Galactic Centre GeV Photons from Dark Technicolor

We present a classically scale-invariant model with a confining dark sector, which is coupled to the Standard Model through the Higgs portal. The galactic centre gamma ray excess can be explained in this model by collision-induced dark matter decays to b-quarks. We discuss the possibility to obtain the dark matter relic density through thermal freeze-out, which however requires excessive fine-tuning. We then instead focus on a freeze-in scenario and perform detailed calculations and a parameter scan. We find that the observed relic density and the gamma ray excess can be explained by a wide range of parameters in this model.


I. INTRODUCTION
During the last decades, much effort has been spent to search for evidence of dark matter annihilation in the Milky Way. Gamma rays offer the most intriguing possible signal, because they travel unperturbed through the interstellar medium from the source to the detectors, and can be unambiguously observed. For some year now it has been know that there is an apparent excess of 1-3 GeV gamma rays in Fermi-LAT data from the galactic centre [1][2][3][4][5][6][7] and the inner regions of the galaxy [8,9].
In this paper we show how the observed photon and antiproton excesses can be explained by the collisioninduced decays of composite dark matter in a classically scale-invariant model. In this setup [23][24][25][26], here referred to as Dark Technicolor, the electroweak scale is generated dynamically in a strongly interacting hidden sector and mediated to the SM via a Higgs portal interaction with a messenger singlet. The stable dark matter candidate is a composite state, i.e. a dark technipion or dark technibaryon.
The coupling between the visible sector and the dark sector through the Higgs portal is necessarily weak to avoid LHC and direct detection constraints. This means that thermal freeze-out is not a viable way to generate a sufficiently small relic density unless the dark matter annihilation process is enhanced by an s-channel resonance, which requires a fine-tuned mass spectrum. Instead, we propose that the observed dark matter can be generated via freeze-in if the couplings of the messenger scalar are sufficiently small.
We will present our setup in section II. In section III we discuss in detail the origin of the gamma-ray signal.
In section IV we discuss dark matter generation through the freeze-in mechanism in the early universe. In section VI we conclude and discuss further implications of the model.

II. THE DARK TECHNICOLOR MODEL
The dark technicolor model consists of a confining dark sector, i.e., dark techniquarks Q charged under a non-Abelian gauge group SU (N ) TC , and a scalar messenger S that is a singlet under all gauge groups. The model, including the SM, is taken to be classically scale invariant, so that any explicit mass terms are absent at tree level. As the dark technicolor sector becomes confining at a scale Λ TC , the messenger field obtains a vacuum expectation value, which in turn generates an effective negative mass squared term for the Higgs via a Higgs portal coupling. Electroweak symmetry breaking then proceeds as in the SM.
The Lagrangian of the model is given by where L SM,mH=0 contains the SM gauge and fermion sectors, including the usual Yukawa couplings with the Higgs, but no explicit Higgs mass term. H is the Higgs is the field strength of the dark technicolor gauge group, and N f is the number of flavors in the hidden sector. The singlet messenger scalar S has Yukawa couplings with the hidden sector quarks, parametrized by y Qi , which are assumed to be flavour diagonal. We have included a parityviolating term in the Yukawa-couplings, parametrized by the constants a i , which is required for the pion decay through the scalar messenger. The confinement and chiral symmetry breaking results in Pseudo-Nambu-Goldstone bosons-massive dark tech-nipions, which are our dark matter candidate. In the absence of explicit chiral symmetry breaking terms these particles would be massless, but here the chiral symmetry of the dark technicolor sector is explicitly broken by the Yukawa couplings. The pion mass is estimated to be [27] where f π is the dark technipion decay constant and Λ T C is the dark technicolor confinement scale. v S is the vacuum expectation value of the messenger field, given by minimising the scalar potential in (1) as where λ Sh is the portal coupling and λ h ≈ 0.13 is the Higgs self coupling. The vev of the Higgs field is given as and the mass matrix of the scalar sector is so that in the zero-mixing limit the Higgs mass returns the SM value and the mass of the messenger scalar is The mixing angle between the messenger and the Higgs is obtained by diagonalising the mass matrix (5) as tan(2θ Sh ) = 4λ We will work in the limit where the mixing angle is small and hence the Higgs potential has the same form as in the SM.

