Gamma rays and positrons from a decaying hidden gauge boson
We study a scenario that a hidden gauge boson constitutes the dominant component of dark matter and decays into the standard model particles through a gauge kinetic mixing. Interestingly, gamma rays and positrons produced from the decay of hidden gauge boson can explain both the EGRET excess of diffuse gamma rays and the HEAT anomaly in the positron fraction. The spectra of the gamma rays and the positrons have distinctive features; the absence of line emission of the gamma ray and a sharp peak in the positron fraction. Such features may be observed by the FGST and PAMELA satellites.
The concept of symmetry has been the guiding principle in modern physics. The structure of the standard model (SM) is dictated by gauge symmetries. The electroweak symmetry, , is spontaneously broken by a non-vanishing vacuum expectation value (vev) of the Higgs boson, and massive and bosons are generated. It is quite natural in the string landscape that there are many other gauge symmetries as well as discrete ones realized in nature, and some of the gauge symmetries may be spontaneously broken, leading to massive gauge bosons, as in the SM.
Suppose that some of the hidden gauge bosons are ‘odd’ under a parity, whereas all the SM particles are ‘even’. If the parity is exact, and if the hidden gauge symmetries are spontaneously broken, the lightest parity-odd gauge boson is stable and can be a good candidate for cold dark matter. Some recent examples are the T-odd gauge boson in the little Higgs model with T-parity Arkani-Hamed et al. (2002); Cheng and Low (2003, 2004); Csaki et al. (2008) and the KK photon in the minimal universal extra dimension model Appelquist et al. (2001). However, it is not known yet which of unbroken or (explicitly or spontaneously) broken discrete symmetries are more common in the string landscape. If a discrete symmetry breaking is a general phenomenon, we may expect that the dark matter is not absolutely stable and ultimately decays into the SM particles.
In this paper we consider a simplest case that there exist a hidden gauge symmetry which is spontaneously broken and a parity under which only the hidden gauge boson changes its sign. We assume that the parity is broken by a kinetic mixing between the and the SM gauge symmetries Holdom (1986); Foot and He (1991). As a result, the hidden gauge boson decays into the SM particles through the parity violating interactions induced by the kinetic mixing. If such a violation is so tiny that the lifetime of the hidden gauge boson is much longer than the age of our universe, the hidden gauge boson can be the dominant component of dark matter. Furthermore, the subsequent decays of the SM particles will form continuous spectra of the gamma rays and the positrons in the high-energy cosmic ray. With an appropriate amount of the parity violation, we see that those gamma rays and positrons could be the source of the excesses of the gamma ray observed by Energetic Gamma Ray Experiment Telescope (EGRET) Sreekumar et al. (1998); Strong et al. (2004) and the positron flux observed by High Energy Antimatter Telescope (HEAT) Barwick et al. (1997), MASS Grimani et al. (2002) and AMS Aguilar et al. (2007) experiments.
Recently, the gravitino dark matter scenario was extensively studied in the framework of supersymmetry with -parity violation Takayama and Yamaguchi (2000); Buchmuller et al. (2007). The decay of the gravitino into the gamma rays was discussed in Refs. Takayama and Yamaguchi (2000); Buchmuller et al. (2007). Furthermore, if the gravitino is heavier than boson, it decays predominantly into a or boson and a lepton. With the gravitino mass of GeV and the lifetime of sec, the gravitino decay can simultaneously explain the EGRET anomaly in the extragalactic diffuse gamma ray background and the HEAT excess in the positron fraction Bertone et al. (2007); Ibarra and Tran (2008a, b); Ishiwata et al. (2008). Our decaying hidden gauge boson has therefore many parallels with the gravitino with -parity violation. However, in our model, the decay branching ratio of the hidden gauge boson to boson is highly suppressed, which is significantly different from the gravitino case.
