Astrophysical signatures of Axino dark matter
Seng Pei Liew (U. of Tokyo) Phys. Lett. B 721 (2013) , pp. 111; (based on...) arXiv: 1301.7536 JCAP 1405 (2014) 044; arXiv: 1403.6621
SUSY 2014, Manchester Axion as a solution of the Strong CP problem
from non-perturbative effects
but from experiments (fine-tuning problem)
Introduce a new field a such that
at the minimum of the potential,
2 for U(1) and SU(2) gauge couplings
supersymmetrize
axino bino wino appears in both KSVZ and DFSZ models 3 Axino can be a DM candidate
L. Roszkowski,(mass: Particle Dark keV~TeV) Matter [hep-ph/0404052]
Figure 1. Aschematicrepresentationofsomewell–motivatedWIMP–type particles 4 for which a priori one can have ω ∼ 1. σint represents a typical order of magnitude of interaction strength with ordinary matter. The neutrino provides hot DM which is disfavored. The box marked “WIMP’ stands for several possible candidates, e.g.,from Kaluza–Klein scenarios. can in principle extend up to the Planck mass scale, but not above, if we are talking about elementary particles. The interaction cross section could reasonably be expected to be of −38 2 −2 the electroweak strength (σEW 10 cm =10 pb) but could also be as tiny as that ∼ 2 −32 −34 purely due to gravity: (mW /MP) σEW 10 σEW 10 pb. What can we put into∼ this vast plane shown∼ in Fig. 1?∼ One obviouscandidateisthe neutrino, since we know that it exists. Neutrino oscillationexperimentshavebasically convinced us that its mass of at least 0.1 eV. On the upper side, if it were heavier than afeweV,itwouldoverclosetheUniverse.Theproblemofcour∼ se is that such a WIMP would constitute hot DM which is hardly anybody’s favored these days. While some like it hot, or warm, most like it cold. Cold, or non–relativistic at the epoch of matter dominance (although not necessarily at freezeout!) and later, DM particles are strongly favored by afewindependentarguments. One is numerical simulations of large structures. Also, increasingly accurate studies of CMB anisotropies, most notably recent results from WMAP [1],implyalargecoldDM (CDM) component and strongly suggest that most ( 90%)ofitisnon–baryonic. In the SUSY world, of course we could add a sneutrino∼ ν,which,likeneutrinos,interacts > weakly. From LEP its mass 70 GeV (definitely a cold DM! candidate), but then Ων 1. Uninteresting and ν does not∼ appear in Fig. 1. " ! The main suspect! for today is of course the neutralino χ.Unfortunately,westillknow little about its properties. LEP bounds on its mass are actually not too strong, nor are they robust: they depend on a number of assumptions. In minimal SUSY (the so-called MSSM)
2 130 GeV γ-line Studies on γ-rays from the Fermi data T. Bringmann, X. Huang, A. Ibarra, S. Vogl, C. Weniger [arXiv:1203.1312] C. Weniger [arXiv:1204.2797] E. Tempel, A. Hektor and M. Raidal [arXiv:1205.1045] and many more... See T. Bringmann, C. Weniger [arXiv:1208.5481] for a review
From the Galactic Center, they found
C. Weniger [arXiv:1204.2797] E. Tempel, A. Hektor and M. Raidal [arXiv:1205.1045] A DM signal?
• Less significant feature from the Fermi Collaboration • γ-lines from the Earth limb? • Instrumental? Still inclonclusive...
Summary & Updates: C. Weniger [arXiv:1303.1798]
From A. Albert’s talk, the Fermi Symposium Nov. 2012 Neutralino
• Annihilating DM
DM γ
DM 130 GeV Neutralino
• γ from loop effects γ DM
DM 130 GeV W boson etc. Neutralino • γ from loop effects Typically... γ DM DM SM << DM DM 130 GeV W boson etc. SM Neutralino • γ from loop effects Typically... γ DM DM SM << DM DM 130 GeV W boson etc. SM Too much continuum γ & antiprotons! Gravitino F. Takayama, M. Yamaguchi [hep-ph/0005214] Decaying DM • Small bilinear R-parity violation 28 • Long lifetime Lifetime ~ 10 sec W. Buchmuller, M. Garny [arXiv:1206.7056] γ W, Z, h
Gravitino << 260 GeV ν l, ν Too much continuum γ & antiprotons! Axino Small bilinear R-parity violation • Decaying DM • Long lifetime
γ W, Z
Axino ~ 260 GeV Branching ratio largeν enough tol, ν explain the 130-GeV γ-line! M. Endo, K. Hamaguchi, SPL, K. Mukaida, K. Results Nakayama [arXiv:1301.7536]
PQ Scale
(Wino NLSP, similar for Bino NLSP)
RPV parameter
13 From Ruchayskiy’s talk
Unidentified spectral line at E 3.5 keV Unidentified spectral line at E⇠ 3.5 keV ⇠ [1402.4119] Boyarsky et al. 2014 [1402.4119] M31 galaxy XMM-Newton,Boyarsky center et al. 2014 & outskirts PerseusM31 cluster galaxy XMM-Newton, XMM-Newton, outskirts center & only outskirts PerseusBlank cluster sky XMM-Newton XMM-Newton, outskirts only Blank sky XMM-Newton [1402.2301] [1402.2301] Bulbul et al. 2014 73 clusters XMM-Newton,Bulbul et central al. 2014 regions 73 clusters XMM-Newton, central regions of clusters only. Up to z =0.35, of clusters only. Up to z =0.35, including Coma, Perseus including Coma, Perseus Perseus cluster Chandra, center only Perseus cluster Chandra, center only VirgoVirgo cluster cluster Chandra, Chandra, center center only only
Position:Position:3.53.5 keV. keV. Statistical Statistical error error for for line line position position 3030 eV.eV. ⇠⇠ SystematicsSystematics ( (50 50eVeV – between – between cameras, cameras, determination determination of of known known ⇠ ⇠ instrumentalinstrumental lines) lines)
28 Lifetime:Lifetime: 10 10sec28 sec (uncertainty (uncertainty(10)(10)) ) ⇠ ⇠ O O
OlegOleg Ruchayskiy Ruchayskiy HNL HNLIN COSMOLOGYIN COSMOLOGY 1717 Stacked X-ray spectrum [1402.2301]
14 ) ) -1 -1
0.8 XMM - MOS 3.51 ± 0.03 (0.05) XMM - PN
3.57 ± 0.02 (0.03) keV keV 1.2 Full Sample -1 -1 Full Sample 0.7 6 Ms 2 Ms 1 0.6 Flux (cnts s Flux (cnts s 0.8 0.015 0.04
0.01 0.02 0.005 0 0 Residuals Residuals -0.005 -0.02
3 3.2 3.4 3.6 3.8 4 3 3.2 3.4 3.6 3.8 4 Energy (keV) Energy (keV) ) ) -1 -1 XMM - MOS XMM - MOS Centaurus+
keV Rest of the Sample keV 3.57 keV 3.57 keV -1 -1 Coma + (69 Clusters) Ophiuchus 4.9 Ms All spectra are525.3 ks blue-shifted0.25 to the Flux (cnts s
Flux (cnts s same frame of reference and 0.01
backgrounds are similarly0.005 subtracted
0 Residuals Residuals -0.005
3 3.2 3.4 3.6 3.8 4 Energy (keV) ) -1 0.75 XMM - PN 3.57 keV Rest of the Sample keV
-1 (69 Clusters) 1.8 Ms
0.5 Flux (cnts s
0.02
0 Residuals
-0.02 3 3.2 3.4 3.6 3.8 4 Energy (keV) Figure 6. 3 4keVbandoftherebinnedXMM-Newton spectra of the detections.The spectra were rebinned to make the excess at 3.57 keV more apparent. (APJ VERSION INCLUDES ONLY THE REBINNED MOS SPECTRUM OF THE FULL SAMPLE).⇠ nax dwarf galaxies (Boyarsky et al. 2010; Watson et al. at a 1 level. Since the most confident measurements 2012), as showin in Figure 13(a). It is in marginal ( 90% are provided by the highest signal-to-noise ratio stacked significance) tension with the most recent Chandra⇠limit MOS observations of the full sample, we will use the flux from M31 (Horiuchi et al. 2014), as shown in Figure at energy 3.57 keV when comparing the mixing angle 13(b). measurements for the sterile neutrino interpretation of For the PN flux for the line fixed at the best-fit MOS this line. energy, the corresponding mixing angle is sin2(2✓)= +1.2 +1.8 11 4.3 1.0 ( 1.7) 10 . This measurement is consistent with that obtained⇥ from the stacked MOS observations 3.2. Excluding Bright Nearby Clusters from the Sample Surface brightness profile 4
M31 surface brightness profile Perseus cluster surface brigtness profile 10 20 on-center NFW DM line, rs = 360 kpc off-center upper limit NFW DM line, rs = 872 kpc NFW, c = 11.7 β-model, β = 0.71, rc = 287 kpc NFW, c = 19 8 15 /sec] /sec] 2 2 Decaying DM 6 density profile 10 R200 [cts/cm [cts/cm 6 6 4
5 Flux x 10 Flux x 10 2
0 0 0 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Radius [deg] Radius [deg]
FIG. 2: The line’s brightness profile in M31 (left) and the Perseus clusterisothermal (right). An beta NFW DMprofile distribution (gas distribution) is assumed, the scale rs is fixed to its best-fit values from [22] (M31) or [23] (Perseus) and the overall normalization is adjusted to pass through the left-most point.
