Simplified Dirac Dark Matter Models and Gamma-Ray Lines
aa r X i v : . [ h e p - ph ] O c t Simplified Dirac Dark Matter Models and Gamma-Ray Lines
Michael Duerr, Pavel Fileviez Pérez, and Juri Smirnov
Particle and Astroparticle Physics Division,Max-Planck-Institut für Kernphysik,Saupfercheckweg 1, 69117 Heidelberg, Germany
We investigate simplified dark matter models where the dark matter candidate is a Diracfermion charged only under a new gauge symmetry. In this context one can understanddynamically the stability of the dark matter candidate and the annihilation through the newgauge boson is not velocity suppressed. We present the simplest Dirac dark matter modelcharged under the local B − L gauge symmetry. We discuss in great detail the theoreticalpredictions for the annihilation into two photons, into the Standard Model Higgs and aphoton, and into the Z gauge boson and a photon. Our analytical results can be used forany Dirac dark matter model charged under an Abelian gauge symmetry. The numericalresults are shown in the B − L dark matter model. We discuss the correlation between theconstraints on the model from collider searches and dark matter experiments. PACS numbers: 95.35.+d, 12.60.Cn
CONTENTS
I. Introduction 3II. Simplified Models 4A. B − L Dirac Dark Matter 51. Relic Density 72. Direct Detection 9B. Upper Bound on the Dark Matter Mass 10III. Gamma-Ray Lines 11A. Loop-Induced Couplings 11B. Gamma-Ray Lines and B − L Symmetry 16IV. Summary 18Acknowledgments 19A. Experimental Constraints on the Annihilation Cross Sections 19B. Loop Functions 19References 21
I. INTRODUCTION
The existence of dark matter (DM) in the Universe has motivated the particle physics communityto investigate extensions of the Standard Model (SM) of particle physics, with countless darkmatter candidates on the market which need to be probed against experiments. There are basicallythree ways to search for these candidates: one looks for possible signals in dark matter directdetection experiments, for gamma-ray lines and other signals from dark matter annihilation inindirect detection experiments, and for large missing energy signatures together with a mono-jet ora mono-photon at colliders. See Refs. [1–4] for a detailed discussion of these possibilities.Most of the effort so far has been focused on studies of complete models and their dark mattercandidates, as for example the neutralino in the Minimal Supersymmetric Standard Model. Alsoeffective field theory (EFT) has been used to constrain the scale of the new physics, particularlyusing the data from the LHC run one. However, given the large center-of-mass energy of the LHCin its second run, it is obvious that the EFT approach is prone to fail in large fractions of parameterspace. Therefore, it is sensible to consider simplified dark matter models which can capture themain features of the dark matter sector with a limited number of parameters. For a discussion ofsimplified models see Refs. [5, 6].A simplified model should contain a mediator and a dark matter candidate, and it should notviolate generically any low energy observables. These criteria are necessary but not sufficient, sincea naive model can result in misleading statements. We argue that the simplified model must beself consistent, i.e., not violate gauge invariance and be anomaly free. Only then is it possible toperform a full study and compute for example higher order processes leading to gamma-ray linesrelevant for indirect dark matter searches.In this article we investigate in detail simplified models for Dirac dark matter, which have thefollowing features: • The dark matter is charged under a local gauge symmetry. This symmetry can be sponta-neously broken at the low scale and a remnant discrete symmetry guarantees the dark matterstability. In the simplest case one has a local U (1) ′ symmetry broken to a Z discrete sym-metry. In the case of an unbroken gauge symmetry one can use the Stueckelberg mechanismand the dark matter stability is ensured by the choice of quantum numbers. • One can generate mass for the new gauge boson using the Stueckelberg or the Higgs mech-anism. In the Stueckelberg scenario the dark matter candidate interacts only with the newneutral gauge boson in the theory. However, if one uses the Higgs mechanism the dark mattercould also have interactions with the new physical Higgs boson. • The existence of a new gauge boson is key for the testability of the mechanism for dark matterstability. Therefore, in the ideal case the new gauge boson should define all the propertiesof the dark matter candidate. In particular, the main annihilation channels must proceedthrough the interaction between the dark matter and the new gauge boson. One can showthat in this case the dark matter annihilation through the new gauge boson is not velocitysuppressed.For a Dirac dark matter, we investigate in detail the relic density constraints, the predictionsfor direct detection and the dark matter annihilation channels producing monochromatic photons, ¯ χχ → Z ′ → γγ, hγ, Zγ . We compute the one-loop generated vertices Z ′ γγ , Z ′ hγ and Z ′ Zγ needed for the annihilation cross sections. We discuss all the technical details for the computationof the loop graphs and we stress the need to