Dark Matter and The Seesaw Scale
DDark Matter and The Seesaw Scale
Pavel Fileviez P´erez , Clara Murgui Physics Department and Center for Education and Research in Cosmology and Astrophysics (CERCA),Case Western Reserve University, Rockefeller Bldg. 2076 Adelbert Rd. Cleveland, OH 44106, USA Departamento de F´ısica Te´orica, IFIC, Universitat de Valencia-CSIC, E-46071, Valencia, Spain
We discuss the possibility to find an upper bound on the seesaw scale using the cosmological bound on thecold dark matter relic density. We investigate a simple relation between the origin of neutrino masses and theproperties of a dark matter candidate in a simple theory where the new symmetry breaking scale defines theseesaw scale. Imposing the cosmological bounds, we find an upper bound of order multi-TeV on the leptonnumber violation scale. We investigate the predictions for direct and indirect detection dark matter experiments,and the possible signatures at the Large Hadron Collider.
INTRODUCTION
The origin of neutrino masses and the nature of the colddark matter in the Universe are two of the most exciting openproblems in particle physics and cosmology. We know todayabout several mechanisms to generate neutrino masses, see forexample Ref. [1], but the so-called seesaw mechanism [2] isconsidered the most appealing and simple mechanism for Ma-jorana neutrino masses. Unfortunately, we only know that theupper bound on the seesaw scale is about GeV, whichis an energy scale very far from any future collider experi-ment. Therefore, it is not clear we could test the mechanismbehind neutrino mass. There are also many possible candi-dates to describe the cold dark matter in the Universe, see forexample Ref. [3]. The weakly interacting massive particles(WIMPs) have been popular dark matter candidates in the lastdecade but the recent experimental results tell us that maybeone should think about other possibilities. However, it is fairto say that the idea of describing the dark matter with WIMPsis so appealing that it is better to understand and revise allconstraints and the different models before abandoning thisidea.The discovery of lepton number violating signatures in lowenergy experiments or at colliders will be striking signals fornew physics beyond the Standard Model. In low energy ex-periments we could discover neutrinoless double beta decay,for a review see Ref. [4], and at colliders different signatureswith same-sign leptons could be seen [5]. These discoverieswill be crucial to establish the origin of neutrino masses.We understand the origin of charged fermion masses in theStandard Model through the spontaneous breaking of the elec-troweak symmetry. In the same way, we could understand theorigin of the seesaw scale if B − L is a local symmetry sponta-neously broken through the Higgs mechanism. Unfortunately,as in canonical seesaw, the upper bound on the B − L is typ-ically very large, M B − L (cid:46) GeV. There are two knownways to establish a much smaller upper bound on the B − L breaking scale: a) In the context of the minimal supersymmet-ric U (1) B − L theory [6] the B − L breaking scale is definedby the supersymmetry breaking scale. Then, if low energysupersymmetry is realized at the multi-TeV scale, we coulddiscover lepton number violating signatures at colliders. b) FIG. 1: Correlation between the origin of neutrino masses, propertiesof cold dark matter candidate and lepton number violating signatures.
The second possibility is to use the cosmological bounds onthe dark matter relic density to impose an upper bound on the B − L breaking scale in the case where the dark matter ischarged under the same gauge symmetry.In this article, we focus on the second possibility mentionedabove in order to find an upper bound on the B − L seesawscale. In this theory, the dark matter candidate is a vector-likefermion which is a SM singlet but charged under the B − L gauge symmetry. We find that, using the constraints on thecold dark matter relic density, the upper bound on the B − L is in the multi-TeV region. Therefore, one can expect ex-otic signatures at colliders with same-sign multi-leptons anddisplaced vertices. This connection between the cosmologi-cal dark matter bounds and exotic signatures at colliders isvery unique and one could hope to test the origin of neutrinomasses at colliders . See Fig. 1 for a simple way to illustratethis correlation.