Blank sky dataset 0.10
For the Perseus cluster the observations can be grouped in 3radialbinsbytheiroff-centerangle.Foreachbinwefix
the line position to its average value across Perseus (3.47 ± [cts/sec/keV] 0.07 keV). The obtained line fluxes together with 1σ errors Normalized count rate -3 are shown in Fig. 2. For comparison, we draw the expected 2⋅10 -3 line distribution from dark matter decay using the NFW pro- 2⋅10 -3 1⋅10 file of [23] (best fit value rs =360kpc, black solid line; 1σ -4 5⋅10 0 upper bound rs =872kpc, black dashed line). The isother- 0⋅10 -4 mal β-profile from [26] is shown in magenta. The surface -5⋅10
[cts/sec/keV] -3
Data - model -1⋅10 brightness profile follows the expected DM decay line’s dis- -3 -2⋅10 -3 tribution in Perseus. -2⋅10 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Finally, we compare the predictions for the DM lifetime from Energy [keV] the two objects. The estimates of the average column den- FIG. 3: Blank sky spectrum and residuals. sity within the central part of M31 give S(rs) ∼ 200 − 2 600 M!/pc [13]. The column density of clusters follows from the c − M relation [27–29]. Considering the uncer- tainty on the profile and that our observations of Perseus go the νMSM. beyond rs,theaveragecolumndensityintheregionofinterest 2 The position and flux of the discussed weak line are inevitably is within S¯ ∼ 100 − 600 M!/pc .Thereforethesignalfrom Perseus can be both stronger and weaker than that of M31, by subject to systematical uncertainties. There are two weak in- 0.2 − 3.0.Thisisconsistentwiththeratioofmeasuredflux strumental lines (K Kα at 3.31 keV and Ca Kα at 3.69 keV), from Perseus to M31 0.7 − 2.7. although formally their centroids are separated by more than 4σ.Additionally,theregionbelow3 keV is difficult to model If DM is made of right-handed (sterile) neutrinos [30], the precisely, especially at large exposures, due to the presence of lifetime is related to its interaction strength (mixing angle): the absorption edge and galactic emission. However, although the residuals below 3 keV are similar between the M31 dataset 4 −8 5 1024π × 29 10 1keV (Fig. 1) and the blank sky dataset (Fig. 3), the line is not de- τDM = 2 2 5 7.2 10 sec 2 . 9αGF sin (2θ)mDM !sin (2θ) "! mDM " tected in the latter. Although the count rate at these energies is 4 times larger for M31, the exposure for the blank sky is 16 times larger. This disfavors the interpretation of the line as due
Using the data from M31 we obtain the mass mDM =7.06 ± to a wiggle in the effective area. The properties of this line are 0.05 keV and the mixing angle in the range sin2(2θ)=(2.2− consistent (within uncertainties) with the DM interpretation. 20) × 10−11.Thisvalueisconsistentwithpreviousbounds, To reach a conclusion about its nature, one will need to find Fig. 4. This means that sterile neutrinos should be produced more objects that give a detection or where non-observation of resonantly [31–33], which requires the presence of significant the line will put tight constraints on its properties. The forth- lepton asymmetry in primordial plasma at temperatures few coming Astro-H mission [34] has sufficient spectral resolution hundreds MeV. This produces restrictions on parameters of to spectrally resolve the line against other nearby featuresand Axino (~ 7 keV)
bilinear R-parity violation [1403.1536] • KC Kong, J-CP & SC Park astrophysicalarXiv:1403.1536 bound
parameter space
BICEP2 result favors fa < HI/2 constraintSee D. Grin’s talk due toVisinelli tree-level & Gondolo, arXiv:1403.4594 contribution to neutrino mass 2 The decay rate in Eq. (7) is proportional to mf . This is due to a chirality flip at the fermionic internal line of the loop diagrams in Figure 1. Thus, loop contributions from (s)fermions of the third generation of (s)lepton or (s)quark have the most significant impact on the decay rate. The decay rate of the conjugate processa ˜ +¯⌫ is the same, and gives a factor of two to the overall decay rate. Note that the overall! decay rate must also be multiplied by the color factor for colored (s)fermions. So far, weAxino have not discussed (~ the7 axino-fermion-sfermionkeV) coupling from the Yukawa interaction, i.e.
c [1403.6621] trilinear ˜ R-parity= C ˜ f˜⇤ aP˜¯ violationLf + C ˜ f ˜⇤aP˜¯ Rf +h.c., (9) • La˜ff a˜fRfL R a˜fLfR L where C is the product of the axino-Higgsino mixing ( v/f ) and the SM Yukawa a˜f˜RfL ⌫ a ⌫ coupling. The strength of C ’s for the third generation of⇠ fermions is comparable with a˜f˜RfL of axino withthose Higgsino from theis gaugev/fa,of where interaction axinov withis the˜ Higgsino as Higgs in Eq. vacuum is (5).v/f expectation However,a, where v inis value the contrast Higgs(VEV). vacuum to the axino-fermion- expectation value (VEV). ⇠ fR R ⇠ fR Hence, sfermion coupling fromHence, the gauge interaction, i.e. Eq. (3), whereR the decay amplitude a˜ fR a˜ f˜R picks up a chirality-flipping mf (seev Eq. (7)), the decay amplitudev for contributions from Ca˜ff˜ fgˆ , Ca˜ff˜ ˜gˆ , (5) (5) the Lagrangian of Eq. (9) picks' Rfa up an m instead [19].' SincefRfa m m for the third a˜ a˜ ⌧ f whereg ˆ isgeneration the gauge coupling of fermions, constantwhere weg suppressedˆ canis the neglect gauge by couplingthe the Higgsino-gaugino contributions constant suppressed from mixing. the by Lagrangian the Higgsino-gaugino of Eq. (9). mixing. To be more concrete, let us consider LLE RPV operators with (s)tau running in the One can compare the strength of OneCa˜ff˜ ’s can for compare KSVZ and the DFSZ strength models of Ca˜ fromff˜ ’s for Eqs. KSVZ (4)-(5). and DFSZ models from Eqs. (4)-(5). 2 (a) 2 (b) gˆ is typicallyloop of ( order i33,i 10=1 ,, and2).gˆ From forisM typically1 Eqs.v, (5) ofKSVZ’s order and (7),C 10a˜ff ˜ ’s the, and are lifetime forapproximatelyM1 of thev, KSVZ’s axino smallerC readsa˜ff˜ ’s are approximately smaller 2 ⇠ 2 ⇠ stau than DFSZ’s by 10 . than DFSZ’s by 10 . Figure 1: Feynman diagrams of2 the processa ˜ + ⌫ via2LLE or LQD RPV operators. 3! 4 2 Here, right-handed27 (s)fermions i33gˆ arem assumeda˜ to befa running/v in them⌧˜ loop.R The (s)fermionm⌧ ⌧a˜ 8flow.7 is in10 thesec opposite direction4 for the conjugate process8 a ˜ +¯⌫. . (10) 2.2 R-parity' violations⇥ 2.210 R-parity 7keV violations10 !100 GeV 1.77 GeV of axino with Higgsino✓ ◆ ⇣ is ⌘ v/f✓ a,◆ where⇣ v⌘is⇣ the Higgs⌘ vacuum expectation value (VEV). The most general formin Figure of RPV 1 are isThe representedcharged most (s)leptons general by the formfor following the ofLLE RPVscenario superpotential is represented and down-type [18]: by4 the (s)quarks following for superpotential the 2 [18]: Here, we have factored out the dimensionful⇠ parameter 1/m from C0(mf ,m˜ ) ,and of axino with Higgsino is v/f , where v is the Higgs vacuum expectation⌧˜R value (VEV).fR Hence,LQD scenario.a The one-loop decay amplitude4 involves2 a axino-sfermion-fermion vertex, used WC0=with(1 .77ijk couplingL⇠ GeViLjE,k 100 constant+ ijk0 GeV)LiQ givenjDk in+W7 Eqs. ijk00 =U10 (4)-(5) iDijk jLDikLGeV for+jEµ KSVZki L+i.H ijk0u and It,LiQ can DFSZjDk be+ models seenijk00 (6)Ui respectively.D fromjDk + Eq.µiLi (10)Hu, that (6) Hence, | 9 11 | ' ⇥can 3avoid1 astrophysical bound fa (10Notice that10 C)GeVforin KSVZ modelsi33 is only(10 an e↵ective10 vertex). This induced parameter at the one-loop region level. is consistent with the summation⇠ O among thea˜withff˜ indices thei, summation j, k =1⇠ O, 2, among3denotingtheleptonandquark the indices i, j, k =1, 2, 3denotingtheleptonandquark with theA observed more rigorous 3.5 calculation keV X-ray ofa ˜ line. v+⌫ would involve the treatment of Feynman diagramsv C ˜ !gˆ , (5) generations assumedwith implicitly. two loops.generations The However, firsta˜ threeff since assumed terms KSVZ’s implicitly. correspondC ˜ is smaller The to trilinear first than three thoseC RPV interms DFSZ and correspond modelsgˆ by, to trilinear RPV and (5) where does 'comefa a˜fromff and whata˜ff˜ is ? the last one correspondstwo orders to bilinear of magnitude,the RPV. last oneL onei, correspondsE cani, Q naivelyi, Di, U deducetoi and bilinear thatHu are RPV. the the decayL usuali, amplitudeEi, matterQi, D ofi, a ˜Ui and +H⌫u are the usual matter ' ! fa wheresuperfieldsg ˆ is the3.2 in gaugethe Minimal Constraintsfor coupling KSVZ Supersymmetric models constantsuperfields is much suppressed smaller in Standard the than Minimal Model those by the Supersymmetric in (MSSM). DFSZ Higgsino-gaugino models. ijk, Henceforth, Standardijk0 , ijk00 mixing.are forModel simplicity, (MSSM). ijk, ijk0 , ijk00 are dimensionless couplingour constants focus is solely whereasdimensionless on axino the couplingDM coupling in DFSZ constants constants models.