Leptophilic Dark Matter from the Lepton Asymmetry
aa r X i v : . [ h e p - ph ] S e p MCTP/09-45
Leptophilic Dark Matter from the Lepton Asymmetry
Timothy Cohen and Kathryn M. Zurek , Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109 Particle Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, IL 60510 (Dated: November 8, 2018)We present a model of weak scale Dark Matter (DM) where the thermal DM density is set by the leptonasymmetry due to the presence of higher dimension lepton violating operators. In these models there is gener-ically a separation between the annihilation cross-section responsible for the relic abundance (through leptonviolating operators) and the annihilation cross-section that is relevant for the indirect detection of DM (throughlepton preserving operators). Due to this separation, there is a perceived boost in the annihilation cross-sectionin the galaxy today relative to that derived for canonical thermal freeze-out. This results in a natural explanationfor the observed cosmic ray electron and positron excesses, without resorting to a Sommerfeld enhancement.Generating the indirect signals also sets the magnitude of the direct detection cross-section which implies a sig-nal for the next generation of experiments. More generically these models motivate continued searches for DMwith apparently non-thermal annihilation cross-sections. The DM may also play a role in radiatively generatingMajorana neutrino masses.
PACS numbers:
In recent decades a canonical model for Dark Matter (DM)utilizing the existence of Weakly Interacting Massive Parti-cles (WIMPs) has emerged. In models which stabilize theHiggs mass at the electroweak scale, the lightest of the newstates introduced in these theories is often “accidentally” sta-ble due to a symmetry which is imposed for other reasons,such as R -parity. The observed DM density, set by thermalfreeze-out, determines the cross-section to annihilate to Stan-dard Model (SM) fields to be a value typical of weak scalephysics, h σ v i ≃ × − cm / s. Within the paradigmof these models, many phenomenological expectations havebeen fixed, including the annihilation modes to the SM inter-action channels with corresponding rates for indirect detectionin the galaxy today.However, the phenomenological successes of thermalWIMP DM can be preserved in other paradigms. For example,the lepton or baryon asymmetry may set the DM density [1].In these so called Asymmetric Dark Matter (ADM) models[2], DM in the GeV-TeV mass scale range naturally generatesthe observed relic abundance without standard thermal freeze-out. When the DM from these models is hidden ( i.e. it carriesno SM charges) [2], its interactions with the SM fields may beset by interactions with new messengers (as in a Hidden Valley[3]) rather than with the SM electroweak fields or their super-partners. And since the DM density is set by the lepton orbaryon asymmetry, the SM-DM interactions are typically lep-tophilic or baryophilic respectively. In addition, because therelic density is not set by the usual thermal freeze-out calcula-tion, the relation between the DM density and the annihilationcross-sections relevant for the indirect observation of the DMtoday is modified.Recent observations provide additional motivation forstudying these models. An excess in cosmic ray positron andelectron signals over the expected background as observed byAMS-01 [4], HEAT [5], PPB-BETS [6], PAMELA [7], Fermi[8] and ATIC [9] may be a signal of annihilating DM. Theannihilation cross-section needed to produce these signals is non-thermal, a factor ∼ − (depending on DM massand