NEUTRINO MASSES AND THE B − L SCALE
The simplest gauge theory where one can understand dy-namically the origin of neutrino masses is based on the B − L gauge symmetry. In this context, we add three copies of right-handed neutrinos to define an anomaly free theory, and onecan easily implement the seesaw mechanism [2] for Majorana a r X i v : . [ h e p - ph ] A ug neutrino masses. The relevant Lagrangian for the generationof neutrino mass is given by −L Seesaw = Y ν ¯ (cid:96) L iσ H ∗ ν R + λ R ν TR Cν R S BL + h . c ., (1)where ν R ∼ (1 , , , − are the right-handed neutrinos, H ∼ (1 , , / , is the Standard Model Higgs, and S BL ∼ (1 , , , is the new Higgs responsible for the spontaneousbreaking of B − L . Using the above interactions one cangenerate masses for the Standard Model neutrinos through thewell-known Type I seesaw mechanism [2], which are given by M Iν = m TD M − R m D , (2)where m D = Y ν v / √ . Here v / √ is the vacuum expec-tation value of the Standard Model Higgs. We note that, inthis case, the masses of the right-handed neutrinos, M R = √ λ R v BL , are defined by the B − L breaking scale. The see-saw scale, in general, is unknown; the only thing we knowis that the upper bound should be the canonical seesaw scale GeV. It is important to mention that if M R is at the TeVscale one can generate masses for Standard Model neutrinos inagreement with the experiments if m D < − GeV, and onecan produce the right-handed neutrinos at the LHC throughthe B − L neutral gauge boson, pp → Z ∗ BL → N i N i [7–11],giving rise to striking lepton number violating signatures withsame-sign leptons and multijets.Now, since the observation of lepton number violation iscrucial to learn about the origin of neutrino masses, it is im-portant to understand the possibility to find an upper boundon the B − L breaking scale which is much smaller than thecanonical seesaw scale. Then, we could hope to test the ori-gin of neutrino masses at current or future experiments. Weknow about two different class of theories where it is possibleto find an upper bound on the B − L breaking scale:• In Ref. [6], one of us (P.F.P.) and collaborators pointedout that in the minimal supersymmetric B − L modelthe gauge symmetry must be broken by the vacuum ex-pectation value of the ‘right-handed’ sneutrinos. Thenone predicts that R − parity must be spontaneously bro-ken, and one expects the existence of lepton number vi-olation. In this context, the R − parity and lepton num-ber violation scales are defined by the supersymmetrybreaking scale. Then, if one has low energy supersym-metry at the multi-TeV scale, and this theory is true,one should discover lepton number violation at currentor future colliders. For detailed studies see Refs. [12–15].• The second possibility is discussed in details in this ar-ticle. We will show that if one has a fermionic colddark matter candidate which is charged under the B − L gauge symmetry, it is possible to find an upper bound onthe B − L breaking scale in the multi-TeV region usingthe cosmological bounds on the dark matter relic den-sity. Therefore, this theory provides a simple scenario which motivates the search for lepton number violationat colliders.These two scenarios provide two major examples of theo-ries where one could expect the discovery of lepton numberviolating processes at the multi-TeV scale. We will focus onthe second example and investigate the impact of all dark mat-ter bounds. DARK MATTER AND THE B − L SCALE
One can write a very simple model to generate Majorananeutrino masses and to explain the presence of cold dark mat-ter in the Universe based on the spontaneous breaking of the U (1) B − L gauge symmetry. The relevant Lagrangian for ourdiscussion is given by L DMν ⊃ − F BLµν F BLαβ g αµ g βν + iχ L γ µ D µ χ L + iχ R γ µ D µ χ R + ( D µ S BL ) † ( D µ S BL ) − ( Y ν ¯ (cid:96) L iσ H ∗ ν R + λ R ν TR Cν R S BL + M χ ¯ χ L χ R + h . c . ) , (3)where F BLµν = ∂ µ Z BLν − ∂ ν Z BLµ defines the kinetic termfor the B − L gauge boson Z BL . Since χ L ∼ (1 , , , n ) and χ R ∼ (1 , , , n ) , the covariant derivates are defined by D µ χ L = ∂ µ χ L + ig BL nZ µBL χ L and D µ χ R = ∂ µ χ R + ig BL nZ µBL χ R . Here, | n | (cid:54) = 1 , in order to avoid the de-cay of χ = χ L + χ R , which must be stable, and