µi whereas’s carry mass the coupling dimension constants µi’s carry mass dimension One canNext, compare we thestudyWe strength assume possible that of the experimentalCa˜ right-handedff˜ ’s for KSVZ sfermion constraints and contributes DFSZ on models dominantly. Trilinear from to Eqs. the RPVs decay (4)-(5). am- generate neu- one. SMwhere gauge symmetriesg ˆ is2 the demandone. gauge the SM indices gauge coupling symmetriesi and j (j and demand constantk)of the iijk33 indices( ijk00 suppressed)tobei and j (j and k)of by ijk the( ijk00 )tobe Higgsino-gaugino mixing. gˆ is typically of orderplitude 10 of,a ˜ and + for⌫. SummingM1 v Feynman, KSVZ’s diagramsC ˜ (1a)’s are and approximately (1b) in Figure⌫ 1, the smaller decay antisymmetric.trino Since masses the atUUD theoperatorantisymmetric. one-loop is irrelevant level. Since The to the neutrino ourUUD study,a˜ffoperator mass 00 =0isimposedby matrix is irrelevant (M )receivesthefollowing to our study, 00 =0isimposedby amplitude2 is! found to be of the⇠ form µ⌫k ✏ , where k ijkand ✏ are the momentumij and the ijk thanassuming DFSZ’s baryon by 10 number . conservation.assuming baryon numberµ conservation.⌫⇤ µ ⌫ contributionsOnepolarization can from compare vector RPV of couplings the photon the respectively,ijk strength[20, 21, as 18]: expected of fromCa˜ gaugeff˜ ’s invariance. for KSVZ Con- and DFSZ models from Eqs. (4)-(5). Let us discuss bilinear RPV in moreLet detail. us discussµiL bilineariHu can RPV be rotated in more away detail. by redefiningµiLiHu can be rotated away by redefining sidering only one trilinear RPV coupling at2 a time, and letting denote ijk and ijk0 ,the e2 2 Li and Hgˆd asisLi0 = typicallyLdecayi ✏iH rated and readsL1H ofi d0and= orderHHdd+as✏iLi0 =with 10Li ✏ i ✏i(˜H,mµdi/µ and,) whereHd0 = forµHisd + themM✏iL Higgsinoi1with ✏i v,µi/µ KSVZ’s, where µ is theC Higgsinoa˜ff˜ ’s are approximately smaller 2.2 R-parity violations ⌫ ⌘ LR ml e˜Rl ⌘ mass parameter originatedMij from= mass the MSSM parameter ikl superpotential2 jmk originatedmek 2 µH fromuH the2d. Inlog MSSM general, superpotential2 SUSY-⇠+(i µHj),uHd. In general,(11) SUSY- 16⇡2 2 2m2 2 m3 m $ than DFSZ’s by 10 . e C e˜˜Rlmf ma˜ e˜Lm e˜Lm breaking soft terms cannot be rotatedbreakingk,l,m away soft along terms with cannot thea˜fR redefinition,f be rotated away and✓ this along2 leads◆ with to the redefinition, and this leads to The most general form of RPV is represented (˜a + ⌫)= by the following C ( superpotentialm ,m˜ ) , [18]: (7) X 5 0 f fR non-zero sneutrino VEVs, ⌫˜i . Thenon-zero value! sneutrino of ⌫˜i signifies VEVs,4096 the⇡⌫˜i strength. The value of bilinear of ⌫˜i RPV,signifies the strength of bilinear RPV, h i h i h i e2 h i and it is oftenwhere parametrized the indices in terms denoteand it of is lepton’si often⌫˜i parametrized/v generation.. in ( termsm ˜ ) ofmlisi the ⌫˜i left-right/v. mixing matrix of where mf and m ˜ are the fermion and right-handed sfermion LR masses respectively. C0(mf ,m˜ ) W = ijkLiLjEkfR+ ijk0 ⌘hLiQjiDk + ijk00 UiDjDk + µiL⌘hiHu,i (6)fR slepton.is we a loop focus function on the defined (s)tau as follows: loop decay scenario as discussed previously. Setting the with the2.2 summationrelated SUSY R-parity among mass the parameters indices violationsi, to j, a k common=1, 2, 3denotingtheleptonandquark mass scale4 mSUSY, as done in [18], the upper 3 Axino dark matter3 with Axino1 R-parity dark matter violationsd q with R-parity3 violations1/2 C0(mf ,mf˜ )= , (8) limit of neutrino mass, mR ⌫ . 1eVimpliesthat2 2 2 2 i33 . 22 10 2(mSUSY2 /100GeV) . This generations assumed implicitly. The firsti⇡ three(q m termsf ) (q correspondp) m ˜ (( toq trilineark) mf ) RPV and Z fR ⇥ the3.1 last one DecayThe corresponds rate most to general bilinear3.1 RPV. Decay formLi, rateE ofi, Q RPVi, Di, Ui isand representedHu are the usual matter by the following superpotential [18]: where pµ is the four-momentum of the axino, and (p k)µ is the four-momentum of the superfields in the Minimal Supersymmetric Standard2 Model (MSSM). ijk, ijk0 , ijk00 are We first calculate theneutrino. one-loopC0( radiativemWef ,m firstf˜ )scalesroughlyas1 decay calculatea ˜ the + one-loop⌫/min˜5 theories, and radiative its mass with decay dimension trilineara ˜ is RPV inverse+ ⌫ in squared. theories with trilinear RPV R ! fR ! dimensionlessinduced by the coupling operators constantsLLE andinduced whereasLQD by. 2 the theThe operators coupling relevantLLE Feynman constantsand diagramsLQDµi’s. carry2 The are massshown relevant dimension Feynman diagrams are shown one.in Figure SM gauge 1. symmetries demandin FigureW the 1.= indices ijkiLandi4Lj E(jkand+ k )ofijk0 LijkiQ( jijk00D)tobek + ijk00 UiDjDk + µiLiHu, (6) Several comments are in order beforeSeveral we present comments the are final in result. order before For the we axino present mass the in final result. For the axino mass in antisymmetric. Since the UUD operator is irrelevant to our study, ijk00 =0isimposedby assumingconsideration, baryon tree-level number three-body conservation.consideration, decays are not tree-level allowed three-body kinematically. decays Hence, are not the allowed axino kinematically. Hence, the axino decays dominantly into and ⌫. Also,decays note dominantly that the (s)fermions into and ⌫ running. Also, note in the that loop the shown (s)fermions running in the loop shown Let uswith discuss bilinear the summation RPV in more detail. amongµiLiHu can the be rotated indices awayi, by j, redefining k =1, 2, 3denotingtheleptonandquark 2 2 L andForH a similaras L0 calculation= L that✏ H involvesand H neutralinoFor0 a= similarH decaying+ calculation✏ L radiativelywith that✏ involves viaµ RPV/µ neutralino operators,, where decaying seeµ is [19]. the radiatively Higgsino via RPV operators, see [19]. i dgenerationsi i i d assumedd d implicitly.i i i ⌘ i The first three terms correspond to trilinear RPV and mass parameter originated from the MSSM superpotential µHuHd. In general, SUSY- 3 3 breaking softthe terms last cannot one be corresponds rotated away along to with bilinear the redefinition, RPV. andL thisi, leadsEi, toQi, Di, Ui and Hu are the usual matter non-zero sneutrino VEVs, ⌫˜i . The value of ⌫˜i signifies the strength of bilinear RPV, and it is oftensuperfields parametrizedh in ini terms the of Minimal h⌫˜ i/v. Supersymmetric Standard Model (MSSM). ijk, ijk0 , ijk00 are i ⌘h ii dimensionless coupling constants whereas the coupling constants µi’s carry mass dimension 3 Axinoone. dark SM matter gauge symmetries with R-parity demand violations the indices i and j (j and k)of ijk ( ijk00 )tobe 3.1 Decayantisymmetric. rate Since the UUD operator is irrelevant to our study, ijk00 =0isimposedby We first calculateassuming the one-loop baryon radiative number decaya ˜ conservation. + ⌫ in theories with trilinear RPV 2 ! induced by theLet operators usLLE discussand LQD bilinear. The relevant RPV Feynman in more diagrams detail. are shownµiLiHu can be rotated away by redefining in Figure 1. SeveralL commentsi and H ared inas orderL beforei0 = L wei present✏iH thed finaland result.Hd0 For= theHd axino+ ✏ massiLi inwith ✏i µi/µ, where µ is the Higgsino consideration, tree-level three-body decays are not allowed kinematically. Hence, the axino ⌘ mass parameter originated from the MSSM superpotential µHuHd. In general, SUSY- decays dominantly into and ⌫. Also, note that the (s)fermions running in the loop shown breaking soft terms cannot be rotated away along with the redefinition, and this leads to 2For a similar calculation that involves neutralino decaying radiatively via RPV operators, see [19]. non-zero sneutrino VEVs, ⌫˜i . The value of ⌫˜i signifies the strength of bilinear RPV, and it is often parametrized3 h ini terms of h⌫˜ i/v. i ⌘h ii 3 Axino dark matter with R-parity violations
3.1 Decay rate We first calculate the one-loop radiative decaya ˜ + ⌫ in theories with trilinear RPV induced by the operators LLE and LQD. 2 The! relevant Feynman diagrams are shown in Figure 1. Several comments are in order before we present the final result. For the axino mass in consideration, tree-level three-body decays are not allowed kinematically. Hence, the axino decays dominantly into and ⌫. Also, note that the (s)fermions running in the loop shown
2For a similar calculation that involves neutralino decaying radiatively via RPV operators, see [19].