check the Ward and Slavnov–Taylor identities to makesure the final results are correct. We point out that the effective coupling Z ′ γγ is possible only inmodels where the charged fermions inside the loop have an axial coupling to the Z ′ . The effectivecouplings Z ′ hγ and Z ′ Zγ are always present when the Standard Model fermions are charged underthe new gauge symmetry. Our results can be used for the study of the gamma-ray lines in anytheory with Dirac dark matter charged under a new gauge symmetry.In order to illustrate the main results, we discuss the simplest possible self-consistent dark mattermodel which is most relevant for studying the connection between direct and indirect dark mattersearches, as its annihilation cross section is not velocity suppressed. In this model, dark matter ischarged under the B − L gauge symmetry and one has only two annihilation channels into photons, ¯ χχ → Z BL → hγ, Zγ . We discuss the parameter space for direct detection in agreement with therelic density and collider constraints, and we show the experimental limits on the indirect searchesfor the gamma-ray lines and b ¯ b . We show that the interplay between the relic density, collidersearches and indirect dark matter detection experiments sets non-trivial bounds on these simplifiedmodels. II. SIMPLIFIED MODELS
We discuss models where the dark matter is a Dirac fermion χ charged only under a new gaugeforce. In this context we can understand why the dark matter is stable. For simplicity, we considerthe case where one has an Abelian force, i.e., a U (1) ′ . The part of the Lagrangian relevant for ourdiscussion is L ⊃ − F ′ µν F ′ µν + i ¯ χ L γ µ D µ χ L + i ¯ χ R γ µ D µ χ R − ( M χ ¯ χ R χ L + h.c. ) + 12 M Z ′ Z ′ µ Z ′ µ , (1)where F ′ µν = ∂ µ Z ′ ν − ∂ ν Z ′ µ , (2) D µ χ L = (cid:0) ∂ µ + ig ′ n L Z ′ µ (cid:1) χ L , (3) D µ χ R = (cid:0) ∂ µ + ig ′ n R Z ′ µ (cid:1) χ R . (4)Here we neglect the kinetic mixing between the new Abelian symmetry and U (1) Y . For the momentwe do not discuss the anomaly cancellation and how the masses are generated but will address theseissues later in a well-motivated model. Notice that in general the Standard Model fermions can becharged under the new symmetry such that new fermions are needed for anomaly cancellation.In these models the relevant interaction of the dark matter candidate χ = χ L + χ R to the Z ′ gauge boson is given by − ig ′ ¯ χ γ µ ( n L P L + n R P R ) χZ ′ µ , (5)where we use the standard projection operators, P L = (1 − γ ) and P R = (1 + γ ) . Theinteractions of all other fermions f in the theory, the SM fermions or new fermions needed foranomaly cancellation, to the Z ′ can be parametrized as − ig ′ ¯ f γ µ (cid:16) g fV − g fA γ (cid:17) f Z ′ µ . (6)As usual, all charged fermions couple to the photon A µ according to their electric charge Q f , − ieQ f ¯ f γ µ f A µ , (7)and the coupling of the fermions to the Standard Model Z can be parametrized as − i g cos θ W ¯ f γ µ (cid:16) g fL P L + g fR P R (cid:17) f Z µ , (8)where g is the SU (2) gauge coupling and θ W is the Weinberg angle. A. B − L Dirac Dark Matter
The local B − L symmetry is anomaly free once we add three copies of right-handed neutrinosto the Standard Model particle content. It is well known that this symmetry could play a majorrole in neutrino physics. Here we focus on a very simple model with Dirac dark matter chargedunder B − L . The relevant part of the Lagrangian is given by L BL ⊃ i ¯ χγ µ D µ χ − M χ ¯ χχ + 12 M Z BL Z BLµ Z µBL , (9)where D µ χ = (cid:0) ∂ µ + ig BL n Z µBL (cid:1) χ , n = ± , and χ = χ L + χ R . One can generate the gauge bosonmass through the Higgs mechanism or the Stueckelberg mechanism. Let us discuss both cases here: • Stueckelberg Mechanism:
The mass of the B − L gauge boson can be generated through theStueckelberg mechanism as discussed in Ref. [7]: L BL ⊃
12 ( M Z BL Z BLµ + ∂ µ σ ) (cid:0) M Z BL Z µBL + ∂ µ σ (cid:1) − (cid:16) Y ν ¯ ℓ L ˜ Hν R + h.c. (cid:17) , (10)where the gauge transformations are given by δZ µBL = ∂ µ λ and δσ = − M Z BL λ. (11)In this case the neutrinos are Dirac fermions because the B − L symmetry is never broken,and the dark matter stability is a result of the choice of the quantum number for the darkmatter candidate. • Higgs Mechanism:
One can generate the mass for the B − L gauge boson through the Higgsmechanism and at the same time we can generate masses for the SM neutrinos through thesee-saw mechanism [8–12] via the following interactions: − L ν = Y ν ¯ ℓ L ˜ Hν R + λ R ν R ν R S BL + h . c . (12)Here S BL is a Standard Model singlet and has B − L charge two. Notice that if the B − L charge of the new Higgs is different from two, the neutrinos will be Dirac fermions. After the U (1) B − L is broken, there is a remnant Z symmetry which is the reason for the dark matterstability.These simple models have only four relevant parameters for the dark matter study: the gaugecoupling g BL , the dark matter mass M χ , the gauge boson mass M Z BL , and n the B − L charge of thedark matter candidate. The relevant interactions needed to compute the dark matter annihilationchannels are − ig BL n ¯ χγ µ χZ µBL and − ig BL n fBL ¯ f γ µ f Z µBL , (13)where n fBL is the B − L charge of the Standard Model fermion f .