can be agood cold dark matter candidate. In the case the proposedtheory is sensitive to UV physics, we note that mixing amongneutrinos and the dark matter candidate could be originatedby non-renormalizable operators only for a choice of n odd,whereas even and fractionally charges would be safe. Thekinetic mixing between the B − L gauge boson and the hy-percharge gauge boson is neglected for simplicity. A simi-lar model has been partially investigated before in Ref. [16],where the main emphasis was the study of the gamma linesfrom dark matter annihilation. Our main goal here is to in-vestigate in detail the connection between the cosmologicalbounds and the lepton number violation scale, and understandthe implications for the search for lepton number violation atcolliders. Higgs Sector:
The Higgs sector of this theory is composedof the SM Higgs H ∼ (1 , , / , and S BL ∼ (1 , , , and the scalar potential is given by V ( H, S BL ) = − µ H H † H + λ H (cid:0) H † H (cid:1) − µ BL S † BL S BL + λ BL (cid:16) S † BL S BL (cid:17) + λ HBL (cid:0) H † H (cid:1) (cid:16) S † BL S BL (cid:17) , (4)where H T = (cid:18) H + , √ v + h + iA ) (cid:19) and S BL = 1 √ v BL + h BL + iA BL ) . (5)In this case the physical states are: h = h cos θ BL + h BL sin θ BL , (6) h = − h sin θ BL + h BL cos θ BL , (7)where tan 2 θ BL = λ HBL v v BL λ H v − λ BL v BL . (8)After symmetry breaking one finds that the mass for the B − L gauge boson is given by M Z BL = 2 g BL v BL . (9)The dark matter candidate χ = χ L + χ R is a Dirac fermionwith mass M χ . We focus on the case where the dark mattercandidate is a Dirac fermion because in the other cases, scalaror Majorana fermion, the annihilation cross section throughthe B − L gauge boson is suppressed. This simple dark mat-ter model has the following relevant parameters for the darkmatter study: n, g BL , M χ , M h M Z BL , and M N i ( i = 1 , , . (10)As we have mentioned above the properties of the DM can-didate χ are very simple since it is a vector-like fermion, χ = χ L + χ R , and it has interactions only with the B − L gauge boson. The two body annihilation channels are ¯ χχ → Z ∗ BL → ¯ u i u i , ¯ d i d i , ¯ e i e i , ¯ ν i ν i , ¯ N i N i , (11) ¯ χχ → Z BL Z BL , Z BL h , Z BL h , (12)where the first one is the dominant channel when M χ
21 2 3 4 5 60.000.050.100.150.20 M χ ( TeV ) Ω h FIG. 3: Relic density predictions for M Z BL = 3 . TeV, g BL = 0 . and different B − L charge n for the dark matter candidate. Thecharges n = 1 / , n = 2 / , and n = 2 are represented by a solidand dashed lines, respectively. The shaded region is excluded by thebound on the relic density Ω h ≤ . ± . [18]. ing ratios of the thermal averaged cross-sections for thechannels ¯ χχ → ¯ f i f i (solid line), and ¯ χχ → Z BL Z BL , ¯ χχ → Z BL h , and ¯ χχ → Z BL h (dashed lines), fordifferent B − L charges. We have used cos θ BL =0 . for the scalar mixing angle, M Z BL = 3 . TeV, g BL = 0 . , and M h = M N = 1 TeV for illustra-tion. For low values of the B − L charge, the anni-hilation channel into two fermions significantly domi-nates over the other channels. The annihilation into twogauge bosons ¯ χχ → Z BL Z BL can be important whenone has large values for the dark matter B − L charge.However, as we will discuss later, perturbativity boundsconstrain this channel in such a way that the annihila-tion into fermions will always dominate over annihila-tion into two gauge bosons, regardless of the choice ofthe B − L charge of the dark matter candidate. Further-more, as we will see, it does not make sense to considerlarger values of n because the direct detection boundsare much stronger and one will only find consistent so-lutions when the gauge boson is very heavy.In Fig. 3, we show the predictions for the relic densitywhen M Z BL = 3 . TeV, g BL = 0 . and different B − L charges for the dark matter candidate. The charges n = 1 / , n = 2 / , and n = 2 are represented by asolid and dashed lines, respectively. One can see thatwhen one has large values of the dark matter B − L charge one can achieve the relic density in agreementwith cosmology even if we are far from the resonance M Z BL ≈ M χ because in this case the annihilation intotwo gauge bosons has a larger contribution. In thesestudies we consider only the main annihilation channelsin the numerical studies, and we will focus on n = 1 / ,and n = 2 as illustrative examples.• Direct DetectionThe elastic spin-independent nucleon–dark matter cross LEP