3 has the axion-like couplings to the quarks, mq sqq¯ ,orleptons, ml s¯ll,sothatitslife-timeis fa fa much shorter than the axino LSP. On the other hand, during the axino regeneration (8), the saxions are also populated by the same amount and thus one finds has the axion-like couplings to the quarks, mq sqq¯ ,orleptons, ml s¯ll,sothatitslife-timeis fa fa 2 much shorter than the−9 axinoms LSP.MeV On the otherΩa˜h hand, during the axino regeneration (8), msYs 10 GeV. (10) the saxions are also populated by the same amountmq and thus one findsm has≈ the axion-likeMeV couplingsm toa˜ the# quarks,0.28 $ sqq¯ ,orleptons, l s¯ll,sothatitslife-timeis ! "! " fa fa much shorter than the axino LSP. On the other hand,2 during the axino regeneration (8), −9 ms MeV Ωa˜h where Ys is the saxion numberthe saxions density arem ins alsoYs unit populated10 of the by entropy the same density. amount and Note thusGeV that one. finds this quantity (10) ≈ MeV >ma˜ # 0.28 $ is strongly constrained by the BBN. In the mass range! m"!s (10)" GeV, the2 above equation −5 −9 ms O MeV Ωa˜h gives msYs > 10 GeV for ma˜ =1MeV.NowthatthesaxiondecaysmainlytobottomandmsYs 10 ∼ GeV. (10) where Ys is the saxion number density≈ in unitMeV of the entropym˜ # density.0.28 $ Note that this quantity ∼ ! "!< a ">−2 charm quarks, oneis strongly finds a constrained strong limit by on the the BBN. saxion In the lifetime: mass rangeτs m10s sec(10) [27]. GeV, Specifically, the above equation >where−5Ys is the saxion number density in unit of∼ the entropy∼ O density. Note that this5 quantity the mass relationgives (6)m givessYs us10 mGeVs 14 for GeVma˜ =1MeV.Nowthatthesaxiondecaysmainlytobottomand for the axino mass ma˜ 1MeVand> Λ =10 is strongly≈ constrained by the BBN. In the mass range≈ms < (10)−2 GeV, the above equation GeV. Then, thecharm saxion quarks, lifetime,∼ one finds a− strong5 limit on the saxion lifetime: τs O10 sec [27]. Specifically, gives m Y > 10 GeV for m˜ =1MeV.Nowthatthesaxiondecaysmainlytobottomand∼ s s a m 5 the mass relation (6)∼ gives us ms 14has GeV the axion-like for the axino couplings mass to∼ thema˜< quarks,1MeVand−2 q sqq¯ ,orleptons,Λ =10ml s¯ll,sothatitslife-timeis charm quarks, one finds a− strong1 limit on the saxion lifetime: τs 10 secfa [27]. Specifically,fa decaying axino DM that2 explains≈ the 3.5 keV X-ray line. Before we conclude,≈ we describe 5 GeV. Then,the the mass saxion relation lifetime,1 m (6) gives usmuchm shorter14 GeV than for the the axino axino LSP. mass∼ Onm˜ the1MeVand other hand,Λ during=10 the axino regeneration (8), cosmological and phenomenologicalb < consequencess ≈−2 and implications of oura ≈ model. GeV.τ Then,s the saxion2 ms lifetime,the10 saxionssec are, also populated by the same amount(11) and thus one finds ≈ %8π fa & ∼ 2 −1 1 mb −2 2 < −1 2 τs 2 m1s m 10 sec , −9 ms MeV Ωa˜h(11) b < ms−Y2s 10 GeV. (10) 2Framework ≈ %8πτsfa &
3.1 Decay rate We first calculate the one-loop radiative decaya ˜ + ⌫ in theories with trilinear RPV induced by the operators LLE and LQD. 2 The! relevant Feynman diagrams are shown in Figure 1. Several comments are in order before we present the final result. For the axino mass in consideration, tree-level three-body decays are not allowed kinematically. Hence, the axino decays dominantly into and ⌫. Also, note that the (s)fermions running in the loop shown
2For a similar calculation that involves neutralino decaying radiatively via RPV operators, see [19].
3 2 2 The decay rate in Eq. (7) is proportionalThe decay to mf rate. This in Eq. is due (7) to is aproportional chirality flip to atmf . This is due to a chirality flip at the fermionic internal line of the loopthe diagrams fermionic in internal Figure 1. line Thus, of the loop loop contributions diagrams in Figure 1. Thus, loop contributions from (s)fermions of the third generationfrom of (s)lepton (s)fermions or of (s)quark the third have generation the most of significant (s)lepton or (s)quark have the most significant impact on the decay rate. The decay rateimpact of the on theconjugate decay rate. process Thea ˜ decay +¯ rate⌫ is the of the same, conjugate processa ˜ +¯⌫ is the same, and gives a factor of two to the overalland2 decay gives rate. a factor Note of that two the to overall the! overall decay decay rate rate. must Note that the overall! decay rate must The decay rate in Eq. (7) is proportional to mf . This is due to a chirality flip at the fermionic internalalso line be of multiplied the loop by the diagrams color factor inalso for Figure colored be multiplied (s)fermions. 1. Thus, by the color loop factor contributions for colored (s)fermions. So far, we have not discussed the axino-fermion-sfermionSo far, we have not discussed coupling from the axino-fermion-sfermion the Yukawa coupling from the Yukawa from (s)fermions of theinteraction, third generation i.e. of (s)leptoninteraction, or (s)quark i.e. have the most significant impact on the decay rate. The decay rate of the conjugate processa ˜ +¯⌫ is the same, c c ˜ = C ˜ f˜⇤ aP˜¯ Lf + C ˜ f˜⇤aP˜¯ Rf !+h˜ =.cC.,˜ f˜⇤ aP˜¯ Lf + C (9)˜ f˜⇤aP˜¯ Rf +h.c., (9) and gives a factor of two to the overallL decaya˜ff a˜ rate.fRfL R Note thata˜fLfR theL overallLa˜ff decaya˜fRfL R rate musta˜fLfR L
also be multiplied by thewhere colorC ˜ factoris the forproduct colored of the (s)fermions. axino-Higgsinowhere C ˜ mixingis the product ( v/fa of) and the the axino-Higgsino SM Yukawa mixing ( v/fa) and the SM Yukawa a˜fRfL a˜fRfL ⇠ ⇠ coupling. The strength of C ˜ ’s forcoupling. the third generation The strength of fermionsof C ˜ ’s is forcomparable the third with generation of fermions is comparable with So far, weAxino have not discussed (~ the7 axino-fermion-sfermionkeV)a˜fRfL coupling froma˜fRfL the Yukawa those from the gauge interaction as inthose Eq. (5). from However, the gauge in interactioncontrast to asthe in axino-fermion- Eq. (5). However, in contrast to the axino-fermion- interaction, i.e. sfermion coupling from the gauge interaction,sfermion coupling i.e. Eq. from (3), where the gauge the decay interaction, amplitude i.e. Eq. (3), where the decay amplitude picks up a chirality-flipping m (see Eq.picks (7)), up the a chirality-flipping decay amplitudem for(see contributions Eq. (7)), the from decay amplitude for contributions from ˜ ¯ f ˜ ¯ c f [1403.6621] trilinear the˜ R-parity Lagrangian= C ˜ off ⇤ Eq.aP˜ violationL (9)f picks+ C up˜ anthef m⇤ LagrangianaP˜a˜ insteadRf +h [19]. of.c Eq.., Since (9)m picksa˜ upmf anform thea˜ instead(9) third [19]. Since ma˜ mf for the third • a˜ff a˜fRfL R a˜fLfR L ⌧ ⌧ Lgeneration of fermions, we can neglectgeneration the contributions of fermions, from we the can Lagrangian neglect the of contributionsEq. (9). from the Lagrangian of Eq. (9). where C is the productTo be of more the concrete, axino-Higgsino let us consider mixingLLETo beRPV more ( operatorsv/f concrete,) and with let us the (s)tau consider SM running YukawaLLE inRPV the operators with (s)tau running in the a˜f˜RfL ⌫ a ⌫ coupling. The strengthloop of C ( i33,i=1’s, for2). From the third Eqs. (5) generation andloop (7), ( thei33,i lifetimeof⇠=1 fermions, 2). of From the axino is Eqs. comparable reads (5) and (7), the with lifetime of the axino reads a˜f˜RfL 2 2 2 2 of axino with Higgsino is v/fa,of where axinov withis the˜ Higgsino Higgs vacuum is v/f expectationa, where v3 is value the Higgs(VEV). vacuum4 expectation32 value (VEV). 