This paper is organized as follows. In Sec. II, we present the effective Lagrangian and Feynman rules used in our calculations. In Sec. III, we calculate the spectra of gamma ray and positron flux from the decay of the hidden gauge boson and compare them with the observed data. Sec. IV is devoted to discussions and conclusions.
We consider a hidden Abelian gauge symmetry and the associated gauge boson , and introduce a hidden parity under which transforms as
The symmetry is assumed to be spontaneously broken, so that the gauge boson has a non-vanishing mass, .
We assume that the low energy effective theory can be written in terms of the SM particles and the hidden gauge boson , and that all the SM particles are neutral under both the and the hidden parity. We would like to introduce a tiny parity violating interaction. Among many possibilities, we will focus on a kinetic mixing term between and , since the kinetic mixing has the lowest dimension #1#1#1It is also possible to write down non-renormalizable parity violating interactions suppressed by a large mass scale. For the magnitudes of kinetic mixing in our work, however, we can safely neglect such interactions if they are suppressed by the Planck scale. . The low energy effective Lagrangian can be written as
where is the coefficient of the hidden parity violation term, and is the gauge boson of , and are the field strength of and , respectively: . In the following discussion we neglect terms, since must be extremely small as for the hidden gauge boson to become dark matter. We will also give a possible origin of such tiny kinetic mixing later.
We can remove the kinetic mixing and bring the kinetic terms into canonical form by redefining the gauge fields as Fukuda et al. (1974)
Note that we omit the Lorentz index hereafter. Taking account of the electroweak symmetry breaking, the mass terms of neutral gauge bosons are
where is the neutral gauge field of , is the vev of the doublet Higgs field , and are weak and hypercharge coupling constants, respectively. After diagonalizing the mass matrix, we can express the gauge eigenstates , and in terms of the mass eigenstates , , and as
where and are the masses of and , respectively. One can easily see that and are reduced to the ordinary photon and boson in the limit of . Since interested values of are extremely small, the kinetic mixing hardly affects any SM predictions of the electroweak measurements.
The interacts with the SM particles through the mixings shown in Eq. (5). The Feynman rules can be derived in a straightforward way by expanding the SM Lagrangian with respect to Chang et al. (2006); Gopalakrishna et al. (2008), and the results are
where , , and represent all the up-type quarks, down-type quarks, charged leptons and neutrinos, respectively, and is the coupling of in the SM.
The hidden gauge boson decays into the SM particles through the above interactions. Fig. 1 shows the decay branching ratios and the lifetime of as a function of . The down-type quark decay modes of , , dominate over the other modes for GeV, whereas they decrease quickly as increases. For GeV, the up-type quark decay modes, , become the largest ones followed by charged lepton decay modes, . This behavior can be understood easily as follows. The becomes more like the boson as approaches , and therefore, the partial decay width of is proportional to , where is the vector (axial) coupling strength of the neutral weak interaction. On the other hand, as becomes much heavier than , tends to not feel the electroweak symmetry breaking, so the branching ratios are insensitive to . For , the decay mode of is allowed. The following discussions on the gamma rays and the positrons are not significantly modified even for .
Iii Gamma ray and positron spectra
The hidden gauge boson decays into a pair of SM fermions and bosons if kinematically allowed, as we have seen in the previous section. Due to the subsequent QCD hadronization processes, a bunch of hadrons are produced, and in particular, continuum spectra for the photon and the positron are formed. Throughout this paper we assume that the constitutes the dominant component of dark matter. Then the produced photons and positrons may be observed in the high-energy cosmic rays. In this section we estimate the fluxes of the gamma rays and the positrons produced from the decay, and see how they may account for the observed excesses.
iii.1 Gamma-ray flux
The gamma-ray energy spectrum is characterized by , the number of photons having energy between and , produced from the decay of one gauge boson. The main contribution to the continuous spectrum of arises from the generated in the QCD hadronization process. To estimate the spectrum, we use the PYTHIA Sjostrand et al. (2006) Monte Carlo program with the branching ratios shown in Fig. 1. We also include the real gamma-ray emission, known as internal bresstrahlung, from the charged particles from the direct decay #2#2#2We are grateful to John Beacom Beacom et al. (2005) for bringing up this fact to us., whose contributions to become important compared to that from when . However, the features of the gamma-ray flux which we discuss below will not change even though the internal bresstrahlung effects are not included. The energy spectra for and GeV are shown in Fig. 2. It is worth noting that there is no line emission of the gamma rays from the decay of , which is present in the case of the gravitino dark matter.