X E N O N T X E N O N n T M Z BL g BL =
20 TeV × - × - × - × - × - M χ ( GeV ) σ χ - N S I ( c m ) FIG. 4: Predictions for the direct detection spin-independent cross-section σ SIχN for points with n = 1 / satisfying the relic densityand LEP bounds. We show the Xenon-1T [20, 21] (the last updatedbounds in orange) and Xenon-nT [22] bounds. The different coloredpoints correspond to different values of the gauge coupling, g BL =0 − . (blue), g BL = 0 . − . (gold), g BL = 0 . − . (green)and g BL = 0 . − . (orange). section is given by σ SI χN = M N M χ π ( M N + M χ ) g BL M Z BL n , (17)where M N is the nucleon mass. We note that σ SI χN isindependent of the matrix elements. The cross sectioncan be rewritten as σ SI χN ( cm ) = 12 . × − (cid:16) µ (cid:17) (cid:18) r BL (cid:19) n cm , (18)where µ = M N M χ / ( M N + M χ ) is the reduced massand r BL = M Z BL /g BL . In our case M χ (cid:29) M N ,and using the collider lower bound M Z BL /g BL > [19] one finds an upper bound on the elastic spin-independent nucleon-dark matter cross section given by σ SI χN < . × − n cm , (19)for a given value of n . In Fig. 4, we show the pre-dictions for the direct detection cross-section σ SIχN forpoints with n = 1 / satisfying the relic density andLEP bounds. We show the Xenon-1T [20, 21] andXenon-nT [22] bounds to understand the available re-gion of the parameter space which is still in agreementwith the direct detection and the expected region whichcould be tested in the near future. The different coloredpoints correspond to the predictions when we use dif-ferent values for the gauge coupling, g BL = 0 − . (blue), g BL = 0 . − . (gold), g BL = 0 . − . (green) and g BL = 0 . − . (orange). Clearly, in thesescenarios the dark matter mass should be above 1 TeVto be in agreement with the direct detection bounds.• Indirect Detection F e r m i L A T
500 1000 1500 2000 2500 300010 - - - - M χ ( GeV ) 〈 σ v 〉 ( c m s - ) b b F e r m i L A T
500 1000 1500 2000 2500 300010 - - - - M χ ( GeV ) 〈 σ v 〉 ( c m s - ) τ + τ - FIG. 5: Allowed parameter space for thermal dark matter annihila-tion into two bottom quarks (upper-panel) and two taus (lower-panel)compatible with the relic density constraint. The dotted-dashed linesshow the predictions on the resonance for n = 1 / (dark blue) and n = 2 (orange). The gray shaded area shows the parameter spaceexcluded by the experimental bounds from FermiLAT [23]. In this model we can have two gamma lines from darkmatter annihilation, ¯ χχ → Zγ , and ¯ χχ → hγ . How-ever, due to the fact that the cross section for the fi-nal state radiation processes ¯ χχ → ¯ f f γ are muchlarger, one cannot identify the gamma line from thecontinuum spectrum. The numerical results for thesegamma lines were studied in Ref. [16]. In Fig. 5, weshow the allowed parameter space for thermal aver-aged dark matter annihilation cross section into two bot-tom quarks (upper-panel), and two tau leptons (lower-panel), compatible with the relic density constraint. Thegray shaded area shows the parameter space excludedby the experimental bounds from FermiLAT [23]. Asone can see, for choices of low n , the allowed parame-ter space is compatible with these bounds.• Upper bound on the Symmetry Breaking ScaleIn Figs. 6, and 7, we show the allowed region in the M Z BL − M χ plane when Ω DM h ≤ . ± . , inagreement with the LEP bounds. In Fig. 6, we show theallowed solutions when the dark matter B − L charge is1/3, and the allowed region for n = 2 is shown in Fig. 7.As we can see, the maximum allowed value for M Z BL is around TeV when n = 1 / , while the upper boundon M Z BL when n = 2 is around 130 TeV. Clearly, each g BL = g BL = g BL = g BL = g BL = g BL = M χ ( TeV ) M Z B L ( T e V ) FIG. 6: Allowed region in the M Z BL − M χ plane for n = 1 / when Ω DM h ≤ . ± . , in agreement with the LEP bounds.Here we use the perturbative bound g BL < √ π . choice of n , and g BL define a theory which is boundedfrom above. However, regardless of the value of the B − L charge, and the choice of g BL , we note that thereis an absolute upper bound at the multi-TeV scale