4 2 those from the gauge interactionfR as27 in Eq. i33 (5).gˆ However,ma˜ inffa contrastR/v 27 m to⌧˜R i the33gˆ axino-fermion-mm⌧ a˜ fa/v m⌧˜R m⌧ ⇠ ⌧a˜ 8.7 10 secR ⇠ ⌧a˜ 8.7 10 sec . (10) . (10) Hence, Hence, ' ⇥ 10 4 7keV ' 10⇥8 100R GeV10 4 1.777keV GeV 108 100 GeV 1.77 GeV sfermion coupling from the gauge interaction,✓ ◆ i.e. Eq.✓ (3), where◆ ✓ the decay◆ amplitude✓ ◆ a˜ fR a˜ ⇣ ⌘ ⇣ f˜R ⌘ ⇣ ⇣ ⌘⌘ ⇣ ⌘ ⇣ ⌘ picks up a chirality-flipping mf (seev Eq. (7)), the decay amplitudev for4 contributions from2 4 2 Here, we have factored out the dimensionfulHere, we parameter have factored 1/m outfrom the dimensionfulC0(mf ,m˜ ) parameter,and 1/m from C0(mf ,m˜ ) ,and Ca˜ff˜ gˆ , Ca˜ff˜ gˆ , (5)⌧˜R fR (5)⌧˜R fR fRf 4 f˜2f 4 2 the Lagrangian of Eq.used (9)C picks0'(1.77a GeV up, an100m GeV)a˜ instead7 used10 [19]. CGeV0'(1 Since. 77R .a GeV Itm can, 100a˜ be GeV) seenmf fromfor7 Eq. the10 (10) thirdGeV that . It can be seen from Eq. (10) that | 9 11 | ' ⇥ 3| 9 1 11 ⌧ | ' ⇥ 3 1 generation of fermions,fa we can(10 neglect10 )GeVfor the contributions i33 f(10a (10 from10 ).10 the This)GeVfor Lagrangian parameter i33 region of(10 Eq. is consistent (9). 10 ). This parameter region is consistent whereg ˆ is the gauge coupling constantwhere⇠g suppressedˆOis the gauge by couplingthe Higgsino-gaugino constant⇠ O ⇠ suppressedO mixing. by the Higgsino-gaugino⇠ O mixing. To be more concrete,with let the us observed consider 3.5 keVLLE X-ray RPV line.with operators the observed with 3.5 (s)tau keV X-ray running line. in the One can compare the strength of OneCa˜ff˜ ’s can for compare KSVZ and the DFSZ strength models of Ca˜ fromff˜ ’s for Eqs. KSVZ (4)-(5). and DFSZ models from Eqs. (4)-(5). 2 (a) 2 (b) gˆ is typicallyloop of ( order i33,i 10=1 ,, and2).gˆ From forisM typically1 Eqs.v, (5) ofKSVZ’s order and (7),C 10a˜ff ˜ ’s the, and are lifetime forapproximatelyM1 of thev, KSVZ’s axino smallerC readsa˜ff˜ ’s are approximately smaller 2 3.2⇠ Constraints2 3.2⇠ Constraintsstau than DFSZ’s by 10 Figure. 1: Feynmanthan diagrams DFSZ’s by of the 10 process. a ˜ + ⌫ via LLE or LQD RPV operators. 2 3! 2 4 2 Here, right-handedNext, (s)fermions i33 wegˆ study are possiblem assumeda˜ experimental to befa running/vNext, constraints inwe the studym on⌧˜ loop. possiblei33. The Trilinear experimental (s)fermionm RPVs⌧ generate constraints neu- on i33. Trilinear RPVs generate neu- 27 R ⌫ ⌫ ⌧a˜ 8flow.7 is in10 thesec oppositetrino direction masses4 at for the the one-loop conjugate level. process The8trino neutrinoa ˜ masses +¯ mass⌫ at. the matrix one-loop (Mij)receivesthefollowing level. The neutrino. (10) mass matrix (Mij)receivesthefollowing 2.2 R-parity' violations⇥ 2.2contributions10 R-parity from7keV RPVviolations couplings10 contributions[20,! 21,100 18]: GeV from RPV1 couplings.77 GeV [20, 21, 18]: ✓ ◆ ⇣ ⌘ ✓ ijk ◆ ⇣ ⌘ ⇣ ijk⌘ The most general form of RPV isThe represented most general by the form following of RPV superpotential is representede2 [18]: by4 the following2 superpotential2 e2 [18]: 2 in Figure 1 are charged (s)leptons⌫ for1 the LLE scenario and(˜m down-typeLR)ml ⌫ (s)quarks1me˜Rl for the (˜mLR)ml me˜Rl Here, we have factored out theM dimensionful= parameter m 1/mM⌧˜ log=from C0( +(mi f ,mjm)f,˜ ) ,and(11) log +(i j), (11) LQD scenario. The one-loop decayij 16 amplitude⇡2 ikl involvesjmk ek m a2 axino-sfermion-fermionm2 Rij 16m⇡2 vertex,ikl $jmk eRk m2 m2 m2 $ k,l,m4 2 e˜Rl e˜Lm e˜Lmk,l,m e˜Rl e˜Lm e˜Lm used WC0=with(1 .77ijk couplingL GeViLjE,k 100 constant+ ijk0 GeV)LiQ givenjDk in+W7 Eqs. ijk00 =U10 (4)-(5) iDijkX jLDikLGeV for+jEµ KSVZki L+i.H ijk0u and It,LiQ can DFSZjDk be+ models seenijk00✓(6)Ui respectively.DX fromj◆Dk + Eq.µiLi (10)Hu, that ✓ (6) ◆ | 9 11 | ' ⇥can 3avoid1 astrophysical bound e2 e2 fa (10Notice that10 C)GeVfora˜ff˜ inwhere KSVZ the modelsi indices33 is denote only(10 an lepton’s e↵ective10 generation. vertex).where This induced the (m parameter˜ indices at) theis denote the one-loop left-right region lepton’s level. is mixing generation. consistent matrix (m ˜ of ) is the left-right mixing matrix of with the summation⇠ O among thewith indices thei, summation j, k =1⇠ O, 2, among3denotingtheleptonandquark the indices i, j, kLR=1ml , 2, 3denotingtheleptonandquarkLR ml with theA observed more rigorous 3.5 calculation keVslepton. X-ray we of focusa ˜ line. on+⌫ thewould (s)tau involve loop theslepton. decay treatment scenario we focus of as Feynman discussed on the (s)tau diagrams previously. loop decay Setting scenario the as discussed previously. Setting the generations assumed implicitly.generations The first three assumed! terms implicitly.light correspond stau! The to trilinear first three RPV terms and correspond to trilinear RPV and with two loops. However,related SUSY since mass KSVZ’s parametersCa˜ff˜ is to smaller a commonrelated than mass SUSY those scale mass in DFSZm parametersSUSY, models as done to by ain common [18], the mass upper scale mSUSY, as done in [18], the upper the last one corresponds to bilinearthe RPV. last oneL , correspondsE , Q , D , U toand bilinearH are RPV. theL usual, E , matterQ , D ,3U and H are the1/2 usual matter 3 1/2 two orders of magnitude,limit of neutrino onei cani naivelymass,i i m deducei 1eVimpliesthat thatu limit the decay of neutrinoi amplitudei 2 mass,i 10 ofi ma ˜(mi 1eVimpliesthat +/100GeV)⌫u . This 2 10 (m /100GeV) . This ⌫ . i33 . ⇥ ⌫ !.SUSY i33 . ⇥ SUSY superfields3.2 in the Minimal Constraintsfor KSVZ Supersymmetric modelssuperfields is much smaller in Standard the than Minimal Model those Supersymmetric in (MSSM). DFSZ models. ijk, Henceforth, Standardijk0 , ijk00 are forModel simplicity, (MSSM). ijk, ijk0 , ijk00 are dimensionless couplingour constants focus is solely whereasdimensionless on axino the couplingDM coupling in DFSZ constants constants models.µi whereas’s carry mass the coupling dimension constants µi’s carry mass dimension We assume that the right-handed sfermion contributes dominantly5 to the decay am- 5 one. SM gaugeNext, symmetries we study possible demandone. the SM experimental indices gauge symmetriesi and constraintsj (j and demandk)of on the iijk33 indices.( Trilinearijk00 )tobei and RPVsj (j and generatek)of ijk neu-( ijk00 )tobe plitude ofa ˜ + ⌫. Summing Feynman diagrams (1a) and (1b) in Figure⌫ 1, the decay antisymmetric.trino Since masses the atUUD the!operatorantisymmetric. one-loop is irrelevant level. Since The toµ the⌫ neutrino ourUUD study,operator mass ijk00 =0isimposedby matrix is irrelevant (Mij)receivesthefollowing to our study, ijk00 =0isimposedby assuming baryon numberamplitude conservation. is foundassuming to be of baryon the form number kµ✏ conservation.⌫⇤, where kµ and ✏⌫ are the momentum and the contributionspolarization from vector RPV of couplings the photon respectively,ijk [20, 21, as 18]: expected from gauge invariance. Con- Let us discuss bilinear RPV in moreLet detail. us discussµiL bilineariHu can RPV be rotated in more away detail. by redefiningµiLiHu can be rotated away by redefining sidering only one trilinear RPV coupling at a time, and letting denote ijk and ijk0 ,the e2 2 Li and Hd as Li0 = Ldecayi ✏iH rated and readsLHi d0and= HHdd+as✏iLi0 =withLi ✏i ✏iHµdi/µand, whereHd0 =µHisd + them✏iL Higgsinoi with ✏i µi/µ, where µ is the Higgsino ⌫ 1 ⌘(˜mLR)ml e˜Rl ⌘ mass parameter originatedMij from= mass the MSSM parameter ikl superpotential jmk originatedmek 2 µH fromuH the2d. Inlog MSSM general, superpotential2 SUSY-+(i µHj),uHd. In general,(11) SUSY- 16⇡2 2 2m2 2 m3 m $ e C e˜˜Rlmf ma˜ e˜Lm e˜Lm breaking soft terms cannot be rotatedbreakingk,l,m away soft along terms with cannot thea˜fR redefinition,f be rotated away and✓ this along2 leads◆ with to the redefinition, and this leads to (˜a + ⌫)= C (m ,m˜ ) , (7) X 5 0 f fR non-zero sneutrino VEVs, ⌫˜i . Thenon-zero value! sneutrino of ⌫˜i signifies VEVs,4096 the⇡⌫˜i strength. The value of bilinear of ⌫˜i RPV,signifies the strength of bilinear RPV, h i h i h i e2 h i and it is oftenwhere parametrized the indices in terms denoteand it of is lepton’si often⌫˜i parametrized/v generation.. in ( termsm ˜ ) ofmlisi the ⌫˜i left-right/v. mixing matrix of where mf and m ˜ are the fermion and right-handed sfermion LR masses respectively. C0(mf ,m˜ ) fR ⌘h i ⌘h i fR slepton.is we a loop focus function on the defined (s)tau as follows: loop decay scenario as discussed previously. Setting the