1. Relic Density
The B − L dark matter candidate χ can annihilate into all the Standard Model particles andthe B − L gauge boson Z BL . Therefore, one can have the annihilation channels ¯ χχ → ¯ qq, ¯ ℓℓ, ¯ νν, Z BL Z BL in both the Stueckelberg and the Higgs scenario discussed above. In the Higgs scenario, one hasthe additional annihilation to right-handed neutrinos. There are two main regimes for our study: • M χ < M Z BL : when the dark matter candidate is lighter than the B − L gauge boson, wehave the following channels, ¯ χχ → Z ∗ BL → ¯ qq, ¯ ℓℓ, ¯ νν, ( ν R ν R ) . • M Z BL < M χ : when the dark matter candidate is heavier than the B − L gauge boson, onehas a new open channel which is not velocity suppressed, ¯ χχ → Z BL Z BL . In order to test this model at the collider, the invisible decay Z BL → ¯ χχ is crucial to establish theconnection between the existence of the new gauge boson and the dark matter candidate. Therefore,we focus on the first regime.The annihilation cross section for ¯ χχ → Z ∗ BL → ¯ f f is given by σ ( ¯ χχ → Z ∗ BL → ¯ f f ) = N fc ( n fBL ) g BL n πs q s − M f q s − M χ (cid:0) s + 2 M χ (cid:1) (cid:16) s + 2 M f (cid:17)h ( s − M Z BL ) + M Z BL Γ Z BL i . (14)Here N fc is the color factor of the fermion f with mass M f , s is the square of the center-of-massenergy, and Γ Z BL is the total decay width of the Z BL gauge boson. In order to compute the relicdensity we use the analytic approximation [13] Ω DM h = 2 . × GeV − J ( x f ) √ g ∗ M Pl , (15)where M Pl = 1 . × GeV is the Planck scale, g ∗ is the total number of effective relativisticdegrees of freedom at the time of freeze-out, and the function J ( x f ) reads as J ( x f ) = Z ∞ x f h σv i ( x ) x dx. (16) Ω D M h M χ [GeV] n = 1 / Planck M Z BL = 3 . g BL = 0 . M Z BL = 1 . g BL = 0 . − − − −
500 1000 1500 2000 2500 3000 Ω D M h M χ [GeV] n = 3 Planck M Z BL = 3 . g BL = 0 . M Z BL = 1 . g BL = 0 . − − − − − Figure 1. Dark matter relic density Ω DM h vs. the dark matter mass M χ for different choices of the darkmatter B − L quantum number n . In the left panel we use n = 1 / , while in the right panel we use n = 3 ,and we give the relic density for two choices of the mass of the Z BL and the gauge coupling g BL that fulfill M Z BL /g BL = 7 TeV . The thin blue band corresponds to the currently allowed relic dark matter densitymeasured by the Planck collaboration, Ω DM h = 0 . ± . [14]. The thermally averaged annihilation cross section times velocity h σv i is a function of x = M χ /T ,and is given by h σv i ( x ) = x M χ K ( x ) Z ∞ M χ σ × ( s − M χ ) √ s K (cid:18) x √ sM χ (cid:19) ds, (17)where K ( x ) and K ( x ) are the modified Bessel functions. The freeze-out parameter x f can becomputed using x f = ln (cid:18) . g M Pl M χ h σv i ( x f ) √ g ∗ x f (cid:19) , (18)where g is the number of degrees of freedom of the dark matter particle. In Fig. 1 we show thenumerical predictions for the relic density vs. the dark matter mass for two values of n . In the left(right) panel we show the results for n = 1 / when M Z BL /g BL = 7 TeV , which is in agreementwith the collider bounds [15]. As expected, for small values of the gauge coupling one needs to relyon the resonance to achieve the right relic density. However, generically one can be far from theresonance and in agreement with relic density constraints.
2. Direct Detection
The direct detection constraints must be considered in order to understand which are the allowedvalues of the input parameters in this theory. The elastic spin-independent nucleon–dark mattercross section is given by σ SI χN = M N M χ π ( M N + M χ ) g BL M Z BL n , (19)where M N is the nucleon mass. Notice that σ SI χN is independent of the matrix elements. The crosssection can be rewritten as σ SI χN ( cm ) = 12 . × − (cid:16) µ (cid:17) (cid:18) r BL (cid:19) n cm , (20)where µ = M N M χ / ( M N + M χ ) is the reduced mass and r BL = M Z BL /g BL .In our case M χ ≫ M N , and using the collider lower bound M Z BL /g BL > [15] one findsan upper bound on the elastic spin-independent nucleon–dark matter cross section given by σ SI χN < . × − n cm , (21)for a given value of n . There is also a simple way to find a lower bound on the spin-independent crosssection. The minimal value of the gauge coupling g BL in agreement with relic density constraintscorresponds to the case when one sits on the resonance, i.e., M Z BL = 2 M χ . Therefore, the lowerbound on the cross section for a given value of the dark matter mass is given by σ SI χN > . × − (cid:18) M χ (cid:19) ( g min BL ) n cm . (22)In Fig. 2 we show the numerical predictions for the direct detection cross section σ SI χN vs. thedark matter mass M χ compatible with the relic density constraints. The colored dashed lines showthe values of σ SI χN for different choices of M Z BL /g BL compatible with