III. THE GAMMA-RAY SIGNAL
The dark matter in our model consists of the stable technicolor mesons and baryons, which carry a conserved quantum number and are therefore protected from decaying. This symmetry charge can be either the dark sector baryon number, or dark sector flavor symmetries if theπ Yukawa couplings to the messenger scalar are strictly diagonal.
As shown in [28], it is possible to stabilise some of the mesons of the theory through dark baryon number if the fermions transform under the adjoint representation of the dark technicolor gauge group. In the simplest realisation of this scenario, the chiral symmetry breaking structure is SU (2N f ) → Sp(2N f ), resulting in 3 unstable and 2 stable pseudo-Goldstone bosons for N f = 2.
Alternatively, QCD-like chiral symmetry breaking with strictly diagonal Yukawa couplings results in 2 stable (π 1 ,π 1 , corresponding to π ± in the SM) and one unstable (π 0 ) meson. For concreteness, we concentrate our numerical analysis on this scenario, i.e., N f = 2 flavours in the fundamental representation of the technicolor gauge group with exactly diagonal Yukawa couplings. However, the general conclusions of the paper are just as valid for other symmetry breaking structures, if one compensates for the different group factors by changing the couplings by corresponding O(1) factors.
A dark matter particle with a mass of 31-40 GeV and an annihilation cross section of (1.4 − 2)10 −26 cm 3 /s into bb can fit the gamma ray signal very well [16]. In our model, however, the annihilation proceeds through the intermediate state of unstable technipions, resulting in four b-quarks per DM collision, π 1π1 → π 0 π 0 → 4b. Here π 1 ∼ |Q 2 Q 1 is the dark matter particle, which is stable due to the unbroken flavour symmetry of the dark technicolor sector and π 0 ∼ 1 √ 2 (|Q 1 Q 1 − |Q 2 Q 2 ) is the unstable pion that decays into bb through the Higgs portal. In order to reproduce the spectrum of the resulting b-quarks, we set the mass of the technipions to ∼ 80 GeV. Since dark matter in the galatic centre is non-relativistic (v/c ∼ 10 −3 ), the expected photon spectrum from the decay of a 80 GeV dark pion to bb is identical to the spectrum of direct s-channel annihilation of 40 GeV dark matter to bb. For a more detailed discussion of the resulting gamma-ray spectrum in a similar scenario, see [29].
In order to reproduce the observed gamma-ray excess, we will then set the self scattering cross section σ(π 1π1 → π 0 π 0 ) to the desired range of (2.8 − 4)10 −26 cm 3 /s. This number is a factor of 2 larger than the number given in [16], since in our model the dark matter mass is larger by a factor of two, as discussed above. Therefore the number density of our dark matter is reduced by a factor of two, resulting in a reduction of the annihilation rate by a factor of four. But as each annihilation results in four b-quarks instead of two, this results in reduction of the final flux of SM particles by a factor of two, which is then compensated by the larger cross section. The thermally averaged pion self scattering cross section is given as [28] σv where x = m π /T is the pion mass divided by the temperature of the DM. Taking m π = 80 GeV, x ∼ 10 3 and requiring σv ∼ (2.8 − 4) × 10 −26 cm 3 /s we can solve for f π , resulting in f π ∼ 280 − 300 GeV. The pion decay constant is not perturbatively calculable, but since it is found to be in the right ballpark between the pion mass and the confinement scale, it is reasonable to assume that by choosing the right gauge group and fermion representations, a theory can be found which reproduces the correct signal strength.
The dominant contribution to the decay width of the unstable pion is the s-channel diagram shown in figure 2. The resulting decay width is given by (10) where m min = min(m H , m S ) and C π is an unknown constant with dimension of mass, describing the strength of the πS coupling induced by the Yukawa coupling. The decay amplitude is proportional to the parity-violating part of the Yukawa-coupling ay Q , since the pseudoscalar pion has to decay through the scalar mediator, which is forbidden if parity is exactly conserved. We will take a ≈ 1 for simplicity. This will induce a small parity violation also to the SM sector through the Higgs mixing, but the effects are suppressed by the small portal coupling and are thus phenomenologically negligible.

IV. FREEZE-IN VS. FREEZE-OUT
We will in the following first discuss the possibility to obtain the correct relic density in a standard freeze-out scenario, and then for the rest of the paper concentrate on thermal freeze-in.