related SUSY mass parameters to a common mass scale4 mSUSY, as done in [18], the upper 3 Axino dark matter3 with Axino1 R-parity dark matter violationsd q with R-parity3 violations1/2 C0(mf ,mf˜ )= , (8) limit of neutrino mass, mR ⌫ . 1eVimpliesthat2 2 2 2 i33 . 22 10 2(mSUSY2 /100GeV) . This i⇡ (q mf ) (q p) m ˜ ((q k) mf ) Z fR ⇥ 3.1 Decay rate 3.1 Decay rate where pµ is the four-momentum of the axino, and (p k)µ is the four-momentum of the 2 We first calculate theneutrino. one-loopC ( radiativemWe,m first˜ )scalesroughlyas1 decay calculatea ˜ the + one-loop⌫/min5 theories, and radiative its mass with decay dimension trilineara ˜ is RPV inverse+ ⌫ in squared. theories with trilinear RPV 0 f fR f˜ induced by the operators LLE andinducedLQD by. 2 theThe operators! relevantLLE FeynmanR and diagramsLQD. 2 The are! shown relevant Feynman diagrams are shown in Figure 1. in Figure 1. 4 Several comments are in order beforeSeveral we present comments the are final in result. order before For the we axino present mass the in final result. For the axino mass in consideration, tree-level three-bodyconsideration, decays are not tree-level allowed three-body kinematically. decays Hence, are not the allowed axino kinematically. Hence, the axino decays dominantly into and ⌫. Also,decays note dominantly that the (s)fermions into and ⌫ running. Also, note in the that loop the shown (s)fermions running in the loop shown
2For a similar calculation that involves2 neutralinoFor a similar decaying calculation radiatively that involves via RPV neutralino operators, decaying see [19]. radiatively via RPV operators, see [19].
3 3 Conclusion
• Axino Dark Matter is interesting! • With R-parity violation, we could “see” axino DM from via indirect detection. • Axino DM can explain several recently observed astrophysical anomalies.
21 Backup Gravitino vs Axino
M. Endo, K. Hamaguchi, SPL, K. Mukaida, K. W. Buchmuller, M. Garny [arXiv:1206.7056] Nakayama [arXiv:1301.7536] Annihilating DM vs. Decaying DM
W. Buchmuller, M. Garny [arXiv:1206.7056] J.C. Park, S.C. Park [arXiv:1207.4981] 2 Axino dark matter with R-parity violation
2.1 Axino decay rate with R-parity violation Let us first introduce R-parity violations. In this letter, we consider a bilinear type of the R-parity violation [21], which is characterized by the superpotential,
W = µiLiHu, (2.1) where Li and Hu are chiral superfields of the lepton doublet and the up-type Higgs doublet, respectively. The index i =1, 2, 3denotesthegeneration,andµi is a parameter with a mass dimension. Here and hereafter, summation over i is implicitly promised. By redefining Li and the down-type Higgs superfield Hd as Li0 = Li ✏iHd and Hd0 = Hd +✏iLi with ✏ µ /µ, where µ is the higgsino mass parameter appearing in the superpotential i ⌘ i as W = µHuHd, the R-parity violating superpotential (2.1) is eliminated. Hereafter, for notational simplicity, the primes on the redefined fields are omitted. After the redefinition, the SUSY breaking potential becomes
2 RPV = BiL˜iHu + m L˜iH⇤ +h.c., (2.2) L LiHd d where L˜i is a scalar2 component Axino dark of matter the superfield with R-parityLi. violation The coe cients are Bi B✏i and 2 2 2 2 2 ' mL H (m˜ m2.1H )✏ Axinoi, where decayB rate, m with˜ and R-paritymH violationrepresent soft SUSY breaking parameters i d ' Li d Li d Let us first introduce R-parity violations. In2 this letter,2 we consider2 a bilinear2 type of2 the ˜ 2 in the MSSM, soft =(BHuHd +h.c.)+mH Hd +mH Hu +m ˜ Li + .Duetothe L R-parity violation [21], which is characterizedd by| the superpotential,| u | | Li | | ···
R-parity violating scalar potential (2.2),W = sneutrinosµiLiHu, obtain non-zero(2.1) vacuum expectation values (VEVs) as where Li and Hu are chiral superfields of the lepton doublet and the up-type Higgs doublet, 2 respectively. The index i =1, 2,m3denotesthegeneration,andcos + Bi sinµi is a parameter with a mass dimension. Here and hereafter,LiH summationd over i is implicitly promised. By ⌫˜i = v, (2.3) redefining L and the down-type Higgs superfield H as2L0 = L ✏ H and H0 = H +✏ L i d i i i d d d i i with ✏ µ /µ, whereh iµ is the higgsino mass parameterm⌫˜i appearing in the superpotential i ⌘ i as W = µHuHd, the R-parity violating superpotential (2.1) is eliminated. Hereafter, for where tan vu/vnotationald is a simplicity, ratio the of primes the on the VEVs redefined of fields the are omitted. up- After and the down-type redefinition, Higgs fields, v v2 + v2 174⌘ GeV,the SUSY and breakingm2 potentialis a becomes sneutrino mass. ⌘ u d ⌫˜i 2 = B L˜ H + m L˜ H⇤ +h.c., (2.2) ' RPV i i u LiHd i d Before proceeding to discuss phenomenological L aspects, several comments are in order. ˜ p where Li is a scalar component of the superfield Li. The coe cients are Bi B✏i and ˜ ˜ It is possible to introduce2 2 the2 bilinear2 R-parity2 violating soft' terms, L H and L H⇤ in mL H (m˜ mH )✏i, where B, m˜ and mH represent soft SUSY breaking parameters i u i d i d ' Li d Li d 2 2 2 2 2 ˜ 2 in the MSSM, soft =(BHuHd +h.c.)+mH Hd +mH Hu +m ˜ Li + .Duetothe addition to (2.1), before the L field redefinition.d | | Theu coe| | cientsLi | | ··· in (2.2) then have additional R-parity violating scalar potential (2.2), sneutrinos obtain non-zero vacuum expectation contributions, butvalues the (VEVs) following as analysis will not be a↵ected as far as the R-parity violation 2 m cos + Bi sin ⌫˜ = LiHd v, (2.3) is parametrized by the the sneutrinoh ii VEV (2.3).m2 Next, trilinear R-parity violating terms, neutrino state at tree level. This can be understood as a kind⌫˜i of seesaw mechanism, in which LLE and LQD, arewhere also tan generatedvu/vd is a ratio of by the the VEVs field of the up- redefinition. and down-type Higgs They fields, v are subdominant and the neutral gauginosv2 + andv2 higgsinos174⌘ GeV, and playm2 theis rˆole a sneutrino of the mass. right-handed neutrinos. Indeed, in the⌘ Neutrinou d ' masses⌫˜i from bilinear will be ignoredlimit of a small in neutrino-neutralino theBefore following proceeding to mixing, discuss study, the phenomenological7 because7 neutrino-neutralino aspects, the several terms mass comments matrix are areM multipliedN inhas order. a by the Yukawa p × “seesaw” structure,It is possiblewith a strong to introduce hierarchy the2 bilinearbetween R-parity the gaugino-higgsino violating soft terms, diagonalL˜ H 4and4L˜blockH in couplings. RPV i u × i d⇤ Mχ and the off-diagonaladdition to3 (2.1),4 beforeblock them: field redefinition. The coe cients in (2.2) then have additional The sneutrinocontributions, VEVs (2.3)× but the induce following analysis mixings