current collider limits. Theblack dash-dotted line shows the minimal direct detection cross section. We show the bounds fromthe LUX [16] and XENON100 experiments [17], as well as the prospects for XENON1T [18]. Onecan see that for n = 1 / , the scenario for M Z BL /g BL = 6 TeV is allowed by the LUX experimentfor a large part of the parameter space. However, for the case n = 3 , the ratio M Z BL /g BL needsto be larger than 20 TeV in order to satisfy the experimental bounds. Therefore, only the scenariowhen n = 1 / could be tested at the LHC. Unfortunately, the minimal value of the cross section isbelow the neutrino background [19] and it is very difficult to test this part of the parameter spacein the current direct detection experiments.0 σ S I χ N [ c m ] M χ [GeV]Ω DM h = 0 . ± . n = 1 / minimal cross section M Z BL /g BL = 6 TeV M Z BL /g BL = 8 TeV M Z BL /g BL = 10 TeVneutrino background − − − − − −
500 1000 1500 2000 2500 3000
XENON100XENON1T LUX σ S I χ N [ c m ] M χ [GeV]Ω DM h = 0 . ± . n = 3 minimal cross section M Z BL /g BL = 6 TeV M Z BL /g BL = 10 TeV M Z BL /g BL = 20 TeVneutrino background − − − − − −
500 1000 1500 2000 2500 3000
XENON100XENON1T LUX
Figure 2. Direct detection cross section σ SI χN vs. the dark matter mass M χ compatible with the relic densityconstraints for n = 1 / (left panel) and n = 3 (right panel). The colored dashed lines show σ SI χN fordifferent choices of M Z BL /g BL compatible with current collider limits. The black dash-dotted line showsthe minimal direct detection cross section. We show the LUX [16] and XENON100 [17] constraints, as wellas the prospects for XENON1T [18]. The orange dash-dotted line shows the coherent neutrino scatteringbackground [19]. B. Upper Bound on the Dark Matter Mass
In this section we show that in these simple models it is possible to derive an upper bound on thedark matter mass. We focus on the case when the Z ′ is heavier than the dark matter because onlythen one can test the main properties of these models. The argument is based on the observationalrequirement that the relic density of dark matter produced in the freeze-out must not overclose theUniverse. Today, we know that Ω DM h ≤ . and since the relic density scales as Ω DM h ∝ / h σv i ,one has a lower bound on the annihilation cross section h σv i & h σv i ≈ × − cm / s . (23)Here h σv i is the minimal value of the cross section compatible with observations. In order toguarantee the validity of a given theory we have to make sure that the maximal value of the crosssection in the theory obeys the condition h σv i max ≥ h σv i . This is a necessary condition, sinceif it is not fulfilled there is no parameter choice in the model which can make it compatible withobservations.1The corresponding cross section has the following structure, σv ( g ′ , M Z ′ , M χ ) = c ( g ′ ) M χ (4 M χ − M Z ′ ) + M Z ′ Γ Z ′ with Γ Z ′ = c ( g ′ ) M Z ′ . (24)It is obvious that the maximum of this expression given a fixed value of g ′ is realized when M χ = M Z ′ . Examining the functional dependence of the obtained expression one finds that h σv i max = c c M χ & × − cm / s . (25)In the model with local B − L and n = 1 / , this leads to an upper bound on the mass of M χ . . and M Z BL . . . (26)Note that this bound is conservative because at the resonance one would need to perform the fullaverage as discussed in Sec. II A 1. This bound is useful to understand the possibility to test thistype of model. III. GAMMA-RAY LINES
In this section, we discuss the predictions for gamma-ray lines in detail. First, we give the generalresults for any simplified model with an Abelian gauge symmetry and a corresponding Z ′ , then wemove on to study numerically the predictions for the minimal B − L model discussed before. A. Loop-Induced Couplings
In order to understand the predictions for the dark matter annihilation into photons, we needto compute the loop-induced effective interactions Z ′ γγ , Z ′ hγ , and Z ′ Zγ shown in Figs. 3–5. • Z ′ γγ coupling : The coupling δ Γ µνσZ ′ γγ is generated by a loop of electrically charged fermions f charged also under U (1) ′ , see Fig. 3, and is given by δ Γ µνσZ ′ γγ = X f (cid:20) ǫ µνσα ( p − p ) α − s ǫ µσαβ p α p β p ν + 2 s ǫ νσαβ p α p β p µ (cid:21) A f + X f (cid:16) ǫ µσαβ p α p β p ν − ǫ νσαβ p α p β p µ (cid:17) A f + X f ǫ µναβ p α p β ( p + p ) σ A f . (27)Since we are interested in processes with two external photons, one has p µ ǫ ∗ µ ( p ) = p ν ǫ ∗ ν ( p ) =0 and the terms in the second line of Eq. (27) proportional to A f do not contribute to theamplitude. The relevant coefficient functions are given by A f = e Q f g ′ g fA N fc π (cid:2) s, M f , M f ) + 2 M f C (0 , , s ; M f , M f , M f ) (cid:3) , (28)2 p p γγZ ′ σ νff fµ Figure 3. Coupling Z ′ γγ generated by a loop of fermions f charged under U (1) ′ and