Extra U(1) as natural source of a monochromatic gamma ray line

Extensions of the Standard Model with an extra U'(1) abelian group generically generate terms coming from loops of heavy fermions, leading to three gauge boson couplings, in particular Z'Z gamma. We show that WMAP data constrains the gauge coupling of the group g_D to values comparable with the electro-weak ones, rather independently of the mass of Z'. Moreover, the model predicts a monochromatic gamma-ray line which can fit a 130 GeV signal at the FERMI telescope for natural values of the Chern-Simons terms and a dark matter mass around 144.5 GeV.


INTRODUCTION
One of the most important issues in particle physics phenomenology is the nature and properties of the dark matter in our universe. The observations made by the WMAP collaboration [1] show that the matter content of the universe is dark, making up about 85 % of the total amount of matter. On the other hand, the XENON collaboration recently released its constraints on direct detection of Dark Matter [2] excluding large regions of several extensions of the Standard Model. These constraints makes it plausible that dark matter sits in a different sector, communicating with our sector through new, weak enough interactions.
Neutral gauge sectors with an additional dark U ′ (1) symmetry in addition to the Standard Model (SM) hypercharge U (1) Y and an associated Z ′ gauge boson are among the most natural extensions of the SM, and give the possibility that a dark matter candidate lies within this new gauge sector of the theory. Extra gauge symmetries are predicted in most Grand Unified Theories (GUTs) and appear systematically in string constructions. Larger groups than SU (5) or SO(10) allow the SM gauge group and U (1) ′ to be embedded into bigger GUT groups. String theory and brane-world U ′ (1)s are special compared to GUT U ′ (1)'s; some of them are hidden, such that SM particles are uncharged under them. For a review of the phenomenology of the extra U ′ (1)s generated in such scenarios see e.g. [3]. In such a framework, the extra Z ′ gauge boson would act as a portal between the dark world (particles not charged under the SM gauge group) and the visible sector. * Emilian.Dudas@cpht.polytechnique.fr † Yann.Mambrini@th.u-psud.fr ‡ Stefan.Pokorski@fuw.edu.pl § Alberto.Romagnoni@uam.es Several papers considered that the key of the portal could be the gauge invariant kinetic mixing (δ/2)F µν Y F ′ µν [4]. One of the first models of dark matter from the hidden sector with a massive additional U ′ (1), mixing with the SM hypercharge through both mass and kinetic mixings can be found in [5]. The Dark Matter (DM) candidate ψ 0 could be the lightest (and thus stable) particle of this secluded sector. Such a mixing has been justified in recent string constructions [6][7][8][9], but has also been studied within a model independent approach [10][11][12] or in a supersymmetric extension [13].
However, there exists another possibility for the Z ′ portal: the diagrams generated by the Chern-Simons terms, usually related to the mechanism of gauge anomaly cancelation. It has been shown [14,15] that these vertices could generate a specific smoking-gun signal for dark matter searches: a monochromatic gamma ray line from the Galactic Center [16]. Moreover, the recent hint for such a line 1 [17,18] raises the hope and interests for such theoretical extensions of the Standard Model (see also [19][20][21] for very recent discussions about this subject). The purpose of the present note is to prove that such models naturally accomodate both the WMAP data on dark matter and the generation of a monochromatic gamma ray line from the Galactic Center 2 .
This note is organized as follows. After defining the model, we present the phenomenological consequences 1 At 3.3 σ if one takes into account the look elsewhere effect, but currently unconfirmed by FERMI collaboration. 2 Very few models can achieve such a signal as its production is one-loop suppressed. However, this monochromatic ray can be enhanced in other scenarios like SUSY ones [22], extra-dimension constructions [23], singlet DM [24], decaying DM [25], including a neutrino sector [26], effective DM models [27] or inert Higgs doublet DM [28]. Internal Bremsstrahlung [29] can also exhibit a spectrum similar to the one produced by the emission of a monochromatic γ-ray line.
and study the parameter space which could respect WMAP and simultaneously explain a monochromatic gamma-ray line signal from the Galactic Center.