will not be between a↵ected as far as the the R-parity SM violationleptons and the gauginos. is parametrized by the the sneutrino VEV (2.3).T Next, trilinear R-parity violating terms, Mχ m The SM neutrinos mixLLE and withLQD, the areM alsoN binotree generated= and by the the field neutral redefinition.. wino, They are and subdominant the(5.4) SM and charged leptons mix | m 03×3 will be ignored in the following study,! because the" terms are multiplied by the Yukawa with the charged winos.couplings. Hence, the R-parity violating parameters are constrained. The The effective massThe matrix sneutrino obtained VEVs by (2.3) integrating2 induce2 mixings out the between neutralinos, the SM which leptons yields and the the neutrino gauginos. neutrinosmass obtain and mixing masses angles, of is givenm⌫ by Mgν ⌫˜ mM/m−˜1 m˜T .Thefactthatonlyoneneutrino, where m ˜ ˜ is a bino (wino) mass [22, The SM neutrinos mix with thetree bino"− and theχ neutralB(W ) wino, and the SM chargedB(W ) leptons mix becomes massivewith is the most charged easily winos. seen⇠ in Hence, a basish thei in R-parity which the violating superpotential parametersR arep mass constrained. parameters7 The 2 2 23, 24]. For gauginoneutrinos masses obtain masses of O of(100)m g GeV,⌫˜ /m ˜ ˜ , wherei m ˜⌫˜˜i0 /vis a$ bino0 (wino)10 massis [22, imposed to satisfy the µ1 and µ2 (together with the sneutrino VEVs⌫ vi)vanish;thenbothB(W ) B(LW1) and L.2 decouple from ⇠ h i ⌘h i7 M ,ascanbeseenfromEq.(2.44).Ofcourse,whenquantumcorre23, 24]. For gaugino masses of O(100) GeV, i ⌫˜i /v . 10 ctionsis imposed are included, to satisfy all the N |tree ⌘h i three neutrinos acquire a mass, as explained in the next sections. 2 2 In order to determine the flavour composition of the single neutrino that acquires a mass at this level, one has in principle to diagonalize the chargino mass matrix. Indeed, since bilinear R-parity violation mixes charged leptons with charginos, thephysicalchargedleptonsarethe three lightest eigenstates of the 5 5 charged lepton-chargino mass matrix MC ,Eq.(2.45).In × e general these do not coincide with the eigenstates of the Yukawa matrix λij.However,inthe limit of a small charged lepton-chargino mixing we are interested in, one can identify the two bases to a good approximation. We therefore choose to write MN and MC in the basis in which e the sneutrino VEVs vi vanish and the charged lepton Yukawa couplings λij are diagonal. In this basis, the effective neutrino mass matrix reads [181]:
2 µ1 µ1µ2 µ1µ3 ν −1 T mνtree 2 Mtree mMχ m = 2 µ1µ2 µ2 µ2µ3 , (5.5) "− − i µi 2 µ1µ3 µ2µ3 µ3 # where mνtree is given by Eqs. (2.46) and (2.47),
M 2 cos2β (M c2 + M s2 ) µ cos ξ m Z 1 W 2 W tan2 ξ . (5.6) νtree " M M µ cos ξ M 2 sin 2β (M c2 + M s2 ) 1 2 − Z 1 W 2 W The misalignment angle ξ,definedinsubsection2.3.1,isabasis-independentquantity that controls the size of the bilinear R effects in the fermion sector (in particular, assuming small $ p neutrino-neutralino and charged lepton-chargino mixings amounts to require sin ξ 1). In 2 % the basis in which we are working, it is given by sin ξ = i µi /µ.Asalreadynoticedin section 2.3.3, phenomenologically relevant values of mν require a strong alignment between tree (# the 4-vectors µ (µ ,µ) and v (v ,v), sin ξ 3 10−6 1+tan2 β m /1 eV. α ≡ 0 i α ≡ 0 i ∼ × νtree Several authors [39, 40] have studied the possibility of obtaining the desired amount of align- ( ( ment from GUT-scale universality in the soft terms. With thisassumptiontheLEPboundon the tauneutrino mass can be satisfied in a relatively large region of the parameterspace, but a significant amount of fine-tuning in the bilinear Rp parameters is necessary in order to reach the eV region (one would typically need µ /µ 10$ −4 at the GUT scale). Another possibility is to i ∼ invoke an abelian flavour symmetry [35]; however rather largevaluesoftheassociatedcharges are necessary to yield the desired alignment (see section 2.5).
2 As in section 2.3, we are workingin a (Hd, Li)basisinwhichthesneutrinoVEVsvanish,vi =0.Theseesaw structure may be less obvious in an arbitrary basis where both µi =0and vi =0,sincelargevaluesofµi and vi are in principle compatible with a small physical neutrino-neutralino$ mixing (see$ subsection 2.3.3). 2 The decay rate in Eq. (7) is proportional to mf . This is due to a chirality flip at the fermionic internal line of the loop diagrams in Figure 1. Thus, loop contributions The decay rate in Eq. (7) isfrom proportional (s)fermions to m2 of. Thisthe third is due generation to a chirality of flip (s)lepton at or (s)quark have the most significant f The decay rate in Eq. (7) is proportional to m2 . This is due to a chirality flip at the fermionic internal line of the loop diagrams in Figure 1. Thus, loop contributions f impact on the decay rate. Thethe decay fermionic rate of internal the conjugate line of the loop process diagramsa ˜ in Figure+¯⌫ is 1. the Thus, same, loop contributions from (s)fermions of the third generation of (s)lepton or (s)quark have the most significant ! and gives a factor of two to thefrom overall (s)fermions decay of rate. the third Note generation that the of (s)lepton overall or decay (s)quark rate have must the most significant impact on the decay rate. The decay rate of the conjugate processa ˜ +¯⌫ is the same, !impact on the decay rate. The decay rate of the conjugate processa ˜ +¯⌫ is the same, and gives a factor of two to thealso overall be decay multiplied rate. Note by that the the color overall factor decay for rate colored must (s)fermions. ! To be more concrete, let usand consider gives a factorLLE ofRPV two to operators the overall decay with rate. (s)tau Note running that the inoverall the decay rate must also be multiplied by the color factor for colored (s)fermions. also be multiplied by the color factor for colored (s)fermions. To be more concrete, let usloop consider ( iLLE33,i=1RPV, 2). operators From with Eqs. (s)tau (5)To andrunning be more (7), in concrete, the the lifetime let us of consider the axinoLLE RPV reads operators with (s)tau running in the loop ( i33,i=1, 2). From Eqs. (5) and (7), the lifetime of the axino reads loop2 ( i33,i=1, 2). From Eqs.2 (5) and (7), the lifetime of the axino reads 3 4 2 2 272 i33gˆ ma˜ fa/v m⌧˜R m⌧ gˆ m 3 f /v m 4 m 2 2 2 27 i33 ⌧a˜ a˜ 8 .7 a10 sec ⌧˜R ⌧ gˆ m 3 f /v m .4 (9)m 2 ⌧a˜ 8.7 10 sec Neutrino4 masses. 27(9) from8 i33 trilineara˜ a ⌧˜R ⌧ 4 ' ⇥ 8 10 ⌧a˜ 7keV8.7 10 sec10 100 GeV 1.77 GeV . (9) ' ⇥ 10 7keV 10 100 GeV 1.77 GeV' ⇥ 10 4 7keV 108 100 GeV 1.77 GeV ✓ ◆ ⇣ ⌘ ✓ ◆ ⇣ ✓ ⌘ ◆⇣ ⇣ ⌘ ⌘ ✓ ◆ ⇣ ⌘ ⇣ ⌘ 2 ✓ ◆ ⇣ ⌘ ✓ ◆ ⇣ ⌘2 ⇣ ⌘ 4 4 2 Here, we have factored out the dimensionful parameter 1/m from C (m ,m˜ ) ,and 4 Here, we have factored⌧ out˜R theHere, dimensionful0 f we havefR factored parameter out the dimensionful 1/m⌧˜ from parameterC0(mf 1,m/m ˜ from) ,andC (m ,m˜ ) ,and RPV R ⌧f˜RR 0 f fR 4 2 used C (1.77 GeV, 100 GeV) 7 10 GeV . It can be seen from Eq. (9)4 that 2 4 2 0 used C0(1.77 GeV, 100 GeV) 7 10 GeV . It can be seen from Eq. (9) that | 9 11 | 'used⇥ C0(13 .77 GeV1 , 100 GeV) 7 10 GeV . It can be seen from Eq. (9) that f (10 10 )GeVfor (10 10 ). This parameter region| is consistent9 11 | ' ⇥ 3 1 a i33 | 9 11 | f'a ⇥(10 103 )GeVfor1 i33 (10 10 ). This parameter region is consistent with⇠ theO observed 7.5 keV X-rayfa line.⇠ O (10 10 )GeVfor i33 ⇠ O (10 10 ). This⇠ parameterO region is consistent with⇠ theO observed 7.5 keV X-raywith line.