carrying electric charge.In the calculation, the crossed diagram has to be taken into account. p p hγZ ′ µ νtt t Figure 4. Coupling Z ′ hγ generated by a top loop. In the calculation, the crossed diagram has to be takeninto account. A f = e Q f g ′ g fA N fc π s [2 + Λ( s, M f , M f )] . (29)Notice that this coupling can be generated only if the axial coupling g fA of the fermions inthe loop to the Z ′ is different from zero. See Appendix B for the explicit form of the loopfunctions Λ( s, M f , M f ) and C (0 , , s ; M f , M f , M f ) . In the above equations s = ( p + p ) , M f is the fermion mass, and N fc is the color factor and Q f the electric charge of the fermion.We checked that the Ward identities δ Γ µνσZ ′ γγ p µ = δ Γ µνσZ ′ γγ p ν = 0 are satisfied. • Z ′ hγ coupling : In Fig. 4 we show the coupling Z ′ hγ generated by a top loop. The otherStandard Model fermions have smaller Yukawa couplings and their contributions are thereforenegligible. For the top quark g tA = 0 , and the coupling is given by δ Γ µνZ ′ hγ = C Z ′ hγ (cid:2) p ν p µ + (cid:0) M h − s (cid:1) g µν (cid:3) + p ν ˜ C µZ ′ hγ , (30)where C Z ′ hγ = ( − i ) 34 π g ′ g tV eQ t M t v ( s − M h ) (cid:8) s (cid:2) Λ( M h , M t , M t ) − Λ( s, M t , M t ) (cid:3) p p ZγZ ′ σ νff fµ Figure 5. Coupling Z ′ Zγ generated by a loop of fermions charged under U (1) ′ and carrying electric charge.In the calculation, the crossed diagram has to be taken into account. + ( M h − s ) (cid:2) s + 4 M t − M h ) C (0 , M h , s ; M t , M t , M t ) (cid:3)(cid:9) . (31)Here v is the vacuum expectation value of the Standard Model Higgs. Notice that forprocesses with an external photon, the ˜ C Z ′ hγ term does not contribute to the amplitude. SeeAppendix B for the explicit form of the loop functions. For this vertex, one can show thatthe Ward identity is satisfied, δ Γ µνZ ′ hγ p ν = 0 . • Z ′ Zγ coupling: The coupling between the Z ′ , the photon, and the Z can be generated atone-loop level as shown in Fig. 5. This coupling can be written as δ Γ µνσZ ′ Zγ = − g ′ g eQ f N fc π cos θ W B µνσ , (32)where B µνσ = X f (cid:26) ǫ µνσα p α B f + B f (cid:20)
12 ( M Z − s ) ǫ µνσα p α + ǫ µσαβ p α p β p ν (cid:21) + ǫ µσαβ p α p β p ν B f + ǫ νσαβ p α p β p µ B f (33) + ǫ νσαβ p α p β p µ B f + ǫ µναβ p α p β (cid:16) p σ B f + p σ B f (cid:17) (cid:27) . For processes with external Z and γ only, the terms in the second line of Eq. (33) proportionalto B f and B f do not contribute and the relevant B fi functions are given by B f = 4 (cid:0) M Z − s (cid:1) n(cid:16) A f g fA + B f g fV (cid:17) × (cid:2) (cid:0) M Z − s + M Z Λ( M Z , M f , M f ) (cid:1) − (2 M Z + s )Λ( s, M f , M f ) (cid:3) (34) + h B f g fV ( M Z − M f ) + A f g fA ( M Z + 2 M f ) i ( M Z − s ) C (0 , M Z , s ; M f , M f , M f ) o ,B f = − (cid:0) s − M Z (cid:1) (cid:16) A f g fA + B f g fV (cid:17) (cid:2) s Λ( M Z , M f , M f ) − s Λ( s, M f , M f ) + (cid:0) M Z − s (cid:1) (cid:0) M f C (0 , M Z , s ; M f , M f , M f ) (cid:1)(cid:3) , (35) B f = 8 (cid:0) s − M Z (cid:1) (cid:16) A f g fA + B f g fV (cid:17) (cid:8) M Z Λ( M Z , M f , M f ) − (2 M Z + s )Λ( s, M f , M f )+ ( M Z − s ) (cid:2) M f + M Z ) C (0 , M Z , s ; M f , M f , M f ) (cid:3)(cid:9) , (36) B f = 8 (cid:0) s − M Z (cid:1) (cid:16) A f g fA + B f g fV (cid:17) (cid:2) Λ( M Z , M f , M f ) − Λ( s, M f , M f ) (cid:3) , (37) B f = 8 (cid:0) s − M Z (cid:1) (cid:16) A f g fA + B f g fV (cid:17) (cid:2) M Z Λ( M Z , M f , M f ) − ( M Z + s )Λ( s, M f , M f )+( M Z − s ) (cid:0) M Z C (0 , M Z , s ; M f , M f , M f ) (cid:1)(cid:3) . (38)Here the coefficients A f and B f are given by A f = 12 (cid:16) g fL + g fR (cid:17) and B f = 12 (cid:16) g fL − g fR (cid:17) . (39)Using the Ward identity one can write B f as a function of B f : B f = − (cid:0) s − M Z (cid:1) B f . (40)In the case of a spontaneously broken gauge symmetry, one has to use the Slavnov–Tayloridentity [20, 21] − ip µ δ Γ µνσZ ′ Zγ = M Z δ Γ νσZ ′ Aγ , (41)which is the generalization of the Ward identity. Here, A is the Goldstone boson. From thisidentity, one finds the relation B f = − M Z B f −
12 ( s − M Z ) B f − B f g fV M f C (0 , M Z , s ; M f , M f , M f ) . (42)We have checked that these identities are satisfied in our calculations. Since for our case wefind B f = 0 , one can write B f = −
12 ( s − M Z ) B f − B f g fV M f C (0 , M Z , s ; M f , M f , M f ) ≡ −