forthe seesaw scale. This statement may not be triviallyseen in Fig. 6 and 7, because it seems that the larger the B − L charge, and the coupling g BL , the larger the upperbound on the B − L breaking scale. However, the cou-pling g BL is bounded by perturbativity, and in the limitof large n , the annihilation into two fermions, whichdefines the upper bound on the resonance, becomes in-sensitive to the B − L charge for large values thereof.We note that the annihilation channel into to new gaugebosons is irrelevant for defining the upper bound sinceit is bounded by perturbativity of the B − L coupling.These results are crucial to understand the testability ofthis theory at colliders. Clearly, we could test at theLarge Hadron Collider only one fraction of the param-eter space.For completeness of the discussion on the upper boundfor the seesaw scale, we would like to mention that theupper bound coming from the cosmological bound onthe relic density is in agreement with partial-wave uni-tarity of the S-matrix. It is well known that, from anaive model-independent study, partial wave unitarityrequires that M χ < TeV [24]. However, in thismodel, the partial wave expansion only becomes rele-vant in regions of the parameter space which are notallowed by cosmology. Therefore, unitarity of the S-matrix does not make any influence on the upper boundfor the seesaw scale.The main implication of these results is that there is ahope to test the existence of lepton number violationsince the upper bound on the B − L breaking scale ismuch smaller than the canonical seesaw scale. g BL = g BL = g BL = g BL = g BL = M χ ( TeV ) M Z B L ( T e V ) FIG. 7: Allowed region in the M Z BL − M χ plane for n = 2 when Ω DM h ≤ . ± . , in agreement with the LEP bounds.Here we use the perturbative bound g BL < √ π . M χ = M Z BL = g BL = M χ = M Z BL = g BL = M χ = M Z BL = g BL =
600 800 1000 1200 140010 - M N ( GeV ) σ ( pp → NN )( f b ) s =
13 TeV N = N = N =
200 400 600 800 10000.00.20.40.60.81.0 M N ( GeV ) B r ( N i → e j ± W ∓ ) ℒ =
50 fb - , s =
13 TeV
FIG. 8: In upper-panel, production cross section for the right-handedneutrinos at the LHC when √ s = 13 TeV, and in different scenariosconsistent with the dark matter relic density. In the lower-panel, ex-pected number of events for the scenarios M χ = 1 TeV, M Z BL = 2 TeV, g BL = 0 . (black), and M χ = 4 TeV, M Z BL = 7 TeV, g BL = 0 . (orange), when √ s = 13 TeV, and L = 50 fb − . LEPTON NUMBER VIOLATION AT THE LHC
In the previous section we have shown the possibility to findan upper bound on the B − L symmetry breaking scale usingthe constraints from the dark matter relic density. Therefore,one can hope to test the existence of a new force associatedto B − L , and observe lepton number violation at the LHC.Unfortunately, the upper bound is large if n = 2 , but still we can hope to test this theory if the symmetry is broken muchbelow the upper bound.The right-handed neutrinos can be produced at the LHCthrough the neutral gauge boson Z BL , i.e. pp → Z ∗ BL → N i N i or through the Higgses present in the theory [7–11].Since the production mechanisms through the Higgses sufferfrom the dependence on the mixing angle in the Higgs sector,we focus our discussion on the production through the B − L gauge boson. The right-handed neutrinos could have the fol-lowing decays: N i → e ± j W ∓ , ν j Z, ν j h , ν j h . Therefore, the lepton number violating signatures can be ob-served when the right-handed neutrinos decay into chargedleptons, and one has the following channels with the same-sign leptons pp → Z ∗ BL → N i N i → e ± j W ∓ e ± k W ∓ → e ± j e ± k j. (20)The number of these events is given by N e ± j e ± k j = 2 × L × σ ( pp → N i N i ) × Br( N i → e ± j W ∓ ) × Br( N i → e ± k W ∓ ) × Br( W → jj ) , (21)where L is the integrated luminosity. Here, the factor two isincluded to discuss the channels with same-sign leptons with-out distinguishing the electric charge of the leptons in the finalstate.In Fig. 8 (upper-panel), we show the predictions for thecross section when √ s = 13 TeV, and for different scenar-ios consistent with relic density bounds. Here we choose n =1 / , and numerical values for M χ , M Z BL , and g BL whichsatisfy the cosmological bounds. In the lower-panel, we showthe number