astrophysical boost factor) larger than the thermal annihi-lation cross-section [10, 11]. Annihilation predominantly toleptons is preferred both by the shape of the PAMELA signaland the lack of excess in the anti-proton data [12]. These factsappear to disfavor an explanation utilizing a canonical neu-tralino (though when combined with an astrophysical flux, itmay be obtained [13]). One possibility is to introduce newGeV scale particles [14]. These light states mediate a Som-merfeld enhancement [15], implying boosted annihilation inthe halo today, while also acting as intermediate final states,thereby providing kinematic constraints on the allowed SMparticles produced from DM annihilations.In this letter we provide a simple paradigm which gives riseto both boosted and leptophilic annihilation of DM, involvingneither Sommerfeld enhancements nor new GeV mass states.When the DM relic density is set by the lepton asymmetry,the annihilation modes are naturally leptophilic. Additionally,this density is derived using lepton number ( L ) violating op-erators that transfer the asymmetry, and not the L -preservingoperators which lead to a signal for indirect detection experi-ments (such as PAMELA and Fermi) at low temperatures [22].Though these models can provide a unique explanation for thecosmic ray excesses, their interest extends beyond this appli-cation.We begin by outlining the general features of this class ofmodels and then turn to constructing a simple model for il-lustration. An initial lepton asymmetry is generated at tem-peratures well above the electroweak scale. We are agnosticabout the source of this asymmetry for the purposes of thispaper. Lepton number violating operators, which connect theSM leptons to dark sector fields, transfer the lepton asymme-try to the dark sector. As in all models of ADM, these opera-tors relate the DM number density to the lepton, and thereforebaryon, density, ( n X − n ¯ X ) ∼ ( n ℓ − n ¯ ℓ ) ∼ ( n b − n ¯ b ) , (1)where the exact proportions are O (1) and are determinedby the particular operator transferring the asymmetries, and ( n X − n ¯ X ) , ( n ℓ − n ¯ ℓ ) and ( n b − n ¯ b ) are the asymmetriesin the DM ( X ), leptons and baryons respectively. As a result m X ∼ Ω DM Ω b m p , where m X is the DM mass, m p is the pro-ton mass, Ω DM is the DM relic density and Ω b is the baryondensity of the universe. This relation implies a DM mass m X ≃ GeV. Though the size of this mass is phenomeno-logically viable, it does not directly link the DM sector to thenew physics which stabilizes the weak scale.If the L -violating operators which transfer the asymmetryhave not decoupled as the DM becomes non-relativistic, thereis a Boltzmann suppression of the DM asymmetry (see [16,17] for a more detailed discussion) ( n X − n ¯ X ) ∼ ( n ℓ − n ¯ ℓ ) e − m X /T d , (2)where T d is the temperature at which the L -violating operatorsdecouple. This implies that the DM mass can be much larger[23] m X = 4529 1 N X f (0) f ( m X /T d ) Ω DM Ω b m p , (3)where N X is the number of DM families and f ( x ) is theBoltzmann suppression factor given by f ( x ) = 14 π Z ∞ y d y cosh ( p y + x ) . (4)The decoupling temperature, T d , is naturally at the elec-troweak scale if the corresponding higher dimensional opera-tors are TeV scale suppressed. Once these L -violating opera-tors decouple, the asymmetric DM density is frozen in.Although the L -violating interactions have frozen out, L -preserving interactions are expected to remain in thermalequilibrium to lower temperatures. This is particularly naturalif the L -violating operators are generated