THE MODEL
Gauge invariance is a fundamental condition to ensure renormalizability and quantum consistency of any extension of the Standard Model. Triangle gauge anomalies cancelation can occur by a consistent field theory content or by cancelation of triangle loops by axionic couplings and Chern-Simons terms, via the string theory Green-Schwarz mechanism. At low-energy, remnants of the anomaly cancelation can lead to generalized Chern-Simon terms [30] containing new three gauge boson couplings. If one extend the Standard Model by an abelian gauge group U ′ (1), at low-energy the particular Chern-Simons terms we are interested in is where CS stands for Chern-Simons, and (α 1 ,α 2 ) are the coefficients (computable exactly once given the fermionic content of the model) generated by the triangle diagrams depicted in Fig.1. In the loops are running heavy fermions charged under both U (1) Y and U ′ (1) (when heavy fermion masses are SM gauge invariant such diagrams are still generated by higher dimensional operators [14,31]). Notice that the CS terms (1) are invariant under electromagnetism 3 U (1) A . The electroweak symmetry breaking then generates Z ′ ZZ, Z ′ Zγ Triangle diagrams whose variation generate counter terms of the form Eq.1 3 The broken SM symmetries and the U (1) ′ are realized in a Stueckelberg phase, as explained in [14]. The CS terms can be written in the manifestly gauge invariant way where θ is the Stueckelberg axion absorbed by the Z ′ gauge boson. Moreover M is a mass scale related to the mass of the heavy fermions. After electroweak symmetry breaking, we recover (1) with α i ∼ c i v 2 /M 2 . We notice here that CS terms are also generic in string constructions, where their gauge non-invariance is compensated by axionic couplings and triangle loops of light fermions charged under the extra U (1) ′ . If light fermions are present, the computations of three gauge boson vertices are changed qualitatively, see e.g. [30], [15], but we expect similar results to hold.
From Eq.1 one can deduce the vertices and Feynman rules after the electroweak breaking: with the obvious notation s W = sin θ W and c W = cos θ W .
If ψ is the lighter of the fermions charged under U'(1) (but not under the SM gauge group), coupling to Z ′ via the vertex where q L = q R = 1 in what follows, it can be considered to be a good dark matter candidate. The diagrams giving the annihilation rate contributing to the relic abundance are shown in Fig.2. Depending on the kinematics and values of the couplings, each of the two diagrams can dominate. Nowadays, the dark matter candidate being mainly at rest, the processψψ → Z ′ → Zγ can generate a monochromatic γ−ray line observable by the FERMI telescope. Depending on the mass of ψ, we describe in detail each possibility in the following section. Recently, it has been argued that the FERMI telescope did observe a monochromatic γ-ray line from the galactic center around a region E γ ≃ 130 GeV, with an annihilation cross section σv ≃ 2 × 10 −27 cm 3 s −1 [17], which would be a tantalizing smoking gun signal for new physics. Without discussing or anticipating the official analysis of the FERMI collaboration, we will try to check if such a clear signal can be produced by the CS terms generated in Eq.1. In what follows, we need to distinguish three different cases: 2m ψ < M Z ′ + M Z , m ψ < M Z ′ < 2m ψ − M Z and M Z ′ < m ψ , for suitable values of the dark matter mass and couplings to fit the supposed γ-ray line. We will show how the first case is strongly disfavored by the astrophysical data and how the latter ones are compatible with very natural values for the parameters of the model. Even if all the discussion in what follows is qualitative, the numerical analysis has been done using a version of Micromegas [32], adapted to include the new features of the model.
