blind, using eqs. (2.17) and (2.18), may be written in the form = v 2 1 + 1 m 2 m 2 sin 2 2 m m sin 2 1 B The LSP (co)annihilation channels In this appendix we will use the following expansion of v around v = 0: v = a + bv2 + O(v4) : With the help of eqs. (2.17) and (2.18), the combination of the LSP components crucial for SD, (N123 N124), may be written as: N123 h 1 (m = )2i (1 1 + (m = ) 2 N125) cos 2 2 (m = ) sin 2 We can see immediately that the cross-section disappears in the limit of tan = 1 or a pure singlino/higgsino LSP. The ratio of the higgsino to the singlino components of the LSP may be calculating from eqs. (2.17) and (2.18): 1 N125 N125 = vh 2 1 + (m = ) 2 2(m = ) sin 2 : h 1 (m = ) 2i2 Using this relation we may rewrite formula (A.5) in the form: Then, the relic density may be written as [58]: 2 9:4 10 12 GeV 2 xf a + 3b=xf ; where xf Resonance with the Z0 boson (unitary gauge) Let us consider the LSP annihilation into the SM fermions (except the t quark14) via Z0 exchange in s channel. The expansion coe cient a and b in eq. (B.1) are equal to: a = b0 = g 4 g 4 32 (4m2 X cF (2 F2 F 2 F + 1) ; (B.2) (B.3) (B.4) b0v2 (B.5) where g p(g12 + g22)=2, cF = 1 for leptons and 3 for quarks, whereas F = 2jqF j sin W2. The 0 index in b0 parameter means that we put fermion masses to 0 (which is a very good approximation for m the t quark) in (B.4) equal 4m2 =m2Z0 ) which means that b mZ0 =2; of course a0 = 0). The sum over the SM fermions (except 14:6. It is worth noting that b0 m2 and a m2F (1 0 (in contrary to naive expectation). Moreover, the terms proportional to higher powers of v2 in v (for m mF ) are suppressed with respect to bv2 term in geometric way by v2=4. Therefore we can approximate v and hence expressed the relic density in the form of eq. (B.2). We will however improve slightly this approach (see appendix C) and write our formula in the following form: h 2 0:1 0:3 N123 N124 Z0 m2Z0 5 : where the term proportional to v2 0:52 stems from the fact that the dark matter particles posses some thermal energy during the freeze-out. Eq. (B.5) reproduces very well the results obtained from MicrOMEGAs far from the resonance (see e.g. gure 1), however very close to the resonance, especially for m . mZ0 =2, the di erence may be sizable ( gure 3). B.2 Annihilation into tt via Z0 In this case the dominant contribution also comes from Z0 exchange in s channel but in contrary to the previous paragraph m mF (= mt). Therefore the statement that b a is now longer true. It becomes clear when we write down the expression for a and b terms 14The e ect from the t quark appears for m discuss this case separately in the next paragraph. mt which is quite far from the resonance | we will One can see that for m mt both terms are comparable whereas for larger m we have a=b 4 and eq. (B.6) su ces (as we would expect, the terms proportional to higher powers of v2 are suppressed for m mt; mZ0 as v2=4). Similarly to eq. (B.5) we can nd the expression for h2. Combining (B.2) with (B.6) and (B.7) we get: 2 0:1 0:05 N123 N124 2 mm2t2 + 3 1 4 xf 1 mt2 2m2 3 1 r 1 1 7 m2 (B.8) The above equation works well for m & 175 GeV (see gure 1), however for m mt we have to be more careful because the expansion in v 2 breaks down. One can see that for mt; mZ0 the square bracket in (B.8) equals roughly 1 and h2 depends on jN123 N124j only. Similarly to the case of the resonance with Z0, the crucial experimental bounds comes from SD direct detection (see right plot in gure 2). It is worth pointing out that both a and b coe cients in (B.6) and (B.7) come purely from into account this term is also crucial for DM annihilation in galactic halos (v2 10 6) for p p =m2Z0 term in Z0 propagator. It was noticed long time ago [59, 60] that taking mZ0 =2. This is because the a coe cient in (B.3) vanishes which causes large dip in the annihilation cross section. Improved formula for h2 near a resonance The method described below may be found e.g. in [58, 61]. Let us consider a general expression for v for scalar dark matter (with mass m) annihilating via s channel exchange of a particle with mass M and total decay width : in the limit m mZ0 : b 3g4 32 3g4 32 (N123 (N123 mt2 m2 : ing dimensionless quantities approximation s = 4m2=(1 For simplicity we assume we assume = const which is generally not the case, however we are mainly focused on the e ect on h2 coming from the denominator in (C.1). Us4m2=M 2 1, =M and considering non-relativistic v = (s M 2)2 + 2M 2 v2=4) 4m2(1 + v2=4) we get: v = =M 4 (B.6) (B.7) (C.1) (C.2) Let us now de ne Y (x) ns , where x = T =m, and write 1 Y (1) 1 Y (xd) = m MPl pg gs r 45 xd Z 1 h vi dx : x2 Parameter xd is de ned as a moment in thermal evolution of DM when the term 1=Y (xd) starts to be small and can be safely neglected. Dark matter relic abundance can be then calculated by double integration over v and x: h2 = GeV MPl gs 1 p 2 Z 1 0 dv ( v)v2 Z 1 xd dx e v2x=4 # 1 p x Note that we changed the usual order of integration. We will now perform the simpler integral over x, obtaining: 1 p 2 Z 1 0 dv ( v)v2 Z 1 xd dx e v2x=4 p x Z 1 0 ( v)v erfc(v=2pxd) dv : Substituting here eq. (C.2) and erfc(v=2pxd) pxd= v + : : : we can easily nd simple expressions for h2 for some hierarchical values of and e.g. etc. In the case of fermionic dark matter our expression (C.1) generalizes to: (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) (C.9) (C.10) v = (s Z 1 f (v)v erfc(v=2pxd) dv : ( + v2=4)2 + 2 1 f (s) For xd = 25 we have v 0:5. The above method e ectively includes the fact that the dark matter particles posses some thermal energy during their freeze-out. Other cases of f (v) can be also easily analyzed and compared with numerical results. Now we have to perform the following integral In order to proceed further we have to specify the formula for f (s). In the case of LSP annihilation into fermions via Z0 exchange the dominant contribution is f (v) v2 { see appendix B.1. Analytical form of the above integral is very complicated even for such simple expression for f (v). The numerator in eq. (C.8) has a maximum for some speci c value of v. Therefore we will take the denominator in front of the integral, substituting v ! v, where v is de ned as a mean value of the numerator. Then we have: : : : M 4 ( + v2=4)2 + 2 0 1 v Z 1 v3 erfc(v=2pxd) dv = : : : R01 v v3 erfc(v=2pxd) dv R01 v3 erfc(v=2pxd) dv 64 15p 3=x2d M 4 ( + v2=4)2 + 2 d x 1=2 : Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] LUX collaboration, D.S. 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Marcin Badziak, Marek Olechowski, Paweł Szczerbiak. Spin-dependent constraints on blind spots for thermal singlino-higgsino dark matter with(out) light singlets, Journal of High Energy Physics, 2017, 1-31, DOI: 10.1007/JHEP07(2017)050