A. Problems with Thermal Freeze-Out
In the freeze-out scenario, contrary to what was discussed above, the annihilation of the dark matter proceeds through the s-channel diagram shown in figure 3, so that the process is π 1π1 → bb, and hence the we set the pion mass to m π = 40 GeV in this section. The corresponding thermally averaged annihilation cross section is given by where C π is a constant with dimension of mass, describing the strength of the ππS coupling. In our analysis we have taken C π = f π , since the magnitude of this coupling is set by the strong dynamics of the dark technicolor sector.
The strength of the portal coupling is restricted by collider constraints from Higgs boson observables and through the non-observation of dark matter at direct detection experiments. If the messenger scalar is lighter than the Higgs, the most stringent limit is set by LEP [31,32], implying sin θ Sh < 0.1. Therefore the annihilation cross section is usually small, so that the relic abundance in the freeze-out scenario is too large. However, if m s ≈ 2m π , there is a resonant enhancement of the scattering cross section. In this case it is possible to enhance the annihilation cross section enough, so that it will produce the correct relic abundance, but this requires fine-tuning between the pion mass and the mass of the messenger scalar, to the precision of O(10 −4 ) GeV as shown in figure 4. If one is willing to accept this level of fine-tuning, the resonant annihilation process can then be used to create the galactic center gamma ray excess. However, there is no a priori reason to expect such finetuned mass spectrum, and we will not consider it further in this paper. In the absence of the resonance, the annihilation process through the Higgs portal is weak and will freeze out early, producing an overabundance of dark matter. In this case, however, the self scattering of the pions dominates over the annihilation. The horizontal green line in figure 4 shows the self scattering cross section for the choice of parameters f π = Λ TC /(4π), λ Sh = 5 × 10 −4 . As is seen in the figure, the pion scattering cross section is stronger than the annihilation process unless m S is very close to the resonance. Therefore, after freeze-out of the annihilation process, the pions will continue to scatter into each other, keeping the stable and unstable pions in thermal equilibrium. Once the self scattering freezes out, the unstable pions will decay into the SM, and the relic abundance will be set by the remaining stable pions. This mechanism could perhaps be used to produce the observed relic abundance. However, the thermally averaged pion self scattering cross section is temperature dependent, as shown in equation (9). Thus, if we select the cross section at the freeze-out temperature so that it can produce the observed relic abundance, the much lower temperature in the galactic centre today will result in a too weak gamma ray signal compared to the observed flux. We will therefore not work out the details of this scenario here.

B. Freeze-In Scenario
We now instead assume that the portal coupling is weak, and that no dark matter is produced through reheating after inflation. The DM abundance is then generated through the freeze-in mechanism, via the scattering processes induced by the Higgs portal, V V → ππ and hh → ππ, where V refers to the Z and W -bosons of the SM, and h is the Higgs boson. The dark technicolor sec-tor then consists of two species of pions, the stable π 1 that is protected by the unbroken flavor symmetry, and the unstable π 0 that can decay into SM via the Higgs portal interaction. Both species are created in equal numbers in the scattering processes described above. The Higgs portal coupling has to be very weak in order to facilitate the freeze-in scenario, and thus the lifetime of the unstable pion is quite long, of the order of O(10 −2 ) s, but still short enough not to interfere with nucleosynthesis. After this time, the unstable pions decay into SM via π 0 → bb, and the dark sector from there on consists only of the stable species π 1 . In the galaxy center, where the local DM density is larger, the DM particles will self scatter into the unstable species, which then decays into bb, resulting in the observed gamma-ray excess.
The freeze-in scenario in the context of the Higgs portal coupling has been discussed in [33]. Our model is slightly more complicated, but following their discussion we can reproduce the relevant parts of the analysis. As the dark matter mass in our model is above 1 2 m H , the relevant processes for producing the DM abundance are the scattering processes V V → ππ and hh → ππ. In our model the relevant cross sections are given as where V = Z, W . Again we have taken C π ≈ f π , and made the approximation that the scattering amplitude is dominated by the lighter of the mass eigenstates S or h in the s-channel, neglecting the interference between the two diagrams. This is a good approximation in the limit of small mixing and large mass splitting between the eigenstates, which is assumed here. For a discussion of the phenomenology in the case of approximate mass degeneracy and nearly maximal mixing see [34]. As discussed in [33], the number density of the dark matter is then given by where c is a constant of order of unity, s is the entropy density, H is the Hubble constant and γ for each process ab → cd is defined as with s min = max((m a + m b ) 2 , (m c + m d ) 2 ). Here K 1 is the modified Bessel function of the second kind and the reduced cross sectionσ is given bŷ where g a,b is the number of degrees of freedom of the particle species a or b, c ab = 2 if a and b are identical, and otherwise c ab = 1, and σ is the cross section given in (12). For further details we refer the reader to [33].