There are galactic and extragalactic contributions from the decay of to the observed gamma ray flux. The flux of the gamma ray from the extragalactic origin is estimated as Ibarra and Tran (2008a); Ishiwata et al. (2008)
where is the speed of light; , and are the density parameters of , matter (including both baryons and dark matter) and the cosmological constant, respectively; is the critical density; is the lifetime of ; is the Hubble parameter at the present time; , where is the redshift, and denotes a value of at the matter-radiation equality. For the numerical results, we use Komatsu et al. (2008)
On the other hand, the gamma ray flux from the decay of in the Milky Way halo is
where is the density profile of dark matter in the Milky Way, is the average of the integration along the line of sight (los). We adopt the Navarro-Frenk-White (NFW) halo profile Navarro et al. (1996)
in our calculation, where is the distance from the center of Milky Way, kpc, and is set in such a way that the dark matter density in the solar system satisfies Bergstrom et al. (1998) with kpc being the distance from the Sun to the Galactic Center.
In order to compare the EGRET results with the above flux from decay, we integrate over the whole sky except for the zone of the Galactic plane (i.e. the region with the galactic latitudes ). For the background, we use a power-law form adopted in Ref. Ishiwata et al. (2008)
where is in units of GeV. Fig. 3 shows our numerical results. Recall that there are only two parameters in our model, i.e. the mass and lifetime of , and the peak position only depends on the mass while the deviation from background is sensitive to both. The signal of GeV (the red line) peaks at around GeV region, which is consistent very well with the observed data. For GeV (the blue line), our prediction can still account for the observed excess, although the fit is not so good in some energy region. However, given the large errors in the observed data, it may be premature to extract any sensible constraint on the model parameters. With more precise data from the FGST experiment, we should be able to have more information about the mass and lifetime, if the excess is indeed from dark matter decay. As mentioned before, there is no line emission of the gamma rays in our model, because the production of a pair of on-shell from is forbidden Yang (1950); Hagiwara et al. (1987). Therefore, there is no secondary peak around the high end of signal region, which is a characteristic difference between our prediction and the gravitino case.
iii.2 Positron fraction
After being produced from the decay, the positron will propagate in the magnetic field of the Milky Way. The typical gyroradius is much smaller than the size of the galaxy, and the positron will propagate along the magnetic field. However, since the magnetic fields are tangled, the motion of the positron can be described by a diffusion equation. Neglecting the convection and annihilation in the disk, the steady state solution must satisfy Ibarra and Tran (2008b)
where is the number density of per unit kinetic energy, is the diffusion coefficient, is the rate of energy loss and is the source of producing from decay. In our case,
The solution of Eq. (13) in the solar system can be expressed as
where , and is approximately given by Ibarra and Tran (2008b)
where is in units of GeV, is related to the properties of the interstellar medium and can be determined mainly from the ratio of Born to Carbon (B/C) Maurin et al. (2001). We adopt parameters, , and Ibarra and Tran (2008b), that are consistent with the B/C value and produce the minimum flux of positrons. Finally, the flux of is given by
In addition to flux from dark matter decay, there exists a secondary flux from interactions between cosmic rays and nuclei in the interstellar medium. The positron flux is considered to be suffered from the solar modulation, especially for the energy below GeV. If the solar modulation effect is independent of the charge-sign, one can cancel the effect by measuring the positron fraction,
Indeed, most of experiments measured the fraction of positron flux. To estimate the positron fraction, it is necessary to include the flux. We use the approximations of the and background fluxes Moskalenko and Strong (1998); Baltz and Edsjo (1999)
where is in units of GeV. Therefore, the fraction of flux is
where is a free parameter which is used to fit the data when no primary source of flux exists Baltz and Edsjo (1999); Baltz et al. (2002). Note also that the primary flux of , in the denominator of Eq. (19) should include the contributions from dark matter decay as well. Our numerical results for GeV (the magenta line) and GeV (the green line) are shown in Fig. 4. The prediction of our model is consistent with the observed excess quite well, and the position fraction starts increasing around GeV and GeV for GeV and GeV, respectively. Another key feature of the signal prediction is that the curve drops off sharply at mainly due to the contribution of from decay channel, i.e. the peak seen in Fig. 2. These characteristics can be checked by the upcoming PAMELA data.