In this case, the only annihilation process kinematically allowed is the s−channel exchange of a Z ′ (see the left Fig.2). However, the two only final states being ZZ and Zγ, from the values of the couplings in Eq.2 one can easily deduce σv ZZ σv Zγ ≃ 0.3 which means that the Zγ final state is always the dominant one. If one wants to fulfill WMAP constraints for a thermal relic ( σv ≃ 3 × 10 −26 cm 3 s −1 ), one should impose σv Zγ ≃ 2 × 10 −26 cm 3 s −1 . Such a huge cross section would have produced a visible monochromatic line, which is already apriori excluded by the FERMI collaboration. Therefore, the mass range 2m Combinations of Chern-Simon coefficients g D × α 1 and g D × α 2 respecting WMAP constraint and producing a monochromatic γ−ray line around 130 GeV with σv Zγ ≃ 2 × 10 −27 [17] for M Z ′ 2m ψ − M Z (case B, see the text for details).
In this case, the opening of a new channelψψ → Z ′ → Z ′ Z allows the possibility to obtain a relic density obeying WMAP constraint and a monochromatic γ−ray flux observable by FERMI at the same time. In fact, the main difference with the case discussed in the previous subsection consists in the fact that now the contribution to thē ψψ → Z ′ Z process is proportional to the second Chern Simons coefficient of the Eq.1. Therefore, it is possible to decouple the two different processes, the annihilation cross section to fulfill WMAP, proportional to α 2 , and the annihilation cross section giving a monochromatic signal, proportional to α 1 .
The result is shown in Fig.3, where we plotted the regions of the parameter space still allowed by WMAP and respecting 0.4 × 10 −27 cm 3 s −1 < σv Zγ < 1.09 × 10 −27 cm 3 s −1 for a dark matter mass m ψ = 144.5 GeV. One can see that the dependence on the parameters can be expressed in terms of the products g D × α i . At the same time, as the WMAP constraints require a relatively larger cross section than the one for the monochromatic line, we require a ratio ≃ 10 between α 2 and α 1 . Therefore, if one consider reasonable values for standard coupling of the Chern Simons terms, α 2 ≃ 10 −2 , from Fig.3 one obtains g D ≃ 1 and even stronger (non-perturbative) values for smaller values of α 2 . This case is therefore compatible with the data, though a small hierarchy between α 1 and α 2 has to be assumed.
In this region the main contribution to the dark matter annihilation process comes from the t-channel ψ exchange depicted in the right panel of Fig.2. In this case, the relic density condition (independent on α 1 or α 2 ) is essentially decoupled from the s−channel diagram which produces the monochromatic line (Fig.2 left). We recall again that it was because the same diagram was responsible for the relic abundance and the monochromatic line that for example the region M Z ′ > 2m ψ − M Z has been excluded in the discussion above. Moreover, interestingly in this case, differently from the scenario in which m ψ < M Z ′ < 2m ψ − M Z , our analysis becomes independent on the parameter α 2 as soon as α 2 g D .
In more details, for a given monochromatic line (and so a given m ψ ), the WMAP condition fixes the coupling g D : the cross-section forψψ → Z ′ Z ′ depends only weakly - through a phase space coefficient -on M Z ′ . Then for a given value of the monochromatic annihilation cross section σv Zγ , one can deduce the value of α 1 fitting the FERMI data. We made a scan on (M Z ′ , g D , α 1 ) and applied the 5σ constraint from WMAP and the annihilation cross section proposed by [17] in the case of an Einasto profile. The result is presented in Fig.4. In Fig.  5 we present an example of the gamma-ray spectrum obtained from the diagrams shown in Fig. 2, for a point in the parameter space respecting WMAP 4 .
There are three striking features in Fig. 4: first, there is a weak dependence on M Z ′ ; secondly, the value of g D takes natural value for a U (1) coupling (we remind that g EW = 0.65); and finally α 1 takes typical one-loop order values ≃ 10 −2 , which are quite consistent with loop contributions generated by triangle diagrams of Fig.1.
This range of values is understandable. Indeed, we know that an annihilation cross section σv ≃ 3×10 −26 cm 3 s −1 leads to values of couplings of the order of the electroweak one for a WIMP mass of 100 GeV. As only the dark coupling g D appears in the annihilation channel ( Fig.2 right), one expect g D ≃ 0.6 for a WIMP mass of ∼ 100 GeV independently on Z ′ mass (except around the threshold) as one can see in Fig.4. Now, if one imposes that σv Zγ ≃ 10 −27 cm 3 s −1 , one can check semianalitically that α 1 ≃ g D /30 which is effectively what we also observe in Fig.4.