V. NUMERICAL RESULTS AND ANALYTIC APPROXIMATIONS
Using the results obtained above, we perform a numerical scan of the parameter space of the model. We use equations (2)-(8) to solve for the remaining parameters in terms of f π , m π , λ Sh and y Q . We fix m π = 80 GeV and use equation (9) to fix f π as explained above. We then vary the parameters λ Sh and y Q in the range [10 −12 , 10 −1 ] and f π in the range given by equation (9), and perform the integral in equation (14) numerically to obtain the dark matter relic density. We keep the parameter space points where the relic density is within 1 2 to 2 times the observed value, to get an estimate of the typical values for the model. The results are shown in figure 5.
The first observation from the upper left plot in figure 5 is that the mixing angle changes sign at around λ Sh ∼ 0.005. This corresponds to change in the ordering of the masses of the messenger scalar and the Higgs. When λ Sh is below the threshold value, the messenger scalar is always lighter than the Higgs, and vice versa. When we approach the threshold, the scalars become degenerate in mass and the mixing angle becomes large, and therefore some of the approximations we have made in our analysis are not valid. Thus, the results in the region λ Sh ∼ [0.001, 0.03] may not be reliable. Furthermore, this region is phenomenologically problematic, since large mixing between the messenger scalar and the Higgs will affect the couplings of the Higgs and would also result in direct production of the messenger scalar in the LHC. This region of parameter space should thus be considered unphysical, or ruled out by experiment.
In the limit of small mixing, there are some features that can be understood simply from the analytical results derived above. First, from the scattering cross sections of equation (9) we see that the relic density is mainly determined by the product sin 2 θ Sh y 2 Q , since C π ≈ f π is effectively fixed by equation (9), and the pion mass is fixed from the observation of the galactic center gamma ray spectrum. Thus, in order to produce the desired relic density, this product has to be nearly constant, sin 2 θ Sh y 2 Q ∼ 6 × 10 −22 . From equation (8) we can then solve, in the limit θ Sh 1 where in the last step we have used λ S λ Sh , which is a good approximation in the small λ Sh limit as we will see below. Now, since sin θ Sh y Q is a constant, the Yukawa coupling must scale like 1/ sin θ Sh , resulting in This relation is visible in the small λ Sh region of the upper right plot in figure 5. Using equations (2) to (4) we can solve for the self coupling of the messenger scalar so that for the region λ Sh < 10 −3 the approximation λ S λ Sh is well justified. From the above equations we can solve for the dark technicolor confinement scale Sh .
Looking at the lower left plot in figure 5, we can clearly see this scaling in the small λ Sh region. The mass of the messenger scalar can similarly be approximated as which in the phenomenologically favored range of λ Sh implies that m S gets values from ∼ 10 −2 GeV to a few hundred GeV, excluding the region near the Higgs mass where the mixing effects are large, as discussed above. Going further towards the high m S would imply larger values for the scalar couplings λ Sh , λ S , and from equation (18) we can see that λ S soon becomes nonperturbative. Therefore we will not explore this region of the parameter space.
The last plot on the lower right in figure 5 shows the lifetime of the unstable pion, resulting from the decay width into bb. Looking at the formula shown in equation (10) we can see that, since sin θ Sh y Q is a constant, the resulting lifetime is essentially a constant in the small λ Sh and hence small m S region. As λ Sh and consequently m S increases and approaches the value m S = m π = 80 GeV there is a resonance in the decay width, resulting in the dip around λ Sh ∼ 0.005. As m S exceeds the Higgs mass, the scattering processes relevant for the freeze-in become dominated by the s-channel Higgs exchange instead of the messenger scalar, resulting in a slight shift in the required value of sin θ Sh y Q and hence in the pion lifetime seen in the plot. In all the plots shown in figure 5 the width of the band of accepted results corresponds to the width of the accepted values for the relic density, which we have taken as (0.5 − 2) times the observed central value, and the width of the accepted values for f π , resulting from equation (9). The mass of the messenger scalar is predicted to be quite small, with the preferred parameter space containing values in the range m S ∈ [10 −2 , 10 3 ] GeV. However, since the couplings between the messenger scalar and the SM are very weak, this scalar resonance is unobservable at the LEP or LHC.

VI. DISCUSSION AND CONCLUSIONS
In this paper we have proposed a mechanism to generate the galactic centre GeV gamma ray signal from self-interacting composite dark matter in a classically scale-invariant model. Because of the small Higgs portal coupling between the dark and visible sectors, thermal freeze-out is not a viable way to generate the observed relic density, if there is no fine-tuned resonance enhancement to the portal interaction. Instead, we propose that the couplings of the messenger scalar to both sectors are small, such that dark matter can be produced through freeze-in.
We notice that the phenomenologically favoured parameter space contains very small values for all of the couplings of the messenger scalar, λ S , λ Sh , y Q , and a moderate hierarchy between the mass scales Λ TC , v EW . It should be noted, however, that taking all these couplings to zero corresponds to a fixed point in the renormalization group running of the messenger scalar, and results in an enhanced symmetry of the model, since the dark sector becomes completely decoupled from the SM in that limit. As was discussed in [35], this scenario is technically natural and does not contain a hierarchy problem.
Due to the composite nature of the dark matter particles, they naturally have sizable self interactions. However, in our model the self-interaction cross section is within O(1) of the annihilation cross section, which is too weak to alleviate the known problems in small scale structure formation. The prospects for direct detection of this type of dark matter particle are weak, since the dark matter-SM scattering cross section is suppressed by the small Higgs portal coupling, and is therefore orders of magnitude below the current experimental sensitivity.
Precision studies of the couplings and decay branching ratios of the Higgs boson could perhaps reveal the presence of the messenger scalar in a future electron-positroncollider.