Iv Discussion and Conclusions
Let us here discuss briefly how such a small kinetic mixing in Eq. (2) can arise. If there is an unbroken parity symmetry under which the hidden gauge boson flips its sign, the kinetic mixing would be forbidden. This parity may arise from a more fundamental symmetry or it may be an accidental one. In order to have a non-vanishing kinetic mixing, we need to break the parity by a small amount and transmit the breaking to the kinetic mixing #3#3#3 With the presence of the parity symmetry, a small breaking is natural in the sense of ‘t Hooft. . To this end we introduce messenger fields and that have both and charges as and , respectively. Under the parity, they transform as:
Note that the charge assignment is consistent with the parity transformation (see the discussion below Eq. (23)). We assume that the messenger mass scale is very large, e.g. the grand unified theory (GUT) scale. Suppose that the parity is broken in the messenger sector in such a way that and obtain different masses. Let us denote the mass difference by , which parametrizes the amount of the parity violation. Integrating out the messengers, we are then left with a small kinetic mixing, , given by
where is the coupling constant of the hidden gauge symmetry. For and , we obtain the tiny kinetic mixing of the right magnitude of .
It is tempting to identify the origin of the parity violation with the spontaneous breaking of the . As an illustration, we will present a toy model below. As we will see later, there are some dangerous couplings which may spoil the stability of the in this model. However, these problems could be solved by embedding the model into a theory with supersymmetry or an extra dimension(s).
Suppose that all the matter fields in the hidden sector are neutral under the parity. Then, the gauge field cannot have any interactions with those hidden matter fields, since they are forbidden by the parity. Let us introduce two scalar fields, and , which transform under the parity as
and we assume that and are neutral under the SM gauge symmetries. Then the following interactions are allowed:
The parity is spontaneously broken if one of the two scalars develops a non-vanishing vev. To this end, we consider the following potential:
where , and are real and positive. For , there are four distinct vacua, and . We take one possibility of them, , as an example. Therefore, the hidden gauge symmetry is spontaneously broken by the vev of , and the associated gauge boson acquires a mass, , by eating the imaginary component of .
In order to transmit the parity violation, we introduce couplings between and the messenger fields,
where is the messenger mass, and is a real and positive constant. After the is spontaneously broken, masses of and are slightly different: and . After integrating out these heavy messengers, we obtain the kinetic mixing in Eq. (2) with given by
For , , and , we obtain . We can therefore realize more or less the correct magnitude of needed to account for the excesses of the gamma rays and the positrons in this toy model.
In the above toy model, we have assumed that the SM particles couple to the hidden sector only through the messenger fields. It is also possible to introduce direct interactions between and the visible sector. For instance, we can couple them to the SM fermions as
where , is the reduced Planck scale, and is the right-(left-)handed fermion and is the Higgs field. Although this interaction does not lead to the gauge kinetic mixing, it induces the decay of the hidden gauge boson into a fermion pair after breaking the parity. However, the decay branching ratio through such interaction is negligible, compared to that through the kinetic mixing with .