by a combinationof the L -preserving interactions and an operator which intro-duces a small amount of L -violation into the theory. While the L -preserving operators may be in thermal equilibrium longerthan the resulting L -violating interactions, they do not changethe relic DM density, which will be dominantly composed of ¯ X s with essentially no X s.If the asymmetry in the DM persisted until today, therewould be no indirect detection signal from X − ¯ X annihi-lation. If, however, there is a small violation of DM numberin the dark sector, as may result from a small DM Majoranamass, X − ¯ X oscillations will erase the asymmetry withoutreducing the relic density, giving rise to a signal for indirectdetection experiments from ¯ X X → ℓ + ℓ − . In some cases thehidden sector may be more complicated, and four lepton finalstates may also result, e.g. ¯ X X → ℓ + ℓ − ℓ + ℓ − . Since this L -preserving interaction is expected to be stronger than the L -violating operator which set the asymmetry, the associatedannihilation cross-section may be large enough to generate thecosmic ray positron excesses. There are many models which exhibit the generic featuresdescribed above. The rest of the letter is devoted to an illustra-tive toy model which reproduces this scenario. Consider the L -violating interaction (from [2]) L asym = 1 M ′ ij ¯ X ( L i H )( L j H ) + h.c. , (5)where L is the lepton doublet, H is the SM Higgs doubletand M ′ is a new L -violating mass scale. This term mediates ¯ X ¯ X ↔ ¯ ν ¯ ν , thereby transferring the lepton asymmetry to an X − ¯ X asymmetry. Consider in addition the L -preservinginteraction L sym = 1 M ij ¯ X X ¯ L i L j + h.c. , (6)where M is a new L -preserving mass scale, which mediates ¯ X X ↔ ℓ + ℓ − , ¯ ν ν . A UV completion of these operators is L ∋ y i L i H ′ ¯ X − λ ′ H † H ′ ) + h.c. , (7)where H ′ is a new Higgs doublet. There is a Z symme-try under which X , ¯ X and H ′ are charged, which is unbro-ken for h H ′ i = 0 . This symmetry ensures that the lightest Z odd state, which we take to be ¯ X , is stable. Upon inte-grating out H ′ , the effective scale of L -violation (Eq. (5)) is M ′ ij = m H ′ / ( y i y j λ ′ ) , and the scale of the L -preserving op-erator (Eq. (6)) is M ij = m H ′ / ( y i y j ) . Also note that whilethe model with N X = 1 does not violate L , it does violate anytwo of electron number, muon number and tau number due tothe first interaction in Eq. (7). For weak scale parameters andassuming that y i = y ≃ , the rate for µ → e γ is ∼ ordersof magnitude above the current bound. One way to avoid thisbound is to assume a hierarchy of O (10 − ) between the firsttwo generations of y i couplings. For N X = 3 the interactionsare expanded to L = y ij L i H ′ ¯ X j + m iX ¯ X i X i . (8)For a generic y ij matrix, the same large rates for µ → e γ are present as describe above for N X = 1 . If y ij =diag( y , y , y ) in this basis (where m X is diagonal), con-tributions to µ → e γ vanish.The λ ′ term is present in Eq. (7) to break a global U (1) X ,under which X, ¯ X and H ′ are charged so that an X asym-metric operator such as Eq. (5) can arise. For M and M ′ ator above the electroweak scale and λ ′ < , ( M ′ ij ) & ( v M ij ) ,implying that the L -violating operators decouple first ( v ≡h H i ). The annihilations through the operator in Eq. (6) (andEq. (12) below) give rise to larger cross-sections than throughEq. (5). The smaller cross-section from the L -violating oper-ators set the DM asymmetry, and hence its relic density.From Eq. (3), m X /T d ≈ − for m X ≈ − GeV(note there is only logarithmic sensitivity to m