12 ( s − M Z ) B f − C f . (43)It can easily be checked that the following additional relation between the coefficients holds, B f = B f − B f − B f . (44)These relations are very useful to cross-check the results and simplify the final expressionsfor the cross sections. We have used Package-X [22] to perform all one-loop calculations andhave cross-checked the results.5Using the above calculations for the loop-induced couplings, we compute the dark matter anni-hilation cross sections for the different channels: • ¯ χχ → γγ : The amplitude for the dark matter annihilation into two photons is given by (cid:12)(cid:12) M ( ¯ χχ → γγ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)P f (cid:16) A f − s A f (cid:17)(cid:12)(cid:12)(cid:12) ( n R − n L ) ( g ′ ) M χ s M Z ′ (cid:0) M Z ′ + Γ Z ′ (cid:1) = α π ( n R − n L ) ( g ′ ) M χ sM Z ′ (cid:0) M Z ′ + Γ Z ′ (cid:1) X f g fA N fc Q f (cid:2) M f C (0 , , s ; M f , M f , M f ) (cid:3) . (45)The cross section times velocity for this channel in the non-relativistic limit is given by σ ( ¯ χχ → γγ ) v = (cid:12)(cid:12)(cid:12)P f (cid:16) M χ A f − A f (cid:17)(cid:12)(cid:12)(cid:12) ( n L − n R ) ( g ′ ) M χ π M Z ′ (cid:0) M Z ′ + Γ Z ′ (cid:1) . (46) • ¯ χχ → hγ : For the dark matter annihilation into the SM Higgs and a photon one finds theamplitude (cid:12)(cid:12) M ( ¯ χχ → hγ ) (cid:12)(cid:12) = ( g ′ ) (cid:12)(cid:12) C Z ′ hγ (cid:12)(cid:12) (cid:0) M h − s (cid:1) h(cid:0) s − M Z ′ (cid:1) + M Z ′ Γ Z ′ i × (cid:2) cos θ ( n L + n R )( s − M χ ) + s ( n L + n R ) + 8 M χ n L n R (cid:3) , (47)where θ is the angle between h and γ in the center-of-mass system. The correspondingannihilation cross section is given by σ ( ¯ χχ → hγ ) = ( g ′ ) | C Z ′ hγ | ( s − M h ) πs / (cid:2) ( n L + n R ) s − ( n L − n L n R + n R ) M χ (cid:3)q s − M χ (cid:2) ( s − M Z ′ ) + Γ Z ′ M Z ′ (cid:3) . (48) • ¯ χχ → Zγ : In general, the explicit form of the amplitude for the dark matter annihilationinto a Z and a photon is very involved and cannot be given here. Here we list the result for n L = n R = n where the integrated amplitude is given by Z d Ω2 π (cid:12)(cid:12) M ( ¯ χχ → Zγ ) (cid:12)(cid:12) = ( g ′ ) g e n π cos θ W × (cid:0) M χ + s (cid:1) (cid:0) M Z − s (cid:1) (cid:0) M Z + s (cid:1) (cid:12)(cid:12)(cid:12)P f Q f N fc (cid:16) C f + B f M Z (cid:17)(cid:12)(cid:12)(cid:12) M Z s (cid:2) ( s − M Z ′ ) + Γ Z ′ M Z ′ (cid:3) , (49)while the cross section is given by σ ( ¯ χχ → Zγ ) = ( g ′ ) g e n (cid:0) M χ + s (cid:1) (cid:0) s − M Z (cid:1) (cid:0) M Z + s (cid:1) (cid:12)(cid:12)(cid:12)P f Q f N fc (cid:16) C f + B f M Z (cid:17)(cid:12)(cid:12)(cid:12) π cos θ W M Z s / q s − M χ (cid:2) ( s − M Z ′ ) + Γ Z ′ M Z ′ (cid:3) . (50)6Notice that these results are very general and can be used for any dark matter model with thefeatures discussed above. For simplicity, we show the results for the ¯ χχ → Zγ only for a vectorcoupling of the Z ′ to dark matter. In Refs. [23–25], some of these effective couplings have beencomputed and the implications for dark matter models have been investigated in detail. B. Gamma-Ray Lines and B − L Symmetry
In the simple B − L dark matter model one has only vector couplings to the B − L gauge bosonand there are only two relevant channels for the annihilation into gamma-ray lines: ¯ χχ → Z ∗ BL → hγ, Zγ. The energy of the line signal for the process ¯ χχ → γX is given by E γ = M χ (cid:18) − m X M χ (cid:19) , (51)where m X is the mass of the particle X . The cross section for the dark matter annihilation intothe Standard Model Higgs and a photon is given by σ ( ¯ χχ → hγ ) = n g BL | C Z BL hγ | ( s − M h ) πs / (cid:0) s + 2 M χ (cid:1)q s − M χ h ( s − M Z BL ) + Γ Z BL M Z BL i . (52)The explicit expression for the one-loop generated coupling C Z BL hγ can be found using Eq. (31)where g tV is replaced by the B − L number 1/3 for the top quark and g ′ → g BL . In the non-relativisticlimit the above cross section times velocity reads as σ ( ¯ χχ → hγ ) v = n g BL | C Z BL hγ | (4 M χ − M h ) πM χ h ( M Z BL − M χ ) + Γ Z BL M Z BL i . (53)The annihilation cross section for ¯ χχ → Z ∗ BL → Zγ is given by Eq. (50).In Fig. 6 we show the allowed parameter space for the indirect detection for the channels ¯ χχ → hγ (left panel) and ¯ χχ → Zγ (right panel) compatible with the relic density constraint. We show theresults in two different scenarios for the dark matter charge n = 3 and n = 1 / . Only in the case n = 3 the bounds from Fermi-LAT rule out a small part of the parameter space in the low massregion. In both panels we show the full range for the annihilation cross sections h σv i ¯ χχ → Zγ and h σv i ¯ χχ → hγ which is defined by the collider limits and the resonance regions. The complete regionbetween the two curves is allowed in the model. However, the region close to the resonance can beruled out in the near future.7 h σ v i ¯ χχ → h γ [ c m s − ] M χ [GeV]Ω DM h = 0 . ± . n = 3 on the resonance n = 3, M Z BL /g BL = 20 TeV n = 1 / n = 1 / M Z BL /g BL = 6 TeV − − − − − − − − −
200 30001000
Fermi-LAT H.E.S.S. h σ v i ¯ χχ → Z γ [ c m s − ] M χ [GeV]Ω DM h = 0 . ± . n = 3 on the resonance n = 3, M Z BL /g BL = 20 TeV n = 1 / n = 1 / M Z BL /g BL = 6 TeV − − − − − − − − −
200 30001000
Fermi-LAT H.E.S.S.