More dangerous direct couplings are those between and the Higgs field, . If there exist such interactions, the could decay into the Higgs bosons immediately. If we extend the model into a supersymmetric one, the direct couplings to the Higgs bosons can be suppressed with the aid of the -symmetry. In a theory with an extra dimension, it is also possible to suppress the direct couplings with appropriate configurations of the branes (e.g. the visible particles on one brane, while the hidden particles on the other).
It is also known that many symmetries appear in the string theory, and the kinetic mixing can similarly arise in the low energy theory by integrating out heavy string states that have charges of two gauge symmetries. It has been extensively studied how large the kinetic mixing can be in e.g. Ref. Dienes et al. (1997) (see also Ref. Abel et al. (2008) and references therein). For instance, in a warped background geometry, we can have an exponentially small kinetic mixing Abel et al. (2008).
Let us also have some comments on the the GUT. So far we have assumed the existence of the kinetic mixing between and the hidden . If one of the gauge symmetries actually sits within an unbroken non-Abelian gauge symmetry, such kinetic mixing is not allowed. This does not necessarily mean that kinetic mixings are incompatible with GUT, because the GUT gauge group is spontaneously broken. Indeed, kinetic mixings between the and the GUT gauge fields can arise below the GUT scale by picking up non-vanishing vevs of the Higgs bosons responsible for the GUT breaking.
Let us also discuss how the hidden gauge boson could be generated in the early universe to account for the observed abundance of dark matter. For simplicity, we neglect a numerical coefficient of order unity in the following discussion. In the presence of the messenger fields, there appears at one-loop level a following interaction between the and gauge fields:
The hidden gauge boson will be produced through the above interaction most efficiently at the reheating. The abundance is roughly estimated to be
where and are the entropy density and the reheating temperature, respectively. For the mass , a right abundance of can be generated from the above interaction for GeV. Also the can be non-thermally produced by the inflaton decay Endo et al. (2006, 2008, 2007).
In the above model, the parity is spontaneously broken by the vev of . If the breaking occurs after inflation, domain walls connecting two of the four vacua in Eq. (24) will be formed Zeldovich et al. (1974); Kibble (1976); Vilenkin (1981), which can be the cosmological disaster. There are several means to get around this problem, and one of which is to introduce a tiny explicit breaking of the parity symmetry. Therefore the domain walls are not stable, and eventually annihilate after collisions Vilenkin (1981); Coulson et al. (1996); Larsson et al. (1997). Another solution is to assume that the breaking occurs before inflation. The last one is to assume that the initial positions of the scalars and are deviated from the origin. In the last case, the domain walls, if formed, will be annihilated eventually #4#4#4 The significant amount of the gravitational waves may be produced in the collisions of domain walls Takahashi et al. (2008). The hidden gauge bosons are also produced by the annihilation processes of the domain walls..
In this paper we have considered a possibility that a hidden gauge boson , which constitutes the dominant component of dark matter, decays into the SM particles through the kinetic mixing term that breaks the parity symmetry. As a result, the branching ratios are solely determined by the mass of the hidden gauge boson. Continuum spectra of photons and positrons are generated from (), and from the decays of hadrons, mainly , produced in the subsequent QCD hadronization process #5#5#5There also exist antiprotons produced from the decay, whose flux can also be calculated and compared with the present data. However, we did not pursue the detailed comparison in this paper due to the large experimental uncertainties and our poor understanding of diffusion models.. If the mass of is about GeV and its lifetime is of order seconds, those gamma rays and positrons from decay may account for the observed excesses in the extra galactic diffuse gamma ray flux and the positron fraction. Interestingly, in our model, the spectra of the gamma rays and the positrons have distinctive features: the absence of line emission of the gamma ray and a sharp peak in the positron fraction. Such features may be observed by the FGST and PAMELA satellites.
Acknowledgements.CRC and FT would like to thank M. Nojiri for useful discussions. This work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
Note added: Very recently the PAMELA group reported a steep rise in the positron fraction Adriani et al. (2008), which is nicely explained by our model.
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