of events assuming a luminosity L = 50 fb − .We have reviewed the current LHC bounds and unfortunatelythey cannot exclude the region of the parameter space studiedhere. A more detailed experimental study will help to un-derstand the testability of this theory with more luminosity.Similar results are shown in Fig. 9, when √ s = 14 TeV. How-ever, in this case one expects a large number of events for thesame-sign leptons for right-handed neutrino masses below 1TeV.In Fig. 10, we show the predictions for the next genera-tion of proton-proton collider at 100 TeV. In this case oneshould be able to test most of the parameter space even whenthe right-handed neutrinos are in the multi-TeV region. SeeRef. [25] for the discovery potential of the 100 TeV collider.We would like to mention that, in a large part of the param-eter space, the right handed neutrino decays can give rise todisplaced vertices [11]. Since the seesaw scale has to be atmost at the multi-TeV scale ∼ O (10 TeV), the Yukawa Dirac Y ν has to be small in order to reproduce the measured ac-tive neutrino masses. A small Y ν enhances the lifetime of theright-handed neutrinos, which then become long-lived parti-cles. Therefore, as a consequence of having a low seesaw M χ = M Z BL = g BL = M χ = M Z BL = g BL = M χ = M Z BL = g BL =
600 800 1000 1200 140010 - M N ( GeV ) σ ( pp → NN )( f b ) s =
14 TeV N = N = N = N = N =
500 1000 15000.00.20.40.60.81.0 M N ( GeV ) B r ( N i → e j ± W ∓ ) ℒ =
300 fb - , s =
14 TeV
FIG. 9: In the upper-panel, production cross section for the right-handed neutrinos at the LHC when √ s = 14 TeV, and in differentscenarios consistent with the dark matter relic density. In the lower-panel, expected number of events for the scenarios M χ = 1 TeV, M Z BL = 2 TeV, g BL = 0 . (black), and M χ = 4 TeV, M Z BL = 7 TeV, g BL = 0 . (orange), when √ s = 14 TeV, and L = 300 fb − . scale, displaced vertices arise as an exotic signatures predictedby the model. For instance, when the right-handed neutrinomass is about GeV, the decay length can be L = (10 − − − )mm . This simple study motivates the search of lepton number vio-lating signatures at the LHC or at future colliders.
FINAL DISCUSSION
We have discussed the simple seesaw mechanism forneutrino masses where the seesaw scale is defined by the B − L symmetry breaking scale and the scenarios whereit is possible to find an upper bound on the seesaw scaleusing theoretical arguments or cosmological bounds. Wehave investigated the relation between the origin of neutrinomasses and the properties of a simple cold dark mattercandidate in the context of a theory based on the B − L gauge symmetry. In this context the upper bound on theseesaw scale is in the multi-TeV region and one predictsthe existence of exotic lepton number violating signatures atcolliders. We use all cosmological constraints and investigate M χ = M Z BL = g BL = M χ = M Z BL = g BL = M χ = M Z BL = g BL =
100 200 500 1000 200010 - M N ( GeV ) σ ( pp → NN )( f b ) s =
100 TeV N = N = N = N = N = N =
500 1000 1500 2000 2500 3000 35000.00.20.40.60.81.0 M N ( GeV ) B r ( N i → e j ± W ∓ ) ℒ =
30 fb - , s =
100 TeV
FIG. 10: Production cross section for the right-handed neutrinos atthe 100 TeV collider and in different scenarios consistent with thedark matter relic density (left panel), and the expected number ofevents for the scenarios M χ = 1 TeV, M Z BL = 2 TeV, g BL = 0 . (black) and M χ = 4 TeV, M Z BL = 7 TeV, g BL = 0 . (orange),when √ s = 100 TeV, and L = 30 fb − (right-panel). the predictions for direct and indirect detection dark matterexperiments. These cosmological bounds motivate the searchfor lepton number violation at the Large Hadron Collider orat future colliders in order to test the origin of neutrino masses. Acknowledgments:
P. F. P. thanks Mark B. Wise for dis-cussions and the Walter Burke Institute for TheoreticalPhysics at Caltech for hospitality. C. M. thanks the theorydivision at IFIC for fruitful comments and discussions. Thework of P. F. P. has been supported by the U.S. Departmentof Energy under contract No. de-sc0018005. The work ofC. M. has been supported in part by the Spanish Governmentand ERDF funds from the EU Commission [Grants No.FPA2014-53631-C2-1-P and SEV-2014- 0398] and ”LaCaixa-Severo Ochoa” scholarship.