X ). Then using H ( T d ) = n ¯ X h σ asym v i to set the L -violating cross-sectionyields λ ′ = 2 × − for m X = 500 GeV, N X = 1 and y = 1 ,or equivalently M ′ ≃ TeV ( m X / GeV ) / N / X . Forreference we include the zero temperature result for the asym-metric annihilation ¯ X ¯ X ↔ ¯ ν ¯ ν h σ asym v i = 116 π v m X M ′ , (9)which results in an O (20 %) error when calculating M ′ .The symmetric annihilation ¯ X X ↔ ℓ + ℓ − , ¯ ν ν throughEq. (6) with cross-section h σ sym v i = 18 π m X M ij , (10)will typically freeze-out at a temperature lower then T d . Theseannihilations do not affect the relic density, which is set by theDM asymmetry.As long as the DM density is asymmetric, there will be noindirect signals for DM in the universe today. However, asmall Majorana mass m M term, L M = m M ¯ X ¯ X, (11)will induce X − ¯ X oscillations which erase the DM asymme-try and give rise to X − ¯ X annihilation signals in the uni-verse today. For m X = 500 GeV and M = 300 − GeV (corresponding to y = 2 − and m H ′ = 600 GeV), h σ sym v i = 10 − − − cm / s which is the size requiredto generate the PAMELA and Fermi signals.One can also generate four lepton final states in this modelwith only a minor modification. For example the Dirac massterm, m X ¯ X X , could result from the vev of a new singletscalar ( Φ ) and the interaction L X = λ X Φ ¯
X X, (12)where m X ≡ λ X h Φ i . Assuming Φ has no direct couplingsto the SM, its decays will occur exclusively to leptonic fi-nal states through a one-loop diagram. Then the interac-tions in Eq. (12) mediate annihilations to ¯ X X → Φ Φ → ℓ + ℓ − ℓ + ℓ − . Note that we do not require kinematic restric-tions to force Φ to decay to leptonic final states.There is a cosmological restriction on the X Majorana mass– to preserve the relic density, we require that no annihilationsrecouple when the X − ¯ X oscillations commence. Otherwisethe relic density would be reduced to the (small) thermal valueset by the symmetric processes. Quantitatively, the symmetric“no-recoupling” temperature ( T nr ), defined by n asym ( T nr )2 h σ sym v i = H ( T nr ) , (13)must be greater than the temperature when oscillations begin( T osc ): H ( T nr ) & H ( T osc ) ∼ m M . (14)For the no-recoupling relation, we have taken equal parts ¯ X and X from oscillations at T nr , and n asym is the relic DM density set by asymmetric annihilations. UsingEq. (3) to find n asym ( T nr ) and Eq. (10) we find T nr ≃ . g − / ∗ (10 − cm / s / h σ a v i ) for m X = 500 GeV.Then Eq. (14) implies m M . O (10 − − − GeV) for h σ sym v i ∼ O (10 − − − cm / s ) . This very small massis natural since X effectively carries lepton number, an un-broken global symmetry in the absence of Majorana neutrinomasses. Then the presence of Majorana neutrino masses in-duces an X Majorana mass: m M ∼ π y λ ′ v m ν m H ′ ∼ O (10 − GeV) , (15)where the last relation is for the parameters described aboveEq. (9). This is a small enough Majorana mass that no washout occurs for h σ sym v i . − cm / s. Also note that sincewe are assuming instantaneous oscillations, even when m M is at the upper bound of the constraint implied by Eq. (14)there will only be an O (1) change in the DM relic density.Thus for the symmetric annihilation cross-sections of interesthere, Majorana neutrino masses are often consistent with theno-recoupling condition. Models with mass varying neutri-nos [18] or where the neutrinos are Dirac will weaken this oreliminate this constraint.The constraints from neutrino masses also do not apply ifthe X Majorana mass induces