Figure 6. Allowed parameter space for the indirect detection for the channels ¯ χχ → hγ (left panel) and ¯ χχ → Zγ (right panel) compatible with the relic density constraint. The limits of Fermi-LAT [26] andH.E.S.S. [27] are shown by the blue and red lines, respectively. See Tab. I in Appendix A for the values ofthe cross section limits. Note that the highest cross section in the indirect detection experiments correspondsto the parameters leading to the lowest cross sections in the direct searches. In Fig. 7 we show the predictions for the cross section of the dark matter annihilation ¯ χχ → b ¯ b .Again, we find the full range for the annihilation cross section in agreement with the relic densityconstraints. The bounds from Fermi-LAT rule out a significant fraction of the parameter space inboth scenarios. As one expects, this bound is much stronger than the bounds from gamma-ray lines.It is important to mention that in the resonance region the direct detection limits are irrelevantand the best way to probe the model is through indirect detection.We have investigated carefully the final state radiation ¯ χχ → ¯ f f γ , which is mediated by the Z BL at tree level and contributes to the continuum gamma-ray spectrum. As we have discussedabove, the gamma-ray line in this model is possible only at the quantum level and the cross sectionsare much smaller than the final state radiation. Therefore, it is very challenging to distinguish thegamma-ray lines from the continuum spectrum. For example, when n = 1 / and M χ = 200 GeV ,the difference between the gamma-ray lines and the continuum is only about one percent. Let usstress that the possibility to distinguish the gamma-ray lines from the continuum is a requirementto actually use experimental limits from line searches to derive bounds on a particular dark mattermodel. In the future one could have a very good energy resolution in experiments such as Gamma-400 [28–30] and one can investigate this issue in more detail.8 h σ v i ¯ χχ → b ¯ b [ c m s − ] M χ [GeV]Ω DM h = 0 . ± . n = 3 on the resonance n = 3, M Z BL /g BL = 20 TeV n = 1 / n = 1 / M Z BL /g BL = 6 TeV − − − − − − − − − −
200 30001000
Fermi-LAT
Figure 7. Allowed parameter space for the dark matter annihilation into two bottom quarks compatiblewith the relic density constraint. The experimental bounds from Fermi-LAT [31] are given.
IV. SUMMARY
We have studied the relic density constraints and the predictions for the direct detection ex-periments in simplified models for Dirac dark matter. In these models the dark matter propertiesare defined mainly by the gauge interactions, and one can understand dynamically the dark matterstability. We discussed the cases where one uses the Stueckelberg or the Higgs mechanism for gen-erating the gauge boson mass. We have presented general results for the three possible annihilationchannels for the Dirac dark matter giving rise to gamma-ray lines, ¯ χχ → γγ, hγ, Zγ . We haveshown that the channel ¯ χχ → γγ is present only when the Z ′ has an axial coupling to fermionsinside the loop generating the effective coupling Z ′ γγ . The channel ¯ χχ → hγ is mainly generatedby the top quark because today one cannot have heavy chiral fermions which would change theHiggs properties. These results can be used for any model with a Dirac dark matter charged undera new Abelian force.In order to illustrate our results numerically we investigated a simple model based on local B − L where the neutrino masses could be generated through the seesaw mechanism and the dark matteris a Dirac fermion charged under B − L . In this case there are only two annihilation channels, ¯ χχ → hγ, Zγ . We have shown the numerical predictions for these channels taking into account therelic density and collider constraints. We have investigated the correlation between the Fermi-LATand H.E.S.S. bounds on gamma-ray lines and the annihilation into bottom quarks to show the9constraints on these models. The results presented in this paper tell us how much the bounds ongamma-ray lines must be improved to be able to rule out or test some well-motivated and simpledark matter models. ACKNOWLEDGMENTS
P.F.P. thanks S. Profumo for discussions. We thank Hiren H. Patel for helpful discussions andsupport while using Package-X [22] for the calculation of one-loop integrals and for pointing out anissue in the calculation of the γγ channel. Appendix A: Experimental Constraints on the Annihilation Cross Sections