Appendix A. Dark Matter Annihilation Cross Sections
The annihilation cross section for ¯ χχ → Z ∗ BL → ¯ f f is given by σ ( ¯ χχ → Z ∗ BL → ¯ f f ) = N fc n f g BL n πs (cid:113) s − M f (cid:113) s − M χ (cid:0) s + 2 M χ (cid:1) (cid:16) s + 2 M f (cid:17)(cid:2) ( s − M Z BL ) + M Z BL Γ Z BL (cid:3) . (22)Here N fc is the color factor of the fermion f with mass M f , s is the square of the center-of-mass energy, and Γ Z BL is the totaldecay width of the Z BL gauge boson. For Majorana neutral fermions, the annihilation cross section into fermions reads as σ ( ¯ χχ → Z ∗ BL → ¯ f f ) = N fc n f g BL n πs (cid:113) s − M f (cid:113) s − M χ (cid:0) s + 2 M χ (cid:1) (cid:16) s − M f (cid:17)(cid:2) ( s − M Z BL ) + M Z BL Γ Z BL (cid:3) . (23)The annihilation cross section for ¯ χχ → Z BL Z BL is given by σ ( ¯ χχ → Z BL Z BL ) = g BL n πE ωv (cid:20) − − (2 + z ) (2 − z ) − v + 6 − z + z + 12 v + 4 v vω (1 + v + ω ) ln (cid:18) v + ω ) v − ω ) (cid:19)(cid:21) , (24)where z , v and ω are defined as z = M Z BL /M χ , v = p/M χ , and ω = k/M χ , (25)being E and p the center-of-mass energy and momentum of the initial particles respectively, E = √ s/ and p = (cid:113) E − M χ ,and k the momentum of the final particles, k = (cid:113) E − M Z BL . The annihilation cross section ¯ χχ → Z BL h i is given by σ ( ¯ χχ → Z BL h i ) = c i g BL n S n πs (cid:32) M χ s (cid:33) ( s + 2 s (5 M Z BL − M h i ) + ( M Z BL − M h i ) )( s − M Z BL ) + Γ Z BL M Z BL × (cid:113) ( s + M Z BL − M h i ) − M Z BL s (cid:114) − M χ s , (26)where c i = (sin θ BL , cos θ BL ) and n S is the B − L charge of the S BL Higgs.
Appendix B. Z BL Decays
For the decays into charged fermions: Γ( Z BL → ¯ f i f i ) = g BL N c n f i πM Z BL (cid:115) − M f i M Z BL (2 M f i + M Z BL ) . (27)For Majorana neutral fermions: Γ( Z BL → νν ) = g BL π M Z BL , (28) Γ( Z BL → N N ) = g BL πM Z BL (cid:115) − M N M Z BL ( M Z BL − M N ) . (29)Total decay width of the Z BL : Γ tot ( Z BL ) = 3 Γ( Z BL → (cid:96)(cid:96) ) + 5 Γ( Z BL → ¯ qq ) + Γ( Z BL → ¯ tt )+ 3Γ( Z BL → νν ) + 3Γ( Z BL → N N ) + Γ( Z BL → ¯ χχ ) . (30) Appendix C. Right-handed Neutrinos Production Cross Section
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