Majorana neutrino masses. Ifthe X Majorana mass results from the vev of a sub-GeV scalarfield ( S ), from the interaction L M = κ αβ S ¯ X α ¯ X β , (16)and the scalar field only obtains a vev at T < T nr , the Ma-jorana mass (( m M ) αβ ≡ κ αβ h S i ) can be arbitrarily largewithout reducing the DM number density. In this case, theneutrino mass is generated at one-loop [19]: ( m ν ) ij = y iα y jβ λ ′ π v ( m M ) αβ m H ′ , (17)where we have taken N X = 3 . Since one must assume that y ij is flavor diagonal to avoid lepton flavor violating decays,the flavor and CP violation in the neutrino sector result fromthe structure of the X Majorana mass matrix. The parame-ters y ∼ O (1) , λ ′ ∼ O (10 − ) and m H ′ ∼ O (600 GeV) require m M ∼ O (10 − GeV) to achieve m ν ∼ O (10 − eV) .The off-diagonal entries in m M lead to µ → e γ but for theseparameters the constraint is satisfied.One might worry that the interaction in Eq. (16) could washout the X asymmetry through, e.g. , ¯ X ¯ X ↔ S S processes.The X asymmetry is safe from wash out provided this processdecouples above T d , which happens for small U (1) X viola-tion, κ . O (10 − ) . The phase transition to the vacuum witha non-zero vev for S obtains if either the temperature dropsbelow the critical temperature associated with the S potentialor the S particles decay. S decays to two neutrinos via a one-loop diagram with rate Γ S − decay ∼ O (10 − GeV) for theparameters discussed above and m S ≃ MeV. The decayhappens just after S becomes non-relativistic but before bigbang nucleosynthesis, avoiding any cosmological problems.This model does not possess any DM-nucleon couplings attree-level. However, the operator in Eq. (6) induces an ef-fective magnetic dipole moment for the DM when couplinga photon to the lepton loop. This leads to a direct detectioncross-section for X scattering off of a nucleon (see [2] andthe references therein for details) σ dd ≃ × − cm (cid:18) Z/A . (cid:19) (cid:18)
600 GeV m H ′± /y (cid:19) . (18)This will be a signal for the next generation of experiments.To conclude, relating the lepton asymmetry to the DM den-sity implies a novel mechanism for obtaining both leptophilicDM and a separation between the freeze-out and present dayannihilation cross-sections. In these models, L -violating oper-ators which transfer the lepton asymmetry set the DM density,while related L -preserving operators set the rates for annihila-tion in indirect detection experiments (such as PAMELA andFermi). The smaller L -violating cross-sections set the relicdensity, while allowing for large cross-sections for indirectdetection experiments through the L -preserving operators. IfDM of this type is responsible for the cosmic ray anomalies,then it will be observed in the next generation of direct de-tection experiments. Non-minimal versions of the model cangenerate the SM neutrino masses and mixings at one-loop.Such classes of Asymmetric Dark Matter will continue to beimportant for both model building and experimental searchesfor DM in the galaxy today.This work has been supported by the US Department ofEnergy, including the grant DE-FG02-95ER40896 (KMZ)and the National Science Foundation, including the NSFCAREER Grant NSF-PHY-0743315 (TC). We thank RoniHarnik, Markus Luty, Ann Nelson, Frank Petriello, and AaronPierce for helpful discussions, and Graham Kribs for hostingthe Unusual Dark Matter Workshop at the University of Ore-gon (under DOE contract DE-FG02-96ER40969) where someof this work was developed. [1] S. Nussinov, Phys. Lett. B , 55 (1985). S. M. Barr,R. S. Chivukula and E. Farhi, Phys. Lett. B , 387(1990). S. B. Gudnason, C. Kouvaris and F. Sannino, Phys.Rev. D , 