It could be interesting to notice that in any of the cases discussed above, the direct detection rate is largely suppressed. Indeed, this rate can become important for a kinetic mixing around δ ≃ 10 −3 [12] and could even explain DAMA/CoGENT excess with δ ≃ 10 −2 . However, 4 Micromegas was used for the calculation.
in the model we are considering, the "portal" between the dark matter sector and the visible one does not go through this kinetic mixing, but through the tri-vectorial couplings generated in Eq.1. In this case, very low values of δ are still allowed, rendering the direct detection (through t-channel Z' exchange) very difficult to observe.
One can also remark that for M Z ′ M Z , the Zγ final state channel is kinematically closed. The main decay channel for the Z ′ is thus through the kinetic mixing with the Z. It is important to check that in this case the kinetic mixing should not be too small to disturb the Big Bang Nucleosynthesis (BBN) problem. A straightforward computation of the Z ′ lifetime leads to (seconds) .
One may deduce that for δ 10 −11 , the Z ′ decays before one minute and will not affect the BBN processes. Thus, the region 10 −11 δ 10 −2 allows for a safely Z ′ decay from the BBN and electro-weak precision tests point of view. Moreover, notice that the natural one-loop contribution to the mixing, due to the diagrams induced by CS interaction terms, stays well inside that windows, for interesting (one-loop) values of the couplings α i and reasonable assumptions 5 for the scale Λ where the theory is completed (for example, the mass M of the heavy fermions).
One should also notice that several constraints from the continuum photons should be taken into account, especially the ones coming from the dwarf galaxies measured by FERMI telescope 6 [33]. Indeed, one have checked that the photons fluxes generated by the subsequent decays of the ZZ ′ final state (case B) or Z ′ Z ′ (the ZZ final state giving weak fluxes, reduced by a factor ≃ sin 2 θ W compared to Zγ) does not exceed the constraints obtained by FERMI collaboration [33]. CONCLUSION We have discussed an extension of the Standard Model with an extra U ′ (1) abelian group, where a three gauge boson couplings Z ′ Zγ is generated from Chern-Simons terms. We studied the different scenarios allowed by this model, for different values of the mass parameters, under the hypothesis that the dark matter candidate is charged only under the extra abelian group. Depending on the ratio between the mass of the dark matter and the mass of the mediator Z ′ , WMAP data constrains, more or less severely, the gauge coupling of the group g D , but always allowing it to have a very natural value comparable with the usual electroweak ones, independently of the absolute value of the Z ′ mass. At the same time, for M Z ′ < 2m ψ − M Z and M Z ′ < m ψ the model can provide a monochromatic γ−ray line which can fit a 130 GeV signal at FERMI telescope for a dark matter candidate mass m ψ = 144.5 GeV, again for rather natural values of the Chern-Simons couplings. Obviously, the model presented is intended to be an effective theory, where the unique effects of the beyond the standard model physics are encoded in the trilinear vector bosons couplings and the presence of a fermionic dark matter candidate. Nonetheless, already at this level it is possible to have a good estimate of the constraints that the new physics should satisfy in order to fit with this dark matter scenario. For example, the CS term in (1) contributes, at the loop level, to the mass of the Z gauge boson, which is experimentally known with an accuracy of 2 − 3 MeV. An order of magnitude estimate gives δM 2 Z ∼ (α 2 i /16π 2 )Λ 2 , where again Λ is an UV cutoff naturally of the order of the fermions generating the CS term. For couplings of order α i ∼ 10 −2 , we find Λ ≤ 500 GeV, which is marginally consistent with limits on vector-like fermions in the Standard Model. It would be interesting to compare the constraints we obtained on α 1 and g D from our combined WMAP/FERMI analysis with the constraints one could find with the LEP searches through the process e + e − → Z → γZ ′ → γγZ which is a 2γ plus Z final state signature. However, such analysis is beyond the scope of the present paper.