⇠ theO observed 7.5 keV X-ray line. 3.2 Constraints 3.2 Constraints 3.2 Constraints Next, we study possible experimental constraints on i33. Trilinear RPVNext, generate we study neutrino possible experimental constraints on i33. Trilinear RPV generate neutrino masses at the one-loop level. The neutrino mass matrix (M ⌫ )receivesthefollowing ⌫ ij masses at the one-loop level. The neutrino mass matrix (Mij)receivesthefollowing Next, we study possible experimental constraints on i33. Trilinear RPV generate neutrino contributions from RPV couplings ijk [20, 21, 18]: contributions from RPV couplings ijk [20, 21, 18]: masses at the one-loop level. The neutrino mass matrix (M ⌫ )receivesthefollowing 1 (˜me2 ) m2 ij e2 2 ⌫ LR ml e˜Rl ⌫ 1 (˜mLR)ml me˜Rl M = ikl jmkme log +(i j), (10) ij 2 contributionsk 2 from2 RPV2 couplings ijk [20,Mij = 21, 18]:2 ikl jmkmek 2 2 log 2 +(i j), (10) 16⇡ me˜ me˜ me˜ $ 16⇡ m m m $ k,l,m RlFigure 5.1:Lm One-loop✓ Lm contributions◆ to neutrino masses andk,l,m mixings inducede by˜Rl the trilineare˜Lm e˜Lm X 2 ✓ ◆ R couplings λ (a) and λ! (b). The cross on thee2 sfermionX line indicates the insertion ofa p ⌫e2 ijk1 ijk (˜mLR)ml me˜Rl e2 where the indices denote lepton’s generation.! (m ˜ LR)ml is the left-rightwhere mixing the matrix indices of denote lepton’s generation. (m ˜ LR)ml is the left-right mixing matrix of left-rightMij mixing= mass term. Theikl arrowsjmkm onek external legs followlog the flow of the lepton+( number.i j), (10) slepton. we focus on the (s)tau loop decay scenario as16 discussed⇡2 previously.slepton. Setting wem focus2 the onm the2 (s)tau loopm2 decay scenario$ as discussed previously. Setting the k,l,m e˜Rl e˜Lm ✓ e˜Lm ◆ related SUSY mass parameters to a common mass scale mSUSYX, as donerelated in [18], SUSY the uppermass parameters to a common mass scale mSUSY, as done in [18], the upper 3 1/2 3 1/2 limit of neutrino mass, m 1eVimpliesthat 2 10 (m limit/100GeV) of neutrino. This mass, me2 1eVimpliesthat 2 10 (m /100GeV) . This ⌫ . where the indicesThei33 massive. denote⇥ neutrino lepton’s isSUSY mainly a generation. superposition of the (m ˜ electrLR⌫ ).oweakml is neutrino the left-right eigenstates,i33 mixing. and matrixSUSY of bound on the neutrino masses can be alleviated by raising the left-handedbound stau on mass.the neutrino For masses can be alleviated by raising⇥ the left-handed stau mass. For slepton.its we flavour focus composition on the is (s)tau given, in loop the basis decay we are scenario considering, as by discussed the superpotential previously.Rp mass Setting the instance, when m =5m =500GeV,theboundisrelaxedto instance,0.02. when m =5m =500GeV,theboundisrelaxedto! 0.02. ⌧˜L ⌧˜R parameters µi [181]: i33 . ⌧˜L ⌧˜R i33 . We now switch our attentionrelated to other SUSY experimental mass parametersconstraints on to We a1. common now, switchare mass our attention scale m toSUSY other, as experimental done in [18], constraints the upper on . , are ν ijk 133(µ ν233+ µ ν + µ ν ) . (5.7) ijk 133 233 3 2 1 e 2 µ 3 τ 3 1/2 constrained to be 133, 233 0.limit07 (m of⌧R neutrino/100 GeV), mass, whichm is derived⌫ "1eVimpliesthatconstrained from theµ measure- to be 133 , i 33233 . 20.07 10( m⌧(Rm/100SUSY GeV),/100GeV) which is derived. This from the measure- . . i i . ⇥ ments of (⌧ µ⌫⌫¯)/ (µ e⌫⌫¯)[22,18].⇥ ments of (⌧ µ⌫⌫¯)/ (µ e⌫⇥⌫¯)[22,18]. ! ! bound onIn the terms neutrino of mixing angles, masses this can gives!" bethe relations alleviated! by raising! the left-handed stau mass. For In addition to i33, i033 could reproduce the observed X-ray line signal asIn well. addition However, to i33, i033 could reproduce the observed X-ray line signal as well. However, instance, when m⌧˜L =5m⌧˜R =500GeV,theboundisrelaxedtoµ1 µ2 i33 . 0.02. this means that sbottom has to be rather light, i.e. (100)sin GeVθ13 = asthis in means Eq. (9)., that Sincesin sbottomθ23 = has to be, rather light, i.e.(5.8) (100) GeV as in Eq. (9). Since ⇠ O 2 2 2 ⇠ O the LHC is capable of producing coloredWe now particles switch copiously our andattention imposingthe to LHC stringenti otherµi is capable experimental mass of producingµ2 + constraintsµ colored3 particles on copiouslyijk. 133 and, imposing233 are stringent mass constrained to be 133, 233 0bounds.07 ( onm colored⌧ /100 particles GeV), ( which100 GeV is derived in general), from (s)bottom-loop the measure- decay scenario is not bounds on colored particles (& 100 GeV inwhile general),sin θ12 (s)bottom-loopis undetermined.. decay!" scenarioR is not ! & 3 ⇥ 3 favored. ments of (⌧ µ⌫⌫¯)/ (µ e⌫favored.⌫¯)[22,18]. 3 0 3Sbottom decays into a quark and a neutrino via the trilinear! RPV couplings.! This mimicsSbottom the decaysq ˜ q into ˜0 a quark and a neutrino via the trilinear RPV couplings. This mimics theq ˜ q ˜ In addition to i33, i033 could reproduce the observed X-ray line signal as well. However, ! ˜ 0 5.1.30 One-Loop Contributionsand ˜b b ˜ Generated0 channels! studied by Trilinear at the LHC,!R whereCouplings ˜0 is the lightest neutralino [23, 24]. The bound on the and b b ˜ channels studied at the LHC, where ˜ is the lightest neutralino [23, 24]. The! bound on the p ! this means that sbottom has tosquark be massrather is 780 light, GeV at i.e. 95% CL when(100)m 0 GeV0 GeV as and in gluino Eq. is (9). decoupled Since [23]. For sbottom, the squark mass is 780 GeV at 95% CL when m ˜0 0 GeV and gluino is decoupled [23]. For sbottom, the ˜ ! ! bound is 620 GeV at 95% CL for⇠m O0 < 120 GeV [24]. ! bound is 620 GeV at 95% CL for m ˜0the< 120 LHC GeVAt [24]. is the capable one-loop of level, producing a variety of diagrams colored involving particles the trilinear copiouslyR ˜ couplings andλ imposingand λ and/or stringent mass ! p bounds oninsertions colored of bilinear particlesRp masses ( 100 contribute GeV to the in neutralino-neutrino general), (s)bottom-loop mass matrix, thus decay correct- scenario is not ing5 Eq. (5.5). In this! subsection,& we concentrate on the diagrams involving trilinear5 R cou- 3 ! p favored. plings only. These diagrams represent the dominant one-loopcontributiontoneutrinomasses 3 and mixings when bilinear R-parity violation is strongly suppressed (i.e. when sin ξ 0 and 0 Sbottom decays into a quark and a neutrino via the trilinear RPV couplings. This" mimics theq ˜ q ˜ sin0 ζ 0 in the language of subsection 2.3.1,0 where the angle ζ formed by the 4-vectors ! and ˜b b ˜ channels" studied at the LHC, where ˜ is the lightest neutralino [23, 24]. The bound on the ! Bα (B0,Bi) and vα (v0,vi) controls the Higgs-slepton mixing ). The one-loop diagrams squark mass is≡ 780 GeV at 95%≡ CL when m ˜0 0 GeV and gluino is decoupled [23]. For sbottom, the involving bilinearRp masses will be discussed! in the next subsection. bound is 620 GeV at 95%! CL for m ˜0 < 120 GeV [24]. The trilinear R couplings λ and λ! contribute to each entry of the neutrino mass matrix ! p ijk ijk through the lepton-slepton and quark-squark loops of Fig. 5.1, yielding [24, 182] 5 e 2 2 ν 1 (˜mLR)ml me˜Rl Mij λ = 2 λiklλjmk mek 2 2 ln 2 +(i j) , (5.9) | 16π me˜Rl me˜Lm me˜Lm ↔ k,l,m# − $ % d 2 2 m ˜ ν 3 ! ! (˜mLR)ml dRl M ! = λ λ m ln +(i j) , (5.10) ij λ 16π2 ikl jmk dk m2 m2 m2 | d˜ d˜ & d˜ ' ↔ k,l,m# Rl − Lm Lm Here the couplings λ (resp. λ! )andtheleft-rightsleptonmixingmatrixm˜ e 2 =(Ae ijk ijk LR ij − µ tan βλe ) v /√2 (resp. the left-right squark mixing matrix m˜ d 2 =(Ad µ tan βλd ) v /√2) ij d LR ij − ij d are expressed in the basis in which the charged lepton masses (resp. the down quark masses) as