115003 (2006) [arXiv:hep-ph/0603014]. S. Do-delson, B. R. Greene and L. M. Widrow, Nucl. Phys. B , 467 (1992). V. Kuzmin, Phys. Part. Nucl. , 257(1998), Fiz. Elem. Chast. Atom. Yadra , 637 (1998),Phys. Atom. Nucl. , 1107 (1998) [arXiv:hep-ph/9701269].M. Fujii and T. Yanagida, Phys. Lett. B , 80 (2002)[arXiv:hep-ph/0206066]. R. Kitano and I. Low, Phys. Rev.D , 023510 (2005) [arXiv:hep-ph/0411133]. R. Kitano,H. Murayama and M. Ratz, arXiv:0807.4313 [hep-ph].G. R. Farrar and G. Zaharijas, Phys. Rev. Lett. , 041302 (2006) [arXiv:hep-ph/0510079]. D. Hooper, J. March-Russell and S. M. West, Phys. Lett. B , 228 (2005)[arXiv:hep-ph/0410114]. L. Roszkowski and O. Seto, Phys.Rev. Lett. , 161304 (2007) [arXiv:hep-ph/0608013]. O. Setoand M. Yamaguchi, Phys. Rev. D , 123506 (2007)[arXiv:0704.0510 [hep-ph]]. M. Aoki, S. Kanemura andO. Seto, arXiv:0904.3829 [hep-ph].[2] D. E. Kaplan, M. A. Luty and K. M. Zurek, Phys. Rev. D ,115016 (2009) [arXiv:0901.4117 [hep-ph]].[3] M. J. Strassler and K. M. Zurek, Phys. Lett. B , 374 (2007)[arXiv:hep-ph/0604261].[4] M. Aguilar et al. [AMS-01 Collaboration], Phys. Lett. B ,145 (2007) [arXiv:astro-ph/0703154].[5] S. W. Barwick et al. [HEAT Collaboration], Astrophys. J. ,L191 (1997) [arXiv:astro-ph/9703192]. J. J. Beatty et al. , Phys.Rev. Lett. , 241102 (2004) [arXiv:astro-ph/0412230].[6] S. Torii et al. , arXiv:0809.0760 [astro-ph].[7] O. Adriani et al. , arXiv:0810.4995 [astro-ph]. O. Adriani et al. ,arXiv:0810.4994 [astro-ph].[8] A. A. Abdo et al. [The Fermi LAT Collaboration],arXiv:0905.0025 [astro-ph.HE].[9] Nature , 362 (2008).[10] I. Cholis, L. Goodenough, D. Hooper, M. Simet and N. Weiner,arXiv:0809.1683 [hep-ph].[11] P. Meade, M. Papucci, A. Strumia and T. Volansky,arXiv:0905.0480 [hep-ph].[12] F. Donato, D. Maurin, P. Brun, T. Delahaye and P. Salati, Phys.Rev. Lett. , 071301 (2009) [arXiv:0810.5292 [astro-ph]].[13] G. Kane, R. Lu and S. Watson, arXiv:0906.4765 [astro-ph.HE].[14] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer andN. Weiner, Phys. Rev. D , 015014 (2009) [arXiv:0810.0713[hep-ph]].[15] J. Hisano, S. Matsumoto, M. M. Nojiri and O. Saito, Phys.Rev. D , 063528 (2005) [arXiv:hep-ph/0412403]. M. Cirelli,M. Kadastik, M. Raidal and A. Strumia, Nucl. Phys. B , 1(2009) [arXiv:0809.2409 [hep-ph]].[16] S. M. Barr, R. S. Chivukula and E. Farhi, Phys. Lett. B , 387(1990).[17] J. A. Harvey and M. S. Turner, Phys. Rev. D , 3344 (1990).[18] R. Fardon, A. E. Nelson and N. Weiner, JCAP , 005 (2004)[arXiv:astro-ph/0309800].[19] P. Fileviez Perez and M. B. Wise, arXiv:0906.2950 [hep-ph].[20] H. Fukuoka, J. Kubo and D. Suematsu, Phys. Lett. B , 401(2009) [arXiv:0905.2847 [hep-ph]]. C. H. Chen, C. Q. Gengand D. V. Zhuridov, arXiv:0906.1646 [hep-ph].[21] P. J. Fox and E. Poppitz, Phys. Rev. D , 083528 (2009)[arXiv:0811.0399 [hep-ph]]. K. M. Zurek, Phys. Rev. D , 115002 (2009) [arXiv:0811.4429 [hep-ph]]. Q. H. Cao,E. Ma and G. Shaughnessy, Phys. Lett. B , 152 (2009)[arXiv:0901.1334 [hep-ph]].[22] Previous works considered DM from the lepton asymmetry asan explanation of the cosmic ray positron excesses, but utilizeddecaying DM with a lifetime tuned to the age of the universe[20]. There have also been other models of leptophilic DM un-related to the lepton asymmetry that have utilized a Sommerfeldenhancement to generate the boost [21][23] In deriving this relation we have assumed that the universe re-heated high enough for the electroweak sphalerons to be activeand that they remain in equilibrium at temperatures below theelectroweak phase transition. Hence, at the sphaleron decou-pling temperature we assume that the top quark and H ′′