Limits on the cross sections are set using the experimental limit on the photon flux, and therelations are different for the various annihilation channels considered in this article. • ¯ χχ → γγ : In the case of the annihilation of a Dirac dark matter into two photons the crosssection is related to the flux by h σv i ¯ χχ → γγ = 8 πJ ann E γ Φ γ . (A1) • ¯ χχ → Xγ : When one has the annihilation into a particle with mass m X and a photon, therelation is h σv i ¯ χχ → Xγ = 4 πJ ann (cid:16) E γ + q E γ + m X (cid:17) Φ γ , (A2)where E γ = M χ (cid:16) − m X M χ (cid:17) has been used.With the help of these relations, one can translate the limits from Fermi-LAT [26] and H.E.S.S. [27]given for the annihilation into two gammas into the hγ and Zγ channels. See Tab. I for theFermi-LAT limits in the range
30 GeV < E γ <
500 GeV . Appendix B: Loop Functions
In this appendix, we define the loop functions used throughout the article. The function Λ( s, m , m ) contains the logarithmic discontinuity of the Passarino–Veltman B function. For0 Table I. Limits on the gamma-ray lines from the Fermi-LAT collaboration for E γ >
30 GeV for the R3 regionwith a NFWc dark matter profile [26]. Using the limits on the γγ channel, we derive the corresponding crosssection limits and dark matter masses for the Zγ and hγ channels. Notice that for the Dirac dark mattercase the cross section limits have to be multiplied by a factor of two. γγ channel hγ channel Zγ channel E γ [GeV] h σv i γγ [ − cm / s ] M χ [GeV] h σv i hγ [ − cm / s ] M χ [GeV] h σv i Zγ [ − cm / s ]31.2 0.661 80.4 8.77 63.8 5.5333.0 0.695 81.5 8.47 65.0 5.3934.9 1.42 82.7 15.9 66.3 10.236.9 3.08 84.0 31.9 67.6 20.739.0 3.87 85.3 37.0 69.1 24.341.3 3.79 86.8 33.5 70.7 22.243.8 6.41 88.5 52.3 72.5 35.146.4 6.54 90.2 49.4 74.4 33.649.1 5.62 92.0 39.5 76.3 27.252.1 3.12 94.1 20.3 78.6 14.255.2 3.38 96.2 20.5 80.9 14.558.6 7.13 98.6 40.4 83.5 29.062.2 6.95 101 36.8 86.3 26.866.0 4.59 104 22.8 89.3 16.870.1 5.18 107 24.1 92.6 18.174.5 5.56 110 24.4 96.1 18.579.2 3.08 114 12.7 100 9.8284.2 2.87 118 11.2 104 8.7889.6 2.87 122 10.6 109 8.4595.4 2.82 127 9.93 114 8.01102 5.77 132 19.3 119 15.8108 5.73 137 18.4 125 15.3115 15.2 143 46.8 131 39.4123 15.1 149 44.6 138 38.0131 10.8 156 30.7 145 26.6140 5.29 164 14.5 154 12.7150 10.6 173 28.2 163 25.0160 8.15 182 21.0 172 18.9171 13.0 192 32.6 182 29.6183 6.68 203 16.4 194 15.0196 13.3 214 31.8 206 29.4210 9.19 227 21.5 219 20.1225 13.0 241 29.9 234 28.1241 18.7 256 42.3 249 40.0259 10.2 273 22.7 267 21.6276 41.2 290 90.7 283 86.8294 41.4 307 90.2 301 86.7321 17.3 333 37.2 327 36.0345 15.0 356 32.0 351 31.0367 48.7 377 103 373 100396 51.7 406 109 401 106427 49.6 436 103 432 101462 39.4 470 81.7 466 80.3 m = m = m it is given by Λ( s, m, m ) = r − m s ln m m − s (cid:18)q − m s + 1 (cid:19) . (B1)For special cases, the Passarino–Veltman C function can be given as C (0 , , s ; m, m, m ) = 12 s ln q − m s − q − m s + 1 (B2)and C (0 , M , s ; m, m, m ) = 12( M − s ) ln q − m M − q − m M + 1 − ln q − m s − q − m s + 1 . (B3) [1] G. Jungman, M. Kamionkowski and K. Griest, “Supersymmetric dark matter,” Phys. Rept. (1996)195 [arXiv:hep-ph/9506380].[2] L. Bergstrom, “Nonbaryonic dark matter: Observational evidence and detection methods,” Rept. Prog.Phys. (2000) 793 [arXiv:hep-ph/0002126].[3] J. L. Feng, “Dark Matter Candidates from Particle Physics and Methods of Detection,” Ann. Rev.Astron. Astrophys. (2010) 495 [arXiv:1003.0904 [astro-ph.CO]].[4] T. Bringmann and C. Weniger, “Gamma Ray Signals from Dark Matter: Concepts, Status andProspects,” Phys. Dark Univ. (2012) 194 [arXiv:1208.5481 [hep-ph]].[5] J. Abdallah et al. , “Simplified Models for Dark Matter Searches at the LHC,” Phys. Dark Univ. (2015) 8 [arXiv:1506.03116 [hep-ph]].[6] O. Buchmueller, M. J. Dolan, S. A. Malik and C. McCabe, “Characterising dark mattersearches at colliders and direct detection experiments: Vector mediators,” JHEP (2015) 037[arXiv:1407.8257 [hep-ph]].[7] D. Feldman, P. Fileviez Perez and P. Nath, “R-parity Conservation via the Stueckelberg Mechanism:LHC and Dark Matter Signals,” JHEP (2012) 038 [arXiv:1109.2901 [hep-ph]].[8] P. Minkowski, “ µ → eγ at a rate of one out of muon decays?,” Phys. Lett. B (1977) 421.[9] R. N. Mohapatra and G. Senjanović, “Neutrino Mass and Spontaneous Parity Nonconservation,” Phys.Rev. Lett. (1980) 912.[10] M. Gell-Mann, P. Ramond and R. Slansky, “Complex spinors and unified theories,” in Supergravity ,eds. P. van Nieuwenhuizen et al., (North-Holland, Amsterdam, 1979), p. 315.[11] S. L. Glashow, “The future of elementary particle physics,” in
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